Hypothesised new phenomena at a scale (Λ) higher than the one accessible by the current
ex-periments are often described by effective theories (L_{eff}) with higher dimension operators (O) as
extensions to the Standard Model Lagrangian (L_{SM}):

L_{eff}=L_{SM}+

"_{∞}
X

i

Ci

ΛOi+h.c.

#

, (1.6)

Motivations 18

Figure 2: Inclusive diﬀerential cross sectiondσ/dptcompared to Monte Carlo predictions and
exponential and power law behaviour for: a) π^{0} production for|η|<0.5 and b) K^{0}_{S} production
for |η| < 1.5. Ratio of the diﬀerential cross section dσ/dp_{t} to Monte Carlo predictions for: c)
π^{0} production and d) K^{0}_{S} production.

12

Figure 2: Inclusive diﬀerential cross section dσ/dpt compared to Monte Carlo predictions and
exponential and power law behaviour for: a)π^{0} production for|η|<0.5 and b) K^{0}_{S} production
for |η|< 1.5. Ratio of the diﬀerential cross section dσ/dpt to Monte Carlo predictions for: c)
π^{0} production and d) K^{0}_{S} production.

12

Figure 1.6: The differential cross section of π^{0} (K_{s}^{0}) for rapidity|y|<0.5 (|y|<1.5) production
at LEP as measured by the L3 Collaboration is shown on the left (right) [48]. Data are compared
to MC predictions from the SM (lines labelled C and D), to an exponential fit (line labelled A)
and to a power low fit (line labelled B). High-p_{T} regions are not well reproduced by simulations.

withC_{i} being constant pre-factors. The leading contributions ofL_{eff} impose to thet¯tγ vertexΓ^{t}_{µ}^{¯}^{tγ}
which, at tree level, in the SM isΓ^{t}µ^{¯}^{tγ} =−ieQ_{t}γ_{µ}, withebeing the proton charge andγ_{µ}the Dirac
matrices.

Ten form-factors F(ˆs^{2}), as a function of the invariants of type ˆs^{2} = (pt+ ¯pt)^{2}, can describe a
most general Lorentz-invariant of theΓ^{t}µ^{tγ}^{¯} vertex [51], which in a low-energy limit, can be assumed
as couplings of dimension-four and -five operators. For on-shell production of γ, or assuming
massless fermions, or when both top quarks are on-shell, the problem is reduced by five degrees of
freedom, and a most general effectivet¯tγ vertex can be written as [12]:

Γ^{t}_{µ}^{tγ}^{¯} sˆ^{2}, q,q¯
are thettγ¯ vector and axial-vector form-factors. The form-factors F_{2,V}^{γ} and F_{2,A}^{γ} are proportional
to the magnetic (gt) and electric dipole-factors (d^{γ}_{t})

F_{2,V}^{γ} =Qt

g_{t}−2

2 , F_{2,A}^{γ} = 2m_{t}

e d^{γ}_{t} (1.8)

and they contribute only at higher order corrections (in one-loop corrections they areO(10^{−3})) [52].

At tree level and for the SM,F_{1,V}^{γ} =Qtand the remainder form-factors are equal to zero.

At high partonic centre-of-mass energies (√ ˆ

s m^{2}_{t}) the unitarity of the S-matrix, via

|M|^{2} ≤Im(M), imposes that anomalous axial and vector-axial couplings have to correspond
asymptotically to the SM values of the couplings, hence they must have a momentum dependance
to ensure such correspondence. Theory, typically, imposes such condition in loop observables by
the implementation of a cut-off Λ in the anomalous couplings, for which the deviations drop to
zero abruptly at√

ˆ

s= Λ. Instead, in order to explore the unitarity constraint withk^{2} dependance
dipole form-factors were used [12] for the restriction of the deviations from the SM couplings
(∆F_{i,V,A}^{γ} ):

Figure 1.7 shows the evolution of such limits as a function of ˆs at fixed scales where the new phenomena are hypothesised. It can bee seen, indeed, that the allowed deviations vanish with large√

ˆ

s, while atO(Λ' TeV) larger are possible in regions of√ ˆ

saccessible by the LHC .

[TeV]

Figure 1.7: Evolution of the anomalous t¯tγ couplings as a function of the the partonic centre-of-mass energy (√

ˆ

s). Allowed regions |∆F_{1,V,A}^{γ} | (|∆F_{2,V,A}^{γ} |) are shown on the left (right). The
unitarity of the S-matrix allows for deviations of the Standard Model couplings in the regions
below the curves that are shown. The curves are parametrised with respect to the scale of new
physics (Λ). Limits are deduced from Eq. 1.10 and Eq. 1.11, which are based on the theoretical
calculation [12]. It can be seen that, asymptotically, with the increase ofˆs, deviations tend to null
values, thus conserving the Matrix Element unitarity.

Motivations 20

At present, stringent experimental limits on∆F_{1,A,V}^{Z} (0) for thet¯tZ vertex restrict the
devia-tions to be of the percent level at the TeV scale. Similarly, anomalous magnetic and electric dipole
form-factors for thet¯tγ vertex are restricted to [3, 12]:

−0.2≤F_{2,V}^{γ} (0)≤0.5 (1.12)

−4.5≤F_{2,A}^{γ} (0)≤4.5 (1.13)

However,F_{1,V}^{γ} and F_{1,A}^{γ} are yet to be constrained by the experiment.

As introduced in Sec. 1.1, the LHC provides a good framework for the study of thetγ couplings compared top¯pcolliders. As can be seen in Fig. 1.8, inp¯pcolliders the domination of initial-state photon radiation from the colliding quarks makes a discrimination between different values of

∆F_{1,V}^{γ} impossible. Moreover, even hypothesising a null electric charge for the top quark (∆F_{1,V}^{γ} =
2/3in the left hand-side distribution of Fig. 1.8), the differential spectrum with respect the photon
transverse momentum in p¯p collisions shows almost no differences in t¯tγ production with respect
to the SM coupling values. On the contrary, atpp colliders, where gg production dominates and
initial-state radiation is suppressed, the discrimination is more prominent.

Figure 1.8: Differential cross section spectra of t¯tproduction in association with a photon in the single-lepton channel as a function of the photon transverse momentumpT(γ) for p¯pcollisions at

√s = 2 TeV (left) and pp collisions at √

s= 14 TeV (right) [12]. The continuous curve labelled

“SM” corresponds to the Standard Model prediction of thet¯tγ cross section, while the dotted and
dashed curves correspond to thettγ¯ cross section with anomalous t¯tγ couplings. For each curve
only, one coupling is allowed to deviate and the labels∆F_{1(2),V}^{γ} (∆F_{1(2),A}^{γ} ) indicate the differences
with respect to the SM of the vector (axial-vector) form-factorsF_{1(2),V}^{γ} (F_{1(2),A}^{γ} ). For p¯pcollisions
the curve labelled ∆F_{1,V}^{γ} = 2/3 corresponds to a null electric charge for the top quark. In that
case, because of the overwhelming photon production from initial-state radiation, it can be seen
that the differences with respect to the SM prediction are small.

Motivations 21