量子開放系の理論と重クォークへの応用

赤松 幸尚（大阪大学）

非平衡物理の最前線 – 素粒子・宇宙から物性まで2017年 12月 6 – 8日@理化学研究所

1 / 23

INTRODUCTION

STOCHASTIC POTENTIAL MODEL

QUANTUM DISSIPATION

2 / 23

クォーコニウム：重いクォークの束縛状態

クォーコニウムの基本的性質

質量 構成 n2S+1LJ JP 束縛エネルギーJ/ψ, ψ(2S) ∼3 GeV c+ c̄ [1, 2]3S1 1− ∼ 0.6-0.05 GeVΥ(1, 2, 3S) ∼10 GeV b+ b̄ [1, 2, 3]3S1 1− ∼ 1.1-0.2 GeV

発見の歴史▶ 1974年 J/ψ 粒子（p+Be/e+ + e−）▶ 1977年 Υ粒子（p+Cu,Pt）

R = σ(e+e−→hadrons)

σ(e+e−→µ+µ−)

電子・陽電子衝突 [Particle Data Group (16)]10

-8

10-7

10-6

10-5

10-4

10-3

10-2

1 10 102

σ[m

b]

ω

ρ

φ

ρ′

J/ψ

ψ(2S)Υ

Z

10-1

1

10

10 2

10 3

1 10 102

R ω

ρ

φ

ρ′

J/ψ ψ(2S)Υ

Z

√s [GeV]

3 / 23

強い相互作用：量子色力学（QCD）

▶ クォークとグルーオンが相互作用する非可換ゲージ理論▶ 漸近的自由性：高エネルギー、短距離の現象ほど弱結合▶ 非自明な真空構造：カラー閉じ込め、カイラル対称性の破れ▶ 量子ゆらぎのスケール：ΛQCD ∼ 0.2GeV ≪MQ ∼ 1.5− 5GeV

▶ 重クォークは非相対論的に扱える▶ 閉じ込めポテンシャル

VCornell(r) = −αeffr

+ σr, αeff ∼ 0.3, σ ∼ (0.4GeV)2

ポテンシャル模型とクォーコニウムスペクトル [Lucha-Schöberl-Gromes (91)]

W. Lucha et a!., Bound states of quarks 165

case in order to reproduce the observed mass spectra of hadrons). Toponium, on the other hand, will bevery sensitive for short distances and will very probably enable us to select the “correct” one fromamong the various proposed potentials.In contrast, light quarks very probably move with relativistic velocities. This circumstance is usually

ignored without a moment of hesitation — just with the phrase “we assume that bound states of lightquarks may also be treated non-relativistically”. Note, however, that the only justification of theapplication of non-relativistic potential models are their phenomenologically successful predictions —especially in the case of light quarks.The possible applications of potential models are numerous. However, for realistic potentials it is, in

general, not possible to find an analytical solution of the Schrödinger equation. Consequently,numerical methods have to be employed in order to find energy eigenvalues and the correspondingwavefunctions [45]. For illustrative purposes we shall only touch upon some applications.

8.1. Mass spectroscopy of hadrons

In the framework of potential models the term “spectroscopy” means the calculation of masses ofbound states of quarks. Historically, potential models have first been applied to heavy quarkonia likecharmonium and bottomonium and, in fact, with noticeable success.We compare in table 8.1 the predictions of some potential models for the masses of mesons

consisting of heavy quarks with experiment. The “center of gravity” (COG) is defined as the averagemass of the (S = 1, 1 = 1) states,

COG(3P1) ~[5M(3P2)+ 3M(

3P,) + M(3P0)]. (8.3)

Perturbatively, for the center of gravity the expectation values of both the spin—orbit and the tensorinteractions separately sum up to zero.

Table 8.2 confronts some predictions for light-meson masses with experiment.

Table 8.1Theoretical against experimental masses (in GeV) of heavy mesons. Numbers in italics indicate input values

Cornell RichardsonMeson Experiment [9] potential [19] potential [46] Buchmüller and Tye [30]

1S) 2.9796 3.031JI~,(1S) 3.0969 3.095 3.1(X) 3.10COG(~,~(1P)) 3.5254 3.522 3.524 3.52js(2S) 3.6860 3.684 3.692 3.70~i(3770) 3.7699 3.81 3.820 3.814i(4040) 4.040 4.11 4.112 4.124,(4160) 4.159 4.19 4.19~,(4415) 4.415 4.46 4.465 4.48

Y(1S) 9.4603 9.46 9.433 9.46COG(xbJ(1P)) 9.9002 9.96 9.875 9.89Y(2S) 10.0233 10.05 9.990 10.02COG(~51(2P)) 10.2607 10.31 10.25Y(3S) 10.3553 10.40 10.337 10.35Y(4S) 10.5800 10.67 10.615 10.62Y(10860) 10.865 10.92 10.86Y(11020) 11.019 11.14

クォーコニウムは強い相互作用の「力」の良いプローブである4 / 23

クォーク・グルーオン・プラズマ（QGP）中のクォーコニウム

QCD相転移：Tc ∼ 160 MeV（約 2兆度）で閉じ込め相から非閉じ込め相へDebye遮蔽：QGP中ではカラー荷が自由に動ける

摂動計算

V (r) = −4αs3r

e−mDr, mD ∼ gsT

格子 QCD シミュレーション [Kaczmarek-Zantow (05)]

e.g. the screening property of the quark gluon plasma[19,20], the equation of state [21,22], and the order pa-rameter (Polyakov loop) [12,23–25]. In all of these studiesdeviations from perturbative calculations and the ideal gasbehavior are expected and were indeed found at tempera-tures which are only moderately larger than the deconfine-ment temperature. This calls for quantitativenonperturbative calculations. Also in this case most oftoday’s discussions of the bulk thermodynamic propertiesof the QGP and its apparent deviations from the ideal gasbehavior rely on results obtained in lattice studies of thepure gauge theory, although several qualitative differencesare to be expected when taking into account the influenceof dynamical fermions; for instance, the phase transition infull QCD will appear as a crossover rather than a ‘‘true’’phase transition with related singularities in thermody-namic observables. Moreover, in contrast to a steadilyincreasing confinement interaction in the quenched QCDtheory, in full QCD the strong interaction below deconfine-ment will show a qualitatively different behavior at largequark-antiquark separations. Because of the possibility ofpair creation, the stringlike interaction between the two testquarks can break, leading to a constant potential and/orfree energy already at temperatures below deconfinement[26].

Thus it is quite important to extend our recently devel-oped concepts for the analysis of the quark-antiquark freeenergies and internal energies in pure gauge theory[12,20,27,28] to the more complex case of QCD withdynamical quarks, and to quantify the qualitative differ-ences which will show up between pure gauge theories andQCD.

For our study of the strong interaction in terms of thequark-antiquark free energies in full QCD, lattice configu-rations were generated for 2-flavor QCD (Nf ! 2) on163 " 4 lattices with bare quark mass ma ! 0:1, i.e.m=T ! 0:4, corresponding to a ratio of pion to rho masses(m!=m") at the (pseudo) critical temperature of about 0.7(a denotes the lattice spacing) [29]. We have usedSymanzik-improved gauge and p4-improved staggered fer-mion actions. This combination of lattice actions is known

to reduce the lattice cutoff effects in Polyakov loop corre-lation functions at small quark-antiquark separations seenas an improved restoration of the broken rotational sym-metry. For any further details of the simulations with theseactions, see [30,31]. In Table I we summarize our simula-tion parameters, i.e. the lattice coupling #, the temperatureT=Tc in units of the pseudocritical temperature, and thenumber of configurations used at each # value. The pseu-docritical coupling for this action is #c! 3:649#2$ [30].To set the physical scale we use the string tension, $a2,

TABLE I. Sample sizes at each # value and the temperature inunits of the (pseudo) critical temperature Tc.

# T=Tc # conf. # T=Tc # conf.

3.52 0.76 2000 3.72 1.16 20003.55 0.81 3000 3.75 1.23 10003.58 0.87 3500 3.80 1.36 10003.60 0.90 2000 3.85 1.50 10003.63 0.96 3000 3.90 1.65 10003.65 1.00 4000 3.95 1.81 10003.66 1.02 4000 4.00 1.98 40003.68 1.07 3600 4.43 4.01 16003.70 1.11 2000

-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]

(a)0.76Tc0.81Tc0.90Tc0.96Tc1.00Tc1.02Tc1.07Tc1.23Tc1.50Tc1.98Tc4.01Tc

-500

0

500

1000

0 0.5 1 1.5 2 2.5

r [fm]

Fav [MeV]

(b)

FIG. 1. (a) The color singlet quark-antiquark free energies,F1#r; T$, at several temperatures close to the phase transition asfunction of distance in physical units. Shown are results fromlattice studies of 2-flavor QCD. The solid line represents in eachfigure the T ! 0 heavy quark potential, V#r$. The dashed errorband corresponds to the string-breaking energy at zero tempera-ture, V#rbreaking$ ’ 1000–1200 MeV, based on the estimate of thestring-breaking distance, rbreaking’ 1:2–1:4 fm [37]. (b) Thecolor averaged free energy, F !qq#r; T$, normalized such thatFav#r; T$ % F !qq#r; T$ & T ln9 [12] approaches the heavy quarkpotential, V#r$ (line), at the smallest distance available on thelattice. The symbols are chosen as in (a).

OLAF KACZMAREK AND FELIX ZANTOW PHYSICAL REVIEW D 71, 114510 (2005)

114510-2

クォーコニウムの束縛状態が高温では消失する▶ J/ψ 抑制が QGP生成のシグナルになる [Matsui-Satz (86)]▶ クォーコニウムの収量は QGPの温度計？[Mocsy (09)]

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相対論的重イオン衝突におけるクォーコニウム収量

Υ崩壊からのミューオン対スペクトル

p+ p −−−−−→w/o QGP

Υ+ anything, Υ → µ+ + µ−

Pb + Pb −−→QGP

Υ+ anything, Υ → µ+ + µ−

pn

Kπ

DB

J/ψ

Y

The CMS Collaboration / Physics Letters B 770 (2017) 357–379 359

3.2. Muon selection

Muons are reconstructed using a global fit to a track in the muon detectors that is matched to a track in the silicon tracker. The offline muon reconstruction algorithm used for the PbPb data has been improved relative to that used previously [18]. The efficiency has been increased by running multiple iterations in the pattern recognition step, raising the number of reconstructed ϒ(1S) candidates by approximately 35%. Background muons from cosmic rays and heavy-quark semileptonic decays are rejected by imposing a set of selection criteria on each muon track. These criteria are based on previous studies of the performance of the muon reconstruction algorithm [28]. The track is required to have a hit in at least one pixel detector layer, and a respective transverse (longitudinal) distance of closest approach of less than 3 (15) cm from the measured primary vertex, primarily to reject cosmic ray muons and muons from hadron decays in flight. To ensure a good pT measurement, more than 10 hits are requested in the tracker, and the χ2 per number of degrees of freedom of the trajectory fits is limited to be smaller than 10 when using the silicon tracker and the muon detectors, and smaller than 4 when using only the tracker. Pairs of oppositely charged muons are considered when the χ2 fit probability of the tracks originating from a common ver-tex exceeds 1%.

For the ϒ(2S) and ϒ(3S) analyses, the transverse momentum of each muon (pµT ) is required to be above 4 GeV/c, as in previ-ous publications [15,17,18], while one of them is relaxed down to 3.5 GeV/c for the ϒ(1S) analysis. Reducing this pT threshold raises the ϒ(1S) yield by approximately 40%, and its statistical signifi-cance by up to 50%, depending on the pT and y of the dimuon system. Relaxing the criterion on the second muon was also con-sidered then discarded, since it did not significantly raise the ac-ceptance for the ϒ states. The resulting invariant mass distribu-tions are shown on Fig. 1 for the entire pp and PbPb data samples.

4. Analysis

4.1. Signal extraction

To extract the ϒ(1S), ϒ(2S), and ϒ(3S) meson yields, unbinned maximum likelihood fits to the µ+µ− invariant mass spectra are performed between 7.5 and 14 GeV/c2. The results for the pT-, y-and centrality-integrated case are displayed as solid lines on Fig. 1. Each ϒ resonance is modelled by the sum of two Crystal Ball (CB) functions [29] with common mean but different widths to account for the pseudorapidity dependence of the muon momentum res-olution. The CB functions are Gaussian resolution functions with the low-side tail replaced by a power law describing final-state ra-diation. This choice was guided by simulation studies, as well as analyses of large pp event samples collected at

√s = 7 TeV [30].

Given the relatively large statistical uncertainties, the only signal model parameters that are left free in the fit are the mean of the ϒ(1S) peak, and the ϒ(1S), ϒ(2S) and ϒ(3S) meson yields. The other parameters, such as the width of the ϒ(1S) peak are fixed in every bin to the corresponding value obtained from simulations. The mean and width of the CB functions describing the ϒ(2S) and ϒ(3S) peaks are set by the fitted ϒ(1S) peak mean and the fixed ϒ(1S) width, respectively, multiplied by the world-average mass ratio [31]. The parameters describing the tail of the CB function are fixed to values obtained from simulations, kept common in the three ϒ states, then allowed to vary when computing the as-sociated systematic uncertainties. The background distribution is modelled by an exponential function multiplied by an error func-tion (the integral of a Gaussian) describing the low-mass turn-on, with all parameters left free in the fit.

Fig. 1. Dimuon invariant mass distributions in pp (top) and centrality-integrated PbPb (bottom) data at √sNN = 2.76 TeV, for muon pairs having one pT greater than 4 GeV/c and the other greater than 3.5 GeV/c. The solid (signal + background) and dashed (background only) lines show the result of fits described in the text.

With one muon having pT greater than 4 GeV/c and the other greater than 3.5 GeV/c, this fitting procedure results in ϒ(1S) me-son yields and statistical uncertainties of 2534 ± 76 and 5014 ± 87in centrality-integrated PbPb and pp collisions, respectively. With both muons’ transverse momenta above 4 GeV/c, it yields 173 ± 41for ϒ(2S) and 7 ± 38 for ϒ(3S) (hence unobserved) in PbPb colli-sions, and 1214 ± 51 for ϒ(2S) and 618 ± 44 for ϒ(3S) states in pp collisions.

4.2. Acceptance and efficiency

To correct yields for acceptance and efficiency in the two data samples, the three ϒ states have been simulated using thepythia 6.412 generator [32] and embedded in PbPb events sim-ulated with hydjet 1.8 [33], producing Monte Carlo (MC) events with the same settings as in Ref. [18], including radiative tails handled by photos [34]. Acceptance is defined as the fraction of ϒ in the |y| < 2.4 range that decay into two muons, each with |ηµ| < 2.4, and pµ2T > 4 GeV/c and p

µ1T > 3.5 or 4 GeV/c for the

ϒ(1S) and ϒ(2S)/ϒ(3S) states, respectively. For the ϒ(1S) state, the acceptance over the analyzed phase space averages to 35%. For all three ϒ states, the acceptance is constant over most of the ra-pidity range, with a drop at large |y|. When the ϒ meson has

1S

1S

2S,3S

2S,3S

The CMS Collaboration / Physics Letters B 770 (2017) 357–379 359

3.2. Muon selection

Muons are reconstructed using a global fit to a track in the muon detectors that is matched to a track in the silicon tracker. The offline muon reconstruction algorithm used for the PbPb data has been improved relative to that used previously [18]. The efficiency has been increased by running multiple iterations in the pattern recognition step, raising the number of reconstructed ϒ(1S) candidates by approximately 35%. Background muons from cosmic rays and heavy-quark semileptonic decays are rejected by imposing a set of selection criteria on each muon track. These criteria are based on previous studies of the performance of the muon reconstruction algorithm [28]. The track is required to have a hit in at least one pixel detector layer, and a respective transverse (longitudinal) distance of closest approach of less than 3 (15) cm from the measured primary vertex, primarily to reject cosmic ray muons and muons from hadron decays in flight. To ensure a good pT measurement, more than 10 hits are requested in the tracker, and the χ2 per number of degrees of freedom of the trajectory fits is limited to be smaller than 10 when using the silicon tracker and the muon detectors, and smaller than 4 when using only the tracker. Pairs of oppositely charged muons are considered when the χ2 fit probability of the tracks originating from a common ver-tex exceeds 1%.

For the ϒ(2S) and ϒ(3S) analyses, the transverse momentum of each muon (pµT ) is required to be above 4 GeV/c, as in previ-ous publications [15,17,18], while one of them is relaxed down to 3.5 GeV/c for the ϒ(1S) analysis. Reducing this pT threshold raises the ϒ(1S) yield by approximately 40%, and its statistical signifi-cance by up to 50%, depending on the pT and y of the dimuon system. Relaxing the criterion on the second muon was also con-sidered then discarded, since it did not significantly raise the ac-ceptance for the ϒ states. The resulting invariant mass distribu-tions are shown on Fig. 1 for the entire pp and PbPb data samples.

4. Analysis

4.1. Signal extraction

To extract the ϒ(1S), ϒ(2S), and ϒ(3S) meson yields, unbinned maximum likelihood fits to the µ+µ− invariant mass spectra are performed between 7.5 and 14 GeV/c2. The results for the pT-, y-and centrality-integrated case are displayed as solid lines on Fig. 1. Each ϒ resonance is modelled by the sum of two Crystal Ball (CB) functions [29] with common mean but different widths to account for the pseudorapidity dependence of the muon momentum res-olution. The CB functions are Gaussian resolution functions with the low-side tail replaced by a power law describing final-state ra-diation. This choice was guided by simulation studies, as well as analyses of large pp event samples collected at

√s = 7 TeV [30].

Given the relatively large statistical uncertainties, the only signal model parameters that are left free in the fit are the mean of the ϒ(1S) peak, and the ϒ(1S), ϒ(2S) and ϒ(3S) meson yields. The other parameters, such as the width of the ϒ(1S) peak are fixed in every bin to the corresponding value obtained from simulations. The mean and width of the CB functions describing the ϒ(2S) and ϒ(3S) peaks are set by the fitted ϒ(1S) peak mean and the fixed ϒ(1S) width, respectively, multiplied by the world-average mass ratio [31]. The parameters describing the tail of the CB function are fixed to values obtained from simulations, kept common in the three ϒ states, then allowed to vary when computing the as-sociated systematic uncertainties. The background distribution is modelled by an exponential function multiplied by an error func-tion (the integral of a Gaussian) describing the low-mass turn-on, with all parameters left free in the fit.

Fig. 1. Dimuon invariant mass distributions in pp (top) and centrality-integrated PbPb (bottom) data at √sNN = 2.76 TeV, for muon pairs having one pT greater than 4 GeV/c and the other greater than 3.5 GeV/c. The solid (signal + background) and dashed (background only) lines show the result of fits described in the text.

With one muon having pT greater than 4 GeV/c and the other greater than 3.5 GeV/c, this fitting procedure results in ϒ(1S) me-son yields and statistical uncertainties of 2534 ± 76 and 5014 ± 87in centrality-integrated PbPb and pp collisions, respectively. With both muons’ transverse momenta above 4 GeV/c, it yields 173 ± 41for ϒ(2S) and 7 ± 38 for ϒ(3S) (hence unobserved) in PbPb colli-sions, and 1214 ± 51 for ϒ(2S) and 618 ± 44 for ϒ(3S) states in pp collisions.

4.2. Acceptance and efficiency

To correct yields for acceptance and efficiency in the two data samples, the three ϒ states have been simulated using thepythia 6.412 generator [32] and embedded in PbPb events sim-ulated with hydjet 1.8 [33], producing Monte Carlo (MC) events with the same settings as in Ref. [18], including radiative tails handled by photos [34]. Acceptance is defined as the fraction of ϒ in the |y| < 2.4 range that decay into two muons, each with |ηµ| < 2.4, and pµ2T > 4 GeV/c and p

µ1T > 3.5 or 4 GeV/c for the

ϒ(1S) and ϒ(2S)/ϒ(3S) states, respectively. For the ϒ(1S) state, the acceptance over the analyzed phase space averages to 35%. For all three ϒ states, the acceptance is constant over most of the ra-pidity range, with a drop at large |y|. When the ϒ meson has

1S

1S

2S,3S

2S,3S

pp衝突に比べて PbPb衝突の Υ(2S, 3S)が相対的に Υ(1S)より減少している

このデータを真空中と QGP中の力の違いとして理解できるか？6 / 23

QGP中のクォーコニウムのダイナミクス

クォーコニウムの変数だけで量子論的に記述したい= 量子開放系

system(!!")

environment(QGP)Hint

▶ 全系の密度行列 & von Neumann 方程式

ρtot(t) =∑

Ψ∈Htot

wΨ|Ψ(t)⟩⟨Ψ(t)|, id

dtρtot = [Htot, ρtot]

▶ 縮約密度行列 & マスター方程式

ρQQ̄

(t) ≡ TrQGPρtot(t), id

dtρQQ̄

= [potential + fluctuation + dissipation]︸ ︷︷ ︸in-medium forces in QGP

ρQQ̄

1. 初期条件：高エネルギー過程 q + q̄, g + g → b+ b̄ など [CasalderreySolana (12)]ρbb̄(x, y, tI) ∝ δ(x)δ(y)

2. QGP 中の時間発展：マスター方程式を解く

id

dtρbb̄ = [. . .]ρbb̄

3. ハドロン化 @T (tF ) = Tc：束縛状態の波動関数へ射影

NΥ(tF ) =

∫x,y

ρbb̄(x, y, tF )PΥ(y, x)︸ ︷︷ ︸projection

→ µ+ + µ−

7 / 23

INTRODUCTION

STOCHASTIC POTENTIAL MODEL

QUANTUM DISSIPATION

8 / 23

Stochastic Potential Model [Akamatsu-Rothkopf (12), Kajimoto et al (17)]

熱浴中のクォーコニウムのエネルギー r/2-r/2

H = − ∇2

MQ+ V (r)︸ ︷︷ ︸

screening

+ θ(t, r/2)︸ ︷︷ ︸noise for Q

− θ(t,−r/2)︸ ︷︷ ︸noise for Q̄

⟨θ(t1, x1)θ(t2, x2)⟩ = δ(t1 − t2)D(x1 − x2)Debye遮蔽ポテンシャルと有限相関長 lcorr を持ったノイズ

V (r) = −αeffre−mDr, D(r) = γ exp[−r2/l2corr]

MQ [GeV] αeff mD γ lcorr摂動論 - 4αs/3 ∼ gsT αsT ∼ 1/gsTモデル計算 4.8 / 1.18 0.3 T 0.3T 1/T

$ ≪ &

→

l

lcorr

lcorr

� l lcorr

⇠ l lcorr

⌧ l

e�i�tH ' 1� i�tH(r, t)� 12(�t⇥(r, t))2 +O(�t3/2)+⇥H ⌘ �rr

2

M+ V (r)

5

波動関数のデコヒーレンスは相関長 lcorr によって決まる9 / 23

確率論的 Schrödinger方程式

1. ユニタリ演算子 exp [−iHdt]を dtまで展開

i∂

∂tΨQQ̄(r, t) = Heff(r, t)ΨQQ̄(r, t)

Heff(r, t) = −∇2rMQ

+ V (r)− i [D(0)−D(r)]︸ ︷︷ ︸positive

+ [θ(r/2, t)− θ(−r/2, t)]

⟨θ(t1, x1)θ(t2, x2)⟩ = δ(t1 − t2)D(x1 − x2)

2. 対応するマスター方程式

ρQQ̄(r, r′, t) ≡ ⟨ΨQQ̄(r, t)ΨQQ̄(r

′, t)∗⟩, ∂∂tρQQ̄(r, r

′, t) = · · ·

▶ 一体系ならもっと単純

∂

∂tρQ(x, x

′, t) = i

(∇2x2MQ

−∇2x′

2MQ

)ρQ(x, x

′, t)−[D(0)−D(x− x′)

]︸ ︷︷ ︸decoherence

ρQ(x, x′, t)

10 / 23

Stochastic Potential Modelと実時間相関関数とのマッチング

クォーコニウムの実時間プロパゲータ（M = ∞）▶ Stochastic Potential Modelでは波動関数の平均

G>(t, 0; r) = ⟨ΨQQ̄(t, r)⟩θ = exp[−it {V (r)− i(D(0)−D(r))}︸ ︷︷ ︸complex potential VRe + iVIm

]

▶ r ∼ 1/gT での Hard-Thermal Loop摂動論（Debye遮蔽、Landau減衰）[Laine et al (07), Beraudo et al (08), Brambilla et al (08)]

VRe(r) = −g2CF4π

[mD +

exp(−mDr)r

],

VIm(r) = −g2TCF

4πϕ(mDr), ϕ(x) = 2

∫ ∞0

dz z

(z2 + 1)2

[1− sin(zx)

zx

],

▶ 格子 QCDシミュレーション [Rothkopf et al (17,. . .,12)]

position and width yield the values of the in-mediumpotential, which are plotted as colored points in Fig. 8.We have shifted the values of Re½V" by hand for better

readability in the top panel of Fig. 8, as indicated by thegray arrow, and plot the statistical errors as colored bars.The systematic errors denoted by the gray error bands havebeen determined only for part of the ensembles, inparticular, however, among the newly generated onesNτ ¼ 60, 64, and 68. We use the variation from changingthe default model amplitude by 2 orders of magnitude aswell as discarding 10% of the small τ and/or 10% of thelarge τ ≈ β data points as an estimate.Just as in the previous analysiswe find that there appears to

exist only a gradual change in behavior of Re½V" withincreasing temperature. Due to limited statistics in the olderanalysis, Nτ ¼ 64 had seemed to exhibit an anomalouslystrong linear rise. This effect vanishes after increasingstatistics, and the slope at Nτ ¼ 64 now lies within the trendof the neighboringNτ ¼ 60 andNτ ¼ 72. On the other hand,it is now Nτ ¼ 68, which is found to show an almost avacuumlike linear rise. The reason for this outlier, however,lies in the fact that the Nτ ¼ 68 simulations show extremelylong autocorrelation times inMonte Carlo time. In turn, evenafter collecting more than 2100 measurements on individualconfiguration, the actual statistics are reduced by arounda factor 10 to 100, making this result rather unreliable.Interestingly this issue does not lead to an increase in thesystematic error bars compared to e.g. Nτ ¼ 64.For the sake of completeness we continue toward

performing the Gauss-law fit by determining first thevacuum parameters that enter into this Ansatz from thelowest temperature result at Nτ ¼ 192. Restricting tothe region of r < 0.3 fm, we obtain

αS ¼ 0.201$ 0.004;σ ¼ 0.186$ 0.008 GeV2

c ¼ 2.58$ 0.01 GeV; ð17Þ

where the errors are again estimated from a variation of theupper and lower ends of the fitting range by six steps each.These agree within errors with our previous values pub-lished in Ref. [18]. The Debye mass parameter we find to fitthe in-medium values of Re½V" at β ¼ 7 best are compiledin Table IV and plotted in Fig. 9. In addition we alsoperform a fit to the extracted values based on Eq. (15),which yields as best fit parameters

κ1 ¼ −0.67$ 0.06; κ2 ¼ 0.34$ 0.06 ð18Þ

and which is plotted as a solid line in Fig. 9. Theuncertainties in the κ values arise from the error inestimating mD itself.Let us discuss in detail the temperature dependence of

the Debye mass parameter in Fig. 9. It significantly differs

FIG. 8. Extracted values of the real (top) and imaginary parts(bottom) of the in-medium potential (colored points) at β ¼ 7.The values of Re½V" are shifted manually in the y direction forbetter readability as indicated by the gray arrow. The values ofIm½V" are plotted individually as a grid from lowest temperature(top left box) to highest temperature (bottom right box). Thecolored error bars denote statistical uncertainty, while the grayerror band denotes systematic uncertainty.

YANNIS BURNIER and ALEXANDER ROTHKOPF PHYSICAL REVIEW D 95, 054511 (2017)

054511-10

GE(τ, 0; r) −−−−−−−−−−−→実時間に “解析接続”

VIm(r) ∼ −γ[1− e−r

2/l2corr

]ノイズ相関関数は実時間ポテンシャルの虚部と対応する

11 / 23

クォーコニウムの 1次元シミュレーション [Kajimoto et al (17)]

1次元方向に膨張する QGPの温度変化 [Bjorken (83)]

T (t) = T0

(t0

t0 + t

)c2s=1/3, T0 = 0.4 GeV, t0 = 1 fm

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8 9

OCCUPATION PROBABILITY

TIME [fm]

BOTTOMONIUM

ground state without noiseground state

1st excited state2nd excited state

数値計算の流れ

1. 初期条件：

ψbb̄(x, t = 0) = ψΥ(x),

ρbb̄(x, y, t = 0) = ψΥ(x)ψ∗Υ(y)

2. QGP 中の時間発展：

id

dtψbb̄ = [stochastic potential]ψbb̄,

id

dtρbb̄ = [potential + fluctuation]ρbb̄

3. 各時刻で ψΥ(x) の存在確率を計算

r1S < r2S < r3S → N1S > N2S > N3S

ノイズは Debye遮蔽に加えてクォーコニウムの解離を促進する

12 / 23

INTRODUCTION

STOCHASTIC POTENTIAL MODEL

QUANTUM DISSIPATION

13 / 23

Stochastic Potential Modelと量子散逸

Stochastic Potential Modelは散逸を記述できず、クォーコニウムを過加熱してしまう

id

dtρQQ̄

= [potential + fluctuation + dissipation]

9

9.5

10

10.5

11

11.5

12

12.5

0 1 2 3 4 5 6 7 8 9

ENERGY [GeV]

TIME [fm]

BOTTOMONIUM

bbbar ground statebbbar 1st excited statebbbar 2nd excited state

励起状態からのフィードダウン

[Nendzig-Wolschin (13)]

ϒ SUPPRESSION IN PbPb COLLISIONS AT . . . PHYSICAL REVIEW C 87, 024911 (2013)

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400

Rpr

elA

A(n

l)

Npart

FIG. 5. (Color online) Preliminary suppression factors RprelAA (nl)from Eq. (8) as functions of centrality for the different bottomiumstates ϒ(1S) (solid), ϒ(2S) (dash-dotted), χb(1P ) (dashed), andhigher excited states (dotted) for the formation time tF = 0.1 fm/cand QGP lifetime tQGP = 6 fm/c.

Now that we have calculated the suppression during theevolution of the fireball we have to consider the feed-downof the remaining bb̄ population to calculate the fraction ofdecays into dimuon pairs, ϒ(nS) → µ+µ−. Figure 6 displaysthe decays within the bb̄ family and into dimuon pairs that aremeasured. Considering first the processes inside the fireballand then performing the decay cascade as a subsequent stepis justified by the very different time scales involved. Atthe LHC the fireball has cooled within less than !10 fm/c,while the subsequent decays take place on time scales∼103 fm/c.

Let us denote bb̄ states by I = (nl) and (CIJ ) (I " J ) thebranching ratio of state J to decay into state I including allindirect decays with intermediate bb̄ states. The initial andfinal bb̄ numbers of state I , Ni(I ), and Nf (I ) in pp and PbPb

Υ(1S)

χb(1P)

Υ(2S)

χb(2P)

Υ(3S)

µ+µ−

0.0218

0.0193

0.0248

0.0660.06410.267

0.21

0.1785

0.0049

0.0237

0.1390.106

0.316

FIG. 6. (Color online) Branching ratios for decays within thebottomium family ϒ(nS) and χb(nP ) and into µ± pairs according toRef. [32].

collisions are then connected by

Nfpp(I ) =∑

I"JCIJ Ni(J ),

(10)N

fPbPb(I ) =

∑

I"JCIJ Ni(J )RprelAA (J ).

Further we define the number of ϒ(nS) states that decay intodimuon pairs

Nfµ±(nS) = B(nS → µ

±)NfPbPb(nS), (11)

where B(nS → µ±) is the corresponding branching ratio.We take Nfµ±(nS) from the 2012 CMS data [12] and

consider that 27.1% and 10.5% of the ϒ(1S) population comefrom χb(1P ) and χb(2P ) decays, respectively [16]. Since theseCDF results are obtained from pp̄ collisions at 1.8 TeV witha transverse momentum cut pϒT > 8.0 GeV/c, it would bedesirable to confirm the ϒ(1S) populations from χb decays innew pp measurements at 2.76 TeV, which are not yet available.

The initial populations are then obtained in units ofB(nS →µ±)Nfpp(1S) as Ni(1S) = 16.2, Ni(1P ) = 43.7, Ni(2S) =20.3, Ni(2P ) = 45.6, and Ni(3S) = 18.8. The final sup-pression factor is now simply calculated as RAA(nS) =N

fµ±(nS)/N

fpp(nS) or

RAA(nS) = B(nS → µ±)∑

nS"J CIJ Ni(J )RprelAA (J )∑

nS"J CIJ Ni(J ). (12)

V. RESULTS

We present the results for screening and collisional dampingderived from the solutions of the Schrödinger equation withthe potential in Eq. (2), and the widths for gluodissociation

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400

RA

A(1

S)

Npart

FIG. 7. (Color online) Suppression factor RAA for the ϒ(1S)ground state calculated for 2.76 TeV PbPb collisions from screening,collisional damping, gluodissociation, and reduced feed-down usingthree QGP lifetimes tQGP = 4, 6, 8 fm/c (dotted, solid, and dashedlines, respectively) for the centrality bins 50–100%, 40–50%, 30–40%, 20–30%, 10–20%, 5–10%, and 0–5%. The dash-dotted upperline is the preliminary suppression factor RprelAA (1S) (tQGP = 6 fm/c)without reduced feed-down. The corresponding CMS data [12] are ingood agreement with the model results for the ϒ(1S) state.

024911-5

量子散逸は現象論的に重要になり得る

14 / 23

Lindblad方程式による Stochastic Potential Modelと量子散逸

1. 密度行列の正定置性を保証する時間発展の一般形 [Lindblad (76)]

d

dtρQQ̄

(t) = −i[H, ρQQ̄

] +

N∑n=1

(LnρQQ̄L

†n −

1

2L†nLnρQQ̄ −

1

2ρQQ̄L†nLn

)2. Stochastic Potential Modelの Lindblad演算子 [Akamatsu (15)]

Lk =√D(k)eikx︸ ︷︷ ︸∆pQ = k

×(1 or ta)︸ ︷︷ ︸U(1) or SU(Nc)

+heavy antiquark

▶ Lindblad 演算子は運動量移行 k の散乱を記述▶ 散乱の頻度は D(k)（ノイズ相関関数の Fourier 変換）に比例する

3. 量子散逸を含んだ Lindblad演算子 [Akamatsu (15)]

Lk =√D(k)eikx/2

[1 +

ik · ∇x4MT︸ ︷︷ ︸

∆xQ ∼ k/MT

]eikx/2 + heavy antiquark

▶ 量子散逸は重クォークの反跳効果で与えられる

15 / 23

Lindblad形式の量子ブラウン運動

1. 影響汎関数 SIF [Feynman-Vernon (63)]▶ 縮約密度行列：ρtot から QGP 環境の変数について平均（トレース）

ρQQ̄

(t, x, y) =

∫dXdY δ(X − Y )︸ ︷︷ ︸

trace out QGP = path closed at t

ρtot(t, x, y,X, Y )

=

∫dx0dy0ρQQ̄(0, x0, y0)

∫ x,yx0,y0

D[x̄, ȳ]eiSQQ̄[x̄]−iSQQ̄[ȳ]+iSIF[x̄,ȳ]

2. SIF を相互作用の二次摂動で評価

iSIF[x, y] =−1

2

∫ t0

dt′dt′′(x, y)(t′)

(G11 −G12−G21 G22

)(t′,t′′)︸ ︷︷ ︸

correlation functions of X, Y

(xy

)(t′′)

+ . . .

3. Lindblad形式と時間の粗視化 [Diosi (93), Akamatsu (15)]

iSIF = iSfluct︸ ︷︷ ︸∝ xx, xy, yy

+ iSdiss︸ ︷︷ ︸∝ xẋ, · · ·

+ iSL︸︷︷︸∝ ẋẋ, · · ·

▶ Lindblad 形式で散逸を考えるときは ẋẋ まで展開（L ∼ x+ ẋ なら L†L ∋ ẋẋ）16 / 23

Quantum State Diffusionによる数値計算１ [Akamatsu et al, in preparation]

Lindblad方程式は非線形確率論的 Schrödinger方程式と同値（stochastic unravelling）[Gisin-Percival (92)]

|dψ⟩ =− iH|ψ(t)⟩dt+∑n

(2⟨L†n⟩ψLn − L†nLn − ⟨L†n⟩ψ⟨Ln⟩ψ

)|ψ(t)⟩dt

+∑n

(Ln − ⟨Ln⟩ψ) |ψ(t)⟩dξn, ⟨dξndξ∗m⟩ = 2δnmdt

重クォーク１個のシミュレーション▶ 波動関数の時間発展（初期条件は k = 0の平面波）

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 20 40 60 80 100 120 140

|Ψ(x

)|2

x

M=10T, lcorr=1/γ=10, L=128, t=10t=50

t=100t=500

各イベントで波動関数は lcorr 程度の波束状態へ（量子測定理論の Pointer状態に類似）17 / 23

Quantum State Diffusionによる数値計算２ [Akamatsu et al, in preparation]

運動量分布の時間変化と熱平衡化▶ 緩和時間 ∼ 100に対応するパラメタセット▶ 重クォークの質量M = 1とする単位系で

1e-08

1e-06

0.0001

0.01

1

100

10000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n(k)

k

M=10T, lcorr=1/γ=10, L=128, t=500t=100

t=50t=10

eq

量子散逸により熱平衡化も記述可能に18 / 23

Quantum State Diffusionによる数値計算３ [Akamatsu et al, in preparation]

量子散逸のない場合（Stochastic Potential Modelに相当）との比較

1e-08

1e-06

0.0001

0.01

1

100

10000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

n(k)

k

M=10T, lcorr=1/γ=10, L=128, t=500t=100

t=50t=10

eq

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120R

e,Im

[Ψ(x

)]x

t=500, w/diss, ReΨImΨ

w/o diss, ReΨImΨ

▶ 熱平衡化せず、過加熱している▶ 波動関数の広がりは lcorr 程度だが構造はより細かい

19 / 23

クォーコニウムのマスター方程式の数値計算１ [DeBoni (17)]

1次元でのマスター方程式

∂tρ(t, r, y) =

[iℏm∂r∂y −

i

ℏ(V (r + y)− V (r − y))

−1

ℏ(2W (y)− 2W (r) +W (r + y) +W (r − y)− 2W (0))

−ℏ

2mT

(∂yW (y)∂y − ∂rW (r)∂r − ∂2rW (r)

)]ρ(t, r, y)

ρ(t, r, y)の時間発展の数値計算▶ Pöschl-Tellerポテンシャル（j = 2）

V (x) = −ω2j(j + 1)sech2

[√mω

2ℏ2x

]▶ ノイズ相関関数

W (x) = −T2exp

[− x

2

2l2env

]

初期条件：有限運動量を持った波束

JHEP08(2017)064

Figure 7. Time evolution of the real part of the density matrix for the thermal scattering statedefined before, propagating through a bath with lenv = 0.25 fm. Top: the medium disturbs thesystem at the very early times. Centre: quantum decoherence appears as a suppression of theoff-diagonal elements of the density matrix. Bottom: the state is squeezed around y = 0 . Noticethe different scale in the y direction.

In order to see how the dissociation and recombination mechanisms depend on the mass

of the heavy quarks, one can find the numerical results for the bound-state probabilities

for a heavier quark with mass (m = 4.7GeV) in appendix D.

8 Summary and outlook

It has been shown how to derive a Lindblad equation for non relativistic heavy quarks and

antiquarks propagating out of equilibrium in a thermalised quark gluon plasma, within

the framework of open quantum systems and starting from the underlying gauge theory.

To achieve this, an abelian model for the plasma has been used, together with some well-

defined approximations, including the perturbative expansion and the Markovian limit. All

– 34 –

20 / 23

クォーコニウムのマスター方程式の数値計算２ [DeBoni (17)]

束縛状態の存在確率の時間変化▶ Stochastic Potential Modelとは異なり存在確率が有限値に収束しそう

JHEP08(2017)064

0 1 2 3 4 50

0.10.20.30.40.50.60.70.80.91

t [fm/c]

Proba

bility

lenv [fm]P0(t) , 0.74P0(t) , 0.25P0(t) , 0.08P1(t) , 0.74P1(t) , 0.25P1(t) , 0.08ρ(0, q, q′) = ψ0(q)ψ∗0(q

′)

0 1 2 3 4 50

0.10.20.30.40.50.60.70.80.91

t [fm/c]

Proba

bility

ρ(0, q, q′) = ψ1(q)ψ∗1(q′)

0 1 2 3 4 50

0.10.20.30.40.50.60.70.80.91

t [fm/c]

SL

lenv [fm]0.74

0.25

0.08

ρ(0, q, q′) = ψ0(q)ψ∗0(q′)

0 1 2 3 4 50

0.10.20.30.40.50.60.70.80.91

t [fm/c]

SL

ρ(0, q, q′) = ψ1(q)ψ∗1(q′)

Figure 3. Top: probabilities P0(t) , P1(t) of having respectively the ground state ψ0 and theexcited state ψ1 at time t . Bottom: time evolution of the linear entropy. In the left (right) panelsthe initial density matrix ρ(0, q, q′) corresponds to the ground (excited) state. Three differentcorrelation lengths of the environment (lenv) have been considered in order to explore the regimeslenv > lψ1 , lψ0 < lenv < lψ1 , lenv < lψ0 , where lψ =

√⟨x2⟩ψ with values lψ0 = 0.162 fm and

lψ1 = 0.384 fm. In all plots of this paper error bars are much smaller than the symbols representingthe data. Some of the points are slightly shifted horizontally to avoid superpositions of symbols.

In this scenario the environment can resolve very well the excited state and also start

affecting the ground state. From the diamond-shaped symbols on the left panels of figure 3

one can see that an initial ground state has survival probability P0(t) ≥ 80% , which is highbut less than in the previous case. The chances of promoting the ground state to the excited

state are slightly higher than before but still fewer than 10% . Similarly, the linear entropy

grows more than before but stays below the value of 0.4 . On the other hand, if the system

is initially in the excited state (diamonds on the right panels of figure 3), the environment

strongly perturbs the bound state, causing the linear entropy to increase abruptly and the

excited state to disappear very rapidly, reaching soon the equilibrium value of about 10%

for the survival probability P1. One might think that the disappearance of the excited

state is simply a signal of its melting due to the kicks received from the bath. However,

the feed-down mechanism is more important here than in the previous case, in which the

excited state receives lighter kicks. In fact the probability of ending up with the ground

state grows rapidly from zero to the equilibrium value of 60% , which leaves a melting

– 29 –

以下は私見▶ 束縛状態の平衡での存在確率は温度だけでなく lenv にも依存しているのでは？

P eq0 (lenv = 0.08) ∼ 0.6, Peq1 (lenv = 0.08) ∼ 0.18, P

eq1 (lenv = 0.25) ∼ 0.09

重イオン衝突でのクォーコニウムの収量は QGPの相関長の情報を持っているかもしれない

21 / 23

重クォーク対の古典極限：一般化 Langevin方程式 [Blaizot et al (16)]

重クォーク対の相関（s ≡ r − r̄ 依存性）は摩擦項とランダム力にも現れる

Mr̈ +βg2

2(H(0)ṙ −H(s) ˙̄r)− g2∇V (s) = ξ(s, t)

M ¨̄r +βg2

2(H(0) ˙̄r −H(s)ṙ)︸ ︷︷ ︸

drag

+ g2∇V (s)︸ ︷︷ ︸potential

= ξ̄(s, t)︸ ︷︷ ︸random

⟨ξ(s, t)ξ(s, t′)⟩ = g2H(0)δ(t− t′), ⟨ξ(s, t)ξ̄(s, t′)⟩ = −g2H(s)δ(t− t′)

▶ Q+ g → Q+ g と Q̄+ g → Q̄+ g の散乱過程が干渉する▶ N 体に拡張してシミュレーション

22 / 23

まとめと展望

クォーコニウムは QGP中の量子開放系である▶ Stochastic Potential Modelのシミュレーション▶ Quantum State Diffusion法による Lindblad方程式のシミュレーション▶ 最近はカラー自由度についての研究も [Akamatsu (15), Brambilla et al (17), Blaizot et al (17)]

応用先▶ 重イオン衝突▶ 初期宇宙における Dark Matter粒子の束縛状態？ [Kim-Laine (17)]▶ 冷却原子気体？ [Braaten-Hammer-Lepage (16)]

23 / 23

Back Up

24 / 23

Classical Langevin equation from Caldeira-Leggett master equation

1. Introduce quantum and classical variables (ra-basis):

xr ≡x+ y

2, xa ≡ x− y

2. Express using noise:

eiSIF ∋ e−2γmT∫dtx2a =

∫Dξe−

∫dtξ2/8γmT+i

∫dtξxa

3. Langevin equation after integrating xa∫D[ξ, xr] e−

∫dtξ2/8γmT︸ ︷︷ ︸

Gaussian white noise

∫Dxa ei

∫dtxa(−mẍr−2γmẋr+ξ)︸ ︷︷ ︸→ Langevin equation

▶ Surface term in partial integration + Wigner transform

fA(t, x, p) =

∫dx0

∫ xr=xxr=x0

D[ξ, xr]fA(0, x0,mẋr0)e−∫dtξ2/8γmT

δ [−mẍr − 2γmẋr + ξ]

Noise term in the Langevin equation originates from the fluctuation of xa

25 / 23