_t63 [Tau] decay and the structure of the a_1tn1 [Elektronische Ressource] / vorgelegt von Markus Wagner

Sujets

• Physik

Informations

• Publié par: justus-liebig-universitat_giessen
• Publié le: 01 janvier 2008
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

τ decay
and the structure of the a1
Dissertation
zur Erlangung des Doktorgrades
der naturwissenschaftlichen Fachbereiche
der Justus-Liebig-Universit¨at Gießen
Fachbereich 7 - Mathematik, Physik, Geographie
vorgelegt von
Markus Wagner
aus Linden
Gießen, 2007Dekan : Prof. Dr. Bernd Baumann
I.Gutachter : Prof. Ulrich Mosel
Ihter : PD Dr. Stefan Leupold
Tag der mundlic¨ hen Prufung¨ : 20.12.2007Contents
1. Introduction 1
2. Chiral Perturbation Theory 5
2.1. Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3. Vector Mesons and Chiral Symmetry 11
3.1. The WCCWZ scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1. Vector-meson couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2. Power counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3. Axial-vector meson couplings . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2. Vector vs tensor formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3. Renormalisation in the presence of spin-1 fields . . . . . . . . . . . . . . . . . . 19
3.3.1. Powercounting for loop diagrams . . . . . . . . . . . . . . . . . . . . . . 20
3.3.2. Crossing symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4. Chiral Unitarity 23
4.1. Unitarity and helicity amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2. N/D method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3. Inverse amplitude method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4. The Bethe-Salpeter equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4.1. Partial wave expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4.2. Onshell reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.3. The kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5. Partial Wave Projectors 39
5.1. Projection of the WT term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2. Connection between helicity states and orbital angular momentum . . . . . . . 46
5.3. Covariant projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6. τ Decay 55
6.1. Weak interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2. The decay width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3. Which diagrams to include? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.4. Calculation of τ decay without a . . . . . . . . . . . . . . . . . . . . . . . . . . 601
6.5. of the τ decay with explicit a . . . . . . . . . . . . . . . . . . . . . 641
6.6. Calculation of τ decay including higher order terms . . . . . . . . . . . . . . . . 71
6.7. W form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.8. Onshell, offshell? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
iContents
7. Results 79
7.1. Spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2. Calculation without a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821
7.3. with explicit a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891
7.4. Higher order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5. Dalitz plot projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8. Summary and Outlook 103
A. Notation and Normalisation 107
A.1. Conventions and γ matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2. Momentum states, helicity states and normalisation . . . . . . . . . . . . . . . 110
A.3. Wigner rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B. Orthogonality Relation of the Projectors 115
B.1. Application to the a loop integral . . . . . . . . . . . . . . . . . . . . . . . . . 1181
C. Construction of the Higher Order Lagrangian 121
D. Vertices 123
D.1. Weinberg-Tomozawa term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D.2. Higher order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
E. Miscellaneous 135
E.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
E.2. Regularisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
E.3. Adding the singular diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Bibliography 145
iiChapter 1.
Introduction
If you have built castles in the air, your work need not be lost; that is where they
should be. Now put the foundations under them.
Henry David Thoreau
In the Fifties and Sixties lots of new particles were discovered in the newly built particle ac-
celerators, and the days of field theory seemed to be over, at least in the theory of the strong
interactions [Gro99]. First of all, one did not know which particles to use as the relevant
degrees of freedom, since they all seemed to be equally qualified at that time. In addition,
the couplings in the strong interactions were too large to admit a perturbative treatment.
In the mid-Sixties the quark model restored the order in the particle zoo, and it allowed a
group-theoretical classification of the observed hadron spectrum [GMN00]. In the Seventies
Quantum Chromo Dynamics (QCD) [GW73b, Wei73] arose as a field theory which could ex-
plain asymptotic freedom [GW73b, GW73a, Pol73]. Thus, one had a field theory at hand,
which in principle was capable to describe the observations, with the quarks as fundamental
fields. At short distances QCD was used to tackle many problems, and no contradiction to
experiment has been found. Unfortunately, the low-energy part of QCD can not be treated
in perturbation theory due to the increase of the coupling at low momentum transfers. To-
day, one still lacks an analytic tool for treating that region and calculating, for example, the
particle masses from QCD directly.
Besides the SU(3) colour gauge symmetry, the QCD Lagrangian possesses an approximate
global symmetry, the so called chiral symmetry. The chiral symmetry leads to conserved
charges and currents with opposite parity, and if that symmetry was realised in nature, every
hadron would have a chiral partner with degenerate mass but with opposite parity. These
parity partners are missing in the observed hadron spectrum, which suggests that the sym-
metry is spontaneously broken. The pseudoscalar mesons are very good candidates for the
Goldstone bosons of the broken symmetry, which are not exactly massless due to the chiral
symmetry breaking quark mass terms in the QCD Lagrangian. Chiral Perturbation Theory
(CHPT) [Wei79, GL84, GL85] describes the interactions among the lightest mesons in terms
ofaneffectivefieldtheory,whichisbasedonthesesymmetryproperties. Althoughonereturns
to a field theory in terms of non-elementary particles, one has a systematic way of treating
interactions in orders of momenta, opposed to an expansion in the coupling. The possible
interaction terms are constrained by the symmetry, which reduces the number of parameters
and endows the theory with predictive power. The momentum expansion, however, restricts
its applicability to energies well below 1GeV. It is also possible to include additional par-
ticles, as for example baryons and vector mesons, in a systematic way into the Lagrangian
[Kra90, CWZ69, Geo84], which leads to a chiral effective field theory with a broader applica-
bility.
1Chapter 1. Introduction
In the energy region between the applicability domains of CHPT and of perturbative QCD
onestillhastorelyonmodels,whichleadsbacktothequestionoftherelevantdegreesoffree-
dom. Is a given hadron, for example, a two- or three-quark state (constituent quark model
[GI85, CI86, PDG06]) or is it a bound state of two different hadrons (’dynamically gener-
ated’,’molecule’ - see below)? In any case, one does not have to start from the beginning, but
the experiences and constraints from QCD and CHPT should be incorporated in these mod-
els. The constituent quark model has been very successful in describing part of the observed
hadron spectrum, especially for the heavy-quark systems, e.g. charmonia and bottomonia
[Swa06]. On the other hand, especially in the light-quark sector, there is still a lively debate
about the nature of many hadronic states. One sector with a lot of activity is, for example,
the light scalar meson sector (σ,a (980),f (980),κ(900)). These states can not be explained0 0
within the naive constituent quark model, and many models have been proposed to explain
the phenomenology of these resonances. The suggestions for the nature of these resonances
vary between qq states, multiquark states, KK bound states and superpositions of them (see
e.g. [PDG06, AT04, Pen06] and references therein). A different route to explain the low-lying
scalars has been taken in [OO97, OO99]. In these works the authors the states as
being dynamically generated by the interactions of the pseudoscalar mesons. The scattering
amplitudes are calculated by iterating the lowest order amplitudes of CHPT, which leads to
a unitarisation of the amp