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International Centre for Theoretical Physics IC/79M INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NON-LEPTONIC RADIATIVE DECAYS OF HYPERONS IN A GAUGE-INVARIANT THEORY Riazuddin and Fayyazuddln INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1979 MIRAMARE-TRIESTE IC/T9M I. INTRODUCTION l) 2l It has recently been shown ' that the contributions from the quark- International Atomic Energy Agency quark scattering processes s+ u •* u + d through the W~ exchange and and s + d ~* q + q through a gluon exchange (yhere one gluon vertex, s -t d + g, is United Nations Educational Scientific and Culturaa Organization weak, while the other, q. + q + g, is strong) to the effective non-leptonic INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Hamiltonian give a good description of non-leptonic decays of hyperons and ff. Such contributions involve four quark operators and for the purpose of calculating the matrix elements ^B |H*IC1B )> . the non-relativistic quark model together with SU(6) wave functions for baryons are used; the low- lying baryona Br are regarded as an a-wave three-quark system. In this T limit <^BaJH*' '|Br y =i 0. The purpose of this paper is to extend the above considerations to non-leptonic radiative decays of hyperona. The effective HOH-LEPTOBIC RADIATIVE DECAYS OF HYPEROKS IB A GAUGE-INVARIANT THEORY • parity-violating Hamiltonian for such decays is obtained in a gauge-invariant way from the corresponding Hamiltonian for ordinary non-leptonic decays, vhile Riozuddln • the parity-conserving radiative decoys are simply given by baryon poles. As International Centre for Theoretical Physics, Trieste, Italy, we shall see,it is possible to get a satisfactory description of non-leptonic radiative decays of hyperons in the above picture. 3) 10 and There have been some recent discussions on non-leptonic radiative decays of baryons but they are quite different from that being presented Fayyazuddin in this paper. In one of them it has been shown that the experimental upper Physics Departemnt, Quald-1-Azam University, Islamabad, Pakistan. limit on the branching ratio for the decay 5~ •+ l" + Y dictates that the contributions to non-leptonic radiative decays of hyperons from two quark ABSTRACT operators must be quite small. This is consistent with the situation for ordinary non-leptonic decays where the matrix elements ^B |h**nB / show The radiative non-leptonic decay* of hyperons are studied in a non- r octet enhancement when effective HP"0' gets contributions from four quark relativistic quark model involving the quark-quark scattering process s + u •+ u + d operators, while the contributions involving tvo quark operators do not show through the w" exchange and 3 + d + q + q through a gluon exchange so that the any such enhancement. effective Hamiltonian involves four quark operators. The effective parity-violating Hamiltonian for Buch decays is obtained in a gauge-invariant way from the corresponding Hamiltonian for ordinary non-leptonic decays, while the parity- conserving radiative decays are simply given by baryon poles. A satisfactory II. EFFECTIVE HAMILT01IIAKS agreement with experimental data is obtained. The quark-quark scattering process s + u * u + d through the H~ exchange gives, in the leading non-relativistic limit,the following effective Hamiltoniana 1'"^' for H*'c* and H.1!"'": MIRAMAKE - TRIESTE May 1979 cos6 c c • To be submitted for publication, T • i -• > J •* On leave of absence from Physics Department, ftuaid-i-Aiam University, Islamabad,Pakistan. -2- = 1 v WF 1 = — sin6 CQSQ — 1 4? sin9c cos9c -^ (6) where m is the proton mass. This estimate is consistent 1'*2' with what is required to obtain a satisfactory description of non-leptonic decays of where and B. are operators which, respectively, transform a u-llke state hyperons and those of Si ; note that n~ •* 2 ¥ get a contribution only Into a d-like state and an s-like state into a u-like state, and m Is the through the effective Hamiltonian given in (3) obtained through the gluon common quark mass which we shall take as m ~ m 6 is the Cabibbo angle. exchange. It may be noted that the Hamiltonians of Eqs.(2) and U) do not The scattering process s + d -» q + q through the gluon exchange gives, In the contribute to the matrix elements /B IHS'^'JB \ when the baryons B are non-reiativistic limit,1' ^ s w r' r regarded as an 3-wave three-quark system. However, from Eqs.(2) and (k), one can. obtain the effective Hamiltonians for the parity-violating non- leptoni c radiative decays of hyperons in a gauge-invariant way by making the following replacements: - eQi 1 - eftj (T) p.T. WG i > J where A is the electromagnetic field and Q. and Q are charges associated with the i and J quarks. The effective Hamiltonians for non-leptonic (B (i ? H ) -. lV .1 J VP.1 , radiative decays (to lowest order in e) relevant to the s-w&ve three-quark system thus obtained from Eqs.(2} and CO are w r-(Y) where B. Is an operator which transforms an s-like state into a d-like state. (8a) Mote that H^'C' as given in (3) is relevant to s-vave fcound states. Here 4 G (6b) a = r* ; g being the strong coupling constant of the quark-gluon inter- •!M* action. Gy is the effective strength of the weak gluon vector a + d + g i > i and is defined as where 6 is the polarization vector for the photon. These are our main theoretical inputs. Hote also that there 1B no contribution from baryon poles to parity-violating radiative non-leptonie decay amplitudes because <B iHg1*1 |B > vanishes. (5) a r where the Gell-Mann matrices X, refer to colour, and indices c and d also la) A refer to colour. €j is the polarization vector of the gluon. U is a Dirac spinor but is a column matrix in flavour space. In Ref.l, an estimate of was obtained in a simple model and is given by -3- Ill. PARITY-VTOIATIHG ABB PAHITY-CQH3EHVMG AHD RESULTS The matrix elements for the process BCp) Then from Eq.s.(91>) and (ll) the p.r. amplitude I Is given \>j where It • p-p' are given by <B'CPI). T 1/2 VB- where /n*'^g ^^ denote the spin and unitary spin uCp')e [C + Dv_] Tt a. u(p) (9a) g (2ir)9/2 parts of the matrix elements of (8a) and (8b) and are summarized in Table I. It is clear from Eqs.(ll) and Table I that the matrix elements of (8a) satisfy In the non-relativistic limit, these matrix elements take the form the relations + e X [iC^.Ck (9b) D(E"p) - 0, D(="l") - 0 , - - 4S D(A°n) . (13) where kQ = |k| and x's are Fauli splnors. The decay rate r and the asymmetry parameter a are given by characteristic 3) of the A, property for the relevant effective Hamiltonlan. The matrix elements of (6b), on the other hand,imply, among others, (10a) 2D(ArO - D(=0 2 Re(CD») (10b) VI" D(An) - D(I° (HO characteristic of the X_ property for the relevant effective Hamlltonian. We first treat the parity-violating (p. V.I amplitudes D for the la order to have numerical estimates of D for various decays, we use the various decays. For this purpose we take the matrix elements of Eqs.(8) estimates of b, a and d found In Ref.l on the basis of the harmonic betveen relevant baryon states B and B' , where B and B' are treated as oscillator potential as a guide for the confining potential and hyperfine an s-wave three-quark system. In other vords, we use SU(6) wave functions for spillings of (I-A) and (N*-H) masses. These estimates together with that these stateB and denote the spatial part of this wave function by t|i . Then s of 0. in Eq.(6) fit the data of non-leptonic decays of hyperons and 0~ the matrix elements of (8a) and (8b), respectively, involve rather well 1) ' 2) and are given by 3 — sinei cose - f- (Ua) b sa MeV, 1.U8 x MeV c c m [IT and a s: 0.U5 (15) ~ (lib) Based on estimates (15) and that of G^ given in Eq.(6), Eq.(l2) and Table I where provide the numerical estimates for the p.v D amplitudes; these are (lie) given in Table II. B = 3 ag b (lid) -6- -5- We now come to the discussion of p.c. amplitudes C. We take these amplitudes to be given by baryon poles . This approach is consistent with the discussion of ordinary non-leptonic decays of hyperona and S3 where also the valiies which fit the p and s amplitudes of ordinary non-leptonio p.c. amplitude* are given by baryon poles. Thus ve have decays of hyperona. For numerical evaluation of amplitudes C from Eqs.(l6), we use £qs.(.17b) and 0-8) and for the magnetic momentB the values given in Ref.7. i.e. (16a.) up = 2-793, vn « -1.86, 2.67, Wj._ - -1.05, -0.6, VJS_ - -0.1*6, 1 39 O.8l - - ' "A£ " " ~t S ' (19) A-n 1& EAJ a o " 1=5- 5£ - The numerical estimates of amplitudes C are summarized in Table II. A n Vn (16c) M Finally, in Table III we present the comparison with experiment! the C(H°A°) 1 PA =oi , Mrtr ^A [SI " 2TJ -no + I=F ImT a_ (I6d) experimental branching ratios are taken from Ref.8. We see from Table III that a satisfactory tmjtumit fa ootainetf.
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