aa r X i v : . [ h e p - t h ] J u l New fermionic soft theorems for supergravity amplitudes
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 “Tor Vergata”,Via della Ricerca Scientifica, 00133 Roma, Italy
Soft limits of massless S-matrix are known to reflect symmetries of the theory. In particular fortheories with Goldstone bosons, the double-soft limit of scalars reveals the coset structure of thevacuum manifold. In this letter, we propose that such universal double-soft behavior is not onlytrue for scalars, but also for spin-1/2 particles in four dimensions and fermions in three dimensions.We first consider Akulov-Volkov theory, and demonstrate the double-soft limit of Goldstinos yieldsthe supersymmetry algebra. More surprisingly we also find amplitudes in 4 ≤ N ≤ N = 16 supergravity in three dimensions behave universallyin the double-soft-fermion limit, analogous to the scalar ones. The validity of the new soft theoremsat loop level is also studied. The results for supergravity are beyond what is implied by SUSY Wardidentities, and may impose non-trivial constraints on the possible counter terms for supergravitytheories. PACS numbers: 04.65.+e, 11.15.Bt, 11.30.Pb, 11.55.Bq
The connection between symmetries of a theory andthe soft behavior of its S-matrix has been previously ex-plored in various examples. The most famous case isWeinberg’s soft graviton theorem [1], which states thatthe leading soft divergence of a gravitational S-matrix isconstrained by Ward identities and is universal. Similarresults, based on other symmetries, were shown to implyuniversality for the subleading divergences in gauge andgravity theories as well [2–6]. Another famous example,which is more relevant to this letter, is in the context ofthe Goldstone bosons of a spontaneous broken symmetry.In particular, taking the momentum of the Goldstone bo-son to near zero, which corresponds to a constant scalarfield, the S-matrix should vanish due to the scalars be-ing derivatively coupled. This is the well-known Adler’szero [7, 8].As discussed in [9], Adler’s zero can also be understoodfrom the structure of the vacuum. Perturbative scatter-ing amplitudes should be identical when computed at anypoint in the vacuum moduli. On the other hand, one canin principle use the operator e iθ · T to relate one vacuum toanother via | θ i = e iθ · T | i , where T a represents the bro-ken generators, and θ a is a constant that is the vacuumexpectation value (vev) of the soft scalar. The amplitudeevaluated in the | θ i vacuum can be written as a pertur-bative series by expanding out the exponent, | θ i = e iθT | i = | i + θ i | π i i + 12 θ i θ j | π i π j i + · · · , (1)where | π i i , | π i π j i e.t.c represent vacuums with one andtwo (or more) additional soft scalars, respectively. Sincethe amplitude is the same in either | θ i or | i vacuum,this implies that amplitudes with one or more soft scalarsmust vanish. The vanishing of the single soft scalar is pre-cisely Adler’s zero. For two soft scalars, it turns out that the amplitude is non-zero due to the non-commutativityof the broken generators. It was shown in [9] that takingtwo Goldstone bosons to have soft momenta, the ampli-tude behaves as: M n (cid:2) φ i ( t p ) , φ j ( t p ) · · · (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) t → = n X a =3 B a f ijK H aK M n − , (2)where B a ≡ p a · ( p − p )2 p a · ( p + p ) and H aK is the generator of theinvariant subgroup in the coset G/H , while f ijK is thestructure constant in [ T i , T j ] = f ijK H K .In extended supergravity theories the scalars also pa-rameterize a coset space, and the double-soft-scalar limitof the S-matrix is given by eq.(2) as well. The fact thatthe non-linearly realized symmetries have non-trivial im-print on the S-matrix is extremely useful in the discussionof ultra-violet behavior of supergravity theories. In par-ticular, modulo quantum anomalies, any possible counterterms for the theory must respect this symmetry, whichcan be verified by checking whether or not the S-matrixelements generated by such counter terms agree witheq.(2) [10, 11].In this letter, we demonstrate that remarkably, thesame single and double-soft behavior also applies tofermions in Akulov-Volkov (A-V) theory [12] as wellas supergravity theories both in three and four dimen-sions. For A-V theory, which is the effective action forthe Goldstinos of spontaneously broken supersymmetry,we show that the amplitudes vanish in the single-soft-fermion limit. In the double-soft limit they exhibit asimilar form as that of the scalars in eq.(2) with H aK replaced by [1 | p a | i , where legs 1 and 2 are +1 / − / The latter precisely reflects the supersymme-try algebra, { Q, ¯ Q } = P , with additional factors propor-tional to the external-line factors of the soft fermions.For the supergravity theories, we show that the ampli-tudes vanish in the single-soft limit of spin-1 / p a · ( p − p ) → [1 | p a | i . Notice the recurrence of thefactor [1 | p a | i , as with the A-V theory. In three dimen-sions, we show that for N = 16 supergravity [13], wherethe 128 scalars parametrize the E /SO(16) coset space,the double-soft limit of any pair of the 128 fermions ex-hibits the same behavior as the spin-1 / DOUBLE-SOFT LIMIT AND SPONTANEOUS(SUPER)SYMMETRY BREAKING
Scattering amplitudes involving Goldstone bosons haveinteresting soft behavior that reveals the details of thevacuum. In [9] it was shown that for N = 8 supergrav-ity, where the 70 scalars parameterize E /SU(8), theamplitude with two soft scalars behaves as: M n (cid:0) φ I I I I ( t p ) , φ I I I I ( t p ) · · · (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) t → (3)= 4 n X a =3 B a ǫ I I I I [ I I I | J ( R a ) I ] J M n − + O ( t ) , where I i , J i = 1 , · · · R a ) I J in eq.(3) are the single site SU(8)generators, where a labels the external leg. The soft limit p , → t p , , is realized on the spinors as λ , → tλ , , ˜ λ , → t ˜ λ , . (4)This analysis was later extended to D = 4, 4 ≤ N <
8, and D = 3, N = 16 supergravity in [14]. One newsubtlety is the presence of U(1) factors in the isotropygroup H , which produces soft-graviton singularities inthe double-soft limit. Such singularities are absent in the N = 8 theory. To extract the finite piece one insteadconsiders the anti-symmetrized double-soft limit M [ i,j ] n , We use standard spinor-helicity formalism in four and threedimensions. For 4D: p α ˙ αi = λ αi ˜ λ ˙ αi , with scalar products as λ αi λ βj ǫ αβ = h ij i , ˜ λ i ˙ α ˜ λ j ˙ β ǫ ˙ α ˙ β = [ ij ], s ij = h ij i [ ji ], and [ i | p a | j i =[ ia ] h aj i . In Minkowski signature, ˜ λ = λ ∗ . Reduction to three-dimensions simply corresponds to λ being real. defined as: M [ i,j ] n ≡ M n (cid:0) φ i ( t p ) , φ j ( t p ) , · · · , n (cid:1) − (1 ↔ (cid:12)(cid:12)(cid:12)(cid:12) t → . (5)Indeed for all 4 ≤ N < N = 16 supergravity in three dimensions, the anti-symmetrized double-soft limit yields: M [ i,j ] n = n X a =3 p a · ( p − p ) p a · ( p + p ) f ijK ( H a ) K M n − + O ( t ) , (6)where the ( H a ) K s are single site U( N ) generators for D =4, 4 < N <
8, U(1) for D =4 N = 4 and SO( N ) for D =3.Another interesting example we would like to consideris the A-V theory [12], which is the low energy effectiveaction of fermions (Goldstinos) associated to spontaneousbreaking of supersymmetry. The action is given by S AV = − g Z d x det(1 + ig ψσ µ ↔ ∂ µ ¯ ψ ) , (7)where the Weyl fermion ψ is the Goldstino and σ µ =(1 , → σ ), with → σ as the Pauli matrices. One can expandthe determinant which generate an infinite series of localoperators. Here we will use the six-point amplitude as anexample. The relevant vertices are the four-point vertex: λ ai ˜ λ ˙ bℓ ˜ λ ˙ aj λ bk g [ V jk − V kj ] ab ˙ a ˙ b − ( i, a ↔ k, b ) − ( j, ˙ a ↔ ℓ, ˙ b )+ [( i, a , j, ˙ a ) ↔ ( k, b , ℓ, ˙ b )] , (8) ,where V ij,ab ˙ a ˙ b ≡ ( p i ) a ˙ a ( p j ) b ˙ b , and the six-point vertex: λ ai ˜ λ ˙ cn ˜ λ ˙ aj λ cm ˜ λ ˙ bℓ λ bk X σ ∈ perm . ( − ) σ V ( σ ( i, a , j, ˙ a , k, b , ℓ, ˙ b , m, c , n, ˙ c )) , (9)where V ( i, a , j, ˙ a , k, b , ℓ, ˙ b , m, c , n, ˙ c ) ≡ i g [2 V ℓmj − V jmℓ − V ℓjm + V jℓm + 2 V kni + 2 V ink + V kin − V ikn ] abc ˙ a ˙ b ˙ c , with V ijk,abc ˙ a ˙ b ˙ c ≡ ( p i ) a ˙ a ( p j ) b ˙ b ( p j ) c ˙ c . A straightfor-ward computation shows that the relative coefficientsbetween the quartic and sextic interactions are pre-cisely needed for the leading terms in the single-softlimit to cancel, such that the six-point amplitude isof order t in the limit. In the double-soft limit one finds: M ( ψ , ¯ ψ , ψ , ¯ ψ , ψ , ¯ ψ ) | λ , → tλ , λ , → t ˜ λ , (10)= t g X a =3 B a [1 | p a | i M ( ψ , ¯ ψ , ψ , ¯ ψ ) + O ( t ) , where M ( ψ , ¯ ψ , ψ , ¯ ψ ) = 2 g s [35] h i . The aboveresults are consistent with the interpretation that thefermions are Goldstinos. From the single- and double-soft behaviors of Goldstone bosons, one would expectthat the single-soft limit of a Goldstino should vanish as O ( t ), while the double-soft limit should be finite and pro-portional to B a p a M n − , due to the fact that the brokengenerators are associated with Q , and satisfy the alge-bra { Q, ¯ Q } = P . The extra factor of t for the single-softlimit, and t λ ˜ λ for the double-soft case, are simply dueto the presence of additional soft external-line factors forfermions. NEW DOUBLE-SOFT THEOREMS IN FOURDIMENSIONS
We now consider soft fermions in four-dimensional su-pergravity theories. As discussed in ref. [14], due to thefact that amplitudes with a soft scalar vanish as O ( t ),SUSY Ward identities [15] require that the amplitudeswith a soft fermion vanish as O ( t ). The same resultcan alternatively be deduced from BCFW recursion [16].However, for the double-soft limit, Ward identities are nolonger sufficient since it implies:0 = h | [ Q, φ ψ , . . . ] | i = [1 q ] M ( ψ ψ , . . . ) + [2 q ] M ( φ φ , . . . )+ P i> [ iq ] M ( φ ψ , . . . ) , (11)where q is a reference momentum. As one can seethe identity yields a linear relation between double-softfermions, scalars and soft-scalar-fermion limits. Thus thedouble-scalar limits by itself is insufficient to determinethe double-fermion limits. Instead we will proceed usingthe recursion relations.To treat all the hard particles democratically, weadd an auxiliary negative-helicity graviton [9], whichat the end is taken away by sending its momentumto be soft . Now, we choose the shifted legs in therecursive formula as one of the soft legs and the addedgraviton. The remaining soft leg will be in one of thefactorized amplitudes which generally vanishes due tothe previous analysis. Thus all diagrams vanish exceptfor the following two special cases: g ˆ1 a n ˆ g ˆ12 n ˆ P ˆ P (I) (II) . where the hatted labels indicate they are the deformedlegs, and must be at opposite side of the propagator forany diagram in the recursion. Unlike the scalars for whichone can use the anti-symmetric extraction scheme in This trick is of course not necessary, a different derivation with-out the auxiliary graviton is presented in [14] eq.(5) to obtain the U(1) part of the symmetry group [14],here U(1) is inaccessible due to the soft-fermions havingopposite helicities and thus cannot be “symmetrized”.Thus we will consider the case where two fermions donot form a singlet, for which only the diagram (I) con-tributes, and to the leading order it is given as [14] (Wehave presented the N = 8 result, from which lower su-perymmetric theories can be obtained via SUSY trunca-tion [17] ): M (ˆ1 , , a, ˆ P ) h ˆ1 P i δ (cid:16) η + h P ih P ˆ1 i η + h P a ih P ˆ1 i η a (cid:17) p a · ( p + p ) (12) × exp − h ˆ12 ih ˆ1 P i η I ∂∂η Ia ! M n − ( g ) . Here the M i ’s are superamplitudes, and M n − ( g ) indi-cates one of the legs being a soft graviton. The superam-plitudes are homogenous polynomial function of the η I swhose coefficients are the component amplitudes. The η I s are Grassmann odd variables carrying fundamentalSU(8) indices. They are used to package all on-shell de-grees of freedom into a superfield, which for N = 8 isgiven by:Φ( η ) = G ++ + η A Λ A + η A η B A + AB + . . . +( η ) G −− , (13)where G ++ is the +2 graviton, Λ A is the + grav-itino, and A + AB the +1 graviphoton and so on. Thehatted labels in eq.(12) indicate the corresponding legsare deformed due to the BCFW shift, and their ex-plicit forms can be found in [14]. For the fermion pair( ψ I I I , ¯ ψ I I I I I ), diagram (I) yields:5[1 | p a | i p a · ( p + p ) ǫ I I I [ I I I I | J ( R a ) I ] J M n − ( g ) . Thus after removing the auxiliary graviton via the soft-graviton theorem, and summing over all relevant BCFWchannels, we obtain: M n (cid:2) ψ I I I ( t p ) , ¯ ψ I I I I I ( t p ) · · · (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) t → = n X a =3 F a ǫ I I I [ I I I I | J ( R a ) I ] J M n − , (14)where F a ≡ [1 | p a | i p a · ( p + p ) . Note the remarkable similar-ity to the double-soft limit of scalars, here one simplyreplaces the factor p a · ( p − p ) in the numerator of B a with [1 | p a | i . NEW DOUBLE-SOFT THEOREMS IN THREEDIMENSIONS
Next we consider amplitudes in three-dimensional N =16 supergravity [13] with two soft fermions. Again theon-shell degrees of freedom, 128 bosons and 128 fermions,can be packaged into a superfieldΦ = ξ + X n =1 ξ I ...I n η I η I · · · η I n , I i = 1 , . . . , . (15)The Grassmann variables η I transform as fundamentalsof SU(8) ⊂ SO(16), thus only part of the SO(16) are lin-early realized in these variables, with the remaining real-ized non-linearly. The 128 bosons parametrize the cosetspace E /SO(16).Due to the presence of U(1) in U(8), the double-softlimit is again polluted by soft-graviton divergences. Thusinspired by [14], we consider the symmetrized double-softlimit: M { i,j } n ≡ M n (cid:0) ψ i ( t p ) , ψ j ( t p ) , · · · , n (cid:1) +(1 ↔ (cid:12)(cid:12)(cid:12)(cid:12) t → . (16)Using the BCFW representation of three-dimensional su-pergravity amplitudes, we find that remarkably ampli-tudes with the symmetrized double-soft fermions also be-have universally and are given by: M { i,j } n = − n X a =3 ( S a ) i,j M n − + O ( t ) , (17)where ( S a ) i,j are the corresponding soft factors act-ing on the ( n − S a ) I ...I v J ...J v +2 = ( v +2)! F a − δv, δ [ I ...I v ][ J ...J v ( R a ) J v +1 J v +2 ] , ( S a ) I ...I v +2 J ...J v = ( v +2)! F a ( − δv, δ [ I ...I v +2 ][ J ...J v ] IJ ( R a ) IJ , ( S a ) I ...I v J ...J v = v ! F a ( − δv, δ [ I ...I v ] I [ J ...J v − [2 v ( R a ) IJ v ] − δ IJ v ] R a ] ,δ I ...I j J ...J j = δ I J δ I J · · · δ I j J j , v = { , } . (18)Here ( R a ) ≡ η Ia ∂ η Ia −
4, ( R a ) IJ ≡ ∂ η Ia ∂ η Ja , ( R a ) IJ ≡ η Ia η Ja and F a ≡ h | p a | i p a · ( p + p ) . Different S a s correspond to differ-ent choices of soft-fermion pairs. Note that although thesoft-fermion limit behaves in a similar fashion as that ofthe bosons, its detailed algebra is different. This is re-flected in the fact that they form distinct representationsunder the on-shell SU(8) symmetry. NEW SOFT THEOREMS AT LOOP LEVEL
It is interesting to see if the new soft theorems in su-pergravity theories are subject to any loop corrections.We begin with N = 16 supergravity in three dimensions,whose one-loop amplitudes can be expressed in terms ofscalar triangle integrals with coefficients determined bygeneralized unitarity cuts. We can then apply tree-levelsoft theorems since it is the tree-level amplitude that en-ters the cuts. Follow the same proof of their scalar part-ner [14], it is straightforward to show that single- and double-soft-fermion theorems do not receive any one-loopcorrections. For the theories in four dimensions, one canalso express the amplitudes in terms of box integrals,which immediately shows that one-loop amplitudes withone soft fermion also vanish. However, for the double-soft limit, unlike the three-dimensional case where am-plitudes only with even-number external legs exist, newcomplication arises due to the discontinuity of the inte-gral functions [18]. Here we provide some evidence thatthe new soft theorems are not corrected by loops by con-sidering the leading IR-divergent part of a L -loop ampli-tude, which is given by: M L − loop n (cid:12)(cid:12) lead . IR = 1 L ! n X i,j s ij log( s ij ) ǫ L M tree n . (19)Thanks to the kinematics factor s ij , applying the tree-level soft theorems to M tree n we find, at least, the leadingIR-divergent part of loop amplitudes satisfies the samesoft theorems as tree-level amplitudes. CONCLUSIONS
In this letter, we propose new soft theorems by study-ing soft fermions for the amplitudes in a wide range oftheories, including Akulov-Volkov theory and supergrav-ity theories in four and three dimensions. We find that allthe amplitudes vanish in the single-soft limit, and behaveuniversally in the double-soft limit, analogous to softGoldstone bosons. The results for Akulov-Volkov theoryprecisely reflect that it is the effective theory for sponta-neously supersymmetry breaking. To our surprises, thedouble-soft-fermion limits in extended supergravity the-ories are also universal and mimic the behavior of thescalars. We also provide evidence that the soft theoremsdo not receive loop corrections. Finally we like to empha-size that the results do not follow from the combination ofdouble-soft scalar limits and supersymmetric Ward iden-tities. It would be of great interest to clarify the implica-tions of all those new soft theorems, in particular whetherthere are new hidden symmetries behind them, and theirpossible application for constraining potential ultravioletcounter terms for supergravity theories.
ACKNOWLEDGEMENTS
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