4 Examples of differences between standard and generalized PT

The difference between standard and generalized PT merely concerns the symmetry breaking sector of the theory. In the chiral limit , observables described by purely derivative terms of the effective Lagrangian are essentially given by soft pion theorems which are identical in both schemes. This concerns, in particular, the electromagnetic radius of the pion, the decay , the scattering in the P-wave, the leading order and form-factors, etc...

The difficulty of disentangling both schemes resides in the fact that symmetry breaking effects are small and not well known experimentally. In this section we are going to give an overview of the main differences between the predictions of standard and generalized PT for symmetry breaking effects which already appear at the leading order . Those have more chances to be observable, at least indirectly.

4a - : Dismiss of the ellipse [14]

We have already stressed the importance of the quark mass ratio as an independent parameter of GPT. When isospin breaking is switched on, GPT leads to a relation between the two quark mass ratios r and R, with

Taking a suitable linear combination of the expansions of and , the terms, the terms and even the terms drop out and one obtains [4]

where is the kaon mass difference in the absence of electromagnetism. Eq. (35) holds independently of the value of r, i.e. both in the standard and in the generalized PT. In the standard case, Eq. (35) should be rather accurate, whereas in the generalized setting the neglected terms - arising from and given by the low energy constants (27) - can easily represent a 30% correction.

To the best of our knowledge, Eq. (35) has never appeared in the literature on standard PT. Instead, one rather expands ratios of pseudoscalar meson masses and one eliminates the terms from this expansion (see e.g. Eqs. (10.11) and (10.17) of the second of Refs. [2]). In this way, one arrives at the well known elliptic relation between and , extensively commented upon in the literature [14]. The difference between the two procedures is instructive : When expanding the linear combinations of pseudoscalar meson masses, one expands in powers of the small parameter . On the other hand, when expanding mass ratios, one assumes, in addition, that the parameter is also small. Actually, the elliptic relation [14] should read

where

The quark mass ratios lie on the ellipse only provided or, equivalently, provided is close to , whereas Eq. (35) has a more general validity.

It is interesting to look at Eq. (35) in the light of recent discussions of a possible large violation of Dashen's theorem [15]. We can write

where the parameter describes the departure from Dashen's theorem : if the theorem is exact. Various recent estimates [15] expect somewhere between 1 and 2. Taking in Eq. (35) , one obtains R=43 for and for . For r=10, Eq. (35) predicts R=66 if , whereas R=53 if . Assuming (as expected from baryon masses and from mixing [9]) we observe that by increasing , we are left with less room for corrections to Eq. (35), which could provide a valuable information on the parameters .

4b - Quark condensates for

The parameter describes the quark condensate in the chiral limit . The leading order Lagrangian allows to express , where is the ground state for , beyond this limit. One gets

where q=u,d,s denotes a given quark flavour. The point is that if , the contribution is Eq. (39) can be relatively important. Comparing for instance the term to one finds, using Eqs. (26a) and (26b), , assuming r=10 and . This should not be surprising : in GPT all contributions of are supposed to be of the same order of magnitude. In practice, it implies that the condensates can exhibit a large flavour dependence. Unfortunately, there is no way to pin down the constant which controls this dependence quantitatively : is not a low energy order parameter, but rather a short distance counterterm (see Ref. [2] for a discussion of this point).

4c - Large corrections to the soft pion theorems

Within generalized PT the corrections to soft pion theorems can sometimes be rather important - of the same order as the soft pion result itself. Consider, for instance, the scalar form factor of the pion at vanishing momentum transfer or, equivalently, the pion -term

The soft pion result for is well known :

Since in the GPT, can be considerably smaller than , one might be tempted to conclude, on the basis of the soft pion result (41), that in GPT is smaller than in the standard theory. This conclusion does not take into account all contributions to described by . Writing , using Eqs. (5) and (26), and neglecting the Zweig rule violating parameter , one obtains the correct result,

It is seen that, when r decreases from to , the pion -term increases from to . In GPT, the soft pion result will receive a large correction, whenever the soft pion theorem result is proportional to the quark condensate . The reason is that both and count as quantities of order . The formalism of GPT automatically takes care of such large corrections. This phenomenon can in principle lead to modifications of the standard evaluation of the non-leptonic K-decay matrix elements in the large limit, in particular, of the penguin-contribution to the ratio [16].

There is another relevant example of a similar nature : The PT prediction for the low behavior of the spectral function associated with the divergence of the axial current,

At low , the continuum part of is dominated by the contribution of intermediate states. Using for the latter the soft pion theorem (and neglecting in the phase space integral (43) the pion mass) one gets the standard result [17]

Within the GPT, this result is considerably modified. Still neglecting the pion mass in the phase space integral (in order to facilitate a comparison with the standard result) one gets [18]

As r decreases from down to , the enhancement factor in Eq. (45) increases from 1 to 13.5. This enhancement would considerably affect the existing estimates of using the QCD sum rules [17].

4d - and scattering

Our last example of a leading order difference between the standard and the generalized PT is of a direct experimental relevance: It concerns the low energy [4] and [5] scattering. The amplitude predicted by the leading order GPT lagrangian reads

where

is the non strange quark antiquark condensate in the SU(2)SU(2) chiral limit (see Eq. (39)). Eq. (46) holds both in the standard case and in the genertalized PT. In the standard case, however, , and one recovers the well known formula first obtained by Weinberg [19]. In GPT, this formula is modified already at the leading order : can be considerably smaller than by an amount which depends on r. For r decreasing from =25.9 to r=6.3, decreases from to zero. The low energy scattering thus provides us with a quasi-unique experimental access to the order parameter . The corresponding GPT predictions, endowed with the necessary loop corrections, are presented in detail in the section on interactions of the present Handbook.

A similar conclusion holds in the case of scattering [5]. The latter gives a contribution to the form factor R, which is measurable in the decay mode. Whereas the leading order predictions for the form factors F, G and H are identical in both schemes, there is a detectable difference in the leading order expression for R [5]. Whether this difference can be observed in practice is presently under investigation [25].