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RENORMALIZATION GROUP PATTERNS AND C-THEOREM

arXiv:hep-th/9109041v1 24 Sep 1991CERN-TH-6201/91August 1991RENORMALIZATION GROUP PATTERNS AND C-THEOREMIN MORE THAN TWO DIMENSIONSAndrea CAPPELLI*, Jos´e Ignacio LATORRE**Theory Division, CERN, Geneva, SwitzerlandandXavier VILAS´IS-CARDONA***Departament d’Estructura i Constituents de la Mat`eriaUniversity of BarcelonaAv. Diagonal 647, 08028 Barcelona, SpainABSTRACTWe elaborate on a previous attempt to prove the irreversibility of the renormalizationgroup flow above two dimensions.

This involves the construction of a monotonically de-creasing c-function using a spectral representation. The missing step of the proof is a gooddefinition of this function at the fixed points.

We argue that for all kinds of perturbativeflows the c-function is well-defined and the c-theorem holds in any dimension. We provideexamples in multicritical and multicomponent scalar theories for dimension 2 < d < 4.

Wealso discuss the non-perturbative flows in the yet unsettled case of the O(N) sigma-modelfor 2 ≤d ≤4 and large N.CERN-TH-6201/91August 1991* On leave of absence from INFN, Sezione di Firenze, Italy. Bitnet: cappelli@cernvm.

** On leave of absence from Departament ECM, Barcelona. Bitnet: latorre@ebubecm1.

*** Bitnet: druida@ebubecm1.

1. IntroductionThere is a common belief which says that the renormalization group (RG) flows areirreversible.Intuitively, short-distance degrees of freedom are integrated out in orderto obtain a long-distance effective description of a physical system and are, therefore,irrecoverable.

Materializing this intuition into a theorem has proven to be quite a hardtask, so far unfinished for dimensions d > 2.Zamolodchikov produced a theorem in two dimensions based on the explicit construc-tion of a function which decreases monotonically along RG trajectories [1]. This is obtainedfrom the two-point function of the stress tensor and it is called “c-function” because it co-incides at fixed points with the central charge of the corresponding conformal field theory.In his original proof, Zamolodchikov assumed Poincar´e invariance, locality, renormaliz-ability and, notably, unitarity.

Actually, the very irreversibility of the RG flow, i.e. themonotonicity of the c-function, is due to unitarity.The c-theorem is by now a valuable tool for non-perturbative field theory in two di-mensions.

A striking example is the proof of spontaneous breaking of supersymmetry inthe flow from the tricritical to the critical Ising model [2]. A similar tool would be veryinteresting in higher dimensions, to investigate long-standing non-perturbative problemslike confinement, chiral symmetry breaking and the Higgs mechanism.

The explicit ingre-dients in Zamolodchikov’s proof are not specific to two dimensions, but so far problemshave been found to extend it to higher dimensions.The efforts to enlarge the validity of the theorem can be roughly divided in two classes.In the first one, some c-numbers characteristic of four-dimensional critical theories arestudied, mainly those parametrizing the gravitational trace anomaly [3][4]*. The ingredientof unitarity is not used, so that the monotonicity of these quantities along the RG flowcannot be proven.

An effort to use positivity along these lines was done in ref. [6].In the second approach, the constraint of unitarity is explicitly built in.

In ref. [7],a refined version of the two-dimensional theorem, originally due to Friedan, was obtainedusing the Lehmann spectral representation for the correlator of two stress tensors.

Thec-function is issued from data of the Hilbert space of the theory, so that renormalization*See also the approach of ref. [5].1

problems are bypassed. This approach can be extended to higher dimensions, except forone point, the meaning of the c-function at the fixed points is unclear.

Thus, we cannotclaim that the theorem is conquered yet.Known counterexamples of RG flows with complex behaviour, like limit cycles andchaos, have been discussed in ref. [8].

They violate some of the assumptions of the theo-rem, namely Poincar´e invariance (spin glasses, hierarchical models) or unitarity (polymers,models with replica trick*).In this paper, we intend to present the state of the art of the spectral approach tothe c-theorem in more than two dimensions. In sect.

2, we recall its main points. Westress that the c-function is actually well-defined for perturbative flows, where it changesinfinitesimally between the ultraviolet (UV) and the infrared (IR) fixed points, ∆c =cUV −cIR ≪1.Therefore the RG flow is irreversible in any perturbative expansion.

This fact is ofgreat practical importance, due to the major role of perturbative expansions in higherdimensions, even if it is far from the non-perturbative goals we mentioned before.In sect. 3, we substantiate this claim by studying the c-function in arbitrary dimensionusing Wilson’s epsilon expansion in scalar theories as a bench-mark [11].

In this framework,we compute the RG patterns of the λϕ4 theory [12], of the multicritical λϕ2r [13] and ofthe multicomponent gijklϕiϕjϕkϕl theories [14]. A typical variation of the c-function is∆c ∼ǫ3 ≪1 in d = 4 −ǫ.

As expected, the c-theorem works perfectly. It supports thebelief that the multicritical pattern of the minimal conformal models in two dimensions[15] extends smoothly to higher dimensions.

Moreover, the c-functions correctly add upfor chains of flows in the multicomponent theory.In sect. 4, we try to understand more difficult flows, which are “non-perturbative”in the following sense.We investigate the O(N) sigma-model in the large N limit for2 ≤d ≤4 [14][16].

In this case, the expansion is perturbative in 1/N (the coupling ofthe theory given by the connected 4-point function), but it is actually non-perturbative forwhat concerns the c-function. Indeed, the flow in the massive phase gives ∆c ∼N ≫1 andis definitely different from that of free massive bosonic particles.

Our results agree with theknown RG pattern of the model for d = 2 and d = 4, but disagree with it for 2 < d < 4. Wesketch some possible explanations of this fact which deserve further investigation.

In the* They are not unitarity in two dimensions [9] and likely not in any dimension [10].2

conclusions, we comment on four-dimensional physical theories like QCD. The Appendixis devoted to setting the technique of conformal perturbation expansion, valid in anydimension.3

2. Steps Towards the Proof of the c-theoremLet us first recall the main features of the c-theorem in two dimensions [7].

We definethe stress tensor Tµν(x) as the response of the action to small fluctuations of the spacetimemetric. Then we consider the spectral representation of the Euclidean correlator< Tµν(x)Tρσ(0) >= π3Z ∞0dµ c(µ)Zd2p(2π)2 eipx (gµνp2 −pµpν)(gρσp2 −pρpσ)p2 + µ2.

(2.1)In this expression, the spectral density c(µ) is a scalar function by Poincar´e invarianceand it is positive definite by unitarity, c(µ) ≥0. In the short-distance limit, the theoryis described by a UV conformal invariant theory, where eq.

(2.1) reduces to the singlecomponent ⟨Tzz(z)Tzz(0)⟩CF T = cUV /2z4, parametrized by the Virasoro central chargecUV . The same happens in the long-distance (IR) limit.

Working out these two limits ineq. (2.1), it follows thatcUV =Z ∞0dµ c(µ)≥cIR = limǫ→0Z ǫ0dµ c(µ).

(2.2)where the inequality stems from unitarity. Thus c(µ)dµ is a dimensionless measure ofdegrees of freedom off-criticality.The proof is completed by considering the RG flow of the spectral density.Theprevious equation implies thatc(µ) = cIRδ(µ) + c1(µ, Λ),∆c ≡cUV −cIR =Z ∞0dµc1(µ).

(2.3)The evolution of c(µ)dµ under the RG flow is governed by the flow of the physical massscale Λ of the theory. As we quit the UV fixed point Λ = 0, the delta term in eq.

(2.3) staysconstant, because it measures the states of the Hilbert space which remains massless forΛ ̸= 0.Instead, the states acquiring mass contribute to the smooth density c1(µ, Λ),roughly bell-shaped and peaked at µ ∼Λ. In the IR limit Λ →∞, c1 is pushed away andcontributes no more to observables.

Thus its integral ∆c gives a quantitative measure ofthe loss of degrees of freedom along the RG flow.In a typical application of the theorem [17][18], cUV and cIR are first determinedfrom ⟨TT⟩CF T in the corresponding conformal field theories. Next, the correlator ⟨ΘΘ⟩4

of the interpolating off-critical theory is considered.By eq. (2.3), the variation ∆c isindependently measured by the sum rule [19]∆c = 34πZ|x|>ǫd2x x2 < Θ(x)Θ(0) >.

(2.4)Thus the data of the critical and off-critical theories are compared and the RG pattern isverified.Let us stress some virtues of this proof. The c-theorem expresses a geometrical prop-erty of the space of theories, parametrized by some coupling coordinates {gi}.

But noticethat our proof was given in a coordinate-free language. We did not need to talk of bareLagrangians and renormalization conditions on fields and couplings, nor care about theirassociated reparametrization invariance.

Actually, the Lehmann representation gives usthe spectral density expressed in terms of matrix elements of the stress tensor, belongingto the Hilbert space of the theory. Moreover, these matrix elements are necessarily non-vanishing, because any matter couples to the stress tensor.

Thus, this is a good measureof degrees of freedom in the theory. Any other current would not couple to all of them(depending on their charges) and would not detect their flow.

Finally, the density c(µ)summarizes all the unitarity conditions on the two-point function, because any positivequantity can be obtained by integrating it against positive smearing functions.We believe that these are rather unique features, which should necessarily appear inany generalization of the theorem to higher dimensions.For later reference, let us also express the theorem in terms of the coordinates {gi}and beta-functions Λ ddΛgi = βi(g) [1]. One has to expand the trace of the stress tensorΘ(x) in the basis of renormalized fields Φi at the UV fixed point,Θ(x) = 2πβiΦi.

(2.5)Next, the c-function c = c(g) and the Zamolodchikov metric Gij(g) ∝⟨Φi(x)Φj(0)⟩||x|=1are introduced by smearing c(µ) against appropriate positive functions, which contain afixed scale [7]. It followsddtc ≡−βi ∂∂gi c(g) = −βiβjGij(g) ≤0,(2.6)Thus, the previous flow of c(µ) driven by a physical mass Λ is now traded for the changeof c(g) along the flow curve of affine parameter t.5

The spectral form of the c-theorem can be generalized to higher dimensions, where itshows a new feature. There are two spectral densities, c(0)(µ), related to spin-zero inter-mediate states, and c(2)(µ) for spin-two ones.

Both densities are, in principle, candidatesfor a c-theorem, since they display some of the properties of the unique density in two di-mensions. c(2)(µ) determines the correlator of two stress tensors at the conformal invariantpoints [20], and it could define an analog of the central charge.

However, by inspectionthis does not correspond to a monotonically decreasing function along RG trajectories andit must be discarded [7].On the other hand, c(0)(µ) is related to changes of scale off-criticality, because< Θ(x)Θ(0) >= AZ ∞0dµ c(0)(µ)Zddp(2π)d eipxp4p2 + µ2(2.7)A =VΓ(d)(d + 1)2d−1,V ≡V ol(Sd−1) = 2πd2Γ( d2). (2.8)Therefore, we can generalize the sum rule (2.4).

Limiting ourselves to theories with van-ishing Θ at fixed points, i.e. conformally invariant fixed points, a dimensional analysis ledus to define∆c =Z ∞ǫdµc(0)(µ, Λ)µd−2= d + 1V dZ|x|>ǫddx xd⟨Θ(x)Θ(0)⟩.

(2.9)The normalization of c(0)(µ) in eq. (2.7) assigns c = 0 to the trivial theory and c = 1to the free bosonic theory in any dimension, the latter being computed by the sum rule(2.9) in the free massive phase.

By smearing c(0)(µ) we can also generalize eq. (2.6) , andget a new c-function c(g) in d dimensions, which is monotonically decreasing along the RGflow and stationary at fixed points, in close analogy with the two-dimensional case [7].However, a point is missing, the characterization of c(g) at fixed points, the would-becentral charge, or c-charge.

A closer inspection shows that this is defined as a limit fromoff-criticality of the spin-zero density,limΛ→0c(0)(µ, Λ)µd−2dµ = c δ(µ)dµ. (2.10)In general, this limit may depend on the path approaching the fixed point, implying unac-ceptable non-universal effects on the c-charge.

If c is monotonic but multivalued, we canstill have closed cycles violating the theorem. On the other hand, the limit is universal if6

the c-charge is related to an observable of the fixed-point theory - we could not prove thisfact so far. Note that ⟨ΘΘ⟩CF T = 0 for d > 2, and is thus independent of c, owing to thefactor µd−2 in eq.

(2.10).As we pointed out above, in two dimensions c has indeed an independent characteri-zation at the fixed point from ⟨TT⟩. The unique density is both responsible for controllingchanges of scale (genuine spin 0 in d ≥2) and for the coefficient of the short distancesingularity of Tzz(x)Tzz(0), (genuine spin 2 in d ≥2).

At the conformal point, the traceof the stress tensor still sees the central charge through contact terms. This is actuallyenforced by Lorentz invariance since Tzz and Θ are just different components of the sameLorentz structure [7].

Above two dimensions, the roles of controlling scale transformationsand the short-distance singularity are related to two different spin structures. These twostructures do not talk to each other since Lorentz invariance acts separately on each one.This is the reason why we could not find an independent characterization of our c-charge.The previous problem can be made quantitative by checking of ∆c in a chain of RGflows.

For three fixed points (fig. 1), this reads(∆c)1→2 + (∆c)2→3 = (∆c)1→3.

(2.11)Additivity of the c-charge also amounts to integrability of the system of beta-functions.Equation (2.11) holds if c at the theory 2 has the same value independently of whether weapproach this theory from 1 or from 3. This is not obvious, because (∆c)i→j is computedin the i-th theory, so that we are comparing calculations in two a priori different off-criticaltheories.Nevertheless, suppose that the three fixed points lie in a region of the space of theoriesparametrized by smooth coordinates.

Indeed, this is the case of perturbative calculations(by definition). ∆c is a polynomial in the renormalized couplings, thus the limit from off-criticality exists, or equivalently, trajectories can be deformed at will to prove additivity.Therefore, we can say that the c-theorem is proven for all kinds of perturbation expansions,namely ∆c ≪1 and polynomial in the couplings.Two examples of this kind will bediscussed in the next section.The drawback to the above comments is that the space of theories has singularitiesand is probably not a manifold.

For example, in fig. 2 we imagine comparing ∆c for twonon-perturbative flows into the massive phase (∆c ∼1).

The trivial theory (c = 0) canappear in several points of the coupling space, thus the two paths can be topologicallyinequivalent. The additivity property, (∆c)1→2 = (∆c)1→3, is not at all trivial in thiscase.

This kind of situation will be discussed in sect. 4.7

3. Perturbative Flows in Scalar Theories for 2 < d < 4In this section, we provide examples of perturbative RG patterns.

Two or more fixedpoints appear in a small region of coupling space, and the d-dimensional c-function variesinfinitesimally while flowing among them. The validity of the c-theorem is verified by usingvarious versions of the ǫ-expansion.First we study the Landau-Ginsburg-Wilson RG pattern of the r-th multicriticalpoints in the λϕ2r theory.

The comparison of the c-charges of the (r) and (r −1) theo-ries suggests that the known multicritical RG flows in two dimensions extend smoothly tohigher dimensions. Next, the multicomponent λϕ4 theory is considered, and eq.

(2.11) foradditivity of the c-charge is verified.3-1. MULTICRITICAL POINTS AND LANDAU-GINSBURG THEORYThe Landau-Ginsburg actionSLG =Zddx 12(∂µϕ)2 −rXk=1λk,02k!

ϕ2k!. (3.1)describes the qualitative features of generic r-multicritical points with parity symmetry(only), which appear for 2 ≤d < 4 [11].

At the multicritical point {λk,0} = {0, 0, ..., λr,0},r minima of the potential merge, which correspond to r coexisting phases of the theory.The flow to lower r′-critical points, r′ < r, is described by switching on the relevantperturbation λr′,0ϕ2r′. At the end of the flow, the higher powers ϕ2k, r′ < k ≤r becomeirrelevant fields and can be neglected in the action.In two dimensions, the minimal conformal theories [21] with central chargec(r) = 1 −6r(r−1) < 1 are the exact renormalization of the above Landau-Ginsburg actions[15].

The renormalized fields ϕ2k, k < r, are the primary conformal fields which appearin the first two diagonals of the Kac table. Off-criticality, this picture has been confirmedalong the flow between the (r) and the (r −1) models, driven by the least relevant fieldϕ2r−2, for r ≫1 [1][9]*.

The dimension of this field is 2 −ε, ε ∼1/r ≪1, thus it is thetypical perturbation situation encountered in the ε-expansion. The IR fixed point, (r −1),* As recalled in example 1 of the Appendix.8

appears infinitesimally close to the UV one, (r), in coupling space. The invariant definitionof distance is given by the Zamolodchikov metric, eq.

(2.6), or, equivalently, by the changeof the central charge, ∆c = cr −cr−1 ∼O(1/r3).The c-theorem is a nice complement to the Landau-Ginsburg picture. Higher multi-critical points have higher values of c, thus flowing downhill corresponds to going to lowermulticritical points.

Vafa has further developed this picture [22]. The c-function can beconsidered as the height function in the space of theories Q, so that Morse theory can beapplied and the Poincar´e polynomial gives some information on the holonomy of Q. Inshort, the qualitative Landau-Ginsburg description together with the c-theorem give somegrasp of the topology of this space of theories.Following Wilson, the Landau-Ginsburg picture holds for all dimensions up to theupper critical dimension dc2 ≤d ≤dc (r) ≡2 +2r −1,(3.2)the dimension for which ϕ2r becomes marginal, i.e.

the r-th multicritical point mergeswith the Gaussian one. A natural question to ask is whether our candidate for a c-theoremextends above two dimensions as well.

The higher multicritical points should continue tohave larger values of c. In such a case, the space of multicritical points and their flowswould have an analytical continuation in dimensions.3-2. THE C-CHARGE OF THE λϕ2r THEORYAbove two dimensions, we compute our candidate c-charge cr of the r-th multicriticalpoint by applying the sum rule eq.

(2.9) to the flow from the Gaussian theory, cr = 1−∆c(fig. 3).

We use the ε-expansion at dimensiond = dc(r) −εrr −1,0 < εr ≪1,(3.3)where εr = dim(λr,0) is the small parameter.Let us start with the familiar example of λϕ4 theory, for d = 4 −ε, and derive thefirst order term of the ε-expansion. The action isS =Zddx12 (∂µϕ0)2 −λ04!

ϕ40,(3.4)9

where λ0 and ϕ0 are the bare coupling constant and field respectively. At d = 4 −ε thedimension of the coupling constant is dim(λ0) = ε, i.e.

the field ϕ40 is slightly relevant andit produces a flow from the Gaussian fixed point λ0 = g = 0 (c = 1, by definition) to theWilson fixed point at g = g∗∼ε, where g is the renormalized coupling constant.The trace of the energy momentum tensor isΘ = −ελ04! V ϕ4.

(3.5)From the computation of the two leading Feynman diagrams, we can extractIm ⟨Θ(p)Θ(−p)⟩|p2=−µ2 = ε2 (λ0S)2V 2Sπ3 · 12816µ4−3ε + λ0Sε µ4−4ε. (3.6)Upon insertion of this imaginary part in eq.

(2.9) and integration, one obtainsc(λ0κ−ε) = 1 −aλ0Sκ−ε2 ε + (λ0Sκ−ε)b,(3.7)a = 548V S,b = 4,S ≡V(2π)d =2(4π)d2 Γ( d2). (3.8)This flowing c-function depends on the bare coupling and the IR cut-offκ, which appearsin the intermediate steps of calculations of massless perturbations, as usual [9].

Since chas no anomalous dimension, its renormalization is simply achieved by replacing λ0κ−εwith the renormalized coupling g. This is a change of coordinates in coupling space whichremoves the unphysical singularity in the Zamolodchikov metric G(λ0) at the IR fixedpoint [23]. Because of eq.

(2.6) , all the information needed to find such a transformationis contained in the c-function itself, and we need not carry out the renormalization of fields.This produces economical formulae for the flow. For one-coupling flows, eq.

(2.6) gives∂∂λ0c(λ0) = G(λ0)β(λ0),(3.9)where the beta-function in terms of the bare coupling constant is β(λ0) = −ελ0, by eq.(3.5). The relation between bare to renormalized couplings can, then, be cast into anelegant geometrical condition – the invariant distance in coupling space remains the samewhatever coordinate system is chosen,ds2 = G(λ0)dλ20 = G(g)dg2.

(3.10)10

The renormalized coupling is defined by requiring that G(g) = 2a, where the specific valuechosen for the constant is of later convenience. Then g(λ0) is obtained by integrating eq.(3.10).

The final form for the c-function readsc(g) = 1 −ag2ε + b4g= 1 −53 · 64g2(ε + g). (3.11)Its derivative is proportional to the beta-function,β (g) = −εg −38bg2 = −εg −32g2,(3.12)which agrees with standard derivations (see e.g.[16]).

In agreement with the general dis-cussion of eq. (2.6)[7], we have obtained a monotonic decreasing c-function, which isstationary at the fixed points g = 0 and g = g∗= −23ε.

These results are an explicitillustration of Zamolodchikov’s ideas in more than two dimensions [1].The value of the c-function at the Wilson (2)-critical point isc2 = c(g∗) = 1 −516 · 81ε3, d = 4 −ε. (3.13)We can generalize this analysis to the flow between the Gaussian and the (r)-criticalpoint.

The action isS =Zddx12 (∂µϕ)2 −λr,02r! ϕ2r,(3.14)where the dimension now is slightly below dc(r), eqs.(3.2),(3.3).

The trace of the stresstensor readsΘ = −εrλr,02r! V ϕ2r.

(3.15)Again the computation of the two-point correlator to leading perturbative order involvestwo Feynman diagrams which have the same singularity structure as the r = 2 case,implying again stationarity of c(g). The only numerical changes arear = 2d−3 (d + 1) VS [r] [1]22r−3[1]3 [r]2[2r −1] 2r!br = 43S [r] [1]2r−2 [1]2 2r!(r!

)3(3.16)11

where the notation [α] = Γαr−1has been introduced*. Again we find a non-trivial fixedpoint at gr = g∗r ≡−8εr/3br, with c-chargecr = c(g∗r) = 1 −3r −13rr!22r!εr3,d = 2r −εrr −1 ,0 < εr ≪1.

(3.17)Notice that the ǫr-expansion to first order is not good enough for reproducingthe known values of the charge in two dimensions, especially for large r.We gotcr ∼1 −O(2−6r) instead of cr ∼1 −O(1/r2). Actually, the first few terms of this asymp-totic expansion give an accurate result for 0 ≤ǫr < O(A−r+1), where A is a positiveconstant, so that we cannot use it for two dimensions (ǫr = 2)**.3-3.

THE HEIGHT OF THE (r)-THEORY VERSUS THE (r −1) ONEWe have now the elements to compare the c-number or “height” of two neighbour mul-ticritical points. From the Landau-Ginsburg picture and the two-dimensional c-theoremwe expect a flow from the (r)-theory to the (r −1) one (see fig.

3). If the c-theorem holdsin any dimension we expect cr > cr−1.

At dimensiond = dc(r) −εrr −1 = dc(r −1) −1r −22r −1 + r −2r −1εr0 < εr ≪1. (3.18)both critical points are present (fig.

3) and we findcr −cr−1 =r!22r!3 "1 −43r 1 −12r31 −1r8r −1 + 4r −2r −1εr3−1 −13rε3r#> 0. (3.19)This bound is satisfied for all εr within the range of validity of the ε-expansion at first order.As shown by fig.

3, both multicritical points exist for εr small at will, and εr−1 ∼O(1/r2)at least, the latter going outside the range of accuracy for large r. Therefore our result ineq. (3.19) is probably numerically good for small r = 2, 3, but only heuristic for large r.Nevertheless, the comparison of the two charges at the same perturbative order is certainly*The corresponding beta-function agrees with ref.

[24] by rescaling g →g/S. **This bound can be obtained from the fact that the k-th term in the expansion growslike (k!

)r−1 [13].12

better than the absolute value of each one. Eq.

(3.19) shows that both c-charges haveexponentially small corrections, but differ in an algebraic factor.Moreover, cr > cr−1 necessarely holds for εr →0. Indeed, the nucleation of multicrit-ical points ordered in dimension (see fig.

3), the fact that c = 1 for the free theory in anydimension and its monotonicity property imply cr−1 < 1 and cr ∼1 in this limit.Let us finally quote Felder’s investigation of this RG pattern in more than two di-mensions [25]. Working in a different perturbative approach, he was also able to build amonotonic function along the RG flows from his system of beta-functions.In conclusion, these are good indications that the topology of the space of multicriticalscalar theories extends smoothly above two dimensions.3-4.

THE MULTICOMPONENT ϕ4 THEORYAnother well-known RG flow pattern is provided by the multicomponent ϕ4 model in4 −ε dimensions [14],S =Zddx12(∂µϕi)2 −14!gijklϕiϕjϕkϕl,(3.20)where the sum over N components ϕi, i = 1, ..., N is implicit. In this theory, Wallace andZia [12] first showed that the RG trajectories are gradient flows to three-loop order in theε-expansion, thus ensuring that the flow is driven to the IR fixed points – a non-trivialfact when we look at the multi-loop β-function!These results can easily be framed into the c-theorem philosophy.

A simple modifica-tion of our previous Feynman rules leads toc (gijkl) = 1 −564 · 3 (εgijklgijkl + gijklgijrsgklrs)(3.21)andβijkl(gmnrs) = 12a∂∂gijklc = −εgijkl −32gijrsgklrs,(3.22)which necessarily agree with the Wallace and Zia result to one loop. To higher orders, ourc-function would correspond to a definite choice of the free parameters in their gradientfunction Φ, c ∼1 −const.Φ.13

The RG flows given by eq. (3.22) correspond to trajectories leaving the Gaussian pointin different directions and reaching IR fixed points.

One of them is stable and, therefore,displays a minimum of c. The other ones are unstable, the stability changing with N. Letus recall the results of ref. [14].

A two-dimensional subspace of the coupling space is givenby the O(N)-symmetric perturbationPi ϕ2i2 leading to the O(N)-symmetric Wilsonpoint, and the hypercubic symmetric one Pi ϕ4i , leading to N decoupled Ising models.This pattern describes the breaking of O(N) symmetry to the hypercubic one in latticeferromagnets. After computing the location of the O(N)- symmetric and hypercubic fixedpoints, it is convenient to introduce rescaled variables x and y,gijklϕiϕjϕkϕl = −x6εN + 8 Xiϕ2i!2−y 2ε3Xiϕ4i,(3.23)such that the O(N) symmetric point is at (x, y) = (1, 0) and the hypercubic symmetric isat (x, y) = (0, 1) (fig.

4).The c-function in this parametrization isc (x, y) = 1 −516 · 3ε3"N (N + 2)(N + 8)23x2 −2x3+2NN + 8xy (1 −x −y) + N273y2 −2y3#. (3.24)Inspection of the extrema of c ( which correspond to βx = 0, βy = 0) shows that there is afourth fixed point at (x∗, y∗) = N+83N , N−4N.

The stability of these points is also deducedfrom c. For any value of N, there are two unstable points besides the Gaussian one and astable fixed point. For N < 4, the O(N) symmetric point is stable; for N > 4, (x∗, y∗) isstable.

An example of the shape of c(x, y) for N = 8 is shown in the level plot fig. 5.3-5.

ADDITIVITY OF THE C-CHARGELet us now verify the additivity property of the d-dimensional charge for compositionsof flows among three fixed points, eq. (2.11),(∆c)1→2 + (∆c)2→3 = (∆c)1→3.

(3.25)In the previous RG pattern, let us consider two possible chains:14

i) For N > 4, we can reach the stable IR fixed point (x∗, y∗) in two different ways,fig. 4a,(0, 0)−→(1, 0)−→(x∗, y∗)or(0, 0)−→(x∗, y∗)(3.26)corresponding to the l.h.s.

and r.h.s. of eq.

(3.25) .ii) For N > 10, we can also compare, fig. 4b,(0, 0)−→(1, 0)−→(0, 1)versus(0, 0)−→(0, 1)(3.27)Remember that each (∆c)i→j in eq.

(3.25) has to be computed using the sum ruleeq. (2.9) applied to the corresponding off-critical theory, having the i-th fixed point asUV limit.

Within the coordinates (x, y) considered so far, the UV fixed point was the freeGaussian theory (0, 0). Thus, without extra work we can particularize eq.

(3.24) for theflowsc(0, 0) −c(1, 0) = 548N (N + 2)(N + 8)2 ε3c(0, 0) −c(0, 1) = 548N27ε3c(0, 0) −c(x∗, y∗) = 548(N + 2)(N −1)27Nε3. (3.28)The other flows appearing in eqs.

(3.26),(3.27) should be described in another coor-dinate patch, having (1, 0) as UV fixed point. A more advanced perturbative technique isneeded, because the starting theory is not free.

Nevertheless, we can make use of the factthat the (1, 0) fixed point is conformal invariant (the trace of the stress tensor vanishes)and use the “conformal perturbation theory” (see the Appendix and ref.[5]). This tech-nique generalizes the one used for flowing offconformal theories in two dimensions [1][17].Conformal invariance in higher dimensions fixes the form of 2- and 3-point functions, upto some coefficients, the conformal dimensions of fields and the structure constants.

Thefirst order perturbative expansion only requires these ingredients, thus it can be given forany flow in any dimension.Let us first build a suitable basis of operators around the (0, 0) fixed point. We choosethe orthonormal basisφ(0,0)s≡A Xiϕ2i!2,φ(0,0)⊥≡BXiϕ4i −3N + 2 Xiϕ2i!2,(3.29)15

A =s3N(N + 2),B =sN + 2N(N −1),which we call “symmetric” and “orthogonal” respectively. It is easy to compute the three-point functions made out of these two operators, and obtain the following structure con-stantsC(0,0)sss= A(N + 8)9C,C(0,0)ss⊥= 0 ,C(0,0)s⊥⊥= A(N −2)N + 2C,C(0,0)⊥⊥⊥= B(N −2)N + 2C,(3.30)where C is the N-independent part of the structure constant.

Note that, to the order in ǫwe are working, the structure constants remain the same at the (1,0) fixed point, i.e.C(0,0)ijk= C(1,0)ijki, j, k = s, ⊥(3.31)and, therefore, we can omit these superindices from now on.Using the orthonormal basis we have just presented, we use the φs field as a perturba-tion offthe Gaussian fixed point. This perturbation leads us to the (1,0) point.

In general,conformal perturbations are subject to RG mixing of operators. Nevertheless, the flowbetween (0,0) and (1,0) is free of this intricacy because ⟨φsφ⊥⟩(0,0) = 0 and Cs⊥s = 0, i.e.φs and φ⊥stay orthogonal all the way along the RG flow*.It is easy to compute the scaling dimensions of the symmetric and the orthogonaloperators which get renormalized,∆(1,0)s= ∆(0,0)s+ 2d −∆(0,0)s= d + ǫ≡d −ys∆(1,0)⊥= ∆(0,0)⊥+ 2d −∆(0,0)s C⊥⊥sCsss= d −N −4N + 8ε ≡d −y⊥.

(3.32)The above equations tell us that the symmetric operator becomes irrelevant at the (1,0)fixed point whereas the orthogonal one is relevant for N > 4, in agreement with thediscussion of the phase diagram.Having established the new basis of fields at (1,0), we can now compute the flows offthis point driven by φ(1,0)⊥for N > 4. In this case, though ⟨φsφ⊥⟩(1,0) = 0, we note that*Actually, in this case the general system of beta-functions given in the Appendixsimplifies to a single equation.16

Cp⊥⊥̸= 0, which implies that φ(1,0)sand φ(1,0)⊥do mix along the new flow. Thus, we areforced to consider a general two-parameter deformation of the (1,0) fixed point, namelyg⊥φ(1,0)⊥+ gsφ(1,0)s.The system of beta-functions one obtains in conformal perturbation theory is (see theAppendix)β⊥= −y⊥g⊥−C⊥⊥⊥g2⊥+ 2C⊥⊥sg⊥gsβs = −ysgs −Cs⊥⊥g2⊥+ Csssg2s.

(3.33)The above set of equations correspond to the new c-function for flows offthe (1, 0) fixedpoint, call it ˜c,˜c(g⊥, gs) = c(1, 0) −y⊥g2⊥2 −ysg2s2 −C⊥⊥⊥g3⊥3 + C⊥⊥sg2⊥gs + Csssg3s3(3.34)It turns out that the system of beta-functions has two solutions,(g∗⊥, g∗s) =−ǫCN−4BN , A(N−4)2(N+8)−ǫC1B, 2AN(N−1)(N+8)(3.35)Finally, we can substitute back the solutions of the βs, β⊥= 0 system in the ˜c-functionand get, for the two solutions,∆˜c =( 548(N+2)(N−4)327N(N+8) ǫ3= c(1, 0) −c(x∗, y∗)548N(N−1)(N−10)(N+8)2ǫ3= c(1, 0) −c(0, 1)(3.36)The comparison of this result to eq. (3.24) shows that the two solutions correspond tothe two flows in eqs.

(3.26),(3.27) and, therefore, the additivity property of our c-charge,eq. (3.25), does hold.We are now confident of the limiting procedure eq.

(2.10) which defines the c-chargeat fixed points from the spectral function c(µ, Λ) away from criticality (Λ ̸= 0).In aperturbative domain, it produces consistent results for inequivalent coupling coordinates,so that the c-charge is indeed a universal quantity attached to each fixed point.These results, though expected, have tought us that, in general, we have to allow forirrelevant fields in the expansion of Θ in eq. (2.5) , if they mix with the relevant onedriving the flow.17

4. The O (N) sigma-model in the Large N ExpansionIn the previous section we considered RG flows between infinitesimally close fixedpoints, ∆c ≪1.

The large N expansion allows, instead, to describe RG flows which runover a large distance in coupling space, e.g. ∆c ∼N, thus non-perturbative with respectto the coupling.

Actually, the saddle point method amounts to the resummation of aninfinite set of diagrams of conventional perturbation theory, and it leads to beta and cfunctions which are non-analytic in the coupling (the mass for the sigma-model). Furthercorrections in the1N expansion are similarly non-analytic for what concerns the c-theoremand the sum rule.

In the following, we study the flow in the symmetric phase of the O (N)sigma-model, for large N and 2 ≤d ≤4.The O (N) symmetric non-linear sigma-model is defined by an action which containsN fields ϕi and a Lagrange multiplier α0Z =ZDα0 Dϕie−SS =Zddx12∂µϕi∂µϕi + α0ϕiϕi −Ng20,(4.1)where the fields and the coupling are conveniently rescaled for the large N expansion. Theintegration over ϕi produces the effective actionSeff = N2 ln det Λ−∂2 + α0(x)−N2g20Zddxα0 (x) .

(4.2)which, for large N, can be estimated using the saddle point approximation. The saddlepoint equation is1g20=Z Λ ddp(2π)d1p2 + m2 ,m2 = ⟨α0⟩s.p.,(4.3)where Λ is the cut-off, and m2 is the translational invariant value of the field α0 at thesaddle point.

The 1/N expansion is obtained by setting α0(x) = m2 + α(x)√N and expandingSeff around the saddle point. The Feynman diagrams and the basic properties of thetheory are discussed in ref.

[26].Investigations of the c-theorem for this theory wereinitiated in ref. [7].For 2 ≤d ≤4, the saddle point defines the physical mass m for ϕ, in terms of thebare coupling g20, while the higher order corrections to the saddle point are weighted with18

the coupling1N . From the heat kernel regularization of the determinant in eq.

(4.2) weobtain1g20=2(4π)d2"Λd−2d −2 −md−2d −2 Γ2 −d2−Λd−4m2d −4+∞Xk=21k!Λd−2k−2 −m2kd −2k −2#. (4.4)In particular, for d = 2,1g20= 14π lnΛ2 + m2m2.

(4.5)The critical point g20,cr is obtained for m = 0g20,cr = 0(d = 2) ,1g20,cr=2(4π)ddΛd−2d −2(d > 2) ,(4.6)and the massive phase corresponds to g20 > g20,cr. On the other hand, for g20 < g20,cr, thereis no solution to the O (N)-symmetric saddle point equation.

There are non-symmetricsaddle points, obtained by integration only N −1 fields in eq. (4.1), giving ⟨ϕ⟩̸= 0 [26].

Inshort, the phase diagram of the sigma-model for 2 ≤d ≤4 contains the O (N)-symmetricphase g20 > g20,cr, with ⟨α0⟩= m2, ⟨ϕ⟩= 0, and (for d > 2), the spontaneously broken oneg20 < g20,cr, with ⟨α0⟩= 0 and ⟨ϕ⟩̸= 0.4-1. THE STRESS TENSOR AND THE SPECTRAL DENSITYLet us consider the RG flow in the symmetric phase which leaves the critical point andreaches the trivial fixed point m = ∞.

We want to compute the associated sum rule eq. (2.9), to leading order in1N .

We start by finding the expression of the trace of the stresstensor for this perturbation in terms of the α field. This is obtained by putting the theoryon a curved space-time background and taking a Weyl variation with respect to the metricof both the effective action and the saddle-point equation [7].

Furthermore, for 2 < d < 4we let the cut-offgo to infinity after one subtraction of the coupling. We obtain*,⟨Θ(x)⟩= β(m) ⟨α(x)⟩s.p.

= 0⟨Θ(x)Θ(0)⟩= (β(m))2 ⟨α(x)α(0)⟩s.p., x2 ̸= 0,(4.7)* Note that Θ vanishes classically for d = 2, but not at the quantum level (saddle point).We renormalize the theory by going to d = 2 + ε, and find a non-vanishing beta-functionto leading order in 1/N.19

whereβ(m) = md−2 √N Γ2 −d22d−1Γ d2. (4.8)The leading contribution to the α propagator is obtained by expanding Seff to secondorder in α⟨α (p) α (−p)⟩s.p.

= −2B (p),(4.9)where B (p) is the well-known bubble diagramB (p) =Zddq(2π)d1(q2 + m2)(p −q)2 + m2 =Z ∞4m2dµ2πImBp2 = −µ2p2 + µ2= md−4Γ 4−d22dπd2F1, 4 −d2; 32; −p24m2ImBµ2=1Γ d−122dπd−321µµ24 −m2 d−32θµ2 −4m2(4.10)where θ (x) is the step function and F the hypergeometric function.4-2. THE SUM RULE IN TWO DIMENSIONSIn two dimensions, the critical point is g0 = 0, and there is only the symmetricphase, in agreement with Coleman’s theorem on the absence of spontaneous symmetrybreaking.

Around the critical point, the theory contains N −1 weakly interacting bosons,asymptotically free [16], thus we can assign the value cUV = N −1. The symmetric phaseis massive and the IR Hilbert space contains N interacting massive scalar particles withpurely elastic scattering, thus cIR = 0.

Therefore, the c-theorem sum rule is expected togive ∆c = N for the flow in the symmetric phase, to leading order in 1/N.In order to compute it, we need the spectral function c (µ) ∝Im⟨ΘΘ⟩, given byeqs. (4.7),(4.9).The imaginary part of the α-propagator is Im −2B= 2Im (B) / |B|2,where B (p) can be expressed in terms of logarithms for d = 2.

Putting all together, weobtainc (µ) =6π2µ3 Im⟨Θ (p) Θ (−p)⟩|p2+µ2=0= 6Nµs1 −4m2µ2π2 + log21 +q1 −4m2µ21 −q1 −4m2µ2−1θµ2 −4m2. (4.11)20

The sum rule is∆c =Z ∞0dµc (µ) = 3NZ 1−1dz1 −z2z2π2 + ln2 1+z1−z. (4.12)After a change of variable, the integral can be computed in the complex plane, giving theexpected result∆c = N,(d = 2,N →∞).

(4.13)This is a rather remarkable check of Zamolodchikov’s theorem, owing to the non-perturbative character of this flow.In this calculation we had no problems with IR singularities, which instead can appearin the perturbation expansion* in g20 . Actually, the large N expansion is better because itcorrectly reproduces the mass of the theory by resumming infinite perturbative diagrams.Let us however stress that IR singularities cannot, in general, appear in the c-theorem,unless the theory itself is sick and plagued with them.As remarked in sect.2, ourformulation of the c-theorem makes use of the states of the Hilbert space, which would beill-defined in the presence of IR singularities.4-3.

THE SUM RULE ABOVE TWO DIMENSIONSThe sum rule can also be computed for 2 < d ≤4 by using the previous formulae.Before that, let us find the value of the c-charge at the critical point of the sigma-model.Having no intrinsic means to compute it, we have to relate it to the value of the Gaussianfree theory, c = N in our conventions. We use a number of arguments known in the liter-ature [16] to embed the sigma-model into its linear realization, the O(N)-symmetric λϕ4theory (fig.

6). The latter theory has a renormalized mass parameter m′, and the additionalinteraction strenght λ, while the unique mass parameter of the non-linear sigma-model hasto be thought as m = m(m′, λ), at least in a region close to the critical point.

The flowin the sigma-model in the symmetric phase m →∞, will correspond to the line drawn inthe (m′, λ) plane of fig. 6 (possibly leaving the plane for large m).

For 2 < d ≤4, thesigma-model critical point m = 0 is believed to fall into the universality class of the O(N)-symmetric Wilson point of λϕ4, discussed in sect. 3.4, i.e.m′ = 0, λ = λ∗= O( d−4N ).

* It has been suggested that the c-theorem could fail because of them [27].21

The other fixed point of λϕ4, the Gaussian one, is close to it for large N, and the corre-sponding flow between them was previously shown to give ∆c = O(1), which is subleadingfor large N. Therefore, we can assign c = N to the critical point of the sigma-model forlarge N.Actually, we are testing the c-charge along a chain of flows as in eq. (2.11), wherethe fixed points are: (1) the Gaussian one, (2) the critical sigma-model, and (3) the trivialtheory c = 0 (see fig.

6). The last one actually sits in two different points of couplingspace, (m′ = ∞, λ = 0) and (m = ∞), thus we are in the situation of non-perturbativeflows of fig.

2.Let us first discuss the sum rule in four dimensions, where the Wilson point mergeswith the Gaussian one for any N. At d = 4 a new logarithmic singularity appears in thesaddle point equation eq(4.3). According to the previous discussion (eqs.

(4.7)), we regulateit by using dimensional regularization. Thus we find,Θ (x) ∼√N4m2−εεα (x),(d = 4 −ε)B (p) ∼1εm−ε2π21 + Oε p24m2.

(4.14)The singularities 1/ε2 cancel in the spectral density c (µ), which has a finite expression inany dimension, as it should. Finally,∆c = NZ ∞0dµc (µ)µ2= NZ ∞2mdµ 120 m4µ5s1 −4m2µ2= N,(d = 4, N →∞) (4.15)Indeed, c (µ) has the same form as in the massive perturbation of the free four-dimensionaltheory, computed in ref.

[7](see eq. (3.31) therein).

Therefore, the sigma-model approachesthe free theory all along the massive flow, and the sum rule confirms the expectations onthe triviality of the model, to leading order in1N . Moreover, the test of addivity of thec-charge eq.

(2.11) is trivial in this case.As an example of the case 2 < d < 4, we shall discuss d = 3, where eqs. (4.10) canbe expressed in terms of elementary functions, and the final integral of c(µ)/µ computednumerically.

This was already done in our previous work [7] , eq. (6.47), and we onlyquote the result,∆c = N (0.5863....),(d = 3, N →∞)(4.16)22

Therefore, we do not find agreement with the expected result ∆c = N. A possible expla-nation of this failure is, of course, that our candidate c-charge does not fulfil the addivityproperty, thus it is not a universal quantity uniquely associated to the fixed points. Indeed,in this case of non-pertubative flows, we do not have arguments in support of additivity,nor can we exclude coordinate singularities in the space of theories of fig.

6.Another explanation could be that the c-theorem, though correct, is not easy to verifyfor non-perturbative flows. In particular, we cannot exclude mixing with irrelevant fieldsin the expansion of Θ = βiΦi, due to our partial understanding of the critical field theoryof the sigma-model.

Actually, the example of sect. 3.4 showed that irrelevant fields canappear, when flowing offinteracting critical theories.

The fact that the sum rule (4.16) hasa value of ∆c lower than expected may be an indication that we are missing contributionsto Θ.23

5. ConclusionIn this paper, we have put forward a candidate for a monotonically decreasing functionalong RG trajectories in d > 2, which is the analogue of the Zamolodchikov c-function intwo dimensions.The analysis of this c-function in the Ginsburg-Landau models usingepsilon expansion as well as conformal perturbation theory displays all the nice propertiesexpected from a c-theorem.

Our c-function behaves as a height function in a perturbativedomain of the space of theories.The study of the sigma-models is less clear. In two dimensions, the theorem worksas it should, proving harmless the fears concerning infrared problems.

In four dimensions,it confirms that the theory coalesces with the free massive one and there are, again, noproblems. Nevertheless, in three dimensions, our computation of ref.

[7] presents a resultwhich is in disagreement with the theorem. Different ways out are sketched in sect.

4,which deserve further investigation.There is an observation we want to emphasize. Physically meaningful theories, likeQCD, behave at short and long distances as free theories*.

Remarkably enough, it wasfound in ref. [7] that the c-charges associated to the spin 0 and 2 spectral densities areequal for free theories of spin 0 and 1/2.

It is, then, natural to conjecture that both c-charges are equal in general, that is c(2) = c(0) for free theories. Since c(2) is well-definedat fixed points, a general proof stating this equality for free theories would be enough toapply the c-theorem to the Standard Model and beyond.As an example, it is easy to see that the long-distance Nambu-Goldstone realizationof QCD in terms of pions is in agreement with the conjectured c-theorem.

Adapting aprevious example [3], we consider QCD with Nf flavors and Nc colours in four dimensions.Using the fact that c0 = 1 for particles with spin 0 and c1/2 = 6 for spin 1/2, we look atthe balance between the short- and long-distance realizations of QCDNfNc c1/2 + (N 2c −1)c1≥(N 2f −1) c0for any value of c1, provided asymptotic freedom holds, i.e. Nf < 112 Nc.

Note that we didnot write the value of c1, because we cannot easily compute this number by a free massive* In fact, there are fewer non-trivial scale invariant theories in four dimensions, incontradistinction with many non-Gaussian fixed points known to exist in less than fourdimensions.24

perturbation, as done in ref. [7] for the spin 0 and 1/2 particles.

The massless limit ofthe massive spin 1, or Proca, particle is not the massless spin-zero particle, because thenumber of degrees of freedom changes. Probably, one has to resort to the Higgs mechanismto give mass to a gauge field in a correct way.To conclude, we would like to mention some lines to progress.

The inclusion of ad-ditional symmetries (e.g. current algebra) remains to be done.

Further analysis of theconstraints imposed by the present form of the c-theorem in theories which go beyond theStandard Model also is left for the future.6. AcknowledgementsWe want to thank J.L.Cardy, P.H.Damgaard, G.Shore and A.B.Zamolodchikov foruseful discussions.

This work has been supported by CYCIT and the EEC Science Twin-ning grant SC1000337. X. V. acknowledges the support of an FPI grant.25

Appendix - Conformal Perturbation Theory in d-DimensionsConformal invariance fixes in any dimension the form of the two- and three-pointcorrelators. This property can be used to set a perturbative expansion around conformalfield theories which are non-Gaussian (See also ref.

[5]).Standard perturbation theoryappears as a particular case of this conformal perturbation theory.Let us consider, then, a quantum field theory, invariant under the conformal group,in an arbitrary dimension d. We denote by φi (x) a generic field and by ∆i its dimension.Conformal invariance fixes the form of the two- and three-point functions to be⟨φi (x) φj (y)⟩CF T =δij|x −y|2∆i⟨φi (x) φj (y) φk (z)⟩CF T =Cijk|x −y|∆i+∆j−∆k |y −z|∆j+∆k−∆j |x −z|∆i+∆k−∆j ,(6.1)where Cijk are constant coefficients called structure constants.A conformally invariant field theory can be perturbed with one of its operators (callit φp, with dimension ∆p). A correlator in the perturbed theory is defined to be⟨φ1 (x1) .

. .

φN (xN)⟩= ⟨φ1 (x1) . .

. φN (xN) eλ0Zddxφp (x)⟩CF T⟨eλ0Zddxφp (x)⟩CF T.(6.2)The subscript CFT means that the vacuum expectation value has to be computed in theconformal theory.

For two point-functions, at first order in λ0, formula (6.1) allows oneto write a general expression for any dimension of space-time,⟨φ (x) φ (0)⟩= ⟨φ (x) φ (0)⟩CF T + λ0Zddy⟨φ (x) φ (0) φp (y)⟩CF T + Oλ20=1(x2)∆1 + λ0Cφφp4A(d −∆)|x|d−∆p + Oλ20. (6.3)The constant A is given byA =πd2 Γ∆p −d2Γ2 1 + d−∆p2Γ2∆p2Γ (1 + d −∆p).

(6.4)26

Remark that, for (d −∆p) →0, A goes to πd2 .This is the case of slightly relevantperturbing fields.For these nearly marginal perturbations, expression (6.3)does notmake sense and needs renormalization.At this point, the procedure mimics the two-dimensional case [1][17], so we will skipthe details, recalling only some important points. We define the renormalized field and therenormalized coupling constant asΦ (x, g) ≡1√Zφ (x) ,g ≡Zgλ0.

(6.5)We then set wave function and coupling constant renormalizations at a scale κ to be⟨Φp (x, g) Φp (0, g)⟩||x|=κ−1 = κ2d. (6.6)Θ (x) = Vdβ (g) Φp (x, g),(6.7)Following the steps described in [17], one can get the expression for the renormalizedcoupling constant in terms of the bare oneg = κ∆−dλ01 + λ0κ∆−dAn −∆Cppp.

(6.8)It is now convenient to rescale the coupling constant g −→gA to simplify our formulae.Next, we compute the beta-functionβ (g) = −(d −∆p) g −Cpppg2,(6.9)and the anomalous dimension for the new scaling dimensions for the perturbing field aswell as for any other field∆p (g) = ∆p −2Cpppg,∆(g) = ∆−2Cφφpg. (6.10)From formulae (6.9), we see that there is a new fixed point at an infinitesimal distancefrom the original conformal point,g∗= −d −∆pCppp.

(6.11)At this point, the scaling dimensions are∆∗p ≡∆p (g∗) = 2d −∆p,∆∗≡∆(g∗) = ∆+ 2 (d −∆p) CφφpCppp. (6.12)27

Note that the IR scaling dimension of the perturbing field is insensitive to the value of thestructure constant.Using formula (6.9), one can also write the Callan-Symanzik equation for the two-pointcorrelators,2|x| ∂∂|x| + 2∆p (g) + β (g) ∂∂g⟨Φp (x, g) Φp (0, g)⟩= 0,(6.13)and find a renormalization group improved two point function for the perturbing field asthe solution of (6.13) that fulfills condition (6.6). Proceeding thus, one has⟨Φp (x, g) Φp (0, g)⟩=κ2d|κx|2∆p11 −gCppp (|κx|d−∆p−1)d−∆p4 .

(6.14)With this expression, ∆c can be computed, using either of the following forms of the sumrule∆c = d + 1dVdZ|x|>ǫddx|x|d⟨Θ (x) Θ (0)⟩=Z ∞0dµ c (µ)µd−2 ,(6.15)where the spin 0 part of the spectral representation isc (µ) = 2dΓ (d) (d + 1)πVd1µ3 ℑm (⟨Θ (p) Θ (−p)⟩)|p2=−µ2. (6.16)In any case, what we find is∆c = 83d + 1dV 2d(d −∆p)3A2C2ppp.

(6.17)which is the change of the c-charge between the UV and IR fixed points.In general, the perturbing field does not close an algebra by itself in the sense of theoperator product expansion. This causes mixing with other fields in the theory.

It is thennecessary to consider a system of beta-functionsβi = −(d −∆i)gi −Xj,kCijk gjgk,(6.18)where {gi} span the coupling space. The c-function is thenc = −12Xi(d −∆i)g2i −13Xi,j,kCijk gigjgk,(6.19)28

which contains the most general case. In practice, the system (6.18) may be simpler andsolvable so that explicit expressions for the variation of c can be obtained, as in sect.

3.5.Example 1. TWO-DIMENSIONAL CONFORMAL FIELD THEORIESOne particular set of two-dimensional conformal field theories is the unitary minimalseries, whose central charges are given by the formula c = 1 −6m(m+1),m ≥3, wherem is an integer.

As discussed in ref. [1], perturbing one such minimal model with its leastrelevant field, if m is large, we obtain the next minimal model at c (m −1).

We can havean inkling of this fact recalling thatd −∆p =4m + 1 = 4m + O 1m2Cppp =4√3 + O 1m. (6.20)Applying (6.17), we correctly obtain∆c = 12m3 + O 1m4= c(m) −c(m −1),(6.21)which is a check for our formulae.Example 2.

THE ε-EXPANSIONThe ε-expansion can be thought as an example of conformal perturbation theory.Recall that it consists in perturbing the massless free theory with λϕ4 theory in 4 −εdimensions. In order to apply the previous formulae, we recall that the propagator of thescalar field is⟨ϕ (x) ϕ (0)⟩=1Vd (d −2)1|x|d−2 .

(6.22)To obtain proper conformal fields, operators have to be normalized. For instance,φ (x) =pVd (d −2)ϕ (x) .29

As a conformal operator, ϕ4 has dimension ∆p = 4 −2ε, so that d −∆p = ε. We seethen that the operator is slightly relevant if ε is small enough.

Other conformal featuresareCppp = 632Cϕϕp = 0(6.23)(from this last equation we see that the two point function ⟨ϕϕ⟩gets no correction at thefirst order). This is just a particular case of the general conformal perturbation expansionand it is elaborated in sect.

3.4.30

References[1]A.B.Zamolodchikov, JETP Lett. 43 (1986) 730; Sov.J.Nucl.Phys.

46 (1987) 1090. [2]D.A.Kastor, E.J.Martinec and S.H.Shenker, Nucl.

Phys. B316 (1989) 590.

[3]J.L.Cardy, Phys.Lett. B215 (1988) 749.

[4]H.Osborn, Phys.Lett. B222 (1989) 97 ;I.Jack and H.Osborn, Nucl.

Phys. B343 (1990) 647.

[5]N.E.Mavromatos, J.L.Miramontes and J.L.S´anchez de Santos, Phys.Rev. D40 (1989)535.[6]G.

Shore, Phys.Lett. B253 (1991) 380 ; B256 (1991) 407.[7]A.

Cappelli, D. Friedan and J.I.Latorre, Nucl.Phys. B352 (1991) 616.

[8]P.H.Damgaard, Stability and Instability of the Renormalization Group Flows, preprintCERN-TH-6073-91. [9]A.W.Ludwig and J.L.Cardy, Nucl.Phys.

B285 (1987) 687. [10]C.Athorne and I.D.Lawrie, Nucl.Phys.

B265 (1985) 577. [11]K.G.Wilson and J.Kogut, Phys.Rep.

12C (1974) 75. [12]D.J.Wallace and R.K.P.Zia, Ann.

Phys. (NY) 92 (1975) 142.

[13]L.N.Lipatov, Sov. Phys.

JETP 44 (1976) 1055. [14]E.Br´ezin, J.C.Le Guillou and J.Zinn-Justin, in Phase Transitions and Critical Phe-nomena, vol.6, ed.

C.Domb and M.S.Green (Academic Press, 1976). [15]A.B.Zamolodchikov, Sov.J.Nucl.Phys.

44 (1986) 529. [16]J.Zinn-Justin, Quantum field theory and critical phenomena (Clarendon, Oxford1989);E.Br´ezin and J.Zinn-Justin, Phys.

Rev. B14 (1976) 3110.

[17]A.Cappelli and J.I.Latorre, Nucl.Phys. B340 (1990) 659.[18]D.Z.

Freedman, J.I.Latorre and X. Vilas´ıs, Mod.Phys.Lett. A6 (1991) 531.

[19]J.L.Cardy, Phys.Rev.Lett. 60 (1988) 2709.

[20]J.L.Cardy, Nucl.Phys. B290 (1987) 355 and preprint UCSB-90-38;A.Cappelli and A.Coste, Nucl.Phys.

B314 (1989) 707. [21]A.A.Belavin, A.M.Polyakov and A.B.Zamolodchikov, Nucl.Phys.

B241 (1984) 333. [22]C.Vafa, Phys.Lett.

B212 (1988) 28.31

[23]M.L¨assig, Nucl. Phys.

B334 (1990) 652. [24]C.Itzykson and J.-M.Drouffe, Statistical Field Theory (Cambridge Univ.

Press, 1989). [25]G.Felder, Com.

Math. Phys.111 (1987) 101.

[26]B.Rosenstein, B.J.Warr and S.H.Park, Nucl.Phys. B336 (1990) 435.

[27]G.Curci, G.Paffuti, Nucl.Phys. B312 (1989) 227.32

7. Figure Captionsfig.

1RG pattern which exemplifies the additivity property of the c-charge.fig. 2Schematic picture of two RG flows leading to the purely massive phase which are notdeformable into each other.fig.

3RG flows for the r-th and (r −1)-th Ginsburg-Landau models between two and fourdimensions.fig. 4Chains of RG flows for the multicomponent ϕ4 theory (see text):a) Gaussian −→Symmetric −→(x∗, y∗), for N > 4;b) Gaussian −→Symmetric −→Decoupled, forN > 10.fig.

5Level map of the c-function for the multicomponent λϕ4 theory with N = 8 and d = 3.8.fig. 6Schematic picture of the RG flows in the linear and non-linear sigma-models.33

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