A thermal neutrino interaction rate at NLO

The interaction rate of an ultrarelativistic active neutrino at a temperature below the electroweak crossover plays a role in leptogenesis scenarios based on oscillations between active neutrinos and GeV-scale sterile neutrinos. By making use of a Euclideanization property of a thermal light-cone correlator, we determine the O (g) correction to such an interaction rate in the high-temperature limit πT (cid:2) m W , ﬁnd-ing a ∼ 15 ... 40% reduction. For a benchmark point, this NLO correction decreases the lepton asymmetries produced by ∼ 1%.

If right-handed neutrinos are added to the Standard Model in a minimal renormalizable way, without introducing any other dynamical fields, then the neutrino sector is fully characterized by the values of Majorana mass parameters and neutrino Yukawa couplings. Even if this amounts to a large-dimensional parameter space, the values of the parameters start to be constrained. Therefore, it is important to scrutinize the precision of the theoretical computations on which the cosmological significance of this model relies. This is the goal of the present study. Specifically, we aim to determine one of the important rates, defined in eq. (2.1), up to next-to-leading order (NLO) in the weak-coupling expansion.
The computation of NLO corrections to real-time rates is a challenging task in the so-called ultrarelativistic regime πT m i , where m i refers to the masses of the plasma particles, hampered as it is by powerlike infrared divergences which lead to the breakdown of the naive loop expansion. 1 In general, a nested resummation of the loop expansion is necessary for generating a consistent weak-coupling expansion. An important breakthrough was achieved in ref. [25], where it was realized that for many ultrarelativistic observables the real-time problem can be reduced to a static one. Then resummations can be implemented in a tractable fashion, which permitted for ref. [25] to recover previous results [26] in a simple way and to push the computation one order higher in the coupling. This insight has subsequently inspired, for instance, NLO determinations of the thermal photon [27] and soft dilepton [28] production rates from a QCD plasma; an estimate of the NLO contribution to its shear viscosity [29]; as well as attempts at incorporating the contribution of soft modes on a non-perturbative level, through numerical simulations of an effective theory [30][31][32][33].
The goal of the present paper is to adapt these techniques to the electroweak theory. Even though the SU(2) gauge coupling g 2 ∼ 2 3 is smaller than the QCD one, the NLO corrections are only suppressed by g 2 2 T /(πm i ) ∼ g 2 /π (assuming m i ∼ g 2 T ), and could come with large prefactors. Previously, they have been determined for so-called susceptibilities, relating chemical potentials to lepton asymmetries. As they were found to be numerically significant [34,35], it appears well motivated to extend the exercise to a genuine rate observable.

Formulation of the problem
Denoting by K = (ω, k) the four-momentum of an active neutrino propagating through a medium at a temperature T and by / its (advanced) self-energy, the chiral nature of gauge interactions (cf. eq. (2.2)) implies that we may write [36] Re / = a / where u ≡ (1, 0) denotes the four-velocity of the plasma in the local rest frame. The function a represents a radiative "wave function correction", whereas b can be interpreted as a thermal correction to a dispersion relation [36] (the full propagator is ∝ (/ K + / ) −1 ). In the following, we are concerned with the interaction rate u . The rate K is relevant for the subleading helicityflipping active-sterile transitions [37], however it was found in ref. [37], drawing upon earlier work in the QCD context [38,39], that only the interaction rate u is susceptible to a simplified Euclideanized treatment à la ref. [25]. We assume that the neutrino is, to a good approximation, ultrarelativistic: k ≡ |k| ∼ πT M, where M = √ ω 2 − k 2 denotes its virtuality. It interacts via weak interactions, where g 1 is the hypercharge coupling, B μ is the corresponding gauge potential, σ a are the Pauli matrices, g 2 is the weak coupling and A a μ are the SU L (2) gauge potentials. In order to isolate the relevant helicity components, we employ the Weyl representation of the Dirac matrices: γ 0 γ i = diag(−σ i , σ i ), a L = diag(1, 0). Going to momentum space, ∂ μ → iK μ , and aligning the momentum in the z-direction, the free part implies that the lower (negative-helicity) components of ν L and e L go on-shell for ω = k z . In the following we denote this component by ψ for ν L , and by χ for e L . For these components, the coefficients of eq. (2.1) appear in the effective action as where ã, b , ˜ K , ˜ u refer to the properties of left-handed electrons. Following ref. [25], the idea now is to expand in fluctuations around the on-shell point, and then to rotate the light-like propagation into a static one. Simultaneously, we go over to Euclidean conventions, viz. A M 0 = iA E 0 , and define a Euclidean Lagrangian as L E ≡ −L M . Linear combinations of gauge potentials are denoted by Then static fluctuations around the on-shell point are described by The "large" ω and k z have cancelled against each other, so the "residual" momentum generated by i∂ z in eq. (2.4) can be taken to be small (it is denoted by k z and is ∼g 2 T ). With the Lagrangian of eq. (2.4), we compute the Euclidean version of eq. (2.3). For nearly on-shell neutrinos, it takes the form 2 It turns out that within the effective theory, the real part of is odd in k z and the imaginary part is even in k z , guaranteeing that correlations decay exponentially (i.e. that the pole is on the imaginary axis). Consequently, only a and u are generated within our computation, as already alluded to above. It can be deduced from eq. (2.5) that the free neutrino and electron propagators have the forms 2 The original ψ has been scaled by a factor T 1/2 so that ψ(x) has the dimension GeV and ψ(k) the dimension GeV −2 . The same dimensions apply to the gauge potentials A a μ (x) and A a μ (k), respectively.
where d = 3 − 2 . The other propagators are those of the dimensionally reduced theory for the Standard Model [40], with the temporal gauge field components A a 0 and B 0 kept as dynamical fields. We carry out computations in a general R ξ gauge, whereby the spatial W ± propagator reads (a , b ∈ {1, 2}) There are similar propagators for the neutral components Z i and Q i which, as usual, are obtained from A 3 i and B i by a rotation with the mixing angle sin(2θ) = 2g 1 g 2 /(g 2 1 + g 2 2 ). For the temporal components Z 0 and Q 0 , the mixing is modified by thermal (Debye) masses. Let us denote the mixing angle of the temporal components by θ . Following ref. [41], the masses of the diagonalized modes are denoted by m 2 Z and m 2 Q . The original gauge fields can be expressed in the new basis as 8) and the corresponding propagators take the forms The masses m 2 W , m 2 Z , m 2 Q and the angles θ, θ satisfy identities which are not always easy to recognize at first sight, e.g.
Let us end this section by commenting upon differences with respect to the observable analyzed in the QCD context [25]. The quantity considered there was a Wilson loop rather than a single propagator (which can be interpreted as a Wilson line). The transverse coordinate of the loop, r ⊥ , served to define a "collision kernel", C(q ⊥ ), and the so-called jet quenching parameter, q, is a weighted integral over C(q ⊥ ). The weighting by q 2 ⊥ implies that the soft contribution to q is UV divergent. In our case there is no such weighting, and u is UV finite. On the other hand, the absence of weighting makes u more IR sensitive than q, and therefore u is perturbatively computable only in the Higgs phase.

Leading-order computation
The leading contribution to u originates at 1-loop level, through the graphs shown in Fig. 1. Inserting the propagators from eqs. (2.6), (2.7), (2.9) and (2.10) and denoting the internal momentum by p = p ⊥ + p z e z , we are faced with integrals of the type (3.1) Fig. 1. Leading-order contributions to self-energy. A double line denotes a neutrino (ψ ) propagator from eq. (2.6); a filled blob an electron (χ ) propagator; a wiggly line a spatial gauge field; and a solid line a temporal gauge field, which in the effective theory has turned into an adjoint scalar field.
Noting that the external momentum k z ∼g 2 T is small compared with the mass scales m i ∼gT , we can expand in k z . Evaluating the integral over p z with the residue theorem, or with where P denotes a principal value, yields 3) The first term contributes directly to the width, whereas the second one gives a wave function correction, proportional to k z (cf. eq. (2.5)). The wave function corrections are gauge dependent and IR-sensitive, but they do play a role in cancelling similar effects from loop diagrams. Therefore they need to be accounted for at NLO, as discussed in Appendix A.7.
Focussing now on the leading-order (LO) width, which originates from the first term in eq. (3.3), it is straightforward to verify that there is no gauge parameter dependence. Summing together the graphs, we obtain It remains to carry out the integral over the transverse momentum p ⊥ . Even if the whole integral is UV finite, it was argued in ref. [37] that it is reasonable to adopt a phenomenological cutoff |p ⊥ | ≤ 2k ∼ 2πT . Adopting this prescription, which amounts to a partial inclusion of higher order contributions in an expansion in ∼ m 2 i /k 2 , transverse integrals evaluate to Thereby eq. (3.4) reproduces the LO result for u as given in eq. (5.23) of ref. [37].

NLO result and some of its features
Proceeding to the NLO level, there are a number of contributions, listed in Appendix A. Summing them together, the result can be expressed as where c i , ċ i , c ij , c T ij , ċ ij are coefficients and i, j = {H, Z, . . .} label particles appearing in the loops. The "photon mass" m Q was introduced as an intermediate IR regulator (see below). Mak-ing use of the notation of Appendix A.1, simplified further by denoting A i ≡ A(m i ), etc., the linear combinations needed in eq. (4.1) read 3  where on the first line B T ,00 indicates that masses can be put to zero. This agrees perfectly with the NLO part of eq. (D.6) of ref. [41], which was obtained by considering the IR asymptotics of a fully relativistic but unresummed computation. It is also good to check that the resummed expression is IR finite. As charged particles appear in the loops, massless photons do affect the NLO result. Some of their contributions are unproblematic, but there are certain terms where it needs to be verified that the limit m Q → 0 can indeed be taken in eq. (4.1). For this we note that Indeed all m Q -dependence of eqs. (4.3) and (4.6) appears in this IR-safe combination. We plot the result from eq. (4.1), both as an integrand and after the integration, in Fig. 2. For this, the parameters have been fixed as in ref. [37]. A partial cancellation between contributions from various domains of p ⊥ can be observed, as a result of which the final magnitude of the negative NLO correction remains at a modest ∼ 15% level at high temperatures. It reaches ∼ 40% at low temperatures, mostly because the LO contribution is anomalously small there due to a cancellation between the spatial and temporal contributions in eq. (3.4), viz. (LO) u ∼g 4 T 3 /m 2 i at gT m i , however in that region the approximation m i πT inherent to our effective theory approach gradually breaks down.  .7) (NLO asymptotics), at T ≈ 140 GeV. Right: the corresponding integrals, as a function of T and k (the latter originates as discussed around eq. (3.5)). At high T the NLO correction is seen to reduce the LO rate by ∼ 15%, at low T by ∼ 40%.

Summary and outlook
We have reported a thermal NLO computation of a neutrino interaction rate u , defined through eq. (2.1), in the temperature range πT m W . The final result is given in eq. (4.1) and plotted numerically in Fig. 2, whereas many technical details are relegated into Appendix A. The result passes a number of crosschecks, in particular it is gauge independent and displays the UV asymptotics predicted by an earlier unresummed computation. By taking g 1 → 0, it can partially also be contrasted with the QCD result of ref. [25], 4 even if the presence of a Higgs expectation value prohibits an unambiguous comparison.
The coefficient u is but one of a set of mass corrections and interaction rates entering a complete GeV-scale leptogenesis framework [37]. 5 It is, however, the first ultrarelativistic rate that has been computed up to NLO for thermal neutrino physics. Even though it will take time before all other coefficients are known at the same level, we hope that the knowledge of one of them helps to motivate such efforts.
A leptogenesis computation typically comes with two goals: determining the baryon asymmetry, which is fixed at T ∼ 130 GeV when sphaleron processes switch off [43], and determining lepton asymmetries, which continue to be produced when T < 130 GeV. As the rate that we computed only concerns processes that are active in the Higgs phase (cf. eq. (5.1)), it is no surprise that it has little effect on the baryon asymmetry: we only observe a variation on the per mille 4 Or better still, with an N = 4 SYM result [42], which includes the contribution of scalar fields. 5 More precisely, u determines the rate of helicity-conserving active-sterile oscillations, as where h ν is a neutrino Yukawa coupling, v is the Higgs expectation value, ω ≡ k 2 + M 2 , M is the mass of a righthanded neutrino, and b is the correction parametrizing the real part of eq. (2.1).
level. The influence on lepton asymmetries is a bit larger, however by considering the benchmark point from ref. [37], chosen because lepton asymmetry production continues for a long time in this case, we found a reduction of lepton asymmetries by ∼ 1%. Hence it appears that for practical applications it is sufficient to use the simple leading-order expression for u from eq. (3.4).
Closing on a conceptual note, our computation can be interpreted as amounting to determining the exponential fall-off of a single light-like Wilson line. Even though we have verified the independence of the result on the gauge fixing parameter up to NLO, the observable itself is not manifestly gauge invariant. Another possible starting point would be to consider a Wilson loop (like in ref. [32] but in the Higgs phase), which is gauge invariant. In this setup, the information relevant for us could be extracted by pulling the sides of the Wilson loop far from each other (r ⊥ → ∞), and interpreting the coefficient of the resulting decay as twice the width u /2 that we are interested in. Our approach has the technical advantage that it could be generalized, graph-by-graph, to determining the complete NLO self-energy of an active neutrino beyond the ultrarelativistic regime, even if the practical implementation of this generalization is challenging, given that a full 4d computation is required.

Acknowledgements
This work was partly supported by the Swiss National Science Foundation (SNF) under grant 200020B-188712.

Appendix A. Contributions from NLO diagrams
In this appendix we record the separate contributions to the function (k z ), defined in eq. (2.5). For momentum dependence, specifically (0), it is sufficient to remain at 1-loop ("LO") level (cf. sec. A.7), whereas (0) is needed up to 2-loop level ("NLO"). For completeness we list results for a general R ξ gauge. The gauge parameter appears in the masses of the longitudinal gauge bosons and Goldstone modes, which are denoted by In terms which are not manifestly IR finite, we denote by m Q a fictitious photon mass, which is taken to zero at the end of the computation.

the NLO result can be expressed in terms of the two master integrals
where 2 q1 ≡ q 2 ⊥ + m 2 1 and 2 pq2 ≡ (p ⊥ + q ⊥ ) 2 + m 2 2 . Sometimes we need the mass derivatives of these functions, defined aṡ In addition it is convenient to define the tensor integral In practice we need B zz at p z = 0, which is then given by B zz | p z =0 = B T . It is possible to express B T in terms of the integrals in eqs. (A.2) and (A.3), as however it is more compact to display results in terms of B T , thereby avoiding inverse powers of p 2 ⊥ and d − 1. We note that B T is symmetric in m 1 ↔ m 2 .
For understanding the UV asymptotics of the result, we need an expansion of the master integrals in inverse powers of p 2 ⊥ . Whereas A and Ȧ are independent of p 2 ⊥ , for B and Ḃ from eqs.

(A.3) and (A.4) these limiting behaviours read
The Z 0 self-energy diagrams can be depicted as where the self-energies are , (A.10) .
Here a wiggly line represents a spatial gauge field, a dotted line a ghost, a dashed line a Higgs field, and a solid line a temporal gauge field. The result reads . (A.12) The transverse spatial self-energy reads whereas the temporal parts can be expressed as (A.14) (A.16)

A.3. W ± self-energy
The contribution of the W ± self-energy diagrams, viz.
, (A.17) can be written as The transverse spatial self-energy reads whereas the temporal part can be expressed as

A.4. Triple gauge vertex
The contribution of triple gauge vertex diagrams, viz. , can be written as  We note that θ zz is symmetric in m 1 ↔ m 2 .

A.5. Crossed fermion self-energy
For the "crossed" NLO self-energy diagrams, viz. The contributions from δχ are closely related to those in sec. A.7.

A.7. Wave function normalization
The final contribution originates from the second term in eq. (3.3), with the various channels of Fig. 1 contributing with coefficients like in eq. (3.4). The tree-level part of the self-energy, k z , gets corrected by this term (cf. eq. (2.5)), and the correction needs to be factored out, in order to determine the location of the pole of the corresponding propagator. In other words, we write the combination appearing in eq. (2.5) as