# Some hadronic parameters of charmonia in \(\varvec{N_{\text {f}}=2}\) lattice QCD

## Abstract

The phenomenology of leptonic decays of quarkonia holds many interesting features: for instance, it can establish constraints on scenarios beyond the Standard Model with the Higgs sector enriched by a light CP-odd state. In the following paper, we report on a two-flavor lattice QCD study of the \(\eta _c\) and \(J/\psi \) decay constants, \(f_{\eta _c}=387(3)(3)\, {\mathrm{MeV}}\) and \(f_{J/\psi }=399(4)(2)\, {\mathrm{MeV}}\). We also examine some properties of the first radial excitation \(\eta _c(2S)\) and \(\psi (2S)\).

## 1 Introduction

The discovery at LHC of the Higgs boson with a mass of 125.09(24) GeV [1] has been a major milestone in the history of Standard Model (SM) tests: the spontaneous breaking of electroweak symmetry generates masses of charged leptons, quarks and weak bosons. A well-known issue with the SM Higgs is that the quartic term in the Higgs Lagrangian induces for the Higgs mass \(m_H\) a quadratic divergence with the hard scale of the theory: it is related to the so-called hierarchy problem. Several scenarios beyond the SM are proposed to fix that theoretical caveat. Minimal extensions of the Higgs sector contain two complex scalar isodoublets \(\Phi _{1,2}\) which, after the spontaneous breaking of the electroweak symmetry, lead to 2 charged particles \(H^\pm \), 2 CP-even particles *h* (SM-like Higgs) and *H*, and 1 CP-odd particle *A*. In that class of scenarios, quarks are coupled to the CP-odd Higgs through a pseudoscalar current. Those extensions of the Higgs sector have interesting phenomenological implications, especially as far as pseudoscalar quarkonia are concerned. For example, their leptonic decay is highly suppressed in the SM because it occurs *via* quantum loops but it can be reinforced by the new tree-level contribution involving the CP-odd Higgs boson, in particular in the region of parameter space where the new boson is light (\(10\, {\mathrm{GeV}} \lesssim m_A \lesssim 100\, {\mathrm{GeV}}\)) and where the ratio of vacuum expectation values \(\tan \beta \) is small (\(\tan \beta <10\)) [2, 3]. Any enhanced observation with respect to the SM expectation would be indeed a clear signal of New Physics. Let us finally note that the hadronic inputs, which constrain the CP-odd Higgs coupling to heavy quarks through processes involving quarkonia, are the decay constants \(f_{\eta _c}\) and \(f_{\eta _b}\).

This paper reports an estimate of hadronic parameters in the charmonia sector using lattice QCD with \(N_f=2\) dynamical quarks: namely, the pseudoscalar decay constant \(f_{\eta _c}\), because of its phenomenological importance, but also the vector decay constant \(f_{J/\psi }\) as well as the ratio of masses \(m_{\eta _c(2S)}/m_{\eta _c}\) and \(m_{\psi (2S)}/m_{J/\psi }\). The two latter quantities are very well measured by experiments and their estimation has helped us to understand how much our analysis method can address the systematic effects, on quarkonia physics, coming from the lattice ensembles we have considered. We present also our findings for the following ratios of decay constants: \(f_{\eta _c(2S)}/f_{\eta _c}\) and \(f_{\psi (2S)}/f_{J/\psi }\).

Parameters of the simulations: bare coupling \(\beta = 6/g_0^2\), lattice resolution, hopping parameter \(\kappa \), lattice spacing *a* in physical units, pion mass, number of gauge configurations and bare charm quark masses

id | \(\quad \beta \quad \) | \((L/a)^3\times (T/a)\) | \(\kappa _{\mathrm{sea}}\) | \(a~(\mathrm fm)\) | \(m_{\pi }~({\mathrm{MeV}})\) | \(Lm_{\pi }\) | \(\#\) cfgs | \(\kappa _c\) |
---|---|---|---|---|---|---|---|---|

E5 | 5.3 | \(32^3\times 64\) | 0.13625 | 0.065 | 440 | 4.7 | 200 | 0.12724 |

F6 | \(48^3\times 96\) | 0.13635 | 310 | 5 | 120 | 0.12713 | ||

F7 | \(48^3\times 96\) | 0.13638 | 270 | 4.3 | 200 | 0.12713 | ||

G8 | \(64^3\times 128\) | 0.13642 | 190 | 4.1 | 176 | 0.12710 | ||

N6 | 5.5 | \(48^3\times 96\) | 0.13667 | 0.048 | 340 | 4 | 192 | 0.13026 |

O7 | \(64^3\times 128\) | 0.13671 | 270 | 4.2 | 160 | 0.13022 |

## 2 Lattice computation

### 2.1 Lattice set-up

This study has been performed using a subset of the CLS ensembles. These ensembles were generated with \(N_f=2\) nonperturbatively \(\mathcal {O}(a)\)-improved Wilson-Clover fermions [9, 10] and the plaquette gauge action [11] for gluon fields, by using either the DD-HMC algorithm [12, 13, 14, 15] or the MP-HMC algorithm [16]. We collect in Table 1 our simulation parameters. Two lattice spacings \(a_{\beta =5.5}=0.04831(38)\) fm and \(a_{\beta =5.3}=0.06531(60)\) fm, resulting from a fit in the chiral sector [17], are considered. We have taken simulations with pion masses in the range \([190\,, 440]~{\mathrm{MeV}}\). The charm quark mass has been tuned after a linear interpolation of \(m^2_{D_s}\) in \(1/\kappa _c\) at its physical value [18], after the fixing of the strange quark mass [19]. The statistical error on raw data is estimated from the jackknife procedure: two successive measurements are sufficiently separated in trajectories along the Monte-Carlo history to neglect autocorrelation effects. Moreover, statistical errors on quantities extrapolated at the physical point are computed as follows. Inspired by the bootstrap prescription, we perform a large set of \(N_{\mathrm{event}}\) fits of vectors of data whose dimension is the number of CLS ensembles used in our analysis (i.e.\(\ n=6\)) and where each component *i* of those vectors is filled with an element randomly chosen among the \(N_{\mathrm{bins}}(i)\) binned data per ensemble. The variance over the distribution of those \(N_{\mathrm{event}}\) fit results, obtained with such “random” inputs, is then an estimator of the final statistical error. Finally, we have computed quark propagators through two-point correlation functions using stochastic sources which are different from zero in a single timeslice that changes randomly for each measurement. We have also applied spin dilution and the one-end trick to reduce the stochastic noise [20, 21]. In our study we have neglected any contribution from disconnected diagrams.

#### 2.1.1 GEVP discussion

*V*is the spatial volume of the lattice, \(\langle \cdots \rangle \) is the expectation value over gauge configurations, and the interpolating fields \(\bar{c} \Gamma c\) can be non-local. As a preparatory work, different possibilities were explored to find the best basis of operators, combining levels of Gaussian smearing, interpolating fields with a covariant derivative \(\bar{c} \Gamma (\vec {\gamma }\cdot \vec {\nabla }) c\) and operators that are odd under time parity. Solving the Generalized Eigenvalue Problem (GEVP) [22, 23] is a key point in this analysis. Looking at the literature we have noticed that people tried to mix together the operators \(\bar{c} \Gamma c\) and \(\bar{c} \gamma _0 \Gamma c\) in a unique GEVP system [4, 6]. In our point of view, this approach raises some questions: let us take the example of the interpolating fields \(\{P=\bar{c} \gamma _5 c\,;\, A_0=\bar{c} \gamma _0 \gamma _5 c\}\). The asymptotic behaviours of the 2-pt correlation functions defined with these interpolating fields read

*i*,

*j*, we can write

*D*’s

*do depend*on

*i*and

*j*: the previous formula for \(\lambda _n(t,t_0)\) is then not correct. Hence, approximating every correlator by sums of exponentials forward in time, together with the assumption that the \(D_{ij}\) are independent of

*i*and

*j*, may face caveats. A toy model with 3 states in the spectrum helps to understand this issue:

#### 2.1.2 Interpolating field basis

*c*quark field reads

*c*quark field is a spinor of type

*u*while the \({\bar{c}}\) antiquark field is of type

*v*, so at this stage we need to express \({\bar{v}}\) in terms of

*u*. Defining the charge conjugation operator by \({{\mathscr {C}}}=-i\gamma ^0 \gamma ^2\) and using the Dirac representation for the \(\gamma \) matrices, we can write

*C*(

*t*) which is compatible with zero.

#### 2.1.3 Summary

*c*, including no smearing, to build \(4\times 4\) matrix of correlators without any covariant derivative nor operator of the \(\pi _2\) or \(\rho _2\) kind [24], from which we have also extracted the \(\mathcal{O}(a)\) improved hadronic quantities we have examined. Solving the GEVP for the pseudoscalar-pseudoscalar and vector-vector matrices of correlators

*n*th excited state as follows:

*L*refers to a local interpolating field while sums over

*i*and

*j*run over the four Gaussian smearing levels.

#### 2.1.4 Decay constant extraction

*R*superscript denotes the renormalized improved operators. The renormalization constants \(Z_A\) and \(Z_V\) have been non perturbatively measured in [26, 27].

^{1}We have also used non-perturbative results and perturbative formulae from [28, 29, 30] for the improvement coefficients \(c_A\), \(c_V\), \(b_A\) and \(b_V\), and the matching coefficient

*Z*between the quark mass \(m_{c}\) defined through the axial Ward Identity

### 2.2 Analysis

Let us first consider the mass and the decay constant of the ground states. Since the fluctuations on effective masses obtained from \(\tilde{C}^1_{A_0P}\) and \(C^1_{VV}\) are large with time, we have decided to use generalized eigenvectors *at fixed time* \(t_{\text {fix}}\), \(v^{P(V)}_ 1 (t_{\mathrm{fix}}, t_0)\), in order to perform the corresponding projections. In practice we have chosen \(t_{\text {fix}}/a = t_0/a + 1\) but we have checked that the results do not depend on \(t_{\text {fix}}\).

For the excited states, the formulae written in Sect. 2.1.3 do apply. Although the fluctuations on effective masses obtained from \(\tilde{C}^2_{A_0P}\) and \(\tilde{C}^2_{VV}\) are larger than for their ground states counterparts, the correlators \(\tilde{C}^2_{A_0P}\) and \(\tilde{C}^2_{VV}\) computed with a projection on \(v^{P(V)}_2(t_{\mathrm{fix}},t_0)\) show a bigger slope than the same correlators computed with a projection on \(v^{P(V)}_2(t,t_0)\). We interpret that observation as a stronger contamination of higher excited states for the \(t_\text {fix}\) projection which can spoil the determination of the \({\eta _c(2S)}\) and \({\psi (2S)}\) masses. Since the masses take part in the computation of the decay constants, such a contamination can propagate into the extraction of \(f_{\eta _c(2S)}\) and \(f_{\psi (2S)}\). Figures 2 and 3 illustrate that point with the correlators \(\tilde{C}^2_{A_0P}\) and \(\tilde{C}^2_{VV}\) on the ensemble F7.

For the ground states, the time range \([t_{\mathrm{min}},t_{\mathrm{max}}]\) used to fit the projected correlators is set so that the statistical error on the effective mass \(\delta m^{\mathrm{stat}}(t_{\mathrm{min}})\) is larger enough than the systematic error \(\Delta m^{\mathrm{sys}}(t_{\mathrm{min}})\equiv \exp [-\Delta t_{\mathrm{min}}]\) with \(\Delta = E_4 - E_1 \sim 2\,{\mathrm{GeV}}\). That guesstimate is based on our \(4\times 4\) GEVP analysis, though we do not want to claim here that we really control the energy-level of the third excited state. Actually our criterion is rather \(\delta m^{\mathrm{stat}}(t_{\mathrm{min}})> 4\Delta m^{\mathrm{sys}}(t_{\mathrm{min}})\) to be more conservative. On the other side, \(t_{\mathrm{max}}\) is set after a qualitative inspection of the data which count for the plateau determination. Since \(\Delta ^{(P)} \sim \Delta ^{(V)}\), fit intervals are identical for pseudoscalar and vector charmonia.

For the first excited states, the time range has been set by looking at effective masses and where the plateaus start and end, including statistical uncertainty. There, we have estimated the systematic error by shifting the fit range to larger times \([t_{\text {min}}, t_{\text {max}}] \rightarrow [t_{\text {min}}+2a,t_{\text {max}}+3a]\).

#### 2.2.1 Ground state results

LO and NLO fit parameters and results for \(f_{\eta _c}\) and \(f_{J/\psi }\), with their respective \(\chi ^2\) per degrees of freedom

Fit | \(X_0\hbox {[MeV]}\) | \(X_1\,\,\hbox {[MeV}^{-1}]\) | \(X_2\,\,\hbox {[MeV]}\) | \(X_3\,\,\hbox {[MeV}^{-1}]\) | \(\chi ^2/dof\) | \(f_{\eta _c}\) |
---|---|---|---|---|---|---|

LO | 386(3) | 0.00005(1) | \(-\)16.7(2.7) | – | 1.99 | 387(3) |

NLO | 391(6) | 0.0006(7) | \(-\)17.1(2.7) | 0.00006(6) | 2.62 | 389(4) |

Fit | \(X_0\,\,\hbox {[MeV]}\) | \(X_1\,\,\hbox {[MeV}^{-1}]\) | \(X_2\,\,\hbox {[MeV]}\) | \(X_3\,\,\hbox {[MeV}^{-1}]\) | \(\chi ^2/dof\) | \(f_{J/\psi }\) |
---|---|---|---|---|---|---|

LO | 397(4) | 0.00010(2) | 37.2(3.8) | – | 0.98 | 399(4) |

NLL | 401(7) | \(-\) 0.0003(9) | 36.4(4.2) | 0.00003(7) | 1.41 | 401(8) |

#### 2.2.2 First radial excitation results

^{2}In addition, a lattice study, performed in the framework of NRQCD, gives \(f_{\Upsilon '}/f_{\Upsilon }<1\) in the bottomium sector [39].

## 3 Conclusion

^{3}Unfortunately, our result for \(f_{\psi (2S)}/f_{J/\psi }>1\) makes the picture less bright, unless one admits that there are very large spin breaking effects. Further investigation to address this issue is undoubtedly required.

Masses and decays constants of \(\eta _c\), \(\eta _c(2S)\), \(J/\psi \) and \(\psi (2S)\), in lattice units, extracted on each CLS ensemble used in our analysis

id | \([t_{\mathrm{min}}, t_{\mathrm{max}}](P)\) | \(am_{\eta _c}\) | \(af_{\eta _c}\) | \([t_{\mathrm{min}}, t_{\mathrm{max}}](V)\) | \(am_{J/\psi }\) | \(af_{J/\psi }\) |
---|---|---|---|---|---|---|

E5 | [11–29] | 0.9836(3) | 0.1246(16) | [11-29] | 1.0202(7) | 0.1499(11) |

F6 | [11–46] | 0.9870(1) | 0.1236(5) | [11-46] | 1.0233(4) | 0.1471(9) |

F7 | [11–45] | 0.9855(1) | 0.1233(3) | [11-45] | 1.0209(3) | 0.1460(5) |

G8 | [12–55] | 0.9861(1) | 0.1231(3) | [12-55] | 1.0217(2) | 0.1454(5) |

N6 | [13–46] | 0.7284(3) | 0.0944(6) | [13-46] | 0.7547(6) | 0.1059(8) |

O7 | [16–55] | 0.7297(1) | 0.0927(3) | [16-55] | 0.7555(3) | 0.1037(4) |

id | \([t_{\mathrm{min}}, t_{\mathrm{max}}](P')\) | \(m_{\eta _c(2S)/m_{\eta _c}}\) | \(f_{\eta _c(2S)}/f_{\eta _c}\) | \([t_{\mathrm{min}}, t_{\mathrm{max}}](V')\) | \(m_{\psi (2S)}/m_{J/\psi }\) | \(f_{\psi (2S)}/f_{J/\psi }\) |
---|---|---|---|---|---|---|

E5 | [6–13] | 1.258(5)(5) | 0.67(10)(6) | [6Th-13] | 1.235(5)(5) | 0.99(6)(7) |

F6 | [6–13] | 1.257(3)(8) | 0.65(4)(5) | [6-13] | 1.233(4)(8) | 0.95(4)(3) |

F7 | [6–13] | 1.254(2)(14 | 0.67(2)(14) | [6-13] | 1.233(3)(19) | 0.98(2)(5) |

G8 | [8–15] | 1.234(12)(4) | 0.61(6)(2) | [8-15] | 1.212(18)(7) | 0.83(10)(8) |

N6 | [8–15] | 1.290(4)(10) | 0.75(6)(9) | [8-15] | 1.270(4)(10) | 1.09(7)(2) |

O7 | [8–15] | 1.257(4)(5) | 0.74(4)(5) | [8-15] | 1.236(5)(5) | 1.09(5)(3) |

Meanwhile, the next step into the measurement of \(f_{\eta _b}\), particularly relevant in models with a light CP-odd Higgs, is underway using step scaling in masses in order to extrapolate the results to the bottom region.

## Footnotes

- 1.
\(Z_A\) is the finite renormalisation constant of a flavour non-singlet axial bilinear of quarks: it is different from 1 because Wilson-Clover fermions break the chiral symmetry. In that work we consider flavour singlet operators: due to the chiral anomaly the renormalisation pattern is different. However the feature of flavour symmetry (singlet vs. non-singlet) is intimately related to the chiral symmetry. As the charm quark is not chiral at all, its sensitivity to the chiral anomaly is negligible. Hence it is acceptable to renormalise the operator \(\bar{c} \gamma _0 \gamma _5 c\) with the “flavour non-singlet” renormalisation constant \(Z_A\).

- 2.
A possible caveat with this approach is that QED effects might be quite large, making, as is done in our work, the encoding in \(f_{\psi (2S)}\) as purely QCD contributions not so straightforward.

- 3.
In the cases \(h \rightarrow \eta _c l^+ l^-\) and \(h \rightarrow \eta _c(2S) l^+ l^-\), the other hadronic quantities which enter the process are the distribution amplitudes of the charmonia.

## Notes

### Acknowledgements

This work was granted access to the HPC resources of CINES and IDRIS under the allocations 2016-x2016056808 and 2017-A0010506808 made by GENCI. It is supported by Agence Nationale de la Recherche under the contract ANR-17-CE31-0019. Authors are grateful to Damir Becirevic and Olivier Pène for useful discussions and the colleagues of the CLS effort for having provided the gauge ensembles used in that work.

## References

- 1.G. Aad et al., [ATLAS and CMS Collaborations], Phys. Rev. Lett.
**114**, 191803 (2015). arXiv:1503.07589 [hep-ex] - 2.E. Fullana, M.A. Sanchis-Lozano, Phys. Lett. B
**653**, 67 (2007). arXiv:hep-ph/0702190 ADSCrossRefGoogle Scholar - 3.D. Becirevic, B. Melic, M. Patra, O. Sumensari. arXiv:1705.01112 [hep-ph]
- 4.L. Liu et al., [Hadron Spectrum Collaboration], JHEP
**1207**, 126 (2012). arXiv:1204.5425 [hep-ph] - 5.C.B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, JHEP
**1509**, 089 (2015). arXiv:1503.05363 [hep-lat]ADSCrossRefGoogle Scholar - 6.D. Becirevic, G. Duplancic, B. Klajn, B. Melic, F. Sanfilippo, Nucl. Phys. B
**883**, 306 (2014). arXiv:1312.2858 [hep-ph]ADSCrossRefGoogle Scholar - 7.C.T.H. Davies, C. McNeile, E. Follana, G.P. Lepage, H. Na, J. Shigemitsu, Phys. Rev. D
**82**, 114504 (2010). arXiv:1008.4018 [hep-lat]ADSCrossRefGoogle Scholar - 8.G.C. Donald, C.T.H. Davies, R.J. Dowdall, E. Follana, K. Hornbostel, J. Koponen, G.P. Lepage, C. McNeile, Phys. Rev. D
**86**, 094501 (2012). arXiv:1208.2855 [hep-lat]ADSCrossRefGoogle Scholar - 9.B. Sheikholeslami, R. Wohlert, Nucl. Phys. B
**259**, 572 (1985)ADSCrossRefGoogle Scholar - 10.M. Lüscher, S. Sint, R. Sommer, P. Weisz, U. Wolff, Nucl. Phys. B
**491**, 323 (1997). arXiv:hep-lat/9609035 ADSCrossRefGoogle Scholar - 11.K.G. Wilson, Phys. Rev. D
**10**, 2445 (1974)ADSCrossRefGoogle Scholar - 12.
- 13.
- 14.
- 15.M. Lüscher, DD-HMC algorithm for two-flavour lattice QCD. http://luscher.web.cern.ch/luscher/DD-HMC/index.html
- 16.
- 17.S. Lottini [ALPHA Collaboration], PoS LATTICE
**2013**, 315 (2014). arXiv:1311.3081 [hep-lat] - 18.J. Heitger, G.M. von Hippel, S. Schaefer, F. Virotta, PoS LATTICE
**2013**, 475 (2014). arXiv:1312.7693 [hep-lat]; J. Heitger, private communicationGoogle Scholar - 19.P. Fritzsch, F. Knechtli, B. Leder, M. Marinkovic, S. Schaefer, R. Sommer, F. Virotta, Nucl. Phys. B
**865**, 397 (2012). arXiv:1205.5380 [hep-lat]ADSCrossRefGoogle Scholar - 20.M. Foster, UKQCD Collaboration. Phys. Rev. D
**59**, 094509 (1999). arXiv:hep-lat/9811010 ADSCrossRefGoogle Scholar - 21.C. McNeile, UKQCD Collaboration. Phys. Rev. D
**73**, 074506 (2006). arXiv:hep-lat/0603007 ADSCrossRefGoogle Scholar - 22.C. Michael, Nucl. Phys. B
**259**, 58 (1985)ADSMathSciNetCrossRefGoogle Scholar - 23.M. Lüscher, U. Wolff, Nucl. Phys. B
**339**, 222 (1990)ADSCrossRefGoogle Scholar - 24.J.J. Dudek, R.G. Edwards, M.J. Peardon, D.G. Richards, C.E. Thomas, Phys. Rev. D
**82**, 034508 (2010). arXiv:1004.4930 [hep-ph]ADSCrossRefGoogle Scholar - 25.D. Mohler, S. Prelovsek, R. M. Woloshyn, Phys. Rev. D
**87**(3), 034501 (2013). arXiv:1208.4059 [hep-lat] - 26.M. Della Morte, R. Sommer, S. Takeda, Phys. Lett. B
**672**, 407 (2009). arXiv:0807.1120 [hep-lat] - 27.M. Della Morte, R. Hoffmann, F. Knechtli, R. Sommer, U. Wolff, JHEP
**0507**, 007 (2005). arXiv:hep-lat/0505026 - 28.M. Della Morte, R. Hoffmann, R. Sommer, JHEP
**0503**, 029 (2005). arXiv:hep-lat/0503003 - 29.
- 30.M. Guagnelli et al., [ALPHA Collaboration], Nucl. Phys. B
**595**, 44 (2001). arXiv:hep-lat/0009021 - 31.A.A. Pivovarov, Phys. Atom. Nucl.
**65**, 1319 (2002)ADSCrossRefGoogle Scholar - 32.
- 33.D. Becirevic, M. Kruse, F. Sanfilippo, JHEP
**1505**, 014 (2015). arXiv:1411.6426 [hep-lat]ADSCrossRefGoogle Scholar - 34.C. Patrignani et al. [Particle Data Group], Chin. Phys. C
**40**(10), 100001 (2016)Google Scholar - 35.D. Becirevic, V. Lubicz, F. Sanfilippo, S. Simula, C. Tarantino, JHEP
**1202**, 042 (2012). arXiv:1201.4039 [hep-lat]ADSCrossRefGoogle Scholar - 36.B. Blossier, J. Heitger, M. Post. arXiv:1803.03065 [hep-lat]
- 37.G.C. Donald, C.T.H. Davies, J. Koponen, G.P. Lepage, Phys. Rev. Lett.
**112**, 212002 (2014). arXiv:1312.5264 [hep-lat]ADSCrossRefGoogle Scholar - 38.V. Lubicz et al. [ETM Collaboration], arXiv:1707.04529 [hep-lat]
- 39.B. Colquhoun, R. J. Dowdall, C. T. H. Davies, K. Hornbostel, G. P. Lepage, Phys. Rev. D
**91**(7), 074514 (2015). arXiv:1408.5768 [hep-lat]

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