On the mass splitting of the charmed mesons

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  • LETT:ERE AL NUOVO CIMENTO VOL. 18, N. 7 12 Febbraio 1977

    On the Mass Splitting of the Charmed Mesons (*).

    J. SA~cm~z GUILLEN

    Department o/ Nuclear Physics, University o/ Zaragoza . Zaragoza, Spain

    (ricevuto il 2 Novembre 1976)

    The explanation of the mass difference between the recently discovered D +, D O doublet ((1876 ~ 15) MeV (1) and (1865 15) MeV (2)) at SPEAR should il luminate our ideas on the various alternative aspects of the standard quark model. As usual, the first western estimate is due to D]~ RUJULA, GEORGI and GLASHOW (3), who obtained re(D+)--m(D o) = 15 MeV applying their nonrclativistic atomic model with a Coulomb term for m(u+)--m(rc ~ and an additional u-d quark mass-difference term for the K and D doublets. W~INB~RG (~) criticized the application of a nourelativistic model to the ~ and used instead information on m(~+)--m(u ~ through the Dashen Theorem (5), which claims that in SUa SUs the electromagnetic splitting should be the same for the squared mass differences. His result for m(D+)--m(D ~ is about 8 MeV. In a later work with K. LAs~ (4) he gets a lower result of 6.5 MeV. FRITZSCn obtains a similar result in an analysis along the same lines (s) when the K-meson mass is used to go from the quadratic splitting (6.6-10 -a GeV ~) to the linear mass difference. C]~LMASTEI~ (~) made also a fit with a gluon perturbed harmonic potential with the result that m(D+)--m(D ~ should be less than 4MeV, suggesting that the discrepancy with the previous results is due to ignoring other effects in the potential. In fact, the results of ref. (a)

    (*) Work suppor ted in par t by the Nat iona l Science Foundat ion under grant No. MPS75-20427. (1) G. GOLDHA~ER, F. M. P IE I~E, G. S. ABRA~IS, ~,~. S. ALA.M, A. 1~r BOYARSKI, ~tl. BREIDENBACH, W. C. C~LRITHERS, W. CHINOWSKY, S. C. COOPER, R. G. DEVOE, J . M. DORFAN, G. J . FELDMAN, C. E. ~'~RIEDBERG, D. FRYBERGER, G. HANSON, J . JAROS, A. D. JOItNSON, J . A. KADYK, R . R . LARSEN, D. LUKE, V. Lt2rg, H . L. LYNCH, R . J . MADARAS, C. C. MOREHOUSE, H. K . NGUYEN, J . M. PATERSOn, M. L. PERL, I . PERUZZI, M. PICCOLO, T. P. PUN, P. RAPIDIS, B. RICItTER, B. SADOULET, R. H. SCHINDLER, R. F. SCBWITTEES, J . SIEGRIST, W. TANENBAUM, G. H . TRILLING, F. VANNUCC], J . S. WIiITAKI,~R and J . E. WlSS: Phys. Rev. LetS., 37, 569 (1976). (') I . PERUZZI, M. PICCOLO, G. J . FELDMAN, H . K . NGUYEN, J . E. WISS, G. S. ABRA.~r ~r S. ALAM, fi-. . BOYA-~SKI, 1VI. BREIDENBACH, W. C. CARITHERS, ~V. CItINO~VSKY, 1={,. G. DEVOE, J . ~[. DORFAN, G. E. FISCttER, C. ~]. FRIDBERG, D. FRYBERGER, G. ~OLDHAEER, G. HANSON, J . A. JAROS, A. D. JOHNSON, J . A. KADYK, ~. R . LARSEN, D. L/JKE, V. LI_TTH, H . L. LYNCH, R. J . MADARAS, C. C. MOREHOUSE, .T.M. PATERSON, M. L. PERL, F. ~ . PIERRE, T. P. PUN, P. ]=~APIDIS, B. RICHTER, R . H . SGIIINDLER, R . F. St)~wJ.TTERS, J . SIEGRIST, ~V. TANENBAUM, G. H. TRILLING, F. VANNUCCI and J . S. VV*HITAKER: Phys. Rev. LetS., 37, 255 (1976). (s) . DE RUJULA, H. GEORGI and S. GbASHOW: Phys. Rev. Lett., 37, 398 (1976). (4) S. W~INBERG: Harvard prepr in t 6]76; S. W. LANE and K . LANE: Phys. Rev. Leit.,37, 717 (1976). (s) R . DASHEN: Phys. Rev., 183, 1245 (1969). (6) I t . FRI2"ZSCH: Phys. LetS., 63 B, 419 (1976). (T) W. CEI~IXSI"ER: Phys. R, ev. LESS., 37, 1042 (1976).



    involve an over-estimate of the effective Coulomb term (1/r) with respect to the Schnit- zer (s) value of 410 MeV from charmonia spectroscopy. In an attempt to clarify the previous results, which in one way or another are based on fits, we present here a direct computation based on a relativistic quark model.

    In our computation the mesons are the solutions of the Bethe-Salpeter equation obtained by BOHM, Joos and KRAMER (9) for a harmonic four-dimensional kernel in the strong binding limit, as a perturbation series in m/2M, where m is the meson mass and M the quark mass. The rules for the matrix elements of those solutions are given by Mandelstam (~0). There are essentially two possible spin structures for the kernel which reproduce the spectrum Y5 Ys(P) and Y5 Y~ ~ Y~-- 1 I(P-~ V- -S ) . The first P choice is appropriate for strong decays, since to the 3rd order the results arc O(M ~ in the triangle approximation (n), but not for current interactions (1~) having as a trivial prediction ]~/]~: = mK/m ~ . On the other hand, the second choice, P ~- V - - S has been shown to be adequate for weak and electromagnetic low-energy phenomenology in a number of applications (~2). It has the unpleasant feature that the quark mass appears explicitly in some of the diagrams, but it can be fixed once and for all from ]K (the simplest diagram) to be 2.45 GeV, a value which is consistent with the approximation O(m~/4M ~) and which, considered merely as an internal parameter, reproduces rather well weak and electromagnetic phenomena. The amplitudes for the pseudosealar mesons in this kernel are

    (1) ) y-P r 2 z(P, r) = ~ 1 + ~- Y5 exp [q~}, where 2 vff i~ 1 GeV ~ is the inverse of the Regge slope, P the meson and r the relative momentum of the quarks. The quadratic mass shift

    1 4~a l"d4q s,~

    2 ( J7 g

    is then given by two kinds of diagrams, according to whether the photon is exchanged in the same or in different quark lines. The graphs of the second type contribute with

    (3) 4za i" d4q I" 4 8a _

    where ~(Q~Qr is the difference of the products of the charges of the i, j quarks in the mesons a and b. For the quark self-energy graphs (~hose of the first type) the Gauss factor is not enough to render finite their contributions


    (a) H. SCItlqITZER: Phys. Rev. D, 13, 7~ (1976). (D M. B6mJ, H. Joos and M. KRAMER: Nucl. Phys., 51 B, 397 (1973). (lo) S. XVJ[ANDELSTAM: Prec. Roy. See., 233 A, 248 (1955). (H) ~. BbH]~I, H. Joos a~d M. KRAM.~IER: Yucl. Phys., 69 B, 349 (1974). (,2) D. FL.~.~I, P. KIELANOWSKI, R. NU~Ez-L.xc~os, A. MORALES and J. S~NCItEZ GUILLt~N: Phys. Rev. D, 13, 2028 (1976) and references,


    where Z is the quark self-energy, which, as shown by ESCOBAI~ (13), can be approximated in this model by

    ~M A 2 ~M (5) X' ~ -~- log ~z

    Now, if the u - -d mass-difference tadpole term is interpreted as the counter term of the first cut-off-dependent contribution, one obtains after trivial renormalization the final result

    (6) 80~ ___ 2 4gM2

    ~(a)- -~qb) = gq,qD~ ~ 2~/~-AQ >~ ~- .

    With the corresponding quark charge differences and the above-mentioned fixed value for M~ 2.45 GeV one obtains



    m~(K + ) - - m~(K ~ = -- 4-10 -3 GeV 2 ,

    m2(D +) - - m2(D ~ = 10.5.10 -3 GeV 2 .

    The result for the K doublet agrees with the experimental value. As for D-mesons, if one divides (Tb) by m x, as was done in ref. (7), we obtain 10 MeV for re(D+)--re(D~ but we see no reason for this procedure: even if the kernel is flavour independent, one can introduce symmetry breaking in a consistent way and account for re(D) (14). With the physical value for re(D), we obtain from eq. (7b) m(D+)--m(D~ MeV in agreement with the relativistic and gluon perturbed analysis of Celmaster (7) supporting his sug- gestion that corrections act substractively on the simple linear-mass behaviour and that one should consider also the gluon terms. The experimental situation is still uncertain, but when the errors become smaller we shall learn how important those terms are and whether to include relativistic corrections is double counting.

    Finally, we note that our heavy model dependent computation does not involve either the m(u)--m(d) mass fit or the Dashen theorem, which has been criticized (15).

    I would like to thank Harvard University, and in particular Prof. COLEMAN, for their hospitality, and the Program for Cultural Cooperation between the USA and Spain for finantial support. Discussions with B. CEI~MASTElZ and Prof. D]~ ROJULA are also acknowledged.

    (xt) C. ESCOBAR: Left. Nuovo Cimento, 10, 741 (1974). (1,) p. BECHEI~ and M. BCi~M: Phys. Left., 60B, 189 (1976). (10 H. PAOEI~: Phys. Rep., 16C, 299 (1975).