into two charmed mesons

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Exclusive decays of bJ and b into two charmed mesonsRegina S. Azevedo,1,* Bingwei Long,1,2, and Emanuele Mereghetti1,1Department of Physics, University of Arizona, Tucson, Arizona 85721, USA2European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), I-38100 Villazzano (TN), Italy3Department of Physics, University of Arizona, Tucson, Arizona 85721, USA(Received 10 September 2009; published 23 October 2009)We develop a framework to study the exclusive two-body decays of bottomonium into two charmedmesons and apply it to study the decays of the C-even bottomonia. Using a sequence of effective fieldtheories, we take advantage of the separation between the scales contributing to the decay processes,2mb mc QCD. We prove that, at leading order in the EFT power counting, the decay rate factorizesinto the convolution of two perturbative matching coefficients and three nonperturbative matrix elements,one for each hadron. We calculate the relations between the decay rate and nonperturbative bottomoniumand D-meson matrix elements at leading order, with next-to-leading log resummation. The phenomeno-logical implications of these relations are discussed.DOI: 10.1103/PhysRevD.80.074026 PACS numbers: 12.39.Hg, 13.25.GvI. INTRODUCTIONThe exclusive two-body decays of heavy quarkoniuminto light hadrons have been studied in the framework ofperturbative QCD by many authors (for reviews, see [1,2]).These processes exhibit a large hierarchy between theheavy-quark mass, which sets the scale for annihilationprocesses, and the scales that determine the dynamicalstructure of the particles in the initial and final states.The large energy released in the annihilation of theheavy-quarkantiquark pair and the kinematics of the de-caywith the products flying away from the decay point intwo back-to-back, almost lightlike directionsallow forrigorously deriving a factorization formula for the decayrate at leading twist (for an up-to-date review of the theo-retical and experimental status of the exclusive decays intolight hadrons, see [3]).For the bottomonium system, a particularly interestingclass of two-body final states is the one containing twocharmed mesons. In these cases the picture is complicatedby the appearance of an additional intermediate scale, thecharm mass mc, which is much smaller than the bottommass mb but is large enough to be perturbative. Thesedecays differ significantly from those involving only lightquarks. The creation of mesons that are made up of purelylight quarks involves creating two quark-antiquark pairs,with the energy shared between the quark and antiquark ineach pair. In the production of two D mesons, however,almost all the energy of the bottomonium is carried awayby 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 andthe intuitive picture of the decay process suggest to tacklethe problem using a sequence of effective field theories(EFTs) that are obtained by subsequently integrating outthe dynamics relevant to the perturbative scalesmb andmc.In the first step, we integrate out the scalemb by describ-ing the b and b with nonrelativistic QCD (NRQCD) [4],and the highly energetic c and c with two copies of soft-collinear effective theory (SCET) [59] in opposite light-cone directions. In the second step, we integrate out thedynamics manifested at scales of order mc by treating thequarkonium with potential NRQCD (pNRQCD) [1012],and the D mesons with a boosted version of heavy-quarkeffective theory (HQET) [1319]. The detailed explanationof why the aforementioned EFTs are employed is offeredin Sec. II. We will prove that, at leading order in the EFTexpansion, the decay rate factors into a convolution of twoperturbative matching coefficients and three (one for eachhadron) nonperturbative matrix elements. The nonpertur-bative matrix elements are process independent and encodeinformation on both the initial and final states.For simplicity, in this paper we focus on the decays ofthe C-even quarkonia bJ and b that, at leading order inthe strong coupling s, proceed via the emission of twovirtual gluons. The same method can be generalized to thedecays of C-odd states and hb, which require an addi-tional virtual gluon. We also refrain from processes thathave vanishing contributions at leading order in the EFTpower counting. So the specific processes studied in thispaper are b0;2 ! DD, b0;2 ! DD, and b ! DD c:c. However, the EFT approach developed in this paperenables one to systematically include power-suppressedeffects, making it possible to go beyond the leading-twistapproximation.The study of the inclusive and exclusive charm produc-tion in bottomonium decays and the study of the roleplayed by the charm mass mc in such processes haverecently drawn renewed attention [2023], in connectionwith the experimental advances spurred in the past fewyears by the abundance of bottomonium data produced at*razevedo@physics.arizona.edulong@ect.itemanuele@physics.arizona.eduPHYSICAL REVIEW D 80, 074026 (2009)1550-7998=2009=80(7)=074026(23) 074026-1 2009 The American Physical Societyhttp://dx.doi.org/10.1103/PhysRevD.80.074026facilities like BABAR, BELLE, and CLEO. The most no-table result was the observation of the bottomoniumground state b, recently reported by the BABAR Col-laboration [24]. Furthermore, the CLEO Collaborationpublished the first results for several exclusive decays ofb into light hadrons [25] and for the inclusive decay of binto open charm [26]. In particular, they measured thebranching ratio BbJ ! D0X, where J is the total angu-lar momentum of the b state, and conclusively showedthat for J 1 the production of open charm is substantial:Bb11P ! D0X 12:59% 1:94%. For the J 0, 2states the data are weaker, but the production of opencharm still appears to be relevant. The measurements ofthe CLEO Collaboration are in good agreement with theprediction of Bodwin et al. [20], where EFT techniques (inparticular, NRQCD) were, for the first time, applied tostudy the production of charm in bottomonium decays.The double-charm decay channels analyzed here havenot yet been observed, so one of our aims is to see if theymay be observable given the current data. Unfortunately,the poor knowledge of the D-meson matrix elements pre-vents us from providing definitive predictions for the decayrates bJ ! DD, bJ ! DD, and b !DD c:c:. As we will show, these rates are indeedstrongly dependent on the parameters of the D- andD-meson distribution amplitudes, in particular, on theirfirst inverse moments D and D : the rates vary by anorder of magnitude in the accepted ranges for D and D .On the other hand, the factorization formula implies thatthese channels, if measured with sufficient accuracy, couldconstrain the form of the D-meson distribution amplitudeand the value of its first inverse moment. In turn, the detailsof the D-meson structure are relevant to other D-mesonobservables, which are crucial for a model-independentdetermination of the Cabibbo-Kobayashi-Maskawa matrixelements jVcdj and jVcsj [27].This paper is organized as follows. In Sec. II we discussthe degrees of freedom and the EFTs we use. In Sec. III Awe match QCD onto NRQCD and SCET at the scale 2mb.The renormalization-group equation (RGE) for the match-ing coefficient is derived and solved in Sec. III B. InSec. IVA the scale mc is integrated out by matchingNRQCD and SCET onto pNRQCD and boosted HQET(bHQET). The renormalization of the low-energy EFToperators is performed in Sec. IVB, with some technicaldetails left to Appendix A. The decay rates are calculatedin Sec. V using two model distribution amplitudes. InSec. VI we draw our conclusions.II. DEGREES OF FREEDOM AND THEEFFECTIVE FIELD THEORIESSeveral well-separated scales are involved in the decaysof the C-even bottomonia b and bJ into two D mesons,making them ideal processes for the application of EFTtechniques. The distinctive structures of the bottomonium(a heavy-quarkantiquark pair) and the D meson (a boundstate of a heavy quark and a light quark) suggest that oneneeds different EFTs to describe the initial and final states.We first look at the initial state. The b is the groundstate of the bottomonium system. It is a pseudoscalarparticle, with spin S 0, orbital angular momentum L 0, and total angular momentum J 0. In what follows wewill often use the spectroscopic notation 2S1LJ, in whichthe b is denoted by1S0. The bJ is a triplet of states withquantum numbers 3PJ. The b and bJ are nonrelativisticbound states of a b quark and a b antiquark. The scales inthe system are the b quark mass mb, the relative momen-tum of the b b pair mbw, the binding energy mbw2, andQCD, the scale where QCD becomes strongly coupled. wis the relative velocity of the quark-antiquark pair in themeson, and from the bottomonium spectrum it can beinferred that w2 0:1. Since mb QCD, mb can beintegrated out in perturbation theory and the bottomoniumcan be described in NRQCD. The degrees of freedom ofNRQCD are nonrelativistic heavy quarks and antiquarks,with energy and momentum E; j ~pj of order mbw2; mbw,light quarks and gluons. In NRQCD, the gluons can be softmbw;mbw, potential mbw2; mbw, and ultrasoft (usoft)mbw2; mbw2. The NRQCD Lagrangian is constructed asa systematic expansion in 1=mb whose first few terms areLNRQCD c yiD0 ~D22mb ~ g ~B2mb . . .c yiD0 ~D22mb ~ g ~B2mb . . .;where c and y annihilate a b quark and a b antiquark,respectively, and denotes higher-order contributions in1=mb. In NRQCD several mass scales are still dynamicaland different assumptions on the hierarchy of these scalesmay lead to different power countings for operators ofhigher dimensionality. However, as long as w 1,higher-dimension operators are suppressed by powers ofw (for a critical discussion on the different power count-ings, we refer to [12]).NRQCD still contains interactions that can excite theheavy quarkonium far from its mass shell, for example,through the interaction of a nonrelativistic quark with a softgluon. In the case mbw QCD, we can integrate outthese fluctuations, perturbatively matching NRQCD ontoa low-energy effective theory, pNRQCD. We are then leftwith a theory of nonrelativistic quarks and ultrasoft gluons,with nonlocal potentials induced by the integration oversoft- and potential-gluon modes. The interactions of theheavy quark with ultrasoft gluons are still described by theNRQCD Lagrangian, with the constraint that all the gluonsare ultrasoft. In the weak coupling regime mbw QCD,the potentials are organized by an expansion in smbw,1=mb, and r, where r is the distance between the quark andantiquark in the quarkonium, r 1=mbw. If we assumeAZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-2mbw2 QCD, each term in the expansion has a definitepower counting in w and the leading potential isCoulombic, V smbw=r.An alternative approach, which does not require a two-step matching, has been developed in the effective theoryvelocity NRQCD (vNRQCD) [2831]. In the vNRQCDapproach there is only one EFT below mb, which is ob-tained by integrating out all the off-shell fluctuations at thehard scale mb and introducing different fields for variouspropagating degrees of freedom (nonrelativistic quarks andsoft and ultrasoft gluons). In spite of the differences be-tween the two formalisms, pNRQCD and vNRQCD giveequivalent final answers in all the known examples inwhich both theories can be applied.We now turn to the structure of the D meson. The mostrelevant features of theDmeson are captured by a descrip-tion in HQET. In HQET, in order to integrate out the inertscale mc, the momentum of the heavy quark is genericallywritten as [15]p mcv k; (1)where v is the four-velocity label, satisfying v2 1, and kis the residual momentum. If one chooses v to be thecenter-of-mass velocity of the D meson, k scales as kvQCD. Introducing the light-cone vectors n 1; 0; 0; 1and n 1; 0; 0;1, one can express the residual mo-mentum in light-cone coordinates, k n kn=2 n k n=2 k? or simply k n k; n k; ~k?. There are tworelevant frames. One is theD-meson rest frame, in which vis conveniently chosen as v0 1; 0; 0; 0, and the other isthe bottomonium rest frame, in which the D mesons arehighly boosted in opposite directions, with v chosen asv vD, the four-velocity of one of the D mesons. By asimple consideration of kinematics and the scaling kvQCD, one can work out the scalings for k in the twoframes. In the D-meson rest frame, kQCD1; 1; 1, andin the bottomonium rest frame (supposing the D mesonmoving in the positive z direction),kQCDn vD; n vD; 1 QCD n vD2; 1; ; (2)where n vD 2mb=mc and mc=2mb 1. It is con-venient for the calculation in this paper to use the botto-monium rest frame, so we drop the subscript in vD and weassume v vD in the rest of this paper. The momentumscaling in Eq. (2) is called ultracollinear (ucollinear), andbHQET is the theory that describes heavy quarks withultracollinear residual momenta and light degrees of free-dom (including gluons and light quarks) with the samemomentum scaling.The bHQET Lagrangian is organized as a series inpowers of QCD=mc and, for residual momentum ultracol-linear in the n direction, the leading term is [18]L bHQET hniv Dhn; (3)where the field hn annihilates a heavy quark and thecovariant derivative D contains ultracollinear and ultrasoftgluons,iD n2i n @ g n An n2in @ gn An gn Aus i@? gAn;?: (4)The ultrasoft gluons only enter in the small component ofthe covariant derivative. This fact can be exploited todecouple ultrasoft and ultracollinear modes in theleading-order Lagrangian through a field redefinition remi-niscent of the collinear-ultrasoft decoupling in SCET[7,18]. The ultracollinear-ultrasoft decoupling is an essen-tial ingredient for the factorization of the decay rate.Therefore, the appropriate EFT to calculate the decayrate is a combination of pNRQCD, for the bottomonium,and two copies of bHQET, with fields collinear to the n andn directions, for theD and Dmesons, symbolically writtenas EFTII pNRQCD bHQET.As we mentioned earlier, we plan to describe the botto-monium structure with a two-step scheme QCD !NRQCD ! pNRQCD. However, at the intermediate stage,where we first integrate out the hard scale 2mb and arrive atthe scale mbw, the D meson cannot yet be described inbHQET. This is because the interactions relevant at theintermediate scale mbw can change the c-quark velocityand leave the D meson off shell of order mbw2 m2c 2QCD. Highly energetic c and c traveling in oppo-site directions can be described properly by SCET withmass. Thus, at the scale 2mb, we match QCD onto anintermediate EFT, EFTI NRQCD SCET, in which theEFT expansion is organized by and w. The degrees offreedom of EFTI are tabulated in Table I.TABLE I. Degrees of freedom in EFTINRQCD SCET. w is the b b relative velocity in the bottomonium rest frame, while mc=2mb is the SCET expansion parameter. We assume mbwmc (or, equivalently, w ) and mbw2 mb2 QCD.NRQCD Field Momentum SCET Field MomentumQuark b, b c b, b mbw2; mbw c, c cn, cn 2mb1; 2; , 2mb2; 1; Gluon Potential A mbw2; mbw Collinear An , An 2mb1; 2; , 2mb2; 1; Soft A mbw;mbw Soft As 2mb; ; Usoft A mbw2; mbw2 Usoft Aus 2mb2; 2; 2EXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-3Then, we integrate out mc and mbw at the same time,matching EFTI onto EFTII at the scale 0 mc. In EFTII,the low-energy approximation is organized by QCD=mcand w. The degrees of freedom of EFTII are summarized inTable II. When no subscript is specified in the rest of thispaper, any reference to EFT applies to both EFTI andEFTII. To facilitate the power counting, we adopt w QCD=mc. As a first study, wewill perform in this paper theleading-order calculation of the bottomonium decay rates.III. NRQCD SCETA. MatchingIn the first step, we integrate out the dynamics related tothe hard scale 2mb by matching the QCD diagrams for theproduction of a c c pair in the annihilation of a b b pair ontotheir EFTI counterparts. The tree-level diagrams for theprocess are shown in Fig. 1. The gluon propagator in theQCD diagram has off-shellness of order q2 2mb2 and itis not resolved in EFTI, giving rise to a pointlikeinteraction.We calculate the diagrams on shell, findingiJQCD iCJEFTI; (5)with, at tree level,JEFTI yb?tac b cnSyn?taSn cn andC 2mb s2mbm2b; (6)where ta are color matrices and the symbol denotes thefour matrices 1; ~, with ~ the Pauli matrices. Thesubscript ? refers to the components orthogonal to thelight-cone vectors n and n. The fields c b and ybaretwo-component spinors that annihilate, respectively, a bquark and a b antiquark. cn; np and cn;np are collineargauge-invariant fermion fields: cn; np Wyn cn np; cn;np Wyn cnnp; (7)where Wn is defined asWn Xpermsexp gn P n An: (8)W n has an analogous definition with n ! n. Collinearfields are labeled by the large component of their momen-tum. Note, however, we omit in Eq. (6) the subscripts n pand n p of the collinear fermion fields, in order to sim-plify the notation. The operator n P in the definition (8) isa label operator that extracts the large component of themomentum of a collinear field, n Pn; np n pn; np,where n; np is a generic collinear field. Sn n is a softWilson line,Sn Xpermsexp gn P n As; (9)where the operator n P acts on soft fields, n Ps n ks.Since in SCET different gluon modes are represented bydifferent fields, we have to guarantee the gauge invarianceof the operator JEFTI under separate soft and collineargauge transformations. A soft transformation is definedby Vsx expias ta, with @V 2mb; ; , while agauge transformation Ux is n collinear if Ux expiaxta and @Ux 2mb2; 1; . It has beenshown in Ref. [7] that collinear fields do not transformunder a soft transformation and that the combinationWyn nis gauge invariant under a collinear transformation. Softfields do not transform under collinear transformations butthey do under soft transformations. For example, theNRQCD quark and antiquark fields transform as c b !Vsxc b. The soft Wilson line has the same transformation,Sn ! VsxSn. Therefore, yb?tac b transforms as anoctet under soft gauge transformations. SincecnSyn?taSn cn behaves like an octet as well, JEFTI isinvariant. It is worth noting that the soft Wilson lines arenecessary to guarantee the gauge invariance of JEFTI . WeTABLE II. Degrees of freedom in EFTIIpNRQCD bHQET. The scale Q in bHQET is Q n v0QCD for the n-collinear sectorand Q n vQCD for the n-collinear sector. n v0 and n v are the large light-cone components of the D-meson velocities in thebottomonium rest frame, n v0 n v 2mb=mc. and w are defined as in Table I.pNRQCD Field Momentum bHQET Field MomentumQuark b, b c b, b mbw2; mbw c, c hcn, h cn Q1; 2; , Q2; 1; u, d n, n Q1; 2; , Q2; 1; Gluon Usoft A mbw2; mbw2 Usoft Aus Q; ; Ucollinear An , An Q1; 2; , Q2; 1; FIG. 1. Matching QCD onto EFTI. On the r.h.s., the doublelines represent the nonrelativistic b ( b) (anti)quark, while thedashed lines represent the collinear c ( c) (anti)quark.AZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-4have explicitly checked their appearance at one gluon bymatching QCD diagrams like the one in Fig. 1, with all thepossible attachments of an extra soft or collinear gluon,onto four-fermion operators in EFTI.B. RunningThe matching coefficient C and the effective operatorJEFTI depend on the renormalization scale . Since theeffective operator is sensitive to the low-energy scales inEFTI, logarithms that would appear in the evaluation ofJEFTI are minimized by the choice mc. On the otherhand, since the coefficient encodes the high-energy dy-namics of the scale 2mb, such a choice would induce largelogarithms of mc=2mb in the matching coefficient. Theselogarithms can be resummed using RGEs in NRQCDSCET.The dependence of JEFTI is governed by an equationof the following form [32],dd lnJEFTI EFTIJEFTI; (10)where the anomalous dimension EFTI is given byEFTI Z1EFTIdd lnZEFTI (11)and ZEFTI is the counterterm that relates the bare operatorJ0EFTI to the renormalized one, J0EFTI ZEFTIJEFTI.Since the left-hand side (l.h.s.) of Eq. (5) is independent ofthe scale , the RGE (10) can be recast as an equation forthe matching coefficient C,dd lnC EFTIC: (12)The counterterm ZEFTI cancels the divergences that appearin Green functions with the insertion of the operator JEFTI .We calculate ZEFTI in the MS scheme by evaluating thedivergent part of the four-point Green function at one loop,given by the diagrams in Figs. 24.Since in NRQCD we do not introduce different gluonfields for different momentum modes, soft and ultra-soft in Figs. 2 and 3 refer to the convention that weimpose soft or ultrasoft scaling to the corresponding loopmomentum. The potential region, which should be consid-ered in the diagrams of Fig. 2, does not give any divergentcontribution.The integrals are evaluated in dimensional regulariza-tion, with d 4 2". We regulate the infrared divergen-ces by keeping the nonrelativistic b and b and the collinearc and c off shell: Eb; b ~p2b; b=2mb b, p2c m2c 2,and p2c m2c 2. We power count the c-quark off-shellness as 2 2 m2b2 and the b-quark off-shellness as b mbw2. We also assume 2, 2 > 0. Toavoid double counting, we define the one-loop integralswith the 0-bin subtraction [33].Even with an off-shellness, the soft diagrams in Fig. 2 donot contain any scale and they are completely cancelled bytheir 0 bin.The divergent part of the ultrasoft diagrams in Fig. 3 isiMusoft i s42CF1"2 1"ln2 2n pc n p c2 1Nc1"ln1 i0 1Nc1"JEFTI ; (13)FIG. 3. Ultrasoft diagrams at one loop.FIG. 2. Soft diagrams at one loop.FIG. 4. Collinear diagrams at one loop.EXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-5where CF N2c 1=2Nc and is the MS unit mass,2 42MS expE. The first term in the curly brack-ets of Eq. (13) corresponds to the sum of the divergences inthe second diagram in Fig. 3, where an ultrasoft gluon isexchanged between the c and c quarks collinear in back-to-back directions, and those in the last four diagrams of thesame figure, which contain ultrasoft interactions betweenthe initial and final states. The second term is an extraimaginary piece generated by the second diagram in Fig. 3.The i0 prescription in the argument of the logarithm,where 0 is a positive infinitesimal quantity, follows fromthe prescriptions in the quark propagators and from thechoice 2, 2 > 0. The divergences arising from the ultra-soft exchanges between the b b pair in the first diagram inFig. 3 are encoded in the last term in Eq. (13).The initial and final states cannot interact by exchangingcollinear gluons because the emission or absorption of acollinear gluon would give the b quark an off-shellness oforder m2b, which cannot appear in the effective theory. Forthe same reason, the c and c cannot exchange n- orn-collinear gluons. The only collinear loop diagrams con-sist of the emission of an n n-collinear gluon from theWilson line Wn n in JEFTI and its absorption by the ccquark, as shown in Fig. 4. The divergent part of the sum ofthe two collinear diagrams isiMcoll i s4 2CF2"2 1"2 ln2 222JEFTI :(14)The collinear diagrams are calculated with a 0-bin sub-traction [33]; that is, we subtract from the naive collinearintegrals the same integrals in the limit in which the loopmomentum is ultrasoft. In this way we avoid double count-ing between the diagrams in Figs. 3 and 4.Summing Eqs. (13) and (14) and adding factors of Z1=2cfor each field,Zc b Zb 11"s2CF;Zn Z n 11"s4CF;the divergent piece becomesiMdiv i s4CF2"2 2"32 lnn pc n p c2 1"Nc i"1NcJEFTI : (15)The counterterm ZEFTI is chosen so as to cancel the diver-gence in Eq. (15),ZEFTI s4CF2"2 2"32 lnn pc n p c2 1"Nc i"1Nc: (16)From the definition in (11), the counterterm in Eq. (16),and recalling that ds=d ln 2"s O2s, theanomalous dimension at one loop isEFTI 2s43CF Nc 4CF lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin pc n p cp i 1Nc: (17)An important feature of the anomalous dimension (17) isthe presence of a term proportional to ln. Because of thisterm, the RGE (12) can be used to resum Sudakov doublelogarithms. As wewill show shortly, the general solution ofEq. (12) can be written in the following form:C C00ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin pc n p cpg0;expU0; ; (18)where g and U depend on the initial scale 0 and the finalscale that we run down to. For an anomalous dimensionof the form (17), U can be expanded as a series,U0; X1n1ns 0Xn1L0un;LlnnL1 0: (19)If =0 1, the most relevant terms in the expansion(19) are those with L 0, which we call leading logs(LL). Terms with higher L are subleading; we call theterms with L 1 next-to-leading logs (NLL), thosewith L 2 next-to-next-leading logs (NNLL), and, ifL m, we denote them with NmLL. The RGE (12) deter-mines the coefficients in the expansion (19). With theanomalous dimensions written asEFTI 2s s lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin pc n p cp; (20)where s and s are series in powers of s,s s40 s421 . . . ;s s40 s421 . . . ;it can be proved that the coefficients of the LL, un0, aredetermined by the knowledge of 0 and of the QCD function at one loop. The NLL coefficients un1 are insteadcompletely determined if and are known at two loopsand s at one loop.In the case we are studying, the ratio of the scales=0 mc=2mb is not extremely small. Indeed, as to beseen shortly, the numerical contributions of the LL andNLL terms in the series (19) are of the same size. It istherefore important to work at NLL accuracy, which re-quires the calculation of the coefficient of ln to two loops.The factors of ln are induced by cusp angles involvinglightlike Wilson lines, and their coefficients are universals / cusps [34]. The cusp anomalous dimensionAZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-6cusps is known at two loops [34],cusps s40cusp s421cusp; (21)with0cusp 4CF; 1cusp 4CF679 23Nc 109 nf;(22)while the constant of proportionality between s andcusps is fixed by the one-loop calculation. Since wehave determined 0,0 3CF Nc i Nc ; (23)and the function is known, we have all the ingredients toprovide the NLL approximation for U0; andg0; . Taking into account the tree-level initial condi-tion in Eq. (6), Eq. (18) determines the leading-ordermatching coefficient, with NLL resummation.The solution (18) can be derived by writing Eq. (12) asd lnC 2 d cuspln0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin pc n p cpZ 0d00; (24)where we have used the definition of the function, d=d ln, to write ln and d ln in terms of .Integrating both sides from0 to and exponentiating theresult, we find the form given in Eq. (18), withU0; 2Z ss0d cuspZ 0d00;g0; 2Z ss0dcusp:(25)At NLL, we findUb; 20cusp20r 1 r lnrs 00Re20cusplnr1cusp0cusp 101 r lnr4 180ln2r 0Im0lnr; (26)andgb; 0cusp0lnr1cusp0cusp 10sb4r 1;(27)where r s=sb, and we have renamed the initialscale b, to denote its connection to the scale 2mb. InEqs. (26) and (27) we have used the two-loop beta func-tion,s 2ss40 s421; (28)with0 11 23 nf; 1 343N2c 103 Ncnf 2CFnf:(29)In Eq. (26) we have kept the contributions of the real andimaginary parts of 0 separated. The imaginary part of0 changes the phase of the matching coefficient C,but this phase is irrelevant for the calculation of physicalobservables like the decay rate, which depend on thesquare modulus of C. In Sec. V the factor Ub;will be evaluated between the scales b 2mb and mc, with nf 4 active quark flavors. The numerical evalu-ation shows that the LL term, represented by the first termin the brackets in Eq. (26), is slightly smaller than and hasthe opposite sign of the term proportional to 0Re , whichdominates the NLL contribution. This observation con-firms, a posteriori, the necessity to work at NLL accuracyin the resummation of logarithms of mc=2mb.The RGE (12) and its solution (18) thus allow us torewrite Eq. (5) asJQCD CJEFTI Cb 2mb expU2mb;mcJEFTI mc;which avoids the occurrence of any large logarithm in thematching coefficient or in the matrix element of the effec-tive operator.IV. pNRQCD bHQETA. MatchingIn the second step, we integrate out the soft modes bymatching EFTI onto EFTII. In NRQCD SCET, contri-butions to the exclusive decay processes are obtained byconsidering time-ordered products of JEFTI and the terms inthe EFTI Lagrangian that contain soft-gluon emissions.The soft gluons have enough virtuality to produce a pairof light quarks traveling in opposite directions with ultra-collinear momentum scaling. These light quarks bind to thecharm quarks to form back-to-back D mesons. The totalmomentum of two back-to-back ultracollinear quarks is2mbQCD=mc1; 1; , and the invariant mass of the pair isq2 2mbQCD=mc2 m2c: in NRQCD SCET, onlysoft gluons have enough energy to produce them. Thetime-ordered products in NRQCD SCET are matchedonto six-fermion operators in pNRQCD bHQET, wherefluctuations of order m2c cannot be resolved.EXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-7We consider the scale 0 mc to be much bigger thanQCD, so the matching can be done in perturbation theory.The Feynman diagrams contributing to the matching areshown in Fig. 5. The gluon and the b-quark propagatorshave off-shellness of order m2c, so the two diagrams on thel.h.s. match onto six-fermion operators on the right-handside (r.h.s.).The amplitude for the decay of a bottomonium withquantum numbers 2S1LJ into two D mesons has thefollowing form:iM iCZ d!!d !!T!; !;;0; 2S1LJF20 hDA;DBjO2S1LJAB !; !;0j bb2S1LJi: (30)A and B, which label the final states and the EFTII opera-tors O2S1LJAB , denote the possible parity, spin, and polariza-tion of the D mesons, A, B fP; VL; VTg, indicating,respectively, a pseudoscalar D meson, a longitudinallypolarized vector meson D, and a transversely polarizedvector meson D. Unlike JEFTI , we have dropped the sub-script EFTII in O2S1LJAB in order to simplify the notation.The EFTII operators that contribute to the decay of theP-wave states areF20O3PJPP !; !;0 yb ~pb ~?c b Hcn6n25 ! n P ln ln! n P y6n25H cn;F20O3PJVLVL!; !;0 yb ~pb ~?c b Hcn6n2 ! n P ln ln! n P y6n2H cn;F20O3PJVTVT !; !;0 ybpb??c b Hcn6n2? ! n P ln ln! n P y6n2?H cn;(31)where pb?? is a symmetric, traceless tensor,pb?? 12pb?? pb?? g? ~pb ~?:At leading order in the EFTII expansion, the b can only decay into a pseudoscalar and a vector meson, with an operatorgiven byF20O1S0PVL!; !;0 ybc bH cn6n25 ! n P ln ln! n P y6n2H cn H cn 6n2 ! n P ln ln! n P y6n25H cn: (32)For later convenience, in the definition of the effectiveoperators (31) and (32) we have factored out the termF20, which is related to the D-meson decay constant.The definition of F20 will become clear when we in-troduce theD-meson distribution amplitudes. The fields lnand ln are ultracollinear gauge-invariant light-quark fields,while H cn Wyn hcn and H cn Wyn h cn are bHQET heavy-quark fields, which are invariant under an ultracollineargauge transformation. The Wilson lines Wn and W n havethe same definition as in Eq. (8), with the restriction toultracollinear gluons. Equations (31) and (32) allow us tointerpret! as the component of the light-quark momentumalong the direction n. Similarly, ! represents the compo-nent of the light-antiquark momentum along n. The minussign in the delta function ! n P is chosen so that! is positive.The tree-level matching coefficients areT!; !;;0 mc; 3PJ CFN2c4smcmb1! ! ;T!; !;;0 mc; 1S0 CFN2c4smcmb12! !! ! :(33)Note that, at leading order in the EFTII expansion, thematching coefficient T!; !;;0; 3PJ is independentof the spin and polarization of the final states, or of thetotal angular momentum J of the b.FIG. 5. Matching NRQCD SCET onto pNRQCDbHQET. On the r.h.s. the double solid lines represent heavy b( b) (anti)quarks, the double dashed lines bHQET c ( c) (anti)quarks, and the single dashed lines collinear light quarks.AZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-8An important feature of bHQET is that the ultracollinearand ultrasoft sectors can be decoupled at leading order inthe power counting by a field redefinition reminiscent ofthe collinear-usoft decoupling in SCET [7,8]. For bHQETin the n direction, the decoupling is achieved by definingh cn ! Ynh cn and ln ! lnYyn , where Yn is an ultrasoftWilson line,Yn Xpermsexp gn P n Aus: (34)An analogous redefinition with n ! n decouples ultrasoftfrom n-ultracollinear quarks and gluons. These redefini-tions do not affect the operators in Eqs. (31) and (32)because all the induced Wilson lines cancel out. As aconsequence, at leading order in the EFTII power counting,there is no interaction between the initial and the finalstates, since the former can only emit and absorb ultrasoftgluons that do not couple to ultracollinear degrees of free-dom. Furthermore, fields in the two copies of bHQET,boosted in opposite directions, cannot interact with eachother because the interaction with an n-ultracollinear gluonwould give an n-ultracollinear quark or gluon a virtualityof order m2c, which, however, cannot appear in EFTII. Thematrix elements of the operators O2S1LJAB !; !;, there-fore, factorize asF20hABjO2S1LJAB !; !;0j bbi h0jybT2S1LJAB c bj bbihAj H cn6n2A ! n P lnj0ihBj ln! n P y6n2BH cnj0i; (35)where A f5; 1; ?g and T2S1LJAB f1; ~pb ~?; pb??g. The charge-conjugated contribution isunderstood in the b case.The quarkonium state and the D mesons in Eq. (35)have, respectively, nonrelativistic and HQET normaliza-tion:hbJE0; ~p0jbJE; ~pi 233 ~p ~p0;hDv0; k0jDv; ki 2v0v;v0 233 ~k ~k0;where v0 is the 0th component of the four-velocity v.TheD-meson matrix elements can be expressed in termsof the D-meson light-cone distribution amplitudes:hPj ln6n25! n P yH cnj0i iFP0 n v2 P!;0;hVLj ln6n2! n P yH cnj0i FVL0n v2VL!;0;hVTj ln6n2?! n P yH cnj0i FVT 0n v2"?VT !;0;(36)where "? is the transverse polarization of the vector me-son. The constants FA0, with A fP; VL; VTg, are re-lated to the matrix elements of the local heavy-lightcurrents in coordinate space. In the heavy-quark limit,where D and D are degenerate, FA is the same for allthe three states: F FP FVL FVT . In this limit,h0j ln6n25hcn0jPi iF0 n v02: (39)At tree level, the matrix element is proportional to theD-meson decay constant fD 205:8 8:5 2:5 MeV[35]. More precisely, F0 fD ffiffiffiffiffiffiffimDp , where the factorffiffiffiffiffiffiffimDpis due to HQET normalization. The scale dependenceof F is determined by the renormalization of heavy-lightHQET currents. At one loop, Ref. [32] showed thatdd ln0F0 FF0 3CF s4F0: (40)The pNRQCD matrix elements can be expressed interms of the heavy quarkonium wave functions. The op-erator yb ~pb ~?c b contains a component with J 0 anda component with J 2 and Jz 0, so its matrix elementhas nonvanishing overlap with both b0 and b2. Theoperator ybpb ?c b instead has only contributions withJ 2 and Jz 2, and therefore it only overlaps withb2. In terms of the bottomonium wave functions, thepNRQCD matrix elements are expressed ash0jyb ~pb ~?c bjb0i 2ffiffiffi3pffiffiffiffiffiffiffiffiffi3Nc2sR0b00; 0; (41)h0jyb ~pb ~?c bjb2i ffiffiffiffiffiffi215s ffiffiffiffiffiffiffiffiffi3Nc2sR0b20; 0; (42)h0jybpb ?c bjb2i "2 "2 ffiffiffiffiffiffiffiffiffi3Nc2sR0b20; 0;(43)where R0bJ 0 is the derivative of the radial wave functionEXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-9of the bJ evaluated at the origin. At leading order, thepNRQCD Hamiltonian does not depend on J, so, up tocorrections of order w2, R0b20 R0b00. The numericalprefactors in Eqs. (41) and (42) follow from decomposing~pb ~? into components with definite Jz. "j is the po-larization tensor of the b2 state, and Eq. (53) states that, atleading order in the w2 expansion, only the particles withpolarization Jz 2 contribute to b2 decay into twotransversely polarized vector mesons. Similarly, one findsh0jybc bjbi ffiffiffiffiffiffiffiNc2sRb0; 0: (44)The factorization of the matrix elements (35) impliesthat the decay rate also factorizes. For the decays of b0and b2 into two pseudoscalar mesons or two longitudi-nally polarized vector mesons, we findb0 ! AA 43m2Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2b0 4m2Dq8mb03Nc2jCj2jR0b00; 0j2F20 n v02n v2Z d!!d !!T!; !;;0; 3PJA !;0A!;02(45)andb2 ! AA 215m2Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2b2 4m2Dq8mb23Nc2jCj2jR0b20; 0j2F20n v02n v2Z d!!d !!T!; !;;0; 3PJA !;0A!;02; (46)where A P, VL. For the decay of b2 into two transversely polarized vector mesons, one finds the decay rate by summingover the possible transverse polarizations:b2 ! VTVT 25m2Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2b2 4m2Dq8mb23Nc2jCj2jR0b20; 0j2F20 n v02n v2Z d!!d !!T!; !;;0; 3PJVT !;0VT !;02: (47)In the case of b decay into a pseudoscalar and a longitudinally polarized vector meson, we findb ! PVL c:c: m2Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2b 4m2Dq8mbNc2jCj2jRb0; 0j212F20n v02n v2Z d!!d !!T!; !;;0; 1S0 VL !;0P!;0 VL!;0P !;02: (48)Note that we are working in the limit mc ! 1, where themD mD mass splitting vanishes.The factorized formulas Eqs. (35) and (45)(48) are themain results of this paper. Each decay rate of (45)(48)depends on two calculable matching coefficients, C and T,and three nonperturbative, process-independent matrix el-ements, namely, two D-meson distribution amplitudes andthe bottomonium wave function. In Sec. V we will providea model-dependent estimate of the decay rates (45)(48)and will discuss the phenomenological implications. Weconclude this section by observing that all the nonpertur-bative matrix elements cancel out in the ratios b0 !PP=b2 ! PP and b0 ! VLVL=b2 ! VLVL,since the spin symmetry of pNRQCD guaranteesR0b00 R0b20, at leading order in EFTII. Neglectingthe b0-b2 mass difference, we find, up to corrections oforder w2,b0 ! AA=b2 ! AA 43152 10; (49)with A P, VL.B. RunningThe dependence of the matching coefficientT!; !;;0; 2S1LJ and of the operators in Eqs. (45)AZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-10(48) on the scale0 is driven by a RGE that can be obtainedby renormalizing the EFTII operators. The RGE for theEFTII operators, which also defines the anomalous dimen-sion EFTII , is similar to Eq. (10),dd ln0F20O2S1LJAB !; !;0 Zd!0Zd !0EFTII!;!0; !; !0;0F20O2S1LJAB !0; !0; 0: (50)To calculate the anomalous dimension at one loop, wecompute the divergent part of the diagrams in Figs. 6 and7. As mentioned in Sec. II, the pNRQCD Lagrangian hasthe following structure,LpNRQCD Zd3xLusoftNRQCD Lpot;where the superscript usoft indicates that the gluons inthe NRQCD Lagrangian are purely ultrasoft mbw2;mbw2, while Lpot contains four-fermion operators, whichare nonlocal in space,Lpot Zd3x1d3x2cyt; ~x1t; ~x2V;~ryt; ~x2 c t; ~x1:At leading order in smbw and r, V is the CoulombpotentialV; smbwr tata:For the explicit form of higher-order potentials, see, forexample, Refs. [12,31]. Vertices from Lpot generate one-loop diagrams as the first diagram in Fig. 6. However, thesediagrams do not give any contribution to the anomalousdimension at one loop. Indeed, the insertion of theCoulomb potential 1=r in Fig. 6 does not produce UVdivergences. Insertions of the 1=mb potentials yield diver-gences, but the coefficient of the 1=mb potential is propor-tional to 2smbw, so it is not relevant if we are contentwith a NLL resummation. Insertions of 1=m2b potentialsgive divergences proportional to subleading operators,which can be neglected. The second diagram in Fig. 6yields a result completely analogous to the last term inEq. (13), with the only difference of a color prefactor,iMpNRQCD i s2CF"O2S1LJAB !; !;: (51)This divergence is completely canceled by the b-quarkfield renormalization constant Zb, and hence thepNRQCD diagrams in Fig. 6 do not contribute to theanomalous dimension at one loop.On the bHQET side, the third diagram in Fig. 7 isconvergent, and hence it does not contribute to the anoma-lous dimension. The first two diagrams giveiMbHQET;n iZd!0d !0!;!0; !; !0O2S1LJAB !0; !0; ; (52)with!;!0; !; !0 s2CF!!0 ! !0 12"2 1"ln0n v0!0 1" 1" ! !01! !0 !0 ! ! !!01!0 !: (53)The diagrams for the bHQET copy in the n direction give aresult analogous to Eqs. (52) and (53), with ! ! !, !0 !!0, and n v0 ! n v. Extracting EFTII from the diver-gence is again standard, just as we did in the case of EFTI .After adding to Eq. (53) the bHQET field renormalizationconstants Zh and Z for heavy and light quarks,Zh 1 1"s2CF; Z 1 1"s4CF;we findFIG. 6. One-loop diagrams in pNRQCD. The first diagramcontains insertions of quark-antiquark potentials. In the seconddiagram the gluon is ultrasoft.FIG. 7. One-loop diagrams in bHQET. There are three analo-gous diagrams for the other copy of bHQET.EXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-11EFTII!;!0; !; !0;0 2F!!0 ! !0 O!;!0; !; !0;0; (54)withO!;!0; !; !0;0 s4 4CF!!0 ! !01 ln0n v0!0 ln0 n v!0 s44CF!!0 ! !01! !0 !0 ! ! !!01!0 ! s44CF ! !0!!01!!0 !0 !! !!01!0 !: (55)The term proportional to F in Eq. (54) reproduces therunning of F20 (40). O is responsible for the running oftheD-meson distribution amplitudes, and it agrees with theresult found in Ref. [36]. Also, in Eq. (36) the coefficient ofln0 is proportional to cusps. Note that, since thebHQET Lagrangian is spin independent, the anomalousdimension does not depend on the spin or on the polariza-tion of theDmeson in the final state, at leading order in thepower counting.Using Eqs. (50) and (54) we find the following integro-differential RGE for the operator O!; !;0:dd ln0O!; !;0 Zd!0Zd !0O!;!0; !; !0;0O!0; !0; 0; (56)where we have dropped both the subscripts A, B, and thesuperscript 2S1LJ does not depend on the quantum num-bers of the initial or final state. Using the fact that theconvolution of F20T!; !;;0; 2S1LJ and the op-eratorO2S1LJAB !; !;0 is0 independent, we can write anequation for the coefficient,dd ln0F20T!; !;;0Zd!0Zd !0!!0!!0F20T!0; !0; ;0 O!0; !; !0; !;0Zd!0Zd !0F20T!0; !0; ;0 O!;!0; !; !0;0; (57)where the last step follows from the property of O at oneloop,!!0!!0O!0; !; !0; !;0 O!;!0; !; !0;0;as can be explicitly verified from the expression inEq. (55).Equation (57) can be solved following the methodsdescribed in Ref. [36]. We discuss the details of the solu-tion in Appendix A, where we derive the analytic expres-sions for T!; !;;0; 3PJ and T!; !;;0; 1S0, withthe initial conditions at the scale 0c mc expressed inEq. (39).V. DECAY RATES AND PHENOMENOLOGYIn Sec. IVA we gave the factorized expressions for thedecay rates (45)(48): b0;2 ! PP, b0;2 ! VLVL,b2 ! VTVT, and b ! PVL c:c:. In Secs. III Band IVB we exploited the RGEs (12) and (57) to run thescales and 0, respectively, from the matching scales 2mb and0 mc to the natural scales that contributeto the matrix elements, mc and 0 1 GeV, resum-ming in this way Sudakov logarithms of the ratiosmc=2mband mc=1 GeV.We proceed now to estimate the decay rates (45)(48).In order to do so, we need to evaluate the followingingredients: the light-cone distribution amplitudes of theD meson and of the longitudinally and transversely polar-ized D mesons, and the wave functions of the states band bJ. In principle, these nonperturbative objects couldbe extracted from other b, b, and D-meson observables.In the case of the b, the value of the wave function at theorigin can be obtained from a measurement of the inclusivehadronic width or of the decay rate for the electromagneticprocess b ! , since they are both proportional tojRb0j2. Unfortunately, at the moment there are not suf-ficient data on b decays. Another way to proceed is to usethe spin symmetry of the leading-order pNRQCD Hamil-tonian, which implies Rb0 R0, and to extract theUpsilon wave function from ! ee 1:280:07 KeV [37]. Using the leading-order expression for ! ee [38], one finds jR0j2 6:920:38 GeV3, where the error only includes the experimentaluncertainty. The above value is in good agreement with thelattice evaluation by Bodwin, Sinclair, and Kim [39], and itfalls within the range of values obtained with four differentpotential models, as listed in Ref. [40].jR0b0;20j2 can be obtained from the electromagneticdecay b0;2 ! . Unfortunately, such decay rates havenot been measured yet. The values listed in Ref. [40] rangefrom a minimum of jR0bJ 0j2 1:417 GeV5, obtainedwith the Buchmuller-Tye potential [41], to a maximumof jR0bJ 0j2 2:067 GeV5, obtained with a Coulomb-plus-linear potential. The lattice value is roughly of theAZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-12same size, jR0bJ 0j2 2:3 GeV5, with an uncertainty ofabout 15% [39]. We use this value in our estimate.For the pseudoscalar D-meson distribution amplitude,we use two model functions widely adopted in the study ofB physics. A first possible choice, suggested, for example,in Ref. [36], is a simple exponential decay:ExpP;0 !;0 1 GeV !!2Dexp !D: (58)Another form, suggested in Ref. [42], isBraunP;0 !;0 1 GeV ~! 4D~!1 ~!211 ~!2 2D 12ln ~!; (59)where ~! !=0. The theta function in Eqs. (58) and (59)reflects the fact that the distribution amplitudes A!;0,with A fP; VL; VTg, have support on !> 0 [43].The subscript 0 indicates that these functional forms arevalid in the D-meson rest frame, with a HQET velocitylabel v0 1; 0; 0; 0. With the definition we adopt inEq. (36), the distribution amplitude is not boost invariant,and in the bottomonium rest frame, in which the D mesonhas a velocity n v; n v; 0 mc=2mb; 2mb=mc; 0, itbecomesP!;0 1n vP;0!n v ;0; (60)as shown in Appendix B. D and D in Eqs. (58) and (59)are, respectively, the first inverse moment and the firstlogarithmic moment of the D-meson distribution ampli-tude in the D-meson rest frame,1D 0 Z 10d!!P;0!;0;D01D 0 Z 10d!!ln!0P;0!;0:Furthermore, we assume that the vector-meson distributionamplitudes VL! and VT ! have the same functionalform asP!, but with different parameters DL ,DL andDT , DT .The D-meson distribution amplitude and its momentshave not been intensively studied unlike, for example, theB-meson distribution amplitude. Therefore, we invokeheavy-quark symmetry and use the moments of theB-meson distribution amplitude in order to estimate thedecay rate. However, the value of B is affected by anoticeable uncertainty. Using QCD sum rules, Braunet al. estimated [42] B0 1 GeV 0:4600:110 GeV, where the uncertainty is about 25%. Otherauthors [4446] give slightly different central values andcomparable uncertainties, so that B falls in the range0:350 GeV< B < 0:600 GeV. The first logarithmic mo-ment D is given in Ref. [42], D B0 1 GeV 1:4 0:4. We assume that the moments of the D-mesondistribution amplitudes fall in the same range as the mo-ments of P!.We evaluate numerically the convolution integrals inEqs. (45)(48). We choose the matching scales b and0c to be 2mb andmc, respectively. Using the RGEs we runthe matching coefficients down to the scales mc and0 1 GeV. For the b and c quark masses we adopt the 1Smass definition [47],mb1S m2 4730:15 0:13 MeV;mc1S mJ=c2 1548:46 0:01 MeV:(61)The values of s at the relevant scales are [37] s2mb 0:178 0:005, smc 0:340 0:020, ands1 GeV 0:5. With these choices, the value of g inEq. (A5) is gmc; 1 GeV 0:12 0:02.The decay rates bJ ! AA with A fP; VL; VTg,(45)(47), depend on the masses of the bJ and of the Dmesons, whose most recent values are reported inRef. [37]. Since the effects due to the mass splitting ofthe bJ and D multiplets are subleading in the EFT powercounting, we use in the evaluation the average mass of thebJ multiplet and the average mass of D and D mesons:mbJ 9898:87 0:28 0:31 MeV and mD 1973:27 0:18 MeV. Therefore, the velocity of the Dmesons in bJ decay is n v n v0 mbJ=mD 5:02, with negligible error. The decay rate b ! PVL c:c: (48) depends on the mass of the b, which has beenrecently measured: mb 9388:93:12:3 2:7 MeV [24].The velocity of the D meson in the b decay is n v n v0 mb=mD 4:76, again with negligible error.The decay rate b0 ! PP (45), obtained with ExpandBraun separately, is shown in Fig. 8. In order to see theimpact of resumming Sudakov logarithms, we show forboth distribution amplitudes the results with (i) the LL andNLL resummations and (ii) without any resummation atall. In the plots, we call the resummed results NLL re-summed, indicating that Sudakov logarithms are re-summed up to NLL. For both distribution amplitudes theresummation does have a relevant effect on the decay rate.In the case of Exp the resummation decreases the decayrate by a factor of 21.5 as D goes from the lowest to thehighest value under consideration. In the case ofBraun thedecay rate decreases too, for example, by a factor 1.5 whenD 1:4. In Fig. 9 we compare the decay rates obtainedwith the two distribution amplitudes. Over the range of Dwe are considering, the two decay rates are in roughagreement with each other.Figures 8 and 9 also describe the relation betweenthe decay rate b0 ! VLVL and DL . According toEqs. (46) and (47), the processes b2 ! PP, b2 !VLVL, and b2 ! VTVT show an analogous dependenceon the first inverse moments of the light-cone distributionEXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-13amplitudes, and they differ from Figs. 8 and 9 by constantprefactors. Therefore, we do not show explicitly their plots.Qualitatively, Figs. 8 and 9 show a dramatic dependenceof the decay rate on the inverse moment D. UsingEqs. (45), (60), and (A16), one can show that whenBraun is used, the decay rate is proportional to 4D , whileit scales as 64gD when we adopt Exp, with g defined inEq. (A5). As a consequence, the decay rate drops by anorder of magnitude when D goes from 0.350 GeV to0.600 GeV. The particular sensitivity of exclusive botto-monium decays into two charmed mesons to the light-conestructure of the D mesonmuch stronger than usuallyobserved in D- and B-decay observablesis due to thedependence of the amplitude on the product of two distri-butions (one for each meson) and to the nontrivial depen-dence of the matching coefficient T on the light-quarkmomentum labels ! and ! at tree level. On one hand,the strong dependence on a relatively poorly known quan-tity prevents us from predicting the decay rate b0 !DD. On the other hand, however, it suggests that, if thedecay rate is measured, this channel could be used to betterdetermine interesting properties of the D-meson distribu-tion amplitude, such as D and D. The viability of thissuggestion relies on the control over the theoretical errorattached to the curves in Fig. 8 and on the actual chances toobserve the process b ! DD at current experiments.The uncertainty of the decay rate stems mainly fromthree sources. First, there are corrections coming fromsubleading EFT operators. In matching NRQCD SCETonto pNRQCD bHQET (Sec. IVA), we neglected thesubleading EFTII operators that are suppressed by powersofQCD=mc andw2, relative to the leading EFTII operatorsin Eqs. (31) and (32). In matching QCD onto NRQCDSCET (Sec. III A), we kept only JEFTI (6) and neglectedsubleading EFTI operators, suppressed by powers of andw2. These subleading EFTI operators would match ontosubleading EFTII operators, suppressed by powers ofQCD=mc and w2. Using w2 0:1 and QCD=mc 0:3,we find a conservative estimate for the nonperturbativecorrections to be about 30%.Second, there are perturbative corrections to the match-ing coefficients C and T. Since s2mb 0:178, we ex-pect a 20% correction from the one-loop contributions inmatching QCD onto NRQCD SCET. In the secondmatching step, similarly, the one-loop corrections toT!; !;;0; 2S1LJ would be proportional to smc 30%. We can get an idea of their relevance by estimatingthe dependence of the decay rate (45) on the matchingscalesb and0c. If the matching coefficients C and T andthe anomalous dimensions EFTI and O!;!0; !; !0;0were known at all orders, the decay rate would be inde- (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)0.000.010.020.030.040.050.060.07NLL resummationNo resummation (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)00.010.020.030.040.05NLL resummationNo resummationFIG. 8. b0 ! PP as a function of D, calculated with the distribution amplitudes Exp (left panel) and Braun (right panel). Thedash-dotted and solid lines denote the NLL-resummed decay rate. For comparison, the decay rate without resummation is also shown,denoted by dashdouble-dotted (left panel) and dashed (right panel) lines. For Braun we vary the parameter D from D 1 (lowercurve) to D 1:4 (middle curve) to D 1:8 (upper curve). (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)0.0000.0050.0100.0150.0200.0250.0300.0350.040ExpBraunFIG. 9. b0 ! PP as a function D. The dash-dotted linedenotes the decay rate calculated withExp, while the three solidlines with Braun. For Braun we vary the value of the parameterD from D 1 (lower curve) to D 1:4 (middle curve) toD 1:8 (upper curve).AZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-14pendent of the matching scales b and 0c. However, sincewe only know the first terms in the perturbative expansions,the decay rate bears a residual renormalization-scale de-pendence, whose size is determined by the first neglectedterms.In Fig. 10 we show the effect of varying b between4mb 20 GeV and mb 5 GeV on the decay rate, usingBraun. The solid line represents the choice b 2mb,while the dashed and dotted lines, which overlap almostperfectly, correspond, respectively, to b 20 GeV andb 5 GeV. The dependence on b is mild, its effectbeing a variation of about 5%. We obtain analogous resultsfor the decay rate computed with Exp, which are notshown here in order to avoid redundancy.On the other hand, even after the resummation, thedecay rate strongly depends on 0c. We vary this scalebetween 1.2 GeV and 2.5 GeV and we observe an overallvariation of about 50%. We expect the scale dependence tobe compensated by the one-loop corrections to the match-ing coefficient T!; !;;0; 3PJ. This observation isreinforced by the fact that the numerical values of therunning factors Ub; and V0c; 0 [defined, respec-tively, in Eqs. (26) and (A6)] at NLL accuracy are smallerthan expected on the basis of naive counting of the loga-rithms. As a consequence, the next-to-leading-order cor-rections to the matching coefficient could be as large as theeffect of the NLL resummation. In the light of Fig. 10, theone-loop correction to T!; !;;0; 3PJ seems to be animportant ingredient for a reliable estimate of the decayrate.A third source of error comes from the unknown func-tional form of the D-meson distribution amplitude. For thestudy of the B-meson shape function, an expansion in acomplete set of orthonormal functions has recently beenproposed and it has provided a systematic procedure tocontrol the uncertainties due to the unknown functionalform [48]. The same method should be generalized to theB- andD-meson distribution amplitudes, in order to reducethe model dependence of the decay rate. We leave such ananalysis to future work.To summarize, the calculation of the one-loop matchingcoefficients and the inclusion of power corrections of orderQCD=mc appear to be necessary to provide a decay ratewith an accuracy of 10%, that would make the decaysbJ ! DD, bJ ! D0 D0 competitive processes to im-prove the determination of D and D, if the experimentaldecay rate is observed with comparable accuracy.We estimate the decay rate b ! PVL c:c: (48)using Exp and Braun for both P and VL . In the limitmc ! 1, spin symmetry of the bHQET Lagrangian wouldimply the equality of the pseudoscalar and vector distribu-tion amplitudes,P VL , and hence the vanishing of thedecay rate b ! PVL c:c:. Assuming spin-symmetryviolations, the decay rate depends on (i) the two parametersD D DL=2 and DL D=D DL, ifExp is used, and on (ii) three parameters D, , andjDL Dj, if Braun is used.The two plots in the left column of Fig. 11 show thedecay rate, computed withExp, as a function of D with adopting various values, and as a function of with Dnow being the parameter. In the right column, the decayrate computed with Braun is shown. Since in this case thedecay rate does not strongly depend on , we fix it at 0and we show the dependence of the decay rate on D andjDL Dj. We normalize the difference between thefirst logarithmic moments by dividing them by 2D.The most striking feature of Fig. 11 is the huge sensi-tivity to the chosen functional form. Though a precisecomparison is difficult, due to the dependence on differentparameters, the decay rate increases by 2 orders of magni-tude when we switch from Exp to Braun. Once again, this (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)0.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022 b = 2 mb = 20 GeVb = 5 GeVb (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)00.0050.010.0150.020.0250.03c = mc = 2.5 GeVc = 1.2 GeVcFIG. 10. Left panel: Scale dependence of b0 ! PP on the matching scale b. We varyb from a central valueb 2mb (solidline) to a maximum of b 20 GeV (dashed line) and a minimum of b 5 GeV (dotted line). The dashed and dotted lines overlapalmost perfectly. Right panel: Scale dependence of b0 ! PP on the matching scale 0c. We varied 0c from a central value of0c mc (solid line) to a maximum of 0c 2:5 GeV (dashed line) and a minimum of 0c 1:2 GeV (dotted line).EXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-15effect hinders our ability to predict b ! PVL c:c:but it opens up the interesting possibility to discriminatebetween different model distribution amplitudes.Using Eqs. (48) and (A17), we know that b !PVL c:c: goes like 44gD when Exp is used or 4Dwhen Braun is used. Figure 11 appears to confirm thisstrong dependence on D. The plots in the lower half ofFig. 11 reflect the fact that the decay rate vanishes if oneassumes P! VL!.We conclude this section with the determination of thebranching ratiosBb0 ! PP b0 ! PP=b0 !light hadrons and Bb ! PVL c:c: b !PVL c:c:=b ! light hadrons. At leading order inpNRQCD, the only nonperturbative parameter involvedin the inclusive decay width of the b is jRb0j2 [4],b ! light hadrons 2 Imf11S0m2bNc2jRb0j2:(62)Therefore, Bb ! PVL c:c: does not depend on thequarkonium wave function, and the only nonperturbativeparameters in Bb ! PVL c:c: are those describingthe D-meson distribution amplitudes.For P-wave states, the inclusive decay rate was obtainedin Refs. [4,49], where the contributions of the configura-tions in which the quark-antiquark pair is in a color-octetS-wave state were first recognized. In pNRQCD the inclu-sive decay rate is written as [50,51]b0 ! light hadrons 1m4b3NcjR0b0j2Imf13P0 19N2cImf83S1E;(63)where the color-octet matrix element has been expressed interms of the heavy quarkonium wave function and of thegluonic correlator E, whose precise definition is given inRef. [50]. E is a universal parameter and is completelyindependent of any particular heavy quarkonium stateunder consideration. Its value has been obtained by fitting (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)00.10.20.30.40.50.6-310 = -0.15 = -0.1 = 0.1 = 0.15 (GeV)D0.35 0.4 0.45 0.5 0.55 0.6 (KeV)00.020.040.060.080.10.120.140.16 = 0.05 |/D - LD*| = 0.1|/D - LD*| = 0.15|/D - LD*|-0.15 -0.1 -0.05 0 0.05 0.1 0.15 (KeV)00.20.40.60.81-310 = 0.300 GeVD = 0.400 GeVD = 0.500 GeVD = 0.600 GeVD|/D - LD*|0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 (KeV)00.050.10.150.20.25 = 0.300 GeVD = 0.400 GeVD = 0.500 GeVD = 0.600 GeVDFIG. 11 (color online). Left panel: b ! PVL c:c: as a function of D and , computed using exponential distributionamplitudes ExpP and ExpVL. Right panel: b ! PVL c:c: as a function of D and jDL Dj=, computed with the Braundistribution amplitudes BraunP and BraunVL.AZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-16to existing charmonium data and, thanks to the universal-ity, the same value can be used to predict properties ofbottomonium decays. It is found in Ref. [50] that E 5:33:52:2. The matching coefficients in Eqs. (62) and (63)are known to one loop. For the updated value we refer toRef. [52] and references therein. For reference, the tree-level values of the coefficients are as follows [4]:Imf11S0 2s2mbCF2Nc;Imf13P0 32s2mbCF2Nc;Imf83S1 nf62s2mb:(64)With the above parameters, we plot Bb0 ! PP andBb ! PVL c:c: as a function of D and D, respec-tively, in Fig. 12. Over the range of D we are considering,Bb0 ! PP varies between 4 105 and 4 106; it isapproximately 1 or 2 orders of magnitude smaller than thebranching ratios observed in Ref. [25] for bJ decays intolight hadrons.Bb ! PVL c:c: depends on the choiceof the distribution amplitude. Choosing the parametriza-tion Braun (59), it appears that, despite the suppression atjDL Dj 0, Bb ! PVL c:c: assumes valuescomparable to Bb0 ! PP even for a small deviationfrom the spin-symmetry limit. If Exp is chosen, thebranching ratio is suppressed over a wide range of jDL Dj. The branching ratio Bb ! PVL c:c: was firstestimated in [53]. The authors of [53] assumed that theexclusive decays into DD dominate the inclusive decayinto charm, b ! PVL c:c: b ! c c X.With this assumption, they estimated the branching ratioto be in the range 103 smc 0:3. The largest nonperturbative contributioncould be as big as QCD=mc, which would amount ap-proximately to a 30% correction. Therefore, corrections tothe leading-order decay rates could be noticeable, as thestrong dependence of the decay rates on the renormaliza-tion scale 0c suggests. However, the EFT approach shownin this paper allows for a systematic treatment of bothperturbative corrections and power-suppressed operators,so that, if the experimental data require, it is possible toextend the present analysis beyond the leading order.For simplicity, we have focused in this paper on thedecays of C-even bottomonia, in which cases the decaysproceed via two intermediate gluons, and both the match-ing coefficients C and T are nontrivial at tree level. Thesame EFT approach can be applied to the decays of C-oddstates, in particular, to the decays ! DD and !DD, with the complication that the matching coefficientT arises only at one-loop level. Moreover, the same EFTformalism developed in this paper can be applied to thestudy of the channels that have vanishing decay rates atleading order in the power counting, such as b ! DD, ! DD c:c:, and b2 ! DD c:c:. Experimentaldata for the charmonium system show that, for the decaysof charmonium into light hadrons, the expected suppres-sion of the subleading twist processes is not seen. It isinteresting to see whether such an effect appears in botto-monium decays into two charmed mesons, using the EFTapproach of this paper to evaluate the power-suppresseddecay rates.Finally, in Sec. V we used model distribution amplitudesto estimate the decay rates. The most evident, qualitativefeature of the decay rates is the strong dependence on theparameters of the D-meson distribution amplitude. Eventhough this feature may prevent us from giving reliableestimates of the decay rates or of the branching ratios, itmakes the channels analyzed here ideal candidates for theextraction of important D-meson parameters, when thebranching ratios can be observed with sufficient accuracy.ACKNOWLEDGMENTSWe would like to thank S. Fleming for proposing thisproblem and for countless useful discussions, N. Brambillaand A. Vairo for suggestions and comments, and R. Briere,V.M. Braun, and S. Stracka for helpful communications.Bw. L. is grateful for the hospitality of the University ofArizona, where part of this work was finished. This re-search was supported by the US Department of Energyunder Grant No. DE-FG02-06ER41449 (R.A. and E.M.)and No. DE-FG02-04ER41338 (R.A., Bw. L. and E.M.).APPENDIX A: SOLUTION OF THE RUNNINGEQUATION IN pNRQCD bHQETThe RGE in Eq. (57) can be solved by applying themethods discussed in Ref. [36] to find the evolution of theB-meson distribution amplitude. We generalize this ap-proach to the specific case discussed here, where twodistribution amplitudes are present. Following Ref. [36],we define!!;!0; s sCF!!01!!0 !0 !! !!01!0 !:Lange and Neubert [36] prove thatZd!0!!;!0; s!0a !aF a;s; (A1)withF a;s sCF c 1 a c 1 a 2E:c is the digamma function and E the Euler constant.Equation (A1) is valid if 1< Re a < 1. Exploiting(A1), a solution of the running equation (57) with the initialcondition T!; !;00 !=00 !=00 at a certainscale 00 isF20T!; !;0 F200f!;0;00;f !;0;00; ;(A2)withf!;0; 00; !00g n vg expU00; 0; ;g g00; 0 Z s0s00dcusp;U00; 0; Z s0s00dcuspZ s00d00 1 F g;;1s 2sCF4 : (A3)The function f !;0; 00; has the same form asf!;0; 00; and is obtained by replacing ! ! !, !, and n v ! n v0 in Eq. (A3). The integrals over canbe performed explicitly using the beta function in Eq. (28).The result isf!;0; 00; f !;0; 00; !00g !00g n vn v0g expV00; 0 1 g1 1 g1 1 g1 1 g1 ;(A4)where, at NLL,AZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-18g00; 0 0cusp20lnr1cusp0cusp 10s004r 1(A5)andV00; 0 0cusp220r 1 r lnrs0 1cusp0cusp 10 1 r lnr4 180ln2r CF02 8E lnr; (A6)with r s0=s00. Notice that in the running from00 mc to0 1 GeV only three flavors are active, so inthe expressions for 0, 1, and 1cusp, we use nf 3.Equation (A4) is the solution for the initial conditionT!; !;00 !=00 !=00. To solve the RGE for ageneric initial condition, we express T as the Fourier trans-form with respect to ln!=00,T!; !;00 122Z 11drds expir ln!00 expis ln !00FTr; s; 00 122Z 11drds!00ir !00isFT r; s; 00;where FT denotes the Fourier transform of T. From thesolutions (A2)(A4) it follows thatF20T!; !;0 F20022Z 11drds!00irg!00isg n vn v0gFT r; s; 00 expV00; 0 1 ir g1 ir1 ir g1 ir 1 is g1 is1 is g1 is : (A7)The Fourier transform of the matching coefficient inEq. (A7) has to be understood in the sense of distributions[55]. That is, we define the Fourier transform of T as thefunction of r and s that satisfies122ZdrdsFTr; s; 0Ar;0Bs;0Z 10d!!d !!T!; !;0A!;0B !;0;(A8)or, more precisely, FTr; s; 0 is the linear functionalthat acts on the test functions Ar and Bs according to122 FTr; s; 0; Ar;0Bs;0Z 10d!!d !!T!; !;0A!;0B !;0:(A9)The function A is the Fourier transform of the D-mesondistribution amplitude,Ar;0 Z 10d!!!0irA!;0; (A10)where the integral on the r.h.s. should converge in theordinary sense because of the regularity properties of theD-meson distribution amplitude. As in Sec. IV, the sub-script A denotes the spin and polarization of the D meson.In the distribution sense, the Fourier transform of thecoefficient 1=! ! isF1! !r; s; 00 221200r s i sech2r s 1222 1200R i sech2S;(A11)where R r s, S r s, and the factor 12 comes fromthe Jacobian of the change of variables. The hyperbolicsecant is defined as sech 1= cosh. Similarly, we findF! !! !R; S;00 i222Rcosech2S i" cosech2S i": (A12)The function in Eq. (A11) has a complex argument.The definition is analogous to the one in real space [55],R i; R i: (A13)Using Eqs. (A11) and (A12), we can perform the integral inEq. (A7), obtaining, respectively, T!; !;;0; 3PJ andT!; !;;0; 1S0. In order to give an explicit example,we proceed using Eq. (A11). Integrating the function weare left withEXCLUSIVE DECAYS OF bJ AND b INTO . . . PHYSICAL REVIEW D 80, 074026 (2009)074026-19F20T!; !;0 F200 expV00; 0100020! !1=2g n vn v0gZ 11dS expi S2ln!! sech2S11 S232 g i2S12 g i2S32 g i2S12 g i2S: (A14)The integral (A14) can be done by contour. The integrand has poles along the imaginary axis. In S i there is adouble pole, coming from the coincidence of one pole of the hyperbolic secant and the singularities in 1=1 S2. The functions in the numerator have poles, respectively, in S i2n 3 2g with n > 0, while the other poles of sech arein S i2n 1 with n 1. We close the contour in the upper half plane for !>! and in the lower half plan for!> !, obtainingF20T!; !;0 F200 expV00; 0 !!1!020 n vn v0! !g1 g2 g1 gg1 ln!! c 1 g c g c 1 g c 2 g X1n1n1!!n 1nn 11 n g2 n gn g1 g n X1n1!!ng n 1! cscg1n g1 n g2 n 2g1 nn 2g ! ! !; (A15)with cscg 1= sing, and c is the digamma function. More compactly, we can express Eq. (A15) using thehypergeometric functions 4F3 and 3F2,F20T!; !;;0; 3PJ F20cCFN2c4s0cmbexpV0c; 002c n vn v0! !g !!!1 g2 g1 gg1 ln!! c 1 g c g c 1 g c 2 g 12!!g 2g 31 g2 g 4F31; 1; g 2; g 3; 3; 1 g; 2 g;!!!!1g4 cosg2 2g2g 2 3F2g 1; 2g 2; 2g 3; 2; g 3;!! ! ! !;(A16)where we have introduced the constants that appear in the initial condition in Eq. (33). In the same way, we obtainF20T!; !;;0; 1S0 F20cCF2N2c4s0cmbexpV0c; 0 !!02c n vn v0! !g21 g2 g1 g2 g!! 3F21; g 1; g 2; 1 g; 2 g;!!21 g21 g!!1g4 cosg1 2g2g 22F12g 2; 2g 1; 2;!! ! ! !: (A17)In Eqs. (A16) and (A17) we renamed the initial scale00 0c to denote its connection to the scale mc. Setting0 0c or, equivalently, g 0, it can be explicitly veri-fied that the solutions (A16) and (A17) satisfy the initialconditions (33).APPENDIX B: BOOST TRANSFORMATION OFTHE D-MESON DISTRIBUTION AMPLITUDEWe derive in this appendix the relation between thedistribution amplitudes in the D-meson and in the botto-monium rest frames, as given in Eq. (60). In the D-mesonAZEVEDO, LONG, AND MEREGHETTI PHYSICAL REVIEW D 80, 074026 (2009)074026-20rest frame, characterized by the velocity label v0 1; 0; 0; 0, the local heavy-light matrix element is definedash0j ln06n25hcn0jDiv0 iF0n v02: (B1)The matrix element of the heavy- and light-quark fields at alightlike separation z0 n z0 n=2 defines the light-conedistribution ~0n z0; 0 in coordinate space:h0j lnn z06n25H cn0jDiv0 iF0 n v02~0n z0; 0: (B2)Equations (B1) and (B2) imply ~00; 0 1. In the defi-nitions (B1) and (B2) the subscript 0 is used to denotequantities in the D-meson rest frame. This convention isused in the rest of this appendix. In the bottomonium restframe, where the velocity label in light-cone coordinates isv n v; n v; 0 and the lightlike separation is z n z n=2, we defineh0j ln06n25hcn0jDiv iF0 n v2 (B3)andh0j lnn z6n25H cn0jDiv iF0 n v2~n z; 0:(B4)Suppose that is some standardized boost that takes theD meson from v, its velocity in the bottomonium restframe, to rest. It is straightforward to find the relationsbetween the D-meson momenta in the two frames:n p0 n vn p and n p0 n v n p:There is a similar relation for the light-cone coordinates,n z0 n vn z:With U, the unitary operator that implements the boost, one can writeUjDiv jDiv0 :We choose such that, for the Dirac fields,UlnxU1 11=2lx andUhcnxU1 11=2hcx;where1=2 cosh2 6n6n 6n 6n4sinh2;with related to v by e n v and e n v.Now we can write the matrix element in Eq. (B3) ash0j ln6n25hcn0jDiv h0jU1U ln0U16n25Uhcn0U1UjDiv h0j l01=26n2511=2hc0jDiv0 n vh0j l06n25hc0jDiv0 iF0n v2n v0 iF0 n v2; (B5)where, in the last step, we have used n v0 1. 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