Using WegnerHoughton equation, within the Local Potential Approximation, we study critical properties of vector models. Fixed Points, together with their critical exponents and eigenoperators, are obtained for a large set of values of , including and . Polchinski equation is also treated. The peculiarities of the large limit, where a line of Fixed Points at is present, are studied in detail. A derivation of the equation is presented together with its projection to zero modes.
UBECMPF 96/21
PACS codes: 11.10.Hi, 64.60.Fr
Keywords: renormalization group, fixed points, critical exponents, large N
1 Introduction
In this article we study linear sigma models near criticality, within the framework of the Exact Renormalization Group (ERG, hereafter), mainly in its simpler nonperturbative approximation, the Local Potential Approximation (LPA). We concentrate mostly in the WegnerHoughton equation [1], although other equations are also considered [2, 3].
Recent results [4] have raised interesting questions about the behaviour of the RG near firstorder phase transitions. They are in some conflict with exact results of WegnerHoughton equation at [5]. A systematic study of this subject at finite is still lacking. Although our interest points in that direction, many aspects of second order phase transitions had to be previously worked out. This paper is meant to fill this gap, as well as review the derivation and projection of the equation with a modern language, paying special attention to some aspects not fully treated in the literature, which we hope will be useful for a subsequent study of models at discontinuous phase transitions. Nevertheless, as this paper is focused in continuous transitions, we leave to the conclusions further comments on firstorder ones.
ERG methods are exact formulations of the RG in differential form (see Ref. [6]), that is, the evolution of the renormalized action along the RG flow is studied in terms of differential equations. There are many such equations, being one of the oldest that of Wegner and Houghton [1], which corresponds to a decimation in momentum space, i. e., a lowering of a sharp momentum cutoff. Unfortunately, those equations are very involved and exact solutions are only available for very simple models. Nevertheless, several feasible nonperturbative approximations are possible, being the most promising an expansion in powers of momenta. The LPA [7] is just the first term of such an expansion [8].
The organization of the paper is as follows. In Section 2 WegnerHoughton equation is derived in some detail. The LPA is discussed in Section 3 and extensively worked out in Section 4, where we compute critical indices of models at finite as well as its eigenoperators. It is also shown that irrelevant eigenvalues, which are related to the anomalous dimensions of higherdimensional composite operators, may be obtained with ease. The comparison with our results from Polchinski equation as well as other determinations is left to Section 5, while the special, exactly solvable, case is studied in Section 6, where some striking results are obtained. Section 7 is devoted to the conclusions. Finally, some peripheral subjects are treated in two Appendices.
2 Derivation
In this section a careful derivation of WegnerHoughton equation is presented, trying to be as selfcontained as possible. Nevertheless, for the sake of simplicity, we concentrate on the case. Once this case is mastered, its generalization for arbitrary is straightforward.
We assume that our action is regulated with a dependent sharp cutoff in momentum space , with all modes with already eliminated, where parametrizes the RG flow and is a fixed scale. A RG transformation consists of two steps, a blocking or elimination of shortdistance degrees of freedom, and a change of length scale. In this way one goes from an action cutoff at to one at , in physical units. From now on, we work with dimensionless quantities, the dimensions being given, if needed, by . We derive WegnerHoughton equation first considering the blocking, and afterwards the rescaling.
Let us parameterize our action as
(1)  
where and . We have assumed symmetry. The field is split as
(2) 
where are the modes with and the ones with . The blocked action is then defined through
(3) 
As we are seeking a differential equation for the renormalized action, we need to consider just an infinitesimal blocking, hence we discard terms of order or higher. For that purpose, it is convenient to rewrite the action, Eq. 1, as
(4) 
with
(5) 
and primes in integrals meaning momenta restricted to the shell .
Written in the above form, the action is suited for treating the integral in a diagrammatic expansion. Its Feynman rules are:

Any legs vertex contributes with , where every leg is labelled by a momentum .

Any propagator between a leg labelled by and another one labelled by is .

All integrals are understood to affect the whole diagram and not only the contribution of its vertex. That is, every delta function always simplifies an integral.

Add the usual symmetry factors.
Before going on we state two technical results.
Lemma 1. For any function , analytic around , it is
(6) 
where stands for a finite number of momenta , with , and for integrals over all ’s.
Proof. Just integrate first over all integrals. QED
Lemma 2. For any function , analytic for any value of and at , it is
(7) 
provided that .
Proof. In spherical coordinates, integrating with respect to ,
with being in the shell and standing for the delta functions that constrain angular variables. The integral over will give a contribution of . To convince ourselves that the rest of the integral is for , let us take to be the angle between and . In this case, the angular contribution of Eq. 2 is
(9)  
where is a shorthand for all the remaining angular variables and is the appropriate Jacobian. From the last two delta functions, recalling that ,
(10) 
QED
Let us now compute the leading contribution, , of a tree diagram assuming that each nvertex is , with being the lowest integer greater or equal than , propagators being because of the delta function. Then,
(11) 
where is the number of vertices and the number of propagators. Let us now put this result in a rigorous basis.
Lemma 3. The leading contribution in of a treelevel diagram, is .
Proof. Consider first a connected tree diagram consisting of one vertex and 1vertices, as in Fig. 1.
Its value is
(12) 
We perform the integrals in pairs, using Lemma 2 to extract the leading dependences in . At the end, we obtain, in agreement with Lemma 3, that it is for even or else for odd, with the final aid of Lemma 1. (Recall the delta function hidden in Eq. 5).
Furthermore, let us assume that Lemma 3 is true for all diagrams with vertices other than 1leg ones. Let us now prove that it is also true for diagrams with such vertices. To this end, note that a generic tree diagram with morethanoneleg vertices contains at least one vertex (for some ) with all but one leg connected to 1vertices.^{1}^{1}1 To pick up one, just eliminate all 1vertices, and take one 1vertex of the remaining diagram (which must have at least one since it is again a tree).
If the selected vertex is a nvertex, with odd, perform all integrals associated with its legs but the one that connects it to the rest of the diagram. Performing the integrals in pairs, and using Lemma 2, it brings about a contribution . It remains a diagram with morethanoneleg vertices, where the former selected vertex enters now as an additional 1vertex (with replaced by some other function). See Fig. 2.
If is even, then integrate over all its legs connected to 1vertices but one. This is, using Lemma 2 again, . There is left over a 2vertex connected both to the remaining subdiagram and to a 1vertex. It can easily be seen that it is of the same order in as one with the 1vertex connected directly to the subdiagram, without the 2vertex. See again Fig. 2.
Therefore,
(13) 
and, by the induction hypothesis,
(14) 
Taking into account that and , a proof of Lemma 3 follows.
QED
Note that every tree diagram must necessarily be^{2}^{2}2 Recall that every diagram must contain at least one propagator. , and, from Lemma 3, it must be at leading order in . Furthermore, in such case, any inclusion of a fourlegs or higher vertex would imply at least one loop, and thus the only vertices allowed, apart from oneleg ones, are an arbitrary number of twolegs ones. Even more, these additional vertices, in order to be , are (Lemma 2)
(15) 
The argument that leads to the proof of Lemma 3 has to be slightly modified for diagrams with loops. Any 1PI part contains one integral per loop, with the only restriction that of being within the shell. After performing those integrals, we end up with a contribution times an effective tree diagram. Therefore, the correct formula is to add the number of loops to the result of Lemma 3, but with Eq. 11 computed considering any 1PI part as a treelevel vertex.
We conclude then that there cannot be diagrams with more than one loop and that oneloop diagrams must have zero external legs. Furthermore, it happens again that diagrams with more than two legs generate further loops and that twolegs ones must have the form of Eq. 15.
All those considerations show that, at leading order in , Eq. 4 is equivalent to
(16) 
The path integral, Eq. 3, is now Gaussian with the result
(17) 
where we assume that is positive definite and “const” means a possible term independent of .
The other step in any RG transformation is a change of scale in all linear dimensions. This amounts to a standard dilatation transformation,
(18) 
which gives,
where the prime in the last partial derivative merely indicates that it does not affect delta functions.
Therefore, WegnerHoughton equation is
To deal with the arbitrary case, we define the partition function as
(21) 
so that couplings are all finite at . The derivation is completely analogous and the ERG equation reads
where the trace is over flavour indices and
(23) 
3 Projection
The Local Potential Approximation (LPA) amounts to projecting the full action into a fixed kinetic term plus a general potential involving only the zero modes,
(24) 
where . The LPA is, thus, just the first term in an expansion in powers of momenta [8].
To project over constant fields, we define the operator [7] which, acting on an arbitrary functional , is
(25) 
The projection of the third term in the second line of Eq. 2 gives zero because of the explicit factor of there. The first two terms in that line are simply
(26) 
with . The second term of the right hand side of Eq. 2 can be seen to cancel whereas the first one is
(27) 
The LPA amounts to
(28) 
Eq. 27 becomes, thus,
(29)  
with ,
(30) 
and the Euler Gamma function.
The equation is further simplified if we perform one more derivative with respect to ,
(31) 
If, instead, the projection is over the kinetic term and its normalization kept fixed under the RG flow,
(32) 
it follows .
Moreover, the delta function in Eq. 31 just reflects the difficulties of selecting one mode out of a continuum set. This problem is not present at finite volume, where the zero mode is neatly separated. In such a case the delta function is just a coefficient proportional to the inverse number of modes. One can then rescale and Eq. 31 becomes
(33) 
Note that, along the same lines, one may absorb .
On projecting, we made just one approximation, that of Eq. 28. Before closing the section, we briefly analyze how, in the limit, this approximation becomes exact. Eq. 28 amounts to,
(34) 
Hence, it becomes exact when couplings do not depend on (except for that has a term). If one expands Eq. 2 in powers of momenta, one realizes that an hypothesis of such a kind does not hold for finite, but there are some chances at .
To proceed, it is necessary to rewrite Eq. 2 introducing invariants,
(35) 
Then, Eq. 2 at becomes closed for the invariants of the form ,
(36) 
with is the piece of that contains only the invariants (called diagonal piece in Ref. [1]), and we set which is a general result on the limit.
An inspection of how couplings of the diagonal piece at are affected by those with increasing powers of suggest that, if all couplings at a certain power of are set to zero, all the rest but the ones at can be simultaneously set to zero, so it should not come as a surprise that an ansatz like
(37) 
with , holds.
Indeed, Eq. 36 leads to a closed equation for ,
(38) 
which turns out to be the same as the large limit of Eq. 33. This is the result we were after. The LPA in the large limit is not only solvable as we will see in Section 6, but also exact. If our initial potential has couplings that do not depend on , the exact evolution for those is governed by Eq. 38.
4 Results
Fixed Points (FPs) of Eq. 33 are the analytical solutions of
(39) 
with some dependent constant (Eq. 30) which equals in . All universal quantities are independent of its precise value. Note that this is an ordinary differential equation of second order and, thus, two initial conditions are required. Nevertheless, at Eq. 39 is singular, since the coefficient of its highest derivative vanishes, and we must supply the analyticity condition
(40) 
which reduces the possible solutions to a one parameter set.
A careful study shows, however, that for any initial value the solution remains negative for any , resulting in an unbounded potential. We, thus, end up with as only possible values. Furthermore, both analytical and numerical considerations show that solutions of Eq. 39 end up with a singularity
(41) 
for all but a finite number of . In Fig. 3 we plot the singular point as a function of the initial condition for different values of at .
For only two initial values are allowed, one corresponding to the Gaussian Fixed Point (GFP) and another one to the Heisenberg Fixed Point (HFP)^{3}^{3}3In fact, there is a third one, . It corresponds to the Trivial Fixed Point with zero correlation length (infinite mass).. We study each in turn.
The GFP is
(42) 
Its eigenoperators are solutions of the linearized version of Eq. 39 around ,
(43) 
with the derivative of the scaling operator and its eigenvalue. For the sake of convenience, we define
(44) 
to obtain
(45) 
which is the generalized Laguerre equation. Polynomial solutions select to be
(46) 
with a nonnegative integer, and, in such a case,
(47) 
with the generalized Laguerre polynomial [9]. Critical exponents are the Gaussian ones, which allow to define massive interacting QFTs in the vicinity of this FP.
The special case of was previously studied in Ref. [7] in terms of the variable . The solutions are the Hermite polynomials of odd degree, . Indeed [9],
(48) 
On the other hand, for the eigenvectors simplify to
(49) 
a result which will be derived independently in Section 6. A direct diagrammatic computation of the above results is left to Appendix A.
The HFP has to be studied numerically. From the asymptotic behaviour of Eq. 39, it is, for ,
with an a priori unknown numerical constant. Imposing this dependency for large enough values of , together with the consistency condition at , selects the true FP out of the whole bunch of nonanalytical solutions.
Recall, however, that Eq. 39 is singular at the origin. This makes it difficult to numerically integrate the equation from large towards the origin, although it is quite easy to integrate it from the origin. We circumvent this difficulty by shooting to a fitting point [10]. That is, we integrate forward to some point from a tiny value of , satisfying the condition in Eq. 40. Which tiny value to take is immaterial as long as one checks it is less than the allowed errors. At we compare it with the backward integration from a large value where the asymptotic condition is imposed. In this way, the constants and from Eqs. 40, 4 are fixed. The precise value of , of course, does not matter, but we found it efficient to take values close to , where the potential is known to have a minimum in the exact solution at . Plots for some values of at are shown in Fig. 4, where they are compared with the corresponding large solution.
Linearizing Eq. 39 around the HFP, its eigenvectors are found to fulfil
(51)  
which can be solved using similar techniques. In general, solutions will grow exponentially,
(52) 
whereas for special values of they are much smoother,
This asymptotic behaviour, together with an arbitrary normalization of eigenvectors, leads to the quantization of . The results for and are summarized in Table 1, where and , , being the first (the relevant) eigenvalue and the second (the first irrelevant) one, respectively.
With nearly the same ease next irrelevant operators can also be studied. In Table 2 we present our results.
The case was first computed in Ref. [7], with results and , and afterwards in Ref. [11]. Our numbers are compatible with the former and coincide with the latter.
Eigenoperators are obtained as an additional bonus. As a matter of example, we plot in Fig. 5 the first eigenvector, normalized to , corresponding to different values of at .
The limit deserves a special remark, as the RG equation 33 is singular at . Nevertheless, one may compute its eigenvalues in an indirect way. Taking values of between and , one finds that they perfectly fit a straight line, with no sign at all of subleading corrections. Critical exponents reported in Table 1 are just their extrapolation to . In Fig. 6 we plot the first eigenvalue for together with its linear extrapolation.
5 Comparison with other equations
In this section we comment on other ERG equations and compare with known results.
Apart from the WegnerHoughton equation studied above, one of the oldest ERG equations appeared in Ref. [6]. It was further modified by Polchinski [2] to put it in the form we now use. The regulator is chosen in such a way that every propagator is multiplied by a function which is analytic everywhere in the finite complex plane, vanishes faster than any polynomial for large real values of its argument and it is standardly normalized to [12]. The evolution of the action takes the form
(54)  
where a dot means both summation over discrete variables and integration over continuous ones. As the nonlinearities are limited to quadratic terms, the projected equation presents a nice form quite amenable for numerical analysis,
(55) 
where a convenient rescaling of the potential and the field variable has been made. It has been previously studied for , using the variable , in Ref. [13].
The GFP is also , with its eigenoperators satisfying Eq. 45 after the rescaling . The solution corresponds to the Trivial Fixed Point,^{4}^{4}4To substantiate it, consider a potential consisting only of a mass term, . Note that at first order in (or, alternatively, for ) it coincides with the relevant mass operator from the GFP, , whereas for . Furthermore, it is easily proved that all linear deviations from are irrelevant. also known as High Temperature Fixed Point.
The analysis of the HFP is analogous to the WegnerHoughton one. There are singular solutions which behave
(56) 
whereas the true FP must fulfil the conditions
(57) 
with a nonvanishing constant. Again we can shoot to a fitting point to obtain the FP. Eigenvalues are obtained from the linearized form of Eq. 55 imposing the eigenvectors to be for . Some of them are shown in Tables 3, 4. We leave to Appendix B the study of its large limit.
Another quite used equation is based on the evolution of an infraredregulated oneparticleirreducible effective action [3],
(58) 
where is some regulating function and is the effective action minus the kinetic term . It presents some attractive features: in the limit where the IR cutoff vanishes (which corresponds to the continuum limit) it is directly observable, as it is the equation of state of the system. Moreover, it presents a natural expansion which can be directly mapped to the usual loop expansion of perturbation theory [14]. Various regulators have been used in connection with this equation. If a sharp momentum cutoff is chosen, its LPA coincides with that of WegnerHoughton’s. On the other hand, it has been extensively worked out using the special polynomial cutoff introduced in Ref. [15], which has the advantage of preserving some symmetries of the exact equation through the derivative expansion. This equation is much harder to numerically analyze than the above two ones. We do not discuss it further, but only quote the results (Tables 3, 4) and refer the reader to the original literature [15, 16].
To assess the systematic errors involved in such calculations would require to compute corrections to the LPA. Instead, we compare our results with some derived from other techniques.
There are accurate determinations for from a variety of methods such as expansion [17], Monte Carlo Renormalization Group [18, 19], Monte Carlo simulations [20, 21] and StrongCoupling series [22]. Unfortunately, estimates for are not so precise. A comparison of our results from WegnerHoughton and Polchinski equations with those from the IRcutoff effective action and other existing estimates are shown in Tables 3, 4.
Critical indices from WegnerHoughton equation are off for , except at , which is quite accurate (). In the interval , they are off by a few per cent, and for , the errors are less than one per mile. Other ERG perform slightly better, specially Polchinski’s equation in which critical indices are a few per cent off at most. This should not come as a surprise, since in spite of its conceptual clarity, the sharp momentum cutoff leads to bad locality properties of the renormalized actions, being then very much sensitive to truncations. In order for the projected ERG equation to deliver more precise outcomes, one should find RG blockings with the property that renormalized actions are as much local as possible, as it is achieved in the lattice regularization with blockings in real space [23].
6
Wegner and Houghton [1] already studied the limit of their equation, encountering both the GFP and the HFP. They also briefly mentioned the possibility of further FPs. Our aim is to discuss in certain detail those not previously studied and present a general framework to deal with this kind of large equations. An application of these methods for the Polchinski equation is given in Appendix B.
The projected RG equation becomes first order in the limit,
(59) 
which is easily solved for once we invert and consider ,
(60) 
The prime now stands for derivative with respect to . Its general solution is
(61) 
with an arbitrary function and
(62) 
which is analytic for . Although derived differently, this is the form of the ERG equation used in Refs. [24, 25] for .
Returning to Eq. 59, FPs verify
(63) 
which can be singular when the denominator of the rhs vanishes. But, if this is the case, then analyticity of the FP would imply , and, therefore, these wouldbe singular points reduce to a unique one,
(64) 
Furthermore, we can Taylor expand the FP around ,
(65) 
an expansion that will be used throughout.
Eigenoperators are
(66) 
with