Neutrino magnetic moments and photo-disintegration of deuterium

Neutrinos with non-zero magnetic moments can dissociate deuterium nuclei by a photon exchange, in addition to the weak neutral current process. We calculate the neutrino-magnetic moment induced photo-dissociation cross section of deuterium using the equivalent photon method. This process would contribute extra events to the neutral current reaction which is observed with high precision in the salt-phase of SNO experiment. Using the SNO data and the recent laboratory measurements of the $^7 Be (p, \gamma) ^8 B$ reaction which give a more precise value of the solar $^8 B$ flux we find that the neutrino effective magnetic moment is $\mu_{eff}^2 = (-2.76 \pm 1.46) \times 10^{-16} \mu_B^2$ which can be interpreted as an upper bound $|\mu_{eff}|<3.71 \times 10^{-9} \mu_B$ (at $95% CL$) on the neutrino magnetic moments.

Enormous progress in neutrino physics has resulted in recent years from the experiments in Super-Kamiokande, SNO, KamLAND, and other experimental sites. We now know beyond any reasonable doubt, for instance, that neutrinos oscillate which implies a non-vanishing neutrino mass spectrum [1]. The wealth of new experimental input should provide also new ways of probing and nailing down neutrino properties other than masses. Electromagnetic static properties and, in particular, transition magnetic moments are obvious quantities to be subject to close scrutiny (precisely because non-zero neutrino masses naturally allow for helicity flip transitions) [2]. In this paper we use the SNO data [3] and the recent laboratory measurements of the 7 Be(p, γ) 8 B reaction [4] to put restrictions on neutrino magnetic moments.
The theoretical calculation of the deuteron break-up cross section induced by electromagnetic static quantities of the neutrino has been carried out in [5]. However, we follow here another approach which is simpler and is specially suited for the magnetic moment case (left-right transition amplitudes do not interfere with Z-exchange). Indeed, we shall use the equivalent photon approximation. Of course, although the method gives only approximate results, they are entirely satisfactory for our purposes, as we have explicitly checked by comparing to the calculation in [5].
The photon-exchange amplitude of the reaction ν i (k) + d(p) → ν j (k ′ ) + n(p ′ n ) + p(p ′ p ) can be written as where J µ is the hadronic current, q = k − k ′ the momentum of the exchanged photon and l µ the neutrino current given explicitly by where µ ij is the transition magnetic moment (in units of Bohr-magneton µ B = e/(2m e )) of the neutrino mass eigenstates involved in the scattering process. The differential cross section can be written as where µ 2 ef f will be defined later, see (27). In (3), I ≃ k · p is the incident flux and we have neglected the neutrino mass in the kinematics. The subscript avg in (3) refers to spin-average.
The spin-averaged neutrino current tensor can be explicitly evaluated and turns out to be The hadronic current tensor can be written in the general form where we have used current conservation and where a and b are in general functions of the invariants q 2 , p 2 and p · q. Contracting the spin averaged currents N µν and D µν we get Using the kinematic relations (8) is times the coefficient of the a term. Since we are dealing with neutrinos in the energy range of (5 − 20) MeV which is much smaller than the deuterium mass, we can drop the b term in (8). Substituting in (3) we find that The amplitude for the photo-dissociation process γ(q) and the deuterium photo-dissociation cross section can be written as where again avg stands for spin average. Using the fact that Comparing the neutrino cross section (10) with the photo-dissociation cross section (13) we see that we have the relation if we assume that in the hadronic current a ≃ a | q 2 =0 . This is totally justified since the transferred momenta are much smaller than the typical hadronic energies that set the scale about and beyond which form factors cease to have a point-like behavior [6]. Using the (14) for the neutrino magnetic moment induced deuterium disintegration reduces to the form For the photo-dissociation cross section we use the expression 1 where the energy dependent factor shown in the square bracket is the theoretical prediction appropriate for the electric dipole transition of deuterium [7]. We have determined the prefactor σ 0 = 19.4 mb by doing a least square fit of the energy dependent function shown in (17) with the experimental results [8] for the photo-disintegration of deuterium in the energy range E γ ≃ (5 − 10) MeV appropriate for the 8 B neutrinos observed at SNO.
The neutrino flux from 8 B in the Sun which is observed at SNO can be represented by where Φ SSM = (5.87±0.44)×10 6 cm −2 sec −1 is the new predicted value of the 8 B neutrino flux in the standard solar model [9] after taking into account the recent laboratory measurements of the 7 Be(pγ) 8 B cross section [4,10]. The spectral shape of the 8 B neutrino flux can be parameterized by the analytical expression [11] ξ(E ν ) = 8.52 × 10 −6 (15 where the neutrino energy E ν is in units of MeV. The total events of deuterium dissociation observed at SNO is the sum of the standard neutral current events plus those due to neutrino magnetic moments 2 where N d is the total number of deuterium atoms in the fiducial volume and T is the exposure time. The neutral current flux reported by SNO [3] assumes that the total dissociation events arise from the standard neutral currents, Combining (20) and (21) we see that the experimental "NC" flux reported by SNO can be related to the magnetic moment cross section as Factoring out the unknown µ 2 ef f from (16), we can evaluate the numerical factor in the second term of (22) by evaluating the integrals over E ν in the range (5.5 − 15.1) MeV. We should add, returning to the comment we made at the beginning of this paper, that in this energy range our cross section σ mag is smaller than the corresponding cross section in [5]. In the higher part of that energy range, where σ mag dominates the integral in (22), the difference is about a factor of 2 and thus the bound on µ ef f we shall obtain below may be considered a conservative limit by roughly a factor of √ 2. We use the numerical tables of σ N C (E ν ) given by Nakamura et al. [12] to evaluate the denominator of the second term of (16). We find that the relation between the experimentally observed flux Φ SN O N C and the SSM prediction Φ SSM is given by Earlier bounds on µ ef f [14,15] were based on the extra electron scattering events that can be accomodated by the SuperK spectrum (|µ ef f | < 1.5 × 10 −10 at 90% C.L. ) [14] and by a combination of all experimental rates (|µ ef f | < 2.0 × 10 −9 at 90% C.L. ) [15]. Notice that in the case of elastic scattering of electrons, since the cross section of ν e e − scattering is different from that of ν µ,τ e − , the extra events due to magnetic moment scattering were adjusted by the uncertainties in δm 2 and (mainly) sin 2 θ 12 . In our case, since in the deuterium dissociation neutral current process the cross sections for all three neutrino flavours are identical, the event rate is independent of the oscillation parameters and therefore the extra events due to possible neutrino magnetic moments cannot be accommodated by shifting the values of the mass squared difference and the mixing angle. The only extra parameter with which the magnetic moment can be adjusted is the theoretical uncertainity in the total 8 B flux.
For the case of the MSW solution of the solar neutrino problem which has been selected by Kamland [16], the 8 B neutrinos undergo resonant adiabatic conversion. The matter mixing angle in the Sun is θ m = π/2. The neutrino mass eigenstate at production is ν e = ν 2 . As the evolution is adiabatic, at the Earth the neutrinos are still in the ν 2 mass eigenstate [14]. The effective magnetic moment for the solar 8 B neutrinos is therefore Our bound on the components of the neutrino magnetic moment tensor can thus be written as (µ 21 2 + µ 22 2 + µ 23 2 ) 1/2 < 3.71 × 10 −9 µ B (95% C.L.) At this point a qualification is in order. In fact, in the present state of affairs one cannot exclude a small contamination of ν 1 in the neutrinos arriving from the Sun that would depend on the neutrino energy (see for example [17]). We should emphasize that our bound on µ ef f would still be valid, but a different interpretation than (27) would follow. In the future, with more data at hand, it may be worth to reconsider the interpretation of µ ef f for solar neutrinos.
Sure enough, our bound here is not much different from other laboratory limits [18] obtained elsewhere and in fact it is definitely worse than the one obtained from the plasma emission argument in globular cluster stars [11]. However, two facts have to be considered when ascribing it its actual relevance. First, as we just mentioned the best limit is derived from energy-loss constraints in stars and hence does rely exclusively on stellar evolution theory. Second, since neutrinos oscillate and as a consequence different flavors mix differently in different settings, reactor, accelerator, solar, and astrophysical data cannot be compared directly when obtaining the bounds on magnetic moments [14,15]. It is the analysis of the various pieces of information coming from a variety of experimental sources that will eventually lead to a separate restriction on each and every µ ij . The SNO data used in this paper, and the better data which will hopefully follow in the future on neutrino initiated deuteron break-up, is just one source of information among other sources that one can use to reach this goal.