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Solving the Bethe-Salpeter equation for positronium
Nuclear Physics B141 (1978) 413-422 © North-Holland Publishing Company
SOLVING THE BETHE-SALPETER EQUATION FOR POSITRONIUM R. BARBIERI Scuola Normale Superiore, Pisa, Italy INFN, Sezione di Pisa E. REMIDDI * Physikalisches Institut der Universitiit Bonn, Germany Received 10 April 1978
We propose a Coulomb-like kernel for the relativistic two-fermion Bethe-Salpeter equation, to be used as the lowest-order approximation in systematic perturbative calculation of bound-state energy levels in QED. The kernel is symmetric in the two fermions and for the exchange of in and out momenta. The resulting equation is exactly soluble, unlike previously considered unperturbed kernels. We give explicitly the Green function and eigenfunctions. We also discuss the problem of the behaviour of the wave functions at zero relative coordinate in connection with the contribution to energy levels from the one-photon annihilation channel in QED.
1. Introduction Why another paper on such a "classical" subject as the bound state problem in quantum electrodynamics? The manifestly good reasons are the following. Experiments on pure QED bound states, positronium and muonium, have reached in recent years a very high degree of accuracy [1,2], almost comparable to the famous (g - 2) experiments of previous years; on the other hand the formulation of the boundstate problem in QED, as given by the relativistic Bethe-Salpeter equation [3] (BSE), although theoretically complete, suffers from the lack of a tractable, systematic computation procedure. This has become more and more clear precisely because of the improvement in the experimental investigation of the bound states, which calls for comparably accurate theoretical predictions. The present situation concerning the ground state splitting of m u o n i u m and positronium is summarized by saying that the experimental determination of these quantities is at the ppm level, whereas the corresponding theoretical predictions are worse * Permanent address: lstituto di Fisica, Universita di Bologna, Italy, and INFN, Sezione di Bologna 413
414
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
by one or two orders o f magnitude, so that the study of muonium and positronium is now perhaps the most interesting open problem in QED. Let us recall more precisely the difficulties encountered in treating systematically the bound states in QED by means of the BSE. As is well known, the BSE is an integral equation whose kernel is formally given as a power series in the fine structure constant or. To find its eigenvalues one has to specify a lowest-order equation - we will see the freedom present in this specification - and carry out on it a perturbative calculation. The main difficulty in pursuing this approach has been the absence of known analytic solutions to the equation even for an approximate Coulomb-like interaction kernel. In general, any equation to be used as a lowest-order approximation to the complete QED BS equation will have a kinetic and a Coulomb-like interaction term, able to produce bound states and supposed to be the largest part of the full BS kernel. As a first obvious condition, in the low-momenta limit both terms must reduce to the kinetic energy and the Coulomb potential of the non-relativistic Schr6dinger equation, with relative corrections o f second order in the momenta, so that the energy eigenvalues o f the bound states are given by the Balmer formula and unwanted o~3 terms in the eigenvalues do not appear. As a second condition it is advisable that in the high-momenta limit, where the interaction vanishes, the kinetic term describes exactly the relativistic free propagation o f the two fermions; this feature guarantees a proper treatment of the free highenergy intermediate states, which appear in the subsequent implementation o f a systematic perturbative expansion to the full BS equation, based on the considered lowest-order equation. The two conditions are indeed fulfilled by the Salpeter [4] approximation to the full BS equation, in which the kinetic term accounts exactly for the free relativistic propagation at all momenta. Unfortunately, exact analytic solutions of the approximated Salpeter equation (as well as of any other similar equation, such as e.g. the Breit equation) are not known; the lowest-order bound-state wave function is known explicitly only in the non-relativistic limit, better approximations to it are obtained through the Salpeter iteration procedure [4] (the iteration procedure is not even a systematic perturbativc expansion in some small parameter, and its use becomes more and more delicate, if not dangerous, when it is pushed to a precision higher than the one envisaged by Salpeter). Despite these shortcomings, in the absence of a better working approach, the Salpeter equation - or some improved version of it [5 ] - has been the starting point of most of the perturbative calculations performed up to now for positronium and muonium. As a last remark let us recall that the Dirac equation for an electron in an external Coulomb potential can be solved exactly and gives the right bound-state energies up to order 0~4 (fine structure), so that one could think of using the Dirac equation as convenient lowest-order equation. Unfortunately this is not the case, for the follov,. ing reasons: the Dirac equation for one fermion in a Coulomb potential is obtained by the complete two-fermion BSE in a rather indirect way, i.e. by retaining as interaction kernel the sum o f all irreducible crossed ladder graphs in the limit of infinite
R. Barbieri, k'. Remiddi / Bethe-Salpeter equation for positronium
415
mass of one of the fermions; no equation for two fermions is known which can be solved exactly and which gives the correct ct4 structure; in the case of positronium, furthermore, as the two constituent fermions have equal mass, spin-spin forces also ~ve ct4 contributions of the same order of magnitude as the fine structure (in hydrogen and muonium, spin-spin forces give hyperfine contributions, of order a 4 times the ratio of the masses). It is not at all necessary, however, that the lowest-order equation we are looking for gives by itself the a4 structure; the requirement is simply that the correct a 4 terms, as well as all higher-order corrections to the energy levels, are obtained in the systematic perturbative expansion to be built starting from the lowest-order equation in question. In sect. 2 of this paper we present an equation which can be solved exactly and which has the required properties in order to be used as the lowest-order equation in a systematic perturbative expansion for the solution of the complete QED BS equation. Our equation is a relativistic equation for the Green function of two fermions with the same kinetic part as the BSE and Coulomb-like kernel which reduces to the usual Coulomb potential in the non-relativistic limit. We give explicitly the relativistic Green function for two fermions which solves our equation; it is symmetric in in and out particles and is expressed in momentum space in terms of the corresponding non-relativistic Coulomb Green function of appropriate arguments times suitable spinors and kinematical factors. From the Green function, the exact properly normalized bound-state wave functions, as well as their conjugates, are easily obtained (without the usual normalization problem arising when only the BSE for the wave function, not the BSE for the Green function is investigated). The systematic perturbative expansion for the energy levels of the complete BSE in terms of such exact wave functions and the Green function is easily obtained. In sect. 3 we then comment on the specific convergence problems one encounters in the ultraviolet when dealing with the one-photon annihilation contribution to the eigenvalues in positronium. We will see how our lowest-order exact solution, which is better behaved at high momentum than the Salpeter wave function, cuts off naturally the loop integrals and allows one to treat the divergences as in the normal Feynman graph expansion o f the complete Green function. In the appendix, finally, we briefly discuss the relation between our work and the results of the interesting recent papers by Caswell and Lepage [6,7], who also obtain exact solutions to suitable equations describing the relativistic two-fermion problem. The main difference between the two approaches is that in refs. [6,7] the BSE forrealism is abandoned, by dropping the relative energy dependence, in favour of equations with a simpler kinematical structure but a more complicated effective kernel.
2. The lowest-order equation Let us consider for simplicity the case of particle-particle scattering, with equal
416
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
~P'p
½ P.*
:s Ip.p
-
l
a
"~P.p
..•pop,
-~P-p
Ip.¢
"J
-~1 P.q 1
Fig. 1. The equation for the Green function.
masses m i
m2 = 1. The equation which we want to solve is
=
. £d4p ' Go(P, P, q) = Go(P , P) (27r)484(p - q) + t J ~ - ~ Kc(P,
p, p') G¢(e,p', ql([)
and is graphically represented in fig. 1. As in the BSE, Go(P, p) is the free two-fermion propagator
Go(P, P)
l i p + ~r',- 1 - ie
il ~ - i~--+ 1 - ie_J
'
where (1), (2) refer to the upper and lower fermion lines of fig. 1. For later use, it is convenient to rewrite Go(P, p), in the c.m.s, w i t h P o = 2W, as
G°(P'P)=-
I X
A+(p)3'~ W+po-E(p)+ie
A+(-p) 3'4 W-Po-E(p)+ie
A__(p)3'._
+
] <')
W+po +E(p)-iel A _ ( - p ) 3'4
w - Voo+ ~ -
1 (2)
~
'
(3)
where, as usual E(p) =x/P~ + 1 ,
&(p) = E ( p ) -+ (1 - i p ' ~ ) 3 " 4 2E(p)
(4)
so that A2(p) = A+(p),
A+(p) A _ ( p ) = A _ ( p ) A+(p) = 0 ,
½(1 + 3'4) A+(p) ½(1 + 3'4) -
E(P) + 1 ½(1 + 3',). 2E(p)
(5) (6)
We define also _
X(p, q)
2E(p) 2E(q) [A+(p)~(1 +3'4)A+(q)] 0) E(p) + 1 E(q) + 1
X [A+(-R) ½(1 + 3'4) A + ( - q ) ] (2)-
(7)
R. Barbieri, E. Remiddi /Bethe-Salpeter equation for positronium
417
On account of (5), (6), one has h(p, p') ;k(p', q) = h(p, q ) ,
(8)
where the r.h.s, is independent ofp'. The Coulomb-like kernel which we propose for eq. (1) is Kc(P, P, q) = ")'(')~/(2)X(P,q)R(W, p) Vc(p - q) R(W, q) ,
(9)
where 2 R(W, p) = E(p) + W '
(10)
and Ve(k) is the scalar Coulomb potential vdk)
47rex - t, 2 .
(1 l )
Eq. (1)is then solved by the ansatz Go(P, p, q) = Go(P, p)[(Z¢r)484(p - q)
+ 7(')70)•(p, q) iR(l¢, p) He(W, p, q) R(W, q) Go(P, q)] ,
(12)
where He(IV, p, q), the unknown, is a scalar function. By inserting eqs. (9), (12) into eq. (1), using eqs. (3), (5), (8), after many simplifications we get Hc(W,p,q)= Ve(p - q ) dp' f~_~ +
--2 I¢ -'" /'dp0 1 V ( p - p ' ) t ~ ( 'P)'/-~ni [ W + p ' o - E f p ) + i e ] [ W - p ' o - E ( p ' ) + i e ]
X Hc(W, p ' , q ) .
(13)
The integration on the relative energy p~) is done and the equation becomes Hc(W'p'q)= V c ( P - q ) + f ~n)3 Vc(P - P ' )p,2 - ( W 21 - l ) - t e. He(W,p,,q) " (14)
Let us now consider the Schr6dinger Coulomb problem. Putting the mass equal to 1 2, the equation for the non-relativistic Green function gc(W, p, q) is
gc(cO,p,q)
p2 _ co - ie
Putting 11(16) gc(w,p, q) - p2 _ coI1 -- ie (27r)383(p - q) + hc(~,p, q)q2 _ co - ie
(15)-
418
R. Barbieri, E. Rerniddi / Bethe-Salpeter equation for positronium
OF
hc(co, p , q ) = _(p2 _ co)[(2n)363(p _ q) _gc(co,p,q)(qZ __ co)],
(17)
the equation for he(co, p, q) reads p
hc(co,p,q) = Vc(P -
q) +f(2@-~) Vc(P
-P')p'2 _co--I-re" hc(co,P',q)
. (18)
By comparing eqs. (14), (18) one finds Hc(I¢, p, q) = he(W: - 1,p, q ) ,
(19)
i.e.
H c ( W , p, q) : - ( E 2 (p) - W2)[(21T)363(p -- q) - - g c ( W 2 -- l , p , q X E 2 ( q )
- W2)] ,
(20)
which gives, when inserted in the ansatz (12), the solution of the relativistic equation (1) in terms of the solution of the non-relativistic Schr6dinger equation (16). Note that the solution Gc(P, p, q) so obtained does depend on relative energies Po, qo, although He(W, p, q) does not. The Schr~Sdinger Green function go(co, P, q) which solves eq. (15) is known, the most convenient way of expressing it being perhaps the Schwinger one-dimensional integral representation [8]. It has bound-state poles at values of co corresponding to con = __.~2/4n2 ' the familiar Balmer formula for mass 1.
go(co,P, q) ~ ~Pn(P)~n( q) w - c o n +ie '
(21)
where the ~Pn(P) are the non-relativistic, hydrogen-like, normalized bound-state Schr6dinger wave functions, for instance a
(22)
for the ground state. Corresponding to any pole ofge at co = COn,we see through eqs. (12), (20) that Go(P, P, q) has bound-state poles at values of W2 given by Wn2 = 1 + con. Writing, at a pole,
Gc(P, p, q) "" i ~sn(W' p) ~;sn(W' q) W2 - Wn2 + ie
(23)
(s refers to any of the four spin states, in which we have complete degeneracy) one obtains by comparison the relativistic bound-state wave function q;sn(W, p), a 4 × 4 spinor depending also on I41,Po [q~(W, p)]o -
~4A÷(p) 3(1 + ~'4
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
X
-'
i ' ~ - i~ r+ I - i~ " Y 4 A + ( - P ) / ( 1
+ "/4)
]
419
(rs)~:
XR(W, p)(/:-2 (p) _ W2) ~Sn(p) '
(24)
where r s is a constant 4 X4 spin wave function:
"
:\0
'
~_+, =,,/~(1,_+i, 0 ) ,
t0 = (0,0, 1).
(2s)
The function ~s(W, q) is the conjugate of ~s(w, p), eq. (24), with the prescription, however, that the sign of - i e in the propagator does not change• For completeness, we give the homogeneous equations satisfied by the boundstate wave functions and their conjugate• d4 ,
Cs(wn, p) = iGo(Pn, P)
c(Pn, P, p') ~n(Wn, P ) .
--s • £ d4q' ~s dgn(Wn, q) = t J ~rr--~ ~un(Wn, q ) Kc(Pn, q', q) Go(Pn, q) . '
(26)
where "°no = 2Wn- One can verify that the Mandelstam normalization condition [9] for the wave functions is indeed satisfied if the Cn(P) in the r.h.s, o f e q . (24) have the usual normalization
fgT
l.
The complete BSE has the form o f e q . (l), with Kc replaced by the full QED kernel K, the sum of all the two-particle irreducible Feynman graphs. Write K =-Kc + 6K ,
(28)
so defining the perturbation 5K as
5K = K - Kc •
(29)
The underlying physics is the assumption that among all the terms appearing in the formal power expansion in ~ which defines K, the most important one, responsible for the generation of bound states, is the Coulomb part present in the non-relativistic limit of the one-photon exchange graph• As also Kc, eq. (9), reduces to the Coulomb potential in the non-relativsitic limit, 6K has a term of order a but regular at low momenta while all other terms have higher powers o f ~ , so that 6K can be treated as a perturbation with respect to Kc. The explicit knowledge of the bound-state wave function and of the Green function of the lowest-order equation allows us to write down a systematic perturbative
420
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
expansion for the energy levels of the complete equation, although Kc depends on W and, due to the presence of relative energies, the concepts of completeness and orthonormality of the solutions do not apply in a straightforward way. Such an expansion is indeed given in ref. [6], by extending a method due to Kato [10], and we do not repeat it here. Typical terms in the expansion for the energy level shifts are of the form anf(a), where an gives the leading a order of the contribution, essentially fixed by the powers of a of the considered terms in 6K, while f(a), usually given as an expansion in a (and possibly lga), is obtained when taking the appropriate covohition integrals of the concerned 6K terms with the unperturbed wave tunction and Green function, which also depend on a but in a non-analytic way. Explicit calculations of positronium hyperfine splitting based on eqs. (1), (9) as the lowest-order equation are in progress. 3. Renormalization of the one-photon annihilation channel and behaviour of the wave function at zero relative coordinate A special problem encountered in the study of positronium is the renormalization of the one-photon annihilation channel. In this case the exact renormalized BSE for the wave function, in the "sidewise" notation proper to the particle-antiparticle case, is • £d4p ' [ (P, p) = S'(½P + p) t 3 ~ /((P, P, P') ~,(e, p ' ) Z] 1 2 Tr(Tu~k(P,p, ))J7 S ' ( - ] lP + p ) , + 47raTu ~-~4W
(30)
3
where K is the renormalized, one-photon irreducible electron-positron kernel, the second term in the square bracket is the one-photon annihilation interaction, S' is the dressed renormalized fermion propagator, and Zt, Z3 are the usual vertex and photon wave function renormalization constants. The one-photon annihilation term in the r.h.s, of eq. (30) is therefore responsible for the presence of counter terms in the renormalized BSE, which are not there to cancel divergences of the kernel itself. A strictly analogous situation is encountered when studying, for instance, the Low equation for the photon propagator, and is related to the overlapping-divergences problem. In this case, one knows how to write down an equivalent equation whose solution is finite when developed in perturbation theory, since the equation itself no longer contains the divergent counter term. It would be very valuable to get something similar (i.e. a counter-term-free equation) also for the positronium eigenvalue problem * even if what we propose in the following is completely satisfactory for practical purposes. • One o f the a u t h o r s (R.B.) acknowledges a correspondence with Professor K. Symanzik on this subject.
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
421
The presence of counter-terms demands that the ultraviolet regulator should be kept also in K and sent to o~ only at the very end of the calculation of the eigenvalues. This problem is typical of positronium; in the case of muonium, for instance, the annihilation term is absent and all the counterterms present in K serve to cancel the divergence in K itself. In other words, in muonium the regularizing parameter can be sent to infinity before solving the equation since no divergence is generated when integrating the equation iteratively. Proper handling of the problems posed by the presence of the divergent counterterms in the one-photon annihilation channel of the positronium kernel requires a precise knowledge of the zero relative coordinate behaviour of the solution of the lowest-Order equation. In fact, when evaluating the one-photon annihilation contribution to the energy levels one gets a term of the type AEccaZ~ ~ T r [ ~ ( 0 )
Tu] Tr[Tu~(0)] ,
(31)
where
f d4p q;(p, ~(0) = d ~ - ~ p).
(32)
When the lowest-order equation is for instance the Salpeter [6] equation, the corresponding wave function develops an anomalous dimension, behaving for large momentum p and small a as p-(,*-ea), e > 0, and ~(0) is divergent. In a systematic expansion in a one should therefore compensate the divergences coming from ~(0) with those from Z 1 (this could perhaps be achieved by introducing an ultraviolet regulator A also in the lowest-order kernel of the Salpeter equation, then trying to solve the equation for large finite A, so obtaining an asymptotic behaviour p-aA2/(p + A 2) for the wave function). In the case of our lowest-order equation, on the contrary, ~(0) is finite, as ~k(p) ~ p -O+lt2). The divergences appearing in the Salpeter wave function are naturally displaced in our approach to the perturbative expansion of the eigenvalues, since the highenergy tail of the one-photon direct exchange is treated as a perturbation of the low-energy binding potential and not kept in the lowest-order interaction kernel itself. This clearly increases the systematicity of the computational procedure, since one keeps control of these divergences in the same way as in the normal Feynman graph expansion of, say, the vacuum polarization amplitude. One of the authors (E.R.) wants to thank W. BuchmiJller for carefully reading the manuscript. Appendix We comment briefly on the relation between the present paper and the recent works by Caswell and Lepage [6,7]. In ref. [6], the BSE for differem m ~ e s is con-
422
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positroniurn
sidered, and the relative energy is frozen by forcing on the mass shell one of the two particles (say the heavier one, but the procedure works for arbitrary masses and is used also for positronium). A three-dimensional integral equation is so derived, whose kernel is obtained by suitably iterating the BS kernel, but with the same eigenvalues as the original BSE. For a proper choice of a lowest-order Coulomb-like kernel, the equation becomes a relativistic Dirac equation for a single fermion in a Coulomb potential and is therefore solved exactly. The wave function is of course less simple than eq. (24) of the present paper, and no attempt is made to obtain the corresponding Green function. The perturbative expansion for the energy levels (already referred to at the end of sect. 2) is then proposed, and successfully applied to t~61g ct terms of muonium and positronium hyperfine splitting. In ref. [7], which became known to us only after the completion of this paper, the starting point is again the BS equation. The relative energy dependence is also eliminated, this time putting the relative energies equal to zero, and the equation is further restricted to the subspace of positive-frequency spinors. By taking in this subspace as lowest-order diagonal interaction the scalar part of the rda.s of our eq. (9), an equation which is essentially our eq. (14) is obtained, solved and used as the starting point of a systematic perturbative expansion for energy levels. Some new a6 terms for muonium and positronium are then evaluated.
References [1] P.O. Egan et al., Phys. Lett. 54A (1975) 412; A.P. Mills, Jr. and G.H. Berman, Phys. Rev. Lett. 34 (1975) 246. [2] D.E. Casperson et al., Phys. Rev. Lett. 38 (1976) 956; H.G.H. Kobrak et al., Phys. Lett. 43B (1973) 526. [3] E.E. Salpeter and H.A. Bethe, Phys. Rcv. 84 (1951) 1232. [4} E.E. S',dpeter, Phys. Rev. 87 (1952) 328. [5] V.K. Cung, T. Fulton, W.W. Repko, A. Schaum and D. Devoto, Ann. of Phys. 98 (1976) 516. [61 G.P. Lepage, Phys. Rev. A16 (1977) 863. [7] W.E. Caswell and G.P. Lepage, Preprint SLAC-PUB-2080 (1978). [8] J. Schwinger, J. Math. Phys. 5 (1964) 1601. [9] S. Mandelstam, Proc. Roy. Soc. A233 (1955) 248. [101 T. Kato, Prog. Theor. Phys. 4 (1949) 154; A. Messiah, Quantum mechanics, vol. 2 (North-Holland, Amsterdam, 1962) p. 712.
SOLVING THE BETHE-SALPETER EQUATION FOR POSITRONIUM R. BARBIERI Scuola Normale Superiore, Pisa, Italy INFN, Sezione di Pisa E. REMIDDI * Physikalisches Institut der Universitiit Bonn, Germany Received 10 April 1978
We propose a Coulomb-like kernel for the relativistic two-fermion Bethe-Salpeter equation, to be used as the lowest-order approximation in systematic perturbative calculation of bound-state energy levels in QED. The kernel is symmetric in the two fermions and for the exchange of in and out momenta. The resulting equation is exactly soluble, unlike previously considered unperturbed kernels. We give explicitly the Green function and eigenfunctions. We also discuss the problem of the behaviour of the wave functions at zero relative coordinate in connection with the contribution to energy levels from the one-photon annihilation channel in QED.
1. Introduction Why another paper on such a "classical" subject as the bound state problem in quantum electrodynamics? The manifestly good reasons are the following. Experiments on pure QED bound states, positronium and muonium, have reached in recent years a very high degree of accuracy [1,2], almost comparable to the famous (g - 2) experiments of previous years; on the other hand the formulation of the boundstate problem in QED, as given by the relativistic Bethe-Salpeter equation [3] (BSE), although theoretically complete, suffers from the lack of a tractable, systematic computation procedure. This has become more and more clear precisely because of the improvement in the experimental investigation of the bound states, which calls for comparably accurate theoretical predictions. The present situation concerning the ground state splitting of m u o n i u m and positronium is summarized by saying that the experimental determination of these quantities is at the ppm level, whereas the corresponding theoretical predictions are worse * Permanent address: lstituto di Fisica, Universita di Bologna, Italy, and INFN, Sezione di Bologna 413
414
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
by one or two orders o f magnitude, so that the study of muonium and positronium is now perhaps the most interesting open problem in QED. Let us recall more precisely the difficulties encountered in treating systematically the bound states in QED by means of the BSE. As is well known, the BSE is an integral equation whose kernel is formally given as a power series in the fine structure constant or. To find its eigenvalues one has to specify a lowest-order equation - we will see the freedom present in this specification - and carry out on it a perturbative calculation. The main difficulty in pursuing this approach has been the absence of known analytic solutions to the equation even for an approximate Coulomb-like interaction kernel. In general, any equation to be used as a lowest-order approximation to the complete QED BS equation will have a kinetic and a Coulomb-like interaction term, able to produce bound states and supposed to be the largest part of the full BS kernel. As a first obvious condition, in the low-momenta limit both terms must reduce to the kinetic energy and the Coulomb potential of the non-relativistic Schr6dinger equation, with relative corrections o f second order in the momenta, so that the energy eigenvalues o f the bound states are given by the Balmer formula and unwanted o~3 terms in the eigenvalues do not appear. As a second condition it is advisable that in the high-momenta limit, where the interaction vanishes, the kinetic term describes exactly the relativistic free propagation o f the two fermions; this feature guarantees a proper treatment of the free highenergy intermediate states, which appear in the subsequent implementation o f a systematic perturbative expansion to the full BS equation, based on the considered lowest-order equation. The two conditions are indeed fulfilled by the Salpeter [4] approximation to the full BS equation, in which the kinetic term accounts exactly for the free relativistic propagation at all momenta. Unfortunately, exact analytic solutions of the approximated Salpeter equation (as well as of any other similar equation, such as e.g. the Breit equation) are not known; the lowest-order bound-state wave function is known explicitly only in the non-relativistic limit, better approximations to it are obtained through the Salpeter iteration procedure [4] (the iteration procedure is not even a systematic perturbativc expansion in some small parameter, and its use becomes more and more delicate, if not dangerous, when it is pushed to a precision higher than the one envisaged by Salpeter). Despite these shortcomings, in the absence of a better working approach, the Salpeter equation - or some improved version of it [5 ] - has been the starting point of most of the perturbative calculations performed up to now for positronium and muonium. As a last remark let us recall that the Dirac equation for an electron in an external Coulomb potential can be solved exactly and gives the right bound-state energies up to order 0~4 (fine structure), so that one could think of using the Dirac equation as convenient lowest-order equation. Unfortunately this is not the case, for the follov,. ing reasons: the Dirac equation for one fermion in a Coulomb potential is obtained by the complete two-fermion BSE in a rather indirect way, i.e. by retaining as interaction kernel the sum o f all irreducible crossed ladder graphs in the limit of infinite
R. Barbieri, k'. Remiddi / Bethe-Salpeter equation for positronium
415
mass of one of the fermions; no equation for two fermions is known which can be solved exactly and which gives the correct ct4 structure; in the case of positronium, furthermore, as the two constituent fermions have equal mass, spin-spin forces also ~ve ct4 contributions of the same order of magnitude as the fine structure (in hydrogen and muonium, spin-spin forces give hyperfine contributions, of order a 4 times the ratio of the masses). It is not at all necessary, however, that the lowest-order equation we are looking for gives by itself the a4 structure; the requirement is simply that the correct a 4 terms, as well as all higher-order corrections to the energy levels, are obtained in the systematic perturbative expansion to be built starting from the lowest-order equation in question. In sect. 2 of this paper we present an equation which can be solved exactly and which has the required properties in order to be used as the lowest-order equation in a systematic perturbative expansion for the solution of the complete QED BS equation. Our equation is a relativistic equation for the Green function of two fermions with the same kinetic part as the BSE and Coulomb-like kernel which reduces to the usual Coulomb potential in the non-relativistic limit. We give explicitly the relativistic Green function for two fermions which solves our equation; it is symmetric in in and out particles and is expressed in momentum space in terms of the corresponding non-relativistic Coulomb Green function of appropriate arguments times suitable spinors and kinematical factors. From the Green function, the exact properly normalized bound-state wave functions, as well as their conjugates, are easily obtained (without the usual normalization problem arising when only the BSE for the wave function, not the BSE for the Green function is investigated). The systematic perturbative expansion for the energy levels of the complete BSE in terms of such exact wave functions and the Green function is easily obtained. In sect. 3 we then comment on the specific convergence problems one encounters in the ultraviolet when dealing with the one-photon annihilation contribution to the eigenvalues in positronium. We will see how our lowest-order exact solution, which is better behaved at high momentum than the Salpeter wave function, cuts off naturally the loop integrals and allows one to treat the divergences as in the normal Feynman graph expansion o f the complete Green function. In the appendix, finally, we briefly discuss the relation between our work and the results of the interesting recent papers by Caswell and Lepage [6,7], who also obtain exact solutions to suitable equations describing the relativistic two-fermion problem. The main difference between the two approaches is that in refs. [6,7] the BSE forrealism is abandoned, by dropping the relative energy dependence, in favour of equations with a simpler kinematical structure but a more complicated effective kernel.
2. The lowest-order equation Let us consider for simplicity the case of particle-particle scattering, with equal
416
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
~P'p
½ P.*
:s Ip.p
-
l
a
"~P.p
..•pop,
-~P-p
Ip.¢
"J
-~1 P.q 1
Fig. 1. The equation for the Green function.
masses m i
m2 = 1. The equation which we want to solve is
=
. £d4p ' Go(P, P, q) = Go(P , P) (27r)484(p - q) + t J ~ - ~ Kc(P,
p, p') G¢(e,p', ql([)
and is graphically represented in fig. 1. As in the BSE, Go(P, p) is the free two-fermion propagator
Go(P, P)
l i p + ~r',- 1 - ie
il ~ - i~--+ 1 - ie_J
'
where (1), (2) refer to the upper and lower fermion lines of fig. 1. For later use, it is convenient to rewrite Go(P, p), in the c.m.s, w i t h P o = 2W, as
G°(P'P)=-
I X
A+(p)3'~ W+po-E(p)+ie
A+(-p) 3'4 W-Po-E(p)+ie
A__(p)3'._
+
] <')
W+po +E(p)-iel A _ ( - p ) 3'4
w - Voo+ ~ -
1 (2)
~
'
(3)
where, as usual E(p) =x/P~ + 1 ,
&(p) = E ( p ) -+ (1 - i p ' ~ ) 3 " 4 2E(p)
(4)
so that A2(p) = A+(p),
A+(p) A _ ( p ) = A _ ( p ) A+(p) = 0 ,
½(1 + 3'4) A+(p) ½(1 + 3'4) -
E(P) + 1 ½(1 + 3',). 2E(p)
(5) (6)
We define also _
X(p, q)
2E(p) 2E(q) [A+(p)~(1 +3'4)A+(q)] 0) E(p) + 1 E(q) + 1
X [A+(-R) ½(1 + 3'4) A + ( - q ) ] (2)-
(7)
R. Barbieri, E. Remiddi /Bethe-Salpeter equation for positronium
417
On account of (5), (6), one has h(p, p') ;k(p', q) = h(p, q ) ,
(8)
where the r.h.s, is independent ofp'. The Coulomb-like kernel which we propose for eq. (1) is Kc(P, P, q) = ")'(')~/(2)X(P,q)R(W, p) Vc(p - q) R(W, q) ,
(9)
where 2 R(W, p) = E(p) + W '
(10)
and Ve(k) is the scalar Coulomb potential vdk)
47rex - t, 2 .
(1 l )
Eq. (1)is then solved by the ansatz Go(P, p, q) = Go(P, p)[(Z¢r)484(p - q)
+ 7(')70)•(p, q) iR(l¢, p) He(W, p, q) R(W, q) Go(P, q)] ,
(12)
where He(IV, p, q), the unknown, is a scalar function. By inserting eqs. (9), (12) into eq. (1), using eqs. (3), (5), (8), after many simplifications we get Hc(W,p,q)= Ve(p - q ) dp' f~_~ +
--2 I¢ -'" /'dp0 1 V ( p - p ' ) t ~ ( 'P)'/-~ni [ W + p ' o - E f p ) + i e ] [ W - p ' o - E ( p ' ) + i e ]
X Hc(W, p ' , q ) .
(13)
The integration on the relative energy p~) is done and the equation becomes Hc(W'p'q)= V c ( P - q ) + f ~n)3 Vc(P - P ' )p,2 - ( W 21 - l ) - t e. He(W,p,,q) " (14)
Let us now consider the Schr6dinger Coulomb problem. Putting the mass equal to 1 2, the equation for the non-relativistic Green function gc(W, p, q) is
gc(cO,p,q)
p2 _ co - ie
Putting 11(16) gc(w,p, q) - p2 _ coI1 -- ie (27r)383(p - q) + hc(~,p, q)q2 _ co - ie
(15)-
418
R. Barbieri, E. Rerniddi / Bethe-Salpeter equation for positronium
OF
hc(co, p , q ) = _(p2 _ co)[(2n)363(p _ q) _gc(co,p,q)(qZ __ co)],
(17)
the equation for he(co, p, q) reads p
hc(co,p,q) = Vc(P -
q) +f(2@-~) Vc(P
-P')p'2 _co--I-re" hc(co,P',q)
. (18)
By comparing eqs. (14), (18) one finds Hc(I¢, p, q) = he(W: - 1,p, q ) ,
(19)
i.e.
H c ( W , p, q) : - ( E 2 (p) - W2)[(21T)363(p -- q) - - g c ( W 2 -- l , p , q X E 2 ( q )
- W2)] ,
(20)
which gives, when inserted in the ansatz (12), the solution of the relativistic equation (1) in terms of the solution of the non-relativistic Schr6dinger equation (16). Note that the solution Gc(P, p, q) so obtained does depend on relative energies Po, qo, although He(W, p, q) does not. The Schr~Sdinger Green function go(co, P, q) which solves eq. (15) is known, the most convenient way of expressing it being perhaps the Schwinger one-dimensional integral representation [8]. It has bound-state poles at values of co corresponding to con = __.~2/4n2 ' the familiar Balmer formula for mass 1.
go(co,P, q) ~ ~Pn(P)~n( q) w - c o n +ie '
(21)
where the ~Pn(P) are the non-relativistic, hydrogen-like, normalized bound-state Schr6dinger wave functions, for instance a
(22)
for the ground state. Corresponding to any pole ofge at co = COn,we see through eqs. (12), (20) that Go(P, P, q) has bound-state poles at values of W2 given by Wn2 = 1 + con. Writing, at a pole,
Gc(P, p, q) "" i ~sn(W' p) ~;sn(W' q) W2 - Wn2 + ie
(23)
(s refers to any of the four spin states, in which we have complete degeneracy) one obtains by comparison the relativistic bound-state wave function q;sn(W, p), a 4 × 4 spinor depending also on I41,Po [q~(W, p)]o -
~4A÷(p) 3(1 + ~'4
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
X
-'
i ' ~ - i~ r+ I - i~ " Y 4 A + ( - P ) / ( 1
+ "/4)
]
419
(rs)~:
XR(W, p)(/:-2 (p) _ W2) ~Sn(p) '
(24)
where r s is a constant 4 X4 spin wave function:
"
:\0
'
~_+, =,,/~(1,_+i, 0 ) ,
t0 = (0,0, 1).
(2s)
The function ~s(W, q) is the conjugate of ~s(w, p), eq. (24), with the prescription, however, that the sign of - i e in the propagator does not change• For completeness, we give the homogeneous equations satisfied by the boundstate wave functions and their conjugate• d4 ,
Cs(wn, p) = iGo(Pn, P)
c(Pn, P, p') ~n(Wn, P ) .
--s • £ d4q' ~s dgn(Wn, q) = t J ~rr--~ ~un(Wn, q ) Kc(Pn, q', q) Go(Pn, q) . '
(26)
where "°no = 2Wn- One can verify that the Mandelstam normalization condition [9] for the wave functions is indeed satisfied if the Cn(P) in the r.h.s, o f e q . (24) have the usual normalization
fgT
l.
The complete BSE has the form o f e q . (l), with Kc replaced by the full QED kernel K, the sum of all the two-particle irreducible Feynman graphs. Write K =-Kc + 6K ,
(28)
so defining the perturbation 5K as
5K = K - Kc •
(29)
The underlying physics is the assumption that among all the terms appearing in the formal power expansion in ~ which defines K, the most important one, responsible for the generation of bound states, is the Coulomb part present in the non-relativistic limit of the one-photon exchange graph• As also Kc, eq. (9), reduces to the Coulomb potential in the non-relativsitic limit, 6K has a term of order a but regular at low momenta while all other terms have higher powers o f ~ , so that 6K can be treated as a perturbation with respect to Kc. The explicit knowledge of the bound-state wave function and of the Green function of the lowest-order equation allows us to write down a systematic perturbative
420
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
expansion for the energy levels of the complete equation, although Kc depends on W and, due to the presence of relative energies, the concepts of completeness and orthonormality of the solutions do not apply in a straightforward way. Such an expansion is indeed given in ref. [6], by extending a method due to Kato [10], and we do not repeat it here. Typical terms in the expansion for the energy level shifts are of the form anf(a), where an gives the leading a order of the contribution, essentially fixed by the powers of a of the considered terms in 6K, while f(a), usually given as an expansion in a (and possibly lga), is obtained when taking the appropriate covohition integrals of the concerned 6K terms with the unperturbed wave tunction and Green function, which also depend on a but in a non-analytic way. Explicit calculations of positronium hyperfine splitting based on eqs. (1), (9) as the lowest-order equation are in progress. 3. Renormalization of the one-photon annihilation channel and behaviour of the wave function at zero relative coordinate A special problem encountered in the study of positronium is the renormalization of the one-photon annihilation channel. In this case the exact renormalized BSE for the wave function, in the "sidewise" notation proper to the particle-antiparticle case, is • £d4p ' [ (P, p) = S'(½P + p) t 3 ~ /((P, P, P') ~,(e, p ' ) Z] 1 2 Tr(Tu~k(P,p, ))J7 S ' ( - ] lP + p ) , + 47raTu ~-~4W
(30)
3
where K is the renormalized, one-photon irreducible electron-positron kernel, the second term in the square bracket is the one-photon annihilation interaction, S' is the dressed renormalized fermion propagator, and Zt, Z3 are the usual vertex and photon wave function renormalization constants. The one-photon annihilation term in the r.h.s, of eq. (30) is therefore responsible for the presence of counter terms in the renormalized BSE, which are not there to cancel divergences of the kernel itself. A strictly analogous situation is encountered when studying, for instance, the Low equation for the photon propagator, and is related to the overlapping-divergences problem. In this case, one knows how to write down an equivalent equation whose solution is finite when developed in perturbation theory, since the equation itself no longer contains the divergent counter term. It would be very valuable to get something similar (i.e. a counter-term-free equation) also for the positronium eigenvalue problem * even if what we propose in the following is completely satisfactory for practical purposes. • One o f the a u t h o r s (R.B.) acknowledges a correspondence with Professor K. Symanzik on this subject.
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positronium
421
The presence of counter-terms demands that the ultraviolet regulator should be kept also in K and sent to o~ only at the very end of the calculation of the eigenvalues. This problem is typical of positronium; in the case of muonium, for instance, the annihilation term is absent and all the counterterms present in K serve to cancel the divergence in K itself. In other words, in muonium the regularizing parameter can be sent to infinity before solving the equation since no divergence is generated when integrating the equation iteratively. Proper handling of the problems posed by the presence of the divergent counterterms in the one-photon annihilation channel of the positronium kernel requires a precise knowledge of the zero relative coordinate behaviour of the solution of the lowest-Order equation. In fact, when evaluating the one-photon annihilation contribution to the energy levels one gets a term of the type AEccaZ~ ~ T r [ ~ ( 0 )
Tu] Tr[Tu~(0)] ,
(31)
where
f d4p q;(p, ~(0) = d ~ - ~ p).
(32)
When the lowest-order equation is for instance the Salpeter [6] equation, the corresponding wave function develops an anomalous dimension, behaving for large momentum p and small a as p-(,*-ea), e > 0, and ~(0) is divergent. In a systematic expansion in a one should therefore compensate the divergences coming from ~(0) with those from Z 1 (this could perhaps be achieved by introducing an ultraviolet regulator A also in the lowest-order kernel of the Salpeter equation, then trying to solve the equation for large finite A, so obtaining an asymptotic behaviour p-aA2/(p + A 2) for the wave function). In the case of our lowest-order equation, on the contrary, ~(0) is finite, as ~k(p) ~ p -O+lt2). The divergences appearing in the Salpeter wave function are naturally displaced in our approach to the perturbative expansion of the eigenvalues, since the highenergy tail of the one-photon direct exchange is treated as a perturbation of the low-energy binding potential and not kept in the lowest-order interaction kernel itself. This clearly increases the systematicity of the computational procedure, since one keeps control of these divergences in the same way as in the normal Feynman graph expansion of, say, the vacuum polarization amplitude. One of the authors (E.R.) wants to thank W. BuchmiJller for carefully reading the manuscript. Appendix We comment briefly on the relation between the present paper and the recent works by Caswell and Lepage [6,7]. In ref. [6], the BSE for differem m ~ e s is con-
422
R. Barbieri, E. Remiddi / Bethe-Salpeter equation for positroniurn
sidered, and the relative energy is frozen by forcing on the mass shell one of the two particles (say the heavier one, but the procedure works for arbitrary masses and is used also for positronium). A three-dimensional integral equation is so derived, whose kernel is obtained by suitably iterating the BS kernel, but with the same eigenvalues as the original BSE. For a proper choice of a lowest-order Coulomb-like kernel, the equation becomes a relativistic Dirac equation for a single fermion in a Coulomb potential and is therefore solved exactly. The wave function is of course less simple than eq. (24) of the present paper, and no attempt is made to obtain the corresponding Green function. The perturbative expansion for the energy levels (already referred to at the end of sect. 2) is then proposed, and successfully applied to t~61g ct terms of muonium and positronium hyperfine splitting. In ref. [7], which became known to us only after the completion of this paper, the starting point is again the BS equation. The relative energy dependence is also eliminated, this time putting the relative energies equal to zero, and the equation is further restricted to the subspace of positive-frequency spinors. By taking in this subspace as lowest-order diagonal interaction the scalar part of the rda.s of our eq. (9), an equation which is essentially our eq. (14) is obtained, solved and used as the starting point of a systematic perturbative expansion for energy levels. Some new a6 terms for muonium and positronium are then evaluated.
References [1] P.O. Egan et al., Phys. Lett. 54A (1975) 412; A.P. Mills, Jr. and G.H. Berman, Phys. Rev. Lett. 34 (1975) 246. [2] D.E. Casperson et al., Phys. Rev. Lett. 38 (1976) 956; H.G.H. Kobrak et al., Phys. Lett. 43B (1973) 526. [3] E.E. Salpeter and H.A. Bethe, Phys. Rcv. 84 (1951) 1232. [4} E.E. S',dpeter, Phys. Rev. 87 (1952) 328. [5] V.K. Cung, T. Fulton, W.W. Repko, A. Schaum and D. Devoto, Ann. of Phys. 98 (1976) 516. [61 G.P. Lepage, Phys. Rev. A16 (1977) 863. [7] W.E. Caswell and G.P. Lepage, Preprint SLAC-PUB-2080 (1978). [8] J. Schwinger, J. Math. Phys. 5 (1964) 1601. [9] S. Mandelstam, Proc. Roy. Soc. A233 (1955) 248. [101 T. Kato, Prog. Theor. Phys. 4 (1949) 154; A. Messiah, Quantum mechanics, vol. 2 (North-Holland, Amsterdam, 1962) p. 712.