SU(3) Chiral Symmetry in Non-Relativistic Field Theory

SU Chiral Symmetry in NonRelativistic Field Theory Thesis by Stephen M Ouellette In Partial Fulllment of the Requirements for the Degree of Do ctor of Philosophy I T U T E O T F S N T I E C A I H N N R O O 1891 L F O I L G A Y C California Institute of Technology Pasadena California Submitted January ii Acknowledgements I recognize three groups of p eople to whom I owe a particular debt of gratitude for the ability to nish the work of this thesis First and foremost are my partnerinlife Heather Frase my parents and my siblings Their unconditional love supp ort and pa tience have b een the glue which holds myworld together Also I wanttoacknowledge my mentors Mark WiseRyoichi Seki and Ubira jara van Kolck and my former ocemate Iain Stewart for the encouragementandvaluable insights they oered during our discus sions To those four p eople I oweeverything that I know ab out the practice of scientic inquiry Finally I want to thank the p eople of the Caltech Theoretical High Energy Physics group and a host of other characters Adam Leib ovich Erik Daniel Torrey Lyons for making my exp erience at Caltechenjoyable as well as educational iii Abstract Applications imp osing SU chiral symmetry on nonrelativistic eld theories are con sidered The rst example is a calculation of the selfenergy shifts of the spin decuplet baryons in nuclear matter from the chiral eective Lagrangian coupling o ctet and de cuplet baryon elds Sp ecial attention is paid to the selfenergy of the baryon near the saturation densityofnuclear matter We nd contributions to the mass shifts from contact terms in the eective Lagrangian with co ecients of unknown value As a second application weformulate an eective eld theory with manifest SU chiral symmetry for the interactions of K and mesons with pions at low energy SU chiral symmetry is imp osed on the eective eld theory by a matching calculation onto threeavor chiral p erturbation theory The eective Lagrangian for the K and sectors is worked out to order Q the eective Lagrangian for the KK sector is worked out to order Q with contact interactions to order Q As an application of the metho d we calculate the KK swave scattering phase shift at leading order and compare with chiral p erturbation the oryWe conclude with a discussion of the limitations of the approach and prop ose new directions for work where the matching calculation may b e useful Thesis Advisor Prof Mark B Wise iv Contents Acknowledgements ii Abstract iii Intro duction Theoretical Background Symmetries of QCD Chiral Perturbation Theory Coupling to Matter Fields NonRelativistic Power Counting Decuplet SelfEnergy in Nuclear Matter Eects of Nuclear Matter The EectiveLagrangian SelfEnergy Shift Results Discussion and Conclusion Corrections at HigherOrder SelfEnergy Near Saturation Density Conclusion Heavy KaonEta Eective Theory Prosp ectus Elements and Principles Reparameterization Invariance Use of Field Redenitions v The Eective Lagrangians The K and Sectors The K K Sector Matching Calculations Application KK Scattering A Symb olic Expansion of L PT B Recursion Relations for F Bibliography vi List of Figures Feynman diagrams for the selfenergy shifts at leading order of decuplet baryons in nuclear matter a mesonnucleon lo op diagrams and b con tact diagrams Double lines represent decuplet baryons single lines repre sentnucleons and dashed lines represent pseudoGoldstone b osons Leadingorder helicity splitting of the decuplet baryon selfenergy in nu h clear matter E E E as a function of neutron Fermi mo n and baryon momentum k a for the baryon and b for mentum p F the baryon Leadingorder resonance widths in nuclear matter as functions of neu n tron Fermi momentum p and baryon momentum k a for helicities F and b for helicities a Infrared divergent baryonmeson b ox diagram and b overlapping meson lo op diagram obtained from the b ox diagram bycontracting nucleon lines and The twolo op diagram in b avoids the infrared divergence of a Selfenergy of the baryon at rest in nuclear matter omitting contact diagrams d d The solid curve is the numerical solution of equa tion the dashed curve is a plot of equation Feynman diagrams contributing to the KK scattering amplitude in the heavy kaoneta eective theory a order Q tree diagram b order Q kaonbubble diagram c order Q tree doublebubble and pion exchange diagrams vii a Leadingorder calculations of the KK swave I scattering phase shift k in degrees as a function of centerofmass momentum k the dashed line is heavy kaoneta theory the solid line is SU PT b Rel PT PT NR ative error b etween the results j jj j viii List of Tables r Phenomenological values of the co ecients L renormalized in MS i at m taken from J Bijnens et al also see reference for dis cussion Transformations of elds and under G chiral symmetry parityPcharge conjugation C and hermitian conjugation yandthechiral index asso ciated with each eld Chapter Intro duction I once heard a wise man say In the history of scientic endeavor no problem has con sumed as much of mankinds resources as the understanding of nuclear forces or some thing like that In any case for all the considerable eort p oured into solving the mysteries of the strong interaction a numb er of signicant problems remain The fundamental the ory of the strong interactions Quantum Chromo dynamics QCD is solidly established as a pillar of the Standard Mo del of particle physics and to the extent that QCD is a renormalizable gauge theoryiswell understo o d in the p erturbative regime As a theory with asymptotic freedom QCD is p erturbative in the highenergy regime for low energies the coupling constant of the theory b ecomes large and p erturbative treatments break down Some features of nonp erturbative QCD which are still not fully understo o d are quark structure of hadrons dynamical symmetry breaking and quark connement The sp ecial diculties accompanying the nonp erturbative regime require sp ecial meth o ds for working within the theory One direct approachistoformulate QCD on a lattice of spacetime p oints and use numerical techniques to p erform the functional integrals The lattice QCD metho d has great p otential for shedding light on manyofthe unanswered questions of QCD and has b ecome an industry unto itself A complementary approach is to fo cus on the longdistance physics and base the eld theory description of the physics on the directly observed degrees of freedom Approaches of the second typ e are generically called eective eld theories and rely on a twopart foundation I refer to the rst imp ortant concept as the Weinb erg Hyp othesis the only content of quantum eld theory apart from the choice of degrees of freedom is analyticity unitarity cluster decomp osition and the assumed symmetry principles As a consequence we can describ e the strong interactions in terms of hadron degrees of freedom provided we write down the most general Lagrangian consistent with the symmetries of QCD The second key concept is that wemust identify some expansion parameter typically a small momentum or energy scale which p ermits us to calculate to anygiven order in the expansion with a nite amountofwork The predictivepower of an eective eld theory arises from the combination of the two underlying concepts the symmetry restricts the parameters of the theory to a meaningful set and the expansion parameter allows a systematic framework in whichwe can include all contributions of a given order and estimate the size of the contributions wehave neglected For a more detailed discussion of eective eld theory in general see references In the case of QCD the hop e is that by exp erimental determination of the param eters of the eective Lagrangian we can learn ab out the underlying theory in the non p erturbative regime p ossibly through lattice QCD as an intermediary An alternative p ossibility will also b e considered in this work If the fundamental theory at higher energy is known and calculable then at a momentum scale where the theories meet the parameters of the eective theory can b e determined bymatching onto the fundamental theory This is sometimes done b ecause certain calculations are more easily p erformed in the eectivetheory either b ecause of additional approximate symmetry in the low energy limit or b ecause a nonrelativistic framework may b e used For instance this sort of matching calculation has b een p erformed and applied with success in nonrelativistic QED NRQED and nonrelativistic QCD NRQCD In this thesis we consider two applications of eective eld theory to exploit the SU SU chiral symmetry of QCD In Chapter wecover the theoretical frame L R work up on which the eective eld theories will b e built We discuss the symmetries of QCD which constrain the eective Lagrangian the principles for constructing an eec tive Lagrangian for the hadron degrees of freedom and the p ower counting schemes that apply to the sectors of the theory with only light elds one heavy eld static case or more than one heavy eld nonrelativistic case In Chapter

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