Products of composite operators in the exact renormalization group formalism. Abstract

Transcription

1 KOBE-TH-7-04 Products of comosite oerators in the exact renormalization grou formalism C. Pagani Institute für Physik WA THEP Johannes-Gutenberg-Universität Staudingerweg 7, Mainz, Germany arxiv: v [he-th] 8 Aug 07 H. Sonoda Physics Deartment, Kobe University, Kobe , Jaan Dated: August, 07 Abstract We discuss a general method of constructing the roducts of comosite oerators using the exact renormalization grou formalism. Considering mainly the Wilson action at a generic fixed oint of the renormalization grou, we give an argument for the validity of short distance exansions of oerator roducts. We show how to comute the exansion coefficients by solving differential euations, and test our method with some simle examles. caagani@uni-mainz.de hsonoda@kobe-u.ac.j

2 I. INTRODUCTION In the framework of the Wilsonian renormalization grou RG, the hysics of a system is comletely characterized by a Wilson action. The momentum cutoff of the action is fixed by rescaling while the corresonding size in hysical units diminishes exonentially under the RG transformation. The Wilson action of a critical theory eventually reaches a fixed oint which is scale invariant with no characteristic length. It is imortant to understand how small deformations of the fixed oint Wilson action grow under the RG transformation. For examle, the exonential growth of deformations is dictated by the critical exonents. There are also sace deendent deformations with nontrivial rotation roerties. These deformations constitute what we call comosite oerators, and their scaling roerties under the RG transformation constitute an essential art of our understanding of critical henomena and continuum field theory. The urose of this aer is to imrove our understanding of comosite oerators using the formalism of the exact renormalization grou ERG or functional RG. The imortance of comosite oerators in ERG was emhasized early by Becchi [], and his results have been extended in some later works such as [ 4]. In this work we use ERG to construct roducts of comosite oerators and study their roerties. We discuss the insertion of two or more comosite oerators in correlation functions of the elementary fields. Particular attention is aid to the ERG differential euation satisfied by comosite oerators and their roducts at the fixed oint. When considering the roduct of two comosite oerators, it is natural to ask about its short distance behavior. At short distances the oerator roduct exansion OPE of K.Wilson[5]isexectedtobevalid. IntheastERGhasbeenusedtorovideanalternative roof of the existence of the OPE in erturbation theory [6 3]; the original erturbative roof goes back to Zimmermann [4]. The urose of examining the OPE within ERG is to fill the ga between ERG and other nonerturbative aroaches to uantum field theory where the OPE forms the backbone structure of the theory. Particularly relevant is the case of conformal field theories, esecially in the two dimensional case. Using ERG we argue the lausibility if not a roof of the existence of OPE. In articular, we derive ERG differential euations a.k.a. flow euations satisfied by the Wilson coefficients and solve them for some simle examles.

3 The aer is organized as follows. In section II we define comosite oerators at a fixed oint of the RG transformation. We introduce three euivalent aroaches using the Wilson action [5, 6], the generating functional of connected correlation functions with an infrared cutoff [7], and its Legendre transform called the effective average action [8, 9], resectively. The three aroaches differ in the natural choice of field variables: φ, J, Φ. In section III we generalize our construction to the roduct of two comosite oerators and consider how the OPE arises. In section IV some working examles are resented. In section V we exlain how to generalize the ERG differential euations to consider the insertion of an arbitrary number of comosite oerators, and in section VI we discuss the ERG differential euations for comosite oerators away from the fixed oint. We summarize our findings in section VII. We confine some technical arts in three aendices. In Aendix A we review the basics of the ERG formalism that this aer is based on. A best edagogical effort has been made for those readers familiar with [6] but not with [5]. In Aendix B we exlain how to construct local comosite oerators in the massive free scalar theories. In Aendix C we derive the asymtotic behavior of a short-range function necessary for the examles of Sec. IV. We shall work in the dimensionless convention, where all dimensionful uantities have been rescaled via a suitable ower of the cutoff. We also adot the following notation; = d D π D, δ = πd δ D, = D µ= µ µ. II. COMPOSITE OPERATORS AT A FIXED POINT At a fixed oint of the exact renormalization grou, the Wilson action satisfies the ERG euation[5, 0] 0 = [ lnk+ D + γ + φ + lnr+ γ K R δ δφ ] δ e S[φ], δφδφ where γ is the anomalous dimension. We have reared Aendix A for the readers who are familiar with [6] but not with [5]. We have introduced two ositive cutoff functions:. K aroaches as 0, and decays raidly for. 3

4 . R must be nonvanishing at = 0 and decays raidly for. The inverse transform of R is a function in sace that is nonvanishing only over a region of unit size. For examle, the choice K = e,r = K = K e original choice made in [5] is K = e,r = e. E. imlies that the modified correlation functions defined by φ φ n satisfy the scaling law n i= for arbitrary momenta.[] K i K ex R δ δφδφ satisfies the criteria. The φ φ n φ e t φ n e t = ex n D + +γ φ φ n 4 In our discussion of comosite oerators we find it more convenient to deal with a functional W[J] defined directly in terms of S[φ] as[7, ] W[J] JJ R where J R K φ. +S[φ], In fact it is even more convenient to deal with the Legendre transform of W[J][8, 3]: RΦΦ +Γ[Φ] W[J] Φ δw[j] δj. J Φ, Γ[Φ] is often called the effective average action. We can interret W[J] as the generating functional of connected correlation functions[7, ] and Γ[Φ] as the effective action[8, 3], both in the resence of an infrared cutoff. The same cutoff is called an ultraviolet cutoff for S and an infrared cutoff for W and Γ. This is because we regard S as the weight of functional integrationover low momenta to bedone, but we regardw andγasconseuences of functional integration over high momenta already done. It has recently been shown that the high momentum limit of W and Γ gives the corresonding functionals without the infrared cutoff.[4] The ERG euations satisfied by W and Γ are given by see [0] and 4 S 3 5a 5b 6a 6b

5 reference therein 0 = where satisfies 0 = J D+ + δγ[φ] δφ + +γ δ δj ew[j] + γr δjδj ew[j], 7 D + +γ Φ + γr G, [Φ], 8 G, [Φ] δ W[J] δjδj δ Γ[Φ] G, [Φ] Rδ r = δ r. 0 δφ δφr Now, comosite oerators can be thought of as infinitesimal changes of S, W, or Γ. Corresondingly, we can regard comosite oerators as functionals of φ, J, or Φ. Let O be a comosite oerator of scale dimension y and momentum. Regarding it as a functional of J, we obtain the following ERG euation: { y + O = J D + δ +γ δj δw[j] δ + + γr δj δj + } δ O. δjδj Similarly, regarding O as a functional of Φ, we can rewrite the above as { y + O = + D+ δ γ Φ δφ + + γr } δ G, r [Φ]G, s [Φ] O, δφrδφs where G is defined by 9. r,s The above two ERG euations are euivalent, and they imly that the modified correlation functions defined by Oφ φ n n K δ O ex K i R δφδφ i= δ φ φ n satisfy the scaling law Oe t φ e t φ n e t = ex t y n D + Oφ φ n. 4 5 S 9 3

6 III. PRODUCTS OF COMPOSITE OPERATORS Given two comosite oerators O,O of scale dimensions y, y, we wish to define their roduct as a comosite oerator of scale dimension y + y. The naive roduct O O will not do because it does not satisfy or. We must define the roduct by adding a local counterterm: [O O ] O O +P,. 5 Otherwise the roduct will not satisfy the scaling law: [ O e t O e t ] φ e t φ n e t [ = ex t y y +n D + ] +γ [O O ]φ φ n. 6 In the ERG formalism we have a dimensionless cutoff of order either in momentum sace or in coordinate sace. In coordinate sace the inverse Fourier transform Or = e ir O 7 is exected to have a suort of unit size around the coordinate r. We exect the same roerty for the roduct of two comosite oerators. Given O and O, we denote their inverse Fourier transforms using the same symbol: O r = eir O, O r = O eir. Both have a distribution of unit size in coordinate sace. The limit 8 O ro r r r O ro r 9 is well defined. If there are short-distance singularities, we cannot find them in O O : we must look for them in the counterterm P,, which is reuired by ERG or euivalently scaling. Even without the hel of ERG, we exect the need for the counterterm in defining the Fourier transform; the integration over the case where the two oerators are dangerously close together reuires secial attention, resulting in a local counterterm. 6

7 Let us further analyze the nature of P,. Regarding comosite oerators as functionals of Φ, we obtain y + DO = 0, 0a y + DO = 0, 0b y +y + + D[O O ] = 0, 0c where D r { r r + D + γ + r r + γrr Φr s,t δ δφr } δ G r, s [Φ]G r, t [Φ]. δφsδφt 0c is euivalent to the scaling 6. From 0, we obtain the following ERG euation for the counterterm: where we have used y +y + + DP, = r r + γrr G r, s [Φ] δo G r, t [Φ] δo r s δφs t δφt = r r + γrr δo δo δjr δj r, r G r, s [Φ] = δw[j] δjrδj s = δφs δjr. 3 Since R is the Fourier transform of a function nonvanishing only over a region of unit size, is local in sace. That means that the inverse Fourier transform, e ix+ix r r r + γrr δo δjr is nonvanishing only when the distance x x is of order or less. δo δj r Therefore, we can exand the counterterm P, using a basis of local comosite oerators: P, = i c,i O i +, 4 where O i is a comosite oerator of scale dimension y i, satisfying y i + DO i =

8 The coefficient c,i deends only on ; we have absorbed all the deendence on + into O i. Similarly, we can exand the right-hand side of as r r + γrr δo δo r δjr δj r = d,i O i +. 6 i Substituting 4 and 6 into, we obtain + +y +y y i c,i = d,i, or euivalently +y +y y i c,i = d,i 7 which determines c,i in terms of d,i. Before discussing the short-distance behavior of c,i for large, we would like to consider the solvability of 7 and uniueness of its solution. We assume analyticity: both c,i and d,i are regular functions of at = 0. 7 can be solved uniuely unless n y +y +y i = 0,,,. 8 If 8 holds, and if d,i contains a constant multile of n, we cannot find an analytic solution. If c,i is a scalar, the following discussion alies only for even n. Even if d,i has no such a term, c,i is ambiguous by a constant multile of n. So, what to do if 8 is the case? If 8 holds, we need to modify 0c so that y +y + + D[O O ] = d i n O i + 9 where the constant d is determined so that d,i +d i n has no term roortional to n. Then, we need to solve nc,i = d,i +d i n 30 instead of 7. This can be solved, but the solution is not uniue. To fix the coefficient of n, we must introduce a convention such as the absence of the n term in c,i : n nc,i = 0. 3 =0 8

9 All this imlies that the scale dimension y + y is extended to a matrix: the roduct [O O ] mixes with a local oerator O i + for which 8 is satisfied. For examle, at the Gaussian fixed oint in D = 4, the roduct [ [ φ ] [ φ ]] y = y = mixes with δ+. See Examle of Sec. IV for more details. So much for the discussion of 8. Now, we consider the short-distance limit of the roduct. We consider [O O ], taking large while fixing +. As has been exlained, a singular behavior is exected not of O O but of P,: [O O ]φ φ n large P,φ φ n. 3 +fixed We can regard P, as a functional of J with momentum +. Since we kee the momenta,, n finite in the above, we can assume δo δjr and δo δj r of to deend only on J s with finite momenta. Hence, r in must be of order by momentum conservation. Therefore, Rr becomes extremely small. Hence, from 6, we exect d,i Thus, we obtain +y +y y i c,i This imlies where C,i is a constant. c,i C,i y y +y i, 35 +fixed We thus obtain a short-distance exansion a.k.a. oerator roduct exansion [O O ] + fixed C,i y y +y i O i i To be able to neglect the contribution of O O, we must restrict the sum over i to y y +y i D, 37 corresonding to singularities in sace. This condition can be rewritten as D y +D y D y i, 38 9

10 where D y i is the scale dimension of the inverse Fourier transform of O i oerator in coordinate sace. The oerator O i with the lowest scale dimension rovides the highest short-distance singularity. IV. EXAMPLES We would like to rovide concrete alications of the general theory we have develoed. In the first subsection we consider a generic fixed oint action, and in the second we take examles from the Gaussian fixed oint in D dimensions < D 4. A. [OΦ] Φ is a comosite oerator corresonding to the elementary field φ. Φ satisfies with scale dimension y Φ = D + +γ. 39 Let O be an arbitrary comosite oerator of scale dimension y. Its roduct with Φ must satisfy This gives[3] This imlies the counterterm [OΦ]φ φ n = Oφφ φ n. 40 [OΦ] = OΦ+ K R δo δφ = OΦ+ δo δj. 4 P OΦ, = δo δj. 4 For the simlest case of O = Φ, we use 6b and 4 to obtain This generalizes to[5] [ΦΦ] = δw[j] δw[j] δj δj + δ W[J] δj δj = e W[J] δ δj δj ew[j]. 43 δ n [Φ Φ n ] = e W[J] δj δj n ew[j], 44 0

11 which can be checked to satisfy 4: e W[J] δ n δj δj n ew[j] = δw[j] δ n δj n e W[J] δj δj n ew[j] δ + e W[J] δ n δj n δj δj n ew[j]. 45 Using 9 we can rewrite 43 as Hence, for + fixed, we obtain From the scaling law we obtain the coefficient of the identity oerator as [ΦΦ] = ΦΦ+G, [Φ]. 46 [ΦΦ] G, [Φ]. 47 +fixed φφ = const δ+, 48 γ [ΦΦ] const δ fixed γ Further coefficients can be comuted by emloying some aroximation scheme. For instance, we have checked in the φ 4 theory in dimension D = 4 that the order λ correction is given by [ΦΦ] + fixed [ δ+ λ φ + ] In coordinate sace the order λ correction is roortional to the logarithm of the distance. B. Examles from the Gaussian fixed oint in D dimensions We now consider the comosite oerators at the Gaussian fixed oint: Γ[Φ] = ΦΦ. 5 There is no anomalous dimension: γ = 0. The high-momentum roagator is given by G, [Φ] = For convenience we introduce δ+ hδ+. 5 +R f + h = R +R, 53

12 which vanishes raidly for. Now, the differential oerator D defined by can be written as D r { r r + D + Φr } δ δφr +fr δ. 54 δφrδφ r In the remaining art of this section we consider the roducts of the comosite oerators [ φ ] Φ Φ δ + +κ δ, 55, [ φ 4 ] Φ Φ 4 δ ! 4!,, 4 +κ Φ Φ δ + +, κ δ, 56 where the constant κ is defined by κ D f. 57 The scale dimensions are and D 4, resectively. See Aendix B for the construction of comosite oerators in the free theory. Examle : [[ φ ][ φ ]] The scale dimension of the roduct is y = 4. Hence, in D = 4, we exect mixing with the unit oerator δ+. Let [ [ φ ] [ φ ]] = [ φ ] [ φ ] +P,. 58 The counterterm must satisfy 4+ + DP, δ [ = fr φ ] δ [ φ ] r δφr δφ r = frφ rφ +r. 59 r To solve this, let us exand P, = Φ Φ δ + u ;,, +u 0 δ+. 60

13 F FIG.. Grahical reresentation of P,: a dark line for h Substituting this into 59, we obtain i i u ;, = f +f, 6a and i=, 4 D + u 0 = r fru ;r, r. 6b The homogeneous solution of 6a is excluded on account of analyticity at zero momentum. Hence, we obtain Substituting this into 6b, we obtain u ;, = h +h. 6 4 D + u 0 = For < D < 4, this is uniuely solved by u 0 = F fh hh See Fig.. Now, for D = 4, 63 does not admit a solution analytic at = 0. Since the left-hand side vanishes at = 0, we must modify it to u 0 = fh + h D = 4 65 by subtracting a constant from the right-hand side. The constant can be evaluated as fh = h+ h = 4+ h = µ h r µ = 4π. 66 3

14 65 determines u 0 u to an additive constant. We define F by F = F0 0. fh+ 4π, 67a 67b As a convention we adot the choice u 0 = F. The subtraction in 67a imlies the mixing of the roduct with the identity oerator, and the roduct satisfies the ERG euation [ [ 4+ + D φ ] [ φ ]] = 4π δ+. 68 Let us find the asymtotic behavior of the roduct as for a fixed +. We find u ;, fixed. 69 From Aendix C, we obtain F c F D 4 + κ < D < 4 4π ln +const+ κ D = 4, 70 where c F is given by C6. Hence, we obtain for < D < 4, and for D = 4. [ [ φ ] [ [ φ ] + fixed [ φ ]] c F D 4 δ++ [ φ + ] 7 +fixed [ φ ]] 4π ln +const δ++ [ φ + ] 7 Examle : [[ 4! φ4 ][ φ ]] The scale dimension of the roduct is y = D 6. Hence, in D = 4, the roduct mixes with [φ +]. Let [ [ φ 4 ] [ φ ]] = [ φ 4 ] [ φ ] +P 4,. 73 4! 4! 4

15 i i κ FIG.. Grahical reresentation of P 4, Solving 6 D + + DP 4, δ [ = fr φ 4 ] δ [ φ ] r δφr 4! δφ r = 4 Φ Φ 4 δ f i 4!,, 4 i= +κ Φ Φ δ + + f +f, 74, we obtain, for < D < 4, P 4, = 4 Φ Φ 4 δ h i 4!,, 4 i= + Φ Φ δ + + κ h i +F, i=, +κ Fδ+. 75 See Fig.. The above exression is valid also for D = 4 excet that the ERG euation for the roduct is modified to [ [ 6 D + + D φ 4 ] [ φ ]] = [ φ + ]. 76 4! 4π Using 70, we obtain [ [ φ 4 ] 4! for < D < 4, and [ [ φ 4 ] 4! for D = 4. [ φ ]] + fixed [ φ ]] c F D 4 [ φ + ] + 4 [ φ 4 + ] 77 +fixed 4! 4π ln +const [ φ + ] + 4 [ φ 4 + ] 4! 78 5

16 V. MULTIPLE PRODUCTS Multile roducts of comosite oerators are defined just like the roduct of two comosite oerators as single nonlocal comosite oerators. To start with, the roduct of three comosite oeratorso i of scale dimension y i i =,,3is defined as a comosite oerator of scale dimension y +y +y 3 as [O O O 3 r] O O O 3 r +P,O 3 r+p 3,rO +P 3,rO +P 3,,r, 79 where the counterterm P is the same counterterm that makes [O O ] = O O +P, a comosite oerator. The extra counterterm P 3,,r has to do with the three oerators close to each other simultaneously. We obtain the ERG euation 3 y i + + +r r D P 3,,r i= δ = fs s δφs P δ, δφ s O 3r + δ δφs P δ 3,r δφ s O + δ δφs P δ 3r, δφ s O. 80 If there is a local comosite oerator O whose scale dimension y satisfies 3 y = y i n n = 0,,, 8 i= the roduct [O O O 3 r] may mix with O+ +r, and we obtain 3 y i + + +r r D [O O O 3 r] = d, ro++r, 8 i= where d, r is a degree n olynomial of, r. Proceeding further, we can define the roduct of four comosite oerators as [O O O 3 ro 4 s] O O O 3 ro 4 s +P,O 3 ro 4 s+5 more terms +P,P 34 r,s+p 3,rP 4,s+P 4,sP 3,r +P 3,,rO 4 s+3 more terms+p 34,,r,s. 83 6

17 In the absence of mixing the last counterterm satisfies 4 y i + + +s s D P 34,,r,s i= δp, = ft δφt t δp 34 r,s δφ t + + δp 3,,r δφt This can be generalized to higher order roducts of comosite oerators. δo 4 s δφ t Let O be a comosite oerator with scale dimension y < 0. If it is a scalar, it is a relevant oerator. The n-th order roduct has scale dimension ny. We can introduce a source J so that where [O O n ] = δ n δj δj n ew[j] J=0, 85 W[J] = J O+ J J P,, + J J J 3 P 3,, !,, 3 If no local comosite oerator has scale dimension as low as y, there is no mixing between multile roducts and local comosite oerators. Since δ has the lowest scale dimension D, there is no mixing if y < D. In the absence of mixing, W[J] satisfies the ERG euation: δ J y + δj ew[j] = J D + δ +γ δj ew[j] δw[j] δ + + γr δj δj + δ e W[J]. 87 δjδj Asthe simlest examle, consider O = Φ with y = D+ +γ. E. 44 imlies[5] W[J] = W [J +J] W[J]. 88 Another simle examle is O = [φ ] with y = at the Gaussian fixed oint < D 4. P is P, obtained in Examle of Sec. IV. P n n 3 are given in the form P n,, n = n ΦΦ δ + i u n,, n ;,, i= n +v n,, n δ i, 89 7 i=

18 σ σ σn σ σ u n v n FIG. 3. Grahical reresentation of u n and v n σ is a ermutation of,,n where u n,v n are exressed grahically in Fig. 3. For examle, we obtain u 3,, 3 ;, = h h +h 3 +, 90 v 3,, 3 = hh h. 9 At D = 4, P, mixes with δ +, and we must change the left-hand side of 87 to δ J + δj + JJ e W[J]. 9 4π VI. AWAY FROM A FIXED POINT Let g be a arameter with scale dimension y E > 0. The Wilson action is arametrized by g. Assuming the anomalous dimension is indeendent of γ, we obtain the ERG euation of the action as y E g g esg[φ] = + lnk+ D + lnr+ γ K R The modified correlation functions defined by n φ φ n g K i ex satisfy the scaling law: i= φ e t φ n e t ge y E t = ex K R Rewriting the ERG euation for Wg[J] JJ R J R K φ, γ + φ δ δφ esg[φ] δ δφδφ esg[φ]. 93 δ δφδφ φ φ n Sg 94 nt D + +γ φ φ n g Sg[φ], 96a 96b

19 we obtain y E g g ewg[j] = J D+ + +γ δ δj ewg[j] + γr δ δjδj ewg[j]. 97 Comaring this with 87, we find that for the constant source J = gδ, 98 the sum Wg[J] W[J]+W[J] 99 satisfies 97. Therefore, W[J] n g=0 e = [Q0 Q0] 00 g newg[j] is the n-th order roduct of the zero momentum comosite oerator Q0 Wg[J] g g=0 of scale dimension y E, defined at the fixed oint. Considering oerator roducts, it is even more convenient to introduce the effective action with an infrared cutoff: Γg[Φ] Φ δwg[j] δj. RΦΦ + Wg[J] Then, a comosite oerator of scale dimension y satisfies J Φ, 0a 0b y +y E g g + DO = 0, 0 where D is given by excet that G is now defined with the g-deendent W as Gg, [Φ] δ Wg[J] δjδj. 03 The discussion of Sec. III goes through as long as we use the g-deendent D. We introduce g-deendent coefficients c,i g; and d,i g; via 4 and 6, resectively. E. 7 is relaced by +y E g g +y +y y i c,i g; = d,i g;. 04 9

20 Since the locality of the cutoff function R gives d,i g; 0, 05 we obtain the asymtotic behavior c,i g; +fixed y y +y i assuming the analyticity of c,i g; in g. The simlest examle is given by the massive Gaussian theory Γm [Φ] = C,i + g g C y,i +O, 06 E y E +m ΦΦ, 07 where the suared mass m lays the role of g. See Aendix B for the construction of comosite oerators. We can show, for 3 < D < 4 the term roortional to m is not singular for < D 3 [ [ φ ] [ φ ]] c F +fixed and for D = 4 [ [ φ ] [ φ ]] + fixed +D 3 m 4π + + m D 4 δ++ [ φ + ], 08 ln +const δ+ [ φ + ]. 09 VII. CONCLUSIONS In this work we have studied the OPE oerator roduct exansions in the framework of ERG the exact renormalization grou. The key concets underlying our analysis are the comosite oerators and their roducts, which we define resectively in Sec. II and Sec. III. We have argued that the ERG differential euation associated with the roduct of two oerators can be exanded in a local basis of comosite oerators, leading to the ERG differential euations for the Wilson OPE coefficients. Particular attention has been aid to the form of the euation at a fixed oint. It is imortant to stress that the Wilson coefficients are defined in the large momentum limit, i.e.,. Taking this limit allows us to eliminate surious contributions deendent on the cutoff. 0

21 We have tested our method by considering some exlicit examles in Sec. IV. At the technical level, we have found it convenient to first solve the ERG differential euation by taking into account the analyticity of the euation at the zero momentum before taking the large momentum limit. In Sec. V we have generalized our discussion to include the definition of multile roducts of a comosite oerator, and in Sec. VI we have considered the ERG differential euations away from the fixed oint. Although the examles we have given are for the Gaussian fixed oint, we would like to stress that the ERG differential euations discussed in Sec. III are nonerturbative and can be emloyed for nonerturbative, albeit aroximate, comutations. In this sense suitable aroximation schemes should be devised. Nonerturbative aroximation schemes, such as the BMW [6], may be emloyed to solve the ERG differential euations for the Wilson coefficients. Note added: after comletion of the resent work, we learned that Prof. H. Osborn had similar ideas as ours about the roducts of comosite oerators. Sec. 3.3 of [7] Aendix A: More background on ERG To make the content of Sec. II easier to understand for the readers already familiar with [6] but not with [5] and the subseuent extensions done more recently, we would like to summarize the basics of the ERG formalism adoted in this aer. We rely on erturbation theory for intuition. In [6] the Wilson action is given in the form S Λ [φ] = K/Λ φφ +S Λ,I[φ], where the first term gives the roagator K/Λ A that dams raidly for > Λ, and the second term gives interactions. To reserve hysics below the momentum scale Λ, the interaction art must obey the following differential euation[6]: Λ Λ S Λ K/Λ Λ Λ,I[φ] = { } δsλ,i [φ] δs Λ,I [φ] + δ S Λ,I [φ]. A δφ δφ δφ δφ

22 n K K G n FIG. 4. Grah for the connected n-oint function n > We can rewrite this euation for the whole action as Λ Λ S Λ[φ] = Λ Λ lnk/λ φ δ δφ S Λ[φ] Λ K/Λ { } Λ δsλ δs Λ + δφ δφ + δ S Λ. A3 δφδφ The correlation functions calculated with S Λ φ φ n Λ [dφ]e SΛ[φ] φ φ n A4 are not entirely indeendent of Λ. Take the sum of diagrams contributing to the connected art of the n-oint function for n >. Fig. 4 Polchinski s euation A guarantees that the shaded blob denoted G n of Fig. 4 is indeendent of Λ. But the external roagators are multilied by K, and we obtain φ φ n connected Λ = G n,, n n i= K i /Λ i. A5 For small momenta i, the cutoff function K i /Λ is almost. We only need to divide the correlation function by a roduct of K s to make this strictly Λ-indeendent: n i= K i /Λ φ φ n connected Λ. A6 We are left with the two-oint function which is given by [ K/Λ φφ Λ = + K/Λ G K/Λ ] δ+, A7 where G is indeendent of Λ. Fig. 5 Again, this is almost indeendent of Λ for < Λ. To make the two-oint function strictly Λ-indeendent, we first modify it by subtracting a high momentum roagator K/Λ K/Λ A8

23 K K K + G FIG. 5. Two-oint function and then divide the result by K/Λ : φφ K/Λ Λ K/Λ K/Λ δ+ = +G δ+. A9 This is indeendent of Λ. We have thus exlained that Λ-indeendent connected correlation functions are given by φφ φφ K/Λ Λ K/Λ K/Λ δ+, A0a n φ φ n connected K i /Λ φ φ n connected Λ, A0b i= where n >. Including the disconnected arts, the Λ-indeendent four-oint correlation function is given by φ φ 4 φ φ 4 connected + φ φ φ 3 φ 4 +t, u-channels 4 [ = φ φ 4 connected Λ K i /Λ i= K /Λ K /Λ δ + φ 3 φ 4 Λ K 3/Λ K 3 /Λ δ φ φ Λ 3 + K /Λ K /Λ δ + K 3/Λ K 3 /Λ δ ] +t, u-channels. A This structure generalizes to higher-oint functions. Using a formal but more convenient notation, we can exress Λ-indeendent correlation functions by φ φ n Λ ex n i= K i /Λ K/Λ K/Λ δ δφδφ 3 φ φ n Λ. A

24 These Λ-indeendent correlation functions were first introduced in [] and termed modified correlation functions. Two Wilson actions are called there euivalent if their modified correlation functions are the same. We find the word modified somewhat misleading since these are the roer correlation functions given by the Wilson action. The ERG differential euation A3 still differs from the ERG euations of the main text. In order to discuss a fixed oint of the renormalization grou, we need to introduce an anomalous dimension and adot the dimensionless convention to fix the momentum cutoff. Let us exlain this one by one.. Anomalous dimension To kee the kinetic term of S Λ indeendent of Λ, we must introduce an aroriate Λ-deendence to the normalization of φ. This introduces an anomalous dimension γ Λ so that where φ φ n Λ = To obtain this we must change A3 to Λ S Λ Λ = + Λ ZΛ Z Λ Λ Λ lnz Λ = γ Λ. Λ lnk/λ γ Λ Λ K/Λ Λ n φ φ n Λ A3 { δsλ δs Λ δφ δφ + φ δs Λ δφ γ Λ K/Λ K/Λ A4 } δ S Λ. A5 δφδφ There are other ways of introducing γ Λ such as the one given in [8]. Here we have followed [0, ]. Rewriting the second integral of the right-hand side, we obtain Λ S Λ Λ = + Λ Λ lnk/λ γ Λ φ δs Λ δφ Λ Λ ln K/Λ K/Λ K/Λ K/Λ γ Λ { } δsλ δs Λ δφ δφ + δ S Λ. A6 δφδφ. Dimensionless convention Finally, to obtain a fixed oint we must adot the dimensionless convention by measuring hysical uantities in owers of aroriate 4

25 owers of the cutoff Λ. This serves the urose of rescaling, fixing the momentum cutoff at an arbitrary but fixed scale. We make the following relacement: Λ µ e t γ Λ γ t Λ Λ D+ φ φ A7 S Λ S t where µ is an arbitrary momentum scale corresonding to the origin t = 0 of the logarithmic momentum scale. The ERG euation becomes t S t [φ] = lnk+ D + γ t + φ δs t[φ] δφ K K K + ln K γ t { } δst [φ] δs t [φ] δφ δφ + δ S t [φ]. A8 δφδφ Introducing we can rewrite the above as t S t [φ] = + R K K lnk+ D + γ t + φ δs t[φ] δφ lnr+ γ t K R { } δst [φ] δs t [φ] δφ δφ + δ S t [φ] δφδφ A9. A0 At the fixed oint, the left-hand side vanishes, and we obtain at the beginning of Sec. II. Though the cutoff function R is given in terms of K in the above summary, we can take K and R indeendently as long as they satisfy the conditions listed in Sec. II.[] Aendix B: Comosite oerators for the massive free theory We consider the massive free theory in D > : Γ[Φ] = +m ΦΦ. B 5

26 The high-momentum roagator is given by where We define G, [Φ] δ Γ[Φ] δφδφ = hm,δ+, hm, +m +R. B B3 fm, +m m + hm, = R +m +R. B4 A comosite oerator of scale dimension y satisfies y +m m + D O = 0, B5 where D is given by D = { + D + Φ δ δφ +fm, } δ. B6 δφδφ A generic even scalar comosite oerator is written as O = N n=0 Φ Φ n δ n!,, n n i= i O n,, n. B7 Substituting this into B5, we obtain m m + m m + = N i= n i= i i +y +ND D i i +y +nd D O N,, N = 0 O n,, n fm,o n+,, n,,. n N B8a B8b The oerator with the lowest scale dimension is the identity oerator δ with y = D. The second lowest dimensional oerator, with y =, is given by [ φ ] = Φ Φ δ + +κ m δ,, B9 where κ m satisfies D+m d κ dm m = fm,. 6 B0

27 We exect κ m to be analytic at m = 0 from the locality of the comosite oerator [φ ]; any non-analytic behavior should result from the integration over the momentum modes below the cutoff. Now, E. B0 has a homogeneous solution roortional to m D. For < D < 4, this is not analytic, and the euation has a uniue analytic solution satisfying κ 0 = f0,. B D For D = 4 and higher even dimensions, we have a roblem: B0 has no analytic solution because the right-hand side has a term roortional to m. We must subtract the linear term and solve instead +m d κ dm m = fm, m where, = m fm m =0, m fm f0,h0, = 4π m =0, Ba Bb as calculated in 66. Ba has an analytic solution, but the solution is not uniue due to the analytic homogeneous solution m. This simly means that [φ ] mixes with m δ under scaling. We remove the ambiguity by adoting an arbitrary convention such as d dm κ m = 0. Bc m =0 The oerator thus defined satisfies +m m + [ D φ ] = m 4π δ, where the right-hand side imlies mixing. Any alternative choice B3 [ φ ] +const m δ B4 is eually good as an element of a basis of comosite oerators. The oerator with y = 4 D can be constructed similarly: [ φ 4 ] = 4 Φ Φ 4 δ i 4! 4!,, 4 i= +κ m Φ Φ δ + +κ 4 m δ,, where κ 4 m is defined by D+m d κ dm 4 m = κ m fm,. 7 B5 B6

28 For < D < 4, we obtain κ 4 m = κ m. B7 For D = 3, the solution is ambiguous by a constant multile of m. No subtraction is necessary; the right-hand side of B6 has no term linear in m. For D = 4, the analyticity at m = 0 demands that we modify the euation to 4+m d κ dm 4 m = κ m fm,+ m 4π where κ m is determined by B. The solution is given by, B8 κ 4 m = κ m +const m 4. B9 Again as a convention, we may imose d κ dm 4 m = 0 B0 m =0 to fix the constant. The oerator thus defined satisfies m m + [ D φ 4 ] = m [ φ ], B 4! 4π which imlies that 4! [φ4 ] mixes with m [φ ] under scaling. The oerator [φ ] has y = 0. The other oerator with y = 0 is the euation-ofmotion comosite oerator[, 3, 5] given by E e S δ K Φ +e S δφ = Φ Φ δ +, R +m +R δ. For < D 4, all the other scalar oerators have y < 0. i +m i= B Aendix C: Asymtotic behavior of F For < D < 4, F hh + 8 C

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