3 Expansion of the effective Lagrangian

The above remarks do not affect the construction of the low energy effective Lagrangian [10]. Its form is merely dictated by the chiral symmetry [11] and by the transformation properties of the symmetry breaking quark mass term, and there is obviously no question to alter these fundamental properties of QCD. is a function of 8 Goldstone boson fields (conventionally collected into an SU(3) element U) and of external sources and p, the scalar source s containing the quark mass matrix . The formalism and notation used here are standard [2], unless otherwise stated. consists of an infinite tower of invariants

where contains k powers of covariant derivatives and l powers of scalar or pseudoscalar sources. In the low energy limit, vanishes like the k-th power of external momenta p and the l-th power of the quark mass ,

Chiral perturbation theory is an expansion of in powers of the pion (kaon) mass assuming that all external momenta are of that size. For sufficiently small quark masses, such that both and hold, one has . In this case, one can write

where [2]

This expansion defines the standard PT. If, on the other hand, for actual values of quark masses one has , both and should count as parameters of the size of the pion mass and, consequently, . This new counting yields a different expansion of ,

where [4]

It should be stressed that Eqs. (20) and (22) represent two different expansions of the same effective Lagrangian. To all orders they are identical, at a given finite order they may differ.

It is straightforward to write down the most general expression [4] of which defines the leading order of GPT :

Here collects the scalar and pseudoscalar sources,

Notice the absence of the factor , which appears in the standard definition of [2]. The meaning of this difference will shortly become obvious. Compared to , Eq. (24) contains additional terms. The constants and are the same as introduced in the previous section : Eq. (5), (with and ) is indeed a straightforward consequence of the Lagrangian (24). The fact that the and terms now appear at the same order as the term reflects the possibility that in Eqs. (5) the first order and second order terms are of comparable size. To this order , the low energy constants in (24) can be expressed in terms of physical masses , of the quark mass ratio and of the Zweig rule violating parameter (14). Expanding Eqs. (12a), (12b) and (10), one gets

where . It is seen that for the particular values of the parameters , Eqs. (26) imply and , i.e. one recovers the standard Lagrangian . Order by order, GPT contains the standard PT as a special case.

The next to the leading term of the expansion (22) is of odd order , which is a new feature, absent in the standard PT. One has [6]

In writing down Eq. (27), we have adopted some conventions which are worth to be specified. The parameters of are finite, (divergences only start at order ) and they may be viewed as independent variables. It is convenient to tag each of these variables by a QCD correlation function. An example is the parameter , closely related to the two-point function (3). This relation may be further specified as

where is defined by Eq. (4). Here, the statement is that the expansion of in powers of does not contain a linear term. This constrains the way one writes :

i) A term which, according to Eq. (23) could be present in , is obviously irrelevant. It can be absorbed into the covariant transformation of sources and of parameters of which has been discovered some times ago by Kaplan and Manohar [12], (see also Ref. [13]). Indeed, Eq. (28) may be viewed as a physical condition fixing the reparametrization ambiguity of pointed out in [12] and in [13]. In general, the dependent terms in Eq. (23) will be introduced only if they are required by renormalization. This is not the case of .

ii) Similarly, a term like which would yield a contribution to Eq. (28) linear in , can be transformed away by a source dependent redefinition of Goldstone boson fields. Notice that this convention has not been used in Ref. [4].

The main physical effect described by the Lagrangian is the splitting of the decay constants , and . One easily finds

This allows to express the constant as

whereas the Zweig rule violating parameter remains at this stage undetermined. The part of , described by the constants generates an contribution to the pseudoscalar masses. Notice that in the standard PT these terms would count as .

provides the simplest example of odd chiral orders characteristic of GPT. They do not correspond to an increase in the number of loops, but to additional corrections in powers of the quark masses. In the standard PT [2], the splitting in the decay constants is a effect arising from loops (tadpoles) and from the corresponding counterterms contained in . Here, the leading contribution (30) counts as , (actually, it can hardly be expressed in terms of the pion mass), and the loop effects only show up at the next, , order. Notice that for , the constant (30) is of the order - a typical size of other effects, such as the deviation from the Goldberger-Treiman relation.

The Lagrangian describing the next order, , consists of several components :

is the part of the standard , which consists of four derivatives and contains no , i.e. no quark mass : is given by the standard five terms [2] described by the low energy constants and . is a new term, which in the standard PT would count as , whereas involves 4 insertions of a quark mass, and in the standard PT it would be relegated up to the order . and involve about 20 independent terms each. An experimental determination of all the corresponding low energy constants is obviously hard to imagine. However, a few particular combinations of these constants which contribute, for instance, to and decays, or to the scattering amplitude at the one loop level, can be estimated and included into the analysis.

The last three terms in Eq. (31) represent -dependent counterterms of order which are needed to renormalize one loop divergences that arise from using the vertices of alone in the loop. They renormalize the constants by higher order contributions, of order , and the constants and by an amount . In generalized PT, renormalization proceeds order by order in the expansion in powers of the constant .

The standard order Lagrangian contains 10 low energy constants [2]. They are all involved in . As already pointed out, and - the constants of - play an identical role in both schemes. and are related to the constants and in :

Finally, the standard constants are related to the constants and , respectively :

On the other hand, involves additional terms, not contained in , and which the standard PT relegates to higher orders d>4. Setting the corresponding additional constants to zero, one recovers the standard PT up to and including order . This phenomenon is general: Order by order, the standard expansion reappears as a special case of GPT.