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RENORMALIZATION GROUP AND TRIVIALITY

arXiv:hep-lat/9205030v1 1 Jun 1992DFTUZ-91.33November 1991RENORMALIZATION GROUP AND TRIVIALITYIN NONCOMPACT LATTICE QED WITH LIGHT FERMIONS.V. AzcoitiDepartamento de F´ısica Te´orica, Facultad de Ciencias, 50009 Zaragoza (Spain)G. Di Carlo and A.F.

GrilloIstituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati,P.O.B. 13 - Frascati (Italy).ABSTRACTIn the framework of noncompact lattice QED with light fermions, wederive the functional dependence of the average energy per plaquette on thebare parameters using block-spin Renormalization Group arguments andassuming that the renormalized coupling vanishes.

Our numerical resultsfor this quantity in 84 and 104 lattices show evidence for triviality in theweak coupling phase and point to a non vanishing value for the renormalizedcoupling constant in the strong coupling phase.1

One of the most succesfull results of lattice regularization of gaugetheories, combined with Monte Carlo simulations has been a deeper un-derstanding of the non perturbative dynamics of asymptotically free gaugetheories, like QCD. Conversely, very little is known about non asymptoti-cally free theories.As a matter of fact, the most important question for lattice regular-ization, i.e.

the existence of a quantum continuum limit with non trivialdynamics, has not (yet) be answered for the simplest and perturbativellymost succesfull gauge theory, namely, Quantum Electrodynamics.In recent years, many efforts have been devoted to the study of thisproblem, firstly using the compact regularization of the abelian model[1,2,3] and more recently within the non compact formulation .The first numerical investigations of the non compact model, in thequenched approximation [4], have shown the existence of a continuos chiraltransition at finite value of the coupling constant. This transition survivesafter the inclusion of dynamical fermions [5-8] so suggesting that the quan-tum continuum physics could be reached there.Having found a candidate point for the continuum limit, two impor-tant questions should be answered: i) Is the theory defined by taking thelimit at the chiral critical point non trivial, i.e.

does this model possess aparticle spectrum with non trivial interactions in this limit? and ii) As-suming answer to i) is positive, has this limiting theory something to dowith standard quantum electrodynamics?Concerning the first point, there exist extensive numerical simulationsperformed by several groups [5-9,11-13].

The first indication of a power-law(as opposed to essential singularity [10]) scaling for the chiral condensatewas suggested by A. Horowitz in [11]. On the other hand, the Illinois groupfound a good quantitative support for a non mean-field power law scaling inthe quenched model [12,13].

Their results ruled out the mean field scenarioand illustrated the degree of difficulty required in extracting the criticalindices in the full theory with dynamical fermions, where larger latticesand a precise determination of the critical coupling are necessary in orderto compute critical exponents [9].On the other hand, G¨ockeler et al. [14,15] computed the renormal-ized charge and fermion mass and found that the corresponding Callan-Symanzik β function is consistent with the prediction of renormalized per-turbation theory.Furthermore, they have not found lines of constantphysics for the matter sector in the two parameters region they explored[15]; on the basis of their results they argue about the non-renormalizabilityof the theory.As for point ii) above, Hands et al.

[9] have shown that the vacuum inthe broken phase of non compact QED is a monopole condensate with U(1)symmetry while the continuum model has no finite action monopoles, andthe gauge group symmetry is ℜ. This means that the lattice model is qual-2

itatively different from standard QED, belonging to a different universalityclass. This result also casts serious doubts on the validity of renormalizationgroup flow calculations as those of refs.

[14,15].In this letter we report a study of the triviality problem of the quantumcontinuum limit of non compact Lattice QED. To this end, we introducea new approach based on a characterization of the behaviour of the meanplaquette energy as a funtion of the bare parameters β and m. The equationdescribing such a behaviour holds only if the continuum limit of the modelconsists of particles without electromagnetic interactions.Consider the action of non compact lattice QEDS = 12Xx,µηµ(x)¯χ(x){Uµ(x)χ(x + µ) −U ∗µ(x −µ)χ(x −µ)}+mXx¯χ(x)χ(x) + β2Xx,µ<νF 2µν(x)(1)Fµν(x) = Aµ(x) + Aν(x + ˆµ) −Aµ(x + ˆν) −Aν(x)where β = 1/e2 and we use staggered fermions coupled to the gauge fieldsAµ(x) through the compact link variable Uµ(x).A problem working in the non compact formulation comes from thefact that the partition function associated to action (1) is not well definedeven in a lattice of finite size.In fact the gauge group integration, incontrast to the compact case, is divergent.

The problem can be overcomeby gauge fixing. We instead factorize the divergency in the density of statesas follows.Define the density of states at fixed non compact normalized energy Ein a lattice of volume VN(E) =Z[dAµ(x)]δ(12Xx,µ<νF 2µν(x) −6V E)(2)N(E) is divergent because of the infinite volume of gauge integration.

How-ever, this divergence can be factorized out and one can easily show thatN(E) = CG(6V E)32 V −1(3)where CG is a divergent constant (the volume of the gauge group).On the other hand it can be shown, following ref. [3], that the partitionfunction can be written as an integral over the normalized non compactenergy EZ =ZdEN(E)e−6βV Ee−SFeff (E,m)(4)3

wheree−SFeff (E,m) =R[dAµ(x)]det∆(m, Aµ(x))δ( 12Px,µ<ν F 2µν(x) −6V E)R[dAµ(x)]δ( 12Px,µ<ν F 2µν(x) −6V E)(5)From eq. (3),(4),(5) we can derive an effective action for the full theoryin the thermodinamical limit V →∞asSeff(E, V, β, m) = −32V lnE + 6βV E + SFeff(E, m)(6)Now let us write the partition function associated to (1) as an integralover the plaquette variables F 2µν in the following wayZ =Z[dAµ(n)][d¯χ(n)][dχ(n)][dEµν(n)]Yδ(F 2µν(n) −Eµν(n))e−S =Z[dEµν(n)]N(Eµν(n))e−S(Eµν(n))(7)wheree−S(Eµν (n)) =R[dAµ(n)][d¯χ(n)][dχ(n)] Q δ(F 2µν(n) −Eµν(n))e−SR[dAµ(n)][d¯χ(n)][dχ(n)] Qδ(F 2µν(n) −Eµν(n))(8)and the denominator in (8) is just the density of states N(Eµν(n)).Next, imagine we apply linear block-spin renormalization group trans-formations to the theory described by the effective action S(Eµν(n)) −lnN(Eµν(n)) .

Our spin variable is the plaquette variable Eµν(n) whichtakes values from 0 to ∞and blocking is performed at each µν plane. Wegenerate in this way a series of effective actions which are equivalent atlarge distances since we are integrating out all the short distance details.If the theory is trivial i.e., if all renormalized couplings vanish, the onlyrelevant parameter at the end of this procedure will be the coefficient ofthe kinetic term Eµν(n).

Then, the renormalized action SR(Eµν(n)) willbe, apart from the density of states contribution, of the formSR(Eµν(n)) = 12¯β(m, β)Xn,µ<νEµν(n) + h(m, β)(9)where ¯β(m, β) and h(m, β) are unknown renormalized constants. DefiningER =16VPn,µ<ν Eµν(n) we get4

SeffR(ER) = −32V lnER + 6¯βV ER + h(m, β)(10)This action and action (6) can differ only by a multiplicative factorX(m, β) in the mean energy E since we have obtained (10) by means oflinear block-spin transformations plus a final linear global transformation.Therefore triviality means that action (6), apart from the logarithmic termcoming from the density of states, must be a linear function of the meanenergy E or equivalently, that SFeff(E, m) in (6) is a linear function of E.We would like to remark at this point that the connection between ac-tions (6) and (10) can be established owing to the use of linear block-spinRenormalization Group transformations, so that we can obtain eq. (10)from (6) through a linear change of variables.

Using non linear transfor-mations or transformations in other kind of variables, we could identify thepartition functions but we would not be able to establish any connectionbetween the corresponding effective actions.Due to the fact that SFeff(E, m) is a linear function of E, we get thatall effects of dynamical fermions can be reduced to a redefinition of thecoupling constant β. Therefore, the mean plaquette energy can be writtenasE(m, β) =14(β + h1(m))(11)The linearity of the effective action (6) as well as equation (11), whichshould hold around the critical point if the theory is trivial, can be comparedwith data obtained by numerical simulations.Following a method that we have recently proposed [3], we calculatedthe mean plaquette energy using the fermionic effective action (5).Wehave obtained the fermionic effective action in 27 values of E in the range0.5−1.7, allowing us to calculate thermodynamical quantities as a functionof β in the range 0.14 ≤β ≤0.40; with these values we go deeply insidethe strong coupling and Coulomb phase respectively.The largest part of the simulations has been performed on a 84 lattice,but from an analysis of the scaling properties of fermionic effective actionin lattices from 44 to 104, we can exclude significant finite volume effectson the mean plaquette energy of the full theory, already in the 84 lattice.For a detailed report of these simulations see [16].In Fig.1 we report the effective fermionic action (5) for vanishingfermion mass as a function of E. Two different regimes, corresponding totwo different phases, can be seen from this figure.Coulomb phase (β > 0.206) which is dominated in the thermodynam-ical limit by energies (E ≤1.0), is characterized by an effective actionlinear in E, meaning that the effect of the inclusion of fermionic degreesof freedom merely reduces to a shift in the coupling constant, indicating5

triviality. In fact, if we try to fit the plaquette energy data in this phasewith a functional form like (11), we obtain a very good fit for h1 = 0.0409( see Fig.

2 ). The fact that (11) is able to reproduce in such a good waythe numerical data is again a strong indication of triviality in this phase.Completely different is the situation in the strong coupling brokenphase (E > 1.0).

Indeed, the behavior of the plaquette energy for β < 0.206deviates from the fit (11), this indicating the existence of a phase transitionat βc ≃0.206. For β < βc we tried to fit our data with a function like (11).However, we have found that we need to give a β dependence to h1(m), ascan be seen in Fig.

3. From the two fits for h1(m) in this Figure, we getβc = 0.206(5).Our results on the effective fermionic action reported in Fig.1 can bevery well understood if a second order phase transition occurs.Indeed,applying the saddle point technique to the computation of the partitionfunction (4) it can be shown that a discontinuity in the specific heat impliesa discontinuity of the second energy derivative of the effective fermionicaction at the energy critical value.Furthermore and as following fromthe main content of this paper, a non vanishing value for the renormalizedcoupling is directly related to a non linear energy dependence of the effectivefermionic action.

Therefore a second order phase transition should producea discontinuity of the renormalized coupling at the critical point.Fitting the points in Fig.1 by two polynomials, one for E < 1.018 andthe other for E > 1.018 (continuous line in the Figure), being Ec = 1.018the mean energy at β = 0.206 m = 0, we get very good fits with a gap of0.38(2) in the second energy derivative normalized by the lattice volume .As a result of the fits we also find that the first energy derivative of theefective fermionic action is continuous at E = Ec and second and higherorder energy derivatives vanish for E < Ec inside the errors of the fits, theseresults being very stable when we encrease the degree of the polynomial fits.The observed approximate scaling of the effective fermionic action with thelattice volume when we go from the 64 to the 104 lattice [16] implies thatfinite size effects does not affect our results in a significant way. In anycase, the important qualitative finding is that the second energy derivativeof effective fermionic action is always discontinuous at the critical energy.In conclusion, our numerical analysis shows the existence of a phasetransition at βc = 0.206(5), Nf = 4, βc = 0.226(5), Nf = 2, in agreementwithin errors with the critical value obtained from the behaviour of thechiral condensate [9, 16].

The behavior of the effective fermionic actionand mean plaquette energy in the broken phase, strongly suggests a nonvanishing value for the renormalized coupling constant in this phase, evenwhen we approach the critical point.Does this result implies the existence of a non trivial (non gaussian)fixed point?.In the general formulation of the Renormalization Groupapproach it is generally assumed that any point at or near the critical6

surface is in the attraction domain of some fixed point, even though singularbehaviour can not be excluded by general arguments [17]. Excluding sucha singular behavior, our numerical results strongly indicate that the fixedpoint is non gaussian.The important question now is: is the quantum theory described bythis fixed point renormalizable?.

The results reported in [15] about nonperturbative renormalizability of the model show that there are no lines ofconstant physics in the (β, m) plane. However, this result does not implynecessarily non renormalizability since it could be that the two dimensionalparameter space is too small.

In fact, in the two parameters action (1) wehave neglected coupling terms such as four Fermi interactions and monopolecontributions which can be generated in the renormalization procedure andwhose associated couplings could become relevant for the continuum limitin the strong coupling phase, as suggested firstly in [18] and also by theresults of ref. [9] (this was also the possibility left open in [15]).

If thisis the real scenario, our numerical results in the broken phase should beregarded as a strong indication for a non trivial continuum limit.On the contrary, when the transition is approached from the Coulombphase, as more appropriate for the definition of continuum QED, the theoryis non interacting. This behaviour is not totally unexpected, since we knowfrom perturbative QED the existence of Landau pole problem.The authors aknowledge J.L.

Alonso and J.L. Cortes for useful discus-sions.

This work has been partly supported through a CICYT (Spain) -INFN (Italy) collaboration.7

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FIGURE CAPTIONS1)Effective fermionic action (5), versus E at m = 0.0 and four flavors,obtained through microcanonical simulation. Statistical errors are in-visible at this scale.2)Mean plaquette energy E(m = 0, β) versus β.Solid line is a fit,equation (11), with h1 = 0.0409.

Errors are of the order of symbolssize.3) h1(m) versus β at m = 0.0. In the weak coupling phase (β > 0.206),h1(m) is well fitted by a horizontal line.

For β < βc equation (11) doesnot hold and we need to give a β dependence to h1(m). The solid linein the strong coupling phase is a polynomial fit.9

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