Eur. Phys. J. C (2009) 62: 481–489DOI 10.1140/epjc/s10052-009-1052-9

Regular Article - Theoretical Physics

Neutrino absorption by W production in the presenceof a magnetic field

Kaushik Bhattacharya1, Sarira Sahu2,3,a

1Physics Department, Indian Institute of Technology Kanpur, Kanpur 208 016, Uttar Pradesh, India2Institute of Astrophysics and Particle Physics and Department of Physics, National Taiwan University, Taipei 106, Taiwan3Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., A. Postal 70-543, 04510 Mexico DF,Mexico

Received: 30 October 2008 / Revised: 31 March 2009 / Published online: 28 May 2009© Springer-Verlag / Società Italiana di Fisica 2009

Abstract In this work we calculate the damping rate of theelectron type neutrinos into W bosons and electrons in thepresence of an external uniform magnetic field. The damp-ing rate is calculated from the imaginary part of the W ex-change neutrino self-energy diagram but in the weak fieldlimit, and we compare our result with the existing one.

PACS 14.60.Lm · 95.30.Cq

1 Introduction

The topic of neutrino propagation in various non-trivial me-dia has been widely studied. Particularly the topic dealingwith the passage of neutrinos through the cosmos or throughsome compact object containing a magnetic field is of par-ticular interest. As magnetic fields are ubiquitous in our uni-verse, when neutrinos travel from their point of creation totheir destiny they may feel the effect of the magnetic fields.As the neutrinos we consider are standard model chiral neu-trinos they do not posses any magnetic moment and themagnetic field affects them through virtual charged parti-cles inside the loops of the neutrino self-energy diagrams[1–3]. It is known that in these circumstances the magni-tude of the magnetic field which is required to affect theneutrino properties is relatively large, O(1013) G. Such amagnitude of the magnetic fields is assumed to exist in neu-tron stars and magnetars. Consequently when neutrinos areproduced in the neutron stars their properties can get mod-ified because of the magnetic field. There have been manyattempts to calculate the dispersion relation of neutrinos inthe presence of a magnetized medium, which employs thereal part of the neutrino self-energy. But if we are trying to

a e-mail: sarira@nucleares.unam.mx

find the absorption rate of the neutrinos then the imaginarypart of the neutrino self-energy becomes important. Therehas been one attempt in the past [4], where the authors triedto calculate the absorption coefficient of the neutrinos in amagnetic field background. In their work, Erdas and Lissia[4] calculated the imaginary part of the self-energy of theneutrino, using the W exchange diagram, and calculated theabsorption coefficient of the neutrino when it decays into anelectron e− and W+ pair, in the presence of a magnetic field.The present paper aims to calculate the damping rate of theneutrino due to its decay into an e− and W+ pair, in thepresence of a magnetic field. The quantities named absorp-tion coefficient by Erdas and Lissia and damping rate by thepresent authors turn out to be the same, both related to theattenuation of a traveling neutrino in the presence of a mag-netic field. Our work estimates the damping rate of neutri-nos when the neutrino propagation is damped due to its de-cay νe → e− + W+ in the presence of a magnetic field. Thethreshold of the pair production process νe → νe + e− + e+in the presence of a magnetic field has also been calculated[5, 6], and it is noted that the pair production threshold issmaller than the neutrino decay threshold. For higher ener-gies of the neutrinos, O(103−4 GeV), and for magnetic fieldmagnitude O(1011−12 G), the neutrino decay rate becomesa prominent phenomenon. The latter reaction is kinemati-cally forbidden in vacuum but can take place in the presenceof a magnetic field as in the presence of a magnetic field thedispersion relation of the electron and positron changes. Intheir work Erdas and Lissia [4] calculated the imaginary partof the neutrino self-energy using the Feynman gauge for theW boson propagator. In the present article we calculate theneutrino self-energy using the unitarity gauge [7]. Our resultis different from the result obtained by Erdas and Lissia [4].

The paper is organized as follows: In Sect. 2, we haveshown the relation between the damping rate and the imagi-nary part of the neutrino self-energy. In Sect. 3, we explain

482 Eur. Phys. J. C (2009) 62: 481–489

the terminology of Schwinger’s proper time method andgive the expression for the neutrino self-energy due to W

exchange. The detailed calculation of the imaginary part ofthe self-energy in the weak field limit and the damping rateare evaluated in Sect. 4. The threshold behavior as well asthe asymptotic limit of the damping is also discussed in thissection. A brief conclusion is given in Sect. 5. In Appendixwe put some of the calculational details for the convenienceof the reader.

2 Damping rate of chiral fermions

In this section we briefly describe the main formula of thedamping rate used in this paper. In general, the dispersion re-lation of a massless fermion with a momentum kμ = (k0,k)

is not given by k0 = |k| and in particular, k0 is not zero atzero momentum. We write

k0 = k0r − i

γ

2, (1)

where k0r and γ are real. The quantity M = k0

r (|k| = 0) isinterpreted as an effective mass and γ as the fermion damp-ing rate. With the above convention the probability of thechiral fermion not to decay in a time period t is ∼ e−γ t . Thephysically interesting regime is when γ � k0

r , since other-wise the system would be overdamped and the concept of apropagating mode is not meaningful. If we decompose theself-energy of the chiral fermion in real and imaginary partsas

Σ(k) = ReΣ(k) + i ImΣ(k), (2)

the damping rate turns out to be [8]

γ = − 1

k0uL ImΣ(k)uL, (3)

where the spinors uL,R satisfy the Dirac equation

(k/ − ReΣ(k)

)uL,R = 0. (4)

Equation (3) can also be written as

γ = − 1

2k0r

Im Tr[k/Σ(k)

], (5)

where we treat the spinors as vacuum spinors and do ne-glect their corrections in the nontrivial media. In the presentwork we calculate the imaginary part of the self-energy ofthe standard model neutrino in the presence of a magneticfield and using that result we find the damping rate of theneutrino. To compare with earlier work we represent γ inunits of m−1.

3 The W exchange diagram

If the weak SU(2) case, the coupling constant is g and theleft-chiral projection operator is L ≡ 1

2 (1 − γ5) and then thecontribution to the neutrino self-energy from the W bosonexchange diagram shown in Fig. 1 is given by

−iΣW(k) =∫

d4p

(2π)4

(−i

g√2

)γμL iSe(p)

×(

−ig√2

)γνL iWμν(q) + c.t. (6)

The c.t. is the counter term which has to be fed in the ex-pression of the self-energy of the neutrino as the self-energydiagram contains an ultraviolet divergence. The form of thec.t. is such that it cancels the divergence of the neutrinoself-energy in the B → 0 limit. As the imaginary part ofthe self-energy in (6) is relevant for the calculation of theνe → W+ + e process so the c.t. will not affect our calcu-lations as it does not contribute to the imaginary part of theself-energy. Consequently in this article we will not mentionc.t. any further and write the self-energy of the neutrino inthe presence of a magnetic field as appearing in (6) withoutthe counter term.

Se(p) is the electron propagator in the presence of a mag-netized medium and assuming the magnetic field to be alongthe z-axis of the coordinate system and Se(p) is given by

iSe(p) =∫ ∞

0ds1 eφ(p,s1)G(p, s1). (7)

The possible phase term in the electron propagator is notwritten as it will disappear in the one-loop calculation of theneutrino self-energy. In the above equation,

φ(p, s1) = is1(p2

0 − m2) − is1

(p2

3 + p2⊥tan z1

z1

)

− ε|s1|, (8)

where m is the mass of the electron and we have definedz1 = |e|Bs1, p2⊥ = p2

1 + p22 and ε is an infinitesimal para-

meter introduced for the convergence of the integral in (7).Henceforth in this article we will not require the ε term ex-plicitly and so it will not appear in the further discussion.For the sake of writing the electron propagator in a covari-ant form two 4-vectors, uμ and bμ, are used. As the mag-netic field is a frame dependent quantity we assume in therest frame of the observer uμ is given by

uμ = (1,0). (9)

Fig. 1 The W exchangediagram. The solid internalvirtual lines represent themodified propagator in thepresence of an externalmagnetic field

Eur. Phys. J. C (2009) 62: 481–489 483

We have a uniform magnetic field along the z-axis. Likewisethe effect of the magnetic field enters through the 4-vectorbμ which is defined in such a way that in the frame in whichuμ = (1,0) we have

bμ = (0, b), (10)

where we denote the magnetic field vector by Bb. With thehelp of these two 4-vectors the other term appearing in (7)can be written as [3]

G(p, s1) = sec2 z1[A/ + iB/γ5 + m

(cos2 z

− iΣ3 sin z1 cos z1)]

, (11)

where Aμ and Bμ stand for

Aμ = pμ − sin2 z1(p · uuμ − p · b bμ), (12)

Bμ = sin z1 cos z1(p · ubμ − p · b uμ), (13)

and

Σ3 = γ5 /b/u. (14)

The momentum dependence of the propagator is shown inFig. 1. The W boson propagator in the presence of a mag-netic field is Wμν(q) where q = k − p. In the unitary gaugeand in the weak field limit i.e. eB � M2

W , the W -propagatoris given by

Wμν(q) = − 1

(q2 − M2W)

[gμν − 1

M2W

(qμqν + ie

2Fμν

)]

+ 2ieFμν

(q2 − M2W)2

. (15)

The q dependence of the terms in the denominator on theright hand side of the above equation can be expressed inintegral form as

1

(q2 − M2W)

= −i

∫ ∞

0ds2 eis2(q

2−M2W ),

(16)1

(q2 − M2W)2

= −∫ ∞

0ds2 s2 eis2(q

2−M2W ).

For convergence of the above integrals we require anotherterm similar to the ε term in (7) which has not been explicitlywritten as it will never appear in the further discussion. Us-ing Schwinger’s proper time method the contribution to theneutrino self-energy coming from the W exchange processcan be expressed as

ΣW(k) = − ig2

2

∫d4p

(2π)4R γμSe(p)γν LWμν(q)

= − ig2

2

∫d4p

(2π)4eφ(p,s1)R γμ

∫ ∞

0ds1

∫ ∞

0ds2

× G(p, s1)γν L

[−gμν + 1

M2W

(qμqν + ie

2Fμν

)

+ 2es2 Fμν

]eis2(q

2−M2W ). (17)

As these integrals are lengthy and complicated to evaluate,we shall calculate them one by one.

4 ΣW(k) and the damping rate γ

In this section we will evaluate ΣW(k) given in (17). Be-fore writing all the terms in ΣW(k) which will be evaluatedseparately at first we write down the phase appearing in theintegrand on the right hand side of (17). Let us define

Δ = φ(p, s1) + is2(q2 − M2

W

)

= −i(s1m

2 + s2M2W

) + i(s1 + s2

)(x2

0 − x23

)

− iλ(x2

1 + x22

) − iz2

1

z1 + z2

k‖2

|e|B + iz2

2

λ

k⊥2

|e|B , (18)

where

x0 = p0 − k0s2

s1 + s2, x3 = p3 − k3s2

s1 + s2, (19)

x1 = p1 − k1s2

λ, x2 = p2 − k2s2

λ, (20)

and

zi = |e|Bsi , λ = s1tan z1

z1+ s2. (21)

Next the expression of ΣW(k) in (17) is decomposed intovarious parts which are evaluated separately. The decompo-sition of ΣW(k) is as follows:

ΣW(k) = Σ(1)W (k) + Σ

(2)W (k) + Σ

(3)W (k) + Σ

(4)W (k), (22)

where

Σ(1)W (k) = ig2

2

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2 eΔ

× R γμG(p, s1)γνLgμν, (23)

Σ(2)W (k) = − ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2 eΔ

× R γμG(p, s1)γνLqμqν, (24)

Σ(3)W (k) = eg2

4M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2 eΔ

× R γμG(p, s1)γνLFμν, (25)

Σ(4)W (k) = −ieg2

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2s2 eΔ

× R γμG(p, s1)γνLFμν. (26)

484 Eur. Phys. J. C (2009) 62: 481–489

Doing the Gaussian integrals in the momenta we can write(23) as

Σ(1)W (k) = − g2

(4π)2

∫ ∞

0ds1

∫ ∞

0ds2 eΔ0

1

(s1 + s2)λ

× R

[s2

(s1 + s2) k‖ + i

s2

(s1 + s2)(k0b/ − k3u/) tan z1

− s2

λ k⊥ sec2 z1

]L, (27)

where

Δ0 = −iz2

2

(z1 + z2)

k2‖|e|B + i

z22

λ

k2⊥|e|B

− i

|e|B(z1m

2 + z2M2). (28)

The expression in (24) can also be written as

Σ(2)W (k) = − ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2

× eΔ sec2 z1R[2q.(A + iB) q − q2(A/ + iB/)

]L

= Σ(2a)W (k) + Σ

(2b)W (k), (29)

where

Σ(2a)W (k) = − ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2

× eΔ sec2 z1R[2q.(A + iB)q/

]L, (30)

Σ(2b)W (k) = ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2

× eΔ sec2 z1R[q2(A/ + iB/)

]L. (31)

The elaborate expressions of the above integrals are givenin Appendix. The evaluation of Σ

(3)W (k) and Σ

(4)W (k) are

straightforward and completing the Gaussian integrals overthe loop momenta we obtain

Σ(3)W (k) = g2

2M2W

1

(4π)2

∫ ∞

0ds1

∫ ∞

0ds2

× eΔ0s2

(s1 + s2)2λR

[i(k0 b − k3 u)

− tan z1 k‖]L, (32)

and

Σ(4)W (k) = 2eBg2

(4π)2

∫ ∞

0ds1

∫ ∞

0ds2

× eΔ0s2

(s1 + s2)2λR

[i(k0 b − k3 u)

− tan z1 k‖]L. (33)

The damping rate of the neutrino is given by

γ = − 1

2k0Im S(k), (34)

where

S(k) = Tr[k/ΣW(k)

]. (35)

In (34) we have used the free dispersion relation of the neu-trinos to write the neutrino energy in the denominator andconsequently here we have k0

r ∼ k0. The one-loop real partof the energy will differ from k0 by factors proportional tothe Fermi coupling, GF . For simplicity, we can change theintegration variables from (s1, s2) to (z, u) where

z = m2

eB(z1 + z2) = 1

β(z1 + z2), (36)

and

u = M2W

m2

z2

z1 + z2= 1

η

z2

z1 + z2. (37)

With these new variables, (28) can be written as

Δ0 = −iz

[1 − uη + u + u2η

k2⊥M2

W

(1 − βz

Ω

)], (38)

where we have defined

Ω = tanβz(1 − uη) + βzuη. (39)

The phase exp(−iz) in exp(Δ0) is oscillating and its maincontribution to the z integral will come from z ≤ 1. Alsofor the weak field limit the condition is eB � m2. So in theweak field limit the product βz � 1 and we can expand theterms in ΣW(k) up to second order in βz (Appendix con-tains the expression of Σ

(2)W (k)). This gives

Δ0 � −iz

[1 − uη + u + 1

3

k2⊥m2

(eB

M2W

)2

× u2z2(1 − uη)3]. (40)

As the Σi , for i = 1,2,3 and 4, are in the form

Σ(i)(k) = f1k/‖(⊥) + f2(k0b/ − k3u/), (41)

from (35) we see that the trace

Tr[k/(k0b/ − k3u/)

] = 0. (42)

Consequently the terms which are proportional to k/‖(⊥) willonly contribute to the damping of the fermion. Also noting

Eur. Phys. J. C (2009) 62: 481–489 485

that for a massless neutrino k = 0 and k2‖ = k2⊥, we can writeS in the weak field limit, where βz � 1, as

S(k) � −g2k2⊥4π2

∫ ∞

0dz

∫ 1/η

0dueΔ0

[β2η2uz(uη − 1)

3M1

− iβ2η3u(uη − 1)

90M2

], (43)

where we have defined

M1 = 2u4z2β2η4 − 4u3z2β2η3 + u2(2z2β2 − 1)η2

+ 5uη + 2 ,

(44)M2 = z2β2(uη − 1)2 (

100u3η3 − 252u2η2 + 129uη + 8)

+ 15(8u2η2 − 22uη + 11

).

From the above equations we can write

Im S(k) = (gβη)2k2⊥12π2

∫ ∞

0dz

∫ 1/η

0duu (uη − 1)

× (F2 cosD + F1 sinD

). (45)

Here

F1 = z(2 + 5 uη − u2η2) + 2u2η2β2z3(1 − 2 uη + u2η2),

F2 = 1

2η(11 − 22 uη + 8u2η2) + 1

30η(uη − 1)2 (46)

× β2z2(8 + 129uη − 252u2η2 + 100u3η3),

and

D = (1 + u − uη)z + 1

3ξ2u2(1 − uη)3z3, (47)

where we have defined

ξ = k⊥m

(eBM2

W

). (48)

By making the substitution

z = y

(ξu)2/3(1 − uη)(49)

we have

D = xy + 1

3y3, where x = (1 + u − uη)

ξ2/3u2/3(1 − uη). (50)

In the new variable, Im S(k) can be written as

Im S = − (gβη)2k2⊥12π2ξ

23

∫ ∞

0dy

∫ 1/η

0duu

13

×[

F2 cos

(xy + 1

3y3

)

+ F1 sin

(xy + 1

3y3

)]. (51)

In terms of the Airy function,

Ai(x) = 1

π

∫ ∞

0dy cos

(xy + 1

3y3

), (52)

we can write (51) as

Im S(k) = − (gβη)2k2⊥12πξ

23

∫ 1/η

0duu

13

× [(W0 − W2 x + W3)Ai(x)

− (W1 − W3 x)Ai′(x)], (53)

where

W0 = 1

2η(11 − 22 uη + 8u2η2),

W1 = (2 + 5 uη − u2η2)

1 + u − uηx,

(54)

W2 = η(uη − 1)2β2(8 + 129uη − 252u2η2 + 100u3η3)

30(1 + u − uη)2x2,

W3 = 2u2η2β2(1 − 2 uη + u2η2)

(1 + u − uη)3x3,

and Ai′(x) ≡ dAi(x)dx

.The numerical integration over u shows that it saturates

much before reaching the value 1/η, so here uη � 1 and wecan neglect the terms with uη or higher power of it in (53).So this gives

Im S(k) � − (gβη)2k2⊥12πξ

23

∫ 1/η

0duu

13

×[(

11η

2− 4

15

ηβ2

(1 + u)2x3

)Ai(x)

− 2

1 + uAi′(x)

], (55)

with

x � (1 + u)

(ξu)2/3. (56)

With these, the damping rate of the neutrino is given as

γ =√

2GF k2⊥6πk0ξ

23

(eBM2

W

)2

M2W

∫ 1/η

0duu

13

×[η

(11

2− 4

15

β2

(1 + u)2x3

)Ai(x)

− 2

1 + uAi′(x)

]. (57)

As η is much smaller than one, we can henceforth ne-glect the part of the integrand proportional to η and assume

486 Eur. Phys. J. C (2009) 62: 481–489

1/η ∼ ∞. Also here we assume that Eν = k0 � k⊥ by takingk3 very small and we write the above equation as

γ ∼ −√

2

3πGF M2

WEν

( BBc

)2(m

MW

)4

× ξ−2/3∫ ∞

0du

u13

1 + uAi′(x), (58)

where Bc = m2/e is the critical magnetic field. For the trans-verse neutrino energy k⊥ above the threshold of e and W

production, we have observed that the function

ξ−2/3f (ξ) = −ξ−2/3∫ ∞

0du

u13

1 + uAi′(x)

=(

ξ−4/3

√3π

)∫ ∞

0

du

u1/3K2/3

(2

3

(1 + u)3/2

ξu

)

� 1.065. (59)

In (59), we express the derivative of the Airy function interms of the modified Bessel function to compare the u de-pendence of the integrand with (23) of [4], which has a dif-ferent u dependence and due to this our result differs fromthe result by Erdas et al., in the above reference. In (59),the contribution to the integral over u is due to the linearorder term in z which can be seen from (43) and (44) andthis comes from the self-energy term in (23), the term pro-portional to gμν . Also let us emphasize that we are usingthe unitary gauge and the weak field limit eB � M2

W forthe vector boson. On the other hand Erdas et al. have usedthe exact propagator for the W boson in a constant magneticmild and have used the Feynman gauge. But after some sim-plification they also use the same weak field limit.

For a neutrino energy much above the threshold, thedamping rate behaves as

γ � 1.065 ×√

2

3πGF M2

WEν

( BBc

)2(m

MW

)4

, (60)

which is same as the absorption coefficient calculated by Er-das et al. (except for the factor 1.065) in [4]. In Fig. 2, wehave shown the behavior of (59) as a function of the neu-trino energy. For large k⊥, it saturates at 1.065 and belowthe energy threshold the function ξ−2/3f (ξ) increases veryrapidly and saturates very fast. Because of this behavior, thedamping rate also increases very rapidly from very smallvalue and after crossing the threshold the behavior becomeslinear in neutrino energy. For small magnetic field, the sat-uration energy is larger than the one with larger magneticfield. As can be seen from the figure, for B/Bc = 0.1, thecurve saturates around k⊥ � 1019 eV, whereas for B/Bc =0.001 the energy is about 1021 eV. For the neutrino energymuch above the threshold the asymptotic behavior is given

Fig. 2 The (59) plotted as a function of neutrino energy Eν = k⊥ fordifferent B. From left to right, the curves are for B/Bc =0.1, 0.01 and0.001 respectively

Fig. 3 The damping rate is expressed in units of meter inverse; it isplotted against the transverse energy of the neutrino Eν . From the topto bottom are the curves for B/Bc = 0.1, 0.01 and 0.001

in (60), where the damping rate is proportional to the neu-trino energy and also it depends quadratically on the mag-netic field as shown in the above equation. The behaviorcan be seen from Fig. 3, where we have plotted the damp-ing rate as a function of energy for three different values(B/Bc = 0.1, 0.01 and 0.001) of the magnetic field. Againthe larger the magnetic field, the larger is the damping rate;and for neutrino energy smaller than the threshold one, thedamping rate is suppressed but increases rapidly by increas-ing the energy.

The behavior of the process νe → e− + W+ solely de-pends on ξ−2/3f (ξ) and the threshold condition satisfiesξ−2/3f (ξ) < 1. We have analyzed the threshold behaviorof the above process by taking into account the behavior ofthe functions ξ−2/3, f (ξ) and their product as a function ofneutrino energy, which are shown in Fig. 4 for B/Bc = 0.1and we accurately find the threshold energy. With increas-ing energy the function ξ−2/3 decreases and f (ξ) increasesand both of them intersect at a point ∼ 0.66 which lies be-fore the saturation point of their product and this is the point

Eur. Phys. J. C (2009) 62: 481–489 487

Fig. 4 The functions ξ−2/3, f (ξ) and ξ−2/3f (ξ) are plotted as func-tion of Eν for B/Bc = 0.1. The extreme left decreasing curve is thefunction ξ−2/3, the second increasing curve is f (ξ) and the third onewhich is saturated around 1 is the function ξ−2/3f (ξ)

of threshold. The estimated neutrino energy at this point isEν = 2.37 × 1017 eV. We have observed that, for B/Bc =0.01 and 0.001, the intersection point of the two functionsas described above remains the same (0.66) but the neutrinoenergy shift towards higher values. For B/Bc = 0.01, we ob-tain Eν = 2.37 × 1018 eV and 2.37 × 1019 eV, respectively.For νμ and ντ decay calculations the electron mass has to bereplaced by the corresponding lepton mass. At threshold weobtain

ξ = 1.87

( BBc

), (61)

or in terms of the above, we can write the threshold neutrinoenergy as

Ethν = 2.36 × 1016

( Bc

B

)eV. (62)

We have compared our result with the result in [4] to showthe difference. The ratio of the damping rates R = γour/γEL

(γEL for Erdas and Lissa of [4]) is plotted in Fig. 5. In this

figure we have also plotted [(1+√3

M2W Bc

mk⊥B )exp(−√3

M2W Bc

mk⊥B )]of (25) of [4] to show the difference with our result. One canclearly see that the two functions are different in both smalland large values of ξ . In the asymptotic limit of ξ (whichimplies large Eν ) our result saturates at 1.065, whereas theresult by Erdas et al. saturates at 1. So the behavior of thefunction ξ−2/3f (ξ) makes our result different from [4] forall the values of ξ .

By knowing the magnetic field, the threshold neutrino en-ergy can easily be calculated from (62). We can also cal-culate the mean free path (l � γ −1) of the neutrino at thethreshold, where the function ξ (−2/3)f (ξ) � 0.66, and ifthe magnetic field is known. For example let us assume thatthe mean free path of the neutrino to be the neutron star ra-

Fig. 5 The ratio R = γour/γEL, which is defined in the text and the

function [(1 + √3

M2W Bc

mk⊥ B ) exp(−√3

M2W Bc

mk⊥ B )] of (25) of [4] are plottedas function of p (where ξ = 10p). The first curve from the top is theratio R

dius (l � 10 km). Then to satisfy this condition, the mag-netic field has to be B ∼ 3 × 109 G which corresponds toEth

ν � 3.4 × 1020 eV.

5 Conclusion

By using Schwinger’s proper time electron propagator wecalculate the imaginary part of the neutrino self-energy ina constant magnetic field background by using the unitarygauge. In this calculation we assume the magnetic field tobe weak, eB � m2. From the imaginary part of the neutrinoself-energy we calculate the damping rate i.e. conversion ofneutrino into an electron and W boson. We explicitly evalu-ated all the contributions in the weak field limit. We foundthat the behavior of the process νe → e− + W+ solely de-pends on the quantity ξ−2/3f (ξ). The threshold energy forthis process can be accurately determined from the intersec-tion of the functions ξ−2/3 and f (ξ) as functions of the en-ergy. Also for the neutrino energy much above the thresholdwe found that the damping rate is proportional to the neu-trino energy and also it depends quadratically on the strengthof the magnetic field. Our calculation gives an alternativeway to arrive at the absorption cross section calculated pre-viously in [4]. Also we have shown that the calculation ofdamping rate in the unitary gauge is different from the oneobtained by Erdas and Lissa [4] in the Feynman gauge.

Acknowledgements We are thankful to the anonymous referee forhis valuable comments and suggestions. This research is partially sup-ported by DGAPA-UNAM (Mexico) project IN101409.

488 Eur. Phys. J. C (2009) 62: 481–489

Appendix: The integral representationof the W exchange diagram

The expression in (24) can also be written as

Σ(2)W (k) = − ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2 eΔ sec2 z1

× R[2q.(A + iB) q − q2(A/ + iB/)

]L

= Σ(2a)W (k) + Σ

(2b)W (k), (A.1)

where

Σ(2a)W (k) = − ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2 eΔ sec2 z1

× R[2q.(A + iB)q/

]L, (A.2)

Σ(2b)W (k) = ig2

2M2W

∫d4p

(2π)4

∫ ∞

0ds1

∫ ∞

0ds2 eΔ sec2 z1

× R[q2(A/ + iB/)

]L. (A.3)

The term q.(A+ iB)q/ in the integrand on the right hand sideof (A.2) can be written in terms of x0, x1, x2, x3 as

q.(A + iB)q/

=[d0 − x2

0 cos2 z1 + λ0x0 + x21 + x2

2 + x23 cos2 z1

+ (x1k1 + x2k2)

λ

(s2 − s1

tan z1

z1

)− λ3x3

]

× (a0 − γ0x0 + γ1x1 + γ2x2 + γ3x3), (A.4)

where

a0 = s1

s1 + s2k‖ − s1

λ

tan z1

z1k⊥,

d0 = s1s2

(s1 + s2)2k‖2 cos2 z1 − s1s2

λ2

tan z1

z1k⊥2,

(A.5)

λ0 = (s1 − s2)

(s1 + s2)k0 cos2 z1 + ik3 sin z1 cos z1,

λ3 = (s1 − s2)

(s1 + s2)k3 cos2 z1 + ik0 sin z1 cos z1.

In a similar fashion the term q2(A/+ iB/) in (A.3) can be writ-ten as

q2(A/ + iB/) =[g0 + x2

0 − x21 − x2

2 − x23 − 2s1

(s1 + s2)x0k0

+ 2s1

λ

tan z1

z1(x1k1 + x2k2) + 2s1

s1 + s2x3k3

]

× (h + h0x0 − γ1x1 − γ2x2 − h3x3), (A.6)

where

g0 =(

s22

(s1 + s2)2− 2s2

(s1 + s2)

)k‖2 −

(s2

2

λ2− 2s2

λ

)k⊥2,

h = s2

s1 + s2 k‖ cos2 z1 − s2

λ k⊥

+ is2

s1 + s2sin z1 cos z1(k0b/ − k3u/), (A.7)

and

h0 = cos2 z1u/ + ib/ sin z1 cos z1,

(A.8)h3 = cos2 z1b/ + iu/ sin z1 cos z1.

Using the expressions of q.(A + iB)q/ and q2(A/ + iB/ ) in(A.2) and (A.3) and doing the xi integrals we obtain

Σ(2a)W (k)

= − g2

M2W

1

(4π)2

∫ ∞

0ds1

∫ ∞

0ds2 eΔ0

sec2 z1

(s1 + s2)λ

× R

[k/‖

{(s2

1s2

(s1 + s2)3k2‖ cos2 z1

− s21s2

(s1 + s2)λ2k2⊥

tan z1

z1

)

− i

(2s1

(s1 + s2)λ+ (3s1 − s2)

(s1 + s2)2cos2 z1

)}

+ k/⊥{(

s21s2

λ3

tan2 z1

z21

k⊥2 − s21s2

(s1 + s2)2λ

sin z1 cos z1

z1

)

+ i

(2s1

(s1 + s2)λ

sin z1 cos z1

z1

+ 1

λ2

(3s1

tan z1

z1− s2

))}

+ (k3u/ − k0b/)sin z1 cos z1

s1 + s2

]L, (A.9)

and

Σ(2b)W (k)

= g2

2M2W

1

(4π)2

∫ ∞

0ds1

∫ ∞

0ds2 eΔ0

sec2z1

(s1 + s2)λ

× R

[k/‖

{(− (2s1 + s2)

(s1 + s2)3k2‖

+ s22

(s1 + s2)λ2

(2s1

tan z1

z1k2⊥

))cos2 z1

Eur. Phys. J. C (2009) 62: 481–489 489

− 2i cos2 z1

(s1 − s2

(s1 + s2)2− s2

(s1 + s2)λ

)}

+ k/⊥{(

− s22

λ3

(2s1

tan z1

z1k2⊥

)k2‖ + s2

2

λ

(2s1 + s2)

(s1 + s2)2

)

− i

(2s2

(s1 + s2)λ+ 2s2

λ2− 2s1

λ

tan z1

z1

)}

+ (k0b/ − k3u/)s2

s1 + s2sin z1 cos z1

×{(

2s1

s2(s1 + s2)− 2

(s1 + s2)− 2

λ

)

− i

(s2(2s1 + s2)

(s1 + s2)2k2‖ − 2s1s2

λ2

tan z1

z1k2⊥

)}]L. (A.10)

This results are used to evaluate Σ(2)W (k). Expressing the

derivative of the Airy function in terms of the Bessel func-tion as

Ai′(x) = − 1

π

x√3K2/3

(2

3x3/2

), (A.11)

we can express the damping rate as

γ =√

2

3√

3

1

π2GF

k2/3⊥k0

(|e|B)2/3(

m2MW

)2/3

×∫ ∞

0

du

u1/3K2/3

(2

3

(1 + u)3/2

ξu

)

=√

2

3πGF M2

Wk⊥( B

Bc

)2(m

MW

)4

×(

1√3πξ4/3

)∫ ∞

0

du

u1/3K2/3

(2

3

(1 + u)3/2

ξu

).

(A.12)

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