(In-)Significance of the Anomalous Magnetic Moment of Charged Fermions for the Equation of State of a Magnetized and Dense Medium

0.5 MB·English

by  E. J. Ferrer

(In-)Significance of the Anomalous Magnetic Moment of
Charged Fermions for the Equation of State of a Magnetized
and Dense Medium
E. J. Ferrer, V de la Incera,1 D. Manreza Paret,2 A. P´erez Mart´ınez,3 and A. Sanchez4
1Department of Physics, University of Texas at El Paso, El Paso, TX 79968, USA
5 2Departamento de Fisica General, Facultad de Fisica,
1 Universidad de la Habana, La Habana, 10400, Cuba
0 3Instituto de Cibern´etica, Matema´tica y F´ısica (ICIMAF), La Habana, 10400, Cuba
2 4Facultad de Ciencias, Universidad Nacional Auto´noma de M´exico,
r Apartado Postal 50-542, M´exico Distrito Federal 04510, M´exico.
p
A We investigate the effects of the anomalous magnetic moment (AMM) in the equation of state
(EoS) of a system of charged fermions at finite density in the presence of a magnetic field. In the
9 regionofstrongmagneticfields(eB>m2)theAMMisfoundfromtheone-loopfermionself-energy.
Incontrasttotheweak-fieldAMMfoundbySchwinger,inthestrongmagneticfieldregiontheAMM

] dependson the Landau level and decreases with it. The effects of the AMM in the EoS of a dense
h
medium are investigated at strong and weak fieldsusing theappropriate AMM expression for each
p
case. In contrast with what has been reported in other works, we find that the AMM of charged
-
p fermions makesno significant contribution to theEoS at any field value.
e
h PACSnumbers: 05.30.-d,12.39.Ki,05.30.Fk,26.60.Kp
[

2
I. INTRODUCTION
v
6
1 The fact that strong magnetic fields populate the vast majority of the astrophysical compact
6
objectsandthatthey cansignificantlyaffectseveralpropertiesofthestarhaveservedasmotivation
6
for many works focused on the study of the Equation of State (EoS) of magnetized systems of
0
fermions and their astrophysicalimplications [1]-[6].
.
1 InthepresenceofamagneticfieldBthedispersionrelationofchargedfermionstakestheformE =
0 p2+2eBl+m2,exhibiting theLandauquantizationofthe cyclotronfrequenciescharacterizedby
5 3
the Landau-level number l = 0,1,2,... [7]. A magnetic field also affects the density of states which
1 p
now becomes proportional to the field, so the three-momentum integrals change as
:
v
i d3p eB
X
2 g(l) dp (1)
(2π)3 → (2π)2 z
r Z l Z
a X
The factor g(l) = [2 (δ )] takes into account the double spin degeneracy of all the Landau
l0
−
levels except l = 0. In addition, a magnetic field breaks the rotational SO(3) symmetry, giving
rise to an anisotropy in the energy-momentum tensor [8] and producing a pressure splitting in
two distinguishable components, one along the field (the longitudinal pressure) and another in the
perpendicular direction (the transverse pressure). As a consequence, a system of fermions in a
constant and uniform magnetic field exhibits an anisotropic EoS [4]-[6].
Besidesmodifiyingtheone-particleDiracHamiltonian,amagneticfieldcanalsoaffecttheradiative
corrections of the fermion self-energy because it introduces an additional tensor F in the theory
µν
that gives rise to new independent structures like 1 σ Fµν, with σ = i[γ ,γ ]. This new term
2T µν µν 2 µ ν
correspondstothecouplingbetweenthe fieldandthe fermionanomalousmagneticmoment(AMM)
[9], which in generalcan be a function of the magnetic field. It induces a Zeeman splitting in the
T
fermiondispersionthatremovesthe spindegeneracy[10],andthe followingchangeinthe density of
2
states
eB
g(l,σ) dp , g(l,σ)=δ +(1 δ ) (2)
(2π)2 z l0 − l0
l Z σ= 1
X X±
with the spin projections σ = 1.
AsshownbySchwinger[9]m±anyyearsago,atweakfields(eB m2),onecanmakeanexpansion
≪
of the fermion self-energy in powers of the magnetic field and find that the leading contribution
proportional to σ is linear in the field and given (in natural units) by B with = (α/2π)µ ,
µν B
T T
where µ = e/2m is the Bohr magneton. In this approximation, is simply independent of the
B
T
magnetic field. At strong field, however, the coefficient of the σ structure varies as a square
µν
logarithm of the field [11] and hence cannot be expanded in powers of the magnetic field. At finite
temperature and/or density, the concept of weak field needs to be revisited, as the field may be
strong with respect to one of the scales, but weak with respect to the others.
Chargedandmassivefermionsalwayspossessmagneticmomentwhichinprinciplecanproducein-
terestingphysicaleffectsviathemodificationoftheself-energy. Moreover,masslesschargedfermions
in the presence of a magnetic field can acquire a dynamical magnetic moment [12]-[16] through the
phenomenon of magnetic catalysis of chiral symmetry breaking (MCχSB) [17]. The mechanism
responsibleforthiseffectisrelatedtothedimensionalreductionoftheinfrareddynamicsofthepar-
ticlesinthelowestLandaulevel(LLL).Suchareductionfavorsthe formationofachiralcondensate
because there is no energy gap between the infrared fermions in the LLL and the LLL antiparticles
in the Dirac sea. This effect has been actively investigated for the last two decades [12]-[19]. In the
original studies of the MCχSB phenomenon [17]-[19], the catalyzed chiral condensate was assumed
to generate only a fermion dynamical mass. Recently, however, it has been shown that in QED
[12]-[13],aswellasinquarksystemswith[15]andwithout[16]finitedensity,the MCχSBinevitably
leads to a dynamicalAMM togetherwith a dynamicalfermionmass. In masslessQED, the dynam-
ical AMM gives rise to a non-perturbative Lande g-factor, a Bohr magneton proportional to the
inverse of the dynamical mass, and to the realization of a non-perturbative Zeeman effect [12]-[13].
TheAMMtermintheHamiltonianchangestheenergyspectrumofthefermionsandcanaffectin
principle the properties of the system. Notice that certain neutral particles which, like the neutron,
are composed by charged particles (charged quarks in this case), can also have nonzero AMM. The
effectsofthenucleons’andquarks’AMMonthestatisticsofmagnetizedmatterhavebeendiscussed
in many works [2, 3]. The AMM has been linked among other strong-field effects to stiffening the
EoS in magnetized stars and to a dramatic variation of the particle fraction, which at very high
magnetic fields would lead, for example, to pure neutron matter (in the papers of Ref. [2, 3] one or
both of these findings are discussed). However, when investigating the effects of the AMM in any
physicalprocessweshouldbe carefulinusingthe analyticexpressionofthe AMMthatis consistent
with the magnetic-field strength under consideration. In particular, as we will discuss in detail
below,consideringalinear-in-Bapproach,whichisthe approximationusedthroughoutallthe Refs.
[2,3], is only consistentinthe regionofweakmagnetic fields wherea largenumberofLandaulevels
are occupied because √eB is smaller than all the other energy scales like mass, temperature, and
chemical potential.
The critical field, below which the weak approximation is reliable then depends on the content
of the stellar matter, the temperature and the density. If one can ignore T and µ, the critical
field separating the weak and strong field regions is determined by the particle mass. For each
particle species, the critical field can then be obtained by equating the magnetic energy ~ω , where
c
ω = qB/mc is the cyclotron frequency in cgs units, to the corresponding rest-energy mc2. The
c
range of critical fields is then quite wide. For example, for electrons, B(e) = 4.4 1013 Gauss, for
c
×
quark matter formed by u and d quarks with current masses m = m = 5 MeV/c2, it is B(u,d) =
u d c
102B(e) = 4.4 1015 Gauss, for protons, whose mass is 938 MeV/c2, one finds B(p) = 1.6 1020
c c
× ×
Gauss, while charged hyperons, which are much heavier, will have a critical field two orders of
3
magnitude larger than protons’. At zero temperature and density, a field larger than the critical
one for that type of particle will constraint them to their LLL. In a system with different types
of particles, a field may be strong, hence over the critical field, for some of them and weak, below
the critical, for others, so care must be taken when using weak and strong field approximations to
consider such subtleties.
In the present paper we are interested in revising the role of the AMM in the EoS of systems
of charged fermions, under both weak and strong magnetic fields. This is a due task given that in
severaloftheworksthathavestudiedtheeffectsoftheAMMinthethermodynamicalpropertiesone
canpointoutseveralissuesinthewaytheresultshavebeenobtained. Oneoftheseissuesisthatthe
strong-fieldregionhasbeenexploredinconsistentlyconsideringSchwinger’sresultfortheAMMofall
the particles, thus ignoringthe existence of different critical fields and the fact that the Schwinger’s
approximationfor the AMM breaksdownfor fields ofthe order ofor largerthan the criticalone, as
pointed out many years ago in Ref. [11]. Second, when calculating the pressure, some works have
ignored the existence of a pressure anisotropy [4]-[6] in a strong magnetic field, so the results were
obtained basically using a single pressure. Third, some papers neglected the contribution of the
Maxwellmagnetic pressureproportionalto B2/2and claimedthatthe SchwingerAMM produceda
significantcontributionto the statisticalquantities, but they concludedthis by consideringa region
of fields where the Schwinger approximation not only breaks down, but the magnetic pressure can
dominate the matter pressure and erase any possible effect of the AMM, as will be shown in this
paper. Fourth,thepressureofthemagnetizedvacuum,thatis,thecontributionthatdoesnotdepend
on temperature or density, was also neglected at strong fields where it can be important. To clarify
all these issues, we shall analyze, through analytical and numerical calculations, the significance of
the AMM contribution to the main statistical quantities of the magnetized system in the weak and
strong-fieldapproximations,aswellastotheEoSofdensesystemswithaninterestforastrophysical
applications. With this goal in mind, we shall investigate the weight of each of the participating
contributions (Maxwell pressure, vacuum pressure, etc.) into the system EoS for different field
values. As will be demonstrated, our thorough analysis leads us to conclude that, when working
consistently, the quantum effect of the AMM of charged particles is negligibly small for the EoS of
the magnetized system at both weak and strong fields.
The paper is organizedas follow. In Sec. II we give the one-loopself-energy of a chargedfermion
system in the presence of a constant and uniform magnetic field using the Ritus’s method [20],
and find the AMM analytical expression for the different LL’s in the strong-field limit. In Sec.
III, it is calculated the one-loop thermodynamical potential depending on the AMM in the strong-
field approximation. The result is given as the sum of the renormalized vacuum contribution,
the contribution at zero temperature and finite density and the thermal contribution. In Sec.
IV, for the sake of completeness, we calculate the AMM in the weak-field approximation using a
combinationofRitus eigenfunctionmethodandproper-timerepresentation. InSec. V,we calculate
theone-loopthermodynamicpotentialintheweak-fieldapproximationincludingtheAMMfoundin
thatapproximation. Therenormalizedvacuumcontributionandthezero-temperaturefinite-density
contribution are presented up to ((eB)4) order. In Sec. VI, we present the numerical results for
O
themainthermodynamicquantities,whichdependontheAMM,intheweak-andstrong-magnetic-
filed limits. There, we make a thoughtful analysis to determine the significance of the AMM for the
EoS of strongly and weakly magnetized systems of charged fermions. Finally, in Sec. VII we state
our concluding remarks. We also include four Appendices. In Appendix A, we give details on the
calculationofthe thermodynamicalpotential in the strong-fieldapproximationatT =0 and µ=0.
6 6
In Appendix B, we discuss some issues in the calculation of the effective potential at B = 0 in the
6
Dittrich’s approach. In Appendix C, we derive the Schwinger propagator at B = 0, starting from
6
theRitus’sformalism. InAppendixD,thedetailsofthecalculationofthethermodynamicpotential
in the weak-field approximation at T =0 and µ=0 are given.
6 6
4
II. AMM IN THE STRONG-FIELD APPROXIMATION
The radiative corrections to the magnetic moment of a charged particle in the presence of a
magnetic field can be found from the one-loop fermion self energy
Σ(x,x)= ie2γµG(x,x)γνD (x x), (3)
′ ′ µν ′
− −
G(x,x) denotes the fermion’s propagator in the presence of a uniform and constant magnetic field
′
and D (x x) is the photon propagator.
µν ′
−
One can transform the self-energy to momentum space by using Ritus’s approach
Σ(p,p′)= d4xd4yElp(x)Σ(x,y)Elp′(y)=(2π)4δ(4)(p−p′)Π(l)Σl(p), (4)
Z
Index l denotes the Landau-level number; Π(l) = ∆(sgn(eBb))δl0 +I(1 δel0) is a projector that
−
separatestheLLL(l=0),withasinglespinprojection,fromtherest(l >0)withtwo;δ(4)(p p)=
′
δll′δ(p p )δ(p p )δ(p p ). The Ritus eigenfuntions [20] are given by −
0− ′0 2− ′2 3− ′3
b
El(x)= Eσ(x)∆(σ), El γ0(El) γ0 (5)
p p p ≡ p †
σ= 1
X±
with
I iγ1γ2
∆( )= ± , (6)
± 2
are spin up (+) and down ( ) projectors, and
−
Ep+(x)=Nle−i(p0x0+p2x2+p3x3)Dl(ρ),
Ep−(x)=Nl 1e−i(p0x0+p2x2+p3x3)Dl 1(ρ) (7)
− −
with normalizationconstant N =(4πeB)1/4/√l!, and D (ρ) are the parabolic cylinder functions of
l l
argument ρ=√2eB(x p /eB).
1 2
−
In momentum space the general structure of the self energy is [19]
Σl(p)=Zlpµγ +Zl pµγ +M I+i γ1γ2, (8)
k k µk ⊥ ⊥ µ⊥ l Tl
Notice the separation between parallel pν = (p0,0,0,p3) and perpendicular pν = (0,0,√2eBl,0)
components due to the spatial symmetrykbreaking in a magnetic field that on⊥ly leaves intact the
subgroupofrotationsalongthefielddirection. In(8),Zl,Zl arethewavefunction’srenormalization
coefficients. The coefficients M and are respectivelky th⊥e radiative corrections to the mass and
l l
T
the magnetic moment. Each of them has to be determined as a solution of the Schwinger-Dyson
(SD) equations of the theory at the given approximation.
In the one-loop approximation, the Schwinger-Dyson equation leads to an infinite set of couple
equations that take the form [12–14]
Σl(p)Π(l) = −ie2(2eB)Π(l) (2dπ4q)4e−q2qb⊥2 [Ll+Ll+1+Ll−1], l=0,1,2,.... (9)
Z
b
b
5
with
L = γ Gl(p q)γ ,
l µk − µk
L = ∆( )γ Gl 1(p q)γ ∆( ) (10)
l±1 ± µ⊥ ± − µ⊥ ±
and fermion propagator
p γ+m
Gl(p)= · Π(l), (11)
p2 m2
−
Here, we introduced the notation q = q / 2eB , p = (p0,0, 2eB l,p3) and (p q) =
µ µ | | µ − | | − µ
(p0 q0,0, 2eB l,p3 q3). Henceforth, we assume eB > 0. As it will become clear below, the
− − | | − p p
representation (9) of the self-energybis particularly convenient for strong-field calculations.
p
FromEqs. (8)and(9), wecanextractthe equationsforthe AMMateachLL.InEuclideanspace
they are
E0 =(M0+T0)=e2m(4eB) (2dπ4q)4e−q2qb⊥2 (p q)12+m2 + (p q)12+m2 , (12)
Z (cid:18) − 0 − 1 (cid:19)
b
b
Tl = −e2m(2eB) (2dπ4q)4e−q2qb⊥2 . (p q)21 +m2 − (p q)21 +m2 , l ≥1 (13)
Z (cid:20) − l+1 − l 1 (cid:21)
−
b
Eq. (12) reflects the single spin orientation of the fermions in the LLL (l = 0) and hence the
b
impossibility of determining M and independently [12, 13]. Thus, E cannot be interpreted as
0 0 0
T
an AMM term, but as the radiative correction to the rest-energy of the LLL particles. Notice, that
E will not produce any Zeeman splitting in the modes of the LLL quasiparticles.
0
In the infrared limit p = 0,p 0, and considering the strong-field approximation, the leading
0 3
→
contributions to (12) and (13) for l =1 are respectively given by
E0 =M0+T0 ≃ e82πm3 dqk2dq⊥2 e−q2qb⊥2 q2+1m2 =m4απ ln2(m2/2eB), (14)
Z
k
b b
b b b
T1 ≃ 1e62πm3 dqk2dq⊥2 e−q2qb⊥2 q2+1m2 =m8απ ln2(m2/2eB), (15)
Z
k
Note that the leading contribution ibn (1b5) comes from the spin-down particles in the first LL (the
b b b
term l 1 in (13) for l =1), since √2eBl acts,for l 1,as a suppressingfactor in the denominator
− ≥
ofthe fermionpropagator. The result(14) coincideswith thatobtainedmany yearsagoinRef. [11]
using a different method. As in massless-QED [12, 13], the relation =E /2 is satisfied here too.
1 0
T
For the remaining , l >1, we have
l
T
αm
Tl = − 16π2 e−Mˆl2+1 −γΓ[0,−Mˆl2+1]+e2γΓ[0,−Mˆl2−1]
iπe2lnMˆ2 ne2E (Mˆ2 )lnMˆ2 +iπlnMˆ2 +E (Mˆ2 )lnMˆ2
− l−1− i l−1 l−1 l+1 i l+1 l+1
G3,0 1,1 Mˆ2 +e2G3,0 1,1 Mˆ2 , (16)
− 2,3 0,0,0 − l+1 2,3 0,0,0 − l−1
(cid:16) (cid:12) (cid:17) (cid:16) (cid:12) (cid:17)o
wfuinthctMioˆnl2±,1Γ=[0,mˆz]2i+s(tlh±e1i)n,cγom≃p0le.5(cid:12)(cid:12)te77g2a1m6misathfuenEcutiloenr’sancodnGst(cid:12)(cid:12)amn,nt,Ea1i,[.z..],adpenzotitnhgetMheeeijxeproGn-efnutniactliionnte[g2r1a].l
p,q b1,...,bq
(cid:0) (cid:12) (cid:1)
(cid:12)
6

0.08
1/2
m/(2eB) =0.1
1/2
m/(2eB) =0.01
0.06
1
0.04

/
1
>
l
0.02
0.00
10 20 30 40 50
l
FIG. 1: Comparison betweenthe AMMsTl and T1 in the strong field region for Landau levels l>1.Theplotshows
a sharp decrease of the ratio Tl>1/T1 with increasing Landau levels for two field values. The largest values of Tl>1
occurforthelowest(l>1)-value,buttheyarestilltwoordersofmagnitudesmallerthanT1. Thestrongerthefield,
thesmallerthevaluesofTl>1forthesameLandaulevel,andthequickertheyapproachtotheirasymptoticnegligible
value.
InFig.1weshowhowtheAMM’sforl >1decreasewithrespectto astheLLincreases. Notice
1
T
thattheAMMatstrongfieldisrelevantonlyforthefirstLandaulevel,whereitgrowsasthesquare
logarithmof the field. As seen in Fig.1, already in the second LL the AMM decreases in two orders
with respect to its value at l=1, / 0.0668 for mˆ =0.1.
2 1
T T ∼
One can explicitly see from (14)-(16) that, in contrast to the AMM found by Schwinger [9] in
the weak-field limit, which was the same for all LL’s and had a linear dependence with the field,
at strong field the AMM does not depend linearly on the field and is different for each LL. Clearly,
using the Schwinger AMM in the strong-field region would be totally inconsistent and care should
be taken not to draw any physical conclusions obtained with such a wrong approach.
III. THERMODYNAMIC POTENTIAL WITH AMM IN THE STRONG-FIELD
REGION
To investigate the effects of the AMM in the EoS we consider an effective theory on which the
fermion propagator is dressed by the one-loop fermion self-energy in the magnetic field, which
dependsontheAMM.ThelackofZeemansplitting(seeEq. (2))intheLLLseparatesthepropagator
in the LLL from those in the rest of the levels, so the dressed propagatorstake the form
G−01(p)=(pk·γk−m)∆(+) (17)
7
and
G−l 1(p)=p·γ−m−iTlγ1γ2 (18)
with p = (ip4,0, 2eB l,p3) for l = 0,1,2,... in Euclidean space. Notice that in (17) we do not
| |
includethe correctionE ,neitherin(18)theone-loopcorrectionstothe mass,astheyarenegligible
0
p
compared to the renormalized mass at B = 0, m. However, since gives rise to a new Lorentz
l
T
structure, it is included in (18).
The fermion contribution to the thermodynamic potential of this effective theory is
Ω(B,µ,T)=−eβB  ∞ (2dπp3)2 lndetG−01(p∗)+ ∞ ∞ (2dπp3)2 lndetG−l 1(p∗) (19)
Xp4 −Z∞ σX=±1Xl=1Xp4 −Z∞
 
Here, β = 1/T denotes the inverse temperature, µ the fermion chemical potential and p = (ip4
∗
−
µ,0,√2eBl,p3).
Performing the sum in Matsubara frequencies and calculating the determinants in Eq. (19) we
obtain
eB 1 β
Ω(B,µ,T)= dp ln cosh (E µ) , (20)
−4π2 3β 2 ησl −
ησl Z (cid:20) (cid:21)
X
which can be rewritten as
Ω(B,µ,T) = −214eπB2 ∞ dp3 |Eησl−µ|− 4eπB2 ∞ dp3 β1 ln(1+e−β|Eησl−µ|) (21)
Z−∞ Xησl Z−∞ Xησl
In these expressions the sum in the energies E include particles/antiparticles (η = ), up/down
ησl
±
spin(σ = ),andLandaulevel l indices. For the LLL,only onespinprojectioncontributes andthe
±
energy becomes
E =η p2+m2 l=0, η = 1 (22)
η,0 3 ±
q
For each l=0, the AMM separates the energies of up and down spin (σ = ) as
6 ±
E =η p2+( 2eBl+m2+σ )2, l >1, σ = 1, η = 1 (23)
ησl 3 Tl ± ±
q
p
Adding and subtracting the vacuum term in (21), one can write the thermodynamic potential as
the sum of vacuum (Ω ), zero-temperature (Ω ), and finite-temperature (Ω ) contributions,
vac µ β
Ω(B,µ,T)=Ω +Ω +Ω (24)
vac µ β
with
eB ∞
Ω =Ω(B,0,0)= dp E , (25)
vac −8π2 3 | ησl|
Z−∞ Xησl
eB ∞
Ω =Ω(B,µ,0)= dp (E µ E ), (26)
µ −8π2 3 | ησl − |−| ησl|
Z−∞ Xησl
8
Ωβ =Ω(B,0,T)=−4eπB2 ∞ dp3 β1 ln(1+e−β|Eησl−µ|) (27)
Z−∞ Xησl
As shown in Section II, in the strong-field approximation all the for l > 1 are very small and
l
T
canbe neglected. Inthis approximationallthe energymodes,exceptfor l=1,reduce to the modes
of the undressed theory.
InAppendixA,wegivedetailsonthecalculationandcarryouttherenormalizationofthevacuum
term (25) in the strong-field region to obtain the following renormalized thermodynamic potential
in the strong-field approximation
ΩS =ΩS(R)+ΩS +ΩS, (28)
R vac µ β
with
ΩS(R) = 1 ∞ dse−sm2 eBscoth(eBs) 1 (eBs)2 + eB 2 ln2eB +2 , (29)
vac 8π2 s3 − − 3 4π2T1 m2
Z1/Λ2 (cid:20) (cid:21) (cid:20) (cid:21)
eB µ+ µ2 m2
ΩS = θ(µ m) µ µ2 m2 m2ln −
µ − 4π2( − " − − pm #
p
µ+ µ2 M+2
+ θ(µ M+) µ µ2 M+2 M+2ln − 1
− 1  − 1 − 1 qM+ 
q 1
 
2
+ θ(µ−M1−)µqµ2−M1−2−M1−2lnµ+qMµ21−−M1− 
 
+ 2 ∞ θ(µ M ) µ µ2 M 2 M 2lnµ+ µ2−Ml2 , (30)
l l l
− " − − pMl #)
l=2 q
X
and
ΩSβ = − 2eπB2β ∞dp3ln(1+e−β|√p23+m2+µ|)(1+e−β|√p23+m2−µ|)
Z0
− 2eπB2β ∞dp3ln(1+e−β|qp23+M1+2+µ|)(1+e−β|qp23+M1+2−µ|)
Z0
− 2eπB2β ∞dp3ln(1+e−β|qp23+M1−2+µ|)(1+e−β|qp23+M1−2−µ|)
Z0
− πe2Bβ ∞ ∞dp3ln(1+e−β|√p23+Ml2+µ|)(1+e−β|√p23+Ml2−µ|), (31)
l=2Z0
X
where
M1± = 2eB+m2±T1, Ml = 2eBl+m2 (32)
The leading vacuum contribpution to Ω in the strong-fielpd approximation (see Appendix A for
details) is then
αB2 eB eB 2eB
ΩS(R) = ln + 2 ln +2
vac − 6π m2 4π2T1 m2
(cid:18) (cid:19) (cid:20) (cid:21)
αB2 eB eB αm 2eB 2 2eB
= ln + ln2 ln +2 , (33)
− 6π m2 4π2 8π m2 m2
(cid:18) (cid:19) (cid:20)(cid:16) (cid:17) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21)
9
where in the second line was evaluated using (15), and the first term was obtained in [22]
1
T
calculating the effective potential at strong field and neglecting the AMM.
Here the following comment is in order. The contribution of the AMM to the vacuum part of
the thermodynamic potential has been previously calculated in [23] with the help of the Green’s
function. In the strong-field region the result reported in [23] was
αB2 eB (eB)2 α 2 eB
ΩˆS(R) = ln ln , (34)
vac − 6π m2 − 32π2 2π m2
(cid:18) (cid:19) (cid:16) (cid:17) (cid:18) (cid:19)
Clearly, there is a discrepancy between (34) and (33), the origin of which can be traced back to
some inconsistencies in the treatment followed in [23] to obtain the strong-field result. On the one
hand, Ref. [23] considered the AMM found by Schwinger and used it in calculations at arbitrary
field strength, including the strong field region, despite that as already discussed, in this region the
AMM becomes a very different function of the magnetic field and depends on the LL. On the other
hand, as described in Appendix B, several steps of the calculations done in [23] were only valid for
weakfields,however,theywereusedindistinctly forweakandstrongfields. The calculationsofRef.
[23] are then reliable in the weak-field region,but fail to describe the AMM-dependent terms in the
strong-field case.
IV. AMM IN THE WEAK-FIELD APPROXIMATION
To get the AMM in the weak-fieldlimit, it will be convenient,for the sake of clarityand to shade
light in our discussion, to use an alternative method that combines the Ritus’s approach, where
the LL contributions are explicit, and the proper time formalism. The result has to coincide with
that found by Schwinger [24] by using an independent method. In doing this, we will stress the
stepswhere the weak-fieldapproximationbecomesabasicelementofthe derivation. Also,itwillbe
apparent the different behavior of the AMM’s for the different LL’s in each approximation.
Due to the fact that the energy separation between consecutive LL’s is proportional to √2eB,
it is expected that at weak field all Landau levels will contribute on equal footing to the fermion
self-energy. Then, we will find convenient, in order to take into account the contributions of all the
LL’s into the fermion self-energy, to work with the self-energy operator in the configuration space
(3) with the field dependent fermion Green’s function given in the Ritus’s approach as
d4p
G(x,y)= E (x)Gl(p)E (y). (35)
(2π)4 p p
Z
X
where
1
Gl(p)= Π(l) (36)
p m
6 −
is the fermion propagator in momentum space and we introduced the notation
d4p ∞ dp0dp2dp3
(37)
(2π)4 ≡ (2π)4
Z l=0Z
X X
We can rewrite the electron propagator in (35) as
d4p E (x)Π(l)E (y)
p p
G(x,y) = (Π +m) (38)
6 x (2π)4 p2 m2
Z −
X
where Π µ = i∂ µ eAµ and we used the property, Π E (x) = E (x)p, satisfied by the Ritus
x x x p p
eigenfunctions, E (x−), defined in (5)-(7). 6 6
p
10
Now, to perform the summation over all Landau levels, we use the proper-time representation
and the integralrepresentationfor the parabolic cylinder functions, so, we get (See Appendix C for
details)
G(x,y) = −(6Πx+m)Φ((4xπ,)y2) ∞ ssienB(edBss) e−is(m2−iǫ)e−i[41s(x−y)2||−4tane(BeBs)(x−y)2⊥]
Z0
eieBs∆(+)+e ieBs∆( ) (39)
−
× −
where Φ(x,y)=exp(cid:2)ieB(x y )(x +y ) is(cid:3)the well known Schwinger’s phase (recallthat F =
2 2− 2 1 1 21
B) [25].
In the weak-field (cid:2)approximation (eB (cid:3)m2), we can perform a Taylor expansion up to linear
≪
terms in eB in the integrand of Eq. (39), to find
G(x,y) ≃ −Φ((4xπ,)y2) ∞ ds2s e−is(m2−iǫ)e−i(x−4sy)2 6x2−s6y + 2eγµFµν(x−y)ν +m
Z0 (cid:18) (cid:19)
[1+ieBs(∆(+) ∆( ))], (40)
× − −
where we used the fact that the Schwinger phase satisfies the identity
e
Πx Φ(x,y) = F (x y)νΦ(x,y). (41)
µ µν
2 −
With the help of the identities eB(∆(+) ∆( )) = eσµνF and [γ ,σ ]FµνXρ =
4iγµF Xρ with Xρ an arbitrary four-vector, −we rew−rite Eq.−(420) as µν ρ µν
µρ
−
G(x,y) = Φ(x,y) ∞ ds e−is(m2−iǫ)e−i(x−4sy)2
− (4π)2 s2
Z0
x y 1 e e
6 −6 + x y , i σµνF +m 1 si σµνF
µν µν
× 2s 4 6 −6 2 − 2
(cid:18) h i (cid:19)h i
Φ(x,y) ∞ ds e−is(m2−iǫ)e−i(x−4sy)2
≃ − (4π)2 s2
Z0
x y 1 e e
6 −6 x y , i σµνF +m msi σµνF (42)
µν µν
× 2s − 4 6 −6 2 − 2
(cid:18) n o (cid:19)
where in the last line we kept up to linear terms in eB, in agreement with the weak-field approxi-
mation.
Tofindthefermionself-energywehavetosubstitutein(3)thephotonpropagatorinconfiguration
space
Dµν(x−y) = −(4iπgµ)2ν ∞ dt2t e−i(x−4ty)2, (43)
Z0
together with the fermion propagator (42). Then, after using the identities γµγργ = 2γρ,
µ
γµγαγβγ =4gαβ and γµγργαγβγ = 2γβγαγρ, the fermion self-energy reduces to −
µ µ
−
Σ(x,y) = ie2Φ(x,y) ∞ dsdt e−is(m2−iǫ)e−i(s1+1t)(x−4y)2
(4π)4 s2t2
Z0
x y 1 e
6 −6 x y , i σµνF +4m (44)
µν
× − s − 2 6 −6 2
(cid:18) n o (cid:19)
Now, we introduce a new variable defined as
1 1 1
= + (45)
w s t