---
abstract: |
    It is pointed out that the decay of the recently observed charmonium $\eta_c^\prime$ resonance, $\eta_c^\prime \to \eta_c \, \pi \, \pi$ is simply related to the well studied decay $\psi^\prime \to J/\psi \, \pi
    \, \pi$ and can thus be used for absolute normalization of other decay modes of the $\eta_c^\prime$. The total rate of the discussed decay should be approximately three to four times the corresponding rate for the $\psi^\prime$ resonance making the channel with charged pions the most probable exclusive decay mode of the $\eta_c^\prime$ with the branching ratio in the range 5-10%. 
---

[**Theoretical Physics Institute\
University of Minnesota\
**]{}

TPI-MINN-02/20-T\
UMN-TH-2105-02\
June 2002\

[**The decay $\eta_c^\prime \to \eta_c \, \pi \, \pi$.\
**]{} Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455\
and\
Institute of Theoretical and Experimental Physics, Moscow, 117259\

The recent observation[@belle] of the charmonium state $\ecp$ with mass $3654\pm 6 (stat) \pm 8 (syst) \, MeV$ illustrates the possibility of studying charmonium resonances produced in the decays of $B$ mesons[@elq].  Despite being known for more than a quarter of a century the charmonium system still leaves an ample space for further study, especially in the spin singlet sector (a recent update on the singlet states is given in Ref.  [@gr]).  Even for the most well established singlet state $\eta_c$ the data on its properties are still quite scarce, and its total decay width is measured with a quite ‘modest’ accuracy $13.2^{+3.8}_{-3.2} \, MeV$[@pdg], which is hardly improved by the recent CLEO result of $26 \pm 6 \, MeV$[@cleo].  For the newly observed $\ecp$ it is unlikely that a direct measurement of its total width can be any easier.  The purpose of the present letter is to point out that in the case of $\ecp$ the differential spectrum and the total rate of the decay $\eta_c^\prime \to \eta_c \, \pi \, \pi$, first discussed in Ref.  [@ok], can be found with absolute normalization through its relation to the well studied decay $\psi^\prime \to J/\psi \, \pi \, \pi$. As simple as this remark is, it still merits being explicitly made, since this decay can provide a useful normalization for other decay channels of $\ecp$ as well as for the total decay rate, at least in the interim, until $\Gamma_{tot}(\ecp)$ is measured directly.  Moreover, due to the known (and, to an extent, understood) behavior of the decay amplitude, a slightly larger energy release in the transition between the singlet states is quite important, so that the total rate $\Gamma(\eta_c^\prime
\to \eta_c \, \pi \, \pi)$ should be three to four times larger than that for the transition between the vector states.  In absolute numbers this corresponds to $\Gamma(\eta_c^\prime \to \eta_c \, \pi^+ \, \pi^-)
\approx  2 \Gamma(\eta_c^\prime \to \eta_c \, \pi^0 \, \pi^0) \approx
300 \pm 50 \, KeV$[^1] which should constitute 5-10% of the expected (with substantial uncertainty) total width of the $\ecp$. Clearly, this implies that the discussed pionic transitions are the most probable exclusive decay channels of the $\ecp$, much in the same way as they are for the $\psi^\prime$.

The outline of the reasoning is as follows.  Hadronic transitions in heavy quarkonium are described in terms of the multipole expansion in QCD[@kg; @mv].  Within this expansion the amplitudes of the decays $\eta_c^\prime \to \eta_c \, \pi \, \pi$ and $\psi^\prime \to J/\psi \, \pi \, \pi$ are exactly equal in the non-relativistic limit for heavy quarks.  Furthermore the amplitude for emission of the pions by the relevant gluonic operator is governed by the current algebra and the trace anomaly in QCD[@vz; @ns].  The presence of the anomaly contribution substantially enhances the amplitude of these decays and gives rise to a rapid growth of the amplitude with the invariant mass of the dipion. 

Proceeding to a more detailed discussion, we start with the remark that the emission of light hadrons in transitions between heavy quarkonium levels goes through quarkonium interaction with soft gluon fields which, in turn, emit the light mesons.  Since at large quark mass the heavy quarkonium is (at least formally) a compact object, its interaction with soft gluon field can be expanded in multipoles[@kg; @mv].  The leading term in the expansion is the $E1$ interaction with the chromo-electric component ${\bf E}^a$ of the gluonic field strength tensor.  The effective Hamiltonian for this interaction has the standard form H\_[E1]{}=-[1 2]{} g \^a ([**r**]{} \^a) , \[he1\] where $g$ is the QCD coupling, $\xi^a=t_1^a-t_2^a$ is the difference of the color generators acting on the heavy quark and antiquark respectively, and ${\bf r}={\bf r}_1-{\bf r}_2$ is the relative position of the quark and the antiquark. 

The emission of two pions in transitions between $S$ states proceeds in the second order in the $E1$ interaction.  Since the heavy quark spin is not involved in this interaction, and since it also factorizes in the quarkonium wave function in the non-relativistic limit, the amplitude of the two pion emission does not depend on the heavy quark spin and thus is the same for transitions between spin-triplet and spin-singlet states.  The expression for the amplitude can be written as[@vz; @vyz] A\_=| \_s ([**E**]{}\^a \^a) | 0 A\_0 , \[app\] where $A_0$ is the quarkonium transition matrix element A\_0=[1 24]{} 1S | \^a r\_i [G]{} r\_i \^a | 2S =[2 9]{} 1S | r\_i [G]{} r\_i | 2S  , \[a0\] with ${\tilde G}$ being the Green function for the quark-antiquark pair in the color octet state. 

The matrix element for the production of two pions by the gluonic operator $\alpha_s \, ({\bf E}^a \cdot {\bf E}^a)$ in eq.(\[app\]) has been understood[@vz; @ns] through its relation to the conformal anomaly in QCD and the current algebra.  Namely, one can write this operator as \_s ([**E**]{}\^a \^a) = -[\_s 4]{} ( F\_\^a )\^2 + [\_s 2]{} ( ([**E**]{}\^a)\^2 + ([**B**]{}\^a)\^2 )=[2 b]{} \_\^+ \_s \_[00]{}\^G , \[theta\] with $\theta_{\mu \nu}$ being the energy-momentum tensor in QCD and $\theta_{\mu \nu}^G$ its gluonic part.  The first term in the last expression arises from the trace anomaly, i.e. in the chiral limit one has $\theta_\mu^\mu=-(b \, \alpha_s/8 \pi) \, (F_{\mu \nu}^a)^2$, where $b=3$ is the coefficient in the QCD beta-function with three quarks. 

Finally, the matrix element of $\theta_\mu^\mu$ over the pions is governed by a current algebra low-energy theorem[@vz], which in the chiral limit yields, e.g. for the pair of charged pions \^+ \^- | \_\^ | 0 =q\^2 , \[let\] where $q=p_++p_-$ is the total 4-momentum of the pion pair.  Therefore the two pion transition amplitude (\[app\]) can be parameterized as A\_[\^+ \^-]{}=[2 \^2 b]{} (q\^2-C) A\_0 .  \[aparam\] The term $C$ stands here for the terms that are smaller than the anomaly contribution in the chiral limit, namely for the terms of order $m_\pi^2$ and those coming from $\alpha_s \, \theta_{00}^G$ in eq.(\[theta\]).  The latter contribution[@ns], formally, is not vanishing in the chiral limit and thus is not parametrically of order $m_\pi^2$. Neither it is a constant, independent of $q^2$. However, it is numerically small and is comparable to terms of order $m_\pi^2$ and also it varies over the physical region of $q^2$ sufficiently slowly, so that the effect of this variation can be neglected. 

The parameterization (\[aparam\]), first considered within the general framework of chiral symmetry in Refs.  [@mv0; @bc], with a small and constant $C$ exceptionally well describes the dipion invariant mass spectrum in the transitions from $\psi^\prime$ and $\Upsilon^\prime$, where no deviation from this parameterization has been observed so far (see e.g. a discussion in the review [@vyz]).[^2] Numerically the fit from the data gives $C=(4.6\pm
0.2) \, m_\pi^2$ for the $\psi^\prime$ decay, and $C=(3.3 \pm 0.2) \,
m_\pi^2$ for the $\Upsilon^\prime$ decay. ( It can be noted that a smaller value of $C$ in the latter decay than in the former had in fact been predicted[@ns] well before the data became available.) 

The absence of dependence of the amplitude in eq.(\[app\]) on the heavy quark spin is generally broken by the spin-spin and tensor interaction between the quark and antiquark, which result in terms of order $v^2/c^2$ in the non-relativistic expansion.  For the $2S$ and $1S$ states of charmonium such terms are generally estimated at the level of 15-20%.  Thus, up to such uncertainty one can consider the amplitude of the decay $\ecp \to \eta_c \, \pi \, \pi$ as being equal to that of the known decay $\psi^\prime \to J/\psi \, \pi \, \pi$, and both amplitudes are parameterized by the linear in $q^2$ expression (\[aparam\]).  One can of course notice that due to larger mass difference between $\ecp$ and $\eta_c$ than between $\psi^\prime$ and $J/\psi$ the physical region in the transition from $\ecp$ extends to somewhat larger values of $q^2$. If the linear in $q^2$ growth of the amplitude in eq.(\[aparam\]) extends over those extra $75-95 \, MeV$ of the dipion invariant mass, the phase space integral for the total decay rate is dramatically enhanced.  Numerically, under this assumption and using $C=4.6 \, m_\pi^2$ for both decays one finds after the phase space integration that = 3.5 0.5 , \[ratio\] where the uncertainty corresponds to the experimental error $\pm 10 \,
MeV$ in the mass of $\ecp$.

The assumption that the linear in $q^2$ behavior of the amplitude persists sufficiently beyond the physical region of the decay $\psi^\prime \to J/\psi \, \pi \, \pi$ does not look unreasonable, given the very high degree of linearity observed in the $\psi^\prime$ and $\Upsilon^\prime$ decays (see a discussion in the review [@vyz]).  In any case, this issue would be an interesting point to study once the discussed decay of $\ecp$ is observed experimentally. 

An estimate of the relative significance of the decay $\ecp \to \eta_c
\, \pi \, \pi$ i.e. of its branching ratio is somewhat less certain because of difficulty of a reliable estimate of the total width of $\ecp$. Given that the width of the radiative decay $\ecp \to h_c \,
\gamma$ is rather small, and is estimated to be approximately $40-50 \,
KeV$[@elq], the total width is dominated by the gluonic annihilation rate, which can be estimated as $\Gamma(\ecp)/\Gamma(\eta_c) \approx
\Gamma(\psi^\prime \to e^+e^-)/\Gamma(J/\psi \to e^+e^-) \approx 0.4$. However, as already mentioned, the total width of $\eta_c$ is known quite poorly.  Under these circumstances, using $4-5 \, MeV$ as a representative value for $\Gamma(\ecp)$, one can expect that the discussed transition with two charged pions should have the branching ratio close to the range of 5-10% and thus should be the exclusive decay channel with the largest branching ratio. 

This work is supported in part by the DOE grant DE-FG02-94ER40823. 

[99]{} S.-K. Choi [*et.al*]{} (Belle Coll.)  hep-ex/0206002, May 2002.  E.J. Eichten, K.Lane, and C. Quigg, Report FERMILAB-PUB-02/104-T, June 2002; \[hep-ph/0206018\] S. Godfrey and J.L. Rosner, Enrico Fermi Inst.  report EFI 02-79, May 2002; \[hep-ph/0205255\] D.E. Groom [*et.al.*]{} ( Particle Data Group), Eur.  Phys.  J., [**C15**]{} (2000) 1.  G. Brandenburg [*et.al.*]{} ( CLEO Coll.),  Phys. Rev.Lett., [** 85**]{} (2000) 3095.  L.B. Okun and A.Yu.  Khodjamirian, Pisma ZhETF, [**23**]{} (1976) 46.  K. Gottfried, .  M.B. Voloshin, .  M.B. Voloshin and V.I. Zakharov, .  V.A. Novikov and M.A. Shifman, .  M.B. Voloshin, Pisma ZhETF, [**21**]{} (1975) 733; \[ JETP Lett., [** 21**]{} (1975) 347\].  L.S. Brown and R.N. Cahn, .  M.B. Voloshin and Yu. M. Zaitsev, .  S. Chakravarty, S.M. Kim, and P. Ko, . 

[^1]: The actual range of uncertainty is somewhat difficult to estimate, since it includes both the experimental errors in $\Gamma(\psi^\prime \to J/\psi \, \pi \, \pi)$ and in the mass of $\ecp$, as well as the theoretical uncertainty of the $v^2/c^2$ terms in the non-relativistic expansion.  Given the number of contributing factors, one should rather treat the quoted error bars as an estimate of “$1 \, \sigma$". 

[^2]: This parameterization fails to correctly describe the observed behavior in the decay $\Upsilon(3S) \to \Upsilon \, \pi \, \pi$, which can be caused by several reasons, generally not directly related to the transitions between $2S$ and $1S$ states.  For a discussion see Ref.  [@vyz] and also [@ckk].