Physics Letters B 716 (2012) 345–349 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb On the low-energy limit of one-loop photon–graviton amplitudes ∗ F. Bastianelli a,b,O.Corradinic, J.M. Dávila d, C. Schubert b,d, a Dipartimento di Fisica, Università di Bologna and INFN Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy b Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Mühlenberg 1, D-14476 Potsdam, Germany c Centro de Estudios en Física y Matemáticas Básicas y Aplicadas, Universidad Autónoma de Chiapas, C.P. 29000, Tuxtla Gutiérrez, Mexico d Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán,Mexico article info abstract Article history: We present first results of a systematic study of the structure of the low-energy limit of the one-loop Received 23 February 2012 photon–graviton amplitudes induced by massive scalars and spinors. Our main objective is the search Received in revised form 27 June 2012 of KLT-type relations where effectively two photons merge into a graviton. We find such a relation Accepted 15 August 2012 at the graviton–photon–photon level. We also derive the diffeomorphism Ward identity for the 1PI Available online 17 August 2012 one-graviton–N-photon amplitudes. Editor: L. Alvarez-Gaumé © 2012 Elsevier B.V. All rights reserved. 1. Introduction Most of this work concerned the case of massless on-shell am- plitudes, for which particularly efficient computation methods are Although graviton amplitudes are presently not of phenomeno- available. Much less has been done on amplitudes involving the logical relevance, their structure has been studied in parallel with interaction of gravitons with massive matter. At the tree level, the more important Yang–Mills amplitudes. The powerful meth- there are some classical results on amplitudes involving gravitons ods that have been developed during the last two decades for the [23,24]. More recently, the tree-level Compton-type amplitudes in- computation of on-shell amplitudes (see, e.g., [1–8]) essentially ap- volving gravitons and spin zero, half and one particles were com- ply equally to both cases. Moreover, the famous relations found in puted in [25], leading to another remarkable factorization prop- 1986 by Kawai, Lewellen and Tye (KLT) between closed and open erty [26] of the graviton–graviton scattering amplitudes in terms string amplitudes [9] in the field theory limit imply relations be- of the photonic Compton amplitudes. tween amplitudes in gravity and Yang–Mills theory that are not From the point of view of (non-)renormalizability, one-loop at all obvious from the field theory Lagrangians or Feynman rules gravity–matter amplitudes were studied in [27,28]. [10–12]. The effort to make these relations transparent also at the There are also a number of results for mixed graviton–gauge field theory level is still ongoing [13–17]. The KLT relations express boson amplitudes involving a matter loop, namely the graviton– gravity amplitudes as sums of squares of Yang–Mills amplitudes. photon–photon vertex [29–33], its non-abelian generalization [34] They hold at the tree level, but can be used together with unitar- and the related amplitude for photon–graviton conversion in an ity methods for the construction of loop amplitudes in gravity. This external field [35–39]. is very interesting considering the different UV behaviour of loop Here we will present first results of a systematic study of the amplitudes in gravity vs. Yang–Mills theory, and has been used as structure of the mixed photon–graviton amplitudes with a massive a tool in the study of the possible finiteness of N = 8 supergravity loop in the low-energy limit, and of the search for KLT-like rela- (see [8,18] and refs. therein). tions for such amplitudes. The great advantage of this limit is that More recently, a different kind of KLT-like relations has been it is accessible through the effective action; let us discuss this first found where the same type of factorization is made manifest even for the purely photonic case. As is well known, the information at the integrand level. Reversing the original flow of information, on the low-energy limit of the QED N-photon amplitudes is con- these relations were first conjectured for the n-graviton tree am- tained in the Euler–Heisenberg Lagrangian (“EHL”) [40].Werecall plitudes in field theory [19], and later extended to and proven in the standard representation of this effective Lagrangian: string theory [20,21]. A multiloop generalization in field theory has ∞ been conjectured in [22]. (EH) 1 dT − 2 L =− e m T spin 8π 2 T 3 0 Corresponding author at: Instituto de Física y Matemáticas, Universidad Michoa- * (eaT )(ebT ) e2 cana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, × − a2 − b2 T 2 − 1 . (1.1) Michoacán, Mexico. tanh(eaT ) tan(ebT ) 3 E-mail address: [email protected] (C. Schubert). 0370-2693/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2012.08.030 346 F. Bastianelli et al. / Physics Letters B 716 (2012) 345–349 2 Here T is the proper-time of the loop fermion, m its mass, (DH) e 2 μα ν L = 5RF − 26Rμν F F α and a, b are the two Maxwell field invariants, related to E, B by spin 180(4π)2m2 μν 2 2 2 2 a − b = B − E , ab = E · B. The subtraction terms implement the 2 + 2R F μν F αβ + 24 ∇α F (1.7) renormalization of charge and vacuum energy. The analogous rep- μναβ αμ resentation for scalar QED was obtained by Weisskopf [41]: (see [47] for our gravity conventions). The corresponding form of ∞ the effective Lagrangian for the scalar loop (using the same opera- tor basis) is [47] (EH) 1 dT − 2 L (F ) = e m T scal 2 3 2 16π T (DH) e 1 2 μα ν 0 L = 15 ξ − RF − 2Rμν F F α scal 180(4π)2m2 6 μν (eaT )(ebT ) 1 × + 2 − 2 2 − a b T 1 . (1.2) μν αβ α 2 sinh(eaT ) sin(ebT ) 6 − Rμναβ F F + 3 ∇ Fαμ (1.8) Obtaining the low-energy (= large-mass) limit of the N-photon (the parameter ξ refers to a non-minimal coupling of the scalar). amplitudes from the effective Lagrangians (1.1), (1.2) is a standard In [49] two of the present authors presented the next order in the procedure (see, e.g., [42]), and the result can be expressed quite expansion of the effective Lagrangians obtained in [47] in powers concisely [43]: of the field strength, i.e. the terms of order RF4 (there are no order 3 + + − − RF terms for parity reasons). A ; ...; ; ; ...; spin ε1 εK εK +1 εN The purpose of the present Letter is twofold. First, we will use 2 4 N K N −K the above effective actions at the RF level to compute the low- m 2ie − − =− (N − 3)! (−1)N K l energy limits of the one-graviton–two-photon amplitudes with a 8π 2 m2 k=0 l=0 scalar and spinor loop, and show that they relate to the four- photon amplitudes in a KLT-like way. Second, as a preparation for Bk+lBN−k−l + − × χ χ − , the study of the higher-point cases we will derive the Ward iden- k!l!(K − k)!(N − K − l)! K N K tities for the one-graviton–N-photon 1PI amplitudes in general. +; ; +; − ; ; − Ascal ε1 ... εK εK +1 ... εN − 2. Ward identities for the 1PI one-graviton–N-photon 4 N K N K m 2ie − − amplitudes = (N − 3)! (−1)N K l 2 2 16π m = = k 0 l 0 We derive the relevant Ward identities, generalizing the discus- 1−k−l) 1−N+k+l (1 − 2 )(1 − 2 )Bk+l B N−k−l + − sion in [37]. There are two types of Ward identities, those derived × χ χ − . (1.3) k!l!(K − k)!(N − K − l)! K N K from gauge invariance and those that follow from general coordi- nate invariance. Here the superscripts ± refer to circular polarizations, and the Bk ± Gauge transformations are defined by are Bernoulli numbers. The invariants χK are written, in spinor helicity notation (our spinor helicity conventions follow [44]), δG Aμ = ∂μλ, δG gμν = 0 (2.1) with an arbitrary local parameter λ. Then gauge invariance of the ( K )! + = 2 [ ]2[ ]2 ··· − 2 + effective action χK K 12 34 (K 1)K all permutations , 2 2 δ Γ [g, A]=0 (2.2) ( K )! G − = 2 2 2 ··· − 2 + χK K 12 34 (K 1)K all permutations . (1.4) implies that 2 2 1 δΓ For the case of the “maximally helicity-violating” (MHV) ampli- ∇μ √ = 0. (2.3) tudes, which have “all +”or“all−” helicities, Eqs. (1.3) simplify g δ Aμ (using Bernoulli number identities) to Similarly, infinitesimal reparametrizations are given by N ν ν ± ± (2e) BN ± δR Aμ = ξ ∂ν Aμ + ∂μξ Aν,δR gμν =∇μξν +∇νξμ (2.4) Ascal k1, ε ; ...;kN , ε =− χ , (1.5) 1 N (4 )2m2N−4 N(N − 2) N μ π with arbitrary local parameters ξ . The invariance of the effective ±; ; ± =− ±; ; ± action Aspin k1, ε1 ... kN , εN 2Ascal k1, ε1 ... kN , εN . (1.6) This relation (1.6) is actually true also away from the low-energy δR Γ [g, A]=0 (2.5) limit, and can be explained by the fact that the MHV amplitudes now implies correspond to a self-dual background, in which the Dirac operator has a quantum-mechanical supersymmetry [45].ForthisMHVcase 2 δΓ 1 δΓ ν 1 δΓ ν ∇μ √ + √ A − √ ∇ Aμ = 0.