{"http:\/\/dx.doi.org\/10.14288\/1.0084976":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Physics and Astronomy, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Murphy, Philip","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2010-11-29T21:26:29Z","type":"literal","lang":"en"},{"value":"1991","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"The radiative annihilation of K\u207bp atoms to \u039b\u03b3 and \u2211\u00b0\u03b3 is investigated using a non-relativistic harmonic oscillator quark model. A nonrelativistic reduction of the first order Feynman diagrams is performed to yield a gauge invariant interaction, which is sandwiched between three quark wave functions. Pseudoscalar and pseudovector coupling schemes are used for the strong vertex and the effects of SU(3)flavour breaking is explored. We obtain results which are in agreement with experiment for the \u2211\u00b0\u03b3 but are somewhat high for the \u039b\u03b3 calculation.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/30167?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"o . NONRELATIVISTIC Q U A R K M O D E L C A L C U L A T I O N OF T H E K~P -+ A 7 AND K-P -> S\u00b07 BRANCHING RATIOS. By Philip Murphy B. Sc. (Physics) University of Otago A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA February 1991 \u00a9 Philip Murphy, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Physics The University of British Columbia 6224 Agricultural Road Vancouver, Canada V6T 1W5 Date: Abstract The radiative annihilation of K~p atoms to A 7 and E \u00b0 7 is investigated using a non-relativistic harmonic oscillator quark model. A nonrelativistic reduction of the first order Feynman diagrams is performed to yield a gauge invariant interaction, which is sandwiched between three quark wave functions. Pseudoscalar and pseudovector cou-pling schemes are used for the strong vertex and the effects of SU(3)flavour breaking is explored. We obtain results which are in agreement with experiment for the E \u00b0 7 but are somewhat high for the A 7 calculation. 11 Table of Contents Abstract i i List of Tables v i List of Figures v i i Acknowledgement v i i i 1 Baryon Wave Functions in the Nonrelativistic Quark Mode l . 1 1.1 Introduction 1 1.2 The Quark Model 2 1.3 Baryon Confinement 3 1.4 Symmetry and the Total Baryon Wave function 4 1.5 517(3) Symmetry 4 1.6 Zeroth Order Eigenstates 5 1.6.1 The case of Unbroken SU(Z)fiavour 6 1.6.2 The S = \u20141 Baryons when 577(3)\/\/ot,our is Broken 13 1.7 Deviations from Harmonic Confinement 17 1.8 Flavour Wave Functions when Sl7(3)\/\/ a\u201e o u r is Unbroken 18 1.8.1 The uds Basis 23 1.9 Spin Wave Functions 25 1.10 Combining Flavour, Spin and Spatial Wave Functions 27 1.10.1 Permutation Group Addition Coefficients 27 iii 1.10.2 The 517(6) Ground State 29 1.10.3 The 517(6) Excited States 29 1.10.4 The uds Basis States 32 1.11 Hyperfine Interactions 32 1.11.1 Hyperfine Mixing in the 517(6) basis 34 1.11.2 Hyperfine Mixing in the uds Basis 37 2 Theory 41 2.1 Method 41 2.2 Symmetry considerations 42 2.3 The Impulse Approximation 44 2.4 Fourier and Jacobi Transformations 44 2.4.1 When 5r\/(3)\/\/owmr is Broken 46 2.4.2 When 5t7(3)\/\/av0ur is Unbroken 49 2.5 The Form of the Interactions 50 2.5.1 Equations satisfied by an interacting field 50 2.5.2 Form of the vertex functions 51 2.5.3 The Feynman Propagators 54 2.5.4 The kaon wave function 55 2.5.5 The photon wave function 56 2.6 Derivation of the explicit form of V 57 2.6.1 Coordinate Representation 57 2.6.2 Momentum Representation 60 2.7 Gauge Invariance 62 2.7.1 Choice of Gauge 62 2.7.2 Proof of Gauge Invariance 63 iv 2.8 Kinematics 67 2.9 Observations 68 2.10 The Nonrelativistic Reduction 70 2.10.1 Validity of the Nonrelativistic Reduction 70 2.10.2 The Nonrelativistic Reduction Prescription 72 2.11 The Problem 76 3 Calculations 79 3.1 Flavour Space 79 3.2 Momentum Space 79 3.2.1 Including the Potentials 83 3.2.2 Plots of the Integrands 86 3.3 Spin Space 95 3.3.1 Evaluation of the 6-j symbols 99 3.3.2 Evaluation of the 3-j symbols 100 3.3.3 Spin summation and squaring the amplitude 100 3.4 Full Amplitude 101 3.5 Determination of the Strong Coupling Constant 101 3.6 Phase space 104 3.7 Results and Discussion 106 A F O R T R A N P R O G R A M S 111 B Details of Integrals 134 C S M P Procedures 137 Bibliography 146 v List of Tables 1.1 Normalized linear harmonic oscillator states V'n r\/ rm l r(Q ;r) {r = p ov r \u2014 A). 10 1.2 Product wave functions 10 1.3 Non-strange baryon space wave functions 14 1.4 Strange baryon space wave functions 16 1.5 Quantum numbers for the it, d and s quarks 18 1.6 Flavour wave functions in the baryon octet 24 1.7 Fully symmetrized flavour wave functions in the baryon decuplet 25 1.8 uds basis states 33 1.9 517(6) excited baryon compositions 37 1.10 uds excited baryon compositions 40 3.11 Flavour matrix elements 80 3.12 Space Matrix Elements 82 3.13 Branching ratio results: parameter set 1 107 3.14 Branching ratio results: main component 107 3.15 Branching ratio results: parameter set 2 107 3.16 Contributions from diagrams to the amplitude 107 3.17 Branching ratio results: with g\\ = g% 108 vi List of Figures 1.1 The effect of the SU(S)flavour operators 21 2.2 Feynman diagrams 78 3.3 Strong vertices 102 vn Acknowledgement Thanks to Harold Fearing. I appreciate his patience and generosity with his time. viii Chapter 1 Baryon Wave Functions in the Nonrelativistic Quark Mode l . 1.1 Introduction When a kaon is captured in the Coulomb field of a proton a K~p atom is formed. Due to strong interaction effects, the K~p system will eventually annihilate. The atom may decay through any one of the following reaction channels [1]: -> E-7T+ (0.47) E\u00b07r\u00b0 (0.27) E+7T- (0.19) \u2014 \u2022 ATT0 (0.07) E \u00b0 7 (0.00144) \u2014 \u2022 A 7 (0.00086) The experimental branching ratio for each channel is given in parenthesis. These reac-tions are interesting in that they provide information on meson-nucleon interactions. In addition the kaon, since it contains a strange quark, has strangeness S = \u2014 1. This enables us to examine effects of the strange quark in strong interactions. Since these are some of the simplest reactions involving strange particles it would appear prudent to explore them further. We will look at the K~p \u2014\u2022 A 7 and K~p \u2014>\u2022 E \u00b0 7 branching ratios in this thesis. 1 Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 2 The small branching ratios of the A 7 and E \u00b0 7 reaction channels make them very difficult to determine experimentally. However, the new experiment by Whitehouse et. al [1] measures the A 7 and E \u00b0 7 branching ratios to better than 1 in 104. In chapter 2 we extract from a gauge invariant set of Feynman diagrams an in-teraction which can be used to act on three-quark wave functions. The form of the three-quark wave functions will be developed in chapter 1. The impulse approximation is used to reduce the interaction to a sum of single quark transition operators. Both pseudoscalar and pseudovector coupling methods are employed at the strong vertices. In chapter 3 we will detail our methods for evaluating the amplitude and compare our results with those of other calculations. 1.2 The Quark Mode l There is overwhelming experimental evidence that baryons and mesons are made up of quarks. Baryons are bound states of three quarks; the mesons are comprised of a quark and an anti-quark. Each quark comes in one of six different flavours, or type : u,d, s,c,t,b. Only the first three of which will be considered in this thesis. The proton is a baryon and consists of the quark combination uud; the neutron has ddu composition. The 7 r + pion has composition ud; the strange meson K~ consists of the combination us. Quarks are spin | particles. Three quarks will combine, by the conventional rules of addition of angular momentum, to a half-integral spin particle. Baryons, therefore, are fermions and obey Fermi statistics. Similarly mesons are integral-spin particles and obey Bose statistics. A quark of a given flavour is in one of three possible colour states. The A + + has Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 3 quaxk composition uuu and therefore is symmetric in flavour. It is also symmetric in spin and in the ground state, symmetric in space. The colour degree of freedom allows one to construct a totally antisymmetric wave function by insisting that the colour part of the wave function is antisymmetric with respect to exchange of quark positions. Such a state will obey Fermi statistics. 1.3 Baryon Confinement Quarks[2],[3] belong to the fundamental triplet representation of the group SU(3)coiour- To form a baryon we combine three of these triplets. This gives a singlet state, two octets and a decuplet [4], 3<8)3<8>3 = 1 0 8 \u00a9 8 e i O . (1.1) The confinement postulate states: All hadrons and all physical states are colour singlets. So the decuplet and octet states in (1.1) are not observed according to the confinement postulate. This rules out the possibility of observing states like diquarks or four-quark states. Since only colour singlet states exist in nature it follows that the confinement forces, between 3 quarks in a baryon, must depend on colour. In the nonrelativistic quark model (NRQM) the quarks are confined in an oscillator potential whose slope is independent of flavour. The assumption [5] that the confine-ment potential is flavour independent (which is supported by studies of Q.C.D. on a lattice) means that the eigenstates of the zeroth order hamiltonian have flavour sym-metry breaking only via explicit appearance of the quark masses in the kinetic energy part of H0. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 4 1.4 Symmetry and the Total Baryon Wave function For a baryon (being a fermion) consisting of identical quarks, the total wave function must be antisymmetric with respect to the interchange of a pair of quarks. The wave function of a hadron has space, spin, flavour, and colour degrees of freedom. In a meson or baryon the colour degree of freedom separates out from the rest. In a baryon, the colour singlet wave function is a (3 x 3) determinant, antisymmetric under the exchange of a pair. This in turn means that the rest of the wave function, containing the space, spin, and flavour coordinates, be symmetric. \\qqq)A \u2014 |colour),i x |space,spin,flavour)s The colour wave function has the same form for all baryons [6] (R=\"red\", G=\"Green\", and B=\"Blue\") |colour)^ = -^=(RGB - RBG + BRG - BGR + GBR - GRB), (1.2) V6 so we will suppress it henceforth. In the S = \u2014 1 baryons1 the strange quark mass differs from the non-strange quark mass. The three quarks are no longer all indistinguishable particles and so it is no longer necessary to construct baryon wavefunctions that are totally antisymmetric in space, spin, flavour, and colour. In this situation we are free to single out the strange quark as quark 3 and only the 1 <-* 2 symmetry of the states remains relevant. This is known as the uds basis and will be discussed further in \u00a71.8.1. 1.5 SU(3) Symmetry The set of the eight traceless, hermitian, Gell-Mann 3x3 matrices generate the unimod-ular, unitary group in three dimensions, denoted Si7(3). For the group 517(3)flavour 1Here S denotes the value of the strangeness quantum number for the state. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 5 these operators act on the fundamental triplet, Here the i t , d, and s quarks are W considered three different quantum states of the same particle. This is analogous to isospin symmetry for the case of S(7(2)\/\/atKmr. However in SU(3)fiavour the symmetry is more approximate due to the significantly larger mass of the strange quark. As a result of this symmetry breaking we treat the strange baryons in two ways: \u2022 mu\/ma \u00ab 1. Here all quarks in the baryon are indistinguishable and it is appro-priate to use the fully symmetrized \"5(7(6) basis\". \u2022 mu\/ma \u00ab 0.6. Here the strange quark is distinguishable by virtue of its larger mass. This gives rise to the \"uds basis\" of Isgur and Karl where the symmetriza-tion is carried out only between the two equal mass, light quarks. The 517(6) and uds bases are two physically distinct descriptions. 1.6 Zeroth Order Eigenstates By taking the instantaneous limit of the Bethe-Salpeter (B.S.) equation one obtains [2] the three-particle Schrodinger equation with Breit-Fermi-type corrections. The Breit-Fermi Hamiltonian can be written (neglecting spin-orbit interactions) H = J2rrii + H0 + U + Hhyp. (1.3) t=i mi is the mass of the ith quark. HQ contains the kinetic energy and a harmonic oscillator potential, which models confinement and 'asymptotic freedom'; U is some unknown potential which is included to incorporate the action of the Coulomb potential at short range and long range deviations from the harmonic oscillator potential; Hhyp is the Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 6 Q.C.D. analog of the hyperfine interaction. The colour hyperfine interaction between two quarks i and j in the same baryon, to order aa is, HhyJ> = \\ \u00a3 Z ^ H T ^ \u2022 ^ ) + \u00b1 [ * S i ' r l S r r 7 i - S \u2022 $\u2022]}. (1.4) Where r,j = |r} \u2014 rj| and r\\- is the position vector of the ith quark in the baryon. 5,- = |cr is the spin vector operator for the ith quark in the baryon. ocs is the effective quark-gluon coupling constant. This piece will be discussed further in \u00a71.11. We now wish to construct eigenstates of the hamiltonian in the case when 517(3) flavour is a good symmetry and when it is broken. 1.6.1 The case of Unbroken SU(3)fiavour In the NRQM the zeroth order basis states are generated by the hamiltonian, Ho = ^ - + ^ - + ^ + lKj2\\r-:-r--\\\\ (1.5) 2mx 2m2 2m3 2 t (1.9) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 8 * \u2014* p~2 = Tn2r~2 substituting these relations into equation (1.5), we find H0 reduces to p2 p2 o 2M ' (1.10) The last term of (1.10) corresponds to the centre of mass motion of the baryon and does not play any role in the intrinsic spectrum of the baryon[7]. The centre of mass motion is a plane wave[8]. The eigenstates of HQ have the form, The elimination of the centre of mass coordinate R is crucial in the correct counting of the states. This is one reason why the nonrelativistic harmonic oscillator approach is so successful in baryon spectroscopy. The hamiltonian has been reduced to that of two independent oscillators each with spring constant 3K. The oscillator energy spacings for the p and A oscillators are identical in the S=0 sector where mu Ri ma,. The zeroth order energy of a state is then specified by the quantum number N *(R,p,\\) = eiP\u2122R$NL(p,\\)e -iEt (1.11) (1.12) EN = 3mu + (TV + 3)nw, (1.13) where N = Np + Nx = (2np + lp) + (2nA + \/A). (1.14) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 9 The principal quantum number of the p, A oscillator, nPi\\, takes on the values 0,1, \u2022 \u2022 \u2022 The orbital angular momentum quantum number of the p, A oscillator, lPt\\, takes on the values 0,1, \u2022 \u2022 \u2022 The wave function of an oscillator, for example an r (where r = p or A) oscillator, is [9], Ki,m l r (ar) = i^ \/ r(ar)y, r m i p(n P), (1-15) where fl\u201er\/r(ar) = Af{arfLn^{oir)e-^T\\ (1.16) a = (mu)i (1.17) and Af = 2 \" 3 \" ' ! (1.18) ^ ^ F ( \u201e , + l r + l ) ( \u201e r + \/ r - ! ) . . . \u00a7 x \u00a7 YirTnir(Qr) are the spherical harmonics. We have used the convention [10] that y,;TO|p(nr) = (-i)m i'i r\/ r-m, P(n P). (1.19) The Lln2(x) axe the associated Laguerre polynomials. Defined in terms of Binomial coefficients these are In+l+l\\x L1:HX)= ^ ( - i r m=0 2m 2 \\ . ml n \u2014 m The states, ipnTirm,r, we require are listed in table 1.1. The total spatial wave function consists of products of these A and p oscillator \u2014\u2022 \u2014\u00bb states. The orbital angular momentum L of the baryon is obtained by coupling lp and h [11], L = i*p + fx, (1.20) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 10 o^oo(ar) = y ^ e - \u00a7 \u00b0 V y 0 0 ( f t r ) Table 1.1: Normalized linear harmonic oscillator states Vvirmi^ ar') (r = \/> or r = A). h L 0 i>oo(p)ipoo(^) 0 0 0 1 V'oiC^ V'ooCA) 1 0 1 V>oo(\u00a3)^ oi(A) 0 1 1 2 V'io(p)V'oo(A) 0 0 0 \u2014* 0oo(p)0io(A) 0 0 0 1 1 0 [V>oiWoi(A)]L=1 1 1 1 [V>oiWoi(A)]i=2 1 1 2 \u2014+ V>o2(p)^ do(A) 2 0 2 ^00(^)^02 (A) 0 2 2 Table 1.2: Product wave functions ^nTiT ( w e have omitted the m). [rp ij}]L indicates coupling lp and l\\ to total orbital angular momentum L. N is obtained via equation (1.14) and the a dependence in the argument of i\/> has been dropped. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 11 To combine the harmonic oscillator wave functions to spatial states of definite per-mutation symmetry ^^LML w e take linear combinations of the products The well known prescriptions to multiply two mixed representations [12] are S = AXA2 + P1P2, A = Aip2 - \/?iA2, M\" = piA2 + \\xp-i, Mx = AXA2 - pip2. The oscillator states must also be coupled to give states of good orbital angular mo-mentum L. The total spatial wave function can be written as the linear combination, $NLM = Z YJ H {h^miP^x,mix\\L,M)^nplpmipipnxlxmix (1.21) The first harmonic oscillator wave function in a product always denotes the p oscillator \u2014\u2022 \u2014* state, and the second the A oscillator state. We will suppress the p or A dependence from now on. m;p; l\\, mix\\L, M) is a Clebsch-Gordan coefficient[13] obtained from the table in ref.[14] and Z is a normalization coefficient. The Condon-Shortley phase convention is used. Up to N = 2 the possible products are listed in table 1.2. For the N = 0 case we can only have (see Table 1.1) = 2(0,0; 0,0|0,0)V>oooV>ooo = (-^)3e-^2('2+A2>. The e-\u00a7* 2<' 2 + A 2) is present in all the product wave functions. It is symmetric since p* + A2 = i((rl - r2f.+ (ri - ftf + {r2 - i=5)A) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 12 is invariant under transposition of quark positions. We write K ' p m i p *nxt\u2122, = K \/ p m 1 ^ n , \/ , m l A ( ^ ) 3 e - \" a 2 ( ' , 2 + A 2 ) (1.22) and the symmetry of the product depends only on the symmetry of ^npipmi^nxixmi^ At JV = 1, L = 1 we can only have $11ML = V'OlMtV'OOO $UMl = ipoootpoiML Since V'oiAfLV'ooo ~ pY\\ML($lp) which transforms as p under permutations. Similarly ^ooo^oiAfi ~ WIML transforms as A under permutations. The states &UML a n d \u00aeIIML are degenerate in the S = 0 sector but cannot be combined to make a state of definite permutation symmetry. At JV = 2 we have the states : L = 0 $ 2 0 0 = U > m ^ ; 1 > m \u00bb A l M ) V W , P V ' 0 1 m , A m ' p ' m | A = -^(V'onV'oi-i + V'oi-iV'on - V'oioV'oio) $ 2 0 0 = 2 E V'npOO V'nxOO = ^(V'IOOV'OOO \u2014 V'oooV'ioo) $ 2 0 0 = 2 2 ^ \" P \u00b0 \u00b0 ^ \u00bb * \u00ab \u00bb n p , n * = \u2014^( '^IOOV'OOO + V'oooV'ioo) L = 1 $211 ' = X ! ( l . r a i p ^ i T O l j M W w ^ O l m , , TO,P'm,A Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 13 = ^ ( V ' o n V ' o i o - V ' o i o V ' o n ) -At L \u2014 2 we can form $ 2 2 2 = ^ ( ^ 0 2 2 ^ 0 0 0 + ^OOuVte) $ 2 2 2 = ^ ( V ' c m V ' o o o - ^ 0 0 0 ^ 0 2 2 ) $ 2 2 2 = V\"011^011-The relative signs of the p-type and A-type wave functions for a given N and L are important. Otherwise, the overall phases are arbitrary. Our definitions differ by a minus sign for the states $A 0 0, $ 2 0 0 $ 2 0 0 a s compared to the phase convention of [15] and [16]. Note that there is an error in equations (A15) of reference [16]. The state [<^ oic6oi]L=1 should have A symmetry and ip2p2 should read [ < \u00a3 o i ^ o i ] L = 2 -The zeroth order eigenstates of HQ, in the S = 0 sector, are listed in table 1.3. 1.6.2 The 5 = \u20141 Baryons when SU(3)jiavour is Broken In this section we consider the case when mu\/ms 0.6. The analogue of equations (1.7) with ms ^ mu are 1 , r 1 . _ O - A D rnu(n + fj) + mar3 9 = ~ = ~ ^ = M ' ^ ^ where M = 2mu + ma. (1-24) Rearranging we get2, 3^ = ^ +4(^-3) V6 V (1.25) 2When SU(3)\/iavour is a good symmetry and m, = mu = mj = mu we get m\\ = m u and (1.25) reduces to (1.9) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 14 $000 = o^oo(\u00a3)V>ooo(A) = 1 $ I I M L = &OO(\u00a3WOIML(A) = a\\Y1ML(\u00a3lx) $ U M L = ^O\\ML{p)^OOQ(X) = 2 ^ apY1ML(tiP) $200 = (^^ ooo(p)V>ioo(A) + V>ioo(\u00a3)V>ooo(A)) $200 = ^(^ioo(\u00a3)<\/>ooo(A) - <\/>ooo(\u00a3)<\/>ioo(A)) = > 2 ( ^ 2 - ^ >2) $200 = (^V>ou(\/d)V>oi-i(A) + V>oi-i(^ )^ ou(A) - o^io(^ )^ oio(A)) $ 2 1 \u00b1 1 = \u00b1^(^01\u00b1 l (^ )^01o ( A ) - ^010(^)V'01\u00b1l(A)) = \u00b1 f x \/2aVA(r 1 \u00b1 1 (n p )r 1 0 ( f iA) - Y10(np)Y1\u00b11(ax)) $210 = ^(V>on(p)V>oi-i(A*) - V>oi-i(p)V>ou(A*)) = f V ^ \u00ab V A ( F I I ( ^ ) F I - I ( ^ A ) - y i - i ( ^ ) F n ( \u00ab A ) ) $22A\/ L = 75(^0 2A\/ t(p)^OOo(A) + $ooo(p)i>02ML(X)) = 2jf5*\\p*Y2ML(tlp) + \\2Y2ML(Slx)) $ 2 2 M L = ^(^02Af L (p )^000(A) -^000(^02W i , ) (A) = 2 ^ a V W f t p ) - A 2F 2 M t(QA)) $ ^ 2 \u00b1 2 = & I \u00b1 I ( P A I \u00b1 I ( A * ) = ?a 2pAyi\u00b1i(flp)yi\u00b1i(ftA) $22\u00b1i = ^(^oio(p)^oi\u00b1i(A) + V>oi\u00b1i(\u00a3)V>oio(A)) = f v\/2^A(r 1 0(ft , , )y 1 \u00b1 1(ft A) + ri\u00b1i(n P)y 1 0(ftA)) $220 = ^ g(^oi-i(p)^oii(A) + ^ on(\/5)^ oi-i(A) + 2V'oio(p)V'oio(A)) = f ^aVA(r i_i(n p )y i i ( f tA) + * i i ( \u00ab , ) * i - i ( f t A ) + 2F 1 0(^)y 1 0(o A)) Table 1.3: Baryon space wave functions $NLML = $ N L M L \/ ( ( ^ ) 3 E ~ * A 2 ( P 2 + A 2 ) ) W H E N 5(7(3)\/\/ a\u201e o u r is unbroken. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 15 \u2022 \u2014* _^ - - (B I P . ^ \/ m A N N = m i r i = m u(ie + - ^ + - ^ ( - ) ) \u2022 \u2014+ =*p2 = m 2 r 2 = m u ( f l + _ + _ ( _ ) ) _ ^ . A . mx P3 = rn3r3 = m s(i\u00a3 + \u2014=( 3)). V6 mu Now the harmonic oscillator hamiltonian, equation (1.5), becomes where PCM = Mi2; and PP = mup and P A = m>A. This hamiltonian generates the same p oscillator states (table 1.1 with r = p) as for the 5 = 0 case. However, due to the higher mass of the strange quark, a \u00bb ax = (ZKmxfl\\ (1.27) in the A oscillator states. The same product states are formed but now the degeneracy between the A and p normal modes has been broken, fiK \\*K u = J , wA = J 1.28 V rnu V m\\ 3 3 andEN = 2mu + ms + (Np +-)huj + (Nx +-)hux (1.29) Here ux < UJ since mu < ms. Because of this frequency splitting, the three states $f0o> $2oo\u00bb $200 which in the degenerate case (5 = 0) have permutation symmetry S,MP, Mx respectively, break into A A , Xp, and pp excitations. For example A A corresponds to a double excitation in the A oscillator (the p oscillator remains in the ground state); that is Nx = 2. Although the complete permutation symmetry of the wave function is lost, we still have permutation symmetry between the u and d quarks. The zeroth order eigenstates of ff0 in the S = \u2014 1 sector for the case of broken SU(3)jiavour are given in table 1.4. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. *000 = V'oooC^ V'oooCA) = 1 $ W = V>OOO(P)^ OIA\/l(A) = 2y^aAAFiML(nA) $ A A v 200 = ^oooWioo(A) = y | a2A ( ^ - A 2 ) v 200 = VWp>000(A) = Vl a 2(2^ - \/=>2) $>PX *200 = ^(^oii(^)^oi-i(A) + V>oi-i(\u00a3)V'oii(A) - o^io(^ )^ oio(A)) = ^ a c A p A C F n ^ ^ - x C ^ ) + yi_i(fl\u201e)rn(ftA) - M ^ o ^ ) ) *21\u00b11 = \u00b1;^ (V>oi\u00b1i(\u00a3)V>oio(A) - ^oio(p)^oi\u00b1i(A)) = \u00b1 f v^ \u00ab a ^ A ( F 1 \u00b1 1 ( Q p)r 1 0 ( f i A ) - r 1 0(ft \u201e ) y 1 \u00b1 1 ( f t A ) ) 3>PX V210 = (^^ on(\u00a3)V'oi-i(A) - $oi-i(p)$oii(A)) = f v^aaApA(y11(ft p)y1_1(nA) - ri-iCft^rnCfiA)) A A A ^ 2 2 ^ = ^000W02M L (A) = ^ \u00ab 2 A A 2 F 2 M i ( ^ A ) S>PP = ^02Mt(p)^ ooo(A) = ^ a V l W ^ p ) *22\u00b12 = ^oi\u00b1i(p)^oi\u00b1i(A) = faaxpXY1\u00b11(ilp)Y1\u00b11(Qx) g>PX *22\u00b11 = ^ (V'oio(\/o)V'oi\u00b1i(A) + ^oi\u00b1i(p)^oio(A)) = f v5aaApA(y1 0(fi p)y 1 \u00b1 1(n A) + y 1 \u00b1 i(fi p )y 1 0(fi A)) &>PX ^220 = (^^ oi-i(\u00a3)V>oii(A) + ^on(\/o)V'oi-i(A) + 2^ oio(\/o)^ oio(A)) = fy\/laa^pXiY^Q^iSlx) + Yll(np)Y1.l(Qx) + 2YW(QP)YW(QX)) Table 1.4: Baryon space wave functions $#\u00a3^ L = $ N L M j ( ( ^ ) 3 \/ 2 e ~ * ( a V + ' the S = \u2014 1 sector when SU(3)fiavour is broken. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 17 1.7 Deviations from Harmonic Confinement The formula for the energy in the 5 = 0 (1.13) sector states that all multiplets with the same N are degenerate and have equal spacing %UJ between multiplets. This is not observed experimentally; these deviations from the harmonic oscillator spectrum are consistent [5] with the action of a short range attractive potential. It can be shown [17] that any potential 17, in first order perturbation theory, will split a harmonic spectrum into the same pattern. This pattern can be described by only three constants. For example EQ = hyperfine unperturbed level of the ground state. = 3mu + 3u> + a = 1135 MeV The ground state refers to the N = 0 zeroth-order eigenstate. Eo is used as a fitting parameter to the baryon spectrum. Where a represents the energy shift due to the deviation from a harmonic potential. By allowing the zeroth order energies of the seven (up to N = 2) supermultiplets3: (notation La: Lis the total orbital angular momentum of the state; a the symmetry of the spatial wave function) Ss, PM, SM,PA,E*S, E*M to be independent parameters, Isgur and Karl [17] found excellent agreement with the energy spacings of the supermul-tiplets predicted by first order perturbation theory. They then take the assumption, due to this confirmation of the first order result, that the harmonic oscillator wave functions remain an adequate approximation to the true zeroth order wave functions even though U is substantial. The U perturbation affects only EQ. We will see however (\u00a71.11) that Hhyp will mix these states. 3 A supermultiplet contains states of various np, n\\, lp, l\\ that combine to N,L and symmetry type (M,A,S) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 18 Jp B Q\/e I Iz Y S u 1\/2+ 1\/3 2\/3 1\/2 1\/2 1\/3 0 d 1\/2+ 1\/3 -1\/3 1\/2 -1\/2 1\/3 0 a 1\/2+ 1\/3 -1\/3 0 0 -2\/3 -1 Table 1.5: Quantum numbers for the u, d and s quarks. 1.8 Flavour Wave Functions when SU(3)fiavour is Unbroken Consider the case of a baryon which consists of three quarks. Each of which may have one of the flavours u,d, or s. The 33 = 27 states decompose into the irreducible representations [4] of 5l7(3)\/\/ouotlT., where the symmetry of the representation is indicated by a subscript. This is the same decomposition into multiplets as for the SU(3)coiour case. Members of these representations must be combined to give the correct charge, strangeness and isospin of the resulting baryon. From table 1.5 it can be seen that the proton must be some permutation of quarks uud. Since the u and d quarks are members of an isospin doublet, they can be coupled together to form states of isospin \/ = | and \/ = | IJ>^>\/l2 = \u00a3 (h,Izx;h, Irf|Ii2, \u00a3 1 2 ) ^ 1 2 ; h, I*\\I,\\h,Izi)\\h, 1,2)\\h, 1,3) 3 eg) 3 (8) 3 = 1 A 0 8 M p 8 8 M a \u00a9 i o s , (1.30) (1.31) For example using table 1.5 we get that \\I,L? = \\\\,W = ( 1 1.11 2 ' 2 ' 2 ' 2 |1,!)(!> Ij 2 ' 2 I 2 ' 2 \u2022)l\u00ab)|t0|e9 + ( |,|;|,-||l,0)(l,0;|,|||,J)|\u00ab)|d)|\u00ab> Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 19 + (l , _ l ; l , | | l , 0 ) ( l , 0 ; | , | | i , | ) | d ) | u ) | u ) , therefore |f, \u00a7)* = \u2014\\=(udu + duu - 2uud) = <\u00a3A. (1.32) V6 The A superscript indicates that this state has M\\ symmetry. It is a member of the M\\ SU(3)fiavour octet. This symmetry arises from choosing the symmetric intermediate isospin state when combining the isospin wave functions. Alternatively, IM) 0 = ( i , | ; | . - | lo.o)(o.\u00b0i | . | l | . l>l u >l d >l\u00ab> + (J , - i ;\u00a7 ,\u00a7 |0 ,0>(0 ,0;l |l|| >\u00a7)|d)|u)|u) = ~y^(uc^ \" du)u = p. (1.33) This state has Mp symmetry and is a member of the Mp SU(3)fiaVour octet. The zeroth order eigenstate for the physical proton contains a mixture of the fiavour states and <\/>A combined to give a totally symmetric space-spin-flavour wave function. The antisymmetric SU(3)flavour singlet state can only be formed by a combination three quarks each with different flavour [4]. It is chosen to be, ^ = -^(sdu \u2014 sud + usd \u2014 dsu + dus \u2014 uds). (1-34) V6 All the states in the M\\ or the Mp octets can be generated by the application of the SU(3)flavour raising and lowering operators [2], U\u00b1, I\u00b1, V\u00b1, on to one member of the SU(3)flavour multiplet. These operators act on states which are SU(2)fiaVour subgroups of 5'l7(3)\/\/a\u201eo u r (see Fig. 1.1). 1+ annihilates a d quark and creates a u quark. U+ annihilates an s quark and creates a d quark. V+ annihilates an s quark and creates a Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 20 u quark. That is l+\\u) = 0, I+\\d) = \\u), I+|a)=0, I-\\u) = \\d), \/ _ | r f )= 0, \/ _ | 3)=0, U+\\u)=0, U+\\d)=0, U+\\s) = \\d), U-\\u) = 0, U-\\d) = \\s), U.\\s) = 0, V+\\u) = \\s), V+\\d) = 0, V+\\s) = 0, V_|u) = 0, V.\\d) = 0, V.\\s) = \\u). When acting on a three particle wave function \\q1q2Q3), such as a Baryon, I\u00b1, U\u00b1, V\u00b1 act on each quark in turn, Jhiqiqa) = J(ki\u00bbl92\u00ab3> + \\qi)J(\\q2))\\q3) + ki^dfe))- (1-35) Since the algebra of SU(2) is the algebra of angular momentum, the standard relation for raising and lowering angular momentum operators applies to these operators, that is, J\u00b1\\l I) = y\/J(J + l)-J,(J*\u00b1l)\\l J* \u00b1 1) (1-36) Here J and Jz represent the total J and its z component for the state. The square root factor in (1.36) is y\/2 when acting on states within an SU(2)fiavour triplet; 1 when acting on states within an SU{2)jiav0UT doublet; and of course zero when stepping out of an SU(3)favour multiplet. So U- acting on the flavour wave function gives the state z+. The other members of the S isospin triplet can then be generated by applying J_ [18]. U~(f>p \u2014 U-(\u2014\\=(udu + duu - 2uud)) v6 = \u2014 \u2014\/=(usu + suu \u2014 2uus) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 21 Y 1\/3 4 Il+ ,ir\\ \/ A v+ v+ u_ v+ u_ u_ ~i 1 1 1 r -\u00bb o 1 (b) Figure 1.1: Illustration of the action of the SU (3) flavour raising and lowering operators in the Iz \u2014 Y diagram (Y = B + S) (a) of the quark triplet and (b) of the baryon octet. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 22 = j=(dsu + usd + sdu + sud \u2014 2dus \u2014 2uds) V6 y\/12 1 (sdu + sud + dsu -f- w.sd \u2014 2(c?M5 + uds)), (sdd + sdd + dsd + dsd \u2014 2(dds + cMs)) (1.38) \\\/l2 =>\u2022 ^ >2_ = \u2014 =^(sdd + dsd \u2014 2dds). V6 (1.39) A is an isospin singlet state at the centre of the M\\ baryon octet. The centre member of the U-spin triplet which contains the neutron (U = 1, Uz = 1) and the S\u00b0 (17 = 1, Uz = \u2014 1) is a linear combination of the isospin eigenstates S\u00b0 and A, \\U = 1,UZ = 0) = \u00ab | ^ 0 ) + \/3|^ > U-\\U = 1,U, = 1) = v5\\U = l,Uz = 0) = v5(a\\'o) + p\\xA)) (1.40) U-t* = U-(^T(dud + udd-2ddu)) (sud + dus + usd + uds \u2014 2sdu \u2014 2dsu) (1.41) V6 I+U-ifc = I+y\/2(a\\fa) + fi\\fi)) = 2a x (1.42) because A is an isospin singlet. I+ commutes with \u00a3\/_. This can be seen from the matrix representation[19] of the raising and lowering operators which act on the SU(3)jiavour triplet d \\ s I l 0 1 0 0 0 0 0 0 0 \\ 1 0 0 0 ^ 0 0 0 0 1 0 (1.43) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 23 I + U - t i = U-I+4>x = U\u2014p = fa (1.44) Comparing (1.42) and (1.44) a = (1.45) Now since |a| 2 + |\/?|2 = 1 we get |\/?|2 = \u00a7 \/* = \u00b1 ^ (1-46) The sign of \/? is not uniquely determined but we fix it to be positive for our flavour wave functions to agree with those of ref.[17]. From (1.40), u - * \u00bb = 7 2 ^ \u00b0 + \/l^ ' (L47) This is the centre member of the U-spin triplet. From (1.38) and (1.41), => \\ = 7:(sud + usd \u2014 sdu \u2014 dsn). (1-48) Similarly for the Mp octet we can generate all the states in the mixed antisym-metric octet. The baryon octet states are listed in table 1.6. The decuplet (fully symmetric) states are simpler than the octet states since there are no mixed states. All the states can be generated by applying the raising and lowering SU(3)flavour operators to one of the members of the decuplet[6], for example uuu. These states are listed in table 1.7. 1.8.1 The uds Basis As mentioned previously, it has been proposed [20],[21] that the uds is a more appro-priate basis than the fully symmetrized 5(7(6) basis when SU(3)flavour is broken. In the uds basis the strange quark is treated as distinguishable and singled out as quark 3. We only symmetrize between the two equal mass (u and d quarks). In this Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 24 x px p \u2014 -^{udu + duu \u2014 2uud) -^{udu \u2014 duu) n -^(dud + udd \u2014 2ddu) -^{udd \u2014 dud) E + \u2014 -^(usu + suu \u2014 2uus) -^(usu \u2014 suu) E\u00b0 \u2014 -j=(sdu + sud + dsu + usd \u20142(dus + uds)) 2(dsu + usd \u2014 sud \u2014 sdu) E - -^(sdd+ dsd- 2dds) -^(dsd - sdd) A ^(sud + usd \u2014 sdu \u2014 dsu) -^=(sdu \u2014 sud + usd \u2014 dsu \u2014 2(dus \u2014 uds)) E~ -^(dss + sds \u2014 2ssd) -^(dss \u2014 sds) E\u00b0 -^(uss + sus \u2014 2ssu) -^(uss \u2014 sus) Table 1.6: Flavour wave functions in the baryon octet. The physical particle X is some mixture of the flavour states x and x to be determined later. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 25 A + + uuu A + -^(duu + udu + uud) A 0 ^(ddu + dud + udd) A\" ddd 1 = 1 E + -^(uus + usu + suu) E\u00b0 -^(dus + uds + dsu + usd + sdu -f- sud) E~ ^(^ds + dsd + sdd) I 2 \u2022^(ssu + sus + uss) \u2022^(ssd + sds + dss) 1 = 0 (7~ sss Table 1.7: Fully symmetrized flavour wave functions in the baryon decuplet. basis the only two flavour states are, \\ = -X=(ud \u2014 du)s, (1-49) v 2 and = \u2014~(ud + du)s. (1.50) v 2 <^A has the u and d quarks coupling to isospin 7 = 0. \\ therefore corresponds to the flavour wave function of a A particle, has 1 = 1 and so corresponds to a E\u00b0. 1.9 Spin Wave Functions Three spin | particles (for example quarks) may give rise to a total spin4 S = | or 5 = | . As in the case of flavour, the spin wave function for a proton can be obtained by 4In this section S denotes the quantum number for the intrinsic spin magnitude. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 26 combining spins of particle 1 and 2 to S = 1 and then coupling to the spin of particle 3 to yield S = \\. This gives a spin wave function with M\\ type symmetry. Alternatively the spins of particles 1 and 2 can be combined to S = 0 and then coupled to the spin of particle 3 to yield S = | . This gives a spin wave function with Mp type symmetry. For the spin up state x\\h = \\\\,W = (i,i;i,-JI\u00a7,J>IT>IT)U> + (\u00a7lJ;J,-||i,0)(il0;|1||i |>|T)I1)IT> + (|,-J;|,J|i,o>(i,0;i,i|i,|)U>|T>IT> therefore X l , i = \" ^ ( U T + ITT \"2 TTD- (1-51) The A superscript indicates that this state has M\\ symmetry. This symmetry arises from choosing the symmetric intermediate spin state when combining the spin wave functions. Similarly, 4.i = (i.|;|.-llo.o>IT>li)IT> + (|,-|;l4lo,o>(o,o ;|,||i,i)U)|T>IT) = ^ ( U - I T ) T - (1-52) This is analogous to the flavour wave function <^ \u00a3. Also, x\u00a7,f = (I b I 111 . i>(i. i ; h i\\l 1)1 T)l T)l T) = I TTT) etc. ( 1 . 5 3 ) States of different ms follow from the Condon-Shortley convention. Note that for the case of coupling three spin | particles together (5C\/(2)sp,n), the restriction to two pos-sible spin directions means that only mixed symmetric and fully symmetric irreducible representations of 5t7(2)sptn can be formed. There is no fully antisymmetric state. That is 2 \u00ae 2 \u00ae 2 = 4 S \u00a9 2 M i \u00a9 2 M p . ( 1 . 5 4 ) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 27 1.10 Combining Flavour, Spin and Spatial Wave Functions. Each quark not only has 3 flavour types but two possible spin directions. We com-bine the SU(2)apin and the SU(3)fiavour multiplet structure into SU(Q)SF (SF ^spin-flavour) multiplets. For baryons, Young diagram techniques[4] give, 6 Cg) 6 \u00ae 6 = 56 s \u00a9 7 0 M a \u00a9 7 0 M p \u00ae 20 A . (1.55) In the SU(6) basis we wish to construct states that are totally symmetric in space, spin and flavour so that when combined with the colour part we get a fully antisymmetric wave function. In the uds basis we construct states that are totally symmetric in space, spin and flavour only with respect to quarks one and two. 1.10.1 Permutation Group Addi t ion Coefficients To combine wave functions of different permutation symmetry it is convenient to in-troduce Permutation group addition coefficients [22]. These are completely analogous to the Clebsch-Gordan coefficients of the rotation group. For example to combine two wave functions of permutation symmetry Pa and P 2 respectively, to permutation symmetry P then, \\ M ,\u00ab2 \\ Ki K.2 K J 1 if P{ = A,S,MX 2 if Pi = Mp. The non-zero coefficients with phase factors chosen such that they are all real are, where Ki = < Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 28 = 1, M M M I In addition the spin must also be coupled to the orbital angular momentum to yield total angular momentum J for the particle. Using the notation[15] where X is the particle n,p, A etc; jx the SU(3)flavour multiplicity (that is whether the flavour wave functions are in the octet, decuplet, or singlet irreducible representations of 5(7(3)favour (see equation (1.30)); L,S,J is the orbital (S,P,D,F,... etc), spin, and total, angular momentum respectively; P is the parity of the spatial wave function; a is the symmetry of the spatial wave function or the symmetry of the SU(6)SF multiplet. P i P 2 \\XF\u00bbL0JP) = \u00a3 \u00a3 Kl,K2,*3 ML,Ms y Ki K2 K\\2 J { a a S Pl,*l J,P2,K2 x (S,Ms]L,ML\\J,Mj)x?^s9'K2m3M-(1.57) For our purposes we will need the Jp = | + states up to N = 2 (that is energy 2hu above the ground state). N = 1 provide only L = 1 states and so have negative parity. For L = 0 all the Clebsch-Gordan coefficients give 1. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 29 1.10.2 The 517(6) Ground State At the ground state (N = 0) where we have only $QOO (X = N, A, S\u00b0), 5 5 5 1 1 1 \\ *x 2'2 $oooXi ix -4$o5oo(xii^ + x i i ^ ) -V2 2 ' 2 2 ' 2 1.10.3 The 517(6) Excited States Radial Excitations At N = 2 we have, KI ,*2 Pi P2 5 \\ \/ 5 5 5 \u00ab2 1 \/ V 1 1 1 \/ <|.|;0,o||,|) 2 '2 V2 2 ' 2 2 2 (1.58) (1.59) The prime on the 5 indicates we are referring to the excited (TV = 2) symmetric spatial state, $foo5 rather than the ground state symmetric spatial wave function, $ooo-\/ D D A\/f \\ \/ l-^swl ) - \u00a3 \u00ab1 .\u00ab2,\u00ab3 M M S \\ K12 K3 1 Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 30 v \/I l . n fill l \\ 6 M ' K 3 v f t . ' \u00ab i i f t , K 2 X \\2 ' 2 > U ' U l 2 ' 2\/*200 X l 1 2 2 ' 2 2 ' 2 2 ' 2 For the SU(Z)flavour decuplet (of which out of X only S \u00b0 is a member) (1.60) PI S PI 12 \/ \u00abi .\u00ab3 \\ Ki 1 K i 2 P 1 2 M S \\ x xX\\ 1 $200 1 1 1 \/ 2'2 + <\u00a3xXl 1$200 2'2 = - 7 i * 5 ( x L * i \u00bb + x i i *Soo)-V2 2 2 2 ' 2 (1.61) For the SU(3)fiav0Ur singlet irreducible representation, \\X?SM\\+) -\u00ab1 .\u00ab3 Pi A P 1 2 V \u00ab 1 1 \u00ab 1 2 J l P X 2 M 5 ^ \u00ab12 K 3 1 x (},i;o,oii>i>*5jir\u00bbx?r^ ' 2 ' 2 M A M \\ \/ M M S 1 1 1 1 1 4>XXl 1$200 2'2 \/ M A M \\ \/ M M S + **X'l 1$200 I 2 1 2 \/ V 2 2 1 2'2 Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 31 ^xixiA-xlA)-y\/2 2'2 2 2 (1.62) Orbital Excitations We combine L = 2, 5 = \u00a7 states to give Jp = | + . In this case the Clebsch-Gordan coefficients are not all unity. ' M M 5 \u00ab 1 2 \u00ab 3 1 A 1 + ^ ^ Pi P* M x (l,M 5;2 ,M L | i , i)$Xx| 1 ,^?\" t 2 + -4xf i (*2Wx + *\u00a72otfr) V5 2 ' 2 - ^ x f . - j C f c ^ + fcWi) + y|xf_|(*2A22^ + $ 2 2 2 ^ ) } . For the 5L7(3) \/ \/ 0 vour decuplet, l ^ 0 4 ^ f > = E Z ML,Ms s s s 1 1 1 5 5 5 1 1 1 (1.63) x ( | , M s ; 2 , M L | i , i ) $ f 2 M t x f M s <^ There exists no fully antisymmetric L = 2 spatial wave function to go with the SU(3)flavour singlet flavour state and the symmetric spin state. Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 32 We neglect the tensor part of the hyperfine interaction (\u00a71.11), and so these or-bital excitations will not contribute to the wave functions we use. We list them for completeness. 1.10.4 The uds Basis States As mentioned previously when we take into account the different mass of the strange quark we require only symmetry with respect to the two light quarks which are in position 1 and 2. This corresponds to the total wave function possessing M\\ symmetry. For example, the ground state lambda has spatial wave function $o00 and flavour wave function (j>\\ (in the uds basis). It therefore must combine with a xp spin wave function to give a total wave function with M\\ symmetry. The strange states in the uds basis are given in Table 1.8. 1.11 Hyperfine Interactions With the zeroth order eigenstates established we can now turn to calculating the effects of the hyperfine interaction. Analogous to Q.E.D., in Q.C.D. one expects a hyperfine interaction of two quarks i and j in the same baryon, which to order aa is, i = $ A O V A X ! i 2 ' 2 |A4A,A> = Em(2, m; | , (I - m)||, | > < O A X| i (i_ m ) | E 2 S ) = ^ooo s^Xi i t n e s ground state 2 ' 2 | S 2 S , A ) = *ft>*EXl 1 2 ' 2 | \u00a3 2 S \u201e P > = ^ Sofoxl i 2 ' 2 |S 25AA) = i 2 ' 2 \\X4DPP) = \u00a3 m ( 2 , m; | , (i - m)|\u00a7, | ) $ \u00ab E X | , ( i _ m ) | \u00a34DA A ) = Em<2, m; | , (\u00a7 - m)|\u00a7, | ) $ ^ s x f i ( i _ m ) Table 18: The uds basis states we will need, combined so that they have Mx symmetry. Notation \\X2S+1 L\u201eia3). Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 34 expectations of one-gluon-exchange [17]. Isgur and Karl speculate that Thomas pre-cession may cancel the spin-orbit effect. The Fermi contact term (the first term in equation (1.65) is operative only when the pair (i & j) have zero orbital angular momentum. The second term is the tensor term. There is an effective 'colour-magnetic' field due to the spin of the quark. It is directly analogous to the tensor term present in the nucleon-nucleon potential. The tensor force couples S and D states of the same Jp. The tensor operator gives zero when acting upon singlet states. We do not include effects of this tensor operator as its effect is negligible in the ground state baryons [23]. As a result there will be no D state admixtures in our wave functions. In the 5 = 0or5 = \u20141 sectors (1.65) can be written *KW = \u00a3 ( ! \" (1 ~ x)(Su + Sja))Hi Jyp (1.66) where Hgp ^ f %{y S \u2022 5 ^ ) + ^ ' ^ ' r \u00ab - S< \u2022 Sj}} (1.67) and x = mu\/ms. (1.68) Hhyp will not connect L = 1 and L = 6 states so matrix elements between the ground state (TV = 0) and N = 1 states will give zero. 1.11.1 Hyperfine Mixing in the 5(7(6) basis In the 5(7(6) basis, Hhyv will lead to mixing within a given 5(7(6) multiplet and also between different 5(7(6) multiplets with the same isospin and Jp[21]. In the 5(7(6) basis x = 1, so the hyperfine matrix elements between the states m and n, take the form: ff^(6) = ( ^ ( 6 ) ( 1 , 2 , 3 ) | \u00ab , + + i f\u00a3 p ) | f (6>(1, 2,3)) = 3,2)|\u00a3&|*f< f l>(l, 3,2)) = ( - l ) a ( ^ ( e ) ( l , 2 ,3 ) | J g p |^ 6 ) ( l , 2 ,3 ) ) Where we have relabeled the quark positions and then used the complete permutational anti-symmetry of the SU(6) baryon wave functions. That is the matrix elements H}^p = H\\lp. Similarly we get H2h% = Jf\u00a32p. Therefore <^)(1,2,3)| \u00a3 fl^f^l, 2,3)) = 3(^<6>(1,2,3)| J^ 0 5 U ( 6 ) (1 ,2 ,3)) . (1.69) 517(6) Basis Compositions. Hhyp has non-diagonal matrix elements between different supermultiplets leading to wave function distortions and second order mass shifts. Using ^ = _ L ( r W 2 ) and the relation in (1.67) with i = 1, j = 2, we get, H\u00bb = (i-7i) 4rv ry 3 with* = (1.72) We also need the relations: (xpJSi \u2022 S2\\Xpm.) = -lsm,m., (1.73) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 36 (xi\\Si \u2022 S2\\xl) = -6m, (*L\\6\\p)\\*loo) = V>ooo(0)<\/>ooo(0) = a Try\/ft' The relevant hyperfine matrix elements are (XgSs\\Hhyp\\X^Ss) = --6, {X^Ss'\\Hhyp\\XgSs} = ~ y ~ ^ ' (X^SM\\Hhyp\\X^Ss) = -^7~S-> y\/Z a3 (1.74) (1.75) (1.76) (1.77) (1.78) {A?SM\\HHyp\\AiSs) = 0, (EwSM\\Hhyp\\%ioSs) = 0, (1.79) (1.80) (1.81) (1.82) (1.83) where X = JV, A, S\u00b0. Using the masses and compositions of the excited states from ref.[17] given in table 1.9, along with the zeroth order ground state nucleon wave func-tion, as a basis and taking 6 = 260 MeV [23] we get, for the nulceon hyperfine mixing matrix, \/ 1005 138.5 132.8 138.5 1405 0 132.8 0 1705 \\ \\ No JV(1405) Y 5 U 1YU0 j \\ JV(1705) j JV0 represents the zeroth order ground state nucleon wave function, iJVg2^). We have used the results (1.84) 1005 = 1135+(N82Ss\\Hhyp\\N82Ss) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 37 X X\u00a3Ss> X\u00a3SsM Xi0SsM JV(1405) -0.99 -0.17 \u2014 JV(1705) +0.15 -0.94 \u2014 \u2014 A(1555) +0.99 +0.02 +0.10 \u2014 A(1740) +0.09 +0.30 -0.95 \u2014 A(1860) -0.04 +0.91 +0.29 \u2014 E(1640) -0.97 -0.23 \u2014 +0.08 S(1910) -0.20 +0.91 \u2014 +0.17 E(1995) -0.08 +0.27 \u2014 -0.23 Table 1.9: Excited baryon compositions, in the SU(6) basis (from ref. [17]), adjusted to our phase convention. 138.5 = ~0.99(X2SS'\\Hhyp\\X2Ss)-0.l7(X2SM\\Hhyp\\X2Ss) 132.8 = +0.15(X82 Ss>\\Hhyp\\X2Ss)-0M{X2SM\\H\\X2Ss) The matrix in (1.84) has 7V(941) = +0.95iV8255 + 0.25NS2SS> + 0.20N2SM (1.85) as its lowest eigenvalue (in MeV) and corresponding eigenvector. Similarly for the A and E\u00b0 in the SU(6) basis we obtain A(957) = +0.97A8255 + 0.18A2S5' + 0.16A25M - 0.01A?SM, (1-86) E(957) = +0.97E255 + 0.17E255' + 0 . 1 7 \u00a3 2 S M - 0.00\u00a32 0SW. (1.87) The low value for the mass is a result of neglecting the mass difference between the two light quarks and the strange quark. 1.11.2 Hyperfine M i x i n g in the uds Basis When SU(3)f^our is broken we must use the uds basis and we can no longer use (1.69). Since the strange quark is always in the third position in this basis, we have using (1.66) #mn = (#f(l, 2,3)\\(Hllp + x{H\\lp + H \u00a3 p ) ) l C s(l, 2,3)) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 38 = 2,3)\\(HH + x{P$3H%pP23 + P}3Hl 2ypP13))\\\u00bbds(l, 2,3)) (1.88) {^ \" (^l, 2,3))} axe the excited uds wave functions up to 7Y = 2, see table 1.8; P,j are permutation operators, transposing quark positions i and j. We have (1,2,-3)1^1^(1,2,3)) = (2,1,3)1^1^(2,1,3)) = (-l)2(tf(l,2,3)K3 p|^(l,2,3)) After relabeling and transposing the quark positions 1 and 2 and noting that the uds wave functions are antisymmetric with respect to exchange of quark positions 1 and 2 only. Therefore Xnt = (^(1 ,2 ,3 )1 (^ + 2 x^^13 ) 1 ^ ( 1 , 2 , 3 ) ) (1.89) uds Basis Compositions The effects of a ^ a\\ are small in the S = \u2014 1 sector [24] and are neglected in the matrix elements of Hhyp. We will need to know how a function F (say) behaves under the permutation operator P 1 3. It can be easily verified that (1.90) (1.91) (1.92) (1.93) \u2022\u00b1 t Along with the relations <*2ool*3(#l$ooo) = 0, (1.94) \u00ab l * 3 ( ^ o o > = 1^^ , (1-95) (^ ool<53(A)l$ooo) = 0, (1.96) PizF\" = 1 \\\/3\\ 2 P ~ T X > Pi3Fpp = -PP--XX- ^ A P13FX = P13FXX = 1 \u201e 3 ^ \\ ~2~P Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 39 we get for the uds hyperfine matrix elements (A2S\\H\\A2S) (A2SPP\\H\\A2S) (A2SXX\\H\\A2S) (A2SPX\\H\\A2S) (E 2 5 | \/ f|S 2 5 ) (\u00a3X|\/f|\u00a32S) (\u00a32S\\\\\\H\\H2S) (Z2SPX\\H\\I:2S) Due to the higher mass of the strange quark the zeroth-order energy (1135 MeV) now becomes5 1135 + ma - mu - (1 - x)J%- = 1295 MeV Using the N = 2 wave functions for the A and E \u00b0 in the uds basis (table 1.10) we can construct the mixing matrix. When diagonalized this gives6, A(1113) \u00bb 0.95A25 + 0.07A25AA + 0.28A25pp + 0.08A2SpA, (1.105) S\u00b0(1209) \u00ab +0.98E25 + 0.17S25AA + 0.02S 25p p + 0.11S25pA. (1.106) for the ground state eigenvectors in the uds basis. We now take these compositions (1.85)-(1.106) and apply them to our problem of calculating the K~p \u2014\u2022 Ay and the K~p \u2014\u2022 E\u00b07 branching ratios. 5 A quark in a harmonic oscillator potential has ground state energy ^ = ^tiu = (H \u2014 1) which gives p3 = a. 6Reference uses a = 0.32, m u = 0.33, m, = 0.55 (GeV) but they point out that if m, \u2014 mu is increased to 0.28GeV with a corresponding increase in a to 0.41 there is little change in the compositions. = o, -x\u20146, -x^6, \u2014x-^-S. 4 (1.97) (1.98) (1.99) (1.100) (1.101) (1.102) (1.103) (1.104) Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 40 X X2SXx x2spp X2SP\\ A(1555) -0.75 -0.66 -0.09 A(1740) +0.56 -0.69 +0.46 A(1860) +0.34 -0.28 -0.85 E(1640) -0.84 -0.53 -0.11 E(1910) +0.23 -0.51 +0.56 E(1995) +0.19 -0.31 +0.02 Table 1.10: Excited baryon compositions, in the uds basis (from ref. [17]), adjusted to our phase convention. Chapter 2 Theory 2.1 Method We wish to calculate the branching ratios for the radiative capture reactions K~p \u2014\u2022 Y-y within the N.Q.M., where Y represents A or E\u00b0. Within the context of this model we picture one of the u quarks of the proton being transformed into an s quark and so creating a A or \u00a3\u00b0. We take the approximation (as in ref.[15]) that the kaon is a point-like particle and ignore its internal quark structure. Also we take the standard hypothesis that photo-emission occurs via the de-excitation of a single quark. Since we are using the nonrelativistic wave functions given in chapter 1, it is appro-priate to develop a nonrelativistic operator from the K~ + u \u2014> 5 + 7 interaction on a single quark. This operator will then be 'sandwiched' between the nonrelativistic wave functions corresponding to the proton and the Y. We will procure this operator from a nonrelativistic reduction of the interaction obtained from the lowest order Feynman diagrams contributing to the process (see Fig. 2.2). In Dirac notation the amplitude for the process is, SYp = (*Y\\Y,V(i)\\*P) (2.107) i=i where the sum is over the three quarks in the baryons and is the single quark transition operator which acts on the space, spin, and flavour of the ith quark. 41 Chapter 2. Theory 42 2.2 Symmetry considerations The Y and proton wave functions are given by three body wave functions, the form of which was described in the previous chapter. In coordinate representation they are dependent on the coordinates of each of the three quarks, suppressing the spin-flavour dependence of the baryon wave functions we get, SYP = j(*y|rllFaif3,*)(ri,ri,r3,t|(VW + V<2> + V<3>)|r* f* t') x ( ^ r ^ r ^ ' l ^ ) ^ ^ ^ (2.108) where rj is the position vector of the ith quark in the Y; f- is the position vector of the ith quark in the proton; and the interaction in coordinate representation, (f\\, f2, r3,t\\(V^+ y(2) -|_ V^)\\f(, f2, r3,t'), is a sum of single quark transition operators {fur2,r3,t\\(V^ + V\u2122 + y(3))|r1',r2',r3',0 = VM(?ur>[,tJ)Pft - r%)83(f3 - f3') + V(r3, rg, *, f ) * V i - r ? ) ^ - r*a')} x * p(r \u2022) dr dr'. In particular \/ Mri,ra , r ^ 1 * ) V W ( f i , r V 1 t > t')Ara \" ^ 3 ( r 3 - r * ) * , ^ , r 2 ' , r 3 ' , f ) Jrc\/r' = J M ^ , r l f Fa,0V\u00ab(i?a.*?a,,t,O\u00abS(*ii - ^ ( r , - ^ ^ ( r j . ^ f g , f ) drdr', where we have just relabeled quarks 1 and 2. Making a transposition of any two quark positions in an antisymmetric wave function will result in a change of sign of the wave function. Transposing fi \u00ab-\u2022 r 2 and f{ \u00ab-+ r 2 in the last expression we get (-1)2 J yY{^,r2,r3,t)V^\\r2,r^^ = J * y ( r i , r 2, r3,*)VW(r1. *, t')*3(r2 - r2')\u00a33(r3 - r%)9p(f{, r2', r3\\ f) Jr dr'. By applying the same procedure, of changing the labels and transposing, to quarks 1 and 3 and quarks 2 and 3 we obtain the result SYP = \/ ^ ( r i H V t V i . ^ t . O ^ - W r s - r g ) (2.109) + V t V a . ^ . ^ O W - ^ C r s - r S ) (2.110) + y(3)(f3,r3',t,i')<53(n - KWf3 - (2-111) x #p(rt')drdr'. = 3 | M n ) y ( 3 W % M 0 < $ 3 ^ (2.112) uds Basis. When we use the uds basis waveo functions the strange quark is always in the third position in the strange baryon. Since the model involves the transformation of one of Chapter 2. Theory 44 the u quarks in the nucleon, to an s quark it follows that only the term J *K(r,-)^ ( 3 ) (r 3 , *, W i - ?{)63(?2 - r*)* p(r$) dr dr' can contribute to the amplitude. Since only the interaction on the third quark is relevant, we drop the superscript on all references to V henceforth. 2.3 The Impulse Approximation By singling out quark 3 in this manner, quarks 1 and 2 are being treated as specta-tor quarks. This amounts to using the impulse approximation of nuclear physics. It states that the interaction occurs over a small enough time period that the momentum-energy absorbed by the third quark has insufficient time to redistribute to quarks 1 and 2. On physical grounds this statement can be justified [25] by noting that the energy uncertainty associated with the absorption of a kaon is of the order of the kaon mass. Therefore the strong interaction takes place over a time interval of the order of l\/rriK ^ 2GeV - 1 . This interval is much shorter than the characteristic periodicity time associated with the binding of the quark which is of the order of l\/(m p \u2014 3m\u201e) \u00ab 2QGeV~l (taking m\u201e = 0.33 GeV). Therefore the interaction may be thought of as an impulse during which the binding forces are unable to play an important role. The target quark is thus regarded as free and the binding forces serve only to determine what momentum components are present in its wave function. 2.4 Fourier and Jacobi Transformations We now Fourier transform the interaction V(r3,r'3) in (2.112) to momentum space V W ) = \/(p\\r3)(r3\\V]r'3)(r'3\\p') d4r'3d4r3. (2.113) Chapter 2. Theory 45 Where the closure relations \/ \\r'3)(r3\\c\/V3 = 1 and J |r3)(r3| d4r3 = 1 (2.114) have been inserted. We get V(p,p') = \/ e - - y ( r 3 , r 3 ) e - \u00ab > ' - 3 ^ ^ (2.115) and the inverse relationship is V(rs, r3\\ t, t') = J e-*\u00bbV(p, P ' ^ ' ^ ^ T - (2.H6) Now using (2.116) in equation (2.112) yields, SYP = T ^ \/ M ^ ) ^ - ^ ^ Where 6 = 3 when using SU(6) wave functions and 6 = 1 when using uds wave func-tions. Separating out the time dependence in the bound state baryon wave functions (equation (1.6)), * r(r.) = Mri)eiEt and*p(rO = ^ 1 ) ^ ' , we obtain x V ' P(^Oe\" , ( E i + E 2 +^ ) <'d*^'dVld 3r; ,dVV^V \/rf(\/)^ (2.117) Since we have a single quark interaction we have E\\ \u2014 E[ and E2 = E'2. Performing the time integrals we get, S Y p = 7 ^ \/ W - P > ( ^ - ( p 7 ) ^ ( n ) ^ ( r l - r Y ) Chapter 2. Theory 46 x *3(r2 - F 2 y r V p V ( p , p y-Wiptfi) d3Ti dzfi d3pd3p'dp0 d(p\u00b0y. (2.118) Ei and E2 have been dropped from the time dependence as po is the zeroth component of the 4-vector, p, which must correspond to the energy of the interacting quark (E3) only. 2.4.1 When SU(3)flavour is Broken We first develop a form for the amplitude when m s ^ mu; appropriate for use with uds wave functions for the Y. We then generalize the amplitude to the case of unbroken SU(3)fiaV0Ur symmetry by letting ms -> mu. We switch to the Jacobi coordinates introduced in \u00a71.7, P 4(ri ' \" *'), A' = -L(rJ' + ri' - 2r1'), = |(r1' + v2' + ri ')\u2022 (r1,r1,r1,r1',r1',r1') i-> (p,\\,R,p',\\\\R'), 0 Vv(\u00bb\"1,r2,r-1) i-> *y(^,A,P) and Vv(r~i', rT, ri') \u2022-\u2022 *P(p\", A', i?'), and using the relations derived in \u00a71.9, Chapter 2. Theory 47 -y\/6 m u 3mum8 with m\\ = 2mu + m s' we get, bJ J' ( 2 * ) ' \/ *(\u00a33 -P\u00b0)S(E'3 -(\/)') x A, R)S3(R -R'+ -j={p- p') + ^ ( ( ^ ) A - A')) x 6\\R -R'- p>) + - ^ ( ( ^ ) A - A'))e i^^e-3)) P V ( p ,y ) x d3pd3\\ d3R d3p' d3\\' d3R! d3pd3p' dp0 d(P\u00b0y x e * ^ + ^ ( - \" 3 ) ) pV(p, p\u00b0 = E3, p', (p0)' = E3) x e-^'-v^')*'*^' = ^ + ^ ( i l - R') + ^ ( ( \u2014 ) A - A'), A',R') x d3pd3\\d3Rd3X'd3R'd3pd3p'. (2.119) Where we have integrated with respect to p\u00b0, (p\u00b0)' and p' and the Dirac delta function property (1.70)) has been invoked. J and J' are the Jacobians J = 3 3n 9fl dp 3A 9r-a dri dp 3A drz dp 3A 1 7 5 1 \" * \u2022y\/6mu 1 \/'\"La 1 0 7 5 ^ - 3 ) (2.120) Chapter 2. Theory 48 \u2014# Evaluating the R' integral we get x e - i l ^ l O J - * ) - V P W 4 ( ( ? , = - J<, # = j? + * ( ( * _ A')) x d3pd3Xd3Rd3X'd3pd3p' (2.122) x c - - ( ^ * ( ^ - v ^ ^ ' w = \u00a3 p , R ' = R+ )A - A')) x d3pd3\\d3Rd3\\'d3pd3p'. (2.123) The Jacobian transformation allows us to separate out the centre of mass motion, since Vy(p,\\,R) = NY*Y(p,Z)e-iP~\u00a5R (2.124) and Vpip', A', R') = Np$p{p',\\')eip>n' (2.125) = i V p $ p ( ^ A ' ) e ^ - ( ^ v f ^ ) X - X ' \u00bb (2.126) \u2014* \u2014\u2022 Where Py is the centre of mass momentum for the Y final state and Pp is the centre of mass momentum for the proton. Ny is a normalization coefficient for the Y and Np is a normalization coefficient for the proton. The $x(p, A) contain normalized flavour and spin wave functions. The harmonic oscillator functions are normalized, over all space, to unity. The normalization coeffi-cient Nx, therefore, contains a from the box normalization of the centre of mass plane wave. It will also include normalization factors associated with the Dirac spinors (\u00a72.10.2). Srp = x e Chapter 2. Theory 49 x d3pd3Xd3Rd3\\'d3p\u00b0 = E3,p\\(p0)' = E 3 )e^ ( ( - ) X - 3 X ' ) ? '$ p (p,A') x e^A&^')^^d3Xd3X'd3pd3p' x e - ^ W ^ ^ - ^ ^ ^ X O e * ^ - ^ d3pd3Xd3X'd3p. (2.127) The delta function * 3 ( i V - P + P ' - PP) = <53((PP - p') - ( iV - P)) tells us that the momenta of quarks 1 and 2 remains constant throughout the process and allows us to perform the integration over p'. 2.4.2 When SU(3)fiavour is Unbroken If 517(6) basis wave functions are used for the Y then (2.127) becomes (with ms \u2014> m, m\\ \u2014* m) x V(p,p0 = E3,p' = Pp+p-PY,(p\u00b0y = E3) x d3pd3Xd3X'd3p. (2.128) We will return to the last two expressions after deriving the explicit form of V(p,p'). Chapter 2. Theory 50 2.5 The Form of the Interactions. We will now derive the form of the vertex functions and propagators for the interacting quarks and the kaon. 2.5.1 Equations satisfied by an interacting field. Spin | particles: Spin | particles such as quarks obey the Dirac equation. The Dirac equation for a free particle is (7Mp\" - m)V> = 0, (2.129) where m is the rest mass and p the 4-momentum of the Dirac particle. When the particle is in the presence of a potential V, the Dirac equation using Bjorken and Drell[26] notation becomes, (7Mp\" - m)V> = Vrl>. (2.130) Integral-spin particles: Spin zero particles such as the kaon obey the Klein-Gordon (KG) equation. The KG equation for a free particle, (\u2022 + m ^ K = 0 , (2.131) where m#- is the kaon rest mass and \u2022 = \u2014p^p^ = g~ gfj:- In the presence of a potential V the KG equation becomes, ( Q + m ^ = - % (2.132) Chapter 2. Theory 51 2.5.2 Form of the vertex functions The electromagnetic interaction for a spin | particle: The minimal substitution1, pM \u2014* p>* \u2014 eA**, introduces the coupling of a Dirac particle, charge e, to the electromagnetic field AM. Using the Feynman slash notation [26], 4 = y^A\" = y\u00b0A\u00b0 - 7 \u2022 A, we get from (2.129) and (2.130) (^ -m)V> = e^V (2.133) =\u2022 V^ac = e4- (2.134) This is the form of the electromagnetic interaction of a Dirac particle such as the spin 1 quark. The electromagnetic interaction for a spinless particle: Again we use the substitution \u2014\u2022 p*4 \u2014 efcA^, this time into (2.131) (ex is the charge on the kaon), giving, [ ( i \/ - - eKA\\x)f - m2K]M*) = 0 [ n + m2K + eK(A\u00bb(x)i\u2014 + A^x)) - e^A^A^x)}^^) = 0. So to first order (neglecting terms 0(e2) oc (y )^) and using (2.132) V$(x) = ieK{A*(x)\u00a3- + -~A,{x)). (2.135) The form of the strong interaction: For the strong (K~p) vertex, consider the meson field in analogy with the electromag-netic potential[26]. 1This substitution is necessary to preserve local phase (gauge) invariance of the QED Lagrangian [6]. Chapter 2. Theory 52 The conventional choice for the electromagnetic transition current that generates A\" in equations (2.134) and (2.135) is, r = e^ V *^'. (2.136) Where xj)x is the wave function of the particle before interacting with the electromagnetic field. ip* is the wave function of the particle after interacting with the electromagnetic field. (2.136) relates to the electromagnetic 4-vector potential, A\", through Maxwell's equations. In the Lorentz gauge Maxwell's equations are DA\" = J\" (2.137) = e^VV>''- (2.138) For example in the case of photon exchange in electron scattering from a Dirac proton, ipf and ij>% would represent the final and initial proton wave functions, and e the charge on the proton. (2.138) would then define the M0ller potential of the Dirac proton. The 'Klein-Gordon' analogy to (2.138) is (A\" ipK), (\u2022 + m2K)i>K = i>YTxpp (2.139) (compare to equation (10.11) of ref.[26]). Also, r = gtcusijs for pseudoscalar (PS) coupling, (2.140) T = 9Kus fits for pseudovector ( P V ) coupling, (2.141) where q = p \u2014 p' and 75 = i7o7i7273 p'(p) represents the up(strange) quark momentum, immediately before(after) the strong interaction. gKus\/idKus) represents the strong coupling Ku \u2014* s constant in the pseu-doscalar \/(pseudovector) coupling scheme for the process K~p \u2014* Y 7 . It is the analogue Chapter 2. Theory 53 of e in (2.138). gnus and 9Kus a i e related by noting that the interactions taken over free quark states must be equal in the PS and PV coupling schemes. That is u(p)gKus7hu(p') = u{p)(p' ) = 0 (2.142) and for the conjugate relation, u(p)O*-m.) = 0, (2.143) and noting the anticommutation relation 7 \u201e 7 s + 757^ = 0, (2.144) we get that 9KUS = - J ^ 2 - . (2.145) Now the equation for a proton absorbing a kaon (analogous to equation (2.133)), can be written, (^ -mp)V>p = (V>\/cr)V>P- (2-146) In the strong (K~p) interaction a u quark in the proton is transformed into an s quark. This corresponds to a V+ operator (see \u00a71.8) acting on the third quark of the proton. It can be thought of as the combination of creation and annihilation operators a\\au, Chapter 2. Theory 54 acting on the third quark of the proton. Where a\\ creates an s quark and au annihilates a u quark. Therefore the form of the strong interaction can be written, from (2.146), VST = TV+i>K. (2.147) 2.5.3 The Feynman Propagators. The Feynman propagator for a spin | particle. The Feynman propagator Sx(y\u2014x) (or Green's function) for a Dirac particle X, satisfies the Green's function equation (\u2022 J7y - mx)Sx(y -x) = 6\\y - x), (2.148) where V y is the 4-vector gradient This operator acts on functions of y only. Sx(y \u2014 x) represents the wavefunction ip(y) at space-time point y produced by a unit source (ip(x) = S4(y \u2014 x)) at space-time point x; the evolution of il>(x) is governed by the free Dirac equation (2.129). By Fourier transforming to momentum space we get [26], f d4q e--\u00ab-(\u00bb-\u00ab) = 7(2^7^7 Now, 1 _ j+rn 4-m {4-m)(4 + m) _ 4+m f-m2' Chapter 2. Theory 55 and by the anticommutation relation T ' V + 7\"7\" = 2flT, (2.149) it is easily found that \u00a3 = q.q = g2. (2.150) r d4o e-\u00ab'<\/(!\/-*) <\u00a34g c w.(v-*)(2TT)4 g2 -m2x +iey The small positive imaginary part added to the denominator in equation (2.151) as-sures that (2.151) meets the desired boundary condition of propagating only the positive frequencies forward in time (particles) and the negative frequencies backward (antipar-ticles). The Feynman Propagator for a Klein-Gordon particle. The Feynman propagator for a Klein-Gordon particle K is the solution of the equation, (\u2022\u201e + m2K)AK(y -x) = -6\\y - x). (2.152) Afc(y \u2014 x) represents the wavefunction ib(y) at space-time point y produced by a unit source (tp(x) = 84(y \u2014 x)) at space-time point x; the evolution of ip{x) is governed by the free Klein-Gordon equation (2.131). By once again, Fourier transforming to momentum space we find [26], ^'-'Wp^-mH - f c - ( 2 1 5 3 ) 2.5.4 The kaon wave function The kaon and proton form a hydrogenic atom immediately prior to capture. Leon and Bethe [27] suggest that the kaon capture by the proton is most likely from an S state. Chapter 2. Theory 56 In fact, the fraction of K mesons reacting from P states is less than 1%. However, the capture can occur from one of many different states with principal quantum number n. The bound state wave function for the kaon can be written M*) = NKxPK(z)e-iEKt, (2.154) where NK is a normalization constant. The probability density for a Klein-Gordon particle is given by which for xp given by equation (2.154) is p = 2EK\\NK\\2\\tJ>K(?)\\2. Normalizing to a box of volume V we get NK = J \u2014 (2.156) with j | ^ ( i ) | 2 d3z = 1. (2.157) J v In terms of its Fourier transform *K(*) = NKe-*\u00abj^^Jj*\u00ab'*4K{pK)#pK = ^T2J^iPK-ZMPK)d3pK. (2.158) Where 4>K(PK) is the bound state momentum space wave function. 2.5.5 The photon wave function AM(iy) has the form, with box normalization [26], - J i g t ' f c . U I = =\/V76Vfc-, (2.159) with 7\\L = . 1 . (2.160) Chapter 2. Theory 57 (2.160) is the photon \"wave function\" and describes the emission of a photon with \u2014* momentum k, energy k , and polarization 4-vector eM. 2.6 Derivation of the explicit form of V We now derive the explicit form of the interactions corresponding to the Feynman diagrams in Fig. 2.2, first in coordinate, and then in momentum representation. From the Feynman graphs (Fig. 2.2) it can be seen that the structure of the interaction Vi, associated with the ith graph is, ViCiirs) = VEM(r3)Sa(r3-r3)VST(r3), (2.161) V2(r'3,r3) = VST(r3)Su(r3-r'3)VEM(r'3), (2.162) V3(r'3,r3) = VST(r3)S\\r3 - r'3), (2.163) V4(r'3,r3) = VST(r3)6\\r3-r3)VEM(r'3). (2.164) VEM is the form of the interaction at the electromagnetic vertex. VST is the form of the interaction at the strong vertex. Sa(r3 \u2014 r3) is the Feynman propagator for a Dirac particle \u2014 the strange quark. ) g J d 4 q d w V ^ i e K d ^ i M w ) M w ) ) by parts. The boundary term is omitted as the potential AM is taken to vanish as t, \\w\\ \u2014\u2022 \u00b1oo. Substituting for V'K'(w) a n d A^(w) into (2.171) and evaluating the derivatives we get, Mr*) = \\^yi\/2 J d'qd^pKd'w ^ _ ^ e ^ j f + q)^e^K{pK) NisN P i s t p-ir3-(PK-k) after performing the integrals over w and q. Therefore, (2.173) Chapter 2. Theory 60 Where m -^, e# is the mass and charge of the kaon. The contact term comes from the minimal substitution in the kaon-nucleon pseudovector Lagrangian [28]. That is, T = igKus P'KIS = igKusip'- p\")75 igicM ~ e* 4~ p\" + 4)75, (2.174) The term i9Kus(-es 4 + e\u00ab 4)75 in (2.174) corresponds to the radiative process, graph (4) of Fig.2.2. Since ea \u2014 eu = we get the result V<(r'3,r3) = -NKN^K ^ j ^^ig^^V+e-^MPK)^ - r'3) (2.175) 2.6.2 Momentum Representation In the next section we wish to examine the effect of a gauge transformation on our interactions (i = 1,2,3). To do this it is convenient to Fourier transform the interactions Vi to momentum space. By inserting the closure relations (2.114) in V\\ (2.165) we get, e \u00bb ' f c . r 3 - \u00ab q . ( r 3 - r J ) - \u00bb P i f . r ^ - j p ' . r J + \u00ab ' p . r 3 KM) = ^ \u00a7 \/ A ^3d S d3p V (2^)19\/27 - * \" ' 3 - ' a - \u2122 ? 2 _ m 2 x f\\4+m.)TV+4>K{pK) = j AdV3d3pVeg 2_m 2 M+m.)rv+tK(pK) Chapter 2. Theory 61 x (2TT)4 S4(k - q + p) = l2^ldpK (k+py-m* S (P\u00ab + P ~ (K + P\u00bbMPK). (2.176) For V2 w \u00ab have, using (2.166), p NisN r f,ik.r'3-iq.(r3-rl3)-ipK.r3-ip'.r'3+ip.T3 x rV +(^+m t t) 9 2 - \u2122 u f 7WJV \/\u2022 e-'(i+k-p')-r3 X (2TT)4 64(q-(p-pK)) = l ^ W ] d p K (p-PKy-ml HPK+p-(k+P))MPK). (2.177) With (2.173), = J * r*d r a d PK {PK-kf-ml x rV + (2p K - fc).e<54(r3 - r3)< (^Ptf) = ^ \u00a7 \/ ^ ( P K _ t ), _ m, rn(2 P , - \u00bb w = J d p K ( p K - k Y - m lS ( P K + p-( k + P\u00bb**M-(2.178) Chapter 2. Theory 62 And for V4 we get from (2.175), x K(PK)6\\r3 - r'3) = ~(2^y J ^ i y j c \u00ab 7 . V + e - ^ ^ ^ - ^ ^ ^ ( f i c ) = -^Jy\/i JD3?K ^9KUS75V+**K(PK)S\\PK +p'-(k+ p)). (2.179) Notice the appearance of the energy-momentum conserving delta function for these processes. From this it can be seen that p corresponds to the 4-momentum of the s quark in the outgoing baryon (Y); and p' corresponds to the 4-momentum of the u quark in the incoming baryon (p). 2.7 Gauge Invariance. 2.7.1 Choice of Gauge In the Lorentz gauge specified by dpA1* = 0 (2.180) we get the constraint, using (2.160), k.e = 0. (2.181) This is a covariant condition which must be true in any frame. Within this gauge we can exploit the residual gauge freedom [29], yiM A\" - d'x, (2.182) provided x satisfies \u2022 X = 0, (2.183) Chapter 2. Theory 63 which keeps us in the Lorentz gauge. For a real photon satisfying the free field equation [29], \u2022O4\" = 0, (2.184) with plane wave solutions of the form (2.160), the photon 4-momentum k must satisfy, the massless condition, jfc^ Jfc\" = 0 k\u00b0 = \\k\\. (2.185) The transformation (2.182) corresponds, by equations (2.181) and (2.185), to e\" ^ e\" + 0Jfc\". (2.186) We choose (3 so that e\u00b0 vanishes. This fixes A*. It is known as the Coulomb gauge and makes manifest the transverse nature of the electromagnetic field. 2.7.2 Proof of Gauge Invariance To check invariance under a gauge transformation (within the Lorentz gauge (2.180)), we note that the amplitude SYp, corresponding to the ith graph in Fig. 2.2, is linear in the polarization vector e. That is under a gauge transformation. Then if the set of Feynman diagrams in Fig. 2.2 is gauge invariant we must have E W = 0, (2.187) i-l where JV is the number of graphs. Chapter 2. Theory 64 We ignore binding between the quarks for the purpose of determining the behaviour of the interactions under a gauge transformation3. Replacing e with k in (2.176) w = u(P)vr\\p,p>(p') = ~(2^jru(p) J d P K (k + pY-m* ^(PK+P~(k+p)Hp), where u(p) and u(p') are Dirac 4-spinors which are solutions of the free Dirac equation for the third quark. Using (2.150) and (2.185) we get, -( M\/e~fc\/ >\\ ( >\\ e * N K N y ^ f J 3 - (m 3 Ji+ P fiTV+JKJpK) \u00ab ( P ) ( p , P )u(P) = -^jruip) J d P K p 2 + 2 k p _ m 2 x 8\\pK + p' - (k + p))u(p'). Using the anticommutation relation for the Dirac gamma matrices (2.149) we find that Ji f> = 2k.p- p1 ft. (2.188) Employing (2.188) and the Einstein mass-energy relation p2 = m 2 , (2.189) and(p')2 = rn-l, (2.190) we get, - \/ N T ^ f c , \/ \\ , eaNKN^_f f 3 ^ (2k.p-{p1-ma) )i)TV+(f>K(pK) u{p)Vx (P,p)u(p) = \u2014\u2014Tu(p)jdpK \u2014 ( 2 7 r )3\/2 J \" 2 k p X S\\pK + p'- (k + p))u(p'). Using (2.143) we get u(p)Vr\\p,P>(p') = e-^^-u(p) J d*pKTV+MPK)u(p')8\\pK-rp'-(k+p)). (2.191) 3However, in general we will lose gauge invariance when we fold the interaction over three quark wave functions. Chapter 2. Theory 65 Replacing e with k in (2.177) \u00ab(P)V2 (P,P)U(P) = -^-fFu{p)jdPK {p_pKy_ml x 64(pK+p'-(k + p))u(p'). Noting that p - pK = p' - k and using (2.150), (2.185), (2.190), and (2.188) we get, euNKN^_, , f _ rV+(\/>K(PK)(2p'.k- }i(f -\u2122u)) (2^)3\/2 v^ yy -2p'.fc x S4(pK + p'- (k + p))u(p'). From the relation (2.142), u(p)VrK(P,P>(p') = - ^ 7 T \" ( P ) \/ ^r^jr(fif)\u00ab(p')^(pJc+p'-(*+p)). (2.192) Replacing e with A; in (2.178) U(P)V 3 (P,P)\u00ab(P) = -wvr\u00bb(p)JdPK i P K _ k y _ m 2 K x \u00a3 4(P*+p'-(fc + p))\u00ab0>')-As before we get, fi(p)v3~ P>(P') = -e^ )I7\"(p) \/ fpx W+MPKMP') 8\\pK + p'-(k + p)). (2.193) W i t h P S Coupling Here we use the substitution (2.140) Chapter 2. Theory 66 From our condition for gauge invariance (2.187), and noting that eu = |e, es = \u2014 |e, e# = \u2014e, where \u2014e is the charge on the electron, we have, J2 w = u(P)(vrkM)+vrk(p,p')+vrk(p,p')Hp') \u00bb=i JS K i v \/ = ^ 2 7 r ^ 3 \/ 2 ^ \u00ab ^ ( p ) ( e 8 - eu - eK) J d3pK 1SV+K - c\u201e i>K- eKifa- P))lhV+K(pK)u(p') S\\pK + p' - (k + p)) \u00b1 0. Therefore we need the contact graph (Fig. 2.2 (4)) in order to obtain a gauge invariant interaction in the PV coupling scheme. Vr\"(P,P') = -^f-2i~9Kusu{p) j d3pKeK V15V+MPKMP')6\\PK+P' ~ (k+p)). (2.194) Chapter 2. Theory 67 2.8 Kinematics In this section we derive the magnitude of the photon momentum in the kaonic atom rest frame. The photon momentum is kinematically constrained to take on the value, = k\u00b0 \u00ab 281MeV when Y = A, and k\u00b0 \u00ab 2l9MeV when Y = \u00a3\u00b0. This can be seen from conservation of energy-momentum in the K~p atom rest frame, P\u00a3 + P?< = k\u00bb + PP = 0. (2.195) The zeroth component gives, mK + mp = k\u00b0 + \\l\\PY\\2 + mY, (2.196) and the 3-vector components give, PY + k = 0 \\Py\\ = \\k\\. (2.197) From (2.196) we get, (mK + mp)2 + (k0)2 - 2k\\mp + mK) = \\PY\\2 + mY, with k\u00b0 = \\PY\\ =\u2022 k\u00b0 = (rnp + mKy-mjr Inserting the values [30], mp = 938.27231Mey, mK = 493.67Mey, mA = 1115.63MeV, mEo = 1192.55MeV, we get, for the reaction K p \u2014\u2022 A 7 , \\k\\ = k\u00b0 = 281MeV and for the reaction K p \u2014> E\u00b07, k\u00b0 = 219MeV. Chapter 2. Theory 68 2.9 Observations It is generally assumed [31],[32] that the kaon momentum is approximately zero prior to the formation of the kaonic atom. This can be seen heuristically by modeling the kaonic atom with the Bohr model[33]. The binding energy of the kaon will then be given by, E a = . l * \u2122 e V = ( 2 . 1 9 9 ) where HK is the proton-kaon reduced mass; fie the proton-electron reduced mass; and n is the principal quantum number of the kaonic orbit. The total energy of the atom is the sum of the kinetic and potential energies, EB = K + U. Taking a circular orbit mv2 e2 2 r 4TT\u20ac0r -e2 1 along with U = and K = -mv2 6 47T\u00a3 0r 2 gives EB = e2 e2 8ire0r 4ir\u20ac0r -\\PK? 2nK \\PK\\ = \\j-2uKEB 5.591 x 1012 T r 2 eV-As mentioned in \u00a72.5.4 almost all captures take place from a relative S state. In addition, half the captures are said to take place for principal quantum number n > 10 and less than 4% survive to n = 4. We take then, as a reasonable estimate, n = 10. Chapter 2. Theory 69 This gives, for the kaon momentum, \\pk\\ \u00ab 0.236MeV. It can be seen that |p#| is far smaller than the other momenta (\\Py\\ = \\k\\ \u00ab 281MeV) and masses (mu = 330MeV, ms = 550MeV) in the problem; we therefore neglect \\p~k\\. Since |p#| w 0 we take the normalized momentum space wave function to be MPK) = \\I{^-63(PK). (2.200) Inserting this into (2.158) we get ipK&t) = NKe-iE^4>K(0). (2.201) with ^ (5) = -~ (2.202) so that it retains the correct dimensions. That is the radial wave function of the kaon is approximately constant and so must go as l\/y\/V to satisfy equation (2.157). From our choice of the Coulomb gauge we had (\u00a72.7.1) k.e = 0 and eo = 0. Inserting these conditions into V3{p,p') (2.178) and letting px \u2014+ 0, we see that the contribution to the amplitude from graph (3) of Fig. 2.2 is zero. However, it was necessary to include this diagram to preserve gauge invariance (\u00a72.7). For Vi(p,p'), ^(PiP 7) a n d V*(P,P') we get V1{p,p') = e3NKN^ Af+J + ro'>rV+ 8\\PK +p'-(k+ p))i>K(0), (2.203) (fc + py \u2014 m* V2(P,P') = euNKN,TV+{pi- m\" 2 } U \\ P K +p'-(k+ p))^(0), (2.204) VAM) = -eKNKN^i~gKusls8\\pK+p' - (k + p))V+ipK(0). (2.205) Chapter 2. Theory 70 2.10 The Nonrelativistic Reduction. In this section we approximate V by an expansion in p\/m, consistent with our use of nonrelativistic wave functions. 2.10.1 Validity of the Nonrelativistic Reduction. The NQM assumes the quarks are approximately at rest in the hadron rest frame. That is the hadron has negligible internal momentum. However most models tend to predict that the momentum of a quark in a hadron, can be sizeable. For example a particle of mass M , localized within a volume of radius R, has momentum p, by the uncertainty relation, (P2)> ~ ^ (2-206) The identity[9], Ap = {(p\u2014 (p))2)* with (p) = 0 in the hadron rest frame, has been used. R is the radius of the ground state wave function. The analysis of various hadronic processes indicates [34] that, R2 ~ 6-12GeV-2 =\u2022 \\ ~ 300MeV. (2.207) R In our calculations we use the constituent quark mass4 [24] mu = 330 MeV ms = 550 MeV So we get, M i < i . (2.208) *Note that 3m\u201e > mp due to the binding energy of the baryon. Chapter 2. Theory 71 This suggests that taking (p\/m)2 and x 8 X 6 two component spinors, (2.209) becomes \/ dt 0 a \u2022 p ^ a \u2022 p 0 & p\\ X | +mq (l o \\ + [ 0 - 1 ) [ ' ) + vint \\-x) { x m\u201e X + Vi int (2.211) We separate out the time dependence in xp xp = \\ X J = e-iE<* \\X ) (2.212) Chapter 2. Theory 73 where (j> and x are constant with respect to time and Eq is the total energy of the state ip. Inserting (2.212) into (2.211) and cancelling the common exponential factor we get, E\u201e \u2014 a \u2022 p int (2.213) The second equation of (2.213) gives, X = a \u2022 p - T, r (2-214) Eq + mq - Vint X are the \"small\" components of the Dirac 4-spinors. They are reduced by v\/c compared to the \"large\" components . Because of relation (2.208) we conclude that the small components of the quark Dirac spinors are of the same order as the large. To treat this relativistic situation we follow Yaouanc et. a\/.'s [34] prescription of replacing the Pauli spinors \\ \\ m the spin wave functions X.+ =T= by Dirac spinors u = Ei + m, 2m,-Xi V Ei+mi Xi J (2.215) The subscript i refers to the iih quark in the baryon. We have choosen normalization so that uu = 1, (2.216) as in Bjorken and Drell [26]. This yields an over all normalization factor, from normal-izing the total baryon wave function, \\N\\2 J tfttf dV = 1 Chapter 2. Theory 74 _ _ L _ \/ m u \/ m \u00ab \/m u A p \" Vvv *?V TV _ _L_ \/ m\" \/ m\" ms Y ~ y\/v\\l Ei^J E2)j E3' Using (2.215) amounts to adopting the spinor structure of free quarks. Since we are using the impulse approximation (\u00a72.3) this seems reasonable. The interaction involves only the third quark. Therefore the spectator quark spinors, as a result of (2.216), give unity when folded together. We are left with x < E'+mu }p(p')d4Pd4p'. (2.217) This is expanded and a nonrelativistic reduction carried out. We \u2022 Expand the interaction and discard terms of order (p\/m)2 or higher, \u2022 treat the denominator of the propagators and phase space factors relativistically. Therefore the expression of the full matrix element is valid up to order (p\/m)2 (p is any momentum and m any mass). Applying this prescription to the { } in (2.217), we get for V = VX (equation (2.203)), f 1 F - z \u00a3 - ) ^ + > + m ' ) ' y \u00ab &3 + ms I \\ \\ 3'v' I \\ E'+mu \/ we get (with e\u00b0 = 0) (1 \u2014a \u2022 p E3 + m3 0 \u2014o \u2022 6 a \u2022 e ( m 3 + k0+p0 - + p \u00b0 ) - \u00b1 J L - + J^l-a \u2022 ea.(k + p) E3 + m3 E'3 + mu E3 + ms a \u2022 pi - a -ea.ik + p)\u2014\u2014 a \u2022 e (ms - k - p ). Now we neglect terms 0(p\/m)2 or higher, and get vPS(n ni\\ _ -NKN^e,igKu3V+ij>K(0)a \u2022 e (m. - k\u00b0 - p\u00b0) 4 ri.,n\\\\ V> {k+pY-ml 6(PK+p-(k+p)) (2.218) (2.219) in PS coupling. With PV coupling we get (1 \u2014cr \u2022 p E3 + ms ( 0 -a \u2022 e \\ I m3 + k\u00b0 + p\u00b0 -a.(k + p) \\ EK 0 a-t 0 \\ 0 \u2014EJ K ) 0 1 1 0 o.(k + p) m3-k\u00b0-p\u00b0 j E'+mu. \/ vr(p,p') = NKN^igKusV+ipK^a \u2022 emK(ms - k\u00b0 - p\u00b0) (k + p)2 \u2014 m2 64(pK+p'-(k + P)) (2.220) 9Kus -trPSf l\\ mK*\\ (P ,P) 9Kus since EK \u2014 mx when PK = 0. Similarly we get from equation (2.204), rps( \/x _ NKN^eJgKu3V^K{Qi)& \u2022 e(mK - p\u00b0 + mu) VriP,p') = (P - PK)2 - ml S\\PK + p'- (k + p)) (2.221) for PS coupling and v2pvM) = NKNyeuigKuaV+^K(0)a \u2022 emK(mK -p\u00b0 + mu) (p-pK)2-ml 64(pK+p' ~(k+p)) (2.222) gKv mKV{sM) Chapter 2. Theory 76 for PV coupling. Recall from equation (2.145) that QKus 1 For the contact graph the interaction (2.205) becomes V4pv(p,p') = NKN^eKigKutV+ipK(0)a \u2022 e8\\pK + p' - (k + p)) (2.223) in the nonrelativistic approximation5. 2.11 The Problem. Now that we know the form of the interactions and their Fourier transforms we can substitute them into (2.127). We write V(p,p') = V(p,p')NKN^K(0)S4(pK+p'-(k + p)). (2.224) V is the total interaction: VPS = + v * s for PS coupling, yPV = yPV + yPV + yPV for p y c o u p H n g > We get, SYp = ZC(2n)4S\\pK + Pp-(k + PY)) x J *Y(p, A ) e i A V ^ ( - - 3 ) t \/ ( p , p \u00b0 = E3,p' = PP + P - PY,(p0)' = E'3) x e ^ ( ( ^ ) ^ ' ) . ( ^ + P - ^ ) $ p ( ^ X ' ) e ^ & - X , ) d3pd3Xd3\\' d3p. (2.225) That is SYp = Z{2it)A8\\pK + Pp-{k + PY))M (2.226) 5If we use the on-shell condition (2.142) before we take the nonrelativistic reduction, we find that the sum Vfv + V2PV + Vfv reduces to V ^ 5 + V2PS. Since we intend to use bound state quark wave functions we discard this approach and the PV amplitude will differ from PS amplitude. Chapter 2. Theory 77 where M Z W,P) and C C jlY{p,fiVIp(^d*pd*p, \/ $ p ( ^ A ' ) e S ^ + 3 ( ^ ) ) d 3 X ' 5 276 (27r)32v '^ The normalization constants are given by 1 N K = 1 Nr, = 1 \/mu frn^ vl; V \u00a3 i V E2 V #3 (2.227) (2.228) (2.229) (2.230) (2.231) Chapter 2. Theory 78 Figure 2.2: Feynman diagrams contributing to the process K p \u2014\u2022 Yj. Chapter 3 Calculations In this chapter we wish to evaluate the integral (2.227) in order to get an expression for the invariant amplitude Ai. This calculation can be divided into evaluating the flavour, space, and spin contributions. 3.1 Flavour Space The only flavour dependent piece in V is V+ (see \u00a71.8). V+ acts in the flavour space of the third quark; transforming a u quark to an s quark. From table 1.6, table 1.7, and equation (1.34) we get the flavour matrix elements listed in table 3.11. 3.2 Momentum Space With the flavour matrix elements determined we have, for the momentum space part of the amplitude, MMS = $YeiX\u2122 d3\\] V [\/ $peiX' d3\\' d3pd3p (3.232) where qY = 4=((~)Py - 3p) (3.233) -y\/6 rnu qp = (3.234) We have already taken care of the flavour dependent part of the potential V+. Also note that the spin dependent term a \u2022 e in the potential, acts only in the spin space of 79 Chapter 3. Calculations 80 - v i ' (tii\\v+\\4>) = 0, V * A | V + | ^ ) = 1, (K\\v+\\**xP) = 0, = o, (4>*\\v+\\4>i) = 0, (ri\\v+\\*i) = 0, (tf\\v+\\) = 0, = 0, (i\\\\v+\\**xP) _ 3 ' ( W W ) 1 _ 1 \u2014 3' (^\\v+\\pp) = 0. Table 3.11: Flavour matrix elements the third quark. Therefore we separate out the spin piece and denote it (SP) = (XJf\\a-e\\Xf). XJ~\u00b0 denotes xp a n d XJ=1 corresponds to %A- This term will be evaluated in \u00a73.3. Since V is a function of p only, we can use the orthonormality of the p oscillator wave functions to write the general momentum space matrix element (3.232) in terms of the A oscillator wave functions. This will involve combinations of the terms aY = J V'ooo(A)e'^ A :d3A, (3.235) ap = \/ V-ooo(A')e -^*'d3A', (3.236) by = \/ V>ioo(A)e'^ -A :d3A, (3.237) bp = \/ V'ioo(A')e''p- (3.238) cy = \/ V>oio(Ay?^ ld3\\, (3.239) cp = \/ ^oio(A')e\u00ab'f'--X'd3X'. (3.240) Integrals involving the harmonic oscillator wave function V'oim with m ^ 0 give zero Chapter 3. Calculations 81 when integrated over all space. Now defining a == J aYVapd3p, (3.241) 6 = J aYVbpd3p, (3.242) c = J byVapd3p, (3.243) d = jbyVbpd3^ (3.244) 9 = J cYVcpd3p, (3.245) and Q = (3.246) a = From table 1.3 and table 1.4, we obtain table 3.12. See Appendix B for details on the evaluation of aY, ap, by, bp, cY, cp. The results are (\u2014Ylu (3-247) \\aat\\\/ ( ^ l )f <'\u00ab - fr* + * M + foM\/,). (3.250) \u201416 \/ 7r \\ r d = (\u2014Y I5. (3.251) \\aa\\J Where Ij = ^ijjk (k is an index which sums over the diagrams), k and Ilk = I e 2al Vke 2q2 d3p, (3.252) J2fc = y|9P|2e M Vke ^ d3p, (3.253) \/a* = y |^|2e ^ I 4 e ^ ^ p , (3.254) Chapter 3. Calculations 82 ( *S\u00bbWI*S\u00bb> a, <*5\u00bbIGI*S\u00bb) (*5\u00bbWI*aoo) = -?56> <*5\u00bbWI*$oo) = 0, (SfoolQWo) = (*fooM*foo> = | (\u00ab + <0, <*fool = <*foolQI*20o) = 0, <*AoolQI*ooo) = -72c' <*$ool = 0, (*UQ\\*ioo) o, (*5oolQI*\u00a3\u00bb) = 0, (*5JQI*$oo> = o, (*SJQI*Soo) = ^\/3, <*SSolQI*S\u00bb> = o, <*SSolQI*S\u00bb) = 7!a' <*5SoWI*W = 72a> (*SSoWI*5oo> = 0, < *& IG I * S\u00bb> = c, <*&IQI*\u00a3x>> (^o A olQI*W \u2014 M , \\\/2 ' <*&IQI*SoO> = 0, = o, (^oolWfoo) = 0, <*&IGI*2bo> = o, (*&IQI*Soo> = 17\/3. Table 3.12: Matrix elements of the harmonic oscillator wave functions in terms of \u2014* \u2014* a,b,c,d,g. The Dirac bracket denotes integration over p, A, A' and p. Chapter 3. Calculations 83 hk = J\\qP\\We~^Vke~^ d3p, (3.255) = \/l?pll9y|e ^ d 8 ^ . (3.256) Since neither V, qY or qp have any dependence the integral over (f> simply yields 27T. The piece which is common to all the momentum integrals can be written where Similarly Aj, = e 1 2 OA , ax = \u2014y, &i = \u2014 T ( \u2014 ) P Y e-%F = e - ^ ( l ^ l 2 + | P l 2 - 2 ? . ^ ) = ^ e - a i P 2 ^ ' . ? where A p = e ^ ' * , oj E i j , 6\/ = \u00b1 P Y . Finally we get h = ax + ai, (3.257) s2 = ft^+^^ + M ) ^ . ( 3 . 2 5 8 ) 3.2.1 Including the Potentials With PS coupling we had, from the previous chapter, yPS = yPS + yPS T , ^ ( -es(ms - k0 - E3) eu(mK + mu - E3) \\ (k + p)2 -m2 + ie (p - pK)2 -m2u + ie Chapter 3. Calculations 84 With PV coupling yPV = Vp v + Vpv + Vpv T \/ - - J es(ms - k0 - E3) eu(mK + m u - E3) eK \\ = iQKusV+a \u2022 emK I v \u20144- + \u2014 i \u2014-L + -H-K [{k + p)2 - m2 + te (p-pK)2 -ml + ie mK J Diagram 1: radiating s quark The denominator of V\\ can be written (k+p)2-m2 + ie = (k\u00b0 + E3)2-\\k + p\\2-m2s + ie = -(\\H\\2-(k\u00b0 + E3)2 + m2s-ie), where we have employed the substitution \u2014\u2022 u = p + k Now we get after integration of equations (3.252)-(3.256) with respect to the 6 and coordinates of vector u. (See Appendix B for details) IJX = *0\u00abAlim f\u00b0\u00b0 \u201e GjM . du J = 1, \u2022 \u2022 \u2022 ,4 (3.259) \u00ab-\u00bbo Jo u2 \u2014 \\{ \u2014 le with A2 = (Jb\u00b0 + \u00a3 3 ) 2 - m 2 and gcc \u2014+ gr u^s in PS coupling gcc -* rnKgKus in PV coupling. Gj(u) has the form1 5 Gj(u) = 2*ApAye-hu2 2\u00bb7.-j(\u00ab'e' l U + (-tO'e^1*), (3.260) i=l where C 2 = e . K - i 0 - ^ - ^ , (3.261) * i = (2fc-ci)fc, (3.262) ,m\\ 3a\\. 1 1For J = 5 the 0 and u integrals must be done numerically. Chapter 3. Calculations 85 The rjij are constants (but depend on J) defined through equations (B.306-B.309) and u* denotes u raised to the power of i. The integrand has a pole at u = Ai. To deal with this we separate the integrand into two parts: one containing the pole, and a part which is bounded over the whole integration range. That is \u201e r \u201eu = f ^MfOAhl i u + u r i u . (3.264) e-o Jo u2 - Af - it Jo u2 \u2014 A? \u00a3-*o Jo u2 - \\ \\ - i e The first integral in (3.264) can be evaluated numerically (Appendix A), the integrand is plotted in \u00a73.2.2. The second integral in (3.264) can be evaluated by contour integration techniques (see Appendix B). The result is 0 so the integrand has no singularity and therefore can be integrated directly. We get after performing the angular integrals Ij2 = iQccCz lim f \u00b0 f ^ . dp, J = 1, \u2022 \u2022 \u2022, 4 (3.269) t-^ o JQ p* + \u2014 e^ Chapter 3. Calculations 86 The integration over 8 in I 5 2 must be carried out numerically. Fj(p) has the form Fj(p) = 2\u00abApAye-h>2J2vu(piea2P + (-p)ie-\u00b02Pl (3.270) (3.271) and the rfu are constants and s2 = \\s2\\> 0. The integrand of the integral in equation (3.269) is plotted in \u00a73.2.2. Diagram 4: the contact term V4 contains no propagator, so we get simply J j 4 = rFj(p)dp. (3.272) rnji Jo We make the replacement, since E3 is the energy of the third quark in the Y baryon, E3 \u2014\u2022 m3. 3.2.2 Plots of the Integrands We perform Gaussian numerical integration on the following integrands. As can be seen from the plots, they are well behaved and converge rapidly to zero above momenta of 1 GeV. The plots labelled (a) show the integrands of I\\k for k = 1,2,4. (b),(c),(d) denote the integrands of J2jt, hk and Zjfc. The following symbols are used : \u2022 = uds basis, Y = A x = SU(6) basis, Y = A \u2022 = uds basis, Y = E\u00b0 + = SU(6) basis, Y = S\u00b0 The I5k must be integrated numerically over 8 and momentum. Surface plots are given for the A uds basis case only. One axis is labelled by cos 8 +1 and so goes between Chapter 3. Calculations 87 0 and 2. 6 is the angle between the vector p(u for the case of the radiating Y diagram) and the z axis. Surface plots for the other cases have the same behaviour and so were omitted. All integrands are multiplied by the constant 1 2wApAy' Chapter 3. Calculations Chapter 3. Calculations Chapter 3. Calculations Momentum in GeV Chapter 3. Calculations Chapter 3. Calculations 92 COS0+1 Chapter 3. Calculations 93 Chapter 3. Calculations 94 cosfl+i Chapter 3. Calculations 95 3.3 Spin Space We wish to calculate the spin piece (Xjf\\f-e\\x?), where xj 1S the spin wave function in the final state in which quarks 1 and 2 are combined to spin J and all three quarks are combined to spin S\/, with z component mS't Xi' is the spin wave function in the initial state in which quarks 1 and 2 are combined to spin J' and all three quarks are combined to spin Si, with z component Now m'a,M' and [xJf\\ = E ( \u2022 7 > M l 2 ' m i ; l^2){Sf,mf\\J,M;\\,mz)(\\,mx\\(\\,m2\\(\\,mz\\. ma,M Where the ma under the summation sign denotes mi,m2,m 3 , similarly m'a denotes m\\, m'2,m'3. We have, since the interaction acts only on the third quark, (X\/l*-e1xf) = E (J,M\\\\,mx;\\,m2)(SJ,mI\\J,M-\\,m3) ma ,M,m'a ,M' x {\\imi\\\\,m\\)(\\,m2\\\\,rn2)(\\,m3\\v \u2022 \u00ab1|'m3) x (Im^lm^J^M'^J^M^lm'^mi) E (J,M\\\\,mu\\,m2)(\\,m\\-,\\,rn'2\\J',M') {Sf,mf\\J,M; \\,mz) x (J',M'; |,m3|5i,mi)(|,m3|o f- el|,m3) E (J,M|i,m 1;i,m 2)(i,m 1;i,m 2 |J',M') x (5 \/,m \/|J,M;|,m 3)(J ,,M';i,m 3|5,-,m i)(|,m 3|a -t\\\\,m'3). Chapter 3. Calculations 96 From the unitarity properties of the Clebsch-Gordan coefficients (equation (3.5.4) of ref.[13]) ]T {J,M\\j1,m1]J2,rn2){JuTn1;J2,m2\\J',M')=6MM>6j>j,, (3.273) 771J , TT12 we get (X \/ I ^ x f ) = D 6M,M'8j,j>{Sf,mf\\J,M]i,m3)(J\\M'^,m3\\Si,rni) T7i3 ,M,m 3,M' x (\\,m3\\& \u2022 t[\\,m3) = D (5\/,m\/|J,M;|,m3)(J \/,M;|,m 3 |5,-,m t)(|,m3|a- e\\\\,m'3)8j,j,. m3,M,m'3 Now a.6-=^e_ f l C r f l (- l) f i (3.274) fl in spherical tensor notation. The Wigner-Eckart theorem (equation (5.4.1) of ref.[13]), (j>,m'\\T(k,q)\\j,m) = (_i)i-~ 0\"> m'^-^K q) (j'\\\\nk)\\\\j), (3.275) (T is a tensor of rank k, component q), in our case becomes, Il I 11 \/\\ \/ i \\ i - m l (I'm 3' 2' _ m 3 | l ) -^ ) \/ l II - . M 1\\ Equation (3.275) defines the reduced or double-bar matrix element. The reduced matrix element is easily calculated from (3.275) with j = |, m = | , j ' = | , m' = | , (and 5 denotes |5|), i 2 ' 2 ' 2 ' 2 I i 2 ' 2 l \" l 2 ' 2 \/ \u2014 V \"V \u2022 r\u2014~~\u2014 \\ 211\"^ 112 > => (\u00a711^ 111) = since (in natural units) 5 = -o\\ Chapter 3. Calculations 97 So, (X\/l^-elxf) = \u00a3 ^\/.m\/IJ.MjI.msX\/.AfjI.mJI^m,-) m3 .M.trij ,R v\/3 \u2022y\/e(-l)Re.RSJvl. y\/2S, + l y\/3 (-1)1\u20143 R , A r m ' \u00ab ( - i ) j - 2 + ^ ( - i ) j - 2 + m ' ( - i ) 2 - 2 + f i ; VEs, 3,3' Si ( 1 2 ^ M m'3 \u2014m,- ^ m3 \u2014 m 3 \u2014R K M m3 \u2014rnj Where we have converted our Clebsch-Gordan coefficients into '3-j symbols'. The definition of the '3-j symbols' is given by (equation (3.7.3) ref.[13]) \/ J \\ 32 J3 \\ mi m2 m3 -J2-\"13 \u2022(Ji,mi;J2,m2\\j3, -m3). V\/2j3~+T Using the symmetry properties of these symbols [13] we obtain (3.276) and Now we have, x \\ \u2014m3 m'3 R j \u00a3 (^2S> + 1)(25,- + l ) ( - l ) - 2 J + f- m\/- m i- m3 e _ f l V6 6J,J, rnj, M,m'3,R ( m.- \u2014mo J -M \\ t 1 2 Sf J ^ m3 \u2014mf M i I 1 \\ 2 2 ^ \u2014 m3 m3 R i (3.277) Chapter 3. Calculations 98 This can be expressed in terms of the '6-j symbol' defined by, J l J2 J3 h h h E 0 'i\u00bbmi;i2,\u00bbw2li3\u00bb\u00bbT\u00bbi+ m2> yf(2j3 + l)(2\/3 + 1) m, ,m2 x { J 3 > m i + m 2 ; \/ i , M - m i - m 2 | \/ 2 , M ) X (j 2, m 2 ; \/ i , M - m a - m 2 | \/ 3 , M - mx)(jx, m a ; \/ 3 , M - m^k, M ) . (3.278) Hence, as a consequence of the reality of our Clebsch-Gordan coefficients, the 6-j symbol is real. Comparing (3.277) with equation (6.2.8) of ref.[13] E (-1) _1 Vl+'2+'3+Ml+M2+M3 \/ i 7 1 \\ t \\ \u2122 1 7^ 2 -7^3 \/ J l J2 J3 mi m 2 m 3 ' l J2 *3 \\ -Pi rh2 7 i 3 7 J l J2 J3 \/ i \/ 2 *3 h h Js (3.279) and noting that rrif = M + m 3 , m, = M + m 3 , _ ^ ( _ 2 ) - 2 J + | - m \/ - m i - m ^ _ ( _ ^ l + J - m 3 - m ^ - M x we get (since J is an integer), (_l)-2-J+!-m; (-1) J+i ( X ^ l ^ ^ l x f ) = (-l)-J+>-miV66JtJ,J(2Sf + l)(2Si + 1) < 5,- 5\/ 1 2 2 J x E e-A 5,- 5, \\ R \\ m,- \u2014m\/ J2 y For the square of the spin amplitude we have, W ' l ^ l x f n x ^ - e l x f ) = ( - l ) - W H 1 - 2 -6(25\/ + l)(25i + 1) Si Sf 1 2 2 J Chapter 3. Calculations 99 x Si Sf 1 1 i 7 2 2 J R> R \\ X Si Sf 1 m,- \u2014my 7?' y 5,- 5\/ 1 m,- \u2014m\/ 72 ) but (\u2014l) 1 - 2 m \u00ab = 1 since the intrinsic spin of the proton 5\",- is half-integral2. Summing over polarization states A and using the identity [26] (3.280) which in the transverse gauge becomes, \u00a3 < ( A ) e j ( A ) = <$,-\u2022 A (3.281) we get (xJs'\\t-Ax?Y(xJ}\\t-Axi) = 6(25, + \\){2Si + 1) Si Sf 1 1 1 7\/ 2 2 J x < Si sf 1 1 i 7 2 2 J E 5,- 5, 1 ^ rrti \u2014trif R j (3.282) 3.3.1 Evaluation of the 6-j symbols Using relation (6.3.4) of ref.[13], Jl 32 J3 1 \u2022 _ 1 \u2022 _ 1 2 J 3 2 '2 2 ' i i i 2 2 A 1 5 0. (-1) h+h+h (jl + J2 + J3 + 1)(J2 + J3 - jl) \\| 2j2(2j2 + l)2j3(2j3 + 1) (3.283) = 1\/4. We consider only S state mixings in our compositions so, since the total angular momentum J = for the Y or proton, 5,- and Sj are both one-half. _ 1 Chapter 3. Calculations 100 In addition relation (6.3.3) of ref.[13] gives us, Jl J2 J3 2 3% 2 2 1 1 1 2 2 1 1 1 2 2 = 1\/36. Ui + J2 - J3 + + J3 - ia) \\ (2j2 + l)(2j3 + 2)2;3(2j3 + 1) (3.284) 3.3.2 Evaluation of the 3-j symbols The 3j-symbol can be evaluated from the relation ((3.7.10) ref.[13]) _ (\u20141 )J1 ~J2 +mi +m2 { ji J2 Ui + h) ^ ma m2 \u2014(mi + m2) 1 (2j1)!(2j2)!(j1 +j2 + mi + m2)l (2ji + 2j2 + l)!(ii +m,)! x \/ i 2 y rrii \u2014rrif R j U1+J2 - (mi + m2))! \\j (ji ~ mi)\\(j2 + m2)\\(J2 - m2)\\ [( ^m^^ + mj-mj^l + Ry. (3.285) 3.3.3 Spin summation and squaring the amplitude Assigning equal a priori probabilities to each of the initial spin states and summing over the possible final spin states i ; W = 2 c T l S , A \" a For example with J = 0, J' = 0, 5, = | and 5\/ = | \u00a3 l ( x ' \/ I^W = 12 < 1 1 1 2 2 1 1 0 2 2 U E m \/ =\u00b1i ,mi=\u00b1if l=\u00b1l \/ 1 2 (3.286) \\ ^ rrii \u2014mj R j (3.287) Chapter 3. Calculations 101 With J = 1, J' = 1 Si = \\ and S, = \\ \\ E K#-e1xA>|2 = 12 j ' 1 1 2 m '\u00ab I I I 1 2 9 ' For the cross term: 5 E (xJI^-ilxfrUjl^-e-X?) = 12 \u00ab 1 1 1 1 2 2 \u00b1 i i O 2 2 u > < m \/ = \u00b1 | , m , = \u00b1 | H = \u00b1 l _2 ~3\" (3.288) i i i 2 2 A 1 1 1 2 2 \/ 1 1 , \\ 2 2 1 ^ m,- \u2014 m , i2 y (3.289) 3.4 Ful l Ampli tude The full invariant amplitude is now obtained in terms of the expressions (3.241)-(3.245), with the assistance of the SMP algebraic manipulation package. The full wave functions were coded, along with the matrix relevant elements, into the file \"WAVEFN.DEF\". These definitions were then used (within the SMP environment) to simplify the ampli-tude as much as possible. A FORTRAN formula for the amplitude was generated. The relevant input data and commands, along with the output, is detailed in Appendix C. 3.5 Determination of the Strong Coupling Constant We need to insert a value for gKus into the invariant amplitude. Unfortunately, the strong coupling constant for interactions at the quark level is not known. However it is possible to express the proton-kaon-Y vertex in terms of quark-kaon-quark vertex. We Chapter 3. Calculations 102 Figure 3.3: Strong vertices for relating gxua to gtcpY follow the treatment in ref.[37], in which the process is assumed to be a single quark transition. We first consider the strong vertex at which a u quark absorbs a kaon and is transformed into an s quark as in Fig.3.3 (a). The amplitude for the process is t=i J in coordinate representation. The sum is over the three quarks in the baryons. $y and $ p include spin-flavour dependence and are given by equation (1.58). We now do a two component reduction and take the nonrelativistic limit (Ea,ms are the energy and mass of the quark in the final state, the s quark; Eu,mu the energy and mass of the initial state quark, the it quark). Neglecting terms of 0(p\/m)2 and disregarding, for the purposes of this calculation, the difference in mass between the strange and up quark, we get, \/ + \\(\u00b0' \u2022 W f ( ^ ) w \/ ( 3 ) x \/ - \u2014\u00bb \u2014\u2022 \\ = 3gj<:uay drxld5x2dix3^\\{x1,X2,xz)(< ^ )V+ '^(^1 ,^2,^3) (3.290) The complete permutational symmetry of the baryon wave functions allows us to write the interaction in terms of the third quark. The V operator in (3.290) arises from taking PK = Pa \u2014 Pu and so acts only on the kaon wave function. Applying this procedure to the analogous baryon-meson-baryon vertex, Fig. 3.3 (b), we find in terms of the baryon centre of mass X HKpy = gKpy J 4 ( I ) ( \" ^ ( i ) V + M i ) *X. Where (F|V+|p) = 1. Neglecting the difference between the centre of mass and the position of the third quark and equating the amplitudes we find 2mp 2m u x 1 1 + z 1 \" Where P and Y are the 3-quark spin-flavour wave functions. The j indicates we are taking the spin up state for the Y and proton. Therefore using table 3.11 and the spin matrix element results (xpM3)\\xx+) = o, (xA>i3)ix;> = o, (X>i3)lxi> = -1\/3, we get 9KUY = A) = (3.291) o mp andgKUS(Y = X\u00b0) = ^ ^ 9 v ^ . (3.292) o mv Chapter 3. Calculations 104 Using the values 9KpA = \u201413.2, 9Kp\u00a3,\u00b0 = 6.0, obtained from ref.[31], we get 9Ku.(Y = A) = -4.83, (3.293) and ^ \u201e , ( y = S\u00b0) = 11.40. (3.294) Form factor corrections to these coupling constants will be small [38] due to the \u2014\u2022 \u2014\u2022 fact that the decay momenta (\\Py\\ and |A;|) are equal. 3.6 P h a s e s p a c e Recall that the amplitude for the process was given by SYp = Z(2v)464(pK + Pp-(k + PY))M Squaring this amplitude, summing over final and averaging over initial spin states, and dividing by VT gives us the transition probability per unit time per unit volume ]\u00a3!! = (2*)4f>4(PK + PP-(k + PY))\\A4l2l^(0)|2m>, (E3 + ms)(E'3 + mu) VT AmKkPV^EiEsE'^E's Amuma 'l ' ' However3 we want the decay rate per decaying K~p atom; therefore we divide by the number of decaying particles per unit volume. Since there are y K~p atoms per unit volume we divide (3.295) by ^ . The decay rate is given by the transition probability per unit time per decaying particle, integrated over the number of final states in volume V, (2K)48\\Pk + Pp-(k + PY))\\M\\2\\ipK(0)\\2m4u(E3 + m3)(E'3 + mu) Vd3k Vd3PY 16mKk0V2EiE2E3Ef1E2E3 (2TT) 3 (2TT) 3 3The bar in S and Ai indicates that the spin summation has been carried out. Chapter 3. Calculations 105 S(mK + m p - (fc\u00b0 + EY))\\M(PY = -k)\\2\\M0)\\2<(E3 + ms)(E'3 + mu) 3~ 6ATr2mKk\u00b0E1E2E3E'lE2E'3 (3.296) Taking Ei = E2 = E[ = E'2 = mu, E3 = ma and using conservation of energy for the third quark E'3 = E3 + \u2014 k\u00b0, we find r = f2* r r S(mK +mp~ (EY + k\u00b0))\\M(PY = -k)\\2\\xpK(0)\\2(ma + mK-k\u00b0 + mu) J=o JB=O Jk=o 32ir2mKk\u00b0(ms + mx \u2014 k\u00b0) x sm6d$d\\k\\2d\\k\\. Substituting \u201er r, ,o dW EY + k\u00b0 W W = * + * * W = - E Y - = E y -we get T = r 6(mK + mp - W)\\M(PY = - f c ) | 2 l ^ ( 0 ) | 2 ( m a + mK - k\u00b0 + mu)k\u00b0EY dW Jw=mY 8TrmK(ms + mK \u2014 k\u00b0)W = + \"\u00ab - k\u00b0 + \">')\\M(?r = -j)?. (3.297) STrmK(mK + mp)(ma + mK - k\u00b0) 1 v \" v ' Where EY = jrriy + (k0)2. As mentioned previously the principal (\u00a72.5.4) quantum number for the kaon wave function from which capture takes place is not precisely known. However, if we assume the kaon wave function at the origin is the same for all decay modes, it cancels in the branching ratio. Unit Conversion The decay rate for the process K p \u2014\u2022 all modes Chapter 3. Calculations 106 is given by Burkhardt et al. [39] as FaU = 2 W P | \u00ab 0 ) | 2 with4 Wp = (0.560 \u00b1 0.135)GeV fm3 Since we have used natural units throughout and taken masses and the confinement parameter to be in GeV, our invariant amplitude is in units of 1\/GeV; as it must in order to yield the dimensions of a decay rate. In order to get T\/\\IPK(0)\\2 to have dimensions GeV fm3 we multiply by ( y ^ ) 3 = (0.19732705359GeV\/m)3. The branching ratio, therefore, for the process K~p \u2014> F 7 is given by BR - k\u00b0EY(m\u00b0 + m K ~ k \u00b0 + mu) \\M(PY = -fc)|2(0.197Gey fm)3 8irmK(mK + mp)(ms + mK - k\u00b0) 2WP 3.7 Results and Discussion Table 3.13 lists the branching ratios to A 7 and E \u00b0 7 in the uds and SU(6) basis with both PS and PV couplings. The confinement parameter and the quark constituent masses are taken from previous analyses [15],[24]. We take the same values as used in the previous NRQM calculation by Darewych et. al [32]. From table 3.13 it can be seen that only the E\u00b0 uds (PS) prediction agrees with the current experimental rangefl] of (1.4 \u00b1 0.2)10~3. The experimental value for the A 7 branching ratio is (0.86 \u00b1 0.07)10-3. The fact that the uds calculation is closer to experiment than SU(6) suggests that taking into account the strange quark mass difference is important. Setting m3 = mu in the uds amplitudes does not yield the same result as the SU(6) amplitude. The two bases are not equivalent physical descriptions. 4It is interesting to note that if the experimental width of the Is level[40] T i , = 620eV is taken, along with a hydrogenic Is orbital we get Wp = 0.565GeV fm3 in good agreement with the result of ref.[39]. Chapter 3. Calculations 107 BRxlO\"3 U<\u00a3s SU(6) PS PV PS PV F = A 7.08 12.54 49.99 52.09 Y = E\u00b0 1.36 3.71 3.50 3.17 Table 3.13: Branching ratios for K~p F 7 (Y = A,E\u00b0). Results obtained from PS and PV couplings within the uds and SU(6) basis are also tabulated. a = 0.41, m u = 0.42, ms = 0.70 GeV (parameter set 1). BRxlO\"3 uds 5*7(6) PS PV PS PV Y \u2014 A 6.39 8.82 68.92 73.44 Y = E\u00b0 1.18 1.57 11.59 14.07 Table 3.14: Calculated branching ratios for K~p \u2014\u2022 F 7 (F = A,E\u00b0). a = 0.41, mu = 0.42, ma = 0.70 GeV. Here only the largest component of the proton and Y wave functions is included. BRxlO\"3 uds SU(6) PS PV PS PV F = A 10.52 18.29 69.24 70.70 F = E\u00b0 2.87 6.30 4.77 4.41 Table 3.15: Calculated branching ratios for K~p -> F 7 (F = A,E\u00b0). a = 0.32, m u = 0.33, ms = 0.55 GeV (parameter set 2). A\/A 0 (%) (%) PS PV PS PV 14 63 7 67 Proton diagram off 40 126 56 115 Y diagram off \u2014 9 \u2014 2 Contact diagram off Table 3.16: Contributions from the various diagrams to the invariant amplitude in the uds basis. A\/A 0 (%) denotes the percentage of the original A 7 branching ratio and E\u00b0\/E\u00b0 (%) the percentage of the original E\u00b07 branching ratio, when one of the diagrams is 'switched off', a = 0.41, mu = 0.42, ms = 0.70 GeV. Chapter 3. Calculations 108 BRxlO\"3 uds SU(6) PS PV PS PV Y = A 1.46 2.59 10.33 10.76 Y = E \u00b0 1.36 3.71 3.50 3.17 Table 3.17: Branching ratios for K~p -> F 7 (Y = A , E \u00b0 ) . a = 0.41, mu = 0.42, ras = 0.70 GeV. Here <7A\"U\u00ab(A) is set equal to <7A-US(E\u00b0). Contributions from the Diagrams In PS coupling graphs (1) and (2) (Fig. 2.2) add constructively. However graph (2), the proton radiation diagram, dominates. As can be seen from table 3.16, turning off the contribution from this graph reduces the amplitude to 14% for the lambda and 7.5% for the sigma of the original. The amplitude is nearly all imaginary; the real part coming entirely from the radiating Y diagram (Y = E \u00b0 , A). Turning off this real part changes the amplitude to only 1-2% (PS) and less than 1% (PV) of the original amplitude. This is contrary to the assumption made in ref.[32] where they neglect graphs (2),(3), and (4). In PV coupling there is destructive interference between graphs (1) and (2) of Fig. 2.2 and the contact term graph (4) dominates. Table 3.16 shows that removing the contact term reduces the uds amplitude, to 9% for A and 2% for the E \u00b0 , of the original amplitude. The large contribution from the contact term is in agreement with that found by Workman and Fearing [31] where they perform an analogous calculation to ours within a pole model. The branching ratios in the PS and PV coupling schemes were found to be roughly the same within the SU(6) basis but the PS\/PV ratio is about 0.56 in the uds basis for both parameter sets. As mentioned in \u00a72.10.2, the PS and PV results should be identical for an interaction taken over free quark states. Presumably, PS\/PV ^ 1 is a result of the off-shell nature of the quarks in a baryon. Chapter 3. Calculations 109 Turning to the ratio K~p -\u00bb S\u00b07 (3.299) K~p \u2014\u2022 A 7 ' which is independent of the uncertainty in Wp and so is perhaps more reliable than either of the individual branching ratios. Here most of the theoretical models do poorly; none predict the experimental result [1] of 1.71 \u00b10.30 for the ratio in (3.299). We obtain values in the range 0.19 - 0.29 for PS and PV in both uds and SU(6). Changing the constituent quark masses and the confinement parameter to ma = 0.55, mu = 0.33, and a = 0.32 GeV we get 0.18 - 0.26 for the E\u00b0\/A ratio. This is in contrast to the other NRQM calculation, ref.[32], where they obtain 0.76 for the ratio (3.299). However our result agrees roughly with ref.[31] where they get 0.16 - 0.18 for (3.299), including only the Born diagrams. When the contribution from the A(1405) resonance is added they find, with a variety of parameters, the E\u00b07 branching ratio to be 'several times larger' [31] than the A 7 branching ratio. The cloudy bag model (CBM) [41] appears to do best here. They obtain 1.1 - 1.2 for the ratio (3.299), but is still below the experimentally observed result. It is interesting to note that when we use the same quark coupling constant for the A and E\u00b0, which is what one would expect from a quark model, we get (see table 3.17) 0.93 (PS) and 1.43 (PV) for (3.299) in the uds basis, close to the experimental result. It can be seen from tables 3.13 and 3.15 that, unfortunately, many of the results appear to be sensitive to the confinement parameter and the quark masses. Comparing table 3.14 and 3.13 it can be seen that adding the excited components has only a small effect on the PS result but gives a large increase in the PV calculation. Differences due to the kinematics (the photon momentum is larger in the A 7 reaction as compared to the E\u00b07) and phase space factors contribute to the value of the A\/E\u00b0 ratio. However, our high value for the A 7 branching ratio may be a result of omitting Chapter 3. Calculations 110 the A(1405) resonance. Since the interaction in our calculations involves two spectator quarks and a freely propagating third quark it would describe broad resonances. How-ever, there is insufficient binding between the three quarks in the intermediate state to generate a sharp resonance such as the A(1405). Its contribution was found to be significant in refs.[32] and [31] (where they found it to interfere destructively with the other Born diagrams). It would be interesting to see a future calculation including the A(1405), along with the graphs we have estimated. However, further terms in the invariant amplitude would be needed to obtain a gauge invariant result for bound propagating quarks. Our value of the ratio in (3.299) can be partially attributed to symmetry consider-ations: the E\u00b0 has isospin one and therefore has (within the uds basis) a flavour wave function with M\\ symmetry. This must be combined with a symmetric (ground state) spatial wave function and a x A spin wave function to yield an overall symmetric space-spin-flavour wave state. On the other hand, the A has isospin zero which gives rise to a Xp spin wave function. These symmetry constraints on the quark spin wave functions lead to an an extra factor of 3 (from the 6-j symbols) in the K~p \u2014\u2022 A 7 amplitude and therefore the branching ratio will increase by a factor of 9. This causes the A to have a larger branching ratio despite gKuaiX = A) < gKua(Y = S\u00b0). Thus our results appear to be qualitatively reasonable but not quantitatively rigor-ous, as was to be expected from the NRQM. Appendix A F O R T R A N PROGRAMS The following program (KAONCAPTURE) calculates the branching ratios K~p \u2014* Yj using the methods outlined in chapter 2. The Gaussian integration routines are called from the routine SETWX. This routines returns the gaussian points and their weights. The boundary points of the intervals and the number of points in an interval can be changed. This was done many times with the same result so we are satisfied that the estimated integrals are reliable. PROGRAM Kaoncapture IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay Jby,spik,qint . , a l i , b l i , b 2 i , c l i , c 2 i , d l i , d 2 i , d 3 i , d 4 i , g l i . , V I ( 5 ) , c l l u , c l 2 u , c l 3 u , c l 4 u , c s l u , c s 2 u , c s 3 u , c s 4 u . , e l l , c l 2 , c l 3 , c l 4 , c s l , c s 2 , c s 3 , c s 4 , c p l , c p 2 , c p 3 .,A(6),I2(6),I3,L,I1, .gt,gs,gl,mp,mtb,msb,mlb,mst,ms,mu,f,b,sixj DIMENSION Mps(2,2),Mpv(2,2) mp=0.93827231 ! proton r e s t mass 111 Appendix A. FORTRAN PROGRAMS 112 mK=0.49367 mlb=l.11563 msb=l.19255 mu=0.42 ms=0.7 pi=3.141592653589793238 e=dsqrt(4*pi\/137.035989561) convfac=(0.1973270539)**3 eu=2*e\/3 es=-e\/3 ed=es eK=-e alps=(0.41)**2 kaon r e s t mass Lambda baryon r e s t mass Sigma baryon r e s t mass d e f a u l t u quark mass d e f a u l t s quark mass elementary charge converts from GeV~-2 to GeV fnT3 charge on u quark charge on s quark charge on d quark charge on kaon d e f a u l t confinement parameter Wp=0.56 ! t o t a l r a t e Kp->Y gamma=2WpIpsiKI\"1 ( i n GeV fm~3) cpl=0.95 cp2=0.25 cp3=0.20 ! proton admixture c o e f f i c i e n t s cllu=0.95 cl2u=0.07 cl3u=0.28 cl4u=0.08 ! Lambda uds admixture c o e f f i c i e n t s cll=0.97 cl2=0.18 ! Lambda SU(6) admixture c o e f f i c i e n t s Appendix A. FORTRAN PROGRAMS cl3=0.16 cl4=-0.01 C cslu=0.98 ! Sigma uds admixture c o e f f i c i e n t s cs2u=0.18 cs3u=0.02 cs4u=0.11 C csl=0.97 ! Sigma SU(6) admixture c o e f f i c i e n t s cs2=0.17 cs3=0.17 cs4=-0.00 C gl=-13.2d0 ! strong coupling constant f o r Kp->sigma gs=6.0d0 ! strong coupling constant f o r Kp->lambda C C a l l INTEGRATE C C *** MAIN LOOP *** C When : C J l = l \u2014 > Yp= Lambda C Jl=2 \u2014 > Yp= Sigma C J2=l \u2014 > Basis= uds C J2=2 \u2014 > Basis= SU(6) C Type *,'Enter 1 to a c t i v a t e proton diagram (0=off)> Appendix A. FORTRAN PROGRAMS 114 Read (5,*) u l Type *,'Enter 1 to a c t i v a t e imag part of Y diagram (0=off)' Read (5,*) u2 Type \u2022.'Enter 1 to a c t i v a t e r e a l part of Y diagram (0=off)' Read (5,*) u3 Type *,'Enter 1 to a c t i v a t e contact diagram (0=off)' Read (5,*) u4 Type *,'Enter 1 f o r parameter d e f a u l t s ' Read (5,*) L i f ( l . e q . l ) go to 2 Type *,'Enter alpha ( i n GeV): ' Read (5,*) alpha alps=alpha**2 Type *,'Enter mu ( i n GeV): \u00bb Read (5,*) mu Type *,'Enter ms ( i n GeV): ' Read (5,*) ms Type *,'Enter 1 f o r f u l l wave f u n c t i o n s ' Read (5,*) L i f ( l . e q . l ) go to 2 cp l = l cp2=0 cp3=0 c l l u = l cl2u=0 cl3u=0 Appendix A. FORTRAN PROGRAMS cl4u=0 c l l = l cl2=0 cl3=0 cl4=-0 c s l u = l cs2u=0 cs3u=0 cs4u=0 c s l = l cs2=0 cs3=0 cs4=-0.00 DO 30 J l = l , 2 DO 15 J2=l,2 q l = J l q2=J2 CALL TRANS(ql,q2) mlt=3*mu*mst\/(2*mu+mst) E3=mst alpst=alps*dsqrt(mlt\/mu) kt=((mp+mK)**2-mtb**2)\/(2*(mp+mK)) ht=0.75*(l\/alpst+l\/alps) Cc=(27*b*f)\/((2*pi)**3*2*dsqrt(2.OdO)) Api=dexp(-(0.75*kt**2)\/alps) Ayi=dexp(-(mlt*kt\/mu)**2\/(12*alpst)) photon momentum h i n t e x t C i n t e x t Ap i n t e x t Ay i n t e x t Appendix A. FORTRAN PROGRAMS 116 cl=(mlt\/mu+3*alpst\/alps)\/(2*alpst) c2=dexp((cl-ht)*kt**2)*es*(mst-kt-E3) c3=(E3-mu-mK)*eu s ( l ) = c l * k t ! s2 i n t e x t s(2)=Abs((2*ht-cl)*kt) ! s i i n t e x t pl=(mu**2-(E3-mK)**2) p2=(kt+E3)**2-mst**2 rtp2=dsqrt(p2) ! pole of integrand 2 (1 i n te x t ) !lambda2 i n t e x t llambdal i n t e x t qint=0 a l i = ( 4 * P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) b l i = 4 \/ d s q r t ( 3 . O d O ) * ( 2 * P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) * ( - 1 . 5 ) b 2 i = 4 \/ d s q r t ( 3 . O d O ) * ( 2 * P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) \/ a l p s c l i = 4 \/ d s q r t ( 3 . O d O ) * ( 2 * P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) * ( - 1 . 5 ) c 2 i = 4 \/ d s q r t ( 3 . O d O ) * ( 2 * P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) \/ a l p s t d l i = 1 2 * ( P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) d 2 i = - 8 \/ a l p s * ( P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) d 3 i = - 8 \/ a l p s t * ( P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) d 4 i = 1 6 \/ ( 3 * a l p s * a l p s t ) * ( P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) g l i = - 1 6 \/ d s q r t ( a l p s * a l p s t ) * ( P i \/ d s q r t ( a l p s * a l p s t ) ) * * ( 1 . 5 ) C C 10 i n t e g r a l ap=l ay=l bp=0 by=0 c a l l s e t y i ( l ) ! evaluate F l ( p ) Appendix A. FORTRAN PROGRAMS c a l l s e t y i ( 2 ) ! evaluate Gl(u) c a l l c a l c ( l ) C C Ip i n t e g r a l ap=3*kt\/Dsqrt(6.OdO) bp=3\/DSQRT(6.0dO) ay=l by=0 c a l l s e t y i ( l ) ! evaluate F2(p) ap=0 c a l l s e t y i ( 2 ) ! evaluate G2(u) c a l l c a l c ( 2 ) C C Iy i n t e g r a l ap=l bp=0 ay=-mlt*kt\/(mu*DSQRT(6.OdO)) by=-3\/DSQRT(6.0dO) c a l l s e t y i ( l ) ay=(3-mlt\/mu)*kt\/DSQRT(6.OdO) c a l l s e t y i ( 2 ) c a l l c a l c ( 3 ) i f (u.eq.l) go to 15 C C Ipy i n t e g r a l ap=3*kt\/DSQRT(6.OdO) Appendix A. FORTRAN PROGRAMS bp=3\/DSQRT(6.0dO) ay=-mlt*kt\/(mu*DSqRT(6.OdO)) by=-bp c a l l s e t y i ( l ) ap=0 ay=(3-mlt\/mu)*kt\/DSQRT(6.OdO) c a l l s e t y i ( 2 ) c a l l c a l c ( 4 ) C C Iq i n t e g r a l q i n t = l ap=3*kt\/DSQRT(6.0dO) bp=3\/DSQRT(6.0dO) ay=-mlt*kt\/(mu*DSQRT(6.OdO)) by=-bp c a l l s e t y i ( l ) ap=0 ay=(3-mlt\/mu)*kt\/DSQRT(6.OdO) c a l l s e t y i ( 2 ) c a l l c a l c ( 5 ) C 17 DO 23 11=1,5 cxpart=c3*GIl(Jl, 3 2,II)*ul+c2*GI2(JI, 3 2 ,II)*u2 r p a r t = p o l e ( I l ) * p i \/ ( 2 * r t p 2 ) * c 2 * u 3 VI(I1)=DCMPLX(rpart,cxpart) 23 CONTINUE Appendix A. FORTRAN PROGRAMS mgamp=DIMAG(VAMPL(ql,q2))**2+DREAL(VAMPL(ql,q2))**2 Mps(JI,J2)=mgamp*(2*pi*gt*Cc*Ayi*Api)**2 DO 24 11=1,5 cxpart=c3*GIl(JI,J2,II)*ul-c2*GI2(JI,J2,II)*u2+ .GI3(JI,J2.I1)*eK\/mK*u4 r p a r t = p o l e ( I l ) * p i \/ ( 2 * r t p 2 ) * c 2 * u 3 VI(I1)=DCMPLX(rpart,cxpart) 24 CONTINUE mgamp=DIMAG(VAMPL(ql,q2))**2+DREAL(VAMPL(ql,q2))**2 Mpv(JI,J2)=mgamp*(gt\/(mst+mu)*mK*Cc*Ayi*Api*2*pi)**2 Ey=dsqrt(mtb**2+kt**2) E3p=mst+mK-kt phase=Ey*kt *(E3p+mu)\/(8*pi*(mK+mp)*mK*E3p) 300 FORMAT(' Y= Lambda, Basis=uds',$) 310 FORMAT(' Y= Sigma, Basis=uds > ,$) 320 FORMATC Y=Lambda, Basis=SU(6) : ' ,$) 330 FORMAT(' Y=Sigma, Basis=SU(6):',$) open(unit=6,carriagecontrol='FORTRAN',status='OLD') i f ( ( j l . e q . l ) .and. (J2.eq.l)) type 300 i f ( ( j l . e q . 2 ) .and. (J2.eq.l)) type 310 i f ( ( j l . e q . l ) .and. (J2.eq.2)) type 320 i f ( ( j l . e q . 2 ) .and. (j2.eq.2)) type 330 TYPE *,'M~2 (PS)=',Mps(Jl,J2) Mps(JI,J2)=phase*convfac*Mps(JI,J2)\/(2*Wp) TYPE *,'Ratio (PS)=',Mps(JI,J2) TYPE *,'M~2 (PV)=',Mpv(Jl,J2) Appendix A. FORTRAN PROGRAMS Mpv(Jl,J2)=phase*convfac*Mpv(Jl,J2)\/(2*Wp) TYPE *,'Ratio (PV)=',Mpv(Jl,J2) 15 CONTINUE 30 CONTINUE STOP END SUBROUTINE s e t y i ( I ) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5) , .s(2),y(5,2) , 31 ,32 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint a0(l,I)=ap**2 aO(2,I)=2*ap*bp aO(3,I)=bp**2 bO(l,I)=ay**2 bO(2,I)=2*ay*by bO(3,I)=by**2 i f ( q i n t . E q . l ) GO TO 34 ! i f doing 15 do cos t h e t a numerica y ( l , I ) = - a O ( l , I ) * b O ( l , I ) \/ s ( I ) + a O ( l , I ) * b O ( 2 , I ) \/ s ( I ) * * 2 .+a0(2,I)*b0(l,I)\/s(I)**2-2*a0(2,I)*b0(2,I)\/s(I)**3 y ( 2 , I ) = a 0 ( l , I ) * b 0 ( 2 , I ) \/ s ( I ) + a 0 ( 2 , I ) * b 0 ( l , I ) \/ s ( I ) .-2*aO(2,I)*bO(2,I)\/s(I)**2 y(3,I)=-aO(l,I)*bO(3,I)\/s(I)-aO(2,I)*bO(2,I)\/s(I) Appendix A. FORTRAN PROGRAMS .+aO(2,I)*bO(3,I)\/s(I)**2-aO(3,l)*bO(l,I)\/s(I) .+aO(3 , I)*bO(2 , I)\/s(I)**2 y(4,I)=aO(2 , I)*bO(3 , I)\/s(I)+aO(3 , I)*bO(2,I)\/s(I) y(5,I)\u2014aO(3,I)*bO(3fI)\/s(I) 34 RETURN END C SUBROUTINE c a l c ( I ) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(lOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint pole(I)=FF(rtp2,2) spik=pole(I) CALL DOINT(I) RETURN END C SUBROUTINE d o i n t ( I ) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , Appendix A. FORTRAN PROGRAMS 122 .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint GI1(J1,J2,I)=0 ! I n t e g r a l f o r proton r a d i a t i o n diagram, GI2(J1,J2,I)=0 ! f o r Y r a d i a t i o n diagram, GI3(J1,J2,I)=0 ! and the contact graph. DO 10 11=1,LIM GI1(J1,J2,I)=GI1(J1,J2,I)+WW(I1)*F1(XX(I1)) GI2(J1,J2,I)=GI2(J1,J2,I)+WW(I1)*F2(XX(I1)) GI3(J1,J2,I)=GI3(J1,J2,I)+WW(I1)*FF(XX(I1),1) 10 CONTINUE 11 RETURN END C FUNCTION F1(X) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint Fl=FF(X,l)\/(X**2+pl) RETURN END C FUNCTION F2(X) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V Appendix A. FORTRAN PROGRAMS COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), .s(2),y(5,2),J1,J2, .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint F2=(FF(X,2)-spik)\/(X**2-p2) RETURN END C FUNCTION FF(X,I) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), .s(2),y(5,2),J1,J2, .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik.qint I f (qint.EQ.O) GO TO 36 FF=sing(X,I) GO TO 37 \u2022 36 tl=0 t2=0 DO 35 11=1,5 t l = t l + y ( I l , I ) * X * * I l t 2 = t 2 + y ( I l , I ) * ( - X ) * * I l 35 CONTINUE tlp=DEXP(-ht*X**2) FF=0 Appendix A. FORTRAN PROGRAMS 124 IF (tlp.Eq.O) GO TO 37 FF=tlp*(DEXP(-s(I)*X)*tl+DEXP(s(I)*X)*t2) 37 RETURN END C FUNCTION sing(X,I) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(lOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint tl=0 DO 10 11=1,LIM2 tl=tl+WWT(Il)*q(XXT(Il),x,I) ! Do numerical cos t h e t a i n t e g r a t i o n 10 CONTINUE s i n g = t l RETURN END C FUNCTION q(u,X,I) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5) , .s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , Appendix A. FORTRAN PROGRAMS 125 .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint tl=dexp(-ht*x*x)*X**2 IF (tl.EQ.O) GO TO 38 t2=u-l tlp=dexp(s(I)*X*t2)*dsqrt(aO(l,I)+aO(2,I)*t2*X+aO(3,I)*X**2) .*dsqrt(bO(1,I)+b0(2,I)*t2*X+bO(3,I)*X**2) 38 q = t l * t l p RETURN END C FUNCTION VAMPL(Yp.bas) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint . , a l i , b l i , b 2 i , c l i , c 2 i , d l i , d 2 i , d 3 i , d 4 i , g l i . , V I ( 5 ) , c l l u , c l 2 u , c l 3 u , c l 4 u , c s l u , c s 2 u , c s 3 u , c s 4 u .,cl1,cl2,cl3,cl4,cs1,cs2,cs3,cs4,cpl,cp2,cp3 C C Main formulas f o r the amplitude as generated by SMP. C Includes s p i n summations, f l a v o u r and space matrix elements. C IF (Yp.EQ.2) GO TO 94 C Yp=Lambda Appendix A. FORTRAN PROGRAMS IF (bas.Eq.2) GD TO 92 C uds LAMBDA VAMPL = (1.DO\/18.D0)*((VI(1)*(18*A1I*CL3U*CP2+18*B1I* $ CL1U*CP2+18*CL2U*CP2*D1I+18*2 ** 0.5D0*A1I $ *CLlU*CPl+9*2 ** 0.5D0*AlI*CL3U*CP3+(-9)*2 $ ** 0.5D0*B1I*CL1U*CP3+18*2 ** 0.5D0*C1I*CL2U $ *CPl+(-9)*2 ** 0.5D0*CL2U*CP3*D1I)+VI(2)*(18* $ B2I*CLlU*CP2+18*CL2U*CP2*D2I+(-9)*2 ** $ 0.5D0*B2I*CLlU*CP3+(-9)*2 ** 0.5D0*CL2U*CP3* $ D2I)+VI(4)*(l8*CL2U*CP2*D4I+(-9)*2 ** 0.5D0* $ CL2U*CP3*D4I)+VI(3)*(18*CL2U*CP2*D3I+18*2 $ ** 0.5D0*C2I*CL2U*CPl+(-9)*2 ** 0.5D0*CL2U* $ CP3*D3I)+(-2)*CL4U*CP3*GlI*VI(5)) \/ 2 ** 0.5D0) GO TO 99 C SU(6) LAMBDA 92 T l = VI(2)*((-36)*B2I*CL1*CP2+18*CL2*CP3*D2I+18* $ CL3*CP2*D2I+(-18)*CL4*CP2*D2I+18*2 ** 0.5D0* $ B2I*CLl*CP3+(-18)*2 ** 0.5D0*CL2*CP2*D2I+(-9) $ *2 ** 0.5D0*CL3*CP3*D2I)+VI(4)*(18*CL2*CP3* $ D4I+18*CL3*CP2*D4I+(-18)*CL4*CP2*D4I+(-18) $ *2 ** 0.5D0*CL2*CP2*D4I+(-9)*2 ** 0.5D0+CL3* $ CP3*D4I) T2 = VI(3)*((-36)*C2I*CL2*CP1+18*CL2*CP3*D3I+18* $ CL3*CP2*D3I+(-18)*CL4*CP2*D3I+18*2 ** 0.5D0* $ C2I*CL3*CPl+(-18)*2 ** 0.5D0*CL2*CP2*D3I+(-9) $ *2 ** 0.5D0*CL3*CP3*D3I)+2 ** 0.5D0*(9*A1I*CL4 Appendix A. FORTRAN PROGRAMS 127 $ *CP3*Vl(l)+(-18)*ClI*CL4*CPl*VI(l)+(-18)*C2I* $ CL4*CP1*VI(3)+9*CL4*CP3*D1I*VI(1)+9*CL4*CP3* $ D2I*VI(2)+9*CL4*CP3*D3I*VI(3)+9*CL4*CP3*D4I* $ VI(4)+2*CL4*CP3*GlI*VT(5))+2*2 ** 0.5D0*CL3* $ CP3*G1I*VI(5) VAMPL = ((-1.DO\/36.D0))*((VI(l)*((-18)*AlI*CL2*CP3+(-18)* $ AlI*CL3*CP2+18*AlI*CL4*CP2+(-36)*BlI*CLl* $ CP2+(-36)*ClI*CL2*CPl+18*CL2*CP3*DlI+18* $ CL3*CP2*DlI+(-18)*CL4*CP2*DlI+(-36)*2 ** $ 0.5D0*AlI*CLl*CPl+(-18)*2 ** 0.5D0*A1I*CL2* $ CP2+(-9)*2 ** 0.5D0*A1I*CL3*CP3+18*2 ** 0.5D0* $ B1I*CL1*CP3+18*2 ** 0.5D0*ClI*CL3*CPl+(-18)* $ 2 ** 0.5D0*CL2*CP2*DlI+(-9)*2 ** 0.5D0*CL3*CP3 $ *D1I)+T1+T2) \/ 6 ** 0.5D0) GO TO 99 C Yp=sigma 0 94 IF (bas.EQ.2) GO TO 96 C SIGMA uds VAMPL = (1.D0\/6.D0)*((VI(1)*(2*A1I*CP2*CS3U+2*B1I*CP2 $ *CSlU+2*CP2*CS2U*DlI+2*2 ** 0.5D0*A1I*CP1 $ *CSlU+(-l)*2 ** 0.5D0*AlI*CP3*CS3U+2 ** 0.5D0 $ *BlI*CP3*CSlU+2*2 ** 0.5D0*ClI*CPl*CS2U+2 $ ** 0.5D0*CP3*CS2U*D1I)+VI(2)*(2*B2I*CP2*CS1U $ +2*CP2*CS2U*D2I+2 ** 0.5D0*B2I*CP3*CSlU+2 $ ** 0.5D0*CP3*CS2U*D2I)+VI(4)*(2*CP2*CS2U*D4I $ +2 ** 0.5D0*CP3*CS2U*D4I)+VI(3)*(2*CP2*CS2U* Appendix A. FORTRAN PROGRAMS $ D3I+2*2 ** 0.5D0*C2I*CPl*CS2U+2 ** 0.5D0*CP3* $ CS2U*D3I)+(-2)*CP3*CS4U*GlI*VI(5)) \/ (2 ** 0.5D0 $ *3 ** 0.5D0)) C type *,cpl,cp2,cp3 C \"tyP e *,cslu,cs2u,cs3u,cs4u C type *,VI(1) C t y p 6 *,Vampl GO TO 99 C SIGMA SU(6) 96 T l = VI(2)*((-4)*B2I*CP2*CSl+(-2)*CP2*CS3*D2I+2* $ CP2*CS4*D2I+(-2)*CP3*CS2*D2I+(-2)*2 ** 0.5D0 $ *B2I*CP3*CSl+(-2)*2 ** 0.5D0*CP2*CS2*D2I+(-l $ )*2 ** 0.5D0*CP3*CS3*D2I+2 ** 0.5D0*CP3*CS4* $ D2I) T2 = VI(4)*((-2)*CP2*CS3*D4I+2*CP2*CS4*D4I+(-2) $ *CP3*CS2*D4I+(-2)*2 ** 0.5D0*CP2*CS2*D4I+(-l $ )*2 ** 0.5D0*CP3*CS3*D4I+2 ** 0.5D0*CP3*CS4* $ D4I)+VI(5)*(2*2 ** 0.5D0*CP3*CS3*GlI+2*2 ** $ 0.5D0*CP3*CS4*GlI)+VI(3)*((-4)*C2I*CPl*CS2+( $ -2)*CP2*CS3*D3I+2*CP2*CS4*D3I+(-2)*CP3* $ CS2*D3I+(-2)*2 ** 0.5D0*C2I*CPl*CS3+2*2 ** $ 0.5D0*C2I*CPl*CS4+(-2)*2 ** 0.5D0*CP2*CS2*D3I $ +(-l)*2 ** 0.5D0*CP3*CS3*D3I+2 ** 0.5D0*CP3* $ CS4*D3I) VAMPL = ((-1.DO\/36.D0))*((VI(l)*(2*AlI*CP2*CS3+(-2)*AlI $ *CP2*CS4+2*AlI*CP3*CS2+(-4)*BlI*CP2*CSl+ Appendix A. FORTRAN PROGRAMS 129 $ (-4)*ClI*CPl*CS2+(-2)*CP2*CS3*DlI+2*CP2* $ CS4*DlI+(-2)*CP3*CS2*DlI+(-4)*2 ** 0.5D0*A1I $ *CPl*CSl+(-2)*2 ** 0.5D0*AlI*CP2*CS2+(-l)*2 $ ** 0.5D0*AlI*CP3*CS3+2 ** 0.5D0*AlI*CP3*CS4+( $ -2)*2 ** 0.5D0*BlI*CP3*CSl+(-2)*2 ** 0.5D0+C1I $ *CPl*CS3+2*2 ** 0.5D0*ClI*CPl*CS4+(-2)*2 ** $ 0.5D0*CP2*CS2*DlI+(-l)*2 ** 0.5D0*CP3*CS3*D1I $ +2 ** 0.5D0*CP3*CS4*D1I)+T1+T2) \/ 2 ** 0.5D0) 99 RETURN END C SUBROUTINE TRANS(Yp,bas) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint . , a l i , b l i , b 2 i , c l i , c 2 i , d l i , d 2 i , d 3 i , d 4 i , g l i . , V I ( 5 ) , c l l u , c l 2 u , c l 3 u , c l 4 u , c s l u , c s 2 u , c s 3 u , c s 4 u . , e l l , c l 2 , c l 3 , c l 4 , c s l , c s 2 , c s 3 , c s 4 , c p l , c p 2 , c p 3 .,A(6),I2(6),I3,L,I1, .gt,gs,gl,mp,mtb,msb,mlb,mst,ms,mu,f,b,sixj IF (Yp.Eq.2) GO TO 110 C Yp=Lambda mtb=mlb Appendix A. FORTRAN PROGRAMS gt=gl\/3*mu\/mp*dsqrt(6.OdO) f= (l.D0\/12.D0)*(2*3 ** 0.5D0*(2*CLl*CLlU+(-l)* $ CL2U+CL3+2 ** 0.5D0*CL2*CL2U+2 ** 0.5D0*CL2* $ CL3U)+2*3 ** 0.5D0*CL3*CL3U+2*6 ** 0.5D0*CL3 $ *CL4U+2*6 ** 0.5D0*CL4*CL4U+12 ** 0.5D0*CL2U $ *CL4+(-l)*12 ** 0.5D0*CL3U*CL4) f = l GO TO 120 C Yp=sigma 0 110 mtb=msb gt=gs\/3*mu\/mp*9*dsqrt(2.OdO) f= (l.D0\/12.D0)*(2*3 ** 0.5D0*(2*CS1*CS1U+CS2U* $ CS3+2 ** 0.5D0*CS2*CS2U+2 ** 0.5D0*CS2*CS3U)+ $ (-2)*3 ** 0.5D0*CS3*CS3U+2*6 ** 0.5D0*CS3* $ CS4U+2*6 ** 0.5D0*CS4*CS4U+(-l)*12 ** 0.5D0* $ CS2U*CS4+12 ** 0.5D0*CS3U*CS4) f = l 120 IF (bas.EQ.2) GO TO 130 C uds b a s i s mst=ms b=l f = l GO TO 140 C SU(6) b a s i s 130 mst=mu Appendix A. FORTRAN PROGRAMS 131 140 RETURN END C FUNCTION FAC(q) IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V IF (q.GE.O) GO TO 40 TYPE *,'Argument of F a c t o r i a l l e s s than zero!' GO TO 55 40 FAC=1 IF (q.LT.2) GO TO 55 DO 45 J=2,q FAC=FAC*J 45 CONTINUE 55 RETURN END C C Subroutine to determine the po i n t s and weights f o r e v a l u a t i n g the i n t e g r a l s . C SUBROUTINE INTEGRATE IMPLICIT REAL*8 (A-H,K,M-U,W-Z) IMPLICIT COMPLEX*16 V COMMON \/WXCOM\/ XX(lOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5), . s ( 2 ) , y ( 5 , 2 ) , J l , J 2 , .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint Appendix A. FORTRAN PROGRAMS 132 . , a l i , b l i , b 2 i , c l i , c 2 i , d l i , d 2 i , d 3 i , d 4 i , g l i . , V I ( 5 ) , c l l u , c l 2 u , c l 3 u , c l 4 u , c s l u , c s 2 u , c s 3 u , c s 4 u ., e l l , c l 2 , c l 3 , c l 4 , c s l , c s 2 , c s 3 , c s 4 , cpl,cp2,cp3 .,A(6),12(6),I3,L,I1 11=4 A(l)=.5 ! I n t e r v a l s f o r numerical cos t h e t a i n t e g r a t i o n I2(l)=16 A(2)=1.0 I2(2)=16 A(3)=1.5 I2(3)=16 A(4)=2.0 I2(4)=16 13=0 42 CALL SETWX(I1,I2,A,I3) LIM2=LIM DO 48,I1=1,LIM2 XXT(I1)=XX(I1) WWT(I1)=WW(I1) 48 CONTINUE C 11=6 A(l)=.25 ! I n t e r v a l s f o r numerical momentum i n t e g r a t i o n I2(l)=16 A(2)=.5 I2(2)=16 Appendix A. FORTRAN PROGRAMS A(3)=.75 I2(3)=16 A(4)=1.0 I2(4)=16 A(5)=1.25 I2(5)=16 A(6)=1.5 I2(6)=12 13=8 43 CALL SETWX(I1,I2,A,I3) RETURN END Appendix B Details of Integrals To evaluate the integrals used in chapter 3 we will need the following results [42] \/ A3sin(Ag)e 2 a A dA = v v * e 2 \u00b0 2 , 7o v 8a5 Using (B.300) we get ay= [Voooe'^A = - L \\ ^ \/ e ^ ^ ^ A Similarly ap \u2014 = j V>io0e,A\" \u00bb yd3A = 2 1 M \/ ( | _ a 2 A 2 ) e - ^ 2 ^ 2 + i X ^ d 3 A y\/lt J 2 (B.300) (B.301) (B.302) (B.303) (B.304) V3v\/4lr\"\\ 4fy^\\ \u00a7 ( ( ^ ) 2 3 V3V\" A \/ U \u00ab A ; 2 ; Similarly 6P = 134 Appendix B. Details of Integrals 135 Finally cY = J ^oioe*X'fy d3A ^ IaxXe-^lx2+iX^cos\\l^-cos9A2sin9dOdd\\ \\ 0 \\ y\/lt J V 47T Ai fy\/ir^ Similarly c\u201e = Angular integrals Ai (sfii^ -A qye K 3 \u2014 \u2014 9Pe ^ . a \\ a I When performing the angular integrals we will need the general formula (u = cos 9) f1 espu(ai + a2pu + a3p2)(6i + b2pu + b3p2)p2 du = \u00a3 (^p'e\"^ + (-p)''eps) Ju=\u20141 (B.305) where m = \u2014 ^ - + + \u2014 \u2014 , (B.306) _ axb2 a2bx 2a2b2 r}2 = + 5\u2014, (B.307) -ai6 3 a262 a2fe3 a3&! a362 773 = + \u2014 + - T - , (B.308) s s s* s sz _ a2b3 a3b2 _ a3b3 rji 1 , 775 = . (B.309) s s s All the angular integrals can be evaluated from this result by appropriate choice of the a,- and 6,-. Contour Integration Since the integrand of the second integral in equation (3.264) is an even function of u, r\u00b0\u00b0 1 G \/ f A i ) r\u00b0\u00b0 1 GJMI 2 x2 \u2022 du = ^ i - ^ l i m \/ . du.- (B.310) Jo r - A i - i e 2 0 J-00 u2 \u2014 \\{ \u2014 le Neglecting terms of 0(e)2 and defining Appendix B. Details of Integrals 136 we have f\u00b0\u00b0 1 fR 1 \/ ~~5 ~du = lira \/ \u2014 du. J-oo Ul \u2014 Af \u2014 It R-KX> J-R (u + z)(u \u2014 z) = y^[l(\u00b1\u00b1*fLdu_l ( H \u00b1 \u00a3 t l d u \\ (B .311) fl-+oo [ y c U \u2014 Z J C R U \u2014 Z J The contour C is to be closed on the top half of the Argand plane. Contour CR represents the semi-circle part of C. Since C is an anti-clockwise contour it is postive. From the Cauchy residue theorem [43] \/(*) i c (x \u2014 a) dx = 2mf(a), (B.312) we get 7T2 lim lim \u00a3 ^ ^\u2014 du \u2014 lim , \u2022 = \u2014 fl-oo \u00a3^ o Jc u - z t-+o 2 A a + f- A i We must now show that the integral over contour CR yields zero. It follows from the triangle inequality [44] that, \\u + z\\\\u-z\\>\\\\u\\-\\z\\\\2. On CR, \\U\\ = R and we get Jc, du < irR \\R-\\z\\\\2 TcR (u + z)(u - z) where irR is the length CR. The desired limit is now evident; that is, 1 lim f R\u2014>oo Jc R-*co JCR (u + z)(u \u2014 z) du = 0. Appendix C SMP Procedures The following commands define the wave functions and rules for evaluating the matrix elements within the SMP environment. They are contained within the file \"WAVEFN.DEF\". The following translations may be useful: p s i p = p s i l = *jv p s i s = N82ss= N^SS etc L2S= A 2 5 etc cr=xp c l = xA P l = A ps= L p r l = 4>PA p l l = \\ etc Ps000= P r l 2 0 f = $^oo etc. Where the \/ denotes final state; that is the spatial wave function for the Y. For the matrix elements: p l * p r p denotes (cf>A\\V+\\4>P) cr~2 denotes (x P\\ amplitude Ex[\/,]; \/* Expand l a s t expression (using d e f i n i t o n s ) Rat [\u00b0\/,] ; \/* R a t i o n a l i z e l a s t expression over a common denominator F a c [ % ] ; \/* F a c t o r i z e l a s t expression CbC'\/.^a.b.c.d.g}] \/* Combine c o e f f i c i e n t s i n l a s t expression 1\/2 a ( 6 c l l u c p l s f r + 3cl3u cp3 s f r + 3 2 c l 3 u cp2 s f r ) 1\/2 + b ( - 3 c l l u cp3 s f r + 3 2 c l l u cp2 s f r ) 1\/2 + d (-3cl2u cp3 s f r + 3 2 c l 2 u cp2 s f r ) 1\/2 Appendix C. SMP Procedures 142 + 6c c l 2 u c p l s f r + 2 c l 4 u cp3 g s f l #0[7]: 1\/2 6 2 p s i s p s i p ; \/* C a l c u l a t e amplitude Ex ['\/.]; Rat ['\/.] ; Fac ['\/,]; Cb[,\/.,{a,b,c,d,g}] 1\/2 - ( a (6cpl c s l u s f l - 3cp3 cs3u s f l + 3 2 cp2 cs3u s f l ) 1\/2 + b (3cp3 c s l u s f l + 3 2 cp2 c s l u s f l ) 1\/2 + d (3cp3 cs2u s f l + 3 2 cp2 cs2u s f l ) 1\/2 + 6c c p l cs2u s f l + 2 cp3 cs4u g s f r ) #0[12]: 1\/2 1\/2 6 2 3 Appendix C. SMP Procedures 143 uds:0; \/* Now do same t h i n g i n SU(6) b a s i s <\"wavefn.def\"; p s i s p s i p ; \/* f o r the Sigma Ex ['\/.]; Rat [*\/.] ; Fac ['\/.]; Cb[,\/.,{a,b,c,d,g}] - ( a (12cpl c s l s f l + 6cp2 cs2 s f l + 3cp3 cs3 s f l - 3cp3 cs4 s f l - 3 2 cp2 cs3 s f l + 3 2 cp2 cs4 s f l 1\/2 - 3 2 cp3 cs2 s f l ) 1\/2 + b (6cp3 c s l s f l + 6 2 cp2 c s l s f l ) 1\/2 + c (6cpl cs3 s f l - 6cpl cs4 s f l + 6 2 c p l cs2 s f l ) + d (6cp2 cs2 s f l + 3cp3 cs3 s f l - 3cp3 cs4 s f l 1\/2 1\/2 1\/2 1\/2 + 3 2 cp2 cs3 s f l - 3 2 cp2 cs4 s f l Appendix C. SMP Procedures 144 1\/2 + 3 2 cp3 cs2 s f l ) + g (2cp3 cs3 s f r + 2cp3 cs4 s f r ) ) #0[19]: 1\/2 36 2 p s i l p s i p ; \/* and f o r the Lambda Ex[\u00b0\/.]; Rat [Fac ['\/.] ] ; Cb[%,{a,b,c,d fg}] - ( a (-24cll c p l s f r - 12cl2 cp2 s f r - 6cl3 cp3 s f r + 6cl4 cp3 s f r 1\/2 1\/2 - 6 2 c l 2 cp3 s f r - 6 2 c l 3 cp2 s f r 1\/2 + 6 2 c l 4 cp2 s f r ) 1\/2 + b ( 1 2 c l l cp3 s f r - 12 2 e l l cp2 s f r ) + c (12cl3 c p l s f r - 12cl4 c p l s f r Appendix C. SMP Procedures 145 #0[25] 1\/2 - 12 2 c l 2 c p l s f r ) + d (-12cl2 cp2 s f r - 6cl3 cp3 s f r + 6cl4 cp3 s f r 1\/2 1\/2 + 6 2 c l 2 cp3 s f r + 6 2 c l 3 cp2 s f r 1\/2 - 6 2 c l 4 cp2 s f r ) + g (-4cl3 cp3 s f l - 4 c l 4 cp3 s f l ) ) 1\/2 24 6 Bibliography [1] D.A. Whitehouse et al, Phys. Rev. Lett, 63, 1352 (1989). [2] D. Flamm and F. Schoberl, \"Introduction to the Quark Model of Elementary Par-ticles, Vol I\", Gordon and Breach, Science Publishers Ltd., London, 1982. [3] O.W. Greenburg, Ann. Rev. Nucl. Part. Sci., 28, 327 (1978). [4] F.E. Close, \"Introduction to Quarks and Partons\", Academic Press, London, 1979. [5] N. Isgur in \"The New Aspects of Sub-Nuclear Physics\", ed. A. Zichichi, Plenum Press, New York, 1980 p 107. [6] F. Halzen and A.D. Martin, \"Quarks and Leptons\", John Wiley and Sons, New York, 1984. [7] R.K. Bhaduri \"Models of the Nucleon: from Quarks to Soliton.\", Addison-Wesley, Advanced Book Program, Redwood City, Calif., 1988. [8] D. Faiman and A.W. Hendry Phys. Rev., 173, 1720 (1968). [9] J.L. Powell and B. Crasemann, \"Quantum Mechanics\", Addison-Wesley Publish-ing Company, Reading, Mass., 1961. [10] Condon and Shortley, \"The Theory of Atomic Spectra\", Cambridge Univ. Press, New York, 1953. [11] S. Capstick and N. Isgur, Phys. Rev., D34, 2809 (1986). [12] G. Karl and E. Obryk, Nucl Phys., B8, 609 (1968). [13] A.R. Edmonds, \"Angular Momentum in Quantum Mechanics\", Princeton Univ. Press, Princeton, New Jersey, 1957. [14] Particle Data Group, Rev. Mod. Phys., 56, No 2, Part II S32 (1984). [15] R. Koniuk and N. Isgur, Phys. Rev., D21, 1868 (1980). [16] M. Brack and R.K. Bhaduri, Phys. Rev., D35, 3451 (1987). [17] N. Isgur and G. Karl, Phys. Rev., D19, 2653 (1979). 146 Bibliography 147 [18] For an interesting discussion on phases see Appendix D of, H.J. Lipkin, \"Lie Groups for Pedestrians \", North-Holland Publishing Company, Amsterdam, 1965. [19] D.C. Cheng and G.K. O'Neill \"Elementary Particle Physics\", Addison-Wesley Publishing Company, Reading, Mass., 1979. [20] J. Franklin, Phys. Rev., 172, 1807 (1968). [21] N. Isgur, G. Karl, and R. Koniuk, Phys. Rev. Lett, 41, 1269 (1978). [22] G. Derrick and J.M. Blatt, Nucl. Phys., 8, 310 (1958). [23] N. Isgur and G. Karl, Phys. Rev., D20, 1191 (1979). [24] N. Isgur and G. Karl, Phys. Rev., D18, 4187 (1978). [25] J.M. Eisenberg and W. Greiner, \"Nuclear Theory Vol. 2 : Excitation Mechanisms of the Nucleus\", North Holland Physics Publishing, Amsterdam, 1988. [26] J.D. Bjorken and S.D. Drell \"Relativistic Quantum Mechanics.\", McGraw Hill, New York, 1964. [27] M. Leon and H. Bethe, Phys. Rev., 127, 636 (1962). [28] I. Blomqvist and J.M. Laget, Nucl. Phys., A280, 405 (1977). [29] I.J.R. Aitchison and A.J.G. Hey \"Gauge Theories in Particle Physics.\", Graduate Student Series in Physics, Adam Hilger Ltd., Bristol, 1982. [30] Particle Data Group, Phys. Lett, B204, April 1984 [31] R.L. Workman and H.W. Fearing, Phys. Rev., D37, 3117 (1988). [32] J.W. Darewych, R. Koniuk and N. Isgur, Phys. Rev., D32, 1765 (1985). [33] D. Halliday and R. Resnick, \"Fundamentals of Physics - Third edition extended\", John Wiley and Sons, New York, 1988. [34] A. Le Yaouanc et al, Phys. Rev., D9, 2636 (1974). [35] D. Gromes in \"Baryon 1980 - Proceedings of the IVth International Conference on Baryon Resonances\", ed. N. Isgur, University of Toronto, 1980 p 195. [36] R. Koniuk, Nucl. Phys., B195, 452 (1982). [37] C. Becchi and G. Morpurgo, Phys. Rev., 149, 1284 (1966); R. Van Royen and V.F. Weisskopf, Nuovo Cimento, 50A, 617 (1967). Bibliography 148 [38] H.J. Lipkin, H.R. Rubinstein, and H. Stern, Phys. Rev., 161, 1502 (1967). [39] H. Burkhardt, J. Lowe, and A.S. Rosenthal, Nucl. Phys., A440, 653 (1985). [40] C.J. Batty in \"Low and Intermediate Energy Kaon-Nucleon Physics\", E. Ferrari and G. Violini editors, D. Reidel Publishing Company, Dordrecht, Holland 1981 p 223. [41] Y.S. Zhong, A.W. Thomas, B.K. Jennings, and R.C.Barrett, Phys. Rev., D38, 837 (1988). [42] I.S. Gradshteyn and I.M. Ryzhik, \"Tables of Integrals, Series, and Products\", Academic Press, New York, 1965. [43] M.R. Spiegel \"Advanced Calculus\", Schaum's Outline Series, McGraw-Hill Book Company, Singapore, 1963. [44] R.V Churchill, J.W. Brown, and R.F. Verhey \"Complex Variables and Applications - third edition\", McGraw-Hill Book Company, New York, 1974. ","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0084976","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Physics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/subject":[{"value":"Quark models","type":"literal","lang":"en"},{"value":"Branching processes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Nonrelativistic quark model calculation of the K-P --> [Lambda gamma] and K-P --> [Sigma]0[gamma] branching ratios","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/30167","type":"literal","lang":"en"}]}}*