Finite Temperature Behavior of the 3D Polyakov Model with Massless Quarks

7 Pages·0.15 MB·English

by  Nikita O. Agasian

IFUP-TH 2001/29
Finite Temperature Behavior of the
3D Polyakov Model with Massless Quarks
Nikita O. Agasian ∗
2
Institute of Theoretical and Experimental Physics,
0
0 B. Cheremushkinskaya 25, RU-117 218 Moscow, Russia
2
and

n Dmitri Antonov †‡
a
J INFN-Sezione di Pisa, Universit´a degli studi di Pisa, Dipartimento di Fisica,

4 Via Buonarroti, 2 - Ed. B - I-56127 Pisa, Italy
2

3
v Abstract
9
8 The(2+1)DGeorgi-Glashow(orPolyakov)modelwiththeadditionalfundamentalmasslessquarks
1 is explored at finite temperature. In the case of vanishing Yukawa coupling, it is demonstrated
9
that the interaction of a monopole and an antimonopole in the molecule via quark zero modes
0
leads to the decrease of the Berezinsky-Kosterlitz-Thoulesscritical temperature when the number
1
0 of quark flavors is equal to one. If the number of flavors becomes larger, monopoles are shown to
/ existonlyinthe molecularphaseatanytemperaturesexceedingacertainexponentiallysmallone.
h
t This means that for such a number of flavors and at such temperatures, no fundamental matter
-
can be confined by means of the monopole mechanism.
p
e
h
:
v 3D Georgi-Glashow model (else called the Polyakov model) is known to be one of the eldest
i
X andthemostfamousexamplesoftheoriesallowingforananalyticaldescriptionofconfinement[1].
However,the phasestructureofthismodelatfinite temperaturehasbeenaddressedonlyrecently.
r
a Namely, firstin Ref.[2]it hasbeen shownthat atthe temperature T =g2/2π the weaklycoupled
c
monopole plasma in this model undergoes the Berezinsky-Kosterlitz-Thouless (BKT) [3] phase
transitionintothemolecularphase. Then,inRef.[4],ithasbeenshownthatapproximatelyatthe
twice smaller temperature, there occurs another phase transition associated to the deconfinement
of W-bosons.
In this paper, we shall be interested in the finite-temperature properties of the monopole en-
semble, rather than the ensemble of W-bosons. Because of that, let us first discuss in some more
details the nature of the above-mentioned BKT phase transition. At high enough temperature,
onecanapplythe ideaofdimensionalreduction. Thedimensionally-reducedtheoryisthenthe2D
XY-model, but with the temperature-depending strength of the monopole-antimonopole (MM¯)
∗E-mailaddress: [e-mail]
†E-mailaddress: [e-mail]
‡Permanentaddress: ITEP,B.Cheremushkinskaya25,RU-117218Moscow,Russia.
1
interaction. Due to this fact, the phase structure of the model becomes reversed with respect
to that of the usual 2D XY-model. Namely, at the temperatures below T , monopoles exist in
c
the plasma phase, that leads to the confinement of fundamental matter [1, 5]. At T > T , the
c
vacuumstateisthe moleculargasofboundMM¯ pairs,andconsequentlyfundamentalquarksare
−
deconfined[2]. Theanalogywiththe 2DXY-modelestablishedinRef.[6]isthatspinwavesofthe
2D XY-model correspond to the free photons of the Polyakovmodel, while vortices correspond to
magnetic monopoles.
Let us briefly discuss the BKT phase transition, occuring at T = T , in the language of the
c
2D XY-model. At T < T , the spectrum of the model is dominated by massless spin waves, and
c
the periodicity of the angular variable is unimportant in this phase. The spin waves are unable
to disorder the spin-spin correlation functions, and those decrease at large distances by a certain
powerlaw. Onthe contrary,atT >T ,theperiodicityofthe angularvariablebecomesimportant.
c
This leads to the appearance of topological singularities (vortices) of the angular variable, which,
contrary to spin waves,have nonvanishing winding numbers. Such vortices condense and disorder
the spin-spin correlationfunctions, so that those start decreasing exponentially with the distance.
Thus,thenatureoftheBKTphasetransitionisthecondensationofvorticesatT >T . Inanother
c
words, at T > T , there exist free vortices, which mix in the ground state (vortex condensate) of
c
indefinite global vorticity. Contrary to that, at T <T , free vortices cannot exist, but they rather
c
mutuallycoupleintoboundstatesofvortex-antivortexpairs. Suchvortex-antivortexmoleculesare
small-sized short-living (virtual) objects. Their dipole-type fields are short-ranged and therefore
cannot disorder significantly the spin-spin correlation functions. However, when the temperature
starts rising, the sizes of these molecules increase, until at T = T they diverge, that corresponds
c
to the dissociationofthe molecules intopairs. Therefore,comingbackto the Polyakovmodel, one
of the methods (which will be employed below) to determine there the critical temperature of the
BKT phase transition is to evaluate the mean squared separation in the MM¯ molecule and find
−
the temperature at which it starts diverging.
In this paper, we shall consider the extension of the Polyakov model by the fundamental dy-
namicalquarks,whichare supposedto be massless. As it will be demonstrated, quarkzero modes
in the monopole field lead to the additional attraction between a monopole and an antimonopole
in the molecule at high temperatures. In particular, when the number of these modes (equal to
the number of massless flavors) is sufficiently large, the molecule shrinks so that its size becomes
ofthe orderofthe inverseW-bosonmass. Another factorwhichgovernsthe size ofthe moleculeis
the characteristic range of localization of zero modes. Namely, it can be shown that the stronger
zero modes are localized in the vicinity of the monopole center, the smaller molecular size is. In
this paper, we shall consider the case when the Yukawa coupling vanishes, and originally massless
quarks do not acquire any mass. This means that zero modes are maximally delocalized. Such
a weakness of the quark-mediated interaction of monopoles opens a possibility for molecules to
undergo eventually the phase transition into the plasma phase. However, this will be shown to
occur only provided that the number of flavors is equal to one, whereas at any larger number
of flavors, the respective critical temperature becomes exponentially small. This means that the
interaction mediated by such a number of zero modes is already strong enough to maintain the
molecular phase at any temperature larger than that one.
LetusstartouranalysiswithconsideringtheEuclideanactionofthePolyakovmodelextended
by the fundamental, originally massless quarks. [Note that such a model can be viewed as the
(2+1)-QCD with the additional adjoint Higgs field.] We shall first consider the general case with
the nonvanishing Yukawa coupling, by means of which quarks acquire a certain mass. The action
under discussion then reads
S = d3x 1 Fa 2+ 1(D Φa)2+ λ (Φa)2 η2 2 iψ¯ ~γD~ +hτaΦa ψ . (1)
4 µν 2 µ 4 − − 2
Z (cid:20) (cid:0) (cid:1) (cid:16) (cid:17) (cid:18) (cid:19) (cid:21)
2
Here,
τa
Fa =∂ Aa ∂ Aa +gεabcAbAc, D Φa =∂ Φa+gεabcAbΦc, D ψ = ∂ ig Aa ψ,
µν µ ν − ν µ µ ν µ µ µ µ µ− 2 µ
(cid:18) (cid:19)
and ψ¯=ψ†β with the Euclidean Dirac matrices defined as~γ = iβα~, where
−
1 0 0 ~τ
β = , α~ = .
0 1 ~τ 0
(cid:18) − (cid:19) (cid:18) (cid:19)
Next, in 3D, the electric coupling g, the Yukawa coupling h, and the vacuum expectation value of
the Higgs field η have the dimensionality [mass]1/2. The Higgs coupling λ has the dimensionality
[mass]. The masses of the W- and Higgs bosons are large compared to g2 in the standard pertur-
bative(else calledweak-coupling)regimeg η andread: m =gη,m =η√2λ. The inequality
W H
≪
g η is necessary to ensure the spontaneous symmetry breaking from SU(2) to U(1). Note also
≪
that for the sake of simplicity, we omit the summation over the flavor indices, but consider the
general case with an arbitrary number of flavors.
One can further see that the Dirac equation in the field of the third isotopic component of the
’t Hooft-Polyakov monopole [7] decomposes into two equations for the components of the SU(2)-
doubletψ. Themassesofthesecomponentsstemmingfromsuchequationsareequaltoeachother
and read m =hη/2. Next, the Dirac equation in the full monopole potential has been shown [8]
q
to possess the zero mode, whose asymptotic behavior at r ~x m−1 has the following form:
≡| |≫ q
e−mqr
χ+ = s+s− s−s+ , χ− =0. (2)
νn N r ν n − ν n νn
Here, χ± are the upper and the lower com(cid:0)ponents of the(cid:1)mode, i.e. ψ = χ+n , next n = 1,2 is
n χ−n
the isotopic index, ν = 1,2 is the Dirac index, s+ = 1 , s− = 0 , and is the normalization
0 1 N(cid:0) (cid:1)
constant.
(cid:0) (cid:1) (cid:0) (cid:1)
Itisawellknownfactthatin3D,the’tHooft-Polyakovmonopoleisactuallyaninstanton[1,5].
Therefore, we can use the results of Ref. [9] on the quark contribution to the effective action of
the instanton-antiinstanton molecule in QCD. Let us thus recapitulate the analysis of Ref. [9]
adapting it to our model. To this end, we fix the gauge Φa =ηδa3 and define the analogue of the
free propagator S0 by the relation S0−1 = i ~γ∂~+mqτ3 . Next, we define the propagator SM
−
in the field of a monopole located at the origin(cid:16), A~aM [AaM(cid:17) εaijxj/ gr2 at r m−1], by the
i → ≫ W
formula SM−1 = S0−1−g~γτ2aA~aM. Obviously, the propagator SM¯ in th(cid:0)e fie(cid:1)ld of an antimonopole
located at a certain point R~, A~aM¯ (~x) = A~aM ~x R~ , is defined by the equation for S−1
− − M
with the replacement A~aM A~aM¯. Finally, one(cid:16)can co(cid:17)nsider the molecule made out of these
→
monopole and antimonopole and define the total propagator S in the field of such a molecule,
A~a =A~aM +A~aM¯, by means of the equation for S−1 with A~aM replaced by A~a.
M
One can further introduce the notation ψ , n = 0,1,2,..., for the eigenfunctions of the
n
| i
operator i~γD~ defined at the field of the molecule, namely i~γD~ ψn = λn ψn , where λ0 = 0.
− − | i | i
This yields the following formal spectral representation for the total propagatorS:
∞
ψ (~x) ψ (~y)
n n
S(~x,~y)= | ih |.
λ im τ3
n=0 n− q
X
Next, it is convenient to employ the mean-field approximation, according to which zero modes
dominate in the quark propagator,i.e.,
3
ψ0(~x) ψ0(~y)
S(~x,~y)≃ | imihτ3 | +S0(~x,~y). (3)
q
−
Indeed, this approximation is valid, since in the weak-coupling regime the monopole sizes, equal
to m−1, are much smaller than the average distance in the MM¯ plasma. This average distance
W −
has an order of magnitude ζ−1/3 (see e.g. Ref. [10] for a discussion). Here, ζ e−4πmWǫ/g2
stands for the so-called monopole fugacity, which has the dimensionality [mass]3,∝and ǫ 1 is
∼
a certain dimensionless function of (m /m ). Obviously, ζ is exponentially small in the weak-
H W
coupling regime under study. The approximation (3) remains valid for the molecular phase near
the phase transition (i.e. when the temperature approaches the critical one from above), we shall
beinterestedin. Thatismerelybecauseinthisregime,moleculesbecomeverymuchinflatedbeing
about to dissociate.
Within the notations adapted, one now has S = SM−1+SM−¯1−S0−1 −1 =SM¯S−1SM, where
(cid:0) ψM ψM (cid:1) ψM¯ ψM¯
0 0 0 0
S =S0−(SM −S0)S0−1(SM¯ −S0)=S0− (cid:12) iEmDτ3 (cid:12)S0−1(cid:12) iEmDτ3 (cid:12),
(cid:12) − q (cid:12) (cid:12) − q (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
and ψM , ψM¯ are the zero modes of the operator i~γD~ defined at the field of a monopole and
0 0
−
an a(cid:12)(cid:12)ntimEono(cid:12)(cid:12)poleE, respectively. Denoting further a = ψ0M¯ g~γτ2aA~aM ψ0M , it is straightforward
(cid:12) (cid:12)
to see by the definition of the zero mode that a = ψD0M¯ (cid:12)(cid:12) i~γ∂~ ψ0M(cid:12)(cid:12) =Eψ0M¯ S0−1 ψ0M . This
−(cid:12) (cid:12)
yields S = S0 + (a∗/m2q) ψ0M ψ0M¯ , where the sDtar st(cid:12)(cid:12)(cid:12)a(cid:16)nds fo(cid:17)r(cid:12)(cid:12)(cid:12)theEcomDplex (cid:12)(cid:12)conju(cid:12)(cid:12)gatioEn, and
(cid:12) ED2 (cid:12)
therefore det = 1+(a(cid:12)/mq) d(cid:12)etS0. Finally, defining the desired effective action as Γ =
S | (cid:12)| · (cid:12)
ln[detS−1/detS−1h], we obtain foir it in the general case with N flavors the following expression:
0 f
Γ = const +N ln m2+ a2 . The constant in this formula, standing for the sum of effective
f q | |
actionsdefinedatthemonopoleandattheantimonopole,cancelsoutinthenormalizedexpression
for the mean square(cid:0)d separatio(cid:1)n in the MM¯ molecule.
−
Let us further set h equal to zero, and so m is equal to zero as well. Notice first of all that
q
althoughinthiscasethedirectYukawainteractionoftheHiggsbosonswithquarksisabsent,they
keep interacting with each other via the gauge field. Owing to this fact, the problem of finding
a quark zero mode in the monopole field is still valid 1. The dependence of the absolute value of
the matrix element a on the distance R between a monopole and an antimonopole can now be
straightforwardlyfound. Indeed, we have a d3r/ r2 ~r R~ = 4πln(µR), where µ stands
| |∝ − −
for NthoewIRwecuswtoitffc.honthetemperatureT β−1,Rsotha(cid:16)tall(cid:12)(cid:12)(cid:12)thebo(cid:12)(cid:12)(cid:12)s(cid:17)onic(fermionic)fieldsshouldbe
≡
supplied with the periodic (antiperiodic) boundary conditions in the temporal direction, with the
period equal to β. The magnetic-field lines of a single monopole thus cannot cross the boundary
of the one-period region and should go parallel to this boundary at the distances larger than β.
Therefore, monopoles separated by such distances interact via the 2D Coulomb law, rather than
1NotethataccordingtoEq.(2)thismodewillbenon-normalizableinthesenseofadiscretespectrum. However,
inthegaplesscasemq =0underdiscussion,thezeromode,whichliesexactlyontheborderofthetwocontiguous
Dirac seas, should clearly be treated not as an isolated state of a discrete spectrum, but rather as a state of a
continuum spectrum. (A similartreatment of the zero mode of a massless left-handed neutrino on electroweak Z-
stringshasbeendiscussedinRef.[11].) Thismeansthatitshouldbeunderstoodasfollows: |ψ0(~x)i∼ lim eipr/r ,
p→0
where p = |p~|. Once being considered in this way, zero modes are normalizable by the standard con(cid:0)dition o(cid:1)f
∞
normalizationoftheradialpartsofsphericalwaves,Rpl,whichreads[12] drr2Rp′lRpl=2πδ(p′−p).
0
R
4
the 3D one. Recalling that the average distance between monopoles is of the order of ζ−1/3, we
conclude that at T ζ1/3, the monopole ensemble becomes two-dimensional (see e.g. Ref. [2]
≥
for a detailed discussion of the dimensional reduction in the Polyakov model). However, at the
temperatures below the exponentially small one, ζ1/3, monopoles keep interacting by the usual
3D Coulomb law,and the monopole confinement mechanismfor the fundamental matter works at
such temperatures under any circumstances.
We are now in the position to explore a possible modification of the standard BKT critical
temperature[2]T =g2/2π duetothezero-modemediatedinteraction. Asitwasdiscussedabove,
c
this can be done upon the evaluation of the mean squared separation in the MM¯ molecule and
−
further finding the temperature below which it starts diverging. In this way we should take into
account that in the dimensionally-reduced theory, the usual Coulomb interaction of monopoles 2
R−1 = +∞ 2+(βn)2 −1/2 goesoverinto 2Tln(µ ), where denotes the absolute value of
n=−∞ R − R R
the 2DvePctor(cid:0)~. This sta(cid:1)tementcanbe checkede.g. byvirtue ofthe Euler- MacLaurinformula.
As far as the nRovel logarithmic interaction, ln(µR) = +∞ ln µ 2+(βn)2 1/2 , is concerned,
n=−∞ R
it transforms into P h (cid:0) (cid:1) i
πT +ln[1 exp( 2πT )] ln2. (4)
R − − R −
Let us prove this statement. To this end, we employ the following formula [13]:
∞
1 1 1
= πcoth(πx) .
n2+x2 2x − x
n=1 (cid:20) (cid:21)
X
This yields
+∞ 1 1 xa2 ∞ 1 a ax
x = + = coth .
x2+(2πn/a)2 x 2π2 n2+(xa/2π)2 2 2
n=X−∞ nX=1 (cid:16) (cid:17)
On the other hand, the L.H.S. of this expression can be written as
+∞ 2
1 d 2πn
ln x2+ .
2dx a
n=−∞ (cid:18) (cid:19) !
X
Integrating over x with the constant of integration set to zero, we get
+∞ 2
2πn ax ax
ln x2+ =a dxcoth =2lnsinh =ax+2ln 1 e−ax 2ln2.
n=X−∞ (cid:18) a (cid:19) ! Z (cid:16) 2 (cid:17) (cid:16) 2 (cid:17) (cid:0) − (cid:1)−
Setting 2π/a=µβ and x=µ we arrive at Eq. (4).
R
Thus, the statistical weight of the quark-mediated interaction in the molecule at high tem-
peratures has the form exp( 2N ln a) [πT +ln[1 exp( 2πT )] ln2]−2Nf. Accounting
f
− | | ∝ R − − R −
for both (former) logarithmic and Coulomb interactions, we eventually arrive at the following
expression for the mean squared separation L2 in the molecule as a function of T, g, and N :
f
2Withoutthelossofgenerality,weconsiderthem(cid:10)olec(cid:11)ulewiththetemporalcomponent ofR~ equaltozero.
5
∞
d 3−8gπ2T [πT +ln[1 exp( 2πT )] ln2]−2Nf
RR R − − R −
L2 =mR−W1 .
∞
d 1−8gπ2T [πT +ln[1 exp( 2πT )] ln2]−2Nf
(cid:10) (cid:11) RR R − − R −
m−1
RW
In this equation, we have put the lower limit of integration equal to the inverse mass of the W-
boson, which acts as an UV cutoff.
At large , ln2 πT and ln[1 exp( 2πT )] exp( 2πT ) πT . Consequently,
R ≪ R − − R ≃ − R ≪ R
we see that L2 is finite at T > T = (2 N )g2/4π, that reproduces the standard result [2] at
N = 0. For N = 1, the plasma(cid:12)(cid:12)cphase −is stfill presen(cid:12)(cid:12)t at T < g2/4π, whereas for N 2 the
f f f
monopole en(cid:10)sem(cid:11)ble may exist only in the molecular phase at any temperature larger tha≥n ζ1/3.
Clearly,atN max 1,4πT/g2 , L2 m−1,whichmeans thatsucha largenumberofzero
f ≫ h i→ W
modes shrinks the molecule to the minimal admissible size. Note finally that both the obtained
critical temperature g2(cid:8)/4π and th(cid:9)e sptandard one (in the absence of quarks), g2/2π, are obviously
much larger than ζ1/3, that fully validates the idea of dimensional reduction.
Inconclusionofthispaper,wehavefoundthecriticaltemperatureofthemonopoleBKTphase
transitionin the weak-couplingregimeof the Polyakovmodel extendedby the masslessdynamical
quarks, which interact with the Higgs boson only via the gauge field. It has been shown that for
N = 1, this temperature becomes twice smaller than the one in the absence of quarks, whereas
f
for N 2 it becomes exponentially small, namely of the order of ζ1/3. The latter effect means
f
≥
that this number of quark zero modes, which strengthen the attraction of a monopole and an
antimonopole in the molecule, becomes enough for the support of the molecular phase at any
temperature exceeding that exponentially small one. Therefore, for N 2, no fundamental
f
≥
matter (including dynamical quarks themselves) can be confined at such temperatures by means
of the monopole mechanism.
Acknowledgments
We are greatful for stimulating and fruitful discussions to A. Di Giacomo, B.O. Kerbikov,I.I. Ko-
gan, V.A. Rubakov,M.A. Shifman, and Yu.A. Simonov. D.A. is indebted to Prof. A. Di Giacomo
and the whole staff of the Quantum Field Theory Division of the University of Pisa for kind hos-
pitality andto INFNfor the financialsupport. And lastbutnotleast,partialfinancialsupportby
the RFBR grant 00-02-17836and the INTAS grantOpen Call 2000,project No. 110, is greatfully
acknowledged.
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