From U(1) to E: soft theorems in supergravity amplitudes
It is known that for supergravity, the double-soft-scalar limit of an -point amplitude is given by a sum of local SU(8) rotations acting on an -point amplitude. For supergravity theories, complication arises due to the presence of a U(1) in the U() isotropy group, which introduces a soft-graviton singularity that obscures the action of the duality symmetry. In this paper, we introduce an anti-symmetrised extraction procedure that exposes the full duality group. We illustrate this procedure for tree-level amplitudes in supergravity in four dimensions, as well as supergravity in three dimensions. In three dimensions, as all bosonic degrees of freedom transform under the E duality group, supersymmetry ensures that the amplitude vanishes in the single-soft limit of all particle species, in contrast to its higher dimensional siblings. Using recursive formulas and generalized unitarity cuts in three dimensions, we demonstrate the action of the duality group for any tree-level and one-loop amplitudes. Finally we discuss the implications of the duality symmetry on possible counter terms for this theory. As a preliminary application, we show that the vanishing of single-soft limits of arbitrary component fields in three-dimensional supergravity rules out the direct dimensional reduction of as a valid counter term.
1 Introduction and motivations
Scattering amplitudes often exhibit universal behaviors in the limit when the momenta of some external particles approach to zero, i.e. so-called soft limit. For instance, it is well known that amplitudes in gauge theories (and gravity) behave universally in the single soft gluon (and graviton) limit, which goes back to the classical work by Weinberg Weinberg (). In particular, the analytic behavior of this limit at tree-level is completely determined by the gauge symmetries of the theory Low (); SoftGravy1 (); BernGauge ().111For the understanding on soft behaviors from other symmetry principles see CS (); SymArg ()
Another famous and well-studied case of soft limit, which will be of our interest in this paper, is the soft-pion theorem. The theorem states that the Goldstone boson decouples at zero momentum, i.e. the amplitude of one soft “pion” with arbitrary number of hard “pions” vanishes AdlerZero (). The full algebra of the symmetry can be exposed by considering the limit where two Goldstone bosons become soft Weinberg:1966kf (). This idea of probing the global symmetries of the theory by studying the single- and double-soft scalar limits was revisited and applied to supergravity theory in four dimensions by Arkani-Hamed et al Simplest (). It is known that the theory contains scalars, which are elements in the coset space E/SU(8), thus according to the soft-pion theorem, the amplitudes vanish in the single-soft-scalar limit, which is indeed the case as shown in Simplest (). The authors of ref. Simplest () then beautifully showed that any -point amplitude in the double-soft-scalar limit has the following universal behavior:
where the superscripts in scalar field are the SU(8) R-symmetry indices, and is the corresponding SU(8) rotation. It might be a surprise that amplitudes vanish in the single-soft limit, but finite in the double-soft limit. As explained in ref. Simplest (), which we will give a brief review in the next section, this is a reflection of the fact that the commutators of the broken generators do not vanish.
For supergravity, one can proceed and derive the corresponding behavior via supersymmetry reduction of the theory. There is one caveat however, in that for , the isotropy group (the of coset ) is U which includes a U(1). In order to generate this U(1) factor, the scalars chosen for the double-soft limit form an SU() 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 two scalars. More precisely, we consider the difference of two distinct amplitudes, one with the (, ) scalars carrying momenta , the other with and exchanged. We will show that
where is the single site U(1) generator and is the diagonal component of the SU() generator . We will refer to such extraction of the double-soft limit as “anti-symmetrized extraction”. Note that due to the fact that we are considering non-maximal supergravity theories, the on-shell degrees of freedom are carried by two distinct multiplets (). As a result, the U(1) generator has a different constant for the two distinct multiplets
We also consider maximal supergravity in three dimensions, which is the theory introduced by Marcus and Schwarz E8 (). The 128 bosonic states now parametrize the coset E/SO(16). We use the three-dimensional recursion formulas 3DRecur (), to derive the double-soft-scalar limit for all multiplicity tree-level amplitudes. Since the on-shell superspace only manifests U(8) SO(16), the other part of the SO(16) generators are non-linearly realized. Thus using the double-soft limit allows us to construct the algebra of in such non-linear realization. Note that the presence of a U(1) again requires us to apply the anti-symmetrized extraction procedure discussed above. We also consider the fate of the duality at loop-level. We demonstrate that at one loop, in the scalar integral basis representation, the integral coefficients are given in such a way that the double-soft behavior is manifest.
One of the many important questions one can ask for a gravitational S-matrix is its ultraviolet behavior. In recent years tremendous progress in computation techniques has allowed us to peer ever deeper into perturbative gravitational S-matrix. Remarkably, explicit computations N8UV (); N4UV (); N5UV () have reveal surprising finiteness in a wide range of supergravity theories with . Although from the viewpoint of four-dimensional divergences, some results can be explained by the constraints imposed by the symmetries of the coset space ElvangR4 (); ElvangFull (), there are examples where finiteness requires explanations that go beyond that explained by traditional symmetry arguments N4UV (); N5UV (); BCJEvid ().
If four-dimensional maximal supergravity is finite, then so must its three-dimensional reduction. Unlike in four dimensions, here all bosonic degrees of freedom transform under the duality group, which implies that coset symmetry imposes stronger constraints on candidate ultraviolet (UV) counter terms. Furthermore, as we will demonstrate, supersymmetric Ward-identities require that amplitudes vanish as well in the fermionic single-soft limits. Thus one can ask whether or not candidate UV counter terms can produce matrix elements satisfying all single- and double-soft behaviors required by the symmetries. As a preliminary step, we consider the direct dimensional reduction of matrix elements of counter terms in four dimensions. We will explicitly show that these matrix elements, which satisfy the E duality symmetry in four dimensions ElvangFull (), do not have the correct single-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 supergravity theories. These theories can be studied from the theory via supersymmetry reduction. However unlike their ancestor, the isotropy group of the duality symmetries for these non-maximal supersymmetric theories contain a U factor. To extract this subtle contribution, we introduce a procedure “anti-symmetrised extraction”, which allows us to throw away unwanted singular parts, and leave behind a beautiful and finite result, corresponding precisely to the U factor. In section 3, we then move on to study supergravity in three dimensions, both at tree and loop level. At tree level, we study the soft limits using BCFW recursion relations in three dimensions, and the same “anti-symmetrised extraction” procedure introduced previously is used to extract the U factor in the symmetry group. After deriving the soft theorems for tree-level amplitudes, we study the possible loop corrections to the theorems. Using generalized unitarity cuts, we show that all one-loop amplitudes satisfy exactly the same soft theorems as the tree-level one. In section 4, we discuss the application of duality symmetry to constrain candidate counter terms for Supergravity. We show that S-matrix generated by many counter terms descendant from four-dimensional ones via direct dimensional reduction do not satisfy the single-soft-scalar theorems. Finally in section 5, we finish the paper with conclusions and remarks.
2 Soft-scalar limits in Supergravity
2.1 Review on single- and double-soft limits on gravity amplitudes
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 corresponding amplitude vanishes, a result that is famously known as “Adler’s zero” AdlerZero (). Consider the coset space , where the generators of the isometry group are represented by , and ’s are the elements of the isotropy group. Schematically they satisfy the following commutation relations:
Since the vacuum expectation values (vev) of scalars spontaneously break the symmetry, thus they can be identified with parameters of the broken generators . The vanishing of the 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 constructed out of covariant derivatives
where Since the scalars are dressed with derivatives, taking the momentum soft results in the vanishing of the amplitude.
In Simplest (), the soft-scalar limits were discussed without relying on any detailed structure of the interactions. Starting with the fact that spontaneous symmetry breaking is a reflection of the presence of continuous set of degenerate vacua, perturbative amplitudes computed at different points on this moduli space must be equivalent. As two different points in the moduli space are connected via the generators , we can schematically write
where represents the vev of the scalar, which parametrizes the vacuum. Assuming that each point in the moduli space can be connected in such fashion, the fact that amplitudes computed in distinct vacua must agree implies that as one expands the exponent in eq.(6), terms beyond the leading term in the expansion must vanish. Since simply corresponds to a constant scalar, i.e. scalars with zero momenta, this leads to the conclusion that amplitudes with any additional soft scalar must vanish.
However, as discussed in Simplest (), the above analysis is not entirely correct. The subtlety lies in the assumption that there is a well-defined path that connects two points. Indeed, one would expect that the difference between two different paths should be proportional to an generator, since . In Simplest () it was argued that this ambiguity leads to the result that in the double-soft-scalar limit, the amplitude is non-vanishing, and behave universally:
For supergravity, whose 70 scalars parametrized E/SU(8) coset, this becomes
where is the single-site SU(8) R-symmetry generator.
For supergravity, the scalars parametrize SO(12)/U(6), SU(1,5)/U(5) and SU(1,1)/U(1) cosets for and respectively. One would expect the double-soft limits for these theories should directly follow from the theory via SUSY reduction. This would indeed be the case if not for the subtle difference between the isotropy group for the theories from the maximal theory: they contain an extra U(1). To see the U(1) in the double-soft limit, the two scalars must form an SU() singlet, which induces singularities in the limit. More precisely, the duality group algebra now involves relations of the form:
where and 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 a graviton:
As the graviton is soft, the amplitude is then proportional to the soft-graviton limit of an -point amplitude which is divergent.
To extract the U(1) part of the duality group, we take the double-soft limit in a manifest anti-symmetric fashion with respect to the two scalars. More precisely, we consider the difference of two distinct amplitudes, one with the (, ) scalars carrying momenta , the other with and exchanged. For this corresponds to considering the following difference
where the pairs , , and indicate the SU() singlet combination of the , and scalars in and supergravity theories respectively. We will refer to such extraction of the double-soft limit as “anti-symmetrised extraction”. Note that due to the fact that we are considering non-maximal supergravity theories, the on-shell degrees of freedom are carried by two distinct mulitplets, each can be considered as a particular truncation of the maximal theory LessSUSY (),
where is the unique superfield for the theory. As a consequence, the U(1)-generator has a different constant for the two distinct multiplets
One can verify that all tree amplitudes vanish under the above refined U(1) generator, i.e
2.2 Double-soft limits of Supergravity
Let us now demonstrate the validity of eq.(1) for Supergravity. We will use the on-shell recursion formula introduced by Britto, Cachazo, Feng and Witten BCFW () to generate the tree amplitudes of supergravity and perform SUSY reduction. To guarantee the presence of a U(1) on the right-hand side of , we choose two scalars from the theory that form a singlet, for example:
In terms of representation, this would correspond to the scalar pairs , and respectively. The double-soft limits of scalar pairs that do not contain 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:
For both cases, the double-soft limits are divergent due to the presence of the soft-graviton pole. To extract the finite term we consider the difference
This anti-symmetrised extraction procedure will allow us to isolate the finger print of the U(1) duality group.
We begin with case in eq.(12). The BCFW shift is given by
We will take the soft limit by setting , . Since the amplitudes vanish in the single-soft scalar limit, legs and must be on the same subamplitude of the BCFW diagram. However, since leg is shifted, plugging the explicit solution for in the generic multiplicity will render the momentum of leg hard. In this case, the subamplitude is again in a single-soft scalar limit, and thus vanishes. The only exception is when both legs and are on a three- or a four-point amplitude. For these diagrams, the propagators vanish in the limit, and one can potentially encounter cancellations.
BCFW diagram with a 4-point subamplitude
We first begin with the latter and consider the following BCFW diagram:
The contribution of this diagram is given as
The explicit solution to the shifted variable is given by
Since is of order , , thus the deformed is still soft in this channel. The spinors for the internal momentum is normalized as and . The fermonic delta-functions are given as
For convenience, we have explicitly written out the dependence.
It is straightforward to see that in the double-soft limit the bosonic pre-factor in eq.(2.2) is of order . We can use to localize the integral and the net effect is in is replaced by . Thus the integrand in eq.(2.2) can be written as
where in the last line is now the unshifted -point amplitude.
Now we want to pick the scalar components on legs and , which entails computing
If all four ’s and four ’s came from , we obtain the singlet contribution, which is divergent as . It turns out that the leading divergent term as well as the subleading contribution are the same for both (a) and (b) in eq.(12), and thus cancel under the anti-symmetrized extraction. To get a non-vanishing result, one must pull down one factor of from the exponent. This will result in finite contributions, as it brings down a factor of , along with the remaining factor associated with in . Thus for finite contribution we can either pull down an or an from the exponent.
Let’s begin with taking an from the exponent. This means that contributes 4 ’s, 3 ’s and left with an unintegrated. What we then get is
Next, we consider bringing down an instead. In this case, contributes 3 ’s and 4 ’s, and leaves an unintegrated. Using the explicit form of in eq.(2.2) we have
Using the explicit representation for , the above can be written as
BCFW diagram with a 3-point subamplitude
Let us now consider the following BCFW diagram with a 3-point subamplitude. The relevant diagram is displayed in diagram (a) of fig.1, which yields
The internal momentum and the solution for is given as
Again, we have explicitly written out the dependence for the exponents. A new feature is that the -point amplitude on the RHS is in fact divergent due to the presence of a soft graviton. In particular since is soft, the RHS of the diagram is an -pt amplitude in the single-soft-limit as illustrated in diagram (b) of fig.(1),. The soft-graviton divergence of the is given by Weinberg (); CS ():
As the three-point amplitude behaves as while the propagator as , the only term we need to consider is . The explicit supersymmetric single-soft operator of an -point supersymmetric amplitude is known to be HHW ()
where we only keep the relevant leading term.
Similar to the previous analysis, in order for there to be a non-vanishing result after the anti-symmetrised extraction, we must take one of the ’s from the exponents. In eq.(2.2) we can only choose and thus contributes 3 ’s and ’s. Again using to localise , the result is:
Using eq.(22) the term above can be rewritten as, to leading order,
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.(20) as
Using the -point super momentum conservation, the numerator can be simplified such that one has
Put everything together, we find that the difference for scenario (a) and (b) in eq.(12) is given by
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.
We now perform the SUSY reduction to . In the reduction, for each leg one needs to choose between integrating away to obtain the multiplet, or setting all s to be zero for the multiplet. Denote the points in two sets, with and . For the legs in , integrating will leave behind:
where we’ve used the identity . On the other hand for the legs in , we have:
The above result is precisely eq.(1). To see this, recall that