Follow-up on non-leptonic Kaon decays at large $N_c$
We report on the status of our dynamical simulations of a $SU (N_c )$ gauge theory with $N_c=3-6$ and $N_f =4$ fundamental fermions. These ensembles can be used to study the Large $N_c$ scaling of weak matrix elements in the GIM limit $m_c=m_u$, that might shed some light on the origin of the $\Delta I=1/2$ rule. We present preliminary results for the $K \to \pi$ matrix elements in the $N_c=3$ dynamical simulations, where we observe a significant effect of the quark loops that goes in the direction of enhancing the ratio of $A_0/A_2$ amplitudes. Finally, we present the relevant NLO Chiral Perturbation Theory predictions for the relation between $K \to \pi $ and $K \to \pi \pi$ amplitudes in…
Kaon decays and other hadronic processes in lattice QCD
This doctoral thesis deals with the study of properties and interactions of light mesons. More specifically, we focus on hadronic decay and scattering processes, which are dominated by effects of the strong interaction in the low-energy regime. Concrete examples that will be addressed are the weak decay of a kaon into two pions, and the scattering of three pions. A peculiarity of the strong interaction is that perturbative expansions fail at hadronic energy scales. For this reason, genuine nonperturbative tools are essential to obtain first-principles predictions. The central technique employed in this work is Lattice Field Theory, which uses a discretized spacetime to stochastically estima…
Two- and three-body interactions in $\varphi^4$ theory from lattice simulations
We calculate the one-, two- and three-particle energy levels for different lattice volumes in the complex $\varphi^4$ theory on the lattice. We argue that the exponentially suppressed finite-volume corrections for the two- and three-particle energy shifts can be reduced significantly by using the single particle mass, which includes the finite-size effects. We show numerically that, for a set of bare parameters, corresponding to the weak repulsive interaction, one can reliably extract the two- and three-particle energy shifts. From those, we extract the scattering length, the effective range and the effective three-body coupling. We show that the parameters, extracted from the two- and thre…
Meson interactions at large $N_c$ from Lattice QCD
We report on the computation of the scaling of QCD observables with the number of colours, $N_c$. For this, we use dynamical configurations with four active flavours, $N_f=4$, and values of $N_c=3-6$. We study the meson masses and decay constants, and compute the leading and subleading contributions to the Low Energy Constants (LECs) of the chiral Lagrangian. We also explore $\pi \pi$ scattering in the $I=2$ channel, and compute the $K \to \pi $ weak decay matrix elements. We comment on the relation of the latter to $K \to \pi\pi$ processes and the $\Delta I=1/2$ rule.
Implementing the three-particle quantization condition including higher partial waves
We present an implementation of the relativistic three-particle quantization condition including both $s$- and $d$-wave two-particle channels. For this, we develop a systematic expansion about threshold of the three-particle divergence-free K matrix, $\mathcal{K}_{\mathrm{df,3}}$, which is a generalization of the effective range expansion of the two-particle K matrix, $\mathcal{K}_2$. Relativistic invariance plays an important role in this expansion. We find that $d$-wave two-particle channels enter first at quadratic order. We explain how to implement the resulting multichannel quantization condition, and present several examples of its application. We derive the leading dependence of the …
The large $$N_{c}$$ limit of QCD on the lattice
We review recent progress in the study of the large $N_c$ limit of gauge theories from lattice simulations. The focus is not only the planar limit but also the size of ${\mathcal O}(N_c^{-1})$ corrections for values of $N_c\gtrsim 3$. Some concrete examples of the topics we include are tests of large-$N_c$ factorization, the topological susceptibility, the glueball, meson and baryon spectra, the chiral dependence of masses and decay constants, and weak matrix elements related to the $\Delta I=1/2$ rule in kaon decays.
A lattice study of ππ scattering at large N c
Abstract We present the first lattice study of pion-pion scattering with varying number of colors, Nc. We use lattice simulations with four degenerate quark flavors, Nf = 4, and Nc = 3 − 6. We focus on two scattering channels that do not involve vacuum diagrams. These correspond to two irreducible representations of the SU(4) flavor group: the fully symmetric one, SS, and the fully antisymmetric one, AA. The former is a repulsive channel equivalent to the isospin-2 channel of SU(2). By contrast, the latter is attractive and only exists for Nf≥ 4. A representative state is $$ \left(\left.\left|{D}_s^{+}{\pi}^{+}\right.\right\rangle -\left.\left|{D}^{+}{K}^{+}\right.\right\rangle \right)/\sqr…
Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states
In this work, we use an extension of the quantization condition, given in Ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the two-particle K matrix that required the absence of two-particle bound states or narrow two-particle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quantity that enters the quantization condition. With this in hand, we extend previous numerical studie…