Anomalous Triple Gauge Couplings

Chapter 2 Theory

2.10 Anomalous Triple Gauge Couplings

The gauge boson self-interactions are a direct consequence of the non-Abelian struc- ture of the electroweak sector of the SM. Elements of a non-Abelian group do not commute. This causes the structure constants f^abc in eq. (2.5) to acquire non-zero values. The last term in eq. (2.6) then leads to non-vanishing terms involving prod- ucts of the gauge field and its derivatives in the Lagrangian. However, Zγγ and ZZγ couplings vanish in the SM at the tree level as discussed in Section 2.6.6.

Such anomalous triple-gauge couplings can be described by an effective theory that includes higher-dimension operators [37–39]. Figure 2.5 shows a Feynman diagram of a resulting generic VZγ ATGC vertex with an off-shell V =γ^∗, Z^∗ and on-shell Z and γ.

Vµ(p)

γβ(q2) Z_α(q1)

=ieΓ^αβµ_{ZV γ}(p, q1, q2)

Figure 2.5: Feynman diagram of a generic VZγ ATGC vertex with an off-shell V = γ^∗, Z^∗ and on-shell Z and γ.

Using the notation from Figure 2.5, the corresponding ZZγATGC vertex function reads [37, 38]:

Γ^αβµ_ZZγ(p, q1, q2) = p² −q₁² m²_Z

"

h^Z₁(q^µ₂g^αβ−q₂^αg^µβ)+

h^Z₂ m²_Zp^α

(p·q2)g^µβ −q₂^µp^β + h^Z₃^µαβρq2ρ+

h^Z₄

m²_Zp^αµβρσpρq2σ

# ,

(2.77)

where h^Z₁ through h^Z₄ are complex couplings characterizing the ZZ^∗γ interaction.

The Zγγ ATGC vertex function is related to the ZZγ one through the symmetry:

p²−q²₁

m²_Z → p²

m²_Z and

h^Z_i →h^γ_i, i= 1, . . . , 4,

where h^γ₁ through h^γ₄ are, again, some complex couplings characterizing the Zγ^∗γ interaction. This Zγγ vertex function vanishes identically when both photons are on the mass shell due to the Landau-Yang theorem [40, 41].

These are the most generic vertex functions for a time-like virtualV = Z^∗, γ^∗ and on-shell Z andγ constrained by:

• Lorentz invariance,

• gauge symmetry,

• the coupling of the off-shell V to essentially massless fermions, which implies that effectively ∂µV^µ= 0, V =Z, A, and

• the omission of terms proportional to pµ and q^α₁ that do not contribute to the cross section.

Using similar constraints, further anomalous couplings can be introduced. They

may involve off-shell photons and Z⁰ bosons [37], W^± bosons [37], or quartic gauge boson vertices [42–49]. Further details on these other couplings would go beyond the scope of this work.

All of the h^V_i , i = 1, . . . , 4 couplings vanish in the SM at leading order, see Section 2.6.6. The i = 1, 2 couplings are P-even, the other two are CP-even. All of the couplings are C-odd. h^V_1,3 receive contributions from operators of dimension ≤6, h^V_2,4 receive contributions from operators of dimension ≥8.

Increasing the timelike-virtual-boson energy √ ˆ

s, the anomalous contributions to the Zγ helicity amplitudes grow like√

ˆ s/mZ

3

forh^V_1,3, and√ ˆ s/mz

5

forh^V_2,4. Tree- level unitarity requires that the anomalous couplingsh^V_i vanish at asymptotically high energies [50–54] since otherwise the Zγ production cross section would grow without bound. The unitarity can be restored by introducing ˆs-dependent form factors [38]:

h^V_i = h^V_i0 1 + _Λ^sˆ2

n, i= 1, . . . , 4, V = Z, γ, (2.78)

whereh^V_i0 = limˆs→0h^V_i are the low energy limits ofh^V_i , Λ mZ is a cut-off scale, and n > 3/2 for h^V_1,3 and n > 5/2 for h^V_2,4. This functional form is not uniquely defined by the unitarity requirement. This particular choice is motivated by the well-known nuclear form factors [55]. It has the feature that the h^V_i are essentially constant at low energies ˆsΛ² and Λ sets the scale at which they start to decrease with growing ˆ

s. This is consistent with a scenario in which non-SM values of h^V_i are a low-energy consequence of some new physics at the mass scale Λ.

The goal of the this work is to constrain, or measure if nonzero, the values of h^V_i0. In order to perform the measurement, we must adopt a particular choice of Λ and n.

We choose to extract the bare couplings without a form factor:

Λ→ ∞, h^V_i →h^V_i0.

(2.79)

This has the advantage that we avoid ad-hoc assumptions about the functional form (2.78), the values of the cut-off Λ, and exponents n. It simplifies the inter-

pretation of the measurements, facilitates their comparison with other results, and avoids possible energy-dependent biases. We argue that we need not worry about the unitarity violation since we assume that we probe energies ˆsΛ². This assumption is based on the lack of an observation of novel interactions at the scale ˆs that we probe.

In full generality, the differential cross section is a real bilinear form in the anoma- lous couplings h^γ,Z_1,...,4:

dσ = dσSM+

{γ,Z}X

V

X4 i=1

<h^V_i dσ_i,<^V +=h^V_i dσ_i,=^V

+

{γ,Z}X

U,V

X4 i,j=1

|h^U_i h^V_j | cos(argh^U_i −argh^V_j + ∆φ^{U V}_ij ) dσ_ij^{U V},

(2.80)

where dσSM is the SM contribution dσ_i,<^V , dσ_i,=^V , and dσ^{U V}_ij are the non-SM contribu- tions after factoring out theirh^V_i dependences, and ∆φ^{U V}_ij are some phases depending on the relative phases of the underlying interfering matrix elements.

Without loss of generality, we are free to redefine the couplings h^γ,Z_1,...,4with some constant phase shifts h^V_i → e^iφVi h^V_i such that most of the phases ∆φ^{U V}_ij in (2.80) vanish. Moreover, the form of (2.80) is such that all of these phases vanish:

∆φ^{U V}_ij = 0 (2.81)

as the anomalous matrix elements have the same relative phases.

All the terms linear in h^V_3,4 vanish to leading order and to next-to-leading order, after summing over the fermion helicities and photon polarizations because of the antisymmetry of ^µαβρ in (2.77):

dσ_i,<^V = dσ_i,=^V = 0 for i= 3, 4,

dσ_ij^{U V} = dσ_ji^{V U} = 0 for (i, j)∈ {1,2} × {3,4},

(2.82)

whereU, V =γ,Z. Therefore, the CP-violating couplingsh^V_1,2 and the CP-conserving couplings h^V_3,4 do not interfere.

All the remaining terms linear in h^V_1,2 are proportional cosθ^∗: dσ_i,C^V ∝cosθ^∗ fori= 1, 2, and

C=<, =,

(2.83)

where θ^∗ is the scattering angle of the photon in the center-of-mass frame of the incoming partons. These terms vanish after the phase-space integration which is symmetric with respect to cosθ^∗ at the LHC. Therefore, only the quadratic terms of (2.80) contribute to the non-SM cross section.

Furthermore, the cross sections, and thus the sensitivities, are nearly identical for the same values of h^V_1,2 and h^V_3,4 [39]:

dσ_ij^{U V} ≈dσ^{U V}_kl for i, j ∈ {1, 2}, and (k, l) = (i+ 2, j+ 2),

(2.84)

where U, V ∈ {γ, Z}. Therefore, in the following, we focus on the CP-conserving couplings h^V_3,4 only.

We continue the discussion of the experimental sensitivity to the couplings h^γ,Z_3,...,4, the method of the limit extraction and the results in Sections 5.2 and 6.2.

Chapter 3

Experimental Apparatus