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Large-N pure gauge critical temperature along a line of constant Physics

Large-N pure gauge critical temperature along aline of constant Physics

Jamie Hudspith Anthony Francis

York University, Toronto

April 21, 2017

,[email protected] 1


Introduction

Plan of the talk

1. BackgroundI DefinitionsI Lattice geometry

2. Scale SettingI Gradient flow for Large NI Planar limit

3. Finite-T simulationsI Determining the critical βI

√t0TC

4. Large-N limit



Introduction

Why?

Theoretical and Technical.

I Various BSM theories rely on SU(N) with some N

I Large-N limit is used in many approaches, simpler than QCD andsome analytic calculations can be performed

I Confinement is a defining feature of SU(N) gauge theories

I No fermions so simulations are cheap, although inevitably thecomputation cost increases

I Some commonly-used techniques are interesting to extend to largematrices

I No studies for a few years, probably worth an update especially withsome different methods



Introduction

Of course I am not the only one to think about this

Lucini, Teper and Wenger (2005)”Properties of the deconfining phase transition in SU(N) gauge theories”

Lucini, Rago and Rinaldi (2012)”SU(Nc) gauge theories at deconfinement”

Nb: Lucini et al use the string tension σ to set the scale and thePolyakov loop susceptibility to measure βC , extrapolating βC to infinitevolume using small volumes.

Francis, A. and Kaczmarek, O. and Laine, M. and Neuhaus, T. andOhno, H. ”Critical point and scale setting in SU(3) plasma: An update”Nb:SU(3) only

Use more or less the same method for pure gauge SU(N).



Introduction

Notation

Uµ

(x + a

µ

2

)= e iag0Aµ(x+a µ

2 ). (1)

We use the Wilson Plaquette action, V = L3s × Lt

Sg = Vβ (N − Up) = a4E 2 + ..

Up =1

V

∑x,µ<ν

<(

Tr

[Uµν

(x + a

µ

2+ a

ν

2

)]),

E 2 =∑x,µ,ν

Fµν(x)Fµν(x).

(2)

l =1

NL3

L3∑x

Tr

[Lt∏t=1

Ut

(x + a

t

2

)]. (3)



Introduction

Deconfinement

I Ls/a ≤ Lt/a = ”Zero Temperature”

I Lt/a < Ls/a = ”Finite Temperature”

Inverse anisotropic length is temperature,

aT =a

Lt(4)

All pure-gauge SU(N) theories show confinement, with an expectedfirst order phase transition for N > 2.

Polyakov loop is the order parameter of this transition,

|l | ≈ e−F . (5)

|l | = 0, F =∞ |l | 6= 0,F = finite.Breaking of center symmetry l ∈ ZN .



Introduction

SU(N) starfish



Introduction

Geometry

Even-dimension checker-boarding

Checker-board updating alternating R and B.B R B R

R B R B

B R B R

R B R B

,

[email protected] 8


Introduction

Geometry

Odd-dimension checker-boarding

Checker-board updating alternating R and B and G.G R B

B G R

R B G

Sometimes NG < NG ,NB but we loop these in a multi-threaded code so

it doesn’t really matter.

https://github.com/RJHudspith/GLU


https://github.com/RJHudspith/GLU


Scale Setting

Scale setting overview

Traditionally lattice spacing a (in QCD) is determined by matching toexperimental quantities amΩ, amK , amπ, afπ ... etc.

There aren’t any for large N, but can match to auxiliary scales such as:

I The string tension a√σ

I The Sommer parameter ar0

I The Gradient Flow scale a√t0



Scale Setting

Gradient Flow

Solve,U = Z (U)U, U(t + ε) = e iεZ(U(t))U(t). (6)

su(N) matrix exponential! should be done by a well-convergent series,trivially,

(eA/(2)n)2n

= eA. (7)

Using a (4,4) pade representation of the exponential,

eA =I + A(C0 + A(C1 + A(C2 + AC3)))

I − A(C0 − A(C1 − A(C2 − AC3))),

C0 =1

2, C1 =

3

28, C2 =

1

84, C3 =

1

1680,

(8)

and n = 3 and 4 to perform this exponential and final reunitarisation stepto remain in the group.



Scale Setting

Walking on a line of constant PhysicsUsually we introduce a reference scale t0/a2 as,

t2E 2|t=t0 = CE . (9)

The beta function diverges in the large-N limit, we must take the planarlimit Λ = Ng 2. But wait! E 2 ∝ g 2

0 so we can define a planar limit scalewhich I will confusingly call t0/a2,

t2E 2|t=t0 = CEN. (10)

Where we will choose CE = 0.1 so that we can compare our SU(3) resultto the literature.Scale set at zero temperature, can form a quantity with constant cut offeffects

L√t0

= CL. (11)

We will tune our ensembles to CL = 10.,

[email protected] 12


Scale Setting

Tuning for SU(3), SU(4) and SU(5)

5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 6.1

9.9

10

10.1 SU( 3 )

10.5 10.6 10.7 10.8 10.9 11 11.1

9.9

10

10.1 SU( 4 )

16.7 16.8 16.9 17 17.1 17.2 17.3 17.4 17.5 17.6

β

9.9

10

10.1 SU( 5 )

CL =

Ls /

( t

0 )

(1/2

)



Scale Setting

Tuning for SU(6), SU(7) and SU(8)

24.3 24.4 24.5 24.6 24.7 24.8 24.9 25 25.1 25.2 25.3 25.4 25.5

9.9

10

10.1 SU( 6 )

33.2 33.4 33.6 33.8 34 34.2 34.4 34.6 34.8

9.9

10

10.1 SU( 7 )

43.5 43.75 44 44.25 44.5 44.75 45 45.25

β

9.9

10

10.1 SU( 8 )

CL =

L /

( t

0 )

(1/2

)



Scale Setting

√t0 as a function of β

0 0.25 0.5 0.75 1 1.25 1.5 1.75

β − βL=10

1

1.25

1.5

1.75

2

2.25(

t 0 )

(1/2

) / a

SU( 3 )

SU( 4 )

SU( 5 )

SU( 6 )

SU( 7 )

SU( 8 )



Finite Temperature

More constants

We define the critical coupling βC as the point where deconfinementoccurs

a(βC )TC =a(βC )

Lt. (12)

We know how our auxiliary scale behaves with β so we can define yetanother dimensionless quantity,

√t0TC =

√t0

Lt= CT . (13)

This is, in principle, a physical quantity that other theorists can use onceit is known in the continuum.



Finite Temperature

The separatrix

Originally considered as a method of determining the phase transition ofthe n-state Potts model with exponentially-small finite volumecorrections.

From SU(3) we see that anisotropy Ls

Lt= CA = 3 is very close to the

infinite volume limit. So we choose to work solely at CA = 3.

With a double-peaked histogram of |l | with W measurements. We countthe number of points below the minimum between the two maxima, thisquantity we call wg ,

S(β) =(q + 1)wg −W

(q − 1)wg + W. (14)

S(β) = 1 confined, S(β) = −1 deconfined.q is the q-fold degeneracy weight, which we set to NC .



Finite Temperature

Separatrix: The Movie



Finite Temperature

SU(3) has small finite volume corrections

Figure: Exponentially-reduced finite volume effects in determining the criticalcoupling as compared to the Polyakov loop susceptibility method. Plot takenfrom Francis-2014.



Finite Temperature

Comparisons 1

Lt/a Francis-2014 Us

SU(3)

5 5.8000(5)† 5.8003(2)6 5.8943(3) 5.8954(5)7 - 5.9810(4)8 6.0624(4) 6.0624(3)

Lt/a Lucini-2005 Us

SU(4)

5 10.6373(5) 10.6398(1)6 10.7898(16) 10.7955(3)7 10.9415(12) 10.9460(6)8 10.0880(22) 11.0880(3)

Table: Comparison of our results with those in the literature for βC for thegauge groups SU(3) and SU(4). The result of † is from Lucini et al 2004hep-lat/0307017.



Finite Temperature

Comparisons 2

Lt/a Lucini-2012 Us

SU(5)

5 16.8762(12) 16.8764(1)6 17.1074(33) 17.1105(2)7 17.3386(31) 17.3294(5)8 17.5585(36) 17.5516(6)

I SU(6), SU(7) and SU(8) data is still being gathered and analyzed.

I Transition becomes more first order as N and Volume increase,requires a large number of ensembles.



Finite Temperature

Continuum extrapolations of√t0TC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

a2 / t

0

0.235

0.24

0.245

0.25

0.255

0.26(

t 0 )

(1/2

) TC

SU( 3 )

SU( 4 )

SU( 5 )



Finite Temperature

Large-N limit?

0 0.02 0.04 0.06 0.08 0.1 0.12

1 / N2

0.235

0.2375

0.24

0.2425

0.245

0.2475

0.25

0.2525

0.255(

t 0 )

(1/2

) TC



Conclusions

Conclusions

I We have accurately set the scale and defined a line of constantPhysics that we can use for large-N extrapolations.

I We have utilised the Separatrix to determine the critical β values forN ≥ 3 and found them more or less in accordance with theliterature.

I We have demonstrated you can accurately obtain the dimensionlessquantity

√t0TC and that the approach to the large-N limit is fairly

mild.


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