In the Born approach, the Feynman diagrams for the LEP2 process are shown in Figure 3.4 [46]. The analytical cross-section formula in the Born approach can be found, for example, in [307] (in the section Cross-section formulas for specific processes).

## The CKM Matrix and Flavour Physics

This is the GIM mechanism [226], which ensures the preservation of the natural flavor of neutral current couplings at the tree level. In SM, in the quark sector, all CP-violating observers must be proportional to J, i.e. to the area of the triangle or to.

## Neutrino Mass and Mixing

The corresponding mass terms are of the order mv2=M, where the Higgs VEV, therefore of the same generic order as the light neutrino masses from (3.77). Given the definition of U and the transformation properties of the effective light neutrino mass matrix in (3.79), viz. The numbering 1,2,3 corresponds to a definition of the frequencies and should in principle not coincide with the ordering from the lightest to the heaviest condition.

Neutrino mixing is important because, in principle, it could provide new clues to understanding the flavor problem. In fact, it is frustrating that there has been no real relief from the problem of flavor.

## Quantization and Renormalization of the Electroweak Theory

The W˙ and Z propagators, as well as those of the scalars w˙andz, have exactly the same general forms as for the Abelian case in with parameters and respectively (and the pseudo-Goldstone bosons w˙ etc have masses mW and mZ) . The pseudo-Goldstone bosons w˙ enzar are directly related to the longitudinal helicity states of the corresponding massive vector bosons W˙ and Z. In fact, the non-singular behavior of propagators at large momentums is very suggestive of the result.

But this problem is avoided in the SM because the quantum numbers of the quarks and leptons in each generation imply a remarkable (and, from the SM point of view, mysterious) cancellation of the anomaly, as originally observed in [109]. Thus, for the chiral anomaly to vanish, all traces of the form trft3QQg, trft3t3Qg, trft3t3t3g (and also trftCtt3g if charged currents are included) must vanish, with the trace extending over all fermions in the theory that can circulate in the loop. .

## QED Tests: Lepton Anomalous Magnetic Moments

An important implication of chiral anomalies together with the topological properties of the vacuum in non-abelian gauge theories is that the conservation of charges associated with baryon (B) and lepton (L) numbers is broken by the anomaly [336], so Conservation B and L are actually violated in standard electroweak theory (but BL remains conserved). the perturbative factor exp.c=g2/, with c a constant and g the gauge relation SU.2/.] The corresponding effect is completely negligible at the temperature zeroT, but becomes important at temperatures close to the degree of breaking of weak symmetry, precisely in T O .TeV/. The non-conservation of BCL and the conservation of B L near the weak scale plays a role in the theory of baryogenesis that aims to quantitatively explain the observed matter–antimatter asymmetry in the Universe (for reviews and references, see, for example, [115] ). However, the estimate of the error in the LBL term, largely a theoretical uncertainty, is not convincing and may be somewhat larger (although perhaps not so much as to completely eliminate the discrepancy).

A small puzzle is the fact that, using current vector conservation (CVC) and isospin invariance, which are well-established low-energy tools, aLO can also be estimated from the decays. Finally, we note that, given the high precision of the measurement and the relative importance of non-QED contributions, it is not unreasonable for a first signal of new physics to appear in this quantity.

## Large Radiative Corrections to Electroweak Processes

Among the one-loop EW radiation corrections, a notable class of contributions are the terms that increase quadratically with the peak mass. Second, the theory without the top quark is no longer renormalizable since the gauge symmetry is broken because.t;b/. More generally, the important consequence of non-coupling is that precision tests of the electroweak theory may a priori be sensitive to new physics, even if the new particles are too heavy for their direct production, but no signal of deviation has subsequently clearly emerged . .

Unfortunately, while radiative corrections are quite sensitive to the top mass, they are much less dependent on the Higgs mass. The difference with the above case is that the splitting m2t m2b is a direct breaking of the gauge symmetry which already affects the 1-loop corrections, while the Higgs couplings are "preserving" SU.2/symmetric in the lowest order.

## Electroweak Precision Tests

TheZ and Wmasses must be precisely defined, for example, in terms of the pole position in the respective propagators. Then, in the first relation, replacing i˛ with the direction coupling in the measure Z˛.mZ/makesrW fully defined in 1-loop by simply weak corrections (GF is protected from the logarithmic direction as an indirect consequence of the conservation of VCurrent in the theory without measure) . In contrast, in the second relation, N m depends on the definition of sin2W beyond the tree level.

In SM, the quantities W,k, for large enough, are all dominated by quadratic terms of order GFm2t. The parameters vanish in the limit where only tree-level SM effects plus pure QED and/or QCD corrections are retained.

## Results of the SM Analysis of Precision Tests

Therefore, the set of partial width and asymmetry results allows the extraction of the effective coupling constants. The2 is boosted by the two most precise measurements of sin2eff, namely those derived from the measurements of Alby SLD, dominated by the left-right asymmetryA0LR, and measurements of the forward-backward asymmetryA0;FBbmeasured inbbN production at LEP, which with about 3 difference. Despite this small sensitivity, the measurements are still accurate enough to obtain a quantitative indication of the mass range.

The precise determination of the associated finite terms would be lost (i.e. the value of the mass in the denominator in the argument of the logarithm). The upper bound shape of the SM from the EW tests depends on the value of the top quark mass.

## The Search for the SM Higgs

This is a clear signal of spontaneous symmetry breaking and the implementation of spontaneous symmetry breaking in a gauge theory is via the Higgs mechanism. The search for the Higgs boson and for possible new physics that can accompany it has been the main goal of the LHC from the start. On the Higgs, the LHC must answer the following questions: do some Higgs particles exist.

So far we have a candidate Higgs boson that really looks like the simplest realization of the Higgs mechanism, as described by the SM Higgs minimum. In the following we will first consider the a priori expectations for the Higgs sector and then the profile of the Higgs candidate discovered at the LHC.

## Theoretical Bounds on the SM Higgs Mass

The Higgs mass comes in because it sets the initial value of the quartic Higgs coupling in its run up to large scales. The upper limit on the Higgs mass in the SM is clearly important for an a priori assessment of the chances of success for the LHC as an accelerator designed to solve the Higgs problem. Of course, in the vicinity of the Landau pole the 2-loop evaluation of the beta function is not reliable.

Indeed, the limit indicates the boundary of the field where the theory is well described by the perturbative expansion. As for a lower limit on the SM Higgs mass, a possible instability of the Higgs potential VŒ is created by quantum loop corrections to the classical expression for VŒ.

## SM Higgs Decays

An improved version of the renormalization group of the corrected potential leads to the exchange4!./04./, where./ is the rolling coupling and0. t0/dt0, with t/anomalous dimensional function, t D log=v and v expected vacuum value v D .2p. In this case, the limit applies to some average mass, but the lightest Higgs particle can be well below, as in the case of the minimal SUSY expansion of the SM (MSSM). For Higgs masses below this range, there may still be a domain where the SM is viable because the vacuum may be unstable, but with a lifetime longer than the age of the universe.

VV dominates over tNt due to threshold factors, which disfavor the fermion channel, and to a large extent mH, from the cubic versus linear behavior with mH of the partial widths for VV versus tNt. Hvertex provides one of the main production channels for the Higgs boson in hadron colliders, while another currently important channel is WHassociate production.

## The Higgs Discovery at the LHC

The possibility of a "conspiracy" (the Higgs is heavy, but it falsely appears to be light due to confusing new physics effects) has been dismissed: the EW precision tests from the SM tell the truth and indeed consistently no "conspirators", namely no new particles , have been seen around. It is also important to search for possible additional Higgs states, such as those predicted in supersymmetric extensions of the SM. We do not expect surprises on the spin-parity assignment, because if they are different, all Lagrangian vertices will be changed and the profile of the SM Higgs particle will be completely changed.

The tree-level couplings of the Higgs are proportional to masses, and are consequently very hierarchical. Finally, the Higgs sector of the minimal SM really appears to be realized in nature, with good approximation.

## Limitations of the Standard Model

This so-called hierarchy problem [219] is due to the instability of the SM with respect to quantum corrections. This term, which appears in the Higgs potential, fixes the scale of the Higgs VEV and all related masses. There is a gap between the mass of the Higgs (corresponding to a pion) and the scale f, where new physics emerges in the form of resonances (corresponding to , etc.).

Personally, I find the application of the anthropic principle to the SM hierarchy problem somewhat exaggerated. None of the alleged evidence for new physics in collisions would hold up (in particular, even the claimed muong2[297] discrepancy would have to be attributed, if not to an experimental problem, then to an underestimation of the theoretical uncertainties, or otherwise to a specific addition to the above model [257]).