KEK—91-13 JP9207322

KEK Report 91-13 February 1992 H

Renormalization Group in Different Fields of Theoretical Physics

D.V. Shirkov

NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS to National Laboratory for High Energy Physics, 1992 KEK Reports are available from: Technical Information & Library National Laboratory for High Energy Physics 1-1 Oho, Tsukuba-shi [baraki-ken, 305 JAPAN Phone: 0298-64-1171 Telex: 3652-534 (Domestic) (0)3652-534 (International) Fax: 0298-64-4604 Cable: KEKOHO RENORMALIZATION GROUP IN DIFFERENT FIELDS OF THEORETICAL PHYSICS

D.V. SHIRKOV

Lab. Thcor. Phys., Joint Inst, for Nucl. Research, Dubna, USSR

Lecture presented at KEK theory division

2 - 3 April, 1991

ABSTRACT

A very simple and general approach to the symmetry that is widely known as a Rcnormalizalion Group symmetry is presented. It essentially uses a functional formulation of group transformations that can be considered as a generalization of self-similarity transformations well known in mathematical physics since last century. This generalized Functional Self-Similarity symmetry and corresponding group transformations are discussed first for a number of simple physical problems taken from diverse fields of classical physics as well as for QED.

Then we formulate the Rcnorm-Gnup Method as a regular procedure that essentially improves the approximate solutions near the singularity.

After that we discuss relations between different formulations of Renormal- ization Group as they appear in various parts of a modern theoretical physics.

Finally we present several topics of RG.M application in modern QFT.

- 1 - CONTESTS

I. Renormalization Group as a Functional Self-Similarity Group (4-33)

$1-1. Introduction 4

§1-2. Mathematical preliminaries 7

§1-3. Dcfiniton of the Renorm-Group 10

§1-1. Simple illustration 14

§1-5. Functional self-similarity 16

§1-6. Other illustrations from classical physics 18

6a. Weak shock wave (IS) 6b. Transfer theory (20)

§1-7. RG in QED 23

7a. Effective electron charge (23) 7b. RG transformation (28)

7c. Dyson transformation (30)

§1-8. The Nature of RG=FSS Symmetry 32

II. Renornialization Group Method (34-47)

§0-1. Basic idea 34

§11-2. Differential formulation 35

§11-3. General solutions 38

§11-4. Technology of RGM 40

§11-5. Illustrations of RGM from classical physics 42

... 5«. Transfer problem (\2) 5b. Shock wave ($3)

§11-6. RGM usage in QFT II §11-7. Essence of RGM 46

III. Different Faces of Renormalization Group (48-63) pi-1. Critical phenomena 48

§111-2. Polymer theory 51

§111-3. Noncoherent radiation transfer 54

§HI-4. Turbulence 56

§M-5. Paths of RG expansion 57

§111-6. Two faces of RG in QFT 59

§HI-7. Summary 63

IV Some RG Applications in QFT (64-82)

§FV-1. General UV analysis 64

... la. One-cc~up!:ng case (64) lb. Multi-coupling case (68)

§IV-2. Perturbative appioach to the UV asymptote 70

... 2a. Structure of RG results (70) 2b. The ghost-pole trouble (72)

§IV-3. Reliability of results in QFT 73

... 3a. Ghost pole in QED (73) 3b. Scalar quariic model (74)

§IV-4. Asymptotic series Borel summation 77

§IV-5. Gauge fantoms in QCD 80

References (83-85)

-3 - I. RENORMALIZATION GROUP as FUNCTIONAL SELF-SIMILARITY GROUP

§ 1-1. Introduction

Specific symmetry underlying so called Renormalization Group (RG) was discovered in 1953 by Stiickelberg and Peterman [1] in the course of analysis of a finite arbitrariness arising in Quantum Field Theory (QFT) after subtraction of UltraViolet(UV) divergencies. They have realized the group structure of fi­ nite renormalizations and stressed the importance of differential group equations which they wrote down in a general form.

Quite independently Gell-Mann and Low by manipulating with Dyson renor- malization transformations have succeeded [2] in obtaining functional equations for QED propagators and performing on their basis a general analysis of short- distance behavior of electromagnetic interaction.

- 4 - 'The next important step was made by Bogoliubov and Shirkov [;i] in 19-"i5.

These authors established close relation between approaches of [1] and [2]. de­

rived functional equations in a massive case and first introduced the differential formulation of RG symmetry. Their most important result is the formulation of a Renormalization Group Method (R.GM), a regular method of improving per­ turbation approximations by combinig them with RG differential Lie equations.

This technique enabled the summation of leading log terms and have been used to deal with UV and IR behaviours in QED and other models. The most pop­ ular results of RG application in QFT obtained in the mid-fifties were related to short-distance behaviour. During about 15 years afterwards, RG and RGM were considered as very specific for only renormalized QFT with its divergencies and renormalizations. It is widely known that in the beginning of 70-ies just the

RG method was used for discovery of a famous asymptotic freedom property in nonabelian quantum fields models.

At the very same time K. Wilson (1971) [4] has managed to transfer RG ideology from QFT into statistical mechanics for the analysis of phase transition phenomena in spin lattices. This new version of RG based on a Kaclanoff's trick of spin blocking appeared to be much more transparent physically and apprehensible for a broad physical audience. Due to this, quite rapidly in the course of 70-ies,

RG algorithms were successively transferred into other theoretical fields : the theory of polymers, turbulence, transfer and some others. Technically researchers in these diverse fields usually used a bit different formulation and terminology.

Such expressions as "real-space R.G", "dynamic RG", "Monte-Carlo RG", 'RG chaos" are in use.

- 5 - In such a situation it was quite natural to suppose that there .should exist

some simple and general basement underlying RG different, faces in various fields

of theoretical physics. An attempt to formulate such an universal picture was

undertaken about ten years ago [5j. It turned out to be possible to discuss the

RG type of symmetry in general terms of a classical mathematical physics. This symmetry can be defined as a symmetry of solution (not of equation!) for a physical system with respect to the transformation involving boundary condition parameters.

In this short set of lectures we present a review of the RG ideas used in modern physics based on a rather simple foundation (we call it junctional self- similarity) which we illustrate by examples from classical physics and discuss the interrelation between RG different formulations.

The idea to create a written version of these lectures belongs to Professor

Yoshimitsu Shimizu. I use this possibility to thank him and Professor Hirotaka

Sugawara, the KEK director, for the hospitality during my stay in Japan. The text of lectures was written with the help of Drs. Chong-Sa Lim, Masato Jirnbo and Professor Nobuya Nakazawa.

-6- 'i J 2. Mathematical preliminaries

Let us start with MMIO simple statements which can be supposed to be widely

known. In QFT the H.(! usually is associated with possibility to present any phys­

ical quantity. F(Q-.g) calculated under definite renonnalization prescription

2 ( in a massless case ) in a form /• '(Q //r, <\;J) with renormalized coupling con­ stant g,, definition attached to some rcriormalization point (or renormalization

momentum scale) Q = //. The differential RG equation is usually said to be driven from the condition that F does not depend on the choice of/( ,

£ = 0. (1-1, d/i.

The coupling constant g^ dependence on /:, normalization momenta value, is described by a specific function Differential Equation(DE)

[*4--0(9)jr]F(*-9) = O • (1-: ox Og

! where x = Q //r, g stands for gti and beta function is defined as

0to,) = --^r ^ .- = ,<- (1-3)

The effective coupling(EC) g also can be considered as a function of two

arguments: g~{Q~j;r,g,L) with the boundary condition ~g{\,g) = g. Besides (2) ii satisfies the nonlinear DE

x r; = .J .'/('•.'/)) • (1-U ox that is nothing else but characteristic equation for (2). To employ this formalism one has to give the /9(//). Usually for this one uses renormalized perturbation theory.

The foregoing' can he considered as a "RG folklore". For brevity we gave it in the simplest massless version that corresponds to the IV case.

A less popular result is the Functional Equation (FE) for the 7j which in I'V case has the form

a(x,9)=ji{x/t,a(t,g)) . (i-5)

This equation which follows from Dyson renormalization transformations repre­ sents a basement of differential RG formulation. Popular DE (!) can be directly obtained from it by differtiating over x and putting / = x. On the other hand by differentiating (5) with respect to / at / = 1 we get partial DE

[x^-/3(g)^}g(xl9) = Q . (1-6)

analogous to (2).

Hence the FE (5) as well as similar FEs for propagators and vertex func­ tions must be considered as the most adequate and general formulation of RG symmetry in QFT. We shall call it the basic RG equation.

- S - However in reality, group 1'1's do not contain any physics at all. being just

the refleciinn of nothing else but. the group composition law! Namely, ,ve can

regard the change of a reference coupling as an operation of group element 1\ defined by I'tfi = Ji(t.'i). If we set .r = -/ , then the l.h.s. of (5) can achieved from

!l by 7Vi. while the r.h.s. may be identified as TTTtc/. The content of Y.<\. (•!) now

is just the group composition law. 7"r< = 7"r7i . Thus the properly of the basic

RG equation (5) is the formal condition for trasformations I!t to form a group.

To make this point more clear, let us show that KG equation (o) can be for­ mally obtained directly from the group law. Generally the mathematical formu­ lation of RG transformation can be presented as an explicit functional realization of the mentioned Lie group.

Consider the transformation 7'(/) of a certain abstract, set. M of elements

Mi into itself depending on a continuous real parameter / (— oc < / < oc) such

1 that for each Gn we have T(I)M = M' (M.M C M) . Suppose that M can be projected onto (he real axis. i.e. , for every Mi there corresponds some real number//,. Then this transformation can be written in the explicit analytic form

T(l)9 = a' = G(l,g), T(\nt) = T,

Here G is a continuous function of two arguments satisfying the normalization condition (7(0,17) = <7 which corresponds to the identity transformation 7^(0) = E.

Transformations 7'(/) form a group if they satisfy the composition law

77/) x 77A) = 7(/+ A) to which there corresponds the functional equation for G:

6'{A,G'(/..7)} = G'(A + /,5) (1-7)

- 9 - A.- ii follows from basic theorems of the Lie group theory, in this equation il is

Miffi'i.-nt to deal with the infinitesimal transformation when ) « 1 , i.e., with tin- l)K

^ = J. (l-S,

Here the .group generator is defined as

d(a) = —o— "l f = n- Of

IViToiming a logarithmic change of variables

7 = In i, / = In r , G(l.g) = g[i,g) ue obtain (1), (5) instead of (7) and (8). These are just RG functional and differential equations for the effective coupling tj in massless QFT with one cou­ pling constant g. Here x = (?"//'" is the ratio of a •1-nionicntuiii Q squared to a

"normalization" momentum // squared ; and g, the coupling constant.

5 1-3. Definition of the Renorm-Group.

In the simplest case, the renormalization group can be defined as a continuous one-parameter group of specific transformations of some solution of the physical problem, solution that is fixed by a boundary condition. The RG transformation involves the boundary condition parameters and corresponds to the change in the way of imposing condition.

for illustration imagine some one-argument solution f(x) that can be fixed by the boundary condition /(^o) = /n. Represent this given solution formally as

- 10- a function of boundary parameters as well: f{x) — f(x, XQ, fo). (This step can be considered as an embedding operation). The RG transformation then corresponds

to the changeover of the way of parametrization, say from {x\, f\ } to {xn, /j}. In other words the position at which the boundary condition is given does not need to be in, but we may choose anothor point a-;. The function / now can be written as F(x/xo,fo) with a property F(l,-/) = 7 . The following equality depends on the fact that under such change of the boundary condition the form of the function itself is not modified (as, e.g., in the case of F(x, 7) = $(ln x + 7 )),

F(—,fo) = F(—,fl)

X0 XI

Noting fi = F(xi/xo, fo) and in terms of £ = X/XQ and / = xt/xo we obtain

F(i, fo) = F(£/t, F(t, fo)) . The group operation is now given as

Rt •• {i^i/i, fo-+fi = F(t,fo)} •

Partial transformations of this kind are known in mathematical physics for a long time. Let x variable be dimensional and abovementioned function be

F(x/xo, fo). Suppose also that it is a power function of the first argument, i.e., has a form F(z,f) = /«*, k being a number. Such a solution is invariant under following simultaneous transformation of its arguments

P, : {z^z'^zt, /-/' = /<-*}. (1-9)

This transformation is well known as a atlf-simiUriiy transformation.

-11 - The renorm-group transformation for a given solution of some physical prob­

lem in the simplest case can be defined as a 1-paiameter transformation of its

two characteristics, say 9 and g , by

R(l) : {q-+j = q-l, g ~ g' = G(l, g) } , (l-lOo)

the first being a translation and the second of a more complicated functional

nature or using "multiplicative" form (x = e', t = e )

Rt : {z' = z/t, g' = g{t,9) } • (1-106)

Equation (7) for the transformation function G provides the group property of the

whole transformation (10a). Sometimes we shall refer to (10a) as to "additive"

version of R.

Usually of interest are some functions 4>{q,g) transforming as a linear repre­ sentation of RG :

M?,<7) - R{l)4> = z(l,g)4>(q\9'). (1-11)

Several simple generalizations of (10) and (11) will be considered below (see (12) and (15)].

If we do not neglect the particle mass m, then we have to insert an additional dimensionless argument into the effective coupling g which now can be considered as a function of three variables: x = Q~/fi~, y = nr/7'3 and 3- The presence of a

- 12- new "mass" argument IJ modifies the group transformation

rt, : {x' = x/l. y' = y/t. g'= g[t. y.g) } (1-12)

and the functional equation

g(',y\g) = n*/t

Here it is important that the new parameter y (that in physical nature must

be close to the x variable as it. scales in the same way) enters also into the

transformation law of

If in the considered QFT model there are several masses, like in QCD, there

will be several mass arguments

y — {y) = j/i.

In what follows we shall use Eqs.( 12,13) keeping the possible change y —• {y} in

mind.

A more serious generalization corresponds to the transition to the case with

several coupling constants: g —> {g} = gi,.. -gk- Here one has to introduce the

"family" of effective couplings

9 — {

M*, y\ {9}) = §i[x/t,

This system is a generalization of (5) and (13) to the case when every element

Mi can be described by k numerical parameters, i.e., by the point {17} in a k-

- 13 - dimensional real parameter space. The RG transformation for this case looks like

Rt {x-x/t, y^y/i, {

§ 1-4. Simple illustration

As can be shown, the symmetry underlying the FSS group transformations can be 'discovered' in many theoretical problems from diverse fields of physics.

Consider a very simple example from statics.

homogeneous external forces

fixation point

Fig. i.i

Imagine an elastic rod with a fixed point (the point "0" in Fig. 1.1) bent by some external force, e. g. gravity or pressure of a moving gas or liquid. The form of the rod can be described by the angle g between the tangent to the rod and the vertical direction considered as a function of distance / along the rod from a fixed point, that is by the function g(l). If the properties of the rod material as well as external forces are homogeneous along its length (i.e., independent of I), then g{l) can be expressed as a function G(l,go) depending also on go the

- 14- deviation angle at the point held fixed from which the distance / is measured.

Naturally, G should depend on some other arguments, like extra forces and rod

material parameters, as well but in this context they are irrelevant.

Consider two arbitrary points on the rod, "1" and "2" (see Fig. 1.1) with

coordinates l\ = A and U = A + /. The angles tji at points "0", "'1" and "2"

can be related via G function :

5i=G(A,.f/o), gj = G(X + l,go) = G[l,gi). (1-16)

Note here that to get the last form of r.h.s. of the second equation, we have to

imagine that the point held fixed is "1" instead of "0" as in Fig. 1.2. Combining

all these Eqs. we come to the group composition law (7) for the function G(l,g).

L— group i. _ operation "0" Fig. 1.2

In course of deriving the second of Eqs. (16) we have tacitly assumed that the rod is of the infinite length. If we introduce a finite length L, then the G(l, g) dependence must be replaced by function G{1, L,g) of three essential arguments, where the second one is the distance between the fixed point and the free end.

- 15 - V V

//•/////////// •// /•////// ////////

Fig. 1.3

Having now

(li=G(\L,g0), g2 = G(\ + l,L,g0) = G(l,L-\,

we come to the functional equation

G{l + \,L,g) = G(l,L-\,G{\,L,g)) (1-13') which is just the "additive" version of (13) and can be transformed to it by an appropriate change of variables used above to get (7) from (5) (Fig.1.3).

Let us stress that in (13') and (16') the second argument is not necessarily the rod length. It can be considered as a distance from a fixed point to the place in which the rod properties undergo a discrete change (say, in thickness or material). Generally, the additional argument L describes the discrete breaking of homogeneity property of the considered system. Such a breaking can take place at several points. Their coordinates must be introduced as G additional arguments: L —> {£}.

§ 1-5. Functional self-similarity

- 16- The !ICi transformations discussed above have close connection with the Self-

Sitnilarify(SS) in mathematical physics. The SS transformations for problems

formulated by nonlinear partial DEs are well known since last, century mainly

in hydrodynamics of liquids and gases. They are simultaneous 1-parameter A

transformations defined as power scaling of variables ; = {r, 1, . ..} and functions

l'(a:,r, ...) , etc.:

5.\ : < I -* t\" ,

To emphasize their power structure we shall use a term Power Selj-Similariiy

= PSS. According to Zel'dovicli and Barenblatt.. the PSS can be divided in two classes:

a/ PSS of the 1st kind with all indices a, u,... being integers ("Integer PSS") that usually could be found from the theory of dimensions;

b/ PSS of the 2nd kind with irrational indices ("Fractal PSS") that should be defined from dynamics.

To relate RG with PSS let us turn to the solution of the basic renorm-group

FE

g{xl.g)=g{x,g(i,g)) . (1-5)

Its general solution is known; it depends on arbitrary function of one argument

see below Eq.(2-12). Let us look for the special SDlution linear in g

9(x,g) =gf{x) .

- IT- Then f{i) should satisfy f(xi) = f(x)f(t) , whose solution is /(j) = x" and

g(x,t) = gx" .

This means that in our special case the RG transformation is just PSS transfor­

mation,

Rt:{z-~xr\ g-gt»} = St • (1-9')

Generally, in RG, instead of a power law, we have arbitrary fuctional dependence.

Hence, we can consider transformations (10), (12) as functional generalizations of usual (i.e., power) self-similarity transformations. It is natural to refer them gen­ erally to the transformations of functional scaling or to functional self-similarity

(FSS) rather than io RG-transformations. In short

RG = FSS ,

where FSS stands for "Functional Self-Similarity".

§ 1-6. Other illustrations from classical Physics

6a. Weak shock wave . Another example can be found in the 1-dimensional weak shock wave. The velocity-distance plot (distance from the creation point) at the given moment has a triangular shape as in Fig. 1.4 and can be described by the expression

v(x) = yx for x < L, =0 for L < x. (1-17)

-18- • vUl)

T

y V *>* I v = 0 x = L Fig. 1.4

where x = x(() is the front position and V = v(x) - the front velocity. In the

absence of viscosity the "conservation law" LV = const, holds. Here L and V

are the coordinate and the velocity at the shock wave front. They are functions

of time. However they can be considered as functions of the front wave position

L = x,V - V(x) as well. If the physical situation is homogeneous then the

front velocity V(x) can be considered as a function of two additional relevant

arguments - its own value V(x0) at some precedent point V0 = V(x0) {x0 < x)

and of to coordinate. In can be written down in the form : V[x) = G(x/xa, t"0) .

If we pick up three points x0, xi and i2,( as in the Fig.1.5) under homogeneity

surrounding the initial condition may be put either at xQ c: xx.

v(x)

A Vo ^ / / ^ 3^^| v2 m Xo X| Xg Fig. 1.5

- 19 - Thus we obtain the FSS equation analogous to (16)

1-2 = (7(jr2/io, V0) = 6'(r2/i-i, V'i) = G{x2/xuG(xi/x0, V0)) .

6b. Transfer theory. A similar argument has been done in the 1-dimensional

transfer problem by Mnatzakanian [6j. Consider a half-space filled with a homo­ geneous medium, on the surface of which some flow (of radiation or particles)

with the intensity go fall from the vacuum (see Fig. 1.6). Let us follow the flow as it moves inside the medium at the distance / from the boundary.

.''HOMOGENEOUS MEDIA /V'/'/'/'/ / ' ' s , . * . * . / J s

y /y/y ////////'y / / ' ' y y y y y ^ y y / /, ^ / y / / ' g y y y / y / y y y y g(f)= G(i,g0) y y y y y / / ~.

, y / / y y y y / y y Q\ = G(X,g0) y .' / / y y y y ' / / = y y y y y / ' / ' ' y 92 G(X+£,g0) y ' "1"-.---- / y y '\/ ' ' ' ' t , / y / y y \ y y y y y y y y — i \y y y y y ' y / y y \ y ' ,' ' ' / y ' y y

y / / / y y y

group x' / y / / , _ fi,. , operation \' ////}* ~G(£'9,)

= G{i,G(X,g0)}

Fig. 1.6

Due to homogeneity along the / coordinate (the distance from boundary) the intensity of the penetrated flow g(l) can be represented as a function of two essential arguments, g(l) — G(l,go). The values of the flow at points "0", 'T"

- 20- and '"2" shown in Fig. 1.6 can lie connected with each other by the transitivity

relations,

.71 = G(A, r/0) , g-i = G'(A + I,g0) = (7(/. <7i) (1-16)

which lead to Eq. (S), i.f., the additive version of FSS FE.

This problem also admits a generalization connected with a discrete inho-

mogeneity, that is with the case of two different kinds of homogeneous materials

separated by the inner boundary surface at / — L as in Fig. 1.7.

x x_x - V x x, /• ////'//, / X X. A A ,K. "• / / / / / x *g(£> ^Xi//'//'/'//// ' ' / ^ \ /N/1 ' / / / / / / / x x /xv^ / / / \ •< x *: x: < V i / / / / / / / /

X X xjX'XX'V,/ / / / / / / /K I \ \ / Two different kinds of material

Fig. I."

As in the elastic rod case the point of breaking / = L may correspond to

the boundary with empty space, and resulting equation coinsides with Eq. (16).

The transfer problem admits another generalization described by the "mul­

tiplication" of the last argument as presented in (12). Consider the case of

radiation of two different frequencies *i\ and ui (or particles of different energies or of different types) as shown in Fig. 1.8.

- 21 - 9o / / / tl^y gm = G(£;g0,h0) / s / / / / / / / //// / / / / / ' / v / / / / / M) / / / h(i)= HU;g0,h0) ho / / / / / / / / / . /^y^^ /

Fie. 1.8

Suppose that the material of the medium lias such properties that the transfer processes of the two flows are not independent. In that case the characteristic functions of these flows G and H depend on three arguments, namely, G(t,g, h) and //(/, g, h). The intensities g and h are defined by the boundary condition of the flows as

g{l) = G(l,go,h0) ,

h(l) = H(l,g0,h0) • After group operation / —* / — A we obtain a coupled set of functional equations:

G(l + \,g,k) = GV,gx,ln) ,

H(l+\,g,h) = H(l,gx,hx) ,

gx = Gl\,g,h) , h3l = H(\lg,k) . which is nothing else but an additive version of system (14) for k = 2.

This problem also admits a generalization connected with the discrete break­ ing of homogeneity. Note that as in the elastic rod case the point of breaking

/ = L may correspond to the boundary with empty space, which yields the prob­ lem of a plane layer of finite thickness. XIAV we ran make the important conclusion I hat a common properly yit-1.1 • mg functional group equal ions is just, the transitivity property of some physical quantity with respect to the way of giving its boundary or initial value. Hen<'<- the K(i or l-'SS symmetry is not a symmetry of equations but the symmetry of equations and boundary conditions considered as ••>. whole.

? I- 7. RG in QED

la. Effective Electron Charge. An essential feature of QFT is the presence of virtual states a,id virtual transitions. In QED, for example, the process of virtual dissociation of a photon into an electron-positron pair 7 <— e+ + c~ and vice versa can take place. The sequence of two such consecutive transitions can be drawn as a simple QED Feynman diagram presented in Fig. 1.9.

Virtual transitions:

photon ^_ Quantum corrections to photon propagation in vacuum

Fig. 1.9

The vacuum polarization processes lead to several specific phenomena and particularly to the notion of effective electron charge. To explain this, let us start, with a classical analogy.

Consider a polarizable medium consisting of molecules that can be imagined as electric dipoles. Let us insert into it an external electric charge Q. Due to the attraction of opposite charges, the elementary dipoles change their position so

- 23 - that this charge Q turns out to be partially screened. As a result, at a distance r from Q the electric potential will be smaller than the vacuum Coulomb value

Q/r and can be presented in the form Q[r)/v where Q(r) < Q. The introduced quantity Q(r) is known as an effective charge. As r decreases, Q(r) increases and as r — 0, Q(r) tends to Q.

In QFT the vacuum, i.e., the interparticle space itself, stand for the "polar- izable medium". Quantum-field vacuum is not physically empty. It is filled with vacuum fluctuations, i.e., with virtual particles. These "zero fluctuation" are a well-known effect of a ground state in quantum world. In QED zero oscillations consist mainly of short-lived virtual e+,e~ pairs that play the role of tiny elec­ tric dipoles. The influence of fluctuations of that sort will be considered here in detail.

a) Electron

Quantum effects of vacuum polarization around given charge (electron E )

Fig. 1.10

Figure 1.10 shows Feynman diagrams corresponding to the process of mea­ suring the electron charge. The electron itself is denoted by the symbol E.

The measurement is made with the help of the external electromagnetic field

-24- ••lf.X( • Figure 1.10a represents just the classical case without quantum effects. In

Fig. 1.101>. the case when the probing photon virtually dissociates into c+. c~ pair is presented. This pair forms a virtual dipole that produces partial screening of the measured charge. The process involves two elementary electromagnetic in­ teractions, its contribution into effective charge being proportional to the small number e~ = o ~ 1/137; and this contribution depends on the distance r ! In the region of r values much smaller than the Compton length of the electron

re = h/wr ~ 19 x 10~"cm it depends on r logarithmically and leads to expression

c - c(r) = e{l - £- In- 4- O(o.2)}.

Here the terms a- correspond to Figs. 1.10 c.d. and so on.

It is important to note that the phenomenon of the electron charge depen­ dence on the distance has a purely quantum origin (as was first discussed by

Dirac [7] in the middle of the 30-ies). The e(r) value grows as r decreases. So, quantitatively the effective-charge behavior corresponds to a classical picture of screening. This dependence is presented in Fig. I.lib by a set of curves.

Q e(r)

Q(r) ei — r a) classical b) quantum

Fig. 1.11

Each curve corresponds to a possible behavior of the effective charge e(r) as obtained from I he theory and considered without any reference lo experiment

(parameter r, a = f" being unspecified numerically).

The point is that in the classical analog the value of an external charge Q that is inserted into the polarizable medium (see Fig. 1.11a) is known from the very beginning. In our quantum case it is not the case and charge value can be measured at not very small distances. The result of measurement must generally be specified by two quantities: the "distance of measurement" r; and the corresponding charge value e; . Hence to make the choice from the set of curves of Fig.1.11 one has to fix the point on the plane with the coordinates r =

>•;, e(7-) = e;. Thus, for this chosen curve e(r;) = e;. Note that the usual definition of the electron charge by classical microscopic type (like Millikan) experiments

corresponds here to very large distances )• 3> re , i.e., 1/137 = e-{r = oo).

As is well known, in microphysics one usually uses the momentum rather than coordinate representation. Correspondingly, instead of e{r) one deals with the quantity d(Q2) i.e., with a Fourier transform of e(r) squared. It is a monotonically increasing function of its argument Q-, that is, of the 4-momenttim transfer squared. Here and below the bar denotes a function (distinct from a, Qf,, a; - its numerical values at some given value of the Q~ argument). The correspondence condition with the classical electrodynamics now takes the form d(0) = 1/137, as in our scale the classical field corresponds to a photon with zero 1-momentum. However, as before, to fix one of possible curves on the plane (Q, a) one has to give the point Q = /(, a = a^ and hence for the selected curve (that is presented in Fig.1.12) one has d(/r) = Q^.

- 26 - Fig. 1.12

The parameter it sometimes is referred to as a scale parameter. As is clear,

it is just the magnitude of photon momentum used for the charge measurement.

The effective coupling function a(Q~) describes the dependence of the electron charge value on the measurement conditions. It is worth noting that in our days

the logarithmic corrections to the Millikan value become essential and can be

measured at big accelerators.

The parameter \i has no analogue in classical electrodynamical Lagrangian.

The phenomenon of its "birth" in QFT is connected with the term dimensional

transmutation. As it is shown above, its appearance is very natural. This is a good place to recall the well-known ideas by Niels Bohr formulated in the middle of the thirties [8] and related to the complementarity principle. The point is that to specify the quantum system, it is necessary to fix its "macroscopical surround­ ing", i.e., to give the properties of macroscopic devices used in the measurement process. Just these devices are described by additional parameters. However this is not the end of the Bohr (i.e., - scale) parameter story. As can be shown, its existence leads to a new symmetry lying in the foundation of the renormalization

-27- group.

76. RG transformaiiovs. To do this, consider again Fig. 1.12 having in mind

that a choice of one curve has been made by the condition a(Q2 = /r) = a,,.

Assume also for simplicity that we deal with a massless QED, more precisely,

with the approximation | Q | ~> mc. This corresponds to the GeV-energy region

or to distances r •< re. Here the dimensionless effective charge function a can

be represented as a function of two dimensionless arguments Q-//r and a^, i.e.,

a(Q3) = &&*/»?,a,).

Now let us take into account that the pair of parameters /i, ap used for iden­

tifying the physical curve may generally correspond to any point on this curve.

Take two points "1" and "2" of that sort with coordinates /,

respectively. It is evident that a can be parameterized by any pair /(;, a,-; i —

1,2, ... so that for arbitrary Q2 values the identity a(Q'!//ij, a\) = a(Q2jpi, a?)

should hold.

At the same time the second argument in the r.h.s., ai which by definition

is equal to a at Q- = fi\ , can be expressed in terms of a parameterized with the

help of point "1" coordinates, i.e.,

a2 = a(Q- - fi]) = a(/4//i;,ai).

Combining the last two equations and introducing a simplified notations

Q2//'2 = x< ai = a< /'2//'i = ' we arrive at the functional equation

a(x,a) = a(x/t,a(t,a)), (1-18)

-28- identical with Eq. (1-5) lying, as was mentioned above in §2 . in the foundations of Hf! mathematical formalism.

Note that the corresponding continuous one-parameter transformation is just

I lie change ("1" — ''2*') of the parameterization point

At : {/'i —*/'2 = \/3 = "~('<*i)} • (1-19)

as explicitly indicated in Fig. 1.13.

a2

137

Pi ^ ' Fig. 1.13

Thus we have shown that in the renormalized QED there exists an invari- ance with respect to continuous transformations of the group type which involve two quantities and contain functional dependence. Going to the dimension less arguments we can write them down in the form (10b).

As can be shown, in QED the effective coupling 6 is equal to product of a and dimentionless function d(x,a) describing the transverse photon propagator with due regard for vacuum polarization effects. Generally, in QFT models with one coupling constant the effective coupling g(x, g) can be expressed as a product

-29- of g. corresponding vortex functions and propagators of different fields , Usually

this can he done on the basis of Dyson renormalization transformations.

7c. Dyson Transformation. As is well known from the QFT renonnalization theory, transition from one renormalization scheme (RS) to another can be for­ mulated in terms of multiplicative Dyson transformations for the vertices V; and

propagators Dk:

l V{ - V{ = ---VI Dk ~D'k = ;kDk (1 - 20)

Here one must treat V and D as renormalised ones and all z as finite numbers.

Dyson transformation is generally equivalent to an effective change o{ op­ erator field functions normalization and at the very end to that of coupling constants. As for a proper definition of these constants, one must specify the kinematic condition of their possible measurements. This means that vertices and propagators corresponding to different renormalization prescriptions can be related by a suitable change of momentum arguments.

With due regards for this momentum rescaling, it is possible to rewrite the

Dyson transformations in the momentum representation as follows (see e.g.. Sec­ tion 47 in the [T-l] or Appendix IX of [T-2])

with

-30- ami {A"'} = A-[,A--i A:;,. Taking into account normalization conditions for

scalar functions .s;. Vj- , which in the RS based upon the momentum subtraction

(MOM-scheme) can be chosen in the very simple form

.'.'(1,!J,.'7) = V)({1}, J,fl) = l (1-21)

one can define finite renormalisation constants ;;, ZJ in terms of s; and Vj-. Using expressions thus obtained it is easy to get equations of the type [compare with

Kq.(ll)]

aii*,y,9) = si(l,y;a)»i(r/t,y/l;g(t,y,g)) , (1-22) where g, the so called invariant coupling, satisfies a more simple closed equation

g(*,v\g) = §Wi,v/Ug{t,v;g)) • (1 - 13)

Note here that the normalisation conditions

*i(li3/>ff) = 1 a"d ail

are "built in" functional equations.

Note also that group FEs for matrix element can be written as follows

*'({*}.»;*) = A/({*/«}.tf/U(<.y;s)) (1-24)

that reflects its independence of reniormalization details.

-31 - Tims the KG invariance is nothing but the invariance with respect, to the

way of parameterization. For instance, instead of using the "Millikan's value" iT(0) = 1/137 we may use a more nice number

acern = a(A/£) = 1/128 = 2"\

the "CERX value" at Q = Mz as well - see Fig.I.ll.

5(0) • . Millikan 100 6eV Fig. 1.14

§ 1-8. The Nature of RG=FSS Symmetry.

What is the meaning of RG = FSS symmetry? It is not a symmetry of a sys­ tem (or symmetry of equation), but is rather a symmetry of solution, considered as a function of physical variable and of its own boundary value. Namely, it is a symmetry with respect to the change of the way of parameterizing the solution.

Let us also note that here the important underlying property of a physical system is the homogeneity that can be broken by the discrete inhomogeneity. The FSS symmetry is a very simple and usual property in physical phenomena.

Now we can answer the question about the physical nature of the symmetry underlying the functional self-similarity and renorm-group transformations. As

-32- can be easily seen from the examples presented in the Sections 6a and 06 this is not a symmetry of equation(s) of a given physical problem but rather the symmetry of solution(s) considered as a function of essential physical variables and of appropriate boundary value(s). It can be defined as invariance property of some physical quantity expressed by such solution with respect to the way of giving its boundary value. The change of this way is just the group operation and generally the group property in this context can be considered as a transitivity property of such operations.

- 33 - n. Reuormalization Group Method it 11-1. Basic idea

Approximate solutions of the physical problems with FSS = RG symmetry usually do not obey this s)-mmetry which is lost, in the course of approximation.

This is essential when solution under consideration posseses a singularity as far as the singularity structure usually is destroyed by approximation.

In QFT, e.g.. the usual way of calculation is based on perturbation method, i.e., on power expansion in g. It is not difficult to see that finite sums of this expansion do not satisfy the functional group Eqs. As the simplest illustration consider the effective coupling j? in the UV region where the one-loop contribution has a logarithmic form

9ptix,9) = g + g'd\nx. (2-1)

By substituting this expression into FEq (1-5) after simple manipulation one arrive to

g+g'18\nx£g + g\8]n x + 2g3i82 In t \n{x/t)

- error of the g3 order. This discrepancy can be liquidated by addition of the next order term g3dJ In" x into the r.h.s. of (1). The "improved" expression would yield the discrepancy of the

On the other we can conclude that functional group equation represents a tool

- 3-1 - for iterative reconstruction of renonn-invariant expression thai lias the form of

infinite series.

This example illustrates a rather general situation. As a rule, approximate solutions do not satisfy a group symmetry. In our case this is happened in the

l:V limit at \n x — co where the observed discrepancy becomes important.

Another illustration is provided by the one-dimensional transfer problem dis­ cussed in jjI-6. Here it is rather simple to get the approximate behavior close to boundary

G(l,g)=g + lG'{0,g), <

which, being considered for arbitrary / values, also does not obey the FSS sym­ metry.

On this basis one can put the problem of "renormalization-invariant improve­ ment" of perturbative results. The key idea is to combine such solutions with group equations. The most simple and convenient way for this "marriage" is the use of Lie equations, i.e., group differential equations. The renormalization group method (RGM) is essentially based on this group equations.

§ H-2. Differential Formulation.

These equations can be obtained from the functional ones in two different ways. Differentiating Eq.(l-13) by x and putting ( = x then, one obtains:

f 7T =0(y/x,g(x,y\g)), (2 - .5)

- 35 with

0(y,g) = —jt— at / = i (2--1)

The nonlinear equation (3) can be considered as a "massive" generalization of

the equation (1-S).

On the other hand, one can differentiate Eq.(l-13) with respect lo /. at the

point, i = 1, which yields

a linear partial differential equation.

Analogous operations applied to Eq.(l-22) lead to:

Ql = y[y/x,g(x,y;g)]s(x,rj\g) (2- 6)

and

+ V **fo T ~ ^y<9)j- - 7(y<9)}s{x, y,a) = 0 (2 - 7)

where

i(y,g) = —jt— at < = i (2-8)

- the so called anomalous dimension of s. For a matrix element M satisfying

functional Eq.(l-24) this dimension is equal to zero. The corresponding partial differential equation looks like

(LXi^VTy~0{V,9)h)M{X,Vi9) = Q (2_9)

Equations (5), (7), (9) express the independence of t variable of the r.h.s. of

the corresponding functional group equations, i.e., a mutual compensation of (

-36- d<'|ieiid'"ii''ii's via three (or more) arguments. They can be called covijicnsaiionul

"'Illations to distinguish them from nonlinear Eqs. (3). (G) which can be referred to as ri-olution group equations.

Stress that compensational as well as evolution differential Eqs. taken to- uvlhci with normalization (i.e., boundary) conditions like g[l,g) = g are equiv­ alent to functional Eqs. and to each other. At. the same time, evolution Lie equations turn out to be more convenient, for practical construction of the solu­ tion, generators 3,y being given.

It is not difficult to formulate group DEs for multi-coupling case by proper differentiating FEs (1-14). For instance, the system of evolution DEs looks like

= /?;(

Let us comment, also that the UV limit of compensational differential Eqs. like e.g.,

{x£-fi[g)^-y{g)}s(x,g) = 0 (2-7') coincides with the UV limit of specific nonclosed Eqs. of the type

{xh ~ ^f/)r| _ ^3)}4x,g) = _\5 (C - S) obtained in the early 70-ies by Callan and Symanzik. The r.h.s. of this equation contains the result of mass counter-term insertion into all internal lines of all dia­ grams for the function o under consideration. For this leason in current literature compensatioiial ECJ.S. are usually related to the Callan-Symanzik equations. How­ ever these Eqs. just in the form (5), (7) were first obtained by L.V.Ovsyannikov

- 37- iii 1050 while solving functional RG equations [9j. Therefore we consider it jus­

tifiable to relate compensational differential Eqs. to the Ovsyannikov's rather

than to some other names.

§ 11—3. General solutions

General solution of the group functional equations was obtained in the men­

tioned paper by applying the general theory of partial differential equations to the

compensational Eqs. (5) and (7). Details of the derivation can also be found in

the Section 48.3 of the monograph [T-l]. The results obtained can be formulated

as follows:

To every solution of the Eq.(5) there corresponds some function oi two argu­

ments F(y,g), reversible with respect to its second argument and connected to

g by the relation

F(y,g) = F(y/x,g(x,y;g)) (2-11)

That is this functional relation represents the general solution of (5) as well as

(1-13). The explicit form of the g can be obtained from (10) by reversing the r.ii.s:

£(*. y. g) = F^][y/x, F(y, g)] , where the suffix (2) indicates the inversion with respect to the second argument.

To determine F it is sufficient to specify the generator P(y,g).

Note also that to get from (11) the solution in the UV limit, that is in a massless case at y = 0, one has to assume for F a rather specific limiting form

F(y,g) = ye\p[f(g)\ or =lny + /(y) as y = 0.

-38- Then

/{$(*.

* = /-'{In * + /(*)} ,

Here f'{g) = if j3(g). Solution (12) was first obtained by T.D.Lee - see Appendix to the Ref.[2].

To every solution of Eq.(7) for an '1-argument' function s, there corresponds some function T,(y,g) related to s by

, , ^[y/x,g{x,y;g)] s(x>y,9) = ™ C • 2-13 2(y,ff)

Let us give also the general Ovsyannikov solution of the system (1-10) at k =

2, i.e., for two effective couplings case. It can be written down in terms of two arbitrary 3-argument functions F\,F-> , that must be reversible simultaneously with respect to last arguments, and defined from the system of two functional relations

Fi(y<9\,9i) = Fi[ylx,9i(*,V\9\>92),9i(*,y\9u9i)], ' = 1,2. (2- 14)

Note here that all presented solutions (10) - (14) satisfy the normalization con­ dition of the Eq.(l-23) type.

- 39 - The transition to the massless limit in expressions (13) -- (14) can be per­

formed by the trick analogous to the given above. Then, e.g.,

Let us also write down the solution for 2 - coupling case in massless limit in the

form analogous to (12)

fi[g,h) = fi{g,h) + \nx, i=l,2. (2-16)

The generalization of this system on a more general case is evident. From the

solutions presented, it follows that imposing group properties one reduces the

number of independent arguments by one unit.

§ 11-4. Technology of RGM

The idea of the marriage of the approximate solution with group symmetry can be realised with help of group DEs. If we define group generators /?, 7 from

these approximate solutions and then solve evolution DEs we obtain RG improved solutions that obey the group symmetry and correspond to approximate solutions

used as input.

Now we can formulate a method of improving approximate solutions. The procedure is given by the following recipe which we illustrate by a massless 1- coupling case (1-4), (1-5):

Qj, Assume some approximate solution Jappr is known.

-40- L On the basis of the Eq. (1-3) define the beia-hmrt ion with the hel|j of

MHO approximate solution,

P{3) = ^tfappr^ffl (2-11 (=1

1L. Integrate the equation (-1), i.e., construct the function

def f d- fig) : (2-IS) *f Pi-!) and

3^ Find the solution,

5RG(x>ff) = /_1{/(3) + l'n} (2 - 19)

4j. Then the solution 5RG, which exactly satisfies the RG symmetry, i.e., it is

an exact solution of Eq. (1-5), corresponds to #appr

Consider the simplest perturbative expression for the effective coupling,

(2 - 20pi) .<7appr(z,tf) =5-/3i

Then the j-function is of the form

& = -fa"- and the solution is

= lux

J9 /%) /?i \a

-41 - g(*.g)= , . ,, • (2-20.-J) 1 + f/O] In J

§ 11-5. Illustrations of RGM from classical physics

5a. Transfer problem . Here we consider an 1-dimentional transfer problem.

The group equation for the transfer problem is

dG{l'3)=B(G) , (1-8) di and the generator BM-W'.*) dl 1=0 is just the infinitesimal response at the boundary

G(e,g) = g+B(g)e .

Suppose we can find it from some simple reasoning (without solving the kinetic Boltzmann equation). Consider two simple cases when this response is linear (Case a) or quadratic (Case b) in the boundary density g,

g — (kg e < 1 (Case a) , (2-21) i g - eng2 c < 1 (Case b) , Hence —kg (Case a) ,

{ — Kg' (Case b) .

-42- and solving l">|.(l-8) we find

hl .<7f - (Case a) 0n.c.{l,9) = (2-22) (Case b) . 1 + K.r/1

These expressions possess the FSS property, i.e., they are solutions of Eq. (1-7).

On the other hand, at small values of / they coincide with the input approximate expressions (21). They, however, describe solutions on the whole positive interval including the asymptotic region as / — oc , as shown in the Fig.II. 1.

• / / / / / / ///

/ / / / / / / />£// / / / , / / / / / / 7~^ ^SL '//////// Fig. U.l

5b. Weak skock wave . Another example of ROM from classical physics is the 1-dimensional weak shock wave. When the wave arrived at a point I (I = /o+f) as in Fig. H.2.

V

— J -—I— distance Fig. fl.2

- 43 - I he front velocity V[r.\ takes the following form:

V(t) = V0--£rt . (2-23)

Hence the generator is

B(V) = ~ . (2-24)

Thus we find the solution,

V(l - /„) = Jvj - a(l - t0) , (2 - 25) which is well known in hydrodynamics.

Note that here we have a rather nontrivial nonperturbative example of j3- fu net ion.

Here there stands an important and interesting problem, which has not yet been solved. It is the problem of transition to a "more strong shock wave" when viscosity becomes essential. There is no analytic solution like (1-17) and (25).

In this case the FSS = RG treatment may be "multi-coupling".

§ ]I-6. RGM usage in QFT

As it was explained above in §11-1 the QFT perturbation expression of finite order does not obey the RG symmetry. On the other hand in §11-4 it was shown that the 1-loop UV approximation for p used as an input in Eq. (17) for the construction of group generator /3(g) yields the expression (20rg) that obeys the group symmetry and exactly satisfies the FE (1-5).

-14 - We propose to the reader to check this fact

Now. using the geometric progression (20rg) as a hint let us represent "2-loop perturbative approximation for "g in the form

3 2 9$ = 3 - a'ft In x + 5 19? In * - 3, In r] + 0(g*) , (2-26) where 3\ and j3-> mean the /3-function coefficients at the 1-loop and 2-loop level, respectively. If we substitute this expression into Eq.(I-o) we obtain the descrep- ancy ~ 0{gA). Meanwhile we can use ji/L, as an input in Eq.(17).

Now the step jL yields

,8{7)(g) = -Ihi - ihg3 and then (step 2^.)

To make the last step we have to solve the equation for g).g :

/'V4'] = /<%) +I"* (2--'7) which, however, is a transcendental one and has no explicit solution. This means that we have to perform the step ^ approximately. For this goal we can account that the second, logarithmic, contribution into f^H:) is a small correction to

- -15 - the first one at bz

expression

instead of grg into this correction and obtain the explicit expression

9r'a l+fagl+{h/0i)g\*[l+gM ' (2 29)

This result (first obtained [3J in mid-fifties) is interesting in several aspects.

First, being expanded in g and gl powers it produces an infinite series contain­

ing "leading", i.e., ~ g"+1/n, and "next-to-leading" ( ~ gn+-l") UV logarithmic

contributions. Second, it contains nontrivial analytic dependence

ln(l + r?j9i0~ln(lnO2)

which is absent in perturbation input. Third, being compared with Eq.(28) it demonstrates algorithm of subsequent accuracy improving, i.e., of RGM regular­

ity.

$ 11-7. Essence of RGM

Now we can resume the RGM properties as follows. The RGM is a regular procedure of combining dynamical information (taken from approximate solu­ tion) with FSS group symmetry. The essence of RGM is the following:

1) The mathematical tool used in RGM is Lie differential equations.

-40- 2) The key element of RGM is possibility of (approximate) determination of group generators from dynamics.

3) The IIGM works effectively in the case when a solution has a singlular be­ haviour. It restoros the structure of singularity. [It does not work at all in

"trivial cases" when dynamical equation coincides with Lie equations.This is the case in our first macroscopic example with elastic rod .]

-47- EI. Different Faces of Reiiorinalizatioii Group

In 70's and SO's RGM was applied to (besides QFT and critical phenomena)

polymers, turbulence, non-coherent radiation transfer, dynamical chaos, and so

on. Simpler and physically transparent motivation in critical phenomena (than

in QFT) makes this "explosion" of RG applications possible.

§ IK-1. Critical phenomena

RG in critical phenomena is based on Kadanoff-Wilson procedure [10.-1] re­ ferred to as "decimation" or "blocking". Consider a 2-diniensiona) spin lattice with spacing a, an elementary spin a silting at every site as in Fig. Ill. 1 - The N, total amount of spin sites is supposed to be very large, of the Avogadro number order.

_ *- _*. _ v - -fh S- - ^~ ' • ' s • '0 ' ' ' s ,' '» f/ t/^ / ' .---At--*-/-/-

Fig. 1.1

The Hamiltonian describing the spin interaction of the nearest neighbours is expressed as

H = k^tr,:-o-,-±i . (3-1) where k is the coupling constant. The statistical sum is obtained from the par­ tition function.

-48- ,S-=< ex p(-///#) >iv,ra6,

'In ivalize )lie blocking or decimation, one has to perform an "averaging" over blocks consisting of n elementary sites. This is a very essential step as far as it diminishes t lie degree of freedom number (from -V to N/n). It destroys the small- range properties of the system under consideration, in the averaging course some information being lost. However, the long-range physics (like critical phenomena of phase transition) is not affected by it and we gain the simplification of our problem.

After this procedure new effective spins S arise on new- effective lattice as shown in Fig. III.2 where the n = 3 and n = 4 blocking are illustrated.

— a

fns3 Kadanoff /\, : " "•'•"•"'• :4;blocking" i:--Vr...:: :./

New Effective - Lattices

Fig. 111.2

- 49 - Wo obtain also new effective Hamiltonian.

uliere AH contains quartic and higher terms,

AH = £ rrr-c + • • • .

Kor 1R properties (long-distance), AH is unessential. Hence we can conclude that the spin averaging leads to a (approximate ) transformation,

A£a.,-A-2£S-S , (3-2) i I or, taking into account the size of "elementary block" change, j;- : erally { a —>

The latter is called Kadanoff-Wilson transformation.

In general the "new" coupling constant A'n is a function of the "old" one and the decimation number n. It is convenient to write it down in the form

A\i = K(l/n, A-). Then we can formulate the KW transformation as follows

KW{n) : la - ^a, k - A'„ = A' (-, Jfc] 1 . (3-3)

These transformations obeys the group composition law

KW{n) • KW(m) = KW(nm)

-50- K{x,k) = R{j,K(t,k)) (3 - 4) nm n

This is just a FIG = FSS symmetry.

We observe the following points:

1. The FSS symmetry is approximate (due to AH).

2. The transformations KW(n)'s are discrete.

3. There exist no reverse transformation.

Hence Kadanoff-Wilson RG is an approximate and discrete semi-group. In a long distance sense (IR limit), however, AH is irrelevant, A(l/n) is close to continuum and it is possible to use differential Lie equations.

In application of these transformations to the critical phenomena the notion of & fixed point is important. As explained below, in the Section IV-1, it is usually associated with power type asymptotic behavior. Note here that, contrary to the QFT case considered in Section IV-1, in phase transition physics we deal with IR stable points.

§ HI—2. Polymer theory

In the polymer physics one consider statistical properties of polymer macro- molecules that can be imagined as a very long chain of identical elements. The number of elements N, called the polymerization rate, could be as big as 10'\ the macromolecular size reaching several hundred Angstroms.

Such a big molecular chain form a specific pattern resembling the pattern of random walk. Indeed these two problems can be formulated in a similar way.

-51 - The central problem of polymer theory is very close to that of the random walk and can be formulated as follows.

Take a very long chain of A'' steps (the length of each "step" = a) as in

Kit;. 111.3

A A

Fig. ffl.3

and define the "chain size" R,\ , i.e., the distance between the "start" and the "finish" points, the distribution function of angles <£; between neighboring elements being given. This function f(

fn reality, polymer molecules are swimming in a solvent and form globulars - see Fisr. HI.4.

R :— Fig. IH.4

52 For huge .V values the molecular size liy follows the power low

A\\- ~ .V [// = Fleury index] , (3 - 5)

- the Fleury law. When N is given, Il,\- is a function of f(

01. external conditions (e.g., temperature T, properties of solvent, etc.). If T increases. IIy increases and at some moment, globulars become touch one another.

This is the polymerization very similar to a phase transition phenomena.

The RG ideology was introduced in polymer physics by De Gennes [J],12]. The key idea is the grouping of n chain subsequent elements into new "elementary blocks" as is shown in Fig. M.5. 2 ••/¥ ^x^==- Fig. in.5

This grouping operation is very close to Kadanoff's blocking. It leads to the transformation,

I " • (3-6) { a - An J which is analogous to one for spin lattice decimation. The second transformation must be specified by direct calculation that gives the explicit form of ,4„ = «(n, a).

Here we have a discrete semi-group. When, by using RG technique one finds the fixed point then one obtains the Fleury power law (5) and can calculate its index

53 Generally the excluded volume account yields some complifications. However

inside the RGM framework it can be made rather simply [13) by introducing

additional argument similar to finite length L in elastic rod or transfer problem

and particle mass m in QFT.

§ "' -3. Noncoherent radiation transfer [15,1G]

Here we consider propagation of light quanta in a media with narrow Breit-

Wigner resonance,

"max > cr0 ~ background . (3-7)

as in Fig. HI.6.

aid))

^^~2 == ~w ——»-• = »• n-— n+1 * Fig. m.6

Suppose that the background mean free path lo ~ cr^"1 is rather large and comparable with the size of the media. Then the given quanta "travel history" can be studied by a Monte-Carlo simulation method with a reasonable total num­ ber of collisions. However photon walking in a media also "walks" in frequency due to heat motion and interactions of gas molecules. Due to this, starting with

-54- M>in>' background frequency value it can happen to enter into resonance zone with very small mean free path.

fr~— • (3-3) °"max

This leads to a specific trouble. The method of successive collisions ceases to be effective as far as the most part of the computer time is "eated" to pass this dangerous resonance neigborhood region as shown in Fig.JII.7.

Fig. in.7

The "blocking idea" in this case is

1. To change the Breit-Wigner curve into the step function with the width P.

2. To calculate separately the effective walk distance across the dangerous

zone as a function of .T: C(f).

Thus we have a pair {T, ((T)}, and can scale and apply the RG ideology and equations [13,11]. The resulting picture is illustrated by Fig.DI.8.

-55- one effective step

Fig. 111.8

§ I1I-4. Turbulence

The formalism of the turbulence problem with the RG ideology is obtained along the following steps [17,18,19]:

1. Define the generating functional.

2. Write the path integral representation.

3. Find the equivalence of the classical statistical system to some quantum

field theory.

4. Construct the system of Schwinger-Dyson equations.

5. Apply the Feynman diagram technique.

6. Perform the finite renormalization procedure.

7. Then the RG ideology and equations are obtained.

The physics of the turbulence problem is displayed in Fig. 111.9.

-56- K wave 0 i- i. / / / / / / / / / / number physical region \ upper

llimiti : Ry (K) -effective K' viscosity —i - i K

group operation • R.(K)

Fig. 1.9

§ M—5. Paths of RG expansion

We see that RG is expanded in different fields of physics by two different ways:

1. the direct analogy with Kadanov-Wilson construction (averaging over some

set of degrees of freedom) on polymers, non-coherent transfer and percola­

tion, i.e., constructing a set of models for a given physical problem.

2. search for the exact symmetry (FSS = RG) by proof of the equivalence

with QFT: e.g., on turbulence (de Dominicis and Martin [17); A. Vasiliev

ei at. [19]), plasma turbulence and phase transition ( continuous spin-field

-57- models l,y K. Wilson [•!]).

Arc there different renormalization groups'.' - The answer is YES:

1. In QFT and macroscopic examples, RG = FSS symmetry is an exact

symmetry of the solution formulated in its natural variables.

2. In turbulence, continuous spin-field models and some others, it is a symme-

try of an equivalent QFT model.

3. In polymers, percolation, etc., (with Kadanov-Wilson blocking), the RG

transformation is a transformation between different auxiliary models

(specially constructed for this purpose).

As we have seen there is no essential difference in mathematical formalism.

There exists, however, a profound difference in physics:

A. In some cases (QFT and macroscopic examples), the FSS = RG is an

exact symmetry of a solution.

B. In Kadanov-Wilson type problem (critical phenomena, polymers, etc.), one

has to construct a set M of models A/,- as in Fig. M.10.

Mi MK

Fig. DI.10

- 58 - The HCi transformation is

«(«)A/(,-) = .Uini-i . (3-1--M

acting inside of the set of models. Thus the FSS symmetry "exis'.s" only inside this set of models.

Let us also make a comment about the mathematical difference between dif­ ferent renormalization groups. In the QFT, as well as in macroscopical examples given above in the Part I, RG (or FSS) is an exact continuous symmetry group.

On the other hand in critical phenomena, polymers and some other similar cases

(with averaging operation) it is approximate, discrete semi-group. We should add that in dynamical chaos theory, where some people use RG ideas and termi­ nology (see, e.g., [20]) the functional iterations do not form any group at all.

§ JII-6. Two faces of RG in QFT

An important notion in QFT is the invariance with respect to RG transfor­ mation, i.e., renorm-invariance. This implies thai the results are independent of the choice of reiiormalization scheme. The symmetry underlying the RG transfor­ mations in QFT is exact. In this case the running parameter being the logarithm of a 4-momentum squared and g, the renormalized coupling constant.

An essential feature of QFT is the presence of virtual states and virtual transitions. In QED, for example, the processes of virtual transition of a photon into an electron-positron pair and vice versa can take place. This process of vacuum polarization leads to several specific phenommena and particularly to the notion of effective charge. In classical electro-magnetism, an external electric

-59- charge in the polarizable medium attracts opposite sign charged particle in the

medium so that the charge is partially screened. In QFT, the vacuum, i.e., the

interparticle space itself, stands for "polarizable medium". Due to the vacuum

fluctuation the charge is screened. It was first shown by Dirac (1933) [T] in

momentum space. Q- dependence of an electron charge is given by

3 ! c(Q ,A ) = e0|l + gln|l + ...} . (3-13)

expressed in terms of bare electron charge eo = x/TJroo and of momentum cutoff

A.

The first approach of using RG idea to this problem was done by Stiickelberg

and Peterman [1]. In their pioneering investigation the RG transformation were

introduced rather formally within a discussion of the structure of finite expres­

sions that arise after subtraction (i.e., renormalization ) of UV infinities and

contain finite arbitrariness. Just this "degree of freedom" in finite renormalized

expressions was used by Bogoliubov and Shirkov [3]. Roughly speaking this cor­

responds to the change (A —» fi) of the degree of freedom parameter. Instead of

(13) we have now

e(OV) = e{l + £l„|I + ...} . (3-14)

Another approach was used by Gell-Mann and Low [2]. Their paper was devoted to the short distance behaviour in QED and the "A degree of freedom"

was used to analyze UV behaviour.

-CO- In the first approach [1,3], the effective charge is given by a finite represen­

tation"

«(Q') = «,[1 + ^I«£ + - (3 - 15)

Here, the FSS = RG symmetry is fonmilatecl in terms of Q scale and the f.i

represents the reference point as shown at Fig.III.11.

5(rH

iW1 Fig. Iff. 11

On the other hand, in the second approach [2] the charge is given by the

"singular representation"

a = a(1 + _,n_ + . (3-16)

We can draw a physically simple picture (as was commented later by Wilson in his Nobel lecture) to this approach. Effectively we can imagine the electron of a finite size, smeared over a small volume with radius R\ = h/cA , ln(.V/mj) >> 1.

The electric charge e(A) of such a non-local electron is considered as depending on the cut-off momentum A so that this dependence accumulates the vacuum polarization effects which take place at distances from the point electron smaller

than RA (see Fig. III.12a).

-CI - e(A)

Fig. HI.12a

We have the set of models with non-local electron correponding to the values of the cut-off A as shown in Fig. HI. 12b. b) Smearing of point electron try Mj nm. R, Rj electron radius = R Fig. M.12b

Here, e,- depends on R{ and the vacuum polarization effects in the excluded volume R^- should be subtracted. In this language the RG transformation is the transition from one value of the smearing radius to another Ri —> Rj, simultane­ ously with a corresponding change of the effective electron charge e(Aj) —* e(\j).

In other words, RG symmetry here is the symmetry of operations in the space of models of non-local QED constructed in such a way that at large distances every model is equivalent to the real local one. It is correct to say that the

Henormalization Groups in two approaches are different from each other.

-62 - § 111-7. Summary

Before going into the discussion of some applications of KG in QFT, wc give the summary of the general properties of RG and FSS. The RG = FSS symmetry is not a symmetry of a physical system, but it is a symmetry of solution with respect to changing a way of parametrization, in terms of natural quantity, or it is a symmetry inside a set of models exploiting the way of their (i.e., models) parametrization. The RG Method is a regular method of improving approximate solution by its "Marriage" with group symmetry. The RGM is effective in the case of solution with singularity. The RG Approach and RGM are general powerful methods in theoretical and mathematical physics.

-63- IV Some RG Applications in QFT

The final lecture is devoted to several actual topics of RG applications in QFT short-distance asymptotic behavior. It is more oriented on particle physicists.

VV'o discuss first the specific features of UV analysis connected with perturbation theory used in its practical applications. Then we consider the problem of gauge dependence of RG results in QCD.

?j IV'-l. General UV analysis

la. One-coupling case. General analysis of UV asymptotic behavior for the

1-coupling QFT model can be performed rather simply on the basis of solution of massless group equation (1-4) for the effective coupling

I 0h)W) =l = ln7 ' (4-1) i here g = g{0) (i.e., g at Q- = /i2) and /i is a renormalization point. As follows from it, the asymptotic at Jna; —• oo corresponds to the divergence at the upper limit, of the l.h.s. integral. Depending on the behaviour of the ft function shown in Fig. IV.1, the resultant UV behaviour of the coupling g differs very much.

Fig. JV.l Suppose first that at very small g values, lieta-funct ion is positive. Then

three eases are possible :

u) Consider first the situation when

(•I-2a) /* /*(--) Jg corresponding to the case when the beta-function has a zero at g, as shown in

Fie. IV.2.

gU) finite renormalization 9.

It Fig. iv.2

Here the UV asymptotic value of effective coupling is finite

that corresponds to the finite renormalization of the coupling constant.

Z = g{oo,g)/g. (4 - 4)

Using the qualitative theory of DEs terminology one can say that at g = g, we have a UV fixed point.

- 65 - If we admit that at g = y, there is a first order zero

0{g) - b(a* - g) 5)

then Eq.(2a) gives

,Q\- 3(l,g)-9*~Cexp(-bl) = C(—f)k as Q~ — oo (4-6)

i.e., in the vicinity of a fixed point we have asymptotic power regime,

b) If ^(p) is (monotonically) increasing but gentler than g2, so that

dz f (4 - 26) then the effective coupling tends to infinity as shown in Fig. IV.3.

g(J) Infinite GO renormalization..

g Fig. 1V.3

lim g(x,g) -» oo (4 - 34) that corresponds to infinite coupling constant renormalization. Formally this is equivalent to the g^ = oo.

-66- c) At

(-1 - •>) xx < oc that happens if

lim 0(q)/q~ > const the theory lias an inner contradiction, as far a.s

(-1 - 3c)

and the momentum region x > xx cannot He described by the theory. We encounter here with a ghost trouble as explained below in the Section JV-2a and illustrated in Fig. IV.4.

Fig. IV.4

Up to now we assumed that the generator 0(g) is positive. In the opposite

d) P(o) = ~Ks) < 0 °"e 'las to consider the equation

(4 - 2d) and study possible divergence of the integral involved on the lower limit.

-67- If this occurs at some finite value g = g^ then the situation is quite analogous

la the case b) considered above and described by Eqs.(3(>), (-1). The only difference

is that now the effective coupling tends to its limiting value gx, from above.

As the most important case we consider the possibility when the singularity

lies at the origin :

m = o. Ad)

That happens in QCD. Then in the UV limit g vanishes 17(00, g) = 0 that corre­ sponds to the asymptotic freedom phenomenon. This situation is illustrated by

Fig. IV.5.

Asymptotic Freedom

i Fig. IV.5

If we assume here that /3(g) = — bg' at g —• 0 then

g(x,g) (4-3c/) film as 1 —* 00

lb. Multi-coupling case. For the quantum field model with several coupling constants one has to consider the system of coupled functional or differential Eqs.

(1-14), (2-10). The last ones can be analyzed by the well known methods of the

-68- theory of differential equations. Let us take the case when there are two coupling constants y and h. The system of evolution differential equations is of the form

9 = P,{gM, Ji = fa(gji). (4-7) with / = df/dlnx. According to (2-16) the general solution of this system can be written in the form

F{g,h.) = F[g,h) + l, *(j, h) = *(

Piig, *)Afo, h) + fa(9, h)Fi{g, h) = 1, (4 - 9)

Pt(g,h)4i(g,h) + Mg,h)*i(g,h) = l. with, e.g., Fi(xx,xi) = dF(x\, X2)/dx{.

As far as argument / = lni does not enter explicitly into generators 0 and b, it can be formally excluded by dividing one equation on other :

§ = F{g,h), F = & (4-10)

This equation can be analyzed on the two dimensional phase plane (g, h) shown in Fig. IV'.6. 0 Fig. IV.G

We shall present an explicit example of such phase portrait below at the end of the Section IV-5 (see Fig. IV. 11).

The essential features now are singular points and singular solutions. Singular points correspond to /?; = 0 (or oo). They can be of different types: stable fixed point that is known as aUracior, unstable fixed point and saddle-type fixed point. In the vicinity of UV attractor one can have a power scaling behavior as in

Eq.(5). Singular solutions join singular points and also can be stable or unstable.

In our case the unstable ones separate the parts of phase plane with different UV asymptotes.

§IV-2. Perturbative approach to the UV asymptote

2a. Structure of RG results. Let us consider a general situation with RG approach to the UV asymptotic behavior based on perturbation calculations in­ put. In a 1-coupling quantum field case group generators entering into differential equations of the Section II — 2 (see pages 35-38) can be written as

flW = 0\92 + fha3 + • • •, T(<7) = Ha) = 4>\g + fag2 + •••• (4 — ll)

- 70- Generally expansion coefficients depend on mass variable

fl{y.3) = J^Mv)9,+\ V-(y,<7) = 2>iM/ (-1-12)

Note that if g is just the expansion parameter (that can be equal to coupling constant or to its square) then usually the first term in perturbation expansion for 0 is quadratic and for ip - linear as is explicitly indicated in (11).

Substituting (11) in (1), and reexpanding the ratio 7"//?(*/) we obtain after integration

1/9 - US - f-Wa) - Ha -g) + 0(sV) = ft in*; PI

In s(x,g) = (4>l/0i)Hg(X,g)/g} + c2(g ~ g) + 0(

63 = ft/A - (ft/A)2, e, = [to//?i](fcM - ft/A)-

As follows, solutions^ and s depend on two arguments g and g In x . By expanding in powers of g we get

x 9( < 9) = 9fi(g I" x) + g-f2(g In x) + ...,

lns(*,j) = y>i{g\nx) + g

where /;- and y>; can be obtained in explicit form. For example

/i(*) = i—T~, Vi(*)~/i(0~ln/i(*). (4-15) 1 — /t»i z

-71 - Comparing expressions (14) with usual perturbative expansions

2 3 4 9Pt(x,g)=g + <7 A In x + g [0l In x + fi2 In x] + 0(

2 2 8 iPt(*,9) = i + gin In * + g {(Mi/2) in * + in in *] + ofo ), (4 - 16) used as input for obtaining our starting generators in the form Eqs. (11), one can see the qualitative effect of RGM using. In the case considered it changes the region of applicability of perturbation method limited by the condition

fflnx < 1 to the more large region defined now by two relations

flr«l, g(x,g)

2b. The ghosi-pok trouble. Let us turn now to the 1-loop RG approximation for the effective coupling g, considered in the UV, i.e., massless, limit. According to (14), (15) it has the form

Suppose now that the numerical coefficient /9; is positive. Such is the case, e.g., in QED where /?i = l/3ir and g stands for expansion parameter a = e2/4jr. This

-72- expression obviously has a pole singularity at

x = x. = exp(l//%) = exp(37r/o). (-1 - 19)

As far as QED effective coupling is proportional to (transverse part of) photon propagator such a pole generally can correspond to some bound state of a system with photon quantum numbers. However the pole describing a physical bound state must have a positive residue, while the l.h.s. of E

This means that it corresponds not to physical but rather to some ghost bound state. The presence of such unphysical singularity can be considered as a signal that a theory is inconsistent. Such claims have been made in the midst of 50-ies when the ghost-pole trouble was first discovered in QED just before the creation of RG method.

This method turns out to be very effective for general discussion of the ghost- pole trouble. The first question that must be answered here is the stability of result of ghost-pole existence with respect to multi-loop correction account.

§ 1V-3. Reliability of UV results in QFT

3a. Ghost pole in QED. It must be noted that in perturbation calculation, the

8 function depends on the adopted renormalization procedure; in the 3-loop or higher level the coefficients of the perturbation series depend on Renormalization

Scheme(RS) used. In QED, the three loop /3 function in MOM (i.e., momentum subtraction) scheme is expressed as

&-<«,-37+4^+ 8^1/(3)-— J • (4"-°)

The numerical value of the last parenthesis is about 0.4. Neglecting it for the

- 73 - moment we start our discussion with 2-loop approximation for the 3 function.

According to Eq.('2-29) the 2-loop RG solution is

21 ^ = l--/ + |ln(l-3-/) ' «- >

This solution has an error of order «'*/ and is interesting from several points of view. As it was mentioned before, its o expansion besides leading logs contains an infinite number of next-to-leading terms a2(al)m, the first of which has been used as an input for /? function construction. Second, in the vicinity of the ghost pole of one-loop RG solution at /( = 3ir/a, the a^) solution differ from a^j considerably.

Hence the infinite sum of the next-to-leading logarithmic contributions in the region al ~ 1 becomes important. It is not trivial because inside the each order of the perturbative input the next-to-leading term is negligible comparing with the leading one (the ratio being of order

This means that the existence problem for the ghost pole in QED cannot be solved by the account of next-to-leading logs and so on. Moreover we can conclude that it is impossible to make any qualitative statement about UV asymptote for

0(ff) > 9 case based on a RG-improved perturbation calculations. Our next example illustrates this thesis.

3b. Scalar quartic model. Consider the nonlinear scalar field with the inter­ action Lagrangian Here considerable progress lias been achieved in the calculation in the higher perturbative orders. The /? fund ion in MS-scheme is known up to 5-loop level

[23,24] and the result is given by

17 0W = o3 ~ Ta + 16-27V - 135V + 14370s (4 - 22)

The number of diagrams in each perturbation order are 1, 2, 7, 23 and ~ 100, respectively. The shape of the /? function up to each loop order is shown in

Fig. IV.7.

Fig. IV. 7

We see from this nice picture that, in contrast with QED case, there is no stability here even on a qualitative level. The odd-order approximations (i.e.,

1-, 3- and 5-loop) have a ghost-pole type behavior, whereas the even ones yield the fixed point (finite charge renormalization) case. Note also that, as shown at Fig.IV.8, the 10% confidence region upper boundaries correspond to g values close to 0.1 .

-75- oop 3 loops 4 loops 2 loops

0.04 0.054 0.06 0.078 0.08 0.10

Fin. IV.8

To understand the "loop dependence" of this boundary it is useful to represent the expression (22) in a slightly different form

5 (4 - 23) 0" (g) = -y- 0 529 \0.303/ V 0.222/ 0.180

This expression looks like a beginning of an asymptotic series of Poincare type. Indeed if we represent the ^-function in a series expansion form

/%) = £>"+1A. . (4-24)

then, as it is well known, coefficients j3n behave like ~ n\ at ;J —• oo .

The method of determining asymptotic estimates of the perturbation expan­ sion coefficients of the Green functions uses a representation in the form of a functional (i.e., path) integral. This integral written down for the mentioned

- 76- expansion coefficient can be calculated by the steepest descent method in the function space first proposed in [25]. To the saddle point there corresponds the

"instanton" type Euclidean classical solution with finite action. In this manner, an asymptotic expression was obtained by Lipatov [26] for the coefficients of 3 function expansion (24). It has the form

7/2 - 3n = ^n! » (l+0(" ')) («-oo), (4-25)

The factorial coefficients growth indicates that this is an asymptotic series with zero radius of convergence and that it. cannot be summed in the usual manner.

We can obtain the information about singularity structure at the origin [

§ IV—4. Asymptotic series Borel summation

Here we give short exposition of the results on the attempt of the summation of the series of (22) type made in [27] (see also [R4]).

Authors of the Ref.[27] used as an input the d-function 4-loop expression in the symmetric MOM-scheme

$°M{9) = \i ~ ~93 + 19 V - 1<%5 . (4 - 26)

The alternate-sign asymptotic series can be summed by Borel method. The idea is to represent the sum in a form of Laplace transform integral. It is not difficult to see that the transition to the Laplace image "kills" the factorial factor

- 77 - ii!. For the modified Borel transformation defined as

i3(a) = r IfLexpl-x/g)^) B[x), (-1-27) Jo

perturbation series can be written down as

*<*> = £ -TV"

Now it has a nonzero circle of convergence and can be summed within the circle.

However, as the integration domain in (27) goes outside the convergency region,

we must make an analytic continuation for the function B{x). It can be done

by a conformal mapping of the z-plane into the w-plane in such a way that the

domain of integration [0, oo] is mapped into the interior of the unit disk and

the cut [—oo,-l] transforms into the boundary of the disk. We also choose

the conformal transformation in such a way that it correctly reproduces the

singularity of the function on the cut. A conformal mapping satisfying these

conditions is

w\x) = . .

Then the Borel image of (26) becomes

<*,2 B(x) = 7—(1 - 0.332a- - 0.127w2 + 0.084a/1) , (4 - 28) 128

at the 4-loop approximation. Analyticity properties in x- and w-plane are ex­

plained in Fig. IV.9.

-78- , Img •^ # (D ® ®

cut • — - --l-~ — - OO Reg

Fig. IV.9

Graphs of the function '3(g) obtained by the Bore! summation with alloivanc

for the 1-, 2-, 3- and 4-loop approximations are shown in Fig. IV. 10.

1 0(g) A loop 500 After Borel Summation y 2 loops 400 /4 loops V/3 loops 300

200 ' 10 % error 100

i i » 0 10 20 30 40 q y Fig. IV. 10

They all lie in a narrow ray below the original parabola and within the limit of our adopted 10% accuracy enables us to advance into the region g ~ 50. This means that the summation procedure adopted enlarges the confidence interval in several hundred times! Besides this it gives us the qualitative stability of results.

All they are in a favour of ghost-type UV asymptote.

Nethertheless the results obtained can be considered only as a support but not the proof of the <^'-model inconsistency. The point is that starting with the

-79- representation (27) we have assumed the definite analyticity properties of /^(r;)

in the whole complex j-plane.

§ IV-5. Gauge faiitoms in QCD

Here we give a brief description of the results obtained recently in papers

[28],[29].

Consider QCD in a relativistic invariant gauge when the gauge-fixing term in the Lagrangian is of the form

with constant gauge parameter a introduced. It is known since the mid-fifties

[30] that in the RG analysis one has to treat the gauge parameter as the second coupling constant. Then we should define the effective gauge coupling

a-» a(<2) = ad"1 (^y;a,aj , (4-29) with the transverse gluon propagator amplitude

involved. This means that if we use constant non-zero gauge a ^ 0 then we are in

2 a two-coupling situation. By setting a = as = g /4ir the RG equations become

(see, e.g., [29]).

da(l;a,a) da(l; a, a)

jf = 0(a,a), j{ = fc(«,a) . (4 - JO)

In MS scheme, the /3 function does not depend on a, however, it depends on a in MOM scheme.

-80- It is possible to have a non-trivial fixed point (at,a*) in (a,a) plane:

/?(<*„ a.) = 0 , 6(a,,a.) = 0 .

Such situation does have place in two-loop approximation for some popular

MOM-schemes. Some examples of trajectories, which occur at two-loop level,

in (a, a) plane are shown in Fig. IV.11.

2-loop level i as

starting points las~0.25)

— a

Feynman gauge

Fig. IV. 11

If we start with some negative gauge parameter low-energy value a < 0, we end in a fixed point (a*,a,). In this solution the asymptotic freedom is not accomplished (a, ^ 0). On the other hand, trajectories starting from positive gauge parameter values show asymptotic free (in ft,) behavior and converge to a fixed gauge parameter value The trajectorie in («, /) plane are also shown in

Fig. IV.12.

-81 - °s ; Q,

// / / / /

£ = £n(07/x2/..2' ,

Fig. IV.12

The main lesson from this exercise is that the effective coupling in QCD shows both sheme dependence and gauge dependence so it cannot be a physical observable. REFERENCES

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