Hypothesised new phenomena at a scale (Λ) higher than the one accessible by the current ex-periments are often described by effective theories (Leff) with higher dimension operators (O) as extensions to the Standard Model Lagrangian (LSM):
Leff=LSM+
"∞ 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) π0 production for|η|<0.5 and b) K0S production for |η| < 1.5. Ratio of the differential cross section dσ/dpt to Monte Carlo predictions for: c) π0 production and d) K0S production.
12
Figure 2: Inclusive differential cross section dσ/dpt compared to Monte Carlo predictions and exponential and power law behaviour for: a)π0 production for|η|<0.5 and b) K0S production for |η|< 1.5. Ratio of the differential cross section dσ/dpt to Monte Carlo predictions for: c) π0 production and d) K0S production.
12
Figure 1.6: The differential cross section of π0 (Ks0) 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-pT regions are not well reproduced by simulations.
withCi being constant pre-factors. The leading contributions ofLeff impose to thet¯tγ vertexΓtµ¯tγ which, at tree level, in the SM isΓtµ¯tγ =−ieQtγµ, withebeing the proton charge andγµthe Dirac matrices.
Ten form-factors F(ˆs2), as a function of the invariants of type ˆs2 = (pt+ ¯pt)2, can describe a most general Lorentz-invariant of theΓtµtγ¯ 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]:
Γtµtγ¯ sˆ2, q,q¯ are thettγ¯ vector and axial-vector form-factors. The form-factors F2,Vγ and F2,Aγ are proportional to the magnetic (gt) and electric dipole-factors (dγt)
F2,Vγ =Qt
gt−2
2 , F2,Aγ = 2mt
e dγt (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,F1,Vγ =Qtand the remainder form-factors are equal to zero.
At high partonic centre-of-mass energies (√ ˆ
s m2t) 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 withk2 dependance dipole form-factors were used [12] for the restriction of the deviations from the SM couplings (∆Fi,V,Aγ ):
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 |∆F1,V,Aγ | (|∆F2,V,Aγ |) 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∆F1,A,VZ (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≤F2,Vγ (0)≤0.5 (1.12)
−4.5≤F2,Aγ (0)≤4.5 (1.13)
However,F1,Vγ and F1,Aγ 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
∆F1,Vγ impossible. Moreover, even hypothesising a null electric charge for the top quark (∆F1,Vγ = 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∆F1(2),Vγ (∆F1(2),Aγ ) indicate the differences with respect to the SM of the vector (axial-vector) form-factorsF1(2),Vγ (F1(2),Aγ ). For p¯pcollisions the curve labelled ∆F1,Vγ = 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