Effective action for Dirac spinors in the presence of general uniform electromagnetic fields

30 April 1998

Physics Letters B 426 Ž1998. 82–88

Effective action for Dirac spinors in the presence of general uniform electromagnetic fields Roberto Soldati a

a,1

, Lorenzo Sorbo

b,2

Dipartimento di Fisica ‘‘ A. Righi’’, Õia Irnerio 46, 40126 Bologna, Italy b S.I.S.S.A., Õia Beirut 2-4, 34013 Trieste, Italy Received 3 December 1997; revised 6 January 1998 Editor: L. Alvarez-Gaume´

Abstract Some new expressions are found, concerning the one-loop effective action of four dimensional massive and massless Dirac fermions in the presence of general uniform electric and magnetic fields, with E P H / 0 and E 2 / H 2. The rate of pair-production is computed and briefly discussed. q 1998 Published by Elsevier Science B.V. All rights reserved.

Non-renormalizable effective field theories turn out to be quite useful in the description of physics below some specific momentum scale. One of the early known examples is provided, taking one-loop corrections into account, by the Žnonlinear. effective lagrangean for the electromagnetic field in the presence of virtual fermions, which enables to describe light-light scattering at low momenta of the order of the electron mass. As it is well known, the use of path-integral techniques allows to obtain the correction to the classical action in terms of the evaluation of the determinant of the Dirac operator. The first efforts in this direction date back to Euler and Heisenberg w1x, who worked out an implicit expression of the effective Lagrange function for Maxwell theory in the context of electron-hole theory. Later on, Schwinger w2x derived a gauge-invariant integral representation of the effective lagrangean by means of the so-called ‘‘proper-time’’ technique. Then, during the next three-four decades, no step was made towards an explicit expression of an effective theory of electromagnetism. The introduction of Hawking’s z-function technique w3x renewed the interest in the subject and some further progress in this direction was put forward: Blau, Visser and Wipf w4x obtained an analytic form for the effective action of a uniform electromagnetic field in any number of space-time dimensions, both for massive and for massless fermions. Unfortunately, none of these expressions is fully satisfactory: on the one hand, Schwinger’s one, besides being implicit, is not valid in a theory with massless fermions. On the other hand, the expressions given in Ref. w4x are explicit and valid also for massless Fermi particles, but they are established only for some particular

1 2

E-mail address: [email protected]. E-mail address: [email protected].

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 2 7 6 - 7

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configurations of the uniform electric and magnetic fields. In this paper a completion will be given to the work of Blau et al., in order to obtain a general and explicit expression for the effective lagrangean of massive and massless QED in the presence of uniform general electromagnetic fields. The use of the path-integral method forces us to work in the euclidean framework: only at the very end of our calculations we shall operate a Wick rotation to Minkowski space-time. The effective euclidean action, in the one-loop approximation, is given by E S Eff Am s SClE Am y logDet Ž Du Am q im .

Ž 1.

where SClE w Am x is the classical euclidean action Hd 4 xFmn Fmnr4 and Du w Am x ' gmŽ Em y ieAm . is the euclidean Dirac operator, m being the fermion mass. As the Dirac operator is normal Žw Du q im, Du † y im x s 0., we have
Ž 2.

and consequently the effective lagrangean is defined Žby means of the z-function regularization. to be E S Eff Am s SClE Am y

1 E 2 Es

z Ž s; Am

. < ss 0

Ž 3.

Since the square of the Dirac operator Du †Du s Ž Em y ieAm . 2 y eFmn Smnr2 is elliptic, its determinant can be evaluated by means of the z-function regularization Žthe matrices Smn are defined as Smn ' iw gm ,gn xr2.: to perform this evaluation, it is necessary to obtain the spectrum of the operator, which can be explicitly calculated in the case of uniform fields. In this particular situation, a frame can always be chosen such that F03 s yF30 s E, F12 s yF21 s B, all the other components of the field-strength tensor vanishing: the spectrum of the operator Du †Du turns out to be of the form

l n E , n B s 2 < eE < n E q 2 < eB < n B

Ž n E ,n B s 0,1,2, . . . .

Ž 4.

Žsee, for instance, w5x.. The first fact we can notice is that E and B play a symmetric role in euclidean space Žthis will not be the case in Minkowski space-time.. Taking into account the degeneracy of each eigenvalue Žwhich can be obtaind by Landau levels counting., we get, for the z-function, the following implicit expression: namely,

z Ž s . s Ž vol . 4

e 2 < EB < 4p 2

`

q2

Ý n Es1

ž

`

4

`

Ý Ý n Es1 n Bs1

2 < eE < n E q m2

m2

ž

2 < eE < n E q 2 < eB < n B q m2

m2

ys

/

`

q2

Ý n Bs1

ž

2 < eB < n B q m2

m2

ys

m2

/ ž / / q

ys

m2

ys

Ž 5.

where Žvol.4 is the Žregularized. volume of the four-dimensional euclidean space and m is a normalization parameter with dimensions of a mass. The z-function defined in Eq. Ž5. is singular in the massless limit, owing to the presence of zero-modes for the squared massless Dirac operator Du †Du ; this limit will be further examinated below. The spectrum of the squared Dirac operator turns out to be quite different when either E or B vanish. Taking, for instance, B s 0, the spectrum will be of the form

l n E , p 1 , p 2 s 2 < eE < n E q p 12 q p 22

Ž p1 , p 2 g R , n E s 0,1,2 . . . .

Ž 6.

Obviously, if B s 0 the z-function will have a different expression with respect to Eq. Ž5.. Here we are not interested in this expression, because the z-function, in the case of vanishing electric – or magnetic – field, may be also calculated exploiting the Žwell known. asymptotic behavior of the heat kernel, as is done in Ref. w4x. There, the authors worked out explicitly the effective action Žboth in theories with m / 0 and m s 0. in the case

R. Soldati, L. Sorbo r Physics Letters B 426 (1998) 82–88

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of vanishing E Žor B . and also, from expression Ž5., in the case < E < s < B <. In what follows, we will obtain an asymptotic expansion of the z-function in powers of < ErB <. In order to simplify the notation, we define the dimensionless quantities a ' < eE
z Ž s;a,b;c . s m4

Ž vol. 4 4p 2

ab

`

H dt t G Ž s. 0

sy1 yct

e

2

ey2 at q ey2 b t

Ž 1 y ey2 at . Ž 1 y ey2 b t .

q1

Ž 7.

One of the main difficulties in the analysis of the z-function for the four-dimensional Dirac theory comes from the fact that it is a function of the two arguments a and b Žfor some fixed values of the parameters s and c .. This difficulty can be overcome by noticing that the z-function Ž7. obeys the scaling property

z Ž s;a,b;c . s a2y sz s;1,

ž

b c ; a a

/

Ž 8.

so that, in order to find the general expression of the effective action, it is sufficient to study the function of the single ratio bra ' y g Ž s; y, z . '

y

`

H dt t G Ž s. 0

sy 1 yz t

e

2

ey2 t q ey2 y t

Ž 1 y ey2 t . Ž 1 y ey2 y t .

q1

Ž 9.

where z ' cra. The explicit formulae reported by Blau et al. are valid for E s 0 and < E < s < B <, namely y s 0 and y s 1. The expansion around y s 1 can be performed quite easily, and is left to the interested reader. We aim now to investigate the behavior of g Ž s; y, z . when y , 0: after some simple manipulations we can rewrite the function g Ž s; y, z . as g Ž s; y, z . s

2ys y

z

`

H dt t G Ž s. 0

sy 1 y

e

2

t

1 q eyt yt

1ye

coth

yt

ž /

Ž 10 .

2

In order to expand the function cothŽ ytr2. in a power series of ytr2, it is necessary to cut the integral in Eq. Ž10. at the upper end: we restrict the integration domain to the interval w0, yya x Žwhere 0 - a - 1.. With such a choice, in the limit y ™ 0 the domain of the t variable extends to the whole set of the real positive numbers. On the other hand, the quantity ytr2 - y 1y ar2 approaches zero in this limit, so that we can expand cothŽ ytr2. in power series of its argument. We must only assure ourselves that the error made when cutting the upper integration limit does itself approach zero when y ™ 0: it can be shown that this is what really happens, since y

`

H G Ž s. y

ya

z

d t t sy 1 ey 2 t

1 q eyt yt

1ye

coth

yt

ž / 2

z

- ey 2 y

ya

f 1Ž y .

Ž 11 .

where f 1Ž y . diverges at most as a power of 1ry when y ™ 0 so that, as long as z / 0, the error made by cutting the integral is exponentially small when y ™ 0. Now we can safely expand the function g Ž s; y, z . in power series of y: the estimate Ž10. allows us to write g Ž s; y, z . s

2 1y s

H G Ž s. 0

yya

z

d t t sy2 ey 2 t

1 q eyt yt

1ye

N

1q

B2 k

Ý Ž 2 k . ! Ž yt . 2 k ks1

z

q O Ž y 2 Nq2 . q ey 2 y

ya

f 1Ž y .

Ž 12 .

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85

where B2 k is the 2 k-th Bernoulli number. This expression, in the limit y ™ 0, yields an asymptotic series in y, whose coefficients can be all explicitly computed: namely, g Ž s; y, z . s

2 1y s

N

B2 k

y 2 kG Ž s y 1 q 2 k . Ý G Ž s . ks0 Ž 2 k . !

ž

2 z H s y 1 q 2 k ;1 q

z

z

/ ž / q

2

2

1y 2 kys

q O Ž y 2 Nq2 .

Ž 13 . z H Ž a; x . ' Ý`ns0 Ž n q x .ya being the Hurwitz z-function. Exploiting the scaling property Ž8., the effective euclidean lagrangean of QED in the presence of uniform fields can be easily calculated as a power series of < BrE < Žor < ErB <, owing to the symmetry between the electric and magnetic field in the euclidean space.. The massless limit unravels a quite different behavior: this fact is a consequence of the explicit exclusion of the zero-modes from the z-function Ž5., whose integral representation will be, in this case, of the form z Ž s;a,b . s m4

Ž vol. 4 4p 2

2

ab

`

H dt t G Ž s. 0

ey2 at q ey2 b t

sy1

Ž 1 y ey2 at . Ž 1 y ey2 b t .

Ž 14 .

The scaling property z Ž s;a,b . s a 2y sz Ž s;1,bra. allows us to extract all the information we need by the analysis of a function of only one variable, namely g Ž s; y . s

2 1y s y

eyt q eyy t

`

H0

G Ž s.

d t t sy 1

Ž 15 .

Ž 1 y eyt . Ž 1 y eyy t .

Now we would like to proceed as in the massive Ž z / 0. theory, but it is apparent from the estimate Ž11. that, for vanishing z, the error made by cutting the integral at yya at the upper end is not negligible in the limit of small y. Yet, it can be easily verified that g Ž s; y . y Ž 2 y .

1y s

zR Ž s . s

2 1y s y

`

H dt t G Ž s. 0

sy1

eyt yt

1ye

coth

yt

ž / 2

Ž 16 .

where z R Ž a. ' z H Ž a;1. is the Riemann z-function. Cutting the integral of Eq. Ž16. at the upper end we get the estimate `

Hy

ya

d t t sy 1

eyt yt

1ye

coth

yt

ž / 2

ya

- eyy f 2 Ž y .

Ž 17 .

where f 2 Ž y . has a behavior analogous to the one of f 1Ž y .. Now we are allowed to expand g Ž s; y . y Ž2 y .1y sz R Ž y . in power series of y, obtaining g Ž s; y . s Ž 2 y .

1ys

zR Ž s . q

2 2y s

N

B2 k

G Ž s y 1 q 2 k . z R Ž s y 1 q 2 k . y 2 k q O Ž y 2 Nq2 . Ý G Ž s . ks0 Ž 2 k . !

Ž 18 .

We have performed all of the calculations necessary to gain an expression of the effective lagrangean for massive and massless QED in an asymptotic series of < ErB <: now we will show how these corrections to the E s 0 case work, paying special attention to the rate of production of fermion-antifermion pairs and to the behavior under parity of the massless effective theory. To this aim, it will be necessary to obtain the effective lagrangean in minkowskian space-time. First of all, we will analyse the ‘‘unperturbed’’ problem Ž E s 0 or, equivalently, B s 0.: in this limit the effective lagrangean can be found in w4x, but in that work the transition to the Lorentz metric is not considered. Denoting by E and B the minkowskian electric and magnetic field strengths respectively Žwhereas in euclidean space we used the symbols E and B ., the transition to the Minkowski space-time is performed by means of the substitution M E L Eff L Eff Ž E , B . s yL Ž E s yi E , B s B .

where the superscripts E and M mean, obviously, euclidean and minkowskian metric.

Ž 19 .

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If E s 0 the effective theory in Minkowski space does not give any new information, being trivially MŽ E Ž L Eff L Eff B . s yL B s B .. It is much more interesting to analyse the case B s 0, since, from Eq. Ž5.1.8. of Ref. w4x, the effective lagrangean is obtained to be E2

M L Eff Ž E ,0 . s

y

2 qi

e2 E 2

½

2p 2

m2 e E

log

8p 2

ž

1 y log y m2

m2

2i e E

m2

/ ž

z H y1;1 q i

y1

m2 2 eE

q z HX y1;1 q i

/ ž

m2 2 eE

/5 Ž 20 .

and turns out to have a nonvanishing imaginary part. The imaginary Žabsorptive. part is interpreted as half of the probability of generation of fermion-antifermion pairs per unit of space-time Žsee also w6x, chapt. 4-3.. The value of this probability cannot be easily obtained from Eq. Ž20., because it is quite hard to evaluate the real and the imaginary part of the Hurwitz z function of complex argument. On the other hand, in w4x both the strong Ž E 4 m2 . and the weak field limits of the effective lagrangean are exhibited in a form which does not contain the Hurwitz special function: so, after having performed the transition to the minkowskian space-time, we can write explicitly the strong- and weak- field approximations of the rate of pair creation in a purely electric uniform field. The exact result is given Žin implicit form. in Ref. w6x, so that we can recover the expressions given by Blau et al. and check the validity of Eq. Ž19. we used to ‘‘Wick-rotate’’ the electromagnetic field: the exact pair production rate per unit space-time is Žw6x, par. 4-3-3. ws

e2 E 2 4p 3

`

1

Ý

n

ns1

½

exp yn 2

p m2 eE

5

Ž 21 .

From the expressions of w4x we have that the weak-field expansion gives w s 0: this fact is in agreement with Eq. Ž21., which shows that the function w Ž E . is not analytic in E s 0. On the other hand, for strong fields we get ws

e2 E 2 24p

y

m2 e E 4p 2

log

eE

ž / p m2

q1 y

m4 16p

4

qm O

p m2

ž / eE

Ž 22 .

By comparison of Eqs. Ž21. and Ž22., we get the estimate `

Ý ns1

1 n2

yn x

e

p2 s 6

y x q x log x y

x2 4

qOŽ x3.

Ž 23 .

that can be verified numerically, showing that Eq. Ž22. gives indeed the correct strong-field limit of the exact expression Ž21.. Eq. Ž23. is not a new result, as it merely follows from the correspondences used to get the minkowskian effective lagrangean. Now, with the aid of Eq. Ž13., we will obtain the corrections to the rate w due to the presence of a uniform magnetic field parallel to the electric one and much weaker than the latter. We find that E . 2 . to the B s 0 effective lagrangean is the first correction Žorder of Ž BrE M d Ž2. L Eff Ž E , B. s

e2 B2 24p

2

where c Ž x . ' dlog G Ž x .rd x.

m2

ž / ž

c i

2 eE

q log yi

2 eE

m

2

/

yi

eE m2

Ž 24 .

R. Soldati, L. Sorbo r Physics Letters B 426 (1998) 82–88

87

An explicit form for the imaginary part of Eq. Ž24. can be achieved by means of the inversion formula w7x for the c-function

c Ž z . y c Ž yz . s yp cot Ž p z . y

1

Ž 25 .

z

so we get Im d Ž2. L Eff s

e2 B2

1

24p e

Žp m 2 .rŽ e E .

Ž 26 .

y1

From Eqs. Ž21. and Ž26. we can compute exactly the total rate of production of fermion-antifermion pairs in E 2 .. a uniform electromagnetic field: we obtain Žat O Ž B 2rE ws

e2 E 2 4p 3

`

Ý ns1

1 n2

q

p 2 B2 3

E2

½

exp yn

p m2 eE

B4

5 ž / qO

Ž 27 .

E4

It is worth noticing that neither Eq. Ž21. nor Eq. Ž26. are analytic in E s 0, owing to the non-perturbative character of this phenomenon. According to a naive interpretation, the phenomenon of particle production in external electromagnetic fields is due to the fact that virtual pairs created Žand, in the absence of the external field, annihilated. in the vacuum, are accelerated by the electric field and may gain energy enough to reach the threshold Ži.e. the fermion mass. and become physical particles. Obviously, the more the electric field is strong, the more pairs are generated. On the other hand, electrically charged particles do not acquire energy from a magnetic field, so that, in the light of the interpretation we have sketched above, the magnetic field should not give any contribution to the rate w. Thus, it is rather surprising that a Žweak. magnetic field gives itself a contribution to the rate of pair creation, as we have shown in Eq. Ž25.. Nevertheless, it is possible to check that the corrections we have obtained to the E . 4 . to the ‘‘unperturbed’’ B s 0 case are the correct ones: considering also the second term Žof order Ž BrE effective lagrangean and retaining only the first terms in the expansion for weak fields, we get the effective lagrangean in the limit < e B < < < e E < < m2r2: M d Ž4. L Eff Ž E , B. ,

e4

2

16p 2 45m

w E 4 q B4 q 5E E 2 B2 x q 4

e2 24p 2

B 2 log

m2

m2

Ž 28 .

which is in perfect agreement with the Euler-Heisenberg effective lagrangean w1x, after setting the arbitrary scale m at the main scale m of the problem. A paper is recently appeared w8x, in which the effective lagrangean of massive QED is obtained within the ™ ™ same approximations of this letter: the corrections to the E P B case shown in w8x are Žapart from some monomial terms coming from a different normalization. the same that can be derived from Eq. Ž7.. The agreement with this result confirms that the effective lagrangean consistently leads to the picture that the rate of pair production does really take a contribution from the magnetic component. B, is real Žin In conclusion, we notice that the effective lagrangean, when expressed as a power series of ErB fact, only even powers of y appear in Eq. Ž13... As usual, anyway, we cannot exclude that an imaginary contribution, exponentially small when E ™ 0, indeed exists: according to the naive interpretation we gave, such a contribution should really exist, even if we have not still been able to achieve a proof of its presence. After having examinated the massive theory, taking special care to the rate of particle production, and bearing in mind that we could extract from Eq. Ž13. the effective lagrangean for massive QED to any order in B or BrE E we turn our attention to the massless theory. ErB

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First, we would like to notice that the most characteristic feature of the massless case is encoded in the first term of the rhs of Eq. Ž18., which appears to be an effect of infrared regularization from the z-function technique. It is easy to obtain, from Eqs. Ž14. and Ž18., the effective euclidean lagrangean in the form E L Eff sy

e2E2 8p 2

½ž 1 3

1q

B2 E2

/

log

2 < eE <

m2

q 4z RX Ž y1 . y 13 y

gE B 2 3E2

B y E

log

< eB <

pm2

4

B

5 žž / / qO

Ž 29 .

E

where the first and the third terms are the relevant ones, whereas the second one may always be subtracted after redefinition of the effective action up to polynomials in the electric and magnetic fields. The above expression turns out to be real, which in fact suggests that the corresponding minkowskian quantity has to exhibit parity invariance. However, owing to our specific Žand frame-dependent. choice of the field variables, an ambiguity concerning the Wick rotation seems to arise, due to the presence of the absolute value of E. This, however, is purely artificial because it is always possible to choose E positive, after suitable frame rotation – we recall that our field configuration is that of parallel electric and magnetic fields of different strengths. Moreover, the very same Wick rotation actually generates a nontrivial imaginary part in the Minkowskian effective action Žjust like in the massive case – see Eqs. Ž20. and Ž26... The outcome that the corresponding pair creation rate is positive – i.e. the probability density to be within zero and one – indeed exhibits the absence of any ambiguity. In particular, we find Žup to polynomials in E and B . L EMf f s

E2 2

y

B2 2

q e2

E 2 y B2 24p 2

½

1 y log

<2 e E <

m2

p qi

2

qi

e2 3

< E B
< e B<

B

4

ž /5 žž / / pm2

qO

E

Ž 30 .

from which it follows, as expected, that the real part of the effective lagrangean as well as the pair creation rate are both parity invariant. To sum up, we have obtained some new expressions for the QED effective action in the presence of general uniform fields which extend the ones given in the literature, both in theories with massive and massless Fermi particles. Concerning the physical consequences, we focused our attention onto the rate of pairs production in a uniform electromagnetic field both in the massive and in the massless theory. For these reasons we think it is worth studying in a more complete way the effective theory of massless QED.

References w1x w2x w3x w4x w5x w6x w7x

H. Euler, W. Heisenberg, Z. Phys. 98 Ž1936. 714. J. Schwinger, Phys. Rev. 82 Ž1951. 664. S.W. Hawking, Comm. Math. Phys. 55 Ž1977. 133. S.K. Blau, M. Visser, A. Wipf, Int. J. Mod. Phys. A 6 Ž1991. 5409. A. Bassetto, Phys. Lett. B 222 Ž1989. 443. C. Itzykson, J.B. Zuber, Quantum field theory, McGraw–Hill, New York, 1980. A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi ŽEds.., The Bateman manuscript project: higher transcendental functions, ´ McGraw–Hill, New York, 1953–1955. w8x J.S. Heyl, L. Hernquist, Phys. Rev. D 55 Ž1997. 2449