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Evidence for the Multiverse in the Standard Model and Beyond

In any theory it is unnatural if the observed parameters lie very close to special values that determine the existence of complex structures necessary for observers. A naturalness probability, P, is introduced to numerically evaluate the unnaturalness. If P is small in all known theories, there is an observer naturalness problem. In addition to the well-known case of the cosmological constant, we argue that nuclear stability and electroweak symmetry breaking (EWSB) represent significant observer naturalness problems. The naturalness probability associated with nuclear stability is conservatively estimated as P_nuc<10^{-(3-2)}, and for simple EWSB theories P_EWSB<10^{-(2-1)}. This pattern of unnaturalness in three different arenas, cosmology, nuclear physics, and EWSB, provides evidence for the multiverse. In the nuclear case the problem is largely solved even with a flat multiverse distribution, and with nontrivial distributions it is possible to understand both the proximity to neutron stability and the values of m_e and m_d - m_u in terms of the electromagnetic contribution to the proton mass. It is reasonable that multiverse distributions are strong functions of Lagrangian parameters due to their dependence on various factors. In any EWSB theory, strongly varying distributions typically lead to a little or large hierarchy, and in certain multiverses the size of the little hierarchy is enhanced by a loop factor. Since the correct theory of EWSB is unknown, our estimate for P_EWSB is theoretical. The LHC will determine P_EWSB more robustly, which may remove or strengthen the observer naturalness problem of EWSB. For each of the three arenas, the discovery of a natural theory would eliminate the evidence for the multiverse; but in the absence of such a theory, the multiverse provides a provisional understanding of the data.


Introduction
The Standard Model, with three gauge forces and three generations of quarks and leptons, has been laboriously constructed from decades of data involving collisions of particles at ever higher energies. It represents a triumph for symmetries, but leaves many questions about nature unanswered. These questions fall into three groups. The first revolves around observations of the universe that are not described by the Standard Model, such as dark matter, dark energy, baryogenesis and inflation. Secondly there is the question of electroweak symmetry breaking: while the Standard Model provides a mathematical description, it does not contain the physical description. The quadratic divergence in the Higgs mass-squared parameter implies that electroweak symmetry breaking is determined by new physics at a scale M at which the Standard Model is embedded into a more fundamental theory. This extraordinary behavior is termed "unnatural" -the Higgs mass-squared parameter, m 2 h , is very sensitive to changes in some parameter x of the fundamental theory, especially as M grows: The final group of questions boils down to "Why the Standard Model?" These include why the forces are the ones we observe, why the quarks and leptons have the masses they do, and why there aren't other forces and particles. Since symmetries were the key to the Standard Model, it is natural to suppose that all three sets of questions will be answered by introducing further symmetries, and this has largely defined beyond the Standard Model physics for the last three decades. Yet there are three clouds on the horizon for this symmetry approach, one for each set of questions • There is no known symmetry to explain why the cosmological constant is either zero or of order the observed dark energy. It appears to take a very unnatural value.
• For the Standard Model to be natural, M must be low; we should already have observed either signs of supersymmetry or signals of new interactions in precision electroweak observables.
• Some of the Standard Model parameters take values in a special region that happen to yield certain nuclear properties.
It is possible that these clouds are the first indications of a breakdown in the power of symmetry to determine fundamental physics. If our universe is just one among very many in an enormous multiverse, then observed universes will be those that contain certain complex structures necessary for observers [1]. Such arguments from environmental selection can potentially solve the cosmological constant problem, and yield a statistical prediction for the dark energy in observed universes [2]. In this paper we consider the extent to which nuclear stability and electroweak symmetry breaking provide evidence for environmental selection.
Many physicists, however, are reluctant to countenance any form of anthropic argument. Why give up on the traditional, extraordinarily successful, methods of physics? How can we hope to understand the conditions for intelligent observers when we struggle even to define what life is? Why make the extraordinary leap of postulating an extra-horizon multiverse, which has the smell of a secular form of God? In short, many believe that appeals to the environment are an escape from true science and that, in the absence of data confirming the conventional symmetry approach, it would be better to change fields than to succumb to the philosophy of anthropics.
The case of the cosmological constant demolishes these arguments. Traditional methods have not given any satisfactory understanding for why the cosmological constant is small. In contrast, the environmental argument not only explains why it must be small, but makes a statistical prediction for a non-zero value. Furthermore, this prediction requires essentially no understanding of what it takes to make an observer. Of course, the prediction does require a multiverse; but with a landscape of vacua provided by string theory [3] and generic inflation eternally creating universes [4], a multiverse no longer appears theoretically unreasonable, and its consequences can be explored under the assumption that we are typical observers [5]. Indeed, since few doubt that environmental conditions explain why we find ourselves on the Earth as opposed to elsewhere in the solar system, the resistance to environmental reasoning is hard to understand. As for any physical theory, the real issue is whether sufficient evidence for it can be found to make it convincing.
Even if there is a multiverse, and environmental selection accounts for the value of the cosmological constant, it is far from clear what the implications are for the rest of physics. One extreme possibility is that all the other questions left open by the Standard Model will be solved within some symmetry framework, for example supersymmetric SO(10) with flavor symmetries. In this case, the environmental solution of the cosmological constant problem simply allows us to ignore this special problem while solving everything else with symmetries -it justifies the model-building approach of the last 30 years. The other extreme is that the Standard Model is as far as we can get with symmetries, all of its parameters are either strongly selected by environmental effects or just reflect random typical values in the multiverse, and similarly for issues outside the Standard Model. Of course there are many intermediate cases between these two extremes. For example, there could be gauge coupling unification within an SO(10) theory, but the flavor sector may involve sufficient scanning parameters that the electron, up and down quark masses are determined by environmental selection. The aim of this paper is to seek evidence for environmental selection both in the measured values of Standard Model parameters, and in electroweak symmetry breaking.
The cosmological constant illustrates an additional crucial point: environmental selection predicts parameters that are, from the conventional viewpoint of symmetries, unnatural. This is a key point, because so much effort of the last 30 years has been expended in trying to understand how new symmetries could naturally lead to an unnatural effective low energy theory, namely the Standard Model. Perhaps the unnaturalness is a hint that symmetries are not the answer. It was realized some time ago that if only the mass parameter of the Standard Model scans, then environmental selection could solve the gauge hierarchy problem [6]. In the context of supersymmetry, differing assumptions about scanning parameters and their distributions could lead to a large hierarchy [7] or a small hierarchy [8]. These papers both illustrate that environmental selection leads to unnaturalness, and that measurements at the LHC could experimentally confirm the presence of unnaturalness in electroweak symmetry breaking. It is an open question whether the landscape favors a low or high supersymmetry breaking scale [9].
The first part of this paper, sections 2 to 5, are devoted to a consideration of the concept of naturalness, and its application to issues of nuclear stability. We are motivated by the belief that environmental selection may lead to precise predictions, and that it is important to evaluate the numerical significance of such predictions. We often hear it said that, in nuclear physics and elsewhere, nature exhibits "amazing coincidences" that give rise to life. How can these be precisely evaluated? In section 2 we formulate the concept of naturalness in a very general way, so that in section 3 we are able to identify naturalness problems associated with the existence of complex structures in the universe. In section 4 we study the stability of neutrons, deuterons and complex nuclei in the parameter space of the Standard Model, identifying how close our universe is to these stability boundaries. In particular, we find a closeness to the neutron stability boundary that has not previously been elucidated. The naturalness problems associated with the closeness to these stability boundaries are evaluated numerically in a variety of theories of flavor in section 5. We stress that the evidence for unnaturalness depends on the ensemble of theories being considered, but we find that there is an irreducible amount of unnaturalness no matter what the theory of flavor.
In the second half of the paper, we investigate the consequences that follow if this unnaturalness arises from environmental selection in a multiverse. It is here that the real utility of our new formulation of naturalness becomes apparent. In section 6 we argue that problems of unnaturalness are indeed solved by environmental selection, and show how the amount of unnaturalness is connected to the probability distribution of the multiverse. We argue that evidence for environmental selection will increase if symmetry arguments fail to solve an accumulation of naturalness problems. In the case of nuclear stabilities, we show in section 7 how this leads to multiverse predictions for the masses of the electron, up and down quarks. In section 8 we embed the Standard Model Higgs sector in a generic theory of electroweak symmetry breaking at mass scale M. Allowing parameters of this sector to scan, as well as parameters of the Standard Model, we consider environmental selection from nuclear stability boundaries. We find that sharply varying multiverse distribution functions generically lead to both large and little hierarchies between M and the scale v of electroweak symmetry breaking. From the viewpoint of the multiverse, a discovery at the LHC that electroweak symmetry breaking requires some amount of fine-tuning would not be surprising. A similar analysis on electroweak symmetry breaking is performed in section 9, assuming that the relevant boundaries are the phase boundary of electroweak symmetry breaking rather than the nuclear stability boundaries. We again find that unnaturalness in electroweak symmetry breaking is expected for sharply varying multiverse distribution functions. In section 10 we study connections between environmental selection for the cosmological constant and for electroweak symmetry breaking. For example, we argue that the driving force on the multiverse for unnaturalness in electroweak symmetry breaking could arise from the distribution for the cosmological constant, the connection being through weakly interacting massive particle (WIMP) dark matter. Finally, our conclusions are given in section 11.

New Definition of Naturalness Problems
The concept of naturalness has often been the driving force for finding fundamental mechanisms in nature. Arguments for naturalness are often phrased in terms of the sensitivity of low energy Standard Model parameters c i to variations of the parameters in the more fundamental theory a j . A simple measure of naturalness is then given by ∆ ≡ |∂ ln c i /∂ ln a j |, with a large value of ∆ signaling a lack of naturalness [10]. This definition, however, could miss some of the important aspects of the naturalness problem. Here we introduce a new, quantitative definition of naturalness that can be applied in much more general situations. In particular, this allows us to identify certain classes of naturalness problems that have not been quantitatively defined. Our definition is also free from some of the problems existing in the simplest definition of naturalness based on the sensitivity of parameters.
First of all, it is very important to notice that in talking about naturalness, we are dealing, either explicitly or implicitly, with an ensemble in which parameters of the theory are varied according to some definite distribution. Consider, for example, that the Higgs mass-squared parameter, m 2 h , is given by the difference of two mass scales of order the fundamental scale, We say that the theory is unnatural if |m 2 h | ≪ M 2 1 , M 2 2 . This statement, however, already assumes that it is unlikely for M 1 and M 2 to be very close or for both to be very small; more specifically, the distribution of possible values for M 1 and M 2 is assumed to be almost structureless in the M 1 -M 2 (or M 2 1 -M 2 2 ) plane. This illustrates that the concept of naturalness is closely related to the distribution of parameters in an ensemble.
A particular member of an ensemble is unnatural if it has parameters very close to special values that are not explained by the symmetries of the theory. The parameters are special if some physical property arises that is not a generic feature of the members in the ensemble, or if they separate regions of parameter space that have differing generic features. Depending on the property considered, we encounter various classes of naturalness problems, whose solutions could point to various different mechanisms in nature. With this definition, the degree of unnaturalness is given by how close the parameters are to the special values, which generically form a special hypersurface in multi-dimensional parameter space. As we will see below, we can quantify this degree in terms of the distribution of parameters within the ensemble. In this section, after presenting a new definition of naturalness, we apply it to well-known situations. In the next section we use it to introduce new types of naturalness problems.

Definition
Let us start describing our precise definition of naturalness by introducing the distribution function f (x) for an ensemble. The function is defined such that the number of members with parameters x i (i = 1, · · · , N) taking a value between x i and x i + dx i is given by Here, x i represent continuous parameters of the theory, e.g. masses and coupling constants. 1 The overall normalization of f becomes relevant if we are interested in the total number of members, which will be the case when we discuss relative likelihoods of different theories. For the present purpose of discussing naturalness of a given theory, however, the overall normalization of f is not important. The distribution function depends on the choice of x i . What variables should we choose as x i ? In general we can choose any variables as x i , depending on the context. To discuss naturalness of the low energy theory, however, it is often most convenient to take "fundamental" parameters, such as masses and coupling constants of the ultraviolet theory, as x i . Now, suppose we have only one such variable x. If the observed value of x, x o , is very close to a special valuex, the level of unnaturalness is given by the following "naturalness probability" where x min ≤ x ≤ x max gives the range of x values in the theory under consideration. 2 Here, we have included only members on one side of the special point, but depending on the situation, a factor of 2 should be added to the numerator to include members on both sides of the special point. With this definition, P ≪ 1 signals the existence of a naturalness problem. The form of the distribution function is modified if we redefine the variable x. For example, if f (x) = 1/x for a variable x, the change of the variable x ′ = ln x can make the distribution function flat for x ′ : f (x ′ ) = 1. It is often convenient to go to the basis in which the distribution function is constant. In that basis, the naturalness probability of Eq. (3) becomes which is simply given by the distance between x o andx divided by the available parameter space. 3 As should be the case, this quantity becomes smaller as x o approachesx. The naturalness probability of Eqs. (3,4) can be extended to the case of multiple parameters x i . The precise definition depends on the dimensionality of x i and the special surface consisting ofx i . If the co-dimension of the special surface is 1, we can choose x to be a linear combination of x i perpendicular to the surface, and then use the definition of Eq. (4) (in the basis where the distribution function is constant). In the case that the co-dimension is higher, we must use an appropriate generalization of Eqs. (3,4) defined using multi-dimensional volumes, rather than simple distances. A useful definition, in the basis where f (x i ) is constant, is P = c n {Σ n a=1 (x a,o −x a ) 2 } n/2 Π n a=1 (x a,max − x a,min ) where a = 1, · · · , n runs over variables whose observed values, x a,o , are close to the special values, x a , and c n = π n/2 /Γ(n/2 + 1) is the volume of the unit ball in n dimensions. Here, the expression in the numerator, v n , is a measure of the accidentally small volume of parameter space required for x a,o to be close tox a , while that in the denominator, V n , is the total volume of parameter space available. The restriction of the variables x a to those with |(x a,o −x a )/(x a,max −x a,min )| ≪ 1, e.g. ≤ 1/5, is important to avoid obtaining P ≪ 1 simply as a result of high dimensionality of the parameter space. In the case that the special surface in question arises as an intersection of co-dimension 1 surfaces, we may restrict the volume of the numerator to be one side of the co-dimension 1 surfaces, depending on the situation. This definition reduces to that of Eq. (4) in the special case of n = 1.
(a) Figure 1: Illustrations of the definition of the naturalness probability P in multi-dimensional parameter space.
The definition of Eq. (5) is illustrated for a 2-dimensional parameter space (x 1 , x 2 ) in Fig. 1(a). The curve C represents special values of these parameters corresponding to some physical phenomenon, and the curve C ′ similarly represents special values for some other phenomenon. If the observed values (x 1,o , x 2,o ), denoted by the dot, are close to the intersection point (x 1 ,x 2 ), the naturalness probability is given by Eq. (5), with a = 1, 2. The numerator represents the area of the shaded region, and the denominator represents a much larger area corresponding to the range of the parameters. Here, we have restricted the area of the numerator to one side (the lower side) of the curves C and C ′ , anticipating an application in later sections.
In Fig. 1(b), it is apparent that the observed point is much closer to C than C ′ ; but the definition of Eq. (5) does not capture the additional unnaturalness associated with this. (The area of the numerator in this expression is represented by the dotted line.) In this situation, we must first consider the naturalness probability P C associated with C, using the definition of Eq. (4), where the axis x is taken normal to C and passing through the observed point at x = x o . We then take the coordinate x ′ along the curve C and consider the naturalness probability P C ′ associated with C ′ in this coordinate. The resulting naturalness probability is then P = P C P C ′ , which is given by the area of the shaded region in the figure divided by that corresponding to the range of the parameters. This tends to zero as the observed point approaches the curve C, as expected. In general, if we find several unnatural features that have a hierarchy in their degrees of unnaturalness, we can obtain the correct estimate for the naturalness probability P by considering it as a product of several naturalness probabilities, each associated with an unnaturalness of some fixed degree. The decomposition of P can be made along the lines presented here.

Illustrations
Let us now illustrate the use of our definition in the case of the conventional gauge hierarchy problem. Consider that the Standard Model is embedded into the fundamental theory at a high scale of M 1 ≈ M 2 ≈ M * and that the Higgs mass-squared parameter is given in terms of M 1 and M 2 by m 2 h = M 2 1 − M 2 2 . In general, we expect that the distribution function is roughly flat in terms of M i (i = 1, 2) and that the range of parameters is 0 ≤ M i ≤ O(M * ). 4 In Fig. 2, we plot the contour of m 2 h in the M 1 -M 2 plane in units of M * . We clearly see that there is a special line in this plane, M 1 = M 2 , where m 2 h = 0. The physical phenomenon making this line special is that members very close to this line have a low energy effective field theory that contains a scalar excitation whose mass is hierarchically smaller than M * . For M * ≈ 10 18 GeV, the observed value of m 2 h ≃ −O(100 GeV) 2 is denoted by the little dot for an arbitrary value of M 1 = 1.2M * . We find that it is located very close to the special line compared with the expected range of M i . In fact, this is always the case regardless of the value of M 1 we choose, and is a manifestation of the lack of naturalness in the Standard Model. The problem is that the special line with m 2 h = 0 is not special in terms of the symmetry structure of the theory. The points with m 2 h = 0 are no more symmetric than those with m 2 h = 0. By choosing a variable x to be a linear combination of M 1 and M 2 perpendicular to the special line, and then using the definition of Eq. (4), we obtain P ≈ 10 −32 . A similar analysis can also be made for the cosmological constant 4 Here, we have assumed that M i are mass parameters associated with "fermions" (including mass parameters in supersymmetric theories). The argument, however, is not affected if they are associated with scalars, in which case the distribution is expected to be roughly flat in problem. In general, the cosmological constant Λ is given by the sum of terms of order M 4 * taking the form of M 2 i M 2 j . The observed value of Λ is then very close to the special hypersurface with Λ = 0, and the points with Λ = 0 are no more symmetric than those with Λ = 0. The physical property making this hypersurface special is that members very close to this hypersurface allow an arbitrary large observable universe. From Eq. (4), the degree of unnaturalness we obtain in this case is P ≈ 10 −120 .
Here we note that our definition does not suffer from the problem existing in the simplest definition of naturalness based on the logarithmic derivative, ∆ ≡ |∂ ln c i /∂ ln a j |. Suppose that a dimensionful parameter µ is given by two dimensionless constants g 2 and b as µ = M * e −8π 2 /g 2 b , where M * is the fundamental scale, and that a natural range for g 2 and b in the fundamental theory is g 2 b = O(0.1). We then naturally obtain µ/M * = e −O(1000) ; there is nothing unnatural with this. The simplest definition, however, gives ∆ = |∂ ln µ/∂ ln g 2 | = O(1000), signaling (incorrectly) the existence of unnaturalness. Our definition does not lead to such a fake signal, since any point with g 2 b = O(0.1) is generic -there is no special value of g 2 b that can be singled out asx in Eq. (3,4).
We now see how theories beyond the Standard Model solve the gauge hierarchy problem in our language. As we have seen, the Standard Model embedded into more fundamental theory at M * ≈ 10 18 GeV leads to the probability of having a small weak scale v ≈ 100 GeV to be P ≈ (v/M * ) 2 ≈ 10 −32 . This is because the Higgs mass-squared parameter, m 2 h , is given by the sum of various contributions M 2 i of order M 2 * , with the distribution function expected to be roughly flat in M i (or M 2 i ). The distribution function, however, varies from one theory to another. For example, if the weak scale arises from dimensional transmutation associated with some gauge coupling g becoming strong, as in technicolor [11] and certain supersymmetric theories [12], then v ≈ M * exp(−8π 2 /|b|g 2 * ). Here, b is the one-loop beta function coefficient for the new gauge force and g * = g(M * ). The distribution function is expected to be roughly flat in the variable g 2 * , 5 giving an approximately flat distribution for 1/ ln(v/M * ). To illustrate this situation, we plot in Fig. 3 the contour of m 2 h = M 2 1 − M 2 2 with M 1 = M * exp(−8π 2 /|b|g 2 * ) and M 2 /M 1 ≡ r in the |b|g 2 * -r plane. This can be regarded as a simplified model for the situation in, e.g., dynamical supersymmetry breaking. Typically, there is considerable uncertainty in the range of |b|g 2 * . Here we take 0 ≤ |b|g 2 * ≤ O(10). As is seen from the figure, the observed value of v can correspond to a completely generic (not special) point in the plane, which is indicated by the little dot for an arbitrary value of r = 1.4. This, therefore, provides a solution to the gauge hierarchy problem; see Fig. 2 for comparison. The important point here is that different theories can lead to very different distribution functions f (x) for physical parameters x i . A distribution flat in v 2 is the origin of the gauge hierarchy problem, while one flat in ln v (or 1/ ln v) provides the solution. The situation with the cosmological constant is similar, although the problem is much more severe because all known theories give distributions that are essentially flat in Λ, while none are known that are flat in ln Λ (or 1/ ln Λ).

Generality of the framework
So far, we have illustrated our definition of naturalness by looking at naturalness problems that have a conventional description: the observed values of parameters are (very) close to special values where the sensitivity of a low energy parameter to high energy ones diverges, for example ∂ ln m 2 h /∂ ln M i , ∂ ln Λ/∂ ln M i → ∞. This is one class of naturalness problems, which we may call "fine-tuning naturalness problems." The real power of our formalism, however, lies in the fact that we can discuss many different types of naturalness problems in a unified manner, simply by extending what we mean by "special values" for parameters. Any physical property that is not a generic feature of the members in the ensemble is a candidate for identifying special values. There can be many classes of naturalness problems, depending on the property considered. The closeness of the observed values to the special values, signaled by P ≪ 1, can then be used as evidence for a new mechanism to understand such accidents.
One application of this general idea is to use naturalness arguments as evidence for the presence of some symmetry beyond the Standard Model. For this purpose it is often convenient to take parameters of the low energy theory to be x i . Imagine that the observed values x i,o of N dimensionless Standard Model parameters x i (i = 1, · · · , N) take values close to the special surface that defines a symmetry relation If the symmetry were in fact absent, nature would be described by some member of an ensemble giving the x i parameters distributed according to f (x i ), which is not generically concentrated on the surface of Eq. (6). After redefining the parameters to make the distribution function flat, we can introduce an axis x normal to the symmetry surface that passes through the observed point at x i = x i,o . The degree of unnaturalness is then given by Eq. (4). The closer the observed point is to the special valuex on the symmetry surface, the less probable it becomes that nature is described by this ensemble, i.e. by the theory without the symmetry. A small value for P can thus be taken as evidence that the symmetry is indeed present in nature, at least in an approximate form. 6 It could be that x o −x is dominated by the experimental uncertainty. In this case improved experiments may provide further evidence for the symmetry. A well known example is provided by a grand unified symmetry, such as SU (5), that gives a symmetry relation between the Standard Model gauge coupling constants S(g 2 1 , g 2 2 , g 2 3 ) = 0 [13]. If in fact there is no unified symmetry, the observed values of α, α s and sin 2 θ W give a naturalness probability of P ≈ 0.1 in an ensemble without supersymmetry, and P ≈ 0.01 in an ensemble with supersymmetry [14]. Here, α, α s and θ W are the fine structure constant, the effective QCD coupling, and the Weinberg angle, respectively. This lack of naturalness can then be regarded as evidence for some form of a unified symmetry in theories with supersymmetry.
In the next section we introduce classes of naturalness problems that arise from the existence of complex structures. We will see that the formalism developed here elucidates the identification of these naturalness problems. As in other naturalness problems, successful solutions to these problems could lead us to find new mechanisms or dynamics in nature, which we consider in later sections.

Complexity and Observer Naturalness Problems
The definition of naturalness introduced in the previous section allows us to identify new classes of naturalness problems. A member in the ensemble is unnatural if it has parameters unusually close to "special" values; but clearly there are many reasons that parameters could be "special." We frequently stress special values that lead to a large hierarchy of mass scales, but in this section we consider special values that lead to the existence of relatively long-lived complex structures, such as nuclei, stars and galaxies.

Complexity naturalness problem
As the parameters x i vary, moving from one member of an ensemble to another, suppose that a physical threshold is crossed that is crucial for the existence of some complex structure. This defines a special surface in the parameter space that divides the volume of parameter space into two regions, one that supports the complex structure, C(x i ) < 1, and one that does not, C(x i ) > 1. In general this surface is not one with enhanced symmetry. Therefore, a member in the ensemble having parameters unusually close to this surface has a "complexity naturalness problem." The degree of unnaturalness is given numerically by Eq. (3), where the single variable x is normal to the "complexity surface" of Eq. (7), where it takes the valuex, and the particular unnatural member in question has a nearby value for this parameter, x o . One caveat is that we have assumed that the physical threshold relevant for the existence of the complex structure is sharp in the parameter space. This can be verified in any particular case, and is certainly true in the examples discussed in the next section. For example, the stability of any particular nucleus gives a sharp boundary corresponding to values of the coupling strengths and quark masses that lead to a surface of zero binding energy. In general a lack of sharpness is due to the time evolution of complex structures in universes corresponding to the different members of an ensemble. As parameters vary from one member to another, complex structures could gradually become less stable. For example, the stability of large scale structure is not completely sharp -as the cosmological constant is gradually increased only the regions with larger statistical fluctuations in the density perturbations are able to collapse. Still, the relevant parameter space for the cosmological constant spans over 100 orders of magnitude, and over this space the transition for the existence of large scale structure is very sharp.
Another caveat is the hidden assumption that these complexity surfaces are not distributed so densely throughout the entire parameter space that a typical member in the ensemble is expected to be close to one or more surfaces. There may indeed be many complexity surfaces; for example, in the Standard Model there are several hundred relatively stable nuclei, each with its own complexity surface in an ensemble that contains the Standard Model. However, if the relevant parameters -the Yukawa couplings, y, the weak scale, v, and the QCD scale, Λ QCDvary by many orders of magnitudes in the ensemble, then these complexity surfaces will all be tightly clustered in a "complexity zone," yv ∼ αΛ QCD ∼ O(0.01)Λ QCD . A complexity naturalness problem now arises because this zone is itself small compared with the entire volume of parameter space. Most members in the ensemble lie in voids far from the complexity zone, and the closeness of the observed parameters to one or more of the complexity surfaces implies that the member describing our universe lies in a special region. There may be several complexity zones for nuclear physics; for example ones with four, five or six quark flavors lighter than the QCD scale, but in certain parameter directions each zone will be small. In the case of a single variable, the degree of unnaturalness can be taken to be P = ∆x z /(x max − x min ) where ∆x z is the width of the complexity zone. If the density of complexity surfaces is very high it might be that a member in the complexity zone is very close to some surface; but this could be a reflection of the density of the surfaces rather than any additional unnaturalness beyond that of being in the complexity zone.
In summary, a theory possesses a complexity naturalness problem if, in the parameter space of an ensemble, the member describing our universe lies very close to a surface corresponding to a physical threshold that allows the existence of some relatively long-lived complex structure. For the problem to exist, there should not be many surfaces of a similar character distributed densely and almost uniformly over the parameter space; otherwise, the closeness to one of these surfaces would simply be a generic phenomenon for the members in the ensemble.

Observer naturalness problem
Now we introduce another, closely related, naturalness problem. Complex structures are required for the existence of "observers." This implies that some complexity boundaries may also act as boundaries that divide the parameter space of the ensemble into those members that may support certain observers and those that can not. These boundaries are harder to define than general complexity boundaries, since we are unable to give a precise definition of an observer. Moreover, the parameter region may not simply be divided into the two regions "with" and "without" observers -the expectation value for the number of observers may in general be a complicated function over the parameter space, with the value significantly varying across a complexity boundary. Nevertheless, since the changes of the expectation value across some of these boundaries are expected to be drastic, caused by drastic changes of complex structures, for certain purposes we may approximate this function to be step-like. This leads to the concept of the "observer boundary" which divides the parameter space into one "with" observers, O(x i ) < 1, and one "without," O(x i ) > 1. In general, there are many complexity boundaries, C a (x) = 1, that are relevant for the existence of observers. The region of parameter space that allows observers, O, is then given by the common set of C a (x) < 1 for all a, and the observer boundary, O(x i ) = 1, is the border of this region. As more boundaries are added as relevant ones for observers, the region O shrinks, but the number of observer boundaries does not generically increase. This is the crucial difference between complexity and observer boundaries. For illustration of these boundaries in an example While the identification of observer boundaries suffers from some ambiguities, there are certain advantages to focusing on these boundaries rather than on general complexity boundaries. In general there can be several disconnected regions, O I , that can support observers. However, since the existence of observers undoubtedly requires certain complex structures, we can be convinced relatively easily that these regions are not distributed densely throughout the entire parameter space. For example, if we take the existence of stable complex nuclei to be one of the required conditions for observers, then we find that almost the entire region with v/Λ QCD > ∼ 10 4 , with the other parameters of the Standard Model fixed, is outside O I [6], regardless of the existence of any other physical thresholds. Increasing the size of the up or down Yukawa coupling more than a factor of a few, with the other parameters fixed, also leads to a sterile universe in which both proton and neutron cannot be stable inside nuclei. The situation is similar for other conditions. For example, the requirement for the existence of complex structures in the universe, such as galaxies, makes the entire region with Λ/Q 3 T 4 eq > ∼ 1 outside O I [2], where Q is the primordial scale of density perturbations and T eq the temperature of matter radiation equality. These imply that the observer regions O I fill only a small fraction of the parameter space. They are small islands in the entire parameter space of the ensemble, and so the observer boundaries are not distributed densely and uniformly over the parameter space.
The sparseness of O I allows us to focus on the region O (⊂ {O I }) that contains our observed point. The unusual closeness of the observed parameters to the relevant observer boundary (the boundary of O) then signals unnaturalness, raising an "observer naturalness problem." The degree of unnaturalness can be quantified using the definition given in the previous section. The estimate for the degree will be conservative if we use only crucial requirements for the existence of observers, such as the existence of nuclei, stars, and galaxies, since this will give a large allowed region for observers, O. Imposing more and more subtle conditions, such as the existence of carbon, oxygen and so on, will only decrease O and therefore increase the degree of unnaturalness. A relevant question is whether we find an observer naturalness problem in the Standard Model and beyond, keeping only relatively robust conditions. This question will be addressed in the next section.

Evidence for Observer Naturalness Problems
In this section we study if there exist observer naturalness problems in the Standard Model and beyond. In general it is easier to study the existence of an observer naturalness problem than that of a complexity naturalness problem, since it is easier there to see that the relevant boundaries are not uniformly and densely distributed over the parameter space. In order to identify and quantify the observer naturalness problem reliably, we take the following approach. We first list complexity boundaries that are candidates for composing the observer boundary, in the relevant parameter space of the Standard Models of particle physics and cosmology. These boundaries have effects of various degrees on the environment, and we take only the ones giving fairly drastic effects. This leads to a large observer volume O and hence to a conservative estimate for the degree of an observer naturalness problem. The actual degree of unnaturalness could only be more severe if other complexity boundaries are important for the development of observers, since this would reduce O.

The relevant parameters
Let us consider some ensemble that leads at low energies to the Standard Model of particle physics (taken to include neutrino masses) and to the standard cosmological model. Different members of this ensemble give different values for the parameters of the Standard Model, x SM , and cosmology, x cos . Some members may give values for this set x = {x SM , x cos } that are so distant from the observed values, x o , that the corresponding physics and astrophysics is completely different from that observed -for example, if five quark flavors were lighter than the QCD scale, or if the baryon asymmetry were of order unity. Here we restrict our discussion to a subset of the ensemble in which variations in the parameters about x o are limited. In fact, we are interested in determining whether small deviations in x from x o could dramatically change certain relatively stable complex structures (atoms, stars ...) that we observe.
The physics and astrophysics of these complex structures depends on only a subset of x; for example, variations in the bottom quark mass, or its mixing to the charm quark, by a factor of 2 have no effect on the complex structures of interest. Hence, we now restrict x to include only and x cos : M Pl , T eq , Q, η B , Λ.
Here, y u,d,t,e are the Yukawa couplings for the up, down and top quarks and the electron, λ h the Higgs quartic coupling, m 2 h the Higgs mass-squared parameter, M Pl the reduced Planck mass, and η B the baryon asymmetry. In this 13-dimensional parameter space there are special surfaces that represent complexity boundaries such that the complex structure A differs drastically from one side of the boundary to the other. Several comments are in order for our choice of the parameter set x.
• λ h and m 2 h can be traded for the electroweak vacuum expectation value, v, and the Higgs boson mass. We include λ h and y t because they typically play an important role in electroweak symmetry breaking. For example, in the Standard Model there is a boundary corresponding to the existence of a electroweak symmetry breaking vacuum with v ≪ M Pl [15] C v (λ h , y t ) = 1. (12) • The SU(2) gauge coupling of the Standard Model is omitted from Eq. (9), since charged current weak interactions at low energies are described by v. Neutral current, and therefore the weak mixing angle, play little role on relevant complex structures.
• We have omitted the strange quark Yukawa coupling y s for simplicity. It does play a role in nuclear physics.
• The only parameter from the lepton sector is y e . Other lepton sector parameters could affect η B via leptogenesis [16]. Since the source of η B is unknown, we prefer to list it as an independent cosmological parameter.
• The 13-dimensional parameter set could be further reduced, since only certain combinations appear in C A of Eq. (11). For example, only dimensionless combinations appear, so that M Pl could be removed by using it as the unit of mass. For the complexity boundaries arising from atomic and nuclear structures, v appears only in the combinations m f ≡ y f v (f = u, d, e) as fermion masses, so that for these boundaries x SM can be reduced to where Λ QCD is the QCD scale, which takes a value of ≈ 100 MeV in our universe. For naturalness arguments, however, we often prefer to use the more basic set of Eq. (9).
• In a theory with two Higgs doublets, such as the minimal supersymmetric standard model, the basic set of Eq. (9) should be expanded. For most purposes, we simply have to replace the electroweak vacuum expectation value v by two vacuum expectation values; for type-II two Higgs doublet theories, we have v u for the up-type Higgs doublet and v d for the down-type Higgs doublet.

The relevant complex structures
A crucial question for the complexity boundaries is what are the relevant complex structures. Our interest in a particular boundary depends on how important the corresponding complex structure is for explaining the structure of the physical world. Consider two extremes. The size of the cosmological constant, relative to the matter density of the universe when density perturbations become non-linear, leads to a boundary that determines whether large scale structure forms: A member on one side of the boundary will lead to formation of galaxies, while one on the other side leads to an inflating universe of isolated elementary particles. This clearly has important effects on the basic structure of the universe. As a second extreme example, consider the complex mesons with a b quark constituent. As the b quark mass, m b , is increased above the W boson mass, m W , the b quark decays so rapidly that B mesons cease to exist. This boundary of m b /m W ≈ 1, however, has a negligible effect on our environment, which is why we did not include the b-quark Yukawa coupling, y b , in x SM . Somewhat arbitrarily, we divide the relevant complexity boundaries into three classes according to how dramatic the environmental change is across the boundary: • Catastrophic boundaries change our universe into one that is essentially unrecognizable. In addition to Eq. (14), and Eq. (12), we would include the case that electroweak symmetry is broken dominantly by the Higgs potential v This is required so that the baryon asymmetry of the universe is not washed out by the sphaleron effects [17]. We also consider that the absence of any complex nuclei is catastrophic. In the simplified case that the only parameter that is varied from its Standard Model value is v, this boundary is [6] v For v larger than this boundary, the only stable nucleus is either p = uud or ∆ ++ = uuu.
• Violent boundaries separate members where a crucial complex structure of our universe is absent. For example, across the boundary where m p and m n are the masses for the proton and neutron, the neutron becomes stable and hydrogen unstable. Such a neutron-stable world would not have dense astrophysical objects fueled by nuclear energy release, such as main-sequence stars in our universe [18]. Another example of violent boundaries is that of vanishing deuteron binding energy Across this boundary, no nuclei form during big bang nucleosynthesis, so that the universe protonizes (or neutronizes). Stars could only burn via exotic triple proton reactions, with extremely high central densities.
• Substantial boundaries separate members where a crucial complex structure of our universe is drastically changed. For example, across certain boundaries stars may exist but are very different from those we see. For example, if the pp reaction pp → De + ν is not available to ignite stars, then protostars would collapse to a higher temperature before igniting via the pep reaction, ppe − → Dν. A more substantial change to stars occurs if the deuteron is beta unstable, D → ppe −ν ; stars could still burn, but only by using the helium produced during big bang nucleosynthesis. The existence of a stable diproton, the 2 He nucleus, would also change stellar nuclear reactions, shortening lifetimes of hydrogen burning stars. Changes of some nuclear energy levels, controlled by quark masses and α, could also lead to substantial changes of the abundances of various nuclear species, such as carbon and oxygen.
Each boundary gives a surface of special values for the parameters,x, which does not correspond to a surface of an enhanced symmetry. Among the ones listed, we find that most boundaries are clustered around a zone y u,d,e v ∼ αΛ QCD ∼ O(0.01)Λ QCD , where our universe also resides. This strongly suggests the existence of a complexity naturalness problem. Here, however, we focus more on the observer naturalness problem, which arises if the member describing our universe lies unusually close to the observer boundary. There are certain ambiguities in identifying which of the complexity boundaries compose the observer boundary. An important point, however, is that by using only the boundaries that certainly have disastrous effects on the environment, we can be on the conservative side in evaluating the existence of an observer naturalness problem. For this reason, we take only boundaries that are catastrophic or violent, rather than just substantial, to compose our observer boundary. We also select only the boundaries that allow us to derive reasonably accurate values forx, allowing a reliable estimate of the naturalness probability P .

Stability boundaries for neutrons, deuterons and complex nuclei
Following work by others, we focus on the boundaries across which we lose complex nuclei, Eq. (16), neutron instability, Eq. (17), and the deuteron, Eq. (18). We take y u , y d , y e , v/Λ QCD and α to be our parameters x i . The choice is motivated by the expectation that the relation of these parameters to those in the ultraviolet theory is relatively direct.
We now represent the boundaries of Eq. (16,17,18) in terms of the deviations of x i from the observed values x i,o . Let us first discuss the neutron stability boundary of Eq. (17). The instability of a neutron, or equivalently the stability of hydrogen, requires where we have neglected the neutrino mass. The neutron-proton mass difference, m n − m p , arises from both the strong isospin violating effect, δ d−u , and the electromagnetic contribution to the proton mass, To a first approximation, these quantities scale as Here we have taken the central value for δ d−u , and the variables with and without the subscript o represent, respectively, the values in our observed universe and those in an arbitrary member of the ensemble. In terms of the parameters x i , Eq. (19) can be written as Here, we have neglected a small dependence of Λ QCD on v as well as logarithmic evolution of the Yukawa couplings. For two Higgs doublet theories, 7 the Yukawa couplings in Eq. (21) should be replaced as where v ≡ v 2 u + v 2 d and tan β ≡ v u /v d . We next discuss the boundary of Eq. (18). This boundary corresponds to the stability of a deuteron under strong interactions, which requires 7 Here and below we assume type-II two Higgs doublet models when we discuss two Higgs doublet theories.
where m D is the deuteron mass. The binding energy B D depends on the quark masses as where B D,o ≃ 2.2 MeV is the observed deuteron binding energy. The parameter a is uncertain, but it is estimated in Ref. [6] to be a ≃ (1.3 -5.5) MeV using models of nucleon binding. This allows us to write Eq. (24) in terms of x i as For two Higgs doublet theories, the Yukawa couplings must be replaced as in Eqs. (22,23). We finally consider the condition for the existence of complex nuclei. The stability of complex nuclei requires the energy release for the β decay, n → pe −ν , to be smaller than the binding energy of nuclei per nucleon, E bin : where we have neglected the neutrino mass. Precisely speaking, E bin varies with a nucleus, receiving contributions both from nuclear forces and the Coulomb repulsion between protons. To a first approximation, however, E bin can be regarded as the same for all nuclei and being controlled purely by Λ QCD , taking the value of E bin ≃ 8 MeV in our universe. 8 Substituting this into Eq. (27), and using Eq. (20), we obtain The expression should be modified according to Eqs. (22,23) for two Higgs doublet theories.

Observer naturalness problem in the Standard Model and beyond
The region inside our observer boundary, O, is defined by Eqs. (21,26,28), with modification by Eqs. (22,23) for two Higgs doublet theories. To visualize the overall shape of this region, in to the three boundaries, especially when viewed on a logarithmic scale. In particular, they are very close to the neutron stability bound of Eq. (19). This implies that we have an observer naturalness problem. The degree of unnaturalness P depends on a theory, since the relation between {m u , m d , m e , α} and the fundamental parameters, as well as the form of the distribution function, depend on the ensemble we consider. However, we can still make a conservative estimate on P , based on the observation that the naturalness probabilityP obtained by fixing all the fundamental parameters except for one generally satisfiesP > ∼ P . In the present context, we can fix y u , y d , y e , α and Λ QCD , as well as tan β for two Higgs doublet theories. The conditions of Eqs. (21,26,28) then where we have used a = 2.2 MeV for illustrative purpose. The probability of v falling in this range,P v , then gives a conservative estimate of the naturalness probability P . Alternatively, we can fix v instead of Λ QCD . In this case, we have The probability of Λ QCD falling in this range,P Λ QCD , can also give a conservative estimate of P , since P < ∼ min{P v ,P Λ QCD }. Which ofP v andP Λ QCD we use is determined by which is easier to estimate and, in case both are easy to estimate, which gives a stronger bound on P . Let us first consider an ensemble in which the distribution of v is flat in v 2 , as in the Standard Model embedded into more fundamental theory at a high scale. In this case, we obtain an extremely small value ofP For M * ≈ 10 18 GeV, we obtainP v ≈ 10 −32 . The origin of this small value, however, is precisely the existence of the gauge hierarchy problem. We thus see the existence of a severe observer naturalness problem in these theories, but it is hard to disentangle from the fine-tuning naturalness problem.
We therefore focus on theories in which the conventional gauge hierarchy problem is solved. In these theories, the distribution of v is expected to be flat in ln v or 1/ ln v within an ensemble. To estimateP v , however, we need to know the range of v, which depends on how v is related to the fundamental parameters of the theory. To avoid model dependence coming from this, we here considerP Λ QCD , instead ofP v . The value of the QCD scale, Λ QCD , is determined from the strong gauge coupling constant at the cutoff scale, g 3 (M * ), through renormalization group evolution. The probabilityP Λ QCD can then be estimated if (i) the theory between Λ QCD and M * , together with the value of M * , is specified, and (ii) the distribution function for g 3 (M * ), including the range of g 3 (M * ), is given. 9 Specifying the theory between Λ QCD and M * is important because the existence of colored states whose masses are associated with v may increase the correlation between v and Λ QCD , enhancingP Λ QCD . The distribution and the range of In this case, we would obtainP Λ QCD estimated using Eq. (30) to be infinitely small. To avoid this unphysical conclusion, we can introduce an arbitrary cutoff c on the distribution, 1/g 2 3 (M * ) < ∼ c. Alternatively, we can consider that the distribution of g 3 (M * ) is flat in g 2 3 (M * ). In this case, restricting the range to be g 2 3 (M * ) < ∼ c ′ gives a finite answer toP Λ QCD . The value ofP Λ QCD depends on c ′ , but we can make a reasonable conjecture on the value of c ′ , e.g. c ′ ≈ 1 or ≈ 16π 2 /b 3 . In Fig. 6, we plot the value of g 2 3 (M * ) as a function of Λ QCD /Λ QCD,o obtained using oneloop renormalization group equations for non-supersymmetric theories (left) and supersymmetric theories (right). In non-supersymmetric theories, M * is taken to be 10 14 GeV, which is the scale where the SU(3) and U(1) gauge couplings almost meet. (This is the scale where the three gauge couplings would almost meet if five Higgs doublet fields or a vector-like fermion with the Higgs quantum numbers were introduced at the weak scale.) In supersymmetric theories, we take M * to be the unification scale, 2×10 16 GeV, and the scale of superparticle masses to be m SUSY ≃ 1 TeV. Dependence of the results on these parameters, however, is weak. From the figure, we find the probability of Λ QCD falling in the range of Eq. (30) to bẽ for non-supersymmetric theories and for supersymmetric theories. Here, we have taken the distribution function to be flat in g 2 3 (M * ). The value of c ′ is uncertain, but we expect c ′ > ∼ 1. The numbers in Eqs. (31,32) depend on a in Eq. (25), through the dependence of the lower value of Eq. (30) on a. For a = 1.3 MeV (5.5 MeV), 0.016 and 0.038 in these equations become 0.020 and 0.046 (0.012 and 0.029), respectively. Note that the analysis here provides the most conservative estimate for the level of an observer naturalness problem existing in theories beyond the Standard Model (in which the conventional gauge hierarchy problem is solved, so thatP v can be larger thanP Λ QCD ): In theories where the gauge hierarchy problem is not solved, the degree of the observer naturalness problem is much more severe, P < ∼Pv ≪ 1. We thus find that the bounds on P derived here provide evidence for an observer naturalness problem in the Standard Model and beyond.
To obtain a conservative estimate on P , we have used here only P < ∼ min{P v ,P Λ QCD }. Physically, the case with P ≈ min{P v ,P Λ QCD } corresponds to the situation where we have the "best" model of flavor, i.e. the observed quark and lepton masses in units of the weak scale are reproduced essentially without any free parameter. This is, however, not the case for many theories of flavor, which will have P < min{P v ,P Λ QCD }, as discussed in the next section.

Naturalness Probabilities and Theories of Flavor
The observed values of the first generation masses, m u,d,e , are special: Fig. 5 shows that they are close to the boundaries of neutron, deuteron and complex nuclei stability. To judge whether this is likely to be accidental we need to evaluate the naturalness probability, but this depends on the theory of flavor. We begin by considering ensembles of theories in which α and v/Λ QCD are fixed, but the flavor observables x F have some probability distribution f (x F ).
In the Standard Model the relevant flavor observables are the Yukawa couplings: x F = y u,d,e . While they are all very small, of order 10 −6 -10 −5 at unified scales, they are all close to the maximal values allowed by the neutron, deuteron and complex nuclei boundaries. Of particular important, the observed values are especially close to the neutron (hydrogen) stability boundary.
While y e is about a factor of 3 from this boundary, in the y u -y d plane the distance from the observed point to this boundary corresponds to a variation in the coupling only of where z ≡ m u /m d and A ≡ (δ d−u −m n +m p +m e )/δ d−u evaluated in our universe. Here, the central value is obtained with z = 0.56 and δ d−u = 2.26 MeV. The electromagnetic interaction raises the proton mass above the neutron mass by about an MeV. Given that quarks can have masses up to 100 GeV or more, it is remarkable that the m d − m u mass difference just overcompensates the electromagnetic shift to make the hydrogen mass only ≃ 0.78 MeV smaller than the neutron mass. The deuteron and complex nuclei stability boundaries are also close by, corresponding to changes in the Yukawa couplings by factors of ≈ (1.5 -3) and ≈ 2, respectively. It is important to numerically evaluate these accidents; for illustration, we consider two simple distribution functions. If f (y u,d,e ) are flat and non-zero in the range of y u,d,e from 0 to 1, then the corresponding naturalness probability for this ensemble, using Eq. (5), is P F ≈ 10 −16 ; while if f (log 10 y u,d,e ) are flat and non-zero in the range of log 10 y u,d,e from −6 to 0, then P F ≈ 10 −4 . As expected from Fig. 5, the volume of parameter space closer to the special points on the observer boundary than the measured point is very small compared with the expected total volume of parameter space in the ensemble. From the viewpoint of neutron instability and deuteron and complex nuclei stability, the Standard Model description of flavor is highly unnatural.
Theories of flavor that go beyond the simple Yukawa coupling parameterization of the Standard Model typically involve further symmetries, such as flavor or unified gauge symmetries. While the set of flavor observables x F in these theories can be smaller than that in the Standard Model, all known models do involve free parameters. If a theory could be found in which y u,d,e are precisely predicted, i.e. if variations in y u,d,e are less than the distances to the special points on the observer boundary when x F vary, then such a theory would have P F ≈ 1. However, if y u,d,e vary significantly (and independently) as x F vary in the ensemble, as in most theories of flavor, then the theory is likely to have P F (much) smaller than of order unity. The best hope for a significant improvement in naturalness is then to obtain a successful prediction for the ratios of the first generation Yukawa couplings -symmetries that successfully predict the y u , y d , and y e ratios could show that our universe lying so close to the observer boundary is just accidental. This is, however, not so easy, especially because of the extreme closeness of the observed point to the neutron stability boundary. Moreover, since the observer boundary involves the combinations y u,d,e v/Λ QCD , the normalization of y u,d,e is coupled to the value of v/Λ QCD . Below, we do not take v/Λ QCD as fixed, but consider it to vary in the ensemble as discussed in the previous section.
In certain theories of flavor, for example with Abelian flavor symmetries, the smallness of y f (f = u, d, e) follows from a single small parameter of the theory where ǫ ≪ 1 while c f are of order unity. The powers p f are not free parameters of the theory, but result from a certain judicious choice of Abelian charges for all the quark and lepton fields [21]. The ratios of the Yukawa couplings can then be just numbers of order unity for a fixed value of ǫ (and tan β for two Higgs doublet theories). Let us consider, for example, that the coefficients c f vary in the range 0 ≤ c f ≤ 2c f,o with a flat distribution on a linear scale, or in the range log 10 c f,o − 1/2 ≤ log 10 c f ≤ log 10 c f,o + 1/2 with a flat distribution on a logarithmic scale. In this case, the extreme closeness of the neutron stability boundary gives P F ≈ 1/18 and 1/36, respectively, for a fixed value of ǫ (and tan β). In addition to this, there is the issue of the normalization of the masses m u,d,e , arising from a variation of v/Λ QCD . Assuming that the distribution function is flat in g 2 3 (M * ), this leads to a further reduction of the probability at least by a factor ofP < ∼ 1/35 and 1/15 for non-supersymmetric and supersymmetric theories, respectively. (These numbers are obtained by requiring that v/Λ QCD should be within a factor of ≈ (2 -3) from the observed value so that its variation can be absorbed into changes of c f , without a large extra cost in P F . The range of g 2 3 (M * ) here has been chosen to be 0 ≤ g 2 3 (M * ) ≤ 1.) Possible variations of ǫ (and tan β) may or may not lead to further reductions of the probability. We thus conclude that theories with Abelian flavor symmetries considered here have from physics of flavor alone. To increase P from flavor physics, it is necessary to predict ratios of y u,d,e to better than the factor of ≈ 2 we used in the above estimates. A more elaborate charge assignment for the quark and lepton fields under an Abelian flavor symmetry is not sufficient, since it will leave the relative coefficients of order unity undetermined. A non-Abelian flavor symmetry, or unified gauge symmetry, could improve P by successfully predicting y u : y d : y e as Clebsch factors. For example, y u : y d : y e = 1 : 2 : 1 at the unified scale agrees with inferred values of m u,d,e , providing the eigenvalues are not affected much by mixing to heavier generations. A simple theory of flavor incorporating such a relation could lessen the significance of the observer naturalness problem of Eq. (36). However, uncertainties from QCD are too large for us to know if the relation really puts us sufficiently close to the neutron stability boundary, so that the problem is simply an accidental consequence of symmetry. Moreover, in conventional theories of flavor we expect simple Clebsch factors to apply to the heaviest generation not the lightest. This is because the lightest generation typically gets contributions from nontrivial matrix diagonalizations, so that the relation would have to involve more than one generation. For example, the inferred values of m u,d,e are consistent with unified boundary conditions y u = y e and y d = 0, provided the down quark mass arises from diagonalizing a symmetric 2 × 2 matrix that fully accounts for the Cabibbo angle. This requires, however, experimental inputs of the strange quark mass and the Cabibbo angle, so that it does not provide a real symmetry solution to the problem. We find it rather difficult to have a symmetry understanding for y u : y d : y e , and even if we had, unnaturalness still arises from the coincidence between v and Λ QCD , given by Eqs. (31,32). We conclude that, in the absence of a convincing and significant progress in flavor physics, we have unnaturalness associated with flavor at the

Environmental Selection
We have seen that theories of particle physics and cosmology are likely to have observer naturalness problems, signaled by P ≪ 1. The problems arise in an ensemble if the observed point is close to the observer boundary. The situations in which this happens can be classified into the following three cases: the observed point is (a) a typical point within a small observer region O, (b) close to the boundary of a small observer region O, and (c) close to the boundary of a large observer region O, which may or may not be a closed region in the parameter space. These cases are depicted schematically in Fig. 7. The heart of the observer naturalness problem is the coincidence: why is the observed point so close to the observer boundary, which is a very special region in the parameter space? To solve this problem, we must make the observer boundary really special -the existence and location of the observer boundary should somehow affect the process of selecting a member in the ensemble as the one describing our universe. This leads us to consider environmental selection, effects sensitive to the existence of an observer. In this section we study how environmental selection on a multiverse can solve observer naturalness problems in general, and under what circumstances (and in what sense) we can identify evidence for it. We also discuss possible implications of this solution in identifying the correct theory describing our universe.
The nature of the observer naturalness problem is dramatically altered if the members in an ensemble are physically realized in a multiverse. Until now, the ensemble has been a useful fictitious mathematical device to study problems of naturalness; now we assume that each member of the ensemble is physically realized as a universe in spacetime. The fundamental theory of nature is assumed to have a huge number of vacua, the landscape of vacua, so that physics at low energy may be described by many possible effective theories. In fact, we are interested in a subset of these theories, T , that lead to physics described by the parameters x i discussed in section 4, with the parameters x i taking different values in different universes. We assume that it is possible to define a distribution function of the multiverse,f T (x i ), such that, averaging over the entire multiverse, we obtain We stress that this is the average value on the entire multiverse, independent of whether universes contain observers. If the population mechanism, including any relevant volume factors, were independent of x i , thenf T (x i ) would represent the distribution of vacua in the theory underlying the landscape. However, in general it includes both relevant population and volume factors. 10 We first focus on a single theory, so that we can drop the index T until later subsections.
Environmental selection on the multiverse involves two key points that give a central role to measurements made by observers • Universes that have parameters outside the region O do not contain observers, hence values of x i outside O cannot be measured. When discussing naturalness of the observed universe, we should not be asking questions about the entire multiverse but only about the observer region O.
• The number of observers in universes in O with x i in the region Each universe is to be weighted by an observer distribution n(x i ), the factor associated with the number of observers that develop in a universe with parameters x i . We make no attempt to define an "observer." These two points are really two aspects of a single selection process, with n(x i ) defined over the entire space of x i . We, however, find it useful to consider that n(x i ) rapidly drops to zero at certain observer boundaries, so that the region outside these boundaries does not affect the naturalness of a multiverse.
In the approximation that the observer boundary is sharp, the naturalness probability of Eq. (3) is then replaced by where the integral in the denominator implies that x is integrated only in the region O. The expression for higher dimensional x space is also obtained similarly. We stress that here and throughout the rest of the paper The entire region of parameter space of the multiverse is now irrelevant; the only question is whether we are typical observers in universes in O. To calculatef it is necessary to know both the landscape of vacua and the population mechanism. On the other hand, n is independent of the landscape and, with sufficient understanding of the physical environment for observers, could be calculated in principle from the low energy theory. We take the practical viewpoint that both f and n are unknown, and hence give ourselves the freedom to assume any reasonable smooth distribution for f . It is then clear that distributions can be found that make P O ≈ 1, solving the observer naturalness problems whether of the form of (a), (b) or (c) of Fig. 7. A more detailed description of each of these three types of solution is given in section 6.1. In section 6.3 we consider the relative probability of different theories, T , solving an observer naturalness problem. It is quite clear that there are many origins for a significant x i dependence of f (x i ) = f(x i ) n(x i ) in the observer region O. Forf these are the distribution of vacua in the underlying theory as well as relevant volume and population factors, while for n a variation in x i could change the density of galaxies, stars, nuclei and so on that are relevant for observers. In section 6.2 we investigate a further effect: it could be that f or the shape of O depends on more variables, x b , than the ones we focus on, x a , and that when these extra variables are integrated out the resulting effective distribution acquires further x a dependence. Finally, a crucial issue is how to evaluate evidence for environmental selection. In section 6.4 we argue that, for any observer naturalness problem, a low value for the naturalness probability, P , provides evidence for environmental selection. More precisely, for the known set of (simple) symmetry theories, T , the evidence is determined by the largest value of P T If P max ≈ 1 there is no evidence, but as P max decreases so the evidence becomes more substantial.

Three manifestations of environmental selection
Using a set of Lagrangian parameters x i of some low energy theory, an observer naturalness problem might appear in the three ways illustrated in Fig. 7. In each case environmental selection may be at work, but the description of the solution to the problem is different. Suppose we find an observer naturalness problem of the form of (a) in Fig. 7. This suggests that the distribution function, f , is (approximately) constant in the Lagrangian basis x i , since the replacement of P → P O then completely eliminates the observer naturalness problem. Assuming a constant f , the naturalness probability P O of Eq. (37) becomes where ∆x is the range of x in the observer region O. Similarly, for multiple parameters the naturalness probability of Eq. (5) is replaced by where V n (O) is the volume of parameter space in the observer region O. The elimination of observer naturalness problems by the replacement P → P O is illustrated in Fig. 8. The enormous increase in the probability here arises simply because all of the universes that lie outside the region O are cut out of the denominator. The situation in Fig. 8 is the case of the overall picture for the cosmological constant Λ. For conventional naturalness, consider an ensemble with a distribution function f that is constant in Λ, which is the case in most theories. The probability that a member of this ensemble has a small value of Λ o near the special value zero is P (Λ) = Λ o /M 4 * , where M * is the fundamental scale. On the other hand, environmental selection, arising from a multiverse withf and n assumed flat in Λ, replaces Λ max ≈ M 4 * by ∆Λ ≈ Q 3 T 4 eq , giving The vast majority of universes have a huge cosmological constant; but that is irrelevant because they contain only a dilute gas of elementary particles undergoing inflation. We should cut all these universes out of the measure, and consider only those where large scale structure forms, allowing the possibility of complex observers. This eliminates (or greatly ameliorates) the problem associated with the cosmological constant: P (Λ) ≪ 1 → P O (Λ) ∼ 1. A more refined analysis includes the effects of nontrivial n(Λ) near the observer boundary [23]. Let us now consider observer naturalness problems in the forms of (b) or (c) of Fig. 7. In these cases, even after selection we are apparently left with some residual naturalness problem P O ≪ 1 in some particular theory T . What does this imply? Does this mean that the observer naturalness problem in these forms cannot be solved by environmental selection?
The answer is clearly no. To understand what can be going on, it is important to realize that our knowledge can often be very incomplete. Imagine that we have assumed some distribution function f for an ensemble and found an observer naturalness problem in the form of (b) or (c) of Fig. 7 in the basis where f is constant. This seems to imply that the observer naturalness problem cannot be solved by a simple cutout procedure illustrated in Fig. 8. However, do we really know that we have identified all the relevant complexity boundaries, and thus identified the correct observer boundary? It could be that we have missed some relevant boundary and, after taking it into account, P O may become order unity. This is illustrated in Fig. 9 for the case of (b) of Fig. 7.
Another possibility is that our initial assumption on the distribution function may not be correct. In practice, when we consider a theory, we start by assuming a "natural" distribution function f (x i ), which is often taken to be constant in "Lagrangian parameters" x i . However, there are many reasons why the distribution function in this parameter basis may not be constant even approximately. First, the Lagrangian parameters x i of the low energy theory may be functions of the Lagrangian parameters y j in the fundamental theory, x i = x i (y j ), and the distribution function may be constant in y j rather than x i . Second, in the multiverse picture, a mismatch between the naive and true distribution functions may also result from the population mechanism depending on x i , or may be induced by environmental selection via a nontrivial n(x i ). Finally, P O ≪ 1 ? environmental selection may depend not only on x i but also on other variables, inducing an extra dependence of f on x i , as we will discuss in the next subsection. Hence, the distribution function f (x i ) may have a strong dependence on the Lagrangian parameters x i of the low energy theory, and in this case the situations of (b) or (c) in Fig. 7 actually correspond to typical observers with P O ≈ 1, expected from environmental selection.
The situation in which naive and true distribution functions differ significantly is illustrated in Fig. 10 for the case of (c) of Fig. 7. (For the discussion here we assume that all relevant parts of the observer boundary have been correctly identified.) There are two (equivalent) ways to describe this situation. One is to go to the basis in which the true distribution function is constant. In this basis, which we may call the "mathematical basis," the observed point is a typical point in O, and the parameters chosen, y i , can in general be nontrivial functions of the "Lagrangian parameters" x i . In practice, the transformation from x i to y i should be inferred from the observed data and a calculated observer boundary. Another way to describe the situation is to keep using Lagrangian parameters, or parameters that have the most direct or intuitive physical meaning at low energies, x i . In this basis, which we may call the "physical basis," we have a nontrivial distribution function f (x i ). The distribution function is peaked towards the region outside O, so that a typical observer in O measures x i close to the observer boundary.
In the physical basis, x i , it is very convenient to introduce a "probability force," which is simply the gradient of the probability distribution: where ∇ ≡ (∂/∂x 1 , ∂/∂x 2 , · · ·). Within the region O, this vector field gives an indication of what values of physical parameters are most likely to be measured in the multiverse. If the vector field indicates a flow towards an observer boundary, observers should be living close to the corresponding boundary. An example of the probability force field is depicted in the physical basis plot of Fig. 10.

Effective distributions from "integrating out" parameters
What are the possible origins of the probability force? One is a nontrivial prior distribution function f prior , which arises from a mismatch between the physical and mathematical bases, as discussed previously. In practice, however, this is not the only source of the probability force. The key is that when we discuss physical predictions in the multiverse picture, we typically choose a subset, x a , of the entire parameter set, x i , and study probabilistic predictions in the space of these chosen parameters x a . An important point here is that when we focus on x a , we cannot simply ignore the other variables x b (i = {a, b}); they must be integrated out. This provides an effect on the distribution function defined in the subspace x a , and thus modifies the probability force in x a space.
To illustrate this idea, let us consider the simplest example of a constant f prior , i.e. the case in which the physical and mathematical bases coincide. In this case the distribution of observers is given by where c = f prior is a constant, and the probability force in x i space is zero, F = 0. This, however, does not mean that the probability force is zero if we consider only a subset of the variables, e.g.
x 1 , · · · , x N −1 . When we focus on the variables x 1 , · · · , x N −1 (as we focused only on y u , y d , y e , v/Λ QCD and α out of the 13 parameters of Eqs. (9,10) in sections 4 and 5), we must integrate out the other variable x N . Now, in the multiverse picture with environmental selection, the only relevant universes are those within the observer region O, implying that integrals should be performed only over O. The domain of x N integration is then determined by the observer boundary, which is generically written as x min 11 giving where The probability force defined in x 1 -· · ·-x N −1 space is then obtained by Eq. (43) with f replaced by f eff : F eff = ∇f eff , where ∇ = (∂/∂x 1 , · · · , ∂/∂x N −1 ). This generically gives a nontrivial force, F eff = 0.
In general, when we consider physics of the landscape in the subspace of x a (a = 1, · · · , n), we must integrate out the other variables x b (b = n + 1, · · · , N) to obtain the effective distribution function, and thus the probability force, defined in x a space. The domain of integration is given by the observer region O, leading to where f prior =f prior n is the prior distribution function defined in the entire parameter space x i , whose x b dependence could arise from bothf prior and n. The important point here is that the effective distribution function f eff in x a space is not obtained simply by "neglecting" the other variables x b in f prior , i.e. by setting x b to the observed values in f prior : Since the observer region O can in general have a complicated shape in x i space, the effective distribution function can have a quite different form than the expression in the right-hand-side of Eq. (47). The effective probability force defined in x a space is then given by where ∇ = (∂/∂x 1 , · · · , ∂/∂x n ).
The argument here suggests that it is a rather common phenomenon to have a nontrivial probability force when we focus only on a subset of the entire parameter set x i , which is almost always the case. Then, if the resulting probability force is strong, we are likely to encounter an observer naturalness problem in the form of (b) or (c) in Fig. 7.

Cut factor and comparisons between different theories
In general there are many theories T that lead to the low energy Lagrangian with parameters x and that are described by a multiverse distribution f T (x). As an example, x may be the Yukawa couplings of the up and down quarks and electron, y u,d,e , and T may label the various theories of flavor. The number of observers in the multiverse who are governed by theory T is A typical observer will be governed by the theory T that has the maximum value of n T (assuming that several T do not have comparable n T ). The absolute normalization of n(x) is unimportant, since we are only interested in relative numbers of observers governed by different theories where N T = f T (x) dx is the total "number" of universes (including volume factor weights) described by theory T andC Suppose now that n(x) is relatively flat over O so that it is a good approximation to take it constant. Multiplying this constant byC T then gives In this case there are a factor C T more universes described by T in the multiverse than in O. Environmental selection solves observer naturalness problems at a cost of removing a factor C T of these universes -we call C T the cut factor for theory T . Consider, for example, the following two theories -the Standard Model where only the Higgs mass-squared parameter scans with a distribution function constant in |m 2 h | and a theory beyond the Standard Model where only the weak scale scans with a distribution function constant in 1/ ln v. Considering selection by Eq. (29), the cut factors in these theories are O(10 32 ) and O(100), respectively. If we take this naively, it seems to suggest that the latter theory is preferred over the former, and we might say that the Standard Model is less likely since it involves the cost of a much larger cut factor. 12 (This result is similar to that of the conventional naturalness argument, although here observable universes always have v ≪ M Pl due to selection effects.) Of course this is not rigorous -our ignorance of N T , as well as other possible selection effects, makes the argument based only on the cut factor unreliable; without knowledge of N T there is 12 Here we do not consider the cut factor arising from selection of the cosmological constant, Λ < ∼ Q 3 T 4 eq . In fact, the size of the cut factor associated with the cosmological constant does not depend, e.g., on the scale of supersymmetry breaking if the constant term in the superpotential, W 0 , scans up to the fundamental scale, M * , with the distribution flat in |W 0 | 2 , as suggest by string theory [24]. This part of the cut factor is then of order M 4 * /Λ ≈ 10 120 regardless of the theory. (An exception for this is given by supersymmetric theories in which both supersymmetry and R symmetry are broken at low energies.) no real reason to prefer a theory beyond the Standard Model over the Standard Model (other than considerations based on other physics such as gauge coupling unification and dark matter). Nevertheless, the relative cut factors for two theories do contribute to the relative number of observers described by each theory, as in Eq. (50), hence we may describe a large cut factor for some theory as a cost in its solution to an observer naturalness problem.

Evidence for environmental selection
We have seen that observer naturalness problems, which appear in the form of (a), (b) or (c) of Fig. 7, can be solved in general by environmental selection. Turning this argument around, we find that effects of environmental selection can show up in one or both of the following forms: (i) Amazing coincidences: we are living in a region of parameter space that admits complex observers, which, however, is only a (very) small portion of the entire parameter space.
(ii) Living on the edge: we are living (very) near an "edge" of the parameter region O that admits complex observers, i.e. physical parameters take values that correspond to a point very close to the observer boundary, compared with the extent of the region O.
In fact, how environmental selection manifests itself depends on the basis -we have seen that a phenomenon that appears as (ii) in one basis can appear as (i) in another basis, or vice versa. However, since we are often presented with a natural basis in which the physical meaning of parameters is most direct and/or intuitive, it is meaningful to consider the manifestation of environmental selection in that particular basis (physical basis). As we have seen, the manifestation then takes the form of (i) if the effective distribution function f eff (x) is (nearly) constant over the observer region O, while it takes the form of (ii) or a combination of (i) and (ii) if f eff (x) is peaked towards a boundary of O. There are many possible origins for a nontrivial form of f eff (x): an x dependence of the landscape vacua, the dynamics of the population mechanism, the observer factor n(x), and environmental selection acting on variables other than x.
Can the observation of one or both of the above two phenomena, (i) and (ii), be viewed as evidence for environmental selection? In a given theory, T , the answer is yes, since otherwise it is very difficult to explain these features. Even though there is an observer naturalness problem associated with a very small value for P T , after environmental selection a typical point within O results, i.e. P T → P O,T ≈ 1.
The smaller the original P T , the more amazing the coincidence, or the closer to the edge we are living. The significance for the evidence for environmental selection is then quantified by the size of the naturalness probability P T , with a smaller value of P T corresponding to stronger evidence.
In practice, however, we do not know beforehand the correct theory describing our universe. Suppose we encounter a situation described by (i) or (ii) above. Do we conclude that we have found evidence for environmental selection, or that the theory we are considering is simply wrong? Whether an observation of the phenomenon described in (i) or (ii) -an observer naturalness problem -can be viewed as evidence for environmental selection depends on whether one can find an alternative theory in which the problem does not arise. A simple alternative theory without the naturalness problem may provide a better description of our universe. On the other hand, it is possible that we cannot find such a theory, or can find only theories that are significantly more complicated. Then we may conclude that environmental selection provides the best explanation of the phenomenon, so that we provisionally accept the multiverse theory.
In principle, with competing theories to describe nature, the evidence for environmental selection at a particular observer boundary is given by the largest value of P T associated with that boundary. However, it may be that a theory with large P T is very complicated or ad hoc, so that it does not appear to be an adequate solution to the observer naturalness problem. In this case it may be judged that the evidence for environmental selection is better represented by a smaller value of P T associated with some simpler theory.
The discussion given above illustrates why the cosmological constant is a powerful argument for environmental selection: in all known quantum field theories the naturalness probability is extremely small (P T ≈ 10 −120 -10 −60 depending on the presence of weak scale supersymmetry and the nature of its breaking), implying that the largest value of P T is of order 10 −60 or smaller. The absence of a simple theory with P T = O(1) is crucial in this argument. A similar argument may also be made for the quark and lepton masses discussed in sections 4 and 5. We have argued that all known theories of flavor are quite inadequate to explain the relevant observer naturalness problems, leading to P T < ∼ (10 −3 -10 −2 ). Although this is numerically less impressive than the case of the cosmological constant, it is nevertheless very important. A single piece of evidence for environmental selection, no matter how significant, could be completely erased by the discovery of a single new theory. The more independent pieces of evidence for environmental selection, the more convincing the overall picture becomes.
One might think it always difficult to "confirm" the absence of alternative theories which do not have the problem. Indeed, the absence of such theories can in general be inferred only from negative results of theoretical search. However, in the case that the observer naturalness problem is related to a fine-tuning naturalness problem, it is possible that we can be convinced rather firmly that the problem does in fact exist. This will be the case, for example, if we do not see any deviation of gravity from Newton's law down to a scale (much) smaller than O(100 µm), since it will tell us the absence of a physical threshold that can control the observed value of the cosmological constant. In sections 8 and 9, we will also argue that the observation (or non-observation) of new physics at the TeV scale may also be viewed, depending on what we will see, as evidence for the existence of an observer naturalness problem, and hence environmental selection.
We stress that environmental selection can provide numerical predictions that are difficult to obtain in other ways. Specifically, this occurs if the effective distribution function f eff has a nontrivial form in the physical basis. In this case the physical parameters take values corresponding to a point close to the observer boundary, giving nontrivial predictions. An example of predictions made possible by environmental selection is given by the stability boundary of the desired electroweak phase of the Standard Model [15]. Suppose that the Standard Model is valid up to some high scale M * near the Planck scale and that the weak scale v results from environmental selection. Suppose further that the Higgs quartic coupling at M * , λ h, * , and the top quark Yukawa coupling at M * , y t, * , vary from one universe to another. There is then an observer boundary O(λ h, * , y t, * ) = 1 corresponding to sufficient stability of the desired electroweak phase. Now, if the distribution function f eff (λ h, * , y t, * ) is strongly peaked towards the phase boundary then our universe is expected to be close to this edge. In particular, if the peaking is stronger in λ h, * than in y t, * then the most probable point on the phase boundary has M Higgs ≃ 107 GeV and m t ≃ 175 GeV. Discovery of the Higgs boson near this mass, together with the absence of any new electroweak physics, would then provide evidence that our universe is near the edge of this observer phase boundary, and hence of environmental selection. In general, if we find ourselves living close to an observer boundary and if we do not have a simple (alternative) theory explaining that fact, then we may regard it as evidence for environmental selection.

Predictions for m u , m d and m e from Environmental Selection
The observer naturalness problem associated with the stability of neutrons, deuterons and complex nuclei was introduced in section 4, and is illustrated in Fig. 5. We showed that, no matter what the theory of flavor, there is always a factor in the naturalness probability,P , from the dependence on v/Λ QCD . In section 5 we argued that there is a factor, P F , in the naturalness probability that is highly dependent on the theory of flavor. In all known theories P = P FP < ∼ (10 −3 -10 −2 ). In this section we argue that this naturalness problem can be solved by environmental selection; furthermore there are several possible solutions with different consequences. We begin by restating the naturalness problem in terms of Standard Model parameters. For the neutron and complex nuclei stability boundaries a crucial quantity is where C I ≈ (0.5 -2) and C α ≈ (0.5 -2) are strong interaction coefficients, and we choose a definition of the QCD scale such that Λ QCD = 100 MeV. Throughout we use approximate ranges for m u,o and m d,o from Ref. [20]. For complex nuclei the binding energy per nucleon is E bin = C B Λ QCD , with C B ≈ 0.08 another strong interaction coefficient. The neutron and complex nuclei boundaries are then described by the inequalities In the approximation that the deuteron binding energy, B D , is linear in m u + m d in the region of interest, the stability boundary for the deuteron can be written as where C 1,2 are two further strong interaction coefficients. These three boundaries involve just four independent combinations of Standard Model parameters, y u,d,e v/Λ QCD and α.
Much of the observer naturalness problem arises because the three terms that contribute to m n − m p − m e in Eq. (54) are comparable, as shown in Fig. 11, even though Yukawa couplings and the ratio of mass scales v/Λ QCD are expected to range over many orders of magnitude. The electron mass, y e v ≃ 0.5 MeV and the electromagnetic mass splitting δ EM ≡ C α αΛ QCD ≃ 1.0 ∓ 0.5 MeV differ by only about a factor of 2. They both stabilize the neutron, and hence an unstable neutron requires y d > y u . Not only is this the case, but the isospin breaking term δ d−u ≡ C I (m d − m u ) = 2.3 ± 0.5 MeV only just over-compensates the negative terms to give a net available energy for neutron beta decay of just 0.8 MeV, an amount that is also very close to the electron mass and electromagnetic terms. In any theory of flavor where y u,d,e are determined by symmetries, these numerical coincidences, allowing neutron instability and the existence of complex nuclei, must simply be accidents. Furthermore, these accidents also involves v/Λ QCD and α. The other Standard Model combination, (y d + y u )v/Λ QCD , which could also range over many orders of magnitude, is numerically within about a factor of 2 of the deuteron stability boundary. Finally, as is apparent from Fig. 5, the observed parameters are actually much closer to the neutron stability boundary than a factor of 2. The perpendicular distance in coupling parameter space is (10 -30)%, depending on the values of C I and y u,o /y d,o . As discussed in previous sections, some aspects of these accidents could arise from a theory of flavor that determines y u,d,e to be small and in the approximate ratio 1 : 2 : 1 at a unified scale; but the accidents associated with the value of v/Λ QCD and the closeness to the neutron stability boundary remain.
Environmental selection cuts out large region of parameter space where neutrons are stable, or deuterons or complex nuclei are unstable, greatly reducing the observer naturalness problem. The only remaining question is whether our observed universe is a typical universe within the observer region O shown in Fig. 5. This depends on the probability distribution for y u,d,e v/Λ QCD and α, and on whether O has been correctly identified, allowing several solutions of the problem with different implications. We begin by taking α fixed to its observed value while allowing y u,d,e , v and Λ QCD to scan. Since there are only two scales v and Λ QCD appearing in the nuclear stability observer boundaries, we can take Λ QCD = 100 MeV without loss of generality, setting the unit of mass. Furthermore, since v always multiplies a Yukawa coupling, environmental selection for the light fermion masses can be discussed in terms of a distribution function f (m u,d,e ) without loss of generality. 13 For a given f we can compute m u , m d and m e and compare them with the values observed in our universe, m u,o , m d,o and m e,o . We can also compute how close a typical universe in O is to the neutron stability boundary.
Predictions for m u,d,e involve the mass scales that arise in the observer boundaries, namely the maximum value of m + = m d + m u allowed by deuteron stability for a = (1.3 -5.5) MeV, the binding energy per nucleon in stable complex nuclei and the electromagnetic contribution to the proton mass While all three mass scales are proportional to Λ QCD , δ EM is significantly smaller than m +max and E bin . Hence it is important to see which of these mass scales enter the predictions for m u,d,e . We consider three situations that solve the observer naturalness problem: I. An important part of the observer boundary is missing.
II. The probability distribution is flat in mass space, dN = A dm u dm d dm e . In this case the closeness to the neutron stability boundary discussed above is accidental. not already excluded, for example using the threshold for the pp reaction. Furthermore, to understand our closeness to the neutron stability boundary, it would be necessary for other cuts to approach our universe with a similar closeness on a logarithmic scale. While we can certainly identify physical processes that would introduce extra boundaries, we are unable to argue that they induce catastrophic changes rather than just the substantial changes discussed in section 4.
For the second case, II, suppose that the distributions of m u , m d and m e are flat on linear scales, so that the effects of environmental selection can be understood from the cuts of the observer boundaries drawn in the right panels of Fig. 5. The observed masses are relatively typical within O, so that P O is not much smaller than unity, and the naturalness problem is largely solved. In fact, the observed masses are still close to the neutron stability boundary, even in the right panels of where m ± = m d ± m u . In these equations the analytic expressions are obtained without including the effect of the complex nuclei boundary, and the numerical range corresponds to a = (1. Of course, case II does imply that the closeness to the neutron boundary is accidental, and the rest of this section is devoted to understanding this closeness. A peaked distribution function can also give m e ∝ δ EM , rather than ∝ m +max , leading immediately to an understanding of the lightness of the electron.
For case III, inside O the distribution f is peaked towards the neutron stability boundary, allowing us to explore the consequences of an environmental explanation for why m u,d,e are so close to this boundary. We call this stability boundary the n surface -it is a 2-dimensional plane in the 3-dimensional space of masses, m u,d,e . Within O, the probability force F = ∇f can be resolved into components parallel, F , and perpendicular, F ⊥ , to the n surface. We assume that F ⊥ points towards the n surface. The F field will determine the most probable location on the n surface.
The n surface has a triangular shape, as shown in Fig. 12. Two of the edges of the triangle correspond to edges of physical space, m u,e = 0, while the other edge corresponds to the intersection of the n surface with the D surface, the boundary for deuteron stability. We call these three edges the u, e and D edges, and label the three vertices of the triangle as ue, uD and eD, as shown in Fig. 12. The F field will determine where on the n surface triangle the distribution f is maximized; there are three possibilities: III-1. In the interior, not close to an edge.

III-2.
On an edge, not close to a vertex.

III-3. On a vertex.
On a linear scale, the n surface is small compared to the expected available parameter space, so that it is unlikely thatf would have a sharp peak in the interior of the triangle. However, the observer factor, n, may vary significantly across the triangle inducing a maximum in the interior. For case III-2, as we move along an edge, f rises to reach a maximum and then falls. As an example consider the distribution in O to be where A is a normalization constant and p is positive. This situation would arise if the populated landscape has a low probability to yield a universe with large breaking of isospin, or if there are more observers in a universe as isospin is restored. In a slice through parameter space at constant m e , the resulting probability force, F − ≡ ∇f , is towards and perpendicular to the n surface. However, in the full 3-dimensional space F − is not perpendicular to the n surface. This is apparent in Fig. 13 which shows the n surface in the 3-dimensional space of m + , m − and m e . Clearly the force will lead to a preference for low values of m e , so that if p is large enough the most probable universes will have m e close to zero, i.e. close to the e edge of the n surface triangle of Fig. 12. From Fig. 13, all points on the e edge are equally probable, so that there is no expectation of being close to either the ue or eD vertices. We can compute the expectation values of m u,d,e by integrating over the 3-dimensional region O. and This explains why the three contributions to m n − m p − m e shown in Eq. (54) are comparable. On the other hand, Fig. 13 shows that m + is uniformly distributed along the n surface from small values to m +max , and hence Notice that, of the three mass scales that enter the observer boundaries, Eqs. (57, 58, 59), E bin does not appear. This is because the probability distribution is peaked away from the complex nuclei boundary, which therefore becomes irrelevant in determining the averages. 14 For case III-3 there are three possible vertices of the n surface triangle to consider. Our universe is certainly not close to the uD vertex, since this gives m e too large, hence we study the remaining two vertices. There are many f that lead to these vertices; we begin by identifying and studying the simplest cases. The ue vertex can be reached by a preference for low values of m u,d,e . However, low values of m u or m e are not sufficient as they lead to the u or e edges, hence as a simple example we study the distribution with q positive. At the ue vertex the value of m d is determined by δ EM . For sufficiently large q this is the only relevant scale so that the average values of m u,d,e over O will all be determined by δ EM : While choices of q and C I can lead to a hierarchy between these averages (for example m e ≃ δ EM , m + ≃ 10 δ EM and m − ≃ 6 δ EM for q = 5 and C I = 0.5), the hierarchy is more readily generated when m e ∝ δ EM and m + ∝ m +max , as in Eqs. (64, 66).
The eD vertex is favored by large m u and small m e ; however large or small m d both lead away from the eD vertex. A simple distribution peaking towards the eD vertex is with r > ∼ m +max /δ EM . Any distribution peaking near this vertex will lead to m − ∝ δ EM and m + ≃ m +max , which agree with observations. The prediction for the electron mass depends on the strength and direction of the force. In the example of Eq. (71), m e ≈ m +max /r. In two Higgs doublet theories a probability distribution for the ratio of vacuum expectation values, tan β, contributes to the distributions for m u,d,e . Suppose that this distribution favors large tan β and that tan β o ≫ 1. In this case low m d and m e are preferred, so that within the observer region the probability distribution is peaked towards the ue vertex of the n surface, leading to m e , m + , m − ∝ δ EM . The probability distribution in the electroweak symmetry breaking sector may be the origin of the force determining m u,d,e .
Finally we consider the possibility that α also scans. In this case the size of the electromagnetic mass difference, δ EM , scans relative to the purely QCD scales of E bin and m +max . This means that there is a shift in the position of the allowed window for (C I m − − m e ) from neutron and complex nuclei stability. If α increases too much this window shifts to a non-physical region where m − > m + , leading to an upper limit on α which is numerically about 0.2, with large uncertainties. Thus α is about an order of magnitude away from the maximum value that it may take anywhere in O.
If α is very small then the range for (C I m − − m e ) increases. The amount of increase is negligible on a linear scale, but is sizable on a logarithmic scale. For example, with α = 10 −4 , (C I m − − m e ) can range from 10 −2 MeV to 8 MeV. However, if gauge couplings unify then α and Λ QCD become related. As α is decreased, so Λ QCD becomes exponentially smaller. It could be that selection effects on the size of Λ QCD compared with the unified scale and/or the electroweak scale dominate over selection effects of α in nuclear physics.
In many circumstances, precise predictions for environmental selection follow from assuming sharply varying distribution functions f . However, in the present example of nuclear physics in the parameter space m u,d,e and α, it is possible thatf is sufficiently slowly varying over O that physical arguments based on the observer factor, n, could lead to predictions without any assumptions of sharply varying f . For example, consider variations in the parameters within O that lead from the neutron surface to the observer boundary with no complex stable nuclei. Moving towards the boundary of no complex stable nuclei, the observer factor may be reduced by successive nuclei becoming unstable. On the other hand, moving closer to the neutron surface than our universe will lead to a longer neutron lifetime and therefore to more primordial helium production; the reduction in primordial hydrogen will result in fewer hydrogen burning stars. The competition of these (and other) effects may determine the location of our universe within O. Of course, even in this case, some assumption aboutf is still needed.

Electroweak Symmetry Breaking selected by Nuclear Stability
The origin of electroweak symmetry breaking is one of the largest mysteries remaining in the Standard Model. The quadratic divergence of the Higgs mass-squared parameter in the Standard Model implies that if the scale of new physics M is (much) larger than the weak scale v, the theory requires fine-tuning. This hierarchy problem was a key motivation for much of the model building in the last 30 years. What do we know about the scale M? In many (non-supersymmetric) theories beyond the Standard Model, precision electroweak data indicates a "little hierarchy problem": v/M is uncomfortably small, typically of order (10 −2 -10 −1 ) or smaller (see e.g. [25]). There is also a similar fine-tuning problem in supersymmetric theories, although its origin is different. A sufficiently heavy Higgs boson typically requires superparticles to be somewhat heavier than the weak scale, leading to some amount of fine-tuning (see e.g. [26]). In either case, we find that some amount of unnaturalness is present, at least for the simplest theories, suggesting that environmental selection may be playing a role. In this section we investigate whether a hierarchy between v and M is to be expected from environmental selection, the size of any such hierarchy, and how the hierarchy depends on which parameters are assumed to scan.
To address the question of the environmental selection of v/M, several issues must be addressed: what is the theory under discussion, which parameters of that theory scan, and what observer boundaries implement the selection? Below we formulate a fairly general class of theories that describes electroweak symmetry breaking, and the nuclear stability boundaries of section 4 are used to implement selection. In section 9 we consider an alternative possibility that selection occurs at the electroweak phase boundary.
The mass scale of the new physics that generates electroweak symmetry breaking is defined to be M, and we assume that the effective theory below M is the Standard Model with the Higgs potential Integrating out the physics of the electroweak symmetry breaking sector at scale M in general leads to several contributions to m 2 h , some positive and some negative, which we write as where the functions g 1,2 are both positive. The dimensionless parameters x i are the set of parameters of the theory above M that substantially affect electroweak symmetry breaking. These parameters are evaluated at the scale M, so that g 1,2 are also functions of M through the logarithmic sensitivity of x i on M: g 1,2 (x i (M)). Note that m 2 h here includes the quadratically divergent radiative corrections in the Standard Model that are regulated by the theory above M. The parameters x i thus include the Standard Model gauge and Yukawa couplings.
The precise nature of the couplings x i and the functional form of g 1,2 are unimportant for our discussion, so that we write g 1 (x i ) = Ax and g 2 (x i ) = Ay, giving where x, y, A > 0. The numerical constant A is chosen such that typical values for y in the observer region are of order unity. The parameters x and y depend logarithmically on M through renormalization group evolution. (A is a one-loop factor in many theories beyond the Standard Model. Since m 2 h contains quadratically divergent contributions in the Standard Model, and thus x and y contain pieces proportional to the SU(2) gauge and top Yukawa couplings squared, respectively, the value of A should not be much smaller than the one-loop factor.) In the context of conventional naturalness criteria, electroweak symmetry breaking is unnatural if x is typically larger than y in an ensemble (only a small fraction of members in the ensemble leads to electroweak symmetry breaking), whereas if x is typically of order y or less then the natural value of the weak scale is √ A M for a quartic coupling of order unity. A key observation is that as experimental limits on physics beyond the Standard Model get stronger so the mass scale M is constrained to be larger. In some (non-supersymmetric) theories of electroweak symmetry breaking, this typically arises from contributions of particles of mass M to the precision electroweak observables. In many supersymmetric models, the increased lower limits on the Higgs boson and superparticle masses have pushed up the mass scale of some superparticles to about a TeV or larger. These little hierarchies between v and M imply that the parameters x i are constrained towards the phase boundary of electroweak symmetry breaking. As experiments push up M, so g 1 and g 2 of Eq. (74) cancel to give v/ √ AM (much) smaller than unity. This is illustrated in Fig. 14 for a 2-dimensional slice through the parameter space. The unusual closeness of the parameters to this boundary can be viewed as an observer naturalness problem.
In this section, we investigate the various possible ways in which environmental selection on a multiverse can play a role in electroweak symmetry breaking. We use the class of theories described above, which contains most non-supersymmetric and supersymmetric theories beyond the Standard Model. The relevant observer boundaries are those of nuclear stability, shown in Fig. 15, so that in general one must consider scanning parameters that affect nuclear physics as well as the parameters of the electroweak symmetry breaking sector. We start by considering only a few parameters scanning, and then progress to more general situations.
First of all, it is natural to expect that some parameters of the theory at M scan in the multiverse, so that there are universes with different values of the weak scale v. Universes with x > y have v = 0, while those with y > x have Scanning of parameters in the electroweak symmetry breaking sector at M may also lead to a nontrivial effective distribution for λ h (v). Since a light Higgs boson has not been discovered, however, this distribution should lead to λ h (v) typically having an order unity value in the observer region. This implies v 2 in our universe, where y is of order unity due to definition of A. The crucial question is if some cancellation between y and x is expected, which cannot be understood in the conventional symmetry approach. If not, then v ≈ √ AM as in the conventional case. If so, however, we obtain an extra hierarchy between v and M that cannot be explained by a symmetry, and the question becomes: is y − x typically of order 10 −2 -10 −1 , with the multiverse generating a little hierarchy, or is it extremely small, for example 10 −30 , giving a large hierarchy?
In addition to v, the nuclear stability observer boundaries depend on parameters y u,d,e , α and Λ QCD . In sections 8.1 and 8.2 we assume that these additional parameters do not scan, so that the only scanning in the Standard Model at low energies relevant for electroweak symmetry breaking is that of v. In this case, environmental selection for neutron instability and deuteron stability defines a fixed observer window for where, from Eq. (29), v − ≃ 0.5 v o and v + ≃ 2v o . (The value of v + depends on the parameter a describing the strength of the deuteron binding; here we take a = 2.2 MeV.) This is illustrated in the m u -m d plane in Fig 15. As v is varied about v o so m u and m d vary, but with a fixed ratio, as shown by the dashed line. (The corresponding variation of m e gives only small effects.) Since v o lies centrally in the observer window, it is consistent with a distribution f v (v) that is slowly varying. Can environmental selection for v generate a little or large hierarchy, and if so is the effective distribution for v mildly varying? Another possibility, studied in section 8.3, is that the entire set y u,d,e , α, Λ QCD and v scans, so that there is a non-zero probability distribution throughout the nuclear observer region. A 2-dimensional slice through the observer region is shown in Fig 15; the scanning is no longer restricted to the dashed line. An interesting question then is whether the closeness to the neutron stability boundary is a statistical accident, or whether it results with high probability due to a strongly varying distribution function. As stressed in section 4, the nuclear observer region depends on only four combinations of these quantities, m u,d,e /Λ QCD and α. For example, a common scanning of v and Λ QCD does not affect nuclear physics. This implies that a numerical value for v is no longer determined by the nuclear observer region alone. Can environmental selection from the nuclear observer boundaries, shown in Fig 15, lead to a little or large hierarchy?

Scanning the mass scale M
The fundamental field theory at the cutoff scale M * (≫ M) will have a certain set of parameters. We assume that the scanning at M * is limited to those parameters that affect the scale M (which may be only M itself), leading to a distribution function f (M). In particular, the dimensionless parameters x i (M * ) do not scan. Assuming that effects on x i from the parameters controlling M are small, this implies that the scanning of the parameters x and y in Eq. (75) comes only through a calculable logarithmic dependence on M, which arises from nontrivial scaling of these parameters under renormalization group evolution.
Suppose now that y − x is positive and of order unity throughout the multiverse (which will be the case if y − x is positive at M * and becomes larger as the renormalization scale is lowered). Environmental selection requires that v lies in the fixed range v − < v < v + . This selects M to be in the range The ratio between v and M is then the same as if there were no selection, v/M ≈ √ A. There is, however, another possibility. It could be that as M varies so y − x passes through zero. In this case, there is a critical value M = M c corresponding to the electroweak phase boundary, y − x = 0. By assumption, neither the critical value M c , nor the nuclear window   [31,32].
We stress that the situation described here is very special: for example, if Λ QCD scans in the multiverse, it is possible that the resulting hierarchy is "little," e.g. v/M ≈ (10 −2 -10 −1 ), since the value of M may be set close to the observed weak scale by environmental selection, as we show below.

Full scan of the electroweak symmetry breaking sector
Next we consider a general scanning of the electroweak symmetry breaking sector, so that all of x, y, λ h and M vary independently, but we keep y u,d,e , α and Λ QCD fixed. In this case it is useful to consider a 2-dimensional observer region in the M-z plane, as shown in Fig. 17, where z ≡ y − x. As before, we assume that λ h (v) has a typical value of order unity, e.g. with its distribution function being strongly peaked at ∼ 1, and hereafter we neglect the effect of its scanning. 15 Electroweak symmetry breaking occurs in universes with z > 0, and the observer boundaries v = v ± are shown in Fig. 17. Other parts of the observer boundary correspond to M = M * , the maximum value of M (the cutoff scale), and z = z max , the maximum value of z determined by the ranges for x and y in the landscape, which we take to be of order unity.
The effective probability distribution in this 2-dimensional observer region O is obtained from the distribution function for x, y, M as With x and y scanning over the multiverse, the logarithmic dependence of these parameters on M frequently does not give a major effect, and hence we neglect it here. When these effects are important they lead to very interesting results, and we defer a discussion of this until section 8.5. If f eff (z, M) were constant over O, it is clear that our universe would be expected to have z ≈ 1 and v ≈ √ AM, since this corresponds to most of the area of O. However, it is also clear that, as f eff (z, M) becomes progressively more peaked towards low z and/or high M, so the typical hierarchy will first grow to a little hierarchy, z ≈ O(10 −2 -10 −1 ), and finally to a large hierarchy, z ≪ O(10 −2 ). A power distribution is easily able to overcome the narrowing of the observer region at large M. The probability distribution for the hierarchy, z, is given by and a general scanning of the electroweak symmetry breaking sector of the theory could lead to a large probability for either a little hierarchy or a large hierarchy. Indeed, since a strong dependence of f eff (z, M) on z and M could result from many sources -the landscape of vacua, the population mechanism, integrating out other parameters and the observer distribution nit would certainly not be surprising if either a little or large hierarchy resulted. Environmental selection of the weak scale can provide a simple generic explanation for our present difficulties in constructing natural theories of electroweak symmetry breaking. Note that this mechanism works both in the context of non-supersymmetric and supersymmetric theories, 16 and that the small value of z implies a cancellation between the positive and negative contributions in Eq. (74) (x and y) that cannot be explained in the conventional symmetry viewpoint. In section 8.4, we study in detail the simplified case that y does not scan, and x and M have power-law or logarithmic distributions. We compute f z (z), find conditions for large and little hierarchies, and obtain an analytic result for the size of the hierarchy when it is little. We also compute the distribution for v 2 , f v (v 2 ), in the observer window.
Before closing this subsection, let us consider the case that only the dimensionless variables in the electroweak symmetry breaking sector, x i , scan, with the value of M fixed. In this case experiments constrain the value of M to be larger than about a TeV. What value should we expect for M? Without any special reason, we expect M to take some "random" value between TeV and fundamental scales; it is unlikely that M is close to the observed weak scale, since it requires an accident of order O(0.01 -0.1), as discussed in section 4.4. Hence we typically expect a large hierarchy with M ≫ v. The values of x i are environmentally selected by Eq. (78) to a very small range: in which there is a large cancellation between x and y.

Scanning over the entire nuclear stability observer region
We continue to study theories of electroweak symmetry breaking that lead to the Standard Model as an effective theory below M, with m 2 h = (x − y)AM 2 . However, as well as having v scan, via the scanning of x, y and M, we now allow the other parameters of the nuclear stability observer boundaries to also scan. This means that instead of exploring a distribution through the observer region corresponding to only varying v, as shown by the dashed line of Fig. 15, we are now exploring a distribution over the entire observer region discussed in section 4. With y u,d,e , α and Λ QCD all scanning, the problem is apparently very complex. However, here we are only interested in two aspects of the problem: the probability distribution for z = y − x that determines the size of the hierarchy between v and M, and the probability force perpendicular to the neutron surface that determines how close typical observers are to the neutron stability boundary. These questions can be addressed by studying an effective distribution over a reduced 2-dimensional projection of the parameter space.
The neutron and complex nuclei stability boundaries of Eq. (55) can be written in the form where and ξ − = C α and ξ + = C α +C B /α. Note the similarity in the form of the equations for ξ, Eqs. (84) and (83), to those for v, Eqs. (76) and (78). This means that the shape of the observer region in the 2-dimensional projection of the parameter space on theM -z plane, shown in Fig. 18, is the same as in the last subsection on the M-z plane. The curved observer boundaries at ξ = ξ − and ξ = ξ + , however, now correspond to neutron and complex nuclei stability, respectively. Our universe thus lies very close to ξ = ξ − . How probable this is will be determined by the effective neglect the effect from this scanning. The effective distributions for z and ξ 2 are given in terms of f eff (z,M ) by and respectively. We can identify three very different situations. The first is that f eff (z,M ) is very mildly varying. In this case, from Fig. 18 we see that a typical universe will lie in the middle of the observer region and hence environmental selection will lead to neither a hierarchy nor a closeness to the neutron boundary. A second situation has a strongly varying f eff (z,M ), but with theM component of the probability force field unable to overcome the narrowing of the observer region at largeM , shown in Fig. 18. In this case, if there is a strong force to low values of z, the multiverse yields a little hierarchy by making low values of z ≈ O(10 −2 -10 −1 ) typical. If in addition there is a significant probability force to lowM, the combined effects of the z andM distributions lead to a closeness to the neutron stability boundary as well as to a little hierarchy, as shown by the dots in Fig. 18. Finally, the probability force towards largeM may be strong enough to give a large hierarchy with extremely small z. In this case the observer boundaries at ξ − and ξ + are very close to each other. At such low values of z, is it reasonable to have f eff sufficiently different on these boundaries to favor the boundary at ξ − ? Here we must recall that f is a product of a populated landscape distribution,f, and an observer distribution, n. It is certainly unreasonable forf to have such a large variation over such a small region of parameter space. However, as we move from the ξ − boundary to the ξ + boundary, nuclear physics changes very significantly, which is in a way independent of how small z is. Hence a closeness to the neutron boundary may be typical with a large hierarchy, but only if it is induced by n, through such arguments as appeared in the last paragraph of section 7.
The closeness to the neutron boundary may be a hint that f eff (z,M ) is not flat. As argued previously, there are many origins for strong probability forces, so that a little or large hierarchy is quite natural in the multiverse. In the next subsection we explore in some detail a subclass of the theories specified by Eq. (74), allowing explicit formulae for the effective distributions and the size of the hierarchy.

An explicit example with power law distributions
In the previous two subsections we assumed that the positive and negative terms in m 2 h /M 2 scan independently. To provide a simple explicit illustration of our results, in this subsection we choose to scan only the positive term, i.e. we fix y = 1 so that and z = 1 − x. The analysis with x fixed and y scanning is very similar. For ease of calculation we assume a polynomial distribution function and the range of x is given by 0 ≤ x ≤ x max , with x max some number larger than 1, while M has some very large maximum value M * . We begin by assuming that the only parameter appearing in the nuclear stability boundary that scans is v, so that the condition of Eq. (78) gives the observer region The observer region O is sketched in the M-z plane in Fig. 19, which is obtained simply by setting y = 1 in Fig. 17. The region is separated by the nuclear physics observer boundaries from regions with v < v − and v > v + , and approaches very close to x = 1, i.e. z = 0, as M approaches M * ≫ v o . The narrowing of the observer region at large M is as we have seen in Fig. 17.
What is the typical value of the scale M? For a logarithmic distribution, q = 0, one might guess that all decades of M are equally probable; but this is not the case, because at large M the observer region narrows. It is useful to compute an effective probability distribution for the hierarchy z = 1 − x = |m 2 h |/AM 2 over the observer region for sufficiently large M * . From the viewpoint of the hierarchy, this shows that the critical value of q is 2 not zero: when q = 2 the distribution for the hierarchy z is flat on logarithmic scales, at least for large M where z is small. Thus q ≥ 2 gives a large hierarchy, shown by the narrow wedge at large M in Fig. 19, and q < 2 gives a little (or no) hierarchy, corresponding to the bulk of the observer region in Fig. 19. For example, if the a priori distribution for M is logarithmic, as might happen if the scale is triggered by dimensional transmutation associated with some gauge dynamics, the most probable observed value of M is small because of the weighting from the narrowing of the observer region O. 17 On the other hand, if dN ∼ dM 2 , as expected in a typical non-supersymmetric perturbative theory with mass scale M, we find q = 2, so that all scales are equally probable, implying a large hierarchy, but not one with M near M * . If q < 2, how large is the little hierarchy? For n > 0, the first factor in Eq. (92) suppresses the probability of having z ∼ 1, i.e. the probability of having the lowest values of M gets suppressed, so that some amount of hierarchy between v and M arises from the scanning in the multiverse. For q < 2 the distribution of Eq. (92) leads to the average value of z: so that for 2n ≫ 2 − q we obtain an extra hierarchy of a factor of ≃ (2 − q)/2n from the multiverse, which cannot be explained in the conventional symmetry approach. Since |q| is expected to be small, this is an appreciable effect for large n. In Fig. 20 we plot f z (z) of Eq. (92) for (n, q) = (3, 0), (10, 0), (3,1). We clearly see that the distribution is peaked towards z ≪ 1.
The probability of us observing fine-tuning of ∆ −1 or more that cannot be understood by the symmetry approach is given by which for ∆ ≫ n ≫ 1 reduces to 17 In the case that M arises as a dimensional transmutation, the expected distribution of M below M * is not exactly ∝ d ln M , since it would lead to an unphysical conclusion that the probabilities of having M between 10 n M * and 10 n+1 M * (n ≤ −1) are equal for all n down to n → −∞. (The distribution should, for example, be cut off at some value M min or in fact be ∝ d(1/ ln M ); see discussions at the end of section 2.2 and in section 4.4.) The assumption/approximation of the exact logarithmic distribution, however, is sufficient for our purposes here. for q = 0 (the case with dimensional transmutation) and for q = 1. For example, for n = 10 and q = 0 (n = 10 and q = 1) we have a ≃ 20% (≃ 50%) probability of observing fine-tuning ∆ −1 < 2% which cannot be explained in the conventional symmetry picture. The observer boundaries of Fig. 19 are the contours for v = v ± so that, by construction, v can only be observed between v − and v + . What are the most probable values of v to be observed? The effective distribution for v 2 is defined by For a large hierarchy with q ≥ 2, we find a flat distribution in v 2 When y u,d,e , α and Λ QCD do not scan, the observed value of the weak scale v o is roughly midway between v − and v + , as shown in Eq. (29), which is consistent with Eq. (98). 18 For a little (or no) hierarchy, with q < 2, we find a distribution 18 Caution, however, is needed in interpreting Eq. (98). This result arises because at large M the contours of v − and v + are extremely close in M -x space, so that the assumed form for f , Eq. (90), implies little variation between the contours. While this is expected for the multiverse distributionf , it may not be true for the observer distribution n. In this case Eq. (90) needs to be modified to incorporate the effect represented by n. This is again consistent with v o centrally located in its observer window, provided that q is not too negative.
Next, we allow the other parameters that appear in the nuclear stability observer boundaries to also scan. As described in section 8.3, the issues of the size of the hierarchy and the closeness to the neutron surface can be addressed by studying a 2-dimensional projection of parameter space. Since y does not scan, z = 1 − x and so here we discuss theM -x plane. We assume the effective probability distribution in this plane has a power law behavior The neutron and complex nuclei boundaries are again given by Eqs. (83, 84, 85), but with z = 1−x, and are represented in Fig. 18 in theM -x plane (with y now set to unity). The observer region as x → 1 again becomes a narrow wedge where the hierarchy is large: The equations describing this setup are very similar to those in the case with only v scanning, except that the observer boundaries are now at ξ ± rather than v ± , andM is not the scale of the new physics, see Eqs. (83 -85). The size of the hierarchy is governed by the effective distribution for z and is given by the analogue of Eq. (92): Thusq ≥ 2 gives a large hierarchy andq < 2 gives a little (or no) hierarchy. The size of the little hierarchy depends on n, and using the distribution Eq. (100) we find The little hierarchy increases with n, as shown by the sequence of dots in Fig. 18 near the neutron stability boundary. What makes the proximity to the neutron boundary typical? The effective distribution for ξ, which determines the probability force perpendicular to the neutron surface, follows immediately from a calculation analogous to that which led to Eqs. (98, 99) giving and As with Eq. (98), caution is necessary in interpreting Eq. (103). At largeM our assumed distribution, Eq. (100), may not adequately account for variations in the observer factor n(ξ) between ξ − and ξ + . In particular, we cannot conclude from Eq. (103) that a large hierarchy is incompatible with a multiverse explanation for the closeness to the neutron stability boundary. With a little hierarchy, sufficient closeness to the neutron stability boundary results with the distribution of Eq. (100) for −10 < ∼q < ∼ −2.
that the distribution for M and z leads to most universes in the observer region having large M, close to M c,max , so that this new part of the observer boundary plays an important role. 19 If y u,d,e , α and Λ QCD do not scan then v ± are fixed numbers that arise from nuclear selection. Since there is no reason for the critical mass M c,max of the landscape to be close to v ± , a large hierarchy is expected. Here we are more interested in the case that v ± varies in the landscape, and for simplicity we accomplish this via a scanning of Λ QCD , which we take to have a mild distribution, for example flat on a logarithmic scale. In this case v ± ∝ Λ QCD , so that the observer boundaries in Fig. 21 labeled by v ± have z ± ∝ Λ 2 QCD , and hence move up and down as Λ QCD scans. For a large hierarchy, M is so much larger than v that the interval of z that can contribute to universes in the observer region is very small ∆z ≈ v 2 /M 2 . Since the distribution for Λ QCD is mild, a gain in probability is accomplished by having the v ± boundary curves of Fig. 21 move upwards, allowing a larger interval of z to contribute. As the v ± curves move up so the values of M that are on the z max observer boundary decrease. At some point this will imply a significant loss of probability via the distribution for M. For example, for f (M)dM ∝ M q d ln M with large q, this loss of probability begins to set in once M ∼ (1 − 1/q)M c,max at which point z max ∼ β/q. The mild distribution for Λ QCD allows the v ± curves of Fig. 21 to rise so that, at this value of M, z ± will be close to z max . A further drop in M causes a significant drop in probability from f (M), which is not offset by a sufficient growth in ∆z, since z c (M) grows logarithmically. Hence we predict that a little hierarchy develops. Specifically, for a distribution function integrating over the observer region we obtain a typical size of the hierarchy where the range of the z integration is from 0 to the maximum value that z takes in the energy interval between ≈ v and M * , and we have taken f (z) to be constant in the last expression for illustration, although the result is not very sensitive to the form of f (z). Note that this result applies to an arbitrary positive value of q > ∼ O(β). Integrating over Λ QCD effectively "removes" the v ± boundaries in the 2-dimensional parameter space of {M, z}, and the resulting effective distribution in the M-z plane is unaltered for a prior distribution flat in ln Λ QCD . Allowing any mild prior distribution for Λ QCD , or allowing other Standard Model parameters such as the Yukawa couplings y u,d,e to scan with mild distributions, would lead to an effective distribution f eff (z, M) that would lead to the basic result of Eq. (107) being unaffected. The physics behind this result is both simple and general: the 1/q factor results from a probability force in the M direction making the most probable universes in the observer region those close to M c,max , while the factor of β arises from the shape of the observer boundary near M c,max , as given in Eq. (105).
We conclude that landscapes having a critical value of M, above which electroweak symmetry breaking is impossible, very easily lead to a little hierarchy with z suppressed by both a loop factor, β, and a probability force factor, for example 1/q. Unlike the case of sections 8.2 and 8.3, large q does not lead to a large hierarchy. Instead it acts to make M close to M c,max and, since z max (M c,max ) = 0, this leads to a little hierarchy. Note that the loop factor can provide an extra hierarchy corresponding to 1 -2 orders of magnitude of fine-tuning, on top of the 1/q factor. Hence, this mechanism typically yields M in the TeV region or larger.
If the physics at M is supersymmetric, this mechanism is the one considered in Ref. [8]. If the only scanning parameters are a uniform scaling of the overall scale of supersymmetry breaking and of the supersymmetric Higgs mass parameter µ, the scale of M c corresponds to the scale where the determinant of the Higgs boson mass matrix, det(M 2 H ), passes through zero. To avoid a large hierarchy, some other parameter, such as Λ QCD should scan. One again needs to assume that, for the landscape as a whole, there is some maximum value of the scale where det(M 2 H ) passes through zero, and this is taken to be M c,max . In this subsection, we have stressed the generality of the idea, and that the loop suppression factor appearing in Eq. (107) does not depend on whether the physics at M is supersymmetric or not. In the supersymmetric case, β is determined by the specific form of the renormalization group equations for the Higgs boson mass parameters, and one finds that β ≈ O(0.1). Nevertheless this is an important accomplishment, since without the landscape the ratio between v and M involves a large logarithm ln(M * /v) (in the case of high scale supersymmetry breaking) so that where M Z is the Z boson mass. The factor of β, together with 1/q, can easily make M at the TeV scale or larger.

Summary and discussion
The conventional naturalness argument implies that the mass scale M of new physics beyond the Standard Model that generates electroweak symmetry breaking should not be much larger than the Higgs vacuum expectation value v. In contrast, we have shown that environmental selection for nuclear stability can lead to M substantially larger than v. We have considered a very general framework that is independent of the model of the new physics, assuming only that it generates both positive and negative contributions to the Higgs mass-squared parameter, m 2 h = (x−y)AM 2 . If the landscape does not possess a maximum critical mass M c,max significantly less than M * , the relevant observer boundaries are shown in Fig. 17 when only x, y and M scan, and in Fig. 18 when y u,d,e , α and Λ QCD also scan. In both cases we find • A large hierarchy is generated by a distribution for M (M) that grows sufficient with M (M ) to overcome the narrowing observer region of Fig. 17 (Fig. 18). While this does not require a very strong peaking, the distribution must continue growing over many orders of magnitude in M (M).
• A little hierarchy is generated when the distribution for M (M ) does not grow sufficiently to generate a large hierarchy, and the distribution for z is strongly peaked to low values of z. This strong peaking need only persist for one or two orders of magnitude in z.
In both cases there is necessarily a large cut factor, either from a mild growth in the distribution for M (M ) over many decades for a large hierarchy, or from a strong distribution for z over a much more limited range for a little hierarchy. In general, a significant variation of a distribution function can arise from many sources -the distribution of vacua in the landscape, the population of these vacua, integrating out parameters, and the number density of observers -so that in the multiverse the existence of a hierarchy, either little or large, is not surprising. For landscapes with a maximum critical mass M c,max • A little hierarchy is generated by a distribution that favors large M near M c,max . The size of the little hierarchy depends on both the strength of this distribution, and also on a loop factor that arises from the beta function for x − y. The combination of these factors makes it probable that M is as large as several TeV.
In section 10 we argue that, in the presence of WIMP dark matter, a strong distribution for M can arise from the probability distribution for the cosmological constant. Until now we have not specified the physics behind x and y; here we consider a few simple schemes, stressing how a cancellation between x and y could lead to a strong distribution for z and hence a little hierarchy. The quadratic divergences of the Standard Model contribute to both x (e.g. the SU(2) gauge contribution) and y (e.g. the top quark contribution). At mass scale M suppose that these quadratic divergences are cut off by particles of mass M W ′ and M t ′ , respectively. If the theory at M is supersymmetric, these are the wino and top squark masses, while if the theory at M is non-supersymmetric (for example, with composite Higgs dynamics) they are some states of the model. In all of these schemes, integrating out the physics at M gives a low energy theory with where g is the Standard Model SU(2) gauge coupling. The numerical coefficients are model dependent and extra contributions to m 2 h are expected, for example those that regulate the hypercharge and Higgs-quartic divergences; but neither of these affects the arguments below.
Whether the theory at M is supersymmetric or not, it may well be that M arises as a dimensional transmutation, in which case it is reasonable that the distribution for ln M is sufficiently flat that a large hierarchy does not develop. In this case, what distribution for the parameters in Eq. (108) would lead to a little hierarchy?
We need a distribution f z (z) peaked at low values, i.e. in Fig. 17 or Fig. 18 the probability force must have a large component downwards. Since z = 0 is not a special point from the fundamental theory point of view (see the discussion in section 2.2), this implies that most universes will be in the phase with z < 0, i.e. m 2 h > 0. In the multiverse, the positive term in Eq. (108) typically dominates over the negative term. Such a distribution could arise in several ways. For example, suppose that M W ′ /M t ′ does not scan, but g and y t do. If g has a distribution peaked at a (much) larger value than y t , then most universes will have negative z, and near the observer boundary the probability force will be towards smaller values of z. The shapes of the distributions for g and y t need not be power law, as assumed for simplicity in section 8.4. For example, they could be Gaussians with peaks atḡ andȳ t , withḡ sufficiently larger thanȳ t , so that most universes have m 2 h > 0. The few universes that have m 2 h negative will typically have low z and therefore a little hierarchy. For narrow Gaussians for g 2 and y 2 t with the standard deviations δ g and δ y , respectively, we find where z c ≡ (A g /A y )ḡ 2 −ȳ 2 t > 0 and δ 2 ≡ (A g /A y ) 2 δ 2 g + δ 2 y , with A g,y positive coefficients defined by m 2 h = (A g g 2 − A y y 2 t )M 2 . Alternatively, it could be that the scanning of the masses M W ′ and M t ′ is more important than that of the couplings, and that the little hierarchy results because the distributions typically give M W ′ (much) larger than M t ′ .
In particular models, it is possible to see other situations that lead to a little hierarchy. For example, in the minimal supersymmetric standard model it could be that the distributions for the supersymmetric Higgs mass parameter µ and the scale of the soft supersymmetry breaking mass parametersm differ. If µ is typically (much) larger thanm, then most universes do not have electroweak symmetry broken by h = 0. This generically leads to a strong distribution preferring low z, and universes in the observer region will have a little hierarchy. More generally, in the minimal supersymmetric standard model the electroweak phase boundary takes the form with the soft Higgs mass-squared parameters m 2 H 1 and m 2 H 2 depending on other parameters of the theory via renormalization group scaling. Any multiverse distribution for the parameters of the model that typically makes the left-hand-side of Eq. (110) larger than the right-hand-side will generically lead to a little hierarchy. The scale of the superparticle masses are then raised significantly above v and the measured values of the parameters should be close to satisfying the critical condition, Eq. (110).

Electroweak Symmetry Breaking as Observer Boundary
In the last section we assumed that the relevant observer boundaries for selecting the electroweak vacuum expectation value, h = v, were those of neutron, deuteron and complex nuclei stability. Of these three boundaries, the requirement that some complex nuclei are stable seems clearly to be the most robust requirement for observers. For example, if the neutron is stable, nuclear energy is produced in diffuse protogalaxies rather than in stars. While this is a drastic change from our universe, some form of observers might still be possible. On the other hand, it is harder to imagine that some complex observers develop in the world in which the only stable nucleus is p or ∆ ++ . In this section, we retain only the complex stable nuclei boundary, dropping the neutron and deuteron (in)stability requirements from the observer boundary. This allows much smaller values for v. How small can v become while remaining in the observer region? If electroweak symmetry is broken dominantly by the QCD condensate, there is a strong washout of the baryon asymmetry of the universe due to sphaleron effects. This implies that complex structures involving baryons do not arise if v < ∼ Λ QCD , so that there is a complexity boundary for v near Λ QCD which is therefore a candidate for being part of the observer boundary. Since Λ QCD is much smaller than the scale of the electroweak symmetry breaking sector, M, we can speak of this boundary as the phase boundary between h = 0 and h = 0 phases. In this section, we use only two boundaries: complex nuclear stability and the Higgs breaking of electroweak symmetry, which were both classified as catastrophic boundaries in section 4.
We consider the generic electroweak symmetry breaking sector discussed in the previous section, with relevant scanning parameters x, y and M and a hierarchy With y u,d,e , α and Λ QCD fixed, the relevant observer region is now as shown in Fig. 22. Compared to Fig. 17, the observer boundary at v + remains while that at v − has been replace by one at v ≈ Λ QCD . This greatly enlarges the observer region, with v varying over about 4 orders of magnitude compared to a factor 4 in section 8. Despite this, we find that there is very little change in the physical picture of environmental selection. For example, in the case that y is fixed and x and M have polynomial distributions as in Eq. (90), the effective distributions for z and v are not changed and given as before (see Eqs. (92, 98, 99)) by and Once again, q ≥ 2 gives a large hierarchy and q < 2 a little (or no) hierarchy. Furthermore the size of the little hierarchy is largely governed by n, as discussed in section 8.4. One difference is that the range of v in the observer region is now much larger; how reasonable is it that our universe is about a factor of 5 from v + and 3 orders of magnitude from Λ QCD ? For 0 < ∼ q < ∼ 1 the observed value of v is quite typical, but for larger q, v becomes progressively more peaked near v + . However, this cannot be viewed as evidence against a large hierarchy. For large q it is unlikely that Eq. (113) is the correct distribution near v + : as v approaches this observer boundary so successively more nuclei become unstable, so that the observer factor, n, is likely to become smaller. Given the observer region of Fig. 22 we can understand the origin of large or little hierarchies from the qualitative features of the distribution f (z, M), without recourse to any particular functional form. The observer boundary at v + causes a narrowing of the observer region at large M, so that a large hierarchy results only if the distribution gives a sufficient preference to large M to overcome this narrowing. If the probability force in the M direction is insufficient, then the size of the little hierarchy depends on the strength of the probability force in the negative z direction. At first sight a force in the negative z direction apparently pushes the electroweak vacuum expectation value close to the value Λ QCD at the baryon washout boundary. This is incorrect: such a force makes the most probable region that with low z, but the value of v that this corresponds to changes with M. This is why Eq. (113) involves q and not n.
We again conclude that the little hierarchy problem (supersymmetric or not) is very easily solved by environmental selection. Strongly varying distribution functions of the electroweak symmetry breaking sector are able to generate either a large or little hierarchy; furthermore this is not sensitive to which observer boundary is used to limit the lower value of v.

Connections to the Cosmological Constant
In this paper we argue that evidence for the multiverse can be found in unnaturalness in the cosmological constant, nuclear physics, and electroweak symmetry breaking. This implies that scanning of parameters is occurring in all three different arenas. Is it correct to consider each scanning problem separately, or should there be a combined treatment? In section 8.3 we provided an analysis of the combined scanning in the Standard Model and electroweak symmetry breaking sectors. In this section we study connections with the scanning of the cosmological constant.
In section 6.2, we argued that "integrating out" a set of parameters x b modifies the distribution function for a set if the observer region O for x b depends on x a . In the present case we consider x b = Λ and x a to be the set of scanning parameters of the Standard Model and the electroweak symmetry breaking sector. Assuming the multiverse probability distribution for Λ to be flat on a linear scale where ρ NL is the energy density of the universe when it goes non-linear. Hence the distribution function for the parameters of the Standard Model and the electroweak symmetry breaking sector is modified by the scanning of the cosmological constant if ρ NL depends on the parameters x a . In our previous analyses, the assumed form of the distribution should apply to f eff , rather than to f prior . This implies that the scanning of the cosmological constant can affect the form of the distribution functions that appeared in the analyses in previous sections.
In this section, we explore some consequences of a nontrivial dependence of ρ NL on x a . Perhaps the simplest origin for such a dependence is the case of WIMP dark matter. The temperature of matter-radiation equality, T eq , depends on the WIMP mass via its annihilation cross section, where Q is the primordial density perturbation. Working with the generic electroweak symmetry breaking sector introduced in section 8, with mass scale M and dimensionless parameters x i , the WIMP mass is proportional to M so that where g(x i ) > 0 is a model dependent function, depending on the mass and interactions of the WIMP. The effective distribution function for the electroweak symmetry breaking sector now becomes Note that here we have assumed that the parameter Q does not scan. If Q also scans, the result of integrating out cosmological parameters is altered from Eq. (117), as we will see below.
If Q does not scan, Eq. (117) shows that the effective distribution for the parameters of the electroweak symmetry breaking sector, {x i , M}, can acquire an important component from integrating out the cosmological constant. In particular, the cosmological constant may provide a strong probability force to larger values of M through the M 8 factor. At first sight it appears that this factor drives a large hierarchy, M ≫ v, but this is not the case. A crucial issue is: what stops the runaway to large M? It is important to separate two cases.
In the first case, the factor T 4 eq ∼ {g(x i )M 2 } 4 in f eff pushes the amount of dark matter up to the maximum allowed by some astrophysical observer boundary, for example that of stellar collisions in galaxies [33], so that T eq is essentially fixed to the boundary value T eq = T eq, * . The distribution function for the parameters relevant for electroweak symmetry breaking, {x, y, λ h , M} where x, y, λ h ⊂ x i , is obtained by integrating out parameters that appear in g(x i ) but not in m 2 h or λ h , within the observer region T eq < T eq, * . This in general leads to a complicated dependence of f eff on {x, y, λ h , M}, not in the simple form of Eq. (117). Note that as M grows beyond the TeV scale, since T eq stays to be T eq, * , cancellations must occur in g(x i ) so that m WIMP ≪ M. This implies that a large hierarchy can develop only if f prior has a strong preference towards larger values of M.
In the second case, the most probable region of scanning parameter space does not lead to the dark matter density being on the edge of its maximal value determined by astrophysics. The values of the parameters {x i , M} are determined by other physics, including electroweak symmetry breaking. The distribution of the parameters relevant for electroweak symmetry breaking may take a form f eff ∼ f T 4 eq , so that the factor of g(x i ) 4 M 8 can play an important role in electroweak symmetry breaking. Some examples of this are given in the following subsection.
A completely different situation arises if the space of scanning parameters is increased to include Q. As illustrated in Fig. 23, the probability force from the cosmological constant no longer acts in the direction of increasing m WIMP , but in the direction of increasing ρ NL , so that now the issue becomes: what stops the runaway in the ρ NL direction? It is again an astrophysical limit on the amount of dark matter, but this is now a limit on ρ NL rather than on T eq : ρ NL < ρ NL, * . The probability distribution along this astrophysical boundary now depends on the distribution Q T eq ρ NL, * ρ NL Figure 23: The probability force from the cosmological constant in the Q-T eq plane. The contours of ρ NL are drawn by dashed lines, while the observer boundary from astrophysics, ρ NL = ρ NL, * is depicted in the solid line.
f Q (Q). Integrating first Λ and then Q over the observer region, we obtain where T eq is given by Eq. (116). The induced contribution to the effective distribution now depends on the form of the original distribution for the density perturbation Q, since the observer boundary for Q depends on T eq = T eq (x i , M). However, in the simple case that f Q ∝ 1/Q, so that the Q distribution is flat on a logarithmic scale, f eff (x i , M) ∼ f prior (x i , M), and the T 4 eq contribution is removed. Hence, a nontrivial contribution to f eff is generated by a nontrivial distribution for ln Q, not by the probability force from the cosmological constant.

Examples of the cosmological constant affecting electroweak symmetry breaking
We have seen that integrating out cosmological parameters with WIMP dark matter can generate a nontrivial component in the distribution of parameters relevant for electroweak symmetry breaking. In general the induced distribution takes a complicated form depending on what parameters scan and how they are determined. In the case that Q does not scan and the dark matter density is not determined by an astrophysical bound, the effective distribution function receives a factor of g(x i ) 4 M 8 , as shown in Eq. (117). Assuming that parameters appearing in g(x i ) but not in m whereg(x, y) is related to g(x i ) in Eq. (116). In this subsection we illustrate how the scanning of cosmological parameters can affect electroweak symmetry breaking, using the example of Eq. (119). Suppose that f prior is a rather mild function of x, y and M. The effective distribution f eff is then determined essentially by the factorg(x, y) 4 M 8 . The fact that the dark matter density does not saturate the astrophysical bound implies that the runaway to larger M due to the M 8 factor should be stopped not by the condition T eq < T eq, * but by some other physics. The simplest possibility is that M does not scan in the multiverse. At the end of section 8.2 we argued that if M is fixed then a large hierarchy is to be expected, as in Eq. (82). However, this conclusion is reversed if Λ QCD scans. Since the nuclear physics boundaries depend on M only through the ratio M/Λ QCD , it makes no difference to the argument on the size of the hierarchy whether M scans or Λ QCD scans. Allowing x, y and Λ QCD to scan, the observer region is that of Fig. 17, with M replaced by 1/Λ QCD . Taking a mild distribution for Λ QCD avoids a large hierarchy. 20 The amount of the little hierarchy is then determined by the factorg(x, y) 4 . For example, forg ∼ x m and a logarithmic distribution for Λ QCD , z ≈ 1/(4m+ 2), so that a little hierarchy of order 10 −2 -10 −1 can be easily obtained for m a factor of a few. In general, withg 4 strongly preferring x > y, we obtain a little hierarchy. The crucial physics here is that the large scale structure observer boundary, Λ ≈ ρ NL , depends sensitively on the electroweak symmetry breaking parameters x and y through WIMP dark matter. The probability distribution for the cosmological constant then generates an effective distribution for x and y, which can be sharply varying.
Another possibility of preventing the runaway to large M arises if the range for the scanning of x and y are such that there is a maximum energy, M c,max , above which the m 2 h parameter is always positive. As shown in detail in section 8.5, a strong probability force to large M leads to a little hierarchy, providing the nuclear observer boundaries of v ± are able to scan, for example via a scanning of Λ QCD . Here we simply point out that the strong force in the M direction can arise from integrating out the cosmological constant, i.e. from the M 8 factor in Eq. (119). This leads to a little hierarchy without the need for any cut factor beyond that for the cosmological constant. In the case of supersymmetry, with dN ∼ M 8 dM (q ≈ O(10)) the multiverse allows an improvement in the naturalness by a factor of ≈ q/β ≈ O(100).

Conclusions
Environmental selection on a multiverse is a radical departure from conventional methods of fundamental physics for explaining physical phenomena. Nevertheless, in light of the cosmological constant problem and the discovery of dark energy, it warrants further exploration. Will sufficient evidence emerge to convince us that the multiverse exists?
Since we cannot directly explore other universes, it may be questioned whether evidence for the multiverse can be found at all. However, theories in physics and cosmology that cannot be directly tested in the laboratory are far from new -one only has to think of unified theories and inflation. The inability to make direct laboratory tests of the new particles and interactions does not put these theories beyond the realm of science; rather, it leads to careful investigations of whether they make successful indirect numerical predictions for data that cannot be satisfactorily explained by other means. Thus for any new theory or framework that cannot be directly probed, two questions are important (a) In the absence of the new theory, is there numerical data that cannot be adequately explained using other known theories? Indeed, does the data represent a problem for existing theories?
(b) Does the new theory provide a numerical understanding of the data, thus solving the previous problem?
For inflation, the data of (a) includes the flatness and isotropy of the universe, and the spectrum of density perturbations. So far, these highly significant cosmological problems have only been solved by inflation. Similarly, unified theories provide an understanding of gauge coupling constant unification. If the numerical understanding of the data is sufficiently precise, and thought to be better than competing theories, then the new theory may become the provisional standard view. As long as the numerical significance of its predictions is not overwhelming, one must stress the provisional nature of the understanding and the importance of seeking both further arenas in which it can be tested and new competing theories. For example, proton decay would provide further evidence for unified theories, but despite intensive searches, such evidence is still lacking.
Seeking evidence for the multiverse is no different in principle than seeking evidence for other theories that cannot be directly probed in the laboratory. In this paper we have argued that evidence for the multiverse can be found in three different arenas: the cosmological constant, nuclear physics and electroweak symmetry breaking. In all three cases the conventional approach based on symmetries has not provided a numerical understanding of the data, rather in each case it leads to naturalness problems. The observed values of parameters are very close to special values that are critical for the formation of some complex structure, yet this closeness is not adequately explained by any symmetry. An observer region in the parameter space of some theory is defined by requiring the existence of certain complex structures necessary for observers. Unnaturalness results if the observer region is very small compared to the entire parameter space, or if we observe values of the parameters very close to the boundary of the observer region. We have introduced a naturalness probability, P , that allows a numerical evaluation of these problems. The value of P can be highly dependent on the theory T under consideration. The evidence for unnaturalness, (a), is governed by the maximal value of P T that can be obtained in simple theories: the lower the maximal P T , the more severe the naturalness problem.
In each arena the multiverse easily and generically solves the naturalness problem. Each arena is somewhat different, and we summarize our results for each below; but there are also some common features. In all three cases, environmental selection elegantly explains why our universe is not to be found in the largest region of parameter space. Furthermore, the multiverse distribution is likely to have a strong dependence on the parameters of the low energy effective theory, through the landscape of vacua of the fundamental theory, the population mechanism, integrating out parameters, and from the physics that determines the density of observers. With a strongly varying distribution, it is most probable to observe a universe close to the observer boundary, solving naturalness problems and leading to predictions. Given the current theoretical status, the multiverse appears to us to provide the most elegant and plausible prediction for several parameters, including Λ and m u,d,e .
The cosmological constant problem is the most severe naturalness problem, with a naturalness probability in the range P Λ ≈ (10 −120 -10 −60 ).
The robustness of the observer boundary is particularly convincing -how could observers form in a dilute gas of inflating elementary particles? The interesting open questions are whether our universe is sufficiently close to the observer boundary, and how runaway behavior can be prevented if T eq and Q scan. These questions, however, are secondary: a notoriously intractable problem has an elegant solution that predicts dark energy.
Astrophysicists have often remarked on the special values of parameters required for a variety of phenomena in nuclear physics. We have obtained the observer region in the 4-dimensional parameter space m u,d,e /Λ QCD , α resulting from the stability boundaries for neutrons, deuterons and complex nuclei, as shown in Fig. 5. We find this observer region to be small, with our universe within (10 -30)% of the neutron stability boundary, so that in the Standard Model the naturalness probability is P nuc,SM ≈ (10 −16 -10 −4 ). Even if we could construct a theory of flavor with successful, precise predictions for the Yukawa couplings y u,d,e and at the same time find a theory that correctly predicted the weak scale, v, there would still be a naturalness probability associated with the value of Λ QCD that we estimate to be ≈ (0.01 -0.04). Despite decades of experiments on flavor physics, with recent increasing levels of accuracy in B meson and neutrino physics, progress on a theory of flavor has been limited. The most promising theories appear to be based on flavor symmetries, with a sequential pattern of transferring symmetry breaking to successive generations of quarks and charged leptons. There are a great number of candidate theories, but none is sufficiently promising to be widely recognized as the standard. We have estimated that such a lack of progress in flavor physics, especially in the first generation masses, decreases the naturalness probability of the nuclear observer region by about an order of magnitude, leading to P nuc < ∼ (10 −3 -10 −2 ).
We think that this estimate is conservative. For example, in theories with Abelian flavor symmetries this estimate does not take into account that there are many simple ways of assigning charges to the three generations. Given the state of theories of flavor, we suspect that the naturalness probability of the nuclear observer boundary is less than this conservative estimate. Suppose that m u,d,e had each been significantly different, for example by an order of magnitude. It would still be possible to accommodate this in theories with Abelian flavor symmetries, with about the same level of success as for the actual observed values, by changing the charges of the first generation. The nuclear naturalness problem certainly requires more than 1% fine-tuning, and has not received sufficient recognition. If symmetries rule flavor physics, it must be viewed as purely accidental; while in the multiverse it can be viewed as a prediction. After decades of studying theories of flavor, the current status is that we are far from a convincing explanation for the masses of the electron, the up quark and the down quark. The multiverse allows simple arguments that relate these masses to the QCD scale. For example, if the multiverse distribution strongly favors isospin restoration, then m e and m d − m u are expected to be close to δ EM ≃ 1.0 ± 0.5 MeV, the electromagnetic mass difference of the proton and neutron. It is true that such multiverse predictions require an assumption on the form of the distribution function; but such assumptions may be much simpler than the choice of flavor group, representations and sequential symmetry breaking of the standard approach.
The multiverse predictions for m u,d,e /Λ QCD can be retained even if unified and/or flavor symmetries describe the overall pattern of quark and lepton masses and mixings. This allows us to preserve particular successful relations, such as θ C ∼ m d /m s or m b = m τ at the unified scale. The only requirement is that the theory contain three independent scanning parameters that allow m u,d,e /Λ QCD to be environmentally selected.
A simple, natural theory of electroweak symmetry breaking is lacking. The natural regions of simple technicolor, supersymmetric and composite Higgs models have been excluded. Precision measurements of electroweak observables, together with direct searches for the Higgs boson, have led to successive increases in unnaturalness, leading to P EWSB < ∼ (10 −2 -10 −1 ), in simple models. This problem in the data is of type (a) and is widely appreciated, so that much recent research has focused on building models that alleviate the problem. We have shown that, no matter what the ultimate physics at mass scale M behind electroweak symmetry breaking, environmental selection on a multiverse leads very easily to M ≫ v, with either a little or large hierarchy. This is a very robust result requiring only that some parameters of electroweak symmetry breaking scan, and that the multiverse distribution is strongly varying. A distribution f (M) ∼ M q gives a large hierarchy if q ≥ 2 and a little hierarchy (or no hierarchy) for q < 2, which includes the important case of M being induced by a dimensional transmutation (q = 0). For q < 2 the little hierarchy, v 2 /M 2 , gains a factor of ≈ 1/n from the multiverse, where n describes the peaking of a distribution in some other parameter of the electroweak symmetry breaking sector. In multiverses where electroweak symmetry breaking is only possible if M is below some critical value, M c,max , a little hierarchy develops from a distribution favoring large M.
The size of the hierarchy, v 2 /M 2 , is enhanced by a loop factor β, gaining a factor of ≈ β/q, so that M is typically in the TeV region or larger. These results are independent of what other Standard Model parameters are scanning, including Yukawa couplings and Λ QCD , and do not even depend on whether selection is happening at the nuclear stability boundaries, or at the phase boundary for electroweak symmetry breaking itself. Hence we stress that the multiverse provides a very general solution to the hierarchy problem, whether little or large.
All three naturalness problems have the common feature of being solved by a multiverse distribution that makes observers typically close to an observer boundary. There may be connections between the three problems, arising from integrating out certain parameters in a more fundamental theory. For example, the probability force driving a little hierarchy may originates from the probability distribution for the cosmological constant, with WIMP dark matter acting as a mediator.
The current status of naturalness in electroweak symmetry breaking, Eq. (122), indicates that either we have not yet arrived at the right theory or that environmental selection is playing an important role. The LHC will determine the correct interpretation of Eq. (122), leading us either to a new natural theory, or to a third arena for multiverse evidence. Even if the naturalness probability is much larger than for the cosmological constant, the pervasive pattern of a finely tuned universe will make the multiverse much harder to dismiss. The importance of the LHC in this regard cannot be overemphasized: for the cosmological constant and nuclear naturalness problems we are stuck -we may not be able to experimentally determine the relevant theory beyond what we already know, in which case increasing the naturalness probability is a theoretical enterprise. However, the LHC will teach us a great deal about the theory of electroweak symmetry breaking and hence will lead to a better determination of P EWSB . One possibility is that the LHC will reveal a completely natural theory that we have not been able to invent. Below we mention a few examples where LHC data could determine a small value for P EWSB .
If the LHC discovers a light Higgs boson of mass m Higgs and sets a limit of M col on new colored particles, then P EWSB < ∼ 0.08 m Higgs 150 GeV This result is true in the vast majority of theories, although some counterexamples are known. Another possibility is that only a light Higgs boson is discovered, with a mass very close to the vacuum instability limit of the Standard Model. While Eq. (123) still applies as a direct consequence, there is the additional implication that the hierarchy is large with a very much smaller P EWSB . Evidence for a large hierarchy could also emerge from the discovery of split supersymmetry. Alternatively, the discovery of weak scale supersymmetry, with a light Higgs boson and a top squark heavier than ≈ 1 TeV would indicate a little hierarchy with a naturalness probability P EWSB < ∼ 0.05 m Higgs 130 GeV where M mess is the messenger scale of supersymmetry breaking. The discovery of dark energy has verified a remarkable prediction of the multiverse; but this could be undermined by the discovery of an alternative solution to the cosmological constant problem. The nuclear stability boundaries imply at least 1% fine-tuning in any known theory, and the multiverse allows a striking understanding of m u,d,e . Our current theories of electroweak symmetry breaking are unnatural; a confirmation by the LHC would solidify evidence for the multiverse in a third arena. Even with P nuc and P EWSB much larger than P Λ , the three arenas together would provide significant, robust evidence for a multiverse.