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Neutrino Electromagnetic Form Factors Effect on the Neutrino Cross Section in Dense Matter

The sensitivity of the differential cross section of the interaction between neutrino-electron with dense matter to the possibly nonzero neutrino electromagnetic properties has been investigated. Here, the relativistic mean field model inspired by effective field theory has been used to describe non strange dense matter, both with and without the neutrino trapping. We have found that the cross section becomes more sensitive to the constituent distribution of the matter, once electromagnetic properties of the neutrino are taken into account. The effects of electromagnetic properties of neutrino on the cross section become more significant for the neutrino magnetic moment mu_nu>10^{-10} mu_B and for the neutrino charge radius R>10^{-5} MeV^{-1}.


I. INTRODUCTION
The demand for precise information on the neutrino transport in the investigations of astrophysical phenomena, such as supernovae explosion and the structure of protoneutron stars, has stimulated several studies on neutrino interactions in matter at high densities [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Even recently, a realistic neutrino opacity in neutron rich matter for the supernovae simulation which takes into account the correlations and weak magnetism of nucleons in the finite temperature calculations has been also considered [4].
In the standard model, massless neutrinos have zero magnetic moment and electronic charge. However, there are evidences that the laboratory bounds on the neutrino-electron magnetic moment (µ ν ) is smaller than 1.0 × 10 −10 µ B at the 90% confidence level [15], where µ B is the Bohr magneton. Stronger bound of µ ν ≤ 3.0 × 10 −12 µ B also exists from astrophysical consideration, particularly from the study of the red giant population in the globular cluster [16]. On the other hand the charge radius of ν e has been bounded by the LAMF experiment [17] to be R 2 = (0.9 ± 2.7) × 10 −32 cm 2 = (22.5 ± 67.5) × 10 −12 MeV −2 , while the plasmon decay in the globular cluster star predicts the limit of e ν ≤ 2 × 10 −14 e [16], where e is electron charge. Recent discussions on the effects of the neutrino electromagnetic properties in astrophysics can be seen in, e.g., Ref. [16], while for the case of the solar neutrino problem one can consult, e.g., Ref. [18].
So far, there has been no calculation of the interaction of neutrinos with dense matter which considers neutrino electromagnetic form factors. To this end we extend our previous report of the interaction of neutrinos with electrons gas [14] to the interaction of neutrinos with the non strange dense stellar matter which takes into account the effects of the trapped neutrinos in matter, but still use the zero temperature approximation. The validity of this approximation is fulfilled by the fact that temperature effects on the equation of state (EOS) of supernovae matter and maximum masses of protoneutron stars are smaller than those without neutrino trapping [23]. We note that considerable efforts have been devoted to study the effective electromagnetic coupling of neutrinos in the thermal background of particles [19,20,21,22]. However, the particles under investigation so far are electron and nucleon gasses. Clearly, the effect of nucleon correlations has not yet been studied.
References [19,22] have shown that the densities and temperature yield enhancements to the form factor with standard charge radius, but they do not give a significant effect on the electric and magnetic dipole form factors. Moreover, they can only give a significant contribution if T /m and µ/m are large, where T , µ and m are the temperature, chemical potential and mass of the corresponding particles in the thermal bath, respectively [22]. Therefore, the expectation that the temperature and hadron correlations can strongly reduce the electromagnetic properties of neutrinos seems to be unlikely. For simplicity, the RPA correlations are still neglected. The importance of the neutrino trapping in the supernova dynamical evolution has been, for example, pointed out in Ref. [24]. We choose this kind of matter, because we suspect that the effects of the neutrino electromagnetic properties would be more pronounced in the protons-and electrons rich matter. Different from Ref. [24], which used the standard relativistic mean field (RMF) model to calculate the EOS, in this work we use the RMF model inspired by the effective field theory (E-RMF model) [25].
The construction of this paper is as follows. Section II consists of a brief review of the matter models used in this work. The analytical results of the neutrino electromagnetic form factors effects on the cross section are given in section III. In section IV, numerical results and their discussions are presented. Finally, the conclusion is given in section V.

II. MATTER MODELS
To describe nucleons interactions, we use the effective Lagrangian density given in Refs. [25,26]. The wide range applications of this model have been discussed in detail in Refs. [27,28]. In our work, leptons are assumed to be free (Fermi gas). The framework of our calculation is the relativistic mean field approximation, which means that we treat the nucleon interactions self consistently. The explicit form of the Lagrangian densities can be seen in appendix A. By solving the Euler-Lagrange equations for the Lagrangian densities given in appendix A, the equations of state for nucleons, mesons and leptons can be obtained, and then, can be used to calculate all matter properties required in this work. The following constraints are used to calculate the fraction of every constituent in matter: • balance equation for chemical potentials • conservation of charge neutrality where the total density of baryon is given by while the fixed electronic-leptonic fraction is defined as Coupling constants for all parameter sets used in this work are shown in Table I.

III. EFFECTS OF THE NEUTRINO ELECTROMAGNETIC FORM FACTORS
In this section we calculate the neutrino-matter cross section, in which the electromagnetic form factors of the neutrino-electron and the weak magnetism of nucleons are explicitly taken into account. We start with the Lagrangian density of neutrino matter interactions for each constituent in the form of where G F and α are the coupling constant of weak interaction and the electromagnetic fine structure constant, respectively, and j= n, p, e − , µ − . The parity violating vertex of neutrino is given by while the electromagnetic properties of Dirac neutrinos are described in terms of four form factors, i.e., f 1ν , g 1ν , f 2ν and g 2ν , which stand for the Dirac, anapole, magnetic, and electric form factors, respectively. The electromagnetic vertex Γ µ EM contains electromagnetic form factors [34,35]. Explicitly, it reads where f mν = f 1ν + (m ν /m e )f 2ν , P µ = k µ + k µ′ , m ν and m e are the neutrino and electron masses, respectively. In the static limit, the reduced Dirac form factor f 1ν and the neutrino anapole form factor g 1ν are related to the vector and axial vector charge radii R 2 where the neutrino charge radius is defined by R 2 = R 2 V + R 2 A . In the limit of q 2 → 0, f 2ν and g 2ν , respectively, define the neutrino magnetic moment and the (CP violating) electric dipole moment [34,36], i.e., where µ 2 ν =(µ m ν ) 2 + (µ e ν ) 2 . The explicit form of J W j µ [4] is given by and for J EM j µ [37] it reads Using the Lagrangian density given by Eq. (5), we can now calculate the differential cross section. Using the standard method we obtain  [4]. Here E ν and E ′ ν are the initial and final neutrino energies, respectively, G F = 1.023×10 −5 /M 2 is the weak coupling, and M is the nucleon mass. The neutrino tensors for the weak contribution is given by while for the electromagnetic contribution, and for the interference contribution, with k the initial neutrino four-momentum and q = (q 0 , q) the four-momentum transfer. The polarization tensors Π µν for the weak (W), electromagnetic (EM) and interference (INT) terms, which define the target particles, can be written as where p = (p 0 , p) is the corresponding initial four-momentum and G(p) is the target particle propagator. The explicit form for nucleons is given by where E * = E + Σ 0 indicates the nucleon effective energy, and M * = E + Σ S is the nucleon effective mass. Σ 0 and Σ S are the scalar and time like self energies, respectively. The lepton propagators have similar expressions, only the starred quantities in Eq. (17) are replaced by the free ones. Explicitly, Eq. (16) for each constituent can be written as where for j = n, p (nucleons), m is equal to M * and M is the nucleon mass, while for Due to the current conservation and translational invariance, the vector polarization Π ImV µν of every contribution consists of two independent components which we choose to be in the frame of q µ ≡ (q 0 , | q|, 0, 0), i.e., The axial-vector and the mixed pieces are found to be The explicit forms of Π T , Π L , Π V A and Π A for nucleons are [32] For leptons, they have also similar expressions, only the starred quantities in Eqs. (22), (23), (24) and (25) are replaced by the free ones. Thus, the analytical form of Eq. (12) can be obtained from the contraction of every polarization and neutrino tensors couple (L µν Π µν ) mentioned previously. The results are where the functions in the front of every polarization terms of Eqs. (26) -(28) are given by for the weak contributions, and for the electromagnetic contributions, and for the interference contributions. The constants a, b and c are defined as where these neutrino form factors, f mν , g 1ν , f 2ν and g 2ν , are related to the neutrino-electron dipole moment and charge radius through Eqs. (8) and (9).

IV. NUMERICAL RESULTS AND DISCUSSIONS
Before investigating the sensitivity of the neutrino-matter differential cross section to the neutrino electromagnetic form factors, we show the predictions of the nuclear model used in the calculation (G2* parameter set) at high density in Figs. 1 and 2. Figure 1 reveals that the G2* parameter set has the softest E sym compared to the other parameter sets (G2 and NL3 The left panel of Fig. 2 shows the relative fraction of every constituent in the case that neutrinos are absent, while the right panel shows the case when neutrinos are trapped. We obtain a similar conclusion as in Ref. [24], i.e., when Fermi momentum of the electrons reaches the muon mass, muon begins to appear, and then the proton and electron fraction curves are splitted into two different paths. The threshold for muon production occurs just below the saturation density. In the case that neutrinos are trapped, for example with Y le = 0.3, it can be seen that the threshold is shifted toward higher densities. This displacement is even larger for the E-RMF model (ρ = 3ρ 0 ) than the result of Ref. [24] (ρ = 2ρ 0 ). This similar finding leads to a similar conclusion drawn in Ref. [24], i.e., the EOS of matter with neutrino trapping is softer than the case where neutrinos are absent. This fact leads to a very important consequence for the physics of supernovae explosions [24].   In order to see the effect more clearly, we take µ ν = 5 × 10 −10 µ B in the case of nonzero neutrino dipole moment. Obviously, this value is larger than its laboratory bound (µ ν = 1. between both cases leads to a difference in the total differential cross sections. It can also be seen in Fig. 6 that a similar situation also happens for the case of R = 5.0 × 10 −5 MeV −1 but with µ ν = 0.
This means that in contrast to the calculation based on the standard weak interaction, the cross section calculated by including neutrino electromagnetic properties is very sensitive to the particle number of each constituent.
To see how sensitive the calculated total cross section to the neutrino electromagnetic properties is, we plot the total differential cross section as functions of the energy transfer It is found that the cross section decreases when q 1 increases. If we see the upper right panel of Fig. 7, it can be seen that this decrement can be slowed down if the charge radius is set to a non zero value (e.g., 5.0 × 10 −5 MeV −1 ). The effect of nonzero neutrino dipole moment appears more dominantly at smaller values of momentum transfer q 1 and energy transfer q 0 , which is due to the role of massless photon propagators in the electromagnetic interaction.
On the contrary, the weak contribution becomes more dominant at larger values of q 1 and q 0 . MeV and q 1 ≃ 2 − 4.5 MeV for low densities (i.e., ρ B = 2.5ρ 0 ), albeit with a different trend.
In addition, if we considered the upper bound of the neutrino-muon dipole moment which is given by µ ν < 7.4 ×10 −10 µ B [43,44], besides the weak magnetism term as investigated by Ref. [4], it seems from the above discussion that the neutrino muon electromagnetic properties might give additional effects to the neutrino muon and its anti-neutrino mean free path difference in the neutron rich matter. How significant the effects should be quantitatively checked by a real calculation.  Fig. 9, but for fixed baryon density (ρ B = 5ρ 0 ).

V. CONCLUSIONS AND OUTLOOK
In conclusion, we have studied the sensitivity of the neutrino cross section to the neutrino electromagnetic properties. In our calculations, we use the G2* parameter set of the E-RMF model to describe matter. This parameter set predicts a soft EOS at high density and has ρ B = 2.5ρ 0 that coincide with the direct URCA process threshold. The calculation has been performed for the cases in which neutrinos are trapped and absent in matter. It is found that in the non strange stellar matter, the electromagnetic form factor has an important role in the neutrino-electron matter cross section if µ ν > 10 −10 µ B and R > 10 −5 MeV −1 . It is also found that the effects of the neutrino electromagnetic form factors on the cross sections are more pronounced at higher densities.
Matters with trapped neutrino, like supernovae ones, are more sensitive to the presence of neutrino electromagnetic properties. This is due to the fact that matters with trapped neutrino have larger fractions of protons and electrons than those without neutrinos.Although we have found that the role of neutrino electromagnetic properties in the neutrino-electron matter interaction is not too crucial, this would not be the case if we considered the bound of the neutrino-muon dipole moment given by Refs. [43,44].
With increasing the density of the protoneutron star, strangeness can be liberated and face up in the the filling of the hyperon Fermi seas and/or in the creation of kaon condensates.
Occurrence of these exotics in protoneutron star interiors will enhance the neutrino scattering rate and it may have also interesting observational consequences, like softening the EOS, possibility of changing nucleon isospin composition in the star matter evolution, as well as enhancing neutrino emission processes in the neutron star matter evolution, as extensively discussed in Refs. [23,45,46,47,48,49,50,51,52]. Further, depending on the temperature and the model used, baryon correlations can also reduce the scattering rate [4,6,8,10,13].
Therefore, an extension of this calculation by including the strange matter, RPA correlation and a more general condition, i.e., finite temperature might also be interesting and quite relevant to consider in the future.
To achieve a softer density dependence of the nuclear matter symmetry energy of the standard RMF model, Refs. [32,33] add isovector-vector nonlinear terms in the Lagrangian density. In this paper, a similar procedure as in Refs. [32,33] is adopted. Since in the E-RMF model the isovector vector nonlinear term is already present, the density dependence of the nuclear matter symmetry energy can be adjusted without adding new isovector nonlinear terms. Thus, we only adjust g ρ and η ρ but maintain the requirement that the symmetry energy at k F = 1.14 fm should have the same value at E sym = 24.1 MeV. The argument behind this procedure is explained in detail in Refs. [32,33]. For leptons, we use the following free Lagrangian density: ACKNOWLEDGMENT We acknowledge P. Danielewicz for his kindness to give us his experimental data.