constant external fields in gauge theory and the spin 0, 12, 1 path integrals

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Annals of Physics � PH5716

annals of physics 259, 313�365 (1997)

Constant External Fields in Gauge Theoryand the Spin 0, 1

2 , 1 Path Integrals

Martin Reuter

Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22603 Hamburg, Germany

Michael G. Schmidt

Institut fu� r Theoretische Physik, Universita� t Heidelberg,Philosophenweg 16, D-69120 Heidelberg, Germany

Christian Schubert*

School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540;and Humboldt Universita� t zu Berlin, Invalidenstr. 110, D-10115 Berlin, Germany

Received December 5, 1996

We investigate the usefulness of the ``string-inspired technique'' for gauge theory calcula-tions in a constant external field background. Our approach is based on Strassler's worldlinepath integral approach to the Bern�Kosower formalism, and on the construction of worldline(super-) Green's functions incorporating external fields as well as internal propagators. Theworldline path integral representation of the gluon loop is reexamined in detail. We calculatethe two-loop effective actions induced for a constant external field by a scalar and spinor loop,and the corresponding one-loop effective action in the gluon loop case. � 1997 Academic Press

1. INTRODUCTION

It is by now well established that techniques from string perturbation theory canbe used to improve on calculational efficiency in ordinary quantum field theory.The relevance of string theory for this purpose is based on the fact that many, andperhaps all, amplitudes in quantum field theory can be represented as the infinitestring tension limits of appropriately chosen (super-) string amplitudes. This is, ofcourse, an intrinsic and well-known property of string theory.

It is a more recent discovery, however, that such representations can lead to aninteresting alternative to standard Feynman diagram calculations. Following earlier

article no. PH975716

3130003-4916�97 �25.00

Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* Supported by Deutsche Forschungsgemeinschaft.

work on the ;-function for Yang�Mills theory [1�4], Bern and Kosower [5] con-sidered the infinite string tension limit of gauge boson scattering amplitudes, for-mulated in an appropriate heterotic string model. By a detailed analysis of thislimit, they succeeded in deriving a novel type of ``Feynman rules'' for the construc-tion of ordinary one-loop gluon (photon) scattering amplitudes. The resultingintegral representations are equivalent to the ones originating from Feynmandiagrams [6], but they lead to a significant reduction of the number of terms to becomputed in gauge theory calculations. This property was then successfullyexploited to obtain both five gluon [7] and four graviton amplitudes [8].

Strassler [9] later showed that, for many cases of interest, the same integralrepresentations can be derived from a first-quantized reformulation of ordinary fieldtheory. In this approach, one starts with writing the one-loop effective action as a(super)particle path-integral, of a type which has been known for many years[10�18]. Those path-integrals are then considered as the field-theory analogues ofthe Polyakov path integral and evaluated in a way analogous to string perturbationtheory (some suggestions along similar lines had already been made in [19]).

The resulting formalism offers an alternative to standard field theory techniqueswhich circumvents much of the apparatus of quantum field theory. It works equallywell for effective action and scattering amplitude calculations, on- and off-shell. Ithas been applied to a number of calculations in gauge theory [20�24] and gener-alized to cases where, besides gauge and scalar self-couplings, Yukawa [25�27] andaxial couplings [28, 29, 27] are present.

Due to its simplicity, it appears also to be well-suited to the construction ofmultiloop generalizations of the Bern�Kosower formalism. Steps in this directionhave already been taken by various authors, and along different lines [30�32]. Inparticular, the original Bern�Kosower program becomes very hard to carrythrough at the two-loop level, due to the complicated structure of genus two stringamplitudes. Nevertheless, recently substantial progress has been achieved in thisline of work [32, 33].

A multiloop generalization following the spirit of Strassler's approach has beenproposed by two of the present authors, first for scalar field theories in [34]. Thisgeneralization uses the concept of worldline Green's functions on graphs [34, 35,33] and leads to integral representations combining whole classes of graphs. Thiswork was extended to QED in [36], and its practical viability demonstrated by arecalculation of the two-loop (scalar and spinor) QED ;-functions. For the case ofmultiloop amplitudes in scalar QED, a more comprehensive treatment along thesame lines was given in [37]. This includes amplitudes involving external scalars.

An important role in quantum electrodynamics is played by calculations involvingconstant external fields. This subject originates with Euler and Heisenberg's classicone-loop calculation of the static limit of photon scattering in spinor quantum electro-dynamics [38]. Schwinger's introduction of the proper-time method in 1951 [39]allowed him to reproduce this result, and the analogous one for scalar quantumelectrodynamics, with considerably less effort. For calculations of this type, it hasturned out generally advantageous to take account of the external field already at the

314 REUTER, SCHMIDT, AND SCHUBERT

level of the Feynman rules, i.e. to absorb it into the free propagators. Appropriateformalisms have been developed both for QED [40�42] and QCD [43].

The purpose of the present paper is threefold. First, we would like to cast theworldline multiloop formalism proposed in [34, 36] into a form suitable for con-stant external field calculations in quantum electrodynamics. This will be achievedin a way analogous to field theory, namely by modifying the worldline Green'sfunctions so as to take the external field into account. At the one-loop level, similarproposals have already been made by several authors [23, 24, 44] (see also [45]).

Second, we will apply this formalism to a two-loop effective action calculation inQED. In [21] the worldline formalism was used for a recalculation of the Euler�Heisenberg Lagrangian. In the present work, we will extend this calculation to thetwo-loop level and calculate the correction to this effective Lagrangian due to oneinternal photon, both for scalar and spinor QED. We discuss how far this calcula-tion improves on previous calculations of the same quantities [46�50].

Third, we reconsider the superparticle path integral representation of the spin 1particle and use this path integral for calculating the effective action induced for anexternal constant pseudo-abelian field by a gluon loop in Yang�Mills theory. Whileexternal gluons pose no particular problems in the worldline formalism, apart fromforcing path-ordering, internal gluons are a much more delicate matter. While it isnot difficult to construct free path integrals describing particles of arbitrary spin,those constructions usually run into consistency problems if one tries to couple apath integral with spin higher than 1

2 to a spin-1 background [51, 52]. A somewhatnonstandard path integral, mimicking the superstring, had been proposed byStrassler [9] in his rederivation of the Bern�Kosower rules for the gluon loop case.To the best of our knowledge, it has not yet been proven that this path integralcorrectly reproduces all the pinch terms implicit in the Bern�Kosower masterformula.

We will first give a rigorous derivation of a spin-1 path integral which, while notidentical with the one used by Strassler, is easily seen to be equivalent. We then useit for calculating the effective action induced by a gluon loop for a constant pseudo-abelian background gluon field. The result will again be in agreement with theliterature [53] and provides a nontrivial check on the correctness of Strassler'sproposal.

The organization of this paper is as follows. In Section 2 we review the worldline-formalism for one-loop photon scattering in scalar and spinor QED, and itsgeneralization to gluon scattering. We then indicate the changes which arenecessary to take a constant external field into account. Section 3 extends thisanalysis to the gluon loop. This so far includes worldline calculations of the one-loop effective actions induced for a (pseudo-abelian) constant external field by ascalar, spinor and gluon loop. For the QED case, we then extend the constant fieldformalism to the multiloop level in Section 4. We apply it to calculations of thecorresponding two-loop effective actions for scalar QED in Section 5 and for spinorQED in Section 6. Section 7 contains some remarks on the various ways ofcalculating the two-loop QED ;-function in this formalism. Our results will be

315CONSTANT EXTERNAL FIELDS

discussed in Section 8. In Appendix A we discuss the path integral representationof the electron propagator in an external field. In Appendix B we derive the variousworldline Green's functions used in our calculations and discuss some of theirproperties. Appendix C contains a collection of some results concerning the deter-minants which we encounter in the evaluation of the spin-1 path integral.

2. ONE-LOOP PHOTON AMPLITUDES IN A CONSTANTBACKGROUND FIELD

We begin with shortly reviewing how one-loop photon (gluon) scatteringamplitudes are calculated in the worldline formalism [9, 21, 36]. In his rederivationof the Bern�Kosower rules for gauge boson scattering off a spinor loop, Strasslersets out from the following well-known path integral representation for the corre-sponding one-loop effective action (see, e.g., [19, 54]):

1[A]=&2 |�

e&m2T |P

_tr P exp _&|T

0d{ \1

4x* 2+

��4 +ieA+x* +&ie�+F+& �&+& . (2.1)

In this formula, T is the usual Schwinger proper-time parameter. The periodic func-tions x+({) describe the embedding of the circle with circumference T into D-dimen-sional Euclidean spacetime, while the �+({)'s are antiperiodic Grassmann functions.The periodicity properties are expressed by the subscripts P, A on the path integral.The colour trace tr and the path ordering P apply, of course, only to the non-abelian case. We have chosen a constant euclidean worldline metric.

The case of a scalar loop is obtained simply by discarding all Grassmann quan-tities and the global factor of &2, which takes care of the difference in statistics anddegrees of freedom.

Analogous path integral representations exist for the scalar and electronpropagators in a background field [10, 11, 55, 56]. The integration is then over aspace of paths with fixed boundary conditions. Those are obvious in the scalar case,while in the fermionic case there has been some debate on the correct boundaryconditions to be imposed on the Grassmann path integral [55, 56, 19]. An ade-quate path integral representation for the gluon propagator seems to be missing inthe literature and will be derived in Section 3. The more familiar cases of the scalarand spin-1

2 propagators are included in Appendix A for completeness.While in the present paper we will make use only of the loop path integrals, we

expect those propagator path integral representations to play an important role infuture extensions of this formalism.

A number of different techniques have been applied to calculating this worldlinepath integral and various generalizations thereof (see, e.g., [57�66]). In the ``string-inspired'' approach, the first step is to split the coordinate path integral into centerof mass and relative coordinates,

| Dx=| dx0 | Dy

x+({)=x +0+y+({) (2.2)

0d{ y+({)=0.

The path integrals over y and � are then evaluated by Wick contractions, as in aone-dimensional field theory on the circle. The Green's functions to be used are

( y+({1) y&({2)) =&g+&GB({1 , {2)=&g +& _ |{1&{2|&({1&{2)2

(�+({1) �&({2)) =12

g+&GF ({1 , {2)=12

g+& sign({1&{2).

We will often abbreviate GB({1 , {2)=: GB12 etc. They solve the differential equations

�{21

GB({1 , {2)=$({1&{2)&1T

(2.4)12

��{1

GF ({1 , {2)=$({1&{2)

with periodic (antiperiodic) boundary conditions for GB(GF). Some freedom existsin the definition of GB , which has been discussed elsewhere [22, 67]; in particular,a constant added to GB would drop out after momentum conservation is applied.

With our conventions, the free path integrals are normalized as

Dy exp _&|T

y* 2&=[4?T ]&D�2

D� exp _&|T

��4 &=1.

The result of this evaluation is the one-loop effective Lagrangian L(x0). Combinedwith a covariant Taylor expansion of the external field at x0 , this yields a new andhighly efficient algorithm for calculating higher derivative expansions in gaugetheory [21, 22].

One-loop scattering amplitudes are obtained by specializing the background to afinite sum of plane waves of definite polarization. Equivalently, one may defineintegrated vertex operators

VA=T a |T

0d{[x* +=+&2i�+ �&k+=&] exp[ikx({)] (2.6)

for external photons (gluons) of definite momentum and polarization and calculatemultiple insertions of those vertex operators into the free path integral (of course,only the first term has to be taken for the scalar loop). T a denotes a generator ofthe gauge group in the representation of the loop particle. In the nonabelian case,for the spinor loop an additional two-gluon vertex operator is required [9].

The path integrals are performed using the well-known formulas for Wickcontractions involving exponentials, e.g.,

(exp[ik1 } x({1)] exp[ik2 } x({2)]) =exp[GB({1 , {2) k1 } k2] (2.7)

etc. (the factors x* +=+ may be formally exponentiated for convenience).Explicit execution of the �-path integral would be algebraically equivalent to the

calculation of the corresponding Dirac traces in field theory. It can be circumventedby the following remarkable feature of the Bern�Kosower formalism, which may beunderstood as a consequence of worldsheet [68] or worldline [9] supersymmetry.After evaluation of the x-path integral, one is left with an integral over theparameters T, {1 , ..., {N , where N is the number of external legs. In the nonabeliancase, the path-ordering leads to ordered {-integrals, � >N&1

i=1 d{i %({i&{i+1). Theintegrand is an expression consisting of an exponential factor,

exp _ :i< j

GB({i , {j) ki kj& ,

multiplied by a polynomial in the first and second derivatives of GB ,

G4 B({1 , {2)=sign({1&{2)&2({1&{2)

T(2.8)

G� B({1 , {2)=2$({1&{2)&2T

(here and in the following, a ``dot'' denotes differentiation with respect to the firstvariable).

All G� B 's can be eliminated by partial integrations on the worldline, leading to anequivalent parameter integral dependent only on GB and G4 B . According to theBern�Kosower rules, all contributions from fermionic Wick contractions may then

be taken into account simply by simultaneously replacing every closed cycle of G4 B 'sappearing, say G4 Bi1 i2 G4 Bi2 i3 } } } G4 Bini1 , by its ``supersymmetrization,''

G4 Bi1 i2 G4 Bi2 i3 } } } G4 Bini1 � G4 Bi1 i2 G4 Bi2 i3} } } G4 Bini1&GFi1 i2 GFi2 i3 } } } GFini1 . (2.9)

Note that an expression is considered a cycle already if it can be put into cycle formusing the antisymmetry of G4 B (e.g., G4 Bab G4 Bab=&G4 BabG4 Bba). Unfortunately thepractical value of this procedure rapidly diminishes with increasing number ofexternal legs (�5), as the number of terms making up the integrand starts tosignificantly increase in the partial integration procedure [69].

The worldline supersymmetry (see Eq. (A.9)) makes it possible to combine thex- and �-path integrals into the following super path integral [13, 19, 58, 70, 36]:

1[A]=&2 |�

e&m2T | DX tr Pe&�0T d{ � d%[&(1�4) XD3X&ieDX+ A+(X )], (2.10)

X +=x++- 2 %� +=x +0+Y +

�%&%

��{

(2.11)

| d% %=1.

The photon (gluon) vertex operator is then rewritten as

&T a |T

0d{ d% =+DX+ exp[ikX ], (2.12)

and we are left with a single Wick contraction rule

(Y +({1 , %1) Y&({2 , %2)) =&g+&G� ({1 , %1 ; {2 , %2)(2.13)

G� ({1 , %1 ; {2 , %2)=GB({1 , {2)+%1%2GF ({1 , {2).

The fermion loop case can thus be made to look formally identical to the scalarloop case and be regarded as its ``supersymmetrization.'' This analogy has its rootsin the fact that the string-inspired technique corresponds to the use of a second-order formalism for fermions in field theory (see [71] and references therein),instead of the usual first-order ones. In practical terms it means that the GrassmannWick contractions are replaced by a number of Grassmann integrals, which haveto be performed at a later stage of the calculation. Ultimately the superfieldformalism leads to the same collection of parameter integrals to be performed;however, we have found it useful for keeping intermediate expressions compact. Inthe nonabelian case it has the further advantage that the introduction of an addi-tional two-gluon vertex operator can be avoided. Instead, one introduces a suitablesupersymmetric generalization of the functions %({i&{i+1) appearing in the ordered{-integrals [70].

Now let us assume that we have, in addition to the background field A+(x) westarted with, a second one, A� +(x), with constant field strength tensor F� +& . We willrestrict ourselves to the abelian case for the remainder of this section. Using theFock�Schwinger gauge centered at x0 , we may take A� +(x) to be of the form

A� +(x)= 12 y&F� &+ . (2.14)

The constant field contribution to the worldline lagrangian may then be written as

2L=&12 iey+F� +& y* &&ie�+F� +&�& (2.15)

in components, or as

2L=&12 ieY+F� +&DY& (2.16)

in the superfield formalism.In any case, it is still quadratic in the worldline fields, and therefore need not

be considered part of the interaction lagrangian; we can absorb it into the freeworldline propagator(s). This means that we need to solve, instead of Eqs. (2.4) forthe worldline Green's functions, the modified equations

�{21

&2ieF��

�{1+ GB({1 , {2)=$({1&{2)&1T

(2.17)

��{1

&2ieF� + GF ({1 , {2)=$({1&{2). (2.18)

These equations will be solved in Appendix B, with the result (deleting the ``bar''again)

GB({1 , {2)=1

2(eF )2 \ eFsin(eFT )

e&ieFTG4 B12+ieFG4 B12&1T+

(2.19)

GF ({1 , {2)=GF12

e&ieFTG4 B12

cos(eFT ).

Equivalent expressions have been given for the pure magnetic field case in [23] andfor the general case in [44]. These expressions should be understood as powerseries in the field strength matrix F (note that Eqs. (2.19) do not assume invertibilityof F ). Note also that the generalized Green's functions are still translation invariantin { and, thus, functions of {1&{2 . They are, in general, nontrivial Lorentzmatrices, so that the Wick contraction rule equation (2.3) have to be rewritten as

( y+({1) y&({2))=&G+&B ({1 , {2),

(2.20)(�+({1) �&({2))= 1

2G+&F ({1 , {2).

Again momentum conservation leads to the freedom to subtract from GB itsconstant coincidence limit,

GB({, {)=1

2(eF )2 \eF cot(eFT )&1T+ . (2.21)

To correctly obtain this and other coincidence limits, one has to apply the rules

G4 B({, {)=0, G4 2B({, {)=1. (2.22)

More generally, coincidence limits should always be taken after derivatives.We also need the generalizations of G4 B , G� B , which turn out to be

G4 B({1 , {2)#2 ({1| (�{&2ieF )&1 |{2)=i

eF \ eFsin(eFT )

e&ieFTG4 B12&1T+

(2.23)

G� B({1 , {2)#2 ({1| (I&2ieF�&1{ )&1 |{2) =2$12&2

eFsin(eFT )

e&ieFTG4 B12.

Let us also give the first few terms of the Taylor expansions in F for those fourfunctions:

GB12=GB12&T6

G4 B12 GB12TeF+\T3

G 2B12&

90+ (eF )2+O(F 3)

G4 B12=G4 B12+2i \GB12&T6 + eF+

G4 B12GB12T(eF )2+O(F 3)

(2.24)

G� B12=G� B12+2iG4 B12eF&4 \GB12&T6 + (eF )2+O(F 3)

GF12=GF12&iGF12 G4 B12TeF+2GF12 GB12 T(eF )2+O(F 3).

Again GB and GF may be assembled into a superpropagator,

G� ({1 , %1 ; {2 , %2)#GB({1 , {2)+%1%2GF ({1 , {2). (2.25)

At first sight this definition would seem not to accomodate the nonvanishing coin-cidence limit of GF (which cannot be subtracted). Nevertheless, comparison with thecomponent field formalism shows that the correct expressions are again reproducedif one takes coincidence limits after superderivatives. For instance, the correlator(D1X({1 , %1) X({1 , %1)) is evaluated by calculating

(D1X({1 , %1) X({2 , %2)) =%1G4 B12&%2GF12 (2.26)

and then setting {2={1 .

It is easily seen that the substitution rule equation (2.9) continues to hold, ifone defines the cycle property solely in terms of the {-indices, irrespectively of whathappens to the Lorentz indices. For example, the expression

=1G4 B12k2 =2G4 B23 =3k3G4 B31k1 (2.27)

would have to be replaced by

=1G4 B12k2 =2G4 B23 =3k3G4 B31k1 &=1G4 F12 k2=2G4 F23 =3k3G4 F31k1 . (2.28)

The only novelty is again due to the fact that, in contrast to G4 B and GF , G4 B andGF have nonvanishing coincidence limits,

G4 B({, {)=i cot(eFT )&i

eFT(2.29)

GF ({, {)= &i tan(eFT ). (2.30)

As a consequence, we now have also a substitution rule for one-cycles,

G4 B({i , {i) � G4 B({i , {i)&GF ({i , {i)=i

sin(eFT ) cos(eFT )&

. (2.31)

This is almost all we need to know for computing one-loop photon scatteringamplitudes, or the corresponding effective action, in a constant overall backgroundfield. The only further information required at the one-loop level is the change inthe path integral determinants due to the external field, i.e. the vacuum amplitudein the constant field. For spinor QED, this just corresponds to the Euler�HeisenbergLagrangian and has, in the present formalism, been calculated in [21]. Let usbriefly retrace this calculation (the fact that the Euler�Heisenberg integrand may berepresented as a superdeterminant was already noted in [72]). In the scalar QEDcase, we have to replace

| Dy exp _&|T

4 y* 2&=Det$&1�2P [&�2

{]=[4?T ]&D�2 (2.32)

Det$&1�2P [&�2

{+2ieF�{]=[4?T ]&D�2 Det$&1�2P [I&2ieF�&1

{ ] (2.33)

(as usual, the prime denotes the absence of the zero mode in a determinant).Application of the ln det=tr ln identity yields

Det$&1�2P [I&2ieF�&1

{ ]=exp _&12

(&1)n+1

n(&2ie)n tr[F n] Tr[�&n

{ ]&=exp _&

n=2n even

Bnn! n

(2ieT )n tr[F n]&

=det&1�2 _sin(eFT )eFT & , (2.34)

where the Bn are the Bernoulli numbers. In the second step Eq. (B.27) was used.The analogous calculation for the Grassmann path integral yields a factor

Det+1�2A [I&2ieF�&1]=det1�2[cos(eFT )]. (2.35)

For spinor QED we therefore find a total overall determinant factor of

[4?T ]&D�2 det&1�2 _tan(eFT )eFT & . (2.36)

Expressing tr[F 2n] in terms of the two invariants of the electromagnetic field,

tr[F 2n]=2[(a2)n+(&b2)n],

a2= 12 [E2&B2+- (E2&B2)2+4(E } B)2], (2.37)

b2= 12 [&(E2&B2)+- (E2&B2)2+4(E } B)2],

we obtain the standard Schwinger proper-time representation of the (unsubtracted)Euler�Heisenberg�Schwinger Lagrangians,

L (1)scal[F ]=

1(4?)2 |

dTT 3 e&m2T (eaT )(ebT )

sin(eaT ) sinh(ebT ),

(2.38)L (1)

spin[F ]=&2

(4?)2 |�

dTT 3 e&m2T (eaT )(ebT )

tan(eaT ) tanh(ebT ).

3. GAUGE BOSON LOOPS IN EXTERNAL FIELDS

In this section we first express the one-loop effective action and also thepropagator of spin-1 gauge bosons in an arbitrary external Yang�Mills field interms of a worldline path integral. We then will evaluate the action in a covariantlyconstant background.

We employ the background gauge fixing technique so that the effective action1[Aa

+] becomes a gauge invariant functional of Aa+ [73, 74]. The gauge fixed classical

action reads, in D dimensions,

S[a; A]=14 | d Dx F a

+&(A+a) F a+&(A+a)+

12: | d Dx(Dab

+ [A] ab+)2. (3.1)

A priori, the background field Aa+ is unrelated to the quantum field aa

+ . The kineticoperator of the gauge boson fluctuations one obtains as the second functionalderivative of S[a, A] with respect to aa

+ at fixed Aa+ . This leads to the inverse

propagator

Dab+&=&Dac

\ Dcb\ $+&&2igF ab

+& (3.2)

and the effective action

1[A]=12

ln det(D)=&12 |

Tr[e&TD]. (3.3)

In writing down Eq. (3.2) we adopted the Feynman gauge :=1. The covariantderivative D+#�++igAa

+ T a and the field strength F ab+&#F c

+&(T c)ab are matrices inthe adjoint representation of the gauge group with the generators given by (T a)bc=&if abc. The full effective action is obtained by adding the contribution of theFaddeev�Popov ghosts to Eq. (3.3). The evaluation of the ghost determinantproceeds along the same lines as scalar QED and we shall not discuss it here.

In order to derive a path integral representation of the heat-kernel

Tr[exp (&TD)] (3.4)

we first look at a slightly more general problem. We generalize the operator D to

h� +&#&D2$+&+M+&(x), (3.5)

where M+&(x) is an arbitrary matrix in color space. In particular, we do not assumethat the Lorentz trace M++ is zero. Given M+& , we construct the following one-particle Hamilton operator:

H� =( p̂++gA+(x̂))2&:�� +M&+�� & : . (3.6)

The system under consideration has a graded phase-space coordinatized by x+ , p+ ,and two sets of anticommuting variables, �+ and �� + , which obey canonicalanticommutation relations:

�� +�� &+�� &�� +=$+& . (3.7)

For a reason which will become obvious in a moment we have adopted the ``anti-Wick'' ordering in (3.6): all �� 's are on the right of all �'s, e.g.,

:�� :�� ; :=�� :�� ;(3.8)

:�� ; �� : :=&�� :�� ; .

We can represent the commutation relations on a space of wave functions 8(x, �)depending on x+ and a set of classical Grassmann variables �+ . The ``position''operators x̂+=x+ , �� +=�+ act multiplicatively on 8 and the conjugate momenta asderivatives p̂+=&i�+ and �� +=��+ . Thus the Hamiltonian becomes [77]

H� =&D2+�&M&+(x)�

��+. (3.9)

The wave functions 8 have a decomposition of the form

8(x, �)= :D

, ( p)+1 } } } +p

(x) �+1} } } �+p . (3.10)

This suggests the interpretation of 8 as an inhomogeneous differential form on RD

with the fermions �+ playing the role of the differentials dx+ [76]. The form-degreeor, equivalently, the fermion number is measured by the operator

F� =�� +�� +=�+�

��+. (3.11)

We are particularly interested in one-forms:

8(x, �)=.+(x) �+ . (3.12)

The Hamiltonian (3.9) acts on them according to

(H� 8)(x, �)=(h� +&.&) �+ . (3.13)

We see that, when restricted to the one-form sector, the quantum system withthe Hamiltonian (3.6) is equivalent to the one defined by the bosonic matrixHamiltonian h� +& [76, 77].

The euclidean proper time evolution of the wave functions 8 is implemented bythe kernel

K(x2 , �2 , t2 | x1 , �1 , t1)=(x2 , �2| e&(t2&t1) H� |x1 , �1) (3.14)

which obeys the Schro� dinger equation

\ ��T

+H� + K(x, �, T | x0 , �0 , 0)=0 (3.15)

with the initial condition K(x, �, 0 | x0 , �0 , 0)=$(x&x0) $(�&�0). It is easy towrite down a path integral solution to Eq. (3.15). For the trace of K one obtains

Tr[e&TH� W ]=|P

Dx({) |A

D�({) D�� ({) tr Pe&�0T d{ L (3.16)

L= 14 x* 2+igA+(x) x* ++�� +[�{$+&&M&+] �& . (3.17)

We have again periodic boundary conditions for x+({), and antiperiodic boundaryconditions for �+({). If we use periodic boundary conditions for the fermionswe arrive at a representation of the Witten index [79] rather than the partitionfunction:

Tr[(&1)F� e&TH� W ]=|P

Dx({) |P

D�({) D�� ({) tr Pe&�0T d{ L. (3.18)

At this point we have to mention a subtlety which is frequently overlooked but willbe important later on. If we regard the Hamiltonian (3.6) as a function of theanticommuting c-numbers �+ and �� + it is related to the classical Lagrangian (3.17)by a standard Legendre transformation. The information about the operator orderingis implicit in the discretization which is used for the definition of the path-integral.Different operator orderings correspond to different discretizations. Here we shalladopt the midpoint prescription [78] for the discretization, because only in thiscase the familiar path-integral manipulations are allowed [80]. It is known [78,80�83] that, at the operatorial level, this is equivalent to using the Weyl orderedHamiltonian H� W . This is the reason why we wrote H� W rather than H� on the l.h.s.of Eqs. (3.16) and (3.18). In order to arrive at the relation (3.13) we had to assumethat the fermion operators in H� are ``anti-Wick'' ordered. Weyl ordering amountsto a symmetrization in �� and � so that

H� W =( p̂++gA+(x̂))2+ 12M&+(x̂)(�� & �� +&�� +�� &)=H� & 1

2M++(x̂). (3.19)

In the second line of (3.19) we used (3.6) and (3.7). (With respect to x̂+ and p̂+ ,Weyl ordering is used throughout.) If we employ (3.19) in (3.16) we obtain therepresentation for the partition function of the anti-Wick ordered exponential:

Tr[e&TH� ]=|P

Dx({) |A

D�({) D�� ({) tr P exp _&|T

0d{[L({)+ 1

2M++(x({))]& .

(3.20)

Let us now calculate the partition function Tr[exp(&Th� )] which is a generaliza-tion of the heat-kernel needed in Eq. (3.3). By virtue of Eq. (3.13) we may write

Tr[e&Th� ]=Tr1[e&TH� ], (3.21)

where ``Tr1'' denotes the trace in the one-form sector of the theory which containsthe worldline fermions. In order to perform the projection on the one-form sectorwe identify M+& with

M+&=C$+&&2igF+& , (3.22)

where C is a real constant. As a consequence,

H� =H� 0+CF� , (3.23)

H� 0#( p̂++gA+(x̂))2&2igF&+(x̂) �� &�� + (3.24)

denoting the Hamiltonian which corresponds to the inverse propagator D. Thefermion number operator F� #�� +�� + is anti-Wick ordered by definition. Its spectrumconsists of the integers p=0, 1, 2, ..., D. Note that M++=DC and that, because ofthe antisymmetry of F+& , the Hamiltonian H� 0 has no ordering ambiguity in itsfermionic piece. It will prove useful to apply Eq. (3.20) not to H� directly, but toH� &C=H� 0+C(F� &1). The result reads then

Tr[e&CT(F� &1)e&TH� 0]=exp _&CT \D2

&1+& |P

Dx({) |A

D�({) D�� ({) tr Pe&�0T d{ L

(3.25)

L= 14 x* 2+igA+(x) x* ++�� +[(�{&C ) $+&&2igF+&] �& . (3.26)

After having performed the path integration in (3.25) we shall send C to infinity.While this has no effect in the one-form sector, it leads to an exponential suppres-sion factor exp[&CT( p&1)] in the sectors with fermion numbers p=2, 3, ..., D.Hence only the zero and the one forms survive the limit C � �. In order toeliminate the contribution from the zero forms we insert the projector[1&(&1)F� ]�2 into the trace. It projects on the subspace of odd form degrees, andit is easily implemented by combining periodic and antiperiodic boundary condi-tions for �+ . In this manner we arrive at a representation of the partition functionof H� 0 restricted to the one-form sector:

Tr1[e&TH� 0]= limC � �

Tr _12

(1&(&1)F� ) e&CT(F� &1)e&TH� 0&= lim

C � �exp _&CT \D

Dx({)12 \|A

&|P+ D�({) D�� ({) Tr Pe&�0

T d{ L. (3.27)

Because Tr[exp(&TD)]=Tr1[exp(&TH� 0)], Eq. (3.27) implies for the effectiveaction

1[A]=&12

limC � � |

exp _&CT \D2

&1+& |P

Dx12 \|A

&|P+ D� D��

_Tr P exp _&|T

0d{ {1

4x* 2+igA+x* ++�� +[(�{&C ) $+&&2igF+&] �&=& .

(3.28)

Several comments are in order here. The factor exp[&CTD�2] in (3.28) is dueto the difference between the Weyl and the anti-Wick-ordered Hamiltonian. It iscrucial for obtaining a finite result in the limit C � �. In fact, for D=4 it convertsthe prefactor exp[+CT ] to a decaying exponential exp[&CT ].1 From the pointof view of the worldline fermions, C plays the role of a mass. Their free Green'sfunction GC({2&{1) $+& with

GC({2&{1)#({2| (�{&C )&1 |{1) (3.29)

reads for periodic and antiperiodic boundary conditions, respectively,

GCP({)=&[3(&{)+3({) e&CT]

1&e&CT

(3.30)

GCA({)=&[3(&{)&3({) e&CT]

1+e&CT .

We observe that for C � � there is an increasingly strong asymmetry between theforward and backward propagation in the proper time. Further details of theGreen's functions (3.30) can be found in Appendix B.

We mention in passing that there exists another simple method for the projectionon the one-form sector. We can insert a Kronecker-delta $1F� into the partition func-tion (3.20) and exponentiate it in terms of a parameter integral over an angularvariable ::

Tr1[e&TH� 0]=T2? |

2?�T

0d: ei:T Tr[e&i:TF� e&TH� 0]. (3.31)

The r.h.s. of (3.31) can be represented by a path integral which, for the fermions,involves antiperiodic boundary conditions only. The corresponding action is similarto the one used above but with C replaced by i:. Instead of the limit C � � one

1 In Ref. [9] the reordering factor was not taken into account and the change of the sign in D=4was attributed to a difference between Minkowski and Euclidean spacetime which is not correct in ouropinion.

has to perform the :-integration now. The computational effort is essentially thesame in both cases.

The representation (3.28) of the effective action does not coincide with the oneused by Strassler [9]. While he uses the same kinetic term in the fermionicworldline Lagrangian, he modifies the interaction term according to

�� +F+& �& � /+F+&/&#�� +F+&�&+ 12F+&(�+ �&+�� +�� &), (3.32)

/+({)#�+({)+�� +({). (3.33)

As a consequence, the modified Hamiltonian H� 0 contains terms which change thefermion number by two units. This means that, during the proper time evolution,the 1-form representing the gauge boson can evolve into a 3-form. However, in thelimit C � � the substitution (3.32) causes no problems. The reason is that forC � � the energy gap between 1- and 3-forms becomes infinite, and therefore a1-form at {=0 will remain a 1-form for all {>0. In the modified formalism, Wickcontractions of the interaction term (3.3) involve the 2-point function of /, i.e.,G/({)#GC({)&GC(&{). From (3.30) we obtain explicitly

G/P({)=sign({)

sinh[C(T�2&|{| )]sinh[CT�2]

(3.34)

G/A({)=sign({)

cosh[C(T�2&|{| )]cosh[CT�2]

These Green's functions do not coincide with the ones given by Strassler [9];however, they become effectively equivalent in the limit C � �. The substitution(3.32) is motivated by the algebraic simplification which it entails in perturbationtheory where one inserts a sum of plane waves for Aa

+(x). We shall not do this inthe present paper but rather calculate the path integral exactly for a covariantlyconstant field strength. In this case the representation (3.28) is more convenientthan the one advocated by Strassler.

An important building block for higher-loop calculations is the gluon propagatorin an external Yang�Mills field. It is given by the proper-time integral of the evolu-tion kernel (3.14). The latter can be represented by the following path integral withopen boundary conditions:

K(x2 , �2 , T | x1 , �1 , 0)=|x(T )=x2

x(0)=x1

Dx({) |�(T )=�2

�(0)=�1

D�({) | D�� ({) e&�0T d{ L. (3.35)

The Lagrangian L is given by (3.17) with M+&=&2igF+& because we do not needthe C-term in the case of open boundary conditions. The reason is that the

Hamiltonian (3.9) preserves the fermion number. Therefore, the kernel K will mapa one-form wave function of the type (3.12) onto another one-form. In fact, in theone-form sector, K is represented by a Lorentz matrix

K+&(x2 , T | x1 , 0)=(x2 , +| e&TD |x1 , &) (3.36)

which is related to the gluon propagator by

(x2 , +| D&1 |x1 , &) =|�

0dT K+&(x2 , T | x1 , 0). (3.37)

(The color indices are kept implicit.) It is easy to express the bosonic matrix K+&

in terms of the kernel K with fermionic arguments. If one writes

8(x2 , �2 , T )=| d Dx1 d D�1 K(x2 , �2 , T | x1 , �1 , 0) 8(x1 , �1 , 0) (3.38)

and inserts a wave function of the type (3.12) at both {=0 and {=T one finds that

K+&(x2 , T | x1 , 0)=�

��2+| d D�1 K(x2 , �2 , T | x1 , �1 , 0) �1& . (3.39)

By combining Eq. (3.37) with Eqs. (3.39) and (3.35) we obtain the desired pathintegral representation of the gluon propagator.

In order to get a better understanding of this representation, let us assume thatthe background A+ is either abelian or quasi-abelian, and that no path-orderingmust be observed therefore. We may then rewrite (3.35) according to

K(x2 , �2 , T | x1 , �1 , 0)

=|x(T )=x2

x(0)=x1

Dx({) KF (�2 , T | �1 , 0) exp _&|T

0d{[ 1

4x* 2+igA+x* +]& . (3.40)

Here the fermionic integral

KF (�2 , T | �1 , 0)=|�(T )=�2

�(0)=�1

D�({) | D�� ({) exp _&|T

0d{ �� +(�{$+&&2igF+&) �&&

(3.41)

is a functional of the bosonic trajectory x+({). It can be evaluated exactly [75�77]and has a remarkably simple structure:

KF (�2 , T | �1 , 0)=$(�2&S(T ) �1). (3.42)

In (3.42) we introduced

S+&(T )=P exp _2ig |T

0d{ F(x({))&+&

, (3.43)

where the path-ordering is necessary because of the Lorentz matrix structure. Alter-natively, Eq. (3.42) can be established by noting that KF solves the Schro� dingerequation corresponding to (3.41):

_ ��T

&2ig�+F+&�

��&& KF (�, T | �1 , 0)=0. (3.44)

Because ST=S&1, it follows that

��2+

| d D�1 KF (�2 , T | �1 , 0) �1&=S+&(T ) (3.45)

and, therefore,

K+&(x2 , T | x1 , 0)=|x(T )=x2

x(0)=x1

Dx({) P exp _2ig |T

0d{ F(x({))&+&

_exp _&|T

0d{[ 1

4x* 2+igA+x* +]& . (3.46)

Later on we shall use the evolution kernel in the form (3.46). Clearly we could havewritten down this representation without going through the original path integral(3.35). However, in multiloop calculations it will be advantageous if the gluon loopsand the corresponding propagators are represented in a coherent framework. Infact, if we recall that �� + amounts to the derivative ��+ in the Schro� dinger picture,the evolution kernel, for any background, may be rewritten in the very elegant form

K+&(x2 , T | x1 , 0)=|x(T )=x2

x(0)=x1

Dx({) | D�({) | D�� ({) $(�(T )) �� +(T ) �&(0) e&�0T d{ L.

(3.47)

In Eq. (3.47), �(0) and �(T ) are integrated independently.Now we evaluate the path integral (3.28) for the case that the background has

a covariantly constant field strength. We assume that the gauge field has the form

Aa+(x)=naA+(x), nana=1, (3.48)

where na is a constant unit vector in color space. The associated field strengthF a

+&=naF+& with F+&=�+A&&�&A+ satisfies Dab: F b

+&=0. Both A+ and F+& enterEq. (3.28) for the gauge boson loop as matrices in the adjoint representation.

Hence, the only nontrivial color matrix which enters the path integral is naT a. If wedenote the eigenvalues of this matrix by &l , l=1, 2, ..., and evaluate the color tracein the diagonal basis, the integral of the path integrand assumes the form,

Tr PI(gA+)=:l

I(g&l A+). (3.49)

The path ordering has no effect here. We observe that we are effectively dealingwith a set of abelian theories whose gauge coupling is given by g&l .

While the condition Dab: F b

+&=0 does not necessarily imply that F+& is constant,we further assume that F+& is an x-independent matrix and that the gauge field isof the form

A+(x)= 12x&F&+ . (3.50)

In the following we keep the diagonalization of the color matrix implicit andcontinue to use the notation Tr I(gA+) rather than �l I(g&l A+). We keep in mind,however, that A+ and F+& may be treated as pure numbers as far as their colorstructure is concerned.

For the field (3.48), (3.50) all path integrals in (3.28) are Gaussian. We separatethe constant mode from the x+ integration as before and obtain

1[A]=&12 | d Dx0 tr |

Det$P[�2{ $+&&2igF+&�{]&1�2 lim

C � �Y(C ) (3.51)

Y(C )=12

exp _&CT \D2

&1+&[DetA[(�{&C ) $+&&2igF+&]

&DetP[(�{&C ) $+&&2igF+&]]. (3.52)

We denote the real eigenvalues of iF+& by f (:), :=1, ..., D, and we use the formulasin Appendix C in order to express Y(C ) in terms of these eigenvalues:

Y(C )=12

exp _&CT \D2

&1+& {DetA[(�{&C ) $+&] `D

cosh[(T�2)(C+2gf (:))]cosh[CT�2]

&DetP[(�{&C) $+&] `D

sinh[(T�2)(C+2gf (:))]sinh[CT�2] = . (3.53)

Note that in the case of the fermionic integration with periodic boundary condi-tions the zero mode of �{ was not excluded from the determinant and that Eq. (C.6)applies therefore. In (3.53) we have factored out the free determinants because the

method of Appendix C can yield only ratios of determinants. The normalizationfactors

`A, P#DetA, P[(�{&C ) $+&]=(DetA, P[�{&C])D (3.54)

are most easily determined by recalling their operatorial interpretation as the parti-tion function and the Witten index of a free Fermi oscillator, respectively,

DetA[�{&C]=Tr[e&CTF� W](3.55)

DetP[�{&C]=Tr[(&1)F� e&CTF� W].

Here F� W=(�� &�� )�2=F� &1�2 is the Weyl-ordered fermion number operator(for D=1) with eigenvalues &1�2 and +1�2. As a consequence,

`A=(2 cosh[CT�2])D

(3.56)`P=(2 sinh[CT�2])D.

Taking advantage of 7: f (:)=0, we can rewrite (3.53) as

Y(C)= 12'&1 _`

(1+'q:)&`:

(1&'q:)& (3.57)

with '#exp(&CT ) and q:#exp[&2gTf (:)]. In the limit C � �, i.e., ' � 0, theleading O(1) terms cancel among the products in (3.57), and one obtains

limC � �

Y(C )= :D

exp[&2gTf (:)]=trL cos[2gTF]. (3.58)

Here we exploited that if f (:) is an eigenvalue, so is &f (:). (trL denotes the tracewith respect to the Lorentz indices.)

In complete analogy with the scalar case, Eq. (2.35), the bosonic determinant in(3.51) gives rise to a factor of

(4?T )&D�2 exp _&12

trL lnsin(gTF )

(gTF) & . (3.59)

Hence, our final result for the gauge boson loop becomes

1[A]=&12 | d Dx0(4?)&D�2

T &D�2 tr exp _&12

trL lnsin(gTF)

(gTF ) & trL cos(2gTF). (3.60)

Equation (3.60) coincides with the result which was found with the help of thetraditional techniques [53, 84�86].

By a computation similar to the previous one, but with open boundary condi-tions for the path integral, one can find the gluon propagator in the background(3.48), (3.50). It is given by (3.37) with

K+&(x2 , T | x1 , 0)=exp(2igTF )+& |x(T )=x2

x(0)=x1

Dx({) e&S[x( } )]. (3.61)

The path integral which remains to be evaluated is just the one for the scalarpropagator. Now in the action functional,

S[x( } )]= 14 |

0d{ x+[&�2

{ $+&+2igF+&�{] x&+ 14 [x+(T ) x* +(T )&x+(0) x* +(0)],

(3.62)

one has to be careful about the surface terms which appear after the integration byparts. For open paths they give rise to a nonzero contribution in general. Since Sis quadratic in x+ , the saddle point approximation of the path integral (3.61) givesthe exact answer. We can solve it by expanding the integration variable about theclassical trajectory connecting x1 and x2:

x+({)=xclass+ ({)+y+({). (3.63)

Here xclass({) obeys

x� class+ =2igF+& x* class

& (3.64)

and it satisfies the boundary conditions xclass(0)=x1 and xclass(T )=x2 . The fluc-tuation y({) satisfies correspondingly y(0)=0=y(T ). Hence the fluctuation deter-minant is almost the same as in the periodic case, the only difference being thatthere is no zero-mode integration in the present case:

K+&(x2 , T | x1 , 0)=exp(2igTF )+& exp(&S[xclass]) Det$P[�2{ $+&&2igF+&�{]&1�2.

(3.65)

The classical trajectory is easily found:

xclass({)=x1+exp(2igF{)&1exp(2igFT )&1

(x2&x1). (3.66)

Its action is entirely due to the surface terms in (3.62). In the Fock�Schwingergauge centered at x1 it reads

S[xclass]= 14 (x2&x1) gF cot(gTF)(x2&x1). (3.67)

Putting everything together we obtain the final result for the propagation kernel:

K+&(x2 , T | x1 , 0)=(4?T )&D�2 exp[2igTF]+&

_ exp _&14

(x2&x1) gF cot(gTF )(x2&x1)&_exp _&

trL lnsin(gTF )

(gTF) & . (3.68)

4. A MULTILOOP GENERALIZATION FOR QUANTUMELECTRODYNAMICS

We return to the case of quantum electrodynamics and proceed to the multiloopgeneralization of the formalism developed in Section 2. This generalization is con-structed in strict analogy to the case without an external field [36], and we referthe reader to that publication for some of the details.

We first consider scalar electrodynamics at the two-loop level, i.e. a scalar loopwith an internal photon correction. A photon insertion in the worldloop may, inFeynman gauge, be represented in terms of the following current�current inter-action term (see, e.g., [87�89]) inserted into the one-loop path integral,

4?*+1 |T

0d{a |

x* ({a) } x* ({b)([x({a)&x({b)]2)* (4.1)

(*=D�2&1). The denominator of this term is again written in the proper-timerepresentation,

1(*)4?*+1([x({a)&x({b)]2)*=|

0dT� (4?T� )&D�2 exp _&

(x({a)&x({b))2

4T� & . (4.2)

It appears then as yet another correction term to the free part of the worldlineLagrangian for the scalar loop path integral. It is convenient to rewrite this termin the form

(x({a)&x({b))2=|T

0d{1 |

0d{2 x({1) Bab({1 , {2) x({2), (4.3)

Bab({1 , {2)=[$({1&{a)&$({1&{b)][$({2&{a)&$({2&{b)]. (4.4)

This allows us to write the new total bosonic kinetic operator as

�2&2ieF�&Bab�T� . (4.5)

To find the inverse of this operator, we write it as a geometric series,

\�2&2ieF�&Bab

T� +&1

=(�2&2ieF�)&1+(�2&2ieF�)&1 Bab

T�(�2&2ieF�)&1+ } } } ,

which can be easily summed to yield the Green's function

G (1)B ({1 , {2)=GB({1 , {2)+

[GB({1 , {a)&GB({1 , {b)][GB({a , {2)&GB({b , {2)]T� & 1

2Cab(4.7)

with the definition

Cab#GB({a , {a)&GB({a , {b)&GB({b , {a)+GB({b , {b)

=cos(eFT )&cos(eFTG4 Bab)

(eFT ) sin(eFT ). (4.8)

Note that this is almost identical with what one would obtain from the ordinarybosonic two-loop Green's function G (1)

B [34] by simply replacing all GBij 's appearingthere by the corresponding GBij 's. The more complicated structure of the denominatoris due to the fact that the GBij 's are not symmetric anymore; rather we have GBij=GT

and, moreover, we have nonvanishing coincidence limits. The denominator is nowin general a nontrivial Lorentz matrix and must be interpreted as a matrix inverse(of course, all matrices appearing here commute with each other).

For the free Gaussian path integral, it is again a simple application of theln det=tr log-identity to calculate

Det$P _&�2+2ieF�+Bab

T� &=Det$P[&�2] Det$P[I&2ieF�&1] Det$P _I&

T�(�2&2ieF�)&1&

=[4?T ]D det _sin(eFT )eFT & det _I&

12T�

Cab& . (4.9)

The generalization from one-loop to two-loop photon amplitude calculations inscalar QED thus requires no changes of the formalism itself, but only of the Green'sfunctions used and of the global determinant factor. Of course, in the end threemore parameter integrations have to be performed.

As in the case without a background field [36], the whole procedure goesthrough essentially unchanged for the fermion loop, if the superfield formalism isused. As a consequence, the supersymmetrization property carries over to the two-loop level, leading to a close relationship between the parameter integrals for thesame amplitude calculated for the scalar and for the fermion loop: They differ onlyby a replacement of all GB 's by G� 's, by the additional %-integrations, and by theone-loop Grassmann path-integral factor, Eq. (2.35).

The generalization to an arbitrary fixed number of photon insertions isstraightforward and leads to formulas for the generalized (super-)Green's functionsand (super-)determinants identical with the ones given in [34, 36] up to a replace-ment of all GB 's (G� 's ) by GB 's (G� 's). The only point to be mentioned here is thatcare must now be taken in writing the indices of the GBij 's appearing. For instance,the bosonic three-loop Green's function must be written

G (2)B ({1 , {2)

=GB({1 , {2)+ 12 :

k, l=1

[GB({1 , {ak)&GB({1 , {bk)] A (2)kl [GB({al , {2)&GB({bl , {2)].

(4.10)

The matrix A appearing here is the inverse of the matrix

\T1& 12 (GBa1a1

&GBa1 b1&GBb1a1

+GBb1 b1)

& 12 (GBa2a1

&GBa2b1&GBb2a1

+GBb2 b1)

& 12 (GBa1a2

&GBa1b2&GBb1 a2

+GBb1b2)

T2& 12(GBa2a2

&GBa2b2&GBb2a2

+GBb2b2)+ .

(4.11)

T1 , T2 denote the proper-time lengths of the two inserted propagators.The discussion of the general case of an arbitrary number of scalar (spinor) loops

interconnected by photon propagators requires no new concepts and will bedeferred to a forthcoming review article [90].

While this multiloop construction is done most simply using the Feynman gaugefor the propagator insertions, other gauges can be implemented as well (the gaugefreedom has also been discussed in [37]). In an arbitrary covariant gauge, thephoton insertion term, Eq. (4.1), would read

4?D�2 |T

0d{a |

0d{b {1+:

2&1+ x* a } x* b

[(xa&xb)2]D�2&1

+(1&:) 1 \D2 +

x* a } (xa&xb)(xa&xb) } x* b

[(xa&xb)2]D�2 = . (4.12)

Here :=1 corresponds to the Feynman gauge, :=0 to the Landau gauge. Theintegrand may also be written as

&1+ x* a } x* b

[(xa&xb)2]D�2&1&1&:

2&2+ �

��{b

[(xa&xb)2]2&D�2. (4.13)

This shows that on the worldline gauge transformations correspond to the additionof total derivative terms, which is another fact familiar to string theorists (see, e.g.,[91]). This form of the photon insertion is also the more practical one for actualcalculations. Again it carries over to the fermion loop in the superfield formalismmutatis mutandis.

5. THE TWO-LOOP EULER�HEISENBERG LAGRANGIANFOR SCALAR QED

We proceed to the simplest two-loop application of this formalism, which is thetwo-loop generalization of Schwinger's formula for the constant field effectiveLagrangian due to a scalar loop, Eq. (2.38). According to the above, we may writethe two-loop correction to this effective Lagrangian in the form

L (2)scal[F ]=(4?)&D \&

2 + |�

e&m2T T &D�2 |�

0dT� |

0d{a |

_det&1�2 _sin(eFT )eFT & det&1�2 _T� &

Cab& ( y* a } y* b). (5.1)

We have now a fourfold parameter integral, with T and T� representing the scalarand photon proper-times and {a, b are the endpoints of the photon insertion movingaround the scalar loop. The first determinant factor is identical with the one-loopEuler�Heisenberg�Schwinger integrand equation (2.35) and represents the changeof the free path integral determinant due to the external field; the second onerepresents its change due to the photon insertion. A single Wick contraction needsto be performed on the ``left over'' numerator of the photon insertion, using themodified worldline Green's function equation (4.7). This yields

( y* a } y* b)=tr _G� Bab+12

(G4 Baa&G4 Bab)(G4 Bab&G4 Bbb)T� & 1

2 Cab & . (5.2)

Care must be taken again with coincidence limits, as the derivatives should not acton the variables {a , {b explicitly appearing in the two-loop Green's function; againthe correct rule in calculating ( y* a y* b) is to first differentiate Eq. (4.7) with respectto {1 , {2 and to put {1={a , {2={b afterwards.

After replacing the G4 Bij 's and Cab by the explicit expressions given in Eqs. (2.23)and Eq. (4.8), we have already a parameter integral representation for the baredimensionally regularized effective Lagrangian.

Alternatively one may, in the spirit of the original Bern�Kosower approach,remove G� B by a partial integration with respect to {a or {b . Using the formula

d det(M )=det(M ) tr(dMM&1) (5.3)

and G4 Bab=&G4 TBba , one obtains the equivalent parameter integral,

L (2)scal[F]=(4?)&D \&

2 + |�

e&m2TT &D�2 |�

0dT� |

0d{a |

_det&1�2 _sin(eFT )eFT & det&1�2 _T� &

12 {tr G4 Bab tr _ G4 Bab

T� & 12Cab&+tr _(G4 Baa&G4 Bab)(G4 Bab&G4 Bbb)

T� & 12 Cab &= . (5.4)

For the further evaluation and renormalization of this Lagrangian, we will spe-cialize the constant field F to a pure magnetic and to a pure electric field in turns.To facilitate comparison with previous calculations [46�50], we will, moreover,switch from dimensional regularization to proper-time regularization. This meansthat henceforth we put D=4 and, instead, introduce a proper-time UV cutoff T0

later on.We begin with a pure magnetic field. The field is taken along the z-axis, so that

F 12=B, F 21=&B are the only nonvanishing components of the field strengthtensor. We also introduce the abbreviations z=eBT and projection matrices

F� #\0 0 0 00 0 1 00 &1 0 00 0 0 0+ , I03#\

1 0 0 00 0 0 00 0 0 00 0 0 1+ , I12#\

0 0 0 00 1 0 00 0 1 00 0 0 0+ . (5.5)

We may then rewrite the determinant factor equations (2.34), (2.36) as

det&1�2 _sin(eFT )eFT &=

zsinh(z)

, (5.6)

det&1�2 _tan(eFT )eFT &=

ztanh(z)

. (5.7)

The Green's function equations (2.19), (2.23) specialize to

G� B({1 , {2)=GB12I03&T2

[cosh(zG4 B12)&cosh(z)]z sinh(z)

I12+T2z \

sinh(zG4 B12)sinh(z)

&G4 B12+ iF�

G4 B({1 , {2)=G4 B12I03+sinh(zG4 B12)

sinh(z)I12&\cosh(zG4 B12)

sinh(z)&

1z+ iF�

(5.8)G� B({1 , {2)=G� B12I03+2 \$12&

z cosh(zG4 B12)T sinh(z) + I12+2

z sinh(zG4 B12)T sinh(z)

GF ({1 , {2)=GF12I03+GF12

cosh(zG4 B12)cosh(z)

I12&GF12

sinh(zG4 B12)cosh(z)

iF� .

In writing GB we have already subtracted its coincidence limit, which is indicated bythe ``bar.'' Cab simplifies to

Cab=&2GBabI03&2G zBabI12 , (5.9)

where we have defined

G zBab#

[cosh(z)&cosh(zG4 ab)]z sinh(z)

=GBab&1

Babz2+O(z4). (5.10)

We will also use the derivative of this expression,

G4 zBab=

sinh(zG4 Bab)sinh(z)

. (5.11)

Similarly, we can rewrite

det&1�2 _sin(eFT )eFT \T� &

Cab+&=z

sinh(z)##z

tr[G� Bab]=8$ab&4&4z cosh(zG4 Bab)

sinh(z)(5.12)

tr G4 Bab tr _ G4 Bab

T� & 12 Cab&=2 _G4 Bab+

sinh(zG4 Bab)sinh(z) &_G4 Bab #+

sinh(zG4 Bab)sinh(z)

tr _(G4 aa&G4 ab)(G4 ab&G4 bb)T� & 1

2Cab &=&#z sinh2(zG4 Bab)+[cosh(zG4 Bab)&cosh(z)]2

sinh2(z)

&G4 2Bab #,

with the abbreviations

##(T� +GBab)&1,

#z#(T� +G zBab)&1.

We rescale to the unit circle, {a, b=Tua, b , and use translation invariance in { to set{b=0. We have then

GB({a , {b)=TGB(ua , ub)=T(ua&u2a),

G4 B({a , {b)=G4 B(ua , ub)=1&2ua.

After performance of the T� -integration, which is finite and elementary, Eq. (5.4)turns into

L (2)scal[B]=&(4?)&4 e2

2 |�

dTT 3 e&m2T z

sinh(z) |1

0dua A(z, ua), (5.13)

ln(GBab �G zBab)

(GBab&G zBab)2+

(G zBab)(GBab&G z

(GBab)(GBab&G zBab)= ,

A1=4[G zBabz coth (z)&GBab],

(5.14)A2=1+2G4 BabG4 z

Bab&4G zBabz coth(z),

A3= &G4 2Bab&2G4 Bab G4 z

GzBab is now given by Eq. (5.10) with T=1. Here and in the following we often use

the identity G4 2Bab=1&4GBab to eliminate G4 Bab in favour of GBab .

Renormalization must now be addressed and will be performed in close analogyto the discussion in [50]. The integral in Eq. (5.13) suffers from two kinds ofdivergences:

1. An overall divergence of the scalar proper-time integral ��0 dT at the lower

integration limit.

2. Divergences of the �10 dua parameter integral at the points 0, 1, where the

endpoints of the photon propagator become coincident, ua=ub .

The first one will be removed by one- and two-loop photon wave function renor-malization, the second one by the one-loop scalar mass renormalization. As is wellknown, vertex renormalization and scalar self-energy renormalization cancel eachother in this type of calculation and need not be considered.

By power counting, an overall divergence can exist only for the terms in the effec-tive Lagrangian which are of order at most quadratic in the external field B.Expanding the integrand of Eq. (5.13), K(z, ua)#(z�sinh(z)) A(z, ua), in thevariable z, we find

K(z, ua)=_ 3GBab

GBab&+_&12

GBab+2& z2+O(z4). (5.15)

The complicated singularity appearing here at the point ua=ub indicates that thisform of the parameter integral is not yet optimized for the purpose of renormalization.In particular, it shows a spurious singularity in the coefficient of the induced Maxwellterm tz2. This comes not unexpected, as the cancellation of subdivergences impliedby the Ward identity has, in a general gauge, no reason to be manifest at theparameter integral level.

We could improve on this either by switching to Landau gauge, or by performinga suitable partial integration on the integrand. The latter procedure is lesssystematic, but easy enough to implement for the simple case at hand: Inspectionof the two versions we have of this parameter integral, the original one, Eq. (5.1),and the partially integrated one, Eq. (5.4), shows that we can optimize theintegrand by choosing a certain linear combination of both versions, namely

L(2)scal[B]= 3

4_Eq. (5.1)+ 14_Eq. (5.4). (5.16)

After integration over T� , this leads to another version of Eq. (5.13),

L (2)scal[B]=&(4?)&4 e2

2 |�

dTT 3 e&m2T z

sinh(z) |1

0dua A$(z, ua), (5.17)

with a different integrand,

A$={A$0ln(GBab �G z

Bab)(GBab&G z

Bab)+A$1

ln(GBab�G zBab)

(GBab&G zBab)2

(G zBab)(GBab&G z

A$3(GBab)(GBab&G z

Bab)= ,

A$0=3 _2z2G zBab&

ztanh(z)

&1& ,(5.18)

A$1=A1& 32[G4 2

Bab&G4 z2Bab],

A$2=A2& 32[G4 Bab G4 z

Bab+G4 z2Bab],

A$3=A3+ 32[G4 2

Bab+G4 Bab G4 zBab].

We have not yet taken into account here the term involving $ab , stemming fromG� Bab , which was contained in the integrand of Eq. (5.17). This term corresponds, indiagrammatic terms, to a tadpole insertion and could, therefore, be safely deleted.However, it will be quite instructive to keep it and check explicitly that it is takencare of by the renormalization procedure. It leads to an integral, ��

0 (dT� �T� 2), whichwe regulate by introducing an UV cutoff for the photon proper-time,

T� 0

dT�T� 2=

1T� 0

. (5.19)

It gives then a further contribution E(T� 0) to L (2)scal[B],

E(T� 0)=&3(4?)&4 e2 1T� 0

dTT 2 e&m2T z

sinh(z). (5.20)

Expanding the new integrand, K$(z, ua)#(z�sinh(z)) A$(z, ua), in z, we find the simpleresult

K$(z, ua)=&61

GBab+3z2+O(z4). (5.21)

In particular, the absence of a subdivergence for the Maxwell term is now manifest.We delete the irrelevant constant term and add and subtract the Maxwell term.Defining

K02(z, ua)= &61

GBab+3z2, (5.22)

the Lagrangian then becomes

L (2)scal[B]=E(T� 0)&

:2(4?)3 |

dTT 3 e&m2T 3z2

2(4?)3 |�

dTT 3 e&m2T |

0dua[K$(z, ua)&K02(z, ua)]. (5.23)

The second term, which we denote by F, is divergent when integrated over thescalar proper-time T. We regulate it by introducing another proper-time cutoff T0

for the scalar proper-time integral:

F(T0) :=&:

2(4?)3 |�

dTT 3 e&m2T 3z2 (5.24)

(we use 2T0 rather then T0 for easier comparison with [50]). The third term is con-vergent at T=0, but it still has a divergence at ua=ub , as it contains negativepowers of GBab . Expanding the integrand in a Laurent series in GBab , one finds

K$(z, ua)&K02(z, ua)=f (z)GBab

+O(G 0Bab),

(5.25)

f (z)=3 _2&z

sinh(z)&

z2 cosh(z)sinh(z)2 & .

Again, the singular part of this expansion is added and subtracted, yielding

L (2)scal[B]=E(T� 0)+F(T0)&

:2(4?)3 |

dTT 3 e&m2T |

1&T0 �T

T0 �Tdua

f (z)GBab

2(4?)3 |�

dTT 3 e&m2T |

0dua _K$(z, ua)&K02(z, ua)&

f (z)GBab& . (5.26)

The last integral is now completely finite. The third term, which we call G(T0), isfinite at T=0, as f (z)=O(z4) by construction. Here we have introduced T0 for thepurpose of regulating the divergence at ua=ub . The ua-integral for this term is thenreadily computed and yields, in the limit T0 � 0, a contribution,

|1&T0 �T

T0 �Tdua

=&2 ln \T0

T +=&2 ln(#m2T0)+2 ln(#m2T ). (5.27)

We have rewritten this term for reasons which will become apparent in a moment(ln(#) denotes the Euler�Mascheroni constant). Next note that we can relate thefunction f (z) to the scalar one-loop Euler�Heisenberg Lagrangian, Eq. (2.38). If wewrite this Lagrangian for the pure magnetic field case and subtract the twodivergent terms lowest order in z, we obtain

L� (1)scal[B]=

1(4?)2 |

dTT 3 e&m2T _ z

sinh(z)+

6&1& . (5.28)

On the other hand, we can write

f (z)=3 _2&z

sinh(z)&

z2 cosh(z)sinh(z)2 &=3T 3 d

dT { 1T 2 _ z

sinh(z)+

6&1&= . (5.29)

By a partial integration over T, we can therefore reexpress

1(4?)2 |

dTT 3 e&m2Tf (z)=3

(4?)2 |�

dTT 2 e&m2T _ z

sinh(z)+

=&3m2 ��m2 L� (1)

scal[B]. (5.30)

To proceed, we need the value of the one-loop mass displacement in scalar QED,computed in the proper-time regularization. This we borrow from [48]2:

$m2=3:4?

m2 _&ln(#m2T0)+76

m2T� 0& . (5.31)

Using this result, we may rewrite

G(T0)=_$m2&72

4?T� 0&�

�m2 L� (1)scal[B]

(4?)3 |�

dTT 3 e&m2T ln(#m2T ) f (z). (5.32)

2 Note that this differs by a sign from $m2 as used in [36]. Here this denotes the mass displacementitself ; there it is the corresponding counterterm.

As expected the (1�T� 0)-term introduced by the one-loop mass renormalizationcancels the tadpole term E(T� 0), up to its constant and Maxwell parts. Moreover,the remaining divergence of G(T0) for T0 � 0 has been absorbed by $m2.

Putting all the pieces together, we can write the complete two-loop approximationto the effective Lagrangian:

L (�2)scal [B0]= &

1(4?)2 |

dTT 3 e&m2

6+L� (1)

scal[B0]+$m20

��m2

L� (1)scal[B0]

��m2

L� (1)scal[B0]&

(4?)3 |�

dTT 3 e&m2

0T ln(#m20T ) f (z)

2(4?)3 |�

dTT 3 e&m2

0 T |1

f (z)GBab&

2(4?)3 |�

dTT 3 e&m2

0T z2 \3&TT� 0+ . (5.33)

We have rewritten this Lagrangian in bare quantities, since up to now we have beenworking in the bare regularized theory. Only mass and photon wave functionrenormalization are required to make this effective Lagrangian finite:

m2=m20+$m2

e=e0Z 1�23 , (5.34)

B=B0Z &1�23 .

Here $m20 has already been introduced in Eq. (5.31), while Z3 is chosen such as to

absorb the diverging one- and two-loop Maxwell terms in Eq. (5.33). Note that thisleaves z=e0B0 T unaffected. The final answer becomes3

L(�2)scal [B]=&

(4?)2 |�

dTT 3 e&m2T _ z

sinh(z)+

:(4?)3 m2 |

dTT 2 e&m2T _ z

sinh(z)+

2(4?)3 |�

dTT 3 e&m2T |

f (z)GBab&

(4?)3 |�

dTT 3 e&m2T ln(#m2T ) f (z). (5.35)

3 Note added in proof: The constant 72 multiplying the third term is incorrect and should be replaced

by 92 . This has now been established both by a detailed comparison with [46] and another recalculation

using dimensional regularization [92].

This parameter integral representation is of a similar but simpler structure than theone given by Ritus [46].

The corresponding result for the case of a pure electric field is obtained from itby the simple substitution

B � &iE. (5.36)

This makes an important and well-known difference. In the electric field case theT-integration acquires new divergences due to the appearance of poles. This leadsto an imaginary part of the effective action and to a probability for pair creation.At the one-loop level, the first such pole appears at the critical field strength

Ecr=m2�e (5.37)

(see Eq. (2.38)). The two-loop contribution to the effective Lagrangian affects alsothe imaginary part and the pair creation probability. A detailed investigation ofthose corrections has been undertaken in [47, 48].

The calculation for the case of a generic constant field would be only moderatelymore difficult, if one uses the Lorentz frame, where the magnetic and electric fieldsare parallel, as in [46].

6. THE TWO-LOOP EULER�HEISENBERG LAGRANGIANFOR SPINOR QED

The corresponding calculation for the spinor loop case proceeds in completeanalogy when formulated in the superfield formalism. This allows us to immediatelywrite down the analogue of Eqs. (5.1), (5.2):

L (2)spin[F]=(&2)(4?)&D \&

2 + |�

e&m2TT &D�2 |�

0dT� |

0d{a d{b | d%a d%b

_det&1�2 _tan(eFT )eFT & det&1�2 _T� &

C� ab& ( &Da ya } Db yb) , (6.1)

with a superfield Wick contraction

(&Da ya } Db yb) =tr _Da DbG� ab+12

Da(G� aa&G� ab) Db(G� ab&G� bb)T� & 1

2C� ab & . (6.2)

The notations should be obvious.Performing the Grassmann integrations � d%a � d%b , and removing G� Bab by partial

integration as before, we obtain the equivalent of Eq. (5.4),

L(2)spin[F]=(4?)&D e2 |

e&m2TT &D�2 |�

0dT� |

0d{a |

_det&1�2 _tan(eFT )eFT & det&1�2 _T� &

12 {tr G4 Bab tr _ G4 Bab

T� & 12Cab&&tr GFab tr _ GFab

T� & 12Cab&

+tr _(G4 Baa&G4 Bab)(G4 Bab&G4 Bbb+2GFaa)+GFab GFab&GFaa GFbb

T� & 12Cab &= . (6.3)

In writing this formula, we have used the symmetry between {a and {b to reduce thenumber of terms. The same expressions could have been obtained starting fromEq. (5.4) and using the generalized one-loop substitution rule.

Specializing to the pure magnetic case and D=4, it is then again a matter ofsimple algebra to calculate the traces and T� -integrals. After rescaling and setting{b=0, the result can be written as

L(2)spin[B]=

:(4?)3 |

dTT 3 e&m2T z

tanh(z) |1

0duaB(z, ua), (6.4)

B(z, ua)={B1

ln(GBab �G zBab)

(GBab&G zBab)2+

G zBab(GBab&G z

GBab(GBab&G zBab)= ,

B1=A1&4z tanh(z) G zBab,

(6.5)B2=A2+8z tanh(z) G z

Bab&3,

B3=A3&4z tanh(z) G zBab+3.

Comparison with an earlier field theory calculation performed by Dittrich and oneof the authors [50] shows that the integrand of Eq. (6.4) allows for a direct iden-tification with its counterpart there, as given in Eqs. (7.21), (7.22). This requiresnothing more than a rotation to Minkowskian proper-time, T � is, a transforma-tion of variables from ua to v :=G4 Bab , and the use of trigonometric identities. Inparticular, our quantities GBab , G z

Bab then identify with the quantities a, b there.The renormalization of this Lagrangian has, for the spinor-loop case, been

carried through in detail in that work. We will therefore not repeat this analysishere, and we just give the final result for the renormalized two-loop contribution tothe Euler�Heisenberg Lagrangian4:

4 Note added in proof. The constant &10, multiplying the third term, is incorrect and should bereplaced by &18 [92] (compare the footnote before Eq. (5.35)).

L (�2)spin [B]= &

(4?)2 |�

dTT 3 e&m2T _ z

tanh(z)&

&10m2 :(4?)3 |

dTT 2 e&m2T _ z

tanh(z)&

(4?)3 |�

dTT 3 e&m2T ln(#m2T ) g(z)

(4?)3 |�

dTT 3 e&m2T |

0dua _L(z, ua)&L02(z, ua)&

g(z)GBab& (6.6)

L(z, ua)=z

tanh(z)B(z, ua),

L02(z, ua)= &12

GBab+2z2, (6.7)

g(z)= &6 _ z2

sinh(z)2+z coth(z)&2& .

For a study of the strong field limit of this Lagrangian see again [50].The first exact calculation of this two-loop Lagrangian for the fermion loop case

is again due to Ritus [49], who also used proper-time methods to arrive at acertain two-parameter integral.

The integral representation given above is equivalent to the one given by Ritus,but simpler. In [50] it was obtained by convoluting a free photon propagator withthe polarization tensor of a fermion in a constant magnetic field. The essential partof this calculation consists of deriving a compact integral representation for thepolarization tensor. To this end, complicated expressions involving Dirac traces andmomentum integrals have to be evaluated. In the string-inspired case, no analogousmanipulations are needed, and the computational effort for doing the parameterintegrals which it introduces instead is much smaller.

Moreover, when applied to spinor QED, the method of the present paper yieldsthe corresponding result for scalar QED with almost no further effort. This wouldnot be the case for the standard field theory techniques.

Concerning the physical relevance of this type of calculation, let us mention theexperiment PVLAS in preparation at Legnaro, Italy, which is an optical experimentdesigned to yield the first experimental measurement of the Euler�HeisenbergLagrangian [93, 94]. It is conceivable that the technology used there may evenallow for the measurement of the two-loop correction in the near future [94, 95].

7. THE 2-LOOP QED ;-FUNCTIONS REVISITED

Finally, let us remark that the method we have employed in this paper for thecalculation of the full two-loop Euler�Heisenberg Lagrangians also improves on thecalculation of the two-loop QED ;-functions as it had been presented in [36]. Forthe extraction of the ;-function coefficients one needs only to calculate the inducedMaxwell terms. Up to the contributions from one-loop mass renormalization, thecorrect two-loop scalar [96] and spinor [97] QED coefficients can thus be read offfrom the expansions (see Eqs. (5.21), (6.7))

K$(z, ua)=&6

GBab+3z2+O(z4),

L(z, ua)=&12

GBab+2z2+O(z4).

Comparing with [36] we see that the use of the generalized Green's functions GB ,GF has saved us two integrations: The same formulas (B.2), which there had beenemployed for executing the integrations over the points of interaction {1 , {2 withthe external field, have now entered already at the level of the construction of thoseGreen's functions. Of course, for the ;-function calculation all terms of order higherthan O(F 2) are irrelevant, so that one could then as well use the truncations of thoseGreen's functions given in Eq. (2.24). Moreover, one would choose an external fieldwith the property F 2

tI.Note that in the fermion loop case a subdivergence-free integrand was obtained

proceeding directly from the partially integrated version, Eq. (6.3). This fact, whichhad already been noticed in [36], is not accidental and can be understood by ananalysis of the quadratic divergences. In the scalar QED case, there are threepossible sources of quadratic divergences for the induced Maxwell term:

1. The contact term containing $ab .

2. The leading order term t(1�G 2Bab) z2 in the (1�GBab)-expansion of the

main term (see, e.g., Eq. (5.15)).

3. The explicit 1�T� 0 appearing in the one-loop mass displacement, Eq. (5.31).

The last one should cancel the other two in the renormalization procedure, if thoseare regulated by the same UV cutoff T� 0 for the photon proper-time, and this wasverified in various versions of this calculation. In the spinor QED case the fermionpropagator has no quadratic divergence (this is, of course, manifest in the first-order formalism, while in the second-order formalism there are various diagramscontributing to the one-loop fermion self energy, and the absence of a quadraticdivergence is due to a cancellation among them). The third term is thus missing,and the other two have to cancel among themselves. In particular, the completelypartially integrated version of the integrand has no $ab-term any more, and conse-quently, the second term must also be absent. However, the (1�GBab)-expansion of

the main contribution to the Maxwell term is, if one does this calculation in D=4,always of the form shown in Eq. (5.15),

_ AG 2

GBab+C& tr(F 2) (7.2)

with coefficients A, B, C. In the partially integrated version, first, consideration of thequadratic subdivergence allows one to conclude that A=0 and, then, considerationof the logarithmic subdivergence that B=0.

Note that this argument does not apply to the scalar QED case, nor does itapply to spinor QED in dimensional renormalization, due to the suppression ofquadratic divergences by that scheme. In both cases one would have only oneconstraint equation for the two coefficients A and B appearing in the partiallyintegrated integrand, and indeed, they turn out to be nonzero in both cases. In thepresent formalism, the fermion QED two-loop ;-function calculation thus becomessimpler when performed not in dimensional regularization, but in some four-dimen-sional scheme such as proper-time or Pauli�Villars regularization.

The reader may rightfully ask why we have gone to such lengths in analyzing thistwo-loop calculation, which is easy to do by modern standards even in field theory.We find this cancellation mechanism interesting in view of some facts known aboutthe three-loop fermion QED ;-function [98�100]. Apart from the well-known can-cellation of transcendentals occurring between diagrams in the calculation of thequenched (one fermion loop) contribution to this ;-function [98, 100], which takesplace in any scheme and gauge, even more spectacular cancellations were found in[99], where this calculation was performed in four dimensions, Pauli�Villarsregularization, and Feynman gauge. In that calculation all contributions from non-planar diagrams happened to cancel out exactly. A recalculation of this coefficientin the present formalism is currently being undertaken [101].

8. DISCUSSION

In this paper, we have extended previous work of two of the authors on themultiloop generalization of the string-inspired technique to the case of quantumelectrodynamics in a constant external field. The resulting formalism has beentested on a recalculation of the two-loop corrections to the Euler�HeisenbergLagrangian for quantum electrodynamics. Several advantages of this calculus overstandard field theory methods have been pointed out. In particular, it treats thescalar and spinor loop cases on the same footing, so that the scalar loop results arealways obtained as a by-product of the corresponding spinor loop calculations.More technically, our parameter integrals are written in a form convenient for partialintegrations. In particular, the integrands are functions that are well defined on thecircle, so that boundary terms do not appear. The usefulness of this property has beendemonstrated in the renormalization of the scalar QED two-loop Lagrangian.

An application to a recalculation of the one-loop QED photon splittingamplitude has been given separately [102].

We have derived a path integral representation of the gluon loop and used it fora recalculation of what is the closest analogon to the one-loop Euler�HeisenbergLagrangian in Yang�Mills theory. More significantly, the analysis of Section 3should be viewed as a first step towards an extension of the worldline technique tomultiloop calculations in nonabilian gauge theory. Our derivation of this pathintegral is entirely nonheuristic, and thus is guaranteed to reproduce the correct one-loop off-shell amplitudes for Yang�Mills theory. From our experience with quantumelectrodynamics, this property alone makes us optimistic about the existence of amultiloop generalization of the worldline method for the Yang�Mills case. We hopeto have more to say about this in the future.

APPENDIX A: PATH INTEGRAL REPRESENTATION OF THEELECTRON PROPAGATOR

Different from the case of the closed loop fermionic worldline lagrangian equa-tion (2.1), for the Dirac propagator besides the einbein field gauged to T, one hasto introduce a gravitino field / in the worldline action. This can be gauged to aconstant, but not to zero. For massive fermions a further field �5({) coupling tomass has to be introduced [13, 14, 19] (its supersymmetric partner x5 is not neededfor the gauge coupling, but it is essential for the worldline implementation ofYukawa couplings [25]).

Integration over / in the path integral leads to a factor (&12�+ x* ++m�5) corre-

sponding to the numerator of the Dirac propagator. This can be demonstratednicely for the free propagator [55, 56, 27] in the coherent state formalism. This``holomorphic representation'' represents operators in terms of their (fermionic)Wick symbols.

In the case with background interaction considered here we prefer a differentapproach in which fermionic operators are represented by their Weyl symbols. Thisformalism is manifestly covariant and, contrary to the holomorphic representation;it treats propagators in external fields and one-loop effective actions on the samefooting. From a canonical point of view we are dealing with the following algebraof hermitian operators �� + and �� 5 :

�� + �� &+�� &�� +=$+& ,(A.1)

�� +�� 5+�� 5 �� +=0, �25= 1

In terms of euclidean Dirac matrices, it can be represented by

�� +=i

- 2#5#+ , �� 5=

- 2#5 . (A.2)

The Weyl symbol map ``symb'' establishes a linear one-to-one map betweenoperators and functions of the anticommuting c-numbers !+ and !5 . In particular,symb(�� +)=!+ and symb(�� 5)=!5 . The inverse symbol map associates the Weyl-ordered (totally antisymmetrized) operator product to strings of !'s:

symb&1(!+!& } } } !\)=[�� + �� & } } } �� \]Weyl . (A.3)

For example,

symb&1(!+!&)= 12 (�� +�� &&�� &�� +)=�� +�� &&

12 $+& (A.4)

and symb(�� + �� &)=!+!&+12$+& . (See Refs. [103, 104] for further details.)

Let us consider the Dirac propagator in the background of an arbitrary abeliangauge field and let us write down a path-integral representation for its kernel(bosonic variables)�symbol (fermionic variables)

GDirac(x2 , x1 ; !)=symb[(x2| (D3 +m)&1 |x1)](!) (A.5)

with D3 ##+D+=2i�� +�� 5D+ . After having integrated out the auxiliary fields / andx5 it reads [13, 27, 104], up to an overall constant:

GDirac(x2 , x1 ;!)B|T

0dT e&m2T |

x(T )=x2

x(0)=x1

_|�(0)+�(T )=2!

D�+ |�5(0)+�5(T )=2!5

�+({) x* +({)+m�5({)= exp[&SB&SF&S5]. (A.6)

The action consists of the following pieces:

0dt[ 1

4x* 2++ieA+(x) x* +]

0dt[ 1

2 �+�4 +&ieF+&(x) �+�&]+ 12�+(T ) �+(0) (A.7)

2 �5�4 5+ 12 �5(T ) �5(0).

Note the surface terms in SF and S5 . They are needed in order to correctlyreproduce the equations of motion [104]. The factor (1�T ) �T

0 d{[ } } } ] in Eq. (A.6)stems from the integration over the worldline gravitino field /. It is important to

realize that the terms inside the curly brackets are actually independent of { andthat we may replace

0d{ {&

�+({) x* +({)+m�5({)=� &12

�+(T ) x* +(T )+m�5(T ). (A.8)

The {-independence of the expectation value of �+ x* + is a consequence of the super-symmetry of the action. In fact, SB+SF is invariant under

$x+=&2'�+(A.9)

$�+='x* +

with a constant Grassmann parameter '. If, instead, a time-dependent parameter isused in (A.9), the action changes according to

$(SB+SF)=|T

0d{ '({)

(�+x* +). (A.10)

Obviously �+x* + is the conserved Noether charge related to the supersymmetry(A.9). If we apply a localized supersymmetry transformation to the path integral� Dx D� exp(&SB&SF) and observe that the measure is invariant, we obtain theWard identity

(�+({) x* +({)) =0. (A.11)

Equation (A.11), together with a similar argument for �5 , justifies the replacement(A.8).

Using (A.8) in (A.6), the insertion &12�+x* ++m�5 is evaluated at the final point

of the trajectory, t=T. Hence it may be pulled in front of the path integral, thenacting as a (differential�matrix) operator on the wave function which was time-evolved by the path integral. If we are dealing with a phase-space path integral ofthe type

|x(T )=x2

x(0)=x1

Dx | Dp exp {i |T

0dt( px* &H)= , (A.12)

we know that

| Dx Dp x(T ) exp[ } } } ]=x2 | Dx Dp exp[ } } } ] (A.13)

| Dx Dp p(T ) exp[ } } } ]=&i�

�x2| Dx Dp exp[ } } } ]. (A.14)

By rewriting Eq. (A.6) in hamiltonian form, it is easy to see that in the case at handx* +(T ) corresponds to the operator &2iD+(x2) acting from the left. With an

analogous reasoning for the fermions this leads us to the representation for theDirac propagator,

GDirac(x2 , x1)B[�� + iD+(x2)+m�� 5]

0dT e&m2T symb&1[KDirac(x2 , T | x1 , 0; !+) I5(!5 , T )]. (A.15)

Here we defined

KDirac(x2 , T | x1 , 0; !+)#|x(T )=x2

x(0)=x1

Dx |�(0)+�(T )=2!

D�+ e&SB&SF (A.16)

I5(!5 , T )#|�5(0)+�5(T )=2!5

D�5 e&S5. (A.17)

Equation (A.15) was obtained from (A.6) by applying the inverse symbol map. Asfor the fermionic degrees of freedom, GDirac(x1 , x2) is an operator now, i.e. a matrixacting on spinor indices.

Up to this point, no assumption about the gauge field A+(x) has been made.From now on we consider fields with F+&=const. In this case the path integral(A.14) factorizes

KDirac(x2 , T | x1 , 0; !+)=KB(x2 , T | x1 , 0) IF (!+ , T ). (A.18)

The bosonic piece,

KB(x2 , T | x1 , 0)=|x(T )=x2

x(0)=x1

Dx e&SB, (A.19)

is the same as in the spin-0 or spin-1 case. Its evaluation is described in detail inSection 3. The result is given by (3.68) with g replaced by e and with the factorexp[2igTF]+& omitted. What remains to be done is to calculate the fermioniccontribution,

IF (!+ , T )=|�(0)+�(T )=2!

D�+ exp[&12�+(T ) �+(0)]

_exp {&12 |

0dt �+[�t$+&&2ieF+&] �&= . (A.20)

Since the �-integral is Gaussian, the saddle point method will yield its exact value.We decompose the integration variable according to

�+(t)=�class+ (t)+.+(t), (A.21)

where �class+ is a solution of the classical equation of motion,

[�t $+&&2ieF+&] �class& (t)=0, (A.22)

subject to the boundary condition �class+ (0)+�class

+ (T )=2!+ . The fluctuation field.+ satisfies antiperiodic boundary conditions. Using (A.21) and (A.22) in (A.20) weobtain

IF (!+ , T )=exp[&12 �class

+ (T ) �class+ (0)]

D. exp {&12 |

0dt .+[�t $+&&2ieF+&] .&=

=exp[&12�class

+ (T ) �class+ (0)] DetA[�t $+&&2ieF+&]1�2. (A.23)

In analogy with Section 3, the determinant in (A.23) is given by 2D detL[cos(eFT )].The solution to (A.22) which satisfies the correct boundary conditions reads

�class+ (t)=2 \ exp(2ieFt)

1+exp(2ieFT )++&!& . (A.24)

Inserting this into (A.23) leads us to the final result for IF ,

IF (!+ , T )=2D�2 detL[cos(eFT )]1�2 } exp[i!+ tan(eFT )+& !&]. (A.25)

Using the same method we can show that I5 equals an unimportant constant whichwe shall drop. Thus, because

symb&1(KDirac)=KB symb&1(IF), (A.26)

our last task is to find out which is the operator corresponding to the symbol(A.25). We shall see that

symb[exp(ieTF+&�� +�� &)](!)=detL[cos(eFT )]1�2 exp[i!+ tan(eFT )+& !&]. (A.27)

In order to prove (A.27), we transform F+& to block-diagonal form. We assume Deven:

F+&=diag _\ 0&B1

0 + , \ 0&B2

0 + , ...& . (A.28)

Since the �� + 's pertaining to different blocks are mutually anticommuting, we mayprove (A.27) for each block separately. Focusing on the first one, it is convenientto define

7� 12#i(�� 1�� 2&�� 2 �� 1). (A.29)

As this operator is Weyl-ordered, we have

symb[7� 12]=2i!1!2 . (A.30)

Because (7� 12)2=1, it follows that

exp[ieTF+& �� +�� &]=exp[eB1T7� 12]=cosh(eB1T )&7� 12 sinh(eB1T ) (A.31)

and, therefore,

symb[exp(ieTF+&�� +�� &)](!)=cosh(eB1 T )[1&2i!1!2 tanh(eB1T )]

=detL[cos(eFT )]1�2 exp[i!+ tan(eFT )+& !&]. (A.32)

In the last line of (A.32) we used that the eigenvalues of the first block are \iB1

and that !21=0=!2

2 . Repeating the same argument for the other blocks establishesEq. (A.27).

Upon inserting (A.26) with (A.25) and (A.27) into (A.15), we obtain the well-known result for the euclidean Dirac propagator in a constant background field[104, 105]:

GDirac(x1 , x2)=[&#+D+(x2)+m] |�

0dT(4?T )&D�2 e&m2T

_exp _&14

(x2&x1) eF cot(eTF )(x2&x1)&_exp _&

trL lnsin(eTF )

(eTF ) & exp _+i2

eTF+& #+ #&& . (A.33)

In (A.33) we used the representation (A.2) for the �� 's and we dropped an overallfactor of #5 which is produced by the path integral, but which is not included in thestandard definition of GDirac. The expression for the bosonic contribution KB wastaken from Section 3. It applies to the Fock�Schwinger gauge centered at x1 . In thegeneral case, KB contains an extra phase factor exp(&ie �x2

x1dx+ A+). We also note

that the scalar propagator (&D2)&1 is obtained from (A.33) by simply deleting theoperator [&D3 +m] and the last exponential involving the #-matrices.

It is remarkable that the above calculation of the propagator is almost identicalto the calculation of the one-loop effective action, the only difference being theboundary condition of the fermionic path integral. For the propagator we need�+(T )+�+(0)=2!+ , whereby the variables !+ give rise to its #-matrix structure.The effective action, on the other hand, is a scalar quantity, and it is obtainedfrom the same path integral with !+=0. Giving a nonzero value to !+ amounts tocreating a fermion line by ``opening'' a loop.

APPENDIX B: DERIVATION OF WORLDLINE GREEN'S FUNCTIONS

The worldline Green's functions appearing in this paper are kernels of certainintegral operators, acting in the real Hilbert space of periodic or antiperiodic func-tions defined on an interval of length T. We denote by H� P the full space of periodicfunctions, by HP the same space with the constant mode exempted, and by HA thespace of antiperiodic functions. The ordinary derivative acting on those functions iscorrepondingly denoted by �P , �� P , or �A . With those definitions, we can write ourGreen's functions as

GB({1 , {2)=2 ({1| (�P2&2iF�P)&1 |{2) ,

GF ({1 , {2)=2 ({1| (�A&2iF )&1 |{2) ,(B.1)

GCP({1 , {2)=({1| (�� P&C)&1 |{2) ,

GCA({1 , {2)=({1| (�A&C )&1 |{2) .

(in this appendix we absorb the coupling constant e into the external field F). Notethat GC

A is, up to a conventional factor of 2, formally identical with GF under thereplacement C � 2iF.

GB and GF are easy to construct using the following representation of the integralkernels for inverse derivatives on the unit circle [36]

(u| �&nP |u$) =&

Bn( |u&u$| ) signn(u&u$)

(u| �&nA |u$) =

12(n&1)!

En&1( |u&u$| ) signn(u&u$).

Here Bn(En) denotes the n th Bernoulli (Euler) polynomial. Those formulas arevalid for |u&u$|�1. For instance, the computation of GB proceeds as follows:

GB(u1 , u2)=2 (u1| (�2P&2iF�P)&1 |u2)

=2 :�

(2iF )n (u1| �&(n+2)P |u2)

=&2 :�

(2iF )n&2 signn(u1&u2)n!

Bn( |u1&u2| )

sign(u1&u2) e2iF(u1&u2)

e2iF sign(u1&u2)&1+

sign(u1&u2)iF

B1( |u1&u2| )&1

2F 2 \ Fsin F

e&iFG4 B12+iFG4 B12&1+ . (B.3)

In the next-to-last step we used the generating identity for the Bernoulli poly-nomials,

et&1= :

Bn(x)tn

n!. (B.4)

This is GB as given in Eq. (2.19) up to a simple rescaling. The computation of GF

proceeds in a completely analogous way.This method does not work for the determination of GC

P , as negative powers of�� P are not even well defined in the presence of the zero mode.

In the following, we will calculate GCP, A in a different, more ``physical'' way,

which corresponds to the usual construction of the Feynman propagator in fieldtheory.

In order to determine GCA({), say, we employ the following set of basis functions

over the circle with circumference T :

fn({)=T &1�2 exp _i2?T \n+

12+ {& , n # Z. (B.5)

They satisfy

0d{ f n*({) fm({)=$nm ,

n=&�

fn({2) f n*({1)= :�

m=&�

$({2&{1&mT ),

and fn({+T )=&fn({). In this basis, the Green's function (3.29) becomes

GCA({1&{2)=

n=&�

exp[i(2?�T )(n+1�2)({1&{2)]i(2?�T )(n+1�2)&C

. (B.7)

By introducing an auxiliary integration in the form ({#{1&{2)

GCA({)=|

&�d|

n=&�

$ \|&2?T \n+

i|&C(B.8)

and using Poisson's resummation formula, the Green's function assumes thesuggestive form [106]

GCA({)= :

n=&�

(&1)n GC�({+nT ) (B.9)

GC�({)#|

i|&C. (B.10)

We verify that

[�{&C] GC�({)=$({) (B.11)

[�{&C] GCA({)= :

n=&�

(&1)n $({+nT ) (B.12)

which shows that GC� is a Green's function on the infinitely extended real line, while

GCA is defined on the circle. The integral (B.10) yields for C>0

GC�({)=&3(&{) eC{. (B.13)

Hence, from (B.9)

GCA({)=&eC{ :

n=&�

(&1)n 3(&{&nT ) enCT. (B.14)

For { # (0, T ) only the terms n=&�, ..., &1 contribute to the sum in (B.14), whilefor { # (&T, 0) a nonzero contribution is obtained for n=&�, ..., 0. Summing up thegeometric series in either case and combining the results we obtain the expressiongiven in Eq. (3.30). It is valid for &T<{< +T. Using a basis of periodic functionsthe same arguments lead to GC

P as stated in (3.30). Note that in the limit of a largeperiod T

limT � �

GCA, P({)=GC

�({), (B.15)

as it should be. For C � 0, both GC� and GC

A have a well-defined limit,

G0�({)=&3(&{)

(B.16)G0

A({)= 12 sign({).

The periodic Green's function GCP blows up in this limit because �� &1

P does not existin the presence of the constant mode. It is important to keep in mind that GC

P isdefined in such a way that it includes the zero mode of �{ .

In the perturbative evaluation of the spin-1 path integral one has to deal withtraces over chains of propagators of the form

_nA, P(C )#TrA, P[(�{&C )&n]. (B.17)

Because

_nA, P(C)=

1(n&1)! \

_1A, P(C ), (B.18)

it is sufficient to know _1A, P(C ). The subtle point which we would like to mention

here is that strictly speaking the sum defining _1A , say,

_1A(C )= :

n=&�

1i(2?�T )(n+1�2)&C

, (B.19)

does not converge as it stands, and it is meaningless without a prescription of howto regularize it. The usual strategy is to combine terms for the positive and negativevalues of n and to replace (B.19) by the convergent series

_1A(C )=&2C :

n=0_\2?

\n+12+

+C2&&1

tanh \CT2 + . (B.20)

It is important to realize that this definition implies a well-defined prescription forthe treatment of the 3 functions in GC

A, P at {=0. In fact,

_1A(C)=|

0d{ GC

A({&{)=TGCA(0), (B.21)

and by combining Eqs. (B.20) and (B.21) we deduce that we must set

lim{z0

3({)=lim{z0

3(&{)= 12 . (B.22)

With (B.20) we obtain

_nA(C )=&

1(n&1)! \

\ ddx+

tanh(x)| x=CT�2 . (B.23)

The analogous relation in the periodic case is

_nP(C )=&

1(n&1)! \

\ ddx+

coth(x)|x=CT�2 (B.24)

if the zero mode of �{ is included in the trace (B.17), and

_$nP (C )=&

1(n&1)! \

\ ddx+

[coth(x)&x&1]| x=CT�2 (B.25)

if the zero mode is omitted. For C sufficiently small one finds the power seriesexpansions

_nA(C )=&

1(n&1)!

k=n�2

(22k&1) B2k

2k(2k&n)!T 2kC2k&n

(B.26)

_$nP (C )=&

1(n&1)!

k=n�2

2k(2k&n)!T 2kC2k&n;

_nA and _$n

P have well defined limits for C � 0:

_nA(0)=&

(2n&1) Bn

n!T n=

(n&1)!T n (n even)

(B.27)

_$nP (0)=&

n!T n (n even).

Those limits vanish for n odd. This brings us, of course, back to Eqs. (B.2).

APPENDIX C: WORLDLINE DETERMINANTS

In this appendix we collect a few results about the determinants which arise inthe computation of the spin-1 wordline path integral. To start with, we consider theoperator �{+|, where | is a real constant and �{ acts on periodic and antiperiodicfunctions of period T, respectively. Its spectrum reads i(2?�T ) n in the former andi(2?�T )(n+1�2) in the latter case, n # Z. Ratios of determinants of the form

DetA[�{+|]DetA[�{]

= `�

n=&�

i(2?�T )(n+1�2)+|i(2?�T )(n+1�2)

are defined by the prescription that terms with positive and negative values of nshould be combined so as to obtain the manifestly convergent product

DetA[�{+|]DetA[�{]

= `�

n=0_1+\|T

2? +2 1

(n+1�2)2&=cosh \|T2 + . (C.2)

In the periodic case we omit the zero mode from the definition of the determinantsand find likewise

Det$P[�{+|]Det$P[�{]

=sinh(|T�2)

(|T�2)(C.3)

(compare Eqs. (2.35), (2.35)). Next we look at the matrix differential operator

(�{&C) $+&+0+& , +, &=1, ..., D.

Here 0 is a constant matrix. We assume that it can be diagonalized and has eigen-values |. For antiperiodic boundary conditions we obtain from (C.2)

DetA[(�{&C) $+&+0+&]DetA[(�{&C ) $+&]

cosh[(T�2)(C&|)]cosh[CT�2]

. (C.4)

The product extends over the spectrum of 0. The corresponding formula for periodicboundary conditions, with the zero mode removed, reads

Det$P[(�{&C ) $+&+0+&]Det$P[(�{&C) $+&]

sinh[(T�2)(C&|)]sinh[CT�2]

. (C.5)

For C{0 we can reinstate the zero mode of �{ . In this case (C.5) is replaced by

DetP[(�{&C ) $+&+0+&]DetP[(�{&C ) $+&]

sinh[(T�2)(C&|)]sinh[CT�2]

. (C.6)

In this paper we use the above determinants for an exact evaluation of theworldline path integral in the background of a constant field F+& . For more com-plicated field configurations only a perturbative calculation of the path integralis possible in general. It is based upon the Green's functions GC

A, P which werediscussed in Appendix B. It can be checked that the determinants given above areconsistent with the perturbative expansion. In perturbation theory, the l.h.s. ofEq. (C.4), for instance, is interpreted as a power series in |:

DetA[1+|GCA]=`

exp Tr ln[1+|GCA]

exp {& :�

n|n_ n

A(C )= . (C.7)

In the last line of (C.7) we have used (B.17). By virtue of Eq. (B.23) one can sumup the perturbation series in closed form:

& :�

A(C)= :�

(&1)n (|T )n

2nn! \ ddx+

tanh(x)| x=CT�2

=_exp \&|T

dx+&1& ln cosh(x)|x=CT�2

=lncosh[x&|T�2]

cosh(x) }x=CT�2. (C.8)

With (C.8) inserted into (C.7) we reproduce precisely the r.h.s. of Eq. (C.4).

ACKNOWLEDGMENTS

We thank S. L. Adler, Z. Bern, D. Broadhurst, A. Denner, W. Dittrich, and E. Zavattini for variousdiscussions and informations. C.S. also thanks the Deutsche Forschungsgemeinschaft for financialsupport and the Institute for Advanced Study, Princeton, for hospitality during the final stage of thisproject.

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