Ultrasoft contribution to quarkonium production and annihilation

We compute the third-order correction to electromagnetic S-wave quarkonium production and annihilation rates due to the emission and absorption of an ultrasoft gluon. Our result completes the analysis of the non-relativistic quarkonium bound-state dynamics in the next-to-next-to-next-to-leading order. The impact of the ultrasoft correction on the Upsilon(1S) leptonic width and the top quark-antiquark threshold production cross section is estimated.


Introduction
The theoretical study of non-relativistic heavy quark-antiquark systems is among the earliest applications of perturbative quantum chromodynamics (QCD) [1]. Perturbation theory applies to the bound-state dynamics of bottomonium, at least within the sum rule approach [2], and top-antitop systems [3], since non-perturbative effects are under control [4,5]. This makes heavy quark-antiquark systems well suited to determine fundamental parameters of QCD, the strong coupling constant α s and the heavy-quark masses m.
The binding energy of a quarkonium state and the value of its wave function at the origin -field-theoretically, the residues of two-point functions of local currents -are of primary phenomenological interest. The former determines the mass of the bound state, while the latter controls its production and annihilation rates. The quarkonium groundstate energy has been computed through O(mα 5 s ) including the third-order correction to the Coulomb approximation [6,7]. This result has been extended to the excited Swave states [8,9]. For the wave function at the origin a complete result is only available including the second-order correction [10,11,12]. This correction is large even for top quarks, and for a reliable perturbative prediction the third-order approximation seems to be needed. This amounts to a difficult calculation, which can be broken into several welldefined pieces, some of which are already available, such as the double-logarithmically enhanced O(α 3 s ln 2 α s ) terms [13,14] and the single-logarithmic O(α 3 s ln α s ) terms [15,16]. The calculation of the most difficult non-logarithmic term has been started in [8,9], where the contribution to the wave function at the origin from the loop corrections to the colour-Coulomb potential have been evaluated. In this and the companion paper [17] the remaining contributions from the non-Coulomb potentials and due to the emission and absorption of an ultrasoft gluon by the quarkonium bound state are presented. The ultrasoft correction discussed below is of special interest, because it constitutes a qualitatively new effect, which shows up for the first time in the third order. No other such effects are expected in higher orders of the perturbative expansion. The complete third-order correction to the wave function at the origin can now be expressed in terms of a few yet unknown matching coefficients, which can be obtained by standard fixed-order loop calculations.
2 Ultrasoft correction to the wave function

Definitions
In non-relativistic bound states the quark velocity v is a small parameter. An expansion in v may be performed directly in the QCD Lagrangian by using the framework of effective field theory [18,19,20], or diagrammatically with the threshold expansion [21]. The relevant momentum regions are the hard region (energy k 0 and momentum k of order m), the soft region (k 0 , k ∼ mv), the potential region (k 0 ∼ mv 2 , k ∼ mv), and the ultrasoft region (k 0 , k ∼ mv 2 ). Integrating out the hard modes amounts to matching onto non-relativistic QCD (NRQCD) [19]. If one also integrates out the soft modes and potential gluons, one obtains the effective theory called potential NRQCD (PNRQCD), which contains potential heavy quarks and ultrasoft gluons as dynamical fields [20] (see also [22]). In this theory the leading colour-Coulomb potential is part of the unperturbed Lagrangian, so that the propagation of a colour-singlet quark-antiquark pair is described by the Green function of the Schrödinger equation with r = |r|, m the heavy-quark pole mass, and C F = (N 2 c −1)/(2N c ), N c = 3. The PNRQCD Lagrangian further contains interactions of quarks with the multipole-expanded ultrasoft gluon field and instantaneous, spatially non-local interactions ("potentials"), which can be treated as perturbations. This constitutes the basic framework for the perturbative analysis of quarkonium bound-state properties. The colour-singlet Coulomb Green function G In this paper, however, we are interested in the leading ultrasoft effect. To connect the concept of a non-relativistic wave function at the origin to a physical quantity, we consider the two-point function of the electromagnetic heavy-quark current j µ =Qγ µ Q in full QCD, whose poles are related to electromagnetic production and annihilation rates of the corresponding quarkonium states. In PNRQCD j µ is represented in terms of operators constructed from the non-relativistic quark and antiquark two-component Pauli spinor fields ψ and χ, The matching coefficients c v (µ) = 1 + c (1) v (µ)α s (µ)/(4π) + . . . and d v (µ) = 1 + O(α s ) represent the contributions from the hard modes with µ a factorization scale that is also implicit in the renormalization convention for the operators on the right-hand side. (We use dimensional regularization with d = 4 − 2ǫ and the MS scheme.) We also introduce the PNRQCD two-point function where E = √ q 2 − 2m. In leading order G(E) coincides with (the correspondingly regularized) G  (4), we obtain which is valid up to the third order. G(E) has Coulomb bound-state poles at energies E n ≈ E (0) n with spin and orbital angular momentum S = 1 and l = 0, respectively, following from the form of the current, ψ † σχ. Near the pole which defines the "wave function at the origin". Beginning from the second-order correction, these wave-functions are factorization-scale dependent, but this dependence cancels in the residues of the poles of Π(q 2 ), which determine the observable electromagnetic production and annihilation rates. The third-order (NNNLO) corrections to |ψ n (0)| 2 originate (i) from single insertions of the third-order potentials in the PNRQCD Lagrangian and multiple insertions of firstand second-order potentials, where the order is determined by the combined suppression in α s and v relative to the leading-order Coulomb potential α s /r ∼ α s v. This contribution has been computed in [8,9,17] and expressed in terms of the yet unknown three-loop correction to the Coulomb potential and four constants related to O(ǫ) parts of loop corrections to other potentials; (ii) from the emission and absorption of an ultrasoft gluon (similar to the Lamb shift for energy levels), calculated below.

Calculation of the ultrasoft correction
The leading ultrasoft interactions in the PNRQCD Lagrangian are g s ψ † (x)(A 0 (t, 0) − xE(t, 0))ψ(x) together with a similar term for the antiquark field. The contribution from the A 0 coupling cancels (or can be gauged away), leaving the chromoelectric dipole interaction, which results in a NNNLO correction [23,24,25]. The PNRQCD diagram = + + Figure 2: The NRQCD decomposition of the PNRQCD vertex. The dashed line corresponds to a potential Coulomb gluon. The black circles on the right-hand side correspond to the NRQCD vertices g s pA/m. The bold arrows correspond to the potential quark and antiquark propagators with any number of the potential gluon exchanges.
representing this correction is shown in Fig. 1. The corresponding correction to G(E) defined in (5) reads with the understanding that one picks up only the pole at k 0 = |k| − iǫ in the gluon propagator. Here G . Only the l = 1 partial wave of the octet Green function contributes to (8), and since the octet potential is repulsive, G C (r, r ′ ; E) does not have bound-state poles. Expression (8) cannot be used in practice, because the ultrasoft correction is divergent. Its definition requires specifying a regulator and subtractions, which must be chosen to be consistent with the calculation of the potential insertions; the potentials themselves; and the hard matching coefficients. In the following we provide the necessary definitions, deferring a detailed discussion of many interesting technical aspects of the calculation to a later publication that will provide results not only for |ψ n (0)| 2 but also for the full correlation function G(E). To apply dimensional regularization we transform (8) to momentum space. We also find it convenient to re-express the PNRQCD vertex in terms of the NRQCD vertices from which it is derived by equation-of-motion relations, see [25] and Fig. 2 for a graphical representation. The formulation of non-relativistic effective theory in dimensional regularization [6,12,26,27] is very convenient, because it allows one to combine bound-state calculations with loop calculations of matching coefficients, which are technically feasible only in dimensional regularization. Furthermore, when loop integrals are expanded in the sense of the threshold expansion [21], the matching of contributions from different regions is automatic.
The integral over the three-momentum k of the ultrasoft gluon is ultraviolet (UV) divergent. The divergence is related to the factorization of the ultrasoft scale from the other scales, and cancels when all pieces of the calculation are added. The UV-divergent part of the ultrasoft integral has the form of a single insertion of a third-order potential and of a one-loop correction to the coefficient d v of the derivative current in (4). In fact, they cancel precisely infrared (IR) divergences in the calculation of these quantities [6,28]. We therefore define the ultrasoft correction by adding counterterms that cancel these ultrasoft subdivergences, where δU follows from (8), and the counterterms read The counterterms added here are subtracted from the other parts of the calculation [17]. Withμ 2 = e γ E µ 2 /(4π) subtracting poles in ǫ corresponds to the MS subtraction. After adding this counterterm the potential loops and Coulomb Green functions can be evaluated in three dimensions unless the potential loop integrations are divergent. Such divergences occur in potential vertex subgraphs up to three loops, and they correspond to IR divergences in the three-loop correction to c v . (There are also over-all divergences in G(E), but they are irrelevant to the calculation of the bound-state pole and residue.) We separate this vertex subdivergence by adding and subtracting the three-loop vertex subdiagram at zero external momentum p, as shown in Fig. 3. The vertex UV divergence is logarithmic, and does not depend on the external momentum. Hence, it is isolated in first term on the right-hand side of the equation of Fig. 3, while the difference in the brackets is finite and can be computed in three dimensions. The first term factorizes into a three-loop diagram at p = 0, which has to be computed in d dimensions, and the leading-order expression for G(E). The divergent part cancels the contribution from c v , and the remainder can again be evaluated in three dimensions. Thus, we do not need the d − 1 dimensional Coulomb Green function, which is unknown.  Table 1: Numerical result for the non-logarithmic part of the ultrasoft contribution, as defined in (13); n denotes the principal quantum number.

Result
We write the perturbative expansion for the wave function at the origin as where the leading-order Coulomb wave function at the origin in three dimensions is given by |ψ C n (0)| 2 = (mα s C F ) 3 /(8πn 3 ), and δ (m) ψ n stands for the mth order correction. For the ultrasoft part of the third-order correction we obtain where H n = ln C F n − 1 n + n−1 k=1 1 k . The most difficult part of the calculation is the nonlogarithmic term δ us n , which we could compute only numerically. For the six lowest states its value is given in Table 1. Note that |ψ C n (0)| 2 in (12) is formally defined in d dimensions, but as explained above, we do not need the explicit d-dimensional expression, because the 1/ǫ pole terms in (13) cancel with pole terms from c 2 v |ψ C n (0)| 2 contained in (6).
We verified that the logarithmic terms in (13) when combined with those from the third-order potentials insertions [17] agree with [15] 1 . The sum of the 1/ǫ poles in the complete third-order correction to |ψ n (0)| 2 from (13) and [17] combined contains a term proportional to the single insertion of the first-order Coulomb potential and a term that must cancel against the infrared pole in twice the three-loop correction to c v . This second term reads where β 0 = 11C A /3 − 4T F n f /3 is the one-loop QCD beta-function, T F = 1/2 and n f is the number of light-quark flavors. The n f -part of the three-loop coefficient c (3) v is known [29], and we checked that the n f -dependent pole parts cancel as required. Eq. (14) is consistent with the scale dependence of the hard matching coefficient given by Eqs. (8), (9) of [15], except for the rational part of the C A C 2 F term in γ ′(3) v , which should be increased by 7/24 to agree with our result. 2 The difference is due to the fact that by using [σ i , σ j ] = iǫ ijk σ k the spin-algebra in [15] is not completely d-dimensional, hence the result for the scale-dependence of c (3) v given there does not correspond to a calculation in conventional dimensional regularization.

Quarkonium phenomenology
We briefly discuss the size of the ultrasoft correction for the two most relevant cases, the leptonic decay of the Υ(1S) and the threshold production of top quark-antiquark pairs in e + e − annihilation. This discussion must necessarily be preliminary, since the ultrasoft correction alone is factorization scheme and scale dependent. For the quarkonium spintriplet ground state, n = 1, we obtain from (13) The scale of the coupling α s is most naturally of order of the inverse Bohr radius mα s C F in two of the three powers of the overall factor α 3 s , and of order of the ultrasoft scale mα 2 s 1 In Eq. (7) of [15] for the excited states the term 4C 3 F (1 − 1/n 2 )/3 is missed and the β 0 ln n term should be multiplied by three. 2 In addition, as noted in [29], the term − 3 2 β 0 γ in Eq. (8) of [15] should read +β 0 γ (2) v .
in the third. However, any other scale choice is formally equivalent at this order. In the following we evaluate α s at µ B = mC F α s (µ B ), wherever it appears. The scale µ in the ln (µ/m) terms is related to scale-dependent potentials and hard matching coefficients. We vary µ/m between α s C F (corresponding to the scale µ B ) and 1 (hard scale).
Υ(1S) leptonic width. Up to a normalization factor the Υ(1S) leptonic decay width is given by the residue of Π(q 2 ) at the ground state pole. The leading-order expression for the decay width follows from (6), resulting in Γ LO , where e q is the electric charge of the quark flavour q and α is the fine-structure constant. The non-perturbative contribution to the width is quite sizeable and out of control for higher resonances [30,31], hence we consider only the case n = 1. Adopting α s = 0.30, which corresponds to µ B ≈ 2 GeV for the Υ(1S), we obtain δΓ 1 /Γ 1 ≈ 3.0 from the non-logarithmic correction δ us 1 alone. Proceeding as described above for the logarithmic terms results in the estimate It therefore appears that the large non-logarithmic term leads to a large enhancement of the width. Whether or not perturbation theory is out of control (as may be suggested by the upper limit of the given range) can be decided only after combining all third-order terms.
Top-quark production near threshold. For top quarks non-perturbative effects are negligible, but its decay width Γ t smears out the Coulomb resonances below the threshold. The NNLO analysis of the cross section [32] shows that only the ground-state pole gives rise to a prominent resonance. Although the calculation of the normalized cross section R = σ(e + e − → ttX)/σ(e + e − → µ + µ − ) requires the full Green function G(E), the height of the resonance can be estimated from the wave function at the origin of the would-be toponium ground state. In the leading-order approximation R LO 1 ≈ 6πN c e 2 t |ψ C 1 (0)| 2 / (m 2 t Γ t ). Adopting α s = 0.14, which corresponds to µ B ≈ 32.5 GeV, we obtain δR 1 /R 1 ≈ 0.31 from the non-logarithmic correction δ us 1 alone. Including an estimate of the logarithmic terms we find Hence, despite the large quark mass, we may anticipate a sizeable third-order correction, unless there are cancellations.

Summary
We evaluated the third-order correction to electromagnetic quarkonium production and annihilation due to the emission and absorption of an ultrasoft gluon. Together with other contributions [8,9,13,14,15,16,17,29] already completed the problem of evaluating the total O(α 3 s ) corrections is now reduced to the calculation of four O(ǫ) terms in the NNNLO heavy quark-antiquark potential [17], the three-loop colour-singlet Coulomb potential, and the three-loop vector current matching coefficient in the MS scheme in dimensional regularization. The previously unknown non-logarithmic ultrasoft contribution is large and significantly increases the production and annihilation rates. It might limit the accuracy of the perturbative analysis of the quarkonium even for top quarks. We should however emphasize that a definite conclusion can only be drawn once the full NNNLO result is available. In this respect the sizable negative third-order correction from the perturbation potentials [8,9,17] should be mentioned.