PHYSICAL REVIEW D 100, 054021 (2019) Novel implementation of the multipole expansion to quarkonium hadronic transitions Antonio Pineda and Jaume Tarrús Castellà Grup de Física Teòrica, Dept. Física and IFAE-BIST, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain (Received 20 May 2019; published 18 September 2019) We compute hadronic transitions between heavy quarkonium states with two, or one, pion/eta particles in the final state. We use the multipole expansion but not the twist expansion. The latter cannot be justified for the energy release of hadronic transitions between heavy quarkonium states with different principal quantum numbers. Instead, we use a counting based on the dimension of the interpolating field of the hybrid. This alternative counting allows us to still use chiral low-energy theorems to compute the pion production by local gluonic operators. We explore the phenomenological impact of this counting. Remarkably enough, for the two-pion transitions, we obtain the same predictions for the normalized differential decay rate as those obtained assuming the twist expansion. We implement this computational scheme using the hadronic representation of the effective theory potential NRQCD. We assume that the inverse Bohr radius of the heavy quarkonium is much larger than ΛQCD but do not impose any constraint on the relative size of ΛQCD and the typical kinetic energy of the bound state. DOI: 10.1103/PhysRevD.100.054021 I. INTRODUCTION hadrons at a relatively long distance. This picture follows from the use of the multipole expansion of the gluon-heavy Hadronic transitions between heavy quarkonium states quark interaction. This allowed Gottfried to give selection have been studied since the middle seventies. Particular rules and decay width rate estimates beyond the approaches attention has been devoted to the transitions with one or two based solely on chiral symmetry. The work of Ref. [6] pions in the final state. The relatively small energy of the connected these nonlocal gluonic matrix elements with outgoing pions makes the analysis of these transitions ideal chiral symmetry by parametrizing them according to chiral for the implementation of (the nonlinear realization of) symmetry. This was used to obtain constrains for higher chiral symmetry. This was first implemented by Cahn and angular momentum channels. Brown in Ref. [1]. Using current algebra, they parametrized At leading order (LO) in the multipole expansion, the amplitude of the two-pion transition in the strict chiral the intermediate heavy quarkonium state is in a color- limit. The observation of the small variation of the width octet configuration. In Ref. [7], Voloshin introduced an distribution with respect to the helicity angle of one of the additional expansion consisting of an operator product pions led them to an approximated simple one-parameter expansion of the nonlocal heavy quarkonium color-octet description of the decay width spectrum. The implementa- two-point function. As a result, the transition amplitude can tion of chiral symmetry using chiral Lagrangians and the be written as a series of matrix elements of local operators, incorporation of the leading light-quark mass corrections which we will refer to as a twist expansion. As before, these was made in Ref. [2]. This gave a good description of local matrix elements can be hadronized and parametrized the width spectrum and also of the pion helicity angle using chiral symmetry but the implementation of the axial distribution [3,4]. [8,9] and energy-momentum tensor anomalies [10–12] Alternatively, in Ref. [5], Gottfried realized that these constraints their general structure. Since then it has become transitions could be though as a two-step process: first a quite customary to describe the two-pion transitions using short-distance gluon emission by the heavy quarks, and the multipole, twist, and the chiral expansion. See, for then the hadronization of the gluons into the light-quark instance, Ref. [13], where the local gluonic matrix elements were obtained up to next-to-leading order (NLO) in the chiral expansions. Unfortunately, this computational Published by the American Physical Society under the terms of scheme is not a model independent derivation of QCD. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to The reason is that there is no kinematic regime where the the author(s) and the published article’s title, journal citation, twist expansion can be justified, as shown in Ref. [14], and DOI. Funded by SCOAP3. because the transfer energy between heavy quarkonium 2470-0010=2019=100(5)=054021(25) 054021-1 Published by the American Physical Society ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) states in these transitions, E, is of order mQ v2, whereas for a series of observables. Finally, we want to mention that the twist expansion requires that E ≪ mQ v2 , as well the pure quarkonium hybrid sector of this theory has as ΛQCD ≪ mv2 . already been developed in Refs. [18–21] for small energy We want to retake this discussion within the context of fluctuations (much smaller than the energy transitions we effective field theories (EFTs), which allows us to make a consider in this paper). systematic analysis of the scales involved in the problem. We organize the paper as follows: in Sec. II, we discuss We will do the analysis using the weak-coupling version of the hadronization of the pNRQCD Lagrangian. In Sec. III, potential NRQCD (pNRQCD) [15,16]. This allows us to we apply it to describe the QQ̄ð2SÞ → QQ̄ð1SÞππ tran- obtain the multipole expansion in a controlled way, since sitions and compare the results for the decay width we assume that mQ v ≫ ΛQCD . The resulting EFT has the spectrum with experiment and the purely chiral description. multipole expansion built in the Lagrangian. The LO Extending our Lagrangian beyond LO we use our approach Lagrangian will also have spin symmetry, which follows to study one-pion transitions, QQ̄ð2SÞ → QQ̄ð1PÞπ in from the heavy quark mass expansion. To simplify the Sec. IV, and QQ̄ð2SÞ → QQ̄ð1SÞπ in Sec. V. Certain problem further, we will also organize the computation uncertainties of our approach cancel out for specific ratios within a 1=N c expansion. If we hadronize this EFT, we of the decay widths. We compute and study them in obtain a Lagrangian in terms of the singlet (standard heavy Sec. VI. Finally, we give our conclusions in Sec. VII. quarkonium), hybrids, and pion fields. B=D mesons and possible tetraquarks are, a priori, subleading in the large II. HADRONIZATION OF THE pNRQCD N c , as we will discuss in more detail later in the paper. LAGRANGIAN Therefore, we will not consider these degrees of freedom (d.o.f.) in this paper and their possible incorporation will be The pNRQCD Lagrangian at LO in 1=mQ (except for the relegated to future work. As their effect could be important kinetic term) and at NLO in the multipole expansion reads for hadronic transitions of states close or above to open Z Z flavor thresholds, we focus on states below threshold in this paper. LpNRQCD ¼ dR3 d3 rðTr½S† ði∂0 − hs ÞS We then write the most general hadronic representation of the weakly coupled pNRQCD Lagrangian made of the þ O† ðiD0 − ho ÞO singlet, hybrids, and pions fields in a combined expansion 1 X f n in the chiral counting and the multipole, 1=mQ and 1=N c − Gaμν Gμνa þ q̄i ði= D − mi Þqi 4 i¼1 expansions. It is formally possible to obtain the coefficients of the Lagrangian by matching to suitable Green functions þ gV A ðrÞTr½O† r · ES þ S† r · EO in the pNRQCD theory in terms of quarks and gluons.  V ðrÞ Nevertheless, these coefficients endure a complicated þ g B Tr½O† fr · E; Og : ð1Þ relation with the elementary fields of the theory, and to 2 determine them would require quite costly lattice simu- lations. Nevertheless, no all hope is lost. In Ref. [16], it was S and O are the quark singlet and octet fields, respectively, pffiffiffiffiffiffi observed a correlation between the dimensionality of the normalized p with ffiffiffiffiffiffi respect to color as S ¼ S1c = N c and interpolating operator of the hybrid/gluelump and its O ¼ Oa T a = T F . They should be understood as functions position in the spectrum. Though not equivalent, we of t, the relative coordinates r, and the center of mass hypothesize in this paper that there is also a correlation coordinates R of the heavy quarks. The trace should be between the dimensionality of the interpolating operator understood as a double trace in color and spin. The singlet, and the strength of the interpolation with the hybrid, such octet, and hybrid fields in the Lagrangians that appear in that higher dimension operators are subleading (in this this paper are organized in SUð2Þ spin multiplets. For respect, it would also be interesting to study whether this instance, S ¼ p1ffiffi2 ðS · σ þ Sη IÞ. All the fields of the light approximation can be related with the analysis made in d.o.f. in Eq. (1) are evaluated at R and t; in particular, [17]). We will see that such hypothesis plus the chiral low- Gμνa ≡ Gμνa ðR; tÞ, qi ≡ qi ðR; tÞ, and iD0 O ≡ i∂0 O − energy theorems (generated by the axial and energy- g½A0 ðR; tÞ; O. hs and ho are the singlet and octet momentum tensor anomalies) lead to the same predictions Hamiltonian densities. They read as for the normalized differential decay rates of the two-pion transitions as using the twist expansion. Moreover, we do ∇2r not need to impose extra conditions on ΛQCD , except the hs ¼ − þ V s ðrÞ; ð2Þ mQ one we already imposed for the multipole expansion mQ v ≫ ΛQCD . In other words, our computational frame- work would still be valid even if ΛQCD ≫ mQ v2 . We ∇2r ho ¼ − þ V o ðrÞ; ð3Þ explore the implications of this computational framework mQ 054021-2 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) where V s ðrÞ and V o ðrÞ are computed in perturbation where summation over index i is implied. Pikλ is a projector theory. Note that we have spin symmetry. Unless stated that acts onto the gluelump angular momentum and otherwise we will work in the isospin limit mi ¼ m̂ ≡ projects it into an eigenstate of K · r̂ (where K is the mu þmd angular momentum operator for the gluelump) with eigen- 2 with i ¼ u, d. At leading log [22] and next to leading log [23] accuracy V A ¼ 1. value λ. We next aim to construct the hadronic version of the It is useful to project the pNRQCD Lagrangian onto the above Lagrangian. We first need to characterize the Fock subspace spanned by the jR; r; k; λi states hadronic d.o.f. relevant to our case. We first work in Z X the static limit. In the short heavy-quark-antiquark distance d3 rd3 R jR; r; k; λiΨkλ ðt; r; RÞ; ð8Þ limit, the gluonic excitations can be characterized by the so- kλ called gluelump operators. They organize themselves in irreducible representations of the Oð3Þ ⊗ C group. The LO where Ψkλ ðt; r; RÞ will represent the hybrid field in the Hamiltonian density in the 1=mQ and multipole expansions hadronic version of the EFT. As we have already men- tioned, the case of most interest to us is that of the spin 1 corresponding to the Lagrangian in Eq. (1) is given by gluelumps. The projectors for this case are Z Z H ¼ d R d3 rTr½S† hs S þ O† ho O 3 Pi10 ¼ r̂i0 ¼ r̂i ; ð9Þ Z  pffiffiffi 1 Pi11 ¼ r̂i ¼∓ðθ̂i  iϕ̂i Þ= 2; ð10Þ þ d3 R ðEa · Ea þ Ba · Ba Þ 2 X ni  and the discrete symmetry transformations for the S, O, and − q̄i ½iD · γ − mi qi : ð4Þ Ψiκ fields are given in Table I. i¼1 Besides the singlet and hybrid fields, we will incorporate pions to our Lagrangian. As a basic building block for the For later convenience, we will restrict the discussion to Goldstone bosons, we use the unitary matrix Uðt; RÞ, L ¼ 1 gluelump states. We define the gluelump operators, which [for SUð3Þ] may be taken as Gia k , as the color-octet gluonic operators that generate the 0 pffiffiffi þ pffiffiffi þ 1 eigenstates of H in the presence of a local heavy-quark- π 0 þ pηffiffi3 2π 2K antiquark octet source B pffiffiffi p ffiffiffi 0 C B η 2K C U ¼ eiΦ=F ; Φ ¼ B 2π − −π 0 þ pffiffi3 C; ð0Þ @ pffiffiffi p ffiffi ffi A HOa† ðR; rÞGia k ðRÞj0i ¼ ðV o þ Λk ÞO ðR; rÞGk ðRÞj0i; a† ia 2K − 2K̄ 0 − p2ffiffi3 η ð5Þ ð11Þ PC where a is the color index, k labels the quantum J numbers of the gluelump, and i labels its vector compo- although final results for observable quantities do not nents. At this stage, we do not have to make explicit the depend on this specific choice. F ¼ 92.419 MeV is the spin content of Oa . We normalize the gluelump operators as pion decay constant. Under chiral symmetry, U transforms g as U→RUL, where R ∈ SU R ðNÞ and L ∈ SU L ðNÞ. 0 0 0 jb k ðR ÞO ðR ; r ÞO ðR; rÞGk0 ðRÞj0i h0jGia† a b† Related useful matrices are u, defined from u2 ¼ U, and ¼ δij δkk0 δðR0 − RÞδðr0 − rÞ: ð6Þ † uμ ¼ iðu† ð∂μ − irμ ÞÞu − uð∂μ − ilμ Þu Þ; ð12Þ Going beyond the LO in the multipole expansion the system is no longer spherically symmetric, instead it is χ  ¼ u† χu†  uχ † u; ð13Þ cylindrically symmetric around the heavy-quark-antiquark axis.1 Representations of the cylindrical symmetry group where χ ¼ 2Bdiagðm̂; m̂; ms Þ, with B being related to the can be constructed by projecting the gluelump operators on vacuum quark condensate. In the isospin limit, the pion various directions with respect to the heavy-quark-anti- mass is m2π ¼ 2Bm̂. The transformation properties can be quark axis. Therefore, we work with states with good found in Table II. transformation properties under the cylindrical symmetry We now construct the hadronic Lagrangian. It is fixed group by the d.o.f., the symmetries, and the parameter expansions we have: 1=N c , r, E, 1=mQ , mi . We emphasize that, at this jR; r; k; λi ¼ Pikλ Oa† ðR; rÞGia level, we do not integrate out extra d.o.f. when going to the k ðRÞj0i; ð7Þ hadronic representation of Eq. (1). Therefore, it is not a different EFT but the very same pNRQCD, including the 1 The symmetry group is D∞h , with P replaced by CP. same d.o.f. and scales. Instead what we do is to write the 054021-3 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) TABLE I. Transformation properties of the heavy quarkonium and gluonic fields, and the projection vectors, under discrete symmetries. The octet field O has the same transformation properties as S. The Ψik transform as O combined with the kPC of the gluelump. The transformations of the projected fields can be obtained by further adding those of the projection vectors; however, these are actually not relevant for the construction of the Lagrangian since the projection vectors always appear in pairs, one explicit in the operator and another implicit in Ψkλ ¼ r̂λ · Ψk . For this reason, we give the transformation properties of the unprojected fields Ψik . Note that the difference in the transformation of r̂λ with respect to Ref. [18] are due to the different definition of r̂þ . P T C Sðt; r; RÞ −Sðt; −r; −RÞ σ 2 Sð−t; r; RÞσ 2 ⊤ σ 2 S ðt; −r; RÞσ 2 Ψi1þ− ðt; r; RÞ −Ψi1þ− ðt; −r; −RÞ −σ 2 Ψi1þ− ð−t; r; RÞσ 2 −σ 2 ðΨi1þ− Þ⊤ ðt; −r; RÞσ 2 Ψi1−− ðt; r; RÞ Ψi1−− ðt; −r; −RÞ σ 2 Ψi1−− ð−t; r; RÞσ 2 −σ 2 ðΨi1−− Þ⊤ ðt; −r; RÞσ 2 Eðt; RÞ −Eðt; −RÞ Eð−t; RÞ −E⊤ ðt; RÞ Bðt; RÞ Bðt; −RÞ −Bð−t; RÞ −B⊤ ðt; RÞ r̂λ −r̂−λ r̂λ −r̂−λ most general Lagrangian consistent with the symmetries in it. Strictly speaking, one should also include glueballs, made out of the heavy quarkonium, hybrids, and pions. We which may interact with the hybrids at LO in the multipole write this Lagrangian at LO in the chiral counting and the expansion, but at NLO in the 1=N c expansion. Such effects 1=mQ expansion (except for the kinetic term) and at NLO would be subleading in our computation of the decays. in the multipole expansion. For the interaction between Therefore, we neglect them. The hadronic version of heavy quarkonium, hybrids, and pions, we also incorporate the pNRQCD Lagrangian projected onto the subspace of the large N c expansion, and consider only the leading terms Eq. (8) reads Z    ∇2r F2 Lhad ¼ d Rd rTr S i∂t − V s ðrÞ þ 3 † 3 S þ ðhuμ uμ i þ hχ þ iÞ pNRQCD mQ 4   X X ð0Þ ∇2r i Ψkλ ði∂t − ðV o þ Λk ÞÞδλλ0 þ r̂λ † r̂ 0 Ψkλ0 i† þ 0 mQ λ PC þ− −− k ¼1 ;1 λλ −− Þ þ ðr · r̂λ S† Ψ1−− λ þ h:cÞtðr1 þ r · δLðΨkλ ; Ψk0 λ0 Þ  ðr1−− Þ ðr1−− Þ ðr1−− Þ þ ðr · r̂λ S† Ψ1−− λ þ h:cÞðtd0 F2 hu0 u0 i þ tdi F2 hui ui i þ tm F2 hχ þ iÞ : ð14Þ Note that for each hybrid channel one should also include states that can be generated by such gluelumps in the the excitations. The fields S and Ψkλ should be under- Lagrangian. According to the operator analysis of Table II stood as depending on t, r, and R. The pion fields depend in Ref. [16], they correspond to Σþ 0 − g , Πg , Σu , Πu , and on t and R. Note that Tr[] now only stands for the trace associated excitations with bigger gluelump masses. The over spin indices. hAi stands for the trace of A in the mixing between hybrids at NLO in the multipole expansion isospin index. is more complicated. It is encoded in r · δLðΨkλ ; Ψk0 λ0 Þ, Since at NLO in the multipole expansion the singlet can where δLðΨkλ ; Ψk0 λ0 Þ is bilinear in the hybrid fields and only mix with k ¼ 1 gluelumps, we only include hybrid transforms as a 1−− vector. Note that in general k ≠ 1 states may contribute to this term. Fortunately, we will not need the details of this interaction for the analysis of this paper. TABLE II. Transformation properties of the basic chiral build- Note also that at NLO in the multipole expansion we do not ing blocks under discrete symmetries. have hybrid bound states but plane waves. We will need to u uμ χ Dμ iterate the OðrÞ vertices to obtain bound states in the hybrid μ sector. P u † −u χ  Dμ The last line in Eq. (14) encodes the interaction of the C u⊤ u⊤ μ χ⊤  D⊤ μ hybrids with the singlet and pions at NLO in the multipole T u uμ χ Dμ expansion and at the leading nonvanishing order in the h.c u† uμ χ  Dμ large N c and chiral expansions. The interaction with two 054021-4 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) pions scales with N c as 1=N 2c . One may consider the since Oðr2 Þ terms in the Lagrangian do not contribute to the incorporation of subleading operators in the chiral counting static energies of the singlet. but still leading in the 1=N c expansion. Since for the In the case of hybrid bound states, mQ v2 ≪ ΛQCD is also process we consider in this paper, the typical energy of the the natural hierarchy between the scales since the bound pions will be of Oðmv2 Þ or smaller (which in general we states are small energy fluctuations around the minima of will consider them to be smaller than ΛQCD ); this will not the hybrid static energies. Therefore, we can also generi- be necessary. This is also the reason we do not include cally write interactions of the pions with the hybrids that are not i suppressed by the multipole expansion. The contribution of V kλλ0 ðrÞ ¼ lim lnðh0jOa ðt; r; R0 ÞP†kλ these operators to heavy quarkonium hadronic transitions t→∞ t would be suppressed by a power of mv2 =ΛQCD compared · Ga† 0 k ðt; R ÞO ð0; r; RÞPkλ0 · Gk ð0; RÞj0iÞ a† a with the contributions considered in this paper. ð0Þ In the above discussion, we have not included tetra- ¼ ðΛk þ V o þ bkλ r2 þ   Þδλλ0 þ Oð1=mQ Þ: quarks (like the Zb or Zc ), nor Qq̄-Q̄q states, in the physical ð16Þ spectrum. One may wonder whether we should do so. Here we will guide our discussion by the 1=N c counting. The Nevertheless, in this case, we cannot guaranty that Oðr2 Þ inclusion of tetraquarks is delicate. They are not stable in terms in the Lagrangian will not contribute to the hybrid the large N c . If the decay width grows like N c , then Γ ≫ E potential. Therefore, we cannot determine bkλ from the and diagrams with intermediate tetraquarks become effec- OðrÞ coefficients of the hadronic Lagrangian alone. We tively local and 1=N c suppressed relative to the accuracy will determine this coefficient by fitting the potential to the for the physical processes we consider in this paper. The lattice data for the static energies. same happens for the Qq̄-Q̄q loops. The suppression of The LO singlet and hybrid spectrum and wave functions tetraquark effects can only be bypassed if, for some reason, will be obtained from solving the Schrödinger equation Γ is much smaller than what is expected by the N c with different variants of the singlet and hybrid potential counting. This could happen if the channels that would based on Eqs. (15) and (16). The details of these solutions contribute at leading order in N c are closed because they can be found in the Appendix B. are below threshold. This is the situation discussed in Let us now discuss the vertices of the hadronic Ref. [24]. Then, indeed, the scaling in N c of Γ is, at most, of Lagrangian. To obtain the coupling in the hadronic order 1=N c. However, we are not in this situation. For Lagrangian, we match equivalent correlation functions instance, Zb states are above threshold, albeit very close to computed in both representations of the theory. The it. Therefore, there is little phase space free, effectively equalities are obtained in the static limit. The singlet-hybrid producing that Γ is very small. In this scenario, the mixing mixing term in the hadronic theory is given by with Qq̄-Q̄q loops is expected to be very large and it does h0jΨ1−− λ ðt; r0 ; R0 ÞS† ð0; r; RÞj0iamp ¼ ir · r̂†λ tðr1 −− Þ not make much sense to consider one without the other. ; ð17Þ In this respect, it is interesting to note that in Refs. [25,26] it has been advocated that tetraquarks and Qq̄-Q̄q loops which is matched to the following correlator of pNRQCD may play an important role for some observables like in term of quarks and gluons: the QQ̄ð3S=4SÞ → QQ̄ð1SÞ decays. Nevertheless, in this paper, we want to specifically study the effect associated h0jr̂†λ · Ga† 0 0 0 † 1−− ðt; R ÞO ðt; r ; R ÞS ð0; r; RÞj0iamp a to the inclusion of hybrids within a context where the sffiffiffiffiffiffi multipole expansion can be applied. Therefore, the possible T F †i j ¼i r̂ r h0jGai† 1−− ðRÞgE ðRÞj0i; aj ð18Þ incorporation of tetraquark and Qq̄-Q̄q states will be Nc λ postponed to future work. In this paper, we will generally consider that the where amp. signals that only amputated contributions are quarkonium binding energies fulfill mQ v2 ≪ ΛQCD . This considered [overall δðr0 − rÞ are also factored out]. To allows us to write the static singlet potential up to Oðr2 Þ as evaluate the matrix element, we consider the interpolating field for Ga1−− to be given by a sum of all possible local i gluonic operators with the same quantum numbers, V Σþg ðrÞ ¼ lim ln ðh0jSðt; r; R0 ÞS† ð0; r; RÞj0iÞ t→∞ t ð0Þ ¼ V s ðrÞ þ bΣþg r2 þ    : ð15Þ Ga1−− ¼ Z−1=2 E gEa þ Z−1=2 D×B ðD × gBÞ þ    : a ð19Þ Now, we then hypothesize that there is a correlation The coefficient bΣþg can actually be determined in terms of between the dimensionality of the interpolating operator −− the coefficients tðr1 Þ from the Lagrangian in Eq. (14), and the strength of the interpolation with the hybrid, such 054021-5 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) that higher dimension operators are subleading, so the III. Q̄Qð2SÞ → Q̄Qð1SÞππ HADRONIC series can be truncated at LO. Though not rigorous, one TRANSITIONS 1−dimðOÞ may consider counting Z−1=2 O ∼ ΛQCD , where dim(O) is The two-pion quarkonium transitions are well described the dimension of the operator O, and the size of the operator using chiral perturbation theory and spin symmetry. The to be given by ∼ðmv2 ÞdimðOÞ, so the series can be truncated most general amplitude for the two-pion transitions at at LO up to corrections of Oðmv2 =ΛQCD Þ. Using this Oðp2 Þ in the chiral counting reads truncation and Eq. (6) in Eq. (18), we obtain Aχ ¼ −a1 p0þ p0− þ a2 pþ · p− − a3 m2π ; ð26Þ sffiffiffiffiffiffiffiffiffiffiffiffi −− Þ T F ZE where pþ , p− are the momentum of the π þ and π − , tðr1 ¼ : ð20Þ Nc respectively. The coefficients ai can be thought as linear combinations of Wilson coefficients of an effective chiral The operators with an even number of pions in Lagrangian made only by heavy quarkonium and pions. Eq. (14) are matched in a similar way. In the hadronic Such effective Lagrangian can be found in Eqs. (4) and (6) pNRQCD, of Ref. [2]. Imposing heavy quark spin symmetry sets the g2 low-energy constant of Ref. [2] to zero. Z Alternatively, we can use the computational scheme d4 xþ d4 x− eipþ ·xþ eip− ·x− developed in this paper. Using Eq. (14), we can compute the leading nonvanishing contribution to the two-pion × h0jπ þ ðxþ Þπ − ðx− ÞSðt; r; RÞΨ†1−− λ ð0; r; RÞj0iamp transitions between quarkonium states. The diagrams are ðr1−− Þ 0 0 ðr1−− Þ ðr1−− Þ drawn in Fig. 1. One vertex mixes singlet and hybrid fields ¼ i4r · r̂λ ð−td0 pþ p− þ tdi pþ · p− − tm m2π Þ; whereas the other produces a two-pion emission vertex. ð21Þ The amplitude for this process reads ðn0 nÞ −− Þ ðr1−− Þ 0 0 ðr1−− Þ ðr1−− Þ and the corresponding correlator in the partonic pNRQCD iA ¼ iβr;Q tðr1 ðtd0 pþ p− − tdi pþ · p− þ tm m2π Þ; reads ð27Þ hπ þ ðpþ Þπ − ðp− ÞjSðt; r; RÞr̂λ · Ga1−− ð0; RÞOa† ð0; r; RÞj0iamp with sffiffiffiffiffiffi X   ¼ ig TF r̂ · rhπ þ ðpþ Þπ − ðp− ÞjEðRÞ · G1−− ðRÞj0i ðn0 nÞ † 1 1 Nc λ βr;Q ¼ hSn0 jr̂λ · rjΨm i þ mn − mm mn0 − mm sffiffiffiffiffiffiffiffiffiffiffiffi   m i TF 8π 2 9 × hΨm jr̂†λ · rjSn i: ð28Þ ¼ r̂λ · r 2 − κ p0þ p0− 3 N c ZE β0 2    The index m sums over all states solution of the 3 2 Schrödinger equation and also for the gluelump excitations. − 2 þ κ pþ · p− þ 3mπ ; ð22Þ 2 For the second term in the brakets of Eq. (27) (correspond- ing to the right diagram in Fig. 1), we have used that where in the last step we use Eq. (19) truncated at the first mn ¼ mn0 þ p0þ þ p0− . It is remarkable that the normaliza- term and hadronize the resulting gluonic matrix element tion factors ZE cancels out, which allows us to completely using Eq. (D15), which uses the anomaly relation of the evaluate Eq. (27) except for the parameter κ that appears in energy-momentum tensor of QCD [11,27]. The derivation the couplings in Eqs. (23)–(25) and has its origin in the of this equation is reviewed in Appendix D 1. hadronization of the gluonic operator. In the next section, Comparing Eqs. (21) and (22), we obtain we will fix κ by fitting the normalized differential decay width spectrum. sffiffiffiffiffiffiffiffiffiffiffiffi  For the case at hand, the quarkonia states are n0 ¼ 13S1 ðr1−− Þ 2π 2 TF 9 td0 ¼− 2− κ ; ð23Þ and n ¼ 23S1 . The intermediate hybrid has 1−− quantum 3β0 N c ZE 2 numbers. Expressions for the hybrid-quarkonia matrix sffiffiffiffiffiffiffiffiffiffiffiffi elements that appear in Eq. (28) are given in Appendix C.  ðr1−− Þ 2π 2 TF 3 tdi ¼− 2þ κ ; ð24Þ 3β0 N c ZE 2 sffiffiffiffiffiffiffiffiffiffiffiffi ðr1−− Þ 2π 2 TF tm ¼− 3: ð25Þ FIG. 1. Single and dashed lines represent quarkonia and pions. 3β0 N c ZE The double line with a curly line inside represents hybrid states. 054021-6 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) FIG. 2. Plot of the normalized differential decay width spectrum. The dots are the experimental data for ψð2SÞ → J=Ψπ þ π − [28] and ϒð2SÞ → ϒð1SÞπ þ π − [29] in blue and yellow, respectively. In the same color scheme, the continuous lines are the fits of the theoretical expression obtained from the amplitude in Eq. (26) computed from an EFT incorporating chiral and spin symmetry. The variable x is ππ −2mπ defined as x ¼ m2Sm−m 1S −2mπ . The wave functions of the bound states that appear in those obtained from the total decay width. Therefore, in the matrix elements can be found in Appendix B. The matrix following subsections, we fit the normalized differential elements in Eq. (28) are diagonal in the spin state of the heavy decay width and total decay width using the formulas quarks. Thus, only spin triplet hybrid states are allowed as discussed above. Detailed formulas can be found in intermediate states. Furthermore, the matrix elements in Appendix A. Here we will analyze the transitions ψð2SÞ → Eq. (28) only receive contribution from the λ ¼ 0 component J=Ψπ þ π − and ϒð2SÞ → ϒð1SÞπ þ π − . The experimental of the hybrid state, which leads to the selection rule l ¼ l data for the former are taken from Ref. [28], and for the for the angular momentum eigenvalues. Therefore, the only latter from Ref. [29].2 The total decay widths are taken possible hybrid intermediate states correspond to m3 S 1 from Ref. [31]. states. Equation (27) yields the following prediction for the A. Line-shape analysis chiral parameters ai : The chiral fit to 1 dΓ produces the following ratios of Γ dmππ   parameters: 8π 2 T F ð12Þ 9 a1 ¼ − βr;Q 2 − κ ; ð29Þ 3β0 N c 2 ac1 ac3   ¼ −0.20þ3.93 −0.98 ; ¼ 3.12þ3.78 −15.05 ; χ 2d:o:f ¼ 0.13; ð33Þ 8π 2 T F ð12Þ 3 ac2 ac2 a2 ¼ − β 2þ κ ; ð30Þ 3β0 N c r;Q 2 ab1 ab3 ¼ 0.26þ5.29 −2.30 ; ¼ 1.84þ8.48 −20.43 ; χ 2d:o:f ¼ 0.14; ð34Þ 8π 2 T F ð12Þ ab2 ab2 a3 ¼ − β : ð31Þ β0 N c r;Q where the superindex c and b label the results for the We now confront the above predictions with experiment. ψð2SÞ → J=Ψπ þ π − and ϒð2SÞ → ϒð1SÞπ þ π − transitions, dΓ Since the experimental data on dm ππ (where mππ ¼ ðpþ þ respectively. These fits are performed using the relativistic 2 kinematics [see Eqs. (A4) and (A5)]. If instead we use p− Þ is the dipion invariant mass) are normalized to an unknown constant, it is convenient to fit the theoretical nonrelativistic kinematics [see Eqs. (A6) and (A7)], fits of expressions to the normalized differential decay width very similar quality are obtained. The results are shown in Fig. 2 together with the experimental data. Note the small 1 dΓ : ð32Þ Γ dmππ 2 In the first version of this paper, the more recent data from Ref. [30] was used. However, these data appear to have some As we will see there is also a strong theoretical motivation problems. This reflects in an strange behavior of the angular to consider this ratio. This object is only sensitive to ratios distributions. We acknowledge conversations on this issue with of the theory parameters. The overall normalization can be the authors. 054021-7 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) FIG. 3. Correlation between the parameters of the fit of the normalized decay width spectrum. Results for charmonium and bottomonium transitions in the left and right sides, respectively. The blue areas indicate the parametric space regions with χ 2d:o:f ≤ 1, and the black dots the best fit whose values are given in Eqs. (33) and (34). χ 2d:o:f obtained in the fits to both transitions. The uncer- functional form depends only on the parameter κ. Thus, tainties quoted in Eqs. (33) and (34) correspond to the the hadronic pNRQCD, which incorporates the multipole range of parameter values with χ 2d:o:f ≤ 1, which is shown expansion, and a dimensional counting for the overlap of in Fig. 3. In this figure, it can be appreciated that there the hybrids with gluonic operators, is more predictive is a strong correlation between aQ Q Q Q than the EFT relying on chiral and spin symmetry only, 1 =a2 and a3 =a2 . Therefore, it would be wrong to consider the error of since the number of free parameters is reduced from two to one. Note that our approach yields the same normalized aQ Q Q Q 1 =a2 and a3 =a2 independently. Instead, the allowed decay width line shape, as the one obtained using the twist region of parameter space is given by the dashed region in expansion. Fig. 3. This corresponds to a particular linear combination We fit the line shapes of the charmonium and bottomo- of aQ Q Q Q 1 =a2 and a3 =a2 . We conclude then that, with the niun data independently. Since the value of κ should be present data on the width spectrum, we cannot simulta- independent of the heavy quarkonium dynamics, we also neously fit both parameter ratios with high accuracy. perform a simultaneous fit to both data sets. Using the If we compare these fits with the ones in Ref. [2], we nonrelativistic kinematics, we obtain the following values observe some differences. The authors in Ref. [2] report for κ: that the best fit yields a3 ¼ 0 (g3 in their notation). This does not coincide with our best fit, but is consistent with the κ c ¼ 0.243þ0.014 −0.013 ; χ 2d:o:f ¼ 0.18; ð35Þ uncertainty. The ratio a1 =a2 corresponds to 1 þ g1 =ð2gÞ in the notation of Ref. [2], and their best fits correspond to κ b ¼ 0.219þ0.016 χ 2d:o:f ¼ 0.14; ð36Þ −0.015 ; ac1 =ac2 ∼ 1.2 and ab1 =ab2 ∼ 1.1. This is compatible with our fits considering the uncertainty. One should keep in mind κ joint ¼ 0.233  0.013; χ 2d:o:f ¼ 0.18: ð37Þ that the experimental data source in Ref. [2] is Ref. [32], whereas we use more recent data [29].3 The range of values correspond to the range with We now use the amplitude in Eq. (27) to obtain the nor- χ 2 − χ 2min ≤ 1. If we use the relativistic kinematics, the fits malized differential decay width and fit it to the exper- yield slightly different values, imental data. Equation (27) is equivalent to Eq. (26) with the parameters ai taken as in Eqs. (29)–(31). The ð12Þ κ c ¼ 0.277  0.015; χ 2d:o:f ¼ 0.17; ð38Þ normalized decay width is independent of βr;Q , and its κb ¼ 0.229  0.016; χ 2d:o:f ¼ 0.14; ð39Þ 3 Very similar results to those in Ref. [2] were obtained in Ref. [33] using more recent experimental data [4,29] including κ joint ¼ 0.247þ0.014 −0.013 ; χ 2d:o:f ¼ 0.25: ð40Þ pion final state interactions through an unitarization of χPT. 054021-8 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) FIG. 4. Plot of the normalized differential decay width spectrum. The dots are the experimental data for ψð2SÞ → J=Ψπ þ π − [28] and ϒð2SÞ → ϒð1SÞπ þ π − [29] in blue and yellow, respectively. In the same color scheme, the continuous lines are the fits of the theoretical expression obtained from the amplitude in Eq. (27) computed with the hadronic pNRQCD Lagrangian. The variable x is defined ππ −2mπ as x ¼ m2Sm−m 1S −2mπ . The differences with the nonrelativistic fit are of the B. Total decay widths order of the difference between the charmonium and The expressions for the total decay width in terms of bottomonium fit. This is reasonable, as they both measure the chiral coefficients ai can be found in Eqs. (A10) and relativistic effects. We will take the difference between (A11). Inserting the values from Eq. (33) or Eq. (34), the Eqs. (37) and (40) as a measure of the size of subleading remaining free parameter, a2 , can be adjusted to reproduce effects. We then combine it in quadrature with the statistical the experimental value of the total decay width. In fact, error quoted in Eq. (40) and give since the total width is a quadratic function in a2 , two solutions are possible, κ ¼ 0.247ð20Þ ð41Þ ac2 ¼ −25.88 GeV−3 ; ac2 ¼ 25.52 GeV−3 ; ð42Þ as our default value. In any case, it is remarkable that all fits yield similar values for κ, which we take as a confirmation ab2 ¼ −10.98 GeV−3 ; ab2 ¼ 10.78 GeV−3 : ð43Þ of the independence of κ of the heavy quarkonium dynamics. For illustration, we show the plots of the fit In our EFT, the total decay width is proportional to ð12Þ using relativistic kinematics for charmonium and bottomo- ðβr;Q Þ2 (note that it is at this level where there is a dif- nium data in Fig. 4. Actually, we can also observe in Figs. 2 ference with predictions using the twist expansion). This and 4 the similarity of the experimental data for the line object is dependent on the wave functions of heavy shapes of the charmonium and bottomonium spectra. This quarkonium and hybrids, as well as on the energy difference can be taken as a reflection of the independence of this among them. Typically, this quantity will suffer from rather observable on the heavy quarkonium dynamics, which is a large uncertainties, as we are forced to make strong prediction of the effective theory. approximations to compute this object. We neglect the effect Previous fits of the decay width spectrum of ψð2SÞ → of higher gluelump excitations with the same quantum J=ψππ have been carried out in Ref. [4] using the transition numbers. We expect that higher gluelumps will give smaller amplitude from Ref. [11], and in Ref. [34] in which the contributions, as they are suppressed by larger energy pion final state interactions were taken into consideration differences. Still, this is an approximation. In some circum- through an Omnès function. The reported values are κc ¼ stances, it can be compulsory to sum all of them to recover 0.186ð3Þ and κ c ¼ 0.135ð5Þ, respectively. We have some high energy logarithms. Nevertheless, compared with checked that the main source of this discrepancy is that other uncertainties, this effect will be small. Therefore, in the transition amplitudes used in those references do not this paper, we neglect the error associated to neglecting include the complete Oðp2 Þ pion mass contribution: in higher gluelump channels. We account for the error asso- Ref. [34], the coefficient a3 is set to zero, and in Ref. [4], ciated to the energy splitting between singlet and hybrid terms proportional to the pion mass have been set to zero. states using the error of Λ1 , the lowest gluelump mass (for 054021-9 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) further details, see Appendix B). For the lowest hybrid, we have Oðαs Þ corrections to Eqs. (D14) and (D15) due to the ð12Þ compute βr with the m3 S 1 hybrid intermediate states beta function. These Oðαs Þ corrections are generated at a m ¼ 1; …; 4 (we observe that the effect of introducing three low-energy scale, which makes their evaluation not fea- or four hybrid states is comparatively small compared with sible. On the other hand, these effects factor out and could ð12Þ other uncertainties). For the static potential of the singlet and be reabsorbed in ðβr;Q Þ2 if we let this object to be a free hybrid, we take two possible parametrization that we explain parameter, not fixed by theory. Alternatively, these effects in further detail in Appendix D 1. One aims for a good are independent of the heavy quarkonium dynamics and description at all distances of the lattice-evaluated static would cancel in the ratio Γψð2SÞ→J=ψπþ π− =Γϒð2SÞ→ϒð1SÞπþ π− . singlet and hybrid potentials, but constrained to have the We will discuss this ratio later in Sec. VI. Other corrections right behavior at short distance. The other parametrization to Eqs. (D14) and (D15) are due to Oðp4 Þ chiral correc- keeps the shape of the potentials predicted by the multipole tions. We expect those not to be very important due to the expansion in the whole fitted range. Throughout the paper, limited phase space available. Other source of error comes we will take the former for our central values and the from neglecting the anomalous dimension of the light- difference with the latter as an estimate of the error quark mass. These two sources of error would affect the associated to the approximate knowledge of the singlet determination of the line shapes. As we have obtained a and hybrid wave functions and binding energies. We obtain pretty good fit for them, we will neglect these sources of ð12Þ the following values for βr;Q : error in the following. The other error we have not incorporated in this analysis, ð12Þ βr;c ¼ 11.46ð −1.36 −3 þ1.81 ÞΛ1 ð3.03Þs:p: GeV ; ð44Þ nor in those we will perform later, is the error due to the working hypothesis we use in this paper (saturation of the ð12Þ interpolating field by those with smaller dimensionality). βr;b ¼ 3.45ð −0.34 −3 þ0.43 ÞΛ1 ð0.97Þs:p: GeV : ð45Þ The reason is that we want to see whether such hypothesis is feasible, and if so what is the expected error, by The uncertainties are labeled according to the source: Λ1 (the comparing our predictions with experiment. lowest lying gluelump mass), and s.p. (the different para- metrization for the singlet and hybrid static potentials). ð11Þ At this point, it is worth mentioning that βr;Q , relevant IV. Q̄Qð2SÞ → Q̄Qð1PÞπ HADRONIC TRANSITIONS for heavy quarkonium-nucleon interaction and the chiral- associated heavy quarkonium energy shift, has also been The Q̄Qð2SÞ → Q̄Qð1PÞπ hadronic transition4 is zero analyzed using hybrid potentials in Ref. [35]. with the LO Lagrangian in Eq. (1) considered this far. The ð12Þ Introducing these values of βr;Q , and κ ¼ 0.247ð20Þ, in first nonzero contribution is generated by spin-isospin Eqs. (29)–(31), the expressions for the total width in breaking effects. The leading spin-dependent operators Eqs. (A10) and (A11) read (the experimental values are originate from the following pNRQCD operators: taken from [31]) Z   gcF δLpNRQCD ¼ d Rd r3 3 Γψð2SÞ→J=ψπþ π− ¼ 46.2ð −10.3 −3.2 þ15.7 ÞΛ1 ð þ3.4 Þκ ð21.2Þs:p: keV; 2mQ Γexp ¼ 102.1ð2.9Þ keV; ð46Þ × Tr½fS† ; σg · BO þ O† B · fσ; Sg: ð48Þ We remind that the singlet, octet, and hybrid fields in the Γϒð2SÞ→ϒð1SÞπþ π− ¼ 3.08ð −0.58 −0.23 þ0.81 ÞΛ1 ð þ0.22 Þκ ð1.49Þs:p: keV; Lagrangians that appear in this paper are organized in Γexp ¼ 5.71ð48Þ keV: ð47Þ SUð2Þ spin multiplets. An alternative representation of Eq. (48) in terms of the spins of the quark and antiquark can Our numbers differ from the experimental ones by about be found in Eq. (105) of [37]. This last representation is the a factor 2. One should keep in mind however that our one we will customarily use for the computation of the estimates suffer from large uncertainties. We find a sig- matrix elements. nificant dependence on variations of the wave function of The renormalization group improved expression for the the hybrid and singlet states. The error generated by the matching coefficient cF is known with next-to-leading uncertainty on the energy difference between singlet and logarithmic accuracy [38,39]. In order to include the hybrid states is somewhat smaller. These error estimates leading isospin violation effects, we should no longer con- are of the right magnitude, though not large enough, to sider the light-quark masses degenerate in Eq. (1): we take completely account for the difference with experiment. One mu ¼ 2.118 MeV and md ¼ 4.690 MeV instead [31]. should keep in mind, however, that besides those errors already estimated, one error that has not been incorporated 4 Let us note that this type of transition was first considered in in this analysis is due to the uncertainties associated to the Ref. [36] for the ϒð2SÞ → hb ð1PÞπ 0 decay. However, this hadronization of the local operator: on the one hand, we particular case turned out not to be kinematically allowed. 054021-10 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) Adding the operator in Eq. (48) to the leading pNRQCD Lagrangian in Eq. (1) and considering isospin violation effects produces the following new terms in the hadronic Lagrangian at LO5: FIG. 5. Single and dashed lines represent quarkonia and pions. The double line with a curly line inside represents hybrid states. δLhad pNRQCD Z þ− Let us now match the operators with an odd number of ¼ d3 Rd3 rTr½ðr · r̂λ S† Ψ1þ− λ þ h:cÞtðr1 Þ iFhχ − i pions in the Lagrangian in Eq. (49). In the hadronic theory, þ− Þ we have þ ðS† fσ · r̂λ ; Ψ1þ− λ g þ h:cÞtðS1 : ð49Þ Z Terms that break spin and isospin symmetry simultane- d4 xeip·x hUSjπ 0 ðxÞSðt; r; RÞΨ†1þ− λ ð0; r; RÞjUSiamp ously are not considered, since they produce subleading þ− Þ md − mu contributions to the transition we are considering. ¼ −itðr1 r · r̂λ 2m2π ; ð54Þ mu þ md Let us match the parton and hadronic description of the spin-dependent mixing operator. In the hadronic EFT, the and in pNRQCD, correlator reads hπ 0 ðpÞjSðt; r; RÞr̂λ · Ga1þ− ð0; RÞOa† ð0; r; RÞj0iamp h0jΨ1þ− λ ðt; r; RÞS† ð0; r; RÞj0iamp ¼ iðσ 1 − σ 2 Þ · r̂†λ tðS1 þ− Þ ; sffiffiffiffiffiffi ð50Þ T F i j 0 aj ¼ ig r̂ r hπ jE ðRÞGai ðRÞj0i Nc λ 1þ− and in the partonic version of pNRQCD, we have sffiffiffiffiffiffiffiffiffiffiffiffi i TF h0jr̂†λ · Ga† ðt; RÞOa ðt; r; RÞS† ð0; r; RÞj0iamp ¼ r̂ · rhπ 0 jg2 E · Bj0i 1þ− sffiffiffiffiffiffi 3 N c ZB λ gc TF sffiffiffiffiffiffiffiffiffiffiffiffi ¼i F ðσ − σ 2 Þi r̂λj† h0jGai† ðRÞBaj ðRÞj0i; ð51Þ i 2 md − mu TF 2mQ N c 1 1þ− ¼ 4π Fmπ 2 r̂ · r; ð55Þ 3 md þ mu N c ZB λ where amp. signals that only amputated contributions are considered. Now we consider Ga1þ− to be given by a sum of where in the last step we have made use of the results of all possible gluonic operators with the same quantum Appendix D 2 for the hadronization of the gluonic operator numbers through the axial anomaly. Comparing Eqs. (54) and (55), we arrive at Ga1þ− ¼ Z−1=2 B gBa þ Z−1=2 D×E ðD × gEÞ þ    : a ð52Þ sffiffiffiffiffiffiffiffiffiffiffiffi ðr1þ− Þ 2π 2 TF Similarly, to the previous section, we hypothesize that there t ¼− F : ð56Þ is a correlation between the dimensionality of the inter- 3 N c ZB polating operator and the strength of the interpolation with the hybrid, such that higher dimension operators are We are interested in investigating the decay ψð2SÞ → subleading, so the series can be truncated at LO. Using hc ð1PÞπ 0 , which is given in the hadronic pNRQCD EFT this truncation and Eq. (6) in Eq. (51), and matching to by the diagrams in Fig. 5. The matrix element reads Eq. (50), we arrive at sffiffiffiffiffiffiffiffiffiffiffiffi 2π 2 cF T F md − mu ð12Þ iA ¼ −Fm2π βσ ; ð57Þ cF T F ZB 3mQ N c mu þ md ðS1þ− Þ t ¼ : ð53Þ 2mQ Nc with ð12Þ X i βσ ≡ hhc ð11 P1 Þjr̂λ · rjΨm i hΨm jr̂†λ · ðσ 1 − σ 2 Þjψð23 S1 Þi m mhc − mm  i þhhc ð11 P1 Þjr̂λ · ðσ 1 − σ 2 ÞjΨm i hΨm jr̂†λ · rjψð23 S1 Þi : ð58Þ mψð2SÞ − mm 5 There is another possible pseudoscalar operator of the same order as hχ − i: hDμ uμ i. However, they are both related through the leading order equations of motion Dμ uμ ¼ iðχ − − hχ − i=2Þ=2. 054021-11 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) In the first term, the ψð2SÞ mixes into a hybrid state with V. Q̄Qð2SÞ → Q̄Qð1SÞπ0 ðηÞ HADRONIC JPC ¼ 1−− but in a spin singlet, then the hybrid decays into TRANSITIONS a hc conserving the spin but changing the angular mo- We now turn our attention to Q̄Qð2SÞ → Q̄Qð1SÞπ 0 ðηÞ mentum. The intermediate hybrid states fulfilling these hadronic transitions. Since these transitions break spin conditions are m1 P 1 . This term turns out to be small symmetry, they are zero at LO in the EFT. The leading because the second vertex only has contributions from the contribution to these decays is generated by λ ¼ 0 piece of the hybrid state, which is small for 1P 1 states. In the second term, the order is switched; in this case, Z gc the allowed hybrid intermediate states are m3 S 1 . Note that δLpNRQCD ¼ d3 Rd3 r F 4mQ all the hybrid states appearing as intermediate states in this transition are associated to the 1þ− gluelump. As in the × Tr½½S† ; σ · ðrl Dl BÞO þ O† ðrl Dl BÞ · ½σ; S; two-pion decays, the normalization factors ZB cancels out. ð61Þ The matrix elements are given in Appendix C. We compute the amplitude for up to m ¼ 4 for the 3S 1 which has to be added to Eq. (1). In the hadronic EFT, these and up to m ¼ 2 for the 1P 1 hybrid intermediate states. We operators correspond to use the renormalization group improved expression of Z cF ð1 GeVÞ ¼ 1.12155 up to next-to-leading logarithmic order, where we have used the values αs ð1 GeVÞ ¼ 0.4798 δLhad pNRQCD ¼ d3 Rd3 rTr½S† ½σ i ; Ψ1−− λ  þ h:c and αs ð1.496 GeVÞ ¼ 0.3522. The decay width can be −− Þ −− Þ obtained using the amplitude of Eq. (57) in Eq. (A12), × ðtðda1 r̂iλ rj þ tðdb1 r̂jλ ri −− Þ þ tðdc1 r̂λ · rδij ÞiF∂j hχ − i: ð62Þ Γψð2SÞ→hc ð1PÞπ0 ¼ 104ð −35 þ80 ÞΛ1 ð21Þl:q: ð1Þs:p eV; Γexp ¼ 255ð39Þ eV: ð59Þ To determine the Wilson coefficients, we compute the transition amplitude both in the hadronic and partonic The error analysis has been performed similarly to the versions of the effective theory. Nevertheless, there is one previous section. In this case, we do not have error extra subtlety to be taken into account. The π 0 and η fields associated to κ, but still have a dependence on Λ1 (the in Eq. (11) mix, and to obtain the physical states, the mass lowest lying gluelump mass), and s.p. (the different para- matrix needs to be diagonalized. The physical states metrization for the singlet and hybrid static potentials). We correspond to also estimate the error due to the value of the light-quark (l.q.) mass ratio, which we take as π 0phys ¼ π 0 þ ϵη; ð63Þ md − mu ηphys ¼ η − ϵπ 0 ; ð64Þ ¼ 0.35  0.07; ð60Þ md þ mu with ϵ the mixing angle with the value of md =mu from the PDG. This observable is interesting. Unlike previous decays, it pffiffiffi 3ðmd − mu Þ does not suffer from the uncertainties associated to the ϵ¼ : ð65Þ hadronization of the local operator: The axial anomaly 4ms þ 2ðmu þ md Þ does not get OðαÞ corrections, nor there are Oðp4 Þ chiral corrections. We also observe a very weak dependence on To compute the decay amplitudes, the physical states must variations of the wave function of the hybrid and singlet be considered. In the transitions with η emission, the states. The major error is generated by the uncertainty on contribution due to the mixing is subleading, but in the the energy difference between singlet and hybrid states. case of π 0 emission both contributions are of the same Once this error is taken into account, the result is roughly order. In practice, this amounts to an extra factor 3=2 in the compatible with experiment. π 0 decay amplitude. In the hadronic EFT, we have In principle, for Q̄Qð2SÞ → Q̄Qð1PÞπ transitions, the Z energy release is small so one can use the twist expansion d4 xeip·x h0jπ 0 ðxÞΨ1−− λ ðt; r; RÞS† ð0; r; RÞj0iamp with no fear if one assumes that ΛQCD ≪ mv2 . We then would have an alternative determination with which one md − mu 2 −− ¼ −3i m ðσ þ σ 2 Þ · ½tðda1 Þ r̂λ ðr · pπ Þ can compare. In this respect, let us note that the only mu þ md π 1 previous theoretical estimate was Γψð2SÞ→hc ð1PÞπ0 ∼ 15 eV, þ tðdb1 −− Þ rðr̂λ · pπ Þ þ tðdc1 −− Þ pπ ðr · r̂λ Þ; ð66Þ from Ref. [27], using the twist expansion (ΛQCD ≪ mv2 and E ≪ mv2 ). and in the partonic pNRQCD, 054021-12 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) hπ 0 ðpÞjr̂†λ · Ga† † 1−− ðt; RÞO ðt; r; RÞS ð0; r; RÞj0iamp a expression of cF with next-to-leading logarithmic accuracy. sffiffiffiffiffiffi The expression for charmonium can be found in the ¼ T F igcF j r ðσ 1 þ σ 2 Þk hπ 0 ðpπ Þjr̂†λ · Ga† previous section. For bottomonium, we have cF ð1 GeVÞ ¼ 1−− ðDj Bk Þ j0i a N c 4mQ ð21Þ 0.87897 using αs ð4.885 GeVÞ ¼ 0.2148. βr;Q is computed sffiffiffiffiffiffiffiffiffiffiffiffi considering up to four hybrid intermediate states. The T F cF 2π 2 md − mu ¼ results obtained for the decays are the following: ZE N c mQ 15 mu þ md × m2π F½3ðσ 1 þ σ 2 Þ · r̂λ r · p − ðσ 1 þ σ 2 Þ · rr̂λ · p; ð67Þ Γψð2SÞ→J=ψπ0 ¼ 40ð −9 þ14 ÞΛ1 ð8Þl:q: ð18Þs:p: eV; Γexp ¼ 373ð14Þ eV; ð73Þ where in the last step we have used Eq. (19) truncated to the first term and the anomaly relation in Appendix D 3 [note Γψð2SÞ→J=ψη ¼ 1.19ð −0.27 þ0.41 ÞΛ1 ð0.5Þs:p: keV; that this relation does not suffer from Oðαs Þ corrections]. Matching Eqs. (66) and (67), we obtain Γexp ¼ 9.91ð30Þ keV; ð74Þ sffiffiffiffiffiffiffiffiffiffiffiffi −− T F cF 2π 2 Γϒð2SÞ→ϒð1SÞπ0 ¼ 0.21ð −0.04 þ0.06 ÞΛ1 ð0.05Þl:q: ð0.10Þs:p: eV; tðda1 Þ ¼ − F; ð68Þ ZE N c mQ 15 Γexp < 1.28 eV; ð75Þ sffiffiffiffiffiffiffiffiffiffiffiffi T F cF 2π 2 Γϒð2SÞ→ϒð1SÞη ¼ 1.58ð −0.80 þ0.42 ÞΛ1 ð0.76Þs:p: eV; ðdb1−− Þ t ¼ F; ð69Þ ZE N c mQ 45 Γexp ¼ 9.3ð1.5Þ eV: ð76Þ −− Þ tðdc1 ¼ 0: ð70Þ The subindices label the source of the uncertainty. The error analysis is equal to the one performed in the previous We are interested in investigating the decays ψð2SÞ → section. Again we do not have error associated to κ, but still J=ψπ 0 , ϒð2SÞ → ϒð1SÞπ 0 , as well as the decays ψð2SÞ → have a dependence on Λ1 (the lowest lying gluelump mass), J=ψη, ϒð2SÞ → ϒð1SÞη. The diagrams involved in the and s.p. (the different parametrization for the singlet and −− decay are drawn in Fig. 5. These contain the tðr1 Þ vertex hybrid static potentials). We also estimate the error due to from Eq. (14) mixing the quarkonia into a hybrid associated the value of the light-quark mass ratio. to the 1−− gluelump, and the P-wave pion emission vertices As in the previous section, these decays do not suffer from Eq. (62). The former vertex conserves the spin state from the uncertainties associated to the hadronization of the and is only nonvanishing for l ¼ l. Therefore, the inter- local operator: the axial anomaly does not get Oðαs Þ mediate hybrid states for the transitions we are considering corrections, nor there are Oðp4 Þ chiral corrections. On must be 3S 1 . The amplitude for a pion emission then reads the other hand, unlike in the previous section, we find a significant dependence on variations of the wave function T F cF 8π 2 md − mu ð12Þ of the hybrid and singlet states. Indeed, the dependence is iA ¼ −i F m2 ðϵ × ϵ2S Þ · pβr;Q : N c mQ 45 md þ mu π π 1S the same as the one we had in Sec. III B. The error generated by the uncertainty on the energy difference ð71Þ between singlet and hybrid states is of the same order. Note that the factor involving the sum over intermediate Nevertheless, these error estimates are not large enough to ð12Þ account for the difference with experiment, particularly for hybrid states: βr;Q , given in Eq. (28), is the same to the one the charmonium case. We later retake this issue when we appearing in the two-pion transitions. ϵmS stands for the consider ratios of decay rates. polarization of a m3S1 quarkonium state. The decays amplitude to one η is VI. RATIOS T c 8π 2 ð21Þ So far we have seen that the hadronic S- to P-wave heavy iA ¼ −i F F pffiffiffi ðm2η − m2π ÞFðϵ1S × ϵ2S Þ · pβr;Q : N c mQ 45 3 quarkonium transitions are roughly compatible with theory ð72Þ within errors. For decays to S-wave heavy quarkonium, we have seen that the overall magnitude of our predictions is We have computed the decay widths by summing over smaller than experiment, specially for the QQ̄ð2SÞ → final polarizations and averaging over the incoming ones QQ̄ð1SÞπ 0 ðηÞ decays. Nevertheless, the normalization of the square of the amplitudes in Eqs. (71) and (72) and the decays are the most uncertain object in our predictions. inserting the result in Eq. (A12). We use relativistic Therefore, we expect several uncertainties to cancel for kinematics for the outgoing quarkonium state in the phase ratios. In fact, we have already seen in Sec. III A that the space calculation and the renormalization group improved normalized differential decay rates are well described by 054021-13 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019)   theory. Going further in this direction, we may try to ψð2SÞ → J=ψπ þ π − Rc;π ≡ R ¼ 1161ð∓232Þl:q: ð −80 þ84 Þκ ; explore the magnitude of the corrections associated to the ψð2SÞ → J=ψπ 0 different approximations we have made in this paper by studying different ratios. This we do in the following. Rexp c;π ¼ 274ð13Þ; ð79Þ The ratios of the QQ̄ð2SÞ → QQ̄ð1SÞπ 0 ðηÞ transitions do   ð12Þ ψð2SÞ → J=ψπ þ π − not depend on βr;Q and are completely determined at LO Rc;η ≡ R ¼ 38.8ð −2.7 þ2.8 Þκ ; ψð2SÞ → J=ψη by chiral symmetry. Indeed, the same result is obtained using a pure chiral approach or by using the twist expansion Rexp c;η ¼ 10.3ð4Þ; ð80Þ (though the use of the latter suffers from the same drawback   as its use in QQ̄ð2SÞ → QQ̄ð1SÞππ transitions [14]), as ϒð2SÞ → ϒð1SÞπ þ π − Rb;π ≡ R studied previously in Refs. [40–42]. What changes is the ϒð2SÞ → ϒð1SÞπ 0 overall coefficient, which cancels in the ratio. The only ¼ 14.4ð∓2.9Þl:q: ð −1.0 3 þ1.1 Þκ × 10 ; uncertainties affecting the ratios are the ones associated Rexp 3 to the light-quark mass values and possible chiral correc- b;π > 4.5ð4Þ × 10 ; ð81Þ tions to the amplitude affecting the ratio Fη =Fπ and the   mixing angle ϵ. Generically, we expect these to be of ϒð2SÞ → ϒð1SÞπ þ π − Oððmπ ; mK Þ=Λχ Þ ∼ 14–50%. We can check this by compar- Rb;η ≡ R ¼ 1.94ð −0.14 3 þ0.15 Þκ × 10 ; ϒð2SÞ → ϒð1SÞη ing the theoretical and experimental ratios (for these and for 3 Rexp b;η ¼ 0.61ð11Þ × 10 : ð82Þ the experimental ratios below we add the error quadratically)   The differences with experiment are large, of the order of ψð2SÞ → J=ψπ 0 Rcχ ≡ R ¼ 3.34ð0.67Þl:q: × 10−2 ; 75%–60% for the charmonium and bottomonium, respec- ψð2SÞ → J=ψη tively. In principle, we expect the most important uncer- Rexp −2 cχ ¼ 3.76ð18Þ × 10 ; ð77Þ tainties of these ratios to be due to the effect of higher order operators in the interpolating function of the hybrids and of   the neglected Oðαs Þ corrections in the hadronization of the ϒð2SÞ → ϒð1SÞπ 0 Rbχ ≡ R ¼ 13.5ð2.7Þl:q: × 10−2 ; local operators (this latter source of error only applies to the ϒð2SÞ → ϒð1SÞη QQ̄ð2SÞ → QQ̄ð1SÞππ transitions). In this respect, it is −2 Rexp bχ < 13.8ð2.2Þ × 10 : ð78Þ interesting to note that we expect the Oðαs Þ corrections in the hadronization of the local operators to be largely Since the theoretical values are compatible experiment, we independent of the bound state dynamics. Along this line conclude that chiral corrections are not needed at this level of of thought, it is rewarding that we can roughly get agree- precision. ment with experiment (within errors) both for charmonium It is also interesting to compare the theoretical and and botttomoniun using the same correcting factor ∼1=3.5. experimental ratios of the two-pion transitions of Sec. III This factor could be understood as evaluating the βðαs Þ over the one-pion or eta transition computed in Sec. V, ð12Þ function at a low scale: − 2πβðα β0 α2s sÞ ∼ 3.5. Still, it is premature since this set of ratios is also independent of βr;Q , or, in to draw any conclusion out of this. Note also that these other words, of the heavy quarkonium dynamics (which is estimates are significantly affected by the uncertainties of κ, also the case with the twist expansion). The hadronization and, for the π 0 case, by the uncertainty of mu =md . On the of the two-pion production using the energy-momentum other hand, the strong dependence on the light-quark tensor anomaly has corrections of Oðαs Þ at a low-energy masses makes these observables interesting for possible scale, as well as contributions of the anomalous dimension determination of the light-quark masses. of the light-quark mass operator, both of which could be For the ratios considered so far, Eqs. (77)–(82), the potentially large.6 On the other hand, the determination of prediction of the twist expansion is the same to the one we the one single pion or eta matrix element through the axial have found here. This is not so for the following ratio: anomaly only has chiral corrections, as discussed in the   previous paragraph, and, in the case of one pion in the final ϒð2SÞ → ϒð1SÞπ þ π − states, the uncertainty on mu =md . Hence, we explore Rbc;ππ ≡ R ψð2SÞ → J=ψπ þ π − whether comparing the ratios of the two- and one-pion (or eta) transitions to their experimental values allows us ¼ 6.65ð þ0.30 −2 −0.38 ÞΛ1 ð∓0.02Þκ ð0.31Þs:p: × 10 ; −2 asses the size of the corrections to Eqs. (D14) and (D15). Rexp bc;ππ ¼ 5.59ð0.50Þ × 10 : ð83Þ 6 In principle, there are also Oðp4 Þ chiral corrections. Never- In principle, we expect that for this ratio most of the theless, we expect those to be comparatively small due to the neglected Oðαs Þ corrections in the hadronization of the limited phase space available. local operators vanish. This observable can be considered a 054021-14 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) ð21Þ ð21Þ VII. CONCLUSIONS rough measure of βr;b =βr;c . The agreement with experi- ment is remarkable: below 20%, and could be accounted for We have studied one- and two-pion transitions between by the quoted errors. quarkonium states below threshold. We have used the We can also consider the following ratios: weakly coupled version of pNRQCD, which assumes that   mv ≫ ΛQCD . On the other hand, our analysis does not need ϒð2SÞ → ϒð1SÞπ 0 to constrain the relative size of ΛQCD with respect to mv2 . Rbc;π ≡ R ψð2SÞ → J=ψπ 0 For definiteness, we have generically considered the sit- ¼ 5.3ð þ0.2 −3 −0.3 ÞΛ1 ð0.25Þs:p: × 10 ; uation mv2 ≪ ΛQCD . This EFT of QCD for heavy quark- −3 antiquark systems incorporates the 1=mQ and multipole Rexp bc;π < 3.4ð1Þ × 10 ; ð84Þ expansions systematically. We then write the hadronic   representation for this EFT in terms of the singlet, repre- ϒð2SÞ → ϒð1SÞη Rbc;η ≡ R senting the heavy quarkonium, hybrids, and pion fields. In ψð2SÞ → J=ψη weakly coupled pNRQCD, the hybrid states consist of a ¼ 1.33ð þ0.06 −3 −0.08 ÞΛ1 ð0.06Þs:p: × 10 ; color-octet heavy quark-antiquark field and a gluonic Rexp −3 excitation operator, called gluelump, characterized by a bc;η ¼ 0.94ð15Þ × 10 : ð85Þ JPC . In order to explicitly compute the matching between Again, the twist expansion would yield a different pre- both versions of the theory, we assume that the gluelump diction for these ratios, and, again, these ratios can be operators overlap predominantly with the lowest dimension ð21Þ ð21Þ gluonic operator with the same JPC . The matching is considered a measure of βr;b =βr;c . The agreement with completed by making use of low-energy theorems gen- experiment is quite reasonable, with difference of the order erated by the axial and energy-momentum tensor anomalies of 30%. In order to assess whether this error and the error of to obtain the local matrix elements for pion production by ð21Þ ð21Þ Eq. (83) are really related to βr;b =βr;c or to something gluonic operators. else, one may consider the double ratios In this framework, we have computed the QQ̄ð2SÞ → QQ̄ð1SÞππ transitions and compared with the description Rbc;π R Rexp c;π obtained solely from chiral symmetry, as well as with ¼ c;π ¼ 0.08; < 0.06ð1Þ; ð86Þ Rbc;ππ Rb;π Rexp b;π experimental data. We do so for the normalized decay width spectra and for the total widths of charmonium and Rbc;η Rc;η Rexp c;η bottomonium. In the case of the normalized decay width ¼ ¼ 2.0 × 10−2 ; ¼ 1.7ð3Þ × 10−2 : Rbc;ππ Rb;η Rexp b;η spectra, both our approach and the purely chiral description fit the data well. However, the chiral description depends on ð87Þ two parameters that cannot be strongly constrained from ð21Þ ð21Þ the experimental data, whereas our approach only depends These double ratios are independent of βr;b =βr;c .We also on one parameter κ. Our best fit to the combined charmo- expect several other uncertainties to cancel: the dependence nium and bottomonium data yields the value quoted in on the light-quark masses cancels; in principle, we would Eq. (41) for κ. also expect most of the effect due to unaccounted for Oðαs Þ Our computational scheme produces the same theoretical corrections in the hadronization of the local operators expression as the twist expansion [7] for the normalized vanish, since the phase space is similar. An indirect decay width spectra computed in Sec. III A. Note, however, reflection of this is that the dependence on κ nearly that the twist expansion requires that ΛQCD ≪ mv2 , and vanishes. Nevertheless, this should be studied with more detail to make this statement more quantitative. For that the energy release E ≪ mv2 , the latter condition is never fulfilled for two-pion transitions. Lower values for κ instance, for the QQ̄ð2SÞ → QQ̄ð1SÞη transition, there is than the ones in Eq. (41) are found in [4,34]. The main little phase space free. As a result, the phase space computation yields very different values for the bottomo- source for the discrepancy is due to Oðm2π Þ terms not nium and charmonium transitions. Overall, we consider included in these analysis. these double ratios as the cleanest objects to perform Our prediction for the total two-pion transition decay dedicated studies of the computational scheme discussed width depends on κ [which we take from Eq. (41)] and ð12Þ in this paper (and of the twist expansion, which yields the on βr;Q . The latter coefficient suffers from large uncer- same prediction for these double ratios). For now, we just tainties. It involves a double sum: one over the gluelump want to highlight that we find remarkably good agreement states with the same quantum numbers and, for each with experiment. This may indicate that the overlap of gluelump, other sum over the states solution of its asso- higher dimensional operators with the hybrids is small. In ciated Schrödinger equation. In this paper, we have made a this respect, the possible incorporation of Bc transitions to first estimate considering only the first gluelump and, for it, these analyses could be of much help. summing the first few states solution of the Schrödinger 054021-15 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) equation. Moreover, the states solution of the Schrödinger expansion. On the other hand, the overall coefficients equation have a significant overlap with long distances. which, in principle, are calculable in both cases, are This makes the result quite dependent on the shape of different for the total transition rates. the potentials. We have estimated this dependence consid- One may consider ratios that yield different predictions ering different fit functions for the singlet and hybrid than the twist expansion. These are Eqs. (83)–(85). We potentials. Overall, our estimates for the total decay widths expect these ratios to be quite independent of the error Γψð2SÞ→J=ψπþ π− and Γϒð2SÞ→ϒð1SÞπþ π− can be found in associated to the hadronization of the local gluonic oper- Eqs. (46) and (47). ator. On the other hand, they depend on the bound state ð12Þ ð12Þ Enlarging our Lagrangian to include spin-dependent and dynamics through the ratio βr;b =βr;c . For Eq. (83), good isospin breaking operators we can use the same procedure agreement is found (below 20%) and is quite reasonable for to compute the width for QQ̄ð2SÞ → QQ̄ð1PÞπ 0 transi- the other two ratios of the order of 30%. This may indicate tions. Our prediction for the total width Γψð2SÞ→hc ð1PÞπ 0 can ð12Þ ð12Þ that our evaluation of βr;b =βr;c is quite reasonable. be found in Eq. (59). This observable is interesting. Unlike Finally, we have considered double ratios in Eq. (86). In previous decays, it does not suffer from the uncertainties principle, these are the cleanest objects to compute [leaving associated to the hadronization of the local operator: the aside Eqs. (77) and (78)]. They are independent of the axial anomaly does not get OðαÞ corrections, nor there are bound state dynamics, and we also expect them to be rather Oðp4 Þ chiral corrections. We also observe a very weak independent of the uncertainties associated to the hadro- dependence on variations of the wave function of the hybrid nization of the local gluonic operator. The agreement with ð12Þ and singlet states: this object depends on βσ . The major experiment is quite good for them. error is generated by the uncertainty on the energy differ- We believe there is room for improvement, and leave for ence between singlet and hybrid states. Once this error is future work more dedicated analysis, in particular of the taken into account, the result is roughly compatible with ratios and, specially, of the double ratio discussed in experiment. Eq. (86). This last object may allow a quantitative analysis Finally, we considered QQ̄ð2SÞ → QQ̄ð1SÞπ 0 ðηÞ tran- of the validity of our computational scheme (and alter- sitions. Our results for the total decay widths Γψð2SÞ→J=ψπ0 , natively of the twist expansion). It would also be interesting Γψð2SÞ→J=ψη , Γϒð2SÞ→ϒð1SÞπ0 , and Γϒð2SÞ→ϒð1SÞη can be found to apply our computational scheme to the two- (or one) in Eqs. (73)–(76). pion hadronic transition of the Bc . On a later stage, the Overall, we find that the S-wave hadronic transitions to possible incorporation of tetraquarks and B-meson loops P-wave heavy quarkonium are roughly compatible with for heavy quarkonium states near, or above threshold, theory within errors. For decays to S-wave heavy quarko- should also be studied in greater detail. Actually, if enough nium, the magnitude of our predictions is smaller than precision is reached, discrepancies with experiment may experiment, especially for the QQ̄ð2SÞ → QQ̄ð1SÞπ 0 ðηÞ hint at the need of incorporating those states. decays. We have also computed the ratios of the above transition ACKNOWLEDGMENTS rates in Sec. VI. We expect a more solid prediction for We thank E. Guido and R. Mussa for sharing the them. Depending on the ratio, different qualitative infor- experimental data of Ref. [30] with us. We thank P. mation on the theory can be obtained. The ratios of Petreczky for sharing the lattice data of Ref. [43] with us. Eqs. (77) and (78) can be considered to be a test of chiral We thank Feng-Kun Guo for providing us the data of [29]. symmetry, as they are independent of the bound state This work was supported in part by the Spanish Grants dynamics, and there is no error associated to the hadroni- No. FPA2017-86989-P and No. SEV-2016-0588 from the zation of local gluonic operators. Good agreement with Ministerio de Ciencia, Innovación y Universidades, and the experiment is found. This may indicate a good behavior of Grant No. 2017SGR1069 from the Generalitat de Catalunya. chiral symmetry for these decays. Alternative ratios one J. T. C acknowledges the financial support from the may consider are Eqs. (79)–(82). These ratios are also European Union’s Horizon 2020 research and innovation independent of the bound state dynamics but on the other programme under the Marie Skłodowska–Curie Grant hand suffer from unquantified errors due to the hadroniza- No. 665919. tion of the local gluonic operators. Large differences with experiment were found, which however could be accom- APPENDIX A: DIFFERENTIAL DECAY WIDTHS modated by a single constant for bottomonium and char- monium. This agrees with expectations that the main The most general amplitude for the two-pion transitions corrections are independent of the bound state dynamics. at Oðp2 Þ in the chiral counting and using spin symmetry is These ratios and the ratios Eqs. (77) and (78) are inde- given in Eq. (26). It is convenient to write it as pendent of the bound state dynamics, i.e., they are a2 2 ð12Þ independent of βr;Q and yield the same result as the twist Aππ ¼ ða2 − a1 Þp0þ p0− − m þ ða2 − a3 Þm2π ; ðA1Þ 2 ππ 054021-16 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) where m2ππ ¼ ðpþ þ p− Þ2 is the dipion invariant mass, with and pþ , p− refers to the momentum of the π þ and π − , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi respectively. In the reference frame of the decaying σ¼ 1 − 4m2π =m2ππ ðA3Þ quarkonia, we find m2Hi − m2Hf þ m2ππ 1 Δ¼ ; ðA4Þ p0þ p0− ¼ ðΔ2 − ρ2 σ 2 cos2 θÞ; ðA2Þ 2mHi 4 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ρ¼ m4Hi þ m4Hf þ m4ππ − 2m2Hi m2Hf − 2m2Hi m2ππ − 2m2Hf m2ππ ; ðA5Þ 2mHi where mHi and mHf are the masses of initial and final 1 a Aππ ¼ ða2 − a1 ÞðΔ2 − ρ2 σ 2 cos2 θÞ − 2 m2ππ quarkonium, respectively. In the nonrelativistic approxi- 4 2 mation of the final quarkonium momentum, the above 2 þ ða2 − a3 Þmπ : ðA8Þ expressions reduce to The differential decay width is 2 Δ ¼ ðmHi − mHf Þ ; nr ðA6Þ  Γππ ρσmππ 1 ¼ ða − a1 ÞðΔ2 − ρ2 σ 2 cos2 θÞ dmππ d cos θ 4ð2πÞ3 4 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ρnr ¼ ðmHi − mHf Þ2 − m2ππ : ðA7Þ a2 2 2 − mππ þ ða2 − a3 Þmπ ; ðA9Þ 2 Using Eq. (A2) in Eq. (A1), we can write and after integrating θ, it reads    dΓππ ρσmππ 2 4 2a2 m2ππ a2 − a1 Δ2 2 ða2 − a1 Þ2 4 4 ¼ 2ða2 − a3 Þ mπ 1 − þ þ ρσ dmππ 4ð2πÞ3 a2 − a3 4m2π a2 − a3 4m2π 40   1 2 2 2 2a2 m2ππ a2 − a1 Δ2 − ða2 − a3 Þða2 − a1 Þmπ ρ σ 1 − þ : ðA10Þ 3 a2 − a3 4m2π a2 − a3 4m2π To obtain the total decay width, we integrate numerically where ϕðnÞ ðrÞ is the solution of Z m −m dΓ   dmππ ππ : Hi Hf Γππ ¼ ðA11Þ 1 lðl þ 1Þ 2mπ dmππ − ∂r r2 ∂ rþ þ V Σþg ðrÞ ϕðnÞ ðrÞ ¼ E n ϕðnÞ ðrÞ: mQ r2 m Q r2 In the one-pion transition, the momenta are fixed by momentum conservation, in particular the final pion ðB2Þ momentum is jpπ j ¼ ρðmπ Þ, and the decay width is given by It is not our aim in this paper to make a detailed analysis ρðmπ Þ and optimization of the solutions of the bound states but Γπ ¼ jAπ j2 ; ðA12Þ 2π rather to make a qualitative analysis. For the numerical with ρðmπ Þ is given in Eq. (A5) replacing mππ by mπ. evaluations of the hybrid and quarkonium spectra, we use the values mc ð1 GeVÞ ¼ 1.496 GeV and mb ð1 GeVÞ ¼ APPENDIX B: SINGLET AND HYBRID 4.885 GeV [44] for the heavy quark masses in the RS0 BOUND STATES scheme [45]. To explore the sensitivity of the matrix elements to the shape of the potential, we have considered The heavy quarkonium wave function reads two versions of the potentials. The first is the potential in X jm Eq. (15) with the nonperturbative constant bΣþg obtained by Snjmj ls ðrÞ ¼ ϕðnÞ ðrÞ Clmljsms Y lml ðθ; ϕÞχ sms fitting to the lattice data from Ref. [43] up to r ¼ 1 fm ml ms together with a free energy offset, boffset Σþ . The second ≡ ϕ ðrÞΦ02sþ1l ðθ; ϕÞ; ðnÞ ðB1Þ g j version of the potential is a fit of the same lattice data 054021-17 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) TABLE III. Quarkonium spectrum obtained from the potential in Eq. (B3) fitted to the lattice data Ref. [43]. The origin of energies is chosen in order to reproduce the spin average of the ground state. All dimension-full entries are in GeV. cc̄ bb̄ nl J PC mH h1=ri Ekin mH h1=ri Ekin 1S f0−þ ; 1−− g 3.068 0.66 0.31 9.442 1.15 0.28 2S f0−þ ; 1−− g 3.686 0.43 0.48 9.916 0.67 0.36 1P f1þ− ; ð0; 1; 2Þþþ g 3.464 0.37 0.39 9.779 0.59 0.30 1D f2−þ ; ð1; 2; 3Þ−− g 3.775 0.27 0.48 10.009 0.42 0.35 FIG. 6. The static singlet potential in Eqs. (B3) and (15) fitted to the lattice data of Ref. [43] (blue dots) as solid and dashed lines, TABLE IV. Quarkonium spectrum obtained from the potential respectively. in Eq. (15) fitted to the lattice data Ref. [43]. The origin of energies is chosen in order to reproduce the spin average of the ground state. All dimension-full entries are in GeV. to a function constrained to reproduce Eq. (15) in the short distance and a to have a linear behavior in the long distance, cc̄ bb̄ while overall adjusting well to the data. The function nl J PC mH h1=ri Ekin mH h1=ri Ekin chosen is 1S f0−þ ; 1−− g 3.068 0.73 0.38 9.442 1.36 0.38 ð0Þ ð0Þ ð1Þ ð2Þ 1=4 2S f0−þ ; 1−− g 3.847 0.51 0.67 10.015 0.76 0.45 V Σþg ¼ V s ðrÞ þ ðcΣþ þ cΣþ r2 þ cΣþ r4 Þ : ðB3Þ 1P f1 ; ð0; 1; 2Þþþ g þ− 3.553 0.42 0.50 9.873 0.64 0.35 g g g 1D f2−þ ; ð1; 2; 3Þ−− g 3.949 0.32 0.65 10.145 0.46 0.41 For the perturbative static singlet potential, we take the Oðα3s Þ result7 evaluated at ν ¼ 5.6 GeV (the shortest 0 1 ðmÞ available scale). The parameters obtained from the fits ψ 0 ðrÞv0lml ðθ; ϕÞ are the following: X B C mjmj ls B ðmÞ C Clmlj sms B p1ffiffi2 ψ þ ðrÞvþ1 jm Ψþ ðrÞ ¼ lml ðθ; ϕÞ Cχ sms @ A bΣþg ¼ 4.07 × 10−2 GeV3 ; boffset ¼ 0.695 GeV; ðB4Þ ml ms p1ffiffi ψ ðmÞ −1 Σþ g 2 þ ðrÞvlml ðθ; ϕÞ 0 ðmÞ 1 ð0Þ cΣþ ¼ 0.222 GeV4 ; ð1Þ cΣþ ¼ 6.91 × 10−2 GeV6 ; ψ 0 ðrÞΦ02sþ1l ðθ; ϕÞ B C j g g B p1ffiffi ψ ðmÞ ðrÞΦþ1 ðθ; ϕÞ C ð2Þ cΣþ ¼ 3.54 × 10−3 GeV8 : ðB5Þ ¼B 2 þ 2sþ1l C; ðB6Þ g @ j A ðmÞ p1ffiffi ψ ðrÞΦ−1 2sþ1l ðθ; ϕÞ In Fig. 6, we plot the lattice data together with the fitted 2 þ j Eq. (15) (with the offset) and Eq. (B3) potentials as dashed 0 1 and solid lines, respectively. The spectra obtained are 0 displayed in Tables III and IV for the potential in X B p1ffiffi ψ ðmÞ C Ψ− mjmj ls ðrÞ ¼ Clmjl sms B jm 2 − ðrÞvþ1 lml ðθ; ϕÞ Cχ sm Eqs. (15) and (B3), respectively. The origin of energies @ A s − p1ffiffi2 ψ ðmÞ −1 ml ms is chosen in order to reproduce the experimental spin − ðrÞvlml ðθ; ϕÞ average of the ground state. 0 1 0 The hybrid wave functions are obtained following the B p1ffiffi ψ ðmÞ ðrÞΦþ1 ðθ; ϕÞ C procedure described in Ref. [18], by solving the coupled ¼B @ 2 − 2sþ1l j C: A ðB7Þ ð0Þ Schrödinger equations involving the potentials V Σ−u ðrÞ − − pffiffi2 ψ − ðrÞΦ2sþ1l ðθ; ϕÞ 1 ðmÞ −1 ð0Þ ð0Þ ð0Þ j V Πu ðrÞ or V Σþ0 ðrÞ − V Πg ðrÞ generated by the 1þ− and 1−− g gluelump at short distances, respectively [see Eq. (16)]. The parity and charge conjugation of these states There are two types of solution of Ψ1λ corresponding to corresponding to the Ψ solutions are mjm ls mjm ls states with opposite parity ðΨþ j Þλ and ðΨ− j Þλ , P ¼ ð−1Þl PG ; C ¼∓ð−1Þlþs CG ; ðB8Þ 7 The Oðαs Þ term was computed in [46], the Oðα2s Þ in [47,48], the Oðα3s Þ logarithmic term in [49], the light-flavor finite piece in with PG and CG the parity and charge conjugation of the [50], and the pure gluonic finite piece in [51,52]. gluelump associated to the states. 054021-18 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) The angular wave functions of the hybrid states are the 1 eigenfunctions of Φλ3P ðθ; ϕÞ ¼ pffiffiffiffiffiffi r̂i† λ σ ; i ðB13Þ 0 8π   λ2 cos θ rffiffiffiffiffiffi † 2 ∂ LQ̄Q þ 2 þ 2λ 2 i θ vλlml ¼ lðl þ 1Þvλlml ; 3 σ · ðr̂λ × êmJ Þ sin θ sin θ Φλ3P ðθ; ϕÞ ¼i pffiffiffi ; ðB14Þ 1 8π 2 ðB9Þ rffiffiffiffiffiffi 3 i ij j† which for λ ¼ 0 corresponds to the usual differential equa- Φλ3P ðθ; ϕÞ ¼ σ h r̂ ; ðB15Þ tion for spherical harmonics and, therefore, v0lml ¼ Y lml . 2 8π 2mJ λ Note that we have chosen to represent the angular where ê are the polarization vectors momentum quantum numbers of the hybrids and quarkonia by l and l, respectively, to highlight the difference in ê0 ¼ ð0; 0; 1Þ; ðB16Þ angular wave functions. When using spectroscopic notation to specify the quarkonium states, we use the usual ∓1 S; P; D; … for l ¼ 0; 1; 2; …, and for hybrid states, we ê ¼ pffiffiffi ð1; i; 0Þ; ðB17Þ use S; P; D; … for l ¼ 0; 1; 2; …. 2 The λ2sþ1 lj spin-angular hybrids wave functions can be shown to be and the tensors hij 2mJ are traceless, completely symmetric, and normalized as 1 1 Φλ1S ðθ; ϕÞ ¼ pffiffiffiffiffiffi pffiffiffi δλ0 ; ðB10Þ 0 4π 2 hij ij δ 2mJ h2m0 ¼ mJ m0J : ðB18Þ J 1 σ · êm Φλ3S ðθ; ϕÞ ¼ pffiffiffiffiffiffi pffiffiffi J δλ0 ; ðB11Þ The 2sþ1lj spin-angular wave functions for quarkonium 1 4π 2 correspond to the λ ¼ 0, 2sþ1lj hybrid wave functions. rffiffiffiffiffiffi The radial wave function in Eq. (B6) is obtained by 3 i† i 1 Φλ1P ðθ; ϕÞ ¼ r̂ ê pffiffiffi ; ðB12Þ (numerically) solving the following coupled radial 1 4π λ mJ 2 Schrödinger equations: " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !  # ! ! lðl þ 1Þ þ 2 −2 lðl þ 1Þ 0 ðmÞ ðmÞ 1 1 VΣ ψ0 ψ0 − ∂r r2 ∂r þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ Em : ðB19Þ m Q r2 m Q r2 −2 lðl þ 1Þ lðl þ 1Þ 0 VΠ ðmÞ ðmÞ ψþ ψþ For the special case l ¼ 0, the equations decouple and in the potential is a fit of the same lattice data up to r ≤ 2 fm ðmÞ fact ψ þ becomes irrelevant since v1 00 does not exist. The by a function constrained to reproduce Eq. (16) in the radial wave function in Eq. (B7) is obtained from the short distance and a to have a linear behavior in the long uncoupled radial Schrödinger equation distance while overall adjusting well to the lattice data.  The function is  1 2∂ þ lðl þ 1Þ − ∂r r r þ V Π ψ ðmÞ ðmÞ − ¼ Emψ − : m Q r2 m Q r2 ð0Þ ð0Þ ð1Þ V Λση ¼ V o ðrÞ þ ðcΛση þ cΛση r2 þ cΛση r4 Þ ð2Þ 1=4 : ðB21Þ ðB20Þ The perturbative static octet potential is taken at Oðα3s Þ The hybrid static potentials, V Πu , V Σ−u , V Πg , and V Σþ0 g , from Ref. [56] and evaluated at ν ¼ 2.6 GeV (the shortest match to the NRQCD heavy quark-antiquark static ener- available scale). The results for the hybrid spectrum for gies, which have been computed on the lattice [53–55] and V Πu -V Σ−u associated to the 1þ− gluelump are discussed at are given up to NLO in the multipole expansion by the length in Ref. [18].8 The parameters obtained from the fits potential in Eq. (16). As in the static potential for the are the following: singlet, we have considered two versions of the potentials in order to assess the sensitivity of the matrix elements to bΣþ0 ¼ 1.26 × 10−2 GeV3 ; boffset ¼ 1.22 GeV; ðB22Þ g Σþ0 the shape of the potential. The first is the potential in g Eq. (16) with the nonperturbative constant bkλ obtained by fitting to the lattice data from Ref. [53] up to r ¼ 1 fm 8 In Ref. [18], the choice of Eq. (B21) was different but the together with a free energy offset. The second version of resulting potentials are completely equivalent in practice. 054021-19 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) bΠg ¼ 1.93 × 10−2 GeV3 ; boffset Πg ¼ 1.22 GeV; ðB23Þ ð0Þ ð1Þ cΣþ0 ¼ 1.58 GeV4 ; cΣþ0 ¼ 12.9 × 10−2 GeV6 ; g g ð2Þ cΣþ0 ¼ 1.94 × 10−3 GeV8 ; ðB24Þ g ð0Þ ð1Þ cΠg ¼ 1.77 GeV4 ; cΠg ¼ 10.6 × 10−2 GeV6 ; ð2Þ ð2Þ cΠg ≡ cΣþ0 : ðB25Þ g The fits of the potentials V Πg -V Σþ0 g , associated to the 1−− FIG. 7. Lattice data for the static energies Σþ0 g and Πg from gluelump are shown in Fig. 7 in solid and dashed lines, and Ref. [53] associated to the gluelump kPC ¼ 1−− plotted as blue the corresponding spectra in Tables V and VI for the dots and orange squares, respectively. The blue (orange) solid and potentials in Eqs. (B21) and (16), respectively. The origin dashed lines are the Σþ0 g ðΠg Þ potentials of Eqs. (B3) and (15) of energies is chosen, as in Ref. [18], such that the short fitted to the lattice data, respectively. distance limit of V Πu and V Σ−u tend to V 0o þ Λ1þ− with the value of Λ1þ− ¼ 0.87ð15Þ GeV computed in Ref. [57]. TABLE V. Heavy hybrid spectrum for kPC ¼ 1−− obtained from the fit of Eq. (B3) to the lattice data of Ref. [53]. All dimension-full R ðmÞ entries are in GeV. PΠ ¼ d3 rjψ þ=− j2 is the integral over the square of the wave function associated with the Πg potential. It can be interpreted as the probability to find the hybrid in a Πg configuration, thus it gives a measure of the mixing effects. cc̄ bb̄ Multiplet l J PC Λση mH h1=ri Ekin PΠ mH h1=ri Ekin PΠ H1 ð1PÞ 4.51 0.42 0.15 0.57 11.14 0.55 0.09 0.44 f1þ− ; ð0; 1; 2Þþþ g Σþ0 g -Πg H01 ð2PÞ 4.86 0.27 0.33 0.20 11.33 0.40 0.19 0.27 1 H2 ð1PÞ 4.74 0.27 0.22 1.00 11.30 0.38 0.13 1.00 f1−þ ; ð0; 1; 2Þ−− g Πg H02 ð2PÞ 5.01 0.24 0.37 1.00 11.51 0.34 0.26 1.00 H3 ð1SÞ 4.63 0.29 0.25 0.00 11.19 0.40 0.15 0.00 0 f0−þ ; 1−− g Σþ0 H03 ð2SÞ g 5.01 0.25 0.40 0.00 11.41 0.35 0.24 0.00 H4 ð1DÞ 4.70 0.28 0.24 0.50 11.24 0.38 0.14 0.39 f2−þ ; ð1; 2; 3Þ−− g Σþ0 g -Πg H04 ð2DÞ 5.06 0.23 0.42 0.24 11.44 0.32 0.25 0.26 2 H5 ð1DÞ 4.92 0.22 0.32 1.00 11.40 0.30 0.18 1.00 f2þ− ; ð1; 2; 3Þþþ g Πg H05 ð2DÞ 5.27 0.19 0.47 1.00 11.61 0.28 0.28 1.00 TABLE VI. Heavy hybrid spectrum for kPC ¼ 1−− obtained from the fit of Eq. (15) to the lattice data of Ref. [53]. All dimension-full R ðmÞ entries are in GeV. PΠ ¼ d3 rjψ þ=− j2 is the integral over the square of the wave function associated with the Πg potential. It can be interpreted as the probability to find the hybrid in a Πg configuration, thus it gives a measure of the mixing effects. cc̄ bb̄ Multiplet l J PC Λση mH h1=ri Ekin PΠ mH h1=ri Ekin PΠ H1 ð1PÞ 4.54 0.44 0.15 0.59 11.18 0.57 0.08 0.58 f1þ− ; ð0; 1; 2Þþþ g Σþ0 g -Πg H01 ð2PÞ 4.87 0.28 0.32 0.20 11.36 0.37 0.18 0.19 1 H2 ð1PÞ 4.79 0.31 0.28 1.00 11.32 0.41 0.15 1.00 f1−þ ; ð0; 1; 2Þ−− g Πg H02 ð2PÞ 5.24 0.28 0.51 1.00 11.57 0.37 0.28 1.00 H3 ð1SÞ 4.68 0.28 0.23 0.00 11.26 0.37 0.12 0.00 0 f0−þ ; 1−− g Σþ0 H03 ð2SÞ g 5.05 0.25 0.41 0.00 11.46 0.33 0.23 0.00 H4 ð1DÞ 4.74 0.29 0.26 0.46 11.29 0.39 0.14 0.46 f2−þ ; ð1; 2; 3Þ−− g Σþ0 g -Πg H04 ð2DÞ 5.06 0.23 0.42 0.25 11.47 0.30 0.23 0.25 2 H5 ð1DÞ 5.02 0.25 0.39 1.00 11.44 0.33 0.22 1.00 f2þ− ; ð1; 2; 3Þþþ g Πg H05 ð2DÞ 5.47 0.23 0.62 1.00 11.69 0.31 0.34 1.00 054021-20 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) APPENDIX C: MATRIX ELEMENTS We now compute the matrix elements that appear in the hadronic transitions. We suppress the labels kPC ¼ 1þ− and k ¼ 1−− on the hybrid fields since it does not affect the calculation. For simplicity, we denote the quarkonia and hybrid PC wave functions just as Sn and Ψm , respectively, X jm Z Z † 3 ðmÞ† Clml s0 m0s Clml sms ss ms ms drr ψ 0 ðrÞϕ ðrÞ dΩv0† jmj hΨm jr̂λ · rjSn i ¼ j δ 0 δ 0 ðmÞ lml ðθ; ϕÞY lml ðθ; ϕÞ; ðC1Þ ml m0s ml ms where we have used that r̂†λ · r ¼ rδλ0 . For λ ¼ 0 v0lm ¼ Y lml , so we can use the spherical harmonics orthogonality relations X jm Z jmj ðmÞ† ðC1Þ ¼ δss0 Clml sms Clml sms δll δml ml drr3 ψ 0 ðrÞϕðnÞ ðrÞ j m0l ml ms X jm Z Z 2 3 ðmÞ† ðmÞ† ¼ δss0 δll ðClmj −ms sms Þ j ðnÞ δ 0 δ drr ψ 0 ðrÞϕ ðrÞ ¼ ss ll drr3 ψ 0 ðrÞϕðnÞ ðrÞ: ðC2Þ ms Let us work out some specific matrix elements necessary for the ψð2SÞ → hc π 0 transition, Z pffiffiffi Z 3 † 1 3 j j i i ðmÞ† hΨðm S 1 Þjr̂λ · ðσ 1 − σ 2 ÞjSðn P1 Þi ¼ dΩ ê 0 r̂ r̂ ê drr2 ψ 0 ðrÞϕðnÞ ðrÞ 2π mJ 0 0 mJ Z 2 ðmÞ† δ ¼ pffiffiffi mJ mJ drr2 ψ 0 ðrÞϕðnÞ ðrÞ; 0 ðC3Þ 3 Z pffiffiffi Z † X 3 j j i i ðmÞ† 1 3 hΨðm P 1 Þjr̂λ · ðσ 1 − σ 2 ÞjSðn S1 Þi ¼ dΩ êm0 r̂λ r̂λ êmJ drr2 ψ λ ðrÞϕðnÞ ðrÞ λ 2π J X 2 Z ¼ pffiffiffi δmJ m0J drr2 ψ ðmÞ† λ ðrÞϕðnÞ ðrÞ; ðC4Þ λ 3 where we have used that hybrid states in l ¼ 0 must have λ ¼ 0 and also that Z 4π ij j λ r̂λ ¼ dΩr̂i† δ : ðC5Þ 3 For P-wave pion (or eta) emission, we need the following matrix element: 1 l k hΨðm3 S 1 Þjðσ 1 þ σ 2 Þ · r̂†λ rj jSðn3S1 Þi ¼ ê 0 ê Tr½½σ i ; σ k σ l  8π mJ mJ Z Z Z 3 ðmÞ† ðnÞ 2  ðmÞ† i j × dΩr̂0 r̂0 drr ψ 0 ðrÞϕ ðrÞ ¼ i ðêm0 × êmJ Þ j drr3 ψ 0 ðrÞϕðnÞ ðrÞ 3 J 2  ¼ i ðêm0 × êmJ Þj hΨðm3 S 1 ÞjΨ†λ r̂†λ · rSjSðn3S1 Þi; ðC6Þ 3 J and due to the symmetry of the angular integral with respect to the indices [see Eq. (C5)], we can also derive 2  † hΨðm3 S 1 Þjðσ 1 þ σ 2 Þ · rr̂j† 3 j 3 3 λ jSðn S1 Þi ¼ i ðêm0J × êmJ Þ hΨðm S 1 Þjr̂λ · rjSðn S1 Þi: ðC7Þ 3 APPENDIX D: HADRONIZATION OF LOCAL GLUCONIC OPERATORS 1. Two-pion production through the anomaly of the energy-momentum tensor The traceless part of the energy momentum tensor for QCD is 1 1 X X1 θμν − gμν θαα ¼ gμν Gαβa Gaαβ − Gaμα Gαa ν þi q̄i γ μ Dν qi − g m q̄ q : ðD1Þ 4 4 i i 4 μν i i i 054021-21 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) The trace of the energy momentum tensor reads normalization, and they correspond to the traceless part X of the energy-momentum tensor, the trace of the energy- 1 βðαs Þ μνa θμμ ¼ G Gμνa þ ð1 − γ i Þmi q̄i qi ; ðD2Þ momentum tensor, and the scalar current, respectively. Let 4 αs i us start by the trace of the energy-momentum tensor where the γ i are the anomalous dimension of the q̄i qi hπ þ ðpþ Þπ − ðp− Þjθμμ j0i operator and β is the QCD β-function   F2 † ∂μ Ui − F2 hχU † þ Uχ † i þ … β0 α2s 11N c 4T F nf ¼ V 1 ðμÞ − h∂μ U βðαs Þ ¼ − þ Oðα3s Þ; β0 ¼ − : 2 2π 3 3 ¼ V 1 ðμÞ½2ðpþ · p− Þ þ 4m2π þ …: ðD11Þ ðD3Þ The first term in Eq. (D2) is generated by the conformal If we compare with Eq. (D5), we determine that V 1 ðμÞ ¼ 1. anomaly, and the second by the explicit breaking of the The matrix element of the scalar current reads scale invariance due to the quark masses. X The matrix element hπ þ ðpþ Þπ − ðp− Þjθμν j0i can be deter- F2 hπ þ ðpþ Þπ − ðp− Þj mi q̄i qi j0i ¼ − hχU† þ Uχ † i þ … mined up to quadratic order in pþ , p− and mπ from [11,27] i 4 (i) Conservation on the mass shell: ðpþ þ p− Þμ × ¼ m2π þ …; ðD12Þ hπ þ ðpþ Þπ − ðp− Þjθμν j0ip2þ ¼p2− ¼m2π ¼ 0. (ii) Normalization: hπ þ ðpÞπ − ð−pÞjθμν j0ip2 ¼m2π ¼2pμ pν . where one can think the parameter B (χ ¼ 2Bm̂1) as taking (iii) Adler zero condition: hπ þ ðpÞπ − ð0Þjθμν j0ip2 ¼m2π ¼ 0. the role of the normalization parameter. Finally, for θgμν, we and reads as have no way to determine the normalization hπ þ ðpþ Þπ − ðp− Þjθμν j0i ¼ ½ðpþ · p− þ p2þ þ p2− Þ hπ þ ðpþ Þπ − ðp− Þjθgμν j0i − m2π gμν − pþμ p−ν − pþν p−μ   2 gμν α † F ∂ U α∂ þ … h∂μ U † ∂ν U þ ∂μ U ∂ν U − 2 † Ui ðD4Þ ¼ V 2 ðμÞ 4 þ hπ ðpþ Þπ − ðp− Þjθμμ j0i ¼ 2ðpþ · p− Þ þ 4m2π : ðD5Þ   1 ¼ −V 2 ðμÞ pþμ p−ν þ p−μ pþν − gμν pþ · p− þ … : The square of chromoelectric and chromomagnetic fields 2 can be written as ðD13Þ 1 E2 ¼ vμ vν θgμν − Gαβa Gaαβ ; ðD6Þ Adding everything together, we arrive at 4 1 β α B2 ¼ vμ vν θgμν þ Gαβa Gaαβ ; ðD7Þ hπ þ ðpþ Þπ − ðp− Þj 0 s E2 j0i 4 2π     9 0 0 3 with vμ ¼ ð1; 0Þ and ¼ 2 − κ pþ p− − 2 þ κ pþ · p− þ 3m2π ; ðD14Þ 2 2 1 θgμν ¼ gμν Gαβa Gaαβ − Gaμα Gαa ν : ðD8Þ β α 4 − hπ þ ðpþ Þπ − ðp− Þj 0 s B2 j0i 2π We can rewrite the chromoelectric and chromomagnetic     9 3 fields in terms of θμμ , ¼ 2 þ κ pþ p− − 2 − κ pþ · p− þ 3m2π ; 0 0 ðD15Þ 2 2  X  2 μ ν g αs μ E ¼ v v θμν − θμ − mi ð1 − γ i Þq̄i qi ; ðD9Þ where we have defined κ ¼ αs β0 V 2 ðμÞ=ð6πÞ as in Ref. [11], βðαs Þ i and we arrive at the results of Refs. [27,58]. We emphasize  X  that in this relation we have neglected higher order αs 2 αs μ corrections to the beta function. These αs corrections are B ¼ vμ vν θgμν þ θμ − mi ð1 − γ i Þq̄i qi : ðD10Þ βðαs Þ i computed at a low-energy scale, which makes their computation unfeasible. Still, they could be parametrized The problem of computing the matrix elements hπ þ ðpþ Þ × as an overall coefficient (redefining κ) of the equation. In π − ðp− ÞjE2 j0i and hπ þ ðpþ Þπ − ðp− ÞjB2 j0i is transformed Eq. (D15), we have also neglected the anomalous dimen- into the computation of hπ þ ðpþ Þπ − ðpP g þ − Þjθμν j0i, hπ ðpþ Þ× sion of the q̄q operator. Otherwise, the coefficient propor- μ π − ðp− Þjθμ j0i, and hπ þ ðpþ Þπ − ðp− Þj i mi q̄i qi j0i. These tional to m2π becomes arbitrary. Whereas this is customary can be evaluated in χPT except for a possible done, this should be further investigated. 054021-22 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) 2. Single pion creation from the axial anomaly which is the result from Ref. [8]. Finally, we note that The axial current matrix elements are defined as αs 1 α m − mu λ h0j E · Bjπ 0 i ¼ h0j s Gμν G̃μν jπ 0 i ¼ d Fm2π0 : h0jq̄ð0Þγ γ 5 i qð0Þjπ j ðpÞi ¼ iFpμ δij ; μ ðD16Þ π 4 π mu þ md 2 ðD22Þ with λ the Gell-Mann matrices and q ¼ ðu; d; sÞ. The divergence of the axial current reads A similar analysis for the η produces αs a a;μν ∂μ ðūγ μ γ 5 uÞ ¼ 2imu ūγ 5 u þ G G̃ ; ðD17Þ pffiffiffi 4π μν 3αs h0j Gμν G̃μν jηi ¼ 3Fm2η : ðD23Þ with 4π 1 We remark that, compared with the prediction of the G̃μν ¼ ϵμναβ Gαβ ðϵ0123 ¼ 1Þ; ðD18Þ previous section, these results do not have neither Oðαs Þ 2 nor Oðp4 Þ corrections. and the same holds replacing u by d or s. Then, we have λ3 3. Single P-wave pion anomaly ∂μ h0jq̄γ μ γ 5 qjπ 0 i 2 We first consider the matrix elements of following 1 gluonic operators: ¼ ∂μ h0jūγ μ γ 5 u − d̄γ μ γ 5 djπ 0 i ¼ ih0jmu ūγ 5 u − md d̄γ 5 djπ 0 i 2 i ¼ h0jðmu þ md Þðūγ 5 u − d̄γ 5 dÞjπ 0 i ¼ Fm2π0 : ðD19Þ ihπ 0 ðpÞjGaμν ðDρ Gλσ Þa j0i 2 ¼ Xpρ ϵμνλσ þ Yðpλ ϵμνρσ − pσ ϵμνρλ Þ; ðD24Þ Since the diagonal flavor current does not interpolate the π 0 , we have ihπ 0 ðpÞjðDρ Gμν Þa Gaλσ j0i 0 ¼ ∂μ h0jūγ μ γ 5 u þ d̄γ μ γ 5 djπ 0 i ¼ X0 pρ ϵμνλσ þ Y 0 ðpμ ϵλσρν − pν ϵλσρμ Þ; ðD25Þ α ¼ h0j2imu ūγ 5 u þ 2imd d̄γ 5 d þ 2 s Gμν G̃μν jπ 0 i 4π where we have used the Schouten identity α ¼ h0jiðmu − md Þðūγ 5 u − d̄γ 5 dÞ þ 2 s Gμν G̃μν jπ 0 i; 4π ðD20Þ pρ ϵμνλσ ¼ pλ ϵμνρσ − pσ ϵμνρλ − pμ ϵνρλσ þ pν ϵμρλσ : ðD26Þ and using Eq. (D19) in Eq. (D20), we obtain Due to the Jacobi identity αs h0j G G̃μν jπ 0 i ¼ −iðmu − md Þh0jūγ 5 u − d̄γ 5 djπ 0 i Dρ Gλσ þ Dσ Gρλ þ Dλ Gσρ ¼ 0; ðD27Þ 2π μν m − mu ¼2 d Fm2π0 ; ðD21Þ mu þ md we have 0 ¼ ihπ 0 ðpÞjGaμν ðDρ Gλσ Þa j0i þ ihπ 0 ðpÞjGaμν ðDσ Gρλ Þa j0i þ ihπ 0 ðpÞjGaμν ðDλ Gσρ Þa j0i ¼ Xðpρ ϵμνλσ þ pσ ϵμνρλ þ pλ ϵμνσρ Þ þ Yðpλ ϵμνρσ þ pρ ϵμνσλ þ pσ ϵμνλρ Þ − Yðpσ ϵμνρλ þ pλ ϵμνσρ þ pρ ϵμνλσ Þ ¼ ðX − 2YÞðpρ ϵμνλσ þ pσ ϵμνρλ þ pλ ϵμνσρ Þ ⇒ X ¼ 2Y; ðD28Þ and similarly X0 ¼ 2Y 0. The sum of the two matrix elements is proportional to the pion momentum ihπ 0 ðpÞjGaμν ðDρ Gλσ Þa j0i þ ihπ 0 ðpÞjðDρ Gμν Þa Gaλσ j0i ¼ ihπ 0 ðpÞj∂ρ ðGaμν Gaλσ Þj0i þ hπ 0 ðpÞjf aeb gAbρ Geμν Gaλσ þ Gaμν f acb gAbρ Gcλσ j0i Z ¼ i d4 xe−ip·x h0jπ 0 ðxÞ∂ρ ðGaμν Gaλσ Þj0i ¼ −pρ hπ 0 ðpÞjGaμν Gaλσ j0i: ðD29Þ 054021-23 ANTONIO PINEDA and JAUME TARRÚS CASTELLÀ PHYS. REV. D 100, 054021 (2019) The sum of the matrix elements in Eqs. (D24)–(D25) is ihπ 0 ðpÞjGaμν ðDρ Gλσ Þa j0i þ ihπ 0 ðpÞjðDρ Gμν Þa Gaλσ j0i ¼ ðX þ X0 þ Y 0 Þpρ ϵμνλσ þ ðY − Y 0 Þðpλ ϵμνρσ − pσ ϵμνρλ Þ; ðD30Þ where we have used Eq. (D26) to reexpress Eq. (D25). In order for Eq. (D30) to be compatible with Eq. (D29), the relation Y ¼ Y 0 must hold. Therefore, we arrive at 5 ihπ 0 ðpÞjGaμν ðDρ Gλσ Þa j0i þ ihπ 0 ðpÞjðDρ Gμν Þa Gaλσ j0i ¼ Xpρ ϵμνλσ 2 ¼ −pρ hπ 0 ðpÞjGaμν Gaλσ j0i: ðD31Þ Contracting both sides of the last equality with ϵμνλσ =2, 1 0 X¼− hπ ðpÞjGaμν G̃μνa j0i; ðD32Þ 30 the matrix element is precisely the one given in Eq. (D21). Now we can write the matrix elements we are interested in, i 1 ihπ 0 ðpÞjEai ðDj Bk Þa j0i ¼ ϵ0klm hπ 0 ðpÞjGai0 ðDj Glm Þa j0i ¼ − X½3pj δki − pi δkj ; ðD33Þ 2 2 i 1 ihπ 0 ðpÞjðDj Bk Þa Eai j0i ¼ ϵ0klm hπ 0 ðpÞjðDj Glm Þa Gai0 j0i ¼ − X½3pj δki − pi δkj : ðD34Þ 2 2 We note that ihπ 0 ðpÞjEai ðDj Bk Þa þ ðDj Bk Þa Eai j0i ¼ −X½3pj δki − pi δkj ; ðD35Þ differs from Refs. [27,59] by a factor 1=2. [1] L. S. Brown and R. N. Cahn, Phys. Rev. Lett. 35, 1 (1975). [14] M. A. Luty and R. Sundrum, Phys. Lett. B 312, 205 (1993). [2] T. Mannel and R. Urech, Z. Phys. C 73, 541 (1997). [15] A. Pineda and J. Soto, Nucl. Phys. B, Proc. Suppl. 64, 428 [3] M.-L. Yan, Y. Wei, and T.-L. Zhuang, Eur. Phys. J. C 7, 61 (1998). (1999). [16] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Nucl. Phys. [4] J. Z. Bai et al. (BES Collaboration), Phys. Rev. D 62, B566, 275 (2000). 032002 (2000). [17] A. L. Fitzpatrick, J. Kaplan, E. Katz, and L. Randall, [5] K. Gottfried, Phys. Rev. Lett. 40, 598 (1978). arXiv:1304.3458. [6] T.-M. Yan, Phys. Rev. D 22, 1652 (1980). [18] M. Berwein, N. Brambilla, J. Tarrús Castellà, and A. Vairo, [7] M. B. Voloshin, Nucl. Phys. B154, 365 (1979). Phys. Rev. D 92, 114019 (2015). [8] D. J. Gross, S. B. Treiman, and F. Wilczek, Phys. Rev. D 19, [19] N. Brambilla, G. Krein, J. Tarrús Castellà, and A. Vairo, 2188 (1979). Phys. Rev. D 97, 016016 (2018). [9] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. [20] R. Oncala and J. Soto, Phys. Rev. D 96, 014004 (2017). Zakharov, Nucl. Phys. B165, 55 (1980). [21] N. Brambilla, W. K. Lai, J. Segovia, J. Tarrús Castellà, and [10] M. B. Voloshin and V. I. Zakharov, Phys. Rev. Lett. 45, 688 A. Vairo, Phys. Rev. D 99, 014017 (2019). (1980). [22] A. Pineda and J. Soto, Phys. Lett. B 495, 323 (2000). [11] V. A. Novikov and M. A. Shifman, Z. Phys. C 8, 43 (1981). [23] N. Brambilla, A. Vairo, X. G. i Tormo, and J. Soto, Phys. [12] R. S. Chivukula, A. G. Cohen, H. Georgi, B. Grinstein, and Rev. D 80, 034016 (2009). A. V. Manohar, Ann. Phys. (N.Y.) 192, 93 (1989). [24] S. Weinberg, Phys. Rev. Lett. 110, 261601 (2013). [13] J.-W. Chen and M. J. Savage, Phys. Rev. D 57, 2837 [25] Y.-H. Chen, J. T. Daub, F.-K. Guo, B. Kubis, U.-G. Meiner, (1998). and B.-S. Zou, Phys. Rev. D 93, 034030 (2016). 054021-24 NOVEL IMPLEMENTATION OF THE MULTIPOLE EXPANSION … PHYS. REV. D 100, 054021 (2019) [26] Y.-H. Chen, M. Cleven, J. T. Daub, F.-K. Guo, C. Hanhart, [42] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. B. Kubis, U.-G. Meiner, and B.-S. Zou, Phys. Rev. D 95, Feruglio, and G. Nardulli, Phys. Lett. B 309, 163 (1993). 034022 (2017). [43] A. Bazavov, P. Petreczky, and J. H. Weber, Phys. Rev. D 97, [27] M. B. Voloshin, Prog. Part. Nucl. Phys. 61, 455 (2008). 014510 (2018). [28] M. Aaboud et al. (ATLAS Collaboration), J. High Energy [44] C. Peset, A. Pineda, and J. Segovia, J. High Energy Phys. 09 Phys. 01 (2017) 117. (2018) 167. [29] J. P. Alexander et al. (CLEO Collaboration), Phys. Rev. D [45] A. Pineda, J. High Energy Phys. 06 (2001) 022. 58, 052004 (1998). [46] W. Fischler, Nucl. Phys. B129, 157 (1977). [30] E. Guido et al. (Belle Collaboration), Phys. Rev. D 96, [47] Y. Schroder, Phys. Lett. B 447, 321 (1999). 052005 (2017). [48] M. Peter, Phys. Rev. Lett. 78, 602 (1997). [31] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, [49] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Phys. Rev. D 030001 (2018). 60, 091502 (1999). [32] H. Albrecht et al. (ARGUS Collaboration), Z. Phys. C 35, [50] A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. 283 (1987). Lett. B 668, 293 (2008). [33] F. K. Guo, P. N. Shen, H. C. Chiang, and R. G. Ping, Nucl. [51] C. Anzai, Y. Kiyo, and Y. Sumino, Phys. Rev. Lett. 104, Phys. A761, 269 (2005). 112003 (2010). [34] Y.-H. Chen, Adv. High Energy Phys. 2019, 7650678 [52] A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. (2019). Rev. Lett. 104, 112002 (2010). [35] O. Lakhina and E. S. Swanson, Phys. Lett. B 582, 172 [53] K. J. Juge, J. Kuti, and C. Morningstar, Phys. Rev. Lett. 90, (2004). 161601 (2003). [36] Y.-P. Kuang and T.-M. Yan, Phys. Rev. D 24, 2874 (1981). [54] G. S. Bali, B. Bolder, N. Eicker, T. Lippert, B. Orth, P. [37] N. Brambilla, D. Eiras, A. Pineda, J. Soto, and A. Vairo, Ueberholz, K. Schilling, and T. Struckmann (TXL and Phys. Rev. D 67, 034018 (2003). T(X)L Collaborations), Phys. Rev. D 62, 054503 (2000). [38] G. Amoros, M. Beneke, and M. Neubert, Phys. Lett. B 401, [55] S. Capitani, O. Philipsen, C. Reisinger, C. Riehl, and M. 81 (1997). Wagner, Phys. Rev. D 99, 034502 (2019). [39] A. Czarnecki and A. G. Grozin, Phys. Lett. B 405, 142 [56] B. A. Kniehl, A. A. Penin, Y. Schroder, V. A. Smirnov, and (1997); 650, 447(E) (2007). M. Steinhauser, Phys. Lett. B 607, 96 (2005). [40] B. L. Ioffe and M. A. Shifman, Phys. Lett. 107B, 371 [57] G. S. Bali and A. Pineda, Phys. Rev. D 69, 094001 (2004). (1981). [58] N. Brambilla, G. Krein, J. Tarrús Castellà, and A. Vairo, [41] J. F. Donoghue and D. Wyler, Phys. Rev. D 45, 892 Phys. Rev. D 93, 054002 (2016). (1992). [59] M. B. Voloshin, Phys. Lett. B 562, 68 (2003). 054021-25