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Double Scaling Limit of Scalar Theories on the Lattice

arXiv:hep-th/9212053v1 8 Dec 1992COLO-HEP/295AZPH-TH/92-43December 1992Double Scaling Limit of Scalar Theories on the LatticeB. S. BalakrishnaDepartment of PhysicsUniversity of Colorado, Boulder, CO 80309andDepartment of PhysicsUniversity of Arizona, Tucson, AZ 85721ABSTRACTScalar field theories regularized on a D dimensional lattice are found to exhibitdouble scaling for a class of critical behaviors labeled by an integer m ≥2.

Thecontinuum theory reached in the double scaling limit defines a universality classand is of a massless scalar with only a m + 1 point self-interaction, but in thepresence of a constant source. The upper critical dimension for this to occur isthe same as that for renormalizability of the m + 1 point interaction.

1/N expansion is a powerful tool that has enabled one to study some of the phenomenonnonperturbative in the coupling constants. However, in most cases, it turns out to be possibleto study only the leading term of the 1/N series, the large N limit.

One way to examine thesubleading terms is to look for their enhancement that maintains the virtues of the large Nlimit. This was possible in the past for matrix models simulating two dimensional quantumgravity and for some low dimensional string theories[1].

There one observed that the higherorder terms of the 1/N series, receiving contributions from higher genus Riemann surfaces,diverged as one approached a critical point. Remarkably, it was found possible to make useof this divergence to overcome the 1/N suppression and elevate all the terms of the 1/Nseries to the same level.

This approach, called double scaling, helped one to sum the 1/Nseries to obtain a nonperturbative solution.This phenomenon is found to occur in other situations as well[2], as for instance in a classof O(N) vector models. Here, the presence of double scaling does not in general help oneto sum the 1/N series, but provides new insights into the problem not available otherwise.Most of these models can be reduced to a scalar field theory with 1/N playing the role ofa loop expansion parameter.

The question then arises as to whether double scaling is moregeneric and is a phenomenon of scalar theories in general. As we will see in this letter,the answer to this question is yes.

A wide class of scalar field theories are found to exhibitdouble scaling as the coupling constants approach a critical point. This occurs for a class ofcritical behaviors labeled by an integer m ≥2.

In the double scaling limit, one approachesthe theory of a massless scalar field with a m + 1 point self-interaction in the presence of aconstant source, defining a universality class reached in the infrared. For even values of m,this theory has problems as the potential is unbounded from below.

For odd values of m, oneexpects it to be well defined. The short distance theory is not unique due to universality andthe one studied here is a lattice model in D dimensions.

The double scaling limit offers aninteresting way of approaching the continuum from the lattice. The upper critical dimensionturns out to be the same as that for renormalizability of a m + 1 point interactionConsider a model given by the following action defined on a D dimensional lattice:S = NX⟨xy⟩φ(x)(φ(x) −φ(y)) + NXxV (φ(x), β),(1)where φ(x) is a real scalar field.

For convenience, we stick to one scalar, but the conclusions1

are extendable to cover more fields. The lattice sites are labeled by x and the links by ⟨xy⟩.1/N plays the role of a loop expansion paraemter, like the Planck’s constant.

The potentialV may depend on a set of coupling constants collectively denoted by β. The theory studiedhere is on the lattice but the conclusions are equally applicable to those regularized witha momentum space cutoff.

Double scaling is however an interesting way to approach thecontinuum from the lattice.The model is easily solved for large N. This is because the partition function is thendominated at the saddle point governed by a translationally invariant φ. Setting the φ(x)’sto be equal to λ gives a potential V (λ, β) whose saddle point equation isV ′(λ, β) = 0.

(2)Here a prime denotes differentiation with respect to λ. With a solution giving an absoluteminimum, V (λ, β) gives the free energy per site for large N.Subleading terms of the 1/N series are better examined in momentum space with thehelp of Feynman diagrams.

Because 1/N is a loop expansion paramter, a graph having Lloops is generated at order N1−L. Starting at an absolute minimum of the potential V , theinteraction potential is obtained by translating φ by an amount λ,V (φ + λ, β) = V (λ, β) +∞Xp=2V p(λ, β)φpp!

. (3)This shows that a vertex of order p in a Feynman graph has strength −V p(λ, β).

V 2(λ, β)is the mass squared term. The lattice propagator is K−1 where K is the matrix appearingin the kinetic part of the action (1).

K, with its indices labeling the sites, has the value2D along the diagonal and −1 for all the 2D neighbors. It is diagonalizable in momentumspace.

Its eigenvalue along a momentum vector k is 2 Pi(1 −cos(kia)), where i runs over allthe D components of k and a is the lattice spacing. For small momenta, this starts offask2a2 where a sum over components is implicit.Double scaling is a phenomenon near the critical point.

One expects in general a criticalbehavior to be present for large N itself. In the space of coupling constants β, there existsin general a class of critical behaviors.

The potential V (λ, β), with λ treated as a variable,leads to a definition of m−criticality,V l(λc, βc) = 0,l = 1, · · ·, m,m ≥2,(4)2

where a superscript l denotes l−th derivative. λc and βc are the critical values of λ and βrespectively.

The potential V (λ, β) near βc is approximatelyV (λ, β) = Vc + (β −βc)∂βVc + (β −βc)∂βV ′c(λ −λc) +V m+1c(m + 1)! (λ −λc)m+1,(5)where a subscript c denotes evaluation at the critical point.

Because λc corresponds to anabsolute minimum of V , V m+1cis positive. A summation over all the coupling constants isimplicit in (β −βc)∂β.

We assume that the critical value βc is approached along ∂βV ′c frombelow. Using the above approximation in the saddle pont equation givesλ −λc =" m!V m+1c(βc −β)∂βV ′c#1/m.

(6)This is nonanalytic as β →βc as is usual in critical phenomenon. We are interested in V p(λ)for p ≥2 near the critical point,V p(λ, β) =V m+1c(m −p + 1)!"

m!V m+1c(βc −β)∂βV ′c#1−(p−1)/m,p ≤m + 1. (7)Higher derivatives of V tend to a constant near the critical point.

As we will see later, thisresults in the suppression of vertices of order p > m + 1 in the double scaling limit.As an example, consider the following potential:V (φ, β) = βe2φ + eφ −φ. (8)The saddle point equation is quadratic in eφ leading to the solution,eφ = 14β−1 +q1 + 8β,(9)where the other root is ignored as it leads to an unphysical value of φ. Nonanalyticity setsin as β →−1/8.

Because the potential is unbounded from below for negative β, this is to beviewed as an analytical continuation. The critical point βc = −1/8 is found to be of orderm = 2.Eq.

(7) shows that the vertices of a Feynman graph of order m+1 and lower go to zero asβ →βc. But, we are looking for an enhancement of the 1/N suppressions.

This comes fromthe infrared divergences that arise because the mass squared term V 2(λ, β) also vanishes inthis limit. To pick up these contributions, one needs to take the lattice spacing a to zero aswell.

The lattice propagator including the mass squared term is [K + V 2(λ, β)]−1. Because3

K is of order a2 as a →0, we require the mass squared term to be also be of order a2. Thisdetermines the rate of approach to the critical point,βc −β ∼a2m/(m−1).

(10)Even if one has begun the discussion with a fixed lattice spacing, there is a need to scaleall the momentum variables, k →ka, in a Feynman graph by a small parameter a to blowup the infrared region. Either way, each of the propagators now contributes a factor 1/a2and each of the loops a factor aD.

A vertex of order p for p ≤m + 1 supplies a factora2(m−p+1)/(m−1). Let us exclude all vertices of order m + 2 and higher for the moment.

Also,let us restrict our attention to graphs having no external legs. If there are Vp vertices oforder p and E propagators in a graph having L loops, the product of a−factors isaDL−2EΠm+1p=3ha2(m−p+1)/(m−1)iVp= aD haDc−Di1−L ,(11)where Dc is the critical dimension,Dc = 2m + 1m −1.

(12)Ignoring the overall factor aD, note that this diverges as a →0 when D is less than thecritical dimension Dc. It is now clear how to enhance the 1/N suppression factors.

LettingN ∼aD−Dc,(13)one can elevate all the loop graphs to the same order with an overall factor aD. The scalinggiven by (10) and (13) as a →0 is what is meant by double scaling in this letter.

The overallfactor aD is what is needed to relate the free energy per site to a free energy density. Verticesof order m + 2 and higher do not contribute any a−factors.

In a graph having such vertices,to obtain the previous result (11), we need to have the factor a2(m−p+1)/(m−1) for each vertexof order p ≥m+2 as well. Allowing for such factors is equivalent to having a−2(m−p+1)/(m−1)for each vertex of order p ≥m + 2 over and above the result (11).

But, a−2(m−p+1)/(m−1)vanishes as a →0 at least as fast as a2/(m−1) for p ≥m + 2. This shows that all the graphshaving vertices of order m + 2 or higher are suppressed in the double scaling limit.

In thepresence of external legs, the external momenta should be scaled by a just like the internalones, which results in an enhancement as before.4

The continuum theory one approaches in the double scaling limit is easily established. Itis determined by the following potential for a continuum field variable ϕ:m+1Xp=2V p(λ, β)ϕpp!∼V m+1c(m + 1)!h(ϕ + λ −λc)m+1 −(m + 1)(λ −λc)mϕi→V m+1c(m + 1)!ϕm+1 −tϕ,(14)where t = (βc −β)∂βV ′c and the factors of a are not included as they are already taken careof.

Also, some constants have been dropped and a translation of ϕ has been made in theend. Vertices of order p > m + 1 are absent as they are suppressed in the double scalinglimit.

What we have reached is the theory of a massless scalar field ϕ with only a m + 1point self-interaction, but in the presence of a constant source t. Because V m+1cis positive,this leads to a well-defined potential for odd values of m. It is problematic for even valuesof m as it is unbounded from below. In this case, the double scaling limit is expected toinvolve some analytical continuation as in the case of the example mentioned earlier.

Thiscan make the absolute minimum we started with only an extremum of the potential. Thedouble scaling limit may still exist as an analytical continuation of the sum of Feynmandiagrams.To justify the analysis based on 1/N requires D to be less than its critical value Dc =2(m + 1)/(m −1).

Dc takes values 6, 4, 3 and 2 respectively for m = 2, 3, 5 and ∞. Becausethis is also the critical dimension for requiring renormalizability of a m+1 point interaction,one is in the superrenormalizable regime.

The combination g = aD−Dc/N survives to playthe role of a loop expansion parameter. In other words, g(m−1)/2 is the strength of the m + 1point interaction.Because the double scaling limit takes one to the infrared region, the resulting theoryis expected to be highly universal.

Already, the previous arguments have suggested thatthe theory (1) governed by a general potential V falls to one of the classes given by (14).But, there is more to this.If the propagator is more general, say [Kf(K)]−1 ignoringthe mass squared contribution, f(K) will tend to a constant f(0) (assumed finite) in theinfrared and the propagator is ∝K−1 as before. This is an indication of the well-known factthat small distance effects like the lattice structure are irrelevant in the continuum limit.Further, if there are derivative couplings (involving lattice derivatives), they contribute atleast a factor k2 →k2a2 ∼a2.

But this vanishes as a →0 faster than a p−point vertex5

strength V p(λ, β) ∼a2(m−p+1)/(m−1) (or constant). Derivative couplings are thus expected tobe suppressed in the double scaling limit.

Universality, being an inherent feature of criticalphenomenon in general, is thus found to be present here as well.Because of the survival of all vertices of order less than m+ 1 in the double scaling limit,like for instance the cubic vertex, there are tadpoles that need to be summed. As shownearlier, a translation of the field variable removes all the vertices except that of order m + 1,but introduces a constant source t. The resulting theory is easier to handle in the effectiveaction approach.

Because the source is a constant, only the effective potential matters. Thesource t is now traded for the expectation value v of the continuum field variable ϕ. Inthis approach only the one particle irreducible diagrams are retained, thus excluding thetadpoles.

The effective potential computed this way gives us the one we are looking for afterthe addition of the source term −tv. Minimization of the resulting effective potential relatesv and t. Because the loop expansion parameter g is dimensionful having mass dimensionDc −D, and v has mass dimension 2/(m −1) in our conventions, an expansion in loops isactually a perturbation in g/v(Dc−D)(m−1)/2.

The tree level potential suggests that v ∝t1/m,hence this is a perturbation in g/t(Dc−D)(m−1)/(2m). Ultraviolet divergences that might appearin the double scaling limit can be handled in the usual way by the addition of counter termsto the bare lattice action we started with.One way to generalize the conclusions for more than one scalars is as follows.Thevariables φ and λ are now given an index to label the various scalars.

Products of φ andλ, and various derivatives of V , receive more indices like tensors. m−criticality could bedefined by requiring all the derivatives of V of order less than m + 1 to vanish.

Let usassume that a solution to the saddle point equation giving an absolute minimum exists nearthe critical point. Now, the order of magnitudes of mass and the vertex strengths are all thesame as before.

All the scalars help in double scaling and hence survive. The final result isthe theory of massless scalars with only m + 1 point interactions, but in the presence of aconstant source.

The source points along a direction given by β∂βV ′c. A simple example isgiven by an O(M) theory where the field variable φ has M components.

The potential V isnow a function of φ†φ, the square of the length of the M−vector φ. It is easily found thatthe double scaling limit of this theory is the same as before, that of one massless scalar withm + 1 point self-interaction in the presence of a constant source, except for the presence of6

M −1 additional free massless scalars.So far, we had treated the coupling parameters as constants. It is possible to generalizethe results to cover space dependent couplings.

The conclusions are the same, except thatthe source t is now replaced by t(x). Space dependent source helps in deriving the Schwinger-Dyson equation.

For the m + 1 theory, this is easily obtained,V m+1cm! gmδmt(x)Z −g∂2δt(x)Z −t(x)Z = 0,(15)where Z is the functional integral of the continuum theory and t(x) = (βc −β(x))∂βV ′c.δt(x) is the functional derivative with respect to t(x).

This differential equation is what oneexpects to emerge in the double scaling limit from an analogous one for the original theoryfor any m ≥2. In zero dimension, the term involving the spatial derivative is absent and thefunctional derivatives are ordinary derivatives.

Then, the solution to the Schwinger-Dysonequation for m = 2 is given in terms of modified Bessel functions,Z ∝t1/2I1/32t3/2/3,(16)where t has been scaled to (V 3c g2/2)1/3t. This sums the series generated by the 1/N expansionin the double scaling limit.

The asymptotic expansion of the Bessel function for large valuesof t recovers the series. Though the potential is unbounded from below for m = 2, theSchwinger-Dyson equation admits a solution that agrees with the Feynman diagrams in thedouble scaling.

This is because one expects the double scaling limit to involve some analyticalcontinuation for even mIn this letter, it is argued that the phenomenon of double scaling exists in a wide class ofscalar field theories. It is discussed here for theories regularized on a D dimensional lattice.The critical behaviors involved here belong to a class ordered by an integer m ≥2.

Thecontinuum theory reached in the double scaling limit turns out to be the theory of a masslessscalar with only a m+1 point self-interaction, but in the presence of a constant source. Thisprovides an interesting way to approach the continuum limit from the lattice.

Only theuniversality class given by the m + 1 theory is expected to matter in the double scalinglimit. The arguments are expected to hold for dimensions less than the critical dimensionfor renormalizability of a m + 1 point interaction.

Besides being interesting in its own right,this work might turn out to be useful in the analysis of double scaling for the scalars that7

are expected to emerge in the 1/N expansion of various theories, like for instance the O(N)models[2] or those constructed to induce QCD[3].I would like to thank Professor Anna Hasenfratz for discussions. This work is supportedby NSF Grant PHY-9023257.References[1] E. Br´ezin and V. A. Kazakov, Phys.

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