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The structure of hadrons containing a heavy quark
Volume 93B, number 1,2
PHYSICS LETTERS
2 June 1980
THE STRUCTURE OF HADRONS CONTAINING A HEAVY QUARK E.V. SHURYAK
Institute of Nuclear Physics, Novosibirsk, USSR Received 25 March 1980
The presence of a heavy quark makes the problem of c.m. motion trivial. Unlike earlier works which ignored it we find reasonable predictions of the MIT bag model for the masses, spin and electromagnetic splittings of D, B, ~c and Ac.
The masses o f light (u, d, s) quarks are rather different, but still there is an approximate SU(3) and a more accurate isotopic SU(2) symmetry on their substitution. The reason for this is that these masses are too small to be important. Something similar occurs for the heavy quarks (c, b .... ), because their masses are too large on the usual hadronic scale. So, some symmetry for their substitution should exist between 0 - and 1 - mesons, 2;. and A-type baryons etc. For each family there exists some limit for M - m h when M h, the mass o f the heavy quark, goes to infinity. In this paper we estimate these limits, as well as O(m - 1 ) corrections, and compare the results with available data in the charm and beauty sectors. As a guide we use the MIT bag model [1,2], which was rather successful for ordinary hadrons. Also important is that its main ingredient, the volume energy term, has recently been explained in the framework of QCD as being due to partial suppression o f instanton-type fluctuations inside hadrons [3,4]. Another motivation was the very strong disagreement between straightforward applications o f this model [5] and the data (see fig. 1 and the discussion below). It is of qualitative character, so one may suspect that some important effect is missing. And indeed, in this work a very simple explanation is proposed b y taking into account the so-called center-ofmass motion. The hamiltonian o f the bag model as it was used in ref. [5] looks as follows: H =E~ + M h + E h +Ebag + E 0 + E E + E M, 134
where E~ = 2 . 0 4 n J R is the kinetic energy of the light quarks, R is the bag radius, E h = rr2/2mh R2 is the kinetic energy o f the heavy quark, Ebag = ~ 7rBR3, where B 1/4 = 145 MeV, E 0 = - 1 . 8 6 / R and the last two terms are gluoelectric and gluomagnetic interactions between quarks. Below we critically discuss this expression and show, that most of its terms should be essentially modified. Let us begin with the problem o f the c.m. motion. This is known to appear if, instead of fixing the total m o m e n t u m of some system, we put it into some Fixed potential well. The arising fluctuations in the total m o m e n t u m are completely fictitious * 1 and the corresponding energy should be subtracted. It is not e.asy to do this in the relativistic case, but still we may estimate this energy Ecru by the projection of the bag wave function to that with zero total momentum. Of course, this is not an exact procedure, but we still may conclude that Ecm ~ ( 1 - 1 . 5 ) / R which is nearly the whole E 0 term given above. Originally it was ascribed to the casimir effect *2 , but now we see that most of it is due to this simple kinematical effect. However, the value 1.86 in E 0 was found from a fit to the masses of the usual hadrons, and as soon
*1 Note, that after quantization they also produce fictitious excited states. *2 The zero-point fluctuations modified by the bag. Note that according to refs. [3,4] the bag itself is of a similar nature, but connected with much stronger zero-point fluctuations, the instantons. The theoretical situation with the E 0 + Ecru term is at the moment very unclear.
Volume 93B, number 1,2
PHYSICS LETTERS
as only such hadrons are considered this observation is not very important. It is more important when one proceeds to a case with different kinematics, say that o f a hadron containing a very heavy quark. The problem of the c.m. motion is trivial in this case, for fixing the center o f mass is just fftxing the heavy quark. This was not done in ref. [5], where it was assumed to move freely inside the fixed bag. So, there is no fictitious energy Ecm in this case, and therefore the corresponding part o f E 0 is not needed. This turns out to be very important numerically as we show below. One should also modify the kinetic f n e r g y of the heavy quark term. Using again the projection idea, we say that the momentum of the heavy quark is just the recoil of the motion of the light ones, so E h = (p2)rght/2m h -~ (2.04)2/ 2mhR2 for mesons and twice thls value for baryons. Also in our "hydrogen-like" picture the interaction terms E E and E M are modified. The gluomagnetic term is now due to the gluomagnetic moment of the heavy quark and it is equal to
The second part of this expression is the estimate of the gluomagnetic field at the origin, B a (0), with the bag model wave functions. This interaction is more short range than that for light quarks, so the corresponding a s value should be smaller. From M(D*) - M ( D ) = 140 MeVwe find a s = 0.4, to be compared with the effective a s = 2.2 for light quarks [2]. So, the use of perturbative expressions is more justified for the interaction with a heavy quark. We find, for example M(Ec) - M(Ac) = 165 MeV to be compared with the experimental value 166 = 16 MeV * s The gluoelectric term is now E E =:-- ~ c~s (l/r). However, we know from charmonium theory that this potential is Coulomb-like only at small distances (r < 0.3 fm). Moreover, we do not agree with the justification o f E E in the original paper, ref. [2], and think that this term is already included in the E 0 term. Also this contribution is small compared with the general uncertainties which are not less than 100 MeV. Therefore, we did not include this term at all. Now we come to our results, which are shown in fig. 1. Note, that we have not done any fitting and that all parameters of the bag, the light quark inter-
g(omXa) E M =-
4m h
Bm
(0)
=
0.3~
mhRZ
(om)ta)(Om~ta)
2 June 1980
. *3 However, the existing evidence for 2;c is very weak, see the Review of particle properties (1978) [ 7 ].
~-ttl h
M - /~r/h
-l.2
\ 5
\
[.0
\ \
o Ac
Z
\ N \ \
.3
\
\
\
\ \
\
..4..
O" s
.~
c
f
2\
\
.6
$
8
t0
\
.t
c 1
6
m~ 10.
Fig. 1. M - m h versus rnh, both in GeV, where M are the masses of hadrons and m h are those of heavy quarks. We use the values used in qJ, T theory, m e = 1300 MeV and m b = 4700 MeV. The solid lines show our calculations and the dashed ones those of refs. [2,5]. The latter give a rather nice fit to strange hadrons, but have the wrong behaviour for large quark mass.
135
Volume 93B, number 1,2
PHYSICS LETTERS
actions etc. are left unchanged. E 0 .~ -0.6/R now, when Ecru is removed. Our results are shown by the solid lines, and that of ref. [5] by the dashed lines. The deviations are largest for mesons, for which the strong compensation in ref. [5] between E~ and E 0 makes them unusually "soft", so the vacuum pressure produces a very high compression. Their bag radius is 0.5 fro, while we find the much more usual value of 0.8 fm. These differences even in the most global features of the charmed mesons can be checked by means of electromagnetic effects, which we understand much better than those of QCD nature. Making use of the proposed symmetry of heavy quark substitution, one is able to separate various QED effects in the data on D and B mesons. For example, the magnetic s p i n spin interaction can be found from the following mass combination:
2 June 1980
in the analysis of other elm splittings), it already contradicts the observed M(D +) - M ( D 0 ) ~ 5 + 0.8 MeV. Our last comment concerns E-type baryons with two heavy quarks. The heavy diquark makes a small "nucleus" in the center with the quantum number h, so for the light quark it does not differ much from the meson case. In coriblusion, we propose significant modifications of the MIT bag model as applied to hadrons containing a heavy quark. The main point is simple: this quark stands in the center rather than moves in the fixed bag. The agreement with the data is significantly improved. I came to these observations during my work on lectures for a group of experimentalists, so I am indebted to their organizer, Prof. A.N. Skrinsky.
E m = M(D +) - M(D 0) - M(D +*) + M(D 0 *),
References
in which we get rid of the u - d quark mass difference and their QED self-energies [6]. It can be directly estimated from the gluomagnetic interaction: E m (a/2ds) [M(D*) - M(D)] ~ 1.2 MeV. The available data are not accurate enough to check this estimate. Another important quantity is the Coulomb interaction which can be found from the relation:
[1] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn andV.F. Weisskopf, Phys. Rev. D9 (1974) 3471. [2] T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [3] G. Callan, R. Dashen and D.J. Gross, Phys. Lett. 78B (1978) 307; Phys. Rev. D19 (1979) 1826. [4] E.V. Shuryak, Phys. Lett. 79B (1978) 135; 81B (1979) 65; QCD and the theory of superdense matter, Phys. Rep., to be published. [5] R.L. Jaffe andJ. Kiskis, Phys. Rev. D13 (1976) 1355; J.F. Donoghue and E. Golowich, Phys. Rev. D14 (1976) 1386. [6] A. Chodos and C.B. Thorn, Nucl. Phys. B104 (1976) 21. [7] Particle data group, Review of particle properties, Phys. Lett. 75B (1978) 1.
E c = M(D +) - M(D 0) + M ( B - ) - M(B 0) - ~ E m. Our estimate in the bag model is E c ~ 3 MeV, while the parameters of ref. [5] result in a nearly twice Iarger value. If one uses m d - m u = 3 - 4 MeV (as found
136
PHYSICS LETTERS
2 June 1980
THE STRUCTURE OF HADRONS CONTAINING A HEAVY QUARK E.V. SHURYAK
Institute of Nuclear Physics, Novosibirsk, USSR Received 25 March 1980
The presence of a heavy quark makes the problem of c.m. motion trivial. Unlike earlier works which ignored it we find reasonable predictions of the MIT bag model for the masses, spin and electromagnetic splittings of D, B, ~c and Ac.
The masses o f light (u, d, s) quarks are rather different, but still there is an approximate SU(3) and a more accurate isotopic SU(2) symmetry on their substitution. The reason for this is that these masses are too small to be important. Something similar occurs for the heavy quarks (c, b .... ), because their masses are too large on the usual hadronic scale. So, some symmetry for their substitution should exist between 0 - and 1 - mesons, 2;. and A-type baryons etc. For each family there exists some limit for M - m h when M h, the mass o f the heavy quark, goes to infinity. In this paper we estimate these limits, as well as O(m - 1 ) corrections, and compare the results with available data in the charm and beauty sectors. As a guide we use the MIT bag model [1,2], which was rather successful for ordinary hadrons. Also important is that its main ingredient, the volume energy term, has recently been explained in the framework of QCD as being due to partial suppression o f instanton-type fluctuations inside hadrons [3,4]. Another motivation was the very strong disagreement between straightforward applications o f this model [5] and the data (see fig. 1 and the discussion below). It is of qualitative character, so one may suspect that some important effect is missing. And indeed, in this work a very simple explanation is proposed b y taking into account the so-called center-ofmass motion. The hamiltonian o f the bag model as it was used in ref. [5] looks as follows: H =E~ + M h + E h +Ebag + E 0 + E E + E M, 134
where E~ = 2 . 0 4 n J R is the kinetic energy of the light quarks, R is the bag radius, E h = rr2/2mh R2 is the kinetic energy o f the heavy quark, Ebag = ~ 7rBR3, where B 1/4 = 145 MeV, E 0 = - 1 . 8 6 / R and the last two terms are gluoelectric and gluomagnetic interactions between quarks. Below we critically discuss this expression and show, that most of its terms should be essentially modified. Let us begin with the problem o f the c.m. motion. This is known to appear if, instead of fixing the total m o m e n t u m of some system, we put it into some Fixed potential well. The arising fluctuations in the total m o m e n t u m are completely fictitious * 1 and the corresponding energy should be subtracted. It is not e.asy to do this in the relativistic case, but still we may estimate this energy Ecru by the projection of the bag wave function to that with zero total momentum. Of course, this is not an exact procedure, but we still may conclude that Ecm ~ ( 1 - 1 . 5 ) / R which is nearly the whole E 0 term given above. Originally it was ascribed to the casimir effect *2 , but now we see that most of it is due to this simple kinematical effect. However, the value 1.86 in E 0 was found from a fit to the masses of the usual hadrons, and as soon
*1 Note, that after quantization they also produce fictitious excited states. *2 The zero-point fluctuations modified by the bag. Note that according to refs. [3,4] the bag itself is of a similar nature, but connected with much stronger zero-point fluctuations, the instantons. The theoretical situation with the E 0 + Ecru term is at the moment very unclear.
Volume 93B, number 1,2
PHYSICS LETTERS
as only such hadrons are considered this observation is not very important. It is more important when one proceeds to a case with different kinematics, say that o f a hadron containing a very heavy quark. The problem of the c.m. motion is trivial in this case, for fixing the center o f mass is just fftxing the heavy quark. This was not done in ref. [5], where it was assumed to move freely inside the fixed bag. So, there is no fictitious energy Ecm in this case, and therefore the corresponding part o f E 0 is not needed. This turns out to be very important numerically as we show below. One should also modify the kinetic f n e r g y of the heavy quark term. Using again the projection idea, we say that the momentum of the heavy quark is just the recoil of the motion of the light ones, so E h = (p2)rght/2m h -~ (2.04)2/ 2mhR2 for mesons and twice thls value for baryons. Also in our "hydrogen-like" picture the interaction terms E E and E M are modified. The gluomagnetic term is now due to the gluomagnetic moment of the heavy quark and it is equal to
The second part of this expression is the estimate of the gluomagnetic field at the origin, B a (0), with the bag model wave functions. This interaction is more short range than that for light quarks, so the corresponding a s value should be smaller. From M(D*) - M ( D ) = 140 MeVwe find a s = 0.4, to be compared with the effective a s = 2.2 for light quarks [2]. So, the use of perturbative expressions is more justified for the interaction with a heavy quark. We find, for example M(Ec) - M(Ac) = 165 MeV to be compared with the experimental value 166 = 16 MeV * s The gluoelectric term is now E E =:-- ~ c~s (l/r). However, we know from charmonium theory that this potential is Coulomb-like only at small distances (r < 0.3 fm). Moreover, we do not agree with the justification o f E E in the original paper, ref. [2], and think that this term is already included in the E 0 term. Also this contribution is small compared with the general uncertainties which are not less than 100 MeV. Therefore, we did not include this term at all. Now we come to our results, which are shown in fig. 1. Note, that we have not done any fitting and that all parameters of the bag, the light quark inter-
g(omXa) E M =-
4m h
Bm
(0)
=
0.3~
mhRZ
(om)ta)(Om~ta)
2 June 1980
. *3 However, the existing evidence for 2;c is very weak, see the Review of particle properties (1978) [ 7 ].
~-ttl h
M - /~r/h
-l.2
\ 5
\
[.0
\ \
o Ac
Z
\ N \ \
.3
\
\
\
\ \
\
..4..
O" s
.~
c
f
2\
\
.6
$
8
t0
\
.t
c 1
6
m~ 10.
Fig. 1. M - m h versus rnh, both in GeV, where M are the masses of hadrons and m h are those of heavy quarks. We use the values used in qJ, T theory, m e = 1300 MeV and m b = 4700 MeV. The solid lines show our calculations and the dashed ones those of refs. [2,5]. The latter give a rather nice fit to strange hadrons, but have the wrong behaviour for large quark mass.
135
Volume 93B, number 1,2
PHYSICS LETTERS
actions etc. are left unchanged. E 0 .~ -0.6/R now, when Ecru is removed. Our results are shown by the solid lines, and that of ref. [5] by the dashed lines. The deviations are largest for mesons, for which the strong compensation in ref. [5] between E~ and E 0 makes them unusually "soft", so the vacuum pressure produces a very high compression. Their bag radius is 0.5 fro, while we find the much more usual value of 0.8 fm. These differences even in the most global features of the charmed mesons can be checked by means of electromagnetic effects, which we understand much better than those of QCD nature. Making use of the proposed symmetry of heavy quark substitution, one is able to separate various QED effects in the data on D and B mesons. For example, the magnetic s p i n spin interaction can be found from the following mass combination:
2 June 1980
in the analysis of other elm splittings), it already contradicts the observed M(D +) - M ( D 0 ) ~ 5 + 0.8 MeV. Our last comment concerns E-type baryons with two heavy quarks. The heavy diquark makes a small "nucleus" in the center with the quantum number h, so for the light quark it does not differ much from the meson case. In coriblusion, we propose significant modifications of the MIT bag model as applied to hadrons containing a heavy quark. The main point is simple: this quark stands in the center rather than moves in the fixed bag. The agreement with the data is significantly improved. I came to these observations during my work on lectures for a group of experimentalists, so I am indebted to their organizer, Prof. A.N. Skrinsky.
E m = M(D +) - M(D 0) - M(D +*) + M(D 0 *),
References
in which we get rid of the u - d quark mass difference and their QED self-energies [6]. It can be directly estimated from the gluomagnetic interaction: E m (a/2ds) [M(D*) - M(D)] ~ 1.2 MeV. The available data are not accurate enough to check this estimate. Another important quantity is the Coulomb interaction which can be found from the relation:
[1] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn andV.F. Weisskopf, Phys. Rev. D9 (1974) 3471. [2] T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [3] G. Callan, R. Dashen and D.J. Gross, Phys. Lett. 78B (1978) 307; Phys. Rev. D19 (1979) 1826. [4] E.V. Shuryak, Phys. Lett. 79B (1978) 135; 81B (1979) 65; QCD and the theory of superdense matter, Phys. Rep., to be published. [5] R.L. Jaffe andJ. Kiskis, Phys. Rev. D13 (1976) 1355; J.F. Donoghue and E. Golowich, Phys. Rev. D14 (1976) 1386. [6] A. Chodos and C.B. Thorn, Nucl. Phys. B104 (1976) 21. [7] Particle data group, Review of particle properties, Phys. Lett. 75B (1978) 1.
E c = M(D +) - M(D 0) + M ( B - ) - M(B 0) - ~ E m. Our estimate in the bag model is E c ~ 3 MeV, while the parameters of ref. [5] result in a nearly twice Iarger value. If one uses m d - m u = 3 - 4 MeV (as found
136