Low lying charmonium states at the physical point
Daniel Mohler, Carleton DeTar, Andreas S. Kronfeld, Song-haeng Lee, Ludmila Levkova, J. N. Simone
LLow lying charmonium states at the physical point
Daniel Mohler ∗ Fermi National Accelerator Laboratory, Batavia, Illinois 60510-5011, USAE-mail: [email protected]
Andreas S. Kronfeld
Fermi National Accelerator Laboratory, Batavia, Illinois 60510-5011, USAInstitute for Advanced Study, Technische Universität München, Garching, GermanyE-mail: [email protected]
J. N. Simone
Fermi National Accelerator Laboratory, Batavia, Illinois 60510-5011, USAE-mail: [email protected]
Carleton DeTar
Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah, USAE-mail: [email protected]
Song-haeng Lee
Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah, USAE-mail: [email protected]
Ludmila Levkova
Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah, USAE-mail: [email protected] (For the Fermilab Lattice and MILC Collaborations)
We present results for the mass splittings of low-lying charmonium states from a calculation withWilson clover valence quarks with the Fermilab interpretation on an asqtad sea. We use five latticespacings and two values of the light sea quark mass to extrapolate our results to the physical point.Sources of systematic uncertainty in our calculation are discussed and we compare our results forthe 1S hyperfine splitting, the 1P-1S splitting and the P-wave spin orbit and tensor splittings toexperiment.
The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] D ec ow lying charmonia Daniel Mohler
1. Introduction
The spectrum of low-lying charmonium states is well determined in experiment [1] and wellunderstood theoretically from potential models. Table 1 lists the 1 S and 1 P states along withtheir masses and widths, as determined from experiment [1]. For all of these states, the massesare determined with a high precision and most states are quite narrow. Lattice QCD calculationsof the low-lying spectrum of charmonium states are therefore an ideal benchmark for the heavy-quark methods used in state-of-the-art simulations. In particular, spin-dependent mass splittingsare extremely sensitive to the charm-quark mass and heavy quark discretization effects. In theseproceedings we report preliminary results on the hyperfine splittings between the triplet and singletstates ∆ M HF = M n L − M n L , on the 1P-1S-splitting ∆ M = M − M , (1.1) M = ( M χ c + M χ c + M χ c ) / , (1.2) M = ( M η c + M J / ψ ) / , (1.3)and on the spin-orbit and tensor splitting among the P-wave states ∆ M Spin − Orbit = ( M χ c − M χ c − M χ c ) / , (1.4) ∆ M Tensor = ( M χ c − M χ c − M χ c ) / . (1.5)This follows the previous efforts of our collaboration [2] and these results supersede previouspreliminary results reported in [3].
2. Methodology
We use the 2+1 flavor gauge configurations generated by the MILC collaboration with theasqtad fermion action [4]. The relevant ensembles are listed in Table 2. The use of 5 differentlattice spacings and two different light sea-quark masses enables us to perform a controlled chiral-continuum extrapolation. Four source time slices per gauge configuration are used, for a total ofmeson mass [MeV] width η c J / ψ χ c χ c χ c h c Table 1:
Mass and width of the 1S and 1P low-lying charmonium states [1]. ow lying charmonia Daniel Mohler ≈ a [fm] m l / m h size κ c κ sim ×
48 2524 0.12237(26)(20) 0.12210.14 0.1 20 ×
48 2416 0.12231(26)(20) 0.12210.114 0.2 20 ×
64 4800 0.12423(15)(16) 0.124230.114 0.1 24 ×
64 3328 0.12423(15)(16) 0.1220/0.1245/0.12800.082 0.2 28 ×
96 1904 0.12722(9)(14) 0.127220.082 0.1 40 ×
96 4060 0.12714(9)(14) 0.127140.058 0.2 48 ×
144 2604 0.12960(4)(11) 0.12980.058 0.1 64 ×
144 1984 0.12955(4)(11) 0.12960.043 0.2 64 ×
192 3204 0.130921(16)(70) 0.1310
Table 2:
MILC ensembles used in this study. In addition to the lattice parameters the number of sources usedin the calculation, the tuned charm-quark hopping parameter κ c and the hopping parameter of our simulation κ sim are given. m l / m h is the ratio of light- (up/down) to strange-quark mass used in the simulation. The firstuncertainty on κ c is statistical, the second is from the uncertainty in the lattice scale. ≈ ≈ κ c has been tuned by demanding that the D s kinetic mass is equal to the physical D s meson mass. The resulting κ c and the (sometimes slightly different) simulation value κ sim aregiven in Table 2.We calculate a matrix of correlators C ( t ) using quark-antiquark interpolators with the quantumnumbers of the states in question. Disconnected contributions from charm-quark annihilation areomitted when calculating the correlators. Our sources are stochastic wall sources with varioussmearings (for more details see [3]). We use the variational method [7 – 9], solving the generalizedeigenvalue problem C ( t ) (cid:126) ψ ( k ) = λ ( k ) ( t ) C ( t ) (cid:126) ψ ( k ) , (2.1) λ ( k ) ( t ) ∝ e − tE k (cid:0) + O (cid:0) e − t ∆ E k (cid:1)(cid:1) . (2.2)The ground state mass can be extracted from the large time behavior of the largest eigenvalue. Forthis we use (multi)exponential fits in the interval [ t min , t max ] . In our analysis the reference time t and t min are kept constant in fm, and t max is chosen such that the eigenvectors (cid:126) ψ ( k ) remain stable.The resulting data are corrected for mistuned charm-quark hopping parameter κ sim . To determinethe necessary correction we measure the κ c dependence of all observables on the ensemble with a = .
114 and m l / m h = .
1. For the 1S hyperfine splitting, autocorrelations in the Markov-chainof gauge configurations are significant and taken into account.
3. Chiral and continuum fits
We perform a combined extrapolation to the continuum values and to physical light- and3 ow lying charmonia
Daniel Mohler a [fm] ∆ E r mBmB’31S hyperfine and 1P tensor a [fm] -0.03-0.02-0.0100.010.020.030.040.050.06 ∆ E r m43w4mE2 1S1P splitting a [fm] -0.03-0.02-0.0100.01 ∆ E r mE2mEE31P spin-orbit Figure 1:
Shapes and size of the expected discretization uncertainties for charmonium splittings (NRQCDpower counting) in the Fermilab approach (using v = . mv ≈
420 MeV ≈ strange-quark masses. Our data indicates a clear sea-quark mass dependence for some of theobservables, which means that we also need to take into account the effect of mistuned strangesea-quark masses. For our combined chiral and continuum fit we use the Ansatz M = M + c ( x l + x h ) + c f ( a ) + c f ( a ) + . . . x l = m ud , sea − m ud , phys m s , phys (3.1) x h = m s , sea − m s , phys m s , phys as our fit model. The functions f i are determined from mass mismatches within the Fermilabprescription [6]. For each observable we determine the most important mismatches arising at v and/or v in NRQCD power-counting. Figure 1 shows the expected discretization uncertaintiesfrom power counting estimates for the splitting indicated in the respective figure. The plottedcurves corresponds to c i = , ∀ i . In some of our fits, we use Bayesian priors centered around 0with a width of 2 as a constraint. In the fit for the 1P-1S-splitting, we also allow for a term fromrotational symmetry breaking ( w term).
4. Preliminary results
For each observable we compare continuum extrapolations with just the leading shape andusing both the leading and subleading shapes. Figure 2 shows the results for the 1S hyperfine split-ting. Including subleading discretization effects significantly enlarges the resulting uncertainty.Notice that significant contributions from charm-annihilation diagrams to this observable are ex-pected [10].Figure 3 shows the 1S-1P splitting. As in the 1S hyperfine splitting, significant effects frommistuned strange-quark masses are visible in our data. The chiral-continuum fits are stable withregard to the number of shapes, provided reasonable priors are used.The P-wave spin-orbit splitting shown in Figure 4 shows small discretization uncertainties,unlike our results for the P-wave tensor splitting (Figure 5) where the dominant uncertainty arisesfrom the choice of fit model.We also show results for the 1P hyperfine splitting which is expected to be very small andwhere experiments measure a value compatible with zero.4 ow lying charmonia
Daniel Mohler a [fm] ∆ M H F r lattice data 0.2 m h lattice data 0.1 m h fit results at lattice parametersleading + subleading shapes χ /d.o.f. aug = 2.84/7 PRELIMINARY a [fm] ∆ M H F r lattice data 0.2 m h lattice data 0.1 m h fit results at lattice parametersleading shape χ /d.o.f. = 4.46/6 PRELIMINARY
Figure 2:
Chiral and continuum fit for the 1S hyperfine splitting using leading and subleading shapes (left)and only the leading shape (right) in the continuum extrapolation. Curves for physical (black), 0 . m s , and0 . m s light-quark masses are plotted. The black crosses show the fit results evaluated at the lattice parametersof the gauge ensemble. a [fm] ( M P - M S ) r lattice data 0.2m h lattice data 0.1m h fit result at lattice parameters leading + subleading shapes χ /d.o.f. aug = 9.79/7 PRELIMINARY a [fm] ( M P - M S ) r lattice data 0.1m h lattice data 0.2m h fit result at lattice parameters leading shape χ /d.o.f. = 7.51/6 PRELIMINARY
Figure 3:
Chiral and continuum fit for the 1P-1S-splitting. For an explanation see caption of Figure 2. a [fm] ∆ M S O r lattice data 0.2 m h lattice data 0.1 m h fit result at lattice parameters leading + subleading shapes χ /d.o.f. aug = 5.79/7 PRELIMINARY a [fm] ∆ M S O r lattice data 0.2 m h lattice data 0.1 m h fit result at lattice parameters leading shape χ /d.o.f. = 5.74/6 PRELIMINARY
Figure 4:
Chiral and continuum fit for the 1P spin-orbit splitting. For an explanation see caption of Figure2. ow lying charmonia Daniel Mohler a [fm] ∆ M T e r lattice data 0.2 m h lattice data 0.1 m h fit result at lattice parameters leading + subleading shapes χ /d.o.f. aug = 9.07/7 PRELIMINARY a [fm] ∆ M T e r lattice data 0.2 m h lattice data 0.1 m h fit result at lattice parameters leading shape χ /d.o.f. = 9.13/6 PRELIMINARY
Figure 5:
Chiral and continuum fit for the 1P tensor splitting. For an explanation see caption of Figure 2. a[fm] -0.02-0.015-0.01-0.00500.0050.010.0150.02 ( M P - M h c )r leading + subleading shapes χ /d.o.f. aug = 4.28/7 PRELIMINARY
Figure 6:
Chiral and continuum fit for the P-wave hyperfine splitting. For an explanation see caption ofFigure 2.
5. Conclusions and Outlook
We have presented preliminary results for the splittings of low-lying charmonium states. Table3 shows our current estimates compared to the experimental values. Our current treatment includesstatistical uncertainties as well as uncertainties from the chiral and continuum extrapolations. Atthis stage uncertainties from our scale-setting procedure are not included, and they will be sig-nificant for the 1S hyperfine and 1P-1S splittings. Further uncertainties, for example from finitevolume effects and from the small shift employed when translating some results to tuned κ c areexpected to be negligible. With the exception of the 1P hyperfine splitting our preliminary resultsshow excellent agreement with experiment. For the 1S hyperfine splitting the uncertainty in thelattice determination is dominated by the poor knowledge of disconnected contributions. Acknowledgments
Computation for this work was done at the Argonne Leadership Computing Facility (ALCF),Bluewaters at the National Center for Supercomputing Applications (NCSA), the National Energy6 ow lying charmonia
Daniel Mohler
Mass difference This analysis [MeV] Experiment [MeV]1P-1S splitting 457 . ± . . ± .
31S hyperfine 118 . ± . − . − . . ± .
71P spin-orbit 49 . ± . . ± .
11P tensor 17 . ± . . ± . − . ± . − . ± . Table 3:
Charmonium mass splittings compared to the experimental values. All numbers are preliminaryand the quoted uncertainties include statistics, chiral and continuum extrapolations only. In particular thescale setting uncertainty remains to be included. The second uncertainty on the 1S hyperfine splitting isbest-estimate for disconnected contributions [10].
Resources Supercomputing Center (NERSC), the National Institute for Computational Sciences(NICS), the Texas Advanced Computing Center (TACC), and the USQCD facilities at Fermilab,under grants from the NSF and DOE. C.D., S.-H.L., and L.L. are supported by the U.S. NationalScience Foundation under grants NSF PHY10-034278 and PHY0903571, and the U.S. Departmentof Energy under grant DE-FC02-12ER41879. A.S.K. acknowledges support by the German Ex-cellence Initiative and the European Union Seventh Framework Programme under grant agreementNo. 291763 as well as the European Union’s Marie Curie COFUND program. Fermilab is operatedby Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the US DOE.
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