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Quark Dynamics and Constituent Masses in Heavy Quark Systems

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Quark Dynamics and Constituent Masses in Heavy Quark Systems QUARK DYNAMICS AND CONSTITUENT MASSES IN HEAVY QUARK SYSTEMS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Nicholas A. Souchlas August, 2009 Dissertation written by Nicholas A. Souchlas B.S., Patras University 2000 Ph.D., Kent State University, 2009 Approved by Dr. Peter Tandy , Chair, Doctoral Dissertation Committee Dr. Declan Keane , Members, Doctoral Dissertation Committee Dr. George Fai, Dr. John Portman, Dr. Anatoly Khitrin, Dr. Luthar Raichel, Accepted by Dr. Bryon Anderson, Chair, Department of Physics Dr. John R. D. Stalvey , Dean, College of Arts and Sciences ii Table of Contents ListofFigures ................................... v ListofTables.................................... xii Acknowledgments.................................. xix 1 Introduction .................................. 1 1.1 QCDasatheoryofstronginteractions . 1 1.2 NonperturbativeQCDstudies . 4 2 QCD essentials ................................ 9 2.1 QCD Lagrangian density and generating functional. ...... 9 2.2 Two Dyson-Schwinger Equations: quark propagator gap equation and Bethe-Salpeter bound states equation. 12 2.3 Normalization and decay constants for pseudoscalar and vector mesons. 16 2.4 Dynamical chiral symmetry breaking (DCSB) and confinement. ... 18 3 Dynamically dressed quark propagator ................. 27 3.1 Chiral symmetry and the axial vector Ward-Takahashi identity (AV WTI). ................................... 27 3.2 Rainbow-Ladder Truncation and the Maris-Tandy model. ..... 31 3.3 Gapequationsolution. .......................... 33 3.4 Quark propagator within meson dynamics. .. 42 3.5 Resultsforquarkoniamesons. 57 iii 4 A dressed quark propagator parametrization ............. 72 4.1 Complex conjugate mass pole parametrization. ..... 72 4.2 Meson masses using the 3ccp propagator representation. ....... 84 4.3 Meson electroweak decay constants using the 3ccp propagator repre- sentation. ................................. 94 5 The constituent mass concept for heavy quark mesons ....... 99 5.1 Definitionsofaheavyquarkmass.. 99 5.2 Studying heavy quark systems. 101 5.3 MT effective kernel and heavy quark mesons. 103 5.4 Mesonmasses................................ 104 5.5 Mesondecayconstants.. 112 5.6 The extreme heavy quark limit. 118 5.7 Roleofheavyquarkdressing. 122 5.8 An effective kernel with quark mass dependence. .... 132 5.9 Recovery of a qQ meson mass shell. 137 6 Remarks and Conclusions ......................... 139 References...................................... 144 iv List of Figures 3.1 σs and σv functions in the chiral limit using the MT and modified MT (mMT)model. .............................. 37 3.2 σs and σv functions for the u/d, s, c quark using modified MT model. Unstable behavior for all quarks starts at about 1 GeV 2, limiting ∼ − the use of the solutions in studying mesons lighter than 2 GeV. ... 38 3.3 σs and σv functions for the c quark on the real axis using the MT and mMT model. Unstable behavior for the c quark starts at about 1 GeV 2, but with the MT model we have stable solution deeper ∼ − inthetime-likeregionontherealaxis. 39 3.4 Re(A) for the c quark in a parabolic region in the complex plane (P 2 = 14 GeV 2, η=0.50) with integration over the gluon momentum (kcp). − There isn’t any unstable behavior or any singularities. In this plot and in the following ones the straight lines connecting the end-points have no significance and should be ignored. 44 3.5 Im(A) for the c quark in a parabolic region in the complex plane (P 2 = 14 GeV 2, η=0.50) with integration over the gluon momentum − (kcp). Im(A)hasasmoothbehaviortoo. 45 3.6 Re(M) for the c quark in a parabolic region in the complex plane (P 2 = 14 GeV 2, η=0.50) with integration over the gluon momentum − (kcp). Like in function A there isn’t any unstable behavior or any singularities................................. 46 v 3.7 Im(M) for the c quark in a parabolic region in the complex plane (P 2 = 14 GeV 2, η=0.50) with integration over the gluon momentum − (kcp). ................................... 47 2 3.8 Re(σs) for the c quark in a parabolic region in the complex plane (P = 14 GeV 2, η=0.50) with integration over the gluon momentum (kcp). − The peak of the parabolic region is at (ηP )2 = 3.5 GeV 2. There − isn’t any unstable behavior but we see indications of the existence of a pair of complex conjugate singularities near the peak of the region, approximately located at (x , y ) (-2.7, 3.5) GeV 2. The type of the o o ∼ ± singularitiesisunknown. 48 2 3.9 Im(σs) for the c quark in a parabolic region in the complex plane (P = 14 GeV 2, η=0.50) with integration over the gluon momentum (kcp). − We have the same singularities as in the real part and no instabilities. 49 3.10 Re(σs) for the c quark in a parabolic region in the complex plane (P 2 = 8 GeV 2, η=0.50) with integration over the gluon momen- − tum(kcp) and keeping only the first IR term of the kernel. The peak of the parabolic region is at (ηP )2 = 2.0 GeV 2. The solution is dif- − ferent but we still have the general characteristics of the solution from the complete MT model. The first pair of singularities approximately appear to be (x , y ) (-1.7, 2) GeV 2. ................. 50 o o ∼ ± vi 2 3.11 Projection of Re(σs) for the c quark onto the real plane xz (x=Re(q+), z=Re(σs)) with integration over the gluon momentum(kcp), with the complete MT model and keeping only the first term in the model. One can clearly see now that in the second case there is a shift of the singularities towards the space-like region. .... 51 3.12 Re(σs) for the c quark in a parabolic region in the complex plane (P 2 = 8 GeV 2, η=0.50) with integration over the gluon momentum − (kcp) and keeping only the UV term in the model. The peak of the parabolic region is at (ηP )2 = 2.0 GeV 2. The first singularity is on − the real axis around (x , y ) (-1.9, 0) GeV 2. ............. 53 o o ∼ 3.13 Re(σs) for the b quark in a parabolic region in the complex plane (P 2 = 72 GeV 2, η=0.50) with integration over the gluon momentum − (kcp) with the IR term of the model. The peak of the parabolic region is at (ηP )2 = 18.0 GeV 2. The first pair of singularities now appear − much closer to the space-like region and closer to the time-like real axis. Roughly their location is at (x , y ) (-17.2, 7.5) GeV 2. ... 55 o o ∼ ± 3.14 First Chebychev moment of the dominant invariant for the ss¯, cc¯, b¯b pseudoscalar systems, from the full MT model solution and the solution with only the IR term. For the b¯b quark system we also have an UV calculated amplitude since we can reach a mass shell even with only theweakerUVtailtermintheinteraction. 58 3.15 Plot of the full MT model and the model with only the IR or UV tail term. The IR term will mostly determine the strong behavior of the model for k2 < 1 GeV 2 and beyond that point the UV dominates. 60 vii 3.16 Plot of the product MT σ2 for the ss¯, cc¯, b¯b quark systems compared · s with the MT model behavior for external momentum p2 = 0. In all cases the propagator amplitude along the real axis is for the on-shell solution of the BSE. One can clearly see the increasing suppression imposed by the amplitude in the IR region and the decreasing sup- pression on the UV tail term, as the current quark mass increase. This IR suppression is mild compared to the other two cases since the am- plitude σs has also the mass function in the numerator, moderating the decrease of the amplitude. Same things are true for every p2 > 0 but there is a significant scale down as p2 increase.. 62 3.17 Plot of the product MT σ2 for the ss¯, cc¯, b¯b quark systems compared · v with the MT model behavior for external momentum p2 = 0 . In all cases the propagator amplitude along the real axis is for the on-shell solution of the BSE. Once again we see the increasing suppression imposed by the amplitude in the IR region and the same, for all cases after certain q2, suppression of the UV tail term, as the current quark 2 mass increase. Since the σv amplitude doesn’t also have the quark mass function in the numerator to moderate the decrease, we notice a much stronger suppression in the IR region as the current quark mass increase. This suppression extents somehow in the UV region, but for q2 > 10 GeV 2 is the same for all cases. Finally notice that for the s quark system we observe an enhancing effect for the MT model in the IR region. Same things are true for every p2 > 0 but there is a significant scale down as p2 increase. .................. 63 viii 3.18 Plot of the product MT σ σ for the ss¯, cc¯, b¯b quark systems compared · s · v with the MT model behavior for external momentum p2 = 0. In all cases the propagator amplitude along the real axis is for the on-shell solution of the BSE. As one would expect, from the two previous plots, the degree of the effects in the IR and UV region is somewhere between the degree of the effects of the other two terms. Same things are true for every p2 > 0 but there is a significant scale down as p2 increase. 64 4.1 Variation of the real parts of parameters mi (i=1,2,3) in the 3ccp rep- resentation, for current masses from the chiral limit up to 5.5GeV.
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