View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server IPNO/TH 94-85 hep-ph/9412261 How to obtain a covariant Breit typ e equation from relativistic Constraint Theory J. Mourad LaboratoiredeModeles de Physique Mathematique, UniversitedeTours, ParcdeGrandmont, F-37200 Tours, France H. Sazdjian Division de Physique Theorique , Institut de Physique Nucleaire, Universite Paris XI, F-91406 Orsay Cedex, France Octob er 1994 Unite de Recherche des Universites Paris 11 et Paris 6 asso ciee au CNRS. 1 Abstract It is shown that, by an appropriate mo di cation of the structure of the interaction p o- tential, the Breit equation can b e incorp orated into a set of two compatible manifestly covariantwave equations, derived from the general rules of Constraint Theory. The com- plementary equation to the covariant Breit typ e equation determines the evolution lawin the relative time variable. The interaction p otential can b e systematically calculated in p erturbation theory from Feynman diagrams. The normalization condition of the Breit wave function is determined. The wave equation is reduced, for general classes of p oten- tial, to a single Pauli-Schrodinger typ e equation. As an application of the covariant Breit typ e equation, we exhibit massless pseudoscalar b ound state solutions, corresp onding to a particular class of con ning p otentials. PACS numb ers : 03.65.Pm, 11.10.St, 12.39.Ki. 2 1 Intro duction Historically, The Breit equation [1] represents the rst attempt to describ e the relativistic dynamics of twointeracting fermion systems. It consists in summing the free Dirac hamiltonians of the two fermions and adding mutual and, eventually, external p otentials. This equation, when applied to QED, with one-photon exchange diagram considered in the Coulomb gauge, and solved in the linearized approximation, provides the correct 4 sp ectra to order [1, 2 ] for various b ound state problems. However, attempts to improve the predictivity of the equation, by developing from it a systematic p erturbation theory or by solving it exactly,have failed, due to its inabilityto incorp orate the whole e ects of the interaction hamiltonian of QED [2]. Also, this equation do es not satisfy global charge conjugation symmetry [3]. Finally, the Breit equation, as it stands, is not relativistically invariant, although one might consider it valid in the c.m. frame. Nevertheless, despite these drawbacks, the Breit equation has remained p opular. The main reason for this is due to the fact that it is a di erential equation in x-space and thus it p ermits the study of e ective lo cal p otentials with standard techniques of quantum mechanics. Improvements of this equation usually transform it into integral equations in momentum space, b ecause of the presence of pro jection op erators [3, 4]. The purp ose of this article is to derive, from relativistic Constraint Theory [5], a covariant Breit typ e equation, where the free part is the sum of individual Dirac hamilto- nians. The latter framework ensures the relativistic invariance of the equations describ- ing two-particle systems with mutual interactions [6, 7] and establishes the connection with Quantum Field Theory and the Bethe-Salp eter equation by means of a Lippmann- Schwinger-Quasip otential typ e equation relating the p otential to the o -mass shell scat- tering amplitude [8]. The fact that a covariant Breit typ e equation can b e obtained from Constraint Theory was already shown by Crater and Van Alstine [9]. In the present pap er we show that, for general classes of interaction, the covariant Breit typ e equation is equivalent to the Constarint Theory wave equations, provided it is supplemented with a second equation which explicitly eliminates the relative energy variable and at the same time ensures Poincareinvariance of the theory. The p otential that app ears in the main equation has a c.m. energy dep endence that also ensures the global charge conjugation symmetry of the 3 system. The pap er is organized as follows. In Sec. 2, we derive the covariant Breit typ e equa- tion from the relativistic Constraint Theory wave equations. In Sec. 3, we determine the normalization condition of the wave function. In Sec. 4, we reduce the wave equation, rel- ative to a sixteen-comp onentwave function, to a Pauli-Schrodinger typ e equation, relative to a four-comp onentwave function. As an application of the covariant Breit equation, we exhibit, in Sec. 5, massless pseudoscalar b ound state solutions, corresp onding to a par- ticular class of con ning p otentials, involving mainly pseudoscalar and spacelikevector interactions. Conclusion follows in Sec. 6. 2 The covariant Breit equation We b egin with the Constraint Theory wave equations describing a system of two spin-1=2 particles comp osed of a fermion of mass m and an antifermion of mass m ,inmutual 1 2 interaction [7] : e e e :p m = :p + m V ; (2.1a) 1 1 1 2 2 2 e e e :p m = :p + m V : (2.1b) 2 2 2 1 1 1 e Here, is a sixteen-comp onent spinor wave function of rank two and is represented as a 4 4 matrix : e e = (x;x ) ( ; =1;:::;4) ; (2.2) 1 2 1 2 12 where ( ) refers to the spinor index of particle 1(2). is the Dirac matrix acting 1 2 1 e in the subspace of the spinor of particle 1 (index ); it acts on from the left. is the 1 2 e Dirac matrix acting in the subspace of the spinor of particle 2 (index ); it acts on 2 e from the right; this is also the case of pro ducts of matrices, which act on from the 2 right in the reverse order : e e e e ( ) ; ( ) ; 1 2 1 1 1 2 1 2 2 2 1 e e ( ) ; = ; (a =1;2) : (2.3) 2 2 a a a 1 2 2 2 2i In Eqs. (2.1) p and p represent the momentum op erators of particles 1 and 2, resp ec- 1 2 e tively. V is a Poincareinvariant p otential. 4 The compatibility (integrability) condition of the two equations (2.1) imp oses con- ditions on the wave function and the p otential. For the wave function, one nds the constraint: 2 2 2 2 e (p p )(m m ) =0; (2.4) 1 2 1 2 which allows one to eliminate the relative energy variable in a covariant form. For eigen- functions of the total momentum op erator P , the solution of Eq. (2.4) is : 2 2 2 iP :X i(m m )P :x=(2P ) T e e 1 2 =e e (x ) ; (2.5) where wehave used notations from the following de nitions : 1 P = p + p ; p = (p p ) ; M = m + m ; 1 2 1 2 1 2 2 1 X = (x + x ) ; x = x x : (2.6) 1 2 1 2 2 We also de ne transverse and longitudinal comp onents of four-vectors with resp ect to the total momentum P : p (q:P ) L T 2 ^ ^ ^ P ; q =(q:P )P ; P = P = P ; q = q 2 P p 2 ^ q = q:P; P = P : (2.7) L L This decomp osition is manifestly covariant. In the c.m. frame the transverse comp onents reduce to the three spacelike comp onents, while the longitudinal comp onent reduces to T 2 2 the timelike comp onent of the corresp onding four-vector. (Note that x = x in the c.m. frame.) Also notice that, with the de nition of the longitudinal comp onents, P , L 2 which is the p ositive square ro ot of P , do es not change sign for negative energy states (under the change P !P); in this case, it is the longitudinal comp onents, q , of those L four-vectors which are indep endentofP that change sign, since these are linear functions ^ of P . For the p otential, one nds the constraint: 2 2 e e p p ;V =0; (2.8) 1 2 e which means that V is indep endent of the relative longitudinal co ordinate x : L T T e e V = V (x ;P ;p ; ; ): (2.9) L 1 2 5 Equations (2.5) and (2.9) show that the internal dynamics of the system is three- dimensional, b esides the spin degrees of freedom, describ ed by the three-dimensional T transverse co ordinate x . e The relationship b etween the p otential V and Feynman diagrams is summarizedby the following Lippmann-Schwinger-Quasip otential typ e [10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 8, 18 ] equation : e e e e V T VG T =0; (2.10) 0 2 3 i 6 7 T 0T 0 e T (P; p ;p ) T(P ; p; p ) ; 4 5 2P L 0 C (p);C (p ) where : i) T is the o -mass shell fermion-antifermion scattering amplitude; ii) C is the constraint (2.4) : 2 2 2 2 2 2 C (p) (p p ) (m m )=2Pp (mm)0; (2.11) LL 1 2 1 2 1 2 in Eq. (2.10) the external momenta of the amplitude T are submitted to the constraint C ; iii) G is de ned as : 0 e e G (p ;p )=S(p)S(p)H ; (2.12) 0 1 2 1 1 2 2 0 e e where S and S are the propagators of the two fermions, resp ectively, in the presence of 1 2 the constraint (2.11), and H is the Klein-Gordon op erator, also in the presence of the 0 constraint (2.11) : 2 2 2 2 P 1 (m m ) 1 2 2 2 2 2 2 2 T2 H =(p m) =(p m) = (m+m)+ +p : (2.13) 0 1 1 2 2 1 2 2 4 2 4P C C In order to obtain the covariant Breit equation, we de ne the covariant Dirac \hamil- tonians" : T T H = m :p ; (2.14a) 1 1 1L 1L 1 1 T T : (2.14b) :p H = m 2 2 2L 2L 2 2 We then multiply Eq.