On-shell constructibility of Born amplitudes in spontaneously broken gauge theories
TTTK-19-43October 29, 2019
On-shell constructibility of Born amplitudesin spontaneously broken gauge theories
Robert Franken a , Christian Schwinn ba Institut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg,D-97074 W¨urzburg, Germany b Institut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University,D–52056 Aachen, Germany
Abstract
We perform a comprehensive study of on-shell recursion relations for Born am-plitudes in spontaneously broken gauge theories and identify the minimal shiftsrequired to construct amplitudes with a given particle content and spin quantumnumbers. We show that two-line or three-line shifts are sufficient to construct allamplitudes with five or more particles, apart from amplitudes involving longitu-dinal vector bosons or scalars, which may require at most five-line shifts. As anapplication, we revisit selection rules for multi-boson amplitudes using on-shellrecursion and little-group transformations. a r X i v : . [ h e p - t h ] F e b Introduction
The development of on-shell recursion relations of Born amplitudes in gauge theories byBritto, Cachazo, Feng, and Witten (BCFW) [1, 2] has motivated an approach to Quan-tum Field Theory that aims at the construction of amplitudes solely in terms of on-shellbuilding blocks. For massless theories, space-time symmetries and factorization propertiesare sufficient to fix the structure of three- and four-point amplitudes and to establish theuniqueness of non-abelian gauge theories [3]. The three-point vertices serve as input tothe BCFW recursion relations, which are based on a continuation of amplitudes into thecomplex plane by a complex shift of two external momenta. This provides a purely on-shellconstruction of Born amplitudes in unbroken gauge theories. The on-shell constructibilityof gauge theories with general matter content or of Effective Field Theories was investigatedusing generalizations of the BCFW construction to shifts of more than two legs [4–6].Concerning theories with massive particles, symmetry constraints on three-point am-plitudes were obtained using supersymmetry [7, 8] and little-group transformations [9, 10].More recently, this analysis was simplified using a manifestly little-group covariant notationfor massive amplitudes [11]. The uniqueness of four-point amplitudes can be argued to arisefrom factorization properties and consistency with the high-energy limit [11]. This line ofargument is expected to reproduce the classic results on the uniqueness of spontaneouslybroken gauge theories (SBGTs) [12–14] within an on-shell approach. All three-point ver-tices and amplitudes for some three- and four-body decays in the electroweak StandardModel (SM) were constructed using this formalism [15, 16], while all three-point verticesincluding contributions from higher-dimensional operators were obtained in [17].In the current paper we investigate the on-shell constructibility of higher-point Bornamplitudes in SBGTs. After initial explorations of BCFW recursion for amplitudes withmassive particles [18–22], it was shown that amplitudes in QCD with massive quarks can beconstructed using at most three-line shifts [23] whereas the on-shell constructibility of am-plitudes in massive, power-counting renormalizable theories using all-line shifts was provenin [4]. The constructibility of massless amplitudes with the matter content of SBGTs us-ing at most five-line shifts was demonstrated in [5]. We extend this analysis to the brokenphase and show that all amplitudes involving at least two transverse vector bosons canbe constructed using three-line shifts, while amplitudes with longitudinal vector bosons orscalars may require four- or five-line shifts. This requires the analysis of the behaviour ofamplitudes for large values of the parameter z , which parameterises the complex contin-uation of the amplitude. Because of the analogy of the large- z limit with a high-energylimit [24], it is anticipated that the result of [5] carries over to the broken phase. However,in the analysis of massive amplitudes we encounter several technical complications com-pared to the massless case. In particular, obtaining shifts with a “good” large- z behaviourappears to make it necessary to break manifest little-group covariance by introducing afixed spin axis. This requires a careful analysis to ensure that amplitudes for arbitraryspin configurations can be constructed recursively. Furthermore the violation of helicityselection rules by mass terms can lead to contributions to amplitudes with a worse large- z behaviour than in the massless case. 1he paper is structured as follows. In Section 2 the little-group covariant spinor for-malism for massive particles [11] is reviewed and related to expressions for Dirac spinorsand polarization vectors for a fixed spin axis. In Section 3 a systematic discussion of shiftsof massive momenta is given, generalizing the extended Risager [25] and BCFW-type shiftsused in [5] to the massive case. After initially constructing shifts of momenta and wave-functions in a little-group covariant form, a suitable choice of spin axes leads to a similar z -dependence of wave functions as in the massless case. All the required shifts with twoto five legs are constructed explicitly. The large- z behaviour of amplitudes in SBGTs isestablished in Section 4. These results are used in Section 5 to identify the minimal num-ber of shifted lines needed to construct a given amplitude. As an application, in Section 6selection rules for amplitudes with massive vector bosons are derived using recursion re-lations and little-group transformations, providing a new perspective on results obtainedusing diagrammatic analysis [26] or supersymmetry [8]. Details on the employed spinorconventions and explicit expressions for little-group transformations changing the spin axisare given in Appendices A and B, respectively. Recursion relations with non-lightlike shiftsof internal lines are briefly discussed in Appendix C. In this section the conventions for the momenta and wave-functions of massive particlesare set up, relating the little-group covariant notation introduced recently in [11] to theconventions used previously for on-shell recursion relations for massive quarks [23]. Theuse of little-group transformations to relate wave-functions with different spin quantumnumbers is also discussed.
At the heart of the spinor-helicity method is the observation that the complex two-by-twomatrix k α ˙ α = k µ σ µα ˙ α associated to a light-like momentum factorizes into a product of two-component Weyl spinors in the two inequivalent fundamental representations of SL (2 , C ), k α ˙ α = k α k ˙ α , (2.1)where we refer to the spinors k α ∈ D ( , as holomorphic and the spinors in the conjugaterepresentation k ˙ α ∈ D (0 , ) as anti-holomorphic. Our Weyl-spinor conventions follow [23]and are summarized in Appendix A. Similarly, a massive momentum satisfying the on-shellcondition k α ˙ α k ˙ αβ = m δ βα (2.2)can be expanded in a basis of two holomorphic and two anti-holomorphic spinors. Variousapproaches have been followed in the literature, e.g. introducing fixed spin vectors [27] orusing helicity eigenstates [28].A notation that makes the transformation properties of spinor variables under thelittle group SU (2) of massive momenta manifest was introduced in [11]. In this notation,2 massive momentum is parameterized as k α ˙ α = k Iα k ˙ α,I = k Iα k J ˙ α ε JI , k ˙ αα = k ˙ αI k α,I = k ˙ αI k αJ ε IJ , (2.3)where the two-component little-group indices I are raised and lowered with the two-dimensional antisymmetric tensor. Using the same conventions as for un-dotted Weyl-spinor indices in (A.3), these relations read k Iα = ε IJ k α,J , k α,J = k Iα ε IJ , ε IJ = ε IJ = (cid:18) − (cid:19) , (2.4)with identical definitions for the anti-holomorphic spinors.The massive spinor variables can be chosen to satisfy the SU (2) and SL (2 , C ) covariantnormalization conventions (cid:104) k I k J (cid:105) = mε IJ , [ k I k J ] = − mε IJ , (2.5) k Iα k β,I = mε αβ , k I ˙ α k ˙ β,I = mε ˙ α ˙ β . (2.6)Holomorphic and anti-holomorphic spinors are related by the Dirac equations k α ˙ α k ˙ αI = − mk α,I , k ˙ αα k Iα = − mk ˙ α,I . (2.7)In the construction of the complex continuation of scattering amplitudes we will beforced to break manifest little-group covariance and fix a particular basis for the decompo-sition of the momenta, as in earlier work on on-shell recursion for massive momenta [23].In this reference a fixed, light-like reference momentum q is used to decompose a massivemomentum into a sum of two light-like vectors, k µ = k (cid:91) ; µ + m q · k ) q µ . (2.8)The associated holomorphic and anti-holomorphic Weyl spinors k (cid:91)α , q α and k (cid:91) ˙ α , q ˙ α provide aparticular example of a basis for the expansion of a massive momentum, which correspondsto the choice k α = k (cid:91)α , k α = m (cid:104) k (cid:91) q (cid:105) q α ,k ˙ α, = k (cid:91) ˙ α , k ˙ α, = m [ qk (cid:91) ] q ˙ α (2.9)in the little-group covariant expressions. External wave-functions of massive particles canbe defined as eigenstates of the corresponding spin operators with respect to the spin vector n µq = k µ m − m q µ q · k , (2.10)as discussed e.g. in [8]. 3 .2 Little-group transformations By definition, little-group transformations R ∈ SU (2) of the Weyl spinors, k I → k (cid:48) I = R I J k J , k I → k (cid:48) I = − R I J k J , (2.11)leave the momenta (2.3) invariant. These definitions hold both for dotted and un-dotted SL (2 , C ) indices, which have been suppressed. Note that spinors with lower little-groupindices transform in the dual representation with the transformations( R T − ) I J = ( ε − ) IK R K L ε LJ = − R I J , (2.12)where indices of the little-group rotations are raised and lowered with the convention (2.4).Infinitesimal little-group transformations, R I J = δ IJ + ω I J + . . . , (2.13)are parameterized by three parameters in the symmetric matrix ω IJ = ω JI . The action ofinfinitesimal little-group transformations on functions of the spinor variables of a momen-tum k induces a representation of the Lie algebra of the little group, δϕ ( k Iα , k I ˙ α ) = ϕ ( k (cid:48) Iα , k (cid:48) I ˙ α ) − ϕ ( k Iα , k I ˙ α ) = − ω I J ( J k ) J I ϕ + O ( ω ) , (2.14)with the differential operators [9, 10]( J k ) J I = − (cid:18) k Jα ∂∂k Iα + k J ˙ α ∂∂k I ˙ α (cid:19) . (2.15)For the determination of spin eigenstates for a fixed spin axis it is useful to form the linearcombinations J − k ≡ ( J k ) = (cid:18) − (cid:104) k (cid:91) q (cid:105) m k (cid:91)α ∂∂q α + m [ qk (cid:91) ] q ˙ α ∂∂k (cid:91) ˙ α (cid:19) , (2.16a) J k ≡
12 (( J k ) − ( J k ) ) = − (cid:18) k (cid:91)α ∂∂k (cid:91)α − q α ∂∂q α − k (cid:91) ˙ α ∂∂k (cid:91) ˙ α + q ˙ α ∂∂q ˙ α (cid:19) , (2.16b) J + k ≡ ( J k ) = (cid:18) − m (cid:104) k (cid:91) q (cid:105) q α ∂∂k (cid:91)α + [ qk (cid:91) ] m k (cid:91) ˙ α ∂∂q ˙ α (cid:19) . (2.16c)These operators satisfy the commutation relations[ J k , J ± k ] = ± J ± k , [ J + k , J − k ] = 2 J k , (2.17)which show that J ± k serve as raising and lowering operators for the eigenstates of J k .4 .3 Massive fermions Solutions to the massive Dirac equation for particle spinors and their conjugates,( /k − m ) u ( k ) = 0 , ¯ u ( k )( /k − m ) = 0 , (2.18)can be constructed in the little-group covariant notation as u I ( k ) = (cid:18) k Iα − k ˙ α,I (cid:19) , ¯ u I ( k ) = (cid:0) k αI , k ˙ α,I (cid:1) . (2.19)Up to different sign conventions these expressions agree with those of [29] where it isshown that they form helicity eigenstates. They satisfy the conventional completeness andnormalization conditions due to the properties of the two-component spinors (2.6): u I ( k )¯ u I ( k ) = (cid:32) k Iα k βI k Iα k ˙ β,I − k ˙ α,I k βI − k ˙ α,I k ˙ β,I (cid:33) = (cid:18) mδ βα k α ˙ β k ˙ αβ mδ ˙ α ˙ β (cid:19) = /k + m, (2.20)¯ u I ( k ) u J ( k ) = (cid:104) k I k J (cid:105) − [ k I k J ] = 2 mδ JI . (2.21)The little-group transformation of the Dirac spinors follows from the index positions, u I → R I J u J , ¯ u I → − ¯ u J R I J . (2.22)Using the translation (2.9) it is seen that the Dirac spinors (2.19) are related to expressionsfor a fixed spin axis in the conventions of [8] by the correspondence u ( k, ) = u ( k ) = (cid:18) m (cid:104) k (cid:91) q (cid:105) q α k (cid:91) ; ˙ α (cid:19) , u ( k, − ) = u ( k ) = (cid:18) k (cid:91)α − m [ qk (cid:91) ] q ˙ α (cid:19) , ¯ u ( k, ) = ¯ u ( k ) = (cid:16) − m (cid:104) k (cid:91) q (cid:105) q α , k (cid:91) ˙ α (cid:17) , ¯ u ( k, − ) = ¯ u ( k ) = (cid:16) k (cid:91) ; α , m [ qk (cid:91) ] q ˙ α (cid:17) . (2.23)Suitable expressions for the corresponding antiparticle spinors are given by v I ( k ) = (cid:18) k α,I k ˙ αI (cid:19) , ¯ v I ( k ) = (cid:0) k α,I , − k I ˙ α (cid:1) , (2.24)which are identified with the expressions for a fixed spin axis as v ( k, ) = v ( k ) = (cid:18) − m (cid:104) k (cid:91) q (cid:105) q α k (cid:91) ; ˙ α (cid:19) , v ( k, − ) = v ( k ) = (cid:18) k (cid:91)αm [ qk (cid:91) ] q ˙ α (cid:19) , ¯ v ( k, ) = ¯ v ( k ) = (cid:16) m (cid:104) k (cid:91) q (cid:105) q α , k (cid:91) ˙ α (cid:17) , ¯ v ( k, − ) = ¯ v ( k ) = (cid:16) k (cid:91) ; α , − m [ qk (cid:91) ] q ˙ α (cid:17) . (2.25)The spinors ¯ u ( k, s ) and v ( k, s ) describe outgoing particles and antiparticles with spin quan-tum number s , while u ( k, s ) and ¯ v ( k, s ) describe incoming particles and antiparticles with5eversed spin label. For the Weyl-spinor conventions of Appendix A the behaviour ofspinors under a reversal of the momentum is given by v ( − k, s ) = i sgn( k + k ) u ( k, s ) , ¯ v ( − k, s ) = i sgn( k + k ) ¯ u ( k, s ) . (2.26)A change of the spin axis corresponds to a little-group rotation of the Dirac spinors, u (cid:48) ( s (cid:48) ) = (cid:88) s = ± R ( ) s (cid:48) s u ( s ) , ¯ u (cid:48) ( s (cid:48) ) = (cid:88) s = ± ¯ u ( s ) R ( ) − ss (cid:48) (2.27)where the matrix R ( ) is given explicitly in (B.4) in Appendix B.The Dirac spinors are eigenstates of the generators J , J k u ( k, s ) = s u ( k, s ) , J k v ( k, s ) = sv ( k, s ) , (2.28)and related by the action of J ± in (2.16), J ± k u ( k, ± ) = 0 , J ± k u ( k, ∓ ) = − u ( k, ± ) ,J ± k v ( k, ± ) = 0 , J ± k v ( k, ∓ ) = v ( k, ± ) , (2.29)which are the expected relations for angular-momentum ladder operators up to a non-standard phase convention for the particle spinors. Polarization vectors of massive spin one particles transform under the three-dimensionalrepresentation of the little group and can be described by symmetric bi-spinors, (cid:15) ( IJ ) α ˙ α ( k ) = 1 √ m k ( Iα k J )˙ α = 1 √ m (cid:0) k Iα k J ˙ α + k Jα k I ˙ α (cid:1) . (2.30)These satisfy the transversality, orthonormality, and completeness relations k ˙ αα (cid:15) ( IJ ) α ˙ α = 0 , (2.31) (cid:15) ( IJ ) α ˙ α (cid:15) ( KL ) ˙ αα = (cid:0) ε IL ε JK + ε JL ε IK (cid:1) , (2.32) (cid:15) ( IJ ) α ˙ α (cid:15) ˙ ββ ( IJ ) = (cid:32) δ βα δ ˙ β ˙ α − k ˙ βα k β ˙ α m (cid:33) . (2.33)The polarization vectors transform under little-group transformations as second-rank ten-sors, (cid:15) ( IJ ) → R I M R J N (cid:15) ( MN ) . (2.34)The expressions in the spinor formalism for the fixed spin axis (see e.g. [8, 28]), (cid:15) α ˙ α ( k, +) = √ q α k (cid:91) ˙ α (cid:104) qk (cid:91) (cid:105) , (cid:15) α ˙ α ( k, − ) = √ k (cid:91)α q ˙ α [ k (cid:91) q ] , (cid:15) α ˙ α ( k,
0) = 1 m (cid:18) k (cid:91)α k (cid:91) ˙ α − m q · k q α q ˙ α (cid:19) , (2.35)6re related to the little-group covariant notation by the identifications (cid:15) ( k, +) = (cid:15) (22) ( k ) , (cid:15) ( k, − ) = − (cid:15) (11) ( k ) , (cid:15) ( k,
0) = −√ (cid:15) (12) ( k ) , (2.36)where the minus sign in the definition of (cid:15) ( k, − ) ensures the conventional normalizationcondition 12 (cid:15) α ˙ α ( λ ) (cid:15) ˙ αα ( − λ (cid:48) ) = (cid:15) ( λ ) · (cid:15) ( − λ (cid:48) ) = − δ λ,λ (cid:48) . (2.37)As a result of these conventions, the action of the generators (2.16) on the polarizationvectors is given by J k (cid:15) ( k, s ) = s(cid:15) ( k, s ) ,J ± k (cid:15) ( k, ± ) = 0 , J ∓ k (cid:15) ( k, ± ) = ±√ (cid:15) ( k, , J ± k (cid:15) ( k,
0) = ±√ (cid:15) ( k, ± ) . (2.38)A change of the spin axis is represented as a little-group transformation, (cid:15) (cid:48) ( s (cid:48) ) = (cid:88) s =+ , , − R (1) s (cid:48) s (cid:15) ( s ) , (2.39)where the matrix R (1) is given in (B.5). The basis of the proof [2] of the original on-shell recursion relation [1] and its generaliza-tions [4, 5, 25] is the construction of a complex continuation A ( z ) of scattering amplitudesobtained by a deformation of a subset S = { k i } , i = 1 , . . . h of the four-momenta of theexternal particles, parameterized by a complex parameter z , k i → ˆ k i ( z ) for k i ∈ S , (3.1)so that the physical amplitude is given by A (0). We consider deformations with the fol-lowing properties: • Four-momenta are deformed by a linear shift in z ,ˆ k i ( z ) = k i + zδk i . (3.2) • The shift does not modify the mass-shell condition of the external momenta,ˆ k i ( z ) = k i = m i . (3.3) • The deformed momenta satisfy momentum conservation if the original momenta doso, i.e. (cid:88) S ˆ k i ( z ) = (cid:88) S k i , (3.4)where all momenta are taken as outgoing. This condition must only hold for the sumover the full set of shifted momenta S , i.e. there should be no subset where (3.4) issatisfied on its own. 7 For all possible “factorization channels”, i.e. all decompositions of the set of externalmomenta P into two subsets P = F ∪ F (cid:48) , the sum over momenta in each subset isdeformed by a light-like vector,ˆ K F ( z ) = (cid:88) F ˆ k i ( z ) = K F + zQ F , Q F = 0 , (3.5)with Q F = (cid:88) S∩F δk i = − (cid:88) S∩F (cid:48) δk i . (3.6)This condition is not necessary (see e.g. [5]) but chosen here to simplify the discussion.Shifts with Q F (cid:54) = 0 do not lead to advantages for our purposes, as discussed inAppendix C.Since poles of Feynman diagrams arise solely through propagators, only simple poles in z appear for Born amplitudes for the above properties of the shift. The integral of thefunction A ( z ) /z over a circle with | z | → ∞ is given by the sum of the residues at the poles, z F and the residue at z = 0, which gives the physical amplitude,12 π i (cid:73) A ( z ) z = A (0) + (cid:88) poles z F Res z F A ( z ) z . (3.7)Factorization properties of Born amplitudes imply that the n -point amplitude at the poles z F factorizes into lower-multiplicity amplitudes according to lim z → z F A ( z ) = (cid:88) s A F ( . . . ˆΦ s F ) i K F + 2 zK F · Q F − M F A F (cid:48) ( ˆΦ − s F (cid:48) , . . . ) , (3.8)where Φ s F denotes a generic particle with spin projection s , momentum ˆ K F and mass M F .In the following, massive vector bosons will be denoted by W s , massive fermions with s = ± by ψ ± , and φ denotes both physical scalars or would-be Goldstone bosons. Thepoles of the complex variable z are located at z F = − K F − M F K F · Q F . (3.9)Provided the condition lim z →∞ A ( z ) = 0 (3.10) If the internal particle Φ F is a fermion, a convention-dependent phase factor arises since one of themomenta K F and K F (cid:48) = − K F corresponds to an incoming particle line [23]. Writing the numerator ofa fermion propagator in terms of the completeness relation and using the convention (2.26) one finds for K F + K F > /K F + M F = (cid:88) s u ( K F , − s )¯ u ( K F , s ) = (cid:88) s ( − i v ( K F (cid:48) , − s )(¯ u ( K F , s )) . A (0) = (cid:88) F (cid:88) s A F ( . . . ˆΦ s F ) i K F − M F A F (cid:48) ( ˆΦ − s F (cid:48) , . . . ) . (3.11)On the right-hand side the shifted momenta in the set S and the momentum ˆ K F of theinternal line are evaluated at the poles (3.9). A prescription for the reference spinors forinternal massive particles is defined in Section 3.3.For internal massive vector bosons, gauge invariance implies that the unphysical degreesof freedom, i.e. the would-be Goldstone bosons and the fourth polarization vector (cid:15) µ ( S ) = p µ /M cancel at the pole (3.9) so that only the sum over the three physical polarizationsneeds to be taken in (3.11).In this section, we derive little-group covariant expressions for the shift of spinor vari-ables and the wave functions of massive spin one-half fermions and vector bosons. However,in order to satisfy the condition (3.10) we will choose a particular spin axis aligned withthe shift [4, 23]. Therefore it must be possible to recover amplitudes for arbitrary spin axesand spin quantum numbers. There are two ways to achieve this: • Construct the amplitude for a fixed spin state for arbitrary and independent spinaxes n q i for all particles. The spin-dependence of the scattering amplitude enters onlythrough the polarization wave functions, while the remaining truncated amplitude islittle-group invariant since it can be expressed in terms of momenta, Lorentz tensorsand Dirac matrices. Therefore amplitudes for arbitrary spin states can be obtainedusing the generators J ± k i (2.16), which only act on the wave-function of a single leg i . • Construct the amplitudes for all spin states for a particular fixed choice of the spinaxes. Results for arbitrary spin axes can be obtained as linear combinations ofthese results using the finite little-group transformations of the polarization wavefunctions (2.27) and (2.39).In the remainder of this Section we construct all possible h -line shifts of massive par-ticles satisfying the properties (3.2)–(3.5). The large- z behaviour (3.10) is investigated inSection 4 while the ability to construct the amplitudes for all spin states is discussed inSection 5. We first consider the shift of a single massive momentum k such that the on-shell condi-tion (3.3) remains satisfied. In the little-group covariant notation, the shift can be definedby introducing pairs of holomorphic and anti-holomorphic spinors η Iα and η ˙ α,I and deform-ing the spinor variables according to k Iα → ˆ k Iα ( z ) = k Iα + zη Iα , k ˙ α,I → ˆ k ˙ α,I ( z ) = k ˙ α,I + zη ˙ α,I . (3.12)9he on-shell condition can be satisfied by demanding that the normalization conditions (2.5)are not modified by the shift, (cid:104) ˆ k I ( z )ˆ k J ( z ) (cid:105) ! = mε IJ , [ˆ k I ( z )ˆ k J ( z )] ! = − mε IJ . (3.13)This implies that the holomorphic spinors η Iα must satisfy the conditions (cid:104) η I η J (cid:105) = 12 (cid:104) η I η I (cid:105) ε IJ = 0 , (cid:104) k [ I η J ] (cid:105) = (cid:104) k I η I (cid:105) ε IJ = 0 , (3.14)where it was used that any two-dimensional antisymmetric tensor is proportional to thetotally antisymmetric symbol. According to the first condition the shift vector factorizesin terms of a light-like Weyl spinor η α and a little-group spinor n I , η Iα = n I η α . (3.15)The second condition then becomes ε IJ n I (cid:104) k J η (cid:105) = 0 , (3.16)which determines the little-group spinor up to a constant c , n I = c (cid:104) ηk I (cid:105) . (3.17)The shift of the anti-holomorphic variable is treated analogously. The general shift of themassive spinor variables is therefore of the formˆ k Iα ( z ) = k Iα + zcη α (cid:104) ηk I (cid:105) , ˆ k ˙ α,I ( z ) = k ˙ α,I + zdη ˙ α [ k I η ] . (3.18)It can be checked that the normalization (2.6) is automatically satisfied by the shiftedspinors as well, ˆ k Iα ( z )ˆ k β,I ( z ) = mε αβ , ˆ k I ˙ α ( z )ˆ k ˙ β,I ( z ) = mε ˙ α ˙ β . (3.19)This follows from identities such as (cid:104) ηk I (cid:105) k β,I = mη β that result from (2.6). Note thatwithin the spin-axis formalism the shift (3.18) corresponds to shifting both the momentumspinors k (cid:91) and the reference spinors q .The requirement (3.2) of a linear shift of the momentum allows only a shift of theholomorphic or anti-holomorphic spinors alone, so there are two possible shifts, ˆ k H α ˙ α ( z ) = k α ˙ α + zc η α (cid:104) ηk I (cid:105) k ˙ α,I = k α ˙ α + zc η α ( (cid:104) η | /k ) ˙ α , (3.20)ˆ k A α ˙ α ( z ) = k α ˙ α + zd k Iα [ k I η ] η ˙ α = k α ˙ α + zd ( /k | η ]) α η ˙ α . (3.21) The only other way to avoid a quadratic term in z is to fix the shift spinors in terms of the momentumspinors such that (cid:104) ηk I (cid:105) [ k I η ] = 0, e.g. | η (cid:105) ∝ | k (cid:105) and | η ] ∝ | k ]. The resulting momentum shift ˆ k α ˙ α ( z ) = k α ˙ α + z ˜ c k α k ˙ α, can be viewed as a special case of both the generic holomorphic and anti-holomorphicshifts and need not be considered separately. c → c/ (cid:104) ηk (cid:105) and d → d/ [ kη ] so one obtains the familiar resultˆ k H α ( z ) = k α + zc η α , ˆ k A ˙ α ( z ) = k ˙ α + zd η ˙ α . (3.22)For shifts of multiple external momenta discussed in Section 3.3, the constants c and d need to be chosen such that the condition of momentum conservation (3.4) is satisfied. Expressions for the Dirac spinors (2.19) and polarization vectors (2.30) for a shifted mo-mentum can be easily defined by replacing the spinors k Iα and k ˙ α,I with the correspondingshifted quantities (3.18). For the example of the holomorphic shift one obtains the Diracwave function ˆ u I, H ( k, z ) = (cid:18) ˆ k Iα ( z ) − k ˙ α,I (cid:19) = u I ( k ) + zc (cid:104) ηk I (cid:105) (cid:18) η α (cid:19) . (3.23)The shifts of the conjugate and anti-particle spinors are defined in complete analogy. Byconstruction, these spinors satisfy the appropriate equations of motion (2.18), completenessconditions (2.20) and normalization conditions (2.21) for the shifted momentum since thenormalization conventions (2.5) and (2.6) are not affected by the shift. Similarly, thepolarization vectorsˆ (cid:15) ( IJ ) , H α ˙ α ( k, z ) = 1 √ m ˆ k ( Iα ( z ) k J )˙ α = (cid:15) ( IJ ) α ˙ α ( k ) + zc √ m η α (cid:104) ηk ( I (cid:105) k J )˙ α (3.24)satisfy the transversality condition (2.31), the normalization (2.32) and the complete-ness relation (2.33) for the shifted momentum. The corresponding results for the anti-holomorphic shift are ˆ u I, A ( k, z ) = u I ( k ) − zd [ k I η ] (cid:18) η ˙ α (cid:19) , (3.25)and ˆ (cid:15) ( IJ ) , A α ˙ α ( k, z ) = 1 √ m k ( Iα ˆ k J )˙ α ( z ) = (cid:15) ( IJ ) α ˙ α ( k ) + zd √ m k ( Iα [ k J ) η ] η ˙ α . (3.26)These results provide little-group covariant expressions for shifted Dirac spinors andpolarization vectors. However, in general the spinors and polarization vectors for all spinorientations receive a linear shift, which is not desirable in the discussion of the scalingof the amplitude A ( z ) for z → ∞ . Nevertheless, the little-group covariant form of theshift (3.18) was recently used for a BCFW shift of one massive and one massless leg [30].In this paper we focus on purely massive shifts and leave a systematic analysis of thelarge- z behaviour of this type of shift for future work.11 .2.1 Choice of spin axis Using a suitable choice of spin axis aligned with the shift spinors η α , η ˙ α it is possible tosimplify the shifted wave functions to a form similar to the massless case [4] so that theystay z -independent for some spin quantum numbers. To this end, it is useful to choosethe holomorphic reference spinor in the light-cone decomposition (2.8) of a holomorphicallyshifted momentum ˆ k H i ( z ) as q i,α = η i,α , (3.27a)while for an anti-holomorphically shifted momenta ˆ k A j ( z ) the choice q j, ˙ α = η j, ˙ α (3.27b)is made. In this way only the light-cone projected momentum spinors are shifted,ˆ k (cid:91) H i,α ( z ) = k (cid:91)i,α + zc i η i,α , ˆ k (cid:91) H i, ˙ α ( z ) = k (cid:91)i, ˙ α , (3.28)ˆ k (cid:91) A i,α ( z ) = k (cid:91)i,α , ˆ k (cid:91) A j, ˙ α ( z ) = k (cid:91)j, ˙ α + zd j η j, ˙ α , (3.29)where the rescaling c i → c i / (cid:104) η i k (cid:91)i (cid:105) and d j → d j / [ k (cid:91)j η j ] was performed. The remainingreference spinors q i, ˙ α and q j,α are still arbitrary at this stage. With this choice, only theDirac spinor with negative spin is affected by the holomorphic shift, as in the masslesscase, ˆ u H ( k i , ) = (cid:32) m (cid:104) k (cid:91)i η i (cid:105) η i,α k (cid:91)i, ˙ α (cid:33) , ˆ u H ( k i , − ) = (cid:32) k (cid:91)i,α + zc i η i,α − m [ q i k (cid:91)i ] q i, ˙ α (cid:33) . (3.30)In the same way, only the polarization vector with negative spin and the longitudinalpolarization are shifted,ˆ (cid:15) H α ˙ α ( k i , +) = √ η i,α k (cid:91)i, ˙ α (cid:104) η i k (cid:91)i (cid:105) , ˆ (cid:15) H α ˙ α ( k i , − ) = √ k (cid:91)i,α + zc i η i,α ) q i, ˙ α [ k (cid:91)i q i ] , ˆ (cid:15) H α ˙ α ( k i ,
0) = 1 m (cid:18) ( k (cid:91)i,α + zc i η i,α ) k (cid:91)i, ˙ α − m (cid:104) η i k (cid:91)i (cid:105) [ k (cid:91)i q i ] η i,α q i, ˙ α (cid:19) . (3.31)For the anti-holomorphic shift, only the Dirac spinors with positive spin are affected,ˆ u A ( k j , ) = (cid:32) m (cid:104) k (cid:91)j q j (cid:105) q j,α k (cid:91)j, ˙ α + zd j η j, ˙ α (cid:33) , ˆ u A ( k j , − ) = (cid:32) k (cid:91)j,α − m [ η j k (cid:91)j ] η j, ˙ α (cid:33) , (3.32) Alternatively, one could consider eliminating holomorphic or anti-holomorphic spinor variables usingthe on-shell condition (2.2) and the Dirac equation (2.7) as advocated in [11] (see also a related discussionat the Lagrangian level [31]). This corresponds to the introduction of higher-dimensional operators andmoves the effect of the shift from the wave functions to the truncated amplitude. This does not appear tosimplify the study of the z → ∞ behaviour but may deserve further study. (cid:15) A α ˙ α ( k j , +) = √ q j,α ( k (cid:91)j, ˙ α + zd j η j, ˙ α ) (cid:104) q j k (cid:91)j (cid:105) , ˆ (cid:15) A α ˙ α ( k j , − ) = √ k (cid:91)j,α η j, ˙ α [ k (cid:91)j η j ] , ˆ (cid:15) A α ˙ α ( k j ,
0) = 1 m (cid:32) k (cid:91)j,α ( k (cid:91)j, ˙ α + zd j η j, ˙ α ) − m (cid:104) q j k (cid:91)j (cid:105) [ k (cid:91)j η j ] q jα η j, ˙ α (cid:33) . (3.33) To construct complex deformations of scattering amplitudes, the set S of shifted momentais split into two subsets, S = H ∪ A , where a holomorphic shift (3.20) is performed for asubset of momenta k i ∈ H while momenta k j ∈ A are deformed by an anti-holomorphicshift (3.21). These candidate shifts must then be constrained so that the conditions (3.4)and (3.5) are satisfied. Momentum conservation (3.4) implies the condition0 = (cid:88) H δk H i,α ˙ α + (cid:88) A δk A j,α ˙ α = (cid:88) H c i η i,α ( (cid:104) η i | /k i ) ˙ α + (cid:88) A d j ( /k j | η j ]) α η j, ˙ α = (cid:88) H c i η i,α k (cid:91)i, ˙ α + (cid:88) A d j k (cid:91)j,α η j, ˙ α , (3.34)where the expression in the second line holds for the choice of reference spinors (3.27) andre-scaled coefficients. For generic shift spinors η i , this identity provides four constraintsfor the h coefficients c i and d j so that a solution always exists for h ≥
4. For the cases h = 2 , F is light-like, i.e. the quantities Q F ,α ˙ α = (cid:88) H∩F c i η i,α ( (cid:104) η i | /k i ) ˙ α + (cid:88) A∩F d j ( /k j | η j ]) α η j, ˙ α = (cid:88) H∩F c i η i,α k (cid:91)i, ˙ α + (cid:88) A∩F d j k (cid:91)j,α η j, ˙ α (3.35)must factorize into a product of two-component spinors, Q F ,α ˙ α = Q F ,α Q F , ˙ α , for all choicesof F . Since the number of factorization channels in general exceeds the number h − η i . Analo-gously to the massless case considered in [5] this leaves two possibilities: Generalizationsof the construction of Risager [25] where only holomorphic or anti-holomorphic shifts areperformed or generalizations of the BCFW construction by performing a holomorphic shiftof h − l > h − l > Q F (cid:54) = 0 and are discussed in Appendix C.In the application of the recursion relation, also the spin states of the internal particleΦ s F in the factorized amplitudes (3.8) need to be defined. It is useful to choose the referencespinors in terms of the factorized spinors of the internal shift (3.35), q F ,α = Q F ,α , q F , ˙ α = Q F , ˙ α , (3.36)which implies the light-cone decomposition of the internal momentum K F ,α ˙ α = K (cid:91) F ,α K (cid:91) F , ˙ α + K F K F · Q F ) Q F ,α Q F , ˙ α . (3.37)This choice results in a simple expression for the shifted internal momentum,ˆ K F ,α ˙ α ( z ) = K (cid:91) F ,α K (cid:91) F , ˙ α + (cid:18) M F K F · Q F ) + ( z − z F ) (cid:19) Q F ,α Q F , ˙ α , (3.38)which has been expressed in such a way that the on-shell condition,ˆ K F ( z F ) = M F , (3.39)at the pole position (3.9) is manifestly satisfied. Note the light-cone projected momentumis not affected by the shift, i.e. ˆ K (cid:91) F = K (cid:91) F . The wave-functions of the internal particles canthen be defined in terms of the reference spinors (3.36) and the momentum spinors K (cid:91) F . For a two-line BCFW-type shift of massive lines, a solution to the on-shell condition (3.3)and momentum conservation (3.34) was constructed in [23] using the fact that two massivemomenta k i/j can be expressed in terms of two light-like vectors l i/j as k i = l i + α j l j , k j = α i l i + l j , (3.40)with the coefficients α j = 2 k i · k j − sgn(2 k i · k j ) √ ∆2 k j , α i = 2 k i · k j − sgn(2 k i · k j ) √ ∆2 k i , (3.41)and ∆ = (2 k i · k j ) − k i k j . (3.42)A two-line shift with the shift spinors η α = l j,α , η ˙ α = l i, ˙ α , (3.43)leads to the momentum shiftsˆ k H i,α ˙ α ( z ) = k i,α ˙ α + z l j,α l i, ˙ α , ˆ k A j,α ˙ α ( z ) = k j,α ˙ α − z l j,α l i, ˙ α , (3.44)14hich manifestly have the desired properties (3.4) and (3.5). This result was recently alsoobtained from the little-group point of view in [32]. Making the choice (3.43) in the resultsobtained in Section 3.2 provides shifted Dirac spinors (3.23) and polarization vectors (3.24)for arbitrary little-group frames. However, eliminating the z -dependence in some of theexternal wave-functions by fixing the spin axes according to (3.27), q i,α = η α = l j,α , q j, ˙ α = η ˙ α = l i, ˙ α , (3.45)reproduces the definition of shifted Dirac spinors in [23]. Also the prescription (3.36) forthe reference spinors of the internal line reproduces the choice of [23], q F ,α = η α = l j,α , q F , ˙ α = η ˙ α = l i, ˙ α . (3.46) In the Risager-type solution to the light-cone condition (3.35) for all factorization channels,all shifts are either exclusively holomorphic or anti-holomorphic and shift spinors η i arechosen identical. In the holomorphic case the shift of the spinors is therefore given byˆ k I, H i,α ( z ) = k Ii,α + zc i η α (cid:104) ηk Ii (cid:105) , ˆ k H i, ˙ α,I ( z ) = k i, ˙ α,I , (3.47)so that the shift of the momentum of internal propagators factorizes according to Q F ,α ˙ α = η α (cid:88) S∩F c i ( (cid:104) η | /k i ) ˙ α ≡ η α Q F , ˙ α . (3.48)Choosing the spin axis according to (3.27) implies that the same reference spinor q i,α = η α is used for all shifted particles. This simplifies the shifted momentum spinors toˆ k (cid:91) H i,α ( z ) = k (cid:91)i,α + zc i η α , ˆ k (cid:91) H i, ˙ α ( z ) = k (cid:91)i, ˙ α , (3.49)and shifted Dirac spinors and polarization vectors are given by (3.30) and (3.31). Thelight-cone projection of the internal momentum is defined using the reference spinors q F ,α = η α , q F , ˙ α = Q F , ˙ α = (cid:88) S∩F c i k (cid:91)i, ˙ α . (3.50)The condition of momentum conservation (3.34) becomes0 = (cid:88) S c i k (cid:91)i, ˙ α . (3.51)For three shifted momenta, i ∈ { i , i , i } the Schouten identity implies the solution c i = [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] , (3.52)in complete analogy to the massless case [25]. In general, a system of h equations for thecoefficients can be obtained by contracting with all of the k (cid:91)i, ˙ α spinors of the shifted legs.15owever, the system is under-determined due to the Schouten identity. In our constructionof scattering amplitudes we will require four-line and five-line shifts, for which solutionscan be written as c i = [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] + [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] + [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] , (3.53)and c i = [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] + [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] , c i = [ k (cid:91)i k (cid:91)i ] . (3.54)The anti-holomorphic Risager-type shift is given analogously by (3.29) with the choice ofreference spinor q j, ˙ α = η ˙ α for all shifted lines, and with corresponding solutions for thecoefficients d j . Generalizations of the BCFW construction are obtained by performing an anti-holomorphicshift for one leg k j ∈ A and holomorphic shifts of h − k i ∈ H , with all shift spinorschosen identical,ˆ k I, H i,α ( z ) = k Ii,α + zc i η α (cid:104) ηk Ii (cid:105) , ˆ k H i, ˙ α,I ( z ) = k i, ˙ α,I , ˆ k I, A j,α ( z ) = k Ij,α , ˆ k A j, ˙ α,I ( z ) = k j, ˙ α,I + zd j η ˙ α [ k j,I η ] . (3.55)The condition of light-like shifts Q F of the internal momenta (3.35) requires the choice η α ∝ ( /k j | η ]) α , (3.56)so that the shift factorizes for all factorization channels, Q F ,α ˙ α = ( /k j | η ]) α Q F , ˙ α . (3.57)The choice of spin axis (3.27) implies that all legs in H share the same holomorphic referencespinor. After re-scaling the coefficients c i and d j , the shift and reference spinors become η α = q i,α = k (cid:91)j,α , η ˙ α = q j, ˙ α . (3.58)The anti-holomorphic reference spinors q i, ˙ α of the legs in H and all reference spinors ofparticle j are kept arbitrary. The generalized BCFW shift therefore takes the simple formˆ k (cid:91) H i,α ( z ) = k (cid:91)i,α + zc i k (cid:91)j,α , ˆ k (cid:91) H i, ˙ α ( z ) = k (cid:91)i, ˙ α , ˆ k (cid:91) A j,α ( z ) = k (cid:91)j,α , ˆ k (cid:91) A j, ˙ α ( z ) = k (cid:91)j, ˙ α + zd j q j, ˙ α . (3.59)The shifted Dirac spinors and polarization vectors are given by (3.30) and (3.31) with η α = k (cid:91)j,α for the holomorphic lines and by (3.32) and (3.33) with η ˙ α = q j, ˙ α for the anti-holomorphic line. According to (3.36) the light-cone projection of the internal momentumis defined using the reference spinors q F ,α = k (cid:91)j,α , q F , ˙ α = Q F , ˙ α = (cid:88) H∩F c i k (cid:91)i, ˙ α + (cid:88) A∩F d j q j, ˙ α . (3.60)16he condition of momentum conservation (3.34) reads0 = (cid:88) H c i k (cid:91)i, ˙ α + d j q j, ˙ α . (3.61)In the case of three shifted momenta, i ∈ { i , i } , a solution is found with help of theSchouten identity c i = [ k (cid:91)i q j ] , c i = [ q j k (cid:91)i ] , d j = [ k (cid:91)i k (cid:91)i ] . (3.62)The solutions for four-line (five-line) shifts can be obtained from the expressions (3.53)and (3.54) for the Risager shift by the replacement c i → d j and k i → q j ( c i → d j and k i → q j ).The BCFW-type shift with one holomorphic line k i and h − k j can similarly be brought to the formˆ k (cid:91) H i,α ( z ) = k (cid:91)i,α + zc i q i,α , ˆ k (cid:91) H i, ˙ α ( z ) = k (cid:91)i, ˙ α , ˆ k (cid:91) A j,α ( z ) = k (cid:91)j,α , ˆ k (cid:91) A j, ˙ α ( z ) = k (cid:91)j, ˙ α + zd j k (cid:91)i, ˙ α , (3.63)with the shift and reference spinors η α = q i,α , η ˙ α = q j, ˙ α = k (cid:91)i, ˙ α . (3.64)The light-cone decomposition of internal momenta is performed using the reference spinors q F ,α = Q F ,α = (cid:88) H∩F c i q i,α + (cid:88) A∩F d j k j,α , q F , ˙ α = k (cid:91)i, ˙ α . (3.65) z behaviour of amplitudes In this section we obtain bounds on the large- z behaviour of the complex continuation of n -point scattering amplitudes, lim z →∞ A n ( z ) ∼ z γ , (4.1)under the shifts constructed in Section 3. Since z enters only through momenta and externalwave-functions, which are all deformed linearly, the exponent γ must be an integer, so thecriterion γ < n -particle scattering amplitude with h shifted external particles can be written in termsof “skeleton amplitudes” ˜ A h,b ( z ) describing the scattering of the shifted particles, dressedby insertions of “background” subdiagrams B i with the unshifted external legs, A n ( z ) = n − h (cid:88) b =1 (cid:88) diagrams ˜ A h,b ( z ) b (cid:89) i =1 B i , (4.2)see Fig. 1 for illustration. More precisely, the background subamplitudes are defined interms of off-shell currents B i , which are given by the off-shell amplitude with b i external on-shell legs and one off-shell leg with attached propagator. The numerator of the propagator17 A h,b ˆ k ˆ k h B B b = X diagrams ˆ k ˆ k h B B b Figure 1: Illustration of the skeleton and background amplitudes appearing in (4.2). Theshifted momentum flows through solid lines while dashed lines denote background lines.of the off-shell leg can be written using a completeness relation in terms of suitably off-shell continued polarization spinors or vectors and possible additional off-shell terms (seee.g. [33]), so that the background currents take the schematic form B i = (cid:88) s χ i ( s ) 1 P i − M χ † i ( − s ) ˜ B i ≡ (cid:88) s χ i ( s ) B i ( − s ) , (4.3)where the sum over s extends beyond the physical polarizations in the off-shell case. In (4.2)the spin sums are implicit and the polarization factors χ i are included in the definition ofthe skeleton amplitudes ˜ A h,b , so that these contain no open spinor or vector indices. Themass dimension of the background subamplitudes is given by[ B i ] = 4 − ( b i + 1) − − b i , (4.4)where the term − h shifted lines and b background insertions can be brokendown into building blocks according to˜ A h,b ( z ) = N h,b ( z ) D h,b ( z ) (cid:89) a g a (cid:89) S ψ ˆ u ( z ) (cid:89) S W ˆ (cid:15) ( z ) b (cid:89) i =1 χ i , (4.5)where g a are coupling constants, D h,b is the product of propagator denominators and N h,b the corresponding numerator function arising from vertex factors and propagator numer-ators. The set of shifted fermion and vector boson lines is denoted by S ψ and S W , respec-tively. The skeleton amplitude is connected, i.e. all propagators in ˜ A h,b are z -dependent,since the shift is assumed to satisfy the property that the condition of momentum conser-vation (3.4) does not hold for a subset of the shifted legs.Dimensional analysis relates the dimension of the scattering amplitude in four space-time dimensions to the mass dimensions of the objects in the ansatz (4.5)[ ˜ A h,b ] = 4 − ( h + b ) = [ g ] + [ N h,b ] − [ D h,b ] + 12 h ψ + 12 b ψ . (4.6)18he term [ g ] denotes the dimension of the product of coupling constants, h Φ denotes thenumber of shifted external legs of particle type Φ and b Φ the corresponding backgroundattachments. Similarly, the number of holomorphically or anti-holomorphically shiftedparticles will be denoted by h H Φ and h A Φ .The scaling exponent of the skeleton amplitudes, lim z →∞ ˜ A h,b ( z ) ∼ z γ h,b , can be decom-posed as γ h,b = γ N − γ D , (4.7)where γ D arises from the propagator denominators and γ N arises from the flow of z throughmomentum-dependent numerators of Feynman diagrams and from external wave functions.Since the background subamplitudes are z -independent, the behaviour of the full amplitudefor z → ∞ is determined by the worst scaling among the skeleton amplitudes, γ = max γ h,b . (4.8)The criterion γ h,b < With the choice of spin axis described in Section 3.2.1, the large-z behaviour of the fermionspinors for holomorphic and anti-holomorphic shifts is given byˆ u H ( k, s ) ∼ z − s + , ˆ u A ( k, s ) ∼ z s + , (4.9)while the vector-boson polarizations behave asˆ (cid:15) H ( k, − ) ∼ z , ˆ (cid:15) H ( k, ∼ z , ˆ (cid:15) H ( k, +) ∼ z , ˆ (cid:15) A ( k, − ) ∼ z , ˆ (cid:15) A ( k, ∼ z , ˆ (cid:15) A ( k, +) ∼ z . (4.10)The naive scaling of the polarization vectors for the “good shifts”, ˆ (cid:15) A ( − ) and ˆ (cid:15) H (+), isworse than in the massless case, where a gauge-dependent 1 /z pole of polarization vectorsimproves the scaling. However, using Ward identities it is possible to establish the 1 /z sup-pression of amplitudes for these polarizations also for a gauge choice with z -independentpolarization vectors [24]. Similarly, gauge cancellations improve the behaviour of ampli-tudes with longitudinal polarization vectors compared to the naive estimate [4].For the case of spontaneously broken gauge invariance, the relevant Ward identityrelates vector-boson to Goldstone-boson amplitudes [34, 35], k µ A µ ( W ( k ) , . . . ) = m W A ( φ ( k ) , . . . ) . (4.11)Using the observation that the shift of the longitudinal polarization vector is proportionalto the momentum shift,ˆ (cid:15) H α ˙ α ( k, − (cid:15) α ˙ α ( k,
0) = 1 m czη α k (cid:91) ˙ α = 1 m (cid:16) ˆ k H α ˙ α ( z ) − k α ˙ α (cid:17) , (4.12)19nd similarly for the anti-holomorphic shift, the identity (4.11) can be used to obtain therelation for amplitudes with shifted longitudinal vector bosons,ˆ (cid:15) µ ( k, A µ ( W (ˆ k ( z )) , . . . ) = A ( φ (ˆ k ( z )) , . . . ) + r k,µ A µ ( W (ˆ k ( z )) , . . . ) , (4.13)with r k,α ˙ α = (cid:18) (cid:15) α ˙ α ( k, − k α ˙ α m (cid:19) = − m q · k q α q ˙ α . (4.14)The amplitudes on the right-hand side depend on z , while the vector r k does not. Inthe usual application of the Ward identity (4.11) in the context of the Goldstone bosonequivalence theorem, the high-energy limit is taken where r k ∼ m/E so that the terminvolving the amputated amplitude A µ ( W, . . . ) is subdominant compared to the Goldstoneboson amplitude. Here we consider the z → ∞ limit where the exact identity (4.13)cannot be simplified further. This is, however, sufficient to see that the scaling of thelongitudinal polarization vectors (4.10) overestimates the z → ∞ behaviour. Note thatthe Goldstone-boson and the vector-boson amplitude on the right-hand side of (4.13) mayhave a different behaviour for z → ∞ and the contraction of the remainder r k with thevector-boson amplitude must be taken into account in the estimate of the large- z behaviour.The Ward identity (4.11) can also be used to bound the large- z behaviour of amplitudeswith vector bosons with positive spin projection in the holomorphic shift [24, 36] by notingthat the holomorphic shift of the momentum can be written in terms of the positive-helicitypolarization vector, ˆ k H α ˙ α ( z ) − k α ˙ α = z c [ k (cid:91) q ] √ (cid:15) H α ˙ α ( k, +) . (4.15)The application of the Ward identity impliesˆ (cid:15) H µ ( k, +) A µ ( W (ˆ k ( z )) , . . . ) = 1 z √ c [ k (cid:91) q ] (cid:16) mA ( φ (ˆ k ( z )) , . . . ) − k µ A µ ( W (ˆ k ( z )) , . . . ) (cid:17) . (4.16)An analogous identity holds for ˆ (cid:15) A ( k, − ) in the anti-holomorphic shift.Altogether these results show that gauge cancellations, as encoded in the Ward iden-tity (4.11), improve the scaling behaviour of massive vector bosons compared to the naiveestimates (4.10) so that the effective scaling is determined by the spin projection s = 0 , ± (cid:15) H ( k, s ) ∼ z − s , ˆ (cid:15) A ( k, s ) ∼ z s . (4.17)Therefore the effective behaviour of amplitudes for the “good shifts” of vector bosons isidentical to the massless case while the longitudinal vector bosons behave like scalars, asintuitively anticipated from Goldstone boson equivalence. The use of the Ward identity is akey place of our analysis where the consequences of spontaneously broken gauge invarianceare employed. 20 .2 Propagator scaling For internal lines with light-like shifts, every propagator denominator in the skeleton am-plitude is linear in z so that γ D = 12 [ D h,b ] = d, (4.18)where d is the number of propagators in the skeleton amplitude. This can be estimatedusing the topological identities of tree diagrams (cid:88) n v n = d + 1 , (4.19) (cid:88) n nv n = 2 d + e, (4.20)where v n is the number of vertices with valency n and e = h + b is the number of externallegs of the skeleton amplitude. The number of propagators is bounded from above andbelow in terms of the smallest and highest valencies n min and n max , h + b − n max n max − ≤ γ D ≤ h + b − n min n min − . (4.21)In the following we limit ourselves to a renormalizable SBGT where n min = 3 and n max = 4so that h + b − ≤ γ D ≤ h + b − . (4.22)This condition can easily be relaxed for the interesting application of on-shell methods ineffective-field-theories of SBGTs with higher-dimensional operators [17, 30, 37–39]. In thederivation of the large- z behaviour in this section, only the upper bound in (4.22) enters,so these results are also valid in the presence of higher-dimensional operators. Q F = 0In addition to the bound on the denominator (4.22), the estimate (4.7) of the large z -scaling of the skeleton amplitude requires a bound on γ N , i.e. on the z -dependence of thenumerator function N h,b in the ansatz (4.5), contracted with the polarization functions ofshifted legs and background insertions. Independent of the structure of the shift, the mostconservative estimate is obtained from the mass dimension of the numerator function [5],which can be expressed in terms of (4.6). Adding the scaling of the external fermions (4.9)and the improved estimate for vector bosons (4.17) gives the bound γ N ≤ [ N h,b ] + (cid:88) H ψ (cid:18) − s i + 12 (cid:19) + (cid:88) A ψ (cid:18) s j + 12 (cid:19) − (cid:88) H W s i + (cid:88) A W s j = 4 − ( h + b ) − [ g ] − b ψ D h,b ] − (cid:88) H s i + (cid:88) A s j . (4.23)21or the concrete examples of the extended Risager and BCFW shifts, better estimates canbe obtained, as discussed below. Using the estimate of the scaling of the propagator (4.22),a conservative bound for the scaling of the amplitude is obtained, γ ≤ − ( h + b ) − min[ g ] − b ψ γ D − (cid:88) H s i + (cid:88) A s j ≤ − min[ g ] − (cid:88) H s i + (cid:88) A s j , (4.24)where b ψ ≥ n min = 3 in the result forthe massless case in Eq. (26) in [5]. However, the first line of (4.24) shows that thisbound can be improved in amplitudes with fermionic background insertions. Note thatthe Feynman diagrams contributing to the amplitude may have different mass dimension[ g ] of the product of coupling constants, so the smallest among these values must be takenfor the bound in (4.24). The generic bound (4.24) can be improved by an analysis of the structure of the contractionof the numerator function in the skeleton amplitude (4.5) with the external and backgroundwavefunctions. Compared to the corresponding discussion of the massless case [5], compli-cations for massive particles arise due to the presence of reference spinors and the need toapply Ward identities to bound the scaling of amplitudes with vector bosons. The followinganalysis does not cover two-line BCFW shifts, where the estimate (4.24) can be used.Using the definition of the polarization wave functions, the numerator can be writ-ten as a polynomial in holomorphic and anti-holomorphic spinor products of momentumspinors and reference spinors. The z -dependence in the terms contributing to the skeletonamplitude is of the schematic form˜ A h,b ( z ) ∼ R h,b (cid:104) ˆ k (cid:91) H i ( z ) · (cid:105) α [ˆ k (cid:91) A j ( z ) · ] β D h,b ( z ) . (4.25)The different contributions to the shifted holomorphic and anti-holomorphic spinor prod-ucts (cid:104) ˆ k H i ( z ) · (cid:105) and [ˆ k A j ( z ) · ] will be analyzed for the extended Risager and BCFW-typeshifts below. The function R h,b includes unshifted spinor products, particle masses andcoupling constants. The analysis will be performed in the ’t Hooft-Feynman gauge wherethe numerators of vector-boson propagators are momentum independent. Note that theexponents α and β are positive since spinor products in the denominator can only arisefrom the definitions of the polarization wave functions, which are not shifted and thereforeincluded in the remainder function R h,b .For all the considered shifts the spinor products involving shifted spinors in the skeletonamplitude (4.25) are linear in z so that the large- z scaling of the numerator function isgiven by γ N = α + β . However, as discussed in Section 4.1, Ward identities improve thelarge- z behaviour for the “good shifts” of vector bosons and for longitudinal bosons in22BGTs compared to naive scaling estimates. Therefore, the estimate of α and β shouldbe based on the right-hand sides of (4.13) and (4.16). The explicit factor of 1 /z in (4.16)can be taken into account by defining the effective scaling exponent, γ N = α + β − h H W (+) − h A W ( − ) , (4.26)with the numbers h H W (+) ( h A W ( − ) ) of holomorphically (anti-holomorphically) shifted vectorbosons with positive (negative) spin projection. In an extended holomorphic Risager-type shift as constructed in Section 3.3.2, the momentaof all particles in the set S are shifted as (3.49) and the spin axis of all shifted particles istaken as q i,α = η α . Therefore the only spinor products that are affected by the shift are (cid:104) ˆ k (cid:91) H i ( z )ˆ k (cid:91) H k ( z ) (cid:105) = (cid:104) k (cid:91)i k (cid:91)k (cid:105) + z ( c i (cid:104) ηk (cid:91)k (cid:105) + c k (cid:104) k (cid:91)i η (cid:105) ) , (cid:104) ˆ k (cid:91) H i ( z ) χ (cid:105) = (cid:104) k (cid:91)i χ (cid:105) + zc i (cid:104) ηχ (cid:105) , (4.27)where χ denotes a generic unshifted spinor, e.g. from background insertions. Due tothe choice of spin axes of the shifted particles, products involving the reference spinors, (cid:104) ˆ k (cid:91) H i ( z ) q k (cid:105) = (cid:104) k (cid:91)i η (cid:105) are not shifted and accordingly contribute to the remainder function R h,b in (4.25). The same holds for all anti-holomorphic spinor products, so β = 0. Thereforethe relevant contributions to the holomorphic spinor products are of the form (cid:104) ˆ k (cid:91) H i ( z ) · (cid:105) α = (cid:104) ˆ k (cid:91) H i ( z )ˆ k (cid:91) H k ( z ) (cid:105) α (cid:104) ˆ k (cid:91) H i ( z ) χ (cid:105) α . (4.28)The most conservative bound is obtained by assuming that the function R h,b does notcontain any holomorphic spinors. Since spinor products of shifted holomorphic spinors arelinear in z , the upper bound on γ N is obtained from half of the number of those holomorphicspinors that contribute to α and α . This receives the following contributions: • The number of holomorphic spinors in the numerator function N h,b , which can bebound by the mass dimension [ N h,b ] since every four-momentum in the numeratorgives rise to one holomorphic spinor. • The number of shifted external holomorphic spinors ˆ k (cid:91) H i,α ( z ), which arise only fromthe wave functions ˆ u H i ( − ) and ˆ (cid:15) H i ( − ) after applying the Ward identity. • The number of shifted vector bosons with positive spin projection. This contributionarises in addition to the explicit term in (4.26) since the term k k,µ A µ ( W, . . . ) in theWard-identity (4.16) can give rise to a spinor product (cid:104) ˆ k (cid:91) H i ( z ) k (cid:91)k (cid:105) in the numerator. • The number b ψ + b W of effective polarization functions χ of the background legs,which contain at most one holomorphic spinor in the numerator. No such contribution arises for the longitudinal gauge bosons, since the vector r k in the Ward iden-tity (4.13) is proportional to the reference spinors η α q k, ˙ α and does not contribute to α . γ N ≤ (cid:0) [ N h,b ] + b ψ + b W + h ψ ( − ) + h W ( − ) − h W (+) (cid:1) = 12 (cid:32) − h − min[ g ] − b φ − b ψ − (cid:88) S s i (cid:33) + γ D , (4.29)where (4.6) was used with b = b φ + b ψ + b W . We have also simplified h ψ ( − ) − h ψ = ( h ψ ( − ) − h ψ (+) ) = − (cid:80) ψ s i . Therefore the full large- z behaviour of the amplitude can bebound by γ H Risager = γ N − γ D ≤
12 (4 − h − min[ g ] − (cid:88) S s i ) , (4.30)since b ψ , b φ ≥
0. Note that for specific examples the estimate can be improved by takingthe concrete structure of the background into account. For the anti-holomorphic Risagershift, this bound holds with the replacement s i → − s j . These results agree with those forthe massless case [5] and for massive all-line shifts [4]. In the generalized BCFW shift (3.59), h − k H i are shifted holomorphically anda single momentum ˆ k A j anti-holomorphically with the shift spinors η α = k (cid:91)j,α and arbitrary η ˙ α = q j, ˙ α . The choice of the reference spinors for the shifted legs is given in (3.58). Toestimate the number of spinor products with anti-holomorphically shifted spinors in (4.25),note that momentum conservation can be used in the numerator function N h,b to eliminateˆ k A j in favour of the remaining external momenta of the skeleton amplitude. Therefore onlythe external wave-functions need to be considered. After application of the Ward identity,only ˆ (cid:15) A j (+) and ˆ u A j (+ ) contribute, β ≤ h A ψ (+) + h A W (+) . (4.31)The nontrivial holomorphic spinor products of the shifted spinors among themselvesand with other spinors are (cid:104) ˆ k (cid:91) H i ( z )ˆ k (cid:91) H k ( z ) (cid:105) = (cid:104) k (cid:91)i k (cid:91)k (cid:105) + z ( c i (cid:104) k (cid:91)j k (cid:91)k (cid:105) + c k (cid:104) k (cid:91)i k (cid:91)j (cid:105) ) , (4.32) (cid:104) ˆ k (cid:91) H i ( z ) χ (cid:105) = (cid:104) k (cid:91)i χ (cid:105) + zc i (cid:104) k (cid:91)j χ (cid:105) , (4.33) (cid:104) ˆ k (cid:91) H i ( z ) q A j (cid:105) = (cid:104) k (cid:91)i q A j (cid:105) + zc i (cid:104) k (cid:91)j q A j (cid:105) , (4.34)while the choice of shift and reference spinors implies that (cid:104) ˆ k (cid:91) H i ( z ) q H k (cid:105) and (cid:104) ˆ k (cid:91) H i ( z )ˆ k (cid:91) A j (cid:105) arenot shifted and in particular (cid:104) q H i ˆ k (cid:91) A j (cid:105) = 0. Therefore the relevant contributions to theholomorphic spinor products in (4.25) are of the form (cid:104) ˆ k (cid:91) H i ( z ) · (cid:105) α = (cid:104) ˆ k (cid:91) H i ( z )ˆ k (cid:91) H k ( z ) (cid:105) α (cid:104) ˆ k (cid:91) H i ( z ) χ (cid:105) α (cid:104) ˆ k (cid:91) H i ( z ) q A j (cid:105) α . (4.35) Here we exclude two-line BCFW shifts, where only one momentum is shifted holomorphically. α can be bound by half the number of theholomorphic spinors contributing to these spinor products. The single anti-holomorphicallyshifted particle can contribute spinors k A j,α and q A j,α , where the former drops out of (4.35)while the letter can contribute to α . The number of relevant spinors receives the followingcontributions: • The number of holomorphic spinors in N h,b , the holomorphically shifted ˆ u H i ( − ),ˆ (cid:15) H i ( − ) and ˆ (cid:15) H i (+), and the background contributions as in the Risager-type shift. • The anti-holomorphically shifted polarization vector ˆ (cid:15) A j (+), since it includes a spinor q A j,α and therefore contributes to α . • For ˆ (cid:15) A j ( − ), the term k µj A µ ( W, . . . ) in the Ward identity (4.16) contains a term in-volving the reference spinor q A j,α in the massive case. Similarly, the term involving r k j ,α ˙ α ∝ q A j,α η ˙ α in the Ward identity (4.13) contributes for ˆ (cid:15) A j (0). • In the massive case, the spinor ˆ u A j (+ ) contains the reference spinor q A j,α and con-tributes one-half to α . However, either only the holomorphic or the anti-holomorphicpart of a Dirac spinor contributes to a given term in the amplitude. Since the pos-sible contribution to α is smaller than the contribution of the anti-holomorphiccomponent of ˆ u A j (+ ) to β already included in (4.31), the former can be dropped.Taking all of the contributions to α into account and adding the explicit powers of z dueto the application of the Ward identities gives the bound α − h H W (+) − h A W ( − ) ≤ (cid:0) [ N h,b ] + b ψ + b W + h H ψ ( − ) + h H W ( − ) − h H W (+) + h A W (+) + h A W (0) − h A W ( − ) (cid:1) = 12 (cid:32) − h − min[ g ] − b φ − b ψ − (cid:88) H s i + (cid:88) A W s j − h A ψ h A W (0) (cid:33) + γ D , (4.36)where the expression for the dimension of the skeleton amplitude (4.6) was used and thespin sums were introduced as discussed above after (4.29). Adding the contribution fromthe anti-holomorphic spinors (4.31), the bound on the scaling of the amplitude can bewritten in a form similar to the bound for the Risager shift and an additional contributiondepending on the spin of the single anti-holomorphically shifted particle, γ H BCFW = γ N − γ D ≤ (cid:32) − h − min[ g ] − b φ − b ψ − (cid:88) S s i (cid:33) + s j W + , A , ψ + , A , W , A s j , W − , A , ψ − , A (4.37)For a dominantly anti-holomorphic BCFW-type shift one obtains analogously γ A BCFW ≤ (cid:32) − h − min[ g ] − b φ − b ψ (cid:88) S s i (cid:33) + − s j W − , A , ψ − , A , W , A − s j , W + , A , ψ + , A (4.38)25he corresponding result for holomorphic shift in the massless case [5] is γ H ,m =0 BCFW ≤ (cid:32) − h − min[ g ] − (cid:88) S s i (cid:33) + 2 s A . (4.39)The better behaviour for the “good shifts” in the massless case can be understood asfollows: • For ψ − , A one can argue in the massless case that the change from the holomorphic tothe anti-holomorphic shift improves the behaviour by one power of z . In the massivecase the spinor ˆ u H i ( − ) also contains a z -independent anti-holomorphic componentso not all contributions to the amplitude are improved. • For W − , A the Ward identity (4.16) gives rise to a contribution involving the referencespinor that does not appear in the massless case. We will illustrate the bounds (4.30) and (4.37) for simple examples in order to illuminatedifferences of the massive and massless cases more concretely. As a first example, considera mostly holomorphic four-line BCFW-type shift (3.59),ˆ k (cid:91) H i,α ( z ) = k (cid:91)i,α + zc i k (cid:91) ,α , ˆ k (cid:91) A , ˙ α ( z ) = k (cid:91) , ˙ α + zd q , ˙ α , (4.40)for the amplitude A ( ¯ ψ − , H , φ H , φ H , ψ − , A ) = ¯ ψ φ ψ φ + ¯ ψ φ ψ φ + ¯ ψ φ ψ φ (4.41)in Yukawa theory. In the massless case, according to the bound (4.39) the shift is allowedwith γ H ,m =0 BCFW ≤ − whereas in the massive case the estimate (4.37) gives γ H BCFW ≤
0. Formassless fermions, due to helicity selection rules, the only contribution to the amplitudearises from the diagram with a triple-scalar vertex, ¯ ψ φ ψ φ ∝ ˆ¯ u H ( k , − )ˆ v A ( k , − ) i(ˆ k H ( z ) + ˆ k H ( z )) − M φ ∼ z , (4.42)since the shift drops out of the product of the Dirac spinors (3.30) and (3.32),ˆ¯ u H ( k , − )ˆ v A ( k , − ) = (cid:104) ˆ k (cid:91) H ( z ) k (cid:91) (cid:105) = (cid:104) k (cid:91) k (cid:91) (cid:105) ∼ z . (4.43)26n the massive case, there is a non-vanishing contribution from the fermion exchangediagrams, which include a z -dependent contribution in the numerator, for example ¯ ψ φ ψ φ ∝ ˆ¯ u H ( k , − ) ˆ /k H ( z ) + ˆ /k A ( z ) + m ψ (ˆ k H ( z ) + ˆ k A ( z )) − m ψ ˆ v A ( k , − ) ∼ z →∞ (cid:104) k (cid:91) H ( z ) | ˆ /k H ( z ) | q ] m ψ (ˆ k H ( z ) + ˆ k A ( z )) − m ψ ∼ z ( c (cid:104) k (cid:91) k (cid:91) (cid:105) + c (cid:104) k (cid:91) k (cid:91) (cid:105) ) [ k (cid:91) q ] m ψ (ˆ k H ( z ) + ˆ k A ( z )) − m ψ ∼ z , (4.44)so that the massive bound (4.37) is saturated. This illustrates how mass-suppressed con-tributions can give rise to a worse large- z behaviour than in the massless case.In this example, this complication is not relevant in practice since the amplitude isconstructible using an anti-holomorphic Risager shift of all lines,ˆ k (cid:91) A i, ˙ α ( z ) = k (cid:91)i, ˙ α + zd i η ˙ α , (4.45)with the reference spinors q i, ˙ α = η ˙ α . In agreement with the anti-holomorphic versionof (4.30) one obtains for the diagram with a fermion propagator ¯ ψ φ ψ φ ∝ ˆ¯ u A ( k , − ) ˆ /k A ( z ) + ˆ /k A ( z ) + m ψ (ˆ k A ( z ) + ˆ k A ( z )) − m ψ ˆ v A ( k , − ) ∼ z , (4.46)since in this case the choice of reference spinors ensures that the shift drops out in thenumerator, ˆ /k A ( z ) | q ] = /k | η ], and similarly for spinor chains involving [ q | . This examplealso illustrates that the large- z behaviour of amplitudes with fixed spin quantum numbersgenerally depends on the choice of the spin axes, in contrast to the massless case wherethe reference spinors are unphysical auxiliary quantities.As an example for the large- z behaviour of amplitudes with massive vector bosons forthe BCFW-type shift (4.40), consider an amplitude with two vector bosons and two scalarsin the Abelian Higgs model in unitary gauge, A ( W − , H H H , H H , W − , A ) = W H W H + W H W H + W H W H + W H W H . In the ’t Hooft-Feynman gauge also diagrams with Goldstone-boson exchange contribute, while the
W W H vertex vanishes in the massless limit. This is, however, irrelevant to the estimate of the large- z behaviour below.
27n the massless case the shift is allowed according to the bound γ H ,m =0 BCFW ≤ −
1, whereas inthe massive case the weaker estimate γ H BCFW ≤ q i in the massless and massivecases. The diagram with the four-point vertex is proportional toˆ (cid:15) H ( k , − ) · ˆ (cid:15) A ( k , − ) = (cid:104) k (cid:91) k (cid:91) (cid:105) [ q q ][ k (cid:91) q ] [ k (cid:91) η ] , (4.47)which can be made to vanish in the massless case by choosing equal reference spinors,without affecting the result for the amplitude. In the massive case the choice of referencespinors determines the spin axis and therefore the physical result. For the choice of referencespinors made in Section 5, the scalar product does not vanish so the bound (4.37) issaturated. The same conclusion is reached using the Ward identity (4.16), A ( W − , H H H , H H , W − , A ) ∝ z (cid:0) m W A ( W − , H H H , H H , φ A ) − k ,µ A µ ( W − , H H H , H H , W A ) (cid:1) . (4.48)Focusing on the second term, the diagram with the four-point vertex is proportional toˆ (cid:15) H ( k , − ) · k = (cid:104) k (cid:91) k (cid:91) (cid:105) [ k (cid:91) q ][ k (cid:91) q ] + m W q · k ) (cid:104) ˆ k (cid:91) ( z ) q (cid:105) [ q q ][ k (cid:91) q ] ∝ z, (4.49)so again equal reference spinors are required to obtain a valid shift. It is obvious that theproblematic term is absent in the massless case. As for the example in Yukawa theory, inthis case a valid Risager shift is available. In this section we establish that the following shifts are sufficient to construct all amplitudesin SBGTs with at least five legs (or six legs for all-fermionic amplitudes): • Three-line and in some cases two-line shifts are sufficient for the following amplitudes: – All amplitudes with at least two transverse vector bosons. – Amplitudes with scalars and at least two vector bosons, at least one of which istransverse. – All amplitudes that contain only fermions; fermions and vector bosons; fermionsand SM-like Higgs bosons; or generic scalars and at least two fermion pairs. • Five-line shifts are required for amplitudes with only scalars. • Four-line shifts are sufficient for all other cases.Here “SM-like” Higgs bosons are defined as scalars that couple to vector bosons throughvertices of type
HW W but not through
HHW vertices. Furthermore, three-line shifts are28ufficient for amplitudes of scalars charged under an unbroken U (1) symmetry [5], whichis, however, not present for the SM Higgs boson in the broken phase. The above conditionsapply when all legs are massive. For at least two massless vector bosons or fermions, thefamiliar conditions for massless BCFW shifts [2,23,24,40] can be used. We do not considerthe case of shifts with one massless particle explicitly; see [23] for a discussion of shifts withmassive and massless quarks and gluons. Compared to the results for an unbroken gaugetheory with fermionic and scalar matter fields [5], the new feature are four-line shifts forlongitudinal gauge bosons, which arise by exploiting little-group transformations.The above statements are derived in the remainder of this section. In the study ofthe z → ∞ behaviour we make use of the ability to perform different types of shifts fordifferent spin states of the external particles. This requires a choice of spin axes for theshifted particles so that all required shifts are feasible. Such a choice is introduced inSection 5.1. The conditions for allowed shifts for this setup are summarized in Section 5.2while the minimal shifts required for the construction of different classes of amplitudes areinvestigated in Section 5.3. To define the spin axes for all particles in a way that allows to perform different typesof shifts depending on the spins of the particles, two “reference particles” will be singledout. For definiteness, these particles will be assigned the momenta k and k n . Since thechoice of a spin axis is not necessary for purely scalar amplitudes, in all relevant cases itis possible to choose at least one particle with spin as reference particle. As in (3.40) thereference momenta can be expressed in terms of two light-like vectors l /n according to k = l + α n l n , k n = α l + l n . (5.1)The reference spinors for the two selected legs are chosen as q ,α = l n,α , q , ˙ α = l n, ˙ α , (5.2) q n,α = l ,α , q n, ˙ α = l , ˙ α , (5.3)while those for all other legs are taken as q i,α = l n,α , q i, ˙ α = l , ˙ α . (5.4)A similar construction was used in [23].Provided the amplitudes for all spin states can be computed recursively with this choiceof spin axes, the amplitudes for arbitrary spin axes follow from little-group transformations.It is even sufficient to keep the spin quantum number of one particle fixed, for instance bytaking s n as negative. This information allows to reconstruct the amplitudes for arbitraryspin axes of the remaining legs using the little-group transformations (2.27) and (2.39), A (Φ s (cid:48) , . . . , Φ s (cid:48) n − n − , Φ − n ) = (cid:88) s · · · (cid:88) s n − R ( S ) s (cid:48) ,s . . . R ( S n − ) s (cid:48) n − ,s n − A (Φ s , . . . , Φ s n − n − , Φ − n ) . (5.5)Since now the spin axes for all particles are independent, the operator J + k n (2.16) can beused to raise the spin of the last particle. 29 .2 Explicit form of shifts The choice of reference spinors of Section 5.1 allows to perform all of the types of shiftsconstructed in Section 3. We summarize the explicit form and the conditions for the shiftedlegs in order to obtain an allowed shift.
Two-line BCFW-type shift
The choice of reference spinors (5.2) and (5.3) allows to shift the two reference particlesby a two-line BCFW shift (3.44),ˆ k (cid:91), H ,α ( z ) = l ,α + zl n,α , ˆ k (cid:91), H , ˙ α ( z ) = l , ˙ α , ˆ k (cid:91), A n,α ( z ) = l n,α , ˆ k (cid:91) A n, ˙ α ( z ) = l n, ˙ α − zl , ˙ α . (5.6)For the internal lines in the recursion relation (3.11), the reference spinors q F , ˙ α = η ˙ α = l , ˙ α and q F ,α = η α = l n,α are the same as for the external legs (5.4). A valid recursion relationis obtained from the bound (4.24) if γ ≤ − s + s n < . (5.7)This allows to shift the following combinations of particles, W + , H W − , A n , ψ + , H W − , A n , W + , H ψ − , A n , ψ + , H ψ − , A n . (5.8)Two fermions may only be shifted if they belong to different fermion lines [23] since inthis case the skeleton amplitude necessarily includes two fermionic background insertionsso that the condition (5.7) improves to γ ≤ − s + s n < . (5.9)For amplitudes with massless particles not all allowed shifts are found by this simple power-counting analysis; in particular shifts of particles with identical helicities are possible [2].We argue at the end of Section 5.3 that similar improvements are not expected for shiftsof massive particles. Holomorphic Risager-type shift An h -line shift, which may include the reference momentum k but not k n , is possible withthe shift spinor η α = l n,α and the shifted momentaˆ k (cid:91), H i,α ( z ) = k (cid:91)i,α + zc i l n,α , ˆ k (cid:91), H i, ˙ α ( z ) = k (cid:91)i, ˙ α . (5.10)Therefore h + 1-point functions can be constructed with an h -line Risager shift for theabove choice of spin axes. The reference spinors for the internal line in the recur-sion (3.11) are fixed according to (3.50), which implies that the holomorphic reference Since a choice of spin axes is not necessary for all-scalar amplitudes, the five-scalar amplitude can beconstructed from a five-line shift. q F ,α = η α = l n,α in the subamplitudes is the same as for the full amplitude. Ac-cording to the bound (4.30), a valid h -line recursion relation for dimensionless couplingconstants is obtained if the spin projections of the shifted legs satisfy (cid:88) S s i > − h. (5.11) Anti-holomorphic Risager-type shift
In an h -line shift with shift spinor η ˙ α = l , ˙ α all legs including the reference momentum k n ,but not k , can be shifted asˆ k (cid:91), A j,α ( z ) = k (cid:91)j,α , ˆ k (cid:91), A j, ˙ α ( z ) = k (cid:91)j, ˙ α + zd j l α . (5.12)The subamplitudes inherit the anti-holomorphic reference spinor, q F , ˙ α = η ˙ α = l , ˙ α . A validrecursion relation is obtained for (cid:88) S s j < − (4 − h ) . (5.13) Mostly holomorphic BCFW-type shift
An holomorphic shift is applied to h − k ) while the referencemomentum k n is shifted anti-holomorphically,ˆ k (cid:91), H i,α ( z ) = k (cid:91)i,α + zc i l n,α , ˆ k (cid:91), H i, ˙ α ( z ) = k (cid:91)i, ˙ α , ˆ k (cid:91), A n,α ( z ) = l n,α , ˆ k (cid:91) A n, ˙ α ( z ) = l n, ˙ α + zd n l , ˙ α . (5.14)As for the holomorphic Risager shift, the subamplitudes have the same common holomor-phic reference spinor q F ,α = η α = l n,α according to (3.60). If leg n has negative spinprojection, a valid recursion relation is obtained from (4.37) for (cid:88) S s i − s A n = (cid:88) H s i + | s A n | > − h. (5.15) Mostly anti-holomorphic BCFW-type shift
Here the reference momentum k is shifted holomorphically and h − k n ) are shifted anti-holomorphically,ˆ k (cid:91), H ,α ( z ) = l ,α + zc l n,α , ˆ k (cid:91), H , ˙ α ( z ) = l , ˙ α , (5.16)ˆ k (cid:91), A j,α ( z ) = k (cid:91)j,α , ˆ k (cid:91) A j, ˙ α ( z ) = k (cid:91)j, ˙ α + zd j l , ˙ α . (5.17)The subamplitudes have the same common anti-holomorphic reference spinor q F , ˙ α = η ˙ α = l , ˙ α . If leg one has positive spin projection, a valid recursion relation is obtained for (cid:88) S s j − s H = (cid:88) A s j − | s H | < − (4 − h ) . (5.18)31 .3 Minimal required shifts It is now possible to identify the minimal number of shifted legs necessary to construct agiven amplitude and verify the claims made in the beginning of this section. Recall thatlittle-group transformations allow to reconstruct amplitudes with general spin quantumnumbers from amplitudes where the reference particle n has spin − or −
1, provided theamplitudes for arbitrary spin configurations of the remaining particles are known for afixed choice of reference legs 1 and n . All amplitudes are constructible either from holomorphic or anti-holomorphic five-line Ris-ager shifts since the condition | (cid:80) S s i | ≥ The following four-line shifts are possible: • Holomorphic (anti-holomorphic) Risager shifts for (cid:80) H s i > (cid:80) A s j < • Mostly holomorphic BCFW-type shifts provided the holomorphically shifted particlessatisfy (cid:80) H s i > −| s n | for s n < • Mostly anti-holomorphic BCFW-type shifts provided the anti-holomorphically shiftedparticles satisfy (cid:80) A s j < s for s > (cid:80) S s i = 0. In thiscase, amplitudes with at least one fermion pair or vector boson can be constructed with aBCFW-type shift with a particle of negative spin projection as reference leg n . The onlyamplitudes that are not four-line constructible are therefore amplitudes with only scalars. The following three-line shifts are possible: • Holomorphic (anti-holomorphic) Risager shifts for (cid:80) H s i > (cid:80) A s j < − • Mostly holomorphic BCFW-type shifts provided the holomorphically shifted particlessatisfy (cid:80) H s i > − | s n | for s n < • Mostly anti-holomorphic BCFW-type shifts provided the anti-holomorphically shiftedparticles satisfy (cid:80) A s j < s − s > Amplitudes with only longitudinal vector bosons can be obtained from the allowed four-line BCFWshift W , H W , H W , H W − , A n using the little group raising operator on the reference leg with negative spin. ector-boson amplitudes Two vector bosons W s and W − n are selected as reference particles. A two-line BCFWshift is possible for the spin configuration W + , H W − , A n . For other spin projections of W ,anti-holomorphic Risager shifts W s i , A i W s j , A j W − , A n can be applied if the amplitude containstwo further vector bosons with s i + s j ≤ − W s i , H i W s j , H j W − , A n are possible for s i + s j ≥
1. These shifts cover all amplitudes with at leastfive vector bosons apart from those where all vector bosons besides the two reference legsare longitudinal, where four-line shifts can be applied as discussed above.
Amplitudes with scalars and vector bosons
A scalar φ and a vector boson W − n can be selected as reference particles. For ampli-tudes with at least one additional transverse vector boson W i , either a mostly-holomorphicBCFW shift φ H W + , H i W − , A n is possible or one can select a further particle Φ j ∈ { W − j , W j , φ j } so that an anti-holomorphic Risager shift W − , A i Φ A j W − , A n can be performed. This covers allamplitudes with scalars and at least two vector bosons, unless all vector bosons besides W − n are longitudinal where again a four-line shift is required.For amplitudes with scalars and only one vector boson, neither the condition (cid:80) A s j < − (cid:80) H s i > Amplitudes with scalars and fermions
A scalar φ and a fermion ψ − n can be chosen as reference particles. For amplitudes withat least two fermion pairs there must be two additional fermions with equal spin quantumnumber so that a mostly-holomorphic BCFW shift ψ + , H i ψ + , H j ψ − , A n or an anti-holomorphicRisager shift ψ − , A i ψ − , A j ψ − , A n are possible.For amplitudes with only one fermion pair, the conditions for a three-line shift cannotbe satisfied for generic scalar particles. However, the situation improves for SM-like Higgsbosons without HHW couplings. In this case there are no internal vector-boson lines andat least one scalar background insertion must appear in the skeleton amplitudes for theshifts φ H ψ + , H ψ − , A n and φ A ψ − , A ψ − , A n . This improves the scaling, for example in case of theBCFW-type shift (4.37), γ H BCFW ≤
12 (1 − b φ ) − | s A n | = − , (5.19)and similarly for the Risager shift (4.30). Therefore, all amplitudes with SM-like Higgsbosons and one fermion pair are three-line constructible. For the case of four-point ampli-tudes, this can be checked in the example given in Section 4.5.33 mplitudes with a fermion pair and vector bosons A fermion ψ s and a vector boson W − n can be taken as reference particles. A two-lineBCFW shift can be performed for the spin configuration ψ + , H W − , A n . Otherwise, a Risagershift Φ s i A i Φ s j , A j W − , A n or a BCFW-type shift Φ s i , H i Φ s j H j W − , A n can be performed if the amplitudecontains two additional particles with s i + s j < s i + s j >
0, respectively. Since theamplitude must contain a second fermion with s = ± , one of these conditions can alwaysbe fulfilled for amplitudes with arbitrary remaining particle content. Amplitudes with only fermions
We consider all-fermionic amplitudes with at least six legs and assume every fermion canbe uniquely assigned to one fermion line for all Feynman diagrams contributing to theamplitude. Then two fermions from different fermion lines can be chosen as referenceparticles ψ s and ψ − n . If reference particle one has positive spin, a two-line BCFW shift ψ + , H ψ − , A n is possible. For negative spin projection of leg one, it is always possible toperform a mostly-holomorphic BCFW shift ψ + , H i ψ + , H j ψ − , A n or an anti-holomorphic Risagershift ψ − , A i ψ − , A j ψ − , A n since two of the remaining (at least four) fermions must have the samespin quantum number. This conclusion agrees with [23], however, here we do not requirethe additional condition that the reference particle ψ and the three shifted legs all belongto different fermion lines.Therefore we have verified all cases of three-line constructible amplitudes given in thebeginning of Section 5. Further improvements?
As mentioned above, for massless particles also two-line BCFW shifts of legs with equal he-licity are allowed, although not obvious from simple power-counting. This may be provenfor instance using an auxiliary three-line recursion [18, 23] or a background field analy-sis [24]. While we have not performed a comprehensive analysis, such improvements donot appear to be possible for massive particles. For massive quarks, it is known thatBCFW shifts of two legs with the same spin are not allowed [23]. As a starting point of theinductive proof using auxiliary shifts, three-point amplitudes involving the two shifted legsmust vanish for z → ∞ [23]. For a two-line shift of two massive vector bosons W − , H W − , A n and the choice of reference spinors as in Section 5.1, this induction assumption is violatedsince three-point functions for an unshifted vector boson W − i and the two shifted vectorbosons do not vanish for z → ∞ , A ( ˆ W − , H , W − i , ˆ W − , A n ) ∝ (cid:0) ˆ (cid:15) H ( k , − ) · ˆ (cid:15) A ( k n , − ) (cid:1) (cid:0) (cid:15) ( k i , − ) · k A n ( z ) (cid:1) ∼ z , (5.20)since ˆ (cid:15) H ( k , − ) · ˆ (cid:15) A ( k n , − ) (cid:54) = 0 as in (4.47). This is another example for the dependence ofthe large- z behaviour on the choice of spin axis discussed in Section 4.5. It is always possible to construct partial amplitudes with this property by assigning the fermions pairsto different, possibly artificial, generations and assuming flavour-mixing matrices to be diagonal. Application to selection rules
Selection rules for helicity amplitudes of massless particles with arbitrary multiplicities arewell known, see e.g. [41, 42] for reviews. In particular, gluonic amplitudes vanish if allgluons have the same helicity or only one has a different helicity. The first nonvanishingamplitudes are the so-called maximally helicity violating (MHV) amplitudes, which havea very simple all-multiplicity expression. The corresponding selection rules for amplitudeswith massive particles are less restrictive and depend on the choice of spin axes. Selectionrules for all-multiplicity amplitudes with massive vector bosons have been derived fromsupersymmetric identities [8, 43] or using diagrammatic arguments [26]. Here we providean alternative derivation of some of these results as an application of on-shell recursionrelations for amplitudes with massive vector bosons. We furthermore discuss the roleplayed by the choice of spin axes.For the derivation of the selection rules, it is useful to deviate slightly from the choiceof spin axes of Section 5.1 and choose the reference spinors of all remaining particles interms of the momentum and spin axis of one reference particle n : q i,α = k (cid:91)n,α , q i, ˙ α = q n, ˙ α . (6.1)Relations with an exchanged role of positive and negative spin projections can be derivedfor reference spinors q i,α = q n,α and q i, ˙ α = k (cid:91)n, ˙ α . On-shell recursion relations can be usedto show that amplitudes with only vector bosons with negative spin projection vanish forthe choice (6.1), as well as those with one additional scalar or longitudinal vector boson, A ( W − , W − , . . . , W − n ) = 0 ,A ( W , W − , . . . , W − n ) = 0 ,A ( φ , W − , . . . , W − n ) = 0 . (6.2)Furthermore, amplitudes where the reference particle has negative spin projection vanish ifall other vector bosons have positive spin projection, or contain one scalar or longitudinalvector boson, A ( W +1 , W +2 , . . . , W − n ) = 0 ,A ( W , W +2 , . . . , W − n ) = 0 ,A ( φ , W +2 , . . . , W − n ) = 0 . (6.3)These identities are derived in Section 6.1. In Section 6.2, similar identities are derived foramplitudes with a fermion pair and vector bosons where all spin projections are negative, A ( ¯ ψ − , ψ − , W − , . . . , W − n ) = 0 , (6.4)or all positive with the exception of the reference particle, A ( ¯ ψ +1 , ψ +2 , W +3 , . . . , . . . W − n ) = 0 , (6.5) A ( ¯ ψ +1 , W +2 , . . . ψ − n ) = 0 . (6.6)35ll amplitudes appearing in these selection rules vanish in the massless limit. Forthe massive case, however, the selection rules are valid only for the spin axes defined bythe choice of reference spinors (6.1). These results are compatible with relations valid inmassive supersymmetric theories for the same choice q i,α = k (cid:91)n,α of holomorphic referencespinors [8]. In section 6.3, little-group transformations are used to transform these selectionrules into a frame with a common spin axis for all particles, as used in [26]. In this frame,the selection rules (6.2) and (6.4) continue to hold exactly, while the amplitudes appearingin the other selection rules are mass-suppressed in the high-energy limit. For the set of amplitudes (6.2) it is possible to perform a three-line anti-holomorphicRisager shift W − , A l W − , A m W − , A n with arbitrary shift spinor η ˙ α so that the amplitudes satisfythe recursion A (Φ , . . . W − l , . . . , W − m , . . . , W − n )= (cid:88) F ,s A F (Φ , . . . ˆ W − , A l . . . ˆΦ s F ) i K F − M F A F (cid:48) ( ˆΦ − s F (cid:48) , . . . , ˆ W − , A m . . . ˆ W − , A n . . . ) + . . . , (6.7)where Φ ∈ { W − , W , φ } and Φ s F ∈ { W s , φ } . The dots indicate similar contributions witha different distribution of the shifted legs over the factorized subamplitudes.For the anti-holomorphic Risager shift, all scalar products of polarization vectors withnegative spin projection vanish for all external and internal lines, (cid:15) ( k i , − ) · (cid:15) ( k j , − ) = 0 , (6.8)since all external and internal lines are defined using the same anti-holomorphic referencespinor, q F , ˙ α = q i, ˙ α = q n, ˙ α . This implies the selection rules for three-point vertices, A (Φ i , W − j , W − k ) = 0 , (6.9)where all particles may be shifted or unshifted, as well as external or internal. The resultfor Φ i = { W − i , φ i } follows trivially since all terms in the vertex involve a scalar productof two polarization vectors. For the longitudinal polarization Φ i = W i , the selection rulefollows from the Ward identity (4.11) since the choice of reference spinors (6.1) implies theidentity (cid:15) ( k j , − ) · r i = 0 (6.10)for all internal or external legs, where the remainder vector r µ ∝ q µ is defined in (4.14).The recursion relation (6.7) allows to construct the four-point amplitudes from three-point amplitudes as input. The selection rules for the three point functions imply that allpossible combinations of three-point building blocks appearing in the numerator for thedifferent factorization channels vanish, i.e. (cid:88) s A (Φ , ˆ W − , A l , ˆΦ s F ) A ( ˆΦ − s F (cid:48) , ˆ W − , A m , ˆ W − , A n ) = 0 . (6.11)36his verifies the selection rules (6.2) for four-point functions. By induction, the argumentgeneralizes to general multiplicities due to the structure of the recursion relation (6.7).For the amplitudes (6.3) it is possible to perform a mostly-holomorphic BCFW three-line shift (3.59) W + , H l W + , H m W − , A n , which leads to a recursion relation of a similar formas (6.7) up to the different spin labels. Internal lines in the recursion relations are definedby the same holomorphic reference spinor q F ,α = q i,α = k (cid:91)n,α as all external line apart fromthe reference leg n . Therefore the polarization vectors with positive spin projection sharethe same reference spinor for internal and external lines. This choice implies that the scalarproducts (cid:15) ( k i , +) · (cid:15) ( k j , +) = 0 , (cid:15) ( k i , ± ) · ˆ (cid:15) A ( k n , − ) = 0 , (6.12)vanish where i and j denote arbitrary vector bosons, including the reference leg n . Thisimplies the selection rules A (Φ i , W + j , W + k ) = 0 (6.13)with Φ i = { W + i , W i , φ i } for three-point functions of generic vector bosons and A (Φ i , W + j , ˆ W − , A n ) = 0 (6.14)for three-point functions involving the reference leg. The selection rules for longitudinalvector bosons hold since the choice of spin axes ensures (cid:15) ( k j , +) · r i = 0 = ˆ (cid:15) A ( k n , − ) · r i (6.15)for all legs i and j . In analogy to the discussion for the Risager shift, it is seen that allpossible combinations of three-point amplitudes contributing to the recursive constructionof the four-point amplitude vanish, (cid:88) s A (Φ , ˆ W + , H l , ˆΦ s F ) A ( ˆΦ − s F (cid:48) , ˆ W + , H m , ˆ W − , A n ) = 0 , (6.16) (cid:88) s A ( ˆ W + , H l , ˆ W + , H m , ˆΦ s F ) A ( ˆΦ − s F (cid:48) , Φ , ˆ W − , A n ) = 0 . (6.17)The selection rules (6.3) for arbitrary multiplicities follow by induction using the recursionrelation. Similar to the bosonic case, the selection rule (6.4) for a fermion pair and an arbitrarynumber of vector bosons with identical spin labels can be established by a three-line anti-holomorphic Risager shift ψ − , A l W − , A m W − , A n . In the recursion relation, contributions frominternal boson and fermion lines must be taken into account, A ( ¯ ψ − , . . . , ψ − l , . . . , W − m , . . . , W − n ) 37 (cid:88) F ,s A F ( ¯ ψ − , . . . ˆ ψ − , A l . . . ˆΦ s F ) i K F − M F A F (cid:48) ( ˆΦ − s F (cid:48) , . . . , ˆ W − , A m , . . . ˆ W − , A n . . . )+ (cid:88) F ,s A F ( ¯ ψ − , . . . ˆ W − , A m . . . ˆ ψ s F ) i K F − M F A F (cid:48) ( ˆ¯ ψ − s F (cid:48) , . . . , ˆ ψ − , A l , . . . ˆ W − , A n . . . ) + . . . , (6.18)where again the dots indicate contributions with a different distribution of the shifted legsover the subamplitudes.Selection rules for the three-point fermion amplitudes can be inferred from the contrac-tions of Dirac spinors (2.23) with the polarization vectors (2.35), /(cid:15) ( k i , − ) u ( k j , − ) ∝ (cid:32) − k (cid:91)i,α m j [ q i q j ][ q j k (cid:91)j ] q i, ˙ α (cid:104) k (cid:91)i k (cid:91)j (cid:105) (cid:33) . (6.19)This implies the selection rule for three-point functions for generic legs, A ( ψ − i , W − j , ψ − k ) = 0 , (6.20)since the same anti-holomorphic reference spinors enter the left-handed polarization vectorsand spinors of all external or internal particles, including the reference leg.In the recursive construction of the four-point amplitude using (6.18), the contributingproducts of three-point amplitudes all vanish for arbitrary spin states of the intermediateparticle, (cid:88) s A ( ¯ ψ − , ˆ ψ − , A l , ˆΦ s F ) A ( ˆΦ − s F (cid:48) , ˆ W − , A m , ˆ W − , A n ) = 0 , (6.21) (cid:88) s A ( ¯ ψ − , ˆ W − , A m , ˆ ψ s F ) A ( ˆ¯ ψ − s F (cid:48) , ˆ ψ − , A l , ˆ W − , A n ) = 0 . (6.22)Note that the selection rules for the bosonic vertices (6.9) ensure that the properties ofthe vertices involving fermions and scalars or longitudinal vector bosons are not requiredto show that the four-point amplitude vanishes. Due to the structure of the recursionrelation (6.18), the selection rule (6.4) follows from induction.The selection rules (6.5) and (6.6) can be derived using a mostly-holomorphic BCFW-type shift ψ + , H l W + , H m W − , A n or W + , H l W + , H m ψ − , A n . In analogy to (6.20), the identities A ( ψ + i , W + j , ψ + k ) = 0 (6.23)hold since the same holomorphic reference spinor enters all wave functions of particleswith positive spin projection. For three-point amplitudes involving the reference particles,selection rules can be inferred from the contractions /(cid:15) ( k i , +)ˆ u A ( k n , − ) ∝ (cid:32) − q i,α m n [ k (cid:91)i q n ][ q n k (cid:91)n ] k (cid:91)i, ˙ α (cid:104) q i k (cid:91)n (cid:105) (cid:33) , / ˆ (cid:15) A ( k n , − ) u ( k j , + ) ∝ (cid:32) k (cid:91)n,α [ q n k (cid:91)j ] − q n, ˙ α m j (cid:104) k (cid:91)n q j (cid:105)(cid:104) k (cid:91)j q j (cid:105) (cid:33) . (6.24) For simplicity vector-like couplings are assumed. The presence of different left- and right-handedcouplings does not change the results, while for purely chiral couplings additional selection rules can arise. A ( ¯ ψ + i , W + j , ˆ ψ − , A n ) = 0 , A ( ¯ ψ + i , ψ + k , ˆ W − , A n ) = 0 , (6.25)if a vector boson or fermion is used as reference particle.The selection rule (6.5) is again derived inductively from the recursion relation, whichhas a similar form as (6.18). For the four-point amplitude this is seen since all products ofthree-point functions appearing in the recursive construction vanish, (cid:88) Φ ,s A ( ¯ ψ +1 , ˆ ψ + , H l , ˆΦ s F ) A ( ˆΦ − s F (cid:48) , ˆ W + , H m , ˆ W − , A n ) = 0 , (6.26) (cid:88) s A ( ¯ ψ +1 , ˆ W + , H m , ˆ ψ s F ) A ( ˆ¯ ψ − s F (cid:48) , ˆ ψ + , H l , ˆ W − , A n ) = 0 . (6.27)The selection rule (6.6) is derived analogously, where the contributions in the four-pointcase are (cid:88) Φ ,s A ( ˆ W + , H l , ˆ W + , H m , ˆΦ s F ) A ( ˆΦ − s F (cid:48) , ¯ ψ +1 , ˆ ψ − , A n ) = 0 , (6.28) (cid:88) s A ( ¯ ψ +1 , ˆ W + , H m , ˆ ψ s F ) A ( ˆ¯ ψ − s F (cid:48) , ˆ W + , H l , ˆ ψ − , A n ) = 0 . (6.29) The selection rules derived so far hold for a different spin axis (6.1) for legs i ∈ { , . . . , n − } compared to leg n . Little-group transformations can be used to transform to a conventionwhere the same reference spinors are used for all particles. Taking the common referencespinors to be those of the reference particle n , the new reference spinors are given by q (cid:48) i,α = q n,α , q (cid:48) i, ˙ α = q i, ˙ α = q n, ˙ α . (6.30)The expressions for the little-group rotation (B.3) imply that the elements of the transfor-mation matrices for legs i satisfy R + − = 0 and R −− = R − . For the comparison with theselection rules for a common spin axis [26], the explicit expressions of the transformationsare not required, but rather the behaviour in the high-energy limit E (cid:29) m W , m ψ . Usingthe scaling | k i (cid:105) ∼ | k i ] ∼ √ E i , the matrix elements behave in the high-energy limit as R −− ∼ , R − + ∼ m i E i , R ++ ∼ . (6.31)Therefore the matrices implementing the transformation of the Dirac spinors (B.4) andvector-boson polarization vectors (B.5) are of upper triangular form, with the off-diagonalelements suppressed by powers of m/E , (cid:18) u (cid:48) (+ ) u (cid:48) ( − ) (cid:19) = (cid:18) O (1) O ( m ψ E )0 O (1) (cid:19) (cid:18) u (+ ) u ( − ) (cid:19) , (6.32) These relations hold independent of the scaling of the reference spinors in the high-energy limit. (cid:15) (cid:48) (+) (cid:15) (cid:48) (0) (cid:15) (cid:48) ( − ) = O (1) O ( m W E ) O (( m W E ) )0 1 O ( m W E )0 0 O (1) (cid:15) (+) (cid:15) (0) (cid:15) ( − ) . (6.33)The resulting expressions for (cid:15) (cid:48) ( − ), (cid:15) (cid:48) (0) and u (cid:48) ( − ) imply that the selection rules (6.2)and (6.4) for equal spin quantum numbers also hold exactly for a common spin axis. Incontrast, the amplitudes (6.3), (6.5), and (6.6) do not vanish for a common spin axis butare suppressed by powers of m/E . For the amplitudes with only vector bosons, one finds A (cid:48) ( W +1 , . . . , W − n ) ∼ (cid:88) i,j (cid:18) m W i m W j E i E j (cid:19) A ( W +1 , . . . W i , . . . , W j , . . . , W − n )+ (cid:88) i (cid:18) m W i E i (cid:19) A ( W +1 , . . . W − i , . . . , W − n ) + . . . , (6.34) A (cid:48) ( W , W +2 , . . . , W − n ) ∼ (cid:18) m W E (cid:19) A ( W − , W +2 , . . . , W − n )+ (cid:88) i (cid:18) m W i E i (cid:19) A ( W , W +2 , . . . W i , . . . , W − n ) . . . , (6.35)where the dots indicate contributions that are suppressed by higher powers of ( m/E ). Theamplitudes with two negative spin labels on the right-hand side of these identities arenon-vanishing in the massless limit, where they are just the MHV amplitudes. The sameholds for the amplitudes A ( φ , W +2 , . . . φ i , . . . , W − n ), which arise from the high-energy limitof amplitudes with two longitudinal vector bosons. Therefore, all suppression factors areexplicit in the above relations. These results agree with Table 1 in [26].For amplitudes with a scalar one obtains A (cid:48) ( φ , W +2 , . . . , W − n ) ∼ (cid:88) i (cid:18) m W i E i (cid:19) A ( φ , W +2 , . . . W i , . . . , W − n ) + . . . (6.36)In the high-energy limit, the amplitude on the right-hand side again tend to MHV-typeamplitudes with two scalars.For the fermionic amplitudes (6.5) the little-group transformation generates two typesof contributions with a ( m/E ) suppression factor, A (cid:48) ( ¯ ψ +1 , ψ +2 , W +3 , . . . W − n ) ∼ (cid:88) i =1 , (cid:18) m ψ i E i (cid:19) A ( ¯ ψ +1 , ψ − i , W +3 , . . . , W − n )+ (cid:88) i (cid:18) m W i E i (cid:19) A ( ¯ ψ +1 , ψ +2 , W +3 , . . . , W i , . . . W − n ) + . . . (6.37)Here the amplitudes in the first term on the right-hand side are of MHV type and there-fore un-suppressed in the high-energy limit. Also the amplitudes in the second line are40on-vanishing in the high-energy limit, as can be seen by applying the Goldstone-bosonequivalence theorem and using the helicity structure of the Yukawa coupling. Transform-ing the selection rule (6.6) to a common spin axis yields A (cid:48) ( ¯ ψ +1 , W +2 , . . . ψ − n ) ∼ (cid:18) m ψ E (cid:19) A ( ¯ ψ − , W +2 , . . . ψ − n )+ (cid:88) i (cid:18) m W i E i (cid:19) A ( ¯ ψ +1 , W +2 , . . . W i , . . . ψ − n )+ (cid:88) i (cid:18) m W i E i (cid:19) A ( ¯ ψ +1 , W +2 , . . . W − i , . . . ψ − n ) + . . . (6.38)Those amplitudes on the right-hand side that are multiplied by a factor of ( m/E ) arenot of MHV type and vanish in the massless limit by helicity conservation, in case of theamplitudes involving longitudinal vector bosons after application of the Goldstone-bosonequivalence theorem. Therefore these amplitudes are themselves suppressed by a factorof m ψ /E relative to the MHV-type amplitudes in the last line, so that all terms on theright-hand side are effectively of order ∼ m i m j /E for different combinations of fermionor vector-boson masses. This is consistent with all-multiplicity results for amplitudes formassive quarks and massless vector bosons [21, 43].The above observations shed some new light on the results of [26]: amplitudes that aremass-suppressed for common spin axes may vanish exactly for some other choice of spinaxes. In fact, the scaling (6.31) is not limited to the particular choice (6.1) of the initialspin axes but holds generically in the high-energy limit. However, in general R + − is non-vanishing but of order m/E , so the sets of amplitudes (6.2) and (6.4) are expected to bemass-suppressed for general choices of the spin axes. Therefore, amplitudes that vanish forone choice of spin axes are mass-suppressed for generic choices. These results are consistentwith the vanishing of these amplitudes in the massless limit, but it is interesting to seethem emerge from the little-group transformations, which also allow to obtain the powerof the suppression factors. We have performed a comprehensive study of complex deformations of Born amplitudesin spontaneously broken gauge theories and analysed the behaviour for large values of thedeformation parameter z , building on previous studies [4, 5, 23]. Based on these resultswe have identified the minimal shifts necessary to obtain valid on-shell recursion relationsfor amplitudes with a given particle content and spin quantum numbers. Spontaneouslybroken gauge invariance has been shown to improve the large- z behaviour through the useof Ward-identities. Since an on-shell construction of three-point and four-point amplitudes Here the high-energy limit is interpreted as the limit ( v/E ) → v . If the limit m ψ → z behaviour and the high-energy limit. However, as demonstrated by some examples,amplitudes with massive fermions and vector bosons may show worse large- z behaviourunder extended BCFW-type shifts than massless amplitudes due to the dependence on thespin axis or the appearance of contributions that are forbidden in the massless case by thehelicity structure.We have presented little-group covariant expressions for shifted momenta and wavefunctions but made a particular choice of spin axes to find valid shifts for all spin configu-rations, so our final prescriptions for the shifts are not manifestly little-group invariant. Thepossibility to use the covariant form for mixed shifts of massive and massless particles [30]deserves further study. The question remains if a “spin blind” recursive construction ispossible, which does not require to specify the spin axes of massive particles. It would beinteresting to explore a possible extension of a definition of shifts for generic polarizationsof massless vector bosons [44] to the massive case.While we have limited the discussion to renormalizable gauge theories with spin s ≤ z behaviour of massive amplitudes couldbe extended towards effective field theories or to higher-spin particles. In the former case,on-shell recursion relations for non-linearly realised effective theories of massless particleshave been studied [6] while the application to the extension of the Standard Model byhigher-dimensional operators has been initiated recently [30]. An initial application ofon-shell recursion relations to amplitudes with massive spin-two particles was given in [45]. Acknowledgments
The work of RF was supported by the DFG project “Precise predictions for vector-bosonscattering at hadron colliders” (project no. 322921231) while CS acknowledges support bythe Heisenberg Programme of the DFG. 42
Conventions
The spinor conventions used in this paper follow [23]. The sigma matrices are defined as σ µα ˙ β = (1 , − (cid:126)σ ), ¯ σ µ ˙ αβ = (1 , (cid:126)σ ), where (cid:126)σ = ( σ x , σ y , σ z ) are the Pauli matrices. Four-vectors x µ are mapped to bi-spinors according to x α ˙ α = x µ σ µα ˙ α , x ˙ αα = x µ ¯ σ µ ; ˙ αα . (A.1)The conventions for the two-dimensional antisymmetric tensor are given by ε αβ = ε ˙ α ˙ β = ε αβ = ε ˙ α ˙ β = (cid:18) − (cid:19) . (A.2)Indices of two-component Weyl spinors are raised and lowered as follows: k α = ε αβ k β , k ˙ α = ε ˙ α ˙ β k ˙ β , k ˙ β = k ˙ α ε ˙ α ˙ β , k β = k α ε αβ . (A.3)In the bra-ket notation spinor products are denoted as (cid:104) pq (cid:105) = p α q α , [ qp ] = q ˙ α p ˙ α , (A.4)while contractions of spinors with “slashed” momenta are given by (cid:104) p | /k | q ] = p α k α ˙ β q ˙ α , [ p | /k | q (cid:105) = p ˙ α k ˙ αβ q β . (A.5)Explicit solutions for the Weyl spinors associated to a light-like momentum can be writtenas k α = e i2 ( χ − φ ) √ k + k (cid:18) − k + i k k + k (cid:19) , k ˙ α = e − i2 ( χ + φ ) √ k + k (cid:18) k + k k + i k (cid:19) , (A.6)where φ is defined by k + k = | k + k | e i φ . In the convention of [23], the arbitrary phase χ is set to zero. With this convention, the complex conjugate spinors for real momentasatisfy ( k α ) ∗ = sgn( k + ) k ˙ α , ( k ˙ α ) ∗ = sgn( k + ) k α , (A.7)with k + = k + k . This implies the relations for the spinor products (cid:104) kq (cid:105) ∗ = [ qk ] , [ qk ] ∗ = (cid:104) kq (cid:105) (A.8)if k + , q + >
0. The relations for the Weyl spinors for crossed momenta, p = − k are givenby p α = i sgn( k + ) k α , p ˙ α = i sgn( k + ) k ˙ α , p α = i sgn( k + ) k α , p ˙ α = i sgn( k + ) k ˙ α . (A.9)43 Little-group transformations
Since we have shown the validity of on-shell recursion relations for a particular choice ofthe spin axes made in Section 5.1, it is necessary to perform little-group rotations to obtainresults in general frames. According to the dictionary (2.9), a transformation of referencespinors { q α , q ˙ α } → { q (cid:48) α , q (cid:48) ˙ α } can be obtained by finding a little-group transformation thattakes k α → k (cid:48) α and k ˙ α, → k (cid:48) ˙ α, (i.e. k α → k (cid:48) α ). This is achieved by the transformationwith components R I = 1 m [ k (cid:48) k I ] , R I = − m (cid:104) k (cid:48) k I (cid:105) . (B.1)It can be verified that the condition ε KL R I K R J L = ε IJ (B.2)is satisfied due to the Dirac equation (2.5) and the normalization condition (2.7). Notethat both the primed and un-primed spinors correspond to the same four-momentum andsatisfy the corresponding Dirac equation. Making the relation to the spin-axis notationexplicit, the components of R are given by R = (cid:104) k (cid:48) (cid:91) q (cid:105)(cid:104) k (cid:91) q (cid:105) = [ q (cid:48) | /k | q (cid:105) [ q (cid:48) k (cid:48) (cid:91) ] (cid:104) k (cid:91) q (cid:105) ≡ R −− , R = − m [ q (cid:48) q ][ q (cid:48) k (cid:48) (cid:91) ] [ k (cid:91) q ] ≡ R + − ,R = − m (cid:104) q (cid:48) q (cid:105)(cid:104) q (cid:48) k (cid:48) (cid:91) (cid:105) (cid:104) k (cid:91) q (cid:105) ≡ R − + , R = [ k (cid:48) (cid:91) q ][ k (cid:91) q ] = (cid:104) q (cid:48) | /k | q ] (cid:104) q (cid:48) k (cid:48) (cid:91) (cid:105) [ k (cid:91) q ] ≡ R ++ , (B.3)where k (cid:48) (cid:91) refers to the light-cone projection of k with respect to the reference vector q (cid:48) .The transformation of Dirac spinors under little group transformations (2.22) impliesthe transformation of the spinors in the spin basis (2.27) with the matrix R ( ) = (cid:18) R ++ R − + R + − R −− (cid:19) , R ( ) − = (cid:18) R −− − R + − − R − + R ++ (cid:19) . (B.4)These results agree with those in [23] with the identifications R ++ = c , R −− = c , R − + = − c , and R + − = − c .The matrix representation of the little-group transformation of the polarization vec-tors (2.39) is found to be R (1) = R −√ R − + R ++ − R − + √ R ++ R + − R ++ R −− + R − + R + − √ R −− R − + − R − √ R + − R −− R −− . (B.5)It satisfies det R = 1 and respects the orthonormalization conditions of the polarizationvectors, R (1) γ R (1) T = γ with γ ss (cid:48) = (cid:15) ( s ) · (cid:15) ( s (cid:48) ) = − δ s, − s (cid:48) . (B.6)44 Q F (cid:54) = 0 shifts In an attempt to improve the large- z behaviour, one may consider giving up the require-ment of light-like shifts for internal lines (3.5) so that propagator denominators scale like z instead of z . The form of the recursion relation valid for this case can be found in [5].Since the shift vectors δk i are always light-like, momentum conservation implies that inter-nal lines are necessarily light-like for two-line or three-line shifts. Shifts with Q F (cid:54) = 0 arethus obtained for at least four shifted lines and are only constrained by the condition ofmomentum conservation (3.34). Note that all shift spinors η i should be chosen differentlysince otherwise the internal shifts Q F ,α ˙ α factorize into two Weyl spinors for some factoriza-tion channels and the corresponding propagator denominators “accidentally” scale like z .However, keeping all η i different implies that spinor products of shifted spinors can scalelike z so that the advantage of an improved large- z scaling of the propagators is partiallycompensated by a worse behaviour of the numerator.A bound for the large- z behaviour for generic Q F (cid:54) = 0 shifts can be obtained using thereasoning of Section 4.3. Despite choosing all η i differently, contributions with Q F = 0can appear due to background insertions into external shifted legs. The large- z scalingof the propagator denominators γ D is therefore constrained by the mass-dimension of thedenominators according to [5] 12 [ D h,b ] ≤ γ D ≤ [ D h,b ] (C.1)instead of (4.18). The value of [ D h,b ] − γ D is given by the number of propagators with Q F = 0. A conservative upper bound is obtained by assuming all background insertionscouple through cubic vertices and lead to a propagator with light-like shift, so that[ D h,b ] − γ D ≤ b. (C.2)Instead of (4.24), the bound on the large- z scaling of the amplitude becomes γ ≤ − ( h + b ) − [ g ] − b ψ D h,b ] − γ D − (cid:88) H s i + (cid:88) A s j ≤ − h − min[ g ] − (cid:88) H s i + (cid:88) A s j . (C.3)Therefore, all amplitudes in renormalizable theories are constructible from five-line shiftswith Q F (cid:54) = 0 [5]. Since this offers no advantage over Risager-type shifts but leads to amore complicated recursion relation, we do not consider such shifts any further. References [1] R. Britto, F. Cachazo, and B. Feng,
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