On the phase diagram of the Higgs SU(2) model

0.11 MB·English

by  C. Bonati

On the phase diagram of the Higgs SU(2) model
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n ClaudioBonati∗
a DipartimentodiFisica&INFN,Pisa,Italy
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E-mail: [email protected]
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2 GuidoCossu

DipartimentodiFisica&INFN,Pisa,Italy
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t E-mail: [email protected]
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- AlessioD’Alessandro
p
e DipartimentodiFisica&INFN,Genova,Italy
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E-mail: [email protected]
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1 MassimoD’Elia
v DipartimentodiFisica&INFN,Genova,Italy
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E-mail: [email protected]
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AdrianoDiGiacomo
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DipartimentodiFisica&INFN,Pisa,Italy
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E-mail: [email protected]
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The Higgs SU(2) model with l =¥ (fixed Higgs length) is usually believed to have two dif-
:
v ferentphasesathighgaugecouplingb , separatedbyalineoffirstordertransitionsbutnotdis-
i
X
tinuguishedbyanytypicalsymmetryassociatedwith alocalorderparameter,asfirstprovedby
r
a FradkinandShenker.Weshowthatinregionsoftheparameterspacewhereitisusuallysupposed
tobeafirstorderphasetransitiononlyasmoothcrossoverisinfactpresent.
TheXXVIInternationalSymposiumonLatticeFieldTheory
July14-19,2008
Williamsburg,Virginia,USA
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
OnthephasediagramoftheHiggsSU(2)model ClaudioBonati
1. Introduction andmotivation
While in pure gauge theories there isa standard way to detect color confinement (the Wilson
criterium [1]), when matter in the fundamental representation of the gauge group is present, large
Wilsonloopsneverobeythearea-law,sothatthereisnoobviouswaytoclearlydefinethemeaning
of“confinedphase”.
Fromthe computational point ofview, the simplest such model isthe HiggsSU(2) model; to
studyonlythegeneralfeaturesofitsphasediagram,thismodelcanbesimplifiedabitmore,fixing
the length ofthe Higgs field. Thisis possible because the scalar fourth-coupling l is irrelevant in
continuum limitandfixingthelengthoftheHiggsfieldisequivalenttousel =¥ (theirrelevance
of l was numerically checked in [2]). In this case the action can be written in the form (see e.g.
[2])
S=b (cid:229) 1−1ReTrPmn (x) −k (cid:229) Tr[f †(x)Um (x+mˆ)f (x+mˆ)] (1.1)
(cid:26) 2 (cid:27) 2
x,m <n x,m >0
where the firstterm is the standard Wilson action and the Higgs field f is written using an SU(2)
matrix. This form of the action is particularly useful because with it standard heatbath ([3], [4])
andoverrelaxation ([5])algorithmscanbeusedtogenerateMonteCarloconfigurations.
Atheorywithaction1.1hasthefollowingimportantlimitingcases
k =0 : puregaugetheory, notransitions inb
b =¥ : O(4)nonlinearsigmamodel,ithasasecondorderphasetransition ink
The general case was studied in [6] using both perturbative and non-peturbative methods: using
a pertubative expansion it was shown that the transition of the O(4) model is not lifted out by
the introduction of the gauge field, but it becames a first order à la Coleman-Weinberg. On the
non-perturbative side, using the methods developed in [7], the authors of [6] were able to prove
the existence of awideregion ofparameter space whereevery local observable isanalytic, theso
called FradkinShenker (FS)theorem. Usingthesetwoinputs theysuggested aphasediagram like
that shown inFig. 1: theregion where theanalyticity isrigorously proven isindicated byARand
is limited by the dotted line, the thick line represents a line of first order transitions and the two
dotsareitssecondorderend-points.
¥
AR
k s
s
0 b ¥
Figure1: PhasediagramoftheHiggsSU(2)modelaspredictedin[6].
Since the FS theorem ensures that no local observable can be used in all the parameter space to
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OnthephasediagramoftheHiggsSU(2)model ClaudioBonati
b=2.5
bility b=2.725
pti
usce
g s 1
n
pli
u
o
gs c
g
Hi
ge-
u
Ga
0.5
10 20 30 40
L
Figure2: Susceptibilitypeakheightsofthegauge-Higgscoupling(thelinesarefitstoa+bx4).
discriminate between a “confined phase” and a “Higgs phase”, there are some efforts to analyze
the behaviour in this model of the most popular proposals for confinement order parameter and
to study how the (possible) singularities in these operators are related to the first order transition
line of the thermodynamical observables (see e.g. [8], [9]). Because of that it is useful to have a
precise location ofthelineoffirstorder transitions andofitscritical end-point, which isabsent in
theliterature onthemodel.
2. Results ofsimulations
Thefirstnumerical results ontheSU(2)Higgsmodelappeared in[10]andseemedtodisplay
the features of the phase diagram in Fig. 1, but they were obtained on a very small 44 lattice,
so that further study was needed to confirm it. Subsequently, in [11], the existence of a double
peakstructurewasclaimedatb =2.3ona124 lattice, stronglysupporting thefirstorderscenario;
however this was probably just a consequence of the poor statistics, since in [12] no double peak
wasobservedatb =2.3and“thesystemexhibitsatransientbehavioruptoL=24alongwhichthe
order ofthe transition cannot be discerned”. Toimprove these results weanalyzed thepoints b =
2.5 andb =2.725, onlattices upto454,looking foraclear firstorder transition. Theobservables
monitored are
• thegauge-Higgs coupling, 1Tr[f †(x)Um (x+mˆ)f (x+mˆ)]
2
• theplaquettes, 1TrPmn
2
• the Z monopoles, M = 1− 1 (cid:229) s , where c stand for the elementary cube and s =
2 Nc c c c
(cid:213) signTrPmn
Pmn ∈¶ c
• thePolyakovloops
Among the observables, the mostsensitive one turned out tobe the gauge-Higgs coupling, whose
susceptibility peak heights and Binder fourth order cumulant minima are shown in Fig. 2 and 3.
The susceptibility of the operator O is defined in the ususal way, c (O)=L4(hO2i−hOi2), while
the Binder fourth order cumulant isV (O)=1−hO4i/(3hO2i2); if in the thermodynamic limit a
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OnthephasediagramoftheHiggsSU(2)model ClaudioBonati
b=2.5 b=2.725
0.66665
0.6665
Binder cumulant 2aB+/+3baxx+bx2 Binder cumulant 0.6666 2aB+/+3baxx
Gauge-Higgs coupling 00.6.666565 00..66666667 Gauge-Higgs coupling 000..66.666666645555 000...666666666666675
0.6665
0 2e-06 4e-06 6e-06 0 2.5e-07 5e-07 7.5e-07
0.6664
0 2.5e-05 5e-05 7.5e-05 0.0001 0 5e-06 1e-05 1.5e-05
1/L4 1/L4
Figure 3: Left, Binder fourthordercumulantminima ofthe gauge-Higgscouplingforb =2.5 (linesare
fitstoexpressionsreportedinfigure).Right,Sameasleftbutwithb =2.725.
discontinuity inhOiispresent atb =b ,the susceptibility c (O) develops maximawhose heigth
c L
scaleasL4,whileV (O) hasminimawhoseasymptotic behaviour is(seee.g. [13])
4 L
2
2 1 O O
V (O) | = − + − − +aL−4+bL−8+o(L−8); O = lim limhOi (2.1)
4 L min ±
3 12(cid:18)O− O+(cid:19) b →bc±L→¥
so that for a first-order transition B=limL→¥ V4(0)L|min is less than 2/3. As is clear from Fig.
2 and 3, both susceptibility and Binder cumulant have twodifferent typical behaviours depending
on the lattice size: for small lattices they have a first order-like scaling, while for larger lattices
cross-over nature of the system appears. Indeed for lattice sizes L≤20 with b =2.5 and lattice
sizesL≤30withb =2.725thesusceptibilities arewelldescribed bythefunctiona+bL4(Fig. 2)
andBindercumulants donotseem toreach2/3(Fig. 3);onlygoing tolarger lattices itispossible
to see the susceptibility saturate and the Binder fourth order cumulant tend to 2/3: the values of
thecostantBobteinedfromthefitsshowninthemagnificationsinFig. 3areB=0.666666(1) and
B=0.666667(1) forb =2.5andb =2.725respectively.
b=2.5 b=2.5
0.05 1.8
n plaquette susceptibility00.0.0445 monopoles susceptibility11..67
mea0.035 Z2 1.5
0.03 1.4
10 15 20 25 30 35 40 10 15 20 25 30 35 40
L L
Figure 4: Left, susceptibility peak heightsof the mean plaquette for b =2.5. Right, susceptibility peak
heightsoftheZ monopoleforb =2.5.
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OnthephasediagramoftheHiggsSU(2)model ClaudioBonati
b=2.725 k=0.709
0.1
p
o
o
v l
o 0.01
k
ya
ol
P
0.001
10 20 30 40 50
L
Figure5:Polyakovloopvaluesforb =2.725andk =0.709(linesarefitstoa+bexp(−cx)).
Asimilartypeofbehaviourisseenalsointhesusceptibilities ofalltheotherobservables, like
the mean plaquette and the Z monopole (Fig. 4) as well as in the way Polyakov loop reaches its
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asymptotic value. In Fig. 5 the Polyakov loop value is shown for fixed b and k and for various
L: the red line is the result of a fit to a+bexp(−cx) using only L ≤ 25 data (a = 0.00244(1)
and c = 0.2893(5)) while the green one is obteinded using only L ≥ 25 (a = 0.00085(2) and
c=0.082(1));inbothcasesa6=0becauseoftheHiggsfield,howeverisclearthatanextrapolation
makinguseonlyofsmallvolumeswouldbebadlywrong.
Thepresenceinalltheobservablesoftwodifferentbehavioursatsmallandbigvolumesisnot
atallnewinlatticegaugetheories,sincetoafinitecubiclatticecorrespondsafinitetemperature,so
thatadeconfinementtransitionistobeexpectedforsomevalueoftheparameters;thenewfeatures
arethepossbility todescribethesmallvolumesregimetohighprecisionusingafirst-orderscaling
andthesurprisinglybiglatticedimensionsneededtorevealthetruethermodynamicalpropertiesof
themodel.
3. Conclusions
Weperformed simulations at b =2.5 and b =2.725, where a first order transition is usually
believed toexist, and wefound that allthe analysed observables have instead smooth infinite vol-
ume limits; we thus conclude that in this region only a smooth cross-over is present. Moreover
wediscovered that inorder to seethecorrect non-singular behaviour itisnecessary touse lattices
muchbiggerthantheonestypically adoptedinstudiesoftheSU(2)Higgsmodel.
Atthis stage it isnotpossible topredict whether the end-point exists atb >2.725 orthe line
of first-order transition is in fact absent; in order to clarify this point it would be necessary either
tostudyevengreaterb values(butbeforeanonperturbative analysisoftheb -functionisneededto
ensure the lattice spacing is big enough) or to keep track of the end-point positions at increalsing
l <¥ (atl =0.5theexistence ofthefirst-order linewasverifiedin[14])andtrytoextrapolate it
tothel →¥ limit.
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OnthephasediagramoftheHiggsSU(2)model ClaudioBonati
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