Is Nc=2 large ?
Norikazu Yamada (KEK / SOKENDAI)
Refs. Otake and NY, JHEP 06 (2022) 044
Kitano, Matsudo, NY, Yamazaki, PLB822, 136657 (2021)
Kitano, NY, Yamazaki, JHEP 02 (2021) 073
Kitano, Suyama, NY, JHEP 09 (2017) 137
Nonperturbative Analysis of Quantum Field Theory and its Applications @Osaka University Hall, Sep. 22, 2022

➡︎

4d SU(Nc) and 2d CP

N−1

4d SU(Nc) YM
1 a aℒ = GμνGμν − iθq
4
2

g
a a
q(x) =
ϵ
G
G
μνρσ
μν
ρσ
2
64π

2d CP

N−1

N
ℒ=
Dμz Dμz − iθq2g〜zi ∈ ℂ (i = 1,⋯, N) with z̄i zi = 1
Dμ = ∂μ + iAμ ,Aμ = iz̄∂μz

1iq(x) =
ϵμν∂μ Aν =ϵμνDμz Dμz2π2πe.g.) asymptotically free, dynamical mass, instanton, 1/N expandable
N−1
2d CP
is good testing ground for 4d SU(Nc) YM
2➡︎

4d SU(Nc) and 2d CP

N−1

4d SU(Nc) YM
1 a aℒ = GμνGμν − iθq
4
2

g
a a
q(x) =
ϵ
G
G
μνρσ
μν
ρσ
2
64π

2d CP

N−1

N
ℒ=
Dμz Dμz − iθq2g〜zi ∈ ℂ (i = 1,⋯, N) with z̄i zi = 1
Dμ = ∂μ + iAμ ,Aμ = iz̄∂μz

1iq(x) =
ϵμν∂μ Aν =ϵμνDμz Dμz2π2πe.g.) asymptotically free, dynamical mass, instanton, 1/N expandable
N−1
2d CP
is good testing ground for 4d SU(Nc) YMθ dependence of vacuum energy density
2

Vacuum energy density f(θ)Z(θ)−Vf(θ)
e
=
Z(0)
where

Z(θ) =∫

Ue−S+iθQ

Q = d x q(x)
∫

𝒟

4

3

⬇︎

Vacuum energy density f(θ)Z(θ)−Vf(θ)e=
Z(0)• In both theories,

Q ∈ ℤ ⇒ f(θ) = f(θ + 2π)
S is CP even ⇒ f(θ) = f(−θ)

whereZ(θ) =∫

Ue−S+iθQf(π − θ′) = f(π + θ′)

Q = d x q(x)
∫

￼

￼

𝒟

4

3

Expected θ dependence: large-Nc vs instanton
Large Nc [Witten (1980, 1998)]⇒ f(θ) = χ/2 min(θ + 2πk)
k∈ℤ

2

2
+ O(1/Nc )

f(θ)

-π

0

2π

π

3π

θ

• many branches• spontaneous CPV (1st order PT) at θ = ± (2n + 1)π• order parameter df(θ)/dθ |θ=π = − i⟨q(x)⟩θ=π4

Expected θ dependence: large-Nc vs instanton
Dilute instanton gas approximation (DIGA)

Large Nc [Witten (1980, 1998)]

370 2⇒ f(θ) = χ/2 min(θ + 2πk) + O(1/Nc )E. WITTEN⇒ f(θ) = χ(1 − cos θ)

2

k∈ℤ

f(θ)
f(θ)

-π

0

2π

π

3π

θ

• many branches• spontaneous CPV (1st order PT) at θ = ± (2n + 1)π• a single branch• smooth everywhere• order parameter df(θ)/dθ |θ=π = − i⟨q(x)⟩θ=π4

Expected θ dependence: large-Nc vs instanton
Dilute instanton gas approximation (DIGA)

Large Nc [Witten (1980, 1998)]

370 2⇒ f(θ) = χ/2 min(θ + 2πk) + O(1/Nc )E. WITTEN⇒ f(θ) = χ(1 − cos θ)

2

k∈ℤ

f(θ)
f(θ)

-π

0

2π

π

3π

θ

• many branches• spontaneous CPV (1st order PT) at θ = ± (2n + 1)π• a single branch• smooth everywhere• order parameter df(θ)/dθ |θ=π = − i⟨q(x)⟩θ=πSigni cant di erence in d f(θ)/d θ around θ = π
ff

fi

4

◻︎

other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þ % "ð0Þ ¼ 2!t # :f(θ)/χ vs θ/2π2059The corresponding results for CP and CP are

Figs. 4 and 5. The solid and dotted curves are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510
4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π2059The corresponding results for CP and CP are

Figs. 4 and 5. The solid and dotted curves are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510
4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.

59The corresponding results for CP and CP are

Figs. 4 and 5. The solid and dotted curves are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510
4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.
Figs. 49 and 5. The solid and dotted curves are• CP consistent with large N

59
The corresponding results for CP and CP are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510
4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.
Figs. 49 and 5. The solid and dotted curves are• CP consistent with large N
95• CP and CP shows CPV at θ = π.

59The corresponding results for CP and CP are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510
4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.
Figs. 49 and 5. The solid and dotted curves are• CP consistent with large N
95• CP and CP shows CPV at θ = π.
1
• CP consistent with the DIGA

5
9The corresponding results for CP and CP are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510

4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.
Figs. 49 and 5. The solid and dotted curves are• CP consistent with large N
95• CP and CP shows CPV at θ = π.
1
• CP consistent with the DIGA

5
9The corresponding results for CP and CP are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

510

4

CP1 E/χt

5

0
0

0.2

0.4

0.6

0.8

1

3

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1f(θ) = χ(1 − cos θ)◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.
Figs. 49 and 5. The solid and dotted curves are• CP consistent with large N
95• CP and CP shows CPV at θ = π.
1
• CP consistent with the DIGA

5
9The corresponding results for CP and CP are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

5

10f(θ) = χ(1 − cos θ)

4

5

1

CP1 E/χt

CP3 ( i.e. N = 2 ) is not large-N like,and gapless and no CPV at θ = π .

0
0

0.2

0.4

0.6

0.8

1

2

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1◻︎other hand, for CP (h’s) and CP (&’s), as shown in
Fig. 2, the coefficient extracted from a simple area-law fit
to the Wilson loops continues to rise beyond # ¼ $, violating the expected symmetry (7). We will argue that the
behavior of CP5 and CP9 for # > $ is an effect of having

Fig. 3. With the topological susceptibility fixed toN−1obtainedfrom the fluctuation of the integer-valu
2
topological charge, !t ¼ h% i=V, the solid line
parameter fit to the dilute instanton gas formulaplotted (dotted line) is the leading-order large N p

Lattice calculation: f(θ) in 2d CP[Keith-Hynes and Thacker (2008)]

1
2"ð#Þstudied.
% "ð0Þ ¼ 2!t # :N − 1 = 9, 5, 1 weref(θ)/χ vs θ/2π

20

• f(θ)/χ decreases as N decreases.
Figs. 49 and 5. The solid and dotted curves are• CP consistent with large N
95• CP and CP shows CPV at θ = π.
1
• CP consistent with the DIGA

5
9The corresponding results for CP and CP are○ : CP 9: CP 5E/χ tf(θ)/χ

15

× : CP 1

5

10f(θ) = χ(1 − cos θ)

4

5

1

CP1 E/χt

CP3 ( i.e. N = 2 ) is not large-N like,and gapless and no CPV at θ = π .

0
0

0.2

0.4

0.6

0.8

1

⟺ 2Haldane conjecture

θ/2π

5

1

FIG. 2 (color online). The free energy density "ð#Þ for CP(!’s), CP5 (h’s), and CP9 (&’s) as measured from fractionally1Lattice calculations of f(θ) in 4d SU(N)
1 a aℒθ = GμνGμν − iθq42⟨Q ⟩θ=0χ=V⟨Q 4⟩θ=0 − 3⟨Q 2⟩2θ=0
b2 = −
12⟨Q 2⟩θ=0• Direct simulation ❌⇨ sign problem⟨Q 6⟩θ=0 − 15⟨Q 2⟩θ=0⟨Q 4⟩θ=0 + 30⟨Q 2⟩3θ=0
b4 =360⟨Q 2⟩θ=0• Taylor expansion around θ = 0

χ 2
2
4
f(θ) = θ (1 + b2θ + b4θ + ⋯)2⋮

• Determine each coe cient on the lattice by

1
If DIGA works, f(θ) = χ(1 − cos θ) ⇒ b2 = −
, ⋯
12

ffi

6

First two coefficients for Nc ≥ 3
c224
χ/σ = C∞ + 2 + O(1/Nc )
Nc
0.04

0.03

C
0.02

0.01

0.00
0.00

N=8 N=6 N=5

0.02

0.04

N=4

0.06

1/N

2

N=3

0.08

0.10

0.12

1: Results for C ≡ χ/σ 2 versus 1/N 2 , for N = 3, 4, 5, 6, 8. We show MC data taken from Refs. [195]rcle, by cooling), [375] (square, by cooling), [379] (diamonds, by cooling), [164] (left triangle, by[191] (cross, by overlap), [30] (triangle, by heating), [222] (open circle, by HYP smoothing). Then the y-axis, indicated by an asterisk, shows the extrapolations to N = ∞ obtained in Ref. [195].
ta have been slightly shifted along the x-axis to make them more visible.
7[Review by Vicari and Panagopoulos (2018)]

First two coefficients for Nc ≥ 3[Review by Vicari and Panagopoulos (2018)]b̄24b2 = 2 + O(1/Nc )
Nc

c
2
2
4
χ/σ = C∞ + 2 + O(1/Nc )
Nc
0.04
0.4

0.03

0.0
2

C

N b2

0.02

-0.4

0.01

N=8 N=6 N=5

N=4

N=3

-0.8

-1.2

0.00
0.00

0.02

0.04

0.06

1/N

2

0.08

0.10

0.12

3

4

5

6

N

2
FigurePlottaken
of Nfrom
b2 Refs.
versus
N , for N = 3, 4, 6. We show MC data taken from Ref. [195] (c1: Results for C ≡ χ/σ 2 versus 1/N 2 , for N = 3, 4, 5, 6, 8. We
show13:MC data[195]
rcle, by cooling), [375] (square, by cooling), [379] (diamonds,
by cooling),[164][199](left (square,
triangle, bycooling),from Ref.
by heating), from Ref. [277] (diamond, by overlap). Some data f[191] (cross, by overlap), [30] (triangle, by heating), [222]have(openbeen
circle,by HYPsmoothing).
The x-axis to make them more visible.
slightly
shiftedalong then the y-axis, indicated by an asterisk, shows the extrapolations to N = ∞ obtained in Ref. [195].
ta have been slightly shifted along the x-axis to make them more visible.
7

First two coefficients for Nc ≥ 3[Review by Vicari and Panagopoulos (2018)]b̄24b2 = 2 + O(1/Nc )
Nc

c
2
2
4
χ/σ = C∞ + 2 + O(1/Nc )
Nc
0.04
0.4

0.03

0.0
2

C

N b2

0.02

-0.4

0.01

N=8 N=6 N=5

N=4

N=3

-0.8

-1.2

0.00
0.00

0.02

0.04

0.06

1/N

2

0.08

0.10

0.12

3

4

5

6

N

These behavior looks smooth ⇒ Nothing special happens down to N = 3

2
FigurePlottaken
of Nfrom
b2 Refs.
versus
N , for N = 3, 4, 6. We show MC data taken from Ref. [195] (c1: Results for C ≡ χ/σ 2 versus 1/N 2 , for N = 3, 4, 5, 6, 8. We
show13:MC data[195]crcle, by cooling), [375] (square, by cooling), [379] (diamonds,
by cooling),[164][199](left (square,
triangle, bycooling),from Ref.
by heating), from Ref. [277] (diamond,by overlap). Some data f[191] (cross, by overlap), [30] (triangle, by heating), [222]have(openbeen
circle,by HYPsmoothing).
The x-axis to make them more visible.
slightly
shiftedalong then the y-axis, indicated by an asterisk, shows the extrapolations to N = ∞ obtained in Ref.c [195].
ta have been slightly shifted along the x-axis to make them more visible.
7

How about N =2 ?

χ and b2 in SU(2) at T = 0
0.1

[14]
[15]
[16]
[37], [43], [44], [45]this work[14]
ph fitχ / σ2

0.08

0.1
0.06

χ/σ

2

[15]
0.04
0.08
[16]
[37], [43], [44], [45]
0.02
0.06
this work
ph fit0.04 0
0
0.1

0.2

0.3

0.2

0.3

0.2

0.3

1/N2

0.02
0
0

0.1

0

b2

1/N2

DIGA
[15]
[16]
this work
ph fit

-0.05

b2

0

-0.1

-0.05

0

DIGA 0.1[15][16]
this work
ph fit2

1/N2
8[Kitano, NY, Yamazaki (2021)]

χ and b2 in SU(2) at T = 0
0.1

[14]
[15]
[16]
[37], [43], [44], [45]this work[14]
ph fitχ / σ2

0.08

0.1
0.06

χ/σ

2

[15]
0.04
0.08
[16]
[37], [43], [44], [45]
0.02
0.06
this work
ph fit0.04 00
0.1• Still smoothly connected down to Nc = 2
0.2

0.3

0.2

0.3

0.2

0.3

1/N2

0.02
0
0

0.1

0

b2

1/N2

DIGA
[15]
[16]
this work
ph fit

-0.05

b2

0

-0.1

-0.05

0

DIGA 0.1[15][16]
this work
ph fit2[Kitano, NY, Yamazaki (2021)]

1/N2
8χ and b2 in SU(2) at T = 0
0.1

[14]
[15]
[16]
[37], [43], [44], [45]this work[14]
ph fitχ / σ2

0.08

0.1
0.06

χ/σ

2

[15]
0.04
0.08
[16]
[37], [43], [44], [45]
0.02
0.06
this work
ph fit0.04 00
0.1• Still smoothly connected down to Nc = 2
0.2

0.3

0.2

0.3

0.2

0.3

1/N2

0.02

1
(i.e. not instanton-like)• b2 ≠ −
12

0
0

0.1

0

b2

1/N2

DIGA
[15]
[16]
this work
ph fit

-0.05

b2

0

-0.1

-0.05

0

DIGA 0.1[15][16]
this work
ph fit2[Kitano, NY, Yamazaki (2021)]

1/N2
8χ and b2 in SU(2) at T = 0
0.1

[14]
[15]
[16]
[37], [43], [44], [45]this work[14]
ph fitχ / σ2

0.08

0.1
0.06

χ/σ

2

[15]
0.04
0.08
[16]
[37], [43], [44], [45]
0.02
0.06
this work
ph fit0.04 00
0.1• Still smoothly connected down to Nc = 2
0.2

0.3

0.2

0.3

1/N2

0.02

[Kitano, NY, Yamazaki (2021)]

1(i.e. not instanton-like)• b2 ≠ −
12

0
0

0.1

0

b2

DIGA
[15]
[16]
this work
ph fit

-0.05

b2

0

-0.1

-0.05

Speculation:
SU(2) YM belongs to large Nc class and
CPV takes places at θ = π.

1/N2

0

DIGA 0.1[15][16]
this work
ph fit
2

0.2

0.3

1/N2
8

Conjectured θ-T phase diagram• θ-vacuum undergoes CPV at θ = π when Nc ≫ 1.
• Above Tc, instanton calc. ⇒ no CPV at θ = π .
[Frison, Kitano, Matsufuru, Mori and NY(2016)]• “For general Nc , CP has to be broken at θ = π if the vacuumis in the confining phase.”[Gaiotto, et al.(2017)], [Kitano, Suyama, NY(2017)]• Tc(θ) is available for SU(3) around θ = 0.
[D’Elia, Negro(2012, 2013)], [Otake, NY (2022)]• Numerical evidences and our speculation ⇒ CPV for Nc ≥ 2[Kitano, NY, Yamazaki(2021)]

9

Conjectured θ-T phase diagram• θ-vacuum undergoes CPV at θ = π when Nc ≫ 1.
• Above Tc, instanton calc. ⇒ no CPV at θ = π .

T[Frison, Kitano, Matsufuru, Mori and NY(2016)]• “For general Nc , CP has to be broken at θ = π if the vacuumis in the confining phase.”[Gaiotto, et al.(2017)], [Kitano, Suyama, NY(2017)]

4d SU(Nc) YM
for Nc ≫ 1

decon neTc
con ne• Tc(θ) is available for SU(3) around θ = 0.
• Numerical evidences and our speculation ⇒ CPV for Nc ≥ 2

CP[D’Elia, Negro(2012, 2013)], [Otake, NY (2022)][Kitano, NY, Yamazaki(2021)]

π

θ
fi

fi

9

2π

Conjectured θ-T phase diagram• θ-vacuum undergoes CPV at θ = π when Nc ≫ 1.
• Above Tc, instanton calc. ⇒ no CPV at θ = π .

T[Frison, Kitano, Matsufuru, Mori and NY(2016)]• “For general Nc , CP has to be broken at θ = π if the vacuumis in the confining phase.”[Gaiotto, et al.(2017)], [Kitano, Suyama, NY(2017)]

4d SU(Nc) YM
for Nc ≫ 1

decon neTc
con ne• Tc(θ) is available for SU(3) around θ = 0.
• Numerical evidences and our speculation ⇒ CPV for Nc ≥ 2

CP[D’Elia, Negro(2012, 2013)], [Otake, NY (2022)][Kitano, NY, Yamazaki(2021)]

We want to check whether SU(2) is large Nc-like.
⇒ How to avoid the sign problem ?

fi

fi

9

π

θ

2π

Sub-volume method[Kitano,Matsudo,NY,Yamazaki(2021)]

c.f. [Keith-Hynes and Thacker (2008)]

does not rely on any expansions
Replace Q with Qsub =4∑q(x) ∉ ℤ

x∈Vsubwhere Vsub = l is a sub-volume.

Z
(θ)
1
sub
−Vsub fsub(θ)−S+iθQsubiθQsub
e
=
=
Ue
= ⟨e
⟩
Z(0)
Z(0) ∫
1
fsub(θ) = −ln⟨ cos(θQsub) ⟩
Vsub

Vsub

s(θ)
2
f(θ) = lim fsub(θ) = lim f(θ)++ O(1/l )}Vsub→∞
l→∞ {
l
𝒟

θ≠0

10

θ=0

Vfull

Some remarks on the sub-volume method
What is the suitable range for Vsub ?

•4
Vsub ≫ ldyn (ldyn: dynamical length scale)•4
As long as Vsub ≫ ldyn , fsub(θ) is expected to show a scalings(θ)
2+ O(1/l ).
behavior, fsub(θ) = f(θ)+
l•As Vsub → Vfull, nite size e ects may appear. ⇒Vsub ≪ Vfull.

4
ldyn

Vsub

θ≠0
θ=0

ff

fi

11

Vfull

Some remarks on the sub-volume method
What is the suitable range for Vsub ?

•4
Vsub ≫ ldyn (ldyn: dynamical length scale)•4
As long as Vsub ≫ ldyn , fsub(θ) is expected to show a scalings(θ)
2+ O(1/l ).
behavior, fsub(θ) = f(θ)+
l•As Vsub → Vfull, nite size e ects may appear. ⇒Vsub ≪ Vfull.

Vsub

4ldyn ≪ Vsub ≪ Vfull

ff

θ≠0
θ=0

11

fi

4
ldyn

Vfull

Lattice parameters and observables• SU(2) YM by Symanzik improved gauge action

Vfull

4
• β = 2 = 1.975 [cf. 1/(aTc) = 9.50]
g•V

3full = 24

θ=0× {48, 8, 6} (T = 0, 1.2Tc, 1.6Tc)• Periodic boundary conditions• # of con gs = { 68000 , 5000 , 5000 }
4

θ≠0

3

• Vsub = l for T=0 and Vsub = l × NT for nite T• After applying APE smearing, we estimate

fi

fi

1
ln⟨ cos(θQsub) ⟩✓ f(θ) = − lim
Vsub→∞ Vsubdf(θ)
1 ⟨ Qsub sin(θQsub) ⟩
= lim
✓
dθ
Vsub→∞ Vsub
⟨ cos(θQsub) ⟩

Vsub

12

θ

limit
at
lFIG.
→3: ∞
T
=
0
The ✓ dependence of f (✓) at T = 1.2 T .
c

θ=π/2, nAPE=20
θ=π, nAPE=20θ=3π/2, nAPE=20• Vsub = l with l ∈ {10, 12, ⋯, 20}• Linear extrapolation with
4

15

as(θ)
fsub(θ) = f(θ) +l10

works well.

4

a fsub(θ) × 10

5

20

5
0
0

0.02

0.04

0.06

0.08

0.1

1/l
FIG. 4:13
The linear extrapolation of fsub (✓) to the infinite

nAPE → 0 limit at T = 0
18

•

16

T=0
1.2determinedTc
Fit range nAPE30= [20, 40]in∫dθ ds1/dθ
[Kitano, NY, Yamazaki (2021)].
20
Linear t works well.

•
10• Monotonic function f(π) < f(3π/2)

12s1(θ) / a χ

4a f(θ) × 10

5

14

40

10
8

θ=π/2
π
3π/2

6
4

0
-10

2
0

10

20

30

40

50

60

0

nAPE

FIG. 5: The linear extrapolation of a f (θ) to nAPE14 = 0.
4

fi

-20
π/2

π

θ
FIG. 7: Comparison of θ dependence os1 (θ) at T = 0 and 1.2 Tc .

θ dependence of f(θ) at T =4 0
141210f(θ) / χ• Succeed to calculate up to θ ∼ 3π/2
• Monotonically increasing function• Inconsistent with DIGA

T=0
2
θ /2
1-cosθ∫dθ df/dθ
full volume

8df• Numerical consistency with ∫dθ dθ

6
4
2
0
0

π/2

π

3π/2

2π

θ
5

T=0

15

at T =π 0
df(θ)/dθ
0
π/2
0

3π/2

2π

θ
5

T=0
θ
sin θ
Δf
full volume

df(θ)/dθ / χ

43• Order parameter is non-zero
df(θ)/dθ2⇒ spontaneous CPV at θ = π

1
0
-1
0

π/2

π

3π/2

2π

θ
FIG. 6:θ=π= − i⟨ q(x) ⟩θ=π ≠ 0

16

f (θ) (top) and df (θ)/dθ (bottom). The results ob-θ dependence of f(θ) at T 3= 1.2Tc
654f(θ) / χ• Large uncertainty due to ambiguity of the

1.2 Tc2
θ /2
1-cosθ∫dθ df/dθ
full volume3

scaling region• Within large uncertainty, consistent with the
DIGA• Similar results at T = 1.6 Tc

2
1
0
-1
0

π/2

π

3π/2

2π

θ
FIG. 3: The θ dependence of f (θ) at T = 1.2 Tc . The results
obtained with different fit ranges in Fig. 1, l ∈ [12, 16], [14, 18]
17
and [16, 20], are shown in triangle-up, circle and triangle-

f(π − θ) = f(π + θ) ?
14

T=0
2
θ /2
1-cosθ∫dθ df/dθ
full volume

1210f(θ) / χ

4

8
6
4
2
0
0

π/2

π

3π/2

2π

θ
5
4

T=0
θ
sin θ

18

f(π − θ) = f(π + θ) ?
14

Interpretation:
Sub-volume method sticks to the original brancheven after passing through the transition point.

T=0
2
θ /2
1-cosθ∫dθ df/dθ
full volume

1210f(θ) / χ

4

8

f(θ)

6
4
2
0

-π

0

π/2

π

3π/2

4

T=0
θ
sin θ

2π

π
θ

2π

θ
5

0

18

3π

f(π − θ) = f(π + θ) ? (Cont’d)• Similar experience in the the static potential on

the dynamical con gs, where“string breaking” isexpected to occur at large separation but …

⇒ Nothing but overlap problem

fi19termined by the energy balance:f(π − θ) = f(π + θ) ? (Cont’d)

2mQ + σRbr = 2mQq̄

(5)(here we neglect a weak Coulomb interaction between
the quarks in the unbroken state and we also disregard
exponentially suppressed van der Waals interaction between the heavy-light mesons in the broken state). If
R > Rbr , then the string breaks and light-heavy mesons
are formed, Fig. 1 (we consider here a simplest case ignoring a multiple pair production via string fragmentation).

• Similar experience in the the static potential on

the dynamical con gs, where“string breaking” isexpected to occur at large separation but …

2e
,
3

e
qd = − .
3(8)In a strong magnetic field the lowest Landau level(LLL) with n = 0 and sz = 1/2 plays a dominant rolein the quark’s motion because the excited states are too
heavy. Indeed, for the soft (low-momentum) fermions thespin flips, sz → −sz and jumps to the higher states withn ! 1 are energetically suppressed by the typical gap√(9)δEq ∼ 2/lq ,!
lq (B) = 1/ |eq B| ,Chernodub (2010)
FIG. 1. The conventional string breaking: the QCD string
spanned between the static quark Q and the static antiquarkQ̄ breaks due to light q q̄ pair creation.

The string breaking was observed in lattice simulationsof QCD with two flavors of equal-mass quarks [16]. An
extrapolation of the lattice results to the real QCD gives
the following string breaking distance [16]:Rbr ≈ 1.13 fm ,(6)

where statistical and systematical errors are of the orderof 0.1 fm each.
QCD string breaking at nonzero magnetic field.
A sufficiently strong background magnetic field shouldfiqu = +

where⇒ Nothing but overlap problem

19

ber n = 0, 1, 2, . . . labels the Landau levels. The signs“±” in front of the square root in Eq. (7) refer to, respectively, particle and antiparticle branches of the energy spectrum. The electric charges of the light u and dquarks are, respectively:

q = u, d(10)is the magnetic length of the quark q.
Thus, at the LLL the motion of quarks becomes essentially (1+1) dimensional. The quark moves along theaxis of the magnetic field and the longitudinal dynamics
of a free quark is governed by one-dimensional relativistic
dispersion relation:!
ωLLL (p! ) = ± p2! + m2q .
The transverse dynamics (i.e., the motion of the quark in
the plane orthogonal to the magnetic field) is restricted
to a region of a typical size of the order of the magnetic
length (10),|δr|q " lq (B) .

(11)

In QCD the influence of the magnetic field should become significant when the magnetic length (10) becomescomparable with a typical QCD length scale ΛQCD ∼1 fm−1 ∼ 200 MeV,
lq (B) ' Λ−1
QCD ' Rbr .

(12)differencein the balance:
simulation details. We therefore carry out atermined by
the energy

quark mass. Other five values of K val,i correspond to then = 0, 1, 2, . . . labels the Landau levels. The signs
set of quenched calculations of the hadron spectrum using bermassof strange quarks treated in the quenched approxima“±”
inof the squarein Eq. (7)‘‘diagonalrefer to, rethe same2m
latticeactionsandsimulation
parameters
as
those
(5)
tion. Infront
the following,
we useroot
the abbreviation
Q + σRbr = 2mQq̄
for full QCD runs.
spectively,
particlehadronand correlators
antiparticle
branches
the endata’’ to represent
or masseswith aofquark
the lat- ergy
Our simulations
are performed
at ! "6.0, where(here we neglecta weak Coulombinteraction
betweenmass
combinationin electric
which allcharges
valence ofquark
spectrum.
The
themasses
light uareand d
0.1074#14$
fm. We takefixed fromm " equals
equal are,to therespectively:
sea quark mass.
the quarksticein spacing
the unbroken
state
and wealso disregard
quarksc SW"1.769 which is the value determined nonWe employ meson operators defined byexponentially suppressed van der Waals interaction beperturbatively by the ALPHA Collaboration %35&. Three late2etween the ticeheavy-lightmesonsin thestate).
If in(=f ) +(g) ,
q
=
−
.
q
sizes 123 !48,
163 !48,and broken
203 !48 are
employeddu#6$ (8)M # x $ "q̄ x 'q x 3 , '"I, ( 5 , ( ) , (5( ) ,
3
R > Rbr , thenstring breaks
andeffects.
light-heavy mesons
order theto investigate
finite-size
Gauge
configurations
are generated
with casea combination
are formed, Fig.
1 (weconsider herea simplest
ignor- of In a strong magnetic field the lowest Landau level
the heat-bathand over-relaxation
We call four (LLL)
wherewithf and ng areis the coordinateson
ing a multiple
pair production
via stringalgorithms.
fragmentation).
= 0flavor
andindicessz = and
1/2x plays† a dominant roleheat-bath sweeps with a succeeding over-relaxation step anthe lattice. Meson correlators * M (x)M (0) + are calculatedin
the quark’s motion because the excited states are too
iteration. We accumulate statistics of 60 000 iterations on
for the following 11 combinations of valence quark masses:f(π − θ) = f(π + θ) ? (Cont’d)• Similar experience in the the static potential on

the dynamical con gs, where“string breaking” isexpected to occur at large separation but …

⇒ Nothing but overlap problem

heavy. Indeed, for the soft (low-momentum) fermions thespin flips, sz → −sz and jumps to the higher states withn ! 1 are energetically suppressed by the typical gap
Static potential√(9)δEq ∼ 2/lq ,where!
lq (B) = 1/ |eq B| ,q = u, d(10)is the magnetic length of the quark q.
Thus, at the LLL the motion of quarks becomes essentially (1+1) dimensional. The quark moves along theaxis of the magnetic field and the longitudinal dynamics
a freequark
by 0.1355,
one-dimensional
FIG. 7. Static quark potential on 203 !48. Left and right offiguresshowdata atisKgoverned
respectively. relativistic
R andsea"0.1340
dispersion relation:
Chernodub (2010)
054502-6!
FIG. 1. The conventional string breaking: the QCD string
ωLLL (p! ) = ± p2! + m2q .
spanned between the static quark Q and the static antiquarkQ̄ breaks due to light q q̄ pair creation.

The string breaking was observed in lattice simulationsof QCD with two flavors of equal-mass quarks [16]. An
extrapolation of the lattice results to the real QCD gives
the following string breaking distance [16]:Rbr ≈ 1.13 fm ,(6)

where statistical and systematical errors are of the orderof 0.1 fm each.
QCD string breaking at nonzero magnetic field.
A sufficiently strong background magnetic field shouldfi

19

The transverse dynamics (i.e., the motion of the quark in
the plane orthogonal to the magnetic field) is restricted
to a region of a typical size of the order of the magnetic
length (10),|δr|q " lq (B) .

(11)

In QCD the influence of the magnetic field should become significant when the magnetic length (10) becomescomparable with a typical QCD length scale ΛQCD ∼1 fm−1 ∼ 200 MeV,
lq (B) ' Λ−1
QCD ' Rbr .

(12)differencein the balance:
simulation details. We therefore carry out atermined by
the energy

quark mass. Other five values of K val,i correspond to then = 0, 1, 2, . . . labels the Landau levels. The signs
set of quenched calculations of the hadron spectrum using bermassof strange quarks treated in the quenched approxima“±”
inof the squarein Eq. (7)‘‘diagonalrefer to, rethe same2m
latticeactionsandsimulation
parameters
as
those
(5)
tion. Infront
the following,
we useroot
the abbreviation
Q + σRbr = 2mQq̄
for full QCD runs.
spectively,
particlehadronand correlators
antiparticle
branches
the endata’’ to represent
or masseswith aofquark
the lat- ergy
Our simulations
are performed
at ! "6.0, where(here we neglecta weak Coulombinteraction
betweenmass
combinationin electric
which allcharges
valence ofquark
spectrum.
The
themasses
light uareand d
0.1074#14$
fm. We takefixed fromm " equals
equal are,to therespectively:
sea quark mass.
the quarksticein spacing
the unbroken
state
and wealso disregard
quarksc SW"1.769 which is the value determined nonWe employ meson operators defined byexponentially suppressed van der Waals interaction beperturbatively by the ALPHA Collaboration %35&. Three late2etween the ticeheavy-lightmesonsin thestate).
If in(=f ) +(g) ,
q
=
−
.
q
sizes 123 !48,
163 !48,and broken
203 !48 are
employeddu#6$ (8)M # x $ "q̄ x 'q x 3 , '"I, ( 5 , ( ) , (5( ) ,
3
R > Rbr , thenstring breaks
andeffects.
light-heavy mesons
order theto investigate
finite-size
Gauge
configurations
are generated
with casea combination
are formed, Fig.
1 (weconsider herea simplest
ignor- of In a strong magnetic field the lowest Landau level
the heat-bathand over-relaxation
We call four (LLL)
wherewithf and ng areis the coordinateson
ing a multiple
pair production
via stringalgorithms.
fragmentation).
= 0flavor
andindicessz = and
1/2x plays† a dominant roleheat-bath sweeps with a succeeding over-relaxation step anthe lattice. Meson correlators * M (x)M (0) + are calculatedin
the quark’s motion because the excited states are too
iteration. We accumulate statistics of 60 000 iterations on
for the following 11 combinations of valence quark masses:f(π − θ) = f(π + θ) ? (Cont’d)• Similar experience in the the static potential on

the dynamical con gs, where“string breaking” isexpected to occur at large separation but …

⇒ Nothing but overlap problem

heavy. Indeed, for the soft (low-momentum) fermions thespin flips, sz → −sz and jumps to the higher states withn ! 1 are energetically suppressed by the typical gap
Static potential√(9)δEq ∼ 2/lq ,where!
lq (B) = 1/ |eq B| ,q = u, d(10)is the magnetic length of the quark q.
Thus, at the LLL the motion of quarks becomes essentially (1+1) dimensional. The quark moves along theaxis of the magnetic field and the longitudinal dynamics
a freequark
by 0.1355,
one-dimensional
FIG. 7. Static quark potential on 203 !48. Left and right offiguresshowdata atisKgoverned
respectively. relativistic
R andsea"0.1340
dispersion relation:
Chernodub (2010)
054502-6!
FIG. 1. The conventional string breaking: the QCD string
ωLLL (p! ) = ± p2! + m2q .
spanned between the static quark Q and the static antiquark• In the present case, the expected transition is the one from domain to a bagQ̄ breaks due to light q q̄ pair creation.

The string breaking was observed in lattice simulationsof QCD with two flavors of equal-mass quarks [16]. An
extrapolation of the lattice results to the real QCD gives
the following string breaking distance [16]:like object although we couldn’t see it …

Rbr ≈ 1.13 fm ,(6)

where statistical and systematical errors are of the orderof 0.1 fm each.
QCD string breaking at nonzero magnetic field.
A sufficiently strong background magnetic field shouldfi

19

The transverse dynamics (i.e., the motion of the quark in
the plane orthogonal to the magnetic field) is restricted
to a region of a typical size of the order of the magnetic
length (10),|δr|q " lq (B) .

(11)

In QCD the influence of the magnetic field should become significant when the magnetic length (10) becomescomparable with a typical QCD length scale ΛQCD ∼1 fm−1 ∼ 200 MeV,
lq (B) ' Λ−1
QCD ' Rbr .

(12)differencein the balance:
simulation details. We therefore carry out atermined by
the energy

quark mass. Other five values of K val,i correspond to then = 0, 1, 2, . . . labels the Landau levels. The signs
set of quenched calculations of the hadron spectrum using bermassof strange quarks treated in the quenched approxima“±”
inof the squarein Eq. (7)‘‘diagonalrefer to, rethe same2m
latticeactionsandsimulation
parameters
as
those
(5)
tion. Infront
the following,
we useroot
the abbreviation
Q + σRbr = 2mQq̄
for full QCD runs.
spectively,
particlehadronand correlators
antiparticle
branches
the endata’’ to represent
or masseswith aofquark
the lat- ergy
Our simulations
are performed
at ! "6.0, where(here we neglecta weak Coulombinteraction
betweenmass
combinationin electric
which allcharges
valence ofquark
spectrum.
The
themasses
light uareand d
0.1074#14$
fm. We takefixed fromm " equals
equal are,to therespectively:
sea quark mass.
the quarksticein spacing
the unbroken
state
and wealso disregard
quarksc SW"1.769 which is the value determined nonWe employ meson operators defined byexponentially suppressed van der Waals interaction beperturbatively by the ALPHA Collaboration %35&. Three late2etween the ticeheavy-lightmesonsin thestate).
If in(=f ) +(g) ,
q
=
−
.
q
sizes 123 !48,
163 !48,and broken
203 !48 are
employeddu#6$ (8)M # x $ "q̄ x 'q x 3 , '"I, ( 5 , ( ) , (5( ) ,
3
R > Rbr , thenstring breaks
andeffects.
light-heavy mesons
order theto investigate
finite-size
Gauge
configurations
are generated
with casea combination
are formed, Fig.
1 (weconsider herea simplest
ignor- of In a strong magnetic field the lowest Landau level
the heat-bathand over-relaxation
We call four (LLL)
wherewithf and ng areis the coordinateson
ing a multiple
pair production
via stringalgorithms.
fragmentation).
= 0flavor
andindicessz = and
1/2x plays† a dominant roleheat-bath sweeps with a succeeding over-relaxation step anthe lattice. Meson correlators * M (x)M (0) + are calculatedin
the quark’s motion because the excited states are too
iteration. We accumulate statistics of 60 000 iterations on
for the following 11 combinations of valence quark masses:f(π − θ) = f(π + θ) ? (Cont’d)• Similar experience in the the static potential on

the dynamical con gs, where“string breaking” isexpected to occur at large separation but …

⇒ Nothing but overlap problem

heavy. Indeed, for the soft (low-momentum) fermions thespin flips, sz → −sz and jumps to the higher states withn ! 1 are energetically suppressed by the typical gap
Static potential√(9)δEq ∼ 2/lq ,where!
lq (B) = 1/ |eq B| ,q = u, d(10)is the magnetic length of the quark q.
Thus, at the LLL the motion of quarks becomes essentially (1+1) dimensional. The quark moves along theaxis of the magnetic field and the longitudinal dynamics
a freequark
by 0.1355,
one-dimensional
FIG. 7. Static quark potential on 203 !48. Left and right offiguresshowdata atisKgoverned
respectively. relativistic
R andsea"0.1340
dispersion relation:
Chernodub (2010)
054502-6!
FIG. 1. The conventional string breaking: the QCD string
ωLLL (p! ) = ± p2! + m2q .
spanned between the static quark Q and the static antiquark• In the present case, the expected transition is the one from domain to a bagQ̄ breaks due to light q q̄ pair creation.

The string breaking was observed in lattice simulationsof QCD with two flavors of equal-mass quarks [16]. An
extrapolation of the lattice results to the real QCD gives
the following string breaking distance [16]:like object although we couldn’t see it …

Rbr ≈ 1.13 fm ,(6)

where statistical and systematical errors are of the orderof 0.1 fm each.

f0(π + ϵ)→ f1(π + ϵ)fi

19

|δr|q " lq (B) .

(11)

In QCD the influence of the magnetic field should become significant when the magnetic length (10) becomescomparable with a typical QCD length scale ΛQCD ∼1 fm−1 ∼ 200 MeV,

4d SU(N) YM has a topologicall (B) ' Λ'R .
object called “glue bag“.
[Luscher (1978)].

QCD string breaking at nonzero magnetic field.
A sufficiently strong background magnetic field shouldf0(π + ϵ)

The transverse dynamics (i.e., the motion of the quark in
the plane orthogonal to the magnetic field) is restricted
to a region of a typical size of the order of the magnetic
length (10),

q

−1
QCD

br

(12)

differencein the balance:
simulation details. We therefore carry out atermined by
the energy

quark mass. Other five values of K val,i correspond to then = 0, 1, 2, . . . labels the Landau levels. The signs
set of quenched calculations of the hadron spectrum using bermassof strange quarks treated in the quenched approxima“±”
inof the squarein Eq. (7)‘‘diagonalrefer to, rethe same2m
latticeactionsandsimulation
parameters
as
those
(5)
tion. Infront
the following,
we useroot
the abbreviation
Q + σRbr = 2mQq̄
for full QCD runs.
spectively,
particlehadronand correlators
antiparticle
branches
the endata’’ to represent
or masseswith aofquark
the lat- ergy
Our simulations
are performed
at ! "6.0, where(here we neglecta weak Coulombinteraction
betweenmass
combinationin electric
which allcharges
valence ofquark
spectrum.
The
themasses
light uareand d
0.1074#14$
fm. We takefixed fromm " equals
equal are,to therespectively:
sea quark mass.
the quarksticein spacing
the unbroken
state
and wealso disregard
quarksc SW"1.769 which is the value determined nonWe employ meson operators defined byexponentially suppressed van der Waals interaction beperturbatively by the ALPHA Collaboration %35&. Three late2etween the ticeheavy-lightmesonsin thestate).
If in(=f ) +(g) ,
q
=
−
.
q
sizes 123 !48,
163 !48,and broken
203 !48 are
employeddu#6$ (8)M # x $ "q̄ x 'q x 3 , '"I, ( 5 , ( ) , (5( ) ,
3
R > Rbr , thenstring breaks
andeffects.
light-heavy mesons
order theto investigate
finite-size
Gauge
configurations
are generated
with casea combination
are formed, Fig.
1 (weconsider herea simplest
ignor- of In a strong magnetic field the lowest Landau level
the heat-bathand over-relaxation
We call four (LLL)
wherewithf and ng areis the coordinateson
ing a multiple
pair production
via stringalgorithms.
fragmentation).
= 0flavor
andindicessz = and
1/2x plays† a dominant roleheat-bath sweeps with a succeeding over-relaxation step anthe lattice. Meson correlators * M (x)M (0) + are calculatedin
the quark’s motion because the excited states are too
iteration. We accumulate statistics of 60 000 iterations on
for the following 11 combinations of valence quark masses:f(π − θ) = f(π + θ) ? (Cont’d)• Similar experience in the the static potential on

the dynamical con gs, where“string breaking” isexpected to occur at large separation but …

⇒ Nothing but overlap problem

heavy. Indeed, for the soft (low-momentum) fermions thespin flips, sz → −sz and jumps to the higher states withn ! 1 are energetically suppressed by the typical gap
Static potential√(9)δEq ∼ 2/lq ,where!
lq (B) = 1/ |eq B| ,q = u, d(10)is the magnetic length of the quark q.
Thus, at the LLL the motion of quarks becomes essentially (1+1) dimensional. The quark moves along theaxis of the magnetic field and the longitudinal dynamics
a freequark
by 0.1355,
one-dimensional
FIG. 7. Static quark potential on 203 !48. Left and right offiguresshowdata atisKgoverned
respectively. relativistic
R andsea"0.1340
dispersion relation:
Chernodub (2010)
054502-6!
FIG. 1. The conventional string breaking: the QCD string
ωLLL (p! ) = ± p2! + m2q .
spanned between the static quark Q and the static antiquark• In the present case, the expected transition is the one from domain to a bagQ̄ breaks due to light q q̄ pair creation.

The string breaking was observed in lattice simulationsof QCD with two flavors of equal-mass quarks [16]. An
extrapolation of the lattice results to the real QCD gives
the following string breaking distance [16]:like object although we couldn’t see it …

Rbr ≈ 1.13 fm ,(6)

where statistical and systematical errors are of the orderof 0.1 fm each.

f0(π + ϵ)→ f1(π + ϵ)fi

19

|δr|q " lq (B) .

(11)

In QCD the influence of the magnetic field should become significant when the magnetic length (10) becomescomparable with a typical QCD length scale ΛQCD ∼1 fm−1 ∼ 200 MeV,

4d SU(N) YM has a topologicall (B) ' Λ'R .
object called “glue bag“.
[Luscher (1978)].

QCD string breaking at nonzero magnetic field.
A sufficiently strong background magnetic field shouldf0(π + ϵ)

The transverse dynamics (i.e., the motion of the quark in
the plane orthogonal to the magnetic field) is restricted
to a region of a typical size of the order of the magnetic
length (10),

q

−1
QCD

br

(12)

Summary and conclusion• We have developed a sub-volume method to
calculate f(θ) up to θ ∼ O(1).

20

Summary and conclusion• We have developed a sub-volume method to
calculate f(θ) up to θ ∼ O(1).

• Our result for the order parameter suggests

Tc

for Nc ≥ 2decon necon neCP

spontaneous CPV at θ = π for the vacuum of
4d SU(2) YM.

T

4d SU(Nc) YM

π

θ
fi

fi

20

2π

Summary and conclusion• We have developed a sub-volume method toTcalculate f(θ) up to θ ∼ O(1).

• Our result for the order parameter suggests

Tc

spontaneous CPV at θ = π for the vacuum of
4d SU(2) YM.

for Nc ≥ 2decon necon ne1CP⇒ SU(2) YM belongs to large N class (not like CPmodel).

4d SU(Nc) YM

π

θ
fi

fi

20

2π

Summary and conclusion• We have developed a sub-volume method toTcalculate f(θ) up to θ ∼ O(1).

• Our result for the order parameter suggests

Tc

spontaneous CPV at θ = π for the vacuum of
4d SU(2) YM.

con ne

1

θ
20

fi

decon neπ

⇒ Nc = 2 is large.

fifor Nc ≥ 2

CP⇒ SU(2) YM belongs to large N class (not like CPmodel).

4d SU(Nc) YM

2πWhen I was a master course
student, he kindly invited
Ishikawa-san and me to theLattice QCD and started toread Rothe’s textbook.
LQCD is still my favorite
field.
I really appreciate yourguidance.

21