講演番号19

格⼦QCDによるカラー超伝導の解明に向けて- 摂動論の予⾔と複素ランジュバン法 -Towards color superconductivity on the lattice- perturbative predictions and the complex Langevin method -Takeru Yokota
Institute for Solid State Physics, Univ. of Tokyo

Collaborators
Yuhma Asano (KEK), Yuta Ito (Tokuyama National College of Technology),
Hideo Matsufuru (KEK), Yusuke Namekawa (YITP), Jun Nishimura (KEK,SOKENDAI), Asato Tsuchiya (Shizuoka Univ.), Shoichiro Tsutsui (RIKEN)
熱場の量⼦論2020, online, 2020/8/25
1Contents• Motivation•QCD phase diagram and sign problem•Possibility of investigating color superconductivity (CSC) on the lattice• Perturbative predictions•

Gap equation•

Prediction of transition points•

Phase diagram on ("! , $)-plane• Current status of our analysis
by the complex Langevin method• Summary2

Contents• Motivation•

QCD phase diagram and sign problem•

Possibility of investigating color superconductivity (CSC) on the lattice• Perturbative predictions•

Gap equation•

Prediction of transition points•

Phase diagram on ("! , $)-plane• Current status of our analysis
by the complex Langevin method• Summary3QCD phase diagram

Fukushima, Hatsuda, RPP (2011)

4

QCD phase diagram

Lattice QCD(Monte Carlo)

Perturbation

Fukushima, Hatsuda, RPP (2011)

5

QCD phase diagram

Lattice QCD(Monte Carlo)

Perturbation

First-principle analysis isprevented bySign problem

Fukushima, Hatsuda, RPP (2011)

6

QCD phase diagram
Methods to overcome sign problems
Lefschetz thimble, tensor renormaliztion group, complex Langevin method (CLM)…

Application of CLM to lattice QCD
at finite density
Sexty ʼ14
Fodor, Katz, Sexty, Török ʼ15
Nagata, Nishimura, Shimasaki ʼ18
Tsutsui, Ito, Matsufuru, Nishimura, Shimasaki, Tsuchiya ʼ18
Ito, Matsufuru, Nishimura, Shimasaki, Tsuchiya, Tsutsui ʼ18
Sexty ʼ19
Kogut, Sinclair ʼ19
Sinclair, Kogut ʼ19
Tsutsui, Ito, Matsufuru, Nishimura, Shimasaki, Tsuchiya ʼ19
Scherzer, Sexty, Stamatescu ʼ20
Ito, Matsufuru, Namekawa, Nishimura, Shimasaki, Tsuchiya, Tsutsui ʼ20
7

Fukushima, Hatsuda, RPP (2011)

Color superconductivity (CSC)
Barrois, NPB (1977); PhD thesis (1979)
Frautschi (1978), Bailin, Love, PR (1984)
Alford, Rajagopal, Wilczek, PLB (1998)

High !! (weak coupling), low "Attractive in color-antitriplet channel

Cooper instability
CSC with various symmetries of
Cooper pairs has been predicted(2SC, CFL…)
8

CSC on the lattice?
Analysis of CSC on the lattice by firstprinciple methods such as CLM?

Reasonable starting point:high ! region (small volume)" = 2%! /'"#

Asymptotic freedom → weak coupling

But the region in which the CSC occurs has not been predicted
so far based on lattice QCD.
c.f.) Calculation in effective models (e.g. NJL) in finite box
Hands, Walters, PLB (2002)
Amore, Birse, McGovern, Walet, PRD (2002)

9

Aim
We give the perturbative prediction of the parameterregion in Lattice QCD for the first time.
• Derive the gap equation for Cooper pair with generalshape and symmetry.
• Derive the condition to determine the critical !
• The shape and symmetry of Cooper pairs also can be
determined with our calculation method.

• Calculate critical ! and give the prediction for thephase diagram on (#$ , !)-plane.
Such a calculation in general setup is possible because only finite number of coupledequations appear on the lattice.
(↔functional equation in continuum case)We also present some preliminary results of the CLM inthe large ! regime.
10

Contents• Motivation•

QCD phase diagram and sign problem•

Possibility of investigating color superconductivity (CSC) on the lattice• Perturbative predictions•

Gap equation•

Prediction of transition points•

Phase diagram on ("! , $)-plane• Current status of our analysis
by the complex Langevin method• Summary11

Nambu-Gorkov formalism
Action for '( = ) Staggered fermion* = *)*+,- + *-./ + *0123.
Staggered fermion !!" " = !" 2" + &[&# = 0 or 1 () = 1~4), ,: color index]

Nambu spinor

Anomalous self-energy

propagator

If %"#(#") ≠ ', CSC emerges (("#(#") ≠ ')12

In the case of continuum QCD: Bailin, Love, PR (1984)…

Gap equation
Gap equation: self-consistent equation to determine Σ45 (54)
We ignore Σ44(55) (higher-order contribution in the gap eq.)
Gap equation (lowest-order, momentum space)!

''
0$$! ,##&,&&!"", ,$$% : gluon propagator-, ,, &-′, ,′, &′'#Π$(),*!,+";! ;*45,9! 9,:! :+2(4-. )'! # !

Π$ !(!)!,*!!!,+!"!

&$

Π"'(,)*,+, : 2quark-gluon vertex

determined by Σ) &'('&)

If this eq. has "!"("!) ≠ $ solution, CSC emerges.

13,+ +#-.,', ',* , * : anomalous propagator

Transition point
since Σ&'('&) → 0

At the transition point,
Gap eq. is reduced to linear equation
Gap eq. for color<antitriplet channel (Σ45)-−, -−),+,+=/ℳ ( should have0>zero eigenvalue.
(- ≠ 0 solution)

cf. another derivation:
Condition for T-matrix
being singular(argument like Thoulesscriterion) Thouless, AP (1960)

1ℳ(!,-..= , )++=

Critical "+()++= ) = 0!% = #)&'( (maximum eigenvalue of ℳ ())
Corresponding eigenvector - determines Σ&'(i.e., shape and symmetry of Cooper pairs)

We use power iteration to calculate *(%&' & +.

＊We omit $ = 0 mode by hand to avoid divergence caused by lattice artifact

14

Prediction of phase diagram in !! , # -plane
8* ×128 lattice

critical beta

3+ = 0150

Normal phaseØ(Σ-. = 0)

CSC phase

10050

Non-perturbativeregion (small .)(Σ-. ≠ 0)

00.00

0.25

0.50

0.75

µq

1.00

1.25

1.50

• Peak structure: CSC phase spreads in perturbative region (large !)at specific 31.

15

Origin of peak structure
The origin of peak is elucidated by comparing with the results of quark number

150
sin3-//01…

!!

0

1500

125
Pass a level=Increase ofquark number

1750

1250
100

1000

75

750

50

500
250

|7| = 2A/C4

144 modes

25|7| = 024 modes (= 0"#$% 0&'(') 0*(+,') )

0
0.00

0.25

0.50

0.75

µq

1.00

1.25

quark number2?

critical beta
quark number (free)Ø

6 7 = arcsinh

2000

175

Energy levels of quarks(discretized because 7 is discretized in finite volume)

01.50• Peak positions correspond to !! at which quark number changes.
• At such !! , modes of fermions exist on the Fermi surface.
These modes form Cooper pairs.
• Consistent with results in NJL model in small box.
•Open of the energy gap at discrete "9 was observed.
16

Amore, Birse, McGovern, Walet,PRD (2002)

Quark-mass dependence
1400 1st peak

Energy levels of quarks(discretized because 7 is discretized in finite volume)
6 7 = arcsinh

1200

2

? sin3-/ + D53/01(?' = 0)critical beta (43 £ 128, mq = 0.1)
critical beta (43 £ 128, mq = 0.5)

1000

…

800

Ø

!!

0

critical beta (43 £ 128, mq = 0)600
|7| = 2A/C4

144 modes

400|7| = 024 modes

200

arcsinh /'1st peak(?' = 0.1)

00.0

0.2

1st peak(?' = 0.5)

0.4

µq

0.6

0.8

The peaks are shifted when 3! is changed.
Consistent with the interpretation that the peak positions are
determined by the energy spectrum of quark.
171.0

Temperature dependence
500
critical beta (43 £ 200)

3+ = 0critical beta (43 £ 152)400

critical beta (43 £ 128)
critical beta (43 £ 100)

Ø

300
200
100
00.0

0.5

1.0µq

1.52.0• ., is sensitive to the temperature.
Low temperature is preferable to observe CSC in the perturbative region.
18Contents• Motivation•QCD phase diagram and sign problem•

Possibility of investigating color superconductivity (CSC) on the lattice• Perturbative predictions•

Gap equation•

Prediction of transition points•

Phase diagram on ("! , $)-plane• Current status of our analysis
by the complex Langevin method• Summary19

Complex Langevin method (CLM)
Parisi ʼ83; Klauder ʼ84;
Aarts, Seiler, Stamatescu ʼ09;
Aarts, James,Seiler, Stamatescu ʼ11;
Seiler, Sexty, Stamatescu ʻ13; Sexty ʼ14;
Fodor, Katz, Sexty, Torok ʼ15;
Nishimura, Shimasaki ʼ15;
Nagata, Nishimura, Shimasaki ʼ15;
Sinclair, Kogut ʼ16

Complexification of gauge field42,3 ∈ SU 3 → :2,3 ∈ SL 3, ℂ
Complex Langevin equation:2,3 I + L = exp(P(−LQ2,3 R :, 31 + LS2,3 (I))):2,3 I
Drift term

Noise term

1
∫ @4A[4]D 45[7,3] = lim A[:(I)] ;Condition for justifying+→:>
The probability of the drift term should be suppressed exponentiallyat large magnitude.
Nagata, Nishimura, Shimasaki, PRD94 (2016)20

CLM on the lattice with a small aspect ratio & $180*

Calculation on 8 ×128, !! = 0.66, . = 20, 30, 40,50

2000
transition beta
quark number (free)

160
140

• We confirm CLM is justified atthese parameters.

1500

quark number

120100

Ø

1000
8060
500

4020

Signal of phase transition in the Polyakov line?

00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.40µq• Polyakov line rapidly changes near 1/.,0 ∼ 2 <D3/E• Normal phase in . > ., ?
• Polyakov line may be enhanced
due to gapless excitation from
Fermi surface.
cf) perturbative calculations on small # 2
Hands, Hollowood, Myers, JHEP (2010)

1/21• CSC in . < ., ?
→ Calculation of CSC order parameteris ongoing.
21

Summary
2000

175

critical beta
quark number (free)

150

1500

1251250
100

Gap equation is only finite number of coupledequations on the lattice with general setup.

1000

Ø•

75

750

50

500

25250

•

Phase diagram on ("9 , $)-plane without assumption
on the form and symmetry of Cooper pairs.
Peak structure is found•• Peak appears when modes on quark exists on the fermi surface
Small aspect ratio (e.g. 8F×128) is preferable to observe CSC.

•00.00

0.25

0.50

0.75

µq1.00

Outlook••

Investigation of CSC with complex Langevin method (CLM) is ongoing.
•CLM works on lattice with a small aspect ratio and $.

•Signal of phase transition in Polyakov line?

Determination of the form & symmetry of the Cooper pair from eigenvector.
•Find better operators for cleaner signals of CSC.
221750

1.25

01.50

quark number

Based on lattice perturbation for QCD,
we explicitly give the prediction for theparameter region in which CSC appears.