On the Feynman Rules of Massive Gauge Theory in Physical Gauges
aa r X i v : . [ h e p - ph ] M a r On the Feynman Rules of Massive Gauge Theory in Physical Gauges
Junmou Chen ∗ School of Physics, Korean Institute for Advanced Study, Seoul, 02455, Korea
For a massive gauge theory with Higgs mechanism in a physical gauge, the longitudinalpolarization of gauge bosons can be naturally identified as mixture of the goldstone compo-nent and a remnant gauge component that vanishes at the limit of zero mass, making thegoldstone equivalence manifest. In light of this observation, we re-examine the Feynmanrules of massive gauge theory by treating gauge fields and their corresponding goldstonefields as single objects, writing them uniformly as 5-component “vector” fields. The gaugegroup is taken to be SU (2) L to preserve custodial symmetry. We find the derivation ofgauge-goldstone propagators becomes rather trivial by noticing there is a remarkable par-allel between massless gauge theory and massive gauge theory in this notation. We alsoderive the Feynman rules of all vertices, finding the vertex for self-interactions of vector(gauge-goldstone) bosons are especially simplified. We then demonstrate that the new formof the longitudinal polarization vector and the standard form give the same results for all the3-point on-shell amplitudes. This on-shell matching confirms similar results obtained withon-shell approach for massive scattering amplitudes by Arkani-Hamed et.al. in ref.([23]).Finally we calculate some 1 → ∗ [email protected] I. INTRODUCTION
In a massive gauge theory with Higgs mechanism, scattering amplitudes involving longitudinalpolarizations have the famous problem of power counting[1–3]: while single Feynman diagramsincrease with energy, the S-matrix is well-behaved when taking into account the contribution ofthe Higgs boson. This failure of power counting causes many complications and confusion bothpractically and conceptually. The origin of the problem is the longitudinal polarization vectorbehave as ǫ µL ∼ k µ m W + O ( m W E ) in high energy limit. Another way to phrase it is that the longitudinalpolarization vector and Feynman diagrams don’t have a smooth limit as m W →
0, thus it’s not clearhow the theory approaches massless limit continuously. In future high energy colliders [4][5], wewill approach energy scales in which the EW symmetry will be effectively restored. This problembecomes even more severe.Practically this problem is often solved by replacing the longitudinal vector bosons with the cor-responding goldstone bosons, according to the so-called goldstone equivalence theorem(GET)[11–15], which states that scattering amplitudes involving longitudinal vector bosons can be approxi-mated by the corresponding goldstone modes in high energy limit: M ( W L , W L , W L , ...., W L ) = ( − i ) n M ( φ, φ, φ, ...., φ ) + O ( m W √ s ) , (1)with √ s being the hard scale of the process. However, this solution is still not completely sat-isfactory, as GET is only an approximation with other terms suppressed in high energy limit.Although the approximation seems to work for naive power counting, it’s not utterly clear if thecontributions of the terms neglected by GET are real subdominant. In fact, it was discovered in[6] that there is a new class of splitting functions contributing to DGLAP evolution of EW PDFsand substructure of EW jets. Those new splitting functions originate precisely from the terms thatare neglected by GET. It then becomes mandatory to account for all the terms that from thelongitudinal polarization vector in calculation. Obviously we need a better solution for the powercounting problem.A physical gauge in a massive gauge theory can be defined by the gauge-fixing condition n · W = The mistake of the naive power counting is that it neglects that a physical process is intrinsically multi-scaled.The terms neglected by GET has soft singularities (with infrared cut-offs provided by the masses) that give riseto contributions up to single logarithms, when the collinear scale λ lies in m W ≪ λ ≪ √ s .
0, with n being any direction other than k , is able to serve this purpose [6–9]. Heuristically we canargue this way: GET is the consequence of gauge symmetry. It can be derived from Ward identitiesof the theory. Nevertheless, there is also an alternative and a more direct way to prove GET, i.e. wecan simply choose another gauge. Since in a physical gauge we only impose gauge-fixing on gaugefields, the gauge-goldstone mixing term in the Lagrangian remains, thus we are forced to identifygauge fields W µ and goldstone fields φ as single fields, which we can denote as W M = ( W µ , φ ).In the resulting gauge-goldstone propagator, goldstone modes and gauge modes obtain the samepole masses, the longitudinal polarization vectors are naturally identified as mixture of gaugecomponents and goldstone components. We can write the longitudinal polarization vector as ǫ ML = ( ǫ µn , − i ), with ǫ µn ∼ − m W E n µ in high energy limit. Its specific form depends on the gaugedirection n . In this way, we obtain a precise formula of scattering amplitudes involving longitudinalvector bosons, which is a generalization of GET in Eq.(1), M ( W L , W L , W L , ...., W L ) = ( − i ) n M ( φ, φ, φ, ...., φ ) + ( − i ) n − M ( W n , φ, φ, ...., φ )+ .... + M ( W n , W n , W n , ...., W n ) (2)The polarization vectors of W n are given by ǫ n , which is usually neglected by GET. Thus physicalgauges are vastly different from R ξ gauge, in which the masses of the goldstone bosons are gauge-dependent. Of course, physical results cannot be gauge-dependent. The author in [19] obtainedsimilar results as Eq.(2) based on Feynman gauge by making use of BRST symmetry to redefinethe physical state. An earlier attempt along this line can be found in [18]. The longitudinalpolarization and related scattering amplitudes in physical gauges agree precisely with those in R ξ gauge in [19][18] if the gauge direction is chosen as n µ = (1 , − ˆ k ), with ˆ k = | ~k | ~k . Thus the twoapproaches are equivalent with each other. Nevertheless, comparing to R ξ gauge, physical gaugesprovide a much more clear physical picture as there is no gauge-dependent goldstone mass, noambiguity in identifying physical states through LSZ reduction formula.Although the power counting problem is overcome in a physical gauge, there is also a drawback:the Feynman rules become complicated, as we need to sum over all the terms from both gaugecomponents and goldstone components. The problem becomes especially severe if the numberof longitudinal states are multiplied. Besides, the derivation of the gauge-goldstone propagators Traditionally a physical gauge is defined for n µ along a fixed direction, e.g. n µ = (1 , , , T . Here we adopt amore general definition, which includes n being momentum dependent, e.g. Coulomb gauge[9]. Other examples ofmomentum dependent “physical” gauge can be found in [6] and [10]. seems also to be complicated due to the gauge-goldstone mixing terms in the Lagrangian. Thegoal of this paper is to investigate and reorganize the Feynman rules of massive gauges in physicalgauges by combining gauge fields and goldstone fields together. The strategy is, as mentionedabove, to treat gauge fields and goldstone fields uniformly as 5-component fields W M = ( W µ , φ ),and exploit possible underlying structures to simplify. The model we choose is the θ W → SU (2) L instead of SU (2) L × U (1).The motivation is that if custodial symmetry is preserved, both the gauge fields and goldstonefields transform as a triplet under SU (2) global symmetry. It then becomes straightforwardlyto combine gauge components and goldstone components. Additionally, it’s noteworthy that thenew Feynman rules can also apply to Feynman gauge, in which goldstone bosons obtain the samemasses as their corresponding gauge bosons. So we can make use of the Feynman rules describehere, if the longitudinal polarization vector is taken to be ǫ ML = ( ǫ µn , − i ).Apart from deriving and documenting the Feynman rules, we also investigate all the 3-pointon-shell amplitudes. In recent years, the on-shell approach of scattering amplitudes using spinor-helicity[21][22] has made remarkable progress. However, the success is still largely confined inmassless particles. There have been many papers[23–26] devoted to the massive case, but the topicstill remains largely unexplored. 3-point on-shell amplitudes are the building blocks of on-shellapproach to scattering amplitudes, thus one might hope that clearer understanding of them canshed some light in the direction. Our basic point is, now that we have two forms of longitudinalpolarization vectors – one from gauge fields only, another from mixture of gauge fields and goldstonefields – the two forms should give the same amplitudes due to gauge invariance. This match betweentwo ways of evaluating amplitudes should also be reflected on the 3-point on-shell amplitudes, whichcan be appropriately called “on-shell match”. This on-shell match gives a way to explain how theinformation of goldstone bosons are “encoded” in gauge fields for the case of 3-point amplitudes.Another motivation for 3-point on-shell amplitudes is the calculation of collinear splitting func-tions, which can be reduced to the calculation of 1 → II. THE MODEL AND FEYNMAN RULESA. Lagrangian
Our goal is to derive the Feynman rules of the Standard Model of Electroweak interactions bytaking the θ W → SU (2) L only. With the custodial symmetry, theHiggs potential has symmetry SU (2) L × SU (2) R . We can parametrize the Higgs field by writingit as H = 1 √ iσ Φ ∗ , Φ)with Φ being Φ = √ − i ( φ − iφ ) h + iφ A more illuminating way to write Higgs doublet H is H = 1 √ iσ Φ ∗ , Φ) = 12 ( h − iσ a φ a ) (3)The would-be goldstone fields are isolated from the would-be Higgs field in this parametrization,which will be more convenient to treat the would-be goldstone bosons as the 5th component of thevector fields/states. The full Lagrangians are written as L Gauge = −
14 ( W aµν W µνa ) + 12 ξ ( n · ∂ n · W a )( n · ∂ n · W a ) ∗ L Higgs = tr[( D µ H ) † D µ H ] − λ h r [ H † H ] − v (4) L Fermion = i X i =1 , , Q ′ iL /DQ ′ iL + i X i =1 , , L ′ iL /DL ′ iL L Yukawa = − X ij =1 , , √ Q ′ iL H Y ′ ijQ Q ′ jR − X i,j =1 , , √ L ′ iL H Y ′ ijL L ′ jR + h.c.Here W aµν = ∂ µ W aν − ∂ ν W aµ − gǫ abc W bµ W cν , D µ = ∂ µ + ig σ a W aµ , n µ can be either a fixeddirection[7][8] (e.g. n µ = (1 , , , ξ is dimensionless, n µ is rescaled by n · ∂ . Ghosts in physicalgauges generally decouple from the theory, but don’t decouple if n is momentum-dependent [10].Nevertheless, we restrict our focus on tree level in this paper.For the Fermion sector and the Yukawa sector, Q ′ iL/R = ( u ′ iL/R , d ′ iL/R ) and L ′ iL/R = ( ν ′ iL/R , e ′ iL/R )denote quarks and leptons in flavor basis respectively. Indices i, j = 1 , , Y ′ ijQ = diag( y ′ iju , y ′ ijd ) is Yukawa matrix for the quark sector in isospin space, Y ′ ijL = diag( y ′ ijν , y ′ ijl )is Yukawa matrix for the lepton sector in isospin space.The Lagrangian terms in Eq.(4) are invariant under H → e iαaσa H Q L → e iαaσa Q L (5) L L → e iαaσa L L Next we expand the Lagrangian terms in terms of W µ , h , φ . After symmetry breaking, theHiggs field has shift: h → h + v . Particles obtain masses, the relations between masses and v are m W = gv m f = y f v √ m h = λ h v L W µ = − ∂ µ W a ∂ µ W a + 12 ξ ( n · ∂ n · W a )( n · ∂ n · W a ) ∗ L W µ + L W µ = gǫ abc ∂ µ W νa W bµ W cν − g ǫ abc ǫ afg W bµ W cν W µf W νg . (7)For the Higgs sector, the covariant derivative on the Higgs doublet H is written as D µ H = ( ∂ µ + igW aµ σ a h + v ) − i σ b φ b )Making use of σ a σ b = δ ab + iǫ abc σ c and separating h and φ , D µ H can be written further as D µ H = ( ∂ µ + igW aµ σ a · ( h + v ) − i σ a ∂ µ δ ac − g ǫ abc W bµ ) φ c + 14 gW aµ φ a . (8)Then we plug in D µ H into tr[( D µ H ) † D µ H ]. Combined with the Higgs potential V (Tr( H † H )),the Lagrangian terms for Higgs sector become L h = 12 ∂ µ h∂ µ h − m h h L φ + φW µ + φ W µ = 12 ∂ µ φ a ∂ µ φ a − g ǫ abc ∂ µ φ a W bµ φ c + g ǫ abc ǫ afg W bµ φ c W fµ φ g + g W aµ φ a W bµ φ b . (9)and L φW µ + hW µ + hφW µ = − m W ∂ µ φ a W µa + ( g h + gm W h + m W W µa W aµ + g ∂ µ hW aµ φ a − ∂ µ φ a W µa h ) L h + h = − λ h h − λ h vh L h φ = − λ h h φ a φ a − λ h vhφ a φ a (10) L φ = − λ h φ a φ a φ b φ b For the fermion sector, after symmetry breaking, Q ′ i and L ′ i are related to the mass basis by Q ′ i = U ijQ Q j = U iju U ijd u j d j L ′ i = U ijL L j = U ijν U ijl ν j l j (11)The Yukawa matrices and mass matrices are diagonalized by the mixing matrices U Q/L , Y ilQ/L = U † ijQ/L Y ′ jk U klQ/L = Y iQ/L δ il (12)as well as m ilQ/L = U † ijQ/L Y ′ jk U klQ/L v √ m iQ/L δ il (13)with Y iQ = diag( y u i , y d i ), Y iL = diag( y ν i , y l i ), m iQ = diag( m u i , m d i ), m iL = diag( m ν i , m l i ). TheLagrangian terms for the fermion sector then become L f = iQ L /∂Q L + iL L /∂L L − Q L m Q Q R − L L m L L R L ffh = − √ Q L Y Q Q R h − (cid:18) √ L L Y L L R h + h.c. (cid:19) L ffW µ + L ffφ = − g Q L γ µ ( U † Q σ a U Q ) Q L W µa + (cid:18) i √ Q L ( U † Q σ a Y Q U Q ) Q R φ a + h.c. (cid:19) (14) − g L L γ µ ( U † L σ a U L ) L L W µa + (cid:18) i √ L L ( U † L σ a Y L U L ) L R φ a + h.c. (cid:19) The generation indices have been suppressed.
B. Propagator
In this section we are deriving the propagator of vector bosons, which has intrinsic mixingbetween gauge modes and goldstone modes. Combining the kinetic terms in Eq.(7), Eq.(9) andEq.(10), the quadratic Lagrangian terms for gauge fields and goldstone fields are L W a = − ∂ µ W νa ∂ µ W aν + 12 ∂ µ W aµ ∂ ν W aν + 12 m W W aµ W aµ + 12 ξ ( n · ∂ n · W a )( n · ∂ n · W a ) ∗ L φ a W a = − m W W aµ ∂ µ φ a (15) L φ a = 12 ( ∂ µ φ a ) We note that Eq.(15) is not only true for the SU (2) L theory, but applies to any model withHiggs mechanism. We write gauge-goldstone fields as 5-component vector fields W Ma = ( W µa , φ a ),then the kinetic Lagrangian terms can be written as following up to terms with total derivative, L W M = − ∂ M W aN ∂ M W Na + 12 ( ∂ M W Ma ) + 12 ξ ( n · ∂ n M W Ma )( n · ∂ n M W Ma ) ∗ (16)with n M = ( n µ , W aM = ( W aµ , φ a ), ∂ M = ( ∂ µ , − m W ), g MN = g MN = diag(1 , − , − , − , − µ becomes M .This similarity is not just a nice way of writing all the Lagrangian terms. Indeed, noticing theFourier transformation of ∂ M = ( ∂ µ , − m W ) gives k M = ( k µ , − im W ) for inwards momentum, and k ∗ M = ( k µ , im W ) for outwards momentum, we could write the dot product of k M as k · k ∗ = g MN k M k ∗ N = k − m W (17)This equals 0 when on-shell, just as k · k = k = 0 when on-shell for massless case. Thus all thealgebra with the tensor g µν and k µ , could be applied straightforwardly to g MN and k M /k ∗ N , with n M = ( n µ , L kinetic = − ∂ µ W aν ∂ µ W νa + 12 ( ∂ · W a ) + 12 ξ ( n · ∂ n · W a )( n · ∂ n · W a ) ∗ + total derivative(18)the propagator of the gauge bosons can be easily evaluated to be < W µa W νb > = − iδ ab ( g µν − n µ k ν + k µ n ν n · k + n k µ k ν ( n · k ) + ξ k ( n · k ) k µ k ν ) k + iǫ (19)Following the arguments above, a direct analogue to the massless propagator in Eq.(19) givesus the gauge-goldstone propagator in massive gauge theory, < W Ma W Nb > = − iδ ab ( g MN − n M k ∗ N + k M n ∗ N n · k + n k M k ∗ N ( n · k ) + ξ k · k ∗ ( n · k ) k M k ∗ N ) k · k ∗ + iǫ (20)By writing gauge components M = µ and M = 4 component separately, the propagator ofvector boson becomes < ( W µa , φ a ) , ( W νb , φ b ) > = iδ ab k − m W + iǫ − ( g µν − n µ k ν + k µ n ν n · k + n k µ k ∗ ν ( n · k ) ) i m W n · k ( n µ − n k µ n · k ) − i m W n · k ( n ν − n k ν n · k ) 1 − n m W ( n · k ) When k = m W or k · k ∗ = 0, the numerator of the propagator can be written as sum of thepolarizations,0 < W Ma W ∗ Nb > = iδ ab P s = ± ,L ǫ Ms ǫ N ∗ s k · k ∗ + iǫ In the 5-component notation, the transverse and longitudinal polarizations are g = − ǫ M ± = ǫ µ ± ǫ ML = 1 r − n m W ( n · k ) − m W n · k ( n µ − n k µ n · k ) i (1 − n m W ( n · k ) ) (21)They satisfy the transverse condition and normalization condition ǫ s · ǫ ∗ s ′ = − · δ ss ′ k ∗ · ǫ s = k · ǫ ∗ s = 0 (22) n · ǫ ( ∗ ) s = ± = 0The amplitudes involving longitudinal vector bosons are evaluated by summing over the con-tributions from both gauge components and goldstone components: i M ( L ) = ig MN M M ǫ N = i M µ ǫ nµ − i M ǫ (23)Notice the “metric” g MN = dig(1 , − , − , − , −
1) induces a minus sign between amplitudesinvolving gauge component and goldstone component. In practical calculations it’s more convenientto absorb this minus sign into the polarization vectors, which become g = 1 : ǫ ML = 1 r − n m W ( n · k ) − m W n · k ( n µ − n k µ n · k ) − i (1 − n m W ( n · k ) ) (24)so that the amplitude is evaluated by simply summing over the diagrams involving gauge com-ponents and goldstone components, i.e. i M ( L ) = i M µ ǫ nµ + i M ǫ (25)Our results agree with Ref.[6] by choosing n µ = (1 , − ~k | ~k | ) up to a global phase which is unphys-ical.1 C. Vertices
In this section we apply the 5-component treatment further to vertices. Our goal is to recombinethe Lagrangian terms for interactions, so that the Feynman rules for vector bosons is given byLagrangian of W aM = ( W aµ , φ a ). We start from the gauge sector and Higgs sector, which give riseto vertices of W-W-W and W-W-W-W. The corresponding Lagrangian terms are written as L W M = gǫ abc ∂ µ W aN W bρ W cK g µρ g ′ NK (26) L W M = − g ǫ abc ǫ aef W bµ W eν W cP W fK g µν g ′ P K + g W aµ W bν φ a φ b g µν − λ h φ a φ a φ b φ b with g ′ NK = diag(1 , − , − , − , − / g ′ NK here is not to be confused with g NK ap-pearing say in Eq.(23): g ′ NK in Eq.(26) is for bookkeeping the relative coefficient between thedifferent Lagrangian terms, whereas g NK in Eq.(23) is to keep track of the relative phase between(sub)diagrams of gauge components and goldstone components, which can be absorbed into thedefinition of polarization vectors.The Lagrangian terms for h-W-W and h-h-W-W are L hW M = g ∂ µ hW aν φ a g µν − ∂ µ φ a W aν hg µν ) + 12 gm W hW aµ W aν g µν − λ h vhφ a φ a (27) L h W M = g h W aµ W aν g µν − λ h h φ a φ a To obtain Feynman rules we need the last step of writing W a in the basis ( W ± M , W M ), with W ± M = 1 √ W M ∓ iW M ) . (28)This identiy are useful σ a W aM = W M T + √ W + M T + + W − M T − ) . (29) T and T ± are separately, T = − T + = T − = V CKM and V PMNS as V CKM = U † u U d V PMNS = U † ν U l (30)as well as W M W N − W M W N = i ( − W + M W − N + W − M W + N ) W M W N + W M W N = W + M W − N + W − M W + N (31)Writing all Lagrangian terms in the phyiscal basis. The Lagrangian terms giving rise to verticesW-W-W and W-W-W-W are L W M = − ig ( ∂ µ W N W + ρ W − K ) g µρ g ′ NK + cyclic in (3, +, -) L W M = g (cid:2) g µν g ρσ W + µ W + ν W − ρ W − σ − g µν g P Σ W + µ W − ν W + P W − Σ (cid:3) − λ h φ + φ − ) − g (cid:2) g µν g ′ P Σ W + µ W − ν W P W − g µν g P Σ W + µ W ν W − P W + ( µ ↔ P, ν ↔ Σ) (cid:3) − λ h φ + φ − ( φ ) (32)+ g W µ W ν φ φ g µν − λ h
16 ( φ ) The Lagrangian terms giving rise to vertices h-W-W and h-h-W-W are L hW M = g ∂ µ hW ν φ − ∂ µ φ W ν h ) g µν + 12 gm W hW µ W µ − λ h vh ( φ ) h g g µν ( ∂ µ hW + ν φ − − ∂ µ φ − W + ν h ) + (+ ↔ − ) i + gm W hW + µ W µ − − λ h vhφ + φ − (33) L h W M = g g µν h (2 W + µ W − ν + W µ W ν ) − λ h h (2 φ + φ − + ( φ ) )The Lagrangian terms for the quark sector become L ffW M = − g u L /W u L − ¯ d L /W d L ) + (cid:20) i √ y u ¯ u L u R φ − y d ¯ d L d R φ ) + h.c. (cid:21) − g √ u L /W + V CKM d L + h.c.) + (cid:20) i √ y u ¯ u L V CKM d R φ + + y d ¯ d L V † CKM u R φ − ) + h.c. (cid:21) (34) L ffh = − √ (cid:0) y u ¯ u L u R h + y d ¯ d L d R h (cid:1) + h.c.3For the lepton sector we simply make the replacement ( u → ν, d → l ) and V CKM → V PMNS .Finally we comment on a subtlety in the derivation of Feynman rules: since W and φ are simplydifferent components of the same physical fields, it’s necessary to sum over all possible contractionsin deriving Feynman rules. This operation gives an additional overal factor to the Feynman rules.For example, the overal factor of vertex hhhh is 4! = 24. However, sometimes it requires writingdifferent terms explicitly. This is especially true in the case of gauge-goldstone fields, in which thegauge components and goldstone components appear to be different fields. For example, in L hW M of Eq.(27), the term g ( ∂ µ hW aν φ a g µν − ∂ µ φ a W aν hg µν ) gives rise to an overal factor 2, which is takeninto account by adding a different term with φ a ↔ W aµ in the Feynman rule. While for W ± M theyhave been written out explicitly, for W M we still need to take into account the term with φ ↔ W µ . III. ON-SHELL MATCH FOR 3-POINT AMPLITUDESA. On-shell Gauge Symmetry
Having derived all the Feynman rules in a physical gauge. We proceed to analyse the on-shell gauge symmetry, especially how they are reflected in 3-point amplitudes. So far we haveconcluded that a longitudinal polarization in a physical gauge is composed of gauge components ǫ µn and goldstone component i/ − i , as in Eq.(21). We choose n = 0, i.e. light-cone gauge, thelongitudinal polarization vector in 5-component becomes: ǫ M L = ǫ µn i (35)for incoming particles, and ǫ ∗ ML for outgoing particles. Here ǫ µn = − m W n · k n µ . On the other hand,we already have the standard form for the longitudinal polarization vector, which can be writtenin the 5-component format as ǫ M L = ǫ µn + k µ m W (36)with the 5th component to be 0 and n µ = (1 , − ~k | ~k | ). If we choose n in Eq. (35) as n = n , ǫ M L and ǫ M L are related with each other by4 ǫ M L = ǫ M L − k M m W (37)Meanwhile since S-matrix is gauge-invariant, the two forms of longitudinal polarizations haveto give the same S-matrix, i.e., ǫ M L ...ǫ M i L S M ...M i ( k ...k i ... ) = ǫ M L ...ǫ M i L S M ...M i ( k ...k i ... ) (38)Plugging in Eq.(37), Eq.(38) is equivalent to k M ...k M i i S M ...M i ( k ...k i ... ) = 0 . (39)Interestingly, Eq.(39) share similar form as the on-shell gauge symmetry for massless gauge theory,with M → µ . Thus, we can appropriately call it “on-shell gauge symmetry” for massive gaugetheory.Ignoring the terms of O ( m W n · k ), Eq.(38) reduced to ǫ M L ...ǫ M i L S M ...M i ( k ...k i ... ) = S ( φ ...φ i ... ) + O ( m W n · k ) (40)So on-shell gauge symmetry reduces to goldstone equivalence theorem as in Eq.(1).In this paper we don’t intend to give a complete proof of on-shell gauge symmetry Rather, weonly set to prove that Eq. (39) is satisfied for all on-shell 3-point amplitudes: W W W , hW W and f f ′ W .The condition of “on-shell” needs some extra comments, as it’s not always kinematically possibleto put all particles on-shell for 3-point amplitudes. This constraint is usually bypassed by analyticalcontinuation of making momenta complex. Nevertheless, for hW W and f f ′ W , we can also putamplitudes on-shell through the analytical continuation of parameters in the theory. For example,for a decay process h → W + W − , the Higgs mass has to satisfy the on-shell condition m h ≥ m W ,with m h ∼ λ h v , m W ∼ gv . We can think of the amplitude as the function of parameters oftheory: M = M ( λ h , v, g ). We can first choose the parameters to satisfy the on-shell condition,but the resulting amplitude will have to be the same for all possible values of parameters withinthe perturbative limits. The same argument can be applied to f → f ′ W , whose amplitude can be5seen as the function of y f , y f ′ , g, v : M = M ( λ f , λ ′ f , g, v ). Notice this argument doesn’t apply to W W W , since the amplitude is controlled by only one coupling g . It’s not possible to adjust g toput the all the particles on-shell.For the convention, we absorb the intrinsic minus sign between gauge components and goldstonecomponents in amplitudes to the definition of polarization vectors, which are given by Eq.(24) with g = 1, n = 0, g = 1 : ǫ ML = ǫ µn − i ǫ ∗ ML = ǫ µn i with ǫ µn = − m W n · k n µ satisfying k · ǫ n = − m W .In proving the on-shell gauge symmetry for 3-point amplitudes, k M /k ∗ M also need to be rede-fined to k ML = k µ im W k ∗ ML = k µ − im W However, in the transverse condition and on-shell condition for k M , the intrinsic minus signbetween gauge components and goldstone component still need to be taken into account, i.e. wehave k M ǫ ∗ LM = k ∗ M ǫ LM = k · ǫ n + m W = 0 k M k ∗ M = k ∗ M k M = k − m W = 0We also extract the factor “i” out for S-matrix by defining S = i M . Our convention is that allparticles are incoming. h-W-W The on-shell gauge symmetry for 3-point amplitudes can be written as i M (1 s s s ) | ǫ Msi → kMimW = 0 (41)6 i denotes any particles being W bosons. To prove it, we start with M ( hW W ), with only one W replaced by k M m W . With one particle being the Higgs and another particle being k M m W , there aretwo cases for the polarizations of particle 2: a) transverse b) longitudinal. We start with case a)with s = ± . In this case ǫ s = 0, so there is no goldstone component contribution from particle2, but k m W = i . i M (1 h s = ± s ) | ǫ Ms → kM mW = igm W ǫ ± · k m W + g k − k ) · ǫ ± )( i )= ig ( k · ǫ ± + 12 ( − k − k ) · ǫ ± )= 0In the second step we used energy-momentum conservation k + k + k = 0, in the third stepwe used the transverse condition for particle 2: k · ǫ ± = 0.Then we turn to case b) s = L , so the polarization vector has both gauge components ǫ µ n andgoldstone component ǫ = − i , the amplitude is i M (1 h s = L s ) | ǫ Ms → kM mW = igm W ǫ n · k m W + g k − k ) · ǫ )( i ) + g k − k ) · k m W ( − i ) − ig m h m W i · ( − i )= ig ( ǫ n · k − ǫ n · k − k · ǫ n − ( k − k )( k + k )2 m W − m h m W )To further simplify, we need first to make use of on-shell condition for k and k :( k − k )( k + k ) = k − k = m h − m W as well as the transverse condition for ǫ M L : k · ǫ n = − m W , which is another expression of k ∗ M · ǫ ML = 0. Plugging in, the amplitude becomes i M (1 h s = L s ) | ǫ Ms → kM mW = ig ( m W m h − m W m W − m h m W )= 0 ,f - f ′ -W f f W , i M (1 s s s ) | ǫ Ms → kM mW = 0with particle 1 and 2 being fermions, particle 3 being W boson. Since only particle 3 is Wboson, the identity is relatively easy to prove. Writing the amplitude with gauge components andthe amplitude with the goldstone component separately, Eq. 41 becomes, i M (1 s s s ) | ǫ Ms → kM mW = M (1 s s s ) | ǫ µs → kµ mW ,ǫ s → + M (1 s s s ) | ǫ µs → ,ǫ s →− i = 0 (42)The first term is the amplitude from the fermion-fermion-gauge vertex, the second term is fromthe fermion-fermion-goldstone vertex. Let’s check Eq. (42) explicitly. First look at the first term,to fix all the momenta to be incoming, we choose particle 1 to be a d-type anti-quark, particle 2to be a d-type quark, then the W boson has to be W + , we also set the CKM matrix to be , theamplitude becomes i M| ǫ µs → kµ mW ,ǫ s → = − i g √ ν s L γ µ u s L k µ m W = i g √ ν s L ( /k + /k ) u s L /m W = i g √ − m d m W ¯ ν s R u s L + m u m W ¯ ν s L u s R ) (43)In the second equality, we made use of the energy-momentum conservation k + k + k = 0, inthe third equality we made use of equation of motion for fermions: /k u L/R ( k ) = m u u R/L ( k ) aswell as ¯ ν L/R ( k ) /k = − ¯ ν R/L ( k ) m d . We then look at the second term in Eq. (42), with ǫ = − i ,we have i M| ǫ µs → ,ǫ s →− i = − · ( − i ) g √ m d m W ¯ ν s R u s L − m u m W ¯ ν s L u s R )= − i g √ − m d m W ¯ ν s R u s L + m u m W ¯ ν s L u s R ) (44)Here we made use of y f = m f √ m W . Combining Eq. (43) and Eq. (44), we finish the proof of Eq. (42).Although we only went through the example of ¯ d u W +3 , it can be checked straightforwardly thatEq. (42) is satisfied for all other cases. Having ignoring the CKM matrix ¯ u d W − is identical8to ¯ d u W +3 . For neutral current, i.e. W being W , the proof is also identical except we need toreplace γ µ with √ γ µ T for the fermion-fermion-gauge vertex, and replace γ with √ γ T for thefermion-fermion-goldstone vertex. It’s also not hard to see that the conclusion also applies to theSM with the gauge group being SU (2) L × U (1) Y , in which case the only difference is the neutralcurrent case. For f f γ , i M| ǫ µ → k µ = 0 according to ward identity. For f f Z , the fermion-fermion-goldstone vertex is identical to the SU (2) case; for the fermion-fermion-gauge vertex, ignoring theoverall factor difference, there is an additional term of vector current Q f sin θ W relative to the SU (2) case. Nevertheless, this term gives 0 when k µ dots into the S-matrix because the interactionis vector-like. So we conclude Eq. (42) is satisfied for the SM too. Indeed, since the argument isvery general, we expect Eq. (42) applies to any massive gauge theory with Higgs mechanism. W-W-W
Next we proceed to prove Eq. (42) for the amplitude of
W W W . Stripping of the overall factorof − ig , the general amplitude of W W W can be written as i M (1 s s s ) = ( ǫ · ǫ − ǫ · ǫ )[( p − p ) · ǫ ] + cyclic (45)Replacing one of the polarizations are replaced by k M m W , there are three different cases for theother two polarizations: a) two transverse; b) one transverse and one longitudinal; c) two longi-tudinal. We start from case a), since both particle 1 and particle 2 are transverse their goldstonecomponents are 0: ǫ = ǫ = 0. Consequently, there is no goldstone contribution in this case, sowe have i M (1 s = T s = T s ) | ǫ Ms → kM mW = i M (1 s = T s = T s ) | ǫ µs → kµ mW ,ǫ s → (46)This means the on-shell gauge symmetry for 1 T T s is directly analogue to the massless case,with only gauge vertex contributing. i M (1 s = T s = T s ) | ǫ µs → k µ ,ǫ s → = ( ǫ T · ǫ T )[( k − k ) · ( − k − k )] + ǫ T · k [( k − k ) · ǫ T ]+ k · ǫ T [( k − k ) ǫ T ]= 0 − ǫ T · k k · ǫ T + 2 k · ǫ T k · ǫ T = 0 (47)9In the second step we made use of energy-momentum conservation, on-shell conditions andtransverse conditions for particle 1 and 2 respectively, k + k + k = 0 k − k = m W − m W = 0 (48) k · ǫ T = k · ǫ T = 0Next we turn to case b) with particle 1 being transverse and particle 2 being longitudinal, weget i M (1 s = T s = L s ) | ǫ Ms → kM mW = ǫ T · ǫ n ( k − k ) · k m W + ( ǫ n · k m W −
12 ) ( k − k ) · ǫ T + k · ǫ T m W ( k − k ) · ǫ n = 0 + ǫ n · k m W ( − k − k ) · ǫ T −
12 ( − k − k ) · ǫ T + k · ǫ T m W (2 k + k ) · ǫ n = ǫ n · k k · ǫ T m W + − k · ǫ T k · ǫ n m W − k · ǫ T + k · ǫ T = 0 (49)Again we have used the conditions of all the particles being on-shell. For the longitudinal state ǫ M L the on-shell condition implies k ǫ n = − m W .Finally, we turn to case c) with both particle 1 and 2 being longitudinal polarizations, we have i M (1 s = L s = L s ) | ǫ Ms → kM mW = ( ǫ n · ǫ n + 12 ) ( k − k ) · k m W + ( ǫ n · k m W −
12 ( − i ) · i ) ( k − k ) · ǫ n + ( k · ǫ n m W −
12 ( − i ) · i ) ( k − k ) · ǫ n = 0 + ǫ n · k m W ( − k − k ) · ǫ n −
12 ( − k − k ) · ǫ n + k · ǫ n m W (2 k + k ) · ǫ n −
12 (2 k + k ) · ǫ n = k · ǫ n ( − k · ǫ n m W + k · ǫ n + − m W + k · ǫ n + k · ǫ n k · ǫ n m W − k · ǫ n + 12 m W − k · ǫ n = 0 (50)Thus combining case a), case b) and case c), we have proved that on-shell gauge symmetryis satisfied for all possible 3-point amplitudes: h - W - W , f - f ′ - W , W - W - W . However, our proof0still has two loop holes: the first one is only one W state is replaced by k M m W , the second oneis particles are assumed to be incoming. Here we demonstrate neither of the two assumptionsaffect the conclusion. Starting with the first one, to prove the general case of arbitrary numberof polarizations of W being replaced by k M m W , we need only notice that by replacing the gaugecomponents ǫ µn with k µ , the transverse condition for the longitudinal polarization k ∗ M ǫ LM = 0turns to the on-shell condition for k M , k ∗ M k M = 0 . Since in the proof above, the only conditions we used are on-shell condition for k M and transverseconditions for ǫ Ms , the proof of Eq. (41) is exactly the same for multiple polarization vectors beingreplaced by k M m W .The second loophole is automatically fixed if crossing symmetry is satisfied. Under k → − k ,the incoming longitudinal state becomes ǫ ML ( − k ) = − ǫ ∗ ML ( k ), as ǫ µn ( − k ) = − ǫ µn ( k ). So we getthe longitudinal polarization vector for the outgoing state up to a minus sign. Energy-momentumconservation becomes X i k i + k = 0 → X i k i + ( − k ) = 0Thus we obtain the amplitude for one particle being outgoing if its momentum is under k → − k .So crossing symmetry is indeed satisfied. Moreover, k M /k ∗ M turns to − k ∗ M / − k M under k → − k .Therefore, we finished our proof of on-shell gauge symmetry for all 3-point amplitudes. B. → Splitting Amplitudes
In this section we are demonstrating how to use the new Feynman rules and the on-shell matchcondition from on-shell gauge symmetry to do calculations. Our examples are 1 → W L → W L W L , h → W L W L and f → f ′ W L . Those splitting amplitudes have been calculated in [6]. However, as we will see, with thenew prescriptions the calculations become largely simplified.When external states of a process become collinear with each other, one internal lines of oneof the Feynman diagrams will approach its pole or mass singularity. The amplitude can then befactorised in the following way i M = X s i M ssplit · ik − m · i M s + power suppressed (51)Thus the splitting amplitude M split should be evaluated on-shell. The collinear splitting am-plitudes M split are related to collinear splitting functions in the following way[6], d P dzdk T ∝ |M split | (52)So evaluating collinear splitting functions is reduced to evaluating splitting amplitudes. h → W + L W − L The splitting amplitude for h ( k ) → W + L ( k ) W − L ( k ) can be more conveniently calculated usingthe polarization vector ǫ µL = k µ m W − m W n · k n µ , and evaluating the splitting amplitude by treating allparticles “on-shell”. i M ( h → W + L W − L ) = igm W (cid:18) k µ m W + ǫ µ n (cid:19) (cid:18) k µ m W + ǫ n µ (cid:19) = igm W (cid:18) k − k − k m W + k · ǫ n + k · ǫ n m W + ǫ n ǫ n (cid:19) onshell = igm W (cid:18) m h − m W m W − k · n k · n − k · n k · n + m W n · n ( n · k )( n · k ) (cid:19) = igm W (cid:18) m h − m W m W − ¯ zz − z ¯ z (cid:19) (53)In the third step we made use of on-shell conditions k = m h , k = m W and k = m W ; in thefinal step we choose n = n = n = n , and define energy fraction of k /k to k /k as z = n · k n · k ¯ z = n · k n · k (54)2In the limit of particles 1, 2 and 3 are massless, as well as k , k , k are collinear with eachother, we have ¯ z = 1 − z .After reorganization, we have i M ( h → W + L W − L ) = igm W z ¯ z (cid:18) m h m W z ¯ z − (1 − z ¯ z ) (cid:19) (55) f → f ′ W + L Similar to h → W + L W − L , the splitting amplitude can be evaluated “on-shell” using the polar-ization vector ǫ µL = k µ m W − m W n · k n µ . For the interaction between fermion current and gauge boson isgiven by L = g √ ¯ ψ γ µ P L ψ W µ , we have the splitting amplitude to be i M ( f s → f ′ s W + L ) = i g √ u s L ( k ) γ µ u s L ( k ) · (cid:18) k µ m W + ǫ n µ (cid:19) = i g √ m W ¯ u s L ( k )( /k − /k ) u s L ( k ) − i gm W √ n · k ¯ u L ( k ) /n u L ( k ) onshell = i g √ m W ( m ¯ u s R ( k ) u s L ( k ) − m ¯ u s L ( k ) u s R ( k )) − i g √ m W n · k ¯ u s L ( k ) /n u s L ( k ) (56)In the second line, we made use of equations of motion for the fermions. The first two termsgive the contribution of the goldstone component, as can be seen by the factor gm fi √ m W = y f i , with i = 2 ,
3. To continue the calculation, we need the explicit form of the fermion wave function: u − L ( k ) = u R ( k ) = √ n · kξ u − R ( k ) = u L ( k ) = m √ n · k ξ ¯ u − L ( k ) = ¯ u R ( k ) = √ n · kξ † ¯ u − R ( k ) = ¯ u L ( k ) = m √ n · k ξ † (57)Here n µ = (1 , − | ~k | ~k ).We take s = s = − as the example, at the collinear limit k ≃ k ≃ k , we have n ≃ n ≃ n = n = (1 , , , −
1) with z direction along ~k , and ξ † ξ ≃ ξ † ξ = 1¯ u − R ( k ) u − L ( k ) = m r n · k n · k ξ † ξ = m √ ¯ z ¯ u − L ( k ) u − R ( k ) = m r n · k n · k ξ † ξ = m √ ¯ z (58)3Here we have used the definition of z/ ¯ z as in Eq. (54). We also need,¯ u − L ( k ) /ǫ n u − L ( k ) = − m n · k p ( n · k )( n · k ) ξ † n · σξ = − m √ ¯ zz (59)Here we have used ξ †− n · σξ − = ξ † (1 − ( − ξ = 2Plug Eq. (58) and Eq. (59) into Eq. (56), we get, i M ( f − → f ′− W + L ) = ig √ √ ¯ zz ( m m W z − m m W z ¯ z − m W ¯ z )= i ( y f m √ ¯ z − y f m √ ¯ z − g √ m W √ ¯ zz ) (60)Similarly, for s = s = , we have i M ( f → f ′ W + L ) = ig √ √ ¯ zz ( m m W z ¯ z − m m W z − m W ¯ z )= i ( − y f m √ ¯ z + y f m √ ¯ z − g √ m W √ ¯ zz ) (61)Based on the results above, it’s also straightforwardly to work out the splitting amplitudes if thegauge boson couples to right-handed fermion current, i.e. L = g √ ¯ ψ γ µ P R ψ W µ . For s = s = − ,and s = s = respectively, the splitting amplitudes are i M ( f − → f ′− W + L ) = ig √ √ ¯ zz ( m m W z ¯ z − m m W z − m W ¯ z ) i M ( f → f ′ W + L ) = ig √ √ ¯ zz ( m m W z − m m W z ¯ z − m W ¯ z ) (62)With splitting amplitudes for gauge boson coupling to left-handed current and right-handed cur-rent, we are able to calculate the splitting amplitudes given by the Lagrangian L = g √ ¯ ψ γ µ ( Q L P L + Q R P R ) ψ W µ , with arbitrary Q L and Q R . W + L → W + L W L W + L → W + L W L are k → k k .The splitting amplitude for is given by the cubic vertex for vector bosons, i M ( W + L → W L W + L ) = − ig { [ ǫ n ( k ) · ǫ n ( k ) − i − k + k ) · ǫ n ( k )+ [ ǫ n ( k ) · ǫ n ( k ) − i ( − i )2 ]( − k − k ) · ǫ n ( k )+ [ ǫ n ( k ) · ǫ n ( k ) − i ( − i )2 ]( k + k ) · ǫ n ( k ) } ǫ n i · ǫ n j ∼ m W E θ is suppressed by both the factor of m W E k and θ , so they are negligible. Indeed,the simplest way is to choose n = n = n = n , which corresponds to the conventional light-conegauge. This choice leads to ǫ n i · ǫ n j = 0.The splitting amplitude then becomes i M ( W + L → W L W + L ) = ig m W (cid:20) − ( k − k ) · nn · k + ( k + k ) · nn · k + − ( k + k ) · nn · k (cid:21) (63)We also write m W = gv , plug all in. After organization, and making use of the definition ofenergy fraction z/ ¯ z in Eq. (54), the amplitude finally becomes i M W + L → W + L W L = ig v z − ¯ zz ¯ z (1 + z ¯ z IV. CONCLUSIONS
In this paper we derived the Feynman rules of massive gauge theory in physical gauges. Themodel is θ W → SU (2) L . The main novelty is thatwe treat gauge fields and goldstone fields uniformly as 5-component vector fields: W M = ( W µ , φ ).Making use of the new notation, we derived the propagator for vector bosons. We noticed there is aremarkable similarity between massless gauge theory and massive gauge theory in the algebra level,making the derivation almost trivial. We also derived the Feynman rules for vertices. Especially,we found that gauge-gauge-gauge vertex and goldstone-goldstone-gauge vertex can be combinedinto single W-W-W vertex with a common factor ǫ abc , which is obviously due to the remainingcustodial symmetry in the scalar potential.5We also investigated the structure of 3-point on-shell amplitudes. We demonstrated that all3-point on-shell amplitudes – W-W-W, h-W-W, f - f ′ -W – satisfy on-shell gauge symmetry, whichis a reflection of on-shell gauge symmetry for general S-matrix. This on-shell gauge symmetryensures that amplitudes calculated with the new Feynman rules and with the usual Feynman rulesare equivalent. We call this equivalence on-shell match condition. Finally, making use of the newFeynman rules and on-shell match condition for 3-point amplitudes, we calculated some collinearsplitting amplitudes in massive gauge theory. Acknowledgements
The author thanks the discussions with Tao Han, Brock Tweedie andKaoro Hagiwara.
Appendix A: Feynman Rules
Convention: g ′ MN = g ′ MN = diag( g µν , − / g MN = g MN = diag( g µν , − k M = k µ − im W k ∗ M = k µ im W n M = n µ n = 0 k · k ∗ = g MN k M k ∗ M Propagators W ± −→ N Mk = − ik · k ∗ + iǫ ( g MN − n M k ∗ N + k M n ∗ N n · k − iǫ + ξ k · k ∗ ( n · k ) k M k ∗ N ) W −→ N Mk = − ik · k ∗ + iǫ ( g MN − n M k ∗ N + k M n ∗ N n · k − iǫ + ξ k · k ∗ ( n · k ) k M k ∗ N ) k = i ( /k + m f ) k − m f + iǫhk = ik − m h + iǫ (A1)7 Gauge-goldstone Sector W M − → k W N − ← − k W K + ← − k = − ig (cid:0) g ′ MN ( k − k ) ρ + g ′ NK ( k − k ) µ + g ′ KM ( k − k ) ν (cid:1) W P + − → W Σ − ← − W M + − → W N − ← − = ig (2 g µρ g νσ − g ′ MN g ′ P Σ − g ′ M Σ g ′ NP ) − i ( λ h − g g M g N g P g Σ4 W P − → W Σ0 ← − W M + − → W N − ← − = − ig (2 g ′ MN g ′ P Σ − g ′ MP g ′ N Σ − g ′ M Σ g ′ P N ) − i λ h g M g N g P g Σ4 W P − → W Σ0 ← − W M − → W N ← − =8 Higgs Sector and “VEV” Sector h − → q W M − ← − k W N + ← − k = − g (cid:0) g N ( k µ − q µ ) + g M ( k ν − q ν ) (cid:1) + igm W g µν − i λ h v g M g N h − → q W M ← − k W N ← − k = − g (cid:0) ( k − q ) µ g N + g M ( k − q ) ν (cid:1) + igm W g µν − i λ h v g M g N h − → W Σ+ ← − h − → W N − ← − = − i g g νσ − i λ h g N g Σ4 h − → W Σ0 ← − h − → W N ← − = − i g g νσ − i λ h g N g Σ4 h h h = − i λ h v h hh h = − i λ h Fermion Sector d i u j W M + = ( − i g √ γ µ P L − ( y d P R − y u P L ) g M ) V ij u i d j W M − = ( − i g √ γ µ P L − ( y u P R − y d P L ) g M ) V ∗ ij f fW N = − igγ µ (cid:0) T f P L (cid:1) − y f γ T g M [1] B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. D , 1519 (1977).[2] B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. Lett. , 883 (1977).[3] J. M. Cornwall, D. N. Levin and G. Tiktopoulos, Phys. Rev. D , 1145 (1974) Erratum: [Phys. Rev.D , 972 (1975)].[4] N. Arkani-Hamed, T. Han, M. Mangano and L. T. Wang, Phys. Rept. , 1 (2016)doi:10.1016/j.physrep.2016.07.004 [arXiv:1511.06495 [hep-ph]].[5] M. Mangano, CERN Yellow Report CERN 2017-003-M doi:10.23731/CYRM-2017-003[arXiv:1710.06353 [hep-ph]].[6] J. Chen, T. Han and B. Tweedie, arXiv:1611.00788 [hep-ph].[7] Z. Kunszt and D. E. Soper, Nucl. Phys. B , 253 (1988).[8] P. Borel, R. Franceschini, R. Rattazzi and A. Wulzer, JHEP , 122 (2012) [arXiv:1202.1904 [hep-ph]].[9] W. Beenakker and A. Werthenbach, Nucl. Phys. B , 3 (2002) [hep-ph/0112030].[10] I. Feige and M. D. Schwartz, Phys. Rev. D , no. 10, 105020 (2014) doi:10.1103/PhysRevD.90.105020[arXiv:1403.6472 [hep-ph]].[11] M. S. Chanowitz and M. K. Gaillard, Nucl. Phys. B , 379 (1985). [12] G. J. Gounaris, R. Kogerler and H. Neufeld, Phys. Rev. D , 3257 (1986).[13] J. Bagger and C. Schmidt, Phys. Rev. D , 264 (1990).[14] H. G. J. Veltman, Phys. Rev. D , 2294 (1990).[15] Y. P. Yao and C. P. Yuan, Phys. Rev. D , 2237 (1988).[16] G. ’t Hooft, Nucl. Phys. B , 173 (1971). doi:10.1016/0550-3213(71)90395-6[17] G. ’t Hooft, Nucl. Phys. B , 167 (1971). doi:10.1016/0550-3213(71)90139-8[18] D. Espriu and J. Matias, Phys. Rev. D , 6530 (1995) [arXiv:hep-ph/9501279].[19] A. Wulzer, Nucl. Phys. B , 97 (2014) [arXiv:1309.6055 [hep-ph]].[20] K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, Nucl. Phys. B , 253 (1987).[21] L. J. Dixon, doi:10.5170/CERN-2014-008.31 arXiv:1310.5353 [hep-ph].[22] H. Elvang and Y. t. Huang, arXiv:1308.1697 [hep-th].[23] N. Arkani-Hamed, T. C. Huang and Y. t. Huang, arXiv:1709.04891 [hep-th].[24] S. D. Badger, E. W. N. Glover and V. V. Khoze, JHEP , 066 (2006) doi:10.1088/1126-6708/2006/01/066 [hep-th/0507161].[25] S. D. Badger, E. W. N. Glover, V. V. Khoze and P. Svrcek, JHEP , 025 (2005) doi:10.1088/1126-6708/2005/07/025 [hep-th/0504159].[26] N. Craig, H. Elvang, M. Kiermaier and T. Slatyer, JHEP1112