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Study of the asymptotic freedom

arXiv:hep-lat/9303010v1 19 Mar 1993February 1993Oxford, OUTP-93-01PJ¨ulich, HLRZ 93-15Study of the asymptotic freedomof 2d Yukawa models on the lattice∗A.K. De1, E. Focht2,3, W. Franzki2,3,J.

Jers´ak2,3 and M.A. Stephanov41Department of Physics, Washington University, St. Louis, MO 63130, USA2Institute of Theoretical Physics E, RWTH Aachen, D-5100 Aachen, Germany3HLRZ c/o KFA J¨ulich, P.O.

Box 1913, D-5170 J¨ulich, Germany4 Theoretical Physics, 1 Keble Rd., Oxford OX1 3NP, UKAbstractWe investigate on the lattice the Yukawa models in 2 dimensions with Z(2) and U(1)symmetries. These models reduce to the usual and chiral Gross-Neveu models, respectively,when the kinetic and the selfcoupling terms of the scalar field are turned off.

The numericaldata and mean field arguments suggest that, at least for some range of the scalar field hoppingparameter, fermion mass is dynamically generated for arbitrarily weak Yukawa coupling. Themodels are asymptotically free in this coupling, like the Gross-Neveu models, even when thescalar quartic selfcoupling is strong.∗Supported by the US Department of Energy grant No.

DE2FG02-91ER40628, by DeutschesBundesministerium f¨ur Forschung und Technologie, by Deutsche Forschungsgemeinschaftand by Jesus College, Oxford.

1Yukawa models in 2 dimensionsThe 2d Yukawa models (Y2) have received little attention since the rigorous establishmentof some of their fundamental field theoretical properties in the seventies (for a summary seeref. [1]).

The superrenormalizability of the Yukawa coupling and the arbitrariness of theselfcoupling terms of the dimensionless scalar field in 2d might have led to a feeling that theY2 models have little relevance for the 4d field theories. This is to be compared with thecontinuing interest in the 2d Gross-Neveu (GN) and nonlinear σ (NLσ) model investigations,shown e.g.

by the recent exact mass gap calculations in these models [2, 3].However, the Y2 models can be chosen so that both the GN and NLσ models are theirspecial cases.This is most obvious on the lattice.For example, the action of the Z(2)symmetric Y2 model on the lattice can be chosen in the formS=−2κXx,µφxφx+µ +Xxφ2x + λXx(φ2x −1)2+Xx,αψαx∂/ψαx + yXx,αψαxφxψαx . (1.1)Here we have introduced N “naive” Dirac fermion fields ψα, α = 1,...,N, on the lattice which,due to the fermion doubling, describe NF = 4N Dirac fermions of zero bare mass.

All thefields and the couplings are made dimensionless by the appropriate rescaling with the latticeconstant a, x enumerates lattice sites and ∂µ is the lattice derivative. The action for theU(1) symmetric Y2 model which we study as well has the form similar to (1.1) with two-component field ⃗φx = (φ1x, φ2x) instead of φx and the Yukawa term y Px,α ψαx(φ1x + iφ2xγP)ψαx(where γP is the 2d analog of γ5).At κ = λ = 0 the action (1.1) describes the Z(2)-symmetric GN model in the auxiliaryscalar field representation of the 4-fermion coupling.

The Yukawa coupling y is related tothe usual GN coupling g by y =√2g. On the other hand, at λ = ∞and y = 0 the action(1.1) describes the Ising model.

The U(1) symmetric Y2 model in similar cases reduces tothe chiral GN model or to the XY model, i.e. the NLσ model with U(1) symmetry, andsimilarly for other symmetry groups.

The Y2 models thus interpolate between the GN andspin or NLσ models.On the lattice, by choosing the above formulation of the scalar field sector using thehopping parameter κ, the kinetic term can be turned on or offgradually, illustrating thesmoothness of the transition from an auxiliary to a dynamical scalar field. As the numericalstudy of lattice Yukawa models in 4d showed, the physical observables behave continuouslywith κ in the vicinity of κ = 0 including a region of negative κ [4, 5, 6].

This fact elucidatesthe relation between the Yukawa and four-fermion theories found in the continuum in theleading order in 1/NF expansion [4, 7].Motivated by these considerations and by the recent discussion of a relationship betweenthe Nambu–Jona-Lasinio type four-fermion theories and the Standard Model [4, 7] (for arecent review see ref. [8]), we address here the question to what extent the Z(2) and U(1)Y2 models still possess the most interesting and important properties of the GN models[9, 10], namely the asymptotic freedom of the Yukawa coupling y, the dynamical fermionmass generation and, in the case of the Z(2) model, the dynamical symmetry breaking.2Expected scaling propertiesIn the GN models the basic scaling properties at y →0 can be derived by means of the 1/NFexpansion.

For Y2 models this expansion is applicable only for small λ, strictly speaking

for λ = O(1/NF) [4, 7]. It gives results very similar to those for the GN models, providedκ < κc(λ).

Here κc(λ) is the critical line of the scalar model at y = 0 (i.e. without fermions)which for −κc(λ) < κ < κc(λ) is in the high-temperature phase.

The fermion mass in latticeunits amF is expected at fixed κ and λ to scale with y asamF ∝exp"−12β0a2m2φZφ1y2#. (2.1)Here β0 is the first coefficient of the β-function of the continuum GN model with NF flavoursof Dirac fermions,β0 = NF −12π(Z(2)),β0 = NF2π(U(1)),(2.2)amφ is the φ-field mass in lattice units and Zφ its renormalization constant at y = 0.

Forλ = 0 we have at y = 0 the free scalar field theory witha2m2φ = 1 −κκc(0)! 1κ ,Zφ = 12κ ,(2.3)and κc(λ = 0) = 1/4.

Note that for small λ the ratio Zφ/a2m2φ appearing in eq. (2.1) is thescalar field propagator at zero momentum, i.e.

the susceptibility χ of the pure scalar modelat the same values of λ and κ. Although a2m2φ and Zφ are not well defined separately atκ ≤0, the ratio χ is.

Thus the scaling law (2.1) is sensible also at κ ≤0. We shall see belowthat for small λ the mean field approximation (MFA) gives the same results as the 1/NFexpansion.For large λ the 1/NF expansion is a priori not applicable.

However, as we shall argue inthe next section, the MFA can be applied to study the Y2 models at all λ ≥0. The effectiveinteraction produced by fermions is very nonlocal and favours ferromagnetic ordering of thefield φx.

Such an interaction can be well described by some effective mean field H acting onφx at each site. Thus using the MFA we reduce the model to a pure scalar model with onlylocal interactions and the external field H. Such a model can be easily studied e.g.

by hightemperature expansion or Monte Carlo (MC) simulation.The results we obtain using this approach are as follows:(i)At any nonnegative λ including λ = ∞the Y2 model with Z(2) symmetry is in thephase with broken symmetry (⟨φ⟩̸= 0) for arbitrarily small y. Similarly, the correspondingU(1) model is in the spin wave phase (analogous to the low temperature phase of the XYmodel).

In both cases fermion mass is dynamically generated for arbitrarily small y andvanishes only at y = 0. This means that the Yukawa coupling is asymptotically free also forany nonnegative λ.

(ii)For κ < κc(λ) the fermion mass and, in the Z(2) case also the magnetization y⟨φ⟩,scale with y according toamF ∝exp"−h(κ, λ)y2#. (2.4)(iii)In the MFA the function h(κ, λ) is given byh(κ, λ) = πNF1χ,(2.5)where χ is the zero-momentum scalar propagator in the pure scalar φ4 model at given κand λ.

If the propagator is dominated by a pole with mass amφ and residue Zφ, then

χ ≃Zφa2m2φ(2.6)Thus according to the MFA the asymptotic freedom and the other mentioned propertiesof the GN models occur also for large λ in the Y2 models. Our Monte Carlo data at λ = 0.5and λ = ∞support (i) and (ii) (see also ref.

[11]), whereas some deviations from (iii) areobserved.3The mean field approximationIntegrating over fermion variables in the partition function for the Z(2) Yukawa model (1.1)we obtain an effective scalar model with the contribution to the action from the fermiondeterminantSdet[φ] = −Ntr ln M[φ];(3.1)whereM[φ]xz = 12γµ(δz,x+ˆµ −δz,x−ˆµ) + yφxδxz ≡Kxz + yφxδxz.We know from the experience with Yukawa models in four dimensions (for a recent reviewsee ref. [6]) that fermions strengthen ferromagnetic ordering: the transition from disorderedto ordered phase occurs at smaller values of κ as y increases from zero.

As was pointed outto us by E. Seiler [12] one can prove also that Sdet[φ] is minimized on the totally orderedconfiguration.We first give a very straightforward derivation of the announced results by the MFAmethod and then discuss its reliability. As infinitely many sites participate in the interactionSdet with a given one, say x, we expect their effect on φx to average into an effective meanfield H whose fluctuations are negligible.

So we obtain a scalar φ4 model with only localinteractions given by the first three terms in (1.1) to which the constant external field H isapplied. The mean magnetization σ of such a model is given at each κ and λ by the responsefunction σ = f(H).The simplest approach to calculate H is to differentiate Sdet with respect to φx at thesite x and then substitute the mean value σ for all φ’s.

This givesH(σ) = 2Ny2σZd2p(2π)2 Xµsin2 pµ + (yσ)2!−1. (3.2)Selfconsistency then requires thatσ = f(H(σ)),(3.3)where H(σ) is from (3.2).

For given y, κ and λ one can find σ as a solution of this equation.At κ < κc(λ) for small σ we can writef(H) = χH + O(H3),(3.4)where χ is the susceptibility of the scalar φ4 model, and (3.2) givesH(σ) = 4πNy2σ ln 1yσ + O(σ). (3.5)Substituting (3.4) and (3.5) into (3.3) one finds that a nonzero solution exists for arbitrarilysmall y.

Taking amF = yσ we arrive at eqs. (2.4) and (2.5) with NF = 4N.

Only the slope χ

of the response function f(H) at the origin was needed to derive this asymptotic scaling law.As we shall see in Sect.4, the eq. (3.3) with the full response function can give a very goodapproximation for mF even when σ is not small.Now let us discuss the MFA method in more detail.

To get an idea of how nonlocalthe interaction Sdet is at small y and whether it can produce the effective mean field let usexpand it in powers of y:Sdet[φ] = const + N y22XxzTrK−1xz K−1zx φxφz + O(y4),(3.6)where Tr is the trace over Dirac indices. The term of order y2 produces a nonlocal interactionbetween φx and φz:S2 = −Ny2 XxzJxzφxφz(3.7)with Jxz = 12TrK−1xz K−1xz ( as K−1xz = −K−1zx ).

One can find out easily that Jxz > 0 when xand z are separated by an odd number of links and Jxz = 0 otherwise. Thus this interactionis ferromagnetic.The MFA works well when fluctuations of the mean field at site x, Hx = 2Ny2 Pz Jxzφz,are small compared to ⟨Hx⟩.

In 2 dimensions Jxz falls offwith |x −z| so slowly that itproduces the infrared logarithmic divergenceXzJxz = 12TrK−2xx =Zd2p(2π)2 Xµsin2 pµ!−1. (3.8)or, in other words, very large number of sites contributes effectively to Hx.

This means thatthe mean value of Hx grows with the size L of the system as Ny2⟨φ⟩lnL if there is somenonzero ⟨φ⟩. As one can check the fluctuations of Hx around its mean value are finite atL →∞and thus at large L the MFA is justified if S2 was the only interaction.

Moreover,the divergence in ⟨Hx⟩means that such a system would be ordered at any y. This can bealso understood if one considers the free energy of the long wavelength modes of φ.

It isnegative and diverges as ln L.The interaction Sdet contains also other nonlocal terms which are of higher order than y2.They obviously contribute to H in the expression (3.2): the mass term amF = yσ can beinterpreted as their effect. However, one can easily realize that the interaction producing themean field H(σ) in (3.2) almost coincides with S2 for the distances smaller than O(1/mF)and falls offexponentially at larger distances.

This means that when mF becomes smallerthe MFA works better as more sites participate effectively in the interaction. The size 1/mFplays the role of the infrared cutoffwhich controls the logarithmic divergence in eq.

(3.5).It is also very useful to look at what happens from the point of view of the free energyof the long wavelength modes, i.e. effective potential.

The effective potential Veff(σ) at theorigin behaves like Veff(σ) ∼y2σ2 ln yσ (this is clear from the fact that H(σ) in (3.2) isminus the derivative of the fermion contribution to the Veff(σ)) and thus σ = 0 is a localmaximum. Approach based on the effective potential was applied in a recent paper [13] andgives the same results as the MFA method, as one would expect.

We think, however, thatthe MFA approach allows us to understand better the nature and the effect of the infraredsingularity, the mechanism of the fermion mass generation in the U(1) case (see below) andthe subtleties of both methods.Although our MFA method as well as the effective potential approach [13] and theirresults might look very consistent one should be cautious and not overestimate their relia-bility. The expression (3.2) has a form of a contribution of one loop with massive fermions.

Higher loop contributions may turn out to be important. For example, for the GN model(κ = λ = 0) the selfconsistency equation (3.3) coincides with the gap equation obtainedin the leading order in 1/NF expansion.

However, we know that in the Z(2) GN case thecoefficient h of 1/y2 in the scaling law given by (2.5) receives 1/NF corrections (comparewith eq. (2.2)).

The same result for h as eq. (2.5) is obtained in [13] and must receive finitecorrections for this reason as well.

The terms in Veff of higher order in y2 neglected in [13]contain in fact powers of y2 ln(1/yσ) and are not negligible as y →0. Nevertheless, one canexpect that H still has the singularity similar to (3.5) at mF →0 as a consequence of thestrong nonlocality of the interaction Sdet in 2 dimensions.

One can also expect that mF ̸= 0if σ ̸= 0. Then the conclusion (i) that the fermion mass is generated for arbitrarily small ystill holds.

It is natural to expect that these features do not depend on the structure of thelocal scalar interactions in (1.1), in particular on the value of λ. The predictions (ii) and inparticular (iii) are less reliable, however.Let us now see how one can apply the MFA in the Y2 model with the U(1) symmetry,when the symmetry cannot be broken spontaneously.

It is known that this fact does notprevent fermions from acquiring a mass [10]. To understand how this happens we considerthe interaction of the scalar field induced by the fermion with a small mass mF in one loopthat we have already discussed.For distances between sites smaller than O(1/mF) onecan neglect the mass and the interaction behaves like (3.7).

Such interaction can produceferromagnetic ordering on the distances smaller than O(1/mF) if mF is small enough. Onthe distances larger than O(1/mF) long wavelength fluctuations (spin waves) destroy theordering as usual in two dimensions.

However, in a finite volume of the linear size O(1/mF)the magnetization is nonzero and its direction is drifting slowly.Now let us imagine a fluctuating scalar field with such properties playing the role ofa background field coupled to a fermion via Yukawa interaction. For small y we expectthe fermion to acquire a mass given by amF ≃y⟨φ⟩.

What ⟨φ⟩enters this formula? It isreasonable to expect that the fermion “feels” only the ordering on the scale of its Comptonwavelength, so that ⟨φ⟩should be averaged over a volume of the size O(1/mF).

This agreeswith the observation made by Witten that “almost long-range order” in chiral GN modelsis sufficient to generate the fermion mass [10].Bearing this in mind we can carry out the mean field considerations in the U(1) casein complete analogy with the Z(2) case. One has to realize, however, that ⟨φ⟩≡σ in theformulas is the average magnetization on the scale of O(1/mF).

We actually see nonzero ⟨φ⟩in a finite volume in MC simulations which agrees with amF = y⟨φ⟩remarkably well (withinfew percent) when L ≈1/amF.We now compare the solutions of the equations (3.2) and (3.3) on finite lattices withnumerical results and use these equations for data analysis.4Numerical resultsWe simulated the Z(2)-symmetric Y2 model defined by the action (1.1) and the analogousmodel with the U(1) symmetry on the lattices L2 = 162, 322 and 642, mostly with NF = 16.The expected scaling behaviour (2.4) suggests to collect data at many y-points at fixed valuesof λ and κ and to study the y-dependence of the fermion mass. We have chosen several κpoints in the interval −0.3 ≤κ < κc(λ) with λ = 0, 0.5 for both Z(2) and U(1) models andλ = ∞for the U(1) model1.

For each pair of (κ, λ) values and L we determined the fermion1The hybrid MC algorithm for simulating dynamical fermions we are using unfortunately does not workin the Z(2) case at λ = ∞because of the discrete nature of the variables.

mass amF at several y values such that 2/L <∼amF ≤1.As has been reported in ref. [11] in detail, the dependence of amF on y at all investigated(κ, λ) points is consistent with the expectations (i) and (ii).

The data analysis on finitelattices has been performed by means of the generalized gap equationhy2 =π2L2X{p}1Pµ sin2 pµ + (amF/s)2(4.1)which gives the scaling law (2.4) at L = ∞and describes quite well the onset of finite sizeeffects and a very slow approach to the asymptotic scaling. The values of the fit parametersh and s depend only on κ and λ but not on L. Note that eq.

(4.1) at s = 1 can be viewedas a selfconsistency equation like (3.3) with a linear response function whose slope is a freeparameter.In the present paper we compare these results with the parameter-free predictions of theMFA, to see how well it can describe the data. To obtain these predictions we solve forgiven κ, λ and y the finite lattice version of eq.

(3.2) and eq. (3.3) numerically.

On a finitelattice we substitute the integral in (3.2) by the discrete sum over the momenta allowed bythe boundary conditions on the fermion fields: periodic in one and antiperiodic in anotherdirection. The response function f(H) is obtained by a MC simulation of the scalar φ4 modelin the external field.

For κ = 0 the response function can be also expressed in terms of easilycomputable one dimensional integrals. So for each (κ, λ) pair and L we obtain a functionamF(y) which we compare with the data.In the case of the Z(2) model we find only a qualitative agreement.

There is a systematicdiscrepancy between the data and the MFA prediction at any κ and λ values.This isconnected with the fact, discussed in the previous section, that in the scaling limit amF →0the one-loop formula for H(σ) (3.2) does not reproduce the correct value of the β0 coefficientin the scaling law (2.1) already for the GN model (κ = 0, λ = 0). We note that this happensin the Z(2) GN model also in the leading order of the 1/NF expansion.

It would be veryuseful to include the higher loop corrections but at the moment we do not know how to dothat at large λ systematically.2 Without these corrections the simple MFA that we usedapparently cannot describe the data in the Z(2) model sufficiently well quantitatively.In the chiral GN model higher loop corrections do not modify the coefficient of 1/y2 inthe exponent of the scaling law. Correspondingly, the agreement is much better in the U(1)Y2 models.

For comparison we present plots for (κ, λ) = (0, 0), (0.2, 0), (0, ∞) and (0.2, ∞).At λ = 0 the 1/NF expansion can be applied and in the leading order coincides with theMFA prediction. The data agree with this prediction well (see figs.

1 and 2) as one canexpect if NF is large. For κ = 0 at λ = 0.5 and λ = ∞(fig.

3) the data agree with the MFAprediction as well as at λ = 0. However, if κ ̸= 0 we observe both at λ = 0.5 and λ = ∞asignificant discrepancy illustrated for λ = ∞and κ = 0.2 in fig.

4.We see that the MFA works not only at λ = 0 but also at large λ as long as κ = 0even better than one could naively expect. The discrepancy at κ ̸= 0 and large λ couldbe due to the crudeness of the simple MFA that we used and more systematic study isneeded to understand it fully.

In any case, as the fits with the generalized gap equation (4.1)performed in ref. [11] demonstrate, all our data are consistent with the scaling of the generalform (2.4).

Here we show such a fit in fig. 4 (dotted-dashed line).

The values of the fitparameters in eq. (4.1) are approximately s = 0.27 and h = 0.86(π/NFχ).

We cannot drawfirm conclusions from these numbers as they are obtained in the region where the asymptotic2Some additional complications in the subleading order in 1/NF arise in the Z(2) model even at λ = 0due to the naive fermion discretization in eq. (1.1).

scaling is not yet achieved and we do not know how well the eq. (4.1) describes the approachto the asymptotic scaling.In conclusion, both our numerical data and mean field considerations suggest that theZ(2) and U(1) Yukawa models in 2d have asymptotically free Yukawa coupling and exhibitdynamical fermion mass generation also for arbitrarily strong scalar field selfcoupling, whenthe usual perturbative and 1/NF expansions are not applicable.

They are thus quite similarto the corresponding Gross-Neveu models.However, at present we cannot say whetherthey belong to the same universality class, i.e. what the precise form of the scaling law is.Our precision is also not yet good enough to compare with the results of the mass gapcalculations [3].Further study of these questions, as well as of the transition from theGross-Neveu-like behaviour to the NLσ or spin model cases is needed.

Finally, we have notyet studied the properties of the 2d lattice Yukawa models at larger negative κ, where theantiferromagnetic scalar field coupling competes with the Yukawa coupling which supportsferromagnetic order.AcknowledgementsWe thank E. Seiler and M.M. Tsypin for helpful suggestions, A. Hasenfratz, P. Hasenfratz,R.

Lacaze, F. Niedermayer and M. Teper for valuable discussions and H.A. Kastrup forcontinuous support.

One of the authors (M.A.S.) would like to thank RWTH Aachen andHLRZ J¨ulich for a kind hospitality during his visits.

The calculations have been performedon the CRAY Y-MP of HLRZ J¨ulich.

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Figure 1: The numerical data for the fermion mass amF on finite lattices in the U(1) Yukawamodel plotted against 1/y2, where y is the Yukawa coupling constant. The dotted, dashed andfull lines are the mean field predictions on the L2 lattices, L = 16, 32 and 64, respectively.

Thestatistical error bars are omitted to make the plot tidy, they do not exceed the symbol size. In thisfigure κ = λ = 0, which means that the U(1) Yukawa model is identical to the chiral Gross-Neveumodel.Figure 2: As in fig.

1, but now for κ = 0.2 and λ = 0. Here both the 1/NF expansion and theMFA are as well applicable as in the GN case.

Figure 3: As in fig. 1, but now for κ = 0 and λ = ∞.

The 1/NF expansion is not applicable butthe MFA still describes the data very well.Figure 4: As in fig. 1, but now for κ = 0.2 and λ = ∞.

The 1/NF expansion is not applicable andalso the MFA does not describe the data very well. The dotted-dashed line represents a fit to the642 data (circles) by means of the generalized gap equation (4.1).

This figure "fig1-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303010v1

This figure "fig2-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303010v1

This figure "fig3-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303010v1

This figure "fig4-1.png" is available in "png" format from:http://arxiv.org/ps/hep-lat/9303010v1

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