From U(1) to E8: soft theorems in supergravity amplitudes
aa r X i v : . [ h e p - t h ] S e p Prepared for submission to JHEP
ROM2F/2014/09
From U(1) to E : soft theorems in supergravityamplitudes Wei-Ming Chen a Yu-tin Huang a,b
Congkao Wen c a Department of Physics and Astronomy, National Taiwan University, Taipei 10617, Taiwan, ROC b School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA c Dipartimento di Fisica, Universit`a di Roma “Tor Vergata” & I.N.F.N. Sezione di Roma “TorVergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
E-mail: [email protected],[email protected],[email protected]
Abstract:
It is known that for N = 8 supergravity, the double-soft-scalar limit of an n -point amplitude is given by a sum of local SU(8) rotations acting on an ( n − N < N ) isotropy group, which introduces a soft-graviton singularity that obscures theaction of the duality symmetry. In this paper, we introduce an anti-symmetrised extractionprocedure that exposes the full duality group. We illustrate this procedure for tree-levelamplitudes in 4 ≤ N < N = 16 supergravityin three dimensions. In three dimensions, as all bosonic degrees of freedom transform underthe E duality group, supersymmetry ensures that the amplitude vanishes in the single-softlimit of all particle species, in contrast to its higher dimensional siblings. Using recursiveformulas and generalized unitarity cuts in three dimensions, we demonstrate the action ofthe duality group for any tree-level and one-loop amplitudes. Finally we discuss the impli-cations of the duality symmetry on possible counter terms for this theory. As a preliminaryapplication, we show that the vanishing of single-soft limits of arbitrary component fieldsin three-dimensional supergravity rules out the direct dimensional reduction of D R as avalid counter term. ontents N ≤ Supergravity 4 ≤ N < N = 16 supergravity 113.2 The double-soft limit: Tree level 133.3 The double-soft limit: One Loop 18 N = 16 SUGRA 21
Scattering amplitudes often exhibit universal behaviors in the limit when the momenta ofsome external particles approach to zero, i.e. so-called soft limit. For instance, it is wellknown that amplitudes in gauge theories (and gravity) behave universally in the singlesoft gluon (and graviton) limit, which goes back to the classical work by Weinberg [1]. Inparticular, the analytic behavior of this limit at tree-level is completely determined by thegauge symmetries of the theory [2–4]. Another famous and well-studied case of soft limit, which will be of our interest in thispaper, is the soft-pion theorem. The theorem states that the Goldstone boson decouplesat zero momentum, i.e. the amplitude of one soft “pion” with arbitrary number of hard“pions” vanishes [7]. The full algebra of the symmetry can be exposed by consideringthe limit where two Goldstone bosons become soft [8]. This idea of probing the globalsymmetries of the theory by studying the single- and double-soft scalar limits was revis-ited and applied to N = 8 supergravity theory in four dimensions by Arkani-Hamed etal [10]. It is known that the theory contains 70 scalars, which are elements in the cosetspace E /SU(8), thus according to the soft-pion theorem, the amplitudes vanish in thesingle-soft-scalar limit, which is indeed the case as shown in [10]. The authors of ref. [10] For the understanding on soft behaviors from other symmetry principles see [5, 6] – 1 –hen beautifully showed that any n -point amplitude in the double-soft-scalar limit has thefollowing universal behavior: M n (cid:0) φ II I I ( ǫ p ) , φ JI I I ( ǫ p ) , , · · · , n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ǫ → = 12 n X a =3 p a · ( p − p ) p a · ( p + p ) ( R a ) I J M n − + O ( ǫ ) , (1.1)where the superscripts in scalar field φ are the SU(8) R-symmetry indices, and ( R a ) I J isthe corresponding SU(8) rotation. It might be a surprise that amplitudes vanish in thesingle-soft limit, but finite in the double-soft limit. As explained in ref. [10], which we willgive a brief review in the next section, this is a reflection of the fact that the commutatorsof the broken generators do not vanish.For 4 ≤ N < N = 8 theory. There is one caveat however, in that for N <
8, the isotropy group (the H of coset G/H ) is U( N ) which includes a U(1). In orderto generate this U(1) factor, the scalars chosen for the double-soft limit form an SU( N )singlet, which is known to be polluted by the singularity from an internal soft graviton.In this paper, to extract the U(1) part of the duality group and subtract the singularity,we take the double-soft limit in a manifest anti-symmetric fashion with respect to the twoscalars. More precisely, we consider the difference of two distinct amplitudes, one with the( φ T , φ ¯ T ) scalars carrying momenta ( p , p ), the other with φ T and φ ¯ T exchanged. We willshow that( N = 4) (cid:20) M n (cid:0) φ ( ǫ p ) , ¯ φ ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ ( ǫ p ) , φ ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → = n X a =3 p a · ( p − p )2 p a · ( p + p ) ( R a ) M n − + O ( ǫ ) , ( N = 5) (cid:20) M n (cid:0) φ I ( ǫ p ) , ¯ φ I ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ I ( ǫ p ) , φ I ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → = n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:18) ( R a ) II + N − N δ II R a (cid:19) M n − + O ( ǫ ) , ( N = 6) (cid:20) M n (cid:0) φ IJ ( ǫ p ) , ¯ φ IJ ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ IJ ( ǫ p ) , φ IJ ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → = n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:18) ( R a ) II δ JJ + ( R a ) JJ δ II + N − N δ IJIJ R a (cid:19) M n − + O ( ǫ ) , (1.2)where R a is the single site U(1) generator and ( R a ) I I is the diagonal component of theSU( N ) generator ( R a ) I J ≡ η Ia ∂∂η Ja − δ IJ N P N l =1 η la ∂∂η la . We will refer to such extraction of thedouble-soft limit as “anti-symmetrized extraction”. Note that due to the fact that we areconsidering non-maximal supergravity theories, the on-shell degrees of freedom are carriedby two distinct multiplets (Φ N , Φ N ). As a result, the U(1) generator R a has a differentconstant for the two distinct multiplets R a = X I η Ia ∂∂η Ia ( a ∈ Φ N ) , R a = X I η Ia ∂∂η Ia − N ( a ∈ Φ N ) . (1.3)– 2 –e also consider maximal supergravity in three dimensions, which is the N = 16theory introduced by Marcus and Schwarz [11]. The 128 bosonic states now parametrizethe coset E /SO(16). We use the three-dimensional recursion formulas [12], to derivethe double-soft-scalar limit for all multiplicity tree-level amplitudes. Since the on-shellsuperspace only manifests U(8) ∈ SO(16), the other part of the SO(16) generators arenon-linearly realized. Thus using the double-soft limit allows us to construct the algebraof E in such non-linear realization. Note that the presence of a U(1) again requires usto apply the anti-symmetrized extraction procedure discussed above. We also consider thefate of the duality at loop-level. We demonstrate that at one loop, in the scalar integralbasis representation, the integral coefficients are given in such a way that the double-softbehavior is manifest.One of the many important questions one can ask for a gravitational S-matrix is itsultraviolet behavior. In recent years tremendous progress in computation techniques hasallowed us to peer ever deeper into perturbative gravitational S-matrix. Remarkably, ex-plicit computations [13–15] have reveal surprising finiteness in a wide range of supergravitytheories with 4 ≤ N ≤
8. Although from the viewpoint of four-dimensional divergences,some results can be explained by the constraints imposed by the symmetries of the cosetspace [16, 17], there are examples where finiteness requires explanations that go beyondthat explained by traditional symmetry arguments [14, 15, 18].If four-dimensional maximal supergravity is finite, then so must its three-dimensionalreduction. Unlike in four dimensions, here all bosonic degrees of freedom transform un-der the duality group, which implies that coset symmetry imposes stronger constraints oncandidate ultraviolet (UV) counter terms. Furthermore, as we will demonstrate, supersym-metric Ward-identities require that amplitudes vanish as well in the fermionic single-softlimits. Thus one can ask whether or not candidate UV counter terms can produce matrixelements satisfying all single- and double-soft behaviors required by the symmetries. As apreliminary step, we consider the direct dimensional reduction of matrix elements of D n R counter terms in four dimensions. We will explicitly show that these matrix elements,which satisfy the E duality symmetry in four dimensions [17], do not have the correctsingle-soft behavior in three dimensions.This paper is organized as following: In the next section, we study the double-soft-scalar limit for four-dimensional N = 4 , , N = 8 theory via supersymmetry reduction. However unlike their ances-tor, the isotropy group of the duality symmetries for these non-maximal supersymmetrictheories contain a U(1) factor. To extract this subtle contribution, we introduce a pro-cedure “anti-symmetrised extraction”, which allows us to throw away unwanted singularparts, and leave behind a beautiful and finite result, corresponding precisely to the U(1)factor. In section 3, we then move on to study N = 16 supergravity in three dimensions,both at tree and loop level. At tree level, we study the soft limits using BCFW recursionrelations in three dimensions, and the same “anti-symmetrised extraction” procedure intro-duced previously is used to extract the U(1) factor in the symmetry group. After derivingthe soft theorems for tree-level amplitudes, we study the possible loop corrections to thetheorems. Using generalized unitarity cuts, we show that all one-loop amplitudes satisfy– 3 –xactly the same soft theorems as the tree-level one. In section 4, we discuss the applicationof duality symmetry to constrain candidate counter terms for N = 16 Supergravity. Weshow that S-matrix generated by many counter terms descendant from four-dimensionalones via direct dimensional reduction do not satisfy the single-soft-scalar theorems. Finallyin section 5, we finish the paper with conclusions and remarks. N ≤ Supergravity
Massless scalars that can be identified as goldstone bosons of spontaneous broken symmetry,exhibit simple behavior in the soft limits. For theories that involve these massless scalars,in the limit where the momentum of one of these scalars becomes soft the correspondingamplitude vanishes, a result that is famously known as “Adler’s zero” [7]. Consider thecoset space
G/H , where the generators of the isometry group G are represented by ( T i , H j ),and H i ’s are the elements of the isotropy group. Schematically they satisfy the followingcommutation relations: [ T, T ] ∼ H , [ T, H ] ∼ T , [ H, H ] ∼ H . (2.1)Since the vacuum expectation values (vev) of scalars spontaneously break the symmetry,thus they can be identified with parameters of the broken generators T i . The vanishing ofthe soft-scalar limit can be understood through the fact that for the non-linear sigma model,which is the effective action for the goldstone bosons, scalar interactions are constructedout of covariant derivatives P µ = ( e ϕ ∂ µ e − ϕ − e − ϕ ∂ µ e ϕ ) , (2.2)where ϕ = φ i T i . Since the scalars are dressed with derivatives, taking the momentum softresults in the vanishing of the amplitude.In [10], the soft-scalar limits were discussed without relying on any detailed structure ofthe interactions. Starting with the fact that spontaneous symmetry breaking is a reflectionof the presence of continuous set of degenerate vacua, perturbative amplitudes computedat different points on this moduli space must be equivalent. As two different points in themoduli space are connected via the generators T i , we can schematically write | θ + ∆ θ i = e i ∆ θ · T | θ i , (2.3)where θ represents the vev of the scalar, which parametrizes the vacuum. Assuming thateach point in the moduli space can be connected in such fashion, the fact that amplitudescomputed in distinct vacua must agree implies that as one expands the exponent in eq.(2.3),terms beyond the leading term in the expansion must vanish. Since ∆ θ simply correspondsto a constant scalar, i.e. scalars with zero momenta, this leads to the conclusion thatamplitudes with any additional soft scalar must vanish.However, as discussed in [10], the above analysis is not entirely correct. The subtletylies in the assumption that there is a well-defined path that connects two points. Indeed,– 4 –ne would expect that the difference between two different paths should be proportionalto an H generator, since [ T, T ] ∼ H . In [10] it was argued that this ambiguity leads tothe result that in the double-soft-scalar limit, the amplitude is non-vanishing, and behaveuniversally: M n (cid:0) φ i ( ǫ p ) , φ j ( ǫ p ) , · · · , n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ǫ → = 12 n X a =3 p a · ( p − p ) p a · ( p + p ) [ T i , T j ] a M n − . (2.4)For N = 8 supergravity, whose 70 scalars parametrized E /SU(8) coset, this becomes M n (cid:0) φ II I I ( ǫ p ) , φ JI I I ( ǫ p ) , , · · · , n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ǫ → = 12 n X a =3 p a · ( p − p ) p a · ( p + p ) ( R a ) I J M n − , (2.5)where ( R a ) I J = η Ia ∂∂η Ja is the single-site SU(8) R-symmetry generator.For N < N = 6 , N = 8 theory via SUSY reduction.This would indeed be the case if not for the subtle difference between the isotropy groupfor the N < N ) singlet, which inducessingularities in the limit. More precisely, the duality group algebra now involves relationsof the form: [ T, ¯ T ] ∼ U (1)where T and ¯ T have opposite charges under the U(1). This implies that the double-soft-limit for such scalars will involve Feynman diagrams where the two-scalars merge into agraviton: ¯ TT . As the graviton is soft, the amplitude is then proportional to the soft-graviton limit of an( n − φ T , φ ¯ T ) scalars carrying momenta( p , p ), the other with φ T and φ ¯ T exchanged. For N = 4 , , N = 4) (cid:20) M n (cid:0) φ ( ǫ p ) , ¯ φ ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ ( ǫ p ) , φ ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → , ( N = 5) (cid:20) M n (cid:0) φ I ( ǫ p ) , ¯ φ I ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ I ( ǫ p ) , φ I ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → , ( N = 6) (cid:20) M n (cid:0) φ IJ ( ǫ p ) , ¯ φ IJ ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ IJ ( ǫ p ) , φ IJ ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → , where the pairs ( φ, ¯ φ ), ( φ I , ¯ φ I ), and ( φ IJ , ¯ φ IJ ) indicate the SU( N ) singlet combination ofthe 2, 10 and 30 scalars in N = 4 , N = Φ N =8 | η , ··· ,η N +1 → , Φ N = Z dη · · · dη N +1 Φ N =8 , (2.6)where Φ N =8 is the unique superfield for the N = 8 theory. As a consequence, the U(1)-generator R a has a different constant for the two distinct multiplets R a = X I η Ia ∂∂η Ia ( a ∈ Φ N ) , ¯ R a = X I η Ia ∂∂η Ia − N ( a ∈ Φ N ) . (2.7)One can verify that all tree amplitudes vanish under the above refined U(1) generator, i.e X a ∈ Φ N R a + X b ∈ Φ N ¯ R b M n = 0 . (2.8) ≤ N < Supergravity
Let us now demonstrate the validity of eq.(1.2) for 4 ≤ N < N = 8 supergravity and perform SUSY reduction. Toguarantee the presence of a U(1) on the right-hand side of [ T, T ] ∼ H , we choose twoscalars from the N = 8 theory that form a singlet, for example:( φ , φ ) . In terms of N = 4 , , φ, ¯ φ ),( φ , ¯ φ ) and ( φ , ¯ φ ) respectively. The double-soft limits of scalar pairs that do notcontain such singlet contribution can be derived similarly without the complication of soft-graviton divergence. We will simply present the final result for these cases.We begin by considering the double-soft limit of the following two amplitudes:( a ) (cid:18)Z d η d η M N =8 n (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p → ǫ p p → ǫ p . ( b ) (cid:18)Z d η d η M N =8 n (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p → ǫ p p → ǫ p . (2.9)– 6 –or both cases, the double-soft limits are divergent due to the presence of the soft-gravitonpole. To extract the finite term we consider the difference M n (cid:0) φ (1) φ (2) · · · (cid:1) − M n (cid:0) φ (1) φ (2) · · · (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) p ,p → ǫ p ,ǫ p . (2.10)This anti-symmetrised extraction procedure will allow us to isolate the finger print of theU(1) duality group.We begin with case ( a ) in eq.(2.9). The BCFW shift is given by | ˆ1 i = | i + z | n i , | ˆ n ] = | n ] − z | , η ˆ n = η n − zη . We will take the soft limit by setting λ , → ǫλ , , ˜ λ , → ǫ ˜ λ , . Since the amplitudesvanish in the single-soft scalar limit, legs 1 and 2 must be on the same subamplitude of theBCFW diagram. However, since leg 1 is shifted, plugging the explicit solution for z in thegeneric multiplicity will render the momentum of leg 1 hard. In this case, the subamplitudeis again in a single-soft scalar limit, and thus vanishes. The only exception is when bothlegs 1 and 2 are on a three- or a four-point amplitude. For these diagrams, the propagatorsvanish in the limit, and one can potentially encounter 0 / • BCFW diagram with a 4-point subamplitude
We first begin with the latter and consider the following BCFW diagram: a ˆ12 ˆ nP . The contribution of this diagram is given as Z d η P M (ˆ1 , , P, a ) 1 p , ,a M n − ( − P, · · · , ˆ n )= s a h ˆ1 P i h ˆ1 a ih a ih P ih P ˆ1 ih ˆ12 ih a ih aP ih P ˆ1 i p a Z d η P δ A δ B M n − ( − P, · · · , ˆ n ) . (2.11)The explicit solution to the shifted variable z is given by z p = − ǫ p a · ( p + p ) h n | a |
1] + O ( ǫ ) . Since z P is of order ǫ , | ˆ1 i ∼ O ( ǫ ), thus the deformed p ˆ1 is still soft in this channel. Thespinors for the internal momentum is normalized as | P i ∼ p a | a and [ P | ∼ − h n | p a h na i . The– 7 –ermonic delta-functions are given as δ A := δ η P + ǫ h ˆ12 ih ˆ1 P i η + h ˆ1 a ih ˆ1 P i η a ! , δ B := δ (cid:18) η + h P ih P ˆ1 i η + ǫ h P a ih P ˆ1 i η a (cid:19) . (2.12)For convenience, we have explicitly written out the ǫ dependence.It is straightforward to see that in the double-soft limit the bosonic pre-factor ineq.(2.11) is of order ǫ − . We can use δ A to localize the dη P integral and the net effect is η − P in M n − is replaced by ( ǫ h ˆ12 ih ˆ1 P i η + η a ). Thus the integrand in eq.(2.11) can be writtenas s a h ˆ1 P i h ˆ1 a ih a ih P ih P ˆ1 ih ˆ12 ih a ih aP ih P ˆ1 i p a × exp − ǫ h ˆ12 ih ˆ1 P i η ∂∂η a ! exp (cid:18) − ǫz P η ∂∂η n (cid:19) exp (cid:18) − ǫ z P ˜ λ ∂∂ ˜ λ n (cid:19) M n − , (2.13)where M n − in the last line is now the unshifted ( n − Z d η d η δ B exp − ǫ h ˆ12 ih ˆ1 P i η ∂∂η a ! exp (cid:18) − ǫz P η ∂∂η n (cid:19) exp (cid:18) − ǫ z P ˜ λ ∂∂ ˜ λ n (cid:19) M n − . (2.14)If all four η ’s and four η ’s came from δ B , we obtain the singlet contribution, which isdivergent as 1 /ǫ . It turns out that the leading divergent term as well as the subleadingcontribution are the same for both (a) and (b) in eq.(2.9), and thus cancel under the anti-symmetrized extraction. To get a non-vanishing result, one must pull down one factor of η from the exponent. This will result in finite contributions, as it brings down a factor of ǫ , along with the remaining ǫ factor associated with η a in δ B . Thus for finite contributionwe can either pull down an η or an η from the exponent.Let’s begin with taking an η from the exponent. This means that δ B contributes 4 η ’s, 3 η ’s and left with an η a unintegrated. What we then get is − s a h ˆ1 P i h ˆ1 a ih a ih P ih P ˆ1 ih ˆ12 ih a ih aP ih P ˆ1 i p a · ( p + p ) (cid:18) h P ih P ˆ1 i (cid:19) h P a ih P ˆ1 i h ˆ12 ih ˆ1 P i X I =5 η Ia ∂∂η Ia M n − = p a · p p a · ( p + p ) X I =5 η Ia ∂∂η Ia M n − . (2.15)Next, we consider bringing down an η instead. In this case, δ B contributes 3 η ’s and 4 η ’s, and leaves an η a unintegrated. Using the explicit form of z p in eq.(2.12) we have − s a h ˆ1 P i h ˆ1 a ih a ih P ih P ˆ1 ih ˆ12 ih a ih aP ih P ˆ1 i h n | p a | (cid:18) h P ih P ˆ1 i (cid:19) h P a ih P ˆ1 i X I =1 η Ia ∂∂η In M n − = [ a h ˆ12 i h a i h n | p a | X I =1 η Ia ∂∂η In M n − . (2.16)– 8 – ˆ nP ˆ nP −→ (a) (b) Figure 1 . (a) The BCFW diagram where the two soft legs are attached to a three-point subamplitude. (b) Due to the soft kinematics, the diagram factorizes into a three-point times an n − Using the explicit representation for | ˆ1 i , the above can be written as[ a h ˆ12 i h a i h n | p a | X I =1 η Ia ∂∂η In M n − = [ a h a ih n | | a ] X I =1 η Ia ∂∂η In M n − . (2.17) • BCFW diagram with a 3-point subamplitude
Let us now consider the following BCFW diagram with a 3-point subamplitude. Therelevant diagram is displayed in diagram (a) of fig.1, which yields M (ˆ1 , , P ) 1 p M n − ( − P, . . . , ˆ n ) (2.18)= δ A ([12] η P + [2 P ] η + [ P η )[12] [2 P ] [ P s exp (cid:18) − ǫz P η ∂∂η n (cid:19) exp (cid:18) − ǫ z P ˜ λ ∂∂ ˜ λ n (cid:19) M n − ( P, . . . , n ) . The internal momentum and the solution for z P is given as | P i = ǫ | i , [ P | = − ǫ (cid:18) [2 | + h n ih n i [1 | (cid:19) , z P = − ǫ h ih n i . (2.19)Again, we have explicitly written out the ǫ dependence for the exponents. A new featureis that the ( n − P is soft, the RHS of the diagram is an ( n − M m +1 is given by [1, 5]: M m +1 (1 , · · · , m, ǫ s ) = 1 ǫ S (0) G M m (1 , · · · , n ) + S (1) G M m (1 , · · · , n ) . (2.20)As the three-point amplitude behaves as ǫ while the propagator as ǫ − , the only term weneed to consider is S (0) G . The explicit supersymmetric single-soft operator of an ( n − M n − ( P, , . . . , n ) → ǫ n X a =3 h aP i [ na ] [ nP ] [ aP ] δ B (cid:18) η P + ǫ [ nP ][ an ] η a + ǫ [ P a ][ an ] η n (cid:19) M n − (3 , . . . , n ) , – 9 –here we only keep the relevant leading term.Similar to the previous analysis, in order for there to be a non-vanishing result after theanti-symmetrised extraction, we must take one of the η ’s from the exponents. In eq.(2.18)we can only choose η and thus δ A contributes 3 η ’s and 4 η ’s. Again using δ B to localise η P , the result is:[12][2 P ] [ P [12] [2 P ] [ P z P s n − X a =3 h aP i [ na ] [ nP ] [ P a ] X I =1 (cid:18) [ nP ][ an ] η Ia + [ P a ][ an ] η In (cid:19) ∂∂η In M n − (3 , . . . , n ) . (2.21)Using eq.(2.19) the term above can be rewritten as, to leading order, − X I =1 n − X a =3 (cid:20) h n ih a i [ na ][12] h n | ( p + p ) | a ] p n · ( p + p ) η Ia − h n i [ na ][12] h a i [ p n · ( p + p )] η In (cid:21) ∂∂η In M n − (3 , . . . , n ) . (2.22)Note that the second term in the soft limit vanishes due to the momentum conservation,and the first term can be combined with the previous BCFW result in eq.(2.17) as n − X a =3 (cid:20) h a i [ a h n | ( p + p ) | a ] − h n ih a i [ na ][12] h n | ( p + p ) | a ] p n · ( p + p ) (cid:21) X I =1 η Ia ∂∂η In M n − (3 , . . . , n )= n − X a =3 h a i [ a p n · ( p + p ) − h n ih a i [ na ][12] h n | ( p + p ) | a ] p n · ( p + p ) X I =1 η Ia ∂∂η In M n − (3 , . . . , n ) . (2.23)Using the n − X I =1 n − X a =3 [ n h a i ( p + p ) · p n η Ia ∂∂η In M n − (3 , . . . , n ) = − X I =1 p n · p ( p + p ) · p n η In ∂∂η In M n − (3 , . . . , n ) , (2.24)Put everything together, we find that the difference for scenario (a) and (b) in eq.(2.9) isgiven by M n (cid:0) φ (1) φ (2) · · · (cid:1) − M n (cid:0) φ (1) φ (2) · · · (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) p ,p → ǫ p ,ǫ p = (cid:20) n − X a =3 p a · p p a · ( p + p ) X I =5 η Ia ∂∂η Ia − p n · p p n · ( p + p ) X I =1 η In ∂∂η In ! − n − X a =3 p a · p p a · ( p + p ) X I =1 η Ia ∂∂η Ia − p n · p p n · ( p + p ) X I =5 η In ∂∂η In ! (cid:21) M n − = (cid:20) n X a =3 p a · p p a · ( p + p ) X I =5 η Ia ∂∂η Ia − X I =1 η Ia ∂∂η Ia ! (cid:21) M n − . (2.25)Thus we see that after anti-symmetrized extraction, the double-soft limit results in single-site U(1)-generators acting on a lower-point amplitude.– 10 –e now perform the SUSY reduction to N <
8. In the reduction, for each leg one needsto choose between integrating away dη N +1 · · · dη to obtain the Φ multiplet, or setting all η N +1 · · · η s to be zero for the Φ multiplet. Denote the ( n −
2) points in two sets, with α ∈ Φ and β ∈ Φ. For the legs in α , integrating dη N +1 · · · dη will leave behind:( α ) : − p a · p p a · ( p + p ) X I =1 η Ia ∂∂η Ia − N X J =5 η Ja ∂∂η Ja − N ! Z dη N +1 · · · dη M n − , (2.26)where we’ve used the identity R dη η ∂∂η ∗ = R dη ∗ . On the other hand for the legs in β , wehave:( β ) : " − p a · p p a · ( p + p ) X I =1 η Ia ∂∂η Ia − N X I =5 η Ia ∂∂η Ia ! M n − η N +1 ··· η → = − p a · p p a · ( p + p ) X I =1 η Ia ∂∂η Ia − N X I =5 η Ia ∂∂η Ia ! [ M n − ] | η N +1 ··· η → . (2.27)The above result is precisely eq.(1.2). To see this, recall that( R a ) I I = η Ia ∂∂η Ia − δ II N N X J =1 η Ja ∂∂η Ja ! , (2.28)where the repeated indices are not summed over. Combined with the definition of the U(1)generators in eq.(2.7), we can see that eq.(2.26, 2.27) are simply:( N = 4) (cid:20) M n (cid:0) φ ( ǫ p ) , ¯ φ ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ ( ǫ p ) , φ ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → = n X a =3 p a · ( p − p )2 p a · ( p + p ) ( R a ) M n − + O ( ǫ ) , (2.29)( N = 5) (cid:20) M n (cid:0) φ I ( ǫ p ) , ¯ φ I ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ I ( ǫ p ) , φ I ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → = n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:18) ( R a ) II + N − N δ II R a (cid:19) M n − + O ( ǫ ) , ( N = 6) (cid:20) M n (cid:0) φ IJ ( ǫ p ) , ¯ φ IJ ( ǫ p ) , · · · , n (cid:1) − M n (cid:0) ¯ φ IJ ( ǫ p ) , φ IJ ( ǫ p ) , · · · , n (cid:1) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ǫ → = n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:18) ( R a ) II δ JJ + ( R a ) JJ δ II + N − N δ IJIJ R a (cid:19) M n − + O ( ǫ ) , where M n − s in each line correspond to the amplitudes in N = 4 , , N = 16 supergravity In three dimensions, the graviton does not have physical degrees of freedom. If one di-mensionally reduces four-dimensional Einstein-Hilbert gravity, the two physical degrees of– 11 –reedom become scalars in three dimensions. Thus the gravity amplitudes under consid-eration correspond to the scattering of scalars, and their supersymmetric partners, withtheir interactions mediated by gravitons. Thus in a sense the system is very much likeChern-Simons matter theories, where the physical matter fields interact through a topo-logical gauge field, and both systems can be considered as a perturbation of a topologicaltheory. Like their higher dimensional parents, the scalars in the supersymmetric theoriesare coordinates of a coset manifold, and the purely scalar part of the action is given bya non-linear sigma model, i.e. three-dimensional supergravity theories are really local su-persymmetric non-linear sigma models. Unlike their higher-dimensional counter part, inthree-dimensions, all bosonic degrees of freedom are governed by this coset structure.The theory we will discuss here is the N = 16 theory constructed by Marcus andSchwarz [11]. 128 scalars and 128 fermions in the theory transform under inequivalentspinor representation of SO(16) R-symmetry. Due to the fact that the physical degrees offreedom are in the spinor representation, they have to come in pairs to form a singlet, andhence only even-multiplicity S-matrix is non-trivial. The scalars parametrize the coset spaceE /SO(16), where the extra 8 in E denotes that the difference between non-compactand compact generators is 8.The on-shell degrees of freedom are encoded in a superfield that is a function of η I where I = 1, 2, . . . , 8 transforms as the fundamental representation of U(8). In thislanguage, the rest 56 of 120 SO(16) generators are non-linearly realized:( ) : X a R IJa ≡ X a η Ia η Ja , ( ) : X a ( R a ) IJ ≡ X a ∂∂η Ia ∂∂η Ja , (3.1)where the numbers in the parenthesis denotes the numbers of generators. For completeness,the linearly realized U(8)=SU(8) × U(1) is given as:( ) : X a ( R a ) I J ≡ X a " η Ia ∂∂η Ja − δ IJ X K η Ka ∂∂η Ka ! , ( ) : X a R a ≡ X a X K η Ka ∂∂η Ka − ! . (3.2)The 128 ⊕
128 states are grouped in a SU(8) singlet scalar super field:Φ = ξ + 12 ξ IJ ( η ) IJ + 14! ξ IJKL ( η ) IJKL + 16! ξ IJKLMN ( η ) IJKLMN + ¯ ξη , (3.3)where we have only denoted the scalar states. Since all scalars participate in the dualitygroup, we anticipate a much richer double-soft structure. In particular we expect:[ ξ I I , ¯ ξ ] s M n ∼ − R I I M n − , [ ξ I I I I I I , ξ J J J J ] s M n ∼ ǫ I I I I I I [ J J R J J ] M n − , [ ξ I I , ξ J J J J ] s M n ∼ ǫ I I J J J J K K R K K M n − , [ ξ I I I I I I , ξ ] s M n ∼ ǫ I I I I I I K K R K K M n − , [ ξ I I I I I I , ξ J J ] s M n ∼ (cid:0) ǫ I I I I I I J [ J R J J ] + ǫ I I I I I I J J R (cid:1) M n − , [ ξ I I I I , ξ J J J J ] s M n ∼ (cid:0) ǫ I I I I J [ J J J R J J ] (cid:1) M n − , (cid:2) ξ , ¯ ξ (cid:3) s M n ∼ RM n − , (3.4)– 12 –here [ , ] s indicates we are taking the antisymmetrised double-soft limit on the amplitudeswith respect to the two scalars indicated in the brackets. As we will show in the following,the double-soft-limit of the tree and one-loop amplitudes indeed behave in the above fashionwith the proportionality factor given by p a · ( p − p ) / p a · ( p + p ) .In the following, we will utilise the BCFW recursion, which has been applied to studyscattering amplitudes in three-dimensional ABJM theory [12]. The applicability of BCFWfor the N = 16 supergravity can be argued by comparing with its four-dimensional parent.Note that in the large- z limit of three-dimensional BCFW correspond to boosting themomenta of the shifted legs along a null direction, with a proportionality given by z ,exactly the same as its four-dimensional parent. Since three-dimensional kinematics issimply a special limit, the large- z asymptotics can be deduced from four dimensions andone concludes that maximal supergravity behaves as z − asymptotically. As we have seen from the previous section, the vacuum structure of supergravity theoriesin four dimensions can be explored by analysing the scattering amplitudes in the double-soft-scalar limit. As a reflection of the coset space structure, an n -point amplitude in thelimit reduces to a rotation operator of R-symmetry acting on an ( n − N = 16 supergrav-ity in three dimensions. As we mentioned earlier, due to the presence of a U(1) in theisotropy group, we will again have divergent contributions from singlet, which requires theantisymmetric extraction procedure introduced previously.Again BCFW recursion relations, now in three dimensions, are the main tool for ouranalysis. We denote the BCFW shifts in 3D as [12],ˆ λ = cλ + sλ n , ˆ λ n = − sλ + cλ n , ˆ η = cη + sη n , ˆ η n = − sη + cη n , (3.5)where c + s = 1 required by the momentum conservation. It is convenient to solve theorthogonal constraint by introducing a parameter z , with c = 12 ( z + 1 /z ) , s = i z − /z ) , (3.6)or z = c − is . Moreover, c ( z ) and s ( z ) are fully determined by the on-shell condition forinternal momentum P in a BCFW diagram. There are four sets of solutions, denoted as z j with j = 1 , , ,
4, but only two of them are linearly independent, the other two arerelated by an overall sign. As a consequence of the fact that there is a single bosonicsuperfield, which implies the factorization of amplitudes A m ( z ) A n ( z ) = A m ( − z ) A n ( − z ), This is also consistent with the observation in [23] that permutation invariance implies that at large- z ,the amplitude behaves as z k . Since for the maximal theory all degrees of freedom are in the same multiplet,permutation invariance is satisfied. In three dimensions the little group is Z , and it acts on the on-shell variable as λ → − λ . Thus thereare only two types of particles, bosons which is a Z singlet and fermions which are Z odd. The N = 16superfield is a bosonic superfield. – 13 –nd the linear dependence of the four set of solutions, it allows us to express the BCFWrepresentation of amplitudes purely in terms of two out of four sets of solutions: A n = X f Z d N η I A L ( z ,f ; η I ) H ( z ,f , z ,f ) P ...i A R ( z ,f ; iη I ) + ( z ,f ↔ z ,f ) , (3.7)where H ( x, y ) ≡ x ( y − x − y . (3.8)Before starting our investigation on the double-soft limit, we like to show that the am-plitudes vanish in the single-soft limit, as the consequence of soft “pion” theorem. This factcan be seen most easily by BCFW recursion relations. First of all, the totally permutationsymmetric four-point amplitude is given as [24], M (1 , , ,
4) = δ ( P i λ i η i ) δ ( P i p i ) h i h i h i . (3.9)Expanding δ ( P i λ i η i ) out, it is straightforward to see that the amplitude vanishes as ǫ atthe single-soft limit, say λ → ǫλ . For a general higher-point amplitude, we can representthe amplitude by BCFW recursion relations with two shifted legs not involving the softparticle. Recursively apply the recursion relation, one can always reduce the amplitudeinto four-point ones, which we have just proved behaving as ǫ , while the propagator inthe BCFW diagram is always finite. Thus we conclude that any amplitudes in N = 16supergravity vanish in the single-soft limit as ǫ in the limit. We will comment on thesingle-soft limit with more details in the next section.Let us now consider the double-soft limit. Because the amplitudes in the single-soft-scalar limit vanish, and only even-point amplitudes are allowed in the theory, the onlyrelevant BCFW diagram is a four-point amplitude with two soft legs glued with an ( n − M n (cid:12)(cid:12)(cid:12) λ → λ → = n − X a =3 Z d η P M (ˆ1 , a, , P ; z ) H ( z , z )( p + p + p a ) M n − ( − P, . . . , ˆ n ; z ) + ( z ↔ z ) , (3.10)where two soft legs are chosen to be 1 and 2. Substituting the four-point amplitude intoeq.(3.10), one gets M n (cid:12)(cid:12)(cid:12) λ → λ → = n − X a =3 H ( z , z ) Z d η P δ ρ ( z ) δ σ ( z ) M n − ( − P, · · · , ˆ n ; z ) + ( z ↔ z ) , (3.11)where δ ρ ≡ δ η P + h ˆ12 ih ˆ1 P i η + h ˆ1 a ih ˆ1 P i η a ! , δ σ ≡ δ (cid:18) ˆ η + h P ih P ˆ1 i η + h P a ih P ˆ1 i η a (cid:19) , – 14 – ( z , z ) ≡ h ˆ1 P i h ˆ1 a i h a i h i H ( z , z )( p + p + p a ) . (3.12)Note that four-point momentum conservation, with all legs on-shell, implies the followingrelations: h P ˆ1 i = ±h a i , h P i = ±h ˆ1 a i , h P a i = ±h i , (3.13)where ± signs correspond to the two on-shell solutions of internal momentum, z and z ,respectively. In the double-soft limit, we parametrize the spinors λ → ǫλ and λ → ǫλ bythe parameter ǫ . Only terms up to O ( ǫ ) are important to us. To get precise contributionup to O ( ǫ ), due to the leading ǫ − contribution from the propagator, we expand remainingfactors in the amplitudes up to O ( ǫ ).With eqs.(3.13), we can solve c j and s j (for j = 1 ,
2) to the order of our interest, c j = 1 − α j ǫ + O ( ǫ ) ,s j = − α j ǫ + " ( α j + α ∗ j ) α j − ( α j − α ∗ j ) β j ǫ + O ( ǫ ) , (3.14)where α j and β j are defined as α ≡ h a i + i h a ih na i , β ≡ h n i + i h n ih na i , α ≡ α ∗ , β ≡ β ∗ . Substituting the above solutions in relevant terms in eq.(3.11), we find δ ρ ( z j ) = δ (8) η P − i η a + ( − j +1 iβ j η ǫ + β j ǫ η a ! + O ( ǫ ) ! , (3.15) δ σ ( z j ) = δ (8) − α j ǫ ! η + ( − j +1 i β j ǫ ! η + β j η a ǫ − α j η n ǫ + O ( ǫ ) ! , H j = f ( z j ) ǫ − + f ( z j ) ǫ − + f ( z j ) ǫ + O ( ǫ ) , f ( z j ) ≡ − ǫ α j − α ∗ j α j β j , where H i ≡ H ( z i , z j ). Explicit forms of f and f are actually irrelevant under antisym-metric extraction. This will be clear shortly. Carrying out the integration of η P on δ ρ ineq.(3.11), the information of δ σ can be recast into an operator acting on the remaining( n − λ ,n and ˆ η ,n . By doingso, we find M n (cid:12)(cid:12)(cid:12) λ → ǫλ λ → ǫλ = n − X a =3 H ( z , z ) δ σ ( z ) exp (cid:2) U ( z ) (cid:3) M n − ( a, . . . , n ; z ) + ( z ↔ z ) , (3.16) U ( z j ) ≡ ( − j +1 iǫβ j η ∂∂η a + ǫ O η a ( z i ) | {z } integration of η P + ǫα j η ∂∂η n + ǫ O η n ( z i ) + ǫ O λ ,n ( z i ) | {z } BCFW shifted ˆ λ ,n , ˆ η ,n + O ( ǫ ) . – 15 –ere, O η a ,η n ,λ ,n are differential operators dependent on ∂ η a ,η n ,λ ,n .To verify eq.(3.4), we will integrate away m number of η ’s and m number of η ’s,with the following possible choices: ( m , m ) = (8 , , , , , , , m ↔ m . To simplify the notations, in the following, theamplitudes with soft scalars ξ ( m A ) and ξ ( m B ) , where ξ s are defined in eq.(3.3), will beabbreviated as Z d η d η η − m B η − m A M n ( ξ ( m A ) ( p ) , ξ ( m B ) ( p ) , . . . ) (cid:12)(cid:12)(cid:12) λ → ǫλ λ → ǫλ ≡ M ( m A ,m B ) n , (3.17)where m A and m B are the number of U(8) indices carried by the soft particles with mo-mentum p and p , respectively, and the subscripts A and B of which are the set of U(8)indices { A i } and { B i } labelled on the fields ξ ’s. To see why we are interested in the afore-mentioned sets of ( m A , m B ), a closer inspection of eq.(3.4) tells us that depending on thesum m A + m B , different SO(16) generators are expected on the RHS. In particular, wehave:( m A + m B ) = 10 : R IJ , ( m A + m B ) = 8 : R I J , R, ( m A + m B ) = 6 : R IJ , (3.18)where we have listed the types of SO(16) generators that can appear in the double-softlimit.As a simple example, let us consider the case with ( m , m ) = (8 ,
0) in detail. In thedouble-soft limit, the amplitude takes the form, M (8 A , B ) n = ǫ I ...I n − X a =3 ( Z a + S a ) M n − + O ( ǫ ) , (3.19)where the operator Z a and S a are defined as Z a ≡ − X j =1 , f ( z j )( α j β j η Ia − α j η In ) ∂∂η In − X j =1 , f ( z j ) α j ,S a ≡ X j =1 , f ( z j ) ǫ + f ( z j ) ǫ + f ( z j ) + X k = λ a ,η n ,η a f ( z j ) O k ( z j ) . Here I is the SU(8) index carried by the soft particle ξ (8 A ) . There are two ways to distribute8 η ’s . One is to take 7 η ’s from δ σ and 1 η from U , which is of order ǫ and correspondsto the first term in Z a . The other case is to extract 8 η ’s purely from δ σ . When oneof the 8 η ’s in δ σ is of order ǫ and the other 7 η ’s are ǫ , we have a term of ǫ whichresults in the second term in Z a . The other terms purely from δ σ with different orders of ǫ give us S a M n − . The first term in Z a can be further simplified, because of momentumconservation and super-momentum conservation, − X j =1 , f ( z j )( α j β j η Iq − α j η In ) ∂∂η In = X j =1 , β j − β ∗ j β j (cid:18) ∂∂η In η In (cid:19) M n − + O ( ǫ )= − p n · p p n · ( p + p ) M n − + O ( ǫ ) . (3.20)– 16 –imilarly for the second term in Z a , by momentum conservation, − n − X a =3 X j =1 , f ( z j ) α j M n − = 4 p n · p p n · ( p + p ) M n − + O ( ǫ ) . (3.21)As a result, we have M (8 A , B ) n = ǫ I ...I " − p n · p p n · ( p + p ) + n − X a =3 S a M n − . (3.22)A similar calculation gives us the following result for M (0 B , A ) n , M (0 B , A ) n = ǫ I ...I " n − X a =3 S a + n − X a =3 p a · p p a · ( p + p ) − n X a =3 p a · p p a · ( p + p ) η Ia ∂∂η Ia M n − . (3.23)By the antisymmetric extraction, the symmetric term S a M n − cancels out, and we are leftwith the U(1) generator of U(8), M (8 A , B ) n − M (0 B , A ) n = ǫ I ...I n X a =3 p a · p p a · ( p + p ) (cid:18) η Ia ∂∂η Ia − (cid:19) M n − . (3.24)Other R-symmetry operators with m A + m B = 8 can be found by the same method. Herewe simply list the results: M (6 A , B ) n − M (2 B , A ) n = n X a =3 p a · p p a · ( p + p ) × (3.25) " ǫ I ...I I [ J η Ia ∂∂η J ] a + ǫ I ...I J J (cid:18) η Ia ∂∂η Ia − (cid:19) M n − , M (4 A , B ) n − M (4 B , A ) n = n X a =3 p a · p p a · ( p + p ) × (3.26) ǫ I ...I I [ J J J η Ia ∂∂η J ] a + ǫ I ...I J ...J η Ia ∂∂η Ia ! M n − . One can immediately see that after taking into account the explicit form of the SO(16)generators in eq.(3.1) and eq.(3.2), the above result can be rewritten as: M (8 A , B ) n − M (0 B , A ) n = − n X a =3 p a · ( p − p )2 p a · ( p + p ) ( R a ) M n − M (6 A , B ) n − M (2 B , A ) n = − n X a =3 p a · ( p − p )2 p a · ( p + p ) × (cid:20) ǫ I ...I I [ J ( R a ) I J ] + 12 ǫ I ...I J J ( R a ) (cid:21) M n − , M (4 A , B ) n − M (4 B , A ) n = − n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:2) ǫ I ...I I [ J J J ( R a ) I J ] (cid:3) M n − . (3.27)– 17 –imilarly we can consider the case with m A + m B = 6 and m A + m B = 10, the resultsare given by M (6 A , B ) n − M (0 B , A ) n = n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:2) ǫ I ...I IJ ( R a ) IJ (cid:3) M n − , M (4 A , B ) n − M (2 B , A ) n = − n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:2) ǫ I ...I J J IJ ( R a ) IJ (cid:3) M n − M (8 A , B ) n − M (2 B , A ) n = − n X a =3 p a · ( p − p )2 p a · ( p + p ) [ − ǫ I ...I ( R a ) IJ ] M n − , M (6 A , B ) n − M (4 B , A ) n = − n X a =3 p a · ( p − p )2 p a · ( p + p ) (cid:2) ǫ I ...I [ J J ( R a ) J J ] (cid:3) M n − . (3.28)Finally for the cases with m A + m B >
10 or m A + m B <
6, it turns out the amplitudesvanish in the limit. Thus, by studying the double-soft-scalar limits of the tree-level ampli-tudes, we have derived all non-trivial R-symmetry operators for three-dimensional N = 16supergravity. Furthermore, we have reconstructed the algebra of E with part of the SO (16) being non-linearly realised. In the next section, we will consider the fate of thesedouble-soft limits at one-loop. In previous section, we have derived the result of double-soft limits for tree-level amplitudesin N = 16 supergravity, in this section we will consider the double-soft limits at one-loopusing generalized unitary cuts in three dimensions [9]. The integral representation of thethree-dimensional theory can be deduced from its four-dimensional parent which is knownto be expressible as linear combinations of scalar box integrals. The four-dimensionalintegral, upon dimensional reduction, can be viewed as the definition of the dimensionallyregulated three-dimensional integral, which captures all subtleties related to − ǫ dependentterms. In three-dimensions, a scalar box-integral can be written as a linear combination ofscalar triangles up to O ( ǫ ) terms, thus one-loop amplitudes for three-dimensional maximalsupergravity do not contain rational terms, and can be expressed in terms of scalar triangleintegrals only.So as we discussed above that an n -point one-loop amplitude can be expressed in termsof scalar triangle integrals I tri with certain coefficients, namely, M n = X i c i I tri i . (3.29)The coefficients c i can be most easily determined by the triple-cut, and written as a productof three tree-level amplitudes with a Jacobian factor from the cuts, c i = 1 q K A K B K C Z d η ℓ d η ℓ d η ℓ M tree1 M tree2 M tree3 , (3.30)where ℓ i ’s are the cut propagators and K A , K B , K C are the external momenta of threecorners in the cuts. First, it is straightforward to see that the coefficients c i vanish in– 18 –he single-soft limit. In what follows, we will mostly study the double-soft limits on thecoefficients. There are three different situations as shown below, BC Aℓ ℓ ℓ (a) BC Aℓ ℓ ℓ (b) BC Aℓ ℓ ℓ (c) ,where the dashed lines indicate the soft legs. It is easy to see that the first two kinds ofdiagrams, namely diagrams (a) and (b), vanish in the double-soft limits: the vanishingof diagram (a) is inherent from the result of the single-soft limit of tree-level amplitudes.One may worry about the degenerate cases where a soft leg is with only one hard leg at acorner, since the Jacobian factor 1 / q K A K B K C is divergent as 1 /ǫ in this case, howeverthe four-point amplitude goes as ǫ in the single-soft limit; as for diagram (b), we first notethat ℓ , ℓ are nothing but BCFW shifted soft legs, namely λ ℓ = cλ p + sλ q , λ ℓ = cλ q − sλ p , (3.31)with c + s = 1 , and p, q are the soft legs. Thus in the double-soft limits, we have not only λ p , λ q → λ ℓ , λ ℓ → ǫ in this case,while the cut Jacobian is divergent only as 1 /ǫ .Let us now consider the interesting case, namely diagram (c), where two soft legs(with some hard legs) are at the same corner of the triple cut. First we note that followingdiagrams lead to the same triangle integral in the soft limit, BC Aℓ ℓ ℓ (c) BC Aℓ ℓ ℓ (d) BC Aℓ ℓ ℓ (e) .So they should be combined together. Taking the diagram (c) as an example, the coefficientis given by the triple-cut, C (c) = J (c) Z d η ℓ d η ℓ d η ℓ M ( A ; p, q, ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) , (3.32)where A, B, C denote the hard legs at the three corners respectively, p, q are the soft legs,and the Jacobian J (c) = 1 / q ( K A + p + q ) K B K C . Using the result of the double-softlimit for the tree-level amplitudes, we have C (c) → C ( A )(c) + C ( ℓ )(c) + C ( ℓ )(c) , (3.33)– 19 –here each term is given by C ( A )(c) = 1 q K A K B K C X j ∈ A S j Z Y i =1 d η ℓ i M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) ! ,C ( ℓ )(c) = 1 q K A K B K C Z Y i =1 d η ℓ i ( S ℓ M ( A ; ℓ , − ℓ )) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) ,C ( ℓ )(c) = 1 q K A K B K C Z Y i =1 d η ℓ i ( S ℓ M ( A ; ℓ , − ℓ )) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) . (3.34)where S i are the double-soft factors, which may be proportional to one of the generatorswe defined in previous section, R IJi , ( R i ) IJ , ( R i ) IJ , R i . (3.35)Similarly one can obtain the results for diagram (d) and (e), summing over all three dia-grams, from the summation over the external legs, we find C ( A )(c) + C ( B )(d) + C ( C )(e) = 1 q K A K B K C × X j ∈{ A,B,C } S j Z Y i =1 d η ℓ i M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) ! , (3.36)which is precisely the result of the double-soft limits at one-loop. So we need to prove thecontributions from internal lines vanish. That is indeed true, they all cancel in pairs, aswe will prove C ( ℓ )(c) + C ( ℓ )(e) = 0 , C ( ℓ )(c) + C ( ℓ )(d) = 0 , C ( ℓ )(d) + C ( ℓ )(e) = 0 . (3.37)Let us take C ( ℓ )(c) + C ( ℓ )(e) as an example. First for the case when the soft factor is proportionalto R IJℓ = η Iℓ η Jℓ , we have C ( ℓ )(c) + C ( ℓ )(e) ∼ h η Iℓ η Jℓ M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ )+ η I − ℓ η J − ℓ M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) i . (3.38)Now, using the fact that η P − ℓ = iη Pℓ , we find the above two terms cancel out precisely. Thesame argument applies to ( R ℓ ) IJ . Let us now consider ( R ℓ ) I J , and we have, C ( ℓ )(c) + C ( ℓ )(e) ∼ Z Y i =1 d η ℓ i h η Iℓ (cid:0) ( R ℓ ) I J M ( A ; ℓ , − ℓ ) (cid:1) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ )+ M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) (cid:0) ( R ℓ ) I J M ( C ; ℓ , − ℓ ) (cid:1) i = Z Y i =1 d η ℓ i ( R ℓ ) I J h M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) i . (3.39)– 20 –hen I = J , ( R ℓ ) I J = η Iℓ ∂ η Jℓ , and the above result is a total derivative, which thenvanishes trivially under the fermionic integration. Whereas when I = J , the vanishing ofthe above result can be seen by using the identity R dη η ∂∂η ∗ = R dη ∗ . Finally, we considerwhen the soft factor is proportional to the U(1) generator R ℓ , we have C ( ℓ )(c) + C ( ℓ )(e) ∼ Z Y i =1 d η ℓ i X K =1 η Kℓ ∂ η Kℓ − ! × h M ( A ; ℓ , − ℓ ) M ( B ; ℓ , − ℓ ) M ( C ; ℓ , − ℓ ) i . (3.40)Due to the same identity, R dη η ∂∂η ∗ = R dη ∗ , we find that the above sum vanishes. Thuswe have proved all possible soft factors in N = 16 supergravity do not receive any one-loopcorrection.It is easy to see that the above discussion, namely the cancellation between internalcut propagators, is actually valid for any multiple cuts, if the lower-point (loop) amplitudesentering in the cuts satisfy the double-soft theorems. This observation as well as the explicitcalculation for one-loop amplitudes we have done strongly suggest that any higher-loopamplitudes should behave in the same way as the tree-level amplitudes in the double-soft-scalar limits, namely duality symmetries do not receive any loop corrections. N = 16 SUGRA
In the previous sections, we have demonstrated both at tree- and one-loop level, in thesingle-soft-scalar limit the n -point scattering amplitude vanishes while the double-soft-scalar limit is given by an SO(16) rotation on the M n − . The same constraint appliesto matrix-elements generated by possible counter terms, and thus provides an on-shellcheck on whether or not potential counter terms respect the duality symmetry of thetheory. This line of approach has been extensively pursued for the four-dimensional N = 8theory [16, 17, 25, 26]. Given that all degrees of freedom in three-dimensional supergravityare subject to duality constraints, we expect that the constraint on counter terms aremuch more stringent compared to its four-dimensional counter part. In particular, in threedimensions, all bosonic soft-limits must vanish. Furthermore, using supersymmetric Wardidentities [27], one can deduce that all single-soft-fermion limits should vanish as well, since: h [ Q, φ · · · ] i = 0 = h q ih ψ · · · i + h φ [ Q, · · · ] i , (4.1)where | q i is an auxiliary spinor and · · · represent a collection of fields. The single-soft-scalar limit vanishes as O ( ǫ ), thus in order for the RHS to vanish, at the very least h ψ · · · i has to be of O ( ǫ ) in the single-soft limit. Thus remarkably, supersymmetry combined withthe duality symmetry implies that all single-soft limits must vanish for three-dimensionalsupergravity!At four points, N = 16 supersymmetry requires the matrix element of any four-pointoperator to be of the form L = δ ( Q ) f ( s, t, u ) , (4.2) This type of argument was first realized in [28] for N = 8 supergravity in four dimensions. – 21 –here the function f ( s, t, u ) is a polynomial and symmetric in s, t, u . This matrix-elementis invariant under the full SO(16) R-symmetry. The fact that it is invariant under R I J istrivial, where as for R IJ let us choose I = 7 , J = 8 in R IJ δ ( Q ) = P i =1 η Ii η Ji δ ( Q ) , andconsider the component η η η η η η . It is given by: Z dη dη dη dη dη dη
84 4 X i =1 η I i η J i δ ( Q ) ∼ ( h ih i + h ih i ) = 0 . (4.3)Similar analysis applies to R IJ . Thus at four-point we can have: δ ( Q ) , δ ( Q )( s + t + u ) , δ ( Q )( s + t + u ) , δ ( Q )( s + t + u ) . (4.4)These counter term elements can be viewed as the descendant of four-dimensional elementsgenerated from R , D R , D R and D R respectively. In three dimensions, an operator O of mass-dimension m corresponds to m = 3 + ( L −
1) loops, and hence the above wouldcorrespond to possible counter terms for 6 − , − ,
12- and 14-loop divergence respectively.Note that leading ultraviolet divergences is automatically ruled out for odd-loops. To test the validity of these operators, we need to see if the corresponding six-point matrixelements satisfy the required soft-behavior. Given that the four-point matrix elementsdiscussed in eq.(4.4) are direct descendants of their four-dimensional counter parts, we willtest whether their six-point descendants satisfy the duality symmetry in three dimensions.In other words, we will consider whether or not direct dimensional reduction of R , D R , D R and D R to three-dimensions yield valid counter terms. Here, by dimensionalreduction, we mean to substitute three-dimensional kinematics to the four-dimensionalamplitudes. Recall that the SO(16) scalars are organized as a representation of SU(8):1 η η η η ξ ξ IJ ξ IJKL ξ IJ ξ . It is then straightforward to identify the states between three and the four-dimensionalcounter part. In particular, using the notation in [26] we can identify h + ↔ ξ, v + IJ ↔ ξ IJ , ϕ IJKL ↔ ξ IJKL , v − IJ ↔ ξ IJ , h − ↔ ¯ ξ . (4.5)In [16, 17] it was shown that the R , D R , and D R operators have non-vanishingsingle-soft limit. In the limit the matrix element becomes proportional to a local quantitythat does not vanish in three-dimensional kinematics. Thus such matrix elements alsoyield incorrect single-soft limit in three-dimensions, and one can rule them out as possiblecounter terms. Here, we use R to denote all R-symmetry generators to distinguish them from the field strength R . By leading we are referring to the first place where ultra-violet divergence is present More precisely, for the amplitude M ( − − + + ϕ ¯ ϕ ) of R D R , D R behaves in the soft limit as h i [34] , h i [34] ( P i 64 which are complex spinor indices. For SO(16) invariants they are pairedwith their complex conjugate. 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