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

The European Physical Journal C December 2018, 78:1018 | Cite as Some hadronic parameters of charmonia in \(\varvec{N_{\text {f}}=2}\) lattice QCD AuthorsAuthors and affiliations Gabriela BailasBenoît BlossierVincent Morénas Open Access Regular Article - Theoretical Physics First Online: 15 December 2018 50 Downloads 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 \(f_{\psi (2S)}/f_{J/\psi }\). This work is an intermediary step before going to the bottom sector, the final target of our program, because it is more promising for the phenomenology of extended Higgs sectors. An extensive study of the spectoscopy of charmonia has been done in [4, 5] while only two lattice estimates of \(f_{\eta _c}\) and \(f_{J/\psi }\) are available so far at \(\mathrm{N}_\mathrm{f}=2\) [6] and \(\mathrm{N}_\mathrm{f}=2+1\) [7, 8]. Table 1 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 The two-point correlation functions under investigation read $$\begin{aligned} C (...truncated)