The Gradient Flow and the Running Coupling
Jarno Rantaharju
RIKEN Advanced Institute of Computational Science

March 7, 2014• The Gradient Flow
• New tool for lattice calculations• Renormalized quantities• Scale setting, chiral condensate• The Coupling Scheme• Several possible schemes• Correction terms• SF results

The Gradient Flow

Gradient Flow
• Flow along the gradient of the an action 1e.g. the gauge action:∂t Bµ,t = Dµ,t Bµν,t ,Bµ,0 = AµBµν,t = ∂µ Bν,t − ∂ν Bµ,t + [Bµ,t , Bν,t ]• Diffusive flow with scale√8t• Produces a finite, smoothed field• Correlators of the flow field renormalized

1M. Luscher and P. Weisz, JHEP 1102 (2011) 051 [arXiv:1101.0963].

Gradient Flow

• On the lattice, Wilson flow

Gµ,t+ = e −Zµ,t (x) Gµ,t (x),Zµ,t (x) = (1/2) M − M † − (1/6)Tr M − M †X
+M=
ν6=µ

• Stout smearing steps, continuous flow
• Cooling, flow generated by the gauge force

Gradient Flow
• For example, the field strenght: 21hGµν,t Gµν,t i42 hi3(Nc2 − 1)gMS
2
4=1+kg+O(g)1 MSMS128π 2 t 2hEt i =k1 = 1/(4π) [ N (11/3γE + 52/3 − 3 ln 3)−Nf (2/3γE + 4/9 − 4/3 ln 2) ]−1/2µ= (8t)• Renormalization scale set by t
2

M. Luscher, JHEP 1008 (2010) 071 [arXiv:1006.4518].

Scale Setting• One of the first applications, scale setting• Find t0 such thatt02 hEt0 i = 0.3√• A physical scale given by l = 8t0• Another scale w0 3d  2W (t) = tt hE (t)i√ dt
w0 = t|W (t)=0.3

3S. Borsanyi et al. JHEP 1209 (2012) 010 [arXiv:1203.4469].

Extension to Fermions• The flow equations can be extended to fermionic fields: 4∂t χt = ∆t χt ,←−∂t χ̄t = χ̄t ∆ t ,χ0 = ψ

χ̄0 = ψ̄∆t = Dµ,t Dµ,t ,Dµ,t = ∂µ + Bµ,t

• Not automatically renormalized, but

χR,t = Zχ 1/2 χt

χ̄R,t = χ̄t Zχ 1/2• Independent of t4

M. Luscher, JHEP 1304 (2013) 123 [arXiv:1302.5246]

Extension to Fermions• Non-zero flow time chiral condensate:Σrrt = − hχ̄r ,t χr ,t i ,rrΣrr
R,t = Zχ Σt• To the first orderΣrr
R,t =2NmR,r(4π)2 t

Z ∞dv
0e −vz,(1 + v )22z = 2tmR,r• Can also be related to the chiral condensate

Other Applications• Anisotropy tuning 5• Energy-momentum tensor 6• Matching improved actions with new observables:

1E (t) = Gµν (t)Gµν (t),4Strs = −χ̄r ,t χs,t ,Ptrs = −χ̄r ,t γ5 χs,t56S. Borsanyi et al. arXiv:1205.0781.
H. Suzuki, PTEP 2013 (2013) 8, 083B03 [arXiv:1304.0533].
L. Del Debbio et al. JHEP 1311 (2013) 212 [arXiv:1306.1173].

Coupling from the Gradient flow

Coupling from the Gradient flow
• E (t) is related to the coupling:7

2gGF=t 2 hE (t)iN• To study scale dependence, need a single scale• Now we have L and t, so fix

t=7(cL)28Z. Fodor et al. JHEP 1211 (2012) 007 [arXiv:1208.1051].
Z. Fodor et al. PoS LATTICE 2012 (2012) 050 [arXiv:1211.3247].

Boundary Conditions• Periodic• Schrödinger Functional boundary conditions8• Unique global minimum: simpler perturbation theory• Breaks the translation symmetry• Twisted boundary conditions 9• Milder than the SF boundary• Restricts the number of fermions in the fundamental
representations to Nf = 2nNc

8

P. Fritzsch and A. Ramos, JHEP 1310 (2013) 008 [arXiv:1301.4388].
P. Fritzsch and A. Ramos, arXiv:1308.4559.
J. Rantaharju, arXiv:1311.3719.
9A. Ramos, arXiv:1308.4558.

Boundary Conditions• Schrödinger Functional boundary conditions:• Periodic spatial boundary conditions, trivial time boundaries

Uk (x) = 1, when x0 = 0, L
Uµ (x + Lk̂) = Uµ (x)• Fermion fields disappear at time boundaries, periodic spatialboundary conditions
ψ(x) = 0, when x0 = 0, L
ψ(x + Lk̂) = ψ(x).

The Model• The Model:

S = (1 − cg )SG (U) + cg SG (V ) + SF (V ) + cSW δSSW (V )• Smeared Wilson fermion action SF (V )• and gauge action SG (V )• HEX Smearing• Bulk correction term δSSW (V )• Here, cg = 0.5 and cSW = 1• N(a/L) in g 2 = t 2 hE i /N(a/L) measured at β = 80

Lattice Results• Large discretization errors in previous results, corrections:• Gradient flow and E from the0.8

Symanzik action

Improved
Unimproved

→53−

110g

20.7

• Correction to the flow time
0.6

t = c 2 L2 /8 − da2 ,c = 0.6, d = 0.5
0.50481216
L/a202428

SU(2) with SF boundaries• Preliminary
β=1.5
β=1.7
β=2
β=3β=4β=5β=6β=8432• SU(2) with 8 flavors• Relatively stable results• Allows for large lattice sizes• Measurements still at lowcoupling
100

10203040SU(2) with SF boundaries• Preliminary

1.1
L=6L=8
L=10
L=12• Scale dependence described by

the step scaling function

1.05

Σ(u, a /L) =2 (g , 2L/a )gGF
0g 2 (g ,L/a )=uGF

10u0.95

σ(u) = lim Σ(u, a /L)
012345a →0• The Gradient Flow
• Renormalizedquantities• Scale setting• Chiral condensate• Space-time symmetries• Improved actions• The Coupling Scheme• Several possible schemes• Correction terms• Promising SF results

The effect of d

• The effect of d in the flow time

d=0.5
d=0

• SU(2) with 2 adjoint fermions,0.6

β=80.59

0.58

t = c 2 L2 /8 − da2 ,0.57

twisted: d = 00.56

0102030

40SF,periodic: d = 0.5

Compared to d = 0

β=1.5
β=1.7
β=2β=3β=4
β=5
β=6β=843322110010203040

β=1.5
β=1.7
β=2β=3β=4β=5
β=6β=840010203040

The effect of d• The effect of d in the flow timed=0d=0.25
d=0.5
d=0.75• SU(2) with 2 adjoint fermions,

β=8SF0.55

t = c 2 L2 /8 − da2 ,TWISTED

twisted: d = 00.5SF,periodic: d = 0.5

PERIODIC(displaced)

0.4502468101214161820

The effect of c0.65• The effect of c in the flow time

c=0.3
c=0.4
c=0.5
c=0.6SF(displaced)• Labels different schemes in thecontinuum

0.6

t = c 2 L2 /8 − da2 ,TWISTED

0.55

twisted: d = 00.5

SF,periodic: d = 0.5

PERIODIC(displaced)

0.4502468101214161820