Meson-meson scattering in the quark-string model

Nuclear Physics North-Holland

A518 (1990) 723-751

MESON-MESON

SCATTERING

C. ALEXANDROU’,

IN THE

T. KARAPIPERIS,

QUARK-STRING

MODEL*

and 0. MORIMATSU’

Institutefor Theoretical Physics III, University of Erlangen-Niirnberg, D-8520 Erlangen, Fed. Rep. Germany Received 5 December 1989 (Revised 7 May 1990) Abstract: We formulate meson-meson scattering in an effective non-relativistic quark model based on the quark-string picture of hadrons. The behaviour of the eigenvaiues of the potential matrix is investigated and the partial-wave phase shifts are calculated using the resonating group method. We compare the potential eigenvalues and the phase shifts to those obtained with the two-body potential and the flip-flop models. In our discussion we draw attention to the behaviour of the phase shift at low energies which reflects the presence of long-range forces, the appearance of resonances at high energies as a result of the coupling to the confined channel and the dependence of the phase shift on the number of colours. Finally we construct equivalent local meson-meson potentials.

1. Introduction In lattice QCD quarks and antiquarks are connected by gluon strings as a consequence of local colour gauge invariance. In the strong coupIing expansion (in l/g, where g is the quark-gluon coupling constant) fluctuations around the configuration of minimal length are treated perturbatively and colour confinement is manifest. Moreover, it is generally expected of an asymptotically free theory that the long-range structure of the continuum theory is qualitatively the same as that of the strong coupling phase of the theory on the lattice. The latter is therefore a good starting point for studying hadronic interactions at low energies where confinement plays an essential role ‘). At the same time systematic studies using phenomenological non-relativistic quark models

have

provided

a good

description

of static

hadron

properties

“). In such

models quark confinement is introduced via two-body potentials, which is satisfactory for isolated hadrons. Given the non-pe~urbative nature of the strong interaction, however, it is not clear that these two-body potentials can be satisfactorily appiied to the description of multi-hadron systems. Indeed once applied to multi-hadron systems two-body potential (TBP) models are plagued by Van der Waals forces for which there is no experimental or theoretical support “). One can trivially regard the successful prediction of hadronic spectra by the TBP models as accidental and l

Supported

by the Bundesministerium fur Forschung und Technologie. Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland. address: Institute for Nuclear Study, University of Tokyo, Tanashi,

’ Present address: ’ Permanent

0375-9474/90/$03.50

@ 1990 - Elsevier

Science

Pubtishers

B.V. (Noah-Holland)

Tokyo

188, Japan.

724

discard

the experience

C. Alexandrou et at. / Meson-meson

scattering

accumulated

studies.

by such spectral

We take a constructive

point of view instead and regard the appearance of unphysical long-range as a signal of a new element in hadronic phenomena, which is not manifest single

hadron

(many-body)

level. flip-flop

Assuming potential

a similar (FFP)

point

model,

of view,

forces at the

Lenz et al. “) proposed

which is equivalent

a

to the TBP model

in the context of the qq system, but is free of long-range forces between hadrons. Here we present the first application of an alternative model, based on the quarkstring picture. We refer to it as the quark-string (QS) model. Again the difference from the TBP model appears in the context of multi-hadron systems. We consider the simplest heavy multi-quark system, q2q2. For given quark positions on the lattice, two kinds of states are considered, corresponding to the two possible ways in which two qq pairs can be connected in a locally gauge invariant fashion by gluon strings. To zeroth order in the strong-coupling expansion the string configuration is given in each case by the product of the ground-state configurations for two isolated mesons. It is shown in ref. “) that, if one works to lowest nonvanishing order in the perturbation, the off-diagonal terms of the overlap and potential matrices damp off exponentially as the separation of the qq pairs increases. This means that, while asymptotically this model avoids the Van der Waals interaction, it can provide a medium-range attraction between mesons. This attraction is expected to be a general phenomenon in hadronic interactions, since it is associated with the confining interaction. Moreover, it may play an important role in the formation of non-trivial bound or resonant systems such as q6, q2q2, etc. In other words, both the asymptotic screening of the force between hadrons and the mediumrange attraction are manifestations of the gluon degrees of freedom, although the latter are not explicitly present in the effective theory. In this paper we show how the scattering problem is formulated for two mesons in the quark-string model. The main step consists introduced in the language of second quantization of a fermion wavefunction in first quantization interpreted as an effective Schrodinger equation plished

that we can then use standard

methods

in showing that a weight function satisfies the exchange symmetry and an equation which can be for the fermions. Having accomto calculate

the S-matrix

as done

in the conventional non-relativistic models. The results obtained with the quark-string model are compared to those obtained with the FFP and TBP models. For a qualitative understanding of the meson-meson interaction we first consider the effective potential for static quarks as a function of the separation between two mesons. In the explicit calculation of the S-matrix we employ the resonanting group method. We first calculate phase shifts by allowing only one intrinsic channel (S-wave, ground-state mesons), In the absence of internal meson excitation long-range colour forces are not possible and the single-channel calculation yields similar results for all three models. Qualitative differences emerge when we extend our channel space to include internally excited qq systems. We will show that in the TBP and QS models the ground-state channel couples directly

C. Alexandrou

to a confined

channel.

et al. / Meson-meson

In the TBP

model

this

scattering

confined

725

channel

mediates

the

long-range force. In the FFP model no direct coupling of the ground state to another intrinsic state exists beyond a certain separation distance. The presence of the confined

channel

three models

leads to a modification

as a sequence

of resonances

of Levinson’s corresponding

theorem.

This is seen in all

to the energies

of the bound

states of the confined channel. Choosing for computational convenience a harmonic confining potential, we show that at low energies the TBP model exhibits pathological phase shift behaviour. Such a confining potential between the quarks leads to a l/R2 tail in the meson-meson interaction. As a consequence, when the scattering energy goes to zero, the phase shift either diverges (for n G 3 and 1= 0, where n is the number of colours and 1 the relative angular momentum of the mesons), or has a finite limit which is in general not a multiple of n (for larger 1 or n). This is to be contrasted with the FFP and QS models, where the phase shift goes with energy to zero (modulo 7r), since in these models no long-range forces are present. Inclusion of the confined channel results in additional attraction between the mesons. The effect is small in the FFP model reflecting weak channel coupling. The differences of the QS from the FFP model are more pronounced for small string tension (weak screening) and low and medium energies, as a result of the different meson-meson interaction at intermediate range. This paper is organized as follows: In sect. 2 we formulate the scattering problem of two mesons in the QS model and relate it to the TBP and FFP models. In sect. 3 we discuss the potential energy eigenvalues. Sect. 4 describes the relevant channels and the method used for the calculation of the phase shifts. We discuss our results in sect. 5. Finally, in sect. 6 we give our conclusions and future prospects. 2. SchrSdinger equation for the q*ij* system The goal of this section is to derive an effective Schrodinger equation for the quarks with antisymmetrization explicitly taken into account. To achieve a consistent formulation

of the problem

we use second

quantisation.

It is instructive,

therefore,

to first formulate the potential model in second quantised form for comparison. particular, we consider the TBP model ‘) and the FFP model of Lenz et al. “). In a potential model one solves the Schrodinger equation Ajw)=E/W),

In

(2.1)

where g is the hamiltonian and (p) the state vector. fi is given by the sum of the kinetic energy operator k and the potential energy operator $ A=ff+?. For a q2q2 system

(2.2)

1W) is given by

IV=4

I dTf(c,

r2,

ri, rz)Mi(r,, ri)MA(r,, rdlo),

(2.3)

C. Alexandrau

726

et at. / Meson-meson

scattering

where dr= d3r, d2rz d3rr d3rI and (0) is the vacuum state with no quarks or antiquarks. f is a weight function which has the symmetry f(r2, rr, rs, ri)=f

and n/ii is the creation operator for a colour-singlet @(5,

*i) =-&IX

rd,

(rr, r2,rit

c

(2.4)

qq pair

4Xr,)xZ(ri)

,

(2.5)

where C$&JJ) creates a quark (antiquark) and the summation is over the colour index c (E is the respective anticolour index). The set of states obtained from (2.3) with a complete set of square integrable functions f satisfying (2.4) is complete in the Hilbert space of the q2q2 system. On multiplying eq. (2.1) from the left by (Oli’WJr,, rr)MP(r2, rz) we obtain drL(rl,

I

r,,

ri,

ri;

r{, r$, ri, ri)f (r:, r&, rf, r&j =0,

(2.6)

where L(rI,r2,

ri,Q;

=(O/M,(r,,

rl,r;, rt,ri)

ri)Mp(r2,

r~)(~-~)~~(r~,

r~)~~(r~,

@IO).

(2.7)

Making use of the fact that for any local operator C? appearing in (2.7) “direct” and “exchange” matrix elements can be separated according to (O(M,(rr,

ri)Mp(rz,rZ)~M~(r:,r~)MIS(r;,

rt)lO)

ri)s(rI_r~)+S(r,-r:)6(r,-r:)

=[s(r,-r;)6(r2-r;)Si(riX8(ri-r$)i3(rg-r~)]t7D(r,,

r2, ri,

rd

-[6(r,-rS)S(rz-r:)6(ri-ri)6(rz-r~)+6(r,-r:)8(r2-r;) X8(ri--r$)6(rt--r$)]O,(r,,r,,ri,rI), we

(2.8)

write (2.6) as [r~~(r,,r2,rir*~)+~~(rl,r2,ri,r~)-E~~(rl,r2,ri,r~)lf(rl,r2,ri,r~) -[&(rI,

r2,ri,rZ)+ClrE(rl,r2,ri,rz)-E~E(rl,r2,ri,rz)l

Xf(r2,rlrri,r5)==0. Here the symmetry off (eq. (2.44)) has also been used. To simplify the notation we define the quantities

(2.9)

C. Alexandrou et al. / Meson-meson

where

scattering

m is the quark mass and u(r) the qq potential

for any potential

127

for an isolated

meson.

Then,

model

ND(~I, rz, ri, 4 = 1

Jdrl,

r2, ri,

5, ri,

%(rl,

r2,ri,rZ)=‘K.

%Jrl,

rd = K

rd=’ n’ (2.11)

n

For the TBP model

(2.12) Eq. (2.12) corresponds

in first quantization

&

vTB=-

where the sum is over all distinct generators we take

of SU( n), A

,C,u(rij)ni

pairs of quarks

a= $A*( -iA

configuration.

’

nj

(2.13)

3

"I

ar) for quarks

yD(rl,

r2, 6,

rd = VI

yE(rl,

r2, rr, rz) = $ min

In the FFP model there is an arbitrariness colour”

to the potential

Eq. (2.14) amounts

and antiquarks (antiquarks).

and, if A” are the For the FFP model

,

{u,, u2} .

concerning to a specific

the potential

(2.14) in the “hidden-

choice for this potential.

order to facilitate comparison we write our flip-flop potential for confinement in the notation of ref. “) [cf. eq. (2.26) of that reference]:

w,=~mw2 $+'-27,

w2=tmw2 $j(z2-y'),

In

harmonic

(2.15)

where P,,(Pr) and pO(F1’,>are projectors to colour singlet (octet) states of (1])(2?) and (12)(2i), respectively. Eq. (2.9) corresponds to the spatial Schrodinger equation for fermions in first quantized form, with exchange terms appearing in the second line. It can be written in compact form as (e-t

?-Efi)F=O,

(2.16)

728

C. Alexandrou

et al. / Meson-meson

scattering

where c_i =

Wc,

r2,

ri,

C)

Wpt,

r2,

ri,

4

C G(r,,

ri,

ri,

r-s>

%(r2,

rl,

ri,

4

(2.17a) > ’

(2.17b) and 6 stands for any of the functions k, ? and fi in (2.16). We now turn to the QS model. The hamiltonian of QCD can be written A 1 A

as

H=HF+HG,

(2.18)

where fir contains the fermion mass and kinetic energy terms as well as the quark-gluon interaction and & is the pure gluonic cont~bution. For a particular path f connecting a quark and an antiquark we define the quark-string operator Mt, by M:.(ri,

ri) = 4r(rl)

which generates the state schematically by the product of operators of parallel Uf(r,,

G(rl,

ri)x’(rJ

(2.19)

,

shown in fig. 1. On the lattice U, is given transport Ui(r), along the path r of links

ri_)= Ui(rl) . . . U,(rf-e,).

(2.20)

Summations over colour indices are understood making Mp locally gauge invariant*. For a single meson we assume that the quarks move adiabatically so that, given any spatial energy

quark configuration,

configuration.

the gluon

field arranges

rr>lo> = 4rl,

rilM%r,,

itself always in the lowest

Then fioM%rl,

ri)b)

(2.21)

,

where Mi is a superposition of M> and the eigenvalue u(r,, ri) is interpreted as a qq potential. In the strong-coupling limit each link contributes a fixed amount of electric flux and the ground state gluon configuration contains the minimal number of links. This corresponds to a linear confining potential, u( rl , rj) = U(rli) = cTJrl- rrl ,

R

rl Fig. 1. A typical

qQ configuration:

(2.22)

a quark

at r, joined

to an antiquark

at ri by a string.

l For the purpose of this work only colour appears explicitly as an intrinsic degree the formalism can trivially be generalized to include spin or other degrees of freedom.

of freedom,

but

C. Alexandrou et al. / Meson-meson

where

u is the string tension

heavy q2q2 system

in this limit. Using

can be approximated

729

scattering

MA, the energy

eigenstates

of the

as

Iw=J d3r,d3rif(r,, rr)M%r,, ri)IO>. We assume

that the energy

eigenstates

jP,=A j- dTf( which square system. E 1W)

of the heavy q2q2 system

rl, r2, rr, rg)M%r,,

rr)M%rz,

(2.23) are given by

rz)lO>,

(2.24)

has the same form as eq. (2.3). The set of states 1!P) obtained for all possible integrable functions f satisfying (2.4) defines our model space for the q2q2 The equation for f is obtained by minimising the expectation value (!P 1fi with respect to f *:

J

d+ L(r,,

r2, ri,

rz;

ri,

r;,

ri,

rt)f

(r:,

r;,

ri,

ri)

=O,

(2.25)

where

L(r 1,

r2,

=

J

+ -E

ri,

4, rk, 4, f-5)

6;

dU(OIMdrl, ri)Mdrz, rd&Mi(rl,, rib@(ri, r#O

J

dU

(O(Mdr,, rr)Mdr2, rz)fi&fA(ri, rf)M%ri, 4)(O)

J

and an integration A Ho do not include

dU (O(M,(

rl,rr)Mdrz, rdM%ri, ri)M84, ri)lO),

over the SU(n) fermion

(2.26)

group

operators,

is performed

application

at every link. Since i and to (2.25) easily

of Wick’s theorem

yields:

J u @bfdrl, d

ri)Mdrz, r&6(4,

rf)M%ri, rt)lO>

=[S(r,-r:)6(r,-r;)6(ri-ri)~(r2-r~)+S(r,-r:)6(r2-r:) X iS(ri-r$)6(rz-ri)]ND(r,,

r,,

ri,

r?)

-[6(r,-r;)6(r2-r;)S(ri-ri)b(rz-r~)+6(r,-r:)6(r2-r:) XS(rf-r~)~(r1-rrt)]~~(r~,r~,ri,r2),

(2.27a)

C. Afexandro~

730

dU

(OIW(rl,

ri)M4rz,

et al. / Meson-meson

rz)&Mi(ri,

rtDf~(rS,

scattering

r’z)lO)

I =[6(r,-r;)S(r,-r~)S(ri_ri)ij(r,-r~)+6(r,-r:)6(r,-r;) X6(ri-r$)6(r~-r~)]“I/D(r1,

r2,

rr,

rj)

r2,

ry,

4,

-[6(r,-r:)S(r,-r:)6(ri-ri)s(rz-rt)+S(r,-r~)6(r,-rb) XfY(ri--r~)6(rg-rf)]~&r,,

(2.27b)

where

J+dr,, r2,

ri , rd =

-CArI, rz, 5, r/‘,(ri,

dU (01Uab(ri, c> V&c,

rz>= dU@IhAri, I

r2,ri,*z)=

I

4 uba(rl,

ri) G,(%,

rdb),

r,)fLd(rz, r2)ubc(r1,r~)ud~(r2, ri)/O),

dU(01U,,(r-i,r,)U,d(rI,r,)~~Uba(rl,ri)Udc(r2,

r&9,

Here colour indices are explicitly displayed and summation over repeated indices is implied. The quantities in (2.28) are investigated, using the strong coupling expansion of lattice QCD, in ref. ‘). It is found there that the essential difference of this model from the TBP model lies in the behaviour of NE and “v;,which decrease exponentially with increasing area of the minimal surface bounded by the straight lines connecting quarks to antiquarks. The precise evaluation of A”Eand YE is extremely hard and is not attempted in this work. Instead, adopting a phenomenological approach, we rather choose an appropriate parametrization. We take for our calculation in the continuum the following

Xdrl,

r2,ri,rd=l,

W;)(rl,r2,ri,rd=fh,

.NE(r,,

r,,

ri,

rT)=iexp

(-aaS)

y;(r,,r2,ri,r~)=~(ti,+U2-Lil)eXp(-a~S)

(2.29) where S is the above mentioned minimal surface area and cx is a dimensionless constant, which is treated as a free parameter. Note that we have chosen the factors multiplying the exponentials of the exchange matrix elements in (2.29) SO that (2.29)

C. Alexandrou et al. / Meson-meson reduces

731

scattering

to the TBP result (2.11) and (2.12) for (Y= 0. We assume that a decomposition

of the type (2.27) is also possible ri)M(**,

(OlM(r,,

for the first term of (2.26) and write

ri)~df+(*:,

r~Mf+(r;,

rk)IO)

=[6(r,-r~)~(r2-r~)~(r~-r~)~(r~-r~)+6(r,-r~)S(r2-r~) Xs(Pi-r~)6(r~-rf)]9~(r~,

r2,

Pi, Pj)

-[S(r,-r~)6(r2-r’,)6(r~-r~)6(r~-r~)+~(r,-r~)6(r2-r~)

r2, Pi, Pz).

X6(r~-r~)b(r~-r~)]~~(r,,

By analogy operator

to the potential

model

we define

(2.30)

an effective

fermion

kinetic

energy

as follows YC,(rl,

r2,

Pi_,rZ)=K,

Yldr

I, r2, Pi, 4

= t(X,K

+ KNE) ,

(2.31)

where we adopted a symmetrized form to make it hermitian. Note that, in contrast to the potential models, JV, depends on the spatial configuration of the quarks. With this definition, $n( gE) can be thought of as a sum of Xn(YC,) and a non-local interaction

term. In this paper we neglect the non-local interaction (2.25) becomes 9 o,a with YG,,,. In terms of these quantities

and identify

[YLo(r, 7 f.2> ~i,~Z)+~~(~~,r2,ri,~I)-~~~(~~,r2,Yi,~~)lf(r,,r2,*i,~Z) -[YC,(*,

, r2,ri,rZ)+CtrE(rl,r2,ri,rS)-E~E(rl,r2,ri,rZ)1

Xf(r2, pl, Pi, rd=O,

(2.32)

or in matrix form as eqs. (2.16), (2.17). Eq. (2.32) or eq. (2.16) with the definitions of eqs. (2.29), (2.31) can be intepreted as an effective Schrodinger equation for fermions with antisymmetrization taken into account. If the fermion fields C$ and x possess additional internal degrees of freedom (e.g. spin) which play no dynamical role, then, depending on the state of the q2q2 system with respect to these degrees of freedom, the exchange terms in eq. (2.32) can have a negative or a positive sign. In the potential model these signs correspond to the cases where the spatial and colour part of the wavefunction is antisymmetric or symmetric, respectively, with respect to quark interchange. Having

obtained

an effective

Schrodinger

equation,

the scattering

problem

can

be formulated on the basis of (2.16) as for the potential model. Eqs. (2.11), (2.12), (2.14), (2.29) and (2.31) fully determine the three models for which (2.16) has to be solved. The method used will be discussed in sect. 4. 3. Potential eigenvalues Before we discuss the meson-meson phase shifts we would like to illustrate the qualitative behaviour of the meson-meson interaction in the three models (see also ref. ‘)). In order to do this we find a matrix A which diagonalizes the potential in eq. (2.16): Al
(3.1)

132

C. Alexandrou _

with A defined goes to infinity.

et al. / Meson-meson

scattering

I_

by ANA= 1 and A+ 1 as the distance Eq. (2.16) can be written as

R between

A($( fiK + Kfi) + c - E$AA-‘F

the two mesons

= 0,

(3.2)

which gives -E+;&K,

G]A+A~~T[K,

A]

A-W=O.

(3.3)

In the limit of very large quark mass m the fourth and fifth terms in the 1.h.s. of eq. (3.3) are suppressed by an order l/m compared to the second term and can be neglected. Therefore, the lower eigenvalue, Al, gives a measure for the meson-meson interaction. We find Ai and A2 by solving the secular equation det(A&-?)=O. For the QS potential and overlap matrices cally (e-2naS < 1) the eigenvalues -2w7.7

h,=Vl-e-

n2 -2ed

h2=

v2+:

n2

(3.4)

p and k (eq. (2.29)) we obtain

(u2-vj)2+

o

asymptoti-

e-4eus

(3.5a) (vl-vv,)2+

vz--1

n4

(

vz-v1 o

>

e-4aus

(

n4

)

’

(3.5b)

where v,, v2 and v3 have been defined in (2.10). Keeping the internal coordinates r,i and r22 fixed and varying the distance R between the two mesons, we expand asymptotically in Irirl/ R and Ir2$ R (Ir,i I, jr221 4 R or equivalently v2, v3 z=-v,). The asymptotic meson-meson potential is then given by the lower eigenvalue after removing the internal energy V(R)

=

_[(r,i *VR)(u *VR)~(R)I~e-ZaoS n2 . u(R)

(3.6)

For u(R) - RP, V(R) behaves as Rpe4 e-2aus. Following ref. ‘) we refer to this as For the TBP model the result looks identical “screened Van der Waals interaction”. to (3.6) except that the exponential factor is not present: a Van der Waals interaction -RPm4 acts between the mesons. In the FFP model the potential eigenvalues are given by Al-v11

A 2

=v2-vl/n2 1 - l/n2

(3.7)

for v, < oz. We note that beyond a separation Rs signalled by v, = v2 the mesons do not interact, i.e. V(R) = 0. In order to display the behaviour of the meson-meson interaction we choose the quark configuration indicated in fig. 2. For this configuration S increases linearly with R. For a typical meson size (-0.5 fm) and string tension (- 1 GeV/fm) the potential is screened over a distance R0-0.2/ (Yfm. The asymptotic potential V(R) RP-4 e-R’RO has an origin different from light meson exchange, since creation of qq pairs is not incorporated in our model.

C. Alexandrou

Fig. 2. Two aligned

qq pairs at a relative

et al. / Meson-meson

distance rri,

scattering

R with /r,~/ = /rzzI =&b 45”.

733

and an angle between

R and

rz? Of

As a specific example we consider a confining potential of the form u = (1/4mb’)(R/b)@. In this case the Schrodinger equation can be cast into a dimensionless form with one relevant dimensionless parameter y = crab’. Therefore, we shall use dimensionless quantities to display our results; all the lengths will be expressed in units of b and all energies in units of (l/2&*). In fig. 3 we show A, - u1 for linear and quadratic confinement. It can be explicitly seen that in the FFP model no interaction exists beyond Rs. In the TBP model the 0

-1.0

-2.0 i

I

5 cy

0

z N

w-

- 1.0

2.0

-3.0 0

I 2

1

3

R/b Fig. 3. The potential eigenvalue in units of 1/2mb2: (a) for linear confinement, (b) for quadratic confinement. The solid and dashed lines correspond to the QS model with y equal to 0.25 and 0.025, respectively. The dotted and the long-dashed lines are the eigenvalues in the TBP and FFP models, respectively. Also shown is the lowest excitation energy of an isolated meson.

C. Alexandrou et al. / Meson-meson

734

scattering

interaction varies asymptotically as l/R’ for linear and as l/R* for quadratic confinement, leading to a Van der Waals potential which is unacceptable from current experimental as well as theoretical considerations. In the QS model two values of y are considered, y = 0.025 and 0.25. For y = 0.025, A1- zll is similar to the TBP for intermediate meson-meson separations but it is damped off exponentially at large distances and leads to no Van der Waals interaction. This is clearer for the larger value of y in which case the interaction is damped much faster. 4. Calculation

of phase shifts

For the numerical evaluation of the scattering wavefunction of two mesons we apply the resonanting group method (RGM) 6). Since the method has been applied extensively to hadron-hadron scattering we give here only the essential ideas which are relevant to the problem at hand. In the q2q2 centre-of-mass frame (which we assume in what follows) we can expand the two-meson weight function f( r, , r2, ry, rz) of eq. (2.17b) in terms of a complete set of intrinsic meson wavefunctions 4p/‘B as follows f(r,,

r2,

ri, rz)

=Cx,(RAs)Cba"(rli)~'B')(r22),

(4.1)

where rri, r2z are the intrinsic spatial coordinates and RAB= i( r, + ry - r2 - rz) the relative separation of the two clusters. We recall from sect. 2 that if we substitute a complete set of functions for f in (2.3) or (2.24) we obtain a set of states which is complete in our q2q2 model space. From (4.1) and (2.3) it can be trivially seen that in potential models an overall eolour-singlet multiquark state can always be expanded in terms of products of colour-singlet clusters. For practical calculations one must truncate the sum in (4.1). Then in choosing the appropriate channels one has to carefully include those that influence the aspects of the system one wishes to address. It is known 3Y6)in particular that a truncation of (4.1) at any finite number of channels cannot accommodate long-range forces, which are of particular significance for the present work. One may still work with a finite number of channels but only by allowing weight functions other than those appearing in the expansion (4.1). In fact, as we shall show, long-range forces arise if one takes into account a particular excited channel to which the ground-state channel directly couples. The weight function for this channel, however, if expanded as in (4.1) requires an infinite number of terms. Let $p’a:,(r> be the meson ground-state wavefunction and define the two-meson state )9J by

IpJ=d 1dT.ft(r,, r2, ri, rdM’(r,, rdMt(r2, rdb, where M’ stands for either of the operators fi(rl,r2, ri, rl)~x,(R,,)~k"(r,i)4SO(r2~).

(4.2)

Mk and MA defined in sect. 2 and

C. Alexandrou

et al. / Meson-meson

scattering

735

For energies below the first inelastic threshold the solution of (2.16)f(r,, r,, ri, rz) approaches as RAB+ COthe ground-state weight function fi(r, , r,, rf, r?) within corrections due to possible long-range forces. For large but finite RAB there is a correction to the asymptotic form of f( r, , r2, ri, rs) and therefore to the two-meson state defined from it. In order to determine this correction which directly couples to )ql) we operate on (4.2) with (A - E) and write the result as a superposition of the states Mt( r, , ri)M’( rz, rz)lO) (g-E)

I

=

dTfi(rl,

ri, ri, W@(r,,

rrM@(rz,

dT&(r,, r,, ri, rz)M+(r,,

rdlO>

ri)M*(rz,

rz)lO).

(4.3)

In the QS model equation (4.3) should be understood within our model space. rq) we obtain as in sect. 2 Operating on both sides with (OlM( r,, ri)M(rz, [X&r,,

r2, ri,Cd-~fAr,,r2,

ri,

cdlfArl,r2,

r2,

4

rdlfi(r~,rl,ri,rd

-[~~(r,,r2,ri,rq)-E~,(r,,r2,ri,

=Ndh,

'i,

(4.4)

~i,~~)~(r~,~~,~i,rI)-~~~~ri,r2,~i,rZ)~(r2~r~,~i,~Z),

where 2KD,E = YtD,E+ 7fD,s.

++[(r2i*

In the TBP and QS models (4.4) becomes asymptotically

VR;J(r12*

V~,&(Kd1S,(r2,

rl,

ri,

rd

(4.5) where Rae=$(r,-rr2-ri+r& KR = -(1/2m)V”, and E, is the ground-state energy of a meson. We use the notation n”= n and neaDS for the TBP and QS models, respectively. In obtaining terms involving gradients

4rl,

r2,

Yi

the equivalent of eq. (4.5) for the QS model of S. We find that

9 f2> = [KRAB-(E--2El)lh(r1,

[(ci

-[(rzi

* VR&)(rlz~

we neglect

r2, *I, ri)-+-& n vR~~)~(~~~)l~(r2,

* VR,&w * VR,B)l~(R!dh(rl,

rl,

r2,

ri,

rd

ri, 4 (4.6)

736

C. Aiexandrou

et al. / Meson-meson

scattering

satisfies (4.4). From (4.6) we identify a state

‘J&

* V~,,Ma”(r~iX~2~

-X~(RAB)(~I~

XM*(h, ri)M+(b,

’

V~,,)4’,“(r2dU(hd

ri)lO),

(4.7)

which directly couples to / 1vJ. We note that asymptotically the coupling is damped exponentially in the QS model, because of the &dependent factors. The term (r- V&(r)(r - v&b(r’) u(R) contains an S- and a D-wave component in R. Neglecting the latter we take

l’y,)=~ d7.f( 2

rl,

r2, ri,

rdM+(r,, ri)Mt(r2,

r2)lO),

(4.8)

with

IzY2)is often referred to in the literature as a hidden-colour state consisting of two qq pairs in the adjoint colour representation coupled to an overall singlet. This nomenclature, however, is not necessary and often misleading since one never has to introduce the concept of hidden-colour states as seen from (4.8). 1q2) together with 1P,) constitute the channels in our two-channel calculation. We point out that if we repeat the above procedure in the case of the FFP model we shall obtain no coupling to a new state beyond a separation determined by the extension of 4 (‘) . We include however, the same second channel with Z = n in the FFP model calculation for ~~rnpa~son with the other models. Since, as can be seen from (4.8), the ground state couples asymptotically directly to a second channel we expect this channel to have a significant influence on the behaviour of the phase shifts in a certain energy regime. We expect this influence to show up in different ways in the three models and therefore this additional channel is impo~ant in studying the dynamical aspects of the different models which become apparent in the context of the meson-meson interaction. One difference is that the inclusion of I!&) leads to a long-range interaction (see appendix A) for the TBP model which must result in a major modification in the behaviour of the phase shift at low energies. A calculation performed with Ip,) as the only channel would contain no long-range force. This is a trivial example of the result quoted above concerning the truncation of (4.1).

C. Alexandrou

et al. / Meson-meson

calculation

the weight

For the two-channel

737

scattering

function

reads

ri,rd=h(r,,r2, ri, rd+fArl,r2, ri, f(r,, r2,

5)

(4.9)

where f2contains a term of the type appearing in (4.1) but with the coordinates and r, interchanged. Expanding fias in (4.1) requires an infinite sum. In the RGM one solves for the relative wavefunction wavefunction fixed. x satisfies the equation

x = {xi} keeping

r,

the intrinsic

(4.10) where the kernels the quantities

Ni, and Hii are given in terms of the intrinsic

ND/E, XDjE and VbiE defined

wavefunctions

and

in sect. 2.

If the meson-meson interaction vanishes for R or R’ greater than R, then the Ith partial wave component of xi has to satisfy the asymptotic condition xl,(R)

=

h',-'(kR) - S,hj+‘(kR)

I

7

Xzl(R)a where k = 2m(E -2~~), by h$+‘= ihj” and hi-‘=

JU,(RIfi)

S, = ezisJ and the spherical -ih, “) . .d%,is the solution

equation for the second channel which length scale is set by the string tension J&( R/6) + 0 as R + 00. The particular of the qq potential. As we have already limit yields a linear qq potential. This lattice QCD and by other theoretical and confinement potential is computationally

(4.11a) R>R, (4.11b)

Bessel functions hj” are given of the asymptotic Schrodinger

is regular at infinity. In this channel the (+ and overall confinement is ensured by function to be used depends on our choice seen lattice QCD in the strong coupling is also supported by numerical results of experimental reasons. However, a quadratic more convenient. Since, for the quantities

of interest in this work, we expect qualitatively similar results for the two choices, we use a quadratic confining potential u(r) = Vor2 with V, = 1/4mb4. For a linear potential, (+ determines the size of the meson. We choose b so that the meson size for the quadratic potential is similar to that obtained for a linear potential. Using a harmonic-oscillator ground-state wavefunction as a variational ansatz for a linear potential

we find b = (&r)1’6(3/4rno)1’3.

For a harmonic

potential,

b sets the length

scale for the problem. In what follows we will discuss the form of (4.11) for the specific case of quadratic confinement. Using a different confining potential amounts to changing the specific functions to be used in the asymptotic region. With this choice of the qq potential the asymptotic condition in the confined channel (4.1 lb) is given by the asymptotically regular solution of the equation

-&v’, +w2R2

3

x2(R) = (E - E2),,y2(R)

(4.12)

C. Aiexandrou et ai. / Meson-meson

738

scaffering

where E2 is a constant and o2 is (2n2/(n2-l))V, and 2V, for the TBP and QS models, respectively. For R > R, the confined channel reads

0

CL

xzt(R)a: ;

e-!lR/d)*

(4.13)

where -When long-range forces are present the asymptotic scattering wavefunctions in (4.1 la) must be modified. It is shown in appendix A that for quadratic confinement the TBP model leads to an asymptotic interaction between the mesons given by* V(R)=

Eq. (4.14) leads to a modification asymptotic form of x,, reads

xtt’3) =

(4.14)

-$-&

of the centrifugal

barrier

& I/--

[Ep(kR) - &l+‘(kR)],

1(1+ 1)/2mR2. The

R>R,

(4.15)

where the Hankel functions HF’ are given by HZ+‘= iHv’ and H1-j = --My’, v = J(l+f)” -3/n2 and R, is chosen so that the meson-meson interaction has reached its asymptotic form (4.14). s, differs from the S-matrix in the Ith partial wave due to the long-range potential. (4.14), whose cont~bution to the phase shift is given by i(t + 4 - ZJ)T. v is imaginary (real and non-integer) for I = 0 and n = 3 (I > 0 or n > 3). The consequences on the phase shift in these two cases will be discussed in detail in sect. 5 and appendix B. Having specified the asymptotic behaviour of the wavefunctions we now give the procedure to obtain xii(R) for the interaction region R < R,. In this region we expand xir in terms of a set of locally peaked gaussians subject to boundary conditions of the same type as xir of eq. (4.11) for the QS and FFP models and of (4.15) and (4.11b) for the TBP model. The expansion coefficients are determined variationally and the S-matrix is then obtained from these coefficients *). Finally, in the QS model we must calculate the minimal surface S for arbitrary quark configurations. This is not possible anal~ically and has to be done numerically. In order to reduce to a tractable level the dimensionality of the integrals involved in the evaluation of the kernels for eq. (4.7) we take the ansatz (4.16)

S(~,Y,Z)=lYxzl, l We expect of course that with, such a Van der Waals force, be enhanced compared with the case of a linear qg potential.

pathologies

at very tow energies

will

C. Alexandrou

et al. / Meson-meson scattering

739

where .Z=$(r,-_*-rf+PT).

x=5(rl+rZ-ri-rI),y=~(r,-r*+ri-rz), This ansatz

is exact for non-twisted

planar

quark

configurations*.

5. Results and discussion The S-wave phase shift for meson-meson scattering is shown for n = 3 in figs. 4 (one channel) and 5 (two channels) as a function of the dimensionless variable kb. Because of the possibility of additional internal degrees of freedom, as explained in sect. 2, we show results with both possible signs of the exchange terms. We refer to eq. (2.17b) as it stands, as the antisymmetric case and the one with the opposite sign as the symmetric case. In what follows we discuss the results with one and two intrinsic channels separately. (‘a) One channel. The phase shift shown

in figs. 4a and 4b have simple

features:

(i) In the antisymmetric (symmetric) case, the phase shift is in all these models negative (positive) at low energies and positive (negative) at high. The cross-over occurs at kb - 1.2 = b/r,, , where r. = v$b is the r.m.s. radius of a meson. The difference in the sign of the phase shifts between the symmetric and the antisymmetric cases can be understood as follows: In the one-channel calculation the scattering of two mesons is entirely due to quark exchange. The exchange terms (see eq. (2.32)) have opposite signs giving rise to the different signs observed in the phase shifts. The change in sign of the phase shift as we go from low to high energy indicates that the overall sign of the exchange term in eq. (2.32) reverses. For low energies the term XE dominates whereas at high energies the energy-dependent term E,NE, is mainly responsible for the interaction thus reversing the sign of the phase shift. (ii) Screening in the QS model results in a smaller phase shift for all energies in comparison with the TBP model and the interaction becomes weaker the stronger the screening.

Again this can be understood

via quark exchange, (2.29)). l

because

exchange

in terms of the interaction

terms become

proceeding

weaker as y increases

For the one-channel calculation we checked the dependence

of our results

on the ansatz

(see eq.

we make

for S. We considered

with g=g,,gz,-g12g2,

?I

%

=

ar ar au, aa,

-

-

and r=Ro+cr,z+ofl+2u,a,w

(RGsi(r*+r.2+ri+r$)).

We found that this leads to an overall reduction in the s-wave phase shift and can be simulated if one employs the initial ansatz with a small effective shift of a to a larger value. This supports our belief that the deviation of either ansatz from the exact result can be simulated by a smaI1 shift of LYwhich anyway we regard as a phenomenological parameter.

C. A~e~a~dro~ et al. / Meson-meson

740

scattering

1.0

a) 0.5 0 f; s

-0.5

UT 0.5 0

- 0.5 81

0

0.5

1.

I,.

.

1,

I.,

1.S

1.0

1

I

,,

2.0

I

$.

2.5

kb Fig. 4. The S-wave meson-meson phase shifts versus kb with one channel, for n = 3: (a) Antisymmetric case, (b) symmetric case. We use the same convention for the curves as in fig. 3.

(iii) result (iv) bound (b) of the (i>

At high energies the phase shifts for the QS and TBP models converge as a of the reduced role of screening at the short distances probed at these energies. The phase shift goes to zero at k = 0 and 00 implying the absence of dimeson states. Two channels. The inclusion of the second channel reveals the different effect confining interaction on meson-meson scattering: For all three models addition of the confined channels has an attractive effect

at all energies as expected in a variational calculation. This additional attraction is much smaller for the FFP model than for the two two models since the coupling to the confined channel is very weak. Thus in the FFP model the results for one and two channels are very close. (ii) The behaviour of the phase shifts is governed by the interplay between exchange symmetry and the confining interaction. The former gives similar energy dependence as for the one channel case and the latter gives an attractive contribution to the phase shifts whose amount depends on the model in question. (iii) At very low energies the phase shifts show a very strong dependence on the confining mechanism. For our quadratic confining interaction the phase shift for the TBP model diverges as k + 0 because of the long-range l/R2 potential, as shown in appendix B. In the QS and FFP models there is no long-range interaction between the mesons and the phase shift is well-behaved at low energies. As seen in fig. 5 the phase shift obtained in the QS model with small y remains close to that of the TBP model down to a low energy, where it turns around and approaches zero. For small y the low-energy maximum occurs at a momentum inversely propo~ional to

C. Alexandrou

X0-

et al. / Meson-meson

741

scattering

a)

2.0-

1.0 -

-+ f;

I 1 ’ I I i f I ’ 1 I

i 2.0-i

I _ --.* i.

0

0.5

1.0

1.5

2.0

2.5

kb Fig. 5. The S-wave meson-meson phase shifts versus kb with two intrinsic channels, for n =3: (a) antisymmetric case, (b) symmetric case. We use the same convention for the curves as in fig. 3.

the range

characteristic

of the screened

interaction.

For the larger

y the exchange

symmetry dominates and the effect of the screened interaction does not show up. (iv) At high energies all phase shifts exhibit resonant structures. This is due to the coupling to the positive energy bound states in the confined channel. The resonances occur at energies which are close to the energies of these bound states and are examples of those found by Castillejo, Dalitz and Dyson9). The relation between the behaviour of the phase shift and these resonances is discussed in appendix C. We note, however, that the resonances appear above the first inelastic threshold and therefore their effect on the S-matrix may be weakened by the coupling to the open channels. (v) For I> 0 we have a centrifugal barrier. For the TBP model the long-range Van der Waals interaction modifies the centrifugal barrier to an effective barrier described by Zefi= J( 1+$’ -3/n2 -+. This modified centrifugal barrier leads to a

142

non-zero

C. Alexandrou et al. / Meson-meson

finite phase

shift at zero energy

scattering

given by

&(k=0)=-;(1,,-1)7r.

(5.1)

Thus a,( k = 0) = 0.11 for n = 3. In the QS and FFP models the phase shift vanishes at k = 0 for I= 0. (vi) As n increases the interaction between fig. 6 we show the S-wave phase shift for n = 10 n = 3 reflecting a weaker interaction. For n > 3 interaction is too weak to produce a divergent

the mesons becomes weaker. In which is generally smaller than for the strength of the Van der Waals phase shift at zero energy. Using

(5.1) the value of the phase shift for 1= 0 and n = 10 at k = 0 is 0.05. In the QS model the phase shift vanishes at zero energy for all n and 1 (fig. 6). The coupling

i 0

0.5

1.0

kb

1.5

2.0

2.5

Fig. 6. The S-wave meson-meson phase shifts versus kb with two intrinsic channels, for n = 10: (a) antisymmetric case, (b) symmetric case. The solid and dashed lines represent the phase shifts in the QS model with y 0.25 and in the TBP model, respectively.

et al. / Meson-meson

C. Alexandrou

scattering

743

to the confined channel decreases with increasing H and in the limit n + fx) the mesons do not interact resulting in zero phase shift. In order to visualize the meson-meson interaction we construct equivalent local potentials which are defined to give the same renormalized relative weight function xR(R) at a given energy E, as follows: V,,(R) = -A--

XR(W (

E+z

0.8

0

1

2

3

>

x/O),

(5.2)

FFP

1,

R/b Fig. 7. Local equivalent potentials versus R at E = 6.25 X 10’ (This corresponds to an energy when m = 500 MeV and b = 0.5 fm): (a) antisymmetric case, (b) symmetric case.

of 10 MeV

744

C. Afexandrou

et al. / Meson-meson scattering

WhHT

xR(R) =

d3R’ N”2(R, R')jy(R')

(5.3)

and N is given in eq. (4.10). In fig. 7 we show the equivalent local potentials at E = 6.25 x 10e2, which corresponds for instance to E = 10 MeV for m = 500MeV and b = 0.5 fm. For all models and for both the symmetric and antisymmetric cases the range of the equivalent local potentials is determined by the meson r.m.s. radius except for possible residual l/R’ tail. For the TBP and FFP models the potential is repulsive in the antisymmetric case and attractive in the symmetric one reflecting the behaviour of the phase shift at this energy. For the QS model however in both symmetric and antisymmetric cases there is a repulsive core and intermediate-range attraction, which is stronger in the symmetric than the antisymmetric case. 6. Summary and conclusions In this work we have formulated meson-meson scattering in the QS model with exchange symmetry fully taken into account. We studied the meson-meson interaction by calculating the S-matrix using the RGM. We compared the results obtained in the QS model to those obtained in the TBP and FFP models. We found that both exchange symmetry and the dynamical aspects of the models play an important role. By appropriately choosing the relevant channels, we studied the effects of the long-range interaction in these models. In the QS model the Van der Waals interaction, which leads to pathologies in the TBP model, is damped exponentially. In comparison to the FFP model, which also cures these pathologies, the QS model in most cases has a stronger attractive component due to the stronger coupling to the confined channel and being based on the confining mechanism has a theoretical foundation. The QS model was obtained by looking at the strong-coupling expansion of lattice QCD. While this work was in progress other groups “-‘“> have considered applying similar ideas to meson-meson scattering. We would therefore like to comment on the relation of this work to theirs. In the work of Miller the transition potential ( V, in eq. (23) of ref. “)) corresponding to our U; was calculated in the strong-coupling expansion as in ref. ‘). It was found there that the effect of string rearrangement is negligible for meson-meson scattering. The Helsinki group used a different approach to calculate the quark-string rearrangement amplitude. Instead of using the strong coupling expansion the amplitude was expressed in terms of the flux-tube breaking amplitude for meson decay. The latter was computed as the overlap of the wavefunctions of the initial and final strings r2). In their actual calculation only simplified geometries were taken into

C. Alexandrou

et al. / Meson-meson

745

scattering

account. In ref. i3) stung-breaking and rejoining at a single plaquette was considered and the same conclusion as in Miller’s work was obtained. In ref. 14) they extended their calculation to include multi-plaquette rearrangement. It was shown that this leads to an enhancement

of up to two orders

plaquette

However,

rearrangement.

a problem

of magnitude of double

compared counting

to the single

exists.

This work differs from the work of Miller and the Helsinki group in essentially two main respects. The first difference is that we take into account the nonorthogonality of our two-meson states to the same order in the strong-coupling expansion as the contribution to ‘YE, while, both of the other two groups seem to assume orthogonality of their two-meson states which can only be true to zeroth order. The second difference arises in the way we choose to incorporate quark-string rearrangement. Both in our and their work the transition potential is given by the product of an exponentially decaying function of the minimal area S and a slowly varying function of the quark positions. For general configurations it is extremely hard to calculate the minimal area exactly. The slowly-varying factor entering in multi-plaquette contributions (C, in ref. I’)) can only be calculated analytically for simple configurations. We feel that only by Monte Carlo simulations where all possible configurations are taken into account can one obtain these quantities reliably. Therefore we chose to treat the slowly varying function and the rate of exponential damping phenomenologically as explained in sect. 2. We thus investigate the effect of quark-string rearrangement on the phase shifts as a function of the exponential damping rate. As we have seen the effect is non-negligible for a certain range of values of this rate and quark-string rearrangement may provide an impo~ant new mechanism in hadronic interactions. The value of the exponential damping rate is therefore crucial for determining the importance of this mechanism. For completeness we would like to mention related work by some other groups. Weinstein

and Isgur have examined

the q2q2 system in a potential

model

and have

found qualitatively similar results to ours I’). Maltman and Isgur have extended it to the q6 system and have obtained intermediate range attraction between two nucleons 16). In both investigations, however the long range tail of the Van der Waals interaction

was cut off by hand.

In ref. “) Isgur has discussed

the effects of

the gluonic string on cutting off the Van der Waals forces. Robson has also presented a similar idea and has applied it to the two nucleon system lx). Finafly, we would like to comment on future extensions of this work. A natural application of these ideas is to consider baryon-baryon interaction. In the case of a two-meson system, the number of string configurations in the QS model coincides with that of colour configurations in the potential model. This is not the case for a two-baryon system. In particular Pauli-forbidden states which exist in the potential model do not appear in the quark-string model. It is interesting to study the implications of the absence of these Pauli-forbidden states. A further application would be to study quark exchange effects in nuclei which recently have received much attention 19). Since exchange terms are suppressed in the QS model by the

746

C. Alexandrou et al. / Meson-meson

scattering

exponential factor, it is interesting to investigate the importance of quark exchanges in this model. Appendix A

We derive here the long-range meson-meson potential in the case of the twochanne1 TBP model with harmonic confinement. Since we are interested in the asymptotic form of the interaction we take in what follows R 4 b. Then exchange terms are negligible, the kernels in (4.10) become local and the equation reads

where cl is the meson ground-state also taken $J12’

d-

3 1 n2-1 2mbZ’

--

energy and E2 is a constant energy. We have

R2 2mp”

q*(R)=-=

*2R2

If we define u,(R) = Rx,i(R); where xi(R) =x/Xi,(R) favour of u,[, we obtain the radial equation

_-- ~ 1 d2 +I(f+I) 2m dR2 2mR2 1

Q(R)+

n:*

pe

3

05 5

n2-f (

n=

YL >

Y,,(R), and eliminate u21 in

dR’ Gl(E’, R, R’)u,i(R’)

= (E

-~E~)Y~I(R)

1

0

(A.21

where E’ = E - E2. The Green function G, satisfies the equation +1(1+1)+ ~ 2mR2

?12 R2 n2-1 2mb4

--

GI(E’, R, R’) = SIR-R’)

_ fA.3)

GI can be written as GJE’,

R,R’)=$$,

($)db(y,

(A-4)

where #+ and & are the solutions of the homogeneous equation which are regular at the origin and at infinity, respectively, and W is their wronskian. They are given in terms of the standard Kummer functions & and U as defined in ref. ‘) 1$11( p)of p’f’ e-p*/2 JCI(#-A),

I++,

P”),

&( p)a-Zp’+’ epL12zJ(~(Z--A), r-t;, p’)

where h =$-g(n’-1)/n2mb2E’. Now takeR. such that R B- R. s b and consider the integral CC dR’ G,(E’, R, R’)u,,(R’) s0 2m =~~RodR~~*(~)u,l(R’)+I~dR’G,(E’,R,R’)U,,(R’). W #z(f)

(A-5)

C. Alexandrou et al. / Meson-meson

The first term on the r.h.s. of (AS)

is proportional

to R” eCRz’2p2 and vanishes

asymptotically.

We consider

the asymptotic

forms of U and M can be used in (A.4). We obtain _mp

next the second

747

scaitering

term. We have chosen

(~)A+‘(~)-A~2e-tR2-R”/2~2,

R. large so that

R>R’sR,,

(A6)

G,(E’, R, R’) = _mp(;)-A-2(;)A+’

Since e

_,,+R’*,,2fi2R-ai

-

(2P2/R)6(

e-,R2-R'2,,2@2,

.

R’>R>R,,.

R - R’), we have

G,(E’, R, RI)==

S(R-R’),

-y

(A-7)

and using this in (A.5) we obtain dR’ G,(E’, R, R’)u,,(R’) Finally

R’”

-T

u,,(R).

(A-8)

(A.2) becomes 1(1+ 1) d2 -- 1 y+---2mR2 2m dR

which displays

a meson-meson

* 2mp2 ‘I2 R2

1

u,,(R)=(E-2q)u,,(R),

R z==b

(A.9)

potential

2mp2 V(R)=-v;~R,=---

3

1

2mn2 R2’

(A.lO)

Appendix B As a result of the long-range meson-meson potential (A.lO) the phase shift does not go to a multiple of n as the energy approaches zero. We show in this appendix that for I= 0, n G 3 the phase shift diverges at zero energy. Intuitively this can be understood as follows. dimensionless variable

Let R, be such that (A.9) holds for R > R,. In terms of the z = kR (A.9) can be written as

-- d2 +1(1+1) ------1 dz2 z2

3 n2z2

q,=o, I

z>z,=kR,.

(B-1)

As k+ 0, z,+O and (B.l) describes scattering by a pure -(3/n’)(l/z’) potential. But the latter is known not to possess well-defined scattering solutions if l/n2 < (1+$2. In the problem at hand this is the case when 1= 0 and n G 3, which leads us to expect some pathology at zero energy.

C. Alexandrou

748

et ai. j Meson-meson

scattering

We now examine explicitly the low energy behaviour R > R, the general solution of (A.9) can be written as

x*rtm= where

v=~J(Z+~)~-~/~*

derivative

of the phase

shift.

[H-I-‘(~R) - S;H~+)(~R)] ,

For

(B-2)

(cf. sect. 4). $,=e2’$ 1is fixed by matching

the logarithmic

LYof xl1 at R = R, : (B.3)

Note that LYis real. For kR + m we can use the asymptotic kR-+co

’ -$

Xl I’

0c -!-

[e-’ s(kR-142)

~ &

[e-i(kR--ls/2)

kR

For low energies

[e-

(kR,-,

expansions

of the Hankel

ikR+i(l/2+u)w/2+~J

ei(kR--Iv/2)

ei(kR--lr/2)]

1

_

0, a + a*) we can expand

Hii’

in (B.3) to obtain

tan (v7r)

tan & =

1 -AI/A,

Then

eikR-i(l/2+v)=/2]

_ g, e i(f+1/2--v)v

_ eti8,

functions.

03.5)

cos (VT) ’

where A,~(tr,R,+t~v)T(1+v)e-“‘og(kRc/2), AZ= (a,R,+~--

V) e”‘og(kRc’2)

and A, = AT = A. Then (B.5) can be written

For I= 0 and n G 3, Y is imaginary e2i5

v)T(l-

0=

em’““-A/A* e Ccv_ A/A*

(B-6)

’

where Y= i,u, p > 0. Taking the absolute value of (B-6) we obtain part of &, Im & = ~~12. The phase shift is given by So= &-&w+$i-. We see immediately that S0 is real as it ought to be by unitarity. of A we finally obtain the result tan ( a0 -$r)

as

for the imaginary

(B.7) Using the definition

= tanh (*JILT) tan [C -g In (+kR,)] ,

o-3.8)

where C is a constant. It follows from (B.8) that as k+O, &, oscillates with finite amplitude around a curve C-f log (kR,/2), where C’ is a constant. We see that the phase shift diverges logarithmically.

C. Alexandrou et al. / Meson-meson

749

scattering

Appendix C We show modification

in this appendix of Levinson’s

We start with the RGM

that

inclusion

equation

differs from the usual coupled-channel

(4.10). The normalisation case, reflecting

basis. Since we intend to apply a result perform the following transformation HR E N-1/2HN-‘/2, Then

(4.10) can be written

of the confined

channel

leads

to a

theorem.

obtained

kernel

N in (4.10)

the use of a non-orthonormal

with an orthonormal

NJ,

basis,

,yR= N’j2,y.

we

(C.1)

as (HR-E)xR=O

(C.2)

Eqs. (C.l) and (C.2) have to be understood as formal Hilbert space. We decompose HR in two parts

equations

in the abstract

HR=H,,+V,

(C.3a)

where H,,( R, R’) =

0

-(1/2m)v2,+2&,

-(1/2rn)VZ,-tE,+~~~(R)

0

> S(R-R’)’ (C.3b)

HR converges asymptotically to Ho, except for the TBP model where HR differs asymptotically from (C.3b) by constant off-diagonal matrix elements*. This determines also uniquely the constants 8, and E2 (the former is the meson ground state energy as before). Use is a linear (quadratic) potential when the qq potential is linear (quadratic). V is by definition short ranged. The spectrum levels associated

of Ho consists of the continuum and an infinite with the confining potential v22. The relation

shift and the number detail

of bound

in ref. lo). It was found

states for a hamiltonian

number of discrete between the phase

of this type was studied

in

there that (C.4)

s(00)-8(0)=(n,-n)~,

where 8 is the phase shift and n,(n) is the number of discrete levels in the spectrum of H,( HR). In ref. lo) no was a finite integer, but in our case it is infinite. This makes equation

(C.4) ill-defined

and hence inappropriate

for the extraction

to give it a precise meaning we proceed as follows. Consider the Lippman-Schwinger equation for the hamiltonian T= V-t VGoT,

of n. In order (C.3a): (W

l Since the consequences of the latter are practically restricted to low energies and have been investigated in appendices A and B, we neglect the off-diagonal matrix elements of HO for the purposes of the present discussion.

750

where

C. Alexandrou

T is the transition

operator

matrix

et al. / Meson-meson

scattering

and Go = (E - HJ’.

We introduce

the projection

P(+J 6(R -R’) J’(“o’(R 3R’) =

0

0

(W

?l 4n(RM:(R’)

where v~~(R)]c#J~(R) = E;&(R)

[-(1/2m)V;+&+

! bb,(R)i* d3R = 1, and define the truncated V’“0’ E p(Q vp(Q The transition

matrix

in the truncated

7

(E: s E;s..

.) ,

operators G$,’ G @,)G,,P(%)

Hilbert

.

(C.7)

space is given by the equation

7-c”,,)= V’“,,‘+ V(n~)G$,)+,)

(C.8)

If S(“o) is the phase shift obtained in the truncated problem and n(“o) the number of bound states of the truncated hamiltonian, the following relation holds ij(“o)@) -6’%,‘(O) = (n,-

n(“o) )n.

(C.9)

But 0 GO( E, R, R’) - G$‘( E, R, R’) =

o

f

(C.10)

Pl=no+l and for large n the overlap

of the wavefunctions

4,(R)

with V will diminish

with

increasing n. Thus by taking no large enough we can make the difference between T(E) and T(“o)( E) arbitrarily small except when E = E&+, , E&+~, . . . It follows that there will always be an no such that the following equation is correct within an arbitrarily small error in S(E), with EL,< E < E&+~ 6(E)-6(O)-(no-n)n.

Eq. (C.ll)

is well defined

of the coupled-channel

and can be used to extract

(C.11) the number

of bound

states

hamiltonian.

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