Charm quark diffusion coefficient and relaxation time on the quenched lattice Atsuro Ikeda, Masayuki Asakawa, Masakiyo Kitazawa Osaka University XQCD2016
Anisotropic flow of open charm • Large elliptic flow of open charm → charm flow ~ medium flow • Rapid thermalization of charm quarks? • Diffusion coefficient is an important quantity
Transport coefficient on the lattice Shear viscosity Karsch and Wyld 1987, Nakamura and Sakai 2005, Meyer 07, Haas 2013, Borsanyi et al. 2014, etc … Electric conductivity Gupta 2004, Aarts et al. 2014, etc … Quark diffusion coefficient Ding et al. 2011, Banerjee et al. 2012, Aarts et al. 2015, Francis et al. 2015 etc… There is a numerical difficulty, called ill-posed problem, and analyses still have uncertainty. Ding et al. arXiv:1504.05274
Measurement of Diffusion coefficient 1. Ansatz for spectral function Depend on ansatz Kubo formula Lattice Euclidean correlator has a lattice artifact 𝐸 = 𝜌 1 𝜍 𝑗𝑗 𝜕, Ԧ 𝑞 χ lim 𝜕→0 lim 2. Maximum entropy method 3 𝜕 Ԧ 𝑞→0 Reconstructed spectral function has the 𝑦 𝑘 𝜈 𝜐, Ԧ † 0, 0 𝐹 𝜐, 𝑞 = න 𝑒 3 𝑦 𝑓 𝑗 Ԧ 𝑞 ⋅ Ԧ strong correlation in whole 𝜕 -space 𝐻 𝜈𝜈 𝑦 𝑘 𝜈 Not sensitive to low energy structure ∞ 𝑒𝜕 cosh (1/2𝑈 − 𝜐)𝜕 = න 𝜍 𝜈𝜈 𝜕, Ԧ 𝑞 Our strategy sinh 𝜕/2𝑈 0 3. Structure of 𝐻 00 (𝜐, 𝑞 2 ) (new) for 𝜈 = 0,1,2,3 𝜕 2 𝜍 00 𝜕, 𝑞 = 𝑞 𝑗 𝑞 𝑘 𝜍 𝑗𝑘 𝜕, 𝑞 = ill-posed problem 𝑞 2 𝜍 𝑀 (𝜕, 𝑞) High energy component of 𝜍 00 (𝜕, 𝑞) is suppressed by 1/𝜕 2 comparing with 𝜍 𝑗𝑗 𝜕, 𝑞
Linear response theory Consider the two relaxation process [Kadanoff and Martin 1963] Perturbative Hamiltonian compare Classical source ℎ(𝑠) 𝐼 𝑠, 𝑨 = 𝐼 0 𝑠 + 𝜀𝐼 𝑠, 𝑢 𝜀 𝑜 𝑠 = 𝜓ℎ(𝑠) 𝜀𝐼 𝑠, 𝑢 = 𝑓 𝜗𝑢 𝜄 −𝑢 ℎ(𝑠) Turn off suddenly 𝑢 𝑒𝑢′ 𝑜 𝑠, 𝑢 , 𝜀𝐼 𝑠 ′ , 𝑢 ′ 𝜀 𝑜 𝑠 = −𝑗 න 𝑓𝑟 −∞ Relaxation process 𝜖 2 𝜖𝑢 2 + 𝜖 𝜖𝑢 𝑘 0 𝑦, 𝑢 = −𝐸𝛼 2 𝑘 0 𝑦, 𝑢 𝜐 relax Response lag caused by heavy quark mass Low energy structure of the spectral function න hydro (𝜕, 𝑙) 𝜓𝐸|𝑙| 2 𝜍 00 = 1 2 𝜕 𝜌 𝜕 2 + 𝐸|𝑙| 2 − 𝜐𝜕 2 𝜕 2 𝜍 00 𝜕, 𝑞 = 𝑞 𝑗 𝑞 𝑘 𝜍 𝑗𝑘 𝜕, 𝑞 Kubo formula 𝜍 𝑗𝑗 𝜕, 𝑙 𝐸 = 𝜌 1 𝜓 lim 𝜕→0 lim 3 𝜕 𝑙→0
Assumptions Structure of the spectral function ℎ𝑧𝑒𝑠𝑝 𝜕, Ԧ ℎ𝑗ℎ 𝜕, Ԧ 𝜍 00 (𝜕, Ԧ 𝑞) = 𝜍 00 𝑞 + 𝜍 00 𝑞 hydro (𝜕, Ԧ 𝜓 𝑞 2 𝐸| Ԧ 𝑞| 2 𝜍 00 𝑞) = 1 ℎ𝑗ℎ 𝜕, Ԧ 𝜍 𝜈𝜈 𝑞 ≥ 0 𝜍 00 (𝜕) 𝜕 2 + 𝐸| Ԧ 𝑞| 2 − 𝜐𝜕 2 2 𝜕 𝜌 Scale high energy part separation hydro part 𝐸𝑞 2 2𝑛 𝑟 𝜕 Quark number susceptibility 2 𝜓 𝑞 2 = 𝜓 + 𝜓 ′ 𝑞 + ⋯ 𝑈 𝜓 ′ ≪ 𝜓 𝐻 00 𝜐, 0 = 𝜓𝑈
Mid-point expansion of 𝐻 00 𝜐, p 2 4 ∞ 𝑒𝜕 1 + 1 1 𝜍 00 𝜕, Ԧ 𝑞 1 𝑈 2 𝜕 2 𝐻 00 𝜐, Ԧ 𝑞 = න 2 − 𝑈𝜐 + 𝑃 2 − 𝑈𝜐 𝜕 2 sinh 0 2𝑈 2 ≡ 𝑁 0 𝑞 2 + 1 1 𝑁 2 𝑞 2 + ⋯ 2 − 𝑈𝜐 2 𝐻 00 (𝜐, p) around mid-point is the most 𝑁 0 𝑞 2 𝐻 00 𝜐𝑈, Ԧ 𝑞 sensitive to the low energy structure of the spectral function 𝜍 00 (𝜕, 𝑞) But 𝑁 0 0 = 𝑈𝜓 and 𝑁 2 0 = 0 M 2 𝑞 2 0.5 𝜐𝑈 𝜖𝑁 0 𝑞 2 𝜖𝑁 2 𝑞 2 Study 𝑏𝑜𝑒 at 𝑞 → 0 𝑞 2 𝑞 2 𝜖 𝜖 ℎ𝑗ℎ 𝑞 2 𝑁 𝑜 𝑞 2 = 𝑁 𝑜 𝑚𝑝𝑥 𝑞 2 + 𝑁 𝑜 𝑞 ≡ 𝑞/𝑈
𝜖𝑁 0 𝑞 2 /𝜖 ǁ 𝑞 2 𝑚𝑝𝑥 𝑞 2 𝜖𝑁 0 = ℎ 0 𝜐 relax 𝑈 𝜓𝐸𝑈 2 + 𝜓 ′ 𝑈 อ 𝑞 2 𝜖 𝑞 2 =0 − 𝑚𝑝2 𝜌 ∞ 𝜖 1 ℎ𝑧𝑒𝑠𝑝 𝜕, 𝑞 ℎ 0 𝜐 relax 𝑈 ≡ lim 𝜖 𝐸𝑞 2 න 𝑒𝜕 𝜍 00 𝜕 𝑞 2 →0 sinh 0 2𝑈 < 0 ℎ𝑗ℎ 𝑞 2 and ℎ𝑗ℎ 𝑞 2 > 0 𝑚𝑝𝑥 𝑞 2 + 𝑁 0 𝜖 with 𝑁 0 𝑞 2 = 𝑁 0 𝑞 2 𝑁 0 𝜖 𝑈 2 𝑁 0 𝑞 2 − 𝜓 ′ 1 𝜖 𝐸 𝑀 𝑈 ≡ อ < 𝐸𝑈 𝑞 2 𝑈 3 𝑈 2 ℎ 0 𝜐 relax 𝑈 𝜓 𝜖 𝑞 2 =0
𝜖𝑁 2 𝑞 2 /𝜖 ǁ 𝑞 2 𝑚𝑝𝑥 𝑞 2 𝜌 𝜖𝑁 2 න = ℎ 2 𝑈𝜐 relax 𝜓𝐸 อ 𝑞 2 𝜖 𝑞 2 =0 ∞ 𝑈 2 𝜖𝜕 2 𝜍 00 𝜕, 𝑞 ℎ 2 𝑈𝜐 relax ≡ lim 𝑞 2 →0 න 𝑒𝜕 𝜕 𝜖 𝐸𝑞 2 sinh 0 2𝑈 > 0 ℎ𝑗ℎ 𝑞 2 , ℎ𝑗ℎ 𝑞 2 > 0 with 𝑁 2 𝑞 2 = 𝑁 2 𝑚𝑝𝑥 𝑞 2 + 𝑁 2 𝜖 𝑞 2 𝑁 2 𝜖 𝑁 2 𝑞 2 1 𝜖 𝐸𝑈 < 𝐸 𝑉 𝑈 ≡ อ 𝑞 2 ℎ 2 (𝑈𝜐 𝑠𝑓𝑚𝑏𝑦 ) 𝜖 𝜓𝑈 𝑞 2 =0 𝐸 𝑀 𝑈 < 𝐸𝑈 < 𝐸 𝑉 𝑈 Opposite sign of h 0 < 0 and ℎ 2 > 0
Lattice set up Quenched lattice N τ T/T c 𝑂 𝜏 Δ p/T Nconf Wilson Fermion and standard 16 4.68 128 0.196 361 Wilson gauge action 20 3.74 128 0.245 229 𝛾 = 7.0, 𝛿 𝐺 = 3.476 24 3.12 128 0.294 240 [Asakawa, Hatsuda 2004] 28 2.67 128 0.344 91 Anisotropic lattice with 32 2.34 128 0.397 100 𝑏 𝜏 32 2.34 64 0.794 304 𝜊 = 𝑏 𝜐 = 4 and 𝑂 𝜏 = 128 36 2.08 128 0.442 100 for high momentum resolution 40 1.87 128 0.491 100 𝑀 𝜏 / 𝑀 𝜐 = 11.5~32 44 1.7 128 0.54 89 Blue Gene/Q@KEK Iroiro++
𝐹 (𝜐, 𝑞) 𝐻 00 𝑂 𝜐 = 24 1. Mid-point correlator at 𝑞 → 0 Momentum dependence of 2. Curvature
ǁ ǁ 𝑞 2 and 𝜖𝑁 2 (𝑞 2 )/𝜖 𝜖𝑁 0 (𝑞 2 )/𝜖 𝑞 2 𝜖𝑁 0 (𝑞 2 )/𝜖 𝑞 2 𝜖𝑁 2 (𝑞 2 )/𝜖 𝑞 2 intercept 𝑞 2 < 1 Fit with linear function where From 𝑂 𝜐 = 32 , finite volume dependence is well suppressed
Result: 𝐸 𝑀 (𝜐 𝑠𝑓𝑚𝑏𝑦 )𝑈 < 𝐸𝑈 < 𝐸 𝑉 (𝜐 𝑠𝑓𝑚𝑏𝑦 )𝑈 Ding et al. arXiv:1504.05274 Consistent with previous works High energy contribution become larger for lower T Information on 𝜐 𝑠𝑓𝑚𝑏𝑦 is needed to determine D
Constraint on 𝐸 and 𝜐 𝑠𝑓𝑚𝑏𝑦 𝑂 𝜐 = 20, 𝑈/𝑈 𝑑 = 3.74 𝐸 𝑉 (𝜐 𝑠𝑓𝑚𝑏𝑦 )𝑈 𝐸 𝑀 (𝜐 𝑠𝑓𝑚𝑏𝑦 )𝑈 𝐸 𝑀 𝑈 at 𝜐 𝑠𝑓𝑚𝑏𝑦 = 0 is still lower limit.
𝜐 𝑠𝑓𝑚𝑏𝑦 from Langevin dynamics 𝜐 𝑠𝑓𝑚𝑏𝑦 kinetic 𝑛 𝑑 on the lattice? 𝑈 = 𝑛 𝑑 𝐸 from Langevin dynamics or heavy quark limit [Petreczky, Teany 2006 Caron-Huot et al. 2009] 𝑂 𝜐 = 20, 𝑈/𝑈 𝑑 = 3.74 𝑈 = 𝑛 𝑑 𝜐/𝐸 𝑀 (𝜐 𝑠𝑓𝑚𝑏𝑦 ) 𝜐/𝐸 𝑉 (𝜐 𝑠𝑓𝑚𝑏𝑦 ) 𝜐 𝑠𝑓𝑚𝑏𝑦 𝐸 We need 𝜐 𝑠𝑓𝑚𝑏𝑦 or 𝜐 𝑠𝑓𝑚𝑏𝑦 /𝐸 on the lattice
Conclusion Constraint on 𝐸 and 𝜐 𝑠𝑓𝑚𝑏𝑦 in ( 𝐸, 𝜐 𝑠𝑓𝑚𝑏𝑦 ) -plane from the p-dependence 1 𝐹 of the mid-point correlator 𝐻 00 2𝑈 , 𝑞 on the lattice with basic assumptions for the spectral function 𝜍 00 (𝜕, Ԧ 𝑞) . 𝑞 2 and 𝜖𝑁 2 (𝑞 2 )/𝜖 𝑞 2 with good statistics. We obtain 𝜖𝑁 0 (𝑞 2 )/𝜖 𝑀 𝜏 Spatial volume dependence was well suppressed even with 𝑀 𝜐 = 8 . Future work 𝜖𝜓 𝑞 2 Can we measure 𝜓 ′ = ฬ on the lattice? 𝜖 𝑞 2 𝑞 2 =0 Other information on D and 𝜐 relax ? Estimate of high energy contribution of 𝜍 𝜈𝜈 𝜕, Ԧ 𝑞 (MEM, ansatz for spectral function): 𝐸 𝑀 (𝜐 𝑠𝑓𝑚𝑏𝑦 )𝑈 < 𝐸𝑈 < 𝐸 𝑉 (𝜐 𝑠𝑓𝑚𝑏𝑦 )𝑈 ֜ equality
ℎ 0 𝜐𝑈 and ℎ 2 (𝜐𝑈) 𝑚𝑝2 1 ℎ 0 𝜐 relax 𝑈 = − 𝜌 + 𝑈𝜐 relax 1 − 𝐺 < 0 𝑈𝜐 relax 1 1 ℎ 2 𝑈𝜐 relax = 𝑈𝜐 relax 𝐺 𝑈𝜐 relax >0 ∞ 𝐺 𝑏 ≡ 𝑏 𝑦 1 = −1 + alog2 − a 𝑏 1 𝜌 න 𝜌 Ψ 4𝜌 − Ψ sinh 𝑏 𝑦 2 + 1 𝜌 2𝜌 2 𝑦 0 Ψ 𝑨 ≡ 𝑒 𝑒𝑨 𝑚𝑝Γ(𝑨) 𝜌 − 𝑚𝑝2 𝜌
Low energy structure of 𝜍 00 𝜕, Ԧ 𝑞 [Kadanoff and Martin 1963] Consider the diffusion eq. as the relaxation process 𝜖 2 𝜖 𝜖𝑢 𝑘 0 𝑦, 𝑢 = 𝐸𝛼 2 𝑘 0 𝑦, 𝑢 𝜐 relax 𝜖𝑢 2 + Response lag caused by heavy quark mass Low energy structure of the spectral function hydro (𝜕, Ԧ 𝑞| 2 𝜍 00 𝑞) = 1 𝜓( Ԧ 𝑞)𝐸| Ԧ 𝜕 2 + 𝐸| Ԧ 𝑞| 2 − 𝜐𝜕 2 2 𝜕 𝜌 න 𝜕 2 𝜍 00 𝜕, 𝑞 = 𝑞 𝑗 𝑞 𝑘 𝜍 𝑗𝑘 𝜕, 𝑞 Kubo formula 𝐸 = 𝜌 1 𝜍 𝑗𝑗 𝜕, Ԧ 𝑞 𝜓 lim 𝜕→0 lim 3 𝜕 𝑞→0 Ԧ
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