Particle production from axial fields

10 Pages·0.19 MB·English

by  Antonio L. Maroto

Particle production from axial fields
Antonio L. Maroto
∗
Astronomy Centre, University of Sussex, Falmer, Brighton, U.K. BN1 9QJ
(February 1, 2008)
Westudytheproductionofmassivefermions inarbitrary vectorandaxial-vectorclassical back-
groundsusingeffectiveaction techniques. Aperturbativecalculation showsthedifferentfeaturesof
each field and in particular it is seen that pure temporal axial fields can produce particles whereas
it is not possible for a pure vector background. We also analyze from a non-perturbative point of
viewaparticularconfigurationwithconstantelectricandaxialfieldsandshowthatthepresenceof
the axial background inhibits theproduction from the electric field.
PACS numbers: 98.80.Cq
9
9
9
I. INTRODUCTION
1

n
a The production of particles from classical backgrounds has become a very active area of researchin the last years.
J We can find it in numerous and disconnected fields of physics such as cosmology [1,2], heavy-ion collision or even

2 plasma physics [3]. The pioneer work of Schwinger[4] mainly focused on the production of electron-positronpairs by
1 strongelectrostaticfields. Sincethenmanyothersourcesofparticlescreationhavebeenstudiedintheliterature. Thus

forexamplewecanmention,timedependentgravitationalfields[2],varyingPlanckmassmodels[5],compactification
2
ofextradimensions[6],dilatonfield[5],inflatonfield,etc. TheoriginalSchwinger’sworkwasbasedontheproper-time
v
technique for the evaluation of the effective action. This method allowed him to obtain an exact non-perturbative
7
4 result for the total number of particles produced. However,although some other electromagnetic configurationshave
4 been studied [7] and some particular cases have also been solved [8], in generalthe number of models for which exact
0 results can be obtained is very limited.
1 In spite of its generalized use, it is probably in cosmology where particle production has been applied more exten-
8
sively. Thus for instance, it is believed that the presence of any small anisotropy in the early universe could have
9
been erased very effectively by particle emssion processes [9]. The theory of reheating after inflation is also based on
/
h the resonant production of particles due to the oscillations of the inflaton field [10]. The reheating of fermions has
p been considered in [11]. In addition, the formation of large scale structures in the early universe is closely related to
-
thegenerationfromvacuumfluctuationofsmallseeddensityinhomogeneitiesthatduetothegravitationalinstability
p
e grew to give rise to the currentgalactic structure. The generationof the vacuum fluctuations canbe studied in much
h the same way as the production of scalar particles in a Robertson-Walker background [12]. Recently the problem of
: particle production has also been extended to the area of the string cosmologyin which, apartfromthe gravitational
v
i background, there is also an additional scalar field, the dilaton, that can also give rise to the generation of particles
X
[5,13].
r In general, the problem of calculating particle production can be approached in two different ways. On one hand,
a
the Bogolyubov technique [1] that allows the calculation of the spectrum of the particle produced. It is based on the
resolution of the harmonic-oscillator equation with variable frequency and only in some particular cases permits the
derivation of exact results. On the other hand we have the already mentioned effective action technique that allows
to obtain the total number of particles produced in a much simpler way [4,14,15], although the difficulties in finding
exact results are also present.
Most of the existing works about the creation of particles from vacuum fluctuations concentrate in the production
of boson fields in the presence of scalar, vector or gravitational backgrounds. In this paper we will study a different
source for the production of fermions, it is the presence of generalvector and axial-vectorbackgrounds. We will thus
extendtheSchwinger’sworkbyincludingtheeffectsofanon-vanishingaxialfield. Classicalaxialbackgroundsappear
naturallyinmoderntheoriesofgravitysuchassupergravity[16]orinlow-energystringeffectiveactions[17,18]. Both
theories contain torsion (or axion) fields as a fundamental ingredient and, in fact, recently several solutions with
non-vanishing axial fields have been found in the context of string cosmology [19,20]. On the other hand, the idea
of modifying General Relativity by introducing an arbitrary metric connection with torsion is an old one [21], and
∗
E-mail:[email protected]
1
in some sense quite natural from the point of view of the gauge theories of gravity [22]. This torsion field is coupled
minimallytofermionsbymeansofitspseudotrace,thusprovidinganewmechanismfortheproductionofparticles. In
factinthisgravitationaltheories,torsionwouldbethedominantmechanismfortheproductionofmasslessfermionsin
cosmologicalRobertson-Walkerbackgrounds. Thisisduetothefactthatwhengravityisminimallycoupledtomasless
fermions, the theory is conformally invariant. This implies the well known result of absence of particle production.
However,thepresenceofadditionalfields,suchastorsionormetricanisotropiesbreaksthatinvariance. Ourworkwill
be based on the effective action method, first from a perturbative point of view and then we will study a particular
case in which a non-perturbative calculation is viable.
The paper is organized as follows: in section 2 we introduce the lagrangian for the model and also give a brief
introduction to the effective action technique. In section3, we perform the perturbative calculation and compare the
result with the pure vector case. Section 4 is devoted to the particular case with constant electric and axialfield and
a non-perturbative result is obtained in the limit of small axial fields and in section 5 we give the main conclusions
of the work. Finally we have included an Appendix with some useful formalae of standard perturbation theory in
quantum mechanics.
II. MODEL LAGRANGIAN AND THE EFFECTIVE ACTION METHOD
We will consider the following interactionlagrangianfor massive fermions minimally coupled to abelian vectorand
axial-vector fields. For simplicity we will use the left-right notation and at the end we will recover the vector-axial
fields:
=ψ(iD m+iǫ)ψ (1)
L 6 −
where iD =iγµ(∂ +iA P +iB P ). As usual the +iǫ factor is introduced to ensure the convergence of the path
µ µ L µ R
6
integralandthe left and rightprojectorsare defined as: P =(1 γ )/2 and P =(1+γ )/2. We will use the chiral
L 5 R 5
−
representation for the Dirac matrices in which γ is diagonal. The coupling constants are included in the own fields.
5
Let us now introduce the effective action (EA) for the A and B fields that is obtained after integrating the
µ µ
fermions out:
0,t 0,t =Z[A,B]=eiW[A,B] =N dψdψei d4xL
h →∞| →−∞i
Z R
=Ndet(iD m+iǫ) (2)
6 −
Here, 0,t denote the initial and final vacuum states that in general will be different due to the presence of
| → ±∞i
the external sources. N is a normalization constant that is taken as usual in such a way that Z[0,0] = 1, this will
allow to discard the vacuum divergences as shown below.
The EA will be in general a complex non-local functional in the external fields. Its real part will contain the
divergences that will be renormalized by adding suitable local counterterms to the action. The imaginary part will
be finite and will contain the information about the particle production probabilities. In fact, the probability that
the vacuum remains stable is given by 0,t 0,t 2. Therefore the probability that the vacuum decays
|h → ∞| → −∞i|
by particle emission will be given by:
P =1 0,t 0,t 2 =1 e−2ImW[A,B] 2ImW[A,B] (3)
−|h →∞| →−∞i| − ≃
When we only have a vector field such that its corresponding electric field is constant, the probability density per
unit time and unit volume p can be obtained exactly and the result is given by [4]:
e2E2 ∞ 1 m2nπ
p= e− eE (4)
4π3 n2
n=1
X
Howeverfornon-constantelectromagneticfields orwhenthe axialpartisswitchedon,the computationbecomesvery
involved and it is neccessary to rely on some perturbative method.
Beforeconcludingthissectionwewillmentionthatalthoughthenotionoftorsionappearsinstringsandsupergravity
in slightly different ways, both can be interpreted as the antisymmetric part of the affine connection in pseudo-
Riemannian geometry [18]. Thus, if the components of the metric connection are: Γˆλ , its antisymmetric part:
µν
Tλ =Γˆλ Γˆλ is knownas the torsiontensor. By means of the Einstein equivalence principle, it is now possible
µν µν− νµ
to minimally couple torsion to fermion fields [23,24], one gets:
2
i
=ψiγµ ∂ +Ω + S γ ψ (5)
µ µ µ 5
L 8
(cid:18) (cid:19)
where S =ǫ Tµνλ is the torsion pseudotrace and Ω the spin-connection.
ρ µνλρ µ
III. PERTURBATIVE METHOD
In this sectionwe presentthe evaluationof the EA in (2) as an expansionin the externalfields, i.e anexpansionin
coupling constants. Let us start by writting:
W[A,B]=iTr log(iD m+iǫ) (6)
6 −
That can formally be expanded as:
W[A,B]=i (−1)kTr (i ∂ m)−1(APL+BPR) k (7)
k 6 − 6 6
k=1
X (cid:0) (cid:1)
where the Dirac propagator is defined as usual by:
q+m
(i6∂−m)−xy1 = dq˜e−iq(x−y)q2 6 m2+iǫ (8)
Z −
The functional traces Tr are evaluated in dimensional regularization with D = 4 ǫ and dq˜= µǫdDq/(2π)D. The
−
lowest order contribution in the expansion is given by the two-point terms, i.e:
i q+m p+m
W[A,B](2) = 2 d4xd4ydp˜dq˜q62 m2e−iq(x−y)(A6 yPL+B6 yPR)p62 m2e−ip(y−x)(A6 xPL+B6 xPR) (9)
Z − −
Expanding these terms and defining k=q p, we obtain the following expression:
−
e ik(x y) (q k)ρ
W[A,B](2) =2i d4xd4ydk˜dq˜ − − qµAy − Ax
(q2 m2)((q k)2 m2) µ 2 ρ
Z − − − (cid:18)
(q k) (q k)ν m2
qµ − µAνAx+qµAxAy − + gµνAyBx+(A B) (10)
− 2 y ν µ ν 2 2 µ ν →
(cid:19)
All the integrals, except for that involved in the term proportional to m2, can now be reduced to a common form
that is evaluated in dimensional regularization:
qα(q k)β i 1 t2 t gαβ 1
dq˜ − = kαkβΓ(2 D/2) dt − Γ(1 D/2) dt D/2−1 (11)
(q2 m2)((q k)2 m2) (4π)D/2 − 2 D/2 − 2 − D
Z − − − (cid:18) Z0 D − Z0 (cid:19)
with =m2 k2t(1 t). The integral proportional to m2 is nothing but:
D − −
1 1 1
dq˜ = Γ(2 D/2) dt D/2 2 (12)
−
(q2 m2)((q k)2 m2) (4π)D/2 − D
Z − − − Z0
The imaginary part of W[A,B] can be easily extracted from the integrals in the Feynman parameter t. They give
rise to:
1 1 0, m2 >k2t(1 t)
Im dt(t2 t)log(m2 k2t(1 t) iǫ)= dt(t2 t) π/2, m2 =k2t(−1 t)
Z0 − − − − Z0 −  −−π, m2 <k2t(1−−t)
π 4m2 2 4m2 k2
= 1 + , >4  (13)
4 − k2 3 3k2 m2
r (cid:18) (cid:19)
and in a similar fashion we obtain:
3
1 4m2 2 k2
Im dt(m2 k2t(1 t))log(m2 k2t(1 t) iǫ)= π 1 m2 (14)
− − − − − − − k2 3 − 6
Z0 r (cid:18) (cid:19)
and
1 4m2 2 k2 k2
Im dtlog(m2 k2t(1 t) iǫ)= π 1 m2 , >4 (15)
− − − − − k2 3 − 6 m2
Z0 r (cid:18) (cid:19)
Putting all the contributions together and changing to the vector and axial-vector fields defined by V = B+A and
S =B A respectively, we obtain the final result for the imaginary part in terms of the Fourier transformed fields:
−
1 4m2 1 2m2
ImW(2)[A,B]= dk˜θ(k2 4m2) 1 1+ (F (k)Fµν( k)
8π2 − − k2 −6 k2 µν −
Z r (cid:18) (cid:18) (cid:19)
+ S (k)Sµν( k))+2m2S (k)Sµ( k) (16)
µν µ
− −
We see that unlike the vector case, the axial contribution to the imag(cid:1)inary part has an additional term proportional
to m2S2. Thistermisprohibitedbygaugeinvarianceinthe vectorcase,howeverasitis well-knownitmayappearin
axialtheories with massive fermions since those theories violate the correspondinggauge invariance. In fact studying
thedivergencesthatappearinthemodelweseethattheyareproportionaltothefollowingoperators: F Fµν,S Sµν
µν µν
andm2S Sµ [25,24]. TheS4 operatoralthoughhavingthesamedimensiondoesnotcontributetothe divergentpart.
µ
Thereforeit is only neccesaryto introduce akinetic anda masscountertermsfor the axialfield inorderto render the
theory finite.
Since the integrand of imaginary part has to be understood as a probability density, it is important to verify that
it is always positive. As far as k2 > 4m2, due to the step function present in (16) it is possible to find a reference
frame in which~k=0. Then we have:
1 2m2
1+ k S0(k)k S0( k) k2S (k)Sµ( k)+S V +2m2S (k)Sµ( k)
3 k2 0 0 − − 0 µ − → µ −
(cid:18) 0 (cid:19)
1 m2 (cid:0) 1 (cid:1) 1
= 1+2 (S 2+ V 2)+2m2(S 2 S 2)= (k2 4m2)S 2+ (k2+2m2)V 2+2m2 S 2 0 (17)
3 k2 | i| | i| | 0| −| i| 3 0 − | i| 3 0 | i| | 0| ≥
(cid:18) 0 (cid:19)
Fromthisexpressionwecanextractsomeofthedifferentfeaturesoftheproductionfromvectorandaxialfields. First
we see that if the axial field is purely spatial in the above reference frame, i.e., S = 0 and we choose S (k)= V (k),
0 i i
then the production from pure axial fields is always supressed with respect to the pure vector case. However when
S (k)=V (k)andS =V =0then,whereasthereisnoproductioninthevectorcase,itispossibletocreateparticles
0 0 i i
in the axial one. In the massless case both fields give rise to the same amount of particles.
IV. CONSTANT ELECTRIC AND AXIAL FIELDS: A NON-PERTURBATIVE RESULT
In the previous section we have obtained the particle production probabilities up to second order in perturbation
theory. This is in general a good approximation for small background fileds, however even in those cases, it does
not contain all the information about the particle production processes. In this section we will study a particular
configuration of vector and axial fields for which it is possible to find an expression for the imaginary part which is
non-perturbative in the electric field, in the limit S2 <<E and m2S2/E2 <1.
LetusstartbyintroducingtheoperatorsX andP actingonstates x and p intheusualform: X x =x x and
µ µ µ µ
P p =p p . Inaddition xP φ =i∂ xφ . Theconumutatorisgiv|einby[X| i,P ]= ig and px| i=eipx/|(i2π)2.
µ µ µ µ µ ν µν
| i | i h | | i h | i − h | i
Following Itzykson and Zuber [26] we recast the Dirac operator in (1) as:
iD =(P AP BP ) (18)
L R
6 6 −6 −6
and taking the transpose we have:
(P AP BP )t = C(P AP BP )C 1 (19)
L R R L −
6 −6 −6 − 6 −6 −6
where C = iγ2γ0 is the charge conjugation matrix that satisfies: Cγ C 1 = γt and Cγ C 1 = γt. The effective
µ − − µ 5 − 5
action in (2) can be written with this notation as:
4
1
W[A,B]= iTrlog (P AP BP m+iǫ) (20)
L R
− 6 −6 −6 − P m+iǫ
(cid:18) 6 − (cid:19)
where we have explicitly introduced the normalization factor N in the last term. The effective action can also be
written in terms of the transposed operators:
t
1
W[A,B]= iTrlog (P AP BP m+iǫ)t
L R
− 6 −6 −6 − (cid:18)P6 −m+iǫ(cid:19) !
1
= iTrlog (P AP BP +m iǫ) (21)
R L
− 6 −6 −6 − P +m iǫ
(cid:18) 6 − (cid:19)
adding both expression we get:
1
2W[A,B]= iTrlog (P AP BP m+iǫ)(P AP BP +m iǫ)
− 6 −6 L−6 R− 6 −6 R−6 L − P2 m2+iǫ
(cid:18) − (cid:19)
i
= iTrlog (P A P B P )2 (A P +B P )[γµ,γν]+m(A B)γ m2+iǫ
µ µ R µ L µν R µν L 5
− − − − 4 6 −6 −
(cid:20)(cid:18) (cid:19)
1
(22)
· P2 m2+iǫ
− (cid:21)
Finally we change to the vector and axial fields, the effective action is then written as:
i 1
2W[V,S]= iTrlog (P V S γ )2 (V +S γ )[γµ,γν] mSγ m2+iǫ (23)
− µ− µ− µ 5 − 4 µν µν 5 − 6 5− P2 m2+iǫ
(cid:20)(cid:18) (cid:19) − (cid:21)
Here V = ∂ V ∂ V and S = ∂ S ∂ S . Let us take the following background fields: Vµ = (0,0,0,Bx1)
µν µ ν ν µ µν µ ν ν µ
− −
and Sµ = (0,0,0,S) with B and S arbitrary constants. This choice correspons to a constant magnetic field B along
the y axisandaconstantaxialfield S in the z direction. Obvisouslythe sameresultwillbe obtainedif wechoosethe
fields in different spatial directions, provided they are orthogonal. By means of the Schwinger proper-time integral
we can write:
2W[V,S]= iTr ∞ dse−is(m2−iǫ) xeis(P02−P12−P22−(P3−BX1−Sγ5)2+2iB[γ1,γ3]−mSγ3γ5) x xeisP2 x (24)
− s h | | i−h | | i
Z0 (cid:16) (cid:17)
The action of the traslation operator will simplify this expression:
(P3 BX1 Sγ5)2 =e−iP1BP3( BX1 Sγ5)2eiP1BP3 (25)
− − − −
Let us now introduce complete sets of momentum eigenstates:
2W[V,S]= iTr ∞ dse−is(m2−iǫ) 1 d4pdp˜1eis(p20−p22)eip3(p˜1−p1)/Be−i(p˜1−p1)x1
− s (2π)4
Z0 (cid:20)(cid:18) Z
p1 eis(−P12−(−BX1−Sγ5)2+2iB[γ1,γ3]−mSγ3γ5) p˜1 d4pd4p˜ei(p˜−p)x peisP2 p˜ (26)
· h | | i − (2π)4 h | | i
(cid:17) Z (cid:21)
Performing the integral in the p3 variable and in the p0 and p2 by means of:
∞ dqe isq2 = π (27)
−
is
Z−∞ r
the above expression reduces to:
2W[V,S]=iTr ∞ dse−is(m2−iǫ) B dp peis(−P2−(BX+Sγ5)2+2iB[γ1,γ3]−mSγ3γ5) p i (28)
s 8π2 h | | i− (4π)2s2
Z0 (cid:20) Z (cid:21)
where for simplicity we have denoted X =X1 and P = P . The integral in p together with the matrix trace can be
1
consideredasthetraceoftheevolutionoperatorcorrespondingtothehamiltonianH =H +H inordinaryquantum
0 1
mechanics with:
5
P2 (BX +S)2 Bσ2 0
H0 = − − 0 − P2 (BX S)2 Bσ2 (29)
(cid:18) − − − − (cid:19)
and
0 mSσ3
H = (30)
1 mSσ3 0
(cid:18) (cid:19)
with σi the corresponding Pauli matrices. The problem of evaluating the effective action is thus reduced to the
calculation of the spectrum of the H operator. However, since we cannot obtain such spectrum in an exact form, we
will consider H as a small perturbation. With that purpose, we will assume that the contributions to the spectrum
1
comingfromtheperturbationaresmallerthantheeigenvaluesofH ,i.e,m2S2 <B2. Wecanthenapplythestandard
0
Kato theory for time-independent perturbations in quantum mechanics [27]. The H operator is made out of two
0
shifted harmonic-oscillatorhamiltonians with mass M =1/2, frequency ω =2B and a new coupling to the magnetic
field. Its spectrum λ(0) and eigenfunctions ψ(0) with n=0.. ,i=1..4 can be easily obtained, they are given
{ n,i} { n,i} { ∞ }
by:
iφ φ
n n
λ(0) =E +B, ψ(0) = 1 φn ; λ(0) =E B, ψ(0) = 1 iφn
n,1 n n,1 √2 0  n,2 n− n,2 √2 0 
0 0
   
(31)
 0   0 
0 0
λ(n0,)3 =En+B, ψn(0,3) = √12iφˆn ; λ(n0,)4 =En−B, ψn(0,4) = √12 φˆn 
 φˆ   iφˆ 
 n   n 
   
where E = 2B(n+1/2) are the energy levels of the ordinary harmonic oscillator and:
n
−
φn(x)=s√π√2Bnn!Hn √B x+ BS e−B2(x+BS)2; φˆn(x)=s√π√2Bnn!Hn √B x− BS e−B2(x−BS)2 (32)
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
with H the Hermite polynomials. Notice the different functional form of these two functions, it reflects the fact
n
that the two harmonic oscillators in (29) are displaced in different ways. We will express the spectrum and the
eigenfuntions of the complete hamiltonian H as perturbative series: λ = λ(p) and ψ = ψ(p). To first
n,i p=0 n,i n,i p=0 n,i
order in the perturbation, the spectrum is given by the solutions of the equation [27]:
P P
det ψ H ψ (λ λ(0))δ =0 (33)
|h n,i| 1| n,ji− n,i− n,i ij|
Thesolutionimplyλ(1) =0, n,i,i.e. thereisnofirstordercorrectiontotheenergies. Thesecondordercontributions
n,i ∀
are given by:
ψ(0) H ψ(0) 2
λ(2) = |h n,i| 1| n,ji| (34)
n,i (0) (0)
λ λ
Xj6=i n,i− n,j
Although,theH spectrumisdoubledegenerated,itis possibleto usetheaboveexpressionvalidfornon-degenerated
0
spectra, since there is no contribution from states in the same multiplet. In any case, the explicit calculation using
Katotheoryyieldsthesameresults. Thevaluesofthesecondorderperturbationcanbeevaluatedinastraightforward
way, we get:
m2S2 m2S2
λ(2) =λ(2) = κ2, λ(2) =λ(2) = κ2 (35)
n,1 n,3 2B n n,2 n,4 − 2B n
where
κn = φn φˆn =e−BS2 Ln 2S2 (36)
|h | i| | B |
(cid:18) (cid:19)
with L the Laguerre polinomials. These polynomials are bounded as n grows, in fact asymptotically we have [28]:
n
L (x) 1 ex/2x 1/4n 1/4cos(2√nx π/4)+ (n 3/4)forx>0andthereforeκ isbounded. InfactforS2/B <<1,
n ∼ √π − − − O − n
6
we can expand: κ2 =1 (S2/B)(4n+2)+ (S4/B2). Therefore, as expected for small values of the axial field with
respect to the mangnetic−one, κ2 = 1 is a goOod approximation. Notice that, although the first correction grows like
n
n, the growth must be controlled by higher order terms since the function is bounded. Let us stress that we have
two different parameters in our problem: on one hand S2/B that we assume to be very small and our perturbative
parameter m2S2/(2B2).
Using (47) we can obtain the first correction to the eigenfunctions, we get:
ψ(1) = imSκ ψ(0) ψ(1) = imSκ ψ(0)
| n,1i 2B n| n,4i | n,2i 2B n| n,3i (37)
ψ(1) = imSκ ψ(0) ψ(1) = imSκ ψ(0)
| n,3i 2B n| n,2i | n,4i 2B n| n,1i
Evaluating now the third order corrections in perturbation theory from (49) they again turn out to be zero. This
is also the case of the second order corrections to the eigenfuntions, i.e. from (47) we get ψ(2) = 0 n,i. Thus
| n,ii ∀
we can calculate fourth order corrections to the energies from (49) which are non-vanishing. Finally, we see that
the fifth order correction again vanishes. In conclusion, our results for the perturbed spectrum, up to sixth order in
perturbations is given by:
1 m2S2 m4S4 m6S6
λ =λ =B 2 n+ +1+ +
n,1 n,3 − 2 2B2 − 8B4 O B6
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
1 m2S2 m4S4 m6S6
λ =λ =B 2 n+ 1 + + (38)
n,2 n,4 − 2 − − 2B2 8B4 O B6
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
Once we know the perturbed spectrum we can readily calculate the traces in (28). We will only perform the Dirac
traces, but not the functional trace that is equivalent to the integration d4x. Thus we obtain the result for the
effective lagrangianw that as expected does not depend on x:
R
2w[V,S]=4i ∞ dse is(m2 iǫ) B cos sB 1+ m2S2 m4S4 + m6S6 eis( 2B(n+1/2))
− − −
s2 8π2 2B2 − 8B4 O B6
Z0 " (cid:18) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19) n
X
i
(39)
− (4π)2s
(cid:21)
Finally performing explicitly the addition of the series in n, we obtain:
cos sB 1+ m2S2 m4S4 + m6S6
2w[V,S]= 4 ∞ dse is(m2 iǫ) B 2B2 − 8B4 O B6 1 (40)
−(4π)2 s2 − −  (cid:16) (cid:16) sin(sB) (cid:16) (cid:17)(cid:17)(cid:17) − s
Z0
 
Itcanbeseenthattheresultispurelyreal[29],i.e,thereisnoparticleproductioninthepresenceofconstantmagnetic
and axial fields. In addition, the integral in s is divergent in the ultraviolet limit s 0. As usual these divergencies
→
has to be removedby adding suitable counterterms. In order to obtain them, let us expand the part of the integrand
in brackets around s=0, we have:
Bcos sB 1+ m2B2S22 − m8B4S44 +O mB6S66 1 =B 1 sB sm2S2 + m6S6 1 + (s3) (41)
(cid:16) (cid:16) sin(sB) (cid:16) (cid:17)(cid:17)(cid:17) − s Bs − 2 − 2B O B6 − s O
(cid:18) (cid:18) (cid:19)(cid:19)
The (s3)giverisetofinitecontributionswhenintegrated. Thefirsttermisexactlycancelledbythe1/ssubstraction.
But wOe will have to include new counterterms proportionalto s(B2+m2S2). Notice that these are exactly the same
operatorsthatwefoundintheperturbativecalculationinsection3. InfactthereisnocontributionfromS4 operators
duetoanexactcancellationofthedifferentcuartictermsintheexpansionin(41). Sincethetheoryisrenormalizable,
it is neccessary that the same kind of cancellation operates on the higher order terms in the r.h.s. of the above
expression.
The presence of the axialfield is known that does not introduce any gauge anomaly in the electromagnetic current
[30], therefore the effective action will be gauge invariant. As a consequence it should be built out of scalar and
gaugeinvariantfunctions. Inourcase,withconstantmagneticandtorsionfields,theonlypossibilitiesare: F Fµν =
µν
2(B~2 E~2) and F Fµν = (4B~ E~)2. Since the second term vanishes in our case with constant magnetic or electric
− µ∗ν ·
fields,the effectiveactionisinvariantunderthe transformationB iE [29]. Inthis waywecanconvertourresults
→−
for constant magnetic field to pure constant electric fields. Taking into account that now sin(sB) isinh(sE)
→ −
7
andcos(sB(1+m2S2/(2B2) m4S4/(8B4))) cosh(sE(1 m2S2/(2E2) m4S4/(8E4))) and integratingusing the
− → − −
residues technique we can obtain the expression for the imaginary part of the effective lagrangian:
p=2Imw[E,S]= 1 E2 ∞ (−1)n cos nπ 1 m2S2 m4S4 + m6S6 e−m2Enπ (42)
4π3 n2 − 2E2 − 8E4 O E6
n=1 (cid:18) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19)
X
When S = 0 we recover the original Schwinger result in (4). Notice that in the absence of electric field there is no
particle production even with non-vanishing axial field. When the mass is zero, we recover the usual result, i.e, axial
fields only contribute in the massive case. In Figure. 1 the upper curve represent the probability density p as a
function of S for E =0.1m2, this value for the electric field ensures that the condition S2 <E is satisfied for all the
values of S in the plot.
2e-18
1.5e-18
p
1e-18
5e-19
0 0.02 0.04 0.06 0.08 0.1
S
Figure 1.-Probabilitydensities pinunits m4 versus axial fieldS inunits m, foranelectric fieldE =0.1m2. Theupper
curve represents the perturbative calculation up to sixth order. The lower curve is the resummation of the perturbative
seriesestimatedinthetext.
We can try to extend further the above result by means of the following observation. Since the only possible
divergences in our model are those mentioned before, it is neccessary that, when expanding the cos function in (41),
the higher order terms in S cancel, this implies:
m2S2 m4S4 2
sB2 1+ +... =s(B2+m2S2) (43)
2B2 − 8B4
(cid:18) (cid:19)
Therefore we can obtain the complete result for the effective action to all orders in perturbation theory in the limit
S2 <<B. Performing the rotation to electric fields, the result is given by :
p=2Imw[E,S]= 1 E2 ∞ (−1)n cos nπ 1 m2S2 e−m2Enπ (44)
4π3 n=1 n2 r − E2 !
X
In Figure.1, this probability is represented by the lower curve.
Weseethattheeffectoftheaxialfieldistosuppressparticleproduction. Eventuallyitcouldmakeittovanish. The
point of vanishing p should indicate the breakdown of the perturbative approximation since the probabilities should
be positive. In order to check whether the perturbative calculation is valid up to the point in which P vansihes, we
will study the convergence of the perturbative series. In particular, there is a general result due to Kato [31] that
8
states thatif therearetwo non-negativeconstantsa andb suchthat: H a H ψ +b ψ for all ψ D(H )
1 0 0
|| ||≤ || | i|| ||| i|| | i∈
and if the operator H is bounded then the eigenvalues perturbative series is absolutely convergent if 2 H < d.
1 1
|| ||
With d being the distance from λ0 to the rest of the spectrum of H . In our case, we take a = 0 and provided
n,i 0
ψ is a complete set of eigenfunction we can expand H ψ = H C ψ . It is then easy to show that:
{| n,ii} 1| i 1 n,i n,i| n,ii
H ψ 2 = m2S2 C2 = m2S2. Therefore if we take b = mS, since H is bounded (it is constant) and in our
|| 1| i|| n,i n,i P 1
cased=2B,wehavethattheseriesisabsolutelyconvergentform2S2/(2B2)<1/2. Transformingtotheelectrostatic
P
case, if we look at the plot we can realize that for those particular values, the point in which the curve crosses the
axis is in fact approximately signalling the breakdown of the perturbative series.
V. CONCLUSIONS
Inthisworkwehavestudiedtheproductionofmassivefermionsfromaclassicalvectorandaxial-vectorbackground.
Using a perturbative evaluation of the effective action we have obtained the contributions to the imaginary part of
the effective action up to second order in the external fields. We have shown that in the reference frame for which
~k =0,the productionfrompurely spatialaxialfields is suppressedwith respectto the thatofthe vectorbackground.
In addition, for purely temporal axial fields it is possible to create particles whereas this is not the case for vector
fields.
Intheparticularcaseofasmallconstantaxialfieldandaconstantelectricfield,itisshownthatanon-perturbative
calculation can be carried out when those fields are orthogonal. In this case, it is shown that in the massless limit
the axialfielddoes notaffect the productionfromthe electric field. However,in the massivecase, the presence of the
axial backgroundinhibits such production.
Finally, let us comparethese resultwith the anisotropydamping phenomenonatthe Planckera. As is well-known,
the presence of smallanisotropies in the earlyuniverse canbe damped in a few Planck times due to the backreaction
of the particles produced on the geometry. An interesting possibility is that a similar mechanism could take place in
the presence of some primordial torsion field. In this case, since torsion can be generated by the initrinsic spin, this
could happen when the fermions are produced in some configuration such that the total spin angular momentum of
the system did not vanish. The use of effective action methods for fermions could be extended to the production of
higher spin fields, such as gravitinos in a straightforwardway. In addition, it is also interesting to study not only the
particle production rate derived from the effective action, but also the spectra and angular distribution of fermions
produced. This could be approached by means of the traditional Bogolyubov technique. Work is in progress in this
direction [32].
VI. APPENDIX
In this Appendix we summarize the main formulae of the standard perturbation theory used in the text. Let us
assume that the hamiltonian of theory can be decomposed in:
H =H +H (45)
0 1
As shown in the text, we denote by λ(0) and ψ(0) the eigenvalues and eigenfunctions of H , which are assummed
n,i | n,ii 0
to be known. The unperturbed spectrum is assumed to be non-degenerated and discrete. The eigenfunctions and
eigenvalues of the complete hamiltonian H are expanded in series: λ = λ(p) and ψ = ψ(p) where:
n,i p=0 n,i | n,ii p=0| n,ii
P P
λ(np,)i =hψn(0,i)|H1|ψn(p,i−1)i (46)
and
p 1
|ψn(p,i)i=Sn,i H1|ψn(p,i−1)i− − λ(nk,i)|ψn(p,i−k)i! (47)
k=1
X
with
(0) (0)
ψ ψ
S = | n,jih n,j| (48)
n,i λ(0) λ(0)
Xj6=i n,i− n,j
9
Some simplified formulae for the lowest order terms in the spectrum are given by:
λ(1) = ψ(0) H ψ(0)
n,i h n,i| 1| n,ii
ψ(0) H ψ(0) 2
λ(2) = |h n,i| 1| n,ji|
n,i λ(0) λ(0)
Xj6=i n,i− n,j
λ(3) = ψ(1) H λ(1) ψ(1)
n,i h n,i| 1− n,i| n,ii
λ(4) = ψ(1) H λ(1) ψ(2) λ(2) ψ(1) ψ(1)
n,i h n,i| 1− n,i| n,ii− n,ih n,i| n,ii
λ(5) = ψ(2) H λ(1) ψ(2) 2λ(2)Re ψ(1) ψ(2) λ(3) ψ(1) ψ(1) (49)
n,i h n,i| 1− n,i| n,ii− n,i h n,i| n,ii− n,ih n,i| n,ii
Acknowledgements: I thank Ed. Copeland and A. Dobado for useful suggestions. This work has been partially
supported by the Ministerio de Educaci´on y Ciencia (Spain) (CICYT AEN96-1634). The author also acknowledges
support from SEUID-Royal Society.
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