Four-point functions in quark flavor dynamics: meson-meson scattering

Volume 253, number 3,4 PHYSICS LETTERS B 10 January 1991

Four-point functions in quark flavor dynamics: meson-meson scattering

V6ronique Bernard Division de Physique Th~orique, Centre de Recherches Nucl~aires et Universitk Louis Pasteur de Strasbourg, B.P. 20, F-67037 Strasbourg Cedex, France

Ulf-G. MeiBner 1 lnstitut f~r Theoretische Physik, Universitiit Bern, Sidlerstrafle 5, CH-3012 Bern, Switzerland

A.H. Blin and B. Hiller Centro de Fisica Te6rica, Universidade de Coimbra, P-3000 Coimbra, Portugal

Received 17 September 1990

We study pion-pion scattering in an extended version of the three-flavor Nambu-Jona-Lasinio model. We show that the leading order contributions of the nn scattering amplitude agree with the standard current algebra results. We evaluate the various scattering lengths and amplitudes. For the regime of "barely broken" chiral symmetry, which is characterized by substantial non- linearities under the introduction of the current quark masses, we find ao ° = 0.26 in agreement with the data. The threshold parameters in the other channels are also satisfactorily described.

1. Introduction

The N a m b u - J o n a - L a s i n i o model [ 1 ] and general izat ions thereof are recently s tudied in various domains of hadron physics. The model gives a s imple way to unders tand the dynamical generation of fermion masses and the dynamics of composi te Golds tone bosons related to broken symmetries. In the hadronic world, these mech- anisms are the dressing of current to const i tuent quarks, the spontaneous breaking of chiral SU (3) L × SU (3) R symmetry down to its vectorial subgroup SU (3 )v together with the appearance o f pseudoscalar Golds tone bosons, the low-lying mesons n, K, and 11. The model is an effective theory of non-l inearly interact ing quarks, and the underlying interact ion is of the four- (mul t i - ) fermion type. Of course, this field theoretical model is not renormal izable in (3 + 1 ) d imensions and needs to be regulated in ha rmony with the underlying symmetries. Fur thermore , the physics of the U ( 1 ) A anomaly can be incorpora ted by an effective six-fermion interact ion [ 2 ] which allows to describe the qq ' mass splitting. In this form, the model has three parameters , the cut-offA and the two coupling constants G and K related to the four- and six-fermion terms, respectively. These can be readily fixed by fitting the propert ies of the low-lying mesons and the vacuum expectat ion values (VEVs) of the scalar quark densities. It is then possible to study the electromagnetic structure of mesons, their behaviour at finite temperatures and densit ies or the admixture of strange matr ix-elements into the proton wavefunction. In the context of the lat ter problem, it was shown in ref. [2] that there are two generically different regimes of the broken chiral symmetry. These regimes are character ized by their very different behav iour under the introduc-

• ~ Work supported in part by Deutsche Forschungsgemeinschaft under contract no. Me 864/2-1 and by Schweizerischer Nationalfonds. t Heisenberg Fellow.

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 443

Volume 253, number 3,4 PHYSICS LETTERS B 10January1991

tion of the current quark masses, in particular of the strange quark mass m~AQcD. In the "barely broken regime" (BBR) with G~> Gcrit #l, non-linearities in ms are important and a considerable quenching of strange matrix elements in the proton takes place. For example this mechanism allows for a large S-term ( ~ 50 MeV) and a small strange quark content of the proton at the same time. In the "firmly broken regime" (FBR) characterized by G >> Gcrit , non-linearities in the quark masses are small and so is the X-term (other observables are discussed in ref. [3] ). Actually, we will use there a slightly more general lagrangian than in ref. [2] so as to properly include the vector meson contributions.

The most fundamental (and simplest) dynamical process to study our understanding of the chiral dynamics of composite Goldstone bosons is indeed elastic rtn scattering, in particular the threshold behaviour as the ex- ternal momenta tend to zero. Rigorous results from chiral perturbation theory have been derived to one-loop order [4], but in particular in the isospin zero ( I = 0 ) , total angular momentum zero ( J = 0 ) partial wave the strong 7tTt final state interactions make a treatment beyond the one-loop approximation mandatory. Here, the NJL model will provide useful information since the Hartree-Fock approximation commonly used to solve the self-consistent equations includes the appearance of various resonances when translated to the language of in- teracting mesons. Furthermore, to leading order one should recover the famous current algebra results of Wein- berg and others [5]. This serves as a non-trivial check on the regularization procedure used, since a general proof of the consistency of the commonly quoted procedures (like e.g. the covariant momentum space cut-ff) with the pertinent symmetries for n-point functions constructed from the underlying quark dynamics has never been given. Indeed, we will show that the familiar Weinberg amplitude A (s, t, u) 2 2 = ( s - M ~ ) / F ~ follows in the limit of infinite constituent quark mass M with the ratio A I M fixed. This result is intuitively expected since by using bosonization methods one can show that this limiting procedure indeed gives (to leading order) the non- linear a-model and therefore the standard current algebra results. The model can now be used to get an estimate about higher order corrections beyond these soft meson theorems. Of particular interest, as already mentioned, the I = J = 0 channel because of the strong nn final state interactions. Furthermore, it is straightforward to extend the formalism to nK, KK, nq, qq or qK scattering. This is important since chiral perturbation theory calculations in these channels have not yet been performed and only the leading order chiral symmetry predictions [ 6,7 ] are known. In the light of the increased interest in kaon interactions (kaon factories), the calculation presented here should be used as a first guide to get an idea about higher order corrections to these processes (see ref. [ 8 ] ). Of course, the NJL model cannot be considered a high-precision theory (a thorough discussion of its limitations is given in ref. [ 2 ] ), but it generally describes the properties of the pseudoscalar mesons well because of their high degree of collectivity. Also, further information on the two regimes of the broken chiral symmetry discussed before is needed, since it allows us to judge the importance of non-linearities in the quark masses when dealing with expansions around the chiral limit.

2. Four-point functions

To be specific, let us consider the generalized NJL model for the sector of the three light quarks, ~T= (U, d, S), i.e.

8 8

~=¢(iO--,J[)~l+Gl ~-~ [ (~,~2a/~/)2+ (~ti~5~a//])2] +G2 ~ [ (¢?,fla{u)2- (ff'i?a?sfl"~) 2] (1) a=O a = l

- K { d e t [#( I + ys) ~v] + det [¢7( 1 - ys)~v] },

with Jg=diag(mu, rod, ms) the current quark mass matrix. 2(fl) are the SU(3) flavor (color) generators. The four-fermion terms are of the S-P and V - A type, with coupling constants G~,2 of dimension [mass ]-2. The

))t Here Gcr~ denotes the value of the coupling constant G beyond which chiral symmetry is spontaneously broken.

444


V - A term proportional to G2 has to be included here since the p-meson plays an important role in various channels of the scattering amplitude. Would one use the minimal type of lagrangian discussed in ref. [ 2 ], the vector mesons would only enter via exchange diagrams and their contribution would be suppressed. The six- fermion term is uniquely specified by requiring that it accounts for the U( 1 )A breaking, with dim K = [mass] -6 (for Nrquark flavors this term had dimensions [mass] 4--3Nf).

Above some critical value of G~.2 and/or K, chiral symmetry is spontaneously broken with the appearance of pseudoscalar Goldstone bosons. Quarks acquire a non-vanishing constituent mass M, which in the first non- trivial approximation to the effective potential constructed along the lines of Cornwall, Jackiw and Tomboulis [9] is momentum-independent (the Hartree-Fock approximation). In this approximation, flavor mixing is entirely given by the six-fermion term [ 2 ]. Naturally, as it stands, this theory is not finite and loop integrals need to be regularized. For the quark propagator, it is most convenient to use a covariant momentum space cut- off which keeps the underlying symmetries intact, i.e.

A

d4k M, (2) Si(k)=i (270 4 k2_M2i ,

0

with Mi the constituent mass of quarks of flavor i which follows from extremizing the effective potential (the so-called self-consistency or gap equation ). For our purpose, it is convenient to bring the interaction part of the lagrangian into the form of an effective two-body operator

28 K 2 5 6 K (Ss + 2Su))( ~/2o75g/)2_G,( l +E_~_~ t Ss)(~2375~')

) K

=V~(g/2~75~)z+V33(~2375~)2+V8s(u228~5~)2+V~"(~2~)~5q~2s75~)+V8~(~2sy5~2~)+ .... (3)

where the dots stand for other terms with 7~ substituted by I (e = + 1 for the pseudoscalar and e = - 1 for the scalar channel) and the vector contributions.

Let us now specialize to the case of ~ scattering. The relevant diagrams are shown in fig. I. First, consider the box diagram (fig. l a). Straightforward algebra leads to the following result for the scattering amplitude A ~ (s, t, u), with s, t and u the conventional Mandelstam variables:

"" V

J, M i ~ ,, S ~, / / \ /. "N

a) b) /~a C ) -~b"

Fig. 1. Meson-meson scattering in the NJL model. ~..b.c.d denotes any pseudoscalar meson (n, K, q ). (a) gives the box diagram, with Mj a constituent quark of flavor i( i = u, d, s ). (b) and (c) display the scalar (S) and vector (V) exchange in the s and t channel, respectively.

445

Volume 253, n u m b e r 3,4 PHYSICS LETTERS B 10 January 1991

AB(s,t,u)=_4Ncg4qq{212(s)+i2(t)+i3(M2, s ,s ) (s_2M2) ~ 2 2 + ~I3(M=, M=, - ( t - 2 M 2 ) / 2 ) ( t - 2 M ~ )

1 2 - ~ I 3 ( M ~ , t, t ) ( s + u - 2 M ~ ) ' 2 - ~I3(M~, s, s) ( t+ u - 2 M 2 )

+½14(s,t ,u, M 2 ) [ M 2 ( t _ s _ u + 2 M 2 ) _ , 2 ( u - 2 M = ) ( s - 2 M 2 )1 + (t*--,u) }, (4)

Here, Nc is the number of colors and gnqq is the quark-pion coupling constant. The integrals •2,3,4 are defined by

d

I2(k 2) = i "J d4p 1 (5a) (2zr) 4 ( p Z - M 2 ) [ ( p - k ) 2 - M 2 ] ,

0 d

d 4 p 1

13(k2, q2, k 'q) =i (2n) 4 ( p 2 _ M : ) [ ( p _ k ) 2 _ M z ] [ ( p _ q ) 2 _ M 2 ] , (5b) 0

A

/4(k 2, q2, l 2, k.q, k.l, q.l) =i -J d4p 1 (5c) (2n) 4 ( p 2 - M 2 ) [ ( p - k ) 2 - M 2 ] [ ( p - q ) 2 - M 2 ] [ ( p - I ) 2 - M 2 ] ,

o

where we have suppressed all flavor indices and work in the isospin symmetry limit mu= mo. The contribution from the scalar exchange shown in fig. lb is given by

A S ( s , t , u ) = 64NcM2 2gnqq4 [i2 (s) + ~t ( s _ 2M~2)13(m=,mr~ 2 - - ( s _ 2 M 2 ) / 2 ) ] 2

X {2J¢oo(S) + ~J/ss (s) + x/~ [;#os (s) +dgso (s) 1}, (6)

with 'J#o the scattering matrix in the scalar channel. Explicitly:

.#o=2G, { [ 1 - J s ( s ) ]ff ' Vo + [ Js(s) [1--Js(s) ] - ' ]ik Vks} . (7)

Here, Js is the fundamental bubble in the scalar channel which produces the pertinent bound states and the effective two-body potential V o is defined in eq. (3). In a similar fashion, one derives the contributions from the vector exchange (fig. lc)

AV(s, t, u ) = 2 4 16G2/9 - 64N¢gnqq ( s - u ) (4M2 _/)2[ 1 - 16GzJv( t ) /9]

× [ ( t_ZMZ=) i2( t )_ 2 2 2M=Iz(M~ ) 4 2 + 2 M J 3 ( M ~ , M 2, - ( t - Z M 2 ) / 2 ) ] 2 + (t*-*u) , (8)

with Jv the fundamental bubble in the vector channel. All together, the 7tn-scattering amplitude follows to be

A(s, t, u )=AB(s , t, u) +AS(s , t, u) + AV (s, t, u) . (9)

It is now a standard procedure to project this amplitude into partial waves with definite isospin I and angular momentum J. Of particular interest is the I = J = 0 amplitude given by

+1

To°(s)= 6 ~ d x P o ( x ) [ 3 A ( s , t , u ) + A ( t , u , s ) + A ( u , s , t ) l , (10) --1

with s = 4 (M~ + q 2 ), t = - 2 q 2 ( 1 - x) and u = - 2 q 2 ( 1 + x). Before proceeding, however, we have to recover the current algebra result for A (s, t, u) to leading order.

3. Current algebra limit

In the chiral limit (m~=0) , and expanding to first order in s, t, and u, we can write the contribution from the box diagram as

446


Aa(s, t, u ) = [AB + BB( t + u ) + Cas] ,

A s = + 16Ncg4qqI2(O), BB= -- 16Ncg4qqI3(O, O, 0) /3 , CB=-8Ncg4qql3(O, O, 0 ) / 3 , ( l l )

using dI2/dp2lp2=o = -13 (0, 0, 0)/3, which is a consequence of the Ward identity. From the scalar exchange,

we analogously find

AS(s, t, u ) = [AS + B S ( t + u ) + C S s ]

AS-- _ 16Ncg4qqI2(O), BS_-0, CS_ - -4Ncg4qq [12(o)/g2q-813(O, 0, 0) ] . (12)

At this order, the vector exchange does not contribute. Now letting the constituents mass go to infinity and simultaneously keeping the ratio A / M fixed ~2, we find that I3(0, 0, 0) = 0 and eqs. ( 11 ) and (12) combine to

give

A(s, t, u ) - 4Ncg2qq s (13) 12M z s = f f 2 '

using the fact that g2qq = [ _ 12Iz (0) ] - ~ and the Goldberger-Treiman relation M~ F~ = g~qq. Similarly, for finite current quark masses and in the SU (2) limit, use of the Gell-Mann-Oakes-Renner relation Mz~ F 2 = _ 2 mu ( au ) (which is exact within the Har t ree-Fock approximation used here but not in QCD) allows one to go through a similar argument with the result

s - M 2 (14) A ( s , t , u ) - 2 ,

F~

which is, of course, the desired result due to Weinberg [ 5 ]. Since the original derivation of eqs. (13), (14) is solely based on chiral symmetry arguments, what we have shown here is that the regularization procedure used to make the propagator finite is also preserving the underlying symmetries for the four-point functions. This is an important check on the model and the way of regularization. We can now proceed to calculate the full scattering amplitude (eq. (9 ) ) , i.e. the deviations from the current algebra predictions. These are expected to be large in particular in the I = J = 0 channel.

4. Resul ts and d iscuss ion

First, we must fix parameters. We use the ones obtained in ref. [ 2 ] from a fit to the properties of the pseudoscalar mesons ( n, K, rl, q ' ) and the scalar quark VEVs, (~]q) ( q = u, d, s ). The pertinent values are A = 1.0 (0.75 ) GeV, G,A2= 3.43 (4.35), KA 5=46.3 (89.1) for the BBR (FBR). Furthermore, we use GaA 2= - 5 . 5 6 ( - 5 . 9 9 ) to fit the p-mass, M p = 7 7 0 MeV ~3. In both cases, we work in the isospin limit m u = m d = 6 ( 9 ) MeV and vary ms between 150 and 200 MeV, as dictated by the standard evaluation of the current quark masses at the renormal- ization scale/~= 1 GeV ~A. One notices that the dimensionless couplings G1,2 A2 and KA s are large, i.e. dynamical chiral symmetry breaking involves strong couplings. Therefore, our results should be considered illustrative since we work within the lowest non-trivial approximation to the effective potential, the one-loop approximation.

The results for the nn threshold parameters are given in table l, with ms = 200 MeV for the regime of barely broken symmetry. The empirical data are taken from refs. [ 12,13 ] since the best determination o f the nn scat-

~2 This procedure amounts to reducing the diagrams shown in fig. 1 to the conventional four-pion point coupling. #3 We have neglected the contribution from the n-A~ mixing. In ref. [ 10] it was shown that this mixing gives minute corrections to

various properties of the mesons. In ref. [ 8 ], the mixing is fully incorporated. It is also important to notice that the bubble sum in fig. l c does not simply represent the exchange of a fundamental vector field with mass Mp. This is different from the bosonized version considered in ref. [ 11 ].

447


tering lengths comes from combining K~4 decay with analyticity constraints (Roy equations). This leads to an essentially unique nn scattering amplitude [ 12 ]. Altogether, the values of the threshold parameters as predicted by the model are in fair agreement with the data. The scattering length a ° is substantially larger than the one- loop chiral perturbation theory predictions and on the mean empirical value. In fig. 2 we show the real part of the isospin zero, S-wave amplitude, T o (s). In the NJL model, T O (s) is real up to the Q 0 threshold when one works in the Hartree-Fock approximation. The NJL prediction lies slightly above the chiral perturbation theory results ~4. For larger energies the real part of T O (s) grows too quickly. Of course, the NJL amplitudes are nec- essarily bounded in applicability by the unitarity limit, [ T/J (s) [ < [ s / ( s - 4M~ ) ] t/2.

The behaviour of the amplitude T O can be understood as follows: the lowest scalar excitation in the bubble sum (cf. fig. lb) has a mass of ~634 MeV (BBR). Of course, no such low-lying scalar particle exists, but it should rather be interpreted as an effective degree of freedom which parameterizes the strong final state interactions in the I = J = 0 channel ~5. It would be false to suggest the existence of such a scalar particle as it is sometimes done in the literature. A similar situation arises when one studies the attractive intermediate range force between two nucleons. It stems from correlated two-pion exchange with a broad spectral distribution peaked around t~ 22M~ 2. Again, in the one-boson-exchange framework one can then choose to parametrize this correlated two-pion exchange by a scalar particle with a mass of 500-700 MeV, however, one always has to keep in mind that this is nothing but a convenient parametrization. This explains the value for a ° and also the rapid increase of T O above x/s = 500 MeV. Higher order corrections not taken into account here will certainly smoothen out the behaviour of the amplitude. One might also consider the possibility of removing the unphysical QQ threshold by use of an appropriately chosen Omnrs function (this will be discussed in ref. [ 8 ] ). Notice also the slope of T O at threshold: the pertinent effective range parameter b ° turns out to be bo o =0.29M~ -2, somewhat larger than the empirical result. In the isoscalar amplitudes one observes large cancellations between the contributions from the box diagram and the scalar exchange accompanied by small corrections from the vector exchange.

In fig. 3, the phase shift difference 5°-51 is shown for center of mass energies smaller than 400 MeV in comparison with the recent chiral perturbation theory prediction of Riggenbach et al. [23] and the data of Rosselet et al. [ 14] ~6. In this range of energy, the NJL model gives a good description of the data.

The regime of firmly broken symmetry is ruled out since it gives a poor description of most of the threshold parameters and amplitudes. This again points towards the importance of non-linearities in the quark masses when one expands around the chiral limit.

Finally, let us briefly summarize the pertinent points of this investigation: - The current algebra prediction for the nn scattering amplitude, eq. (14), can be recovered in the NJL model.

~4 Notice that we have taken the central values of the low energy constants determined by Gasser and Leutwyler in ref. [4]. A more detailed discussion of this can be found in ref. [20].

as Compare also the discussion of the broad "e-meson" e.g. by Basdevant, Froggatt and Petersen [ 21 ]. In the more recent analysis of Au, Morgan and Pennington [22] ~o passes n/2 at x/s-~850 MeV.

~6 Here, we have assumed that the amplitudes are properly unitarized by the next-to-leading order contributions.

Table 1

nn scattering threshold parameters. We show the results for the NJL model (for BBR) in comparison with the soft meson theorems (SMT) and the one-loop chiral perturbation theory predictions (CHPT) [ 4 ]. The empirical values are taken from Froggatt and Petersen [ 12 ]. All scattering lengths and effective ranges are given in appropriate units of the pion mass.

a ° b ° a~ bo ~ al a ° a~

NJL 0.26 0.29 - 0 . 0 6 2 - 0 . 0 1 2 4 0.047 13× l0 -4 - 0 . 8 × 10 -4 SMT 0.16 0.18 - 0 . 0 4 5 - 0 . 0 8 9 0.030 CHPT 0.20 0.24 - 0 . 0 4 2 - 0 . 0 7 5 0.037 input input exp. 0 .26+0.05 0.25_+0.03 -0.028_+0.012 - 0 . 0 8 2 + 0 . 0 0 8 0.038+0.002 (17_+3)×10 -4 ( 1 . 3 + 3 ) × 1 0 -4

4 4 8


0.8

0.6

%_° 0.t~

t~

0.2

! I!i J / /

l I i 0 0.2 0.4 0.6

[GAY]

Fig. 2. Real part of the l = J = 0 scattering amplitude. The solid line shows the NJL model result in comparison with the current algebra (dash-dotted line) and chiral perturbation theory (using the central values of ref. [4 ], dashed line) predictions. The data are taken from refs. [ 14-19].

o

o l o tO

10

01

280 300 320 340 360

4~ [MeV]

Fig. 3. The phase differences 3 o -~I - The NJL model result (solid line) is displayed together with the data of ref. [ 14] and the recent chiral perturbation theory prediction of ref. [23] (dashed line).

- The par t ia l wave ampl i tudes are un ique ly fixed once one has d e t e r m i n e d the few paramete rs of the model , e.g. f rom the meson spec t rum an d the scalar quark VEVs. Altogether, one f inds a sat isfactory descr ip t ion of the var ious scat ter ing lengths an d effective range parameters . - The l = J = 0 ampl i tude , which exhibi ts s t rong f inal state in teract ions , is well descr ibed in the low energy re- gion. In par t icular , we f ind a ° = 0.26, somewhat larger t han the one- loop chiral pe r tu rba t ion theory result.

In a fo r thcoming pub l i ca t ion [ 8 ], we will cons ider nK, nq, KK, ... scat ter ing and present correct ions b e y o n d the cur ren t algebra predic t ions . Th i s will serve as a first o r i en ta t ion for the more complex d e t e r m i n a t i o n s o f these quan t i t i es which should soon be feasible at kaon factories.

A c k n o w l e d g e m e n t

We thank John D o n o g h u e for p rov id ing us with the da ta for the n n ampl i tudes . The work of V.B., A.B., B.H. was par t ia l ly suppor ted by a C N R S / I N I C exchange p rogram grant.

R e f e r e n c e s

[ 1 ] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 ( 1961 ) 345; 124 ( 1961 ) 246. [2] V. Bernard, R.L. Jaffe and U.G. MeiBner, Nucl. Phys. B 308 (1988) 753. [ 3 ] V. Bernard and U.G. MeiBner, Phys. Lett. B 223 ( 1989 ) 439. [4] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142. [5] S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. [6] R.W. Griffith, Phys. Rev. 176 (1968) 1705. [7] H. Osborn, Nucl. Phys. B 15 (1970) 501. [ 8 ] V. Bernard, A. Blin, B. Hiller and U.-G. MeiBner, in preparation. [9] J. Cornwall, R. Jackiw and T. Tomboulis, Phys. Rev. D 10 (1974) 2428.

449


[ I 0 ] V. Bernard and U.-G. MeiBner, Nucl. Phys. A 489 ( 1988 ) 647. [ 11 ] M.K. Volkov and A.A. Osipov, Sov. J. Nucl. Phys. 39 (1984) 440. [ 12 ] C.D. Froggatt and J.L. Petersen, Nucl. Phys. B 129 ( 1977 ) 89;

J.L. Petersen, CERN yellow report 77-04 ( 1977 ). [ 13 ] O. Dumbrajs et al., Nucl. Phys. B 216 ( 1983 ) 277. [ 14] L. Rosselet et al., Phys. Rev. D 15 (1977) 574. [ 15 ] N. Cason et al., Phys. Rev. D 28 ( 1983 ) 1586. [ 16 ] V. Srinivasan et al., Phys. Rev. D 12 ( 1975 ) 681. [ 17 ] A. Belkov et al. JETP Lett. 29 (1979) 1597. [ 18] B. Hyams et al., Nucl. Phys. B 73 (1974) 202. [ 19 ] W. Hoogland et al., Nucl. Phys. B 69 ( 1974 ) 266. [20] J.F. Donoghue, C. Ramirez and G. Valencia, Phys. Rev. D 38 (1988) 2195. [21 ] J .L Basdevant, C.D. Froggatt and J.L. Petersen, Nucl. Phys. B 72 (1974) 413. [22] K.L Au, D. Morgan and M.R. Pennington, Phys. Rev. D 35 (1987) 1633. [23 ] C. Riggenbach, J. Gasser, J.F. Donoghue and B.R. Holstein, CERN preprint CERN-TH.5755/90.

4 5 0

Documents

Four-point functions in quark flavor dynamics: meson-meson scattering