o(mα7ln2α) corrections to positronium energy levels

1 July 1999

Ž .Physics Letters B 458 1999 143–151

ž 7 2 /OO ma ln a corrections to positronium energy levels

Kirill Melnikov a,1, Alexander Yelkhovsky b,2

a Institut fur Theoretische Teilchenphysik, UniÕersitat Karlsruhe, D-76128 Karlsruhe, Germany¨ ¨b Budker Institute for Nuclear Physics, NoÕosibirsk 630090, Russia

Received 15 April 1999Editor: P.V. Landshoff

Abstract

Ž 7 2 .We present a calculation of the OO ma ln a corrections to positronium energy levels. The result is used to estimate thecurrent uncertainty in theoretical predictions of the positronium spectrum. q 1999 Published by Elsevier Science B.V. Allrights reserved.

PACS: 36.10.Dr; 06.20.Jr; 12.20.Ds; 31.30.Jv

1. Recently, a calculation of the positronium spec-Ž 6. w xtrum with OO ma accuracy was completed 1,2 .

The remaining uncertainty in theoretical predictionsw xwas estimated in 2 using the value of the

Ž 7 2 .OO ma ln a leading logarithmic corrections topositronium energy levels:

ma 7 ln2aX499

dEsy q7ss d . 1Ž .Ž . l015 332p n

The purpose of this Letter is to present a detailedderivation of this result.

In general, an appearance of logarithms of thefine structure constant in QED bound state problemsis related to the fact that several momentum scalesare involved in bound state calculations. Contribu-

1 E-mail: [email protected] E-mail: [email protected]

tions that are logarithmic in a , usually appear asintegrals of the form:

d3kF k , 2Ž . Ž .H 32pŽ .

Ž . y3with F k ;k for the values of k such that2 w xma<k<m or ma <k<ma 3 . Given the

inequality k<m, it is possible to determine theleading logarithmic corrections in the framework ofthe nonrelativistic Quantum Mechanics by expandingall perturbations in series of krm and by using thetime-independent perturbation theory.

Ž 7 2 .In order to find OO ma ln a corrections, we firstcalculate all effective operators which deliverŽ n .OO ma lna contributions for n-7. Such contribu-

tions first appear for ns5 and ns6. Then, theŽ 7 2 .OO ma ln a corrections to energy levels are found

by calculating relativistic corrections to these lowerorder operators. Some new operators arising atŽ 7.OO ma order should be also taken into account. For

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00566-3

( )K. MelnikoÕ, A. YelkhoÕskyrPhysics Letters B 458 1999 143–151144

completeness we consider here the bound state oftwo different particles with masses m and M.

2. Let us first consider the self energy operator ofone particle in the Coulomb field of the other. Thescattering amplitude 3 reads:

d3k 4p XX X yi k r eyA sya j p , p yk eŽ .HSE i3 2k2pŽ .

=

k ki jd yi j 2k i k r ej pyk , p e . 3Ž . Ž .jkqHyE

The leading order logarithmic contribution is ob-Ž .y1tained by expanding the denominator kqHyE

Ž .in HyE rk and by using the leading non-relativis-tic approximation for the currents j . Only spin-inde-i

pendent part of the currents should be consideredsince the spin-dependent part contains additionalpowers of k which destroy the logarithmic integra-tion. One obtains:

d3k 2pa p eyi k r e p ei k r ek kLOV ™ HyE ,Ž .HSE 3 3 m mk2pŽ .

p 'pyk pk rk 2 . 4Ž . Ž .k

It is easy to see that the integration over k in theabove equation is already logarithmic. For this rea-son we can neglect the non-commutativity of the

Ž .Hamiltonian H with the exponent exp ikr . Fusinge

two exponents together, we integrate over k cuttingthe integral by the reduced particle mass ms

Ž .mMr Mqm from above and by the particle energy;p2rm from below. Including the self energy ofthe second particle, we obtain an effective operator:

2a m2LOV p ,r ™y 1y2Ž .SE 2 ž /mM3pm

=p2

p HyE pln , 5Ž . Ž .2m

Ž 2 2 .Here and below we consider ln p rm as commut-ing with all other quantities.

3 w xThe notations of 2 are used throughout this paper.

By averaging this operator over nS states, oneobtains the well-known non-recoil logarithmic con-tribution to the Lamb shift of the nS levels:

22 28a c 0 mŽ .LOd Esy 1y2 lna . 6Ž .SE 2 ž /mM3m

Ž 7 2 .In order to obtain OO ma ln a corrections in-duced by the operator V LO it is necessary to calcu-SE

late relativistic corrections to the expectation valueof this operator. There are several sources of suchcorrections; below we analyze them.

The simplest one is the relativistic correction tothe currents. It is obtained by the substitution

pX qp pX 2 qp2 pX qpXj p , p f ™y 7Ž . Ž .22m 2m4m

Ž .performed for one of the currents in Eq. 4 . Again,only the orbital part of the currents contributes. Thecorrection to the energy levels reads:

3 yi k r i k r 2e ed k 4pa p e p e pk kcurrd Es H ,HSE 3 3 2¦ ;m m2k m2pŽ .

d3k 8pam™y C r p ,C r pŽ . Ž .H 3 4 3¦ ;3m k2pŽ .

24 2 416a c 0 m mŽ .2™ 1y4 q2 ln a .2 2 2ž /mM3m m M

8Ž .

At the final stage of the calculation the self energy ofthe second particle was included.

The next-to-next-to-leading order effect of theretardation is described by the operator

3 yi k r ed k pa p ekretV sy H ,HSE 3 5 mk2pŽ .

=

i k r ep ekH , H , qh.c. 9Ž .

m

Though there is the fifth power of k in the denomi-nator, the spin-dependent part of the currents stilldoes not contribute to the double logarithmic correc-tion. It can be easily seen, that the commutator

( )K. MelnikoÕ, A. YelkhoÕskyrPhysics Letters B 458 1999 143–151 145

w Ž .x Ž 2 .H,s=kexp ikr is OO k as k™0, so that thee

resulting integral over k for the spin-dependent partof the currents is non-logarithmic. We therefore cal-

Ž .culate the commutators in Eq. 9 and keep thereonly such terms that are quadratic in k. We obtain:

d3k 4pa mretd Es 3qHSE 3 3 3 ž /¦ m3m k2pŽ .

=C r p ,C r p q mlMŽ . Ž . Ž .;24 2 48a c 0 m mŽ .

2™y 4y13 q2 ln a .2 2 2ž /mM3m m M

10Ž .

Next, we consider relativistic corrections to theCoulomb interaction and to the dispersion law of theparticles in the intermediate state:

3 yi k r 4 4ed k 2pa p e p pkqH 3 3 3 3mk 8m 8 M2pŽ .

i k r epa 1 1 p ekq q d r . 11Ž . Ž .2 2ž /2 mm M

Taking the average value of this operator, one seesthat the double logarithmic contribution is absent.

Accounting for additional magnetic exchange be-tween the particles requires some care. There existeight irreducible diagrams. Only four of them, shownin Fig. 1, deliver double logarithmic contributions.One finds, that the double logarithmic contributions

Ž .Fig. 1. Corrections to self-energy operator Eq. 5 due to addi-tional magnetic exchange.

from the diagrams Fig. 1a and Fig. 1b compensateeach other, while that of Fig. 1c and Fig. 1d sum upto the following energy shift:

a d3k 4pa p 1 pmagn 2d Esy pHSE 3 3¦ ;2mM m r m3k2pŽ .

q mlMŽ .

4m2a 4 m2 ln mrŽ .™y 1y2 ¦ ;3ž /3p mM mM r

24 28a c 0 mŽ .2™y 1y2 ln a . 12Ž .ž /3mM mM

Integration over k is performed in the limitsŽ 2 .y1mr -k-m.

Ž 2 .Finally, there exists OO a correction to the wavefunction of the bound state, i.e. the correction due tothe iteration of the Breit Hamiltonian and the LO

LO Ž . Ž Ž ..operator V p,r cf. Eq. 5 . The calculation ofSE

this correction is described in the final part of thisLetter, where all lowest order logarithmic operatorsare considered simultaneously.

3. We consider now an exchange of a singlemagnetic photon between the two particles. Thecorresponding scattering amplitude is similar to Eq.Ž .3 :

d3q 4p XX X yi qr pyA sa J yp ,yp yq eŽ .HM i3 2 q2pŽ .

=

q qi jd yi j 2q i qr ej pyq , p e qh.c.Ž .jqqHyE

13Ž .

w xWe used this amplitude in 2 when discussed theretardation effects. Considering the next-to-leadingorder retardation, one finds the operator

d3q 4paLO i qrV ™y e p HyE p , 14Ž . Ž .HM q q3 3mMq2pŽ .


where again the non-commutativity of ei qr and Hhas been neglected. Because of the presence of the

Ž . Ž .exp iqr , the logarithmic integral over q in Eq. 14w xis cut at 1rr;p from above 3 . We arrive at the

following effective operator:

2a p2LOV p ,r sy p HyE pln . 15Ž . Ž . Ž .M 23p mM m

The sources of double logarithmic corrections tothe single magnetic exchange are the same as in thecase of the self energy operator. The only essentialdifference in two calculations consists in a change ofthe upper cut-off. For this reason we skip a detaileddiscussion and present the results of the calculation.We obtain:

24 28a c 0 mŽ .curr 2d Es 1y2 ln a , 16Ž .M ž /3mM mM

24 216a c 0 5 mŽ .ret 2d Esy 1y ln a , 17Ž .M ž /15mM 2 mM

24 28a c 0 mŽ .magn 2d Esy ln a . 18Ž .M 2 23m M

4. The next source of the double logarithmiccorrections is the double magnetic exchange withtwo seagull vertices. The corresponding potentialreads

a 2 d3k 4p 4pi qr pV sy eH XSS 3mM 2k 2k2pŽ .

=

2XeeŽ .yi qr ee . 19Ž .Xkqk qHyE

Here kX sqyk, e e sd yk k rk 2 and similarlyi j i j i j

for eX. It is sufficient to consider the leading nonrela-tivistic approximation for the seagull vertex. From

Ž .Eq. 19 one sees, that the logarithmic contribution< < < <comes from the region of momenta where q < k

<m. Neglecting HyE in comparison with k andintegrating over k from q to m one obtains:

2a 2 qLOV s ln . 20Ž .SS mM m

We consider now relativistic corrections to theŽ .operator in Eq. 20 . In this case there are other

sources of corrections, as compared to two casesconsidered above. The following corrections shouldbe considered – relativistic corrections to the seagullvertex, an appearance of the magnetic-magnetic-Coulomb vertex, an expansion of the ‘‘heavy’’ inter-mediate energy denominators in powers of< < X 4p rm,krm,k rm and the usual retardation effects.We find that none of these effects produces thedouble logarithmic correction.

For example, relativistic corrections to the seagullvertex are polynomial in the momenta p, k and q.On the other hand, the structure of the denominator

Ž .in Eq. 19 does not change. In this case the logarith-mic contribution comes only from the region of large< <k , so that the resulting operator is of the form of

Ž .Eq. 20 times a polynomial in external momenta.Such operators are too singular to produce a doublelogarithmic contribution.

Ž 7 2 .Hence we conclude that the OO ma ln a correc-tions to the expectation value of the operator Eq.Ž .20 appear only due to relativistic corrections to thewave functions. As we have mentioned already,these corrections are considered at the end of thisLetter.

5. We now turn to logarithmic operators whichŽ 6. Ž .arise at the OO ma order. The OO a correction to

such operators can be caused only by the effect ofretardation. There are two effects of this sort. Let usdenote by k the momentum of the photon in theintermediate state. Then, for the ‘‘light’’ eqeyg

intermediate state the energy denominators are ex-Ž .panded in HyE rk, whereas for the ‘‘heavy’’

eqeyeqeyg intermediate state the expansion pa-rameter is krm. In contrast to these, all other rela-tivistic corrections are of relative a 2 order. Retarda-tion corrections clearly require that the correspond-ing effective operators are induced by an exchangeof at least one magnetic photon. Inspecting such

w xoperators in 2 , we observe that only two of them,the ‘‘one-loop’’ operator and the operator induced by

4 By ‘‘usual’’ we mean here such retardation effects, that donot resolve the point-like seagull vertex.


Fig. 2. Double magnetic exchange with one-seagull vertex. Thedashed lines, representing Coulomb exchanges, show that the

Ž .exact Green function for the system positronium plus photon sshould be used in the calculation.

a double magnetic exchange with one seagull vertex,Ž 7 2 .can give rise to OO ma ln a corrections on one

hand, and were not considered yet on the other.Retardation correction to the one-loop operator

cannot produce the double logarithmic contribution.Indeed, both effective vertices entering the one-loopoperator, are p-independent, so that their commuta-tors with H produce only positive powers of mo-menta. One can easily check that resulting operatorsare too singular to produce two logarithms.

For the same reason considering the double mag-netic exchange with one seagull vertex we keep

Žorbital currents only. The sum of three diagrams see.Fig. 2 then gives:

a 2 d3k 4p 4p2exV s H XS 3m 2k 2k2pŽ .

=1 1

Õ Õ ÕX2 1 3½ k qkqHyE kqHyE

1 1qÕ Õ ÕX1 2 3k qHyE kqHyE

1 1qÕ Õ Õ ,X X1 3 2 5k qHyE k qkqHyE

21Ž .

where

pXeX peX X Xi k r i qr i k rp e pÕ s e , Õ se ee , Õ s e .1 2 3M M22Ž .

The contribution of three diagrams with the seagullvertex on the second particle line is obtained from

Ž .Eq. 21 by the substitution mlM. An account ofthe retardation to first order gives the operator:

a 2 d3k 4p 4p2exV ™y H XS 3m 2k 2k2pŽ .

=1 1

w xÕ Õ H ,ÕX2 1 32½ k qk k

1 1w xqÕ Õ H ,ÕX1 2 32k k

1 1w xqÕ Õ H ,Õ qh.c. 23Ž .X1 3 22X 5k k qkŽ .

For k<q this operator reduces to

a 3 d3k 4p 4p2exV s p p ,C rŽ .HS q q2 3 3 23mM k q2pŽ .

qh.c.q mlM . 24Ž . Ž .

Double logarithmic correction to the energy arises ifone integrates over k from q2rm to q, and then overq from ma to m. The result is

248a c 0Ž .2ex 2d Esy ln a . 25Ž .S 3mM

The last ‘‘irreducible’’ correction arises due to thesingle-seagull diagrams with one of the magnetic

Žquanta absorbed by the same charged particle see.Fig. 3 . In contrast to the previous calculation, there

is a double logarithmic contribution in the diagramsFig. 3a,b coming from the region of q<k<m. In

Fig. 3. Seagull corrections to single magnetic exchange.


the sum of Fig. 3a and Fig. 3b the contribution ofthis region cancels out. The result is:

248a c 0Ž .1ex 2d Esy ln a . 26Ž .S 3mM

Ž .Eq. 26 completes the analysis of the ‘‘irreduci-Ž 7 2 .ble’’ OO ma ln a corrections.

Ž 7 2 .6. The last source of the OO ma ln a correctionsis related to the wave function modification by rela-tivistic effects:

² :d Es VGUqUGV . 27Ž .c

Here G is the reduced nonrelativistic Green function,U is the Breit Hamiltonian projected on S-states:

m2

221y3 pmMU p ,r syŽ . ž /2m 2m

m p2

q ,C rŽ .½ 5mM 2m

pa m2X2q 1q2 1q ss d r ,Ž .32 ž /mM2m

28Ž .

and the operator V is the sum of the lowest-orderŽ . Ž . Ž .logarithmic operators Eqs. 5 , 15 and 20 :

2a p2 m2

V p ,r sy ln 1y p HyE pŽ . Ž .2 2 ½ ž /mM3pm m

3pa m2

y . 29Ž .52 mM

Ž .The sum of the first two terms in U p,r can berepresented as a linear combination of the operators

2 � Ž .4 Ž .2 Ž w x.H , H,C r , and C r see 2 . The last two

terms induce the following double logarithmic cor-rections:

m2

1ymM ² :d Es H ,C r GV� 4Ž .c ,1

m

m2

1ymM ² :™y C r VŽ .

m

222a m™y 1y3 ž /mM3pm

=p2

ln C r p p ,C rŽ . Ž .2¦ ;m

224 216a c 0 mŽ .™ 1y2 ž /mM3m

=d3p p 2 p2

lnH 3 3 2p m2pŽ .224 28a c 0 mŽ .

2™y 1y ln a ; 30Ž .2 ž /mM3m

m2

1q2mM

d Esy C r GV² :Ž .c ,2m

m2

21q p 2mM™y ln C r GŽ .3 2¦m m

=m2 3pa m2

1y p p ,C r qŽ . ;½ 5ž /mM 2 mM

m2

21q p 2mM™ ln C r GŽ . 03 2¦m m

=m2 3pa m2

4pa 1y y ;½ 5ž /mM 2 mM

24 216a c 0 3 mŽ .™ 1y2 ž 8 mM3m

11 m42y ln a . 31Ž .2 2 /8 m M


In a similar manner we consider the contributionŽ .of the d r operator from the Breit Hamiltonian. We

obtain:

pa m2X2 ² :d E™ 1q2 1q ss d r GVŽ .c ,3 32 ž /mMm

24 24a c 0 mŽ .X2™y 1q2 1q ss32 ž /mM3m

=7 m2

21y ln a . 32Ž .ž /4 mM

7. In order to check that we have accounted for allŽ 7 2 .sources of the OO ma ln a corrections, it is useful

to consider the problem from a more formal point ofview. As we mentioned above, logarithms of a arisebecause of the hierarchy of scales in QED boundstates calculations. In the framework of NRQEDsuch logarithms may be detected by analyzing theon-shell scattering amplitude of two particles andidentifying such contributions to this amplitude thathave zero index in the nonrelativistic region. There-

Ž 7 2 .fore, to get the OO ma ln a corrections to energyŽ 4.levels, we have to consider the OO a scattering

amplitude and find all zero index graphs..To do so, we proceed in the following way: i

divide all graphs into classes according to the num-Žber of magnetic photons e.g., class 0 contains all

graphs with four Coulomb exchanges, class 1 con-tains all graphs with three Coulomb and one mag-

. .netic photon, and so on ; ii in each class, weconsider only such graphs that in the leading nonrel-ativistic approximation have negative or zero in-

.dices; iii the negative indices are shifted to zero byaccounting for retardation corrections as well as allpossible relativistic corrections to the elements of agiven graph.

Consider first the class 0. In the leading non-rela-tivistic approximation the only contribution comesfrom the four-Coulomb ladder graph with both elec-tron and positron staying in positive-energy interme-diate states. This graph has the index 3=3y3=2y4=2sy5, where the first term comes fromthree-loop integration volume, the second one fromthree energy denominators, corresponding to eqey

intermediate states and the last term arises from thefour Coulomb propagators. To shift this index tozero, we have to consider relativistic corrections with

the fifth net power of momenta. However, all rela-tivistic effects in pure Coulomb graphs bring in

Ž 2 .corrections which scale like OO Õ , i.e. they increasethe index by 2. Therefore, we conclude that the class0 is free from potentially logarithmic graphs.

Turning to other classes, we notice that one cansignificantly decrease the number of graphs by con-sidering only such where all magnetic and seagullvertices belong to one and the same fermion line. Itis easy to see that the equal-time transfer of a vertexfrom one fermion line to the other does not changethe index of the graph. Hence, for the purpose ofindex counting we can consider the simplified prob-

Ž 4.lem of the OO a scattering of one particle at theexternal Coulomb field. After identifying potentiallylogarithmic graphs with magnetic and seagull ver-tices on one fermion line one must consider allpossible equal-time transfers of those vertices be-tween two fermion lines and all possible relativisticcorrections.

To make further discussion more transparent, weintroduce the following notations. The Coulomb,magnetic, and seagull vertices are denoted by C, M,and S, respectively. For the class 1, in the leadingnonrelativistic approximation we have the followinggraphs: MMCCC, CMMCC, CCMMC, and CC-CMM 5. Their index, y3, is the sum of 9 from theintegration volume, 2 from two Pauli currents inmagnetic vertices, y6 from three Coulomb propaga-tors, y6 from three energy denominators related tointermediate states without photon, y1 from theeqeyg intermediate state, and y1 from the normal-ization factor of the propagator of the magneticphoton. To increase this index to zero it is necessaryto account for the effects of retardation. Since eachorder of perturbation theory for the retardation in-creases the index by 1, we have to consider either thethird order retardation effects or the first order retar-

Ž 2 .dation with additional OO Õ relativistic corrections.Note that an account of the retardation includes not

Žonly an expansion of energy denominators kq.y1DE , but also permutations of the Coulomb and

magnetic vertices, e.g., MMCCC™MCMCC. By

5 The notations are such, that e.g. MMCCC represents a graphwere first the magnetic photon is emitted, then it is absorbed andthen three Coulomb exchanges occur.


inspection we find that all these effects were consid-ered in our previous analysis of single-magneticcontributions.

An analysis of the class 2 graphs is slightly moreinvolved because of an appearance of the seagullvertex. The lowest index, y2, arises in the leadingnonrelativistic approximation for the graphs with twoseagull vertices, CCSS, CSSC, and SSCC. To com-pensate this index we have to account for the next-

Ž 2 .to-next-to-leading order retardation or some OO Õ

relativistic correction. The index y1 graphs includeSMM, MSM, MMS, and M M M M with twoi i j j

additional Coulomb vertices, none of which is lo-cated either between S and M or M and M . Thei i

next-to-leading order retardation shifts the index ofthese graphs to zero. Finally, there are index 0graphs, M M M M and M M M M with two ad-i j j i j i j i

ditional Coulomb photons, each located either to theleft or to the right of all M-vertices. One can easilyfind the one-to-one correspondence between any ofthese graphs and some parts of the double-magneticcontributions considered in our previous analysis.

In the class 3 there are only index 0 graphs,SSMM and MMSS with one C outside pairs SS andMM. After equal-time transfer of one seagull andone magnetic vertex to the second fermion line, weget the graphic representation for the correction to

Ž .the average value of the operator Eq. 20 , inducedby the magnetic part of the Breit Hamiltonian.

Finally, an inspection of the class 4 graphs showsthat all of them have positive indices.

Ž 7 2 .8. It remains to sum up all OO ma ln a contribu-tions. We obtain:

24 2a c 0 149 mŽ .2dE sy 4y ln a , 33Ž .aver 2 ž /15 mMm

24 264a c 0 7 mŽ .2dE sy 1y ln a . 34Ž .hfs ž /9mM 4 mM

In the limiting case mrM™0, the old result for thew xhydrogen 5 is reproduced. Considering pure recoil

corrections, one finds:2411a c 0Ž .

rec 2dE sy ln a ,aver 15mM24 216a c 0 mŽ .

rec 2dE s ln a . 35Ž .hfs 2 23m M

Ž 2 Ž .2 .It is interesting to note, that the OO m r mMŽ .terms mutually canceled not only in Eq. 33 , but

also in individual contributions, i.e. those induced bythe self energy, the single magnetic exchange and thedouble magnetic exchange with one seagull vertex.

For the positronium, it is necessary to add thecontribution caused by the virtual annihilation which

Ž .is easily extracted from Eq. 32 . One obtains:

2X 43qss a c 0Ž .2d Esy3 ln a . 36Ž .ann 24 m

Ž .2 3 3 Ž 3.Using c 0 sd m a r 8p n , we arrive at thel0Ž 7 2 .final result for the OO ma ln a contribution to the

positronium energy levels:

ma 7 ln2aX499

dEsy q7ss d . 37Ž .Ž . l015 332p n

Its spin-dependent part is in agreement with thew xresult of Ref. 4 .

Ž 7 2 .Numerically, the OO ma ln a contribution to thetriplet energy levels equals to

1.33dE n S sy MHz. 38Ž .Ž .1 3n

Ž 3 . Ž 3 .The correction to the difference E 2 S yE 1 S1 1

amounts therefore to 1.16 MHz. For the singletŽ 7 2 .states, the OO ma ln a correction gives:

0.401dE n S sy MHz. 39Ž .Ž .0 3n

Ž 7 2 .We then find that the OO ma ln a correction to thepositronium ground state hyperfine splitting amountsto y0.9 MHz.

Ž 7 2 .We conclude, that the OO ma ln a correctionsturn out to be of the order of 1 MHz and hencesomewhat enhanced, as compared to the naive esti-mate of the magnitude of the ma 7 effects. We note,

Ž 6.that at O ma the remaining, non-logarithmic cor-rections, contributed approximately one half of theleading logarithmic ones. Extrapolating this situation

Ž 7.to order OO ma , we conclude that the currenttheoretical uncertainty in the predictions for the

w xpositronium energy levels 1,2 should be approxi-mately 0.5 MHz.


Acknowledgements

We are grateful to K. Pachucki for communicat-ing to us some of the results prior to publication andfor comments on the manuscript. We are grateful toA. Czarnecki for providing the figures. The researchof K.M. was supported in part by BMBF under grantnumber BMBF-057KA92P, and by Graduiertenkol-leg ‘‘Teilchenphysik’’ at the University of Karl-sruhe. A.Y. was partially supported by the RussianFoundation for Basic Research under grant number98-02-17913.

Note added. We are indebted to K. Pachucki andŽ .S. Karshenboim for pointing out an error in Eq. 26

in the preprint version of this Letter. After correctingŽ 7 2 .this error, our result for the OO ma log a correc-

tions to positronium energy levels agrees with that ofw xRef. 6 .

References

w x Ž .1 K. Pachucki, S.G. Karshenboim, Phys. Rev. Lett. 80 19982101.

w x2 A. Czarnecki, K. Melnikov, A. Yelkhovsky, hep-phr9901394.w x3 I.B. Khriplovich, A.I. Milstein, A.S. Yelkhovsky, Phys. Scr. T

Ž .46 1993 252.w x Ž .4 S.G. Karshenboim, JETP 76 1993 41.w x Ž .5 A.J. Layzer, Phys. Rev. Lett. 4 1960 580; H.M. Fried, D.R.

Ž .Yennie, Phys. Rev. Lett. 4 1960 583.w x6 K. Pachucki, S.G. Karshenboim, to be published.

o(mα7ln2α) corrections to positronium energy levels

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