# Isospin Symmetry Breaking

Through - - Mixing

###### Abstract

Mixing of the pseudoscalar mesons is discussed in the quark-flavor basis. The divergences of the axial vector currents which embody the axial vector anomaly, combined with the assumption that the decay constants follow the pattern of particle state mixing in that basis, determine the mixing parameters for given masses of the physical mesons. These mixing parameters are compared with results from other work in some detail. Phenomenological applications of the quark-flavor mixing scheme are presented with particular interest focussed on isospin symmetry breaking which is generated by means of and admixtures to the pion. Consequences of a possible difference in the basis decay constants and for the strength of isospin symmetry breaking are also examined.

PACS Nos.: 12.38.Aw, 13.25.Gv, 14.40.Aq

## 1 Introduction

That the character of the approximate flavour symmetry of QCD is determined by the pattern of the quark masses is a well-known fact that has been extensively discussed in the literature for decades. Isospin symmetry in particular which would be exact for identical and quark masses, , holds to a rather high degree of accuracy empirically, although the ratio is about 1/3 and not, as one would expect for a true symmetry, much smaller than unity. Violations of isospin symmetry for pseudoscalar mesons within QCD are generated by admixtures of the and to the pion. On exploiting the divergences of the axial vector currents but neglecting mixing, Gross, Treiman and Wilczek [1] obtained for the mixing angle between the pion and the flavor-octet state

(1) |

a result that also follows from lowest order chiral perturbation theory [2]. We learn from (1) that due to the effect of the U(1) anomaly which is embodied in the divergences of the axial vector currents, isospin symmetry breaking (ISB) is of the order of instead of the expected . Isospin symmetry is thus partially restored and amounts to only a few percent for pseudoscalar mesons. It is therefore to be interpreted rather as an accidental symmetry which comes about as a consequence of the dynamics. For hadrons other than pseudoscalar mesons the strength of ISB is not necessarily set by the mass ratio (1); for comments on ISB in the vector meson sector see Ref. \refcitefritzsch.

Due to a number of recent experiments the interest in ISB has been renewed. It therefore seems opportune to review recent progress in our understanding of mixing. In an analysis of mixing [4] the quark-flavor basis has been used and assumed that the decay constants in that basis follow the pattern of particle state mixing. With the help of the divergences of the axial vector currents the basic parameters of that mixing scheme can be determined for given masses of the physical mesons. It has been found in Refs. \refciteFKS1 and \refciteFKS2 that this approach leads to consistent results and explains many empirical features of mixing. This mixing scheme will be presented in Sect. 2 and, in Sect. 3, the phenomenology of mixing briefly reviewed. Sect. 4 is devoted to a discussion of mixing in the flavor octet-singlet basis and to a detailed comparison with other recent work on mixing.

The inclusion of the in the quark-flavor mixing scheme is straightforward [5]. Here, in this work, mixing will be discussed in great detail and, as a new aspect, the role of possible differences in the decay constants and will be investigated. It is important to realize that the decay constants which represent wavefunctions at zero spatial quark-antiquark separation, are functions of the quark masses. Hence, ISB generated through is related to the quark-mass difference, too. It is only due to our inability of calculating the decay constants to a sufficient degree of accuracy at present that we have to consider them as independent soft parameters. ISB will be discussed in Sect. 5. The vacuum- matrix element of the topological charge density is calculated in Sect. 6 and phenomenological consequences are presented. Finally, in Sect. 7, isospin symmetry violating processes, which are not directly controlled by the U(1) anomaly, are examined. The conclusions are provided in Sect. 8.

## 2 Mixing in the quark-flavor basis

The quark-flavor basis is constructed by means of the states , which are understood to possess the parton compositions

(2) |

in a Fock expansion. Here denotes a (light-cone) wavefunction. The higher Fock states, whose presence are indicated by the ellipses, also include two-gluon components. Because of the fact that light-cone wave functions do not depend on the hadron’s momentum, one can define decay constants

(3) |

associated with the states . The variable denotes the usual (light-cone plus) momentum fraction the quark carries and its transverse momentum with respect to its parent meson’s momentum. The definition (3) is exact, only the component contributes to the decay constants. The basic decay constants are assumed to possess the property

(4) |

where denotes the
axial-vector current for quarks of flavor . Eqs. (2),
(3) and (4) form the basis of the quark-flavor
mixing scheme proposed in Ref. \refciteFKS1. This mixing scheme
holds to the extent that Okubo-Zweig-Iizuka (OZI)-rule violations,
except those mediated by the U(1) anomaly enhanced
transitions, can be neglected ^{1}^{1}1
OZI-rule violations are of order and become negligible in the
large limit.. Flavor symmetry
breaking, on the other hand, is large and is to be taken into account
in any mixing scheme for the pseudoscalar mesons.

Since mixing of the with the and is weak while mixing is strong it is appropriate to employ isoscalar and isovector combinations

(5) |

as basis states instead of and . The unitary matrix that transforms from the basis

(6) |

with . The sector of is identical to the parameterization utilized in Ref. \refciteFKS1. It is also of advantage to introduce isoscalar and isovector axial vector currents

(7) |

The matrix elements () then define new decay constants which can be collected in a decay matrix

(8) |

In contrast to (4), is non-diagonal. The decay constants and are the basic decay constants in the - sector while the parameter occurs in mixing; it is obviously of order . The decay constants are in principle renormalization scale dependent [6]. Ratios of decay constants or mixing angles are, on the other hand, scale independent. Since the anomalous dimension controlling the scale dependence of the decay constants is of order , this effect is tiny and discarded here. This simplification is consistent with the neglect of OZI-rule violations [7].

Taking vacuum-particle matrix elements of the current divergences, one finds with the help of (4), (6) and (8) (; )

(9) |

where is the particle mass matrix which appears necessarily quadratic here. Next, I recall the operator relation [8]

(10) |

which holds as a consequence of the U(1) anomaly. The topological charge density is given by where denotes the gluon field strength tensor and its dual. Inserting (10) into (9) and neglecting terms of order , one obtains a set of equations which can be solved for the mixing parameters

(11) |

and

(12) |

The pion’s mass and decay constant fix to more parameters

(13) |

up to corrections of order . Finally, the symmetry of the mass matrix forces relations between the decay constants and the matrix elements of the topological charge density

(14) |

The quark mass terms in the above equations are defined as matrix elements of the pseudoscalar currents

(15) |

The quark-flavor mixing scheme can readily be extended to the case of the [4].

## 3 mixing

The three relations (11), taken from Ref. \refciteuppsala, fix the mixing parameters for given masses of the physical mesons if the current algebra result

(16) |

is adopted for the strange quark mass term. This theoretical
estimate of the mixing parameters ^{2}^{2}2Note that the theoretical estimate presented
here differs slightly from the one presented in Ref. \refciteFKS1
where an additional constraint on has been employed.
provides the values quoted in Tab. 3.

In order to take full account of flavor symmetry breaking a phenomenological determination of the mixing parameters has also been attempted in Ref. \refciteFKS1 by using experimental data instead of the theoretical result (16). This analysis includes a number of processes like radiative transitions between light vector () and pseudoscalar () mesons or scattering processes like , which all rely on the validity of the OZI rule. The ratios of corresponding processes provide the mixing angle . Other dynamical effects are in general expected to cancel in the ratios, only phase space corrections have to be considered. An example is set by the radiative decays of the meson. These decays proceed through the emission of the photon from the strange quark and a subsequent transition into the component, see Fig. 1. Hence, the ratio of and decay widths reads

(17) |

where

(18) |

is the three momentum of the final state particles in the rest frame of the decaying meson . A small correction due to vector meson mixing [5] is to be taken into account here; the mixing angle amounts to . The PDG [10] average for the ratio (17) leads to a value of for the mixing angle. A new still preliminary result on that ratio from KLOE [11] provides an angle with an even smaller error namely . Both these values have not been used in the determination of the phenomenological mixing parameters quoted in Tab. 3.

Other reactions like the two-photon decays of the and [12] or the photon-meson transition form factors [12, 13] are sensitive to the decay constants. Thus, one finds for the two-photon decay width from PCAC [4]

(19) |

and for the transition form factor to next-to-leading order (NLO) of perturbative QCD and leading-twist accuracy (in the scheme) [13]

(20) |

where the effective decay constants read ()

(21) |

These processes fix the decay constants quite well [12]. In the form factor data there is a little hint at contributions from the two-gluon Fock component of the which occurs to NLO perturbative QCD [13].

The radiative decays of the into the and
provide another interesting piece of information ^{3}^{3}3
These are examples of OZI-rule violating decays mediated by the
U(1) anomaly..
According to Novikov et al [14] the photon is here emitted
from the charm quarks which subsequently annihilate into lighter quark
pairs through the effect of the U(1) anomaly (see Fig. 1). This mechanism leads to the following result
for the ratio of the decay widths

(22) |

Using Eqs. (6), (11) and (14), one can express the ratio of the vacuum-particle matrix elements of by the mixing angle

(23) |

a relation that has been derived by Ball et al [15] independently on the quark-flavor mixing scheme. From the PDG [10] averages for the radiative decays one obtains .

The analysis of these reactions and a number of others leads to the set of phenomenological mixing parameters quoted in Tab. 3. These values absorb corrections from higher orders of flavor symmetry breaking. As the comparison with the theoretical results reveals these effects are not large, they are on the level.

## 4 The octet-singlet basis

Transforming from the quark-flavor basis to the SU(3) octet-singlet one by an appropiate orthogonal matrix, one easily notices that the decay matrix in the octet-singlet basis has a more complex structure than (8) which is diagonal in the sector. In addition to the state mixing angle where the ideal mixing angle is , two more angles, and , are needed in order to parameterize the weak decays of a physical meson through either the action of a singlet or an octet axial vector current [6, 12]:

(24) |

The mesons may also decay through strange and non-strange axial vector currents. The corresponding decay constants can be parameterized in a similar fashion

(25) |

In the quark-flavor mixing scheme described in the preceding sections, the three angles are assumed fall together, , and, hence, the decay constants follow the pattern of state mixing. For the sake of comparison I will keep the three different angles for the remainder of this section. The two sets of decay constants are related to each other by

(26) |

and

(27) |

In the quark-flavor mixing scheme these relations simplify drastically:

(28) |

Depending on the strength of flavor symmetry breaking, embodied in the ratio , the mixing angles and may differ from each other and from the state mixing angle, , substantially. The angle is smaller than if . Only in the flavor symmetry limit, i.e. if , the three angles fall together.

Evaluating the mixing parameters in the octet-singlet basis from the theoretical and phenomenological sets of mixing parameters compiled in Tab. 3, one obtains the results shown in Tab. 4. Indeed the three mixing angles occuring in the octet-singlet basis, differ markedly. At the best mixing is simple in the quark-flavor basis which is favored because of the smallness of those OZI-rule violations which are not induced by the U(1) anomaly. On the other hand, SU(3) breaking is large and cannot be ignored.

It should be clear from the above discussion that any mixing scheme which takes into account the U(1) anomaly and includes the proper masses of the physical mesons will lead to similar results for the mixing parameters provided different values of the mixing angles, , and , are allowed for. This assertion is indeed confirmed by the results found in many papers on this subject. For a detailed comparison of various mixing schemes it is refered to the reviews [7, 9]. With regard to the limited space available for this work I refrain from repeating that and concentrate on the comparison with recent work in which particular attention is paid to the issue of the diverse mixing angles.

Results from two NLO chiral perturbation theory analyses [6, 16] are listed in Tab. 4 in which the meson is consistently incorporated by means of the large limit. The meson acts as a ninth Goldstone boson in this limit. For Ref. \refcitegoity where the experimental widths the two-photon decays into and are used as input in addition to the masses and decay constants of the physical mesons, the averages of their three fits results are quoted. In general fair agreement of the results is to be seen. Only the state mixing angle , quoted in Ref. \refcitegoity, seems to be too small in magnitude. It corresponds to with an uncertainty of about which is rather large as compared with the above quoted values obtained from radiative and decays.

De Fazio and Pennington [17] calculated the decay constants from QCD sum rules. The octet-singlet mixing parameters, evaluated through Eqs. (26) and (27), differ quite a bit from the results obtained within the quark-flavor mixing scheme. The errors in the sum rule analysis are however large. The typically amount to about for the mixing angles. Despite the large error the result on is in conflict with the experimental value on the ratio (22). Since the ratio of the matrix elements (23) can also be expressed as

(29) |

experiment [10] leads to directly.

Escribano and Frère [18] performed a phenomenological analysis of a subset of the data exploited in Ref. \refciteFKS1, namely , and , the vector mesons include the . Their fits also provide clear evidence for substantially different values of and , see Tab. 4. The large value of (along with a large ) they obtained, seems to be forced by the width [10] which has not been included in the analyses of Refs. \refciteFKS1,kaiser,goity and \refcitepenn.

The angle exhibits the largest uncertainties of the mixing parameters since none of the present data is really sensitive to it. Helpful in pinning down its value would be more and better data on the transition form factors or the form factors [5] which may for instance be obtained from the process in the central rapidity region [19].

Inspection of Tab. 4 reveals strong flavor symmetry
breaking effects as characterized by the large values of
. In contrast to this the
ratio which can be evaluated from
the information given in Tab. 4 with the help of the inverse
of Eqs. (26) and (27), is tiny; the values lie in the
range and, provided errors are available, are compatible
with zero ^{4}^{4}4
Admission of three mixing angles in the analysis
of the form factor leads to
[12]..
I.e. within the present accuracy of the data, the three mixing
angles , and fall together. This observation
strongly supports the central assumption (4) of the
quark-flavor mixing scheme. As the comparison with OZI-rule violating
terms in the chiral Lagrangian reveals [7]
implies that OZI-rule violations except those mediated by the
U(1) anomaly, are negligible. It is important to realize in
this context that the validity of the OZI rule is a prerequisite for
the existence of process-independent mixing parameters.

## 5 Isospin symmetry breaking

Having discussed mixing in some detail and shown that mixing is well understood, let me now turn to ISB induced by mixing. Defining the isospin-zero admixtures to the by

(30) |

one finds with the help of the matrix (6)

(31) |

The limits of these results, termed and in the following, coincide with the results reported in Ref. \refciteFKS2. The numerical value of the quark mass term in (31) may be estimated from the mass difference corrected for mass contributions of electromagnetic origin

(32) |

According to Dashen’s theorem [20], the electromagnetic
correction is given by ^{5}^{5}5
The QCD contribution to the mass difference [1]
is and amounts to about 0.2 MeV which is much smaller
than the experimental value of 4.6 MeV.

(33) |

in the chiral limit and amounts to . This leads to . However, finite quark masses increase substantially. The exact size of this enhancement is subject to controversy. Different authors [21] obtained rather different values for the electromagnetic mass splitting of the K mesons. For an estimate of the values of the and admixtures to the I take the average of the results for quoted in Ref. \refcitedonoghue (). This way I obtain

(34) |

Due to the electromagnetic mass corrections the value for is larger than that one quoted in Refs. \refciteFKS2,feld and \refciteuppsala.

It is elucidating to turn the masses occuring in (31) into quark masses. With the help of the Gell-Mann-Oakes-Renner relations [22] and the use of Dashen’s theorem (33) one finds

(35) |

with defined in (1). The additional factor of () would be unity if , i.e. if the physical and mesons are pure flavour octet and singlet states, respectively. In this case there is no mixing, i.e. . We now see that the small value of () obtained by Gross, Treiman and Wilczek [1], is a consequence of the disregard of mixing and the use of Dashen’s result (33) for the electromagnetic contribution to the kaon mass splitting.

Chao [23] has also investigated mixing on exploiting the axial anomaly but he has used the PCAC hypothesis instead of diagonalizing the mass matrix. Despite his assumption of equal mixing angles, , a supposition that has been shown to be inadequate and theroretically inconsistent (see Sect. 4), his results on and agree with the values (34) within errors. NLO chiral perturbation theory in the large limit [16] leads to a slightly smaller value for the mixing angle () than (34).

If the decay constants and differ from each other the mixing angles may deviate from the values quoted in (34) substantially. This potentially large effect is a source of considerable theoretical uncertainty of our understanding of ISB in the pseudoscalar meson sector. In the absence of theoretical estimates for one has to rely on phenomenological estimates of its value. Even this difficult as will turn out in Sect. 7.

## 6 The vacuum- matrix element of the topological charge density

ISB as a consequence of mixing is accompanied by a non-zero vacuum- matrix element of . From the mixing formulas derived in Sect. 2, one readily finds

(36) |

Any dependence cancels in this matrix element. As an immediate consequence of (36) the pion is contaminated by strange quarks

(37) |

This result implies a tiny violation of the OZI rule through the anomaly. As a consequence the can decay through the strange axial-vector current with a decay constant defined by . Using the operator relation (10) and Eqs. (36), (37), one obtains for this decay constant

(38) |

It is very small, about , and will likely have no experimental consequences. For numerical estimates of the quantities just introduced one may use (16) and the mixing parameters quoted in the preceding sections.

The decays allow for an immediate test of the prediction (36). As has been suggested in Ref. \refcitevol:80 these decays are dominated by a mechanism where the pseudoscalar meson is coupled to the system through the effect of the anomaly. Hence, the ratio of these processes is given by

(39) |

in analogy to (22). The ratio of the gluonic matrix elements follows from (36) and results presented in Sect. 2. It reads

(40) |

It is to be stressed that Eq. (40) includes mixing. It is responsible for the factor which amounts to about 1.7. This is so because the small mixing angle is enhanced by the large matrix element , see (29). The importance of mixing in Eq. (39) has been pointed out by Genz [25] long time ago. Using the recent, rather accurate BES measurement [26] of the ratio (39), one extracts the mixing angle

(41) |

which is large as compared to the theoretical estimate (34). The origin of this discrepancy is not understood. Neglected electromagnetic contributions to the transitions are likely not the reason [27]. A possible explanation could be that the coupling of the pseudoscalar mesons to the system is not or only partially controlled by the anomaly. This possibility has been suggested in Ref. \refcitecasa and discussed whithin the framework of the heavy quark effective theory. Since the strength of this new mechanism has not been predicted in Ref. \refcitecasa its bearing on the determination of cannot be estimated. A better understanding of the decay may also affect the determination of quark masses [29].

One may wonder whether the isospin-violating radiative decay of the into the is also under control of the U anomaly. This is, however, not the case in contrast to the and channels which are well described by the anomaly contribution. An estimate of the decay width using (40) and the analogue of (22), provides values that are too small by more than an order of magnitude. In fact the radiative transition into the is mediated by the higher order electromagnetic process and by a vector meson dominance contribution .

## 7 Phenomenological determination of the mixing angles

In this section ISB-violating processes which are not under control of particle-vacuum matrix elements of , will be discussed. The full mixing angle and not just its value, occurs under these circumstances. Care is to be taken that only ISB within QCD are analyzed, eventual electromagnetic effects have to be subtracted.

Clear signals for ISB and/or charge symmetry breaking have been observed in a number of hadronic reactions. The extraction of the mixing angle from these data is however difficult and model dependent. The ratio of the and deviates from unity, the charge symmetry result, experimentally [30]. On the basis of a rather simple model that includes state mixing and a number of corrections which take into accounts effects such as differences in the meson-nucleon coupling constants or the proton-neutron mass difference but ignores mixing with the , the authors of Ref. \refcitetippens:01 extracted a mixing angle of

(42) |

from their data. The non-zero forward-backward asymmetry in measured at TRIUMF [31] is in conflict with charge symmetry. The phenomenological analysis of this data suffers from large ambiguities. A combination of contributions from mixing, from the quark-mass difference and from electromagnetic effects in the nucleon-mass difference, controls the asymmetry [32]. A further complication arises from the fact that the mixing contribution is in fact given by a product of the mixing angle and the badly known -nucleon coupling constant [33]. This complicated situation prevents an extraction of the mixing angle with a significant accuracy. But the measured asymmetry in is compatible with the present knowledge of the various quantities occuring in the theoretical result. The cross section data [34] for have not yet completely been analyzed theoretically [35] while the COSY measurement of the ratio of the and cross sections provides only a very weak signal for ISB [36]. More precise data on the latter cross sections are required before one can draw a definite conclusion here.

The and decays into violate G-parity and hence isospin symmetry. Since electromagnetic contributions are strongly suppressed [37] ISB is here mainly of QCD origin and can be estimated through mixing. Recent analyses of the decays within NLO chiral perturbation theory [38, 39] can be summarized as

(43) |

where . This result matches the experimental value of if

(44) |

This value corresponds to the result obtained within the quark-flavor mixing scheme provided mixing is ignored. In fact (44) is a typical result of NLO chiral perturbation theory in which the is not considered as an explicit degree of freedom in the effective Lagrangian. Its effect is, however, partially embodied in the effective coupling constants of the theory. The NLO terms of chiral perturbation theory whith which a number of low energy processes can be calculated rather precisely, comprise contributions to ISB of various origin. Hence, the comparison with results from the quark-flavor mixing scheme which allows for an investigation of many processes at low and high energies, is not straightforward.

The ratio of the and decay widths can be written as [1]

(45) |

if electromagnetic contributions can be neglected. Coon et al [40] estimated the factor relying on PCAC ideas and taking into account the experimentally observed mild dependence of the process amplitudes on kinematics [41]. Their result on combined with the experimental value of for the ratio of the decay widths [10] leads to

(46) |

More accurate experimental data and a revision of the theoretical analysis is advisable. A calculation of the decays within the framework of chiral perturbation theory is not available.

There are several decays into vector and pseudoscalar mesons which violate isospin symmetry, e.g. . As compared to the corresponding isospin allowed decays, , they are typically suppressed in experiment by a factor of about in accord with expectations for an electromagnetic decay mechanism. Contributions from ISB mechanisms within QCD are negligible small. Indeed, within the mixing scenario, ratios like would be proportional to and would therefore amount to only . As is the case for the process the contribution from mixing to the radiative decay of the meson into the is negligible small; it involves the component of the (see Fig. 1) which is proportional to . The process proceeds through mechanisms similar to those occuring in .

Recently a new meson, the , has been observed [42, 43, 44]. Its experimental properties are consistent with a state interpretation. It decays into , no other decay channel have been observed as yet. This ISB decay likely proceeds through the production of the component of the pion and consequently its width is , see (6), (12). Estimates of the decay width [45] using chiral perturbation theory and the heavy quark effective theory provides values way below the present experimental upper bound for the total width. A second new meson which also decays into the isospin symmetry violating channel has been observed too [43, 44]. Once the decay widths of these two processes will be measured more precisely the mixing picture of ISB can again be probed.

Isospin violations within QCD are also of relevance in CP-violating processes whose analyses are intricate and do not allow a simple extraction of the mixing angles. An example is set by the decays [46]. In a recent analysis [47] of this process whithin NLO chiral perturbation theory the value (44) for the mixing angle has been used. The above comments on the comparison of results from this theory with others apply here as well. Another example of ISB effects in weak decays is the process . ISB spoils the triangle relation obeyed by the amplitudes for the three processes , and in the isospin symmetry limit and consequently affects the determination of the CKM angle [48, 49]. As estimated by Gardner [48] the mixing angles (34) lead to a correction of which is substantial but still smaller than the error of the present experimental result: [50].

## 8 Summary

A detailed theoretical and phenomenological analysis showed that the quark-flavor mixing scheme provides a consistent description of mixing. On exploiting the divergencies of the axial vector currents all basis mixing parameters can be determined for given masses of the physical mesons. It turned out that flavor symmetry breaking manifests itself differently in the mixing properties of states and decay constants, their parameterization requires three different mixing angles in general. Only in the quark-flavor basis the three angles fall together approximately and mixing is simple. Other approaches to mixing lead to similar results provided the U(1) anomaly and the masses of the physical mesons are taken into account.

Inclusion of the into the quark-flavor mixing scheme induces ISB of about on the amplitude level. Extraction of the mixing angle from experiment, on the other hand, provide values in the range 0.02 - 0.03. Thus, the exact magnitude of ISB through mixing, is not yet determined and more work is clearly needed. There are other sources of ISB within QCD besides mixing. Their estimate is however difficult and often not or only partially taken into account in analyses. One of these sources of ISB is a possible difference between the basic decay constants and or, respectively, the parameter in Eq. (14). From the results (42), (44), (46) there is no clear evidence for a non-zero value of . One may only conclude that is smaller than about 0.015. This entails a difference between and of less than .

Acknowledgements It is a pleasure to thank A. Bernstein, F. De Fazio, R. Escribano, W. Kluge, G. Nardulli and H. Neufeld for discussions.

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