g2 lattice gauge theory at finite density and temperature · g 2 lattice gauge theory at finite...

G2 Lattice Gauge Theory at Finite Density andTemperature

Andreas Wipf

Theoretisch-Physikalisches InstitutFriedrich-Schiller-University Jena

collaboration withA. Maas, L. von Smekal, B. Wellegehausen (C. Wozar)

5th International Conference on New Frontiers in Physics , Kolymbari, July 2016

Andreas Wipf (Jena) G2 Lattice Gauge Theory at Finite Density and Temperature

1 G2-Gluodynamics

2 G2 Yang-Mills-Higgs Theory and SSB

3 G2 Gauge Theory with Dynamical Fermions

4 Spectroscopy

5 Finite Temperature and DensityPhase diagram – deconfinement transitionPhase diagram – baryon numberHigh-precision simulations for two-flavor SU(2) in two dimensions


Bjoern Wellegehausen,Jena/Gießen

Axel MaasGraz

Lorenz von Smekal,Gießen

eff. Polyakov Loop dynamicsPhys. Rev. D80 (2009) 065028

Casimir Scaling/string breakingPhys. Rev. D83 (2011) 016001

phase diagram YMHPhys. Rev. D83 (2011) 114502

phase diagram with fermions Phys. Rev. D86(2012) 111901

phase diagram, spectroscopyPhys. Rev. D89 (2014) 056007

earlier papers with C. Wozar.


Expected QCD phase diagram 4 / 57

various condensates, quark density, Polyakov-loops, . . .almost no first principle results for large baryon density


In this talk: Lattice approach 5 / 57

space-time latticefinite volume

V = (aNt )× (aNs)d−1

β = aNt , L = aNs

functional integrals→finite-dimensional integralswithout quarks:accurate resultson large lattices ≈ 1284

with quarks:fermionic determinantnon-local integrandextrapolation (a,mF )sign problem for µ 6= 0


Gluodynamics and QCD 6 / 57

zero baryon-density: accessible to simulationswithout quarks⇒weakly first order PT with order parameterMC method applicable, relatively cheapwith quarks⇒masses of hadrons, glueballs, . . . , matrix elements, life-times,equation of state, . . .expensive simulations

large baryon density: not accessible to simulationsnot much known (effective models)ideas/proposals on exotic phases of cold dense matterrelevant for n∗ (equation of state)


Why G2-Gauge Theory 7 / 57

no sign problem⇒ simulations at finite density feasiblerepresentations are realbaryon and meson spectrum similar to QCDsimilar glueball spectrum as SU(3) Wellegehausen, Wozar, AW

G2 contains SU(3) as subgroup⇒smooth interpolation G2 −→ SU(3) by SSBconfinement↔ center of gauge group? Hollands, Minkowski, Pepe, Wiese

smallest (simply connected) Lie group with trivial center

only gauge theory with fermionic baryons whichhas been simulated at high baryon density


Color-singlet states and branching to SU(3) 8 / 57

expected colour-singlet bound 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} ⊗ {14} = {1} ⊕ . . .

SSB G2 → SU(3) (cp. GUTS)

fermions: {7} −→ {3} ⊕ {3} ⊕ {1},gauge bosons: {14} −→ {8} ⊕ {3} ⊕ {3}.


G2 Yang-Mills theory has 9 / 57

quark gluon plasma at high Tconfinement of color at low Tteaches us about role of center for confinementtests ideas/proposals on exotic phases of cold dense matterallows for a test quality of frequently used approximations:MF-type, hopping-parameter expansions, strong-coupling expansions,effective models, renormalization group, . . .condensates and nB distinguish between different phasessuggestions are welcome


G2 gauge theory without fermions: what is known 10 / 57

1st order confinement/deconfinement PT Hollands, Minkowski, Pepe, Wiese 2003

Cossu, D’Elia, di Giacomo, Lucini, Pica 2007

Casimir scaling Greensite, Langfeld, Olejnik, Reinhardt, Tok 2007; Liptak, Olejnik 2008

effective Polyakov loop dynamics Wellegehausen, AW, Wozar 2009

chiral restoration in quenched theory at same Tc Danzer, Gattringer, Maas 2009

G2 monopoles (Shnir), semiclassical dyon picture (Diakonov, Petrov 2010), . . .string breaking, gluelump spectrum Wellegehausen, AW, Wozar 2011

SSB G2 →SU(3) Wellegehausen, AW, Wozar 2012

topological properties, instantons Maas, Olejnik, Ilgenfritz 2013

equation of state, 〈Tµµ 〉T ∝ T 2

Bruno, Caselle, Panero, Pellegrini 2015


simulations: first order PT (local HMC, moderate 163 × 6 lattice)

rapid change with β ∝ 1/g2

histogram in vicinity of βc ≈ 9.65first order transition


Casimir scaling 12 / 57

static quark-antiquark potential representation R

VR(R) = γR −αRR

+ σRR

e−βV (R) varies over ≈ 50 orders of magnitude until string breaksCasimir scaling hypothesis for string tensions:

σRcR

=σR′

cR′

−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

(scaled) string tensions (µ2 = σ7) Liptak, Olejnik; WWW: N = 48, β = 30

aAndreas Wipf (Jena) G2 Lattice Gauge Theory at Finite Density and Temperature

Phase diagram of finite-temperature G2 Higgs Model 13 / 57

lattice action with normalized 7-component Higgs ϕ

SYMH[U , ϕ] = −β∑2

(1− 1

7tr U2

)− κ

∑ϕT

x Ux,µϕx+µ

spontaneous symmetry breaking G2 → SU(3):{14} −→ {8} ⊕ {3} ⊕ {3} gluons ⊕ massive Vector bosonsκ = 0: pure G2 gauge theory: 1st order deconfinement transitionκ =∞: pure SU(3) gauge theory: 1st order deconfinement transitioncomplete phase diagram of G2 YMH theory:average plaquette action, Higgs action and Polyakov loopsusceptibilities (and higher derivatives), finite size analysis≤ 5× 105 configurations, on grid ≤ β = 5 . . . 10, κ = 0 . . . 104


Phase diagram: global picture 14 / 57

βcrit = 9.55(5) , κcrit = 1.50(4)

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

163 × 6-lattice


Gap in first order line? 15 / 57

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

Summarizing the phase diagram of G2 YMH-theoryWellegehausen, Wozar, AW


G2 QCD with dynamical fermions 16 / 57

collaboration with Axel Maas, Lorenz von Smekal and Bjoern Wellegehausen

det D(U, µ,m] ≥ 0 for any Nf

⇒ simulations at finite T and µ possiblemeasure chiral/di-quark condensate, baryon-density, Polyakov loop,. . .interpretation of phase transitions and condensates⇒ need particle spectrum at T = 0 Tglueballsmesonsbaryons: nq = 1,2,3exotic particles


global symmetries and possible breakings for Nf = 1 17 / 57

SU(2) U(1)Bm

U(1)B U(1)Bm

µ µ

enlarged chiral symmetry (Pauli-Gürsey)Goldstone bosons

d(0++) ∼ ψCγ5ψ + ψγ5ψC and d(0+−) ∼ ψCγ5ψ − ψγ5ψ

C


Spectroscopy – lattice setup 18 / 57

Wilson fermions, Symanzik improved gauge action, RHMC-algorithmβ = 0.96, κ = 0.159 fixedensemble for Spectroscopy16× 83 latticeProton mass mN = 938 MeV, diquark mass md(0+) = 247 MeV

Lattice spacing a = 0.343 fm ∼ (575 MeV)−1

ensembles at finite temperature and densityLattice size: Nt × 83 with Nt = 2 . . . 16T = 36 . . . 287 MeV, 9 valuesµ = 0 . . . .354 MeV, 60 values below lattice saturation


Interpolating operators: Mesons nq = 0 19 / 57

{7} ⊗ {7} = {1} ⊕ . . .

Name Operator Pos. Spin Colour Flavour J Pη uγ5u S A S S 0 -f uu S A S S 0 +ω uγµu S S S A 1 -h uγ5γµu S S S A 1 +π uγ5d S A S S 0 -a ud S A S S 0 +ρ uγµd S S S A 1 -b uγ5γµd S S S A 1 +

unquenched or partially quenched in one-flavor simulations


Interpolating operators: Baryons nq = 1 20 / 57

{7} ⊗ {7} ⊗ {7} = {1} ⊕ . . .{7} ⊗ {14} ⊗ {14} ⊗ {14} = {1} ⊕ . . .

Name Operator Pos. Spin Col. Flav. J PHybrid εabcdefguaF p

µνF qµνF r

µνT bcp T de

q T fgr S S A S 1/2 ±

∆ T abc(uaγµ287ub)uc S S A S 3/2 ±N T abc(uaγ5db)uc S A A A 1/2 ±

hybrid-spectroscopy difficultmη −mdiquark = disconnected contributions



{7} ⊗ {7} = {1} ⊕ . . .

Name Operator Pos. Spin Colour Flavour J P Cd(0++) uCγ5u + uγ5uC S A S S 0 + +d(0+−) uCγ5u − uγ5uC S A S S 0 + -d(0−+) uCu + uuC S A S S 0 - +d(0−−) uCu − uuC S A S S 0 - -

d(1++) uCγµd + uγµdC S S S A 1 + +d(1+−) uCγµd − uγµdC S S S A 1 + -d(1−+) uCγ5γµd + uγ5γµdC S S S A 1 - +d(1−−) uCγ5γµd − uγ5γµdC S S S A 1 - -

diquark masses are degenerate (cc)no sources for diquarks needed



{7} ⊗ {7} ⊗ {7} = {1} ⊕ . . .

Name Operator Pos. Spin Colour Flavour J P∆ T abc(uC

a γµub)uc S S A S 3/2 ±N T abc(uC

a γ5db)uc S A A A 1/2 ±


Masses of mesons and baryons 23 / 57

Wellegehausen, Maas, Smekal, AW (2013)


Low temperature: baryon density vs. chemical potential 24 / 57

silver blazediquark BEcondensationmd(0+) = 247 MeVplateaus visible forlarger nq

three transitionshadronic phase atm∆ ≈ mN

Wellegehausen, Maas, Smekal, AW (2013)


(Preliminary) interpretation 25 / 57

low density: in accordance with silver blazetwo small jumps at diquark thresholds⇒ two (probably) second order PT?two plateaus after thresholdsBose-condensates of diquarks?one (probably) first order PT at ≈ ∆ thresholdhadronic phase for higher nq (under investigation)aµ & 1: lattice artifacts, e.g. saturation effects


Phase diagram – Deconfinement transition (µ = 0) 26 / 57

Polyakov loop 〈P〉 and susceptibility ∂〈P〉∂T at µ = 0

Deconfinement transition at Tc(µ = 0) ≈ 137 MeV(cubic splines, no thermalization at 80 MeV)


Phase diagram – deconfinement transition (selected µ’s) 27 / 57

Polyakov loop 〈P〉 and susceptibility ∂〈P〉∂T

deconfinement transition shifts to smaller T for larger µpeak of susceptibility increases


Phase diagram – deconfinement transition 28 / 57

Polyakov loop 〈P〉 and susceptibility ∂〈P〉∂T (60 µ’s)

deconfinement transition shifts to smaller T for larger µpeak of susceptibility increases, consistent with T ′c(µ)

∣∣µ=0 = 0

Tc(µ)

Tc(0)∼ 1 + 0.001(8)

µ

Tc(0)− 0.031(6)

(µ

Tc(0)

)2


scanning (T , µ) space 29 / 57

temperatures: T = 287, 192, 144, 115, 96, 82, 72, 48, 36 MeVchemical potential: 60 µ-values between 0 and 354 MeV for each T

100 200 300 400 500 600

50

100

150

200

250

300

00

Tin

MeV

µ in MeV

deconf

conf

vacuum

diquark

cond

nuclear

matter

underway

latticesaturation

?


Phase diagram – baryon number 30 / 57

quark number density

T = 287 MeV




T = 192 MeV




T = 144 MeV




T = 115 MeV, µdeconf = 317 MeV




T = 96 MeV, threshold effects?




T = 82 MeV

estimater problem for small µ when 〈nB〉 ≈ 10−3 − 10−4




T = 72 MeV

〈nB〉 may jump at thresholds


Phase diagram – baryonic matter transition 37 / 57


T = 48 MeV


Phase diagram – baryonic matter transition 38 / 57


T = 36 MeV


Simulation at higher µ and lower T required 39 / 57

G2-QCD phase diagram with V = (2.7 fm)3 and md(0+) = 247 MeV


High-precision model calculations: two-flavor SU(2) in two dimensions 40 / 57

4-dimensions: parameter study in (µ,T ) plane very expensiveas today low T and finite µ not accessible in 4-dimensional theoriesdetailed studies of µ-dependence for interpretation⇒ lower-dimensional theories as test-bedshere: two-flavor SU(2) in two dimensionsno sign problembehaves (not only qualitatively) similar to G2 QCDin two dimensions: no breaking of continuous symmetry!

earlier studies in 4d: Simon Hands et al., von Smekal et al.


no SSB of continuous symmetry, π not lightest particle,lightest particle: vector diquark (as for G2)scale: µc defined by silver blaze, µc = 1

2 md

mass of vector diquark in lattice units 0.22complete mass-spectrum: in progressSimulation parameters:

simulations at fixed β = 3.8 and κ = 0.27 and Ns = 16Nt ∈ {2, 4, 6, 8, 10, 12, 14, 16, 24, 32, 48, 128}, 12 temperaturesµ from 0 to 1.15 in steps ∆µ = 0.002, fine gridcolor-plots in T -variable interpolatedtypical 1000 configurations, HMC, Wilson fermions,


µ-dependence of chiral condensate for different T 42 / 57

steps are washed out at higher temperaturesvarious types of diquarks can condensevery high resolution in µ, small statistical errors


color-plot 〈ψψ〉, gobal picture 43 / 57

µc defined by silver blazeµ ≥ 7µc : lattice artifacts, saturation


color-plot 〈ψψ〉, intermediate µ 44 / 57

µ < µc : explicit breaking of chiral symmetryµ ' µc : chiral condensate rotates into diquark condensate


color-plot 〈ψψ〉, small µ 45 / 57

line emanating from µ = µc bends ’to the right’


color-plot ∂µ〈ψψ〉, small µ 46 / 57


T -dependence of 〈P〉, several µ 47 / 57

staggered fermions 〈P〉 = 0, not true for Wilson fermionscan probe very low temperature (not yet feasible in 4 dimensions)


color-plot 〈P〉, all µ 48 / 57

lattice artifacts for µ > 7µc


color-plot 〈P〉, intermediate µ 49 / 57


µ-dependence of nq for different T 50 / 57

only states with vanishing total momentum contributesaturation for large µ, Nsat = 2× 16 = 32onset at µ = µc , steps of 2steps washed out with increasing T


color-plot nq , global picture 51 / 57


color-plot nq , intermediate µ 52 / 57


color-plot nq , small µ 53 / 57


color-plot ∂µnq , intermediate µ 54 / 57


color-plot ∂T nq , intermediate µ 55 / 57


Conclusions and outlook 56 / 57

finite size/low temperature for high-density G2 in four dimensions?

evidence for first order nuclear matter transition

chiral symmetry restoration vs. confinement/deconfinement:

to this day not conclusive

deconfinement transition at lower temperatures / critical endpoint?

hybrid spectroscopy (difficult)

diquark condensation in 4d G2 and 2d two-flavour SU(2) at work

equations of state for µ ≤ 350 MeV


Summary 57 / 57

G2 QCD is a useful laboratoryinterpolation G2-QCD→ QCD with Higgs-mechanism possiblefull phase diagram (in principle) accessible to simulationsfirst order(?) transition line (critical end point?)deformation vs. sign problem?testbed for model buildingtestbed for alternative approaches (expansions, renormalization group)?comparison with strong coupling / hopping parameter expansionsome of these questions can be answered in lower dimensions


g2 lattice gauge theory at finite density and temperature · g 2 lattice gauge theory at finite...

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