G2-QCD at Finite Density

A. Wipf

Theoretisch-Physikalisches Institut, FSU Jena

collaboration with

Axel Maas (Jena)

Lorenz von Smekal (Darmstadt/Gießen)

Bjoern Wellegehausen (Gießen)

Christian Wozar (Jena)

Swansea, 07.06.2014

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 1 / 37

1 Introduction

2 G2 and G2 Gauge Theory

3 Confinement in pure G2 gauge theory

4 Breaking of G2 to SU(3) Gauge Theory

5 G2-QCD at Finite Density

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 2 / 37

Water

different phases, transition lines, critical point, triple point, . . .

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 3 / 37

that is not all:

crystalline andamorphous phasesorder of H-bindingcritical pointtriple pointsstructural transitionssmall p:hex/cub ice, ice XIhigh p:ice VII, VIII, X

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 4 / 37

Some facts about exceptional Lie group G2

smallest exceptional Lie grouprank = 2, dimension = 14real group (not only pseudo-real!)subgroup of SO(7)G2/SU(3)∼ S7 → efficient parametrizationfundamental representations {7}, {14} (= adjoint)7 quarks instead of 3 (cp. GUTS)can be broken to SU(3) with scalars in {7}fermions: {7} → {3}+ {3}+ {1}, gauge bosons: {14} → {8}+ Xsmallest (simply connected) Lie group with trivial center

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 5 / 37

singlet representation colorless states

{7} ⊗ {7} = {1} ⊕ {7} ⊕ {14} ⊕ {27}{7} ⊗ {7} ⊗ {7} = {1} ⊕ 4 · {7} ⊕ 2 · {14} ⊕ . . .{14} ⊗ {14} = {1} ⊕ {14} ⊕ {27} ⊕ . . . ,

{14} ⊗ {14} ⊗ {14} = {1} ⊕ {7} ⊕ 5 · {14} ⊕ . . . ,{7} ⊗ {14} ⊗ {14} = {1} ⊕ . . .

branching G2 → SU(3)

{7} −→ {3} ⊕ {3} ⊕ {1},{14} −→ {8} ⊕ {3} ⊕ {3}.

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 6 / 37

Same facts about G2 gauge theory

asymptotically freefirst order confinement/deconfinement PT Pepe et al.; Greensite; Cossu et al.

no center, no order parameterchiral restoration in quenched theory at same Tc Graz group

qualitatively similar glueball spectrum as SU(3) Wellegehausen, Wozar, AW

Casimir scaling to high accuracy Wellegehausen, Wozar, AW; Liptat et al.; Greensite et al.

topological properties, instantons Maas, Olejnik, Ilgenfritz

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Why consider G2 Theories

G2 has trivial center:confinement models based on center?G2 contains SU(3) as subgroup:smooth interpolation between G2- and SU(3)-gauge theoriesparticle spectrum comparable to QCD (mesons and baryons + . . . )G2 has no sign problem:simulations at finite T and finite possible µ possibleG2-QCD has baryons and mesons:can build a ”neutron star”

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 8 / 37

understanding of G2 under extreme conditionsphases and phase-transition at finite T and nB

distinguish phases:densities, pressure, energy density, condensates, symmetries, . . .

order parameters⇔ symmetriesvary control parameters:temperature, chemical potentials, fields,physics question:what are the relevant degrees of freedom in a given phase?

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Lattice simulations for SU(3)

large baryon density:sign problem⇒ not accessible to simulations based on important samplingeffective models, functional methods, ....proposals/speculations on exotic phases of cold dense matterrelevant of n∗?simulations of theories without sign problemeven better: solve sign problem?here: fast implementation of (local)HMC, no low acceptence rate,

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Confinement in pure G2 gauge theory

finite T ⇒lattice = cylinder withcircumferenceβT = 1/kT = aN0.

space

time

Px

approximate order parameter: Polyakov loop

P(x ) = tr P exp[i∫ β

0dτA0(τ, x )

]static potential

〈P(x )〉β = e−βF (x ) , 〈P(x )P†(y)〉β = e−βVqq(R)

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 11 / 37

Polyakov loop in fundamental representation

rapid change with β = 1/g2 histogramm in vicinity of βc

Polyakov loop approximate order parameterfirst order PT as in SU(3) gluodynamics

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Vqq(R) static potential

confinement: 〈P〉 = 0 , de-confinement: 〈P〉 6= 0

confinement: V → σR ⇒ 〈P(x )P†(y)〉β ∝ e−σ·Area

e σ·Area varies over 100 orders of magnitudebrute force approach does not workLüscher and Weisz method: exponential error reductionsplit lattice in time slicescalculate 〈. . . 〉 with fixed bc on each slicefull result: integral over boundary conditionsiteration→ multilevel algorithm

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 13 / 37

static potential for charges in representation R:

VR(R) = γR −αRR

+ σRR

Casimir scaling hypothesis for string tensions:

σRcR

=σR′cR′

from ratios of Wilson(Polyakov) loops

VR(R) =1τ

ln〈WR(R,T )〉〈WR(R,T + τ)〉

.

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 14 / 37

−6

−4

−2

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0 2.5

VR/µ

µR

R = 7R = 14R = 27R = 64R = 77R = 77′R = 182R = 189

linear potential for static quarks in different G2 representations

Wellegehausen, AW., Wozar

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String-breaking

meson, diquark qq →2 mesons, diquarks

Click here

energy scale = 2 mglueball

decay products: glue-lumps

−2

−1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6

VR/µ

µR

R = 7, β = 30, N = 483R = 14, β = 30, N = 483R = 7, β = 20, N = 323R = 14, β = 20, N = 323

Wellegehausen, AW., Wozar (2011)

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 16 / 37

G2 Yang-Mills-Higgs theory

breaking G2 → SU(3)

lattice action with normalized Higgs ϕ = (ϕ1, . . . , ϕ)T in {7}

SYMH[U , ϕ] = − 1g2

∑U2 − κ

∑ϕT

x Ux ,µϕx+µ

Higgs-mechanism for v = 〈ϕ〉 6= 0:{14} −→ {8} ⊕ {3} ⊕ {3}{8}: SU(3) gluons{3}+ {3}: massive Vector bosonsscalars 7→ 1

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κ = 0: pure G2 gauge theory:first order deconfinement transitionκ =∞: 6 vector bosons decouple, pure SU(3)

first order deconfinement transitionfirst order transition line connecting two theories?calculatePolyakov loop and plaquette actions, susceptibilitieson grid in β ∝ 1/g2, κ-plane (β = 5 . . . 10, κ = 0 . . . 104)⇒

Phase diagram of G2 YMH theory

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average actions and susceptibilities (small lattice)

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Results

average plaquette action, Higgs action and Polyakov loopsusceptibilities (and higher derivatives) and finite size analysislarge β ∝ 1/g2 ⇒ Higgs transition linecluster algorithm for SO(7) nonlinear sigma modelline of second order PT O(7)→ O(6)

line of first order PT G2 →SU(3) with small gap in betweentriple point

βcrit = 9.55(5) , κcrit = 1.50(4)

first order (almost?) line hits second order lineconfining phase meets two deconfining phases

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0

100

101

102

103

∞

5 10 15 30 ∞

κ

β

SO(7) broken

SU(3) deconfinement

SO(7) unbroken

G2 deconfinement

confinement

3

456

phase diagram (163 × 6): global picture

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1

1.2

1.4

1.6

1.8

2

9 9.2 9.4 9.6 9.8 10

κ

β

1

2

7

phase diagram (163 × 6): where the lines almost meet

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 22 / 37

00.5 6 7 8 900.5 6 7 8 9 9.39.49.59.69.79.89.9 10β

20 30 60 ∞20 30 60 ∞

11.21.41.61.82κ

101001000∞101001000∞ SO(7) brokenSU(3) de on�nement

SO(7) unbrokenG2 de on�nement on�nement

phases of G2 YMH-theoryWellegehausen, Wozar, AW

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 23 / 37

G2 QCD with dynamical fermions

collaboration with Axel Maas, Lorenz von Smekal und Bjoern Wellegehausen

fermionic determinant real and positiveno sign problem: simulations at finite T and µexpected particle spectrum:glueballsbosonic quark-quark bound states (mesons, diquarks)fermionic 3 quark states (baryons)fermionic 1 quark - 2 gluon bound states (fermion hybrids)

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 24 / 37

confinement-deconfinement transition (Polyakov loop)chiral symmetry breaking (chiral condensate)quenched: same critical temperatures Maas, Gattringer

G2 has fermionic baryonsdegenerate Fermigas at large ρB

relevant for physics of compact ’stellar objects’Bose-condensates of diquarks, . . .results from (expensive) numerical simulations

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G2-QCD: global symmetries and breaking

anomalousspontaneous 〈ψψ〉explicit m, µ

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first results on 〈ψψ〉, nB and 〈P〉

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Andreas Wipf (FSU Jena) G2-QCD at Finite Density 28 / 37

Spectroscopy I

mesons (baryon number 0)Name O T J P Cπ uγ5d SASS 0 - +η uγ5u SASS 0 - +a ud SASS 0 + +f uu SASS 0 + +ρ uγµd SSSA 1 - +ω uγµu SSSA 1 - +b uγ5γµd SSSA 1 + +h uγ5γµu SSSA 1 + +

exotic particles (baryon number 1)Name O T J P C

N ′ T abc(uaγ5db)uc SAAA 1/2 ± ±∆′ T abc(uaγµub)uc SSAS 3/2 ± ±

Hybrid εabcdefguaF bcµνF de

µνF fgµν SSSS 1/2 ± ±

T: (x , s,C,F )

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 29 / 37

diquarks (baryon number 2)Name O T J P Cd(0++) uCγ5u + c.c. SASS 0 + +d(0+−) uCγ5u − c.c. SASS 0 + -d(0−+) uCu + c.c. SASS 0 - +d(0−−) uCu − c.c. SASS 0 - -d(1++) uCγµd − dCγµu + c.c. SSSA 1 + +d(1+−) uCγµd − dCγµu − c.c. SSSA 1 + -d(1−+) uCγ5γµd − dCγ5γµu + c.c. SSSA 1 - +d(1−−) uCγ5γµd − dCγ5γµu − c.c. SSSA 1 - -

baryons (baryon number 3)Name O T J P C

N T abc(uCa γ5db)uc SAAA 1/2 ± ±

∆ T abc(uCa γµub)uc SSAS 3/2 ± ±

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spectroscopy II

diquark masses are degeneratecontain only connected contributions (as pions in QCD)mη −mdiquark = disconnected contributionstree level improved Szymanzik actionWilson fermions (chiral properties?)no sources for diquarks neededNF complex-valued pseudo-fermions plus RHMCtwo time-scale integration (Sexton-Weingarten and leapfrog)further optimizations (preconditioning, adaptive mesh, . . . )

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 31 / 37

masses of mesons, diquarks, baryons

heavy ensemble light ensemble

Wellegehausen, Maas, Smekal, AW (2013)

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zooming in in: baryon density vs. chemical potential

nq grows rapidly at half of(0−) mass (silver blaze)plateaus visible for larger nq

three transitionsphase betweenµq = 300− 600 MeV:hadronic phase→ quasi particle picture

Wellegehausen, Maas, Smekal, AW (2013)

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comparison with fermi gas of (Wilson) fermions

fit above aµ = 1⇒ κ = 0.162, nsatf = 14.4

cp. with free quarks: κ = 0.147, nsatf = 14⇒ saturation regime

fit below aµ = 1⇒ κ = 0.211, nsatf = 4.02

cp. lattice gas of freee ∆−baryons: nsatf = 4

0.6 ≤ aµ ≤ 1: nq due to fermionic baryons (same as spectroscopy)

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(preliminary) interpretation

low density: in accordance with silver blazeclean signal (no diquark sources)two small jumps at diquark thresholds⇒ two (probably) second order PT?two plateaus after thresholdsBose-condensates of diquarks?admixture with gas of diquarks?one (probably) first order PT at ≈ ∆ thresholdsimulations slow down near PThadronic phase for higher nq (under investigation)aµ & 1: lattice artifacts, e.g. saturation effects

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 35 / 37

Summary

G2 QCD is a useful laboratoryaccessible at finite density by lattice methodsphases and transition at high densities and temperaturescondensates, access to hadronic phaseaccess to dense baryonic matter (as in n∗)shares many features with real QCDinterpolation G2-QCD→ QCD with Higgs-mechanism possiblefull phase diagram in principle accessible to simulations

Andreas Wipf (FSU Jena) G2-QCD at Finite Density 36 / 37

Outlook

clarify further nature of dense and cold phasesinclude finite temperature effects beyond rough overviewfollow first order transition line (critical end point?)break G2-QCD→ QCD with quarks via Higgs-mechanismdeformation vs. sign problem?testbed for model buildingtestbed for alternative approaches (eg. renormalization group)?

Thanks for your attention

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