Carnegie Mellon Physics Dept., Pittsburgh PA Abstract. the Skyrme model. Unitarity requires only that the usual normalization conditions

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1 November, 995 CMU-HEP95- DOE/ER/ he-h/95309 Large-N c Relations Among Isgur-ise Functions David E. Brahm and James alden Carnegie Mellon Physics Det., Pittsburgh PA 53 Abstract e investigate the relations that must hold among baryonic Isgur-ise functions in the large- limit from unitarity constraints, and comare to those found by Chow using the Skyrme model. Unitarity requires only that the usual normalization conditions hold at w =, and that the Isgur-ise functions vanish away from threshold. These constraints are consistent with, but less owerful than, the Skyrme model relations. brahm@fermi.hys.cmu.edu walden@fermi.hys.cmu.edu

2 Introduction QCD simlies greatly in the chiral [], heavy-quark [], and large-n c [3] limits (where N c is the number of colors). The Skyrme [4] model of heavy baryons [5, 6, 7] incororates all of these, and makes owerful redictions [8] about the baryonic Isgur-ise functions,, and. Since it has been conjectured that all arameter-indeendent Skyrme model redictions can be derived just from large-n c unitarity constraints, weinvestigate what these constraints alone can tell us about Isgur-ise functions. Our model-indeendent large-n c results turn out to be less owerful than Chow's Skyrme model redictions. e nd that unitarity requires Isgur-ise functions to vanish away from w v v 0 =. This result is consistent with the rediction of Jenkins, Manohar and ise [7] that exf 3= (w )g. Baryon Isgur-ise Functions The weak transition b! c is characterized by a single Isgur-ise function, which reresents the overla of the sin-0 light degrees of freedom (\brown muck): hl(v 0 )jl(v)i = (w) () Q is an isosin singlet (I = J =0). The weak transition () b! () c is characterized bytwo other functions, since in this system the brown muck has sin : hl (v 0 )jl (v)i = (w)g + (w)v 0 v () The Q and Q can be treated together in the heavy quark limit, as a single \suereld [9], since they dier only in the relative sin orientation of the heavy quark and the brown muck. is an isosin trilet (I = J =). The normalization of these functions at w = is: () = ()= (3) Heavy quark symmetry makes no rediction for the value of (). Chow [8] found the following relations among baryon Isgur-ise functions using the Skyrme model: 3 (w) = ( + w) (w) =(w) (4) These relations are consistent with the normalizations in eq. (3), and additionally redict that () = =. 3 Chow writes ( ; ) for ( ; ) and uses \east coast metric g = diag( ; ; ; ).

3 3 -Loo Renormalization of (w) In Fig. we show the -loo renormalization (vertex and wavefunction) of b (v)! c (v 0 ) (i.e. of ), which is calculated by Cho [9, eq. (3.4)]. Since (g =f) N c, the term that it multilies must vanish at least as fast as =N c. The relevant iece is h 3 (r + w) +(w ) i O( N c ); r ln w + w w At threshold, this is consistent with the normalizations of eq. (3), but gives no additional rediction. Away from threshold, it would be inconsistent with Chow's result, eq. (4), unless the Isgur-ise functions vanish for (w ) > =N c. (5) Fig. : -loo renormalization of b (v)! c (v 0 ) One might be temted to use the renormalization of! 0 (i.e. of and ), also calculated by Cho, to derive more relations. However, in the large-n c limit there exists an I = J tower of states above the and. In articular, the state with I = J = contributes to the -loo renormalization of! 0. It introduces 3 new Isgur-ise functions [0, eq. (.6)], only one of which is normalized at w =. Thus no useful new information is obtained. 4 Single Pion Emission In Fig., welookatweak decay accomanied by single ion emission: b (v)! c (v 0 ) l (q). The sum of the two diagrams gives an invariant amlitude where M = g f g V cb h ij = ( 5 ) ih ki (T l ) j k + kj (T l ) i k i (fq + gv ) (6) f v 0 q v q ; g v 0 q v q + w (7) and the T l 's are avor SU() generators. e used Cho's [9] Feynman rules restricted to SU(), so fi; j; kg f;g, and l f;;3g; the grou theory factor is just the Clebsch- Gordan coecient h;;; 0 j0;0i. These rules automatically obey unitarity constraints for! analogous to those derived elsewhere [] for N! N.

4 Λ (v) Λ (v ) (v ) Fig. : Single ion emission b (v)! c (v 0 )(q). Since (g =f) N c, the last factor of eq. (6) must vanish at least as fast as = N c when contracted with any nal state, which is in turn constrained only by v 0 =0. This imlies that for any q satisfying q (m v 0 (fq + gv )=v 0 + O( ) (8) m v). By contracting eq. (8) rst with a 4-vector X satisfying v X = v 0 X = 0, and then with a 4-vector Y satisfying q Y = v 0 Y =0,we nd f O( ); g w O( ) (9) The w-deendence of the second relation arises because as w!, (v Y )vanishes like w, so the constraint ongdisaears. e conclude that all three Isgur-ise functions must vanish away from threshold: f; ; go( whereas near threshold we only require O( ) for (w ) > =N c (0) ) always () This is consistent with the normalization conditions eq. (3), but we nd no additional rediction for (). 5 Double Pion Emission Double ion emission, b (v)! c (v 0 ) l () m (q), arises from the 3 diagrams of Fig. 3, lus 3 \crossed diagrams related by fl; g $fm; qg. As long as we restrict our indices to SU() as before, the grou theory factor of the crossed diagrams equals that of the uncrossed diagrams. Since (g =f) N c, the remaining term must vanish at least as fast as =N c : + ( w ) + w v0 v v v 0 q (v 0 q)(v ) + q (v 0 )(v q) v 0 q v q 3 v q v 0 q q (v )(v q) + q (v 0 )(v 0 q) (v 0 )(v q) (v 0 q)(v ) + q)(v ) (v0 (v 0 )(v q) O( N c ) ()

5 This equation, exanded around w =, is of the form A 0 ( )+A w ( )+A (w ) f[; ; ]+O( N c ) (3) where the vanishing of f[; ; ] imlies the vanishing of f; ; g individually. conclude So we f; ; go( N c ) for (w ) > =N c (4) O( N c ) always (5) Again, we cannot continue with n-ion emission because higher states in the I = J tower come into lay for n>. l() m (q) l() m (q) Λ l() Λ m (q) Fig. 3: Two-ion emission, b (v)! c (v 0 ) l () m (q) (3 crossed diagrams not shown). 6 Conclusions e have analyzed three weak-decay rocesses ( b! c at one loo, b! c, and b! c ) in the chiral/heavy/large-n c limits. These are the only rocesses that do not involve higher states in the I = J tower. In this diagrammatic aroach, unitarity requires certain constraints on the baryonic Isgur-ise functions. At w = we only nd that () = (), which holds just by heavy quark symmetry. Away from threshold (i.e. for (w ) > =N c )we nd the functions must all vanish at least as fast as =N c. These unitarity constraints are consistent with, but not as owerful as, Chow's Skyrme model relations [8]. In articular, unitarity constraints give no rediction for (), whereas the Skyrme model analysis redicts () = =. The vanishing of all the i away from threshold is also consistent with the form exf N 3= c (w )g given in ref. [7]. e have shown that there are arameter-indeendent redictions of the Skyrme model that cannot be derived just from large-n c erturbative unitarity constraints. To make such redictions, the Skyrme model must incororate non-erturbative information about the N c!limit. 4

6 Acknowledgments The authors thank Martin Savage and Mark ise for helful discussions. This work was artially suorted by the U.S. Det. of Energy under Contract DE-FG0-9-ER4068. References [] J. Gasser & H. Leutwyler, Ann. Phys. 58:4 (984). [] N. Isgur & M.B. ise, Nucl. Phys. B348:76 (99); H. Georgi, Nucl. Phys. B348:93 (99). [3] G. 't Hooft, Nucl. Phys. B7:46 (974); E. itten, Nucl. Phys. B60:57 (979). [4] T.H.R. Skyrme, Proc. Roy. Soc. A60:7 (96). [5] E. itten, Nucl. Phys. B3:433 (983). [6] M. Rho, D.O. Riska & N.N. Scoccola, Phys. Lett. B5:597 (990); Z. Phys. A34:343 (99); Y. Oh, D. Min, M. Rho & N.N. Scoccola, Nucl. Phys. A534:493 (99); E. Jenkins, A.V. Manohar & M.B. ise, Nucl. Phys. B396:7 (993); Z. Guralnik, M. Luke & A.V. Manohar, Nucl. Phys. B390:474 (993). [7] E. Jenkins, A.V. Manohar & M.B. ise, Nucl. Phys. B396:38 (993). [8] C.K. Chow, Phys. Rev. D5:4 (995). [9] P. Cho, Nucl. Phys. B396:83 (993); ibid. B4:683 (994)(E). [0] A. Falk, Nucl. Phys. B378:79 (99); H.D. Politzer, Phys. Lett. B50:8 (990). [] J.-L. Gervais & B. Sakita, Phys. Rev. Lett. 5:87 (984), Phys. Rev. D30:795 (984); R. Dashen & A. Manohar, Phys. Lett. B35:45 (993); E. Jenkins, Phys. Lett. B35:43 (993). 5