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Wilson Renormalization Group and Continuum Effective Field Theories* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SNUTP 98-100 KIAS-P98026 hep-th/9810056 Wilson Renormalization Group and Continuum E ective Field Theories y Chanju Kim Center for Theoretical Physics Seoul National University Seoul, 151-742, Korea and z 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 z present address y 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 4 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 0 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 N 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 1 among them wemaycho ose to keep explicitly some minimal numb er of irrelevant terms in N 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 0 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 0 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 0 the unnaturally large co ecients would be ampli ed by some p ositive power of =M as 0 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. 1 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 0 bare action is written as D E 1 1 0 0 0 S ; P []= + S [] ; (1) int 2 0 where P is the free-particle cuto propagator and hf; gi is de ned by the momentum-space integral of f and g , Z 4 d p f (p)g (p) ; (2) hf; gi 4 (2 ) 0 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 0 functions of the theory. To account for this, we de ne S []as a formal p ower series in , tot 2 0 which has S []as its zeroth order term, viz., int N X ;0 ;N 0 0 0 0 [] S []) (3) S [] : (S []= S int int int tot N ! jNj0 Also, for later conveniences, we will denote N X ;N 0 0 0 0 [] : (4) []= [] S S S [] = S int int C tot N ! jNj1 ;N 0 S 's may be regarded as additional lo cal vertices app ended to the original Lagrangian int 0 S or as comp osite op erators in which one is interested.
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