The Ultraviolet Properties of N=4 Supergravity at Four Loops
Zvi Bern, Scott Davies, Tristan Dennen, Alexander V. Smirnov, Vladimir A. Smirnov
aa r X i v : . [ h e p - t h ] D ec UCLA/13/TEP/107
The Ultraviolet Properties of N = 4 Supergravity at Four Loops
Zvi Bern a , Scott Davies a , Tristan Dennen b , Alexander V. Smirnov c and Vladimir A. Smirnov d a Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA b Niels Bohr International Academy and Discovery Center,The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark c Scientific Research Computing Center, Moscow State University, 119992 Moscow, Russia d Skobeltsyn Institute of Nuclear Physics of Moscow State University, 119992 Moscow, Russia
We demonstrate that pure N = 4 supergravity is ultraviolet divergent at four loops. The formof the divergence suggests that it is due to the rigid U (1) duality-symmetry anomaly of the theory.This is the first known example of an ultraviolet divergence in a pure ungauged supergravity theoryin four dimensions. We use the duality between color and kinematics to construct the integrandof the four-loop four-point amplitude, whose ultraviolet divergence is then extracted by standardintegration techniques. PACS numbers: 04.65.+e, 11.15.Bt, 11.25.Db, 12.60.Jv
Recent years have seen enormous advances in our abil-ity to obtain scattering amplitudes in gauge and grav-ity theories. Using these advances we can address basicquestions on the ultraviolet properties of quantum grav-ity that had seemed relegated to the dustbin of undecid-able questions. Power-counting arguments suggest thatall point-like theories of gravity should be ultraviolet di-vergent. However, such arguments can be misleading ifthere are additional hidden symmetries or structures. Inparticular, the duality between color and kinematics [1, 2]has been shown to be responsible for improved ultravi-olet behavior in the relatively simple two-loop case ofhalf-maximal supergravity in five dimensions [3]. Thisexample emphasizes the importance of carrying out moregeneral investigations of the ultraviolet properties of su-pergravity theories to ascertain the full implications ofnew structures.Pure Einstein gravity has long been known to be fi-nite at one loop [4] but divergent at two loops [5]. Italso diverges at one loop under the addition of genericmatter [4, 6]. However, the situation with pure un-gauged supergravity is less clear. Such theories areknown not to diverge prior to three loops [7]. The consen-sus from studies in the 1980s was that all supergravitytheories likely diverge at three loops (see for example,Ref. [8]), although with appropriate assumptions tighterbounds are possible [9]. However, it was not possibleto check these arguments until the advent of the unitar-ity method [10, 11]. For the most supersymmetric case of N = 8 supergravity [12], explicit calculations have shownthat the four-point amplitudes are finite at three loops fordimensions D <
D < / E duality symmetry of the theory [15, 16].However, a D R counterterm appears to be valid un-der all standard symmetries, leading to predictions of aseven-loop divergence in N = 8 supergravity in D = 4. While seven loops is at present out of reach of di-rect computations, reducing the supersymmetry lowersthe loop order at which nontrivial ultraviolet cancella-tions can be studied. As discussed in ref. [16], the sametype of symmetry argument used for N = 8 supergrav-ity at seven loops also implies the existence of an ap-parently valid three-loop R counterterm in N = 4 su-pergravity [17]. This suggests that pure N = 4 super-gravity should diverge at three loops. This is consistentwith speculations based on the pattern of cancellationsat one loop, suggesting that at least N ≥ N = 4supergravity actually vanishes [19]. (See Ref. [20] fora string-theory argument.) Another related exampleis the unexpected finiteness of the two-loop four-pointamplitude of half-maximal supergravity in five dimen-sions [3, 20]. By assuming the existence of appropri-ate 16-supercharge superspaces, the observed finitenesscan be understood as a consequence of standard symme-tries [21]. However, these superspaces also lead to predic-tions in direct contradiction to explicit calculations whenmatter multiplets are added [22], implying that the as-sumption needs to be altered. There are also conjecturesthat certain structures or hidden symmetries may play arole [23]. In any case, these examples remain unexplainedfrom standard symmetry considerations. This makes itimportant to investigate the next loop order. If there areno additional cancellations at four loops beyond the onesalready identified at three loops, either in string theoryor in field theory, it should diverge [20, 21].In this Letter, we compute the four-loop four-point di-vergence of N = 4 supergravity following the same basicmethods used in the corresponding three-loop computa-tion [19] and described in some detail in Ref. [22]. Wefind that although N = 4 supergravity does have an ul-traviolet divergence, its form suggests that it is special (1) (48) (59) (85)
12 34
FIG. 1: Four of the 85 diagrams with cubic vertices used toorganize the N = 4 super-Yang-Mills amplitudes into a formthat respects the duality between color and kinematics. Theremaining diagrams are listed in Ref. [24]. and tied to the U (1) duality anomaly of the theory.Our construction of the four-loop four-point ampli-tude of N = 4 supergravity starts with the correspondingpure Yang-Mills Feynman diagrams in Feynman gauge.To obtain N = 4 supergravity, we also need the N = 4super-Yang-Mills diagram kinematic numerators listed inRef. [24] that obey the duality between color and kine-matics. In this form the kinematic-numerator factors n i satisfy algebraic relations in one-to-one correspondencewith relations satisfied by the color factors c i . These fac-tors are associated with 85 diagrams (plus permutationsof external legs) containing only cubic vertices, as illus-trated in Fig. 1. The N = 4 supergravity integrands areobtained simply by replacing the color factors c i in thepure-Yang-Mills integrand with the corresponding N = 4super-Yang-Mills kinematic-numerator factors, c i → n i . (1)The construction of the supergravity integrand via theduality between color and kinematics automatically sat-isfies the D -dimensional unitarity cut constraints, giventhat the input gauge-theory amplitudes have the correctcuts.The N = 4 super-Yang-Mills numerators [24] used inthe construction are proportional to the color-ordered N = 4 super-Yang-Mills tree-level amplitudes A tree N =4 ,which can be conveniently expressed in an on-shell su-perspace formalism in four dimensions [25]. As an ex-ample, diagram (1) in Fig. 1 has a numerator given by n = s tA tree N =4 , where s = ( k + k ) and t = ( k + k ) arestandard Mandelstam invariants. The remaining numer-ator factors are specified in Ref. [24] and are, in general,somewhat more complicated, depending also on loop mo-menta.Using Feynman diagrams for the nonsupersymmetricpure Yang-Mills amplitude might seem inefficient, butfor the problem at hand it is a reasonable choice. Itautomatically gives us local covariant expressions withno spurious singularities that could complicate loop in-tegration. Moreover, only the relatively small subsetof diagrams containing color factors matching those ofthe nonvanishing diagrams in the corresponding N = 4super-Yang-Mills theory are needed, otherwise the con-tribution vanishes as well in N = 4 supergravity. Feyn-
341 2 1 2
FIG. 2: The two basic vacuum graphs. man diagrams also avoid subtleties associated with thebubble-on-external-leg diagrams, such as diagram (85) ofFig. 1. After integration all such pure Yang-Mills Feyn-man diagrams are smooth in the on-shell limit, cancelingthe 1 /k propagator as k →
0. In N = 4 supergrav-ity such contributions vanish because the color factors inthe pure Yang-Mills diagrams are replaced by vanishingnumerator factors independent of loop momentum [24].The logarithmic ultraviolet divergence may be ex-tracted by series expanding in small external momenta,or equivalently large loop momenta [26]. The resultingtensor integrals are then reduced to scalar integrals viaLorentz invariance. We regularize the integrals using di-mensional reduction [27]. Further details of the proce-dure are given in Ref. [22].The small-momentum expansion has the undesired ef-fect of introducing new unphysical infrared singularities.To separate out all resulting infrared divergences fromthe ultraviolet ones, we use a mass regulator. A partic-ularly convenient choice is to introduce a uniform massinto all Feynman propagators prior to expanding in ex-ternal momenta [28]. For the case of pure N = 4 super-gravity with no matter multiplets, with this regulator,the subdivergences should all cancel amongst themselvesbecause there are no one-, two- or three-loop divergences.This can be used to greatly simplify the computationsince we do not need to compute subdivergences. How-ever, we compute them regardless, using their cancella-tion as a nontrivial consistency check. More generally,the issue of infrared regularization is delicate because ofregulator dependence. For example, if the mass regula-tor were introduced after the expansion in external mo-menta, it would ruin the cancellation of subdivergencesbetween different integrals, and one would need to in-clude all subdivergence subtractions to remove the regu-lator dependence.At the end of this process, we obtain a large number ofvacuum integrals with the two basic diagrammatic struc-tures shown in Fig. 2. These are of the form, Z Y j =1 d D p j (2 π ) D P ( m , p · p ) Q i =1 ( p i − m ) a i , (2)where P is a numerator polynomial in the mass and theirreducible dot product formed from the momenta flowingthrough propagators 1 and 2, indicated in Fig. 2. (By ir-reducible we mean that it cannot be expressed as a linearcombination of inverse propagators and masses.) The 9 p i correspond to the 9 propagators in each of the vacuumdiagrams of Fig. 2, with the first four being independentloop momenta. The indices a i are integers.The standard modern way to evaluate these vacuum in-tegrals is to use integration-by-parts relations [29] withindimensional regularization. This allows us to write downany given integral as a linear combination of so-calledmaster integrals which can then be evaluated. Forfour-loop Feynman vacuum integrals, this was done inRef. [30]. In our calculation, the reduction to master in-tegrals turns out to be complicated because high powersof numerator loop momenta are involved. To deal withthis, we use the C++ version of the code
FIRE [31], im-plementing the Laporta algorithm [32]. We use the samemaster-integral basis set as in Ref. [33]. (See Ref. [34] fora high-precision numerical evaluation.)Each state of pure N = 4 supergravity is a direct prod-uct of a color-stripped state of N = 4 super-Yang-Millstheory and of pure nonsupersymmetric Yang-Mills the-ory. Pure N = 4 supergravity contains two multipletsthat do not mix under linearized supersymmetry: onecontains the negative-helicity graviton and the other thepositive-helicity graviton. We find that all amplitudes inpure N = 4 supergravity are divergent at four loops, M - loop (cid:12)(cid:12)(cid:12) div . = 1(4 π ) ǫ (cid:16) κ (cid:17) − ζ ) T , (3)where ǫ = (4 − D ) / T = stA tree N =4 ( O − O − O ) , (4)where O = X S ( D α F µν ) ( D α F µν ) F ρσ F ρσ , O = X S ( D α F µν ) ( D α F νσ ) F σρ F ρµ , (5) O = X S ( D α F µν ) ( D β F µν ) F α σ F σβ . The sum runs over all 24 permutations of the externallegs. The linearized field strength for each leg j is givenin terms of polarization vectors for that leg, F µνj ≡ i ( k µj ε νj − k νj ε µj ) ,D α F µνj ≡ − k αj ( k µj ε νj − k νj ε µj ) . (6)We have also included contributions from N = 4 mat-ter multiplets in the loops. As discussed in Refs. [22,35], amplitudes with matter multiplets are straightfor-wardly obtained via dimensional reduction from higher-dimensional pure half-maximal supergravity withoutmatter. After including the contribution of n V mattermultiplets, with all four external states belonging to the two graviton multiplets, the divergence is M - loop n V (cid:12)(cid:12)(cid:12) div . = 1(4 π ) (cid:16) κ (cid:17) n V + 22304 h n V + 2) n V ǫ (7)+ ( n V + 2)(3 n V + 4) − − n V ) ζ ǫ i T . In this expression n V is independent of ǫ , a restrictionthat arises from imposing this on subdivergence subtrac-tions. The two- and three-loop subdivergences, and sub-divergences thereof, all cancel amongst themselves whenwe use a uniform mass regulator, as happened for the n V = 0 case. These cancellations are analogous to similarcancellations that occur at three loops and are surprisingbecause there are subdivergences when matter multipletsare included [22, 36]. However, the one-loop subdiver-gences do not cancel when n V = 0. Instead, these enternontrivially to make the divergence gauge invariant andproportional to T .By taking linear combinations, O −− ++ = O − O , O − +++ = O − O , O ++++ = O , (8)each of the obtained operators are nonvanishing only forthe indicated helicity configurations and their parity con-jugates and relabelings. Here the helicity labels refer tothose of the polarization vectors used in Eq. (6) and notthe supergravity states which are obtained by tensoringthese states with those of N = 4 super-Yang-Mills theory.Using explicit helicity states in D = 4, we have O −− ++ = 4 s t h i h i h i h i h i , O − +++ = − s t [2 4] [1 2] h i h i [4 1] , (9) O ++++ = 3 st ( s + t ) [1 2] [3 4] h i h i , using spinor-helicity notation. (See Ref. [37] for a recentreview.) The divergence is thus present in all nonvan-ishing four-point amplitudes of N = 4 supergravity. Lin-earized supersymmetry acts only on the A tree N =4 factor inEq. (4), so each of these three configurations will not mixunder this symmetry.The appearance of the divergences in all three in-dependent helicity configurations in Eq. (8) is surpris-ing. In general, the analytic structure of amplitudes inthe −− ++ sector is rather different from those of theother two sectors. This follows from generalized uni-tarity, where we decompose the supergravity loops intosums of products of tree amplitudes. In the − +++ and++++ sectors, all generalized cuts vanish in four dimen-sions because at least one tree amplitude will vanish. Thesame does not hold in the −− ++ sector. In particular,at one loop this implies that amplitudes in the −− ++sector contain logarithms while amplitudes in the othertwo sectors are pure rational functions. The rationalfunctions appearing in these sectors have been directlyinterpreted [35] as due to the U (1) duality-symmetryanomaly [38]. We can understand the similarity of thefour-loop ultraviolet divergence in all three sectors if weassume that it is due to the anomaly. As already noted inRef. [35], unitarity implies that the anomaly contributesto higher-loop divergences in the −− ++ sector as well(unless canceled from another source). The similarity ofthe divergence in all three sectors would be a consequenceof it arising from the same source. Another helpful cluecomes from the fact that the divergence in Eq. (7) isproportional to n V + 2. As explained in Ref. [35], theanomaly terms are proportional to this factor, providingfurther nontrivial evidence that the four-loop divergenceis due to the anomaly.We can re-express the divergences in terms of coun-terterms involving the Riemann tensor. If we restrict theexternal states to four dimensions, numerical analysis re-veals that the four-external-graviton counterterm can bereduced to a rather simple expression, C = − π ) (cid:16) κ (cid:17) ǫ (1 − ζ )( T + 2 T ) , (10)where T ≡ ( D α R µνλγ )( D α R λγρσ ) R νρδκ R σµδκ ,T ≡ ( D α R µνλγ )( D α R λγρσ ) R µνδκ R ρσδκ . (11)Using the divergence given in Eq. (3), one can also obtainthe explicit counterterms for any other external states ofthe theory.In any calculation of this type, it is important to havenontrivial consistency checks on the results. The mostobvious one is the gauge invariance of the results (3)and (7). This requires intricate cancellations among theterms. We also find a required cancellation of poles in ǫ ,as well as an expected [29] cancellation of various tran-scendental constants. Because there are no lower-loopdivergences in pure N = 4 supergravity, only a 1 /ǫ polecan remain at four loops. As an illustration, considerthe basis integral corresponding to the first integral inFig. 2, with all propagators having unit indices, exceptfor the ones labeled by 3 and 4 which have vanishing in-dices (PR9 in the notation of Ref. [33]). Up to an overallfactor, the divergent parts of this basis integral arePR9 = 14 ǫ + 73 ǫ + 1 ǫ (cid:16) −
272 S2 + 12 ζ + ζ (cid:17) (12)+ 1 ǫ (cid:16) − − T1ep + 16 ζ − ζ + 32 ζ (cid:17) , where S2 and T1ep are transcendental constants speci-fied in Ref. [33]. Besides finding the required cancellationof all poles down to the 1 /ǫ level in Eq. (3), the transcen-dental constants other than ζ also cancel. Another cross check on our procedure comes from com-puting the coefficient of an analogous potential diver-gence in pure Yang-Mills theory. By renormalizability,the divergences are proportional to tree-level color ten-sors, so all divergences containing independent color ten-sors other than the tree-level ones must vanish. Usingidentical methods as for the supergravity case, we haveconfirmed the ultraviolet finiteness of terms multiplyingthe two independent four-loop color tensors listed in Ap-pendix B of Ref. [39].Instead of providing definitive answers for the ultra-violet behavior of supergravity theories, our calculationraises additional interesting questions. We showed thatthe nonvanishing four-loop divergence of N = 4 super-gravity has a form suggesting that it is caused by the U (1) duality-symmetry anomaly. It would be importantto demonstrate this directly either via the countertermstructure or by tracking the contributions of the anomalyto the amplitudes. One may also wonder whether it ispossible to remove the divergence by adding a finite termto the action so that an appropriate symmetry is pre-served. A key issue is to find the higher-loop ultravio-let behavior of N ≥ N = 4 supergravityor at two loops in five-dimensional half-maximal super-gravity. It would be desirable to investigate this further.If history is any guide, further surprises await us as weprobe supergravity theories to ever deeper levels.We especially thank Guillaume Bossard, Kelly Stelleand Radu Roiban for detailed discussions and for impor-tant comments on the manuscript. 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