- Email: [email protected]
A note on the constituent quark resonances
19 January 1995 PHYSICS
LETTERS
B
Physics Letters B 343 ( 1995) 3777380
A note on the constituent quark resonances Kuang-Ta Chao a, Hanqing Zheng b~l B Center of Theoretical Physics, CCAST (World Laboratory) and Department of Physics, Peking University, Beijing 100871, People’s Republic of China b International Center for Theoretical Physics, &34100Trieste, Italy
Received 29 September 1994 Editor: R. Gatto
Abstract If there exist a constituent quark resonance, we find by using the Adler-Weisberger relation and the finite-energy sum rule for pion quark scattering that it must be very massive and must have a large decay width. Our results indicate that the presence of the constituent quark excitations does not contradict the observed low lying hadron spectrum. Keywords: Constituent quark; Resonance;
Sum rules; SA; Regge behaviour
The successfulness of the simple constituent quark model introduced two decades ago [ 11 still remains somewhat mysterious, since deep inelastic scattering experiments reveal that there is an infinite number of quarks, anti-quarks and gluons inside a nucleon. It is still not very clear whether the constituent quark is just a rough approximation of the valence quark plus sea quark and gluon effects or that it can be considered as a quasi-particle, which is highly implied by the success of the model in many circumstances; examples are from baryon magnetic moments to the ratio of the total cross-section a( TN) /u( NN) at high energies. If the constituent quark is really a composite quasi-free particle2 we then can discuss the intrinsic properties of the quark itself, such as its mass, anomalous magnetic moment, gA, etc., which should not vary in different surroundings while in the former case this may have to ’ Address after Nov. I, 1994: P. Scherrer Institut. CH 5232 Villigen, Switzerland. * This is the basic assumption we adopt in the present note. If it is not correct then the whole discussion will be invalid. 0370.2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(94)01425-6
vary in order to fit different hadronic properties. The constituent quark as a composite quasi-particle may be explained by the fact that the spontaneous breaking of chiral symmetry in strong interactions occurs at the quark level, as illustrated in Ref. [ 21. Because the chiral symmetry breaking scale (A, N 1 GeV) is larger than the confinement scale (A, N 100 MeV), once we run down from the high energy scale (where the strong interactions are described by perturbative QCD of current quarks and gluons) to A,, chiral symmetry breaks down. As a consequence, quarks get their constituent masses and can still be considered as quasifree particles because the energy scale is still above the confinement scale. Pions also appear as Goldstone excitations. In the picture of the constituent quark model, the physical hadronic states only contain quarks and pions, and maybe glueballs, but no colored gluons or their excitations. This picture explains very well the low lying hadron spectrum. For example, all the meson states below 2 GeV with nonvanishing flavor quantum number are well identified [ 31.
318
K.-T. Chao, H. Zheng / Physics Letters B 343 (1995) 377-380
If we believe that constituent quarks are composite particles, in general we should also expect the existence of resonances of the constituent quark. Then there is a natural question: why must the resonances be heavy? To explain the success of the constituent quark model it is necessary to answer this question. If there are low lying resonances, then the hadron spectrum would be more complicated then it shows. Another relevant question is that if there exist a heavy constituent quark resonance, how large is its decay width to the constituent quark? We know from experiments that the low lying spectrum is rather simple and that complex. For this reason, it is usually believed that the constituent quark has no resonances, or in other words, it is not necessary to assume the existence of quark resonances. The main purpose of this note is in trying to answer this question more seriously, in a more quantitative way. Indeed, from simple physical arguments, we know that the quark excitation must have a large energy gap because the size of the quark is small (compared with the scale of the nucleon). For the rotational excitations, the energy gap may obey a simple relation, 1
AE--,
(1)
I
where I is the momentum of inertia of a composite object. Roughly, we may expect IQ
N
r=<
MN
ri
M [email protected]'
N
AEN& > 1.8 GeV, 1,
where I is the partial decay width of the resonance into a quark and a pion, MR is the mass of the resonance.
F,=
2X?+
1
(2SI + 1)(2s2+
1)
IN
is the usual statistical factor. Taking Eq. (3) into the Adler-Weisberger sum rule 5 ,
Taking AEN - 300 MeV, we obtain AEQ
cuts. In the spirit of the constituent quark model, we assume that all these contributions are saturated by the resonance contributions (and also 7r quark continuum) . However, not only the mass of a resonance but also another parameter, its coupling to the quark and pion, will be involved. This can be resolved by using another relation, the finite energy sum rule which has been used in understanding the short distance contribution to the g.4 of the quark [ 63, For simplicity, we assume that there is a single resonance that dominates the r quark scattering process below the Regge scale. From the large NC argument we know that the intermediate states in the pion quark scattering process must have isospin l/2 4 . So when we consider more than one resonance their contributions to the sum rules will have the same sign - there will be no large cancellations among their contributions. This is correct to leading order in the l/NC expansion and enables us to make a confidential estimate. In the narrow resonance approximation, we have
(2)
where we have made use of the fact that the sum of the volume of three quarks is less than the volume of the nucleon. If we use the size of the quark adopted in Ref. [4], rQ E 0.2 fm, more or less the same value of AEQ can be obtained. Except for the above naive estimate, we also have another method which contains more information about the dynamics of the strong interaction. That is, if quarks have resonances they will contribute to the gA of the quark through the Adler-Weisberger relation for Z- quark scattering3 . In the picture of QCD the contribution to the imaginary part of the scattering amplitude will be given by the quark and gluon
3Assuming the forward two-two quasi-elastic scattering amplitudes vanish rapidly at high energies, Weinberg has derived a super-convergence relation which leads him to conclude that even if there are constituent quark resonances they will not contribute to gA through the Adler-Weisberger sum rule [ 51. However, we know that for diffraction dissociation processes, even though they have much smaller cross-sections than the elastic one they will not vanish at high energies because they are also dominated by the Pomeron exchange. 4 See for example the second paper in Ref. [ 51. 5In the non-linear chiral quark model this relation needs to be modified because such a model does not have a good high energy behavior. An additional constant term, -2 [ rni/(4~rf~)* 1X should be added on the right hand side. This term log (A;s*l’n;) includes the main contributions to 8~ from the low energy chiral quark model which is I/NC suppressed. See the discussions in Ref. [ 71 for details.
K.-T. Chao, H. Zheng / Physics Letters B 343 (1995) 377-380
319
LS GeV L=.lD.CeV ...-c. ,5 ($”
0.75
we find that the resonance
contribution
L=?!p
to gA is 0.7 0.65 0.6
The lowest moment finite energy sum rule reads
,,
,:’
L
0.55
s
vdvAu=
$$$--&)ltrrO.
where & and /Sq are (p) Reggeon couplings to pions and quarks, respectively. & = 2/?Iq N gPrr N 6. L is the scale above which the scattering amplitude has a proper Regge behavior and s = 2mqL + rni. 6 Taking SO N rns, my = 350 MeV and the intercept of the Regge trajectory, LYO= 0.44, the right hand side of the finite energy sum rule can be estimated and will be compensated by the resonance contribution to the left hand side (the r quark continuum contribution is much smaller than the Regge contribution in here). We have L
s
vdvA&
= 4.1r2FJ rni(Mi
M:, - n$) ’
(6)
0
We find from Eqs. (5) and (6) that
,;,‘+ao,s, I-N
47r2(1 + ~o)MR
I
* 2.5 ’ I = !,&$5
1
(5)
4
0
1.5
, :’ ’ ! 2
Fs.
The lower bound on MR obtained in our approach depends on the value of gA, the Regge scale L and other parameters we use and will therefore suffer from some ambiguities. In the non-relativistic quark model ’ The explicit value of L (pion excess energy in the laboratory frame) should be at the order of a few GeV but is not well determined. In Ref. [ 61, L is taken to be 3.5 GeV which is a little bit small, the upper bound of go is in turn found to be 0.6. To estimate L, we consider the case for rr nucleon scattering, where the Regge behavior starts to work at the center of mass energy square s = 2MNL + Mi + tn$ N 10 GeV2. Assuming the same L (Y 4.85 GeV), rather than s, works also for r quark scattering; using the method in Ref. [6] gives SA 5 0.65. Taking L = 14 GeV (then the s for rr quark scattering is about 10 GeV2) gives RA 5 0.75.
Fig. I. The value of go as a function of the resonance mass, MR. for different choice of the Regge scale L. The vertical lines indicate the maximal value of MR (5 4) can be reached for different L.
of the nucleon gA = 0.75. However, in the heavy-light meson (Qq) system where there is evidence that the light quark is relativistic, the value of gA is found to be significantly smaller, gA N 0.6 [ 8 J , although there is no difficulty in increasing this value by reducing the mass parameter of the constituent quark [9] 7 . Here we take the conservative value gA > 0.6. We plot the g,J of the constituent quark as a function of MR in Fig. 1 for several values of L. The Reggeon contribution to gi has a Lao-’ dependence, whereas the resonance contribution has a Lao+’ dependence. We may also choose the parameter L in such a way that the value of g$ is not sensitive to L. This leads to L= (Mi-mi)/2m,,
(8)
which means that the Regge behavior starts just from s = Mi. This is however rather artificial because of the single resonance dominance assumption. If more resonances are added, in general no stable point ( dgA/dL = 0) exist. We see from Fig. 1 that the larger the used Regge scale L is the larger the lower bound on MR will be, and the larger the resonance mass is the larger the value that gA can take. However, it is difficult for g: to reach the value 0.75, within a reasonable range of the various parameters. A conservative lower bound on the mass can be obtained, ’ As we have emphasized earlier, the intrinsic properties of the constituent quark as a composite particle should not vary in different surroundings. However, the status of the constituent quark parameters are not very satisfactory.
380
MR > 1.67 GeV.
K.-T. Chao, H. Zheng /Physics
(9)
Further, Eq. (7) tells that the decay width of the resonance into a quark and a pion is very large, i.e., larger than N 650 MeV. (If the resonance has spin 312, this value will be reduced by half. And this value is roughly proportional to L as L increases.) We mention that we have made use of the narrow resonance assumption in obtaining the large width, but the result should make sense qualitatively. Considering more resonances will not change the above results much. We have mentioned earlier that to leading order in l/NC all the possible resonances should have isospin l/2 and therefore contribute with equal sign to the sum rules. The values obtained above can therefore be considered as a smeared effect with the larger weight on the lowest lying resonance. The constituent quark resonances can form hadron resonances, like Q*Q or Q*QQ, etc. We conclude from the above analysis that these states have a large mass and a large decay width into a normal hadron and a pion (pions) and will therefore be difficult to observe. The presence of these states will not contradict the observed low lying hadron spectrum. Our results confirm the self-consistency of our present understanding on the strong interaction physics.
Letters B 343 (1995) 377-380
One of us (H.Z.) would like to thank Professors D. Amati, C.S. Huang and M.L. Yan for helpful conversations. He also acknowledges Professor Abdus Salam, the International Atomic Energy Agency, UNESCO and the international Center for Theoretical Physics, Trieste, for support.
References [II H. Fritzsch and M. Gell-Mann, Proc. of XVI International Conference on High Energy Physics, Vol. 2 (Chicago, 1972) p. 135; S. Weinberg, Physica A 96 (1979) 327, Section 6. 121 A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189. [31 S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189. 141 S.M. Troshin and N.E. Tyurin, Phys. Rev. D 49 ( 1994) 4427. [51 S. Weinberg, Phys. Rev. Left. 65 ( 1990) 1177, 118 1. [61 M. Jacob and H. Zheng, Phys. Lea. B 321 (1994) 105. 171 S. Perk and E. de Rafael, Phys. L&t. B 309 (1993) 309. [81 See, e.g.. M.B. Wise, Phys. Rev. D 45 (1992) R2188; T.M. Yan et al., Phys. Rev. D 46 ( 1992) 1148; P Cho and H. Georgi, Phys. Lett. B 296 (1992) 408; J.F. Amundson et al., Phys. Lett. B 296 (1992) 415; H.Y. Cheng et al., Phys. Rev. D 47 (1993) 527; Y.B. Dai, C.S. Huang and H.Y. Jin, Z. Phys. C 60 (1993) 527. [91 N.G. Chen, K.T. Chao, PUTP-94-10.
LETTERS
B
Physics Letters B 343 ( 1995) 3777380
A note on the constituent quark resonances Kuang-Ta Chao a, Hanqing Zheng b~l B Center of Theoretical Physics, CCAST (World Laboratory) and Department of Physics, Peking University, Beijing 100871, People’s Republic of China b International Center for Theoretical Physics, &34100Trieste, Italy
Received 29 September 1994 Editor: R. Gatto
Abstract If there exist a constituent quark resonance, we find by using the Adler-Weisberger relation and the finite-energy sum rule for pion quark scattering that it must be very massive and must have a large decay width. Our results indicate that the presence of the constituent quark excitations does not contradict the observed low lying hadron spectrum. Keywords: Constituent quark; Resonance;
Sum rules; SA; Regge behaviour
The successfulness of the simple constituent quark model introduced two decades ago [ 11 still remains somewhat mysterious, since deep inelastic scattering experiments reveal that there is an infinite number of quarks, anti-quarks and gluons inside a nucleon. It is still not very clear whether the constituent quark is just a rough approximation of the valence quark plus sea quark and gluon effects or that it can be considered as a quasi-particle, which is highly implied by the success of the model in many circumstances; examples are from baryon magnetic moments to the ratio of the total cross-section a( TN) /u( NN) at high energies. If the constituent quark is really a composite quasi-free particle2 we then can discuss the intrinsic properties of the quark itself, such as its mass, anomalous magnetic moment, gA, etc., which should not vary in different surroundings while in the former case this may have to ’ Address after Nov. I, 1994: P. Scherrer Institut. CH 5232 Villigen, Switzerland. * This is the basic assumption we adopt in the present note. If it is not correct then the whole discussion will be invalid. 0370.2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(94)01425-6
vary in order to fit different hadronic properties. The constituent quark as a composite quasi-particle may be explained by the fact that the spontaneous breaking of chiral symmetry in strong interactions occurs at the quark level, as illustrated in Ref. [ 21. Because the chiral symmetry breaking scale (A, N 1 GeV) is larger than the confinement scale (A, N 100 MeV), once we run down from the high energy scale (where the strong interactions are described by perturbative QCD of current quarks and gluons) to A,, chiral symmetry breaks down. As a consequence, quarks get their constituent masses and can still be considered as quasifree particles because the energy scale is still above the confinement scale. Pions also appear as Goldstone excitations. In the picture of the constituent quark model, the physical hadronic states only contain quarks and pions, and maybe glueballs, but no colored gluons or their excitations. This picture explains very well the low lying hadron spectrum. For example, all the meson states below 2 GeV with nonvanishing flavor quantum number are well identified [ 31.
318
K.-T. Chao, H. Zheng / Physics Letters B 343 (1995) 377-380
If we believe that constituent quarks are composite particles, in general we should also expect the existence of resonances of the constituent quark. Then there is a natural question: why must the resonances be heavy? To explain the success of the constituent quark model it is necessary to answer this question. If there are low lying resonances, then the hadron spectrum would be more complicated then it shows. Another relevant question is that if there exist a heavy constituent quark resonance, how large is its decay width to the constituent quark? We know from experiments that the low lying spectrum is rather simple and that complex. For this reason, it is usually believed that the constituent quark has no resonances, or in other words, it is not necessary to assume the existence of quark resonances. The main purpose of this note is in trying to answer this question more seriously, in a more quantitative way. Indeed, from simple physical arguments, we know that the quark excitation must have a large energy gap because the size of the quark is small (compared with the scale of the nucleon). For the rotational excitations, the energy gap may obey a simple relation, 1
AE--,
(1)
I
where I is the momentum of inertia of a composite object. Roughly, we may expect IQ
N
r=<
MN
ri
M [email protected]'
N
AEN& > 1.8 GeV, 1,
where I is the partial decay width of the resonance into a quark and a pion, MR is the mass of the resonance.
F,=
2X?+
1
(2SI + 1)(2s2+
1)
IN
is the usual statistical factor. Taking Eq. (3) into the Adler-Weisberger sum rule 5 ,
Taking AEN - 300 MeV, we obtain AEQ
cuts. In the spirit of the constituent quark model, we assume that all these contributions are saturated by the resonance contributions (and also 7r quark continuum) . However, not only the mass of a resonance but also another parameter, its coupling to the quark and pion, will be involved. This can be resolved by using another relation, the finite energy sum rule which has been used in understanding the short distance contribution to the g.4 of the quark [ 63, For simplicity, we assume that there is a single resonance that dominates the r quark scattering process below the Regge scale. From the large NC argument we know that the intermediate states in the pion quark scattering process must have isospin l/2 4 . So when we consider more than one resonance their contributions to the sum rules will have the same sign - there will be no large cancellations among their contributions. This is correct to leading order in the l/NC expansion and enables us to make a confidential estimate. In the narrow resonance approximation, we have
(2)
where we have made use of the fact that the sum of the volume of three quarks is less than the volume of the nucleon. If we use the size of the quark adopted in Ref. [4], rQ E 0.2 fm, more or less the same value of AEQ can be obtained. Except for the above naive estimate, we also have another method which contains more information about the dynamics of the strong interaction. That is, if quarks have resonances they will contribute to the gA of the quark through the Adler-Weisberger relation for Z- quark scattering3 . In the picture of QCD the contribution to the imaginary part of the scattering amplitude will be given by the quark and gluon
3Assuming the forward two-two quasi-elastic scattering amplitudes vanish rapidly at high energies, Weinberg has derived a super-convergence relation which leads him to conclude that even if there are constituent quark resonances they will not contribute to gA through the Adler-Weisberger sum rule [ 51. However, we know that for diffraction dissociation processes, even though they have much smaller cross-sections than the elastic one they will not vanish at high energies because they are also dominated by the Pomeron exchange. 4 See for example the second paper in Ref. [ 51. 5In the non-linear chiral quark model this relation needs to be modified because such a model does not have a good high energy behavior. An additional constant term, -2 [ rni/(4~rf~)* 1X should be added on the right hand side. This term log (A;s*l’n;) includes the main contributions to 8~ from the low energy chiral quark model which is I/NC suppressed. See the discussions in Ref. [ 71 for details.
K.-T. Chao, H. Zheng / Physics Letters B 343 (1995) 377-380
319
LS GeV L=.lD.CeV ...-c. ,5 ($”
0.75
we find that the resonance
contribution
L=?!p
to gA is 0.7 0.65 0.6
The lowest moment finite energy sum rule reads
,,
,:’
L
0.55
s
vdvAu=
$$$--&)ltrrO.
where & and /Sq are (p) Reggeon couplings to pions and quarks, respectively. & = 2/?Iq N gPrr N 6. L is the scale above which the scattering amplitude has a proper Regge behavior and s = 2mqL + rni. 6 Taking SO N rns, my = 350 MeV and the intercept of the Regge trajectory, LYO= 0.44, the right hand side of the finite energy sum rule can be estimated and will be compensated by the resonance contribution to the left hand side (the r quark continuum contribution is much smaller than the Regge contribution in here). We have L
s
vdvA&
= 4.1r2FJ rni(Mi
M:, - n$) ’
(6)
0
We find from Eqs. (5) and (6) that
,;,‘+ao,s, I-N
47r2(1 + ~o)MR
I
* 2.5 ’ I = !,&$5
1
(5)
4
0
1.5
, :’ ’ ! 2
Fs.
The lower bound on MR obtained in our approach depends on the value of gA, the Regge scale L and other parameters we use and will therefore suffer from some ambiguities. In the non-relativistic quark model ’ The explicit value of L (pion excess energy in the laboratory frame) should be at the order of a few GeV but is not well determined. In Ref. [ 61, L is taken to be 3.5 GeV which is a little bit small, the upper bound of go is in turn found to be 0.6. To estimate L, we consider the case for rr nucleon scattering, where the Regge behavior starts to work at the center of mass energy square s = 2MNL + Mi + tn$ N 10 GeV2. Assuming the same L (Y 4.85 GeV), rather than s, works also for r quark scattering; using the method in Ref. [6] gives SA 5 0.65. Taking L = 14 GeV (then the s for rr quark scattering is about 10 GeV2) gives RA 5 0.75.
Fig. I. The value of go as a function of the resonance mass, MR. for different choice of the Regge scale L. The vertical lines indicate the maximal value of MR (5 4) can be reached for different L.
of the nucleon gA = 0.75. However, in the heavy-light meson (Qq) system where there is evidence that the light quark is relativistic, the value of gA is found to be significantly smaller, gA N 0.6 [ 8 J , although there is no difficulty in increasing this value by reducing the mass parameter of the constituent quark [9] 7 . Here we take the conservative value gA > 0.6. We plot the g,J of the constituent quark as a function of MR in Fig. 1 for several values of L. The Reggeon contribution to gi has a Lao-’ dependence, whereas the resonance contribution has a Lao+’ dependence. We may also choose the parameter L in such a way that the value of g$ is not sensitive to L. This leads to L= (Mi-mi)/2m,,
(8)
which means that the Regge behavior starts just from s = Mi. This is however rather artificial because of the single resonance dominance assumption. If more resonances are added, in general no stable point ( dgA/dL = 0) exist. We see from Fig. 1 that the larger the used Regge scale L is the larger the lower bound on MR will be, and the larger the resonance mass is the larger the value that gA can take. However, it is difficult for g: to reach the value 0.75, within a reasonable range of the various parameters. A conservative lower bound on the mass can be obtained, ’ As we have emphasized earlier, the intrinsic properties of the constituent quark as a composite particle should not vary in different surroundings. However, the status of the constituent quark parameters are not very satisfactory.
380
MR > 1.67 GeV.
K.-T. Chao, H. Zheng /Physics
(9)
Further, Eq. (7) tells that the decay width of the resonance into a quark and a pion is very large, i.e., larger than N 650 MeV. (If the resonance has spin 312, this value will be reduced by half. And this value is roughly proportional to L as L increases.) We mention that we have made use of the narrow resonance assumption in obtaining the large width, but the result should make sense qualitatively. Considering more resonances will not change the above results much. We have mentioned earlier that to leading order in l/NC all the possible resonances should have isospin l/2 and therefore contribute with equal sign to the sum rules. The values obtained above can therefore be considered as a smeared effect with the larger weight on the lowest lying resonance. The constituent quark resonances can form hadron resonances, like Q*Q or Q*QQ, etc. We conclude from the above analysis that these states have a large mass and a large decay width into a normal hadron and a pion (pions) and will therefore be difficult to observe. The presence of these states will not contradict the observed low lying hadron spectrum. Our results confirm the self-consistency of our present understanding on the strong interaction physics.
Letters B 343 (1995) 377-380
One of us (H.Z.) would like to thank Professors D. Amati, C.S. Huang and M.L. Yan for helpful conversations. He also acknowledges Professor Abdus Salam, the International Atomic Energy Agency, UNESCO and the international Center for Theoretical Physics, Trieste, for support.
References [II H. Fritzsch and M. Gell-Mann, Proc. of XVI International Conference on High Energy Physics, Vol. 2 (Chicago, 1972) p. 135; S. Weinberg, Physica A 96 (1979) 327, Section 6. 121 A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189. [31 S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189. 141 S.M. Troshin and N.E. Tyurin, Phys. Rev. D 49 ( 1994) 4427. [51 S. Weinberg, Phys. Rev. Left. 65 ( 1990) 1177, 118 1. [61 M. Jacob and H. Zheng, Phys. Lea. B 321 (1994) 105. 171 S. Perk and E. de Rafael, Phys. L&t. B 309 (1993) 309. [81 See, e.g.. M.B. Wise, Phys. Rev. D 45 (1992) R2188; T.M. Yan et al., Phys. Rev. D 46 ( 1992) 1148; P Cho and H. Georgi, Phys. Lett. B 296 (1992) 408; J.F. Amundson et al., Phys. Lett. B 296 (1992) 415; H.Y. Cheng et al., Phys. Rev. D 47 (1993) 527; Y.B. Dai, C.S. Huang and H.Y. Jin, Z. Phys. C 60 (1993) 527. [91 N.G. Chen, K.T. Chao, PUTP-94-10.