Gauge Theories of Weak Decays

Abstract

We have mentioned already at the beginning of the book that the fundamental role in elementary particle physics, that is, the SM and its extensions, is played by Lagrangians. They encode the information about the particle content of a given theory and of fundamental interactions between these particles that are characteristic for this theory. Therefore, it is essential to start our presentation by discussing the general structure of various Lagrangians that we will encounter in this book.The theories we will discuss are relativistic quantum field theories, and it would appear at first sight that this first step of our expedition is extremely difficult. Yet, the seminal observation of Feynman that a given classical theory can be quantized by means of the path integral method simplifies things significantly. We can formulate the quantum field theory with the help of a Lagrangian of a classical field theory without introducing operators as done in canonical quantization. Having it, a simple set of steps allows us to derive the so-called Feynman rules and use them to calculate the implications of a given theory for various observables that can be compared with experiment.A very important role in particle physics is played by symmetries. They increase significantly the predictive power of a given theory, in particular by reducing the number of free parameters. In this context a very good example is quantum chromodynamics (QCD), the theory of strong interactions. With eight gluons and three colors for quarks, there is a multitude of interactions that, without the SU(3) symmetry governing them, could be rather arbitrary. But the SU(3) symmetry implies certain conservation laws, and at the end there is only a single parameter in QCD: the value of the strong coupling evaluated at some energy scale that can be determined in experiment. Once this is done, all effects of strong interactions can be uniquely predicted, even if this requires often very difficult calculations.The case of QCD is however special as it is based on an exact nonabelian symmetry. We will be more specific about this terminology later. In quantum electrodynamics (QED), which is based on an exact abelian symmetry U(1), in addition to the value of the electromagnetic coupling also the electric charges of quarks and leptons and generally fermions, scalars, and vector particles in a given theory are free. They have to be determined in experiment. In QCD all color charges are fixed by the SU(3) symmetry.Yet QED, similar to QCD, is a very predictive theory because it is based on an exact symmetry. This is generally not the case in nature, and on many occasions the symmetries that we encounter in particle physics are only approximate, and the manner in which 25

SMEFT atlas of ∆F = 2 transitions

“…and D 0 −D 0 mixings have been already for many years the stars among the flavour-changing neutral-current processes (FCNC) [1,2]. This is in particular the case of the parameter ε K , of the B s,d −B s,d mass differences ∆M B s,d and of mixing-induced CP asymmetries in the latter systems.…”

Another SMEFT story: Z′ facing new results on ε′/ε, ∆MK and K → $$ \pi \nu \overline{\nu} $$

“…are given by the gauge-independent functions X 0 (x t ), Y 0 (x t ) and Z 0 (x t ) [78] that can be found in [2]. The NP contribution to the Wilson coefficients of the ∆F = 1 semi-leptonic operators in (3.20) and (3.21) at µ EW in terms of the semi-leptonic SMEFT Wilson coefficients at µ EW is given as follows [49,77,79]…”

“…The ratio ε /ε that measures the size of direct CP violation in K L → ππ decays relative to the indirect CP violation described by ε K and the rare decays K + → π + νν and K L → π 0 νν have been already for many years together with the ∆I = 1/2 rule, K L,S → µ + µ − and K L → π 0 + − decays the stars of Kaon flavour physics [1,2]. The K L -K S mass difference ∆M K remained due to large theoretical uncertainties until recently under the shadow of these decays although it played a very important role in the past in estimating successfully the charm quark mass prior to its discovery [3].…”

“…However, recently progress in evaluating ∆M K within the SM has been made by the RBC-UKQCD collaboration [4][5][6] so that ∆M K JHEP12(2020)097 begins to play again an important role in phenomenology, not only to bound effects of NP contributions [7][8][9][10][11][12], but also to help identify what this NP could be. But as stressed in [1,2] and in particular in [13] such an identification is only possible by considering all the stars of Kaon physics simultaneously and also invoking observables from other meson systems.…”

“…As noted already in [13] the suppression of ∆M K is only possible in the presence of new CP-violating couplings. This could appear surprising at first sight, since ∆M K is a CP-conserving quantity but simply follows from the fact that the BSM shift (∆M K ) BSM is proportional to the square of a complex g sd coupling so that (∆M K ) BSM = c Re[g 2 sd ] = c (Re[g sd ]) 2 − (Im[g sd ]) 2 , c > 0. (1.5) The required negative contribution implies automatically NP contributions to ε /ε and also to rare decays K → πνν, K S → µ + µ − and K L → π 0 + − , provided this NP involves JHEP12(2020)097…”