Soft Photon and Graviton Theorems in Effective Field Theory
Extensions of the photon and graviton soft theorems are derived in 4d local effective field theories with massless particles of arbitrary spin. We prove that effective operators can result in new terms in the soft theorems at subleading order for photons and subsubleading order for gravitons. The new soft terms are unique and we provide a complete classification of all local operators responsible for such modifications. We show that no local operators can modify the subleading soft graviton theorem. The soft limits are taken in a manifestly on-locus manner using a complex double deformation of the external momenta. In addition to the new soft theorems, the resulting master formula yields consistency conditions such as the conservation of electric charge, the Einstein equivalence principle, supergravity Ward identities, and the Weinberg-Witten theorem.
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In this letter, we show that in a 4d local effective field theory of only massless particles, the tree-level soft photon and graviton theorems receive modifications at subleading and subsubleading orders, respectively. Specifically, in effective field theory, the soft theorems for positive-helicity soft photons or gravitons take the form
where and are the standard soft factors, well-known from the work of Low (1954); Gell-Mann and Goldberger (1954); Low (1958); Weinberg (1965); Burnett and Kroll (1968); Gross and Jackiw (1968); Jackiw (1968); Cachazo and Strominger (2014), and given explicitly in (4) and (5) below. The new soft terms are
where denotes the couplings of the associated effective operators. The tilde and superscript on the -point amplitude indicate that the particle type of the th leg of may differ from that in . Thus, the new soft terms are different from the factorized form of the traditional soft theorems. Only a small set of effective operators can modify the soft theorems and we provide a complete classification. We show that no matter which operator is responsible for the modification, the kinematic soft factor is uniquely fixed to take the form (3). Our result for the photon soft theorem naturally generalizes to non-abelian gauge theory.
Only effective operators with 3-point interactions can affect the single-particle soft theorems in (1) and (2). If an operator has too many derivatives, its interaction is too soft to affect the soft theorems at these orders. For example, does not modify the soft theorem, but the Pauli dipole operator does. All effective operators that can modify the soft theorems (1)-(2) are listed in (21) and (23). Note that our results imply that the soft graviton theorem is not corrected at subleading order in effective field theory. This is important for recent proposals Kapec et al. (2016); Cheung et al. (2016) connecting soft graviton theorems to asymptotic symmetries.
To investigate the soft limits, we present a novel approach based on a double complex deformation of the amplitudes. Combining a “soft shift” with two BCFW shifts allows us to identify the parts of the amplitude responsible for the soft theorems as factorization poles. Note that we are not deriving new recursion relations and the results are independent of which lines are shifted along with the soft line. The method allows us to take the soft limit in a manifestly on-locus fashion that emphasizes the path dependence of the soft theorems at subleading order.
The approach yields not only the well-known soft theorems and new soft terms, it also implies non-trivial, though well known, consistency conditions, such as charge conservation, the equivalence principle, and the supersymmetric Ward identities which state that a spin 3/2 particle must be coupled supersymmetrically to a graviton. We also demonstrate the Weinberg-Witten theorem Weinberg and Witten (1980) in the form that no massless spin particle can couple consistently to massless particles with spin 2 or less 111Similar conclusions were also reached for example in McGady and Rodina (2014) using factorization..
Soft theorems have been connected to asymptotic symmetries Cachazo and Strominger (2014); Kapec et al. (2014); He et al. (2014); Lysov et al. (2014); Kapec et al. (2016); Cheung et al. (2016) and this has led to a recent cascade of soft limit investigations, especially at tree level. Our work was motivated by the question of possible loop corrections. This is subtle for loops of massless particles because of IR divergences, but loops of massive particles can be integrated out to leave effective operators. As we show here, certain local operators can indeed modify the soft theorems at subleading orders. It would be interesting to know if these new universal modifications are associated with asymptotic symmetries.
Ii Complex deformations
We work with spinor helicity formalism in 4d following the conventions of Elvang and Huang (2015, 2013). Momenta are assumed to be complex so that angle and square spinors are independent. The momentum is taken soft holomorphically: and , with a small parameter. The standard soft theorems for soft positive-helicity photons and gravitons then take the form (1)-(2) (without the tilde’d modifications), where for a soft photon
and for a soft graviton
Here and are arbitrary reference spinors and . When the amplitudes have their momentum-conserving delta functions stripped off, the derivatives are taken with a prescription where one uses momentum conservation to eliminate a choice of two square spinors Cachazo and Strominger (2014).
In this note we use a prescription in which the soft limit is taken along a path on the algebraic locus in momentum space defined by requiring that the external momenta are on-shell and satisfy -particle momentum conservation. Start with unshifted momenta satisfying -particle momentum conservation, . Introduce the soft momentum such that the shifted momenta , defined as
with no other spinors shifted, satisfy -particle momentum conservation, . The spinor is completely arbitrary. The complex deformation (6) can be viewed as the combination of a soft -shift Cheung et al. (2015) and two BCFW shifts with parameters and , and . The choice of the two lines and is arbitrary and does not affect the physics conclusions.
For any momentum we have
Evaluating at , we obtain with
We are interested in poles at in the -particle amplitude. With , there are multiple contributions to such poles, since (as is obvious from (7)) all 2-particle channels with a soft line contribute. The role of is to separate these poles to different locations in the complex -plane and exploit that the amplitude factorizes on simple poles. Since the only possible poles in come from the 2-particle channels, we can write
because when a propagator goes on shell, the amplitude factorizes into a product of on-shell amplitudes 222In this and subsequent formulae we leave implicit any additional minus signs which may arise from reordering and crossing fermion lines during factorization or from rearrangements of spinor brackets. These signs are important for getting correct results, particularly for the relative signs in the supersymmetry Ward identities (24) but we will not spell them out in this Letter.. The sum is over all relevant momentum channels as well as over the spectrum of particles on the internal line, as indicated with the helicity label and a collective index of other quantum numbers. The superscript on the -point amplitude indicates that it in general depends on the channel momentum : .
Little-group scaling fixes the 3-particle amplitude up to a constant which we absorb in the associated coupling , where labels helicities and possible quantum numbers:
where , , and . In special 3-particle kinematics, another option is that could depend on angle brackets only; however, the shifted angle brackets vanish. The mass dimension of the coupling is
Using the kinematics above, (9) becomes
This is the “master formula” for the following analysis. (For comments about signs, see footnote .) We work with the Laurent expansion (12) for sufficiently small and, as we shall see, the soft theorems then follow from the terms.
Iii Photon and graviton consistency conditions
At tree level, locality requires that an amplitude can be singular only on a factorization channel. For and generic there is no associated channel, so the appearance of such a pole violates locality. Therefore, if the value of is greater than 1 in (12), the sum of residues of the apparent poles at must vanish. This imposes non-trivial constraints on the amplitudes.
Two non-trivial constraints arising from this requirement are
The first condition is simply charge conservation. The second condition can be satisfied only when the graviton couples identically to all particles; we recognize this as the equivalence principle. These results were first obtained by a different argument by Weinberg Weinberg (1964).
We now prove that a unitary local theory can have no interactions with . Let the highest value of in a theory be . The kinematic structure of the corresponding 3-particle amplitude is uniquely determined by little group scaling as in (10). Denote the coupling by , where are collective indices for all internal quantum numbers. CPT invariance requires that the theory also includes the amplitude of the CP conjugate states; its coupling is . Consider the soft limit of the 4-particle amplitude , whose -channel diagram includes the 3-particle interaction with and its conjugate, as well as the - and -channel diagrams, if relevant. The consistency condition arising from the absence of the pole in (12) implies
where and similarly for and (if present). Importantly, the are independent of . Applying the operator to (14) gives
Since and are arbitrary, we can choose them to be and in which case (15) requires . (Similarly, one can show .) Since
it can vanish only if . This shows that any couplings of interactions with must vanish.
For , the above argument fails because the power of in (15) is no longer strictly positive. Indeed, is perfectly fine for gravitons. For soft photons, however, we have proven that there are no interactions with .
Consider two examples of excluded interactions:
A 3-particle interaction with and gives . It may appear strange that such an interaction is excluded here, since the gluon amplitude certainly exists and is non-vanishing in Yang-Mills theory. However, this 3-gluon amplitude is non-vanishing in terms of angle brackets only. To produce such an amplitude in terms of square brackets only would require a non-local interaction Elvang and Huang (2015, 2013).
Consider a soft photon case of : take , , . This matrix element can be obtained from the operator which clearly is not local.
Since implies , we conclude from the above bounds on that no 3-point interactions involving photons, gravitinos, or gravitons are allowed if the sum of the three helicities vanishes.
Iv Soft photon theorems
Standard soft photon theorem. Set in the master formula (12) for a soft positive-helicity photon (). Expanding the -point amplitude and the denominator factor in small , there are two contributions at order . One goes as and takes the form
where is the unshifted amplitude, which is a function of the momenta that satisfy -particle momentum conservation. The result (17) is the standard leading soft factor .
The other contribution is order :
The shifted amplitude depends on through the momentum line as well as potentially through the shifted momenta and . In the momentum channel with , one uses the chain rule to find with
The first term gives the familiar subleading soft factor . The two other terms are consequences of our prescription for taking the soft limit. In contrast to Cachazo and Strominger (2014) where the soft limit is taken by defining an extrinsic continuation of the amplitude off-locus (away from the support of the momentum conserving delta function), our soft limit is calculated along an on-locus path defined by the deformation (6). The corresponding soft theorems can therefore depend only on intrinsic on-locus data. The modified differential operators can be understood as an element of tangent space of the momentum conserving locus. Our soft limit prescription can be shown to be equivalent to that of Cachazo and Strominger (2014).
Modification of the subleading soft photon theorem. The only other contributions from (12) for arise from interactions with . These give
which yields the new subleading soft factor in (3). By (11), the coupling must have mass dimension and . The new contribution to the subleading soft theorem involves an -point amplitude whose external states may differ from the hard states of . To determine which theories can have these corrections, one simply goes through the options to find that the only possible operators are
where is a spin 1/2 field and is the gravitino field. The operator is shorthand for the 3-particle interaction that arises from the metric expansion of .
To summarize, we have shown that in effective field theory the soft theorem for a positive-helicity soft photon takes the form (1), where and are as given in (4) with on the momentum conserving locus. The new soft factor in (3) is unique no matter which of the possible effective operators in (21) are responsible for the modification of the soft theorems.
V Soft graviton theorems
Standard soft graviton theorem. The familiar terms (5) of the graviton soft theorem (2) follow from the master equation (12) by setting and . As we have already seen in (13), the absence of the pole in this expression implies the equivalence principle: the graviton couples uniformly to all particles with a universal coupling . Using this, the terms can be rewritten in terms of the Lorentz generators and terms that vanish by momentum conservation. Since annihilates the on-shell amplitudes, the residue of the pole vanishes without imposing further constraints. The terms give the soft theorem (2) in a form with
Using the Schouten identity to write e.g. as well as using momentum conservation and annihilation of the amplitude by , one can show that the soft factors (22) are equivalent to those in (5) with the replacement as discussed for the photon soft theorem above.
Subleading soft graviton theorem unchanged. The only way to get a modification to the soft graviton theorem at order is via interactions with . The responsible local operators would have couplings of mass dimension and give rise to 3-particle amplitudes with . Restricting to spin2, the options are . The requirement that the pole in (12) vanishes implies that no such local operators exist. For the case , consider the 4-graviton amplitude at quadratic order in the non-standard effective coupling . Only one factorization channel contributes to in (12), namely
with implicit sum over possible internal quantum numbers of the exchanged spin-1 state. CPT invariance requires the couplings of the two interactions to be conjugate, so the pole in (12) will be proportional to . Absence of this pole requires . The three other cases of interactions can be similarly excluded. In conclusion, in a unitary CPT-invariant theory there can exist no local operators that modify the subleading soft graviton theorem.
This result may have relevance to recent discussions of asymptotic symmetries. In Kapec et al. (2014) it was shown that the universality of the subleading soft graviton theorem (2) is equivalent to the Ward identity of a Virasoro symmetry of the quantum gravity S-matrix. Our result implies that the subleading soft graviton theorem, and consequently the Virasoro symmetry, is unmodified at tree-level in the presence of local effective operators. In particular this includes curvature corrections.
Modification of the subsubleading soft graviton theorem. The only other contributions from (12) for arise from interactions with . By (11), the coupling must have mass dimension and . The corresponding operators in effective field theory are
All of these operators, up to constants, give the same correction in (3) to the soft theorem. The modification due to the operator was previously noted by Bianchi et al. (2015); Di Vecchia et al. (2016).
Vi Weinberg-Witten and supergravity
Weinberg-Witten. The equivalence principle mandates that any theory containing a massless spin 2 boson and a particle of spin must include a coupling with the universal coupling constant. Taking the soft limit of the helicity particle, this coupling has . As discussed in Section III, the condition for the vanishing of poles in has no non-trivial solutions for , which implies . Thus, by demanding locality and unitarity in the soft limit, we find that massless higher spin particles cannot interact in any way with particles of spin in a theory of gravity. This is an on-shell version of the gravitational Weinberg-Witten theorem Weinberg and Witten (1980).
Supergravity. We learned above that the usual soft graviton theorems arise from interactions with , for which . Such interactions include the standard graviton self interactions, and if we have spin 3/2 massless fields, the equivalence principle implies that the coupling of must be the same as that of . Let us explore the soft limit of a positive-helicity spin 3/2 particle. With , the interaction has , for which (12) yields a non-trivial constraint from the absence of a pole. Consider for example The -pole in (12) has three contributions with , namely from . (Lines give .) The sum over the three channels gives the consistency condition
This is precisely the MHV version of the supersymmetric Ward identities Grisaru et al. (1977); Grisaru and Pendleton (1977a) (see also 333The supersymmetric Ward identities have previously been derived from soft limits Grisaru and Pendleton (1977b), though assuming supersymmetry.). Thus we reached the well known conclusion that spin 3/2 to couple to gravity supersymmetrically. The role of the usual reference spinor in the SUSY Ward identities is here played by the soft momentum.
Acknowledgements.We thank Thomas Dumitrescu for discussions which initiated this project. We also thank Ratin Akhoury for useful discussions. HE was supported in part by the US Department of Energy under Grant No. DE-SC0007859. CRTJ was supported in part by an award to HE from the LSA Associate Professor Support Fund at the University of Michigan. SGN is supported by the National Science Foundation under Grant No. PHY14-16123, and he also acknowledges sabbatical support from the Simons Foundation (Grant No. 342554 to Stephen Naculich). SGN also thanks the Michigan Center for Theoretical Physics and the Physics Department of the University of Michigan for generous hospitality and for providing a welcoming and stimulating sabbatical environment.
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