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Strong decays of the glueball in a scaled effective Lagrangian
Nuclear Physics A 662 Ž2000. 157–174
Strong decays of the glueball in a scaled effective Lagrangian M. Jaminon b
a,1
, M. Mathot b, B. Van den Bossche
b
a Instituto de Fısica Teorica, UniÕersidad Autonoma de Madrid, Madrid, Spain ´ ´ ´ UniÕersite´ de Liege, ` Institut de Physique B5, Sart Tilman, B-4000 Liege ` 1, Belgium
Received 26 January 1999; received in revised form 27 July 1999; accepted 10 August 1999
Abstract We calculate the scalar glueball decays into pp , hh , hhX and KK within a modified version of the NJL model which implements the QCD trace anomaly. The glueball is assumed to be the f 0 Ž1500.. It is coupled to the quarks due to its mixing with the scalar isoscalar mesons. The decay widths then contain a contribution coming from the triangle Feynman diagram. Moreover, they contain two additional contributions corresponding to the local decay of the glue component of the f 0 Ž1500.: one comes from the process of regularization of the quark loop, the other from the interaction term a2x 2f 2. Except for the hhX case, this local decay reduces the value of the width. The axial anomaly is introduced via a mass term for the hX. As far as the lightest scalar glueball is identified with the f 0 Ž1500., the model cannot reproduce at the same time the dynamical properties of scalars and pseudoscalars. Since we do not include a model of confinement, we need to use a large quark mass Ž725 MeV. in our calculations. This leads to unphysical results for the two-pion decay width. A definitive test of our model awaits the introduction of a model of confinement, which we hope to include in a future work. q 2000 Elsevier Science B.V. All rights reserved. PACS: 12.39.Fe; 12.39.Mk; 14.40.yn
Keywords: Scalar mesons; Chiral Lagrangian; NJL; Glueball; hh , hhX decays
1. Introduction Up to very recently, the f 0 Ž1500. appeared to decay into pp , hh , hhX and KK w1–4x with the corresponding branching ratios, corrected for phase space effects,
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:0.27 " 0.11:0.19 " 0.08:0.24 " 0.09 , Ž 1. and a total width Gf 0 Ž1500. s 112 " 10 MeV w5x. 1 On leave of absence from Universite´ de Liege, ` Institut de Physique B5, Sart Tilman, B-4000 Liege ` 1, Belgium.
0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 3 5 8 - 9
158
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However, the results quoted in Ž1. do not refer to the same form factor, being then in some sense inconsistent. The very recent report w6x of the latest data for the f 0 Ž1500. suggests that the calculated branching ratios, corrected for phase space effects and form factor, are
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:0.195 " 0.075:0.320 " 0.114:0.138 " 0.038
Ž 2.
for a mass of 1505 " 9 MeV and a total decay width of 111 " 12 MeV. Taking the widths Ga 0 s 265 " 13 MeV and GK 0) s 287 " 23 MeV as a scale for the members of the scalar nonet this small value of Gf 0 Ž1500. seems incompatible with a nonet structure. Moreover, Amsler and Close w7,8x have argued that the branching ratios could not be explained with a f 0 Ž1500. assumed as to be a pure qq state. The latter was identified with a nn s 1r '2 Ž uu q dd . state, regarding to the strong coupling of the f 0 Ž1500. to the pp channel. In contrast, Ritter et al. w9x have reproduced the experimental branching ratios assuming a ss content for the f 0 Ž1500. with, however, a value of the total width which remains of the order of magnitude of GK 0) that is twice larger than the experimental value. In the same way, the valence approximation of Ref. w10x assigns a mass to the scalar ss that lies around 200 MeV below the lightest scalar glueball identified with the f 0 Ž1710.. Moreover, lattice calculations w10,11x indicate a very small value Žf 100 MeV. for the decay of the lightest glueball into two pseudoscalars, value still reduced Žf 30 MeV, with a large hh component. if dressing of the glueball due to some qq content is taken into account w12x. The experimental value is there recovered by introducing the opening of multi-pion channels, mimicked by a 4p channel. In a previous paper w13x, two of us have followed the idea of Ref. w7,8x, identifying the glueball with f 0 Ž1500. and have calculated its decay into two photons. In the present paper we go on this line and we compute the strong decays of the glueball f 0 Ž1500. into two pseudoscalars, using a modified version of the SUŽ3. NJL model w14,15x which implements the trace anomaly of QCD w16x. This is done by introducing one single point-like scalar dilaton field x whose mean value x 0 is identified with the vacuum gluon condensate w17x. Keeping in mind the limitations of the model – large breaking of scale invariance assigned to a single pole w13,18x, nonrenormalizability, nonconfinement – it has however the merit to provide a qq content to the glueball. It can therefore describe processes as the one shown in diagram a of Fig. 1. In our model, it is the value of x 0 which fixes the mixing with the scalar mesons: the smaller x 0 the larger the mixing. In order to regularize the quark loop, one has to introduce a cutoff Lx where the x field accounts for the scale invariance of the trace. In a similar way, the scale invariance requires to divide by x the strength of the 4-quark coupling. This provides an additional contribution to the f 0 Ž1500. decays, a contribution that reduces the numerical value of the respective widths except for the hhX case. This contribution involves the coupling constant of the f 0 Ž1500. to the pure glue Žsee diagram b of Fig. 1.. However, it has nothing to do with a process where one would calculate the overlap of the glue content of the f 0 Ž1500. with some pseudoscalar glue content of the pseudoscalar mesons. The latter could be achieved following the idea of Ref. w19x where the axial anomaly is introduced by mixing the h and hX with a pseudoscalar glueball. We plan to look for this effect in a forthcoming paper w20x.
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Fig. 1. Schematic diagrams for the decay of the glueball into two pseudoscalars.
Our paper is organized as follows. Section 2 contains the useful tools of the scaled NJL model. Section 3 derives the strong decay widths of the f 0 Ž1500.. Section 4 insists on the problems raised by the model and on its merits. Our results are presented in Section 5. Section 6 contains our conclusions.
2. The model The scaled NJL model is a modified version of the NJL model that has been constructed w17x in order to implement the QCD trace anomaly. It entails the introduction of a scalar field x whose mean value is intimately related to the vacuum gluon condensate. Since the model has already been intensively described in preceding papers w14,15,17,21x, we only recall here the useful tools for the understanding of the present paper. We use the following bosonized Euclidean action: Ieff Ž w , x . s yTrL x ln Ž yi Em gm q m q Ga wa . q d 4 x
H
q d 4 x Lx q d 4 x LUA Ž1. ,
H
H
a 2x 2 2
wa wa
Ž 3.
with the meson fields wa s Ž sa ,pa . and Ga s Ž l a ,ig 5 l a . for a s 0, . . . , 8. The l a are the usual Gell–Mann matrices with l0 s 2r3 1. We work in the isospin symmetry
'
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limit and the quantity m stands for the diagonal matrix diagŽ m u ,m u ,m s .. The Lagrangian which implements the scale anomaly has been originally introduced in Ref. w16x, 2
Lx s 12 Ž Em x . q 161 b 2 x 4 ln
x4 xG4
y Ž x 4 y xG4 . ,
Ž 4.
introducing two additional parameters b 2 and xG with respect to the usual NJL. The axial anomaly is introduced via a mass term for the field p 0 which would be the hX if the mixing between p 0 and p 8 were neglected, a2 LUA Ž1. s
2
j x 2p 02 .
Ž 5.
In a forthcoming paper w20x we plan to introduce the anomaly by coupling the p 0 and p 8 fields with a pseudoscalar glueball field. This would be consistent with the fact that the quadridivergence of the axial current is proportional to the dual tensor F˜mna w22x. Moreover, the model so constructed would be quite elegant due to the analogy between the mixing of the scalars with a scalar glueball and the one of the pseudoscalars with a corresponding pseudoscalar glueball. Our model contains seven parameters: the current quark masses Ž m u ,m s ., the strengths Ž a2 ,b 2 ., the gluon parameter xG , the cutoff L and the axial anomaly strength j . Five of these parameters are adjusted to reproduce the pion mass Ž mp ., the weak pion decay constant Ž fp ., the kaon and hX masses and that of the glueball f 0 Ž1500.. Using the usual gap equation w14,15x, the strength a 2 can be connected to the constituent up quark mass Mu which is here chosen rather large Ž Mu s 725 MeV. to reproduce the a 0 Ž1450. w13x. One remains with one free parameter x0.
3. The strong decay widths In the limit of isospin symmetry, the widths for the strong decays of the glueball f 0 Ž1500. are given by
Gf 0 Ž1500. a b s
NI
1
N 2 m f 0 Ž1500. Ž 2p . NI
s
)
a , b s p ,h ,h X , K ,
2
HH 2 E
a
1
16 Np m f 0 Ž1500. =
d 3 k1 d 3 k 2
1
1y
ž
Tf 0 Ž1500. a b
ma q mb m f 0 Ž 1500 .
2
2
/
2 Eb
2
Tf 0 Ž1500. a b d Ž q y k 1 y k 2 .
1y
ž
ma y mb m f 0 Ž 1500 .
2
/
,
Ž 6.
with NI s Ž2 I q 1.. N s 2 for identical emerging particles, N s 1 otherwise. In Eq. Ž6., m f 0 Ž1500. denotes the mass of the glueball, q its quadrimomentum while k 1 and k 2 stand for the quadrimomenta of the emerging mesons. The glueball is assumed to be at rest
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and the pseudoscalar mesons on their mass shell. To obtain the various amplitudes arising in Eq. Ž6., one has to expand the action Ž3. up to the third order into the meson fields. Writing Ieff Ž w , x . s yTrL x ln Ž Gy1 ˜a . q d4 x 0 . y TrL x ln Ž 1 q G 0 Ga w
H
a 2x 2 2
Ž wa2 q jp 02 .
q d 4 x Lx ,
Ž 7.
H
with Ž was corresponding to the stationary point of the action. s Gy1 0 s yi Em gm q m q Ga wa ,
wa s was q w˜ a ,
Ž 8.
one gets, using the scaling of the strong coupling constant with the number of colors Ž a2 s a˜ 2 Nc ., Nc
3 Ieff Ž w,x . sy
q
3 Nc 2
3
TrLX x 0Ž G 0 Ga w˜ a . q Nc d 4 x
E Ex
H
X TrLx Ž G 0 Ga w˜ a .
2
a˜ 2 2
2 x 0 x˜ Ž w˜ a2 q jp˜ 02 .
x˜ PPP ,
Ž 9.
x0
with Tr s NcTrX , TrX being the trace over space-time and flavor and spin indices and x s x 0 q x˜ . In Eq. Ž9. only the terms that can give the decay of the glueball into two pseudoscalar mesons have been specified. The first term corresponds to the triangle Feynman diagram Žsee Fig. 1, diagram a.. It is studied in Subsection 3.1. The second and third terms correspond to the local decay of the glue component of the f 0 Ž1500. Žsee Fig. 1, diagram b.. They are investigated in Subsection 3.2. 3.1. Triangle diagram We calculate here the contribution to the decay widths coming from the triangle diagram depicted in Fig. 1, diagram a. It writes . ) )2 Tf Ž0tŽ1500. p 0 p 0 s 2 2 g f 0 Ž 1500 . u u gp u u Ju u u Ž q,qp . ,
Ž 10 .
. ) )2 Tf Ž0tŽ1500. hh s 2 2 g f 0 Ž 1500 . u u gh u u Ju u u Ž q,qh .
q '2 g f)0 Ž 1500 . s s
ž
/ ž '2 g / ) hss
2
Js s s Ž q,qh . ,
Ž 11 .
. ) )2 Tf Ž0tŽ1500. h X h X s 2 2 g f 0 Ž 1500 . u u gh X u u Ju u u Ž q,qh X .
q '2 g f)0 Ž 1500 . s s
ž
/ ž '2 g / ) hX s s
2
Js s s Ž q,qh X . ,
Ž 12 .
. ) ) ) Tf Ž0tŽ1500. hh X s 2 g f 0 Ž1500.u u gh u u gh X u u Ju u u Ž q,qh . q Ju u u Ž q,qh X .
q '2 g f)0 Ž1500. s s
ž
/ ž '2 g / ž '2 g / ) hss
) hX s s
Js s s Ž q,qh . q Js s s Ž q,qh X . ,
Ž 13 .
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2
. ) ' ) Tf Ž0tŽ1500. K q K y s g f 0 Ž1500.u u Ž 2 g K u s . Ju su Ž q,q K q .
q '2 g f)0 Ž1500. s s
ž
2 ) K us
/ Ž'2 g .
Jsu s Ž q,q K y . ,
Ž 14 .
with Ž i, j,k s u, s . Ji jk Ž q,qa . s yNc Trg
d4 k
H Ž 2p .
g i Ž k q qa . ig 5 g j Ž k . ig 5 g k Ž k y q q qa . ,
4
Ž 15 . gi Ž k .
y1
s km gm q Mi ,
Ž 16 .
or Ji i i Ž q,qa . s 4 Nc Mi =
d4 k
H Ž 2p .
4
k 2 q Mi2 q qa Ž q y qa . 2 2 Ž k q qa . q Mi2 Ž k 2 q Mi2 . Ž k y q q qa . q Mi2
Ž 17 .
and
Ji ji Ž q,qa . s4Nc Mi
d4 k
H Ž 2p .
4
ž
k 2 q Mi2 q qa Ž q y qa . q 1 y =
Mj Mi
/
Ž k q qa y q . Ž k q qa . y Mi2
2 2 Ž k q qa . q Mi2 Ž k 2 q M j2 . Ž k q qa y q . q Mi2
Ž 18 .
calculated for q 2 s ym2f 0 Ž1500. , qa2 s yma2 , Ž q y qa . 2 s ymb2 . The factor 2 in front of the squared brackets in the r.h.s. of Eqs. Ž10. – Ž12. comes from the cross term of the diagrams of Fig. 1. The factors 2 in the brackets come from the flavor trace and the factors '2 from the normalization of our wave functions. Eqs. Ž10. – Ž14. reflect the fact that due to the mixing between scalar mesons and dilaton, the glueball f 0 Ž1500. shows a qq excitation content implying nonvanishing values of g f)0 Ž1500.uu and g f)0 Ž1500. s s . In a similar way, the axial anomaly Ž j / 0. yields gh)X u u / 0 and gh)s s / 0. For completeness, we have written Žsee Eq. Ž12.. the width for the process glueball hXhX . It is however not allowed within our model since m f 0 Ž1500. - 2 mh X . The calculation of the decays involves the coupling strengths of the mesons and of the glueball to the quarks u and s as well as their mass. The problem is quite simple for the pion and the kaon. Their respective coupling constant, corresponding to the residue at the pole of the propagator, writes
™
E gy2 p uu s
E q2 E
gy2 K us s
E q2
Ž q 2 Zu u .
,
Ž 19 .
2 ym p
Ž q 2 q Ž Mu y M s . 2 . Zu s
, ym 2K
Ž 20 .
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with Ž i, j s u, s . Zi j s 4 Nc
d4 k
1
H Ž 2p . Ž k 4
2
2
q Mi2 . Ž k y q . q M j2
.
Ž 21 .
The problem of the glueball has been largely investigated in a preceding paper w13x. Due to the mixing with the scalars, the coupling constants do not take a simple expression as Ž19. and Ž20. Žsee Section 4 of Ref. w13x.. We recall here their expression, g f)0 Ž1500.u u s Ž l0 . 11V13 q Ž l8 . 11V23 Gf 0 Ž1500. , s Ž l0 . 33 V13 q Ž l8 . 33 V23 Gf 0 Ž1500.
g f)0 Ž1500. s s 1
'2
s V33 Gf 0 Ž1500. ,
,
g f)0 Ž1500. x
Ž 22 .
with
E Gy2 f 0 Ž1500. s
E q2
L f 0 Ž1500.
.
Ž 23 .
ym 2f 0 Ž1500.
Vi3 Žym2f 0 Ž1500. . denotes the element i,3 of the eigenvector matrix for the glueball Ž i s 3. and the scalars Ž i s 1,2. calculated at q 2 s ym2f 0 Ž1500. , and L f 0 Ž1500.Ž q 2 . the eigenvalue associated with the f 0 Ž1500. w L f 0 Ž1500.Žym2f 0 Ž1500. . s 0x. We proceed in the same way for the pseudoscalar mesons. One then has gh)u u s Ž l 0 . 11 T11 q Ž l8 . 11 T21 Gh , gh)X u u s Ž l0 . 11T12 q Ž l8 . 11T22 Gh X , gh)s s s Ž l0 . 33 T11 q Ž l8 . 33 T21 Gh
1
'2
gh)X s s s Ž l 0 . 33 T12 q Ž l8 . 33 T22 Gh X
1
'2
, ,
Ž 24 .
with
E Ghy2 s
Eq
E 2
Lh
,
X s Ghy2
ym h2
E q2
Lh X
, ym h2 X
Ž 25 .
Lh and Lh X being the eigenvalues of the coupling matrix between the fields p˜ 0 and p˜ 8 . T is the eigenvector matrix h Gh p˜ 0 sT X . h GhX p˜ 8
ž / ž /
Ž 26 .
It is not written in the usual form
ž
cos u sin u
ysin u , cos u
/
because the model provides a matrix T which is not orthogonal due to the q 2 dependence of its elements. h and hX denote the physical fields, corresponding to the usual kinetic terms.
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M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
3.2. Local contribution In addition to the previous triangle contribution, the decay amplitudes contain a component corresponding to the local decay of the glue component of the f 0 Ž1500.. This component is given by the second and third terms of the r.h.s. of Eq. Ž9.. The second term is a reminiscent of the local 4-quark interaction while the third one is due to the regularization of the quark loop. Indeed, the scale invariance of this quark loop requires to introduce the dilaton field in the cutoff Lx which upon expansion of the Tr ln up to third order in the fields provides a contribution to the amplitude for the studied decays. A priori, there is no reason for dropping it since it is this process which has yielded g f)0 Ž1500. q q / 0 on which the whole study is relied on. These local contributions write . ) )2 Tf Ž0lŽ1500. p 0 p 0 s 2 2 g f 0 Ž1500. x gp u u Hu Ž qp , j s 0 . , . ) )2 Tf Ž0lŽ1500. hh s 2 g f 0 Ž1500. x 2 gh u u Hu Ž qh . q
ž '2 g / ) hss
Ž 27 . 2
Hs Ž qh . q gh)u u
ž '2 g / H ) hss
j
,
Ž 28 . . ) )2 Tf Ž0lŽ1500. h X h X s 2 g f 0 Ž1500. x 2 gh X u u Hu Ž qh X . q
qgh)X u u
ž '2 g / H ) hX s s
ž '2 g / ) hX s s
2
Hs Ž qh X .
,
j
Ž 29 .
. ) ) ) Tf Ž0lŽ1500. hh X s g f 0 Ž1500. x 2 gh u u gh X u u Hu Ž qh . q Hu Ž qh X .
½
q '2 gh)s s
ž
q gh)u u
/ ž '2 g / H Ž q . q H Ž q . ž '2 g / q g ž '2 g / H 5 , ) hX s s
s
) hX s s
. ) ' ) Tf Ž0lŽ1500. K q K y s g f 0 Ž1500. x Ž 2 g K u s .
) hX u u
2
h
s
) hss
hX
j
Hu s Ž q K q . q Hsu Ž q K y . ,
Ž 30 . Ž 31 .
with Hu , Hs , Hu s , Hsu and Hj defined in Appendix A. These expressions involve at the same time the coupling of the glueball to the x field and the coupling of the pseudoscalars to the quarks but without any factor which would account for the propagation of quarks. It looks like the glue part of the f 0 Ž1500. was decaying instantaneously into the mesons.
4. Limitations and merits of the model The scaled NJL model used in the present paper presents the same diseases as the usual NJL. Firstly, it does not confine the quarks, generating unphysical qq pairs whenever the mesons lie above their respective threshold. Secondly, it is not renormalizable, the divergencies of the quark loop having to be regularized with a cutoff which has to be seen as defining itself the model. In the spirit of constructing effective models that reproduce the symmetries of QCD, the scaled NJL model implements the scale anomaly into the NJL model. This is done by introducing one single point-like dilaton field x whose mean value can be identified with the vacuum gluon condensate. This is a drastic simplification since the large
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breaking of the scale invariance should prevent to assign to a single pole the matrix elements of the energy-momentum tensor w18x. In spite of these deficiencies, the scaled NJL model provides a very simple way to yield a mixing between the scalar glueball – whatever it is, f 0 Ž1500. or f 0 Ž1710. – and the scalar isoscalar mesons. In this paper, the glueball, described by the dilaton field, is assumed to be f 0 Ž1500. while the scalars are f 0 Ž1370. and f 0 Ž1710.. All of them have qq and glue content. Playing on the value of the free parameters Mu and x 0 , it is possible to reproduce their mass as far as Mu is large Ž Mu f 700 MeV.. Moreover, the sign of the components in glue or in quark excitations are in agreement with the results of w23x. For instance, the uu and ss components of f 0 Ž1500. have opposite sign Žsee below Eq. Ž41... Finally, as far as strong decays are concerned, our model can account for two distinguishable contributions that can be seen as modelization of the processes depicted in Figs. 7a and 7b of w24x.
5. Results Due to the large value of Mu , the mesons p , K,h and hX are always below their respective threshold, whatever the value of the parameter x 0 . Since f 0 Ž1500. only lies slightly above the threshold 2 Mu , the fact that the NJL model is a nonconfining model is then far from being a problem in the present context and it is treated in the same way as in Ref. w13x for consistency. As explained in Section 2, three parameters of the model are fixed to reproduce mp ,mh X and m K . The last pseudoscalar mass is independent of x 0 and is found to be mh s 482 MeV. The coupling constants of the pseudoscalar mesons to the quarks are also x 0 independent. One has gp)u u s 7.65 ;
gh)u u s 5.23 ,
gh)s s s y4.08 ,
gh)X u u s 1.82 ,
gh)X s s s 4.29 ,
g K)u s s 6.38 .
Ž 32 .
™
On the contrary, the coupling constants of the scalars strongly depend on x 0 . For instance, in the large x 0 limit, g f)0 Ž1500.u u and g f)0 Ž1500. s s 0. In Fig. 2, we plot the x 0
Fig. 2. Coupling constants of the scalar glueball to the x field Žfull curve., to the u-quark Ždotted curve, absolute value of the coupling constant. and to the s-quark Žplain circle curve..
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behavior of the three coupling constants g f)0 Ž1500.u u , g f)0 Ž1500. s s and g f)0 Ž1500. x , showing that, even for large values of x 0 , the quark content of f 0 Ž1500. remains large. ŽNote that for x 0 ` the glueball decouples from the theory. In that limit, f 0 Ž1370. and f 0 Ž1710. become pure nn and ss excitations, respectively w13x.. Let us turn to the strong decay widths of f 0 Ž1500. and consider the local contribu. ŽEqs. Ž27. – Ž31... They can be identified with the contributions given by tions Tf Ž0lŽ1500. the graph 6c of Ref. w24x, provided the vertex is reduced to a single point. Thinking in terms of coupling constants and introducing the quark mass effect via the parameter l, one gets
™
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž hXhX . :gŽ2l . Ž KK . s 1:
:
1 3 1 3
ž ž
gh)u2u gp)u2u gh)X u2u gp)u2u
ql
ql
gh)s2s gp)u2u gh)X s2s gp)u2u
2
2
gh)u u gh)X u u
3
gp)u2u
/ ž / ž / :
2
:
4
g K)u2s
3
gp)u2u
ql
gh)s s gh)X s s 2 gp)uu
2
/
2
l.
Ž 33 .
In Eq. Ž33., the quantity 'l has been introduced each time a ss pair is created in the final channel. This l is the flavor-symmetry violation parameter which is fixed by the ratio of strange to nonstrange hadrons produced in the central region of high energy hadron collisions andror by the branching ratios of the radiative decays JrC ghrghX and JrC gffrgvv : l s 0.4–0.5 w24,25x. It is related to the production probabilities uu:dd: ss s 1:1: l and to the ratio of the quark masses l s Mu2rM s2 w26x. Finally, making the rough approximation according to which the coupling constants can be seen as the qq content of the meson physical fields, Eq. Ž33. reduces to
™
™
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž hXhX . :gŽ2l . Ž KK . 2
2
2
s 1: 13 Ž cos 2u q lsin2u . : 23 Ž 1 y l . sin2u cos 2u : 13 Ž sin2u q lcos 2u . : 43 l ,
Ž 34 .
with p 0 s uu y dd, pq s '2 ud, py s '2 d u, h s '2 Žcos u nn y sin u ss ., hX s '2 Žsin u nn q cos u ss ., K s '2 us, K s '2 su where we have reconciled the normalization with the present work. One then recovers the results of Ref. w24x. The exact . expressions for the amplitudes Tf Ž0lŽ1500. , Eqs. Ž27. – Ž31., would lead to the relations Ž33. if Hu : Hu s : Hs s 1:'l : l ,
Ž 35 .
whatever j and if Hj could be neglected. In fact, one has Hu : Hu s : Hs s 1:0.753:0.546 ,
Ž 36 .
which is coherent with Ž35. provided that l f 0.55. The values Ž36. are obtained for x 0 s 400 MeV, at q 2 s 0. This value of x 0 has been chosen because it leads to quite
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acceptable values for the masses of the scalar nonet w13x. An exact calculation, i.e. with the contribution of Hj and with each H evaluated at its respective on-shell mass yields
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž KK . s 1:0.193:0.011:0.641 ,
Ž 37 .
which is indeed in rather good agreement with the results obtained from Ž34. with u s 378 and l s 0.55 w25x,
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž KK . s 1:0.233:0.031:0.73 ,
Ž 38 .
justifying in some sense the assumptions done above. However, neither the exact calculation nor the approximated one give results in agreement with the experimental one Ž2.: one gets a too small value for hhX and a much too large one for KK. . As explained before, the decay amplitudes receive additional contributions Tf Ž0tŽ1500. Žsee Fig. 1, diagram a. due to the fact that the f 0 Ž1500. exhibits some qq content. Following the same steps as for the local contribution, one has
gŽ2t . Ž pp . :gŽ2t . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž hXhX . :gŽ2t . Ž KK . s 1:
:
:
:
1 3 2 3 1 3 1 3
gh)u2u
ž ž ž ž /ž gp)u2u
q '2 'l
gh)u u gh)X u u gp)u2u
gh)X u2u gp)u2u
g K)u2s
g f 0 Ž 1500 . s s gh)s2s g f 0 Ž 1500 . u u gp)u2u
q '2 'l
q '2 'l
gp)u2u
gp)u2u
g f 0 Ž 1500 . u u
g f 0 Ž 1500 . u u gp)u2u
'l q '2
/
g f 0 Ž 1500 . s s gh)s s gh)X s s
g f 0 Ž 1500 . s s gh)X s2s
2
2
g f 0 Ž 1500 . s s g f 0 Ž 1500 . u u
2
/
2
/
2
/
.
Ž 39 .
If one assumed that the coupling strengths could be in the ratio g f 0 Ž1500. s s g f 0 Ž1500.u u
s
(
l 2
,
Ž 40 .
one would recover exactly the various ratios of Eq. Ž34.,
gŽ2t . Ž pp . :gŽ2t . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž hXhX . :gŽ2t . Ž KK . 2
2
2
s 1: 13 Ž cos 2u q lsin2u . : 23 Ž 1 y l . sin2u cos 2u : 13 Ž sin2u q lcos 2u . : 43 l .
Ž 41 .
However, the assumption Ž40. is far from being satisfied here. One has g f 0 Ž1500. s srg f 0 Ž1500.u u s y0.46 Žsee Fig. 2, where we have changed the sign of g f 0 Ž1500.u u . whose minus sign is in agreement with results of w23x and imposes very small values for
168
M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
. . the branching ratios. The amplitudes Tf Ž0tŽ1500. and Tf Ž0lŽ1500. will then lead to completely different results. We now turn to our model predictions, using the exact expressions Ž10. – Ž14.. One has
gŽ2t . Ž pp . :gŽ2t . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž KK . s 1:2.2 P 10y2 :0.8 P 10y2 :1.1 P 10y2 ,
Ž 42 . showing that the relative strengths of the decays are modified with regard to Ž37.. The full curves of Fig. 3 represent the x 0 behavior of the Žtriangle. decay widths: Ža. pp , Žb. hh , Žc. hhX and Žd. KK. It is clear that, whatever x 0 , the value of Gf 0 Ž1500. ™ p p has no physical meaning. The origin of these awfully large values lies in the large 2 value of Mu which affects at the same time the values of g f 0 Ž1500.u u , gp)u u and Ju u u . It is important to stress that this value was forced to us in order to consider the a 0 Ž1450. as a member of the scalar nonet: in the NJL model, this mass is exactly 2 Mu w13x. In order to reduce the value of the constituent quark mass, one has to consider a confining model w20x. One can relax the constraint of considering a 0 Ž1450. as being a member of the scalar nonet. Taking a 0 Ž980. as the new candidate gives a constituent quark mass Mu s 490 MeV. The trouble we have now is that the scalar isoscalar sector is well above the threshold if we try to reproduce a glueball of mass 1.5 GeV. The small width approximation Žfor the unphysical decay of the scalar mesons into two free quarks. is no longer valid. This is once again a problem linked to the nonconfining nature of the NJL model. One can say that, within the present model, a high value of the constituent quark mass is necessary not only because a0 Ž1450. is taken as a member of the scalar nonet, leaving the a 0 Ž980. as a candidate for a KK molecule, but also because of the lack of confinement of the model. These considerations can be checked numerically: it is observed that multiplying the value of Mu by a factor 2 multiplies the width by roughly a factor 2 5. In fact, it can be verified analytically that the width behaves roughly as Mu5. For instance, would we take the mass of the scalar around 800 MeV, i.e. Mu f 400 MeV, we would obtain results in agreement with w27x. ŽIn this comparison, we relax the constraint of a mixing with the glueball. Then the nonet lies around 800 MeV and the model sits just above the threshold, making the small unphysical decay width into two free quarks irrelevant.. The problem is much less severe for the other decays. This can be traced back to the negative value of the ratio g f 0 Ž1500. s srg f 0 Ž1500.u u . There is then destructive interference between the amplitudes u and s, which strongly reduces the values of the widths. Note that, before trying to apply the model to the scalar nonet, it may be worth testing it by studying the well-known two-pseudoscalar decay modes in the conventional
2
Note that this value does not affect the usual properties of the NJL model. This is due to the fact that the parameters are fixed to reproduce common observables. The pion mass and decay constant being fixed to their physical value, one can for instance obtain a two photon decay width of the neutral pion of 7.64 eV. Although it is often assumed that the r mass is roughly twice the constituent quark mass, it does not play any role in our fitting procedure: the rho mass is fixed, and this is the vectorial four quark coupling constant G V which is fitted accordingly.
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X Fig. 3. Decay widths of the scalar glueball into pp , hh , hh and KK, for the quark loop contribution Žfull . Ž . curves and for the total contribution dotted curves .
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M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
meson sector, for example the f 2 Ž1270. and the f 2X Ž1525.. However, a tensor structure is not included in the NJL model, and this study is beyond the scope of the paper. . . Let us now turn to the total contribution Tf Ž0tŽ1500. q Tf Ž0lŽ1500. . Except for the hhX . decay, the total widths are reduced with regard to the quark loop contribution Tf Ž0tŽ1500. whatever the value of x 0 . This feature is exhibited by the dotted curves of Fig. 3. However, even for values of x 0 as large as 1 GeV, the case of the 2p decay is hopeless. The branching ratios keep tremendous small values independently of x 0 ,
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:1.3 P 10y2 :1.2 P 10y2 :0.7 P 10y3 ,
Ž 43 . reflecting again the problem of the pion. For completeness, we give below the equivalent of Eqs. Ž37., Ž42. and Ž43. for the value Mu s 490 MeV. They are given for illustrative purpose only, to show that taking a lower quark mass leads to less unphysical branching ratios. We stress again that these numbers have to be taken with caution since the scalar isoscalar are well above the threshold. Moreover, the nonet would no longer be the one considered here but would consist of a0 Ž980., f 0 Ž975., f 0 Ž1370., K 0) Ž1430. Žstill with the glueball mass fixed at 1.5 GeV.. With these cautions, the equivalent of Eq. Ž37. is
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž KK . s 1:0.2045:0.1:0.947 ,
Ž 44 .
while the equivalent of Eq. Ž42. is
gŽ2t . Ž pp . :gŽ2l . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž KK . s 1:1.326:0.325:5.679 .
Ž 45 .
Finally, the total contribution would be Žwith the same cautions.
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:0.84:0.046:3.224 .
Ž 46 .
6. Summary and conclusion The model of the present paper uses a dilaton field x to implement the QCD trace anomaly in the Nambu–Jona-Lasinio model. Since the latter is not renormalizable, all the ultraviolet divergences have to be regularized using a cutoff Lx which entails a mixing between the three scalar isoscalar fields. One then gets three scalar ‘‘hybrids’’ f 0 which are all a mixing of glue and uu and ss excitations. Identifying these hybrids with f 0 Ž1370., f 0 Ž1500., f 0 Ž1710., the constituent u-quark mass used in the model has to be large: Mu f m f 0r2. Here Mu s m a 0r2. Such a large mass does not destroy the rather good results obtained up to now for the properties of the pseudoscalar mesons within the NJL model. For instance, the two photon decay of p 0 is not affected by the value of Mu . In the present paper, we focused on the f 0 Ž1500. which is identified with the glueball in the sense that it is the state that would yield pure glue if there was no mixing. We have studied to what extent a large value of Mu can reproduce its possible strong decays into two pseudoscalar mesons. The main point of the paper is that the strong decays receive contribution from two distinguishable processes. Due to its qq content, the f 0 Ž1500. can decay via processes
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described by triangular quark loops Žsee Fig. 1, diagram a.. These processes are similar to the one shown in Fig. 7a of w24x. The corresponding amplitude goes as 1r Nc . f 0 Ž1500. can also decay via two local processes schematically represented in Fig. 1, diagram b: one is the reminiscent of the 4-quark interaction Žterm %a2x 2 in Eq. Ž7.., the other comes from the regularization of the quark loop. These processes can be seen as the local limit of the diagram 7c of w24x. Our contribution is of order O Ž1. in Nc while the diagram 7c of w24x behaves as 1rNc . This can be explained by the way the glueball is incorporated in our formalism. Indeed, it is not seen as a two gluon composite. Note that provided some assumptions described below, both contributions Ždiagrams a and b of Fig. 1. satisfy the sum rule w24x
(
Ý g t2,l Ž i . % Ž 2 q l. 2 .
Ž 47 .
i
These assumptions are however not fulfilled by our model, mainly due to the quark loop contribution. For the local contribution, one has to assume that the various meson–quark coupling constants can be identified with the qq contents and that the quark mass effect Ž Mu / M s . is introduced via the flavor symmetry violation parameter l. Our calculation is consistent with a value of l f 0.55, see Eqs. Ž35. and Ž36.. Our results for the local contribution are then similar to those of Ref. w25x but in complete disagreement with the experimental data Žsee Eqs. Ž2. and Ž37... For the quark loop contribution, one has to add the assumption Ž40. which is not satisfied here. Our results for this contribution are then at variance with those of w24x: the sum rule Ž47. is not verified. Our results allow to stress that, due to the large value of Mu , the calculated pp decay width takes awfully large values Ž1–10 GeV. and has therefore no physical meaning. This problem is partly suppressed for the other decays due to the destructive interferences between the amplitudes associated with the u- and s-quarks. Adding the two contributions, the results remain qualitatively the same as for the quark loop alone. Except for the hhX mode, the widths are reduced with regard to the quark loop contribution. The various widths decrease with an increasing value of the vacuum parameter x 0 . Whatever x 0 , the case of pp is hopeless. However, for value of x 0 f 1 GeV, the other widths take acceptable value Ž Gf 0 ™ hh q Gf 0 ™ hh X q Gf 0 ™ K K f 50 MeV.. The results obtained for these latter modes are strongly dependent on the value and on the sign of the ratio g f)0 s srg f)0 u u . In the present model, we have g f)0 s srg f)0 u u s y0.46. Since the minus sign is in agreement with the result of w23x but in contradistinction with w24x, it would then be worthwhile to investigate further the qq content of the glueball. One can draw the following conclusions. If one trusts in the idea of w7,8x, according to which the f 0 Ž1500. is the glueball, the scaled NJL model cannot reproduce at the same time the dynamical properties of pseudoscalars and scalars. One way to get a way out would be to introduce confinement as in Ref. w28x. Indeed, such a confinement has as effect to increase the scalar meson mass with regard to the nonconfining model. The masses of the scalars could then be reproduced with a smaller value of the constituent up quark mass giving better results for the decay width of the glueball into pions. Preliminary results are promising and will be published in a subsequent publication w20x. However, one could think in another way, following the results of w10x which clearly identify the lightest scalar glueball with f 0 Ž1710.. In that case, our f 0 Ž1500., even if
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mixed with the others ‘‘hybrids’’, would have its larger component in the ss channel w21x. Its decay into pp would be spectacularly reduced. The way the other decays would be modified is not trivial due to the destructive interferences between the uu and ss channels. This problem requires further investigation.
Acknowledgements The work of M.M. and B.VdB. is supported by the Institut Interuniversitaire des Sciences Nucleaires de Belgique. B.VdB. acknowledges Dr. J.-R. Cudell for several ´ discussions during the completion of the work. M.J. thanks the Departamento de Fısica ´ Teorica of the Universidad Autonoma de Madrid for hospitality. ´ ´ Appendix A . In order to get the contributions Tf Ž0lŽ1500. , one has to calculate the second and third terms of Eq. Ž9.. Keeping only terms which couple the x˜ field to two pseudoscalars, one gets Ž a,b s 0,1,2,3,8. 3Ž l . Ieff s
1
Ž bV .
2
Ý Ý x˜ q ½ p˜ a q yqp˜ byq X
X
X
q
Ž3. ˜y X Ž2. X DaŽ b2 . q DaŽ b3 . q 2 K˜q q yq Kyq Du s q Du s
q
Ž3. ˜q X Ž2. X q2 K˜y q yq Kyq Dsu q Dsu
5,
Ž A.1 .
with Žsecond term. Ž2. Ž2. DiŽ2. ˜ 2x 0 d i j j s Du s s Dsu s Nc a
Ž i , j s 1,2,3,8 . ,
D Ž2. ˜ 2x 0 Ž 1 q j . , 00 s Nc a
Ž A.2 .
and Žthird term. 2 D Ž3. 00 s Nc L x 0
4 3
X
Bu u Ž q . q
B s s Ž qX . 2
,
4
'2 Bu u Ž qX . y Bs s Ž qX . , 3 4 Bu u Ž q X . Ž3. D 88 s Nc L 2x 0 q B s s Ž qX . , 3 2 Ž3. D 08 s Nc L 2x 0
X 2 DiŽ3. i s Nc L x 0 2 Bu u Ž q .
Ž i s 1,2,3 . , 2
DuŽ3.s s Nc L2x 0 Bu s Ž qX . Ž '2 . , 2
Ž3. Dsu s Nc L2x 0 Bsu Ž qX . Ž '2 . ,
Ž A.3 .
where Ž i, j s u, s . Bi j Ž q . s trg
d4 k
H Ž 2p .
4
g i Ž k . ig 5 g j Ž k y q . ig 5 d Ž k 2 y L2x 02 . ,
Ž A.4 .
M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
173
l
g i Ž k . being defined in Eq. Ž16.. The factor 2 in Eq. ŽA.1. comes from isospin symmetry Ž K 0 Kq. . In Eq. ŽA.3., the trace in flavor space has been done. We introduce the physical pion Ž i s 1,2,3. and kaon fields Žwith g K)u s s g K)su .
p i) s gp)y1 ˜i , uu p
˜q Kq) s g K)y1 us K , ˜y Ky) s g K)y1 Ž A.5 . su K , X Ž Ž . Ž .. as well as h and h see Eq. 24 and Eq. 26 . Due to its glue component x˜ , the glueball f 0 Ž1500. noted here x ) feels the following interaction term with the pseudoscalar mesons: 1 3Ž l . ) X Ieff s x ) g f)0 Ž1500. x p i)qXyq p iyq Lp p 2 Ý Ý q Ž bV . q qX X X X L X Xqh X X X qhqXyq hyqX Lhh q hqX Xyq hyq hh q yq hyq Lhh )y )y )q X X X X q2 K q)q yq Kyq L K q K yq 2 K q yq Kyq L K y K q ,
where g f)0 Ž1500. x is defined in Eq. Lpp s gp)u2u R u qX , j s 0 , Lhh s gh)u2u R u qX q gh)s2s R s
Ž Ž .
Ž A.6 .
Ž22. and
.
Ž qX . q gh)u u gh)s s Rj ,
LhX h X s gh)X u2u R u Ž qX . q gh)X s2s R s Ž qX . q gh)X u u gh)X s s Rj , Lhh X s gh)u u gh)X u u R u Ž qX . q gh)s s gh)X s s R s Ž qX . q Ž gh)u u gh)X s s q gh)s s gh)X u u .
Rj 2
,
2
L K q K ys Ž '2 g K)u s . R u s Ž qX . , 2
L K y K qs Ž '2 g K)u s . R su Ž qX . ,
Ž A.7 .
R u Ž qX . s 13 Ž 2 D 00 q '2 D 08 q D 88 . 2j s Nc x 0 a˜ 2 1 q q 2 L 2 Bu u Ž q X . , 3
ž
/
R s Ž qX . s 13 Ž D 00 y '2 D 08 q 2 D 88 . j s Nc x 0 a˜ 2 1 q q 2 L 2 B s s Ž qX . , 3
ž
/
R u s Ž qX . s Nc x 0 L2 Bu s Ž qX . q R su Ž qX . s Nc x 0 L2 Bsu Ž qX . q
a˜ 2x 0 Nc 2 a˜ 2x 0 Nc 2
, ,
Rj s 13 Ž 2'2 D 00 y D 08 y 2'2 D 88 . 2'2 s 3
j Nc x 0 a˜ 2 .
Ž A.8 .
M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
174
One then gets the amplitudes Ž27. – Ž31. with Hu Ž q X . s H s Ž qX . s
Ru 2 Rs 2
s Nc x 0
s Nc x 0
a˜ 2
2j
ž / ˜ ž / 2
a2 2
1q
3
j
1q
3
q L 2 Bu u Ž q X . ,
q L 2 B s s Ž qX . ,
Hu s Ž qX . s R u s Ž qX . s Nc x 0 L 2 Bu s Ž qX . , Hsu Ž qX . s R su Ž qX . s Nc x 0 L 2 Bsu Ž qX . , Hj s
Rj
'2
s 23 j Nc x 0 a˜ 2 ,
Ž A.9 .
once one has introduced the same conventions of notation for the factors 2 and '2 as in Ž3. Eqs. Ž10. – Ž14.. In Eq. ŽA.8., the Di j stand for the sum of DiŽ2. j and Di j .
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x
D. Alde et al., Phys. Lett. B 201 Ž1988. 160. S. Abatzis et al., Phys. Lett. B 324 Ž1994. 509. D.V. Bugg et al., Phys. Lett. B 353 Ž1995. 378. Crystal Barrel Collaboration, S. Spanier, Nuc. Phys. B ŽProc. Suppl.. 54A Ž1997. 362. Particle Data Group, Eur. Phys. J. C 3 Ž1998. 1. C. Amsler, Rev. Mod. Phys. 70 Ž1998. 1293. C. Amsler, F.E. Close, Phys. Lett. B 353 Ž1995. 385. C. Amsler, F.E. Close, Phys. Rev. D 53 Ž1996. 295. C. Ritter, B.C. Metsch, C.R. Munz, ¨ H.R. Petry, Phys. Lett. B 380 Ž1996. 431. W. Lee, D. Weingarten, Nucl. Phys. B ŽProc. Suppl.. 63 Ž1998. 194. J. Sexton et al., Nucl. Phys. B ŽProc. Supp.. 47 Ž1996. 128. M. Boglione, M.R. Pennington, Phys. Rev. Lett. 79 Ž1997. 1998. M. Jaminon, B. Van den Bossche, Nucl. Phys. A 636 Ž1998. 113. M. Jaminon, B. Van den Bossche, Nucl. Phys. A 582 Ž1995. 517. B. Van den Bossche, Ph.D. thesis ŽUniversity of Liege, ` 1996.. H. Gomm, P. Jain, R. Johnson, J. Schechter, Phys. Rev. D 35 Ž1987. 2230. G. Ripka, M. Jaminon, Ann. Phys. 218 Ž1992. 51. M. Birse, J. Phys. G 20 Ž1994. 1287. M.L. Nekrasov, Phys. Rev. D 56 Ž1997. 5681. M. Jaminon, M. Mathot, B. Van den Bossche, in preparation. M. Jaminon, B. Van den Bossche, Nucl. Phys. A 619 Ž1997. 285. J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, 1st ed., Cambridge University Press, Cambridge, 1992. F.E. Close, G.R. Ferrar, Z. Li, Phys. Rev. D 55 Ž1997. 5749. A.V. Anisovich, V.V. Anisovich, A.V. Sarantsev, Z. Phys. A 359 Ž1997. 173. V.V. Anisovich, Phys. Lett. B 364 Ž1995. 195. V.V. Anisovich, M.G. Huber, M.N. Kobrinsky, B.Ch. Metsch, Phys. Rev. D 46 Ž1990. 3045. K. Nakayama, S. Krewald, Phys. Lett. B 273 Ž1991. 199. L.S. Celenza, Xiang-Dong Li, C.M. Shakin, Phys. Rev. C 55 Ž1997. 3083.
Strong decays of the glueball in a scaled effective Lagrangian M. Jaminon b
a,1
, M. Mathot b, B. Van den Bossche
b
a Instituto de Fısica Teorica, UniÕersidad Autonoma de Madrid, Madrid, Spain ´ ´ ´ UniÕersite´ de Liege, ` Institut de Physique B5, Sart Tilman, B-4000 Liege ` 1, Belgium
Received 26 January 1999; received in revised form 27 July 1999; accepted 10 August 1999
Abstract We calculate the scalar glueball decays into pp , hh , hhX and KK within a modified version of the NJL model which implements the QCD trace anomaly. The glueball is assumed to be the f 0 Ž1500.. It is coupled to the quarks due to its mixing with the scalar isoscalar mesons. The decay widths then contain a contribution coming from the triangle Feynman diagram. Moreover, they contain two additional contributions corresponding to the local decay of the glue component of the f 0 Ž1500.: one comes from the process of regularization of the quark loop, the other from the interaction term a2x 2f 2. Except for the hhX case, this local decay reduces the value of the width. The axial anomaly is introduced via a mass term for the hX. As far as the lightest scalar glueball is identified with the f 0 Ž1500., the model cannot reproduce at the same time the dynamical properties of scalars and pseudoscalars. Since we do not include a model of confinement, we need to use a large quark mass Ž725 MeV. in our calculations. This leads to unphysical results for the two-pion decay width. A definitive test of our model awaits the introduction of a model of confinement, which we hope to include in a future work. q 2000 Elsevier Science B.V. All rights reserved. PACS: 12.39.Fe; 12.39.Mk; 14.40.yn
Keywords: Scalar mesons; Chiral Lagrangian; NJL; Glueball; hh , hhX decays
1. Introduction Up to very recently, the f 0 Ž1500. appeared to decay into pp , hh , hhX and KK w1–4x with the corresponding branching ratios, corrected for phase space effects,
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:0.27 " 0.11:0.19 " 0.08:0.24 " 0.09 , Ž 1. and a total width Gf 0 Ž1500. s 112 " 10 MeV w5x. 1 On leave of absence from Universite´ de Liege, ` Institut de Physique B5, Sart Tilman, B-4000 Liege ` 1, Belgium.
0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 3 5 8 - 9
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However, the results quoted in Ž1. do not refer to the same form factor, being then in some sense inconsistent. The very recent report w6x of the latest data for the f 0 Ž1500. suggests that the calculated branching ratios, corrected for phase space effects and form factor, are
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:0.195 " 0.075:0.320 " 0.114:0.138 " 0.038
Ž 2.
for a mass of 1505 " 9 MeV and a total decay width of 111 " 12 MeV. Taking the widths Ga 0 s 265 " 13 MeV and GK 0) s 287 " 23 MeV as a scale for the members of the scalar nonet this small value of Gf 0 Ž1500. seems incompatible with a nonet structure. Moreover, Amsler and Close w7,8x have argued that the branching ratios could not be explained with a f 0 Ž1500. assumed as to be a pure qq state. The latter was identified with a nn s 1r '2 Ž uu q dd . state, regarding to the strong coupling of the f 0 Ž1500. to the pp channel. In contrast, Ritter et al. w9x have reproduced the experimental branching ratios assuming a ss content for the f 0 Ž1500. with, however, a value of the total width which remains of the order of magnitude of GK 0) that is twice larger than the experimental value. In the same way, the valence approximation of Ref. w10x assigns a mass to the scalar ss that lies around 200 MeV below the lightest scalar glueball identified with the f 0 Ž1710.. Moreover, lattice calculations w10,11x indicate a very small value Žf 100 MeV. for the decay of the lightest glueball into two pseudoscalars, value still reduced Žf 30 MeV, with a large hh component. if dressing of the glueball due to some qq content is taken into account w12x. The experimental value is there recovered by introducing the opening of multi-pion channels, mimicked by a 4p channel. In a previous paper w13x, two of us have followed the idea of Ref. w7,8x, identifying the glueball with f 0 Ž1500. and have calculated its decay into two photons. In the present paper we go on this line and we compute the strong decays of the glueball f 0 Ž1500. into two pseudoscalars, using a modified version of the SUŽ3. NJL model w14,15x which implements the trace anomaly of QCD w16x. This is done by introducing one single point-like scalar dilaton field x whose mean value x 0 is identified with the vacuum gluon condensate w17x. Keeping in mind the limitations of the model – large breaking of scale invariance assigned to a single pole w13,18x, nonrenormalizability, nonconfinement – it has however the merit to provide a qq content to the glueball. It can therefore describe processes as the one shown in diagram a of Fig. 1. In our model, it is the value of x 0 which fixes the mixing with the scalar mesons: the smaller x 0 the larger the mixing. In order to regularize the quark loop, one has to introduce a cutoff Lx where the x field accounts for the scale invariance of the trace. In a similar way, the scale invariance requires to divide by x the strength of the 4-quark coupling. This provides an additional contribution to the f 0 Ž1500. decays, a contribution that reduces the numerical value of the respective widths except for the hhX case. This contribution involves the coupling constant of the f 0 Ž1500. to the pure glue Žsee diagram b of Fig. 1.. However, it has nothing to do with a process where one would calculate the overlap of the glue content of the f 0 Ž1500. with some pseudoscalar glue content of the pseudoscalar mesons. The latter could be achieved following the idea of Ref. w19x where the axial anomaly is introduced by mixing the h and hX with a pseudoscalar glueball. We plan to look for this effect in a forthcoming paper w20x.
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159
Fig. 1. Schematic diagrams for the decay of the glueball into two pseudoscalars.
Our paper is organized as follows. Section 2 contains the useful tools of the scaled NJL model. Section 3 derives the strong decay widths of the f 0 Ž1500.. Section 4 insists on the problems raised by the model and on its merits. Our results are presented in Section 5. Section 6 contains our conclusions.
2. The model The scaled NJL model is a modified version of the NJL model that has been constructed w17x in order to implement the QCD trace anomaly. It entails the introduction of a scalar field x whose mean value is intimately related to the vacuum gluon condensate. Since the model has already been intensively described in preceding papers w14,15,17,21x, we only recall here the useful tools for the understanding of the present paper. We use the following bosonized Euclidean action: Ieff Ž w , x . s yTrL x ln Ž yi Em gm q m q Ga wa . q d 4 x
H
q d 4 x Lx q d 4 x LUA Ž1. ,
H
H
a 2x 2 2
wa wa
Ž 3.
with the meson fields wa s Ž sa ,pa . and Ga s Ž l a ,ig 5 l a . for a s 0, . . . , 8. The l a are the usual Gell–Mann matrices with l0 s 2r3 1. We work in the isospin symmetry
'
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limit and the quantity m stands for the diagonal matrix diagŽ m u ,m u ,m s .. The Lagrangian which implements the scale anomaly has been originally introduced in Ref. w16x, 2
Lx s 12 Ž Em x . q 161 b 2 x 4 ln
x4 xG4
y Ž x 4 y xG4 . ,
Ž 4.
introducing two additional parameters b 2 and xG with respect to the usual NJL. The axial anomaly is introduced via a mass term for the field p 0 which would be the hX if the mixing between p 0 and p 8 were neglected, a2 LUA Ž1. s
2
j x 2p 02 .
Ž 5.
In a forthcoming paper w20x we plan to introduce the anomaly by coupling the p 0 and p 8 fields with a pseudoscalar glueball field. This would be consistent with the fact that the quadridivergence of the axial current is proportional to the dual tensor F˜mna w22x. Moreover, the model so constructed would be quite elegant due to the analogy between the mixing of the scalars with a scalar glueball and the one of the pseudoscalars with a corresponding pseudoscalar glueball. Our model contains seven parameters: the current quark masses Ž m u ,m s ., the strengths Ž a2 ,b 2 ., the gluon parameter xG , the cutoff L and the axial anomaly strength j . Five of these parameters are adjusted to reproduce the pion mass Ž mp ., the weak pion decay constant Ž fp ., the kaon and hX masses and that of the glueball f 0 Ž1500.. Using the usual gap equation w14,15x, the strength a 2 can be connected to the constituent up quark mass Mu which is here chosen rather large Ž Mu s 725 MeV. to reproduce the a 0 Ž1450. w13x. One remains with one free parameter x0.
3. The strong decay widths In the limit of isospin symmetry, the widths for the strong decays of the glueball f 0 Ž1500. are given by
Gf 0 Ž1500. a b s
NI
1
N 2 m f 0 Ž1500. Ž 2p . NI
s
)
a , b s p ,h ,h X , K ,
2
HH 2 E
a
1
16 Np m f 0 Ž1500. =
d 3 k1 d 3 k 2
1
1y
ž
Tf 0 Ž1500. a b
ma q mb m f 0 Ž 1500 .
2
2
/
2 Eb
2
Tf 0 Ž1500. a b d Ž q y k 1 y k 2 .
1y
ž
ma y mb m f 0 Ž 1500 .
2
/
,
Ž 6.
with NI s Ž2 I q 1.. N s 2 for identical emerging particles, N s 1 otherwise. In Eq. Ž6., m f 0 Ž1500. denotes the mass of the glueball, q its quadrimomentum while k 1 and k 2 stand for the quadrimomenta of the emerging mesons. The glueball is assumed to be at rest
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and the pseudoscalar mesons on their mass shell. To obtain the various amplitudes arising in Eq. Ž6., one has to expand the action Ž3. up to the third order into the meson fields. Writing Ieff Ž w , x . s yTrL x ln Ž Gy1 ˜a . q d4 x 0 . y TrL x ln Ž 1 q G 0 Ga w
H
a 2x 2 2
Ž wa2 q jp 02 .
q d 4 x Lx ,
Ž 7.
H
with Ž was corresponding to the stationary point of the action. s Gy1 0 s yi Em gm q m q Ga wa ,
wa s was q w˜ a ,
Ž 8.
one gets, using the scaling of the strong coupling constant with the number of colors Ž a2 s a˜ 2 Nc ., Nc
3 Ieff Ž w,x . sy
q
3 Nc 2
3
TrLX x 0Ž G 0 Ga w˜ a . q Nc d 4 x
E Ex
H
X TrLx Ž G 0 Ga w˜ a .
2
a˜ 2 2
2 x 0 x˜ Ž w˜ a2 q jp˜ 02 .
x˜ PPP ,
Ž 9.
x0
with Tr s NcTrX , TrX being the trace over space-time and flavor and spin indices and x s x 0 q x˜ . In Eq. Ž9. only the terms that can give the decay of the glueball into two pseudoscalar mesons have been specified. The first term corresponds to the triangle Feynman diagram Žsee Fig. 1, diagram a.. It is studied in Subsection 3.1. The second and third terms correspond to the local decay of the glue component of the f 0 Ž1500. Žsee Fig. 1, diagram b.. They are investigated in Subsection 3.2. 3.1. Triangle diagram We calculate here the contribution to the decay widths coming from the triangle diagram depicted in Fig. 1, diagram a. It writes . ) )2 Tf Ž0tŽ1500. p 0 p 0 s 2 2 g f 0 Ž 1500 . u u gp u u Ju u u Ž q,qp . ,
Ž 10 .
. ) )2 Tf Ž0tŽ1500. hh s 2 2 g f 0 Ž 1500 . u u gh u u Ju u u Ž q,qh .
q '2 g f)0 Ž 1500 . s s
ž
/ ž '2 g / ) hss
2
Js s s Ž q,qh . ,
Ž 11 .
. ) )2 Tf Ž0tŽ1500. h X h X s 2 2 g f 0 Ž 1500 . u u gh X u u Ju u u Ž q,qh X .
q '2 g f)0 Ž 1500 . s s
ž
/ ž '2 g / ) hX s s
2
Js s s Ž q,qh X . ,
Ž 12 .
. ) ) ) Tf Ž0tŽ1500. hh X s 2 g f 0 Ž1500.u u gh u u gh X u u Ju u u Ž q,qh . q Ju u u Ž q,qh X .
q '2 g f)0 Ž1500. s s
ž
/ ž '2 g / ž '2 g / ) hss
) hX s s
Js s s Ž q,qh . q Js s s Ž q,qh X . ,
Ž 13 .
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2
. ) ' ) Tf Ž0tŽ1500. K q K y s g f 0 Ž1500.u u Ž 2 g K u s . Ju su Ž q,q K q .
q '2 g f)0 Ž1500. s s
ž
2 ) K us
/ Ž'2 g .
Jsu s Ž q,q K y . ,
Ž 14 .
with Ž i, j,k s u, s . Ji jk Ž q,qa . s yNc Trg
d4 k
H Ž 2p .
g i Ž k q qa . ig 5 g j Ž k . ig 5 g k Ž k y q q qa . ,
4
Ž 15 . gi Ž k .
y1
s km gm q Mi ,
Ž 16 .
or Ji i i Ž q,qa . s 4 Nc Mi =
d4 k
H Ž 2p .
4
k 2 q Mi2 q qa Ž q y qa . 2 2 Ž k q qa . q Mi2 Ž k 2 q Mi2 . Ž k y q q qa . q Mi2
Ž 17 .
and
Ji ji Ž q,qa . s4Nc Mi
d4 k
H Ž 2p .
4
ž
k 2 q Mi2 q qa Ž q y qa . q 1 y =
Mj Mi
/
Ž k q qa y q . Ž k q qa . y Mi2
2 2 Ž k q qa . q Mi2 Ž k 2 q M j2 . Ž k q qa y q . q Mi2
Ž 18 .
calculated for q 2 s ym2f 0 Ž1500. , qa2 s yma2 , Ž q y qa . 2 s ymb2 . The factor 2 in front of the squared brackets in the r.h.s. of Eqs. Ž10. – Ž12. comes from the cross term of the diagrams of Fig. 1. The factors 2 in the brackets come from the flavor trace and the factors '2 from the normalization of our wave functions. Eqs. Ž10. – Ž14. reflect the fact that due to the mixing between scalar mesons and dilaton, the glueball f 0 Ž1500. shows a qq excitation content implying nonvanishing values of g f)0 Ž1500.uu and g f)0 Ž1500. s s . In a similar way, the axial anomaly Ž j / 0. yields gh)X u u / 0 and gh)s s / 0. For completeness, we have written Žsee Eq. Ž12.. the width for the process glueball hXhX . It is however not allowed within our model since m f 0 Ž1500. - 2 mh X . The calculation of the decays involves the coupling strengths of the mesons and of the glueball to the quarks u and s as well as their mass. The problem is quite simple for the pion and the kaon. Their respective coupling constant, corresponding to the residue at the pole of the propagator, writes
™
E gy2 p uu s
E q2 E
gy2 K us s
E q2
Ž q 2 Zu u .
,
Ž 19 .
2 ym p
Ž q 2 q Ž Mu y M s . 2 . Zu s
, ym 2K
Ž 20 .
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with Ž i, j s u, s . Zi j s 4 Nc
d4 k
1
H Ž 2p . Ž k 4
2
2
q Mi2 . Ž k y q . q M j2
.
Ž 21 .
The problem of the glueball has been largely investigated in a preceding paper w13x. Due to the mixing with the scalars, the coupling constants do not take a simple expression as Ž19. and Ž20. Žsee Section 4 of Ref. w13x.. We recall here their expression, g f)0 Ž1500.u u s Ž l0 . 11V13 q Ž l8 . 11V23 Gf 0 Ž1500. , s Ž l0 . 33 V13 q Ž l8 . 33 V23 Gf 0 Ž1500.
g f)0 Ž1500. s s 1
'2
s V33 Gf 0 Ž1500. ,
,
g f)0 Ž1500. x
Ž 22 .
with
E Gy2 f 0 Ž1500. s
E q2
L f 0 Ž1500.
.
Ž 23 .
ym 2f 0 Ž1500.
Vi3 Žym2f 0 Ž1500. . denotes the element i,3 of the eigenvector matrix for the glueball Ž i s 3. and the scalars Ž i s 1,2. calculated at q 2 s ym2f 0 Ž1500. , and L f 0 Ž1500.Ž q 2 . the eigenvalue associated with the f 0 Ž1500. w L f 0 Ž1500.Žym2f 0 Ž1500. . s 0x. We proceed in the same way for the pseudoscalar mesons. One then has gh)u u s Ž l 0 . 11 T11 q Ž l8 . 11 T21 Gh , gh)X u u s Ž l0 . 11T12 q Ž l8 . 11T22 Gh X , gh)s s s Ž l0 . 33 T11 q Ž l8 . 33 T21 Gh
1
'2
gh)X s s s Ž l 0 . 33 T12 q Ž l8 . 33 T22 Gh X
1
'2
, ,
Ž 24 .
with
E Ghy2 s
Eq
E 2
Lh
,
X s Ghy2
ym h2
E q2
Lh X
, ym h2 X
Ž 25 .
Lh and Lh X being the eigenvalues of the coupling matrix between the fields p˜ 0 and p˜ 8 . T is the eigenvector matrix h Gh p˜ 0 sT X . h GhX p˜ 8
ž / ž /
Ž 26 .
It is not written in the usual form
ž
cos u sin u
ysin u , cos u
/
because the model provides a matrix T which is not orthogonal due to the q 2 dependence of its elements. h and hX denote the physical fields, corresponding to the usual kinetic terms.
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3.2. Local contribution In addition to the previous triangle contribution, the decay amplitudes contain a component corresponding to the local decay of the glue component of the f 0 Ž1500.. This component is given by the second and third terms of the r.h.s. of Eq. Ž9.. The second term is a reminiscent of the local 4-quark interaction while the third one is due to the regularization of the quark loop. Indeed, the scale invariance of this quark loop requires to introduce the dilaton field in the cutoff Lx which upon expansion of the Tr ln up to third order in the fields provides a contribution to the amplitude for the studied decays. A priori, there is no reason for dropping it since it is this process which has yielded g f)0 Ž1500. q q / 0 on which the whole study is relied on. These local contributions write . ) )2 Tf Ž0lŽ1500. p 0 p 0 s 2 2 g f 0 Ž1500. x gp u u Hu Ž qp , j s 0 . , . ) )2 Tf Ž0lŽ1500. hh s 2 g f 0 Ž1500. x 2 gh u u Hu Ž qh . q
ž '2 g / ) hss
Ž 27 . 2
Hs Ž qh . q gh)u u
ž '2 g / H ) hss
j
,
Ž 28 . . ) )2 Tf Ž0lŽ1500. h X h X s 2 g f 0 Ž1500. x 2 gh X u u Hu Ž qh X . q
qgh)X u u
ž '2 g / H ) hX s s
ž '2 g / ) hX s s
2
Hs Ž qh X .
,
j
Ž 29 .
. ) ) ) Tf Ž0lŽ1500. hh X s g f 0 Ž1500. x 2 gh u u gh X u u Hu Ž qh . q Hu Ž qh X .
½
q '2 gh)s s
ž
q gh)u u
/ ž '2 g / H Ž q . q H Ž q . ž '2 g / q g ž '2 g / H 5 , ) hX s s
s
) hX s s
. ) ' ) Tf Ž0lŽ1500. K q K y s g f 0 Ž1500. x Ž 2 g K u s .
) hX u u
2
h
s
) hss
hX
j
Hu s Ž q K q . q Hsu Ž q K y . ,
Ž 30 . Ž 31 .
with Hu , Hs , Hu s , Hsu and Hj defined in Appendix A. These expressions involve at the same time the coupling of the glueball to the x field and the coupling of the pseudoscalars to the quarks but without any factor which would account for the propagation of quarks. It looks like the glue part of the f 0 Ž1500. was decaying instantaneously into the mesons.
4. Limitations and merits of the model The scaled NJL model used in the present paper presents the same diseases as the usual NJL. Firstly, it does not confine the quarks, generating unphysical qq pairs whenever the mesons lie above their respective threshold. Secondly, it is not renormalizable, the divergencies of the quark loop having to be regularized with a cutoff which has to be seen as defining itself the model. In the spirit of constructing effective models that reproduce the symmetries of QCD, the scaled NJL model implements the scale anomaly into the NJL model. This is done by introducing one single point-like dilaton field x whose mean value can be identified with the vacuum gluon condensate. This is a drastic simplification since the large
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breaking of the scale invariance should prevent to assign to a single pole the matrix elements of the energy-momentum tensor w18x. In spite of these deficiencies, the scaled NJL model provides a very simple way to yield a mixing between the scalar glueball – whatever it is, f 0 Ž1500. or f 0 Ž1710. – and the scalar isoscalar mesons. In this paper, the glueball, described by the dilaton field, is assumed to be f 0 Ž1500. while the scalars are f 0 Ž1370. and f 0 Ž1710.. All of them have qq and glue content. Playing on the value of the free parameters Mu and x 0 , it is possible to reproduce their mass as far as Mu is large Ž Mu f 700 MeV.. Moreover, the sign of the components in glue or in quark excitations are in agreement with the results of w23x. For instance, the uu and ss components of f 0 Ž1500. have opposite sign Žsee below Eq. Ž41... Finally, as far as strong decays are concerned, our model can account for two distinguishable contributions that can be seen as modelization of the processes depicted in Figs. 7a and 7b of w24x.
5. Results Due to the large value of Mu , the mesons p , K,h and hX are always below their respective threshold, whatever the value of the parameter x 0 . Since f 0 Ž1500. only lies slightly above the threshold 2 Mu , the fact that the NJL model is a nonconfining model is then far from being a problem in the present context and it is treated in the same way as in Ref. w13x for consistency. As explained in Section 2, three parameters of the model are fixed to reproduce mp ,mh X and m K . The last pseudoscalar mass is independent of x 0 and is found to be mh s 482 MeV. The coupling constants of the pseudoscalar mesons to the quarks are also x 0 independent. One has gp)u u s 7.65 ;
gh)u u s 5.23 ,
gh)s s s y4.08 ,
gh)X u u s 1.82 ,
gh)X s s s 4.29 ,
g K)u s s 6.38 .
Ž 32 .
™
On the contrary, the coupling constants of the scalars strongly depend on x 0 . For instance, in the large x 0 limit, g f)0 Ž1500.u u and g f)0 Ž1500. s s 0. In Fig. 2, we plot the x 0
Fig. 2. Coupling constants of the scalar glueball to the x field Žfull curve., to the u-quark Ždotted curve, absolute value of the coupling constant. and to the s-quark Žplain circle curve..
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behavior of the three coupling constants g f)0 Ž1500.u u , g f)0 Ž1500. s s and g f)0 Ž1500. x , showing that, even for large values of x 0 , the quark content of f 0 Ž1500. remains large. ŽNote that for x 0 ` the glueball decouples from the theory. In that limit, f 0 Ž1370. and f 0 Ž1710. become pure nn and ss excitations, respectively w13x.. Let us turn to the strong decay widths of f 0 Ž1500. and consider the local contribu. ŽEqs. Ž27. – Ž31... They can be identified with the contributions given by tions Tf Ž0lŽ1500. the graph 6c of Ref. w24x, provided the vertex is reduced to a single point. Thinking in terms of coupling constants and introducing the quark mass effect via the parameter l, one gets
™
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž hXhX . :gŽ2l . Ž KK . s 1:
:
1 3 1 3
ž ž
gh)u2u gp)u2u gh)X u2u gp)u2u
ql
ql
gh)s2s gp)u2u gh)X s2s gp)u2u
2
2
gh)u u gh)X u u
3
gp)u2u
/ ž / ž / :
2
:
4
g K)u2s
3
gp)u2u
ql
gh)s s gh)X s s 2 gp)uu
2
/
2
l.
Ž 33 .
In Eq. Ž33., the quantity 'l has been introduced each time a ss pair is created in the final channel. This l is the flavor-symmetry violation parameter which is fixed by the ratio of strange to nonstrange hadrons produced in the central region of high energy hadron collisions andror by the branching ratios of the radiative decays JrC ghrghX and JrC gffrgvv : l s 0.4–0.5 w24,25x. It is related to the production probabilities uu:dd: ss s 1:1: l and to the ratio of the quark masses l s Mu2rM s2 w26x. Finally, making the rough approximation according to which the coupling constants can be seen as the qq content of the meson physical fields, Eq. Ž33. reduces to
™
™
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž hXhX . :gŽ2l . Ž KK . 2
2
2
s 1: 13 Ž cos 2u q lsin2u . : 23 Ž 1 y l . sin2u cos 2u : 13 Ž sin2u q lcos 2u . : 43 l ,
Ž 34 .
with p 0 s uu y dd, pq s '2 ud, py s '2 d u, h s '2 Žcos u nn y sin u ss ., hX s '2 Žsin u nn q cos u ss ., K s '2 us, K s '2 su where we have reconciled the normalization with the present work. One then recovers the results of Ref. w24x. The exact . expressions for the amplitudes Tf Ž0lŽ1500. , Eqs. Ž27. – Ž31., would lead to the relations Ž33. if Hu : Hu s : Hs s 1:'l : l ,
Ž 35 .
whatever j and if Hj could be neglected. In fact, one has Hu : Hu s : Hs s 1:0.753:0.546 ,
Ž 36 .
which is coherent with Ž35. provided that l f 0.55. The values Ž36. are obtained for x 0 s 400 MeV, at q 2 s 0. This value of x 0 has been chosen because it leads to quite
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acceptable values for the masses of the scalar nonet w13x. An exact calculation, i.e. with the contribution of Hj and with each H evaluated at its respective on-shell mass yields
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž KK . s 1:0.193:0.011:0.641 ,
Ž 37 .
which is indeed in rather good agreement with the results obtained from Ž34. with u s 378 and l s 0.55 w25x,
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž KK . s 1:0.233:0.031:0.73 ,
Ž 38 .
justifying in some sense the assumptions done above. However, neither the exact calculation nor the approximated one give results in agreement with the experimental one Ž2.: one gets a too small value for hhX and a much too large one for KK. . As explained before, the decay amplitudes receive additional contributions Tf Ž0tŽ1500. Žsee Fig. 1, diagram a. due to the fact that the f 0 Ž1500. exhibits some qq content. Following the same steps as for the local contribution, one has
gŽ2t . Ž pp . :gŽ2t . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž hXhX . :gŽ2t . Ž KK . s 1:
:
:
:
1 3 2 3 1 3 1 3
gh)u2u
ž ž ž ž /ž gp)u2u
q '2 'l
gh)u u gh)X u u gp)u2u
gh)X u2u gp)u2u
g K)u2s
g f 0 Ž 1500 . s s gh)s2s g f 0 Ž 1500 . u u gp)u2u
q '2 'l
q '2 'l
gp)u2u
gp)u2u
g f 0 Ž 1500 . u u
g f 0 Ž 1500 . u u gp)u2u
'l q '2
/
g f 0 Ž 1500 . s s gh)s s gh)X s s
g f 0 Ž 1500 . s s gh)X s2s
2
2
g f 0 Ž 1500 . s s g f 0 Ž 1500 . u u
2
/
2
/
2
/
.
Ž 39 .
If one assumed that the coupling strengths could be in the ratio g f 0 Ž1500. s s g f 0 Ž1500.u u
s
(
l 2
,
Ž 40 .
one would recover exactly the various ratios of Eq. Ž34.,
gŽ2t . Ž pp . :gŽ2t . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž hXhX . :gŽ2t . Ž KK . 2
2
2
s 1: 13 Ž cos 2u q lsin2u . : 23 Ž 1 y l . sin2u cos 2u : 13 Ž sin2u q lcos 2u . : 43 l .
Ž 41 .
However, the assumption Ž40. is far from being satisfied here. One has g f 0 Ž1500. s srg f 0 Ž1500.u u s y0.46 Žsee Fig. 2, where we have changed the sign of g f 0 Ž1500.u u . whose minus sign is in agreement with results of w23x and imposes very small values for
168
M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
. . the branching ratios. The amplitudes Tf Ž0tŽ1500. and Tf Ž0lŽ1500. will then lead to completely different results. We now turn to our model predictions, using the exact expressions Ž10. – Ž14.. One has
gŽ2t . Ž pp . :gŽ2t . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž KK . s 1:2.2 P 10y2 :0.8 P 10y2 :1.1 P 10y2 ,
Ž 42 . showing that the relative strengths of the decays are modified with regard to Ž37.. The full curves of Fig. 3 represent the x 0 behavior of the Žtriangle. decay widths: Ža. pp , Žb. hh , Žc. hhX and Žd. KK. It is clear that, whatever x 0 , the value of Gf 0 Ž1500. ™ p p has no physical meaning. The origin of these awfully large values lies in the large 2 value of Mu which affects at the same time the values of g f 0 Ž1500.u u , gp)u u and Ju u u . It is important to stress that this value was forced to us in order to consider the a 0 Ž1450. as a member of the scalar nonet: in the NJL model, this mass is exactly 2 Mu w13x. In order to reduce the value of the constituent quark mass, one has to consider a confining model w20x. One can relax the constraint of considering a 0 Ž1450. as being a member of the scalar nonet. Taking a 0 Ž980. as the new candidate gives a constituent quark mass Mu s 490 MeV. The trouble we have now is that the scalar isoscalar sector is well above the threshold if we try to reproduce a glueball of mass 1.5 GeV. The small width approximation Žfor the unphysical decay of the scalar mesons into two free quarks. is no longer valid. This is once again a problem linked to the nonconfining nature of the NJL model. One can say that, within the present model, a high value of the constituent quark mass is necessary not only because a0 Ž1450. is taken as a member of the scalar nonet, leaving the a 0 Ž980. as a candidate for a KK molecule, but also because of the lack of confinement of the model. These considerations can be checked numerically: it is observed that multiplying the value of Mu by a factor 2 multiplies the width by roughly a factor 2 5. In fact, it can be verified analytically that the width behaves roughly as Mu5. For instance, would we take the mass of the scalar around 800 MeV, i.e. Mu f 400 MeV, we would obtain results in agreement with w27x. ŽIn this comparison, we relax the constraint of a mixing with the glueball. Then the nonet lies around 800 MeV and the model sits just above the threshold, making the small unphysical decay width into two free quarks irrelevant.. The problem is much less severe for the other decays. This can be traced back to the negative value of the ratio g f 0 Ž1500. s srg f 0 Ž1500.u u . There is then destructive interference between the amplitudes u and s, which strongly reduces the values of the widths. Note that, before trying to apply the model to the scalar nonet, it may be worth testing it by studying the well-known two-pseudoscalar decay modes in the conventional
2
Note that this value does not affect the usual properties of the NJL model. This is due to the fact that the parameters are fixed to reproduce common observables. The pion mass and decay constant being fixed to their physical value, one can for instance obtain a two photon decay width of the neutral pion of 7.64 eV. Although it is often assumed that the r mass is roughly twice the constituent quark mass, it does not play any role in our fitting procedure: the rho mass is fixed, and this is the vectorial four quark coupling constant G V which is fitted accordingly.
M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
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X Fig. 3. Decay widths of the scalar glueball into pp , hh , hh and KK, for the quark loop contribution Žfull . Ž . curves and for the total contribution dotted curves .
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M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
meson sector, for example the f 2 Ž1270. and the f 2X Ž1525.. However, a tensor structure is not included in the NJL model, and this study is beyond the scope of the paper. . . Let us now turn to the total contribution Tf Ž0tŽ1500. q Tf Ž0lŽ1500. . Except for the hhX . decay, the total widths are reduced with regard to the quark loop contribution Tf Ž0tŽ1500. whatever the value of x 0 . This feature is exhibited by the dotted curves of Fig. 3. However, even for values of x 0 as large as 1 GeV, the case of the 2p decay is hopeless. The branching ratios keep tremendous small values independently of x 0 ,
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:1.3 P 10y2 :1.2 P 10y2 :0.7 P 10y3 ,
Ž 43 . reflecting again the problem of the pion. For completeness, we give below the equivalent of Eqs. Ž37., Ž42. and Ž43. for the value Mu s 490 MeV. They are given for illustrative purpose only, to show that taking a lower quark mass leads to less unphysical branching ratios. We stress again that these numbers have to be taken with caution since the scalar isoscalar are well above the threshold. Moreover, the nonet would no longer be the one considered here but would consist of a0 Ž980., f 0 Ž975., f 0 Ž1370., K 0) Ž1430. Žstill with the glueball mass fixed at 1.5 GeV.. With these cautions, the equivalent of Eq. Ž37. is
gŽ2l . Ž pp . :gŽ2l . Ž hh . :gŽ2l . Ž hhX . :gŽ2l . Ž KK . s 1:0.2045:0.1:0.947 ,
Ž 44 .
while the equivalent of Eq. Ž42. is
gŽ2t . Ž pp . :gŽ2l . Ž hh . :gŽ2t . Ž hhX . :gŽ2t . Ž KK . s 1:1.326:0.325:5.679 .
Ž 45 .
Finally, the total contribution would be Žwith the same cautions.
g 2 Ž pp . :g 2 Ž hh . :g 2 Ž hhX . :g 2 Ž KK . s 1:0.84:0.046:3.224 .
Ž 46 .
6. Summary and conclusion The model of the present paper uses a dilaton field x to implement the QCD trace anomaly in the Nambu–Jona-Lasinio model. Since the latter is not renormalizable, all the ultraviolet divergences have to be regularized using a cutoff Lx which entails a mixing between the three scalar isoscalar fields. One then gets three scalar ‘‘hybrids’’ f 0 which are all a mixing of glue and uu and ss excitations. Identifying these hybrids with f 0 Ž1370., f 0 Ž1500., f 0 Ž1710., the constituent u-quark mass used in the model has to be large: Mu f m f 0r2. Here Mu s m a 0r2. Such a large mass does not destroy the rather good results obtained up to now for the properties of the pseudoscalar mesons within the NJL model. For instance, the two photon decay of p 0 is not affected by the value of Mu . In the present paper, we focused on the f 0 Ž1500. which is identified with the glueball in the sense that it is the state that would yield pure glue if there was no mixing. We have studied to what extent a large value of Mu can reproduce its possible strong decays into two pseudoscalar mesons. The main point of the paper is that the strong decays receive contribution from two distinguishable processes. Due to its qq content, the f 0 Ž1500. can decay via processes
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described by triangular quark loops Žsee Fig. 1, diagram a.. These processes are similar to the one shown in Fig. 7a of w24x. The corresponding amplitude goes as 1r Nc . f 0 Ž1500. can also decay via two local processes schematically represented in Fig. 1, diagram b: one is the reminiscent of the 4-quark interaction Žterm %a2x 2 in Eq. Ž7.., the other comes from the regularization of the quark loop. These processes can be seen as the local limit of the diagram 7c of w24x. Our contribution is of order O Ž1. in Nc while the diagram 7c of w24x behaves as 1rNc . This can be explained by the way the glueball is incorporated in our formalism. Indeed, it is not seen as a two gluon composite. Note that provided some assumptions described below, both contributions Ždiagrams a and b of Fig. 1. satisfy the sum rule w24x
(
Ý g t2,l Ž i . % Ž 2 q l. 2 .
Ž 47 .
i
These assumptions are however not fulfilled by our model, mainly due to the quark loop contribution. For the local contribution, one has to assume that the various meson–quark coupling constants can be identified with the qq contents and that the quark mass effect Ž Mu / M s . is introduced via the flavor symmetry violation parameter l. Our calculation is consistent with a value of l f 0.55, see Eqs. Ž35. and Ž36.. Our results for the local contribution are then similar to those of Ref. w25x but in complete disagreement with the experimental data Žsee Eqs. Ž2. and Ž37... For the quark loop contribution, one has to add the assumption Ž40. which is not satisfied here. Our results for this contribution are then at variance with those of w24x: the sum rule Ž47. is not verified. Our results allow to stress that, due to the large value of Mu , the calculated pp decay width takes awfully large values Ž1–10 GeV. and has therefore no physical meaning. This problem is partly suppressed for the other decays due to the destructive interferences between the amplitudes associated with the u- and s-quarks. Adding the two contributions, the results remain qualitatively the same as for the quark loop alone. Except for the hhX mode, the widths are reduced with regard to the quark loop contribution. The various widths decrease with an increasing value of the vacuum parameter x 0 . Whatever x 0 , the case of pp is hopeless. However, for value of x 0 f 1 GeV, the other widths take acceptable value Ž Gf 0 ™ hh q Gf 0 ™ hh X q Gf 0 ™ K K f 50 MeV.. The results obtained for these latter modes are strongly dependent on the value and on the sign of the ratio g f)0 s srg f)0 u u . In the present model, we have g f)0 s srg f)0 u u s y0.46. Since the minus sign is in agreement with the result of w23x but in contradistinction with w24x, it would then be worthwhile to investigate further the qq content of the glueball. One can draw the following conclusions. If one trusts in the idea of w7,8x, according to which the f 0 Ž1500. is the glueball, the scaled NJL model cannot reproduce at the same time the dynamical properties of pseudoscalars and scalars. One way to get a way out would be to introduce confinement as in Ref. w28x. Indeed, such a confinement has as effect to increase the scalar meson mass with regard to the nonconfining model. The masses of the scalars could then be reproduced with a smaller value of the constituent up quark mass giving better results for the decay width of the glueball into pions. Preliminary results are promising and will be published in a subsequent publication w20x. However, one could think in another way, following the results of w10x which clearly identify the lightest scalar glueball with f 0 Ž1710.. In that case, our f 0 Ž1500., even if
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mixed with the others ‘‘hybrids’’, would have its larger component in the ss channel w21x. Its decay into pp would be spectacularly reduced. The way the other decays would be modified is not trivial due to the destructive interferences between the uu and ss channels. This problem requires further investigation.
Acknowledgements The work of M.M. and B.VdB. is supported by the Institut Interuniversitaire des Sciences Nucleaires de Belgique. B.VdB. acknowledges Dr. J.-R. Cudell for several ´ discussions during the completion of the work. M.J. thanks the Departamento de Fısica ´ Teorica of the Universidad Autonoma de Madrid for hospitality. ´ ´ Appendix A . In order to get the contributions Tf Ž0lŽ1500. , one has to calculate the second and third terms of Eq. Ž9.. Keeping only terms which couple the x˜ field to two pseudoscalars, one gets Ž a,b s 0,1,2,3,8. 3Ž l . Ieff s
1
Ž bV .
2
Ý Ý x˜ q ½ p˜ a q yqp˜ byq X
X
X
q
Ž3. ˜y X Ž2. X DaŽ b2 . q DaŽ b3 . q 2 K˜q q yq Kyq Du s q Du s
q
Ž3. ˜q X Ž2. X q2 K˜y q yq Kyq Dsu q Dsu
5,
Ž A.1 .
with Žsecond term. Ž2. Ž2. DiŽ2. ˜ 2x 0 d i j j s Du s s Dsu s Nc a
Ž i , j s 1,2,3,8 . ,
D Ž2. ˜ 2x 0 Ž 1 q j . , 00 s Nc a
Ž A.2 .
and Žthird term. 2 D Ž3. 00 s Nc L x 0
4 3
X
Bu u Ž q . q
B s s Ž qX . 2
,
4
'2 Bu u Ž qX . y Bs s Ž qX . , 3 4 Bu u Ž q X . Ž3. D 88 s Nc L 2x 0 q B s s Ž qX . , 3 2 Ž3. D 08 s Nc L 2x 0
X 2 DiŽ3. i s Nc L x 0 2 Bu u Ž q .
Ž i s 1,2,3 . , 2
DuŽ3.s s Nc L2x 0 Bu s Ž qX . Ž '2 . , 2
Ž3. Dsu s Nc L2x 0 Bsu Ž qX . Ž '2 . ,
Ž A.3 .
where Ž i, j s u, s . Bi j Ž q . s trg
d4 k
H Ž 2p .
4
g i Ž k . ig 5 g j Ž k y q . ig 5 d Ž k 2 y L2x 02 . ,
Ž A.4 .
M. Jaminon et al.r Nuclear Physics A 662 (2000) 157–174
173
l
g i Ž k . being defined in Eq. Ž16.. The factor 2 in Eq. ŽA.1. comes from isospin symmetry Ž K 0 Kq. . In Eq. ŽA.3., the trace in flavor space has been done. We introduce the physical pion Ž i s 1,2,3. and kaon fields Žwith g K)u s s g K)su .
p i) s gp)y1 ˜i , uu p
˜q Kq) s g K)y1 us K , ˜y Ky) s g K)y1 Ž A.5 . su K , X Ž Ž . Ž .. as well as h and h see Eq. 24 and Eq. 26 . Due to its glue component x˜ , the glueball f 0 Ž1500. noted here x ) feels the following interaction term with the pseudoscalar mesons: 1 3Ž l . ) X Ieff s x ) g f)0 Ž1500. x p i)qXyq p iyq Lp p 2 Ý Ý q Ž bV . q qX X X X L X Xqh X X X qhqXyq hyqX Lhh q hqX Xyq hyq hh q yq hyq Lhh )y )y )q X X X X q2 K q)q yq Kyq L K q K yq 2 K q yq Kyq L K y K q ,
where g f)0 Ž1500. x is defined in Eq. Lpp s gp)u2u R u qX , j s 0 , Lhh s gh)u2u R u qX q gh)s2s R s
Ž Ž .
Ž A.6 .
Ž22. and
.
Ž qX . q gh)u u gh)s s Rj ,
LhX h X s gh)X u2u R u Ž qX . q gh)X s2s R s Ž qX . q gh)X u u gh)X s s Rj , Lhh X s gh)u u gh)X u u R u Ž qX . q gh)s s gh)X s s R s Ž qX . q Ž gh)u u gh)X s s q gh)s s gh)X u u .
Rj 2
,
2
L K q K ys Ž '2 g K)u s . R u s Ž qX . , 2
L K y K qs Ž '2 g K)u s . R su Ž qX . ,
Ž A.7 .
R u Ž qX . s 13 Ž 2 D 00 q '2 D 08 q D 88 . 2j s Nc x 0 a˜ 2 1 q q 2 L 2 Bu u Ž q X . , 3
ž
/
R s Ž qX . s 13 Ž D 00 y '2 D 08 q 2 D 88 . j s Nc x 0 a˜ 2 1 q q 2 L 2 B s s Ž qX . , 3
ž
/
R u s Ž qX . s Nc x 0 L2 Bu s Ž qX . q R su Ž qX . s Nc x 0 L2 Bsu Ž qX . q
a˜ 2x 0 Nc 2 a˜ 2x 0 Nc 2
, ,
Rj s 13 Ž 2'2 D 00 y D 08 y 2'2 D 88 . 2'2 s 3
j Nc x 0 a˜ 2 .
Ž A.8 .
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One then gets the amplitudes Ž27. – Ž31. with Hu Ž q X . s H s Ž qX . s
Ru 2 Rs 2
s Nc x 0
s Nc x 0
a˜ 2
2j
ž / ˜ ž / 2
a2 2
1q
3
j
1q
3
q L 2 Bu u Ž q X . ,
q L 2 B s s Ž qX . ,
Hu s Ž qX . s R u s Ž qX . s Nc x 0 L 2 Bu s Ž qX . , Hsu Ž qX . s R su Ž qX . s Nc x 0 L 2 Bsu Ž qX . , Hj s
Rj
'2
s 23 j Nc x 0 a˜ 2 ,
Ž A.9 .
once one has introduced the same conventions of notation for the factors 2 and '2 as in Ž3. Eqs. Ž10. – Ž14.. In Eq. ŽA.8., the Di j stand for the sum of DiŽ2. j and Di j .
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