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 String Tensions Mass Spectrum Deconfining Temperature Conclusions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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-

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

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

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-

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

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

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-

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

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

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

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

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

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

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|

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

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

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

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

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

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

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

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)

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

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

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)

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

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

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

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

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

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 Richard Lau and Michael Teper