Evolution of Primordial Neutrino Helicities in Astrophysical Magnetic Fields and Implications for their Detection

Since decoupling in the early universe in helicity states, primordial neutrinos propagating in astrophysical magnetic fields precess and undergo helicity changes. In view of the XENON1T experiment possibly finding a large magnetic moment of solar neutrinos, we estimate the helicity flipping for relic neutrinos in both cosmic and galactic magnetic fields. The flipping probability is sensitive both to the neutrino magnetic moment and the structure of the magnetic fields, thus potentially a probe of the fields. As we find, even a magnetic moment well below that suggested by XENON1T could significantly affect relic neutrino helicities and their detection rate via inverse tritium beta decay.

The early universe was bathed with thermal neutrinos which decoupled from matter around 1 s after the big bang. Detection of these neutrinos, e.g., through inverse beta decay capture on tritium [1] in the PTOLEMY experiment [2,3], remains a major challenge. Relic neutrinos carry information about the early universe at a much earlier epoch than that of photon decoupling. In addition, neutrinos propagating through the universe acquire information about the gravitational and magnetic fields they encounter en route to Earth. We focus here on the evolution of the helicity of primordial neutrinos and implications for their detection rates. Neutrinos in the early universe decoupled essentially in chirality eigenstates at temperatures orders of magnitude larger than neutrinos masses, leaving these highly relativistic neutrinos essentially in helicity eigenstates as well. Were neutrinos to travel freely from decoupling to the present, we would expect neutrinos to be left handed, and antineutrinos right handed. However, two effects modify this conclusion.
The first is that as the neutrino trajectory is bent gravitationally by density fluctuations in the universe, the deflection of its spin vector lags behind that of its momentum vector; gravitational fields do not conserve neutrino helicity [3][4][5][6][7]. Second, neutrinos of finite mass are expected to have a non-zero magnetic moment [8][9][10][11][12][13][14][15][16], so that propagation in galactic and cosmic magnetic fields [17,18] rotates their spins with respect to their momenta, again allowing neutrinos and antineutrinos to have an amplitude to be flipped in helicity, as first noted in Ref. [9]. Neutrinos are potential probes of cosmic and galactic magnetic fields as well as density fluctuations in the expanding universe.
We explore here consequences of large neutrino magnetic moments on the time evolution of primordial neutrino helicities. Helicity modification by slowly varying astrophysical magnetic fields occurs via diagonal magnetic moments and is thus limited to Dirac neutrinos. In contrast, both Dirac and Majorana helici-ties are modified by gravitational fields [4]. The recent XENON1T report of an excess of low energy electron events [19] has triggered interest in the possibility that a large magnetic moment of solar neutrinos, of order ∼ 1.4 − 2.9 × 10 −11 µ B (≡ µ 1T ), where µ B is the Bohr magneton, could account for these excess events [20,21]. While beyond-the-Standard Model physics. which is required for large magnetic moments, generally favors sizeable moments for Majorana rather than Dirac neutrinos [13,14,20,21], the XENON1T data, which does not distinguish diagonal from transition moments, can accommodate both neutrino types. We do not assess here the possibility of moments exceeding estimated theoretical bounds [13,22,23].
We find that even a moment several orders of magnitude smaller than µ 1T could lead to significant helicity changes as Dirac neutrinos propagate through the cosmos, as well as the Milky Way. As we discuss, detection rates for primordial neutrinos are sensitive to both their helicity structure as well as whether they are Dirac or Majorana fermions. We assume = c = 1 throughout.
We first briefly recall some properties of primordial neutrinos from standard cosmology. At temperatures, T , small compared with the muon mass but well above 1 MeV, muons and tau are frozen out and the only charged leptons present are electrons and positrons; neutrinos are held in thermal equilibrium with the ambient plasma through neutral and charged current interactions. As estimated in Ref. [15], ν τ and ν µ freeze out at temperature T µ ∼ 1.5 MeV, while ν e freeze out at T e ∼ 1.3 MeV. However, the temperature differences at freezeout do not effect the present temperature, T ν0 = 1.945 ± 0.001K = (1.676±0.001)×10 −4 eV, of the various neutrino species.
The observation of neutrino oscillations establishes well that neutrino flavor eigenstates α are linear superpositions of mass eigenstates i with PMNS flavor-mass mixing matrix elements U αi [24]. While neutrinos decouple in flavor eigenstates, the velocity disperson, δv = 1 2 ∆m 2 /p 2 , among different mass components of momentum p soon separates a given flavor state into three effectively decoherent wave packets of mass eigenstates [25]. The distribution of the present momenta p 0 of primordial neutrinos of each mass state |ν i is independent of the neutrino mass, with total number density n = 3η(3)T 3 ν0 /2π 2 = 56.25 cm −3 . The magnetic moment of a non-zero mass Dirac neutrino is estimated in the (extended) standard model to be [8][9][10] µ B = 1.40 MHz/gauss is the Bohr magneton, and m −2 the neutrino mass in units of 10 −2 eV. Diagonal moments of Majorana neutrinos must vanish, although transition moments connecting different mass eigenstates are non-zero [16]. Magnetic moments could be substantially larger than Eq. (2) predicts. According to the most recent Review of Particle Physics [26], the most sensitive upper bounds for µ ν are given by the GEMMA and Borexino experiments. The GEMMA reactor experiment [27] gives an upper limit µ ν < 2.9 × 10 −11 µ B , and Borexino [28] reports upper bounds from solar neutrinos, µ νe < 2.8 × 10 −11 µ B . These bounds are comparable to the moment µ 1T that could explain the XENON1T lowenergy electron-event excess, which does not distinguish diagonal from transition magnetic moments. As a neutrino with a magnetic moment propagates through magnetic fields its spin precesses, see, e.g., [29]. To set the scale, we first neglect relativistic effects; then since the neutrino magnetic moment vector is µ BŜ = 2µ B S, the rotation rate of the spin is ω s = 2µ ν B, where B is a characteristic field strength. For example, for B ∼ 10 −12 G, of order present intergalactic magnetic fields, the rotation rate with (2) becomes ω s ∼ 8 × 10 −27 m −2 Hz. Over the total age of the universe, t 0 ∼ 4.3 × 10 17 s, the spin would rotate by a net angle 2µ ν Bt 0 ∼ 4 × 10 −9 m −2 , and more generally, ∼ 10 12 (µ ν /µ B )(B/10 −12 G). Owing, however, to magnetic fields being considerably larger in the early universe, this result underestimates the spin rotation. Transition moments do not lead to such spin rotation, and thus Majorana neutrinos would not be affected [30,31].
We calculate the neutrino spin S and its rotation in the neutrino rest frame, measuring transverse and longitudinal spin components S ⊥ and S with respect to the axis of the neutrino "lab" momentum, where the lab frame is that of the "fixed stars." For rotation from an initial helicity state, for which S ⊥ = 0, by angle θ, one has | S ⊥ |/| S| = sin θ. The helicity changes from ±1 to ± cos θ, and the probability of observing the helicity flipped is then P f = sin 2 (θ/2); for θ ≪ 1, The spin precesses in its rest frame according to where τ is the neutrino proper time, and B R is the magnetic field in the rest frame.
In terms of the lab frame magnetic field and time, t, the equations of motion of the rest frame spin are [32], since in the absence of an electric field in the lab frame, We neglect the ν e -e matter effect [33], important only for very dense matter or vanishingly small µ ν . For small deviations, |S ⊥ | ≪ |S|, from a pure helicity state, the S ⊥ × B term in Eq. (4) is negligible; thus a neutrino of velocity v and helicity ±1 experiences a cumulative spin rotation with respect to its momentum, One of the larger magnetic fields a relic neutrino encounters en route to local detectors is that of our galaxy, B g ∼ 10µG. Galactic fields do not point in a uniform direction, but rather change orientation over a coherence length, Λ g , of order kpc [34][35][36][37]. The spin orientation undergoes a random walk through the changing directions of B, reducing the net rotation by a factor ∼ ℓ g /Λ g , where ℓ g is the mean crossing distance of the galaxy, of order the galactic volume V g divided by σ g , its crosssectional area. Thus the mean square spin rotation of a neutrino passing through a galaxy (g) is All quantities (except µ ν ) nominally depend on the epoch t. The spin rotation is larger for more massive neutrinos since 1/v 2 = 1 + m 2 ν /p 2 , with p the neutrino momentum. The spin rotation for non-relativistic neutrinos (m ν ≫ p ≃ T ν0 ), evaluated with parameters characteristic of the Milky Way, B g ∼ 10 µG, ℓ g ∼ 16 kpc, Λ g ∼ kpc, is A moment ∼ 1.5 × 10 −15 µ B , a factor 10 −4 smaller than what would account for the XENON1T excess, would yield a helicity flip probability P f of order unity for m −2 (B g /10µG)(Λ g /1kpc) 1/2 itself of order unity.
Neutrinos propagate past distant galaxies before reaching the Milky Way. The effective number of galaxies a neutrino sees per unit path length is ∼ n g σ g , where n g is the number density of galaxies. Integrated over the neutrino trajectory from early galaxies to now, the effective number, N eff , of galaxies a neutrino passes through is ∼ n g σ g R u ∼ n g V g (R u /ℓ g ), where R u is the present radius of the universe. Since n g V g ∼ 10 −6 and R u /ℓ g ∼ 10 6 , a neutrino would pass through N eff of order unity before reaching the Milky Way. The cumulative rotation of a neutrino prior to reaching our galaxy is comparable to the spin rotation it would undergo within the Milky Way.
We now estimate the net rotation a relic neutrino experiences from cosmic magnetic fields in the expanding universe, from decoupling to now. We work in the metric ds 2 = −a(u) 2 (du 2 − d x 2 ), where x are the co-moving spatial coordinates, and a(u) is the increasing scale factor of the universe (with a = 1 at present); the conformal time u is related to the coordinate time by dt = a(u)du. Over the evolution of the universe from decoupling, where a(t d ) ≡ a d ∼ 10 −10 , to now the cosmic magnetic field decreases; assuming that the field lines move with the overall expansion, flux conservation implies that globally Ba 2 should remain essentially constant in time. As with galaxies, the coherence length, Λ, of the cosmic magnetic field is not well determined, but expected to be on Mpc scales [38][39][40]; the coherence length reduces the net spin rotation by a factor ∼ Λ/R u .
In order of magnitude, the ratio of the helicity flip probability from the present cosmic field to that from a galactic field, is: The magnetic field ratio is of order of at least microgauss vs. picogauss, while the ratio of length scales is of order (kpc) 2 /(Gpc Mpc) ∼ 10 −9 , which would indicate a scale of neutrino spin rotation in galaxies up to three orders of magnitude larger than in cosmic magnetic fields. However, in assessing whether cosmic rotation is competitive with the rotation from the galactic magnetic field, it is necessary, in addition to determining better the cosmic and galactic magnetic fields and correlation lengths, to take into account the larger cosmic fields as well as smaller coherence lengths at earlier times.
We turn now to this latter task. We start from the squared rotation in Eq. (6), written in terms of u for relativistic neutrinos, with c denoting "cosmic," with the expectation value in the cosmic background. The correlation function of the cosmic magnetic field, in an otherwise isotropic background, has the structure where r = | x − x ′ |, F is the normal and G the helical field [41] correlation. The latter does not contribute to the spin rotation since ∇ z G(r) is odd in x − x ′ , and thus its contribution for transverse spin components, ∼ dudu ′ ∂ z G(r), vanishes by symmetry.
The normal correlation has the Fourier structure [42], where in another convention for the correlation function [18], P B (k) = (2π) 2 E M (k)/k 2 . Equation (12) implies that The schematic structure of P B is a power law ∼ αk s at small k out to a wavevector k * (called k i in Ref. [18]) , followed by a sharper falloff, ∼ βk −q beyond k * , with q > 3 and β = αk s+q * . The sign of s is uncertain [18,43] but infrared convergence of the integral in Eq. (15) requires s > −2. With this approximate form Eq. (13) implies With Eq. (12) and taking the z-axis along the neutrino velocity, Eq. (10) becomes Since the scale of k is ≫ 1/u, the u integrals are vanishingly small except in the neighborhood of k z = 0, and to a first approximation we set k z = 0 in (1 − k 2 z /k 2 )P B (k). Then the k z integral gives a factor 2πδ(u − u ′ ), and (15) where 0 denotes present values, u 0 = 3t 0 , and d denotes neutrino decoupling. Here With conservation of flux, B 2 (u) ≃ B 2 0 /a(u) 4 , and k * (u) ∼ 2π/Λ 0 a(u) 1/2 [18]. The factor η = (s + 3)(q − 3)/(s+2)(q −2) is not strongly dependent on the spectral indices, and for simplicity we take η = 1/2 (corresponding to s = 2 and q = 2 + 5/3). Then The main contribution to the integral is from the radiation-dominated era, from the time of neutrino decoupling, u d , to the time of matter-radiation equality, u eq , where a(t eq ) ≡ a eq ∼ 0.8 × 10 −4 . In this era a ∝ u, and since u eq is related to u 0 by u ∝ a 1/2 in the matterdominated era. By comparison, in the matter-dominated era, a factor a d /a eq /4 ∼ 10 −4 smaller. Altogether independent of the neutrino momentum. To within uncertainties in magnetic fields, correlation lengths, and neutrino masses, the estimated spin rotation in the cosmos is basically comparable to that in galaxies.
If the low-energy electron-event excess found in the XENON1T experiment [19] does arise from a neutrino magnetic moment, and the neutrino is a Dirac particle, its diagonal moment could lead to a significant spin rotation. A magnetic moment of order 10 −2 µ 1T would still produce a spin rotation in the range of detectability. On the other hand, if the neutrino is a Majorana particle, the excess would occur entirely from transition magnetic moments, with no helicity changes from magnetic fields.
Having described the expected spin rotation of relic neutrinos we turn to their detection. The most promising approach is to capture neutrinos on beta unstable nuclear targets. Particularly favorable for detecting primordial neutrinos is the inverse tritium beta decay (ITBD) [1,3], ν e + 3 H→ 3 He + e − the reaction inverse to tritium beta decay, 3 H → 3 He + e − +ν e . The ITBD would yield a distinct signature of a mono-energetic peak separated from the endpoint of the tritium beta decay by 2m ν .
The cross section for capture of a neutrino in mass state i on tritium is [3] σ h i (p, p e ) with V ud the up-down quark element of the CKM matrix, the U ei are the neutrino mixing matrix elements, and F (Z, E e ) the Fermi Coulomb correction for the electron-3 He system. Thef andḡ are the nuclear form factors for Fermi and Gamow-Teller transitions, and the neutrino helicity-dependent factor is A ± i = 1∓β i , where β i = v i /c. The total ITBD rate is given by σ h i v i integrated over the distribution (1) of neutrinos and summed over mass states i. For Dirac neutrinos with spin rotated by θ i , both negative and positive helicity states, weighted by 1 2 (1 ∓ cos θ i ), contribute and yield the neutrino dependence in the rate, The dash-dot curve shows the result for Dirac NH neutrinos with θ 2 given by the Milky Way estimate (7), with B = 10 µG, Λg = 1 kpc, and µν = 5 × 10 −14 µB. The present neutrino temperature, Tν0 (arrow), demarcates the transition of the lightest neutrino from relativistic to non-relativistic.
The subscript T includes the thermal average over the distribution (1) as well as the average of the spin rotation over the neutrino's history. Majorana neutrinos, as noted, have no diagonal magnetic moments and cannot flip spin in a slowly varying magnetic field, so that cos θ = 1. Since the ITBD measures both Majorana neutrinos and antineutrinos, independent of the neutrino masses, and spin rotation by cosmic gravitational fluctuations [4]. Figure 1 shows the A eff as a function of the mass of the lightest neutrino, for both Dirac and Majorana neutrinos with normal and inverted mass hierarchies. For neutrinos maintaining their original helicity (θ i = 0), the A eff are the solid curves. As the mass of the lightest neutrino approaches zero, A eff,D approaches 1 + |U e1 | 2 = 1.6794 in the normal and 1 + |U e3 | 2 = 1.0216 in the inverted hierarchy. When the lightest neutrino mass rises and all neutrinos become nonrelativistic, A eff,D = 1+(7π 4 T ν0 /180ζ(3)) i |U ei | 2 /m i eventually approaches unity independent of the mass hierarchy; A eff is always larger for Majorana than Dirac neutrinos, independent of the mass hierarchy and the mass of the lightest neutrino.
The dashed curves in Fig. 1 show the dependence of A eff,D on the lightest neutrino mass for complete helicity flip, θ i = π. For partial spin rotation, A eff,D lies between the solid and dashed curves. When θ i = π/2, the amplitudes to be left and right handed are equal and A eff,D = 1. To illustrate the qualitative dependence of the helicity-flip probability on µ ν in Fig. 1, we show A eff,D for Dirac neutrinos passing through the Milky Way as the dash-dot curve, calculated from Eq. (7) for small angle bending with B g = 10 µG and Λ g = 1 kpc, and with µ ν = 5 × 10 −14 µ B , two orders of magnitude smaller than the magnetic moment XENON1T would need to explain their event excess. The value of µ ν = 5 × 10 −14 µ B is also below the upper bound derived from the analysis of solar neutrino data [44,45], and is consistent with the upper bound deduced from the stellar energy loss [46]. If the magnetic moment of normal hierarchy Dirac neutrinos is of order that suggested by XENON1T, then for the characteristic parameters assumed for cosmic or galactic magnetic fields the neutrino spin rotations would no longer be small; the mean cos θ would decrease A eff,D to essentially unity, with a concomitant decrease in the ITBD detection rate. A magnetic moment of the standard model prediction of Eq. (2) would affect A eff,D insignificantly. In contrast, a value of µ ν = 10 −14 µ B , the naturalness upper bound obtained from an EFT analysis [13,14], would have a significant effect on A eff,D . Figure 1 illustrates how measurements of the rate of relic neutrinos can distinguish Dirac from Majorana neutrinos, with an accuracy that will improve as knowledge of the correct hierarchy as well as the lightest mass come into sharper focus. As the dash-dot curve indicates, the interesting regime is of bending not too small to be indistinguishable and not so large that all spins have comparable probability of being left and right handed. This regime is characterized by a falloff in A eff,D and the ITBD detection rate with increasing light neutrino mass. Unfortunately, it becomes increasingly difficult to resolve the relic neutrino events from the tritium beta decay background for smaller neutrino mass. Inventing novel techniques to probe the region of interest shown in Fig. 1 remains a challenge.
In conclusion, investigating the implications of a possible large neutrino magnetic moment beyond that in the standard model on the helicities of relic neutrinos as they propagate through the cosmic and galactic magnetic fields, we find significant helicity modifications even if µ ν is two orders of magnitude smaller than that suggested by the XENON1T result. The present estimates of neutrino spin rotation can be sharpened by using detailed maps as well as numerical simulations of the astrophysical magnetic fields, e.g., [47][48][49][50]. In addition, the spin rotation of MeV energy neutrinos from the diffuse supernova background [51] as well as from neutron stars [9] is also potentially detectable, although using different experimental techniques than for relic neutrinos (e.g., with the Gd-doped Super-K detector and the inverse beta decay reaction) [52].