A description of χ cJ → V V decays within the effective field theory framework

: We study χ cJ → V V decays using the QCD effective field theory approach. The helicity suppressed decay amplitudes are also considered. The colour-singlet contributions of these amplitudes suffer from the endpoint singularities, it is shown that they can be absorbed into renormalisation of the nonfactorisable colour-octet matrix element. The latter can be associated with the colour-octet component of the charmonium wave function. The heavy quark spin symmetry makes it possible to establish the relationships between colour-octet matrix elements for different states χ cJ up to higher order corrections in small velocity v . This allows us to estimate the polarisation parameters for χ c 2 → V V using data for χ c 0 , 1 → V V . This analysis is carried out for the available data on the χ cJ → ϕϕ decays.

The branching fractions of χ cJ decays into different pairs of vector mesons like ωω, ϕϕ and K * K * have been measured in last two decades by BES [1][2][3] and BELLE [4,5] collaborations, see Table 1.The current BES measurements are based on the accumulated 448 million ψ(2S) decays [7] that provided access to χ cJ decays through radiative decays ψ(2S) → χ cJ + γ.The increasing of statistics to several billion ψ(2S) events by BESIII in future [7] opens up opportunities for a more accurate and detailed analysis of rare exclusive χ cJ decays.In particular, sufficiently accurate data can provide a more detailed information about the various helicity amplitudes, which describe χ cJ → V V decays.Such information can be very useful for a better understanding of the underlying hadronic dynamics in charmonium decays.The first step in this direction has already been taken recently in the work [8], where the helicity amplitudes of χ cJ → ϕϕ decays were studied for the first time.
Various approaches have been proposed to describe χ cJ → V V decay dynamics.The hadronic loop mechanism is considered in refs.[9][10][11], perturbative QCD and quark pair creation model in refs.[12,13].The QCD factorisation framework based on the collinear factorisation has been used in ref. [14].However, these predictions can not describe the available experimental data on χ cJ → ϕϕ decays [8].
Existing data on χ cJ hadronic decays indicate about large effects associated with the violation of QCD helicity selection rule [20].As a rule, a description of corresponding colour-singlet amplitudes involves the higher Fock states for the final state hadrons and therefore such contributions are usually suppressed by additional powers of 1/m c comparing to helicity conserving amplitudes.In addition, the contribution of the colour-octet component of the charmonium wave function can mix with the colour-singlet contribution, which complicates a theoretical description.There is an evidence that the octet contribution provides a large numerical effect and therefore must be included into the theoretical description.
Attempts to build a theoretical formalism for the colour-octet mechanism in exclusive hadronic decays were considered in refs.[21] for χ cJ → ππ and in ref. [22] for χ cJ → pp decays.However, suggested approach represents a kind of a phenomenological model and can not be related with a systematic effective field theory framework.It is interesting to note that endpoint divergences are also one of the problematic features of this approach.
The mixing of the colour-octet and colour-singlet amplitudes in the context of effective field theory was discussed in B → χ cJ K-decays [23] and χ cJ → KK * decays [24].It was demonstrated that the endpoint divergences in the colour-singlet contribution can be absorbed into renormalisation of the colour-octet matrix element.In ref. [24] it is also shown that the heavy quark spin symmetry allows to relate the octet matrix elements for different states χ cJ .This makes it possible to obtain a relationship between the amplitudes with different spin J and calculate the well defined spin symmetry breaking corrections.The latter allows one to get relations for different observables, which can be verified experimentally.The goal of this work is to carry out a similar analysis for the amplitudes of χ cJ → V V decays.

Decay amplitudes and observables
In order to describe χ cJ → V V decays we define the following amplitudes V (k, e V )V (k ′ , e ′ V ) i T |χ cJ (P, ϵ χ )⟩ = i(2π) 4 δ (4) The particle momenta satisfy on-shell relations where m V and M χ denote the masses.For simplicity, we do not take into account the difference in masses of the quarkonium states and assume For the polarisation vectors of different mesons we use short notations where In order to decode the structure of the amplitudes A J defined in (2.1) it is convenient to introduce the following auxiliary tensors ) where V µ denotes an arbitrary Lorentz vector and iε µναβ is Levi-Civita symbol.
Using these definitions we define the following scalar amplitudes: ) where we use short notation for the symmetric traceless combination All scalar amplitudes A (... ) J have dimension of mass and can be associated with the different helicity amplitudes χ cJ (i) → V (λ)V (λ ′ ).The coefficients in front of different Lorentz structures in Eqs.(2.8)-(2.10)are chosen in order to get more simple analytical expressions for the corresponding decay rates ) where χ and δ V V = 1 for identical states like ωω, ϕϕ and zero otherwise.The superscripts L and T correspond to longitudinally and transversely polarised mesons in the final state V V .Using the various decays modes for the vector resonances V one can get information about vector meson decays with definite helicities χ cJ → V (λ)V (λ ′ ).The ratios of the different helicity amplitudes can be extracted from the angular distributions of the cascade process e + e − → ψ(2S) → χ cJ +γ → (V V )+γ where the vector mesons decay into stable hadrons V → f f ′ .A more detailed formalism for such helicity analysis was already considered in refs.[11,13,25].The helicity amplitudes F J λ,λ ′ defined in refs.[11,13] can be easily related to the scalar amplitudes defined in Eqs.(2.12)-(2.14) and the corresponding formulas will be considered later.
The total widths are given by the sum of the different partial rates ) where each term in the rhs includes the appropriate amplitude from Eqs. (2.12) and (2.14).QCD analysis and helicity conservation in the perturbative QCD interactions suggest the following hierarchy of helicity amplitudes where λ ∼ Λ/m c and Λ is the typical hadronic scale Λ ∼ Λ QCD .The power suppressed amplitudes are sensitive to the higher Fock components of meson wave functions.Using the terminology of the collinear factorisation, the nonperturbative coupling to mesons is described by the higher-twist operators that gives additional powers of the soft scale.
In the formal heavy quark limit m Q → ∞ the power suppressed amplitudes are expected to be small.However, various existing data indicate that the mass m c is not large enough and effects from the helicity suppressed amplitudes are sufficiently large.As an example, we refer to the value of the branching fraction χ c1 in Table 1.Therefore a study of such amplitudes can be important for a consistent description of the data.
3 The leading-power amplitudes A LL 0 , A LL 2 and A T T 2t In the limit m c → ∞ decay amplitudes are dominated by the colour-singlet contribution, which can be described within the collinear QCD factorisation framework, see e.g.
refs.[26,27].Such description allows one to unambiguously separate the physics associated with the short-and long-distance dynamics.The short distance subprocess cc → 2g * → (q q)(q q) is described in pQCD and can be computed systematically order by order in α s .Corresponding calculations are well known in the literature and therefore we skip the technical details in this section.The nonperturbative dynamics is described by the various long-distance matrix elements defined in the collinear QCD and NRQCD.The collinear matrix elements are parametrised in terms of the light-cone distribution amplitudes (LCDAs).To the leadingpower accuracy the required LCDAs are defined by the leading twist quark-antiquark operators on the light-cone z 2 1 = z 2 2 = 0 where Γ denotes some Dirac structure and χ n (z) denotes the field describing the quark jet with the momentum ∼ k ′ , which is collinear to light-like vector n, see more details in Appendix A.
Corresponding LCDAs can also be obtained from Bethe-Salpeter wave functions at (almost) zero transverse separations of the constituents (assuming the valence quark-antiquark component of the wave function) In Appendix B we provide the definitions of various LCDAs, which are used in this work.
For the NRQCD matrix elements of χ cJ states we use standard definitions, see e.g.[28], which corresponds where R ′ 21 (0) denotes the radial wave function at the origin.The expressions for the amplitudes can be written as follows where the different dimensionless convolution integrals are given by (x = 1 − x) The soft couplings f V and f ⊥ V have dimension of mass.All the integrals over the momentum fractions x and y in Eqs.(3.5)-(3.7)are well defined because the LCDAs have smooth endpoint behaviour ϕ ∥,⊥ 2V (x) ∼ xx that compensates the singularities in the integrand denominators.The running QCD coupling α s is defined at the scale µ ∼ m c .A more detailed discussion of the various parameters and the models for the LCDAs are considered in Appendix B.

Helicity suppressed amplitudes
Collinear factorisation for helicity suppressed amplitudes is violated by the endpoint divergencies in the collinear integrals ref. [14].Such divergencies indicate about the overlap of collinear and ultrasoft (usoft) modes.The latter describes the particles with p us ∼ m c v 2 ∼ Λ. Below it will be shown that the structure of the endpoint divergencies is closely associated with the configuration where one of the gluons in the annihilation subprocess cc → g * g * → (qq)(qq) becomes usoft: g * g * → g * us g * h .Therefore the subprocess with the two hard gluons (g * h g * h ) transforms into the subprocess with the one hard gluon cc → g * h → (qq).Such subprocess is closely associated with the annihilation of the colour-octet configuration of cc pair.The endpoint divergencies indicate about the overlap of colour-singlet and colour-octet contributions.
In such a situation, the factorisation is described by the sum of colour-singlet and the colour-octet LDMEs.Let us consider, as example, the amplitude A 1 .Schematically, the factorisation formula for this amplitude can be written as where the first term in the rhs denotes colour-singlet collinear integral with the hard kernel T 1 and twist-2 and twist-3 LCDAs denoted as ϕ 2,3V .The hard kernel T 1 corresponds to the hard transition cc → g * h g * h → (qq)(qq).The second term in the rhs of Eq.(4.1) includes the coefficient function C h associated with the hard subprocess cc → g * h → (qq) and the octet nonfactorisable LDME a The operator O oct ∼ (cc) 8 (q q) 8 consists of the two colour-octet components, which are constructed from the NRQCD fields (cc) 8 and from the hard-collinear fields in SCET-I (q q) 8 .The LDME a (oct) 1 is nonfactorisable because of usoft exchanges between the initial and final states.The formula (4.1) implies that the infrared endpoint singularities in the collinear convolution ϕ 2V * T 1 * ϕ 3V can be absorbed into the renormalisation of the octet contribution C h a (oct) 1 . The described picture makes it possible to develop the following interpretation.The charmonium colour-octet configuration can be viewed as a compact cc colour-octet pair of size ∼ 1/(m c v) surrounded by an ultrasoft cloud of size ∼ 1/Λ.Such a configuration is in some sense similar to heavy-light systems, such as B-mesons.Following to this analogy, the operator O oct in the LDME in Eq.(4.2) plays the role of an external source, similar to the low-energy effective Lagrangian in the Standard Model.This source instantly annihilates the colour-octet cc pair into light energetic quark-antiquark pair (q q) 8 .This quark-antiquark pair hadronises into a final hadronic state interacting with the usoft cloud from the parent charmonium.
The similar factorisation formula as in Eq.(4.1) can also be written for the decay χ c2 into the same final state V V , i.e. for A LT 2 amplitude where the octet LDME is defined analogously a Keeping in mind the analogy with the heavy-light system noted above, one can expect that defined nonfactorisable LDMEs do not depend on the quantum numbers of the heavy cc-pair in the limit m c → ∞.In particular, the ultrasoft modes can not resolve the total angular momentum J cc of the heavy quark-antiquark pair and therefore the nonperturbative dynamics does not depend on J cc .These arguments leads to an idea of approximate heavy quark spin symmetry (HQSS).This symmetry make it possible to get the following relation between the nonfactorisable LDMEs where λ 12 is some number.The Eq.( 4.4) can be used in order to establish relationship between the physical amplitudes.Excluding from the expressions (4.1) and (4.3) the nonfacrorisable LDMEs, with the help of Eq.( 4.4), one finds where the second term on rhs is well defined and represents the HQSS breaking correction, which occurs due to the interactions at short distances due the subprocess cc → g * g * → (qq)(qq).The infrared endpoint singularities must cancel in the combination T 2 − λ 12 T 1 , hence this correction can be calculated in collinear factorisation scheme.The relation (4.5) can be used in order to restrict the ratio , which can be measured experimentally.
The program described above is close in spirit to the well-known approach in decays of B mesons [43].This scheme was used for description of χ cJ → K K * decays in ref. [24].In the following we use the similar approach for helicity suppressed amplitudes in χ cJ → V V decays.describing the coloursinglet contribution.At the next step, we will analyse the structure of the emerging IRsingularities and propose the additional contribution that need to be added to the factorisation formula.

Hard kernels and HQSS relation for amplitudes A LT
To calculate the amplitudes 1,2 , various twist-3 LCDAs are needed. 2The set of the corresponding twist-3 light-cone operators can be divided on valence (quark-antiquark) and non-valence (quark-antiquark-gluon) operators.In this work we consider only the contributions associated with the valence operators and neglect the non-valence ones, for simplicity.Schematically, the matrix elements of the valence operators can be described as where ∂ ⊥ is the transverse derivative.The presence of the derivative provides additional power of the small scale λ ∼ Λ/m c , and the corresponding operators have the kinematical twist-3.This transverse derivative can be associated with the quark-antiquark angular momentum L = 1 as required by the total angular momentum conservation in the hard subprocess.The full set of required valence matrix elements and their parametrisation in terms of LCDAs are presented in Appendix B.
The Lorentz symmetry and QCD equations of motions make it possible to derive the valence twist-3 LCDAs in terms of the twist-2 LCDAs ϕ 2V .Such an approximation is also also known in literature as Wandzura-Wilczeck one.It has been verified in many various calculations that twist-3 valence contributions provide a consistent theoretical description preserving all basic symmetries of the QCD.
The analytical expression for the colour-singlet contributions to the amplitudes A 1,2 can be written as For a convenience, we define the prefactor consisting of the colour factor C F /(2N 2 c ) = 2/27 and the vertex factors (ig) 4 , the multiplier 2M χ f χ is associated with the NRQCD matrix element.
The Dirac trace in the first line (4.7)describes the light quark subdiagram with the twist-2 and twist-3 projections M2,3 for the light-cone matrix elements and the matrices γ α,β from the QCD vertices, see Fig 1.The second line in (4.7) includes the two gluon propagators (in Feynman gauge) and the trace involving the Dirac matrices from the heavy quark line.The outgoing collinear quarks and antiquarks are defined with the following momenta The transverse components must be treated carefully because the twist-3 projections M 3 in the first line (4.7)involve derivatives with the respect to k ⊥ and k ′ ⊥ , which act on the analytical expression in the second line.The explicit expressions for the different projections M i are given in Appendix C.
The heavy quark trace is described by the heavy quark subdiagram, it involves the differentiation with respect to relative heavy quark momentum that gives two terms The Dirac projectors P µ J for different charmonium states with J = 1, 2 are given by where ω denotes the velocity P = M χ ω, ω 2 = 1 and Performing the calculation of the traces and contractions of the indices in Eq.(4.7) and matching to the definitions in Eqs.(2.9) and (2.10) gives the following results where the convolution integrals can be conveniently written as with Remind, the twist-3 LCDAs G ⊥ 3V , G ∥ 3V , G∥ 3V describe the different valence light-cone matrix elements as in Eq. (4.6) and explicitly defined in Appendix B.
The integrals with the superscripts (s) and (r) are singular or regular, respectively.The occurring divergencies are closely associated with the singularities of the hard kernels in the integrands.From the structure of the integrands in Eqs.(4.17)-(4.19) it follows that the singular behaviour is associated with the endpoint limits x → 0(1) and y → 1(0), respectively.In the collinear region the momentum fractions are considered to be of order one x ∼ O(1), which implies, see for instance Eq.(4.8), that x(kn) ∼ xm c ∼ m c .However, if the collinear region overlaps with the usoft one, then the integration domain with the small fractions x ∼ λ ∼ Λ/m c is not suppressed by λ and this usually leads to logarithmical divergencies in the collinear convolutions.Therefore our task is to inspect the regions where x(x) ∼ λ and y(ȳ) ∼ λ and to find the configurations of particle momenta leading to divergent integrals.
In order to perform such analysis one has also to take into account the properties of the various LCDAs.The different twist-2 LCDAs, which enters into the integrals, have the following endpoint behaviour where the prime denotes the derivative with respect to the argument f ′ (x) = d/dxf (x).This behaviour is closely associated with the properties of the evolution kernels for these LCDAs.
The twist-3 LCDAs in the valence approximation have the following endpoint behaviour The explicit expressions for the first derivatives in these formulas read The higher order terms shown in Eqs.(4.22)-(4.25) will be required later.
Let us regularise the integrals in Eq.(4.17)-(4.19)setting infrared cut-off δ IR < x, y and δ IR < x, ȳ3 , which helps to work with the divergent integrals.Consider as example the region δ IR < x < η U V and δ IR < ȳ < η U V , where the parameter η U V plays the role of the boundary, which separates the usoft region x, ȳ ∼ λ from the collinear region x, ȳ ∼ 1. Obviously, it is assumed that δ IR ≪ η U V .Expanding the integrand with respect to small fractions x and ȳ and keeping the leading term one finds The leading order usoft integral scales as [J (s) The similar consideration gives [J (r) The similar results can be obtained for the symmetric usoft region x ∼ y ∼ λ [J (r) All other possible endpoint regions give power suppressed contributions and therefore can not be associated with the logarithmic endpoint singularities.This also can be seen subtracting the usoft contributions from the regularised integrals, the results will not depend on the parameter δ IR .The contributions, which appear from the endpoint regions, can be associated with the different factorisation scheme, which involves usoft modes.Such contributions must be included into the description along with the collinear singlet contribution described in Eq. (4.14).The structure of these usoft contributions can be understood from the analysis of the endpoint regions in the diagrams.Consider again the usoft limit x ∼ ȳ ∼ λ, then the external quark with momentum k 1 and antiquark with momentum k ′ 2 can be understood as the usoft particles.Similarly the virtual gluon producing this pair, see Fig. 1, is also usoft At the same time the second gluon remains hard Emission of the usoft gluon does not change the virtuality of the potential heavy quark 2 , therefore in the usoft regions the momentum of the heavy quark, see Fig. 1, remains unchanged which implies that Λ ∼ m c v 2 as it was assumed.The similar picture also holds for the usoft limit x ∼ y ∼ λ.Therefore the hard subprocess in this case is associated with the annihilation of the octet Q Q pair into light collinear-anticollinear quark-antiquark pair: cc → g * → q q.The created light quark and antiquark interact with the usoft particles in order to hadronise into the hadronic final state.Due to the the soft-overlap between the initial and final states, see Fig. 2 a), corresponding matrix element are not factorised after integration over the hard modes.Factorising the hard modes gives the four-fermion operator, which can be represented as product of two colour-octet operators, which are built from NRQCD and SCET-I hard-collinear fields χ n,n where we assume that σ µ = {1, σ}.The hard coefficient function C h = πα s /m 2 c is independent of the hadronic states.The resulting matrix elements still depends on the mass m c but this dependence is defined by the interactions of hardcollinear particles with momenta p 2 hc ∼ m c Λ. Since the charm mass is not sufficiently large, the realistic hard-collinear scale m c Λ is too small in order to proceed with the hardcollinear factorisation.However, it is useful to consider, at least schematically, how the matrix element (4.43) can be described in the limit m Q → ∞.
In this case, a proposal for the factorisation of this contribution is shown in Fig. 2 b).The corresponding factorisation formula can be schematically described as where ω i denotes the momentum fractions of the usoft quark fields, J n,n are the appropriate jet functions, which are convoluted with the collinear LCDAs ϕ 2,3V (these convolutions are shown by the asterisks ).It is also supposed that the charmonium LDME is described by the LCDA Φ χ (ω 1 , ω 2 ), which is defined by the light-cone matrix element where the operator O µ,a (q, q) includes usoft light quark q(z q+ ) and antiquark q(z q− ) fields describing the usoft charmonium cloud.The structure of the operator in Eq.(4.45) indicates that the corresponding state consists of the octet-charm component and light degrees of freedom.Notice, that interactions of the usoft particles with the hard-collinear ones make the NRQCD operator in Eq.(4.45) to be nonlocal.In some sense, such charmonium LCDA reminds the B-meson LCDAs φ ± , which describe valence heavy-light components of b-meson in the SCET factorisation framework.This very schematic picture is only discussed in order to stress the relation of the matrix element in (4.43) with the colour-octet component of the charmonium wave function and to illustrate the common features of such configuration with heavy-light systems.
The colour-octet matrix element can be parametrised in the same way as the relevant amplitudes in Eqs.(2.9) and (2.10).Therefore the final result for the amplitudes A (LT ) J is given by the sum of singlet and octet contributions where a (LT ) J denotes the contribution of the colour-octet matrix element.We suppose that the endpoint singularities in the collinear integrals J (J) V ∥,⊥ can be absorbed into UVrenormalisation of the octet amplitude a (LT ) J .The power counting 4 for the colour-singlet contribution in Eq.(4.46) gives that it scales as A (LT ) J ∼ v 4 λ 3 .Obviously, the octet amplitudes a (LT ) J must have the same behaviour.The heavy quark component in the colour-octet operator scales as v 3 only.The additional factor of velocity v, which reproduces the balance appears from the interaction of the cc pair with the usoft gluon.The interaction of the potential heavy quarks with ultrasoft gluons is the subject of potential NRQCD (pNRQCD).The contributions of order v in the effective Lagrangian are given by the dipole interactions, which read g s ψ † (x)(−x • E(t, 0))ψ(x) together with a similar term for the antiquark field [23].These interactions are sensitive to the colour charge only and conserve the spin and angular momentum of heavy quarks, which leads to the HQSS.The pNRQCD can not be used for the real calculations because of relatively small value of m c .However this EFT gives us a possibility to better understand the power counting in the non-relativistic sector and the dynamical origin of the HQSS.
The colour-octet amplitudes a (LT ) J differ by the spin of quarkonium only and therefore they must satisfy the symmetry relation up to higher order corrections in velocity v a with a certain numerical coefficient λ 12 .This symmetry coefficient can be directly calculated in pNRQCD in the Coulomb limit m Q v 2 ≫ Λ, this analysis is carried out Sec. 5. On the other hand, λ 12 can also be derived from the colour-singlet contribution by comparing the collinear integrals in the usoft limit.
The simplest way to get λ 12 is to derive the relation between the physical amplitudes as described above, see Eq.(4.5).The divergent contributions must cancel in the symmetry breaking correction, therefore using Eq.(4.14) gives where V ∥ − λ 12 J (1) V ∥ − λ 12 J (1) The coefficients in front of divergent integrals J (s) V ∥,⊥ must vanish that gives Notice, that this results holds for the two independent integrals simultaneously.In Sec. 5 we reproduce this result using pNRQCD calculation of the octet matrix element.Therefore the final expression for the correction ∆A (LT ) in Eq.( 4.48) reads where the regular integral J (r) V ∥ is defined in Eq.(4.19).And finally, let's add a comment to the case K * K * channel.In this case the LCDAs are not symmetrical with respect to x ↔ x because of SU (3)-symmetry breaking effects.In addition, the twist-3 projections M 3 and twist-3 LCDAs explicitly involve the contributions ∼ m s /m K * .The expression for the amplitude in Eq.( 4.46) and relation (4.48) are obtained taking into account such terms.We find that the contributions ∼ m s occur in the final result only through the twist-3 LCDAs.For simplicity, such terms are not shown in the Eqs.(4.22)-(4.26)describing the endpoint behaviour.The inclusion of such terms complicates the formulas, but does not affect the main results.For instance, the LCDA G ⊥ 3K * in (4.22) in the endpoint region is described as where the first term in the rhs is given in Eq.(4.27).The endpoint behaviour of the correction in (4.55) is not analytic ∼ x ln x, which gives the double logarithmic contribution ∼ ln 2 η U V /δ IR for the corresponding usoft integrals.It is interesting to note, that the similar behaviour is also observed for the amplitudes in χ cJ → K * K decays [24], which are proportional to SU (3) breaking contributions.However, in K * K * decays, the numerical effect from such corrections is relatively small and their detailed discussion is omitted for simplicity.

The hard kernels and HQSS relations for amplitudes
The calculation of these amplitudes involves the twist-3 and twist-4 LCDAs.The matrix elements of the valence twist-4 LCDAs G 4V are defined as the matrix elements of the chiral-odd operator see details in Appendix B. The analytical expression for the amplitudes is similar to one in Eq.(4.7) but with the appropriate projections M i for the collinear matrix elements.This calculation is also very similar to the previous ones, so we omit the technical details and present the result , (4.57) The colour-singlet contributions are described by G 3V ⊗ G 3V and ϕ ⊥ 2V ⊗ G 4V configurations, which define the collinear integrals I 33 and I 24 , respectively.Their explicit expressions read With the help of these formulas one finds (4.70) 5 For simplicity, we assume that LCDAs in Eqs.(4.66)-(4.68)satisfy the symmetry [ϕ , which does not hold for LCDAs of K * mesons, however the final results also valid for K * K * case. (4.72) The similar result also holds for the symmetric domain x ∼ y ∼ λ.The obtained structure of the IR-divergencies is the same as in the previous section: the usoft integrals yields the simple logarithms.This result differs from one of ref. [14] where the double logarithmic and even a power divergencies are obtained.The occurence of the endpoint power divergence in ref. [14] indicates about the problem in the calculations: it can either be an error or an invalid operation on divergent collinear convolution integrals, see e.g. a discussion in ref. [30].
Because the usoft structure is similar to one in the previous section, we follow the same factorisation scheme by introducing the appropriate colour-octet matrix elements.This suggests where a again denotes the contributions of the colour-octet matrix element.Following the way as in the previous section and using results (4.69) and (4.71) for the usoft integrals, we can conclude that the colour-octet contributions satisfy the HQSS relation This result yields the desired relationship between the decay amplitudes where the symmetry breaking corrections are described as with the well defined integrals (D ≡ xy + xȳ) with the kernels Figure 3. Two diagrams, which describe the leading-order pNRQCD contribution to the colouroctet matrix element in the Coulomb limit.The crossed vertex denotes the dipole interaction (5.4), the dashed lines correspond to collinear particles with small fractions ∼ v 2 , the double line denotes the Coulomb Green function G c and the black square is associated with the octet operator vertex.
function G c is known and its analytical expression is somewhat complicated and can be found in refs.[36,37].Two diagrams in Fig. 3 (a, b) describe two different regions x 1 ∼ y 2 ∼ v 2 and x 2 ∼ y 1 ∼ v 2 , respectively 6 .The calculation of the both diagrams is very similar and therefore we only consider the diagram (a) for simplicity.The analytic expression for this contribution can be written as where all colour traces are calculated giving the factor 2/27.This expression includes the usoft gluon propagator The integration region over the usoft fractions x 1 ∼ y 2 ∼ v 2 is restricted by UV-cut off η.
The momentum space radial wave function of P -wave state reads where R ′ 21 (0) is the derivative of the position radial wave function at the origin.The angular integration yields The factor D αβ q in Eq.(5.5) describes the trace associated with the collinear projectors and LCDAs.It can be computed in the same way as in Eq.(4.7) but now one must also expand the obtained expression with respect to the small usoft fractions.
The traces tr[Λ J (1 − / ω)γ ⊥α (1 + / ω)] describe the contractions of the Dirac indices associated with the heavy quarks.The projections onto P -wave states read where the projectors P µ J are given in Eq.(4.12).The transverse matrix γ ⊥ in these traces occurs from the colour-octet operator, see Eq. (4.42).
The function D Q β (E, ∆) in Eq.(5.5) is described by the interaction vertex (5.4) and by octet Coulomb Green function where k g is the outgoing ultrasoft gluon momentum.The expression for the Green function is quite complicated, but we need only those terms, which give the UV-divergent integrals (∼ ln η) in the diagrams in Fig. 3.It turns out that in this case the Green function can be simplified where E Q is the energy of heavy quark in the rest frame of Q Q pair, ∆ denote the heavy quark relative momentum.The dots denote the contributions, which provide U V -regular expressions and therefore can be ignored for now.Substituting G c from Eq.(5.11) into Eq.(5.10) one finds where E denotes the binding energy E = M χ − 2m Q for P -wave triplet. 7Using Eqs.(5.9) and (5.13) and rotation invariance (∆ λ ⊤ ∆ µ ⊤ → −g µλ ⊤ ∆ 2 /3) one can rewrite the integrand This gives a (a) (5.15) In order to get expressions for the scalar amplitudes a , it is enough to consider the contribution to the factor D αβ q with the projections of twist-3 for the state V (k, e) and 7 To our accuracy the value of E is degenerated and therefore does not depend on the total angular momentum J of the perturbative state χQJ .
twist-2 for V (k ′ , e ′ ) only.This yields with Substituting this into Eq.(5.15) one finds a (a) where the universal integral reads From this expressions it follows that a (LT,a) 2 which is in agreement with the result (4.53).The same result also valid for the contributions of the second diagram in Fig. 3.
In order to demonstrate the overlap of the IR-divergencies in Eqs.(4.31) and (4.33) with UV-divergencies in Eqs.(5.21) and (5.22) let us consider the integral in Eq.(5.20).Taking the formal limit η → ∞ and assuming that m can neglect small terms in the denominator of the integrand in (5.20) that gives where we used Eq.(5.8).Substituting this result into Eqs.(5.21) and (5.22) one can easily observe that obtained expression correctly reproduces the corresponding IR-asymptotic (4.31) and (4.33) of the colour-singlet contribution in Eq. (4.14).
Let us also note, that the integrals d 3 ∆ in Eqs.(5.21) and (5.22) are well defined.The denominator of the Coulomb Green function has the pole at ∆ 2 /(2m Q ) = E − (x 1 + y 2 )m Q , which leads to the nontrivial imaginary phase.The obtained in this way imaginary part does not have UV-divergencies at η → ∞ as it must be.This observation illustrates an interesting mechanism leading to the nonperturbative phases in QCD.
In order to find expressions of F (J) λ,λ ′ in terms of scalar amplitudes A (... ) J we consider the rest frame of the parent charmonium χ cJ and define the momenta as The polarisation vectors for different particles cab be defined as A simple calculation yields . (6.9) . (6.10) From Eq.(2.17) one finds which implies 12) The available experimental results read [8] x = 0.30 ± 0.03, (6.13) y 0 = 1.45 ± 0.10, y 1 = 1.265 ± 0.054, y 2 = 0.81 ± 0.05, ( where only the statistical uncertainties are shown.Taking the conservative estimate Λ/m c ∼ 0.3, one finds that expected hierarchy (6.12) is violated.In particular, the value of y 0 is almost the same order as y 1 , even despite the large numerical coefficient 3 √ 2 in the definition (6.9).The value of y 2 also looks much larger than might be expected.These observation indicates about the possible scaling violation effects in these decays.
At the same time the value of the parameter x in (6.13) is still small indicating that the leading amplitude A where we used the value Br[χ c0 → ϕϕ] = 8.48 × 10 −4 from ref. [8] and the total width Γ[χ c0 ] = 10.5 MeV [6].
On the other hand Γ (LL) 0 can be estimated using result for the amplitude A (LL) 0 in Eq. (3.4).In order to get numerical estimates we use the following input for different parameters.For the charmonium matrix element in (3.3) we use the value, which follows from a Buchmüller-Tye potential model in ref. [38] The averaged charmonium mass is fixed as M χ = 3.5 GeV.For quark mass m c there is well known estimate m c = 1.4 ± 0.2 GeV.The errors in this expression give large numerical effect because the expressions in (3.4) are proportional to 1/m 4 c .For simplicity, we take the value of m c = 1.5 GeV.Various LCDA models are described in Appendix B. For the QCD running coupling we fix the value α s = 0.25.Then we find Γ (LL) 0 = (8.2± 2.7) KeV, (6.18)The relations of these LCDAs with ones in refs.[39,42] can be easily obtained rewriting the Lorentz covariant light-cone operators in the definitions of the matrix elements in [39,42] through the SCET fields χ n.Neglecting the quark-gluon contributions we obtained the following expressions The symbol δ V K * is the Kronecker delta with respect to symbol V .The twist-4 valence matrix element is defined as 4V ] , (B.12) where we use short notation The first term in Eq.(B.12) is irrelevant for the calculations of amplitudes A (T T ) 0,2s because this LCDA describes the quark-antiquark pair with L z = 0 and therefore corresponding trace vanishes.Hence, we only need to derive the analytical expression for LCDA G (2) 4V .In order to do this we proceed as follows.
At first step we use the light-cone matrix element [40] ⟨V (k, e)| ψ(z  3 and G 4V .Excluding G 4V with the help of Eq.(B.16) yields the following differential equation for the h (q) 3 2 2h (u) = uūh This solution agrees with the pure quark contribution for h 3 in case of ρ-meson, which was derived in ref. [40] using a different technical method.

C Projection operators for collinear matrix elements
The projections of the higher-twist light-cone matrix elements are often derived using the covariant parametrisation of the correlation function, see e.g.discussion ref.[43].In this work we use a different technique, which is based on the light-cone cone expansion of the external collinear fields to a given accuracy and reduction of the light-cone operators to the valence ones.Such approach involves the QCD equation of motions in order to integrate the small components of the collinear fields and allows one to deal with the light-cone operators only.This approach has many advantages, in particular, one can apply the field redefinition in order to get the collinear Wilson lines in the operators and therefore better maintain the colour gauge invariance.The valence LCDAs have simple analytical properties, which simplify the analysis of infrared singularities.The technical details was described in ref. [44] and we will not repeat them here.
In the following we just provide the analytical expressions for the different projection operators M i , which are used for the calculation of the colour-singlet contributions in the collinear factorisation framework, see Eq.(4.7).Below we assume that these projection operators are associated with the off light-cone matrix element ⟨V (k, e)| χα 1 (x) χ α 2 (y) |0⟩ → where we assume that x 2 ̸ = 0 and y 2 ̸ = 0 and we also skip the colour structure for simplicity.
In the following expressions we assume We also write all Dirac matrices in the square brackets [. . .] and do not show the indexes α i explicitly.Some terms in M i do not depend on transverse derivatives ∂/∂k ⊥ then the momenta k i in e i(k 1 x)+i(k 2 y) in Eq.(C.1) are defined as in Eq.(B.3).For the terms with transverse derivatives the momenta k i must be modified as The auxiliary momentum l for the term with derivative ∂/∂l α is defined through the momenta k i in Eq.(C.1) as

1 Introduction 1 2amplitudes and observables 2 3 5 4suppressed amplitudes 6 4 . 1 2 7 4 . 2 26 A 27 B Definitions and models of LCDAs 29 C
Decay The leading-power amplitudes A LL 0 , A LL 2 and A T T 2t Helicity Hard kernels and HQSS relation for amplitudes A LT 1 and A LT The hard kernels and HQSS relations for amplitudes A T T Set up, useful notation and power counting Projection operators for collinear matrix elements 32 1 Introduction

1 and A LT 2
Let us start our analysis from the calculations of hard kernels T

Figure 1 .
Figure 1.The pQCD diagrams, which describe the hard kernels in the decay amplitudes.The heavy quark momenta (double lines) are incoming and all the light quark momenta are outgoing.The relative quark momentum q in Eqs.(4.10) and (4.11) is set to zero after differentiation.

Figure 2 .
Figure 2. a) The non-factorisable contribution, associated with the colour-octet amplitude.The red gluon line denotes the hard gluon with momentum p 2 h ∼ m 2 c .b) A proposal for factorisation of the colour-octet matrix element in the limit m Q → ∞.The black square represent the colour-octet operator O 8 , the dotted fermion lines denote the usoft light quarks.The dark blobs denote the hard-collinear jet functions associated with the momenta p 2 hc ∼ Λm Q , the light-grey blobs denote the different LCDAs.
.65) For simplicity, the corrections associated with the explicit SU (3)-breaking contributions ∼ m s /m V are neglected in these formulas .The convolution integrals I singular in the endpoint regions where one of the gluons is soft x ∼ ȳ ∼ λ or x ∼ y ∼ λ, see Fig.1.This can be seen in the same way as in the previous section.The endpoint behaviour of the twist-3 LCDAs was already discussed, using Eqs.(4.22)-(4.25)one finds5

22 )
Using the twist-2 LCDA from Eq.(B.4) one finds the solution of this equation for h