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S.Yu.Shmakov, A.Mizadorozhny1 Щ2) GAUGE THEORY in the MAXIMALLY ABELIAN GAUGE WITHOUT MONOPOLES Submitted to «Physics Letters

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S.Yu.Shmakov, A.Mizadorozhny1 Щ2) GAUGE THEORY in the MAXIMALLY ABELIAN GAUGE WITHOUT MONOPOLES Submitted to «Physics Letters E2-95-531 S.Yu.Shmakov, A.MiZadorozhny1 Щ2) GAUGE THEORY IN THE MAXIMALLY ABELIAN GAUGE WITHOUT MONOPOLES Submitted to «Physics Letters B» 'li-mail address: /adorC» Icta32.jinr.dubna.su 1 Introduction In (bo hut ten year» there have been extcmivc studies of abclian monopole dynamics Injpired by conjectured dual-superconductor confinement mechanism (1, 2j. The maximal/ abelian (MA) gauge projection (3) *u found having abetian dominance and monopole condensation [i] in SU{2) gauge theory. It suggest* the existence of ал effectives (/(1) theory of confinement. Out questions about role of Gribov'* copies (5) and lattice artifacts are not clear yet. Calculation of monopole condensate like [0] and direct investigation of the model without rfldfiepoles are also interesting- For the»? tc.votn a simulation of SU{2) gauge theory under I ho MA gauge constraint* if desirable. This Is the purpose of this work to present an algorithm for simulation of £7/(2) gauge theory under the MA projection and first results of studying topology of the field manifold. In Clio QOKl section we briefly discos* the Fa/Meev-Popov operator for the MA projection of 51/(1) lattice gauge theory and a partial solution of the gauge con* ,stralnt*. We abo give there & short description of л hybrid Motile Carlo д)до)()м» of the simulation |7]. The ftuffierie&l results end concluding remark» ere presented in the lait two section*. 2 SI/(2) under the MA gauge The МЛ'projection for a lattice gauge field configuration {0) in Si/(2) theory is defined by [3J maxfi({0),{a})«max £ Гг(<т,^<г,^) where Ui '=» g,Otjt are gauge transformed link variable* and gt, g, are arbitrary SV(2) matrixes of the gauge transformation al sites i and / я t + ft. Written a» a stationary condition it.ieads || O (1) tly X? » 0 wbete 5» b aa Stf(2) ft;a{ri?c with jjj uJtrn to be ie*o. A' e otXt 4- U a tracclcss 2x2 antihermilian matrix and .Y* я *tXt + ffjAj. • Partition function for the theory with gauge conditions (I) could be written in the form Zш Jt(X*) dtt[F) WVtiert-SwlU)) (2) «,'tiere (l(i(U) is itifc invanaut integration measure, Fi* the FsuMecv-Popov operator for tlic МЛ gauge and Sw Is the Wilson action. follow Gtlbav'e We* (8j »utf«ntitUy «is.(l) eWltd be supplemental will» llie «(Ability condition that the I'addccV-Popov operator be positively defined /' > 0. On the lattice F «a /^ » -|^ 1» « square symmetric matrix with nonzero diagonal clement» (/ a j) and elements for site* i and j connected by a link. On surface (1) fti reads 0 (3) fori^j where Xt is defined by (1) and fof we suppose t/y == Iffi. So F \% A sptime c N U 2.1 Partial solution of the MA gauge constraints Having in mind a hybrid scheme [9] to guide a MC simulation of theory with partition function (2) we now define a set of independent, with respect to constraints (1), variables {IJ). bet a field configuration CAM = W) be л solution of (I). We consider three link variables S = {Uti,Utj,U)k} of the configuration CM A such that links Ls = {(«i')« ('<j)< 0Д')) f°rIn я continual patli (3/-path) on the lattice. Equations (1), which include link variables of set 5, read where the terms independent on 5 arc omitted. Substitution t/J = and 1/Д « cxp(i<rjt')»fj* solves the Tint and last of equations (-I). Two others are equivalent to 4 algebraic equations with \independent variables, say >4jjtut4j<u3jij and v, and allow to find A single solution U1t and 11(7). So, for any I/set 5 we can locally define a new set of variables Qs - {7, <?,\ <?,\ <J,\ <?|}5 = (7. <?lb (5) where <J,1 is the rijht-hand side of the ith equation (4). If the link variables apart from S Are kept fixed, then the gauge-fixing condition* require Qk to be <onstMt, whereas 7 is a free parameter. Choosing a ect of 3/-pnths {5} such tlinl b't П Sj я 0 for i ф j we enn locally define ft set of iti<lc|)C(idi-iit Variables {</}, It allow* to make transition (!щл -* £'лм where the both configurations lay on the surface defined by the gauge constraints (1). 2.2 Hybrid Monto Carlo method for SU(2) gauge theory under tho MA gauge Ну following the well known procedure we en» rlitnlimtr the determinant in (2) tind write the partition function in the form 7. » y<(Ar1)(r//i((/)</(*))cxp(-6V{f/} - Ф'/"Ч) (С) where Ф Is <in Additional complex scalar field. It look* very яипПпг to л partition function of gauge theory with pswidefennion* except tin4 ^-function* AIM! form of the interaction matrix tor the scalar field. For numerical «imillation of the nyntctri n Markov ргоссяя PIIOIIII) l»r conxtrilctrd with the fixed-point distribution deli lied by (G), mid the only requirements for the transition probability of the process Arc detailed balance and ergodicity. The Algorithm suggested in this paper includes two steps. First, we choose л set (5) of N,nt nonintcrsecting 3/-pnths on the lattice. The choice is made at random from a uniform distribution- Second, we generate a new field configuration by hybrid Monte Carlo method [9], so the detailed balance and the gauge-fixing conditions are satisfied. We also repeat the transition for dilfcrrnt {5}< to gain ergodicity. To specify the second step of the Algorithm, let us take A configuration C,MA{U) satisfying the constrains (1) And select A set (5} of nonlntersceting Л/-paths on the lattice. Restricted to the link variables associated with {.9}, the desired probability distribution reads where Qo ~ Q_ And Jacobian Д is given by b(Q,Us) m Тт{а3ич<тли+)ТгЫ After integration over QL the probability distribution 1дкгя th<* form S.Jj) (7) where the sum in the exponent is taken over AII 3f-pAths of set {S). We apply the hybrid Monte Carlo Algorithm [9] with evolution in psctidotime defined by S,n to generate я new MA configuration С'мл. It is ему to вес that Mnrkov proem converging to (7) In variables Q yields A correct simulation of system (6). Moreover, if step-size in the integration of the «filiations of motion is taken MIIAII, a qtiasteoutiinious evolution In Langevin time keeps the. configuration within the G'ribov horizon (10]. 3 Numerical simulation The algorithm has been implemented for simulation of SU(2) gauge theory on the \* lattice. We started with a configuration very close, to zero field localise matrix F Tor the completely ordered configuration has zero eigenvalues. The usual relaxation algorithm has been employed for initial MA gauge fixing. For the described in the previous section hybrid Monte Carlo algorithm, we chose duration 6т ш 0.000" for Langcvin time steps and -100 steps for every run. The choice wiis not (iplliufacU but ^lultlwl ftrtt!j>Uu»'e tfcte •«• 0.7. The generated configuration» wrrc separated by 12 Lnngrvln runs with dHIert'Jit (глшЬш) Ml» of 250 3/-|)atlis. It approximately reproduces a real number of ilrgrees of freedom of the system. The first 500 configurations were skipped for Ihrrmolisalion At /i values <1.0, 3.0. 2.5, 2.3 and ~ 2000 configurations at /) a '2.2b. The next 100 configurations were used to calculate average plaquclte, l-'very configuration was searched for abelian monopoles |11] but no tnonopoles were found. The results arc summarized in Table I, Here we also presented d.ita on я one- loop calculations of the ptaquclles (12] for 8* lattice And values of the plaqurttr calculated by the hcatbatb algorithm. From the collected data we could observe that the calculations under the MA gauge are in an excellent agreement with the one-loop pcrturhative results at all 0, whereas the hcathath data essentially differ at (i <, 2.5. Table 1 li Am« < /'/AM > /'/w/^-t < I'hm > 'i.'Vt 0 0.6-12(2) 0.037 o.oaa 2.3 0 O.GI7(3) 0.GI5 0.609 2.5 0 0.672(2) 0.C7G 0.C55 .1.0 0 0.720(3) 0.733 0.721 •t.o 0 0.790(2) 0.80.1 0.600 We eddilionally lesletl the stability of the generated configurations with respect to я small random gauge transformation with A further gauge fixing by the rrl< xalion Algorithm. In all rasrs the configurations Appear to lie stable. The situation changes crucially if matrix F \% replaced by /" with /,» в /„ and fu в 0 for i £ j. Al 0 s 2,25 After ** 200 «WcHfp* «tarting from лп almost ordered MA configuration we receive я nonstable one »oni<-timm having abelian пюпп|и>1ет. So wo cotne to the conclusion that it is the fmiuWPopov determinant that kcc|» us within the Gribov llOfi/OII, 4 Conclusion We Imve presented an algorithm for tlic numerical simulation of &U('l) gauge the- ory under tlic МЛ gniigci The first numerical results give evidence thnt Gribov region for the МЛ gauge projection Is split into At lefl.it two separated sectors. The configurAtion* laying within the locus of the first /<т<н of dtl[l') have MO nbclian monopolcs. Without inonopoles the smallest Wilson loop • pl,v|tictte become» very close to its one-loop pcrturbativc value. It supports the idea tli.it nn effective theory could be built with Monpcrturhfttivc interaction of the residunl nlxlinn firld with nionopolcs nnd the other Interactions nrc perturbntive. This work l.< supported in pnrt by I SI* CJrntit of l<ong Terin Itr^cnrrh I'rograrti (No.
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