UCLA/09/TEP/09/47 SLAC–PUB–13608 The Ultraviolet Behavior of N = 8 Supergravity at Four Loops Z. Berna, J. J. Carrascoa, L. J. Dixonb, H. Johanssona, and R. Roibanc aDepartment of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA bSLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA cDepartment of Physics, Pennsylvania State University, University Park, PA 16802, USA We describe the construction of the complete four-loop four-particle amplitude of N =8 super- gravity. The amplitude is ultraviolet finite, not only in four dimensions, but in five dimensions as well. The observed extra cancellations provide additional non-trivial evidence that N =8 super- gravity in four dimensions may beultraviolet finiteto all orders of perturbation theory. 9 0 PACSnumbers: 04.65.+e,11.15.Bt,11.25.Db,12.60.Jv 0 2 An often-expressed sentiment is that point-like quan- ermcomposedoffourappropriatelycontractedRiemann y tum field theories based on Einstein’s theory of General tensors(the squareofthe Bel-Robinsontensor),denoted a M Relativity, including supersymmetric extensions thereof, byR4. Arecentstudy[19]explainstheknownlackofthis are quantum mechanically inconsistent, due to either a counterterm [1, 2], both in terms of non-renormalization 4 proliferation of divergences associated with the dimen- theorems and an algebraic formalism for constraining 1 sionful nature of Newton’s constant, or absence of uni- counterterms. However, it does predict divergences at tarity. A series of recent computations has challenged L = 5 loops in dimension D = 4 and at L = 4 loops ] h thiswidelyheldbelief. Inparticular,thethree-loopfour- in D =5 [31], unless additionalcancellationmechanisms t graviton amplitude [1, 2] in N =8 supergravity [3] ex- beyondsupersymmetryandgaugeinvariancearepresent. - p poses cancellations beyond those needed for ultraviolet e (UV) finiteness at that order. Novel cancellations occur In contrast,explicit computations of the four-graviton h [ already in this theory [4, 5] at one loop, related [6, 7] amplitude at successive loop orders have consistently to the remarkably good behavior of gravity tree ampli- revealed unexpected UV cancellations. Results at 1 tudes under large complex deformations of external mo- two loops strongly suggested [20], and at three loops v 6 menta [7, 8], and to the unordered nature of gravity proved [1, 2] that the R4 divergence is absent in N =8 2 amplitudes [5]. The modern unitarity method [9] im- supergravity. In addition, UV divergences are absent at 3 plies that extensive UV cancellations occur to all loop threeloopsinD <6. ThetheoryfirstdivergesinD =6, 2 orders [10], for a class of terms obtained by isolating andthecountertermhastheschematicformD6R4,where 5. one-loop sub-amplitudes via generalized unitarity [11], D is a space-time derivative acting on the Riemann ten- 0 leading to the proposal [6] that the multiloop UV can- sors[2]. The computationdescribedinthis letter reveals 9 cellations trace back to the tree-level behavior. These no UV divergences at four loops in both D = 4 and 0 surprising cancellations point to the possible perturba- D = 5, specifically ruling out a counterterm of the form : v tive UV finiteness of the theory. D6R4 in D = 5. The origin of the observed UV proper- i X Interestingly, M theory and string theory have also ties is, however, not yet properly understood. r been used to argue both in favor of the finiteness of a N =8 supergravity [12], and that divergences are de- Itis worthnotingthatmorespeculativefield-theoretic layed through nine loops [13, 14]; issues involving the studies have suggested further delays to the onset of decoupling of certain massive states [15] remain in ei- divergences. For example, if off-shell superspaces with ther case. The non-compact E7(7) duality symmetry of manifest N = 6,7 or 8 supersymmetries were to exist, N =8 supergravity [3, 16] may also play a role [7, 17], D =4 divergences would be delayed to at least L =5,6 though this remains to be demonstrated. A mechanism or 7 loops, respectively [21, 22]. Locality of countert- rendering a point-like theory of quantum gravity ultra- erms in N = 8 light-cone superspace has also been used violet finite would be novel and should have a profound to argue [17] for an L = 7 bound. With the additional impact on our understanding of gravity. speculation that all fields respect an eleven-dimensional Indeed, all studies to date conclude that supersymme- gauge symmetry, one can even delay the first potential try and gauge invariance alone cannot prevent the on- divergence to nine loops [19]. Interestingly, this bound set of UV divergences to all loop orders in four dimen- coincideswiththeonesuggested[14]fromastringtheory sions. In fact, it had been a longstanding expectation non-renormalizationtheorem [13]. that, in generic supergravity theories, four-graviton am- plitudes diverge at three loops in four dimensions [18]. In this letter, we describe the four-loop four-particle Such a divergence would be associated with a countert- amplitude of N =8 supergravity, denoted by M4-loop, 4 2 (a) (b) (c) (d) (a) (b) (c) (d) (e) FIG. 1: Vacuum graphs from which one can build the con- (e) (f) (g) (h) tributing four-point graphs by attaching external legs. They are also useful for classifying theUV divergences. 12 1 (i) (j) (k) 2 3 11 4 2 2 3 2 6 1 4 FIG. 3: Evaluating these 11 cuts, along with 15 two-particle 1 4 1 5 7 389 59 367810 rfoeudru-cpiboilnetcaumtsp,listuuffidec.esEtaochubnlioqbuedlyendoetetesramtinreeetahmepfloituurd-leo.op I1 I25 I32 4 I50 FIG. 2: Four of the 50 distinct graphs corresponding to the forfourintegrals,labeledI1,I25,I32 andI50 intheonline integrals composing theresult for M4-loop. expressions [23], are shown in fig. 2. 4 To determine the amplitude, we first construct an ansatz with numerator polynomials N (l ,k ) that con- i j j which we representas a sum of 50 four-loop integrals I , i tain undetermined coefficients. Then we consider gen- eralized unitarity cuts decomposing thee four-loop ampli- 50 M4-loop = κ 10stuMtree c I . (1) tude into a product of tree amplitudes Mtree, as shown 4 2 4 i i (i) in fig. 3. Equating the cuts of the ansatz to the corre- (cid:16) (cid:17) XS4 Xi=1 sponding cuts of the amplitude, Here S is the set of 24 permutations of the massless ex- 4 ternal legs {1,2,3,4} with momenta ki, the ci are com- M4-loop = MtreeMtree···Mtree, (3) binatorial factors depending on the symmetries of the 4 cut (1) (2) (n) integrals, κ is the gravitational coupling, and M4tree is (cid:12)(cid:12) sXtates the corresponding four-point tree amplitude. (All 2564 constrains the u(cid:12)ndetermined coefficients in the ansatz. four-point amplitudes of N =8 supergravity are related As only tree amplitudes enter eq. (3), we follow the to each other by supersymmetry, which enforces the strategy[20]ofre-expressingtheN =8supergravitycuts proportionality of M4-loop to its tree-level counterpart in terms of sums of products of related cuts of the four- 4 Mtree.) The Mandelstam invariants are s = (k +k )2, loop four-gluon amplitude in N =4 super-Yang-Mills 4 1 2 t = (k +k )2, u = (k +k )2. Each integral I corre- (sYM) theory[24,25]. The strategyreliesonthe Kawai- 2 3 1 3 i sponds to a four-loop graph with 13 propagatorsand 10 Lewellen-Tye(KLT)relationsbetweengravityandgauge cubic vertices. The 50 graphs may be obtained by at- theorytreeamplitudes[26],facilitatedbytheirrecentre- tachingfourexternallegstotheedgesofthefivevacuum organizationintermsofdiagrams[27]. Whilewesuspect graphsinfig.1. Notallpossibilitiescontribute,however; that a representation of the N =8 amplitude exists in diagrams containing nontrivial two- or three-point sub- which each N is at most of degree four in the loop mo- i graphs, such as all those obtained from fig. 1(a), do not menta, it is natural, given the squaring nature of the appear in the amplitude. Every integral takes the form KLTrelations,tofirstsolvethe cutconstraintswiththis condition relaxed. We present a solution in which each I = 4 dDlnp Ni(lj,kj), (2) Ni is at most of degree eight [23]. This representationis i Z "p=1(2π)D# 1n3=1ln2 sufficientforourpurposeofdemonstratingUVfiniteness Y in D =4,5. where the propagator momenta lQare linear combina- The KLT relations are valid in arbitrary dimensions. n tions of four independent loop momenta l and the ex- Thus, if the N =4 amplitudes are valid in D dimen- np ternalmomentak . ThenumeratorpolynomialN (l ,k ) sions, then so are the N =8 amplitudes derived from j i j j is of degree 12 in the momenta, by dimensional analysis. them. While we do not yet have a complete proof of the Generically, we denote loop momenta by l and external (D >4)-dimensionalvalidity ofthe non-planarcontribu- momenta by k. tionstothefour-loopN =4amplitudes,wehavecarried Thefullamplitudeistoolengthytopresentinthislet- outextensivechecks. Inparticular,ordinarytwo-particle ter. Rather,weoutlineits constructionanddemonstrate cuts and cuts isolating four-point subamplitudes extend some of the relevant UV cancellations. Explicit expres- easily to D dimensions [24, 25, 27]. The full N =4 sYM sionsforthenumerators,symmetryfactorsandpropaga- amplitude, the details of its calculation, and non-trivial tors may be found online [23]. As examples, the graphs consistency checks will be presented elsewhere [25]. 3 Following the method of maximal cuts [2, 28], we (e) of fig. 1. For example, the k4l8 terms in the numera- firstfixthosecoefficientsoftheNi(lj,kj)thatcontribute tors of the integrals I25 and I32 in fig. 2 are when the number of cut propagators is maximal—13 in 1 this case. We then considercutsewith 12 cut lines, fixing N(8) = l2l2l2 (30s2+13t2+13u2)l2 25 8 5 6 7 9 the coefficients that appear in terms proportionalto sin- h gle inverse propagatorsl2 (i.e., contact terms). We con- −(32s2+19t2+19u2)l2 , n 8 tinue this procedure down to nine cut lines, considering, 1 i in total, 2906 distinct cuts. At this point, the resulting N(8) = 2(7s2+7t2+6u2)l2l2l2 l2 32 8 5 8 10 12 expressioniscomplete,whichwedemonstrateusingaset n of 26 cuts, sufficient to completely determine any four- +l2 12(2s2−t2+2u2)l2l2l2 9 6 7 12 loop four-point amplitude in any massless theory. The h −(24s2+19t2+19u2)l2l2l2 . (6) 11 cuts that cannot be straightforwardly verified using 5 8 11 lower-loopfour-pointamplitudes in two-particlecuts are io shown in fig. 3. All of the l2 factors in eq. (6) cancel propagators in the n The UV properties of the amplitude are determined integrals. Thus,toleadingorderintheexpansioninsmall by the numerator polynomials Ni. We decompose them external momenta, the k4l8 terms in I25 and I32 reduce into expressions N(m) containing all terms with m pow- to the vacuum diagram V(d) of fig. 1(d), i ers of loop momenta (and 12−m powersin the external I →−14(s2+t2+u2)V(d)+O(k5), momenta), 25 I →+14(s2+t2+u2)V(d)+O(k5). (7) 32 N =N(8)+N(7)+N(6)+...+N(0). (4) i i i i i Here we have summed over the S permutations of ex- 4 ternallegsineq.(1). Becausetheircombinatorialfactors There is some freedom in this decomposition, including c and c are equal [23], the I and I contributions thatinducedbythechoiceofindependentl intheloop 25 32 25 32 np cancel at leading order. Similarly, all k4l8 contributions integral (2). The overall scaling behavior of eq. (2) im- in the remaining diagrams cancel, independent of D. (m) pliesthatanintegralwithN inthenumeratorisfinite i Asthek5l7termscannotgeneratealeadingdivergence, when4D−26+m<0. Formodd,byLorentzinvariance, we need only inspect the k6l6 term to determine the UV the leading divergence trivially vanishes under integra- properties of the amplitude in D = 5. It is necessary to tion, effectively reducing m by one. Our representation expand all integrands down to k6l6. For the 12 the in- has m ≤ 8 for all terms; hence the four-loop amplitude tegrals starting at O(k4l8), two derivatives are required is manifestly UV finite in D =4. withrespecttotheexternalmomentak ,actingonprop- i Demonstrating UV finiteness in D =5 is more subtle. agators of the form 1/(l +K )2 (where K denotes a j n n Itrequiresthecancellationofdivergencesform=6,7,8. sum ofexternalmomenta). The numeratorsobtainedby We employ a systematic procedure for extracting diver- expanding the integrals to this order have the schematic gences from multiloop integrals by expanding in small form, external momenta [29]. We find that the numerator terms with m = 8 can K ·l K2 K ·l K ·l all be expressed solely in terms of inverse propagators Ni(6)+Ni(7) nl2 j +Ni(8) l2n + n lj2l2 q p . (8) l2; those with m = 7 have six powers of loop momenta j (cid:18) j j p (cid:19) n carried by inverse propagators; and those with m = 6 Theadditionaldenominatorscanleadtodoubledoreven have four powers; schematically, tripled propagators for the graphs in fig. 1. Vacuum in- tegrals with lµlν in the numerator can be reduced using Ni(8) ∼sasblj2ln2lp2lq2, Lorentzinvariianjce,liµljν →ηµνli·lj/D,withD =5. After N(7) ∼s s (k ·l )l2l2l2, (5) this reduction, the potential UV divergence is described i a b j n p q r by30vacuumintegrals. Ofthese,23possessnoloopmo- Ni(6) ∼sasb(kj ·ln)(kp·lq)lr2lw2 +sasbsc(lj ·ln)lp2lq2, menta in the numerator, while seven have an (li +lj)2 numeratorfactorthatcannotbereducedtoinverseprop- where each sa denotes s, t or u. After expanding in agators using momentum conservation. There are many small external momenta, potential UV divergences enter ways to expand the original 50 integrals I . Shifting the i throughvacuumintegrals,justasatthreeloops[1]. Vac- loop momenta in eq. (2) by dDl → dD(l +k ) leads np np j uum integrals also exhibit infrared singularities, which to different representations of the terms proportional to we regularizeby injecting two fictitious off-shellexternal N(7) and N(8) in eq. (8), and hence to different forms of i i momenta at appropriate locations in the graph. the UV divergences in terms of the 30 vacuum integrals. Only 12 of the 50 integrals have a nonvanishing N(8); Requiring that the different forms are equal generates i allofthemareassociatedwithvacuumdiagrams(d)and identities between vacuum integral divergences. These 4 l2 3 =2 l12,2 =5 −2 l1 [1] Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, (a) (b) D. A. Kosower and R. Roiban, Phys. Rev. Lett. 98, 161303 (2007) [hep-th/0702112]. FIG. 4: Two of the vacuum relations used to analyze the [2] Z.Bern,J.J.M.Carrasco,L.J.Dixon,H.Johanssonand D = 5 divergence. They are valid in D = 5−2ǫ to order R. Roiban, Phys. Rev. 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