UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Nonrelativistic quark model calculation of the K-P --> [Lambda gamma] and K-P --> [Sigma]0[gamma] branching… Murphy, Philip 1991

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1991_A6_7 M87.pdf [ 6.43MB ]
Metadata
JSON: 831-1.0084976.json
JSON-LD: 831-1.0084976-ld.json
RDF/XML (Pretty): 831-1.0084976-rdf.xml
RDF/JSON: 831-1.0084976-rdf.json
Turtle: 831-1.0084976-turtle.txt
N-Triples: 831-1.0084976-rdf-ntriples.txt
Original Record: 831-1.0084976-source.json
Full Text
831-1.0084976-fulltext.txt
Citation
831-1.0084976.ris

Full Text

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 A N D K-P -> S°7 B R A N C H I N G RATIOS. By Philip Murphy B. Sc. (Physics) University of Otago A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R O F SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES PHYSICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH COLUMBIA  February 1991 © 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 ° 7 is investigated using a nonrelativistic 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 E ° 7 but are somewhat high for the A 7 calculation.  11  Table of Contents  Abstract  ii  List of Tables  vi  List of Figures  vii  Acknowledgement  viii  1 Baryon Wave Functions i n the Nonrelativistic Quark M o d e 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  5  Zeroth Order Eigenstates 1.6.1  The case of Unbroken SU(Z)fi  6  1.6.2  The S = —1 Baryons when 577(3)// ,  avour  ot our  is Broken  1.7 Deviations from Harmonic Confinement 1.8 Flavour Wave Functions when Sl7(3)// „ a  1.8.1 1.9  13 17  our  is Unbroken  The uds Basis  18 23  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)//  46  2.4.2  When 5t7(3)//avur is Unbroken  2.5  2.6  2.7  owmr  is Broken  49  0  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  Derivation of the explicit form of V  57  2.6.1  Coordinate Representation  57  2.6.2  Momentum Representation  60  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 FORTRAN PROGRAMS  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 / m  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)  r  excited baryon compositions  r  (Q r) ;  lr  {r = p ov r — A). 10  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  2.2  Feynman diagrams  3.3  Strong vertices  operators  21 78 102  vn  Acknowledgement  Thanks to Harold Fearing. I appreciate his patience and generosity with his time.  viii  Chapter 1 Baryon Wave Functions i n the Nonrelativistic Quark M o d e l .  1.1  Introduction  When a kaon is captured in the Coulombfieldof 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°7r°  (0.27)  E+7T-  (0.19)  ATT  (0.07)  E° — •  0  7  (0.00144) (0.00086)  A 7  The experimental branching ratio for each channel is given in parenthesis. These reactions are interesting in that they provide information on meson-nucleon interactions. In addition the kaon, since it contains a strange quark, has strangeness S = — 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 —• A 7 and K~p —>• E ° 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 ° 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 ° 7 branching ratios to better than 1 in 10 . 4  In chapter 2 we extract from a gauge invariant set of Feynman diagrams an interaction 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 M o d e 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 differentflavours,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 inflavour.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  To form a baryon we combine three of these triplets. This gives a singlet  SU(3)coiour-  state, two octets and a decuplet [4], 3<8)3<8>3 = 1 0 8 © 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 confinement potential isflavourindependent (which is supported by studies of Q.C.D. on a lattice) means that the eigenstates of the zeroth order hamiltonian haveflavoursymmetry breaking only via explicit appearance of the quark masses in the kinetic energy part of H . 0  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, andflavourcoordinates, be symmetric. \qqq) — |colour),i x |space,spin,flavour)s A  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), V6  (1.2)  so we will suppress it henceforth. In the S = — 1 baryons the strange quark mass differs from the non-strange quark 1  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 §1.8.1. 1.5  SU(3) Symmetry  The set of the eight traceless, hermitian, Gell-Mann 3x3 matrices generate the unimodular, unitary group in three dimensions, denoted Si7(3). For the group Here S denotes the value of the strangeness quantum number for the state.  1  517(3)flavour  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  5  Here the it, d, and s quarks are  these operators act on the fundamental triplet,  W considered three different quantum states of the same particle. This is analogous to isospin symmetry for the case of S(7(2)//  . However in SU(3)fi  atKmr  avour  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: • m /m u  a  « 1. Here all quarks in the baryon are indistinguishable and it is appro-  priate to use the fully symmetrized "5(7(6) basis". • m /m u  a  « 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 symmetrization 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 BreitFermi Hamiltonian can be written (neglecting spin-orbit interactions) H = J2rrii + H + U + H . 0  (1.3)  hyp  t=i  mi is the mass of the i quark. HQ contains the kinetic energy and a harmonic oscillator th  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; Hh  yp  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 a is, a  H > = \ £ Z ^ H T ^ •^ )  +±[*  hyJ  S  i  ' l r  S  r  r  7  - S • $•]}.  i  Where r,j = |r} — rj| and r\- is the position vector of the i  th  5,- = |cr is the spin vector operator for the i  th  (1.4)  quark in the baryon.  quark in the baryon. oc is the effective s  quark-gluon coupling constant. This piece will be discussed further in §1.11. 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  SU(3)fi  T h e case o f U n b r o k e n  avour  In the NRQM the zeroth order basis states are generated by the hamiltonian, Ho = ^ - + ^ - + ^  2m  2m  x  + lKj2\r-:-r--\\ 2 t<j  2m  2  3  (1.5)  pi, r~i, mi are the momentum, position and mass of the i quark. th  In the 5 = 0 sector, or in 5 = —1 when 5t7(3)//  avour  is unbroken, the quark masses  in the baryon are taken to be identical. m  \ = 2 = 1^3 = u m  m  We assume the eigenstates are of the form, =^ r V I K ™  *(ri,r ,r ) 2  3  (1.6)  That is we are assuming the baryon is in an eigenstate of total energy.  In order  to separate out the centre of mass motion it is convenient to transform to Jacobi coordinates, ? =  7  1  rl)  vV*"* " '  X=  2rl)  76  {fl  +  r  " ~ 2  '  &  =  l  (rl  +  *  +  ^  (  L  7  )  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 7  p is antisymmetric with respect to interchange of the space coordinates of quarks 1 and 2 and is a representation of mixed permutation symmetry of type M , while A is p  symmetric with respect to interchange of the space coordinates of quarks 1 and 2 and corresponds to symmetry M\. By symmetry we mean with respect to quark positions: • S Fully symmetic. Exchanging any two quark positions gives the same state. • A Fully antisymmetic. Exchanging any two quark positions gives the same state times —1. • M The state transforms as p. That is antisymmetric with respect to exchange p  of quarks 1 and 2 but has no definite symmetry with respect to exchange of other quark pairs. We will denote this symmetry with a superscript p on the state. • M\ The state transforms as A. That is symmetric with respect to exchange of quarks 1 and 2 but has no definite symmetry with respect to exchange of other quark pairs. We will denote this with a superscript A on the state. Define M  3m m  =  u  •  m p, dX and X = dt ' u  —t  P\  —*  = m\X,  —•  PM C  (1.8)  3  —*  = MR  In the case of unbroken 517(3)//,'avour m\ = m . From equations (1.7) we get u  >  r-3 = - R - y | A  (1.9)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  *  p~2  8  —*  = Tn2r~2  substituting these relations into equation (1.5), we find H reduces to 0  p2  p2  o  (1.10)  2M '  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 H have the form, Q  e ™ $ (p,\)e -iEt  *(R,p,\) =  iP  (1.11)  R  NL  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 (1.12) for the p and A oscillators are identical in the S=0 sector where m Ri ma,. The zeroth u  order energy of a state is then specified by the quantum number N = 3m + (TV + 3)nw,  E  (1.13)  u  N  where N = N + N = (2n + l ) + (2n + / ). p  x  p  p  A  A  (1.14)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  9  The principal quantum number of the p, A oscillator, n \, takes on the values 0,1, • • • Pi  The orbital angular momentum quantum number of the p, A oscillator, l \, takes on Pt  the values 0,1, • • • The wave function of an oscillator, for example an r (where r = p or A) oscillator, is  [9],  Ki,m (ar) lr  = i^ (ar)y, /r  rmip  (n ),  (1-15)  P  where fl„ / (ar) = Af{arfLn^{oir)e-^ \  (1.16)  T  r r  a  = (mu)i  and Af =  (1.17)  +  r+  3  r  y,;  r +  r  (nr) = (-i) 'i / -m, (n ). mi  TO|p  r  r  P  (1.18)  !  are the spherical harmonics. We have used the convention  Yi (Q ) rTnir  " "' l)(„ / -!)...§ x§ 2  ^ ^F(„, l  [10]  that (1.19)  P  The L n (x) axe the associated Laguerre polynomials. Defined in terms of Binomial l  2  coefficients these are In+l+l\x  L :HX)= ^ ( - i r 1  m=0  The states, ip  nT  i ,, rm r  \  2  2m  . ml  n—m we require are listed in table 1.1.  The total spatial wave function consists of products of these A and p oscillator —•  —»  states. The orbital angular momentum L of the baryon is obtained by coupling l and p  h [11], L = i* + fx, p  (1.20)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  10  ^ooo(ar) = y ^ e - § ° y ( f t ) V  0 0  r  Table 1.1: Normalized linear harmonic oscillator states Vvirmi^ar') (r = /> or r = A).  h 0  L  0 0 0  i>oo(p)ipoo(^)  1 0  1 V'oiC^V'ooCA)  1  V>oo(£)^oi(A)  0 1 1  2 V'io(p)V'oo(A)  0 0 0  —*  0 0 0  0oo(p)0io(A)  1 1 0 [V>oiWoi(A)]  L=1  [V>oiWoi(A)] —+  i=2  1 1 1 1 1 2  V>o2(p)^do(A)  2 0 2  ^ 0 0 ( ^ ) ^ 0 2 (A)  0 2  2  Table 1.2: Product wave functions ^n i ( have omitted the m). [rp ij}] indicates coupling l 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. w e  T  p  T  L  Chapter  1.  Baryon Wave Functions  in the Nonrelativistic  Quark  11  Model.  To combine the harmonic oscillator wave functions to spatial states of definite permutation symmetry  ^^LM  w  e  L  take  linear combinations of the products  The well known prescriptions to multiply two mixed representations [12] are S = A A + P1P2, X  2  A  = Aip - /?iA ,  M"  = piA + \xp-i,  M  = AA -  2  2  2  x  X  pip . 2  2  The oscillator states must also be coupled to give states of good orbital angular momentum L. The total spatial wave function can be written as the linear combination, $NLM  Z  YJ  =  H  {h^ i ^x,mi \L,M)^ m  P  x  ip  nplpmip  (1.21)  nxlxmix  The first harmonic oscillator wave function in a product always denotes the p oscillator —•  —*  state, and the second the A oscillator state. We will suppress the p or A dependence from now on.  m; ; p  l\, mi \L, x  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 The e-§* <' 2  2+A2  = (-^) -^ (' >. 3  2  2+A2  e  ) is present in all the product wave functions. It is symmetric since p* +  A = i((rl 2  A  r f.+ 2  (ri - ftf  + {r 2  i=5))  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  12  is invariant under transposition of quark positions. We write = K/ m ^n,/,m (^) e-" 3  K ' p m i p *nxt™,  p  1  l A  a 2 (  '  (1.22)  , 2 + A 2 )  and the symmetry of the product depends only on the symmetry of  ^npipmi^nxixmi^  At JV = 1, L = 1 we can only have $11M  = V'OlMtV'OOO  $UM  =  L  l  ipoootpoiML  Since V'oiAfV'ooo ~  pY\M ($lp)  ^ooo^oiAfi ~  transforms as A under permutations. The states  L  WIM  L  which transforms as  L  p  under permutations. Similarly &UM  a n L  d  ®IIM  L  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  $200  =  U> ^; > »AlM)VW, V'01m, m  =  $200  =  =  ' p '  m  |  m  A  A  -^(V'onV'oi-i + V'oi-iV'on - V'oioV'oio)  E  2  V'npOO V'nxOO  ^(V'IOOV'OOO  =  $200  1  P  m  2  2  ^ " P ° °  —  V'oooV'ioo)  ^ » * « »  n ,n* p  =  —^(^'IOOV'OOO + V'oooV'ioo)  L =1 $211 ' =  X ! (l.raip^iTOljMWw^Olm,, TO,  P' A m,  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 13  =  ^ ( V ' o n V ' o i o  -  V'oioV'on)-  At L — 2 we can form $222  =  ^ ( ^ 0 2 2 ^ 0 0 0  + ^OOuVte)  $222  =  ^(V'cmV'ooo  -  $222  =  V"011^011-  ^000^022)  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 00  ,  $200  $200  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] should have A symmetry and ip should read L=1  2 2 p  [<£oi^oi]  L  =  2  -  The zeroth order eigenstates of HQ, in the S = 0 sector, are listed in table 1.3.  1.6.2  T h e 5 = —1 B a r y o n s w h e n SU(3)ji  is B r o k e n  avour  In this section we consider the case when m /m u  0.6. The analogue of equations  s  (1.7) with m ^ m are s  u  1 , 9  =  r ~  =  1  . _ O - A D rn (n + fj) + m r ~ ^ M ' where M = 2m + m . u  a  3  ^ ^ (1-24)  =  u  a  Rearranging we get , 2  (1.25)  ^3 = ^+4(^-3) V6 V  When SU(3)/iavour is a good symmetry and m, = m = mj = m we get m\ = m and (1.25) reduces to (1.9) 2  u  u  u  14  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  = ^ooo(£)V>ooo(A) = 1  $000 $IIM $ U M  a\Y (£l )  = &OO(£WOIML(A) =  L  = ^O\M {p)^OOQ(X)  L  L  =  1ML  2 ^  x  apY (ti ) 1ML  P  $200  = ^(^ooo(p)V>ioo(A) + V>ioo(£)V>ooo(A))  $200  = ^(^ioo(£)</>ooo(A) - </>ooo(£)</>ioo(A)) = >  2  ( ^  2  - ^>) 2  $200  = ^(V>ou(/d)V>oi-i(A) + V>oi-i(^)^ou(A) - ^oio(^)^oio(A))  $21±1  = ±^(^01±l(^)^01o(A) -  = ± f x/2aVA(r  $210  1±1  (n )r (fiA) p  Y (n )Y (a ))  10  10  p  1±1  x  = ^(V>on(p)V>oi-i(A*) - V>oi-i(p)V>ou(A*)) = f V ^ « V A ( F I I ( ^ ) F I - I ( ^ A ) - yi-i(^)Fn(«A))  $22A/  =  L  75(^0 A/ (p)^OOo(A) 2  t  = 2jf *\p*Y (tl ) 5  $22M  ^010(^)V'01±l(A))  L  2ML  + $ooo(p)i>02M (X)) L  +  p  \ Y (Sl )) 2  2ML  x  =  ^(^02Af (p)^000(A)-^000(^02W ,)(A)  =  2^aVWft )  L  i  p  - A F (Q )) 2  2Mt  A  = ?a pAyi±i(flp)yi±i(ftA) 2  $^ ±2  = &I±I(PAI±I(A*)  $22±i  = ^(^oio(p)^oi±i(A) + V>oi±i(£)V>oio(A))  $220  = f v/2^A(r (ft,,)y (ft ) + ri±i(n )y (ftA)) = ^g(^oi-i(p)^oii(A) + ^on(/5)^oi-i(A) + 2V'oio(p)V'oio(A))  2  10  =  f  1±1  our  10  ^aVA(ri_i(n )yii(ftA) + * i i ( « , ) * i - i ( f t A ) p  Table 1.3: Baryon space wave functions 5(7(3)// „ is unbroken. a  P  A  $ LM N  L  2F (^)y (o ))  +  10  10  = $NLM /((^) L  3  E  ~*  A  A  2  (  P  2  +  A  2  )  )  W  H  E  N  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  •  _^  -  -  =  (B  I  —*  .  P  ^  =  2  m 2 r 2 = m u  _  m  A N N  —+  (fl+_+_(_))  ^  .  A  .  P3 = rn r = m (i£ + —=( 3  /  = m u (ie+ - ^ + - ^ ( - ) )  m i r i  •  =*p  15  3  s  V6  m m  3)).  x  u  Now the harmonic oscillator hamiltonian, equation (1.5), becomes  where PCM = Mi2; and P = m p and P A = m>A. P  u  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 » a = (ZKm fl\ x  (1.27)  x  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, u  Jfi , w = J m\* V rn V \ K  =  K  1.28  A  u  3 andE  N  =  3  2m + m + (N +-)huj + (N +-)hu u  s  p  x  (1.29)  x  Here u < UJ since m < m . x  u  s  Because of this frequency splitting, the three states $fo> $2oo» $200 which in the 0  degenerate case (5 = 0) have permutation symmetry S,M , M respectively, break into P  x  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 N = 2. Although the x  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 ff in the 0  S = — 1 sector for the case of broken SU(3)ji  avour  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/(A) = 2y^aAAFiM (n )  $ AA v  200  = ^oooWioo(A) = y | a  200  = VWp>000(A) = Vl (2^ - /=>)  v  $>P *200  l  L  2  (^-A ) 2  A  a2  X  A  2  ^(^oii(^)^oi-i(A) + V>oi-i(£)V'oii(A) - ^oio(^)^oio(A)) = ^ a c A p A C F n ^ ^ - x C ^ ) + yi_i(fl„)r(ft) - M ^ o ^ ) ) =  n  *21±1  = ±;^(V>oi±i(£)V>oio(A) - ^oio(p)^oi±i(A)) = ± f v^«a^A(F (Q )r (fiA) - r (ft„)y (ft )) 1±1  3>P 210  X  V  ^22^  10  p  1±1  10  A  = ^(^on(£)V'oi-i(A) - $oi-i(p)$oii(A)) = f ^aa pA(y (ft )y _ (n ) - ri-iCft^rnCfiA)) v  AAA  A  A  p  11  1  = ^000W02M (A) =  1  A  ^« A F 2  L  2  A  2 M i  (^A)  S>PP = ^02M(p)^ooo(A) = ^ a V l W ^ p ) t  *22±2  = ^oi±i(p)^oi±i(A) =  g>P *22±1  = ^(V'oio(/o)V'oi±i(A) + ^oi±i(p)^oio(A)) = f v5aa pA(y (fi )y (n ) + y i(fi )y (fi ))  X  A  &>P ^220  X  10  faa pXY (il )Y (Q ) x  p  1±1  A  1±1  p  1±  1±1  p  x  10  A  = ^(^oi-i(£)V>oii(A) + ^on(/o)V'oi-i(A) + 2^oio(/o)^oio(A)) = fy/laa^pXiY^Q^iSlx) + Y (n )Y . (Q ) + 2Y (Q )Y (Q )) ll  p  Table 1.4: Baryon space wave functions $ # £ ^ the S = — 1 sector when SU(3)fi is broken. L  avour  1 l  =  x  $NLMj((^)  W  3 / 2 e  P  W  ~*  X  ( a V +  '  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, infirstorder 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. =  3m + 3u> + a = 1135 MeV u  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) supermultiplets : 3  (notation L : Lis the total orbital angular momentum of the state; a the symmetry of a  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 supermultiplets 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 (§1.11) that Hh will mix yp  these states. A supermultiplet contains states of various n , n\, l , l\ that combine to N,L and symmetry type (M,A,S) 3  p  p  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  J  B  p  u 1/2+ d 1/2+ a 1/2+  Y  S  1/3 2/3 1/2 1/2 1/3 1/3 -1/3 1/2 -1/2 1/3 1/3 -1/3 0 0 -2/3  Q/e  I  I  0 0 -1  z  18  Table 1.5: Quantum numbers for the u, d and s quarks. 1.8  Flavour Wave Functions when SU(3)fi  avour  is Unbroken  Consider the case of a baryon which consists of three quarks. Each of which may have one of theflavoursu,d, or s. The 3 = 27 states decompose into the irreducible 3  representations [4] of 5l7(3)//  .,  ouotlT  3 eg) 3 (8) 3 = 1 0 8 A  M p  88  M a  (1.30)  © io , s  where the symmetry of the representation is indicated by a subscript. This is the same decomposition into multiplets as for the SU(3) i  co our  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  =  £  (h,Izx;h, Irf|Ii2, £ 1 2 ) ^ 1 2 ; h, I*\I,\h,I i)\h, z  1,2)\h, 1,3)  (1.31) For example using table 1.5 we get that \I,L?  = \\,W  11.11  =  ( 2 ' 2 ' 2 ' 2 |1,!)(!> Ij  +  (  2 ' 2 I 2' 2•)l«)|t0|e9  |,|;|,-||l,0)(l,0;|,|||,J)|«)|d)|«>  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  +  19  (l,_l l,||l,0)(l,0;|,||i,|)|d)|u)|u), ;  therefore |f, §)* = —\=(udu + duu - 2uud) = <£ . V6  (1.32)  A  The A superscript indicates that this state has M\ symmetry. It is a member of the M\ SU(3)fi our octet. This symmetry arises from choosing the symmetric intermediate av  isospin state when combining the isospin wave functions. Alternatively, IM)  0  =  (i,|;|.-|lo.o)(o.°i|.|l|.l>l >l >l«>  +  (J,-i;§,§|0,0>(0,0;l l|| §)|d)|u)|u)  =  u  |  d  >  ~y^( ^ " du)u = <f>p.  (1.33)  uc  This state has M symmetry and is a member of the M SU(3)fi our octet. p  p  aV  The zeroth order eigenstate for the physical proton contains a mixture of the fiavour states  and </> combined to give a totally symmetric space-spin-flavour wave function. A  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, <f>^ = -^(sdu — sud + usd — dsu + dus — uds). V6  (1-34)  All the states in the M\ or the M octets can be generated by the application of the p  SU(3)flavour raising and lowering operators [2], U±, I±, V±, on to one member of the SU(3)flavour multiplet. These operators act on states which are SU(2)fi our subgroups aV  of 5'l7(3)// „ a  our  (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.\d) = 0,  V_|u) = 0,  V \s) = 0, +  V.\s) = \u).  When acting on a three particle wave function  \q1q2Q3),  such as a Baryon, I±, U±, V±  act on each quark in turn, Jhiqiqa) = J(ki»l92«3> + \qi)J(\q2))\q ) + k i ^ d f e ) ) 3  (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±\l  I) = y/J(J + l)-J,(J*±l)\l  J* ± 1)  (1-36)  Here J and J represent the total J and its z component for the state. The square z  root factor in (1.36) is y/2 when acting on states within an SU(2)fi  avour  acting on states within an SU{2)ji  av0UT  triplet; 1 when  doublet; and of course zero when stepping out of  an SU(3)favour multiplet. So U- acting on theflavourwave function  gives the state  <f>z+. The other members of the S isospin triplet can then be generated by applying J_ [18]. U~(f>p =  — U-(—\=(udu + duu - 2uud))  v6  — —/=(usu + suu — 2uus)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 21  Y  I<V  1/3 4  \ -2/3 "  -1—  -1— 1/2  -1/2  (a)  u  H  u  +  l-<  v  v_ u  +  ^  ft  ,ir\  /A  v  +  ~i -»  v_  +  X  u_  u_ v u_  +  +  1  1 o  1  >l+  r 1  (b)  Figure 1.1: Illustration of the action of the SU (3) flavour raising and lowering operators in the I — Y diagram (Y = B + S) (a) of the quark triplet and (b) of the baryon octet. z  22  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  =  j=(dsu + usd + sdu + sud — 2dus — 2uds) V6  y/12  1 \/l2  =>• ^>2_  (1.38)  (sdu + sud + dsu -f- w.sd — 2(c?M5 + uds)),  (sdd + sdd + dsd + dsd — 2(dds + cMs))  (1.39)  = —^=(sdd + dsd — 2dds). V6  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, U = 1) and the S° z  (17 = 1, U = — 1) is a linear combination of the isospin eigenstates S° and A, z  \U = 1,U = 0) = « | ^ ) + /3|^> 0  Z  U-\U = 1,U, = 1) = U-t* =  v5\U = l,U = 0) = v5(a\<f>'o) + p\<f> )) (1.40) x  z  A  U-(^ (dud + udd-2ddu)) T  V6  (sud + dus + usd + uds — 2sdu — 2dsu) (1.41)  x I+U-ifc = I+y/2(a\fa) + fi\fi)) = 2a<j>  (1.42)  because A is an isospin singlet. I commutes with £/_. This can be seen from the matrix representation[19] of the +  raising and lowering operators which act on the SU(3)ji  avour  triplet  d \  l  0 10\  1  s  I  0 0 0 ^  0 00  0 00  0 00  0 10  (1.43)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  I+U-ti  =  U-I+4>  x  =  U—<f>p =  Comparing (1.42) and (1.44)  23  (1.44)  fa  a=  (1.45)  Now since |a| + |/?| = 1 we get |/?| = § 2  2  2  /* = ± ^  (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  - * »  =  72^°  +  /l^'  (L47)  This is the centre member of the U-spin triplet. From (1.38) and (1.41), => <f>\ = 7:( d + usd — sdu — dsn).  (1-48)  su  Similarly for the M octet we can generate all the states p  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 appropriate 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  <t>  p x  p  — -^{udu + duu — 2uud)  n  -^(dud + udd — 2ddu)  -^{udd — dud)  — -^(usu + suu — 2uus)  -^(usu — suu)  E  +  E°  — -j=(sdu + sud + dsu + usd —2(dus + uds))  E  -  -^{udu — duu)  -^(sdd+ dsd- 2dds)  (dsu + usd — sud — sdu)  2  -^(dsd - sdd)  A  ^(sud + usd — sdu — dsu)  -^=(sdu — sud + usd — dsu — 2(dus — uds))  E~  -^(dss + sds — 2ssd)  -^(dss — sds)  E°  -^(uss + sus — 2ssu)  -^(uss — sus)  Table 1.6: Flavour wave functions in the baryon octet. The physical particle X is some mixture of the flavour states <f>x and <j>x to be determined later.  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  A  + +  A  +  -^(duu + udu + uud)  A  0  ^(ddu + dud + udd)  25  uuu  A"  ddd  1=1 E  -^(uus + usu + suu)  +  E°  -^(dus + uds + dsu + usd + sdu -f- sud)  E~  ^(^ds + dsd + sdd) I 2  •^(ssu + sus + uss) •^(ssd + sds + dss) 1=0 (7~  sss  Table 1.7: Fully symmetrizedflavourwave functions in the baryon decuplet. basis the only twoflavourstates are, <f>\ and  = =  -X=(ud — du)s, v2  (1-49)  —~(ud + du)s. v2  (1.50)  <^A has the u and d quarks coupling to isospin 7 = 0. <f>\ therefore corresponds to the flavour wave function of a A particle, 1.9  has 1 = 1 and so corresponds to a E°.  Spin Wave Functions  Three spin | particles (for example quarks) may give rise to a total spin S = | or 5 = 4  | . As in the case offlavour,the spin wave function for a proton can be obtained by 4  In 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 M type symmetry. p  For the spin up state  x\ = \\,W = <J,§;i,ili,i>(i,i;i,-JI§,J>IT>IT)U> h  + (§ J;J,-||i,0)(i 0;| ||i |>|T)I1)IT> l  1  l  + (|,-J;|,J|i,o>(i,0;i,i|i,|)U>|T>IT> therefore l , i = " ^ ( U T + ITT "2 TTD-  (1-51)  X  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><o.o;MI|.i>IT>li)IT> + (|,-|;l4lo,o>(o,o |,||i,i)U)|T>IT) ;  = ^(U-IT)T-  (1-52)  This is analogous to theflavourwave function <^£. Also, x§,f  =  (I  b I  1 1 1 . i>(i.  i ; h i\l  1)1 T)l T)l T) =  I TTT)  etc.  (1.53)  States of different ms follow from the Condon-Shortley convention. Note that for the case of coupling three spin | particles together (5C/(2) , ), the restriction to two possp n  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 ® 2 ® 2 = 4  S  © 2  M i  © 2 . Mp  (1.54)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  1.10  27  Combining Flavour, Spin and Spatial Wave Functions.  Each quark not only has 3flavourtypes but two possible spin directions. We combine the  SU(2) i  ap n  and the  multiplet structure into SU(Q)SF (SF ^spin-  SU(3)fi  avour  flavour) multiplets. For baryons, Young diagram techniques[4] give, 6 Cg) 6 ® 6 = 5 6 © 7 0 s  M a  ©70  M p  ® 20 . A  (1.55)  In the SU(6) basis we wish to construct states that are totally symmetric in space, spin andflavourso 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 andflavouronly with respect to quarks one and two. 1.10.1  Permutation Group A d d i t i o n Coefficients  To combine wave functions of different permutation symmetry it is convenient to introduce 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 P and P respectively, to permutation a  2  symmetry P then,  \ M,«2  \  Ki  K.2  where Ki = <  K  J  1 if P = {  A,S,M  X  2 if Pi = M . p  The non-zero coefficients with phase factors chosen such that they are all real are,  28  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  = 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. \XF»L J ) P  0  =  £  Pi P  £  Kl,K2,*3  M ,Ms  y  L  Ki  { a  2  K2  K\2  Pl,*l  a  J  J,P2,K2 K2  x (S,M L,M \J,Mj) ?^ <f>9' m Ms]  L  For our purposes we will need the J = | p  x  +  S  3  (1.57)  s  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°),  5 5 5\ 1 1 1  <i|;0,o|| i)  $oooXi  i<f>x  $oooXi  i<f>x  2'2  (1.58)  -4$o oo(xii^ + x i i ^ ) 5  V2 1.10.3  2  '  2  2  '  2  The 517(6) Excited States  Radial Excitations At N = 2 we have, Pi  P  5 \ /5 5 5  «2  1/  2  KI ,*2  V1 1 1 /  <|.|;0,o||,|)  2 '2  V2  2  '  2  2  (1.59)  2  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, $ooo/  l-^swl ) -  £  «1 .«2,«3  D  D  A/f \  /  M  \ 12 K  M S K  3  1  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.30  /I l . n fill l \ 6 ' 3 f t . ' « i i f t , K 2 \ 2 ' 2 > ' l 2 ' 2/*200 X l 1 <PN M  v X  K  v  U  U  2 '2  = ^{$oo(xl M -xiijx) + *WxIi^i + xi i^)}. A  2  2>2  /  2'2  2'2  2'2  (1.60)  For the SU(Z)flavour decuplet (of which out of X only S ° is a member)  «i.«3 \  x  PI  S PI12  Ki  1  Ki  /P  M S \  12  2  <M;0,o|§,|)^b xr r 4 K3  M  i  i  M S  i  1 1 1 /  1 $200  <£xXl 1$200 2'2  (1.61)  - 7 i * 5 ( x L * i » + xii*Soo)V2 2'2 2  For the SU(3)fi r  x  2'2  + =  s  <i>xX\  2  singlet irreducible representation,  av0U  Pi \X?SM\ ) +  « 1 .«3  V  A P 1  «1  l P  1 2  M  X2  «12 J^ «12  K  3  5 1  x (},i;o,oii i>*5jir»x?r^ >  2'2  '  M  1  +  /  M  I  2  A  M \ / M  M  S  1  1  M  M  S  2  2  1  1 1 A M \ 1  2  /  /  V  4>XXl  2'2  **X'l  1$200  1$200 2'2  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.31  (1.62)  ^xixiA-xlA)'  y/2  2 2  2  2  Orbital Excitations We combine L = 2, 5 = § states to give J = | . In this case the Clebsch-Gordan p  +  coefficients are not all unity. 1  A  +  ^  ^  Pi  P*  '  M  M 5  M «12  x (l,M ;2,M |i,i)$Xx| ,^?" 1  5  1  t2  L  +  - 4 x f i ( * 2 W x + *§2otfr) V5 '  -  ^ x f . - j C f c ^ + fcWi)  +  y|xf_|(*2 22^  2  «3  2  A  +  (1.63)  $222^)}.  For the 5L7(3)// vour decuplet, 0  4  0  Z  s  5 5 5  1 1 1  1 1 1  s  l^ ^f > = E ML,Ms  s  x (|,M ;2,M |i,i)$f s  L  2 M t  xf  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 (§1.11), and so these orbital excitations will not contribute to the wave functions we use. We list them for completeness. 1.10.4  T h e 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 $o and flavour wave 00  function (j>\ (in the uds basis). It therefore must combine with a x spin wave function p  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 a is, a  i<j  Where  m  »  m  j  °  "ij  ij  r  = |rj — fj\, Si = \o is the spin vector operator for the i quark in the baryon. th  a is the effective quark-gluon coupling constant, analogous to the electromagnetic cous  pling constant a = 1/137, and is determined by calculating the TV — A mass difference. a depends on how large an oscillator basis is chosen[7], we take up to N = 2 states. s  The spin-orbit part of the potential has been neglected. Since it has been found experimentally that spin-orbit effects are reduced to a level of less than 10% of naive  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  5)  2  IA  = ^OOO^AXI i  t  h  e  A  ground state  2'2  = *&*Axi i  |A S A) 2  P  2'2  \^ s ) = *5WAX1 i 2  PP  2 '2  |A 5AA> = $ O V A X ! i 2  A  2 '2 4  |A A,A>  = E (2, m; | , (I - m)||, m  | E S ) = ^ooo^sXi i 2  t  n  e s  |><OAX| i_ i(  m)  ground state  2'2 | S  S ,  2  *ft>*EXl 1  )  =  >  = ^Sofoxl i  A  2 '2 | £  S „  2  P  2'2  |S 5AA) =  i  2  2 '2  \X D ) = £ ( 2 , m; | , (i - m)|§, | ) $ « E X | , i _ 4  PP  4  |£ DAA)  m  (  = E <2, m; | , (§ - m)|§, | ) $ ^ s x f i _ m  i (  m )  m )  Table 18: The uds basis states we will need, combined so that they have M symmetry. Notation \X L„ ). x  2S+1  ia3  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  34  expectations of one-gluon-exchange [17]. Isgur and Karl speculate that Thomas precession 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 J . p  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 = 0 o r 5 = —1 sectors (1.65) can be written *K  where Hg  p  = £ ( ! " (1 ~ x)(S + S ))Hi  W  ^ f %{y S • 5 ^ )  and x =  ja  u  +^  (1.66)  J yp  '^  ' « - S< • Sj}} r  m /m . u  (1.67) (1.68)  s  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, Hh  yv  will lead to mixing within a given 5(7(6) multiplet and also  between different 5(7(6) multiplets with the same isospin and J [21]. p  In the 5(7(6) basis x = 1, so the hyperfine matrix elements between the states <f> and <f> , take the form: n  ff^  (6)  = (^  ( 6 )  (1,2,3)|«, +  + i f £ ) | < C ( l , 2,3)). (6)  p  m  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model. 35  (1,2,3) denotes quark positions. In particular,  (<eU, 2 , 3 ) | i ^ > f (>(1, 2,3)) ( 6 )  3,2)|£&|*f< >(l, ,2))  =  6  =  fl  3  (-l) (^ a  ( e )  (l,2,3)|Jg |^ )(l,2,3)) 6  p  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\l .  Similarly we get H % = Jf£ . Therefore 2  p  2  h  p  fl^f^l,  <^)(1,2,3)| £  2,3))  = 3(^< >(1,2,3)| J ^ 6  5U(6) 0  (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»  =  (i-7i) 4rv ry  3  with* =  (1.72)  We also need the relations: p  (x JSi • S2\X m.) p  = -ls , ., m  m  (1.73)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  (xi\Si • S2\xl)  =  36  (1.74)  -6 >, m<m  a (*L\6\p)\*loo) = V>ooo(0)</>ooo(0) = Try/ft'  (1.75)  y/Z a  3  (1.76) (1.77) (1.78)  The relevant hyperfine matrix elements are (1.79)  (XgSs\H \X^S )  =  {X^Ss'\H \XgSs}  =~y~^'  (1.80)  (X^SM\Hh \X^Ss)  = -^7~S->  (1.81)  hyp  s  hyp  yp  --6,  {A?S \H \AiS ) = 0, M  Hyp  (1.82)  s  (1.83)  (EwSM\Hhyp\%ioSs) = 0,  where X = JV, A, S°. 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 function, as a basis and taking 6 = 260 MeV [23] we get, for the nulceon hyperfine mixing matrix,  / 1005 138.5 132.8 138.5 Y 132.8 132.5  1405  \  No  \ (1.84)  JV(1405)  0  U 1705 1YU0 j \ JV(1705) j 0  JV represents the zeroth order ground state nucleon wave function, iJVg ^). 2  0  We have used the results 1005 =  1135+(N S \H \N S ) 2  8  2  s  hyp  8  s  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  X X£Ss> X£Ss -0.17 JV(1405) -0.99 -0.94 JV(1705) +0.15 A(1555) +0.99 +0.02 +0.09 +0.30 A(1740) A(1860) -0.04 +0.91 E(1640) -0.97 -0.23 S(1910) -0.20 +0.91 E(1995) -0.08 +0.27  37  Xi Ss  M  0  M  — —  —  +0.10 -0.95 +0.29  — —  —  +0.08 +0.17 -0.23  — —  —  Table 1.9: Excited baryon compositions, in the SU(6) basis (from ref. [17]), adjusted to our phase convention. ~0.99(X S '\H \X S )-0.l7(X S \H \X Ss)  138.5 =  2  2  S  132.8 = +0.15(X  hyp  2  2  M  s  hyp  S >\H \X S )-0M{X S \H\X S )  2  2  8  s  hyp  2  2  M  s  s  The matrix in (1.84) has 7V(941) = +0.95iV 5 + 0.25N S > + 0.20N S 2  8  2  5  S  (1.85)  2  S  M  as its lowest eigenvalue (in MeV) and corresponding eigenvector. Similarly for the A and E° in the SU(6) basis we obtain A(957) = +0.97A 5 + 0.18A S ' + 0.16A 5 - 0.01A?S ,  (1-86)  E(957) = +0.97E 5 + 0.17E 5 ' + 0 . 1 7 £ S - 0.00£ SW.  (1.87)  2  2  8  5  2  5  2  M  2  5  2  5  M  2  M  0  The low value for the mass is a result of neglecting the mass difference between the two light quarks and the strange quark. Hyperfine M i x i n g i n the uds Basis  1.11.2  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) #n m  = (#f(l, 2,3)\(Hll  p  + x{H\l + H £ ) ) l C ( l , 2,3)) s  p  p  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  2,3)\(HH + x{P$ H% P  =  p  3  23  + P} Hl P ))\<f>» (l, 2  3  yp  ds  13  38  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) (tf(l,2,3)K |^(l,2,3)) 2  3  p  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 ^ ^ 1 3 ) 1 ^ ( 1 , 2 , 3 ) )  (1.89)  uds Basis Compositions The effects of a ^ a\ are small in the S = — 1 sector [24] and are neglected in the matrix elements of Hh . We will need to know how a function F (say) behaves under yp  the permutation operator P . It can be easily verified that 13  PizF"  Pi F  =  pp  =  PF  =  3  X  13  PF  =  XX  13  1  \/3\  (1.90)  2 ~T > P  X  -PP--XX-  ^  A  (1.91) (1.92)  1„  3  ^ \ ~2~ P  •±  t  (1.93)  Along with the relations <*2ool* (#l$ooo) = 0, 3  «l* (^oo> 3  = ^1^,  (^ool<5(A)l$ooo) = 0, 3  (1.94) (1-95) (1.96)  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  39  we get for the uds hyperfine matrix elements (1.97)  (A S\H\A S) 2  2  (1.98)  (A S \H\A S) 2  2  PP  = o,  (A S \H\A S) 2  2  XX  -x—6,  (A S \H\A S) 2  (1.99)  2  PX  (1.100)  (E 5|/f|S 5)  (1.101)  (£X|/f|£ S)  (1.102)  2  2  2  (£ S\\\H\H S)  -x^6,  (1.103)  (Z S \H\I: S)  —x-^-S.  (1.104)  2  2  2  2  4  PX  Due to the higher mass of the strange quark the zeroth-order energy (1135 MeV) now becomes  5  1135 + m - m - (1 - x)J%- = 1295 MeV a  u  Using the N = 2 wave functions for the A and E ° in the uds basis (table 1.10) we can construct the mixing matrix. When diagonalized this gives , 6  A(1113) » 0.95A 5 + 0.07A 5AA + 0.28A 5 + 0.08A S , 2  2  2  S°(1209)  «  +0.98E 5 + 0.17S 5AA + 0.02S 5 2  2  (1.105)  2  pp  pA  2  + 0.11S 5 A.  (1.106)  2  pp  p  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 —• Ay and the K~p —• E°7 branching ratios. A quark in a harmonic oscillator potential has ground state energy ^ = ^tiu = (H — 1) which gives p3 = a. Reference uses a = 0.32, m = 0.33, m, = 0.55 (GeV) but they point out that if m, — m is increased to 0.28GeV with a corresponding increase in a to 0.41 there is little change in the compositions. 5  6  u  u  Chapter 1. Baryon Wave Functions in the Nonrelativistic Quark Model.  X  A(1555) A(1740) A(1860) E(1640) E(1910) E(1995)  XS x xs 2  -0.75 +0.56 +0.34 -0.84 +0.23 +0.19  XS\ 2  2  X  40  pp  -0.66 -0.69 -0.28 -0.53 -0.51 -0.31  P  -0.09 +0.46 -0.85 -0.11 +0.56 +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 —• Y-y within the N.Q.M., where Y represents A or E°. 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 £°. 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 appropriate to develop a nonrelativistic operator from the K~  +  u  —>  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, S  = (*Y\Y,V \* )  (2.107)  (i)  Yp  i=i  P  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 i quark. th  41  Chapter 2. Theory  2.2  42  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, S  YP  = j(*y|rl F f3,*)(ri,ri,r3,t|(VW l  + V<> + V<>)|r* f* 2  ai  t')  3  x (^r^r^'l^)^^^  (2.108)  where rj is the position vector of the i quark in the Y; f- is the position vector of the i th  th  quark in the proton; and the interaction in coordinate representation, (f\, f ,  r3,t\(V^+  2  y( ) -|_ V^)\f(, f , r ,t'), is a sum of single quark transition operators 2  2  {f r2,r ,t\(V^ u  3  + V™ + y( ))|r ',r ',r ',0 = VM(? r>[,tJ)Pft  - r%)8 (f - f ')  3  3  1  2  3  3  u  3  *. O ^ i - ri)S (r  3  +  V<Va.  +  V&(r , r ', t, 0 * ( n - f^)S (f - r ').  3  3  3  3  - r ')  3  3  3  2  2  5(7(6) Basis. When we use the 5(7(6), or fully antisymmetrized, description of the baryon wave functions we can use symmetry considerations to simplify the interaction. Using the condensed notation,  tf(r.) =  *(Fi,f ,r ,0, 2  3  dr = d r\ d f d r dt, 3  3  3  2  3  and dr' = d ^ d 7 d% dt', 3  3  2  we have S  YP  =  /Mr.OfVWCn.^t.O^-^^-rS)  Chapter 2. Theory  43  +  W\r ,  f* t, O ^ C r i - ri)6 (f - r ') 3  2  3  3  + V< >(r , rg, *, f ) * V i - r ? ) ^ - r* ')} 3  3  a  x * (r •) dr dr'. p  In particular M r i , r a , r ^ * ) V W ( f i , r V t t ' ) A r a " ^ ( r - r * ) * , ^ , r ' , r ' , f ) Jrc/r'  / =  3  1  J M ^ , r l  1  >  3  2  3  Fa,0V«(i a.* ,t,O« (* i - ^ ( r , - ^ ^ ( r j . ^ f g , f ) ?  f  ? ,  S  drdr',  i  a  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 «-• r and f{ «-+ r in the last expression we get 2  (-1)  y {^,r ,r ,t)V^\r ,r^^  J  2  2  Y  2  3  2  J * y ( r i , r 2 , r ,*)VW(r1.  =  *, t')* (r - r ')£ (r - r%)9 (f{, r ', r \ f) Jr dr'. 3  3  2  3  2  3  p  2  3  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 S  YP  = / ^ ( r i H V t V i . ^ t . O ^ - W r s - r g )  (2.109)  +  (2.110)  V t V a . ^ . ^ O W - ^ C r s - r S )  + y( )(f ,r ',t,i')<5 (n 3  3  3  3  KWf  3  -  x  # (r ')drdr'.  =  3|Mn)y(3W%M0<$3^  p  (2-111)  t  (2.112)  uds Basis. When we use the uds basis wave functions the strange quark is always in the third o  position in the strange baryon. Since the model involves the transformation of one of  Chapter 2.  44  Theory  the u quarks in the nucleon, to an s quark it follows that only the term J *K(r,-)^ (r , (3)  3  *, W  i  - ?{)6 (? - r*)* (r$) dr dr' 3  2  p  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  T h e Impulse Approximation  By singling out quark 3 in this manner, quarks 1 and 2 are being treated as spectator 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 momentumenergy 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 — 3m„) « 2QGeV~ (taking m„ = 0.33 GeV). Therefore the interaction may l  p  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(r ,r' ) in (2.112) to momentum space 3  VW)  3  = /(p\r3)(r \V]r' )(r' \p') 3  3  3  d r' d r . 4  4  3  3  (2.113)  Chapter 2.  Theory  45  Where the closure relations / \r' )(r \c/V 3  3  = 1  3  and J |r )(r | d r = 1 4  3  3  (2.114)  3  have been inserted. We get V(p,p') = / e - - y ( r , r ) e - « > ' - 3 ^ ^ 3  (2.115)  3  and the inverse relationship is V(rs, r \ t, t') = J e-*»V(p,  (2.H6)  P ' ^ ' ^ ^ T -  3  Now using (2.116) in equation (2.112) yields,  SY = P  T ^ / M ^ ) ^ - ^ ^  Where 6 = 3 when using SU(6) wave functions and 6 = 1 when using uds wave functions. Separating out the time dependence in the bound state baryon wave functions (equation (1.6)), * (r.)  =  r  and* (rO = p  Mri)e  iEt  ^1)^',  we obtain  x V' (^Oe" P  ,(Ei+E2+  ^ 'd*^'dVld r; dVV^V rf(/)^ )<  3  ,  /  (2.117)  Since we have a single quark interaction we have E\ — E[ and E = E' . Performing 2  the time integrals we get, S Y p  =  7^/W-P>(^-(p7)^(n)^(rl-rY)  2  Chapter 2.  46  Theory  x  * (r 3  2  F y 2  rVp  V(p,py-Wiptfi)  d(p°y.  d Ti dzfi d3pd3p'dp0 3  (2.118) Ei and E have been dropped from the time dependence as po is the zeroth component 2  of the 4-vector, p, which must correspond to the energy of the interacting quark (E ) 3  only. 2.4.1  When SU(3)  flavour  is Broken  Wefirstdevelop a form for the amplitude when m ^ m ; appropriate for use with uds s  u  wave functions for the Y. We then generalize the amplitude to the case of unbroken SU(3)fi r aV0U  symmetry by letting m -> m . s  u  We switch to the Jacobi coordinates introduced in §1.7, P  4 ( r i ' " *'), A' = -L(rJ' + ri' - 2r1'),  (r1,r1,r1,r1',r1',r1') i->  = |(r1' + v ' + ri ')•  (p,\,R,p',\\R'),  Vv(»"1,r2,r-1) i-> *y(^,A,P) r  and Vv( ~i', rT, ri') •-• *P(p", A', i?'), and using the relations derived in §1.9,  2  0  47  Chapter 2. Theory  -y/6 m  u  3m m with m\ = 2m + m ' u  8  u  s  we get, bJ J'  / *(£ -P°)S(E' 3  (2*)'  A, R)S (R -R'+ -j={p- p') + ^ ( ( ^ ) A - A'))  x  3  x 6\R -R'-  x  x  p>) + - ^ ( ( ^ ) A -  A'))e ^^e-3)) V ,y i  P  (p  )  d pd \ d R d p' d \' d R! d pd p' dp d( °y 3  3  3  3  3  3  3  3  0  P  *^ ^ - " +  x  -(/)')  3  e  (  3))p  V(p, p° = E , p', (p )' = E ) 0  3  e-^'-v^')*'*^' = ^  x  +  3  ^ ( i l - R') + ^ ( ( — ) A - A'), A',R') (2.119)  d pd \d Rd X'd R'd pd p'. 3  3  3  3  3  3  3  Where we have integrated with respect to p°, (p°)' and p' and the Dirac delta function property (1.70)) has been invoked. J and J' are the Jacobians 9fl  J  =  9r-a  dp dri dp drz dp  3n  3  3A  1  3A  1  3A  1  75 "  *  0  •y/6m  u  1 /'"La  75^-3)  (2.120)  Chapter 2. Theory  48  —#  Evaluating the R' integral we get  x e-il^lOJ-*)-VPW x  x  x  4  ((?  ,  - J<, #  =  j? + * ( ( * _ A'))  =  d pd Xd Rd X'd pd p' 3  c  3  3  3  3  (2.122)  3  - - ( ^ * ( ^ - v ^ ^ '  w  £p,R'  =  = R+  )A - A'))  d pd \d Rd \'d pd p'. 3  3  3  3  3  (2.123)  3  The Jacobian transformation allows us to separate out the centre of mass motion, since Vy(p,\,R)  = N *Y(p,Z)e- ~  ¥R  (2.124)  and Vpip', A', R') = N $ {p',\')e > '  (2.125)  iP  Y  ip  p  =  n  p  iV $ (^A') ^- ^vf^) - '» (  p  —*  p  X  (2.126)  X  e  —•  Where Py is the centre of mass momentum for the Y final state and P is the centre p  of mass momentum for the proton. Ny is a normalization coefficient for the Y and N  p  is a normalization coefficient for the proton. The $x(p, A) contain normalizedflavourand spin wave functions. The harmonic oscillator functions are normalized, over all space, to unity. The normalization coefficient 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 (§2.10.2). Sr  p  =  x  e  Chapter 2.  49  Theory  x d pd Xd Rd \'d p<Fp' 3  =  3  3  3  3  21bN N {2ir) / _ , s — - y * (*V-p + p ' - P ) $ K ( / 9 , A ) e Y  p  R  3  /  R  ?  (=*-3)  R  p  p  ^ -« ' l  x F ( p > ° = E ,p\(p )' = E )e^ - - ' '$ (p,A') ((  0  e  3X  )?  p  ^A&^')^^ Xd X'd pd p' 3  x  )X  3  3  3  3  3  d  e - ^ W ^ ^ - ^ ^ ^ X O e * ^ - ^ d pd Xd X'd p. 3  x  3  3  3  (2.127)  The delta function * ( i V - P + P ' - P ) = <5 ((P - p') - ( i V - P)) 3  3  P  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) i  f avour  is Unbroken  If 517(6) basis wave functions are used for the Y then (2.127) becomes (with m —> s  m,  m\ —* m)  x V(p,p = E ,p' = P +p-P ,(p°y = E ) 0  3  x  p  d pd Xd X'd p. 3  3  3  3  Y  3  (2.128)  We will return to the last two expressions after deriving the explicit form of V(p,p').  Chapter 2. Theory  2.5  50  T h e F o r m 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 ( p" - m)V> = 0, 7M  (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, (7 p" - m)V> = Vrl>. M  (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, (• + m ^ K = 0 ,  (2.131)  where m#- is the kaon rest mass and • = —p^p^ = g~ gfj:- In the presence of a potential V the KG equation becomes, ( Q + m ^ =- %  (2.132)  Chapter 2. Theory  2.5.2  51  F o r m of the vertex functions  The electromagnetic interaction for a spin | particle: The minimal substitution , p —* p>* — eA**, introduces the coupling of a Dirac particle, 1  M  charge e, to the electromagnetic field A . Using the Feynman slash notation [26], M  4 = y^A" = y°A° - 7 • A, we get from (2.129) and (2.130)  (^-m)V> = e^V =• V^  ac  (2.133) = 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  —• p* — efcA^, this time into (2.131) (ex is the charge 4  on the kaon), giving, [ ( i / - - e A\x)f K  - m ]M*) = 0 2  K  [ n + m + e (A»(x)i—  +  2  K  K  A^x)) - e^A^A^x)}^^)  = 0.  So tofirstorder (neglecting terms 0(e ) oc (y^)) and using (2.132) 2  V$(x)  = ie {A*(x)£K  + -~A,{x)).  (2.135)  The form of the strong interaction: For the strong (K~p) vertex, consider the mesonfieldin analogy with the electromagnetic potential[26]. 1  This 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) is the wave function of the particle before interacting with the electromagnetic x  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)  =  (2.138)  e^VV>''-  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"  ip ), K  (• + m )i> = i> Txp  (2.139)  2  K  K  Y  p  (compare to equation (10.11) of ref.[26]). Also, r  =  T =  gtcusijs  9Kus  for pseudoscalar  (PS)  fits for pseudovector  coupling,  (PV)  coupling,  (2.140) (2.141)  where q = p — p' and 75 = i o7i 273 7  7  p'(p) represents the up(strange) quark momentum, immediately before(after) the strong interaction.  gKus/idKus)  represents the strong coupling Ku —* s constant in the pseu-  doscalar /(pseudovector) coupling scheme for the process K~p —* Y 7 . It is the analogue  Chapter 2.  53  Theory  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)<JKus ilbu(p')  u(p)gKus7hu(p')  = u(p)g (p'Kua  p*)y u(p'). s  Since the u(p) and u(p') are solutions of the free Dirac equation (/-m>(p') = 0  (2.142)  u(p)O*-m.) = 0,  (2.143)  and for the conjugate relation,  and noting the anticommutation relation 7 „ 7 s + 757^  = 0,  (2.144)  we get that 9K = - J ^ 2 - .  (2.145)  US  Now the equation for a proton absorbing a kaon (analogous to equation (2.133)), can be written,  (^-m)V> = (V>/cr)V>p  p  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 §1.8) acting on the third quark of the proton. +  It can be thought of as the combination of creation and annihilation operators a\a , u  Chapter 2. Theory  54  acting on the third quark of the proton. Where a\ creates an s quark and a annihilates u  a u quark. Therefore the form of the strong interaction can be written, from (2.146), V 2.5.3  = TV+i> .  ST  (2.147)  K  T h e Feynman Propagators.  The Feynman propagator for a spin | particle. The Feynman propagator Sx(y—x) (or Green's function) for a Dirac particle X, satisfies the Green's function equation (• J7 - m )S (y y  x  x  -x) = 6\y - x),  (2.148)  where V is the 4-vector gradient y  This operator acts on functions of y only. Sx(y — x) represents the wavefunction ip(y) at space-time point y produced by a unit source (ip(x) = S (y — x)) at space-time point 4  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 d q --«-(»-«) 4  e  =  7(2^7^7  Now, 1 4-m  _ _  j+rn {4-m)(4 + m) 4+m f-m ' 2  Chapter 2. Theory  55  and by the anticommutation relation T ' V + 7"7" = 2flT,  (2.149)  £ = q.q = g .  (2.150)  it is easily found that 2  r <£ d go  -«'</(!/-*) -w.(v-*)  44  ce  (2TT) g -m 4  2  2  +ie  y  x  The small positive imaginary part added to the denominator in equation (2.151) assures that (2.151) meets the desired boundary condition of propagating only the positive frequencies forward in time (particles) and the negative frequencies backward (antiparticles). The Feynman Propagator for a Klein-Gordon particle. The Feynman propagator for a Klein-Gordon particle K is the solution of the equation, (•„ + m )A (y 2  K  K  -x) = -6\y - x).  (2.152)  Afc(y — x) represents the wavefunction ib(y) at space-time point y produced by a unit source (tp(x) = 8 (y — x)) at space-time point x; the evolution of ip{x) is governed 4  by the free Klein-Gordon equation (2.131). By once again, Fourier transforming to momentum space we find [26],  ^'-'Wp^-mH-fc2.5.4  ( 2 1 5 3 )  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.  56  Chapter 2. Theory  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*)  = N xP (z)e- ,  (2.154)  iEKt  K  K  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 = 2E \N \ \tJ> (?)\ . 2  K  2  K  K  Normalizing to a box of volume V we get N  =  K  J —  (2.156)  with j | ^ ( i ) | d z = 1. Jv 2  (2.157)  3  In terms of its Fourier transform *K(*) = =  N e-*«j^^Jj*«'*4 {p )#p K  K  K  K  ^T J^ - MPK)d p . iPK Z  (2.158)  3  K  2  Where 4>K(PK) is the bound state momentum space wave function. 2.5.5  T h e p h o t o n wave function  A (iy) has the form, with box normalization [26], M  -Jigt'fc.UI  = with 7\L =  =/V 6V -, fc  7  .  1  .  (2.159) (2.160)  Chapter 2. Theory  57  (2.160) is the photon "wave function" and describes the emission of a photon with —*  momentum k, energy k , and polarization 4-vector e . M  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 i  th  graph is,  ViCiirs) = V (r )S (r -r )V (r ),  (2.161)  V (r' ,r )  = V (r )S (r -r' )V (r' ), 3  (2.162)  V (r' ,r )  = V (r )S\r  - r' ),  (2.163)  V (r' ,r )  = V (r )6\r -r )V (r' ).  (2.164)  EM  2  3  4  3  3  3  3  3  ST  3  3  ST  3  a  3  u  3  3  ST  3  3  3  3  3  ST  3  EM  3  3  EM  3  VEM is the form of the interaction at the electromagnetic vertex. VST is the form of the interaction at the strong vertex. S (r — r ) is the Feynman propagator for a Dirac particle — the strange quark. a  3  3  <Su(r — r'3) is the Feynman propagator for a Dirac particle — an up quark. 3  Note that all the interactions act only on the quark in the third position, as was shown in §2.2. 2.6.1  Coordinate Representation  Using equations (2.134), (2.151), and (2.147), and since the interactions act only on the third quark of the Y and proton, we get , 2  ^i(r ,r ) 3  3  =  V^ \r )S (r -r' )Vs (r' ) a  3  a  3  3  T  Suppressing the +ie term in the denominator.  3  58  Chapter 2. Theory  f}4  /  =  p-*<l( 3-r' )  (finv  r  3  (2TT) (2TT) /  2  3  g -m2 2  4  T2^7f / * * * * *  +  rn )TV MPK), a  +  (2.165) where m , e is the mass and charge of the strange quark. s  s  V (r' ,r ) 2  3  =  3  V (r )S (r ST  3  u  -  3  r' )V^ (r' ) ac  3  =  3  cPnis  t da A  ~ - e~ -( ~ 3) ipK r3  e  (2^7? / * * * * *  iq  r3  r  •,  rV+ (4 + m )  qTZ^  u  ,  &k (PK),  (2.166) where m , e is the mass and charge of the up quark. Using equation (2.147), u  u  V (r' , 3  The kaon wave function,  3  r)= 3  V (r )6\r ST  3  (2.167)  - r' )  3  3  to be inserted into  IPK(?Z)I  VST  of equation (2.167) for  graph (3) is a solution of the Klein-Gordon equation in the presence of a potential  Vgjf,  equation (2.132). (•  +m  r3  2 K  = -V^(r )xp (r ).  )tp (r ) K  3  3  K  (2.168)  3  From equation (2.152) we have, (• , 4- m ) J 2  r  K  A (r K  3  - w)V™(w)xp (w) K  dw = 4  -Vg°(r )ip {r ) 3  K  3  (2.169)  comparing equations (2.168) and (2.169), we get that the kaon wave function at r after 3  scattering off the potential V£M at w is, xp (r ) = J A (r K  3  K  3  - w)V{w)Tp (w) d w. 4  K  (2.170)  Chapter 2. Theory  59  Where the form of the interaction between the kaon and the electromagnetic field at w is known from equation (2.135). V(w) = ieK(A"(w)£-  +  £-Mv,))  Using this along with (2.170) and (2.153) we get,  (2.171) Where  indicates the derivative operator acts to the left and  indicates the  derivative operator acts to the right. The minus sign arises from integrating /•  J  e  d  4  q  d  -ig-(r -v>)  g  3  w  V ^  i  e  K  d ^  i  M  w  )  M  w  )  )  by parts. The boundary term is omitted as the potential A is taken to vanish as M  ±oo. Substituting for V'K'(w)  t, \w\ —•  a n  d A^(w) into (2.171) and evaluating the derivatives we  get, Mr*)  =  \^yi/2  NisN  ^ _ ^  J d'qd^pKd'w  P i s  t  p  e  ^  j  f +  q)^e^<f> {p ) K  K  -ir -(PK-k) 3  after performing the integrals over w and q. Therefore,  (2.173)  60  Chapter 2. Theory  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  =  ig us  P'KIS = igKusip'-  K  igicM  p")75  (2.174)  4)75,  ~ * 4~ p" + e  The term 4 + « 4)75  i9Kus(-e  e  s  in (2.174) corresponds to the radiative process, graph (4) of Fig.2.2. Since e — e = a  u  we get the result V<(r' ,r ) = -NKN^K 3  j  ^  3  ^^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,  KM)  = x  =  ^(2^)19/27 § / A ^- *3 " d' 3S - d'  3  e  » ' f c . r  3  - « q . ( r  3  - r J ) - » P i f . r ^ - j p ' . r J + « ' p . r 3  p V  a - ™  ?  2  _  m  2  f\4+m.)TV+4> {pK) K  j  3  AdV3d pV  e g2  _m2  M+m.)rv t (p ) + K  K  61  Chapter 2. Theory  x  (2TT) S (k - q + p)  =  l2^l  4  4  dpK  (k+py-m*  S  (P« + P ~ ( +  P»MPK).  K  (2.176) For V « have, using (2.166), w  2  p  NisN  r  ,ik.r' -iq.(r -r )-ipK.r3-ip'.r' +ip.T3 l  f  3  3  9  2  3  3  - ™u  rV (^+m )  x  +  X  tt  -'(i+ -p')- 3  f 7WJV  /•  (2TT)  6 (q-(p-p ))  4  k  e  r  4  K  = l^W]  (p- y-ml  d p K  HPK+p-(k+ ))MPK).  PK  P  (2.177) With (2.173), r  J * *  =  d  r a d PK  {PK-kf-ml  x r V ( 2 p -fc).e<5 (r - r )<^(Ptf) 4  +  = ^  =  K  3  § /^  J  d  p  3  ( P K  K  ( p  K  _t ) , _m , r n ( 2 , -  - k Y - m l  » w  P  S ( P K + p  -  ( k +  P»**M-  (2.178)  Chapter 2. Theory  62  And for V we get from (2.175), 4  x <f>K(PK)6\r - r' ) 3  =  ~ 2^y (  = -^Jy/i  3  ^iyjc«7.V e-^^^-^^^(fic)  J  +  ^9KUS75V+<I>K(PK)S\PK +p'-(k+  J ? D3  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 2.7.1  Gauge Invariance. Choice of Gauge  In the Lorentz gauge specified by 1  dpA * = 0  (2.180)  k.e = 0.  (2.181)  we get the constraint, using (2.160),  This is a covariant condition which must be true in any frame. Within this gauge we can exploit the residual gauge freedom [29], yi  A" - d'x,  M  (2.182)  provided x satisfies •  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], •O4" = 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  (2.185)  k° = \k\.  The transformation (2.182) corresponds, by equations (2.181) and (2.185), to  e" ^ e" + 0Jfc".  (2.186)  We choose (3 so that e° vanishes. This fixes A*. It is known as the Coulomb gauge and makes manifest the transverse nature of the electromagnetic field. 2.7.2  P r o o f of Gauge Invariance  To check invariance under a gauge transformation (within the Lorentz gauge (2.180)), we note that the amplitude S , corresponding to the i Yp  th  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 i-l  where JV is the number of graphs.  = 0,  (2.187)  Chapter 2. Theory  64  We ignore binding between the quarks for the purpose of determining the behaviour of the interactions under a gauge transformation . Replacing e with k in (2.176) 3  w  = u( )vr\p,p>(p') P  =  ~(2^jr  J  u(p)  d  P  ^(PK+P~(k+p)Hp),  (k + pY-m*  K  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/ « ( P ) ( p , P  *  e  K  N  N  f  y ^  Jd  )u( ) = -^jruip) P  x 8\p  J 3 - (m P  3  K  Ji+ P p 2 +  2  k  fiTV+JKJpK) p  _  m  2  + p' - (k + p))u(p').  K  Using the anticommutation relation for the Dirac gamma matrices (2.149) wefindthat Ji f> = 2k.p- p ft.  (2.188)  1  Employing (2.188) and the Einstein mass-energy relation p  2  = m ,  (2.189)  = rn-l,  (2.190)  2  and(p')  2  we get, - /  N T ^ f c ,  u{p)V  x  /\  e N N^_ f = —— u(p)jdp  ,  (P,p)u(p)  a  (  X  K  f  T  27r  )3/2  S\p  K  3  ^  (2k.p-{p -m ) 1  a  K  J "  — 2  k  )i)TV f> (p ) +(  K  K  p  + p'- (k + p))u(p').  Using (2.143) we get u(p)Vr\p,P>(p')  = -^^-u(p) J e  d*p TV MPK)u(p')8\pK-rp'-(k+p)). K  +  (2.191) However, in general we will lose gauge invariance when we fold the interaction over three quark wave functions. 3  Chapter 2. Theory  65  Replacing e with k in (2.177) «(P)V  (P,P)U(P) = -^- u{p)jd  2  fF  _ y_  PK  {p  x 6 (p +p'-(k  pK  ml  + p))u(p').  4  K  Noting that p - p = p' - k and using (2.150), (2.185), (2.190), and (2.188) we get, K  e N N^_, u  K  (2^)3/2  , f _ v^yy  rV /> (PK)(2p'.k- }i(f -™u )) -2p'.fc +(  K  x S (p + p'- (k + p))u(p'). 4  K  From the relation (2.142), u(p)Vr (P,P>(p') = - ^ 7 T " ( P ) / K  ^r^ (fi )«(p')^(p c+p'-(*+p)). jr  J  f  (2.192) Replacing e with A; in (2.178) U(P)V  3  (P,P)«(P)  = -wvr»(p)J PK d  i  P  K  _  k  _  y  m  2  K  x £ (P*+p'-(fc + p))«0>')4  As before we get,  fi(p)v ~ 3  - ^)I "(p) / fpx  P>(P') = e  7  W+MPKMP')  8\p  K  + 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 e = |e, e = — |e, e# = u  s  —e, where —e is the charge on the electron, we have,  J2 w = u( )(vr M)+vr (p,p')+vr (p,p')Hp') k  k  k  P  »=i  JS K i v =  ^  /  ^ / ^ « ^ ( p ) ( e - e - e ) J d p 1SV+<J)K(PK)U(P') 3  2 7 r  x 6( 4  PK  3  2  8  u  + p'-(k+p))  K  K  = 0.  Thus our interaction is gauge invariant at the level of free quarks, in PS coupling, using only the first three diagrams of Fig. 2.2. With P V Coupling Here we use the substitution (2.141) T i-¥ g  j#7 . 5  Kus  For graphs (1) and (2) of Fig. 2.2 q = PK- However for graph (3), q = PK — k, since the kaon radiates prior to capture. Now we have J2 I=I  w = *(p){vr (p, fc  =  ^yJ igKu u(p)  x  8\pK+p'-(k  =  ^^WKUMP)  ±  2  S  p')+vr (p,  p') + vr (p,  k  k  P')HP')  J d pK (es i>K - c„ i>K- eKifa3  P))lhV <j)K{PK)u{p') +  + p)) J PK e  h5V+(f>K(pK)u(p') S\p  D3  K  K  + p' - (k + p))  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- i~9Kusu{p) 2  j  dp e 3  K  K  V V MPKMP')6\PK+P' 15  +  ~ (k+p)). (2.194)  Chapter 2. Theory  2.8  67  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° « 281MeV when Y = A, and k° « 2l9MeV when Y = £°. This can be seen from conservation of energy-momentum in the K~p atom rest frame, P£ + P?< = k» + PP = 0.  (2.195)  The zeroth component gives, m + m = k° + \l\P \  2  K  p  Y  +m,  (2.196)  Y  and the 3-vector components give, P + k= 0  \Py\ = \k\.  Y  (2.197)  From (2.196) we get, (m  + m ) + (k ) 2  K  p  0  2  - 2k\m  p  + m) K  =  with k° = \P \ =• k° = Y  \P \ + m , 2  Y  Y  (rn + p  m y-mj K  r  Inserting the values [30], m  =  m  = 493.67Mey,  p  K  m  938.27231Mey,  A  = 1115.63MeV,  mo  = 1192.55MeV,  E  we get, for the reaction K p —• A 7 , \k\ = k° = 281MeV and for the reaction K p — > E°7, k° = 219MeV.  Chapter 2. Theory  2.9  68  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 * ™  =  e  V  =  ( 2  .  1 9 9 )  where HK is the proton-kaon reduced mass; fi the proton-electron reduced mass; and e  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, E  = K + U.  B  Taking a circular orbit mv  e  2  2  4TT€ r 2  r  0  -e  1 and K = -mv  2  along with U = 6  47T£ r 0  e = 8ire r 2  gives E  B  0  2  2  e 4ir€ r 2  0  -\PK?  2n  K  \PK\ =  \j-2u E K  B  5.591 x 10  12  2  T r eV  -  As mentioned in §2.5.4 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\ « 0.236MeV. It can be seen that |p#| is far smaller than the other momenta (\Py\ = \k\ « 281MeV) and masses (m = 330MeV, m = 550MeV) in the problem; we therefore neglect \p~k\. u  s  Since |p#| w 0 we take the normalized momentum space wave function to be MPK)  =  \I ^-6 (PK).  (2.200)  = N e- ^4> (0).  (2.201)  {  3  Inserting this into (2.158) we get ipK&t)  iE  K  K  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 (§2.7.1) k.e = 0 and eo = 0. Inserting these conditions into V {p,p') (2.178) and letting px —+ 0, we see that the 3  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 (§2.7). For Vi(p,p'), ^(PiP ) 7  a  n  d  V*(P,P') we get  V {p,p') = e N N^ 1  3  K  f+J + '>  A  ro  (fc +  V (P, ') 2  P  py  —  = e N N, + TV  u  {pi  +8\  PK  +p'-(k+  p))i> (0), K  (2.203)  m* m  K  rV  "  } 2  U\  P K  +p'-(k+  p))^(0), (2.204)  VAM)  = -eKN N^i~g s8\p +p' K  Kusl  K  - (k + p))V ip (0). +  K  (2.205)  Chapter 2. Theory  2.10  70  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, (P )> ~ ^  (2-206)  2  The identity[9], Ap = {(p— (p)) )* with (p) = 0 in the hadron rest frame, has been used. 2  R is the radius of the ground state wave function. The analysis of various hadronic processes indicates [34] that, R ~ 6-12GeV =• \ ~ 300MeV. R 2  -2  (2.207)  In our calculations we use the constituent quark mass [24] 4  m  u  = 330 MeV  m  = 550 MeV  s  So we get,  Mi<i. *Note that 3m„ > m due to the binding energy of the baryon. p  (2.208)  Chapter 2. Theory  71  This suggests that taking (p/m) <C 1 may be reasonable, although not (p/m) <C 1. 2  In addition Capstick and Isgur [11] point out that the hadronic wave functions (of chapter 1) derived from the harmonic confinement potential depend only on the quark coordinates. In QCD however, the hadronic wave function must also depend on the state of the glue. Despite these shortcomings in the NQM, the agreement with experiment (predicting masses, magnetic moments, and decay amplitudes) has been excellent. It is claimed [11], that this can be attributed to choices of effective parameters such as quark constituent masses and ot (which can absorb the effect of relativistic modifications of spin a  dependent interactions). Looking at the derivation of the effective Hamiltonian one can see that the typical approximations like V ™ + |p| - m + H£ 7  2  2  are not that bad for (p/m) = 1 (6% difference). Furthermore, some of the approximations will probably be compensated by a renormalization of parameters [35]. Koniuk[36] points out that relativistic models, once their parameters have been chosen, give essentially the same results as nonrelativistic models. In addition, it is argued [5], that neglected terms in the Hamiltonian equation (1.3)  i  such as relativistic corrections and other one-gluon exchange effects, seem to be relatively unimportant at the level of 10-20%. From its considerable success in the past it appears the model can give good qualitative agreement with the observed properties of low-lying baryons. With the caveat, however, that the model should not be take too seriously quantitatively. It is still a  Chapter 2. Theory  72  source of some debate as to why the NQM works as well as it does. We believe that it is a acceptable model of confinement; suitable for our purposes. 2.10.2  T h e Nonrelativistic Reduction Prescription.  In chapter 1 we treated spin nonrelativistically, that is we introduced it in a purely 'ad hoc' fashion. Had we used the Dirac equation for our wave functions, spin would have arisen naturally. The Dirac equation for a quark of mass m , in the presence of a g  potential, V{ is, nt  (2.209)  = (* • p + Pm + V )xp. q  Vi t n  int  represents the internal confinement with relativistic corrections experienced by the  quarks in the baryon. In the representation 1 0 \  ec= I ° ° |,/9 = S 0  (2.210)  0 -1  where o are the Pauli spin matrices and <f> and x  8 X 6  two component spinors, (2.209)  becomes  <f> dt  /  0  a •p  ^ a •p  & p\  X  +  0 | +m  q  (l  \  o  [0-1)  +v [ ' ) \-x)  m„  int  <f>  X  + Viint  (2.211)  { x  We separate out the time dependence in xp (2.212)  = e- <*  xp =  iE  \ X J  \X  )  Chapter 2. Theory  73  where (j> and x are constant with respect to time and E is the total energy of the state q  ip. Inserting (2.212) into (2.211) and cancelling the common exponential factor we get, E„  <f>  —a•p  int  (2.213)  The second equation of (2.213) gives, a•p  T, -<f> r X= E + m - V q  q  (2-214)  int  X are the "small" components of the Dirac 4-spinors. They are reduced by v/c compared to the "large" components <f>. 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= The subscript i refers to the i  ih  Ei + m,  2m,-  Xi  (2.215)  V Ei+mi Xi J  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 normalizing the total baryon wave function, \N\ J tfttf dV = 1 2  Chapter 2. Theory  74  / « /m m  _  _  L  _  /  u  m  " Vvv *?V  Ap  _ _L_ / " / " m  TV  ~  Y  m  u  s  m  y/v\l Ei^J E2)j E ' 3  Using (2.215) amounts to adopting the spinor structure of free quarks. Since we are using the impulse approximation (§2.3) 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  }<f> (p')d d p'. 4  x <  p  E'+m  4  P  u  (2.217) This is expanded and a nonrelativistic reduction carried out. We • Expand the interaction and discard terms of order (p/m) or higher, 2  • 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) (p is 2  any momentum and m any mass). Applying this prescription to the { } in (2.217), we get for V = V (equation (2.203)), X  \  I f  F-z£-)^ >  1  +  &3 +  + m  ')'y«  s  m  \ \  3  ''  I  v  E'+m  /  u  we get (with e° = 0)  (1  —a • p E +m 3  3  0 a•e  —o  •  6  (  m  ^  + k +p 0  3  0  a.(k + p)  -<?.(£ + p )  0 1  m -k°-p°  1 0  a  ~~/  Chapter 2. Theory  E  3  +m  75  ? ( , + *<> + p ° ) - ± J L - + J^l-a • ea.(k + p) m  3  a • pi a -ea.ik + p)——  -  >  i\  (n  n  3  +m  s  (2.218)  s  or higher, and get  2  V  E  u  a • e (m - k - p ).  Now we neglect terms 0(p/m) vPS  +m  E'  3  _ -NKN^e,ig V ij> (0)a • e (m. - k° - p°) {k+pY-ml Ku3  +  K  ri., \\ 6(PK+p-(k+p))  4  n  (2.219) in PS coupling. With PV coupling we get —cr • p E +m  (1  3  E  (  0  3  a-t  s  o.(k + p)  0  \  0  K  -a.(k + p) \  -a • e \ I m + k° + p°  m -k°-p°  j  3  0 1 1 0  0 —E K ) J  E'+mu.  • em (m  NKN^igKusV+ipK^a (k + p) — m  vr(p,p') =  2  K  /  - k° - p°) 6 (pK+p'-(k  s  + ))  4  2  P  (2.220) 9Kus 9Kus  -trPSf l\ K*\ (P,P)  m  since EK — mx when K = 0. P  Similarly we get from equation (2.204), rps /x _ N N^eJg V^ {Qi)& VriP,p') = (  K  Ku3  K  (P -  • e(m - p° + m ) S\ PK) - ml K  u  PK  2  + p'- (k + p)) (2.221)  for PS coupling and  v M) = pv  2  N N e ig V ^ (0)a • em (m (p-p ) -ml K  y  u  Kua  +  K  K  2  K  -p° + m ) u  6 (p +p' 4  K  ~(k+p))  K  (2.222)  m V{ M) s  gKv  K  76  Chapter 2. Theory  for PV coupling. Recall from equation (2.145) that QKus  1  For the contact graph the interaction (2.205) becomes V (p,p') = N N^e ig V ip (0)a  • e8\p  pv  4  K  K  Kut  K  +  + p' - (k + p))  K  (2.223)  in the nonrelativistic approximation . 5  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')N N^ (0)S (p +p'-(k  + p)).  4  K  K  K  (2.224)  V is the total interaction: PS  + v * forPS coupling, s  V  =  yPV  =  yPV  +  yPV  +  yPV  for  p y  c  o  u  p  H  n  g  >  We get, S  = ZC(2n) S\p  + P -(k  4  Yp  K  x J * (p, A ) e Y  x  e  + P ))  p  i A V  Y  ^ - - t / ( p , p ° = E ,p' = P + - P ,(p )' (  3 )  ^ ( ( ^ ) ^ ' ) . ( ^ + P - ^ )  = E' )  0  3  $  p  (  X')e^&-  ^  P  X  ,  P  Y  3  d pd Xd \' d p.  )  3  3  3  3  (2.225)  That is S  = Z{2it) 8\p A  Yp  K  + P -{k + P ))M p  Y  (2.226)  If we use the on-shell condition (2.142) before we take the nonrelativistic reduction, we find that PV v v PS the sum Vf + V2 + Vf reduces to V ^ + V2 . Since we intend to use bound state quark wave functions we discard this approach and the PV amplitude will differ from PS amplitude. 5  5  Chapter 2. Theory  77  where M  C  jl {p,fiVI (^d*pd*p, Y  p  Z  (2.227) (2.228) (2.229)  W,P)  /$ (^A')eS^+ (^))d3X' 3  p  276 (27r) 2v^'  and C  The normalization constants are given by  K  1  =  1  Nr,  =  1  (2.230) (2.231)  3  N  5  /m  vl; V  u  £i  frn^  VE V 2  #3  Chapter 2. Theory  Figure 2.2: Feynman diagrams contributing to the process K  78  p —•  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 onlyflavourdependent piece in V is V+ (see §1.8). V+ acts in theflavourspace 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 theflavourmatrix elements listed in table 3.11. 3.2  Momentum Space  With theflavourmatrix elements determined we have, for the momentum space part of the amplitude, MMS where q  Y  =  $e ™  =  4=((~)Py -y/6 rn  iX  Y  d \] V [/ $ e ' 3  iX  p  3p)  d \' d pd p 3  3  3  (3.232) (3.233)  u  q  p  =  (3.234)  We have already taken care of theflavourdependent part of the potential V . Also +  note that the spin dependent term a • e in the potential, acts only in the spin space of  79  Chapter 3. Calculations  80  - vi'  (tii\v+ \4>) = 0,  V * A | V | ^ ) = 1,  (K\v \<i>= )0,  = o,  = 0, (4>*\v+ \4>i)  x  +  (ri\v \*i)  +  (tf\v \<n)  = 0,  +  <<&\v+ \4>)= (<h\v+ \4$  P  1  +  0,  = 0,  = 0,  {&\v \<H) = o, +  _ (<i>i\\v \<i>3 ') ( W W )  1  x  +  P  _  = 0. (^\v \<t> )  1  p  3'  —  +  p  Table 3.11: Flavour matrix elements the third quark. Therefore we separate out the spin piece and denote it (SP) = X ~° J  denotes x  p  a n  dX  J=1  ( \a-e\ f). J  X f  X  corresponds to % - This term will be evaluated in §3.3. A  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 a  = J V'ooo(A)e'^ d A,  (3.235)  = / V-ooo(A')e^- *'d A',  (3.236)  A:  Y  a  3  3  p  by = / V>ioo(A)e'^- d A,  (3.237)  b = / V'ioo(A')e'' -  (3.238)  A:  3  p  p  cy = / V>oio(Ay ^ d \, ?  c  p  =  l  (3.239)  3  / ^oio(A')e«' '-- 'd X'. f  X  (3.240)  3  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  a Va d p,  (3.241)  6 = J  a Vb d p,  (3.242)  c = J  byVa d p,  (3.243)  3  Y  p  3  Y  p  3  p  d = jbyVbpd ^  (3.244)  3  c Vc d p,  9 = J  (3.245)  3  Y  p  and Q =  (3.246)  From table 1.3 and table 1.4, we obtain table 3.12. See Appendix B for details on the evaluation of a , a , by, b , c , c . The results Y  p  p  Y  p  are a = (—Ylu \aat\/  (3-247)  f  d =  ( ^ l ) <'« -  fr*  +  * M  + foM/,).  —16 / 7r \ r (—Y I. \aa\J  (3.250) (3.251)  5  Where Ij = ^ i j j k (k is an index which sums over the diagrams), k and I  lk  =  I e l Ve 2a  J = y|9 | e 2  2fc  d p,  2q2  P  (3.252)  3  k  M Ve ^ k  d p, 3  /a* = y |^| e ^ I 4 e ^ ^ p , 2  (3.253) (3.254)  82  Chapter 3. Calculations  a,  (*S»WI*S»>  <*5»IGI*S»)  (*5»WI*aoo)  = -?5 >  <*5»WI*$oo) = 0,  (SfoolQWo)  =  (*fooM*foo> = | ( « + <0,  <*fool<?l*200>  =  <*foolQI*20o) = 0,  <* oolQI*ooo)  = -72 '  (*^00lQI*U  = i(a + rf), <*2\)oM*200> = 0,  A  6  c  o,  (*UQ\*ioo)  <*$ool<?l*£») = J(a - rf),  (*5oolQI*£») = 0,  (*5JQI*$oo>  = o,  (*SJQI*Soo) = ^/3,  <*SSolQI*S»>  = o,  <*SSolQI*S»)  <*5SoWI*W  = 72>  (*SSoWI*5oo> = 0,  <*&IGI*S»>  = c,  <*&IQI*£x>>  (^o olQI*W A  <*&IGI*2bo>  a  —  M, \/2 '  = 7! ' a  <*&IQI*SoO> = 0,  = o,  ( ^ o o l W f o o ) = 0,  = o,  (*&IQI*Soo> = 17/3.  Table 3.12: Matrix elements of the harmonic oscillator wave functions in terms of —*  —*  a,b,c,d,g. The Dirac bracket denotes integration over p, A, A' and p.  Chapter 3. Calculations  83  hk = J\q \We~^V e~^ P  d p,  (3.255)  3  k  ^ d  = /l?pll9y|e  8  ^ .  (3.256)  Since neither V, q or q have any <f> dependence the integral over (f> simply yields Y  p  27T. The piece which is common to all the momentum integrals  can be written  where Aj, = e  12O  , a = —y, &i = — ( — ) P  A  T  x  Y  Similarly =  e-%F  -^(l^l +|Pl -2?.^) 2  e  ^e-aiP ^'.?  2  where A = e ^ ' * ,  2  =  oj E i j , 6/ = ± P  p  Y  .  Finally we get h = a + ai,  (3.257)  x  s  2  3.2.1  =  ft^+^^  +M ) ^ .  (  Including the Potentials  With PS coupling we had, from the previous chapter, yPS  =  yPS  yPS  +  T  ,  ^ ( -e (m - k - E ) (k + p) -m + ie s  s  0  2  2  3  e (m + m - E ) \ (p - p ) -m + ie u  K  u  2  K  3  2  u  3.  2 5 8 )  Chapter 3. Calculations  84  With PV coupling yPV  p =  v  V  +V  + V  pv  - -  T /  pv  J e (m - k - E ) s  s  = iQKusV+a • em I  0  e (m + m - E )  3  u  K  u  3  —4- + — i  v  K  [{k + p) - m + te D i a g r a m 1: radiating s quark 2  K  —-L + -H-K  (p-p )  2  e \  -ml + ie  2  K  m J K  The denominator of V\ can be written ( +p) -m 2  + ie = (k° + E ) -\k  2  + p\ -m  2  k  2  3  = -(\H\ -(k° 2  + E) + 2  3  + ie  2 s  m -ie), 2  s  where we have employed the substitution —•  u =p+ k  Now we get after integration of equations (3.252)-(3.256) with respect to the 6 and <f> coordinates of vector u. (See Appendix B for details) IJX = *0«Alim f°° „ M «-»o Jo u — \{ Gj  2  with A  =  and g  — + gr^ in PS coupling  2  cc  2  . du J = 1, • • • ,4 le  (3.259)  2  3  us  g  -* rn g  cc  Gj(u)  (Jb° + £ ) - m  —  K  in PV coupling.  Kus  has the form  1  5  Gj(u)  = 2*A A ep  2»7.-j(«'e' + (-tO'e^ *),  hu2  lU  y  1  (3.260)  i=l  where C *i  1  2  = e . K - i  0  - ^ - ^ ,  = (2fc-ci)fc, ,m\ 3a\. 1  For J = 5 the 0 and u integrals must be done numerically.  (3.261) (3.262)  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  „ r e-o Jo  „u =  u - Af - it 2  f ^MfOAhl  Jo  u  i u +  u — A? 2  r £-*o Jo u  2  i u  -\\-ie  . (3.264)  The first integral in (3.264) can be evaluated numerically (Appendix A), the integrand is plotted in §3.2.2. The second integral in (3.264) can be evaluated by contour integration techniques (see Appendix B). The result is <W.) Jo f ui, 2-XI, —it •it* • = ^ V — \Af 2A 2Aiq  (3-265)  M  2  X  Therefore —  IJI  ' / c o Gj(u) - GJM Jo u —A  ig C2  2  cc  G^AiK 2A  DU  2  (3.266)  X  Diagram 2: radiating u quark The denominator of V  c  a  be written  n  2  (P ~ PK)  2  -m  2 u  + it = -(\pf + (ml - (E - m f) 3  K  -  it).  Now defining A  2  and C  3  EE ml-(E -m f z  (3.267)  K  = - e „ K + m - E ). u  3  (3.268)  A > 0 so the integrand has no singularity and therefore can be integrated directly. We 2  get after performing the angular integrals Ij2  =  iQccCz  lim f ° t-^o  JQ  f ^  p* +  . dp, — ^e  J  = 1, • • •, 4  (3.269)  86  Chapter 3. Calculations  The integration over 8 in I Fj(p)  52  must be carried out numerically. Fj(p) has the form  2«A A e- > J2vu(p e h  =  p  2  i  a2P  y  + (-p) e-° l i  2P  (3.270) (3.271)  and the rf are constants and s = \s \> 0. u  2  2  The integrand of the integral in equation (3.269) is plotted in §3.2.2. Diagram 4: the contact term V4 contains no propagator, so we get simply J j 4  r (p)dp.  =  Fj  rnji Jo  (3.272)  We make the replacement, since E is the energy of the third quark in the Y baryon, 3  E —• m . 3  3.2.2  3  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 J jt, hk and Zjfc. The following symbols are used : 2  •  = uds basis, Y = A  x = SU(6) basis, Y = A •  = uds basis, Y = E°  + = SU(6) basis, Y = S° The I k must be integrated numerically over 8 and momentum. Surface plots are 5  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 2wA Ay' p  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  94  Chapter 3. Calculations  cosfl+i  Chapter 3. Calculations  3.3  95  Spin Space  We wish to calculate the spin piece (X \f-e\x?), j  f  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 m  S'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' ,M'  l^ ){Sf,m \J,M;\,m )(\,m \(\,m \(\,m \.  a  J  and [x f \ =  E  (• > l2' i; 7  M  m  2  m ,M  f  z  x  2  z  a  Where the m  a  under the summation sign denotes mi,m2,m , similarly m' denotes 3  a  m\, m' ,m' . We have, since the interaction acts only on the third quark, 2  3  (X/l*-e1xf)  =  E m  x  a  (J,M\\,m ;\,m )(S ,m \J,M-\,m ) 2  x  ,M,m' ,M'  I  J  3  a  {\i i\\, \)(\, 2\\,rn )(\,m \v m  m  m  x  • «1|' 3) m  3  2  (Im^lm^J^M'^J^M^lm'^mi)  (J,M\\,mu\,m )(\,m\-,\,rn' \J',M')  E  2  2  {S ,m \J,M; \,m ) f  f  x  (J',M'; |,m |5i,mi)(|,m |o - el|,m ) f  3  3  3  (J,M|i,m ;i,m )(i,m ;i,m |J',M')  E x  z  1  2  1  2  (5 ,m |J,M;|,m )(J ,M';i,m |5,-,m )(|,m |a -t\\,m' ). ,  /  /  3  3  i  3  3  Chapter 3. Calculations  96  From the unitarity properties of the Clebsch-Gordan coefficients (equation (3.5.4) of ref.[13]) ]T {J,M\j ,m ]J2,rn ){JuTn ;J2,m2\J',M')=6 >6j j,, 1  1  1  2  MM  (3.273)  >  771J , TT12  we get  (X/I^xf) =  D  6 ,M'8j,j>{Sf,m \J,M]i,m )(J\M'^,m \Si,rni) M  f  3  3  T7i3 ,M,m 3 ,M'  x  (\,m \& • t[\,m ) 3  =  3  (5/,m / |J,M;|,m3)(J ,M;|,m |5,-,m )(|,m3|a- e\\,m' )8j,j,.  D  /  3  m3,M,m'  t  3  3  Now a.6-=^e_ r (-l) fl flC  (3.274)  fi  fl  in spherical tensor notation. The Wigner-Eckart theorem (equation (5.4.1) of ref.[13]), (j>,m'\T(k, )\j,m) = (_i)i-~ q  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-ml  (I'  m3  '  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|),  i2'2l"l2'2/  —  V  "V  => (§11^111) =  since (in natural units) 5 = -o\  i2' 2' 2' 2I • r—~~—  \ 211"^ 112 >  Chapter 3. Calculations  97  So, (X/l^-elxf)  =  £  m3 .M.trij ,R  ^/.m/IJ.MjI.msX/.AfjI.mJI^m,-) •y/e(-l) e. S . R  R  v/3 y/2S, + l  ,Ar '«(-i) j  2 +  m  Jvl  y/3  ^(-i) j  2 + m  (-1)1—3  '(-i) 2  2 + f i  ;  (  Si  VEs, 3,3' K  M m —rnj  ^ M m' —m,-  3  R  1  2  ^m  3  — m —R  3  3  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]) /  32 J3  J\  \ mi m  2  -J2-"13  V2j3~+T /  m  3  •(Ji,mi;J2,m \j , 2  -m ).  3  (3.276)  3  Using the symmetry properties of these symbols [13] we obtain  \  and  —m m' R j 3  3  Now we have, £  ^(2S> + 1)(25,- + l ) ( - l ) -  2J+  f - / - - 3 _ V 6 6J,J, m  mi  m  e  fl  rnj, M,m' ,R 3  (  x  J m.- —mo  -M  \  t  1 2  ^m  3  S  f  J  —mf M i  1\  I 2  2  ^ —m  3  m  3  Ri (3.277)  Chapter 3. Calculations  98  This can be expressed in terms of the '6-j symbol' defined by, Jl  h  J2  J3  h  h  E  m  yf(2j + l)(2/ + 1) m, ,m 3  0'i» i;i2,»w2li3»»T»i+ m >  3  2  2  x  {J3> i + m 2 ; / i , M - m i  -m |/ ,M)  X  (j , m ; / i , M - m - m | / , M - m )(jx, m ; / , M - m^k, M ) .  m  2  2  a  2  2  2  3  x  a  3  (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  (- )  \  /  i  7  \  ™1  7^2  -7^3  Jl  J2  J3  mi  m  1  t  'l  \  -Pi  _1 1 Vl+'2+'3+Ml+M2+M3  2  m  /  3  J2  rh  2  Jl  J2  J3  /i  /  *3  2  h  *3  h  Js  7i 7 3  (3.279)  and noting that rrif =  M + m , 3  m, = M + m , 3  _^  (_2)-2J+|-m/-mi-m^  _  ( _ ^ l + J - m  3  - m ^ - M  (_l)-2-J+!-m; x  (-1) J+i we get (since J is an integer), (X^l^^lxf)  =  (-l)- >- V66 ,J(2S J+  + l)(2Si + 1) <  mi  JtJ  f  5,- 5/ 1 2  x  5,E -A  2  J  Si  S  \  5,  e  R  \  m,- —m/ J2 y  For the square of the spin amplitude we have, W'l^lxfnx^-elxf)  =  ( - l ) - W H - - 6(25/ + l)(25i + 1) 1  2  2  f  2  1 J  Chapter 3. Calculations  99  1  Si S  f  x  2  2  5,-  X  7  i  1  R> R  J  1 \  5/  m,- —m/ 72 but (—l)  1-2m  Si  1  S  f  m,- —my 7?' y  )  « = 1 since the intrinsic spin of the proton 5",- is half-integral . Summing 2  over polarization states A and using the identity [26] (3.280) which in the transverse gauge becomes, £ < ( A ) ( A ) = <$-,•,  (3.281)  e j  A  we get J  x <  Si s  f  1  2  3.3.1  Si S  J  (x s '\t-Ax?Y(x } \t-Axi) = 6(25, + \){2Si 1 E  7  i  2  +  1) 2  1 2  5,-  5,  1  1  f  7/ J  1  ^ rrti —trif  (3.282)  R j  J  Evaluation of the 6-j symbols  Using relation (6.3.4) of ref.[13], Jl  1  2  J3  32  (-1) h+h+h \|  •_ 1 •_1  J3  2  '2  ' 2 i i 2i  (jl  + J2 +  J3 + 1)(J2 + J3 - jl)  2j (2j + l)2j (2j + 1) 2  2  3  (3.283)  3  2  A  =  1/4.  1 5 0. We consider only S state mixings in our compositions so, since the total angular momentum J _ =1 for the Y or proton, 5,- and Sj are both one-half.  Chapter 3. Calculations  100  In addition relation (6.3.3) of ref.[13] gives us, Jl  J2  2  3%  2  Ui  2  2  1  1  1  1  1  2  2  + J2 2  2  1  3.3.2  - J3 + + J3 - ia) (3.284) \ (2j + l)(2j + 2)2; (2j3 + 1)  J3  =  3  3  1/36.  Evaluation of the 3-j symbols  The 3j-symbol can be evaluated from the relation ((3.7.10) ref.[13]) {  ji  ^m  a  h)  J2  Ui +  m  —(mi +  2  _  (—1 ) ~ + J1  J2  +  mi  m2  m) 2  (2j )!(2j )!(j +j + mi + m )l (2ji + 2j + l)!(ii +m,)!  1  3.3.3  [(  —rrif  1  2  2  2  \j (ji ~ mi)\(j  i 2  y rrii  2  U1+J2  x /  1  - (mi + 2  m ))! 2  + m )\(J2 2  -  + mj-mj^l  ^m^^  m )\  (3.285)  2  + Ry.  R j  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 W  i;  = 2cTl  S  , A  "  (3.286)  a  For example with J = 0, J' = 0, 5, = | and 5/ = | 1  £  l ( x ' / I ^ W = 12 <  1  2  2  1  1  2  2  /  1 0 U  1  2  \  E ^ rrii —mj R j m =±i,mi=±ifl=±l /  (3.287)  101  Chapter 3. Calculations  With J = 1, J' = 1 Si = \ and S, = \  \ E 2m  '«  A  2  K#-e1x >|  = 12 j ' 2  1  II I 1 1  1  (3.288)  9'  For the cross term: 5 E (xJI^-ilxfrUjl^-e-X?) = 12 «  1  1  2  2  1 ±  i  2  ><  i i O 2  2  1  m =±|,m,=±|H=±l  2 1 2  ^ m,-  _2 ~3" 3.4  2 1  2  u  /  /  i  i A  1 1 2  —m ,  ,  \  1  i2 y  (3.289)  Full A m p l i t u d e  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 amplitude 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 $ include spin-flavour dependence and are given by equation (1.58). We now do p  a two component reduction and take the nonrelativistic limit (E ,m a  and mass of the quark in the final state, the s quark; E ,m u  s  are the energy  the energy and mass of  u  the initial state quark, the it quark). Neglecting terms of 0(p/m) and disregarding, 2  for the purposes of this calculation, the difference in mass between the strange and up quark, we get, /  +  <?M • p  ,• J  &3 + m  x V $ (xi, x , x ) d x d x d x 3  +  p  2  •p  s  3  3  a  3  2  3  s  JtLu +  m  u  u  Chapter 3. Calculations  f j3-  o  =  103  3gj<: y ua  j 3 -  f-  ,3-  -  -> \(°'  • W f ( ^ ) w  ^  drx d x d x ^\{x ,X2,x )( 5  i  l  2  1  3  z  <  ( 3 ) x  /  / -  —» —• \  )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 — 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 H y =g y J Kp  Kp  4(I) "^ (  ( i )  V Mi) +  *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 2m  2m  p  x  1 1  +  z  "  1  u  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  (x M \x ) = o, p  3)  x  +  (x >i ix;> = o, A  3)  (X>i lxi> = -1/3, 3)  we get 9KUY = A) = andg (Y KUS  o  m  (3.291) p  = X°) = ^ ^ 9 v ^ . o  m  v  (3.292)  Chapter 3. Calculations  104  Using the values = —13.2,  9KpA 9Kp£,°  =  6.0,  obtained from ref.[31], we get = A) = -4.83,  (3.293)  and ^ „ , ( y = S°) = 11.40.  (3.294)  9Ku.(Y  Form factor corrections to these coupling constants will be small [38] due to the —•  —•  fact that the decay momenta (\Py\ and |A;|) are equal. P h a s e space  3.6  Recall that the amplitude for the process was given by S  = Z(2v) 6 (p 4  Yp  + P -(k  4  K  + P ))M  p  Y  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 ]£!! (2*) f> (PK + P -(k + P ))\A4l l^(0)| m>, (E + m )(E' + m ) 4  2  4  =  P  VT  2  Y  3  s  AmKkPV^EiEsE'^E's  3  Am m u  u  ' ' l  a  '  However we want the decay rate per decaying K~p atom; therefore we divide by the 3  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) 8\ + P -(k + P ))\M\ \ip (0)\ m (E + m )(E' + m ) Vd k 16mKk V EiE2E E E E (2TT) 4  2  Pk  p  Y  0  2  2  u  1  2  Vd P  3  3  3  3  f  3  3  4  K  3  The bar in S and Ai indicates that the spin summation has been carried out.  3  u  3  (2TT)  3  Y  Chapter 3. Calculations  S(m  + m  K  = -k)\ \M0)\ <(E3 6ATr m k°E E E E' E E'  (fc° + E ))\M(P  -  p  105  Y  2  2  + m )(E' + m )  Y  s  3  ~  u  3  2  1  K  2  3  l  2  3  (3.296) Taking Ei = E = E[ = E' = m , E = m and using conservation of energy for the 2  2  f* 2  r  =  3  r r  (  S  a  — k°, we find  third quark E' = E + 3  3  u  + p~  mK  (Y  m  E  J<t>=o JB=O Jk=o x  = -k)\ \xp (0)\ (m  + k°))\M(PY  2  2  K  32ir mKk°(m  a  + m -k° K  +  + mx — k°)  2  s  sm6d$d<f>\k\ d\k\. 2  Substituting „r  ,o  r,  = *  W  dW  + *  E  Y  + k°  * W = - E Y -  W E y -  =  we get T  r  =  6  + m  (K m  W)\M(P  -  p  Jw=m  =-fc)|  2  l^(0)| (m  8Trm (m  Y  =  Y  K  2  + m  s  + "« - ° + ">')\M(? k  STrm (m K  Where E  Y  K  + m )(m p  a  + m  K  - k°)  1  v  r  K  a  —  = -j)?. "  + m  K  - k° + m )k°E u  Y  dW  k°)W  v  (3.297) '  = jrriy + (k ) . 0  2  As mentioned previously the principal (§2.5.4) 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 —• all modes  m) u  Chapter 3. Calculations  106  is given by Burkhardt et al. [39] as FaU = 2 W | « 0 ) |  2  P  with W = (0.560 ± 0.135)GeV fm 4  3  p  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)\ to have 2  dimensions GeV fm  3  we multiply by (y^)  3  = (0.19732705359GeV/m) . 3  The branching ratio, therefore, for the process K~p —> F 7 is given by BR -  ° (° 8irm (m k  EY m  K  3.7  +m) \M(P = -fc)| (0.197Gey fm) + m )(m + m - k°) 2W +  K  m  2  K ~ k °  p  u  s  3  Y  K  P  Results and Discussion  Table 3.13 lists the branching ratios to A 7 and E ° 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° uds (PS) prediction agrees with the current experimental rangefl] of (1.4 ± 0.2)10~ . The experimental value for the A 7 3  branching ratio is (0.86 ± 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 m = m 3  u  in the uds amplitudes does not yield the same result as the SU(6) amplitude. The two bases are not equivalent physical descriptions. It 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 W = 0.565GeV fm in good agreement with the result of ref.[39]. 4  3  p  Chapter 3. Calculations  BRxlO"  3  U<£s  PS 7.08 1.36  F = A Y = E°  PV 12.54 3.71  107  SU(6) PS PV 49.99 52.09 3.50 3.17  Table 3.13: Branching ratios for K~p F (Y = A,E°). Results obtained from PS and PV couplings within the uds and SU(6) basis are also tabulated. a = 0.41, m = 0.42, m = 0.70 GeV (parameter set 1). 7  u  BRxlO"  s  uds PS PV 6.39 8.82 1.18 1.57  3  Y — A Y = E°  5*7(6) PS PV 68.92 73.44 11.59 14.07  Table 3.14: Calculated branching ratios for K~p —• F 7 (F = A,E°). a = 0.41, m = 0.42, m = 0.70 GeV. Here only the largest component of the proton and Y wave functions is included. u  BRxlO"  a  uds PS PV 10.52 18.29 2.87 6.30  3  F =A F = E°  SU(6) PS PV 69.24 70.70 4.41 4.77  Table 3.15: Calculated branching ratios for K~p a = 0.32, m = 0.33, m = 0.55 GeV (parameter set 2). u  A/A (%) PS PV 14 63 40 126 — 9  ->  F 7 (F  =  A,E°).  s  0  PS 7 56 —  (%) PV 67 Proton diagram off 115 Y diagram off 2 Contact diagram off  Table 3.16: Contributions from the various diagrams to the invariant amplitude in the uds basis. A / A (%) denotes the percentage of the original A 7 branching ratio and E°/E° (%) the percentage of the original E ° 7 branching ratio, when one of the diagrams is 'switched off', a = 0.41, m = 0.42, m = 0.70 GeV. 0  u  s  Chapter 3. Calculations  BRxlO" Y =A Y = E°  3  uds PS PV 1.46 2.59 1.36 3.71  108  SU(6) PV PS 10.33 10.76 3.17 3.50  Table 3.17: Branching ratios for K~p -> F 7 (Y = A , E ° ) . a = 0.41, m ra = 0.70 GeV. Here <7A"U«(A) is set equal to <7A- S(E°). s  u  = 0.42,  U  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 ° , 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 ° , 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 §2.10.2, the PS and PV results should be identical for an interaction taken over free quark states. Presumably, PS/PV result of the off-shell nature of the quarks in a baryon.  ^ 1 is a  Chapter 3. Calculations  109  Turning to the ratio K~p -» S°7 K~p —• A 7 '  (3.299)  which is independent of the uncertainty in W and so is perhaps more reliable than p  either of the individual branching ratios. Here most of the theoretical models do poorly; none predict the experimental result [1] of 1.71 ±0.30 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 m = 0.55, m = a  u  0.33, and a = 0.32 GeV we get 0.18 - 0.26 for the E°/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°7 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°, 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°7) and phase space factors contribute to the value of the A/E° 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. However, 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 considerations: the E° has isospin one and therefore has (within the uds basis) aflavourwave function with M\ symmetry. This must be combined with a symmetric (ground state) spatial wave function and a x spin wave function to yield an overall symmetric spaceA  spin-flavour wave state. On the other hand, the A has isospin zero which gives rise to a X spin wave function. These symmetry constraints on the quark spin wave functions p  lead to an an extra factor of 3 (from the 6-j symbols) in the K~p —• 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°).  Thus our results appear to be qualitatively reasonable but not quantitatively rigorous, as was to be expected from the NRQM.  Appendix A  FORTRAN PROGRAMS  The following program (KAONCAPTURE) calculates the branching ratios K~p —* 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 K a o n c a p t u r e 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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 2 , 2 , 5 ) , .s(2),y(5,2),Jl,J2, .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay by,spik,qint J  .,ali,bli,b2i,cli,c2i,dli,d2i,d3i,d4i,gli .,VI(5),cllu,cl2u,cl3u,cl4u,cslu,cs2u,cs3u,cs4u .,ell,cl2,cl3,cl4,csl,cs2,cs3,cs4,cpl,cp2,cp3 .,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 rest  111  mass  Appendix A. FORTRAN  PROGRAMS  112  mK=0.49367  kaon r e s t  mass  mlb=l.11563  Lambda b a r y o n r e s t  msb=l.19255  Sigma b a r y o n r e s t  mu=0.42  d e f a u l t u quark  mass  ms=0.7  default  mass  s quark  mass mass  pi=3.141592653589793238 e=dsqrt(4*pi/137.035989561)  elementary  charge  convfac=(0.1973270539)**3  c o n v e r t s f r o m GeV~-2 t o GeV f n T 3  eu=2*e/3  c h a r g e on u quark  es=-e/3  c h a r g e on s quark  ed=es  c h a r g e on d quark  eK=-e  c h a r g e on kaon  alps=(0.41)**2  default  confinement  parameter  Wp=0.56  ! t o t a l r a t e Kp->Y gamma=2WpIpsiKI"1  cpl=0.95  ! p r o t o n admixture  coefficients  cp2=0.25 cp3=0.20  cllu=0.95  ! Lambda uds a d m i x t u r e  coefficients  cl2u=0.07 cl3u=0.28 cl4u=0.08  cll=0.97 cl2=0.18  ! Lambda SU(6) a d m i x t u r e  coefficients  ( i n GeV fm~3)  Appendix A. FORTRAN  PROGRAMS  cl3=0.16 cl4=-0.01 C cslu=0.98  ! Sigma uds a d m i x t u r e  coefficients  cs2u=0.18 cs3u=0.02 cs4u=0.11 C csl=0.97  ! Sigma SU(6) a d m i x t u r e  coefficients  cs2=0.17 cs3=0.17 cs4=-0.00 C gl=-13.2d0  ! strong coupling  c o n s t a n t f o r Kp->sigma  gs=6.0d0  ! strong coupling  c o n s t a n t f o r Kp->lambda  C Call  INTEGRATE  C C  *** MAIN LOOP ***  C When : C  Jl=l  —>  Yp=  Lambda  C  Jl=2 — >  Yp= Sigma  C  J2=l — >  B a s i s = uds  C  J2=2 — >  B a s i s = SU(6)  C Type * , ' E n t e r  1 t o a c t i v a t e p r o t o n diagram  (0=off)>  Appendix A. FORTRAN  Read  (5,*) u l  Type * , ' E n t e r Read  if  1 t o a c t i v a t e contact diagram ( 0 = o f f ) '  ( 5 , * ) u4  Type * , ' E n t e r Read  1 t o a c t i v a t e r e a l p a r t of Y diagram (0=off)'  (5,*) u3  Type * , ' E n t e r Read  1 t o a c t i v a t e imag p a r t o f Y d i a g r a m ( 0 = o f f ) '  ( 5 , * ) u2  Type • . ' E n t e r Read  PROGRAMS  1 f o r parameter d e f a u l t s '  (5,*) L  (l.eq.l)  go t o 2  Type * , ' E n t e r a l p h a Read  ( i n GeV): '  (5,*) a l p h a  alps=alpha**2 Type * , ' E n t e r mu Read  ( 5 , * ) mu  Type * , ' E n t e r ms Read  if  ( i n GeV): '  ( 5 , * ) ms  Type * , ' E n t e r Read  ( i n GeV): »  1 forfull  (5,*) L  (l.eq.l)  cpl=l cp2=0 cp3=0 cllu=l cl2u=0 cl3u=0  go t o 2  wave f u n c t i o n s '  114  Appendix A. FORTRAN  PROGRAMS  cl4u=0 cll=l cl2=0 cl3=0 cl4=-0 cslu=l cs2u=0 cs3u=0 cs4u=0 csl=l cs2=0 cs3=0 cs4=-0.00 DO 30  Jl=l,2  DO 15 J2=l,2 ql=Jl 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))  p h o t o n momentum  ht=0.75*(l/alpst+l/alps)  h i n text  Cc=(27*b*f)/((2*pi)**3*2*dsqrt(2.OdO))  C i n text  Api=dexp(-(0.75*kt**2)/alps)  Ap i n t e x t  Ayi=dexp(-(mlt*kt/mu)**2/(12*alpst))  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)=cl*kt  ! s2 i n t e x t  s(2)=Abs((2*ht-cl)*kt)  ! s i i n text  pl=(mu**2-(E3-mK)**2)  !lambda2  i n text  p2=(kt+E3)**2-mst**2  llambdal  i n text  rtp2=dsqrt(p2)  ! pole of integrand  2 (1 i n t e x t )  qint=0 ali=(4*Pi/dsqrt(alps*alpst))**(1.5) bli=4/dsqrt(3.OdO)*(2*Pi/dsqrt(alps*alpst))**(1.5)*(-1.5) b2i=4/dsqrt(3.OdO)*(2*Pi/dsqrt(alps*alpst))**(1.5)/alps cli=4/dsqrt(3.OdO)*(2*Pi/dsqrt(alps*alpst))**(1.5)*(-1.5) c2i=4/dsqrt(3.OdO)*(2*Pi/dsqrt(alps*alpst))**(1.5)/alpst dli=12*(Pi/dsqrt(alps*alpst))**(1.5) d2i=-8/alps*(Pi/dsqrt(alps*alpst))**(1.5) d3i=-8/alpst*(Pi/dsqrt(alps*alpst))**(1.5) d4i=16/(3*alps*alpst)*(Pi/dsqrt(alps*alpst))**(1.5) gli=-16/dsqrt(alps*alpst)*(Pi/dsqrt(alps*alpst))**(1.5) C C  10  integral  ap=l ay=l bp=0 by=0 call  setyi(l)  ! evaluate F l ( p )  Appendix A. FORTRAN  call  setyi(2)  call  calc(l)  PROGRAMS  ! evaluate Gl(u)  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 call  setyi(l)  ! evaluate F2(p)  call  setyi(2)  ! e v a l u a t e G2(u)  call  calc(2)  ap=0  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) call  setyi(l)  ay=(3-mlt/mu)*kt/DSQRT(6.OdO) call  setyi(2)  call  calc(3)  if  ( u . e q . l ) go t o 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 call  setyi(l)  ap=0 ay=(3-mlt/mu)*kt/DSQRT(6.OdO) call  setyi(2)  call  calc(4)  C C  Iq integral qint=l ap=3*kt/DSQRT(6.0dO) bp=3/DSQRT(6.0dO) ay=-mlt*kt/(mu*DSQRT(6.OdO)) by=-bp call  setyi(l)  ap=0 ay=(3-mlt/mu)*kt/DSQRT(6.OdO) call  setyi(2)  call  calc(5)  C 17  DO 23 11=1,5 cxpart=c3*GIl(Jl,32,II)*ul+c2*GI2(JI,32,II)*u2 rpart=pole(Il)*pi/(2*rtp2)*c2*u3 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 rpart=pole(Il)*pi/(2*rtp2)*c2*u3 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, B a s i s = u d s ' , $ )  310  FORMAT(' Y= Sigma, B a s i s = u d s  320  FORMATC  330  FORMAT(' Y=Sigma, B a s i s = S U ( 6 ) : ' , $ )  > ,$)  Y=Lambda, Basis=SU(6) : ' ,$)  open(unit=6,carriagecontrol='FORTRAN',status='OLD') if  ((jl.eq.l)  .and. ( J 2 . e q . l ) ) t y p e 300  if  ((jl.eq.2)  .and. ( J 2 . e q . l ) ) t y p e 310  if  ((jl.eq.l)  .and. ( J 2 . e q . 2 ) ) t y p e 320  if  ((jl.eq.2)  .and. ( j 2 . e q . 2 ) ) t y p e 330  TYPE *,'M~2 ( P S ) = ' , M p s ( J l , J 2 ) Mps(JI,J2)=phase*convfac*Mps(JI,J2)/(2*Wp) TYPE * , ' R a t i o TYPE *,'M~2  (PS)=',Mps(JI,J2)  (PV)=',Mpv(Jl,J2)  Appendix A. FORTRAN  PROGRAMS  Mpv(Jl,J2)=phase*convfac*Mpv(Jl,J2)/(2*Wp) TYPE * , ' R a t i o ( P V ) = ' , M p v ( J l , J 2 ) 15  CONTINUE  30  CONTINUE STOP END  SUBROUTINE  setyi(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 if  ( q i n t . E q . l ) GO TO 34  ! i f doing  15 do c o s t h e t a n u m e r i c a  y(l,I)=-aO(l,I)*bO(l,I)/s(I)+aO(l,I)*bO(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)=a0(l,I)*b0(2,I)/s(I)+a0(2,I)*b0(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)—aO(3,I)*bO(3 I)/s(I) f  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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 2 , 2 , 5 ) , .s(2),y(5,2),Jl,J2, .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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 2 , 2 , 5 ) , .s(2),y(5,2),Jl,J2,  Appendix A. FORTRAN  PROGRAMS  .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 p r o t o n 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 t h e c o n t a c t g r a p h .  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 F 1 ( X ) IMPLICIT  REAL*8 (A-H,K,M-U,W-Z)  IMPLICIT  COMPLEX*16 V  COMMON /WXCOM/ XX(IOO), .XXT(IOO),  WWT(IOO),  WW(IOO), WXXVEC(IOO), LIM,  LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5),  .s(2),y(5,2),Jl,J2, .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 F 2 ( X ) IMPLICIT  REAL*8 (A-H,K,M-U,W-Z)  IMPLICIT  COMPLEX*16 V  122  Appendix A. FORTRAN  PROGRAMS  COMMON /WXCOM/ XX(IOO), WW(IOO), WXXVEC(IOO), LIM, .XXT(IOO), WWT(IOO), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 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 F F ( 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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 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 If  ( q i n t . E Q . O ) GO TO 36  FF=sing(X,I) GO TO 37 36  tl=0 t2=0 DO 35 11=1,5 tl=tl+y(Il,I)*X**Il t2=t2+y(Il,I)*(-X)**Il  35  CONTINUE tlp=DEXP(-ht*X**2) FF=0  •  Appendix A. FORTRAN  IF  PROGRAMS  124  ( t l p . E q . 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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 2 , 2 , 5 ) , .s(2),y(5,2),Jl,J2, .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) 10  ! Do n u m e r i c a l  cos t h e t a  integration  CONTINUE sing=tl 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),Jl,J2,  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=tl*tlp 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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 2 , 2 , 5 ) , .s(2),y(5,2),Jl,J2, .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint .,ali,bli,b2i,cli,c2i,dli,d2i,d3i,d4i,gli .,VI(5),cllu,cl2u,cl3u,cl4u,cslu,cs2u,cs3u,cs4u .,cl1,cl2,cl3,cl4,cs1,cs2,cs3,cs4,cpl,cp2,cp3 C C  Main f o r m u l a s  f o r the amplitude  C  I n c l u d e s s p i n summations,  C IF  (Yp.EQ.2) GO TO 94  C Yp=Lambda  as g e n e r a t e d  f l a v o u r and space  by SMP. matrix  elements.  Appendix A. FORTRAN  IF  PROGRAMS  ( b a s . E q . 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  $  * C P l + ( - 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  $  D 2 I ) + V I ( 4 ) * ( l 8 * C L 2 U * C P 2 * D 4 I + ( - 9 ) * 2 ** 0.5D0*  $  **  ** 0.5D0*CL2U*CP3*  CL2U*CP3*D4I)+VI(3)*(18*CL2U*CP2*D3I+18*2  $  ** 0.5D0*C2I*CL2U*CPl+(-9)*2  $  C P 3 * D 3 I ) + ( - 2 ) * C L 4 U * C P 3 * G l I * V I ( 5 ) ) / 2 ** 0.5D0) GO TO  ** 0.5D0*CL2U*  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  $  B 2 I * C L l * C P 3 + ( - 1 8 ) * 2 ** 0.5D0*CL2*CP2*D2I+(-9)  $ $ $ $  *2 **  ** 0.5D0*  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  $  C 2 I * C L 3 * C P l + ( - 1 8 ) * 2 ** 0.5D0*CL2*CP2*D3I+(-9)  $  *2 ** 0.5D0*CL3*CP3*D3I)+2 **  ** 0.5D0*  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*  $  V I ( 4 ) + 2 * C L 4 * C P 3 * G l I * V T ( 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 . 5 D 0 * A l I * C L l * C P l + ( - 1 8 ) * 2 **  $  CP2+(-9)*2  $  B1I*CL1*CP3+18*2 ** 0 . 5 D 0 * C l I * C L 3 * C P l + ( - 1 8 ) *  $  2 **  $  0.5D0*A1I*CL2*  ** 0.5D0*A1I*CL3*CP3+18*2 ** 0.5D0*  0.5D0*CL2*CP2*DlI+(-9)*2  ** 0.5D0*CL3*CP3  *D1I)+T1+T2) / 6 ** 0.5D0) GO TO  C Yp=sigma 94  99 0  I F (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  $  * C S l U + ( - l ) * 2 ** 0.5D0*AlI*CP3*CS3U+2 ** 0.5D0  $  *BlI*CP3*CSlU+2*2 **  $ $ $ $  **  ** 0.5D0*A1I*CP1  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 ** +2 **  0.5D0*CP3*CS2U*D2I)+VI(4)*(2*CP2*CS2U*D4I 0.5D0*CP3*CS2U*D4I)+VI(3)*(2*CP2*CS2U*  Appendix A. FORTRAN  PROGRAMS  $  D3I+2*2 ** 0.5D0*C2I*CPl*CS2U+2  $  CS2U*D3I)+(-2)*CP3*CS4U*GlI*VI(5)) / (2 ** 0.5D0  $  ** 0.5D0*CP3*  *3 ** 0.5D0))  C  type  *,cpl,cp2,cp3  C  "tyP  *,cslu,cs2u,cs3u,cs4u  C  type  *,VI(1)  C  typ  *,Vampl  6  e  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*  $  D 4 I ) + V I ( 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  129  $  (-4)*ClI*CPl*CS2+(-2)*CP2*CS3*DlI+2*CP2*  $  C S 4 * D l I + ( - 2 ) * C P 3 * C S 2 * D l I + ( - 4 ) * 2 ** 0.5D0*A1I  $  * C P l * C S l + ( - 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  $ 99  PROGRAMS  ** 0.5D0*CP3*CS3*D1I  +2 ** 0.5D0*CP3*CS4*D1I)+T1+T2) / 2 ** 0.5D0) 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), L I M 2 , G I 1 ( 2 , 2 , 5 ) , G I 2 ( 2 , 2 , 5 ) , G I 3 ( 2 , 2 , 5 ) , .s(2),y(5,2),Jl,J2, .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint .,ali,bli,b2i,cli,c2i,dli,d2i,d3i,d4i,gli .,VI(5),cllu,cl2u,cl3u,cl4u,cslu,cs2u,cs3u,cs4u .,ell,cl2,cl3,cl4,csl,cs2,cs3,cs4,cpl,cp2,cp3 .,A(6),I2(6),I3,L,I1, .gt,gs,gl,mp,mtb,msb,mlb,mst,ms,mu,f,b,sixj IF C  (Yp.Eq.2) GO TO 110  Yp=Lambda mtb=mlb  Appendix A. FORTRAN  PROGRAMS  gt=gl/3*mu/mp*dsqrt(6.OdO) f=  ( l . D 0 / 1 2 . D 0 ) * ( 2 * 3 ** 0 . 5 D 0 * ( 2 * C L l * C L l U + ( - 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  $  * C L 4 + ( - l ) * 1 2 ** 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 . D 0 / 1 2 . D 0 ) * ( 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  I F (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  140  PROGRAMS  131  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 o f F a c t o r i a l l e s s t h a n z e r o ! ' 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 S u b r o u t i n e t o d e t e r m i n e t h e p o i n t s and w e i g h t s f o r e v a l u a t i n g t h e 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/ X X ( l O O ) , WW(IOO), .XXT(IOO), WWT(IOO),  WXXVEC(IOO), LIM,  LIM2,GI1(2,2,5),GI2(2,2,5),GI3(2,2,5),  .s(2),y(5,2),Jl,J2, .a0(3,2),b0(3,2),pole(5),ht,pl,p2,rtp2,ap,bp,ay,by,spik,qint  Appendix A. FORTRAN  PROGRAMS  132  .,ali,bli,b2i,cli,c2i,dli,d2i,d3i,d4i,gli .,VI(5),cllu,cl2u,cl3u,cl4u,cslu,cs2u,cs3u,cs4u ., 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),12(6),I3,L,I1 11=4 A(l)=.5  ! Intervals  f o r numerical cos t h e t a  integration  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 I2(l)=16 A(2)=.5 I2(2)=16  ! Intervals  f o r n u m e r i c a l momentum  integration  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] (B.300) (B.301) /  A sin(Ag)e 2 3  a  A  dA =  v  (B.302)  e 2° ,  *  v  2  (B.303) 7o  8a  v  (B.304)  5  Using (B.300) we get ay=  = - L \ ^  [Voooe'^A  / e ^ ^ ^ A  Similarly a — p  = j V>ioe"» d A = A ,  y  3  0  2 1 M V3v/4lr"\  /(|_ A )e-^ ^ J 2 a  y/lt  4fy^\ ^ §  ((  V3V"A/ Similarly 6  P  =  134  U  «  2  3  )2 A  2  ;  2  ;  2  2 + i X  ^d A 3  Appendix B. Details of Integrals  Finally c  135  = J ^oioe* ' d A X fy  Y  3  Ia Xe-^l ^ \l^-cos9A sin9dOd<f>d\  ^  x2+iX  \ y/lt J  \ 0  2  cos  x  V 47T  -A  Ai fy/ir^  qye K Ai (sfii^ Similarly c„ = — —  3  a \ a 9I e  ^ .  P  Angular integrals When performing the angular integrals we will need the general formula (u = cos 9) f  1  e ( spu  ai  + a pu + a p )(6i + b pu + b p )p du = £ ^(p'e"^ + (-p)''e ) 2  2  2  3  2  2  ps  3  Ju=—1  (B.305) where  = —^- +  m  _ r} =  ab x  2  2  773 = rji  _  +  +—  ab 2  —,  (B.306)  2a b 5—,  x  2  2  (B.307)  -ai6 a6 a fe a &! a 6 +— +- -, s s s* s s ab ab _ ab 1 , 775 = . s s s 3  2  2  2  3  3  3  2  T z  2  3  3  2  3  3  (B.308) (B.309)  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, GJMI  r°°  1  G/fAi)  r°°  1  2 x2 • du = ^ i - ^ l i m / . du.Jo r - A i - i e 2 0 J-00 u — \{ — le 2  Neglecting terms of 0(e) and defining 2  (B.310)  Appendix B. Details of Integrals  136  we have f°° / ~~5 J-oo U  l  f lira /  1  ~du =  — Af — It  R  1  —  R-KX> J-R  du.  (u + z)(u  — z)  y^[l(±±*fLdu _l  =  fl-+oo [ y c  (H±£tl \ d u  U —Z  JC  R  U —Z  (B.311)  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 — a)  dx = 2mf(a),  (B.312)  we get lim lim £ ^  fl-oo ^o Jc £  7T2 • =—  ^ — du — lim ,  u - z  t-+o 2 A + f a  Ai  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  Tc (u + z)(u Jc, R  z)  du  irR  < \R-\z\\  2  where irR is the length CR. The desired limit is now evident; that is, 1  lim f  R-*co  R—>ooJCJc R  (u  +  z)(u  —  z)  du = 0.  Appendix C  SMP Procedures  T h e following commands define the wave functions a n d rules for evaluating the matrix elements within the  SMP environment.  T h e y are contained within the file  "WAVEFN.DEF".  T h e following translations may be useful: psip=  p s i l = *jv  psis=  N 8 2 s s = N^S  L2S=  A 5 2  cl=  x  ps=  <f>  cr=x  etc  A  S  etc  p  A <f> p r l = 4>  Pl=  P  L  p l l = <f>\ etc  A  Ps000=  P r l 2 0 f = $^oo etc. W h e r e the / denotes final state; that is the spatial wave function for the Y. For the m a t r i x elements: p l * p r p denotes  (cf> \V \4> ) P  A  P  +  p  c r ~ 2 denotes (x \<?• t\x ) PsOOf *Ps000 denotes  ($ooolQI$ooo)  psip:cpl*N82ss+cp2*N82ssp+cp3*N82sm  /* P r o t o n wave f u n c t i o n  psil:If[uds=l,cllu*L2S+cl2u*L2Sll+cl3u*L2Srr+cl4u*L2Srl,\ cll*L82ss+cl2*L82ssp+cl3*L82sm+cl4*L12sm]  137  /* Lambda wave f u n c t i o n  Appendix C. SMP Procedures  138  psis:If[uds=l,cslu*S2S+cs2u*S2Sll+cs3u*S2Srr+cs4u*S2Srl,\ csl*S82ss+cs2*S82ssp+cs3*S82sm+cs4*S102sm]  /* Sigma wave f u n c t i o n  N82ss:(cr*prp+cl*plp)PsOOO/Sqrt[2] N82ssp:(cr*prp+cl*plp)Ps200/Sqrt[2] N82sm:(P1200*(cr*prp-cl*plp)+Pr200*(cr*plp+cl*prp))/2  L82ss:(cr*prl+cl*pll)PsOOf/Sqrt[2] L82ssp:(cr*prl+cl*pll)Ps20f/Sqrt[2] L82sm:(P120f*(cr*prl-cl*pll)+Pr20f*(cr*pll+cl*prl))/2 L12sm:(P120f*cr-cl*Pr20f)pal/Sqrt[2]  S82ss:(cr*prs+cl*pls)PsOOf/Sqrt[2] S82ssp:(cr*prs+cl*pls)Ps20f/Sqrt[2] S82sm:(P120f*(cr*prs-cl*pls)+Pr20f*(cr*pls+cl*prs))/2 S102sm:(P120f*cl+cr*Pr20f)pss/Sqrt[2]  L2S:PsOOf*pl*cr L2Sll:P1120f*pl*cr L2Srr:Prr20f*pl*cr L2Srl:Prl20f*pl*cl  S2S:Ps00f*ps*cl S2Sll:P1120f*ps*cl S2Srr:Prr20f*ps*cl S2Srl:Prl20f*ps*cr  Appendix C. SMP Procedures  'Flavour  Matrix  element  pl*prp:1 pl*plp:0  prl*prp:Sqrt[2/3] pll*prp:0 pal*prp:-l/Sqrt[3] prl*plp:0 pll*plp:0 pal*plp:0  ps*prp:0 ps*plp:-l/Sqrt[3]  prs*prp:0 prs*plp:0 pls*prp:0 pls*plp:-Sqrt[2]/3 pss*prp:0 pss*plp:-l/3  'Space M a t r i x  elements  Ps00f*Ps000:a PsOOf*Ps200:b/Sqrt[2] PsOOf*P1200:-b/Sqrt[2]  results  Appendix C. SMP Procedures  Ps00f*Pr200:0 Prl20f*Ps000:0 Prl20f*Ps200:0 Prl20f*P1200:0 Prl20f*Pr200:g/3 Prr20f*PsOOO:0 Prr20f*Ps200:a/Sqrt[2] Prr20f*P1200:a/Sqrt[2] Prr20f*Pr200:0 P1120f*Ps000:c P1120f*Ps200:d/Sqrt[2] P1120f*P1200:-d/Sqrt[2] P1120f*Pr200:0 Ps20f*PsOOO:c/Sqrt[2] Ps20f*Ps200:(a+d)/2 Ps20f*P1200:(a-d)/2 Ps20f*Pr200:0 P120f*PsOOO:-c/Sqrt[2] P120f*Ps200:(a-d)/2 P120f*P1200:(a+d)/2 P120f*Pr200:0 Pr20f*PsOOO:0 Pr20f*Ps200:0 Pr20f*P1200:0 Pr20f*Pr200:g/3  Appendix C. SMP Procedures  'Spin  Matrix  141  elements  cr~2:sfr cr*cl:0 cl*cr:0 cl-2:sfl  SMP  1.6.2  Mon Feb  4 16:14:28  1991  uds:l;  /* uds b a s i s  <"wavefn.def";  /* Load  psil  /* C a l c u l a t e  psip;  definitions <Lambda|V|proton>  Ex[/,];  /* Expand l a s t e x p r e s s i o n  Rat [°/,] ;  /* R a t i o n a l i z e l a s t e x p r e s s i o n  Fac[%];  /* F a c t o r i z e  last  amplitude  (using  definitons)  o v e r a common d e n o m i n a t o r  expression  CbC'/.^a.b.c.d.g}] /* Combine c o e f f i c i e n t s  i n last  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  cllu  cp2 s f r )  1/2 + d (-3cl2u cp3 s f r + 3 2  1/2  c l 2 u cp2 s f r )  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  psis psip;  /* C a l c u l a t e <Sigma 0|V|proton>  amplitude  Ex ['/.]; Rat ['/.] ; F a c ['/,]; Cb[ /.,{a,b,c,d,g}] ,  1/2 -(a  ( 6 c p l c s l u s f l - 3cp3 c s 3 u s f l + 3 2  cp2 c s 3 u 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 c s 2 u s f l + 3 2  cp2 c s 2 u s f l )  1/2 + 6c c p l c s 2 u s f l + 2 #0[12]: 1/2 6 2  1/2 3  cp3 c s 4 u g s f r )  Appendix C. SMP Procedures  143  /* Now do same t h i n g i n SU(6) b a s i s  uds:0; <"wavefn.def"; psis  /* f o r t h e Sigma  psip;  Ex ['/.]; Rat [*/.] ; F a c ['/.]; Cb[ /.,{a,b,c,d,g}] ,  -(a  (12cpl  c s l s f l + 6cp2 c s 2 s f l + 3cp3 c s 3 s f l - 3cp3 c s 4 s f l  1/2  1/2 - 3 2  cp2 c s 3 s f l + 3 2  cp2 c s 4 s f l  1/2 - 3 2  cp3 c s 2 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 c s 2 s f l + 3cp3 c s 3 s f l - 3cp3 c s 4 s f l 1/2  1/2 +  32  cp2 c s 3 s f l - 3 2  cp2 c s 4 s f l  Appendix C. SMP Procedures  144  1/2 +32  cp3 c s 2 s f l )  + g (2cp3 c s 3 s f r + 2cp3 c s 4 s f r ) ) #0[19]: 1/2 36 2  psil  psip;  /* and f o r t h e Lambda  Ex[°/.]; Rat [Fac ['/.] ] ; Cb[%,{a,b,c,d g}] f  -(a  ( - 2 4 c l l c p l s f r - 12cl2  cp2 s f r - 6 c l 3 cp3 s f r + 6 c l 4 cp3 s f r  1/2 - 6 2  1/2 c l 2 cp3 s f r - 6 2  c l 3 cp2 s f r  1/2  +62  c l 4 cp2 s f r )  1/2 + b  (12cll  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  1/2 - 12  + d  (-12cl2 cp2  2  cl2 cpl sfr)  s f r - 6 c l 3 cp3  s f r + 6 c l 4 cp3 s f r  1/2  1/2  +62  c l 2 cp3  sfr +62  c l 4 cp2  sfr)  1/2  - 6 2  #0[25]  + g  (-4cl3  cp3  s f l - 4 c l 4 cp3 s f l ) )  1/2 24  6  c l 3 cp2  sfr  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 Publishing 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., D 9 , 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.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084976/manifest

Comment

Related Items