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Volume 84B, number 3 PHYSICS LETTERS 2 July 1979

A QCD RELATION BETWEEN ELECTROMAGNETIC MASS DIFFERENCES

OF CHARMED MESONS AND STRANGE BARYONS

Isaac COHEN 1 and Harry J. LIPKIN 2

Weizmann Institute of Science, Rehovot, Israel

Received 7 May 1979

Relations between hadron mass splittings are obtained from the assumption that the ratio of the electromagnetic and. strong contributions to badron spin splittings depends only upon the ratio of the electromagnetic to the strong couplings of the quark-quark or quark-antiquark pairs involved. This ratio is the same in charmed mesons and strange baryons when the values of color and electric charges of the standard colored quark model are used. Predictions for D*÷-D *° and Z*- - :~*+ mass splittings obtained are in good agreement with present experimental data and can be tested much better when better data on D and D* masses become available.

Hadron electromagnetic mass differences have been difficult to calculate in the quark model [ 1,2] because there are several independent contributions all of the same order of magnitude. These include the mass difference between the u and d quarks and changes in the strong hyperfine interactions due to the quark mass difference as well as the purely electromagnetic static coulomb and magnetic hyperfine interactions. Since each of these contributions depends in a different manner on details of hadron wave functions, which are different for mesons and baryons, it is difficult to obtain simple relations between baryon and meson mass splittings.

We wish to point out that the QCD description of the spin-dependent part of the mass splittings as due to colored gluon exchange [3] enables a simple relation to be obtained between masses of charmed mesons and strange baryons:

I On leave of absence from Departamento de Fisica, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela.

2 Supported in part by the Israel Commission for Basic Research and the United States-Israel Binational Science Foundation.

(D*- - ~ , 0 ) - ( D - - ~0)

(D* - D)

= ( z * - _ z . + ) _ ( z - _ z + )

( Z , _ ~ )

_ ( z * - - ~ , o ) _ ( z - - ~ o ) - ( z * - ~ ) ( 1 )

This relation is in good agreement with experiment with present experimental errors and can be tested much better in the near future with better data for the D and D* masses which should soon be available.

The colored gluon exchange derivation of eq. (1) applies only to charmed mesons and to strange baryons and not to other hadrons. In particular there is no simple equality relating masses of strange mesons to either charmed mesons or strange baryons. Relation (1) is not a universality relation for all hadrons, but only for those whose electromagnetic and strong couplings satisfy relations predicted by colored gluon exchange. The success of this relation and the fact that kaon masses do not satisfy any similar simple relation is a sensitive test of the colored gluon exchange description of the strong spin splittings [3].

Relations between strange baryons and charmed mesons are also particularly easy to test experimentally

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because of the narrowness of the D* and -'-* states due to the small phase space available for decays. In general spin splittings are difficult to obtain experimentally with accuracies of 1 MeV necessary to test relations of type (1), because the vector mesons and spin 3/2 baryons decay by strong interactions and have characteristic widths of 100 MeV.

The essential features of the QCD description which lead to this relation (1) are:

(1) The spin splittings are assumed to be due to a hyperfine interaction. The dominant component is a strong (color) hyperfine interaction, but there is also an electromagnetic contribution. The numerators of eq. (1) are the changes in these spin splittings resulting from changing all u quarks to d quarks. There are two contributions: (A) The change in the electromagnetic interaction due to the change in the quark charge. (B) The change in the strong interaction due to the change in the quark mass. Both contributions are assumed to depend upon the hadron wave functions in the same way. Thus the ratios used in eq. (1) depend only on the ratio of the electromagnetic to the color couplings and are independent of the detailed properties of the hadron wave functions.

(2) The denominators of eq. (1) are dominated by the strong hyperfine interaction; the numerators con- tain approximately equal contributions from the strong and electromagnetic interactions. The relative magni- tudes of the strong and electromagnetic contributions vary from one hadron to another, because the ratio of the strong to the electromagnetic coupling constant varies. However, this ratio happens to be the same for charmed mesons and strange baryons, and the unknown ratio of the strong to the electromagnetic coupling cancels out of eqs. (1). This is because the interaction between a u or d quark and a charmed antiquark is exactly double the interaction between a u or d quark and a strange quark for both strong and electromagnetic interactions. The ratio of the color exchange force between a quark-antiquark pair and a quark- quark pair is exactly two [4], and the electric charge of the charmed antiquark(-2/3) is exactly double that of the strange quark.

In most previous treatments of electromagnetic mass splittings [1 ] the effect of the quark mass on the strong hyperfine splitting has been omitted. The pres- ence of this contribution and its dependence on quark masses and hadron wave functions is a distinctive fea-

tur~ of the QCD description of the spin splittings by one-gluon exchange.

The essential physics in relation (1) can be seen by noting that the two strange quarks in the -~ and "--* are in the same state in both baryons. Thus the hyperfine splitting is entirely due to the interaction between the third nonstrange quark and the two strange quarks. The ~ system differs from the -'- system only in revers- ing the roles of the strange and nonstrange quarks. Thus the hyperfine splittings are determined by the same products of strange and nonstrange coupling constants. The hyperfine splitting in the D mesons is due to the interaction between a nonstrange quark and a charmed antiquark with exactly the same color charge and electromagnetic charge as the "strange diquark" in the ~. Thus the ratio of the strong to electromagnetic hyperfine splitting is the same in the ,E, ~ and D systems.

Relation (1) is obtained explicitly as follows. The spin-dependent part of the interaction hamiltonian is assumed to have the form [3]

V s = -.i~>/(ctsK c + otaiQj )oi] s i • s / /mim] , (2)

where Qi, mi and s i are the electric charge, mass and spin of quark (or antiquark) i, oij gives the spatial dependence of the two-body interaction, a and a s are the electromagnetic and strong coupling constants and the expression is summed over all quark-quark or quark- antiquark pairs in the hadron. K c is a color charge factor which is flavor independent and is larger by a factor of two for the color singlet quark-antiquark system than for the color triplet quark-quark system found in baryons.

A simple unified relation can be obtained for the hyperfine splittings in quark-antiquark and three- quark systems with only two flavors, i.e. all mesons and the baryons containing one pair of quarks of the same flavor. Let (qx)~ denote a hadron containing n constituents of flavors q and x and with maximum spin (n/2) and let (qX)n denote the corresponding hadron with spin (n/2) - 1. We consider only the cases n = 2 and n = 3 where (qx)* and (qx) denote spin 1 and 0 mesons and spin 3/2 and 1/2 baryons. The hyperfine splittings M(qx)n - M(qx)n between corresponding meson or baryon states are given by the expectation values of the interaction (2). This can be expressed in a general form valid for both mesons and baryons as

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I ) mq m x (3)

where (Oqx) is the expectation value in the hadron state of the interaction between constituents with different flavors and -4 /3 (n - 1)is the color charge factor K c in baryons and mesons. The interaction between two quarks of the same flavor in the baryon does not contribute to the hyperfine splitting and does not appear in eq. (3). The factor (n/2) is the number of pairs in the spin singlet state in the (qX)n state, i.e. 1 for mesons and (3/2) for baryons. The mass splittings are multi- plied by this factor since the (qx)~ state has all pairs in the spin triplet state for both mesons and baryons. The factors (n/2) and 1/(n - 1) express the difference between meson and baryon systems.

To obtain relation (1) we consider the case when q is either u or d and x is either s or ~. We are interested in the change in expression (3) when q is changed from u to d. By the Feynman-Hel lman theorem this is given to first order by differentiating eq. (3) with respect to the parameters Qq and mq. The expectation value (Oqx) depends upon the hadron wave function and is not known. It is eliminated in obtaining expressions of the form (1 )by taking logarithmic derivatives of the expression (3):

(d/dmq) log[M(qx)* - M ( q x ) n ] = - 1 / m q , (4a)

(d/dQq) log [M(qx)n - M(qX)n ]

[ 40t s _ Qq] - 1

= -[ .3a(n 7-1) Qx (4b)

We immediately note that the only dependence upon the hadron state appearing in eqs. (4) is in the factor (n - 1) Qx in eq. (4b). Relation (1) then follows because this factor has the same value - 2 / 3 for anti- charmed mesons (n = 2, Qe = - 2 / 3 ) and for strange baryons (n = 3, Qx = -1 /3 ) .

The numerical values of the ratios (1) can be calcu- lated from the values of the parameters on the right- hand side of eqs. (4):

6 (u -+ d) log [M(qx)~ - M(qx)n ]

m d - m u 3a(n - 1) Qx

m q 4or s (5)

where 6(u ~ d) denotes the change when all u quarks are changed to d quarks, and eq. (4b) has been simpli- fied by neglecting the term -Qq on the right-hand side as small compared with %/a. Any model which gives values for the parameters on the right-hand side of eq. (5) can then be tested against the experimental ratios (1). However, certain qualitative predictions can be made without knowing the values of these parameters.

The sign of the ratios (1) is seen to be predicted to be negative, since Qx is negative for this case and both terms on the right-hand side of eq. (5) are negative. For the case x = ~ which describes the kaon system, Qx = +1/3. The second term on the right-hand sidg of eq. (5) is positive and half the magnitude of the corresponding term for charmed mesons or strange baryons. This leads immediately to the inequality

1 ( r - - - r - 0 ) _ ( ~ - . - _ r - * 0 )

2 (r-* - ,~)

> (K,O _ K*+) _ (KO - K +)

(K* - K)

> ( r - , - _ r- ,o) _ (r-- _ r.o) (r-, _ r-) (6a)

For the nucleon and A systems we obtain the equality

(A0 _ A +) _ (n -- p) = 0 . (6b)

This follows because the spin splittings come from the quark pairs which are coupled to spin zero in the nucleon. These are the ud pairs which are the same in the neutron and proton. The difference between the neutron and proton is in the uu and dd pairs which are coupled to spin one and do not contribute to spin splittings.

We now discuss the quantitative comparison of the relations (1), (5) and (6) with experiment. This is most conveniently done by transforming them into predictions for the mass differences between the high-spin (1 and 3/2) states, since these have the largest errors. The results are shown in table 1.

The results for the ~ and D masses are in good agree ment with experiment, although the D results are not yet convincing because of the large errors. New data giving better values of the D and D* masses will reduce the errors of both the theoretical and experimental

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Table 1 Comparison of predicted electromagnetic mass differences with experiment.

butiohs are of the same order of magnitude. Relation (1) is thus non-trivial and checks that the ratio of the two couplings satisfies QCD predictions.

Mass difference Theoretical Experimental

~*-- _ ~*+

D*+ _ D*0 A0 _ 4 +

K*0 _ K *+

5.1 ± 0.6 MeV 5.2 ± 0.7 MeV 2.9 ± 0.9 MeV 2.6 ± 1.8 MeV 1.3 MeV from - l . 9 to +7 MeV 4.1±0.6MeV

One of us (I.C.) wishes to acknowledge the support of CDCH Universidad Central de Venezuela, which permits his stay at the Weizmann Institute.

References

values in table 1, since the theoretical error is mainly in the D mass. This should provide a conclusive test of the predictions. The kaon results are interesting because they are roughly midway between the bounds which correspond to purely strong and purely electromagnetic contributions. They thus indicate that the two contri-

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[2] W. Celmaster, Phys. Rev. Lett. 37 (1976) 1042; L. Chan, Phys. Rev. D15 (1977) 2478.

[3] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147.

[4] H.J. Lipkin, Phys. Lett. 45B (1973) 267.

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