Quantitative corrections are estimated to three of Franklin's sum rules
involving the masses of baryons containing at least one charmed quark and to
three analogous sum rules for baryons containing at least one bottom quark. The
corrections arise from three-body contributions to baryon interaction energies
and are calculated from a semiempirical formula for the colormagnetic
contributions to baryon masses.Comment: 4 pages, plainte

Lichtenberg, D. B.

English

arXiv

Quantitative corrections are estimated to three of Franklin's sum rules involving the masses of baryons containing at least one charmed quark and to three analogous sum rules for baryons containing at least one bottom quark. The corrections arise from three-body contributions to baryon interaction energies and are calculated from a semiempirical formula for the colormagnetic contributions to baryon masses

Lichtenberg, D B

CERN Document Server

arXiv:hep-ph/9506435   27 Jun 95
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2that the rst two quarks have spin 1 and the other so that the rst two quarks have spin 0. It has
been shown [10] that in the physical baryons the mixing is small between the two ideal states of
spin 1/2 dened above, provided the rst two quarks are the two lightest ones. If the symbol [9] for
the two spin-1/2 baryons is otherwise the same, we put a prime on the state in which the lightest
two quarks have spin 1. We let q stand for either a u or d quark.
We neglect isospin breaking terms. Then we can take linear combinations of Franklin's sum
rules [1] to obtain the following relations among baryon masses:


c
(qqc)  
c
(qqc) = 

cc
(ccq)  
0
cc
(ccq); (1)



c
(ssc)   

c
(ssc) = 


cc
(ccs)  

cc
(ccs); (2)


c
(qqc)   
c
(qqc) + 


c
(ssc)   

c
(ssc) = 2[

c
(qsc)   
0
c
(qsc)]; (3)
where the symbol for a baryon is followed by its quark content in parentheses. In subsequent
equations, we omit repeating the quark content of these baryons.
Franklin did not consider sum rules for baryons containing b quarks, as the b quark had not yet
been discovered when he wrote his paper. However, as is clear from his paper, equations analogous
to (1){(3) hold for b-quark baryons simply by the replacement of all c quarks by b quarks in the
formulas.
In Ref. [5], the following expressions were derived relating the masses of baryons with dierent
spin wave functions but having the same quark content:
B

(123)  B
0
(123) = 3R(132) + 3R(231); (4)
2B

(123) + B
0
(123)  3B(123) = 12R(123) (5)
where the R(ijk) are three-body colormagnetic interaction energies. If the rst two quarks have
the same avor, the prime should be omitted on B
0
(123) in Eq. (4), and Eq. (5) is absent. Our
notation diers from that in Ref. [5] in several respects, including the fact that we use the prime on
the baryon whose rst two quarks have spin 1, whereas in [5] the prime is used for the baryon whose
rst two quarks have spin 0. If we have the values of R(132) +R(231) for dierent baryons, we can
use these values to obtain corrections to Eqs. (1){(3). In order to correct the sum rules (1){(3), we
need Eq. (4) but not Eq. (5).
For some baryons, the values of R(132) + R(231) can be obtained from experiment [5]. For
other baryons, the experimental data are not available. In the latter cases, we use a semiempirical
mass formula for the spin splittings in baryons in the form given in Ref. [7]. Application of this
formula leads to the values of R(ijk) given in Table I. For completeness, we have included the values
of R(123) in Table I, even though these values are not necessary for our considerations. The values
of the R(ijk) in Table I are rounded to the nearest MeV. We have also included in Table I the values
of R(ijk) from Ref. [5] where the data are known.
The interaction energies R(ijk) result from the colormagnetic force between quarks i and j. In
deriving his sum rules, Franklin assumed that the R(ijk) depend only on the avors of the quarks i
and j, but not on the avor of the \spectator" quark k. However, it can be seen from the values in
Table I that there is a small dependence on the avor of the spectator quark. It is this dependence
that leads to corrections to Franklin's sum rules. Using the values of R(ijk) from the semiempirical
mass formula rather than the rounded values in Table I, we obtain the following formulas instead
of Eqs. (1){(3):


cc
  
0
cc
  

c
+
c
= 15 MeV; (6)



cc
  

cc
  


c
+

c
= 10 MeV; (7)


c
  
c
+


c
 

c
  2(

c
  
0
c
) = 0; (8)
3where we have rounded to the nearest 5 MeV. These formulas give the corrections to Franklin's
formulas (1){(3). We see from the above equations that three-body interaction energies lead to
small corrections in two of the sum rules and the third survives essentially unchanged.
The corrections to Franklin's sum rules are still smaller in the case of baryons containing b
quarks. Although it may be a long time before mass formulas for baryons containing b quarks can
be tested, we nevertheless give these sum rules below:


bb
  
0
bb
  

b
+ 
b
= 10 MeV; (9)



bb
  

bb
  


b
+

b
= 5 MeV; (10)


b
  
b
+


b
 

b
  2(

b
  
0
b
) = 0; (11)
again rounding to 5 MeV.
Although the semiempirical mass formula reproduces the observed spin-dependent splittings
rather well in baryons containing no heavy quarks, it is not known how accurately it gives the
splittings in baryons containing heavy quarks, as the necessary measurements have not yet been
carried out. However, even if the formula is in error by 10 MeV or more, Eqs. (6){(11) should
have much smaller errors; we estimate these errors to be less than 5 MeV. The reason is that these
equations involve only dierences in spin-dependent splittings, and therefore systematic errors in
the semiempirical mass formula will tend to cancel.
In conclusion, taking into account three-body interaction energies leads to small, but signicant
departures from Franklin's baryon sum rules. In principle, these eects can be tested by future
measurements of baryon masses.
It is a pleasure to thank Jerry Franklin for valuable discussions. This work was supported in
part by the Department of Energy.
References
[1] J. Franklin, Phys. Rev. D 12, 2077 (1975).
[2] I. Cohen and H. J. Lipkin, Phys. Lett. 106B, 119 (1981).
[3] J. M. Richard and P. Taxil, Ann. Phys. (N.Y.) 150, 267 (1983).
[4] D. B. Lichtenberg, Phys. Rev. D 35, 2183 (1987); Phys. Rev. D 40, 3675 (1989).
[5] M. Anselmingo, D. B. Lichtenberg, and E. Predazzi, Z. Phys. C 48, 605 (1990).
[6] Yong Wang and D. B. Lichtenberg, Phys. Rev. D 42, 2404 (1990).
7] R. Roncaglia, A. Dzierba, D. B. Lichtenberg, and E Predazzi, Phys. Rev. D 51, 1248 (1995)
[8] R Roncaglia, D. B. Lichtenberg, and E. Predazzi, Phys. Rev. D 52, Aug (1995)
[9] Particle Data Group: L. Montanet et al., Phys. Rev. D 50, 1171 (1994).
[10] J. Franklin, D. B. Lichtenberg, W. Namgung, and D. Carydas, Phys. Rev. D 24, 2910 (1981).
4Table I. Contributions to baryon masses arising from the colormagnetic interaction. The values
of the colormagnetic interaction energy R(ijk) are obtained from a baryon semiempirical mass
formula given in Ref. [7]. For comparison, where known, we give in parentheses the values of R(ijk)
using experimental data as input with the prescription of Ref. [5].
Quark content R(123) (MeV) R(132) (MeV) R(231) (MeV)
qqq 49 (49) 49 (49) 49 (49)
qqs 51 (51) 34 (32) 34 (32)
ssq 25 35 (36) 35 (36)
sss 26 26 26
qqc 54 12 12
qsc 38 13 10
ssc 28 11 11
qqb 55 4 4
qsb 39 5 4
ssb 29 4 4
ccq 5 14 14
ccs 6 12 12
qcb 15 5 2
scb 13 5 3
bbq 2 6 6
bbs 2 5 5


Quantitative corrections to mass sum rules involving baryons containing heavy quarks

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