Exclusive decays of χ_{bJ} and η_b into two charmed mesons
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Exclusive decays of χ bJ and η b into two charmed mesons Regina S. Azevedo, ∗ Bingwei Long,
2, 1, † and Emanuele Mereghetti ‡ Department of Physics, University of Arizona, Tucson, AZ 85721, USA European Centre for Theoretical Studies in Nuclear Physicsand Related Areas (ECT*), I-38100 Villazzano (TN), Italy
Abstract
We develop a framework to study the exclusive two-body decays of bottomonium into twocharmed mesons and apply it to study the decays of the C -even bottomonia. Using a sequence ofeffective field theories, we take advantage of the separation between the scales contributing to thedecay processes, 2 m b ≫ m c ≫ Λ QCD . We prove that, at leading order in the EFT power counting,the decay rate factorizes into the convolution of two perturbative matching coefficients and threenon-perturbative matrix elements, one for each hadron. We calculate the relations between thedecay rate and non-perturbative bottomonium and D -meson matrix elements at leading order,with next-to-leading log resummation. The phenomenological implications of these relations arediscussed. PACS numbers: 12.39.Hg, 13.25.GvKeywords: quarkonium decay; non-relativistic QCD; soft-collinear effective theory ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION The exclusive two-body decays of heavy quarkonium into light hadrons have been studiedin the framework of perturbative QCD by many authors (for reviews, see [1] [2]). Theseprocesses exhibit a large hierarchy between the heavy quark mass, which sets the scale forannihilation processes, and the scales that determine the dynamical structure of the particlesin the initial and final states. The large energy released in the annihilation of the heavyquark-antiquark pair and the kinematics of the decay — with the products flying awayfrom the decay point in two back-to-back, almost light-like directions— allow for rigorouslyderiving a factorization formula for the decay rate at leading twist (for an up-to-date reviewof the theoretical and experimental status of the exclusive decays into light hadrons, see [3]).For the bottomonium system, a particularly interesting class of two-body final statesis the ones containing two charmed mesons. In these cases the picture is complicated bythe appearance of an additional intermediate scale, the charm mass m c , which is muchsmaller than the bottom mass m b but is large enough to be perturbative. These decaysdiffer significantly from those involving only light quarks. The creation of mesons that aremade up of purely light quarks involves creating two quark-antiquark pairs, with the energyshared between the quark and antiquark in each pair. In the production of two D mesons,however, almost all the energy of the bottomonium is carried away by the heavy c and ¯ c ,while the light quark and antiquark, which bind to the ¯ c and c respectively, carry away(boosted) residual energies.The existence of well-separated scales in the system and the intuitive picture of the decayprocess suggest to tackle the problem using a sequence of effective field theories (EFTs)that are obtained by subsequently integrating out the dynamics relevant to the perturbativescales m b and m c .In the first step, we integrate out the scale m b by describing the b and ¯ b with Non-Relativistic QCD (NRQCD) [4], and the highly energetic c and ¯ c with two copies of Soft-Collinear Effective Theory (SCET) [5] [6] [7] [8] [9] in opposite light-cone directions. Inthe second step, we integrate out the dynamics manifested at scales of order m c by treat-ing the quarkonium with potential NRQCD (pNRQCD) [10] [11] [12], and the D mesonswith a boosted version of Heavy-Quark Effective Theory (HQET) [13] [14] [15] [16] [17][18] [19]. The detailed explanation of why the aforementioned EFTs are employed is of-2ered in Sec. II. We will prove that, at leading order in the EFT expansion, the decayrate factors into a convolution of two perturbative matching coefficients and three (one foreach hadron) non-perturbative matrix elements. The non-perturbative matrix elements areprocess-independent and encode information on both the initial and final states.For simplicity, in this paper we focus on the decays of the C -even quarkonia χ bJ and η b that, at leading order in the strong coupling α s , proceed via the emission of two virtualgluons. The same method can be generalized to the decays of C -odd states Υ and h b , whichrequire an additional virtual gluon. We also refrain from processes that have vanishingcontributions at leading order in the EFT power counting. So the specific processes studiedin this paper are χ b , → DD , χ b , → D ∗ D ∗ , and η b → DD ∗ + c.c. However, the EFTapproach developed in this paper enables one to systematically include power-suppressedeffects, making it possible to go beyond the leading-twist approximation.The study of the inclusive and exclusive charm production in bottomonium decays andof the role played by the charm mass m c in such processes have recently drawn renewedattention [20] [21] [22] [23], in connection with the experimental advances spurred in thepast few years by the abundance of bottomonium data produced at facilities like BABAR,BELLE, and CLEO. The most notable result was the observation of the bottomoniumground state η b , recently reported by the BABAR collaboration [24]. Furthermore, theCLEO collaboration published the first results for several exclusive decays of χ b into lighthadrons [25] and for the inclusive decay of χ b into open charm [26]. In particular, theymeasured the branching ratio B ( χ bJ → D X ), where J is the total angular momentumof the χ b state, and conclusively showed that for J = 1 the production of open charm issubstantial: B ( χ b (1 P ) → D X ) = 12 . ± . J = 0 , et al. [20],where EFT techniques (in particular NRQCD) were for the first time applied to study theproduction of charm in bottomonium decays.The double-charm decay channels analyzed here have not yet been observed, so one ofour aims is to see if they may be observable given the current data. Unfortunately, the poorknowledge of the D -meson matrix elements prevents us from providing definitive predictionsfor the decay rates Γ( χ bJ → DD ), Γ( χ bJ → D ∗ D ∗ ), and Γ( η b → DD ∗ + c.c.). As we willshow, these rates are indeed strongly dependent on the parameters of the D - and D ∗ -meson3istribution amplitudes, in particular on their first inverse moments λ D and λ D ∗ : the ratesvary by an order of magnitude in the accepted ranges for λ D and λ D ∗ . On the other hand,the factorization formula implies that these channels, if measured with sufficient accuracy,could constrain the form of the D -meson distribution amplitude and the value of its firstinverse moment. In turn, the details of the D -meson structure are relevant to other D -mesonobservables, which are crucial for a model-independent determination of the CKM matrixelements | V cd | and | V cs | [27].This paper is organized as follows. In Sec. II we discuss the degrees of freedom and theEFTs we use. In Sec. III A we match QCD onto NRQCD and SCET at the scale 2 m b . Therenormalization-group equation (RGE) for the matching coefficient is derived and solved inSec. III B. In Sec. IV A the scale m c is integrated out by matching NRQCD and SCET ontopNRQCD and bHQET. The renormalization of the low-energy EFT operators is performedin Sec. IV B, with some technical details left to App. A. The decay rates are calculated inSec. V using two model distribution amplitudes. In Sec. VI we draw our conclusions. II. DEGREES OF FREEDOM AND THE EFFECTIVE FIELD THEORIES
Several well-separated scales are involved in the decays of the C -even bottomonia η b and χ bJ into two D mesons, making them ideal processes for the application of EFT techniques.The distinctive structures of the bottomonium (a heavy quark-antiquark pair) and the D meson (a bound state of a heavy quark and a light quark) suggest that one needs differentEFTs to describe the initial and final states.We first look at the initial state. The η b is the ground state of the bottomonium system.It is a pseudoscalar particle, with spin S = 0, orbital angular momentum L = 0, and totalangular momentum J = 0. In what follows we will often use the spectroscopic notation S +1 L J , in which the η b is denoted by S . The χ bJ is a triplet of states with quantumnumbers P J . The η b and χ bJ are non-relativistic bound states of a b quark and a ¯ b antiquark.The scales in the system are the b quark mass m b , the relative momentum of the b ¯ b pair m b w ,the binding energy m b w , and Λ QCD , the scale where QCD becomes strongly coupled. w isthe relative velocity of the quark-antiquark pair in the meson, and from the bottomoniumspectrum it can be inferred that w ∼ .
1. Since m b ≫ Λ QCD , m b can be integrated outin perturbation theory and the bottomonium can be described in NRQCD. The degrees4f freedom of NRQCD are non-relativistic heavy quarks and antiquarks, with energy andmomentum ( E, | ~p | ) of order ( m b w , m b w ), light quarks and gluons. In NRQCD, the gluonscan be soft ( m b w, m b w ), potential ( m b w , m b w ), and ultrasoft (usoft) ( m b w , m b w ). TheNRQCD Lagrangian is constructed as a systematic expansion in 1 /m b whose first few termsare L NRQCD = ψ † iD + ~D m b + ~σ · g ~B m b + . . . ! ψ + χ † iD − ~D m b − ~σ · g ~B m b + . . . ! χ , where ψ and χ † annihilate a b quark and a ¯ b antiquark respectively, and · · · denotes higher-order contributions in 1 /m b . In NRQCD several mass scales are still dynamical and differentassumptions on the hierarchy of these scales may lead to different power countings foroperators of higher dimensionality. However, as long as w ≪
1, higher-dimension operatorsare suppressed by powers of w (for a critical discussion on the different power countings werefer to [12]).NRQCD still contains interactions that can excite the heavy quarkonium far from itsmass shell, for example, through the interaction of a non-relativistic quark with a soft gluon.In the case m b w ≫ Λ QCD , we can integrate out these fluctuations, matching perturbativelyNRQCD onto a low-energy effective theory, pNRQCD. We are then left with a theory of non-relativistic quarks and ultrasoft gluons, with non-local potentials induced by the integrationover soft- and potential-gluon modes. The interactions of the heavy quark with ultrasoftgluons are still described by the NRQCD Lagrangian, with the constraint that all the gluonsare ultrasoft. In the weak coupling regime m b w ≫ Λ QCD , the potentials are organized by anexpansion in α s ( m b w ), 1 /m b , and r , where r is the distance between the quark and antiquarkin the quarkonium, r ∼ /m b w . If we assume m b w ∼ Λ QCD , each term in the expansionhas a definite power counting in w and the leading potential is Coulombic V ∼ α s ( m b w ) /r .An alternative approach, which does not require a two-step matching, has been developedin the effective theory vNRQCD [28] [29] [30] [31]. In the vNRQCD approach there is onlyone EFT below m b , which is obtained by integrating out all the off-shell fluctuations at thehard scale m b and introducing different fields for various propagating degrees of freedom(non-relativistic quarks and soft and ultrasoft gluons). In spite of the differences betweenthe two formalisms, pNRQCD and vNRQCD give equivalent final answers in all the knownexamples in which both theories can be applied.We now turn to the structure of the D meson. The most relevant features of the D meson5re captured by a description in HQET. In HQET, in order to integrate out the inert scale m c , the momentum of the heavy quark is generically written as [15] p = m c v + k , (1)where v is the four-velocity label, satisfying v = 1, and k is the residual momentum. Ifone chooses v to be the center-of-mass velocity of the D meson, k scales as k ∼ v Λ QCD .Introducing the light-cone vectors n µ = (1 , , ,
1) and ¯ n µ = (1 , , , − k µ = ¯ n · k n µ / n · k ¯ n µ / k µ ⊥ or simply k = ( n · k, ¯ n · k, ~k ⊥ ). There are two relevant frames. One is the D -meson rest frame, in which v is conveniently chosen as v = (1 , , , D mesons are highly boosted in opposite directions, with v chosen as v = v D ,the four-velocity of one of the D mesons. By a simple consideration of kinematics and thescaling k ∼ v Λ QCD , one can work out the scalings for k in the two frames. In the D -mesonrest frame, k ∼ Λ QCD (1 , , D mesonmoving in the positive z -direction), k ∼ Λ QCD ( n · v D , ¯ n · v D , ∼ Λ QCD ¯ n · v D (cid:0) λ , , λ (cid:1) , (2)where ¯ n · v D ∼ m b /m c and λ = m c / m b ≪
1. It is convenient for the calculation in thispaper to use the bottomonium rest frame, so we drop the subscript in v D and we assume v = v D in the rest of this paper. The momentum scaling in Eq. (2) is called ultracollinear(ucollinear), and boosted HQET (bHQET) is the theory that describes heavy quarks withultracollinear residual momenta and light degrees of freedom (including gluons and lightquarks) with the same momentum scaling.The bHQET Lagrangian is organized as a series in powers of Λ QCD /m c and, for residualmomentum ultracollinear in the n -direction, the leading term is [18] L bHQET = ¯ h n iv · Dh n , (3)where the field h n annihilates a heavy quark and the covariant derivative D contains ultra-collinear and ultrasoft gluons, iD µ = n µ i ¯ n · ∂ + g ¯ n · A n ) + ¯ n µ in · ∂ + gn · A n + gn · A us ) + (cid:0) i∂ µ ⊥ + gA µn, ⊥ (cid:1) . (4)The ultrasoft gluons only enter in the small component of the covariant derivative. Thisfact can be exploited to decouple ultrasoft and ultracollinear modes in the leading-order6 RQCD field momentum SCET field momentumquark b , ¯ b ψ b , χ ¯ b ( m b w , m b w ) c , ¯ c ξ c ¯ n , ξ ¯ cn m b (1 , λ , λ ), 2 m b ( λ , , λ )gluon potential A µ ( m b w , m b w ) collinear A µ ¯ n , A µn m b (1 , λ , λ ), 2 m b ( λ , , λ )soft A µ ( m b w, m b w ) soft A µs m b ( λ, λ, λ )usoft A µ ( m b w , m b w ) usoft A µus m b ( λ , λ , λ )TABLE I: Degrees of freedom in EFT I (NRQCD + SCET). w is the b ¯ b relative velocity in thebottomonium rest frame, while λ ∼ m c / m b is the SCET expansion parameter. We assume m b w ∼ m c (or, equivalently, w ∼ λ ) and m b w ∼ m b λ ∼ Λ QCD . Lagrangian through a field redefinition reminiscent of the collinear-ultrasoft decoupling inSCET [7] [18]. The ultracollinear-ultrasoft decoupling is an essential ingredient for thefactorization of the decay rate.Therefore, the appropriate EFT to calculate the decay rate is a combination of pNRQCD,for the bottomonium, and two copies of bHQET, with fields collinear to the n and ¯ n direc-tions, for the D and ¯ D mesons, symbolically written as EFT II ≡ pNRQCD + bHQET.As we mentioned earlier, we plan to describe the bottomonium structure with a two-stepscheme QCD → NRQCD → pNRQCD. However, at the intermediate stage, where we firstintegrate out the hard scale 2 m b and arrive at the scale m b w , the D meson cannot yet bedescribed in bHQET. This is because the interactions relevant at the intermediate scale m b w can change the c -quark velocity and leave the D meson off-shell of order ∼ ( m b w ) ∼ m c ≫ Λ . Highly energetic c and ¯ c travelling in opposite directions can be described properlyby SCET with mass. Thus, at the scale µ = 2 m b , we match QCD onto an intermediateEFT, EFT I ≡ NRQCD + SCET, in which the EFT expansion is organized by λ and w . Thedegrees of freedom of EFT I are tabulated in Tab. I.Then, we integrate out m c and m b w at the same time, matching EFT I onto EFT II at thescale µ ′ = m c . In EFT II , the low-energy approximation is organized by Λ QCD /m c and w .The degrees of freedom of EFT II are summarized in Tab. II. When no subscript is specifiedin the rest of this paper, any reference to EFT applies to both EFT I and EFT II . To facilitatethe power counting, we adopt w ∼ λ ∼ Λ QCD /m c . As a first study, we will perform in thispaper the leading-order calculation of the bottomonium decay rates.7 NRQCD field momentum bHQET field momentumquark b , ¯ b ψ b , χ ¯ b ( m b w , m b w ) c , ¯ c h c ¯ n , h ¯ cn Q (1 , λ , λ ), Q ( λ , , λ ) u , d ξ ¯ n , ξ n Q (1 , λ , λ ), Q ( λ , , λ )gluon usoft A µ ( m b w , m b w ) usoft A µus Q ( λ, λ, λ )ucollinear A µ ¯ n , A µn Q (1 , λ , λ ), Q ( λ , , λ )TABLE II: Degrees of freedom in EFT II (pNRQCD + bHQET). The scale Q in bHQET is Q = n · v ′ Λ QCD for the ¯ n -collinear sector and Q = ¯ n · v Λ QCD for the n -collinear sector. n · v ′ and¯ n · v are the large light-cone components of the D -meson velocities in the bottomonium rest frame, n · v ′ ∼ ¯ n · v ∼ m b /m c . λ and w are defined as in Tab. I.FIG. 1: Matching QCD onto EFT I . On the r.h.s., the double lines represent the non-relativistic b (¯ b ) (anti)quark, while the dashed lines represent the collinear c (¯ c ) (anti)quark. III. NRQCD + SCETA. Matching
In the first step, we integrate out the dynamics related to the hard scale 2 m b by matchingthe QCD diagrams for the production of a c ¯ c pair in the annihilation of a b ¯ b pair onto theirEFT I counterparts. The tree-level diagrams for the process are shown in Fig. 1. The gluonpropagator in the QCD diagram has off-shellness of order q = (2 m b ) and it is not resolvedin EFT I , giving rise to a point-like interaction.We calculate the diagrams on shell, finding iJ QCD = iC ( µ ) J EFT I ( µ ) , (5)with, at tree level, J EFT I = χ † ¯ b σ µ ⊥ t a ψ b ¯ χ c ¯ n S † ¯ n γ µ ⊥ t a S n χ ¯ cn and C ( µ = 2 m b ) = α s (2 m b ) πm b , (6)8here t a are color matrices and the symbol σ µ denotes the four matrices σ µ = (1 , ~σ ), with ~σ the Pauli matrices. The subscript ⊥ refers to the components orthogonal to the light-cone vectors n µ and ¯ n µ . The fields ψ b and χ † ¯ b are two-component spinors that annihilaterespectively a b quark and a ¯ b antiquark. χ ¯ cn, ¯ n · p and χ c ¯ n, n · p are collinear gauge-invariantfermion fields: χ ¯ cn, ¯ n · p ≡ ( W † n ξ ¯ cn ) ¯ n · p , χ c ¯ n, n · p ≡ ( W † ¯ n ξ c ¯ n ) n · p , (7)where W n is defined as W n ≡ X perms exp (cid:16) − g ¯ n · P ¯ n · A n (cid:17) . (8) W ¯ n has an analogous definition with n → ¯ n . Collinear fields are labelled by the largecomponent of their momentum. Note, however, we omit in Eq. (6) the subscripts n · p and¯ n · p of the collinear fermion fields, in order to simplify the notation. The operator ¯ n · P inthe definition (8) is a label operator that extracts the large component of the momentum ofa collinear field, ¯ n · P φ n, ¯ n · p = ¯ n · p φ n, ¯ n · p , where φ n, ¯ n · p is a generic collinear field. S n (¯ n ) is asoft Wilson line, S n ≡ X perms h exp (cid:16) − gn · P n · A s (cid:17)i , (9)where the operator n · P acts on soft fields, n · P φ s = n · k φ s .Since in SCET different gluon modes are represented by different fields, we have toguarantee the gauge invariance of the operator J EFT I under separate soft and collinear gaugetransformations. A soft transformation is defined by V s ( x ) = exp ( iβ as t a ), with ∂ µ V ∼ m b ( λ, λ, λ ), while a gauge transformation U ( x ) is n -collinear if U ( x ) = exp ( iα a ( x ) t a ) and ∂ µ U ( x ) ∼ m b ( λ , , λ ). It has been shown in Ref. [7] that collinear fields do not transformunder a soft transformation and that the combination W † n ξ n is gauge invariant under acollinear transformation. Soft fields do not transform under collinear transformations butthey do under soft transformations. For example, the NRQCD quark and antiquark fieldstransform as ψ b → V s ( x ) ψ b . The soft Wilson line has the same transformation, S n → V s ( x ) S n . Therefore, χ † ¯ b σ µ ⊥ t a ψ b transforms as an octet under soft gauge transformations.Since ¯ χ c ¯ n S † ¯ n γ µ ⊥ t a S n χ ¯ cn behaves like an octet as well, J EFT I is invariant. It is worth notingthat the soft Wilson lines are necessary to guarantee the gauge invariance of J EFT I . Wehave explicitly checked their appearance at one gluon by matching QCD diagrams like theone in Fig. 1, with all the possible attachments of an extra soft or collinear gluon, ontofour-fermion operators in EFT I . 9 . Running The matching coefficient C and the effective operator J EFT I depend on the renormalizationscale µ . Since the effective operator is sensitive to the low-energy scales in EFT I , logarithmsthat would appear in the evaluation of J EFT I are minimized by the choice µ ∼ m c . Onthe other hand, since the coefficient encodes the high-energy dynamics of the scale 2 m b ,such a choice would induce large logarithms of m c / m b in the matching coefficient. Theselogarithms can be resummed using RGEs in NRQCD + SCET.The µ dependence of J EFT I is governed by an equation of the following form [32], dd ln µ J EFT I ( µ ) = − γ EFT I ( µ ) J EFT I ( µ ) , (10)where the anomalous dimension γ EFT I is given by γ EFT I = Z − I dd ln µ Z EFT I (11)and Z EFT I is the counterterm that relates the bare operator J (0)EFT I to the renormalized one, J (0)EFT I = Z EFT I ( µ ) J EFT I ( µ ). Since the l.h.s. of Eq. (5) is independent of the scale µ , theRGE (10) can be recast as an equation for the matching coefficient C ( µ ), dd ln µ C ( µ ) = γ EFT I ( µ ) C ( µ ) . (12)The counterterm Z EFT I cancels the divergences that appear in Green functions with theinsertion of the operator J EFT I . We calculate Z EFT I in the MS scheme by evaluating thedivergent part of the four-point Green function at one loop, given by the diagrams in Figs.2 - 4. FIG. 2: Soft diagrams at one loop.
Since in NRQCD we do not introduce different gluon fields for different momentum modes,“soft” and “ultrasoft” in Fig. 2 and Fig. 3 refer to the convention that we impose soft or10
IG. 3: Ultrasoft diagrams at one loop.FIG. 4: Collinear diagrams at one loop. ultrasoft scaling to the corresponding loop momentum. The potential region, which shouldbe considered in the diagrams of Fig. 2, does not give any divergent contribution.The integrals are evaluated in dimensional regularization, with d = 4 − ε . We regulatethe infrared divergences by keeping the non-relativistic b and ¯ b and the collinear c and ¯ c off-shell: E b, ¯ b − ~p b, ¯ b / m b = ∆ b , p c − m c = ∆ and p c − m c = ¯∆ . We power count the c -quark off-shellness as ∆ ∼ ¯∆ ∼ m b λ and the b -quark off-shellness as ∆ b ∼ m b w . Wealso assume ∆ , ¯∆ >
0. To avoid double counting, we define the one-loop integrals withthe 0-bin subtraction [33].Even with an off-shellness, the soft diagrams in Fig. 2 do not contain any scale and theyare completely cancelled by their 0-bin.The divergent part of the ultrasoft diagrams in Fig. 3 is i M usoft = − i α s π (cid:26) C F (cid:20) ε − ε ln (cid:18) ∆ ¯∆ n · p c ¯ n · p ¯ c µ (cid:19)(cid:21) + 1 N c ε ln( − − i − N c ε (cid:27) J EFT I , (13)where C F = ( N c − / N c and µ is the MS unit mass, µ = 4 πµ exp ( − γ E ). The firstterm in the curly brackets of Eq. (13) corresponds to the sum of the divergences in thesecond diagram in Fig. 3, where an ultrasoft gluon is exchanged between the c and ¯ c quarks11ollinear in back-to-back directions, and those in the last four diagrams of the same figure,which contain ultrasoft interactions between the initial and final states. The second term isan extra imaginary piece generated by the second diagram in Fig. 3. The − i , ¯∆ >
0. The divergencesarising from the ultrasoft exchanges between the b ¯ b pair in the first diagram in Fig. 3 areencoded in the last term in Eq. (13).The initial and final states cannot interact by exchanging collinear gluons because theemission or absorption of a collinear gluon would give the b quark an off-shellness of order m b , which cannot appear in the effective theory. For the same reason, the c and ¯ c cannotexchange n or ¯ n -collinear gluons. The only collinear loop diagrams consist of the emissionof a n (¯ n )-collinear gluon from the Wilson line W n (¯ n ) in J EFT I and its absorption by the ¯ c ( c )quark, as shown in Fig. 4. The divergent part of the sum of the two collinear diagrams is i M coll = i α s π C F (cid:20) ε + 1 ε (cid:18) − ln (cid:18) ∆ ¯∆ µ µ (cid:19)(cid:19)(cid:21) J EFT I . (14)The collinear diagrams are calculated with a 0-bin subtraction [33], that is, we subtract fromthe naive collinear integrals the same integrals in the limit in which the loop momentum isultrasoft. In this way we avoid double counting between the diagrams in Figs. 3 and 4.Summing Eqs. (13) and (14) and adding factors of Z / ψ for each field, Z ψ b = Z χ b = 1 + 1 ε α s π C F , Z ξ n = Z ξ ¯ n = 1 − ε α s π C F , the divergent piece becomes i M div = i α s π (cid:26) C F (cid:20) ε + 2 ε (cid:18) − ln (cid:18) n · p c ¯ n · p ¯ c µ (cid:19)(cid:19)(cid:21) + 1 ε N c + iπε N c (cid:27) J EFT I . (15)The counterterm Z EFT I is chosen so as to cancel the divergence in Eq. (15), Z EFT I = α s π (cid:26) C F (cid:20) ε + 2 ε (cid:18) − ln (cid:18) n · p c ¯ n · p ¯ c µ (cid:19)(cid:19)(cid:21) + 1 ε N c + iπε N c (cid:27) . (16)From the definition (11), Eq. (16), and recalling that dα s /d ln µ = − εα s + O ( α s ), theanomalous dimension at one loop is γ EFT I = − α s ( µ )4 π (cid:26) C F + N c + 4 C F ln (cid:18) µ √ n · p c ¯ n · p ¯ c (cid:19) + iπ N c (cid:27) . (17)An important feature of the anomalous dimension (17) is the presence of a term propor-tional to ln µ . Because of this term, the RGE (12) can be used to resum Sudakov double12ogarithms. As we will show shortly, the general solution of Eq. (12) can be written in thefollowing form: C ( µ ) = C ( µ ) (cid:18) µ √ n · p c ¯ n · p ¯ c (cid:19) g ( µ , µ ) exp U ( µ , µ ) , (18)where g and U depend on the initial scale µ and the final scale µ that we run down to. Foran anomalous dimension of the form (17), U can be expanded as a series, U ( µ , µ ) = ∞ X n =1 α ns ( µ ) n +1 X L =0 u n,L ln n − L +1 µµ . (19)If µ/µ ≪
1, the most relevant terms in the expansion (19) are those with L = 0, which wecall “leading logs” (LL). Terms with higher L are subleading; we call the terms with L = 1“next-to-leading logs” (NLL), those with L = 2 “next-to-next-leading logs” (NNLL), and,if L = m , we denote them with N m LL. The RGE (12) determines the coefficients in theexpansion (19). With the anomalous dimensions written as γ EFT I = − (cid:26) γ ( α s ) + Γ( α s ) ln (cid:18) µ √ n · p c ¯ n · p ¯ c (cid:19)(cid:27) , (20)where γ ( α s ) and Γ( α s ) are series in powers of α s , γ ( α s ) = α s π γ (0) + (cid:16) α s π (cid:17) γ (1) + . . . , Γ( α s ) = α s π Γ (0) + (cid:16) α s π (cid:17) Γ (1) + . . . , it can be proved that the coefficients of the LL, u n , are determined by the knowledge of Γ (0) and of the QCD β function at one loop. The NLL coefficients u n are instead completelydetermined if Γ and β are known at two loops and γ ( α s ) at one loop.In the case we are studying, the ratio of the scales µ/µ ∼ m c / m b is not extremelysmall. Indeed, as to be seen shortly, the numerical contributions of the LL and NLL termsin the series (19) are of the same size. It is therefore important to work at NLL accuracy,which requires the calculation of the coefficient of ln µ to two loops. The factors of ln µ areinduced by cusp angles involving light-like Wilson lines and their coefficients are universalΓ( α s ) ∝ Γ cusp ( α s ) [34]. The cusp anomalous dimension Γ cusp ( α s ) is known at two loops [34],Γ cusp ( α s ) = α s π Γ (0)cusp + (cid:16) α s π (cid:17) Γ (1)cusp , (21)with Γ (0)cusp = 4 C F , Γ (1)cusp = 4 C F (cid:20)(cid:18) − π (cid:19) N c − n f (cid:21) , (22)13hile the constant of proportionality between Γ( α s ) and Γ cusp ( α s ) is fixed by the one-loopcalculation. Since we have determined γ (0) , γ (0) = 3 C F + N c + i πN c , (23)and the β function is known, we have all the ingredients to provide the NLL approximationfor U ( µ , µ ) and g ( µ , µ ). Taking into account the tree-level initial condition in Eq. (6), Eq.(18) determines the leading-order matching coefficient, with NLL resummation.The solution (18) can be derived by writing Eq. (12) as d ln C = − dαβ ( α ) (cid:26) γ ( α ) + Γ cusp ( α ) (cid:20) ln (cid:18) µ √ n · p c ¯ n · p ¯ c (cid:19) + Z αα ( µ ) dα ′ β ( α ′ ) (cid:21)(cid:27) , (24)where we have used the definition of the β function, β ( α ) = dα/d ln µ , to write ln µ and d ln µ in terms of α . Integrating both sides from µ to µ and exponentiating the result wefind the form given in Eq. (18), with U ( µ , µ ) = − Z α s ( µ ) α s ( µ ) dαβ ( α ) (cid:26) γ ( α ) + Γ cusp ( α ) Z αα ( µ ) dα ′ β ( α ′ ) (cid:27) ,g ( µ , µ ) = − Z α s ( µ ) α s ( µ ) dαβ ( α ) Γ cusp ( α ) . (25)At NLL, we find U ( µ b , µ ) = 2 π Γ (0)cusp β " r − − r ln rα s ( µ ) + β γ (0)Re π Γ (0)cusp ln r + Γ (1)cusp Γ (0)cusp − β β ! − r + ln r π + β πβ ln r (cid:21) + γ (0)Im β ln r , (26)and g ( µ b , µ ) = Γ (0)cusp β " ln r + Γ (1)cusp Γ (0)cusp − β β ! α s ( µ b )4 π ( r − , (27)where r = α s ( µ ) /α s ( µ b ) and we have renamed the initial scale µ b , to denote its connectionto the scale 2 m b . In Eqs. (26) and (27) we have used the two-loop beta function, β ( α s ) = − α s (cid:18) α s π β + (cid:16) α s π (cid:17) β (cid:19) , (28)with β = 11 − n f , β = 343 N c − N c n f − C F n f . (29)In Eq. (26) we have kept the contributions of the real and imaginary part of γ (0) separated.The imaginary part of γ (0) changes the phase of the matching coefficient C ( µ ), but this phase14s irrelevant for the calculation of physical observables like the decay rate, which depend onthe square modulus of C ( µ ). In Sec. V the factor U ( µ b , µ ) will be evaluated between thescales µ b = 2 m b and µ = m c , with n f = 4 active quark flavors. The numerical evaluationshows that the LL term, represented by the first term in the brackets in Eq. (26), is slightlysmaller than and have the opposite sign of the term proportional to γ (0)Re , which dominatesthe NLL contribution. This observation confirms, a posteriori , the necessity to work at NLLaccuracy in the resummation of logarithms of m c / m b .The RGE (12) and its solution (18) thus allow us to rewrite Eq. (5) as J QCD = C ( µ ) J EFT I ( µ ) = C ( µ b = 2 m b ) exp U (2 m b , m c ) J EFT I ( µ = m c ) , which avoids the occurrence of any large logarithm in the matching coefficient or in thematrix element of the effective operator. IV. pNRQCD + bHQETA. Matching In the second step, we integrate out the soft modes by matching EFT I onto EFT II . InNRQCD + SCET, contributions to the exclusive decay processes are obtained by consideringtime-ordered products of J EFT I and the terms in the EFT I Lagrangian that contain soft-gluon emissions. The soft gluons have enough virtuality to produce a pair of light quarkstravelling in opposite directions with ultracollinear momentum scaling. These light quarksbind to the charm quarks to form back-to-back D mesons. The total momentum of two back-to-back ultracollinear quarks is 2 m b Λ QCD /m c (1 , , λ ) and the invariant mass of the pair is q ∼ (2 m b Λ QCD /m c ) ∼ m c : in NRQCD + SCET, only soft gluons have enough energy toproduce them. The time-ordered products in NRQCD +SCET are matched onto six-fermionoperators in pNRQCD + bHQET, where fluctuations of order m c cannot be resolved.We consider the scale µ ′ = m c to be much bigger than Λ QCD , so the matching can be donein perturbation theory. The Feynman diagrams contributing to the matching are shown inFig. 5. The gluon and the b -quark propagators have off-shellness of order m c , so the twodiagrams on the l.h.s. match onto six-fermion operators on the r.h.s.The amplitude for the decay of a bottomonium with quantum numbers S +1 L J into two15 IG. 5: Matching NRQCD + SCET onto pNRQCD + bHQET. On the r.h.s. the double solid linesrepresent heavy b (¯ b ) (anti)quarks, the double dashed lines bHQET c (¯ c ) (anti)quarks, and thesingle dashed lines collinear light quarks. D mesons has the following form: i M = iC ( µ ) Z dωω d ¯ ω ¯ ω T ( ω, ¯ ω, µ, µ ′ ; S +1 L J ) F ( µ ′ ) h DA, DB |O S +1 L J AB ( ω, ¯ ω, µ ′ ) | ¯ bb ( S +1 L J ) i . (30) A and B , which label the final states and the EFT II operators O S +1 L J AB , denote the possibleparity, spin, and polarization of the D mesons, A, B = { P, V L , V T } , indicating respectivelya pseudoscalar D meson, a longitudinally-polarized vector meson D ∗ , and a transversely-polarized vector meson D ∗ . Unlike J EFT I , we have dropped the subscript EFT II in O S +1 L J AB in order to simplify the notation.The EFT II operators that contribute to the decay of the P -wave states are F ( µ ′ ) O P J P P ( ω, ¯ ω, µ ′ ) = χ † ¯ b ~p b · ~σ ⊥ ψ b ¯ H c ¯ n / n γ δ ( − ¯ ω − n · P ) χ ¯ l ¯ n ¯ χ ln δ (cid:0) ω − ¯ n · P † (cid:1) /¯ n γ H ¯ cn ,F ( µ ′ ) O P J V L V L ( ω, ¯ ω, µ ′ ) = χ † ¯ b ~p b · ~σ ⊥ ψ b ¯ H c ¯ n / n δ ( − ¯ ω − n · P ) χ ¯ l ¯ n ¯ χ ln δ (cid:0) ω − ¯ n · P † (cid:1) /¯ n H ¯ cn ,F ( µ ′ ) O P J V T V T ( ω, ¯ ω, µ ′ ) = χ † ¯ b p ( µb ⊥ σ ν ) ⊥ ψ b ¯ H c ¯ n / n γ µ ⊥ δ ( − ¯ ω − n · P ) χ ¯ l ¯ n ¯ χ ln δ (cid:0) ω − ¯ n · P † (cid:1) /¯ n γ ν ⊥ H ¯ cn , (31)where p ( µb ⊥ σ ν ) ⊥ is a symmetric, traceless tensor, p ( µb ⊥ σ ν ) ⊥ = 12 ( p µb ⊥ σ ν ⊥ + p νb ⊥ σ µ ⊥ − g µν ⊥ ~p b · ~σ ⊥ ) . At leading order in the EFT II expansion, the η b can only decay into a pseudoscalar and avector meson, with an operator given by F ( µ ′ ) O S P V L ( ω, ¯ ω, µ ′ ) = χ † ¯ b ψ b (cid:20) ¯ H c ¯ n / n γ δ ( − ¯ ω − n · P ) χ ¯ l ¯ n ¯ χ ln δ (cid:0) ω − ¯ n · P † (cid:1) /¯ n H ¯ cn + ¯ H c ¯ n / n δ ( − ¯ ω − n · P ) χ ¯ l ¯ n ¯ χ ln δ (cid:0) ω − ¯ n · P † (cid:1) /¯ n γ H ¯ cn (cid:21) . (32)16or later convenience, in the definition of the effective operators (31) and (32) we havefactored out the term F ( µ ′ ), which is related to the D -meson decay constant. The definitionof F ( µ ′ ) will become clear when we introduce the D -meson distribution amplitudes. Thefields χ ln and χ ¯ l ¯ n are ultracollinear gauge-invariant light-quark fields, while H c ¯ n = W † ¯ n h c ¯ n and H ¯ cn = W † n h ¯ cn are bHQET heavy-quark fields, which are invariant under an ultracollineargauge transformation. The Wilson lines W n and W ¯ n have the same definition as in Eq. (8),with the restriction to ultracollinear gluons. Eqs. (31) and (32) allow us to interpret ω as thecomponent of the light-quark momentum along the direction n . Similarly, ¯ ω represents thecomponent of the light-antiquark momentum along ¯ n . The minus sign in the delta function δ ( − ¯ ω − n · P ) is chosen so that ¯ ω is positive.The tree-level matching coefficients are T ( ω, ¯ ω, µ, µ ′ = m c ; P J ) = C F N c πα s ( m c ) m b ω + ¯ ω ,T ( ω, ¯ ω, µ, µ ′ = m c ; S ) = C F N c πα s ( m c ) m b ω − ¯ ωω + ¯ ω . (33)Note that, at leading order in the EFT II expansion, the matching coefficient T ( ω, ¯ ω, µ, µ ′ ; P J ) is independent of the spin and polarization of the final states, or of thetotal angular momentum J of the χ b .An important feature of bHQET is that the ultracollinear and ultrasoft sectors can bedecoupled at leading order in the power counting by a field redefinition reminiscent of thecollinear-usoft decoupling in SCET [7] [18]. For bHQET in the n direction, the decouplingis achieved by defining h ¯ cn → Y n h ¯ cn and ¯ ξ ln → ¯ ξ ln Y † n , where Y n is an ultrasoft Wilson line, Y n = X perms h exp (cid:16) − gn · P n · A us (cid:17)i . (34)An analogous redefinition with n → ¯ n decouples ultrasoft from ¯ n -ultracollinear quarks andgluons. These redefinitions do not affect the operators in Eqs. (31) and (32) because allthe induced Wilson lines cancel out. As a consequence, at leading order in the EFT II powercounting, there is no interaction between the initial and the final states, since the formercan only emit and absorb ultrasoft gluons that do not couple to ultracollinear degrees offreedom. Furthermore, fields in the two copies of bHQET, boosted in opposite directions,cannot interact with each other because the interaction with a ¯ n -ultracollinear gluon wouldgive a n -ultracollinear quark or gluon a virtuality of order m c , which, however, cannot appear17n EFT II . The matrix elements of the operators O S +1 L J AB ( ω, ¯ ω, µ ), therefore, factorize as F ( µ ′ ) h AB |O S +1 L J A B ( ω, ¯ ω, µ ′ ) | ¯ bb i = h | χ † ¯ b T S +1 L J AB ψ b | ¯ bb i h A | ¯ H c ¯ n / n A δ ( − ¯ ω − n · P ) χ ¯ l ¯ n | ih B | ¯ χ ln δ (cid:0) ω − ¯ n · P † (cid:1) /¯ n B H ¯ cn | i , (35)where Γ A = { γ , , γ µ ⊥ } and T S +1 L J AB = { , ~p b · ~σ ⊥ , p ( µb ⊥ σ ν ) ⊥ } . The charge-conjugated contribu-tion is understood in the η b case.The quarkonium state and the D mesons in Eq. (35) have respectively non-relativisticand HQET normalization: h χ bJ ( E ′ , ~p ′ ) | χ bJ ( E, ~p ) i = (2 π ) δ (3) ( ~p − ~p ′ ) , h D ( v ′ , k ′ ) | D ( v, k ) i = 2 v δ v,v ′ (2 π ) δ (3) ( ~k − ~k ′ ) , where v is the 0th component of the 4-velocity v µ .The D -meson matrix elements can be expressed in terms of the D -meson light-conedistribution amplitudes: h P | ¯ χ ln / ¯ n γ δ (cid:0) ω − ¯ n · P † (cid:1) H ¯ cn | i = iF P ( µ ′ ) ¯ n · v φ P ( ω, µ ′ ) , (36) h V L | ¯ χ ln / ¯ n δ (cid:0) ω − ¯ n · P † (cid:1) H ¯ cn | i = F V L ( µ ′ ) ¯ n · v φ V L ( ω, µ ′ ) , (37) h V T | ¯ χ ln / ¯ n γ µ ⊥ δ (cid:0) ω − ¯ n · P † (cid:1) H ¯ cn | i = F V T ( µ ′ ) ¯ n · v ε µ ⊥ φ V T ( ω, µ ′ ) , (38)where ε µ ⊥ is the transverse polarization of the vector meson. The constants F A ( µ ′ ), with A = { P, V L , V T } , are related to the matrix elements of the local heavy-light currents incoordinate space. In the heavy-quark limit, where D and D ∗ are degenerate, F A is the samefor all the three states: F ≡ F P = F V L = F V T . In this limit, h | ¯ ξ ¯ ln /¯ n γ h cn (0) | P i = − iF ( µ ′ ) ¯ n · v ′ . (39)At tree level, the matrix element is proportional to the D -meson decay constant f D =205 . ± . ± . F ( µ ′ ) = f D √ m D , where the factor √ m D is dueto HQET normalization. The scale dependence of F is determined by the renormalizationof heavy-light HQET currents. At one loop, Ref. [32] showed that dd ln µ ′ F ( µ ′ ) = − γ F F ( µ ′ ) = 3 C F α s π F ( µ ′ ) . (40)The pNRQCD matrix elements can be expressed in terms of the heavy quarkoniumwavefunctions. The operator χ † ¯ b ~p b · ~σ ⊥ ψ b contains a component with J = 0 and a component18ith J = 2 and J z = 0, so its matrix element has non-vanishing overlap with both χ b and χ b . The operator χ † ¯ b p ( µb σ ν ) ⊥ ψ b instead has only contributions with J = 2 and J z = ± χ b . In terms of the bottomonium wavefunctions, thepNRQCD matrix elements are expressed as h | χ † ¯ b ~p b · ~σ ⊥ ψ b | χ b i = 2 √ r N c π R ′ χ b (0 , µ ′ ) , (41) h | χ † ¯ b ~p b · ~σ ⊥ ψ b | χ b i = − r r N c π R ′ χ b (0 , µ ′ ) , (42) h | χ † ¯ b p ( µb σ ν ) ⊥ ψ b | χ b i = ( ε (2) µν + ε ( − µν ) r N c π R ′ χ b (0 , µ ′ ) , (43)where R ′ χ bJ (0) is the derivative of the radial wavefunction of the χ bJ evaluated at the origin.At leading order, the pNRQCD Hamiltonian does not depend on J , so, up to corrections oforder w , R ′ χ b (0) = R ′ χ b (0). The numerical pre-factors in Eqs. (41) and (42) follow fromdecomposing ~p b · ~σ ⊥ into components with definite J z . ε ( j ) µν is the polarization tensor of the χ b state, and Eq. (43) states that, at leading order in the w expansion, only the particleswith polarization J z = ± χ b decay into two transversely-polarized vectormesons. Similarly, one finds h | χ † ¯ b ψ b | η b i = r N c π R η b (0 , µ ′ ) . (44)The factorization of the matrix elements (35) implies that the decay rate also factorizes.For the decays of χ b and χ b into two pseudoscalar mesons or two longitudinally-polarizedvector mesons, we findΓ ( χ b → AA ) = 43 m D q m χ b − m D πm χ b N c π | C ( µ ) | | R ′ χ b (0 , µ ′ ) | (cid:20) F ( µ ′ ) n · v ′ n · v Z dωω d ¯ ω ¯ ω T (cid:0) ω, ¯ ω, µ, µ ′ ; P J (cid:1) φ A (¯ ω, µ ′ ) φ A ( ω, µ ′ ) (cid:21) (45)andΓ ( χ b → AA ) = 215 m D q m χ b − m D πm χ b N c π | C ( µ ) | | R ′ χ b (0 , µ ′ ) | (cid:20) F ( µ ′ ) n · v ′ n · v Z dωω d ¯ ω ¯ ω T (cid:0) ω, ¯ ω, µ, µ ′ ; P J (cid:1) φ A (¯ ω, µ ′ ) φ A ( ω, µ ′ ) (cid:21) , (46)19here A = P, V L . For the decay of χ b into two transversely-polarized vector mesons, onefinds the decay rate by summing over the possible transverse polarizations:Γ ( χ b → V T V T ) = 25 m D q m χ b − m D πm χ b N c π | C ( µ ) | | R ′ χ b (0 , µ ′ ) | (cid:20) F ( µ ′ ) n · v ′ n · v Z dωω d ¯ ω ¯ ω T (cid:0) ω, ¯ ω, µ, µ ′ ; P J (cid:1) φ V T (¯ ω, µ ′ ) φ V T ( ω, µ ′ ) (cid:21) . (47)In the case of η b decay into a pseudoscalar and a longitudinally-polarized vector meson, wefindΓ ( η b → P V L + c . c . ) = m D q m η b − m D πm η b N c π | C ( µ ) | | R η b (0 , µ ′ ) | (cid:20) F ( µ ′ ) n · v ′ n · v Z dωω d ¯ ω ¯ ω T (cid:0) ω, ¯ ω, µ, µ ′ ; S (cid:1) ( φ V L (¯ ω, µ ′ ) φ P ( ω, µ ′ ) − φ V L ( ω, µ ′ ) φ P (¯ ω, µ ′ )) (cid:21) . (48)Note that we are working in the limit m c → ∞ , where the m D ∗ − m D mass splitting vanishes.The factorized formulas Eqs. (35) and (45) - (48) are the main results of this paper. Eachdecay rate of (45) - (48) depends on two calculable matching coefficients, C and T , and threenon-perturbative, process-independent matrix elements, namely, two D -meson distributionamplitudes and the bottomonium wavefunction. In Sec. V we will provide a model-dependentestimate of the decay rates (45) - (48) and will discuss the phenomenological implications.We conclude this section by observing that all the non-perturbative matrix elements cancelout in the ratios Γ( χ b → P P ) / Γ( χ b → P P ) and Γ( χ b → V L V L ) / Γ( χ b → V L V L ), sincethe spin symmetry of pNRQCD guarantees R ′ χ b (0) = R ′ χ b (0), at leading order in EFT II .Neglecting the χ b - χ b mass difference, we find, up to corrections of order w ,Γ( χ b → AA ) / Γ( χ b → AA ) = 43 152 = 10 , (49)with A = P, V L . B. Running
The dependence of the matching coefficient T ( ω, ¯ ω, µ, µ ′ ; S +1 L J ) and of the operators inEqs. (45) - (48) on the scale µ ′ is driven by a RGE that can be obtained by renormalizing20he EFT II operators. The RGE for the EFT II operators, which also defines the anomalousdimension γ EFT II , is similar to Eq. (10), dd ln µ ′ h F ( µ ′ ) O S +1 L J AB ( ω, ¯ ω, µ ′ ) i = − Z dω ′ Z d ¯ ω ′ γ EFT II ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ ) F ( µ ′ ) O S +1 L J AB ( ω ′ , ¯ ω ′ , µ ′ ) . (50)To calculate the anomalous dimension at one loop, we compute the divergent part of thediagrams in Figs. 6 and 7. As mentioned in Sec. II, the pNRQCD Lagrangian has thefollowing structure, L pNRQCD = Z d x L usoft NRQCD + L pot , where the superscript usof t indicates that the gluons in the NRQCD Lagrangian are purelyultrasoft ( m b w , m b w ), while L pot contains four-fermions operators, which are non-local inspace, L pot = Z d x d x ψ † α ( t, ~x ) χ β ( t, ~x ) V αβ,γδ ( ~r ) χ † γ ( t, ~x ) ψ δ ( t, ~x ) . At leading order in α s ( m b w ) and r , V is the Coulomb potential V αβ,γδ = α s ( m b w ) r t aαδ t aγβ . For the explicit form of higher-order potentials, see, for example, Refs. [12] [31]. Verticesfrom L pot generate one-loop diagrams as the first diagram in Fig. 6. However, these diagramsdo not give any contribution to the anomalous dimension at one loop. Indeed, the insertionof the Coulomb potential 1 /r in Fig. 6 does not produce UV divergences. Insertions of the1 /m b potentials yield divergences but the coefficient of the 1 /m b potential is proportional to α s ( m b w ), so it is not relevant if we are content with a NLL resummation. Insertions of 1 /m b potentials give divergences proportional to subleading operators, which can be neglected.The second diagram in Fig. 6 yields a result completely analogous to the last term in Eq.(13), with the only difference of a color pre-factor, i M pNRQCD = − i α s π C F ε O S +1 L J AB ( ω, ¯ ω, µ ) . (51)This divergence is completely cancelled by the b -quark field renormalization constant Z b ,and hence the pNRQCD diagrams in Fig. 6 do not contribute to the anomalous dimensionat one loop. 21 IG. 6: One-loop diagrams in pNRQCD. The first diagram contains insertions of quark-antiquarkpotentials. In the second diagram the gluon is ultrasoft.FIG. 7: One-loop diagrams in bHQET. There are three analogous diagrams for the other copy ofbHQET.
On the bHQET side, the third diagram in Fig. 7 is convergent, and hence it does notcontribute to the anomalous dimension. The first two diagrams give i M bHQET , ¯n = i Z dω ′ d ¯ ω ′ ∆( ω, ω ′ , ¯ ω, ¯ ω ′ ) O S +1 L J AB ( ω ′ , ¯ ω ′ , µ ) , (52)with∆( ω, ω ′ , ¯ ω, ¯ ω ′ ) = α s π C F δ ( ω − ω ′ ) (cid:26) δ (¯ ω − ¯ ω ′ ) (cid:20) − ε − ε ln (cid:18) µ ′ n · v ′ ¯ ω ′ (cid:19) + 1 ε (cid:21) + 1 ε (cid:20) θ (¯ ω − ¯ ω ′ ) (cid:18) ω − ¯ ω ′ (cid:19) + + θ (¯ ω ′ − ¯ ω ) θ (¯ ω ) ¯ ω ¯ ω ′ (cid:18) ω ′ − ¯ ω (cid:19) + (cid:21)(cid:27) . (53)The diagrams for the bHQET copy in the n -direction give a result analogous to Eqs. (52)and (53), with ¯ ω → ω , ¯ ω ′ → ω ′ , and n · v ′ → ¯ n · v . Extracting γ EFT II from the divergence isagain standard, just as we did in the case of γ EFT I . After adding to Eq. (53) the bHQETfield renormalization constants Z h and Z ξ for heavy and light quarks Z h = 1 + 1 ε α s π C F , Z ξ = 1 − ε α s π C F , we find γ EFT II ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ ) = 2 γ F δ ( ω − ω ′ ) δ (¯ ω − ¯ ω ′ ) + γ O ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ ) , (54)22ith γ O ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ )= α s π C F δ ( ω − ω ′ ) δ (¯ ω − ¯ ω ′ ) (cid:20) − (cid:18) µ ′ n · v ′ ¯ ω ′ (cid:19) + ln (cid:18) µ ′ ¯ n · vω ′ (cid:19)(cid:21) − α s π C F δ ( ω − ω ′ ) (cid:20) θ (¯ ω − ¯ ω ′ ) (cid:18) ω − ¯ ω ′ (cid:19) + + θ (¯ ω ′ − ¯ ω ) θ (¯ ω ) ¯ ω ¯ ω ′ (cid:18) ω ′ − ¯ ω (cid:19) + (cid:21) − α s π C F δ (¯ ω − ¯ ω ′ ) (cid:20) θ ( ω − ω ′ ) (cid:18) ω − ω ′ (cid:19) + + θ ( ω ′ − ω ) θ ( ω ) ωω ′ (cid:18) ω ′ − ω (cid:19) + (cid:21) . (55)The term proportional to γ F in Eq. (54) reproduces the running of F ( µ ′ ) (40). γ O isresponsible for the running of the D -meson distribution amplitudes and it agrees with theresult found in Ref. [36]. Also, in Eq. (55) the coefficient of ln µ ′ is proportional to Γ cusp ( α s ).Note that, since the bHQET Lagrangian is spin-independent, the anomalous dimension doesnot depend on the spin or on the polarization of the D meson in the final state, at leadingorder in the power counting.Using Eqs. (50) and (54) we find the following integro-differential RGE for the operator O ( ω, ¯ ω, µ ′ ): dd ln µ ′ O ( ω, ¯ ω, µ ′ ) = − Z dω ′ Z d ¯ ω ′ γ O ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ ) O ( ω ′ , ¯ ω ′ , µ ′ ) , (56)where we have dropped both the subscripts A , B , and the superscript S +1 L J , since γ O does not depend on the quantum numbers of the initial or final state. Using the factthat the convolution of F ( µ ′ ) T ( ω, ¯ ω, µ, µ ′ ; S +1 L J ) and the operator O S +1 L J AB ( ω, ¯ ω, µ ′ ) is µ ′ -independent, we can write an equation for the coefficient, dd ln µ ′ (cid:2) F ( µ ′ ) T ( ω, ¯ ω, µ, µ ′ ) (cid:3) = Z dω ′ Z d ¯ ω ′ ωω ′ ¯ ω ¯ ω ′ F ( µ ′ ) T ( ω ′ , ¯ ω ′ , µ, µ ′ ) γ O ( ω ′ , ω ; ¯ ω ′ , ¯ ω ; µ ′ )= Z dω ′ Z d ¯ ω ′ F ( µ ′ ) T ( ω ′ , ¯ ω ′ , µ, µ ′ ) γ O ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ ) , (57)where the last line follows from the property of γ O at one loop, ωω ′ ¯ ω ¯ ω ′ γ O ( ω ′ , ω ; ¯ ω ′ , ¯ ω ; µ ′ ) = γ O ( ω, ω ′ ; ¯ ω, ¯ ω ′ ; µ ′ ) , as can be explicitly verified from the expression in Eq. (55).Eq. (57) can be solved following the methods described in Ref. [36]. We discuss the detailsof the solution in App. A, where we derive the analytic expressions for T ( ω, ¯ ω, µ, µ ′ ; P J )and T ( ω, ¯ ω, µ, µ ′ ; S ), with the initial conditions at the scale µ ′ c = m c expressed in Eq. (33).23 . DECAY RATES AND PHENOMENOLOGY In Sec. IV A we gave the factorized expressions for the decay rates (45) - (48): Γ( χ b , → P P ), Γ( χ b , → V L V L ), Γ( χ b → V T V T ), and Γ( η b → P V L + c . c . ). In Secs. III B and IV B weexploited the RGEs (12) and (57) to run the scales µ and µ ′ , respectively, from the matchingscales µ = 2 m b and µ ′ = m c to the natural scales that contribute to the matrix elements, µ = m c and µ ′ ∼ m c / m b and m c / D meson andof the longitudinally- and transversely-polarized D ∗ mesons, and the wavefunctions of thestates η b and χ bJ . In principle, these non-perturbative objects could be extracted from other η b , χ b , and D -meson observables. In the case of the η b , the value of the wavefunction at theorigin can be obtained from a measurement of the inclusive hadronic width or of the decayrate for the electromagnetic process η b → γγ , since they are both proportional to | R η b (0) | .Unfortunately, at the moment there are not sufficient data on η b decays. Another way toproceed is to use the spin symmetry of the leading-order pNRQCD Hamiltonian, whichimplies R η b (0) = R Υ (0), and to extract the Upsilon wavefunction from Γ(Υ → e + e − ) =1 . ± .
07 KeV [37]. Using the leading-order expression for Γ(Υ → e + e − ) [38], one finds | R Υ (0) | = 6 . ± .
38 GeV , where the error only includes the experimental uncertainty.The above value is in good agreement with the lattice evaluation by Bodwin, Sinclair, andKim [39] and it falls within the range of values obtained with four different potential models,as listed in Ref. [40]. | R ′ χ b , (0) | can be obtained from the electromagnetic decay χ b , → γγ . Unfortunately,such decay rates have not been measured yet. The values listed in Ref. [40] range from aminimum of | R ′ χ bJ (0) | = 1 .
417 GeV , obtained with the Buchmuller-Tye potential [41], to amaximum of | R ′ χ bJ (0) | = 2 .
067 GeV , obtained with a Coulomb-plus-linear potential. Thelattice value is roughly of the same size, | R ′ χ bJ (0) | = 2 . , with an uncertainty of about15% [39]. We use this value in our estimate.For the pseudoscalar D -meson distribution amplitude we use two model functions widelyadopted in the study of B physics. A first possible choice, suggested for example in Ref.2436], is a simple exponential decay: φ Exp P, ( ω, µ ′ = 1GeV) = θ ( ω ) ωλ exp (cid:18) − ωλ D (cid:19) . (58)Another form, suggested in Ref. [42], is φ Braun P, ( ω, µ ′ = 1 GeV) = θ (˜ ω ) 4 λ D π ˜ ω ω (cid:20)
11 + ˜ ω − σ D − π ln ˜ ω (cid:21) , (59)where ˜ ω = ω/µ ′ . The theta function in Eqs. (58) and (59) reflects the fact that thedistribution amplitudes φ A ( ω, µ ′ ), with A = { P, V L , V T } , have support on ω > D -meson rest frame,with a HQET velocity-label v = (1 , , , D meson has a velocity ( n · v, ¯ n · v, ∼ ( m c / m b , m b /m c , φ P ( ω, µ ′ ) = 1¯ n · v φ P, (cid:16) ω ¯ n · v , µ ′ (cid:17) , (60)as shown in App. B. λ D and σ D in Eqs. (58) and (59) are, respectively, the first inversemoment and the first logarithmic moment of the D -meson distribution amplitude in the D -meson rest frame, λ − D ( µ ′ ) = Z ∞ dωω φ P, ( ω, µ ′ ) ,σ D ( µ ′ ) λ − D ( µ ′ ) = − Z ∞ dωω ln (cid:18) ωµ ′ (cid:19) φ P, ( ω, µ ′ ) . Furthermore we assume that the vector-meson distribution amplitudes φ V L ( ω ) and φ V T ( ω )have the same functional form as φ P ( ω ), but with different parameters λ D ∗ L , σ D ∗ L and λ D ∗ T , σ D ∗ T .The D -meson distribution amplitude and its moments have not been intensively studiedunlike, for example, the B -meson distribution amplitude. Therefore, we invoke heavy-quarksymmetry and use the moments of the B -meson distribution amplitude in order to estimatethe decay rate. However, the value of λ B is affected by a noticeable uncertainty. Using QCDsum rules, Braun et al. estimated [42] λ B ( µ ′ = 1 GeV) = 0 . ± .
110 GeV, where theuncertainty is about 25%. Other authors [44] [45] [46] give slightly different central valuesand comparable uncertainties, so that λ B falls in the range 0 .
350 GeV < λ B < .
600 GeV.The first logarithmic moment σ D is given in Ref. [42], σ D = σ B ( µ ′ = 1 GeV) = 1 . ± .
4. We25ssume that the moments of the D ∗ -meson distribution amplitudes fall in the same rangeas the moments of φ P ( ω ).We evaluate numerically the convolution integrals in Eqs. (45) - (48). We choose thematching scales µ b and µ ′ c to be 2 m b and m c respectively. Using the RGEs we run thematching coefficients down to the scales µ = m c and µ ′ = 1 GeV. For the b and c quarkmasses we adopt the 1S mass definition [47], m b (1 S ) = m Υ . ± .
13 MeV ,m c (1 S ) = m J/ψ . ± .
01 MeV . (61)The values of α s at the relevant scales are [37] α s (2 m b ) = 0 . ± . α s ( m c ) = 0 . ± . α s (1 GeV) ∼ .
5. With these choices, the value of g in Eq. (A5) is g ( m c , − . ± . χ bJ → AA ) with A = { P, V L , V T } , (45) - (47), depend on the masses ofthe χ bJ and of the D mesons, whose most recent values are reported in Ref. [37]. Since theeffects due to the mass splitting of the χ bJ and D multiplets are subleading in the EFT powercounting, we use in the evaluation the average mass of the χ bJ multiplet and the averagemass of D and D ∗ mesons: m χ bJ = 9898 . ± . ± .
31 MeV and m D = 1973 . ± . D mesons in χ bJ decay is ¯ n · v = n · v ′ = m χ bJ /m D = 5 . η b → P V L + c . c . ) (48) depends on the mass of the η b , which has been recently measured: m η b = 9388 . +3 . − . ± . D meson in the η b decay is ¯ n · v = n · v ′ = m η b /m D = 4 .
76, again with negligible error.The decay rate Γ( χ b → P P ) (45), obtained with φ Exp and φ Braun separately, is shownin Fig. 8. In order to see the impact of resumming Sudakov logarithms, we show for bothdistribution amplitudes the results with ( i ) the LL and NLL resummations and ( ii ) withoutany resummation at all. In the plots, we call the resummed results NLL-resummed, indi-cating that Sudakov logarithms are resummed up to NLL. For both distribution amplitudesthe resummation does have a relevant effect on the decay rate. In the case of φ Exp theresummation decreases the decay rate by a factor of 2 − . λ D goes from the lowest tothe highest value under consideration. In the case of φ Braun the decay rate decreases too, forexample, by a factor 1.5 when σ D = 1 .
4. In Fig. 9 we compare the decay rates obtained withthe two distribution amplitudes. Over the range of λ D we are considering the two decayrates are in rough agreement with each other.26 (GeV) D l ( K e V ) G NLL resummationNo resummation (GeV) D l ( K e V ) G NLL resummationNo resummation
FIG. 8: Γ( χ b → P P ) as a function of λ D , calculated with the distribution amplitudes φ Exp ( left )and φ Braun ( right ). The dash dotted and solid lines denote the NLL-resummed decay rate. Forcomparison, the decay rate without resummation is also shown, denoted by dash double-dotted( left ) and dashed ( right ) lines. For φ Braun we vary the parameter σ D from σ D = 1 (lower curve)to σ D = 1 . σ D = 1 . (GeV) D l ( K e V ) G Exp f Braun f FIG. 9: Γ( χ b → P P ) as a function λ D . The dash dotted line denotes the decay rate calculatedwith φ Exp , while the three solid lines with φ Braun . For φ Braun we vary the value of the parameter σ D from σ D = 1 (lower curve) to σ D = 1 . σ D = 1 . Figs. 8 - 9 also describe the relation between the decay rate Γ( χ b → V L V L ) and λ D ∗ L .According to Eqs. (46) and (47), the processes χ b → P P , χ b → V L V L , and χ b → V T V T show an analogous dependence on the first inverse moments of the light-cone distributionamplitudes, and they differ from Figs. 8 - 9 by constant pre-factors. Therefore, we do notshow explicitly their plots.Qualitatively, Figs. 8 - 9 show a dramatic dependence of the decay rate on the inverse27oment λ D . Using Eqs. (45), (60) and (A16), one can show that when φ Braun is used, thedecay rate is proportional to λ − D , while it scales as λ − − gD when we adopt φ Exp , with g definedin Eq. (A5). As a consequence, the decay rate drops by an order of magnitude when λ D goesfrom 0 .
350 GeV to 0 .
600 GeV. The particular sensitivity of exclusive bottomonium decaysinto two charmed mesons to the light-cone structure of the D meson —much stronger thanusually observed in D - and B -decay observables— is due to the dependence of the amplitudeon the product of two distributions (one for each meson) and to the non-trivial dependenceof the matching coefficient T on the light-quark momentum labels ω and ¯ ω at tree level. Onone hand, the strong dependence on a relatively poorly known quantity prevents us frompredicting the decay rate Γ( χ b → DD ). On the other hand, however, it suggests that,if the decay rate is measured, this channel could be used to better determine interestingproperties of the D -meson distribution amplitude, such as λ D and σ D . The viability of thissuggestion relies on the control over the theoretical error attached to the curves in Fig. 8and on the actual chances to observe the process χ b → DD at current experiments.The uncertainty of the decay rate stems mainly from three sources. First, there arecorrections coming from subleading EFT operators. In matching NRQCD + SCET ontopNRQCD + bHQET (Sec. IV A), we neglected the subleading EFT II operators that aresuppressed by powers of Λ QCD /m c and w , relative to the leading EFT II operators in Eqs.(31) and (32). In matching QCD onto NRQCD + SCET (Sec. III A), we kept only J EFT I (6) and neglected subleading EFT I operators, suppressed by powers of λ and w . Thesesubleading EFT I operators would match onto subleading EFT II operators, suppressed bypowers of Λ QCD /m c and w . Using w ∼ . QCD /m c ∼ .
3, we find a conservativeestimate for the non-perturbative corrections to be about 30%.Second, there are perturbative corrections to the matching coefficients C and T . Since α s (2 m b ) = 0 . T ( ω, ¯ ω, µ, µ ′ ; S +1 L J ) would be proportional to α s ( m c ) ∼ µ b and µ ′ c . If the matching coefficients C and T and the anomalous dimensions γ EFT I and γ O ( ω, ω ′ , ¯ ω, ¯ ω ′ ; µ ′ ) were known at all orders, the decay rate would be independent of thematching scales µ b and µ ′ c . However, since we only know the first terms in the perturbativeexpansions, the decay rate bears a residual renormalization-scale dependence, whose size is28 (GeV) D l ( K e V ) G b = 2 m b m = 20 GeV b m = 5 GeV b m (GeV) D l ( K e V ) G c = m c ’ m = 2.5 GeV c ’ m = 1.2 GeV c ’ m FIG. 10:
Left : Scale dependence of Γ( χ b → P P ) on the matching scale µ b . We vary µ b from acentral value µ b = 2 m b (solid line) to a maximum of µ b = 20 GeV (dashed line) and a minimumof µ b = 5 GeV (dotted line). The dashed and dotted lines overlap almost perfectly. Right : Scaledependence of Γ( χ b → P P ) on the matching scale µ ′ c . We varied µ ′ c from a central value of µ ′ c = m c (solid line) to a maximum of µ ′ c = 2 . µ ′ c = 1 . determined by the first neglected terms.In Fig. 10 we show the effect of varying µ b between 4 m b ∼
20 GeV and m b ∼ φ Braun . The solid line represents the choice µ b = 2 m b , while the dashedand dotted lines, which overlap almost perfectly, correspond respectively to µ b = 20 GeVand µ b = 5 GeV. The dependence on µ b is mild, its effect being a variation of about 5%. Weobtain analogous results for the decay rate computed with φ Exp , which are not shown herein order to avoid redundancy.On the other hand, even after the resummation, the decay rate strongly depends on µ ′ c .We vary this scale between 1 . . T ( ω, ¯ ω, µ, µ ′ ; P J ). This observation is reinforced by the factthat the numerical values of the running factors U ( µ b , µ ) and V ( µ ′ c , µ ′ ) (defined respectivelyin Eqs. (26) and (A6)) at NLL accuracy are smaller than expected on the basis of naivecounting of the logarithms. As a consequence, the next-to-leading-order corrections to thematching coefficient could be as large as the effect of the NLL resummation. In the light ofFig. 10, the one-loop correction to T ( ω, ¯ ω, µ, µ ′ ; P J ) seems to be an important ingredient29or a reliable estimate of the decay rate.A third source of error comes from the unknown functional form of the D -meson distri-bution amplitude. For the study of the B -meson shape function, an expansion in a completeset of orthonormal functions has recently been proposed and it has provided a systematicprocedure to control the uncertainties due to the unknown functional form [48]. The samemethod should be generalized to the B - and D -meson distribution amplitudes, in order toreduce the model dependence of the decay rate. We leave such an analysis to future work.To summarize, the calculation of the one-loop matching coefficients and the inclusion ofpower corrections of order Λ QCD /m c appear to be necessary to provide a decay rate withan accuracy of 10%, that would make the decays χ bJ → D + D − , χ bJ → D ¯ D competitiveprocesses to improve the determination of λ D and σ D , if the experimental decay rate isobserved with comparable accuracy.We estimate the decay rate Γ( η b → P V L + c . c . ) (48) using φ Exp and φ Braun for both φ P and φ V L . In the limit m c → ∞ , spin symmetry of the bHQET Lagrangian would imply theequality of the pseudoscalar and vector distribution amplitudes, φ P = φ V L , and hence thevanishing of the decay rate Γ( η b → P V L + c . c . ). Assuming spin-symmetry violations, thedecay rate depends on ( i ) the two parameters ¯ λ D = ( λ D + λ D ∗ L ) / δ = ( λ D ∗ L − λ D ) / ( λ D + λ D ∗ L ), if φ Exp is used, and on ( ii ) three parameters ¯ λ D , δ , and | σ D ∗ L − σ D | , if φ Braun is used.The two plots in the left column of Fig. 11 show the decay rate, computed with φ Exp , asa function of ¯ λ D with δ adopting various values, and as a function of δ with ¯ λ D now beingthe parameter. In the right column, the decay rate computed with φ Braun is shown. Since inthis case the decay rate does not strongly depend on δ , we fix it at δ = 0 and we show thedependence of the decay rate on ¯ λ D and | σ D ∗ L − σ D | . We “normalize” the difference betweenthe first logarithmic moments by dividing them by σ = 2 σ D .The most striking feature of Fig. 11 is the huge sensitivity to the chosen functional form.Though a precise comparison is difficult, due to the dependence on different parameters,the decay rate increases by two orders of magnitude when we switch from φ Exp to φ Braun .Once again, this effect hinders our ability to predict Γ( η b → P V L + c . c . ) but it opens up theinteresting possibility to discriminate between different model distribution amplitudes.Using Eqs. (48) and (A17), we know that Γ( η b → P V L + c . c . ) goes like ¯ λ − − gD when φ Exp used or ¯ λ − D when φ Braun used. Fig. 11 appears to confirm this strong dependence on ¯ λ D .The plots in the lower half of Fig. 11 reflect the fact that the decay rate vanishes if one30 (GeV) D l ( K e V ) G -3 · = -0.15 d = -0.1 d = 0.1 d = 0.15 d (GeV) D l ( K e V ) G = 0.05 s |/ D s - L D* s | = 0.1 s |/ D s - L D* s | = 0.15 s |/ D s - L D* s | d -0.15 -0.1 -0.05 0 0.05 0.1 0.15 ( K e V ) G -3 · = 0.300 GeV D l = 0.400 GeV D l = 0.500 GeV D l = 0.600 GeV D l s |/ D s - L D* s |0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ( K e V ) G D l = 0.400 GeV D l = 0.500 GeV D l = 0.600 GeV D l FIG. 11:
Left : Γ( η b → P V L + c . c . ) as a function of λ D and δ , computed using exponentialdistribution amplitudes φ Exp P and φ Exp V L . Right : Γ( η b → P V L + c . c . ) as a function of λ D and | σ D ∗ L − σ D | /σ , computed with the Braun distribution amplitudes φ Braun P and φ Braun V L . assumes φ P ( ω ) = φ V L ( ω ).We conclude this section with the determination of the branching ratios B ( χ b → P P ) =Γ( χ b → P P ) / Γ( χ b → light hadrons) and B ( η b → P V L +c.c.) = Γ( η b → P V L +c.c.) / Γ( η b → light hadrons). At leading order in pNRQCD, the only non-perturbative parameter involvedin the inclusive decay width of the η b is | R η b (0) | [4],Γ( η b → light hadrons) = 2Im f ( S ) m b N c π | R η b (0) | . (62)Therefore, B ( η b → P V L + c . c . ) does not depend on the quarkonium wavefunction and theonly non-perturbative parameters in B ( η b → P V L + c . c . ) are those describing the D -mesondistribution amplitudes.For P -wave states, the inclusive decay rate was obtained in Refs. [4] [49], where thecontributions of the configurations in which the quark-antiquark pair is in a color-octet S -wave state were first recognized. In pNRQCD the inclusive decay rate is written as [50]31 (GeV) D l -6 · Braun f Exp f B (GeV) D l -6 · = 0.05 s |/ D s - L D* s | = 0.1 s |/ D s - L D* s | = 0.15 s |/ D s - L D* s | B FIG. 12: Branching ratios B ( χ b → P P ) ( left ) and B ( η b → P V L + c . c . ) ( right ). The latter iscomputed using the distribution amplitude φ Braun . [51] Γ( χ b → light hadrons) = 1 m b N c π | R ′ χ b (0) | (cid:20) Im f ( P ) + 19 N c Im f ( S ) E (cid:21) , (63)where the color-octet matrix element has been expressed in terms of the heavy quarkoniumwavefunction and of the gluonic correlator E , whose precise definition is given in Ref. [50]. E is a universal parameter and is completely independent of any particular heavy quarkoniumstate under consideration. Its value has been obtained by fitting to existing charmoniumdata and, thanks to the universality, the same value can be used to predict properties ofbottomonium decays. It is found in Ref. [50] E = 5 . +3 . − . . The matching coefficients inEqs. (62) and (63) are known to one loop. For the updated value we refer to Ref. [52] andreferences therein. For reference, the tree-level values of the coefficients are as follows [4]:Im f ( S ) = α s (2 m b ) π C F N c , Im f ( P ) = 3 α s (2 m b ) π C F N c , Im f ( S ) = n f α s (2 m b ) π . (64)With the above parameters, we plot B ( χ b → P P ) and B ( η b → P V L +c . c . ) as a function of λ D and ¯ λ D , respectively, in Fig. 12. Over the range of λ D we are considering, B ( χ b → P P )varies between 4 · − and 4 · − ; it is approximately one or two orders of magnitudesmaller than the branching ratios observed in Ref. [25] for χ bJ decays into light hadrons. B ( η b → P V L + c . c . ) depends on the choice of the distribution amplitude. Choosing theparameterization φ Braun (59), it appears that, despite the suppression at | σ D ∗ L − σ D | = 0, B ( η b → P V L + c . c . ) assumes values comparable to B ( χ b → P P ) even for a small deviationfrom the spin-symmetry limit. If φ Exp is chosen, the branching ratio is suppressed over a32ide range of | σ D ∗ L − σ D | . The branching ratio B ( η b → P V L + c . c . ) was first estimated in[53]. The authors of [53] assumed that the exclusive decays into DD ∗ dominate the inclusivedecay into charm, Γ( η b → P V L + c.c.) ∼ Γ( η b → c ¯ c + X ). With this assumption, theyestimated the branching ratio to be in the range 10 − < B ( η b → P V L + c . c . ) < − . Ouranalysis shows that such an assumption does not appear to be justified in the range of ¯ λ D considered in Fig. 12, while it would be appropriate for smaller values of the first inversemoments, for example for ¯ λ D ∼ .
200 GeV if the distribution amplitudes are described by φ Braun .Our estimates indicate that observing the exclusive processes η b → DD ∗ + c.c. and χ b → DD would be extremely challenging. A preliminary analysis for χ b → D ¯ D [54]suggests that the number of Υ(2 S ) produced at BABAR allows for the measurement of abranching ratio B ( χ b → D ¯ D ) ∼ − , which is two or three orders of magnitude biggerthan the values in Fig. 12. An even bigger branching ratio would be required for the smallerΥ(2 S ) sample of CLEO. However, we stress once again the strong dependence of the decayrates on the values of the first inverse moments. In particular, our estimates rely on therelation λ D = λ B , which is valid in the limit of m b , m c → ∞ ; even small corrections to theheavy flavor symmetry, if they had the effect of shifting the value of λ D towards the range0 . − .
350 GeV, could considerably increase the branching ratios.
VI. CONCLUSIONS
In this paper we have analyzed the exclusive decays of the C -even bottomonia into apair of charmed mesons. We approached the problem using a series of EFTs that lead tothe factorization formulas for the decay rates (Eqs. (45) - (48)), valid at leading order inthe EFT power counting and at all orders in α s . We improved the perturbative results byresumming Sudakov logarithms of the ratios of the characteristic scales that are germane tothe dynamics of the processes.The decay rates (45) - (48) receive both perturbative and non-perturbative corrections.Perturbative corrections come from loop corrections to the matching coefficients C and T , which are respectively of order α s (2 m b ) ∼ . α s ( m c ) ∼ .
3. The largest non-perturbative contribution could be as big as Λ
QCD /m c , which would amount approximatelyto a 30% correction. Therefore, corrections to the leading-order decay rates could be notice-33ble, as the strong dependence of the decay rates on the renormalization-scale µ ′ c suggests.However, the EFT approach shown in this paper allows for a systematic treatment of bothperturbative corrections and power-suppressed operators, so that, if the experimental datarequire, it is possible to extend the present analysis beyond the leading order.For simplicity, we have focused in this paper on the decays of C -even bottomonia, in whichcases the decays proceed via two intermediate gluons and both the matching coefficients C and T are non-trivial at tree level. The same EFT approach can be applied to thedecays of C -odd states, in particular, to the decays Υ → DD and Υ → D ∗ D ∗ , with thecomplication that the matching coefficient T arises only at one-loop level. Moreover, thesame EFT formalism developed in this paper can be applied to the study of the channelsthat have vanishing decay rates at leading order in the power counting, such as η b → D ∗ D ∗ ,Υ → DD ∗ + c.c., and χ b → DD ∗ + c.c.. Experimental data for the charmonium systemshow that, for the decays of charmonium into light hadrons, the expected suppression of thesubleading twist processes is not seen. It is interesting to see whether such an effect appearsin bottomonium decays into two charmed mesons, using the EFT approach of this paper toevaluate the power-suppressed decay rates.Finally, in Sec. V we used model distribution amplitudes to estimate the decay rates.The most evident, qualitative feature of the decay rates is the strong dependence on theparameters of the D -meson distribution amplitude. Even though this feature may preventus from giving reliable estimates of the decay rates or of the branching ratios, it makes thechannels analyzed here ideal candidates for the extraction of important D -meson parameters,when the branching ratios can be observed with sufficient accuracy. Acknowledgments
We would like to thank S. Fleming for proposing this problem and for countless usefuldiscussions, N. Brambilla and A. Vairo for suggestions and comments and R. Briere, V. M.Braun and S. Stracka for helpful communications. BwL is grateful for hospitality to theUniversity of Arizona, where part of this work was finished. This research was supportedby the US Department of Energy under grants DE-FG02-06ER41449 (RA and EM) andDE-FG02-04ER41338 (RA, BwL and EM).34
PPENDIX A: SOLUTION OF THE RUNNING EQUATION IN PNRQCD +BHQET
The RGE in Eq. (57) can be solved by applying the methods discussed in Ref. [36] tofind the evolution of the B -meson distribution amplitude. We generalize this approach tothe specific case discussed here, where two distribution amplitudes are present. FollowingRef. [36], we define ω Γ( ω, ω ′ , α s ) = − α s C F π (cid:20) θ ( ω − ω ′ ) (cid:18) ω − ω ′ (cid:19) + + θ ( ω ′ − ω ) θ ( ω ) ωω ′ (cid:18) ω ′ − ω (cid:19) + (cid:21) . Lange and Neubert [36] prove that Z dω ′ ω Γ( ω, ω ′ , α s )( ω ′ ) a = ω a F ( a, α s ) , (A1)with F ( a, α s ) = α s C F π [ ψ (1 + a ) + ψ (1 − a ) + 2 γ E ] .ψ is the digamma function and γ E the Euler constant. Eq. (A1) is valid if − < Re a <
1. Exploiting (A1), a solution of the running equation Eq. (57) with initial condition T ( ω, ¯ ω, µ ′ ) = ( ω/µ ′ ) η (¯ ω/µ ′ ) ξ at a certain scale µ ′ is F ( µ ′ ) T ( ω, ¯ ω, µ ′ ) = F ( µ ′ ) f ( ω, µ ′ , µ ′ , η ) f (¯ ω, µ ′ , µ ′ , ξ ) , (A2)with f ( ω, µ ′ , µ ′ , η ) = (cid:18) ωµ ′ (cid:19) η − g (¯ n · v ) g exp U ( µ ′ , µ ′ , η ) ,g ≡ g ( µ ′ , µ ′ ) = Z α s ( µ ′ ) α s ( µ ′ ) dαβ ( α ) Γ cusp ( α ) ,U ( µ ′ , µ ′ , η ) = Z α s ( µ ′ ) α s ( µ ′ ) dαβ ( α ) " Γ cusp ( α ) Z αα s ( µ ′ ) dα ′ β ( α ′ ) + γ ( α ) + F ( η − g, α ) ,γ ( α s ) = − α s C F π . (A3)The function f (¯ ω, µ ′ , µ ′ , ξ ) has the same form as f ( ω, µ ′ , µ ′ , η ) and is obtained by replacing ω → ¯ ω , η → ξ , and ¯ n · v → n · v ′ in Eq. (A3). The integrals over α can be performedexplicitly using the beta function in Eq. (28). The result is f ( ω, µ ′ , µ ′ , η ) f (¯ ω, µ ′ , µ ′ , ξ ) = (cid:18) ωµ ′ (cid:19) η − g (cid:18) ¯ ωµ ′ (cid:19) ξ − g (¯ n · v n · v ′ ) g exp [ V ( µ ′ , µ ′ )]Γ(1 − η + g )Γ(1 + η )Γ(1 + η − g )Γ(1 − η ) Γ(1 − ξ + g )Γ(1 + ξ )Γ(1 + ξ − g )Γ(1 − ξ ) , (A4)35here, at NLL, g ( µ ′ , µ ′ ) = − Γ (0)cusp β ( ln r + Γ (1)cusp Γ (0)cusp − β β ! α s ( µ ′ )4 π ( r − ) , (A5)and V ( µ ′ , µ ′ ) = − Γ (0)cusp πβ ( r − − r ln rα s ( µ ′ ) + Γ (1)cusp Γ (0)cusp − β β ! − r + ln r π + β πβ ln r ) + C F β (2 − γ E ) ln r , (A6)with r = α s ( µ ′ ) /α s ( µ ′ ). Notice that in the running from µ ′ = m c to µ ′ = 1 GeV only threeflavors are active, so in the expressions for β , β , and Γ (1)cusp we use n f = 3.Eq. (A4) is the solution for the initial condition T ( ω, ¯ ω, µ ′ ) = ( ω/µ ′ ) η (¯ ω/µ ′ ) ξ . To solvethe RGE for a generic initial condition, we express T as the Fourier transform with respectto ln ω/µ ′ , T ( ω, ¯ ω, µ ′ ) = 1(2 π ) Z + ∞−∞ drds exp (cid:18) − ir ln ωµ ′ (cid:19) exp (cid:18) − is ln ¯ ωµ ′ (cid:19) F [ T ]( r, s, µ ′ )= 1(2 π ) Z + ∞−∞ drds (cid:18) ωµ ′ (cid:19) − ir (cid:18) ¯ ωµ ′ (cid:19) − is F [ T ]( r, s, µ ′ ) , where F [ T ] denotes the Fourier transform of T . From the solution (A2)-(A4) it follows that F ( µ ′ ) T ( ω, ¯ ω, µ ′ ) = F ( µ ′ )(2 π ) Z + ∞−∞ drds (cid:18) ωµ ′ (cid:19) − ir − g (cid:18) ¯ ωµ ′ (cid:19) − is − g (¯ n · v n · v ′ ) g F [ T ]( r, s, µ ′ )exp [ V ( µ ′ , µ ′ )] Γ(1 + ir + g )Γ(1 − ir )Γ(1 − ir − g )Γ(1 + ir ) Γ(1 + is + g )Γ(1 − is )Γ(1 − is − g )Γ(1 + is ) . (A7)The Fourier transform of the matching coefficient in Eq. (A7) has to be understood inthe sense of distributions [55]. That is, we define the Fourier transform of T as the functionof r and s that satisfies1(2 π ) Z drds F [ T ]( r, s, µ ′ ) ϕ A ( r, µ ′ ) ϕ B ( s, µ ′ ) = Z + ∞ dωω d ¯ ω ¯ ω T ( ω, ¯ ω, µ ′ ) φ A ( ω, µ ′ ) φ B (¯ ω, µ ′ ) , (A8)or, more precisely, F [ T ]( r, s, µ ′ ) is the linear functional that acts on the test functions ϕ A ( r )and ϕ B ( s ) according to1(2 π ) ( F [ T ]( r, s, µ ′ ) , ϕ A ( r, µ ′ ) ϕ B ( s, µ ′ )) = Z + ∞ dωω d ¯ ω ¯ ω T ( ω, ¯ ω, µ ′ ) φ A ( ω, µ ′ ) φ B (¯ ω, µ ′ ) . (A9)36he function ϕ A is the Fourier transform of the D -meson distribution amplitude, ϕ A ( r, µ ′ ) = Z ∞ dωω (cid:18) ωµ ′ (cid:19) ir φ A ( ω, µ ′ ) , (A10)where the integral on the r.h.s. should converge in the ordinary sense because of the reg-ularity properties of the D -meson distribution amplitude. As in Sec. IV, the subscript A denotes the spin and polarization of the D meson.In the distribution sense, the Fourier transform of the coefficient 1 / ( ω + ¯ ω ) is F (cid:20) ω + ¯ ω (cid:21) ( r, s, µ ′ ) = (2 π ) µ ′ δ ( r + s + i ) sech h π r − s ) i = 12 (2 π ) µ ′ δ ( R + i ) sech h π S i , (A11)where R = r + s , S = r − s , and the factor comes from the Jacobian of the change ofvariables. The hyperbolic secant is defined as sech = 1 / cosh. Similarly, we find F (cid:20) ω − ¯ ωω + ¯ ω (cid:21) ( R, S, µ ′ ) = i π ) δ ( R ) (cid:16) cosech h π S + iε i + cosech h π S − iε i(cid:17) . (A12)The δ function in Eq. (A11) has complex argument. The definition is analogous to theone in real space [55], ( δ ( R + i ) , ϕ ( R )) = ϕ ( − i ) . (A13)Using Eqs. (A11) and (A12), we can perform the integral in Eq. (A7), obtaining respectively T ( ω, ¯ ω, µ, µ ′ ; P J ) and T ( ω, ¯ ω, µ, µ ′ ; S ). In order to give an explicit example, we proceedusing Eq. (A11). Integrating the δ function we are left with F ( µ ′ ) T ( ω, ¯ ω ; µ ′ ) = F ( µ ′ ) exp [ V ( µ ′ , µ ′ )] 1 µ ′ (cid:18) µ ′ ω ¯ ω (cid:19) / g (¯ n · v n · v ′ ) g Z ∞−∞ dS exp (cid:20) − i S ω ¯ ω (cid:21) sech h π S i
11 + S Γ (cid:0) + g + i S (cid:1) Γ (cid:0) − g − i S (cid:1) Γ (cid:0) + g − i S (cid:1) Γ (cid:0) − g + i S (cid:1) . (A14)The integral (A14) can be done by contour. The integrand has poles along the imaginaryaxis. In S = ± i there is a double pole, coming from the coincidence of one pole of thehyperbolic secant and the singularities in 1 / (1 + S ). The Γ functions in the numerator havepoles respectively in S = ± i (2 n + 3 + 2 g ) with n >
0, while the other poles of sech are in S = ± i (2 n + 1), with n ≥
1. We close the contour in the upper half plane for ¯ ω > ω and in37he lower half plan for ω > ¯ ω , obtaining F ( µ ′ ) T ( ω, ¯ ω, µ ′ ) = F ( µ ′ ) exp [ V ( µ ′ , µ ′ )] θ (¯ ω − ω ) 1¯ ω (cid:18) µ ′ ¯ n · v n · v ′ ω ¯ ω (cid:19) g (cid:26) Γ(1 + g )Γ(2 + g )Γ(1 − g )Γ( − g ) h − ln ω ¯ ω + ψ (1 − g ) − ψ ( − g ) + ψ (1 + g ) − ψ (2 + g ) i + ∞ X n =1 ( − ) n +1 (cid:16) ω ¯ ω (cid:17) n n ( n + 1) Γ(1 − n + g )Γ(2 + n + g )Γ( − n − g )Γ(1 − g + n ) − ∞ X n =1 (cid:16) ω ¯ ω (cid:17) n + g π ( n − gπ ) 1( n + g )(1 + n + g ) Γ(2 + n + 2 g )Γ(1 + n )Γ( − n − g ) ) + ( ω → ¯ ω ) , (A15)with csc( gπ ) = 1 / sin( gπ ) and ψ is the digamma function. More compactly, we can expressEq. (A15) using the hypergeometric functions F and F , F ( µ ′ ) T ( ω, ¯ ω, µ, µ ′ ; P J ) = F ( µ ′ c ) C F N c πα s ( µ ′ c ) m b exp [ V ( µ ′ c , µ ′ )] (cid:18) µ ′ c ¯ n · vn · v ′ ω ¯ ω (cid:19) g θ (¯ ω − ω )¯ ω (cid:26) Γ(1 + g )Γ(2 + g )Γ(1 − g )Γ( − g ) h − ln ω ¯ ω + ψ (1 − g ) − ψ ( − g ) + ψ (1 + g ) − ψ (2 + g ) i + 12 ω ¯ ω Γ( g + 2)Γ( g + 3)Γ(1 − g )Γ(2 − g ) F (cid:16) , , g + 2 , g + 3; 3 , − g, − g ; − ω ¯ ω (cid:17) − (cid:16) ω ¯ ω (cid:17) g gπ ) Γ(2 + 2 g ) g + 2 F (cid:16) g + 1 , g + 2 , g + 3; 2 , g + 3; − ω ¯ ω (cid:17)(cid:27) + ( ω → ¯ ω ) , (A16)where we have introduced the constants that appear in the initial condition in Eq. (33). Inthe same way, we obtain F ( µ ′ ) T ( ω, ¯ ω, µ, µ ′ ; S ) = F ( µ ′ c ) C F N c πα s ( µ ′ c ) m b exp [ V ( µ ′ c , µ ′ )] θ (¯ ω − ω ) (cid:18) µ ′ c ¯ n · vn · v ′ ω ¯ ω (cid:19) g (cid:26) g )Γ(2 + g )Γ(1 − g )Γ(2 − g ) ω ¯ ω F (cid:16) , g + 1 , g + 2; 1 − g, − g ; − ω ¯ ω (cid:17) + Γ (1 + g )Γ (1 − g ) − (cid:16) ω ¯ ω (cid:17) g gπ )Γ(1 + 2 g )Γ(2 g + 2) F (cid:16) g + 2 , g + 1; 2; − ω ¯ ω (cid:17)(cid:27) − ( ω → ¯ ω ) . (A17)In Eqs. (A16) and (A17) we renamed the initial scale µ ′ = µ ′ c to denote its connection tothe scale m c . Setting µ ′ = µ ′ c or, equivalently, g = 0, it can be explicitly verified that thesolutions Eqs. (A16) and (A17) satisfy the initial conditions Eq. (33).38 PPENDIX B: BOOST TRANSFORMATION OF THE D -MESON DISTRIBU-TION AMPLITUDE We derive in this Appendix the relation between the distribution amplitudes in the D -meson and in the bottomonium rest frames, as given in Eq. (60). In the D -meson rest frame,characterized by the velocity label v = (1 , , , h | ¯ ξ ¯ ln (0) /¯ n γ h cn (0) | D i v = − iF ( µ ′ ) ¯ n · v . (B1)The matrix element of the heavy- and light-quark fields at a light-like separation z µ = n · z ¯ n µ / φ ( n · z , µ ′ ) in coordinate space: h | ¯ χ ¯ ln ( n · z ) /¯ n γ H cn (0) | D i v = − iF ( µ ′ ) ¯ n · v φ ( n · z , µ ′ ) . (B2)Eqs. (B1) and (B2) imply ˜ φ (0 , µ ′ ) = 1. In the definitions (B1) and (B2) the subscript 0is used to denote quantities in the D -meson rest frame. This convention is used in the restof this Appendix. In the bottomonium rest frame, where the velocity label in light-conecoordinates is v = ( n · v, ¯ n · v,
0) and the light-like separation is z µ = n · z ¯ n µ /
2, we define h | ¯ ξ ¯ ln (0) /¯ n γ h cn (0) | D i v = − iF ( µ ′ ) ¯ n · v h | ¯ χ ¯ ln ( n · z ) /¯ n γ H cn (0) | D i v = − iF ( µ ′ ) ¯ n · v φ ( n · z, µ ′ ) . (B4)Suppose that Λ is some standardized boost that takes the D meson from v , its velocityin the bottomonium rest frame, to rest. It is straightforward to find the relations betweenthe D -meson momenta in the two frames: n · p = ¯ n · vn · p and ¯ n · p = n · v ¯ n · p . There is a similar relation for the light-cone coordinates, n · z = ¯ n · v n · z . With U (Λ), the unitary operator that implements the boost Λ, one can write U (Λ) | D i v = | D i v .
39e choose Λ such that, for the Dirac fields, U (Λ) ξ ¯ ln ( x ) U − (Λ) = Λ − / ξ ¯ l (Λ x ) and U (Λ) h cn ( x ) U − (Λ) = Λ − / h c (Λ x ) , where Λ / = cosh α n / n − / n /¯ n α , with α related to v by e α = ¯ n · v and e − α = n · v .Now we can write the matrix element in Eq. (B3) as h | ¯ ξ ¯ ln /¯ n γ h cn (0) | D i v = h | U − (Λ) (cid:16) U (Λ) ¯ ξ ¯ ln (0) U − (Λ) (cid:17) /¯ n γ (cid:0) U (Λ) h cn (0) U − (Λ) (cid:1) U (Λ) | D i v = h | ¯ ξ ¯ l (0)Λ / /¯ n γ Λ − / h c (0) | D i v = ¯ n · v h | ¯ ξ ¯ l (0) /¯ n γ h c (0) | D i v = − iF ( µ ′ ) ¯ n · v n · v = − iF ( µ ′ ) ¯ n · v . (B5)where, in the last step, we have used ¯ n · v = 1. Eq. (B5) is thus in agreement with thedefinition in Eq. (B3). Applying the same reasoning to Eq. (B4), one finds h | ¯ χ ¯ ln ( n · z ) /¯ n γ H cn (0) | D i v = ¯ n · v h | ¯ χ ¯ l (¯ n · v n · z ) /¯ n γ H c (0) | D i v = − iF ( µ ′ ) ¯ n · v φ ( n · z , µ ′ ) . (B6)Comparing Eq. (B6) with (B4), we see that ˜ φ ( n · z, µ ′ ) = ˜ φ (¯ n · v n · z, µ ′ ). Note that in thebottomonium rest frame the normalization condition for the distribution amplitude is also˜ φ (0 , µ ′ ) = 1.In the main text of this paper we have used the D -meson distribution amplitudes inmomentum space, φ ( ω , µ ′ ) ≡ π Z dn · z e iω n · z ˜ φ ( n · z , µ ′ ) ,φ ( ω, µ ′ ) ≡ π Z dn · z e iωn · z ˜ φ ( n · z, µ ′ ) . Using Eq. (B6), we can relate the two distributions: φ ( ω, µ ′ ) = 12 π Z dn · z e iωn · z ˜ φ ( n · z, µ ′ ) = 12 π Z dn · z e iωn · z ˜ φ (¯ n · v n · z, µ ′ )= 12 π n · v Z dn · z e i ω ¯ n · v n · z ˜ φ ( n · z, µ ′ ) = 1¯ n · v φ (cid:16) ω ¯ n · v , µ ′ (cid:17) ,
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