Vacuum Polarization Function of Heavy Quark near Threshold and Sum Rules for b b System in the Next-to-Next-to-Leading Order

Transcription

1 Vacuum Polarization Function of Heavy Quark near Threshold and Sum Rules for System in the Next-to-Next-to-Leading Order A. A. Penin Institut für Theoretische Teilchenphysik Universität Karlsruhe D-7618 Karlsruhe, Germany and A. A. Pivovarov Institut für Physik, Johannes-Gutenerg-Universität Staudinger Weg 7, D Mainz, Germany Astract A correlator of the vector current of a heavy quark is computed analytically near threshold in the next-to-next-to-leading order in perturative and relativistic expansion that includes α s, α s v and v corrections in the coupling constant and velocity of the heavy quark to the nonrelativistic Coulom approximation. Based on this result, the numerical values of the -quark pole mass and the strong coupling constant are determined from the analysis of sum rules for the Υ system. The next-to-next-to-leading corrections are found to e of the order of the next-to-leading ones. Insufficiency of the ordinary PT for description the near threshold ehavior of vacuum polarization function was noted long ago in the context of On leave from Institute for Nuclear Research, Moscow, Russia 1

2 Coulomic resummation in nonrelativistic QED [1, ]. Recently a considerale progress has een made in studying the near threshold production of heavy quark-antiquark pair within perturation theory of QCD with resummation of threshold singularities. Both perturative and relativistic corrections have een taken into account in the next-to-next-to-leading order in the coupling constant and velocity of the heavy quark to the leading nonrelativistic approximation ased on Coulom potential [3, 4, 5]. This theoretical development provides more accurate description of the heavy quark vacuum polarization function in the threshold region necessary for such applications as the top quark production [6] and the precise quantitative investigation of the Υ system [7, 8]. In the latter case higher order corrections to leading Coulom ehavior in the threshold region are essential oth numerically for extracting the -quark mass and the strong coupling constant [3] and qualitatively for justifying the perturative expansion around Coulom solution. The analytical calculation of the next-to-next-to-leading order corrections has not een completed yet though some results are availale 1. In this paper we present the complete analytical expression for a correlator of the vector current of heavy quarks near threshold in the next-to-next-toleading order resumming all O[α s /v n α s,α s v, v ] terms, with v eing the heavy quark velocity. The correlator is further used for determination of the ottom quark pole mass m and the strong coupling constant α s from sum rules for the Υ system. We study the near threshold ehavior of the polarization function Πs of the -quark vector current j μ = γ μ qμ q ν g μν q Πq =i dxe iqx 0 Tj μ xj ν 0 0 within the nonrelativistic expansion [9] which in the next-to-next-to-leading order reads Πs = N c C m h α s G0, 0,k+ 4 k G 3 m C 0, 0,k 1 with k = m s/4 eing a natural energy variale near threshold. First term in rackets gives the representation for the correlator within NRQCD 1 Semi-analytical analysis of the complete next-to-next-to-leading order corrections to the heavy quark polarization function near the two-particle threshold has een done in the context of the photon mediated t quark pair production [4, 5].

3 with C h α s eing a perturative coefficient matching correlators of relativistic and nonrelativistic vector currents. The coefficient C h α s is computale in full QCD and y now is known to the second order in α s expansion with C 1 h = 4 [10] and C h α s =1 C 1 h C F C h = 39 4 ζ3 + 4π 3 α s π + C h C F αs π 35π ln C F π 179π ζ3 + ln C A π n f T F + β 0 + π 3 C F + π C m A ln μ with α s defined in MS renormalization scheme [4, 5, 11]. Here the group invariants for QCD are C A =3,C F =4/3, T F =1/, and γ E = is the Euler constant, ζz istheriemannζ-function, n f is the numer of light flavors, and β 0 =11C A /3 4T F n f /3. The quantity Gx, y,kisthe nonrelativistic Green function GF of the following Schrödinger equation Δ x m Δ x 4m 3 + V C x+ α s 4π V 1x+ αs V x 4π +V NA x+v BF x, s+ k Gx, y,k=δx y 3 m where V C x = C F α s /x is the Coulom potential which is supposed to dominate the whole QCD interaction in the energy region of interest, x = x, V NA x = C A C F αs /m x is the non-aelian potential of quark-antiquark interaction [1], V BF x, s is the standard Breit-Fermi potential up to the color factor C F containing the quark spin operator s, e.g. [13]. The terms V i i =1, represent first and second order perturative QCD corrections to the Coulom potential [14, 15] V 1 x =V C xc C 1 1 lnxμ, 3

4 where a = V x =V C xc 0 + C 1 lnxμ+c ln xμ, 4 C0 1 = a 1 +β 0 γ E, C1 1 =β 0, π C0 = 3 +4γ E β0 +β 1 +β 0 a 1 γ E + a, C1 =β 1 +β 0 a 1 +8β0γ E, C =4β0, a 1 = 31 9 C A π π ζ3 9 T F n f, ζ3 C F T F n f + C A ζ3 C A T F n f 0 9 T F n f β 1 = 34 3 C A 0 3 C AT F n f 4C F T F n f. The second term in eq. 1 is generated y the operator of dimension five in the nonrelativistic expansion of the vector current see, for example, [16]. It contains the GF of the pure Coulom Schrödinger equation [17] at the origin G C 0, 0,k= C F α s m 4π +γ E +Ψ 1 1 C F α s m k, k +ln C F α s m k μ where Ψ 1 x =Γ x/γx andγx is the Euler Γ-function. The solution to eq. 3 can e found within the standard nonrelativistic perturation theory around the Coulom GF G C x, y,k. The leading order corrections to the Coulom GF at the origin due to Δ, V NA and V BF terms are known analytically [4, 5]. After including these corrections the approximate GF of eq. 3 at the origin takes the form [4] G0, 0,k= C F α s m 1 5 4π 8 k m k + 1 k C F α s m m The term V NA can e fully accounted for the Coulom GF ecause the corresponding differential equation is exactly solvale in standard special functions. Numerically this is not important for applications though. 4 5

5 k ln + γ E +Ψ 1 1 C F α s m + 11 C F α s k Ψ 1 C F α s m 6 μ k 16 m k + 4π C F α s 1+ 3 C A G 3 m C 0, 0,k C F where Ψ x = Ψ 1x. Note that in ref. [4] the shift of the spectrum of intermediate nonrelativistic Coulom ound states was treated exactly i.e. without expanding of the energy denominators. This accounts for a part of the higher order corrections. We, however, consistently work in the next-tonext-to-leading order and keep only the second order terms in eq. 6. Since this part of the corrections is relatively small the difference etween these two approaches is really negligile for the numerical analysis of the sum rules. The correction ΔG 1 to eq. 6 due to the first iteration of V 1 term of the QCD potential has een found in ref. [3] where the consistent analysis of sum rules for system in the next-to-leading order has een performed ΔG 1 0, 0,k= α s C F α s m F mm +1 C0 1 +L k +Ψ 1 m +C1 1 4π 4π m=0 m 1 m=1 n=0 F mf n n +1 m n C1 1 + where L k =ln μ k and m=0 +L k C γ E L k + 1 L k F m C 1 0 +L k γ E Ψ 1 m +1C 1 1 C1 1 F m = C F α s m m +1 C F α s m 1. m +1k k The correction ΔG to eq. 6 due to V part of the potential is also known [3] ΔG 0, 0,k= αs C F α s m F m m +1 C0 4π 4π + L kc1 + L k C m=0 +m +1Ψ 1 m + C 1 +L kc + ImC 7 5

6 m 1 + F mf n n +1 C m=1 n=0 m n 1 +L k C + Jm, nc 8 + F m C0 + L kc1 +L k + KmC γ E +Ψ 1 m +1 C1 +L kc m=0 +L k C0 γ + E L k + 1 L k C1 + NkC where Im =m +1 Ψ 1m + Ψ m ++ π 3 m +1 Jm, n = n +1 m n Ψ 1 m +1+γ E, Ψ 1 m n 1 n +1 +γ E + m +1 m n Ψ 1m n +1 Ψ 1 m +1, Km =Ψ 1 m +1+γ E +Ψ m +1 Ψ 1 m +1+γ E, Nk = γ E + π 6 L k γ E L k L3 k. In this paper we complete these results y computing the correction ΔG 1 due to the second iteration of V 1 term which of the proper next-to-nextto-leading order according to counting of smallness in nonrelativistic QCD with respect to α s and v. The result reads ΔG 1 0, 0,k= αs C F α s m 3 H 3 mm +1 4π 4π k m=0 C 1 0 +Ψm ++L k C 1 1 m 1 n +1 m=1 n=0 m n C1 1 H mhn C 10 + Ψm ++L k 1 1 C1 1 m n +HmH n C 10 + Ψn ++L k 1 n +1 C1 1 9 m nm +1 6

7 +C m 1 l 1 m= l=1 n=0 m 1 n 1 m= n=1 l=0 n 1 m 1 n= m=1 l=0 n +1 HmHnHl l nm n l +1 HmHnHl n lm n l +1m +1 HmHnHl n +1n ln m where Hm = m +1 C F α s m 1. k We are going to descrie the details of this rather cumersome calculation elsewhere. One remark is in order though. Because ultraviolet divergences in eq. 6 depend on k one has to match the calculation of these corrections to the calculation of the Wilson coefficient C h α s eq. [4, 5]. Such matching is not necessary for the calculation of ΔG 1 and ΔG i terms ecause their divergent parts are k independent. Thus eqs. 1, 5-9 give the complete analytical expressions for the vacuum polarization function of heavy quarks near the two-particle threshold in the next-to-next-to-leading order up to inessential additive renormalization constant 3. Eqs. 5-9 look awkward and they can e rendered into more readale form y using Ψ functions for expressing some of the sums entering the formulae. However for direct numerical analysis of sum rules for system this form is most suitale with respect to applicaility of efficient numerical algorithms of a symolic system. Otained formulae are applied to the analysis of the Υ system for extraction of the -quark pole mass m and the coupling constant α s. The sum rules are formulated in the literature [3, 8] and we will use the latest version [3] with correct large n ehavior. The moments M n M n = 1π 4m n! n dn ds Πs n =4m n s=0 0 Rsds s n+1 3 In refs. [4, 5] the corrections to the Coulom GF due to V i i =1, terms of the potential were treated numerically for complex values of energy far from the real axis. 7

8 of the spectral density Rs = 1πImΠs + iɛ are compared with experimental ones M exp n = 4m n R sds Q 0 s n+1 under the assumption of quark-hadron duality. The experimental moments M exp n are generated y the function R s which is the normalizad cross section R s =σe + e hadrons /σe + e μ + μ. Here Q = 1/3 is the -quark electric charge. Numerical values are otained asically y saturating the experimental moments with the contriution of the first six Υ resonances see [3] for details. Their leptonic widths Γ k and masses M k k =1...6 are known with good accuracy [18] M exp n = 4m n 9π Q αqedm 6 k=1 Γ k M n+1 k + s 0 ds R s s n+1 The rest of the spectrum eyond the resonance region for energies larger than s GeV continuum contriution lies far from threshold and is safely approximated y the ordinary PT expression for the theoretical spectral density, so there R s Rs. The influence of the continuum on high moments is almost negligile numerically and in any case under strict control 4. Electromagnetic coupling constant is renormalized to the energy of order m with the result αqedm =1.07α [18]. We work with moments for 10 <n<0 that simultaneously guarantees the smallness of oth the continuum contriution and the nonperturative power corrections due to the gluonic condensate [8]. The first one is not well known experimentally and has to e suppressed to make results independent of s 0. The second one should e small ecause the value of gluonic condensate and higher order condensates is not known well numerically. The normalization point μ = m is used throughout the computation 5.Fora 4 The expressions for the first few moments of the spectral density are now availale in ordinary perturation theory with α s accuracy [19], however, they cannot e used in theoretical formulas for sum rules directly ecause the spectrum is well known experimentally only for energies close to threshold due to existence of sharp resonances while the contriution of the continuum to these low moments is large in comparison with the resonance contriution. 5 We work strictly in the next-to-next-to-leading order approximation and, therefore,. 8

9 lower scale oth the hard and soft corrections ecome large and the peturative series for the moments is strongly divergent. We found that at μ m the μ dependence of the results is minimal which is a solid indication that at this point the higher order corrections are also small. The result of the fit is α s m =0. ± 0.0, or α s M Z =0.118 ± The sum rules are much more sensitive to the -quark mass than to the strong coupling constant so it is instructive to fix α s M Z =0.118 to the world average value [18] and then to extract m n. In this way we otain the following estimate for the mean value over the considered range of n m =4.80 GeV. This value is in a good agreement with the results of the first order analysis [3] where at α s M Z =0.118 we otained m =4.75 GeV. Note that the optimization procedure [0] was used to improve convergence of perturation theory in the previous analysis [3]. As we see this procedure turns out to e a powerful tool to estimate the higher order contriutions. For comparison, the leading order result is m =4.70 GeV and in the nextto-leading approximation without optimization one gets m =4.7 GeV. Main uncertainties of numerical values for considered parameters stem from the same sources that were identified in ref. [3]. The error coming from n distriution for the mass at fixed value of the coupling constant is negligile while the μ dependence for μ = m ± 1.5 GeV where this dependence is minimal introduces the main uncertainty. Thus our final estimate of the ottom quark pole mass is m =4.80 ± 0.06 GeV. Note that the uncertainty originated from the n and μ dependence is not reduced in comparison with the next-to-leading order. This means that the use the same normalization point for soft and hard corrections in contrast to [4, 5] where different normalization points were chosen for these two parts. 9

10 contriution of the higher order corrections which has to cancel n and μ dependence of the results is still important. Let us emphasize that the convergence of the perturation theory for the vacuum polarization function of heavy quark near threshold is not fast. We have found the next-to-nextto-leading order corrections to e of the order of the next-to-leading ones. Furthermore, in the case of -quark the corrections due to the perturative modification of the Coulom instantaneous potential i.e. related to ΔG 1 and ΔG i terms dominate the total correction in the next-to-leading and next-to-next-to-leading orders. Inclusion of these corrections is quite important for consistent analysis of sum rules for the Υ system. To conclude we have constructed an expression for the vacuum polarization function of the vector current of a heavy quark near threshold. It is completely analytic in the next-to-next-to-leading order in perturative and relativistic expansion up to α s, α sv and v corrections. The polarization function was used for determination of the -quark pole mass and the coupling constant from sum rules for the Υ system that are saturated y contriutions near threshold. In fact, there is no much hope for improving our results: next order approximation seems to e too complicated for analytical treatment within the regular perturation theory for NRQCD. The analysis showed a remarkale staility with respect to the next-to-leading one supplied with an optimization procedure in a variational spirit. Having in mind the considerale technical difficulty of computing next approximation and recognizing the necessity of improving the theoretical predictions in view of new high quality experimental data we think that the next step in the near future will e connected with optimization of the present approximation. Acknowledgements We thank J.H.Kühn for support, encouragement, and discussions. A.A.Penin gratefully acknowledges discussions with K.Melnikov. This work is partially supported y Volkswagen Foundation under contract No. I/ A.A.Pivovarov is supported in part y the Russian Fund for Basic Research under contracts Nos and The work of A.A.Penin is supported in part y the Russian Fund for Basic Research under contract

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