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Baryon Chiral Perturbation Theory in Its Manifestly Covariant Forms and the Study of the Πn Dynamics & on the Y (2175) Resonance

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Baryon Chiral Perturbation Theory in Its Manifestly Covariant Forms and the Study of the Πn Dynamics & on the Y (2175) Resonance Baryon Chiral Perturbation Theory in its manifestly covariant forms and the study of the πN dynamics & On the Y (2175) resonance Jose Manuel Alarc´on Soriano Departamento de F´ısica Universidad de Murcia Tesis Doctoral Director: Prof. Jos´eAntonio Oller Berber . Dedicado a todos aquellos que siempre confiaron en m´ı. Contents 1 Introduction 9 2 Chiral Perturbation Theory 21 2.1 PropertiesofnonlinearLagrangians . 23 2.2 Standard form of nonlinear realizations . 24 2.3 Classification of all nonlinear realizations . 28 2.4 Relations between linear and nonlinear transformations . 31 2.5 Covariant derivatives and invariant Lagrangians . 36 2.6 Application to the chiral group . 38 2.7 GaugeFields ........................... 41 2.8 ThePowerCounting ....................... 43 2.9 ConstructionofEffectiveLagrangians . 44 3 πN Scattering 49 3.1 PartialWaveAnalyses . .. .. 50 3.2 ChiralPerturbationTheory . 53 3.2.1 Miscellaneous . 55 3.2.2 ChiralLagrangians . 59 3.2.3 The Power Counting in Covariant Baryon Chiral Per- turbationTheory . .. .. 64 3.3 Isospinbreakingcorrections . 66 3.4 Unitarization Techniques . 68 4 Infrared Regularization 71 4.1 Recovering the Power Counting . 71 4.2 Calculation of the Chiral Amplitude . 74 4.3 Fits................................. 75 4.3.1 Strategy-1......................... 75 5 4.3.2 Strategy-2......................... 81 4.4 ConvergenceoftheChiralSeries. 86 4.5 The Goldberger-Treiman Relation . 88 4.6 Unitarized amplitudes and higher energies . 90 4.7 SummaryandConclusions . 94 5 Extended-On-Mass-Shell 97 5.1 The Extended-On-Mass-Shell renormalization . 97 5.1.1 Chiral corrections to the nucleon mass . 99 5.1.2 The Axial Coupling of the Nucleon . 101 5.1.3 The ScalarFormFactoroftheNucleon . 102 5.1.4 ScaleIndependence . .103 5.2 Calculation of the Scattering Amplitude . 104 5.3 Fits.................................105 5.3.1 Strategy-I .........................106 5.3.2 Strategy-II.........................111 5.3.3 ConvergenceoftheChiralseries . 115 5.4 TheGoldberger-Treimanrelation . 116 5.5 Subthresholdregion. .119 5.6 Thepion-nucleonsigmaterm . .122 5.7 Unitarized amplitudes . 129 5.8 SummaryandConclusions . .131 6 φ(1020)f0(980) and φ(1020)a0(980) S-wave scattering and the Y (2175) resonance 133 6.1 Formalism.............................135 6.2 φ(1020) f 0(980)scattering. .137 6.2.1 Extracting the f 0(980)poles. .141 6.2.2 φ(1020) f 0(980)resonantstates . 145 6.3 Derivation of the φ(1020)a0(980) scattering amplitude . 151 6.3.1 Results and discussion . 157 6.4 Summaryandconclusions . .164 7 Summary and Outlook 167 A The Goldstone Theorem 171 B Generalization of IR 175 C Tree Level Calculations 177 C.1 (q)................................177 O C.2 (q2)................................178 C.3 O(q3)................................178 C.4 The∆(1232)contributionO . .179 D Loop Level Calculations 181 D.1 Definitions.............................181 D.2 Coefficients of the Passarino-Veltman Decomposition ..........................185 D.3 Resultsfortheloopdiagrams . .186 E Identifying the power counting breaking terms 193 F Low-Energy Constants Renormalization 195 F.1 (q2)LECs ............................195 F.2 O(q3)LECs ............................196 O G Partial Wave Projections 197 H Suppression of diagrams in Fig. 6.1 201 I Resumen en espa˜nol 209 Chapter 1 Introduction The word “hadron” was introduced in the sixties to name particles that in- teract strongly. Nowadays it is known that these particles are part of two different groups: mesons and baryons, which have integer and semi-integer spin, respectively. In 1961, Murray Gell-Mann [1] and, independently, Yuval Ne’eman [2] proposed a model to explain the properties of these hadrons. According to this model, the hadrons were composed by more fundamental particles called quarks and their antiparticles. This model was very success- ful in explaining properties as the relations between the masses of different hadrons as well as to predict the existence of new ones. It also allowed to group the large amount of discovered hadrons into multiplets of defined spin and parity. This model was known as the Eightfold-Way or, in more technical terms, SU(3) flavor symmetry (SU(3)F ). Quarks also need to have an extra degree of freedom called color charge. This is so because in the quark model the ∆++ baryon is composed by only one type of quark (the up quark) with parallel spins, so as to to have total spin equal to 3/2. The quarks need an additional degree of freedom in order to build from them an antisymmetric wave function [3]. That additional degree of freedom was proposed via an additional SU(3) gauge symmetry associated with a conserved charge called color.♯1 This gauge symmetry associated with color predicts the existence of an octet of gauge bosons that are known as gluons, which carry the strong interaction. Despite the great success of the quark model, the existence of quarks was a mere hypothesis since they had not been experimentally observed before. It was not until 1968 that deep inelastic scattering experiments at the Stanford Linear Accelerator Center (SLAC) showed that the proton contained much ♯1 From now on we will refer to this gauge group as SU(3)C . 9 1 Introduction smaller point-like particles that could be identified with the hypothesized quarks. However, this experiment showed puzzling results since while many theorists of that time suspected that field theory became infinitely strong at short distances (Landau pole), the SLAC results showed that the strong interaction was weaker at high energies. The puzzle was solved in 1973 with the discovery by Gross, Wilczek and Politzer that non-Abelian gauge theories ♯2 like SU(3)C exhibit a property known as asymptotic freedom. This property implies that at high energies the quarks behave as free particles, explaining the deep inelastic scattering results obtained at the SLAC. On the other hand it also pointed out a possible reason of why quarks were not observed as isolated states, because the same calculation lead to a stronger interaction in the low energy regime. This is known as the confinement hypothesis.♯3 A Lagrangian formulation that deals with quarks as the fundamental ♯4 degrees of freedom describing their interaction through their SU(Nc) gauge symmetry exist and is known as Quantum Chromodynamics (QCD). The Lagrangian of QCD can be written in the following compact form: 1 = q¯ iD/ m q Ga Gµν (1.1) LQCD i − qi i − 4 µν a u,d,... X Where qi denotes the Nc-multiplet corresponding to the quark u,d,. , Dµ is the covariant derivative in the colour space, mqi is the mass of the quark qi (the same in the Nc-multiplet) and Gµν is the gluon field. In order to study the symmetries of the strong interaction we will take the following part of the QCD Lagrangian: 0 = iq¯Dq/ = iu¯Du/ + id¯Dd/ (1.2) LQCD i i Xu,d It was pointed by Heisenberg [4] in the thirties than protons and neutrons (also called nucleons) have almost the same mass, and the strength of the strong interaction between them is the same, independently of whether they are protons or neutrons. Because the quark content of the proton and neutron was determined to be uud and udd respectively, at the quark level it means that the u and d quarks are almost identical from the strong interaction 0 point of view. In the massless limit because of QCD they behave in the same way. Due to this fact, is a good idea to groupL them in a doublet, called ♯2This discovery led them to win the Nobel Price in Physics in 2004. ♯3Confinement is an hypothesis since there is no formal proof nowadays. ♯4 SU(Nc) is the generalization of the SU(3)C symmetry to an arbitrary number of colors Nc. 10 1 Introduction in the literature isospin-doublet or isodoublet, and rewrite (1.2) in terms of this isodoublet: 0 = iψ¯Dψ/ (1.3) LQCD u Where ψ = is the isospin-doublet.♯5 The Lagrangian (1.2) is d invariant under rotations in the space defined by the quarks that constitute the isodoublet, also called isospin space. Mathematically, the transformation can be defined by: u u exp iθV τ i + iγ θAτ i (1.4) d → i 5 i d Where the τ i are the so-called isospin matrices (Pauli matrices divided by two) and obey the isospin algebra of SU(2) [τi, τj] = iǫijkτk. One can check easily that (1.3) is invariant under (1.4). We can take τ and γ τ τ˜ as the generators of these transformations, i 5 i ≡ i satisfying the following algebra: [τi, τj] = iǫijkτk [τi, τ˜j] = iǫijkτ˜k [˜τi, τ˜j] = iǫijkτk (1.5) This Lie algebra can be decoupled into two commuting SU(2) subalge- bras that act independently on the left- and right-handed part of the quark fields.This is the so called chiral symmetry. The generators of this SU(2) SU(2) algebra are: L × R 1 1 τ i = (1 + γ )τ i, τ i = (1 γ )τ i (1.6) L 2 5 R 2 − 5 And satisfy: [τLi, τLj] = iǫijkτLk (1.7) [τRi, τRj ] = iǫijkτRk (1.8) [τLi, τRj ] = 0 (1.9) As we argue later, this SU(2) SU(2) symmetry is spontaneously bro- L × R ken into the isospin SU(2)V subgroup, that remains unbroken, according to ♯5 This procedure can be generalized to Nf -multiplet, where Nf is the number of quark flavours. We would have for (1.2) a sum over the s,c,. quarks. 11 1 Introduction the Vafa-Witten theorem [5]. On the other hand, an insertion of a mass term in (1.2), like the one of Eq.(1.1), explicitly breaks the symmetry generated by axial transformations associated to the generatorsτ ˜i = γ5τi, but this will be considered later.
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