Lattice energy-momentum tensor from the Yang-Mills gradient
flow.
.
Hiroshi Suzuki
Kyushu University

2013/10/16 JICFuS seminar on non-perturbative physics @ Osaka University

H.S., Prog. Theor. Exp. Phys. (2013) 083B03 [arXiv:1304.0533 [hep-lat]].
H.S., work in progress

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

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1 / 39

Lattice field theory and the energy-momentum tensor (EMT)
Lattice field theory

x

^
aµ

a
µ

best successful non-perturbative formulation of QFT; keeps internal gauge
symmetries exactly. . . but quite incompatible with spacetime symmetries (translation, rotation, SUSY,
conformal, . . . )Ward–Takahashi (WT) relation associated with translational invariance (Tµν (x):energy-momentum tensor (EMT))h∂µ Tµν (x)O(y )O(z) · · · i = −δ(x − y ) h∂ν O(y )O(z) · · · i + · · ·conservation law is a special case of this:h∂µ Tµν (x)O(y )O(z) · · · i = 0,

for x 6= y , x 6= z, . . .

can we construct lattice EMT which reproduces these relations in a → 0?
if this is possible, the application will be vast (thermodynamics, viscosities,
conformal field theory, dilaton physics, vacuum energy, . . . )Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

2 / 39Naive construction will not work . . .
naive EMT for the pure Yang–Mills theory
–
»
1
1
a
a
a
a
(x)Fρσ(x)Tµν (x) = 2 Fµρ(x)Fνρ
(x) − δµν Fρσ4g0
a correct WT relation: h∂µ Tµ1 (x)T01 (0)i = C∂0 δ(x)Monte Carlo computation of LHS (x = (x0 , 0, 0, 0))0.1"correlators16x16beta2.64t0.00" u ($1*0.8971):($4*2.657*0.4356):($5*2.657*0.4356)"correlators24x24beta2.80t0.00" u ($1*0.5660):($4*167.7*0.49):($5*167.7*0.49)"correlators32x32beta2.91t0.00" u ($1*0.4125):($4*2892*0.52925625):($5*2892*0.52925625)

0.05

0

-0.05

-0.1

0

2

4

6

8

10

12

14

extremely noisy . . .
(although consistent with 0) it appears diverging as a → 0
after all, there is no guarantee that the naive expression is conserved for a → 0,since lattice regularization breaks translational invariance
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

3 / 39Possible approaches
Invent somehow a lattice formulation that is invariant under the desired symmetry(in the present case, translation) as the lattice chiral symmetry on the basis of theGinsparg–Wilson relation
this is certainly ideal, but seems formidable for spacetime symmetries . . .
(eventually, SLAC derivative?)

What the general argument says is that a linear combination of dim. 4 operators
being consistent with lattice symmetry
!
X a a
X a a
X a a
X a a
1
Tµν (x) = C1
Fµρ Fνρ − δµνFρσ Fρσ +C2 δµνFρσ Fρσ +C3 δµνFµρ Fνρ4ρρσρσρ
is conserved in a → 0; we may determine ratios of these coefficients by theconservation law (Caracciolo et al. (1989))
overall normalization should be fixed separately (expectation value in a
one-particle state? current algebra?)no one yet studied whether this construction generates correct translations oncomposite operators!
approach on the basis of SUSY algebra and Ferrara–Zumino supermultiplet (H.S.
(2012))
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

4 / 39Our approach

Use a UV finite quantity that can be related with EMT in a translationally invariant
regularization
Any regularization (including lattice) will produce the same number for such a UV
finite quantity
To define this UV finite quantity, we employ the so-called Yang–Mills gradient flow
Yang–Mills gradient flow (a diffusion equation wrt a fictitious time t ∈ R)
∂t Bµ (t, x) = Dν Gνµ (t, x) = ∆Bµ (t, x) + · · · ,Bµ (t = 0, x) = Aµ (x),where Gµν is the field strength of the flowed gauge potential:Gµν (t, x) = ∂µ Bν (t, x) − ∂ν Bµ (t, x) + [Bµ (t, x), Bν (t, x)],Dµ = ∂µ + [Bµ , ·]Note: the mass dimension of t is −2

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

5 / 39

Yang–Mills gradient flow or the Wilson flow

Yang–Mills gradient flow (continuum theory)∂t Bµ (t, x) = Dν Gνµ (t, x) = ∆Bµ (t, x) + · · · ,Bµ (t = 0, x) = Aµ (x)Wilson flow (lattice theory)∂t V (t, x, µ)V (t, x, µ)−1 = −g02 ∂SWilson ,V (t = 0, x, µ) = U(x, µ)Applications (Lüscher):definition of the topological charge
scale setting (just like the Sommer scale r0 )˛p
˛t 2 hE(t, x)i˛
= 0.3,and set (for instance) 8t0 = 0.5 fm
t=t0

where

1 a
a
G (t, x)Gµν
(t, x)
4 µν
define UV finite quantities ← Our usage here
computation of the chiral condensate
E(t, x) ≡Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

6 / 39Perturbative expansion of the gradient flow
Yang–Mills gradient flow∂t Bµ (t, x) = Dν Gνµ (t, x) + α0 Dµ ∂ν Bν (t, x),Bµ (t = 0, x) = Aµ (x),where the second term in RHS was introduced to suppress the gauge mode; it
can be seen that gauge invariant quantities are independent of α0 . This can besolved formally as
»
–
Z
Z t
Bµ (t, x) = d D y Kt (x − y )µν Aν (y ) +ds Kt−s (x − y )µν Rν (s, y ) ,0where K is the heat kernel and R is non-linear terms
Z ipz hi22eKt (z)µν =(δµν p2 − pµ pν )e−tp + pµ pν e−α0 tp2p p
Rµ = 2[Bν , ∂ν Bµ ] − [Bν , ∂µ Bν ] + (α0 − 1)[Bµ , ∂ν Bν ] + [Bν , [Bν , Bµ ]]Pictorially (cross: Aµ ; open circle: flow vertex R),+

Hiroshi Suzuki (Kyushu University)+

Lattice energy-momentum tensor . . .

++2013/10/16 @ Osaka Univ.

7 / 39Perturbative expansion of the gradient flow
quantum correlation function of the flowed gauge field
hBµ1 (t1 , x1 ) · · · Bµn (tn , xn )i ,is obtained by taking the expectation value of the initial value Aµ (x). For example,
the contraction of two Aµ ’s"!

=

produces the propagator of the flowed field
»
–
1
1
ab 2
2
−(t+s)p2−α0 (t+s)p2
δ g0 2 2 (δµν p − pµ pν )e+pµ pν e,(p )λ0(where t and s are flow times at the end points; λ0 is the conventional gaugeparameter). Similarly, forconsidering the contraction with the usual Yang–Mills vertex (the full circle)

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

8 / 39Gauge invariance of the gradient flow
Under the infinitesimal gauge transformation,Bµ (t, x) → Bµ (t, x) + Dµ ω(t, x),
the flow equation∂t Bµ (t, x) = Dν Gνµ (t, x) + α0 Dµ ∂ν Bν (t, x)changes to∂t Bµ (t, x) = Dν Gνµ (t, x) + α0 Dµ ∂ν Bν (t, x) − Dµ (∂t − α0 Dν ∂ν )ω(t, x)
Therefore, by choosing ω(t, x) as the solution of(∂t − α0 Dν ∂ν )ω(t, x) = −δα0 ∂ν Bν (t, x),ω(t = 0, x) = 0,α0 can be changed as

α0 → α0 + δα0That is, Bµ (t, x)’s corresponding to different α0 ’s are related by a gaugetransformation
Also, by choosing ω(t, x) as the solution of(∂t − α0 Dν ∂ν )ω(t, x) = 0,

ω(t = 0, x) = ω(x),the D dimensional gauge transformation ω(x) can be extended to a
D + 1 dimensional gauge transformation ω(t, x) that leaves the flow equation
unchanged
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

9 / 39

UV finiteness of the gradient flow I
Correlation function of the flowed gauge field
hBµ1 (t1 , x1 ) · · · Bµn (tn , xn )i ,t1 > 0, . . . , tn > 0,

when expressed in terms of renormalized parameters, is UV finite without thewave function renormalization
tree-level two-point function
DE
B̃µa (t, p)B̃νb (s, q) ∼ δ ab g02

»
–
1
1
2
−(t+s)p2−α0 (t+s)p2
(δ
p
−
p
p
)e
+
p
p
e
µν
µ
ν
µ
ν
(p2 )2λ01-loop two point function (those containing only Yang–Mills vertices)

The last counter term comes from rewriting to renormalized parameters
g02 = µ2 g 2 Z ,λ0 = λZ3−1

Usually, this becomes UV finite only by taking the wave function renormalization
factor into account . . .
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

10 / 39

UV finiteness of the gradient flow I. . . here, we have also diagrams containing flow vertices

which give rise to the precisely same effect as the wave function renormalization
factorAll order proof (Lüscher–Weisz (2011))

when a loop contains a vertex in the bulk (t > 0), the loop integral contains theflow-time evolution factor2∼ e−t`
which makes the loop integral finite; no bulk counterterm is necessary
by using a BRS symmetry, it can be shown that all boundary (t = 0) counterterms
are those of the Yang–Mills theory
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

11 / 39

UV finiteness of the gradient flow II
Correlation function of the flow gauge field
hBµ1 (t1 , x1 )Bµ2 (t2 , x2 ) · · · Bµn (tn , xn )i ,t1 > 0, . . . , tn > 0,remains finite even for the equal-point productt1 → t2 ,x1 → x2 ,2the new loop always contains the flow-time evolution factor ∼ e−t` and this makes
integral finite; no new UV divergence arises
This is an extremely powerful property!
Bµ (t, x)Bν (t, x)|Dimensional Regularization = Bµ (t, x)Bν (t, x)|Lattice
On the other hand, the difficulty in the present problem comes from(AR )µ (x)(AR )ν (x)|Dimensional Regularization 6= (AR )µ (x)(AR )ν (x)|lattice
Using this property of the gradient flow, we relate a certain quantity defined by thegradient flow and EMT in the dimensional regularization
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

12 / 39EMT in the dimensional regularization
SU(N) Yang–Mills theory in D = 4 − 2 dimensions
Z
1
a
a
S=
d D x Fµν(x)Fµν
(x)
4g02
Assuming the dimensional regularization, since it preserves the translational
invariance, the naive expression
–
»
1
1
a
a
a
a
Tµν (x) = 2 Fµρ(x)Fρσ(x)(x)Fνρ
(x) − δµν Fρσ
4g0
fulfills the correct WT relationh∂µ Tµν (x)O(y )O(z) · · · i = −δ(x − y ) h∂ν O(y )O(z) · · · i + · · ·It follows from this that Tµν (x) does not receive the multiplicative renormalization
So, with dimensional regularization, we define a renormalized (finite) EMT bysubtracting VEV,
{Tµν }R (x) = Tµν (x) − hTµν (x)i

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

13 / 39

Local product in finite flow time and EMT
We consider the following dim. 4 gauge invariant combinationsaaUµν (t, x) ≡ Gµρ
(t, x)Gνρ
(t, x) −1aaδµν Gρσ(t, x)Gρσ(t, x)
4

1 aa
Gµν (t, x)Gµν
(t, x)
4
These are quite similar to 4 dimensional EMT (Itou–Kitazawa, 2012 ∼), but can we
make the relationship precise?
√
The flow equation is a diffusion equation whose diffusion length is ∼ 8t. So, in
t → 0 limit, Uµν (t, x) and E(t, x) can be regarded as local operators in
D dimensional x space
Moreover, from the UV finiteness of the gradient flow, these are UV finite
From these facts, for t → 0, above local products can be expressed by anasymptotic series of D dimensional renormalized operators (coefficients will be
finite too):»–1Uµν (t, x) = αU (t) {Tµν }R (x) − δµν {Tρρ }R (x) + O(t),4
E(t, x) ≡E(t, x) = hE(t, x)i + αE (t) {Tρρ }R (x) + O(t),Here, we have used the fact that Uµν (x) is traceless for D = 4. O(t) is thecontribution of operators with dim. 6 or higher
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

14 / 39

Local product in finite flow time and EMT

By eliminating the trace part {Tρρ }R (x) from the above expansion,»–1Uµν (t, x) = αU (t) {Tµν }R (x) − δµν {Tρρ }R (x) + O(t),4
E(t, x) = hE(t, x)i + αE (t) {Tρρ }R (x) + O(t),we have{Tµν }R (x) =1
1Uµν (t, x) +δµν [E(t, x) − hE(t, x)i] + O(t)αU (t)
4αE (t)Therefore, if we know the t → 0 behavior of the coefficients αU (t) and αE (t), theEMT can be obtained by t → 0 limit of the combination in RHS

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

15 / 39

Renormalization group argument
We apply
„
«
∂
µ
,
∂µ 0

µ: renormalization scale, 0: bare quantities fixed

to both sides of»–1Uµν (t, x) = αU (t) {Tµν }R (x) − δµν {Tρρ }R (x) + O(t),4
E(t, x) = hE(t, x)i + αE (t) {Tρρ }R (x) + O(t)Expressed in terms of bare quantities, LHS does not contain µ. So,
«
»
„
–
∂
1
αU (t) {Tµν }R (x) − δµν {Tρρ }R (x) = 0,
µ
∂µ 0
4
„
«
∂
µ
αE (t) {Tρρ }R (x) = 0∂µ 0
Further, the EMT is not renormalized,„«∂µ
αU,E (t) = 0∂µ 0
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

16 / 39

Renormalization group argument
Introducing the β function by
β≡

«
„
∂
µ
g
∂µ 0

the above relation becomes
„
«
∂
∂
µ
+β
αU,E (t)(g; µ) = 0∂µ∂gThis implies that using the running coupling ḡ defined byqd ḡ(q)= β (ḡ(q)) ,dq

ḡ(q = µ) = g,

the coefficients do not depend on the renormalization scale:
αU,E (t)(ḡ(q); q) = αU,E (t)(ḡ(q 0 ); q 0 ).
So, we may set
q = µ,

1
q0 = √
8t

and then√√
αU,E (t)(g; µ) = αU,E (t)(ḡ(1/ 8t); 1/ 8t)√
Because of the asymptotic freedom, ḡ(1/ 8t) → 0 for t → 0 and coefficients canevaluated by the perturbation theory! (a sort of factorization)
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

17 / 39Perturbative calculation of coefficients
To 1-loop, we have to evaluate following flow-line Feynman diagrams

In terms of the renormalized gauge coupling in the MS scheme,
n
h √
i
o
αU (t)(g; µ) = g 2 1 + 2b0 ln( 8tµ) + s1 g 2 + O(g 4 ) ,o
1 nαE (t)(g; µ) =1 + 2b0 s2 g 2 + O(g 4 ) ,
2b0
where
s1 = ln√π+7' 1.00986,
16

s2 =

109
b1
−
' 0.197831,
176
2b02

and b0 = 11N/(48π 2 ) and b1 = 17N 2 /(384π 4 ) are the first two coefficients of theβ function; we see that αU,E (t) are actually UV finite
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

18 / 39Perturbative calculation of coefficients

By the above RG argument,no√√αU (t) = ḡ(1/ 8t)2 1 + 2b0 s1 ḡ(1/ 8t)2 + O(ḡ 4 ) ,o√
1 nαE (t) =1 + 2b0 s2 ḡ(1/ 8t)2 + O(ḡ 4 ) ,
2b0
where
ḡ(q)2 =1
b1 ln[ln(q 2 /Λ2 )]
− 3 2
+O
2
2
b0 ln(q /Λ )b0 ln (q 2 /Λ2 )„ln2 [ln(q 2 /Λ2 )]
ln3 (q 2 /Λ2 )«Therefore,
11√=− 2b0 s1 + O(ḡ 2 ),αU (t)ḡ(1/ 8t)2
and

Hiroshi Suzuki (Kyushu University)

i
√
b0 h
1
=
1 − 2b0 s2 ḡ(1/ 8t)2 + O(ḡ 4 )
4αE (t)2

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

19 / 39

Master formula

Gathering all the above arguments,
("
#
1
t→0+
√
{Tµν }R (x) ∼− 2b0 s1 Uµν (t, x)ḡ(1/ 8t)2
+

)
i
√
b0 h
1 − 2b0 s2 ḡ(1/ 8t)2 δµν [E(t, x) − hE(t, x)i] ,2and we obtained a formula that extracts a correctly normalized conserved EMTfrom local products defined through the gradient flow
Correlation functions of the quantities in RHS can (in principle) be computed
non-perturbatively by using lattice regularization
Practically, we have to take sufficiently small t in the window
a

√

8t 

1
Λ

and the applicability is not quite obvious . . .

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

20 / 39

Study of the feasibility by numerical experiment (gauge group SU(2))
Configuration: Wilson plaquette action, pseudo-heat bath (+ overrelaxation)
Wilson flow: 3rd order Runge–Kutta method (Lüscher),  = ∆t/a2 = 0.01,
t/a2 ∈ [0, 6]
afield strength Gµν(x) is clover-type, symmetric difference is usedsimulation parameters
latticeβNconfig

√
a/ t0

164
244
324

2.64
2.80
2.91

100
100
100

0.8971(63)0.5660(48)0.4125(40)a ' 0.036 fm for√

σ = 440 MeV

Here, we have introduced a reference flow time t0 by using the expectation valueof the “energy density”
1 aa(t, x)Gµν (t, x)Gµν4˛˛
t 2 hE(t, x)i˛
= 0.045E(t, x) =as

t=t0

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

21 / 39

Study of the feasibility by numerical experimentt 2 hE(t, x)i as a function of t/a2
0.12

"coefficients16x16beta2.64" u 1:2:3"coefficients24x24beta2.80" u 1:2:3"coefficients32x32beta2.91" u 1:2:3

0.1

0.08

0.06

0.04

0.02

0
0

1

2

3

4

5

6

From these values and the perturbative calculation (Lüscher (2010))
2
√ 2h
√ 2i
t→0+ 3(N − 1)t 2 hE(t, x)i ∼
ḡ(1/
8t)
1
+
2b
c
ḡ(1/
8t) ,
0
128π 2where (in the MS scheme)√
926c ≡ ln(2 π) +−ln 3 ' 1.60396,
33
22√
we estimate the perturbative running coupling ḡ(1/ 8t)2
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

22 / 39

Study of the feasibility by numerical experiment√
Perturbative running coupling ḡ(1/ 8t)2
8"coefficients16x16beta2.64" u 1:4:5"coefficients24x24beta2.80" u 1:4:5"coefficients32x32beta2.91" u 1:4:5

7

6

5

4

3

2

1

0
0

1

2

3

4

√

5

6

2

We may trust this perturbative computation of ḡ(1/ 8t) for the region of t inwhich the following “effective Λ parameter” is (almost) constanti−b1 /(2b02 )
√ 2
√
1 h
Λ(t) ≡ √b0 ḡ(1/ 8t)2
e−1/[2b0 ḡ(1/ 8t) ] ← 2-loop
8t
»
–
√
√ 4
−b2 + b0 b2
b3 − 2b0 b1 b2 + b02 b3
× exp − 1 3ḡ(1/ 8t)2 − 1
8t)← 4-loop
ḡ(1/
4b04
2b0
Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

23 / 39

Study of the feasibility by numerical experimentaΛ(t) as a function of t/a2
0.06

"coefficients16x16beta2.64" u 1:6:7"coefficients16x16beta2.64" u 1:8:9"coefficients24x24beta2.80" u 1:6:7"coefficients24x24beta2.80" u 1:8:9"coefficients32x32beta2.91" u 1:6:7"coefficients32x32beta2.91" u 1:8:9

0.05

0.04

0.03

0.02

0.01

0
0

1

2

3

4

5

6

(Ideally) we should use t/a2 in the almost-flat region

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

24 / 39

Study of the feasibility by numerical experiment
The order of the limits is importantFirst, while keeping the flow time t fixed in physical units, take the continuum
limit a → 0. This gives flowed values in the continuum Yang–Mills theory
Then, to extract EMT, take a small flow time limit t → 0
We may fix t in physical units by setting
t 2 hE(t, x)i = const.
We considered 10 combinations, t 2 hE(t, x)i = 0.045, 0.040, 0.035 with 3 different
lattice spacings and t 2 hE(t, x)i = 0.030 on 324 lattice
0.12"coefficients16x16beta2.64" u 1:2:3"coefficients24x24beta2.80" u 1:2:3"coefficients32x32beta2.91" u 1:2:3
0.045
0.040
0.035
0.030
"points.txt"

0.1

0.08

0.06

0.04

0.02

0

Hiroshi Suzuki (Kyushu University)

0

1

2

3

4

Lattice energy-momentum tensor . . .

5

6

2013/10/16 @ Osaka Univ.

25 / 39

Study of the feasibility by numerical experiment
Example of the correlation function
hU01 (t, x)U01 (t, 0)ix = (x0 , 0, 0, 0)

For t hE(t, x)i = 0.045 (the upper horizontal line) and for t 2 hE(t, x)i = 0.040 (the2nd horizontal line)
20.000160.0006"correlators16x16beta2.64t1.24" u ($1*0.8971):($2*2.384):($3*2.384)"correlators24x24beta2.80t3.12" u ($1*0.5660):($2*94.94):($3*94.94)"correlators32x32beta2.91t5.88" u ($1*0.4125):($2*1193):($3*1193)

0.00014"correlators16x16beta2.64t0.92" u ($1*0.8971):($2*2.384):($3*2.384)"correlators24x24beta2.80t2.20" u ($1*0.5660):($2*94.94):($3*94.94)"correlators32x32beta2.91t4.04" u ($1*0.4125):($2*1193):($3*1193)

0.00050.000120.00040.00010.0003

8e-05

6e-05

0.00024e-05
0.00012e-05
00-2e-05
0

2

4

6

8

10

12

14

-0.0001
0

2

4

6

8

10

12

14

For t 2 hE(t, x)i = 0.035 (the 3rd horizontal line)
0.003"correlators16x16beta2.64t0.65" u ($1*0.8971):($2*2.384):($3*2.384)"correlators24x24beta2.80t1.43" u ($1*0.5660):($2*94.94):($3*94.94)"correlators32x32beta2.91t2.55" u ($1*0.4125):($2*1193):($3*1193)

0.00250.002

0.00150.001

0.00050

Hiroshi Suzuki (Kyushu University)-0.000502
4
68
Latticeenergy-momentum
tensor. 10. .

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Study of the feasibility by numerical experiment
Example of the correlation functionh∂µ Uµ0 (t, x)E(t, 0)ix = (x0 , 0, 0, 0)

For t hE(t, x)i = 0.045 and for t hE(t, x)i = 0.040
22

0.000150.0006"correlators16x16beta2.64t1.24" u ($1*0.8971):($8*2.657):($9*2.657)"correlators24x24beta2.80t3.12" u ($1*0.5660):($8*167.7):($9*167.7)"correlators32x32beta2.91t5.88" u ($1*0.4125):($8*2892):($9*2892)

0.00010.00045e-05

0.000200-5e-05-0.0002-0.0001-0.0004-0.00015
0

2

4

6

8

10

12

-0.000614"correlators16x16beta2.64t0.92" u ($1*0.8971):($8*2.657):($9*2.657)"correlators24x24beta2.80t2.20" u ($1*0.5660):($8*167.7):($9*167.7)"correlators32x32beta2.91t4.04" u ($1*0.4125):($8*2892):($9*2892)

0

2

4

6

8

10

12

14

For t hE(t, x)i = 0.035
2

0.003

"correlators16x16beta2.64t0.65" u ($1*0.8971):($8*2.657):($9*2.657)"correlators24x24beta2.80t1.43" u ($1*0.5660):($8*167.7):($9*167.7)"correlators32x32beta2.91t2.55" u ($1*0.4125):($8*2892):($9*2892)

0.002

0.001

0

-0.001-0.002-0.003Hiroshi Suzuki (Kyushu University)

0

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Study of the feasibility by numerical experiment
Example of the correlation functionh∂µ δµ0 E(t, x)E(t, 0)ix = (x0 , 0, 0, 0)

For t hE(t, x)i = 0.045 and for t hE(t, x)i = 0.040
22

0.00040.0015"correlators16x16beta2.64t1.24" u ($1*0.8971):($10*2.657):($11*2.657)"correlators24x24beta2.80t3.12" u ($1*0.5660):($10*167.7):($11*167.7)"correlators32x32beta2.91t5.88" u ($1*0.4125):($10*2892):($11*2892)"correlators16x16beta2.64t0.92" u ($1*0.8971):($10*2.657):($11*2.657)"correlators24x24beta2.80t2.20" u ($1*0.5660):($10*167.7):($11*167.7)"correlators32x32beta2.91t4.04" u ($1*0.4125):($10*2892):($11*2892)

0.00030.001
0.00020.00050.000100-0.0001-0.0005-0.0002-0.001-0.0003-0.0004
0

2

4

6

8

10

12

-0.0015
14

0

2

4

6

8

10

12

14

For t 2 hE(t, x)i = 0.035
0.006"correlators16x16beta2.64t0.65" u ($1*0.8971):($10*2.657):($11*2.657)"correlators24x24beta2.80t1.43" u ($1*0.5660):($10*167.7):($11*167.7)"correlators32x32beta2.91t2.55" u ($1*0.4125):($10*2892):($11*2892)

0.004

0.002

0

-0.002-0.004-0.006Hiroshi Suzuki (Kyushu University)

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Study of the feasibility by numerical experiment
Although next we should make an extrapolation to a → 0, here we proceed byregarding the 324 results are sufficiently close to the continuum
Remembering the t → 0 behavior»–
1Uµν (t, x) = αU (t) {Tµν }R (x) − δµν {Tρρ }R (x) + O(t),4
E(t, x) = hE(t, x)i + αE (t) {Tρρ }R (x) + O(t),for example,
1t→0+h∂µ Uµ0 (t, x)E(t, 0)i −→αU (t)αE (t)

fi »
–
1
∂µ {Tµ0 }R (x) − δµ0 {Tρρ }R (x) {Tρρ }R (0)4

and RHS is obtained as an t → 0 extrapolation of LHS (Thanks, Aoki-san!)Similarly,¸
1 1
t→0+ 1 ˙h∂µ δµ0 E(t, x)E(t, 0)i −→∂µ δµ0 {Tρρ }R (x) {Tρρ }R (0)4 αE (t)2
4
Sum of these two is

t→0+ ˙−→ ∂µ {Tµ0 }R (x) {Tρρ }R (0)¸and should be 0 for x 6= 0 (conservation law of EMT!)
Hiroshi Suzuki (Kyushu University)

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2013/10/16 @ Osaka Univ.

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Study of the feasibility by numerical experiment
To obtain,on√√αU (t) = ḡ(1/ 8t)2 1 + 2b0 s1 ḡ(1/ 8t)2 + O(ḡ 4 ) ,o√
1 nαE (t) =1 + 2b0 s2 ḡ(1/ 8t)2 + O(ḡ 4 )2b0
we have
to
know
Λ
that
corresponds to the initial value of the running coupling√ḡ(1/ 8t)
0.025

"coefficients32x32beta2.91" u 1:6:7"coefficients32x32beta2.91" u 1:8:9
0.01880.0200

0.02

0.015

0.01

0.005

0
0

1

2

3

4

5

6

Here, as a rough estimate,0.0188 ≤ aΛ ≤ 0.0200
and used the 2-loop running formula
Hiroshi Suzuki (Kyushu University)

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2013/10/16 @ Osaka Univ.

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Study of the feasibility by numerical experiment(in what follows, only show plots with aΛ = 0.0188; not much difference foraΛ = 0.0200)
fi »
–
1
1
t→0+
h∂µ Uµ0 (t, x)E(t, 0)i −→ ∂µ {Tµ0 }R (x) − δµ0 {Tρρ }R (x) {Tρρ }R (0)αU (t)αE (t)42e-05
lators32x32beta2.91t5.88" u ($1*0.4125):($8*2892/57.9468275300718):($9*2892/57.9468275300718)
lators32x32beta2.91t4.04" u ($1*0.4125):($8*2892/52.3929495042157):($9*2892/52.3929495042157)
rrelators32x32beta2.91t2.55" u ($1*0.4125):($8*2892/46.955568940936):($9*2892/46.955568940936)
lators32x32beta2.91t1.43"
u ($1*0.4125):($8*2892/41.6043501721344):($9*2892/41.6043501721344)
1.5e-05

2e-05

1e-05

1e-05

5e-06

5e-06

0

0

-5e-06-5e-06-1e-05-1e-05-1.5e-05-2e-05"dUE_interpolation" index 5 u 1:2:3"dUE_interpolation" index 6 u 1:2:3"dUE_interpolation" index 7 u 1:2:3"dUE_interpolation" index 8 u 1:2:3"dUE_interpolation" index 9 u 1:2:3

1.5e-05-1.5e-05

0

2

4

Hiroshi Suzuki (Kyushu University)

6

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-2e-050Lattice energy-momentum tensor . . .

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Study of the feasibility by numerical experiment¸
1 1
t→0+ 1 ˙h∂µ δµ0 E(t, x)E(t, 0)i −→∂µ δµ0 {Tρρ }R (x) {Tρρ }R (0)2
4 αE (t)42e-05
32x32beta2.91t5.88" u ($1*0.4125):($10*2892/132.363913095121/4):($11*2892/132.363913095121/4)
32x32beta2.91t4.04" u ($1*0.4125):($10*2892/131.113157921798/4):($11*2892/131.113157921798/4)
rs32x32beta2.91t2.55" u ($1*0.4125):($10*2892/129.84558647364/4):($11*2892/129.84558647364/4)
ors32x32beta2.91t1.43"
u ($1*0.4125):($10*2892/128.551749454383):($11*2892/128.551749454383)
1.5e-05

2e-05

1e-05

1e-05

5e-06

5e-06

0

0

-5e-06-5e-06-1e-05-1e-05-1.5e-05-1.5e-05-2e-050

2

4Hiroshi Suzuki (Kyushu University)

6

8

10

12

"dEE_interpolation" index 5 u 1:2:3"dEE_interpolation" index 6 u 1:2:3"dEE_interpolation" index 7 u 1:2:3"dEE_interpolation" index 8 u 1:2:3"dEE_interpolation" index 9 u 1:2:3

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Study of the feasibility by numerical experimentt→0+ ˙−→ ∂µ {Tµ0 }R (x) {Tρρ }R (0)2e-05
0718+$10*2892/132.363913095121/4):($9*2892/57.9468275300718+$11*2892/132.363913095121/4)
2157+$10*2892/131.113157921798/4):($9*2892/52.3929495042157+$11*2892/131.113157921798/4)
8940936+$10*2892/129.84558647364/4):($9*2892/46.955568940936+$11*2892/129.84558647364/4)
1721344+$10*2892/128.551749454383):($9*2892/41.6043501721344+$11*2892/128.551749454383)
1.5e-05

2e-05

1e-05

5e-06

5e-06

0

0

-5e-06-5e-06-1e-05-1e-05-1.5e-05-1.5e-05

0

2

4

6

8

10

12

"dTE_interpolation" index 5 u 1:2:3"dTE_interpolation" index 6 u 1:2:3"dTE_interpolation" index 7 u 1:2:3"dTE_interpolation" index 8 u 1:2:3"dTE_interpolation" index 9 u 1:2:3

1.5e-05

1e-05-2e-05¸

14-2e-05
0

1

2

3

4

5

6

Good indication for the EMT conservation?!!

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

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Study of the feasibility by numerical experiment

But the situation is not so clear for . . .
fi »
–
fl
1
1
t→0+
h∂
U
(t,
x)U
(t,
0)i
−→
∂
{T
}
(x)
−
δ
{T
}
(x)
{T
}
(0)
µ µ1
µ
ρρ R
01
µ1 R
µ1
01 R
αU (t)24
2e-05
lators32x32beta2.91t5.88" u ($1*0.4125):($4*2892/25.3682045376433):($5*2892/25.3682045376433)
lators32x32beta2.91t4.04" u ($1*0.4125):($4*2892/20.9362752088432):($5*2892/20.9362752088432)
lators32x32beta2.91t2.55" u ($1*0.4125):($4*2892/16.9803650200663):($5*2892/16.9803650200663)
correlators32x32beta2.91t1.43"
u ($1*0.4125):($4*2892/13.46478722065):($5*2892/13.46478722065)
1.5e-05

2e-05

1e-05

1e-05

5e-06

5e-06

0

0

-5e-06-5e-06-1e-05-1e-05-1.5e-05-1.5e-05-2e-050

2

4Hiroshi Suzuki (Kyushu University)

6

8

10

12

"dUU_interpolation" index 5 u 1:2:3"dUU_interpolation" index 6 u 1:2:3"dUU_interpolation" index 7 u 1:2:3"dUU_interpolation" index 8 u 1:2:3"dUU_interpolation" index 9 u 1:2:3"dUU_interpolation" index 10 u 1:2:3

1.5e-0514-2e-050Lattice energy-momentum tensor . . .

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Study of the feasibility by numerical experiment. . . and¸
1
1
t→0+ 1 ˙h∂µ δµ1 E(t, x)U01 (t, 0)i −→∂µ δµ1 {Tρρ }R (x) {T01 }R (0)4 αE (t)αU (t)42e-05
rs32x32beta2.91t5.88" u ($1*0.4125):($6*2892/57.9468275300718/4):($7*2892/57.9468275300718/4)
rs32x32beta2.91t4.04" u ($1*0.4125):($6*2892/52.3929495042157/4):($7*2892/52.3929495042157/4)
ators32x32beta2.91t2.55" u ($1*0.4125):($6*2892/46.955568940936/4):($7*2892/46.955568940936/4)
rs32x32beta2.91t1.43"
u ($1*0.4125):($6*2892/41.6043501721344/4):($7*2892/41.6043501721344/4)
1.5e-05

2e-05

1e-05

1e-05

5e-06

5e-06

0

0

-5e-06-5e-06-1e-05-1e-05-1.5e-05-2e-05"dEU_interpolation" index 5 u 1:2:3"dEU_interpolation" index 6 u 1:2:3"dEU_interpolation" index 7 u 1:2:3"dEU_interpolation" index 8 u 1:2:3"dEU_interpolation" index 9 u 1:2:3"dEU_interpolation" index 10 u 1:2:3

1.5e-05-1.5e-05

0

2

4

Hiroshi Suzuki (Kyushu University)

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Study of the feasibility by numerical experiment

An example of 2 point correlation function (relevant for the shear viscosity)¸
1
t→0+ ˙hU01 (t, x)U01 (t, 0)i −→ {T01 }R (x) {T01 }R (0)αU (t)2
2e-05
lators32x32beta2.91t5.88" u ($1*0.4125):($2*1193/25.3682045376433):($3*1193/25.3682045376433)
lators32x32beta2.91t4.04" u ($1*0.4125):($2*1193/20.9362752088432):($3*1193/20.9362752088432)
lators32x32beta2.91t2.55" u ($1*0.4125):($2*1193/16.9803650200663):($3*1193/16.9803650200663)
correlators32x32beta2.91t1.43"
u ($1*0.4125):($2*1193/13.46478722065):($3*1193/13.46478722065)
1.5e-05

2e-05

1e-05

1e-05

5e-06

5e-06

0

0

-5e-06-5e-06-1e-05-1e-05-1.5e-05-1.5e-05-2e-050

2

4Hiroshi Suzuki (Kyushu University)

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"UU_interpolation" index 5 u 1:2:3"UU_interpolation" index 6 u 1:2:3"UU_interpolation" index 7 u 1:2:3"UU_interpolation" index 8 u 1:2:3"UU_interpolation" index 9 u 1:2:3

1.5e-0514-2e-050Lattice energy-momentum tensor . . .

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Future direction: numerical experiment

It seems that we had a good indication (!) although we still have to carry out . . .
systematic extrapolation to the continuum a → 0
systematic extrapolation to t → 0 (hopefully) using data with smaller flow times
clear demonstration of the conservation of EMT”O(t) improvement” might be usefulaaGµρ(t, x)Gνρ
(t, x)

ˆ
˜
a
a
a
a
a
a
→ Gµρ
(t, x)Gνρ
(t, x) − t Dσ Dσ Gµρ
(t, x)Gνρ
(t, x) + Gµρ
(t, x)Dσ Dσ Gνρ(t, x)

this replacement removes O(t) terms in the tree levelalso 1-loop improvement will not be impossible (presumably)
step size scaling for small t?

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

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Future direction: theoretical aspects
Inclusion of matter fields: flowed matter field requires the wave function
renormalization (Lüscher (2013))
χ(t, x) = Zχ−1/2 χR (t, x),χ̄(t, x) = Zχ−1/2 χ̄R (t, x)

To avoid the determination of Zχ in lattice/continuum theory, we may define anoperator by normalizing it by the “condensation” as, for example,
χ̄(t, x)Dχ(t,
/
x)
3/2

t0 hχ̄(t0 , x)χ(t0 , x)iwhere t0 is an arbitrary fixed flow time. This is a dim. 4 UV finite quantity to which
our argument is applied
1-loop mixing coefficients (to be computed)

Non-perturbative determination of mixing coefficients? (Del Debbio–Patella–Rago,arXiv:1306.1173 [hep-th])Hiroshi Suzuki (Kyushu University)

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2013/10/16 @ Osaka Univ.

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Future direction:

Physical application? (thermodynamics, viscosities, conformal field theory, dilaton
physics, vacuum energy, . . . )For bulk thermodynamical quantities, for instance, for the so-called “trace anomaly”˙¸hε − 3piT = − {Tµµ }R (x) T ,or for the entropy densityhε + piT =fi
fl1− {T00 }R (x) + {Tii }R (x) ,3
T

our definition should coincide with the traditional one (Engels–Karsch–Scheideler,(1982)) in the continuum limit
This is the case also for other off-diagonal components (Giusti–Meyer (2013))?
Can we define the chiral current and/or SUSY current from the gradient flow?

Hiroshi Suzuki (Kyushu University)

Lattice energy-momentum tensor . . .

2013/10/16 @ Osaka Univ.

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