HUSEFT R 199505 (for ZPC)
UNDERSTANDING THE SCALAR MESON NONET
NILS A. TÖRNQVIST
University of Helsinki, Research Institute for High Energy Physics
PB 9, Siltavuorenpenger 20, Fin00014 Helsinki, Finland
ABSTRACT
It is shown that one can fit the available data on the , , and mesons as a distorted nonet using very few (56) parameters and an improved version of the unitarized quark model. This includes all light twopseudoscalar thresholds, constraints from Adler zeroes, flavour symmetric couplings, unitarity and physically acceptable analyticity. The parameters include a bare or mass, an overall coupling constant, a cutoff and a strange quark mass of MeV, which is in accord with expectations from the quark model.
It is found that in particular for the and the component in the wave function is large, i.e., for a large fraction of the time the state is transformed into a virtual pair. This component, together with a similar component of for the , and and components for the , causes the substantial shift to a lower mass than what is naively expected from the component alone.
Mass, width and mixing parameters, including sheet and pole positions, of the four resonances are given, with a detailed pedagogigal discussion of their meaning.
1 Introduction.
As has often been stated in many reviews[1] our present understanding of the light meson mass spectrum is in a deplorable state, especially when one considers the vast amount of data that has been available already for quite some time, and that QCD in principle should solve the hadron spectrum. This is mainly because of the fact that the expectations of most ”QCD inspired quark models” fail so dramatically for the scalar mesons,  the Higgs bosons of the hadronic sector.
For the other nonets such as the , , and even the the naive quark model works reasonably well as a rough first approximation. Therefore, few authors doubt that they should be classified as states. Certainly, also here there are some ”second order effects” such as the observed deviations from ideally mixed states (i.e., mass splittings like or and mixing angles like or ), for which one has not yet reached consensus as to their full origin. Many authors believe these deviations are mainly due to gluonic intermediate states, while others[2], including myself, believe the dominant effects come from hadronic loops like etc.
But, for the lightest scalars the , , , and one has not even reached a clear consensus as to their true nature. Are some of these or bound states? Or is one of them possibily a glueball? Many authors today believe the and the to be bound states[1, 3]. This seems, at first, to be a natural assumption, since they lie just a little below the threshold. If so, the I=1 and the state must be sought for at higher masses. And indeed, there are now candidates for such states: the of Crystal barrel[4] (or possibly the questionable of GAMS[5]) for the I=1 state, and the LASS for the state. However, flavour symmetric couplings (which works rather well for the established nonets) would require[6] that their widths should be at least 500MeV, which is much larger than the observed widths of these candidates. Furthermore, these can have other interpretations (as radial excitations, mesonmeson bound states, glueball or threshold effects) like many of the other observed scalars in the much too overpopulated 13701720MeV region, where many I=0 candidates[7], , , do not find a place in the quark model. One of these could be a glueball, another a deuteronlike state (deuson)[9] etc. In order to reach a better conclusion of their true nature it would of course be very helpful if at least the lightest scalar nonet would be resolved.
It is fundamental also in many other respects to have a good model of the scalars, in particular for understanding chiral symmetry breaking, nuclear forces and of course confinement. For chiral symmetry breaking it would be important to know: Where is the sigma meson? Is it the , or the , or must its mass be pushed to infinity? How does the Nambu–JonaLasinio mechanism[10] work for the light spectrum? Is the pion both the NambuGoldstone boson and the I=1 meson, as most authors believe, or does one have to look for the pion at higher masses, as questioned e.g., by Georgi and Manohar[11].
For nuclear forces, and for the understanding of the enhancement in and decay[12], one would like to have a very light , in the range of 600900 MeV, coupling strongly to ; a meson which does not seem to exist. The lightest scalar is the , which behaves more like an or a state coupling weakly to , while the seems too heavy. In this paper I shall suggest a new solution to this old question.
Finally, for confinement the scalars play an important role in building through tadpole diagrams the hadronic bag, within which the quarks reside. They are important for the confinement energy, for the condensate, and for the difference between constituent and chiral quark masses. Thus, they are crucial for the understanding of all hadronic masses.
Many authors have recently studied some of the scalars we discuss here, but generally these have tried to fit at the same time only one or two of them and not the whole flavour nonet simultaneously. Bugg et al.[13] and Morgan and Pennington[14] have made detailed fits with many parameters to amplitudes using general techniques of unitarity, analyticity and the Kmatrix. Janssen et al.[15] studied the and in a meson exchange model for and interactions, Achasov[16] studied the low energy data also with meson exchange and Adler zero constraints. Kaminsky et al. [17] studied the and using a coupled channel model concluding that the is a ” molecule”. Earlier many authors[18] have discussed unitarity and analyticity for the hadron spectrum.
The results presented here is a very much improved calculation and discussion of a short letter[19] 14 years ago. In particular, I have included constraints from Adler zeroes, which considerably improve the agreement with data close to thresholds without increasing the number of parameters.
In the following I first discuss the general ingredients and properties of the unitarized quark model (UQM) in Sec.2. Although this section does contain new results and new material to resonance phenomena and the UQM, it can perhaps be skimmed by those who just want to understand the fits. It is written in a rather pedagogical way and emphasizes some facts, which are often forgotten by many model builders. Sec.3 is the central chapter of this paper, where the actual application and fits to the scalar nonet are presented while in the concluding remarks, Sec.4, some comments on these results are discussed.
2. The unitarized quark model.
The UQM incorporates unitarity and physically acceptable analyticity with resonances in a way, by which one maintains a simple and transparent physical interpretation of the introduced resonance parameters. It is a kind of advanced form of the classical work of Weisskopf and Wigner[20]. The UQM was applied to many different hadrons[21], and it can explain the signs[6] and magnitudes of deviations from ideal mixing, and many mass splittings. Particularily significant is the large splitting between and , which cannot be understood in single channel potential models (where the predicted splitting is over 50 MeV too small) nor by gluonic exchange. At present there is no other mechanism than hadronic shifts from the loops etc. [22], which can account for this large splitting.
2.1 General formulation In the UQM one writes for the partial wave amplitudes (PWA) in ”Argand units” a factorized matrix form:
(1) 
where one sums over the resonance indices and (for e.g., , etc.) and where and denote the twobody thresholds (e.g., etc.). The matrices include coupling constants (), phase space (and angular momentum) factors, and form factors. We write for real :
(2) 
where is the cm momentum of the two intermediate particles and . The form factors are at this stage still quite arbitrary, except for the fact that we shall require them to vanish sufficiently fast at (In the following we generally suppress the index on the , since in our application in this paper we assume this does not depend on the resonance). One expects them to be smooth functions of , which include angular momentum barriers, radial nodes, and in principle the left hand cuts. In the quark pair creation model[23] they are given by an overlap of the three hadronic wave functions multiplied by a matrix element for the pair creation. In fact, the include most of the model depencence of our scheme.
For resonances the propagator matrix depends on the bare mass parameters , and on the mass shifts , and the widthlike functions , which together determine the analytic vacuum polarization functions .
(3) 
The unitarity condition, takes a very simple form determining the imaginary part of :
(4) 
Since the functions are analytic functions with only right hand cuts, we can write dispersion relations for the real parts :
(5) 
These need no subtractions, since we require that the hadronic form factors make go to zero sufficiently fast at infinity because hadrons have finite size. Thus the integrals are finite and we need not add any polynomial to apart from the term, which we already included in in order to have the resonances as CDD[24] poles. By defining through the dispersion relation one automatically satisfies physically correct analytic properties, i.e., one gets no spurious poles nor cuts and right asymptotic behaviour. (See discussion in Sec. 2.7. below). Thus, once we have a model for the and the bare masses, the PWA can be calculated.
By having the functions in the inverse propagator one automatically sums over all iterated loop diagrams of the Born terms (see the diagrams in Fig. 1a,b), such that the amplitude includes an infinite set of diagrams of the form shown in Fig. 1c. In Fig. 1 we have included, in addition to the resonance terms also contact terms (Fig. 1b) to be discussed in subsection 2.5.
2.2 A single resonance
Some special cases are instructive. For a single resonance is onedimensional, and assuming real (as can quite generally be done for particle amplitudes) the PWA takes the form:
(6) 
One recovers a generalization of the familiar BreitWigner form in the second expression of eq.(6), where one has put . Here is the ”running squared mass”, which is given by the bare squared mass plus the generally negative mass shift . This function is approximately constant only in special situations, e.g., if one is far from all thresholds. For Swaves the dependence is particularily important, since has square root cusps at each threshold, and near such a threshold there is a dramatic dependence. See Fig. 2a,b where we display the running mass and for the and for the as obtained in the fit presented in the next section. Note in particular the strong cusp at the threshold. Since the lies sufficiently below this threshold the dependence of is here not so crucial as it is for the and for the resonances, which lie essentially right at the threshold. One sees from Fig. 2b that for the all three thresholds , , and lie close to each other. They all contribute to a large mass shift and a strong dependence in the running mass.
2.3 Two and more resonances
When one considers two resonances the propagator matrix becomes twodimensional and has important offdiagonal elements given by . In general, one gets an dependent complex mixing angle between the two states, since the mass and propagator matrices are diagonalized by a complex orthogonal matrix. This generates e.g., OZI violation although the bare states are assumed ideally mixed states.
A special, instructive case is obtained if one has only one threshold, or if the coupling constants of the two resonances are proportional to each other, . Such a proportionality can hold when the resonances have the same flavour content, say two ’s with flavour symmetric couplings. Then both bare states have couplings proportional to the same ClebschGordan coefficients, which are given by the general trace formulae:
(7) 
where stand for the three mesons involved at the vertex , and is the 3x3 flavour matrix for meson . Either the symmetric or the antisymmetric trace is to be taken, depending on the sign of the product the three charge conjugation quantum numbers. For the vertices, which we discuss in this paper it is thus the symmetric trace. Then after some algebra one can reduce^{*}^{*}*The result looks almost like magic if one starts from eq. (1), but it is easily found starting with the or even better with the matrix defined in Sec. 2.7. the expression (1) to:
(8) 
where
(9)  
(10)  
(11)  
(12) 
This shows explicitely how resonances must be ”added multiplicatively” because of unitarity. Wheras the single resonance form (6) gives one loop in the Argand diagram the two resonance form (8) gives two loops with a zero in the amplitude at . Note that the zero is unshifted, which could be phrased as a theorem: A zero in the PWA in the physical region remains a zero after unitarization.
I find it rather surprising that this quite simple generalization (8) of the BW formula to the case of two resonances is not found in the literature, although it could be very useful phenomenologically e.g., when studying two resonances in channels with nonzero flavour like , where eq. (9) can hold.
A special case of eq.(8) is also instructive, and this bears some resemblance to the ”S effect”, i.e. the discussed below: Imagine two nearly degenerate resonances both with large couplings to common channels. Unitarity will shift both masses and the phase shift will pass 90 for the first and 270 for the second resonance, close to each other in energy. Thus, one gets a narrow resonance structure in the form of a dip between two broad bumps. How can this this come about? In fact, when looking at the original form, eq.(1), the mass matrix in the propagator, when diagonalized [cf. next subsection 2.4, eqs.(1719)] results in a mixing between the two resonances, such that one of them nearly decouples from the thresholds, while the other gets very large couplings (cf. eq.(18) below). Thus in terms of eigenvalues one has one very broad and one very narrow resonance, the latter producing a strong dip in the broad resonance bump. Or in other words, two inherently broad resonances can together produce one very narrow one!
For the this mechanism is of course not the whole story. Here the couplings are not proportional to each other, i.e., they do not satisfy eq.(9), and the threshold plays a crucial role in bringing the two resonances close to each other, but loosely speaking a variant of this mechanism is operative. A somewhat similar phenomenon appears also for the two axial mesons, which are near mixtures of the and , (which belong to the and nonets) such that one of them nearly decouples from and the other decouples from (cf. Katz and Lipkin[2]). Also in the late sixties there were discussions of similar effects in the connection with split resonances[25] (in particular the now well forgotten ”split ”).
For the more general case with two (or more) resonances and many thresholds with different kinds of couplings, like and states, one does not gain much insight by trying to reduce the matrix form of the propagator of eq. (1) algebraically. The ”reduced” formula becomes quite complicated because of the energy dependent complex mixing induced. But, fortunately eq. (1) is as it stands already quite transparent physically, and easy to compute numerically.
2.4 BreitWigner masses and widths, pole positions and resonance mixings.
For the single resonance case eq.(6) the BreitWigner () mass is given by the value where vanishes or by
(14) 
while the widths could be defined as as in eq.(6). However, if the slope of at the resonance is large, one should correct for this and absorb the slope into the term by dividing both numerator and denominator in eq.(6) by the same term, and define the BW widths by
(15) 
This renormalization of the widths and couplings (which is also familiar in field theory) has a clear physical interpretation: Below each threshold the pair produces virtual pairs of the two mesons, and the probability for such pairs is proportional to . The wave function obtains mesonmeson components :
(16) 
Each component has in configuration space an exponential radial tail, whose slope is inversely proportional to how much below the threshold the state is. Thus for the and the size of the component grows both in spatial size and in absolute magnitude the closer to the threshold from below the resonance is. The reduction of the width and coupling in eq. (15) is physically due to the fact that only the component annihilates directly to respectively . The part is rather inert, since it must first transform into near the origin and then into or . Above the threshold the situation is very different; the component vanishes since it can simply fall apart, and gives an absorptive part to the wave function. The slope of is here generally positive implying that the BW width is in fact enhanced.
In the case when one has many resonances one can define the generalization of BW masses and widths by first diagonalizing the mass matrix and propagator by a complex orthogonal matrix , which also rotates the couplings which now become complex:
(17)  
(18)  
(19) 
One can then define the BW masses in the multiresonance case as the energies where vanishes. This definition has the advantage that this mass in principle is the same in all channels . We shall here use this definition. Other definitions, such as the energies where the phase of each term in the sum (19) is 90, or the energies where their absolute value of each term is maximal, do not have this property. But, the analogy with the single resonance case is not simple, because the coupling constants are now complex and energy dependent and furthermore, there is a background from other resonance tails. As an example the mixing angle between the and components in the or is complex and is strongly dependent. It has a different value at the than at , and furtermore it will be different when evaluated at the BW mass, than at the pole position for the same data and model.
The pole positions have the advantage that only here does the process factorize into production and decay independently of the background. These are determined by the complex value and sheet number, where the the whole inverse propagator vanishes, or more generally where vanishes. For each threshold the number of sheets is doubled, and the sheet number is determined by the signs of . The same resonance can, in general, have several image poles on different sheets, which considerably complicates matters since the same resonance can have more than one nearby pole (See Morgan and Pennington[14] and [26] for a discussion). In our model for the and we have 5 thresholds, which means there are 32 sheets, some of which could even have 2 poles each! To find the nearest poles one must analytically continue defined above at the first sheet only just above the real axis. At the first sheet we can calculate by generalizing eq.(5) to complex values on the first sheet () by the Cauchy integral around the cut on the real axis:
(20) 
This is discontinuous across the cut. To get to the second sheet () one must add a term:
(21) 
where is given by essentially the same expression as eq. (2) or eq.(4), but now without the theta function, and defined also for complex values of . Near the real axis has the opposite sign compared to in the imaginary part above threshold. Again below threshold, the cusp in the real part changes sign. Explicitly the additional term is given by:
(22) 
or with the Adler zero constraints and flavour symmetric couplings (See sec. 2.5):
(23)  
(24) 
In order to get to the third sheet () two terms from the first and second threshold must be added
(25) 
while to get to the fourth sheet one should only add the second term. In general, with the signs of given by the sheet number () one should add for each threshold with negative :
(26) 
2.5 Background and and channel exchange terms.
For the case of a nonresonant background term as given e.g., by a contact term (Fig. 1b) one still has an expression very similar to the one above, but with the terms in the propagator replaced by a constant, which could be chosen = 1. But, in order to have the same dimensions for the coupling constants of the background as for the resonances we put this constant = (with dimension GeV), and with the sign allowing for constructive or destructive interference. Then for the contact terms we have:
(27) 
The same constraints from unitarity and analyticity still apply as given by eqs. (2,4,5), i.e., is given by the same formulas as before although the interpretation of the coupling constants now refers to the background.
We can thus add a background term to each resonance by doubling the dimension for the propagator, such that for the case of one resonance and a background one writes a inverse propagator matrix
(28) 
The same formulas for still apply, but since has off diagonal terms the background mixes with the resonance in a complex although specified way just as in the two resonance case. An important special case appears when the background couplings are proportional to the resonance couplings, as is the case when one assumes the same flavour structure for resonance and background (). In this case when one algebraically reduces the dimension of the propagator matrix one gets the same resonance formula, but with a modified . With the same original for resonance and background one needs only to substitute . This new linear factor implies that there is a zero in the amplitudes, which we identify with the Adler zeroes near . Thus by adding the resonance and contact term with a definite relative weight one introduces the Adler zeroes[27] needed from current algebra. We let the Adler zeroes be at . All this is accomplished by the simple substitution:
(29)  
(30) 
where we now have introduced dimensionless coupling constants , which are related by flavour symmetry eq.(24).
One can of course in a similar way include more complicated dynamics coming from  and channel exchanges. By making a partial wave projection of the Born term one first determines the and the function, which should replace or above, which will contain logarithms etc. Then one sums the iterated higher order diagrams by the above unitary formalism.
2.6 The UQM when two pseudoscalars are produced by some other reaction.
Much of the physics of and systems can be learnt from other reactions than the twobody reactions discussed above. Such reactions are , or central production of in collisions. The UQM can easily be modified in order to be applicable also to such processes. All one needs to do (provided the strong interactions in the final state is only between the two pseudoscalars) is to replace the vertex functions at one of the two vertices by some other real functions , parametrized in an appropriate way for the production, while the propagator matrix including the bare masses and the functions, and at the second vertex for the decay remain as above:
(31) 
This guarantees that the Watson final state theorem, and unitarity in general, is automatically satisfied and the Adler zeroes will appear only at one vertex in the remaining .
I believe this formalism should be much easier to apply, and physically more transparent, than e.g., the formalism of refs.[13, 14] where one multiplies the whole PWA’s by real functions . I have made some initial calculations along these lines, and seen that it is easy with appropriate choices of the to obtain the either as a dip or as a peak in the cross section. But more quantitative applications are left for further work.
2.7 Comparison with the N/D method and the Kmatrix. Physically acceptable analyticity.
For a single resonance and one threshold our function is the N function, while our is the D function of the N/D method. For the more general case of many channels and many resonances there is no clear connection between the parameters in eq. (1) and those of a matrix form of N/D[28]. But, much of the same philosophy of N/D methods is also present in the UQM: Given a model for analyticity and unitarity constrain the form of the propagator .
As to the relation between the Kmatrix formalism and the UQM, one obtains the Kmatrix corresponding to the matrix (1) by simply putting :
(32) 
where .
The expression (1) is regained from the familiar formula:
(33) 
Note that the mass shift term remains in our Kmatrix. In particular, thresholds which have not yet opened contribute to . Thus the UQM can be looked upon as a particular way of parametrizing the Kmatrix in a way which is consistent with analyticity and dispersion relations, whereby one has a simple physical interpretation of the parameters.
By adding the terms to the propagator one unitarizes the model and sums over the imaginary parts of an infinite series of loop diagrams (Fig. 1c). But to add the terms is equally important. In many models such terms from nearby closed thresholds are omitted, wheras I find it essential to include a complete set of nearby flavour related thresholds. Through these one includes the mixing and mass shifts of the bare states with the continuum of meson pairs.
By omitting also the terms from the denominator one obtains the bare matrix:
(34) 
Since the bare masses and the functions determine the PWA through the scheme described above, there is a one to one correspondence between the bare and the physical masses, once the functions are given.
One can gain some physical intuition for the UQM formalism, by viewing the dispersion relation for as the limiting case of the familiar second order perturbation formula for a mass shift . Each piece of the continuum shifts the state at the same time as the continuum is mixed into the state, with an amplitude . The sum of the squared mixing amplitudes again corresponds to the above . But, in contrast to second order perturbation theory the dispersion relation formulas are ”exact” in the sense that they solve the coupled channel model exactly.
It is important to emphasize that the function must in general satisfy a dispersion relation like the one in eq.(5), whereby one automatically has physically correct analytic properties for an arbitrary form factor. But in the literature one often finds violations of this rule. A few examples should clarify this point:
If one would put , i.e. make it proportional to the relativistic phase space factor, (which would be reasonable for Im if one had pointlike hadrons) and use it both for the real and imaginary parts, one would have a spurious pole and cut at in the physical region. The correct procedure is to calculate the real part from a dispersion relation, whereby one gets the ChewMandelstam function, which is more complicated and has a logaritmic large behaviour (one needs one subtraction constant because of the logaritmic divergence). Then one has no spurious pole nor cut at =0.
An even simpler example is given by the function , which also has a spurious pole. But, defining the function through the dispersion relation and its cut (now one needs no subtraction constant) one gets i.e., one automatically subtracts the spurious pole at from the physical sheet. [On the second sheet the pole remains, but does no harm: There the function is .] As a third example we take a finite cut as given by the function . The dispersion relation automatically subtracts a polynomial, which guarantees that the function vanishes asymptotically in the physical region: One gets .
These added terms are not small nor insignificant. They alter the model predictions in a very crucial way, giving e.g., the sharp rise in above the threshold (see Fig. 2). But as already mentioned, in the literature one often encounters models, which either disregard the real part entirely, or uses some variant of the physically unacceptable forms discussed above. Such forms might work within a limited energy region, if the spurious singularities are very far away, but would certainly fail if one considers several thresholds and a large energy region from threshold to 1.5 GeV, like in the present model.
3. Comparison with data on the lightest scalar mesons.
3.1 Parameters and form factor
As discussed in the previous section the PWA’s of eq. (1) are defined once one has a model for the vertex functions in eq. (2) or (30) and the bare masses of eq. (3). For the latter it is natural to assume an ”ideal” and isospin symmetric structure such that the bare or mass is a free parameter, while the and bare masses are given by and the bare mass by , where is the bare strange constituent mass.
For the bare couplings we use the OZI rule giving connected flavour symmetric couplings of eq. (24). The actual numbers are given in Table 1 for general values of the pseudoscalar mixing angle , which measure the deviation from the ideal states. The value of is fixed to (which is equivalent to a mixing angle for the angle measuring the deviation from pure states). Such a value is close to what has been measured in other contexts. (E.g. linear mass matrix formulas give , quadratic , a recent crystal ball experiment[29] quotes , Akers et al.[30] has , while Gilman and Kaufman[31], Baghi et al.[32] and Donoghue et al. find [33] . The results are not too sensitive to its actual value as long as it is in this ball park. All the above determinations assume the mixing angle to be the same at the as at the mass. This need not exactly be the case, since mixing angles are in general mass dependent; cf. our below. This same simplifying assumption for is made also in this paper. With our sign conventions the and states are given by:
(35)  
(36) 

Table 1. The relative ClebschGordan coefficients for the coupling constants as given by the trace of eq. (24). The pseudoscalar mixing angle relative to the ideal frame is fixed at , while is the ideal mixing angle.
More generally, one could allow for contributions from disconnected, OZI rule violating diagrams for the bare couplings, like or , which would introduce more parameters, and whereby the bare couplings involving the neutral members () would be different. Note however, that our physical couplings do not obey the OZI rule, because of the nonzero value of the pseudoscalar mixing angle , and because the loops included in our formalism generate a scalar meson mixing angle . It is possible that most of the OZI violation found is of this nature, coming from mixing in the mass matrix; therefore for simplicity, we assume as a starting point that the disconnected diagrams in the bare couplings can be put = 0. Of course, if e.g. a nearby gluonium state would be present, this assumption would have to be relaxed.
Since the data start only a few 100 MeV above the thresholds, they are not sensitive to the exact positions of the Adler zeroes near , provided these are small enough or of the order 0.1 GeV. Thus we put all , except in the case of the K and the thresholds. For we put it equal to the current algebra value (although would work just as well). For we let the fit find the best value, which turns out rather large and negative GeV. Current algebra including corrections from chiral perturbation theory[34] would predict this to be closer to zero. By modifying the flavour symmetry prediction for the K coupling, such that it is 1015% larger than predicted by the ClebschGordan coefficient of of table 1, one could fit the data with a smaller . Thus this determination of should not be taken too seriously.
Finally for the form factor one assumes the simple Gaussian form:
(37) 
This, no doubt, is the most drastic assumption of the model. It is used because it works, and because forms of this kind are obtained in the quark pair creation model (QPCM) [23]. There, it is given by the overlap of the three wave functions of the three hadrons at the vertex times a matrix element for the quark pair creation. With wave functions of Gaussian shape one gets a similar Gaussian factor, multiplied by a polynomial in . The cutoff parameter is then related to the hadron size through , from which with fm one would estimate to be of the order 0.60.8 GeV/c. In the fit one finds GeV/c. Including another cutoff parameter at the quark pair creation vertex the Orsay group[35] can account for a smaller cutoff parameter .
There is another crude phenomenological argument for a form factor of this form: If one plots the elastic branching ratios for resonances on the leading trajectory (i.e the branching ratios for the , see Fig. 3, or the branching ratio for the or trajectory[6]) one finds that they fall exponentially with squared mass or with as the square of eq.(37) with GeV/c. Of course this is not the same thing, but assuming that angular momentum barriers approximately cancel in the branching ratio, that the total widths grow much slower, perhaps linearly with , and that each resonance has an exponential form factor similar to (37) in their partial widths to exclusive twobody channels etc., one can argue that there is some experimental support for such a shape and that the parameter is about the same for all mesons.
No doubt, is in reality much more complicated than this. It should in principle include all the left hand cuts. But since the left hand singularities lie much farther away than the unitarity cuts, which we have included in detail, one can expect that the form (37) need not be too bad an approximation.
In table 2 below the parameters of the model are summarized:

Table 2. The parameters of the model. In addition, the pseudoscalar mixing angle is fixed to a conventional value of , and the Adler zero for is given the current algebra value , while for the remaining channels the Adler zeroes are put = 0. Other constants used in the model include the pseudoscalar masses and the ClebschGordan coefficients of Table 1.
3.2. The and the Swave.
It is natural to start the data comparison with the Swave, since this is the simplest to understand, having only one resonance, the , and in addition the experimental data are rather good. The comparison of the fit to the LASS data[36] is shown in Figs. 4a,b. The older data of Estabrooks et al.[37] are very similar but have larger error bars. The error bars of the LASS data are of the same magnitude as the dots in the figures. Fig. 4a shows the Swave phase shift, while Fig. 4b shows the absolute magnitude of the amplitude (in Argand units). The corresponding Argand diagram is shown in Fig. 5. As can be seen the model has no difficulty in fitting the data rather well. These data essentially fix four of the parameters in Table 2: and . We say essentially, since some of the parameters are strongly correlated, in particular and , and in practice because of the time consuming numerical integration in eq. (5) one must keep fixed.
The parameters of the also turn out to be rather close to the conventional ones (cf. Table 3). The BW mass is at 1349 MeV where the phase shift passes , and the BW width is 398 MeV with negligible correction from eq. (15), since is almost flat at 1350 MeV. The nearest pole is also where one normally expects it on the third sheet at MeV, with an imaginary part MeV (Since the coupling almost vanishes this threshold is very weak and, in fact, there is an image pole on the second sheet at almost the same position).