monopole condensation in two-flavour adjoint qcd: an update · introduction simulation details...

IntroductionSimulation details

Results and conclusions

Monopole condensationin two-flavour Adjoint QCD: an update

Giuseppe Lacagnina1, 2

1Dipartimento di Fisica “E. Fermi”Pisa

2INFN Pisa

Lattice 2007, Regensburg

Giuseppe Lacagnina An update in monopole condensation in Nf = 2 aQCD

Work in collaboration with:

G. Cossu, Scuola Normale Superiore and INFN, PisaM. D’Elia, University of Genova and INFNA. Di Giacomo, University of Pisa and INFNC.Pica, BNL

Plan of the talk

1 Introductionconfinement and Lattice QCDmonopole condensation basicsQCD with adjoint fermions

2 Simulation detailssimulation algorithmssimulation parameters

3 Results and conclusions

confinementmonopole condensation basicsaQCD

Plan of the talk

the ABC of confinement

isolated quarks have never been observedit seems that only colour-less states exist in nature, i.e.states that transform as singlets under local SU(3)rotationshowever, at high temperatures, one expects to find a newstate of matter, the Quark Gluon Plasma, in which quarksare deconfinedexperimental results are expected from RHIC and LHC

are we facing two phases with different realizations ofsome symmetry?if so, which symmetry is involved?what is the order of such a transition?

the quenched case

In the quenched case (no quark loops, det D = const.),equivalent to the limit

m →∞

for the quark mass

we have a weak first order transition (order-disorder);the symmetry is Z3 (center of SU(3))...and one possible order parameter is the Polyakov loop

the quenched case

m →∞

for the quark mass

the quenched case

m →∞

for the quark mass

the quenched case

m →∞

for the quark mass

the Polyakov loop is defined in the continuum as

L(~x) = TrcP exp

∫ 1/T

0dτA4(τ, ~x)

]and on the lattice as

L(~x) =1

Nt−1∏τ=0

U4(τ, ~x)

the Polyakov loop is defined in the continuum as

L(~x) = TrcP exp

∫ 1/T

0dτA4(τ, ~x)

]and on the lattice as

L(~x) =1

Nt−1∏τ=0

U4(τ, ~x)

under the transformation

U4(τ0, ~x) → zU4(τ0, ~x)

where z ∈ Z3 and all other links are unchangedthe gauge action is invariant, while the Polyakov loop

L(~x) → zL(~x)

under the transformation

U4(τ0, ~x) → zU4(τ0, ~x)

where z ∈ Z3 and all other links are unchangedthe gauge action is invariant, while the Polyakov loop

L(~x) → zL(~x)

if the ground state is invariant under the abovetransformation, it must be

〈L〉 = 0

otherwise

〈L〉 6= 0

if the ground state is invariant under the abovetransformation, it must be

〈L〉 = 0

otherwise

〈L〉 6= 0

however, the expectation value of the Polyakov loop is relatedto the free energy of an isolated quark

〈L〉 ' exp(−F/T )

and the Polyakov loop is an order parameter for confinement:

it is zero in the confined phase: F = ∞and non-zero in the deconfined phase: F < ∞

see L.D. McLerran, B. Svetitsky, Phys. Rev. D24 (1981)

〈L〉 ' exp(−F/T )

quarks however...

finite mass quarks present a problem: their presencebreaks the Z3 symmetry explicitlyand the Polyakov loop is not a good order parameterthe nature of the confinement/deconfinement transition forrealistic quark masses and two dynamical flavours isargument of debate

quarks however...

Plan of the talk

a theoretical explanation of confinement in QCD from firstprinciples is still lackingbut models exist that relate it to some property of thefundamental state of the theoryone of these models is based on the dualsuperconductivity of the QCD vacuum

the dual superconductor

a magnetic symmetry is spontaneously brokengiving rise to a non-vanishing magnetically chargedHiggs condensatethe QCD vacuum acts as a dual superconductor of typeIIand chromo-electric fields between static colourcharges are squeezed into Abrikosov tubeswhose energy grows linearly with the separation, givingrise to confinement

we therefore need to define some magnetically chargedoperator in such a way that it adds a monopole field to a givenfield configuration

µ|~A〉 = |~A + ~b〉

〈µ〉 6= 0

signals condensation of magnetic charges

we therefore need to define some magnetically chargedoperator in such a way that it adds a monopole field to a givenfield configuration

µ|~A〉 = |~A + ~b〉

〈µ〉 6= 0

signals condensation of magnetic charges

such an operator can be constructed (see D’Elia, M. et al.,Phys. Rev. D71, 114502(2005))and its vacuum expectation value is given by

〈µ〉 =ZM

Zwhere ZM is the partition function for the action in presence of amonopole

such an operator can be constructed (see D’Elia, M. et al.,Phys. Rev. D71, 114502(2005))and its vacuum expectation value is given by

〈µ〉 =ZM

Zwhere ZM is the partition function for the action in presence of amonopole

to better cope with fluctuations, one instead calculates thequantity

dβln〈µ〉 = 〈S〉S − 〈SM〉SM

in which 〈SM〉SM is the average of the action with a monopoleinsertion weighted with the modified action itself

to better cope with fluctuations, one instead calculates thequantity

dβln〈µ〉 = 〈S〉S − 〈SM〉SM

in which 〈SM〉SM is the average of the action with a monopoleinsertion weighted with the modified action itself

the parameter 〈µ〉 should be different from zero in theconfined phase where magnetic charges condense anddrop to zero at deconfinement where magnetic symmetryis restoredthis drop corresponds to a negative peak of ρ

In the vicinity of the critical temperature, β ' βc , ρ isexpected to scale as

ρN−1/νs = f

(N1/ν

s (βc − β))

the critical exponent ν should be equal to 1/3 for a first ordertransition

ρN−1/νs = f

(N1/ν

s (βc − β))

ρN−1/νs = f

(N1/ν

s (βc − β))

ρN−1/νs = f

(N1/ν

s (βc − β))

ρN−1/νs = f

(N1/ν

s (βc − β))

Plan of the talk

Quarks in the adjoint rep. of QCD have 8 colour degrees offreedom and can be described with 3× 3 hermitian, tracelessmatrices:

q(x) = qa(x)λa

using Gell-Mann’s λ matrices

q(x) = qa(x)λa

to accommodate adjoint fermions, gauge link variables must berewritten in terms of their 8− dimensional rep.:

Uab(8) =

Trc(λaU(3)λ

bU†(3))

which is real

Uab(8) =

Trc(λaU(3)λ

bU†(3))

which is real

Uab(8) =

Trc(λaU(3)λ

bU†(3))

which is real

The full action for aQCD therefore reads:

S = Sgauge[U(3)] +(q̄, D(U(8))q

)what is the interest one could have in studying this model?

The full action for aQCD therefore reads:

S = Sgauge[U(3)] +(q̄, D(U(8))q

)what is the interest one could have in studying this model?

why aQCD?

in QCD, the chiral and the deconfinement transition takeplace at the same temperaturein aQCD the two transitions seem to take place at differenttemperaturesfermions in aQCD do not break the Z3 symmetry and thePolyakov loop is again a good order parameter

why aQCD?

our plan (still in progress...)

study monopole condensation in aQCDlook at the Polyakov loopstudy the properties of the monopole parameter ρ

study the chiral transition and its effects on ρ

simulation algorithmssimulation parameters

Plan of the talk

some technical details...

C∗ boundary conditions had to be used to accommodatemonopoles on the lattice;we simulated 2 flavours of staggered adjoint quarks usingthe RHMC and Φ ( with Metropolis test ) algorithmswith trajectories of unit length and δt = 0.02giving an acceptance of around 90%

the code was written in TAO and run on ApeMille andApeNEXT facilities (Pisa, Roma)

number of crates 12number of processors per crate 256peak performance per crate 250 GFlopsmemory per crate 64 GBytesaggregated peak performance 2.5 TFlops

Plan of the talk

some numbers

previous simulations had a smaller volume of 83 × 4

two different lattice sizes: 123 × 4, 163 × 4values of β in the range 3...7bare quark masses am = 0.01, 0.04deconfinement: βd ' 5.2; chiral transition: βχ ' 5.8

some numbers

the Polyakov loop

4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 β

4x123, m = 0.04

4x163, m = 0.04

5 5.25 5.5 5.75 6 6.25 6.5 6.75β

163 x 4, m = 0.01

the monopole order parameter ρ

2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6 6.3β

-15000

-10000

123x4, m = 0.04

163x4, m = 0.04

5 5.2 5.4 5.6 5.8 6β

123 x 4, m = 0.01

163 x 4, m = 0.01

scaling of ρ with spatial volume

-600 -500 -400 -300 -200 -100 0V (β−βc)

123x4, m = 0.04

163x4, m = 0.04

-1000 -500 0 500 1000(β-βc)xVolume

chiral condensate susceptibility

5.2 5.4 5.6 5.8 6 6.2 6.4β

m = 0.01, 163x4

m = 0.01, 83x4

plaquette susceptibility

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6β

m = 0.01, 163x4

the Polyakov loop and the monopole order parameter areconsistent with a first order deconfinement transitionfor both the simulated adjoint quark masses;the monopole order parameter has the correct scalingproperties with respect to variations of spatial volume;the chiral transition is located only for the smallest value ofthe quark mass and is consistent with a cross-over (seechiral cond. susc.): further studies could clarify this point;furthermore, it does not have an effect on the monopoleorder parameter (see plaquette susc.)

monopole condensation in two-flavour adjoint qcd: an update · introduction simulation details...

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