a molecule?

Is Xð3872Þ a molecule?

C. E. Thomas* and F. E. Close+

Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom(Received 2 July 2008; published 8 August 2008)

We show that the literature on pion exchange between charm and bottom mesons is inconsistent. We

derive the formalism explicitly, expose differences between papers in the literature, and clarify the

implications. We show that the Xð3872Þ can be a bound state but that results are very sensitive to a poorly

constrained parameter. We confirm that bound states in the B �B sector are possible. The circumstances

whereby exotic combinations can bind with cc or bb quantum numbers are explored.

DOI: 10.1103/PhysRevD.78.034007 PACS numbers: 14.40.Gx, 21.30.Fe

I. INTRODUCTION

The nature of the enigmatic charmonium mesonXð3872Þ, which appears at the D0D�0 threshold with JP ¼1þ, has been the subject of intense debate ever since itsdiscovery. Historically, Ericson and Karl [1] consideredpion exchange in hadronic molecules. Tornqvist predicted[2,3] that one pion exchange between charmed mesonsgives an attractive force in the I ¼ 0 channel such thatmolecular or resonant D �D� states might arise near thresh-old. Following the discovery of the Xð3872Þ several paperssuggested that it could be a 1þþ state, driven by pionexchange [4,5] and/or quark exchange [6] where flavorsymmetry breaking was associated with its affinity forthe neutral D0D�0 threshold.

More recently, several papers have appeared assessingthe potential attractive forces and asking whether a boundstate is dynamically realizable. These notably includeSuzuki [7] who has argued that the one pion exchangeforces are only able to make a feeble attraction at best, andmost recently Liu et. al. [8,9] who have claimed that abound state does not exist for reasonable values ofparameters.

All of these papers [3,5–9] make different assumptionsof detail, are not always self-consistent, and do not allagree on the mathematical expressions even where theirassumptions are the same. Hence the purpose of the presentpaper is to attempt a unified treatment of this problem,enabling comparison between the various approaches to bemade. In particular, we shall make explicit the calculationof some critical signs, upon which attraction or repulsioncan depend, and whose derivation is not described in theexisting literature. As a result we shall find that expressionsin the various papers are mutually incompatible. We shallthen propose a consistent formulation, discuss its conse-quences, and compare with the existing literature.

Tornqvist [3] initially assumed isospin symmetry andfound I ¼ 0 attraction. Following the experimental discov-ery, Close and Page [5] showed that the d� u mass dif-

ference can lead to substantial breaking of the flavorsymmetry enabling attraction in the neutral D0 �D�0 con-figuration. Tornqvist also studied isospin symmetry break-ing [4]. Recently Liu et. al. [8,9], using the empirical factthat the state is at neutral threshold, have focused solely onthe neutral channel without discussion of isospin symmetryor its breaking. In addition, they assume that any phenome-non at the BB� threshold also involves only one chargechannel. As we shall discuss here, the mechanism of flavorsymmetry breaking can be critical in deciding which chan-nels if any are attractive, and, in particular, whether attrac-tive forces are strong enough to bind. Wewill also note thatthe B �B� and D �D� situations can be very different. As weshall argue, the symmetry breaking and dynamics in theD �D� relative to B �B� cases depend on the mass splittingsbetween vector and pseudoscalar masses and whether theyare larger or smaller than the � mass. This has been notedclearly in the work of Suzuki [7] and Liu et. al. [8,9] wherethe Fourier transform gives different potentials in positionspace, but is not apparent in the original work of Tornqvist[3]. Furthermore, we find differences in some critical signsrelative to Tornqvist in Ref. [3]. These have potentialimplications for the attraction or repulsion in the D �D�and B �B� systems which differ from that reference.In the present paper, we shall first derive the expression

for the �-exchange potential along the lines of the originalpaper [3]. This will expose the origin of the signs thatdetermine the overall attraction and repulsion and howthe vector-pseudoscalar mass gap is critical. We shallconcentrate on making contact with existing literature,showing where there are differences of assumption, sensi-tivity to inputs, and possible errors of calculation. Finally,we shall assess the implications.In Sec. II A, we make pedagogic comments about differ-

ent conventions for charge conjugation eigenstates, in or-der to clarify discussions in the literature and to define ourformalism. In Sec. II B, we give a simple illustration of thespin expectation values for P �V (V �P) and V �V. This exposesa relative sign between these quantities that disagrees withRef. [3]. In Sec. II C, we give the overall spin and flavorfactor. We calculate the effective potential in position

*[email protected][email protected]

PHYSICAL REVIEW D 78, 034007 (2008)

1550-7998=2008=78(3)=034007(12) 034007-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.78.034007

space in Sec. III; our result in Eq. (23) exposes the differ-ences in the existing literature [3,7–9].

In Sec. IV, we move on to applications of the formalism:we discuss normalization in Sec. IVA, show the shape ofthe potentials in Sec. IVB, and apply the formalism to theD �D� system in Sec. IVC. In Sec. IVD, we study the B �B�and flavor exotic DD� and BB� systems. We finish withsome general comments and conclusions in Sec. V.

II. BASICS

The effective one pion exchange potential between twolight quarks can be split into a central term proportional toð�i � �jÞð�i � �jÞ and a tensor term proportional to SijðrÞ�ð�i � �jÞ with SijðqÞ � 3ð�i � qÞð�j � qÞ � ð�i � �jÞ. �i are

the isospin matrices acting on light quark i, and �i are thePauli spin matrices acting on light quark i. The matrixelements of these operators capture all the spin and flavordependence. The remaining dependence on kinematics,normalization of the potential, and more detailed modelassumptions is discussed in Secs. III and IV.

We obtain the effective potential between two hadronsby summing over all the interactions between light u and dquarks/antiquarks. In heavy-light mesons such as D and Bthere is only one possible interaction to consider.

A. Charge conjugation and conventions

There has been some confusion in the literature as to thecorrect definition of the C ¼ þ state made of D �D� � c:c:.In Ref. [3], the state with definite C parity is defined byðP �VÞ� ¼ ½P �V � CðP �VÞ�. As C2 � 1 the above equation isself-consistent but does not specify the wave function untilone chooses a convention whether CVðPÞ ¼ � �Vð �PÞ. Onlythe neutral state is an eigenstate of C and so one has CV ¼� �V ¼ �V, which is consistent with C �V ¼ �V ¼ � �V.The intermediate step CVðPÞ ¼ � �Vð �PÞ is arbitrary (i.e.convention dependent).

In Refs. [2,6], the C ¼ þ state is defined to be D �D� þ�DD�, whereas in Refs. [8–10], it is claimed that the ‘‘cor-rectly argued’’ form is D �D� � �DD�. As the overall sign ofattraction versus repulsion depends on this sign, it is im-portant to understand the origin of these alternate forms.As a particular example of our discussion above, theD andD� are not eigenstates of C, and so the eigenvalues for theeigenstates D �D� � �DD� depend on what conventions areused to define the states.

Hence first we define our flavor states. For a q and �q ofgiven flavor and spin at positions 1 and 2, eigenstates of Cfor P � 0�þ and V � 1�� are

P: ðqð1Þ �qð2Þ þ �qð1Þqð2ÞÞ= ffiffiffi2

p

V: ðqð1Þ �qð2Þ � �qð1Þqð2ÞÞ= ffiffiffi2

p:

(1)

For a meson made of a heavy quark, Q, and light flavor q,we define the meson and ‘‘antimeson’’ to be

P: ðqð1Þ �Qð2Þ þ �Qð1Þqð2ÞÞ= ffiffiffi2

p;

�P: ð �qð1ÞQð2Þ þQð1Þ �qð2ÞÞ= ffiffiffiffi2;

pV: ðqð1Þ �Qð2Þ � �Qð1Þqð2ÞÞ= ffiffiffi

2p

;

�V: ð �qð1ÞQð2Þ �Qð1Þ �qð2ÞÞ= ffiffiffiffi2:

p(2)

With these definitions the C eigenstates are

CjP �V � �PVi ¼ �jP �V � �PVi: (3)

As Tornqvist [2] used this convention, we shall do so inorder to make most immediate comparison with his results.We agree however that there are possible advantages inusing the opposite convention, for example, in makingcontact with approaches that use interpolating currents inQFT as argued in Refs. [8–10].

B. Quark states and one pion exchange

Our plan in this section is to calculate the sign of the spinexpectation values of the central term in two cases: (i) P �Vor V �P in Jz ¼ þ1 and (ii) V �V in Jz ¼ 2. We will show thatthe C ¼ þ combination of ðP �V þ V �PÞ has the same signfor the spin operator as does V �V in Jz ¼ 2. This is oppositeto the results in Tables 2 and 5 of Ref. [3].As the � vertex connects D $ D� (antiparticles under-

stood here also) we will calculate

hP �V � �PVjHjP �V � �PVi ¼ �hP �V � �PVjHj �VP� V �Pi(4)

so that the sign of the effective matrix element will begiven by

� �hPjhjVih �Vjhj �Pi: (5)

The overall sign is therefore dependent on �.From general arguments, we have � ¼ 1 for the JP ¼

1þ state: because L is even, there is no phase change fromthe spatial wave function on swapping �Vð �PÞ and PðVÞ.There is no phase from the spin wave function becausespin 1 coupling with spin 0 to give spin 1 has the same signas spin 0 coupling with spin 1 to give spin 1. To show thisexplicitly we need to define the quark content of themesons, including their spin orientation. For simplicity,we consider the vector to be in state Jz ¼ þ1, and forshorthand write ½ �Qq� � �Q"q" (and similar for �qQ etc.),

while the S ¼ 0 state is denoted ð �QqÞ �ffiffi12

qð �Q"q# � �Q#q"Þ.

This implies that jP �Vi and jV �Pi are as follows:

jP �Vi ¼ ð �QqÞ½ �qQ� � ð �QqÞ½Q �q� � ðq �QÞ½Q �q� þ ðq �QÞ½ �qQ�(6)

and

C. E. THOMAS AND F. E. CLOSE PHYSICAL REVIEW D 78, 034007 (2008)

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jV �Pi ¼ �½ �Qq�ð �qQÞ � ½ �Qq�ðQ �qÞ þ ½q �Q�ðQ �qÞþ ½q �Q�ð �qQÞ (7)

and thus � ¼ 1 in Eqs. (4) and (5).It is sufficient to consider one ordering as long as

we are consistent in our conventions. We chooseAð �Q1q2ÞBðQ3 �q4Þ ! A0ð �Q1q

02ÞB0ðQ3 �q

04Þ where q and �q are

the light u and d quarks and antiquarks, and Q and �Q arethe heavy quarks and antiquarks. In this convention thespin and isospin operators act on the light quarks/anti-quarks i ¼ 2 and j ¼ 4.

For simplicity of presentation, just consider the termswhere the heavy �Q and Q are in positions 1 and 3, respec-tively; the analysis trivially applies to all other combina-tions with the same conclusion. The relevant terms are then

jP �Vi ¼ ð� �Q"q#Q" �q" þ �Q#q"Q" �q"Þ (8)

and

jV �Pi ¼ ð� �Q"q"Q" �q# þ �Q"q"Q# �q"Þ: (9)

The � exchange leaves the spins of theQð �QÞ unchangedand the V $ P spin transition comes from the qð �qÞ. Thenonzero transitions are then between the first term inEq. (8) and the first term in Eq. (9). In each case, theoperator ��þ gives þ1 and hence the overall sign fromthe spin contributions to hP �VjHjV �Pi ¼ þ.

Hence the sign is

hP �V � �PVjHjP �V � �PVi ¼ hP �V � �PVjHj �VP� V �Pi¼ � � SignðCÞ: (10)

The case of V �V with all spins aligned, S ¼ Sz ¼ 2, inthe conventions above is

jV �Vi ¼ ½ �Qq�½Q �q� � ½ �Qq�½ �qQ� � ½q �Q�½Q �q� þ ½q �Q�½ �qQ�:(11)

Here again, focusing on the terms where the heavy �Q andQ are in positions 1 and 3, respectively, and noting that theQð �QÞ spins do not flip, it is immediately obvious that thesign is positive:

hV �VjHjV �Vi ¼ þ; (12)

and hence the same as that for the ðP �V þ �PVÞ, C ¼ þchannel 1þþ.

The explicit inclusion of flavor (isospin) for the q �q �d �d� u �u (and appropriate charge conjugated form) intro-duces further signs, causing I ¼ 0 and I ¼ 1 channels tohave opposite behaviors. However these factors are com-mon to all of the above and do not change the generalconclusion that for a given isospin the C ¼ þ combinationof ðP �V þ V �PÞ has the same sign as does V �V in Jz ¼ 2.

It is possible to deduce the overall sign as follows. Startfrom NN with S ¼ 1 and I ¼ 0: the deuteron. This in-volves � exchange between q and q; there is no q� �qinteraction here. Now consider the case of VV with S ¼ 2

and I ¼ 0, which is like the deuteron in that, again, there isno q� �q interaction. The h�i � �ji has the same sign in

both cases. The � exchange (G-parity) gives opposite signbetween q� q and q� �q, and hence V �V with S ¼ 2 andI ¼ 0 has the opposite overall sign. So far everythingagrees with the calculations in Ref. [3]. It is only now,where Eqs. (10) and (12) imply that the C ¼ þ combina-tion of ðP �V þ V �PÞ has the same sign as does V �V in 5S2,and hence opposite to the deuteron, in contrast to Ref. [3].We shall show that when the spatial matrix elements are

calculated, we agree with the formulation in Refs. [7–9]and disagree with [3]. This ironically introduces a furtherrelative sign in the spatial contribution of Ref. [3] in thecharm sector, which, as we shall show, will eventuallycause the D �D� þ �DD� state to be mildly attractive, inpart as a result of two sign errors mutually cancelling.However, the spatial sign flip does not occur in the heavyquark limit, and hence some care is required in comparingB �B� þ �BB� to D �D� þ �DD�.

C. Overall sign

The overall sign is determined, inter alia, by the expec-tation values of ð�i � �jÞð�i � �jÞ and SijðqÞð�i � �jÞ. We

now consider the spin and isospin parts separately.The spin matrix element of the central term, ð�i � �jÞ,

can be calculated explicitly as above or using generalangular momentum theory. General expressions are givenin Appendix A. It is straightforward to show that the spinmatrix elements for PP ! PP, PV ! PV and VP ! VPvanish. For PV ! VP and VP ! PV we get þ1, and forVV ! VV we get 1

2 ðSðSþ 1Þ � 4Þ. These results agree

with the calculation given above (Sec. II B). The spinmatrix element of the tensor term, SijðqÞ � 3ð�i � qÞð�j �qÞ � ð�i � �jÞ, can be calculated in a similar way, see

Appendix A.The isospin factor is trivial to evaluate:

ð�i � �jÞ ¼ 1

2ðð�i þ �jÞ2 � �2

i � �2jÞ: (13)

For two isospin half mesons interacting, this is 2IðI þ 1Þ �3, i.e. �3 in total isospin I ¼ 0 or þ1 in I ¼ 1. Theinteraction potential between a quark and an antiquark isopposite to that between two quarks (or two antiquarks)because of the G-parity of the pion.Adopting the above results, for total spin S, isospin

states with isospin I and charge conjugation parity C, theflavor and spin factors for the central term are(i) VV: ðSðSþ 1Þ � 4ÞðIðI þ 1Þ � 3=2Þ.(ii) V �V: � ðSðSþ 1Þ � 4ÞðIðI þ 1Þ � 3=2Þ.(iii) PV: ð2IðI þ 1Þ � 3Þ.(iv) P �V: � Cð2IðI þ 1Þ � 3Þ.The VV and V �V expressions agree with those of

Tornqvist [3]. However, there are different overall minussigns in the PV and P �V expressions compared to those ofTornqvist.

Is Xð3872Þ A MOLECULE? PHYSICAL REVIEW D 78, 034007 (2008)

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For reference we note that the matrix elements for D �D�with JP ¼ 1þ, charge conjugation parity C, and isospin I,in the basis L ¼ 0, L ¼ 2 are

� Cf2IðI þ 1Þ � 3g"

1 00 1

� �VC þ 0 � ffiffiffi

2p

� ffiffiffi2

p1

!VT

#;

(14)

where VC and VT distinguish the central and tensor terms.When isospin symmetry is broken, we replace the ð�1 �

�2Þ factor by a 2 by 2 matrix

�1 �2�2 �1

� �;

in the basis of charged/neutral states 00, þ� . In theisospin limit, it is easy to see that this has eigenvalues�3 and 1 with the respective isospin eigenvectors. Closeand Page [5] and Tornqvist [4] discuss isospin symmetrybreaking.

If one only considers the particular interactionD0 �D�0 !D�0 �D0 and not every charge combination, the interactionstrength is reduced by a factor of 1=3 compared to theisospin symmetry limit with I ¼ 0. Liu et. al. [8] onlyconsider this particular charge interaction.

We can not really say whether the potentials are attrac-tive or repulsive until we know something about the kine-matic dependence which we discuss in the next section.

III. EFFECTIVE POTENTIAL IN POSITION SPACE

If the interaction between � and a light quark is taken tobe

L ¼ g

f��qðxÞ��5�qðxÞ � @��ðxÞ; (15)

the effective potential between two light quarks due to onepion exchange in the static limit is

VðqÞ ¼ g2

f2�

ð�i � qÞð�j � qÞq2 �m2

�

ð�i � �jÞ: (16)

Herem� is the �mass, f� is the � decay constant (defined

byffiffiffi2

p h0jA�ð0Þj�0ðqÞi ¼ if�0q� or h0jA�ð0Þj��ðqÞi ¼if��q� as in the PDG Review 2006 [11]), g is a dimen-

sionless coupling constant, and q is the four-momentumtransfer. N.B. in this section, for clarity, we omit the �1factor in V �V interactions and the �C factor in P �V inter-actions. These factors were discussed above in Sec. II C.For elastic scattering, this reduces to the form given inEricson and Weise [12]:

VðqÞ ¼ � g2

f2�

ð�i � qÞð�j � qÞjqj2 þm2

�

ð�i � �jÞ; (17)

VðrÞ ¼ g2

f2�ð�i � �jÞð�i � rÞð�j � rÞ e

�m�r

4�r; (18)

where q is the three-momentum transfer.Following Tornqvist [3], we define �2 � m2

� �ðmV �mPÞ2; mV is the vector meson mass and mP thepseudoscalar meson mass. Close to the static limit (bothinitial and final 3-momenta are zero) we have q20 � ðmV �mPÞ2 and so q2 �m2

� � �jqj2 ��2. In the limit of elasticscattering, � ¼ m�.We write the overall scale in terms of V0:

V0 � m3�g

2

12�f2�; (19)

and so the expression for the effective potential is

VðqÞ ¼ � 12�V0

m3�

ð�i � qÞð�j � qÞjqj2 þ�2

ð�i � �jÞ: (20)

This is the same as Tornqvist’s except for an overall minussign—this missing sign in his momentum space form turnsout not to be important because his expression for thepotential in position space does have the correct sign.Note that there are other definitions of� in the literature.

Suzuki [7] and Liu et. al. [8] use 2ðmV �mP �m�Þm� ��2, where the correspondence, with a minus sign, isexact in the static limit. As mentioned by Suzuki, theform of the potential in position space depends on thesign of �2.Following Ericson and Weise [12], it is useful to high-

light the different spin dependences in the potential. Thecentral potential is proportional to ð�i � �jÞ (hyperfinelikeinteraction) and the tensor term is proportional to SijðqÞ:

VðqÞ ¼ 4�V0

m3�

��1þ �2

jqj2 þ�2

�ð�i � �jÞ

� jqj2jqj2 þ�2

SijðqÞ�ð�i � �jÞ (21)

with SijðqÞ � 3ð�i � qÞð�j � qÞ � ð�i � �jÞ. The literature

[2–4,6–8,12] has focused on the potential in position spacebut disagree on its form. Hence we take the Fourier trans-form and compare with the literature.For �2 > 0:

VðrÞ ¼ V0

�� 4�

m3�

�ðrÞ þ �2

m2�

e��r

m�r


þ �2

m2�

e��r

m�r

�1þ 3

�rþ 3

�2r2

�SijðrÞ

�ð�i � �jÞ;

(22)

and for �2 � � ~�2 < 0 the real part is


034007-4

VðrÞ ¼ V0

�� 4�

m3�

�ðrÞ � ~�2

m2�

cos ~�r

m�r


� ~�2

m2�

1

m�r

�cos ~�r� 3 sin ~�r

~�r� 3 cos ~�r

~�2r2

�SijðrÞ

�� ð�i � �jÞ; (23)

where we have kept the terms in the same order.The expression with �2 > 0 reproduces that of

Tornqvist [3], except that he ignores the �ðrÞ term. Heallows for a varying �2 but his quoted potential appearsimplicitly to assume that �2 is always positive. In caseswhere �2 < 0, our potential agrees with that of Liu et. al.[8] (except that they have ignored the tensor term) and withthat of Suzuki [7] apart from some relative minus signs(where we confirm the form of Liu et. al.).

Regularization

The potential is singular at small distances and so has tobe regularized. Following Tornqvist and Liu et. al., we dothis by introducing a form factor at each � vertex which

leads to an extra factor of ð�2�m2�

�2�q2Þ2 multiplying VðqÞ

[Eq. (21)]. This gives the � an effective rms radius offfiffiffiffiffiffi10

p=� [3]. The value of � has to be determined phenom-

enologically. Tornqvist mentions that in nucleon-nucleoninteractions values between 0.8 and 1.5 GeV have beenused depending on the model and application, but thatlarger values (�> 1:4 GeV) are required for nucleon-nucleon phase shifts. He says that for heavy mesons whichhave a smaller size than nucleons, one would expect asmaller effective radius of the � source corresponding toa larger �. A larger � gives a stronger potential at shortdistances; we shall find in Sec. IV that the results dependstrongly on �.

The central terms (i.e. all but the tensor SijðrÞ) for

�2 > 0 are

V0

��Xð�2 �m2

�Þ2m3

�

e�Xr þ �2

m3�r

ðe��r � e�XrÞ�

� ð�i � �jÞð�i � �jÞ (24)

with X2 � �2 þ�2 �m2�. Tornqvist omits the ‘‘� func-

tion’’ piece (more precisely the piece that gave the deltafunction when no regularization was used), ‘‘which fromthe phenomenological point of view will be included in theshort range potential and regularization scheme’’ [3]. Thisresults in all his terms having a common �2 factor. If forcomparison we also ignore the � function piece we get

V0

�� 1

2

�2ð�2 �m2�Þ

Xm3�

e�Xr þ �2

m3�r

ðe��r � e�XrÞ�

� ð�i � �jÞð�i � �jÞ: (25)

Note that this is not quite the same as the expression inRef. [3], namely,

V0

��2ð�2 ��2Þ

�m3�

e��r þ �2

m3�r

ðe��r � e��rÞ�

� ð�i � �jÞð�i � �jÞ: (26)

The most significant difference (e.g. see Figs. 1 and 2) isdue to the missing factor of 1=2 in the first term. The otherdifferences are a m2

� ! �2 in one place and the approxi-mation X � �.We now comment on the case �2 � � ~�2 < 0. The real

part of the full central term (including the � function piece)is

FIG. 1. Central potential for P �V in I ¼ 0, L ¼ 0, JPC ¼ 1þþwith � ¼ 1 GeV, V0 ¼ 1:3 MeV, and �2 ¼ m2

�. (1) is ourpotential [Eq. (24)]; (2) is our potential without the � functionpiece [Eq. (25)], to be compared with Tornqvist’s analogous (i.e.no � function) potential [Eq. (26)] shown as curve (3).

FIG. 2. Central potential for P �V in I ¼ 0, L ¼ 0, JPC ¼ 1þþwith � ¼ 1 GeV, V0 ¼ 1:3 MeV, and �2 ¼ �2000 MeV2.(1) is our potential [Eq. (27)]; (2) is our potential without the� function piece [Eq. (28)]; (3) is Tornqvist’s potential modifiedfor negative �2.


034007-5

V0

��Xð�2 �m2

�Þ2m3

�

e�Xr � ~�2

m3�r

ðcosð ~�rÞ � e�XrÞ�

� ð�i � �jÞð�i � �jÞ: (27)

This expression agrees with Eq. (19) in the paper of Liuet. al.. If we ignore the � function piece we get

V0

�þ ~�2ð�2 �m2

�Þ2Xm3

�

e�Xr � ~�2

m3�r

ðcosð ~�rÞ � e�XrÞ�

� ð�i � �jÞð�i � �jÞ: (28)

The tensor terms for �2 > 0 are

VðrÞ ¼ V0

��2

m2�

e��r

m�r

�1þ 3

�rþ 3

�2r2

�

� X2

m2�

e�Xr

m�r

�1þ 3

Xrþ 3

X2r2

�

� ð�2 �m2�Þ

2m2�

e�Xr

m�rð1þ XrÞ

�SijðrÞð�i � �jÞ: (29)

This agrees with Eq. (26) in the paper of Tornqvist apartfrom one place in the last term where m2

� ! �2. For �2 �� ~�2 < 0 the real part of the tensor terms are

VðrÞ ¼ V0

�� ~�2

m2�

1

m�r

�cos ~�r� 3 sin ~�r

~�r� 3 cos ~�r

~�2r2

�

� X2

m2�

e�Xr

m�r

�1þ 3

Xrþ 3

X2r2

�

� ð�2 �m2�Þ

2m2�

e�Xr

m�rð1þ XrÞ

�SijðrÞð�i � �jÞ: (30)

We collate some useful relationships and Fourier trans-forms in Appendix B from which the derivation of theabove expressions may be checked. We shall use the formsEqs. (24) and (27) for the central term, and Eqs. (29) and(30) for the tensor term in our analysis. We shall illustratethe implications for binding by comparing with other ex-pressions in the previous literature.

IV. APPLICATIONS

A. Normalization

We have defined V0 in the same way as Tornqvist [3] andso we can write it in terms of the �-nucleon couplingconstant f�N:

f2�N4�

¼ 25

9

g2m2�

f24�¼ 25

9

3V0

m�

� 0:08: (31)

Tornqvist finds V0 � 1:3 MeV, which is consistent withEricson and Weise [12] and Ericson and Karl [1].

V0 can also be related to the D� ! D0�þ transition:

�ðD� ! D0�þÞ ¼ g2

6�f2�p3� ¼ 2V0

p3�

m3�

: (32)

The PDG Review 2006 [11] gives �ðD� ! D0�þÞ ¼ð65� 15Þ keV and so V0 ¼ ð1:5� 0:3Þ MeV which isconsistent with the above.Suzuki [7] appears to have an effective V0 � 0:73 MeV

and Liu et. al. [8] have V0 � 0:68 MeV. Note that there arevarious definitions of the � decay constant used in theliterature. We take the definition of the PDG Review 2006[11] where f� � 130 MeV. Compare this with the defini-

tion used in Swanson [6] where g2

f2�! g2

2f2�and f� ¼

92 MeV.We will take V0 ¼ 1:3 MeV throughout this work.

B. Shape of the potentials

In Figs. 1 and 2, we plot the central part of the potentialP �V in I ¼ 0, L ¼ 0, JPC ¼ 1þþ with � ¼ 1 GeV, V0 ¼1:3 MeV, and �2 ¼ m2

� or �2 ¼ �2000 MeV2, respec-tively. As well as our expression for the full central poten-tial [Eqs. (24) and (27)], for comparison we plot thepotential without the � function piece [Eqs. (25) and(28)] and Tornqvist’s expression for the potential[Eq. (26)]. Note that in all three cases we use our expres-sion for the spin and flavor factors as discussed in Sec. II C.Figure 1 shows that for positive�2 the � function term is

attractive, dominates at short distances, and is opposite insign to the other central terms. The effect of the differentfactor of 2 in Tornqvist’s potential compared to ours whenwe ignore the � function term can also be seen, in particu-lar, the sign change as r ! 0. The three potentials differ atshort distances but become indistinguishable for r * 2 fm.Figure 2 shows that for negative �2 the � function term

remains attractive, dominates at short distances, and is thesame sign as the other central terms. The effect of thedifferent factor of 2 in Tornqvist’s potential can again beseen, as can the fact that the three potentials differ at short

FIG. 3. Central potential for P �V in I ¼ 0, L ¼ 0, JPC ¼ 1þþwith V0 ¼ 1:3 MeV, �2 ¼ �2000 MeV2, and with our expres-sion for the potential [Eq. (27)] for various values of �.


034007-6

distances but are the same at long distances. As is apparentfrom comparing Figs. 1 and 2, the � function term does notchange sign when �2 ! ��2 whereas the other centralterms do. Thus the phenomenological conclusions maydepend significantly on how the � function term is treated.

In Fig. 3 we plot our expression for the central potentialwith the above parameters, �2 ¼ �2000 MeV2 and showthe effect of varying �. It can be seen that increasing �increases the strength of the potential at short distances andthat the potential is quite sensitive to �.

C. Application to D �D�

In this section, we discuss application of the aboveformalism to the D �D� system with JPC ¼ 1þþ. The me-sons can be in two different partial waves (L ¼ 0 or 2) andmixing between these may be important.

We solve the potential by discretizing the time-independent Schrodinger equation with the one pion ex-change potential and then diagonalizing the resulting ma-trix. We vary the number of points and the maximum radiusto check for discretization and finite volume effects.

To get some handle on the size of �, we first calculateand solve the effective potential for the deuteron. Weassume isospin symmetry and include both the centraland tensor terms. For comparison, we consider three differ-ent expressions for the central potential: (1) our full ex-pression [Eqs. (24) and (27)], (2) Our expression withoutthe � function piece [Eqs. (25) and (28)], and(3) Tornqvist’s expression [Eq. (26)]. We vary � to fixthe binding energy to � 2:2 MeV.

For potentials (1), (2), and (3) we find the values re-quired are � ¼ 960 MeV, 750 MeV, and 760 MeV, re-spectively. More details of the calculations and results aregiven in Appendix C. � gives an effective radius of thepion 1=� which is expected [3] to depend on the inter-acting hadron and so be smaller for the D �D� systemcompared to the deuteron. This argument implies that the� we have found from solving the deuteron is a lowerbound on the value of � that should be used for the D �D�system. The value of� required for the B �B� system shouldbe larger again.

Moving to the D �D� system, we verify the results ofRef. [8] within the assumptions made there (noting thepossible difference in normalization discussed, above, inSec. IVA). Specifically, we initially considered only theD0 �D�0 þ �D0D�0 charge combination, used our full expres-sion for the central term (1) and ignored the tensor term. Inthis case, the isospin factor is �1 [from the matrix belowEq. (14)] instead of �3 for an I ¼ 0 state. The overallsign is therefore þ1 (Sec. II C) and one pion exchange isslightly inelastic with a negative�2 (�2 ¼ �2000 MeV2).For a reasonable range of� (we checked up to 2.5 GeV) wefind no bound state, supporting the results of Ref. [8]within their assumptions. However, for a larger � of3 GeV, we do find a bound state with energy� 3868 MeV.

However, this shows how sensitive the results are to suchassumptions. For example, if we now assume isospin sym-metry with I ¼ 0 (isospin factor is �3) and do not changeanything else, we do find a bound state. For � in the range960 to 1200 MeV we find E ¼ 3871–3863 MeV. Thishighlights the importance of including all the possiblecharge combinations.We then performed a more complete calculation includ-

ing the tensor term. We also allow for isospin symmetrybreaking through different charged/neutral meson massesand �2’s. The �2’s are all negative except for D�0 ! Dþwhich has a positive �2. Because we can only use one�2 for each charge mode, there is ambiguity as to whetherwe should calculate the charged-neutral �2 from(A) D�þ ! D0 or (B) D�0 ! Dþ. We will compare theresults obtained with both choices (A) and (B).The detailed results are given in Table III of

Appendix D. With our full expression for the potential(1), we find a bound state with E ¼ 3870–3806 MeV for� � 960–1400 MeV. For � ¼ 800 MeV we find nobound state. This shows that the binding energy is verysensitive to �. The choice of �2 (A) or (B) modifies thebinding energy by at most 1 MeV. In particular, Table IIIshows that isospin symmetry breaking is important, espe-cially when the state is close to threshold. This is notparticularly surprising: the charged-neutral mass differ-ence gets relatively more important when the state isonly just bound. The state is mostly L ¼ 0 with a smallL ¼ 2 component.Using potentials (2) and (3), we require a larger � for a

bound state (� * 1750 MeV) because these potentials donot have the strong attraction of the � function term. Wefind E ¼ 3871–3863 MeV for � � 1750–2000 MeV. Thedifferences between potentials (2) and (3) do not signifi-cantly change the binding energy or other properties, asshown in Table III.In summary, we have found a bound state for reasonable

values of � using all three potentials. The binding energyis sensitive to the potential and the value of� used. Clearlythe exact values of the binding energies calculated are notsignificant, because of their sensitivity to parameters andthe potential. Nonetheless, the general conclusion thatwithin one pion exchange potentials a bound state can beformed appears to be robust. Conversely, there is no theo-retical reason to expect that it must be formed, nor that itcannot occur. Basically, the phenomena of the Xð3872Þ,within the assumption of one � exchange, would constrainthe parameter � more than present knowledge of thisparameter can constrain the DD� dynamics.

D. Further applications

We can also apply a similar analysis to the analogousB �B� system. Furthermore isospin breaking is expected tobe less important here because of the relatively smallercharged/neutral differences. The larger B meson masses


034007-7

mean that the kinetic energy is less of a hindrance tobinding.

Results are given in Table IVof Appendix D which showthat a bound state is found for reasonable values of �, thebinding energy is sensitive to �, and isospin breaking isless important than in the D �D� system. We note that thereare larger L ¼ 2 components here compared to the D �D�system, especially when using potentials (2) and (3).Because �2 is positive here, the central terms other thanthe � function term are repulsive and so the tensor term isimportant for forming a bound state.

The same analysis again can be applied to flavor exotics,that is states such as DD� and BB� with two charm orbottom quarks. The one pion exchange interaction in suchstates has opposite overall sign to that in the D �D� and B �B�systems (Sec. II C). However, mixing with the tensor termscomplicates the situation and we must do more than justconsider the sign of the central term.

For the DD� system, results are given in Table V ofAppendix D. With potential (1) we find a bound state withenergy � 3870 MeV for � ¼ 2000 MeV. There is nobound state for � ¼ 1700 MeV. The importance of thetensor term is apparent from the relatively large L ¼ 2component. Theþ0 and 0þ components have the oppositesign and so the state would be isoscalar in the isospinsymmetry limit. Using potential (2) or (3) we find a boundstate with energy � 3870 MeV for � ¼ 1500 MeV, abound state with energy � 3810 MeV for � ¼2000 MeV, and no bound state for � ¼ 1000 MeV.Again, there are relatively large L ¼ 2 components.

For the BB� system, results are given in Table VI ofAppendix D. With potential (1), we find a bound state withenergy� 10600 MeV with� ¼ 1000 MeV, a bound statewith energy � 10560 MeV with � ¼ 1500 MeV, and nobound state for � ¼ 700 MeV. There are again relativelylarge L ¼ 2 components and the þ0 and 0þ componentshave opposite sign. Using potential (2) or (3) we find abound state with energy � 10600 MeV for � ¼700 MeV, a bound state with energy � 10590 MeV for� ¼ 1000 MeV, and no bound state for � ¼ 500 MeV.Again there are relatively large L ¼ 2 components.

In summary, we find great sensitivity to parameters, butqualitatively confirm that B �B� and exotic states can bind in

one pion exchange. As can be seen from Table I, using ourexpression for the potential (1), we find D �D�, B �B�, andBB� can bind with � 1000 MeV. However, DD� re-quires a larger � 2000 MeV. Hence, within our poten-tial and assumptions, the parameters that allow Xð3872Þ toemerge as a bound state preclude binding the exotic DD�channel. However, binding in both B �B� and exotic BB� arepossible.

V. CONCLUSIONS

In summary, we agree with the qualitative results ofTornqvist, but as a result of various differences cancellingout. Swanson [6] found that quark exchange alone did notbind within the one gluon exchange contact approximation.This led him to include one pion exchange based upon thework of Ref. [2].We have quantified the arguments of Ref. [7] that a small

�2 leads to a small binding energy. In doing so, we haveincluded the tensor term, and also flavor factors, neither ofwhich were discussed in that reference. Our calculationsshow that the sensitivity to � is the overriding factor.Liu et. al. [8] considered only D0 �D0�; charged modes

were ignored, as was the tensor term. Within their assump-tions we confirm their results, though there is the questionof overall normalization. Our work highlights the impor-tance of taking into account all charged modes. They havealso considered � exchange, and argue that this makes itharder for the D �D� to bind.We have discussed flavor exotic states and find that the

overall sign of the central term alone does not determinewhether or not a bound state is formed; mixing due to thetensor term is also important. It can be dangerous to ignorethe tensor term and its contribution to different processescan be important: this is especially true for the case offlavor exotics.

ACKNOWLEDGMENTS

We are indebted to S.-L. Zhu, V. Lyubovitskij,E. Swanson, and N. Tornqvist for discussions of theirwork, and to C. Downum, R. C. Johnson, and Q. Zhaofor other useful discussions. This work is supported bythe Science & Technology Facilities Council (UK).

APPENDIX A: SPIN MATRIX ELEMENTS

We choose Að �Q1q2ÞBðQ3 �q4Þ ! A0ð �Q1q02ÞB0ðQ3 �q

04Þ

where q and �q are the light u and d quarks and antiquarks,and Q and �Q are the heavy quarks and antiquarks. In thisconvention, the spin operators act on the light quarks/antiquarks i ¼ 2 and j ¼ 4.In general, for initial state mesons with spin SA and SB

coupled to total spin ST going to final state mesons S0A and

S0B coupled to total spin S0T , the matrix element of thecentral term, ð�2 � �4Þ is given by

TABLE I. Approximate minimum values of � required tobind.

System Potential Approximate �=MeV

D �D� 1 960

2 and 3 1750

B �B� 1 <8002 and 3 1000

DD� 1 2000

2 and 3 1500

BB� 1 1000

2 and 3 700


034007-8

hS0AS0BðS0TM0Þjð�2 � �4ÞjSASBðSTMÞi ¼ �S0T ;ST�M0;MX

S13S24

26642ðS24ðS24 þ 1Þ � 3=2Þ�SASBS

0AS0BS13S13S24S24

8>><>>:1=2 1=2 SA

1=2 1=2 SB

S13 S24 ST

9>>=>>;

�

8>><>>:1=2 1=2 S13

1=2 1=2 S24

S0A S0B S0T

9>>=>>;3775; (A1)

where fg are Wigner 9j symbols and �ABC... �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Aþ 1Þð2Bþ 1Þð2Cþ 1Þ . . .p

.From this expression, it is straightforward to show that the spin matrix elements for PP ! PP, PV ! PV, and VP !

VP vanish. For PV ! VP and VP ! PV, we get þ1 and for VV ! VV, we get 12 ðSTðST þ 1Þ � 4Þ. These results agreewith the direct calculation given above.

The matrix element of the tensor term, S24ðqÞ � 3ð�2 � qÞð�4 � qÞ � ð�2 � �4Þ, can be calculated in a similar way [13].With the spins defined above and with the total spin JAB, J

0AB, relative orbital angular momentum L, L0 and total angular

momentum J, J0 ¼ J in the initial and final states, respectively, the matrix element is

hA0B0jS24ðqÞjABi ¼ 4

ffiffiffi5

2

s X~JS13S24

2664�S24;1ð�1ÞL0þJABþJ0ABþ~JþS24�SASBS

0AS0BS13S13S24S24S24JABJ

0AB

~J ~J L

8>><>>:1=2 1=2 SA

1=2 1=2 SB

S13 S24 JAB

9>>=>>;

�

8>><>>:1=2 1=2 S0A1=2 1=2 S0BS13 S24 J0AB

9>>=>>;(L S24 ~J

S13 J JAB

)(L0 S24 ~J

S13 J J0AB

)(L0 L 2

S24 S24 ~J

)hL0; 0jL; 0; 2; 0i

3775; (A2)

where fg are Wigner 6j and 9j symbols and hL0; 0jL; 0; 2; 0i is a Clebsch-Gordan coefficient.

APPENDIX B: USEFUL EXPRESSIONS FOR CALCULATING THE POTENTIAL IN POSITION SPACE

We collate useful Fourier transforms used in Table II where

VðrÞ ¼ 1

ð2�Þ3Z

VðqÞeiq�rd3q: (B1)

TABLE II. Summary of Fourier transforms used.

VðqÞ VðrÞ1 �ðrÞ

1jqj2þ�2

12�2

R10

jqj2j0ðjqjrÞjqj2þ�2 djqj

14�

e��r

r14�

ei ~�r

r

S12ðqÞ jqj2jqj2þ�2 �S12ðrÞ 1

2�2

R10

jqj4j2ðjqjrÞjqj2þ�2 djqj

�S12ðrÞ �2

4�e��r

r ð1þ 3�r þ 3

ð�rÞ2ÞS12ðrÞ ~�2

4�ei ~�r

r ð1þ 3i~�r � 3

ð ~�rÞ2ÞjFðjqj2Þj2 ð�2 �m2

�Þ2 18�

e�Xr

XjFðjqj2Þj2jqj2þ�2

12�2

R10

jqj2jFðjqj2Þj2j0ðjqjrÞjqj2þ�2 djqj

14� ½e

��r

r � e�Xr

r � ð�2�m2�Þ

2X e�Xr�14� ½e

i ~�r

r � e�Xr

r � ð�2�m2�Þ

2X e�Xr�S12ðqÞ jqj2jFðjqj2Þj2jqj2þ�2 �S12ðrÞ 1

2�2

R10

jqj4jFðjqj2Þj2j2ðjqjrÞjqj2þ�2 djqj

�S12ðrÞ 14� ½�2 e��r

r ð1þ 3�r þ 3

ð�rÞ2Þ � X2 e�Xr

r ð1þ 3Xr þ 3

ðXrÞ2Þ � ð�2 �m2�Þ e�Xr

2r ð1þ XrÞ��S12ðrÞ 1

4� ½� ~�2 ei ~�r

r ð1þ 3i~�r � 3

ð ~�rÞ2Þ � X2 e�Xr

r ð1þ 3Xr þ 3

ðXrÞ2Þ � ð�2 �m2�Þ e�Xr

2r ð1þ XrÞ�


034007-9

A useful decomposition is

3ð�1 � qÞð�2 � qÞ

jqj2 ��2¼ ð�1 � �2Þ ð�1 � �2Þ �2

jqj2 ��2

þ S12ðqÞ jqj2jqj2 ��2

(B2)

with S12ðqÞ � 3ð�1 � qÞð�2 � qÞ � ð�1 � �2Þ.The form factor is jFðjqj2Þj2 � ð�2�m2

�

jqj2þX2Þ2 with X2 ��2 þ�2 �m2

� ¼ �2 � ~�2 �m2� and ~�2 � ��2.

APPENDIX C: DEUTERON POTENTIAL

The deuteron is a combination of two nucleons in anisosinglet state with J ¼ 1, S ¼ 1, L ¼ 0 or 2. Because wehave to sum over interactions between all the light quarks,relative to the heavy-light mesons the spin factors arechanged to

� 25

3

1 00 1

� �

for the central term and

� 25

30

ffiffiffi8

pffiffiffi8

p �2

!

for the tensor term [3]. The mixing between the L ¼ 0 andL ¼ 2 states is important in binding the deuteron.If we follow Tornqvist and assume that pion exchange is

the only binding mechanism, we can require the bindingenergy to be � 2:22 MeV and so fix the scale �. We takeV0 ¼ 1:3 MeV throughout.We use three different expressions for the potential:(1) our full expression given in Eq. (24),(2) our expression without the � function piece,

Eq. (25),(3) Tornqvist’s expression [3].We solve the potential by discretizing the time-

independent Schrodinger equation and then diagonalizingthe resulting matrix. We take N ¼ 500 or N ¼ 1000 pointsfor each L and use maximum radii R0 ¼ 0:10 MeV�1 and0:15 MeV�1 to check for finite volume effects. We set� ¼m� ¼ 135 MeV and the reduced mass � ¼ 1=ð1=mn þ1=mpÞ with mn ¼ 939:57 MeV and mp ¼ 938:27 MeV.

Our results are(i) Using potential (1) we obtain a binding energy E ¼

2:23 MeVwith� ¼ 962 MeV and we find a relativeamplitude squared of 0.93 in L ¼ 0 and 0.07 in L ¼2 with a rms radius in L ¼ 0 of 0:02 MeV�1.

(ii) With potential (2) we obtain a binding energy E ¼2:20 MeV with � ¼ 752 MeV and we find a rela-tive amplitude squared of 0.94 in L ¼ 0 and 0.06 inL ¼ 2 with a rms radius in L ¼ 0 of 0:02 MeV�1.

(iii) While potential (3) gives a binding energy E ¼2:22 MeV with � ¼ 760 MeV and we find a rela-

TABLE III. D �D� results. The amplitudes squared of the different components are given along with the rms radius of the 00, L ¼ 0component. The þ� and 00 L ¼ 0 components have the same sign (i.e. would be an isoscalar state in the isospin symmetry limit).

Potential �2 Set �=MeV E=MeV 00, L ¼ 0 þ� , L ¼ 0 00, L ¼ 2 þ� , L ¼ 2 rms Radius=MeV�1

(1) (A) and (B) 800 (not bound)

(1) (A) 962 3870 0.76 0.22 0.01 0.01 0.02

(1) (B) 962 3870 0.79 0.19 0.01 0.01 0.02

(1) (A) 1000 3868 0.69 0.28 0.02 0.02 0.01

(1) (B) 1000 3869 0.70 0.27 0.02 0.02 0.01

(1) (A) 1400 3806 0.50 0.46 0.02 0.02 0.003

(1) (B) 1400 3807 0.50 0.46 0.02 0.02 0.003



(2) (A) 1750 3871 0.86 0.11 0.02 0.02 0.03

(2) (B) 1750 3871 0.90 0.08 0.01 0.02 0.03

(2) (A) 1800 3870 0.78 0.17 0.02 0.03 0.02

(2) (B) 1800 3870 0.81 0.14 0.02 0.02 0.02

(2) (A) 2000 3863 0.59 0.33 0.04 0.04 0.007

(2) (B) 2000 3863 0.60 0.32 0.04 0.04 0.008



(3) (A) 1750 3871 0.86 0.11 0.02 0.02 0.03

(3) (B) 1750 3871 0.90 0.08 0.01 0.01 0.03

(3) (A) 2000 3863 0.59 0.33 0.04 0.04 0.007

(3) (B) 2000 3863 0.60 0.32 0.04 0.04 0.008


034007-10

tive amplitude squared of 0.94 in L ¼ 0 and 0.06 inL ¼ 2 with a rms radius in L ¼ 0 of 0:02 MeV�1.

Although the results are sensitive to �, once this is fixed

the other properties do not depend strongly on the details ofthe potential.

APPENDIX D: TABLES OF RESULTS

TABLE VI. Exotic BB� results. The amplitudes squared of the different components are given. The 0þ and þ0 L ¼ 0 componentshave the opposite sign (i.e. would be an isoscalar state in the isospin symmetry limit).

Potential �=MeV E=MeV 0þ , L ¼ 0 þ0, L ¼ 0 0þ , L ¼ 2 þ0, L ¼ 2

(1) 700 (not bound)

(1) 1000 10 600 0.33 0.38 0.15 0.15

(1) 1500 10 560 0.27 0.28 0.23 0.23

(2) 500 (not bound)

(2) 700 10 600 0.38 0.46 0.08 0.08

(2) 1000 10 590 0.35 0.36 0.15 0.15

(3) 500 (not bound)

(3) 700 10 600 0.38 0.47 0.08 0.08

(3) 1000 10 590 0.35 0.36 0.15 0.15

TABLE V. Exotic DD� results. The amplitudes squared of the different components are given. The 0þ and þ0 L ¼ 0 componentshave the opposite sign (i.e. would be an isoscalar state in the isospin symmetry limit).

Potential �=MeV E=MeV 0þ , L ¼ 0 þ0, L ¼ 0 0þ , L ¼ 2 þ0, L ¼ 2

(1) 1700 (not bound)

(1) 2000 3869 0.46 0.36 0.09 0.09


(2) 1500 3871 0.49 0.39 0.06 0.06

(2) 2000 3814 0.39 0.38 0.12 0.12


(3) 1500 3871 0.49 0.39 0.06 0.06

(3) 2000 3814 0.39 0.38 0.12 0.12

TABLE IV. B �B� results. The amplitudes squared of the different components are given along with the rms radius of the 00, L ¼ 0component. The þ� and 00 L ¼ 0 components have the same sign (i.e. would be an isoscalar state in the isospin symmetry limit).

Potential �=MeV E=MeV 00, L ¼ 0 þ� , L ¼ 0 00, L ¼ 2 þ� , L ¼ 2 rms Radius=MeV�1

(1) 800 10 580 0.47 0.48 0.03 0.03 0.004

(1) 962 10 540 0.47 0.48 0.03 0.03 0.002

(1) 1000 10 530 0.47 0.48 0.03 0.03 0.002

(1) 1400 10 270 0.47 0.47 0.03 0.03 0.001

(2) 752 (not bound)

(2) 1000 10 600 0.40 0.50 0.06 0.05 0.010

(2) 1400 10 580 0.41 0.42 0.08 0.08 0.004

(2) 1750 10 520 0.41 0.41 0.09 0.09 0.003

(3) 760 (not bound)

(3) 1000 10 600 0.40 0.49 0.06 0.05 0.010

(3) 1400 10 580 0.41 0.42 0.08 0.08 0.004

(3) 1750 10 520 0.41 0.41 0.09 0.09 0.003


034007-11

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[12] T. Ericson and M. Weise, Pions and Nuclei (ClarendonPress, Oxford, 1988).

[13] R. C. Johnson (private communication).


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