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On the Martin potential
Volume
109B. number
4
PHYSICS
LETTERS
25 February
1982
ONTHEMARTINPOTENTIAL K.J. MILLER and M.G. OLSSON Physics Department, Universityof Wisconsin,Madison, WI 53706, USA Received 26 October 1981 Revised manuscript received
18 December
1981
The interquark potential V(r) = A + Bra accounts for all spin-averaged levels and ratios of leptonic widths within each heavy-quark spectroscopy. We find, however, that ratios of leptonic widths between different spectroscopies are inconsistent with flavor-independence. Modification of the Martin potential by a Coulomb short-range part restores complete flavor-independence.
A number of interquark potentials [l] have been generally successful in reproducing the observed energy levels and leptonic widths for the J/ and 7’ spectroscopies. Most of these potentials are QCD-motivated in that they exhibit asymptotic freedom at small radii and linear confinement at large radii. Recently, Martin [2] has advocated an interesting example of a nonQCD-motivated potential seemingly quite proficient at accounting for the observables. From a phenomenological point of view, the existence of such a potential considerably diminishes the significance of quarkonia potential predictions as a test of QCD. We show here that the Martin potential fails in one respect and that restoration is accomplished by modification to include a Coulomb short-range part. Successes of the Martin potential. Quarkonia potentials of a simple power type have been considered [3] because of their simple scaling properties. The phenomenological advantages of the simple potential
V(r) = A t BP
(1)
were emphasized by Martin [2], who showed that by properly choosing the three potential parameters and two quark masses, a considerable amount of data could be understood. We begin by verifying this result in an overall fit. The data [4] used in our analysis is summarized in table 1. The spin-dependence is eliminated by using 314
spin-averaged level values. Since the two lowest s-wave CChyperfine splittings have now been measured, the spin-averaged levels msl and ms2 are directly calculated. The spin-averaged 3PJ level mpl is commonly taken degenerate with the unobserved IP, state due to the assumed short-range dominance in hyperfme splitting. To allow for possible uncertainty in this assumption, we increase the error on mpl to 10 MeV. The ‘T, 7)b hyperiine splitting has variously been estimated to range from SO to 120 MeV. Taking this splitting to be 80 f 40 MeV gives the bb state msl in table 1. For the higher b6 mass differences the hyperfine splitting largely cancels. We now adjust the five Martin parameters by nonlinear regression so that solving the Schrodinger equation yields observables which equal, as closely as possible, their measured values. The resulting observables calculated from the Martin potential are given in table 1. We see that everything is quite nicely accounted for with the possible exception of mp 1, where the predicted value lies 29 MeV below experiment. The absolute $ and ‘T leptonic widths were not used in the fit. Our adjusted values for the parameters are mc=l.75-+0.32GeV, A = -6.3 + 0.6 GeV , a=0.13-+0.02.
mb=5.12?0.29GeV, B = 5.2 f 1 .O GeV , (2)
Although the quark masses are somewhat large, they
Volume
109B, number
4
PHYSICS
LETTERS
25 February
1982
Table 1 The experimental results of ref. [4] are summarized. The levels are spin-averaged with the hyperfine splittings either measured or estimated, as discussed in the text. The right-hand column contains the predicted values using Martin’s potential. The two absolute leptonic widths, in parentheses, were not used in the fit but are predicted using eq. (3) with h = 1. --
I# spectroscopy
State
Experiment
msr
3067.2 3523 i 3662 r 4.6
mPr ‘nsz r($ + e+e-) p(J/’ -t e+e-) I($ r spectroscopy
msl
ms3
-
msl
ms4
-
ms1
r(T
--, e’e-)
T(T’+ r (T r (T” r(T r (V” F (T
e+e-) +
* 1.0 10 1.3 * 0.39
3068 3494 3662 (8.95) 0.40
9439 + 14 560 ? 3 889 f 4 1114* 5 1.08 f 0.06
msr -
Martin
0.45 i- 0.06
* e’e-)
ms2
a)
9439 558 887 1125 (1.07)
0.48 + 0.03
e+e-) +
e+e-)
0.29 f 0.03
0.35
0.22 + 0.03
0.26
--t e’e-) --f e+e-) +
e+e-)
- -
a)
Levels are in MeV, leptonic
widths
in keV.
are not well determined. Forcing them both down by 300 MeV does not markedly change the results. Trouble with the Martin potential. We have seen that Martin’s potential does remarkably well for energy levels and relative leptonic widths. The leptonic widths are computed using the corrected van RoyenWeisskopf formula [ 51
ree = (4rra~Q2/M~)I~,(O)l2X
)
(3)
where h is a correction factor representing QCD and relativistic effects. In the lowest-order static limit we have [6] x = 1 - 16as/3n )
tion is needed in the ‘Y’case, while h 2: 0.5 is required to obtain the observed J/ leptonic width. The above violation of flavor independence observed with the Martin potential follows generally from the scaling properties of power-law potentials [3]. By resealing the Schrodinger equation, we find directly that (5) 4ree(T)jre,($) = (mb/mc)3’(2+Cl)(m$ /mT)2X’r& With little loss in accuracy, we take mb/???, z giving 1+2a”(2+a’4re,(r)lr,e(~) AT/&~= (mT/mQ)(
mT/m$,
. (6)
By table 1 and using CY = 0.13, we obtain (4)
however, higher-order corrections are unknown and possibly large. We expect h to be approximately the same for all states of a given spectroscopy so that ratios of leptonic widths eliminate the unknown factor A. This property was used in our fit to the data. In addition, as indicated in eq. (4), we expect h to be approximately flavor-independent. To check this, we compute the $ and Tleptonic widths using eq. (3) with h = 1. The result from table 1 is that no correc-
h,/h$
= 1.94(0.94)
= 1.82
(7)
which is consistent with the ratio obtained from our fit. Relativistic [7] and running coupling constant corrections are not thought to alter this ratio significantly. The deviation of eq. (7) from unity is thus a violation of flavor independence, which is unacceptable in a potential claiming to be flavor-independent. Fixing up the Martin potential. We next investigate
315
Volume 109B, number 4
PHYSICS LETTERS
which aspect of the Martin potential must be changed to reduce the ratio (7) to unity. In fig. 1 we show two potentials which work equally well in explaining the data of table 1. One one hand, we have the Martin potential, and on the other a modified Cornell [8,9] potential with QCD-motivated limiting behavior at small and large radii. The modified [9] Cornell potential is given by V(r) = (0.22/r + 0.065 r)F(t)
,
(8) F(t) = 1 + 1.83 cos nt + 0.54 cos 2nt + 0.34 cos 3nt , where t = l/( 1 t r) and coordinate units are GeV-I In fig. 1 we have forced the Martin quark masses to be equal to the Cornell values (m, = 1.36 GeV, mb = 4.77 GeV) in order to obtain similar normalization for intermeidate radii. As previously mentioned, this does not appreciably change the results. Between 0.5 and 10 GeV-’ these potentials are essentially identical and in both cases level energies and relative leptonic widths are correctly given. For the modified Cornell potential the ratio (7) turns out to be nearly unity and h, z h, 20.7. To determine how the Martin potential should be changed to restore flavor independence, we consider four potentials generated by fig. 1: Martin, Cornell, Upper and Lower. Upper and Lower are the largest and smallest values respectively of the two curves in fig. 1. In each case the energy levels and relative lep-
-
Martin
-
---
Modified
Cornell
, //
5-
/
‘7 s ‘;
o-
s/ -5/
I -10
/
/’
t
i
I
/
Fig. 1. Two potentials which equally well fit the energy levels and relative leptonic widths for CCand bk states.
316
25 February 1982
tonic widths are correctly predicted, but the ratio (7) only comes out unity for the Lower and Cornell potentials. We conclude that at large radii the Martin potential cannot be distinguished from linear confinement. At small radii flavor independence requires a stronger singularity than eq. (1). A Coulomb singularity not only restores complete flavor independence, but the van Royen-Weisskopf correction is consistent with that expected in QCD. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the Department of Energy under contract DE-AC02-76ER0088 1. [II Some reviews of the development of the subject are: T. Appelquist, R.M. Barnett and K.D. Lane, Ann. Rev. Nucl. Part. Sci. 28 (1978) 387; V.A. Novikov et al., Phys. Rep. 41C (1978) 1; M. Krammer and H. Krasemann, Acta Phys. Austr. Suppl. XXI (1979) 259; K. Gottfried, Comm. Nucl. Part. Phys. 9 (1981) 141. L21 A. Martin, Phys. Lett. 93B (1980) 338; Proc. XXth Intern. Conf. on High energy physics (Madison, WI, July 1980), eds. L. Durand and L.G. Pondrom, AIP Conference PIOceedings, No. 68, (AIP, New York). 131 C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 167. I41 D. Scharre, 1981 Intern. Symp. on Lepton and photon interactions at high energies (Bonn, August 1981); Intern. Conf. on Physics in collision (Blacksburg, VA, May 1981), SLAC-PUB-2761; K. Bcrkelman, PIOC. XXth Intern. Conf. on High energy physics (Madison, WI, July 1980), eds. L. Durand and L.C. Pondrom, AIP Conference Proceedings, No. 68 (AIP. New York). [51 H. Pietschmann and W. Thirring, Phys. Lett. 21 (1966) 713; R. van Royen and V.F. Weisskopf, Nuovo Cimento 50 (1967) 617; 51 (1967) 583. [61 R. Barbieri et al., Nucl. Phys. B105 (1976) 125; W. Celmaster, Phys. Rev. D19 (1979) 1517; E. Poggio and H.J. Schnitzer, Phys. Rev. D20 (1979) 1175. [71 L. Bergstrom, H. Snellman and G. Tengstrand, 2. Phys. C4 (1980) 215; Phys. Lett. 82B (1979) 419; 80B (1979) 242. [81 E. Eichten, Phys. Rev. Lett. 34 (1975) 369; E. Eichten and K. Gottfried, Phys. Lctt. 66B (1977) 286; E. Eichten, K. Gottfried, K.D. Lane and T.M. Yan, Phys. Rev. 21D (1980) 203. 191 K.J. Miller and M.C. Olsson, Quarkonia potential by an inverse method with given endpoint behavior, Wisconsin preprint MAD/PH/19 (October 1981).
109B. number
4
PHYSICS
LETTERS
25 February
1982
ONTHEMARTINPOTENTIAL K.J. MILLER and M.G. OLSSON Physics Department, Universityof Wisconsin,Madison, WI 53706, USA Received 26 October 1981 Revised manuscript received
18 December
1981
The interquark potential V(r) = A + Bra accounts for all spin-averaged levels and ratios of leptonic widths within each heavy-quark spectroscopy. We find, however, that ratios of leptonic widths between different spectroscopies are inconsistent with flavor-independence. Modification of the Martin potential by a Coulomb short-range part restores complete flavor-independence.
A number of interquark potentials [l] have been generally successful in reproducing the observed energy levels and leptonic widths for the J/ and 7’ spectroscopies. Most of these potentials are QCD-motivated in that they exhibit asymptotic freedom at small radii and linear confinement at large radii. Recently, Martin [2] has advocated an interesting example of a nonQCD-motivated potential seemingly quite proficient at accounting for the observables. From a phenomenological point of view, the existence of such a potential considerably diminishes the significance of quarkonia potential predictions as a test of QCD. We show here that the Martin potential fails in one respect and that restoration is accomplished by modification to include a Coulomb short-range part. Successes of the Martin potential. Quarkonia potentials of a simple power type have been considered [3] because of their simple scaling properties. The phenomenological advantages of the simple potential
V(r) = A t BP
(1)
were emphasized by Martin [2], who showed that by properly choosing the three potential parameters and two quark masses, a considerable amount of data could be understood. We begin by verifying this result in an overall fit. The data [4] used in our analysis is summarized in table 1. The spin-dependence is eliminated by using 314
spin-averaged level values. Since the two lowest s-wave CChyperfine splittings have now been measured, the spin-averaged levels msl and ms2 are directly calculated. The spin-averaged 3PJ level mpl is commonly taken degenerate with the unobserved IP, state due to the assumed short-range dominance in hyperfme splitting. To allow for possible uncertainty in this assumption, we increase the error on mpl to 10 MeV. The ‘T, 7)b hyperiine splitting has variously been estimated to range from SO to 120 MeV. Taking this splitting to be 80 f 40 MeV gives the bb state msl in table 1. For the higher b6 mass differences the hyperfine splitting largely cancels. We now adjust the five Martin parameters by nonlinear regression so that solving the Schrodinger equation yields observables which equal, as closely as possible, their measured values. The resulting observables calculated from the Martin potential are given in table 1. We see that everything is quite nicely accounted for with the possible exception of mp 1, where the predicted value lies 29 MeV below experiment. The absolute $ and ‘T leptonic widths were not used in the fit. Our adjusted values for the parameters are mc=l.75-+0.32GeV, A = -6.3 + 0.6 GeV , a=0.13-+0.02.
mb=5.12?0.29GeV, B = 5.2 f 1 .O GeV , (2)
Although the quark masses are somewhat large, they
Volume
109B, number
4
PHYSICS
LETTERS
25 February
1982
Table 1 The experimental results of ref. [4] are summarized. The levels are spin-averaged with the hyperfine splittings either measured or estimated, as discussed in the text. The right-hand column contains the predicted values using Martin’s potential. The two absolute leptonic widths, in parentheses, were not used in the fit but are predicted using eq. (3) with h = 1. --
I# spectroscopy
State
Experiment
msr
3067.2 3523 i 3662 r 4.6
mPr ‘nsz r($ + e+e-) p(J/’ -t e+e-) I($ r spectroscopy
msl
ms3
-
msl
ms4
-
ms1
r(T
--, e’e-)
T(T’+ r (T r (T” r(T r (V” F (T
e+e-) +
* 1.0 10 1.3 * 0.39
3068 3494 3662 (8.95) 0.40
9439 + 14 560 ? 3 889 f 4 1114* 5 1.08 f 0.06
msr -
Martin
0.45 i- 0.06
* e’e-)
ms2
a)
9439 558 887 1125 (1.07)
0.48 + 0.03
e+e-) +
e+e-)
0.29 f 0.03
0.35
0.22 + 0.03
0.26
--t e’e-) --f e+e-) +
e+e-)
- -
a)
Levels are in MeV, leptonic
widths
in keV.
are not well determined. Forcing them both down by 300 MeV does not markedly change the results. Trouble with the Martin potential. We have seen that Martin’s potential does remarkably well for energy levels and relative leptonic widths. The leptonic widths are computed using the corrected van RoyenWeisskopf formula [ 51
ree = (4rra~Q2/M~)I~,(O)l2X
)
(3)
where h is a correction factor representing QCD and relativistic effects. In the lowest-order static limit we have [6] x = 1 - 16as/3n )
tion is needed in the ‘Y’case, while h 2: 0.5 is required to obtain the observed J/ leptonic width. The above violation of flavor independence observed with the Martin potential follows generally from the scaling properties of power-law potentials [3]. By resealing the Schrodinger equation, we find directly that (5) 4ree(T)jre,($) = (mb/mc)3’(2+Cl)(m$ /mT)2X’r& With little loss in accuracy, we take mb/???, z giving 1+2a”(2+a’4re,(r)lr,e(~) AT/&~= (mT/mQ)(
mT/m$,
. (6)
By table 1 and using CY = 0.13, we obtain (4)
however, higher-order corrections are unknown and possibly large. We expect h to be approximately the same for all states of a given spectroscopy so that ratios of leptonic widths eliminate the unknown factor A. This property was used in our fit to the data. In addition, as indicated in eq. (4), we expect h to be approximately flavor-independent. To check this, we compute the $ and Tleptonic widths using eq. (3) with h = 1. The result from table 1 is that no correc-
h,/h$
= 1.94(0.94)
= 1.82
(7)
which is consistent with the ratio obtained from our fit. Relativistic [7] and running coupling constant corrections are not thought to alter this ratio significantly. The deviation of eq. (7) from unity is thus a violation of flavor independence, which is unacceptable in a potential claiming to be flavor-independent. Fixing up the Martin potential. We next investigate
315
Volume 109B, number 4
PHYSICS LETTERS
which aspect of the Martin potential must be changed to reduce the ratio (7) to unity. In fig. 1 we show two potentials which work equally well in explaining the data of table 1. One one hand, we have the Martin potential, and on the other a modified Cornell [8,9] potential with QCD-motivated limiting behavior at small and large radii. The modified [9] Cornell potential is given by V(r) = (0.22/r + 0.065 r)F(t)
,
(8) F(t) = 1 + 1.83 cos nt + 0.54 cos 2nt + 0.34 cos 3nt , where t = l/( 1 t r) and coordinate units are GeV-I In fig. 1 we have forced the Martin quark masses to be equal to the Cornell values (m, = 1.36 GeV, mb = 4.77 GeV) in order to obtain similar normalization for intermeidate radii. As previously mentioned, this does not appreciably change the results. Between 0.5 and 10 GeV-’ these potentials are essentially identical and in both cases level energies and relative leptonic widths are correctly given. For the modified Cornell potential the ratio (7) turns out to be nearly unity and h, z h, 20.7. To determine how the Martin potential should be changed to restore flavor independence, we consider four potentials generated by fig. 1: Martin, Cornell, Upper and Lower. Upper and Lower are the largest and smallest values respectively of the two curves in fig. 1. In each case the energy levels and relative lep-
-
Martin
-
---
Modified
Cornell
, //
5-
/
‘7 s ‘;
o-
s/ -5/
I -10
/
/’
t
i
I
/
Fig. 1. Two potentials which equally well fit the energy levels and relative leptonic widths for CCand bk states.
316
25 February 1982
tonic widths are correctly predicted, but the ratio (7) only comes out unity for the Lower and Cornell potentials. We conclude that at large radii the Martin potential cannot be distinguished from linear confinement. At small radii flavor independence requires a stronger singularity than eq. (1). A Coulomb singularity not only restores complete flavor independence, but the van Royen-Weisskopf correction is consistent with that expected in QCD. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the Department of Energy under contract DE-AC02-76ER0088 1. [II Some reviews of the development of the subject are: T. Appelquist, R.M. Barnett and K.D. Lane, Ann. Rev. Nucl. Part. Sci. 28 (1978) 387; V.A. Novikov et al., Phys. Rep. 41C (1978) 1; M. Krammer and H. Krasemann, Acta Phys. Austr. Suppl. XXI (1979) 259; K. Gottfried, Comm. Nucl. Part. Phys. 9 (1981) 141. L21 A. Martin, Phys. Lett. 93B (1980) 338; Proc. XXth Intern. Conf. on High energy physics (Madison, WI, July 1980), eds. L. Durand and L.G. Pondrom, AIP Conference PIOceedings, No. 68, (AIP, New York). 131 C. Quigg and J.L. Rosner, Phys. Rep. 56 (1979) 167. I41 D. Scharre, 1981 Intern. Symp. on Lepton and photon interactions at high energies (Bonn, August 1981); Intern. Conf. on Physics in collision (Blacksburg, VA, May 1981), SLAC-PUB-2761; K. Bcrkelman, PIOC. XXth Intern. Conf. on High energy physics (Madison, WI, July 1980), eds. L. Durand and L.C. Pondrom, AIP Conference Proceedings, No. 68 (AIP. New York). [51 H. Pietschmann and W. Thirring, Phys. Lett. 21 (1966) 713; R. van Royen and V.F. Weisskopf, Nuovo Cimento 50 (1967) 617; 51 (1967) 583. [61 R. Barbieri et al., Nucl. Phys. B105 (1976) 125; W. Celmaster, Phys. Rev. D19 (1979) 1517; E. Poggio and H.J. Schnitzer, Phys. Rev. D20 (1979) 1175. [71 L. Bergstrom, H. Snellman and G. Tengstrand, 2. Phys. C4 (1980) 215; Phys. Lett. 82B (1979) 419; 80B (1979) 242. [81 E. Eichten, Phys. Rev. Lett. 34 (1975) 369; E. Eichten and K. Gottfried, Phys. Lctt. 66B (1977) 286; E. Eichten, K. Gottfried, K.D. Lane and T.M. Yan, Phys. Rev. 21D (1980) 203. 191 K.J. Miller and M.C. Olsson, Quarkonia potential by an inverse method with given endpoint behavior, Wisconsin preprint MAD/PH/19 (October 1981).