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Cosmic Four-Fermion Neutrino Secret Interactions, Enhancement and Total Cross Section

The scattering of neutrinos assuming a"secret"interaction at low energy is considered. To leading order in energy, the two-body potential is a $\delta$-potential, and it is used as a motivation to study generic short-range elastic interactions between neutrinos. The scattering cross section and Sommerfeld enhancement depend on two phenomenological parameters deriving from the exact form of the potential, akin to"renormalized"coupling constants. Repulsive potentials lead to a decrease in the total cross section, resulting in an enhancement of the neutrino density. For attractive potentials of the right form, substantial Sommerfeld enhancement can appear.


I. INTRODUCTION
In the last twenty years the nature of dark matter, needed to address the missing mass problem, has been extensively investigated [1]. Specifically, there is a large body of evidence from astronomical observations indicating that there is more matter than can be associated with the luminous part of galaxies. This problem remains unresolved, and opens a window to new physics now called astroparticle physics [2]. It is not yet fully understood what kind of particle dominates the dark matter and what are the mechanisms of production/annihilation for them. Neutrinos would be one possibility, but presently we know that only a small portion of neutrinos (1%) of the CNB (cosmic neutrino background) could be considered as dark matter. Specifically, in the CNB the number of neutrinos per cubic centimeter is about 56 per flavor, and this suggests that a small fraction of the dark matter could be attributable to neutrinos. Therefore, cosmic neutrinos would be responsible for only a small part of dark halos that would explain the flatness of galaxy rotation curves [5,6], i.e. the constant value of the velocity of rotation of a galaxy as a function of distance rather than decreasing as expected. Even though dark neutrinos are very few, they are nevertheless important, as their mass bounds are very robust and can be used as phenomenological data for estimating the total cosmological neutrino flux and cross sections [3,4].
In the present state of our knowledge the cosmic neutrino background (CNB) is a prediction of the standard cosmological model. There are, however, known data which, when consistently interpreted, might lead to new insights beyond the standard model. In this direction, there are several interesting questions; for example, if there are any enhancement mechanisms for dark halo neutrinos, and how they would affect physical observable processes.
In reference [7] (see also [8]) an enhancement mechanism for weakly interacting particles was proposed, inspired by the Sommerfeld enhancement effect, and several examples illustrating how this mechanism works were presented. A thorough treatment of the effective non-relativistic theory of dark matter long-range interactions was also presented in [9], including their renormalization properties. In spite of the intense interest in this field in the last few years, however, to our knowledge there is no discussion on the role played by very short-range potentials (invisible at long distances), nor relevant extensions of the results presented in [7]. This would be especially relevant for low energy scattering of bosons or fermions with a short-range two-body potential. The dynamics of the CNB neutrinos at very low energy, in particular, could be considered by modeling the neutrino-neutrino interaction with a contact potential.
The goal of this paper is to provide a justification for the short range potential as stemming from possible "secret" neutrino interactions, to give a derivation of the scattering amplitude for such short range potentials, and to show how Sommerfeld enhancement can emerge by computing the corresponding enhancement factor. This work is organized as follows. In section II we calculate the two-body potential for neutrinos from secret interaction in inverse powers of the neutrino mass and show that the leading term is a delta potential (contact interaction). In section III we solve the associated scattering problem for a general short-range potential and show that physics depends on a set of phenomenological parameters. In section IV we consider the Sommerfeld enhancement and derive its strength in various domains of the scale of the annihilation processes. In section V we compare and contrast our approach with standard regularizations procedures for the delta-potential and elucidate its connection to renormalization in effective potentials, and briefly comment on bound states. In section VI we present our conclusions, pointing out that the neutrino density can be substantially enhanced if the potential is repulsive, and that there is Sommerfeld enhancement if the potential is attractive. Several calculations and justifications of approximations valid for short-range potentials are presented in the appendix.

II. EFFECTIVE TWO-BODY POTENTIAL FOR INTERACTIONS BETWEEN NEUTRINOS
Effective four-fermion interactions between neutrinos with nonstandard coupling (also called "secret" neutrio interactions) have been considered for quite some time [10], as a way to include new classes of matter as the particles mediating this interaction [11]. Clearly the four-fermion interaction is only a first approximation, capturing the lowenergy physics of an otherwise more general interaction, valid for distances larger than some characteristic scale for the interaction. (A pure four-fermion interaction would not even be renormalizable.) Its form is taken to be where a sum over the three neutrino species is understood, α = ±1 corresponds to an attractive/repulsive potential and M is a given mass scale. This contact neutrino interaction leads to the Born scattering amplitude for two distinct neutrinos (see Fig. 1) where w α 's denote the positive energy spinors in the non-relativistic limit and U αβ,γδ is the two-body potential which will be computed in detail below. To extract the low-energy dynamics, we work in the non-relativistic regime p m, with m the neutrino mass. For a free massive fermion, the Foldy-Wouthyusen transformation that diagonalizes the Hamiltonian in the non-relativistic limit is given by where and We use the normalization w * α (p)w α (p) = 1, Therefore, to order 1/m 2 we have and as a consequence This leads to The spinor u(p) can now be expanded up to order 1/m 2 as follows Thus by restricting to the positive energy spinor space, we have to order 1/m 2 where q = p 1 − p 1 = −(p 2 − p 2 ) is the transferred momentum. Similarly for the other spinor we havē Bilinears involving γ i can also be calculated and lead tō where the indices 1, 2 correspond to the two particle species and the greek indices α, β, · · · denote spinor components. Using the identity the scattering amplitude can now be written as where the two-body potential has the form This shows that even though the particles are fermions, at leading order the dominant contribution is a contact potential given by where x is the relative coordinate x 1 − x 2 and the potential can be repulsive or attractive depending of the sign of α.
Momentum-dependent higher-order corrections in this potential can also be calculated from (14) and are, generically, spin-dependent. We can use the above contact interaction as a motivation to consider momentum-independent short-range interactions with a range shorter than the neutrino Compton wavelength. In that regime, the full details of the interaction potential are expected to become irrelevant and be subsumed in some macroscopic phenomenological parameters.

III. THREE-DIMENSIONAL SHORT-RANGE POTENTIALS AND THEIR SCATTERING PROPERTIES
We turn our attention to the scattering properties of short-range potentials of arbitrary form in the regime where their range is much smaller than the de Broglie wavelength of the incident neutrinos. Such potentials would macroscopically look like delta-functions with strength equal to their space integral. Their details, however, will in general matter for scattering, so that more than one parameter may be relevant We assume that the neutrinos interact with a central two-body interaction potential of the form V (r) where r is the distance between the neutrinos, with the property that V (r) is nonzero only within a very short range 0 < r < a. We are interested in scattering properties for wavenumbers much smaller that a −1 . Clearly, not all the details of the form of V (r) will be relevant. The question is: what are the relevant parameters that fix physics for such wavelengths?
The motion in the above potential is described by the time independent Schrödinger equation for the relative coordinate wavefunction where k 2 = 2mE and m = m1m2 m1+m2 is the reduced mass of scattering neutrinos. The integral scattering equation, in this case, is where ϕ(x) = e ik·x represents the incident plane wave. The retarded (outgoing) Green's function satisfies and is explicitly given by with r = |x|.
In the kinematical regime of interest, only the spherically symmetric s-wave ( = 0) part of the vavefunction will matter for scattering and Sommerfeld enhancement, since higher-partial waves vanish at the origin and thus are much smaller over the range of the potential than the s-wave part. (This assertion is fully justified in the appendix.) In fact, restricting to the s-wave sector is exact for the calculation of ψ(0) since the integrand in the scattering equation in that case is spherically symmetric for a central potential V (r) and the higher angular momentum part of ψ(x) drops out. Specifically, putting x = 0 in (17) we obtain The s-wave part of the wavefunction, ψ s (r), satisfies the standard radial fixed-energy Schrödinger equation As usual, φ(r) vanishes at r = 0 and ψ(0) = φ (0). We write the full wavefunction in terms of a normalized u(r) = φ(r)/ψ(0) as where ψ ns (x) is the non-s-wave part (nonzero angular momentum) of the wavefunction, satisfying ψ ns (0) = 0, and the function u(r) is normalized to satisfy (primes are r-derivatives). With these initial conditions at r = 0 the solution u(r) is unique. The coefficient ψ(0) is fixed by matching the solution for ψ(x) to the appropriate boundary conditions at r → ∞.
Plugging the above form of ψ s (r) in (20) we obtain where we used the fact that V (r) vanishes for r > a. This fixes ψ(0) as Using equation (23) for u and integrating by parts we eventually obtain the exact expression for the wavefunction at the origin Similarly, to find the scattering amplitude we put |x| |x | in the scattering equation (17) and keep only the s-part of the wavefunction ψ s (r ) to obtain where θ is the polar angle of x measured with respect to x. The factor multiplying the outgoing spherical wave e ikr /r is the scattering amplitude f . Performing the angular integral and using (23) and integration by parts as before we obtain from which we obtain the scattering cross-section as The above expressions are exact. We are interested, however, in the limit ka 1. In this regime, in general k 2 2mV and we can treat k 2 as a perturbation parameter. The full solution for u(r) can then be expressed as a series in k 2 in a way analogous to the Lippman-Schwinger expansion. Specifically, we define the unique function u 0 satisfying: and the sequence of functions u n , n = 1, 2, . . .
Then the full solution for u can be written as The above u satisfies equation (23), and also has the correct boundary conditions at r = 0 due to the boundary conditions of u 0 and u n , n ≥ 1. Formally, the u n can be written The point of the above expansion is that u 0 represents the zero mode of the operator − d 2 , however, is defined on the set of states with boundary conditions u(0) = u (0) = 0, on which this operator has no zero modes (the unique solution with these boundary conditions is 0), so the inverse exists.
We can now use this expansion in the expressions for ψ(0) and f . We have The values are an (infinite) set of phenomenological parameters that determine the scattering properties of the potential as a function of k. u 0 behaves linearly near r = 0 (u 0 (r) ∼ r), while u n (r) ∼ r 2n+1 and u n (r) ∼ r 2n . For a generic potential such that V a 2 1 So the leading term λ 0 = 1 + O(a) becomes the only relevant one in the limit a → 0, the rest being negligible. Such potentials have no interesting scattering dynamics, leading to |ψ(0)| 1 and σ 0. Physically interesting potentials are these that lead to at least the first couple of the above parameters to be of the same order of magnitude. Achieving this needs potentials whose magnitude scales as a −2 or faster and have nontrivial profiles. (Examples of such potentials and the scaling of u n are given in the appendix.) Keeping only λ 0 and λ 1 and assuming all higher λ n are negligible in the limit ka 1, we have for the wavefunction and the scattering amplitude and cross section

IV. SOMMERFELD ENHANCEMENT
For short-range annihilation processes, Sommerfeld enhancement is given by S = |ψ(0)| 2 . In our case, however, the elastic interaction potential is also short-range, so the above formula is not automatically true. We have to distinguish different regimes: a) If the range of the annihilation process b is much smaller than the range of the potential a, then ψ(0) determines Sommerfeld enhancement: with a maximal value for k = 0 b) If the range of the annihilation process is comparable to the range of the potential a, then the values of the wavefunction within a range a are relevant. The exact formula would involve an integral of the profile of the annihilation amplitude over the wavefunction. But as an order of magnitude: with an approximate maximal value for k = 0 c) If the range of the annihilation process is much bigger than the range of the potential a, then the values of the wavefunction outside the range a are relevant. The exact formula would again depend on the exact profile of the annihilation amplitude, but as an order of magnitude: with an approximate maximum value for k = 0 We observe that the maximun enhancement is obtained when the range of the annihilation process is comparable to the range of the interaction.

V. RENORMALIZATION
The concept of renormalization appears quite often in quantum mechanics in the context of effective interactions or singular potentials. In the first approach, the unknown UV properties of a potential with known long-range behavior is parametrized by adding short-range regularization terms and fitting with known physical properties (spectra, phase shifts etc.) (For a nice pedagogical review see [12], and for recent applications related to our considerations see [9].) Since the same physical properties can be obtained for various values of the regularization terms, this leads to the idea of renormalization group.
Renormalization also appears in the context of singular potentials, such as the delta-function potential: they are regularized with a form of cutoff and their coupling constants are renormalized such that they give finite physical results. Again, "bare" constants vary with the cutoff parameter such that they give the same physical observables.
In our approach, the issue of renormalization is, a priori, moot: we start with a physical short-range interaction and we work out the physical effects. There is no need to regularize anything, nor do we need to fit any known physical data. Nevertheless, there are parallels with renormalization that are worth pointing out.
We found that the effects of a short-range interaction can be effectively described in terms of a set of phenomenological parameters λ n . For generic potentials only λ 0 matters; for more interesting potentials λ 1 becomes important, while for potentials with even more nontrivial behavior higher lambdas may become relevant. This is in the spirit of renormalization: physics is parametrized in terms of a set of macroscopic parameters, with the full details of the short-range interaction becoming irrelevant.
In fact, in our parameters λ 0 and λ 1 we see analogs of both coupling constant and wavefunction renormalization: the long-range properties of the potential, such as the scattering cross-section (38) and the Somerfeld enhancement for scales much larger than the scale of the potential (43), depend only the ratio λ 1 /λ 0 . This ratio can be considered as a renormalized coupling constant, and is the only one relevant for macroscopic physics. The wavefuction at the origin (37), however, depends on both λ 0 and λ 1 , thus involving an additional parameter. This can be considered as a wavefunction renormalization at short ranges.
The three-dimensional δ-potential and its scattering properties have also been examined as an example of regularization and renormalization in quantum mechanics [13]. The scattering equation for an exact delta function potential of strength g reads Putting x = 0 in (45) we find G(0) is infinite, and we proceed by regularizing it. It has the momentum integral form The integral (47) is linearly divergent and can be regularized with a momentum cutoff Λ as in [13,14], i.e.
Note that we would have obtained essentially the same result if we had, instead, imposed a short-distance regularization a and defined where we omitted terms of order a or smaller, which identifies Λ = π 2a . So we obtain From (45) for large |x| we identify the scattering amplitude as We observe that formulae (50) and (51) have the same functional form as our previous formulae (37) and (38) upon identifying Note, however, that there is no choice of scaling of the "bare" parameter g with the cutoff Λ that can make both λ 0 and λ 1 finite. The best one can do is demand for the long-range physical properties of the system, such as f , to be finite. Thus we define which defines a renormalized coupling constantg. In terms of the phenomenological variables of last section,g is essentially their ratio:g Written in terms ofg, f and σ are finite: ψ(0), however, would still diverge as Λ → ∞ andg remains finite. This calls for wavefunction renormalization and introduces a new parameter into the problem. We define the renormalized wavefunction at the origiñ The above renormalizedψ(0) is finite.λ is a new, finite parameter that plays the same role as λ 1 in the previous section.g andλ define the physical parameters of the system,g being relevant for macroscopic quantities andλ for short-distance ones. The bare parameters g and Z are cutoff-dependent (as manifest in (53) and (56) for fixedg and λ) and go to zero as Λ becomes large. The moral of the above is that a formal renormalization procedure of a singular delta function potential can reproduce the physics of a short-range interaction upon proper identification of renormalized parameters. The true physics of the situation are in the details of the short-range interaction, and its physical content is encoded in the phenomenological parameters λ n that remain relevant for wavelengths much larger than the interaction scale.
If the short-range potential is sufficiently attractive there can be bound states (see [13][14][15] for a discussion in the renormalized delta-potential case). These are easily found through our formula (37). For a bound state at E = −B (B > 0) we put ik = −κ, 2mB = κ 2 , and look for a resonance in (37) where the denominator vanishes: In the presence of a bound state, the total cross section can be expressed as We see that the bound state energy is determined by the same macroscopic parameter λ 1 /λ 0 that determines the scattering properties (and defines the renormalized strength in the delta-potential case). This is sensible, as a bound state energy can be detected macroscopically without the need to probe short-distance physics at the scale of the potential.

VI. DISCUSSION AND CONCLUSIONS
In this paper we studied the neutrino scattering problem in the low energy regime by assuming that the interaction between neutrinos is short range. As in this limit we are considering only tree level processes, the only renormalization involved is related to the effective physics at long distances as arising from short-range properties of the potential.
We would like to emphasize two important facts. First, for the scattering cross section, a unique "renormalized" coupling constant is relevant and plays a role analogous to an effective Fermi constant G X . In principle, bounds for that constant can be found as in [10]. The total cross section has a maximum, as can be seen from where m ν represents the reduced neutrino mass. This behavior is independent of the potential being atractive or repulsive.
Using an appropriate rescaling of the total cross section (σ tot /σ max tot ) and the energy (k 2 /B), the energy dependence of the cross section is as plotted in figure 2. The second important point is that the Sommerfeld enhancement factor depens on additional short-range properties of the potential, and can be quite large. So, if dark matter scattering processes occur via the mechanism proposed in [7], Sommerfeld enhancement could be significant.

ACKNOWLEDGMENTS
We would like to thank the anonymous referee for remarks that prompted a much more comprehensive analysis and led to substantial improvements in the content and presentation of this paper. This work was supported by FONDECYT/Chile grants 1130020 (J.G.), 1140243 (F.M.) and Conicyt/21140036(D. C.), by NSF grant 1213380 (A.P.) and by a PSC-CUNY grant (A.P.).

APPENDIX
We first point our that unless the potential scales as a −2 , that is, unless 2mV a 2 is not of negligible magnitude, it will have no effect on scattering. We simply rescale the variable in the equation for u(r) to s = r/a, thus expanding the range of the potential to the interval s ∈ [0, 1], and also definē u(s) = 1 a u(as).
For small a, the effect of the k 2 − 2mV term is negligible and only the zero-order solutionū = s survives, the deviation from it being of order a 2 ; that is, leading to negligible scattering effects. Even a generic potential scaling like a −2 would have limited effect:ū 0 (s) would be a function of order 1 and we would get u 0 (a) = aū 0 (1) , u 0 (a) =ū 0 (1) ( = 1).
So, generically, λ 1 would still be neglibible, although λ 0 would now be different than 1 and thus would produce a (momentum-independent) Sommerfeld enhancement factor. The other interesting possibility would arise ifū 0 (1) is small (so s = 1 would be near a zero of the functionū 0 (s)) and of order a. In that case both λ 0 and λ 1 are small, but their ratio is of order 1, thus leading to nontrivial scattering effects but divergent Sommerfeld enhancement. (This is, essentially, what the standard "renormalization" of the delta-potential [13] achieves.) Nontrivial potentials where both λ 0 and λ 1 are finite and non-negligible require a behavior that is stronger than a −2 for a range of values of r, as well as a change of sign in the range (0, a). To illustrate the possibility, we will present a "proof of concept" potential, of no particular physical significance. We start by giving the desired u 0 (r), from which the appropriate potential 2mV = u 0 /u 0 can be obtained. Take For small a, both u 0 (a) and u 0 (a) remain nonzero and finite. The potential corresponding to the above u 0 is The above potential in nonsingular everywhere, of order a −2 for most of the range 0 < r < a but becoming of order a −3 as r nears 0. It is positive for r < a/ √ 2 + O(a 2 ) and becomes negative for r > a/ √ 2 + O(a 2 ). These features are generic: a purely positive potential cannot produce u 0 (a) and u 0 (a) that are both nonzero and finite in the limit of small a.
We can also show that u n (a) and u n (a) (n ≥ 1) will go to zero as a → 0. To start, we point out that higher u n can be expressed recursively in terms of integrals u n (r) = u 0 (r) The argument for the smallness of higher u n is based on the fact that where f 0 (s) is a function that remains finite in the limit a → 0. By changing variables r = as in (68) we see that Since f 0 remains finite in the limit a → 0, u 1 will scale like a 2 unless the integrals above develop a singularity as a → 0. This, however, is not the case: the only singularity could arise at s → 0, where f 0 → as and the denominator diverges, but we can check that the s -integral does not diverge. Therefore, u 1 (a) ∼ a 2 . A similar argument shows that u 1 (a) ∼ a, and recursively u n (a) ∼ a 2n , u n (a) ∼ a 2n−1 , so the only survining parameters are u 0 (a) and u 0 (a). Clearly, to get potentials with nonvanishing u 1 or higher, we need to pick u 0 that becomes divergent (of order a −1 or higher) somewhere in the range 0 < r < a. Such behavior is highly unphysical and provides a reason why we do not expect potentials with nonvanishing higher λ n to arise.
We conclude by showing that for the scattering cross-section higher angular momentum sectors have a negligible contribution in the limit ka → 0. (For the Sommerfeld factor their contribution is exactly zero.) The physical reason is that the radial part of the wavefunction for such sectors satisfies u + k 2 u = 2mV u + ( + 1) the same as for the s-wave sector but with an additional centrifugal potential. The classical inflection point of a particle with energy k 2 /2m off the centrifugal potential is at Since ka 1, r c a. So the region where the potential V is nonzero is deep inside the classically forbidden region of the particle. The potential is effectively shielded by the centrifugal barrier, accessible only through tunneling effects. Only for = 0 there is no barrier and the wavefunction can access the potential and feel its effects. In what follows we will back this intuition with a calculation.
We will define, again, the unique solution of the radial equation (71) with boundary conditions u ∼ r +1 as r → 0 , or u ( +1) (0) = ( + 1)! , u (n) (0) = 0 for n < + 1, (exponents in parenthesis indicate derivatives). Then the radial wavefunction in this sector is with A a scale parameter that is fixed by boundary conditions at r → ∞. For = 0 (the s-wave), A = ψ(0), but for higher , A does not contribute to ψ(0).
To determine the bounday conditions at r → ∞ we write the plane wave decomposition in terms of spherical harmonics and spherical Bessel functions e ikx = i (2 + 1)j (kr)P (θ), with θ measured with respect to the axis k. So the radial part of this plane wave in the u-parametrization (u = re ikx ) in the sector is u plane (r) = i (2 + 1)rj (kr).
To isolate the incoming and outgoing part, we write it in terms of Hankel functions h = j + iy .
In fact, we define the modified Hankel functions in terms of which the radial -plane wave function is u plane (r) = 1 k i 2 + 1 2i U (kr) −Ū (kr) (79) In the limit kr 1 the functions U behave as so U is the outgoing part andŪ is the incoming part. U and its conjugateŪ separately satisfy the free radial equation d 2 U (kr) dr 2 + k 2 U (kr) = ( + 1) r 2 U (kr).
In the limit ka → 0 this gives as before.
We can now tackle the issue of whether higher angular momenta contribute to the scattering amplitude for a very small. We will work out the case = 1, the others being qualitatively similar. For = 1 we can substitute the explicit form of U 1 from (82) and obtain for b b = e −2ika (1 + ika − k 2 a 2 )u(a) + a(1 + ika)u (a) (1 − ika − k 2 a 2 )u(a) + a(1 − ika)u (a) .
So the scattering phase shift is −2ka plus twice the phase of the complex number in the numerator. We observe that for ka 1 this phase will be negligible (and in fact of order k 3 a 3 ) no matter what the scaling of u(a) and u (a) with a. So, completely generically, the contribution of the = 1 sector to the scattering amplitude is vanishingly small. The only possibility to get an appreciable phase shift and scattering amplitude is for the real and imaginary parts of the numerator to be of comparable magnitude. For this to happen we need au (a) = −(1 − k 2 a 2 + λk 3 a 3 )u(a), for some finite λ. Thus, not only do we need au (a) to be close to −u(a), but we need it tuned to such a value to an accuracy of order k 3 a 3 (!). This is a tremendous level of fine-tuning. Moreover, it is k-dependent. So even if it were to hold for some k, it would stop holding as soon as k moves away from that value by the tiniest amount. For higher , a similar pattern emerges: the dominant terms are such that the scattering amplitude vanishes, and an increasingly accurate fine tuning is required to have an "accidental" scattering resonance that would happen for only one specific value of k, if at all. The final result is that only the = 0 sector contributes.