hyperfine splitting in positronium and muonium

HYPERFINE SPLITTING IN POSITRONIUMAND MUONIUM

Geoffrey T. BODWIN and Donald R. YENNIE

LaboratoryofNuclearStudies,Cornell University,Ithaca,New York14853,U.S.A.

INORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICSREPORTS(SectionC of PhysicsLetters)43, No. 6(1978)267-303.NORTH-HOLLAND PUBLISHING COMPANY

HYPERFINE SPLITTING IN POSITRONIUM AND MUONIUM*

GeoffreyT. BODWIN andDonaldR. YENNIE**

LaboratoryofNuclearStudies,Cornell University,Ithaca,New York 14853, U.S.A.

ReceivedDecember1977

Contents:

1. Introduction 269 5.3. Calculationof L~E(D) 2892. Perturbationtheory 270 5.4. The correctionsi~E(B) 2913. Applicationto quantumelectrodynamics 275 5.5. The remainingcorrections 294

3.1. Coulombgaugeperturbationtheory 275 6. Othercontributionsto thehfs 2963.2. Strategyfor determiningtheordersof a 277 7. Summaryandconclusions 2983.3. Gaugeinvariance 278 Acknowledgements 299

4. ThepureCoulombsplitting 279 Appendix1. Normalizationof Salpeterwavefunctions 2995. Theonetransversephotonkernels 280 Appendix2. Someusefulintegrals 301

5.1. Calculationof theleadingtermAE(L) 284 References 3035.2. Higher iterationsof thewavefunction 288

Abstract:

We calculate correctionsto the groundstatehyperfinestructureof relativeorder a2 ln a 1 in positroniumand relativeorder

(m,/m~)a2Ina~in muoniumdueto theexchangeof virtualphotons.Ourresultsarein agreementwith thoseof Lepage.ContributionsarisingfromtheCoulombpotentialandfromtheexchangeofonetransversephotonalongwith anynumberofladderCoulombphotonsarediscussedin detaiLIn treatingthesingletransversephoton-multipleCoulombphotonexchanges,we sumthecontributionsinvolvingdifferentnumbersof Coulombphotonsand reexpandthe resultingexpressionin termsof a quantity that is inherently smallerthantheCoulombpotentialin thenon-relativisticregion. This procedureenablesus to takeinto accountfrom the beginningimportantcancellationsthat occurbetweenthe various termsin anexpansionin powers of the CoulombpotentiaLThe techniquesdevelopedheremaybeusefulin calculatinghigherordercorrections.

Singleordersfor this issue

PHYSICSREPORT(SectionC of PHYSICSLETTERS) 43, No. 6 (1978) 267—303.

Copiesofthis issuemaybeobtainedatthepricegivenbelow.All ordersshouldbesentdirectlyto thePublisher.Ordersmustbeaccompaniedby check.

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* Supportedin part by theNationalScienceFoundation.

** This author expresseshis appreciationto Willis Lambfor his earlyencouragementandsubsequentfriendship.

G.T. Bodwinand DR. Yennie,Hyperfinesplitting in positroniumandmuonium 269

1. Introduction

The differencebetweenthe singlet andtriplet levelsin a two fermion systemis denotedby theterm hyperfine structure(hfs). Measurementsof the hfs in theground statesof hydrogen,positro-nium andmuoniumcurrentlyprovidethe besttestsof relativistic two-bodytheory.* As aconse-quence,considerableexperimentaland theoreticaleffort has beendevotedto the study of thegroundstatehfs in thesesystems.Thegroundstatehfshasbeenmeasuredmostpreciselyin hydro-gen(to an accuracyof aboutonepart in l0 12) [1,2]. However, thecalculationof thetheoreticalvalueof the hydrogenhfs is limited by uncertaintiesin the stronginteractioneffects atthe level ofaboutthreepartspermillion (ppm)[3,4]. In contrast,atthepresentlevel of experimentalaccuracy,positroniumandmuoniumareessentiallypurequantumelectrodynamicalsystems.**Thegroundstatehfs hasbeenmeasuredto anaccuracyof 0.12 ppmin muonium [5] and6 ppmin positronium[6, 7]. In order to obtain theoreticalresultsof comparableaccuracy,it is necessaryto carrytherelativistic two body calculationat leastto termsof relativeorder ~2 in positroniumandrelativeorder ct

3 and c~2me/m,.in muonium.The lowest orderground statehfs due to exchangeof a transversephotonwas first calculated

by Fermi [8]. It is

EF =

3cx m1m2/(m1 + m2)3.*** (1.1)

An additionalrelativistic correctionin muoniumof O(ot2 EF) was foundby Breit andco-workers[9]. In order to progressfurther,it was necessaryto developaconsistentfully-relativistic two-bodyformalismfor boundstatecalculations.This wasdonein 1951 by Schwinger[10] andBetheandSalpeter[11]. Subsequently,Kroll and Pollock [12] and Karplus, Klein and Schwinger [13]evaluatedthe termsof O(ot2EF) in muonium,andKarplus andKlein [14] evaluatedthe termsofO(cxEF) in positroniumandalsofound acorrectionmct4/4 to the Fermi splitting due to virtualannihilationof thee~epair. In the Karplus—Kleincalculationa cut-off was introduced— withlittle justification — in order to handlespuriouslogarithmic divergencesin the infrared region.Later calculationsby Salpeterand Newcomb [15] and Fulton and Martin [16] avoidedthisdeficiencyandconfirmed the Karplus—Klein result. In addition, the terms of O(~(me/mp)EF)inthe muoniumhfs werefound. Thesetermswereverified by Arnowitt [17] andGrotchandYennie[18] — the latter using an effective potential formalism. Electron self energy correctionsofO(x3 1n2x EF) andO(cc3 ln ~ EF) in the muoniumhfs werecalculatedby Layzer[19] andZwanziger[20] andverified by BroskyandErickson[21], who alsoestimatedtheterm of O(ot3EF).

A firststeptowardthecalculationof thehfs termsof O(cc2EF)in positroniumandO(0t2(me/mp)EF)in muoniumwas madeby Fulton, Owen,andRepko[22]. Theyattemptedto calculatethe termsof O(X2ln0C’EF) in positronium and O(ct2(mC/m,L)lnc(’EF)in muonium. Additional terms ofthis orderdueto three-photonprocesseswerefoundby Barbieri andRemiddi[23]. Cung,Fulton,Repko,andSchnitzler[24] foundan O(x2 ln c~- 1EF) contç~butionto thehfs comingfrom the pure

* Other high-accuracytestsof quantumelectrodynamicsthclude theanomalousmagneticmomentsof the electronand muon,theHe fine structure,andtheLambshift in hydrogen.Theanomalousmoments,of course,do not test theboundstatetheory.

** We estimatethat modificationsof thephoton propagatordueto thep mesonenterin relativeordera2(m,/m~)2 10b0. Theneutral currenteffectsin theweakinteractiongivea splitting of relativeorder~/~mlm

2GF/1ra, which is relativeorder10b0 for posi-tronium and 10-8for muonium.

~ ni1 and In2 arethemassesof thetwo particlesanda 1/137 is thefinestructureconstantThroughoutthis paperwe useunitsin which Ii = c = 1.

270 G.T. Bodwin andDR. Yennie,Hyperfinesplitting in positroniumandmuonium

Coulombinteraction.This contributionwasverified by Lepage[25] aswas the Fulton—Owen—Repkoresult for two transversephotonprocesses.However,the result of Lepagefor processesinwhich onetransversephotonandanynumberof Coulombphotonsareexchangedis in disagree-mentwith theresultof Fulton, Owen,andRepko.The latter resultwas obtainedby conventionalBethe—Salpetertechniques,whereasthe Lepagecalculationwasbasedon the Grossequation[26].Thus,althoughthe two approacheshavethe samephysicalcontent,it is difficult to resolvethediscrepancyby direct comparison.

In the presentwork we concernourselveswith the contributionto the hfs arising from theexchangeof virtual quanta. We calculate the terms of O(oc2 ln ~ 1EF) in positronium andO(~x2(m

8/m,5)ln ~ -

1EF) in muonium— makinguseof noveltechniquesfor handlingCoulombphotonexchanges.Our resultsprovidean independentcheckon the work of Lepage[25] andFultonetal. [22]. The techniquesdevelopedheremay be usefulin extendingthe hfs calculationto higherorders.

2. Perturbationtheory

In this sectionwegiveabriefderivationof theperturbationtheoryexpressionfor the two bodyenergylevelsin quantumfield theory.Thisdiscussionalsoservesto introducemuchof thenotationthatis usedin the remainderof the paper.We beginby discussingthe perturbationproblemin fullgenerality.Then, in order to relate the four-dimensionalwave functionsandmatrix elementstotheir ordinarythree-dimensionalcounterparts,we specializeto thecasein which theunperturbedinteractioncan be representedby an instantaneouspotential. Our result, eq. (2.21), resemblesordinary non-relativistic perturbationtheory. It includesthe first order energyshift derived bySalpeter[27] andalsothe secondordercorrections.Ultimately,in carryingout calculations,wereducethe matrixelementsin eq. (2.21)to matrix elementsbetweennon-relativisticstates.Implicitin this treatmentis the assumptionthat the spectrumof statesin the non-relativistic problemhasthe samebasicstructureas the spectrumof statesin theoriginal four-dimensionalrelativisticproblem. Conceivably,in a fully-relativistic treatment,new types of statescould appear.Ourtreatmentwould fail to accountfor such states.

The two body problemin quantumfield theory may be formulated in terms of the Bethe—Salpeter(BS) equationfor the four-point function:

G(x1,x2,x3,x4) = S~’~(x1,x3)S~

2~(x2,x4)

+ Jdx5dx6dx7dx8S~’~(1,x5)S~2~(x

2,x6)K(x5,x6,x7,x8)G(x7,x8, x3,x4). (2.1)

Here S~is the Feynmanpropagator*and G is the complete four-point function (S~(x1,x2) =

<01 Tiji(x1)~(x2)10> andG(x1, x2, x3, x4) = <01 T~(x1)~i(x2)~(x3)i~i(x4)10> for Dirac particles).**K is the sum of all two-particleirreduciblegraphs,namely thosegraphsthat cannotbe divided

* In principleS~is thecompletetwo pointfunction, but for our purposesonly theunperturbedpropagatorS~is used.** Notice thatweusea two particle formalismratherthanaparticle—antiparticleformalism.In orderto treata particle—antiparticle

systemsuchaspositroniuniwemustchargeconjugateoneof theparticlesin our two particlesystem.Thisamountsto includinga factor— 1 in theexpressionfor each vertexinvolving two antiparticles,anda factorC(C_1) on theincoming(outgoing)antiparticlesideof eachvertexinvolving a particleandanantiparticle.C = iy0y2 is thechargeconjugationmatrix.

G.T. Bodwinand D.R. Yennie,Hyperflnesplitting in positroniumandmuonium 271

in two by cutting two legs of the sametypesas the two incoming (or outgoing)legs.Eq. (2.1) is

representedgraphicallyin fig. 1.(I) (I) (I) (I)

(2) (2) (2) (2)

Fig. 1. Graphicalrepresentationof theBethe—Salpeterequationfor thefour pointfunction.

Fromthepropertyof translationinvarianceweknowthat G is amatrixfunctionof the variables= x1 — x2, ~ = x3 — x4 and p = ~(x1 + x2 — x3 — x4) alone, and is independent of= ~(x1 + x2 + x3 + x4). Thus,we maydefine theFouriertransformof G as follows:

Jdxi dx2dx3 dx4exp[+i(p1 . x1 + P2 — p3 x3 — p4~x4)]G(x1, x2,x3,x4)

= ~ — q~ + P~p)]exp[+i~(p1+ P2 — P3 —

(2n)4ö(p

1 + P2 — — p4)G(p,q, P), (2.2)

where p = ~ — P2), q = ~(p~ — p4) and P = Pi + P2 = p3 + ~4.* Lurié, MacFarland and

Takahashi[28] haveshownthat

G ~ — .[~ 1 ~~(p)~(q) ~ xp(p)~(q)(p,q, )—1[~-~ —(P

2 + M~)”2+ic~ ~2~P0 +(P

2 + M~)”2+ termsregularin P

0. (2.3)

The symbolS denotesa sum overdiscretestatesandan integrationovercontinuumstates.The

x1~arethe BS wave functionsdefinedby

= Jd(xi —x2)exp{+ip~(x1—x2)} <01 ~ exp{iP~~x1+x2)},

~(q) = fd(x3 — x4)exp{ —iq~(x3—x4)} ~<iPl T~’~(x3)~2~(x

4)I0>exp{ —iP1~x3+x4)}.

(2.4)

I iP> + and I iP> — are,respectively,the two-particleand two-antiparticlephysical statesof four-momentumP. = (~,P). ~ = (P

2 + M~)~2.M1 is the massof the state.

The singularitiesin P0 associatedwith thediscretestatesarepoles.In the vicinity of thepoleat

P0 = ±(P2+ M~)~2,eq. (2.1) becomes

th(xi, x2) = Jdx5dx6 dx7dx8S~(x1,x5)S~(x2,x6)K(x5,x6,x7,x8)~~(x7,x8), (2.5)

* Notation: dx indicatesa four-dimensionalintegral;three-dimensionalintegralsareindicatedexplicitly: d3x; we use thesame

symbolfor functionsin coordinateor momentumspace.

272 G.T. Bodwinand DR. Yennie,Hyperfinesplitting in positroniumand muonium

the boundstateBS equation.(For continuumstates,an inhomogeneousterm describingtwo freeparticlesmustbe addedto the rhs of eq. (2.5).)

We now sketchaderivationof the perturbationexpressionfor the energyshifts. Wedivide Kinto a pieceK°whosesolutionis assumedknown anda remainderAK:

K= K0 + AK,

with

G0 = S ~1F2~+ S~’~S~K

0G° (2.6a)

and

= S~’~S~Ko~°. (2.6b)

Here, in order to simplify thenotationwe suppressthe coordinatesand integrations.Now if wemultiply eq. (2.1) on the left by GO(S~S~Y’andusethe adjoint of eq. (2.6a),the result is

G = G0 + GØAKG. (2.7)

Iterating this, we obtaintheperturbationexpansionfor G:

G = G0 + GØAKG + GØAKGØAKG0 +.... (2.8)

The energylevels of theboundstatesaredeterminedby the positionsof the polesin G. In orderto find the positionsof thesepolesin termsof a powerseriesin AK, we displaythe polestructurein G0 explicitly:

G0 = ~ i~+~ ~ + termsregularin P0. (2.9)

Eq. (2.6) is assumedto be solved,so that the M~andx~°~are known.The total four-momentumlabel of the wave functions has beensuppressedin eq. (2.9) andsubsequentexpressions.Nowsupposethat we areinterestedin the shift in the position of the pole in G0 at P0 = * due tothe effect of theperturbationAK. We single out this pole by writing

0—0

G —G’’ lxjxj 2100 — 2~~,°(P0— ff~,°+ is)~ . a

Substitutingeq. (2. lOa) into eq. (2.8) andretainingonly thepart of G0 associatedwith thepole at

P0 = ~9, we obtain the geometricseries0—0 / \ 2 —0 Axix~ + I 0 X~ X~Xj +

2~r(P0 — g~,P+ i~) k~2~)XJ (P0 — ~ + ii)2

— 1XJXJ (2 lOb)— 26~°(P

0— — i~AK~?/2~P+ is)’ .

* In theremainderof this paper,we specializeto thecaseof particlewavefunctionsanddropthesuperscript+. Theequationsfor

anti-particlesmay beobtainedfrom thecorrespondingparticleequationsby reversingthesignsof 8~andic.

G.T. Bodwinand D.R. Yennie,Hyperfinesplittingin positroniumand muonium 273

So, to first order in AK, thepolein P0 at~ is shiftedto the location

~(1) — ~0 j ‘(—OA 0~— .~ 2~~ ~ p~g

0~J

In order to obtainthe energyshift throughnth order,we retain the termsin eq. (2.8) in whichnot more than n — 1 factors of G

0 appear without at least one intervening factor ofix?x?/2~f~(P0— ~f+ ii~).In the outermostfactorsof G0 we alwaysretain the term ixYx?/2~f(Po—~ + ic). (TheG~termsproducecorrectionsto the wave functionsdue to AK. Thesecorrectionscanaffect theresidue,but not the positionof the pole.) Thisprescriptionleads,in general,to ageo-metric series.In the particularcaseof theenergyshift throughsecondorder,weobtaina seriesthatsumsto

x~°~[~0 — — ~-~is~AKx?— ~~YAK~o(i)AKx?]. (2.12)

Now,

= (~AK~?) + —~— (~AK~) (P0 — ~9)+Po=g° ap0 PO~°

So, to secondorder in AK, the polein P0 atSf is shiftedto the locationg(

2) = g,p + [1 — ~ 1 [(~°AK~°)I + (~?AK~oAKx?)Ip0=~o]

g39 + {(~?AKx?)I~0~~+ (~AKGoAKx~9)I~0~9

+ ~ ~ (~?AK~?)]~ }. (2.13)0 p0go

Eq. (2.13) is reminiscentof theperturbationexpansionin non-relativistictheory.Note, however,thatthereareadditionaltermsthatarisein secondandhigherordersowingto theP0 dependenceofthe matrix elements.In practice,we evaluateeq. (2.13) in the rest frame so that S~,= M3 and

In order to relatethe resultsof the precedingdiscussionto the morefamiliar threedimensionalwavefunctionsandmatrix elements,we follow Salpeter[27] andspecializeto the casein whichK0 is aninstantaneousinteraction.That is,

K0(x1,x2, x3,x4) = ó(x~— x20)ö(x~— x2)5(x? — x~)~

0(x1,x2,x3,x4).

Thenthe momentumspaceequivalentof eq. (2.6b)in the c.m.frameis

= 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’)isthethreedimen-sional Fourier transformof ~(x1, x2,x3,x4) with the overall momentumconservingö-function

274 G.T. Bodwin andD.R. Yennie,Hyperfinesplitting in positroniumandmuonium

deleted.Next we definetheequaltime wavefunction

~ ~_ 1 f~O(p)* (215)

i v~J27tLAn equationfor 4’(p) canbe obtainedby integratingeq. (2.14)with respectto p°.For this purposeit is convenientto usethenon-covarian.tform of thefermionpropagators:

5(O(p) = _________ — ~F A~(i,) + A~(1,) (2 16)F — — m1 + is — [p° — E.(p) + is p°+ E1(p) —

where E.(p) = (p2 + rn?)”2 and A~)(p)= [E~(p)±(~p + fl

1m~)]/2E~(p).The p°integration in(2.14) is theneasilycarriedout andit yields the result

— F A~(p)A~(—p) A~kp)A~(—p) 1— LM? — E,(p) — E2(p) + is — M? + E,(p) + E2(p) — is]fhfl2

X (27)~Jd3p!~0(p,p~)4J(pF), (2.17)

which is knownasSalpeter’sequation.The matrix elementsappearingin eq. (2.12) still containwave functionshaving relativetime

dependence.We wishto replacethemby newexpressionswith modified kernelsin whichthetimecomponentsof relativemomentumareintegratedout.This is accomplishedby usingtheintegralequationfor x?

= S~S~Ko~?= \/~ S~)S~)K0~~, (2.18)

wherein the secondform athree-dimensionalratherthanafour-dimensionalmomentumintegra-tion is implied. Thus

~?MXJ°= (2)6 JdP d3p~~4p’)M(p’,p,P

0)4~31p), (2.19a)

where

M(p’,p, P0) ~ Jd~dq’~0(,p’,q)S~(q+ P)S~(— q)M(q+ P, — q, q’ + P, — q’)

x S~’)(qI + P)S~(— q’)~0(q’, p). (2.19b)

We maywrite eq. (2.19) morecompactlyin termsof theDirac notation:

= 2M~<~MI 4’.,>. (2.20)

Incorporatingtheseresultsinto eq. (2.13), we obtain the energyshift throughtermsof secondorder for thecaseof aninstantaneousK0:

* Note that ~/J = ii,,/~ J(dp°/2a)~(p)=

4j fl1fl2, where~, andfi, aretheusualDirac matrices.This follows from eqs.(2.4) and

thefact that timeorderingandanti-timeorderingareequivalentin theequaltime limit.

G.T. Bodwin andDR. Yennie,Hyperfinesplitting in positroniumand muonium 275

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

- ~ (2.21)

G0 canbe expressedin termsof the three-dimensionalGreen’sfunctionas follows:

G0 = S1F1~S~F2~+ ~ + ~

= S~’~S~+ ~ + ~ (2.22a)

wheretheproductI~0g0I~0involvesonly 3-dimensionalintegrations,and

Cd ‘° d 0

g0(p’,p, P0) = J —f— -~— G0(p’, p, P0). (2.22b)

Eq. (2.22) is representedgraphicallyin fig. 2.

+II+I~~~iL3-d integration

Fig. 2. Reductionof thefour-dimensionalGreen’sfunction G0 to theform usefulfor three-dimensionalcalculations.

It follows from the BS equationthatwhenP0 = SfG0(j) = ~ + ~ + ~ (2.23a)

where

~j) = g0 — i4’~~~/(P0— Sf + is). (2.23b)

3. Application to quantum electrodynamics

In this sectionwe specializethe generalperturbationformalism to electromagneticallyboundstates,describeastrategyfor determiningordersin ~ of differentcontributions,andgiveademon-stration that the boundstateenergiesareindependentof the choiceof gauge.

3.1. Coulombgaugeperturbationtheory

It is mostconvenientto usetheCoulombgaugefor photonsexchangedbetweenthe two par-ticles. Exceptwhen annihilationcontributionsareto be takeninto accountin positronium,it ispossibleto “mix” the gaugesby using the covariantgaugefor photonsassociatedentirely withoneparticle.This makesthe treatmentof renormalizationeffects easier.In the presentwork wetreat only the contributionsfrom exchangedphotons.We proceednow to apply theboundstateperturbationtheorydevelopedin theprecedingsection.Firstcomesadiscussionof theunperturbedwave functions 4’; this is followed by a descriptionof the Feynmanrules for the perturbationkernels.


We choosefor the unperturbedkernel K0 the single Coulombphotonexchange:

K0(p’,p) = (p~~,)2 fl1fl2. (3.1)

With this choiceof kernel,Salpeter’sequationisr A

te’ ~A~2~’— A~’~’~A~2~’— \ 1 C —

— I + ~PJ + ‘~ — W/ —‘. ~ I I d~k k 3 2— [E — E

1(p) — E2(p) — E + E1(p) + E2(p)] J 2it

2(p — k)2

E — H”~ — H~2~~ — A~’~A~~)V4’, (3.3)

whereE = M°,H~(p)m;p + f3~m,,E~(p)= (p2 + rn?)1’2. WenotethatH”~(p)A~(p)= ±E1(p)A~kp).

The is in thedenominatoris irrelevantfor boundstates.Now wecanrationalizethedenominatorsin eq. (3.2) to obtain

4’(p) = ~N+p)A~(p)A~(—p) — N(p)A~(p)A~(—p)}2 Vçb, (3.4a)p

2mRswhere

N~(p) [E ±(E1 + E2)] [E

2 — (E1 — E2)

2]/8rnRE2, (3.4b)

2rnRs [E2 — (m1 + m2)

2] [E2 — (m, — m2)

2]/4E2, (3.4c)

andwherethe reducedmassrnR is rn1m2/(m, + m2).

Unfortunately,eq. (3.4) is too difficult to solvedirectly. However,agoodstartingpoint canbefound by studying the limit of small binding energy (I~l4 (rn1 + m2)) and small momentum(I’l 4 m1 andm2). In this limit we find

E m1 + m2 + t, (3.5a)

N~~ 1, (3.5b)

N_ ~ (2mRs — p2)/(4rn,m

2), (3.5c)

~ ~ ~(l ±~j. (3.5d)

Ignoringcorrectionsto eq. (3.5), we find asimplerintegralequation4’NR = 2 -~(1+ ~,)(1 + $2)V4’NR, (3.6)

p —2mRc04

whichis our startingpoint for all calculations.Obviously4’NR isnon-vanishingonlyfor the“large—large” componentsof the wave function. For thosecomponents,the equationis simplythe non-relativistic Schrödingerequation,whosesolutionsare well known. For the normalizedgroundstate,we have,for example,

4’NR = {8ir”2y512/(p2 + y2)2} ~ (3.7a)

where~2 = —2mRs0= (rnRot)2and

= (~)® (~)® k~~>.* . 1 0 (3.7b)

* We usethestandardrepresentationof theDirac matrices: = , = .

\0 —1i \~ 0

G.T. Bodwinand DR. Yennie,Hyperfinesplitting in positroniumand muonium 277

The small difference betweenthe eigenvaluesof (3.4) and (3.6), correspondingto s — ~o’ willbe discussedin section4. The main difference,of order c~

4mR,is associatedwith relativistic andspin-orbitinteractions(for non-Sstates);but thereis alsoasmallcontributionto hfs.

Therearetwo wayswe can use(3.6) as the startingpoint for calculationsof the hfs. Oneis toreplacethe G

0 of section2 by anon-relativisticpropagatorhavingonly large-largecomponents,and to treat thecorrectionsdueto that replacementas additionalperturbations.The otherwayis to use

4’NR as an initial approximationin eq. (3.4), and then iteratethat equationas oftenasnecessaryto producea sufficiently accurateapproximation.The singly-iteratedwave function,accuratefor mostpurposes,is

4’sI = [N+(p)A~kp)A~(—p) — N_(p)A~(p)A~(—p)]4’NR(p). (3.8)

Higher iterationsarediscussedin section5.At this stagewecaneasilywrite down the“Feynmanrules” for thematrix elementsin eq. (2.21).

We initially factor the/31’s out of ~ = 4’~/3152 andincorporatetheminto AK. Then,usingthegraph

rules for quantumelectrodynamicsgivenby BjorkenandDrell [29], we obtainthe factors4’t(p’) for the final wavefunction,4’(p) for theinitial wavefunction,4~wti/q

2for eachCoulombphotonpropagatorin AK,

4itcti 7 ~i q~ q\ 4~ T T

02 2 (~1 — 2 ) —~--- ~ ~ for eachtransversephotonpropagatorin AK,

iLK :E(~+ A+(:) + K0 + — ~ A(~] for eachfermion propagatorin AK,

d k/(2ir)4 for eachclosedioop in AK,

d3p/(2ir)3,d3p’/(2ir)3 for thewavefunction integrations.The graphsymbolsusedfor thesequantitiesareshownin fig. 3. As usual,momentumis conservedat eachvertex.

(111i11 +~(P) 11111)————-~—— Coulomb Photon Pt’opogotoi’

q

.‘wi~i.4,w~. Transverse Photon Propagator

4 Fermion PropagatorIt

Fig. 3. Feynmangraph symbolsfor quantitiesappearingin perturbationtheorymatrix elements.

3.2. Strategyfor determiningthe ordersofc~

Becausethe fine structureconstantoccursin the wavefunction as well asin the Coulombandtransversephotonpropagators,the expansionin powersof ~ is not particularlystraightforward.

278 G.T. Bodwinand DR. Yennie,Hyperfinesplitting in positroniumand muoniurn

Thereforeit is not possibleto specifyasetof rulesthatis guaranteedto generateall thetermsof agiven order.However, it is not too hard to identify the leadingorder of anygiven contribution,andto makeasequenceof approximationsuntil theremainderbecomesnegligible.Thisprocedureis certainlyfar from uniquewhenoneis interestedin higherordercontributions,asonecanarrangethe sequenceof approximationsin variousways.We triedavariety of approachesuntil we foundone that is relatively economical.While we believethata better approachprobablyexists,thisoneseemssufficiently compactto be convenientfor the calculationat hand.

In analyzingthe order of acontribution,avery importantconsiderationis whethertherelevantmomenta(we alwaysanalyzethe integralsin momentumspace)aresmall (—.‘ ccm~)or large(oneor bothparticlesrelativistic) in the dominantregion of integration.If initially acontributionisdominatedby smallmomenta,onecanobtainthe order by replacingeachE1(p) by rn andby re-scaling eachmomentumby p —~ yt. In many casesthe powersof cc theneasily factorout, sincethe wavefunctions,Coulombpotentials,andmanyof the energydenominatorsarehomogeneousin y. In casean energydenominatoris not homogeneousin y, we caninitially neglecttheportionsof it thatareof higherorderin cc. An exampleof thisis given in section5.

As we proceedto higher order corrections,arising both from more complicatedgraphsandfrom the remaindersof the approximationsjust mentioned,the dimensionalityof the integrationsincreasesand their structurebecomesmore complicated.As long as the integralsconvergeforsmallmomenta,additional powersof ~Iyield additional powersof cc. At somepoint, however,the integralsno longerconvergein the smallmomentumrangeand the whole integral or somesubintegralneedsthe convergenceprovidedby factorsof E1 in the denominator.At that stage,additionalpowersof momentumno longerincreasetheorder,andoneshouldavoidsuchspuriousexpansions.In this highmomentumregime,expansionin powersof Vusuallyleadsto increasedordersof cc. However,in thesmallmomentumregion,the extrafactorof cc in eachV canbe com-pensatedby the momentumintegration.Thusat eachstage,onemustexercisejudgmentso as tominimize thefalsestepsandto obtainintegralsthat aretractable.

The correctionsinvolving a factor ln cc’ arisefrom integralswhose relevantmomentaspanthe region from non-relativistic(‘-.~ccm~()to relativisticvalues.

3.3. Gaugeinvariance

We discussherethe gaugeinvarianceof the positionsof the poles of the four point function.To makea gaugetransformationon the four point function,we addto eachphotonpropagator

a gaugecontributionwith numeratorstructureK,~5or ~ whereK is themomentumlabel ofthe photon. Thus, we can write the gaugetransformationas a perturbationexpansionby re-groupinggraphsaccordingto numberof gaugephotons,as shownin fig. 4.

L~1gauge = + +transformed g g

Denotes a gauge photon

Fig. 4. Illustration of theexpansionof thegauge-transformedfour pointfunction.

G.T. Bodwinand DR. Yennie,Hyperfinesplitting in positroniumandmuoniurn 279

At this stagewe canusetheFeynmanidentity

1 1_ 1 1

to effect the usualpairwise cancellationbetweenthe graphs in which a given gaugephoton isinsertedin all possibleways. (Note that this procedureis unaffectedif we performa differentgaugetransformationon eachgauge-invariantsetof graphs.)In the surviving terms,at leastoneendof eachgaugephotonis connectedto an externalleg of the four-point function. Thus,noneof the termsin the perturbationseriesin fig. 4 hasthe structureof an iteration of thefour-pointfunction.Thereis no way to build up ageometricof the typein eq. (2.10), so the gaugetransforma-tion cannotshift the positionsof the four-point functionpoles— althoughit canaffect the residuesat thosepoles.

4. The pure Coulomb splitting

We now calculatethecontributionto the hfs containedin the Salpeterequationwith Coulombpotential,that is, thecontributionto the hfs due to laddergraphswith Coulombphotonrungs.We take the Schrodingerequation(eq. (3.6)) as our starting point. Of coursethe Schrodingerequationhasno fine or hyperfinestructure.Thesearisewhenwe calculatethe differencebetweentheeigenvaluesof eqs.(3.4) and (3.6). Thatdifferenceis 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)

To adequateaccuracy,we mayapproximate<

4’NR 4’> by 1 and in the matrix elementreplace4”by its singly iteratedapproximation~ given in eq. (3.8). In the resultingexpression,only thosetermsthatcontainanevennumberof cc,’sandcc

2’s arenon-zero.Furthermore,aterm mustcontainan cc1 andan cc2 if it is to contributeto the hfs.* So, the only contributionto the hfs comesfromthe small—smallcomponentsof the wave function. Retainingonly the lowestpowersof p in thenumeratorstructure,we seethat theleadingcontributionto the hfs from eq. (4.2) is

~ 1AE(Coul) = 64iry J ~~12+ y2)2 2E,(p’)2E

2(p’)(p’ — p)22E,(p)2E

2(p)

X (p2 ±p2)2 [N~(p’) — N_(p’)] [N+(p) — N(p)]~3~—~3. (4.3)

* In addition to thehfs, which is our current interest,eq. (4.1) containsrelativisticand spin-orbit corrections.Thesearegivenby

the contributionsthat havetwo powers of p in addition to thefactor of V. Thesecome either from expandingfactorsof E, in H’ orfrom pairingonefactorof ~ p in H’ with onein 4s~.(Onecannotpair~,p with x2 p because4~hasonly large-largecomponents.)It is easyto seethat thesecontributionsareof order526g. While theydo not contributedirectly to thehfs, theycanmodify thewavefunction slightly andproducea correctionof relativeordera

2, whichis beyondthepresentorderof interest.


Now,

2,.a21b22c22.d=a1.aa,.ba2~ca2.d=[a~b+ia1.(aXb)][Cd+ia2~(CX11)],

whosehfs part is equivalentunderangularaveragingto

a2(a x b) . (c x d).

As can be seenby power counting,the integral in eq. (4.3) haszerodegreeof convergence.This meansthat if we replacethe E~by m1 (including in N±)‘ it would divergelogarithmicallyatthe upper limit. Since the exact behaviorin the relativistic region affects the additive termsofrelative order cc

2 but not the coefficient of the logarithm,we canmakethe replacementE, -+ m,andcut off the integralsat li’ I ~ m

1,m2. The integralcannow be expressedasacombinationofthe onesin table 4.

Theln cc’ contributionis

AE(Coul) ~ <a~ a2> (m~/6m~m~)cc6ln cc~= +(m~/m,m

2)cc2ln cc’EF. (4.4)

Here,EF is theFermi splitting:

EF = ~(cc4m~/m,m2)<a, a2>.

It is obviousfrom countingpowersof momentumin theintegrandthatthetermswehavedropped,which containadditionalpowersof p, cannotleadto logarithmicintegrals.Thus,eq.(4.4) containsthe completecontributionof O(cc

2 ln cc ‘AEF) of thepureCoulombpotentialto the hfs. Thisresultis in agreementwith that of Cunget al. [24] andLepage[25].

It is interestingto notethat the part of H’ thatproducesthehfs alsomixesthe3S and3D states(in secondorderperturbationtheory).Thus,the part of H’ that splits the singletand triplet levelsis associatedwith an effective tensorforce. The connectionbetweenthe pure Coulombhfs andan effective tensorforce in the non-relativisticreductionof Salpeter’sequationwas first pointedout by Cungeta!.

5. The one transversephotonkernels

In thissectionweinvestigatethekernelsin AK in which onetransversephotonandanynumberof Coulomb photons are exchanged.We ignore kernelsinvolving crossedCoulomb photonssincethesecontributeat most in O(cc2EF)(seesection6).* The remainingone transversephotonkernelsareof the typeshownin fig. 5. We calculatethecontributionof thesekernelsto thehfs inO(cc2ln cc- ‘EF) andalsoobtaintheleadingcontribution,EF. Termsof O(ccEF)havebeencalculatedpreviously,so we bypassthem.The variousordersare identified by studyingthe expressionforthe energyshifts in momentumspace.

The kernelsin fig. 5 are treatedin the first orderperturbationtheory developedin section2.In turns out that the secondorder energyshifts contributeto the hfs at most in O(cc2EF). Thispoint is discussedin detail in section6.

Our basicstrategyin evaluatingthe BS matrix elementsof the kernelsin fig. 5 is to useformaloperatormanipulationsto sumthe seriesof Coulombexchanges,andtherebyobtainan expression

* In specifyingorderswe displayonly thea dependence.All thecontributionsdiscussedherearerecoil corrections,sotheycontain

anadditionalfactor in themuoniumcase. .

G.T. BodwinandDR. Yennie,Hyperfinesplitting in positroniunl and muoniun, 281

FH + + +. ..

+ + S S S

Fig. 5. Onetransversephotonkernel.

analogousto that for the energyshift in non-relativisticperturbationtheory. It is known that, ifonecalculatesthetermsof fig. 5 separately,variousimportantcancellationsoccurwhentheyareadded(see,for example,KarplusandKlein [14]). Bykeepingtheserelatedtermsin asingleexpres-sion from the start,we hopeto obtain asimplerandmorecompactformalism.

In the processof manipulatingthe expressionfor the energyshift, we identify andset asidevarious terms thatappearto be small comparedto the leading term. Thesecorrectionsto theleadingterm arediscussedattheendof thesection.It is not clearthat we haveidentified the mostconvenientchoiceof correctionterms. However,our proceduredoessystematicallyaccountforthevariouspiecesin the first orderenergyshift.

The first stepin ourprocedureis to set asidetheA termsin thefermionpropagators.(Weshowlater that termsthat contributeto the hfs in O(cc2in cc— 1EF) containat most onefactor of A .)In particular,we retain theA~A~term in the Salpeterequation,so that we usea wavefunction4” satisfying

4” = E - E1(p) - E2(p)A+(~~

4’. (5.1)

The correctiontermsinvolving A - A_ give anenergyshift denotedby AE(A). In addition,therearecorrectionsthatarisefrom the A_ partsof propagatorsin the kernelsthemselves.We denotetheenergyshift dueto thesecorrectionsby AE(B).

Next, we usepartial fractionsandeq. (5.1) to rewrite the factorsadjacentto the wavefunctionsin the matrix elementsof thekernelsin fig. 5:

+ Po —E1~+ is Po — E2(p) + i~]4’

+ 1 1 A~(p)A~(—p)

— [E + Po — E1(p) + ~ Po — E2(p) + is] E — E1(p) — E2(p)1 1

— [E + Po — E,(p) + is + Po — E2(p) + ic]4’. (5.2)

We representthis graphicallyas shownin fig. 6.Wecangiveausefulinterpretationof thegraphsin fig. 6in termsof timeorderingsin configura-

tion space.EachCoulombphotonrepresentsaninstantaneousinteraction,andsoconnectspointsat equaltime. Similarly, 4” is independentof the time componentof the relative momentum,sobothparticlesareatequaltimesin thewavefunction.Thechoiceof theA~pieceof eachfermion

282 G.T. BodwinandDR. Yennie,Hyperfinesplitting in positroniumand muonium

çz~c~~c~c~

Fig. 6. Partialfractionrearrangementof thematrixelements.A hashmark ona fermion line meansthat thepropagatordenominatorassociatedwith that line hasbeen removed.

propagatorguaranteesthat eachfermionpropagatesforwardin time. A hashmark on a fermionline meansthat the endsof that line areat equaltimes. Usingthis interpretation,it is easyto seethat, of the graphscontainedin fig. 6, only two classesare non-zero.Theseare shownin fig. 7.Note that the graphsof fig. 7 are, in fact, time-ordered,with time increasingfrom bottom to top.Of course,this resultcan alsobe obtainedby writing out the expressionfor the matrix elementsandperformingtheintegrationsoverthetime componentsof the1oop momenta.In thetermsthatvanish,it turnsout that it is alwayspossibleto label themomentain away such that, if Po is thetime componentof the relativemomentumin oneof the wave functions, then the polesin Poin theintegrandareall on the samesideof the real axis.Thus,the Po integrationgiveszero.

(Il()(2) (t)()(2) —+ ~ + +

V I p1+q~fq p~qi,’q I

~1 ~——-~.j~i ~i 4-...4-~i p1

(a)

(_;~ (_;~ (~(I~4(2) (I1~4~2) (II~~..fI

2)

1~,. I p~t.q IP2q P~~ql-P2-q

I ~\ I-P-~ + V~td-p -q + P2~~1-p2-q +.I N-p1-qPI1~~PI

(b)Fig. 7. Partialfractionrearrangementof theonetransversephotonterms.

Next, we useformal operatortechniquesto sumthe graphsin fig. 7. We work in theCM frame,using the indicatedmomentumlabels. Hereit is to be understoodthata momentumlabelp~onaparticle 1 line indicatesthat the line carriesfour-momentum(p~+ E,ps). The integration overp? gives the factor

~dp~ A~(p1+ q) A~(—p1)

— J2ir p~+ q°+ E — E1(p1 + q) + is —p~’—E2(p,) + is

— . A~kp1 +q)A~(—p1) (3)— lqO + E — E1(p1 + q) — E2(p,) + ic~


In graphscontainingCoulombphotons,wemustincludethe factor4iricc/(,p1 — P2) in theintegrandof the d

3p,/(2ir)3 integral. We recognizethis as the momentumspaceequivalentof applying theoperator —iV = icc/r, where r = — x

2 is the relativecoordinatein the two particlesystem.Proceedingin the samefashion for P2, P3’ ..., we obtaina factor analogousto that in eq. (5.3)from each dp~integrationanda factor iV from eachd

3p, integration.By making useof~,theoperatorfor the momentumof particle 1, we can write all the factorsof the type in eq. (5.3) asiA~i + q)A~(—~)/{q°+ E — E

1Cp + q) — E2(~i)+ is}. Then, the sum of the contributionsrepresentedin fig. 7a is

~ d4q —4ircc T jq.rF A~(~+ q)A~(—~)J ~4q2 +is<4’ [q0 +E —E

1~+q)—E2~)+is

A~(~+ q)A~(~ A~(~+ q)A~(—~) 1. T+q0+E_Ei~+q)_E2~p)+ic qo±EE,~p+q)E2~p)+is~~] 8~

(5.4a)which sumsformally to

~ —4itcc <4” 2T ~iq.r 1J(2ir)

4q2 + is 2 q°+ E — E1~+ q) — E2(j~)— A~(~+ q)A~(—~i)V+ is

x ~ (5.4b)

Thefactore~r accountsfor the momentumtransferfrom particle 1 to particle2 dueto the trans-versephoton. The q0 integrationis most convenientlyperformedby closingthe contourin theupper half-plane.If we carry out this integrationin expression(5.4b), addin the analogouscon-tributionfrom fig. 7b,andmultiply thematrix elementsby thefactori as requiredby theperturba-tion theoryexpressionin eq. (2.21),we obtainthe first orderenergyshift

(~3 r 1

—~-— 4”I I~~I T iqr A~U)A(2).

2T2ir2 J 2q [22 q + E

1(~+ q) + E2(j~)+ A’4PA~~V— E + + ~

+ 2T ~iq~r 1 A~’~A’12~. 4”> (5 5)

1 q+E1p)+E2~p+q)+A(~i)A~V_E + + 2]

Here ~ = A~(~),A’.~F= A~(~+ q), ~ = A~(—~),A’421 = A~(—~ — q). For future use,we

alsodefinethe operatorj~’=~ + q.Intuitively, we expectthat q is typically of the order of the expectationvalue of p I (~ccmR)

andV andE — E1 — E2 aretypically of theorderof the bindingenergy(~ccmR).Therefore,we

approximatethedenominatorsin eq. (5.5) by q anddenotethe correctionby AE(D). The leadingterm is then

AE(L) = ~ J~q<4”I V E — E1(~)— E2(~)

x ~ E — E,Cp) — E2(~)V 4”>. (5.6)

Herewe haveusedeq. (5.1) andtheidentity

=

284 G.T. Bodwinand D.R. Yennie,Hyperfinesplitting in positroniumand muonium

We notethat eq. (5.6) canalsobe written in themorecompactform

AE(L) = ~ <4”I JL~e1~~~T2~ 4”>. (5.7)

In calculatingAE(L) from eq. (5.6),we approximate4” by the ~ part of the singly-iteratedSalpeterwave function 4’s~(eq. (3.8)). Correctionsresulting from further iterationsof the wavefunctionarediscussedlater.

Beforeproceedingwith the calculationof AE(L), let us dispensewith the transversalityterm,2 qq/q2, in

8T Using 2 q = 2 (~‘— ~),we find

A~(~’)21. qA~(~)= [E1(~’)— E1~)]A~A~= A~’~A~ (5.8a)

and

A~(—~‘)22. qA~( .) = — (~‘)2?i2 ~ (5.8b)

whereE = E1~’).From eq. (5.6), we thenseethat thecontributionto AE(L) of the transversality

term is

Cd3 1 f~”\2—

-_~--. I___~<4”~V ~iqrkP) ~ A’”~A”~2ir2 J q’~ E — E

1(~i)— E2~p) E’1 + E1 ÷ +

F A’~2~A~2~ V 4”> (5 9)

+ ~E—E1~p)—E2~p)

Now, following the argumentgiven in section4, we know that theremust beat leasttwo factorsof 8i andtwo factorsof 22 in anyterm thatis to contributeto thehfs. Associatedwith eachfactorof; is apowerof momentum.Thus,if we approximate4” by

4’NR’ usethe~ partof eq. (3.8),andevaluateexpression(5.9)in momentumspace,thenumeratorhasatleast 14 powersof momen-tum (countingthe elementsof integration),andthe denominatorhas12 powersof momentuminthe non-relativisticregion.Hence,expression(5.9) doesnot containa ln cc’. The correctionsto~ clearlybring in additionalpowersof momentum.If we takeinto accountthe cc5 in the wavefunctionnormalization,weseethatthecontributionof the transversalityterm to thehfs is O(cc2EF).

5.1. Calculationofthe leading term AE(L)

First we rewrite the cc-matrix structurein eq. (5.6), groupingtermsaccordingto whethertheycontainan evenor an oddnumberof factorsof ~:

= [~(l + ~ ~)21~~ + 121 ~i’ (1 +

i~ m,\ / m1\ 12,~f~’2~2,~

+—(l +fl,—-)2i(1 +fl,—)+—4\ iJ \ ,,, 4

G.T. Bodwinand D.R. Yennie,Hyperfinesplitting in positroniumand muonium 285

Xf2’

= [2~ ia1 x q 1 4E1E~(E1+ m1)________________ Fia~XP4EEi(E~+m)+4E(l+$1~)+4E~(1+fl1~)l

12j . p’2

121 ~(5.lOa)+ [~(l + si?)2i(l + ~ ~ E~E~

Similarly,

m2 . 1+J32 ________

A~( ‘)2~2~A~(_~ = [ 2E

2E’2Ia2 X q 2 — ia2t24E

2E’2(E2+ rn2)____________ ~ (1 m~’\ ~ (~ m2\1

+ ia2 X ~ 4E2E’2(E’2 + rn2) — 4E2 +$2 — +f3~

1 22 P2222 •f1

+ [~(l + $2~T)22(l + fl2~) +~ E’2E2 (5.lOb)

The productof thea x q termsgivesan energyshift

AE(L1) = ~Tmim2 j’—r <4”I v (1 + fl~)/2~iq.r°1 x q~a2x q (1 + 52)/2 VI4”>.E — E1 — E2 4E1E~E2E~E — E1 — E2

(5.11)

Note that the factor ~(l + $~}~(l+ /12) is non-zeroonly for the large—largecomponentof 4”.Initially, we approximate4” by

4’NR• Thena, x q~a2 x q goes to ~a1 a2q

2underangularaver-agingof thespin wavefunction. Also,

— E2 VINR = N+4’NR 4’+. (5.12)

E—E

In this approximation,the energyshift is

ccm’m2Jd3q<4’+ I e~r a1 a2 I 4’+>. (5.13)

3~2 4E1E’,E2E’2In momentumspacethis is

ccm,m2 ___________________

3~2 ~ 4’~’~4E1~E1(p’)E2~E2(p’)~ (5.14)

Clearly the integrationsover the two wave-functionscompletelydecouple.Neitherof theseinte-grationscan givealn cc~sinceN÷4’~containsonly evenpowersof IpI. However,the expression(5.13) cangive cc

4, cc5 and cc6 contributionswith no ln cc- ~.If we approximateN~4’~,by 4’~,andreplaceE. by m~,we obtainthe well-knownFermi formula:

E 16cc6m~ 1 1 d3p’ I d3p 2cc4m3______ ________ ________ R <a,a2>. (5.15)~ +y2)2J(p2 +Y2)2<~~a2> = 3m1rn2


It is easyto see that the correctionsto the approximationsof the precedingparagraphhaverelativeorder cc, cc2, etc. For example, the differencebetweenE and m

1 + m2 is of order cc2mR,

andthe differencebetweens andso in the energydenominatorsis of ordercc4mR.Correctionsfrompowersof either of thesesmall quantititesareclearly of relativeorder cc2. The correctionsfromletting E

1 —~ m, in eq. (5.14) give integralsthatare dominatedby the relativistic region andhaverelativeorderscc, cc

2, cc3, etc.Next we studythe correctionto theapproximation~(1+ f3,)~(1+ /12)4” = 4’NR~From eq. (3.8)

wefind the large—largecomponentof 4” (once-iterated)is

(1 in1 m2 ~[l(~ + E1\(1 + E2~— ~2 1 8~y5/24’LL ~2+ 2E1)~2+ 2E2)[4~ m1)~ m2) 4mim2](p2 + y

2)2

— p2 ~ 8\/~y5/2 (516)‘~ 4m,m

2)(p2+y2)2~

In the last stepwe have madean expansionin powersof p2 and retainedonly terms to 0(p2)in thenumerator.The term 1 in the parantheseshasalreadybeentakeninto account.The termp2/4m

1m2canappearas a correctionin either wave function. Substitutingin eq. (5.11), we findthat the total correctionof this typeis

AE L WF — 2 cc 64ny5 (‘d3pd3p’d3p” p”2 —4itcc —2mR <a

1~a2>— — —( ~ 16m~m~J (2ir)

6 (p”2 + y2)2 (p” — p’)2 (pf2 + y2) (p2 + y2)2•

(5.17)

Here we have used 1/(E — E, — E2) ~ — 2m~/(p

2+ y2) and replaced E, by rn elsewhere.Wenotethat, as it stands,the integral is logarithmically divergent.This is, of course,dueto the factthat we havenot retainedthe full l~I dependencein the integrand.Had we doneso, the integralwould haveconvergedat p ~ m. This fact is all that weneedto know aboutthe largep behaviorof theintegrandin order to computethe coefficientof ln cc~’.The logarithmic part of theintegralin eq. (5.17) is computedin appendix2. The result is

AE(L-WF1) ~ ~ <a,~a2> —2EFcc

2lncc’m~Jmlm2. (5.18)

Now considerthe contributionsto eq. (5.6) of the termsin eq. (5.lOa)or eq. (5.lOb) thatcontainno spin dependence.Thesetermscangive acontributionto the hfs only in conjunctionwith thesmallcomponentsof bothwavefunctions.Actingon the smallcomponents,thefactors(1 + fim/E)become(1 — rn/E) ~ p

2/2m2.So, writing the hfs contributionsin momentumspacewe find thatthe overall power countis logarithmic or higher.Thereis an explicit factorcc8 (cc5 from the wavefunctionnormalization,cc2 from thefactorsof V. and cc1 from the transversephoton).Hencethesetermsgive an O(cc4lncc’EF)contributionto the hfs.

Nextwe examinethe contributionsto eq. (5.6) of the remainingproductsof terms in eq. (5.10)thatareevenin 2. Theseproductscontainatleastonefactorofpandonefactorofp’ in thenumera-tor, so the integrationsover the wave function momentacan no longer give the factors of l/ythatproducecc” andcc5 contributions.In order to give an cc6 ln cc ~,aproductof termsin eq. (5.10)must containfour powersof momentum.The energyshift from such productsis

G. T. Bodwin andDR. Yennie,Hyperfinesplitting in positroniumandmuonium 287

1 C.-i3cc .. ~~2rn2AE(L2) = ~ i~rn~rn~J ~r <

4’NR le” r(ai x p a2 x

x ~a2 x q~-~--+ (1 +-~2))I4’NR>. (5.19)

We havereplacedV4’/(E — E1 — E2) by 4’~since the correctionsinvolve additionalpowersofmomentumin the numeratorand so cannot yield ln cc ~. Under angularaveragingof the spinwavefunction, thespin factorsbecome

and

a, x~a2x q—~~a1~a2[~’2~2 —4q2].

Usingthe resultsin table4 we seethat

A L 64iry5cc<a1~a2>(‘d

3p’d3p p’2 1 p2 (m2 m1

E( 2) ‘~‘ 3~2 16rn~m~J (2ir)3 (p~2 + y12 (p’ — p)2 (p2 + y2)2 k,~m +

= EFcc2ln cc’ m~(~-~-± (5.20)

m1m2\m1 m2j

Thisis an exampleof a“wrongmass”term.Weexpectthat suchtermsmustcancelin thecompletecalculationsincetheywould give anon-zerorecoil correctionto the hfs in the limit m2 —~ ~.

Now considerthe termsin eq. (5.10) that areodd in 2. Theseconnectthe largecomponentofthe initial wavefunction in eq. (5.6) to the smallcomponentsof thefinal wavefunction,andviceversa.So we seethat thesetermsare non-zeroby virtue of thecorrectionsto

4’NR containedin 4”.Let usdispensewith the termsthat do not lead to contributionsof O(cc2 ln cc ‘Er). Considerfirsttermsof the form 2 ~‘22 ~ timesa x q. An exampleof suchacontributionis

613 3 ~ 1 1 1cc Jd pd pd ~ (p~2 +y2)2(p’ —p)2p2 +y2

x <2122~P22 ~l (p’ — p)22 ~ 222 ~ 12> (p~’2

Herewe haveusedeq.(5.1), 1/(E — E1 — E2) ~ —2m~/(p2+ v2)’ andhaveretainedonly the lead-

ing momentumdependence.Now, by making the changeof variablesp,p”, q —÷ yp, vi-”~yq, we seethat this expressionis equalto cc6 timesan integralthat is independentof cc. Correctionsfrom thesubstitutionE~—~ m• are of orderO(cc3EF). Similarly, the productof an rL~pruzp term with anyof the remainingtermsin eq. (5.10) leadsto integralswith powersof momentumin excessof alogarithmicpowercountandproducescorrectionsof O(cc3EF).It is also easyto seethat the productof an 2 term with any of the termsin eq. (5.10) exceptthea x q term leadsto O(cc3EF)contribu-tions.

The productof an 2 termwith aa x q term hasthe correctmomentumdependenceto give aln cc 1, and yields the energyshift


cc 1 Id3q

AE(L-WF2) = ~ 4m1m2j -~- <4’NRI e” r {(_ ia2 x q~8~)

2 + ~2 2m~] [2m~ ~~n;~) (E~+ $imi)]x [(E1 + him1)(~ 2mR)Vr~-i--rfl+ [8~ ~‘ _____

x (—ia2 x q~81)+ [1 ~2]}I4’NR>. (5.21a)

Replacing/1~between8-matricesby — 1, usingE1(p) — m1 ~ p2/2rn

1,anddoingthe usualangularaverage,we find

cc Id3q qe ~2 ( —2mR

AE(L-WF2) ~ — 8m~m2~

2j~<4’NRIe1 ~ +2)V(3)ai ~a2q~p~4’NR> + (1 ~2)

—cc 264~y5Jd3Pd3P~d3P~~1 1 2~ —2mR “ / —4ircc \

= 24m~m2~ (2~)6 (p~2+ ~2)2 (p~_p)2P ~2 + y2)~(p —

1

x (p’ — p) .p” (p~~2+ ~2~2<a

1 a2> + (1 i-÷2)

d3 “ p2 1 ~cc6m~ > Jd3p p (p2 + 2)2 (P _~)2 (p~~2+ V2)2~ <a,a

2— 3ir”m~m2 (5.21b)

Thenfrom table4 we seethat

26 rn~(m1 m2’\ rnAE(L-WF2)~ — cc 2 2 ~ + — i <a~a2> ln cc’ = ~ (~-+ -~-~-~)cc

2 ln cc ‘EF.3 m,m

2\m2 m1) m1m2\m2 m1 /

(5.22)

The total O(cc2 ln cc ‘EF) contributionfrom eq. (5.6) is

AE(L-WF1) + AE(L2) + AE(L-WF2) = E~cc

2lncc’(m~/m1m2)(—2+ 2m1/m2 + 2m2/m,).

(5.23)

5.2. Higher iterationsof the wavefunction

We can now discussconvenientlythe possibility that higher iterationsof the wave functioncontributein O(cc

2 In cc~EF).The doubly-iterated wavefunction is given by

cc [2(m1+ E1)(m2+ E2) — 2p

2]A~kp)A~(—p)I d3k4’DI 2it~ 4(m

1 + m2)(p2 + y2) j(k_p)2

x{(1+81~(1_82~ k2 ~\ 2m1 ) 2m2) — 4 ¶~ 4’1.~(k). (5.24)m1m2J

The term 1 in braces,of course,gives the singly-iteratedwavefunction, andthe remainingtermsgive the correctiondueto the doubleiteration. Imagine substitutingthesewavefunctioncorrec-

G.T. Bodwinand DR. Yennie,Hyperfinesplitting in positroniumandmuonium 289

tions for oneof the wavefunctions4” in eq. (5.6). In the resultingexpressionwe can isolatethetermswith theleadingmomentumdependenceby using

l/(E — E1 — E2) ~ —2m~/(p2+ y2)

andsettingE, = m, elsewhere.This leadsto an energyshift of the form

-~J~-~<4’~IV2

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

x (m, + $1E+ 8~~~)]. [(m2 + /12E~-22 ~p)2~(m2+ 52E2 -22 .~)]fl v~

(5.25)

where

cc —2rnR 1

4’DI=~22+V24mm(m1 +$1E, +21p)(m2 +$2E2 —82~p)

~“ d3k f8

1.k x2k 81k22k k2_~ k

X j (k — p)2 ~ 2rn1 — 2rn2 — 4m1in2 — 4mim2f

4’NR( ).

Now, if we write this expressionin momentumspaceand rescaleall the variablesof integrationby a factor y, we obtainfor theleadingorder cc6 timesan integralindependentof cc.

5.3. Calculationof AE(D)

We examinenext the correctionsto retaining only q in the denominatorof eq. (5.4b). In theusualtreatmentsof the one transversephotonkernels,onemakesan expansionin termsof thenumberof Coulombphotonsexchanged.This is equivalentto expandingthe denominatorofeq. (5.4b) in powersof V. Insteadof making such a V-expansion,we write the denominatorasfollows:

1 — 1 + 1 1 ‘E—E’ —E —A’’~A~2~V

q±E~+E2+A~~A~V—Eqqq+E~+E2+A~1)A~V_E’~ 1 2 + ÷

(5.26)

We havealreadytreatedthefirst term.In order to analyzethe secondterm we notethat, by virtueof 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— E~)Ai~1~2

14”+ A’ A~[A’~81A~,V]q5’.

(5.27)

It turnsout that the commutatorterm in eq. (5.27) is negligible.* In order to demonstratethis,

* This is thecounterpartin our formalism to thecompensationthat occursbetweenthe varioustermsin a V-expansion.See,for

example,KarplusandKlein [14].

290 G.T. BodwinandDR. Yennie.Hyperfinesplitting in positroniumandmuonium

we substitute the expression for A~’~81A~given in eq. (5.10). Consider first the termi(m,/2E1E~)a,x q. Expandingthe factorsof E. in the denominator,we have

m1 _____

‘2E1E~,a, X q = ~—a1 X q —‘ 4m~ a1 x q + ... . (5.28)

The commutatorof the first term with V is zero.Theremainingtermslead to momentumspaceintegralsthat contain logarithmic andhigher power counts.Sincethe matrix elementcontainsan explicit factor cc

7 (cc5 from the wave function normalization,cc1 from the transversephoton,cc1 from the factor V), we conclude that such terms can give a contribution to the hfs inO(cc3lncc~E~)or O(cc2EF),but not in O(cc2lncc’EF).The remainingtermsin eq. (5.10) lead toexpressionscontainingat leastonepowerofp andonepowerof p’ in the numerator.Thus,neitherof the integrationsoverwavefunction momentumcan give a factor i/y. We concludethat thesecontributionsare at most O(cc3 ln cc - ‘EF).

Havingshownthatthecommutatortermin eq.(5.27)is negligible,weneedonlyconsidertheterm

1 1 ‘E E’qq+E~+E

2+A~1)A~V_E’~1 1

For this termwe expandthedenominatoronceagain.

1 1 (E — E’) — E1 — E1 +~LE—E’ —E —A’”~At2~V)

qq + E’1 + E2 + A1’~A~~V— E 1 1 — q

2 q2” 1 2 + +

X q + E~+ E2 +A~

1~A~V— E (E1 — Ei). (5.29)

The first term integratesto zerobecauseof the symmetrybetweenp andp’ in theremaininginte-grandfactors.By analogywith the analysisof eq. (5.27), we concludethat the importantpart ofthe secondterm is

1 E’ E 1 E E’ _ (E’2 — E2)(E1 — E’1) 302 — 2)q + E~+ E2 + A’wAt

2)V — E~1 — 1) q3 . (5. )The termswehavedroppedin eq. (5.30) areeitherof higherorder becauseof an additionalpowerof Vor fail to givealn cc’ becauseof additionalpowersof momentumin thenumerator.In writingdown the energyshift, weagainmakeuseof eq. (5.10). It is necessaryto retain only theproductof a x q termssinceall otherscontaintoo many powersof momentumto givea !n cc ~. We alsouse

E — E. = (p~2— p2)/(E + E.) ~ (p~2— p2)/2m,. (5.31)

So theln cc~contributiondue to the expression(5.30) is

AE(D) = — j~ ~ J—’~<4’NRI ~ — p2)2 I4’NR>

— 4cc6m~<a1 . a2> ~d

3pd3p’ 1 (p~2— p2)2 1

— — 3ir m~m~J (2ir)3 (pt2 + y2)2 (p — ~)2 (p2 + y2)24 5 2~ 6 mR —1 mR 2 -1~ —cc 2 2 <a, ~a

2>lnx = 2 cc lncc EF. (5.32)3 m1m2 rn1m2

G. T. BodwinandD.R. Yennie.Hyperfinesplitting in positroniumand muoniurn 291

5.4. ThecorrectionsAE(B)

Wedivide thecorrectionsinvolving the A partsof the fermionpropagatorsinto threeclasses.The first class,denotedby AE(B1), containsall correctionswith oneA. betweenthe transversephotonand an externalCoulombinteraction.The secondclass,denotedby AE(B2), containsallcorrectionswith oneA inside the region spannedby the transversephoton.The AE(B1)andAE(B2)correctionsare shown in fig. 8. Note that, becauseof time-orderingconsiderations,agraphwith oneA betweenCoulombphotonrungsmustvanish.

H + ~f’:1 ±

4” 4” 4”(a)

o 00+ (.)~�_1 + (_)~~ + .‘

0 00(b)

Fig. 8. Examplesof correctionsinvolvingonefactorof A_. The line containingtheA_ is indicatedby (—)andall othershave a factorA~.(a) AE(B,) corrections.(b) AE(B

2) corrections.

Considerthe classof graphsin fig. 8a. From time ordering arguments(or examinationof theintegrationsover time componentsof momentum)we see that thesegraphsgive the energyshiftshownin fig. 9.

± ± + S S

Fig. 9. Examplesof AE(B,)correctionsthat arenon-vanishingafter thepartialfractionrearrangement.

In eachgraph thelower legsgive 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 transversephoton vertex gives a factorA’.~P/(E+ Po + q0 — E’1 + is). The momentumroutingcanbe chosensothat all otherpropaga-tors areindependentof p0. Now


1 [ 1 + 1 1 1J 2ir E + p0 + q0 — E~+ is[p0 + E + E1 — is ~ — E2 + is]E + E1 — E2

—i r 1 1 1= E+E1 _E2[E’1 + E1 —q0 ~is+ E+q0 — E~—E2 + is]~ (5.33)

if we sumthe Coulombphotonladdersas in eq. (5.4), the secondterm in eq. (5.33) leadsto theenergyshift

“d3

iqr 1 A~~)A(2).2T — + V

2ir2 \‘V

2e q + E~+ E2 + A2~’A~~V— E + + 1 E + E1 — E2(5.34)

Note that this is identical to expression(5.5) exceptfor the last factor in the matrix element.*Thefirst termin eq. (5.33) givesan energyshift

cc d3q . 1

— ~ ~ + E’, + E2 + A’~

1~A~~V— E(cl + E~+ E2 — E)

1 A(l)A(2)

X A’(1)A(2L T — + V 4”>q+E~+E1 + ÷

81E+E1E2

The expression(5.34) can be analyzedin the samemanneras eq. (5.5). The conclusionis thattheleadingcontributionto the hfs dueto correctionsof this typeis given by

cc d3 At1~A12~

+ E E V14i>. (5.36)

We againmakeuseof eq. (5.10) to reducethe 8-matrix structurefor particle2. Only the 82 X q

termcontributesin leadingorder.The othertermscanbeeliminatedby powercounting.Notethatthe a

2 x q term is manifestlyorthogonalto q, so that we can drop thetransversalityterm in thedot product.Then,it is easyto seethatthe minimumnumberof powersof momentumcomesfromtaking4’~for the left-handwavefunction andusingeither the 8~~ppart of A~Pin conjunctionwith the

4’NR part of the right-handwave functionor the E1 — fl,rn1 part of ~ in conjunction

with the8~~ppart of the right-handwavefunction. So, the leadingcontributionto the hfs fromexpression(5.36) is

~ fd3q jq.r(~72_X_q~8

1\A~PV4’>4~2J q

2 <4’NR e k~ 2m2 ) 2m1

~ 14~~ ~iqr~(1(72_X_q.81”\[ ~ + ~~i!l ~4~2J q

2 <‘I’NR 2m1 ~ 2m2 )[ 2m1 2m1] ‘i’NR

= - ~ J~-~<4’NRI e~r[_q ~pV+ Vq .~] I4’NR>

* In fact,analternativeapproachwould be to treatexpressions(5.34) and(5.4b) togetherby writing

/ ~ \ / ~ A~’~A~2~\

I 1 + V 114”> = I + I vI4”> = VI4”>.1, E+E,E

2 J \E_E,_E2 E+E1E21 EH,E2

G.T. BodwinandDR. Yennie,Hyperfinesplitting in positroniumand muonium 293

4cc6m~(‘d3pd3p’d3p” 1 1 [ , ,, ,, (—4ircc)= — 3irm~rn

2J (2ir)6 (p~2+ ~2)2(p~ ~~~)2[(P _~ ).~

+ (p~~)2 (p’ — i”) P] (p2 ±~2)2 <a~a2>. (5.37)

If we makethe substitutionp” —÷p’ + p — p” in the last term andassociatedfactors,expression(5.37) becomes

16cc7rn~1d3pd3p’ d3p” 1 1 13m~m

2J (2n)6 ~ + y2)2 (p” — p)2 (p’ — p”)2

x [(p’ — p”) Op” + (p” — p) .~] (2 2)2 <a1 . a2>

— cc6rn~ 1d~‘d3 [ —l ~‘ ~ 1

— 12m~m2ir4J P p [(p~2 + ~2)2 (p’ — p)

2 (p2 + V2)

1 (p’~4.~ 1 ~, 538+ (p~2+ ~,2) (p’ .,)2 (p2 + ~

2)2]\al a2

Theln cc’ contribution is

— 1cc6m~lncc1<a

1~a2>=_~( rn2 ~cc2lncc1Er. (5.39)3m~m2 2 \m, + m2j

The expression(5.35) containsat least oneadditional power of momentumin the integrandcomparedwith expression(5.34).Sincetheleadingpartof expression(5.34) hasthecorrectmomen-tum dependenceto give a In cc 1, the leading part of expression(5.35) cannot be logarithmic.The non-leadingpartsof expression(5.35) either havetoo many powersof momentumto givea ln or areof higherorder in cc, so they are negligible.We concludethat the hfs contributionofAE(B1) in O(cc

2 ln cc1EF)may be obtainedby adding to expression(5.39) the correspondingexpressionwith m

1 andm2 interchangedanddoublingthe result.(Thisaccountsfor thepossibilityof havinga A - on theupperpartsof line2 andfor theoppositeorderof transversephotonemissionandabsorption.)Hence,

AE(B1) ~ (—2 — m1/m2— m2/ml)(rn~/mlrn2)cc2lncc1EF. (5.40)

The contributionAE(B2)is representedgraphicallyin fig. 10. Onceagain,time-orderingargu-

mentsshowthat the graphswith otherarrangementsof the hashmarksvanish.Thep0 integration

Fig. 10. Exampleof AE(B2) correctionsthat arenon-vanishingafter the partial fraction rearrangement.The shadedblob indicatesanynumber(includingzero)of ladderCoulombexchanges.


is

1 1 - -iJ 2ir E + p0 + q0 + E’1 — isE + Po — E1 + is — q0 + E1 + E~— is

= 2~(E1+ E~+ q0) — 1 (5.41)q0 + E1 + E~,+ is

By the usualprocedure,wefind that thesecondterm leadsto anenergyshift

cc Cd3 1 A’~’~<4” ~ (l~iqr T~ A’11~A~2~V — T 4”

4I~2 J q 2 q + E~+ E2 + A1’~A~~V— E + + E1 + E~ q 1

Cd3 A’(l)

iqr T.A(2)v TJq2 2 + 2m

1 1

-~~<4”~ ~‘~ 4~2 Jq

2 2m1 ~ 2m2 ) 8~ >

~~$~.<4’NRIet~[_ V~’~+ ~ V](w2 ~<~.2l) I4’NR>. (5.42)

Clearly, this leadsto the sameresultas in eq. (5.37). The 5-functionterm in eq. (5.42) leadsto anexpressioncontainingtwo additionalpowersof p in the non-relativisticregion comparedwitheq. (5.42). (The factorsthat go like 1/q are replacedby 1/2m.) It cannot give a ln cc

1. Thus,weconcludethat, to O(cc2ln cc

AE(B2) ~ AE(B1). (5.43)

5.5. Theremainingcorrections

Considerfirst the correctionAE(B3), which arisesfrom taking the A part of morethanonepropagator.The non-vanishingkernelsinvolving two A’s areshownfor oneorderof transversephotonemissionandabsorptionin fig. 11. We setaside,temporarily,the possibilityof A’s inside

(a) (b) (c)

(d) (e) (I)Fig. 11. Examplesof AE(B3) contributions.

G.T. Bodwin andDR. Yennie,Hyperfinesplitting in positroniumandmuonium 295

the Coulombphotonblob. The otherarrangementsof two A’s vanishbecauseof time-orderingconsiderations.The graphsin fig. 11 maybe obtainedby changinga A÷to a A.... in the kernelsalreadyconsideredin conjunctionwith the A(B1) and A(B2) corrections.(Seefigs. 9 and 10.) Ingeneral,the effect of this changeis to convertapotentially smalldenominatorto adenominatorof order rn1 or m2. For example,in fig. 1 la the Po integralis

11d 1 1 1

2iriJ ~~°p0+q0+E—E’1+isp0+E+E1—is—p0+E2—is

— 1 1— q0 — E1 — E~,+ isE + E1 + E2 — is~

In comparison,the correspondingintegral from the AE(B1) kernelsis

11d 1 1 1

~J POPo+qo+E_E~+ispo+E+Ei_is_po_E2+is

— 1 1

— — —q0+E, +E~—isq0±E—E~—E2+ie~

We see that the denominatorq0 + E — E’, — E2 ~ q1~ has been replaced by E + E1 + E2~ 2(m1 + m2).(This canalsobe seenby writing the variousgraphsin a time-orderedfashionandexamining the old-fashioned perturbation theory denominatorscorrespondingto the inter-mediatestates.)Similarly, the kernelsinvolving threeor moreA — ‘s containat leastonelesssmalldenominatorthanthe correspondingAE(B1)or AE(B2)kernels.Sincethe integralsin the AE(B1)and AE(B2) calculationshaveat least a logarithmic power count, the additional A .‘s eliminatethepossibility of obtainingaln cc’.

Now considerthe effect of changingAk’s to A’s in the Coulombphotonblob. If a graph isto be non-vanishing,eachA betweenCoulombphotonson agiven fermionlegmust be pairedwith a A directlyoppositeit on theotherfermionleg. The propagatordenominatorsassociatedwith suchapair leadto the loop integral

it’ 1 1 1 1~ —p0+E2—is E+q0+E~+E2_is’~ 2(m1+m2)’

as comparedto the integral

lId 1 1 — 1~T~J POPo+E+qo_E~+is_Po_E2+is~ q0+E—E~—E2+is

that occursin the A÷case.Thus,we see that the net effect of aA - pair in the Coulombphotonblob is to replacethe expressionl/(q + E’1 + E2 — E + A’4PA~V)in eq. (5.5) by

1 A’t1~At2~ I

A’~A12~V — — Vq + E~+ E

2 — E + A’4.’~A~~V+ + 2(m1 + m2) q + E~+ E2 — E + A’.~1~A~V

We recover the leading term by replacingthe denominatorsq + E’1 + E2 — E + A’

11~A~2~Vwith q. In the resultingintegral,thereareat leastthreepowersof momentumtoo manyto obtaina ln. Thereis an explicit factor cc8 outsidetheintegral, andeachof theintegrationsover wavefunc-


tion momentumcan give a factor i/y. We concludethatthe contributionof AE(B3) to the hfs isat mostin O(cc

2EF).

Table 1Onetransversephotoncontributionsto thehfs in O(a2In a— 1EF)

LiE Eq. Coefficientof (ml/m1m2)a

2In a ‘EF

L, (5.11) 0L — WF, (5.18) —2L

2 (5.20) m2/m1 + m1/m2L — WF2 (5.22) m2/m1 + m,/m2D (5.32) 2B1 (5.37) —2 — m1/m2— m2/m1B2 (5.42) —2 — m1/m2— m2/m,B3 0A — 0Total —4

The correctionAE(A) is obtainedby replacingoneor both of the wave functionsin eq. (5.5)with A~~A~

1V/(E— E1 — E2) timesthe ~ part of the Salpeterwavefunction (seeeq. (3.8)).

Supposewe makethisreplacementfor oneof the wavefunctionsin eq. (5.5). Then,in the leadingcontributionto thehfs, theintegrandcontainstoo manypowersof momentumto givealogarithm,andthereis anexplicit factorcc

7 outsidetheintegral.Theintegrationoverthemomentumassociatedwith uncorrectedwavefunction4” givesafactor i/y. The remainingintegrationoverwavefunctionmomentumcontainstoo manypowersof momentumto give afactor i/y, so the leadingcontribu-tion is pureO(cc2EF).Thenon-leadingcontributionscontainadditionalpowersof cc or momentum,as does the contributionin which the ~ correctionis usedin both wave functions.Thus,AE(A) doesnot contributeto the hfs in O(cc2 ln cC 1EF).

The calculationof the one-transversephotonkernelcontributionto the hfs in O(cc2ln cc - 1EF)

is summarizedin tablei. It is reassuringthatall the“wrongmass”termshavecancelled.Our result,AE(i — transverse)= —4(m~/rnlm

2)cc2lncc~EF, (5.44)

is in agreementwith the result of Lepage[25] for the single transversephotonkernels.

6. Othercontributionsto the hfs

In this sectionwediscussotherkernelsthatcould conceivablygivean O(cc2 ln cc- 1EF) contribu-tion to the hfs.* Many of thesekernelshavebeeninvestigatedpreviouslyby otherauthors;here,wemerelyreport their results.

Kernels involving two transversephotonscan arise from both the first order terms and thesecondorder terms in the perturbationexpressioneq. (2.21). The two-particle irreduciblekernelof fig. 12ais, of course,presentin AK. The two-particlereduciblekernelsof fig. 12b andc aretheresult of substitutingthe first andsecondtermsof eq. (2.23) in placeof G

0(j) in eq. (2.2i). For our

* We remind thereaderthat all contributionsunderdiscussionarerecoil correctionsandhave an additionalfactorm,/m,. in the

caseof muonium.


~1 FR F~ ~~ ~ ~

(a) (b) Cc) Cd) (e)

Fig. 12. Additionkernelsthat arisein perturbationtheorythroughsecondorder.

purposesthe remainingtwo transversephotonpiecesof the perturbationexpansionarenegligible.We proceedto demonstratethis point. The~(j) term in eq. (2.23) leadsto the kernelin fig. i2d.In the non-relativisticlimit, this gives the usual secondorderperturbationtheoryexpressionfortheenergyshift

‘cç~ <4’31 AKI 4’s> <4’1IA1(I4’~>

Since thematrix elementsareat least O(cc”) and the energydenominatoris O(cc2), we expectthat

this givesan O(cc2EF)contribution.*The last term in eq. (2.21)containstheproductof an O(cc4) matrix elementwith thederivative

of thatmatrixelementwith respectto P0.The effect of the derivativeis to squarethedenominator

in the expression(eq. (5.5)) for the matrix element,or to inserta factor i/(E — E1 — E2) next tothe wavefunction. By countingpowersof momentum,we concludethat

<4’~IAK

is at most O(cc2).Thus, thelast term in eq. (2.2i) is alsoO(cc2EF).

The kernelof fig. 12c hasbeenexaminedby BarbienandRemiddi [23] in theequalmasscase,andby Lepage[25] in the unequalmasscase.Its leadingcontributionto thehfs isof O(cc2 ln cC‘EF).BarbieriandRemiddialsoconsiderthecontributionof annihilationgraphsto thepositroniumhfs.The termsof O(cc2 ln cc — ‘E) thatderive from thesesourcesare

AE(2-transverse,i Coulombladder) = ~cc2in cC ‘(rn~/mlm2)EF (6.la)

AE(annihilation)= —~cc2ln cc’(m~/rn,rn

2)EF. (6.lb)

Theresult(6.la) hasbeenverified by Lepage[25].The kernelsin figs. 12a andb havebeeninvestigatedby LepageandFultonet al. [22]. They

find that the leadingcontributionto the hfs is of relative order cc for the kernel of fig. 12a andrelativeorder cc

2 ln cc~for thekernelof fig. 12b. Thetotalcontributionof relativeordercc2 ln cC’ is

AE(two-transverse)= ~{m~/m1rn2)cc

2ln cC1EF. (6.2a)

Fultoneta!. alsoshowthat theleadinghfs contributionarisingfrom thecrossedCoulombgraph(fig. 1 2e) is of O(cc2EF).Althoughnot reportedhere,we haveconfirmedtheseresultsfor exchangedphotons.

* There is, of course,the possibility that the sum over statesproducesa logarithm. We have investigatedthis possibility using

thenon-relativisticCoulombpropagator.In thatcase,it turnsoutthat thevariablesof integrationcanbe resealedso asto produceanoverall factora6 timesan integralthat is independentof a.


If we insert a Coulombphoton into the kernelsof fig. 12 in sucha way that it crossesotherphotons,theresultingkernelsgenerallycontributein higherorder.To seethis,wenotethatthereisafactorcc associatedwith eachCoulombphoton.It can,in general,becancelledby asmalldenomina-tor arisingfrom thetwo fermionpropagatorsthatmustbeinsertedalongwith the Coulombphoton.(This is, of course,the mechanismwherebyaboundstatepoledevelopsin thefour point function.)However,if the Coulombphotoncrossesanotherphoton,the momentumof thatphotonentersinto oneof thefermionpropagators.The smalldenominatoris then—~(crossedphotonmomentum)ratherthan (p2 + V

2) (p beingthetwo particlerelativemomentum).*Hence,momentumpowercounting indicates that the explicit factor of cc is not cancelled.The consequenceis that termsoriginally of O(cc2EF) and thoseof O(cc2ln cC1EF) becomeat most O(cc3in cC 1EF).

Kernelsinvolving threetransversephotonsare at most relativeO(cc2EF).This is becausethe

8T’5 associatedwith the third transversephotonmustbe multiplied by factors8 p from the wavefunctionor thefermionpropagatorsif theyareto connectthe largecomponentsto largecompo-nents.The additionalpowersofp comingfrom the factors8~p guaranteethatnoneof the momen-tumspaceintegralsarelogarithmicandthattheadditionalfactorof cc associatedwith thetransversephotoncannotbe cancelledby asmalldenominator.

Thus,we concludethat the total contributionto the hfs in relativeO(cc2ln cC1EF) is given by

eqs.(4.4), (5.44), (6.i) and(6.2). Thesplitting in this order is

AE(cc2lncC’) = —*cc2lncC1EFfor positronium (6.3a)positronium

AE(cc2In cC 1) = 2cc2(rne/m,,) ln cC 1EF for muonium. (6.3b)muonium

7. Summaryandconclusions

We havecalculatedthecontributionsto the hfs in O(cc2(m~/m,rn2)ln cC ‘Er) thatarisefrom the

Coulombinteraction,the onetransversephoton-multipleCoulombphotonkernels,and thetwotransversephotonkernels.(The latterarenot reportedin detail heresinceour techniquesarenotverydifferentfrom thoseof others,andtheresultsarethesame.)Our resultsarein agreementwiththoseof Lepage[25]. In investigatingthe onetransversephoton-kernels,wedevelopedtechniquesthat enabledus to treatall numbersof Coulombphotonexchangesat once.This permitted anapproximation procedurein which intermediatecalculation of spurious contributions wasavoided.

The total theoretical result for the hfs is comparedwith the experimentalvalue for muoniumin table2 andfor positroniumin table3. Numericalvaluesaretakenfrom Lepage[25].

We see that theory and experimentare in agreementto within the estimateduncertainties.However, the theorycannotbetestedat the presentlevel of experimentalaccuracyuntil the termsof O(cc

2EF)in positroniumandthetermsof O(cc3EF,cc2(m~/m)EF)in muoniumhavebeencalculated.A first stepin this directionwould be the calculationof the O(cc2(m~/m,,)ln (m,,/mC)EF)contri-

bution to the muonium hfs. Sucha calculationwould reducethe theoreticaluncertaintyin the

* In thecaseof crossedCoulombphotons,time orderingargumentsshowthatonly theA partsof thefermionpropagatorscon-

tribute,so thesmall denominatoris in fact m.

G.T. BodwinandDR. Yennie,Hyperfinesplitting in positroniumandmuonium 299

Table2 Table3Comparisonof theory andexperimentfor muoniumhfs Comparisonof theoryandexperimentfor positroniumhfs

Theory Theory

Total excludingO(a2(m,/m

6)In a ‘EF) 4463 293.(6) kHz6 Totalexcluding O(a2In a ‘EF) 203381.2 MHz

2a2(m,/m6)ln a’EF 11 kHz —~a

2In a ‘E~ — 3.8 MHz

Total 4463304(6) kHz Total 203 377.4 MHz

Uncertaintydueto termsof Uncertaintydueto termsof O(a2EF) 10 MHzO(a2(m,/m

6)In (rn,/m,)EF) .~.10 kHz

ExperimentExperiment Mills and Bearman[7] 203387.0(16) MHz

(Caspersonet al. [53]) 4463 302.35(52) kHz Egan etal. [6] 203384.9(12) MHz

* This uncertaintyis dueto theuncertaintyin g,/g,.

muoniumhfs by abouta factorof 5. The techniquesdevelopedin this paperseemwell suited tothe calculation of logarithmic terms, so some variant of them may be useful in obtaining thein (mp/me)contribution.

If one attemptsto calculate additional non-logarithmiccontributions to the muonium andpositroniumhfs, new technicaldifficulties are encountered.For example,onemustdeal with theinfinite sum over intermediatestatesin secondorder perturbationtheory. Also if oneuses

4’NR

as astartingpoint, thenindicationsarethat no finite numberof iterationsof theSalpeterequationproducesasufficiently accuratewavefunction [30]. Assumingthat suchdifficulties canbe over-come,it appearsthat thecompleteO(cc2EF)calculationis tedious,but doable.

Acknowledgements

We wish to thank ProfessorKinoshita for suggestingthis problemand for useful discussionsduring the courseof the calculation.We alsohadnumeroushelpful discussionswith ProfessorsWayne Repko,andHans BetheandDr. G. PeterLepage.We especiallythank Velma Ray forher patiencein typing themanuscriptthroughseveralrevisions.

Appendix 1. Normalizationof Salpeterwave functions

Our approachis similar to that of Lurié et a!. [28], who derivedthe normalizationconditionfor the Bethe—Salpeterwavefunctions.Werewrite eq.(2.22b),whichdefinesthethree-dimensionalGreen’sfunction g

0, in the c.m.frame

Cd ‘°d °g0(p’,p,P0) = J -~—— -i-— G0(p , p, P0). (Al.l)

In the caseof an instantaneouskernel, we can integratethe momentumspaceversionof theinhomogeneousBS equation(eq. (2.1)) with respectto p’°andp°anduseeq. (Al.i) to obtainthe


inhomogeneousSalpeterequation

g0(p’,p,P0) = p0 H,(p’) — H2(—p’) + is F(p’)fi,fl2[(2ir)3~(r’ —

d3

+ j’(21r)3 K0(p’,p”)g0(p”,p, P0)],, (A1.2)

whereF(p) A~(p)A~(—p) — A~(p)A~(—p). We canwrite thismorecompactlyby suppressingthe integrations:

— H, — H2) — i14]g0 = (2ir)3iö(p’ — p). (A1.3a)

Similarly, if we carryout this procedurefor theBS equationin theform

G0 = S~F’~S~~+ GOKOS~F’~S~?~,

we obtainthe alternativeform of theinhomogeneousSalpeterequation

g0[/3,/32F(P0 — H1 — H2) — i~0]= (2it)3i~5(p’— p). (Al.3b)

Eqs.(A 1.3), togetherwith the earlierdefinition of 4’~,can be usedto fix the normalizationof theSalpeterwavefunctions.We work in the subspaceof statesspecifiedby the projectionoperatorA~(p)A~(—p) + A~(—p)A~(—p). In thisspacef2 = 1. In order to pickout the poleat P

0 =

we multiply (Al .3a) by P0 — ~, anddifferentiatewith respectto P0. The result is

fi,/32Fg0(P0— ~)+ [fl,132F(Po— H, — H2) — iK0] ~ [g0(P0— ~~)]= (2ir)3iö(p’ p). (Al .4)

Fromeqs. (2.3) and(2.15) we havethat

lim (P0 — ~g0 = i ~ 4’~><4’~~IfluI~2 (Al.5)

Po-.~~j k

wherethe index k labels the degeneratestates.Also, the homogeneousversionof eq. (A1.3b) is

— H1 — H2) — il?0] = 0.*

Thus,if wemultiply eq. (Ai.4) on the left by 4’’fl,132 andtakethe limit P0 —÷~, weobtain

~ <4’k’ I FI4’,> <4’JkI /31132 = <4’Jk’ I 131132. (Ai.6)

Hence

<4’JkIFI4’Jk> = ~kk’ (Al.7)

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

F — ifl1fl~l?0/~P0Ip0,,,~.)

* Note that for thetwo formsof thehomogeneousSalpeterequationto be consistentit is necessarythat i~7J1fl2= — i/31$2R0.

This is a consequenceof theHermitiannatureof a field theoreticinteractioncorrespondingto theinstantaneouspotential.

G.T. Bodwinand D.R. Yennie,Hyperfinesplitting in positroniumand muonium 301

The orthogonality betweeneigenfunctionsof different eigenvaluefollows by consideringtherelation

~J’<4’JkIFI4’J’k’>= <4’JkIF(Hl +H2 -ifl1fl2Fl?0)I4’~~,~>

=

Thus

<4’JkIFI4’Jk> = ~jj’~kk’~ (Al.8)

In the non-relativisticlimit F —÷ 1, so eq. (Al .8) reducesto the familiar normalizationcondition

= ~jj’~kk’~ (A 1.9)

Appendix2. Someusefulintegrals

We areconcernedwith integralsof the type

d3 ,[~.‘2p’2~p4~p2(p— P’)2 ~ — ~~)4] (A2 1)J ~ ‘7 (p2 + y2)(p — p’)2(p’2 + .~,2)2

which havea logarithmic power count. We assumethat the integralshavea large momentumcutoffat p, p’ ~ m

1 or m2. For ourpurposestheprecisenatureof this cutoff is unimportantsinceit cannotaffect the coefficientof ln cC

1.Considerfirst the integral

Jd3pd3p’p2p’2/(p2 + y2)2(p — p’)2(p’2 + y2)2.

We performthe p’ integrationto obtain

~2 Jd3P[VP2/(p2 + ~2) + 2ptan’ (p/y)]/(p2 + y2)2.

(We ignorethehigh momentumcutoff in the p’ integrationsincetheintegralconvergeswithout it.)Now, the ln cC’ contributionscomefrom thoseintegralsthat divergelogarithmicallyin thelimit

—~ 0. In this limit thefirst termin bracketsleadsto aconvergentintegralwhich is 0(1). However,in the limit V —. 0 thesecondterm in bracketsgives the integral

~J(p22)2dP ~ ~ = 4~~[lnp]~~ 4~

4lncC1.

Hence,2 ‘2

J’d3~d3p’ (p2 + y2)2(p_p~)2(p~2+ p2)2 4~4In cC’. (A2.2)

302 G.T. Bodwinand DR. Yennie,Hyperjmnesplitting in positroniumandmuonium

Now considerthe integrals

Jd3Pd3p’~4;p2(p— p’)2]/(p2 + V2)2(p — p’)2(p’2 + V2)2. (A2.3)

The p’ integrationis easilyperformed,with the result

(R2/V)Jd3P~4/(P2 + V2)3 p2/(p2 + V2)2].

These integralsare linearly divergent so we may not ignore the convergencefactors.With theconvergencefactor included,they are O(cc 1), but not 0(ln cC’). They are the origin of O(ccEF)corrections,which we havebypassedin this paper.

Finally, we considertheintegral

I — ~‘~2 d3 d3 ‘ — I — 2P ~P’+ .~ d3 d3J (p2 + V2)2(p’2 + V2)2 ~ P — J(p2 + y2)2(p’2 + y2)2 P P.

The term proportional to p p’ vanisheson performingthe angularintegration.The remainingtwo termsgive equalcontributions.We performthe p’ integrationfirst for the term proportionalto p2 andthe p integrationfirst for the term proportionalto p~2The net result is

‘)2C 2

~ _______d~yJ(p2+V2)2 P.

As in the casejust described,the correctconvergencefactorsmust be includedin the integrand,andthe integralis of 0(1/cc),but containsno 0(!n cC’).

Thus,we concludethat, in expression(A2.l), only the integralwhoseintegrandis proportionaltop2p’2 givesa in cC’ contribution.In theotherintegralsthe leadingcontributionis —~l/y. Theseresultsaresummarizedin table4.

Table4

1 f(p,p’)Contributionsfrom integralsof theform jd pd p (p2 + .~,2)2(1, p’)2(p’2 + 1,2)2

f (p,p’) Leadingcontribution Coefficientof In a

4it4lna~ 4iz4p4 .-. rn/7 0

p2(.p — p’)2 rn/y 0

(p —p’)’ m/y 0

Noteaddedin proof: CaswellandLepage(seenoteaddedin proofin ref. [25]) havediscoveredan additional O(cc2 ln cc1EF) contributionto the positroniumhfs arising from the kernel witha transversephotonexchangefollowed by an annihilationphoton.We haveverified their~result.Its net effect is to changethecoefficientof the ln cC’ term in table3 from —~ to +~,making thiscontribution + 19.1 MHz. The weak contribution to the hfs has beenexaminedby Beg andFeinberg[Phys. Rev.Lett. 33 (1974)606; 35 (1975) 130(E)],andour estimatein sectioni agreeswith theirs.

G.T. BodwinandD.R. Yennie,Hyperfinesplitting in positroniumand muonium 303

References

[1] H. Hellwig, R.F.C.Vessot,MW. Levine, P.W. Zitzewitz,D.W. Allan andDi. Glaze,IEEETrans.Instrum.1M-19 (1970)200.[2] L. Essen,R.W. Donaldson,M.J. Banghamand E.G.Hope,Nature229(1971) 110;

MJ. BanghamandR.W. Donaldson,NationalPhysicalLaboratoryReportQu 17 (March 1971).[3] S.D. Drell andiD. Sullivan,Phys.Rev.138 (1965)B446.[4] E. deRaphael,Phys.Lett. 37B (1971)201.[5] D.E. Casperson,T.W. Crane,A.B. Denison,P.O.Egan,V.W. Hughes,F.G. Mariam,H. Orth,H.W. Reist,PA. Souder,R.D.Stam-

baugh,PA. ThompsonandG. zuPutlitz,Phys.Rev.Lett. 38 (1977)956.[6] P.O. Egan,WE. Frieze,V.W. Hughesand M.H. Yam, Phys.Lett. 54A (1975)412.[7] A.P. Mills Jr.and G.H.Bearman,Phys.Rev.Lett. 34 (1975)246.[8] E. Fermi,Z. Physik60 (1930)320.[9] G. Breit andRE. Meyerott,Phys.Rev.72(1947)1023;

G. Breit, G.E.Brown andGB. Ariken, Phys.Rev.76 (1947) 1299.[10] J. Schwinger,Proc.Nat. Acad.Sci. US 37 (1951)452, 455.[11] E.E. Salpeterand HA. Bethe,Phys.Rev.84 (1951) 1232.[12] N. Kroll and F. Pollock,Phys.Rev.84 (1951)594; 86 (1952)876.[13] R. Karplus,A. KleinandJ.Schwinger,Phys.Rev. 84 (1951)597.[14] R. KarplusandA. Klein, Phys.Rev.87 (1952)848.[15] WA. NewcombandE.E.Salpeter,Phys.Rev.97(1955)1146.[16] T. FultonandP.C.Martin, Phys.Rev.95 (1954)811.[17] R. Arnowitt, Phys.Rev.92 (1963) 1002.[18] H. GrotchandDR. Yennie,Rev.Mod. Phys.41(1969)350;Zeitschr.für Physik202 (1967)425.[19] A.J. Layzer, Bull. Am. Phys.Soc. 6(1961)514; NuovoCimento33 (1964) 1538.[20] D.W. Zwanziger,Bull. Am. Phys.Soc. 6 (1961)514;NuovoCimento34(1964)77.[21] S.J. BrodskyandG.W. Erickson,Phys.Rev.148 (1966) 26.[22] T. Fulton,D.A. OwenandW.W. Repko,Phys.Rev.A4 (1971)1802; Phys.Rev.Lett. 26(1971)61.[23] R. Barbieri andE. Remiddi,Phys.Lett. 65B (1976)258.[24] V.K. Cung,T. Fulton, W.W. RepkoandD. Schnitzler,Ann. Phys. (N.Y.)96(1975)261.[25] G.P. Lepage,Phys.Rev.A16 (1977)863.[26] F. Gross,Phys.Rev.186 (1969) 1448.[27] E.E. Salpeter,Phys.Rev.87 (1952)328.[28] D. Lurié, A.J. MacFarlaneandY. Takahashi,Phys.Rev. 140B (1965) 1091.[29] J.D.Bjorken andS.D. Drell, Relativisticquantumfields (McGraw-Hill Book Company,1964).[30] G.P. Lepage,privatecommunication.

hyperfine splitting in positronium and muonium

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pure coulomb splitting

transverse photon kernels

higher order corrections

corrections i

study of theground state

remaining corrections

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