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PHYSICAL REVIEW D 92, 125020 (2015) Majorana magnetic moment and neutrino in nucleosynthesis

† ‡ N. Vassh,1,* E. Grohs,2, A. B. Balantekin,1, and G. M. Fuller3,§ 1Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA 2Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 3Department of Physics, University of California, San Diego, La Jolla, California 92093, USA (Received 1 October 2015; published 22 December 2015)

We examine the physics of the early when Majorana (νe, νμ, ντ) possess transition magnetic moments. These extra couplings beyond the usual couplings alter the way neutrinos decouple from the of / and . We calculate how transition magnetic moment couplings modify , and then use a full weak, strong, and electromagnetic reaction network to compute corresponding changes in abundance yields. We find that light element abundances and other cosmological parameters are sensitive to −10 magnetic couplings on the order of 10 μB. Given the recent analysis of sub-MeV Borexino data which −11 constrains Majorana moments to the order of 10 μB or less, we find that changes in cosmological parameters from magnetic contributions to neutrino decoupling temperatures are below the level of upcoming precision observations.

DOI: 10.1103/PhysRevD.92.125020 PACS numbers: 13.15.+g, 26.35.+c, 14.60.St, 14.60.Lm

I. INTRODUCTION Such processes alter the primordial abundance yields which can be used to constrain the allowed sterile neutrino mass In this paper we explore how the early universe, and the and magnetic moment parameter space [4]. weak decoupling/big bang nucleosynthesis (BBN) epochs Since the magnetic moment interaction always changes in particular, respond to new neutrino sector physics in the the helicity of the incoming neutrino, Dirac neutrinos in form of beyond-the- neutrino magnetic particular have been of interest in BBN research due to their moments. Small electromagnetic couplings of neutrinos, ability to populate the “wrong” helicity states. For instance e.g., magnetic moments, may have effects in the early studies of the spin-flip rate in a primordial magnetic field universe, where neutrinos determine much of the energetics [5] and primordial plasmon decay into νν¯ pairs [6] are and composition (e.g., isospin). Examinations of the role centered around the possibility of generating right-handed that neutrino magnetic channels can play in astrophysics neutrino states. If the neutrino is a Dirac , magnetic and began long before laboratory experiments, interactions between the neutrinos and charged leptons such as GEMMA [1], ever attempted to detect beyond-the- could keep the right-handed states in thermal, Fermi-Dirac standard model magnetic moments. Any plasma environ- equilibrium with the left-handed states. The higher energy ment in which neutrinos are produced in copious amounts, ∼3 ∼6 density would increase Neff from to which is in such as the Sun and supernovae, is susceptible to the disagreement with current observations [7]. This has lead possibility of small neutrino magnetic moments having many authors to examine limits on the magnetic moments observable consequences. Since neutrinos are a dominant of Dirac neutrinos by constraining the production of right- constituent of the early universe during the BBN era, this handed states during times earlier than BBN such as the environment is sensitive to the additional interactions that QCD epoch [8–11]. Morgan [8] and Fukugita et al. [9] use the magnetic channels provide. For example, changes in the an approximate neutrino- scattering cross section 4 νν¯ primordial He abundance due to annihilation into to quantify the production of these helicity states. Elmfors electron- pairs were used to provide limitations et al. [10] and Ayala et al. [11] perform numerical treat- on the mass and magnetic moment of tau neutrinos before ments of the right-handed production rate by considering experiment was able to exclude tau neutrino mass of order the proper propagator in an electron-positron MeV [2,3]. Similarly, massive sterile neutrinos have the plasma. These analyses then require that these helicity ability to magnetically decay into a light neutrino while states have decoupled prior to BBN. The connection with injecting a nonthermal photon into the primordial plasma. BBN comes through either comparisons of the expansion rate and right-handed interaction rate or in the case of [9] through constraining the number density of right-handed *[email protected] † [email protected] states. Therefore the limits on the magnetic moment ‡ [email protected] of neutrinos that these works produce are necessarily §[email protected] functions of the right-handed decoupling .

1550-7998=2015=92(12)=125020(13) 125020-1 © 2015 American Physical Society N. VASSH et al. PHYSICAL REVIEW D 92, 125020 (2015) −11 Fukugita et al. [9] obtain μν < 7 × 10 μB for a right- difficult to solve in the general case, but become simpler in ≃ handed neutrino decoupling temperature of Tdec the homogeneous and isotropic conditions of the early 100 MeV. Elmfors et al. [10] and Ayala et al. [11] obtain universe [24–34]. In contrast to the solar and supernovae −11 −10 μν < 6.2 × 10 μB and μν < 2.9 × 10 μB respectively environments, the literature on the role of Majorana neutrino ≃ 100 (also for Tdec MeV). However, limitations on the transition moments in BBN is sparse. To the best of our magnetic interaction of neutrinos which appeal to the need knowledge the only investigation of Majorana moments in to avoid right-handed states are only applicable to Dirac the early universe appeals to an active-sterile transition neutrinos. A right-handed Majorana neutrino behaves as an moment [35] in the presence of a primordial magnetic field. antineutrino and so would not cause a sizable increase in In this paper, we examine effects of the Majorana the effective number of neutrino species. Since for this neutrino transition moments in the early universe without paper we will consider the case of Majorana neutrinos, invoking sterile neutrinos or primordial magnetic fields. inclusion of the magnetic channels can keep the neutrinos If its transition magnetic moments are large enough, a given interacting with the primordial plasma into the BBN era. active neutrino species can remain coupled to the electro- This allows us to examine explicitly the connection magnetic plasma into the BBN epoch. In the early universe, between neutrino magnetic moment and BBN observables a Majorana neutrino can magnetically interact with elec- such as the primordial abundances and Neff. trons and positrons as well as photons and other neutrinos. In addition to helicity considerations, magnetic inter- However magnetic neutrino-neutrino scattering and inter- actions of Majorana neutrinos differ from the Dirac case in actions between neutrinos and photons (such as neutrino that Majorana dipole moments are necessarily transition Compton scattering) are proportional to the fourth power of moments. Dirac neutrinos can have both diagonal and the magnetic moment [36] and so are suppressed relative to nondiagonal moments, but Majorana neutrino diagonal interactions with electrons and positrons which are propor- moments are identically zero [12]. This implies that for tional to the second power of the magnetic moment. Here we Majorana neutrinos the magnetic channels must occur with explore effects on the primordial abundances when enhanced − − flavor changing currents such as νe þ e → ν¯μ þ e and interactions between electrons/positrons and Majorana neu- þ − νe þ νμ → e þ e . Constraints on the effective magnetic trinos via magnetic scattering and annihilation channels are moment of neutrinos measured by experiments such as taken into account. In Sec. II we demonstrate how magnetic TEXONO [13] and GEMMA are not readily applicable to neutrino-electron scattering can play a significant role in a the transition magnetic moments of Majorana neutrinos. relatively low temperature environment such as the early Experimental upper bounds need to be converted into limits universe. In Sec. III we find the neutrino interaction rate for the case of thermal equilibrium by introducing expressions on transition moments, μxy, where subscripts x and y denote the neutrino states coupling to the photon. Such constraints for the thermally averaged cross section times Moller were first found in [14] using available solar þ reactor velocity which make use of Fermi-Dirac statistics. We use these interaction rates to find neutrino decoupling temper- scattering data from experiments such as SNO, SuperK, μ ROVNO and MUNU. A recent analysis of sub-MeV atures as a function of the transition magnetic moments eμ, μ τ,andμμτ. In Sec. IV we explore the corresponding change Borexino scattering data [15] found constraints on the e in the predicted abundances and N when these decoupling three Majorana transition moments in the mass basis to be eff temperatures are implemented in a modified version of μ ≤ ½3.1–5.6 × 10−11μ [16]. These new limits given by ij B the Wagoner-Kawano (WK)code[37,38]. We conclude in Borexino data are stronger than those found from MUNU Sec. V. Appendix A discusses the inverse Debye screening and TEXONO data, and secondary only to constraints length at all temperatures for an electron-positron plasma from GEMMA data which for Majorana moments are μ ≤ ½2 9–5 0 10−11μ which is used here to demonstrate that the magnetic ij . . × B [16]. scattering cross section is finite. Appendix B shows the The most restrictive constraint on Dirac and Majorana derivation of the interaction rate expressions employed here dipole moments comes from astrophysics by considering which are functions of the cross section. All relevant cross how energy loss due to plasmon decay in red giant stars sections are listed in Appendix C. Throughout this paper, we affects the core mass at the flash [17]. This analysis use natural units where ℏ ¼ c ¼ k ¼ 1. was updated using the red-giant branch in the globular B −12 cluster M5 and found μν ≤ 4.5 × 10 μ at the 95% C.L. B II. LOW TEMPERATURE ENHANCEMENT TO [18]. The previous constraint is applicable to the sum of MAGNETIC SCATTERING neutrino magnetic and electric dipole moments in the mass basis. When considering Majorana transition moments It is well known that for neutrino-electron scattering the exclusively, previous astrophysical examinations have magnetic channel can dominate over the weak channel mainly focused on the spin flavor precession of solar when the recoil energy of the outgoing electron is suffi- neutrinos [19–21] and neutrinos in supernovae [22,23]. ciently small [39]. Experiments which aim to measure the The behavior of active neutrino flavor and spin in dense magnetic moment of electron antineutrinos from reactors, media is governed by quantum kinetic equations which are such as TEXONO and GEMMA, exploit this property to

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FIG. 1 (color online). The weak and magnetic scattering cross sections as a function of temperature for different values of the invariant − þ variable s. The weak cross sections for νe − e and νe − e scattering are represented by the black dashed and black dotted lines respectively. The magnetic cross sections for neutrino-electron (or positron) scattering are shown as the solid lines for two possible −10 −11 effective magnetic moments (red—10 μB and blue—10 μB). obtain bounds on neutrino moments. By continuously push- Here s is the Mandelstam variable (related to the incoming 2 ing their threshold energies lower, these experiments are able neutrino energy in the electron’s rest frame by s ¼ meþ 2 j j to correspondingly obtain more restrictive upper bounds on meEν)and tmax is the magnitude of the upper bound of the the measured effective magnetic moment given by Mandelstam variable t (related to the minimum value of the ’ electron recoil energy, Te;min, in the electron s rest frame by X X 2 t ¼ −2m T 2 −iEiL max e e;min). In this paperp weffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cut off the infrared μν¯ ¼ Ueie μij ; ð2:1Þ 2 2 e;eff divergence by using t ¼ −2m ð k þ m − m Þ where j i max e sc e e ksc is the inverse Debye screening length. For a homo- where L is the distance between the neutrino source and geneous and isotropic electron-positron plasma in the absence of an external magnetic field, dynamic screening detector, Uei is the appropriate vacuum Pontecorvo-Maki- need not be considered and the static inverse screening length Nakagawa-Sakata matrix element, and μij are the values of neutrino magnetic moments in the mass eigenstate basis since can be found to be this is the appropriate interaction basis for the magnetic Z 4α ∞ 1 channel. The sum over j is performed outside the square in 2 2 ksc ¼ dpp ; ð2:3Þ Eq. (2.1) since reactor experiments which search for neutrino πT 0 1 þ coshðE=TÞ magnetic moment do not measure the final state of the scattered neutrino. The most recent upper limit given by where we have taken the chemical potential of electrons and μ 2 2 10−10μ TEXONO is ν¯e;eff < . × B at the 90% confidence positrons to be negligible (see Appendix A). In Fig. 1 we level [13]. The lowest experimental upper limit comes from compare the weak and magnetic cross sections for electron μ 2 9 10−11μ GEMMAwith ν¯e;eff < . × B at the 90% confidence neutrino scattering with electrons and positrons. Figure 1 level using a threshold energy of ∼2.8 keV [1].Aspreviously shows explicitly that at low temperature the magnetic channel mentioned, these constraints must be translated into bounds can dominate over the weak channel particularly for the low on the transition magnetic moments. The low energy energy portion of the thermal distribution of the neutrino. enhancement of the magnetic scattering channel over the The treatment of the divergent behavior of the scattering σ ∝ ð 2 2 Þ weak channel can play a role in environments other than the cross section ln qmax=qmin is what led to the re- laboratory such as the early universe during big bang examination of the result of Morgan [8] in Refs. [9–11]. nucleosynthesis. To demonstrate this we examine the low Fukugita et al. [9] take the minimum photon momenta → 2π −1 ¼ð 4π αÞ1=2 temperature behavior of the magnetic neutrino-electron transfer to be qmin lD where lD T= n is the scattering cross section: Debye screening length in the classical limit. Elmfors et al. [10] and Ayala et al. [11] perform more proper numerical πα2μ2 j j − 2 ð − 2Þ2 σð Þ¼ ν tmax − s me þ s me ð Þ treatments of the plasma effects on the interaction rate by s 2 2 ln j j ; 2:2 me s − me s s tmax modifying the photon propagator explicitly. Simple Debye screening of the Coulomb divergence using the inverse where α is the fine structure constant and μν is the neutrino’s screening length will only slightly underestimate the effective magnetic moment in units of Bohr magnetons. contribution of the photon’s longitudinal mode to the

125020-3 N. VASSH et al. PHYSICAL REVIEW D 92, 125020 (2015) cross section [40]. Here we extend the approach of [9] by we adopt the approach of Gondolo and Gelmini [44],but implementing the proper expression for the inverse Debye introduce expressions which make use of the proper Fermi- length, Eq. (2.3), at all temperatures. Dirac distributions (see Appendix B). For massless neutrinos interacting via the annihilation channel the thermally aver- III. DECOUPLING TEMPERATURE AND aged cross section can be written as INTERACTION RATE CALCULATION Z Z 4π2 2 ∞ ∞ −x To examine magnetic effects of massless Majorana hσ i ¼ T σ e vMol ann 2 sds pffiffi dx −x neutrinos in BBN, flavor changing currents must be con- nν 4m2 s=T 1 − e 2 e 0 pffiffiffiffiffiffiffiffiffiffi13 sidered. To treat this behavior correctly would require pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2− 2 2 2 − x x s=T solving the Boltzmann and quantum kinetic equations 6 x − s=T B1 þ e 2 C7 × 4 þ ln@ pffiffiffiffiffiffiffiffiffiffiA5: [24–34] to properly evolve the number densities of each 2 − 2− 2 − x x s=T neutrino flavor, while simultaneously following a BBN 1 þ e 2 network. A code in development which incorporates neu- ð3:3Þ trino transport when calculating primordial abundances and other cosmological parameters, BURST [41,42],willeven- tually be able to properly treat flavor changing currents while For the scattering channel we obtain self-consistently calculating the primordial abundances. For this paper, we proceed by approximating neutrino decou- Z Z 4π2T2 ∞ ∞ e−x pling as a sharp event for each flavor. hσ i ¼ σð − 2Þ vMol scatt s me ds pffiffi dx −x nνn 2 2 1 − e Within the decoupling temperature approximation we 2 e me s=T treat flavor changing processes such as scattering, e.g., pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν þ − → ν¯ þ − ν þ ν → 6 x2 − s=T2 1 − 2m2=s e e μ e , and annihilation, e.g., e μ 4 e þ þ − ≈ × 2 e e , as equilibrium channels and take nνe nν ≈ nν . This means the flavor dependence for the μ τ 0 pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi13 magnetic rates only appears in the magnetic moment þ 2− 2 1−2 2 − x x s=T me=s contained in the cross sections. With the equilibrium B1 þ e 2 C7 þ ln@ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiA5; ð3:4Þ 2 2 2 approximation of equivalent number densities, the mag- x− x −s=T 1−2m =s − e netic flavor changing rates can be added directly to the 1 þ e 2 weak flavor preserving rates in order to obtain the total Γ ¼ Γweak þ Γmag þ Γweak þ Γmag interaction rate scatt scatt ann ann for an and so the interaction rates can now be found by numerical incoming neutrino. To find the decoupling temperature, evaluation of the previous two-dimensional integrals using the total interaction rate must be compared with the Hubble the cross sections as functions of s given in Appendix C. expansion rate: In Fig. 2, we show the rates for weak and magnetic sffiffiffiffiffiffiffiffiffiffiffiffiffi 8π scattering on both electrons and positrons. The low temper- ¼ ρ ð Þ ature enhancement of the magnetic scattering channel over H 3 2 ; 3:1 mpl the weak channel discussed in Sec. II is evident. For a magnetic moment of 10−10μ the magnetic scattering rate ρ B where mpl is the mass, and is the energy density. dominates over the weak rate starting at a temperature of A particle species can be roughly considered to be coupled about 0.4 MeV. Figure 3 shows the weak and magnetic Γ ≥ Γ when H or decoupled when

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FIG. 2 (color online). The weak and magnetic rates for neutrino FIG. 4 (color online). Decoupling temperature of the three scattering from electrons and positrons are given as a function of flavors as a function of their effective magnetic moment. The temperature. The weak channel scattering rates are shown as black solid line represents electron neutrinos which undergo black dashed (for electron neutrinos) and black dotted (for muon weak decoupling at ∼1.27 MeV. The blue dashed line represents and tau neutrinos) lines. Magnetic scattering rates for three muon and tau neutrinos which decouple weakly at ∼1.48 MeV. possible values of the effective magnetic moment are represented −9 −10 by the solid lines (indigo—10 μB, red—10 μB, and blue— −11 μ ≃ ð2–9Þ 10−11μ 10 μB). magnetic moment. For the range x;eff × B, the magnetic interaction has a small effect on the decou- pling temperature. The magnetic channels start to play a μ2 ¼ μ2 þ μ2 eμ eτ; μ 10−10μ e;eff more significant role when x;eff > B (roughly 2 2 2 μμ ¼ μμ þ μμτ; expected from an examination of the rates in Fig. 2). If ;eff e μ 2 2 2 the μτ transition magnetic moment is sufficiently high in μτ ¼ μτe þ μτμ: ð3:5Þ ;eff comparison to the transition magnetic moments with a νe component, the νμ and ντ species could remain coupled to The effective transition moments defined above are not the electron-positron plasma longer than the νe. identical to the effective electron antineutrino magnetic moment measured by reactor based experiments such as GEMMA. Here we assume that the incoming and outgoing IV. CHANGES IN PRIMORDIAL N massless neutrinos populate definite flavor states in the ABUNDANCES AND eff early universe. Figure 4 shows the decoupling temperatures The WK code assumes at the start of the computation that for electron and mu=tau neutrinos as a function of effective all flavors of neutrinos have already decoupled from the plasma. To alter the neutrino-decoupling epoch, we modify the summed energy density of neutrinos and antineutrinos such that

2 7 π 4 ρν ¼ Tν ; ð4:1Þ x 8 15 x

where the temperature of neutrino species x, Tνx ,isnot necessarily related to the inverse of the . We define the comoving temperature parameter as an energy scale such that the product Tcma is a comoving invariant, where a is the scale factor. To calculate Tcm, we set Tcm equal to the plasma temperature at an early epoch where the photons, electrons, positrons, and all flavors of neutrinos and antineutrinos are in thermal equilibrium. If we denote FIG. 3 (color online). The weak and magnetic rates for νν¯ this epoch with the subscript h, we obtain the expression þ − ¼ ð Þ annihilation into e e pairs. The line assignments are the same as Tcm Th ah=a . The temperature of each neutrino species in Fig. 2. evolves as

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 0.2452 0.2464 0.2456 Tγ if Tνx >Tνx;dec 0.2448 Tν ¼ a ð4:2Þ x dec Tν Tν

0.2455 decoupling. Equation (4.2) states that Tνx is equal to the plasma temperature until the decoupling epoch, after which Tν scales with the comoving temperature. x 0.2466

The late neutrino decoupling epoch necessitates a 0.2458 0.2462 modification to the WK temperature derivative. To calculate 0.2460 the plasma temperature derivative in the presence of 0.2435 0.244 coupled neutrinos, we need the quantities 0 ρ νx pν ¼ ; ð4:3Þ x 3

dρν 4ρν x x FIG. 5 (color online). Contours of constant Y in the Tν ¼ ; ð4:4Þ P τ;dec dTγ Tγ versus Tν ;dec (left) and Tνμ;dec (right) parameter spaces. The left e ¼ 0 245 panel uses a constant Tνμ;dec . MeV. The right panel uses ρ ¼ 0 245 where p is the pressure, and d =dTγ is the temperature a constant Tνe;dec . MeV. derivative of the energy density. Initially, all three neutrino species stay coupled to the plasma and contribute to the The hotter νe and ν¯e spectra will keep n=p in chemical relevant components of the temperature derivative: equilibrium longer, thereby reducing the ratio and YP. P P However, n=p is not solely determined by the ν and ν¯ ρ þ ρ þ 3 ρ þ þ þ 3 e e dTγ γ e ¼1 ν pγ pe ¼1 pν spectra. The comparison of the expansion rate H and the ¼ −3H x x P x x ; dργ ρ dρν dt þ d e þ 3 x n ↔ p rates determines the epoch of weak freeze-out, dTγ dTγ x¼1 dTγ and thus n=p. The hotter νe and ν¯e spectra imply a larger ð4:5Þ neutrino energy density, a faster expansion rate, and an earlier epoch of weak freeze-out. The earlier epoch leads to where the subscript e denotes the sum of electron and a larger n=p. Therefore, lower neutrino decoupling temper- positron components. In writing Eq. (4.5), we have omitted atures induce two competing effects with regards to 4He the terms associated with baryons for the sake of brevity. production. We will adopt this theory as our initial The actual numerical derivative does include baryons and paradigm for the behavior of YP. related derivatives. As each individual neutrino species We demonstrate the interesting behavior of the primor- decouples, we remove it from the summation for ρ, p, and dial 4He mass fraction as a function of neutrino decoupling dρ=dTγ. Eventually, all three neutrino species decouple and temperatures in Fig. 5. When holding Tνe;dec fixed, the we obtain the default derivative in the WK code. − Tνμ;dec Tντ;dec parameter space shown in the right panel The electron neutrino and antineutrino distributions are gives the expected behavior. A decreasing decoupling inputs into the weak interaction rates: temperature for either νμ or ντ precipitates an earlier epoch of weak freeze-out, yielding a larger n=p and Y . þ ν ↔ − þ ð Þ P n e e p; 4:6 Nucleosynthesis is insensitive to the flavor of neutrinos which are not of νe type and so the right panel of Fig. 5 is n þ eþ ↔ ν¯ þ p; ð4:7Þ ¼ e symmetric about the line Tνμ;dec Tντ;dec. No such sym-

metry exists in the left panel of Fig. 5. If we hold Tντ;dec ↔ − þ ν¯ þ ð Þ n e e p: 4:8 fixed and examine the change in YP, we observe that YP

increases with decreasing Tνe;dec. The monotonicity would For this paper we alter the n → p and p → n rates given seem to indicate that the faster expansion rate dominates ν ν¯ in the WK code to use the hotter e and e spectra. The weak over the effect of the hotter νe spectrum on the n ↔ p rates. interaction rates influence the to ratio, n=p, Conversely, if we hold Tν ;dec fixed and examine the change 4 e which results in the prediction of the primordial He mass in YP, we observe that YP decreases with decreasing Tντ;dec ∼ 0 4 fraction: until a turn-around temperature Tντ;dec . MeV. The turn-around temperature is independent of Tν , however, 4 4ð 2Þ 2 e;dec ≡ nHe ≃ nn= ¼ n=p ð Þ we do notice that we traverse more contours when YP þ 1 þ : 4:9 nb np nn n=p decreasing Tντ;dec with larger Tνe;dec than we do with

smaller Tνe;dec. Once Tντ;dec becomes smaller than the Equation (4.9) is only an approximation as not all of the turn-around temperature, YP then begins to increase with 4 remaining are incorporated into He nuclei. decreasing Tντ;dec. This behavior implies that the faster

125020-6 MAJORANA NEUTRINO MAGNETIC MOMENT AND … PHYSICAL REVIEW D 92, 125020 (2015) expansion rate does not precipitate an earlier epoch of ∼0 4 weak freeze-out until Tντ;dec falls below . MeV. This phenomenon is unlike the behavior described earlier for constant Tντ;dec on the left panel, and the behavior as seen on the right panel. We conjecture that our naive paradigm of n=p as a function of only the νe, ν¯e spectra and expansion rate neglects the effect of important nuclear physics rates involving nuclides with mass number two and three. We would like to note that constraints from neutrino decoupling are more encompassing than just having impli- cations for neutrino transition moments. The effect of nonstandard neutral current (NC) neutrino-electron inter- actions on the neutrino decoupling temperature in BBN has been previously considered in Ref. [46]. This work found that laboratory constraints on the nonstandard NC cou- plings do not permit such processes to significantly effect cosmological parameters [46]. Since the neutrino charge radius generates an additive term to the weak neutrino- electron scattering cross section [47], studies of nonstand- ard NC neutrino couplings also have implications for the magnitude of the effective charge radius in the primordial plasma. Other electromagnetic properties of neutrinos can be probed through examinations of the magnetic channel. Limits on neutrino magnetic moments from existing experimental neutrino-electron scattering data have been used to obtain bounds on the neutrino electric millicharge [48] and generic tensoral couplings of neutrinos to charged fermions [49]. Since the magnetic interaction coupling is tensoral in nature, it should not be readily assumed that the insensitivity of BBN to nonstandard NC couplings translates to the magnetic channel. Nonstandard tensoral couplings are being further probed by the TEXONO experiment [50], and so a proper treatment of neutrino decoupling could be a complimentary approach to experiment in studying these interactions. Since for this paper we are concerned with the connection between neutrino decoupling and transition magnetic moments, we proceed with the assumption that the magnetic interaction is the only new physics affecting the neutrino FIG. 6 (color online). The change in relative abundances of decoupling temperatures. In Fig. 6 we show the dependence 4He, D, 3He, and 7Li as a function of the transition neutrino μ magnetic moment μeμ. The solid lines show the results when of the primordial abundances on the transition moment eμ. −11 −11 −10 μeτ ¼ 10 μB (black—μμτ ¼ 10 μB, red—μμτ ¼ 4 × 10 μB, It is interesting to note that increases in magnetic moment −10 and blue—μμτ ¼ 6 × 10 μB). The dashed lines are the results cause a decrease in the predicted lithium abundance but μ ¼ 6 10−10μ do not have a large enough effect to be a potential solution to for eτ × B with the colors representing the same values of μμτ that they did in the solid case. the lithium problem since even relatively high magnetic −10 moments ∼6.3 × 10 μB can only reduce lithium to ∼3.94 × 10−10 [far from the observationally inferred value rate] [52]. The values found from quasar absorption-line of ð1.6 0.3Þ × 10−10 [51]]. The known 2σ tension systems [D=H ¼ð2.53 0.04Þ × 10−5] and indirect deter- between the deuterium abundance found with codes such minations from CMB data therefore do not allow for this as PArthENoPE and the observationally inferred value [52] is work to place exclusions on Majorana neutrino magnetic also a factor in this paper. With the integrated n → p and moments through observational constraints. The 4He abun- p → n rates, our modified WK code gives a value of D=H ¼ dance is also modified by nonzero neutrino transition 2.61 × 10−5 for small neutrino magnetic moment which moments, however the largest found 4He abundance for −11 −10 agrees well with the value given by PArthENoPE of D=H ¼ magnetic moments in the range of ð10 –6.3 × 10 ÞμB is −5 3 ð2.65 0.07Þ × 10 [with the updated dðp; γÞ He reaction YP ∼ 0.2465. This is well within the observationally inferred

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0 .246 0.2460 0.2475 0.2450 5 0.2470 0.2455 0.2465 2.660

0.2460 0.2455

0.2450

2.650 0.2445 2.615

0.2440

0.244

2 2.680 2.73 2

. .

2.619 2.660 2.630 2.64 8 7

0 3

5

0 0 0

0.2440 5

μ 5 FIG. 7 (color online). Contours of constant YP in the eτ versus FIG. 8 (color online). Contours of constant 10 ×D=Hin μ μ eμ (left) or μτ (right) planes. The third transition magnetic the μμτ versus μeμ plane. The solid contours correspond to ∼10−10μ −10 moment is set to be B in both planes. μeτ ¼ 10 μB and the dashed contours correspond to μeτ ¼ −10 4.9 × 10 μB. range of 0.2465 0.0097 [53]. Future observations of Y P μ μ may provide constraints on neutrino transition magnetic for increasing eμ as compared to increasing μτ. The larger moments if errors can be reduced to the 1% level. D/H relative change of D/H as compared to YP (7% vs 1% over could be probed to the sub-one percent level with thirty-meter the same parameter space) is not due to the n=p ratio, but class telescopes [54–56]. However, transition magnetic instead a result of a larger initial entropy per baryon. If moments would tend to increase D/H, thereby causing more neutrinos remain coupled to the plasma after the initiation tension with current observations [57] than in the case of electron-positron annihilation, then entropy is trans- without magnetic moments. ferred from the plasma into the neutrino seas. Therefore, a late neutrino-decoupling epoch implies a loss of entropy in In Fig. 7 we explore contours of constant YP as a function of transition magnetic moments. The effective the photon sector. Reference [7] gives the baryon density as ω ¼ 0 022068 magnetic moment for electron neutrinos defined in b . , which is inversely related to the entropy ω Eq. (3.5) is a symmetric function of μeμ and μeτ. per baryon. To match the value of b from Ref. [7],we Therefore, in the left panel the plot is symmetric about must begin BBN with a larger entropy per baryon. Figure 8 the line μeμ ¼ μeτ. The general behavior is that a larger validates the exquisite sensitivity of the deuterium abun- magnetic moment implies a larger value of n=p with a dance to the initial entropy per baryon. concurrent change in YP. However, this is not universally Although examinations of the primordial abundances true as the YP ¼ 0.2440 contour does not uphold this trend. could not yield approximate constraints on neutrino tran- An example of an exception to the general trend occurs for sition moments, there is an additional parameter with high −10 −10 values of μeτ ∼ 5 × 10 μB with μeμ ≲ 5 × 10 μB. In this sensitivity to neutrino decoupling, namely Neff, defined by narrow range an increase in μeμ results in a decrease in YP,     7 4 4=3 π2 implying that here the n → p rate is faster than H. The 4 ρ ¼ 1 þ N Tγ ; ð4:10Þ general trend of larger magnetic moment producing a larger rel 8 11 eff 15 value of n=p seen in the left panel applies to the right panel ρ without exception. For the right panel of Fig. 7, there is no where rel is the radiation energy density, i.e. the sum of symmetry along the μeτ ¼ μμτ line. The asymmetry in the the photon and neutrino energy densities. We examine space implies there still exists a competition between contours of constant Neff as a function of neutrino transition the n → p rate and H, but the effect is not as dramatic moments in Fig. 9. The axes and contour sets of Fig. 9 are as in the parameter space of the left panel. the same as Fig. 8. As defined in Eq. (4.10), Neff is The deuterium abundance is less sensitive than YP to the independent of neutrino flavor and therefore symmetric and n=p ratio. Figure 8 shows the primordial relative abun- monotonic in any decoupling temperature parameter space. dance of deuterium (with respect to ) multiplied However, Fig. 9 shows Neff in the transition magnetic 5 ¼ 3 050 by 10 as a function of neutrino transition moments μeμ and moment space. The solid blue Neff . contour is not μμτ. The different sets of contours correspond to different symmetric in this parameter space. The asymmetry is the −10 ν values of μeτ, namely μeτ ¼ 10 μB for the solid contours result from different decoupling temperatures for τ com- −10 and μeτ ¼ 4.9 × 10 μB for the dashed contours. For both pared to νe for equivalent effective magnetic moments. The sets of contours, the value of D/H increases more rapidly magnetic moment channel conduces a rate similar in value

125020-8 MAJORANA NEUTRINO MAGNETIC MOMENT AND … PHYSICAL REVIEW D 92, 125020 (2015) 4 3.400 3.800 freeze-out of BBN events. He exhibited the most exotic

3.400 3.600 3.200 behavior of any observable quantity we investigated. The 3.300 results for YP when changing μeτ and μμτ invariably showed a larger n=p and larger YP for increasing magnetic moment strength. When investigating the effect on YP with chang- ing μeμ and μeτ, the larger magnetic moment did not always engender a larger value of YP. However, the general trend of larger magnetic moments yielding larger values of n=p

3.300 3 was still apparent. In all cases, the change in Y is still

. 3.20 3.10 P 6

3.050 0

0 0 0 consistent with the observational bounds of Ref. [53]. The trends for D/H, and to a lesser extent 3He=H and 7Li=H, follow from changes in the initial entropy. A larger initial entropy-per-baryon in the plasma causes increases in deuterium and 3He, and a decrease in 7Li. Changes in Neff from the resultant extra energy density μ −10 FIG. 9 (color online). Contours of constant Neff in the μτ of a coupled neutrino limit μ ≲ 6 × 10 μ .Our −10 xy B versus μeμ plane. The solid contours correspond to μeτ ¼ 10 μB −10 approximate constraint was limited by current observatio- and the dashed contours correspond to μeτ ¼ 4.9 × 10 μB. nal errors in Neff measurements. Future advances in CMB astronomy could limit the error in Neff to percent levels, to the weak scattering rates of ντ as compared to the weak and the baryon density to sub-percent levels. However the projected sensitivities would still only allow for this work scattering rates of νe, as shown in Figs. 2 and 3.The magnetic channel contributes more to lowering the decou- to obtain constraints on magnetic moments at the order of 10−10μ . For instance the projected sensitivity on N of pling temperature of ντ than it does for νe at these particular B eff magnetic moment values. For larger magnetic moments, the σ ∼ 0.021 for CMB-S4 [58] applied about the base value magnetic channel dominates over the weak channels regard- found using our modified WK code in the absence of 3 100 magnetic moments yields an N of 3.038 0.021. This less of flavor, as seen in Fig. 4. Therefore, the Neff > . eff contours of Fig. 9 are symmetric. We note that it is possible range would imply a transition magnetic moment constraint ¼ 3 30 0 27 μ ≲ 2 5 10−10μ 2σ to use the one-sigma range Neff . . [7] from the of xy . × B to . This is still roughly an order most recent Planck data to constrain the values of neutrino of magnitude above the recently found constraints on magnetic moments. We find that Neff is able to exclude transition moments using sub-MeV Borexino data [16]. −10 transition moments larger than ∼6 × 10 μB to two sigma. If we consider values of magnetic moments closer to the recent laboratory bounds, we find percent changes in D/H to be 0.021%, YP to be 0.0021%, and Neff to be 0.068% for V. CONCLUSIONS −11 the case that all three transition moments are ∼7 × 10 μB. Our analysis revealed that Majorana neutrino transition BBN observations are not forecast to be at this level of magnetic moments can alter the nucleosynthesis of the precision in the near future. Thus if experiments such as primordial elements by keeping the neutrinos coupled during SPT-3 [59], the Simons array [60], or CMB-S4 [58] the BBN epoch. Investigating active neutrinos which are determine Neff is statistically larger than the predicted Majorana in nature allowed us to explicitly look at the value of 3.046 [61], it is unlikely that beyond-the-standard cosmology associated with this physics by quantifying the model neutrino magnetic moments are a contributing connections between transition magnetic moment and factor. Although the calculated abundance changes are observables. However, here we were limited by the tendency below current observational sensitivities, the technique of BBN codes to oversimplify the neutrino physics. Since a outlined in this paper shows the way toward BBN probes transition magnetic moment necessarily changes flavor and of other beyond-the-standard model neutrino sector issues. parity, a proper analysis will use the Boltzmann solvers in Since we find that transition moments begin to play a BURST to account for the out-of-equilibrium neutrino dis- role in neutrino decoupling at the now excluded order of −10 tributions and the spectral swaps associated with changes in 10 μB, this work suggests that effects on cosmological flavor. To circumvent these issues and perform a preliminary parameters from neutrino magnetic channels are imma- investigation of neutrino transition moments in BBN, we terial. However the instantaneous decoupling approxima- employed the decoupling temperature approximation to tion used here neglects out-of-equilibrium effects which are quantify how magnetically enhanced interaction rates would known to distort neutrino spectra. The energy dependence maintain thermal equilibrium between the neutrinos and of standard weak neutrino interactions implies that the charged leptons until late epochs. high energy tail of the neutrino distribution interacts most We expected the late neutrino decoupling epochs to strongly which in turn causes energy dependent spectral beget a faster expansion rate and result in quicker weak distortions. However as demonstrated in Sec. II, the energy

125020-9 N. VASSH et al. PHYSICAL REVIEW D 92, 125020 (2015) dependence of magnetic interactions significantly differs Note that in the relativistic limit T ≫ me the above inverse 2 from those of weak interactions (also evident from cross screening length reproduces the well-known relation ωp ¼ 1 2 sections in Appendix C). Thus magnetically modified 3 ksc [62], where the plasma frequency in the relativistic limit 2 4απ 2 spectral distortions may have more pronounced effects is given by ωp ¼ 9 T at zero chemical potential [63]. on Neff than what was found here with decoupling con- siderations alone. Additionally given the sensitivity of 4He to electron neutrinos demonstrated by Figs. 5 and 7, APPENDIX B: INTERACTION RATES WITH changes in the electron neutrino number density from FERMI-DIRAC DISTRIBUTIONS spectral swaps associated with flavor changing currents Gondolo and Gelmini [44] outline the procedure for should be carefully considered. In order to fully understand 3 3 turning the integral of Eq. (3.2) over d p1 and d p2 into an possible effects of neutrino magnetic moment in BBN, integral over the Mandelstam variable s. Using the change the inclusion of neutrino spectral distortions and swaps of variables, deserves to be studied by a more rigorous treatment. ¼ þ ACKNOWLEDGMENTS Eþ E1 E2; E− ¼ E1 − E2; ðB1Þ N. V. would like to thank the organizers of the TALENT NT4AsummerschoolRichardCyburt,MortenHjorth-Jensen, 3 3 2 gives d p1d p2 ¼ 2π E1E2dEþdE−ds and we can write and Hendrik Schatz for encouraging the pursuit of this work. We thank Joe Kapusta, Charles Gale, Yamac Pehlivan, Chad − Eþ 1 1 e 2T 1 Kishimoto, Mark Paris and Pat Diamond for useful discus- ¼ : ðB2Þ 1 þ eE1=T 1 þ eE2=T 2 ðEþÞþ ðE−Þ sions. We also thank the referee for useful comments. We cosh 2T cosh 2T would like to acknowledge the Institutional Computing Program at Los Alamos National Laboratory for use of their For massless neutrinos interacting via the annihilation HPC cluster resources. This work was supported in part by channel the numerator of Eq. (3.2) becomes U.S. National Science Foundation Grants No. PHY-1205024 Z 3 3 and No. PHY-1514695 at the University of Wisconsin, σ d p1 d p2 vMol No. PHY-1307372 at UC San Diego, and in part by the 1 þ eE1=T 1 þ eE2=T Z Z University of Wisconsin Research Committee with funds 2 π ∞ ∞ Eþ − 2 granted by the Wisconsin Alumni Research Foundation. ¼ σsds pffiffi dEþe T 2 4m2 s Z peffiffiffiffiffiffiffiffiffi 2 − APPENDIX A: INVERSE SCREENING LENGTH Eþ s dE− × pffiffiffiffiffiffiffiffiffi : ðB3Þ Eþ − 2 − ð Þþ ðE−Þ Following the discussion given by Kapusta and Gale [62] Eþ s cosh 2T cosh 2T the exact inverse screening length is found from the longitudinal polarization in the static infrared limit and The last integral of Eq. (B3) is known analytically to is given in natural units by be [64] ∂ Z 2 ¼ 2 n ð Þ dx ksc e ∂μ ; A1 coshðaÞþcoshðxÞ     where we must use the net electron density n ¼ n ≡ x þ a x − a e ¼ cosechðaÞ ln cosh − ln cosh ; ne− − neþ given by 2 2 Z 1 ∞ 1 1 ðB4Þ n ¼ dpp2 − ðA2Þ e π2 ðE−μÞ=T þ 1 ðEþμÞ=T þ 1 0 e e which allows the thermally averaged cross section for the because we have an electron-positron plasma [63]. For the annihilation channel to be reduced to a two-dimensional case of zero chemical potential integral and written as Z Z Z 4π2T2 ∞ ∞ e−x ∂n 2 1 ∞ eE=T hσ i ¼ σ e 2 vMol ann 2 sds pffiffi dx −x ¼ dpp : ðA3Þ 4 2 1 − 2 E=T 2 nν me s=T e ∂μ μ→0 π T 0 ðe þ 1Þ 2 0 pffiffiffiffiffiffiffiffiffiffi13 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2− 2 2 2 − x x s=T 6 x − s=T B1 þ e 2 C7 Then the inverse screening length is × 4 þ ln@ pffiffiffiffiffiffiffiffiffiffiA5; 2 − 2− 2 Z − x x s=T ∞ 1 þ e 2 2 4α 2 1 ksc ¼ dpp : ðA4Þ πT 0 1 þ coshðE=TÞ ðB5Þ

125020-10 MAJORANA NEUTRINO MAGNETIC MOMENT AND … PHYSICAL REVIEW D 92, 125020 (2015) 2 2 which is the expression given in Eq. (3.3). For the scattering t ¼ q ¼ðp1 − p4Þ ¼ −2p1 · p4 channel we obtain Z Z ¼ 2p1 · p3 − 2p1 · p2; 2 2 ∞ ∞ −x 4π T 2 e 2 2 hσ i ¼ σð − Þ s ¼ðp1 þ p2Þ ¼ m þ 2p1 · p2: ðC2Þ vMol scatt s me ds pffiffi dx −x e nνn 2 2 1 − e 2 e me s=T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 Then the cross section can be found to be 6 x − s=T 1 − 2m =s × 4 e 2 2 ð − 2Þ2 2 GF s me 2 me 0 pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi13 σðsÞ¼ ðgA þ gVÞ þ 2gAðgA − gVÞ þ 2− 2 1−2 2 4π s s − x x s=T me=s   B1 þ e 2 C7 1 2 2 þ ln@ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiA5; ðB6Þ 2 me 2 2 2 þ ð − Þ 1 − ð Þ x− x −s=T 1−2m =s gA gV C3 − e 3 s 1 þ e 2 which is the expression given in Eq. (3.4). The kinematic after the differential cross section has been integrated from conditions needed to derive the interaction rate for the ð − 2Þ2 ¼ − s me ¼ 0 j j ≤ tmin to tmax . pannihilationffiffiffiffiffiffiffiffiffiffiffiffiffiffi channel given by Eq. (3.3) are E− s 2 1 Eþ − s and E1E2v ¼ s for massless neutrinos. Mol 2 þ The results for the scattering channel given by Eq. (3.4) b. Weak ν − e scattering Making use of crossing symmetry with p2 ↔ p3 we make use of the kinematic scattering conditions jE−j ≤ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi have 2 2 1 2 − 1 − me ¼ ð − 2Þ Eþ s s and E1E2vMol 2 s me . Note that hjM j2i¼16 2 ½ð 2 − 2 Þ 2ð Þ in using Eqs. (3.3) and (3.4) for rate calculations the phase weak G g g me p1 · p4 3 3 F A V space integrals d p3 and d p4 contained in the cross 2 þðgA þ gVÞ ðp1 · p3Þðp2 · p4Þ section must be multiplied by g3 and g4 (for the internal 2 degrees of freedom of the outgoing particles). The results þðgA − gVÞ ðp3 · p4Þðp1 · p2Þ for all rates in this paper have included these multiplicative 2 2 2 2 2 ¼ 4G ½2ðg − g Þm t þðg þ g Þ factors. F V A e A V 2 2 2 2 2 × ðs þ t − meÞ þðgA − gVÞ ðs − meÞ ; APPENDIX C: CROSS SECTIONS ðC4Þ The following cross sections assume the incoming neutrino to be polarized. Polarization makes use of the 1 and so the cross section is projection operator 2 ð1 − γ5Þ and ultimately leads to a factor of 1=2 in the magnetic cross sections. However it should be noted that this polarization does not modify the G2 ðs − m2Þ2 m2 σðsÞ¼ F e ðg − g Þ2 þ 2g ðg þ g Þ e results for the weak interaction due to the structure of the 4π s A V A A V s weak interaction vertex.   1 m2 2 þ ðg þ g Þ2 1 − e : ðC5Þ 1. Neutrino-electron/positron scattering cross 3 A V s sections: νðp1Þþe ðp2Þ → νðp4Þþe ðp3Þ a. Weak ν − e− scattering c. Magnetic The invariant amplitude squared is It is well known that the differential cross section for hjM j2i¼16 2 ½ð 2 − 2 Þ 2ð Þ Coulomb scattering diverges at t ¼ q2 → 0. Here we will weak GF gA gV me p1 · p4 implement a cutoff which can be generically written as þð þ Þ2ð Þð Þ gA gV p1 · p2 p3 · p4 → −j j tmax tmax . The averaged matrix element squared is 2 þðgA − gVÞ ðp2 · p4Þðp1 · p3Þ given by ¼ 4G2 ½2ðg2 − g2 Þm2t þðg þ g Þ2ðs − m2Þ2 F V A e A V e 32π2α2μ2 1 2 2 2 hjM j2i¼ ν ð Þð Þ þðgA − gVÞ ðs þ t − meÞ ; ðC1Þ γ 2 p1 · p2 p1 · p3 me p1 · p4 ¼ 1 ¼ 1 þ 2 2θ 4 2κ2 where for electron neutrinos gA 2, gV 2 sin w and ¼ e ν ð 2 − Þð þ − 2Þ ð Þ 1 1 2 me s s t me ; C6 for muon=tau neutrinos gA ¼ − 2, gV ¼ − 2 þ 2sin θw. The t Mandelstam variables in the second expression of Eq. (C1) are defined by where κν ¼ μνμB. The cross section is

125020-11 N. VASSH et al. PHYSICAL REVIEW D 92, 125020 (2015) πα2μ2 j j − 2 ð − 2Þ2 ¼ 2 ¼ð þ Þ2 ¼ 2 ¼ 2 þ 2 σð Þ¼ ν tmax − s me þ s me ð Þ s q p1 p2 p1 · p2 p1 · p3 p1 · p4; s 2 2 ln j j : C7 me s − me s s tmax 2 2 t ¼ðp1 − p4Þ ¼ me − 2p1 · p4; ðC9Þ The cross section for neutrino-electron scattering applies and so the cross section can be found to be to neutrino-positron scattering as well since using crossing rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi symmetry with p2 ↔ p3 we can see from Eq. (C6) that 2 4 2 the invariant amplitude squared is symmetric under this σð Þ¼GF 1 − me s 2π interchange. s 1 × ðg2 − g2 Þm2 þ ðg2 þ g2 Þðs − m2Þ ; ðC10Þ 2. νν¯ annihilation cross sections: V A e 3 A V e ¯ − þ νðp1Þþνðp2Þ → e ðp3Þþe ðp4Þ a. Weak after the differentialp crossffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi section has been integrated from t ¼ 1 ½2m2 − s − sðs − 4m2Þ to t ¼ 1 ½2m2 − sþ For this process the invariant amplitude squared is pminffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 e e max 2 e 2 sðs − 4meÞ. hjM j2i¼16 2 ½ð 2 − 2 Þ 2ð Þ weak GF gV gA me p1 · p2 2 b. Magnetic þðgA þ gVÞ ðp1 · p4Þðp3 · p2Þ 2 The invariant amplitude squared is þðgA − gVÞ ðp2 · p4Þðp1 · p3Þ 2 2 2 2 2 2 2 2 2 2 64π α μν 1 ¼ 4G ½2ðg − g Þmes þðgA þ gVÞ ðt − meÞ hjM j2i¼ ð Þð Þ F V A γ 2 p1 · p4 p1 · p3 þð − Þ2ð þ − 2Þ2 ð Þ me p1 · p2 gA gV s t me ; C8 2 2 8e κν ¼ ðm2 − tÞðs þ t − m2Þ; ðC11Þ where the first expression of Eq. (C8) can be obtained from s e e 2 2 a p2 ↔ p4, m → −m switch to the scattering invariant e e 1 2 amplitude given by the first expression of Eq. (C1). Use of where the above includes the factor of = due to this switch requires that the annihilation process is then polarization of the incoming neutrino and antineutrino. − þ So the cross section is defined by νðp1Þþν¯ðp2Þ → e ðp3Þþe ðp4Þ with four- momentum conservation p1 þ p2 ¼ p3 þ p4. The second rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2πα2μ2 4 2 2 2 expression of Eq. (C8) comes from a s ↔ t switch in the σð Þ¼ ν 1 − me 1 þ me ð Þ s 2 : C12 second expression of Eq. (C1) where here 6me s s

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