LU TP 97/03

NORDITA-97/22 N/P

hep-ph/9704212

March 1997 (revised July 1997)

Vector Meson Masses in Chiral Perturbation Theory

[2cm] J. Bijnens, P. Gosdzinsky and P. Talavera

[1cm] Department of Theoretical Physics, University of Lund

Sölvegatan 14A, S-22362 Lund, Sweden.

[0.5cm] NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark.

PACS: 12.39.Fe, 12.40.Yx, 12.15.Ff, 14.40.-n

We discuss the vector meson masses within the context of Chiral Perturbation Theory performing an expansion in terms of the momenta, quark masses and . We extend the previous analysis to include isospin breaking effects and also include up to order . We discuss vector meson chiral perturbation theory in some detail and present a derivation from a relativistic lagrangian. The unknown coefficients are estimated in various ways. We also discuss the relevance of electromagnetic corrections and the implications of the present calculation for the determination of quark masses.

## 1 Introduction

The relevance of vector meson masses and mixings in the determination of the quark masses is well known. A review is in [1]. Virtually all the information used in that paper was at the lowest order in Chiral Perturbation Theory and at most the leading nonanalytic corrections are included. The main parts of the information used for quark mass ratios were pseudoscalar, baryon and vector meson masses and mixings. The full one-loop corrections to the pseudoscalar meson masses and mixings was performed in [2] and a recent review on their relevance to quark mass ratios is in [3]. The subject of baryon masses in chiral perturbation theory is a subject of a lot of current research. For recent results see [4] and references therein. This paper is devoted to a similar study for the vector meson masses.

Chiral Perturbation Theory for Vector Mesons using a heavy meson formalism was first introduced in [5]. We will refer to this as the Heavy Meson Effective Theory (HMET). There the first correction of order was evaluated. This was later extended to include isospin breaking effects in [6]. In that reference an estimate of the electromagnetic effects using a short-long-distance matching calculation was also performed. The result for the quark masses from mixing agreed well with standard expectations, but the – mass difference did not. The size of the chiral corrections of order is rather large so a more complete calculation seemed necessary. We therefore perform in this paper the full vector meson mass and mixing corrections to order . We also discuss the electromagnetic corrections.

The paper is organized as follows. In Sect. 2 we give the main notation and conventions. In the next section we explain the HMET and list all of the terms necessary for the calculation of the vector meson masses. Then in Sect. 4 we explain the connection between the relativistic descriptions and the HMET. In Sect. 5 we then give the model estimates for the various parameters in the HMET, followed in Sect. 6 by a description of the calculation of the main result in this paper, the vector meson masses to order . We then discuss how we include the electromagnetic correction in Sect. 7 and the numerical results in Sect. 8. We then give the main conclusions and the implications for ratios of quark masses in Sect. 9.

## 2 Basic Ingredients

In this section we put some of the basic notation. The notation is essentially the same as in [7] and [8].

The pseudo-Goldstone boson fields can be written as a special unitary matrix

(1) |

where

(2) |

and MeV.

The relativistic vector meson fields are introduced as a nonet matrix
(in the limit)

(3) |

In what follows denotes trace of . We also use the quantities

(4) |

In (4), we have ignored the vector and axial external sources. We will also set the pseudoscalar source, , equal to zero and will be the diagonal matrix . The parameter is given by

(5) |

The transformations under a chiral symmetry transformation are:

(6) |

Eq. (2) also serves as the definition of , which depends on , and .

## 3 Vector Meson Chiral Perturbation Theory

The Chiral Corrections we are interested in are long distance corrections
to a propagating ‘‘bare’’ vector meson^{1}^{1}1A similar formalism is of
course possible for axial vector mesons. The relevance for quark masses is
much smaller, since masses and mixings are much less well known in this sector.
We can therefore calculate these corrections in a systematic fashion. We
cannot in a similar way calculate the chiral corrections to vector meson
decays of the kind: since that is inherently shorter distance.
“Hard” loop corrections
to the vector meson masses will be included in the constants in the
effective lagrangians we will write. An example of this already at tree
level will be given in Sect. 4.

It will also be nice to have explicit power counting present in the Lagrangian. We therefore use a formalism similar to the Heavy Quark Effective Theory[9] and the one used for Baryons[10, 11]. The main part of the mass, the vector meson mass in the chiral limit, we will remove explicitly by introducing a velocity with and the chiral mass . Vector momenta are then treated as residual momenta compared to . I.e. and we only refer to in the space time dependence. We introduce the matrix for the effective vectors:

(7) |

Under the chiral symmetry group transforms as with defined in Eq. (2). The fields in (7) are to be understood as only containing annihilation operators. The creation operators are contained in . In order to have a proper spin one field we impose the condition

(8) |

This condition is enforced by putting in the Lagrangian always the combination and its Hermitian conjugate. Here

(9) |

In the remainder this projector should be understood. For the calculation presented in this paper it can be forgotten, since the component never contributes in any of the diagrams.

Let us now proceed to construct the relevant Lagrangians. The terms at order are:

(10) |

The first term, though allowed by the symmetry, can be removed from the action through the choice of the reference momentum. The second term is small because of Zweig’s rule, we will therefore treat it as the same order as the quark masses, . The term at order is:

(11) | |||||

For the purpose of counting, for constants in the lagrangian that are suppressed by , we will count them as order . So the coefficient needs to be kept in the loop diagrams but not and . Similarly, for the Lagrangian below, the suppressed coefficients appearing in loops we neglect.

For the higher orders we can use the lowest order equation of motion to remove terms. So we will never encounter and . We will also not explicitly mention terms that vanish as the external vector and axial vector fields vanish. These never contribute to the masses. In addition, since we treat as a small quantity the effects of this term can be reabsorbed in the other parameters by the field redefinition

(12) |

The field still transforms under the chiral group in the same way as . In fact the remainder can also be removed by slightly changing the coefficient in (12).

At order there are several terms. First there are those whose coefficients are fixed by Lorentz-invariance. At the level of the HMET Lagrangian these constraints can be directly derived using reparametrization invariance[12]. The Lagrangian at order is then

(13) | |||||

In addition to the terms leading in of Eq. (13) we also have

(14) | |||||

contributing to the vector masses directly. The terms of Eqs. (10), (11) and (13) are all those that will appear within loops in the approximations used here.

The terms fixed by reparametrization invariance are:

(15) |

We have not performed a full classification of the and terms, this is work in progress, but only of those that can contribute to the vector meson masses. We have shown the subleading ones in . These are all the terms needed to absorb the infinities occurring in the present calculation.

There are no terms at tree level of order that contribute to the vector masses. The argument is as follows: In order to have , we need three derivatives or one insertion of quark masses and one derivative. To have an invariant term contributing to the masses we also need a , and Lorenz invariance requires an extra . If the is contracted with the vectors it vanishes because . If is contracted with a derivative, it has to act on one of the vectors. This term then vanishes by the equations of motion as mentioned above.

At we have four terms contributing to the masses:

(16) | |||||

The last two terms in fact do not contribute to the masses at order . The reason is that the external momentum is proportional to a mass difference that is itself of order . So to the masses these terms only contribute to order . For a similar reason there are no contributions from terms with four derivatives.

The suppressed terms are:

(17) | |||||

## 4 Relation to Relativistic Vector Meson Lagrangians

The usual parametrizations of vector mesons in chiral Lagrangians is done in a relativistic formalism. There are various popular versions of this. A review of some of them exists in [13]. A discussion relevant to the relation between the different ways of parametrizing them can be found in [7]. The relation in the context of the functional integral approach can be found in [14].

Let us first treat the case of a single noninteracting Vector Meson. The relevant Lagrangian is:

(18) |

with . This Lagrangian produces three propagating modes and one constraint equation. The latter makes the non relativistic limit a little more tricky than just a naive identification. We define first a parallel and perpendicular component of the vector field with respect to the velocity :

(19) |

This is similar to the Heavy Quark Effective Theory where a projector is introduced[9]. The Lagrangian of (18) then becomes

(20) | |||||

We can now split both and in its creation and annihilation parts (keeping in mind the projector acting on the effective field):

(21) |

We can then identify where the dots are a small quantity. We also assume that the residual momentum dependence described by , and Hermitian conjugates, is small compared to . This is where we restrict to the one vector meson sector.

The only terms in the action that are still proportional to is . So, in this limit we precisely have or the constraint we assumed in the previous section . The symbol used in the previous section always included the projector of (9); i.e. it is . One could also alternatively integrate out the component in a functional integral language. Here we will just remove it using the equations of motion:

(22) |

This equation can simply be solved iteratively. If applied to the kinetic terms this leads to the constraints given in Eq. (15).

In practice we can choose slightly different values of to do the reduction to the HMET. This is precisely the freedom that leads to the presence of the term proportional to in Eq. (10).

An alternative procedure, that is in practice somewhat easier to implement, is to use tree level matching. We calculate certain processes with the relevant kinematics both in the relativistic formulation and in the HMET formulation. We then determine the HMET constants in terms of the constants in the relativistic Lagrangian by requiring equality between the two formulations. Here one has to watch out somewhat. The equivalence is at the level of -matrix elements so when comparing Green functions we can only easily compare for the perpendicular components and one should remember the extra factors of because the relativistic and HMET fields are differently normalized.

## 5 Coefficients in the Lagrangian

The total number of parameters that contribute to order to the vector meson masses is very large. It is obviously too large to be fitted simply from the data on the masses alone. In our numerical results we will therefore use several estimates of the parameters.

The most important ones are the extra assumptions used. Here we
add a suppression factor of for each factor
of . The loops themselves are in principle also suppressed by factors
of . But large logarithms and combinatorial factors can in
practice make up for this extra loop suppression. We therefore still take
the loops into account.
For the parameters this means:^{2}^{2}2As mentioned earlier,
derivatives on external vector field count as for the calculation
of the masses.

(23) |

Here equal to zero means that they do not contribute to the masses when the expression is taken into account. should be included inside loops and at tree level. is not physically relevant as described earlier. and only contribute at tree level. is like a vector meson sigma term. On the masses it only contributes just like and is thus of no relevance here. In the end there are two possibly relevant parameters, and .

First the terms. Here we only need . It was estimated in [5] using the chiral quark model with a value of . Assuming , this changes to . In [6] the value of was fixed using a VMD argument for , this yielded in reasonable agreement with the previous estimate. A double VMD estimate from leads to essentially the same result. The ENJL prediction for this vertex[15] is lower . Fitting to , and , together with various assumptions about VMD and the KSRF relation, leads also to values roughly within the above range. We will discuss the implications for the vector meson masses of these and other values in Sec. 8.

The terms with more derivatives and vector fields only are as described earlier fully determined by Lorenz invariance, and . never contributes to the masses.

The interaction terms are more difficult to obtain. In [7] a good description of Vector meson phenomenology is given by the model III:

(24) |

Here transforms nonlinearly under the chiral group as and

(25) |

In this model there are three diagrams possibly contributing to scattering. These are depicted in Fig. 1. The effects of these have to be described by the pointlike interaction in the HMET. We have chosen this model because most properties of Vectors and Axial vectors are well described by the Yang-Mills type models. The type “hard” contributions of Fig. 1a cannot be simply described using equations of motion as was done in the previous section. For these we use the matching procedure. The internal propagator can both be a pseudoscalar or an axial vector. Similarly in Fig. 1c the vector in the vertical line has a small momentum, not a small deviation from . Diagram Fig. 1c reproduces the pion interactions from the kinetic term and the term of the HMET in addition to:

(26) |

This doesn’t contribute to the masses. The diagram of Fig. 1a is reproduced in the HMET via

(27) |

Higher momentum dependences of the scattering can be produced by the axial vector intermediate states. The vertex in this model can be described by the two terms

(28) |

in addition to the kinetic terms for the axial nonet. The field transforms here as in Eq. (2). In the HMET they can be described via

(29) |

The coefficients and should then be chosen to reproduce the rather uncertain width.

The above model did not contain much internal vector meson properties. As an example of a model that does have some internal structure we will use the ENJL model. For a review see [18]. The coefficients needed here were obtained in [8] and [15]. In order to obtain the from there we have to perform the HMET reduction. We also have to include the diagrams of Fig. 1, keeping in mind that in [8, 15] a different vector representation was used where the vector-pseudoscalar-pseudoscalar vertex contains three derivatives. What it in the end corresponds to is that we have the above contributions in terms of and ; and in addition terms coming from the pointlike relativistic contribution of Fig. 1b. These extra terms are

(30) |

The symbols , , and are defined and their values in terms of the ENJL parameters given in [15]. For the contributions to the masses, and only appear in the combination . To obtain the numerical values we can use the ENJL relations:

(31) |

The remaining constants to be estimated are , and . For these we will use an ENJL model estimate. The Yang-Mills like models normally just assume these terms to be zero. These are the vector meson masses in the large limit. We could in principle determine those from the measured meson masses but we will here also produce an ENJL calculation of them. In the ENJL model the two-point functions were well described by a VMD picture, see [18, 17]. We will therefore do the same thing. We will fit the two-point function to . We do this fit in the Euclidean region to remove the artifacts due to non-confinement in the ENJL model. We observe that a good fit can be obtained with independent of the quark masses and will use the fitted to obtain the coefficients , and . Notice that we fit the squared of the mass and not the mass itself, like one should expect in an effective theory. With this we take into account the resummation of the terms suppressed inside these coefficients. The values chosen for the ENJL parameters are , and which gives a good fit to a large number of hadronic parameters. We have also used the way suggested in [17] to determine the vector masses. This is another recipe to remove the pole at in . The differences are within the quoted errors. We have also done the fit using various ranges of the quark masses and various ranges of . The results are all within the quoted values. We will use the values

(32) |

The vector mass in the chiral limit obtained from the ENJL model with the
above parameters is about . The model uncertainty on these
estimates increases the error on to a somewhat larger value.

## 6 Determination of the Masses

The method we will use to determine the masses is to compute the inverse propagator and then we look for its zeros. As mentioned earlier we perform the calculation fully in the HMET. We are free to choose the external momentum in a way that simplifies the calculation. Our choice is that the external momentum is always proportional to the chosen velocity . This together with the constraints removes a lot of irrelevant contributions. A disadvantage of the HMET are that there are more possible diagrams, corresponding to the extra operators with fixed coefficients. In the relativistic formulation we only have two possible diagrams. Then we have no simple powercounting because the large mass appears in the nonanalytic parts. In the nonrelativistic case we have more diagrams. These are depicted in in Fig. 2.

As argued before we only have to include the effects of the terms up to inside the loops. In practice this means that we have to diagonalize the masses including the terms proportional to and inside the sunrise type diagrams, depicted in the first three diagrams of Fig. 2. We also diagonalize the lowest order pseudoscalar mass term to take mixing into account.

We proceed in the following way. We define a set of external vector fields with that corresponds to the usual isospin basis. We then define a 2nd set of “internal” vector fields that diagonalize the lowest order mass terms, i.e. the , , and (see next Section) terms. Similarly we define a set of pseudoscalar meson fields that diagonalize their lowest order mass term, .

In terms of these the sunrise diagrams depicted the first three diagrams in Fig. 2 as a function of the incoming flavour and outgoing vector with residual momentum can be described via:

(33) |

The function is defined as follows:

(34) |

can be easily constructed from the integrals given in App. A. is the vector meson mass minus the chiral limit value, i.e. . is the mass of the pseudoscalar. We have disregarded in these expressions terms of higher orders found in the integrals, i.e and , their numerical effects are found to be small and mainly affects de . For the we actually find a second zero in the inverse propagator for some of the parameter sets. Adding the omitted higher order terms in the sunrise diagrams moves this extra zero beyond the range of validity of CHPT as discussed in Sect. 8. These extra terms correspond to replacing in the last term of , Eq. (34), by .

The tadpole contributions, last diagram in Fig. 2 can be given by

(35) | |||||

The traces can be easily performed but lead to rather lengthy expression. In App. B we have given the expressions in the limit where internal lines are the nonmixed states. The numerical difference, especially the part due to the mixing, is rather small, less than half an MeV for all masses.

To equations (34) and (35) we still have to add all tree level contributions to obtain the inverse propagator to order . The corrections to the masses of the are given by values of for which the corresponding amputated two point function vanishes. For the the situation is slightly more complicated, because these mix with each other. Here the corrections to the masses are given by the values of for which the determinant of the matrix of the amputated two point function vanishes. Once we have found the three solutions, we have to identify the particle to which they correspond. This can be done by analyzing the null eigenvectors at the value of where the determinant vanishes. These null eigenvectors define the physical basis for the and , and they do not necessarily form an orthogonal set since the physical states appear at different values of .

## 7 Electromagnetic contributions to the Masses

The subject of the electromagnetic contributions to masses is in fact very old, an early attempt is in [19]. In [6] a matching calculation was performed. We will use the results of that paper here. The main electromagnetic effects can be described by the three terms

(36) |

with as numerical values[6]

(37) |

The numerical results from [20] cannot be easily compared. The model used there makes rather drastic assumptions about the high energy behaviour of the various formfactors needed. The bad matching found in [6] should however be kept in mind. There are several effects included in [19, 20] that in the language of [6] are higher order. The main effect of these is the intermediate state. The effect of this we model in the present work by including the term in the diagonalization used in the previous section and by using the measured vector meson masses inside the loops.

Another possibly large effect is the electromagnetic part of the pseudoscalar
meson masses in the loops. This is in fact the leading quark mass effect
in the electromagnetic corrections[6]. Here it is included
by using the physical pseudoscalar masses within the loops^{3}^{3}3The
effects on the mixings of pseudoscalars is higher order and we neglect it..

## 8 Numerical Results

As a general inputs we choose MeV, MeV and as well as . These masses are compatible with the small deviation from the Gell-Mann-Oakes-Renner relation and the quark mass determined in [22]. The other values are found via the ratios quoted in [3]. We also take the coefficients given in Eqs. (5), (37) as fixed parameters. We also fix the mass appearing in Eqs. (29) via and choose as a subtraction point GeV. That leaves 7 parameters to vary, , , , , , and . We evaluate the masses in different scenarios:

Scenario , we use the ENJL estimates for , , , and the set of parameters given in Eq. (32). We also use MeV and [6], with . We found a large mixing with a large dependence. There is also a very large negative correction to the -mass resulting in the being the lightest state. Most of these problems are caused by the very large contribution from the sunrise diagrams. We can use and to try to solve these problems. would lead to a very large mass difference so we will use to cancel the mixing. Requiring the mixing to be small in a reasonable range of leads to GeV.

A very similar result can be obtained with as suggested by the ENJL model. It has the masses somewhat better than scenario I but there is no qualitative improvement, we therefore do not discuss it further.

This result is not in good agreement with the experimental data as shown in Table 2. We should keep in mind that the values for , , and we have used are rough estimates, and that MeV is a tree level value which can receive corrections mainly from the sunrise diagram at .

In the remainder we therefore use a least squares fit to determine viable sets of parameters. We have minimized the following function:

(38) | |||||

The masses should be understood here as the difference of the CHPT calculation and the measured values. All data are from Ref. [21] except for the mass difference[23]. Here we have chosen the errors so that the quantities mainly determined by are allowed a larger variation. We will then have to check afterwards if the found sets have a reasonable convergence in the various orders.

Scenario | I | II | III | IV | V | VI |

(GeV) | 0.7685 | 0.84304 | 0.85801 | 0.84250 | 0.74516 | 0.75935 |

(GeV) | 0.0065 | 0.00578 | 0.00756 | 0.00581 | 0.0195 | 0.0198 |

(GeV) | 0.15 | 0.16371 | 0.12484 | 0.16926 | 0.12166 | 0.11990 |

(GeV) | 0.08 | 0.13582 | 0.15531 | 0.13718 | 0.17327 | 0.17307 |

(GeV) | 0.06 | 0.12964 | 0.08406 | 0.13457 | 0.0373 | 0.03873 |

(GeV) | 0.16 | 0.10284 | 0.10070 | 0.10279 | 0.01937 | 0.00937 |

0.32 | 0.25749 | 0.26507 | 0.25863 | 0.11926 | 0.14352 |

Scenario | I | II | III | IV | V | VI |

(GeV) | 0.1666 | .00001 | 0.0001 | - | - | - |

(GeV) | 0.1658 | +0.0009 | 0.0023 | 0.0004 | 0.0003 | - |

(GeV) | 0.1708 | +0.0009 | 0.0025 | 0.0004 | 0.0003 | - |

(GeV) | 0.1665 | 0.0005 | 0.0001 | 0.0005 | 0.0001 | - |

(GeV) | 0.1638 | - | 0.0001 | 0.0005 | 0.0001 | - |

(GeV) | 0.3177 | - | 0.0003 | 0.0009 | +0.0003 | +0.0001 |

() (GeV) | 0.0023 | 0.0024 | 0.0016 | 0.0025 | 0.0016 | 0.0014 |

() (GeV) | +0.0004 | +0.0009 | +0.0008 | +0.0008 | +0.0006 | +0.0005 |

() (GeV) | 0.035 | 0.042 | 0.048 | 0.043 | 0.071 | 0.072 |

() (GeV) | 0.056 | +0.015 | +0.009 | +0.011 | 0.055 | 0.050 |

18 | 0.49 | 0.19 | 0.52 | 0.06 | 0.017 |

We will present five different fits, all of which are acceptable. The input parameters can be found in Table 1 and the resulting deviation from the observed masses in Table 2. In that table we also present the predictions for the various mixings. These are evaluated as the two-point function in the isospin basis at the relevant value of the momentum. Notice that in most fits these were not input so they present genuine predictions. The mixing has a large dependence. We have given it both at the -mass and the -mass. The mixing is more stable and we quote it only at the -mass. For definiteness we have quoted the mixing at the -mass. Using the -mass instead is not visible with the precision quoted.

Scenario II is a minimum found with a fairly large value of . All masses can be fit within the errors assumed in Eq. (38). All parameters are of the expected order of magnitude here, the main change is the flip in sign of . As mentioned earlier, there is a second spurious zero in the determinant of the inverse two-point functions of the neutral states about 60 MeV above the -mass. In order to check the sensitivity to higher orders of this phenomenon and the possible variations of the parameters we have also performed a fit with the higher order parts of the sunrise diagrams included. This pushes the spurious zero to about 230 MeV above the -mass and varies the input parameters within less than one standard deviation from the previous fit. That case is in the tables as scenario III.

A much better minimum can be obtained for a value of around . This is quoted in scenario V and VI. The latter is again with the higher order parts of the sunrise diagrams included.

As can be easily seen we obtain reasonable agreement with the observed values for all the mixings as derived in [6]. These were:

(39) |

We should keep in mind that the latter two have possibly large final state corrections.

The agreement with the mixing in scenario III is not so good. We have therefore added to Eq. (38) also

(40) |

Including this then gave a minimum within the errors expected of the input parameters. This we have given as scenario IV.

We have not discussed the dependence in the two-point function because this can be absorbed in a change in the parameters in principle if we include the -suppressed terms as well. In practice we have to fit the parameters anyway. We have checked that putting GeV instead we can get as good as a fit as scenario III and IV with similar changes in the input parameters.

To see the convergence of the series for the various quantities we show in Table 3 the various contributions corresponding to scenario IV. They are ordered by the order of the contributions. The sunrise diagrams contain pieces of order and higher. The term is a term with two derivatives but as explained earlier these count as for the calculation of the vector masses. For the masses of the neutral states we have quoted the various contributions to the unmixed inverse two-point function at the correct mass. The effect of the mixing on the masses can be seen by comparing the total line to the real masses as obtained in Table 2 for scenario IV.

12.3 | 154.5 | 158.0 | 12.3 | 12.3 | 300.1 | 3.55 | 0 | 0 | |

0 | 0 | 0 | 0 | 11.6 | 5.8 | 0 | 0 | 8.2 | |

Sun | 140.8 | 126.2 | 127.2 | 140.8 | 143.6 | 329.5 | 3.26 | 2.6 | 90.3 |

-tad | 26.3 | 53.2 | 53.6 | 26.3 | 26.3 | 80.5 | 0.44 | 0 | 0 |

-tad | 30.1 | 70.7 | 71.7 | 30.1 | 29.0 | 58.1 | 0.28 | 0.4 | 41.1 |

0 | 0 | 0 | 0 | 15.0 | 182.3 | 2.16 | 1.5 | 134.2 | |

0.2 | 3.2 | 5.7 | 0.2 | 0.2 | 107.8 | 0.10 | 0 | 0 | |

0.4 | 105.9 | 106.1 | 0.4 | 0.4 | 211.6 | 0.20 | 0 | 0 | |

3.2 | 1.5 | 1.7 | 3.3 | 2.2 | 18.4 | 0 | 0 | 0 | |

em | 1.5 | 1.5 | 0.06 | 1.5 | 0.3 | 0.4 | 0.54 | 0.6 | 0.2 |

Total | 74.0 | 49.5 | 54.0 | 74.0 | 51.4 | 175.5 | 2.53 | 0.8 | 10.8 |

As can be seen the convergence for most quantities is acceptable but the -mass is very slowly converging.

## 9 Conclusions

We have in this paper extended the calculation of the vector meson masses in CHPT beyond [5] and included also the isospin breaking effects. The problem that the