Exploring soft constraints on effective actions
We study effective actions for simultaneous breaking of space-time and internal symmetries. Novel features arise due to the mixing of Goldstone modes under the broken symmetries which, in contrast to the usual Adler’s zero, leads to non-vanishing soft limits. Such scenarios are common for spontaneously broken SCFT’s. We explicitly test these soft theorems for sYM in the Coulomb branch both perturbatively and non-perturbatively. We explore the soft constraints systematically utilizing recursion relations. In the pure dilaton sector of a general CFT, we show that all amplitudes up to order are completely determined in terms of the -point amplitudes at order with . Terms with at most one derivative acting on each dilaton insertion are completely fixed and coincide with those appearing in the conformal DBI, i.e. DBI in AdS. With maximal supersymmetry, the effective actions are further constrained, leading to new non-renormalization theorems. In particular, the effective action is fixed up to eight derivatives in terms of just one unknown four-point coefficient and one more coefficient for ten-derivative terms. Finally, we also study the interplay between scale and conformal invariance in this context.
./Figs/ a] Massimo Bianchi, a]Andrea L. Guerrieri, b] Yu-tin Huang, b]Chao-Jung Lee, a]Congkao Wen \affiliation[a]Dipartimento di Fisica, Università di Roma “Tor Vergata” I.N.F.N. Sezione di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy \affiliation[b]Department of Physics and Astronomy, National Taiwan University, Taipei 10617, Taiwan, ROC \emailAddmassimo.firstname.lastname@example.org \emailAddandrea.email@example.com \emailAddyutinyt@gmail.com \emailAddr02222074@ntu.edu.tw \emailAddCongkao.Wen@roma2.infn.it
Effective actions in general contain an infinite number of higher dimensional operators whose precise coefficients require detailed understanding of their ultra-violet (UV) completion. In particular, except for low energy global symmetries and some positivity constraints , these coefficients are in principle arbitrary. On the other hand for effective theories associated with spontaneous symmetry breaking, it has long been known that soft theorems associated with the broken symmetries can be exploited to constrain the S-matrix, and in turn the effective action. Famous examples include Adler’s zero for single U(1) Goldstone boson (GB) , as well as its non-abelian extension . Recently it has been shown that a class of effective field theories, including non-linear sigma models, Dirac Born-Infeld (DBI) and a special Galileon, can be completely determined through the use soft theorems .
When spacetime, or both spacetime and internal symmetries are spontaneously broken, the soft-limits of GB’s in general will no-longer vanish and are proportional to lower point amplitudes.
If leads to a state in the physical spectrum, then the RHS can lead to a non-vanishing result upon LSZ reduction and thus a non-vanishing soft limit. The conventional vanishing soft-pion limits simply reflect the fact that pions shift under the broken symmetry, and hence does not lead to a physical state under infinitesimal transformations.
For broken conformal symmetry, the Goldstone modes that arise from dilatation and conformal boost are not independent, leading to a single dilaton . This implies that the soft-dilaton limit can be non-vanishing, as the broken symmetries relate the dilaton to itself. Indeed the plurality of broken generators is reflected in the universality of the single soft dilaton behaviour. In particular expanding the -pt amplitude involving one dilaton in terms of its soft momentum leads to leading and sub-leading terms that are simply proportional to the -point amplitude [6, 7]. In the presence of other global symmetries, the broken generators can rotate the dilaton into the new GB’s and vice versa. This is a common situation for super conformal field theories on the Coulomb or Higgs branch, where both conformal and R-symmetry are broken. Consider for example , super Yang-Mills (SYM) in the Coulomb branch, where the massless scalars comprise one dilaton and 5 GB’s for R-symmetry breaking SO(6)SO(5). As the broken R-symmetry generators mix the GB’s and the dilaton, we will find non-vanishing soft limits. In this perspective, the Coulomb branch effective action of maximal SCFTs not only enjoys maximal supersymmetry but also exposes “maximal broken symmetry”.
Note that these soft theorems must be respected both in the UV where massive degrees of freedom are present, and in the infrared (IR) where they are integrated out. In this paper we verify this perturbatively by computing the one-loop effective action of SYM up to six fields. This is done by considering the one-loop amplitude of maximal SYM in higher dimensions with the extra component of loop momenta identified as the mass of the massive multiplet. Expanding the integrand around the large mass limit, the integral yields the matrix element of the effective action. For non-perturbative tests, we examine the amplitudes from the instanton effective action obtained in . We have verified the validity of the new soft theorems to order at six points and at five points for one-loop amplitudes, where generically denotes Mandelstam invariants . While for the amplitudes generated from the one-instanon effective action  are always of order for the scalar sector, we have confirmed the soft theorems for pure-dilaton amplitudes to nine points and for dilaton and pion mixed amplitudes up to seven points. In [9, 10] leading and sub-leading soft theorems have also been checked against the amplitudes generated by the dilaton effective action, related to the trace anomaly in the recent study of the -theorem [11, 12, 13, 14, 15, 16, 17].
Soft theorems provide additional information on the analytic structure of scattering amplitudes, which can be combined with factorization constraints to recursively construct higher multiplicity results. Armed with the dilaton soft theorems, one can show that the matrix elements of the pure dilaton effective action are fully determined by a subset of operators via on-shell recursion . In particular, at -derivative order, the S-matrix for any multiplicity, i.e. any number of dilaton insertions, is completely determined in terms of operators of the form for . For maximal susy, the dilaton effective action for arbitrary number of dilatons are fixed up to ten derivatives in terms of three parameters: the coefficients of four-point operators at orders and . For , , we find that the dilaton amplitude at and are one and two-loop exact respectively for arbitrary multiplicity. At orders and , amplitudes with arbitrary multiplicity are completely determined in terms of the four-point coefficient. Beyond higher point coefficients are necessary to determine the -point amplitude.
Dilaton soft theorem is separated in two pieces, reflecting the fact that there are two kinds of generators being broken, scale and conformal boost. A theory endowed with only scale invariance will satisfy the leading soft theorem but not the sub-leading one. Thus the question of scale vs conformal symmetry becomes to which extent sub-leading soft theorem follows from leading. We study this question beginning with five-point amplitudes to very high order in (until ), and show that amplitudes satisfying the leading soft theorems automatically satisfy sub-leading soft theorem. Similar statements hold if one considers the amplitudes determined by recursion relations using the leading soft behaviour alone, for which we have verified the statements with many non-trivial examples. This can be viewed as supporting evidence for the equivalence of scale and conformal symmetry.
This paper is organized as follows: in section 2, we give a review of soft theorems for spontaneous symmetry breaking, and show that the mixing of GB modes under the broken symmetry can lead to non-vanishing soft limits, in contrast to the usual Adler’s zero. Explicit tests for the new soft theorems were conducted in subsection 2.1 on the one-loop and 2.2 for the instanton effective action. In section 3, we consider to which extent the matrix element of the dilaton effective action is fixed via soft and factorization constraints. In section 4, we consider further constraints from maximal supersymmetry. In section 5, we study scale vs conformal symmetry in the context of soft-theorems. We conclude in section 6.
2 Soft theorems
Soft behaviour of amplitudes with massless particles are often dictated by Ward identities of the underlying symmetries. Here we follow the discussion in , and clarify where one departs from the usual Adler’s zero. Spontaneous broken symmetry implies that the current associated with the broken generators excite GBs from the vacuum:
where label the generators. Inserting the current between a set of incoming and out going asymptotic states (), one finds, with
where the RHS is understood as an expansion in and we’ve separated out the pole term for the emission of a GB, which corresponds to fig.1(a), and is the transition amplitude.
Contracting on both sides of eq.(3), the LHS vanishes since the current is conserved:
This implies that
Thus in the limit where , the soft limit of the amplitudes involving a GB would vanish unless is finite. This requires non-vanishing contributions from diagrams associated with fig.1(b). Note that for the latter to yield non-trivial contribution, there must be more than one massless state in the spectrum that is charged under the current, and thus form the necessary three-point vertex.
The explicit form of can be directly read off from the Ward identity:
Fourier transform on both sides leads to
where represents Fourier transformed field. We now perform LSZ reduction on legs on both sides by multiplying and taking the momenta on-shell. The RHS vanishes for generic , due to one uncanceled inverse propagator from the reduction. Taking the limit , the RHS develops the requisite inverse propagator if yields a physical state in the spectrum. At the same time, the LHS is simply the amplitude with one soft GB. Thus we see that if does not correspond to another particle in the spectrum, then the RHS will not survive the LSZ reduction and hence vanishes. This is the Adler’s zero for soft pion emission . Indeed in these classical examples, the Goldstone bosons transforms non-linearly under the broken symmetry, and hence its infinitesimal transformation (a shift) does not yield a particle in the spectrum. On the other hand, if does produce a particle in the spectrum then the RHS is non-zero, and is given by the sum of Fourier transformed amplitude with the -th field transformed under the generator of the broken generator. This would be .
For broken conformal symmetry, one has the latter case. The broken dilatation symmetry constrains the leading term whilst the conformal boost generators constrain the sub-leading term in the soft momentum expansion. Thus amplitudes with single soft dilaton () satisfy the following universal soft theorem [6, 7]:
where the superscript indicates the degree in and is the vacuum expectation value of the dilaton field. The explicit form of are given by
where is the space-time dimension.
For spontaneously broken superconformal theories, the set of massless scalars comprise the dilaton as well as the GB’s for the spontaneous breaking of R-symmetry. If the dilaton is identified with one of the scalars that transforms non-trivially under the broken R-symmetry generator, following the above discussion the soft limit of the R-symmetry GB is non-vanishing. For instance, in SYM, the scalars form a 6 of SO(6), any one of the scalars taking a vev (say ) breaks R-symmetry down to SO(5), with 5 GB’s associated with the broken rotation generators with . Under this broken generator, the GB’s is rotated into , while is rotated into with a relative minus sign due to the antisymmetry of . Thus the soft limit of R-symmetry GB’s are given as:
where represents either a dilaton or , with and . In the following subsections we will verify the soft theorems by explicitly computations of scattering amplitudes one-loop and one-instaton effective action of SYM in the Coulomb branch.
We should add a comment at this point. In SYM one can define a different dilaton that represents the radial direction in holographic contexts and coincides with the above (up to a sign) if the other GB’s are set to zero. Moreover, the orthogonal ‘angular’ directions of SO/SO would behave as bona fide pions and satisfy Adler’s theorem, since they would transform non-linearly into one another and would not mix with the radial dilaton, that is a singlet of SO. While this is not particularly useful in the SYM context, since it would spoil the beautiful symmetry among the various scalars, for SCFT’s with lower supersymmetry, such as theories holographically dual to D3-branes at Calabi-Yau singularities (CY cones), the reduced R-symmetry would not allow such a ‘linear’ representation of the dilaton and pions as above but only the standard non-linear one, whereby the dilaton is an R-symmetry singlet (radial direction) and the pions are the angular directions of the Sasaki-Einstein base of the CY cone.
2.1 The one-loop verification
As discussed in the introduction, soft theorems hold both in the presence of the massive states and in the low energy limit where the massive states are integrated away. To verify this, we construct the one-loop effective action of SYM on the Coulomb branch.
with the super charge . Notice that the prefactors containing fermionic ’s in both four and five points are permutation symmetric. The integrals and are scalar one-loop box and pentagon integrals with massive propagators and we sum over non-cyclic permutations, and
In the above formulae the breaking of SU(4) to Sp(4) is manifest in the choice of R-symmetry indices in , which correspond to taking the anti-symmetric Sp(4) metric to be . In this notation, the dilaton represents fluctuations around the vev . With this choice the dilaton is and the other five real scalars corresponding to the pions of R-symmetry breaking are
One can straightforwardly verify that five and four-point amplitudes do satisfy the soft theorems. Six-point amplitudes are more involved, we utilize the integrand of 10D YM obtained in  (especially equation in the reference) and campactify to 4D. In particular, to distinguish the dilaton from other five scalars, we set if is dilaton and if is one of the R-symmetry pions, here denotes the loop momentum and is the 10D polarization vector which becomes a scalar after compactification. We computed six-point amplitudes up to the order from the integrands by performing the integrals in the large-mass expansion, and checked the six-point amplitudes also obey the soft theorems. We have done the same computation by obtaining the corresponding integrand for SYM using the generalized unitary cuts. Some of the results will be summarized in what follows in the form of the effective action.
Although the SU(4) R-symmetry is broken down to Sp(4) on the Coulomb branch, the effective action can be conveniently decomposed into SU(4) singlet and non-singlet sectors. The one-loop effective action up to six field strengths reads
where represents super-local operators that contain . In the Coulomb branch . Including an overall , the explicit form of the superfunctions reads
The Grassmann odd parameters appear in the super-polynomials
For the non-singlet part, we will only list the results of scalar operators which are relevant for the soft theorems we will discuss momentarily. Note that since the SO(5)Sp(4) subgroup of R-symmetry is preserved, the pion fields must come with even multiplicity. In the following we list the result of one-loop effective action with mixed dilaton and pions,
where the on-shell matrix elements corresponding to the higher-dimensional operators are given by
and the ’s are independent symmetric polynomials, they are given by
here denotes summing over permutations of elements, while denotes summing over permutations of and elements.
2.2 Non-perturbative checks
Relying on (unoriented) open strings and D-brane instantons
where is the complexified coupling and the on-shell superfields can be expanded in terms of the component fields and their conjugate according to
For the study of soft-dilaton and soft-pion theorems, we will turn on just the scalar fields so that
In the last step we have expanded the denominator and only kept the term which is non-vanishing after Grassman integration if one takes into account that the super-field becomes
when only scalars are turned on as in the case of interest here. Switching to 4-vector indices may be decomposed into a symmetric traceless tensor in the of SU(4)SO(6) and an anti-symmetric tensor in the of SU(4)SO(6). For instance, for pure dilaton sector only symmetric tensor contributes, and after performing the fermionic integration the action is given as,
and the dilaton has a non-vanishing vev
With the one-instanton action at hand, we have computed amplitudes up to seven points for dilaton and pion mixed amplitudes and pure-dilaton amplitudes up to nine points. We find that they indeed satisfy all the soft theorems. Here we list a few pure dilaton amplitudes
where the six-point amplitude is expanded in the basis given by eq.(36), while for the higher-point amplitudes
In appendix A, we have also listed the higher-dimensional vertices that generate the above amplitudes. As we mentioned we have verified that all these amplitudes indeed satisfy the soft theorems. In fact, as we will discuss in section 4.1, at four, five and six points, amplitudes (with both dilatons and pions) at order are fully fixed by SUSY and the soft theorems. Furthermore, for the pure-dilaton amplitudes all the higher-point amplitudes at this order are fully determined by the soft-dilaton theorems from the knowledge of the five-point amplitude, as we will discuss in the next section. Thus consistency with the conformal symmetry and SUSY (which fixes the form of the five-point amplitude), the pure-dilaton amplitudes in fact must take the unique form given in (51), and the same holds true for higher-point ones.
3 Constraining the effective actions by means of soft theorems
An immediate consequence of the dilaton soft theorem is its constraint on the effective action. A systematic way to explore soft constraints is the recently constructed on-shell recursion relations [4, 10]. On-shell recursive methods are constructed using the fact that under complex deformation of the momenta, the only allowed singularities are propagator singularities whose residues are determined by lower point data. Using the fact that S-matrix elements are analytic functions, we start with [29, 30]:
where the contour encircles the origin, and is the -point amplitude with deformed momenta and is the undeformed amplitude which we would like to compute. If is meromorphic, via the residue theorem, we can recast the amplitude as a sum over residues at finite values in the complex plane plus the one at infinity. The poles at finite values in the complex plane are simply due to factorization and their residues are determined by lower-point amplitudes. The usefulness of the recursion then relies on whether one can avoid contributions from the point at infinity or one can determine that contribution a priori. Effective theories in general do receive contributions at infinity. In [4, 10], it was shown that if it is known that the amplitude has universal behaviour in some kinematic regime, then one can instead consider