Abstract
We derive the socalled Barbieri–Remiddi solution of the Bethe–Salpeter
equation in QED in its general form and discuss its application
to the bound state energy spectrum.
PACS: 12.20.m, 11.10.St, 36.10.Dr
HDTHEP9730
The Barbieri–Remiddi solution of the bound state problem in QED
[2cm] Antonio Vairo^{1}^{1}1Alexander von Humboldt Fellow ^{2}^{2}2
Institut für Theoretische Physik, Universität Heidelberg
Philosophenweg 16, D69120 Heidelberg, FRG
1 Introduction
The Bethe–Salpeter (BS) equation [1] is usually considered the rigorous framework in which to approach the bound state problem in QED. The increasing precision of the experimental data concerning QED bound states (e. g. decay rate, energy levels, etc. for some recent reviews we refer the reader to [2]) makes more and more urgent to effort the evaluation of physical quantities by handling the BS equation with a systematic and unified formalism.
In this paper we will focus our attention to the bound state energy levels in QED and will discuss the socalled Barbieri–Remiddi (BR) formalism. This formalism was first suggested for positronium [3, 4] (for a clarifying quantomechanical example see also [5]), but has been used in recent years also for hydrogenic atoms [6], QCD bound states [7] and scalarscalar bound states [8]. The main idea is to solve exactly the BS equation for a suitable zerothorder kernel containing the relevant binding interaction (i. e. the Coulomb potential) and then to perform a perturbative expansion in terms of the difference between the complete twobody kernel and the zerothorder one. What is appealing in this approach is that the zerothorder solution is completely known in analytic closed form. Therefore the perturbative expansion obtained in this way is completely selfcontained and does not need to be improved for higher correction in the fine structure constant .
In the following the BR formalism will be derived in the general case of muonium (i. e. different masses). This result is new and contains the positronium and hydrogenic case as a particular one. Moreover it furnishes a way to treat radiative and recoil corrections in the same theoretical framework, which seems to be very promising.
The paper contains two main sections. In section 2 we derive the perturbative expansion of the energy levels from the BS equation in the socalled Kato formalism. In section 3 we derive in some detail the BR solution for muonium. Section 4 is devoted to some comments and conclusions.
2 The Bethe–Salpeter equation
In this section we review some basics concerning the Bethe–Salpeter equation in QED and set up the theoretical background for the next section. The main result will be the perturbative expansion of the energy levels of the two fermion bound state given by Eq. (12).
Let us consider a system of two fermions (of mass and and electric charge and respectively) like muonium. The four point Green function is the sum of the Feynman graphs shown in Fig. 1 (notice that for a particleantiparticle system, like positronium, one has to add the annihilation graphs). Let us define the two fermions free propagator:
(1) 
where and refer to the two fermion lines, , is the bound state energy and , are the relative momenta of the outcoming and incoming particles in the centreofmass reference frame ^{1}^{1}1 Let and be the momenta of the outcoming particles, and
(2) 
This equation (for simplicity we have neglected the spinor indices) is known as the Bethe–Salpeter equation [1]. The kernel , describing the interaction between the two fermions, is not known in analytic closed form and is given by all the two particle irreducible graphs without external legs shown up to two loop in Fig. 2. Graphically the BS equation can be represented as in Fig. 3.
, as a function of , has simple poles in the bound state energy levels [9] ( is a convenient set of quantum numbers which classifies the levels). Since the Coulomb interaction is contained in , (without mass terms) has to coincide with the Bohr levels at the leading order in . Therefore for any , we can write
(3) 
where is the residuum at the pole, is non singular in the limit and is the reduced mass of the two particle system. From now on we will neglect the explicit indication of the momenta in the argument of the functions, where considered not strictly necessary.
Inserting (3) into the BS equation and comparing the residua, we obtain:
(4) 
which is known as the homogeneous Bethe–Salpeter equation. Moreover, from the comparison of the non singular parts in we obtain the normalization condition [10]:
(5) 
The BS equation (2) is, up to now, not solvable in analytic closed form. Let be an interaction kernel satisfying the following two properties:

reproduces the same non relativistic limit of , i.e. the Coulomb potential times some spinorial factors;

the BS equation for :
(6) is analytically solvable in closed form.
With these assumptions it is possible to solve the BS equation for at least perturbatively in terms of and to give a perturbative expansion for the bound state energy levels (the poles of ).
From the property it follows that has simple poles for . These poles are surely more degenerate than the poles of the complete Green function and give back, at the leading order in , the Bohr levels:
(7) 
where is the residuum at the pole and is non singular in the limit . The sum is extended over all the degenerate states for each . The residuum satisfies the analogous of equations (4) and (5):
(8)  
(9) 
From the definition of and from (2) and (6) we obtain the perturbative expansion of in terms of :
(10) 
In order to obtain from (10) the perturbative expansion of the poles we will use the socalled Kato perturbation theory [11]. Since the energy levels are the poles of we can write:
(11) 
where is a closed curve in lC which contains only the poles and of and respectively, is an operator which does not vanish on and and means the trace over the spinor indices. A convenient choice is
Inserting (10) in (11) integrating in and taking into account (9), we obtain (up to order ):
(12)  
where
is the degeneracy of the level and is defined to be
(13) 
Equation (12) expresses the bound state energy as an expansion in . Since is the difference between the sum of the infinit series of Feynman graphs drawn in Fig. 2 and the kernel , is not known in closed form. Each graph of Fig. 2 contributes to (12) with a series of powers of , because the dependence of the residuum on the fine structure constant (like in the wellknown nonrelativistic case where the hydrogen wavefunctions depend on ). For consistency with the explicit calculation must exhibit that to an increasing order in it corresponds an increasing leading order in in the contributions to the energy levels. In this sense expansion (12) can be interpreted as a perturbative expansion in the fine structure constant. Once is explicitly given and the corresponding BS equation is solved (this means we have an analytic expression for , and ) the expansion (12) allows to obtain without any ambiguity the energy levels up to a given order in for all the twofermions bound states in QED. We emphasize that, in absence of an exact solution of Eq. (6), expression (12) could be evaluated only for an approximate choice of to improve at any increasing of the requested precision.
3 The Barbieri–Remiddi solution
In this section we work out with some detail the socalled Barbieri–Remiddi solution of equation (6) (for and exhaustive description see [12]). With this name we mean a zeroth order kernel which satisfies the previous given properties and (in other words should describe correctly at the leading order in the bound state and make solvable the corresponding BS equation (6)) as well as the solution of the corresponding BS equation. The BR solution was first given for the positronium [3]. In the following we will give the generalization of that solution for a bound state of two fermions with different masses, i. e. muonium. Once is given, we solve the equation for and work out the poles and the residua . At that point the perturbative expansion of the energy levels (12) will be completely defined.
Let us define the energy projectors
(14) 
with . In terms of the free fermion propagator can be written as
Moreover, we define
The zeroth order BR interaction kernel for the muonium is
(16) 
with
(17) 
We assume (16) as a definition. In the following we will verify that this choice satisfies the properties and given in the previous section.
In the static limit (, and ),
i. e. reproduces the Coulomb potential times some spinorial factors.
In order to verify that the choice (16) makes solvable the BS equation (6) it is useful to express the Green function in terms of a new function ^{2}^{2}2 In general could depend on each component of the momenta and . The explicit calculation, however, will show that does not depend on and (see Eq. (20)).:
(18) 
Including (18) and (16) in (6), we have
Integrating on we obtain
(19) 
where
Equation (19) is nothing else than the Schrödinger equation for the propagator of a non relativistic particle in an external Coulomb field. Therefore its solution is known. A way to express it is by means of the Gegenbauer polynomia (for the definition and some properties see [13]) [14]:
(20) 
where . Substituting Eq. (20) in (18) we obtain the explicit analytic expression of the Green function corresponding to the kernel given by (16). As we will see, once is given, it is straightforward to work out the poles , the residua and .
From (20) we have immediately that has poles in
(21) 
Notice that up to order
The poles of give back the mass terms plus the Bohr levels, i. e. the physically correct levels up to order . Moreover we have
The residuum at the pole , as defined in (7), is
(22)  
where
(23) 
are the wellknown hydrogen atom wave functions [15]. Moreover, we can write
(24) 
where we have used the Fiertz identity:
with the definitions:
Eqs. (23) and (24) allow to identify the quantum numbers with the principal quantum number , with the numbers describing the spin of the bound state and with the numbers describing the angular momentum. The corresponding states are
(25)  
These states are not degenerate (how it is possible to verify directly by calculating ). Sometimes in the literature the residua at the poles of the Green function are written as
the functions and are called the BR wave functions of the bound state:
(26)  
(27) 
Actually there are no reasons to introduce the bound state wave functions: from the formalism developed in the previous section it is clear that all the physical quantities can be expressed in terms of residua.
4 Conclusions
From the above given expressions we can recover some interesting limiting cases.
Putting and ( and ) the above given zerothorder solution of the BS equation reduces to the original BR solution given in [3] for positronium. The main difference with the muonium case is that for positronium also annihilation graphs contribute to the interaction kernel . In the literature the singlet state () is usually referred as parapositronium and the triplet state () as orthopositronium. Some applications can be found in [12, 16, 17, 18, 19].
Taking the limit of one particle mass to infinity the case of a particle in an external Coulomb field is recovered. In this case (e. g. ) , the difference is finite and the bound state energy is given by
This case has been extensively studied in [6, 12, 20] for the evaluation of the pure radiative corrections to the energy levels of hydrogenic atoms.
As a conclusive remark, we stress that all these systems and muonium can be studied now in the same framework. In particular the formalism provides a powerful tool in dealing simultaneously with radiative, recoil and radiativerecoil corrections. Extremely interesting seems also to be the study of the infinit mass limit of one particle in the energy expansion (12) evaluated on muonium states. This should eventually clarify how this limit works in a purely off massshell context.
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