Wilson Renormalization Group and Continuum Effective Field Theories*

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SNUTP 98-100

KIAS-P98026

hep-th/9810056

Wilson Renormalization Group and Continuum E ective Field

Theories

Chanju Kim

Center for Theoretical Physics

Seoul National University Seoul, 151-742, Korea

and

Korea Institute for Advanced Study

207-43 Chungryangri-dong Dongdaemun-ku, Seoul, 130-012, Korea

Abstract

This is an elementary intro duction to Wilson renormalization group and con-

tinuum e ective eld theories. We rst review the idea of Wilsonian e ective

theory and derive the ow equation in a form that allows multiple insertion of

op erators in Green functions. Then, based on this formalism, we prove decou-

pling and heavy-mass factorization theorems, and discuss how the continuum

e ective eld theory is formulated in this approach.

Lecture presented at The 11th Summer Scho ol and Symp osium on Nuclear Physics (NuSS'98),

\E ective Theories of Matter", Seoul National University, June 23{26, 1998, Korea

present address

e-mail: [email protected] 1

I. INTRODUCTION

The purp ose of this lecture is twofold. First, we will give an elementary intro duction

to Wilson renormalization group in eld theories develop ed recently. Then based on this

formalism, we discuss some basic asp ects of continuum e ective eld theories.

The concept of e ective eld theories has played an imp ortant role in mo dern theoretical

physics and it acquires its natural physical interpretation in the Wilson renormalization

group formalism [1]. In the latter, one integrates out the high frequency mo des scanned by

a cuto and then considers lowering the cuto . This generates the renormalization group

ow, and the di erential equation governing this ow is called the exact renormalization

group equation or the ow equation. After Wilson's formulation, this ow equation has

b een applied to wide area of physics, from condensed matter physics to particle physics [2],

esp ecially for nonp erturbative problems which are dicult to treat in the other approaches.

It has also greatly enhanced the understanding of renormalization in quantum eld theory.

In spite of the understanding of renormalization with Wilsonian approach, the pre-

cise connection to the conventional approach has, however, not b een explicitly given until

Polchinski was able to prove the p erturbative renormalizability of a renormalizable quantum

eld theory within this framework [3], taking the -theory as his mo del eld theory. His

metho d was remarkably simple and could di use all diculties of the conventional approach.

More recently, direct physical meaning of Wilsonian e ective action was given in the frame-

work of conventional eld theory that it is nothing but the generating functional of Green

functions with an infrared cuto [4]. After this imp ortant observation, Wilsonian e ective

action and its renormalization group equation b ecame a powerful to ol to study wide range

of nuclear and high-energy physics problems [5]- [26].

If we take seriously the ab ove p oint of view on e ective eld theory and renormalization,

it should be natural also to have a simple connection to the conventional formulation of

e ective theory. Here, one of the most basic results is the decoupling theorem [27] which

states that, in a generic renormalizable quantum eld theory with heavy particles of mass M ,

heavy-particle e ects decouple from low-energy light-particle physics except for renormal-

ization of couplings involving light elds and corrections of order 1=M . Thus the resulting

renormalizable e ective eld theory describ es low energy physics to the accuracy 1=M . In

other words, to the zeroth order in 1=M there are no observable e ects due to the existence

of virtual heavy particles. If one wishes to understand the low-energy manifestations of

heavy particles, then irrelevant (nonrenormalizable) terms should b e considered to incorp o-

rate higher order e ects in 1=M . Investigation of this issue shows that the virtual heavy

particle e ects can be isolated via a set of e ective lo cal vertices with calculable couplings.

It implies that low-energy light-particle physics can be describ ed by a suitable lo cal e ec-

tive eld theory when combined with appropriate calculation rules to deal with irrelevant

vertices. This heavy-mass factorization was proved to the lowest in 1=M adapting Zimmer-

mann's algebraic identities [28] in the BPHZ formulation [29], and, recently, to all orders in

p erturbation theory and to to any given order in p owers of 1=M using the ow equation [7]

(see also Refs. [23,24]). Actually the pro of is almost as simple as the renormalizability pro of

itself and the whole scheme allows a natural physical interpretation from the viewp oint of

the renormalization group ow as we will see later.

There is, however, an issue to clarify regarding the di erence b etween Wilsonian e ective 2

theories and e ective theories in continuum. We have three widely separated scales here|

light particle mass m, heavy particle mass M , and the ultraviolet (UV) cuto , with

m M . And our hop e is to nd a low-energy e ective theory with the UV cuto

0 0

whichinvolves only the light eld of mass m and describ es low-energy light-particle physics

accurately up to the order 1=M (N is any xed integer). Then the e ective theory must

include a nite number of irrelevant terms (restricted by the dimensional argument). Here

arises a problem. In the original Wilson's view, the cuto scale of the e ective theory may

well be identi ed with the heavy mass scale M , ab ove which it is no more e ective. The

presence of irrelevant terms in the e ective Lagrangian at scale M is then very natural

as discussed ab ove and there is no need to worry ab out their presence in particular. The

\natural scale" of those terms will be around M , i.e., they are order one at scale M , and

among them wemaycho ose to keep explicitly some minimal numb er of irrelevant terms in

our e ective Lagrangian for the accuracy of order 1=M . But, in the conventional discussion

of quantum eld theory, the UV cuto is supp osed to go to in nity eventually. So, to

connect Wilson's view with this conventional formulation, we may supp ose scaling up the

cuto of the ab ove Wilsonian e ective Lagrangian from M to . It will then generate

in nitely many irrelevant terms which are unnaturally large. Also, during the scaling, all

of them get mixed together and so we are forced to work with the Lagrangian consisting of

in nitely many terms all the time. Any kind of truncation for the bare Lagrangian to some

nite number of terms would yield divergences in physical quantities as ! 1, b ecause

the unnaturally large co ecients would be ampli ed by some p ositive power of =M as

the cuto is scaled down. This is nothing but the statement of nonrenormalizability in

the language of renormalization group ow. To avoid this problem we need to deal with

irrelevant terms carefully, i.e., give suitable rules to obtain unambiguous nite results with

only a nite number of terms included in the bare Lagrangian. It is achieved through the

mo di cation of the ow equation when irrelevant vertices are inserted, in the more-or-less

same way as one treats the renormalization of comp osite op erators and their normal pro ducts

[28] in the ow equation approach [11]. We will discuss this pro cedure later.

The plan of the pap er is as follows. In section 2, starting from the generating functional,

we identify the Wilsonian e ective action and give a rather lengthy derivation of the Wilson

renormalization group equation [31,21]. This derivation will show clearly how the Wilsonian

e ective action is interpreted as a generating functional of Green functions. Then we de-

rive the ow equation in a form applicable to the case that multiple insertion of comp osite

op erators is allowed in Green functions. In section 3, after a review on p erturbative renor-

malizability,we show decoupling and heavy-mass factorization taking as our mo del a scalar

theory involving two real scalar elds of masses m and M with m M . Then we discuss

how the continuum e ective eld theory is formulated to any desired order of accuracy in

the ow equation approach. We conclude in section 4.

For example, the continuum version of Symanzik's improved action [30] in lattice theory have

b een discussed in [9] by adding suitable irrelevant terms in Lagrangian. 3

II. THE FLOW EQUATION

For simplicity let us consider a theory of single scalar eld in four Euclidean dimensions

with a momentum space cuto . Generalization to several elds is straightforward. The

bare action is written as

D E

0 0

S ; P []= + S [] ; (1)

int

where P is the free-particle cuto propagator and hf; gi is de ned by the momentum-space

integral of f and g ,

d p

f (p)g (p) ; (2) hf; gi

(2 )

and S [] represents the interaction part of the bare action. For later use, we will also

int

allow the insertions of some additional lo cal vertices or certain comp osite op erators in Green

functions of the theory. To account for this, we de ne S []as a formal p ower series in ,

tot

which has S []as its zeroth order term, viz.,

int

;0 ;N

0 0

[] S []) (3) S [] : (S []= S

int int int

tot

N !

jNj0

Also, for later conveniences, we will denote

0 0

[] : (4) []= [] S S S [] = S

int int

tot

N !

jNj1

S 's may be regarded as additional lo cal vertices app ended to the original Lagrangian

int

S or as comp osite op erators in which one is interested. For example, if one wishes to

int

consider a single or twice insertion of a comp osite op erator O (x), one may consider

S = d x (x)O(x);

int

4 4 0

S = d xd y (x) (y)O (x; y ) ; (5)

int

where (x) is the source for O (x) and O (x; y )is a suitable counterterm for the pro duct of

op erators O (x)O (y ), which can be determined through renormalization conditions and the

ow equation derived b elow [11]. In this case, S is equal to the original interaction part

tot

of the action plus the comp osite-op erator source terms.

S []

The generating functional, with the insertion of the op erator e , is

More generally, one can consider p ower series of many parameters ;::: ; . Equation (3) and

subsequent discussion can accommo date this case with the interpretation that = ( ;::: ; ),

. N isamulti-index N =(N ;::: ;N ), jN j = N , N !=N ! N !, and =

1 i 1

k k

i=1

k 4

h;P iS []+hJ;i

tot

Z [J ]= De : (6)

Following the idea of Wilson and Polchinski [1,3], we wish to integrate out the high-

momentum comp onents of and reduce the cuto to a lower scale . For this, we

divide the propagator P into the high-frequency part P and the low-frequency part P ,

the b orderline b eing set by momentum p =:

P =P +P ; (7)

Then it is not dicult to show that the generating functional may be written in terms of

two elds and rather than alone as

H L

1 1

0 0

i h ;P h ;P [ + ]+hJ; + i iS

H H L H L H L L

2 2

tot

; (8) Z [J ]= D D e

H L

up to a multiplicative factor. (The equivalence of Eq. (6) and Eq. (8) may be seen if one

substitutes = into Eq. (8) and p erforms the integral over which is Gaussian.)

L H H

Since the eld ( ) has P (P ) as its propagator, only the low-(high-)frequency mo des

L H

of ( ) will now propagate e ectively. The integral over may b e p erformed to obtain

L H H

[21]

h ;P i W [ ;J ]

L L L

tot

Z [J ]= D e e ; (9)

where W is given by

tot

0 0

W [ ;J ] h ;P iS [ + ]+hJ; + i

L H H H L H L

tot tot

e D e : (10)

Notice that W [0;J] is nothing but the generating functional of connected Green functions

tot

S []

(with the op erator e inserted) in the theory with both UV cuto and IR cuto .

Substituting = into Eq. (10), we may write

H L

1 1 1

0 0 0

W [ ;J ] h;P iS []+hJ +P ;i

h ;P i

L L

tot tot

e = e D e

1 1

0 0 0 0

hJ;P J i+hJ; i hJ +P ;P (J +P )i

L L L

2 2

= e e

1 1

0 0 0

S [ ]

hJ +P ;P (J +P )i

L L

tot

e e

0 0

W [0;0] hJ;P J i+hJ; iS [P J + ]

L L

tot tot

= e e ; (11)

for some S [] satisfying S [0] = 0. (Here we factored out the eld indep endent part as

tot tot

W [0;0]

tot

e whichgoesto1as! .) Therefore, the generating functional can b e written,

up toamultiplicative factor, as

There will be in nite ways in doing the separation; cho osing one way of separation may be

considered as cho osing a \renormalization scheme" in the Wilson renormalization group approach,

and physical quantities are indep endent of suchchoices. It is not necessary to sp ecify a particular

scheme at this stage. 5

1 1

0 0

h;P hJ;P J i iS [P J +]+hJ;i+

2 2

tot

Z [J ]= De ; (12)

where we have replaced by . Supp ose that J (p) = 0 for p> so that J couples to the

low-frequency mo des only. Then, since P has only high-frequency mo des, all J 's drop out

from the expression except for hJ;i and S coincides with the Wilsonian e ective action

tot

[1]. However, we do not have to insist on such a restriction for J (p); for general J (p), Eq.

(11) connects directly S with Green functions as will b e discussed shortly.

tot

Nowwe derive the ow equation [3] satis ed by S . Di erentiating Eq. (10) with resp ect

tot

to while cho osing =0, we have the result

* +

W [0;J ] W [0;J ]

tot tot

@ e = ;@ P e ; (13)

2 J J

W [0;J ]

tot

which, on using Eq. (11) for e , gives the ow equation,

" #

4 2

d p S 1 S S

tot tot tot

@ S []= @ P (p) : (14)

tot

2 (2 ) (p)(p) (p) (p)

eld-dep: part

Equation (14) is integrable; the formal solution with the appropriate b oundary condition

satis ed at = is [4]

S []

S []I

h i

tot tot

e ; (15) e = e

where I is supp osed to collect precisely all -indep endent pieces from the right hand side so

that wemayhave S [0] = 0 as we required in (11). Equation (15), together with Eq. (11),

tot

repro duces a well-known expression for the generating functional e . In fact, S b ears a

tot

simple connection with physical amplitudes. To see this, notice that wehave, from Eq. (11),

0 0

W [0;J] W [0; 0] = hJ;P Ji + S [P J ] ; (16)

tot tot tot

and hence

n n

W S []

tot tot

(p ) = P ; n>2: (17)

(p ) (p ) J (p ) J (p )

1 n 1 n

i=1

J =0

Therefore, we arriveatavery interesting result. Namely, Wilsonian e ective action S with

tot

UV cuto can also be interpreted as the generating functional of amputated connected

S []

Green functions (with the op erator e inserted) with IR cuto s . Physical Green

functions are then obtained in the limit ! 0 [4,20,21]. Actually, the reason b ehind this

result is rather simple. The generating functional of a eld theory is obtained byintegrating

out al l mo des, all informations of the theory b eing enco ded in the dep endence on the external

source coupled to the eld. If a cuto is intro duced and integration is p erformed only over

high frequency mo des, then we will get the generating functional with IR cuto b ecause

low frequency mo des remain unintegrated. On the other hand, from the p oint of view of

low frequency mo des, the resulting functional gives, by de nition, the Wilsonian e ective

action. Equation (16) precisely expresses this relation. 6

Now that Wilsonian e ective action is essentially the same as the generating functional

of connected Green functions, it is also interesting to p erform a Legendre transformation

to obtain the ow equation for the generating functional of one-particle irreducible Green

functions with IR cuto . As usual, we de ne

['] W [0;J]+hJ;'i

= [']+ h'; P 'i ; (18)

int

with

= J; = ': (19)

' J

Then from Eq. (13), we obtain [17]

4 2

d p 1 W

@ = @ P (p)

int

2 (2 ) J (p)J (p)

" #

4 2

d p 1

1 1

int

0 0

: (20) @ P (p) + P =

2 (2 ) '(p)'(p)

It is often more convenient to work with 1PI quantities than connected ones, esp ecially for

the purp ose of practical calculations [18,21]. However, we will not discuss it further. See

Refs. [8,20,21] for details.

Let us go backtoS . Expanding S in p owers of ,we de ne S 's as the expansion

int

tot tot

co ecients, i.e.,

;N ;0

S []= S [] ; S [] S [] : (21)

int int

tot int

N !

jNj0

As we mentioned ab ove, S [] is the generating functional of amputated connected Green

int

functions with an insertion of an op erator O which is identi ed as the N -th order co ecient

S []

of e , i.e.,

N N

X X

O e = exp ( S ) : (22)

int

N ! N !

jN j1 jN j1

;1 ;2 ;1 ;3 ;2 ;1 ;1

0 0 0 0 0 0 0

2 3

For example, O = S , O = S +(S ) , O = S +3S S +(S ) and

1 2 3

int int int int int int int

so on. At the zeroth order of , we obtain the ow equation for the e ective Lagrangian

S from Eq. (14), which assumes an identical form as Eq. (14) other than the replacement

int

S ! S . At the N -th order of ,onthe other hand, we have

tot int

Expression for general O is also available in [7].

N 7

;N ;N

4 2

d p S 1 S S

int int

int

@ S = @ P (p) 2

int

2 (2 ) (p)(p) (p) (p)

;N I ;I

S S

int int

;

(p) (p)

eld-dep: part

(23)

For a single insertion of op erator through S , the last term of the right hand side in the

int

second line of Eq. (23) vanishes and so wehave a linear and homogeneous equation in S .

int

This equation has b een used in discussing the renormalization of comp osite op erators [10,11].

Equation (23) with multi index N =(1;1) is useful in discussing the short-distance expansion

of two comp osite op erators [11]. Notice that, if N > 1, Eq. (23) contains inhomogeneous

terms which may lo ok dicult to deal with. However, in spite of those terms, it is still

of the rst order in S while the homogeneous part remains exactly the same for all N .

int

Therefore the general solution will be a sum of a particular solution and a solution to the

homogeneous equation (which is just the N =1 equation for Green functions with a single

insertion of an op erator). Though we will not explicitly use this prop erty here, it is imp ortant

in considering Zimmermann's normal pro duct and multiple insertion of lo cal op erators of

which the formulation is crucial in understanding the structure of e ective theories [7].

A few remarks are in order.

(i) As noted ab ove, choice of the cuto function in the propagator is quite arbitrary. For

example, the propagator P of a particle with mass m is given by

R (;p)

P = ; (24)

2 2

p + m

where R (;p) is a cuto function. If wecho ose sharp cuto , R (;p)= ( p). Sometimes

2 2

p =

it is more convenient to cho ose smo oth cuto such as R (;p)= 1e . More generally,

even the separation of the action S into the free part and the interaction part S is

int

arbitrary. One may write, for example,

W [J ]

h;P iS []+hJ;i

tot

e D e ; (25)

where S is the full bare action of the theory with UV cuto and

0 0

0 e

(26) P P P

is the pure cuto term added to the action. Then all the previous equations are valid with

0 0

S and P replaced by S and P , resp ectively. The corresp onding ow equation for

int

is the one used by Berges in this volume.

(ii) The ow equation (14) is to o complicated to solve exactly. For practical purp oses, it

is therefore necessary to nd suitable approximation metho ds. An obvious way is to p erform

a derivative expansion

 

4 2 4

S = d x V ('; ) + (@ ') Z ('; ) + O (@ ) : (27)

2 8

Then the ow equation reduces to simple di erential equations of co ecient functions

V ('; ), Z ('; ) etc. and one can study their prop erties by various means. There are some

problems with approximation metho ds. As mentioned ab ove, physical quantities should be

indep endent of the choice of cuto functions. However, this scheme indep endence is lost af-

ter approximations. Another problem is ab out reparametrization invariance: physics should

not dep end on the reparametrization of elds in the partition function. This is re ected in

the ow equation in some complicated way [22] and is broken in general by approximations.

For the discussion of these issues see, e.g., [12,13].

(iii) When the system has a gauge symmetry, integrating out high momentum mo des

do es not preserve gauge symmetry and this could be a p otentially serious problem. There

are a few ways with which this problem can b e cop ed with. The simplest way is to insist on

using the gauge-symmetry-violated ow equation derived here. Even if the gauge symmetry

is not manifest for nite cuto , it is eventually restored at the physical p oint = 0 up to

O (1= ) [5,20,14,15]. Also, one maywork in the background gauge in which the background

gauge invariance may b e maintained in the e ective action [14] though it may not necessarily

mean quantum gauge invariance [16].

III. CONSTRUCTION OF EFFECTIVE FIELD THEORIES

As discussed in section 1, one of the most natural application of the ow equation would

be to understand basic results of e ective eld theories in continuum. Here we discuss

decoupling and heavy-mass factorization, and so construct the continuum e ective theory

to any desired order of accuracy in ow equation approach. As a rst step, let us review

Polchinski's pro of of p erturbative renormalizability [3].

A. Perturbative Renormalizability

A theory is said to b e p erturbatively renormalizable if Green functions of the theory are

b ounded and converge to nite limits at each order of p erturbation as UV cuto of the theory

go es to in nity. Now knowing that Wilsonian e ective action gives physical Green functions

in the limit ! 0 and that it is controlled by the ow equation, we may exp ect that the

b oundedness and convergence of Green functions can be easily shown by estimating the

ow equation. Indeed, this line of argument can b e implemented in a quite straightforward

way. Since basically the same reasoning is rep eatedly used throughout in discussing e ective

theories, we rst explain the metho d in rather detail. It consists of the following steps:

(i) Write down the ow equation for vertex functions and identify b oundary conditions

which follow from the form of the bare action (at = ) and also from the renor-

malization conditions on Green functions (at = 0).

(ii) Integrate the ow equation over from the b oundary at which b oundary conditions

are given (either at =0orat= ) and estimate the resulting expression.

(iii) Prove b oundedness and convergence order by order using induction. 9

Let us illustrate this with a -theory. The bare action with UV cuto reads

D E

0 0

S ; P []= + S ; (28)

int

where P is the free-particle cuto propagator of mass m and S is the interaction part

int

of the bare action as b efore. Explicitly,

 

1 1 1

2 2 4 4

 (x)+ (@ (x)) + (x) : (29) S = d x

1 2 3

int

2 2 4!

In conventional way, the bare couplings may b e written as

2 2 2 0

 = Z m m ; = Z 1; = Z g ; (30)

1 2 3

0 

where m , Z , and g are the bare mass, the wavefunction renormalization, and the bare

0 

coupling of the real scalar eld . Propagator P is de ned by

R ( ;p)

m 0

P = ; (31)

2 2

p + m

where R (;p) is a smo othed variant of the sharp cuto function ( p). As we indicated

in section 2, cho osing R corresp onds to cho osing a \renormalization scheme" and physical

quantities are not a ected. Here, for convenience' sake, we cho ose

i h 

K R (;p)= 1K (1+=m) ; (32)

where K (t) is a smo oth function such that K (t) ! 1 (exp onentially) as t ! 1 and K (t) ! 0

(also exp onentially) as t ! 4 (detailed form of K (t) is not imp ortant).

Now, the Wilsonian e ective action S is de ned to satisfy the ow equation (14).

int

Expanding S in p owers of in momentum space, we write

int

1 1 2n1

XX Y

d p

S []= (p ) (p )G (p ;:::;p ); (33) g

1 2n 1 2n1

int r;2n

(2 )

r=0 n=1

j =1

 

2n1

p p ;

2n j

j=1

where g is the p erturbation-expansion parameter which may be identi ed as renormalized

coupling constants. Then as we noted ab ove, at = 0, vertex functions G can be

r;2n

identi ed with amputated connected Green functions of the theory. Furthermore, we have

G = 0 if n > r +1 b ecause just the connected diagrams contribute to G . Therefore

r;2n r;2n

the sum over n in Eq. (33) actually extend only over a nitely many n for given r .

If we insert the expansion (33) into the ow equation (14) we arriveat

@ G (p ;:::;p )

1 2n1

r;2n

 

d q

2n +2

= @ P (q )G (q; q; p ;:::;p )

1 2n1

m r;2(n+1)

(2 )

r 1 n

X X

+2 l (n l +1)@ P (P)

r =1

l =1

h i

G (p ;:::;p )G (P; p ;:::;p ) ;

1 2l1 2l 2n1

r;2l rr ;2(nl +1)

sy mm

 

2n1

P = p (34)

j =1 10

 

@ P

 H H H H

 

2(n l + 1) 2(n + 1)

FIG. 1. Graphical representation of the ow equation (34).

where [] implies symmetrization with resp ect to momenta p ;:::;p . Denoting @ P

sy mm 1 2n

by a straight line, we can represent this equation by the diagram shown in Fig.1.

If suitable b oundary conditions are given, the ow equation (34) will pro duce the ampu-

c =0

tated connected Green functions of the theory, G = G . First, we know that the bare

r;2n r;2n

action (29) has a simple form: at = ,

@ G =0; 2n+jwj>4;

p r;2n

@ G (0) bare couplings in Eq. (30); 2n + jw j 4; (35)

p r;2n

where w is the multi index fw g fw ;:::;w g with w = (w ;:::;w ) and @ 

1 2n1 j j 1 j4

Q P

w w w w w

1 2n1

i i i

@ @ , @ @ =(@p ) . We will also use the notations jw j jw j.

i i

i;

p p p =1

1 2n1

We do not know the value of bare couplings ; they are xed by the renormalization

conditions imp osed on Green functions at = 0 for 2n + jw j 4. For example, we may

cho ose the conditions at zero momenta,

=0 =0 =0

 : (36) G (0) = @ @ G (0) = 0; G (0) =

r 1 p p

 

r;2 r;2 r;4

Now that the b oundary conditions have b een completely sp eci ed, weintegrate Eq. (34)

and estimate it. For that, weintro duce a set of norms [11], kk , where a and b are p ositive

(a;b)

real numb ers:

z w

k(@ f )g k max j(@ f )g (p ;:::;p )j: (37)

1 n

(a;b)

jp jmax fa;bg

jw j=z

Then using the prop erty of cuto function in (32) one can nd [7] that, for a xed constant

 of order m ( maybe considered as the momentum scale),

@ @ G

r;2n

(2;)

(

 const ( + m) @ G

r;2n+1

(2;)

)

z z

2 3

+ @ G @ G ;

0 0

r ;2l r r ;2(nl +1)

3+z

(2;) (2;)

( + m)

r ;l

z +z +z =z

1 2 3

(38) 11

where \const" stands for some nite number which is indep endent of and (but may

have dep endence on m and through (=m)). This estimate shows that the derivative of

vertex functions with resp ect toiswell b ounded for all 2 [0; ].

As a next step, we apply this estimate to the integral form of the ow equations. For

irrelevantvertices, 2n + z>4, b oundary conditions are given at = and so weintegrate

the ow equation from = down to :

z z

@ G = @ (G G )

r;2n r;2n r;2n

(2;) (2;)

z 

 d @ @ G : (39)

r;2n

(2;)

For the relevantvertices, i.e., in the case of 2n + z 4, weintegrate the ow equation from

0to (at p = 0) instead,

w w =0 z 

j@ G (0) @ G (0)j d @ @ G : (40)

p r;2n p r;2n r;2n

(2;)

The right hand side of (39) and (40) are now estimated and integrated using (38). Finally,

applying a suitable induction argument over the p erturbation order r and the number of

legs n, we obtain the following b ound on vertex functions [7]

 

z 42nz

; 0 ; (41) @ G ( + m) Plog

r;2n

(2;)

where Plog(( + m)=m) is some p olynomial in log(( + m)=m) whose co ecients are inde-

p endent of and . In particular, at = 0, (41) tells us that amputated connected Green

functions and their derivatives of the theory are b ounded by

w c 42njw j

j@ G (p ;:::;p )jconst m ; for jp j : (42)

1 2n1 i

p r;2n

Convergence of Green functions as !1 can also b e proved in a similar way and the

result is

 

z 32nz

@ @ G ( + m) Plog ; 0 : (43)

0 r;2n

(2;)

w c

Consequently, amputated connected Green functions @ G converge to nite limits as fast

p r;2n

as O [( ) ] (mo dulo p owers of log( =m)).

Before we move to the next topic we make a few remarks.

(i) When estimating the relevant vertices, we have had to integrate up from =0 and

not just down from . If we had integrated down from assuming \natural values" like

0 0

 

G Plog the irrelevant-vertex case, wewould have had simply etc. That

r;2

(2;)

is, by imp osing Eq. (36) wehave forced the initial bare values of to b e nely adjusted so

as to pro duce a scalar with mass m . This is the famous naturalness problem in scalar

theories [32].

(ii) In this pro of of p erturbative renormalizability, we have not encountered any com-

plication usually found in diagrammatic metho ds, for example, overlapping divergences,

cancellation between diagrams and so on. This is b ecause, here, we always work with the

whole Green function directly and so problems in considering only a part of a Green function

do not app ear. 12

B. The Full Theory

We are now going to discuss decoupling and heavy-mass factorization of e ective eld

theories using the renormalization group ow equation. We will try to avoid going into

technical details but explain only main line of the argument. As our mo del for the full

theory,We consider a scalar theory (- theory) whichinvolves two real scalar elds and

of which the masses are m and M ( ), resp ectively, and interact via quartic couplings.

Let us write the bare action as

D D E E

1 1

(f)

(f )

0 0

S = ; P ;P + S : (44) +

int

2 2

Here, P and P are resp ective free-particle cuto propagators, and the interaction part

(f )

S is given by

int

1 1 1 1

(f )

f f f f

2 2 4 2 4

 (x)+ (@ (x)) + (x)+ (x) S = d x

1 2 3 4

int

2 2 4! 2

1 1 1

f f f

2 4 2 2

+ (@ (x)) + (x)+ (x) : (45)

5 6 7

2 4! 4

The sup erscript f is used for couplings of the full theory. As in the previous section, the

bare couplings (a =1;2;;7) maybe written as

f f f

2 2 2 0

 = Z m m ; = Z 1; = Z g ;

 

1 2 3

0 1

f f f

2 2 2 0

(46)

 = Z M M ; = Z 1; = Z g ;

4 5 6

0 2

 = Z Z g ;

2 2 0 0 0

where (m ;M ), (Z ;Z ) and (g ;g ;g ) are bare masses, wave-function renormalizations

0 0 1 2 3

and bare coupling constants, resp ectively.

As p ointed out in section 2, there is a freedom to cho ose the cuto functions in propaga-

tors P and P . Here, we take advantage of this exibility and cho ose a \mass-dep endent

m M

scheme," i.e., take di erent cuto functions for the light-particle propagator P and the

heavy-particle propagator P : P is chosen to be the same as Eq. (31) while, for P , we

M m M

cho ose

R ( ;p)

M 0

P = ; (47)

2 2

p + M

with

h i

2 2

R (;p)= 1K =M : (48) K

The reason for this choice is b ecause, if

is very natural for our purp ose, since it implies that at the scale =M we have all mo des

of the heavy eld integrated out and there remains only the light eld b elow the scale

M ; it explicitly implements Wilson's p oint of view on e ective eld theory. Therefore, in 13

the ow equation, terms involving the heavy-particle propagator drop out if < M and

it will lo ok like the ow equation for the theory having the light particle eld only. One

may regard this prop erty sp eci c to our choice (32) and (48) as the analogy of the so-called

\manifest decoupling" in the conventional approach [33]. On the contrary, if one chose a

\mass-indep endent scheme", i.e. if R and R were chosen indep endently of their masses

m M

m and M , a substantial part of heavy particle mo des would not be integrated out even

b elow the scale M , thus making subsequent discussions rather complicated.

(f )

Now, we de ne S [; ] following the general line discussed in section 2. It will then

int

satisfy the ow equation,

2 3

(f ) (f ) (f )

4 2

1 d p S S S

(f )

int int int

4 5

@ S = @ P (p)

int

2 (2 ) (p)(p) (p) (p)

)

+(m !M; ! ) : (49)

eld-dep: part

(f )

Expanding S in p owers of and in momentum space, we write

int

4 0

2n 1 2n

2 1

Y X X Y

d p

(f )

0 0 r

(p ) (p ) (p ) (p ) S [; ]= g

1 2n

int

1 1 2n 1

4 4

(2 ) (2 )

j =1 j =1

jrj0 jnj1

(f )

0 0

G (p ;:::;p ;p ;:::;p ): (50)

1 2n

r;2n

1 1 2n 1

 

P P

2n 1 2n

0 0

2 1

p p p 

j=1 j =1

j 2n

Some explanations on our notations are in order. g (g ;g ;g ) are p erturbation-expansion

1 2 3

parameters which may be identi ed as renormalized coupling constants, r and n represent

(r ;r ;r ) and (n ;n ), resp ectively, with jnj n + n , jr j r + r + r and nally,

1 2 3 1 2 1 2 1 2 3

r r r

1 2 3

g g g g . The vertex functions satisfy the ow equation

1 2 3

(f )

0 0

@ G (p ;:::;p ;p ;:::;p )

1 2n

r;2n

1 2n 1

 

d q

2n +2

(f )

0 0

= @ P (q )G (p ;:::;p ;q;q;p ;:::;p )

1 2n

m 1 2n 1

r;2(n +1;n )

1 2

(2 )

(f )

0 0 0

l l @ P (P ) G (p ;:::;p ;P;p ;:::;p ) +2

1 1 2l 1

1 m 1 2l 1

r ;2(l ;l )

1 2

0 00

r +r =r

=n +1 l +l

1 1

(f )

0 0

=n l +l

G (p ;:::;p ;P;p ;:::;p )

0 0

2 2

2l 2n

2 1 1

2l +1 2n 1

r ;2(l ;l )

2 2

1 2

sy mm

 

d q

2n +2

(f )

0 0

@ P (q )G (p ;:::;p ;p ;:::;p ;q)

1 2n

M 1 2n

r;2(n ;n +1)

1 2

(2 )

(f )

0 0 0 0

+2 l l @ P (P ) G (p ;:::;p ;p ;:::;p )

2 1 2l

2 M 1 2l 1

r ;2(l ;l )

1 2

0 00

r +r =r

l +l =n

1 1

(f )

0 0

l +l =n +1

G (p ;:::;p ;p ;:::;p ) ; (51)

0 0

2 2

2l +1 2n

1 1 2l 2n

r ;2(l ;l )

2 2

1 2

sy mm

P P P P

2l 1 2l 2l 2l 1

0 0 0

1 2 1 2

where P = p p , P = p p . This equation can be rep-

j j

j =1 j =1 j =1 j =1

j j

resented by the diagram shown in Fig. 2, where we denote ( )-legs by thin(thick)-lines.

Also, With our choice of propagators, (see Eq. (48)), the last two terms in Eq. (51) vanish 14

2(n + 1) 2n 2n

1 1 1

@ P

 

@ @ @

 H H 

H H H

 

@ P

2n 2n 2(n + 1)

2 2 2

2(n l + 1) 2(n l )

2l 2l

1 1 1 1

1 1

 

@ @ @ @

X X

@ @ @ @

H H H H 

@ P @ P

M m

+ +

H H H H 

l l

H H H H 

 

2(n l ) 2(n l + 1)

2l 2l

2 2 2 2

2 2

FIG. 2. Graphical representation of the ow equation (51). Thin (thick) lines represents light

(heavy) eld.

for < M . Therefore the form of this ow equation coincides with that of a eld theory

for a single scalar eld. Finally, in relation with the low-energy e ective theory, it should b e

(f )

noted that, for G (i.e., no external heavy particles), the last term in Eq. (51) is iden-

r;2(n ;0)

tically zero and the heavy eld enters the ow equation only through the -di erentiated

propagator in the third term.

(f )

The b oundary conditions which G 's ob ey are given as in the previous section. First,

r;2n

at = , irrelevantvertices vanish, i.e.,

(f )

@ G =0; 2jnj+jwj>4: (52)

r;2n

At = 0, we imp ose renormalization conditions on relevant terms:

(f )=0 (f )=0 (f)=0

 ; G (0) = 0; @ @ G (0)=0; G (0) =

r;(1;0;0) p p

 

r;(2;0) r;(2;0) r;(4;0)

(f )=0 (f)=0 (f)=0

0 0 0 0 0

(53)

G (p )= 0; @ @ G (p )=0; G (p ; p ; p )= ;

p p

r;(0;1;0)

 

1 1 2 3 4

r;(0;2) r;(0;2) r;(0;4)

(f )=0

G (0; 0;p )= :

r;(0;0;1)

r;(2;2)

Here, normalization momenta p (i =1;:::;5) are chosen to be constants of magnitude M ;

i.e., wehavechosen the renormalization p oints for the light-particle Green functions at zero

momentum, and those for the heavy particles at momenta of magnitude M .

With the ow equation (51) and the b oundary conditions (52) and (53), now one can

demonstrate the p erturbative renormalizability (i.e., b oundedness and convergence of Green

functions as ! 1) of the full theory following the line of arguments given in section

3.1, with the help of our choice of cuto functions. The result is [7,24]: for vertices with no

external heavy-particle leg,

 

+ m

(f )

42n z z

 ( + m) Plog ; (54) @ G

r;2(n ;0)

(2;;M )

m 15

while, for irrelevantvertices (2jnj + z>4) with external heavy-particle legs,

 

+ M

(f )

z 42jnjz

@ G ( + m) Plog ; (55)

r;2n

(2;;M )

and, nally, for relevant vertices (2jnj +4 4) with external heavy-particle legs,

 

(f )

z 42jnjz

@ G ( + M ) Plog ; (56)

r;2n

(2;;M )

where the co ecients in p olynomials of logarithms are indep endent of M , , and , and

the norm kk is de ned by

(2;;M )

(f ) (f )

z w 0 0

@ G max j@ G (p ;:::;p ;p ;:::;p )j: (57)

1 2n

p 1 2n 1

r;2(n ;n ) r;2(n ;n )

1 2 1 2

jp jmaxfa;bg

(a;b;b )

jp jmaxfa;b g

z =jw j

(If there is no external heavy-particle leg (n = 0), this de nition reduces to that of the

norm kk in Eq. (37).)

(a;b)

Actually these b ounds are not unexp ected | they just show the right scaling b ehaviors.

That is, the e ect of imp osing the renormalization conditions at momenta of magnitude M

for heavy-particle legs shows with appropriate p owers of M (up to logarithmic corrections)

in the ow equation of vertices with external heavy-particle legs. On the other hand, in the

light-particle sector, the b ounds have the same form as those of the single scalar theory (see

Eq. (41)): the large M corrections do not show up in the b ounds b ecause the light-particle

mass m is forced to be (unnaturally) small by hand. Also, it should be noted that the

b ounds (54) on the light particle sector have exactly the same form as that of the theory

with a single scalar eld. Of course, this should be the case for decoupling to o ccur in the

rst place.

C. Low-Energy E ective Theory

Now supp ose that we are interested in physics at scale much smaller that M . What is the

low-energy e ective theory? Decoupling theorem states that, to the zeroth order of 1=M ,

it is simply given by the -theory with heavy eld removed from the Lagrangian of the

full theory. To establish this, we should show that the di erence between Green functions

of the - theory and those of the theory are at most of order 1=M (no O (1=M ) term

b ecause of the Z symmetry). In our approach, it is given by the following b ound:

(f )

G ) @ (G

r;(r ;0;0)

r ;2n 1

r;2(n;0)

(2;)

 

+m M

42nz

( + m) Plog ; 0 M

M m

 (58)

 

42nz

Plog ; M ;

where r =(r ;r ;r ). Notice that the di erence in the vertex functions is no longer small if

1 2 3

b ecomes comparable to the heavy particle mass M , which implies that low-energy e ective

theory is not useful ab ove the heavy particle mass scale. 16

The strategy to show this b ound is almost straightforward. As we have seen, the form

of the ow equation of - theory is the same as that of theory for < M . Moreover,

the b oundary conditions of the two theories are the same by de nition. Consequently,

considering the ow of the di erence of vertex functions,

(f)

D G G : (59)

r;(r ;0;0)

r;2n r ;2n 1

r;2(n;0)

we can exp ect that D is almost zero since b oundary conditions for D is zero and the

r;2n r;2n

ow equation for D is homogeneous for < M . Following the steps explained in the

r;2n

previous section, one can easily establish the b ound (58). As we exp ect, decoupling theorem

is proved more or less trivially in this approach.

The zeroth order e ective theory is just renormalizable theory. In other words, it

describ es the light particle physics accurately to the zeroth order of 1=M at low energy.

Since it is p erturbatively renormalizable, the theory itself do es not know the scale M be-

low which the theory is e ective. In a sense the e ective theory nature b ecomes manifest

only when we raise the accuracy to higher order in 1=M . It is then necessary to include

irrelevant (nonrenormalizable) op erators in the e ective Lagrangian. In addition, we must

also supply appropriate calculation rules to obtain unambiguous nite results b ecause the

presence of such irrelevant terms can makephysical quantities diverge if naively calculated,

as we mentioned b efore. If these tasks are systematically p erformed with lo cal op erators,

we will have a lo cal e ective quantum eld theory where virtual heavy-particle e ects are

isolated into the co ecients of irrelevant op erators in the Lagrangian to any desired order

of accuracy. Let us supp ose that we want to describ e low-energy physics in the full theory

accurately up to order 1=M . (N is some p ositive integer.) We will show b elow that we

can then factorize all heavy-particle e ects to the given order theory by making use of the

ow equation (23) and by appropriately cho osing the op erators S , N =1;2;:::;N .

int

Here, S 's maybe assumed to have the form

int

 

lo cal, Euclidean invariant, even p olynomials

; N =1;2;:::;N : (60) S = d x

int

in and its derivatives, of dim. 4+2N

The co ecient of the p olynomials will be chosen later so that S carries information

int

appropriate to O (1=M )-e ects from the full theory. Then we claim that the sum of

S 's, viz.,

int

;N ;N

0 0 0

 

S S

;N ;N 1

int int

0 0 0 0

e e

S S + = S + ; (61)

int int int

N ! N !

N =1

will repro duce the predictions of the full theory at low energy with an accuracy of order

1=M . As we noted in section 2, S will describ e amputated connected Green functions

int

with the insertion of the op erator O de ned in Eq. (22). As usual, the quantity S , which

int

satis es the ow equation (23), maybe expanded in p owers of :

1 1 2n1

XX Y

d p

;N ;N

(p ) (p )G (p ;:::;p ); (62) S []= g

1 2n 1 2n1

r;2n

int

(2 )

r=0 n=1

j =1

 

2n1

p p :

2n j

j=1 17

We may also de ne G 's through the same kind of relation as in Eq. (61). To complete

r;2n

;N ;N

the de nition of S , we must also state the b oundary conditions for G . At = ,

r;2n

int

Eq. (60) implies that

@ G =0; for 2n + jw j > 4+2N: (63)

r;2n

For 2n + jw j 4+ 2N, we imp ose the following condition, recursively in N :

(f )=0

=0;N =0;N 1

w w

@ G (0) = N ! @ (G (0) G (0)) ; (64)

r;2n r;2n r;2n

p p

where G is the vertex function obtained by summing over the couplings involving heavy

elds in Eq. (50), i.e.,

(f ) (f )

r r

2 3

G g g G : (65)

r ;2n

2 3

(r ;r ;r );2n

1 2 3

r ;r

2 3

=0;N

As a consequence, @ G (0) = 0 for the case 2n + jw j < 4+ 2N. The meaning of this

r;2n

b oundary condition is clear. To obtain the N -th order e ective theory,we add new (higher

dimensional) op erators to the e ective action which comp ensate the di erence between the

full theory and the N 1-th order e ective theory.

As we discussed in section 1, this seemingly natural pro cedure has a p otential problem

b ecause new op erators are nonrenormalizable and naive treatmentof those op erators make

Green functions divergent (rather than making the e ective theory more accurate). However

one can show that, if one demands the ow of the e ective theory follow Eq. (23), divergences

are automatically taken care of and virtual heavy particle e ects are correctly incorp orated.

Indeed, straightforward extension of the argument used in proving decoupling shows that

(f )

[7] the di erence between two vertex functions, G of the - theory and G of the

r;2n

r;2(n;0)

 theory, satis es the b ound

 

2N+2

+ m M

42nz

( + m) Plog ; 0 M

M m

(f )

@ (G G ) (66)

r;2n

r;2(n;0)

 

(2;)

42nz

Plog ; M ;

M m

which establishes the factorization of virtual heavy particle e ects to any given order of

accuracy in 1=M ,

(f )c

c;N

j@ (G (p ;:::;p ) G (p ;:::;p ))j

1 2n1 1 2n1

r;2n

r;2(n;0)

 

2N +2

m M

42nz

 m Plog ;

M m

(for jp j): (67)

;N ;N =0

w w

e e

If is larger than M , Eq. (66) says that the vertex function @ G is larger than @ G

r;2n r;2n

p p

by a factor (=M ) . Therefore, in this scale, increasing N (or adding more irrelevant

terms, equivalently) makes the vertex functions unnatural ly large and the theory b ecomes

\less e ective". This is b ecause wehave imp osed unnatural renormalization conditions (64) 18

on irrelevantvertices. If wechose natural values as the bare irrelevant couplings, they would

give only O ((m= ) ) contributions to Green functions; but, in Eq. (64), we imp osed the

values of order (m=M ) to G as the renormalization conditions, which are natural only

r;2n

if the cuto is around M . This arms that the e ective theory is useful only b elow the

heavy particle mass scale M .

So far we have shown that, at low energy, connected amputated Green functions of the

full theory are repro duced by considering those of the theory plus those with insertion

of op erators O , N = 1; ;N , de ned in Eq. (22). Such op erator insertions have a

N 0

simple interpretation if one uses the notion of renormalization of comp osite op erators and

their normal pro duct notation. As mentioned in section 2 this sub ject can also be easily

describ ed using the ow equation but we will not discuss it in this lecture. Interested reader

can consult Refs. [4,7]. Results are the following. For N =1, S can b e written as

int

R;1 ;1

b [M ] ; (68) S =

2n;fw g

int

2n;fw g

2n+jw j=6

(f )c

R;1

w c

where b = @ (G G )(0), N is a combinatorial factor de ned by

fw g

p 2n

2n;fw g 2(n;0)

fw g

w w w

2n 1

] = N ; (69) (ip ) @ [(ip )

2n 1

fw gfw g fw g

sy mm

p ==p =0

1 2n

4 w w

1 2n1

and [M ] is the normal pro duct of a lo cal vertex M = d x@ @ (x):

2n;fw g 2n;fw g

x x

In terms of the lo cal Euclidean-invariantvertices of dimension six, the ab ove equation can

be cast into a more attractive form:

Z Z Z

(6) (6) (6)

4 2 2 4 3 2 4 6

S = a d x [(@ ) (x)] + a d x [ @ (x)] + a d x [ (x)]

1 2 3

int

(6) (6)

 a [ ] ; (70)

i i

i=1

with

(6) (f )c

2 2 c

a = (@ ) (G G )(0) ;

(2;0)

8 4!

(6) (f )c

2 c

a = @ (G G )(0) ;

(4;0)

(6) (f)c

=(G G )(0) : (71) a

(6;0)

The 1=M -order e ects of the full theory are then describ ed by the insertion of the op erator

(6) (6)

O = a [ ], viz.,

i i

 

82n

M m

(f )c (6) (6)

c;1

Plog ; (72) G = G + a G ([ ]) + O

i i

2(n;0)

M m

i=1

(6) (6)

c;1 ;N

where G ([ ]) denotes the Green function with [ ] inserted. For general N , S

i i int

(N =1;:::;N ) can be identi ed as

0 19

R;N ;N ;N

0 0

b [M ]+S ; (73) S =

2n;fw g

int

2n;fw g

2n+jw j=4+2N

where

N !

(f )c

R;N c;N 1

b = @ (G G )(0) ; (74)

2n;fw g 2(n;0)

fw g

2n+jw j=4+2N

and S denote counterterms which are needed to cancel the new divergences due to the

;I ;N

0 0

multiple insertion of S 's, I =1;:::;N 1. (See [7] for explicit forms of S .) Now let

int

(4+2N )

[ ]'s form a complete set of mutually indep endent Euclidean-invariant lo cal vertices

(4+2N )

of dimension (4 + 2N ) and let a 's b e appropriate expansion co ecients as in Eq. (70),

so that we may write

(4+2N ) (4+2N )

;N ;N

0 0

S = a [ ]+S : (75)

int i i

0 0

Then the e ective action S may b e written as

int

(4+2N )

N N

0 0

X X X

a 1

(4+2N )

;N ;N

0 0 0

+ [ ]+ S S = S

int i int

N ! N !

N =1 N=2

(4+2N )

a 1

(4+2N )

;N 1 ;N

0 0 0 0

= S + [ ]+ S ; (76)

int i

N ! N !

0 0

(4+2N )

R;N

where a 's are appropriate linear combinations of b 's (all of which are O (1=M )).

2n;fw g

(4+2N )

Thus the 1=M -order information of the full theory is factorized into the co ecient a 's

of lo cal vertices of dimension (4 + 2N ). If wewant to increase the accuracy from O (1=M )

(4+2(N +1))

0 0

2(N +1)

to O (1=M ), wehave only to include in S lo cal vertices [ ] of dimension

int i

4+ 2(N + 1). (The lower dimensional vertices need not be mo di ed.) The co ecients

4+2(N +1)

a are calculable as the di erence of appropriate Green functions of the full theory

0 0

and those of the e ective theory (i.e., S ) at renormalization p oint. New divergences

int

;N +1

0 0

are subtracted away by S . Given this bare Lagrangian, the ow equation guarantees

2(N +1)

that all Green functions are nite and accurate up to order 1=M .

The interpretation of this result is clear from the viewp oint of the renormalization group

ow in the in nite dimensional space of p ossible Lagrangians. As we reduce the cuto , the

bare Lagrangian of the full theory, which is sp eci ed by the b oundary conditions (52) and

(53), ows down to a submanifold parametrized by relevant couplings only. It has b oth

heavy and light op erators. Now the ow may be pro jected onto the subspace of op erators

consisting of the light eld only. Then nding a lo cal low-energy e ective eld theory of

light particles is equivalent to nding a renormalization group ow with a few (relevant or

irrelevant) lo cal op erators which can b est approximate the pro jected ow of the full theory.

To the zeroth order in 1=M , it is done with relevant terms only by cho osing the same

renormalization conditions as those of the full theory. The deviation from the full-theory

ow is corrected by reading o values of the remaining irrelevant co ordinates at the = 0

p oint. At rst, comp onents of dimension six op erators may be read, which tell us the 20

2 2

e ects of order 1=M ; the ow of the full theory is now approximated to the order 1=M

at < M . If one wants to increase the accuracy, it is necessary to read more and more

irrelevant co ordinates (at = 0) of the pro jected ow of the full theory and mo dify the

e ective-theory ow with the corresp onding ow equation given in Eq. (23). Our equation

(23) automatically takes care of p ossible divergences due to the unnaturally large irrelevant

comp onents in such a way that the bare Lagrangian may contain only a nite number of

irrelevant terms.

IV. CONCLUSIONS

In the context of a simple scalar eld theory,we have demonstrated that, at low energy,

virtual heavy particle e ects on light-particle Green functions are completely factorized

via e ective lo cal vertices to any desired order. For this, we have used the powerful ow

equation approach which is a di erential version of Wilson's renormalization group trans-

formation. In applying this metho d, we have seen that irrelevant terms in the Wilsonian

e ective Lagrangian (which has a nite UV cuto = M ,acharacteristic scale representing

heavy-particle thresholds) are replaceable by the corresp onding higher dimensional comp os-

ite op erators plus counterterms for their pro ducts in the continuum limit (where UV cuto is

supp osed to go to in nity). The latter can b e dealt with with the help of the normal pro duct

concept. Once this fact is noticed, factorization is straightforward: all 1=M -order e ects

can b e isolated in terms of lo cal vertices involving op erators of dimension (4 + 2N ). Thereby

we arrive at a lo cal continuum e ective eld theory which describ es low-energy light-particle

physics accurately up to any desired order in 1=M with appropriate calculation rules for ir-

relevant (nonrenormalizable) vertices. Since the arguments here are essentially dimensional

and largely theory-indep endent, it should not b e dicult to generalize them to di erent eld

theories such as gauge theories or theories with sp ontaneous symmetry breaking. Indeed,

decoupling was proved for these theories in [23] within this approach.

Since the couplings for e ective lo cal vertices contain p owers of log(M=m) in p erturbation

theory, it is often desirable to sum such logarithms in a systematic way. This has b een

achieved by utilizing a set of improved Callan-Symanzik equations [29]. Also, here we

limited our attention to vertex functions at momentum range = O (m). As one increases 

to suciently high energy scale, the (=m)-dep endence we neglected will b ecome imp ortant.

Keeping such -dep endence and establishing b ounds more carefully, one may study the high-

momentum b ehavior of Green functions (even in the exceptional momentum region) within

the e ective eld theory context. This b ound has b een obtained in [24] for zeroth order

e ective theory (corresp onding to decoupling).

In summary, the ow equation, which is based on Wilsonian viewp oint of e ective theory,

provides clear understanding of e ective theories in continuum.

ACKNOWLEDGMENTS

It is my great pleasure to thank Prof. D. P. Min for inviting me to the summer scho ol.

This work was supp orted in part by Korea Science and Engineering Foundation through

SRC program. 21

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