Low's Subleading Soft Theorem as a Symmetry of QED
aa r X i v : . [ h e p - t h ] J u l Low’s Subleading Soft Theorem as aSymmetry of QED
Vyacheslav Lysov, Sabrina Pasterski and Andrew Strominger
Center for the Fundamental Laws of Nature, Harvard University,Cambridge, MA 02138, USA
Abstract
It was shown by F. Low in the 1950s that the subleading terms of soft photon S -matrix elements obey a universal linear relation. In this paper we give a new interpre-tation to this old relation, for the case of massless QED, as an infinitesimal symmetryof the S -matrix. The symmetry is shown to be locally generated by a vector fieldon the conformal sphere at null infinity. Explicit expressions are constructed for theassociated charges as integrals over null infinity and shown to generate the symmetry.These charges are local generalizations of electric and magnetic dipole charges. ontents → symmetry 64 Charges 8 Soft theorems can be reinterpreted as symmetries of the S -matrix for which the soft particlesare Goldstone modes [1–3]. A priori, there is no guarantee that the resulting symmetry takesany simple or local form. However, for the case of the soft graviton theorem [4], the symmetryturns out to be a diagonal subgroup of the product group of BMS [5] diffeomorphisms actingon past and future null infinity [3]. There is also a subleading soft graviton theorem [6],which is equivalent to a Virasoro symmetry at null infinity [7–9]. For the leading soft photontheorem, the resulting symmetry was very recently shown to be the infinite-dimensionalsubgroup of U (1) gauge transformations which approach the same angle-dependent constantat either end of any light ray crossing Minkowski space [10]. In this paper we consider thesubleading soft photon theorem, specializing to massless QED. It has been known since the work of Low [11, 12], Burnett-Kroll [13] and Gell-Mann-Goldberger [14] that the subleading, as well as the leading, term of soft photon absorptionor emission is universal; see equation (3.2) below. In the massless case loop corrections arein general expected [15–17], but we will not consider their effects here. We re-express thesubleading soft relation as a symmetry acting on in- and out-states. However, unlike allthe cases mentioned above, the resulting symmetry is not a subgroup of the original gaugesymmetry. It acts locally on the conformal sphere at I where it is parameterized by avector field Y . However it is bilocal in advanced or retarded time. As already noted in [16],the bilocal form is reminiscent of the Yangian appearing in N = 4 gauge theories, but we This specialization is made, as in [10], to avoid dealing with singularities in the conformal compactifica-tion of past and future timelike infinity in the massive case. We expect this also to be the case for the subsubleading soft graviton theorem. Y is one of the global SL (2 , C ) rotations, thesymmetry implies global magnetic dipole charge conservation. Having a generic Y is a localgeneralization of this, in the same sense that supertranslations (superrotations) are localgeneralizations of global translations (rotations) in the gravity case.We wish to stress that, despite the precise formulae presented, the nature and significanceof the symmetry remains largely mysterious to us. It is not a subgroup of the gauge groupand, unlike the cases considered in [1,10] does not come under the usual rubric of asymptoticsymmetries. Moreover, the infinitesimal symmetry generators do not commute and theircommutators give yet more symmetries. We do not know whether or not a finite version ofthe symmetry transformation exists. The presence of so many symmetries would ordinarilyimply integrability, but it is highly implausible that all abelian theories with massless chargesare integrable. Another possibility is that there is no simple extension to massive QED, andloop corrections in the massless case somehow eliminate the symmetries. Despite all theseuncertainties, our formulae seem of interest and are presented here in the hope that furtherinvestigations can put them into proper context!This paper is organized as follows. In section 2 we give our conventions, the modeexpansion for the U (1) gauge field and define both the leading and subleading soft operators.In section 3 we review the subleading term in Low’s soft photon theorem, and then rewriteit as a symmetry of the S -matrix. We construct the associated charges and show that theiractions on the fields reproduce the infinitesimal symmetries. The charges are first presentedas integrals over all of past or future null infinity, and then, in section 4, are shown to reduceto boundary expressions after using the gauge constraints. This is surprising as they are notgauge symmetries! Finally we discuss the connection to dipole charges. In this section we collect essential formulae and introduce our conventions. For more detailssee [10].
Flat Minkowski coordinates ( x , x , x , x ) are given by x = u + r = v − r, + ix = 2 rz z ¯ z ,x = r (1 − z ¯ z )1 + z ¯ z . (2.1)where u ( v ) is retarded (advanced) time. In retarded (advanced) coordinates, the metric is ds = − du − dudr + 2 r γ z ¯ z dzd ¯ z = − dv + 2 dvdr + 2 r γ z ¯ z dzd ¯ z, (2.2)with γ z ¯ z is a round metric on the unit S . In terms of F µν = ∂ µ A ν − ∂ ν A µ and mattercurrent j Mν the Maxwell equations in retarded coordinates are − γ z ¯ z r ∂ u F ru + ∂ z F ¯ zu + ∂ ¯ z F zu + ∂ r ( γ z ¯ z r F ru ) = e γ z ¯ z r j Mu ,∂ z F ¯ zr + ∂ ¯ z F zr + ∂ r ( γ z ¯ z r F ru ) = e γ z ¯ z r j Mr ,r ∂ r ( F rz − F uz ) − r ∂ u F rz + ∂ z ( γ z ¯ z F ¯ zz ) = e r j Mz . (2.3)A similar expression applies to advanced coordinates. The mode expansion for the outgoing free Maxwell field is A outµ ( x ) = e X α = ± Z d q (2 π ) ω q (cid:0) ε α ∗ µ ( ~q ) a outα ( ~q ) e iq · x + ε αµ ( ~q ) a outα ( ~q ) † e − iq · x (cid:1) , (2.4)where q = ω q = | ~q | , α = ± are the two helicities and[ a outα ( ~q ) , a outβ ( ~q ′ ) † ] = 2 ω q δ αβ (2 π ) δ ( ~q − ~q ′ ) . (2.5)Outgoing photons with momentum q and helicity α correspond to final-state insertions of a outα ( ~q ). They arrive at a point w on the conformal sphere at I + . It is convenient toparametrize the photon four-momentum by ( ω q , w, ¯ w ) q µ = ω q w ¯ w (1 + w ¯ w, w + ¯ w, i ( ¯ w − w ) , − w ¯ w ) , (2.6)with polarization tensors ε + µ ( ~q ) = √ ( ¯ w, , − i, − ¯ w ) ,ε − µ ( ~q ) = √ ( w, , i, − w ) . (2.7)3hese obey ε ± µ q µ = 0 and ε +¯ z ( ~q ) = ∂ ¯ z x µ ε + µ ( ~q ) = √ r (1 + z ¯ w )(1 + z ¯ z ) , ε − ¯ z ( ~q ) = ∂ ¯ z x µ ε − µ ( ~q ) = √ rz ( w − z )(1 + z ¯ z ) . (2.8)We define the boundary field on I + by A ¯ z ( u, z, ¯ z ) = lim r →∞ A out ¯ z ( u, r, z, ¯ z ) = lim r →∞ ∂ ¯ z x µ A outµ ( u, r, z, ¯ z ) . (2.9)This is related to the plane wave modes by A ¯ z = e lim r →∞ ∂ ¯ z x µ X α = ± Z d q (2 π ) ω q (cid:0) ε α ∗ µ ( ~q ) a outα ( ~q ) e − iω q u − iω q r (1 − cos θ ) + h.c. (cid:1) (2.10)where θ is the angle between between the ~x and ~p . At large r the leading saddle pointapproximation near θ = 0 gives A ¯ z = − ie ˆ ε +¯ z π ∞ Z dω q ( a out − ( ω q ˆ x ) e − iω q u − a out + ( ω q ˆ x ) † e iω q u ) . (2.11)Here, ˆ x is parameterized by ( z, ¯ z )ˆ x ≡ ~xr = 11 + z ¯ z ( z + ¯ z, i (¯ z − z ) , − z ¯ z ) (2.12)and ˆ ε +¯ z = ∂ ¯ z x µ r ε + µ = √
21 + z ¯ z . (2.13)One may also check that in the gauge (2.7), A u = lim r →∞ ∂ u x µ A outµ vanishes on I + and hence F u ¯ z ( u, z, ¯ z ) = ∂ u A ¯ z ( u, z, ¯ z ). Using (2.11), a similar mode expansion for A z , and the commu-tation relations (2.5) the I + commutator is (cid:2) F u ¯ z ( u, z, ¯ z ) , F u ′ w ( u ′ , w, ¯ w ) (cid:3) = ie δ ( z − w ) ∂ u δ ( u − u ′ ) . (2.14)Similarly, defining the field A − ¯ z on I − by A − ¯ z = − ie ˆ ε +¯ z π ∞ Z dω q ( a in − ( ω q ˆ x ) e − iω q v − a in + ( ω q ˆ x ) † e iω q v ) , (2.15)4ives (cid:2) G v ¯ z ( v, z, ¯ z ) , G v ′ w ( v ′ , w, ¯ w ) (cid:3) = ie δ ( z − w ) ∂ v δ ( v − v ′ ) , (2.16)where G vz = ∂ v A − z . We would now like to construct the operators corresponding to soft photon insertions on I + and I − . To examine the soft limit of the above mode expansions, we define F ωu ¯ z ≡ R due iωu ∂ u A ¯ z = − e π ˆ ε +¯ z ∞ R ω q dω q [ a out − ( ω q ˆ x ) δ ( ω − ω q ) + a out + ( ω q ˆ x ) † δ ( ω + ω q )] . (2.17)For ω > ω < F ωu ¯ z = − e π ˆ ε +¯ z ωa out − ( ω ˆ x ) ,F − ωu ¯ z = − e π ˆ ε +¯ z ωa out + ( ω ˆ x ) † , (2.18)with ω > I − G ωv ¯ z = − e π ˆ ε +¯ z ωa in − ( ω ˆ x ) ,G − ωv ¯ z = − e π ˆ ε +¯ z ωa in + ( ω ˆ x ) † . (2.19)The zero mode of F u ¯ z is defined as F u ¯ z ≡ lim ω → ( F ωu ¯ z + F − ωu ¯ z )= − e π ˆ ε +¯ z lim ω → [ ωa out − ( ω ˆ x ) + ωa out + ( ω ˆ x ) † ] , (2.20)while on I − G v ¯ z ≡ lim ω → ( G ωv ¯ z + G − ωv ¯ z )= − e π ˆ ε +¯ z lim ω → [ ωa in − ( ω ˆ x ) + ωa in + ( ω ˆ x ) † ] . (2.21)As in [7], it is useful to define operators which create subleading soft photons, insertions ofwhich automatically have the soft pole projected out. These are given on I + by F (1) u ¯ z ≡ R du u∂ u A ¯ z = − lim ω → i ( ∂ ω F ωu ¯ z + ∂ − ω F − ωu ¯ z )= ie π ˆ ε +¯ z lim ω → (1 + ω∂ ω )[ a out − ( ω ˆ x ) − a out + ( ω ˆ x ) † ] , (2.22)5hile at I − G (1) v ¯ z ≡ R dv v∂ v A − ¯ z = − lim ω → i ( ∂ ω G ωv ¯ z + ∂ − ω G − ωv ¯ z )= ie π ˆ ε +¯ z lim ω → (1 + ω∂ ω )[ a in − ( ω ˆ x ) − a in + ( ω ˆ x ) † ] . (2.23) → symmetry In this section we rewrite the subleading soft theorem as an asymptotic symmetry actingon in- and out-states. Let us denote a state with n massless hard particles of energies E k ,charges eQ k and momenta p µk = E k z k ¯ z k (1 + z k ¯ z k , z k + ¯ z k , i (¯ z k − z k ) , − z k ¯ z k ) , (3.1)by | z , ... i , and hard S -matrix elements by h z n +1 , ... |S| z , ... i . The Low-Burnett-Kroll-Goldberger-Gell-Mann soft theorem [11–14, 19–21] then states that if we add to the out-state a positivehelicity photon with energy ω →
0, the first two terms in the soft expansion are h z n +1 , ... | a out − ( ~q ) S| z , ... i = ( J (0) − + J (1) − ) h z n +1 , ... |S| z , ... i + O ( ω ) . (3.2)Here J (0) − = e X k Q k p k · ε − p k · q ∼ O ( ω − ) , J (1) − = − ie X k Q k q µ ε − ν J µνk p k · q ∼ O ( ω ) , (3.3)with J kµν the total angular momentum operator of the k th particle. In [10] it was shownthat the leading J (0) term implies a symmetry under large gauge transformations whichapproach an arbitrary angle dependent gauge transformation at null infinity. Here we wishto understand the subleading J (1) term. For this purpose it is convenient to eliminate the J (0) − contribution using the projection operator (1 + ω∂ ω )lim ω → (1 + ω∂ ω ) h z n +1 , ... | a out − ( ~q ) S| z , ... i = J (1) − h z n +1 , ... |S| z , ... i . (3.4)From (2.20) one than has e ˆ ε +¯ z J (1) − h z n +1 , ... |S| z , ... i = e ˆ ε +¯ z lim ω → (1 + ω∂ ω ) h z n +1 , ... | a out − ( ~q ) S| z , ... i = − πi h z n +1 , ... | F (1) u ¯ z S| z , ... i . (3.5)6or the special case of a scalar field with J kµν = − i (cid:16) p kµ ∂∂p νk − p kν ∂∂p µk (cid:17) , rewriting ( p µk , q µ ) interms of ( E k , z k , ¯ z k ) in (3.3) gives for the right hand side of (3.5) J (1) − = − e X k Q k √ z k − ¯ z ) (cid:2) (1 + z ¯ z k ) ∂ E k + E − k ( z − z k )(1 + z k ¯ z k ) ∂ z k (cid:3) . (3.6)This is nonlocal on the conformal sphere. However acting with two covariant derivativesgives the local expression D z (ˆ ε +¯ z J (1) − ) = 2 πe X k Q k (cid:0) D z δ ( z − z k ) ∂ E k + E − k δ ( z − z k ) ∂ z k (cid:1) . (3.7)Acting with D z on both sides of the soft theorem and integrating the result against anarbitrary vector field Y z gives R d z D z Y z e ˆ ε +¯ z lim ω → (1 + ω∂ ω ) h z n +1 , ... | a out − ( ~q ) S| z , ... i = − πe P k Q k (cid:0) D z Y z ( z k ) ∂ E k − E − k Y z ( z k ) ∂ z k (cid:1) h z n +1 , ... |S| z , ... i . (3.8)For spinning fields we need to replace Y z ∂ z by the Lie derivative L Y . For a hermitian actionwe should include ˆ ε − z and Y ¯ z but we suppress this for notational brevity. Similarly for theinsertion of an incoming soft photon − R d z D z Y z e ˆ ε +¯ z lim ω → (1 + ω∂ ω ) h z n +1 , ... |S a in + ( ~q ) † | z , ... i = − πe P k Q k (cid:0) D z Y z ( z k ) ∂ E k − E − k Y z ( z k ) ∂ z k (cid:1) h z n +1 , ... |S| z , ... i . (3.9)Let us define soft photon operators Q + S = − e Z d zdu u∂ u A ¯ z D z Y z , (3.10) Q − S = 2 e Z d zdv v∂ v A − ¯ z D z Y z . (3.11)Hard particle symmetry operators Q ± H are defined by their action h E, z |Q + H = − iQ ( D z Y z ∂ E − E − Y z ∂ z ) h E, z | , (3.12) Q − H | E, z i = iQ ( D z Y z ∂ E − E − Y z ∂ z ) | E, z i . (3.13) Various conditions at the boundaries of I may lead one to impose constraints such as D ¯ z D z Y z = 0. Q ± = Q ± S + Q ± H . (3.14)Then the subleading soft theorem for massless QED takes the form h z n +1 , ... |Q + S − SQ − | z , ... i = 0 . (3.15)This expresses the subleading term in Low’s theorem as an infinitesimal symmetry of themassless QED S -matrix. In this section we express the operators Q ± , for the case of scalar charged fields, as integralsover local fields on I ± . The fact that this is possible is perhaps surprising as the factor of E − in (3.12) suggests time nonlocality.A massless scalar field has an expansion near I + Φ( u, r, z, ¯ z ) = φ ( u, z, ¯ z ) r + ∞ X n =0 φ n ( u, z, ¯ z ) r n +2 . (4.1)The commutation relation for the boundary field at I + is[ φ ( u, z, ¯ z ) , ¯ φ ( u ′ , w, ¯ w )] = − iγ z ¯ z u − u ′ ) δ ( z − w ) , (4.2)where Θ( x ) is the sign function. The boundary charge current is J Mµ = iQ lim r →∞ r ( ¯Φ ∂ µ Φ − Φ ∂ µ ¯Φ) = iQ ( ¯ φ∂ µ φ − φ∂ µ ¯ φ ) . (4.3)Expressing Q + H in terms of current operators gives Q + H = Z I + d zdu ( uD z Y ¯ z J Mu + Y ¯ z J Mz ) . (4.4)Using (4.2) as well as iπ Z e − iEu E + iε + dE = 1 + Θ( u ) , (4.5)8ne finds the desired action on the Fourier transform φ E = R du e iEu φ of φ [ Q + H , φ E ( z, ¯ z )] = iQ (cid:0) D z Y z ∂ E − E − Y z ∂ z (cid:1) φ E ( z, ¯ z ) . (4.6)Similarly on I − Q − H = − Z I − d zdv ( vD z Y ¯ z J Mv + Y ¯ z J Mz ) (4.7)generates the hard action (3.13) on incoming massless scalars. It is likely possible to gener-alize the construction to spinning fields but we have not worked out the details.Using the constraint equations (2.3), one can eliminate the matter charge currents andexpress the combined hard and soft charges as a boundary term. On I + Q + = lim r →∞ e Z I + dud z∂ u (cid:0) uD z Y z ( r F ur γ z ¯ z + F z ¯ z ) + 2 r Y ¯ z F zr (cid:1) . (4.8)For the field configurations that revert to vacuum at I ++ this reduces to the S integral Q + = − lim r →∞ e Z I + − d z (cid:0) uD z Y z ( r F ur γ z ¯ z + F z ¯ z ) + 2 r Y ¯ z F zr (cid:1) . (4.9)Similarly on I − Q − = lim r →∞ e Z I − + d z (cid:0) vD z Y z ( r F − vr γ z ¯ z − F − z ¯ z ) + 2 r Y ¯ z F − zr (cid:1) . (4.10)It is interesting to compare these to the expressions for the electric and magnetic charges Q and ˜ Q and the dipole moments ~℘ and ~µ : e Q + 2 πi ˜ Q = lim r →∞ Z d z ( r F ru γ z ¯ z + F z ¯ z ) (4.11) − e ~℘ + 2 πi~µ = lim r →∞ Z d z r F zr ∂ ¯ z ˆ x. (4.12)We see that if we take Y to be a global SL (2 , C ) rotation and use the boundary condition F z ¯ z =0 from [10], Q ± are nothing but the total magnetic dipole charge. This is ‘conserved’in the sense that, given that the system begins and ends in the vacuum, the total incomingdipole charge must equal the total outgoing dipole charge. More generally, Q ± are localgeneralizations of dipole charge in the same sense that supertranslations (superrotations) For such rotations, Y is real and hence entails nonzero Y ¯ z which we have been suppressing. We notethat the particular restriction on Y mentioned in footnote 3 would eliminate these rotations. Acknowledgements
We are grateful to F. Cachazo, T. He, P. Mitra, and M. Schwartz for useful conversations.This work was supported in part by DOE grant DE-FG02-91ER40654 and the FundamentalLaws Initiative at Harvard.
References [1] A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” arXiv:1308.0589 [hep-th].[2] A. Strominger, “On BMS Invariance of Gravitational Scattering,” arXiv:1312.2229 [hep-th].[3] T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’ssoft graviton theorem,” arXiv:1401.7026 [hep-th].[4] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. , B516 (1965).[5] H. Bondi, M. G. J. van der Burg, A. W. K. Metzner, “Gravitational waves in generalrelativity VII. Waves from isolated axisymmetric systems”, Proc. Roy. Soc. Lond. A269, 21 (1962); R. K. Sachs, “Gravitational waves in general relativity VIII. Waves inasymptotically flat space-time”, Proc. Roy. Soc. Lond. A 270, 103 (1962).[6] F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,”arXiv:1404.4091 [hep-th].[7] D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro Symmetryof the Quantum Gravity S-Matrix,” arXiv:1406.3312 [hep-th].[8] T. Adamo, E. Casali and D. Skinner, “Perturbative gravity at null infinity,”arXiv:1405.5122 [hep-th].[9] Y. Geyer, A. E. Lipstein and L. Mason, “Ambitwistor strings at null infinity and sub-leading soft limits,” arXiv:1406.1462 [hep-th].1010] T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “New Symmetries of MasslessQED.”[11] F. E. Low, “Scattering of light of very low frequency by systems of spin 1/2,” Phys.Rev. , 1428 (1954).[12] F. E. Low, “Bremsstrahlung of very low-energy quanta in elementary particle collisions,”Phys. Rev. , 974 (1958).[13] T. H. Burnett and N. M. Kroll, “Extension of the Low soft photon theorem,” Phys.Rev. Lett. , 86 (1968).[14] M. Gell-Mann and M. L. Goldberger, “Scattering of low-energy photons by particles ofspin 1/2,” Phys. Rev. , 1433 (1954).[15] V. Del Duca, “High-energy Bremsstrahlung Theorems for Soft Photons,” Nucl. Phys.B , 369 (1990).[16] S. He, Y. -t. Huang and C. Wen, “Loop Corrections to Soft Theorems in Gauge Theoriesand Gravity,” arXiv:1405.1410 [hep-th].[17] Z. Bern, S. Davies and J. Nohle, “On Loop Corrections to Subleading Soft Behavior ofGluons and Gravitons,” arXiv:1405.1015 [hep-th].[18] A. R. Exton, E. T. Newman and R. Penrose, “Conserved quantities in the Einstein-Maxwell theory,” J. Math. Phys. , 1566 (1969).[19] E. Casali, “Soft sub-leading divergences in Yang-Mills amplitudes,” arXiv:1404.5551[hep-th].[20] S. L. Adler and Y. Dothan, “Low-energy theorem for the weak axial-vector vertex,”Phys. Rev.151