Hyperfine splitting in positronium and muonium

HYPERFINE SPLITTING IN POSITRONIUM AND MUONIUM

Geoffrey T. BOD WIN and Donald R. YENNIE Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853, U.S.A.

I

NORTH-HOLLAND PUBLISHING COMPANY

-

AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters) 43, No. 6(1978)267-303. NORTH-HOLLAND PUBLISHING COMPANY

HYPERFINE SPLITTING IN POSITRONIUM AND MUONIUM*

Geoffrey T. BODWIN and Donald R. YENNIE** Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853, U.S.A. Received December 1977

Contents: 1. Introduction 2. Perturbation theory 3. Application to quantum electrodynamics 3.1. Coulomb gauge perturbation theory 3.2. Strategy for determining the orders of a 3.3. Gauge invariance 4. The pure Coulomb splitting 5. The one transverse photon kernels 5.1. Calculation of the leading term AE(L) 5.2. Higher iterations of the wave function

269 270 275 275 277 278 279 280 284 288

5.3. Calculation of L~E(D) 5.4. The corrections i~E(B) 5.5. The remaining corrections 6. Other contributions to the hfs 7. Summary and conclusions Acknowledgements Appendix 1. Normalization of Salpeter wave functions Appendix 2. Some useful integrals References

289 291 294 296 298 299 299 301 303

Abstract: 2 ln a 1 in positronium and relative order We calculate corrections to the ground state hyperfine structure of relative order a (m,/m~)a2In a~ in muonium due to the exchange of virtual photons. Our results are in agreement with those of Lepage. Contributions arising from the Coulomb potential and from the exchange ofone transverse photon along with any number ofladder Coulomb photons are discussed in detaiL In treating thesingle transverse photon-multiple Coulomb photon exchanges, we sum the contributions involving different numbers of Coulomb photons and reexpand the resulting expression in terms of a quantity that is inherently smaller than the Coulomb potential in the non-relativistic region. This procedure enables us to take into account from the beginning important cancellations that occur between the various terms in an expansion in powers of the Coulomb potentiaL The techniques developed here may be useful in calculating higher order corrections.

Single ordersfor this issue PHYSICS REPORT (Section C of PHYSICS LETTERS) 43, No. 6 (1978) 267—303. Copies ofthis issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 18.00, postage included.

*

Supported in part by the National Science Foundation. This author expresses his appreciation to Willis Lamb for his early encouragement and subsequent friendship.

**

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

269

1. Introduction The difference between the singlet and triplet levels in a two fermion system is denoted by the term hyperfine structure (hfs). Measurements of the hfs in the ground states of hydrogen, positronium and muonium currently provide the best tests of relativistic two-body theory.* As a consequence, considerable experimental and theoretical effort has been devoted to the study of the ground state hfs in these systems. The ground state hfs has been measured most precisely in hydrogen (to an accuracy of about one part in l0 12) [1,2]. However, the calculation of the theoretical value of the hydrogen hfs is limited by uncertainties in the strong interaction effects at the level of about three parts per million (ppm) [3,4]. In contrast, at the present level of experimental accuracy, positronium and muonium are essentially pure quantum electrodynamical systems.** The ground state hfs has been measured to an accuracy of 0.12 ppm in muonium [5] and 6 ppm in positronium [6, 7]. In order to obtain theoretical results of comparable accuracy, it is necessary to carry the relativistic two body calculation at least to terms of relative order ~2 in positronium and relative order ct3 and c~2me/m,.in muonium. The lowest order ground state hfs due to exchange of a transverse photon was first calculated by Fermi [8]. It is EF

3.*** (1.1) 3cx m1m2/(m1 + m2) An additional relativistic correction in muonium of O(ot2 EF) was found by Breit and co-workers [9]. In order to progress further, it was necessary to developa consistent fully-relativistic two-body formalism for bound state calculations. This was done in 1951 by Schwinger [10] and Bethe and Salpeter [11]. Subsequently, Kroll and Pollock [12] and Karplus, Klein and Schwinger [13] evaluated the terms of O(ot2EF) in muonium, and Karplus and Klein [14] evaluated the terms of O(cxEF) in positronium and also found a correction mct4/4 to the Fermi splitting due to virtual annihilation of the e~e pair. In the Karplus—Klein calculation a cut-off was introduced with little justification in order to handle spurious logarithmic divergences in the infrared region. Later calculations by Salpeter and Newcomb [15] and Fulton and Martin [16] avoided this deficiency and confirmed the Karplus—Klein result. In addition, the terms of O(~(me/mp)EF)in the muonium hfs were found. These terms were verified by Arnowitt [17] and Grotch and Yennie [18] the latter using an effective potential formalism. Electron self energy corrections of O(x3 1n2x EF) and O(cc3 ln ~ EF) in the muonium hfs were calculated by Layzer [19] and Zwanziger [20] and verified by Brosky and Erickson [21], who also estimated the term of O(ot3EF). A first step toward the calculation of the hfs terms of O(cc2EF) in positronium and O(0t2(me/mp)EF) in muonium was made by Fulton, Owen, and Repko [22]. They attempted to calculate the terms of O(X2ln0C’EF) in positronium and O(ct2(mC/m,L)lnc(’EF) in muonium. Additional terms of this order due to three-photon processes were found by Barbieri and Remiddi [23]. Cung, Fulton, Repko, and Schnitzler [24] found an O(x2 ln c~1EF) contç~butionto the hfs coming from the pure =

—

—

—

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* Other high-accuracy tests of quantum electrodynamics thclude the anomalous magnetic moments of the electron and muon, the He fine structure, and the Lamb shift in hydrogen. The anomalous moments, of course, do not test the bound state theory. ** We estimate that modifications of the photon propagator due to the p meson enter in relative order a2(m,/m~)2 10b0. The neutral current effects in the weak interaction give a splitting of relative order ~/~mlm 2GF/1ra, which is relative order 10b0 for positronium and 10-8 for muonium. ~ ni1 and In2 are the masses of the two particles and a 1/137 is the fine structure constant Throughout this paper we use units in which Ii = c = 1.

270

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

Coulomb interaction. This contribution was verified by Lepage [25] as was the Fulton—Owen— Repko result for two transverse photon processes. However, the result of Lepage for processes in which one transverse photon and any number of Coulomb photons are exchanged is in disagreement with the result of Fulton, Owen, and Repko. The latter result was obtained by conventional Bethe—Salpeter techniques, whereas the Lepage calculation was based on the Gross equation [26]. Thus, although the two approaches have the same physical content, it is difficult to resolve the discrepancy by direct comparison. In the present work we concern ourselves with the contribution to the hfs arising from the exchange of virtual quanta. We calculate the terms of O(oc2 ln ~ 1EF) in positronium and O(~x2(m 1EF) in muonium making use of novel techniques for handling Coulomb photon 8/m,5) l n ~ exchanges. Our results provide an independent check on the work of Lepage [25] and Fulton et al. [22]. The techniques developed here may be useful in extending the hfs calculation to higher orders. —

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2. Perturbation theory In this section we give a brief derivation of the perturbation theory expression for the two body energy levels in quantum field theory. This discussion also serves to introduce much of the notation that is used in the remainder of the paper. We begin by discussing the perturbation problem in full generality. Then, in order to relate the four-dimensional wave functions and matrix elements to their ordinary three-dimensional counterparts, we specialize to the case in which the unperturbed interaction can be represented by an instantaneous potential. Our result, eq. (2.21), resembles ordinary non-relativistic perturbation theory. It includes the first order energy shift derived by Salpeter [27] and also the second order corrections. Ultimately, in carrying out calculations, we reduce the matrix elements in eq. (2.21) to matrix elements between non-relativistic states. Implicit in this treatment is the assumption that the spectrum of states in the non-relativistic problem has the same basic structure as the spectrum of states in the original four-dimensional relativistic problem. Conceivably, in a fully-relativistic treatment, new types of states could appear. Our treatment would fail to account for such states. The two body problem in quantum field theory may be formulated in terms of the Bethe— Salpeter (BS) equation for the four-point function: G(x 1, x2, x3, x4)

=

2~(x x4) 2,

S~’~(x1, x3)S~

2~(x x6)K(x5, 2, x6, x7, x8)G(x7, x8, x3, x4).

+ Jdx5 dx6 dx7dx8S~’~( 1,x5)S~

(2.1)

Here S~is the Feynman propagator* and G is the complete four-point function (S~(x1,x2) = <01 Tiji(x1)~(x2)10> and G(x1, x2, x3, x4) = <01 T~(x1)~i(x2)~(x3)i~i(x4) 10> for Dirac particles).** K is the sum of all two-particle irreducible graphs, namely those graphs that cannot be divided *

In principle S~is the complete two point function, but for our purposes only the unperturbed propagator S~is used. Notice that we use a two particle formalism rather than a particle—antiparticle formalism. In order to treat a particle—antiparticle

**

system such as positroniuni we must charge conjugate one of the particles in our two particle system. This amounts to including a factor — 1 in the expression for each vertex involving two antiparticles, and a factor C(C _1) on the incoming (outgoing) antiparticle side of each vertex involving a particle and an antiparticle. C = iy0y2 is the charge conjugation matrix.

G.T. Bodwin and D.R. Yennie, Hyperflne splitting in positronium and muonium

271

in two by cutting two legs of the same types as the two incoming (or outgoing) legs. Eq. (2.1) is represented graphically in fig. 1. (I)

(I)

(I)

(I)

(2)

(2)

(2)

(2)

Fig. 1. Graphical representation of the Bethe—Salpeter equation for the four pointfunction.

From the property of translation invariance we know that G is a matrix function of the variables x1 x2, ~ = x3 x4 and p = ~(x1 + x2 x3 x4) alone, and is independent of

=

—

=

—

—

—

~(x1 + x2 + x3 + x4). Thus, we may define the Fourier transform of G as follows: Jdxi dx2 dx3 dx4 exp [+i(p1

=

~

.

x1 + P2

—

—

p3 x3

—

p4~x4)]G(x1, x2, x3, x4)

q~ + P~p)]exp[+i~(p1 + P2

—

P3

—

4ö(p 1 + P2

(2n)

where p

~

=

—

P2),

—

q

—

=

~(p~

p4)G(p, q, P), —

p4) and P

(2.2) =

Pi + P2

=

p3 +

Lurié, MacFarland and

~4.*

Takahashi [28] have shown that

~ .[~ 1

G (p,q,

—

)—1[~-~ + terms

~~(p)~(q) 2 + M~)”2+ic~ ~~2~P —(P 0 regular in P 0.

xp(p)~(q) 2 + M~)”2 +(P (2.3)

The symbol S denotes a sum over discrete states and an integration over continuum states. The x1~are the BS wave functions defined by

~(q)

=

Jd(xi —x2)exp{ +ip~(x1—x2)} <01 ~

=

fd(x3

—

{

exp {iP~~x1 +x2)}, 2~(x

x4)exp —iq~(x3—x4)} ~
{

4)I0> exp —iP1~x3+x4)}. (2.4)

I iP>

I

+ and iP> are, respectively, two-particle and two-antiparticle physical states of four2 +the M~)~2.M momentum P. = (~, P). ~ = (P 1 is the mass of the state. The singularities in P0 associated with the discrete states are poles. In the vicinity of the pole at 2+ M~)~2, eq. (2.1) becomes P0 = ±(P —

th(xi, x 2)

=

Jdx5 dx6 dx7 dx8S~(x1,x5)S~(x2,x6)K(x5, x6, x7, x8)~~(x7, x8),

(2.5)

3x; we use the same * Notation: dx indicates a four-dimensional symbol for functions in coordinate or momentumintegral; space. three-dimensional integrals are indicated explicitly: d

272

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

the bound state BS equation. (For continuum states, an inhomogeneous term describing two free particles must be added to the rhs of eq. (2.5).) We now sketch a derivation of the perturbation expression for the energy shifts. We divide K into a piece K°whose solution is assumed known and a remainder AK: K= K0 + AK, with 1F2~ +

G0

S

=

S~’~S~K 0G°

~

(2.6a)

and S~’~S~Ko~°.

=

(2.6b)

Here, in order to simplify the notation we suppress the coordinates and integrations. Now if we multiply eq. (2.1) on the left by GO(S~S~Y’ and use the adjoint of eq. (2.6a), the result is G

G0 + GØAKG.

=

(2.7)

Iterating this, we obtain the perturbation expansion for G: G = G0 + GØAKG + GØAKGØAKG0

(2.8)

+....

The energy levels of the bound states are determined by the positions of the poles in G. In order to find the positions of these poles in terms of a power series in AK, we display the pole structure in G0 explicitly: G0

=

~

i~+~ ~

+ terms regular in P0.

(2.9)

Eq. (2.6) is assumed to be solved, so that the M~and x~°~ are known. The total four-momentum label of the wave functions has been suppressed in eq. (2.9) and subsequent expressions. Now suppose that we are interested in the shift in the position of the pole in G0 at P0 = * due to the effect of the perturbation AK. We single out this pole by writing 0—0

lxjxj 210 2~~,°(P0 ff~,°+ is)~ a Substituting eq. (2. lOa) into eq. (2.8) and retaining only the part of G0 associated with the pole at P0 = ~9, we obtain the geometric series G —G’’ 0

—

—

/

xi0—0 x~ 2~r(P0

—

—0 A —

~

X~ Xj 2 + + ii) (2 lOb)

1XJXJ

26~°(P 0

—

*

\2

+ I 0 X~ g~,P+ i~) k~2~)XJ (P0

— —

.

—

.

i~AK~?/2~P + is)’

In the remainder of this paper, we specialize to the case of particle wave functions and drop the superscript

anti-particles may be obtained from the corresponding particle equations by reversing the signs of 8~and ic.

+.

The equations for

G. T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

So, to first order in AK, the pole in P0 at ~(1)

~0

‘(—OA

~

273

is shifted to the location

0~

0~

J ~ 2~ ~ p~g In order to obtain the energy shift through nth order, we retain the terms in eq. (2.8) in which not more than n 1 factors of G 0 appear without at least one intervening factor of ix?x?/2~f~(P0 + ii~).In the outermost factors of G0 we always retain the term ixYx?/2~f(Po ~ + ic). (The G~terms produce corrections to the wave functions due to AK. These corrections can affect the residue, but not the position of the pole.) This prescription leads, in general, to a geometric series. In the particular case of the energy shift through second order, we obtain a series that sums to — —

j

.~

—

—

~f

x~°~ [~0

—

—

—

~-~is~AKx?

—

~~YAK~o(i)AKx?].

(2.12)

Now, =

(~AK~?)

+

Po=g°

—~—

(~AK~)

ap0

(P0

—

~9) +

PO~°

So, to second to the location + (~?AK~oAKx?)Ip 2) = g,porder + [1 in AK, ~ the pole in P0 at Sf is shifted 1 [(~°AK~°)I g( 0=~o] —

{(~?AKx?)I~0~~ + (~AKGoAKx~9)I~0~9

g39 + +

~

~

0

(~?AK~?)]~ }. p0go

(2.13)

Eq. (2.13) is reminiscent of the perturbation expansion in non-relativistic theory. Note, however, that thereare additional terms that arise in second and higher orders owing to the P0 dependenceof the matrix elements. In practice, we evaluate eq. (2.13) in the rest frame so that S~,= M3 and In order to relate the results of the preceding discussion to the more familiar three dimensional wave functions and matrix elements, we follow Salpeter [27] and specialize to the case in which K0 is an instantaneous interaction. That is, 0)ö(x~ x2)5(x? x~)~ K0(x1, x2, x3, x4) = ó(x~ x2 0(x1, x2, x3, x4). —

—

—

Then the momentum space equivalent of eq. (2.6b) in the c.m. frame is =

S~(p1)S~(p2)

1~o(P’P’)X?(P)’

(2.14)

wherep1 =p = —P2’P3 =p’ = —p4,p? + p~?= p~+ p~= M~,andI~0(p,p’)isthethreedimensional Fourier transform of ~(x1, x2, x3, x4) with the overall momentum conserving ö-function

274

G. T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

deleted. Next we define the equal time wave function ~

f~

1

~_

(215)

O(p)*

v~J27tL

i

An equation for 4’(p) can be obtained by integrating eq. (2.14) with respect to p°.For this purpose it is convenient to use the non-covarian.t form of the fermion propagators: 5(O(p) F

= _________ —

—

m1 + is

~F[p°

— —

—

A~(i,) + A~(1,) E.(p) + is p°+ E1(p)

(2 16)

—

2 + rn?)”2 and A~)(p)= [E~(p)±(~p + fl where E.(p) = (p 1m~)]/2E~(p).The p° integration in (2.14) is then easily carried out and it yields the result —

—

F

LM? X

A~(p)A~(—p) E,(p) E2(p) + is

—

—

—

A~kp)A~(—p) M? + E,(p) + E2(p)

—

1 is]fhfl2

(27)~Jd3p!~0(p,p~)4J(pF),

(2.17)

which is known as Salpeter’s equation. The matrix elements appearing in eq. (2.12) still contain wave functions having relative time dependence. We wish to replace them by new expressions with modified kernels in which the time components of relative momentum are integrated out. This is accomplished by using the integral equation for x? =

S~S~Ko~? = \/~

S~)S~)K0~~,

(2.18)

where in the second form a three-dimensional rather than a four-dimensional momentum integration is implied. Thus ~?MXJ°= where

3p~~4p’)M(p’,p, P 0)4~31p),

M(p’,p, P

0)

9a)

JdP d

(2)6

(2.1

Jd~dq’~0(,p’,q)S~(q+ P)S~( q)M(q + P,

~

—

x S~’ )(qI + P)S~(

—

q’)~0(q’,

—

q, q’ + P,

p).

—

q’) (2.1 9b)

We may write eq. (2.19) more compactly in terms of the Dirac notation: =

MI 4’.,>.

2M~<~

(2.20)

Incorporating these results into eq. (2.13), we obtain the energy shift through terms of second order for the case of an instantaneous K0:

J

4j

fl

Note that ~/J = ii,,/~ (dp°/2a)~(p) = 1fl2, where ~, and fi, are the usual Dirac matrices. This follows from eqs. (2.4) and the fact that time ordering and anti-time ordering are equivalent in the equal time limit. *

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium =

275

~.,9+ i<~~IAKl4’~>ip0=~y + i<~JIAKGo(j)AK)i4’J>Ip0=gQ -

~

(2.21)

G0 can be expressed in terms of the three-dimensional Green’s function as follows: 1F1~S~F2~ +~ +~ G0 = S = S~’~S~ + ~ +~

(2.22a)

where the product I~ 0g0I~0 involves only 3-dimensional integrations, and Cd

g0(p’, p, P0)

=

‘° d 0

J —f—

-~—

G0(p’, p, P0).

(2.22b)

Eq. (2.22) is represented graphically in fig. 2.

+II+I~~~iL 3-d

integration

Fig. 2. Reduction of the four-dimensional Green’s function G0 to the form useful for three-dimensional calculations.

It follows from the BS equation that when P0 G0(j)

=

~

+ ~

+ ~

=

Sf (2.23a)

where ~j)

=

g0

—

i4’~~~/(P0Sf + is). —

(2.23b)

3. Application to quantum electrodynamics In this section we specialize the general perturbation formalism to electromagnetically bound states, describe a strategy for determining orders in ~ of different contributions, and give a demonstration that the bound state energies are independent of the choice of gauge. 3.1. Coulomb gauge perturbation theory It is most convenient to use the Coulomb gauge for photons exchanged between the two particles. Except when annihilation contributions are to be taken into account in positronium, it is possible to “mix” the gauges by using the covariant gauge for photons associated entirely with one particle. This makes the treatment of renormalization effects easier. In the present work we treat only the contributions from exchanged photons. We proceed now to apply the bound state perturbation theory developed in the preceding section. First comes a discussion of the unperturbed wave functions 4’; this is followed by a description of the Feynman rules for the perturbation kernels.

276

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

We choose for the unperturbed kernel K0 the single Coulomb photon exchange: K0(p’,p)

= (p~~,)2

fl1fl2.

(3.1)

With this choice kernel, Salpeter’s equation te’of~A~2~’ A~’~’ ~A~2~’is 1 C Ir A + ~PJ + ‘~ W/ ~ I I d~k [E E 2(p 1(p) E2(p) E + E1(p) + E2(p)] 2it —

—

—

—

\

J

—

—

—

—

E

—

—

H”~ H~2~ ~ —

—

k

—‘.

—

32

k)2

A~’~A~~)V4’,

where E = M°,H~(p)m;p + f3~m,,E~(p)=

(p2

(3.3)

+ rn?)1’2. We note that H”~(p)A~(p) = ±E 1(p)A~kp).

The is in the denominator is irrelevant for bound states. Now we can rationalize the denominators in eq. (3.2) to obtain 4’(p)

=

~N+p)A~(p)A~(—p)

—

where N~(p) [E ±(E

N(p)A~(p)A~(—p)} 2 p

2 (E 1 + E2)] [E 1 2rnRs [E2 (m 2] [E2 (m, 1 + m2) and where the reduced mass rnR is rn

2]/8rnRE2,

—

—

2mRs Vçb,

(3.4b)

E2)

m 2]/4E2, 2) 1m2/(m, + m2).

—

—

(3.4a)

(3.4c)

—

Unfortunately, eq. (3.4) is too difficult to solve directly. However, a good starting point can be found by studying the limit of small binding energy 4 (rn1 + m2)) and small momentum 4 m1 and m2). In this limit we find E m1 + m2 + t, (3.5a)

(I~l

(I’l

N~~ 1, 2mRs N_ ~ ( ~

(3.5b) —

p2)/(4rn,m 2),

(3.5c)

~ ~(l ±~j.

(3.5d)

Ignoring corrections to eq. (3.5), we find a simpler integral equation = 2 -~(1+ ~,)(1 + $2) V4’NR, p —2mRc04 4’NR

(3.6)

which is our starting point for all calculations. Obviously 4’NR is non-vanishing only for the “large— large” components of the wave function. For those components, the equation is simply the nonrelativistic Schrödinger equation, whose solutions are well known. For the normalized ground state, we have, for example, 4’NR

where ~2

=

{8ir”2y512/(p2 + y2)2}

=

—2mRs0

= *

=

(rnRot)2

~

(3.7a)

and

(~)® (~)® k~~>.*

1

.

We use the standard representation of the Dirac matrices:

=

,

\0

(3.7b)

0 —1i

=

.

\~ 0

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

277

The small difference between the eigenvalues of (3.4) and (3.6), corresponding to s ~o’ will be discussed in section 4. The main difference, of order c~4mR,is associated with relativistic and spin-orbit interactions (for non-S states); but there is also a small contribution to hfs. There are two ways we can use (3.6) as the starting point for calculations of the hfs. One is to replace the G 0 of section 2 by a non-relativistic propagator having only large-large components, and to treat the due to that replacement additional perturbations. The asother way 4’NR as corrections an initial approximation in eq. (3.4),asand then iterate that equation often as is to use necessary to produce a sufficiently accurate approximation. The singly-iterated wave function, accurate for most purposes, is —

4’sI

=

[N+(p)A~kp)A~(—p)

—

N_(p)A~(p)A~(—p)]4’NR(p).

(3.8)

Higher iterations are discussed in section 5. At this stage we can easily write down the “Feynman rules” for the matrix elements in eq. (2.21). We initially factor the /3 1’s out of ~ = 4’~/3152 and incorporate them into AK. Then, using the graph rules for quantum electrodynamics given by Bjorken and Drell [29], we obtain the factors 4’t(p’) for the final wave function, 4’(p) for2for theeach initialCoulomb wave function, photon propagator in AK, 4~wti/q 4itcti 7 ~i q~ q\ 4~ T T 02

iLK

:

2 (~1

—

E(~+

2

)

—~---

~

~

for

each transverse photon propagator in AK,

A+(:) + K

0 + ~ A(~] for each fermion propagator in AK, 4 for each closed ioop in AK, dd3p/(2ir)3, k/(2ir) d3p’/(2ir)3 for the wave function integrations. The graph symbols used for these quantities are shown in fig. 3. As usual, momentum is conserved at each vertex. —

(111i11

+~(P)

————-~——

11111)

Coulomb Photon Pt’opogotoi’

q .‘wi~i.4,w~.

4

Transverse Photon Propagator

Fermion Propagator

It Fig. 3. Feynman graph symbols for quantities appearing in perturbation theory matrix elements.

3.2. Strategyfor determining the orders ofc~ Because the fine structure constant occurs in the wave function as well as in the Coulomb and transverse photon propagators, the expansion in powers of ~ is not particularly straightforward.

278

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muoniurn

Therefore it is not possible to specify a set of rules that is guaranteed to generate all the terms of a given order. However, it is not too hard to identify the leading order of any given contribution, and to make a sequence of approximations until the remainder becomes negligible. This procedure is certainly far from unique when one is interested in higher order contributions, as one can arrange the sequence of approximations in various ways. We tried a variety of approaches until we found one that is relatively economical. While we believe that a better approach probably exists, this one seems sufficiently compact to be convenient for the calculation at hand. In analyzing the order of a contribution, a very important consideration is whether the relevant momenta (we always analyze the integrals in momentum space) are small (—.‘ ccm~)or large (one or both particles relativistic) in the dominant region of integration. If initially a contribution is dominated by small momenta, one can obtain the order by replacing each E1(p) by rn and by rescaling each momentum by p yt. In many cases the powers of cc then easily factor out, since the wave functions, Coulomb potentials, and many of the energy denominators are homogeneous in y. In case an energy denominator is not homogeneous in y, we can initially neglect the portions of it that are of higher order in cc. An example of this is given in section 5. As we proceed to higher order corrections, arising both from more complicated graphs and from the remainders of the approximations just mentioned, the dimensionality of the integrations increases and their structure becomes more complicated. As long as the integrals converge for small momenta, additional powers of yield additional powers of cc. At some point, however, the integrals no longer converge in the small momentum range and the whole integral or some subintegral needs the convergence provided by factors of E1 in the denominator. At that stage, additional powers of momentum no longer increase the order, and one should avoid such spurious expansions. In this high momentum regime, expansion in powers of V usually leads to increased orders of cc. However, in the small momentum region, the extra factor of cc in each V can be compensated by the momentum integration. Thus at each stage, one must exercise judgment so as to minimize the false steps and to obtain integrals that are tractable. The corrections involving a factor ln cc’ arise from integrals whose relevant momenta span the region from non-relativistic (‘-.~ccm~()to relativistic values. —~

~I

3.3. Gauge invariance We discuss here the gauge invariance of the positions of the poles of the four point function. To make a gauge transformation on the four point function, we add to each photon propagator a gauge contribution with numerator structure K,~5or ~ where K is the momentum label of the photon. Thus, we can write the gauge transformation as a perturbation expansion by regrouping graphs according to number of gauge photons, as shown in fig. 4.

L~1

gauge = transformed

+

g

+

g

Denotes a gauge photon Fig. 4. Illustration of the expansion of the gauge-transformed four point function.

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muoniurn

279

At this stage we can use the Feynman identity 1

1_

1

1

to effect the usual pairwise cancellation between the graphs in which a given gauge photon is inserted in all possible ways. (Note that this procedure is unaffected if we perform a different gauge transformation on each gauge-invariant set of graphs.) In the surviving terms, at least one end of each gauge photon is connected to an external leg of the four-point function. Thus, none of the terms in the perturbation series in fig. 4 has the structure of an iteration of the four-point function. There is no way to build up a geometric of the type in eq. (2.10), so the gauge transformation cannot shift the positions of the four-point function poles although it can affect the residues at those poles. —

4. The pure Coulomb splitting We now calculate the contribution to the hfs contained in the Salpeter equation with Coulomb potential, that is, the contribution to the hfs due to ladder graphs with Coulomb photon rungs. We take the Schrodinger equation (eq. (3.6)) as our starting point. Of course the Schrodinger equation has no fine or hyperfine structure. These arise when we calculate the difference between the eigenvalues of eqs. (3.4) and (3.6). That difference is given by (s

—

s~)

<4’NR

I 4’>

=

<4’NR

H’

I 4’>

(4.1)

where

H’

[N+A~)A~)

N.A~PA~~~(1 + /1~)(1+ $2)]V. (4.2) 4’> by 1 and in the matrix element replace 4” To adequate accuracy, we may approximate by its singly iterated approximation ~ given in eq. (3.8). In the resulting expression, only those terms that contain an even number of cc,’s and cc 2’s are non-zero. Furthermore, a term must contain an cc1 and an cc2 if it is to contribute to the hfs.* So, the only contribution to the hfs comes from the small—small components of the wave function. Retaining only the lowest powers of p in the numerator structure, we see that the leading contribution to the hfs from eq. (4.2) is —

—

4’NR

<

AE(Coul)

=

64iry

J ~~12+1 y2)2 2E,(p’)2E2(p’) (p’

X

±p2)2 [N~(p’)

(p2

22E,(p)2E

~

—

—

p)

N_(p’)] [N+(p)

—

2(p) N(p)]~3~—~3.

(4.3)

* In addition to the hfs, which is our current interest, eq. (4.1) contains relativistic and spin-orbit corrections. These are given by the contributions that have two powers of p in addition to the factor of V. These come either from expanding factors of E, in H’ or from pairing one factor of ~ p in H’ with one in 4s~.(One cannot pair p with x2 p because 4~has only large-large components.) It is easy to see that these contributions are of order 526g. While 2, which theyisdobeyond not contribute the present directly order to ofthe interest. hfs, they can modify the wave function slightly and produce a correction of relative order a

~,

280

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

Now, 2,.a21b22c22.d=a1.aa,.ba2~ca2.d=[a~b+ia1.(aXb)][Cd+ia2~(CX11)], whose hfs part is equivalent under angular averaging to a2(a x b) . (c x d). As can be seen by power counting, the integral in eq. (4.3) has zero degree of convergence. This means that if we replace the E~by m1 (including in N ±)‘it would diverge logarithmically at the upper limit.2 Since thethe exact behavior relativistic affects additive terms of but not coefficient of in thethe logarithm, we region can make the the replacement E, m, relative order cc and cut off the integrals at ~ m 1, m2. The integral can now be expressed as a combination of the ones in table 4. The ln cc’ contribution is ln 6 cc~= +(m~/m,m 2ln cc’EF. (4.4) AE(Coul) ~ (m~/6m~m~)cc 2)cc Here, EF is the Fermi splitting: -+

li’ I

EF

=

~(cc4m~/m,m

2) .

It is obvious from counting powers of momentum in the integrand that the terms we have dropped, which contain additional powers 2oflnp,cc cannot logarithmic integrals. Thus, eq.hfs. (4.4) contains ‘AEF) lead of thetopure Coulomb potential to the This result the complete contribution of O(cc is in agreement with that of Cung et al. [24] and Lepage [25]. It is interesting to note that the part of H’ that produces the hfs also mixes the 3S and 3D states (in second order perturbation theory). Thus, the part of H’ that splits the singlet and triplet levels is associated with an effective tensor force. The connection between the pure Coulomb hfs and an effective tensor force in the non-relativistic reduction of Salpeter’s equation was first pointed out by Cung et a!. 5. The one transverse photon kernels In this section we investigate the kernels in AK in which one transverse photon and any number of Coulomb photons are exchanged. We ignore kernels involving crossed Coulomb photons since these contribute at most in O(cc2EF) (see section 6).* The remaining one transverse photon kernels are of the type shown in fig. 5. We calculate the contribution of these kernels to the hfs in O(cc2 ln cc ‘EF) and also obtain the leading contribution, EF. Terms of O(ccEF) have been calculated previously, so we bypass them. The various orders are identified by studying the expression for the energy shifts in momentum space. The kernels in fig. 5 are treated in the first order perturbation theory developed in section 2. In turns out that the second order energy shifts contribute to the hfs at most in O(cc2EF). This point is discussed in detail in section 6. Our basic strategy in evaluating the BS matrix elements of the kernels in fig. 5 is to use formal operator manipulations to sum the series of Coulomb exchanges, and thereby obtain an expression -

* In specifying orders we display only the a dependence. All the contributions discussed here are recoil corrections, so they contain an additional factor in the muonium case. .

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positroniunl and muoniun,

FH

+

+

+

+

+.

S

S

281

..

S

Fig. 5. One transverse photon kernel.

analogous to that for the energy shift in non-relativistic perturbation theory. It is known that, if one calculates the terms of fig. 5 separately, various important cancellations occur when they are added (see, for example, Karplus and Klein [14]). By keeping these related terms in a single expression from the start, we hope to obtain a simpler and more compact formalism. In the process of manipulating the expression for the energy shift, we identify and set aside various terms that appear to be small compared to the leading term. These corrections to the leading term are discussed at the end of the section. It is not clear that we have identified the most convenient choice of correction terms. However, our procedure does systematically account for the various pieces in the first order energy shift. The first step in our procedure is to set aside the A terms in the fermion propagators. (We show later that terms that contribute to the hfs in O(cc2 in cc 1EF) contain at most one factor of A .) In particular, we retain the A~A~ term in the Salpeter equation, so that we use a wave function 4” satisfying —

4”

E

4’. (5.1) 1(p) E2(p)A+(~~ The correction terms involving A A_ give an energy shift denoted by AE(A). In addition, there are corrections that arise from the A_ parts of propagators in the kernels themselves. We denote the energy shift due to these corrections by AE(B). Next, we use partial fractions and eq. (5.1) to rewrite the factors adjacent to the wave functions in the matrix elements of the kernels in fig. 5: =

-

E

-

-

+ Po —E 1~+ is +

—

[E + Po

—

—

[E

Po

—

+

1 E1(p) +

~

E,(p) + is +

Po

—

Po

—

Po

—

E2(p) + 1

i~]4’

1

E2(p) + is] E 1 4’. E2(p) + ic]

A~(p)A~(—p)

—

E1(p)

—

E2(p) (5.2)

We represent this graphically as shown in fig. 6. We can give a useful interpretation of the graphs in fig. 6 in terms of time orderings in configuration space. Each Coulomb photon represents an instantaneous interaction, and so connects points at equal time. Similarly, 4” is independent of the time component of the relative momentum, so both particles are at equal times in the wave function. The choice of the A~piece of each fermion

282

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

çz~c~~ c~c~ Fig. 6. Partial fraction rearrangement of the matrix elements. A hash mark on a fermion line means that the propagator denominator associated with that line has been removed.

propagator guarantees that each fermion propagates forward in time. A hash mark on a fermion line means that the ends of that line are at equal times. Using this interpretation, it is easy to see that, of the graphs contained in fig. 6, only two classes are non-zero. These are shown in fig. 7. Note that the graphs of fig. 7 are, in fact, time-ordered, with time increasing from bottom to top. Of course, this result can also be obtained by writing out the expression for the matrix elements and performing the integrations over the time components of the 1oop momenta. In the terms that vanish, it turns out that it is always possible to label the momenta in a way such that, if Po is the time component of the relative momentum in one of the wave functions, then the poles in Po in the integrand are all on the same side of the real axis. Thus, the Po integration gives zero. (Il()(2)

+ V

I

~1 ~——-~.j~i

+

~

+

p

1+q~fq

~i

—

(t)()(2)

p~qi,’q I

4-...4-~i

p1

(a)

(_;~ 1~,. I

(I~4(2)

(~

(_;~

2) (II~~..fI P~~ql-P2-q

(I1~4~2) p~t.q IP2q

I ~\ I-P-~ +

V~td-p -q

+

P2~~1 -p 2-q N-p1-q PI1~~PI

I

+.

(b)

Fig. 7. Partial fraction rearrangement of the one transverse photon terms.

Next, we use formal operator techniques to sum the graphs in fig. 7. We work in the CM frame, using the indicated momentum labels. Here it is to be understood that a momentum label p~on a particle 1 line indicates that the line carries four-momentum (p~+ E, ps). The integration over p? gives the factor ~dp~ —

J2ir

A~(p1+ q) A~(—p1) p~+ q°+ E E1(p1 + q) + is —p~’—E2(p,) + is —

— —

.lqO + EA~kp1 +q)A~(—p1) E1(p1 + q) E2(p,) + ic~ —

—

(3)

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

283

In graphs containing Coulomb photons, we must include the factor 4iricc/(,p1 P2) in the integrand 3p,/(2ir)3 integral. We recognize this as the momentum space equivalent of applying the of the d —iV = icc/r, where r = operator x 2 is the relative coordinate in the two particle system. Proceeding in the same fashion for P2, P3’ ..., we obtain factor analogous to thatuse in of~,the eq. (5.3) 3p,aintegration. By making from eachfordp~integration factor 1, iVwe from operator the momentumand of aparticle caneach writed all the factors of the type in eq. (5.3) as iA~i + q)A~(—~)/{q° + E E 1Cp + q) E2(~i)+ is}. Then, the sum of the contributions represented in fig. 7a is 4q —4ircc T jq.rF A~(~ + q)A~(—~) +is<4’ [q0 +E —E ~~4q2 d 1~+q)—E2~)+is —

—

—

—

J

1. 8~

A~(~ + q)A~(~ +q0+E_Ei~+q)_E2~p)+ic

A~(~ + q)A~(—~) qo±EE,~p+q)E2~p)+is~~]

T

(5.4a) which sums formally to

~

4q2—4itcc + is <4”

2T2

~iq.r

q°+ E

—

E

J(2ir)

1~+ q)

—

1 E2(j~) A~(~ + q)A~(—~i)V + is —

x~

(5.4b)

The factor e~r accounts for the momentum transfer from particle 1 to particle 2 due to the transverse photon. The q0 integration is most conveniently performed by closing the contour in the upper half-plane. If we carry out this integration in expression (5.4b), add in the analogous contribution from fig. 7b, and multiply the matrix elements by the factor i as required by the perturbation theory expression in eq. (2.21), we obtain the first order energy shift 2

—~-—

2ir

r

4”I J 2q (~3

I~~I

+

[22 T

iqr

1

2). A~U)A( 2T 1(~+ q) + E2(j~)+ A’4PA~~V E 12~ . 4”> 1 A~’~A’

q+E

+

—

2T

1

~iq~r

q+E

+

~

(5 5)

1p)+E2~p+q)+A(~i)A~V_E 2] 21 = A~( q). For future use, we Here ~ =the A~(~), A’.~F = A~(~ also define operator j~’= ~ + q.+ q), ~ = A~(—~),A’4 Intuitively, we expect that q is typically of the order of the expectation value of p (~ccmR) and V and E E 1 E2 are typically of the order of the binding energy (~ccmR). Therefore, we approximate the denominators in eq. (5.5) by q and denote the correction by AE(D). The leading term is then +

+

—~

—

I

—

—

AE(L)

=

~

J~q<4”I

x ~

VE

—

E1(~) E2(~) —

E

—

Here we have used eq. (5.1) and the identity

=

E,Cp)

—

E2(~)V

4”>.

(5.6)

284

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

We note that eq. (5.6) can also be written in the more compact form AE(L)

~

=

<4”I JL~e1~~~T 2~ 4”>.

(5.7)

In calculating AE(L) from eq. (5.6), we approximate 4” by the ~ part of the singly-iterated Salpeter wave function 4’s~(eq. (3.8)). Corrections resulting from further iterations of the wave function are discussed later. Before proceeding with the calculation of AE(L), let us dispense with the transversality term, 2 qq/q2, in Using 2 q = 2 (~‘ ~),we find 8T

—

A~(~’)21 . qA~(~)= [E1(~’)

—

E1~)]A~A~ =

A~’~A~

(5.8a)

and A~(—~‘)22. qA~( .) where E

=

2?i2 =

~

(5.8b)

(~‘)

—

E 1~’).From eq. (5.6), we then see that the contribution to AE(L) of the transversality

term is -_~--.

2ir2

3

1

I___~<4”~ V Cd

J

E

q’~

f~”\2— ~iqrkP)

A’”~A”~

~

E

—

1(~i) E2~p) E’1 + E1 ÷ 2~A~2~ V 4”> A’~ ~E—E 1~p)—E2~p)

+

—

F

(5 9)

+

Now, following the argument given in section 4, we know that there must be at least two factors of 8i and two factors of 22 in any term that is to contribute to 4’NR’ the hfs. with use Associated the ~ part each of eq.factor (3.8), of; is a power of momentum. Thus, if we approximate 4” by and evaluate expression (5.9) in momentum space, the numerator has at least 14 powers of momentum (counting the elements of integration), and the denominator has 12 powers of momentum in the non-relativistic region. Hence, expression (5.9) does not contain a ln cc’. The corrections to ~ clearly bring in additional powers of momentum. If we take into account the cc5 in the wave function normalization, we see that the contribution of the transversality term to the hfs is O(cc2EF). 5.1. Calculation ofthe leading term AE(L) First we rewrite the cc-matrix structure in eq. (5.6), grouping terms according to whether they contain an even or an odd number of factors of ~: =

[~(l

+ ~

i~

+ 121

~)21~~

m,\

/

+—(l +fl,—-)2i(1 4\ iJ \

~i’

(1 +

1\

12,~f~’2~2,~

m

+fl,—)+—4 ,,,

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

285

Xf2’

[2~ ia1________________ x q1 F 4E1E~(E1+ m1) ia~XP4EEi(E~+m)+4E(l+$1~)+4E~(1+fl1~)l

=

2j

1 + [~(l

+ si?)2i(l

+ ~

~

. p’2

121 ~ E~E~

(5. lOa)

Similarly, 2~A~( = _~

A~(

‘)2~

[

1+J32 2Em2 . 2E’2 Ia2 X q 2 ____________ + ia2 X ~ 4E2E’2(E’2 + rn2)

—

________ t24E ia2 2E’2(E2 + rn2) ~ (1 m~’\ (~ m2\1 4E2 +$2 +f3~ ~

—

—

1 22 + fl2~) +~ E’2E2 P2222

+ [~(l

+

$2~T)22(l

•f1

(5.lOb)

The product of the a x q terms gives an energy shift AE(L1)

=

~Tmim2

j’—r

<4”I v E (1

—

+ fl~)/2~iq.r°1 E1 E2 —

x q~a2x q (1 + 52)/2 4E1E~E2E~E E1 E2 —

VI4”>.

—

(5.11) Note that the factor ~(l + $~}~(l + Then /12) is a,non-zero component 4”. 4’NR• x q~a only for the large—large 2 under angular of averInitially, we spin approximate 4” by Also, 2 x q goes to ~a1 a2q aging of the wave function. E 2 VINR = N+4’NR 4’+. (5.12) —

E—E

In this approximation, the energy shift is <4’+ I e~r

ccm’m2Jd3q 3~2

I 4’+>.

a1 a2

(5.13)

4E1E’,E2E’2

In momentum space this is ccm ,m2 3~2

___________________

~

4’~’~4E1~E1(p’)E2~E2(p’) ~

(5.14)

Clearly the integrations over the two wave-functions completely decouple. Neither of these integrations can give4,acc5ln and cc~cc6 since N÷4’~ contains only of IpI. However, thebexpression contributions with no ln even cc ~. powers If we approximate N~4’~, y 4’~,and (5.13) can give cc replace E. by m~,we obtain the well-known Fermi formula: -

16cc6m~ ______

E ~

1

1

d3p +y2)2J(p2 +Y2)2<~~a2>

d3p’

________

I

2cc4m3R
________

=

3m1rn2

2>.

(5.15)

286

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

It is easy to see that the corrections to the approximations of the preceding paragraph have relative order cc, cc2, etc. For example, the difference between E and m1 + m2 is of order cc2mR, and the difference between s and so in the energy denominators is of order cc4mR. Corrections from powers of either of these small quantitites are clearly of relative order cc2. The corrections from letting E 1 m, in eq. (5.14) 2, cc3, etc. give integrals that are dominated by the relativistic region and have relative orders cc, cc Next we study the correction to the approximation ~(1 + f3,)~(1+ /12)4” = 4’NR~From eq. (3.8) we find the large—large component of 4” (once-iterated) is —~

(1 in ~2 + 2E1

4’LL

—

‘~

m2 ~[l(~ + E1\(1 + E2~ 1 8~y5/2 2)2 1)~2+ 2E2)[4~ m1)~ m2) 4mim2](p2 + y p2 ~ 8\/~y5/2 4m,m 2+y2)2~ 2)(p —

~2

(516)

In the last step we have made an expansion in powers of p2 and retained only terms to 0(p2) in the numerator. The term 1 in the parantheses has already been taken into account. The term p2/4m 1m2 can appear as a correction in either wave function. Substituting in eq. (5.11), we find that the total correction of this type is AE L WF —

2

— —

—(

cc

~

5 (‘d3pd3p’d3p” p”2 —4itcc 6 (p”2 + y2)2 (p” p’)2 64ny 16m~m~ (2ir)

J

—

+ y2)2•

—2mR (pf2

(5.17)

Here we have used 1/(E E, E 2+ y2) and replaced E, by rn elsewhere. We 2) ~ 2m~/(p note that, as it stands, the integral is logarithmically divergent. This is, of course, due to the fact that we have not retained the full l~ I dependence in the integrand. Had we done so, the integral would have converged at p ~ m. This fact is all that we need to know about the large p behavior of the integrand in order to compute the coefficient of ln cc~’.The logarithmic part of the integral in eq. (5.17) is computed in appendix 2. The result is —

—

—

AE(L-WF

2lncc’m~Jmlm 1) ~ ~



—2EFcc

2.

(5.18)

Now consider the contributions to eq. (5.6) of the terms in eq. (5.lOa) or eq. (5.lOb) that contain no spin dependence. These terms can give a contribution to the hfs only in conjunction with the small components of both waveSo,functions. Acting the small components, the factors (1 find + fim/E) 2/2m2. writing the hfs on contributions in momentum space we that become (1 power rn/E) count ~ p is logarithmic or higher. There is an explicit factor cc8 (cc5 from the wave the overall function normalization, cc2 from the factors of V. and cc1 from the transverse photon). Hence these terms give an O(cc4lncc’EF) contribution to the hfs. Next we examine the contributions to eq. (5.6) of the remaining products of terms in eq. (5.10) that are even in 2. These products contain at least one factor ofp and one factor ofp’ in the numerator, so the integrations over the wave function momenta can no longer give the factors of l/y that produce cc” and cc5 contributions. In order to give an cc6 ln cc ~, a product of terms in eq. (5.10) must contain four powers of momentum. The energy shift from such products is —

G. T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

~ cc

AE(L2)

=

C.-i3 i~rn~rn~ ~r

J

1

x ~a2

le”.. r(ai

4’NR

x p a 2 x

<

287

~~2rn2

x q~-~-+ (1 +-~2))I4’NR>.

(5.19)

We have replaced V4’/(E E1 E2) by 4’~since the corrections involve additional powers of momentum in the numerator and so cannot yield ln cc ~. Under angular averaging of the spin wave function, the spin factors become —

—

and

2

—4q2].

~2

a, x~a2 x q—~~a1~a2[~’ Using the results in table 4 we see that A L E( 2)

64iry5cc
=

3p’d3p 1~a2>(‘d 3 3~2 16rn~m~ (2ir) EFcc2 ln cc’ m~ ±

J(~-~-

(p~2

p’2 + y12

1 (p’

—

p2 (m 2 m1 p)2 (p2 + y2)2 k,~m + (5.20)

m 1m2\m1 m2j This is an example of a “wrong mass” term. We expect that such terms must cancel in the complete calculation since they would give a non-zero recoil correction to the hfs in the limit m2 ~. Now consider the terms in eq. (5.10) that are odd in 2. These connect the large component of the initial wave function in eq. (5.6) to the small components of the final wave function, andinvice contained 4”. versa. we see with that the these terms aredonon-zero of the corrections Let us So dispense terms that not leadbytovirtue contributions of O(cc2 lntocc ‘Er). Consider first terms of the form 2 ~‘22 ~ times a x q. An example of such a contribution is —~

4’NR

cc 613 Jd pd 3 pd ~

~

x <2122

~P22

(p~2

~l

1

1 1 —p)2p2 +y2

+y2)2(p’

(p’

—

p)2 2

~

222

~

12>

(p~’2

2+ v2)’ and have retained only the leadHeremomentum we have used eq. (5.1), 1/(E E1 making E2) ~the—2m~/(p ing dependence. Now, by change of variables p, p”, q —÷ yp, vi-”~yq, we see that this expression is equal to cc6 times an integral that is independent of cc. Corrections from the substitution E~ m• are of order O(cc3EF). Similarly, the product of an rL~pruzp term with any of the remaining terms in eq. (5.10) leads to integrals with powers of momentum in excess of a logarithmic power count and produces corrections of O(cc3EF). It is also easy to see that the product of an 2 term with any of the terms in eq. (5.10) except the a x q term leads to O(cc3EF) contributions. The product of an 2 term with a a x q term has the correct momentum dependence to give a ln cc and yields the energy shift —

—~

1,

—

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

288

AE(L-WF 2)

=

Id3q

cc

1

~

4m1m2

j

-~-

<4’NRI e”

{(_ ia2 x q~8~)

r

x [(E1 + him1) (~2 2mR)Vr~-i--rfl+ + ~2 2m~] [2m~ [8~

~~n;~) _____

~‘

(E~+ $imi)]

~2]}I4’NR>.

x (—ia2 x q~81)+ [1

(5.21a)

2/2rn Replacing /1~between 8-matrices by 1, using E1(p) m1 ~ p 1, and doing the usual angular average, we find 3q qe ~2 ( —2mR cc Id AE(L-WF j~<4’NRIe1 2 ~ + 2) ~ 8m~m2~ 2)V(3)ai ~a2q~p~4’NR> + (1 ~2) 64~y5Jd3Pd3P~d3P~~ 1 1 2~ —2mR “ / —4ircc \ —cc 2 = 24m~m 2~ (2~)6 (p~2+ ~2)2 (p~_p)2P ~2 + y2)~(p —

—

—

—

1 x (p’

—

—

p) .p”

2~2
(p~~2 +

~

1 a2> + (1 i-÷2)

cc6m~ ~ Jd3p 3 p 3ir”m~m2 2 d

“

(p2

p2 +

2)2

(P _~)2 1

(p~~2 ~ +

V2)2 (5.21b)

Then from table 4 we see that 26 rn~(m1 m2’\ AE(L-WF2) ~ 3 cc m,m 2 2 + i ln cc’ 2 \m2 m1) —

~

—

=

2 ln cc rn ~ (~-+ -~-~-~) cc m1m2 \m2 m1 /

(5.22)

2 ln cc ‘EF) contribution from eq. (5.6) is The total O(cc AE(L-WF

‘EF.

2lncc’(m~/m 1) + AE(L2) + AE(L-WF2)

=

E~cc

1m2)(—2 + 2m1/m2 + 2m2/m,). (5.23)

5.2. Higher iterations of the wavefunction We can now discuss conveniently the possibility thatfunction higher isiterations 2 In cc~EF).The doubly-iterated wave given by of the wave function contribute in O(cc cc [2(m 2]A~kp)A~(—p)I d3k 4’DI 2it~ 4(m + E2) 22p + y2) j(k_p)2 1 + E1)(m2 1 + m2)(p —

\

x{(1+81~(1_82~ 2m

k2

~

1 ) 2m2) 4 m1m2J¶~ 4’1.~(k). (5.24) The term 1 in braces, of course, gives the singly-iterated wave function, and the remaining terms give the correction due to the double iteration. Imagine substituting these wave function correc—

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

289

tions for one of the wave functions 4” in eq. (5.6). In the resulting expression we can isolate the terms with the leading momentum dependence by using 2+ l/(E E1 E2) ~ —2m~/(p and setting E, = m, elsewhere. This leads to an energy shift of the form y2)

—

—

-~J~-~ <4’~I

V 2~ 2

x (m, + $ 1E +

8~

~iqr

16rn~m~ [(rn~+ $,E~+81

~~)]. [(m2 + /12E~-22

~p)2~(m2+ 52E2

-22

.~)]fl

v~ (5.2 5)

where

2rnR

1

cc — 4’DI=~22+V2

+$1E, +21p)(m2 +$2E2 2_~—82~p) k 1.k x2k 81k22k k X j p)2 2rn 4’NR( ). 1 2rn2 4m1in2 4mim2f Now, if we write this expression in momentum space and rescale all the variables of integration by a factor y, we obtain for the leading order cc6 times an integral independent of cc. 4mm(m1 3k f8

~“

(k d

—

~

—

—

—

5.3. Calculation of AE(D) We examine next the corrections to retaining only q in the denominator of eq. (5.4b). In the usual treatments of the one transverse photon kernels, one makes an expansion in terms of the number of Coulomb photons exchanged. This is equivalent to expanding the denominator of eq. (5.4b) in powers of V. Instead of making such a V-expansion, we write the denominator as follows: 1 1 +1 1 ‘E—E’ —E —A’’~A~2~V q±E~+E 2+A~~A~V—Eqqq+E~+E2+A~1)A~V_E’~ 1 2 + ÷ (5.26) We have already treated the first term. In order to analyze the second term we note that, by virtue of eq. (5.1), —

[E

—

E~ E2 —

—

A

A~V]A~’~A~2,4”= A~’~A~21(E1 + E2 + A~A~V)4” —

=

(E~+ E2 ~

(E1

—

1~2

E~)Ai~

14” + A’ A~[A’~81A~, V]q5’. (5.27)

It turns out that the commutator term in eq. (5.27) is negligible.* In order to demonstrate this, * This is the counterpart in our formalism to the compensation that occurs between the various terms in a V-expansion. See, for example, Karplus and Klein [14].

G.T. Bodwin and DR. Yennie. Hyperfine splitting in positronium and muonium

290

we substitute the expression for A~’~81A~ given in eq. (5.10). Consider first the term i(m,/2E1E~)a,x q. Expanding the factors of E. in the denominator, we have m1 ‘2E1E~,a,

_____

X

q

~—a1

=

X

q

4m~ a1 x q + ... .

—‘

(5.28)

The commutator of the first term with V is zero. The remaining terms lead to momentum space integrals that contain logarithmic and higher power counts. Since element contains 7 (cc5 from the wave function normalization, cc1 the frommatrix the transverse photon, an explicit factor cc cc1 from the factor V), we conclude that such terms can give a contribution to the hfs in O(cc3lncc~E~) or O(cc2EF), but not in O(cc2lncc’EF). The remaining terms in eq. (5.10) lead to expressions containing at least one power ofp and one power of p’ in the numerator. Thus, neither of the integrations over wave function momentum can give a factor i/y. We conclude that these contributions are at most O(cc3 ln cc ‘EF). Having shown that the commutator term in eq. (5.27) is negligible, we need only consider the term 1 1 ‘E E’ qq+E~+E 2+A~1)A~V_E’~1 1 -

For this term we expand the denominator once again. 1 + E’ qq

1 (E 1 + E2 + A1’~A~~VE 1

t2~V) E1 2 E1 +~LE—E’ q2” 1 —E 2 —A’”~A + 1 q q + E~+ E 1~A~VE (E 2 +A~ 1 Ei). (5.29)

E’)

—

—

—

—

+

—

X

—

—

The first term integrates to zero because of the symmetry between p and p’ in the remaining integrand factors. By analogy with the analysis of eq. (5.27), we conclude that the important part of the second term is 1 E’ E 1 E1 E’1) (E’2 E2)(E1 E’1) . 2)V E~ q3 (5.30 ) 2 + E~+ E2 + A’wAt The terms we have dropped in eq. (5.30) are either of higher order because of an additional power of V or fail to give a ln cc’ because of additional powers of momentum in the numerator. In writing down the energy shift, we again make use of eq. (5.10). It is necessary to retain only the product of a x q terms since all others contain too many powers of momentum to give a !n cc ~. We also _

—

—

—

—

—

2)q

use

E

—

E.

(p~2 —

=

p2)/(E + E.)

~

p2)/2m,.

(p~2 —

(5.31)

So the ln cc~contribution due to the expression (5.30) is AE(D)

=

—

<4’NRI

— 4

~ 6 ~ —cc

3

3pd3p’ 1 . a2> ~d m~m~ (2ir)3

~

—

4cc6m~
—

—

J—’~

~

j~

3ir

J

5

mR 2 2
m1m2

1 (pt2

2>lnx

=

2

rn1m2

I4’NR>

(p~2—

+ y2)2 (p 2 mR

—1

~a

p2)2

2

cc lncc

—

1

p2)2

~)2

(p2

+ y2)2

-1

EF.

(5.32)

G. T. Bodwin and D.R. Yennie. Hyperfine splitting in positronium and muoniurn

291

5.4. The corrections AE(B) We divide the corrections involving the A parts of the fermion propagators into three classes. The first class, denoted by AE(B1), contains all corrections with one A. between the transverse photon and an external Coulomb interaction. The second class, denoted by AE(B2), contains all corrections with one A inside the region spanned by the transverse photon. The AE(B1) and AE(B2) corrections are shown in fig. 8. Note that, because of time-ordering considerations, a graph with one A between Coulomb photon rungs must vanish.

H

1

+

±

~f’:

4”

4”

4”

(a)

o (.00 )~_1 +

+

0

(_)~~

+ .‘

00 (b)

Fig. 8. Examples of corrections involving one factor of A_. The line containing the A_ is indicated by (—) and all others have a factor A~.(a) AE(B,) corrections. (b) AE(B 2) corrections.

Consider the class of graphs in fig. 8a. From time ordering arguments (or examination of the integrations over time components of momentum) we see that these graphs give the energy shift shown in fig. 9.

±

±

+

S

S

Fig. 9. Examples of AE(B,) corrections that are non-vanishing after the partial fraction rearrangement.

In each graph the lower legs give a factor [l/(p0 + E + E,

—

is) + i/(Po

—

E2 + is)]A~A~/(E+ E1

—

E2)

and the propagator on line 1 just above the transverse photon vertex gives a factor A’.~P/(E+ Po + q0 E’1 + is). The momentum routing can be chosen so that all other propagators are independent of p0. Now —

292

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

J

1 2ir E + p0 + q0

[

—

1 E~+ is[p0 + E + E1

1

+ —

~

is

r

=

—i 1 E+E1 _E2[E’1 + E1 —q0 ~is+

E+q0

—

—

1 1 E2 + is]E + E1

—

1 1 E~—E2 + is]~

E2 (5.33)

if we sum the Coulomb photon ladders as in eq. (5.4), the second term in eq. (5.33) leads to the energy shift 3 “d iqr 1 A~~)A(2). 2T V 2 \‘V 2ir 2e q + E~+ E2 + A2~’A~~VE + 1 E + E1 E2 (5.34) Note that this is identical to expression (5.5) except for the last factor in the matrix element.* The first term in eq. (5.33) gives an energy shift 3q . 1 ~ cc ~ d + E’, + E 1~A~~V E(cl + E~+ E 2 + A’~ 2 E) 2) 1 A(l)A( X q+E~+E A’(1)A(2L 81E+E T V 4”> 1 + ÷ 1E2 The expression (5.34) can be analyzed in the same manner as eq. (5.5). The conclusion is that the leading contribution to the hfs due to corrections of this type is given by 3 At1~A12~ cc d +E E V14i>. (5.36) —

—

—

+

+

—

—

—

—

+

We again make use of eq. (5.10) to reduce the 8-matrix structure for particle 2. Only the 82 X q term contributes in leading order. The other terms can be eliminated by power counting. Note that the a 2 x q term is manifestly orthogonal to q, so that we can drop the transversality term in the dot product. Then, it is easy to see that the minimum number of powers of momentum comes from taking 4’~ for the left-hand wave function and using either the 8~~p part of A~Pin conjunction 4’NR part of the right-hand wave function or the E with the 1 fl,rn1 part of ~ in conjunction with the 8~ ~p part of the right-hand wave function. So, the leading contribution to the hfs from expression (5.36) is —

~ 4~2

3q fd 2

J

k~

2m

=

~ J14~ q

2

4~2

~

-

1\A~PV

)

2

q

~

*

jq.r(~72_X_q~8 <4’NR e

<‘I’NR ~iqr~(1(72_X_q.81”\[ 2m

1 ~

4’>

2m1 2m2

J~-~

)[

~

2m1

+

~~i!l ~ 2m1]

<4’NRI e~r[_q ~pV + Vq .~] I4’NR>

In fact, an alternative approach would be to treat expressions (5.34) and (5.4b) together by writing \2~ I/ 1 + V\114”> = I/ ~ + VI4”>. ~ A~’~A~ I vI4”> = 1, E+E,E 2

J

\E_E,_E2

E+E1E21

EH,E2

‘i’NR

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

=

4cc6m~ (‘d3pd3p’d3p” 3irm~rn 6 2 (2ir)

J

—

+

(p~~)2

(p’

If we make the substitution p” (5.37) becomes 16cc7rn~ 1d3p d3p’ d3p” 3m~m 6 2 (2n)

J

x [(p’

P]

—

i”)

—÷

p’ + p

(p2

—

[

1

~~~)2[(P

~2)2(p~

±~2)2 .

p” in the last

term

,

,, _~

,,

(—4ircc)

).~

(5.37) and associated factors, expression

1 1 1 + y2)2 (p” p)2 (p’ p”)2

~

—

— p”) Op” + (p” 6rn~

— —

1 (p~2+

293

1d~‘d3

cc 12m~m

[

—

—

p) .~] (2 —l

1 . a2> 1 2 (p2 + V2)

~

~‘

2ir4J P p [(p~2 + ~2)2 1 (p’~4.~ 1 + (p~2+ ~,2) .,)2 (p2 + ~


2)2

(p’

—

p)

~,

538

(p’

2)2]\al

a2

The ln cc’ contribution is 6m~lncc1=_~( rn2 ~cc2lncc1Er. 3m~m2 2 \m, + m2j —

(5.39)

The expression (5.35) contains at least one additional power of momentum in the integrand compared with expression (5.34). Since the leading part of expression (5.34) has the correct momentum dependence to give a In cc 1, the leading part of expression (5.35) cannot be logarithmic. The non-leading parts of expression (5.35) either have too many powers of momentum to give a ln or are of higher order may in cc, be so they are negligible. that the hfs of 2 ln cc1EF) obtained by addingWe to conclude expression (5.39) the contribution corresponding AE(B1) in O(cc expression with m 1 and m2 interchanged and doubling the result. (This accounts for the possibility of having a A on the upper parts of line 2 and for the opposite order of transverse photon emission and absorption.) Hence, 2lncc1EF. (5.40) AE(B1) ~ (—2 m1/m2 m2/ml)(rn~/mlrn2)cc The contribution AE(B 2) is represented graphically in fig. 10. Once again, time-ordering argu-

—

—

ments show that the graphs with other arrangements of the hash marks vanish. The p0 integration

Fig. 10. Example of AE(B2) corrections that are non-vanishing after the partial fraction rearrangement. The shaded blob indicates any number (including zero) of ladder Coulomb exchanges.

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

294

is

J

1

1

-

2ir E + p0 + q0 + E’1 isE + Po E1 + is = 2~(E1+ E~+ q0) 1 q0 + E1 + E~,+ —

—

—

q0

-i + E1 + E~ is —

(5.41)

—

is

By the usual procedure, we find that the second term leads to an energy shift 1 <4” ~Cd3 (l~iqr T~ A’11~A~2~V A’~’~ cc 4I~2 J q + E~+ E 2 + A1’~A~~VE E1 + E~ q 3 iqr T.A(2)v A’(l) T Cd Jq2 2m 1 —

2

T

4”

q

—

2

+

+

1

+

1

2m

-~~<4”~

2

‘~

4~2

1

Jq

~~$~.<4’NRI et~[_

~

)

2m2

V~’~ +

~

8~

~

>

V](w2

~<~.2l)

I4’NR>.

(5.42)

Clearly, this leads to the same result as in eq. (5.37). The 5-function term in eq. (5.42) leads to an expression containing two additional powers of p in the non-relativistic region compared 1. Thus,with we eq. (5.42). (The factors that go like 1/q are replaced by 1/2m.) It cannot give a ln cc conclude that, to O(cc2 ln cc AE(B 2) ~ AE(B1).

(5.43)

5.5. The remaining corrections Consider first the correction AE(B3), which arises from taking the A part of more than one propagator. The non-vanishing kernels involving two A’s are shown for one order of transverse photon emission and absorption in fig. 11. We set aside, temporarily, the possibility of A’s inside

(a)

(b)

(c)

(d)

(e)

(I)

Fig. 11. Examples of AE(B3) contributions.

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

295

the Coulomb photon blob. The other arrangements of two A’s vanish because of time-ordering considerations. The graphs in fig. 11 may be obtained by changing a A÷to a A.... in the kernels already considered in conjunction with the A(B1) and A(B2) corrections. (See figs. 9 and 10.) In general, the effect of this change is to convert a potentially small denominator to a denominator of order rn1 or m2. For example, in fig. 1 la the Po integral is 1d 1 1 1 1 2iriJ ~~°p 0+q0+E—E’1+isp0+E+E1—is—p0+E2—is 1

— —

q0

—

E1

—

1 E~,+ isE + E1 + E2

—

is~

In comparison, the corresponding integral from the AE(B1) kernels is 1d 1 1 1 1 POPo+qo+E_E~+ispo+E+Ei_is_po_E 2+is 1 1 —q0+E, +E~—isq0±E—E~—E2+ie~

~J

—

—

—

We see that the denominator q0 + E E’, E2 ~ q1~ has been replaced by E + E1 + E2 ~ 2(m1 + m2). (This can also be seen by writing the various graphs in a time-ordered fashion and examining the old-fashioned perturbation theory denominators corresponding to the intermediate states.) Similarly, the kernels involving three or more A ‘s contain at least one less small denominator than the corresponding AE(B1) or AE(B2) kernels. Since the integrals in the AE(B1) and AE(B2) calculations have at least a logarithmic power count, the additional A .‘s eliminate the possibility of obtaining a ln cc’. Now consider the effect of changing Ak’s to A’s in the Coulomb photon blob. If a graph is to be non-vanishing, each A between Coulomb photons on a given fermion leg must be paired with a A directly opposite it on the other fermion leg. The propagator denominators associated with such a pair lead to the loop integral —

—

—

it’

1

~

1 —p0+E2—is

1 1 E+q0+E~+E2_is’~ 2(m1+m2)’

as compared to the integral lId ~T~J

1 1 POPo+E+qo_E~+is_Po_E2+is~

—

1 q0+E—E~—E2+is

that occurs in the A ÷case. Thus, we see that the net effect of a A pair in the Coulomb photon blob is to replace the expression l/(q + E’1 + E2 E + A’4PA~V)in eq. (5.5) by t1~At2~ I A’~A12~VA’ V 1 q + E~+ E 1~A~V 2 E + A’4.’~A~~V + + 2(m1 + m2) q + E~+ E2 E + A’.~ -

—

—

—

—

—

We recover the leading term by replacing the denominators q + E’ 11~A~2~V 1 + E2too many E + A’ with q. In the resulting integral, there are at least three powers of momentum to obtain a ln. There is an explicit factor cc8 outside the integral, and each of the integrations over wave func—

296

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

tion momentum can give a factor i/y. We conclude that the contribution of AE(B3) to the hfs is 2EF). at most in O(cc Table 1 One transverse photon contributions to the hfs in O(a2In a — 1EF) LiE

Eq.

Coefficient of (ml/m

2 In a

‘EF

1m2)a

L,

(5.11)

0

L — WF, L 2 L — WF2 D B1 B2 B3 A Total

(5.18)

—2

(5.20)

m2/m1 + m1/m2 m2/m1 + m,/m2

(5.22) (5.32) (5.37) (5.42) —

2 —2 —2 0 0 —4

— —

m1/m2 m1/m2

— —

m2/m1 m2/m,

The correction AE(A) 1V/(E E is obtained by replacing one or both of the wave functions in eq. (5.5) with A~~A~ 1 E2) times the ~ part of the Salpeter wave function (see eq. (3.8)). Suppose we make this replacement for one of the wave functions in eq. (5.5). Then, in the leading contribution to the hfs, the integrand contains too many powers of momentum to givea logarithm, 7 outside the integral. The integration over the momentum associated and thereis an explicit cc 4” gives a factor i/y. The remaining integration over wave function with uncorrected wavefactor function momentum contains too many powers of momentum to give a factor i/y, so the leading contribution is pure O(cc2EF). The non-leading contributions contain additional powers of cc or momentum, as does the contribution in which the ~ correction is used in both wave functions. Thus, AE(A) does not contribute to the hfs in O(cc2 ln cC 1EF). The calculation of the one-transverse photon kernel contribution to the hfs in O(cc2 ln cc 1EF) is summarized in table i. It is reassuring that all the “wrong mass” terms have cancelled. Our result, AE(i transverse) = —4(m~/rnlm 2lncc~EF, (5.44) 2)cc is in agreement with the result of Lepage [25] for the single transverse photon kernels. —

—

-

—

6. Other contributions to the hfs In this section we discuss other kernels that could conceivably give an O(cc2 ln cc 1EF) contribution to the hfs.* Many of these kernels have been investigated previously by other authors; here, we merely report their results. Kernels involving two transverse photons can arise from both the first order terms and the second order terms in the perturbation expression eq. (2.21). The two-particle irreducible kernel of fig. 12a is, of course, present in AK. The two-particle reducible kernels of fig. 12b and c are the result of substituting the first and second terms of eq. (2.23) in place of G 0(j) in eq. (2.2i). For our -

* We remind the reader that all contributions under discussion are recoil corrections and have an additional factor m,/m,. in the case of muonium.

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

297

~1 FR F~ ~

~

~

~

(a)

(b)

Cc)

Cd)

(e)

Fig. 12. Addition kernels that arise in perturbation theory through second order.

purposes the remaining two transverse photon pieces of the perturbation expansion are negligible. We proceed to demonstrate this point. The ~(j) term in eq. (2.23) leads to the kernel in fig. i2d. In the non-relativistic limit, this gives the usual second order perturbation theory expression for the energy shift ‘cç~

<4’31 AKI 4’s> <4’1IA1(I4’~>

2), we expect that Sincegives the matrix elements are at least O(cc”) and the energy denominator is O(cc this an O(cc2EF) contribution.* The last term in eq. (2.21) contains the product of an O(cc4) matrix element with the derivative of that matrix element with respect to P 0. The effect of the derivative is to square the denominator in the expression (eq. (5.5)) for the matrix element, or to insert a factor i/(E E1 E2) next to the wave function. By counting powers of momentum, we conclude that —

—

<4’~IAK 2). Thus, the last term in eq. (2.2i) is also O(cc2EF).

is atThemost O(ccof fig. 12c has been examined by Barbien and Remiddi [23] in the equal mass case, kernel and by Lepage [25] in the unequal mass case. Its leading contribution to the hfs is of O(cc2 ln cC ‘EF). Barbieri and Remiddi also consider the contribution of annihilation graphs to the positronium hfs. The terms of O(cc2 ln cc ‘E) that derive from these sources are —

AE(2-transverse, i Coulomb ladder) 2ln

AE(annihilation)

=

—~cc

=

~cc2in cC ‘(rn~/mlm 2)EF

cc’(m~/rn,rn 2)EF.

(6.la) (6.lb)

The result (6.la) has been verified by Lepage [25]. The kernels in figs. 12a and b have been investigated by Lepage and Fulton et al. [22]. They find that the leading contribution of relative order cc forofthe kernel of fig. 12acC’ and 2 ln cc~ for the kerneltoofthe fig.hfs 12b.is The total contribution relative order cc2 ln is relative order cc AE(two-transverse) = ~{m~/m ln2 cC 1EF. (6.2a) 1rn2)cc Fulton et a!. also show that the leading hfs contribution arising from the crossed Coulomb graph (fig. 1 2e) is of O(cc2EF). Although not reported here, we have confirmed these results for exchanged photons. * There is, of course, the possibility that the sum over states produces a logarithm. We have investigated this possibility using the non-relativistic Coulomb propagator. In that case, it turns out that the variables of integration can be resealed so as to produce an overall factor a6 times an integral that is independent of a.

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium

298

If we insert a Coulomb photon into the kernels of fig. 12 in such a way that it crosses other photons, the resulting kernels generally contribute in higher order. To see this, we note that thereis a factor cc associated with each Coulomb photon. It can, in general, be cancelled by a small denominator arising from the two fermion propagators that must be inserted along with the Coulomb photon. (This is, of course, the mechanism whereby a bound state pole develops in the four point function.) However, if the Coulomb photon crosses another photon, the momentum of that photon enters into one of the fermion propagators. The small denominator is then —~ (crossed photon momentum) rather than + V2) (p being the two particle relative momentum).* Hence, momentum power counting indicates that the explicit factor of cc is not cancelled. The consequence is that terms originally of O(cc2EF) and those of O(cc2 ln cC 1EF) become at most O(cc3 in cC 1EF). Kernels involving three transverse photons are at most relative O(cc2EF). This is because the (p2

associated with the third transverse photon must be multiplied by factors 8 p from the wave function or the fermion propagators if they are to connect the large components to large components. The additional powers ofp coming from the factors 8~p guarantee that none of the momentum space integrals are logarithmic and that the additional factor of cc associated with the transverse photon cannot be cancelled by a small denominator. 2 ln cC 1EF) is given by conclude that(6.2). the total contribution the hfs eqs.Thus, (4.4),we (5.44), (6.i) and The splitting in thistoorder is in relative O(cc 8T’5

AE(cc2lncC’) positronium AE(cc2 In cC muonium

1)

=

=

—*cc2lncC1EF for positronium

(6.3a)

2cc2(rne/m,,) ln cC 1EF for muonium.

(6.3b)

7. Summary and conclusions We have calculated the contributions to the hfs in O(cc2(m~/m,rn 2)ln cC ‘Er) that arise from the Coulomb interaction, the one transverse photon-multiple Coulomb photon kernels, and the two transverse photon kernels. (The latter are not reported in detail here since our techniques are not very different from those of others, and the results are the same.) Our results are in agreement with those of Lepage [25]. In investigating the one transverse photon-kernels, we developed techniques that enabled us to treat all numbers of Coulomb photon exchanges at once. This permitted an approximation procedure in which intermediate calculation of spurious contributions was avoided. The total theoretical result for the hfs is compared with the experimental value for muonium in table 2 and for positronium in table 3. Numerical values are taken from Lepage [25]. We see that theory and experiment are in agreement to within the estimated uncertainties. However, theory cannot bethe tested at of theO(cc3EF, presentcc2(m~/m)EF) level of experimental accuracy untilcalculated. the terms 2EF)the in positronium and terms in muonium have been of O(cc A first step in this direction would be the calculation of the O(cc2(m~/m,,)ln (m,,/mC)EF) contribution to the muonium hfs. Such a calculation would reduce the theoretical uncertainty in the * In the case of crossed Coulomb photons, time ordering arguments show that only the A parts of the fermion propagators contribute, so the small denominator is in fact m.

G.T. Bodwin and DR. Yennie, Hyperfine splitting in positronium and muonium Table 2 Comparison of theory and experiment for muonium hfs

Table 3 Comparison of theory and experiment for positronium hfs

Theory 2(m,/m

Total excluding O(a 2a2(m,/m Total 6) ln a’EF

6 6) In a ‘EF) 4463 293.(6) 4463 11 304(6)

Uncertainty due to terms of O(a2(m,/m 6) In (rn,/m,)EF)

kHz kHz kHz

(Casperson et al. [53])

203 381.2 — 3.8 203 377.4

MHz MHz MHz

10

MHz

Experiment Mills and Bearman [7]

203 387.0(16)

MHz

Egan et al. [6]

203 384.9(12)

MHz

kHz

Experiment

*

Theory Total excluding O(a2 In a ‘EF) 2In a ‘E~ —~a Total Uncertainty due to terms of O(a2EF)

.~.10

299

4463 302.35(52) kHz

This uncertainty is due to the uncertainty in g,/g,.

muonium hfs by about a factor of 5. The techniques developed in this paper seem well suited to the calculation of logarithmic terms, so some variant of them may be useful in obtaining the in (mp/me) contribution. If one attempts to calculate additional non-logarithmic contributions to the muonium and positronium hfs, new technical difficulties are encountered. For example, one must deal with the infinite sum point, over intermediate states second order perturbation theory. Also if one uses as a starting then indications areinthat no finite number of iterations of the Salpeter equation produces a sufficiently accurate wave function [30]. Assuming that such difficulties can be overcome, it appears that the complete O(cc2EF) calculation is tedious, but doable. 4’NR

Acknowledgements We wish to thank Professor Kinoshita for suggesting this problem and for useful discussions during the course of the calculation. We also had numerous helpful discussions with Professors Wayne Repko, and Hans Bethe and Dr. G. Peter Lepage. We especially thank Velma Ray for her patience in typing the manuscript through several revisions.

Appendix 1. Normalization of Salpeter wave functions Our approach is similar to that of Lurié et a!. [28], who derived the normalization condition for the Bethe—Salpeter wave functions. We rewrite eq. (2.22b), which defines the three-dimensional Green’s function g 0, in the c.m. frame g0(p’,p, P0)

=

° JCd ‘°d -i-— G0(p , p, P0). -~——

(Al.l)

In the case of an instantaneous kernel, we can integrate the momentum space version of the inhomogeneous BS equation (eq. (2.1)) with respect to p’°and p°and use eq. (Al.i) to obtain the

G. T. Bod win and DR. Yennie, Hyperfine splitting in positronium and muonium

300

inhomogeneous Salpeter equation g0(p’,p, P0)

=

p0

H,(p’) d3

3~(r’

—

H2(—p’) +

is

—

F(p’)fi,fl2[(2ir)

+ j’(21r)3 K

0(p’,p”)g0(p”,p, P0)],,

where F(p) A~(p)A~( —p) the integrations:

—

(A1.2)

A~(p)A~( —p). We can write this more compactly by suppressing

3iö(p’ p). i14]g0 = (2ir) Similarly, if we carry out this procedure for the BS equation in the form

(A1.3a)

—

—

H,

—

H2)

—

G 0

=

S~F’~S~~ + GOKOS~F’~S~?~,

we obtain the alternative form of the inhomogeneous Salpeter equation 3i~5(p’ p). (Al.3b) g0[/3,/32F(P0 H1 H2) i~0] = (2it) Eqs. (A 1.3), together with the earlier definition of 4’~,can be used to fix the normalization of the Salpeter wave functions. We work in the subspace of states specified by the projection operator A~(p)A~( —p) + A~(—p)A~(—p). In this space f2 = 1. In order to pick out the pole at P —

—

—

—

0

we multiply (Al .3a) by P0

—

~, and differentiate with respect

fi,/32Fg0(P0 ~) + [fl,132F(Po H, From eqs. (2.3) and (2.15) we have that —

lim (P 0 Po-.~~j

—

—

~g0

=

i

—

H2)

—

iK0]

~

=

to P0. The result is

[g0(P0

—

~~)] =

3iö(p’ p).

(Al .4)

(2ir)

~ 4’~><4’~~I fluI~2

(Al.5)

k

where the index k labels the degenerate states. Also, the homogeneous version of eq. (A1.3b) is —

H1

—

H2)

—

il?0]

Thus, if we multiply eq. (Ai.4) on the left by

~ <4’k’ I FI4’,>

<4’JkI /31132

=

=

0.*

4’’fl,132

<4’Jk’ I 131132.

and take the limit P0

—÷

~, we obtain (Ai.6)

Hence <4’JkIFI4’Jk> (Note:

F

—

=

~kk’

(Al.7)

In more general cases where i?~ has a P0 dependence, F must be replaced by

ifl1fl~l?0/~P0Ip0,,,~.)

* Note that for the two forms of the homogeneous Salpeter equation to be consistent it is necessary that i~7J1fl2= — i/31$2R0. This is a consequence of the Hermitian nature of a field theoretic interaction corresponding to the instantaneous potential.

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

301

The orthogonality between eigenfunctions of different eigenvalue follows by considering the relation ~J’<4’JkIFI4’J’k’> = <4’JkIF(Hl +H2 -ifl1fl2Fl?0)I4’~~,~> =

Thus

<4’JkIFI4’Jk>

=

(Al.8)

~jj’~kk’~

In the non-relativistic limit F =

—÷

1, so eq. (Al .8) reduces to the familiar normalization condition (A 1.9)

~jj’~kk’~

Appendix 2. Some useful integrals We are concerned with integrals of the type

J

3 ‘7 ,[~.‘2p’2~p4~p2(p P’)2 ~+ .~,2)2 (A2 1) (p2 + y2)(p d which have a logarithmic power count. We assume that the integrals have a large momentum cutoff at p, p’ ~ m 1 or m2. For our purposes the precise nature of this cutoff is unimportant since 1. it cannot affect the coefficient of ln cC Consider first the integral ~

—

— p’)2(p’2

Jd3p d3p’p2p’2/(p2 + y2)2(p

—

—

~~)4]

p’)2(p’2 +

y2)2.

We perform the p’ integration to obtain ~2

Jd3P[VP2/(p2

+

~2)

+ 2ptan’ (p/y)]/(p2 +

y2)2.

(We ignore the high momentum cutoff in the p’ integration since the integral converges without it.) Now, the ln cC’ contributions come from those integrals that diverge logarithmically in the limit 0. In this limit the first term in brackets leads to a convergent integral which is 0(1). However, in the limit V —. 0 the second term in brackets gives the integral —~

~J(p2

2)2dP ~

~

4lncC1. =

4~~[lnp]~ ~ 4~

Hence, J’d3~d3p’ (p2 +

2 ‘2 y2)2(p_p~)2(p~2 + p2)2

4~4In

cC’.

(A2.2)

G.T. Bodwin and DR. Yennie, Hyperjmne splitting in positronium and muonium

302

Now consider the integrals Jd3Pd3p’~4;p2(p

—

+ V2)2(p

2]/(p2 p’)

—

p’)2(p’2

+ V2)2.

(A2.3)

The p’ integration is easily performed, with the result (R2/V)Jd3P~4/(P2

+ V2)3 p2/(p2 + V2)2].

These integrals are linearly divergent so we may not ignore the convergence factors. With the convergence factor included, they are O(cc but not 0(ln cC’). They are the origin of O(ccEF) corrections, which we have bypassed in this paper. Finally, we consider the integral 1),

JI

—

(p2

~‘~2

+ V2)2(p’2 + V2)2

d3 d3 ~ P

‘

— —

I + y2)2(p’2 2P ~P’+ d3 d3 J(p2 + y2)2 P P. .~

—

The term proportional to p p’ vanishes on performing the angular integration. The remaining two terms give equal contributions. We perform the p’ integration first for the term proportional to p2 and the p integration first for the term proportional to p~2The net result is ‘)2C 2 ~ yJ(p2+V2)2

_______ d~

P. As in the case just described, the correct convergence factors must be included in the integrand, and the integral is of 0(1/cc), but contains no 0(!n cC’). Thus, we conclude that, in expression (A2.l), only the integral whose integrand is proportional top2p’2 gives a in cC’ contribution. In the other integrals the leading contribution is —~l/y. These results are summarized in table 4. Table 4

1

Contributions from integrals of the form jd pd

f (p, p’) p4 p2(.p — p’)2 (p —p’)’

p (p2

+

f(p,p’) p’)2(p’2 +

.~,2)2(1,

Leading contribution

Coefficient of In a

4it4lna~

4iz4 0

.-. rn/7 rn/y

m/y

1,2)2

0 0

Note added in proof: Caswell and Lepage (see note added in proof in ref. [25]) have discovered an additional O(cc2 ln cc1 EF) contribution to the positronium hfs arising from the kernel with a transverse photon exchange followed by an annihilation photon. We have verified their~result. Its net effect is to change the coefficient of the ln cC’ term in table 3 from to +~, making this contribution + 19.1 MHz. The weak contribution to the hfs has been examined by Beg and Feinberg [Phys. Rev. Lett. 33 (1974) 606; 35 (1975) 130(E)], and our estimate in section i agrees with theirs. —~

G.T. Bodwin and D.R. Yennie, Hyperfine splitting in positronium and muonium

303

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