String-inspired representations of photon/gluon

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Published for SISSA by Springer Received: November 15, 2012

Accepted: December 26, 2012

Published: January 21, 2013

Naser Ahmadiniaz,a,b Christian Schuberta,b,c and Victor M. Villanuevaa

aInstituto de Fsica y Matemticas, Universidad Michoacana de San Nicols de Hidalgo, Edicio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacn, Mxico

bDipartimento di Fisica, Universit di Bologna and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy

cInstitutes of Physics and Mathematics, Humboldt-Universitat zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

E-mail: [email protected], [email protected], [email protected]

Abstract: The string-based Bern-Kosower rules provide an e cient way for obtaining parameter integral representations of the one-loop N-photon/gluon amplitudes involving a scalar, spinor or gluon loop, starting from a master formula and using a certain integration-by-parts (IBP) procedure. Strassler observed that this algorithm also relates to gauge invariance, since it leads to the absorption of polarization vectors into eld strength tensors. Here we present a systematic IBP algorithm that works for arbitrary N and leads to an integrand that is not only suitable for the application of the Bern-Kosower rules but also optimized with respect to gauge invariance. In the photon case this means manifest transversality at the integrand level, in the gluon case that a form factor decomposition of the amplitude into transversal and longitudinal parts is generated naturally by the IBP, without the necessity to consider the nonabelian Ward identities. Our algorithm is valid o -shell, and provides an extremely e cient way of calculating the one-loop one-particle-irreducible o -shell Greens functions (vertices) in QCD. It can also be applied essentially unchanged to the one-loop gauge boson amplitudes in open string theory. In the abelian case, we study the systematics of the IBP also for the practically important case of the one-loop N-photon amplitudes in a constant eld.

Keywords: Scattering Amplitudes, Gauge Symmetry, Bosonic Strings, QCD

ArXiv ePrint: 1211.1821

String-inspired representations of photon/gluon amplitudes

JHEP01(2013)132

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP01(2013)132

Web End =10.1007/JHEP01(2013)132

Contents

1 Introduction 1

2 The P-representation 5

3 The R-representation 6

4 The Q and Q-representations 7

5 The QR representation 10

6 The S representation 12

7 Alternative version of the two-tail 13

8 The case of spinor QED 14

9 The nonabelian case 15

10 The constant external eld case 17

11 Conclusions 21

1 Introduction

It is by now well-known that techniques originally developed for the computation of amplitudes in string theory can be used also for the simplication of calculations in ordinary quantum eld theory. Already in 1972 Gervais and Neveu observed that the eld theory limit of string theory generates Feynman rules for Yang-Mills theory in a special gauge that has certain calculational advantages [1]. Actual calculations along these lines were done, however, only much later [25]. A systematic investigation of the eld theory limit at the tree and one-loop level was undertaken by Bern and Kosower, and led to the establishment of a new set of rules for the construction of the one-loop gluon amplitudes in QCD [68]. These Bern-Kosower rules were used for a rst calculation of the ve-gluon amplitudes [9]. Their relation to the usual Feynman rules was claried in [10].

Shortly afterwards, a simpler approach to the derivation of Bern-Kosower type formulas was initiated by Strassler [11] based on the representation of one-loop amplitudes in terms of rst-quantized path integrals. For the case of QED representations of this type had been known for a long time [12] although they had rarely been considered as a tool for state-of-the-art calculations. Subsequently such representations were derived also for Yukawa and axial couplings [1316], and generalized to higher loop orders [1723] as well as to the

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inclusion of constant external elds [2426], gravitation [2735] and nite temperature [36 39]. For a review see [40].

The central formula in the Bern-Kosower formalism is the following master formula:

scal[k1, 1; . . . ; kN, N]

= (ie)N(2)D X

Yi=1

T (4T )

0 di

2 em2T

exp

(

Xi,j=1

12GBijki kj iGBiji kj + 12GBiji j ) |lin("1,...,"N)

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(1.1)

As it stands, this formula represents the one-loop N-photon amplitude in scalar QED, with photon momenta ki and polarisation vectors i. m denotes the mass, e the charge and T the total proper time of the scalar loop particle.1

Each of the integrals

di represents one photon leg moving around the loop. The integrand is written in terms of the bosonic worldline Greens function GB and its derivatives,

GB(1, 2) = |1 2| (1 2)2T

GB(1, 2) = sign(1 2) 2(1 2)

GB(1, 2) = 2(1 2) 2 T(1.2)

Dots generally denote a derivative acting on the rst variable, GB(1, 2) @

@1 GB(1, 2),

and we abbreviate GBij = GB(i j) etc. In deriving the master formula a formal ex

ponentiation has been used that needs to be undone by expanding out the exponential in (1.1) in the polarization vectors, and keeping only the terms linear in each of them (in this paper our interest will be mostly in the o -shell case, but it will still be useful to keep the polarization vectors as book-keeping devices).

We will not dwell here on the derivation of this formula, which can be obtained either from the innite string tension limit of string theory [68, 42] or using the worldline path integral formalism [11, 40]. Its role in the Bern-Kosower formalism is central, since it provides the input for the Bern-Kosower rules, which allow one to obtain, from the scalar QED integrand, the corresponding integrand for the photon amplitudes in fermion QED, as well as for the (on-shell) N-gluon amplitude in QCD. However, those rules do not apply to the master formula as it stands. Writing out the exponential in eq. (1.1) one obtains an integrand

exp |multilinear = (i)NPN( GBij, GBij) exp

12N

Xi,j=1GBijki kj

(1.3)

1We work in the Euclidean throughout. With our conventions a Wick rotation k4i ! ik0i, T ! is yields the N-photon amplitude in the conventions of [41].

with a certain polynomial PN depending on the various GBij, GBij and on the kinematic invariants. The application of the Bern-Kosower rules requires one to now remove all second derivatives GBij appearing in PN by suitable integrations by parts in the variables i.

That this removal of all GBs is possible for any N was shown in appendix B of [7]. The new integrand is written in terms of the GBijs and GBijs alone, and serves as the

input for the Bern-Kosower rules. Those allow one to classify the various contributions to the N-photon/gluon amplitude in terms of 3-diagrams, and moreover lead to simple relations between the integrands for the scalar, spinor and gluon loop cases. A complete formulation of the Bern-Kosower rules is lengthy, and we refer the reader to [8, 40, 42, 43]. For our present purposes, the most relevant part of the rules is that, up to a global factor of 2 correcting for the di erences in degrees of freedom and statistics, the integrand for the spinor loop case can be obtained from the one for the scalar loop simply by replacing every closed cycle of GBs appearing in QN according to the replacement rule

GBi1i2 GBi2i3 GBini1 GBi1i2 GBi2i3 GBini1 GFi1i2GFi2i3 GFini1 (1.4)

where GF12 sign(1 2) denotes the fermionic worldline Greens function. Note that an expression is considered a cycle already if it can be put into cycle form using the antisymmetry of GB (e.g. GB12 GB12 = GB12 GB21). A similar cycle replacement rule

holds for the gluon loop case. Of course, in the nonabelian case there will also be many other modications.

As was discussed already in [7], the IBP procedure is generally ambiguous. However, this does not constitute an impediment to the application of the Bern-Kosower rules, whose application requires only that all second derivatives GBij have been removed. For this reason, in the application to the computation of gluon amplitudes in [8, 9] the partial integration had been performed in an essentially arbitrary way.

A closer look at the IBP was taken by Strassler in [44], who noted that this procedure bears an interesting relation to gauge invariance. For each photon leg, dene the corresponding eld strength tensor,

F i ki i ik i (1.5)

Remove all GBijs and combine all terms contributing to a given -cycle GBi1i2 GBi2i3 GBini1. Then the sum of their Lorentz factors can be written as a Lorentz cycle Zn(i1i2 . . . in), dened by

Z2(ij) 12tr

(n 3)

(1.6)

Thus Zn generalizes the familiar transversal projector. However, in [44] no systematic way was found to perform the partial integrations at arbitrary N, and also the absorption of polarisation vectors into eld strength tensors (a process to be called covariantization in

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FiFj

Yj=1

Fij

= i kjj ki i iki kj

Zn(i1i2 . . . in) tr

the following) worked only partially; after the IBP and the sorting of the resulting integrand in terms of cycle content some terms are just cycles or products of cycles, but, starting from the three-point case, there are also terms with left-overs, called tails in [44], and the polarisation vectors in them were not absorbed yet into eld strength tensors.

The IBP procedure was further studied in [45], where a denite partial integration prescription was given which works for any N, preserves the full permutation symmetry and is suitable for computerization. This algorithm is completely satisfactory as far as the application of the Bern-Kosower rules is concerned. However, it is obviously an interesting question whether some IBP algorithm exists which leads to an integrand where all polarisation vectors would be contained in eld strength tensors, thus making gauge invariance,i.e. transversality, manifest at the integrand level. It is the purpose of the present work to present such an algorithm.2

In detail, we will do the following: in sections 2 and 3 we nd a surprisingly simple way of using IBPs to covariantize the Bern-Kosower master formula itself; the resulting representation will be called the R-representation. In 4 we summarize the symmetric IBP procedure of [45], leading to the Q- and Q-representations. The next two sections 5 and 6 dene our new algorithm, which combines elements of both the R- and the Q-

representation, and results in what we will call the S-representation of the N photon/gluon amplitudes. As an aside, in section 7 we present a further improvement of the two-tail which leads to a particularly compact integrand at the four-point level (but does not seem to generalize to the N-point case). In section 8 we shortly comment on a direct treatment of the spinor QED case in the worldline super formalism. Section 9 is devoted to our main application, which is the calculation of the one-loop o -shell one-particle-irreducible N-gluon amplitudes (or N-vertices). While in the abelian case there are never any boundary terms in our IBPs, since all integrations run over the full loop and the integrand is written in terms of worldline Greens functions with the appropriate boundary conditions, in the nonabelian case the color ordering of the gluon legs leads to the restriction of the multiple parameter integrals to ordered sectors, and to the appearance of such boundary terms [11, 44]. Those generally can be combined into color commutators and, in x-space, would in principle allow one to achieve a complete nonabelian extension of the covariantization, namely to rewrite the nal integrand in terms of full nonabelian eld strength tensors, and to complete all derivatives to covariant ones. This is not possible in momentum space, but here instead the IBP procedure generates a natural form factor decomposition of the N-vertices, where the bulk terms are manifestly transversal and all non-transversality has been pushed into boundary terms. Finding such a decomposition by standard methods usually involves a tedious analysis of the nonabelian Ward identities, and so far has been completed only for the three-point case [47].

In section 10 we return to the abelian case, but now with a constant external eld added. Here the replacement rule (1.4) applies as well [25, 26], however the IBP procedure is complicated by the fact that the worldline Greens functions become nontrivial Lorentz

2An IBP algorithm for the general worldline integrand was also developed in [46], but for the unrelated purpose of tensor reduction.

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matrices. We will point out here the necessary modications, and also nd a way of using the IBP procedure to e ectively eliminate the nonvanishing coincidence limits of the generalized Greens functions GB, GF (see (10.2), (10.5) below) which otherwise would lead to a proliferation of terms as compared to the vacuum case.

In the conclusions section we summarize the properties of our new IBP algorithm, and discuss various applications, some of which have already been published or are actually in progress. We shortly comment on possible generalizations to gravity and string theory.

2 The P-representation

We will call P-representation the integrand obtained directly from the expansion of the Bern-Kosower master formula,

scal[k1, 1; . . . ; kN, N] = (ie)N(2)D X

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T (4T )

2 em2T

Yi=1

GBij, GBij) exp

0 diPN(

Xi,j=112GBijki kj

(2.1)

Explicitly, the polynomial PN is given by

PN = GB1i11ki1 GB2i22ki2 GBNiN N kiN

Xa,b=1a<bGBabab GB1i11ki1 [hatwider]GBaiaakia [hatwider]GBbibbkib GBNiN N kiN

Xa,b,c,d=1 a<b<c<d

GBabab GBcdcd + GBacac GBbdbd + GBadad GBbcbc

GB1i11ki1 [hatwider]

GBaiaakia [hatwider]

GBbibbkib [hatwider]

GBcicckic [hatwider]

GBdiddkid GBNiN N kiN

. . .

(2.2)

Here and in the following the dummy indices i1, i2, . . . should be summed over from 1 to N, and a hat denotes omission. Note that all terms in PN are obtained from the rst one by a simultaneous replacement of pairs of GBrirr kir GBsiss kis by GBrsr s, which

has to be done in all possible ways. Note also that GBii = 0 by antisymmetry.

These P-representation integrals are still directly related to the ones arising in a standard Feynman or Schwinger parameter calculation of the N photon amplitude [10, 11]. The exponential factor will, after a multiple rescaling and performance of the global T -integration, turn into the standard one-loop N-point Feynman denominator polynomial. The -function contained in GBij will bring together the photons i and j, corresponding to a quartic vertex, and the contributions of such terms match the ones from the seagull vertex of scalar QED.

3 The R-representation

Before coming to the old IBP procedure of [11, 45] and its intended improvement, it will be useful to solve a simpler problem, namely how to covariantize the Bern-Kosower master formula itself. In the following we will often abbreviate

PGBijkikj e() (3.1)

Consider rst the case of N = 2, where

P2 = GB121 k2 GB212 k1 GB121 2 (3.2)

We choose two vectors r1, r2 that fulll

ri ki 6= 0 (3.3)

but are arbitrary otherwise. Adding to the integrand P2 e() the following sum of total derivative terms,

r1 1r1 k1 1

GB212 k1 e()

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r2 2r2 k2 2

GB121 k2 e()

+ r1 1 r1 k1

r2 2

r2 k2 12 e() (3.4)

(i @

@i ) the total result is a change of P2 into R2,

R2 := GB12 r1 F1 k2

r1 k1

GB21 r2 F2 k1r2 k2 +

GB12 r1 F1 F2 r2r1 k1r2 k2 (3.5)

Thus we have managed to absorb the polarization vectors into eld strength tensors. And this procedure can be immediately generalized to the N-point case: let us abbreviate

i := ri i

ri ki (3.6)

Tr(i) :=

XjGBij ri Fi kjri kj (3.7)

Wr(ij) := GBij ri Fi Fj rjri kikj rj (3.8)

and choose vectors r1, . . . , rN fullling (3.3). Then it is a matter of simple combinatorics to verify that

PN e() +

Ya=1(1 aa a) 1

PN e()

= PN

GBaiaa kia Tr(a), GBaba b Wr(ab)

e()

(3.9)

where the operator a is dened as follows: each term in PN e() either involves the index a in a second derivative factor GB, or it carries a factor of GBaiaa kia. In the former case the term will be annihilated by a, in the latter case the action of a is to replace the factor of GBaiaa kia by 1.

We can then reexponentiate the new integrand, and arrive at the following covariantized version of the Bern-Kosower master formula (1.1):

scal[k1, 1; . . . ; kN, N] = (ie)N(2)D X

12GBijki kj iGBij ri Fi kjri ki 12GBij ri Fi Fj rjri ki rj kj ) lin(F 1,...,FN )

(3.10)

Thus we have achieved manifest gauge invariance at the integrand level, with a large freedom of choosing the vectors r1, . . . , rN. We will call this the R-representation of the N-photon amplitudes. Note that it reduces to the original master formula (1.1) if ri i = 0 for all i.

4 The Q and Q-representations

Next, we review the IBP procedure motivated by the Bern-Kosower rules, whose primary purpose is to get rid of all second derivatives GB [7, 44, 45].

For the two-point case P2 has been written down already in (3.2). After an IBP of the second term in either 1 or 2, and using GB12 = GB21, it turns into

Q2 = GB12 GB21 1 k22 k1 1 2k1 k2

GB121 2 GB3i3 ki + (1 2 3) + (1 3 2)

(4.3)

This term together with ve similar ones removes all the GBs. Decomposing the new integrand according to its cycle content, P3 gets replaced by Q3 = Q33 + Q23, where

Q33 = GB12 GB23 GB31Z3(123) ,

Q23 = GB12 GB21Z2(12)T (3) + GB13 GB31Z2(13)T (2) + GB23 GB32Z2(23)T (1)(4.4)

Note that Q33 contains a cycle of length three and Q23 a cycle of length two, as indicated by the upper indices, and that each -cycle appears together with the corresponding Lorentz-cycle. This motivates the further denition of a bicycle as the product of the two:

G(i1i2 in) := GBi1i2 GBi2i3 GBini1Zn(i1i2 in) (4.5)

Yi=1

T (4T )

0 di

2 em2T

exp

(

Xi,j=1

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= GB12 GB21Z2(12) (4.1)

Proceeding to the three-point case, here (2.2) becomes

P3 = GB1i1 ki GB2j2 kj GB3k3 kk

i (4.2)

In this three-point case it is still possible to remove all GBs in a single step. To remove, e. g., the term involving GB12 GB31 in the second term of P3, we can add the total derivative term

GB121 2 GB313 k1e()

But the terms of Q23 have, apart from the cycle, also a one-tail, dened by

T (a) := GBaia ki (4.6)

This tail still has a polarisation vector that is not absorbed into a eld strength tensor. It is easy to see that, nonetheless, each term in Q23 is individually gauge invariant; if one replaces in, e.g., the term

GB12 GB21Z2(12) GB3k3 kk e()

3 by k3, then it becomes proportional to

GB12 GB21Z2(12) e()

However, our aim here is to make gauge invariance manifest even at the integrand level. Now in the three-point case there are already various chains of IBP that can be used to remove all the GBs, but if one assumes that the corresponding total derivative terms are added with constant coe cients (i.e., they involve no dependences on momentum or polarization other than the ones already present in the term which one wishes to modify), then it is easy to convince oneself that they all lead to the same Q3 of (4.4). Thus we have to look for a more general type of IBP. We will now essentially apply the procedure of the previous section to the tails. Consider again the rst term in Q23 above, eq. (4.4). Choose a momentum vector r3 such that r3 k3 6= 0, and add the total derivative

r3 3r3 k3 Z2(12)3

GB12 GB21e()

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(4.7)

The addition of this term to the rst term in Q23, and of similar terms to the second and third one, transforms Q23 into

R23 := GB12 GB21Z2(12) GB3k r3 F3 kk

r3 k3 +

GB13 GB31Z2(13) GB2j r2 F2 kj

r2 k2

+ GB23 GB32Z2(23) GB1i r1 F1 ki

r1 k1

(4.8)

Thus we have completed the covariantization of the integrand.

In the abelian case the 3-point amplitude must, of course, vanish, which we can see by noting that the integrand is odd under the orientation-reversing transformation of variables i = T i, i = 1, 2, 3.

Proceeding to the four-point case, here even using only total derivative terms with constant coe cients (in the above sense) there are already many ways to remove the GBs

by IBP, with a large ambiguity for the nal integrand, and it is not obvious how one should proceed. But certainly one would like to preserve the manifest permutation symmetry of the N-photon amplitudes, and in [45] this was used as a guiding principle to develop the following algorithm:

1. In every step, partially integrate away all the second derivative factors GBijs ap

pearing in the term under inspection simultaneously. This is possible since di erent GBijs never share variables.

2. In the rst step, for every factor of GBij present use both i and j for the IBP, and take the mean of the results.

3. At every following step, any GBij appearing must have been created in the previous step. Therefore either both variables i and j were used in the previous step, or just one of them. If both were used, then both should be used again in the actual IBP step, and the mean of the results be taken. If only one of the variables was used in the previous step, then the other variable should be used in the actual step.

In the four-point case, applying this algorithm to P4 and decomposing the resulting integrand according to cycle content, leads to the following version of the numerator polynomial Q4 [45]:

Q4 = Q44 + Q34 + Q24 Q224

Q44 = G(1234) + G(1243) + G(1324)

Q34 = G(123)T (4) + G(234)T (1) + G(341)T (2) + G(412)T (3)

Q24 = G(12)T (34)+ G(13)T (24)+ G(14)T (23)+ G(23)T (14)+ G(24)T (13)+ G(34)T (12) Q224 = G(12) G(34) + G(13) G(24) + G(14) G(23)

(4.9)

Here we have now further introduced the two-tail,

T (ij) :=

Xr,s

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GBiri kr GBjsj ks + 12

GBiji j

GBirki kr GBjrkj kr

(4.10)

Thus the nal representation of the (still o -shell) four-photon amplitude in scalar QED becomes

scal[k1, 1; . . . ; k4, 4] = e4 (4)

(2)D

T T

2 em2T

0 d1 d4 Q4(

GBij) exp

T24

Xi,j=1GBijki kj

(4.11)

We shortly summarize the advantages of this representation compared to a standard Feynman/Schwinger parameter integral representation (see [40] for details):

First, the r.h.s. of (4.11) represents already the complete amplitude, with no need to add crossed terms. The summation over crossed diagrams which would have to be done in a standard eld theory calculation here is implicit in the integration over the various ordered sectors.

Second, the IBP procedure has homogenized the integrand; every term in QN has N factors of GBij and N factors of external momentum. In the four-point case this has the additional advantage of making the UV niteness of the photon-photon scattering amplitude manifest before integration. While the original numerator P4 contains terms involving products of two GBijs which lead to spurious divergences in the T -integration, after the IBP the integrand is nite term by term.

Third, It allows one to obtain the corresponding spinor QED amplitude by the application of the replacement rule (1.4). In applying the rules it must be observed, though, that the form of the integrand given aboven still contains, apart from the explicit cycle factors, additional cycles from the tail factors for certain values of the dummy indices. In the four-point case this occurs for Q24 only: the two-tails contained in Q24 as given in (4.9)

above each contain a two-cycle, since the content of the braces on the r.h.s. of (4.10) for r = j, s = i turns into

GBij GBji

i kjj ki i jki ki = G(ij) (4.12)

For the application of the replacement rules it is therefore convenient to decompose Q4 in a slightly di erent way [45]. Namely, note that

Q24 = Q24 + 2Q224 (4.13)

where Q24 is obtained from Q24 by eliminating the term with r = j, s = i from the sum over dummy indices. With this denition, and setting Q

()4 = Q()4 for the remaining

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components, we can write

Q4 = Q44 + Q34 + Q24 + Q224 (4.14)

In this form all cycle factors are explicit, and moreover all the coe cients in the decomposition turn out to be unity. Generally, we will denote by T (i1 . . . in) a tail whose cycles have been removed by the appropriate restrictions on the multiple dummy sums appearing in it.

Thus the four-photon amplitude for the spinor loop case can now be obtained from the scalar loop formula (4.11) simply by multiplying with a global factor of 2 from spin and statistics, and by replacing, simultaneously, each bicycle G(i1 . . . in) by the corresponding super-bicycle

GS(i1 . . . in) := ( GBi1i2 GBi2i3 GBini1 GFi1i2GFi2i3 GFini1)Zn(i1 in) (4.15)

(the notation refers to the worldline supersymmetry underlying the replacement rule (1.4), see [40]). This symmetric partial integration procedure has been worked out explicitly for up to the six-photon case; see [40, 45] for the explicit formulas.

5 The QR representation

So far we have established two seemingly unrelated IBP procedures, the rst one leading to the manifestly gauge invariant R-representation, the second one to the Q-representation

that is suitable for the application of the Bern-Kosower rules, but manifestly gauge invariant only in the cycle factors, not in the tails. We will now combine the two IBP strategies, using the following three simple observations:

First, it had been noted in the appendix C of [40] that the Q-representation is recursive, in the following sense: each term in the cycle decomposition of QN contains at least one cycle [40], so that any tail appearing in the N-point amplitude has at most N 2 arguments. And a tail of length, say, M, is related to the (undecomposed) lower-order QM simply by writing QM in the tail variables, and then extending the range of all dummy variables occurring in it to run over the full set of indices 1, . . . , N. For example, writing out (4.10) for the two-tail T (12) in the last term of Q24 in (4.9) gives

T (12) =

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Xr,s=1GB1r1 kr GB2s2 ks + 12GB121 2

4 GB1rk1 kr GB2rk2 kr

(5.1)

which can also be obtained by writing Q2, dened in (4.1), as

Q2(12) GB121 k2 GB212 k1 + 12

GB121 2

GB12k1 k2 GB21k2 k1

(5.2)

and introducing appropriate dummy index summations. That this property holds in general can be easily seen by considering those terms in the cycle factors of the decomposition of QM that do not contain factors of i j, and thus can have involved IBPs only in the tail and not in the cycle variables; see the appendix C of [40] for more details.

Second, for each N the symmetric IBP procedure denes a unique QN and thus a total derivative term

SN e() QN e() PN e() (5.3)

Consider now an arbitrary term in the cycle decomposition of QN. It will have the form C(i1 . . . iL)T (j1 . . . jM), where M N 2, C(i1 . . . iL) is a bicycle or product of bicycles involving the variables i1, . . . , iL, and T (j1 . . . jM) is the unique (in the symmetric

IBP scheme) tail of M variables, written in the remaining variables j1, . . . , jM . Consider C(i1 . . . iL)SM(j1 . . . jM) e(), where possible dummy variable summations in SM are extended to run over the full range of variables 1, . . . , N as above. This is still a total derivative term (involving only derivatives in the tail variables), and the above simple relation between the M-tail and QM implies, that

C(i1 . . . iL)T (j1 . . . jM) e() C(i1 . . . iL)SM(j1 . . . jM) e() = C(i1 . . . iL)Tp(j1 . . . jM) e()

(5.4)

with a new version Tp of the M-tail which relates to PM in the same way as the standard tail T to QM, i.e. by an extension of the dummy index sums.

Continuing with our example above, here we have

S2 eGB12k1k2 (Q2 P2) eGB12k1k2

12GB121 2

GB12k1 k2 GB21k2 k1

+ GB121 2

eGB12k1k2

= 1

2(1 2)

GB121 2 eGB12k1k2

(5.5)

and (5.4) becomes

G(34)T (12) e() 12(1 2)

G(34) GB121 2 e()

= G(34)Tp(12) e() (5.6)

where T (12) was given in (5.1) and Tp(12) is given by

Tp(12) =

Xr,s=1GB1r1 kr GB2s2 ks GB121 2 (5.7)

Finally, the new tail Tp( ) can now be covariantized by a simple extension of (3.9):

Ya=1(1 jaja ja)

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C()Tp(j1 jM) e()

= C()Tp

GBaiaa kia Tr(a), GBaba b Wr(ab)

e()

(5.8)

In our two-tail example, this lead to the following result:

Tr(12) = GB1r r1 F1 kr

r1 k1

GB2s r2 F2 ksr2 k2 +

GB12 r1 F1 F2 r2r1 k1k2 r2 (5.9)

where the sums over r and s run from 1 to 4. This should be compared with R2 of (3.5).

6 The S representation

Thus in the QR-representation we have the usual decomposition into cycle and tail factors, with the tails already covariantized, and in a form that generalizes the (lower order) R-representation by an extension of the dummy index sums to run over all N variables, including those belonging to the cycle factors of the term under consideration. To nish our quest for an integrand that would be both covariant and suitable for the application of the Bern-Kosower rules, two more steps are needed: rst, we need to remove the remaining GBs; this can be done by reapplying the symmetric IBP procedure of section 4, without any modications. And nally all cycles still contained in the tails have to be separated out. We will call this nal, in some sense optimized result for the integrand of the N photon/gluon amplitudes, the S-representation, and denote the corresponding tails by T s().

Continuing with our example of the two-tail in Q24, the rst step transforms Tr(12) of (5.9) into

Ts(12)

Xr,s=1GB1r r1 F1 krr1 k1GB2s r2 F2 ks r2 k2

GB12

Xr=1GB1rk1 kr

Xs=1GB2sk2 ks

r1 F1 F2 r2

r1 k1k2 r2

(6.1)

This form of the two-tail, like the original two-tail T (12) of (4.10), still contains a cycle - the terms on the r.h.s. with r = 2 and s = 1 combine to form a G(12), as in (4.12). Eliminating these terms from the tail one arrives at the nal form,

T s(12)

Xr,s=1r,s6=(2,1)GB1r r1 F1 krr1 k1GB2s r2 F2 ks r2 k2

GB12

Xr=1r6=2GB1rk1 kr

Xs=1s6=1GB2sk1 ks

r1 F1 F2 r2

r1 k1k2 r2

(6.2)

For the one-tail there is no di erence between Tr(i), Ts(i) and T s(i), being all given by (3.7).

We can now write down a covariantized version of the four photon amplitude, simply by taking over (4.9), (4.11), and (4.14) and replacing all one-tails by (3.7) and all two-tails by (6.2). Explicit formulas for higher point amplitudes will be given elsewhere.

7 Alternative version of the two-tail

For the two-tail, there is actually yet another form which is covariant, free of GBs and at the same time more compact than (6.1). Starting again with Tp(12) of (5.7) we add the following total derivative term to Tp(12) (omitting now the inert cycle factors C())

1(k1 k2)2 tr (F1F2) 12 e()

+ 1

k1 k2

h1 2 12 e() 1 k22 kj1

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GB2j e()

2 k11 ki2

GB1i e()

(7.1)

One obtains the new two-tail

TH(12) GBi1 GB2jki H12 kj (7.2)

where we have introduced the tensor

H 12 (F1F2) k1 k2 k1k 2tr (F1F2)

(k1 k2)2 (7.3)

Note that tr H12 = 0 and HT12 = H21. Note also that the term with i = 2, j = 1 in (7.2) as before produces a G(12), since

k2 H12 k1 = 1

2tr (F1F2) tr (F1F2) =

2tr (F1F2) = Z2(12) (7.4)

Thus TH(ij) can be used as well as T (ij) and Ts(ij) in the construction of the four-point amplitudes, including the application of the replacement rule (1.4) and the simple sign change in passing from (4.9) to (4.14). However, contrary to Ts(ij) it appears that TH(ij)

has no natural generalization to the higher-point tails.

8 The case of spinor QED

One of the main purposes of the IBP procedure is to trivialize the transition to the spinor QED case though the replacement rule (1.4). Still, it is interesting to note (and of possible practical relevance for the generalization to the case of open fermion lines, where the IBP is less attractive due to the existence of boundary terms) that in the worldline formalism there is also a more direct treatment of the spin 12 case using an approach based on explicit worldline supersymmetry [18, 40, 46, 48, 49]. It allows one to write down a master formula for N-photon scattering [48] which is formally analogous to the one for the scalar loop, eq. (1.1):

spin[k1, 1; . . . ; kN, N] = 2(ie)N(2)D X

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T (4T )

2 em2T

Yi=1

0 di Z

di exp

(

Xi,j=1

12ijki kj + iDiiji kj + 12DiDjiji j

# lin("

1,...,"N )

(8.1)

Here we have further introduced integrals over the Grassmann variables 1, . . . , N, such

that

dii = 1, and the super derivative

D =

(8.2)

The two worldline Greens functions GB,F now appear combined in the super Greens function

(1, 1; 2, 2) GB(1, 2) + 12GF (1, 2) (8.3)

Now also the polarization vectors 1, . . . , N are to be treated as Grassmann variables. The overall sign of the master formula refers to the standard ordering of the polarization vectors 12 . . . N.

Starting with this master formula, all the manipulations which we have applied in the previous sections to the scalar loop integrands can, mutatis mutandis, also be used in the spinor loop case starting from (8.1). Here we will be satised with pointing out that the covariantized Bern-Kosower master formula (3.10) generalizes to the spinor QED case as follows:

spin[k1, 1; . . . ; kN, N] = 2(ie)N(2)D X

T (4T )

2 em2T

Yi=1

0 diZ

di exp

(

Xi,j=1

12ijki kj + iDiij ri Fi kj ri ki

12DiDj

ij ri Fi Fj rj ri ki rj kj

)

lin(F1,...,FN )

(8.4)

where now F1, . . . , FN have to be treated as Grassmann variables.

9 The nonabelian case

As far as concerns the calculation of the on-shell QED photon amplitudes, or of the on-shell gluon S-matrix elements via the Bern-Kosower rules, the availability of a representation that is manifestly transversal at the integrand level is satisfying, but the signicance of this fact for practical calculations is not obvious. To the contrary, it is easy to recognize the advantages of such a representation when it comes to the gluon amplitudes o -shell. Here the natural objects to consider in QCD are the N-vertices, that is the one-particle-irreducible N-point functions, and for applications of those it is often essential to decompose them into a basis of transversal and longitudinal tensor structures (see, e.g., [47, 5052]). Such a transversality-based form factor decomposition in the present approach emerges essentially automatically in the IBP procedure through the appearance of eld strength tensors. We have seen how this happens for the abelian case, but it is true also for the nonabelian case; here in principle one would like to see the full nonabelian eld strength tensor emerging,

F F a T a = F 0 + ig[AbT b, Ac T c] (9.1)

where by

F 0 (Aa Aa)T a (9.2)

we now denote its abelian part; and indeed Strassler demonstrated already for some simple cases how this happens [11, 44]: when the external particles are gluons, the various ordered sectors of the integral

d1 dN need to be considered separately, since they carry di erent color factors. Therefore boundary terms now arise in the IBP procedure, and the commutator terms are generated as di erences of boundary terms between adjacent sectors that in the abelian case would cancel, but cannot do so any more in the presence of color since two of the color matrices appear in di erent orders. In an x-space calculation of the e ective action, those commutator terms could then be combined with the abelian parts of the eld strength tensor, but this is not possible in a momentum space calculation of the N-point function at xed N, since any term in the nonabelian e ective action after Fourier transformation contributes to amplitudes with various numbers of external particles; e.g., the term tr (DF DF ) will contribute to the N-point functions with

N between two and six. Generally, each term in the nonabelian e ective Lagrangian has a core term, which has a counterpart already in the abelian case (in the example this would be F 0 F 0 ) and a number of covariantizing terms that all involve commutators, and belong to amplitudes with more legs than the core term. In this section, we will explain the essentials of how to calculate the scalar, spinor and gluon loop contributions to the one-loop N-gluon vertex. The details and a full recalculation of the three-gluon-vertex will be left to a separate paper [53].

Starting with the scalar loop case, here the master formula (1.1) generalizes to the nonabelian case simply by supplying a global color factor, and keeping the gluons in a

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xed order:

a1...aN1PI,scal[k1, 1; . . . ; kN, N] = (ig)Ntr(T a1 . . . T aN )(2)Di

0 dT (4T )D/2em

0 d1 Z

0 d2 . . . Z

0 dN1 exp (

Xi,j=1

12GBijki kj

i GBiji kj + 1

GBiji j

) lin("

1,...,"N )

(9.3)

Here the T a are the generators of the gauge group in the representation of the loop scalar. This treatment of color correponds to a color-ordered representation (although not necessarily in the usual sense, where the T a would be in the fundamental representation, see, e.g., [54, 55]). Note that we have not only xed the ordering of the gluons along the loop but also used the translation invariance in to set N = 0. Summation over all (N 1)! inequivalent orderings of the N vertex operators is implied. Also, has been given an index 1PI to indicate that the r.h.s. gives only the one-particle-irreducible part of the N-gluon amplitude, not including the reducible part that now also exists, di erently from the abelian case. Starting from (9.3) one can apply the IBP procedure leading from the P-representation to the S-representation as before, the only novelty being the boundary terms. Since for the bulk terms all the polarization vectors i get absorbed into the transversal structures (1.5) (which now, however, represent only the abelian part of the eld strength tensor), in the nal representation the non-transversal part of the N-vertex must be entirely in the boundary terms, given by lower-point integrals. For those one still has to choose the vectors ri, preferably in a way that is consistent with the cyclic invariance of the nonabelian amplitudes. In the three-point case a convenient cyclic choice is r1 = k2 k3, r2 = k3 k1, r3 = k1 k2, and indeed it turns out [53] that, with this choice, the resulting form factor decomposition matches precisely the standard Ball-Chiu decomposition [47] of the three-gluon-vertex.

Coming to the spinor loop case, here the only issue is whether the replacement rule (1.4) can be applied also to all the boundary terms now arising in the IBP procedure. This is indeed the case, as we can see as follows: it su ces to show the corresponding statement for the e ective action, rather than the momentum space Greens functions. Now, the e ective Lagrangian can in principle be written as an innite series of terms that are Lorentz scalars formed using any number of eld strength tensors and covariant derivatives. As was already mentioned, each such term has a core term, whose calculation is not di erent from the abelian case for either the scalar or spinor loop, such that the replacement rule applies to it. All covariantizing terms of a core term must share its coe cient, and low-order calculations show that, as one would expect, the way this works is that they all involve the same parameter integral [44, 53]. And for the whole structure to continue to be gauge invariant for the spinor loop case it is necessary that the same replacement rule applies to all the covariantizing terms as well as for the core term.

Similarly one can convince oneself that also the replacement rules that connect the scalar with the gluon loop cases [8, 11, 25, 40, 42] can be extended from the core terms to the ones involving boundary contributions [53]. An additional issue with the gluon loop

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contribution to the N-vertex is that one has to choose a gauge for the gluon propagator. The application of the gluonic replacement rules gives the N-vertex corresponding to the use of the background eld method with Feynman gauge for the quantum part [11, 25], which is also known to coincide with the result of the application of the pinch technique [5658]. This version of the gluonic vertex is also the one that leads to SUSY sum rules [59].

10 The constant external eld case

The Bern-Kosower master formula (1.1) has the following straightforward extension to the Nphoton amplitude in a constant external eld [2426]:

scal[k1, 1; . . . ; kN, N]

= (ie)N(2)D X

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sin Z Z

0 [4T ]

2 em2T det

Yi=1

0 di

exp

(

Xi,j=1

12ki GBij kj ii GBij kj + 12i GBij j

)

|lin("1,...,"N)

(10.1)

Here we have introduced the matrix Z eT F , GBij GB(i, j) denotes the bosonic

worldline Greens function in the constant eld background, and GBij, GBij its rst and

second derivatives with respect to i. Explicitly, they are given by [25, 26]

GB12 = T 2Z2

Zsin Z eiZ GB12 + iZ

GB12 1

GB12 = i Z

Zsin Z eiZ GB12 1

GB12 = 2(1 2) 2

Zsin Z eiZ GB12

(10.2)

while the fermionic Greens function GFij generalizes to

GF12 = GF12 eiZ GB12cos Z (10.3)

These expressions are to be understood as power series in the matrix Z. We note the symmetry properties

GBji = GTBij, GBji = GTBij, GBji = GTBij, GFji = GTFij (10.4)

and the coincidence limits

GB(, ) = T

2Z2 (Z cot Z 1) GB(, ) = i cot Z i

GF (, ) = i tan Z

(10.5)

Note that those are independent of . For the following, it will also be convenient to introduce the subtracted Greens function

GB(1, 2) := GB(1, 2) GB(, ) = T

eiZ GB12 cos Z

sin Z + i

GB12

(10.6)

The corresponding spinor QED amplitude is obtained from this master formula in the same way as in the vacuum case, with the following minor modications: rst, note from (10.2), (10.5) that the worldline Greens functions for the constant eld are generally nontrivial Lorentz matrices. Thus the denition of a cycle must now be slightly generalized; for example, a term

1 GB12 k2 2 GB23 3 k3 GB31 k1

would have to replaced by

1 GB12 k2 2 GB23 3 k3 GB31 k1 1 GF12 k2 2 GF23 3 k3 GF31 k1

Second, the above nonvanishing coincidence limits of GB and GF must be considered as cycles of length one, and included in the replacement rule (1.4):

GB(i, i) GB(i, i) GF (i, i) (10.7)

Third, there is now also a replacement rule for the determinant factor:

det

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sin(Z) Z

det

tan(Z) Z

(10.8)

This algorithm yields the one-loop N - photon amplitudes in a constant eld for spinor QED.

Although the worldline formalism in a constant eld has already found extensive applications [25, 6067] a systematic investigation of the IBP procedure for the constant eld case has so far been lacking. Inspection of the symmetric partial integration algorithm of section 4 shows that it applies as well as to this case, and it leads to essentially the same decomposition of the integrand into cycles and tails, with only two modications: rst, one-cycles have to be included; and second, the denition of the bicycle of length n, now to be denoted by G(i1i2 in), has to be changed to

G(i) := 1

2tr(Fi

GBii) = i GBii ki

GBi1i2 Fi2 GBi2i1)

= 1 GB12 k22 GB21 k1 1 GB12 2k2 GB21 k1

G(i1i2 . . . in) := tr(Fi1 GBi1i2 Fi2 GBi2i3 Fin GBini1) (n 3) (10.9)

These bicycles generalize the vacuum ones (4.5), and will turn into them in the limit F 0.

However, it turns out that the presence of one-cycles which would lead to a signi-cant proliferation of terms as compared to the vacuum case can be altogether avoided,

G(i1i2) := 12tr(Fi1

by the following slight modication of the IBP: rst, we observe that we have the freedom to shift both the Greens function GBij and its derivatives GBij by arbitrary constant matrices. For GBij, which appears only in the rst term in the exponent of the master formula (10.1), this is an obvious consequence of momentum conservation,

Pj kj = 0. For GBij, we note that it can appear in a i GBij kj, ki GBij kj or i GBij j. The rst type of terms comes directly from the second term in the master formula, and here again a constant matrix added to GBij drops out immediately because of momentum conservation.

The second type of terms arises in the integration-by-parts procedure as a

Xl,j=112GBljkl kj =

Xj=1ki GBij kj

with j running, so that again we can modify GBij by a constant. The third type arises as the integral of a i GBij j in the IBP procedure, and here the constant matrix can be added as an integration constant.

For the scalar QED case, we can use this freedom to directly eliminate all one-cycles by subtracting from GBij its (constant) coincidence limit, that is, replacing GBij by GBij

GBij GBii throughout. For the spinor QED case, this would not make sense, since there are still the fermionic one-cycles. Here instead one should anticipate the application of the replacement rule for one-cycles (10.8), and use the freedom of modifying GBij to replace it by

GBij := GBij + GFii = GBij GBii + GFii

= i eiZ GB12sin Z cot Z tan Z

(10.10)

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This eliminates all one-cycles, since now by construction GBii GFii = 0.

With these modications, we can write down the one-loop N-photon amplitudes in a constant eld for scalar and spinor QED in a way which is almost as compact as in vacuum. For example, the formula (4.11) for the four-photon amplitude in scalar QED generalizes to

scal[k1, 1; . . . ; k4, 4] = e4

(4)D/2 (2)D X

dTT 1+D/2 em2T det

sin Z Z

exp

0 d1d2d3d4 Q44 + Q34 + Q24 + Q224

Xi,j=1

12ki GBij kj

(10.11)

where

Q44 = G(1234) + G(1243) + G(1324)

Q34 = G(123) T(4) + G(234) T(1) + G(341) T(2) + G(412) T(3)

Q24 = G(12) T(34) + G(13) T(24) + G(14) T(23)

+ G(23) T(14) + G(24) T(13) + G(34) T(12)

Q224 = G(12) G(34) + G(13) G(24) + G(14) G(23)

(10.12)

Here the bicycles G(i1 in) di er from the original ones (10.9) only by the replacement of GBij by GBij, and the tails T(i1i2 . . . in) are isomorphic to the one of the vacuum case (4.6), (4.10),

T(i) :=

Xai GBia ka

T(ij) :=

Xa,bi GBia kaj GBjb kb + 12

Xai GBij j

ki GBia ka kj GBja ka

(10.13)

As before, a prime on a tail means that its cycle have been removed; note that now this is necessary already for the one-tail.

The corresponding representation for spinor QED is obtained from (10.11), (10.12) by the change of determinants (10.8), multiplication by the global factor of 2, and replacement of the bicycles G(i1 . . . in) by the super-bicycles GS(i1 . . . in)

GS(i1i2) := 12tr

Fi1 GBi1i2 Fi2 GBi2i1

12tr (Fi1 GFi1i2 Fi2 GFi2i1)

GS(i1i2 in) := tr

Fi1 GBi1i2 Fi2 GBi2i3 Fin GBini1

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tr (Fi1 GFi1i2 Fi2 GFi2i3 Fin GFini1) (n 3)

(10.14)

Moreover, the GBijs in the tails (10.13) must also be replaced by GBijs.

The covariantization of the tails does not seem to extend to the constant eld case in a natural manner, except for the modication of the one-tail (3.8), which generalizes to

Tr(i) :=

Xari Fi GBia kari ki (10.15)

Also the optimized form of the two-tail (7.2) does not seem to generalize to the case of the general constant eld. It does so, however, for the important special case of a self-dual eld, which obeys F 2 1 [26, 66, 67]. Here one can generalize the total derivative (7.1) to

1(k1 k2)2 tr(F1F2)12e()

+ 1

k1 k2

h1 212e() 1 k21

2 GB2j kje()

2 k12

1 GB1i kie()

(10.16)

and adding this to the (unsubtracted) two-tail T(12) of (10.13), one arrives at

TH(ij) :=

Xa,bka GBai Hij GBjb kb (10.17)

with the same tensor H ij as in (7.3).

11 Conclusions

We have continued here the systematic investigation of the Bern-Kosower partial integration procedure, initiated in [44] and continued in [45]. We have presented an IBP algorithm that unambiguously leads to a form of the integrand of the one-loop N photon amplitude in scalar QED which is manifestly gauge invariant (transversal) at the integrand level, and suitable for an application of the Bern-Kosower rules. We have worked out this integrand explicitly at the four-point (two-tail) level. The N-point integrand contains N vectors r1, . . . , rN which are constrained only by the condition (3.3). Further study will be needed to nd out what is the signicance of this ambiguity, and how to make the best use of it. As far as concerns the on-shell N-photon/gluon amplitudes, one would surmise that the dependence of the integrand on the vectors ri is related to the usual dependence on the reference vectors q1, . . . , qN which one would normally have in the application of the spinor helicity formalism, but does not exist any more once all polarization vectors are absorbed into eld strength tensors. And indeed, there is clearly a relation: for example, consider the case of the the on-shell N-photon amplitudes with all helicities positive. When using the P representation together with the standard spinor helicity formalism (see, e.g., [54]), one can remove all terms involving a GBiji j by choosing, in the spinor helicity formalism, the same reference vector qi = q for all legs, since then +i +j = 0. Similarly, in the

S representation one could make disappear all the factors ri Fi Fj rj by choosing all ri = r equal, and r on-shell, since then

r Fi Fj r = 12r {Fi, Fj} r =

on account of the identity [68]

2[ij]2 (11.2)

However, it is clear that this match between the freedom of choosing the ris and the qis cannot be a perfect one, since the ris need not be chosen as on-shell.

All our representations are valid o -shell. This makes them relevant for state-of-the-art calculations already at the four-point level, since neither the four-photon nor the four-gluon amplitudes are presently available in the literature fully o -shell (for any spin in the loop). This fact is particularly conspicuous in the case of spinor QED, where the on-shell four-photon scattering amplitude was obtained already in 1951 by Karplus and Neumann [69], and the extension to the case of two o -shell legs in 1971 by Costantini et al. [70]. Our integral representations for the QED four-photon amplitudes are, with any of the various denitions of the tails, manifestly nite term by term and thus suitable for a numerical evaluation as they stand. For analytical purposes one would still like to reduce the various parameter integrals appearing in them to scalar box, triangle and bubble integrals. This could be done using existing tensor reduction algorithms (see [71] and refs. therein), but in a companion paper [72] we will rather perform this tensor reduction in a way that is specically adapted to the structure of the worldline integrals.

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14[ij]2r2 = 0 (11.1)

{F +i, F +j} =

Due to this validity o -shell our representations can also be used to construct, by sewing, all higher-loop N-photon amplitudes. From the calculation of the two-loop QED function [18] it is clear that when calculating those multiloop amplitudes in the worldline formalism signicative simplications can be expected from a judicious application of IBP.

The manifest transversality is probably a more signicant issue in the nonabelian case. Here we have in mind not so much the calculation of on-shell gluon amplitudes, for which other extremely powerful methods have been developed in recent years (see, e.g., [73, 74] and refs. therein), but rather of the one-particle-irreducible o -shell N-gluon vertices, for which there is presently still a dearth of e cient methods. For the use of these amplitudes, e.g. in Schwinger-Dyson equations, it is usually important to have them in a form that separates them into transversal and non-transversal parts, which normally requires a tedious analysis of the nonabelian Ward identities. This is one of the reasons why, using standard methods, the explicit calculation of these o -shell vertices has been completed so far only for the N = 3 case [47, 59, 75]. In another companion paper [53] we use the S representation to recalculate the three-gluon vertex, for the scalar, spinor, and gluon loops, achieving a drastic reduction in computational e ort compared to earlier attempts (a summary of this calculation was given in [76]). The main advantages of our approach are that the gluon and spinor loop cases can be e ortlessly obtained from the scalar loop one through the (o -shell extended) Bern-Kosower replacement rules, and that there is no necessity to solve the Ward identities, rather the decomposition into transversal and non-transversal pieces emerges automatically in the IBP procedure. In relation with the latter point it must also be mentioned that the boundary terms in the IBP procedure applied to the N-vertex always involve color-commutators, and are always connected to some lower-point term, even appearing with the same integral. Thus the algorithm makes it also easy to separate the genuinely new structures appearing in the N-vertex from those that, in terms of the e ective action, only serve the completion of lower-point expressions to fully gauge-invariant ones; see [53].

In the abelian case, we have also extended the IBP procedure to the QED N-photon amplitudes in a constant external eld. Here our main motivation is that these amplitudes can be used for the construction of higher-loop Euler-Heisenberg Lagrangians, and those as a tool to obtain insight about the asymptotic behavior of the QED N-photon amplitudes at high loop orders and large photon number [66, 67, 7779]. In that context it would be highly desirable to calculate the three-loop Euler-Heisenberg Lagrangian in various dimensions, and indeed the IBP procedure presented in section 10 has made it possible to nally achieve this goal at least for the 1+1 dimensional case [80].

It should be possible, and very interesting, to generalize our approach to the inclusion of gravity. On-shell, the gauge theory Bern-Kosower rules were generalized to the construction of string-based representations for the one-loop N-graviton amplitudes in [81], and these gravity rules were then successfully applied at the four-point level in [82]. They also involve an IBP and replacement rules connecting the amplitudes with di erent spins in the loop. Worldline path integral representations of the one-loop e ective actions for gravity have so far been constructed for spin zero [28], spin half [29, 30] and spin one [31, 32] in the loop, and can be used to obtain parameter integral representations of the corresponding

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o -shell one-particle irreducible N-graviton amplitudes that are closely related to those string-based representations. They are written in terms of the same worldline Greens functions as the ones for gauge theory, and the challenge is again to nd an IBP algorithm that would allow one to apply the replacement rules and at the same time make covariance manifest, where the latter now means the emergence of full Riemann tensors in the IBP. This algorithm can, however, not be a simple extension of the one which we have presented here, as one can see from the fact that in gravity there are no boundary terms in the IBP, so that the nonlinear terms in the Riemann tensor now have to be created by -functions; therefore a complete removal of all GBs in the IBP is not called for, and in fact also not possible, as one can easily see already from the case of the graviton propagator.

Finally, in open string theory the one-loop gauge boson amplitudes can be written in terms of a master formula that is analogous to the master formula (1.1), only that the variables i parametrize positions along the boundary of the string worldsheet, and the

Greens functions are worldsheet Greens functions (see, e.g., [83]). Since in all of our manipulations we have, apart from the translation invariance in proper-time, not used any specic properties of the worldline Greens functions, our IBP procedure could as well be applied at the string level to achieve a form factor decomposition based on gauge invariance.

Acknowledgments

N. A. and C. S. gratefully acknowledge the hospitality of the Dipartimento di Fisica, Universit di Bologna and INFN, Sezione di Bologna, as well as of the Institute of Physics and Mathematics of Humboldt University Berlin and of the AEI, Potsdam. C. S. andV.V. thank CONACYT for nancial support through Proyecto CB2008-101353, and N.A. for a PhD fellowship.

References

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JHEP01(2013)132

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SISSA, Trieste, Italy 2013

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The string-based Bern-Kosower rules provide an efficient way for obtaining parameter integral representations of the one-loop N -photon/gluon amplitudes involving a scalar, spinor or gluon loop, starting from a master formula and using a certain integration-by-parts ("IBP") procedure. Strassler observed that this algorithm also relates to gauge invariance, since it leads to the absorption of polarization vectors into field strength tensors. Here we present a systematic IBP algorithm that works for arbitrary N and leads to an integrand that is not only suitable for the application of the Bern-Kosower rules but also optimized with respect to gauge invariance. In the photon case this means manifest transversality at the integrand level, in the gluon case that a form factor decomposition of the amplitude into transversal and longitudinal parts is generated naturally by the IBP, without the necessity to consider the nonabelian Ward identities. Our algorithm is valid off-shell, and provides an extremely efficient way of calculating the one-loop one-particle-irreducible off-shell Green's functions ("vertices") in QCD. It can also be applied essentially unchanged to the one-loop gauge boson amplitudes in open string theory. In the abelian case, we study the systematics of the IBP also for the practically important case of the one-loop N -photon amplitudes in a constant field.

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Title

String-inspired representations of photon/gluon amplitudes

Author

Ahmadiniaz, Naser; Schubert, Christian; Villanueva, Victor M

Pages

1-28

Publication year

2013

Publication date

Jan 2013

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP01(2013)132

ProQuest document ID

1652943893

SISSA, Trieste, Italy 2013

String-inspired representations of photon/gluon amplitudes

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