electromagnetic vertex of neutrinos in an electron background and a magnetic field

PHYSICAL REVIEW D 68, 113003 ~2003!

Electromagnetic vertex of neutrinos in an electron background and a magnetic field

JoseF. NievesLaboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, P.O. Box 23343, Rı´o Piedras,

Puerto Rico 00931-3343, USA~Received 19 September 2003; published 15 December 2003!

We study the electromagnetic vertex function of a neutrino that propagates in an electron background in thepresence of a static magnetic field. The structure of the vertex function under the stated conditions is deter-mined and it is written down in terms of a minimal and complete set of tensors. The one-loop expressions forall the form factors are given, up to terms that are linear in the magnetic field, and the approximate integralformulas that hold in the long wavelength limit are obtained. We discuss the physical interpretation of some ofthe form factors and their relation with the concept of theneutrino induced charge. The neutrino acquires alongitudinal and a transversecharge, due to the fact that the form factors depend on the transverse andlongitudinal components of the photon momentum independently. We compute those form factors explicitly invarious limiting cases and find that the longitudinal and transverse charges are the same for the case of anonrelativistic electron gas, but not otherwise.

DOI: 10.1103/PhysRevD.68.113003 PACS number~s!: 12.20.Ds, 13.15.1g

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I. INTRODUCTION AND SUMMARY

In the study of the electromagnetic properties of neutrinin a medium, the two quantities of particular interest areneutrino self-energy in the presence of an external magnfield and the neutrino electromagnetic vertex function. Tformer quantity determines the neutrino dispersion relatas it propagates in a magnetized medium, while the ladetermines the relevant couplings in the calculation ofneutrino electromagnetic processes that occur in the medsuch as plasmon decay and Cherenkov radiation, amongers. The electromagnetic couplings of the neutrino canviewed in terms of effective dipole moments and an eleccharge induced by the medium@1,2#. Although such identi-fications reveal some of the properties of the neutrino, bthe intrinsic ones as well as those that result from its inaction with matter, for the practical calculation of the trantion rates of the various processes the full electromagnvertex function is needed. Since the original calculationsthat quantity more than a decade ago@3–5#. Some of thepossible physical effects have been considered over thefew years@6–16#. In addition, the calculations have beeimproved and refined using a variety of methods and teniques@17–20#.

Recently, there has been interest on another closelylated quantity, namely the electromagnetic coupling ofneutrino when it propagates in a medium in the presencemagnetic field. The main issue here is to determine howneutrino electromagnetic processes take place in a medbut in the absence of a magnetic field, are modified whemagnetic field is present. From a technical point of view,relevant quantity is the neutrino electromagnetic vertex fution, which must be calculated taking the presence ofmagnetic field into account.

This problem has been approached from two pointsview. On one hand, the vertex function has been calculataking the magnetic field into account, but neglectingeffects of the matter medium@21#. On the other hand, theeffect of both the magnetic field and the medium have b

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taken into account@22#, but only for calculating the couplingthat was identified as the induced neutrino charge, andfor the calculation of the full vertex function. Thus thedoes not exist a complete calculation of the full vertex funtion that takes into account the simultaneous presence omagnetic field and the background medium.

In this work we have taken precisely this problem. Ogoal is to calculate the complete one-loop electromagnvertex function, including the effects of the medium and tlinear terms in the magnetic field. To this end, we determthe most general form of the neutrino vertex function in tphysical environment and under the conditions that weconsidering. The vertex function is written in terms ofcomplete and minimal set of tensors, which are consiswith the requirements that follow from the chiral naturethe neutrino interactions and electromagnetic gauge invance. Such a decomposition is generally useful in its oright, and it is also a useful reference point for the actcomputation of the vertex function. Thus, we give the fomulas for all the corresponding form factors in terms of mmentum integrals over the background electron distributfunctions. For arbitrary values of the photon momentuand/or general distribution functions the integrals are nmally not doable, but they can be evaluated for various liiting cases and approximations that correspond to realsituations. A particularly useful one that we consider is tso-called long wavelength limit, which is valid when thephoton energy and momentum are smaller than the typenergy of an electron in the gas. In this limit the integrathat appear in the formulas for the form factors simplify byet they remain valid for general forms of the momentudistributions, and for a wide range of values of the photenergy and momentum subject only to the restriction staabove. The simplified integral expressions so obtained caused to compute explicitly the form factors for different knematical regimes, such as the static limit~taking the photonenergy to zero!, and for various conditions of the backgrounsuch as the classical or degenerate gas.

We observe that, apart from the photon energy, the fo

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JOSEF. NIEVES PHYSICAL REVIEW D68, 113003 ~2003!

factors depend separately on the perpendicular and parcomponents of the photon momentum. That is to say, ifdenote those three quantities byv,Q' ,Qi , respectively, theform factors depend on those three variables and not jusv and Q25Q'

2 1Qi2 as it is the case in the absence of t

magnetic field. A consequence of this is that, in contrasthe latter case, there is no unique meaning to the concethe induced neutrino charge, or in fact any other electromnetic moment, when the magnetic field is non-zero. Briethe reason is that the zero~photon! momentum limit of thestatic form factors~i.e., evaluated forv50) can be taken intwo different ways, according to whether we setQ'50 firstandQi→0 afterwards or the other way around. If we inson identifying a quantity such as the induced charge by loing at the static vertex function in the zero~photon! momen-tum limit, we are thus forced to define two different quanties, which in the specific case of the neutrino induced chawe denote byen

i anden' . We stress that the form factors hav

a different value in the two limits, and this is not a matematical ambiguity but actually a reflection of the fact thin the presence of the magnetic field, the two ways of takthe zero momentum limit correspond to different physisituations. As an illustration, and to further clarify somethese issues, we compute explicitly the form factors thatrelated toen

i anden' . As we show,en

i anden' have the same

value for the case of a nonrelativistic electron gas, but tare given by different formulas otherwise. In order to estlish contact with previous calculations@22#, the quantity cal-culated there is identified with what we callen

i . However,some differences in the results are found and are noted.

In what follows we present the details of the calculatiodivided as follows. An essential ingredient is thelinearizedform of the thermal propagator for an electron in a magnefield. In Sec. II, the linearized form is obtained from thSchwinger formula for the electron propagator, generalito include the effects of the background medium, andproximating it up to terms that are linear in the magnefield. In Sec. III, the linearized propagator is used to obtthe one-loop formula for the neutrino electromagnetic verup to terms that are linear in the magnetic field. In Sec.we determine the structure of the vertex function and decpose it in terms of a minimal set of tensors. The end resulthis procedure is a set of formulas for the corresponding fofactors, expressed as integrals over the momentum distrtion functions of the electrons in the medium. In Sec. V, tapproximate formulas that are valid in the long wavelenlimit are derived first, and they are then used to evaluexplicitly the form factors that are related toen

i and en' for

various specific cases. Section VI contains our conclusioIn addition there are two Appendixes. The general decomsition of the vertex function is guided by several requiments, such as the transversality condition which of coursa general consequence of the electromagnetic gauge inance. However, it is neither obvious to see nor trivialprove that the one-loop expression for the vertex functsatisfies the transversality condition. The proof that it in fdoes so is supplied in those Appendixes.

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II. THE LINEARIZED ELECTRON PROPAGATOR

In the notation of Ref.@23#, the Schwinger propagator fothe electron in a magnetic field can be written in the form

iSF~p!5E0

`

dtG~p,s!ei tF(p,s)2et, ~2.1!

where the variables is defined by

s5ueuBt, ~2.2!

and the exact formulas forG(p,s) andF(p,s) are given inEq. ~3.4! of that reference. The linear approximation that wpropose to use consists in retaining up to terms that are linin B. To this order,

F5p22m2,

G5p”1m1 i tueuBGB , ~2.3!

where

GB5g5@~p•b!u”2~p•u!b”1mu”b” #, ~2.4!

so that

SF5S01SB , ~2.5!

where

iS05 ip”1m

p22m2,

iSB52 i

~p22m2!2~ ueuBGB!. ~2.6!

The vectorum that appears in Eq.~2.4! is the velocity four-vector of the medium which, in the frame of referencewhich the medium is at rest, takes the form

um5~1,0!. ~2.7!

We have also introduced the vectorbm which is such that, inthat frame,

bm5~0,b!, ~2.8!

where the magnetic field is given by

BW 5Bb. ~2.9!

The thermal electron propagator which, in addition to tmagnetic field, incorporates the effect of the electron baground, is given by

Se5SF2@SF2SF#he , ~2.10!

where

he~p•u!5u~p•u! f e~p•u!1u~2p•u! f e~2p•u!,~2.11!

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ELECTROMAGNETIC VERTEX OF NEUTRINOS IN AN . . . PHYSICAL REVIEW D68, 113003 ~2003!

with

f e~x!51

eb(x2me)11,

f e~x!51

eb(x1me)11. ~2.12!

Here b stands for the inverse temperature andme the elec-tron chemical potential. Using Eqs.~2.5!, ~2.6!, ~2.4!, thecomplete propagator can be expressed as

Se5S01SB1ST1STB , ~2.13!

where

iST522pd~p22m2!he~p”1m!,

iSTB522pd8~p22m2!he~ ueuBGB!, ~2.14!

whereGB is given in Eq.~2.4! and the notationd8 denotesthe derivative of the delta function with respect to its argment.

Although we will not give the details here, it is reassurito mention that when this linearized propagator is usedcompute the self-energy diagrams for the neutrino in an etron gas, the usual expression for the magnetic field conbution to the neutrino dispersion relation, or equivalentlyindex refraction, is reproduced.

III. CALCULATION OF THE VERTEX FUNCTION

The off-shell neutrino electromagnetic vertex functionGm

is defined such that the matrix element of the electromnetic current between neutrino states with momentak andk8is given by

^k8u j m~0!uk&5u~k8!Gmu~k!. ~3.1!

To order 1/MW2 , the one-loop diagrams that are relevant

the calculation ofGm are shown in Fig. 1. Their contributionare

2 iGn(W)~q!52S eg2

2MW2 D E d4p

~2p!4gmL

3 iSe~p1q!gniSe~p!gmL,

2 iGn(Z)~q!5S eg2

4cos2uWMZ2D gmLE d4p

~2p!4

3Tr@ iSe~p1q!gniSe~p!gm~ae1beg5!#, ~3.2!

where

q5k82k ~3.3!

is the momentum of the incoming photon,L5 12 (12g5) and,

in the standard model

11300

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ae52 12 12sin2uW , ~3.4!

be5 12 . ~3.5!

Using the identity

gmLMgmL52Tr~MgmL !gmL, ~3.6!

which holds for any 434 matrix M, the charge current contribution can be put in the same form as the neutral currone, namely

2 iGn(W)~q!5S eg2

2MW2 D gmLE d4p

~2p!4

3Tr@ iSe~p1q!gniSe~p!gmL#. ~3.7!

When Eq.~2.13! is substituted in the above expressionsGn

(W,Z) , several terms are generated and we want to reonly those that, schematically, involve the productsSTBS0 orSTSB , i.e., the terms that depend both onB and the tempera-ture. Proceeding in this way, we then have

Gn(W)52S eg2

4MW2 D gmL~Tmn

(V)1Tmn(A)!,

Gn(Z)52S eg2

4MW2 D gmL~aeTmn

(V)2beTmn(A)!, ~3.8!

FIG. 1. One-loop diagrams for the neutrino electromagnetic vtex in an electron background, to order 1/MW

2 .

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where

Tmn(V)5 i E d4p

~2p!4Trgm@S0~p8!gnSTB~p!1STB~p8!gnS0~p!

1ST~p8!gnSB~p!1SB~p8!gnST~p!#, ~3.9!

Tmn(A)5 i E d4p

~2p!4Trg5gm@S0~p8!gnSTB~p!

1STB~p8!gnS0~p!1ST~p8!gnSB~p!

1SB~p8!gnST~p!#, ~3.10!

and we have defined

p85p1q. ~3.11!

The vertex function for the electron neutrino isGn(W)

1Gn(Z) , while for the other neutrino flavors it is jus

Gn(Z) . Denoting a given neutrino flavor by,5e,m,t, then

the neutrino vertex function is

Gn(,)52S eg2

4MW2 D gmL~xV

(,)Tmn(V)1xA

(,)Tmn(A)!, ~3.12!

where

xV(,)5d,,e1ae ,

xA(,)5d,,e2be . ~3.13!

All the four terms in Eq.~3.9! involve the traces

4Lmn(1)5Tr gm~p

”81m!gnGB~p!,

4Lmn(2)5Tr gmGB~p8!gn~p

”1m!,

~3.14!

and analogously in Eq.~3.10!,

4Kmn(1)5Tr g5gm~p

”81m!gnGB~p!,

4Kmn(2)5Tr g5gmGB~p8!gn~p

”1m!.

~3.15!

It is useful to observe thatLmn(2)5Lnm

(1)(p8↔p) and Kmn(2)

5Kmn(1)(p↔p8). Then, by standard Dirac matrix algebra w

find

Lmn(1)5 i emnab~p•up8a bb2p•bp8aub2m2uabb!,

Lmn(2)5 i emnab~2p8•u pa bb1p8•b paub1m2uabb!,

Kmn(1)5p•u~pm8 bn1bmpn82p8•bgmn!

2p•b~pm8 un1umpn82p8•ugmn!

2m2~umbn2bmun!,

11300

Kmn(2)5p8•u~pmbn1bmpn2p•bgmn!

2p8•b~pmun1umpn2p•ugmn!

2m2~umbn2bmun!. ~3.16!

Going back to Eqs.~3.9! and ~3.10! and using the aboveformulas for the traces, we obtain

Tmn(V)54ueuBE d4p

~2p!3he~p•u!H 2Lmn

(1)d8~p22m2!

~p1q!22m2

1Lmn

(2)d~p22m2!

@~p1q!22m2#22~q→2q!J ,

Tmn(A)54ueuBE d4p

~2p!3he~p•u!H 2Kmn

(1)d8~p22m2!

~p1q!22m2

1Kmn

(2)d~p22m2!

@~p1q!22m2#21~q→2q!J . ~3.17!

In arriving at this expression we have used the standard tof making the change of variablesp→p2q in those termswhere the factorhe(p8) appears, together with the propertie

Lmn(1)~p→p2q!52Lmn

(2)~q→2q!,

Lmn(2)~p→p2q!52Lmn

(1)~q→2q!,

Kmn(1)~p→p2q!5Kmn

(2)~q→2q!,

Kmn(2)~p→p2q!5Kmn

(1)~q→2q!, ~3.18!

which can be seen by inspection of Eq.~3.16!.

IV. STRUCTURE OF THE VERTEX FUNCTION

In the context of the one loop calculation that we aconsidering, the conservation of the electron vector currimplies that the tensorsTmn

(V,A) satisfy the transversality relations

qnTmn(V)5qmTmn

(V)50,

qnTmn(A)50. ~4.1!

It is neither immediately evident nor trivial to prove that thone-loop formulas given in Eq.~3.17! satisfy such relationsbut, as we show explicitly in Appendixes A and B, they dSimilarly, there we also verify that, in contrast to the sitution without the magnetic field, in the present caseqmTmn

(A)

50.To exploit Eq. ~4.1! we introduce the following set o

mutually orthogonal vectors:

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um5gmnun,

bm5gmnbn2q•u

q2~q•ubm2q•bum!,

t m5emabguabbqg, ~4.2!

where

gmn[gmn2qmqn

q2. ~4.3!

These vectors are transverse toqm and satisfy the closurerelation

umun

u21

bmbn

b21

t m t n

t 25gmn . ~4.4!

It is useful to note thatbm and t m also satisfy

b•u5 t •u50. ~4.5!

A. The structure of Tµn„V…

The transversality ofTmn(V) with respect to both indices

implies that it can be expanded in terms of the nine bilinproducts that can be formed out of the set of vectors$u,b, t %.Therefore, we define the following three diagonal tensors

Qmn5umun

u2,

Bmn5bmbn

b2,

Rmn5gmn2Qmn2Bmn , ~4.6!

the three symmetric combinations

S1mn5bm t n1~m↔n!,

S2mn5um t n1~m↔n!,

S3mn5umbn1~m↔n!, ~4.7!

and the three antisymmetric ones

A1mn5bm t n2~m↔n!,

A2mn5um t n2~m↔n!,

A3mn5umbn2~m↔n!. ~4.8!

A1mn andA2mn can be expressed in a more convenient foby using the relations

11300

r

unemabguabbqg2~m↔n!5u2emnabbaqb

bnemabgbaubqg2~m↔n!5b2emnabuaqb, ~4.9!

which follow from contracting the identity

glnemgab2glmengab1glgenmab2glaenmgb1glbenmga50~4.10!

with ulunbaqb in one case and withblbnuaqb in the other.Therefore, instead of the antisymmetric combinationsA1,2,we will use instead

P1mn[ i emnabuaqb,

P2mn[ i emnabbaqb. ~4.11!

Two additional properties of the one-loop formula fTmn

(V) , which follow from inspection of Eq.~3.17!, are~i! it isantisymmetric and~ii ! it transforms as a pseudo-tensor undparity. Out of the set of nine tensors defined above, onlyP1

and P2 share both properties. It is then clear that, to torder, the structure ofTmn

(V) is simply

Tmn(V)5T1

(V)P1mn1T2(V)P2mn . ~4.12!

The form factorsT1,2(V) are, in general, functions of the thre

scalar variablesv, Qi andQ' which are defined by

v5q•u,

Qi52q•b,

Q'5AQ22Qi2, ~4.13!

where

Q25uQW u25v22q2. ~4.14!

These variables correspond to the photon energy in theframe of the medium and the components of the photon mmentumQW that are parallel and perpendicular, respectiveto BW in this frame. It is important to stress that in generbesidesv, the form factors introduced in Eq.~4.12! dependon the variablesQ' andQi , separately. This contrasts witthe situation without the magnetic field, in which case tform factors depend only onv andQ. As we will see in Sec.V B, this requires some extra care if we want to interpretform factors in terms of the static electromagnetic momeof the neutrino.

The form factors are determined by contracting both siof Eq. ~3.17! with the tensorsP1,2 and using the relations

P12522Q2,

P22522Q2

Q'2

q2,

P1P250, ~4.15!

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with Pi Pj5 PimnPj mn . In this way, we obtain

T1(V)5

21

Q2T1

(V) ,

T2(V)5

1

Q'2 S T2

(V)1vQi

Q2T1

(V)D , ~4.16!

whereT1,2(V) are the integrals

Ti(V)54ueuBE d4p

~2p!3he~p•u!H 2Li

(1)d8~p22m2!

~p1q!22m2

1Li

(2)d~p22m2!

@~p1q!22m2#21~q→2q!J , ~4.17!

with

L1(1)5~pu1qu!K1pb~p•q1q2!2m2qb ,

L1(2)52puK2~pb1qb!p•q1m2qb ,

L2(1)5~pb1qb!K1pu~p•q1q2!2m2qu ,

L2(2)52pbK2~pu1qu!p•q1m2qu .

~4.18!

In writing the last equation we have introduced the facto

K5puqb2pbqu , ~4.19!

and used the shorthand notation

pu5p•u,

pb5p•b, ~4.20!

and similarly forqu andqb .

B. The structure of Tµn„A…

SinceTmn(A) satisfiesqnTmn

(A)50 but it is not transverse inthe indexm, it can in principle contain additional terms thare proportional to the tensorsqm t n,qmbn,qmun. However,an inspection of Eq.~3.17! reveals thatTmn

(A) is a true tensor

rather than a pseudo-tensor and therefore the termqm t n can-not be present, and in addition that there is no term proptional to bmbn , which is a consequence of the calculatibeing up to linear terms inB. ConsequentlyTmn

(A) is of theform

Tmn(A)5TL

(A)Qmn1TT(A)~ gmn2Qmn!1TA

(A)A3mn1TS(A)S3mn

1Tu(A)qmun1Tb

(A)qmbn , ~4.21!

where the integral formulas for the form factors are obtainby projecting Eq.~3.17! with the tensors that appear in thexpansion. In this way, the form factors are given by

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r-

d

TX(A)54ueuBCXE d4p

~2p!3he~p!H 2KX

(1)d8~p22m2!

~p1q!22m2

1KX

(2)d~p22m2!

@~p1q!22m2#21~q→2q!J ~X5L,T,S,A!,

TX(A)54ueuBCXE d4p

~2p!3he~p!H 2KX

(1)d8~p22m2!

~p1q!22m2

1KX

(2)d~p22m2!

@~p1q!22m2#22~q→2q!J ~X5u,b!,

~4.22!

where

CA5CL52CT51

u2,

CS5S 1

u2b2D ,

Cu5S 1

q2u2D ,

Cb5S 1

q2b2D , ~4.23!

and

KL(1)52puK82u2K,

KL(2)52pu~K82qb!1u2K,

KT(1)522puK82u2K,

KT(2)522pu~K82qb!1u2K,

KS(1)5pbK81pupub•b,

KS(2)5pb~K82qb!1pu~pu1qu!b•b,

Ku(1)5~p•q1q2!K81puK1m2qb ,

Ku(2)5p•q~K82qb!1puK1m2qb ,

Kb(1)5pbK1b•b@pu~p•q1q2!2m2qu#,

Kb(2)5pbK1b•b@~pu1qu!p•q2m2qu#,

KA(1)5KA

(2)52m2, ~4.24!

whereK is defined in Eq.~4.19!,

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K85u•bpu2u2 pb , ~4.25!

and, in addition to Eq.~4.20!, we have used the shorthannotation

pb5p•b,

pu5p•u. ~4.26!

In writing Eq. ~4.24! we have used the fact thatum and bm

are orthogonal toqm, as well as the relationsu•b50 andu2 b•b5b2.

C. Generic formulas

The integrals involved to evaluate the form factors arethe generic form

I 15~21!E d4p

~2p!3he~p!

F1~p,q!d8~p22m2!

~p1q!22m2,

I 25E d4p

~2p!3he~p!

F2~p,q!d~p22m2!

@~p1q!22m2#2. ~4.27!

as-

-

wefri-

algte

-he

11300

f

The various form factors differ only in the choice of thfunctionsF1,2. In I 2, the integration overp0 can be carriedout trivially, and

I 25E d3p

~2p!32EF F2~p,q! f e

@q212p•q#21

F2~2p,q! f e

@q222p•q#2GUp05E(upW u)

,

~4.28!

where

E~p!5Ap22m2. ~4.29!

Using the relation

]md~p22m2!52pmd8~p22m2!, ~4.30!

the variablep0 can also be integrated out inI 1 by writing

d8~p22m2!51

2p•uum]md~p22m2!, ~4.31!

followed by an integration by parts. Thus,

I 15umE d4p

~2p!3d~p22m2!]mF S 1

2p•uD heF1~p,q!

~p1q!22m2G5umE d3p

~2p!32EUH ]mF S 1

2p•uD S F1~p,q! f e

~p1q!22m21

F1~2p,q! f e

~p2q!22m2D G J Up05E(upW u)

. ~4.32!

est

y-

In this fashion, we can write down the integral formulfor all the form factors in terms of a pair of threedimensional momentum integrals of the form ofI 1,2, withthe appropriate choices of the functionsF1,2 that can be readoff directly from Eqs.~4.17! and~4.22!. There are no furthermanipulations that can be made to the integralsI 1,2 that willallow us to evaluate the form factors in a general way. Hoever, as we show below, the above formulas are a usstarting point to compute explicitly the form factors in vaous limiting cases.

V. APPROXIMATE FORMULAS

A. Long wavelength limit

A useful approximation, which is valid in many practicsituations, results from taking the so-called long wavelenlimit. The approximation is valid in the regime in which thphoton momentumq!^E&, where^E& denotes a typical average energy of the particles in the background. The metto obtain the approximate formulas in this limit is the samthat was used in Ref.@4# and we will therefore omit heresome of the details.

-ul

h

od

For the remainder of this section we setum5(1,0W ), sothat all the kinematical variables refer to the medium rframe as usual. Let us consider firstI 2. The method involves

making the substitutionspW →pW 7 12 QW in the two terms in the

integrand of Eq.~4.28!, respectively, and then making a Talor expansion in powers ofq/E. A useful auxiliary formulathat is obtained as we have described is

E d3p

~2p!3

F~p!

@q21l2p•q#n

5lnE d3p

~2p!3

S F2lQW

2•

dFdpW

2lv

EFD S 12

lnv

E D@2~Ev2pW •QW !#n

,

~5.1!

where p05E(upW u) and l561, and the total derivative onthe right-hand side is defined as

3-7

d

t

x-

rmbin

-

titho

tifythe

p-

m-c-

d

ralitionof

on

ortter,re-fi-

fol-

rally

on


d

dpW5

]

]pW1

pW

E

]

]E. ~5.2!

In order to write the resulting formula forI 2 concisely, itis useful to define

F26~p,q!5F f e~p0!F2~p,q!6 f e~p0!F2~2p,q!

2p0 GUp05E(upW u)

.

~5.3!Then using Eq.~5.1!, the procedure that we have outlineyields

I 25E d3p

~2p!3

F212

QW

2•

dF22

dpW2

v

EF2

2

@2E~v2vW •QW !#2, ~5.4!

where

vW 5pW

E. ~5.5!

The result forI 1 is slightly more involved algebraically, bustraightforward to derive also. In analogy with Eq.~5.3! it isuseful to define

F16~p,q!5F f e~p0!F1~p,q!6 f e~p0!F1~2p,q!

2p0 GUp05E(p)

,

F186~p,q!5

1

2E

]F16~p,q!

]E~5.6!

in terms of which the long wave limit formula can be epressed in the form

I 15E d3p

~2p!3H F18

22QW

2•

dF181

dpW2

v

2EF18

1

2E~v2vW •QW !

2

F112

QW

2•

dF12

dpW

@2E~v2vW •QW !#2J . ~5.7!

Since we are concerned only with the real part of the fofactors, the singularity of the integrand is to be handledtaking the principal value part of the integral, as usualthese circumstances.

The formulas in Eqs.~5.4! and ~5.7! are particularly use-ful for computing explicitly the form factors in various limiting cases, as we now illustrate.

B. Static limit and the neutrino induced charge

Similar to the situation in the absence of the magnefield @1#, it is possible to some extent to interpret some ofform factors, or certain combinations of them, in terms

11300

y

cef

electromagnetic moments of the neutrino. In order to identhe static electromagnetic moments we have to look atvertex function in the static limitv→0 first, and then takethe long wavelength limitQ→0. It is well known that takingthe limit in the reverse order is not equivalent, since it reresents a different physical condition@24#.

However, in the present case there is the additional coplication that the vertex function and whence the form fators, are not isotropic functions ofQW and instead they depenon the componentsQi andQ' separately. Therefore, theQ→0 limit can be taken in two different ways, and in genethe form factors will have different corresponding limvalue. In order to keep this clear, we use the notatGm(v,Q' ,Qi) to denote the vertex function for any valueq, andGm(0,0,Qi→0) or Gm(0,Q'→0,0) for its value in thetwo limits that we have indicated. We use a similar notatifor the form factors.

If we insist in identifying a neutrino induced chargesome other induced electromagnetic moment for that mawe are thus forced to define two different quantities, corsponding to the two different limits. For example, the denition of the neutrino induced charge used in Ref.@25# forthe case of an isotropic medium, must be amended aslows:

en,

i 51

2EnTr Lk”G0

(,)~0,0,Qi→0!,

en,

' 51

2EnTrLk”G0

(,)~0,Q'→0,0!, ~5.8!

where km5(En ,kW ) is the neutrino momentum vector. Foillustrative purposes we consider the evaluation specificof the form factors that enter in these two formulas.

We consideren,

i first, which corresponds to settingQW '

50 and then taking the limit asQi→0 afterwards. As apractical trick, this amounts to evaluating the vertex functifor

qm5Qibm, ~5.9!

and then take theQi→0 limit. We therefore look at

G0(,)~0,0,Qi!5S 2eg2xA

(,)

4MW2 D $TL

(A)g02QiTu(A)gW •b%L,

~5.10!

where the form factors are evaluated atv50 andQ'50.From the above formula foren,

i , Eq. ~5.10! represents a

contribution to that quantity given by

en,

i 5S 2eg2xA(,)

4MW2 D lim

Qi→0@TL

(A)2QiTu(A)k•b#uv50, Q'50 ,

~5.11!

wherek is the neutrino momentum unit vector.In order to evaluateTL

(A) andTu(A) in this limit we proceed

as follows. First, from Eq.~4.24!, for qm5Qibm,

3-8

n

s

-io

nily

e

-ds

sence

to

of

q.-icin-The

sing.

of-atonavemo-ors

le-

the


KL(2)5KL

(1)52p0Pi1p0Qi ,

Ku(2)5Ku

(1)5~2Qi!@p0 21Pi21m21QiPi#,

~5.12!

where we have decomposed

pW 5PW'1Pib. ~5.13!

For the particular kinematic configuration that we are cosidering, the integrands do not depend explicitly onPW' , butonly implicitly through E(upW u). Therefore, the derivativethat appear in Eq.~5.6! can be rewritten using

1

2E

]

]E5

]

]P'2

. ~5.14!

Applying the results given in Eqs.~5.4! and ~5.7!, the inte-gral formula forTL

(A) becomes

TL(A)~0,0,Qi!5~4ueuB!E d2PW'dPi

~2p!3 S 21

2PiD ]

]P'2

3F S 12Pi]

]PiD ~ f e1 f e!G

50, ~5.15!

by symmetric integration overPi . Similarly,

Tu(A)~0,0,Qi!5

21

Qi~4ueuB!E d2PW'dPi

~2p!3

]

]P'2

3F S 1

2E22

1

2Pi

]

]PiD E~ f e1 f e!G .

~5.16!

Putting d2PW'5pdP'2 the integral over the transverse com

ponents can be performed trivially, and only the contributfrom P'

2 50 survives. The remaining integral overPi can bewritten in a simple form yielding

Tu(A)~0,0,Qi!5

~4ueuB!

Qit i , ~5.17!

where

t i521

2E0

` dp

~2p!2 F ]

]E@ f e~E!1 f e~E!#GU

E5E(p)

.

~5.18!

Using Eqs.~5.15! and~5.17! in Eq. ~5.11!, we finally obtain

en,

i 5S 2e2g2xA(,)

MW2 D ~ k•BW !t i . ~5.19!

11300

-

n

The integralt i cannot be evaluated in closed form for aarbitrary distribution, but for particular cases it is readevaluated.

1. Nonrelativistic classical gas

For example, in the classical and nonrelativistic limit wuse

] f e,e

]E52b f e,e , ~5.20!

whereb is the inverse temperature, and obtain

t i5~ne1ne!b

2

8me. ~5.21!

2. Degenerate gas

In this case, neglectingf e ,

t i5S EF

8p2pFD , ~5.22!

wherepF5(3p2ne)1/3 is the Fermi momentum of the elec

tron gas andEF the corresponding energy. This result holwhether the gas is relativistic or not.

A formula that resembles Eq.~5.19! was obtained in Ref.@22#. Although it was not explicitly stated there, upon cloinspection it becomes clear that the authors of that referetook theQW →0 limit by settingQ'50 first, and then lettingQi→0 afterwards. Whence their calculation correspondsthe calculation of the quantity that we have identified asen

i .However, our result given in Eq.~5.19! differs from the for-mula that is inferred from the results obtained in Ref.@22#@Eqs.~60! and ~61!#. In fact, we can reproduce the resultthat reference if we use the relation given in Eq.~5.20! toeliminate the derivatives of the distribution functions in E~5.18!. However, Eq.~5.20! is not valid for a general FermiDirac distribution, but only for the case of a nonrelativistand classical gas. For other cases, the formula for theduced charge obtained in that reference cannot be used.appropriate formula for a degenerate gas is obtained by uEq. ~5.22! or, for an arbitrary distribution, by using Eq~5.18!.

A particular feature of Eq.~5.19!, which was first noted inRef. @22#, is the termk•BW , which indicates a dependenceen

i on the direction of propagation of the neutrino. It is important to keep in mind that this is a kinematic factor tharises from taking the matrix element of the vertex functibetween the neutrino spinors. The observation that we hmade above regarding the non-uniqueness of the zeromentum limit refers to the dependence of the form factthemselves on the variablesQ' and Qi , independently ofsuch kinematic terms that may enter in specific matrix ements.

For completeness, we summarize below the results ofcalculation ofen,

' which we have carried out following a

similar procedure.

3-9

e-

es

e


C. The induced chargeenø

�

For v50 andQi50, we have

Tm0(V)5

1

Q'2

T2(V)i emabguabbqg,

Tm0(A)5~TS

(A)2TA(A)!bm1TA

(L)um1Tu(A)qm . ~5.23!

Either from the complete formulas given in Eq.~4.22! orfrom their long wavelength limit versions, it is easy to dduce that

TL(A)~0,Q',0!5Tu

(A)~0,Q',0!50, ~5.24!

by focusing on the integration overPi and observing that inboth cases the integrand is an odd function of that variablthis configuration. A less trivial result, but which followstraightforwardly by using the formulas given in Eqs.~5.4!and ~5.7! is

yor

nhe

roai

ll

11300

in

T2(V)~0,Q'→0,0!5Q'

2 3const. ~5.25!

Therefore,

G0(,)~0,Q'→0,0!

5S 2eg2xA(,)

4MW2 D lim

Q'→0@~TS

~A)2TA(A)!b”L#uv50, Qi50 .

~5.26!

Applying Eqs. ~5.4! and ~5.7! once again to evaluate thcombination of form factorsTS

A)2TA(A) we find

TS(A)~0,Q'→0,0!2TA

(A)~0,Q'→0,0!54ueuBt' ,~5.27!

where

t'5E d3p

~2p!3 F 1

2E

]

]E H 1

2E

]

]E S ~E21Pi21m2!~ f e1 f e!

2E D J GUE5E(upW u)

, ~5.28!

sestion

r infor-g-msithndi-in

ionatedc-

vi-he. Inhe

eec-

heur-e

rtexheced

and therefore, from Eq.~5.8!,

en,

' 5S 2e2g2xA~,)

MW2 D ~ k•BW !t' . ~5.29!

For the case of a nonrelativistic electron gas, Eq.~5.28! canbe manipulated to yield

t'521

2E0

` dp

~2p!2 F ]

]E@ f e~E!1 f e~E!#GU

E5E(p)

~nonrelativistic case!, ~5.30!

which is exactly the same as the integral in Eq.~5.18!. There-fore in this limit,

en,

' 5en,

i ~nonrelativistic case!. ~5.31!

The result given in Eq.~5.30!, and consequently the equalitbetweenen,

i anden,

i , is valid whether the gas is classical

degenerate, or in fact for any Fermi-Dirac distribution cosistent with the nonrelativistic limit. For a relativistic gas, tintegrals t' and t i are different, and Eq.~5.31! no longerholds.

VI. CONCLUSIONS

In this work we have been concerned with the electmagnetic properties of a neutrino that propagates in a mnetized electron background. Our goal has been to determthe neutrino electromagnetic vertex function systematica

-

-g-ney,

and in a way that it be useful to study the neutrino procesthat may occur in such media, such as Cherenkov radiaand plasmon decay.

Our starting point was to use the electron propagatothe presence of a magnetic field to obtain the one-loopmula for the vertex function up to linear terms in the manetic field. We then decomposed the vertex function in terof the minimal and complete set of tensors, consistent wbasic physical requirements such as the transversality cotion, and obtained the expressions for the form factorsterms of a set of integrals over the momentum distributfunctions of the background electrons. Simpler approximformulas that are valid in the long wavelength limit, anwhich are useful for practical calculations of the form fators, were given.

For illustrative purposes, and to make contact with preous work, the calculation of the form factors that enter in teffective neutrino charge was considered in some detailconnection with this we pointed out that, in contrast with tsituation in an isotropic medium~i.e., in the absence of themagnetic field!, the static form factors do not have a uniquvalue in the zero~photon! momentum limit. We stress oncmore that this is not a mathematical ambiguity, but it is atually a physical effect that results from the fact that tmedium in the present case is essentially anisotropic. Fthermore, it is a dynamical effect quite different from thkinematic dependence that the matrix elements of the vefunction may have on the direction of propagation of tneutrino. Thus, for the specific case of the neutrino inducharge, the two quantitiesen

i and en' were introduced and

3-10

Astehetlar

narat

ohat

heveiebe

.S

e

t

e


their expressions in terms of the form factors were given.we showed, they have the same value for a nonrelativigas, but are given by different formulas otherwise. In a givapplication all the form factors are in principle relevant. Tsame method that we have used to evaluate those tharelated to the neutrino induced charge can be used simito yield the corresponding formulas for all the others.

In this work we have taken into account only the electroin the background. While the effects of the nucleonssometimes suppressed due to their mass, it is known ththe presence of a magnetic field their contribution to somethe form factors is important. The method and formulas twe have presented here provide a firm basis, both concepand practical, to take those effects into account as well. Tprovide a consistent basis to continue and extend the intigation of problems in which the electromagnetic propertof the neutrino and its coupling to a magnetic field arelieved to be important.

ACKNOWLEDGMENTS

This material is based upon work supported by the UNational Science Foundation under Grant No. 0139538.

APPENDIX A: PROOF OF THE TRANSVERSALITYRELATIONS FOR Tµn

„V…

We consider in some detail the proof of the relation

qnTmn(V)50, ~A1!

whereTmn(V) is given in Eq.~3.17!. SinceTmn

(V) is antisymmet-ric, this implies thatqmTmn

(V)50 also.We write Tmn

(V) in the following form:

Tmn(V)5~4ueuB!E d4p

~2p!3he~p!I mn

(TB) , ~A2!

where

I mn(TB)[H 2Lmn

(1)d8~p22m2!

d1

Lmn(2)d~p22m2!

d2 J 2~q→2q!

~A3!

and

d[p822m2. ~A4!

Let usdefinethe quantity

I mn(T)[d~p22m2!FLmn

d1~q→2q!G , ~A5!

where

Lmn52pmpn1pmqn1qmpn2~p•q!gmn . ~A6!

We now state the following result, which is a purely algbraic identity. Defining

Pmn[ i emnabbaub , ~A7!

11300

sicn

arely

seinft

ualys-s-

.

-

then

qmI mn(TB)5

i

2Plm]lI mn

(T) , ~A8!

where ]l[]/]pl. We prove Eq.~A8! below but, for themoment, notice that it is all that we need. Observing tha

Plm]lh~p!50 ~A9!

~because]lh}ul), then from Eqs.~A2! and ~A8! we have

qmTmn(V)5~4ueuB!

i

2PlmE d4p

~2p!3]l@he~p!I mn

(T)#

50. ~A10!

Proof of Eq. „A8…

To prove Eq.~A8!, let us consider first its right-hand sid

Plm]lI mn(T)5PlmH ]lS d~p22m2!Lmn

d D1~q→2q!J .

~A11!

Using

]ld~p22m2!5d8~p22m2!~2pl!,

]lS 1

dD522

d2pl8 ,

]lLmn52gmlpn12gnlpm1gmlqn1gnlqm2gmnql ,

~A12!

we obtain

Plm]lI mn(T)5

2id8~p22m2!

d@pnemlabqmpluabb

2~p•q!enlabpl uabb#

22id~p22m2!

d2@pn8e

mlabpmqluabb

2~p•q!enlabp8l uabb#

22id~p22m2!

denlabp8l uabb1~q→2q!.

~A13!

The third term can be rewritten thus,

d~p22m2!

d5

d~p22m2!

d2~q212p•q! ~A14!

3-11


and then combining it with the second term,

Plm]lI mn(T)5

2id8~p22m2!

d@2pnemlabpmqluabb

2~p•q!enlabpluabb#

22id~p22m2!

d2@pn8e

mlabpmqluabb

1~p8•q!enlabp8l uabb#1~q→2q!. ~A15!

This is all we will do with the right-hand side of Eq.~A8!.Now let us consider the left-hand side. From Eq.~3.16!,

we get

qmLmn(1)52~p•u!emnabqmpabb1~p•b!emnabqmpaub

1m2emnabqmuabb,

qmLmn(2)5~p8•u!emnabqmpabb2~p8•b!emnabqmpaub

2m2emnabqmuabb ~A16!

Using the identity given in Eq.~4.10!, we obtain the follow-ing two identities,

11300

~p•u!emnabqmbapb2~p•b!emnabqmuapb

1p2emnabqmuabb

52pnemlabpmqluabb2~p•q!enlabpluabb ~A17!

and

~p8•u!emnabqmbapb2~p8•b!emnabqmuapb

1p8 2emnabqmuabb

52pn8emlabpmqluabb2~p8•q!enlabpluabb,

~A18!

and using them we can rewrite Eq.~A16! in the form

qmLmn(1)52pnemlabpmqluabb2~p•q!enlabpluabb

2~p22m2!emnabqmuabb,

qmLmn(2)5pn8emlabpmqluabb1~p8•q!enlabp8luabb

1~p8 22m2!emnabqmuabb. ~A19!

Therefore,

qmI mn(TB)5H 2qmLmn

(1)d8~p22m2!

d1

qmLmn(2)d~p22m2!

d2 J 2~q2q!

52d8~p22m2!

d@2pnemlabpmqluabb2~p•q!enlabpluabb2~p22m2!emnabqmuabb#

1d~p22m2!

d2@pn8emlabpmqluabb1~p8•q!enlabp8luabb1~p8 22m2!emnabqmuabb#1~q→2q!. ~A20!

The two terms that contain (p22m2) and (p8 22m2) com-bine to give

1

d@d8~p22m2!~p22m2!1d~p22m2!#50, ~A21!

where we have used the relation

xd8~x!52d~x!. ~A22!

The remaining terms precisely coincide with Eq.~A15!,which therefore proves Eq.~A8!.

APPENDIX B: PROOF OF THE TRANSVERSALITYRELATIONS FOR Tµn

„A…

The relation

qnTmn(A)50 ~B1!

is proven similarly. From Eq.~3.17!,

qnTmn(A)5~4ueuB!E d4p

~2p!3he~p!H 2qnKmn

(1)d8~p22m2!

~p1q!22m2

1qnKmn

(2)d~p22m2!

@~p1q!22m2#22~q→2q!J , ~B2!

where, from the defining equation Eq.~3.16!, we obtain

qnKmn(1)5pm@2~p•b!~u•q!1~p•u!~b•q!#

1um@2~p•b!~p8•q!2m2b•q#

1bm@~p•u!~p8•q!1m2u•q#,

3-12


qnKmn(2)5pm8 @2~p•b!~u•q!1~p•u!~b•q!#

1um@2~p8•b!~p•q!2m2b•q#

1bm@~p8•u!~p•q!1m2u•q#. ~B3!

n

o

a

11300

Then, in analogy with Eq.~A5!, we define

Jmn(T)5d~p22m2!

emnabpaqb

d, ~B4!

and by straightforward calculation of]lJmn(T) it follows that

2qnKmn(1)

dd8~p22m2!1

qnKmn(2)

d2d~p22m2!5S 21

2 D F1

iPnl]lJmn

(T)1d~p22m2!

d~2Ym!G

1@Xm~q212p•q!1Ym~p22m2!#Fd~p22m2!

d22

d8~p22m2!

d G5S i

2D Pnl]lJmn(T)2Xmd8~p22m2!. ~B5!

In Eq. ~B5! we have defined

sorweal-s-

e,

Xm5bm~u•p!2um~b•p!,

Ym5um~b•q!2bm~u•q!, ~B6!

and to arrive at the last equality we have used the relatioEq. ~A22! once more. Thus, using Eq.~B5! in Eq. ~B2!,

qnTmn(A)5~4ueuB!

i

2PnlE d4p

~2p!3]l@he~p!Jmn

(T)2~q→2q!#

50. ~B7!

On the other hand, by separating the tensorsKmn(1,2) into

their symmetric and antisymmetric parts, it is easy to shthat

@1# J.F. Nieves and P.B. Pal, Phys. Rev. D40, 1693~1989!.@2# V.B. Semikoz and Y.A. Smorodinsky, Zh. Eksp. Teor. Fiz.95,

35 ~1989! @Sov. Phys. JETP68, 20 ~1989!#.@3# V.N. Oraevsky, V.B. Semikoz, and Y.A. Smorodinsky, Pis’m

Zh. Eksp. Teor. Fiz.43, 549 ~1986! @JETP Lett. 43, 709~1986!#.

@4# J.C. D’Olivo, J.F. Nieves, and P.B. Pal, Phys. Rev. D40, 3679~1989!.

@5# R.F. Sawyer, Phys. Rev. D46, 1180~1992!.@6# J.C. D’Olivo, J.F. Nieves, and P.B. Pal, Phys. Rev. Lett.64,

1088 ~1990!.@7# V.B. Semikoz and J.W.F. Valle, Nucl. Phys.B425, 65 ~1994!.@8# V.B. Semikoz and J.W.F. Valle, Nucl. Phys.B485, 585~1997!.@9# J.C. D’Olivo and J.F. Nieves, Phys. Lett. B383, 87 ~1996!.

@10# J.C. D’Olivo, J.F. Nieves, and P.B. Pal, Phys. Lett. B365, 178~1996!.

@11# A. Kusenko and G. Segre´, Phys. Rev. Lett.77, 4872~1996!.@12# A. Kusenko and G. Segre´, Phys. Lett. B396, 197 ~1997!.@13# A. Kusenko and G. Segre´, Phys. Rev. D59, 061302~1999!.

of

w

qmTmn(A)5~4ueuB!~4m2Yn!E d4p

~2p!3he~p!

3Fd~p22m2!

d21~q→2q!G , ~B8!

which explicitly verifies that, as expected,Tmn(A) is not trans-

verse with respect to its first index. In this respect, the tenTmn

(A) , calculated in the presence of the magnetic field asare doing here, differs from the corresponding quantity cculated without the magnetic field, in which case it is tranverse in the first index as well.

@14# A. Kusenko and G. Segre´, Phys. Rev. Lett.79, 2751~1997!.@15# Y.Z. Qian, Phys. Rev. Lett.79, 2750~1997!.@16# H. Nunokawa, V.B. Semikoz, A.Y. Smirnov, and J.W.F. Vall

Nucl. Phys.B501, 17 ~1997!.@17# S. Esposito and G. Capone, Z. Phys. C70, 55 ~1996!.@18# A. Erdas, C.W. Kim, and T.H. Lee, Phys. Rev. D58, 085016

~1998!.@19# P. Elmfors, D. Grasso, and G. Raffelt, Nucl. Phys.B479, 3

~1996!.@20# J.C. D’Olivo and J.F. Nieves, Phys. Rev. D56, 5898~1997!.@21# A.N. Ioannisian and G.G. Raffelt, Phys. Rev. D55, 7038

~1997!.@22# K. Bhattacharya, A. Ganguly, and S. Konar, Phys. Rev. D65,

013007~2002!.@23# J.C. D’Olivo, J.F. Nieves, and S. Sahu, Phys. Rev. D67,

025018~2003!.@24# J.F. Nieves and P.B. Pal, Phys. Rev. D51, 5300 ~1995!, and

references therein.@25# J.F. Nieves and P.B. Pal, Phys. Rev. D49, 1398~1994!.

3-13

electromagnetic vertex of neutrinos in an electron background and a magnetic field

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