LatticeQCD study of the heavy quark potentialwith the multilevel algorithm

Yoshiaki Koma

NumazuCollege of Technology

Miho Koma

NihonUniversity, College of International Relations

— 滞在型研究会「チャームバリオンの構造と生成」—

Abstract

直ちにチャームバリオンに応用できる話題ではないが... ，I クォークの閉じ込め機構の説明や，クォーク模型やpNRQCDで使われる重いクォーク間ポテンシャルについて，マルチレベルアルゴリズムを用いた格子QCDシミュレーションによって新たに明らかにされた性質について示す。

I マルチレベルアルゴリズムを用いると，Polyakovループの相関関数から，クォークソースの形やゲージ固定に依らずに，精密かつ容易に基底状態のポテンシャルを計算できることを示す。

I マルチレベルアルゴリズムを用いた，３クォーク系やクォーク-ダイクォーク系などのバリオン系ポテンシャルの計算結果を示す。

Y. Koma (Numazu) 2 / 23

Heavy quark potential in QCD

I Hierarchy of energy scales, mq � mqv � mqv2 vs. ΛQCD

I Integration of the higher energy scales leads to effective field theo-ries, HQET, NRQCD, pNRQCD, ... (Application of Wilson’s renor-malization group method to QCD)

I The matching coefficients in effective field theories are to be de-termined by perturbative or nonperturbative QCD depending onthe matching scale

I pNRQCD contains matching coefficients corresponding to theheavy quark potential with relativistic corrections, which must bedetermined nonpertrubatively[pNRQCD: Brambilla etal., Rev.Mod.Phys77(2005)1423]

Y. Koma (Numazu) 3 / 23

Lattice QCD

I Define QCD in four dimensional Euclidean space-time on a hyper-cubic lattice (lattice spacing a, lattice volume L3 × T )

I Periodic boundary conditions are imposed in all space-time direc-tions

I The Wilson gauge action

S =−β

Nc

ReTr∑x,µν

[Uµ(x)Uν(x+aµ)U †µ(x+aν)U †

ν(x)]

I Lattice QCD partition function and the expectation value ofphysical operators are computed numerically by the Monte-Carlomethod

Y. Koma (Numazu) 4 / 23

Wilson loop

I The heavy quark potential can be extracted from the expectationvalue of the Wilson loop 〈W (r, t)〉, the trace of the path-orderedproduct of link variables Uµ(x)

W (2, 3) = Tr

6

6

6?

?

?

- -

��

Y. Koma (Numazu) 5 / 23

Wilson loop

I W(r, t) = Tr L(0, ~x1,2)∗αγ{T(0)T(a)· · ·T(t−a)}αβγδL(t, ~x1,2)βδ

I L(0, ~x1,2)αγ and L(t, ~x1,2)βδ

products of spatial link variables from ~x1 to ~x2 at x0 = 0 and t(quark-antiquark source/sink)

I T(x0)αβγδ ≡U4(x0, ~x1)αβU4(x0, ~x2)∗γδ (9 × 9 complex matrices)

the two-link correlators (direct product of two link variables sepa-rated by a distance r = |~x1−~x2|), which act on the color states inthe 3⊗3 rep. of SU(3) group |n; ~x1,2〉αβ :

T(x0)αλγε|n; ~x1,2〉αγ =e−En(r)a|n; ~x1,2〉λε

Y. Koma (Numazu) 6 / 23

Wilson loop

I Transfer matrix theory:

inserting the complete set of eigenstates 1=∞∑

m=0

|m; ~x1,2〉〈m; ~x1,2|

at each time slices

⇒ 〈W (r, t)〉=∞∑

n=0

wn(r, t)e−En(r)t

where wn(r, t)=〈0|L(0, ~x1,2)∗|n; ~x1,2〉〈n; ~x1,2|L(t, ~x1,2)|0〉

I −1

tln〈W(r, t)〉=E0(r) −

1

tln w0(r,t)+O(

1

te−(E1−E0)t)︸ ︷︷ ︸

unwanted terms

⇒ V (r) ≡ E0(r) = − limt→∞

1

tln〈W(r, t)〉

Y. Koma (Numazu) 7 / 23

Wilson loop

I high simulation cost for large t, where t < T/2

I It is not straightforward to extract the ground state potential (atlong distances) due to contamination from excited states as t can-not be large practically

⇒ good choice of the spatial quark source, smearing technique forbetter overlap with the ground state w0 → 1 needed

Y. Koma (Numazu) 8 / 23

PLCF

I The heavy quark potential can also be extracted from the ex-pectation value of the Polyakov loop correlation function (PLCF)〈TrP (~x1)TrP (~x2)

∗〉, a pair of Polyakov loops separated by a dis-tance r ≡ |~x1−~x2|

TrP (~x1)TrP (~x2)∗ = Tr Tr

~x1 ~x2

6

6

6

6

6?

?

?

?

?

f

f

v

vI TrP (~x1)TrP (~x2)

∗ = {T(0)T(a)· · · T(T −a)}ααγγ

Y. Koma (Numazu) 9 / 23

PLCF

I Transfer matrix theory:

inserting the complete set of eigenstates 1=∞∑

m=0

|m; ~x1,2〉〈m; ~x1,2|

at each time slices

⇒ 〈TrP (~x1)TrP (~x2)∗〉=

∞∑n=0

wne−En(r)T (w0 = 1)

I −1

Tln〈TrP (~x1)TrP (~x2)

∗〉=E0(r) +O(1

Te−(E1−E0)T )︸ ︷︷ ︸

negligible

⇒ V (r) ≡ E0(r) = −1

Tln〈TrP (~x1)TrP (~x2)

∗〉

I Theoretically clean, but impossible to compute the PLCF accu-rately with ordinary simulations

Y. Koma (Numazu) 10 / 23

The multilevel algorithm

I The multilevel algorithm is one of the most powerful noize reduc-tion techniques in lattice QCD, which is applicable when the latticeQCD action consists of the sum of local variables

(ex.) The Wilson gauge action (sum of plaquette variables)

I The multilevel algorithm computes the correlation functions of ex-tremely small expectation values from the product of relativelylarge “sublattice average” of its components

⇒ “ hierarchical integration scheme of the partition function ”

[Luscher&Weisz, JHEP0109(2001)010, JHEP0207(2002)049]

Y. Koma (Numazu) 11 / 23

PLCF with the multilevel algorithm

I (ex.) case T/a = 4, Ntsl = 2, Nsub =T/a

Ntsl= 2

1. construct the Polyakov line correlators from the two-link correla-tors:T(0)αλγεT(a)λβεδ = {T(0)T(a)}αβγδ

α

β

γ

δ

6

6?

?

λλ

εε

⊗

⊗⇒

6

?

⊗

α

β

γ

δ

⇒ compute the sublattice averages of the Polyakov line correlators,[T(0)T(a)]αβγδ and [T(2a)T(3a)]αβγδ, by internal update of gaugeconfigurations, where the spatial links at the sublattice boundariesremain intact during the internal update (iupd)

Y. Koma (Numazu) 12 / 23

PLCF with the multilevel algorithm

2. construct the PLCF from the sublattice averages of the Polyakovline correlators:

[T(0)T(a)]αβγδ[T(2a)T(3a)]βαδγ =[T(0)T(a)T(2a)T(3a)]ααγγ

α

β

γ

δβ δ

α γ

66

66

??

??

⊗

⊗

⇒

66

??

α γ

α γ

⊗

I The number of internal updates Niupd and the number of timeslices in a sublattice Ntsl must be tuned appropriately dependingon β

Y. Koma (Numazu) 13 / 23

Efficiency of the multilevel algorithm

I (ex.) q-q potential and force from the PLCF

4

3

2

1

0

-1

-2

-3

V0

(r)

[1/r

0]

2.52.01.51.00.50.0

r / r0

β=5.85 β=6.0 β=6.2 β=6.3 fit curve(β=6.0)

8

6

4

2

0

d V

0 (r

) / d

r [

1/r 0

2 ]

2.52.01.51.00.50.0

r / r0

β=5.85 β=6.0 β=6.2 β=6.3

fit curve (β=6.0)

[Koma&Koma, Nucl.Phys.B769(2007)79 etc.]

Y. Koma (Numazu) 14 / 23

Other applications

I relativistic corrections to the q-q potential (input in pNRQCD)(field strength correlators)

I qqq potential (multi-quark system) ?

I q-qq potential ?

I flux-tube profile

I glueball spectrum

I energy momentum tensor

I ...

Y. Koma (Numazu) 15 / 23

Why the multilevel algorithm works ?

I (ex.) The PLCF for the q-q system with 2 sublattices:

⇒ fixed spatial links at x0 =0, T/2 correspond toinserting the two fixed intermediate states,

|φ1〉αβ =∞∑

n=0

an|n; ~x1,2〉αβ, |φ2〉αβ =∞∑

m=0

bm|m; ~x1,2〉αβ,

where an and bn are unknown

I TrP (~x1)TrP (~x2)∗

=Tr

[〈φ1|T(0)· · ·T(

T

2−a)|φ2〉

][〈φ2|T(

T

2)· · ·T(T−a)|φ1〉

]=∑αγλε

( ∞∑n=0

a∗nbne−En(r)(T/2) ·

∞∑m=0

b∗mame−Em(r)(T/2)

)

Y. Koma (Numazu) 16 / 23

Why the multilevel algorithm works ?

I If O(e−(E1−E0)(T/2)) is already negligible:

TrP (~x1)TrP (~x2)∗ =

∑αγλε

|a0|2|b0|2e−E0(r)T︸ ︷︷ ︸independent of αγλε

⇒ E0(r) can be extracted by choosing the number of time slicesNtsl in a sublattice appropriately

⇒ Each color component of the intermediate states equally con-tributes to the potential (practically, |a0|2 = |b0|2 = 1)

⇒ Possible to obtain a gauge-invariant potential from the gauge-variant PLCF with partial intermediate states ?

Y. Koma (Numazu) 17 / 23

Demonstration 1

I q-q potential (β = 6.0, 244 lattice, Nsub = 6)

[T]αβγδ(full) vs. [T]ααγγ(diagonal)

full : diag = 59049 : 1 = (3Nsub−1)2 : 1

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

PP*

1032 3 4 5 6 7 8 9

1042 3 4 5 6 7 8 9

105

# of sublattice update

full diagonal

r/a = 1

r/a = 2

r/a = 3

r/a = 4r/a = 5r/a = 6r/a = 7r/a = 8r/a = 9r/a = 10

1.4

1.2

1.0

0.8

0.6

0.4 -

(1

/T)

ln (

PP

*)

121086420

r / a

full V(r)=-0.296(6)/r + 0.0463(5)r + 0.749(4)

diagonal fit_fullPotential

Y. Koma (Numazu) 18 / 23

Demonstration 2

I q-q potential (β = 6.0, 164 lattice, Nsub = 4)

U4αβU4∗γδ(full) vs. U4ααU4

∗γγ(diag) vs. U4α6=βU4

∗γ6=δ(offdiag)

full : diag : offdiag= 205891132094649 : 1 : 477247716

=(3T/a−1)2 :1 :((2T/a+2(−1)T/a)/3

)2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

PP*

104

105

106

# of sublattice update

full diagonal off-diagonal

r /a = 1

r /a = 2

r /a = 3

r /a = 4 r /a = 5 r /a = 6 r /a = 7

1.4

1.2

1.0

0.8

0.6

0.4

- (

1/T

) ln

(PP

*)

14121086420

r / a

full V(r) = -0.30(2)/r + 0.044(2)r + 0.80(1)

diagonal offdiagonal

Y. Koma (Numazu) 19 / 23

Application: qqq potential

I qqq potential (β = 6.0, 244 lattice, Nsub = 6)

~x1 = (r, 0, 0), ~x2 = (0, r, 0), ~x3 = (0, 0, r)

[T]αβγδεζ(full) vs. [T]ααγγεε(diagonal)

full : diag = 14348907 : 1 = (3Nsub−1)3 : 1

10-23

10-22

10-21

10-20

10-19

10-18

10-17

10-16

10-15

10-14

10-13

10-12

10-11

10-10

PPP

4 5 6 7 8 9

104

2 3 4 5 6 7 8 9

105

2 3 4 5 6

# of sublattice update

full diagonal

r/a = 1

r/a = 2

r/a = 3

r/a = 4

r/a = 5

r/a = 6

r/a = 7

r/a = 8

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

- (

1/T

) ln

(PP

P)

1086420

r / a

full V(r) = -0.26(1)/r + 0.109(1)r + 0.151(9)

diagonalTakahashi et al. (PRD65)

Y. Koma (Numazu) 20 / 23

Application: qqq potential

I RY =√

6r : (ex.) r = 8 corresponds to RY = 1.8 fm

⇒ σY a2 = 0.0445 ' σqqa2

I R∆ = 3√

2r : (ex.) r = 8 corresponds to R∆ = 3.2 fm

⇒ σ∆a2 = 0.0257

I 3-quark Wilson loop vs. PLCF[Takahashi etal., PRD65(2002)114509]

���

����HHHj

6 6

6

���*HHHY

���

vs.

6

6 6

hh h

hh h

Y. Koma (Numazu) 21 / 23

Application: q-qq potential

I q-qq potential (β = 6.0, 244 lattice, Nsub = 6)

3 ⊗ (3 ⊕ 6) ⇒ 3 ⊗ 3

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

P(0

)PP(

r)

103

2 3 4 5 6 7 8 9

104

2 3 4 5 6 7 8 9

105

2

# of sublattice update

r/a = 1

r/a = 2

r/a = 3

r/a = 4 r/a = 5 r/a = 6 r/a = 7 r/a = 8 r/a = 9 r/a = 10 r/a = 11 r/a = 12

1.4

1.2

1.0

0.8

0.6

0.4

V(r

)

121086420

r / a

Q-diQ: - (1/T) ln (P(0)PP(r)) V(r) = -0.296(6)/r + 0.0463(5)r + 0.795(4)

Q-Qbar: - (1/T) ln (P(0)P*(r)) V(r) = -0.296(6)/r + 0.0463(5)r + 0.749(4)

I q-q and q-qq potential are exactly the same (up to constant)[cf. Bissey,Signal&Leinweber, PRD80(2009)114506]

Y. Koma (Numazu) 22 / 23

Summary

I By employing the multilevel algorithm, we have investigated thestatic inter-quark potentials from the Polyakov loop correlationfunction (PLCF) (constructed from the selected intermediatestates)

I While the use of partial intermediate states cannot guaranteegauge invariance of the PLCF, we have found that the functionalforms of the PLCF and the static potential are unchanged fromthe gauge invariant ones, which is due to the fact that each colorcomponent of the intermediate states “equally” contributes to thePLCF

I The method is applicable not only to the q-q potential but also toother inter-quark potentials such as the qqq and q-qq systems

Y. Koma (Numazu) 23 / 23