LAPTH758/99
Vacuum Polarisation Tensors in Constant Electromagnetic Fields: Part I
Christian Schubert
[1.5ex] Laboratoire d’AnnecyleVieux de Physique Théorique LAPTH
Chemin de Bellevue, BP 110
F74941 AnnecyleVieux CEDEX
FRANCE
Abstract
The stringinspired technique is used for the calculation of vacuum polarisation tensors in constant electromagnetic fields. In the first part of this series, we give a detailed exposition of the method for the case of the QED oneloop Nphoton amplitude in a general constant electromagnetic background field. The twopoint cases are calculated explicitly, leading to compact representations for the constant field vacuum polarisation tensors for both scalar and spinor QED.
1 Introduction: QED Processes in Constant Electromagnetic Fields
Processes involving constant electromagnetic fields play a special role in quantum electrodynamics. An obvious physical reason is that in many cases a general field can be treated as a constant one to a good approximation. In QED this is expected to be the case if the variation of the field is small on the scale of the electron Compton wavelength. Mathematically, the constant field is distinguished by being one of the very few known field configurations for which the Dirac equation can be solved exactly, allowing one to obtain results which are nonperturbative in the field strength. An early and wellknown example is the Euler – Heisenberg Lagrangian [1, 2, 3], the oneloop QED vacuum amplitude in a constant field,
(1.1) 
where , . This Lagrangian encodes the information on the low energy limit of the oneloop photon S  matrix in a form which is convenient for the derivation of nonlinear QED effects such as photon – photon scattering and vacuum birefringence [4, 5, 6]. Vacuum birefringence is a subject of actual interest [7] since, due to recent improvements in laser technology, the first measurement of this effect in the laboratory seems now imminent [8].
Schwinger’s ingenious use of Fock’s proper–time method in 1951 [9, 10] allowed him to reproduce this result, as well as the analogous one for scalar QED, with considerably less effort. Shortly later Toll [11] initiated the study of the effect of a background field on the oneloop photon propagator. This subject was then over the years investigated by a number of authors, first for the pure magnetic field [12, 13, 14, 15, 16, 17] and crossed field cases [18, 19]. The vacuum polarisation tensor in a general electromagnetic field was first obtained in [20] and given in more explicit form in [21, 22, 23]. The recent [24] contains another recalculation of this quantity, as well as a detailed analysis of the implications for light propagation.
Another consequence of the presence of a background field is the invalidation of Furry’s theorem; already the threephoton amplitude is nonvanishing in a constant field. Moreover the modification of the photon dispersation relation through the background field can, depending on the photon polarisations, lead to the opening up of phase space for the photon splitting process . For the case of a magnetic field this process was calculated in the low photon energy limit in [25] and for general photon energies in [26]. The amplitude turns out to be very small for the magnetic field strengths presently attainable in the laboratory. Nevertheless, the photon splitting process is believed to be of relevance for the physics of neutron stars which are known to have magnetic fields approaching, and even surpassing, the “critical” magnetic field strength Gauss [27, 28, 29]. It seems also not impossible that, with some further improvements in laser technology, photon splitting may be observable in the laboratory in the near future [30, 31].
For QED calculations in constant external fields it is possible and advantageous to take account of the field already at the level of the Feynman rules, i.e. to absorb it into the free electron propagator. Suitable formalisms have been developed decades ago [32, 33, 21]. However beyond the simplest special cases they lead to exceedingly tedious and cumbersome calculations.
A different and more efficient formalism for such calculations has been developed during the last few years, using the socalled “stringinspired” technique. The idea of using string theory methods as a practical tool for calculations in ordinary quantum field theory was advocated by Bern and Kosower [34]. Their work led to the formulation of new computation rules for QCD amplitudes which made it, for example, feasible to perform a complete calculation of the oneloop five gluon amplitudes [35]. A parallel line of work led to the formulation of analogous computation rules for quantum gravity [36, 37]. The relation between the stringderived rules and ordinary Feynman rules was clarified in [38].
Later it was found that even in abelian gauge theory significant improvements over standard field theory methods can be obtained along these lines [39, 40]. Moreover, in this formalism the inclusion of constant external fields turned out to require only relatively minor modifications [41, 42, 43, 44, 45]. For this reason it has been extensively applied to constant field processes in QED. This includes a recalculation of the photonsplitting amplitude [46] as well as high order calculations of the derivative expansion of the QED effective action [42, 43, 47, 48]. A generalization to multiloop photonic amplitudes [49, 40, 50, 51, 45] was applied to a calculation of the twoloop correction to the EulerHeisenberg Lagrangian, using both propertime [45] and dimensional regularisation [52, 53, 54]. We will not discuss here the involved history of this subject but rather refer the interested reader to the review articles [55, 56]. For other work on QED amplitudes similar in spirit to the stringinspired approach see [57, 58, 59, 60, 61, 62].
In the present paper we first give, in chapter 2, a detailed and selfcontained exposition of the stringinspired technique for the calculation of oneloop scalar/spinor QED photon amplitudes in vacuum. In chapter 3 we extend this formalism to the inclusion of constant external fields along the lines of [45]. As a technical improvement on [45] we derive a decomposition of the generalised worldline Green’s functions in terms of the matrices with coefficients that are functions of the two standard Maxwell invariants. This will allow us to arrive at explicit results in a manifestly Lorentz covariant way. In chapter 4 we apply the formalism to a calculation of the scalar and spinor QED vacuum polarisation tensors in a constant field. Chapter 5 contains our conclusions.
2 The QED OneLoop Photon Amplitude in Vacuum
In [39] it was shown that, for the QED case, the full content of the BernKosower rules can be captured using an approach to quantum field theory based on firstquantized particle path integrals (‘worldline path integrals’).
2.1 Scalar Quantum Electrodynamics
For the case of scalar QED the basic formulas needed go back to Feynman [63]. The oneloop effective action due to a scalar loop for a Maxwell background can (in modern notation) be written as ^{1}^{1}1We work initially in the Euclidean with a positive definite metric . The Euclidean field strength tensor is defined by , , its dual by with . The corresponding Minkowski space amplitudes can be obtained by replacing , .
Here denotes the usual propertime for the loop fermion. For fixed denotes an integral over the space of all closed loops in spacetime with periodicity .
This path integral can be used for the calculation of the effective action itself as well as for obtaining the corresponding scattering amplitudes. As a first step in any evaluation, one has to take care of the zero mode contained in it. This is done by fixing the average position of the loop, i.e. one writes
where
(2.3) 
The remaining path integral is, in the ‘stringinspired formalism’, performed using the Wick contraction rule
(2.4) 
where
A “dot” always refers to a derivative in the first variable.
The free Gaussian path integral determinants are, in our conventions, given by
(2.6) 
Here denotes the spacetime dimension. Although in this paper we will consider only the fourdimensional case in this factor must be left variable in anticipation of dimensional regularization.
We can use this path integral for constructing the scalar QED photon amplitude as follows [39]. Expanding the ‘interaction exponential’,
the individual terms correspond to Feynman diagrams describing a fixed number of interactions of the scalar loop with the external field. The corresponding – photon scattering amplitude is then obtained by specializing to a background consisting of a sum of plane waves with definite polarizations,
(2.8) 
and picking out the term containing every once. This immediately yields the following representation for the  photon amplitude,
Here denotes the same photon vertex operator which is also used in string perturbation theory (see, e.g., [64]),
(2.10) 
At this stage the zeromode integration (LABEL:split) can be performed, yielding the energymomentum conservation factor
(2.11) 
The reduced path integral is Gaussian. Its evaluation therefore amounts to Wick contracting the expression
(2.12) 
using the correlator (2.4). For the performance of the Wick contractions it is convenient to formally exponentiate all the ’s, writing
(2.13) 
This allows one to rewrite the product of photon vertex operators as an exponential. Then one needs only to ‘complete the square’ to arrive at the following closed expression for the oneloop  photon amplitude [39]
Here it is understood that only the terms linear in all the have to be taken. Besides the Green’s function also its first and second deriatives appear,
With ‘dots’ we generally denote a derivative acting on the first variable, , and we abbreviate etc.
The expression (LABEL:scalarqedmaster) is identical with the corresponding special case of the ‘BernKosower Master Formula’ [34]. Let us consider explicitly the vacuum polarisation case, . For the expansion of the exponential factor yields the following expression,
After performing a partial integration on the first term of eq. (2.1) in either or , the integrand turns into
(2.16) 
(). Thus we have
(2.17)  
Note that the transversality of the vacuum polarization tensor is already manifest. We rescale to the unit circle, , and use translation invariance in to fix the zero to be at the location of the second vertex operator, . We have then
After performing the trivial  integration one arrives at
The result of the final integration is, of course, the same as is found in the standard field theory calculation for the sum of the corresponding two Feynman diagrams (see, e.g., [65]).
2.2 Spinor Quantum Electrodynamics
The worldline path integral representation (LABEL:scalarqedpi) can be generalized to the spinor QED case in various different ways. The formulation most suitable to the ‘stringy’ approach uses Grassmann variables [66, 67, 68],
Thus we have, in addition to the same coordinate path integral as in (LABEL:scalarqedpi), a Grassmann path integral representing the fermion spin. The boundary conditions on the Grassmann path integral are antiperiodic, , so that there is no new zero mode. The appropriate correlator is
(2.21) 
Our normalization for the free Grassmann path integral is
(2.22) 
The photon vertex operator (2.10) acquires an additional Grassmann piece,
(2.23) 
Looking again at the vacuum polarization case, we need to Wickcontract two copies of the above vertex operator. The calculation of is identical with the scalar QED calculation. The additional contribution from is
(2.24) 
Taking the free Grassmann path integral normalization (2.22) into account, eq.(2.17) for the scalar QED vacuum polarisation tensor generalises to the spinor QED case as follows,
(2.25)  
Proceeding as before one obtains
Remarkably, the explicit calculation of the Grassmann path integral can be circumvented, and replaced by the following simple pattern matching rule [34]. Writing out the exponential in eq.(LABEL:scalarqedmaster) one obtains an integrand
(2.27) 
with a certain polynomial depending on the various and on the kinematic invariants. Now one removes all second derivatives appearing in by suitable partial integrations in the variables ,
(2.28) 
This is possible for any [34]. The result is an alternative integrand for the scalar QED amplitude involving only and . The integrand for the spinor loop case can then, up to the global factor of , be obtained from the one for the scalar loop simply by replacing every closed cycle of ’s appearing in by its “worldline supersymmetrization”,
Note that an expression is considered a cycle already if it can be put into cycle form using the antisymmetry of (e.g. ). The replacement is done simultaneously on all cycles.
For the result of the partial integration procedure is not unique, however the above replacement rule is valid for all possible results. In [69] a certain standardized way was found for performing the partial integrations which leads to a canonical, permutation symmetric and gauge invariant decomposition of the QED  photon amplitudes.
3 The QED NPhoton Amplitude in a Constant Field
3.1 Generalization of the BernKosower Master Formula
The presence of an additional constant external field, taken in FockSchwinger gauge centered at [41], changes the path integral Lagrangian in eq.(LABEL:spinorpi) only by a term quadratic in the fields,
(3.1)  
(3.2)  
(3.3) 
where we have defined . These expressions should be understood as power series in the field strength matrix . Note that the generalized Green’s functions are still translationally invariant in , and thus functions of . By writing them as functions of the vacuum worldline Green’s functions we have left the  dependence implicit. This allows us to avoid making a case distinction between and that would become necessary otherwise [44]. Note also the symmetry properties
Since are, in general, nontrivial Lorentz matrices, the Wick contraction rules eqs.(2.4),(2.21) have to be replaced by
(3.5)  
(3.6) 
Another slight complication compared to the vacuum case is that, in contrast to their vacuum counterparts, , and have nonvanishing coincidence limits. Those are  independent:
(3.10) 
This is almost all we need to know for computing oneloop photon scattering amplitudes, or the corresponding effective action, in a constant overall background field. The only further information required at the one–loop level is the change in the free path integral determinants due to the external field. This change is [41]
(3.11)  
(3.12) 
Since those determinants describe the vacuum amplitude in a constant field they turn out to be, of course, just the propertime integrands of the EulerHeisenbergSchwinger formulas (see (1.1)).
Retracing our above calculation of the  photon path integral with the external field included we arrive at the following generalization of eq.(LABEL:scalarqedmaster), representing the scalar QED  photon scattering amplitude in a constant field [44, 45]:
From this formula it is obvious that adding a constant matrix to will have no effect due to momentum conservation. We can use this fact to get rid of the coincidence limit of , (3.7), namely instead of one can work with the equivalent Green’s function , defined by
(3.14) 
No such redefinition is possible for or .
The transition from scalar to spinor QED is done as in the vacuum case, again with only some minor modifications. The spinor QED integrand for a given number of photon legs is obtained from the scalar QED integrand by the following generalization of the BernKosower algorithm:

Partial Integration: After expanding out the exponential in the master formula (LABEL:scalarqedmasterF), and taking the part linear in all , remove all second derivatives appearing in the result by suitable partial integrations in .

Replacement Rule: Apply to the resulting new integrand the replacement rule (2.2) with substituted by . Since the Green’s functions are, in contrast to their vacuum counterparts, nontrivial matrices in the Lorentz indices, it must be mentioned here that the cycle property is defined solely in terms of the – indices, irrespectively of what happens to the Lorentz indices. For example, the expression
would have to be replaced by
The only other difference compared to the vacuum case is due to the nonvanishing coincidence limits of , eqs.(3.8),(3.9). Those lead to an extension of the “cycle replacement rule” to include onecycles [45]:
(3.15) 
The scalar QED EulerHeisenbergSchwinger determinant factor must be replaced by its spinor QED equivalent,
(3.16) 
Multiply by the usual factor of for statistics and degrees of freedom.
3.2 Lorentz covariant decomposition of the generalized worldline Green’s functions
For the result to be practically useful it will be necessary to know in more explicit form. In the calculations performed in [45, 46, 52] for a purely magnetic field pointing along the  direction the explicit matrix form of had been used. This would also be possible in the generic case, where, excepting the case , one could use the Lorentz invariance to choose both and to point along the  axis. For this case the Green’s functions can be easily written out explicitly. However, it is possible to directly express them in terms of Lorentz invariants, without specialisation of the Lorentz frame. This can be done in the following way. Defining the Maxwell invariants
we have the relations
(3.18)  
(3.19) 
Define
(3.20)  
(3.21)  
(3.22) 
Then one has
(3.23)  
(3.24)  
(3.25) 
With the help of these relations one easily derives the following formulas,
where () are arbitrary even (odd) functions in the field strength matrix regular at ,
(3.27) 
Decomposing into their even (odd) parts (),
(3.28) 
and applying the above formulas we obtain the following matrix decompositions of ,
Here we have further introduced
Note that , . The scalar, dimensionless coefficient functions appearing in these formulas are given by
The nonvanishing coincidence limits are in ,