The effective field theory approach allows a rigorous disentangling of high
and low energy effects in the heavy quarkonium dynamics. Focusing in particular
on the spectrum, we describe the nature of the non-perturbative effects and
discuss our present knowledge of them.Comment: Invited talk presented at the Fifth Workshop on Quantum
  Chromodynamics, Villefranche-sur-Mer, France, 3-7 January 2000; 9 pages,
  sprocl.st

Brambilla, N.

Vairo, A.

English

arXiv

The effective field theory approach allows a rigorous disentangling of high and low energy effects in the heavy quarkonium dynamics. Focusing in particular on the spectrum, we describe the nature of the non-perturbative effects and discuss our present knowledge of them.The effective field theory approach allows a rigorous disentangling of high and low energy effects in the heavy quarkonium dynamics. Focusing in particular on the spectrum, we describe the nature of the non-perturbative effects and discuss our present knowledge of them

Brambilla, Nora

Vairo, Antonio

CERN Document Server

arXiv:hep-ph/0004192v1  20 Apr 2000The Role of the QCD Vacuum in the Heavy-Quark Bound StateDynamicsaNORA BRAMBILLA and ANTONIO VAIROInstitut fu¨r Theorestische Physik, Universita¨t HeidelbergPhilosophenweg 16, D-69120 Heidelberg, GermanyandTheory Division CERN, 1211 Geneva 23, SwitzerlandE-mail: nora.brambilla@cern.ch, antonio.vairo@cern.chThe effective field theory approach allows a rigorous disentangling of high and lowenergy effects in the heavy quarkonium dynamics. Focusing in particular on thespectrum, we describe the nature of the non-perturbative effects and discuss ourpresent knowledge of them.1 IntroductionIt is well known that the complex structure of the QCD vacuum can, in someregimes, be parametrized by local vacuum condensates, i.e. the expectationvalues of operators where all the perturbative vacuum fluctuations are takenout at the best of our ability 1. The first attempt to calculate the effect ofnon-perturbative vacuum condensates on the energy levels of heavy quarkoniawas performed in 2. The leading contribution (in αs and ΛQCD, the scale ofnon-perturbative physics) of the vacuum condensates to Coulombic n, l heavyquarkonium states reads 2:δEV−Lnl = mǫnn6π2G2(mCFαs)4, (1)where ǫn is a (known) number of order 1 and G2 ≡ 〈(αs/π)Faµν(0)Faµν(0)〉 isthe gluon condensate. The above expression for the corrections to Coulombicenergy levels displays two relevant characteristics: 1) it is a correction of non-potential type, like the Lamb shift in QED; 2) it grows like ∼ n6, thus being outof control for levels beyond the ground state. It was soon realized 3,4 that thestrong growth in n could be corrected to some extent by considering non-localgluon condensatesG2(x) ∼〈αsπF aµν(x)φ(x, 0)adjab Fb µν(0)〉, (2)aTalk given by N.B. at the Fifth Workshop on Quantum Chromodynamics, Villefranche-sur-Mer, France, 3-7 January 2000.1where φ(x, 0)adj is the Schwinger line (in the adjoint representation) connectingx with 0. These were understood as due to the presence of a fluctuating gluonicbackground with a characteristic time length Tg ∼ Λ−1QCD. It is apparent thatthe local condensates stem from an expansion of the non-local ones in cases inwhich the correlation length is large with respect to the other physical scalesof the system.Heavy quarkonium is a non-relativistic bound system. Besides ΛQCD, it ischaracterized by at least three hierarchically ordered scales, mv2 ≪ mv ≪ m,where m is the mass of the heavy quark and v its velocity. Therefore, itis only under the condition ΛQCD ≪ mv2 that the use of local condensatesand, hence, the Voloshin–Leutwyler formula (1) can be justified. In the otherregimes where heavy quarkonium states can sit, the non-perturbative dynamicswill be encoded into more extended objects: non-local condensates and Wilsonloop operators. This is the case for higher quarkonium levels (n > 1).In the following we will consider non-perturbative effects in heavy quarko-nium systems in different kinematic regimes, in an effective field theory context.This approach has not only the considerable practical advantage of disentan-gling the different dynamical scales of the system, but also the conceptuallyrelevant feature of disentangling perturbative effects from non-perturbativeones. This allows us to fully exploit the predictive power of QCD.2 Local and non-local condensates: ΛQCD ≪ mvFor the lower lying quarkonium states, it can be expected that the inverse ofthe typical size of the system is larger than ΛQCD. If this condition is fulfilled,both scales m and mv can be integrated out perturbatively from QCD, leadingto an effective field theory where only the degrees of freedom of order ΛQCDor mv2 remain dynamical. This effective field theory is known as potentialNRQCD, pNRQCD 5,6.The infrared sensitivity of the quark–antiquark static potential at threeloops 7 signals that it may become sensitive to non-perturbative effects if thenext relevant scale after mv is ΛQCD. Indeed, in the situation mv ≫ ΛQCD ≫mv2, the leading non-perturbative contribution (in αs and in the multipoleexpansion) to the static potential reads 6,8V0(r)non−pert = −ig2NcTFr23∫∞0dt e−itCAαs/(2r)〈Ea(t)φ(t, 0)adjab Eb(0)〉(µ), (3)where r is the quark–antiquark distance. This term explicitly cancels, up tothe considered order, the dependence of the perturbative static potential on theinfrared scale µ. It is interesting to note that the leading contribution in the2ΛQCD/mv2 expansion of V non−pert (obtained by putting the exponential equalto 1) cancels the order Λ3QCDr2 renormalon that affects the static potential(the leading-order renormalon, of order ΛQCD, cancels against the pole mass).Therefore, also in renormalon language, the above operator is the relevant non-perturbative contribution to the static potential in the considered kinematicsituation.If ΛQCD <∼ mv2 the static potential is purely perturbative and its explicitdependence on the infrared scale µ is reabsorbed in a physical observable bynon-potential contributions. In the specific case of the quarkonium energylevels up to order α5s lnµ, these contributions are9δEn,l,j = −ig23NcTF∫∞0dt 〈n, l|reit(En−Ho)r|n, l〉〈Ea(t)φ(t, 0)adjab Eb(0)〉(µ), (4)En ≡ −mC2Fα2s4n2, Ho ≡p2m+12Ncαsr,where |n, l〉 are the Coulomb wave functions. It is worth while to notice that,if ΛQCD ≪ mv2, the scale mv2 can be integrated out perturbatively from theabove formula and the non-perturbative contributions reduce to the Voloshin–Leutwyler formula (1), as can easily be seen by recognizing that:mǫnn6(mCFαs)4=13NcTF〈n, l∣∣∣∣r 1En −Ho r∣∣∣∣n, l〉.The above outline leads to the following conclusion. For quarkonium ofa typical size smaller than 1/ΛQCD, the most relevant operator of the non-perturbative dynamics is the bilocal gluon condensate 〈Ea(t)φ(t, 0)adjab Eb(0)〉,which belongs to the class of non-local gluon condensates considered in theintroduction. In the following section we will discuss our present knowledge ofit.2.1 The non-local condensate 〈F aµν(x)φ(x, 0)adjab Fbλρ(0)〉The correlator 〈F aµν(x)φ(x, 0)adjab Fbλρ(0)〉 is perturbatively known at the next-to-leading order in αs10. However, here we are interested in its non-perturbati-ve behaviour. Different parametrizations have been proposed 12,13,14,15,16.Because of its Lorentz structure, the correlator is in general described by twoform factors. A convenient choice of these consists in the chromoelectric andchromomagnetic correlators:〈Ea(x)φ(x, 0)adjab Eb(0)〉 , 〈Ba(x)φ(x, 0)adjab Bb(0)〉.3The strength of the correlators is of the order of the gluon condensate. In thelong range (x2 →∞) they fall off exponentially (in the Euclidean space) withsome typical correlation lengths. In the following we will concentrate on thesecorrelation lengths.The lattice calculation 15, using cooling techniques, obtains the same cor-relation length (Tg) for both form factors, and this isTg = 0.34± 0.02± 0.03 fm (4 flavours, am = 0.01), (5)Tg = 0.22± 0.01± 0.02 fm (quenched). (6)The less accurate, but traditional (quenched) lattice calculation done in 16obtains two different correlation lengths for the chromoelectric (TEg ) and thechromomagnetic correlators (TBg ):TEg 6= TBg ≃ 0.1– 0.2 fm (quenched). (7)Finally, a recent sum-rule estimation 14 obtains TEg < TBg . The sum ruleturns out not to be stable for the chromoelectric correlator, while for thechromomagnetic correlation length it givesTgB = 0.13+0.05−0.02 fm (3 flavours), (8)TgB = 0.11+0.04−0.02 fm (quenched). (9)The correlation lengths TEg and TBg have a precise physical interpretation.Their inverses correspond to the masses of the lowest-lying vector and pseu-dovector static quark–gluon hybrids, respectively. This can be explicitly seenin the short-range limit, x → 0, where the hybrids (in this case also calledgluelumps) operators can be explicitly constructed 6,17. The suitable effectivefield theory is pNRQCD in the static limit 6. Gluelump operators are of thetype Tr{OH}, where O = OaT a corresponds to a quark–antiquark state in theadjoint representation (the octet) and H = HaT a is a gluonic operator. Bymatching the QCD static hybrid operators into pNRQCD, we get the staticenergies (also called potentials) of the gluelumps. At leading order in themultipole expansion, they readVH(r) = Vo(r) +1THg, (10)〈Ha(t)φ(t, 0)adjab Hb(0)〉non−pert. ≃ h e−it/THg + . . . .Since hybrids are classified in QCD according to the representations of D∞,h,while in pNRQCD, where we have integrated out the length r, their classifi-cation is done according to the representations of O(3)× C, the static hybrid4short-range spectrum is expected to be more degenerate than the long-rangeone 18,17. The lattice measure of the hybrid potentials done in 19 confirmsthis feature. In 6 it has been shown that the quantum numbers attribution ofpNRQCD to the short-range operators, and the expected O(3)×C symmetryof the effective field theory match the lattice measurements. By using only Eand B fields and keeping only the lowest-dimensional representation we mayidentify the operator H for the short-range hybrids called Σ+′g (and Πg) withr · E (and r × E) and the operator H for the short-range hybrids called Σ−u(and Πu) with r ·B (and r×B). Hence, the corresponding static energies forsmall r areVΣ+′g ,Πg(r) = Vo(r) +1TEg, VΣ−u ,Πu(r) = Vo(r) +1TBg.The lattice measure of19 shows that, in the short range,VΣ+′g ,Πg(r)>VΣ−u ,Πu(r).This supports the sum-rule prediction14 that the pseudovector hybrid lies lowerthan the vector one, i.e. TEg < TBg .3 Wilson loop operators: ΛQCD ∼ mvFor higher quarkonium levels, ΛQCD is expected to be comparable withmv. Wecannot match into pNRQCD perturbatively, since the scale associated to thequarkonium size r is already non-perturbative. The relevant non-perturbativedynamics is therefore contained in more extended objects than (local or non-local) gluon condensates: Wilson loops and field insertions on these.In particular, disregarding effects due to scales lower than ΛQCD, the staticpotential is given by 20V0(r) = limT→∞iTln〈W✷〉, (11)where W✷ is the static Wilson loop of size r × T and 〈 〉 means an averageover the gauge fields. Lattice studies tell us that at distances r ≃ 1/ΛQCD thepotential is no longer Coulombic but rises linearly (V0(r) ≃ σr). Higher-ordercorrections in the 1/m expansion have been calculated over the years 21 andare given by field strength insertions on the Wilson loop. For instance thenext-to-leading potential in the 1/m expansion is 22V1m=1mlimT→∞(−g24T∫ T/2−T/2dt∫ T/2−T/2dt′|t− t′|[〈〈E(t) · E(t′)〉〉✷ (12)−〈〈E(t)〉〉✷ · 〈〈E(t′)〉〉✷]), (13)5where 〈〈 〉〉✷ means a normalized gauge average in the presence of the staticWilson loop.Wilson-loop operators of the above type may be interpreted as a superpo-sition of states, describing gluonic excitations between static sources 22. Theyhave been so far evaluated only inside QCD vacuum models or by lattice sim-ulations (for some reviews, see 23 and also 24).3.1 The stochastic expansionWe can now ask if some relation can be established between the Wilson loop(and field strength insertions on it) and the non-local gluon condensates dis-cussed above. In fact we can express the Wilson loop as a formal expansionin terms of gluonic correlation functions by means of the so-called stochasticexpansion 11,12ln〈W✷〉 =∞∑n=0(ig)nn!∫S(✷)dSµ1ν1(u1) · · · dSµnνn(un)〈φ(0, u1)×Fµ1ν1(u1)φ(u1, 0) · · ·φ(0, un)Fµnνn(un)φ(un, 0)〉cum, (14)where S(✷) denotes a surface whose contour the rectangular Wilson loop. Thecumulants 〈 〉cum are defined as〈φ(0, u1)F (u1)φ(u1, 0)〉cum = 〈φ(0, u1)F (u1)φ(u1, 0)〉 = 0,〈φ(0, u1)F (u1)φ(u1, u2)F (u2)φ(u2, 0)〉cum =〈φ(0, u1)F (u1)φ(u1, u2)F (u2)φ(u2, 0)〉−〈φ(0, u1)F (u1)φ(u1, 0)〉 〈φ(0, u2)F (u2)φ(u2, 0)〉 . . . =〈F (u1)φ(u1, u2)adjF (u2)〉,· · · · · ·It is important to realize that the expansion of Eq. (14) is substantially differ-ent with respect to the expansions in 1/m and r, which led to the constructionof the low-energy effective field theories discussed above. Those expansionswere justified by the dynamics of the system under study, and each term ofit is, indeed, suppressed by powers of the highest dynamical scale left dividedby the scale (which is larger) that has been integrated out. Instead, from apower counting point of view, each term of the expansion (14) is of the samesize (for instance each term of the series is in general expected to contributeto the string tension σ). As soon as we truncate the series, e.g. up to thebilocal cumulant (the first non-vanishing one), we introduce an uncontrolledapproximation and define a model. This model is known as the model of the6stochastic vacuum12. Indeed, this model turns out to be quite successful in thestudy of processes that involve quark systems that may be described by almoststatic Wilson loops (for some reviews, see 25). Let us only mention that themodel predicts, in agreement with the lattice data, a long-range linear staticpotential with slope σ ∼ TE 2g G2. It would be highly desirable to have a fieldtheoretical justification of the expansion (14); however, such a justification ismissing up to now (for a recent investigation on higher cumulants, see 26).4 ConclusionsIn the previous sections we have shown how the non-perturbative QCD vacuumenters the dynamics of heavy quarkonium. As much as the non-perturbativescale ΛQCD is bigger than the dynamical scales of the non-relativistic system(m, mv and mv2), as extended the relevant non-perturbative operators are.In the situation ΛQCD ≪ mv2 these operators reduce to local condensates,i.e. some numbers. This is the most favourable situation for a theoreticalinvestigation. The bottomonium and charmonium ground states have beeninvestigated in this framework (for a recent review, see27). In the limiting caseof tt¯ threshold production the non-perturbative corrections can be neglectedand the investigation is completely accessible to perturbative QCD 28. In thesituation mv2 <∼ ΛQCD ≪ mv the non-perturbative physics is encoded intonon-local condensates. The dominant one is the bilocal gluon condensate.As we have discussed above it essentially depends on its strength, i.e. thegluon condensate, and on some correlation lengths, which can be related to themasses of the lowest-lying (static) quark–gluon hybrid resonances. Althoughthis situation seems quite interesting, it has been poorly investigated in heavyquarkonium phenomenology, and mainly in the framework of models 8,4. Themain reason is, in our opinion, that, while the quarkonium ground state seemsto be accessible by a purely perturbative treatment (plus local condensates)and the potential describing higher excited quarkonium states seems entirelydominated by non-perturbative effects, it is, a priori, not clear to which statesthe situation mv2 <∼ ΛQCD ≪ mv is applicable. We will come back to thispoint at the end of this section. Finally, in the situation ΛQCD ∼ mv therelevant non-perturbative objects are Wilson-loop operators. A lattice studyof them and the subsequent quarkonium spectroscopy have been done in 29.The outlined study is rigorous and allows a systematic disentanglementof the high from the low energy scales of the heavy quarkonium system understudy. In this specific sense also perturbative and non-perturbative effects turnout to be disentangled.We conclude by mentioning a somehow delicate point, connected with the7scales of heavy quarkonium systems: the difficulty to state a priori in whichkinematic situation a particular quarkonium state is. This is mainly due to thefact that the scalesmv andmv2 are not so well defined, nor so widely separatedthat different kinematic situations cannot overlap in a preliminary analysis (fora related discussion on the energy scales also in quarkonium physics, see 30).Therefore, there are situations in which scales can only be fixed a posteriori,i.e. by assuming a particular situation and by checking that the final result isconsistent with it and with the experimental data.AcknowledgementsN.B. would like to thank for the invitation Jan Rafelski, chairman and orga-nizer of the Stochastic Vacuum Session and the organizers of the conference,Herbert Fried, Bernd Mu¨ller and Yves Gabellini. N.B. is grateful to the theorygroup of the University of Milan for supporting her participation to the con-ference. N.B. and A.V. gratefully acknowledge the Alexander von HumboldtFoundation.References1. M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B 147, 385and 448 (1979).2. M. B. Voloshin, Nucl. Phys. B 154, 365 (1979); H. Leutwyler, Phys.Lett. B 98, 447 (1981).3. D. Gromes, Phys. Lett. B 115, 482 (1982); A. Kramer, H. G. Dosch andR. A. Bertlmann, Phys. Lett. B 223, 105 (1989).4. U. Marquard and H. G. Dosch, Phys. Rev. D 35, 2238 (1987); Yu. A.Simonov, S. Titard and F. J. Yndura´in, Phys. Lett. B 354, 435 (1995).5. A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. 64, 428 (1998).6. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B 566, 275(2000).7. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Phys. Rev. D 60, 091502(1999).8. I. I. Balitsky, Nucl. Phys. B 254, 166 (1985).9. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Phys. Lett. B 470, 215(1999).10. M. Eidemu¨ller and M. Jamin, Phys. Lett. B 416, 415 (1998).11. N. G. Van Kampen, Phys. Rep. C 24, 171 (1976).12. H. G. Dosch, Phys. Lett. B 190, 177 (1987); H. G. Dosch and Yu. A.Simonov, Phys. Lett. B 205, 339 (1988).813. H. G. Dosch, E. Ferreira and A. Kramer, Phys. Rev. D 50, 1994 (1992).14. H. G. Dosch, M. Eidemu¨ller and M. Jamin, Phys. Lett. B 452, 379(1999).15. M. D’Elia, A. Di Giacomo and E. Meggiolaro, Phys. Lett. B 408, 315(1997).16. G. S. Bali, N. Brambilla and A. Vairo, Phys. Lett. B 421, 265 (1998).17. N. Brambilla, hep-ph/9909211.18. M. Foster and C. Michael, Phys. Rev. D 59, 094509 (1999).19. K. J. Juge, J. Kuti and C. Morningstar, hep-lat/9809015.20. K. G. Wilson, Phys. Rev. D 10, 2445 (1974); L. Susskind, Les Houcheslectures (1976); W. Fischler, Nucl. Phys. B 129, 157 (1977).21. E. Eichten and F. L. Feinberg, Phys. Rev. D 23, 2724 (1981); D. Gromes,Z. Phys. C 26, 401 (1984); A. Barchielli, E. Montaldi and G. M. Prosperi,Nucl. Phys. B 296, 625 (1988); (E) ibid. 303, 752 (1988); A. Barchielli,N. Brambilla and G. Prosperi, Nuovo Cimento 103, 59 (1990); N. Bram-billa and A. Vairo, Nucl. Phys. Proc. Suppl. 74, 201 (1999).22. N. Brambilla, A. Pineda, J. Soto and A. Vairo, hep-ph/0002250.23. N. Brambilla and A. Vairo, hep-ph/9904330; G. S. Bali, hep-ph/0001312.24. N. Brambilla and A. Vairo, Phys. Rev. D 55, 3974 (1997); N. Brambilla,hep-ph/9809263.25. H. G. Dosch, Prog. Part. Nucl. Phys. 33, 121 (1994); O. Nachtmann,Schladming lectures (Springer, Berlin, 1996); H. G. Dosch, in these pro-ceedings; see also E. Berger, in these proceedings.26. W. Kornelis and H. G. Dosch, hep-th/0003207.27. F. J. Yndura´in, hep-ph/9910399.28. A. Hoang, in these proceedings.29. G. S. Bali, K. Schilling and A. Wachter, Phys. Rev. D 56, 2566 (1997).30. E. Shuryak, hep-ph/9911244.9

The Role of the QCD Vacuum in the Heavy-Quark Bound State Dynamics

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