It is shown that the corrections of $\alpha^2$ order to the total cross
sections for interaction of elementary hydrogen-like atoms with target atoms,
reported in the previously published paper [S.Mrowczynski, Phys.Rev. D36, 1520
(1987)], do not include some terms of the same order of magnitude. That results
in a significant contribution of these corrections in particular cases. The
full $\alpha^2$-corrections have been derived and it is shown that they are
really small and could be omitted for most practical applications.Comment: 5 page

A. Tarasov

A.V. Kuptsov

D.Yu. Ivanov

G Molière

G.H. Gillespie

I. Amirkhanov

K.G. Denisenko

L. Afanasyev

L.G. Afanasyev

L.S. Dul’yan

O. Voskresenskaya

O.O. Voskresenskaya

S. Mrówczyński

Z. Halabuka

English

arXiv

Crossref

Physical Review D

Contribution of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>terms to the total interaction cross sections of relativistic elementary atoms with atoms of matter

It is shown that the corrections of $\alpha^2$ order for the total interaction cross sections of elementary hydrogen-like atoms with target atoms, found in the previous paper [S.Mrowczynski, Phys.Rev. {\bf D36}, 1520], does not include some terms of the same order of magnitude. This results to significant contribution of these corrections in particular cases. It is shown that the full $\alpha^2$-correction is really small and could be omitted for most practical applications.It is shown that the corrections of $\alpha^2$ order to the total cross sections for interaction of elementary hydrogen-like atoms with target atoms, reported in the previously published paper [S.Mrowczynski, Phys.Rev. D36, 1520 (1987)], do not include some terms of the same order of magnitude. That results in a significant contribution of these corrections in particular cases. The full $\alpha^2$-corrections have been derived and it is shown that they are really small and could be omitted for most practical applications

Afanasyev, L.

Tarasov, A.

Voskresenskaya, O.

CERN Document Server

arXiv:hep-ph/0109208   22 Sep 2001Contributionof2-termstothetotalinteractioncrosssectionsofrelativisticelementaryatomswithatomsofmatterL.Afanasyev,A.Tarasov,yandO.VoskresenskayazJointInstituteforNuclearResearch,141980Dubna,MoscowRegion,Russia(Dated:January29,2002)Itisshownthatthecorrectionsof2orderforthetotalinteractioncrosssectionsofelementaryhydrogen-likeatomswithtargetatoms,foundinthepreviouspaper[S.Mrowczynski,Phys.Rev.D36,1520],doesnotincludesometermsofthesameorderofmagnitude.Thisresultstosignicantcontributionofthesecorrectionsinparticularcases.Itisshownthatthefull2-correctionisreallysmallandcouldbeomittedformostpracticalapplications.PACSnumbers:11.80.Fv,34.50.-s,36.10.-kTheexperimentDIRAC[1],whichnowunderwayatPSCERN,aimstomeasurethelifetimeofhydrogen-likeelementaryatoms(EA)consistingof+and mesons(A2)withaccuracyof10%.Theinteractionof+ -atomswithmatterisofgreatimportancefortheexper-imentasA2dissociation(ionization)insuchinterac-tionsisexploitedtoobserveA2andtomeasureitslife-time.Intheexperimenttheratiobetweenthenumberof+ -pairsfromA2dissociationinsideatargetandthenumberofproducedatomswillbemeasured.Thelifetimemeasurementisbasedonthecomparisonofthisexperimentalvaluewithitscalculateddependenceonthelifetime.Accuracyoftheinteractioncrosssectionsofrel-ativisticEAwithordinaryatoms,whicharebehindallthesecalculations[2],isessentialfortheextractionofthelifetime.Studyofinteractionsoffasthydrogen-likeatomswithatomshasalonghistorystartingfromBethe.Oneoftheresentcalculationsforhydrogenandone-electronionswaspublishedin[3].InteractionsofvariousrelativisticEAconsistingofe,,,Kwereconsideredindierentapproaches[4,5,6,7,8,9,10,11,12,13,14,15,16].Inthispaperwereconsidercorrectionsof2ordertotheEAtotalinteractioncrosssectionsobtainedin[7].(Throughthispaperisthene-structureconstant.)Asshownin[7]analysisoftherelativisticEAinterac-tionwiththeCoulombeldoftargetatomscanbeper-formedconvenientlyintherestframeoftheprojectileEA(anti-labframe).Asthecharacteristictransfermomen-tumisoforderoftheEABohrmomentum,inthisframeaftertheinteractionEAhasanon-relativisticvelocityandthusinitialandnalstatesofEAcanbetreatedintermsofthenon-relativisticquantummechanics.Inthismannerthewell-knowndiÆcultiesoftherelativisticAlsoatCERNEPDivision,CH-1211Geneva23,Switzerland;Electronicaddress:Leonid.Afanasev@cern.chyCurrentaddress:InstituteforTheoreticalPhisics,UniversityofHeidelberg,Philosophenweg19,D-69120,Heidelberg,Germany;Electronicaddress:tarasov@nu.jinr.ruzElectronicaddress:voskr@cv.jinr.rutreatmentofboundstatescanbegetround.AsintheEArestframeatargetatommoveswiththerelativisticvelocityitselectromagneticeldisnolongerpureCoulomb.Itisdescribedbythe4-vectorpoten-tialA=(A0;A)withcomponentsrelatedtoitsrestCoulombpotentialU(r):A0=UA=U:(1)Here=v=c,visthetargetatomvelocityintheEArestframeandisitsLorentz-factor.Thetime-likecom-ponentA0ofthe4-potentialinteractswiththechargesofparticlesformingEAandthespacecomponentwiththeircurrents.InthispaperweconsideronlyEAconsistingofspinlessparticles(,K-mesonsetc.)whichareofinterestfortheDIRACexperiment.IntheBornapproximationtheamplitudesoftransitionfromtheinitialstateitothenalfduetotheinteractionwithAcanbewrittenas:Afi=U(Q)afi(q);(2)U(Q)=21Z0U(r)sinQrQrdr;(3)afi(q)=fi(q) jfi(q):(4)Thetransitiondensitiesfi(q)andtransitioncurrentsjfi(q)areexpressedviathetheEAwavefunctions iand fforthetheinitialandnalstates:fi(q)=Zfi(r) eiq1r e iq2rd3r;(5)jfi(q)=Zjfi(r)m1eiq1r+m2e iq2rd3r;(6)fi(r)= f(r) i(r)(7)jfi(r)=i2 i(r)r f(r)  f(r)r i(r):(8)TheEAwavefunctions i;fandthebindingenergies"i;fobeytheSchrodingerequationH i;f="i;f i;f(9)2H =  2+ V (r)with the Hamiltonian H. It is worth noting that theexplicit form of the potential V (r) of the interaction be-tween the EA components have no inuence on the nalresult of this paper.In the above equations m1;2are masses of EA com-ponents, q = (q0; q) is the transfer 4-momentum, andall other kinematic variables are related by the followingequations:q1=m1q ; q2=m2q ;  =m1m2M; M = m1+m2;q = (q0; q) ; q = (qL; qT) ;q0= !fi+Q22M= q = qL; !fi= "f  "i; (10)Q =pQ2; Q2= q2  q20= q2T+ q2L(1  2) :The dierential and integral cross sections of the EAtransition from the initial state i to the nal f due tointeraction with the electromagnetic eld of the targetatom are related to the amplitudes (2):d fid qT=12jAfi(q)j2fi=12ZjAfi(q)j2d2qT: (11)Formulae (2{11) allow to calculate the transition (par-tial) cross sections in the Born approximation. But forapplications (for example see [2]) the total cross sectionsof the EA interaction with target atoms are required also.Because the Born amplitudes of the EA elastic scatteringare pure real values the optical theorem can not be usedto calculate the total cross sections. Thus they should becalculated as the sum of all partial cross sections:toti=Xffi: (12)Usually to get a closed expression for the sum of thisinnite series (the so-called \sum rule") the transitionamplitudes (2) are rewrite as:Afi(q) = hf jbA(q)jii ; (13)where the operatorbA(q) does not contain an explicit de-pendence on the EA nal state variables (for example itsenergy "f, see bellow). Then using of the completenessrelationXfjfihf j = 1 ; (14)the sum (12) can be written in the form:toti=12ZhijbA(q)bA(q)jiid2qT: (15)However one should to take some caution while goingfrom the exact expressions (2{10) for the transition am-plitudes, with explicitly dependence on the "f(throughthe time-like q0and longitudinal qLcomponents of 4-vector q), to the approximate one without such depen-dence. Otherwise, it is possible to obtain a physicallyimproper result as it has happened to the authors of thethe paper [7] at deriving of the sum rules for the totalcross section of interaction of ultrarelativistic EA ( = 1)with target atoms. Below we discuss this problem in de-tail.The most essential simplication, that arises in thecase of  = 1 is that Q2= q2T. Thus U (Q) = U (qT)[see (10)] and only Afiin (2) depends on "fthrough theexponential factors exp (iq1r) and exp ( iq2r) in (5) and(6)q1;2r =m1;2qr =m1;2(qLz + qTrT) ; (16)where qL= !fi+ q2T=2M if  = 1.Now let us take into account that the typical valueof z in these expressions is of order of the Bohr radiusrB= 1= and the typical qL !fi 2, thus theproduct qLz is of order of . Then it seems natural toneglect the qL-dependence of afi:afi(q)  afi(qT) : (17)and consider this case as the zero order approximationto the problem [7]. It corresponds to the choice of theoperatorbA in the form:bA(q) = U (qT)eiq1TrT  e iq2TrT (eiq1TrT=m1+ e iq2TrT=m2)bp:(18)Herebp =  ir is the momentum operator.Substituting (18) in (15) results in the following sumrules [7], where the total cross sections is expressed asthe sum of the \electric" eland \magnetic" magcrosssections:tot= el+ mag; (19)el=ZU2(qT)M (qT)d2qT; (20)M (qT) = 2(1  S(qT))S(qT) =Zj (r)j2eiqTrd3r ;mag=ZU2(qT)K(qT)d2qT; (21)K(qT) =Z12+2m1m2(eiqr  1)j r  i(r)j2d3r :These results dier from the sum rules used in [2] bythe additional term mag. For beginning let us considerits contribution qualitatively. For this purpose the target3atom potentialU (r) can be approximated by the screenedCoulomb potential:U (r) =Zre r;   meZ1=3; (22)where meis the electron mass and Z is the atomic num-ber of the target. The pure Coulomb wave function canbe used for  i(i.e. the contribution of the strong inter-action between the EA components is neglected see [17]).For the ground state it is written as: i(r) =3=2pe r: (23)Under such assumptions for the ground state the follow-ing results can be easy obtained:el=8Z22ln2Z1=3me 34; (24)mag=43Z2+ O(2el)=4Z4=323m2e+ O(2el) : (25)It is seen that in spite of 2in the numerator of magthe electron mass square in the denominator makes thecontribution of the \magnetic" term in (19) not negligiblewith respect to the \electric" one, especially for the caseof EA consisted of heavy hadrons and low Z values.To obtain exact numerical values we have preciselyrepeated the calculations made in [7]. More accuratepresentation of the target atom potential, namely, theMoliere parameterization of the Thomas-Fermi potential[18] was used as in [7]:U (r) = Z3Xi=1cie irr; (26)c1= 0:35; c2= 0:55; c3= 0:1 ;1= 0:30; 2= 1:20; 3= 60; 0= meZ1=3=0:885:The values of \electric" (el) and \magnetic" (mag) totalcross sections (in units of cm2) and their ratio (mag/el)are presented in Table I for various EA and target mate-rials. The values published in [7] are given in the paren-theses. It is seen that the \electric" cross sections arecoincide within the given accuracy, but the \magnetic"ones are underestimated in [7]. It is worth noting that thecorrect values of magdoes not depend on EA masses asit follows from the simplied approximation result (25).The ratio values conrm the above estimation about the\magnetic" term contribution. Thus inaccuracy in thecalculations did not allow the authors of [7] to observe sosignicant contribution of magin their results.It is clear, that such strong enhancement of the mag-netic term in (19) is the consequence of its inverse powerdependence (25) on the small screening parameter . ItTABLE I: The \electric" (el) and \magnetic" (mag) totalcross sections in units of cm2and their ratio (mag/el) in % forEA consisting of  and K mesons (A2, AK, A2K) and targetmaterials with the atomic number Z. The values publishedin [7] are given in the parentheses.Z A2AKA2K6 el 3:03  10 221:37  10 223:08  10 23(3:1  10 22) (1:4  10 22) (3:0  10 23)6 mag 6:73  10 246:73  10 246:73  10 24(2:5  10 24) (1:3  10 24) (0:3  10 24)6 mag/el 2.22% 4.90% 21.9%13 el 1:33  10 216:08  10 221:37  10 22(1:3  10 21) (6:2  10 22) (1:4  10 22)13 mag 1:89  10 231:89  10 231:89  10 23(0:96  10 23) (0:55  10 23) (0:15  10 23)13 mag/el 1.41% 3.10% 13.7%29 el 6:17  10 212:84  10 216:48  10 22(6:1  10 21) (2:9  10 21) (6:7  10 22)29 mag 5:50  10 235:50  10 235:50  10 23(3:6  10 23) (2:3  10 23) (0:68  10 23)29 mag/el 0.891% 1.94% 8.49%47 el 1:55  10 207:15  10 211:64  10 21(1:5  10 20) (7:3  10 21) (1:7  10 21)47 mag 1:05  10 221:05  10 221:05  10 22(0:79  10 22) (0:52  10 22) (0:17  10 22)47 mag/el 0.676% 1.46% 6.37%82 el 4:46  10 202:07  10 204:81  10 21(4:4  10 20) (2:1  10 20) (5:1  10 21)82 mag 2:20  10 222:20  10 222:20  10 22(1:9  10 22) (1:3  10 22) (0:48  10 22)82 mag/el 0.493% 1.06% 4.58%is also easily to see that the origin of such unnatural de-pendence is in the behaviour of the factor K(qT) at smallvalues of qTin (21). This factor, contrary to M (qT) in(20), does not approach to zero at qT! 0. But at  = 1such behaviour of K(qT) is in contradiction with somegeneral properties of the transition amplitudes (4), whichfollow from the continuity equation:!fifi(q)  qjfi(q) = 0 : (27)(The later can be derived from the Schrodinger equation(9)). Indeed, rewriting the continuity equation in theform:!fifi(q)  qLjfi(q)  qTjfi(q) =!fi[fi  jfi(q)]  q2Tjfi(q)=2M   qTjfi(q) = 0 ;(28)it is easily obtain, thatafi(q) = fi(q)  jfi(q)=1!fiq2Tjfi(q)=2M + qTjfi(q):(29)4That is all transition amplitudes become zero at qT= 0.It follows, that any transition cross section (11) can de-pend on the screening parameter  at least only logarith-mically, but never like inverse power of this parameter.The same is valid for the sum (12) of this quantities, i.ethe total cross section.Since the -dependence of the magnetic term in (25)is contradictory to the general result, we must concludethat there is a fallacy in the deriving of sum rules (19)somewhere. To understand the origin of the error, madeby authors of [7], let us go back to the quantities (5),(6)and expand them over powers of the longitudinal mo-mentum transfer qL:fi=1Xn=0(n)fi; (n)fi=qnLn!dndqnLfiqL=0; (30)jfi=1Xn=0j(n)fi; j(n)fi=qnLn!dndqnLjfiqL=0: (31)It is easily shown that terms of these expansions obeythe following estimation:(n)fi/ n; j(n)fi/ n+1: (32)The additional power of  in the current expansion coef-cients, in comparison with the density one, reects theordinary relation between the values of current and den-sity in the hydrogen-like atoms.Expanding the value (4) and taking into account (32) itseems reasonable to group terms with the same order of rather than qLas it was done in [7]. Then the successiveterms of the afiexpansion over  powers areafi=Xna(n)fia(n)fi= (n)fi  j(n 1)fi:(33)From above it is clear that in the \natural" approxi-mation (17) includes a(0)fiand the only one part of theterm a(1)fiof the expansion (33), namely:j(0)fi=  iZ fE(qT; rT)@ i@zd3r : (34)While the second one(1)fi= iqLZ fE(qT; rT)z id3r (35)was omitted according to the reasoning of the approxi-mation (17). In the late equations (34), (35) E(qT; rT)denotes:E(qT; rT) =m1eiq1TrT+m2e iq2TrT: (36)Let us consider this neglected part in detail. As it isproportional to qL= !fi+ q2T=2M and therefore explic-itly depends on "f, one can not use completeness relation(14) to calculate its contribution to the total cross sec-tion directly. Before we need to transform it to the formfree of such dependence. It can be done with help of theSchrodinger equation (9)."fiZ f(r)E(qT; rT)z i(r)d3r =Z f(r) f"fE(qT; rT)z   "iE(qT; rT)zg i(r)d3r =Z f(r)[H;E(qT; rT)z] i(r)d3r (37)The commutator in this relation is easily calculated andafter simple algebra we get the following result:(1)fi(q) =  iZ f(r)E(qT; rT)@ i(r)@zd3r +(1)fi(q)(38)(1)fi(q) =iZ f(r)m1eiq1TrTbO1+m2e iq2TrTbO2z i(r)d3r(39)bO1;2=q2T 2qTbp2m1;2;bp =  ir : (40)It is seen, that \large" (nonvanishing at qT= 0) partsof two terms (34) and (38), contributing to a(1)fi, are equaland opposite in sign, so that in the resulting expressionthey cancel each other, leaving only the term with the\correct" behaviour at small qT:a(1)fi= (1)fi(q) (41)The same is valid for any a(n)fi. Applying the Schro-dinger equation (9) to exclude one power of qLfrom theexpression:(n)fi(q) =(iqL)nn!Z f(r)m1neiq1TrT++ ( 1)n+1m2ne iq2TrTzn i(r)d3r ; (42)one can represent it in the form:(n)fi(q) = j(n 1)fi(q) + (n)fi(q) (43)(n)fi(q) =i(iqL)n 1n!Z f(r)m1neiq1TrTbO1++m2ne iq2TrTbO2zn i(r)d3r :(44)So thata(n)fi= (n)fi(q) : (45)5That conrms the qualitative result (29), derived withhelp of continuity equation (27).The remaining "f-dependence of right side of (44) canbe removed by repeated applying of the Schrodingerequation (9), that allows to represent the transition am-plitudes in the form (13).From z-dependence of the integrand in (44) it is easilyto derive, that a(2k)fi= 0 for the odd values of lmfi, anda(2k+1)fi= 0 for even lmfi, where lmfi= (lf  li)  (mf mi), and li, lf, mi, mfare the values of the orbitaland magnetic quantum numbers of the initial i and nalf states (the quantization axis is supposed to be z-axis).Thus \odd" and \even" terms of the expansion (33) donot interfere and therefore in the expansion of the totover the powers of tot=1Xn=0(n); (n)/ n(46)only even powers are present.The structure of the zero order term of this expansionis well established [see (20)]. In view of the above dis-cussion one may be sure that the higher order terms arenumerically negligible and may not be discussed in de-tail. Nevertheless, for completeness of the considerationwe present the expression for contribution of 2-term tothe total cross section which includesa(1)fi2term andthe interference term a(0)fia(2)fi.(2)=  ZU2(qT)W (qT)d2qT+ O(4) ; (47)W (qT) =14m1m2Zz2q4j i(r)j2   j2qTbp i(r)j2ieiqrd3r :The \correct" qT-dependence of the last integrand ex-cludes a possibility of arising some extra -dependence,that could dramatically enhance the contribution of thisterm (like it happened to magterm in [7]). This canbe illustrated by the explicit expression for the case ofthe screened Coulomb potential (22) and the EA groundstate (23):(2)=  8(Z)25Mln2Z1=3me 45: (48)Because numerical smallness of the value 2this termcan be successfully neglected compared to (24) in thepractical applications.This result warrants the usage of the simple expression:tot= 2ZU2(qT) [1  S(qT)]d2qT(49)for the total cross section calculation in Born approxima-tion in [2] and for the Glauber extensions in [15].The authors express their gratitude to professorsS.Mrowczynski, L.Nemenov and D.Trautmann for help-ful discussions.[1] B. Adeva et al., Lifetime measurement of + atoms totest low energy QCD predictions, Proposal to the SPSLC,CERN/SPSLC 95{1, SPSLC/P 284, Geneva, 1995.[2] L.G.Afanasyev and A.V.Tarasov: Yad.Fiz. 59, 2212(1996); Phys.Atom.Nucl. 59, 2130 (1996).[3] G.H.Gillespie, Phys.Rev. A18, 1967 (1978);G.H.Gillespie, M.Inokuti, Phys.Rev. 22A, 2430 (1980).[4] L.S.Dul'yan and A.M.Kotsinyan: Yad.Fiz. 37, 137(1983); Sov.J.Nucl.Phys. 37, 78 (1983).[5] A.S. Pak, A.V. Tarasov: JINR-P2-85-903, Dubna 1985.[6] S.Mrowczynski: Phys.Rev. A33, 1549 (1986).[7] S.Mrowczynski: Phys.Rev. D36, 1520 (1987);K.G.Denisenko and S.Mrowczynski: Phys.Rev. D36,1529 (1987).[8] A.V.Kuptsov, A.S.Pak and S.B.Saakian: Yad.Fiz. 50,936 (1989); Sov.J.Nucl.Phys. 50, 583 (1989).[9] L.G.Afanasyev: JINR-E2-91-578, Dubna 1991.[10] L.G.Afanasyev and A.V.Tarasov: JINR E4-93-293,Dubna, 1993.[11] L.G.Afanasyev: Atomic Data and Nuclear Data Tables,61, 31 (1995).[12] Z.Halabuka et al.: Nucl.Phys. 554, 86 (1999).[13] A.V.Tarasov and I.U.Christova, JINR P2-91-10, Dubna,1991.[14] O.O.Voskresenskaya, S.R.Gevorkyan and A.V.Tarasov:Yad.Fiz. 61, 1628 (1998); Phys.Atom.Nucl. 61, 1517(1998).[15] L.Afanasyev A.Tarasov and O.Voskresenskaya: J.Phys.G 25, B7 (1999).[16] D.Yu.Ivanov and L.Szymanowski: Eur.Phys. J.A5, 117(1999).[17] Amirkhanov I. et al., Phys. Lett. B 452, 155 (1999).[18] Moliere G., Z. Naturforsch. 2A, 3 (1947).