Lectures on the Functional Renormalization Group

CEJP 1(2003)1{71

Lectures onthe functional renormalization groupmetho d

1;2 Janos Polonyi ¤

1 Laboratoryof Theoretical Physics, Louis PasteurUniversity, Strasbourg,France 2 Departmentof Atomic Physics, L.EÄotvÄosUniversity, Budapest,Hungary

Received 20June 2002; revised 7December2002

Abstract: Theseintroductory notes areabout functional renormalization group equations andsome of their applications. It is emphasised that the applicability ofthis methodextends well beyondcritical systems, it actually provides us ageneralpurpose algorithmto solve strongly coupledquantum eld theories. Therenormalization group equation ofF.Wegnerand A. Houghtonis shownto resum the loop-expansion.Another version, dueto J.Polchinski, is obtainedby the methodof collective coordinates andcan beused for the resummationof the perturbation series. Thegenuinely non-perturbative evolution equation is obtainedby amannerreminiscent ofthe Schwinger-Dysonequations. Twovariants ofthis schemeare presented wherethe scale whichdetermines the order of the successive elimination ofthe modesis extracted fromexternal andinternal spaces. Therenormalization ofcomposite operators is discussed brie®y asan alternative way to arrive atthe renormalization groupequation. Thescaling lawsand xed points are considered fromlocal andglobal points ofview. Instability induced renormalization and newscaling lawsare shown to occurin the symmetry broken phaseof the scalar theory. The® attening ofthe e¬ective potential ofa compactv ariable is demonstrated in caseof the sine-Gordonmodel. Finally,amanifestly gaugeinvariant evolution equation is given for QED. c Central EuropeanScience Journals. All rights reserved. ® Keywords:str ongly coupledtheories, non-perturbative approximation, universality PACS(2000):25.75.-q

¤ E-mail:[email protected] 2 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Contents 1Introduction 3 2Functional RGequations 4 2.1Resumming the loopexpansion 5 2.1.1Blocking in continuous space-time 5 2.1.2Local potential approximation 7 2.1.3Gradient expansion 10 2.2Resumming the perturbation expansion 14 2.2.1Polchinski equation 15 2.2.2Gradient expansion 18 2.3Composite operator renormalization 19 2.3.1T oymodel 20 2.3.2Quantum Field Theory 21 2.3.3Parallel transport 23 2.3.4Asymptotical scaling 24 2.3.5Perturbative treatment 25 2.4Continuous evolution 26 2.5Blocking in the internal space 29 2.5.1Wilsonian action 30 2.5.2E¬ ective action 31 3Applications 33 3.1Fixed points 34 3.1.1Rescaling 34 3.1.2Reparametrizing 35 3.1.3Local potential approximation 36 3.1.4Anomalous dimension 37 3.2Global RG 38 3.2.1RG in Statistical andHigh Energy Physics 38 3.2.2Scalar model 42 3.2.3RG microscope 46 3.2.4The Theory of Everything 50 3.3Instability induced renormalization 50 3.3.1Unstable e¬ective potential 52 3.3.2T ree-level WHequation 53 3.3.3Plane wavesaddle points 53 3.3.4Correlation functions 56 3.3.5Condensation as crossover 56 3.4Sine-Gordon model 58 3.4.1Zoology of the sine-Gordonmodel 58 3.4.2E¬ ective potential 60 3.4.3Breakdown of the fundamentalgroup symmetry 61 3.4.4Lower critical dimension 62 3.5Gauge models 63 3.5.1Evolution equation 63 3.5.2Gauge invariance 64 4Whathas been achieved 65 References 66 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 3

1Introduction

The origin of renormalization goesback to hydrodynamics, the de¯nition of the mass and other dynamicalcharacteristics of bodies immersed into °uids. The more systematic elaboration ofthis concept isbased on the (semi) group property ofchanging the observational scalein atheory.The renormalization group (RG) method has already been used with anumber ofdi®erent goals.Some of the more important directions are (i) to remove U.V.divergences [1], (ii) to describe the scaledependence of physicalparameters and to classifythe parameters of atheory around acriticalpoint according to their impact on the dynamics [2],(iii) to express the highly singular product of local¯ eldvariables [3], (iv) to resum the perturbation expansion inQuantum FieldTheory [4]and inthe caseof di®erential equations [5]and ¯nally(v) to solvestrongly coupled theories. Furthermore, the RG method o®ers asystematicway oforganizing our understanding ofacomplicated dynamics by successivelyeliminating degrees of freedom. The distinctivefeature of the method isthat it retains the in°uence of the eliminatedsubsystem onthe rest. The main subject of these lectures isapplication (v) in the framework ofthe functional formalism, the search ofageneral purpose algorithm to handle non-perturbative and not necessarilycritical theories. The strategy of the RG schemefor the resummation ofthe perturbation expansion isto introduce and to evolvee® ective vertices instead ofdealing with higher order corrections. The result isan iterativeprocedure where the contributions to the e®ective vertices computed at agivenstep in the leading order of the perturbation expansion and inserted in agraph at the next step reproduce the higher order corrections. In the traditional implementation ofthe RG procedure one follows the evolution offew coupling constants only.Our concern willbe animprovement onthis method inorder to follow the evolution of alargenumber of coupling constants. This isrealized by means of the functional formalism where the evolution isconstructed for the generator function for allcoupling constants. Before starting letus reviewthe conventional multiplicativeRG schemesas used in Quantum FieldTheory .This schemeis based onthe relation 2 2 (n) n=2 (n) · pj GR (p1; ; pn; gR; · ) = Z ¡ GB (p1; ; pn; gB; ¤) + + ; (1) ¢ ¢ ¢ ¢ ¢ ¢ O Ã ¤2 ! O Ã ¤2 ! for the Green functions, where ¤and · denote the U.V.cut-o® and the substraction

(observational) scales,respectively . Z depends on either gR or gB and the ratio ¤=· (the mass istreated asone ofthe parameters g).The ignored terms stand for non-universal, cut-o®dependent interactions. Bare RG equation :The renormalized theory isindependent of the choiceof the cut-o®, d d ¤ G(n)(p ; ; p ; g ; ¤) = ¤ Z n=2(g ; ¤)G(n)(p ; ; p ; g ; ¤) = 0; (2) d¤ R 1 ¢ ¢ ¢ n B d¤ B B 1 ¢ ¢ ¢ n B h i where the renormalized coupling constants and the substraction scaleare kept ¯xedin computing the derivatives.The equation involvingthe renormalized quantities isnot 4 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 too useful. It isthe multiplicativerenormalization schemewhich leadsto the second equation, expressing the possibility ofcompensating the change of the substraction scale by the modi¯cation of the coupling constants gB(¤) arising from the solution of the di®erential equation by the method of characteristics. Noticethat the compensation ispossible onlyif the listof the coupling constants g includes allrelevant coupling f Bg constants of the appropriate scalingregime, around the U.V.¯xedpoint. RenormalizedRG equation :The bare theory isindependent of the choiceof the substraction scale,

d (n) d n=2 (n) · G (p ; ; p ; g ; · ) = · Z ¡ (g ; · )G (p ; ; p ; g ; · ) = 0; (3) d· B 1 ¢ ¢ ¢ n R d· R R 1 ¢ ¢ ¢ n R h i where the bare coupling constants and the cut-o®are kept ¯xed. Callan-Symanzik equation :The change of the mass inthe propagator isgoverned by the expression d 1 1 1 = (4) dm2 p2 m2 p2 m2 ¢ p2 m2 ¡ ¡ ¡ which can beused to ¯nd out the dependence on the renormalized mass when the cut-o® and allbare parameters exceptthe bare mass are kept ¯xed.The resulting evolution equation [6]is similar to the two previous RGequations exceptthat the derivativeis with respect to the bare mass instead of the cut-o®or the substraction scaleand the right hand sideis non-vanishing. This latter feature indicates that contrary to the ¯rst 2 two schemesthe evolution gR(m )inthe Callan-Symanzik equation isnot arenormalized trajectory,it connects theories with di®erent mass. The serious limitation of these equations isthat they are asymptotic, i.e.,are appli- cablein the regime · 2; p2 << ¤2 only.In fact, the omission ofthe non-universal terms in the multiplicativerenormalization scheme(1) requires that we stay away from the cut-o®. In models with IRinstability (spontaneous symmetry breaking, dynamicalmass gener- 2 2 2 ation,...)another limitation arises, mdyn << · ; p ,to ensure that we stay in the U.V. scalingregime. This isbecause the IRscalingregime may have relevant operators which are irrelevant on the U.V.side [7]. In more realisticmodels with severalscaling regimes which havedi® erent relevantoperator sets we need non-asymptotic methods which can interpolate between the di®erent scalingregimes. This isachieved by the functional extensions of the RGequation based onthe in¯nitesimal blocking steps.

2FunctionalRG equations

There are di®erent avenuesto arriveat afunctional RG equation. The simplest isto follow Wilson-Kadano®blocking in continuous space-time.It leadsto afunctional di®erential equation describing the cut-o®dependence of the bare action. Better approximation schemescan be worked out when the modes are suppressed smoothly as the cut-o® changes. The parameters of the bare action which are followed by these RGequations are related to observables qualitativelyonly .It ismore advantageous to transform the RG equation for the e®ective action whose parameters havedirect relation with observables. J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 5

For the sakeof simplicitywe consider anEuclidean theory in d dimensions for ascalar

¯eld ¿ x governed by the action SB[¿ ]. An O(d)invariant U.V.cut-o® ¤ isintroduced in the momentum spaceby requiring ¿ = 0 for p > ¤to render the generator functional p j j 1 W [j] 1 S [Á]+ 1 j Á e h· = D[¿ ]e¡ h· B h· ¢ (5) Z ¯nite.W eshall use the following notation conventions, stated below:In order to render the functional manipulations wellde¯ ned we alwaysassume discrete spectrum, i.e.,the presence of U.V.and IRregulators, saya latticespacing a and system volume V = adN d. d d The space-timeintegrals are d x = a x = x, x fxgx = f g, d d 1 ipx ¢ ipx d p=(2º ) = V ¡ p = pR, fp = xPe¡ fRx, andR fx = p e fp.The Dirac-deltas d K K K R¯ x;y = a¡ ¯ x;y and ¯ p;qP = V ¯Rp;q are expressedR in terms of theR Kronecker-deltas ¯ x;y and K ¯ p;q.

2.1 Resumming theloop expansion

Westart with the simplest form of in¯nitesimal blocking-step RG equations which isthe functional extension ofthe bare RGschemementioned above[8].

2.1.1Blocking in continuous space-time

Weshall denote the moving U.V.cut-o®by k.Its lowering k k ¢k leads to the ! ¡ blockingtransformation of the action which preserves the generator functional (5). Due to the presence of the source this blockingintroduces an explicitsource dependence in the action, 1 1 1 1 Sk[Á;j]+ j Á Sk dk[Á;j]+ j Á Dk[¿ ]e¡ h· h· ¢ = Dk ¢k[¿ ]e¡ h· ¡ h· ¢ ; (6) Z Z ¡ where D [¿ ]stands for the integration measure overthe functional space consisting k Fk of functions whose Fourier transform isnon-vanishing for p k.In order to avoid j j µ this complication one usually assumes j k ¢k.In this casethe blockingbecomes 2 F ¡ a mapping Sk[¿ ] Sk ¢k[¿ ]and it isenough impose the invarianceof the partition ! ¡ function 1 1 Sk ¢ k[Á] Sk[Á+Á~] e¡ h· ¡ = D[¿ ~]e¡ h· ; (7) Z ~ where ¿ k ¢k and ¿ k k ¢k. 2 F ¡ 2 F nF ¡ The evaluation ofthe path integral by means ofthe loop-expansion givesimmediately the functional RG equation, 2 ~ ~ ¹h ¯ Sk[¿ + ¿ cl] 2 Sk ¢k[¿ ] = Sk[¿ + ¿ cl] + tr ln + ¹h ; (8) ¡ 2 ~ ~ O ¯ ¿ ¯ ¿ ³ ´ ~ ~ where ¿ cl denotes the saddle point. The trace isoverthe functional space ¿ k k ¢k 2 F nF ¡ and h¹2 represents the higher loop contributions. Wewrite this equation as O ³ ´ 2 ~ ~ ¹h ¯ Sk[¿ + ¿ cl] 2 Sk[¿ ] Sk ¢k[¿ ] = Sk[¿ ] Sk[¿ + ¿ cl] tr ln + ¹h : (9) ¡ ¡ ¡ ¡ 2 ¯ ¿~¯ ¿~ O tree level ³ ´ loop contributions | {z } | {z } 6 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

(a)

(b)

Fig. 1 Graphscontributing to the blocking,(a): tree-level, (b): one-loop corrections. The dashedline stands for a particleof momentum p = k andthe solidlines represent ¿ . j j

~ When can one safelyassume that the saddle point istrivial, ¿ cl =0? Let us suppose that ¿ isweakly and slowlyvarying, ie ¿ = © + ² where ² 0and the characteristic x x x x º momentum of ² issmall with respect to the cut-o® k.Then the °uctuating component ² appears as an external source breaking translation invariance.The saddle point, ~ ¿ k k ¢k,being inhomogeneous existsas ² 0onlyif the external space-time 2 F nF ¡ ! symmetry isbroken dynamically. But a ¢k-dependent smallsaddle point, ¿~ = (¢kn), n > 0mayoccur with- cl O out breaking external symmetries.In this casewe expand in ¿~cl and ¯nd the Wegner- Houghton (WH) equation [8]

2 1 1 ¯ Sk[¿ ] ¯ Sk[¿ ] ¡ ¯ Sk[¿ ] Sk[¿ ] Sk ¢k[¿ ] = (10) ¡ ¡ 2 ¯ ¿~ ¢ Ã ¯ ¿~¯ ¿~ ! ¢ ¯ ¿~ ¹h ¯ 2S [¿ ] tr ln k + ¹h2 + ¢k2 ; ¡ 2 ¯ ¿~¯ ¿~ O O ³ ´ ³ ´ where allfunctional derivativesare taken at ¿~ =0.The saddle point can be omitted in the argument of the logarithmic function because the trace brings afactor of ¢k as we ~ shallsee below. The discussion ofthe tree-levelrenormalization when ¿ cl isnon-vanishing and ¯nite as¢k 0isdeferred to section 3.3. ! Onecan understand the loop contribution better by splitting the action into the sum 1 1 of the quadratic part and the rest, S = ¿ G¡ ¿ + S and expanding in S , 2 ¢ 0 ¢ i i 2 n 2 n ¯ Sk[¿ ] 1 1 ( 1) ¯ Sk[¿ ] tr ln = tr ln G0¡ ¡ tr G0 : (11) ¯ ¿~¯ ¿~ ¡ n Ã ¯ ¿~¯ ¿~ ¢ ! nX=1 Werecoveredthe sum of one-loop graphs. The loop corrections closeon allpossible ~ pair of ¿ legswith the propagator G0.The tree-levelpiece describes the feedbackof the change of the cut-o®on the dynamics ofthe classicalbackground ¯eld ¿ .This isclassical contribution because the cut-o®controls the con¯guration space,the number ofdegrees of freedom. Some of the graphs contributing to the right hand sideare shown inFig.1. The point of central importance isthat ¢k serveshere as anew smallparameter to suppress the higher-loop contributions. As an exampleconsider the simplest case, J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 7

~ ¿ x = ©, ¿ cl =0with the ansatz

1 2 Sk[¿ ] = Zk(¿ x)(@¹¿ x) + Uk(¿ x) ; (12) Zx · 2 ¸ for the action with Z(¿ ) = 1,

¹h 2 2 Uk ¢k(©) = Uk(©) + ln[p + Uk00(©)] + (¹h¢k) : (13) ¡ 2 k ¢k< p

2.1.2Local potential approximation

In order to relatethe WHequation to conventional perturbation expansion we shall use the ansatz (12) with Z =1,in the localpotential approximation. In order to determine the onlynon-trivial pieceof this action the potential Uk(¿ )it issu± cient to consider a homogeneous ¯eld ¿ (x)=©in (9). The saddle point isvanishing as mentioned above, ~ ¿ cl =0,so long as the external space-timesymmetries are unbroken. The blocking equation simpli¯ed in this manner yieldsEq. (13), the projection of the WHequation into the restricted spaceof actions, h¹ kd U_ (©) = d ln[k2 + U (2)(©)] (14) k ¡ 2(2º )d k 8 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Fig. 2 The rstfour one-loopgraphs contributing to the WHequationin the localpotential approximationwhen the potential U(¿ )istruncated to the terms ¿ 2 and ¿ 4.The dashedline correspondsa particleof momentum p = k. j j

d=2 where the dot stands for k@k and d = 2º =¡(d=2) denotes the solid angle. The expansion of the logarithmic function leadsto the series

n d n (2) 2 h¹ k 1 ( 1) U (©) m U_ (©) = d ln[k2 + m2] ¡ k ¡ ; (15) k ¡ 2(2º )d 8 ¡ n 0 k2 + m2 1 9 < nX=1 = @ A : ; where m2 = U (2)(0), the sum of the one-loop graphs whose external legshave vanishing momentum and the internal linescarry momentum p = k,cf Fig.2, the leading order j j contributions to the renormalization of the coupling constants gn in the loop expansion. The partial resummation of the perturbation seriesperformed by the WHequation can easilybe seenby comparing the solution of Eq.(14) with those obtained in the independent mode approximation, where the k-dependence isignored in the right hand sideof the equation,

h¹ 2 (2) Uk(©) = U¸(©) ln[p + U¤ (©)]: (16) ¡ 2 k< p <¤ Z j j

Onerecovers the one-loop e®ective potential at the IR¯xedpoint, Ve® (©) = Uk=0(©). This isnot by accident and remains validfor the completesolution, as well,because V V (©) e¡ e¬ isthe distribution of the homogeneous mode and the blocking(7) at k = 0 retains the homogeneous mode only, Sk=0[©] = V Uk=0(©). Weshallsee in section 3.3 that this simpleargument isvalid in the absence of largeamplitude inhomogeneous instabilities only. Let us consider as an examplethe di®erential equation

x_ k = f (xk); x¤ = xB ; (17) where f (x)isweakly varying analyticfunction givenin its expanded form around the base point x = 0, (n) f (0) n x_ k = xk : (18) n n! X The RGmethod allows us to perform the expansion at the current point,

2 xk ¢k = xk ¢kf(xk) + ¢k ; (19) ¡ ¡ O ³ ´ J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 9 cf Eq.(8). The k-dependence of the right hand siderepresents the accumulation of the information obtained during the integration. In asimilarmanner, the virtue of the functional RG scheme(9) isthe expansion ofthe path integral around around the current action, instead of the Gaussian ¯xedpoint. Amore detailed picture ofthe RG°ow can beobtained by inspecting the beta functions. Forthis end we parametrize the potential as g (k) U (¿ ) = n (¿ ¿ )n; (20) k n! ¡ 0 Xn and write the beta functions corresponding to ¿ = ¿ as 0 h i h¹ kd = k@ g (k) = k@ @nU (¿ ) = @nk@ U (¿ ) = d @n ln[k2 + @2U (¿ )] (21) n k n k Á k 0 Á k k 0 ¡ 2(2º )d Á Á k

It iseasy to seethat n is an n-th order polynomial

¹h kd = d [G ; ; G ]; (22) n ¡ 2(2º )d Pn 3 ¢ ¢ ¢ n+2

2 of the expression Gn = gn=(k + g2), eg

= G G2; P2 4 ¡ 3 = G 3G G + 2G3; P3 5 ¡ 3 4 3 = G 4G G + 12G2G 3G2 6G4; (23) P4 6 ¡ 3 5 3 4 ¡ 4 ¡ 3 = G 5G G + 20G2G 10G G 60G3G + 30G G2 + 24G5; P5 7 ¡ 3 6 3 5 ¡ 4 5 ¡ 3 4 3 4 3 = G 6G G + 30G2G 5G G 120G3G + 120G G G P6 8 ¡ 3 7 3 6 ¡ 5 6 ¡ 3 5 3 4 5 270G2G2 10G2 10G G + 360G4G + 30G3 120G6; ¡ 3 4 ¡ 5 ¡ 4 6 3 4 4 ¡ 3 etc.The correspondence between these expressions and Feynman graphs contributing to the beta functions in the traditional renormalization schemeis indicated in Fig.3. Weshall need later the beta functions corresponding to coupling constants made (d 2)=2 dimensionless by the running cut-o®. Bymeans of the parametrization ¿ = k ¡ ¿~, [gn] gn = k g~n, d [gn] = d + n 1 ; (24) Ã ¡ 2! U (¿ ) = kdU~(¿ ~) one ¯nds

[gn] ¹h d ~ (¿ ~) = g~_ (k) = k¡ (¿ ) [g ]~g = [G~ ; ; G~ ] [g ]~g ; (25) n n n ¡ n n ¡ 2(2º )d Pn 3 ¢ ¢ ¢ n+2 ¡ n n where G~n = g~n=(1 + g~2). An instructive way to read Eqs.(20) and (14) isthat the potential U (¿ ) is the generator function for the coupling constants. The functional RGmethod isan econom- icalbook-keeping method for the computation of graphs and their symmetry factors by keepingtrack of their generator function(al). 10 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

(a)

(b)

(c)

Fig. 3 Graphscontributing to n, (a): n = 2, (b): n = 3 and (c): n = 4.

Fig. 4 Atwo-loopcontribution to g2.

The higher loop contributions to the running coupling constants are generated by the integration of the di®erential equations k@kgn = n.The us consider the two-loop graph depicted in Fig.4 and expand the subgraph inthe square in the momenta ofits external legs.The zeroth order contributions isthe last graph of Fig.3b. Bygoing higher order in the gradient expansion one can,in principle,generate the full momentum dependence of the subgraph. In general,the n-th loop contributions appear in the integration after n step k k + ¢k and allloops are resummed inthe limit¢ k 0. ! ! 2.1.3Gradient expansion

The natural expansion for the long distance properties of systems with homogeneous ground state isthe expansion in the inversecharacteristic length of the °uctuations, in the gradient operator acting on the ¯eldvariable. Functional derivativesof localfunctionals: Weconsider the following generalization of the (@2)ansatz (12), O

1 1 S[¿ ] = Z(¿ x)@¹¿ xK ¡ ( h )@¹¿ x + U (¿ x) : (26) Zx · 2 ¡ ¸

Weshalluse the notation Zx = Z(¿ x), Ux = U (¿ x)for the coe±cient functions and (n) n (n) n Zx = @Á Z(¿ x), Ux = @Á U (¿ x)for their derivatives.The ¯rst two functional derivatives J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 11 of the action are ¯ S ¯ S V = eipy ¯ ¿ p ¯ ¿ y Xy ipy (1) 1 (1) 1 = e ¯ x;yUx + ¯ x;yZx @¹¿ xK ¡ @¹¿ x (27) Zx;y " 2 1 1 1 + Zx @¹¯ x;yK ¡ @¹¿ x + @¹¿ xK ¡ @¹¯ x;y ; 2 # ³ ´ and

2 2 ¯ S ipy+iqz 1 (2) 1 V = e ¯ x;y¯ x;zZx @¹¿ xK ¡ @¹¿ x ¯ ¿ p¯ ¿ q Zx;y;z (2 1 1 1 + Z @ ¯ K ¡ @ ¯ + @ ¯ K ¡ @ ¯ 2 x ¹ x;y ¹ x;z ¹ x;z ¹ x;y ³ ´ 1 (1) 1 1 + Z ¯ @ ¯ K ¡ @ ¿ + @ ¿ K ¡ @ ¯ 2 x x;y ¹ x;z ¹ x ¹ x ¹ x;z · ³ ´ 1 1 + ¯ x;z @¹¯ x;yK ¡ @¹¿ x + @¹¿ xK ¡ @¹¯ x;y ³ ´¸ (2) +¯ x;y¯ x;zUx : (28) )

Wesplit the ¯eldinto ahomogeneous background and °uctuations, ¿ x = © + ² x and ¯nd up to (² 2), O

¯ S ipy (1) (2) 1 (3) 2 1 (1) 1 V = e ¯ x;y U + U ² x + U ² x + ¯ x;yZ @¹² xK ¡ @¹² x ¯ ¿ p Zx;y ( µ 2 ¶ 2" (1) 1 1 +(Z + Z ² x) @¹¯ x;yK ¡ @¹² x + @¹² xK ¡ @¹¯ x;y #) ³ ´ ipy (1) (2) ir x 1 (3) i(r+s) x = e ¯ x;y U + U e ¢ ² r + U e ¢ ² r² s Zx;y (µ Zr 2 Zr;s ¶ 1 (1) i(r+s) x 1 Z e ¢ r s² rKs¡ ² s ¡ 2" Zr;s ¢ (1) ir x is x 1 1 + Z + Z e ¢ ² r p se ¢ ² s Ks¡ + Kp¡ r s ¢ #) µ Z ¶ Z ³ ´ (1) (2) 1 2 1 1 = ¯ p;0U + U ² p + p Z(K ¡p + Kp¡ )² p ¡ 2 ¡ ¡ 1 (3) (1) 1 1 1 + ¯ p+r+s;0² r² s U Z r sKs¡ + p rKr¡ + p sKp¡ (29) 2 r;s ¡ ¢ ¢ ¢ Z h ³ ´i and

2 2 ¯ S ipy+iqz (2) (3) 1 (4) 2 V = e ¯ x;y¯ x;z U + U ² x + U ² x ¯ ¿ p¯ ¿ q Zx;y;z ( µ 2 ¶ 1 (2) 1 + ¯ x;y¯ x;zZ @¹² xK ¡ @¹² x 2" (1) (2) 1 1 +¯ x;y(Z + Z ² x) @¹¯ x;zK ¡ @¹² x + @¹² xK ¡ @¹¯ x;z ³ ´ 12 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

(1) (2) 1 1 +¯ x;z(Z + Z ² x) @¹¯ x;yK ¡ @¹² x + @¹² xK ¡ @¹¯ x;y ³ ´ (1) 1 (2) 2 1 1 + Z + Z ² x + Z ² x @¹¯ x;zK ¡ @¹¯ x;y + @¹¯ x;yK ¡ @¹¯ x;z 2 #) µ ¶ ³ ´ ipy+iqz 1 (2) irx 1 isx = e ¯ x;y¯ x;z Z r se ² rKs¡ e ² s Zx;y;z (¡ 2" Zr;s ¢ (1) (2) irx isx 1 1 1 1 + Z + Z e ² r e ² s q s(Ks¡ + Kq¡ ) + p s(Ks¡ + Kp¡ ) r s ¢ ¢ µ Z ¶ Z ³ ´ (1) irx 1 (2) i(r+s)x 1 1 +p q Z + Z e ² r + Z e ² r² s (Kp¡ + Kq¡ ) ¢ µ Zr 2 Zr;s ¶ # (2) (3) irx 1 (4) i(r+s)x + U + U e ² r + U e ² r² s µ Zr 2 Zr;s ¶) (2) 1 1 1 = ¯ p+q;0 U Zp q(Kp¡ + Kq¡ ) · ¡ 2 ¢ ¸ (3) 1 (1) 1 1 + ¯ p+q+r;0² r U Z p q(Kp¡ + Kq¡ ) Zr · ¡ 2 µ ¢ 1 1 1 1 +q r(Kr¡ + Kq¡ ) + p r(Kr¡ + Kp¡ ) ¢ ¢ ¶¸ 1 (4) (2) 1 1 1 1 + ¯ p+q+r+s;0² r² s U Z p q(Kp¡ + Kq¡ ) + r sKs¡ 2 Zr;s · ¡ µ 2 ¢ ¢ 1 1 1 1 +q s(Ks¡ + Kq¡ ) + p r(Kp¡ + Kr¡ ) ; (30) ¢ ¢ ¶¸ (n) (n) (n) (n) 2 where Z = Z (©), U = U (©) and Ks = K(s ). WHequation in (² 2) : We set K =1in (26), use sharp cut-o®, i.e.all momentum O integral isfor k ¢k < p < k, ¡ j j 2 2 ¯ S (2) (3) (1) V = ¯ p+q;0 U Zp q + ¯ p+q+r;0² r U Z (p q + q r + p r) ¯ ¿ ¯ ¿ ¡ ¢ r " ¡ ¢ ¢ ¢ # p q ³ ´ Z 1 (4) (2) + ¯ p+q+r+s;0² r² s U Z (p q + r s + q s + p s + q r + p r) 2 Zr;s " ¡ ¢ ¢ ¢ ¢ ¢ ¢ # 1 = Gp;q¡ + Ap;q + Bp;q; (31) with

1 (2) 1 2 2 Gp;q¡ = ¯ p+q;0 U + Z(p + q ) ; · 2 ¸ (3) 1 (1) 2 2 2 Ap;q = ¯ p+q+r;0² r U + Z (p + q + r ) ; Zr " 2 # 1 (4) 1 (2) 2 2 2 2 Bp;q = ¯ p+q+r+s;0² r² s U + Z (p + q + r + s ) : (32) 2 Zr;s " 2 # As emphasised above,the logarithmic function on the right hand sideof the WH equation iscomputed within the spaceof ¯eldcon¯ gurations whose Fourier transform is non-vanishing for k ¢k < p < k only.In order to keepthis constrain in an explicit ¡ j j manner we introduce the projection operator corresponding to this spaceand write P the rest of the (² 2)evolution equation as O h¹ 1 @ S[¿ ] = tr log[G¡ + A + B] k ¡ 2¢k J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 13

h¹ 1 h¹ h¹ = tr log G¡ tr[ G(A + B)] + tr[ GA GA]: (33) ¡ 2¢k P ¡ 2¢k P 4¢k P P Sincethe propagator does not leadout from the function space,[ ; G]=0,it issu± cient P to makethe replacement G G on the right hand sideof the evolution equation. ! P The term (² )drops owing to the vanishing of the homogeneous component of the O °uctuation, ² p=0 = 0 and we ¯nd

d 1 2 (2) h¹k 2 (2) ² r² r Z_ r + U_ + V U_ = de V log[Zk + U ] 2 r ¡ ¡ 2(2º )d ( Z h i Z 2 U (3) + 1 Z (1)(k2 + (ek r)2 + r2) 2 ¡ ² r² r r ek;ek r · ¸ ¡ r ¡ P ¡ ¡ 2(Zk2 + U (2))(Z(ek r)2 + U (2)) Z ¡ U (4) + Z (2)(k2 + r2) + ² r² r 2 (2) ; (34) Zr ¡ 2(Zk + U ) ) including anintegration overthe direction of k,the unit vector e. The identi¯cation ofthe terms (² 0r0)results the equations O ¹hkd U_ = d log[Zk2 + U (2)]; (35) ¡ 2(2º )d c.f.Eq. (14). The terms (² 2r2) give O d ¹hk ² r² r _ 2 ¡ (2) 2 Z ² r² rr = d 2 (2) de Z r Zr ¡ ¡ 2(2º ) Zr Zk + U Z ( (3) (1) 2 (3) (1) 2 2 2 (1) U + Z k 2 (U + Z k ) r ek;ek r 2r Z 2 (2) Zr 2 (2) 2 ¡ P ¡ ¡ · Zk + U ¡ (Zk + U ) (1)2 (3) (1) 2 2 2 Z (1) U + Z k +k (er) 2 (2) ZZ 2 (2) 2 : (36) Ã Zk + U ¡ (Zk + U ) !¸) Finally, in (² 2r0) we ¯nd O d h¹k ² r² r _ (2) ¡ (2) 2 U ² r² r = d 2 (2) de Z k Zr ¡ ¡ 2(2º ) Zr Zk + U Z · (1) 2 (3) 2 (4) (Z k + U ) +U r ek;ek r 2 (2) : (37) ¡ P ¡ ¡ Zk + U ¸

Let us makethe replacement r k;k r 1 + [ r k;k r 1]in Eqs.(35)-(37) and call P ¡ ¡ ! P ¡ ¡ ¡ the contributions corresponding to 1and k r;k r 1regular and irregular, respectively. P ¡ ¡ ¡ The regular contribution can be obtained from the one-loop graphs. This has already been seenfor Eq.(35) and the regular part of Eq.(37) isjust the second derivativeof Eq. (35). But there are problems with the irregular contributions which represent the cut-o®in the multi-loop integrals. Oneobvious problem isthe inconsistency between Eqs.(35) and (37). Another problem isthat the irregular contributions include atruncated r-integration and asaresult the left and right hand sides ofthe equations havedi® erent ² -dependence. 14 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

This could beavoided by imposing r << ¢k for the momentum ofthe °uctuations. But the priceis unacceptable high sinceit would indicate that the radius of convergencefor the gradient expansion issmallerthen the in¯nitesimal ¢ k. Another problem the projection operator leadsto isnon-locality .It isknown that the bare action isnon-local at the scaleof the U.V.cut-o® , i.e.it contains higher order derivativesup to a¯nite order. In fact, the gradient @n=@x¹n represents couplings up to n latticespacing distances when latticeregularization isused. But terms in the action which are non-polynomial in the gradient induce correlations at ¯nite,i.e. cut-o® independent distances and are not acceptable.Returning to the gradient expansion we note that the operator r k;k r restricts the integration domain for r in such anon-isotropical manner P ¡ ¡ that not only r2 but r appears, too. This latter corresponds to the non-local operator j j ph in the gradient expansion. This problem, inherent in the sharp cut-o®procedure in the momentum spaceis rather general.The one-loop correction for the two-particle scattering amplitude in the ¿ 4 model,represented by the fourth graph in Fig.3c has anon-local contribution when the momenta inboth internal linesare properly restricted by the U.V.cut-o®. Such non- locale® ects, disregarded inthe usual textbooks, might wellbe negligiblein arenormalized theory because they are represented by non-renormalizable, irrelevant operators such as ¿ 3ph ¿ .Therefore one hopes that their e®ects become weak when the cut-o®is removed. The more careful analysisis rather involvedsince the details of the cut-o®procedure within the momentum regime(1 ° )k < p < (1 + ° )k in°uences the dynamics on the ¡ distance scale¹h =(2° k)which isaquantity ofthe form 0 . ¢ 1 But itisnot so easyto dispel the doubts. First, the irrelevanceand unimportance ofa coupling constant are quitedi® erent concepts. An irrelevant coupling constant approaches its I.R.¯xedpoint valueas we movein the I.R.direction. But its valueat the ¯xedpoint maybe strong, indicating that the coupling constant in question isimportant. It isthe scaledependence onlywhat becomesunimportant for an irrelevant coupling constant and not its presence.The higher order verticesinduced atlow energiesin astrongly coupled selfinteracting scalar¯ eldtheory in d < 4mayserve as an exampleof this di®erence. Second, the global viewof the renormalization group, outlined in Section 3.2suggest that acoupling constant which isirrelevant at the U.V.¯xedpoint does not necessarilyremain so at around the I.R.¯xedpoint. Finally,this problem isobviously present in e®ective theories where the cut-o®reaches physical scales.

2.2 Resumming theperturbation expansion

The basicidea leading to the WHequation isthe successiveelimination of the degrees of freedom. This procedure produces the blockedaction, the integrand ofthe functional integral of the blockedtheory .This integrand must be wellde¯ ned, ieeach mode must either beleft intact or becompletelyeliminated during the blocking.This isthe positive sideof the sharp cut-o®in momentum space.The negativeside is that itgenerates non- localinteractions and spoils the gradient expansion. The simplest cure,to smear out the J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 15 regulator and to use smooth cut-o®procedure, isnot avalidalternative. This isbecause the blockingbased on smooth cut-o®suppresses the modes partially and the integrand for the functional integral ofthe blockedtheory isill de¯ ned. Despite this general sounding argument one can proceed and generalizethe successiveelimination process for smooth cut-o®in arather surprising manner.

2.2.1Polchinski equation

Let us start with the partition function

1 1 1 I ©G¡© S [©] Zk = D[©]e¡ 2·h k ¡ h· k ; (38) Z I where the interaction functional Sk [©]corresponds to the U.V.cut-o® k.Wesplit the propagator and the ¯eldvariable into IRand U.V.components,

Gk = Gk ¢k + G~k; © = ¿ + ¿ ~; (39) ¡ with the intention that the ¯elds ¿ and ¿~ should propagate with Gk ¢k and G~k, respec- ¡ tively.In other words, the kineticenergy contribution to the action issupposed to beof 1 1 ~ ~ 1 ~ ~ the form (¿ Gk¡ ¢k¿ + 2 ¿ Gk¡ ¿ )=2when written in terms of ¿ and ¿ . ¡ The introduction of the ¯elds ¿ and ¿ ~ separated by asmooth cut-o®makes both ~ ¿ p and ¿ p non-vanishing and appears as adouble counting of the degrees of freedom. In order to ensure the proper integration measure we introduce adummy ¯eld, ©~, in such amanner that asuitable, momentum dependent linear combination of ¿ p and ¿ ~p reproduces ©p and ©~ p,

© A1;1(h ) A1;2(h ) ¿ = ~ (40) Ã ©~ ! Ã A2;1(h ) A2;2(h ) ! Ã ¿ ! and decouple ©~, 1 1 1 ~ ~ 1 ~ 1 1 1 ~ ~ 1 ~ ¿ Gk¡ ¢k¿ + ¿ Gk¡ ¿ = ©Gk¡ © + ©GD¡ ©; (41) 2 ¡ 2 2 2 with afreelychosen dummy propagator G~D.Owing to the second equation in (39)

A1;1 = A1;2 =1.The condition (41) gives

1 1 Gk¡ ¢k 0 1 A2;1(h ) Gk¡ 0 1 1 ¡ ~ 1 = ~ 1 ; (42) Ã 0 Gk¡ ! Ã 1 A2;2(h ) ! Ã 0 GD¡ ! Ã A2;1(h ) A2;2(h ) ! whose solution is A2;1(h ) = G~DG~k=Gk ¢kGk, A2;2(h ) = G~DGk ¢k=G~kGk. The trans- ¡ ¡ formation (40) isnon-singular q so long as G~k = Gk ¢k,theq propagations of ¿ and ¿ ~ are 6 ¡ distinguishable. Weare ¯nallyin the position to describe the blockingprocedure which consists of the following steps: (1) Redouble the number of degrees offreedom, © (©; ©~). ! (2) Rotate the new, dummy degrees offreedom into the dynamics in such manner that the rotated ¯eldssplit the original ¯eldvariable and follow the prescribed propagation. 16 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

(3) Integrate out ¿~. In order to completestep 3.we insert a¯eldindependent constant into the partition function, 1 1 1 1 1 I ©~ G~¡©~ ©G¡© S [©] Zk = D[©]D[©~]e¡ 2·h D ¡ 2·h k ¡ h· k : (43) Z The redistribution of the degrees of freedom by the inverseof the transformation (40) yields 1 1 1 ~ ~ 1 ~ 1 I ~ ~ 2·hÁGk¡¢ kÁ 2·h ÁGk¡Á h· Sk[Á+Á] Zk = D[¿ ]D[¿ ]e¡ ¡ ¡ ¡ ; (44) Z up to aconstant. This motivates the introduction of the blockedaction Sk ¢k[¿ ] de¯ned ¡ as 1 1 1 I Sk ¢ k[Á] Á~G~¡Á~ S [Á+Á~] e¡ ¡ = D[¿~]e¡ 2·h k ¡ h· k : (45) Z The higher loop corrections were suppressed in the WHequation by restricting the functional spaceof the blocking,the volumeof the loop integration inmomentum space. But there isanother way to suppress the radiative corrections, to decreasethe propagator. Such asuppression method isbetter suited for smooth cut-o®where the U.V.¯eldcan not be constrained into arestricted, ’small’functional space.This strategy isimplemented by requiring that in the so far arbitrary split (39) G~ (p) = (¢k).The obvious choice k O is

G~k(p) = ¢k@kGk(p): (46) Byassuming the convergenceof the perturbation expansion we expand the action in ¿ ~ and perform the integration over ¿~.Sincethe new smallparameter isthe propagator we retain graphs where the propagator appears in aminimalnumber. Byassuming that the saddle point isatmost (¢k)we can expand the action in ¿~, O ¯ SI[¿ ] ¯ 2SI[¿ ] 1 I 1 1 1 I 1 k 1 k 3 Á~G~¡Á~ S [Á] Á~ Á~ Á~+ Á~ h· Sk ¢ k[Á] ~ 2·h k h· k h· ¯ ¿ 2·h ¯ ¿ ¯ ¿ ( ) e¡ ¡ = D[¿ ]e¡ ¡ ¡ ¢ ¡ ¢ ¢ O

Z 1 ¯ SI[¿ ] ¯ 2SI[¿ ] ¡ ¯ SI[¿ ] ¯ 2SI[¿ ] 1 I 1 k 1 k k 1 1 k 2 S [Á]+ G~¡+ tr log G~¡+ + ¢k ¡ h· k 2·h ¯ ¿ ¢ k ¯ ¿ ¯ ¿ ¢ ¯ ¿ ¡ 2 k ¯ ¿ ¯ ¿ O( ) = e µ ¶ µ ¶ ¯ SI[¿ ] ¯ SI[¿ ] ¯ 2SI[¿ ] 1 SI[Á]+ ¢ k 1 k @ G k h· tr @ G k + ¢k2 ¡ h· k h· 2 ¯ ¿ ¢ k k¢ ¯ ¿ ¡ 2 k k ¯ ¿ ¯ ¿ O( ) = e · µ ¶¸ ¹h¢k ¯ ¯ 1 SI[Á] 2 = 1 @kGk e¡ h· k + ¢k : (47) " ¡ 2 ¯ ¿ ¢ ¢ ¯ ¿ # O ³ ´ The third equation was obtained by expanding the logarithmic function and givesthe di®erential equation [9],[10]

I I 2 I I 1 ¯ Sk[¿ ] ¯ Sk[¿ ] ¹h ¯ Sk [¿ ] @kSk [¿ ] = @kGk tr @kGk : (48) 2 ¯ ¿ ¢ ¢ ¯ ¿ ¡ 2 " ¯ ¿ ¯ ¿ # The fourth lineleads to alinear equation,

1 SI[Á] 1 SI[Á] 1 ¯ ¯ k@ e¡ h· k = h¹ e¡ h· k ; = k@ G : (49) k B B 2 ¯ ¿ ¢ k k ¢ ¯ ¿ It isinstructive to noticethe similaritiesand di®erences with the WHequation. Eq.(45) looksas the starting point for the derivation of the WHequation and could J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 17 havebeen obtained in atrivialmanner for sharp cut-o®. The crux of the argument leading to this relation isthat this simple-lookingequation isactually valid for smooth cut-o®. But in asurprising manner it isjust the sharp cut-o®limit when the rest of the derivation of Polchinksiequation isinvalid. This isbecause G~k as givenin (46) is 1 0 ¢k (¢k¡ ) = (¢k ). O O Eq.(48) isa leading order equation in the perturbation rather than the loop expansion. It isformally similarto the WHequation, (10) when onlythe leading order is retained in the expansion (11) which corresponds to keepinggraphs with asinglevertex onlyin Fig.1(b) and using the dashed linesto denote the propagation of ¿ ~.Noticethat evenif the higher order contributions of the perturbation expansion are suppressed in the limit ¢k 0,the convergenceof the perturbation expansion was assumed. ! The di®erence between WHequation and (48) isthat the tree-level¯ rst term on the right hand sideis always non-vanishing in Eq.(48), contrary to the caseof the WH equation. It iseasier to understand the reason of the non-trivial saddle point when the blocking(6) isconsidered in the presence of the source term. The source can beomitted and the cut-o®independence ofasinglequantity ,the partition function can beconsidered as the basicequation when the source has vanishing component inthe functional subspace to integrate overin the blocking.In caseof asmooth cut-o®the background ¯eldhas arbitrary momentum component and we can not reallyomit the source. Oneought to follow the more cumbersome blockingwhich keepsthe generator functional RG invariant. When classicalphysics (action) ismodi¯ ed by changing the cut-o®, the ¯eldinduced by the source changes, as well.This isthe origin of the tree-levelrenormalization. But we made ashort-cut and omitted the source. The priceis that the tree-levelrenormalization can onlybe understood by letting the background ¯eld, ¿ varyfreely . The understanding how the ¿ -dependence maygive rise the tree-levelrenormalization in (47) isan interesting demonstration of the wellknow theorem stating that the expansion in loops and inh ¹ are equivalent.In fact, the ¯rst equation in (47) introduces an e®ective theory for ¿~ and the ¯rst two terms in the exponent of the second linegive the e®ective action corresponding to agivenbackground ¯eld ¿ in the classicalapproxi- mation.In other words, the action with lowered cut-o®which issupposed to reproduce the action with the original cut-o®for arbitrary con¯guration ¿ requires tree-leveladjust- 1 ment. The second term in the exponent has two factors of ¹h¡ coming from the ’source’ 1 I ~ ~ ¹h¡ ¯ Sk [¿ ]=¯ ¿ of the ¯eld ¿ and afactor ofh ¹ from the ¿ propagator. The overall1 =h¹ disappears in the di®erential equation (48) which displaysh ¹ times the exponent. This is the way the graphs of Fig.1(b) appear on the h¹0 tree-levelin Eq.(48). Another O reason isthat they are graphs with no freelychanging ³ ´ loop-momentum variable.One can understand in asimilarmanner that both the tree- and the loop-levelgraphs appear with acommon factor ofh¹ in Eq. (49). 18 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

2.2.2Gradient expansion

Weshall use here the results obtained insection 2.1.3with the ansatz

I 1 1 2 S [¿ ] = (Z(¿ x) 1)@¹¿ xK ¡ ( h =k )@¹¿ x + U (¿ x) (50) Zx · 2 ¡ ¡ ¸ to derivethe gradient expansion for Eq.(48). For the left hand sidewe ¯nd in the given order ofthe gradient expansion

_ 2 _ (2) Z p Z 1 2 _ 1 _ U ² p² p + ¡ ² p² pp Kp¡ + V U + ² p² p 2 Zp ¡ Kp 2 Zp ¡ 2 Zp ¡ Z_ p2 U_ (2) ² p² p + V U_ + ² p² p: (51) º 2 Zp ¡ Kp 2 Zp ¡

2 2 2 Bymeans of the relation k@ G = 2K 0 =k where K 0 = dK(z)=dz, z = p =k the right k ¡ p p hand sidegives

2 (1) (2) p 1 (3) (1) r (p r) p r p (p r) ¯ U + ² U + (Z 1) + ² ² U Z ¢ ¡ ¢ ¢ ¡ p p;0 p Kp 2 r r p r Kp r Kr Kp ¡ ( ¡ ¡ ¡ ¡ ¡ ¡ ) R ³ ´ R h ³ í 2 Kp0 (1) (2) p 1 (3) (1) p s (p+ s) s p (p+ s) 2 ¯ U + ² U + (Z 1) + ² ² U Z ¢ ¢ ¢ k p;0 p Kp 2 s p s s Kp Ks K p s £ ( ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ) ³ ´ R h ³ í 2 2 2 Kp0 (2) p 1 (4) (2) p r +·h 2 V U + (Z 1) + ² ² U + Z + p k Kp 2 r r r Kp Kr ( ¡ ¡ ) R ³ ´ R h ³ í 2 2 2 2 2 (1)2 K0 K0 + K0 0 K0r =(k Kr) (2) r K0 (1) (3) (1) r V U 2 ² r² r 2 U + (Z 1) + 2 U U + Z º ¡ k ¡ r ¡ k ¡ Kr k Kr · ¸ R ³ ´ ³ ´ 2 2 2 Kp0 (2) p 1 (4) (2) p r +·h 2 V U + (Z 1) + ² ² U + Z + p k Kp 2 r r r Kp Kr (52) ( ¡ ¡ ) R ³ ´ R h ³ í This equation isconsistent sincethe terms (² 2)are comparable thanks to the smooth- O ness of the cut-o®. Wecompare the coe±cients of the same ² -dependent expressions and ¯nd

2 2 _ (1)2 (2) p k U = U K00 + ¹h Kp0 U + (Z 1) ; ¡ Zp Ã ¡ Kp ! 2 (2) (1) (1) K000K0 (2)2 (2) _ 0 0 0 k Z = 4K0U (Z 1) 2K0U Z 2 2 U + ¹hZ Kp; (53) ¡ ¡ ¡ ¡ k Zp and 2 2 _ (2) (2)2 (1) (3) (4) (2) p k U = 2K00 U 2K 0 U U + h¹ Kp0 U + Z (54) ¡ ¡ Zp Ã Kp ! Wehavetwo independent equations for two functions because the Eq.(54) follows from the ¯rst equation of(53). It isinstructive to checkthe ¯rst equation in the localpotential approximation Z = 1,

(1)2 (2) _ U k@kKp ¹hU k@kKp U = 2 2 : (55) 2 p p=0 ¡ 2 p p j Z J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 19

The ¯rst term describes the tree-leveladjustment of the bare action in keepingthe physics cut-o®independent and someof the corresponding graphs are depicted inFig.1(a). The present form, characteristic of the gradient expansion, isnon-vanishing when the change of the cut-o®at p = k isfelt at the base point of the gradient expansion, p = 0. The second term isthe leading order perturbative contribution to the WHequation (14) when the mass term istreated perturbatively,asexpectedand the beta functions,

(n+2) h¹U k@kKp n(¿ ) = 2 ; (56) ¡ 2 Zp p for n > 2correspond to the leading order ¯rst column in Eqs.(23) with g2 = 0. The independent mode approximation reproduces the leading order renormalization of the coupling constants and the numerical integration resums the perturbation expansion in the approximation where the e®ective vertices are replaced by their valuesat p = 0. The leading order perturbative contributions, retained in the RG equation are linearin the coupling constants and the classical,non-linear terms drop out in the linearizedevolution around the Gaussian ¯xedpoint therefore the usual criticalexponents are reproduced.

2.3 Composite operatorrenormalization

The renormalization ofcomposite operators [11]seems to beahighly technicaland subtle issue,dealing with the removalof the U.V.divergences from Green functions where composite operators are inserted. But it becomesmore elementaryand general[12] as soon as we are ready to giveup the perturbative approach and can reformulate the procedure in ageneral,non-perturbative manner. In order to seethis letus go backto the remark made at Eq.(6). It isnot enough to impose the RG invarianceof the partition function onlysince it yieldsa singleequation for in¯nitely many coupling constants. Instead one should require the RGinvarianceof the generator functional for Green functions because any observable can be reconstructed from this functional. This turns out to be rather cumbersome due to the source dependence generated for the blockedaction [13].W e took another direction by imposing the RG invarianceof another functional, the blocked action, cf Eqs.(7) and (45). Instead of this strategy we shall now use the matching of observables computed at di®erent valuesof the cut-o®to rederivethe RGequation. This isa non-perturbative, blockinginspired generalization of the multiplicativeRG schemes mentioned in the Introduction in what the scaledependence of the observables isfollowed instead ofthose ofnon-physical bare coupling constants. Westart with atoy modelto demonstrate the natural relation between blockingand composite operator renormalization and the ¯eldtheoretical application follows next by generalizingthe source term inthe generator functional. This term which usually involves the elementary¯ eldonly is extended here for any localoperator with inhomogeneous source and we consider it as part of the action which now contains inhomogeneous coupling constants. Our main point isthat the beta functions corresponding to this extended action provide abridge between the blockingand the composite operator renormalization. 20 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

2.3.1T oymodel

Consider the two dimensional integral

S(x;y) Z = dxdye¡ (57) Z over x and y which playthe roleof low- and high-frequency variables,respectively ,where the bare action isgiven by the expression

1 1 1 S(x; y) = s x2 + s y2 + g (x + y)n (58) 2 x 2 y n nX=0 in terms of bare coupling constants gn. We use h¹ =1in this section.The blockedaction is de¯ned as S(x) S(x;y) e¡ = dye¡ : (59) Z The elementary,bare operators are ( x + y)n with n = 0; 1; 2; : : : and their expectation values are @S(x) S(x) n S(x;y) dx e¡ dxdy(x + y) e¡ @gn S(x;y) = S(x) : (60) R dxdye¡ R dxe¡ The right hand sideof this equationR expressesthe expectationR valueof the bare operator in terms ofanoperator ofthe blocked,’thinner’ system.The action S(x)can beexpanded in the terms ofthe operators xn,

1 2 1 m S(x) = s x + g0 x ; (61) 2 x m mX=0 givingrise the composite operators

n @S(x) 1 m x = = x Sm;n (62) f g @gn mX=0 where @gm0 Sm;n = : (63) @gn Wehavechosen abasis for operators inthe bare theory and searched for operators in the blockedtheory .The opposite question, the construction of the composite operators [(x + y)n]of the bare theory which reproduce the expectation valuesof the blocked operators gives n S(x;y) n S(x) dxdy[(x + y) ]e¡ dxx e¡ S(x;y) = S(x) ; (64) R dxdye¡ R dxe¡ with R R n 1 m 1 [(x + y) ] = (x + y) (S¡ )m;n: (65) mX=0 It isinstructive to generalizethe toy model for three variables,

1 1 1 1 S(x; y; z) = s x2 + s y2 + s z2 + g (x + y + z)n; (66) 2 x 2 y 2 z n nX=0 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 21 with

S(x;y) S(x;y;z) e¡ = dze¡ ; Z S(x) S(x;y) e¡ = dye¡ ; (67) Z where the operator mixinglooks like

n @S(x; y; z) n x z = = (x + y + z) ; f g @gn @ S(x;y;z) dze¡ n @S(x; y) @gn x y = = S(x;y;z) ; f g @gn ¡ dRze¡ @ S(x;y;z) dydze¡ n @S(x) @gn R x x = = S(x;y;z) : (68) f g @gn ¡ dRydze¡ These relations yield R

n S(x;y;z) n dz x ze¡ x y = f g S(x;y;z) ; f g R dze¡ n S(x;y) n dRy x ye¡ x x = f g S(x;y) ; (69) f g R dye¡ indicating that the evolution of the operatorsR comesfrom the elimination of the ¯eld variablein their de¯nition. Wecan compute the expectation valueof xn at any level, f g n S(x;y;z) n S(x;y) n S(x) dxdydz x ze¡ dxdy x ye¡ dx x xe¡ f g S(x;y;z) = f g S(x;y) = f g S(x) : (70) R dxdydze¡ R dxdye¡ R dxe¡ The lessonR of this toy modelis twofold. R First, one seesthat theR formal de¯nition of the renormalized operator at two di®erent valueof the cut-o®di® ers because these operators are supposed to reproduce the sameaverages by means of di®erent number ofdegrees of freedom. Second, it shows that the cut-o®dependence of the renormalized operators can be obtained in anatural and simplemanner by considering the bare coupling constants at one valueof the cut-o®as the functions of the coupling constants givenat another valueof the cut-o®.

2.3.2Quantum FieldTheory

Wegeneralizethe operator mixingof the toy model for the scalar ¯eldtheory givenby the bare action

S¤[¿ ] = Gn;x(¤)On(¿ x) = Gn~ (¤)On~(¿ x); (71) x n Z X Xn~ where On(¿ x)represents acompleteset oflocaloperators (functions of ¿ x and its space- timederivatives) and Gn;x(¤)denotes the coupling constant. Wesimplifythe expressions by introducing asingleindex n~ for the pair ( n; x)labelingthe basis elementsfor the local ~ operators and n~ = n x.The decomposition ¿ x = ¿ k;x + ¿ k;x ofthe scalar ¯eldinto a low- and high-frequencyP P R parts iscarried out asinEqs.(7) and (39). 22 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

The Kadano®-Wilson blocking

Sk[Ák] S¤ [Ák+Á~k] e¡ = D[¿~k]e¡ (72) Z for the action

Sk[¿ k] = Gn~(k)On~(¿ k) (73) Xn~ generates the RG°ow

k@kGn~ (k) = n~(k; G): (74) The blockedoperators are de¯ned by

~ ~ ~ S¤ [Ák+Ák] ¯ Sk[¿ k] D[¿ k]On~(¿ k + ¿ k)e¡ On~(¿ k) k = = (75) S [Á +Á~ ] f g ¯ Gn~ (¤) R D[¿ ~k]e¡ ¤ k k R in agreement with Eq.(68). They satisfy the equation

±Sk[Ák] S [Á ] ~ D[¿ ] e¡ k k ~ ~ S¤ [Ák+Ák] k ±Gn~(¤) D[¿ k]D[¿ k]On~(¿ k + ¿ k)e¡ = ~ ; (76) Sk[Ák] ~ S¤ [Ák+Ák] R D[¿ k]e¡ R D[¿ k]D[¿ k]e¡ R R c.f.Eq. (70), showing that they are represented inthe e®ective theory by the functional derivativeof the blockedaction with respect to the microscopicalcoupling constants. The form (73) of the action givesthe operator mixing

¯ Gm~ (k) ¯ Sk[¿ k] On~(¿ k) k = = Om~ (¿ k)Sm~ n~ (k; ¤); (77) f g ¯ Gn~ (¤) ¯ Gm~ (k) Xm~ Xm~ in amanner similarto Eq.(69). Weintroduced here the sensitivitymatrix

¯ Gm~ (k) Sm;~ n~(k; ¤) = : (78) ¯ Gn~(¤) It isthe measure of the sensitivityof an e®ective strength of interaction on the initial condition of the RG trajectory,imposed in the U.V.domain, c.f.Eq. (153). The composite operator O (¿ ) introduced in Eq.(77) replacesthe bare operator O (¿ ) f n~ k gk f n~ g¤ in the Green functions of the e®ective theory for the I.R.modes ¿ k. The di®erential equation generating the operator mixing(77) isobtained in the following manner. The relation ¢k G (k ¢k) = G (k) (k; G) (79) n~ ¡ n~ ¡ k n~ isused to arriveat the sensitivitymatrix,

¯ Gm~ (k ¢k) ¯ Gm~ (k ¢k) ¯ G`~(k) Sm;~ n~(k ¢k; ¤) = ¡ = ¡ ¡ ¯ Gn~(¤) ¯ G~(k) ¯ Gn~(¤) X`~ ` ¢k ¯ m~ (k; G) = ¯ m~ `~ S`;~ n~(k) (80) " ¡ k ¯ G~(k) # X`~ ` J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 23 and to ¯nd ¯ m~ (k; G) k@kSm;~ n~(k; ¤) = S`;~ n~(k; ¤): (81) ¯ G~(k) X`~ ` The scaledependence ofthe operator mixingmatrix isgoverned by

1 ¯ n~ (k; G) ® n~m~ (k) = : (82) k ¯ Gm~ (k)

Summary: Eq.(74) represents the linkbetween Kadano®-Wilson blocking and composite operator renormalization. On the one hand, n~ describes the evolution of the action in the traditional blockingscheme, Eq. (72). Onthe other hand, interpreting the coupling constants inthe action as sources coupled to composite operators n~ determines the mixingof composite operators in Eq.(82). Therefore allconepts and results of the blocking,e.g. ¯ xedpoint, universality,etc.has acounterpart in composite operator renormalization.

2.3.3Parallel transport

The operator mixingdiscussed abovehas anicegeometrical interpretation, aparallel transport of operators along the RGtrajectory [14]with the connection ¡n;~ m~ (k). The independence of the expectation valuefrom the cut-o®,

S [Á +Á~ ] Sk[Ák+Á~k] ~ ~ k0 k0 k0 ~ ~ D[¿ k0]D[¿ k0]On~(¿ k0 + ¿ k0)e¡ D[¿ k]D[¿ k]On~ (¿ k + ¿ k)e¡ ~ = ~ (83) ~ Sk [Ák +Ák ] ~ Sk[Ák+Ák] R D[¿ k0]D[¿ k0]e¡ 0 0 0 R D[¿ k]D[¿ k]e¡ R R de¯nes the paralleltransport of composite operators. The linearityof the mixing(77) assures that this paralleltransport isindeed linearand can therefore becharacterized by acovariant derivative D O = (@ ¡)O (84) k k k ¡ k 1 where ¡ = (S¡ ® S) in such amanner that D O =0along the RG °ow. m;~ n~ ¢ ¢ n;~ m~ k k The dynamicalorigin of the connection comesfrom the fact that there are two di®erent sources the scaledependence of O comesfrom: from the explicit k-dependence of the h ki operator and from the implicit k-dependence due to the cut-o®in the path integration. The operator mixingis to balance them. In fact, the covariant derivativecould havebeen introduced by the relation @ O = D O ; (85) kh ki h k ki requiring that the operator mixinggenerated by the connection isto makeup the implicit k-dependence of the expectation valuecoming from the cut-o®. It isobvious that ¡isvanishing inthe basis O , f n~ gk @ c (k) O = @ c (k) O = @ c (k) O : (86) kh n~ f n~gki k n~ hf n~gki h k n~ f n~gki Xn~ Xn~ Xn~ The connection can inprinciple be found in any other basis by simplecomputation. 24 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

2.3.4Asymptotical scaling

Welinearizethe beta-functions around a¯xedpoint Gm¤~ ,

¡¤ (G G¤ ); (87) n~ º n~m~ m~ ¡ m~ Xm~ and write the scalingcoupling constant, the left eigenvectorsof ¡

left left cn;~ m~ ¡m;¤~ r~ = ¬ n~ cn;~ r~; (88) Xm~ as sc left G = c (G G¤ ) : (89) n~ n;~ m~ m~ ¡ m~ Xm~ They display the scaledependence

Gsc k n~ : (90) n~ ¹ Furthermore letus ¯xthe overallscale of the localoperators. This can be done by using the decomposition

O(¿ x) = bnOn(¿ x) (91) n X where O (¿ (x))isthe product of the terms @ @ ¿ m(x)with coe±cient 1. The norm n ¹1 ¢ ¢ ¢ ¹` O = b2 isintroduced with the notation jj jj n n qP O O = ; (92) O jj jj for the operators ofunit norm. Weshalluse the convention that the coupling constants

Gn~ (¤) alwaysmultiply operators of unit norm in the action. The scalingoperators sc right On~ = cn~m~ Om~ (93) Xm~ are obtained by means of the right eigenvectorsof ¡¤,

right right ¡r¤~m~ cm~ n~ = ¬ n~ cr~n~ ; (94) Xm~ and they satisfying the conditions of completeness cright cleft =1and orthonormality ¢ cleft cright =1.The coupling constants of the action ¢

sc Sk = Gn~ (k)On~ (¿ k) (95) Xn~ obviouslyfollow (90). The operator

O = b Osc ; (96) n~ f n~ gk Xn~ J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 25 written atscale k inthis basis yieldsthe paralleltransport trajectory

n~ sc k0 sc O k = bn~ On~ k k = bn~ On~ k (97) f g 0 ff g g 0 Ã k ! f g Xn~ Xn~ in the vicinityof the ¯xedpoint. The beta function introduced by Eqs.(72)-(74) agreeswith the usual one.One ¯ nds that onlyrelevant operators havenon-vanishing paralleltransport °ow. Universality manifests itselfat agivenscaling regime in the suppression of the paralleltransported irrelevant operators.

The blockedaction Sk[¿ k;x; Gn~(k); Gn~(¤)] possesses two interpretations: The valueof the coupling constant at the running cut-o®, G (k),re°ects the scale ° n~ dependence ofthe physicalparameters. The dependence onthe bare coupling constants, G (¤), the initialcondition ofthe ° n~ RG °ow provides us the generator functional for composite operators. Finally,the composite operator renormalization represents an alternative way to arrive at the RGequation. Infact, the beta functions arising from the blocking(72) are obtained in this schemeby the paralleltransport, the matching (76) of the expectation values.

2.3.5Perturbative treatment

Let us ¯nallycompare the matching of the observables, described abovewith the traditional method of perturbative composite operator renormalization. The inversion of Eq.(77) gives 1 [On~ (¿ k + ¿~k)]k = Om~ (¿ k + ¿~k)(S¡ (k; ¤))m;~ n~: (98) Xm~ ~ It isnot di±cult to seethat the operator [ On~(¿ k + ¿ k)]k of the bare theory which corresponds to the operator O (¿ ) of the e®ective theory agreeswith the result of the f n~ k gk usual composite operator renormalization. Infact, the bare action (71) can be split into the sum of the renormalized part and the counterterms the framework of the renormalized perturbation expansion, giving GnB~ = GnR~ + GnCT~ , Gn~(¤) = GnB~ and Gn~(k) = GnR~ . The counterterms are introduced just to render certain Green functions with composite operator insertions cut-o®independent [13].The composite operator corresponding to the renormalized (ieblocked) operator On~(¿ k)in this perturbative framework is

~ ¯ GmB~ Om~ (¿ k + ¿ k) (99) ¯ GnR~ Xm~ which agrees with (98). It isworthwhile recallingthat the parameters and operators of abare theory correspond to the cut-o®scale in anatural manner. Therefore On~(¿ k) k represents the bare ~ f g operator [On~ (¿ k + ¿ k)]k in the e®ective description atthe scale k and reproduces the observational scaledependence by construction inamanner similarto the scaledependence of the hadronic structure functions obtained in the framework ofthe composite operator renormalization conveythe sameinformation [15]. 26 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

2.4 Continuous evolution

The RG equations obtained so far dealwith the evolution of the bare action during the gradual lowering ofthe U.V.cut-o®. There are two di®erent reasons to lookfor alternative schemeswhere the evolution ofthe Green functions rather than the bare action isfollowed. Onereason isusefulness. In the traditional RGstrategy followed so far the renormalizedtra jectories givethe bare coupling constants as functions of the running cut-o®. Though one can extract number ofuseful information from the trajectories they remain somehow qualitativebecause the bare parameters ofthe theory with the running cut-o® are not physicalquantities. Though this remark does not apply to the multiplicativeRG schemewhere the e®ective parameters are constructed by means of Green functions this schemeis seriously limited as mentioned inthe Introduction. Returning to blocking,one can sayat most that the di®erence between bare parameters and physicalquantities arises due to the °uctuations inthe path integral and this latter decreaseswith the number of degrees of freedom as we approach the IRend point of the trajectories.Therefore it is the blockedaction with verylow cut-o®only which issupposed to bedirectlyrelated to physicalquantities (in the absence ofIRinstability). Another more formal point ofviewis to construct astrictly non-perturbative scheme. The RG equations (10), (48) represent the completeresummation of the perturbation expansion but they are not reallynon-perturbative equations. This situation isremi- niscent of the Schwinger-Dyson equations. They were ¯rst obtained by resumming the perturbation expansion and onlylater by agenuine non-perturbative method, by the in¯nitesimal shift of the integral variablein the path integral formalism. The fact that the naiveresult derived by assuming the convergenceof the perturbation expansion is correct ispresumably related to the unique analyticcontinuation ofthe Green functions in the coupling constants. Can we ¯nd in asimilarmanner the truly non-perturbative RGequations? If possible,it willcome by adi®erent route: by relating two path integral averagesinstead of computing them by brute force.The quantities in question are one-particle irreducible (1PI) amplitudes and we shall giveup to follow the evolution of the bare coupling constants, the parameters in the path integral. This willallows us to avoidany reference to the perturbation expansion. Itremains to beseenif (10) and (48) can bederived inasimilarmanner orthey remain validfor strongly coupled models. Webegin this program by writing the generator functional for the connected Green functions inthe form 1 W [j] 1 (S [Á]+C [Á] j Á) e h· k = D[¿ ]e¡ h· B k ¡ ¢ ; (100) Z where the term Ck[¿ ]isintroduced to suppress °uctuations. What we require isthat (i) for k = the °uctuations be suppressed, C [¿ ] = ,(ii) the original generator 1 1 1 functional berecoveredfor k = 0, C0[¿ ]=0and (iii)the °uctuations are suppressed only, ie C [¿ 0]=0for somecon¯ gurations closeto the vacuum expectation value, ¿ 0 ¿ . k º h i The simplest choicefor models without condensate isaquadratic functional, 1 C [¿ ] = ¿ C ¿ (101) k 2 ¢ k ¢ J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 27 but incertain caseshigher order terms in the ¯eldvariables are necessaryin the suppression. Wedistinguish two kinds of suppression. The length scaleof the modes ’released’ `(k)iswellde¯ ned when

2 2 ¯ Ck >> 1 p < 1=`(k), k(p) = j j (102) M ¯ ¿ p¯ ¿ p Á= Á ( 0 p > 1=`(k). ¡ j h i º j j The evolution generated by such suppressions shows the scaledependence in the spirit of the Kadano®-Wilson blocking, the contribution of modes with agivenlength scaleto the dynamics.Examples are,

2 f(p) ap 1 f(p) [16] ¡ 2 b Ck(p) = 8 a k [17] (103) > p2 < a(³k2 ´ p2)£(k2 p2) [18] ¡ ¡ > b(p2=k2)c : 2 where f(p) = e¡ and a; b; c > 0.Another kind of suppression for which (p) Mk shows no clearstructure willbe considered insection 2.5below. The evolution equation for W [j]iseasyto obtain,

1 1 Wk[j] (Ck[Á]+SB[Á] j Á) @kWk[j] = e¡ h· D[¿ ]@kCk[¿ ]e¡ h· ¡ ¢ ¡ Z 1 1 Wk[j] ¯ Wk[j] = e¡ h· @kCk ¹h e h· : (104) ¡ " ¯ j #

This isalready aclosedfunctional di®erential equation but of littleuse. The reason is that it isdi± cult to truncate W [j]being ahighly non-local functional. Inorder to arrive at amore localfunctional we shallmake a Legendre transformation and introduce the e®ective action ¡k[¿ ] as

¯ W [j] ¡ [¿ ] + W [j] = j ¿ ; ¿ = ; (105) k k ¢ ¯ j with ¯ W [j] @ ¡ [¿ ] = @ W [j] @ j + @ j¿ = @ W [j]: (106) k k ¡ k k ¡ ¯ j k k ¡ k k

It isadvantageous to separate the auxiliarysuppression term Ck[¿ ]o®the e®ective action by the replacement ¡[ ¿ ] ¡[¿ ] + C [¿ ]resulting in ! k

1 1 Wk[j] ¯ Wk[j] @k¡k[¿ ] = e¡ h· @kCk h¹ e h· @kCk[¿ ]: (107) " ¯ j # ¡

This relation takesa particularly simpleform for the quadratic suppressions (101). Since the quadratic part of the e®ective action isthe inverseconnected propagator we ¯nd [16], [10],[19], [20], [21],

h¹ h¹ 1 @k¡k[¿ ] = tr [@kCk ¿ ¿ conn] = tr @kCk 2 : (108) 2 ¢ h i 2 2 ¢ ± ¡k 3 Ck + ±Á±Á 4 5 28 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Two remarks are in order incomparing the evolution equation obtained here with the pervious RGequations. First, the Kadano®-Wilson blocking is constructed to preserve the generator functional W [j]for the thinner system and this isrealized by eliminating modes at the U.V.side, by changing Sk[¿ ].Oneis not aiming at keepinganything ¯xedin the continuous evolution scheme.The 1PI generator functional does depend on k which appears as the lowest momentum ofmodes considered inthe theory whose e®ective action is ¡k[¿ ].The second remark isthat the right hand sideis proportional to ¹h. But the apparent absence of the mixingof tree and loop-levelsis misleadingsince the propagator maybe the sum of h¹0 tree-leveland (¹h)°uctuation contributions. The explicit O O tree-levelterm ismissing ³ ´ because the modi¯cation of the bare action was carried out in the part (¿ 2)and this correction to the free propagator was removedby the step O ¡[¿ ] ¡[¿ ] + C [¿ ],the substraction ofthe tree-levelsuppression term from the e®ective ! k action. It isilluminating to compare the evolution and the Schwinger-Dyson (SD) equations. Though the formal aspects appear similarthe content ofthe equations di®ers. A similarity appearing immediatelyis that both set ofequations determine the Green functions in a hierarchicalmanner. The derivation of the equations shows somesimilarity ,as well.In fact, the crucialstep in the derivation of the evolution equation isthe ¯rst equation in (104). This step, the expression of the derivativeof the functional integral formally, without its actual evaluation isthe hallmark ofthe non-perturbative demonstration of the SDequations. Both the SDand the evolution equations are genuinelynon-perturbative because we actuallydo not evaluatethe functional integral, instead we relateto another one by bringing aderivativeinside the path integral. In both schemeson compares two functional integrals which di®er slightly,either in the in¯nitesimal shift of the integral variablesor in the in¯nitesimal change in the action. Let us write the ¯rst equation in (104) for ¯nite ¢k,

1 1 (Ck[Á]+SB[Á] j Á) (Ck ¢ k[Á]+SB[Á] j Á) D[¿ ] e¡ h· ¡ ¢ e¡ h· ¡ ¡ ¢ ¡ Z ³ ´ 1 1 C [Á] 2 1 (S [Á] j Á) = D[¿ ] ¢k@ C [¿ ]e¡ h· k + (¢k) e¡ h· B ¡ ¢ : (109) ¹h k k O Z ³ ³ ´´ The smallparameter ¢k isused to suppress the insertion of more ¯eldvariables in the evolution equation, to cut o®the higher order Green functions from the evolution,the strategy common with the non-perturbative proof ofthe SDequation. The higher order contributions in¢k bring in higher order Green functions and we suppress them by the smallnessof the step in ’turning on’ the °uctuations. This ishow the RGidea,the replacement of the higher loops by running e®ective coupling constants and the dealing with smallnumber of modes ateachstep, isrealizedin the evolution equation scheme. The obvious di®erence between the evolution and the SDequations isthat the latter express the invarianceof the functional integral in amanner similarto the RGequations but the former issimply an expression of the derivativeof the functional integral with respect to aparameter. The similarityof the evolution equation and the RGstrategy leadsto another di®erence. The subsequent eliminationof modes and taking into account J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 29 their dynamics by the introduction/modi¯cation of e®ective vertices generates a’universal’evolution equations which does not depend on the theory in question. In fact, the evolution equation (108) and the functional RGequations Eqs.(6), (45) donot contain anyreference to the model considered .The e®ective or the bare action of the model appear in the initialcondition only.The SDequation isbased on acareful application of the equation of motion within the expectation valuesand therefore contains the action of the model inanobvious manner. The evolution equation method seemsto bebetter suited to numerical approximations then the SDequations. This isbecause the ’dressing’, the summing up the interactions is achievedby integrating out di®erential equations rather then coupling the Green functions in the SDhierarchy,anumerical problem we can control easier.

2.5 Blocking in theinternal space

Wepresent now ageneralization the RGmethod. The traditional RG strategy isaiming at the scaledependence of observables and consequentlyfollows the cut-o®dependence in the theories. The results are obtained by the successivemodi¯ cation ofthe cut-o®and the accumulation of the resulting e®ects. One can generalizethis method by replacing the cut-o®by any continuous parameters ofthe theory onwhich the observables depend upon in adi®erentiable manner. Such ageneralization replacesthe RGequation by an evolution equation corresponding to the parameter orcoupling constant in question. Onegains and loosesin the sametime during such ageneralization.One looses intuition of and insight into the dynamics sincethe trajectories generated by the evolution equation haveno relation to scaledependence. Tomakethings evenmore complicatedthe trajectories do not correspond anymore to a¯xedphysical content, instead they trace the dependence of the dynamics in the parameter considered. But we gain in °exibility.In fact, the parameter we selectto evolvecan bethe Plank constant oracoupling constant and the integration ofthe resulting evolution resums the semiclassicalor the perturbation expansion. Wemayapply this method inmodels with non-trivial saddle point structure or where the cut-o®would break important symmetries,e.g. in gauge theories. Amore technicalaspect of this generalization touches the U.V.divergencesin the theory.The traditional RGprocedure isbased on the tacit assumption that the U.V. divergencesare properly regulated by the blocking,the moving cut-o®. This seemsto be anatural requirement ifthe RG °ow isinterpreted asscaledependence. In fact, anU.V. divergenceleft-over by the blockingwould indicate the importance of the processes at the regulator and would leadto the appearance of asecond scale.In the generalization of the RGprocedure we introduce below there maynot beawellde¯ ned scalerelated to the evolution and we haveto regulate the U.V.divergences. There are two di®erent kind of physicalspaces in FieldTheory .The space-timeor the momentum-energy spacewhere the ’events’are taking placewill be calledexternal space.The ’events’are characterized by further quantities, the ¯eldamplitudes. The spacewhere the ¯eldamplitudes belong willbe calledinternal space.Therefore the ¯eld 30 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 con¯guration ¿ realizesa map ¿ : external space internal space.The traditional x ! Kadano®-Wilson blocking orders the degrees of freedom to be eliminatedaccording to their scalein the external space.W emayrealize a similarscheme by ordering the modes according to their scalein the internal space,their amplitude [22].W epresent two di®erent implementations of this idea,one for Wilsonianand another for the e®ective action.

2.5.1Wilsonian action

Weshalluse the °exibilityof the method of deriving the Polchinksiequation in section 2.2to separate o®and eliminatethe °uctuation modes with the largest amplitude inthe path integral. Sincethe °uctuation amplitude iscontrolled by the mass this leadsto the description ofthe dynamics by highly massivemodes, small° uctuations atthe ¯nal point of the evolution. Let us consider the model(38) where the roleof the parameter k isplayed by the 1 2 2 2 2 2 mass M and GM¡ 2 = p + M .Weshall follow the evolution from M = 0 to M >> ¤ where ¤isthe U.V.cut-o® . The decomposition

~ ~ GM 2 = GM 2+¢M 2 + GM 2; © = ¿ + ¿ ; (110) willbe used with

2 2 1 2 2 1 (M h ) 1 2 1 1 ¿ M + ¢M h ¿ + ¿~ ¡ ¿~ = © M h © + ©~G~¡ ©~: (111) 2 ¡ 2 ¢M 2 2 ¡ 2 D ³ ´ ³ ´ Wereplacethe degrees offreedom ©~ by the ¯eld ¿~ with in¯nitesimal ° uctuations. After eliminating ¿ ~ the remaining ¯eld ¿ has smaller° uctuations than ©.Weuse the ansatz

I 1 SM [¿ ] = (ZM (¿ x) 1)@¹¿ x@¹¿ x + UM (¿ x) (112) Zx · 2 ¡ ¸ where the U.V.regulator with smooth cut-o®is not shown explicitly.Wefollow the steps outlined insection 2.2.2which givesrise the evolution equation with left hand side

1 2 (2) @M 2 ² p² p (Z 1)p + U + V U : (113) 2 p ¡ ¡ ½ Z h i ¾ The right hand sidereads as

1 (1) 2 (2) 1 (3) (1) 2 2 ¯ p;0U + ² p (Z 1)p + U + ² r² p r U Z r p r + p ¡ 2 p( ¡ 2 r ¡ ¡ ¢ ¡ ) Z h i Z h ³ ´i 1 £(p2 + M 2)2

(1) 2 (2) 1 (3) (1) 2 2 ¯ p;0U + ² p (Z 1)p + U + ² p s² s U + Z s + p s + p £( ¡ ¡ 2 s ¡ ¡ ¢ ) h i Z h ³ ´i h¹ 1 2 (2) 1 (4) (2) 2 2 + V (Z 1)p + U + ² r² r U + Z (p + r ) 2 p (p2 + M 2)2 ( ¡ 2 r ¡ ) Z h i Z h i J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 31

2 (Z 1)r2 + U (2) (1)2 1 1 1 (1) (3) (1) 2 V U ² r² r 8 ¡ + U (U + Z r )9 º ¡ 2M 4 ¡ 2 r ¡ h (r2 + M 2)2 i M 4 Z <> =>

h¹ 1 :> 2 (2) 1 (4) (2) 2 2 ;> + V (Z 1)p + U + ² r² r U + Z (p + r ) (114) 2 p (p2 + M 2)2 ( ¡ 2 r ¡ ) Z h i Z h i The system of evolution equation projected onto the di®erent ² -dependent terms is

2 (2) 4 (1)2 4 (Z 1)p + U 2 2M @M U = U + ¹hM ¡ 2 2 2 ¡ Zp (p + M ) (2)2 4 (2) U (1) (1) 4 (2) 1 2 2M @M Z = 4U (Z 1) + 4 2 2U Z + h¹M Z 2 2 2 ¡ ¡ M ¡ Zp (p + M ) (2) 2 (4) 4 (2) (2)2 (1) (3) 4 Z p + U 2 2M @M U = U U U + ¹hM 2 2 2 : (115) ¡ ¡ Zp (p + M )

The ¹h0 terms onthe right hand sideare to keepthe tree-levelobservables M -invariant O within³ the´ gradient expansion ansatz. As far as the loop corrections are concerned, the beta functions 2 (n+2) 1 n = M @M gn = h¹M U 2 2 2 ; (116) Zp (p + M ) obtained from the ¯rst equation with Z =1agreewith the leading order contribution to the one-loop renormalized potential,

1 loop ¹h 2 2 (2) UM¡ = UM + ln[p + M + UM ]; (117) 2 Zp excepttheir sign.This isbecause we intend to keepthe partition function unchanged as opposed to Eq.(117) where the mass isevolvingtogether with the dynamics.

2.5.2E® ective action

Let us consider for the sakeof simplicityagain the scalarmodel given by the bare action

2 1 2 mB 2 SB = (@¹¿ x) + ¿ x + UB (¿ x) (118) Zx " 2 2 # and the suppression with no structure inthe external space[22],

M 2 C (p) = (119) k 2 which acts asa’smooth cut-o®’for the amplitude of the °uctuations. The corresponding evolution equation is 1 2 ¡ h¹ 2 ¯ ¡M [¿ ] @M 2 ¡M [¿ ] = tr M + : (120) 2 " ¯ ¿ ¯ ¿ # The e®ective action ofthe theory isobtained by integrating this equation from the initial 2 2 2 2 condition ¡M0 [¿ ] = SB[¿ ] imposed at M = M0 >> ¤ >> mB to M = 0. 32 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Let us consider how this schemelooks like for the ansatz

1 2 ¡M [¿ ] = ZM (¿ x)(@¹¿ x) + UM (¿ x) : (121) Zx · 2 ¸ One ¯nds h¹ 1 2 @M U = 2 2 (2) 2 Zp Zp + M + U 2 p Z (1) + 2 Z (1)p2 + U (3) (2) h¹ (1) d Z @M 2 Z = 2Z 2 ³ 2 (2) 3 ´ 2 2 (2) 2 2 Zp" (Zp + M + U ) ¡ (Zp + M + U ) 2 Z (1)p2 + U (3) 8p2 Z (1)p2 + U (3) 2Z ZZ (1) ¡ (Z³p2 + M 2 + U (2)´ )4 ¡ d (Z³p2 + M 2 + U (2)´ )4 2 8p2 Z (1)p2 + U (3) + Z 2 : (122) d (Z³p2 + M 2 + U (2)´ )5 #

The asymptotic form of the ¯rst equation for M 2 >> U (2) inthe localpotential approximation, 2 (2) M M @M UM UM 2 2 2 (123) º ¡ Zp (p + M ) up to ¯eldindependent constant and itagreeswith asymptotic form ofthe ¯rst equation in (115) exceptthe sign. The integrand in (123) corresponds to transformation rule (4) of the propagator under the in¯nitesimal change of the mass for the ¯rst, leading order graphs in Figs.3. The numerical integration resums the higher orders in the loop expansion. In short, we seethe Callan-Symanzik schemeat work in the functional formalism. The suppression (119) freezesout modes with momentum p if p2 < M 2 and generates 2 2 (2) 2 (2) acharacteristic scale pcr where (Zp + M + U )=(Zp + U )deviatesfrom 1.So long this scaleis far from the intrinsic scalesof the model, M 2 >> U (2) in the U.V.scaling regime,such aninternal spaceblocking generates the external scale p M=pZ. In other cr º words, the universal part of the beta functions obtained in the U.V.regime by means of the Callan-Symanzik schemeshould agreewith the samepart of the beta functions comingfrom other schemes.This agreement has been observed in the framework of the scalar model[23]. W eshallcheck this quicklyin the localpotential approximation. By comparing Eq.(123) with the asymptotic form of the WHequation (14),

(2) d 2 h¹U k ¡ k@ U = k d ; (124) k k ¡ 2(2º )d we ¯nd 2 d 1 dk ¤=k y ¡ dy 2 = 2 2 2 2 2 : (125) dM Z0 (y + M =k ) Belowthe upper criticaldimension, d < 4,the right hand sideis ¯ nite and the external (WH) and the internal (CS) scalesare proportional. At the criticaldimension the M dependence of the right hand sideis through the proportionality factor ln(1 +¤2=M 2). J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 33

The two scalesbecome approximately proportional onlyfor much higher valueof the cut-o®¤ ,sincethe RG°ow spends more ’time’close to the non-universal short distance regimedue to the tree-levelmarginality of g4.There isno agreement beyond d = 4 where the cut-o®scale is always important. The agreement between the two schemesis violated by higher order terms in the perturbation expansion sincethey represent the insertion of irrelevant e®ective vertices and by leavingthe asymptotical U.V.regimes. This explains that similarargument does not hold when Z displays important M -dependence. It isworthwhile noting that the non-trivial wavefunction renormalization Z and the corresponding anomalous dimension ² of the ¯eldvariable can bethought asthe re°ection of the mismatch between the scaledependence in the external and internal spaces around the U.V.¯ xedpoint. Infact, the phenomenological form

2 d ´ ¿ ¿ c x y ¡ ¡ (126) h x yi º j ¡ j cf Eq.(128) below connects the fundamental dimensional objects of the internal and external spaces.

The °exibilityof choosing the suppression functional Ck[¿ ]maybe important for certain models.By choosing k C [¿ ] = S[¿ ] (127) k ¤ where ¤isthe U.V.cut-o® the evolution in k resums the loop expansion sincethe e®ective 1 1 Planck-constant, ¹h¡ (k) = ¹h¡ + k=¤evolvesfrom 0toh ¹.This schemeis advantageous for models with inhomogeneous saddle point egsolitons or instantons because their space- timestructure is’RG invariant’, being independent of the gradual control ofthe amplitude of the °uctuations. Another advantage o®ered by the °exibilityin choosing the suppression functional isthe possibility ofpreserving symmetries,the point considered in section 3.5 below.

3Applications

Weshall brie°y reviewa few applications of the functional evolution equations. An incompletelist of the developments not followed due to limitation in timeis the following. The excitingcompetition for the ’best’ criticalexponents haveled to severalworks using this method [24],[53], [17], [50], [51], [25], [57], [26]. Such kind of application opens up the issue of understanding the impact of truncation on the blocking[27], [28], [29] [25],[30], [31] and lookingfor optimization [32],[33], [18]. The phase structure and the nature of phase transitions are natural subjects [21],[34], [35]. The incorporation of fermions isessential to arriveat realisticmodels in High Energy [36]and Condensed Matter Physics [37].The computation of the quenched averageof Green functions, one of the essentialobstacle ofprogress inCondensed Matter Physics can beapproached inanew fashion by applying the internal spacerenormalization group method [38].A promising application isin generalrelativity [39]. Finally ,the infamous problem of bound states can bereconsidered in this framework [40],[41]. 34 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Much more to be found in reviewarticles [42], conference proceedings [43]and PhD thesis [44].

3.1 Fixed points

3.1.1Rescaling

`The original form of the RGprocedure consists oftwo steps, the blocking,followed by a rescaling.The latter can beomitted if the RGstrategy isused to solvemodels only.But it becomesimportant when we try to understand the scaledependence of the theories. Our physicalintuition isbased on classicalphysics and particles.After losing much ofthe substance during the quantization procedure there isachanceto recoversome clarity by the introduction of quasiparticles,localized excitations with weakresidual interaction. The rescalingstep of the RGschemeis to checka givenquasiparticle assumption by removing the corresponding scaledependence. The remaining scaledependence isamea- sure of the qualityof the quasiparticle picture. The deviation from this non-interactive system isparametrized in terms of anomalous dimensions. The particle content of atheory isusually ¯xedby the quadratic part of its action. Therefore the rescalingis de¯ ned in such amanner that this part of the action stays invariant during the blocking.F oran O(d)invariant Euclidean orrelativisticallyinvariant real-timesystem the resulting rescalingre° ects the classicaldimension ofthe dynamical variablesand coupling constants. For non-relativistic systems the ¯xedpoint scalingis an arti¯cial device only to identify the non-interacting part ofthe action. When smooth cut-o®is used its details mayin° uence of the scalingproperties of the quadratic action, as well,and mayinduce deviations from classicaldimensions.. Consider the scaletransformation

p p0 = (1 + ° )p; x x0 = (1 ° )x; ¿ ((1 + ° )x0) (1 ° d )¿ (x) (128) ! ! ¡ ! ¡ Á in an O(d)invariant scalarmodel where d = (d 2 + ² )=2.The parameter ² re°ects the Á ¡ deviation of from classicaldimensional analysis.The e®ect of this rescalingon the action

S[¿ ] = up1; ;pn¯ p1+ +pn;0¿ p1 ¿ pn; (129) n p1; ;pn ¢¢¢ ¢¢¢ ¢ ¢ ¢ X Z ¢¢¢ can befound inthe following manner [45]: d d The momentum integral measure changes as d p (1 d° )d p0, giving ° ± ! ¡ S [1 ° d ¿ p ]S. ! ¡ p ±Áp The couplingR constants change as up1; ;pn u(1 ²)p ; ;(1 ²)p .This can be written ¢¢¢ ¡ 10 ¢¢¢ ¡ n0 ° !± as the transformation S [1 ° ¿ pp @0 ]S of the action where the prime on ! ¡ p ¢ p ±Áp the gradient @p indicates that the derivativeR acts onthe coupling constants onlyand not onthe Dirac-deltas.

The Dirac-deltas change as ¯ p1+ +pn;0 ¯ (1 ²)p + +(1 ²)p ;0,amounting to the trans- ° ¢¢¢ ! ¡ 10 ¢¢¢ ¡ n0 formation S [1 + d° ]S ofthe action. ! J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 35

Finally,the ¯eldtransforms as ¿ p [1 ° (dÁ d)]¿ p, inducing ° ± ! ¡ ¡ S [1 ° (dÁ d)] ¿ p S. ! ¡ ¡ p ±Áp Adding up these contributionsR we ¯nd the rescalinggenerator

¯ ¯ = ¿ pp @p0 + dÁ¿ p : (130) G ¡ Zp " ¢ ¯ ¿ p ¯ ¿ p # The RG equation for the rescaledaction can be obtained by adding acting on the G (e®ective) action to the right hand side,eg the evolution (49) turns out to be

1 SI[Á] 1 SI[Á] k@ e¡ h· k = ( + ¹h ) e¡ h· k : (131) k G B The rescalingof the ¯eldmay be viewedas atransformation of the action without changing the physics at the ¯xedpoint. The terms generated on the ¯xedpoint action by the rescaling, S¤[¿ ]are calledredundant atthe ¯xedpoint in question. G 3.1.2Reparametrizing

Weconsidered linear rescalingof the ¯eldvariable but one can easilygeneralize the rescaling to non-linear reparametrization. The in¯nitesimal change ¿ ¿ 0 = ¿ + ° ª[¿ ; x] x ! x x performed inside the path integral gives

¯ S[¿ ] S[Á] S[Á ] ¯ ª[¿ ; x] S[Á] ² ª[Á;x] D[¿ ]e¡ D[¿ 0]e¡ 0 = D[¿ ] 1 + ° e¡ ¡ x ¯ ¿ x : (132) Z ! Z Z " Zx ¯ ¿ x # R The reparametrization invarianceof the integral assures that the modi¯cation

S[¿ ] S[¿ ] + ° S[¿ ] (133) ! G ª of the action where ¯ S[¿ ] ¯ ª[¿ ; x] ª = ª[¿ ; x] (134) G Zx " ¯ ¿ x ¡ ¯ ¿ x # isan invarianceof the partition function. It has furthermore been noted [46]that some of the in¯nitesimal blocking relations for the (e®ective) action can bewritten in the form (133). Though it iscertainly very interesting to ¯nd acommon structure for the functional RGequations the connection with reparametrization invarianceis not clear.The point is that non-linear reparametrizations can not becarried out inside ofthe path integral as in ordinary integrals. The problem arises from the fact that the typicalcon¯ gurations are rather singular inthe path integration. Calculus known from classicalanalysis is replaced by Ito-calculusin Quantum Mechanicsdue to the nowhere-di®erentiable nature of the quantum trajectories [47].The quantum ¯eldcon¯ gurations are evenmore singular. Consider afree massless¯ eld ¿ in d dimensions and its partition function

ad 2 2 1 2 ¡ (¢· Áx) (¢· Á~x) Z = d¿ xe¡ 2 x = d¿ xe¡ 2 x (135) x x Y Z P Y Z P 36 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

where ¢¹¿ x = ¿ x ¿ x ¹ and the dimension of the ¯eldwas removedin the second ¡ d=2 1¡ 0 equation, ¿ ~ = a ¡ ¿ .The typicalcon¯ gurations have¢ ¿~ = (a ),iethe discontinuity 1 d=2 O of the original ¯eldvariable is ¢¿ = a ¡ . O This simplescaling, the basis of the³ usual U.V.´ divergences in quantum ¯eldtheory , givesnon-di® erentiability in Quantum Mechanics,for d =1,¯nite discontinuity in d = 2 and diverging discontinuities for d > 2.The non-di®erentiability can be represented by e®ective potential verticesin Quantum Mechanics,[48], a re°ection of the unusual quantization rules in polar coordinates, sensitivityfor operator ordering and quantum anomalies[49]. In two dimensions the discontinuity is¯ nite and the continuous structure can either be preserved or destroyed by quantum °uctuations, cf.section 3.4.In higher dimensions the singularities remain alwayspresent. It isthis singular nature of the trajectories which requires that one goesto unusually high order inthe smallparameter ° characterizing the in¯nitesimal change ofvariablesin the path integral. This problem, composite operator renormalization, renders the non-linear change of variablesa poorly controlled subject in quantum ¯eldtheories.

3.1.3Local potential approximation

Weshall search for the ¯xedpoints of the Polchinskiequation in the localpotential 1 d=2 approximation, [50][51]. W eintroduce dimensionless quantities ¿ = k ¡ © and d uk(¿ ) = k Uk(©). The RG equation with rescalingfor u is

² + d 2 (1) (1)2 (2) u_ = du + ¡ ¿ u u K 0 + ¹hK¹ 0u (136) ¡ 2 ¡ 0

¹ d where K 0 = k¡ p Kp0 and we set ² =0in the approximation Z =1.Notice the explicit dependence on theR cut-o®function K.This re°ects the fact that the RG °ow depends on the choiceblocking transformation. Onlythe qualitative,topological features of the renormalized trajectory and the criticalexponents around a¯xedpoint are invariant under non-singular rede¯nitions of the action.

Gaussian ¯xedpoint :The ¯xedpoint equation,u _ ¤ =0has two trivialsolutions,

u¤(¿ ) = 0; (137) and 2 ¿ K¹ 0 u¤(¿ ) = + h¹ ; (138) 2 K 0 dK 0 j 0j 0 2 where K 0 ; K¹ 0 < 0.F or any other solution u = (¿ ) as ¿ . 0 O ! 1 In order to ¯nd the scalingoperators we introduce aperturbation around the ¯xed ¸ point by writing u = u¤ + ° k¡ v(¿ ) where ° isin¯ nitesimal and solvethe linearized eigenvalueequation

¹ (2) 2 d (1) (1) h¹K 0v = (d ¶ )v + ¡ ¿ + 2u¤ K00 v : (139) ¡ Ã 2 ! J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 37

Havinga second order di®erential equation one can construct aone-parameter familyof solution after having imposed say v(1)(0) =1in atheory with the symmetry ¿ ¿ . ! ¡ But the polynomialsolutions n 2` vn(¿ ) = vn;`¿ ; (140) X`=1 which are parametrized by adiscrete index n correspond to discrete spectrum [52]. The criticalexponents are identi¯ed by comparing the terms (¿ 2n)in Eq.(139), O ¶ = d + (2 d)n, and ¶ = d (d+2)n,for the ¯xedpoints (137) and (138), respectively n ¡ n ¡ [53].The leading order criticalexponents are givenentirely by the tree-levelcontributions. The dimensionful coupling constants are cut-o®independent in this caseand we have ¶ n = [g2n]for the ¯xedpoint (137) where the dimension ofthe coupling constant [ g] isgiven by Eq.(24). The dominant, largest exponent ¶ T and the corresponding coupling constant playdistinguished rolein the scalinglaws. The dimensionless coupling constant º can beidenti¯ed with the ’reduced temperature’ t.According to the de¯nition ¹ t¡ of º the criticalexponent ¸ we havethe mean-¯eld exponent, ¸ = 1=¶ T = 1=2.The mass is 2 relevantat the point (137) as expected,as shown by the beta function = 2K 0 g~ 2~g , 2 ¡ 0 2 ¡ 2 according to Eq.(55). The scalingpotentials with exponential growth for large ¿ [54] correspond to the continuous spectrum and their physicalinterpretation isunclear. Allexponents are negativeat (138), this isanIR¯xedpoint. Wilson-Fischer¯ xedpoint :There are non-Gaussian ¯xedpoint solutions for Eq.(136) when 2 d < 4.One can construct aone-parameter familyof solutions but the ¯xed µ point potentials which remain non-singular for arbitrary ¿ correspond to adiscrete set [55],[52]: the Wilson-Fischer¯ xedpoint for 3 < d < 4and as many ¯xedpoints as relevantwhen 2 < d < 3.The perturbation around these ¯xedpoints isa one-parameter familyof scalingpotentials with continuous spectrum ofcriticalexponents. The restriction for solution which are ¯nite and non-singular everywhereproduces adiscrete spectrum [56]in good agreement with other methods of determining the criticalexponents [53], [57]. The truncation of the ¯xedpoint solution at a¯nite order of ¿ introduces error and spurious solutions [27],[28] which can partially beeliminatedby expanding the potential along the cut-o®dependent minimum [29].

3.1.4Anomalous dimension

The RGequation for the wavefunction renormalization constant z(¿ ) = Z(©) is

(2) (1) (1) K000K0 (2)2 z_ = 4K 0 u (z 1) 2K 0 u z 2 u ¡ 0 ¡ ¡ 0 ¡ k2 (2) d + ² 2 (1) +¹hz K¹ 0 ² (z 1) + ¡ ¿ z : (141) ¡ ¡ 2 The lesson of the case Z =1isthat the requirement of the existenceand ¯niteness of u¤(¿ )introduces discrete number of ¯xedpoint solutions. Let us try to follow the same strategy again. The dominant terms ofsuch ¯xedpoint solutionsu _ = z_ =0of Eqs.(136) 38 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 and (141) are 2 ² d 2+² 4 d+2¡ ² u¤(¿ ) ¡ ¿ + A¿ ¡ ; z¤(¿ ) B: (142) º 4 º Together with the conditions u(1)(0) = z(1)(0) =0imposed for the ¿ ¿ symmetrical ! ¡ models the solution are welldetermined in terms of A and B.The problem isthat such kind ofargument does not ¯xthe valueof ² . There isanother condition to be ful¯lled by the ¯xedpoints, the criticalexponents should beinvariant under rescaling.This issu± cient to determine ² .Unfortunately the rescalinginvariance of the ¯xedpoint islost unless sharp orspeciallychosen polynomial smooth cut-o®is used [56],[58]. F urthermore the truncation of the gradient expansion contributes to the violationof this invariance,too. An approximation to ¯nd the ’best’ solution when the rescalinginvariance is not respected isthe following [51]:Introduce afurther condition which violatesrescaling invariance, say ¯ xthe valueof z(0). This allowsthe determination of ² which would be unique if rescalingcould be used to relax our last condition. Weare as closeas possible to the invariant situation within our parametrized problem when the dependence of ² on z(0) isthe slowest. Therefore the condition d² =dz(0) =0selectsthe ’best’ estimate of ² . The numerical error due to the truncation of the gradient expansion has been the subject of extensivestudies in the framework of the scalarmodel, c.f. Ref. [42]. The general trend isthat the truncation of the gradient expansion at the localpotential approximation (zeroth-order) or at the wavefunction renormalization constant level(¯ rst order) yieldsapproximately 10% or3%di®erence in the criticalexponents compared with Monte-Carlo simulations, seven-loopcomputations in ¯xeddimensions or the ¯fth-order results of the expilon-espansion.

3.2 Global RG

The usual application of the RG method can be calledlocal because as in the determination of the criticalexponent it isperformed around apoint in the spaceof coupling constants. Models with more than one scalemay visit several scaling regimes as the observational scalechanges between the U.V.and the IR¯xedpoints. The determination of the ’important’ coupling constants ofsuch models which parametrize the physics goes beyond the one-by-one, localanalysis of the scalingregimes. It requires the careful study of crossovers,the overlapbetween the scalinglaws of di®erent ¯xedpoints, aproblem considered inthis section.

3.2.1RG in Statistical and High Energy Physics

It isimportant to realizethe similaritythe way RGisused in Statistical and High Energy Physics despite the super¯cial di® erences. The most obvious di®erence is that whilethe running coupling constants are introduced inParticle Physics by means of Green functions or scattering amplitudes at acertain scale,the parameters ofSolid State Physics models are de¯ned at the cut-o®, the latter being a¯nite scaleparameter, eglattice spacing. J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 39

Sincethe bare coupling constants characterizethe strength of the physicalprocesses at the scaleof the cut-o®, the two ways of de¯ning the scaledependence are qualitatively similarand reproduce the sameuniversal scalinglaws. The U.V.¯xedpoint where the correlation length divergesin units of the latticespacing in Statistical Mechanicscorresponds to renormalized theory in High Energy Physics where the U.V.cut-o® is removed. There are two classi¯cation schemesof operators, one comesfrom Statistical and the other from High Energy Physics:if the coupling constant ofanoperator increases,stays constant or decreasesalong the renormalized trajectory towards the IRthe operator is calledrelevant, marginal or irrelevant inStatistical Physics.The coupling constants which can be renormalized as the U.V.cut-o® is removed in such amanner that observables convergeare calledrenormalizable inHigh Energy Physics.The important point isthese classi¯cation schemesare equivalent,in particular the set of irrelevant operators at an U.V.¯ xedpoint agrees with the set ofnon-renormalized ones. Onecan easilygive a simplequantitative argument in the leading order of the perturbation expansion. First letus establish the power counting argument about renormalizability.Suppose for the sakeof simplicitythat there isonly one coupling constant, g, playing the roleof smallparameter and aphysicalquantity isobtained as

n = g In (143) hOi n X where In issum ofloop integrals. This seriesgives rise the expression

[I ] = [ ] n[g] (144) n O ¡ for the (mass) dimension of the loop integral. Let us recallthat the degreeof the overall (U.V.) divergenceof aloop integral isgivenby its dimension. Wedistinguish the following cases: [g] < 0:The higher powers of g decreasethe dimension in (143) which iscompen- ° sated for by increasing the degreeof the overalldivergence of the loop integrals, cf Eq.(144). Graphs with arbitrary high degreeof divergenceappear inthe perturbation expansion and g iscallednon-renormalizable. g[g] > 0:The higher order loop integrals are lessdivergent, there are ¯nite number of ° divergent graphs inthe perturbation seriesof any observable.The coupling constant issuper-renormalizable. [g]=0:The maximaldegree of (U.V.) divergenceis ¯ nite for any observable but ° there are in¯nitely many U.V.divergent graphs. g isa renormalizable coupling constants. Sincethe IRdegreeof divergenceof masslessloop integrals isjust [I ]the perturbation ¡ n expansion ofasuper-renormalizable or non-renormalizable masslessmodel is IRunstable (divergent) or stable (¯nite), respectively.The compromise between the U.V.and the IR behaviors isattained by dimensionless,renormalizable coupling constants. The IRsta- bilitycan berealizedby apartial resummation ofthe perturbation expansion in massless super-renormalizable models [59]. 40 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

It israther cumbersome but possible to prove by induction that the de¯nition of renormalizabilityoutlined aboveaccording to the overalldivergence of the loop integrals remains validin everyorder ofthe perturbation expansion. In order to separate o®the trivialscale dependence one usually removesthe classical dimension of the coupling constants by means of the cut-o®in Statistical Physics.On the tree-level,in the absence of °uctuations classicaldimensional analysisapplies giving the relation g = k[g]g~ between the dimensional and dimensionless coupling constants, g and g~,respectively.Assuming that there isno evolution on the tree-levelwe havethe scaling law [g]+ (¹h) k ¡ O g~(k) = g~(¤) (145) Ã ¤! for the dimensionless coupling constants, showing that the non-renormalizability of a coupling constant, [ g] < 0isindeed equivalentto irrelevance,the decreaseof the coupling constant towards the IRdirection. Higher loop contributions do not change this conclusion so long as the loop expansion isconvergent and the anomalous dimension, the contribution (¹h)in the exponent can not overturn the sign of the classicaldimension O [g].In caseof amarginal coupling constant, [ g]=0on has to carry on with the loop expansion until the ¯rst non-vanishing contribution to the anomalous dimension. Onecan construct amuch simpler and powerful argument for the equivalenceof the irrelevant and the non-renormalizable set of coupling constants inthe following manner. Consider the U.V.¯xedpoint P and the region around it where the blockingrelations can be diagonalized,as depicted in Fig.5. The solid lineshows arenormalized trajectory of amodelwhich contains relevantoperator onlyat the cut-o®. The Lagrangian ofanother modelwhose trajectory isshown by the dotted linecontains an irrelevant operator, as well.The di®erence between the two models becomessmall as we movetowards the IR direction and the physics around the end of the U.V.scaling regime is more independent on the initialvalue of the irrelevant coupling constants z longer the U.V.regimeis. This iswhat iscalled universality of the long range, low energyphenomena. Bylooking "backwards" and increasing the cut-o®energy ,as done in Particle Physics,the non- vanishing irrelevant coupling constants explodeand the trajectory isde° ected from the ¯xedpoint. As aresult we cannot maneuver ourselvesinto the U.V.¯ xedpoint in the presence of irrelevant operators in the theory.Sincethe in¯nite valueof the cut-o® corresponds to the renormalized theory,represented by the ¯xedpoint, the irrelevant operators are non-renormalizable. Despite its simplicity,this argument isvalid without invokingany smallparameter to use in the construction of the perturbation expansion. Renormalizablemodels were thought to be essentialin High Energy Physics.These models in which asu±ciently large class of observables can be made convergent are distinguished by their simplicity,their interactions existwithout any cut-o®. Examples are Yang-Millsmodels in four dimensions and the sine-Gordon, Thirring and the X-Y models in two dimensions. The non-renormalizable models are those whose existence requires a¯nite valueof the U.V.cut-o®, likeQED, the ¿ 4 modelin four dimensions, the z Moreprecisely all the irrelevantcoupling constants can modify isan overallscale. J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 41

Fig. 5 An U.V. xed pointand its vicinity .The x and the y axiscorrespond a relevantand anirrelevantoperator, respectively .The circledenotes the vicinityof the xed pointwhere the blockingrelation is linearizable.

Standard Modeland models in Solid State Physics.It isbelieved that asymptotically free models are renormalizable only. The traditional reason to discard non-renormalizable models was their weakpredictive power due to the in¯nitely many renormalization conditions they require.Since we can neverbe sure what kind ofLagrangian to extrapolate up to high energieswe need another point of view.According to the universalityscenario the non-renormalizable models can beexcludedbecause they cannot produce anything di®erent from renormalizable theories. The cut-o®independent physics of the latter isparametrized by means of renormalized coupling constants. But the subtle point to study below isthat this reasoning assumes the presence of asinglescaling regime with non-trivial scalinglaws in the theory which isa rather unrealistic feature [7]. Thus renormalizabilityis the requirement of asimpleextrapolation to the U.V.regime without encountering "new physics".The evolution of High Energy Physics shows that this isa rather unrealistic assumption, any modelwith such afeature can be an approximation at best, to study agiveninteraction and to sacrify the rest for the sake of simplicity.The goals are lessambitious in Statistical Physics and apart of the case of second order phase transitions the renormalizabilityof models isrequired seldom,for convenienceonly ,not to carry the regulator through the computation. Allmodels in Solid State Physics are e®ective ones givenwith aphysicallymotivated cut-o®. Let us start the discussion of the possible e®ect of the co-existenceof severalscaling regimesalong the RG°ow with the simplest case,a modelwith agap in the excitation spectrum abovethe ground state, iewith ¯nite correlation length, ¹ < .Suppose that 1 the RG °ow ofthe model starts inthe vicinityof anU.V.¯xedpoint and reaches an IR ¯xedpoint region as shown in Fig.6. Universality ,the parametrizability of the physics beyond the U.V.scalingregime by the relevantcoupling constants of the U.V.scaling laws,tacitly assumes the absence of any new relevant coupling constants aswe movealong the renormalized trajectory towards the IR,asshown inFig.5. This assumption isinfact correct for massivemodels whose IRscalinglaws are trivial,meaning that that onlythe Gaussian mass term isrelevant. T oseewhy letus consider the RG°ow inthe IRsideof the crossover,when the U.V.cut-o® , saythe latticespacing a >> ¹ and makea blocking step, a a0 which generates the change g (a) g (a0)in the coupling constants. The ! n ! n 42 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

UV IR

Fig. 6 The RG®owof amodelwith massiveparticles. The renormalizedtra jectoryconnects the U.V.and IR scalingregimes, indicated by circleswhich areseparated by acrossoverat the intrinsicscale of the model, p 1=¹ . º

change ¢g (a) = g (a0) g (a)isdue to °uctuation modes whose characteristic length n n ¡ n is a < ` < a0.Owing to the inequality ¹ << a < ` these °uctuations are suppressed a=» by e¡ and the °ow slows down, g (a) g (a0).This impliesthe absence of run-away n º n trajectories,the absence of relevantnon-Gaussian operator. But more realisticmodels with masslessparticles or with dynamicalor spontaneous symmetry breaking occurring at ¯nite or in¯nite length scales,respectively ,or with condensate in the ground state the IRscalingmay generate new relevant operators and universality,as stated in the introduction, referring to asinglescaling regime is not auseful concept any more.

3.2.2Scalar model

In order to understand better the genericcase shown in Fig.6 we return to its simplest realization,the scalar modelin the localpotential approximation, Eq.(14), [7].One does not expectanything unusual here sincethe modelpossesses agap and the IRscaling regimeis trivial in either the symmetricalor symmetry broken phase. But we shallsee that the modelhas soft largeamplitude °uctuations and the true vacuum isjust there where this non-perturbative and the wellknown perturbative domains join. The second derivativeof the localpotential isdiscontinuous there, depends on the direction from which the vacuum isapproached. Weshallarrive at someunderstanding of the soft modes in two step, by separating the loop contributions from the tree-levelstructure. We start with the more traditional loop contributions. Tofacilitatethe dealing with soft modes we introduce an external constraint which controls the expectation value ¿ = V ©.The usual realizationof this constraint xh xij isthe introduction of an externalR source as in Eq.(5). Weshall consider the model in the symmetry broken phase where ¿ = V © =0and choosehomogeneous xh xi0 vac 6 external source jx = J which can destabilizeR the naivesymmetry broken vacuum whenever J© < 0and can induce © < © .Note that such acontrol of the ¯eldexpectation vac j j vac valueby anexternal source preserves the localityof the action. The beta-functions, as givenby Eq.(23), depend strongly onthe propagator 2 2 G(k ) = 1=(k + g2(k)), represented by the the internal linesin Figs.3. The qualitative 1 2 features of the function G¡ (k )are shown in Fig.7 for ©=0.According to the loop J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 43 expansion

2 (2) (2) 2 1 2 2 (2) k + U¤ (©) + (¹h) Uk << k , G¡ (k ) = k + Uk (©) = O (146) ( k2 + U (2)(©) + (¹h) k2 << U (2), 0 O k the bending of the linesin the ¯gure isdue to radiative corrections, as far as the per- 1 turbative region, G¡ >> 1isconcerned. The dashed linecorresponds to the massless Coleman-Weinberg case[23]. The °ow aboveor below this separatrix isin the symmetricalor symmetry broken phase, respectively.This qualitativepicture isvalid so long © < © . j j vac It isimportant to recallthat the argument of the logarithmic function in the WH 1 2 equation in Eq.(14), G¡ (k ),isthe curvature of the action at the cut-o®and serves as ameasure of the restoring force driving °uctuations backto their trivialequilibrium position. Thus perturbative treatment isreliable and the loop contributions are calculable 1 2 1 2 2 so long G~¡ (k ) = G¡ (k )=k >> g~4. In order to havea qualitativeides if what ishappening letus consider the model where 2 2 4 U¤(©) = m ¿ =2 + g¿ =4!at the cut-o®, k =¤.Bysimply ignoring the loop corrections 2 2 we have g2(k) = m , g4(k) = g and gn(k) = 0 for n > 4onthe tree-level,ie m and g can be identi¯ed with renormalized parameters and m2 < 0.F or an isolated system ( j = 0) 2 where © = ©vac = 6m =g and the use of the perturbation expansion appears to be j j 1 2 ¡ 2 2 d 4 justi¯ed for G~¡ (k ) = 1 m =k > g k ¡ .Spontaneous symmetry breaking, or the ¡ 4 appearance of acondensate inthe ground state in general,is characterized by instability of °uctuations around the trivial,vanishing saddle point. When anexternal source iscou- pled to ¿ to dial © < © in the symmetry broken phase then non-perturbative e®ects x j j vac set in as k decreases.W ecan seethis easilywhen the ¯eldexpectation valueis squeezedin 2 1 the concavepart ofthe potential, © < © = 2m =g = © =3. In fact, G¡ approaches j j in° ¡ vac zeroand the amplitude of the °uctuations explodesat k = m2 g©2=2.Below this ¡ ¡ 1 2 non-perturbative regimethe plane wavemodes becomeunstable q since G¡ (k ) < 0 and a coherent state isformed, re°ected in the appearance of anon-homogeneous saddle point with characteristic scale1 = m2 g©2=2. ¡ ¡ The mechanism responsibleq of spreading the instability overthe whole region © < © j j vac comesfrom the tree-levelstructure and willbe discussed later,in section 3.3. Weidentify atthis point four di®erent scalingregimes: A:In the U.V.scaling regime k2 g (k): G g =k2,the mass term isnegligible. ¾ 2 n º n B:The explicitscale dependence disappears in the IRregime k2 g (k) of the sym- ½ 2 metricalphase, where G g =g .This isa trivialscaling, there are no interactive n º n 2 relevant operators. 1 2 2 C:The onset of the condensation isat G¡ (k ) = k + g (k) 0and the higher loop 2 º e®ects start to dominate. D:The spinodal unstable region where the homogeneous ground state becomesunstable 2 against in¯nitesimal ° uctuations, k +g2(k) < 0.The leading scalinglaws are coming from tree-level,classical physics. There are two crossover regimes.One is between the scalingregimes A and B at k2 g (k).Another one isbetween A and C when © < © .This willbe extended to º 2 j j in° 44 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

A G 1

2 C p

1 2 Fig. 7 The inversepropagator G¡ (p )of the scalarmodel as the function of the momentum square p2 for© =0.The dottedand solid lines correspond to the symmetricaland the symmetry broken phase,respectively .The masslesscase is shown by dashedline. One candistinguish four di¬erent scaling regimes, namely A: U.V.,B: symmetricalmassive, C: precursorof condensation andD: spinodialphase separation regime. the region © < © by tree-levelcontributions. Weshall lookinto the crossover ofthe j j vac symmetry broken phase. The best isto follow the evolution of the localpotential the plane (©; k2), as shown in Fig.8. The potential Uk(©) should be imagined as asurface abovethe plane 2 _ ~ (©; k ).The RGequations for the dimensionless coupling constants, g~n = n where ~ n isgiven by Eq.(25) are integrated in the direction of the arrows. The initialcondi- 2 2 2 d 4 4 tion U~¤(¿~) = m ¿ ~ =2k + gk ¡ ¿~ =4!, set at k =¤,along the horizontal lineon the top. As the running cut-o® k isdecreased and the RGequations are integrated the potential becomesknown along horizontal lineintercepting the ordinate at k2.The spontaneous breaking of the symmetry in the vacuum impliesthat for su±ciently small © the prop- j j agator explodesby approaching asingularity at k = kcr(©) as k decreases.This happens ¯rst at©=0when the running horizontal linetouches the two curvesshown inthe Figure. These curvescorrespond to the generalization of the minimum and the in°ection point of the potential for k2 > 0and are givenby the relations k2 + m2 + g©2 =6 + (¹h) = 0 B min O and k2 + m2 + g©2 =2 + (¹h)=0,respectively .Weshallargue in section 3.3that the B in° O singularity liesclose to the extension of the minimum,

k (© (k)) k: (147) cr min º The numerical integration ofthe WHequation requires atruncation ofthe summation in Eq.(20) for the localpotential. Byusing n 22 the singular linecan be located max µ with reasonable accuracy.Closeto the singular lineon the plane (©; k2)the coupling constants increaserapidly and the truncation of the potential isnot acceptable.In the U.V.scaling regime (denoted by Ain Fig.7) the propagator issmall, G = (k2), and O the ¯rst term on the right hand sideof Eq.(23) isdominant. As we approach kcr the propagator starts to increase,all term contribute equallyin the beta functions and we enter into anew scalingregime (C in Fig.7). In order to connect the two scalingregimes we shall consider ~ fm;n(k=¤) = @ m;k(¿ )=@g~¤;n,the dependence of the beta functions at k on the micro- J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 45

k 2 2 2 k = L

2 Fig. 8 The (©; k )planewhere the localpotential Uk(©) isde ned. The potentialshould be imaginedas a surfaceabove the thisplan. The initialcondition gives U¤ (©) onthe horizontal dashedline and the RGequation is integratedfrom this line in the directionof the arrows.The extensionof the minimumand the in®ection point of the potentialfrom k2 = 0 to k2 > 0 is shown by solidand dotted lines, respectively .

scopicalinitial condition, imposed at k =¤.The simplest way to estimate fm;n(x) is to start with f (1) 1which can be inferred from the form (23). The scalinglaw (145) m;n º allows us to write

~ ~ [gn] [gn] k @ m(¿ ) @ m(¿ ) k ¡ k ¡ fm;n = = = (148) Ã ¤ ! @g~n(¤) @g~n(k) Ã ¤ ! O 0Ã ¤ ! 1 @ A up to corrections (¹h),con¯rming the universalityof the U.V.scaling regime. In fact, O initial,microscopical value of the the irrelevant,non-renormalizable coupling constants ([g] < 0) leavevanishing trace onthe dynamics as ¤=k . ! 1 The numerical determination ofthe function fm;n(x) in d =4supports the prediction (148) in the U.V.scaling regime, namely the impact of anon-renormalizable coupling [gn] constants g (¤) on the beta functions weakensas ( k=¤)¡ for x 1.But this trend n º changes for x < 1as the criticalvalue of the cut-o®, kcr,found to be in agreement with Eq.(147), isapproached in what f starts to increase.The potential must not be truncated closeto the singularity and the numerical results appear inconclusive.But the increase of fm;n(x)at the crossover actuallystarts already far enough from kcr when all couplings are small,g~ n << 1and the truncation issafe.The lesson ofthis numerical result isthat universalityas used around the U.V.scalingregime alone is not validanymore. The instability at the onset of the condensation at k k introduces divergenceswhich º cr are strong enough to overwrite the U.V.scalinglaws and generate new relevantoperators. Such an instability ofthe RG °ow enhances the sensitivityof the physics at ¯nite length scaleson the microscopicalparameters. The need of generalizinguniversality in aglobal manner has been shown for © < © . j j vac It remains to seeif such aphenomenon can be observed at the highly singular point © = © ,inthe true vacuum. j j vac Asimilarphenomenon has already been noticed inconnection with the BCS ground state. It has been pointed out that kinematicalfactors turn the four electron operator which isirrelevant according to the power counting into marginal for processes closeto 46 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 the Fermilevel [60], [61] [62]. The collineardivergences at the Fermilevel drive the instability ofthe non-condensed vacuum and generate new scalinglaws. Another similar mechanism isthe origin of the strong long range correlations in the vacuum state of Yang-Millsmodels. They appear asaconsequenceof non-renormalizable, U.V.irrelevant Haar-measure term of the path integral [63]. Wenote ¯nallythat the viewof the RG °ow as aparalleltransport, mentioned in section 2.3is particularly wellsuited for the studies of crossovers.This isbecause the sensitivitymatrix (78) isa global quantity displaying clearlythe sensitivityof the IR physics onthe U.V.parameters by construction.

3.2.3RG microscope

Wenow embark on arather speculativesubject, playing with the possibility ofmatching di®erent scalinglaws. Let us start with the genericcase, a modelwith U.V.and IRscaling regimes,separated by acrossover,as sketched in Fig.6. W ehavea classi¯cation of the verysame operator algebra atboth scalingregimes. One can write any localoperator A as alinearsuperposition of scalingoperators, A = c ,the latter being the eigen- n A;nOn operators of the linearizedblocking relations of ascalingregime, (k) (¤=k)º (¤) . P On º nO n Let us denote by ¸ A the largest scalingdimension ofscalingoperators which occur in the alinear decomposition of A, ¸ A = maxn ¸ n.Weshallsimplify matters by callinga local

A operator relevant,marginal or irrelevant if ¸ A > 0, ¸ A = 0 or ¸ A < 0,respectively . Weignore marginal caseby assuming that radiative corrections alwaysgenerate non- vanishing scalingdimensions. Onecan distinguish between the following four cases:

(a) rU:V:rIR:relevant in both regimes,

(b) rU:V:iIR:relevant in the U.V.,irrelevant inthe IR,

(d) iU:V:iIR:irrelevant in both regimes. The qualitativedependence of the corresponding coupling constants isshown inFigs.9.

Let us consider for exampleQED containing electrons,muons and ions with metallic density.There isan U.V.scalingregime for energieswell above the electron mass. Sup- pose that the ground state isa BCStype superconductor where the IRscalinglaws are driven by long range phonon mediated interactions. The electron mass isan exampleof the class(a). The muon mass belongs to class(b) owing to the decoupling of muons at low energy.Byassuming that radiative corrections turn the four fermion vertexrelevant it servesas an exampleof class(c). Finallythe sixfermion vertexis doubly irrelevant and belongs to class(d). The interesting classis (c). Suppose that we modify the valuesof the bare coupling constants at the initialcondition and ¯nd the dashed line.Due to the universalityof the U.V.scaling regime the two renormalized trajectories convergetowards eachother for 0 < x < xuv and run veryclose between the two scalingregimes, xuv < x < xir. But in the IRscalingregime where the coupling constant isrelevant the two trajectories start to J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 47

g g

xuv xir xuv xir (a) (b) g g

xuv xir xuv xir (c) (d)

Fig. 9 The qualitativedependence ofthe four types ofcoupling constants on the observational lengthscale. The U.V.and the IRscalingregimes are 0 < x < xuv and xir < x,respectively. The dashedline in (c) corresponds to a di¬erent initial condition for the RG®ow. divergefrom eachother. The question iswhether the di®erence of the trajectories inthe U.V.scaling regime has any impact on the di®erence found atthe IR.Itmayhappen that the IRphenomena isindependent ofthe U.V.regime and the crossover smear completely out the smalle® ects of the U.V.initial condition ofthe trajectory.But one can imagine that the extremelysmall di® erences left at the crossover remain important and leadto an initialcondition dependent divergenceat the IR.The answer ismodel dependent and can begivenby the detailed analysisof the set ofcoupled di®erential equations. In order to makethis point clearerlet us imaginea modelwith two parameters, saya mass m and acoupling constant g,both expressed in adimensionless manner by means of the U.V.cut-o®, the latticespacing a.Suppose the following rather simplescaling laws:The beta function ofthe mass isapproximately constant, a @m ® (m; g) = > 0 (149) m @a º m and of the coupling constant g can be written as

@g m mcr (m; g) = a = À ¸ U:V: + À ¸ I:R: g (150) @a · µ mcr ¶ µ m ¶ ¸ where À (z)isinterpolating smoothly between À (0) = 1 and À ( )=0.The running mass 1 is given by a ¯m m(a) = m(a0) ; (151) µ a0 ¶ and the length scaleof the crossover between the U.V.and the I.R.scalingregimes is at a a (m =m )1=¯m.The asymptotical scalingfor the coupling constant is cr º U:V: cr U:V: ºU:V: aU:V: ¡ g(aU:V:) cU:V:g(acr) ; aU:V: acr; º acr ½ ºI:R: ³ aI:R:´ g(aI:R:) cI:R:g(acr) ; aI:R: acr: (152) º acr ¾ ³ ´ 48 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Let us furthermore assume that ¸ U:V: < 0 and ¸ I:R: > 0,i.e. this coupling constant belongs to the class(c). The sensitivityof the coupling constant in the asymptotical regions, g(aU:V:) and g(aI:R:) on g(acr) is

ºU:V: @g(aU:V:) aU:V: ¡ cU:V: ; @g(acr) º µ acr ¶ ºI:R: @g(aI:R:) aI:R: cI:R: (153) @g(acr) º µ acr ¶ therefore @g(a ) c aºI:R: aºU:V: I:R: I:R: I:R: U:V: : (154) ºI:R:+ºU:V: @g(aU:V:) º cU:V: acr If the IRscalingregime is long enough then the di®erence between renormalized trajectories with di®erent initialconditions imposed in the U.V.regimecan beas largeor even larger than at the initialcondition. When the U.V.and the I.R.observational scalesare related by º =º acr ¡ I:R: U:V: aU:V: acr (155) º µ aI:R: ¶ then (154) is (a0) c =c and despite the ’focusing’ of the universalityin the U.V. O I:R: U:V: regimethe ’divergence’of the trajectories in the I.R.can amplify the extremelyweak dependence on the initialcondition at ¯nite scales.The increaseof aI:R: in this case enhances the sensitivityon the initialcondition and we can ’see’the initialvalue of the non-renormalizable coupling constants at smallerdistance a. The lesson of the numerical results mentioned aboveis that the scalar theory in the unstable regimerealizes such aRG’microscope’.The lossof the U.V.based universality as we approach the criticalline on the plane (©; k2)suggests that there isat leastone relevant operator at the new scalinglaws which isirrelevant in the U.V..W edo not know the eigen-operators of the linearizedblocking relations in this region but this new n relevantoperator must contain the localmonomials ¿ x with n > 4.T omakeit more di±cult, it isnot obvious that this isalocaloperator as opposed to the scalingoperators of anU.V.¯xedpoint. In order to parametrize the physics of the scalar model we haveto use this new coupling constant at the U.V.cut-o®, as an additionalfree parameter .This israther un-practical not onlybecause we do not know the operator in question but mainlydue to the smallnessof this coupling constants in the U.V.and the crossover region. Itseems more reasonable to use amixedparametrization, consisting of the renormalizable coupling constants at the U.V.cut-o®and the new coupling constant taken closeto the singularity where it has largeenough value,ie k k .In the scalar model k 0as the external º cr cr ! source isturned o®. Weshallcall the new coupling constants appearing in this manner hidden parameters. What operator corresponds to the hidden parameter of the scalarmodel? It seems reasonable to assume that the onlyobservables whose valueis determined at the onset of the condensate isjust the magnitude ofthe condensate. Therefore the conjecture isthat the strength of the condensate, ©vac,isdynamically independent of the renormalizable J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 49

parameters g2, g4 in d =4.W ehavetherefore the following possibilitiesin parametrizing the scalar model with spontaneously broken symmetry: Weidentify anon-renormalizable coupling constant, g ,which in°uences the value ° nr of the condensate and use the bare valuesof g2(¤); g4(¤); gnr(¤) at the U.Vcut-o®. Oneuses the renormalizable bare coupling constants from the U.V.end and the ° strength ofthe condensate from the I.R.regime, g2(¤); g4(¤); ©vac. Onemay use I.R.quantities only,say g (0); g (0); © . ° 2 4 vac Our point isthat the scalar modelwith spontaneously broken symmetry has three free parameters instead of two. This rather surprising conjecture can be neither supported nor excludedby perturbation expansion because one can not connect the U.V.and the I.R.regimesin areliablemanner. The numerical studies of the scalar model carried out so far are inconclusive,as well,because they were constrained to the quartic potential and the possible importance of higher order verticesin forming the condensate was not considered. The modi¯cation of the scalinglaws of the scalar model,necessary to generate a hidden parameter comesfrom the onset ofacondensation. Asdiscussed below in Section 3.3there are non-trivial tree-levelscaling laws in acondensate which overwrite the loop- generated beta functions and mayprovide the dynamicalorigin of the hidden parameter. In asimilarmanner one can speculate about the roleof the Cooper-pair condensate in the BCS vacuum. Ifthere turns out to beahidden parameter as in the one-component scalar modelthen the non-renormalizable e®ective coupling constants of the Standard Model would in°uence the supercurrent density in ordinary metals!Another possible example isthe Higgs sector of the Standard Model,where the hidden parameter, if exists,would be anadditional free parameter of the model. Onemay go further and inquire if dynamicalsymmetry breaking can modify the scalinglaw in asimilarmanner. The bound state formation, responsible to the generation of largeanomalous dimensions in strong coupling QED[64],in strong extended technicolor scenario [65,66], in the Nambu-Jona-Lasinio model[67] or in the top-quark condensate mechanism [68,66] may be asource ofhidden parameters, as well. Hidden parameters represent anunexpected ’coupling’between phenomena with very di®erent scalesand question of our traditional strategy to understand acomplexsystem by analyzingits constituents ¯rst. The axial[70] or scale[71] anomalies represent a wellknow problem of this sort, exceptthat the regulator independence of the anomaly suggests that for eachclassical IR ¯xedpoint there isa single’anomalous’ one without continuous ¯ne tuning. Universalityis expressed in the context of High Energy Physics by the decoupling theorem [69].Let us start with arenormalizable modelcontaining aheavyand alight particle and consider the e®ective theory for the light particle obtained by eliminating the heavyone. There are two possible classi¯cation schemesfor the e®ective vertices for the light particle,generated by the eliminationprocess. Oneis according to the U.V. scalinglaws in the e®ective theory ,i.e.there are renormalizable and non-renormalizable vertices.Another schemeis based on the strength of the e®ective interactions, i.e.there 50 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 are verticeswhich are stay constant ordivergewhen the ratio ofthe light and the heavy particle mass tends to zeroand there are verticeswhose coe±cients tend to zeroin the samelimit. The decoupling theorem asserts that these two classi¯cation schemesare equivalent,i.e. the non-renormalizable e®ective coupling constants are vanishing when the heavymass diverges.The existenceof the hidden parameter is,at the ¯nal count is aviolationof this theorem.

3.2.4The Theory ofEverything

The appearance of ahidden coupling constant renders our goalof understanding the hierarchy of interactions in Nature extremelycomplicated. In fact, the RG °ow of the Theory ofEverything must beimagined in the spaceof allcoupling constants inPhysics, including the parameters ofthe Grand Uni¯ed Models down to di®erent models in Solid State Physics or hydrodynamics inthe classicalregime, as indicated inFig.10. Di® erent ¯xedpoints are approached by the trajectory when the scaledependence isdominated by asingleinteraction. But all¯ xedpoint exceptthe U.V.one isavoided due to the non-renormalizable, irrelevant vertices[69] generated by the dynamics of the next ¯xed point inthe U.V.direction. Such awandering in the spaceof coupling constants echoesthe ageold disagreement between High Energy and Solid State Physics.It isusually taken for granted inthe High Energy Physics community that the su±ciently precise determination ofthe microscopical parameters of the Theory of Everything would ’¯x’ the physics at lower energies.The obvious di±culties to extract the high energyparameters from experiments which render eachnew experiment expensive,long and largescale operation indicate that this might not beapractical direction to follow.In other words, the relevantcoupling constants have positiveLyapunov exponent and render the trajectory extremelysensitive on the initial conditions. Therefore the characterization of the RG °ow of The Theory of Everything by the initialcondition, though being possible mathematically,isnot practical due to the ¯nite resolution ofthe measurements. The other sideof the coin,the physicalparameters of a¯xedpoint are clearlythe relevantcoupling constants of the givenscaling regime. They are the object ofSolid State Physics,as far asthe scalingregimes QED, CMand IR are concerned in Fig.10. One looses sight ofthe fundamental laws but gains predictive power by restricting oneself to localstudies of the RG°ow.

3.3 Instabilityinduced renormalization

In the traditional applications of the RG method which are based on the perturbation expansion the RGtrajectory issought inthe vicinityof the Gaussian ¯xedpoint. Onecan describe in this manner crossovers which connect scalingregimes which share the same smallparameter. Wecan enlargethe classesof accessiblecrossovers by relyingon the saddle point expansion, the onlysystematical non-perturbative approximation scheme. The genericway to induce new crossover in this schemeis by passing acondensation at a¯nite length scale[72]. J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 51

GUT QCD IR’ CM

TOE QED IR

Fig. 10 The renormalizationgroup ® owof the Theory ofEverything. The branchingdrawn at anIRscaleis a phasetransition driven by the environment,such asparticle or heatreservoirs.

Onemay ob ject that there isnothing surprising ornew in¯nding that the physics is fundamentally changed by acondensation mechanism.What additional knowledge can then begained by lookinginto condensation by the method ofthe RG? The point isthat the classi¯cation ofoperators around a¯xedpoint which was achievedin the framework of the perturbation expansion changes essentiallyby the condensation, iethe appearance of anon-trivial saddle point. Toseehow this happens we return to the strategy followed at Eq.(2). Let us consider anobservable A inamodelwith asinglecoupling constant g for simplicity,computed in the saddle point expansion, A = F (g; ¤) + h¹F (g; ¤) + ¹h2 , h i 0 1 O where F`(g; ¤)denotes the `-loop contribution. Wecan obtain the beta function³ by´ taking the derivativeof this equation with respect to cut-o®¤ , k@ (F + h¹F ) k@ F @ F @ F = k@ g = k 0 1 + h¹2 = k 0 1+h¹ k 1 g 1 + ¹h2 : (156) k ¡ @ (F + ¹hF ) O ¡ @ F @ F ¡ @ F O g 0 1 ³ ´ g 0 · µ k 0 g 0 ¶¸ ³ ´ The loop corrections F (g; ¤), ` 1are polynomials of the coupling constants g. But ` ¶ the leading order, tree-levelpiece usually has stronger dependence on g and maybecome singular as g 0.The tree-levelcontribution mayinduce qualitativelynew scalinglaws ! with new set of relevant operators. Due to the singularity at g =0the condensation actuallyrealizes the dangerous irrelevant variablescenario [73]. At the onset of an instability,region Cin Fig.7 certain modes experiencestrong nonlinearity and developlarge amplitude °uctuations. But as we enter in the unstable regimeone mayhope to recoverquasiparticles with weakresidual interaction after having settled the dynamics of the unstable modes. At leastthis iswhat happens within the framework of the semiclassical,or loop expansion where the °uctuations remain small after the proper condensate isfound. There are models where the vacuum state isjust at 52 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 the edgeof the instability,likethe scalarmodel discussed abovein the symmetry broken phase. The tree-levelcontributions to the blockingtransformation, if exist,are more important then the loop corrections. This isthe reason that we shall consider the tree-level, h¹0 RG °ow. There isno evolution in this order so long the system isstable and O the³ saddle´ point istrivial. But as soon as we arriveat the unstable region the RG°ow becomesnon-trivial.

3.3.1Unstable e®ective potential

Before turning to the actual tree-levelblocking let us reviewbrie° y the kind of instabilities 2 one expects.As we enter into the unstable regime k + g2(k) < 0the trivialsaddle point,

¿ k =0becomesunstable in the blocking(7). Tounderstand this instability better we introduce the constrained functional integral,

SB[Á] 1 Z© = D[¿ ]e¡ ¯ ¿ x © (157) Z µV Zx ¡ ¶ corresponding to aconserved order parameter. Byfollowing the RG trajectory until the IRend point one eliminatesall but the homogeneous mode by keepingthe partition function unchanged, Sk=0[©] V Uk=0(©) Z© = e¡ = e¡ : (158) To¯nd another localrepresentation for this constrained partition function we consider the generator functional density w[j] = W [j]=V de¯ned by means ofEq.(5) and compute its

Ve® (©) = w[J] J© = min[ Uk=0(¿ ) + J(¿ ©)]: (159) ¡ ¡ J;Á ¡ ¡ The minimization with respect ¿ and J yields J = dw(J)=dJ and and ¿ =©,respectively.

Finallywe have Ve® (©) = Uk=0(©), the result announced after Eq.(16) so long the mean ¯eldapproximation isreliable,namely in the thermodynamical limitand in the absence of largeamplitude °uctuations. It isfurther known that Veff (©) isconvex. The instabilities ofthe kind mentioned aboveare wellknown inthe caseof ¯rst order phase transitions. But similarinstabilities mayappear at higher order phase transitions, as well.In fact, the magnetization of the Ising model as the function of the external magnetic ¯eldshows discontinuous behavior below the criticaltemperature. Therefore the free energyconstrained into asector with agivenmagnetization displays such kind of instabilities.

Suppose that the Uk=0(©) determined perturbatively has degenerate minima and a concavepart. Then it isadvantageous to introduce two curveson the (©; k) plane of 2 (1) 2 (2) Fig. 8, ©in°(k) and ©min(k) de¯ned by k ©min + Uk (©min) = 0, and k + Uk (©in°) = 0. The stable region in the mean-¯eld approximation is © > © (0) = © . j j min vac For © (0) < © < © (0) there are two minima in ¿ for the last equation in (159). in° j j min Oneof them ismetastable, ieis unstable against su±ciently large amplitude modes. The spinodal phase separation, the instability against in¯nitesimally small amplitude J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 53

°uctuations occurs when someeigenvalues of the second functional derivativeof the action becomesnegative. This isregion DinFig.7, © < © (0). It isunreachable by the mean- j j in° ¯eldtreatment because no localminimum isfound in ¿ which would satisfy ¿ < © (0). j j in° Tounderstand better the nature of these unstable regions one has to go beyond the mean-¯eld approximation and to follow the dynamics of the growing inhomogeneous instabilities.W eshall use the tree-levelWH equation to dealwith such largeamplitude, inhomogeneous °uctuations.

3.3.2T ree-levelWH equation

The construction ofthe condensate within the unstable region impliesthe minimization of the action with respect to alargenumber of modes. The RG strategy o®ers an approximation for such arather involvedproblem, it dealswith the modes one-by-one during the minimization [75],[74]. The negativecurvature of the potential makesthe saddle point in(9) non-trivial,

Sk ¢k[¿ ] = min Sk[¿ + ¿~0] = Sk[¿ + ¿ ~cl[¿ ]] = Sk[¿ ]: (160) ¡ ~ Á0 6 ~ ~ Sincethe saddle point depends in general on the background ¯eld, ¿ cl = ¿ cl[¿ ],the action ismodi¯ ed during the blockingand we ¯nd anon-trivial RG °ow [74].The localpotential approximation to the tree-levelblocking is

1 ~ 2 ~ Uk ¢k(©) = min (@¹¿ cl;x) + Uk(© + ¿ cl;x;) : (161) ¡ ~ Ácl;x Zx · 2 ¸ Oneencounters here aconceptual problem. The k-dependence of the tree-levelRG °ow might wellbe singular sincethere isno obvious reason that the saddle points which are usually rather singular functions of the parameters evolvesmoothly .This problem has already been noticed asapossible ’¯rst order phase transition’ inthe blockingwhich induces discontinuous RG°ow [76],[77], [78]. It was later shown by rigorous methods that the RG °ow iseither continuous or the blockedaction isnon-local [79].The resolution of this apparent paradox isthat the saddle point actuallydevelops in acontinuous manner as we shall shown below.As the RG °ow approaches the onset ofthe condensate, k k ! cr then the new scalinglaws generate such anaction that the saddle point turns out to be continuous in k.

3.3.3Plane wavesaddle points

The saddle points, the minima in (160) satisfy the highly non-linear Euler-Lagrange equations whose solutions are di±cult to ¯nd. The use of sharp cut-o®slightly simpli¯ es the problem sinceit reduces the functional spacein which the minimum issought to

¿~ k k ¢k.Weshall retain the the plane wavesaddle points only, 2 F nF ¡ » k ~ iµk iµk ~ ¿ cl;p = e ¯ p;ke + e¡ ¯ p; ke ; ¿ cl;x = » k cos(kek x + ³ k): (162) 2 k ¡ k ¢ h i The parameter ³ k and the unit vector ek correspond to zeromodes, they control the breakdown oftranslational and rotational symmetries. 54 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

This isa keyto what happens at the unstable line: Despite thediscreteness of theinternal symmetry ¿ ¿ thereare soft modesbecause the inhomogeneous sad- ! ¡ dlepoints break the continuous externalsymmetries .The condensation ofparticles with non-vanishing momenta automatically generates Goldstone modes. This phenomenon is wellknown for solids where the saddle point isa crystal ofsolitons which breaks external symmetriesand there isno integration overthe zeromode to restore the symmetrical ground state. Our point isthat the soft modes maketheir appearance evenif the symmetry ofthe ground state isrestored by the integration overthe zeromodes.

The amplitude » k isdetermined by minimization, ¼ 1 2 2 1 Uk ¢k(©) = min k » + dyUk(© + » k cos y) ; (163) ½ k ¡ k µ 4 º Z0 ¶ in the localpotential approximation. The numerical implementation of this iterative procedure to ¯nd the RG °ow with an initialcondition imposed on the potential at k =¤gavethe following results [74]: (1) The recursiveblocking relation, Eq.(163) isnot a¯nite di®erence equation. Despite of this the °ow convergesas ¢k 0. ! (2) The amplitude ofthe saddle point satis¯es the equation ©+ » (k) = ©min(k). (3) The potential obtained by the tree-levelblocking is U (©) = 1 k2©2. k ¡ 2 (4) The tree-levelresults abovehold independently ofthe choiceof the potential atthe cut-o®. The keyis point 3,point 1follows immediatelyfrom it.The lesson ofthis result isthat the action isdegenerate for the modes at the cut-o®, the kineticand potential energies cancel.The ’best’ e®ective theory for agivenplane wavemode isthe one whose cut-o® k isslightly above the wavevector of the mode.This result suggests the degeneracyof the action within the whole unstable region. Conversely,if we can show that the action isdegenerate at the cut-o®within the unstable regimewe established this potential. Weprove by induction in the number of steps k k ¢k that the variation of the ! ¡ action density within the unstable regimeis (¢k).This result protects the consistency O of the saddle point expansion for d > 1 since ¢k 2º =L where L isthe sizeof the system ¶d 1 and therefore the variation ofthe action is L ¡ .Recallthat k denotes the cut-o® O cr where the kineticsnd the potential energiescancel. ³ ´

First step: Let us denote by k0 the valueof the cut-o®at the ¯rst occurrence of non- trivialsaddle point in the numerical implementation of Eq.(163). It obviouslysatis¯ es the inequality k ¢k < k0 < k .The trivialsaddle point ¿~ =0becomesunstable for cr ¡ cr cs blockingswith © < © (k0)in which case © + ¿~ < © (k0), ie ¿ ~ = p¢k . j j in° j cs;xj in° j cs;xj O The term ¿~ iscanceled in the action on ahomogeneous background ¯eld³ so the´ O ¿~2 contribution³ ´ gives (¢k)variation. O cs O ³Induction:´ Suppose that the variation of the action is (¢k)in the unstable region O and we lower the cut-o®, k k ¢k.At the new cut-o®the balance between the kinetic ! ¡ and the potential energyis lost by an amount of (¢k)sincethe potential energyis O stillthe givenby k but the kineticenergy corresponds to the lowered cut-o®, k ¢k. ¡ Thus the negativecurvature potential energywins and the action bends downward in the J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 55 unstable region as the function of the amplitude of the plane wave.By assuming that the amplitude isstabilized at an (¢k0)value(will be checkedlater) the variation of O the action density (¢k)in the unstable region. O Point 2can be understood in the following manner: During the minimization of the action » slidesdown on the (¢k)slope until the potential starts to increaseagain. We k O can ¯nd where this happens by equating the slopeof the original,bare potential with that stated in point 3.One would expectthat the deepest point isreached at » k = ©in°(k). But the (¿ 4) term in U (¿ )isnot yetstrong enough to makethe potential increase O k strong enough here.The two slopes agreejust at » k = ©min(k). Result 4which follows from 3,as well,can becalledsuper universalitysince it re°ects scalinglaws where allcoupling constants are irrelevant.Note that the k-dependence is continuous through the whole RGtrajectory. Point 3isa generalizedMaxwell construction. It reduces to the traditional Maxwell construction for k =0,to the degeneracyof U (©) for © < © (0) = © . The naive k=0 j j min vac Maxwellcut, applied for the concavepart of the potential would givethe degeneracy for © < © only.The problem with this argument isthat it produces an e®ective j j in° potential which isconvex everywhere except at © = © (0). The second derivative j j in° of the potential isill de¯ ned and the ¯rst derivativeis discontinuous at this point. By placing the system into a¯nite box the singularity isrounded o®and the second derivative becomes¯ nite,but turns out to be negativein avicinityof © = © (0). Convexity j j in° regained onlyif the cut isextended between the minima, © < © (0). j j min The generalizedMaxwell construction in the mixedphase ofa¯rst order phase transition can be understood by the dynamics of the domain walls.The °atness of certain thermodynamical potentials in the mixedphase re°ects the presence of zero-modes,the location of domain walls.Such arather simplekinematical mechanism which isindependent of microscopicdetails isthe source of the `super universality’,point 4above. The roleof the domain wallsis played by saddle points in our computation, the cosine function in (162) realizesin¯ nitely many equallyspaced, paralleldomain walls.The integration overthe zeromodes ³ k and ek according to the rules ofthe saddle point expansion restores translational and rotational symmetriesof the ground state and reproduces the mixedphase. Onemight object our independent treatment of the plane wavesaddle points. The RGequation (163) handles the plane wavesin the consecutivemomentum spaceshells independently which seemsas avast oversimpli¯cation. But one should recallat this point that according to the general strategy ofthe RG method the dynamics ofthe modes eliminatedduring the blockingis retained by the modi¯cation of the e®ective coupling constants. This isnot alwayspossible sincethere are more modes than coupling constants, the problem mentioned at the end ofsection 2.1.1.More precisely,the generalframework of the blockingwhat isin principle alwaysapplicable is turned into apowerful scheme when an approximation ismade. One truncates the e®ective action and assumes that the solution of this over-determined problem exists.This isthe main,and so far unproven, assumption of the RG method. Accepting this point we can determine the blocked 56 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 action by means of ahomogeneous background ¯eldand use itfor non-homogeneous ¯eld con¯gurations atthe next blockingstep as written inEq.(163).

3.3.4Correlation functions

Onecan compute the tree-levelcontributions to the correlation functions inacombination of the mean-¯eld and the saddle point approximation [80].Let us split the complete functional integral into the sum of contributions which comefrom the unstable and the stable regions. The °uctuations from the stable regions are governed by anon- trivialaction and they are taken into account as loop corrections. The °uctuations in the unstable region experiencea °at action and their contributions must be taken into account on the tree-level.The complicationof integrating up these contributions isthe determination of the region where the action is° at. Our approximation consists of estimating this region for eachplane waveindependently ,ieextending the integration overthe amplitude rp ofthe degenerate plane wave

~ ¿ cl(x) = rp cos(pepx + ³ p) (164) overthe interval » < r < » , where » isgiven by (162). Let us denote the integration ¡ p p p overthe resulting domain by D©[r]and write the correlation function in momentum space in this single-modedetermined °at region as

1 © 1 ¡ Gtree(p; q) = D[³ ]D[e]D©[r] D[³ ]D[e]D©[r]rprq 4 ·Z ¸ Z iµp iµp iµq iµq e ¯ p;kep + e¡ ¯ p; kep e ¯ q;keq + e¡ ¯ q; keq : (165) £ ¡ ¡ ³ ´ ³ ´ Due to the °atness of the action these integrals are purely kinematicaland can easily be carried out. Allintegration whose variabledoes not show up in the integrand drops out. The integration overthe shift of the plane waves,the phase angle ³ ,restores the translation invariance,

1 1 where ©min¡ (©) isde¯ ned as ©min(©min¡ (©)) =©.The Fourier transform of this propaga- 1 tor describes adi®raction type oscillationwith characteristic length scale ¹ = kcr¡ , the characteristic feature ofdomains in ahomogeneous state.

3.3.5Condensation ascrossover

The scalinglaws of the scalarmodel change already approaching the condensation in the stable region C ofFig.7. Inside the instable region D the action with the potential given by point 3ofsection 3.3.3becomes invariant under blocking,ie the whole region D is an IR¯xedpoint. J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 57

The vacuum of the scalar modelis a single,homogeneous coherent state consisting of zeromomentum particles when © > © . When © < © we encounter the sin- j j vac j j vac gularities as seenabove. This singularity maynot be real,it might be smoothened out by higher order verticesand asaddle points appearing for k < kcr.But the vacuum is the superposition of inhomogeneous coherent states with characteristic scales ` > 1=kcr. Each ofthem breaks space-timesymmetries but their sum remains symmetrical. Onemay wonder ifany ofthe discussion applies to the true vacuum © = © . For a j j vac weaklycoupled system the °uctuations are smalland one mayhope that the °uctuations around ©vac are stable and un-in°uenced by what ishappening with the unstable modes

¿ x < ©vac.But the answer to this question israther uncertain and isopen because the true vacuum isjust at the border ofthe instabilities and the typical° uctuations around the true vacuum ( ¿ © ) g2(0)=g (0) penetrate into the unstable region. x ¡ vac º 4 The issue of the modi¯cation of the scaledependence by acondensate [64]is rather general and points far beyond the simplescalar model. The vacuum of asymptoticallyfree models supports strong correlations at long distance.In the caseof 4dimensional Yang-Millsmodels the naive,perturbative vacuum is destabilizedby the one-loop levele® ective action which predicts that the vacuum ismade up by acoherent state of gluons [82].But this state cannot be the true vacuum. The problem isnot onlythat the ¯eldgenerated by the condensate isstrong and spoils the saddle point expansion but itturns out that there isan unstable mode. The true vacuum issupposed to befound at evenlower energydensities where the long range °uctuations restore the the external and colorsymmetry broken by ahomogeneous condensate ofthe charged spin one gluons [83]and the vacuum isthought to contain domains of homogeneous ¯eldin astochastic manner [84].This scenario isclose to the viewof the mixed phase of the scalar model developedhere with the di®erence that the instability comes from the loop or the tree-levelrenormalized action in the Yang-Millsor scalar model, respectively.Furthermore the instability in the Yang-Millsmodel can beavoided by ex- treme environment only,by immersing the system into strong external ¯eldor bringing into contact with heat orparticle reservoir. Similar instability occurs in vacuacontaining condensate of bound states. The BCS vacuum ismade homogeneous in anon-trivial manner when represented in terms of the electrons making up the Cooper pairs. The spontaneous breakdown of the chiral invariancein QCDmanifests itselfin the condensate of quark-anti quark pairs which is homogeneous after integrating out the instanton zeromodes only[85]. The crossover from the U.V.scaling regime to the instability takesplace at k 1=` where ` is the size º of the bound states. Higher order derivativeterms appearing as e®ective vertices may generate similarcrossover, as well[72], [86]. Finallyit isworthwhile mentioning the tunneling phenomena, the dynamicalextension of the instabilities considered above.The interesting feature of the dynamicalrealization of the tunneling by means of time-dependent, tree-levelinstabilities isthat they take placein conservativesystems but the long timedistribution agrees with the equilibrium predictions coming from the canonicalensemble [81]. 58 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

3.4 Sine-Gordonmodel

It isnot unusual that one arrives at periodic or variablesin the construction of e®ec- tivetheories, cf non-linear sigma models or non-Abelian gauge theories. Such variables represent adouble challenge.One problem which can be considered as localin the internal spaceis that the perturbation expansion around aminima of the potential should keepin¯ nitely many verticesin order to preserve the periodicity.Another, more di±cult, global problem isa con°ict between the two requirements for the e®ective potential for periodic variables,namely periodicity and convexity.Onecan ful¯ll both requirements in atrivialmanner only,by constant e®ective potential. If true, this conclusion has far reaching consequencefor the phenomenological description of such systems.As acase study we shall consider the simplest model with periodic variable,the two dimensional sine-Gordon model [87],[88].

3.4.1Zoology of the sine-Gordon model

The sine-Gordon model 1 L = (@ ¿ )2 + u cos ¿ (167) SG 2 ¹ has been shown to be equivalentwith the X-Ymodel [89],[90], [87], the Thirring model [91],[92] and aCoulomb gas [93].The methods ofRefs.[90], [91] and [92],approximate duality transformation, bosonization and semiclassicalapproximation are validin certain regions of the coupling constant space.The maps used in Refs.[87] and [93]are exact. ¯Áx The mapping [87] Ãx = e transforms the model (167) into the compacti¯ed sine-Gordon model 1 u L = @ Ã¤@ Ã + (Ã + Ã¤): (168) CSG 2 2 ¹ ¹ 2 The models (167) and (168) are equivalentin any order of the perturbation expansion in continuous space-timewhere the con¯gurations are assumed to beregular. The X-Ymodel which ischaracterized by the action

1 S = cos(³ ³ ) + h cos(³ ) ln z m2 ; (169) XY 2 x x0 x 3 x ¡ T ¡ x ¡ x X X 4 5 2 where T = , h=T = u, z isthe vortexfugacity and mx denotes the vortexdensity .The RGequation of the X-Ymodel obtained in the dilute vortexgas limit[89], [90]

dT dh T dz º a = 4º 3z2 º T 2h2; a = 2 h; a = 2 z: (170) da ¡ da µ ¡ 4º ¶ dz µ ¡ T ¶ The X-Ymodelappears as ageneralization of the compacti¯ed sine-Gordon model sincethe latticeregularization transforms the Lagrangian (168) into (169) with z = 1. What issurprising here isthat the vortexfugacity isa relevantoperator in the high temperature phase of the X-Ymodelbut itisentirelymissed by the latticeregularization! Does that mean that the sine-Gordon modelLagrangians (167) and (168) are incomplete? J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 59

The vortexterm could be missed by the regulator because the plane z = 0 is RG invariant. Therefore it isconsistent to excludevortices from the path integral in the sine-Gordon model givenin continuous space-time.When we recast the model in lattice regularization without everthinking about vorticesour latticeaction neither suppresses, nor enhances the singular con¯gurations and sets z =1.But oncethe vorticesare not suppressed, z =0the renormalized trajectories movesaway from from the plane of ¯xed 6 z.There isstill no problem in the low temperature phase because allwhat happens is that an irrelevant coupling constant became¯ xed.This ishow regulators work. But the problem ismore serious inthe high temperature phase. The Lagrangians (167) or(168) can not giveaccount of this phase because they lacka relevant,renormalizable coupling constant. The perturbative continuum theory without vorticesis consistent but once vorticesare availablekinematically their density evolvesin anon-trivial manner. There isno problem with the latticeregulated X-Ymodelbecause the vortexfugacity gets renormalized due to the cut-o®dependence of the vortexaction. In other words, we can movealong the RG °ow by adjusting T and h only,the renormalization of z is carried out ’automatically’in the partition function. The renormalization of z isa problem in the continuous formalism onlywhere the vorticesare introduced formally as point-like charges. This isan unexpected mechanism which brings the singular nature of the ¯eldcon- ¯gurations into playduring renormalization and mayplague any quantum ¯eldtheory . The knowledge of the classicalaction in continuous space-timeleaves open the possibility that we haveof adjust the fugacity of certain localizedsingularities ortopologicaldefects not considered inthe continuum. It isinteresting to speculate about similarphenomenon inQCD.LatticeQCD isconstructed with ln z = 0 where z isthe fugacity of sometopological defects, likeinstantons, monopoles, merons etc.These topologicaldefects and singularities are supposed to play an important rolein the con¯ning vacuum, asvorticesdo inthe high temperature phase of the X-Ymodel.The renormalization of the fugacities of these objects should ¯rst be studied in latticeQCD inorder to construct acontinuum description of the vacuum.

The sine-Gordon model possesses atopologicalcurrent, j¹;x = ° ¹º @º¿ x=2º which isobviously conserved when the path integral issaturated by ¯eldcon¯ gurations with analyticspace-time dependence. Its °ux,the vorticity,givesthe soliton number. The world linesof the sine-Gordon solitons end at the X-Ymodel vortices,cf Fig.11. The distance between the bound vortex-anti vortexpairs shrinks with the latticespacing in the low temperature continuum limit.Any measurement with ¯nite resolution loosesight of the instability of solitons in the renormalized theory.The averagedistance between vorticesstays ¯nite,cut-o® independent inthe continuum limitof the high temperature phase and the soliton decaycan be observed. Oneexpects the breakdown or at least important modi¯cation of the bosonization transformation inthis phase. Infact, the non- conservation of the topologicalcurrent requires fermion number non-conserving terms in the fermionic representation, afundamental violationof the rules inferred from the weak coupling expansion. 60 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71

Fig. 11 The worldline of asolitonstarts at avortex (V) andends atananti-vortex (A) inthe twodimensional space-time.

3.4.2E® ective potential

In order to understand better the dynamics ofthe long wavelength modes we compute the e®ective potential in the sine-Gordon model.W efollow the WHmethod truncated to the localpotential approximation which picksup the tree levelevolution, too. It isobvious that period length of the potential 2 º = remains RGinvariant in this approximation. The tree ands the loop levelRG equations are

2 1 k 2 Uk ¢k(¿ ) = min » + duUk(¿ + » cos(º u)) ; (171) ¡ ½ " 4 1 # Z¡ and 2 k 2 (2) kUk ¢k(¿ ) = kUk(¿ ) + ¢k ln[k + Uk (¿ )] (172) ¡ 4º in the plane z = 0. Weuse the Fourier expanded form for the potential

1 Uk(¿ ) = un(k) cos (n ¿ ) (173) nX=0 and compute the leading order contribution to the WHequation when expanded in the potential, 2n2 k@ku~n = 2 u~n; (174) Ã 4º ¡ ! 2 in terms of the the dimensionless coupling constantsu ~ n = un=k .This agreeswith the second equation in(170). The solution of(174) is

2 2 n 2 k 4º ¡ u~n(k) = u~n(¤) : (175) Ã ¤ !

2 The coupling constants un are irrelevant in the disordered phase T = > 8º and the e®ective potential obtained for k =0is° at. The coupling constants n < 8º =T are relevantin the ordered phase and the e®ective potential isnon-trivial. At this pointe one suspects that Maxwellconstruction interferes with the evolution because anon-trivial periodic functions necessarilyhas concaveregions. Tosettlethis question one has to relyon the numerical solution of the evolution equation (171). Wefollowed the loops- generated evolution (172) from the initialcondition set at k =¤until the stability islost J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 61

at k = kcr for background ¯eldvalues which lieat localmaxima of the periodic potential.

For k < kcr equation (171) was used. The result isthat the coupling constants approach zero as k 0due to the plane wavesaddle points. The e®ective potential istrivial in ! both phases.

3.4.3Breakdown of the fundamental group symmetry

The localpotential Uk(¿ )°attens without developingsingularities in the high temperature disordered phase. The transformation

2º ¿ ¿ + (176) x ! x isa discrete symmetry of the action and ispreserved in the vacuum sincethe potential barrier between the minima isvanishing for long range modes. Sincewe are at the lower criticaldimension, d =2,the largeamplitude long range modes realizethe ’tunneling’ between the minima of the periodic potential. On the contrary to this situation, the potential developsdiscontinuous second derivativesin the ordered, low temperature phase and the instability driven °attening ofthe potential Uk(¿ )re°ects the survivalof barriers between the minima of the potential. Allthis lookslike a spontaneous symmetry breaking, therefore our conclusion isthat the transformation (176) isnot asymmetry ofthe vacuum in the ordered, low temperature phase. The usual circumstance under which such aphenomenon arises isthe multiple con- nectedness of the internal space.In the present context the non-linear U (1) ¼ -model parametrization, Eq.(168), isbased on the internal space U (1), with the fundamental group Z generated by the transformation (176). The dynamicalbreakdown ofthe fundamental group symmetry isa genuine quantum e®ect. In fact, the path integral formally extends overall homotopy classes,this isthe symmetricalphase. When the dynamics de- velopssu± ciently high barriers between the homotopy classesthe path integral becomes restricted to asinglehomotopy class. It isimportant to recallthat the timeevolution described by the Schrodinger equation can be derived from the path integral by performing in¯nitesimal variations on the end point of the trajectory.Therefore the consistencyof the dynamics can be maintained when the path integration isrestricted into any functional subspace which isclosed under continuous deformation ofthe trajectories.The lossof the interference between homotopy classesis the characteristic feature of the symmetry broken phase. Such symmetry breaking isthe keyto understanding the way quarks becomedecon- ¯ned at high temperature or the droplet phase isformed for quantum liquids[94]. The con¯guration spacefor global gauge transformations is SU (3)=Z3 for gluons, with the fundamental group Z3.The quark propagator vanishes in the symmetricalphase due to the destructive interference between the three homotopy classes.In the Z3 symmetry broken decon¯ned phase there isno interference and quarks can beobserved. Quantum liquidsin the ¯rst quantized formalism display similarsymmetry breaking pattern. The 3N coordinate spacefor N particles is R =SN where SN consists of permutations of the 62 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 particles.The absence of the overlap among states belonging to di®erent droplets suppresses the (anti)symmetrization ofthe states and reduces the exchangesymmetry of the ground state. How can we recognizethe dynamicalbreakdown of the fundamental group symmetry? The simplest strategy,to lookfor the minima of the e®ective potential failsbecause Maxwellconstruction hides any structure in aperiodic potential. The answer to this question liesin the topologicalstructure of excitationsand the order parameter willbe atopologicalsusceptibility . This israther natural sincethe phase transition isthe restriction of the path integral into asinglehomotopy class,the imposition of atopologicalconstraint. Let us assume that the analysissketched abovefor the plane z =0remains qualitativelyvalid for z =0,iethe e®ective potential isalways° at, due to the loop-generated evolution or the 6 Maxwellconstruction in the disordered or ordered phase, respectively.The topological invariant characterizing the homotopy classesis the soliton number. Inthe weakcoupling expansion (ordered phase) the path integral isconstrained into asinglehomotopy class hencethe soliton number isconserved, its susceptibility isvanishing in the continuum limit.The stability of the soliton, based on the continuity ofthe timeevolution islost in the disorder phase because the short distance, largeamplitude °uctuations of the typical ¯eldcon¯ guration extends the path integral allsoliton number sector. Therefore the susceptibility of the soliton number isnon-vanishing in this phase.

3.4.4Lower criticaldimension

It isworthwhile noting the manner the Mermin-Wagner-Coleman theorem [95]appears in the framework of the localpotential approximation. Bysetting g2 = 0 for d = 2 the linearpart of in Eqs.(23) provides a¯nite evolution but the higher order terms Pn makethe renormalized trajectory to divergeas k 0unless the coupling constants ! approach zero.W ecan construct interactive,massless models intwo dimensions so long the running coupling constants approach zerosu± ciently fast in the IRregimeas inthe caseof the sine-Gordon model,presented above.The retaining of the terms (@2) in O the gradient expansion provides another mechanism to suppress the IRdivergencesby generating singular wavefunction renormalization constant, Z as k 0 [96]. k ! In both casesthe way to avoidthe non-interactive system predicted by the Mermin- Wagner-Coleman theorem isto use the RG improved perturbation expansion. In the ordinary perturbation serieswe havethe contributions like

n g fp1; ;pk (177) p1; ;p ¢¢¢ Z ¢¢¢ k and the integral isIR divergent in d =2.One has running coupling constants, g g , ! P in the RG improved perturbation expansion where the analogous contribution is

n gP (p1; ;pk)fp1; ;pk: (178) p1; ;p ¢¢¢ ¢¢¢ Z ¢¢¢ k When the IR¯xedpoint isGaussian, lim P 0 gP =0,then the theory remains interactive ! and the running coupling constants suppress the IRdivergences. J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 63

3.5 Gauge models

The blockingstep of the RG scheme,being scaledependent, violateslocal symmetries which mixmodes with di®erent length scales. The most natural way to dealwith this breakdown of gauge invarianceis to cancel the gauge non-invariant terms by ¯ne tuning non-invariant counterterms [97].But this ispossible in perturbation expansion only.Another, better suited strategy to the RG method isto argue that the way Ward identities are violatedby blockingshows that the BRST invarianceis broken by the running cut-o®only and the BRST symmetry is recoveredat the IR¯xedpoint when the cut-o®is removed [20], [98]. This were certainly agood procedure if it could be implemented without further truncation. In aweakly coupled modelsuch asQEDone could construct approximations with controllable errors. In asymptoticallyfree models likeQCD non-perturbative long range correlations make any schemewhich imposes gauge invariancein an approximate manner unreliable [63]. The demonstrate this point consider the static force law between two test charges in a Yang-Millstheory in aschemewhere the gauge invariance,Gauss’ law isimplemented approximately.The gauge non-invariant components of the vacuum state appear charged, by de¯nition. Thus we havean uncontrollable colorcharge distribution around the test charges. When the distance between the test charges are largeenough then it willbe energeticallyfavorable for the uncontrollable charges to break the °ux tube. Weloose the string tension in amanner similarhow it happens in the true QCDvacuum due to the virtual quark-anti quark polarizations. In more formal words, an arbitrarily weak gauge-dependent perturbation can change the long range features of the vacuum as in the ferromagnetic phase ofthe Ising model. The most natural way to guarantee gauge invarianceis to achieveindependence on the choiceof gauge by using gauge invariant quantities only.The proposal ofRef.[99] goes along this linebut the use of loop variablesrenders the computation rather involved. Wediscuss now aversion of the RGschemein the internal spacewhich at leastfor Abelian models produces the e®ective action without any gauge ¯xing.The electron mass combined with chiraltransformation has already been used in generating evolution equation in Ref.[100]. W eshall follow asimpler and more general by avoiding gauge dependence in an explicitmanner for the photon Green functions [101].The gauge invariant electron composite operators are not di±cult to include [41].

3.5.1Evolution equation

Let us consider the generator functional

1 1 1 · ¸ ¬ · 2 · W [j;j;J¹ ] [ F F· ¸ (@· A ) +Ã¹(iD= m)Ã+j¹ Ã+Ã¹ j+J A· ] e h· = D[Ã¹]D[Ã]D[A]e h· x ¡ 4e2 ¡ 2 ¡ ¢ ¢ ¢ ; (179) Z R where D¹ = @¹ + iA¹ and the dimensional regularization isused to render W ¯nite. We control the quantum °uctuations by modifying the action, S S + S with ! ¸ ¶ h ¹º S¸ = 2 A¹;x T Aº (x); (180) 4e Zx 64 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 where T = g @ @ =h .The in¯nitesimal change ofthe control parameter ¶ ¶ +¢¶ ¹º ¹º ¡ ¹ º ! modi¯es the free photon propagator

D¹º (x y) D¹º(x y) + ¢¶ dzD¹½(x z)h T D·º(z y) + (¢¶ )2 ; (181) ¡ ! ¡ ¡ z ½· ¡ O Z ³ ´ in amanner reminiscent ofthe Callan-Symanzik scheme. The evolution equation (108) with the present suppression takesthe form

1 h¹ ¯ 2¡~[A] ¡ h ¹º @¸¡¸[A] = 2 tr T (182) 2e 2 Ã ¯ A¹¯ Aº ! 3 4 5 in terms ofthe photon e®ective action ¡~¸[A] = ¡¸[A] + S¸[A].Weproject the evolution equation on the functional spacegiven by the ansatz

1 1 ¹º ¡~[A] = A¹;xD¡ (i@)T Aº;x + [A] (183) 2 Zx C where 1¹º 1 + ¶ ¹º ¹º D¡ = h T ¬ h L ; (184) ¡ e2 ¡ L¹º = ¯ ¹º T ¹º and [A]isa gauge invariant functional. ¡ C The control parameter ¶ ’turns on’ the °uctuations ofthe photon ¯eld.Therefore the electrons loop contributions to the e®ective action must already be present atthe initial condition which ischosen to be [A] = itr ln(iD= m) at ¶ = ¶ . C ¡ ¡ 0 The second functional derivativematrix iswritten as

2 ~ 2 ¯ ¡[A] 1¹º ¯ [A] = D¡ + C (185) ¯ A¹¯ Aº ¯ A¹¯ Aº and the inversion iscarried out by expanding in the non-diagonal piecesto write the evolution equation as

2 n h¹ 1 ¯ [A] h ¹º n @¸ [A] = 2 tr T D ( 1) C D : (186) C 2e " ¡ Ã ¯ A¹¯ Aº ! # nX=0 3.5.2Gauge invariance

Weshow now that the limit ¬ 0can betaken inthe evolution equation without hitting ! any singularity.The gauge ¯xingparameter ¬ enters through the photon propagator D in the Neuman-expansion of the right hand sideof Eq.(186). The ¬ -dependent longitudinal contributions ofthe ¯rst and the last D factor are suppressed by the gauge invarianceof the suppression term, represented by the transverse projection T ¹º in (186). The longitudinal photon contributions ofthe internal propagators are suppressed by the gauge invarianceof the e®ective action,

¯ [A] @¹ C = 0: (187) ¯ A¹ J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 65

According to this equation ¯ 2 [A]=¯ A ¯ A does not mixthe longitudinal and the trans- C ¹ º versemodes, ienon-longitudinal contributions appear in the evolution equation. Wechoosea gauge in the argument abovewhat was relaxedat the end of the computation. But the steps followed remain wellde¯ ned evenif we start with ¬ = 0, without any gauge ¯xing.Our argument about the decoupling ofthe longitudinal and transverse contributions to the evolution equation stillapplies but it isnot clearif the longitudinal part was wellde¯ ned. This subtle issueis settled by Feynman’s ° parameter, devisedto lift in¯nitesimally the degeneraciesof the action. It iseasy to seethat it givesa weak variation of the action along gauge orbits by breaking gauge invariance.As long as our truncation of the evolution equation, the functional [A]isexplicitly gauge invariant ° C playsthe roleof an in¯nitesimally external magnetic ¯eldin the Isingmodel, ie helps the breakdown ofthe gauge symmetry onlyif it isreally broken in the true vacuum.

4What hasbeen achieved

The traditional implementation of the RG ideahas proven to beessentialin Statistical and High Energy Physics,starting with the understanding ofcriticalphenomena [102],¯ nite sizescaling [103], ° -expansion [104],dynamical processes with di®erent timescales [105], continuing with partial resummation of the perturbation expansion [4],parametrizing the scaledependence at high energies[106] and ending with the construction of e®ective theories inParticle Physics [107,108, 109]. This isan inexcusableshort and incomplete list,its roleis to demonstrate variabilityand the importance of the method only.Our main concern here was the functional form of the RG method and its generalizations. From this point of viewone maydistinguish conceptual and more technicalachievements as the power of the functional formalism ismore exploited. Weshould consider the RGmethod as ’meta-theory’, or in more practical terms as alanguage, as Feynman graphs are used inparticle physics.But this language isnot bound by smallparameters and can provide us ageneral,non-perturbative approximation method beyond the semiclassicalexpansion and numerical simulations. The path integration was ¯rst viewedas apowerful book-keepingdevice for perturbation expansion and the truly non-perturbative application camelater, after having gained someexperience with the formalism. This issimilarto the development ofthe functional RGmethod where the RGequation refers to the generator function of the e®ective coupling constants and provides us with asimpleprocedure to keeptrack of the Feynman graphs. The steps in functional calculusare more cumbersome because they dealwith generator function(al)s, with in¯nitely many coupling constants. The ultimate goalis to go beyond this leveland to use this formalism in agenuinelynon-perturbative manner. The semiclassicalexpansion isour ¯rst step inthis direction. It willbe important to checkhow the RGcontinues to beapplicablewhere the saddle point expansion ceasesto be reliable. Another ¯eldthe RGschememight becompared with islattice regulated ¯eldtheory . Both are general purpose tools to dealwith non-perturbative systems.The bottle-neck of 66 J.Polonyi/ CentralEuropean Journal of Physics 1 (2003)1{71 the numerical simulations onthe latticeis the need to send the U.V.and the IRcut-o®s su±ciently far from eachothers and the restriction to Euclidean space-time.The RG strategy isset up incontinuous, Minkowskispace time and there isno particular problem with keepingthe U.V.and IRcut-o®s far from eachother. But the drawback of the RG strategy isthat it israther lengthy to extend the spaceof (e®ective) action functionals used in the computation. Sincethe limitations of the two methods are quitedi® erent they might beused in acomplementary manner. The functional formalism ispromising because of the possibility offollowing the mixing ofamuch larger number ofoperators as inthe traditional strategy.This feature gives the hope ofextending the applicabilityof the method from asinglescaling regime to the whole range ofscalescovered by the theory.Such an extension mayprovide us valuable information about the competition ofinteractions inrealistictheories.

Acknowledgments

Ithank Jean Alexandre and Kornel Sailer for severaluseful discussions during our col- laboration.

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