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<ul><li><p>ANNALS OF PHYSICS 150, 15&171 (1983) </p><p>The Properties of Charmed Baryon (Act) and </p><p>the Proton Revisited </p><p>S. N. BANERJEE AND APARAJITA CHAKRABORTY </p><p>Departmenl of Physics. Jadaopur University. Calcutta-700032, India </p><p>Received September 13, 1982: revised March 8. 1983 </p><p>A statistical model for baryons was suggested leading to an analytical expression for the </p><p>square modulus iv(r)/ of baryon wavefunctions. which is not only free of constituent quark </p><p>masses but also devoid of the interaction parameters of the effective linear potential in which </p><p>a quark moves. Using awn as an input. the lifetimes of the newly established charmed baryon (,I, ) and the proton were studied and compared with other existing theoretical </p><p>estimates. </p><p>1. INTRODUCTION </p><p>Recent experimental findings [ 1, 2 1 suggest that the average lifetime r of the charmed baryon (A,! ) is r(A? ) = (1.1 [I::) x lo- sec. Barger et al. 13 1, in their studies on the mass spectator quark interactions and $A,! ), have obtained a substantial enhancement of its decay rate due to the non-spectator interactions between c and d quarks. Thus, they have computed r(Af ) in the usual quark model and have derived a relation between $4 </p></li><li><p>CHARMEDBARYONANDPROTON 151 </p><p>It is well known that the most striking implications of the sort of grand unification of strong, weak and electromagnetic interactions is that there exist interactions violating baryon and lepton conservation. Superheavy gauge basons mediate such interactions and occur in second order in the grand unified coupling constant or could proceed via Higgs boson. The SU(5) grand unified theory of strong, weak and electromagnetic interactions has an interesting and practical appeal and advantage over all other currently proposed models. There are no new parameters beyond those already measured apart from the masses of Higgs scalar particles and that of the top quark which is yet to be discovered. The most exciting result which is amenable to test is the prediction of proton decay. Because of the availability of sufficiently sensitive detectors to observe this process at a rate in the range suggested by the approximate calculations using an Sli(5) theory, it has become of immense impor- tance to predict this rate as accurately as possible so that one may specify exactly where do the uncertainties exist and set an upper bound on r,, (the proton lifetime) so that we can state at what level the failure to observe this decay rules out the theory. It may perhaps be pertinent to point out that several experiments have been planned and are well under way for observing nucleon decay. Krishnaswamy et al. (9 1 have reported results of experiments in Kolar gold mines and have recorded three events with tracks fully confined to the detector volume and have tentatively suggested a mean life of about 7 x 10 years for nucleons bound in iron nuclei. The most interesting prediction of the idea of a grand unification of colour SU(3) and elec- troweak SU(2) x 1/(l) gauge theories is the instability of a proton through very weak interaction violating both lepton (L) and baryon number (B) and conserving B-L. Several theoretical works have been carried out in proton decay. An excellent review of the proton decay has been presented by Langacker ] 10 ]. Buras et al. Ill ] have investigated proton decays in a model in which two quarks annihilate freely into a lepton and an anti-quark. They have estimated r,, with second-order baryon number violation and found it to be 0 (102-10) times greater than that suggested from dimensional counting. Jarlskog and Yndurain ] 12 ] have also studied the stability of matter through baryon-number violating nucleon decays by considering all possible diagrams that contribute to the process and have obtained T, w r,, v 1Oj years. Machacek ] 13 ] has also investigated the decay modes of the proton (bound neutron) in the SU(5) and SO(10) grand unified theories of weak, electromagnetic and strong interactions and has estimated branching fractions to specific two-body final states using non-relativistic SU(6) arguments. Ellis et al. ] 14 ] have studied the effective Lagrangian for baryon decay and have discussed the flavour structure and short- distance renormalisation of the effective Lagrangian for this phenomenon. They have reevaluated the lifetime formula using Machaceks phase space assumptions and have obtained 1 w(O)i by adjusting it against R and S wave hyperon decays. With the lepto-quark mass m, = 6 x lOi GeV, they [lo] have estimated the proton lifetime in grand unified theories based on SU(5) equal to 8 x 103* years. Din et al. [ 15 ] have performed a bag model calculation of the nucleon lifetime in grand unified theories. Using bag model wavefunctions for the quark in the effective baryon-number violating four-fermion interaction of the SU(5), they have estimated r,, z lo- years if </p></li><li><p>152 BANERJEE AND CHAKRABORTY </p><p>m,y = 4.2 x lOI GeV. Golowich [ 161 has also computed branching ratios for proton and neutron decay for SU(5), SO(10) and SU(2), x SU(2), X U(1) unification schemes. Using quark wavefunctions from the MIT bag model to describe hadron structure, he has estimated r,, equal to 8 x 103 years and has argued that although it falls beneath current experimental limits, it is close enough to them to be detectable in experiments now under consideration, although these results have been found to differ from those of non-relativistic SU(6). Utilising the bag model, he neglected mixing and obtained a total two-body decay rate by summing the partial rates. He asserted that the large recoil momentum would lead to a suppression of the pionic modes by a factor of 3 in amplitude, whereas Kane and Karl [ 17 1 in their investigations have estimated the branching ratios for proton and neutron in the SU(5) with a simple SU(6) quark model and found much smaller effects. Donoghue 1181 has studied proton lifetime and branching ratios in SU(5). using hadronic wavefunctions of the bag model. Summing only the two-body modes, one gets r,, = 8 x 10 years for a heavy boson mass of 3.8 x lOI GeV. To explore the accuracies on the estimates of rP in Sc1(5) grand unified theory, Goldman and Ross [ 19 1 have found that major uncertainties originate solely out of QCD coupling constant, mass of Higgs boson and proton wavefunction at the origin and have suggested an upper bound on rP. Silverman and Soni 1201 have also investigated the decay of proton, i.e., p + e + 1~ in grand unified gauge theories and have provided constraints on the proton wavefunction at the origin and have observed that the grand unified parameters are independent of the hadron final state uncertainties. Gavela et al. (21 ) have also computed two-body branching ratios in proton decay in the SU(5) grand unification scheme and have found that rp v 103 years for m,y = 6 x lOI GeV and r, = 5 X lo* years for m, = 3 x lOr4 GeV. Rolnick [22] has also studied proton and neutron decay to a meson and anti-lepton as a function of the momentum of the anti-quark produced, considering two different Higgs structures. Arisue 1231 has also studied two-body decay amplitudes of nucleon in grand unified theories in a relativistic quark model of hadrons and has presented results for nuclear lifetime and branching ratios in SU(5) and SO(10) unification schemes and has obtained rP = (1.54.1) X 103 years for m, = 6 x lOI GeV in the SU(5) model. Hara [24] has investigated the lifetime and branching ratios of the proton (neutron) in the standard SU(5) by making use of a simple SU(6) symmetric quark model for nucleon structure including the form factor effects due to the finite size of the initial nucleon and final mesons. Most recently, he [25] has also investigated the same problem in SO( 10) grand unified theory. Further, Labastida and Yndurain 1261, Estrada and Gomez 127 1 also investigated proton decay in a nucleus. Tomozawa [28 ] has also investigated the decay of proton in SU(5) grand unified theory with the Bethe-Saltpetre wavefunction of proton as an input. </p><p>In the present work, we have suggested a statistical model for baryons where we have derived an expression for 1 v(r)i analytically. This 1 Al is free of several uncertain parameters like the constituent quark masses and the parameters of the average linear potential in which a quark or anti-quark moves. Consequently, it would be interesting to study and to compute r(/l,t) and r, for proton in this </p></li><li><p>CHARMED BARYONANDPROTON 153 </p><p>perspective, by using j v/(0)( * as an input. In other words. our main contribution becomes a new estimate of 1 v,(O)j which plays an important role in various baryon decays. In Section 2, we present the theory and formulation of our statistical model for baryons. In Sections 3 and 4, we cite our results and discuss them. Section 5 contains the conclusions of our work. </p><p>2. THEORY AND FORMULATION </p><p>Unlike the conventional quark model for baryons. we assume a baryon like /i,t(cud) or a proton (uud) to consist of three valence quarks 9, q2q3 in addition to a sea or a cloud of virtual q, 4,) q2q2 and q3ij3 pairs so that the valence quarks only determine the quantum numbers of a hadron. We further assume that it has a finite radius r0 within which a quark moves so that all the quarks containing colour are confined within it and thereby colour is also confined for a colour-singlet baryon. A quark inside a baryon may now be assumed to move in an average linear effective potential of the type V(r) = ar + b where a and b are the interaction parameters. </p><p>Let AI, represents the Fermi momentum of the q, quark in a small volume 6V (at a distance Y from the origin) and it is related to the number densities of quarks (anti- quarks) in a baryon, n,,(r) [n,,(r)) through the relation </p><p>n,,(r) = 13(P;,>/3~2 I. (1) </p><p>Ignoring the internal motions of the quarks we employ a nonrelativistic picture for the description of the baryons. This would receive some justification for the newly flavoured, like the charmed baryon, /i: as at least one of the constituent quarks, charmed quarks c, is very heavy so as to justify its motion in the relativistic language. </p><p>Assuming that a quark (ql) experiences an effective potential V,,(r) within the sea, we get </p><p>(2) </p><p>where both V,,(r) and q, are positive, r0 corresponds to the radius of the baryon (q,q2q3 system) and m4, is the mass of a q1 type of quark. Using (1) and (2) and defining U,(r) = V,(r,) - VJr), we arrive at </p><p>Similar relation is obtained for the q1 quark. With the effective potential Vg,(r) = ar + b where a and b are the parameters of the potential, we get from (3) </p></li><li><p>154 BANERJEE AND CHAKRABORTY </p><p>n,,(r) = A ,(r, - r)32, r ro, (4d) </p><p>where A, and A, are constants. For a baryon containing three distinct types of quarks q, . q2 and q3 we have from the normalisation condition j34nrZ[nql(r) - n,-,(r)] dr = 1 that </p><p>A=A,-A2=?!?r-Qi:, </p><p>64~ o (5) </p><p>If MB represents the mass of a baryon (q, qzq3 system) such that if E is the corresponding binding energy we can write </p><p>J </p><p>.ro M,-E= o 1 b,,(r) - %,W %I </p><p>+ { nq,(r) - ny,(r) I my, I 4nr dr, (6) </p><p>where q, , q2, q3 represent in general three types of quarks, so that for proton and A,! we come across uud and cud systems and we get </p><p>(MB- E)=j7A( r,, - r)32(my, + m,,, + m,J 4xr dr (7) </p><p>or </p><p>! </p><p>.ro 1 = A(r, - r)12 4nr dr (8) </p><p>-0 </p><p>with A = (3 15/64~) rdy12. Let u/(r), the usual Schrodinger wavefunction, describe the baryon (q,q2q3) where </p><p>the normalisation condition becomes </p><p>1 = I 1: / ty(r)l 4x? dr. (9) </p><p>Hence. comparing (8) and (9) we get </p><p>1 ty(r)l* = 2 (r. - r)3 . r;, r< r. </p><p>= 0, r > r,,. (lob) </p></li><li><p>CHARMEDBARYONANDPROTON 155 </p><p>In our previous work [29-321, the normalisation integrals were carried out approx- imately to estimate order of magnitude of the decay properties of the newly discovered flavoured mesons. In the present work, the normalisation integrals have been carried out exactly to yield </p><p>1 Iy(O)l = 315 G$ </p><p>From (1 l), it appears that j 1+~(0)1 is not only independent of the masses of the constituent quarks in a baryon but it is also independent of the parameters a and b of the average linear potential in which a quark or an anti-quark moves. So far. no theoretical estimate of It,~(r)( has yet been suggested which possesses the aforesaid properties. In other words, our expression for 1 v(O)j in (11) is devoid of many uncertain parameters but for the size parameter r. of a baryon. </p><p>3. LI, DECAY </p><p>Barger et al. [3] in their studies on the spectator quark interactions have obtained a substantial enhancement of the decay rate of A,! due to the non-spectator interactions between c and d quarks. It is customary to assume that inclusive weak decays of a meson with a new flavour proceed through the heavy quark, with the light quarks acting as spectators so that the lifetimes r of D(cd), D(cti) and A ,t (cud) are equal whereas recent experimental findings [ 1, 21 suggest r(A,:) = (1.1 :I,) X lo-l3 set s(D) = (2.3 Ti:E) x lo-l3 set and r(D+) = (9.1 f:g) x </p><p>FIG. la. The non-spectator contribution to .4,: decay where the weak interaction. </p><p>b d </p><p>black circle represents the effective </p><p>FIG. lb. The spectator contribution to A,7 decay. </p></li><li><p>156 BANERJEE AND CHAKRABORTY </p><p>lo- I3 sec. The discrepancy in agreement of the experimental results for s(/i,f ), r(D ) and r(DO) probably hints at the suggestion that interactions involving light quarks in the decay may play a role. As in Barger et al. [3], we shall study presently the non- spectator transition cd + su of Fig. la in the context of our statistical model for baryon in addition to the spectator process displayed in Fig. lb. All charmed particles, in the spectator quark model, have the same lifetime as inclusive weak decays of a hadron with a new flavour are usually assumed to proceed through the heavy quark with the light quarks acting as spectators. </p><p>As in Barger et al. [ 31,. we assume that the non-spectator contribution to the tran sition cd + su for the /i,? decay leads to </p><p>l- non-spec = f. I v/WI*(G:/W - /K,rl12 . IA*l(m,. + m,,) I </p><p>. [,/y - 4m; . my*, (12) </p><p>where A* = (m, -t md)* - rnt - mi andf, are the short distance enhancement factors and H is the Kobayashi-Masakawa mixing matrix. With m,, = 1.65 GeV, m, = 540 MeV, m, = md = 0.33 GeV, we obtain f = 2.09 and f + = 0.7. the usual short- distance enhancement factor. Hence we get </p><p>(13) </p><p>where ro(A,) stands for the radius of the charmed baryon ii, . Barger et al. [ 3 1, on the other hand, have obtained rnon.spec = 21 X 10-l GeV with (w(O)l:,: = 7.4 x lop3 GeV3 as an input obtained from C: -/i,: mass splitting. They have asserted that the major source of uncertainty rests with / w(O)l and rnonmbpec is fairly independent of the light quark masses. In our model, we find / v(O)\ is independent of the constituent quark masses and the parameters of the effective linear potential and I- nOn~bpec is proportional to ri3 of Arv , 1 = rspxv , > + m-spec(~ , 1. (14) The consideration of s(D+) automatically incorporates all gluon corrections to the free-quark decay for both sem...</p></li></ul>