SO(2N) and SU(N) gauge theories - Lattice 2013

SO(2N) and SU(N) gauge theoriesLattice 2013

Richard Lau and Michael Teper

Rudolf Peierls Centre for Theoretical Physics,University of Oxford

2 August 2013

xford

hysics

SO(2N) Gauge Theories String Tensions Mass Spectrum Deconfining Temperature Conclusions

Talk Structure

1 SO(2N) Gauge Theories

2 String Tensions

3 Mass Spectrum

4 Deconfining Temperature

5 Conclusions

SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


SO(2N) Gauge Theories: Why SO(2N)?

• Group equivalence

SU(2) ∼ SO(3)

SU(4) ∼ SO(6)

• Large-N equivalence12

SU(N →∞) = SO(2N →∞)

g2∣∣SU(N→∞)

= g2∣∣SO(2N→∞)

• No sign problem

1C. Lovelace, Nucl. Phys. B201 (1982) 3332A. Cherman, M. Hanada, and D. Robles-Llana, Phys. Rev. Lett. 106,

091603 (2011)SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


Going between SU(N) to SO(2N)

SU(N →∞) SO(2N →∞)

SU(N) SO(2N)

-�large-N equivalence

?

6

O(

1N2

)corrections

?

6

O( 1N ) corrections

Our approach

• Continuum limit at specific SO(2N)

• Large-N extrapolation



SO(2N) Gauge Theories: Lattice Setup

• D = 3 + 1I Bulk transition occurs at very small lattice spacingI Very large lattices required to get continuum extrapolation.3

• D = 2 + 1I Bulk transition occurs at larger lattice spacing

• Pure gauge theories

S = β∑p

(1− 1

NTrUp

): β =

2N

ag2

3e.g. P. de Forcrand and O. Jahn, Nucl. Phys. B651 (2003) 125SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


String Tensions: Extracting string tensions

• Polyakov loop operators lP(t)

• Correlators

C (t) ≡ 〈lP(t)lP(0)〉 ∝ e−mP t

• Nambu-Goto model4

mP(l) = σl(

1− π

3σl2

) 12

4A. Athenodorou, B. Bringoltz, and MT, JHEP 1102 (2011) 030SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


String Tensions: SO(2N →∞) and SU(N →∞)Large-N extrapolation of continuum limitsSO(N) : N = 6, 8, 12, 16⇒ N = 3, 4, 6, 8SU(N) : N = 2, 3, 4, 5, 6, 8

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0.0 0.1 0.2 0.3 0.4 0.50.10

0.12

0.14

0.16

0.18

0.20

1

N�

Σ

g2 N�

SOHNL: N�

=N�2SUHNL: N

�=N



String Tensions: SO(2N →∞) and SU(N →∞)

Large-N extrapolation of SO(2N) and SU(N)5 string tensions

Gauge group√σ

g2N

∣∣∣N→∞

SO(2N) 0.1981(6)SU(N) 0.1974(2)

5B. Bringoltz and MT, Phys. Lett. B645: 383−388 (2007)SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


Mass Spectrum: Extracting glueball masses

• Operators for JP glueballs

φ(t) =∑~x

∑n

e ijθnTr{UR(θn)C ± UPR(θn)C

}: θn =

nπ

2

• Variational method6

• Correlation functions

C (t) ≡ 〈φ(t)φ(0)〉〈φ(0)φ(0)〉

∝ e−mG t

6MT, Phys. Rev. D59 (1999) 014512SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


Mass Spectrum: Continuum extrapolation

SO(6): 0+/− glueball masses

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0.00 0.02 0.04 0.06 0.082

4

6

8

10

a2Σ

MG

Σ

0+

0+*

0+**

0+***

0+****

0-



Mass Spectrum: SO(2N →∞)Large-N extrapolation: 0+/− glueball massesN = 6, 8, 12, 16

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0.00 0.05 0.10 0.15 0.202

4

6

8

10

1N

MG

Σ

0+

0+*

0+**

0+***

0+****

0-



Mass Spectrum: SO(2N →∞)Large-N extrapolation: 1+/−, 2+/− glueball massesN = 6, 8, 12, 16

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0.00 0.05 0.10 0.15 0.205

6

7

8

9

10

11

1N

MG

Σ

2+

2+*

1+

1-

2-



Mass Spectrum: SO(2N →∞) and SU(N →∞)Large-N extrapolation of 0+ glueball mass for SO(2N) andSU(N)7.

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0 5 10 153.0

3.5

4.0

4.5

5.0

N

M0+

Σ SOH2NLSUHNL

7B. Lucini and MT, Phys. Rev. D 66, 097502 (2002)SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


Mass Spectrum: SO(2N →∞) and SU(N →∞)

Large-N extrapolation of SO(2N) and SU(N)8 mass spectra

JP SO(2N →∞) SU(N →∞)

0+ 4.14(3) 4.11(2)0+∗ 6.51(4) 6.21(5)2+ 6.95(9) 6.88(6)2− 7.03(8) 6.89(21)

0+∗∗ 8.31(20) 8.35(20)0− 9.44(22) 9.02(30)2+∗ 8.74(12) 9.22(32)1+ 9.65(41) 9.98(25)1− 9.56(48) 10.06(40)

8B. Lucini and MT, Phys. Rev. D 66, 097502 (2002)SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


Deconfining Temperature: Finding the deconfining phasetransition

• Finite temperature theory

T =1

a(β)Lt

• Deconfining temperature

Tc =1

a(βc)Lt

• Order parameters O

Ut , |lp|



Deconfining Temperature: Deconfining phase transition

Histograms of 〈lp〉 in SO(16) on a 823 lattice

-0.3 -0.2 -0.1 0 0.1 0.2 0.30.

0.01

0.02

Β=148.0

-0.3 -0.2 -0.1 0 0.1 0.2 0.30.

0.01

0.02

Β=149.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.30.

0.01

0.02

Β=150.5



Deconfining Temperature: Reweighting

• Order parameters O

Ut , |lp|

• Susceptibility χO

χO∼ 〈O2〉 − 〈O〉2

• Reweighting9

Z (β) ≡∑i

D(Si )e−βSi

9A. Ferrenberg and R. Swendsen, Phys. Rev. Lett. 63, 11951198 (1989)SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper


Deconfining Temperature: Reweightingχ|lp | on SO(16): 823 lattice

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148.0 148.5 149.0 149.5 150.0 150.5 151.0

0.006

0.007

0.008

0.009

Β

Χ l p¤

|lp| susceptibility⇒ βc = 149.32(1)



Deconfining Temperature: Finite volume extrapolationSO(16) : βc(V →∞) at Tc = 1

3a , Ls = 8, 10, 12, 14

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0. 0.05 0.1 0.15148

149

150

151

HLt �LsL2

Βc

XU t2\-XU t \2

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0. 0.05 0.1 0.15148

149

150

151

HLt �LsL2

Βc

X l p¤2\-X l p¤\2

ut susceptibility⇒ βc(V →∞) = 150.21(3)

|lp| susceptibility⇒ βc(V →∞) = 150.16(2)



Deconfining Temperature: SO(2N →∞) andSU(N →∞)

Large-N extrapolation of SO(2N)10 and SU(N)11 deconfiningtemperatures

Gauge group Tc/√σ

SO(2N →∞) 0.924(20)SU(N →∞) 0.903(23)

10F. Bursa, RL, and MT, JHEP 1305:025,201311J. Liddle and MT, arXiv:0803.2128



Conclusions

• There are large-N equivalences between SO(2N) and SU(N)gauge theories.

• Their pure gauge theories in 2+1 dimensions have matchingphysical properties at large-N.

I String tensionsI Mass spectraI Deconfining temperature

• SO(2N) theories may provide a starting point for answeringproblems with SU(N) QCD theories at finite chemicalpotential.


SO(2N) and SU(N) gauge theories - Lattice 2013

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