\subsection{Effective Field Theory}
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The concept of effective theory \cite{eff-Th}
has become a very important tool in
theoretical physics; it is widely used in particle
physics phenomenology, in particular in the field of weak interaction.

The presently known spectrum of particles spans a mass range of
severel orders of magnitude, ranging from the eV region up to
the  mass of the top quark of about 170 GeV, and new accelerators
will be able te explore masses in the TeV regime. Due to this large
variation of mass scales a frequently occurring situation in particle
physics phenomenology is that a calculation involves very disparate
mass scales. The standard example are weak decays of leptons and
hadrons, the masses of which lie in the MeV or GeV range and which set
the typical scales involved in this type of process. On the other
hand, weak interactions are mediated by bosons with masses of about
100 GeV and thus these processes involve also mass scales much
larger than the mass of the decaying particle.

Intuitively it is clear that in such a situation a description should
be possible which does not explicitly involve the degres of freedom
associated with the large mass scale. In other words, one should
switch to an effective theory in which the heavy degrees of freedom
have been removed.

In general this is possible due to the so called
decoupling theorem \cite{Decoup}
which states that in general a heavy particle
decouples from the lighter ones in the limit in which its mass
tends to infinity;  the only remnant of the heavy particle will
be on one hand a logarithmic dependence of the coupling parameters 
of the effective low energy theory on the mass of the heavy particle 
and terms suppressed by inverse powers of the heavy mass on the other 
hand.

Technically this corresponds to an expansion of the Greens functions
of the full theory (i.e. the one still containing the heavy particle)
in inverse powers of the heavy mass up to logarithms, which may be
accounted for by renormalization group techniques. This expansion 
of the Greens functions may be obtained from a Lagrangian field 
theory, which is called an effective theory.  

The leading term in the heavy mass expansion corresponds in
general to dimension four contributions to the effective
theory Lagrangian and thus defines a renormalizable theory.
From these pieces of the Lagrangian one obtains the leading term
in the heavy mass expansion of the full theory Greens functions;
higher order terms in this expansion may be included by adding
higher dimensional operators (i.e. dimension five operators and
higher) to the effective Lagrangian. Up to some given order in
the heavy mass expansion only a finite number of insertions of
these higher dimensional operators is needed and any finite number
of insertions of operators of dimension five or higher may be
renormalized and hence the apearance of these terms in the
effective theory does not spoil the renormalizability of the
effective theory up to a finite order in the heavy mass expansion.

The construction of the effective theory Lagrangian proceeds in
three steps. After having identified the massive degree of freedom
this degree of freedom is removed by integrating it out from the
functional integral of the full theory. This step in general leads
to a nonlocal action functional; the nonlocality is related to the
mass of the heavy particle which has been integrated out in the
previous step and which may be replaced at scales much lower than
the mass of the heavy particle by local interactions. Technically
this step is an operator product expansion of the nonlocal terms into
a series of operators with increasing dimension; the dimension of the
effective Lagrangian is four and thus these higher dimensional
operators have to have dimensional coupling constants, the scale
of which is set by the heavy mass. The terms with dimension four
have dimensionless couplings and define the renormalizable piece
of the effective Lagrangian; in other words, this part is used
to calculate the renormalization group flow in the effective theory.
 
Alternatively, one may proceed slightly differently in constructing
the effective theory Lagrangian and obtain it from matching the
full to the effective theory. As a first step one writes down all
possible operators up to a given dimension which involve only the
light degrees of freedom and which are allowed by the symmetries of
the system. The effective theory Lagrangian is then written as a sum
of these operators with coefficients which have to be determined
from comparing the effective Lagrangian with the one of the full
theory.

In the conventional sense one would say that an  effective theory 
is not renormalizable due to the higher dimensional operators. 
However, up to a finite order in the expansion in inverse powers 
of the heavy mass one has only a finite number of insertions of 
this type of operators in the Greens functions, and for this case
the renormalization of these contributions is known once the 
renormalizable piece, i.e.\ the dimension four operators are 
known. In this sense the concept of renormalizability becomes
irrelevant for effective theories. 

The comparison of the full theory with the effective theory is 
called matching. At some scale $\mu$ of the order of the heavy mass 
the Greens functions of the full and the effective theory are 
compared, thereby fixing the coupling constants of the effective 
theory in terms of the parameters of the full one. Changing the 
scale $\mu$ will induce changes in the Greens functions in such a way
that physical quantities remain independent of the scale. This 
in general involves also appropriate changes of the couplings in 
the effective theory, turning them into ``running'' quantities 
depending on the scale $\mu$.   
The running, i.e.\ the renormalization group flow, is determined 
by the renormalization group equations. 

The choice of a scale may also be interpreted as a seperation 
of long and short distance contributions: The effect of physics
originating from scales above $\mu$ is parametrized in the 
couplings while any effect from scale below $\mu$ is still 
explicit in the dynamics described by the Lagrangian. Of course, 
any observable quantity has to be independent of the arbitrary 
choice of $\mu$, which is assured by the renormalization group 
equations. 

At that point one may rise the question, why the effective theory
description has any advantage over performing the calculations in 
the full theory. First of all, there are examples where the full 
theory is not known or where the matching to the effective theory 
is technically not feasible (such as chiral perturbation theory 
and QCD). In these cases it is obvious that one directly performs 
the calculations in the effective theory. But even in cases where
the matching may be performed explicitly (such as weak interactions 
or Heavy Quark Effective Theory) it may be advantageous to 
switch to an effective theory description, mainly for two reasons.  
Firstly the strong interactions become weak at large  scales
and the matching and running may be done perturbatively. The 
renormalization group then automatically takes care of large 
logarithms appearing in perturbation theory and resums them properly.
Secondly, it may happen that in the effective theory additional 
symmetries appear which are not present in the full theory. This 
is the case for Heavy Quark Effective Theory where these additional 
symmetries restrict the non-perturbative input quite severely. 

 


