Slide 1

Effective Polyakov line actions from the relative weights method

Jeff Greensite and Kurt Langfeld Lattice 2013Mainz, GermanyJuly 2013

arXiv: 1301.4977, 1305.0048, and in progressStart with lattice gauge theory, and integrate out all d.o.f. subject to the constraint that the Polyakov lines are held fixed. In temporal gauge:

Motivation:

possible insight into the deconfinement transition

address the sign problem

PLAs with a chemical potential solve by dual representation, stochastic quantization, reweighting, mean field

The problem is to find the PLA corresponding to lattice QCD. Effective Polyakov Line Action (PLA) Previous work:

Strong coupling + hopping parameter expansions: Langelage et al. (2012)

Inverse Monte Carlo: Heinzl et al. (2005)

Demon method: Gocksch & Ogilvie (1985) , Velytsky (2008), Wozar et al (2008)

A crucial test: compare Polyakov line correlators

in both the effective PLA and the underlying lattice gauge theory.

We do not believe that an accurate agreement has been demonstrated, at leastup to now, in any of these approaches.

The method lets us compute dSP/d , where parametrizes a path through the space of all Polyakov line holonomies {Ux(), all x}.

From such derivatives, we try to deduce SP itself. Relative Weights MethodLet Ux and Ux be two configurations, corresponding to 0- and 0 + respectively. Define the PLA difference

and lattice actions (in temporal gauge) with fixed holonomies

so that, e.g.

Then

where the notation refers to a VEV in the probability distribution

We then have

But which derivatives will help us to determine SP ? Start with SU(2). The PLA can only depend on the Polyakov lines

Fourier expand

We compute

via relative weights at a typical point in field configuration space, i.e. a thermalized configuration generated by lattice Monte Carlo.

Differentiation wrt Fourier componentsProcedure

generate a thermalized lattice configuration U(x) by the usual lattice Monte Carlo, and set

Fourier decompose Px= Tr[Ux] and set ak=0. Call this . Construct

Compute

by the relative weights method.

and corresponding holonomies Ux , Ux , with Here is a plot of

Note that it is independent of SP is quadratic in ak SP is bilinear in Px . SP is bilinear in Px=2.2 , 243 x 4 lattice volume take

with

so

(relative factor of 2: ) lattice momentum

large kL data fits a straight line

which determines c1 , c2 , and at large kL .

If it were true that at all kL , it would mean that

which is long-range (violates Svetitsky-Yaffe assumption). make the simplest modification:

and determine rmax from a fit to the kL 0 data. rmax = 3 works nicely.

with c1 , c2 , rmax determined from the dSP/dak data.

So, how well does it work? The obvious test: compare correlators

in both the effective theory, and the underlying lattice gauge theory. Effective Polyakov Line Action=2.2, 243 x 4

Agreement to O(10 -5) !

The data fits c1 - 2c2kL2 over the whole range, which implies

a nearest-neighbor coupling. A test at strong-coupling: = 1.2 The result, compared to SP determined from a lowest-order strong coupling expansion at =1.2, is

We now add a fixed-modulus scalar field to the action, breaking the Z2 center symmetry:

we work at = 0.75 and = 2.2 on a 243 X 4 lattice as before.

This time, the PLA picks up a Z2-symmetry breaking term which is linear in Px

The first term is linear in a0 , which we can determine from the dS/da0 data. Adding a Matter Field

This is the dS/da0 data at

=2.2 , = 0.75 , 243 X 4

shown over a large (upper) and small (lower) range of .

The coefficient c0 of the linear term in SP is determined from the intercept with the y-axis.

Our effective theory gives the blue triangles c0 = 0.0236(14). We get a perfect fit if we use a value c0 = 0.02165 (red circles) about 1.4 from our estimate. As before, generate a thermalized configuration, extract the Polyakov lines, Fourier decompose, and set one momentum mode, ak=0. The resulting configuration is . Then construct

and from these the corresponding holonomies Ux and Ux . This is not quite as straightforward as for SU(2), but it can be done.

Then we compute by relative weights.

A complication

The data indicates, in addition to the bilinear term, also the presence of a quadrilinear term. Preliminary results for SU(3)

SU(2) SU(3) we find

with kernel

and constants c1 , c2 , rmax , q1 , q2 determined from the data. Results including off-axis points, at

= 5.6 , t = 6 and = 5.5 , t = 4 (includes off-axis points)

Once we have obtained the PLA SP at =0 , we get the PLA at 0 by a simple substitution:

uestion: Why is that the right thing to do?

nswer: This is the correct substitution to all orders in the gauge coupling and hopping parameter h . It is reasonable to assume that it holds in general.

Given SP at finite chemical potential, we can apply any of the methods used so far on Polyakov line actions:

dual representation, stochastic quantization, reweighting, mean field..

to solve the theory. Relevance to the Sign Problem in a strong coupling/hopping parameter expansion:

where

In the same way:

comparing the two expansions at = 0 and 0 , we see that to all orders in coupling and hopping parameter h

It means that once we have found the PLA at = 0 , the PLA at 0 follows from a simple substitution. Other values of and Nt in SU(3) for the pure gauge theory.

Add in a scalar field breaking the Z3 symmetry. Find the corresponding action SP , and determine transition points in the T- plane by one of the applicable methods.

Throw in quark fields. First heavy, then light. Again try to solve the resulting PLA at 0 to find the phase diagram. Next Steps

expSP [Ux]

=

DU0(x, 0)DUkD

x

[Ux U0(x, 0)]eSL

G(x y) = PxP y

xtypicalconfiguration

Configuration Space of U

SP = SP [Ux] SP [U x ]

SL[U] SL

U0(x, 0) = U

x

SL[U

] SLU0(x, 0) = U

x

eSP =

DUkD eS

L

DUkD eSL

=

DUkD exp[SL SL]eS

L

DUkD eSL

=exp[SL SL]

exp[SL]DUkD eS

L

dS

d

=0

S

Px = a0 +1

2

q =0

aq cos(q x) + bq sin(q x)

SPak

ak=

Px =1

2Tr[Ux]

SPak

ak=

SPak

Px

f = 1

Ux = U0(x, 0)

P x = (1

2ak) cos(k x) + f Px

P x = (+1

2ak) cos(k x) + f Px

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5

(L-3

dS P

/da k

)/_

kL

L=24 data with _ rescaling

_=.05_=.10_=.15_=.20

1

1

L3

SPak

ak=

vs.

SP =1

2c1x

P 2x 2c2xy

PxQ(x y)Py

Q(x y) = 1L3

k

Q(kL)eik(xy)

1L3

dSP [Ux(ak)]

dak

ak=

=

(12c1 2c2 Q(kL)) kL = 0

2( 12c1 2c2 Q(0)) kL = 0 .

x

cos2(k x) = 12L3 ,

x

1 = L3

Q(kL) = kL

Q(x y) =

2Lxy

-0.06-0.04-0.02

0 0.02 0.04 0.06 0.08

0.1 0.12

0 0.5 1 1.5 2 2.5 3 3.5

L-3 d

S P/d

a k

kL

L=24 data at _=0.05

datafit

kL =

4 3i=1

sin2(1

2ki)

(1

2c1 2c2kL)

Q(kL) kL

Q(x y) = 2L

xy|x y| rmax

0 |x y| > rmax

-0.08-0.06-0.04-0.02

0 0.02 0.04 0.06 0.08

0.1

0 0.5 1 1.5 2 2.5 3 3.5

L-3 d

S P/d

a k

kL

L=24 data and finite range kernel

datafit using Q(k)

SP =1

2c1x

P 2x 2c2xy

PxQ(x y)Py

Q(x y) = 2L

xy|x y| rmax

0 |x y| > rmax

G(R) = PxPy with R = |x y|

0 4 8 12 16 20 24R

1e-06

0.0001

0.01

1

G(R)

lattice YMeffective theory

On-axis Polyakov line correlators, L=24

= 2.25

= 2.3

0 2 4 6 8 10 12 14 16R

0.0001

0.001

0.01

0.1

1

G(R)

lattice YMeffective theory

SU(2) 163x4 beta=2.25

0 2 4 6 8 10 12 14 16R

0.01

0.1

G(R)

lattice YMeffective theory

SU(2) 163x4 beta=2.30

rmax =10

rmax =13

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.5 1 1.5 2 2.5

(L-3

dS P

/da k

)/_

kL

`=1.2, L=16

datak2 fit

Q(x y) =2L

xy

SP =

0.02859(3)

x

3i=1 PxPx+ relative weights

0.02850

x

3i=1 PxPx+ strong coupling

( = 1.2)

SL = plaq

1

2Tr[UUUU ] +

x,

1

2Tr[(x)U(x)(x+ )]

SP = c0x

Px +1

2c1x

P 2x 2c2xy

PxQ(x y)Py ,

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

L-3 d

S P/d

a 0

_

gauge-Higgs k=0 data, L=24

datalinear fit

-0.2-0.15

-0.1-0.05

0 0.05

0.1 0.15

0.2 0.25

-0.04 -0.02 0 0.02 0.04

L-3 d

S P/d

a 0

_

gauge-Higgs k=0 data, L=24

datalinear fit

0 4 8 12 16 20 24R

0.01

1

G(R)

eff theory: c0=0.021654latt SU(2) + Higgseff theory: c0=0.0236

On-axis Polyaov line correlators, L=24

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05 0.1 0.15 0.2

L-3 d

S P/d

a 0

_

kL=0 data, linear fit

datafit

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

0 0.05 0.1 0.15 0.2 0.25 0.3

L-3 d

S P/d

a 0

_

kL=0 data, linear fit

datafit

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4

G(R)

R

Polyakov line correlators, `=5.6, L=16, T=6

eff actionlattice SU(3)

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5

G(R)

R

Polyakov line correlators, `=5.5, L=16, Nt=4

eff action on axisLuscher-Weisz

SP [Ux, Ux] = S

=0P [e

NtUx, eNtUx]

expS=0P [Ux, U

x]=

DUkDD exp

SL[Ux, U

x, Ui(x),,]

=D

cDNpD

2N2

pDhp

DPD[Ux, U

x]

PD[Ux, Ux] = TrU

w1x1 TrU

w2x2 ...TrU

wnxn TrU

w1y1 TrU

w2y2 ...TrU

wmym

expSP [Ux, U

x]=

DUkDD exp

SL[e

NtUx, eNtUx, Ui(x),,]

=D

cDNpD

2N2

pDhp

DPD[e

NtUx, eNtU x]

SP [Ux, Ux] = S

=0P [e

NtUx, eNtUx]