so(2n) and su(n) gauge theories - lattice 2013
TRANSCRIPT
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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
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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
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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
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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
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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
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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
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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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SOHNL: N�
=N�2SUHNL: N
�=N
SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper
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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
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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
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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
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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
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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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9
10
11
1N
MG
Σ
2+
2+*
1+
1-
2-
SO(2N) and SU(N) gauge theories Richard Lau and Michael Teper
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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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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
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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
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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
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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
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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
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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
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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
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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
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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