###### Abstract

We show that isospin/ breaking terms can be introduced
to the anomalous coupling in the hidden local symmetry
scheme without changing Wess-Zumino-Witten term
in the low-energy limit.
We make the analysis for anomalous processes of
2-body and 3-body decays;
radiative vector meson decays(),
conversion decays of photon into a lepton pair()
and hadronic anomalous decays().
The predictions successfully reproduce all experimental data
of anomalous decays.
In particular, we predict the decay widths of
and as
and ,
respectively, which will be tested in the DANE
-factory.
Moreover, prediction is also made for ,
, and so on,
for which only the experimental upper bounds are available now.

number(s): 12.39.Fe, 12.40.Vv, 13.25.-k, 13.65.+i, 14.40.Aq, 14.65.Bt

DPNU-96-27

hep-ph/9605422

May 1996

Hidden Local Symmetry

for Anomalous Processes

with Isospin/SU(3) Breaking Effects

Michio Hashimoto
^{*}^{*}*e-mail address:

Department of Physics Nagoya University, Nagoya 464-01 Japan

## 1 Introduction

Anomalous processes involving vector mesons are interesting probes to test the effective theories of QCD through the low-energy and high-luminosity collider experiments in near future. In particular, the DANE -factory is expected to yield -meson decays per year [1], which will provide us with high quality data for decays of pseudoscalar() and vector mesons() in the light quark sector. It is expected to obtain the branching ratio of [1] for which only the upper bound is known today [2]. Moreover, uncertainty of the data on will be much reduced [1].

These radiative decays are associated with the flavor anomaly of QCD and are described by the Wess-Zumino-Witten(WZW) term[3] in the low energy limit. Based on the hidden local symmetry(HLS) [4][5][6] for the vector mesons, Fujiwara et al.[7] proposed a systematic way to incorporate vector mesons into such a chiral Lagrangian with WZW term without affecting the low-energy theorem on etc. Bramon et al.[8] studied extensively the radiative vector meson decays by introducing breaking into the anomalous Lagrangian of Fujiwara et al.[7]. However, the method of Bramon et al. is not consistent with the low-energy theorem, especially on , which are essentially determined by the WZW term. Thus, if isospin breaking effects were introduced through their method, successful low-energy theorem on and the coupling of would be violated. Furthermore, the breaking effects (and - interference effect) are important to account for the difference between and .

In the previous paper[9], we proposed isospin/-broken anomalous Lagrangians without changing the low energy theorem. These were obtained by eliminating direct and coupled terms, which were absent in the original Lagrangian[7], from all possible isospin/-broken anomalous Lagrangians with the smallest number of derivatives. Then we found a parameter region which was consistent with all the existing data on radiative decays of vector mesons. In this paper, we give a full description of our analysis and -fitting. We also include the analysis of in addition to the previous results on and .

The paper is organized as follows: In section 2, a review of HLS Lagrangian is given for both non-anomalous and anomalous terms. breaking terms are introduced into the non-anomalous HLS Lagrangian à la Bando et al.[5]. In section 3, we construct the most general isospin/-broken anomalous Lagrangians with the lowest derivatives in a way consistent with the low energy theorem. This is systematically done through spurion method for the breaking term. In section 4, the phenomenological analysis of these Lagrangians will be successfully done for radiative decays of vector mesons. In section 5, conversion decays of photon into a lepton pair are analyzed. In section 6, we make the analysis for hadronic anomalous decays. Section 7 is devoted to summary and discussions.

## 2 Hidden Local Symmetry

Here we give a brief review of HLS approach[6]. A key observation is that the non-linear sigma model based on the manifold is gauge equivalent to another model having a symmetry . Vector mesons are introduced as the gauge fields of a hidden local symmetry . The photon field is introduced through gauging a part of .

The HLS Lagrangian is given by :[4][5]

(2. 1) | |||||

(2. 2) | |||||

(2. 3) |

where is the decay constant of pseudoscalar mesons, , with

The fields and transform as follows;

(2. 4) | |||||

(2. 5) |

where . To do a phenomenological analysis, we take unitary gauge:

(2. 6) | |||||

(2. 7) | |||||

(2. 8) |

where we assumed that - mixing angle is degrees and - mixing angle is the ideal mixing (35 degrees). If we take in (2. 1), we have the celebrated KSRF relation , the universality of the -meson coupling and the vector meson dominance for the electromagnetic form factor(VMD) [4].

For obtaining the pseudoscalar meson mass terms, we introduce the quark mass matrix() as,

(2. 9) |

where is related to the mass of , and , and is the mass of due to breaking by the gluon anomaly. Analogously, we may add appropriate breaking terms to (2. 1) [5],

(2. 10) | |||||

(2. 11) |

Even if those breaking terms are introduced, we can show the successful relations [5]:

(2. 12) |

We will use this relations, when we consider radiative decays of vector mesons and conversion decays of photon into a lepton pair.

Further improvements of (2. 1) have been elaborated in Ref.[10]. Here we will not discuss the non-anomalous sector (2. 1) any furthermore, because we are only interested in the anomalous sector. We simply assume that the parameters of the non-anomalous Lagrangian have been arranged so as to reproduce the relevant experimental data. Thus we use the experimental values as inputs from the non-anomalous part.

In addition to (2. 1) there exists an anomalous part of the HLS Lagrangian. Fujiwara et al.[7] proposed how to incorporate vector mesons into this part of the Lagrangian without changing the anomaly determined by WZW term[7]. They have given the anomalous action as follows:

(2. 13) |

where

(2. 14) | |||||

(2. 15) | |||||

(2. 16) | |||||

(2. 17) | |||||

(2. 18) | |||||

(2. 19) | |||||

(2. 20) | |||||

(2. 21) |

Notice that have no contribution to anomalous processes such as and at soft momentum limit, because these Lagrangian are constructed with hidden-gauge covariant blocks such as and [7].

We take in (2. 13) for phenomenological reason[7]. Then we obtained the Lagrangian of anomalous sector as follows:

(2. 22) | |||||

Here, it is important that the amplitude such as at low energy limit are determined only by the non-Abelian anomaly of the chiral symmetry. The Lagrangian is, of course, consistent with the low energy theorem related to the anomaly.

## 3 Isospin/-breaking Terms in the Anomalous Sector

We now consider how to modify by introducing isospin/-breaking parameters, ’s, treated as “spurions”[11]. The spurion transforms as . Then we define the hidden-gauge covariant block . We construct Lagrangians out of the hidden-gauge covariant blocks such as , , and so as to make them “ invariant ” under as well as parity()-, charge conjugation()- and -transformations. After hidden-gauge fixing, they become explicit breaking terms of the symmetry. Then, in general, we obtain isospin/-broken anomalous Lagrangians with the lowest number of derivatives.

(3. 1) | |||||

(3. 2) | |||||

(3. 3) | |||||

(3. 4) | |||||

(3. 5) | |||||

(3. 6) | |||||

(3. 7) | |||||

(3. 8) | |||||

(3. 9) | |||||

(3. 10) | |||||

(3. 11) |

Here , , transform under and transformations as

(3. 13) | |||||

(3. 15) | |||||

We could introduce another “spurion” , which, however, is not relevant to the following analysis.

There still exist too many parameters. However, we may select the combination of so as to eliminate the direct -, -coupling terms, which do not exist in the original Lagrangian . Then the isospin/-broken anomalous Lagrangians consist of only the following two terms:

(3. 16) | |||||

(3. 17) | |||||

We can also understand these in a more straightforward way: We can introduce the breaking terms to the first -term in the original via two possible ways, which correspond to the first terms of . Next, we determine -, -terms so as to eliminate them at soft momentum limit by using the relation as well as to make them invariant under , and -transformations. These terms correspond to the second and third terms of .

Our resembles the -broken anomalous Lagrangian introduced by Bramon et al.[8], but is conceptually quite different from the latter. In fact the prediction on decay width in the latter disagrees with the low energy theorem. On the other hand, our obviously do not change the low energy theorem by construction.

## 4 Phenomenological Analysis for Radiative Decays

We now discuss phenomenological consequences of our Lagrangian . For convenience, we define relevant coupling constant as

(4. 1) |

considering that these decays proceed via intermediate vector mesons . Then we obtain each radiative decay width

(4. 2) | |||||

(4. 3) |

where , and are anomalous coupling constant, - mixing and mass of the vector meson, respectively. In we take a parametrization for convenience:

(4. 4) |

Thus each is given in terms of the parameters in :

(4. 5) |

where and we used the relations of (2. 12).

The parameters appearing in the expression of stand for the deviation of -, - mixing angles from the ideal mixing and - mixing, respectively. The parameter comes from the - interference effect arising from the small mass difference of and .

For reproducing the experimental value of , we took . The sign comes from the observed - interference effects in [2].

The mixing angle arcsin(1/3)) has been supported in - phenomenology[13], thus, it is admitted to take .

Similarly, we consider the decay of , which is -parity violating process. If the isospin were not broken, such process would not exist. The experimental value of is reproduced for . We calculated from is the final state pion momentum. The ambiguity of the sign has been resolved recently through the decays of produced in [12], in which the constructive interference has been supported. , where

There are essentially five free parameters from in (4. 5), because is negligible. We determine these parameters by fitting, using the data of the radiative vector meson decays and . Then we obtain , , , and .

We take from , and [2], and from [2]. Then we obtained the results listed in Table I.

In Table I, (i)(iii) mean:

These parameters suggest that isospin/-breaking effects for the anomalous sector cannot be given by the quark mass matrix in a simple manner. We discuss this point later.

In the previous paper[9], we determined a parameter region so as to reproduce all experimental data of radiative vector meson decays as , , , and . The parameters by -fitting are slightly different from the above region, which then yield the prediction of from the experimental value. However, the difference of the former maximum value from the latter minimum value is about . If we consider that the experimental data[2] is determined only by Ref.[12], where large momentum transfer events have been selected in order to eliminate - interference contribution, our result is not inconsistent with the experiments. In fact, if we take the experimental value of Ref.[14], where the decay width of has been reported as keV based on the assumption of constructive - interference, our prediction by -fitting also reproduces the experimental value. ,

The results for , and in Table I suggest that isospin breaking terms are very important. Both (i) and (ii) in Table I do not have isospin breaking terms. These values differ substantially from the experiments, which cannot be absorbed by the ambiguity of the hidden-gauge coupling whose value is determined either by or by . In order to avoid this ambiguity, let us take some expressions cancelling , i.e. , and . Then we find that predictions of the original and Bramon et al.[8] still do not agree with the experiments. These Lagrangians without isospin breaking terms yield

Finally, we pay attention to , which are given by

(4. 9) | |||||

(4. 10) | |||||