Relativistic correction to the static potential at O(1/m)

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1 Relativistic correction to the static potential at O(1/m) Miho Koma (Inst. f. Kernphysik, Mainz Univ.) Yoshiaki Koma, Hartmut Wittig Lattice 2007, Regensburg, 30 July 2007 We investigate the relativistic correction to the QCD static potential at O(1/m) with the multi-level algorithm. We find that the correction is long ranged, raises linearly at r fm. [Y.Koma, M.Koma & H. Wittig, Phys.Rev.Lett.97( 06)122003] + α

2 INTRODUCTION Heavy quarkonia ( QQ) is one of the hot topics in high energy physics laboratory to study QCD both in perturbative and in nonperturbative region a lot of experiments are ongoing Theoretical study Effective field theories (EFTs) for heavy quarks Spectroscopy on lattice Phenomenological potential models EFTs for QQ non relativistic QCD (NRQCD), potential NRQCD (pnrqcd) key idea: assume hierarchy of energy scales m Λ QCD and v 1 (quark velocity) = m mv mv 2... integrate out the scales above m ( NRQCD) and mv ( pnrqcd)

3 EFT FOR HEAVY QUARKS pnrqcd [Brambilla,Pineda,Soto&Vairo( 99-), hep-ph/ ] potential picture appears Interquark potential V (r) is parametrized as inverse power of m Q, m Q V (r) = V (0) (r) + 1 V (1,0) (r) + 1 V (0,1) (r) + O(1/m 2 ) m Q m Q Once V (r) is obtained from QCD, one can compute full spectrum and wavefunctions decay, production, etc.. 2 Large predictive power! For low-lying Q Q states (m m mv mv 2 1 Λ QCD ) the potential can be computed High-lying states perturbatively For higher excited Q Q 0 states PT NPT (m m mv Λ QCD mv 2 Low-lying ) states the potential should be computed nonperturbatively r [fm] V (0) (r) [GeV]

4 OUR PROJECT Nonperturbative determination of the interquark potential including relativistic corrections (Previous lattice studies) V (0) (r): The static potential V (0) (r) can be computed from the expectation value of the Wilson loop. Many studies, the result is established, well parametrized by the Coulomb plus linear term V (1) (r): Nonperturbative expression is given by pnrqcd. [Brambilla,Pineda,Soto&Vairo( 01)] one study[koma,koma,wittig( 06)] = This talk V (2) (r): Contains spin-dependent and velocity-dependent corrections. some studies but large numerical error = Next talk

5 RELATIVISTIC CORRECTIONS V (1) Nonperturbative representation of V (1) from pnrqcd [Brambilla,Pineda,Soto&Vairo( 01)] V (1) (r) = 1 2 = 1 2 lim τ,t 0 τ 0 dτ τ dτ τ g E(0, 0) g E(0, τ) c W (r,t)/ W (r, t) +2 c / (sum over all possible leading fluctuation of quark propagation) = 1 0(r) ge(0) n(r) 2 (Spectral representation) 2 ( E n=1 n0 (r)) 2 where ˆT n(r) = e ae n(r) n(r), E n0 (r) = E n (r) E 0 (r), E 0 (r) = V (0) 0 (r)

6 RELATIVISTIC CORRECTIONS V (1) Nonperturbative representation of V (1) from pnrqcd [Brambilla,Pineda,Soto&Vairo( 01)] V (1) (r) = 1 2 = 1 2 lim τ,t 0 τ 0 dτ τ dτ τ g E(0, 0) g E(0, τ) c W (r,t)/ W (r, t) +2 c / Compute field strength correlators (FSCs) on Wilson loop and perform integral then extrapolate to τ, T... Too naive! Statistical error from the ratio of large Wilson loops Systematic errors from numerical integral and extrapolation

7 RELATIVISTIC CORRECTIONS NEW STRATEGY Spectral representation of V (1) from pnrqcd V (1) (r) = 1 2 0(r) g E(0) n(r) 2 ( E n0 (r)) 2 n=1 Field strength correlators (FSCs) on Polyakov line correlation function (PLCF) E x (0, 0) E x (0, τ) c = c /

8 RELATIVISTIC CORRECTIONS NEW STRATEGY Spectral representation of V (1) from pnrqcd V (1) (r) = 1 2 0(r) g E(0) n(r) 2 ( E n0 (r)) 2 n=1 Field strength correlators (FSCs) on Polyakov line correlation function (PLCF) E x (0, 0) E x (0, τ) c = (2 0(r) E x (0) n(r) 2 e E n0(r) T2 cosh ( E n0 (r)( T2 )) τ) + O(e E10(r)T ) n>0 Finite temporal size T and the periodic boundary condition are taken into account Error term is small Extract the matrix element and the energy gap from the fit. (No numerical integration, No extrapolation)

9 RELATIVISTIC CORRECTIONS NEW STRATEGY Spectral representation of V (1) from pnrqcd V (1) (r) = 1 2 0(r) g E(0) n(r) 2 ( E n0 (r)) 2 n=1 Field strength correlators (FSCs) on Polyakov line correlation function (PLCF) E x (0, 0) E x (0, τ) c = (2 0(r) E x (0) n(r) 2 e E n0(r) T2 cosh ( E n0 (r)( T2 )) τ) + O(e E10(r)T ) n>0 Finite temporal size T and the periodic boundary condition are taken into account Error term is small Extract the matrix element and the energy gap from the fit. (No numerical integration, No extrapolation) Measurement of FSC on PLCF is more difficult than that on Wilson loop

10 MULTILEVEL ALGORITHM Modified version for PLCF with two field strength operators (1) Compute the component of the Polyakov loops with the field strength insertion in each time slice (2) Compute sublattice correlators (3) Take average of sublattice correlators through internal update (iupd) (Large memory is required) (4) Construct correlation functions from sublattice correlators. (Average over all spatial points, all possible combinations of two fields insertion for given τ) = Measurement from 1 conf. FSC can be measured with high accuracy through the product of stabilized sublattice correlators (How good does this method work? Next talk) N4 E E r1 r r2

11 SIMULATION DETAILS Setting of the simulation Wilson gauge action, 1 Heatbath + 5 Overrelaxation β = 6/g 2 a Volume N tsl N iupd fm fm NEC SX8@RCNP Osaka University N conf Electric field strength operator: ga2 2 F 4i (U 4i U 4i )/(2i) (traceless, with two-leaf-type modification) Huntley-Michael (HM) factor [Huntley & Michael 87]: Z Fµν = P P / Re U µν P P (cancel most of self energies in FSCs at O(g 2 )) How to optimize the parameters for the multi-level algorithm N tsl : dependent on β, N tsl a = 0.3 fm N iupd : dependent on the magnitude of the signal, fixed from the iupd history of PLCF

12 STATIC POTENTIAL & FORCE V (0) (r I ) = 1 T ln P (0)P (r) + O(e ( E10)T ) V (0) ( r) = 1 a {V (0) (r) V (0) (r a)} V (0) (r) [ GeV ] β = 6.0 β = 5.85 V (0) '(r) [ GeV 2 ] β = 6.0 β = r [fm] r [ fm ]

13 STATIC POTENTIAL & FORCE V (0) (r I ) = 1 T ln P (0)P (r) + O(e ( E10)T ) V (0) ( r) = 1 a {V (0) (r) V (0) (r a)} V (0) (r) [ GeV ] β = 6.0 β = 5.85 V (0) '(r) [ GeV 2 ] β = 6.0 β = r [fm] r [ fm ] Fit to V (0) (r) = σ + c r 2 (β=5.85) σa2 2 = 802(2), c = 0.306(4) σ=1.05[gev/fm]

14 V (1) (R) V (1) (r) [ GeV 2 ] β = 6.0 β = 5.85 Normalized at r = 0.5 fm. Good scaling behavior Clear long range contribution r [ fm ]

15 V (1) (R) V (1) (r) [ GeV 2 ] r [ fm ] β = 6.0 β = Normalized at r = 0.5 fm. Good scaling behavior Clear long range contribution Preliminary fit result (β=5.85) V (1) (r) = A r + Br + C 2 A = 45(5), B=94(7) [GeV 2 /fm] cf. perturbation theory 1/r 2 [Melnikov etal( 98), Hoang( 99), Brambilla etal( 01)] Estimate of the correction For charm δσ = 2 94 = 0.144, δσ 1.3 = σ For bottom δσ = 2 94 = 40, δσ 4.7 = 38 σ

16 SUMMARY We have investigated the relativistic corrections to the heavy quark potential at O(1/m) We have developed a new simulation procedure which is suitable for the computation of the relativistic corrections Current observation... Measured at r 0.9 fm with high accuracy. Linearly raising behavior at r fm. A few to 15 percent correction to the string tension of the static potential. = mass dependence of the potential. The correction at O(1/m 2 ):... to be continued on the next talk

17 V (1) (R) [PRL] fit result V (r) = V (0) (r) + 2 m V (1) (r) + O( 1 m ) 2 V (0) fit (r) = c + σr + µ c = 97(1) r V (1) fit (r) = c r + µ ac = 81(4), a 2 µ = 17(1) V (1) For m c = 1.3 GeV 2c /m c = 6(1) For m b = 4.7 GeV 2c /m c = 73(4) cf. perturbation theory 1/r 2 [Melnikov etal( 98), Hoang( 99), Brambilla etal( 01)] a 2 V (1) (r) c' / r c'' / r r / a [Koma,Koma&Wittig,PRL97( 06)] 6 8