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Measures on Mixing Angles

We address the problem of the apparently very small magnitude of CP violation in the standard model, measured by the Jarlskog invariant J. In order to make statements about probabilities for certain values of J, we seek to find a natural measure on the space of Kobayashi-Maskawa matrices, the double quotient U(1)^2\SU(3)/U(1)^2. We review several possible, geometrically motivated choices of the measure, and compute expectation values for powers of J for these measures. We find that different choices of the measure generically make the observed magnitude of CP violation appear finely tuned. Since the quark masses and the mixing angles are determined by the same set of Yukawa couplings, we then do a second calculation in which we take the known quark mass hierarchy into account. We construct the simplest measure on the space of 3 x 3 Hermitian matrices which reproduces this known hierarchy. Calculating expectation values for powers of J in this second approach, we find that values of J close to the observed value are now rather likely, and there does not seem to be any fine tuning. Our results suggest that the choice of Kobayashi-Maskawa angles is closely linked to the observed mass hierarchy. We close by discussing the corresponding case of neutrinos.


Introduction
A traditional attitude to theoretical physics has been that the main problem is to discover the fundamental laws of physics and leave it to experiment and observation to decide what particular implementation best describes "Our Universe." Thus traditionally a "physical theory" is often thought of in terms of a local Lagrangian including certain "coupling constants," "mass ratios," and "mixing angles," all of which, since Planck's introduction of Planck units [1], may be taken to be dimensionless numbers. In addition, the local Lagrangian must be supplemented with an account of the general class of boundary conditions for which the variational principle is valid. Different classes of boundary conditions are usually thought of as different "superselection sectors" of the theory, and describe qualitatively different types of situations which traditionally are not thought of as having any relation to one another.
Within each sector, there are many solutions of the equations of motion, each of which may be specified by providing suitable "initial conditions." Classically these conditions may be thought of as the space of classical histories, and given in terms of Cauchy data modulo the relation that two sets of Cauchy data giving the same history are taken to be equivalent. Quantum mechanically one thinks in terms of some initial, and thus in the Heisenberg picture, eternal state.
The hope has frequently been expressed in the past that eventually theorists will hit upon a unique theory, with all coupling constants determined by consistency or symmetry considerations, and with just one superselection sector. Even given such a "theory of everything" (TOE), there remains the issue of boundary conditions or initial state, as emphasized by Hawking [2]. Recently, however, there has been a considerable decline in optimism and few now seem to believe in a single TOE with a single superselection sector, and many refer to a "landscape" of theories.
One approach to this perceived crisis in theoretical physics is to resort to "anthropic" considerations and invoke the idea that there may indeed exist, in the Platonic sense, an enormous number of "possible universes," of which only very few will allow the development of sentient beings, and even fewer will allow sentient beings like ourselves. Thus one is led to contemplate the ensemble of all possible universes, sometimes referred to as a "multiverse" [3]. This ensemble is sometimes thought of non-Platonically 1 as an ensemble of connected subsets of a much bigger physically existing universe, referred to as a meta-universe [4].
At this point it may be helpful to remark, lest the daunting task of thinking about and making more precise, the nebulous notion of such a multiverse should not be thought entirely a problem for theorists seeking credit for making predictions about the world we see about us: that the observers and experimenters must also face up to that task when assessing the reliability of their measurements or the extent to which they can confirm theoretical predictions. All such activities are essentially Bayesian in character [5] and require some notion of "priors," that is, some sort of a priori measure of the space of possibilities. This problem has been addressed, with admittedly only partial success, in a previous paper [6] 1 in an Aristotelian sense?
where the multiverse, for concreteness, was identified with the set of classical histories of a minisuperspace cosmological model. A well-defined and natural local measure on the space of classical histories is easily constructed, but unfortunately the total measure of all histories, even in this finitedimensional truncation of the full set of solutions of Einstein's equations, is infinite. The problem was recently revisited in [7].
In the present paper, we shall turn to the problem of finding a natural measure on the space of coupling constants. Thus the multiverse in the present paper is a set of Lagrangians parametrized by a manifold X or "moduli space," whose coordinates consist of masses, mixing angles, coupling constants, etc., and we wish to place a natural measure on this space. We hope this will be useful for anthropic considerations such as those of [8], where X ≡ S 1 , the circle parametrizing the phase of the axion. In that case the issue of a measure was trivial, but in more complicated cases such as we shall consider in the present paper, the situation is more complicated. We also hope that the work in this paper will help in clarifying the notion of "fine-tuning," which is so prevalent in phenomenological discussions.
The structure of this paper is as follows. After introducing the notion of geometric probability and outlining the Kobayashi-Maskawa theory of CP violation in the standard model, we discuss metrics on SU (3) and its quotients in Sec. 2, starting with a left-invariant metric on SU (3) which induces a metric on the flag manifold SU (3)/U (1) 2 . We perform a Kaluza-Klein type reduction on the left phases and discuss different possible metrics on the double quotient U (1) 2 \SU (3)/U (1) 2 , the space of Kobayashi-Maskawa matrices. We also discuss the metric used by Ozsváth and Schücking [9], and argue that it lacks a geometrical justification.
We then use all metrics we have discussed to compute statistics of the Jarlskog invariant J. While the measure on SU (3)/U (1) 2 is independent from the choice of left-invariant metric, the measure on the double quotient is nonunique. We find that in each case the standard deviation ∆J (with J = 0) is about three orders of magnitude greater than the experimentally observed value of J, which appears to be finely tuned. In Sec. 4, we do a closer numerical analysis of the probability distribution of |J| on the double quotient, using several possible choices for the measure. We quantify the statement that a magnitude of CP violation as small as observed appears unlikely.
In Sec. 5, we take a different viewpoint: We now consider random distributions on the space of mass matrices in the standard model. We therefore need to find a measure on the space of 3 × 3 Hermitian matrices. We find that the simplest choice which gives convergent integrals over this space, and expectation values for squared quark masses which reproduce the observed values, is a Gaussian weighting function with four free parameters, which can be chosen appropriately. We then find that the standard deviation ∆J is much smaller for this measure, making the observed value of J appear typical. We conclude that with an appropriate distribution which respects the known quark mass hierarchy there is no need for fine-tuning in J.
We briefly discuss the case of neutrinos in Sec. 6, explaining the general theory, and the difference between Dirac and Majorana masses. We cannot give reliable predictions for ∆J for neutrinos, due to the absence of known values for their masses.

Geometric Probability
The construction of appropriate measures over spaces of geometric objects goes back to the 18th century cosmologist Buffon and his celebrated needle problem [10]. The reader may find a general account of the subject in [11]. The simplest case to consider is when the space of coupling constants X may be regarded as a finite-dimensional homogeneous space with respect to some Lie group G of symmetries, and the stabilizer or little group is H ⊂ G. Thus X = G/H. If dimX = n, our aim is to construct an n-form on X which is invariant under the action of G. In the case that X = G, as in the example of the axion circle, this is completely unambiguous. We pick any n-form at the unit element e ∈ G and spread it over G by left or right translation. On a unimodular group, such as a compact group or a semisimple group, left or right translation will give identical results. The original n-form, being a top degree form, is unique up to a multiple. This multiple can be fixed by normalizing the total measure to unity. The normalized measure is therefore unique.
We could, if we wished, construct the measure as the Riemannian volume element of any left or right-invariant metric on G. The result would be the same. In practice, a convenient procedure for calculating the measure could be to construct an invariant metric on X = G and then calculate its Riemannian volume element. Often, the bi-invariant or Killing metric is the most convenient choice.
In the case of a coset, X = G/H, the measure can again be taken to be any n-form at some arbitrarily chosen point x ∈ X, which is then spread around using the group action. Since any n-form at x will be H-invariant, the result is again unique and invariant under all the symmetries of the problem. Of course it is possible that one may express X = G/H in more than one way. This could in principle give rise to some discrete nonuniqueness, but in practice this seems not to be important.
Although the situation when coupling constants may be regarded as belonging to a homogeneous space is quite satisfactory, it is often the case that coupling constants belong to an inhomogeneous space. In particular, in the case of "mixing angles," they typically belong to a double coset, or bi-quotient, of the form H 1 \G/H 2 , where H 1 and H 2 are (not necessarily identical) Lie subgroups of G. The reason for this is that mixing angles relate two unitary bases for the same space of physical states. The two unitary bases may not be unique. In particular, it is often the case that the individual basis vectors can be multiplied by arbitrary phases. In this case, H 1 and H 2 may belong to U (1) k , where k is the number of states in the basis. In the case of the Kobayashi-Maskawa matrix the states are quarks, and one basis diagonalizes the strong Hamiltonian while the other basis diagonalizes the weak interaction quantum numbers.
A biquotient, or double coset, H 1 \G/H 2 , is typically not a homogeneous space. This is because the left action of G will not in general commute with H 1 , and similarly, the right action of G will not commute with H 2 . As a consequence, one cannot, in the case of biquotients, use group invariance to construct an unambiguous measure on the space of mixing angles. Later in this paper, we shall explore in detail some available options, and the extent to which they affect the probability distribution of mixing angles.

CP Violation and the Jarlskog Invariant
In this section we shall review the Kobayashi-Maskawa theory of CP violation in the quark sector of the standard model.
If m and m ′ are the (Hermitian) mass matrices for the charge 2 3 and − 1 3 quarks, respectively, then there exist unitary matrices U and U ′ such that The Kobayashi-Maskawa matrix V is defined by The normalized mass eigenstates are only defined up to a phase, and changing these phases changes the matrices U and U ′ according to Hence the Kobayashi-Maskawa matrix changes according to where P L and P R are diagonal matrices belonging to SU (3). In other words, P L and P R may each be thought of as belonging to is an example of a flag manifold. The maximal torus U (1) 2 acts on the flag manifold via left actions of SU (3), but its action on SU (3)/U (1) 2 is not free, and as a consequence, the biquotient is not a smooth compact manifold without boundary. Rather, it is a stratified set whose boundary consists of components at which either or both of the left-acting U (1) factors has fixed points.
In the standard notation and it is customary to choose the phases so that where s 12 = sin θ 12 , c 12 = cos θ 12 , etc., and the angles θ 12 , θ 13 , and θ 23 are taken all to lie in the first quadrant (i.e. between 0 and 1 2 π). One can take the angle δ as a measure of CP violation, but its definition depends on the choice of phases. Jarlskog [12,13] introduced a formalism that eliminates this arbitrariness. She defined a Hermitian tracefree matrix C by and took det C as a measure of CP violation. She showed that where and the Jarlskog invariant J is given by Despite appearances, J is independent of the arbitrary phases. In other words, it is invariant under (4). In fact, it has an extremely elegant geometrical interpretation. Since V is a unitary matrix, its three rows and columns are orthogonal. Thus, for example, there are three relations of the form The three complex numbers a = V 11 V * 21 , b = V 12 V * 22 , and c = V 13 V * 23 , satisfying a + b + c = 0, may be thought of as the three sides of a unitarity triangle in the complex plane. The absolute value of J is twice the area of this triangle: The effect of the transformation (4) is to rotate this triangle in the complex plane, but the area 1 2 |J| is unchanged. Less obviously, the same area results from taking either of the two other possible inner products, (V V † ) 13 = or (V V † ) 23 = 0. Thus J is an invariant, and so it is well defined on the space of mixing angles.
In terms of the standard parametrization (6), the Jarlskog invariant is given by J = c 12 c 23 c 2 13 s 12 s 23 s 13 sin δ .
One could choose to take a different quantity as a measure of CP violation. Jarlskog [14] suggested appropriately normalizing the determinant (8) and using which takes values between +1 and −1 and is zero if and only if CP is conserved. Written out explicitly in terms of the quark masses and mixing angles, this is a complicated expression that we do not give here. As in the present paper the observed quark mass hierarchy is assumed, we shall not consider the case of coinciding quark masses, and we concentrate on J as a measure of CP violation.
Another possible source of confusion is the assumption of general, not necessarily Hermitian, mass matrices. In this case the commutator (7) is replaced by (15) in order for C to be Hermitian. The use of C or C can lead to ambiguous "orders of magnitude" estimates for CP violating processes, e.g. when discussing baryogenesis. We will assume that m and m ′ are Hermitian, and as our calculations only involve J these considerations will not be relevant.

Metrics on SU (3) and its Quotients
A generic element U of SU (3) is conveniently parametrized by eight real coordinates (p, q, r, t, x, y, z, w), so that where and Here, we are using the standard Gell-Mann representation for the generators of SU (3): Explicitly, the matrices W , T L , and T R are given by where we use the notation s x = sin x, c x = cos x, etc.
If we identify W as the Kobayashi-Maskawa matrix V in the standard conventions (6), then If we define left-invariant one-forms σ a by then the general left-invariant metric on SU (3) takes the form where g ab is a constant symmetric matrix. in [15].
In the generic case (i.e. when K is the identity), 28 = 36 − 8 parameters are required to specify the metric. One of these parameters sets the overall scale. For the intermediate cases there are correspondingly fewer parameters [15]. The bi-invariant metric has the smallest number, namely just the overall scale. In all cases, the invariant measure on the group SU (3) is the same and given where N is a constant normalization factor.
One of the intermediate cases given in [15] corresponds to This has the symmetry SU (3) L × SO(3) R , where the SO(3) R is generated by λ 2 , λ 5 and λ 7 . Remarkably, there is a second Einstein metric in this class [16], in addition to the standard bi-invariant metric that arises when a = b. The nonstandard Einstein metric occurs when b = a/ √ 11.
However, the measure on the biquotient U (1) 2 \SU (3)/U (1) 2 of SU (3) is not unique. In particular, if one constructs the measure from an invariant metric it will depend upon the metric that is used. One of the cases enumerated in [15], which is of particular interest for purposes, is when . One may denote these U (1) subgroups by U (1) 3 and U (1) 8 , indicating that they are generated by λ 3 and λ 8 . The possible U (1) 3 × U (1) 8 × SU (3) L invariant metrics on SU (3) are The induced metric on the right coset SU (3)/U (1) 2 is then given by The normalized invariant measure on this coset is given by There is no similarly unique construction of a measure on the biquotient U Locally, the biquotient U (1) 2 \SU (3)/U (1) 2 is a fiber space whose fibers are orbits of U (1) 2 L × U (1) 2 R , whose action has fixed points. The biquotient is therefore not a smooth manifold. Nevertheless, any metric on SU (3) will induce on any local section a metric, and hence a Riemannian measure. However, the metric and the measure will in general depend upon the choice of section. In the language of Kaluza-Klein theory, such metrics will in general depend upon the choice of gauge.
One way to resolve this ambiguity is to project the initial metric on SU (3) orthogonally to the  There is a general construction showing that every quotient of a compact Lie group G by its maximal torus may be regarded as an Einstein-Kähler manifold. A physical application of this result would be to the modulus space of vacua of a Yang-Mills theory with Higgs in the adjoint. We shall describe the special case of G = SU (3), following the construction described in [17]. This makes use of the fact that where B is the Borel subgroup of SL(3, C). In other words, we can express an SU (3) matrix U in the Iwasawa form Substituting the expression for U given in (16), we find in particular that z 3 = e −3ip+iq (e iw sin x tan y + cos x sec y tan z) .
These expressions can be inverted to give the real coordinates in terms of the z α : with p and q then obtained using As discussed in [17], the z α can be viewed as complex holomorphic coordinates on the flag manifold. The Kähler function is given by It is easy to check that the Kähler metric, given by has a determinant given by which can therefore be written as Thus K satisfies the Monge-Ampère equation, implying that the Kähler metric g αβ is Einstein.
Substituting (34) into (38), one obtains the Einstein-Kähler metric on the flag manifold written in terms of the real coordinates (p, q, x, y, z, w). It is straightforward to verify directly that it satisfies In terms of the real coordinates, the Kähler function (37) is given by The Einstein-Kähler metric (38) is invariant under the left action of SU (3), and in particular, under the T 2 action generated by ∂/∂p and ∂/∂q. From (34), this action corresponds to phasing the complex coordinates z α in such a way as to leave the Kähler function (37) invariant. It is possible, therefore, to perform a Kaluza-Klein reduction on the two angles p and q, to obtain a metric on the The resulting metric is extremely complicated, and we shall not give it explicitly. However, the metric we obtain is not the same as the one discussed in Sec.

2.2.
This difference is connected with the fact that the Einstein-Kähler metric given by (37) and (38) is not the "round" metric on SU (3)/U (1) 2 , but rather, it is a particular member of a one-parameter family of homogeneous squashed metrics. (It corresponds to the only other member of the family, other than the round metric, that is Einstein.) The Einstein-Kähler metric constructed in (38) is given, in terms of the left-invariant one-forms σ a defined in (25), by This illustrates the remarks we made previously about the ambiguity of measures on bi-quotients.
This construction, while not providing us with a "simple" metric on the double quotient, has the virtue of being possible for any coset SU (N )/U (1) N −1 ; we shall see in Sec. 6.2 that the case N = 6 may be of relevance to neutrinos.
As expected, this metric on SU (3) does not depend on p, q, r and t, which are the arbitrary quark phases appearing in the Kobayashi-Maskawa matrix.
It is perhaps worth remarking here that the first expression in (46) is well defined for any complex matrices U , unitary or not. For general complex matrices, it defines a flat metric on the space of matrix elements, which may be identified with C 9 ≡ E 18 , the 18-dimensional Euclidean space. 2 SU (3) may be regarded as a real eight-dimensional submanifold of E 18 , defined by the nine real unitary constraints U U † = 1 together with the one real unimodularity constraint det U = 1. The bi-invariant metric on SU (3) is the induced metric on this submanifold.
One approach to placing a measure on mixing angles would be to give a uniform measure on the unconstrained mixing angles, and then to obtain a measure on the mixing angles by implementing the unitarity and unimodularity conditions. The left-invariant measure on SU (3) is unique up to a scale. Thus, any construction which respects SU (3) invariance will result in a measure which is a constant multiple of the Riemannian measure constructed from the bi-invariant metric.
Note that one can alternatively obtain the four-dimensional metric on the bi-quotient by means of a T 2 Kaluza-Klein reduction of the round flag-manifold metric which differs from the one used in Sec. 2.1.
We have calculated the Riemannian metrics of the two different metrics on the bi-quotients, and confirmed that these two four-dimensional measures are indeed different. Later, we shall demonstrate the dependence of the mean-square value of the Jarlskog invariant on the choice of squashing.
Note that this expression reduces to (53) if β = 1, which is the special case of the reduction of the round flag metric. The nontrivial dependence of (57) on the squashing parameter β shows that the measure √g dxdydzdw on the double coset is also nontrivially dependent on the choice of squashing.

Ozsváth-Schücking metric
The previous calculations give rise to rather complicated formulae. It is striking, therefore, that the metric obtained by Ozsváth and Schücking [9] is so much simpler. Their choice of section consists of simply setting p = q = r = t = 0 in the metric (47). This results in the metric This metric is manifestly invariant under translating the coordinates x and z. Remarkably, there is one further commuting Killing vector. If one defines new coordinates u and v by (sin y cos w, sin y sin w, cos y) = (cos u, sin u cos v, sin u sin v) , then the metric (58) takes the form which has the three commuting Killing vectors ∂/∂x, ∂/∂z and ∂/∂v.
Geometrically, we can understand this if we note that the metric (58) may recast as which exhibits it as a T 2 fibration (having coordinates x and z) over a round hemisphere (having coordinates y and w). (The colatitude y lies in the interval 0 ≤ y ≤ 1 2 π.) The hemisphere can be embedded isometrically into Euclidean three-space as (sin y cos w, sin y sin w, cos y). Equation (59) then gives a different embedding such that while ∂/∂w generates a rotation around the first axis, ∂/∂v generates a rotation around the third axis. The projection along the third axis is given by cos y, while the projection along the first axis is given by cos u.
Note that the extra Killing vector ∂/∂v is purely local, since rotations about the first axis do not preserve the hemisphere.
The metric (60) can be recast in the form which is locally of the form of a U (1) fibration (with coordinate v) over S 3 .
The un-normalized measure is given by the remarkably simple formula µ = sin y (1 − sin 2 y cos 2 w) 1/2 dxdydzdw , Despite its appealing simplicity, the Ozsváth-Schücking construction lacks a geometrical justification and introduces a spurious U (1) 3 symmetry into the problem. A simpler example, which makes this clear, is provided by considering the lower-dimensional example of quotients of SU (2).
The bi-invariant metric on SU (2) is where ∂/∂φ generates U (1) L and ∂/∂ψ generates U (1) R . Projecting the metric orthogonally to the orbits of right translations,à la Kaluza-Klein, gives the round metric on S 2 . By contrast, simply setting dψ = 0 (the analog of the construction of Ozsváth and Schücking) instead gives the flat metric The round metric (65) is invariant under SO(3). The flat metric (66) appears to be invariant under the Euclidean group, with ∂/∂θ and ∂/∂φ having the appearance of translations, but these are only local symmetries since φ is a periodic coordinate and θ lies in an interval.
The example of SU (2) also illustrates the difference between taking the flag-manifold measure and the Kaluza-Klein measure on a biquotient. The biquotient U (1)\SU (2)/U (1) = U (1)\S 2 is just an interval. Its metric becomes, after performing another Kaluza-Klein reduction of (65), The measure would be dθ, and not dθ sin θ as obtained by integrating a function f (θ) over the coordinate φ. It is apparent from this simple example that there are inequivalent ways of calculating integrals of a function on a right quotient that is invariant under the left group action; namely, one can either reduce the metric to obtain a measure on the double quotient or take the measure on the single quotient and integrate out the left phases.
3 Statistics of the Jarlskog Invariant J Expressed in terms of the coordinates (x, y, z, w), the Jarlskog invariant (13) is given by The average of a function f on a space with metricg µν is defined by The experimental value of the Jarlskog invariant J of the Kobayashi-Maskawa matrix is which is very small compared with its maximum value In the following subsections 3.1 to 3.5, we calculate the moments of J for the Kobayashi-Maskawa matrix, using the various measures we have introduced, and compare them with the experimental value. We will see that the average values one obtains are rather insensitive to the choice of measure.
In subsection 3.1, we start with SU Since the Jarlskog invariant (10), or (68), is independent of all phasing angles, the averaging of its moments over the flag manifold will give the same results regardless of whether one constructs the manifold as the left quotient or the right quotient of SU (3) by U (1) 2 . This is convenient because we have already presented detailed results for the metrics on the right cosets SU (3)/U (1) 2 .
A straightforward calculation shows that for the general class of SU (3)-invariant flag metrics (30), √ g = 3αβγ sin 2x sin 2z sin y cos 3 y .
Since an overall constant factor in the measure cancels out in the normalized averaging process, we see therefore that in contradistinction to the situation for the double coset U (1) 2 \SU ( Since it is only V = U U ′ † that enters into the CP violating parameters, one could consider U and V as independent variables, i.e. write U ′ = V † U for some matrix V . Then the Maurer-Cartan form on the second SU (3) is For the measure (72) the evaluation of the necessary integrals is very simple and we find that all odd powers of J average to zero, and Thus we find that ∆J for the Jarlskog invariant is given by which is about three orders of magnitude larger than the experimental value (70).

The Kaluza-Klein measure from the bi-invariant metric
For the metric on the biquotient discussed in Sec. 2.2, the expression for the measure is too complicated to allow us to perform the integrations analytically. Using numerical integration, we find that with the odd powers of J again averaging to zero. Thus we find which is very close to the previous result.
Naively, one might have thought that since J is independent of all the U (1)  and J 4 seem to be maximized by the round case (77).)

The Kaluza-Klein measure from squashed metrics
We can repeat the calculations of Sec. 3.2 using the measure given by (56) and (57) with the odd powers of J averaging to zero. Thus we find This is smaller than the value of ∆J we obtained in (78) for the averaging over the Kaluza-Klein reduction of the bi-invariant metric, but only by about 0.7%. This does not bring it significantly closer to the experimental value of J, given in (70).
One might wonder whether, for some sufficiently large or small choice for the squashing parameter β, it might be possible to obtain a result for ∆J that was comparable with the experimentally observed value. In fact, it appears that J 2 is a rather slowly varying function of β. The value of J 2 appears to be maximized by the choice β = 1, and to fall off monotonically in both directions as β is taken to zero or to infinity.
For example, if we choose β = 1 2 we find while if we take β → 0 we find Taking β = 1000, we find while for β = 10 6 we find Even quite extreme values for the squashing parameter only bring about small reductions in ∆J.

The Ozsváth-Schücking measure
Using the Ozsváth-Schücking measure (63), we find that J = 0 and and that the standard deviation is and Again, the results are rather similar to the previous cases.

The uniform measure
Assuming a uniform distribution over the angles, and hence treating the double coset as a flat four-dimensional manifold so that the measure is simply µ = 1, would give Hence, this simplest possible choice gives 4 Fine-tuning of J In the previous section we saw that different measures on the space of mixing angles all seem to lead to expectation values for J which are about three orders of magnitude larger than the observed value. The value for J that we observe hence appears to be finely tuned. In this section we shall do a closer, mainly numerical, analysis of the fine-tuning involved. We compare results obtained by taking the SU (3)-invariant and Kaluza-Klein measures, which seem natural from a geometric perspective, with a uniform distribution which is just the simplest possible choice.

Probability distribution of J
The observed value for the Jarlskog invariant J is In order to obtain a probability distribution for J we have used Mathematica to numerically compute integrals of the form using Monte Carlo methods. The SU (3)-invariant flag-manifold measure and the Kaluza-Klein measure disfavor small values of J more strongly than a uniform distribution would. For example, we obtain Taking a uniform distribution √g ≡ 1, we get finally for the uniform measure (Fig. 3), the best fit to the data in the region below |J| = 10 −3.4

Wolfenstein parametrization
A different parametrization of the Kobayashi-Maskawa matrix which is frequently used was introduced by Wolfenstein and is based on the experimentally observed hierarchy in the mixing angles. One rewrites [18] sin z = λ , sin x = Aλ 2 , sin ye −iw = Aλ 3 (ρ − iη) and treats λ as a small parameter while A, ρ, and η are supposed to be parameters of order unity. In the modern literature one also frequently usesρ,η instead of ρ and η because then the combinationρ+ iη is independent of the phase convention in the Kobayashi-Maskawa matrix [19]. These parameters are defined by One viewpoint on the Wolfenstein parametrization is that it is adapted to the values for the Kobayashi-Maskawa matrix entries that we observe and has no deeper significance; but often the viewpoint is expressed that this parametrization expresses some kind of "natural hierarchy" in the mixing angles coming from physics beyond the standard model (see e.g. [20]). Treating the other parameters as "naturally of order unity" reduces our calculations to a one-dimensional problem as everything is only expanded in terms of λ. We find that the SU (3)-invariant measure on the flag manifold is now, to leading order in λ, and the Jarlskog invariant J is Inverting this expression to leading order gives the probability distribution for J which is incompatible with the numerical results. Trying to improve this approximate result by letting A,ρ andη take all possible values leads to inconsistencies since the expansion in powers of J contains poles of arbitrary order in A. From our present viewpoint, where no mechanism for fixing these parameters close to one is known, the Wolfenstein parametrization seems rather misleading when discussing geometric probability.

Quark Mass Matrices and Gaussian Weighting Functions
In the previous sections we have focussed on U (1) 2 \SU (3)/U (1) 2 , the space of Kobayashi-Maskawa matrices, as the space of CP violating parameters. Since SU (3) is compact, this space has finite volume for a natural measure. But the Kobayashi-Maskawa matrix is derived from the Hermitian quark mass matrices, which could be viewed as more fundamental and more directly determined by physics beyond the standard model. In this section, we try to obtain statistics of the Jarlskog invariant J from a random distribution on the space of 3 × 3 Hermitian matrices.

Distributions on Hermitian matrices
We follow Sec. 1.2 and write the quark mass matrices as where U and U ′ should be thought of as elements of U (1) 2 \SU (3). Following [12] we normalize the mass matrices by dividing by mass scales Λ and Λ ′ (often taken to be the top and bottom quark mass, respectively) which may be chosen for convenience: The matrices D and D ′ are now dimensionless quantities, and it is clear that U and U ′ are only defined up to left multiplication by elements of U (1) 2 . We consider Λ and Λ ′ as arbitrary mass scales, and so we will allow arbitrary eigenvalues for both matrices, instead of fixing one of them to be one.
A natural measure on the space of Hermitian matrices is induced by the metric which is invariant under conjugation under U (3). If we define right-invariant one-forms τ a by The corresponding volume form is As explained above, the measure on the coset U (1) 2 \SU (3) is unique and equal to the measure induced from the bi-invariant metric on SU (3). We obtain a Riemannian measure where S 3 is the symmetric group of degree 3 (the dihedral group of order 6, sometimes denoted by D 3 or D 6 ) which permutes the canonical basis vectors of R 3 . The set of matrices with coinciding eigenvalues has zero measure and hence can be ignored in the present discussion.
Thus we need to consider the space (3) instead, restricting the coordinates on the flag manifold to an appropriate range to pick one of the six matrices related by the S 3 action.
We can use the fact that the S 3 action permutes the rows of an SU (3) matrix to demand that the elements of the third column (see Sec. 2) satisfy the relation | sin y| ≤ | sin x cos y| ≤ | cos x cos y| , which restricts the coordinates x and y to Using the natural measure on the flag manifold, we see that this region has precisely one-sixth of the total volume of the flag manifold: An integral over R 3 with the given measure diverges. We could introduce a cutoff for the quark masses, but then any expectation values for quark masses would strongly contradict observation, as there is no way to explain the observed mass hierarchy.
We therefore choose to introduce a weighting function in the measure which decays sufficiently fast for large positive or negative eigenvalues and is able to reproduce the known hierarchy. The simplest assumption is to take a weighting function of the form An integral of a quantity such as J 2 becomes 5 Here DU and DU ′ are the measures on (U (1) 2 × S 3 )\SU (3) and dD : , and the normalization factor N is defined by which commutes with A and A ′ . We find that this symmetry breaking is necessary to obtain a distribution that reproduces different expectation values for squared quark masses.
It should be clear from (116) that multiplying A (or A ′ ) by a constant is the same as rescaling the eigenvalues D i (or D ′ i ) and so amounts to a rescaling of Λ (or Λ ′ ). We can therefore, without any loss of generality, choose where µ c , µ u , µ s , and µ d are dimensionless parameters that we are free to choose so as to reproduce the observed quark masses as expectation values. (In the case of an exponential exp(−Tr(D 2 A)), these would of course be equal to the respective quark masses, expressed in units where Λ = m t and Λ ′ = m b .) Because of experimental uncertainties in the up and quark masses, one can modify this distribution to reproduce different values for these masses.
It seems practically impossible to evaluate the integral (116), as the expression for J in terms of coordinates on ((U (1) 2 × S 3 )\SU (3)) 2 is too complicated to be given explicitly. However, since with ξ 1 = A 1 cos 2 y cos 2 z+A 2 cos 2 y sin 2 z+A 3 sin 2 y , ξ ′ and we assume A 3 ≫ 1 and A ′ 3 ≫ 1, the integrand is negligibly small unless y ≈ 0 and y ′ ≈ 0. We use this to approximate the integrals over y and y ′ : It turns out that this is independent of w and w ′ . Constant prefactors such as 1/4A 3 A ′ 3 appearing in both numerator and denominator can be dropped, and so we have where and similarly for d 4 x ′ . Now we can integrate over both copies of R 3 in (123), using The explicit expression for J at y = y ′ = 0 is where s x = sin x, c z ′ = cos z ′ , etc.,r = r − r ′ , andt = t − t ′ . Integrating (126) over r, r ′ , t, and t ′ indeed gives zero, which is why we choose to use J 2 .

Results and dependence on quark masses
We need to determine the parameters appearing in the matrices A and A ′ in (119). We first observe that expectation values for squared mass matrices take the relatively simple form The denominator is explicitly where ξ 1 = A 1 cos 2 z + A 2 sin 2 z , ξ 2 = A 1 cos 2 x sin 2 z + A 2 cos 2 x cos 2 z + A 3 sin 2 x , with A 3 ≫ A 2 ≫ A 1 . We notice that all ξ a are nonzero for all values of x and z. Furthermore, the integral is dominated by very small x and z (we cannot have x = π 2 ), and we can approximate I D well by only keeping the terms of leading order in x and z in the trigonometric functions, and which are the leading terms (as we shall see, the first of these is effectively also of order A 2 3 A 2 ): Similarly, we find hence Redoing the same calculation for D 2 and D 3 gives There is a relative factor of 3 which has to be taken into account when determining A and A ′ .
Because of the dependence of masses on the energy scale in quantum field theory, described by the renormalization group, there is some ambiguity in what is meant by the "quark masses" we want to reproduce. Following [22], for example, we take all the quark masses evolved to the scale of the Z boson mass. These are given in [23]: The mass scales Λ and Λ ′ are fixed by setting D 2 1 = (m t /Λ) 2 and (D ′ 1 ) 2 = (m b /Λ ′ ) 2 . By comparing the results obtained by numerical integration with the values we want to reproduce, we can then fix the parameters µ c , µ u , µ s and µ d .
In the case of the positively charged top, charm and up quarks, which exhibit a more extreme quark mass hierarchy, we find that numerical calculations (using Mathematica) reproduce the results we have obtained analytically very well (see Table 1). For the negatively charged quarks, we find numerically that we have to use relative factors different from 3 to reproduce the observed masses.
Comparing the numerical results with (136), we fix the parameters appearing in A and A ′ to As a brief side remark, we see that the dominant contributions to these integrals come from the and these values are all small compared to one. We can therefore give rough estimates for magnitudes of individual elements of the Kobayashi-Maskawa matrix.
Our rough estimates reproduce the right ordering of the three parameters and are accurate to factors of order a few. A more careful analysis would involve computing expectation values for these parameters in the distribution we have assumed.
We return to the task of computing the expectation value of J 2 . In order to obtain an analytical expression, we use the fact that the main contribution to the integral (123) will come from small z to only take the term in (126) that is nonzero at z = 0. Averaging over r, t, r ′ ,and t ′ gives a factor of 1/2, as one might have expected, and therefore we use for our calculations. The numerator of (123) is the product (using again that only small z contributes) sin 2x sin 2 x cos 2 x(A 3 3 cos 2 x sin 2 x + A 2 3 A 2 (cos 6 x + 2 cos 2 x sin 2 x + sin 6 x)) (A 2 cos 2 x + A 3 sin 2 x) 5/2 (A 3 cos 2 x + A 2 sin 2 x) 5/2 (143) The first two factors are 2/(3A , respectively; for the other two (which are identical) we change variables to X = cos 2 x to obtain where we are dropping corrections of order A2 A3 . Putting everything together, we obtain where the numerical factors appearing in the last line come from the different factors chosen in (137).
Note that the top quark mass does not appear in this approximate result.
For numerical calculations we use both the simplified expression J 2 small z and the expression for J given in (126). We find that for the first quantity, the numerically evaluated expectation value small z is about 7/6 of (145), and the numerical result for J 2 (taken at y = y ′ = 0) is which gives which is much closer to the observed value than any of the previously obtained results. Assuming a Gaussian distribution for J which is peaked at zero, the probability of finding a small J, in the sense of Sec. 4, is now whereas the probability of finding a J which is even smaller than the observed value is The observed value for J can no longer be viewed as being finely tuned if the distribution used in our calculations is assumed.  To test the sensitivity of our results to changes in the parameters, we take values at the upper or lower limit in (135) and try to find the highest and lowest values for J 2 . We find that setting Since the former choice makes J appear more typical, these results are perhaps an indication that the correct values for the (up, down, and strange) quark masses are probably closer to the lower than to the upper bounds given in (135). Also, even the greatest possible value for ∆J is significantly lower than any of the values obtained in previous sections.
In this section, we have established that assuming the observed hierarchy in quark masses in a Gaussian distribution over the space of mass matrices gives expectation values for J 2 which are small enough to regard the observed value as "natural" and not finely tuned. This statistical observation seems to open up the possibility that the same mechanism that is responsible for the apparently unlikely hierarchy in quark masses might also explain why the observed value for |J| is so small.
A more detailed analysis including a probability density for |J| for this distribution is left to future work, since the numerical methods used here do not give sufficiently accurate results.

Extension to Neutrinos
In this section we review the case of neutrino masses, highlighting the difference between Majorana and Dirac masses and commenting on some recently made suggestions in the literature that one could distinguish the two cases by gravitational effects. In contrast to the physical situation which is at present rather unclear, the mathematical problem of obtaining a measure on the space of mixing matrices is in this case simpler, since one considers the flag manifold U (1) 2 \SU (3).

Neutrino masses
The spectrum of neutrinos and their masses and their nature, Majorana or Dirac, is currently not well known. Since not all of the material may be familiar to all readers we shall review some basic facts about Dirac and Majorana masses in a framework which is sufficiently general to encompass all likely possibilities. We find it helpful to use a Majorana notation, but no loss of generality thereby results since, if one starts with complex Weyl notation, one may always take real and imaginary parts. Alternatively, given a treatment in terms of Majorana spinors, one may always transcribe it into Weyl notation. In order to simplify the analysis we depart from common practice in phenomenological particle physics and adopt the spacetime signature (− + ++) for all spinors. This has the advantage that all gamma matrices may be taken to be real, t denotes transpose. C = −C t is the charge conjugation matrix and γ 5 = −(γ 5 ) t = γ 0 γ 1 γ 2 γ 3 so that (γ 5 ) 2 = −1. If required, a concrete representation is given by In this representation we may take C = γ 0 , and it is often useful to note that γ 0 is antisymmetric while γ 1 , γ 2 and γ 3 are symmetric.
The mass matrix of a set of fermions is defined by imagining setting to zero that part of the effective or large distance Lagrangian containing couplings to all gauge interactions except gravity.
In fact this is how information about neutrino masses is obtained. One observes mixing as they pass from the sun to the earth and upper bounds on their masses have been obtained by observing their arrival times from distant supernova 1987a.
The most general effective Lorentz-invariant Lagrangian for a system of k free Majorana fermions ψ i , i = 1, 2, . . . , k, is where ψ should be thought of as a k-dimensional column vector all of whose entries are four component real Majorana spinors, and m 1 and m 2 are real symmetric k × k matrices. Note that at this stage k may be even or odd. We have made use of the fact that we may diagonalize the kinetic term using GL(k, R) transformations. We are still allowed SO(k) ⊂ GL(k, R) transformations where The kinetic term, but not the mass term is also invariant under chiral rotations Combining these two sets of transformations we see that the kinetic term, but not the mass term is The U (k) invariance is perhaps more obvious if one uses a Weyl basis. Since one may regard γ 5 as providing a complex structure on the space of 4k real dimensional Majorana spinors, converting it to the 2k complex dimensional space of positive chirality Weyl spinors for which Clearly S then becomes the exponential of the k × k anti-Hermitian matrix Thus The mass matrix is then a complex symmetric matrix and under a U (k) transformation At this point we invoke the result of Zumino [24] that S may be chosen to render the matrix m diagonal with real non-negative entries m i .
In the general case, all the masses m i are distinct. They are then said to be of Majorana type.
However, it may happen that two masses, m 1 and m 2 say, coincide. One may then combine ψ 1 , ψ 2 into a Dirac spinor. One then has the case of a Dirac mass. For quarks and charged leptons all masses are of Dirac type. For neutrinos, however, it is not yet known of what type they are, nor indeed how many. A simple assumption is that k = 6, with three having very heavy masses and three having very light masses. This corresponds to the so-called "seesaw mechanism". Of course the very light neutrinos may be combined into three Weyl neutrinos and are taken to be massless in the standard model.
From the analysis above it follows that in the general case when all masses m i are distinct, there is a unique basis for the neutrino states, determined by their inertial motion. If, however, two or more masses coincide, then the basis becomes ambiguous up to rotations of the components with equal masses. In the case that k is even and there are [ k 2 ] distinct pairs of coincident masses the basis is arbitrary up to the action of . Any mixing matrix taking one to a basis which is preferred from the point of their nongravitational gauge interactions will be ambiguous to the extent that the inertial basis and the gauge basis are ambiguous. For that reason, in general, a mixing matrix belongs to a double quotient.

The Universality of free fall
In our discussion above we have referred to the interactions of neutrinos with gravitational fields.
Of course neutrinos observed to be coming from the sun, or the supernova 1987a are traveling so fast that the effects of gravity on them are negligible. However, it has been suggested that it is in principle possible to distinguish Majorana from Dirac masses by their behavior in the gravitational fields of rotating objects [25][26][27]. Our analysis above shows that unless there are gauge interactions such as might correspond to neutrino magnetic or electric dipole moments this is not so, as long as the coupling to gravity is "minimal."If so one simply uses for ∇ the standard Levi-Civita covariant derivative acting on spinors.
Assuming that the mass matrix m is independent of position, we may take it to be everywhere real and diagonal. Thus each component ψ i of the inertial basis propagates independently. One may iterate the Dirac equation and use the cyclic Bianchi identity in a curved space to get (reinstating powers of ) If the effects of curvature are negligible on the scale of the Compton wavelength, the second term may be dropped and one obtains the Klein-Gordon equation for each component.
As is well known, there is no "gyro-magnetic" coupling between the spin and the Ricci or Riemann tensors [28]. To proceed, one may pass to a Liouville-Green-Jeffreys-Wentzel-Kramers-Brillouin (L-G-J-W-K-B) approximation of the form One obtains from the original Dirac equation and It follows from (173) that Evaluation of the determinant in (175) gives the Hamilton-Jacobi equation This shows that the orthogonal trajectories defined by are timelike geodesics. The same conclusion follows by applying the L-G-J-W-K-B approximation to the second order iterated Dirac equation (170).
The iterated Dirac equation also gives The same result may be obtained by differentiating (173) and using (174). Thus the spinor χ i is parallelly propagated along the timelike geodesics up to direction in spin space. The amplitude of the spinor χ i is governed by the expansion u µ ;µ of the hypersurface timelike congruence whose tangent vector is given by We also have from (173) that χ i is an eigenspinor of u µ γ µ . As in flat space it follows that the spin tensor Since χ i is parallelly propagated along u µ in direction and since S µν depends only on the direction of χ i , it follows that the spin tensor S µν is parallelly transported along the timelike congruence, just like any other perfect gyroscope. The geodesics are independent of the mass eigenvalue m i and the polarization state given by χ i . Indeed if the fermion starts off in a given polarization state (with the associated mass), it remains in it. In other words, at the L-G-J-W-K-B level, the weak equivalence principle, in the form of the universality of free fall, i.e. the statement that all particles fall in the same way in a gravitational field independently of their mass, polarization, charge, etc., continues to hold. Thus there should be no unusual behavior in the vicinity of a spinning black hole, or indeed in the neighborhood of any spinning system due to the Lense-Thirring effect as suggested in [25,26], denied in [29] and maintained in [27].

Neutrino mixing matrix
Here, we briefly review the theory of the neutrino mixing matrix, assuming that the neutrinos are Majorana.
One conventionally fixes the phases so that M takes the form where the three angles θ 12 , θ 13 , and θ 23 lie in the first quadrant.
The Jarlskog invariant for the neutrino mixing matrix, defined as in (10) but with V now replaced by M , is again given by (13). Note, in particular, that it is independent of the phases α 1 and α 2 .
Experimentally, parameters of the neutrino mixing matrix are not completely known. According to [19], sin 2 2θ 12 = 0.86 +0.03 −0.04 , 0.92 < sin 2 2θ 23 ≤ 1 , sin 2 2θ 13 < 0.19 , and there is no experimental information about the Dirac angle δ. Thus, we can certainly deduce that there is an upper bound on the Jarlskog invariant for the neutrino mixing matrix, given by For six different neutrino mass eigenstates, as in the seesaw mechanism, a general mixing matrix would be an element of U (1) 5 \SU (6), since one would diagonalize a 6 × 6 Hermitian matrix.

Statistics of J
We have seen that the parameter space for the neutrino mixing matrix is the six-dimensional single quotient U (1) 2 \SU (3), and that the Jarlskog invariant for the neutrino mixing matrix takes the same form (10), or (68), as it does for the Kobayashi-Maskawa matrix. Therefore, all results obtained in This can be compared with the experimental bound given in (185).
One could also repeat the calculations of Sec. 5, assuming particular values for the neutrino masses. A strong hierarchy in the neutrino masses would then presumably again lead to "naturally" small CP violation from the corresponding mixing matrix. Alternatively, an experimental observation of small CP violation for neutrinos would perhaps be an indication of a mass hierarchy in neutrinos. At present, neither the magnitude of CP violation nor any values of neutrino masses have been measured sufficiently accurately to allow predictions.

Conclusions and Outlook
In this paper, we analyzed the problem of finding a natural measure on a space of coupling constants, which in our case was the space of Kobayashi-Maskawa matrices, the double quotient U (1) 2 \SU (3)/U (1) 2 . We saw that the measure on this double quotient is nonunique, and we ana- When calculating expectation values for J, we found that all of the measures we considered led to rather similar statistics of J. In each case, the observed value was about three orders of magnitude below what one would normally expect; the observed value appears to be finely tuned. The same applied to the Ozsváth-Schücking measure, an extremely squashed Kaluza-Klein measure, or a flat measure, which is just the simplest choice and not justified geometrically.
In Sec. 5, we adopted the different viewpoint that the Kobayashi-Maskawa matrix should not be viewed as separate from the quark masses, but that it is really the mass matrices which are "chosen" by a yet unknown physical mechanism. We took the observed values for the quark masses as an input and chose the simplest distribution which was able to reproduce these observed values, while inducing a different measure on the space of Kobayashi-Maskawa matrices. Assuming such a distribution, we found that the observed value of J now appears very natural and not finely tuned at all. In this statistical approach, regarding the Yukawa couplings determining the mass matrices as randomly chosen seems more appropriate than separating quark masses and mixing angles. (On submittal of this article to the archive we were informed of an earlier work [31], similar in spirit to ours but using different assumptions and methods, which reaches broadly similar conclusions).
Our analysis also applies to the case of massive neutrinos, where the predictions will conceivably be tested by future experiments. In the standard theory, the Maki-Nakagawa-Sakata matrix [32] which appears is naturally an element of the single quotient U (1) 2 \SU (3). Since the right phases do not play any role in neutrino oscillations and the relevant J is independent of these phases, the calculations are identical to the ones presented here, although with the appropriate values of the µ parameters appearing in A and A ′ .
In the seesaw mechanism one adds very heavy right-handed neutrinos, and the most general mixing matrix would be an element of U (1) 5 \SU (6). This is naturally a Kähler manifold, and the measure induced by the Kähler metric can be obtained from the analysis in [17]. We leave a detailed treatment of this case, following our approach here, to future work.
Finally, one could analyze the effects of a fourth generation of quarks on CP violation by repeating the calculations for 4 × 4 Hermitian matrices. If this generalization spoils the agreement with the observed J, one might obtain interesting lower bounds on the masses of a hypothetical extra generation of quarks.