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Description of Bound States and Scattering Processes Using Hamiltonian Renormalization Group Procedure for Effective Particles in Quantum field Theory

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Description of Bound States and Scattering Processes Using Hamiltonian Renormalization Group Procedure for Effective Particles in Quantum field Theory Description of bound states and scattering processes using Hamiltonian renormalization group procedure for effective particles in quantum field theory Marek Wi˛eckowski June 2005 Doctoral thesis Advisor: Professor Stanisław D. Głazek Warsaw University Department of Physics Institute of Theoretical Physics Warsaw 2005 i Acknowledgments I am most grateful to my advisor, Professor Stanisław Głazek, for his invaluable advice and assistance during my work on this thesis. I would also like to thank Tomasz Masłowski for allowing me to use parts of our paper here. My heartfelt thanks go to Dr. Nick Ukiah, who supported and encouraged me during the writing of this thesis and who helped me in editing the final manuscript in English. Finally, I would like to thank my family, especially my grandmother, who has been a great support to me during the last few years. MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... ii MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... iii Abstract This thesis presents examples of a perturbative construction of Hamiltonians Hλ for effective particles in quantum field theory (QFT) on the light front. These Hamiltonians (1) have a well-defined (ultraviolet-finite) eigenvalue problem for bound states of relativistic constituent fermions, and (2) lead (in a scalar theory with asymptotic freedom in perturbation theory in third and partly fourth order) to an ultraviolet-finite and covariant scattering matrix, as the Feynman diagrams do. λ is a parameter of the renormalization group for Hamiltonians and qualitatively means the inverse of the size of the effective particles. The same procedure of calculating the operator Hλ applies in description of bound states and scattering. The question of whether this method extends to all orders in QFT is not resolved here. The relativistic Hamiltonian formulation of QFT is based on a global regularization of all terms in a relativistic operator H∆ (a canonical Hamiltonian with an ultraviolet cutoff ∆, plus counterterms). The renormalization group procedure for effective particles (RGPEP) makes it possible to find the structure of the counterterms in H∆ and calculate the effective Hamiltoni- ans Hλ for λ ranging from infinity down to λ on the order of masses of bound states. I investigate bound states of two relativistic fermions using Yukawa theory as an example. I give an explicit form of the effective Hamiltonian Hλ in the second order, and discuss the reduction of its eigenvalue equation to a Schrödinger equation for the wave function of the con- stituents. Every interaction term in the Hamiltonian Hλ contains a form factor fλ generated by RGPEP, which eliminates overlapping divergences in the bound-state eigenvalue problem expressed in terms of effective particles. The overlapping divergences appear in the eigenvalue problem expressed in terms of pointlike particles and without the form factors fλ, and result from relativistic relative motion of fermions. Such divergences appear in all Hamiltonian the- ories of pointlike particles with spin, and in particular in quantum chromodynamics (QCD). The advantage of the Yukawa theory is that it allows one to investigate the ultraviolet behav- ior in bound states of fermions without additional complications of QCD. The form factors fλ also cause the bound state to be dominated by the lowest sectors in the Fock-space basis built with effective particles. The ultraviolet complications of local QFT are contained in a complex structure that emerges in the effective particles as a result of dressing of the bare particles of the initial canonical theory. My description of scattering in an asymptotically free scalar field theory of φ3 type in 5+1 ∆ dimensions starts from constructing explicitly counterterms in H by calculating Hλ . I then use the Hamiltonian H∆ to calculate a scattering amplitude for a process analogous to e+e− → hadrons in perturbation theory up to the order e2g2, i.e., in one loop (e is an analogue of the electric charge of electrons in QED, and g of the color charge of quarks in QCD). I show that counterterms found using RGPEP without referring to the S matrix, remove the divergent regularization dependence from the calculated amplitude. I also give the explicit form of the finite parts of the counterterms in H∆ that lead to a covariant result for the scattering amplitude. I show that the dependence of the amplitude calculated this way on the momenta of the colliding particles, is the same as the dependence derived from Feynman diagrams (the diagrams are regularized covariantly and without defining a regularized Hamiltonian ab initio). I prove a theorem that states that the scattering amplitude obtained using H∆ is the same ∆ as the scattering amplitude obtained using Hλ . Note that in the calculation using H physical states of colliding particles are expressed in terms of bare particles and one uses the renor- ∆ malized interaction Hamiltonian HI . In the calculation using Hλ physical states of colliding particles are expressed in terms of non-pointlike effective particles and one uses the effective MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... iv interaction Hamiltonian Hλ I with form factors in all vertices. In the case of the considered am- plitude of the type e+e− → hadrons in a one-loop approximation, this theorem implies that the effective Hamiltonian Hλ leads to the same predictions for the scattering matrix as the Feynman diagrams. I also present an alternative, simplified procedure for deriving the Hamiltonian counterterms needed for the description of the scalar analogue of e+e− → hadrons. I illustrate the simpli- fied procedure by giving mass and some vertex counterterms in QCD coupled to QED, in the Appendices. MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... v Streszczenie Niniejsza praca podaje przykłady perturbacyjnej konstrukcji takich Hamiltonianów Hλ dla cz ˛a- stek efektywnych w kwantowej teorii pola (KTP) na froncie swietlnym,´ które (1) maj ˛adobrze zdefiniowany (ultrafioletowo skonczony)´ problem własny dla stanów zwi ˛azanych relatywistycz- nych fermionów-składników, oraz (2) w skalarnej teorii z asymptotyczn ˛aswobod ˛aw rachunku zaburzen´ do trzeciego i cz˛esciowo´ czwartego rz˛eduprzewiduj ˛aultrafioletowo skonczon´ ˛ai ko- wariantn ˛amacierz rozpraszania, tak ˛ajak diagramy Feynmana. λ jest parametrem grupy renor- malizacji dla hamiltonianów i jakosciowo´ oznacza odwrotnos´c´ rozmiaru cz ˛astekefektywnych. Ta sama procedura obliczania operatora Hλ stosuje si˛ew przypadku opisu stanów zwi ˛azanych i rozpraszania. Dowód stosowalnosci´ metody do wszystkich rz˛edóww KTP nie jest rozwa˙zany. Relatywistyczne hamiltonowskie sformułowanie KTP opiera si˛ena globalnej regularyzacji wszystkich wyrazów w relatywistycznym operatorze H∆ (hamiltonian kanoniczny z ultrafiole- towym obci˛eciem ∆, plus kontrczłony). Procedura grupy renormalizacji dla cz ˛astekefektyw- nych (RGPEP) pozwala znale´zc´ struktur˛ekontrczłonów w H∆, a nast˛epnieobliczyc´ hamiltonian efektywny Hλ dla λ z zakresu od nieskonczono´ sci´ do małych wartosci´ rz˛edumas stanów zwi ˛a- zanych. W przypadku stanów zwi ˛azanych dwóch relatywistycznych fermionów (zbadanym na przy- kładzie teorii Yukawy) podana jest jawna postac´ hamiltonianu efektywnego Hλ w drugim rz˛e- dzie rachunku oraz redukcja jego równania własnego do równania Schrödingera na funkcj˛efa- low ˛aefektywnych składników. Hamiltonian Hλ zawiera w oddziaływaniach czynniki kształtu fλ wygenerowane przez RGPEP, które usuwaj ˛anakrywaj ˛acesi˛erozbie˙znosci´ (overlapping di- vergences) w problemie własnym stanu zwi ˛azanego. Nakrywaj ˛acesi˛erozbie˙znosci´ pojawiaj ˛a si˛ew problemie własnym zapisanym za pomoc ˛acz ˛astekpunktowych bez czynników fλ, i s ˛a spowodowane relatywistycznym ruchem wzgl˛ednym fermionów. Takie rozbie˙znosci´ pojawiaj ˛a si˛ewe wszystkich hamiltonowskich sformułowaniach teorii cz ˛astekpunktowych ze spinem, a w szczególnosci´ w chromodynamice kwantowej (QCD). Zalet ˛ateorii Yukawy jest mo˙zliwos´c´ zba- dania ultrafioletowego zachowania fermionów w stanie zwi ˛azanym niezale˙znieod dodatkowych komplikacji w QCD. Czynniki fλ powoduj ˛arównie˙z,˙zew stanie zwi ˛azanym dominuj ˛anajni˙z- sze sektory Focka cz ˛astekefektywnych. Ultrafioletowe komplikacje lokalnej KTP zawarte s ˛aw skomplikowanej strukturze, która powstaje wewn ˛atrzcz ˛astekefektywnych na skutek ubierania si˛ecz ˛astekgołych wyjsciowej´ teorii kanonicznej. W przypadku teorii rozpraszania, jawna konstrukcja kontrczłonów w H∆ na podstawie obli- czen´ Hλ jest przeprowadzona w przypadku asymptotycznie swobodnej skalarnej teorii pola typu φ3 w 5+1 wymiarach. Nast˛epniehamiltonian H∆ jest u˙zytydo obliczenia amplitudy rozpraszania dla procesu analogicznego do e+e− → hadrony w rachunku zaburzen´ do rz˛edu e2g2, tj. w jednej p˛etli(e jest analogiem ładunku elektronów w QED a g ładunku kolorowego kwarków w QCD). Kontrczłony znalezione za pomoc ˛aRGPEP bez odwoływania si˛edo ma- cierzy rozpraszania, usuwaj ˛arozbie˙zn˛azale˙znos´c´ od regularyzacji w obliczonej amplitudzie. Podane s ˛ajawne wzory na skonczone´ cz˛esci´ kontrczłonów w H∆, prowadz ˛acedo kowariant- nego wyniku na macierz rozpraszania. Zale˙znos´c´ otrzymanej w ten sposób amplitudy od p˛edów zderzaj ˛acych si˛ecz ˛astekjest taka jak otrzymana z diagramów Feynmana (zregularyzowanych kowariantnie i bez definicji ab initio zregularyzowanego hamiltonianu). Podane jest twierdzenie, które mówi, ˙zeamplituda rozpraszania otrzymana za pomoc ˛a H∆ ∆ jest taka sama, jak amplituda otrzymana za pomoc
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