PHYSICAL REVIE% A VOLUME 51, NUMBER 4 APRIL 1995

Contribution of light-by-light scattering to the orders O(orms) and O(m, ns 1n cx)orthopositronium decay rate

Gregory S. AdkinsFranklin and Marshall College, Lancaster, Pennsylvania 17804

Marios LymberopoulosDepartment of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 78019

(Received 29 September 1994)

The gauge invariant contribution of the six radiatively corrected light-by-light scattering processesto the decay rate of orthopositronium into three photons is calculated. The infrared-safe Yennie

gauge is used for the loop photons. The relative order n is part of our result, 0.0074(9)(o. /7r )I'r.o,where I'I,& is the lowest-order decay rate. This makes a 0.000350 ps contribution to the decayrate, thus reducing slightly the apparent discrepancy between the theoretical and experimentalresults. The analytically obtained relative order n ln n part of our result, —n ln(1/n)I'«, is in

agreement with earlier evaluations.

PACS number(s): 36.10.Dr, 12.20.Ds

I. INTRODUCTION

I',„=7.0514(14)ps

I,„=7.0482(16) ps

(1.1a)

(1.1b)

Due to charge conjugation invariance of @ED, or-thopositronium may decay into an odd number of pho-tons, primarily into three. The lowest-order expressionfor the decay rate, first obtained by Ore and Powell [3,4],1S

2nsmc2 (vr2 —9)9~

= 7.211 169(4)ps (1.2)

Using natural units from now on [5], we can write, forthe complete theoretical expression giving I'

o-PS '

Positronium is an attractive testing ground for @EDcalculations and in particular for the bound state for-malisms in quantum field theory. The decay of or-thopositronium (o-Ps), however, is even more interesting,since currently there is a discrepancy between the theo-retically calculated and the experimentally measured val-ues of the decay rate. The most recently measured valuesare [1,2]

1 th = 7.038204(10) ps + r„.(c, — + ".)(1.4)

The discrepancy is seen to be [setting Cs ——0 and usingEq. (1.1b)]

r,„ t —rii„,„=0.0100(16) ps-', (1.5)

(1.6)

which is 6.2 times the experimental uncertainty. ThecoeKcient C3 would have to be of the order of 250 inorder to bring theory and experiment into agreement,since (n/m) I'« ——0.000039 ps

The five-photon decay channel contributes 0.187(11)to Cs [13,14]. Another contribution 28.8(2) comes fromthe "square" of the relative order cx corrections [15].

In this work we present the calculation of a part of C3resulting &om radiative corrections of the light-by-lightscattering process. A known part of C2 is also obtainedanalytically &om one of the graphs. The Gnite, gaugeinvariant [16] set of the six graphs under consideration isshown in Fig. 1. We regulated the ultraviolet divergencesthat appear in all six graphs by use of dimensional reg-ularization, working in n = 4 —2z dimensions of space-time. The divergences were isolated in the form of polesin e, at e = 0 . They were generated by factors of I (2e)and/or terms such as

2+c3 — +c~a (inn) +. ).3 2

We used the Yennie gauge photon propagator [17]

k'g

k' (1.7a)

The coeKcient C~ has been calculated by several authors[6—9]. The constants C2 and C4 have been found to beC2 ——1/3 [7,10] and C4 ———3/2' [11,12]. Using themost accurate value —10.2866(6) for Ci [9], we find

1 —28'(1.7b)

The Yennie gauge does not contain in&ared divergences,

1050-2947/95/51(4)/2908(11)/$06. 00 51 2908 1995 The American Physical Society

CONTRIBUTION OF LIGHT-BY-LIGHT SCATTERING TO THE. . .

(c)

FIG. 1. The 6nite, gauge invariant setof radiatively corrected light-by-light scatter-ing graphs, contributing to the three-photondecay of orthopositronium: (a) the outerself-energy, (b) the inner self-energy, (c) thedouble vertex, (d) the outer vertex, (e) theinner vertex, and (f) the annihilation vertexgraphs.

(e)

which makes it convenient for the study of bound states.The choice P = 2/(1 —2e) instead of P = 2 results insimpler expressions for the self-energy and vertex func-tion one-loop corrections [18,19]. One also avoids possiblecomplications when higher order graphs are considered.

II. CENERAL METHOD

quired properties in order to be used as the lowest-orderequation in a systematic perturbative expansion for thesolution of the Bethe-Salpeter equation [20,21]. The wave

function that appears in Fig. 1, used in M, is the refer-ence wave function one finds after solving this referenceproblem. For the purposes of our work the following ap-proximation was sufBcient:

The decay rate of orthopositronium can be written as eo, (I) =—(2~)'d(&)42myo~

1 d kg d k2 d k3 4F(OPs MS+)M()3()s()s(27r) y, = (mn)'~'/(2~2~), (2.3)1 1

xb(P —ki —k2 —ks)24py 247224)3 3

x ) —, ) ~M(o-Ps m Sp)~2,&rn

A A

8'1 qE'g )8'3

(2.1)

8I'(o-Ps -+ Sp) =~ dzi dzs—

2 'K p

x ) —, ) iM(o-Ps m Sp)i . (2.2)A A

Cy qCg qC3

where 2;,k, are the polarization and four-momentumvectors of the ith external photon with u; = k, =

~k, ~,

is the orthopositronium polarization vector, to beshown later, and P is the bound system four-momentumvector.

We now go to the rest &arne, where the four-momentum is P" = (2W, O) with M = 2W being themass of the system. Using the energy-momentum con-serving b function and rotational symmetry we can re-duce the dimensionality of the integral &om nine to two.%le may choose the energies of two of the final photons tobe our two variables. Using the normalized final photonenergies x, = o~;/W, one finds

with cp ——(0, 0, 1) and e-~ = ~(~1, i, 0) . —Here p~ is

the relative four-momentum of the two particles, whichis set to zero by the b function, since inclusion of eitherpo or p would result in higher powers of n. Also, Pp 1s

the nonrelativistic ground state wave function at contact.The mass of the reference bound state is

0!M„= 2Th'„= 2m 1—4n2

(2.4)

A Q A-+&v —+ —gpv (2.5)

where n is the principal quantum number.

Prom now on only the ground state n = 1 will be con-sidered. Furthermore, we can ignore the binding energyand use the approximation W = m. This approximationis adequate for most of the calculation; only the loga-rithmic contribution &om the annihilation graph requiresbetter accuracy.

Since the light-by-light scattering tensor is the mainpart of all graphs to be considered and since it is gaugeinvariant, we can perform the polarization sum by use of

The bound state formalism we have used is based onan exactly soluble reference problem, which has the re-

Furthermore, the spin average can be performed as fol-lows:

2910 GREGORY S. ADKINS AND MARIOS LYMBEROPOULOS 51

('0 -.s ') f0 -.--40 0 J ~&0 o )

k3,c3

-1/2 P+1+k3

(1+Y)Y S (1+Y)Y ( gu+NNu)k2, e&

1/2 P+1-ki

'F(p), e

(2.6)

where N = (1,0 ), since 0 8 is real and 8 has onlyspatial components. In general, M will consist of variousparts of all orders in a. So

kl, Cl -i'(1/2 P+1)

FIG. 2. One of the OSE graphs.

+2Re (M, )') M, +, (2.7)

where i stands for any graph that we can writedown. From Fig. 1 we have contributions &om

MosE MIsE &Dv &ov &Iv &Av where thesubscripts denote the following: OSE, outer self-energy;ISE, inner self-energy; DV, double vertex; OV, outer ver-tex; IV, innver vertex; AV, annihilation vertex. The con-tributions to the decay rate are included as follows:

In the last three relations, S3 is the group of permuta-tions for the external photons. Since the phase space issymmetric under permutations of the external photons,we do not need to symmetrize both in ML~ and M~ .We simply symmetrize in M~~ and. multiply by 3!.Fur-thermore, K; = k;/m. Finally, =„„„,which is itself

a trace and is actually defined by (2.9), is the fourth ranktensor that describes the fermion loop of a particular ra-diatively corrected light-by-light scattering graph.

I' = I'„+) I'; + (2.8)III. CALCULATION

By writing the amplitude M for a particular graph as

M~ ———inurn

~6 S3

0 I T'(3) ~w(2) ~T(1) ) (2.9)

1 1

I~ = — dz] dx32:lX2X3

Tx, (2.10)

where

1T» = ) » (a)

—D(v"'" (—I)I+ 4 (s) + I)

a&S3

we obtain the corresponding decay rate correction by useof

In this section the calculation of the contributionsof the six radiatively corrected light-by-light scatter-ing graphs to the orders O(mn ) and O(mns inn) or-thopositronium decay rate will be discussed. The ap-proximation R' m was used for all but the infraredsensitive parts of the annihilation graph. Our gauge usedP = 2/(1 —2e) . A (—1) factor was included in all ex-pressions for JM~ because of the existence of one morefermion loop compared to the lowest order process. Theelectron charge in n = 4 —2e dimensions is e(n) = p'e,where p is an arbitrary mass parameter and e is the usualdimensionless electron charge parameter. We used e(n)for all vertices except those associated with the three 6-nal state photons, since the 6nal state photons are thesame for all graphs, and an overall factor of (const)' hasno effect on the final finite result. Finally, the contribu-tions of the OSE, ISE, DV, and OV graphs were doubledsince these graphs occur in pairs.

» I ()Y S (1) + I)Y ( + I)IIA. Self-energy graphs

4w g P + +P (2.11)From Fig. 2 we write for the OSE amplitude

d"8(—1) ) Tr [

—ie(n)] [—ie(n)p"] ( i)

(—iep"')~qS3 + m

Z zx i (—iep"') i (—iep"I)

(—if+ g+ g (s) —m) (2g+g —g (i) —m) (2/+ g —m)

F1 &0 e.eI

-&+e I, T ~I 0 I

&2 4'o[-()) .['-()1 .['-()1 .(2 ) —,' + m—

51 CONTRIBUTION OF LIGHT-BY-LIGHT SCA'I I'BRING TO THE. . . 2911

where Po ——ps/~/~sr and p = ma/2. The renormalized self-energy function in the dimensionally regularized Yenniegauge is [18]

Z (p) = C(p)($™) (3.2)

where C(p) is given by the equivalent expressions

a (4vr p2 ) (3 —2s) (1 —c) g s(1 —s)Cp = —2 —i

I'1+a dsdt4' i m2 J 1 —2e m2 o s'G i+~

or

(3.3a)

(3.3b)

where

(m2 —p')G = s+ (1 —s)t,

m' —p')@=s+ (1 —s).

(3.4a)

(3.4b)

poth expressions for C(p) were used in calculating I'oss. As an example, we read oK:-o«&om (3.1) using (3.3a)

3 —28' 1 —E' 1

8(l —2s)=;,"'""= (4 ')'I'(1+ ) d dt ' '(1 — )

d"E p ( g+ g+ m)p —'(—1)f+ g+ ps+ m)p"'(1)f+ g —pi+ m)p"'(++ g)iver !2 - 1+a

[(E—N)2 —m2] [(/+ks —N)2 —m2] [(E—ki+N)2 —m ] sm —(l2 + 2/. Nm)(1 —s)t

(3 5)

The 8 integral was done by use of three Feynman parameters. The traces were performed by the computer programREDUCE [22]. The divergent term was isolated and done analytically. In the process, a fiiute contribution was foundanalytically as well. The rest of the terms were integrated over numerically by use of the adaptive Monte Carlointegration routine vEGAs [23]. The results, using (3.3a) and (3.3b), were

ar...= ( 0 'I"(2s)(—I) + 0.080 288(5) ),( n"I'(2s) (—I) + 0.080 289(3) ),

(3.6a)

(3.6b)

where 0 = 4m@, /m2 and I = s ((2) ——s . Our final result, which is a composite of these two, is given in Table I.

The inner self-energy graph was done in exactly the same way. Prom Pig. 3 the amplitude is

TABLE I. Results from the calculation of the radiatively corrected light-by-light scatteringcontributions to the three-photon orthopositronium decay rate. In the divergent part the fac-tor 0 = 4s y, /m, results from dimensional regularization; finally, I = g(2)/3 —31/48. The unitsfor each of the last four columns are indicated.

OSEISEDVOVIVAV

n* r{2.) (- )—1—1—6

22

(n'I;o)00

0—ln(1/a. )

Graph Divergent part Logarithmic part Numerical part

(-')0.080 289(3)0.233 046(6)1.596 125(108)

—1.047 442(72)—0.417 101(40)

1.295 719(102)

Numerical part

ILO

0.415 48(2)1.205 96(3)8.259 57(56)

—5.420 27(37)—2.158 40(21)6.705 05(53)

Totals —ln(l/n) 1.740 636(170) 9.007 39(88)

2912 GREGORY S. ADKINS AND MARIOS I YMBEROPOULOS

~ISEd"1

(—1) ) Tr [—ie(n)]~ASS

1X —iz„

I

P——+ E+(—-', /+/+A ~, l

—m) ( 2

(-:~+&-&-&.i--) (-:~+&--)

[—ie(n)p"] (, )(—iep"')

k (s) I (—iep"')~. ( ', I'+-4-+I/-(. &- ).&o ~ e ~2m A [e-(.)]p. [s-(.)]„[s-(.)]„(0 o

(3.7)

The resulting contributions to the decay rate were

I'rsE =

6I'zsE =

(n"r(2e) (—I) + O.233 O52(14) ),mo. s

(0"r(2e)(—I) + 0.233045(6) j

(3.8a)

(3.8b)

The composite result appears in Table I.

B. Double vertex graphs

One of these graphs is shown in Fig. 4 with momentum labeling. The amplitude is

(3.9)

(—1) ) Tr [—ie(n)] . [—ie(n)p"] (—iep"')

7.q S3 2 +Z z zx, . [—ie(n)p ], [

—iep"'](——,'P+ g+ g (3) m) i~"~ (

—2/+»/+P ~z) m) — (2//+»/ —P ~)) m)—i—(»

—&)~(p —&)& ~& (—ie~"'), [—ie(n)~'1,

I a~~+ P( ,'//+u -) -(-p-&)' & (p- &)' & ( ,'r+// )--.&0 ~e

&& T v I 0 I~2m''e[& ()1~.[&-()]~.[&-()]~.(0 0

(f —g+ I/, —m) (y' —g+ g&—m)

(u' —4)

1

(g —g+ I/Is —m) (y' —g+ g&—m)

(3.1o)

where the Yennie propagator is used for the virtual pho-ton in the loop. We calculated the contribution of MD~by dividing it in two parts: the "Feynman" part, coming&om the g~q part of the virtual photon propagator, andthe "gauge" part, coming from the P part of the propa-gator. Furthermore, the gauge part was divided into twoparts by use of

I

This produced two gauge parts GB and GA, which wereconsidered separately. This method reduced the amountof work and made the traces less complex as well. Con-sequently, the computer memory requirements were keptrelatively low. A similar decomposition of the second(»/

—g) term with its adjacent propagators was not used,since in one of the pieces that would be created thereis no direct way of isolating the divergences; one wouldhave to solve analytically rather complicated integrals toextract the pole coefficients. Both the p integral and theI. integral in (3.9) were done using Feynman parameters.The resulting contributions from the three pieces were

k, ,e3

k2,e2

I I

-1/2 P+1+k3

V2

I[1/2 P+1-ki

-iXR(-1/2 P+1+k3)

k3,e3I(-1/2 P+1+k3

k),gl

FIG. 3. One af the ISE graphs.

kl, e,1

FIG. 4. One af the DV graphs.

51 CONTRIBUTION OF LIGHT-BY-LIGHT SCA I IERING TO THE. . . 2913

mo. 'I'~ = ( 02'I'(2s)( 2—I) + 0.703875(48) )

(3.11a)mo. '

I'a~ = ( 0 'I'(2e)( 2I—) + 1.594164(75) )(3.11b)

mn81&~ =, ( n'r(2s)( —2I ) —0.701914(61)) .

(3.11c)

Putting these together we have, for the total contributionof the DV graphs,

mo. sI' = ( 0 'I'(2s)( 6I) +—1.596125(108) ),

(3.11d)

C. The one-loop vertex function

For the vertex graphs the use of the renormalized one-loop correction to the vertex function A",(,&

is required.An expression for the latter is given in Ref. [19]. De-pending on the choice of the parameters a, b, and c, interms of which A",(» is expressed in [19], it may take ondifFerent but equivalent forms. We calculated the contri-bution to the decay rate &om the vertex graphs by useof two difFerent choices of a, b, and c:

(i)a=b=0, c= —P(2 —e); (3.12a)

(ii)a = 2(1 —s) —P, b = 0 c = —P(2 —c) . (3.12b)

The final results in the two approaches were in agree-ment. The approach that used the first choice will nowbe presented in more detail. Using (3.12a), A",(,) can bewritten as

2(1 —c)2 + P(2 —s)A",(,)(P, p) = —(4mp, )' dx du dtx ' —I'(1+a)7"x(H —xm )s(1) ) 4 ~ 1+@

—I'(2+ s)p&x2(H —xm )m + 2+(

2(1 —x) G" 2(A" + sB") + PC"(3.13)

with

xm2 —xu(1 —u)k2+(1 —x)u(m —p )+(1—x)(1—u)(m —p ),H = xm'+ t(H —xm'),

1~&(p, p) = —p" (g —g' + m)p" (y' —g'+ m)p„—(2 —2x —x )p"m

2

(3.14a)

(3.14b)

(3.14c)

(3.14d)

c"(p p) = ( (1 —x) [—yg~" —~"gd2(r u)v" +~"@~+d~"~l f'0&" —&"&( + ~"&"—

-2e —.(y."y+u."+."e)+-""(-2+5*-*'))

G"(r, v) = [~'"(0('u —Ã'~ —f'd~+(('~') + (~ ~')(~ K)f'&"—(3.14e)

(3.14f)

where the p .p„contraction in A" is carried out in four dimensions, k:—(p —p) represents the incoming photonfour-momentum, and

E' = (kux+ px) = [pxu+ px(1 —u) ] . (3.14g)

D. Outer anck inner vertex graphs

The amplitude for the outer vertex graph of Fig. 5 is

2914 GREGORY S. ADKINS AND MARIOS LYMBEROPOULOS 51

d"E

v.g Sg 2

z Z Z

x (—iep"'), (—i«":( ))( 1p+g+g

~ ) m, ) ('P+ g —t(( i)) —m) (2P+g —m).&0 ~.sqo 0

(3.15)

In A",'&» the following were used:

P = (K+/ —kl),

&= (w+e),I' = x(I(I+ E —k)u),

(3.16a)

(3.16b)

(3.16c)

mo. '( 0 'r(2e') I + 0.073 209(24) }

r...=0,mo, '

+0.884178(47) ),

(3.17a)

(3.17b)

(3.17c)

H = —Z 1 —x —2Z N —kgu 1 —x

+++ 2xiu(1 —x) ), (3.16cl)

I'ov, = mo. '

—1.504 828 (181) ) . (3.17d)

H =x+t —Z 1 —x —2Z N —kgu 1 —x

+2m, u(1 —x) ) . (3.16e)

Furthermore, the contributions Rom each one of the fourpieces of A",'&» were calculated separately. The four

pieces are (1) the term that includes (H), (2)

the term that includes (H), (3) the term thatinclud. es G"' H ~ +'), and (4) the term that includes

2(A"' + eB"') + PC"' H ~ +') . The correspondingcontributions to I' will be denoted by I', , I', , I'and I'~~, , respectively. Following the general methodthat has already been discussed we obtained the results

Putting these together yields the total contribution of theouter vertex graphs

mnsr = ( 0 'r(2z) (4I) —1.047441(142) }(3.18a)

The result with the second choice of a, t), and c (3.12b)1s

mo. sr.'.=, ( n"r(2s) (4I) —1.047443(83) } .

(3.18b)The composite result in listed in Table I.

The inner vertex graph was evaluated in the exact sameway. Prom Fig. 6 we have

(—1) ) Tr [—ie(n)] [—ie(n)p"]

& z)

(—icy"3)v gSg

L7l ( 2 + fll

z 2 2—ie "'( lP+((+((-() — ) — '' (2P+(7 —)'-() — ) (l&+& — )).&0 ~.s &2~ do [sT(3)]gk3 [sT(2)]p2 [sT(1)]gl1 .(0 0

(3.19)

k3,e3

-1/2 P+1+k3

k3,e3I (

-1/2 P+1+k3

kZ, e2 1/2 P+1-k1

+(p),&

k2, eZ

-1CA&( i)

kl, el

1cA (&)

kl, el

II

1/2 P+1-k

P1

FIG. 5. One of the OV graphs. FIG. 6. The IV graph.

51 CONTRIBUTION OF LIGHT-BY-LIGHT SCA i I'BRING TO THE. . . 2915

p = ( N+—l+ ks),p=(N+e-k, ),E' = x( —k2u + N + E —kq ),H = x+ (1—x)(2(xsu+xg —xgu) —8

—2&[N(1 —2u)+ ksu+ kq (u —1) ] $,H = x+ t(1—x)(2(xsu+xy —xyu) —&

—2&7[ N (1—2u) +k u+ k (u —1) ]).

(3.2Oa)

(3.20b)

(3.20c)

(3.20d)

(3.20e)

The expression that gives A~2&» [see (3.13) ] involvesthe following: k3,e3

k2, e,2

k1,81

-1/2 P+1+k3I II

1I2 P+1-kiII

FIG. 7. The AV graph.

By isolating the divergent terms and evaluating the nu-merical integrals we obtained

F, , =~

12 'F&2e)~

—Il + 0.066528&16) ),mes 2, (1

7I &, 2 j(3.21a)(3.21b)

mo. 8

The total contribution of the inner vertex graph is

I', = ( 0 'I (2s) (2I ) —0.417076(51) ) . (3.22a)

The result using (3.12b) is

mo. 'I', = ( 0 'I'(2 ) (2I) —0.417139(63) j . (3.22b)

+0.221477&21) ), (3.21c)The composite result appears in Table I.

F,„,= Q~'F&2e)~

—+ —~

I —0.705081&43) ) .

(3.2ld)

E. The annihilation graph

The annihilation graph is shown in Fig. 7. Its ampli-tude is

(—1) ) Tr [—ie(n)] . ( [—ie(n)A", &, )] ( ~)

(—iep"')&6Sp

m

t(--,'~+~+~-~i--) '

(-,'&+~-~-~i--) (-,'&+&--)&0 esx Tr &

I I~2m40[s &3)],.[s &2)],.[s &»]„.)

The following were used in A",(» .'

p=E+WN,p = 8 —R'N,e' = * e —(1 —2u)WN,H = f —E (1 —x) +2WE N(l —x)(1 —2u) +m —W +x(1 —2u) W )H = xm + t j —E (1 —x) + 2WE N(1 —x)(l —2u) + m —W + x(1 —2u) W

(3.23)

(3.24a)

(3.24b)

(3.24c)

(3.24d)

(3.24e)

—{1+a) —(2+& )The parts of A, &, ) involving the factors (H) and (H) were done as before (in the limit W = m) withthe results

mnsF, = 12~'F&2e)~

—I~

—0.238172&14) I7r

~~V, = o .

(3.25a)

(3.25b)

The parts of A",(,&involving the factors H ~ +'~ and H ~ + ~, however, are problematic in the in&ared. The

following method was implemented. Prom (3.23) and (2.9) it is easily seen that

2916 GREGORY S. ADKINS AND MARIOS LYMBEROPOULOS 51

dX 1nr"&2 '"

[{E—WN)2 —m2] [(/+ WN)2 —m2] D{E)' (3.26)

where ( is some constant and

T(I) = (g —Wg+ m)p"'(g —Wg+ gs + m)p"'(f + Wg —t|i + m)p"'(f + Wg+ m), (3.27a)

D(I) = (l —WN+ ks) —m {8+WN —ki) —m2

From (3.26) we can now write

(3.27b)

d7Lg

,. „/, ([ ()— (o)] '()())[(g WN), ,][(g+WN), ,]D(g)

1 1

D(E) D(0) [(e —WN)2 — 2] [(I. + WN)2—1 1

D(0) ](5 —WN) —m ]](P. +WN)~ —m~] )(1) i (~) ~ (~)~AV ~ ~AV ~ ~AV (3.28)

Now some comments about the three di8'erent partsare due.

(i):-(~$ has the usual uv divergence, but no ir prob-lem. Therefore, we can set W' = m in that part.

(ii):"(„'$ and:-(„'$ combined do not have a uv diver-gence. So when done together one can use c = 0 .

(iii):-„$ has no ir problem, so we can set W = mhere as well.

(iv) =(~$ does have an ir problem, so we kept W g m.(v) When the contribution to I' of the third part is

worked out, the O(n2lna) term that will be extractedwill be directly proportional to lL

(vi) The D(0) to be used in all expressions is

I = dx x du

1

[(E —WN) —m ][(/+ WN) —m ]' (3.31)

the terms with enough factors of E in the numerator togive rise to ultraviolet divergences were done in the previ-ous step. For part (2), we set W = m and used standardtechniques to set up and evaluate the integral numeri-cally. In the infrared sensitive part (3), we kept W g m.The E integral for part (3) is simple. For example, theterm I'(", is proportional to the integral

D(0) = (—2m zs)( —2m zi) = 4zszim {3.29)We first used a parametric identity to combine the lasttwo denominators as

The contribution to the rate from =(~$ when the~ + ~ and H ~ + & parts of A" (z) are considered was

done by isolating the divergent terms and integrating nu-merically. The result was

I = dx x dzL dv

1

[ P+ 2W—E Nb+ W2o.]2 ' (3.32)

mo. s12*'I'(2 ) ]

—I]

—0.501597(92) )mrs q2

(3.30)

We now outline the evaluation of terms lA(v)„andI'(g„. We set e' = 0, since the ultraviolet divergenceswere already identi6ed, and the remainder is 6nite in theultraviolet. As can be seen by an examination of theform of H [see (3.24d)] the logarithmic singularity comes&om the region in parameter space where E M 0 anda = (1 —2u) + 0. Terms in A" + C" containing factorsof 1 or a and terms in G" with more than two powersof E or a do not contribute to the logarithm. For theseterms, we simply set W = m and calculated parts (2) and(3) together. The remaining terms contain the logarithmand here we calculated pieces (2) and (3) separately. Noultraviolet divergences arose in these contributions since

where b = (1 —2v), cr = (m —W )/W = n /4+0(o. ),and

H = (1 —z) ( E+ 2W—I. . N a + W (o.

+xa')/(1 —x)) . (3.33)

On performing the 1 integral, we obtained

1dx du dv dz 1 —z

(3.34)

where

(1 —z)b. = ((1 —x)[za+ (1 —z)b] + zza+o.[(1 —x) (1 —z) + z] )W (3.35)

Now it is clear that the logarithmic singularity appears

CONTRIBUTION OF LIGHT-BY-LIGHT SCATTERING TO THE. . . 2917

when a, b ~ 0. We rewrote I in terms of the a and b

variables and used the notation T(a, b) = (1 —2:)4/Wto get

dx x dz 1 —z

1 1

dx f (z) dz(1 —z)0 0

o Tab Ta, —b(3.36)

We broke the unit square 0 & a, b & 1 into a triangle withone vertex at the origin and one side at a + b = 1, andanother triangle with a vertex at a = b = 1 and a sideat a+ b = 1. The logarithmic singularity comes &om thetriangle which includes the origin. On breaking I = Iq +I~ into contributions &om the triangles defined above,we found that the nonsingular piece I2 can be evaluatedwith W = m. Using the parameters a = A(1 —s) andb = As for the first triangle, we have

In all, the contibution of the AV graph is

r:.= — 'l (1/ )r„.+ ( 0 'r(2s)(2I) + 1.295607(145) ) .

(3.42a)

The result with the second choice of a, 6, and c (3.12b)is

r~ = —n ln(1/a)I'„

( 0 'r(2s)(2I) + 1.295 827(142) ) .

(3.42b)

The composite result is given in Table I.

IV'. CONCLU SIONS

1 1

Ig ——— dx f(x) dz(1 —z)0 0

A2/ + o. A2+

(3.37)

The total contribution to the three-photon decay rateof orthopositronium resulting &om all six radiatively cor-rected light-by-light scattering graphs is given by the lastrow in Table I. It is

r~ ~$ (light-by-light)

R~ = (1 —z) z(1 —s) + (1 —z)s + xz(1 —s)(3.38a)

Q = (1 —x)(1 —z) + z . (3.38b)

Now we have

1 1

Ii ——— dx f (x) dz(1 —z)2 0 0

1

2R+ go.Q

1 (R +crQIln

I2R ( oQ )The logarithmic part is clearly

1 1

Iq (ln) = — dx f (x) dz(1 —z)0

(1 15 /11x ds

I+

I

lnI

&R+

(3.39)

(3.40)

= —o. 1n(1/n)r + ( 1.835376(ill) ) .

(3.41)

and in the remainder we set n = 0. We performed thethree-dimensional integrals for the logarithmic contribu-tions analytically.

The I'&'&„and I'(')„contributions were combined togive

= —n in(1/n)r + 1.740636(170)

1n(1/n) r„~ + 9.0074(9) — r„~ . (4.1)

We see that the sign of the outcome is the "right" one,in order to bring theory and experiment closer. How-

ever, due to the relative smallness of the numerical value,it obviously does not reduce the 6.2' discrepancy dras-tically. In order to bridge the gap between theory andexperiment more work is definitely required in the futuretowards calculating contributions &oni other sets of dia-grams and relativistic corrections. Given the magnitudeof our result, it is reasonable to hope that the rest of thecontributions to the decay rate of orthopositronium tothis order may very well eliminate the discrepancy.

ACKNOWLEDGMENTS

It is a great pleasure for M.L. to thank Professor Kim-ball A. Milton for all his support, encouragement, andguidance toward the completion of this project. G.S.A.would like to acknowledge the support of the NationalScience Foundation, through Grants Nos. PHY9008449and PHY9408215, and of the Franklin and Marshall Col-lege Grants Committee. The work of M.L. was sup-ported by the U.S. Department of Energy Grant No.DE-FG05-91ER40636. We also acknowledge Grant No.PHY930009P &om the Pittsburgh Supercomputer Cen-ter.

2918 GREGORY S. ADKINS AND MARIOS LYMBEROPOULOS 51

[1]

[21

[31

[4]

[71

[8][9]

[10]

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