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AEI-2010-146, MCTP-10-40, MIT-CTP-4176, MPP-2010-122, PUPT-2348

E7(7) constraints on counterterms in N = 8

N. Beiserta, H. Elvangbc, D. Freedmande, M. Kiermaierf , A. Moralesd, and S. Stiebergerg

aMax-Planck-Institut f¨urGravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Potsdam, Germany bMichigan Center for Theoretical (MCTP), Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA cInstitute for Advanced Study, Princeton, NJ 08540, USA dDepartment of and eCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA f Department of Physics, Princeton University, Princeton, NJ 08544, USA , and gMax-Planck-Institut f¨urPhysik, Werner-Heisenberg-Institut, 80805 M¨unchen,Germany (Dated: October 25, 2018) We prove by explicit computation that 6-point matrix elements of D4R4 and D6R4 in N = 8 supergravity have non-vanishing single-soft scalar limits, and therefore these operators violate the continuous E7(7) symmetry. The soft limits precisely match automorphism constraints. Together 4 with previous results for R , this provides a direct proof that no E7(7)-invariant candidate coun- terterm exists below 7-loop order. At 7-loops, we characterize the infinite tower of independent supersymmetric operators D4R6, R8, ϕ2R8,... with n > 4 fields and prove that they all violate E7(7) symmetry. This means that the 4- amplitude determines whether or not the theory is finite at 7-loop order. We show that the corresponding candidate counterterm D8R4 has a non- linear supersymmetrization such that its single- and double-soft scalar limits are compatible with E7(7) up to and including 6-points. At loop orders 7, 8, 9 we provide an exhaustive account of all independent candidate counterterms with up to 16, 14, 12 fields, respectively, together with their potential single-soft scalar limits.

INTRODUCTION trix elements vanish in the ‘single-soft limit’ pµ → 0 for each external scalar line [12–14]. The scalars of N = 8 su- N = 8 supergravity has maximal , and pergravity are the ‘pions’ of this soft-pion theorem since the classical theory has global continuous E7(7) symme- they are the 70 Goldstone bosons of the spontaneously try which is spontaneously broken to SU(8). Explicit broken generators of E7(7). It was recently proven [15] calculations have demonstrated that the 4-graviton am- that the soft scalar property fails for 6-point matrix el- 4 plitude in N = 8 supergravity is finite up to 4-loop order ements of the operator R (see also [16]). Thus E7(7) [1]. Together with - and -based obser- excludes R4 and explains the finite 3-loop result found in vations [2, 3], this spurred a wave of renewed interest in [1]. the question of whether the loop computations based on In the present paper we show first that the 5- and 6- generalized unitarity [4] could yield a UV finite result to loop operators D4R4 and D6R4 are incompatible with all orders1 — or at which loop order the first divergence E7(7) symmetry because their 6-point matrix elements might occur. have non-vanishing single-soft scalar limits. Previous In gravity, logarithmic UV divergences in on-shell L- [17] and superspace [18] arguments sug- loop amplitudes are associated with local counterterm gested this E7(7)-violation. Our results mean that no operators of mass dimension δ = 2L + 2 composed of UV divergences occur in N = 8 supergravity below the fields from the classical theory. The counterterms must 7-loop level. arXiv:1009.1643v2 [hep-th] 22 Sep 2010 respect the non-anomalous symmetries of the theory. It was shown in [7–9] that below 7-loop order, there are We then survey the candidate counterterms for loop or- only 3 independent operators consistent with linearized ders L = 7, 8, 9 using two new algorithmic methods: one N = 8 supersymmetry and global SU(8) R-symmetry program counts monomials in the fields of N = 8 super- [10]. These are the 3-, 5- and 6-loop supersymmetric gravity in representations of the superalgebra SU(2, 2|8), candidate counterterms R4,D4R4, and D6R4. the other applies Gr¨obnerbasis methods to construct The perturbative S-matrix of N = 8 supergravity their explicit local matrix elements. Our analysis shows that at each loop level 7, 8, 9, there is an infinite tower of should respect E7(7) symmetry [11], so one must subject R4,D4R4, and D6R4 to this test. A necessary condition independent n-point supersymmetric counterterms with n ≥ 4. At 7-loop order we find that none of the n-field for a counterterm to be E7(7)-compatible, is that its ma- operators with n > 4 are E7(7)-compatible. This leaves D8R4 as the only candidate counterterm at L = 7. We show that its matrix elements are E7(7)-compatible at 1 This question is well-defined whether or not N = 8 supergravity least up to 6-points. We observe that it requires remark- is sensible as a full quantum theory [5, 6]. able cancellations for E7(7) to be satisfied to all orders 2 for any L ≥ 7 candidate counterterm. 5-loop counterterm D4R4 At order α05, the SU(8)-average (2) of the string the- 4 4 6 4 ory amplitudes directly gives the matrix elements of the E7(7)-VIOLATION OF D R AND D R unique SU(8)-invariant supersymmetrization of D4R4. The result is a complicated non-local expression, but its To investigate E we study the soft scalar limit of the 7(7) single-soft scalar limit is very simple and local, viz. 6-point NMHV matrix elements h++−−ϕϕ¯iD2kR4 . The external states are two pairs of opposite helicity gravi- 6 4 4 X 2 lim h++−−ϕϕ¯iD4R4 = − 7 [12] h34i sij . (4) tons and two conjugate scalars. These matrix elements p6→0 i

This fixes the coefficient of the polynomial (7) to vanish Here, ∆ is the Laplacian on E7(7)/SU(8); in terms of the and determines the SUSY completion of the desired pole scalars ϕabcd of N = 8 supergravity, its leading terms are diagram uniquely! Finally we project out the scalar-graviton matrix el- h ∂ ∂ i ement from this superamplitude and take its single-soft ∆ = + 34 inequiv. perms + .... (12) scalar limit to find ∂ϕ1234 ∂ϕ5678

1 4 4 X 3 It is easy to see that the function (10) with the above lim h++−−ϕϕ¯i(R4)2 = − 70 [12] h34i sij . (8) p6→0 i

6 4 This non-vanishing result shows that the operator D R 2 λR4  2 is also incompatible with continuous E7(7) symmetry. (∆ + 60)fD6R4 (ϕ) = − fR4 (ϕ) . (13) λ 6 4 R4, D4R4 and D6R4 are the only local supersymmet- D R ric and SU(8)-symmetric operators for loop levels L ≤ 6 From (5) and (9), we can reconstruct the coefficient [7–9]. Hence N = 8 supergravity has no potential coun- a in (10) of the functions associated with the SU(8)- terterms that satisfy the continuous E symmetry for 7(7) averaged string-theory operator (e-12φD6R4) and with L ≤ 6. We stress that string theory is used as a tool to avg the supergravity operator D6R4. We find extract SU(8)-invariant matrix elements that must agree with the matrix elements of the N = 8 supergravity op- 66 60 a -12φ 6 4 = , a 6 4 = . (14) erators R4, D4R4 and D6R4 because each of these oper- (e D R )avg 35 D R 35 ators is unique. No remnant of string-specific dynamics -12φ 6 4 remains in the final results. For (e D R )avg, the couplings λ in (13) must take 03 their string theory values λR4 = −2α ζ(3) and λD6R4 = 2 α06ζ(3)2. The N = 8 operator D6R4, on the other Matching to automorphism analysis 3 hand, must satisfy (13) with λR4 = 0 because the opera- The non-vanishing single-soft scalar limits found above tor R4 does not appear in the action of N = 8 supergrav- have their origin in local 6-point interactions of the ity. Indeed, our results for f for both operators satisfy schematic form ϕ2D2kR4 which appear in the non-linear the Laplace equation with the expected choice of λ’s. completion of D2kR4. Let us encode this completion as f(ϕ)D2kR4, with CONSTRUCTION AND COUNTING OF  1234 5678  f(ϕ) = 1 − a ϕ ϕ + 34 inequiv. perms + .... COUNTERTERMS (10) The “. . . ” indicate higher order terms. The constant a 6 4 We now discuss the techniques used to classify and con- depends on the operator; for example a 4 = for R R 5 struct local supersymmetric operators, especially those [15]. We can determine a for D4R4 by taking a further needed for L ≥ 7. We are interested in SU(8)-invariant single-soft limit p →0 on (4) and comparing the resulting 5 operators, which are candidate counterterms, and in op- s, t, u-polynomial with the 4-point normalization of (3). 12 erators transforming in the 70 of SU(8). The latter The result is a 4 4 = . D R 7 are candidate operators for local single-soft scalar limits Ref. [17] used supersymmetry and duality consider- (SSL’s) of the matrix elements of singlet counterterm op- ations in d dimensions to constrain the moduli depen- erators. First we use of the super- dent functions f(ϕ) of the BPS operators R4, D4R4 and algebra SU(2, 2|8) to determine the spectrum and mul- D6R4. Specifically, for R4 and D4R4 in 4 dimensions, tiplicity of these operators. The spectrum is classified they found that f(ϕ) should satisfy the Laplace equation by the number n of external fields, the scale dimension, and the order k of the NkMHV type. Then we construct matrix elements of several operators explicitly using al- (∆ + 42)f 4 (ϕ) = 0 , (∆ + 60)f 4 4 (ϕ) = 0 . (11) R D R gorithms which incorporate Gr¨obnerbasis techniques. 4

Spectrum of local operators Note that locality requires that the exponents in (16) are non-negative numbers. In particular, 3p − q − n ≥ 0 Counterterms of N = 8 supergravity are supersymmet- implies 2L ≥ 6 + 2n − 4k using (17). This bound on the ric, SU(8)-invariant, Lorentz scalar local operators C in- existence of local non-BPS operators was conjectured in tegrated over spacetime. These local operators involve n- [8] and confirmed very recently in [9]. fold products of the fundamental fields and their deriva- As a simple illustration, consider (16) with n = 4 and tives. We restrict to diffeomorphism-covariant combina- p = q. We find C = D2p−4ϕ4, and after application of tions of the fields, such as the Riemann tensor R. In 0 D16 it becomes D2p+4R2R¯2; this is just the 4-point MHV enumerating all local operators of a given order n (up to local counterterm D2p+4R4. Locality of C requires p ≥ covariance), the equations of motion set the Ricci tensor 0 2, so the first available non-BPS operator is D8R4. equal to a combination of fields of quadratic order (and In practice, we use a C++ program to enumerate all lo- higher), which is automatically included at order > n in cal operators with 2L ≤ 30 − n amounting to ∼ 4.8 · 1022 the enumeration. The remaining 10 components of the terms.2 These are decomposed into irreps of SU(2, 2|8) on-shell Riemann tensor group into fields with Lorentz by iteratively removing the lowest weights and their cor- spin (2, 0) and (0, 2). The collection of all on-shell super- responding supermultiplets [23]. Special attention needs gravity fields span a representation of the N = 8 super- to be paid to BPS and short supermultiplets [24]. In to- Poincar´ealgebra as well as an ultrashort representation tal we obtained around 8.8 · 105 types of supermultiplets of N = 8 superconformal symmetry (see [9] for a recent along with their multiplicities.3 Finally we extract su- discussion). Using the SU(2, 2|8) permultiplets with scalar SU(8) singlets and 70’s as top –⊗– – – – – – – –⊗– , (15) supersymmetry components. The results at L ≤ 9 are SU(2)L | {z } SU(2)R SU(8) presented in Table I. 1 1 1 Our analysis shows that there are unique 2 , 4 , 8 the Dynkin labels of this lowest-weight representation BPS counterterms R4, D4R4 and D6R4, in agreement read [0,0,0001000,0,0], where the SU(8) labels [0001000] with earlier results [7–9]. They correspond to the lowest describe a 70 and the SU(2)L × SU(2)R Lorentz spins weights indicate a scalar. The graded symmetric tensor product of n copies of [0,0,0004000,0,0], [0,0,0200020,0,0], [0,0,2000002,0,0] . the above multiplet provide all local operators with n (18) fields. We are interested in supersymmetric operators: In the previous section, we showed that their 6-point ma- there is typically one such operator C in each irrep of trix elements have non-vanishing single-soft limits origi- the tensor product. For long supermultiplets it is the nating from the non-linear completion of the operators. 4 unique top component, obtained by acting with SUSY The limits correspond to local 70 BPS operators ϕR , 16 16 4 4 6 4 1 1 generators Q Q˜ on the lowest-weight component C0. ϕD R and ϕD R , which are descendants of the 2 , 4 , 1 5 (In superspace approaches, this is equivalent to the full 8 BPS superconformal primaries ϕ with SU(8) Dynkin superspace measure R d32θ.) For short or BPS supermul- labels [0005000], [0201020], [2001002]. The relationship tiplets fewer are needed to get from C0 between BPS operators are illustrated in Table I. to the top component(s). Hence it is sufficient to enu- merate the lowest superconformal weights C0. Its super- Explicit matrix elements and superamplitudes conformal transformation properties determine the spin, The matrix elements of potential counterterms such as k SU(8) representation as well as loop and N MHV level. D2kRn must be polynomials of degree δ = 2(k + n) in More concretely, Dynkin labels translate to scalar local angle and square brackets hi ji, [kl] which satisfy several operators as follows (assume q ≥ p) constraints. If ai and si denote the number of angle |ii [0,p,0000000,q,0] → D3p−q−nϕn+p−qRq−p (singlet) , and square |i] spinors for each particle i = 1, 2, . . . , n, then the total number of spinors is fixed by the dimen- [0,p,0001000,q,0] → D3p−q−n+1ϕn+p−qRq−p (70) . sion of the operator to be P (a + s ) = 4(k + n). For (16) i i i each particle i, of helicity h , there is a helicity weight Note that we display only prototypical terms, mixture i constraint a − s = −2h . We need polynomials which with other fields is implied: E.g. D4ϕ2 ' RR¯. (Here i i i are independent under the constraints of momentum con- we distinguish between the chiral and anti-chiral compo- servation and the Schouten identity, nents of the Riemann tensor.) To get from C0 to the su- 16 ˜16 16 X persymmetric C in long multiplets, apply Q Q ' D . hiji[jk] = 0 , hijihkli+hjkihili+hkiihjli = 0 , (19) k A lowest weight as in (16) then corresponds to a N MHV j counterterm at L loops with n 14 + p + q − n (singlet), 2k = n + p − q − 4 , 2L = 15 + p + q − n (70). 2 The computation took 3.5 hours on a desktop PC. (17) 3 The decomposition took 42 hours. 5 and a similar Schouten identity for square brackets. obtain These polynomials must satisfy bose and fermi symme- -14φ 8 4 tries when they contain identical particles. Finally, the lim h++−−ϕϕ¯i(e D R )avg p6→0 polynomials must satisfy SUSY Ward identities. This 2 4 4h 3 X 4 1  X 2  i (20) was ensured in [8] by packaging n-point matrix elements = −2[12] h34i 4 sij + 16 sij . into the manifestly SUSY- and SU(8)-invariant superam- i

3-loop 4-pt 5-pt 6-pt 5-loop 4-pt 5-pt 6-pt 6-loop 4-pt 5-pt 6-pt  2 ¨4  2 44  2 64 R4 ϕ2D2R3 ϕ ¨R D4R4 ϕ2D6R3 ϕ D R D6R4 ϕ2D8R3 ϕ D R singlet  ¨ singlet   singlet   1×MHV R4 non-linear 1×MHV D4R4 non-lin. 1×MHV D6R4 non-lin.

.soft .soft .soft 70 1×ϕR4 70 1×ϕD4R4 70 1×ϕD6R4

7-loop 4-pt 5-pt 6-pt 7-pt 8-pt 9-pt 10-pt 11-pt 12-pt 13-pt 14-pt 15-pt 16-pt 8 4 65 4 6 27 8 227 2 8 427 4 8 627 6 8 827 8 8 singlet D R D R D R D R R ϕ D R ϕ R ϕ D R ϕ R ϕ D R ϕ R ϕ D R ϕ R 1×MHV 2×NMHV 3×N2MHV 4×N3MHV 6×N4MHV 8×N5MHV 10×N6MHV

.soft .soft .soft .soft .soft .soft 8 4 4 6 8 3 8 5 8 7 8 70 ϕD R ϕD R ϕR ϕ R ϕ R ϕ R 2× 4× 6× 9× 14× 19×

8-loop 4-pt 5-pt 6-pt 7-pt 8-pt 9-pt 10-pt 11-pt 12-pt 13-pt 14-pt

10 4 8 5 6 6 4 7 2 8 9 2 2 8 2 9 4 2 8 4 9 6 2 8 singlet D R D R D R D R D R R ϕ D R ϕ R ϕ D R ϕ R ϕ D R 1×MHV 1×MHV 3×NMHV 3×NMHV 8×N2MHV 8×N2MHV 25×N3MHV 22×N3MHV 66×N4MHV 51×N4MHV 153×N5MHV ...... 10 4 8 5 6 6 4 7 2 8 9 3 2 8 3 9 5 2 8 70 ϕD R ϕD R ϕD R ϕD R ϕD R ϕR ϕ D R ϕ R ϕ D R 3× 4× 17× 16× 81× 63× 232× 211× 1033×

9-loop 4-pt 5-pt 6-pt 7-pt 8-pt 9-pt 10-pt 11-pt 12-pt D12R4 D10R5 D8R6 D6R7 D4R8 D2R9 R10 ϕ2D2R9 ϕ2R10 singlet 2×MHV 1×MHV 12×NMHV 14×NMHV 117×N2MHV 123×N2MHV 780×N3MHV 783×N3MHV 4349×N4MHV 2×MHV 7×NMHV 36×N2MHV 169×N3MHV ...... ϕD12R4 ϕD10R5 ϕD8R6 ϕD6R7 ϕD4R8 ϕD2R9 ϕR10 70 5×N0.5MHV 8×N0.5MHV 122×N1.5MHV 194×N1.5MHV 1814×N2.5MHV 2317×N2.5MHV 16485×N3.5MHV 5×N0.5MHV 52×N1.5MHV 469×N2.5MHV

TABLE I. Supersymmetric SU(8)-singlet L-loop counterterms and the SU(8) 70 operators which describe their potential single-soft scalar limits. When the singlet operator is in the NkMHV classification, the single-soft scalar limit operator belongs (k− 1 ) to the N 2 MHV sector. For L < 7, there are no independent singlet operators with n > 4, but the non-vanishing single-soft scalar limits arise from the non-linear completions of the 4-point operators D2kR4. one can choose the basis two contractions given in (23). A similar construction 2 ˜   can be carried out for the three 8-point N MHV super- AD4R6 = δ(Q)δ(Q) (ϕ1, ϕ2)(ϕ3, ϕ4)(ϕ5, ϕ6) + perms , amplitudes of R8. Again, one can immediately propose ˜   8 AD4R60 = δ(Q)δ(Q) (ϕ1, ϕ2, ϕ3, ϕ4, ϕ5, ϕ6) + perms . three superamplitudes that span the space of R coun- (23) terterms, for example by considering three order-8 con- tractions involving the ϕ-products (25) and their 8-scalar Here Q, Q˜ are the usual supercharges that act on the generalizations. Grassmann η-variables of the superamplitude as differ- entiation and multiplication, respectively, and thus The infinite tower of 7-loop counterterms

8 8 We now examine the multiplicity of potential 7-loop Y X ∂2 Y X δ(Q) = [ij] , δ(Q˜) = hijiηiaηja . n-point counterterms [8] ∂ηia∂ηja a=1 i 4 points, the number nS of SSL op- 7-loop operators that are potential counterterms. Opera- erators D16ϕn−1 is always at least as large as the number 16 n tors corresponding to their SSL are also listed in Table I. nC of potential counterterms D ϕ . Generically, one Their construction is similar, and their multiplicities is therefore expects the soft limits of the potential coun- the number of 70’s in a product of (2q−1) 70’s. terterms to span an nC -dimensional subspace in the nS- dimensional space of SSL operators. It would follow that E7(7) violation of higher-point 7-loop operators all potential counterterms with n > 4 violate E7(7). In- deed, this is what we explicitly proved above for the 7- Consider the leading n-point matrix elements of a local loop case. counterterm C. If non-vanishing, the SSL produces a By the same logic, it is not at all surprising that there local (n − 1)-matrix element, which can be generated by is a non-linear supersymmetrization of D8R4 that pre- an (n − 1)-point local operator dC in the 70. Locality serves E at the 6-point level. The number of 5-point of C ensures that the SSL operation C → dC commutes 7(7) SSL operators precisely matches the number of D4R6 op- with the SUSY generators Q and Q˜. For a long-multiplet erators. Therefore, the 6-point soft limit of D8R4 can be (L ≥ 7) counterterm C = Q16Q˜16C we can therefore 0 made to vanish after adding an appropriate combination write of the two D4R6 operators, just as we found above. For 16 ˜16 dC = Q Q dC0. (28) the 8-point soft limits of D8R4, however, there are 4 SSL operators available; more than the 3 potential 8-point E7(7) requires that the SSL vanishes. Now there are two 8 8 4 ways to obtain dC = 0: either the single-soft limit of counterterms R . If the 8-point soft limits of D R take 16 ˜16 a generic value in the 4-dimensional space of SSL op- C0 vanishes (dC0 = 0), or dC0 is annihilated by Q Q . 8 2q erators, no linear combination of R operators can be At the seven-loop level, C0 = ϕ consists of only scalars chosen to give an E7(7)-preserving supersymmetrization with no derivatives, and consequently dC0 6= 0. C0 is also 8 4 16 ˜16 2q−1 of D R ; a remarkable cancellation is thus required for not annihilated by Q Q because dC0 = ϕ in a 70 8 4 satisfies a shortening condition only for n = 2q ≤ 4 [24]. D R to be compatible with E7(7). Therefore, all 7-loop linearized counterterms with n > As Table I illustrates, E7(7) becomes more and more 4 have non-vanishing single-soft scalar limits and thus constraining as we increase the number of points and loops. For example, the 14-point soft limits of D10R4 violate E7(7). (This was also observed in [28]; see [29] for discussion of non-perturbative aspects.) These operators have to lie in a specific 153-dimensional subspace of the 1033-dimensional space of SSL operators in order for may, however, play an important role as dependent terms 10 4 in the non-linear completion of the D8R4 operator, as we D R to satisfy E7(7) after an appropriate addition of demonstrated above at the 6-point level. independent 14-point operators. It follows that E7(7) is a very constraining symmetry even for L = 7 and be- yond. Although there is an infinite tower of independent SSL STRUCTURE: 7-, 8- AND 9-LOOPS counterterms at each of loop L ≥ 7, we cannot expect any of these operators to preserve E7(7) ‘accidentally’. We now show that all our findings on E7(7)-(in)compa- There may, however, be a very good reason for the can- tibility of operators have a natural explanation in terms cellations of terms that is needed for E7(7)-invariant op- of the multiplicities of SSL operators in the 70 that is dis- erators to exist for L ≥ 7; namely, when there is a con- 4 4 4 played in Table I. Let us first revisit the case of D R struction of a manifestly E7(7)-invariant supersymmetric and D6R4. At the 5- and 6-loop level, there are no poten- operator [30, 31]. At the 7-loop level, for example, we can tial 3-point or 4-point SSL operators available. Therefore only expect the 8-point single-soft limits of D8R4 to van- the matrix elements of D4R4 and D6R4 must have van- ish after an appropriate addition of R8, if the manifestly ishing soft limits at 4- and 5-points, and this is indeed E7(7)-invariant superspace integral that was proposed as the case. There exists, however, one potential 5-point a candidate counterterm in [30] is indeed non-vanishing. SSL operator at L = 5 and L = 6. Generically, one One new feature that emerges at L = 8, 9 is the exis- tence of n > 4 operators that have vanishing soft-limits at the linearized level. This holds for the MHV operators D8R5, D10R5 and 2 × D8R6 as well as for at least 7 of 4 Throughout this section we are only concerned with the lowest- the 12×D8R6 NMHV operators. The latter follows from point non-vanishing SSL’s of an operator. If an operator has non-vanishing SSL’s at n-point, its higher-point matrix elements the multiplicity 5 of 5-point SSL operators at 9 loops. will generically have non-local SSL’s, which are not classified by E7(7)-invariance beyond the linearized level, however, is our analysis. a highly non-trivial constraint on all of these operators. 8

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