Exotic phases of Finite Temperature SU(N) gauge theories withmassive fermions: F, Adj, A/S

Joyce Myers and Michael Ogilviejcmyers@wustl.edu

mco@wuphys.wustl.edu

Washington University in St. Louis

July 17, 2008Lattice 2008

1 / 17

Outline

Motivation: To understand the effects of massive fermions on the phase diagram ofSU(N) gauge theories for fermions in various representations.

Phase diagram of a simple deformed Yang-Mills theory formulated on the lattice

Phase diagram of a more complicated deformed Yang-Mills theory (compare withQCD(Adj))

Phase diagram of SU(N) gauge theories with massive fermions from the one-loopeffective potential

◮ Fermion representations: Fundamental (F), Antisymmetric (A), Symmetric (S), andAdjoint (Adj)

◮ Boundary conditions: periodic (PBC), antiperiodic (ABC)◮ Nc = 2 through 9.◮ various Nf

One-loop contribution to 〈ψ̄ψ〉R (R = F ,A,S ,Adj)

2 / 17

Lattice modelLast year we analyzed a simple deformed Yang-Mills theory on the lattice (Myers andOgilvie 2008):

Slat,def = SW +∑x

Vlat,def [P(x)]

Vlat,def [P(x)] ≡ HATrAP(x) = HA

(

|trP (x)|2−1)

Simulations in SU(3) and SU(4) revealed two interesting new phases.The simulations also showed that confined phase could be accessed perturbatively inSU(3).

-0.1

-0.08

-0.06

-0.04

-0.02

0

HA

5.8 6 6.2 6.4 6.6 6.8β

deconfined

U(1) × SU(2)

confined

Phase diagram in SU(3)

−0.3

−0.2

−0.1

0

0.1

0.2

0.3ℑ〈P

(x)〉

−0.3 0 0.3ℜ〈P (x)〉

〈P(x)〉 in SU(3), β = 6.5,

HA =−0.055

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

ℑ〈TrP

(x)〉

−0.15 0 0.15ℜ〈TrP (x)〉

〈P(x)〉 in SU(4), β = 11,

HA = −0.12

3 / 17

Deformed Yang-Mills theory

To keep the confined phase accessible for N > 3 additional terms were required in thedeformation potential (Ogilvie et al 2007, Unsal and Yaffe 2008):

Vdef (P) ≡1

β

⌊N/2⌋

∑n=1

an |tr(Pn)|2 =

1

β

⌊N/2⌋

∑n=1

an

N

∑i , j=1

cos[

n(

vi −vj

)]

where ⌊N/2⌋ is the integer part of N/2.Including the boson contribution from pure Yang-Mills theory

Vmodel (P) =1

β4

[

1

24π2

N

∑i ,j=1

[vi −vj ]2(

2π − [vi −vj ])2

−π2

45

(

N2 −1)

]

+1

β

⌊N/2⌋

∑n=1

an

N

∑i , j=1

cos[

n(

vi −vj

)]

We minimize this potential to determine the phase diagram for a range of values ofthe an

4 / 17

One-loop effective potentialThe one-loop effective potential for Nf Majorana fermions (Nf ,Dirac = 1

2Nf ) of mass m in

a background Polyakov loop P = diag{e iv1 ,e iv2 , ...,e ivN} gauge field is (Meisinger andOgilvie 2001):

Veff (P,m) ≡−1

βV3lnZ (P,m)

=1

βV3

[

−Nf lndet(

−D2R (P)+m2

)

+lndet(

−D2adj(P)

)]

=m2Nf

π2β2

∞

∑n=1

(±1)n

n2Re [TrR (Pn)]K2 (nβm)

+1

β4

[

1

24π2

N

∑i ,j=1

[vi −vj ]2 (

2π − [vi −vj ])2

−π2

45

(

N2 −1)

]

where we have (+1)n for periodic boundary conditions (PBC)and (−1)n for antiperiodic boundary conditions (ABC) applied to fermions.

Chiral Condensate:

〈ψ̄ψ〉1−loop(m) = − limV4→∞

1

V4Nf

∂∂m

lnZ (m) =1

Nf

∂∂m

Veff(P,m)

5 / 17

Possible phases of QCD for PBC and ABC

ABC◮ confined phase

v = {0,2π3

,4π3}

◮ deconfined phase

v = {0,0,0},{2π3

,2π3

,2π3},{

4π3

,4π3

,4π3}

PBC◮ confined phase◮ deconfined phase◮ C -breaking phase (P is not invariant under P → P∗.)

v = {2π3

,2π3

,2π3}

Note: In QCD(F) with PBC on fermions, C -symmetry is only broken for N odd. For N

even, TrF P is magnetized along the negative real axis (v = {π,π, ...}).

6 / 17

VEFF and 〈ψ̄ψ〉 in perturbative QCD

We calculate Veff for fermions in the fundamental representation to which ABC areapplied.

−3

−2

−1

0

1

VFUNDβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONFDECONFC-breaking

VF ,−, Nc = 3, Nf = 2 (1 Dirac flavour)

0

0.1

0.2

〈ψ̄ψ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈ψ̄ψ〉F ,1−loop for Nc = 3

Only the deconfined phase is accessible in the perturbative limit.

The fermion contribution to Veff vanishes as mβ → ∞.

The inflection point in VEFF at mβ ≈ 1.4 implies a large one-loop contribution to〈ψ̄ψ〉.

7 / 17

Phases of adjoint QCD: Nc = 3, Nc = 4, PBC, Nf > 1

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0ℑ

[TrP

]

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

ℜ[TrP ]

confined

deconfinedU(1) × SU(2)

Nc = 3

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

ℑ[TrP

]

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

ℜ[TrP ]

confined

deconfined

SU (2) confined

Nc = 4

confined: v = {0, 2π3 , 4π

3 }

U(1)×SU(2): v = {0,π,π}

deconfined: v = {0,0,0}

confined: v = {0, π4 , 3π

4 , 5π4 }

SU(2) conf: v = {− π2 , π

2 ,− π2 , π

2 }

deconfined: v = {0,0,0,0}

8 / 17

Phases of adjoint QCD: Nc = 5, Nc = 6, PBC, Nf > 1

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

ℑ[TrP

]

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

ℜ[TrP ]

confineddeconfinedSU (2) × SU (3) deconfmoving {0,−φ,−φ, φ, φ}

Nc = 5

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

ℑ[TrP

]

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

ℜ[TrP ]

confineddeconfinedSU(2) confSU(3) conf

Nc = 6

confined: v = {0, 2π5 , 4π

5 , 6π5 , 8π

5 }

inconst: v = {0,−φ ,−φ ,φ ,φ}

SU(2)×SU(3) dec: v = {π,π,0,0,0}

deconfined: v = {0,0,0,0,0}

confined: v = { π6 , 3π

6 , 5π6 , 7π

6 , 9π6 , 11π

6 }

SU(3) conf: v = {0, 2π3 ,− 2π

3 ,0, 2π3 ,− 2π

3 }

SU(2) conf: v = {− π2 , π

2 ,− π2 , π

2 ,− π2 , π

2 }

deconfined: v = {0,0,0,0,0,0}

9 / 17

SU(3) Adjoint QCD (PBC) Nf = 2

−2

−1

0

VADJβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONF

DECONF

U(1) × SU(2)

VADJ , Nc = 3, Nf = 2

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

ℑ[TrP

]

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

ℜ[TrP ]

confined

deconfinedU(1) × SU(2)

VADJ , Nc = 3 phases

The data points(black dots) werefound by minimizingVeff with respect tothe Polyakov loopeigenvalues vi .

The confined phaseis accessibleperturbatively formβ ≤ 1.6.

−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

〈λλ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈λλ〉ADJ

−2

−1.5

−1

−0.5

0

0.5

1

Vm

odel

0 0.1 0.2 0.3 0.4 0.5a1

CONF

DECONF

U(1) × SU(2)

Vmodel , deformed YM, Nc = 3

There is a dramaticjump in 〈ψ̄ψ〉corresponding tothe deconfinementtransition

The model has thesame phases asQCD(Adj)

10 / 17

SU(4) Adjoint QCD (PBC) Nf = 2

−3

−2

−1

0

VADJβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONF

DECONF

SU(2) CONF

VADJ , Nc = 4, Nf = 2

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

ℑ[T

rP

]

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

ℜ[TrP ]

confined

deconfined

SU (2) confined

VADJ , Nc = 4 phases

The confined phaseis accessibleperturbatively formβ ≤ 1.0.

−1

0

〈λλ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈λλ〉ADJ Vmodel , deformed YM, Nc = 4

The model has thesame phases asQCD(Adj), andmore, but theadditional phasescan becircumnavigated.

11 / 17

SU(5) Adjoint QCD (PBC) Nf = 2

−1

0

VADJβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONFDECONFSKEW

VADJ , Nc = 5, Nf = 2

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

ℑ[T

rP

]

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

ℜ[TrP ]

confineddeconfinedSU (2) × SU (3) deconfmoving {0,−φ,−φ, φ, φ}

VADJ , Nc = 5 phases

The confined phaseis accessibleperturbatively formβ ≤ 0.8.

A moving phase isfound between theconfined andSU(2)×SU(3)-decphases.

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6−0.4

−0.20

0.2

0.4

0.6

0.8

1

〈λλ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈λλ〉ADJ Vmodel , deformed YM, Nc = 5

The model includesthe phases ofQCD(Adj).

The(non-C -breaking)moving phase of themodel is the sameas that ofQCD(Adj).

12 / 17

Accessibility of the confined phase as N → ∞, or as Nf is increased

00.20.40.60.8

11.21.41.61.8

2

mβ

2 4 6 8 10 12 14 16 18Nc

confined

critical mβ

Range of mβ for which the

confined phase is accessible in

QCD(Adj) with Nf = 2

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

VA

DJβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONF

DECONF

SU(2) CONF

SU(3) CONF

Veff for Nc = 6, Nf = 2

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

VA

DJβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONF

DECONF

SU(2) CONF

SU(3) CONF

Veff for Nc = 6, Nf = 3

As N → ∞ the maximum mβ for which the confined phase is accessible, (mβ )crit ,decreases.

However, as Nf increases, (mβ )crit increases (we must have Nf ≤ 5 Majoranaflavours to preserve asymptotic freedom).

13 / 17

Orientifold Planar Equivalence

The story:

Armoni, Shifman, and Veneziano(2003 - 2004) prove non-perturbatively theequivalence of the bosonic sectors of QCD(Adj) with Nf Majorana fermions andQCD(AS/S) with Nf Dirac fermions, in the planar limit.

Unsal and Yaffe (2006) show that on S1×R3 C -symmetry is broken in QCD(A/S)

when PBC are applied to fermions.

DeGrand and Hoffman (2007), Lucini et al (2007) showed using lattice simulationsthat the C -breaking is lifted as S1 is decompactified

Lucini et al. (2008) non-perturbatively prove orientifold equivalence in the quenchedapproximation (in the absence of C -breaking) using lattice simulations and calculatethe quark condensate in QCD(A/S/Adj)

We compare (to 1-loop order) the phase diagrams of QCD(A),QCD(S) with Nf = 2 (1Dirac flavour), to QCD(Adj) with Nf = 1 (Majorana flavour), for massive fermions withPBC.

14 / 17

SU(6) QCD(A) (left), (S) (middle), and (Adj) (right) for PBC on fermions

−10

−5

0

VASβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONFDECONFC-breaking

VA,+, Nc = 6, Nf = 2

-14

-12

-10

-8

-6

-4

-2

0

2

VSYMMβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONF

DECONFC-breaking

VS,+, Nc = 6, Nf = 2

−8

−7

−6

−5

−4

−3

−2

−1

0

1

VADJβ

4

0 1 2 3 4 5 6 7 8 9 10mβ

CONF

DECONF

SU(2) CONF

SU(3) CONF

VAdj ,+, Nc = 6, Nf = 1

0

0.1

0.2

0.3

0.4

0.5

0.6

〈ψ̄ψ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈ψ̄ψ〉A for Nc = 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

〈ψ̄ψ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈ψ̄ψ〉S for Nc = 6

−3

−2

−1

0

〈λλ〉β

3

0 1 2 3 4 5 6 7 8 9 10mβ

〈λλ〉Adj for Nc = 6

15 / 17

The C breaking phase of QCD(AS/S)

The C -breaking phase is favoured in the case where PBC are applied to fermions inthe A and S representations (When ABC are used the deconfined phase is favoured).

For example, when Nc = 6 the C -breaking phase has the Polyakov loop eigenvalues

v = {2π6

,2π6

,2π6

,2π6

,2π6

,2π6}

P is clearly not invariant under P → P∗.

16 / 17

Conclusions

One-loop PT:

In QCD(Adj) for Nf ≥ 2 there are several exotic phases occuring between theconfined and deconfined phases

In QCD(Adj) for Nf ≥ 2, as N increases, (mβ )crit , below which the confined phase isaccessible, decreases.

In QCD(Adj) for Nf ≥ 2, as Nf is increased, the confined phase is accessible for alarger (mβ )crit .

In QCD(A/S) with PBC for fermions the C -breaking phase is favoured for all mβ .

For all representations there is a clear one-loop contribution to 〈ψ̄ψ〉 for small mβ .◮ In QCD(AS), QCD(S), QCD(F) there is an inflection point in Veff at which 〈ψ̄ψ〉 6= 0,

in the deconfined phase.◮ In QCD(Adj) for Nf ≥ 2 (with PBC on fermions) the chiral condensate peaks at the

transition to the deconfined phase

In QCD(Adj) for Nf = 1 the deconfined phase is favoured for all mβ .

The deformed Yang-Mills theory finds all the phases of QCD(Adj), and the an can beslowly varied to go through the phases in the same order.

17 / 17