Next-to-next-to-leading order vacuum polarization function of heavy quark near threshold and sum rules for bb system

Transcription

1 0 Septemer 998 Ž Phyic Letter B Next-to-next-to-leading order vacuum polarization function of heavy uar near threhold and um rule for ytem AA Penin a,, AA Pivovarov, a Intitut fur Theoretiche Teilchenphyi UniÕeritat Karlruhe, D-768 Karlruhe, Germany Intitut fur Phyi, Johanne-Gutenerg-UniÕeritat, Staudinger Weg 7, D Mainz, Germany Received 7 May 998 Editor: PV Landhoff Atract A correlator of the vector current of a heavy uar i computed analytically near threhold in the next-to-next-to-leading order in perturative and relativitic expanion that include a, aõ and Õ correction in the coupling contant and velocity of the heavy uar to the nonrelativitic Coulom approximation Baed on thi reult, the numerical value of the -uar pole ma and the trong coupling contant are determined from the analyi of um rule for the F ytem The next-to-next-to-leading correction are found to e of the order of the next-to-leading one 998 Elevier Science BV All right reerved PACS: 465Fy; 38Bx; 38Cy; 55Hx Inufficiency of the ordinary PT for decription the near threhold ehavior of vacuum polarization function wa noted long ago in the context of Coulomic reummation in nonrelativitic QED w, x Recently a coniderale progre ha een made in tudying the near threhold production of heavy uar-antiuar pair within perturation theory of QCD with reummation of threhold ingularitie Both perturative and relativitic correction have een taen into account in the next-to-next-to-leading order in the coupling contant and velocity of the heavy uar to the leading nonrelativitic approximaw3 5 x Thi theoret- tion aed on Coulom potential ical development provide more accurate decription of the heavy uar vacuum polarization function in On leave from Intitute for Nuclear Reearch, Mocow, Ruia the threhold region neceary for uch application a the top uar production wx 6 and the precie uantitative invetigation of the F ytem w7,8 x In the latter cae higher order correction to leading Coulom ehavior in the threhold region are eential oth numerically for extracting the -uar ma and the trong coupling contant wx 3 and ualitatively for jutifying the perturative expanion around Coulom olution The analytical calculation of the next-to-next-to-leading order correction ha not een completed yet though ome reult are availale In thi paper we preent the complete analytical expreion for a correlator of the vector current of Semi-analytical analyi of the complete next-to-next-to-leading order correction to the heavy uar polarization function near the two-particle threhold ha een done in the context of the photon mediated t uar pair production w4,5 x r98r$ - ee front matter 998 Elevier Science BV All right reerved Ž PII: S X

2 44 ( ) AA Penin, AA PiÕoÕaroÕrPhyic Letter B heavy uar near threhold in the next-to-next-to- a leading order reumming all OwŽ rõ n Ž a, a Õ,Õ x term, with Õ eing the heavy uar velocity The correlator i further ued for determination of the ottom uar pole ma m and the trong coupling contant a from um rule for the F ytem We tudy the near threhold ehavior of the polarization function P Ž of the -uar vector current jmgm yg P Ž m n mn Ž H m n ix i dxe ² 0< Tj Ž x j Ž 0 < 0: wx within the nonrelativitic expanion 9 which in the next-to-next-to-leading order read N c 4 GC Ž 0,0, Ž 3 m with ( m y r4 eing a natural energy variale P Ž ChŽ a GŽ 0,0, m near threhold Firt term in racet give the repreentation for the correlator within NRQCD with C Ž a h eing a perturative coefficient matching correlator of relativitic and nonrelativitic vector current The coefficient C Ž a h i computale in full QCD and y now i nown to the econd order in a expanion a a ChŽ a ychcf ChC p F p with C 4 w0xand h 39 4p 35p Ch yz Ž 3 ln y C F p p m 0 F A 3 m y 5 3 8p 79p z Ž 3 ln y C 44 4p y nf TF C C ln Ž A with a defined in MS renormalization cheme w4,5, x Here the group invariant for QCD are CA3, CF4r3, TFr, and ge05776 i the Euler contant, z Ž z i the Riemann z-function, n f i the numer of light flavor, and 0 C r3y4t n r3 The uantity GŽ x, y, A F f i the nonrelativitic Green function Ž GF of the following Schrodinger euation Dx Dx a y y V Ž x V Ž x 3 C m 4m 4p a VŽ x VNAŽ x VBFŽ x, 4p m GŽ x, y, d Ž xyy Ž 3 where V Ž x C ycfarx i the Coulom potential which i uppoed to dominate the whole QCD interaction in the energy region of interet, x< x <, Ž V x ycc a ržm x NA A F i the non-aelian potential of uar-antiuar interaction w x, V Ž x, i the tandard Breit-Fermi potential Ž BF up to the color factor C F containing the uar pin operator, eg w3 x The term V Ž i, i repreent firt and econd order perturative QCD correction to the Coulom potential w4,5x VŽ x VC Ž x Ž C0 C lnž xm, VŽ x VC Ž x Ž C0 C lnž xm C ln Ž xm, Ž 4 where C0 a0g E, C 0, p 0 E 0 Ž 0 E C 4g a g a, 3 C Ž 0a 80 g E, C 40, 3 0 a CAy TFn f, p 4 a 6p y z Ž 3 C y z Ž 3 CTn A F f y y6z Ž 3 CFTFnf 3 A 0 F f 9 T n, 34 0 CAy CTny4C A F f FTFn f 3 3 The econd term in E Ž i generated y the operator of dimenion five in the nonrelativitic ex-

3 ( ) AA Penin, AA PiÕoÕaroÕrPhyic Letter B panion of the vector current Žee, for example, w6 x It contain the GF of the pure Coulom Schrodinger euation w7x at the origin CFam GC Ž 0,0, y ln 4p C a m m F CFam g C y Ž 5 E where C Ž x G X Ž x rg Ž x and G Ž x i the Euler G-function The olution to E Ž 3 can e found within the tandard nonrelativitic perturation theory around the Coulom GF G Ž x, y, C The leading order correction to the Coulom GF at the origin due to D, V w x NA and VBF term are nown analytically 4,5 3 After including thee correction the approximate GF of E Ž 3 at the origin tae the form wx 4 CFam 5 GŽ 0,0, y 4p y 8 C a m m F m F y y ln g m E C a m C CFa CFam C y 6 m 4p CFa 3 CA GC Ž 0,0, 3 m CF Ž 6 where C Ž x C X Ž x Note that in Ref wx 4 the hift of the pectrum of intermediate nonrelativitic Coulom ound tate wa treated exactly ie without expanding of the energy denominator Thi account for a part of the higher order correction We, however, conitently wor in the next-to-nextto-leading order and eep only the econd order term in E Ž 6 Since thi part of the correction i 3 The term V can e fully accounted for the Coulom GF NA ecaue the correponding differential euation i exactly olvale in tandard pecial function Numerically thi i not important for application though relatively mall the difference etween thee two approache i really negligile for the numerical analyi of the um rule The correction DG to E Ž 6 due to the firt iteration of V term of the QCD potential ha een found in Ref wx 3 where the conitent analyi of um rule for ytem in the next-to-leading order ha een performed DGŽ 0,0, a CFam 4p 4p Ý F m m m0 Ž Ž Ž 0 Ž C L C Ž m C my n Ý Ý y FŽ m FŽ n C m n0 Ý Ž Ž 0 F m C m0 Ž E L yg yc Ž m C LC 0 ygel L C Ž 7 m where L ln and y C F a m C F a m F m my Ž m Ž The correction DG to E 6 due to V part of the potential i alo nown wx 3 a CFam Ž DG Ž 0,0, 4p 4p Ž Ž Ý F m Ž m m0 Ž Ž Ž 0 C LC L C Ž m C Ž m Ž C LC IŽ m C my Ý Ý m n0 FŽ m FŽ n n y C LC Ž

4 46 AA Penin, AA PiÕoÕaroÕrPhyic Letter B 435 ( 998) JŽ m,n C H Ž m HŽ n C0 Ý FŽ m m0 Ž 0 Ž Ž C LC L K m C y g C Ž m Ž E Ž C LC LC 0 Ž E yg L L C N C where IŽ m Ž m C Ž m ycž m p y 3 Ž m yž CŽ m g E, n JŽ m,n CŽ y ge n m ŽCŽ ycž m, KŽ m Ž CŽ m ge CŽ m yc Ž m g, E Ž 8 p 3 NŽ ge LygEL L 6 3 In thi paper we complete thee reult y computing the correction DG Ž due to the econd iteration of V term which of the proper Ž next-to-next-to-leading order according to counting of mallne in nonrelativitic QCD with repect to a and Õ The reult read 3 a Ž CFa m Ž DG Ž 0,0, 4p 4p 3 Ý H Ž mž m m0 Ž C0 Ž C Ž m L C my n Ý Ý y C m n0 where Ž C m L y C HŽ m H Ž n C C Ž n L 0 n y C Ž Ž m Ž C my ly Ý Ý Ý m l n0 n Ž lynž my ny Ý Ý Ý m n l0 l Ž nylž ny my Ý Ý Ý n m l0 HŽ m HŽ n HŽ l HŽ m HŽ n HŽ l HŽ m HŽ n HŽ l Ž lž m Ž nž nylž nym y Ž 9 C F a m H m my We are going to decrie the detail of thi rather cumerome calculation elewhere One remar i in order though Becaue ultraviolet divergence in E Ž 6 depend on one ha to match the calculation of thee correction to the calculation of the Wilon coefficient C Ž a ŽE Ž w4,5 x h Such matching i not neceary for the calculation of DG and DG Ži term ecaue their divergent part are independent Thu E ŽŽŽ, 5-9 give the complete analytical expreion for the vacuum polarization function of heavy uar near the two-particle threhold in the next-to-next-to-leading order up to ineential addi- Ž

5 ( ) AA Penin, AA PiÕoÕaroÕrPhyic Letter B tive renormalization contant E ŽŽ 5-9 loo awward and they can e rendered into more readale form y uing C function for expreing ome of the um entering the formulae However for direct numerical analyi of um rule for ytem thi form i mot uitale with repect to applicaility of efficient numerical algorithm of a ymolic ytem Otained formulae are applied to the analyi of the F ytem for extraction of the -uar pole ma m and the coupling contant a The um rule are formulated in the literature w3,8x and we will ue the latet verion wx 3 with correct large n ehavior The moment M n n p n d Mn Ž 4m P Ž n n! d 0 n RŽ d 4m Ž H 0 n of the pectral denity Rp Ž Im P Ž ie are compared with experimental one Ž n 4m RŽ d exp n n Q 0 M H under the aumption of uar-hadron duality The experimental moment Mn exp are generated y the function R Ž which i the normalizad cro ection R Ž Ž y Ž y y e e hadron r e e m m Here Q yr3 i the -uar electric charge Numerical value are otained aically y aturating the experimental moment with the contriution of the firt ix F reonance Žee wx 3 for detail Their leptonic width G and mae M Ž 6 are nown with good accuracy w8x Ž n 4m 9p 6 G exp n n Q aqedž m M M Ý R Ž d H n 0 The ret of the pectrum eyond the reonance region for energie larger than fž GeV Žcon- 4 In Ref w4,5x the correction to the Coulom GF due to V i Ž i, term of the potential were treated numerically for complex value of energy far from the real axi 0 tinuum contriution lie far from threhold and i afely approximated y the ordinary PT expreion for the theoretical pectral denity, o there R Ž f R Ž The influence of the continuum on high moment i almot negligile numerically and in any cae under trict control 5 Electromagnetic coupling contant i renormalized to the energy of order m with the reult a Ž m 07a w8 x QED We wor with moment for 0- n- 0 that imultaneouly guarantee the mallne of oth the continuum contriution and the nonperturative power correction due to the gluonic condenate wx 8 The firt one i not well nown experimentally and ha to e uppreed to mae reult independent of 0 The econd one hould e mall ecaue the value of gluonic condenate Žand higher order condenate i not nown well numerically The normalization point m m i ued throughout the com- putation 6 For a lower cale oth the hard and oft correction ecome large and the peturative erie for the moment i trongly divergent We found that at m;m the m dependence of the reult i mini- mal which i a olid indication that at thi point the higher order correction are alo mall The reult of the fit i až m 0"00, or a Ž M 08" 0006 Z The um rule are much more enitive to the -uar ma than to the trong coupling contant o it i intructive to fix a Ž M Z 08 to the world average value w8x and then to extract m Ž n In thi way we otain the following etimate for the mean value over the conidered range of n m 478 GeV Thi value i in a good agreement with the reult of 5 The expreion for the firt few moment of the pectral denity are now availale in ordinary perturation theory with a accuracy w9 x, however, they cannot e ued in theoretical formula for um rule directly ecaue the pectrum i well nown experimentally only for energie cloe to threhold due to exitence of harp reonance while the contriution of the continuum to thee low moment i large in comparion with the reonance contriution 6 We wor trictly in the next-to-next-to-leading order approximation and, therefore, ue the ame normalization point for oft and hard correction in contrat to w4,5x where different normalization point were choen for thee two part

6 48 ( ) AA Penin, AA PiÕoÕaroÕrPhyic Letter B the firt order analyi wx 3 where at a Ž M Z 08 we otained m 475 GeV Note that the optimization procedure w0x wa ued to improve convergence of perturation theory in the previou analyi wx 3 A we ee thi procedure turn out to e a powerful tool to etimate the higher order contriution For comparion, the leading order reult i m 470 GeV and in the next-to-leading approximation without optimization one get m 47 GeV Main uncertaintie of numerical value for conidered parameter tem from the ame ource that were identified in Ref wx 3 The error coming from n ditriution for the ma at fixed value of the coupling contant i aout "05% for 0-n-0 The m dependence for mm " GeV Ž where thi de- pendence i minimal introduce another "05% of uncertainty Thu our final etimate of the ottom uar pole ma i m 478"004 GeV Note that the uncertainty originated from the n and m dependence i not reduced in comparion with the next-to-leading order Thi mean that the contriution of the higher order correction which ha to cancel n and m dependence of the reult i till important Let u emphaize that the convergence of the perturation theory for the vacuum polarization function of heavy uar near threhold i not fat We have found the next-to-next-to-leading order correction to e of the order of the next-to-leading one Furthermore, in the cae of -uar the correction due to the perturative modification of the Coulom intantaneou potential Žie related to DG Ži and DG term dominate the total correction in the next-to-leading and next-to-next-to-leading order Incluion of thee correction i uite important for conitent analyi of um rule for the F ytem To conclude we have contructed an expreion for the vacuum polarization function of the vector current of a heavy uar near threhold It i completely analytic in the next-to-next-to-leading order in perturative and relativitic expanion up to a, a Õ and Õ correction The polarization function wa ued for determination of the -uar pole ma and the coupling contant from um rule for the F ytem that are aturated y contriution near threhold In fact, there i no much hope for improving our reult: next order approximation eem to e too complicated for analytical treatment within the regular perturation theory for NRQCD The analyi howed a remarale taility with repect to the next-to-leading one upplied with an optimization procedure in a variational pirit Having in mind the coniderale technical difficulty of computing next approximation and recognizing the neceity of improving the theoretical prediction in view of new high uality experimental data we thin that the next tep in the near future will e connected with optimization of the preent approximation We than JH Kuhn for upport, encouragement, and dicuion AA Penin gratefully acnowledge dicuion with K Melniov Thi wor i partially upported y Volwagen Foundation under contract No Ir736 AA Pivovarov i upported in part y the Ruian Fund for Baic Reearch under contract No and The wor of AA Penin i upported in part y the Ruian Fund for Baic Reearch under contract Reference wx MA Braun, ZhETP Lett 7 Ž wx R Barieri, P Chritillin, E Remiddi, Phy Rev A 8 Ž wx 3 JH Kuhn, AA Penin, AA Pivovarov, Preprint TTP-98-0, hep-phr wx 4 AH Hoang, T Teuner, Preprint UCSDrPTH 98-0, hepphr wx 5 K Melniov, A Yelhovy, Preprint TTP-98-0, hepphr wx 6 VS Fadin, VA Khoze, Pi ma Zh Ep Teor Fiz 46 Ž ; Yad Fiz 48 Ž ; W Kwong, Phy Rev D 43 Ž ; MJ Straler, ME Pein, Phy Rev D 43 Ž ; M Jezae, JH Kuhn, T Teuner, Z Phy C 56 Ž ; Y Sumino, K Fujii, K Hagivara, H Murayama, C-K Ng, Phy Rev D 47 Ž ; K Fujii, T Matui, Y Sumino, Phy Rev D 50 Ž wx 7 VA Noviov et al, Phy Rev Lett 38 Ž ; VA Noviov et al, Phy Rep C 4 Ž 978 ; MB Volohin, Yad Fiz 36 Ž 98 47; MB Volohin, Yu M Zaitev, Up Fiz Nau 5 Ž wx 8 M Volohin, Int J Mod Phy A 0 Ž wx 9 WE Cawell, GE Lepage, Phy Lett B 67 Ž w0x G Kallen, A Sary, K Dan Viden Sel Mat-Fi Medd 9 Ž 955 N7, wx AH Hoang, Phy Rev D 56 Ž ; AH Hoang, JH Kuhn, T Teuner, Nucl Phy B 45 Ž ; A Czar-

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