Vector Boson Scattering in the Standard Model-An Overview Of

Home , Vector boson

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2 Scattering Amplitudes

Calculation of the scattering amplitudes of the gauge bosons by hand is a tedious task. The use of function SpecificPolarization of the Mathematica package FeynCalc [9] has substantially reduced algebraical manipulations. Longitudinal polarization picks up a speciﬁc direction in space. Thus the amplitudes written in terms of Mandelstam variables s,t,u do not describe scattering of longitudinally polarized particles in all Lorentz frames. For the process V (p1)+ V (p2) V (k1)+ V (k2) the function SpecificPolarization uses the following representation of the longitudinal→ polarization vectors ε(p,L)

µ µ ε (p1,L) = FL (p1, k1, k2) µ µ ε (p2,L) = FL (p2, k1, k2) µ µ ε (k1,L) = FL (k1, p1,p2) µ µ ε (k2,L) = FL (k2, p1,p2) (1) where rµ(b r + a r) (a + b)µr2 F µ(r, a,b)= · · − (2) L r2[(b r + a r)2 r2(b + a)2] · · − Using this function, we get the longitudinalp polarization only in the CM system, where p1 + p2 = k1 + k2 = 0. Table 1 summarizes tree-level contributions (contact graphs are not listed) to the scattering amplitudes of the gauge bosons in the SM in the U-gauge. Contributions to the process #n from

process # process s t u references − 1 W +W Z Z H W W [4, 5] 1 2 → 3 4 2 W +Z Z W + W W H [5] 1 2 → 3 4 3 W +W + W +W + Z,γ,H Z,γ,H [6] 1 2 → 3 4 − − 4 W +W W +W Z,γ,H Z,γ,H [7] 1 2 → 3 4

Table 1: Individual processes together with exchanged particles in diﬀerent channels (s,t,u). W,Z,γ are electroweak gauge bosons and photon and H is the SM Higgs boson. Contact graphs contribute to each of the process and are not listed. graph with V -boson exchange in the k-channel are denoted as g g (n) = 1 2 A(n) (3) MkV −k m2 k − V and from the contact graph (n) = g A(n) (4) Mc c

2 with g1,g2 and g denoting relevant coupling constants. Amplitudes given by the low-energy the- (n) orems are denoted as LET . Note that in the paper of Bento and Llewellyn Smith [5] the WZ WZ amplitudeM is claimed to be (in the current notation) → (2) = (2) + (2) + (2) + (2) (wrong) M Mc MuH MuW MtW and is also calculated in this way. This is wrong, because W is exchanged in s and t (or u) channel2. Explicit formulae for the scattering amplitudes are somewhat difficult to read and are postponed to the appendix A. Table 2 summarizes relations among different A parts of gauge amplitudes as defined in (3) and (4). These relation are almost obvious, nevertheless they can be used as a check. (1) (2) (2) (1) Besides these relations we have AtW = AtW but AtW = AtW . Note that the mZ =mW − mZ =mW 6 −

process # Gauge boson exchange Contact graph

(1) (1) (1) 1 A (cos θcm)= A ( cos θcm) Ac tW uW − (2) (2) (2) 2 AsW , AtW Ac

(3) (1) (3) (1) 3 A = A Ac = Ac (t,u)(γ,Z) − (t,u)W mZ =mW mZ =mW (4) (2) (4) (2) 4 A = A Ac = Ac (s,t)(γ,Z) − (s,t)W mZ =mW mZ =mW

Table 2: Relations among the A parts of the scattering amplitudes.

(n) (n) α β Aγ = AZ because the longitudinal term in propagator ( q q ) does not contribute to the amplitudes with exchange of neutral gauge boson. ∼

2.1 W +(k )+ W −(k ) Z(k )+ Z(k ) 1 2 → 3 4 On the tree-level the pure gauge contributions are from W -exchange in t and u channels and the contact graph 2 2 g cos θW (1) = − A(1) k = t,u (5) MkW k m2 kW − W (1) 2 2 (1) = g cos θW A (6) Mc − c Complete gauge amplitude grows linearly with s

(1) = (1) + (1) + (1) Mgauge MtW MuW Mc (1 x2) 2cos2 θ ρ(1 + x2) m2 (1) g2 W O W = LET − 2 − 2 2 + (7) M − 2ρ cos θW (1 x ) s − where g2 s m2 (1) , ρ W x θ . LET = 2 = 2 2 and cos cm M 4 ρmW mZ cos θW ≡ with θcm an angle between k1 and k3 in CMS. In the SM this growth is canceled by exchange of the Higgs boson in the s-channel

2 ∗ ∗ (1) g mW mZ (ε1 ε2)(ε3 ε4) sH = · 2 · (8) M − cos θW s m + imH ΓH − H (1) The amplitude sH has, for longitudinal polarizations, the high-energy (E mH ,mW ) expansion (for exact formulaeM see appendix) ≫

g2m m s g2√ρ (1) = W Z + O(s0) = s + O(s0) MsH − cos θ 4m2 m2 − 4m2 W Z W W 2There is the interchange t ↔ u in [5] in comparison with this paper.

3 25

20 W+W- → Z Z 15

5 (1) M 0

-5 mH = 500 GeV

mH = 1500 GeV -10 (1) Mgauge M (1) -15 LET

200 400 600 800 1000 √s [GeV]

Figure 1: Tree-level amplitudes of the process WW ZZ as a function of √s for diﬀerent values of → 2 mH . The Higgs boson width, ΓH , is set to zero, cos θcm = 0.5, ρ = 1, mZ = 91.2 GeV, sin θW = 0.231, 2 4παem . g = 2 = 0.42. sin θW so the cancellation occurs for ρ = 1. (1) (1) (1) Figure 1 shows dependence of the complete amplitude = gauge + sH on the Higgs M M (1) M (1) boson mass, mH , in the region mW < √s< 1TeV and compares it with gauge and LET . Note (1) (1) M M that gauge diﬀers from LET in the limit s by a constant term. Numerical values of all parametersM are the same throughoutM the paper.→ ∞

2.2 W +(k )+ Z(k ) Z(k )+ W +(k ) 1 2 → 3 4 On the tree-level the pure gauge contributions are from W -exchange in s and t channels and the contact graph 2 2 g cos θW (2) = − A(2) k = s,t (9) MkW k m2 kW − W (2) 2 2 (2) = g cos θW A (10) Mc − c The asymptotic behaviour of the complete tree-level gauge amplitude is

(2) = (2) + (2) + (2) Mgauge MsW MtW Mc g2(2x(1 x)+ ρ cos2 θ (3+2x x2)) m2 (2) W O W = LET − 2 2 − + (11) M − 4ρ cos θW (1 x) s − where a low-energy amplitude is usually deﬁned as

2 2 (2) g u g s 0 LET = 2 = 2 (1 + cos θcm)+ O(s ) . M 4ρmW −8ρmW The exchange of the Higgs boson in the u-channel ∗ ∗ (2) 2 mW mZ (ε1 ε4) (ε2 ε3) uH = g · 2 · (12) M − cos θW u M − H has for longitudinal polarizations the high-energy expansion 2 2 (2) g √ρ 0 g √ρ 0 uH = 2 (1 + cos θcm) s + O(s )= 2 u + O(s ) . M 8mW − 4mW

4 0 W+ Z → Z W+ -2

-4 (2) -6 M

mH = 500 GeV m = 1500 GeV -8 H (2) Mgauge M (2) -10 LET

-12 200 400 600 800 1000 √s [GeV]

Figure 2: Tree-level amplitudes of the process W Z W Z. →

(2) (2) Figure 2 shows complete tree-level amplitude for diﬀerent mH and compares it with gauge M M and (2) . MLET 2.3 W +(k )+ W +(k ) W +(k )+ W +(k ) 1 2 → 3 4 On the tree-level graphs with γ and Z exchange in t and u channels and contact graph contribute. In the Standard Model, Higgs boson exchange in t and u channels gives the desired high energy behaviour. 2 2 2 g cos θW e (3) = − A(3) (3) = − A(3) k = t, u . MkZ k m2 kZ Mkγ k kγ − Z Because longitudinal term in the Z boson propagator does not contribute we have

(3) (3) Akγ = AkZ .

Contact graph amplitude for longitudinally polarized gauge bosons has the form

2 (3) 2 (3) g s 2 2 c = g Ac = 4 ( 8 mW +3 s s cos θcm) (13) M 8 mW − − The gauge amplitude can be written as

(3) (3) (3) (3) (3) 2 2 AtZ AuZ 2 2 Atγ Auγ (3) = g cos θW + g sin θW + + (14) Mgauge − t m2 u m2 − t u Mc " − Z − Z # " # Expanding this expression in powers of s gives

2 2 (3) 2 2 3 cos θcm 2 3mZ 0 = g cos θW − s s + O(s ) Mgauge − 8m4 − 4m4 W W 2 2 2 3 cos θcm 2 0 g sin θW − s + O(s ) − 8m4 W 2 3 cos θcm s + g2 − s2 (15) 8m4 − m2 W W

5 0

W+ W+ → W+ W+ -2

-4

(3) -6 M

-8

m = 500 GeV -10 H

mH = 1500 GeV M (3) -12 gauge (3) MLET -14 200 400 600 800 1000 √s [GeV]

Figure 3: Tree-level amplitudes of the process W +W + W +W +. →

Quadratic (in s) divergencies are canceled and including constant terms of the order O(s0) we get

2 2 2 2 2 3+ x ρ cos θW (6 4 ρ + 10x 12 ρ x m (3) = (3) g2 − − − + O W (16) gauge LET 2 2 2 2 M M − 2 ρ (1 x ) cos θW s − where g2 s 3 (3) = 4 . (17) MLET −4m2 − ρ W This linear divergence should be canceled by Higgs exchange in t and u channels (ε ε∗)(ε ε∗) (ε ε∗)(ε ε∗) (3) = g2m2 1 · 3 2 · 4 + 1 · 4 2 · 3 (18) MH − W t m2 u m2 − H − H with high-energy expansion 2 (3) g s 0 H = 2 + O(s ) . (19) M 4mW Comparing (17) and (19) we see that ρ = 1 ensures desired cancellation.

2.4 W +(k )+ W −(k ) W +(k )+ W −(k ) 1 2 → 3 4 In this case we have to consider Z, γ and Higgs boson exchange in s and t channels and contact graph. As in the previous sections we write

2 2 2 g cos θW e (4) = − A(4) (4) = − A(4) k = s,t MkZ k m2 kZ Mkγ k kγ − Z The results for the A parts of the amplitude can be obtained directly from corresponding formulae + + for the process W Z W Z by setting mZ = mW → A(4) = A(2) (s,t)(Z,γ) − (s,t)W mZ =mW (4) (3) (4) (4) or we can also notice that AtZ = AtZ . Again we have AkZ = Akγ . In the case of longitudinally polarized gauge bosons −

2 (4) 2 (4) g s 2 2 2 c = g Ac = 4 (8 mW 3 s 24 mW x +6 s x + s x ) (20) M 16 mW − −

6 25

20 + - → + - 15 W W W W

5 (4)

M 0

-5

-10 mH = 500 GeV

mH = 1500 GeV -15 (4) Mgauge -20 (4) MLET -25 200 400 600 800 1000 √s [GeV]

− − Figure 4: Tree-level amplitudes of the process W +W W +W . →

Let us examine high-energy expansion of gauge amplitude

(4) (4) (4) (4) (4) 2 2 AsZ AtZ 2 2 Asγ Atγ (4) = g cos θW + g sin θW + + . (21) Mgauge − s m2 t m2 − s t Mc " − Z − Z # " #

2 2 2 2 2 (4) 2 2 x 2 mZ x x +2x 3 2 3mZ 16mW x + mZ x = g cos θW s + s + − s + − s Mgauge − 4m4 4m4 16m4 8m4 W W W W 2 2 2 x 2 x +2x 3 2 2x 0 g sin θW s + − s s + O(s ) − 4m4 16m4 − m2 W W W x2 +6x 3 1 3x + g2 − s2 + − s (22) 16m4 2m2 W W After simpliﬁcation and including terms of order O(s0) we get

(4) = (4) Mgauge MLET (3 + x2 ρ cos2 θ (12 12 ρ + 12 x 16 ρ x 8 x2 + 12 ρ x2)) g2 W + − −2 2− − 4 ρ (1 x) cos θW − m2 + O W s where as in the case of the process #1 I denote g2u 3 (4) = 4 MLET −4m2 − ρ W in accordance with [8]. High-energy expansion of the Higgs boson contribution (ε ε )(ε∗ ε∗) (ε ε∗)(ε∗ ε∗) (4) = g2M 2 1 · 2 3 · 4 + 1 · 3 2 · 4 (23) MH − W s m2 t m2 − H − H is 2 2 (4) g s 0 g u 0 H = 2 (1 + cos θcm)+ O(s )= 2 + O(s ) M −8mW 4mW

7 A Exact formulae

The following Feynman rules and notation is used (all momenta are outgoing) [10]

− + γ(Z),ρ W , σ W ,ν

q ✻ ■ ✒ ♣ ♣ 2 ie(ig cos θW )Vνµρ(k,p,q) ig Vµνλσ ✠ k ❘ p ✠ ❘

− W +, µ W ,ν W +, µ W −, λ

Z,σ Z,ν H

♣ ♣ 2 ig cos θW Vνσµλ igmW gµν −

+ − W +, µ W −, λ W , µ W ,ν

♣ ig mZ g cos θW µν

Z,µ Z,ν

where Vλµν (k,p,q) = (k p)ν gλµ + (p q)λ gµν + (q k)µ gλν , k + p + q = 0 (24) − − − Vµνρσ =2gµν gρσ gµρgνσ gνρgµσ . − − Propagators of the gauge bosons are

αβ qαqβ g + 2 αβ αβ αβ m P (q) g Pγ Dαβ(q)= − V V ,Dαβ = − . V q2 m2 ≡ q2 m2 γ q2 ≡ q2 − V − V Mandelstam variables are deﬁned as

2 2 s = (k1 + k2) = (k3 + k4) t = (k k )2 = (k k )2 1 − 3 4 − 2 u = (k k )2 = (k k )2 1 − 4 3 − 2 The expression containing polarization vectors has in all amplitudes the form

∗ ∗ µνρσ = εµ εν ε ρ ε σ E 1 2 3 4

where εi = ε(ki). I use the abbreviation x = cos θcm, where θcm is the angle between k1 a k3 in CMS.

A.1 W +(k )+ W −(k ) Z(k )+ Z(k ) 1 2 → 3 4

(1) (1) (1) 2 2 AtW AuW (1) (1) = g cos θW + + A + M − t m2 u m2 c MsH " − W − W #

8 (1) αβ µνρσ A = Vµαρ( k ,q,k ) Vβνσ( q, k , k )P (q) tW − 1 3 − − 2 4 W E (1) αβ µνρσ A = Vµασ ( k ,q,k ) Vβνρ( q, k , k )P (q) uW − 1 4 − − 2 3 W E (1) µνρσ A = Vρσµν c E After contraction (1) ∗ ∗ ∗ ∗ AtW = 4 (k1 ε3) (k2 ε4) (ε1 ε2)+4(k1 ε3) (k4 ε2) (ε1 ε4) · ∗ · · ∗ · · ∗ · ∗ + 4 (k2 ε4) (k3 ε1) (ε2 ε3)+4(k3 ε1) (k4 ε2) (ε3 ε4) · · · ∗ ∗ · · ∗ · ∗ 2 (k4 ε2) ((k1 + k3) ε4) (ε1 ε3) 2 (k2 ε4) ((k1 + k3) ε2) (ε1 ε3) − · ∗ · · ∗ − · · ∗ · ∗ 2 (k1 ε3) ((k2 + k4) ε1) (ε2 ε4) 2 (k3 ε1) ((k2 + k4) ε3) (ε2 ε4) − 2 · 2 2 · · − · · · (mW mZ ) ∗ ∗ ∗ ∗ + −2 (ε1 ε3) (ε2 ε4) + (s u) (ε1 ε3) (ε2 ε4). (25) mW · · − · ·

A(1) = A(1) (3 4,u t) uW tW ↔ → A(1) = 2 (ε ε ) (ε∗ ε∗) (ε ε∗) (ε ε∗) (ε ε∗) (ε ε∗) c 1 · 2 3 · 4 − 1 · 4 2 · 3 − 1 · 3 2 · 4 Replacing all polarization vectors εi by εi(L) given in (1) we get amplitudes for longitudinally polarized gauge bosons in CMS

(1) 1 4 4 2 6 4 2 2 4 AtW (s, x) = 4 2 [ 96 mW mZ + 32 mW mZ +8 mW mZ s + 16 mW mZ s 32 mW mZ − 6 4 2 2 2 2 4 2 2 3 8 mZ s 4 mW s 10 mW mZ s +2 mZ s +3 mW s − 4 − 2 − 4 2 2 2 2 + 16 mW mZ βW βZ s x + 12 mW βW βZ s x + 24 mW mZ βW βZ s x 4 2 2 3 6 2 4 mZ βW βZ s x 5 mW βW βZ s x + 32 mW s x − 4 2 2 − 2 4 2 4 2 2 + 96 mW mZ s x + 32 mW mZ s x 16 mW s x 2 2 2 2 4 2 2 −2 3 2 2 3 3 22 m m s x +2 m s x + m s x + m βW βZ s x ] (26) − W Z Z W W A(1) (s, x)= A(1) (s, x) uW tW − and 2 2 2 (1) s ( 4 mW 4 mZ +3 s s x ) Ac = − − 2 2 − 8 mW mZ Kinematical variables are related by

2 2 s s t = m + m + βW βZ cos θcm W Z − 2 2 2 2 s s u = m + m βW βZ cos θcm W Z − 2 − 2 where 2 2 4mW 4mZ βW = 1 βZ = 1 r − s r − s 2 2 2 (1) (1) g mZ cos θW C gauge = 4 (1) M 4 mW J

C(1) = 96 m4 m2 32 m2 m4 48 m4 s +8 m2 m2 s W Z − W Z − W W Z + 8 m4 s +4 m2 s2 6 m2 s2 + s3 + 128 m6 x2 + 32 m4 s x2 Z W − Z W W 64 m2 m2 s x2 +8 m2 s2 x2 +6 m2 s2 x2 s3 x2 − W Z W Z −

J (1) = 4 m4 4 m2 s + s2 16 m2 m2 x2 Z − Z − W Z + 4 m2 s x2 +4 m2 s x2 s2 x2 W Z − For the Higgs boson amplitude (8) we have

2 2 2 (1) g mW mZ (2mW s) (2mZ s) sH = 2 2 − 2 − M − cos θW 4m m (s m + imH ΓH ) W Z − H

9 A.2 W +(k )+ Z(k ) Z(k )+ W +(k ) 1 2 → 3 4

(2) (2) (2) 2 2 AsW AtW (2) (2) = g cos θW + + A + M − s m2 t m2 c MuH " − W − W #

(2) αβ µνρσ A = Vµαν ( k , q, k ) Vβσρ( q, k , k )P (q) sW − 1 − 2 − 4 3 W E (2) αβ µνρσ A = Vµαρ( k ,q,k ) Vβσν ( q, k , k )P (q) tW − 1 3 − 4 − 2 W E (2) µνρσ A = Vµσνρ c E After contraction

(2) ∗ ∗ ∗ ∗ AsW = 4 (k1 ε2) (k3 ε4) (ε1 ε3) 4 (k1 ε2) (k4 ε3) (ε1 ε4) · · ∗ · ∗ − · · ∗ · ∗ 4 (k2 ε1) (k3 ε4) (ε2 ε3)+4(k2 ε1) (k4 ε3) (ε2 ε4) − · ∗ · · ∗ · · ∗ · ∗ 2 (k3 ε4) (ε1 ε2) ((k1 k2) ε3)+2(k4 ε3) (ε1 ε2) ((k1 k2) ε4) − · ∗ · ∗ − · · ∗ · ∗ − · 2 (k1 ε2) (ε3 ε4) ((k3 k4) ε1)+2(k2 ε1) (ε3 ε4) ((k3 k4) ε2) − 2 · 2 2· − · · · − · (mW mZ ) ∗ ∗ ∗ ∗ + −2 (ε1 ε2) (ε3 ε4) + (u t) (ε1 ε2) (ε3 ε4) (27) mW · · − · ·

∗ A(2) = A(2) (k k ,ε ε ) tW − sW 2 ↔− 3 2 ↔ 3

A(2) = 2 (ε ε∗) (ε ε∗) (ε ε∗) (ε ε∗) (ε ε ) (ε∗ ε∗) c 1 · 4 2 · 3 − 1 · 3 2 · 4 − 1 · 2 3 · 4 2 2 2 2 2 s (mZ mW ) 2 t = m + m + − +2 k cos θcm W Z − 2 2s 2 u = 2k (1 + cos θcm) − where in CMS

2 1 2 2 2 2 2 2 2 k = s + (m m ) 2s (m + m ) = ki i =1, 2, 3, 4 4s W − Z − W Z | | g2 m2 cos θ 2 C(2) (2) = Z W Mgauge 8 m4 s (s m2 ) J (2) W − W

C(2) = 3 m10 12 m8 m2 + 18 m6 m4 12 m4 m6 +3 m2 m8 W − W Z W Z − W Z W Z + 17 m8 s 32 m6 m2 s + 10 m4 m4 s +8 m2 m6 s W − W Z W Z W Z 3 m8 s + 26 m6 s2 + 32 m4 m2 s2 30 m2 m4 s2 − Z W W Z − W Z + 4 m6 s2 50 m4 s3 + 16 m2 m2 s3 +2 m4 s3 +3 m2 s4 Z − W W Z Z W 4 m2 s4 + s5 +6 m10 x 24 m8 m2 x + 36 m6 m4 x − Z W − W Z W Z 24 m4 m6 x +6 m2 m8 x 16 m8 s x + 36 m6 m2 s x − W Z W Z − W W Z 28 m4 m4 s x + 12 m2 m6 s x 4 m8 s x + 20 m6 s2 x − W Z W Z − Z W 44 m4 m2 s2 x 20 m2 m4 s2 x + 12 m6 s2 x 16 m4 s3 x − W Z − W Z Z − W 4 m2 m2 s3 x 12 m4 s3 x +6 m2 s4 x +4 m2 s4 x +3 m10 x2 − W Z − Z W Z W 12 m8 m2 x2 + 18 m6 m4 x2 12 m4 m6 x2 +3 m2 m8 x2 − W Z W Z − W Z W Z 33 m8 s x2 + 68 m6 m2 s x2 38 m4 m4 s x2 +4 m2 m6 s x2 − W W Z − W Z W Z m8 s x2 + 122 m6 s2 x2 + 12 m4 m2 s2 x2 6 m2 m4 s2 x2 − Z W W Z − W Z + 34 m4 s3 x2 4 m2 m2 s3 x2 +2 m4 s3 x2 +3 m2 s4 x2 s5 x2 W − W Z Z W −

J (2) = m4 2 m2 m2 + m4 +2 m2 s s2 + m4 x 2 m2 m2 x W − W Z Z Z − W − W Z + m4 x 2 m2 s x 2 m2 s x + s2 x Z − W − Z

10 Higgs boson amplitude (12)

2 (2) (2) g CuH uH = (2) M 8 mW mZ cos θW s JuH

C(2) = m8 4 m6 m2 +6 m4 m4 4 m2 m6 + m8 4 m6 s +4 m4 m2 s uH W − W Z W Z − W Z Z − W W Z + 4 m2 m4 s 4 m6 s +6 m4 s2 +4 m2 m2 s2 +6 m4 s2 4 m2 s3 W Z − Z W W Z Z − W 4 m2 s3 + s4 +2 m8 x 8 m6 m2 x + 12 m4 m4 x − Z W − W Z W Z 8 m2 m6 x +2 m8 x 4 m6 s x +4 m4 m2 s x +4 m2 m4 s x 4 m6 s x − W Z Z − W W Z W Z − Z + 4 m4 s2 x 8 m2 m2 s2 x +4 m4 s2 x W − W Z Z 4 m2 s3 x 4 m2 s3 x +2 s4 x + m8 x2 4 m6 m2 x2 +6 m4 m4 x2 − W − Z W − W Z W Z 4 m2 m6 x2 + m8 x2 2 m4 s2 x2 +4 m2 m2 s2 x2 2 m4 s2 x2 + s4 x2 − W Z Z − W W Z − Z

J (2) = m4 2 m2 m2 + m4 +2 m2 s 2 m2 s 2 m2 s + s2 + m4 x uH W − W Z Z H − W − Z W 2 m2 m2 x + m4 x 2 m2 s x 2 m2 s x + s2 x − W Z Z − W − Z A.3 W +(k )+ W +(k ) W +(k )+ W +(k ) 1 2 → 3 4

(3) (3) (3) (3) (3) 2 2 AtZ AuZ 2 2 Atγ Auγ 2 (3) (3) (3) = g cos θW + g sin θW + + g A + + . M − t m2 u m2 − t u c MtH MuH " − Z − Z # " #

(3) αβ µνρσ A = Vµρα( k , k , q) Vνσβ ( k , k , q, )P (q) tZ,γ − 1 3 − 2 4 − Z,γ E (3) αβ µνρσ A = Vµσα( k , k , q) Vνρβ ( k , k , q)P (q) uZ,γ − 1 4 − 2 3 − Z,γ E

(3) µνρσ A = Vµνρσ . c E Kinematical variables are given by s s t =2 m2 (1 x) u =2 m2 (1 + x) W − 4 − W − 4

A(3) = 2 (k ε∗) ((k ε ) + (k ε )) (ε ε∗) 4 (k ε∗) (k ε∗) (ε ε ) tZ 2 · 4 1 · 2 3 · 2 1 · 3 − 1 · 3 2 · 4 1 · 2 + 2 ((k ε∗) + (k ε∗)) (k ε ) (ε ε∗) 4 (k ε∗) (k ε ) (ε ε∗) 1 · 4 3 · 4 4 · 2 1 · 3 − 1 · 3 4 · 2 1 · 4 + 2 (k ε∗) ((k ε ) + (k ε )) (ε ε∗) 4 (k ε∗) (k ε ) (ε ε∗) 1 · 3 2 · 1 4 · 1 2 · 4 − 2 · 4 3 · 1 2 · 3 + 2 (k ε ) ((k ε∗) + (k ε∗)) (ε ε∗) 3 · 1 2 · 3 4 · 3 2 · 4 (s u) (ε ε∗) (ε ε∗) 4 (k ε ) (k ε ) (ε∗ ε∗) . (28) − − 1 · 3 2 · 4 − 3 · 1 4 · 2 3 · 4 A(3) = A(3)(3 4) uZ tZ ↔ (3) (3) A(t,u)γ = A(t,u)Z For relations among diﬀerent As see table 2 and below.

(3) 1 6 4 2 2 3 6 AtZ (s, x)= 4 ( 64 mW 16 mW s + 12 mW s 3 s + 64 mW x (32 mW ) − − + 112 m4 s x 52 m2 s2 x +5 s3 x 160 m4 s x2 W − W − W + 36 m2 s2 x2 s3 x2 +4 m2 s2 x3 s3 x3) (29) W − W − A(3) (s, x)= A(3)(s, x) uZ tZ −

11 2 2 2 2 2 (3) 2 2 (3) g s 8 m +3 s s x g cos θ C g sin θ Cγ (3) = − W − + W Z W Mgauge 8 m4 8 m4 (3) − 8 m4 (3) W W JZ W Jγ

C(3) = 256 m8 128 m6 m2 128 m6 s + 32 m4 m2 s + 64 m4 s2 Z W − W Z − W W Z W 24 m2 m2 s2 24 m2 s3 +6 m2 s3 +3 s4 + 256 m8 x2 − W Z − W Z W 256 m6 s x2 + 320 m4 m2 s x2 16 m4 s2 x2 − W W Z − W 72 m2 m2 s2 x2 + 32 m2 s3 x2 +2 m2 s3 x2 − W Z W Z 4 s4 x2 + 16 m4 s2 x4 8 m2 s3 x4 + s4 x4 − W − W J (3) = 4 m2 2 m2 s 4 m2 x + s x 4 m2 +2 m2 + s 4 m2 x + s x Z W − Z − − W − W Z − W C(3) = 64 m6 16 m4 s + 12 m2 s2 3 s3 + 64 m6 x2 γ W − W W − W 48 m4 s x2 16 m2 s2 x2 +4 s3 x2 +4 m2 s2 x4 s3 x4 − W − W W − J (3) = s 4 m2 x2 1 γ − W − Higgs boson amplitude (18) C(3) (3) = g2 H MH (3) JH

C(3) = 32 m2 m4 64 m6 16 m2 m2 s + 48 m4 s H H W − W − H W W + 2 m2 s2 12 m2 s2 + s3 32 m4 s x2 +2 m2 s2 x2 H − W − W H + 12 m2 s2 x2 s3 x2 (30) W − J (3) =4 m2 2 m2 4 m2 + s +4 m2 x s x 2 m2 4 m2 + s 4 m2 x + s x H W H − W W − H − W − W A.4 W +(k )+ W −(k ) W +(k )+ W −(k ) 1 2 → 3 4

(4) (4) (4) (4) (4) 2 2 AsZ AtZ 2 2 Asγ Atγ 2 (4) (4) (4) = g cos θW + g sin θW + + g A + + . M − s m2 t m2 − s t c MsH MtH " − Z − Z # " #

(4) αβ µνρσ A = Vµνα( k , k , q) Vσρβ (k , k , q, )P (q) sZ,γ − 1 − 2 4 3 − Z,γ E (4) αβ µνρσ A = Vµρα( k , k , q) Vσνβ (k , k , q)P (q) tZ,γ − 1 3 4 − 2 − Z,γ E (4) 2 µνρσ A = g Vµσρν . c E

g2 s 8 m2 3 s 24 m2 x +6 s x + s x2 (4) = W − − W gauge m4 M 16 W (4) (4) (4) (4) 2 2 CtZ 2 2 Ctγ + + g cos θW g sin θW MsZ Msγ − (4) − (4) JtZ Jtγ

2 2 2 (4) 2 2 4 mW s 2 mW + s x = g cos θW − MsZ − 4 m4 (m2 s) W Z − 2 2 (4) 2 2 4 mW x s x = g sin θW 3 x + Msγ − − − s 4 m4 W

C(4) = 64 m6 + 16 m4 s 12 m2 s2 +3 s3 64 m6 x 112 m4 s x + 52 m2 s2 x tZ − W W − W − W − W W 5 s3 x + 160 m4 s x2 36 m2 s2 x2 + s3 x2 4 m2 s2 x3 + s3 x3 − W − W − W

12 J (4) = 16 m4 4 m2 2 m2 s 4 m2 x + s x tZ W W − Z − − W (4) (4) (4) (4) C = C J = J m tγ tZ tγ tZ | Z =0 Higgs boson amplitude (23) C(4) (4) = g2 H MH (4) JH

C(4) = 32 m2 m4 32 m6 24 m2 m2 s + 24 m4 s H H W − W − H W W + 5 m2 s2 8 m2 s2 + s3 + 32 m6 x +8 m2 m2 s x H − W W H W 40 m4 s x 2 m2 s2 x +8 m2 s2 x + m2 s2 x2 s3 x2 − W − H W H − J (4) =8 m2 m2 s 2 m2 4 m2 + s +4 m2 x s x H W H − H − W W −

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