Propagation of Neutrinos through Magnetized Gamma-Ray Burst Fireball

# Propagation of Neutrinos through Magnetized Gamma-Ray Burst Fireball

Sarira Sahu111 Email address: sarira@nucleares.unam.mx , Nissim Fraija222Email address: nissim.ilich@nucleares.unam.mx and Yong-Yeon Keum333Email address: yykeum@korea.ac.kr Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Department of Physics and BK21 Initiative for Global Leaders in Physics, Korea University, Seoul 136-701, Korea
School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea
###### Abstract

The neutrino self-energy is calculated in a weakly magnetized plasma consists of electrons, protons, neutrons and their anti-particles and using this we have calculated the neutrino effective potential up to order . In the absence of magnetic field it reduces to the known result. We have also calculated explicitly the effective potentials for different backgrounds which may be helpful in different environments. By considering the mixing of three active neutrinos in the medium with the magnetic field we have derived the survival and conversion probabilities of neutrinos from one flavor to another and also the resonance condition is derived. As an application of the above, we considered the dense and relativistic plasma of the Gamma-Ray Bursts fireball through which neutrinos of 5-30 MeV can propagate and depending on the fireball parameters they may oscillate resonantly or non-resonantly from one flavor to another. These MeV neutrinos are produced due to stellar collapse or merger events which trigger the Gamma-Ray Burst. The fireball itself also produces MeV neutrinos due to electron positron annihilation, inverse beta decay and nucleonic bremsstrahlung. Using the three neutrino mixing and considering the best fit values of the neutrino parameters, we found that electron neutrinos are hard to oscillate to another flavors. On the other hand, the muon neutrinos and the tau neutrinos oscillate with equal probability to one another, which depends on the neutrino energy, temperature and size of the fireball. Comparison of oscillation probabilities with and without magnetic field shows that, they depend on the neutrino energy and also on the size of the fireball. By using the resonance condition, we have also estimated the resonance length of the propagating neutrinos as well as the baryon content of the fireball.

###### pacs:
98.70.Rz, 14.60.Pq
preprint: ICN/000-03-HEP

## I Introduction

The particle propagation in a heat bath with or without magnetic field has attracted much attention due to its potential importance in plasma physics, astrophysics and cosmology. The processes which are forbidden in vacuum can take place in the medium and even massless particles acquire mass when they propagate through the medium. Studying the behavior of particles in such environments requires the technique of thermal field theory. Therefore, in connection with these astrophysical and cosmological scenarios it has become increasingly important to understand the quantum field theory of elementary processes in the presence of a thermal heat bath. The neutrino self-energy is studied in the magnetized medium by many authors, where the effective potential of neutrino is calculated and applied in the physics of supernovae, early Universe and physics of Gamma-ray bursts (GRBs)Elmfors:1996gy (); Koers:2005ya (); Dessart:2008zd (); Sahu:2009iy (); Langacker:1982ih (); BravoGarcia:2007uc ().

Gamma-ray bursts (GRBs) are the most luminous objects after the Big Bang in the universePiran:1999kx () and believed to emit about erg in few seconds. During this few seconds non-thermal flashes of about 100 keV to 1-5 MeV photons are emitted. The isotropic distribution of GRBsMeegan:1992xg () in the sky implies that they are of cosmological origin Piran:1999kx (); Zhang:2003uk (); Piran:1999bk (). The GRBs are classified into two categories: short-hard bursts () and long-soft bursts. It is generally accepted that long gamma-ray bursts are associated with star forming regions, more specifically related to supernovae of type Ib and Ic. The observed correlations of the following GRBs with supernovae GRB 980425/SN 1998bw, GRB 021211/SN 2002lt, GRB 030329/SN 2003dh and GRB 0131203/SN 2003lw show that long duration GRBs are related to the core collapse of massive starsDellaValle:2005cr (). The origin of short-duration bursts are still a mystery, but recently there has been tremendous progress due to accurate localization of many short bursts by the SwiftGehrels:2005qk (); Barthelmy:2005bx () and HETE-2Villasenor:2005xj () satellites. The afterglow observation of GRB 050709 at z=0.1606Hjorth:2005 () by HETE-2 and the Swift observation of afterglow from GRB050709b at z=0.225Gehrels:2005qk () and GRB 050724 at z=0.258Berger:2005 () seems to support the coalescing of compact binaries as the progenitor for the short-hard bursts although definite conclusions can not be drawn at this stage. Very recently millisecond magnetars have been considered as possible candidates as the progenitor for the short-hard burstsUsov:1992zd (); Uzdensky:2007uf (). For a future study of short-hard GRBs, the ultra-fast flash observatory (UFFO) project is proposeduffo:2009 ().

Irrespective of the nature of the progenitor or the emission mechanism of the gamma-rays, these huge energies within a very small volume imply the formation of and fireball which would expand relativistically. In the standard fireball scenario; at the first, a radiation dominated plasma is formed in a compact region with a size - kmPiran:1999kx (); Waxman:2003vh (). This creates an opaque fireball due to the process . However, in addition to , pairs, fireball also contain a small amount of baryons, both from the progenitor and the surrounding medium and the electrons associated with the matter (baryons), that increase the opacity and delay the process of emission of radiation. The average optical depth of this process is very high. Because of this huge optical depthGoodman:1986az (), photons can not escape freely and even if there are no pairs to begin with, they will form very rapidly and will Compton scatter lower energy photons. In the fireball the and pairs will thermalize with a temperature of about 3-10 MeV.

In this stage, a phase of acceleration begins and the fireball expands relativistically with a large Lorentz factor, converting internal energy into bulk kinetic energy. As the fireball shell expands, the baryons will be accelerated by radiation pressure. The fireball bulk Lorentz factor increases linearly with radius, until reaching the maximum Lorentz factor, so the photon number density and typical energy drop. At certain radius, the photons become optically thin (the optical depth is ) to both pair production and to Compton scattering off the free electrons associated with baryons. At this radius, although much of the initial energy is converted to the kinetic energy of the shell, some energy will be radiate away with an approximately black body spectrum. This is the first electromagnetic signal detectable from the fireball. For an intermittent central engine with typical variability timescale of , appears adjacent mini-shells with different Lorentz factor, which will collide with each other and will form strong ”internal” shocks. Later, the fireball shell is eventually decelerated by successive strong external shocks with the ambient medium (ISM), propagates into the mediumZhang:2003uk (). As in each shell exists a non thermal population of baryons and electrons through Fermi acceleration and during each shock the system behaves like an inelastic collision between two or more shells converting kinetic energy into internal energy, which is given to the non thermal population of baryons and electrons cool via synchrotron emission and/or inverse Compton scattering to produce the observed prompt emission. Piran:1999kx (). The synchrotron spectrum can be calculate if we know the detailed physical conditions of the radiating region. For internal shocks, the so-called ’equipartition’ hypothesis is often used, which assumes that the energy is equally distributed between protons, electrons and the magnetic fieldVedrenne ().

As is well known, during the final stage of the death of a massive star and/or merger of binary stars copious amount of neutrinos in the energy range of 5-30 MeV are produced. Some of these objects are possible progenitors of GRBs Lee:2007js (). Apart from the beta decay process there many other processes which are responsible for the production of MeV neutrinos in the above scenarios: for example, electron-positron annihilation, nucleonic bremsstrahlung etc, where neutrinos of all flavor can be producedRaffelt:2001kv (). Within the fireball, inverse beta decay as well as electron-positron annihilation will also produce MeV neutrinos. Many of these neutrinos has been intensively studied in the literature Ruffert:1998qg (); Goodman:1986we () and may propagate through the fireball. However, the high-energy neutrinos created by photo-meson production of pions in interactions between the fireball gamma-rays and accelerated protons have been studied tooWaxman:1997ti (). The accretion disc formed during the collapse or merger is also a potentially important place to produce neutrinos of similar energy. Fractions of these neutrinos will propagate through the fireball and they will oscillate Volkas:1999gb (); Dasgupta:2008cu () if the accreting materials survive for a longer period. Although neutrinos conversions in a polarized medium have been studied Nunokawa:1997dp (), resent we have studied the neutrino propagation within the fireball environment with and without magnetic field where resonant neutrino oscillate from one flavor to another are studied by considering the mixing of two flavors only. In this paper we have calculated the neutrino effective potential in the weak field and done a complete analysis of the three neutrino mixing within the magnetized fireball and studied the resonant oscillation of it. By considering the best fit neutrino parameters from different experiments, we found that electron neutrino can hardly oscillate to other flavor, whereas muon and tau neutrinos can oscillate among themselves with almost equal probability and their oscillation probabilities depends on neutrino parameters as well as the fireball parameters.

The organization of the paper is as follows: In sec. 2, we have derived the neutrino self-energy by using the real time formalism of finite temperature field theory Nieves:1990ne (); Weldon:1982aq (); D'Olivo:2002sp (); Erdas:1990gy () and Schwinger’s proper-time method Schwinger (). By considering the weak-field approximation we have derived the neutrino effective potential and compare it with the effective potential for case. We also calculate the effective potential for matter background as well as for neutrino background. A brief description about the Gamma-Ray Burst and the fireball model is discussed in sec. 3. The case of three-neutrino mixing is considered in sec. 4, where we have calculated the survival and conversion probabilities of neutrinos and also the resonance condition. In sec. 5, we discuss our results for GRB fireball and a brief conclusions is drawn in sec. 6.

## Ii Neutrino Effective Potential

As is well known, the particle properties get modified when it is immersed in a heat bath. A massless neutrino acquires an effective mass and an effective potential in the medium. The resonant conversion of neutrino from one flavor to another due to the medium effect is important for solar neutrinos which is well known as the MSW effect. Similarly the propagation of neutrino in the early universe hot plasmaEnqvist:1990ad (), supernova mediumSahu:1998jh () and in the GRB fireballSahu:2005zh () can have also many important implications in their respective physics. In all the astrophysical and cosmological environment, magnetic field is entangled intrinsically with the matter and it also affect the particle properties. Although neutrino can not couple directly to the magnetic field, its effect can be felt through coupling to charge particles in the backgroundErdas:1998uu (). Neutrino propagation in a neutron star in the presence of a magnetic field and also in the magnetized plasma of the early universe has been studied extensively.

We use the field theory formalism to study the effect of heat bath and magnetic field on the propagation of elementary particles. The effect of magnetic field is taken into account through Schwinger’s propertime methodSchwinger (). The effective potential of a particle is calculated from the real part of its self energy diagram.

The most general decomposition of the neutrino-self energy in presence of a magnetized medium can be written as:

 Σ(k)=R(a∥to0.0pt/k∥+a⊥to0.0pt/k⊥+bto0.0pt/u+cto0.0pt/b)L, (1)

where , and stands for the 4-velocity of the center-of-mass of the medium given by . The projection operators are conventionally defined as and . The effect of magnetic field enters through the 4-vector which is given by . The background classical magnetic field vector is along the -axis and consequently . So using the four vectors and we can express

 to0.0pt/k∥=k0to0.0pt/u−k3to0.0pt/b, (2)

and the self-energy can be expressed in terms of three independent four-vectors , and . So we can write ()

 ~Σ=a⊥to0.0pt/k⊥+bto0.0pt/u+cto0.0pt/b. (3)

The determinant of , i.e

 det[to0.0pt/k−~Σ]=0, (4)

gives the dispersion relation up to leading order in , and as:

 k0−|k|=b−ccosϕ−a⊥|k|sin2ϕ=Veff,B, (5)

for a particle, where is the angle between the neutrino momentum and the magnetic field vector. One has to remember that the scalars and in this case are not the same if one expresses the self-energy in the form given in Eq. (1), but the is independent of how we express . Now the Lorentz scalars , and which are functions of neutrino energy, momentum and magnetic field can be calculated from the neutrino self-energy due to charge current and neutral current interaction of neutrino with the background particles.

### ii.1 Neutrino self-energy

The one-loop neutrino self-energy in a magnetized medium is comprised of three piecesBravoGarcia:2007uc (), one coming from the -exchange diagram which we will call , one from the tadpole diagram which we will designate by and one from the -exchange diagram which will be denoted by . The total self-energy of the neutrino in a magnetized medium then becomes:

 Σ(k)=ΣW(k)+ΣZ(k)+Σt(k). (6)

Each of the individual terms appearing in the right-hand side of the above equation can be expressed as in Eq. (1) and the Lorentz scalars , and have contributions from all the three pieces as described above.The individual terms on the right hand side of Eq. (6) can be explicitly written as:

 −iΣW(k)=∫d4p(2π)4(−ig√2)γμLiSℓ(p)(−ig√2)γνLiWμν(q), (7)
 −iΣZ(k)=∫d4p(2π)4(−ig√2cosθW)γμLiSνℓ(p)(−ig√2cosθW)γνLiZμν(q), (8)

and

 −iΣt(k)=−(g2cosθW)2RγμiZμν(0)∫d4p(2π)4Tr[γν(CV+CAγ5)iSℓ(p)]. (9)

The subscripts in correspond to W-exchange, Z-exchange and Tadpole diagrams. In the above expressions is the weak coupling constant and is the Weinberg angle and can be expressed in terms of the Fermi coupling constant as . The quantities and are the vector and axial-vector coupling constants which come in the neutral-current interaction of electrons, protons (), neutrons () and neutrinos with the boson. Their forms are as follows,

 CV=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−12+2sin2θWe12ν12−2sin2θWp−12n, (10)

and

 CA={−12ν,p12e,n. (11)

Here and stand for the -boson propagator and charged lepton propagator respectively in presence of a magnetized plasma. The is the -boson propagator in vacuum and is the neutrino propagator in a thermal bath of neutrinos. The form of the charged lepton propagator in a magnetized medium is given by,

 Sℓ(p)=S0ℓ(p)+Sβℓ(p), (12)

where and are the charged lepton propagators in presence of an uniform background magnetic field and in a magnetized medium respectively. In this article we will always assume that the magnetic field is directed towards the -axis of the coordinate system. With this choice we have,

 iS0ℓ(p)=∫∞0eϕ(p,s)G(p,s)ds, (13)

where,

 ϕ(p,s)=is(p2∥−m2ℓ−tanzzp2⊥). (14)

In the above expression

 p2∥ = p20−p23, (15) p2⊥ = p21+p22, (16)

and where is the magnitude of the electron charge, is the magnitude of the magnetic field and is the mass of the charged lepton. In the above equation we have not written another contribution to the phase which is where is an infinitesimal quantity. This term renders the integration convergent. We do not explicitly write this term but implicitly we assume the existence of it and it will be written if required. The above equation can also be written as:

 ϕ(p,s)=ψ(p0)−is[p23+tanzzp2⊥], (17)

where,

 ψ(p0)=is(p20−m2ℓ). (18)

The other term in Eq. (13) is given as:

 G(p,s)=sec2z[to0.0ptA/+ito0.0ptB/γ5+mℓ(cos2z−iΣ3sinzcosz)], (19)

where,

 Aμ = pμ−sin2z(p⋅uuμ−p⋅bbμ), (20) Bμ = sinzcosz(p⋅ubμ−p⋅buμ), (21)

and

 Σ3=γ5to0.0pt/bto0.0pt/u. (22)

The second term on the right-hand side of Eq. (12) denotes the medium contribution to the charged lepton propagator and its form is given by:

 Sβℓ(p)=iηF(p⋅u)∫∞−∞eϕ(p,s)G(p,s)ds, (23)

where contains the distribution functions of the particles in the medium and its form is:

 ηF(p⋅u)=θ(p⋅u)eβ(p⋅u−μℓ)+1+θ(−p⋅u)e−β(p⋅u−μℓ)+1, (24)

where and are the inverse of the medium temperature and the chemical potential of the charged lepton.

The form of the -propagator in presence of a uniform magnetic field along the -direction is presented in Erdas:1998uu () and in this article we only use the linearized (in the magnetic field) form of it. The reason we assume a linearized form of the -propagator is because the magnitude of the magnetic field we consider is such that . In this limit and in unitary gauge the propagator is given by

 Wμν(q)=gμνM2W(1+q2M2W)−qμqνM4W+3ie2M4WFμν, (25)

where is the -boson mass. Here we assume that and keep terms up to in the propagator.

Let us assume that an electron neutrino is propagating in the medium (generalization to other neutrinos is straight forward) which contain electrons and positrons, protons, neutrons and all types of neutrinos and anti-neutrinos.

By evaluating the Eq. (7) explicitly we obtain

 Re ΣW(k)=R[aW⊥to0.0pt/k⊥+bWto0.0pt/u+cWto0.0pt/b]L, (26)

where the Lorentz scalars are given by

 aW⊥ = −√2GFM2W[{Eνe(Ne−¯Ne)+k3(N0e−¯N0e)} (27) +eB2π2∫∞0dp3∞∑n=0(2−δn,0)(m2eEn−HEn)(fe,n+¯fe,n)],
 bW = bW0+~bW (28) = √2GF[(1+32m2eM2W+E2νeM2W)(Ne−¯Ne)+(eBM2W+Eνek3M2W)(N0e−¯N0e)

and

 cW = cW0+~cW (29) = −eB2π2M2W∫∞0dp3∞∑n=0(2−δn,0){2Eνe(En−m2e2En)δn,0+2k3(En−32m2eEn−HEn)}(fe,n+¯fe,n)].

The electron energy in the magnetic field is given by,

 E2e,n=(p23+m2e+2neB)=(p23+m2e+H). (30)

From Eqs. (28) and (29), we have defined

 ~bW=−√2GFeB2π2M2W∫∞0dp3∞∑n=0(2−δn,0){2k3Enδn,0+2Eνe(En+m2e2En)}(fe,n+¯fe,n)], (31)

and

 ~cW=−√2GFeB2π2M2W∫∞0dp3∞∑n=0(2−δn,0){2Eνe(En−m2e2En)δn,0+2k3(En−32m2eEn−HEn)}(fe,n+¯fe,n)]. (32)

In the above equations, the number density of electrons is defined as

 Ne=eB2π2∞∑n=0(2−δn,0)∫∞0dp3fe,n (33)

and the number density of electrons for the Lowest Landau (LL) state which corresponds to is

 N0e=eB2π2∫∞0dp3fe,0 (34)

We can express the Eq. (8) for Z-exchange as

 ReΣZ(k)=R(aZto0.0pt/k+bZto0.0pt/u)L, (35)

and explicit evaluation gives,

 aZ=√2GF[EνeM2Z(Nνe−¯Nνe)+231M2Z(⟨Eνe⟩Nνe+⟨¯Eνe⟩¯Nνe)], (36)

and

 bZ=√2GF[(Nνe−¯Nνe)−8Eν3M2Z(⟨Eνe⟩Nνe+⟨¯Eνe⟩¯Nνe)]. (37)

In Eq. (35) we have a term proportional to , because there is no magnetic field. But using the four vectors and the parallel component of the four vector can be decomposed as in Eq. (2). In the calculation of the potential the contribution from these terms will cancel each other and only one which will remain is .

From the tadpole diagram Eq. (9) we get,

 ReΣt(k) = √2GFR[{CVe(Ne−¯Ne)+CVp(Np−¯Np)+CVn(Nn−¯Nn)+(Nνe−¯Nνe) (38) +(Nνμ−¯Nνμ)+(Nντ−¯Nντ)}to0.0pt/u−CAe(N0e−¯N0e)to0.0pt/b]L.

So the different contributions to the neutrino self-energy up to order are calculated in a background of , nucleons, neutrinos and anti-neutrons.

### ii.2 Weak field limit eB≪m2e

In the above subsection, the result obtained is weak compared to the W-boson mass i.e. . But here we would like to use another limit that is magnetic field much weaker compared to the one done in the above subsection. We also assume that the chemical potential of the background electron gas is much small than the electron energy (). The implies CP symmetric medium where number of electrons equals number of positrons. So by taking we assume that . In a fireball medium this condition can be satisfied because the excess of electrons will come from the electrons associated with the baryons which will come from the central engine.

In the weak field limit () and , the electron distribution function can be written as

 fe,n=1eβ(Ee,n−μ)+1≃∞∑l=0(−1)le−β(Ee,n−μ)(l+1). (39)

Also we shall define

 α=βμ(l+1), (40)

and

 σ=βme(l+1). (41)

Using the above distribution function, the electron number density and other quantities of interest are given below:

 N0e−¯N0e=1π2BBcm3∞∑l=0(−1)lsinhαK1(σ)]=m3eπ2(BBc)Φ1, (42)
 Ne−¯Ne=m3π2∞∑l=0(−1)lsinhα[2σK2(σ)−BBcK1(σ)]=m3eπ2Φ2, (43)
 eB2π2∫∞0dp3E0(fe,0+¯fe,0)=m4eπ2(BBc)∞∑l=0(−1)lcoshα(K0(σ)+K1(σ)σ), (44)
 eB2π2∫∞0dp31E0(fe,0+¯fe,0)=m2eπ2(BBc)∞∑l=0(−1)lcoshαK0(σ), (45)
 eB2π2∞∑n=0(2−δn,0)∫∞0dp3En(fe,n+¯fe,n) =m2eπ2∞∑l=0(−1)lcoshα (46) [(6σ2−BBc)K0(σ)+(2−BBc+12σ2)K1(σ)σ],
 eB2π2∞∑n=0(2−δn,0)∫∞0dp31En(fe,n+¯fe,n)=m2eπ2∞∑l=0(−1)lcoshα[2σK1(σ)−BBcK0(σ)] (47)

and

 eB2π2∞∑n=0(2−δn,0)∫∞0dp3HEn(fe,n+¯fe,n)=m4eπ2∞∑l=0(−1)lcoshασ2[4K0(σ)+8σK1(σ)]. (48)

All the above quantities are necessary to evaluate the effective potential.

### ii.3 Neutrino Potential without Magnetic field

In the absence of magnetic field the neutrino self-energy and the neutrino effective potential is calculated earlierGarcia:2007ij (). In this case the neutrino self-energy is decomposed as

 Re~Σ(k)=ato0.0pt/k+bto0.0pt/u, (49)

and the neutrino effective potential for a massless neutrino is given by

 Veff=b=14EνTr(to0.0pt/kRe~Σ(k)). (50)

By evaluating the right hand side (RHS) up to order gives,

 Veff = √2GF[(1+32m2eM2W)(Ne−¯Ne)B=0 (51) −4π2(m2eMW)2Eνe∞∑l=0(−1)lcoshα{4K0(σ)σ2+(1+8σ2)K1(σ)σ}].

Also we have

 (Ne−¯Ne)B=0=m3eπ2∞∑l=0(−1)lsinhα2σK2(σ). (52)

This is the result obtained in ref.Garcia:2007ij () up to order for neutrino propagating in a medium with only electrons and positrons in it.

### ii.4 Comparison of Veff with and without magnetic field

The neutrino effective potential in a magnetic field is given in Eq. (5). To simplify our calculation we assume that, the magnetic field is along the direction of the neutrino propagation so that and the term does not contribute. Also one has to remember that by taking , we should get back the result obtained in Eq. (51) and this is only possible when we take in our calculation. Then the effective potential should be defined as (independent of the angle is zero or not),

 Veff,B=(b−c)/k3=−Eν. (53)

Hence forth we shall replace by in our calculation. This gives

 Veff,B = √2GF[(1+32m2eM2W−eBM2W)(Ne−¯Ne)−(1+m2e2M2W−eBM2W)(N0e−¯N0e) (54) +eB2π2M2W∫∞0dp3∞∑n=0(2−δn,0){2EνeEnδn,0−2Eνe(2En−m2eEn−HEn)}(fe,n+¯fe,n)]

With simplifications this gives,

 Veff,B = √2GF[m3eπ2∞∑l=0(−1)lsinhα{(1+32m2eM2W−eBM2W)(2σK2(σ)−BBcK1(σ)) (55) −BBc(1+m2e2M2W−eBM2W)K1(σ)}

We can write this in a simpler form as

 Veff,B=√2GFm3eπ2[ΦA−2meEνM2WΦB], (56)

where the functions and are defined as,

 ΦA = ∞∑l=0(−1)lsinhα[(1+32m2eM2W−eBM2W)(2σK2(σ)−BBcK1(σ)) (57) −BBc(1+m2e2M2W−eBM2W)K1(σ)],

and

 ΦB=∞∑l=0(−1)lcoshα[(8σ2−52BBc)K0(σ)+(2−4BBc+16σ2)K1(σ)σ]. (58)

By taking in Eq. (55) it reduces to Eq. (51). So in the weak field limit we get back the potential for in the medium. Here we have shown only for the W-boson contribution. In Z-exchange diagram we do not have magnetic field contribution. In the tadpole diagram only electron loop will be affected by the magnetic field. But as the momentum transfer is zero, there will not be any higher order contribution. As the magnetic field is weak, the protons and neutrons will not be affected by the magnetic field.

In Fig. 1 we have plotted the potential Eq. (55) as a function of temperature in the range to 10 MeV for a fixed value of the magnetic field . This shows that the potential is an increasing function of temperature. We have also shown in Fig. 2, only the magnetic field contribution, i.e. by subtracting the part from Eq. (55), which shows that, the magnetic field contribution is opposite compared to the medium contribution and also order of magnitude smaller.

### ii.5 Matter Background

Let us consider the background with electrons, positrons, protons, neutrons, neutrinos and anti-neutrinos in the background. As we are considering the magnetic field to be weak, the magnetic field will have no effect on protons and neutrons. For an electron neutrino propagating in this background, we have

 aW⊥ = −√2GFM2W[Eνe{(Ne−¯Ne)−(N0e−¯N0e)} (59)
 be = bW+bZ+bt=b0e+~bW (60) = √2GF[(1+32m2eM2W+E2νeM2W+CVe)(Ne−¯Ne)+(eBM2W−E2νeM2W)(N0e−¯N0e) +CVp(Np−¯Np)+CVn(Nn−¯Nn)+2(Nνe−¯Nνe) +(Nνμ−¯Nνμ)+(Nντ−¯Nντ)−83EνeM2Z(⟨Eνe⟩Nνe+⟨¯Eνe⟩¯Nνe)]+~bW,

and the coefficient of is,

 ce = cW+ct=c0e+~cW (61) = √2GF[(1+m2e2M2W−E2νeM2W−CAe)(N0e−¯N0e)+(eBM2W+E2νeM2W)(Ne−¯Ne)]+~cW,

where and are given in Eqs. (31) and (32). In the weak field limit these two functions are given as

 ~bW=−√2GF2π2(m2eMW)2Eνe∞∑l=0(−1)lcoshα[(6σ2−52BBc)K0(σ)+(3−2BBc+12σ2)K1(σ)σ] (62)

and

 ~cW=√2GF2π2(m2eMW)2Eνe∞∑l=0(−1)lcoshα[2σ2K0(σ)−(1+2BBc−4σ2)K1(σ)σ]. (63)

Similarly for muon and tau neutrinos,

 bμ=b0μ = √2GF[CVe(Ne−¯N