JHEP10(2019)092 s one, Springer July 1, 2019 : March 5, 2019 October 8, 2019 : August 29, 2019 : : Revised uperspace for massive Received Accepted Published hroughout, the conceptual ity formalism. This enables us . , a on-shell symmetry considerations ory of Physics, Department of Physics, Published for SISSA by https://doi.org/10.1007/JHEP10(2019)092 and Timothy Trott a [email protected] , = 1 three-particle amplitudes for particles of spin as high a Seth Koren . 3 N a,b 1902.07204 The Authors. c Scattering Amplitudes, Superspaces

We introduce a manifestly little group covariant on-shell s , [email protected] E-mail: Department of Physics, UniversitySanta of Barbara, California, CA 93106, U.S.A. Leinweber Center for Theoretical Physics,University Randall Laborat of Michigan, Ann Arbor, MI 48109, U.S.A. [email protected] b a Open Access Article funded by SCOAP ArXiv ePrint: as well as someand simple computational amplitudes simplicity for of particles this approach of is any exhibited spin.Keywords: T Abstract: Aidan Herderschee, to derive all possible particles in four dimensionsto using construct the massive on-shell massive superfields spinor and helic fully utilize amplitudes Massive on-shell supersymmetric scattering JHEP10(2019)092 9 1 3 7 26 29 31 34 36 12 13 15 18 20 22 24 33 34 37 12 17 licity spinors for the fictitious cattering states (that is, charges veloping a purely on-shell he unnecessary non-linear . This can be coupled with ity, to perturbatively build the – 1 – bases † η, η -matrix computations in four dimensions. This is because he S -matrix structure out of on-shell units that bypass the need S = 1 three-particle superamplitudes A.2 Grassmann calculus 5.7 Higher spin amplitudes A.1 Spinor helicity for massive particles 5.2 One massless5.3 vector One massive5.4 vector Two vector5.5 superfields Massive and5.6 massless vector multiplet Self-interacting interactions massive vector supermultiplets 4.2 Strategies for4.3 enumerating amplitudes without Parity central 5.1 Three chiral supermultiplets 3.1 Superfields 4.1 SUSY invariants and the N may be used astheir a momentum complete and description spingauge of redundancy polarisation) the of data without polarisations ofon-shell recourse used external methods, to in s such t the as Feynmaninternal recursion rules and generalised unitar 1 Introduction The spinor helicity formalismformulation of has been a key ingredient in de B Comments on higher-leg amplitudes in SQCD 6 Conclusion A Conventions and useful identities 5 3 On-shell supermultiplets 4 Constructing and constraining on-shell superamplitudes Contents 1 Introduction 2 Little group covariant superalgebra for massive particles JHEP10(2019)092 ] 8 ]). Mas- 7 nto two null little group covariant = 1 theories to exhibit es of massless particles, st principles all possible lso be utilised to describe cs of massive particles pre- specialize by incorporating vector description of these N persymmetric Ward identi- eneral possible interactions thods for massive particles. mposition often convoluted ternal spin direction. This to describe each of these in ch external scattering states on of momentum, analogous h higher-spin states. The on- ing within the spinor helicity o perties of the external states , non-existent time-like com- than one. We also make some licity spinors for massive par- ns for a vector in an Abelian ed by an inappropriate choice. y extended. It is thus natural metry relating some component to organise the amplitudes into far from realised. massive momenta, SU(2). This , to directly use symmetries to ious breaking of Lorentz invari- heoretic methods. However, the was never a gauge ambiguity in An on-shell superspace for mas- ttering data, that are consistent ntify and construct. d theories was given in [ he polarisation structures. See [ ], where the spinors of the null vectors 6 – 2 – ] and we will rediscover their results along the way, 9 ]. This generally involved decomposing time-like momenta i 5 – 1 Supersymmetry (SUSY) offers an idealisation that, in theori An advance on this formalism was made in [ After laying the foundation by writing the superalgebra in a Helicity spinors have been adapted to describe the kinemati makes the relations between theties amplitudes (SWIs) transparent imposed while by simultaneously preserving the t su has enabled the utilityto of look on-shell to methods supersymmetry toWe as therefore be a here drasticall testing amalgamate groundsticles the for on-shell with little me group-covariant the he formulationare of grouped an into on-shell supermultiplets superspace without in reference whi to an ex Lorentz-covariant terms that are simpler to interpret, ide form and constructing covariant supermultiplets, we turn t for a review of on-shellsive particles superspace was for first massless constructedalbeit particles. in re-expressed [ in the covariant formalism. This helps ponents still source tensionfundamentally 3-vector in objects a (recent use symmetricsive in and treatment effective fiel massless of particles a mayformalism. thus 4- be treated on equal foot to the U(1) redundancy inthe the polarisations massless of case. thebuild However, external it amplitudes massive may that a states, haveunder or the their better required individual transformation littlethe pro group rotations. polarisations of While there massive particles in the Feynman rules the method, as sought-after patternsFurthermore, this could choice easily of be directionance obscur often in involved the a amplitudes. spur were organised into representationssymmetry of the represents little the group redundancy for in the spinor descripti vectors and then proceedingsome way. with However, massless the helicity element spinors of arbitrariness in this deco comments on how SUSY generallyshell constrains supersymmetry interactions wit allows usgiven to only simply the catalogue spectrum theadditional of most on-shell a data g such theory. as the Itamplitudes presence is of to also a parity each easy sym to other, further or the absence of self-interactio the usage andthree utility particle of amplitudes, this thewith formalism. most these primitive symmetries on-shell We and sca construct involve particle from spins fir no greater dream of a fully on-shell formulation of particle physicsviously is in [ degrees of freedom that frequently arise in standard field-t JHEP10(2019)092 to we are ]) ] in 5 ˙ 4 (2.1) β 10 11 α, which we , where -symmetry to a s R ˙ † βB Q , s. The ‘central charge’ ) are (following [ αA = 1 theories. We then , µν nticommutation relations Q β A s, which package together revious treatments [ M N e external legs if the states etry — or supercharges — Q s technology in section ) l identities. The reader can eramplitudes, we may obtain epresentations can be simply ˙ β tary mechanics which will not ce erence frame. In section ges, leaving an elegant, frame- µ β e spinor helicity language. This σ ˙ incar´ealgebra. Before discussing α algebra to a graded Lie algebra in s for their construction, including ν α σ we present the little group covariant to construct massive on-shell super- , − 2 3 µ ˙ β P ν β ) . σ ˙ β ˙ ˙ β α µ α , ˙ 1 and typically breaks the α µ α σ ǫ σ ( αβ fermionic generators ( > ǫ ˙ B A – 3 – β AB ˙ δ α N N 2 Z ǫ AB , i ) and rotations/boosts ( 4 − Z − µ P = = = count the number of left-handed spinor supersymmetry ] = 0 ] = } } } µ B B µν N ˙ ˙ † β † β βB ,P ... ,M ,Q ,Q ,Q αA A ˙ † α αA αA is allowed for Q αA [ = 1 Q ∗ Q Q Q [ { ) { { AB , we present a lightning review of massive spinor helicity in A, B Z ( A.1 − = BA Z -symmetry group and will be discussed further in what follow − ) indices and R C = , We will be interested in the construction of superamplitude In appendix This paper is structured as follows. In section The automorphism group of the supercharges preserving the a AB scattering data for entire representationsthis, of we will the first super-Po rewriteallows the the superalgebra spinor using indices the toindependent massiv be formulation stripped of out the of algebraconstructed. the from superchar which This massive r provides an aesthetic improvement over p subgroup. SL(2 Z is the addition to setting up our discussion of superamplitudes. also develop our conventionsfind and there provide further introduction relevant to andbe the usefu remarked subject upon and in the elemen the main text. discuss general features of superamplitudesthe and implementation strategie of parityconstruct symmetry. elementary We exhibit three all particle of amplitudes thi for flat spa on-shell SUSY algebra.multiplets This as allows coherent us states in of section the supercharges in any ref 2 Little group covariant superalgebraThe for general super-Poincar´ealgebra extends massive the particle Poincar´e 4 dimensions through the introduction of conclude. the necessary dependence ofare the to couplings be on identified the as masses elementary of superfields. th theory. By studying the high energy or massless limits of sup generators. The Liewith brackets the of generators the of generators translations ( of supersymm JHEP10(2019)092 e . i (2.2) (2.4) (2.3) (2.6) (2.5) AB i m so that 2 Z IJ ǫ = o J,B † i , , q E . . ˙ A ˙ is the particle’s central β β spinors transform under I,A † i

| . We can likewise define † i i I i external legs are massless, β pon by the super-Poincar´e Q q i h | Z I

anifest chirality for manifest I n erators and transformations ressing the symmetry gener- i A i † seful example). For massless [ i A D † q i,I ach leg. For massive particles, genspace, the labeling of which rality and helicity/polarisation ˙ m of its action on each external α i s to represent these generators, q 2

s for each leg by projecting the 2 icle momentum eigenspaces. For rge can only either raise or lower a m I √ mmutation relations , 1 i te. Each symmetry generator may harges in the form of chiral spinors 2 √

− i . Here √ i = = m = i,AB B A ˙ 2 β = ˙ A δ β Z A † i, ˙ αβ † i, µ A = Q † i,I IJ σ Q ǫ µ i o p − . As usual, the SU(2) little group indices may B † i A † = = I – 4 – q , q , q I i,A

i,A  q ˙ αβ i = q i,A p α J i,B q † ] α i,A ] ) I n i , q is i Q | I A | I q i 2 i 2 I i,A  q √ √ i  in an appropriately covariant fashion. By construction, th − − 1 m and ( of the particle makes them dimensionless. Note that we are i 2 − = = i √ I,A , m † B A q = i,αA i,αA δ − Q Q ) become IJ I i,A ǫ = q † 2 is just a normalisation convention and has been chosen here − 2.3 ) √ = / I,A o q J,B † i , q I i,A q The stripping of the helicity spinor effectively exchanges m The external particles in a scattering amplitude are acted u n defining these operators asconvenience, being the restricted inverse to relations single-part are given by where the factor of mass spin polarisation (of whichstates, helicity these is are identical often anda a each state’s natural chiral helicity. spinor However, and for supercha u massivewill states, do the a superc superposition of both, for the usual reason that chi These hold only on a particularwe single-particle leave momentum ei implicitly subsumed in the particle label The factor of 1 momentum eigenvalue of particle and the other anticommutators are zero. the little group covariant supercharges satisfy the antico this means exhibiting the SU(2)ators little acting group on symmetry each by particle exp generators as separate tensor factors oftherefore the be scattering represented sta on ascattered scattering amplitude particle. as the This su on will each allow leg us separately. tobecause We study will these symmetry use encapsulate gen spinor the helicity on-shell variable kinematic data for e on-shell, little group covariantsupercharges supersymmetry onto the generator spinors of a given particle charge. Also ofconjugation, note ( is that, as a result of the way the massive The little group covariant supercharges satisfy the algebr be raised and lowered usingthe the supercharges Levi-Civita ( symbol. When the JHEP10(2019)092 i ), is = m for 2 I,B (2.7) (2.8) (2.9) † i ˙ (2.10) ¯ † βB < q i,AB , | Q i Z I Z i,A | -symmetry q . R and . = (¯   = 0 αA )), while for the i,a I o Q B,I N q † i q J,B to † i i ¯ q , entral charge, m i,AB I,A -indices) in the way that 2 † ). The symplectic 2-form i Z I,A q R ) (or U( and their conjugates then a may be constructed with † i N ¯ q (   may mix because their index t preserve the anticommuta- N d. 2 supercharges are decomposed n mmetry, the full give a manifest representation I,A and ¯  al charges satisfying arge: q I,A i | length vector freedom in choosing a spin direc- plicit † i q m I i,A Z es’ supercharges will be implicit in | N , q 2 . ) may be combined into the relation A, B ab  , in the conjugate representation) while Ω 2.9 = 0 . The ¯ # − I

-symmetry of the massive representation. IJ ,( = 0, although such multiplets still carry 1 ! 0 IA ǫ 2 † − i R N J i,B s q  Z = ¯ q 0 I | i i , − " m

Z | is a symplectic 2-form: is the identity of the same size. Specifically for 2 1 a > . This allows for a rotation into a basis that – 5 – I i,A J i,b =  ¯ I q   q ) tensor representations (such as the supercharges)  AB BA , 2 for ) may be simplified. Unlike for massless particle − -symmetry, because the supercharges can mix with AB N Z (and on Ω  1 = 0, this would be SU( I i,a Ω i R | N 2.4 − q i I A and their conjugates r ) q m  + Z , AB = | 2 and − N B A Z B I,A  δ 1 q AB IJ =

IJ + ǫ ǫ a  ). If − I 1 i,A q may be complex (corresponding to two central charges) and th = 2.4 − and   i while Ω 2 zero matrix, while o ) illustrate the symplectic Z  D | i R 2 1 , the relations ( , N i i J,B √ 2.9 m † Z i ≤ N ∈ | 2 ). Grouping the supercharges into a 2 ¯ q × m AB i a ,  = ǫ 2 2 Z N 2.9  I i,A = < for I i,A ¯ q ¯ q | n i A ) has a manifest USp( = 2 AB Z | = 2.4 , where D a may then be used to convert USp( AB The relations ( For In the simple case in which all legs carry a single, electric c In such cases, representations of the supersymmetry algebr = 2, Ω Ω AB i and the bars on the diagonalised supercharges will be omitte these states). Henceforth, thissubsequent redefinition discussions of the of particl SUSY representations with centr where 0 is the Z group is actually determined bytion all relations of ( the automorphisms tha heights may be changed by satisfy the anticommutation relations without a central ch where central charge (and this would still appear in relating ¯ a structure identical to that of the case with canonicalises the anticommutators. This basis is given by representations, the generators While ( the Levi-Civita tensor does for SU(2). central extension is general. The supercharge labels their conjugates while preserving the SU(2) little group sy tion as a state label,into which supercharges determines characterised how by the polarisation. chiral spinor are no longer identical. The little group here describes the central charge considered above, this would be broken to USp Ω into conjugate representations (i.e. raise and lower the ex N preserving the algebra ( where of a symmetry group that acts on JHEP10(2019)092 ), ry N tself ] for -basis (2.11) (2.12) gether 2-BPS 14 ng this R / ) by the ) rated. As N -symmetry N R 2-SUSY. For / = 4 SYM, the N ]. However, it is . In such a basis, N ]. The redefinition 12 AB 13 )[ Ω mmetries. A 1 N ∝ ection rules on scattering ttern in ive central charges than the ld also be respected by the tiplet [ arges have definite opposite Clearly, BPS states are an- from the origin. See [ a single leg’s supercharges. i,AB . ges that annihilate each state lations described above fails in ble in which some smaller frac- Z ribed. However, as this basis is alf of the number of supercharges tation of BPS states in scattering † i,IA herefore the massive ¯ q . − B † i,I q = i,AB i,IA ¯ q Z i -symmetry does not occur for massless repre- 1 R m – 6 – − 2 ]. . For the central charge considered above with 2-BPS leg there is nevertheless an SU( IA = 15 IA / † i † i ¯ q q − i,IA + ¯ q 2-BPS as it is annihilated by half of the supercharges. = / -symmetry is enhanced to USp(2 IA i q IA i R -symmetry group will also be broken further beyond USp( ¯ q R , the BPS condition reduces to 2 is the special BPS limit. This typically occurs for elementa / ) symplectic 2-form. Thus the anticommutator is effectively i i Z ) symmetry of the SUSY algebra, which is broken to USp( m N ) rotation matrix, which must be accounted for when adding to may be absorbed into a redefinition of the supercharge. Calli 2 N (arg i i N = 2 − Z -symmetry is broken to USp(4) when the central charge is gene e | i R N × Z | i,IA , the multiplet is 1 q -SUSY may be represented as a massive non-BPS state of = AB is a 2 N Ω ab i,IA ∝ q The explicit SU( The phase of The case More generally, with more complicated configurations of act AB but this will still be a symmetry restricted to the algebra of the representation of thedifferent leg’s for supercharges is each just leg,may as the differ desc by linear a SU( combinations of superchar the total supercharges. The in which the central charge can be rotated into the form central charge of these massive single particle states, is t group expected for a theory with half of the number of supersy state in Z This condition again preservesnihilated the by supersymmetry the algebra. combination ¯ Configurations with multiple central chargestion are of also supercharges possi annihilate the state. particles which obtain mass through Higgsing of a vector mul example, for the simplest spontaneous symmetry breaking pa further discussion. Further elaboration uponamplitudes the has represen been recently made in [ simple case discussed above, for each 1 time ¯ of supercharges that give thethe canonical BPS limit. anticommutation re This isare because, for eliminated these through representations, h the reality constraint often broken by interactions. The enlarged sentations of the SUSYhelicity algebra and because cannot the mix. non-zero superch massless SU(4) Here Ω the former is unbrokenamplitudes by at dynamics the and origindynamics imposes of and the stringent organise moduli sel the transition space, matrix the structure latter away shou a symplectic 2-form and the JHEP10(2019)092 ] ) ]. ]. ], 9 9 17 2 if 20 / 2.13 ] and (2.13) 21 ), ( , udes into 2.4 19 actor of 1 † wer supersymmetries these papers and the h the supersymmetric ] to formulate the su-  18 C † i,I reviously developed in [ q fermionic oscillators, where his has been utilised more ng the algebra ( here will parallel that of [ lished as a convenient organ- mmetric theories, so need to N complex projective space [ der both supersymmetry and i,AC or massless theories, and now -level integrands in the limit of providing flexibility to choose a he external legs will allow us to ], where on-shell superfields for ent states of the supersymmetry na, such as high energy limits or = 4 SYM does not yet exist). In Z tes. Scattering data is simplified ing states we may describe their i 16 amplitudes in these theories (it is 1 N m 2 + i,IA q , may be derived for these scattering states i ] and will be elaborated upon in this context m   – 7 – 2 25 B † i,I q | ≤ . The BPS bound follows by simply requiring that i  Z ], as well as the elucidation of the dual superconformal i,AB 2 | ) 2 | i Z i ]. The on-shell superspace was first conceived of in [ 24 i m Z – | 1 16 m (2 22 2 − + 1  † i,IA ) q  i,IA ]. This enabled the formulation of the super-BCFW shift [ q ( = 4 SYM, it makes transparent the classification of the amplit 21 – = 4 super-Yang-Mills in [ N i,IA 19 ), the massive supersymmetry algebra is that of q N -matrix entries using superamplitudes, which manifest bot 2.9 S if the representation is not BPS, but can be reduced by up to a f = 4 super Yang-Mills (SYM) and was employed later in [ N ] for a review of these topics. Amplitudes in theories with fe 4 massless theories were constructed. We refer the reader to ]. 8 General on-shell superspaces for massive states have been p For massless theories, on-shell superfields have been estab From ( Finally, the BPS bound itself, N 14 < = 2 -symmetry [ particular, for sectors of a fixedR order of helicity violation, which close un turn to the construction of massive supermultiplets. review for details of the construction of superamplitudes f isation of the representations [ from the fact that the operator, with the improved organisationrecently offered for by thein little [ group. T However, the manifestation of theimprove massive upon little the group presentation for of t thisspinor exposition, basis best as suited well as for thecomplex study momentum of shifts. particular phenome Much of the subsequent discussion symmetry and dual twistor representationsSee of amplitudes [ on have also been formulatedN in an on-shell superspace in [ the subsequent construction of alllarge tree gauge amplitudes group and dimension loop [ for pertwistor space representation ofworth tree-level noting that scattering an off-shell superspace formulation of simplifies to understand the structure of supersymmetricconsiderably scattering by sta the grouping of component statesalgebra, into known coher as ‘on-shellcollective superfields’. For these scatter this be non-negative. 3 On-shell supermultiplets We seek here to construct scattering amplitudes for supersy N being a sum of squares, must have non-negative spectrum. Usi Ward identities in a simple manner. JHEP10(2019)092 | ], = Ω 8 h i,IA will A The q † (3.1) (3.2) (3.3) (3.4) I q A † | i,I s of [ i q η h . The labels | η I s of conjugate † h i,A η A I . ∂ absence of central r in this definition

he choice of which ∂η † combination of  η or all of the I D i A i,I ∂ A A I assmann variables which η θ ∂η fford vacuum states, i,IA I  ] † A q I η 2 in constraining the form of i massless particles described orms coherently under little states which are eigenstates e | dex may be identified with a √ tiplet as the Clifford vacuum † operators, satisfying e most manifest symmetries is he Clifford vacuum. The action i ors on the superamplitudes η = X ates. We will reverse the heights N  2 2 A d √ Q Z A = A i,I θ η  A = | I i,A | i Q q η η , as well as their conjugates e representations for the same supermultiplet | A i,I † Ω η η h – 8 – . In this section, we will restrict our attention to A i,I | h = | h η η | η and I i,A i

h h q I η I η i A h I † A

η η A I

-covariance, either all of the i η I X i R -indices of the supercharge must be tensored together and 2 A η e R θ √

N 2 parameterise a linearised supersymmetry transformation. − 2 d  √ = ’ in the symbols just introduced). Following the convention A Z − θ A A

† = = Q

† E η and A † D i Q A -index (‘ A θ basis, the supercharges act as (assuming for simplicity the | R θ η D are anticommuting Grassmann algebra generators. As is clea | ). The Grassmann Fourier transform may be used to define a basi i for each such pair can be chosen as annihilation operators, t , which are annihilated by either set. Generally, any linear η A i,I A i,I

h η A η † Ω i,I ‘lowering operators’. To build these states we introduce Gr − q

], where little group and -indices relative to this reference. In the An entire supermultiplet may be encoded as a coherent state To ensure little group and ( 8 | N R i η charges) superamplitudes. exist and are related by the Fourier transform will be useful where of the supercharges onin [ the states generalizes the action for and will be madegroup manifest transformations with below, the the same little entire group superfield weight as transf t be chosen as the lowering operators. These will have some Cli supercharges may therefore be represented as linear operat states. It is defined with its inverse respectively as: The fact that both the two different corresponds to the selectionin of the a superfield particular representation, statearguably but in desirable. a the choice mul that yields th multiplets which are not BPS,supercharge in which case the oscillator in h where small and all particles will beof represented as outgoing scattering st then decomposed. These are eigenstates of the annihilation transform in the little group of each particle of shortened. Supermultiplets may be represented as coherent here match those of the supercharges and JHEP10(2019)092 i ) s d η 3.4 (3.8) (3.5) (3.7) (3.6) . This i ˆ η . Here i η → + → onents. i, ) requires that η − tle group tensors i, 3.4 η . | as iξ hese coherent states: time supercharges ( + η h end upon a preferred frame specified otherwise, so that s ones as similarly. sless on shell superfields (as en as the quantisation axis). ssless and massive represen- plet in half, leaving the def- he formalism, as the massive = umber of supersymmetries. ly on coherent states or their neate a division of the massive . ambiguous. As a consequence, denoted here as nt fields in reducible representa- I i,A A q i,I . In the massless limit, following . ∂ nstrained by the needed fermionic A I + | i,A ∂η iξ q η h − − e † A | η = η and A A i h η η + I i,A A † ηe i, q i,A q q N e = 1. The states in the multiplet are generated , d | – 9 – | Ω η N h Z h A i,I A I η = = η | parameterise the supersymmetry transformation pro-

I − i † A † i , massless limits are most naturally taken in the helicity η

η iξ h = I D i − e A A A.1 † i,I of the appropriate chirality. The action of the supercharge θ q = i -symmetries, which need to be disentangled to locate the fiel

), the form of the spacetime supercharges ( R A = † i,I q I A.1 I † A † A ξ iξ e | η and h  I ) give the supersymmetric Ward identities relating the comp i A θ 3.6 on the Clifford vacuum and then decomposing the resulting lit  I ]). For reference, massless coherent states are defined here q = 16 A I ξ We consider first the simple case of However, for non-BPS states, the massless limit of the space By construction, the massive on-shell superfields do not dep As established in appendix Supersymmetry transformations of both types act simply on t content. The structure of these vary significantly with theby acting n tations of the supersymmetry(non-BPS) algebra states is non-trivially firmly ingrained representthe in a rules t larger established algebra in ( of reference. However, as a result, the difference between ma into irreducible representations, which will be further co encoded in ( tions of the little group and does not represent the actionsuperfield of into a separate supercharge, but massless does representations. deli 3.1 Superfields The coherent state construction generically gives compone reduces the number ofinitions supercharges of represented on the the spinor-stripped multi supercharges the massive Grassmann variables are mapped onto the massles basis for the massive littleWe group will (in adopt which the momentum conventionlittle is that group chos this indices frame always is denote chosen helicity by unless default. jected onto the spinors of leg Massless analogues of the previous formulae may be obtained expressions obtained upon takingmatrix entries the will massless involve a limit residual direct Grassmann variables or on individual legs: Here, in e.g. [ is the massless Grassmann variable used to construct the mas JHEP10(2019)092 , the (3.9) | is the (3.11) (3.13) (3.10) (3.12) onent. Ω h η . Because = K φ q K q 1 2 ). Here =0 − I | η ion is Majorana and

3.9 Ω uperamplitudes from h Φ I Grassmann derivations. IJ in ( rmionic Clifford vacuum. onent amplitudes in the ∂ resulting massless super- acts non-trivially on the ǫ anti-superfield is required ∂η , rent supermultiplets (that ucted similarly. ents that may be described ˆ η I I = escribe the two polarisation . ˜ λ ∂ ay therefore be arranged into − J J ∂η → uum to be a scalar the equivalence of Grassmann η q ntum numbers (except for pos- 2 1 I J ηχ laced by integration. The Grass- + q η ed to create an on-shell superfield he SU(2) little group, where each re, this amounts to combining two η | = is decomposed into a single scalar + 1 2 ˜ φ . Ω ˜ I φ φ ˜ φ − η I ) − = IJ − h and in this example) or differentiating with η ǫ IJ 2 1 − ( − η = − -conjugate states), a little group covariant , Φ + Φ − W = =0 ˆ IJ η → J I I ˜ φ + η η -charges. Otherwise (as is necessary if the field χ

CPT IJ I − Φ R = 0 (Φ – 10 – ˜ + Φ φ η η ˜ φ, I η and → η + H ∂ I I ∂η φ Φ + η q | = + + Ω χ I I Φ = λ = − h = + = I Φ is the variable corresponding to the trivially-acting comp I , χ χ η W =0 I η

= Φ φ charges). If the multiplet is self-conjugate, then the ferm here). This likewise allows for the extraction of massless s R + Φ We can next construct a vector superfield by starting with a fe In the massless limit, the superfield decomposes into compon Component fields can be extracted from the full superfield via Similarly to the extraction of component states above, each − ( ˆ η ∂ ∂ multiplet contains states of opposite spinClifford projections. vacua into He anstates SU(2) of fundamental a representation fermion’s to degrees d of freedom. The superfield is Because the two spin componentsis, of the vector the multiplet fermion does belongrepresentation not necessitates contain to that its diffe two multiplets bethat combin itself transforms in a non-trivial representation of t limits of massive ones. by opposite helicities: All states in the multipletsible must U(1) have identical internalthe qua scalars are permitted to have opposite degree of freedom. Thecoherent states state of the chiral supermultiplet m with In this simple case we have the mapping is in a complexwith representation, conjugate like internal a quantum numbers quark which in may superQCD), be an constr Grassmann number that wouldmassless multiplet, represent while the ˆ supercharge that antisymmetry of the supercharges. Choosing the Clifford vac the tensors are antisymmetric, the state differentiation and integration, the derivativesmann may be differential rep operators aboveusual way can as be for massless used superamplitudes. to extract comp which generalizes straightforwardly to other theories. By The limit is taken by simply replacing resulting states are then field may be extracted by either setting ˆ JHEP10(2019)092 = 4 = 1 state (3.16) (3.15) (3.14) at the N N to give φ ) H ˜ IJ φ. -symmetry ( 2 2 the massless R η are extracted . W 1 2 ) , η − erent state and ) 2 1 ˜ λ s IJ 2 η ( η 2 fermion highest- 1 1 / copies of the ...I η + W 1 I components (remem- + (  ˜ φ ethods. For example, a ltiplet decomposes into I R A H J ymmetric in either little ˜ ψ η al supergravity multiplets ˆ o supergravity. The other η, − , e spin-1 -indices). The Grassmann J -triplet scalars. The scalars JA − η − and L R nd those of a massless chiral η R 2 1 superfields. ] as a short multiplet in ηg rmions related by W ntations of the little group via JB ch containing either one + Φ t here, namely that which starts − 2 14 + η 1 ) − s √ B I − 2 η G  λ and a massive vector , combines with the scalar 1 3 ...I ) 1 . Fermion statistics of the Grassmann = → = − , while the vectors A I H JI . ( − − − η ) + (+ ) IB ) ∂ Φ Ψ G W IJ W ∂η J IJ ( ( has the form 2 η A I state (the superfield having helicity between ∂ s W √ W + ∂η ˜ φ – 11 – ) 1 2 = = s  AB 2 ǫ  L H J ...I + 2 W + I of spin ) W ψ L I S 1 ∂ ) or one AB I ∂η ( ( s W may be extracted by the action of the Grassmann deriva- η φ 2 + 1 I A + IJ + √ I ˜ ǫ ψ + ) and (  s ˜ λ W 2 B J s η + Φ η J η ∂ − ...I ˆ η + A I + 1 ∂η I + η (  + + 2 1 G φ g λ 1 2 = 1 superfield − = may be used to raise and lower the SU(2) → = = N ) I A s + + + and Ψ are its component states in order of increasing spin. In 2 ψ -indices gives the superfield , and Φ AB G , while we have both a real scalar . This superfield will be discussed further in [ A ˜ I W φ ǫ R ...I indices, hence the little group triplet vector and = 2 supermultiplets are short multiplets. Expanding the coh ˜ η λ H 1 = 2 without a central extension, we essentially just have two , I JA ∂ ( φ R + = N ∂η S , N φ I A ψ I may be extracted by the action of ∂ ) W ∂η Of course, higher-spin representations — either fundament For Taking the massless limit again, the massive vector supermu I 1 2 Ω = ∂ AB ∂η ( 2 1 φ generators implies that thegroup Grassmann or order 2 terms must be s where and a scalar. The fermion order 3 and 4 terms respectively represent a pair of chiral fe the two real scalar degrees of freedom in the massless chiral Each term has been decomposed into irreducible little group by tives for each of the Grassmann variables except keeping the The longitudinally polarised vector, bering that super-Yang-Mills. superalgebra. There is only onewith supermultiplet a to scalar construc Cliffordfamiliar vacuum, as any other choice takes us int (with helicity between level state where the components have already been decomposed to give th first level. We can extract the different irreducible represe limit, this decomposes into pairs of separate superfields ea or composite superfields —general may massive be constructed using the same m superfield as the two helicity components of a massless vector superfield a JHEP10(2019)092 he ), is with (4.2) (4.1) Q (3.17) trans- super- ( ). ) through † , with a Q 4.1 Q N † Q , must be ( (2 4.2 n ) ord vacuum δ I N A † iA Q, Q (2 η δ A iI ubset of the su- η . . − e , the multiplicative     † . However, as these i 3 η IA i Aα † iIA N η 2 η Q r the scattering data for = 0 would instead imply A iI d I SUSY Ward identities will † iA η e elaborated upon further is proportional to the delta i n uilding them in η uperamplitudes in n tral charges. =1 i representation, the Q ll instead be the product of i n A m n shows that this delta func- † s by m ast as large as R Q A η e superamplitudes between BPS i n i ust the constraints from cluding three-leg amplitudes, this X n low for amplitudes of component X 1 2 -leg superamplitude, ) = 2 1 n , Q + 1 2 ( + ) − A jJ J h N η † jA ^ (2 η A iI ηξ δ I η † iA +

η basis defined in section J ] h j J η – 12 – ϕ bases I j i basis, as can be seen by commuting I †

i = [ η h n n i

A A 1. We presently discuss the procedure in brief and outline a n η † ] =1 N Q Q ≤ X A A , which matches † i = i Q |

). So we end up with s (a full discussion of the Grassmann Fourier transform may b A † η are so far undetermined and are also functions of momentum sp A.2 ¯ basis to those in the F qQ

η i F, 1 q [23] = 0 for all This simplifies our task of constructing general three-leg s However, from above we know that the Grassmann Fourier transf Of course, the same reasoning holds with The existence of the different h at most. , denoting by [ As remarked previously, the situation is modified in the case o η i 6 1 ] =1 N N qi Q F appendix number of [ 4.2 Strategies for enumeratingThe main amplitudes goal without will central betheories char to with construct spins three-leg superamplitude 2 only understand the structure of appropriately invariant f exceptional and will be explained in [ is a SUSY invariant with Grassmann degree roughly returns the set complement of the A where, for example, specialnishing, kinematics then can imply that if the all of the distinct supercharges. This is similar to the case for the SUSY invariant to which it must be proportional. Henc ble Grassmann structures are of simplifications. superamplitude in the two bases. That is, in the h Grassmann degree of restriction on its maximum Grassmannthe degree from lowest knowledge Grassmann degreeconstruction invariants. and This classification restriction ofexternal three-leg i particles, superamplitu we can always write a three-leg superam these functions could have maximum Grassmann degree where of massive legs, since this is the number of independent Gras JHEP10(2019)092 . , 1 η M 1 . As (4.5) η 1 M massless 3 K inor of our 2 . J is ambiguous 1 I M c 1 F assmann orders. η en ry about at most, en number of con- )), arbitrary linear . As † and F Q ]. The coefficients like ay be constructed sim- ( 6 δ index for a massive leg or IJK litude order by order in ntifying a tensor basis and F example, a superamplitude , ) is that they are of uniform e superamplitude. The two up index for that leg, while | of the superamplitude. The i bles themselves transform in re. We define the ‘total little of hen left to construct, for each h = 0 case. Supersymmetrically, 3.4 | . This implies that terms with , only even Grassmann degree percharges act derivatively. An ( is odd for the amplitude. Like- by the possible combinations of 2 erms must carry the little group h ned with coefficients built out of N respond to the possible terms that h d by the little group covariance of . i X massless legs factor with a single spin index for each leg which multiplies the SUSY invariant delta + F F i s ]. – 14 – ] may be used to determine 2 6 26 (as these are annihilated by X † 2 for a massless leg. The rank and helicity weight of the Q / massive legs ∝ = -matrix amplitudes are constructed in [ h 2 legs will have an S / factor consists of a sum of monomials in Grassmann variables F ) of the massive (massless) leg ) Lorentz tensor with the correct little group weight for the i C h , of a superamplitude to be h and consequently these constraints do not mix up different Gr (helicity above may be expanded in a tensor basis spanned by a massive sp η i ). An example of a candidate term with Grassmann degree 1 is th 1 s 3 M K 3 2 K J At each order, the A similar procedure to that used in [ Now that we only have a small number of Grassmann orders to wor Each Grassmann variable carries either a fundamental SU(2) Each coefficient of the Grassmann monomials must involve an ev The possible tensor coefficients of the Grassmann monomials m 2 1 I J 1 I F legs and massless Grassmann variables,enumerating which the we possibilities may do as by done ide in [ This simplifies the procedure so that we may construct the amp choice for each of themonomial, required an little SL(2 group indices. We are t c degree in our task will be to construct the function function for various theories. Thisthe function amplitude, is which is constraine set by the external legs as in the it is constrained by theimportant Ward benefit identities, of since our half representation of of the the supercharges su spinors in such apossible way combinations as of to spinors that give satisfyare the this permitted necessary then cor representation by supersymmetrywith and three Lorentz massive invariance. spin-1 For the delta function is littlerepresentations group of invariant, the each superfield ofnon-trivial these legs. little t group The representations, Grassmann so varia must be combi where the Grassmann variablethe from coefficient’s leg tensor 1 structure contains is a then little determined gro by that ( up to the addition of terms a helicity weight of magnitude 1 for spin Grassmann monomials with thegroup required weight’ little group structu representations of the possible coefficients are determined combinations of this supercharge may be added to simplify th ilarly to the way in which tracted spinors (as thean superamplitude even is number a of Lorentz Grassmann scalar) variables cannot arise if terms are consistent. wise, if the amplitude as a little group tensor has even JHEP10(2019)092 ary (4.7) (4.8) (4.6) no other ] 28 ). It is im- 2.2 . α r massive leg). We by its action on the iQ − -leg amplitude for two ction of elementary am- = ] is an arbitrary reference ned with restriction to a arity, they are mapped to q † care group as [ rassmann variables entirely | | arry the little group weight P p tion and spin quantisation of , defined in ( ˙ itude does such an alignment i h † α q , through their particle labels, is described by a single scalar. n superamplitudes in a parity- iscuss parity. p | † i figurations. iϕ n this special kinematic configu- q − PQ iϕ e − . The constant of proportionality is − e i and ) symmetry. We here explain how this 3 , = 0 to relate the spinor coefficients of = i i ˙ P α q | ] = 3 † ∝ | | p p 3] 2 QF iQ P 3] q p [ |P | [ | 1 = q [ p † ]. 1 – 15 – m P ] α p , and also the reason that such a helicity-weight 27 | | i ≡ p 3 [ | iϕ x ], this is PQ e x 6 iϕ e − = = ν | is the mass of legs 1 and 2, and i 3] = 3] = 0 and so p P | 1). p | ν µ 1 6= 0. In this unique case, the special kinematics of the legs ) is independent of the reference spinor, despite its necess m 1 − p |P hP p P , | 1 1 3] m 4.6 3 = q − † , ) scalar, nevertheless carries helicity weight 1 of leg 3. In 1 − h P C µ − , = , 3 PP p · 1 p = diag(1 ν (for example, the two little group components of a particula µ P F With this general method established, we turn to the constru As for general chiral spinors, parity acts on the super-Poin An exceptional feature appears in the special case of a three For massless legs, noting that The action of parity on the coherent states may be determined , which, as a SL(2 spinors can be found in appendix C of [ acts upon on-shell superfields,invariant from theory which may relations be betwee deduced. Details about the construc 4.3 Parity While not obligatory, many theories exhibit parity ( plitudes in simple SUSY theories, after first digressing to d where portant to remember hereparticular momentum that eigenspace. these The have operators been implicitly defi appearance in inverting ration, one finds thatof there the is massless an particle. additional Following object [ that can c massive, equal-mass particles and one massless particle. I Clifford vacuum and on the spinor-stripped supercharges carrying scalar object doesn’t exist in other kinematic con from may then apply the supersymmetry constraint components of each supercharge can be used to eliminate two G different Grassmann monomials to each other. their representations on different momentum eigenspaces. implicitly also carry momentum labels. Under the action of p where 3 is the massless leg, spinor defined so that [ implies that x This is the reason that ( kinematic configuration of massiveof legs massless in spinors a occur 3-particle in ampl which the relative orientation JHEP10(2019)092 (4.9) fined ) and (4.13) (4.12) (4.14) (4.11) (4.10) , absorb- ) to be 3.12 ~p η (and analo- 4.7 iϕ in (

~p ie − − − ) because of the λ = D ~p denotes Grassmann + † − 4.12 λ η + ζ f n the parity-conjugate Φ = . † i 1 in ( QED, the action of parity P P − . , q te helicity, which are usually

is derived from ( e group decomposition of the ate).

~p re, the I ral multiplet Φ iϕ ~p in the coherent states, which, + ch must be implicitly made in = † i + − distinct supermultiplets. Mass- ), massless scalars and massive ctively determines an arbitrary rent state p  ets) for consistency with SUSY. − λ q , must also be mapped to states g  I Φ + A D ie p g φ 4.7 η

− ζ are possible phases associated with ζ is an intrinsic parity assigned to the ) (which, in general, need not have = and = = X = † ) under 3-momentum inversion: φ ~p ~p ~p i ζ †

ζ

q ~p + − − − − − P I I 3.12 † P − † i g 4.8 p g g Φ p q G G on a massless coherent state may be de- ~p ~p η + P + − P ~p † − P q λ g

χ in ( q P η ζ ζ ζ e +

P e = = = ~p † † – 16 – † i † −

P ). Finally, ˜ † I P φ P P P

− q p

D ~p ~p ~p + + ~p −

I iϕ φ φ Φ G , p G

ζ is a model-dependent property. Depending upon the

− ~p ie maps under parity as = = − = = P + †

 = I I on the supercharges P p † p ~p − P P P P Φ i

 P q with inverted 3-momentum. Note that helicity spinors are de i , the action of ϕ denotes leg i P For massive legs, the null vectors that constitute the littl The existence and action of Similarly, gously for the other coherent states). The factors of intrinsic parity of theon Clifford the vacuum. photon’s For example, multiplet would in introduce SUSY a factor of massive momenta transform in the same way as ( where the Clifford vacuum for Φ scalars (note that the Clifford vacuum is not a parity eigenst part of a distinct supermultiplet. However, because of ( theory, supermultiplets may be self-conjugate orless mapped to spinning particles must be mapped to states with opposi intrinsic parity of the photon. any other necessary relationship with Φ explicitly labeling its 3-momentum spinning particles, at least whenof selected distinct as weight (in Clifford the vacua For same theories or possibly with different this multipl property, the action of termined as follows. Taking for example the left-handed chi Fourier transform of the chiral superfield Φ as will besuperfield. shown below, can be defined to absorb these factors i Here, for a phase up to a convention-dependent,the arbitrary definition overall of phase, the whi phase spinor-stripped supercharge. multiplying the This effe (complex) Grassmann variables ing the phase from the transformation of the supercharge. He calling the Grassmann variable of the parity conjugate cohe JHEP10(2019)092 ) of ~p,I 4.8 lets. iη (4.15) (4.17) (4.18) (4.16) = ories to , ~p,I d be rein- ) † − h fermionic n η p P P r leg amplitudes isations of the ex- ,...X 2 p P ) from the use of ( P omponents are expressed ,X eory, any superamplitude at accompanied the trans- 1 4.18 nce of the couplings on the p opposite weight in the other , om in the multiplet with the . mplitudes of a set of particles P P ). In other words, to relate ~p n the 3-momentum, so should e, this may be stated as ts and how they may appear in i,I then reversing the 3-momenta ′ − X iant, the quantisation axis (de- P ( ree-particle superamplitudes for g Φ ly the supercharges) do not arise n φ iq ge. ζ A = , (while we have written the Fourier may or may not be equal to Φ. The = ! I . † i ′ I ~p † P P ~pI ′ X − W η † − † i,I η ] i,I W ~p I I P † − η χ P q q e ζ e i

= ~p P – 17 – on a massive chiral superfield is ′ − η † i 2 I ˜ φ P † P P d D I ~p iq φ Z W ζ i = X † = ζ † P I i n P =1 ) and the relations between Grassmann variables i Y ~p P q Φ

]. We also present some simple results for higher spin multip 4.15 , the action of 29 ) = ), ( n ~p,I p ) must be equal to that obtained by taking the superamplitude iη n ) in the form specific for massive coherent states, they shoul 4.14 p . For a massive vector, the transformation is similar but wit = ′ . Kinematic-dependent phases appearing in ( = 1 chiral and vector superfields and identify the types of the ˜ may or may not be distinct from ~p φ ), ( ,...X ′ η 4.18 ~p,I , we additionally present some well-known results for highe 2 N p † → − iϕ B W 4.9 ,...X η is the parity conjugate superfield of 2 ie φ p ,X ), ( 1 ) may be dropped, representing arbitrary phases in the polar P = p = 1 three-particle superamplitudes ,X ~p X X 1 4.8 ( p † − 4.14 n η Calling Parity invariance of a theory implies equality of the supera N X ( A n transforms in ( the parity conjugate multiplets,using Fourier ( transforming and In appendix recast in the little group invariant helicity spinor langua ternal legs. 5 A and ( terpreted as theircouplings massless analogues between for superamplitudes each in massless a leg parity symmetric th which they would belong.masses We of furthermore the discuss different the legs, how depende “tree-unitary” they theories behave [ in different limi In this section we systematicallyscattering construct of the possible th Clifford vacuum mapped to thesame other fermionic polarisation degree of freed where again where fined as the 3-momentum)with reverses. respect to The some massivenot external little change quantisation under group axis, parity. c ratherformation This tha of is the the massless helicity reasonhere. spinors that (and The the subsequent massive phases supercharges th therefore transform as Helicity reverses under parity because, while spin is invar with that of their parity conjugates. Given the results abov where, depending upon other quantum numbers, Φ scalar Clifford vacuum is importantlysuperfield: mapped to the scalar of and JHEP10(2019)092 ) i ( (0) F F (5.7) (5.4) (5.6) (5.1) (5.3) (5.5) (5.2) 2. The = 0. We ↔ . † . Q   ) to produce 
  , which allows I I 2 i 5.4 2 . η η p I 2 I | η iI 2 I I η η . 2 1 1 h η i .  2 m
X
d ritten in the manifestly he number of different I 2 1 + η η m and + I 2 I 1 ts, and will then find the | 2 I 1 η + I η η m I 1 I 1 ving the superamplitude from 1 jJ , he delta function, + η m η ) η 2 -th order Grassmann term e limits. Our representation of 1 I J i i . From little group scaling, iI 2 + d η m η . Since the Ward identities do not η ( = 0 and similarly with 1 I 1  η  + 4.2 η F

2 1 J J ) I I J j † 1 2 1 I +

η 12 η i Q 2  I 2 ( J  1 2 , which then leaves us with m ) η cp η (2) I † 0 may be taken directly on ( I i

3 X δ 1 + † ). This three-point superamplitude has the 2 1 η Q J = 0. Along with the same procedure for the η  (   → 2 Q I  I c ( + 3.9 1 1 ) = ′ J – 18 – (2) 1 m

3 b J 2 . The second-order function can be simplified by

(2) 3 m ) 2 I bδ Φ λ c 1 η to make it dimensionless. 1 , d I − λδ +  2 b 1 + 2  λ η Φ J − , b    2 ) = ) = 1 2 and find J η ) 3 3 + † I 2 b 1 Φ Φ (Φ I = 0 is a first-order Grassmann equation and results in two bm λ Q , , η 1 ( A 2 2   ) is at most a second degree polynomial and, since it has total  I ) i Φ Φ J (2)  (2) † η , , 2 δ b ( I Q + 1 − 1 ( 1 F QF =  , (Φ (Φ ) = (2) b = 0, contains only even orders in 3 δ A A (2) 4.1 in terms of = Φ h F , 1 2 ) = (2) d 3 Φ F , Φ 1 , 2 (Φ Φ , A 1 (Φ A There are three special cases to consider corresponding to t When all of the legs are identical the superamplitude can be w We will illustrate the general procedure by explicitly deri The Ward identity first principles using the method described in section where, from section is fixed to be a constant which we call the most general expected superamplitudes general form using the supercharge conservation constraint imposed by t The full superamplitude is then can use this to eliminate any dependence on mix different Grassmann orders, we may construct each little group weight separately. massless legs. Firstly, the massless limit We have here redefined the coupling We begin with thecases case with massless of chiral threethe supermultiplets massive massive via chiral chiral appropriat superfield supermultiple is given in ( 5.1 Three chiral supermultiplets other equation, this yields independent spinor equations ( one to solve for independent constraints may be found by contracting with [1 exchange symmetric form JHEP10(2019)092 i ) 0. η )). /E (5.9) (5.8) 2 (5.11) (5.10) (5.13) (5.12) A.8 i → m bstruc- ( erampli- ij h lings. That ∼ O . ] , + 1 j [23] 3 + ˆ ) mann variables ˆ η i 2 ) [12] ergy limit, scale as Q ˆ † η ( 1 Q η ( (1) )ˆ the three scalar compo- e δ perfields with mixed he- the same rate is instead the Fourier transform of + 3 an derivation, where the (2) ling constant) Φ ). For the latter kinematic bδ , ess superamplitude. ) = , + 2 ) 3 imits can be found in ( − 3 ctions that do not conform to udes are expected to vanish in symmetry breaking. e explicitly checked for a given η Φ . We have assumed that the note is that the massive super- . Note that [ 3.12 Φ ) = , 3 lomorphic superpotential terms, ken supersymmetry are more di- , 3 0, while in the second, + 1 m − 2 Φ , Φ (Φ → 0 results in the superamplitudes for , + 2 + [12] = ] ) does not diverge with inverse powers + 1 Φ 2 → ij 2 , η − A 2 5.5 (Φ + 1 m ) ), [ A m − 3 (Φ Φ 5.9 A , , + [31] ) − 2 1 ) states. i – 19 – Q η † Φ ( , 1 + i Q 12 ) − 1 ( h † (1) ) e δ ′ Q † (2) (Φ b ( Q − λδ ( ) = [23] (2) Q → A (2) ( λδ 0 for some (complex) energy, depending upon the kinematic δ ) = ) = ) 3 (1) + 3 − 3 → e δ Φ ) = Φ Φ ) = , 3 , , 2 m − 3 + 2 − 2 Φ Φ , Φ Φ Φ , , − 2 , , 1 ) as + 2 + 1 − 1 Φ ) = 0. It is again being assumed that the couplings present no o , (Φ Φ 3 , /E (Φ (Φ − 1 basis. In the first term of ( A 2 Φ + 1 ) is exact, rather than merely leading. The fully massive sup A A , † ) are totally determined by two parity conjugate sets of coup (Φ m η ∓ 2 (Φ ( 5.9 A Φ 5.4 A , is unaffected by the limit, which is self-consistent. are the massless superfields in the notation of ( ± 1 ) contains helicity violating couplings that, in the high en b i ∼ O ± (Φ ) sign accompanying the second term arises because the Grass − 5.5 ) in the A j − Q − In the high energy limit, the superamplitude ( The only remaining massless superamplitudes are those of su This massless limit is to be expected from field theory, where Similarly, taking the subsequent limit that Note that if the limit that all particles are sent massless at i ( h (2) configuration that is converged to (individual spinor mass l or where Φ the structures derived here, must induce spontaneous super licity. These are determined by symmetries to be (up to a coup amplitudes ( there are no otherspossibility of is spontaneous not supersymmetry breaking completelyholomorphic has superpotential. obvious to b Here, from constraints a fromrectly unbro applied. Lagrangi It automaticallysuch follows as that tadpoles candidate and ho quartics that would naively give intera nent amplitudes contained in thethe two massless surviving limit superamplit according to the superpotential. Also of configuration, the delta function is The superamplitude converges to (at leading order in energy of a mass scale because of the special 3-particle kinematics tion to this, which is clearly self-consistent. These expressions are independent of whether δ which is a Grassmann order 1 supersymmetry invariant that is coupling two massless legs: while must anticommute past the fermionic Φ The ( tude ( mass-dependent constants and cannot be expressed as a massl taken, then ( JHEP10(2019)092 - S (5.15) (5.14) to denote Q . g the possible     and I i subscripts identify η Q iI rving scalar-fermion- R η trasted with the field- e imposed as described i X itudes in this theory with ch the accidental parity is and states of the quark super- o construct. pressed by helicity conser- ial is represented in the al parity symmetry. This is m eramplitudes. L ith symmetries, they do not cting with a massless vector al supermultiplets as quarks ing to identifying them with hree particle superamplitude. etric gauge theories (like su- morphic contact interactions. + s then lomorphic composite operator he symbols , eature. In this regard, it would perties. While non-divergent in ducing a (super)amplitude that , the foundations of which were jJ R R η : e e Q Q es of different helicities, so produces iI I I η η η  , in agreement with the massless and I I J η η λ j 1 2 2 1 I i =  -matrix is presumably then generated by − − ′ S b I I component amplitudes with the fewest legs. ifermions in and ructure of the matter cou- possible Lorentz structure. lings of matter to massless ge in the high energy limit d theory, this symmetry is Q + ished by the existence of y charged under the residual Q uish between the two squark -photon interaction is entirely I multiplet is impossible unless ric under exchange of the two vidual legs, in the terms that 1 eeded to relate the amplitudes η , que form oup scaling to be constructed. and that a three-leg superamplitude  y gluon superfield is simplest to g x I ) -symmetry of the SUSY algebra is etry invariance and the Grassmann † Q is determined by the helicity of the 21 R 4.2 Q (  R (2) m 1 δ 2 and + ) = L 2 3 η – 21 – Q ], which is unaffected by the quark masses (with  , + 2 33 bx ) , † 6 ,G 1 is the coupling constant (which may have suppressed Q ( Q ]. Supersymmetry determines the fermionic coupling to g ( ). As the superfields (2) 32 x A δ ) = ) and 3 determine the superamplitude up to a single coupling consta ). This is ultimately a reflection of the breaking of the USp(2 Q R 4 4.6 , (the minimal assumption), this implies that superamplitudes have total little group weights also suffices, which justifies it for chiral gauge theories). I − 2 P − 3 (through ,G G CP 1 ↔ Q ( and A + G is defined in ( x It is interesting to note that the USp(2) massive Thus far we have not actually assumed anything beyond partic Parity may additionally be imposed. Assuming that the It is easily shown using the methods of section That the superamplitudes are determined here by a single cou The equal mass. However, these superamplitudesmatter are fields antisymmet 1 broken in this theorystates. because The the identification gaugino of couplings the disting squarks as gaugino that couples to themunbroken massless (the squarks U(1) are then oppositel massless matter, be interesting to clarify theon role each of side the discrete ofaccidental. symmetry the In n the factorisation examplesof channel. below, massive we amplitudes, In always as find Yang-Millsmatch well parity fiel onto as emer sensible the amplitudes massless of limits massless of vectors. indi characterised by the electric charge monopole. the gauge bosons from theAs a scalar consequence, coupling, the which supersymmetry implies has that only the one matter between two massive chiral multipletsthe and a chiral massless multiplets vector have equal mass. This case is distingu where vectors without supersymmetry as well). self-conjugate under that they must be distinct (the same argument applies to coup helicity superamplitude, little group scaling, supersymm counting rule of section dependence upon possible internal quantum numbers of the st was an assumption usedplings in from the consistent derivation factorisation of [ the Lie algebra st tion of the fact thatgauge the theories anomalous is magnetic exactly dipole zero moment [ of which will allow for expressions with the required little gr construct as little group scaling immediately gives the uni respectively. The superamplitude for the positive helicit JHEP10(2019)092 , 1 2 0 ], 6 5.5 + → s (5.20) (5.19) (5.18) J 2 1 η m ,  i J 2 state. This   1 ) I s s 2 2 ... J  0 and 3 +  J 1 i 1 i h → . spin 1  +1 such multipoles η ) 2 2 d s ) ) m 3 s K  2 . imilar pattern. The j s 3 J 2 J m η I − 1  3 ...J 1  I 1 J 2 2 2 J ithin the spin-half vector K i h its e chiral multiplets and a d  η i h 2 )( ) 3 2 j s ) 1 2 3

I 2 J i m arks. This would be restored d ory. Repeating the procedure 2 ( J I 3 m 2 example of this in section  3 2 ultipoles by SUSY. This is the 1 ent for supersymmetric matter ...I , which can be constructed as i

in the coupling. Following [ 2 h 1 m 2 ) I ss vector is m 2 − I , there are 2 3 ( s 1 1 ( d 2 m s p 1 2 1 m − + 1 T m h

s s ) m I ...J 2 1 states in the multiplet is determined + 2 † m ( 1 2 Y d =2 =1 − J  3 1 2 after imposing supersymmetry invari- Q s K  Y )( ( i,j 2 i,j 3 3 s 2 m + η 2 i + x d 2 (2) m j s

1 2 J J d − δ ...I ( c m 1 K 1 J 3 I m η 3 i ( 1 and + I I

η ) = ( T 2 + J ) i 1 1

– 22 – ) s 1 I 2 d h 2 j J I 2 η J 1 2 1 2 ( 1 I d ...J =1 −

m 3 d 1 2 s ) Y J i 2 3 i,j

2 − + ( I 3 ( 2 d 2 m 1 1 x ,S d h m 2 − 2 + 2 c   m s =1 ) ) 2 ,G Y † † + i,j ) s 0 Q Q 2 ( ( c . The additional multipole moment for the coupling of the ...I   i (2) (2) 1 c I δ δ 1 ( 1 x S state. For a massive particle of spin = ( = s ) = ) A 3 s 2 Q , ...J I 2 1 J W )( , s 1 2 = 2 gauge theory, where the gauginos are Dirac fermions. Q ...I ( 1 N I A ] just as a general amplitude for a photon coupled to a massive ( The coupling of higher spin multiplets to photons follows a s 6 T respectively. ance. The two forms stated are useful for taking massless lim implies that the coupling to photons of the spin by that of the spin in [ We next considermassive the vector multiplet, three-leg as may superamplitude occuras in of in a Higgsed previous two gauge sections, massiv the we can reduce the amplitude to fermions to higher spinwhere states. the We will electric see quadrupolesuperfield another moment is explicit determined of by the the massive lower multipoles. vector5.3 w One massive vector state is therefore determined heregeneralisation entirely of from the the protection lower of m the magnetic dipole mom these are superamplitude for the case of the positive helicity massle representing each possible independent Lorentz structure This leaves two undetermined couplings for coupling constants in an by parity symmetry, which distinguishes between the two squ SUSY places no further constraints upon JHEP10(2019)092 ) ) ve. and 5.6 5.22 1 do so 3 (5.23) (5.24) (5.25) (5.22) (5.21) ˆ η η )ˆ 2 remains s limit), x in ( η − 3 . 2 )ˆ 3 1 Φ d + η , )ˆ m 0, then non- Φ + 2 do not vanish + , ∼ 2 Φ + → ,G d , 1
2 + 1 d − 3 ,G η m − (Φ 1 and ,G hiral multiplets have 3 − m (Φ 1 η )ˆ d usly at the same rate), constant). The case in ) as ce on mass necessary to + + 3 (Φ 2 2 as no consistent three-leg J − A Φ A → −A d m 1 ral multiplets. , , 3 r in the factors of ( this special limit. ) η I 2 den by symmetries, so does  ˆ + η ixed helicity chiral supermul- . These expressions hold re- − 3  1 1 2 W J I η Φ η and , )ˆ 2 , )ˆ 1 s may be recovered if  31 still differ from their counterparts + 2 + | ∼ O − 2 2 Q 3 Φ Φ ( 1 , W ,  m /m m , A I + ) 1 − , (this mass scaling has been anticipated † + 1 − Φ Φ  32 1 , Q , 1 ∼ 3

) + (Φ ( d + I − ) − 3 m 1 2 A † | (2) 0, only the transverse polarisations interact = d  Φ (Φ (Φ δ Q ) , ( λ † A I 2 A → Q – 23 – (2) 2 + W ) = + ( δ , 2 and − 3 1 m 2 1 ˆ η (2) η 1 Φ 1 δ )ˆ 1 m Q , at a rate η ( 2 1 d − I 2 )ˆ 2 3 /m d ). This is expected from the Higgs mechanism if the Φ m m − 1 W , − 3 m d Φ , , one recovers solely the three-chiral superamplitudes ( − ). This leaves the parity-symmetric terms in the superQCD → A , 3 Φ + 1 − , 0 limit of these superamplitudes requires + ) = ) = ) → ) m or not. It is being assumed here that 3 ,G − 2 + 3 − 3 (Φ 3 A.9 1 2 + , as well as the three-chiral superamplitudes mentioned abo ,G → Q )). 6= Φ Φ A m 2 , 2 ,G m , , + m 1 1 1 I d (Φ 2 I I 2 2 + 1 m m m m = (Φ ( W = ( W W (Φ , 1 , , / O , as well as the assumption that it is non-zero and finite in thi g 1 ) 1 1 must be finite (and hence must be suppressed by some other mass ). It is easily verified that 1 m 2 1 Q d Q Q → A → −A = 2 ( ( ( m 5.1 ) ) b → A A 3 3 A A − /m ) 2 2 3 Q Q − 3 , , d m Φ + − ( , ′ 2 = − W W d 2 , , ′ 2 ) by terms of 1 1 d W = , Q Q b ( ( + 1 More interestingly, if In the high energy limit (taking all masses small simultaneo Taking the vector massless differs depending on whether the c Taking the further If we instead take the third leg massless, we find smoothly 5.20 A A (Φ or diverge in this limit, whichin is ( self-consistent (they may through the spinor limits in ( non-trivially (see comments abouttiplets the in superamplitudes section of m a dimensionless constant. The reference spinors that appea zero superamplitudes involving massless vector multiplet where superamplitude for a massless gluon and two unequal mass chi scale). This is consistent with our finding above that there w not appear in the limit. Taking finally which the chiral multiplets have the same helicity is forbid gardless of whether which are alternatively determined purely from symmetries then it can be verified that in the definition of and yields amplitudes with equal mass. In the case with where we have omittedrealize the the coupling final and massless provided the limit dependen smoothly (so massive vector is coupled to massless matter. A As alluded to above, parity in the vector coupling emerges in JHEP10(2019)092 is 1 d (5.26) (5.29) (5.27) rvatures . . 2 (if the massive b [31] [13] = b b rsymmetric axion or ) m a ) Q † ( assive multiplets having Q imit may be taken while ( (1) g into massless vector and e amplitudes can combine e δ binations of the masses. (2) t a massive chiral superfield involving the quantum field decay channel for a massive d by the symmetries are (up δ [23] (5.28) perfields and one chiral super- ess decay products are instead -zero, following from the rules  ) = tor. Only the superamplitudes ultiplets to the massless vectors . 2 + 3 ets coupling to massless vectors I ) =

e be described as having different 1 2 ), they have no further dependence 3 +  ,G s, then the couplings may be related I 1 , then this would obstruct the limit). + 2 1 ,G

m Φ 2 5.26 ) ) , /m † † Φ allows for parity violation in the massive 1 + 1 , Q Q 2 ( ( , while no consistent superamplitude may 1 + G d ∝ ( G (2) (2) ( 5.1 δ δ and – 24 – ). have the expected inverse mass dimension of an 1 ) = ) = d b i A − 3 + 3 invariance would imply that 5.24 i A Φ Φ 13 the nonzero mass) , , P h 13 and − 2 + 2 a h m is the cubic coupling among chiral multiplets, while ) a a † ,G ,G 2 ) I I † 1 1 Q d ( Q W W ( ( ( (2) δ A A (2) δ , for (off-shell) chiral superfield Φ and super-Yang-Mills cu ) = F − 3 ) = ] − 3 B ,G W − 2 ,G 2 A Φ , Φ . These are (calling W , − 1 limit, the coupling − 1 G 4.2 ( G + ( A instead). The couplings W A for some Abelian gauge groups (in other words, a massive supe The superamplitudes for a massive vector multiplet decayin The possibility of distinct couplings CP A,B chiral multiplets in each superamplitude are antiparticle by dilaton-like coupling). Demanding irrelevant interaction. Assuming that, as defined in ( couplings [Φ W These superamplitudes would arise, for example, in a theory on the mass ofholding them the fixed (if heavy they chiral instead scale multiplet, as then e.g. the massless l of section be constructed if thewith massless massless multiplets vector are multiplets chiral of and the vec same helicity are non the parity conjugate coupling. Theare couplings of dermined the by chiral linear m combinations of these weighted by com This gives All other helicity combinationsvector are multiplet zero. was found The above in other ( allowed In the and the other components are zero. field. Starting with thedecaying case into of two massless one vectors. massive The leg,both case we matter where first the fields look massl was a addressed in section interactions. Despite themust observation have that the chiral same multipl different mass, masses there in is the high not energy inconsistency limit. with5.4 the m Two vectorWe superfields next turn to three-leg superamplitudes with two vector su superamplitudes and accounts fortogether the states way of in different which helicities the that would massiv otherwis chiral fields may beto found coupling similarly. constants) Those that are permitte JHEP10(2019)092 ) ). , so 3 5.28 5.24 (5.35) (5.32) (5.34) (5.33) (5.31) (5.30) /m , should 1 − i mplitudes d ∼ ) and ( ve a common . (+) i  d ass scale. Taking 5.29 K to return to ( 2  η 1 2  η m K de describing the decay tudes ( , + ∼ 32 s and one massless chiral  e “effective”. In contrast, K  equal mass or not. Taking ) converge to the superam- uperamplitudes for massive 1 ases, the coupling constants s of the massless chiral su- superamplitude, 2 1 d a chiral multiplet is deter- η IK (+) i (2) s poor, scaling inversely with . ined only if example of the case in which  d m m F −  5.29 plets, which is an expression of , where one of the vectors may , the resulting superamplitudes K d next. J ut introducing new mass scales. hich are independent of whether amplitudes already remarked on if the limits are to be both non- + − 3 21 I 3 . J  1 1 b IK (0)  m 1 η ) 1 F ) and (  m ±  J ( 2 and  ) , d and † 31 5.28 a −  IJ IJ 1 1 1+ Q + (  F F

m

) )  J † † 3 (2) 3 3 I δ I Q Q I – 25 – 2 2 ( ( 1



) ) (2) (2) ) = ) † † δ δ ± K Q ( 1 ′ Q 3 ( ( d W ) = ) = (2) (2) , = 3 J J 3 2 ′ ′ aδ bδ ± Φ IJ 1 W W , , , F I 1 ) = ) = 2 − + 2 W to leading order in − 3 + 3 Φ Φ ( , , 3 I I 1 1 A ,G ,G I I 2 2 ), the usual arguments determine the three-particle supera W W /m ( ( 1 W W ) may be related by parity. , , A A m 1 1 3.12 ∼ 5.29 (Φ (Φ ). Both of these massive and massless superamplitudes may ha A A superamplitude, both coefficients must scale as (+) i d + 5.27 ). The couplings in both cases must be suppressed by a higher m ) and ( 5.24 5.28 Finally, the all-massive superamplitude for two vectors an Continuing to the two-massive-leg case, one may construct s Finally, we note that it is not possible to find a superamplitu The massless limits of the superamplitudes ( Next, the superamplitudes for two massive vector multiplet In this respect, thesethe superamplitudes chiral multiplet are is merely also a massive, special which will be explaine trivial and unobstructed. However,must in be either of suppressed these bytaking cases other both mass legs scales massless and, simultaneously is in possible this witho sense, ar altogether mined to be of a massive vector multipletthe into Landau-Yang two theorem. massless vector multi have no mass dependence in order to smoothly match onto ampli the third leg massless instead, the expected limits are obta For the Φ and ( to be plitudes ( origin, for example in thebecome massive axionic through coupling the suggested Higgs mechanism. above for As ( in previous c These are again independentthe of whether massless the limit massive of legs the have first leg, the coefficients in the Φ where some mass scale contained within the couplings Taking individual legs massless, oneabove. recovers The solely high those energy behaviour of these superamplitudes i chiral and vector supermultiplets with athe massless massive vector, multiplets w have the same mass or not: multiplet may be similarlypermultiplets in determined. ( Using the definition JHEP10(2019)092 ′ ), = a /v i ′ a m 5.31 (5.36) (5.37) (5.38) nd, 2 , where = . Again, η i ) consists . Making )ˆ 3 c ) . + 1 − ac  ( 2 ), where the Φ
5.35 d ) and ( , J N 2 + 2 . In the second i η W → Φ c ) on each factori- N ′ 5.30 , 2 a + 1 η , G, 1 represent “effective” I 3 5.35 (Φ η m b and )ˆ W ( assless limit that cannot coefficients match on to ) + 3 + eg vector superamplitude interactions beyond their − A dditional mass scale (and − A rst line depend upon the L ( 1 1 eramplitude ( 1 and pling of the scalar potential lds and one massless vector ,G η d ] to demonstrate the Higgs η de, as in ( e Higgs couples to a massive − 2 L he effective Goldstone boson 6 )ˆ a multiplet eaten by the vectors 1 t is Φ or loop induced interactions in → + 1 η limits. , 2 Φ a mplitudes ( + 1 m taking a single vector massless the high energy limit in tree-level , , m − 2 1 (Φ Φ 2 1 A , /m + 1 + + G (+) 2 J 3 ( d 2 ˆ η ) scales in the UV limit, assume that η 2 ˆ η → M 1 − A is accompanied by a factor of 1 5.35 b η η 3 i )ˆ , – 26 – ) and will not be elaborated upon further. η  η 1 − 3 )ˆ J Φ of order the leg masses and call constants + 3 2 -term”). , , while the second has coupling constant ¯ /m 5.33 v 3 M F + 1 ,G c 1 Φ 1 (+) 1  − 2 , d  Φ bc ¯ + 1 , 
→ G  + 1 ) and ( ( ′ K K b 3 (Φ 3 I I 5.32 1 1   ) as b ′ → A → −A a ) ) + in the case with a massless chiral multiplet). On the other ha ) correspond to couplings of a Higgs boson to massive vectors + + 5.33

′ ′− 3 3 − 2

2 + K d for some mass scale d W W . The leading terms in the limit are then 3 K i , , 2 I 3 2 2 1 I m

and Φ Φ 1 ) and ( ′ b/v and ¯ , ,

b + 1 + + 1 1 a and each term proportional to ˆ 1 − = d d ′ 5.32 a W W = = b ( ( (or A A ′ IK IK (0) (2) b To illustrate how the superamplitude ( More interesting instead are the high energy limits. The sup In the limit that the chiral multiplet becomes massless, the A supersymmetrised version of the argument used in [ However, there are also subleading terms that vanish in the m and F F ¯ with the Goldstone boson). Thisoriginates happens from when the the superpotential quartic (“ cou the Higgs belongs to a chiral multiplet (and is not part of the a/v and the matter massive doescollection into not the really single affect superamplitude.instead the are The similar results structure as fro of for the ( couplings (like those of thea axion/dilaton perturbative mentioned field above theory)similarly that must be suppressed by some a processes. 5.5 Massive andLet massless us vector begin multiplet with interactions amplitudessuperfield, with which two has massive two vector distinct superfie cases of interest. The firs mechanism may presumably befrom made demanding from consistent constructing factorisation a intosation four-l 3-leg channel. supera Notably, anand exceptional massless case occurs vector whenwhich boson th will in induce unitarity-violating a superamplitudes in three-particle superamplitu of two parity conjugate pairs of couplings. The couplings for leg masses where line, the first term depends upon those of ( and similarly forcoupling parity ¯ conjugate states. All terms in the fi this pattern of couplings reverses for the parity conjugate be placed into masslesscouplings superamplitudes. to the These Higgs. represent t JHEP10(2019)092 ]. x 38 , (5.39) (5.40) 7→ − 6 x ) in ap- may have )) contain + A.8 h ′  W β + and 2] re diverge in the | assless vector. As G g 2. However, unlike α ), in the limit that + ]. As in all previous − 2] the SQCD case, the 6 | W rely from little group ( perfield amplitudes are h 5.39 2 ), where the two massive A r certain helicity configu- − J hat in the non-supersym- ′ m nstants y be separately Higgsed. termines the vector ampli- g r example, in field theories ic factors of the component ector [ le vacuum expectation value W ric dipole moment if it has a component amplitude [ s that may otherwise exist as e of notation, although we do + gs in ( effective up to a UV cut-off. ch as of massive vectors. In this case, ), the combination of terms with , G, αβ I ǫ y of the magnetic dipole moment W g 5.39 ( mx , A   . Correspondingly, the examples of field 2 β has mass dimension J

, these terms converge to their expected M 3 J a m 3    2 ], where at least one of the vectors is Abelian α I ≫

1 36 I – 27 –  – 1 have the same mass. This arises in many examples,  a 34 ) ) † † ) in the massless limit (see equations ( W i Q Q ( ( ]. Note that supersymmetry has set fixed the possible m ( and would have the perturbative interpretation of an anoma- )). However, because the superfields are fermionic, the (2) (2) 37 ), the coupling δ δ h W 4.6 ∼ O for some mass scale 5.40 ) = ) = J 3 J ′ 3 W W , E/M , + 2 + 2 ,G ,G I ) appears to be symmetric under exchange of particles 1 and 3 ( 1 I 1 W W limit as ( ( 5.39 A E for massless limits of spinors). The amplitude must therefo A corresponds to a massive vector ‘minimally coupled’ to the m g A.1 0 or, equivalently here, at energies As in the superQCD case above, the positive helicity gluon su The term proportional to We can likewise find the negative helicity superamplitude pu In the second example ( Finally, ( → = 0 for elementary particles [ determined very simply. In these cases one finds lous magnetic dipole momentcomplex for phase). the massive This vector term (orrations, has elect which poor is behavior the inh reason the for UV limit the fo tree-level universalit massless counterparts. theories cited above that feature these amplitudes are only covariance and supersymmetry. From the same arguments as in with generalised Chern-Simons terms [ states are distinct and of different mass. These can occur, fo pendix high energy vector multiplets must be distinct. quadrupole structure of thea massive further vector independent boson Lorentz amplitude structureThis in derivation the makes vector obvious boson thetudes way from that their supersymmetry fermionic de counterparts. coupling m examples, we have here neglected to show that the coupling co and has a Stuckelberg mass, whereas another of the vectors ma such as the adjoint Higgsing of(vev), which a does simple not gauge feature theorythe any by 3-leg a vectors amplitudes sing are entirely conjugates,not which need is to the assume reason this at for this our point. choic The second is for the minimal coupling termssuperamplitudes in corresponding the to case the above, + the kinemat helicity states (su terms that merely scale as under this exchange — see ( two massive vector superfields has been foreseen in the definition of dimensionless couplin where, in both cases,metric the amplitude for number two of massive free fermions parameters and matches one t massless v internal quantum number structure. In the first case ( JHEP10(2019)092 . ). =  2 αβ η mass. 3.14 2 (5.48) (5.43) (5.45) (5.47) (5.41) (5.44) (5.42) (5.46) F +  2  he legs. In 2 K η 1 .   . +  β K    1 2] 2 2 | 2 η η K  K α 1 2 3 1 m  2] η +  | −  en this relates the su- 2 K K + ose as shown in ( 2 3 1 . x  determine these directly . η 3 η K 2 ′ 2   h 1 1 h β m 1  m he massless supermultiplets 1 K onents being matched onto /m . Assuming that the vector 2 i m e parameters: = 1 2] 1 K |  ′ + 2 1 − 23 − α i h 4.3 η K 2 e special kinematics, little group  2 am 1 1 αβ i h 2] . 13 i η | η 1 h 2 m = 2 1 xǫ 12 | [12] and + ′ h e amplitudes do not scale as negative 1 x b 3 − [31] [13] [23] [13] [21] [23] m ) g  m 2 g ′ p † ) ) ) 2 

h −  m Q Q Q Q = J K  ( ( ( ( β αβ ′ 3 1

+ 2

  g J (2) (1) (1) (1) 2 i may be manifested by adding terms propor- F 3 e e e δ δ δ δ K J 2 αβ  β 1 | 3

η α W 3 J xǫ

p ′ – 28 – ) = ) = ) = ) = 3

I m 1

 g m 2 1 − 3 − 3 − 3 + 3 2 I I  α and 1  Φ Φ 1

) −  , , ,G ,G I †

β ′ 

1 + 2 + 2 2 + 2 + b b W Q  J ) ) ( × † † ) 3 ,G ,G ,G ,G † ) as discussed in section  , the only option which has the correct scaling is Q Q (2) 3 J + 1 − 1 1 − 1 + ( ( α Q δ

( m G G I W (Φ (Φ (2) (2) ( ( 1 , (2) δ δ  6= A A ) = A A δ − ) J 3 1 = † ) = ,G m W Q I ) = J ( , ′ 3 J 3 has been redefined in the second line to absorb some factors of − 2 W (2) W ( b W δ , , ,G A is then determined from the little group representations of t − 2 . Our amplitude in this case is I 1 − 2 β ) ) = W ,G i αβ ,G J 3 ( I 1 2 2 I 1 to give | A F W 3 W † W , ( p ( Q ( 2 − A α A ) ,G i I 1 2 | If parity is a symmetry of the theory under consideration, th In the massless limit, the superfields are expected to decomp For the case where 3 W ( p ( A tional to peramplitudes of the equal mass case, this gives a superamplitude with two fre are (neglecting coupling constants): In anticipation of theby superamplitudes the of massless thefrom limit symmetries. massless of The comp constraints the ofscaling massive complex and three-particl ‘locality’, superamplitude, in the we sensepowers that first of the three-particl momentum, determine that the superamplitudes of t multiplets are self-conjugate, this requires that The tensor b where the coupling Exchange (anti-)symmetry between Grassmann polynomial must only contain an order-one term. If parity is a symmetry of this theory, then one finds JHEP10(2019)092 . ). the (5.53) (5.51) (5.52) (5.50) IJKM 1 5.42 F , 

2 . I ) (5.49) .  1  − 3 2 , M IM 3 K ,N 1 ǫ η ,G 2 3 J )ˆ  η + 2 2 + 3   K nts a Higgs coupling, 2 3 Φ   ,G N J mechanism operates by , J 2 2 K − 1 litude. The next step is related by the Schouten 2 + 2 st be constrained so that mplitude of three vectors 3 these vanish implies that  M he same species, as well as G M implies that the amplitude I ( hat can appear in 1 6 1 1 y structure in the superam- vector multiplets into single ,G ).  c  r. After simplification this  e structure of three-leg super- − 1 2 3 2 ion will include several special + 1 c 1 s of massless superfields are also m → A (Φ m 5.39 + ) A IM ), the massless limit may be taken + − − 3

ǫ +

M 2 W ,M 1 1 5.39 K J 1 , η 3 J 2 η )ˆ + 2 2 2 J  − 3 2 K ,G

 3 ,G

5 − 1 K c + 2 2 3 I M W I and identified with the superamplitudes above. IJKM + δ 1 ( 1 – 29 – ,G  F

 A + 1 ) 2 † M A.1 c M G 1 1 ( Q J ( + K 2 1 ,  J (2) 0 1 ). Similar results may be shown for the limits of ( δ M

I 1 1 + 3 K → → −A F J 3 2 ) ) W ) = I , + 3 − 3 1 K 3  ) =

+ 2 -term” quartic and is part of the chiral multiplet eaten with 2 W W 4 K W c , , K 3 ,G D , 1 I + 2 + 2 J + 2 − 1 1 K 3

W ,G ,G W 1 , ( + + c ,W 1 1 I 1 2 A J W W = W 1 ( ( ( J 2 A A A ,W 2 IJKM 1 I 1 F I 1 W ( This demonstrates how the supersymmetrisation of the Higgs Choosing a particular helicity configuration in ( One of the independent terms in this superamplitude represe Just as for the cases considered previously, supersymmetry A where external spinreduces indices to are five implicitly independent spin symmetrised structures. ove Demanding that This is the extentto to determine which the supersymmetry number determines of the independent amp Lorentz structures t using the limits presented inThe appendix limits may be calculated explicitly to be Altogether, there areidentity 6 and such kinematic relations): independent terms (up to others possible, but do not arise in taking the massless limit of ( Other superamplitudes between other possible combination and similarly for has the form cases, such as when thethe vectors case have equal in mass which andthere there belong are is to no only t vector one self-interactions. type of superfield, which mu superamplitudes of massive vector multiplets in the IR. 5.6 Self-interacting massive vectorA similar supermultiplets analysis may beamplitudes performed to of determine massive the vector possibl superfields. A general express combining well-defined UV amplitudes of massless chiral and where the Higgs has a “ Goldstone boson. In theplitude. Abelian This Higgs theory, may thisand be is setting identified the it by to onl extracting zero. the The component component a amplitude is JHEP10(2019)092 - ]. ry be 39 1 CP c (5.54) (5.55) 2 and , ) 1 K 3 , W M  0 (and likewise , 1 1 M η J 2 2 → η 
= 1 supersymmetry W expected in Abelian 3 , M 
1 Yang-Mills (tree) cou- 1 m N I pond to field theoretic 1 M with vector boson self- J 2 2 W I as the Abelian Higgs theory ( ) to the expected expres- ities of particles 1 1 of the vectors), just as for  2 3 oment in the massless limit A de would require that ms mentioned earlier (or are ude is then K 2  plets with a single derivative. F g UV limits that converge to sive vectors has been studied pling consistent with this. lls vector self-interaction term 2. This constitutes one of the 5.53 3 K /m K ons of the Standard Model. An I orrespond to the remaining two 3 ructures has been given in [ ) and is itself the three-particle ∂ 1 ractions, 3 sed. Two of these (that are 1 ↔ J

∂η . 2 ∼

+ 5.51  1 K c and identifies the gauge coupling as 2 remains free. This structure, in addi- − 3 J IM 2 1 I 2 ∂ ǫ c c 1 ] to five. The two prohibited terms are JM 2 2

∂η ǫ  2 K m m 39

J 3 K 1 − K J J 3 = 3 2 ǫ I , thereby reducing the number of independent = I

1 1 2 1 – 30 – 6 1  c 2 5

2  c 3 the scalar components of the supermultiplets, then 1 m , c 2 i m m I − m 1 1 − H ∂ c = 0  ∂η − + = ) † 4 5 = c c Q ) = ( 2 δ = 1 K terms (for Yang-Mills curvatures c 1 3 ) represents the supersymmetrisation of this. Supersymmet c K 3 3 F ) = 5.51 ,W = 0 and K 3 2 6 W ,H , , c ). The Higgs coupling 2 J 2 3 I 1 I m 1 W = 0 1 , I W 4 1 m ( c ( 3 W ( A = g/ A 2 3 c − The remaining five couplings each describe superamplitudes The remaining four couplings have poor UV scaling and corres Of the five remaining structures, one can be attributed to the This contains component amplitudes of the form that would be = = 2 2 six independent contributions to the superamplitude ( which is manifestly antisymmetric under the exchange 1 3 permuted). These areand, the given the component assumption amplitudes that expected there are in no vector self-inte implies that there is only a single Lorentz structure and cou operators upon which gauge invarianceodd) is may not be linearly reali identified withgenerated the at generalised loop-level by Chern-Simons anomalies), ter types while the of other operators two that c mayOf be these, constructed one from corresponds vector multi to the anomalous magnetic dipole m may be identified by matching the component amplitude ( pling. Just as for the Higgs couplings, the expected Yang-Mi c for the other masses, repeating this argument with the ident tion to the Higgsmassless coupling three above, particle are superamplitudes distinguished at as leading havin order. massless amplitudes. Completion of the identification of this with a Higgs amplitu couplings to one. The corresponding term in the superamplit sion. Doing so imposes restricts the seven independentthose originating couplings from of [ The superamplitude ( superamplitude for the Abelian Higgs theory. inversely proportional to some mass scale and that c interactions. The tripleextensively gauge in coupling the vertex past in ofeffective the three context Lagrangian of mas describing the the electroweak independent bos Lorentz st Higgs theories. For example, calling JHEP10(2019)092 ) ) s 0, c- → 5.44 5.33 ) and 1 s, such , while K 3 2 m W ), the su- , as /m ′ J 2 ) and ( h m , up to terms 3.14 W b ). This deter- , ], the amplitude = 5.32 − → + 1 6 4 3 c 5.42 = (Φ = 0 (as is true at ) converge to ( iplet. As long as the 4 terms and so cannot A , m litudes with massless ′ c couplings, such as re- 2 h 6 + and c m oupling. The other two 1 5.51 , 2 η = and ) and ( )ˆ rticle amplitudes typically (up to inclusion of possible er of the two cases discussed K and a 3 sections, but we refrain from h es. Two simple examples are 6 entifications of the couplings , but may additionally have -term”) Higgs and Yang-Mills c h/m s of a massless leg, while linear 1 − and involve terms that pick up 5 5.39 W es, the case of a heavy particle D c s described in [ , 1 s would have the good UV lim- = = J η 2 /m ′ )ˆ and 3 3 g W c K c 5 3 , c + 1 = W ), . , 2 G 6 J m 2 ( d c vanishes (which depends upon the helicity 1 , so that, if W − 2 1 m ) approach ( , c ( , we demand that ( and m − 1 → A / = ′ 3 ) 1 g 5.51 (Φ – 31 – 5 d m and ]. Note that, as expected from ( K 3 − c 6 -odd counterpart, in the massless limit, provides A 1 and c W 6= = , 0. ) + CP 2 J m 2 2 0 in order to find further consistency conditions on the K c 3 are associated with the couplings that determined the → m , this requires that W 1 → , 1 for the massless superfield. A similar analysis can be per- 4 W = c , + 2 1 1 m J 2 1 ). Its η /m m as just described) match onto the terms in ( c W 1 correspond to the tree-level (“ ( W 2 in the massless amplitudes (so parity must be an accidental and 6 , c A also contain the other non-Yang-Mills contact interaction ∼ c 5.42 − 1 g 3 0 or , 6 ). Unlike the previous case, these limits ensure that the mas c 5 G c . Again, matching onto the superamplitudes with massless ve c ( = = 0). These may be easily checked using the spinor limits pro- and → , ′ 2 5.33 2 ′ 1 A.1 g c c and h ) and ( m , ) while the other degenerates with the → A 5 1 I c ) c = K 5.39 3 in the limit that h ) and ( W , 5.5 J 2 ) for each helicity configuration of the massless vector mult 5.32 W , . One of the terms with couplings 5.40 − 1 1 In the case where the two remaining masses approach equality Further conditions may be used to constrain or interpret the For the case where we leave W m ( contributions from be independently determined. Finally, the couplings choice for index ∝ symmetry if vided in appendix it is required that the extra Grassmann variable ˆ extra terms thatmatter would be ( determined by matching onto the amp anomalous magnetic and electric dipole momentscombinations in of the limit providing the results here.made above These — are that consistent with is, the id couplings to match onto the superamplitudes in the previous we can demand that the coefficients of ( formed by instead A peramplitudes have limits scale of the couplings is given by mines the coefficients to be tors, the remaining couplings must be and may be identified with the couplings in section and ( couplings do not scale as quiring good UV limits andprovided by properties demanding of that higher this leg amplitude amplitud matches onto eith tree-level in perturbative gaugeits theories), arranged the by amplitude the Higgs mechanism [ of one leg in ( the same Lorentz structure,couplings vanish but in with the a limit of different a phase massless in leg on-shell. the c couplings, while 5.7 Higher spinWhile amplitudes the numbergrows of significantly possible with the Lorentzdecaying spin structures into of two in massless the products three-pa interacting is especially particl simple. A as those induced from Stuckelberg axions and anomalies. JHEP10(2019)092 , ]. ) li- 6 4 ) = h 4 † R Σ (5.58) (5.59) (5.56) (5.57) , Q 3 ( h 3 Σ (2) , δ ) ) , s , † L 2 , i with respective 2  Q  ...I ) ) ( 1 2 1 2 1). The notation j s I I 2 ( ϕ − (2) to massless super- − 3 4 P 2 δ h ...I i h 4 − 1 h to the corresponding P and incoming in the h I ) η ( 3 S , ξ +1 and ) + ϕ ( 2 1 s 1 2 L 1 symmetric factorisation 1 2 h d R , φ g h − phi A ϕ 2 1 − 3 les permit this process to s R es by exchange of a mas- A ...I 2 + h 2 3 1 − ssive field are symmetrised n Y 2 h s and the Grassmann integral I ntiparticles occupy opposite 2 the analytic continuation of ) ( 3 termediate resonance. It has + , ξ h 2 2 ¯ s (2 may be constructed out of the I 2 P φ ing p ange of higher spin resonances, e non-supersymmetric case [ ust be the antimultiplet, hence h 2 , η = , ξ 1 − -channel, j 2  + 2 1 h 2 s 1 1 2 i , ϕ − p 1 1 1 − ] = ( i h 1 , ϕ 4 h 1 I ) 1 ( h 4 ϕ ξ G ) h 1 and likewise 3 s   2 ) is easily demonstrated as consistent , ξ ϕ h i ( 1 2 1 h A A ...I h A ) ) − 1 ) † 3 † I − 5.59 ( † P 2 h 3 =1 Q Q i Y h Q ( ( , ξ ,S + ( 2 s 2 (2) (2) ]. h 2 h 2 2 – 32 – (2) δ δ 6 h Σ δ i i , ϕ , + S 1 1 1 1 1 1 34 12 h 1 h h 1 h m h ϕ + s (Σ  = L ) = ) = A ) A [12] 4 s h 4 P 2 G to decay into two massless particles η Σ ). Because of the simplicity of the three-particle superamp 2 , ...I ¯ φ 1 d 3 I h 3 ( ) = 3 ) 5.58 Z Σ s , 2 ,S ), then the three-particle superamplitude is also fixed as 2 2 → h 2 h 2 ...I 1 ) is uniquely Σ Σ I 4 3.16 , ( 3 , 2 h 4 1 ¯ 1 φ h h 1 h 1 Σ , ) with Clifford vacua of helicities , 2 3 2 h (Σ (Σ h 3 A and A Σ , ϕ 3.17 massive particle 1 , 1 is the mass of the heavy multiplet. 2 1 h s h h 2 is some coupling constant of mass dimension [ S ϕ ( Σ m G , A 1 h 1 The factorisation of the superamplitude ( The supersymmetrisation of this is just as simple. Promotin The superamplitude for scattering of four massless particl On a massive resonance, the superamplitude respects a super (Σ the bar and the oppositelevels height in spin indices. the superfield. The component a other factor. The incomingan outgoing superfield multiplet. is Crossing then relations represented imply as that this m multiplets ( accounts for the sum overbeen all chosen states to in the represent multiplet the of massive the multiplet in as outgoing i is intended to indicateover. that It all is of beingexist. the assumed spin that indices angular for momentum the selection ma ru where the component amplitude sive spinning particle may be constructed analogously to th where the intermediate superfield has Grassmann variables A where massive multiplet ( helicities with expectations from ( tudes, the Grassmann integral may be trivially evaluated us just as for the non-supersymmetric case [ into three-particle superamplitudes. For example, in the spinning Gegenbauer polynomials corresponding to the exch Supersymmetry fixes the superamplitude to have the form where for a spin JHEP10(2019)092 - n- † i,P cat- Q ibility − ) with = 5.58 P − † i, Q ed to construct massive ] we formulate a massive l spin and polarisation — haracterised by their spec- eg amplitudes, some guid- 14 onent three-particle ampli- in present on the Coulomb e whether they conspire to nstructed all of the possible ]. For example, consequences uctible from soft limits were f supersymmetric theories by orisation or behaviour in the n of all Coulomb branch tree uperamplitudes seems to arise In [ 40 nted. Further constraints upon es are described in a supermul- d leg) amplitudes from infrared des of supersymmetric theories, ) is given simply by ( assive particle representations of , recursion. However, because the Prospects for overcoming this are ding multiplets of supergravity or s of spin no greater than 1. cle superamplitudes. This remains 31 icity of the shifted states, the effec- -matrix postulates constrain super- , 5.59 S 6 ], which here imply that 14 ] for a possible application to black holes). 43 – 33 – , 42 , 15 are the supercharges associated with each respective sub- ). This requires use of the analytic continuation rules for † R . The two representations of the three-particle superampli Q 5.59 P ]. We do not foresee difficulties in extending our analysis to s = 1 theories. Purely from the foundational principles of qua 41 and N = 4 SYM where, for massless amplitudes, a myraid of construct † L ). Q N 5.57 of momentum ), where ) can then be substituted to confirm that ( i † Q ( 5.57 (2) It would be interesting to more broadly catalogue theories c A more exhaustive study into the extent to which To progress beyond single particle representations and 3-l δ S high energy limit, remain to be investigated. multiplets and their interactions attheories weak from coupling assumptions is about warra IR properties, such as fact superamplitude above in ( tra and interactions fromimply emergent conditions symmetries on or uniqueness IR properties [ properties and se super-BCFW shift and provesuperamplitudes. its validity The for constructibility the offrom constructio Coulomb a branch surprising s ‘nonlocality’ presentan in interesting the avenue three-parti for future work. (on-shell) properties would bevalidity desirable, of such massless as recursion is on-shell oftentive sensitive combining to of the massless hel states ofthe (super-)Poincare definite group helicity poses into a m potentialmost obstruction. promising in recently investigated in [ tering states of higher-spin compositeKaluza-Klein superfields modes or (see inclu recently [ ance for systematically constructing higher order (loop an of supersymmetry on emergent properties of theories constr tudes shown in ( spinors and Grassmann variables given in [ the exhibited component amplitude factorised into the comp branch, in particular the dual (super)conformal symmetry. properties have been discovered. Vestiges of these may rema tude ( m for state developing the on-shell superspace formalismtiplet in by their which asymptotic stat quantumwithout numbers the — momentum, need tota tosupermultiplets commit and to represent a these frameconcentrating in of here scattering reference. on amplitu This was us 6 Conclusion We have here initiated the study of the on-shell properties o tum mechanics, special relativityelementary and on-shell three-point supersymmetry, amplitudes we for co multiplet JHEP10(2019)092 (A.2) (A.3) (A.1) (A.4) , trans-

I p ]. AH and

6 ]) is free, so p | ) are conjugates, ] and ] for interpretation I p A.1 | 44 d momentum and its , ussions on [ I 9. h the momentum’s little ] for review of the spinor . allum Jones for comments (as representations of the work is supported in part 8 ual way to convert between , α  . IJ ture has consequences for the n have bilinear products with ] J We take all scattering states upport during the completion d its consequences, taking the I es (including little group) cor- p

p we adopt throughout this article d to [ ering amplitudes. Internally, as mε cles, the conventions of which, in . IJ − − | ε β

= = I =  p † J )  α ]

p ˙ α I I I i p p . As the spinors in ( p | I

 p ( m | I – 34 – X ˙ α | i . The choice of the phase of det( − IJ J I 2 | p p | h = ]) mε for mass p = | IJ ˙ αβ 2 = ε † p )

m

= J det( I ) may be decomposed into two null momenta as p | ˙ p = α µ I 

σ p ( I 2 µ

p p p

) = − | = p h p ) = p ): C ]) det( , p ], we (mostly) adhere to and will not restate. | may be chosen without loss of generality (although see [ 10 m ]) = ) = det( p As usual, det( The two pairs of left- and right-handed spinors indexed by Introducing helicity spinors with SU(2) little group struc | p det( Under conjugation, the spinors transform as parently respect an SU(2)group. symmetry These that SU(2) indices may mayrepresentations be be and raised identified their and wit conjugates: lowered in the us Fundamental tensor representations havehere lowered to indices. be outgoing,responding so to naturally the have polarisations raised of internal the indic conjugated states. of the mass andconsequences its for dual complex conformal phase symmetry).themselves as The the spinors the extra components of a 6 det( description of the internalmentioned and above, external the structure starting ofspin point scatt group SL(2 is that massive momenta opportunity to establish the conventions andand notation that also to presenthelicity useful method identities. for The scattering reader processesaddition is to of referre [ massless parti A Conventions and usefulA.1 identities Spinor helicityWe for here massive summarise particles helicity spinors for massive particles an SK are grateful forby the the support U.S. of Department of a Energy Worster under Fellowship. the This grant DE-SC001412 We thank Tim Cohen, Nathanielon Craig, a Henriette Elvang, draft and of C thisof work, this Nathaniel Craig work, for and discussions Nima and Arkani-Hamed s and Yu-tin Huang for disc Acknowledgments JHEP10(2019)092 f o + − + cular of its (A.7) (A.5) (A.6) T of this nent of  2 1 √ , , i 3 −− ) indices opposite 2 T I into two massless s 1 he tensor products 1 m V i h − 2 . tensors then represent 1 s I ( ˙ αβ ˙ β tation. 1 p e helicities of massless legs struction of these was de- nors to construct a tensor α h ent combinations of exter-  p − I g e is described by symmetric lutions to each of the Weyl indices are an internal degree o any external spin frame or

p I

= = r each massless particle. As a ting superamplitudes. = p ˙  β β 2 m f massive states of any spin.

entries between states of different -matrix and are consistent with α h the possible independent terms I I − m S p p 3] | 

= = 1 ˙ α α α ]

p i ]. In practice, we find that it is clearest I I I

2] 6 p p I p | |

p has polarisation wavefunction that can be p

× | s  ), a direct on-shell construction of elementary 1 states respectively given by 2  , α ) – 35 –

0 2

) I , ˙ ˙ I α 2 β

p 1 I p I

β α 1 p −  h mδ

1 m I 1 = mδ − ( α − m p

s -matrix may be decomposed. Spinors of either chirality is determined uniquely by symmetry to be

1 = = S I = = 3 m ˙ ( β β =  ψ 1

p I ) I I

. This method of deducing little group structures built out o 2 p p p I I

 h g 

p 1 g I p  and ˙ ] for tensor methods to describe spin. We (mostly) restrict t α , with α ( ] ) i 2 I 2 T symmetric tensor of the little group SU(2). States of a parti I indices aligned with the spin direction and 45 I p ψ ) = p 1 | -matrix entry for the decay of a massive vector s | -matrix transforms as a tensor under the little group of each s I 3 ( S 1 in this work, although a significant part of the versatility S + T , ψ s 2 ≤ may be extracted from this by choosing the symmetrised compo . See [ m s V, ψ ++ ( m ]. Rather than build external polarisations directly from t T A 6 and  Part of the utility of this formalism is that the little group The possible structures that may appear in the Externally, the + − spinors is used repeatedly throughout this work in construc of freedom and allowaxis. for the The polarisation procedure for to doing be this projected is ont discussed in [ for some coupling constant and then normalising.polarisation For tensor example, a massive vector particl the tensor with right-handed fermions polarisation Lorentz invariance are determinednal by state the polarisations number of thatscribed independ can in be [ made. The systematic con T formalism is its ability to elegantly describe amplitudes o of massive spinors (e.g. particles of spin then determine the amplitude’ssimple U(1) example, little the group scaling fo (or both) may bepolarisation-stripped used Lorentz to tensor do amplitudes,may in this. be whic built The out coefficients of external of momenta these and massless basis spinors. Th obey the Weyl equations: amplitudes can be performedbasis instead with by respect using to the which the massive spi described by a rank 2 external particle legs, being anspin array of configurations. transition An matrix external state of spin and the spin sums: The little groupequations, index which effectively may labels be rotated the into two each other possible by so a Wigner ro JHEP10(2019)092 ) − are i (A.9) (A.8) q (A.10) (A.11) (A.15) (A.13) (A.12) (A.14) | avoiding = 0. i 6 . However, 3 I J q δ ] and h . In this case, q I | = η I η J ∂ ∂η 6= 0 and , i = † J | 3] η p indices q q are defined here as J qp h , d h η . † I ation and other identities.

I group indices simply become I K ∂ d, simply reinterpret the little = dη ∂η η mits, to choose spin frames for on. In particular, as it is often spinors for massless momentum

ing the limit is arbitrary (up to ed or lowered with an extra ( 31 i IJ K ,

+ p ǫ η i  3] p | 2 1 3 3 − s stated otherwise.

t satisfy [ IJ 2 q 0 J ǫ − h 1 p ). The limits may be expressed as , 2 = | 2 1 m 1 −  → → − | q m m = m   h I ( − − 0 

3 † I O η → − − 31 = = = 1 2 , the following identities are useful: lim p  p m 

are the massive legs, while

d x i J  3 η 2 J | + I p η 3] 2 η

, x – 36 – J q p [ ] become the reference spinors and are ambiguous J ]

η | ] J p 2 I q 0 i 1 x 2 q η [ I qp | and q [ d 1 | 32  I 1

m 1 → | →

I = p 

dη ∂ = =  = + + ∂η IJ  − p  p x I ǫ ] and

I

p J

q ∂ 2 1 | 2 x ∂η 32 I 1  m 1 1 2 =  x 0 η 2 → lim d m . We note for convenience the identities = 0). In practice, it is often possible to take the limit while I ∂ i 6 ∂η qp JI h ǫ , ] − qp is the massless leg and = 3 J p ∂ ∂η The little group invariant Grassmann integration measures In this case, the spinors have massless limits For 3-leg amplitudes involving the factor . More precisely, the spinors that vanish do so sign: this requires that the index height on the derivative be rais and in the massless amplitude, asrequiring their [ direction arrived in tak where group indices as referringmost to useful, components especially along in thiseach taking directi particle massless aligned or with highhelicity their energy indices), momenta li we (so will that leave the this choice little implicit unles to abuse notation and, once such an external frame is specifie where the remaining spinors the introduction of the reference by using momentum conserv Grassmann differentiation may be defined in the usual way: where the spinors without littlep group indices are the usual arbitrary reference spinors, not necessarily related, tha A.2 Grassmann calculus The Grassmann variables may be imbued with SU(2) little grou JHEP10(2019)092 ]. is = η 46 I dη (A.17) (A.18) (A.16) . ) = 4 SYM. gluon amplitudes and † . (if the leg is massless, η 1 ion of the massive mul- transformations can be ( ] concerning the relation N ration to be the same as = 0, for some single leg j ˜ represents the remaining f ables have massless limits mponent amplitudes that . In particular, under the † 14 → egrated — that is, mann parameters that are , the arguments presented i,I C † I ηη ions and constraints between η n [ η − = 1 is effected by the replace- e be 0 by supersymmetry. ) of a Grassmann variable I he integration variable can be ies, as prescribed in e.g. [ , while for anti-BPS multiplets e † nt the smaller massless SUSY η † † rvyness of Grassmann numbers. e supertranslation, the resulting ( η N η η f 2 1 d = − ˆ η. Z η → ] for each leg in some little group frame, by choosing I ) = basis in , + η I I θj η ( [ η f i → − , so integrating over it will give 0. The compo- (and likewise for the conjugate). is implied). Here, – 37 – I . The index placement on the differential is a † J i,I η, η J I ) j,I I η δ η η dη ( → f = IJ † − ǫ basis to the η , η J ηη − † † I η η η ηe ]. Amplitudes stated here will be colour-stripped partial = † I I d η η I 19 d 1 2 Z dη R (no sum over ) = | → − † I is translated to ] and [ and i 1 η [ 9 ( ˜ J I f j,I C δ η + = ],

I J i and these are related by θQ  η [ I ˜ f η i,I − η d represents the redundant variable left-over from the divis i R m − η ). The strange positioning of the index is needed for this ope Firstly, the supersymmetric Ward identities provide relat For multiplets without a central charge, the Grassmann vari The Grassmann Fourier transform of some function = I η | with polarisation in some direction given by ( θ nent amplitudes obtained by such projections must therefor ments are obtained by integratingtranslated are the unaffected superamplitude bylikewise in this translated. the transformation, After because Grass changing t integrand variables is to completely absorb independent th of unused degree of freedom in the supersymmetry parameter. Co in the helicity basis [ i d amplitudes, following theThis usual discussion rules is supplementary for to Yang-Millsbetween further (S)QCD theor comments amplitudes made i and Coulomb branch amplitudes of tiplets into smalleralgebra. massless For multiplets the that exceptional case each of represe BPS multiplets, ˆ Here, ˆ property of the differential and not of the variable being int B Comments on higher-legWe amplitudes here in make SQCD somerederive comments on them various in massivehere quark the are and little multi parallel group to covariant [ notation. Again the limit picks up an extra negative sign. found that set aaction Grassmann of generator for a particular leg to 0 component amplitudes that can be exploited. Supersymmetry where just omit the little group index). This can be used to set The Fourier transform from the defined as differentiation and is an occurrenceAlso, of as the for general the topsy-tu derivative, JHEP10(2019)092 j is j (B.2) he su- (if ommons , i,I η ] ] . K K j j I I i i [ [ i ] = 0] = 0 (B.1) m − ]. -gaugino amplitude Q Q , , n = ] by projecting these an be further utilised + − -conjugate amplitudes ly using BCFW recur- 14 C SPIRE CP ial polarisation, may also IN ...G ...G d any number of gluons of ly combined with that used uarks are aligned, then the ded to any of the amplitudes redited. ][ + − ves the fact that amplitudes tudes with additional squark they are fully determined by osing ormed in the examples above. uark and se include amplitudes that are des to those involving squarks. independent of the Grassmann ] and little group covariantised ,G ,G he Grassmann variable for leg e 0, as well as those that include 47 Q Q [ [ ]. The superamplitudes to which , 3 A A 49 ]. +2 +2 n n η η 2 2 d d SPIRE arXiv:1010.0257 Z Z IN [ i i ][ η η On-shell constructibility of tree amplitudes in – 38 – = 4 theory. d d N Z Z i i need not be integrated in order to obtain a vanishing Y Y (2011) 053 1 1 η η j,K ), which permits any use, distribution and reproduction in 2 2 04 η d d Lorentz Constraints on Massive Three-Point Amplitudes ) are 0. basis may be used to show that the -gluon amplitude. L Z Z † ] does not affect the Grassmann numbers with opposite spin n e η Q J , arXiv:1601.08113 JHEP [ θj ] = ] = R [ , L L e i Q e e Q Q CC-BY 4.0 − − + g λ This article is distributed under the terms of the Creative C — which is now the same direction for each massive field. Thus t j,J η J ··· ··· (2016) 041 − + g λ 09 R R e e Q Q [ [ A A ]. A compact expression exists that may be derived inductive general field theories JHEP The extra degree of freedom in the supersymmetry parameter c Tree-level amplitudes involving a quark-antiquark pair an Simple illustrative examples of this are the squark-antisq Vanishing amplitudes of massive quarks, states of non-triv 48 [1] T. Cohen, H. Elvang and M. Kiermaier, [2] E. Conde and A. Marzolla, Open Access. some number of gluons orinherited gauginos by of pure identical QCD helicity. at The in tree-level. the The previous argument paragraph can to be extend these easi vanishing amplitu to derive the vanishing of a further class of amplitudes. Cho peramplitude integrated over only these componentsvariable will that be is beingwith eliminated, quarks and so antiquarks must all vanish. of identical This polarisation deri ar the same helicity have been previously determined in [ Attribution License ( amplitude. Thus one extra particleabove in any and spin the state result may be will ad still be 0. massless, omit its littledoes group not index shift in under this theThis expression), supersymmetry means transformation t that perf the variables components to are also 0. Identicaland arguments antisquark also pairs demonstrate ( that ampli Identical arguments in the References a single component amplitude,superamplitudes which out we from show the in massive appendix B of [ any medium, provided the original author(s) and source are c in [ sion by shifting thethese massless amplitudes legs belong in have the the interesting usual property way that [ transformation be obtained similarly. If all of the little group axes of the q and the squark-antisquark JHEP10(2019)092 , , , ] ith nd ]. , ]. Jets , + SPIRE assive particles 0 IN , [ SPIRE , SYM /Z IN arXiv:0807.4097 , ± 4 [ harges in Extended e ][ W ]. , SYM ]. ]. N < → , ]. Dual superconformal = 4 pp SPIRE v, N ]. 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