Explicit Bound states and Resonance fields in Effective Field Theories

Abstract

The first three chapters of this thesis have been devoted to the theoretical background. We presented the novel work of this thesis in chapters 4 and 5. In chapter 4 we have constructed a Chiral effective field theory for the nucleon--nucleon system which contains dibaryon fields as fundamental degrees of freedom. The large scattering lengths in the 1S0 and the 3S1 channels force the dibaryon residual masses to be much smaller than the pion mass. Since no counterterm has to be enhanced like in the KSW approach, naïve dimensional analysis is sufficient to assess the size of the effective field theory low--energy constants, keeping the perturbative expansion under control. We organized the calculation in a series of effective theories, which are obtained by sequentially integrating out higher energy and momentum scales. We first integrate out energy scales of the order of the pion mass. This leads to an effective theory with dibaryon and nucleon fields, pNNEFT. For three momenta much smaller than the pion mass, it is convenient to further integrate out three momenta of the order of pion mass, which leads to the npNNEFT.

We have calculated the nucleon--nucleon scattering amplitudes for energies smaller than the pion mass in the 1S0 and the 3S1-3D1 channels at NNLO. The numerical results for the phase shifts and mixing angle are also similar to the ones obtained in the KSW approach. A good description of the 1S0 channel is obtained, but for the 3S1-3D1 channel our results also fail to describe data. The reasons of this failure can be traced back to the iteration of the one potential pion exchange potential.

We have calculated the matching of the dibaryon residual masses and dibaryon-nucleon couplings up to NLO. We have showed that, certain class of diagrams that contribute to the residual mass, involving n potential pion exchanges in loops with radiation a pion, have to be summed in the 3S1 channel. In the 3S1 channel the resummation can be carried out. However in the 1S0 channel the resummation is not possible, but it is very likely that loop contributions are still large. Using the results for the matching for residual masses and dibaryon--nucleon coupling for npNNEFT we have given Chiral extrapolation formulas for scattering lengths of the scalar channels up to corrections of order mq(3) We have fitted these expressions to lattice data and compared the results to previous studies of the quark mass dependence of the scattering lengths.

In chapter 5 we have considered the possibility that the spectrum of QCD in the Chiral limit contains an isosinglet scalar with a mass much lower than the typical hadronic scale, and have constructed the corresponding effective theory that includes it together with the standard pseudo-Goldstone bosons, ChiPTs. In the purely scalar sector of the theory we argued that the scalar self interactions can be ignored. Demanding that the scalar does not mix with the vacuum together with Chiral symmetry imposes that two of the low--energy constants should be taken as zero.

We have presented the calculation of pion mas and decay constant at NLO. The dynamical scalar field introduces new non-analyticities in the quark mass dependence of these observables. We have used lattice data from the ETM collaboration to fit the low--energy constants. The chi-squared per degree-of-freedom delivered by the ChiPTs fits are similar to ChiPT ones indicating that lattice data does not favor any of the theories over the other.

The ChiPTs expressions for the S-wave pion-pion scattering lengths differ from those of ChiPT already at leading order. Furthermore ChiPTs allows for the calculation of the sigma decay width. Neither the value of the scattering lengths for the I=0 and I=2 channels are close to the experimental numbers. Although the value of I=0 is slightly closer to it than the one obtained in tree-level ChiPT, the value of the I=2 channel is much further away. We argue, using the decoupling limit that this is due to the sizable NLO corrections because of the large value of l1. We also show a different approach in which we fit the scattering length expressions with all parameters free to lattice data and use the results to provide predictions for the sigma mass and decay width.

En el marc de teories efectives per a Cromodinàmica Quàntica a baixes energies, una situació interessant es presenta quan els graus de llibertat de baixes energies poden formar estats lligats, estats virtuals o ressonàncies pròximes al llindar. Com que aquest estats estan a prop del llindar afecten a les amplituds de dispersió, però tan mateix no poden ser descrites utilitzant teoria de pertorbacions, ja que les series polinòmiques finites en el moment no poden generar un pol en l’amplitud. Aquest pols es poden obtenir resumant certes classes de diagrames, per exemple usant tècniques d’unitarització, que no són consistents amb el comptatge de la teoria efectiva, o alternativament assumint un augment de l’importància de certs acoblaments. En aquest últim cas s’han de calcular les equacions del grup de renormalització per a tots els acoblaments per tal de determinar-ne el tamany correcte, el que dificulta mantenir la sèrie pertorbativa sota control. És una vella observació de Weinberg que la inclusió explícita d’estats lligats i ressonàncies com a graus de llibertat de la teoria efectiva millora la convergència de la teoria de pertorbacions. Es pot entendre fàcilment aquesta millora de la convergencia ja que les amplituds de dispersió tindran la estructura analítica correcta. Un dels temes principals d’aquesta tesi ha sigut explorar aquest fet dins d'un marc modern de teories efectives.

El treball original d’aquesta tesi és als capítols 4 i 5. Al capítol 4 hem construït una teoria efectiva Quiral pel sistema nucleó–nucleó que conté camps dibariònics com a graus de llibertat fonamentals. Les longituds de dispersió grans en els canals 1S0 i 3S1 poden ser representades de forma natual gràcies a les petites masses residuals dels dibarions. Em calculat les amplituds de dispersió per aquesta teoria fins a NNLO per als canals 1S0 i 3S1-3D1, i em donat fòrmules d'extrapolació quiral per a les longituds de dispersió d'ona S fins a NLO.

Al capítol 5 hem considerat la possiblitat de que l’espectre de QCD en el limit Quiral contingui un isosinglet escalar amb massa molt mes petita que l’escala hadroníca típica, i hem construït una teoria efectiva que l’inclou conjuntament amb els pseudo–bosons de Goldstone. Hem calculat la massa i la constant de decaïment del pion fins a NLO i hem ajustat els resultats a dades en el reticle. També hem estudiat les longituds de dispersió de les col•lisions pió-pió per a ona S en isospin I=0 i I=2 i les hem comprat amb dades al reticle.