Effective Neutrino Photon Interaction in a Magnetized Medium

8 Pages·0.17 MB·English

by  Kaushik Bhattacharya

Neutrino Interactions in a Magnetized Medium
Kaushik Bhattacharya1, Avijit K. Ganguly1, Sushan Konar2
1 Saha Institute of Nuclear Physics, 1/AF, Bidhan-Nagar, Calcutta 700064, India
2IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India
e-mail : [email protected], [email protected],[email protected]
Neutrino-photonprocesses,forbiddeninvacuum,cantakeplaceinpresenceofathermalmediumor
an external electro-magnetic field,mediated bythecorresponding charged leptons(real orvirtual).
Theeffectofamediumoranelectromagneticfieldistwo-fold-toinduceaneffectiveν−γvertexand
tomodify thedispersion relations ofall theparticles involvedtorendertheprocesses kinematically
viable. It has already been noted that in presence of a thermal medium such an electromagnetic
interaction translates into the neutrino acquiring a small effective charge. In this work, we extend
thisconcepttothecaseofathermalmediuminpresenceofanexternalmagneticfieldandcalculate
the effective charge of a neutrino in the limit of a weak magnetic field. We find that the effective
2
0 charge is direction dependent which is a direct effect of magnetic field breaking the isotropy of the
0 space.
2
n I. INTRODUCTION up the mantle of the supernova progenitor [6].
a Inthiswork,weshowthattheeffectivechargeacquired
J Processesinvolvingneutrinosandphotonsareofgreat by a neutrino in a magnetised medium is, in fact, direc-

7 importance in astrophysicsand cosmology[1]. In partic- tion dependent which should affect these processes sig-
1 ular, reactions which are forbidden (or are highly sup- nificantly. Though it should be noted here that except

2 pressed) in vacuum, notably plasmon decay (γ νν) or for in the very early universe or in the recently discov-
v the Cherenkov process (ν νγ) and the cross-→processes ered ‘magnetars’ (newly born neutron stars with mag-
9 (e.g,γγ νν¯,νν¯ e+e−→etc.) playa significantrolein netic fields in excess of 1015 Gauss) [7] or perhaps in
5 regions p→ervaded b→y dense plasma and/or large-scaleex- the central region of core-collapse supernovae, the mag-
72 ternal magnetic fields. White Dwarfs, Neutron Stars or netic fields are smaller than the QED limit (eB < m2e
thefinalphasesofstellarevolution(Supernovae)arepar- i.e, B 1013 Gauss). This allows for a weak-field treat-
0 ≤
1 ticularexampleswheresuchprocessesbecomeimportant mentoftheplasmaprocessesrelevanttoalmostallphys-
0 by virtue of the large material density and the presence ical situations. Moreover,this treatment is also valid for
h/ of strong magnetic fields. Prompted by these objectives compactastrophysicalobjects(viz. white dwarfsorneu-
p we calculate the effective neutrino charge in an external tronstars)for which the Landaulevel spacingsare quite
- magnetic field in presence of a thermal medium. small compared to the electron Fermi energy [8]. This
p
Ithasalreadybeenshownthattheν γ interactionin ensures that the magnetic field does not introduce any
e
h presence of a thermal medium induces−a small effective spatial anisotropy in the collective plasma behaviour.
: chargetotheneutrinoandthattheneutrinoelectromag- Inthestandardmodel,theabovementionedν γ pro-
v −
i netic vertex is related to the photon self-energy in the cesses appear at the one-loop level. They do not occur
X medium [2,3]. We re-investigate this problem consider- in vacuum because they are kinematically forbidden and
r ingnotonlyathermalmediumbutalsoanexternalmag- also because the neutrinos do not couple to the photons
a
netic field for a neutrino coupled to a dynamical photon at the tree-level. In the presence of a medium or a mag-
having q = 0 and ~q 0. This calculation is perti- netic field, it is the charged particle running in the loop
0
nent, for example, i|n|th→e case of a type-II supernovae which, when integrated out, confers its electromagnetic
collapse [4]. It is conjectured that the neutrinos, pro- properties to the neutrino. Therefore, processes involv-
duced deep inside the stellar core, deposit some fraction ing two neutrinos and one photon which are forbidden
of their energy to the surrounding medium through dif- in vacuum can become important in the presence of a
ferentkindsofelectromagneticinteractions,e.g,ν νγ, mediumand/orexternalfields[9–11]. Thesechargedpar-
νν¯ e+e− [23]. But it is important to note tha→t the ticles could be the electrons of the background thermal
amp→litudes ofsuchprocessesareproportionaltoG2 and medium or the virtual electrons and positrons in pres-
F
the amount of energy transferred to the mantle of the enceofanexternalmagneticfieldorboth. Theprocesses
proto-neutron star is barely sufficient to blow the stellar also become kinematically allowed since the photons ac-
envelope out. Recently in a series of papers, it has been quire an effective mass in a thermal medium. This, for
argued that the freely streaming neutrinos from the su- example, opens up the phase space for the Cherenkov
pernova core interact with the non-relativistic electrons process ν νγ [10,12,13]. The presence of a magnetic
→
present in the outer part of the core through collective field would also modify the photon dispersion relation
interactions(knownasthe‘two-streaminstability’inthe and then the Cherenkov condition would be satisfied for
contextofplasmaphysics)andisresponsibleforblowing significant ranges of photon frequencies [14–16]. A ther-
1
mal medium and an external magnetic field, thus, fulfill where, Π5 represents the vector-axial vector coupling
µν
thedualpurposeofinducinganeffectiveneutrino-photon and Π is the polarisation tensor arising from the dia-
µν
vertex and of modifying the photon dispersion relation gram in fig. 2. In an earlier paper [17] (paper-I hence-
such that the phase-space for various neutrino-photon forth) we have analysed the structure of Πµν and cal-
processesis openedup(see [11]andreferencesthereinfor culated the photon dispersion relation, in a background
a detailed review). mediuminpresenceofauniformexternalmagneticfield,
Theorganizationofthe paperisasfollows. Insection- in the weak-field limit by retaining terms up-to ( ),
O B
IIwediscussthebasicformalismforcalculatingtheeffec- calculated at the 1-loop level. We shall use the results
tivechargeofaneutrino. Section-IIIcontainsthe details of paper-I here to obtainthe total effective chargeof the
of the calculation of the 1-loop diagram in presence of a neutrinos under equivalent conditions. Because of the
magnetisedmedium. Insection-IVwediscussthegeneric electromagneticcurrentconservation,forthepolarisation
form of the neutrino effective charge in different back- tensor,wehavethefollowinggaugeinvariancecondition:
ground conditions. Finally, in section-V we present the
expressionfor the effective chargeof a neutrino. And we qµΠµν =0=Πµνqν. (6)
conclude with a discussion on the possible implications
Same is true for the photon vertex of fig. 1 and we have
of our result in section-VI.
Π5 qν =0. (7)
µν
II. FORMALISM
Therefore, the effective charge of the neutrinos is de-
fined in terms of the vertex function by the following
The off-shell electromagnetic vertex function Γ is de-
λ relation [2]:
fined in such a way that, for on-shell neutrinos, the ννγ
amplitude is given by: 1
e = u¯(k)Γ (q =0,q 0)u(k). (8)
eff 0 0
2k →
= iu¯(k′)Γ u(k)Aλ(q), (1) 0
λ
M −
FormasslessWeylspinorsthisdefinitioncanberendered
where, q,k,k′ are the momentum carried by the photon
into the form:
and the neutrinos respectively and q = k k′. Here,
u(k)isthe neutrinowave-functionandAµ st−andsforthe e = 1 tr Γ (q =0,q 0)(1+λγ5)k/ (9)
eff 0 0
electromagnetic vector potential. In general, Γλ would 2k0 (cid:2) → (cid:3)
dependonk,q,thecharacteristicofthe mediumandthe
where λ= 1 is the helicity of the spinors.
external electromagnetic field. We shall, in this work, ±
ItcanbeseenfromEq.(9)that,ingeneral,theeffective
consider neutrino momenta that are small compared to
neutrino chargedepends onΠ (q) aswellas onΠ5 (q).
the masses of the W and Z bosons. We can, therefore, µν µν
Now,inamagnetisedmedium,thedispersivepartofΠ
neglectthemomentumdependenceintheWandZprop- µν
to liner order in has the following form [paper-I]:
agators,whichis justified if we areperforming a calcula- B
tion to the leading order in the Fermi constant, GF. In Π ε qβ, (10)
µν µνα β
this limit four-fermioninteraction is given by the follow- ∝ k
ing effective Lagrangian: where α stands for either 0 or 3 (to be explained in
k
detail in the next section). This evidently vanishes in
1
Leff = √2GFνγµ(1+γ5)νlνγµ(gV+gAγ5)lν, (2) the limit q0 = 0,q → 0. Therefore, in a magnetised
medium,thenon-zerocontributiontotheeffectivecharge
where, ν and l are the neutrino and the corresponding of the neutrinos come solely from Π5 (q). In section-
ν µν
lepton field respectively. For electron neutrinos, IV we shall discuss, from a more general point of view,
why the effective charge of a neutrino, in a magnetised
gV =1−(1−4sin2θW)/2, (3) medium, comes only from Π5µν(q) to linear order in B.
g =1 1/2; (4) Therefore, the effective charge of the neutrinos is given
A
−
by:
where the first terms in g and g are the contributions
V A
from the W exchange diagram and the second one from e = 1 GF g Π5 (q =0,q 0)
the Z exchange diagram. Then the amplitude effectively eff −2k0 √2e A µ0 0 →
reduces to that ofa purely photonic casewith one ofthe tr γµ(1+γ )(1+λγ5)k/ . (11)
5
photons replaced by the neutrino current, as seen in the ×
(cid:8) (cid:9)
diagram in fig. 1. Therefore, Γ is given by:
λ
1
Γ = G γν(1+γ )(g Π +g Π5 ), (5)
µ −√2e F 5 V µν A µν
2
p+q p′
≡
k′
q σ =iγ γ = γ γ γ , (19)
→ ν µ z 1 2 − 0 3 5
k and we have used,
p eieBsσz =cos e s+iσ sin e s. (20)
z
B B
FIG.1. One-loopdiagramfortheeffectiveelectromagnetic
OfcourseintherangeofintegrationindicatedinEq.(12)
vertex of the neutrino in the limit of infinitely heavy W and
s is never negative and hence s equals s. It should be
Z masses. | |
mentioned here that we follow the notation adopted in
p+q p′ paper-I to ensure continuity. In the presence of a back-
≡
groundmedium,theabovepropagatorismodifiedto[21]:
q q
→ → iS(p)=iSV(p)+Sη(p), (21)
B B
where
p
FIG.2. One-loop diagram for thepolarisation tensor. SBη(p)≡−ηF(p)hiSBV(p)−iSVB(p)i , (22)
and
III. CALCULATION OF THE 1-LOOP DIAGRAM
SV(p) γ SV†(p)γ , (23)
B ≡ 0 B 0
A. The Propagator
for a fermion propagator,such that
Sinceweinvestigatethecaseofapurelymagneticfield, ∞
Sη(p)= η (p) dseΦ(p,s)C(p,s). (24)
itcanbetakeninthez-directionwithoutanyfurtherloss B − F Z
−∞
ofgenerality. Wedenotethemagnitudeofthisfieldby .
B
Ignoringatfirstthepresenceofthemedium,theelectron And ηF(p) contains the distribution function for the
propagatorin sucha field can be written downfollowing fermions and the anti-fermions:
Schwinger’s approach [18–20]:
η (p)=Θ(p u)f (p,µ,β)
F F
·
∞
iSBV(p)= dseΦ(p,s)C(p,s), (12) +Θ(−p·u)fF(−p,−µ,β). (25)
Z
0
Here, f denotes the Fermi-Dirac distribution function:
F
where Φ and C are defined below. To write these in a
1
compact notation, we decompose the metric tensor into
f (p,µ,β)= , (26)
two parts: F eβ(p·u−µ)+1
and Θ is the step function given by:
g =gk +g⊥ , (13)
µν µν µν
Θ(x)=1, for x>0,
where
=0, for x<0.
gk =diag(1,0,0 1)
µν −
g⊥ =diag(0, 1, 1,0). (14)
µν − −
B. Identifying the Relevant Terms
This allows us to use the following definitions,
p2 =p2 p2 (15) The amplitude of the 1-loop diagram of fig. 1 can be
k 0− 3 written as:
p2 =p2+p2. (16)
⊥ 1 2 d4p
iΠ5 (q,β)= (ie)2 tr [γ γ iS(p)γ iS(p′)] ,
Using these notations we can write: µν −Z (2π)4 µ 5 ν
tan(e s) (27)
Φ(p,s) is p2 B p2 m2 ǫs , (17)
≡ (cid:18) k− e s ⊥− (cid:19)− | |
B where,forthesakeofnotationalsimplicity,wehaveused
eieBsσz e−ieBsσz
C(p,s)≡ cos(e s) (cid:18)/pk+ cos(e s)/p⊥+m(cid:19) p′ =p+q. (28)
B B
= (1+iσ tane s)(p/ +m)+(sec2e s)p/ , (18) The minus sign on the right side is for a closed fermion
z k ⊥
h B B i loop and S(p) is the propagator given by Eq. (21). This
where implies:
3
Π5 (q,β)= ie2 d4p tr [γ γ iS(p)γ iS(p′)] . and
µν − Z (2π)4 µ 5 ν
sec2e stan2e s′
= ε B B q2
(29) Rµν − µν12 tane s+tane s′ ⊥
B B
ε (q p)(tane s+tane s′)
Now using Eq.(24) we have the terms containing the ef- − µν12 · B B
fects of medium and the external field (non-absorptive) +2εµ12αk(p′νkpαktaneBs+pνkp′αktaneBs′)
Π5 (q,β)= ie2 d4p tr γ γ Sη(p)γ iSV(p′) +gµαkqν⊥pαk(taneBs−taneBs′)
µν −+γµ γZ5i(S2BVπ)(4p)γνhSµBη(p5′)B. ν B (30) −gµαkqν⊥qeαek tsaenc2eeBBss+tatnan2eeBBss′′
i +ngµν(p·q)k+gναkpαkqµ⊥o (taneBs−taneBs′)
Scyucblsictitpurtoipnegrtpyboyf t−rapc′eisn, wtheecsaencownrditteerEmq.a(n3d0)uassing the +gναkqαkpeµ⊥sec2eBsetaneBs′. (36)
e
d4p Inwritingthisexpression,wehaveusedthenotationpαk,
Π5 (q,β)= ie2 tr γ γ Sη(p)γ iSV(p′)
µν − Z (2π)4 h µ 5 B ν B forexample. Thissignifiesthatαkcantakeonlythe‘pear-
allel’indices,i.e.,0and3,andismoreoverdifferentfrom
+γνSBη(−p)γµγ5iSBV(−p′)i. (31) the index α appearing elsewheree in the expression. We
performthe calculationsintherestframeofthe medium
UsingnowtheformofthepropagatorsfromEqs.(12)and where p u = p . Thus the distribution function does
0
(24), we obtain not depe·nd on the spatial components of p and is given
simply by η (p ).
d4p ∞ ∞ + 0
Π5 (q,β)=ie2 dseΦ(p,s) ds′ eΦ(p′,s′)
µν Z (2π)4 Z Z
−∞ 0
µν(p,p′,s,s′,B), (32) IV. TENSORIAL STRUCTURES OF Πµν AND Π5µν
×G
with, ItcanbeseenfromEq.(9)that,ingeneral,theeffective
neutrino chargedepends onΠ (q) aswellas onΠ5 (q).
µν µν
µν =ηF( p)tr γνC( p,s)γµγ5C( p′,s′) Now, in vacuum we have,
G − h − − i
+η (p)tr γ γ C(p,s)γ C(p′,s′) . (33) Π (q)=(g q2 q q )Π(q2), (37)
F µ 5 ν µν µν µ ν
h i −
whereΠ(k2)vanishesforq =0,q¯ 0. Theothercontri-
It should be mentioned here that the effective charge of 0
→
bution to the effective charge, coming from Π5 (k), also
the neutrinos come from the dispersive part of the axial µν
vanishes in vacuum. This can be understood from the
polarisation tensor. Therefore, we work with the real
general form factor analysis. We should be able to ex-
partofthe 11-componentofthe axialpolarisationtensor
press Π5 (k), in vacuum, in terms of g , ǫ and q .
throughoutandfornotationalsimplicitysuppressthe11- µν µν µνλσ λ
The parity structure of the theory forbids the appear-
index everywhere.
ance of g . Therefore, the only possible combination,
µν
to obtain a second rank tensor is, ǫ q q . Since, this
µνλσ λ σ
C. Extracting the Gauge Invariant Piece is identically zero, there can be no effective charge of a
neutrino in vacuum.
Π5 (k,β)inoddpowersof -Noticethatthephase On the other hand, in the presence of a medium the
µν B polarisationtensorcanbeexpandedintermsofthe form
factors appearing in Eq. (32) are even in . Thus, we
B factors as follows [10]:
need consider only the odd terms from the traces. Per-
formingthetracesthegaugeinvariantexpression,oddin
Π (k)=Π T +Π L +Π P , (38)
powers of , comes out to be: µν T µν L µν P µν
B
where
d4p ∞
Π5 (q,β)= 4e2 η (p) dseΦ(p,s)
µν − Z (2π)4 + Z T =g L (39)
−∞ µν µν µν
−
×Z ∞ ds′eΦ(p′,s′) Rµν; (34) Lµν =euuµu2ν (40)
−∞
e e
i
where Pµν = eεµναβqαuβ (41)
Q
η (p)=η (p)+η ( p) (35) and,
+ F F
−
4
q q
g =g µ ν (42) only terms that would survive would contain the combi-
µν µν − k2 nationqµF (alsoborne outby the explicitcalculations
euµ =gµρuρ (43) ofpaper-I).µHνence,inthezerofrequency,longwavelength
= (q u)2 q2 (44) limititvanishesleavingonlyΠ5 tocontributetotheef-
eQ e · − µν
p fective charge of the neutrino. Now, in a magnetized
intherestframeofthemediumwhereuµ =(1,0,0,0). It medium Π5 can be written, in terms of general basis
µν
iseasytoseethatneitherthelongitudinalprojectionLµν vectors available, as follows:
or Π is non-zero in the limit q = 0,q¯ 0. This then
L 0
provides a non-zero contribution to the→effective charge Π5 =ǫ (q2f +q2f +(q.u)f )
µν µν12 ⊥ 1 k 2 3
onfonth-zeernoeucotrnitnroib.uAtinodnitnot[h2e]eitffehcatsivbeecehnarsgheo,winnpthreastenthcee +ǫµ12αk(qνkuαkf3+uνkuαkf4+qαkuνkf5
of a thermal medium, comes only from the Πµν(q) part. +qνkqαkf6+uνkqαkf7+qαkqνkf8)
Tεhe teqnαsuoβr satnrducdtuorees onfoΠt 5µcνonitnriabumteedtioumtheiszoefrotthhecfoormm- +gµαkqν⊥(uα˜kf9+qα˜kf10)
pµoνnαeβnt of Γν. For a more physical understanding of the +gµνqαkqα˜kf11+gµνuαkqα˜kf12+gναkuα˜kqµ⊥f13
appearance of the effective charge of the neutrinos, in a +gναkqα˜kqµ⊥f14+gναkqα˜kqµ⊥f15, (48)
medium, see [2].
wheref saretherespectiveformfactors. Itcanbeeasily
Now, at the 1-loop level Π is invariantunder charge i
µν
seenthatthetermsproportionaltotheproductofu’sare
conjugation, i.e., if we calculate the vacuum polarisation
non-zerointhestaticlongwavelengthlimitgivingafinite
inamediumwithacertainbackgroundfield,itshouldbe
contribution to the effective charge of a neutrino.
thesameasthatobtainedinacharge-conjugatedmedium
Inthis connection,it shouldbe mentionedthat the ef-
with an opposite background field. This means that, in
fectivechargeoftheneutrinosbearasimplerelationwith
the polarization tensor, the terms are either even in the
the Debye screening length in the case of an unmagne-
background field or even in µ or odd in both. There-
tized plasma. As has been shown by [2] the contribution
fore,inabsenceofamedium(whichcanbethoughtofas
containing µ0), the terms containing odd powers of the to the effective charge comes only from ΠL which corre-
spondstotheDebyescreeninginthelimitq =0,q 0.
backgroundfield should vanish (see paper-I for a discus- 0
→
InthecaseofamagnetisedplasmaΠ5 ,ingeneral,would
sion). Therefore, the contribution to the effective charge µν
havemanymoretensorialformsinitdue tothepresence
ofaneutrino,inpresenceofabackgroundmagneticfield
in vacuum comes only from Π5 (q), to linear order in . of the electromagnetic tensor (paper-I0. Hence, there
Now, Π5 (q) in a magnetisedµvνacuum is given by [9]: B may not exist a simple correspondence between the De-
µν
bye screening and the effective charge of a neutrino in a
i ∞ 1 magnetised medium.
Π5 (q)= ds dv/2e−isχ
µν (4π)2 Z Z
0 −1
(1 v2)qµeF˜q V. EFFECTIVE CHARGE OF A NEUTRINO
n − k ν
+R q2eF˜µν +qµeF˜q +qνeF˜q , (45)
(cid:16)− ⊥ ⊥ ν ⊥ µ(cid:17)o It is evident from Eq.(11) that to find the effective
charge of the neutrino we need only to calculate Π5 in
µ0
with, the limit (q = 0,q 0). From Eq.(36) it can be seen
0
→
that inthis limit the only survivingterms in Π5 are the
(1 v2) cosvz cosz 00
χ=m2+ −4 qk2+ 2zs−inz (46) ones containing odd powers of p0. Now, η+(p0) is even
in p and so is the exponent in the zero frequency limit.
1 vsinvzsinz cosvzcosz 0
R= − sin2z− , (47) Hence, p0 integration makes Π500 vanish. We also have,
R =p q (tane s tane s′)
wmheterriec,uzse=d ienBEsqa.n(d45F)˜µisνg=µν12=ǫµdνiλaσgF(λσ,+(n,o+te,+t)h)a.tItthies 10 +3q31p1sec2Be −staneBs′, (49)
B B
evident that in the zero frequency and−long wavelength R =p q (tane s tane s′)
20 3 2
B − B
limit Π5µν vanishes resulting in zero effective charge of a +q3p2sec2e stane s′. (50)
neutrino in magnetised vacuum. B B
The tensorialform of Π has been discussedin detail Itcanbeseenthatafterpintegrationweshallhaveterms
µν
in paper-I. It can be seen from that discussion that in proportionaltoq q andq q inΠ5 andΠ5 respectively,
3 1 3 2 10 20
a magnetised medium, the electromagnetic field always as the integrals in p and p are Gaussian. In the limit
1 2
appears in the combination uµF or qµF in Π , to q 0 these terms would vanish. Therefore, in the rel-
µν µν µν
→
linear order in the field strength. Since, we consider the evant limit of vanishing external photon momenta only
caseofapuremagneticfieldandastationarymediumthe Π5 , given by,
30
5
Π5 (q =0,~q 0) d2p (p2 m2)
30 0 → lim k (2π) k− η (p )(q p)
= lim 4e2 d4p ∞ dseΦ(p,s) ∞ds′eΦ(p′,s′) q0=0,q~→0Z (2π)2 (p′k2−m2)+iε + 0 · k
q0=0,q~→0 Z (2π)4 Z−∞ Z0 1 dp η+(Ep)
= , (58)
×η+(p0) (q·p)k−2p20 (taneBs+taneBs′) (51) 2Z 2π Ep
(cid:0) (cid:1)
has a non-zero contribution to the effective charge. It and,
can be seen that in the above expression, except for the
d2p (p2 m2)
exponents,theintegrandiffreeoftheperpendicularcom- lim k (2π)p 2 k− η (p )
ponents ofthe momentum. Therefore,the perpendicular q0=0,q~→0Z (2π)2 0 (p′k2−m2)+iε + 0
componentsoftheloopmomentumcanbeintegratedout.
1 dp η (E )
+ p
Now, the exponential factors can be written as, = [ βη+(Ep)], (59)
4Z 2π Ep −
Φ(p,s)+Φ(p′,s′)=Φ +Φ , (52)
k ⊥ where E2 =p2+m2. Therefore, we have,
p
where,
(e3 ) dp
Π5 (q =0,~q 0)= B β η (E ). (60)
Φ =is(p2 m2)+is′(p′2 m2) ε s ε s′ , (53) 30 0 → 2π Z 2π + p
k k− k − − | |− | |
Φ⊥ =−itaeneBsp2⊥− itaneeBs′ p′⊥2. (54) doNmoinwanitticsoemvpidoennetntthoaftΠin5 thiseΠli5m.itT(hq0er=efo0r,eq,t→he0i)ndthexe
B B µν 30
µ in Eq.(11) can only take the value 3. Incorporating
Therefore, integration of the perpendicular part of the
this fact and after taking the trace, Eq.(11) takes the
momentum gives us,
following form:
d2p
Z (2π)⊥2 eΦ⊥ eeff =−√G2FegA(1−λ)Π530(q0 =0,q→0)cosθ, (61)
i tane stane s′ d2p
=exp B B k2 ⊥
(cid:18)−e tane s+tane s′ ⊥(cid:19) Z (2π)2 where, θ is the angle between the magnetic field and the
B B B direction of the neutrino propagation. Therefore, in the
tane stane s′ tane s′
exp i B B (p + B q )2 limit of m µ we obtain :
× (cid:18)− e ⊥ tane s+tane s′ ⊥ (cid:19) ≥
B B B
=−41πtane si+eBtane s′ ; (55) eνeff = 8√12π3 e2GF gA(1−λ)B cosθ
B B
∞
wcuhlearteewtheeheaffveecntievgelecchtaerdgeteorfmtshuepn-etuotOrin(qo⊥2s)wseinuclet,imtoatceally- × nX=0(−1)n(1+n)βmK1[(n+1)βm]
have to take the limit k 0. Hence, eq.(51) can be cosh (n+1)βµ , (62)
written as → × { }
whereK isthen-thordermodifiedBesselfunctionofthe
n
Π530(q0 =0,~q→0) second kind. It should be noted here that even though
=−q0=l0im,q~→0ieπ3B Z (d22πp)k2η+(p0)Z−∞∞dseis(p2k−m2)−ε|s| tehxiasctretsoulatllisodlidnepaorwienrsthoef Bfie.ldInsttrheenzgetrhoBfr,etqhuiesnrceys,ullotnigs
∞ wavelengthlimit it is only the term linear in that sur-
×Z ds′eis′(p′k2−m2)−ε|s′| (q·p)k−2p20 vives. To get a feeling of the magnitude of thBe effective
0 (cid:0) (cid:1) charge of a neutrino in presence of a background mag-
= lim 2e3 d2pk η (p ) (p2k−m2) netic field, we compare this with that in absence of a
k0=0,q~→0 BZ (2π)2 + 0 (p′k2−m2)+iε magnetic field. In the limit of vanishing chemical poten-
tial, the ratio between the effective charge in a magne-
(q p) 2p2 , (56)
× · k− 0 tisedmediumandinasimple thermalmedium turnsout
(cid:0) (cid:1)
where we have used the following relation, to be :
∞ dseis(p2k−m2)−ε|s| ∞ds′eis′(p′k2−m2)−ε|s′| eνeνef(f(B=)0) = 4cπA3 B (mβ)3K1(mβ)cosθ (63)
Z−∞ Z0 eff B (cid:16)Bc(cid:17)
=2πi δ(p2k−m2) . (57) It can be seen from the equation above that the ratio
(p′2 m2)+iε is proportional to B . Since, in almost all astrophysical
k − Bc
situations encountered so far, this ratio is less than one,
The expression for Π5 given by Eq.(56) contains two intheweakfieldlimitthechargeduetotheunmagnetized
30
parts which can be integrated separately to obtain, plasmaislargerthaninthecaseofamagnetisedplasma.
6
VI. CONCLUSION ACKNOWLEDGMENT
To conclude, we note that only left-handed neutrinos WewouldliketothankN.D.HariDassandPalashB.
acquireaneffectivecharge. Since wehaveperformedour Pal for helpful discussions. We also thank Patrick Au-
calculation for massless, standard-model neutrinos that rencheforgoingthroughthepreliminarydraftanddraw-
should automatically come out of the theory. But re- ingoutattentiontosomeimportantaspects. AKGwould
cent observations indicate that the neutrinos have mass like to thank Laboratoire de Physique Theoretique, An-
(whichallowsforbothleft-handedandright-handedneu- necy, France for supporting a visit where a part of this
trinos). Our treatment can be modified for massive neu- work was carried out.
trinosfollowingthemethodadoptedin[2]andweexpect
that the qualitative aspects of our result should remain
the same.
More importantly, we notice that the presence of a
magnetic field breaks the isotropy of space and intro-
duces a preferential direction. As a consequence neu-
[1] G. G. Raffelt, Stars as Laboratories for Fundamental
trinos propagating along the direction of the magnetic
Physics, (University of Chicago Press, 1996)
field acquire a positive charge whereas those propagat-
[2] J.F.NievesandP.B.Pal,Phys.Rev.D49,1398(1994)
ing in the opposite direction acquire a negative charge. [3] T. Altherrand P. Salati, Nuc.Phys. B 421, 662 (1994)
The net effect of this would then be the creation of a [4] J. Cooperstein, Phys.Rep.163, 95 (1988); H. A.Bethe,
charge current along the direction of the field. Interest- Rev. Mod. Phys. 62,801 (1990); S. Bludman, Da Hsuan
ingly,neutrinospropagatinginadirectionperpendicular Feng,Th.GaisserandS.Pittel,Phys.Rep.256,1(1995)
to the field wouldacquire zeroeffective charge. Whereas [5] R. C. Duncan, S. L. Shapiro and I. Wasserman, Astro-
in an unmagnetized thermal medium neutrinos acquire phys.J.309,141(1986);S.A.ColgateandR.H.White,
Astrophys. J. 143, 626 (1986); H. A. Bethe and J. R.
aneffective chargeirrespectiveoftheir directionofprop-
Wilson ibid 295, 14 (1985); M Ramp and H. T. Janka,
agation. Therefore for isotropic neutrino propagationno
astro-ph/0005438
net current is generated in that case unlike in the pres-
[6] L. O. Silva et. al, Phys. Rev. Lett. 83, 2703 (1999) and
ence of a magnetic field.
references therein
It is worthmentioning here thatthe generationofthis
[7] C. Kouveliotou, T. Strohmayer, K. Hurley, J. van
charge current may play a significant role in the mag- Paradijs,M.H.Finger,S.Dieters,P.Woods,C.Thomp-
netic field generationof neutron stars being produced in son, R.T. Duncan,Astrophys.J. 581, L103 (1999)
supernova explosion. Another possible application could [8] G.Chanmugam,Ann.Rev.Astron.&Astrophys.30,143
be tothe neutrino winddriveninstabilityresponsiblefor (1992)
blowinguptheoutermantleofsupernovaasproposedby [9] L. L. DeRaad Jr., K. A. Milton and N. D. Hari Dass,
[6]. The basic mechanism of this instability can be un- Phys. Rev.D 14, 3326 (1976)
[10] J. C. D’Olivio, J. F. Nievesand P. B. Pal, Phys.Rev.D
derstoodasfollows. Considerthe collisionoftwoplasma
40, 3679 (1989)
fluids. If we consider the motion of one plasma with
[11] A. N. Ioannisian and G. G. Raffelt, Phys. Rev. D 55,
respect to the centre of mass of the other then the dis-
7038 (1997), hep-ph/9612285
persionrelationoftheparticlesofthe firstwoulddepend
[12] S. J. Hardy and D. B. Melrose, Publ. Astron. Soc. Aus.
on the relative velocity between the two plasma. Now 13, 144 (1996)
this velocity dependent dispersion relation can give rise [13] V.N.Oraevsky,V.B.SemikozandYa.A.Smorodinsky,
to damping or instability of the plasma modes depend- JETP Lett. 43, 709 (1986)
ing on the relative velocity between the two media. The [14] R. Shaisultanov, Phys. Rev. D 62, 113005 (2000), hep-
growth rate of the plasmons, in such systems, has been th/0002079
estimatedusingformalismoffinitetemperaturefieldthe- [15] H. Gies and R. Shaisultanov, Phys. Rev. D 62, 73003
ory[22,23]aswellasusingplasmaphysicstechniques[6]. (2000), hep-ph/0003144
[16] H. Gies and R. Shaisultanov, Phys. Lett. B 480, 129
It is important to note that the finite temperature field
(2000), hep-ph/0009342
theory techniques show the damping/growth to be pro-
[17] A.K.Ganguly,S.KonarandP.B.Pal,Phys.Rev.D60,
portionaltoG2 whereasthatcalculatedusingtheplasma
F 105014 (1999), hep-ph/9905206
physics techniques show a scaling as G [6].
F [18] J. Schwinger, Phys. Rev. 82, 664 (1951)
In conclusion, we have calculated the effective charge [19] W. Y. Tsai, Phys. Rev.D 10, 1342 and 2699 (1974)
of neutrinos in a weakly magnetized plasma,in the limit [20] W. Dittrich, Phys. Rev.D 19, 2385 (1979)
m > µ. It is observed that the neutrino charge acquires [21] P.Elmfors,D.GrassoandG.Raffelt,Nucl.Phys.B479,
a direction dependence as a result of the presence of an 3 (1996)
external magnetic field and it is also proportional to the [22] L. Bento, Phys.Rev.D 61, 13004 (2000)
magnitude of the field strength present in the system. [23] J. F. Nieves, Phys. Rev. D61, 113008, 2000; for a simi-
larconclusionusingRelativisticQuantumKinetictheory
7
see: H.T.Elze,T.KodamaandR.Opher,Phys.Rev.D
63, 13008 (2000)
8