Symanzik improvement of Yang-Mills gradient flow observables
Stefan Sint (Trinity College Dublin)in collaboration with Alberto Ramos (CERN)

ALPHA
Collaboration

HHIQCD 2015, Yukawa Institute for Theoretical Physics
6 March 2015

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Outline
The Yang-Mills gradient flow equation & its properties
Some current applications
The gradient flow equation on the lattice
Large cutoff effects and their tree-level anatomy
4+1 dimensional formulation & Symanzik improvement
Classical a-expansion of observables & the flow equation
Conclusions & Outlook

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The Yang-Mills gradient flow equation
Starting point: Yang-Mills theory in 4-dimensions:Z1
d4 x tr {Fµν (x)Fµν (x)},Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ]Sg [A] = − 22g0
Add extra (flow) time coordinate t and define the gauge field Bµ (t, x)
Gµν (t, x)=∂t Bµ (t, x)=∂µ Bν (t, x) − ∂ν Bµ (t, x) + [Bµ (t, x), Bν (t, x)]δS [B]
Dν Gνµ (t, x)= − δBµg(t,x) ,Bµ (0, x) = Aµ (x)The linearized gradient flow equation reduces to the heat equation (use gauge
freedom to diagonalize RHS):∂t Bµ (t, x) = −∂ν ∂ν Bµ (t, x),Bµ (t, x) = (4πt)−2Z(x − y )24t
d ye
Aµ (y )
4−

⇒ The gauge field Bµ (t, x) is smoothed over a range with radius r (t) =(2σ range of the Gaussian smoothing function).

√

8t

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Properties of the Yang-Mills gradient flow
Correlation functions of (gauge invariant) observables at t > 0 are
renormalizedZ
1hO[B]i =D[A]O[B] exp (−Sg [A])Z
once the coupling g0 is renormalized as usual!
Local gauge invariant composite fields at t > 0 such as
E (t, x) = − 21 tr {Gµν (x, t)Gµν (x, t)}
are renormalized; no mixing with other fields of same or lower dimensions!
Established to all orders in perturbation theory [Lüscher & Weisz ’2012 ];Explict one-loop calculation (infinite volume, dimensional regularization)[Lüscher 2010 ]:
3g 2 (µ)
11.0978 + 0.0075 × Nf 2
hE (t, x)i = MS2 2 1 +gMS (µ) + O(g 4 ) , µ = √
16π t
4π
8t
⇒ E (t, x) is, for t > 0, a renormalized field; unlike E (0, x) which has aquartic and a logarithmic divergence!

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Applications of the gradient flow I
Convenient (implicit) definition of reference scale, e.g. t0 [Lüscher 2010 ]easy to measure, small statistical fluctuations, mild quark massdependence also in χPT [Bär & Golterman 2013 ]t 2 hE (t, x)i t=t = 0.30

Non-perturbative√ definition of a renormalized “gradient flow coupling” atscale µ = 1/ 8t:2
def 16π2gGF(µ) =t 2 hE (t, x)i3√
Coupling at scale µ0 = 1/ 8t0 :2gGF(µ0 ) =

16π 2
× 0.3 = 15.8
3⇒

αGF (µ0 ) =15.8= 1.257
4π

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Applications of the gradient flow II
Consider hE (t, x)i in a finite box of dimension L4 , fix the ratio c =and define
2ḡGF(L) = N (c)−1 t 2 hE (t, x)i,

lim N (c) =

c→0

√
8t/L

3
16π 2

defines family of renormalized couplings, with parameter c.
(typical range from 0.2 to 0.5; c > 0.5 implies that 2r (t) > L, i.e.
“smearing around the universe”)
The normalization constant is calculable in lowest order perturbation
theory; depends on b.c’s for the gauge field; periodic in spatial directions,and in the time direction
periodic b.c.’s [Fodor et al. 2012 ]SF (Dirichlet) b.c.’s [Fritzsch & Ramos 2012 ]twisted periodic b.c.’s [Ramos 2013 ]open-SF (Neumann-Dirichlet) b.c.’s [Lüscher 2013 ]QCD, αs determination: advantage of gradient flow coupling at low
energies, but loses to SF coupling at high energies⇒ pursue mixed approach [Fritsch et al. (ALPHA coll.), 2014 ]6/ 23

Applications of the gradient flow III
Small flow time expansion & operator renormalization [Lüscher 2013 ]Definition of renormalized energy-momentum tensor [Suzuki 2013ff; Del
Debbio et al.,2013; Patella et al, 2014 ]enormalized energy-momentum tensor & SU(3) thermodynamics [Asakawa
et al. (FlowQCD collaboration), 2014 (cf. Hatsuda on 4 March) ]Extension to fermions possible [Lüscher 2013 ]Use flow quantities to check for autocorrelations in Monte-Carlosimulations; significant coupling to slow modes [Lüscher & Schaefer,2012 ]Assess quality of lattice actions, new improvement conditions,...
...
The lattice community has only just begun to explore the possibilities! Expect
much more to come;However: Improvements and/or combination with other techniques may berequired;
Here: systematic reduction of O(a2 ) cutoff effects.

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The gradient flow on the latticeconsider generic SU(3) lattice action with 4-link and 6-link Wilson loops(normalization: c0 + 8c1 + 16c2 + 8c3 = 1):(a)S[U; ci ] =

1 X
tr
g02 x

(a)
1 − c0

r
r

r
r
(a)
−
c
1
r
r

r
r

r
(a)
r − c2

r r
rp !
r
p
pr p p p r− c3(a) r p ppr p p p rpr rrp p r

Expectation values defined by integral over Uµ .
Gradient flow equation, choose the gradient of a lattice action:(f )∂t Vµ (t, x) = −g02 ∂x,µ S[V ; ci ]Vµ (t, x),(a)Vµ (t = 0, x) = Uµ (x)(f )⇒ ci 6= ci in general!
Observables: we focus on E (t, x); two options:
1define a lattice version of Gµν (t, x) (e.g. clover leaf of plaquettes in µ − νplane), then formclcl(t, x)Gµν(t, x)}
E cl (t, x) = − 21 tr {Gµν

2

E (t, x) is an action density; choose a lattice action density such thatX(o)a4
E (t, x) = g02 S[V ; ci ]x(o)⇒ yet another set of ci !
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The gradient flow on the lattice
Popular choices for action parameters & terminology:
Wilson action, Wilson flow, plaquette observable:
c0 = 1, c1 = c2 = c3 = 0;
Tree-level Lüscher-Weisz action, Symanzik flow:
c0 = 5/3, c1 = −1/12, c2 = c3 = 0;
a = 0.05 fm

0.07 fm

0.1 fm0.96

Wilson action & flow;t0 from clover (upper points)
and plaquette observable[Lüscher 2010 ]

√8t0
r0

0.94

0.92

0.90

0

0.01

0.02

0.03

0.04

(a/r0 )2

O(a2 ) cutoff effects are surprisingly large & and even larger in QCD!
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Anatomy of tree-level O(a2 ) effects in hE (t, x)i I

t 2 hE (t, x)i

g2

=

Z π/a

h
i
(o)
d4 p tr Kµν(p, 0)D̄µν (p, λ, α) ,−π/aD̄µν (p, λ, α)=(e−tK (f) (p,α)

)µρ K (a) (p, λ)−1

(e−tK

(f)

(p,α)

)σν ,

ρσ

λ, α: gauge fixing parameters for the action and flow equation,
respectively.
Observable, gradient flow and action characterized by kernels Kµν (p) of”free lattice actions”:
Z π/a(a,o,f)
S (a,o,f) = 12
d4 p Abµ (−p)Kµν(p, λ)Abν (p) + O(A3 ),−π/a(a,o,f)Kµν(p, λ)=

cont(a,o,f)Kµν(p, λ) + a2 Rµν(p, λ) + O(a4 )

cont
Kµν
(p, λ)=

p 2 δµν + (λ − 1)pµ pν

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Anatomy of tree-level O(a2 ) effects in hE (t, x)i II
Extend momentum integrals to infinity, then evaluate traces:





a2  (o)
3g 2
(a)
(o)
(a)
1
+
d1 − d1J4,−2 + d2 − d2
J2,0hE (t, x)i =
2
2
16π t
t


(f)
(f)
− 2d1 J4,0 − 2d2 J2,2 + O(a4 ) + O(g 4 )
where
R∞
2
t (m+n)/2 −∞ d4 p e−2tp p n p m
R∞Jn,m =,
d4 p e−2tp2
−∞

pn

=
n=2,4,...

X

pµn ,

p −n = 1/p n .

µAll momentum integrals can be evaluated:
J2,0 = 1,

J2,2 = 3/2,

J4,0 = 3/4,

J4,−2 = 1/2

Hence:hE (t, x)i =
a2  (o)
3g 2
(a)
(f)
4
1
+
d
−
d
−
3d
+O(a
)
.
16π 2 t 2t |{z}
= d total

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Anatomy of tree-level O(a2 ) effects in hE (t, x)i III
For each d-coeffcient we may choose Wilson-plaquette, Lüscher-Weisz or theclover kernel or combinations thereof:3
plaquette (pl),− 72 ,
(a,o,f)
1
1
2
d
=
,
=
−
−
c
Lüscher-Weisz
(lw),
72
24
3 1

 15
− 72 ,
clover (cl).
Popular combinations, d total = d (o) − d (a) − 3d (f) :Clover observable, Wilson action & flow:
d total = (−15 + 3 + 9)/72 = −3/72
Wilson observable & action & flow:
d total = (−3 + 3 + 9)/72 = 9/72
Clover observable, Lüscher-Weisz action & flow:
d total = (−15 − 1 − 3)/72 = −19/72
Lüscher-Weisz observable & action & flow:
d total = (1 − 1 − 3)/72 = −3/72
N.B.: not improved!

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Qualitative understanding the O(a2 ) effects in t0plaq vs. t0cl
Strategy: keep the standard definition of t0 fixed (clover definition) andlook at cutoff effects in t0 from the Wilson/plaquette definition.
√
Define r (t) = 8t and the coupling ḡ (r ) in the GF scheme at scale r (t0 )t 2 hEcl (t, x)i|t=t0 =3
ḡ 2 (r (t0 )) = 0.3
16π 2⇒ ḡ 2 (r (t0 )) = 15.8

While this relation is (by definition) exact we otherwise have


a2
3
2
plaq
4
ḡ
(r
(t
))
1
+
∆d
+
O(a
)
= 0.3
t 2 hEplaq (t, x)i|t=t plaq =
0
0
16π 2
t0
Here,
∆d = d pl − d cl =−3 + 15
12
1
=
=
72
72
6

Use 1-loop or 2-loop evolution of the GF coupling to obtain the O(a2 )shift in t0plaq w.r.t. the reference (clover) definition.

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Integration of the RG equation
Recall the definition of the β-function:r∂g (r )= −β(g ) = b0 g 3 + b1 g 5 + ...
∂rwith the universal coefficients (N = 3)b0 = 11 − 32 Nf /(4π)2 ,
b1 = 102 − 28
Nf /(4π)43

Writing r (t0plaq ) = r (t0 ) + ∆r with ∆r small (an O(a2 ) effect):ḡ 2 (r (t0 ))a2
1 + ∆dt0qExpanding both sides and using that ∆r /r (t0 ) = t0plaq /t0 − 1, one obtainsḡ 2 (r (t0 ) + ∆r ) =t0plaq∆da2= 1−+O(a4 ),t0{−β(ḡ )/g } t0−β(g )/g |g 2 =15.8,Nf =0 = 1.10+1.02+O(g 6 )Nonetheless one obtains qualitatively correct results:
The sign is correctly obtained: the plaquette definition comes from below!
Switching on Nf decreases the β-function and increases the cutoff effect.

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GF coupling in finite volume with twisted periodic b.c.’s
fix the relation between
flow time t and space-time volume L4 by choosing√
a value for c = 8t/L.
The trace algebra same as before, however: the numbers Jn,m become
linearly independent functions of c!
⇒ more conditions for improvement: each coefficient must vanish separately!
Cannot be satisfied with LW/Symanzik type flow, need to be more general:
include bent rectangles/chairs with coefficient c2 & define the chair flow
c0 = 1,c1 = −1/12,

c2 = 1/24

2⇒ complete tree-level O(a ) improvement of hE (t, x)i
However, we then checked that the connected 2-point function
X −ip(x−y )def
G (t, s, p) = a4e[hE (t, x)E (s, y )i − hE (t, x)ihE (s, y )i]x2

is not tree-level O(a ) improved for p 6= 0!
⇒ need to be more systematic, i.e. apply Symanzik’s procedure!

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Symanzik’s effective theory I
Systematic expansion of lattice correlation functions in terms of arenormzalized effective continuum theory;
Terms are organized by increasing canonical dimensions⇒ reformulate as 4 + 1 dimensional local theory [Lüscher & Weisz ]:Z ∞X S[V , L] = Slat [U] + a4
dt a4
tr Lµ (t, x)
0

x


o
× (∂t Vµ (t, x)) Vµ (t, x)† + ∂x,µ g02 Slat [V ]nFlow quantities as expectation values of local fields [in coordinates (t, x)]:Z
hO[V , L]i = Z −1 D[V ]D[L] O[V , L] exp (−S[V , L])
D[V ] includes integration over t = 0 gauge field Vµ (0, x) ≡ Uµ (x)Note: Vµ (t, x) is not constrained to be a solution to the flow equation;
The gradient flow equation is enforced by the integration over L.
For observables O = O[U]: definition reduces to usual 4-d expectation
values!
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Symanzik’s effective theory II
Symanzik’s effective theory, effective action and observables:
Seff [V , L]=

S0cont [B, L] + a2 S2,v [B, L] + a2 S2,b [B, L] + O(a4 )

Oeff

=

O0 + a2 O2 + O(a4 )S2,v , S2,b : 4+1 dimensional volume action and t = 0 boundary action.
Expansion of lattice expectation values:
hOilatt = hO0 icont + a2 hO2 icont − a2 hO0 S2,v icont − a2 hO0 S2,b icont + O(a4 )Possible terms appearing in O(a2 ) contributions:
A priori any terms with the correct dimension which share all symmetriesof the lattice theory;
Z
Z ∞ Z
X
X
Oi (t, x)|t=0
S2,v [L, B] =dt d 4 x
Ti (t, x),
S2,b [L, B] = d 4 x0iiwith gauge invariant Oi , Ti of dimension 8 and 6 respectively, polynomialin Bµ and Lµ and derivatives ([Lµ ] = 3,[Bµ ] = 1 )Similarly, O2 contains local terms of dimension [O0 ] + 2 sharing all thesymmetries of the lattice observable.
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Symanzik’s effective theory III
BUT: This is a rather special 4 + 1 dimensional theory!
only tree diagrams are generated in the bulk [Lüscher & Weisz, 2012 ]⇒ counterterms for bulk action and observables in the 4+1 dim. volume aredetermined by classical expansion!
Need to expand classically the lattice bulk action & flow observables(s. below)⇒ Full Symanzik analysis only required for S2,b .
List of dimension 6 terms (reduced by use of flow equation):
O1 = tr {Jµνρ Jµνρ }O4 = tr {Lµ (0, x)Jνµν }O2 = tr {Jµµρ Jννρ }O5 = tr {Lµ (0, x)Lµ (0, x)}

O3 = tr {Jµµρ Jµµρ }where Jµνρ = Dµ Fνρ18/ 23

Symanzik’s effective theory IV
the counterterms O1,2,3 are already fixed in the standard 4-d theory⇒ cannot be modified!
The only new counterterms are O4,5
Find by explicit calculation (tree-level) and directly from functional
integral (general):ZZ
d 4 x hO4 (0, x)O[B]i ∝ d 4 x hO5 (0, x)O[B]ifor observables O[B] defined at t > 0.
⇒ O5 is redundant
Effect of O4 insertion: change of boundary conditions at t = 0, possible
implementation:Vµ (t = 0, x) = Uµ (x) exp c4 g02 ∂x,µ Sg [U]
In the continuum this corresponds to the addition of a term
∝ a2 Dν Fνµ (x)to Bµ (t = 0, x)
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Classical a-expansion of gradient flow observables
Smooth underlying continuum gauge field Bµ (x) (t-dependence suppressed),
link variables are induced:
 Z 1
Vµ (x) = P exp adλBµ (x + (1 − λ)aµ̂)
0

Z 1
=

dλ Bµ (x + (1 − λ)aµ̂)

1l + a+ a20
Z 1

dλ2 Bµ (x + (1 − λ1 )aµ̂) Bµ (x + (1 − λ2 )aµ̂) + . . .

1l + aBµ (x) + a2 21 ∂µ Bµ (x) + Bµ2 (x)+ 61 a3 ∂µ2 Bµ (x) + 2Bµ (x)∂µ Bµ (x) + (∂µ Bµ (x)) Bµ (x) + Bµ3 (x) + . . .
0

=

Z λ1
dλ1

0

Note: gauge transformations in the continuum and on the lattice arecompatible:Bµ (x) → g (x)Bµ (x)g (x)−1 +g (x)∂µ g (x)−1⇔Vµ (x) → g (x)Vµ (x)g (x+aµ̂)−1

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Expansion of small Wilson loops, plaquettes etc.
Plaquette field:
Pµν (x)=Vµ (x)Vν (x + aµ̂)Vµ (x + aν̂)† Vν (x)†Using the gauge fixing tricks by Lüscher & Weisz ’85 one obtains relatively
quickly:Pµν (x)=1l + a2 Gµν (x) + a3 12 (Dµ + Dν )Gµν (x)no2+ a4 61
Dµ2 + 32 Dν Dµ + Dν2 Gµν (x) + 3Gµν(x) + O(a5 )Dµ=∂µ + [Bµ , ·],[Dµ , Dν ] = [Gµν , ·]similar expansions for rectangles, chairs, etc.
Obtain O(a2 ) improved lattice expressions for E (t, x) either from LW
action density or
defElat (t, x) =4
1
E (t, x)|plaq − E (t, x)|clover = Econt (t, x) + O(a4 )33however, up to total derivative terms (often irrelevant, but not always!).

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Classical expansion of flow equation
Lattice flow equation:
a2 (∂t Vµ (t, x)) Vµ (t, x)† = −∂x,µ g02 Slat [V ] ,Vµ (0, x) = Uµ (x)The O(a2 ) term for the LW flow has a simple structure
1 2 2∂t Bµ = Dν Gνµ − 12a Dµ Dν Gνµ + O(a3 )as expected: not O(a2 ) improved
This suggests a simple modification of the lattice flow equation (Zeuthenflow):




1 2 ∗
a2 (∂t Vµ (t, x)) Vµ (t, x)† = − 1 + 12a ∇µ ∇µ ∂x,µ g02 Slat [V ]This removes all O(a2 ) effects from the flow equation, corrections are infact O(a4 ).

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Conclusions & Outlook
In principle, O(a2 ) effects can be removed from all gradient flow observables:
use a non-perturbatively O(a2 ) improved 4-dim. pure gauge actionuse of classically improved flow observables, e.g. choose E (t, x) as densityof LW action;
impose an O(a2 ) modified t = 0 boundary condition on Vµ (t, x);
integrate the modified lattice flow equation, e.g. the Zeuthen flow;
In practice...
4-d gauge action and initial boundary conditions determined perturbatively⇒ residual O(a2 ) artefacts even in pure gauge theory.
Still important to completely eliminate the relatively large O(a2 ) effects
from the gradient flow and observables.
Future:implementation of the Zeuthen flow in openQCD (done)
application to the QCD running coupling (started)
quenched/unquenched scaling tests for t0 (planned)

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