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Register a new userSU(3) analysis of four-quark operators: $Ktopipi$ and vacuum matrix elements
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Antonio Rodr\\'iguez S\\'anchez
Publication date
2021
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English
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Hadronic matrix elements of local four-quark operators play a central role in non-leptonic kaon decays, while vacuum matrix elements involving the same kind of operators appear in inclusive dispersion relations, such as those relevant in $tau$-decay analyses. Using an $SU(3)_Lotimes SU(3)_R$ decomposition of the operators, we derive generic relations between these matrix elements, extending well-known results that link observables in the two different sectors. Two relevant phenomenological applications are presented. First, we determine the electroweak-penguin contribution to the kaon CP-violating ratio $varepsilon/varepsilon$, using the measured hadronic spectral functions in $tau$ decay. Second, we fit our $SU(3)$ dynamical parameters to the most recent lattice data on $Ktopipi$ matrix elements. The comparison of this numerical fit with results from previous analytical approaches provides an interesting anatomy of the $Delta I = frac{1}{2}$ enhancement, confirming old suggestions about its underlying dynamical origin.
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We perform for the first time a direct calculation of on-shell $Ktopipi$ hadronic matrix elements of chromomagnetic operators (CMO) in the Standard Model and beyond. To his end, we use the successful Dual QCD (DQCD) approach in which we also consider off-shell $K-pi$ matrix elements that allows the comparison with lattice QCD calculations of these matrix elements presented recently by the ETM collaboration. Working in the SU(3) chiral limit, we find for the single $B$ parameter $B_{rm CMO}=0.33$. Using the numerical results provided by the ETM collaboration we argue that only small corrections beyond that limit are to be expected. Our results are relevant for new physics scenarios in the context of the emerging $epsilon^prime/epsilon$ anomaly strongly indicated within DQCD and supported by RBC-UKQCD lattice collaboration.
The study of neutrinoless double beta decays of nuclei and hyperons require the calculation of hadronic matrix elements of local four-quark operators that change the total charge by two units Delta Q=2 . Using a low energy effective Lagrangian that induces these transitions, we compute these hadronic matrix elements in the framework of the MIT bag model. As an illustrative example we evaluate the amplitude and transition rate of Sigma- -> p e- e-, a decay process that violates lepton number by two units (Delta L=2). The relevant matrix element is evaluated without assuming the usual factorization approximation of the four-quark operators and the results obtained in both approaches are compared.
If physics beyond the Standard Model enters well above the electroweak scale, its low-energy effects are described by Standard Model Effective Field Theory. Already at dimension six many operators involve the antisymmetric quark tensor $bar q sigma^{mu u} q$, whose matrix elements are difficult to constrain from experiment, Ward identities, or low-energy theorems, in contrast to the corresponding vector and axial-vector or even scalar and pseudoscalar currents. However, with normalizations determined from lattice QCD, analyticity and unitarity often allow one to predict the momentum dependence in a large kinematic range. Starting from recent results in the meson sector, we extend this method to the nucleon case and, in combination with pole dominance, provide a comprehensive assessment of the current status of the nucleon form factors of the quark tensor.
We determine the non-perturbative gluon condensate of four-dimensional SU(3) gauge theory in a model independent way. This is achieved by carefully subtracting high order perturbation theory results from non-perturbative lattice QCD determinations of the average plaquette. No indications of dimension two condensates are found. The value of the gluon condensate turns out to be of a similar size as the intrinsic ambiguity inherent to its definition.
The pressure of QCD admits at high temperatures a factorization into purely perturbative contributions from hard thermal momenta, and slowly convergent as well as non-perturbative contributions from soft thermal momenta. The latter can be related to various effective gluon condensates in a dimensionally reduced effective field theory, and measured there through lattice simulations. Practical measurements of one of the relevant condensates have suffered, however, from difficulties in extrapolating convincingly to the continuum limit. In order to gain insight on this problem, we employ Numerical Stochastic Perturbation Theory to estimate the problematic condensate up to 4-loop order in lattice perturbation theory. Our results seem to confirm the presence of large discretization effects, going like $aln(1/a)$, where $a$ is the lattice spacing. For definite conclusions, however, it would be helpful to repeat the corresponding part of our study with standard lattice perturbation theory techniques.
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