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## Material Information- Title:
- Relativistic parametrization of the Pâ‚ƒâ‚ƒ resonance in pion-nucleon scattering
- Creator:
- Tschang, Yin-po, 1944-
- Publication Date:
- 1972
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- English
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- vii, 63 leaves. : ; 28 cm.
## Subjects- Subjects / Keywords:
- Energy ( jstor )
Kinematics ( jstor ) Mathematical constants ( jstor ) Nucleons ( jstor ) Particle resonance ( jstor ) Phase shift ( jstor ) Physics ( jstor ) Pions ( jstor ) Resonance scattering ( jstor ) Scattering amplitude ( jstor ) Meson resonance ( lcsh ) Mesons -- Scattering ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaves 60-62.
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- Typescript.
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- Vita.
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ocm14106089 00582428 ( ALEPH )
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RELATIVISTIC PARAMETRIZATION OF THE P' RESONANCE IN PION-NUCLEON SCATTERING 33 By Yin-po Tschang A Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1972 RELATIVISTIC PARAMETRIZATION OF THE P' RESONANCE IN PION-NUCLEON SCATTERING By Yin-po Tschang A Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1972 ACKNOWLEDGMENTS The author wishes to thank his dissertation advisor, Professor M.T. Parkinson, for suggesting this interesting problem and for providing guidance with immense patience and diligence throughout this work. He would also like to ex press his deep gratitude for the help and advice of two dear teachers, Professor A.A. Broyles and Dr. J. McEnnan. They have helped build a foundation from which the present en deavor has started. Thanks are extended to the Department of Physics for assistantships and an instructorship, to the Graduate School for a fellowship, and to the College of Arts and Sciences for a research grant at the University of Florida Computing Center. The technical help received from Dr. F.E. Riewe and the U.F.C.C. staff is also to be thanked. ii TABLE OF CONTENTS page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi ABSTRACT vii CHAPTERS I. INTRODUCTION 1 II. PION-NUCLEON SCATTERING 4 1. Introduction 4 2 Kinematics 5 3. The Scattering Amplitude 9 4. The Phase Shift 12 5. The Br eit-Wigner Formula 15 6. The Chew-Low Model 16 7. The Effective Range Approximation l8 III. THE K-MATRIX FORMALISM 20 IV. PHASE SPACE FUNCTION 24 1. The Regular Two-Body Phase Space Function 24 2. Kinematic Singularities of Helicity Amplitudes 25 3.The Phase Space Function 29 V.PARAMETRIZATION OF THE P* RESONANCE 31 iii TABLE OF CONTENTS (CONTINUED) page 1. Introduction 31 2. Basis of Computation j 32 3. Phase Shifts.. 34 4. Left Half-Plane Singularities.. 36 5. Contributing Channels.. 37 6. Computation 39 VI. RESULTS AND CONCLUSIONS 41 1. Comparison with Breit-Wigner Formula 41 2. General Description of Phase Shift Fits.. 42 3. The Resonance Mass 47 4. The Subtraction Constant 47 5. The Coupling Constants 48 6. General Remarks 51 APPENDICES A. MECHANICS OF CALCULATION ...... 53 B. COMPUTER PROGRAM 54 REFERENCES 60 BIOGRAPHICAL SKETCH 63 iv LIST OF TABLES Table page I. Characteristics of Contributing Channels 33 II. Parameters from Phase Shift Fits 43 III.Phase Shifts Fitted According to Carter et al. 44 IV.Phase Shifts Fitted According to Compilation Data.. 45 v LIST OF FIGURES Figure page 1. Diagrams 17 vi Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RELATIVISTIC PARAMETRIZATION OF THE P^3 RESONANCE IN PION-NUCLEON SCATTERING By Yin-po Tschang December, 1972 Chairman: Professor M. T. Parkinson Major Department: Physics and Astronomy A relativistic generalization of the Breit-Wigner for mula is further improved and applied to the study of the P^3 resonance in pion-nucleon scattering. Reasonable fits with the P^3 phase shift data are obtained. The new parametriza- tion reduces to a simple Breit-Wigner form in the neighbor hood of a resonance. Away from the resonance, however, the new parametrization has been seen to be far superior to the Breit-Wigner formula. This parametrization opens up a pos sible way of attacking the problem of hadron dynamics. In addition, values of the coupling constants C^^ and C^^A are predicted by this parametrization from phase shift fits. vii CHAPTER I INTRODUCTION ' Direct channel resonances have been assumed to dominate the scattering amplitudes of strong interactions over a wide energy range. Empirically, this idea seems to be well sup ported.1 Theoretically, it is now believed that the duality between direct channel resonances and cross channel exchanges allows the resonance-dominance assumption to coexist with the 2 usual exchange mechanisms. In particular, the dominance of a nearby pole, plus the requirements of unitarity, have been used to describe the behavior of the scattering amplitude in the neighborhood of that pole. The most commonly used parametrization for a pole is the Breit-Wigner formula. This treatment, while exact at o the pole, has the following deficiencies: (1) It is non-relativistic. (2) Only open channels are used, nearby singularities can thus be omitted. (3) Poles are produced on all sheets of the scattering am plitude. (4) The position of the pole is unrelated to partial widths, thus the parametrization does not say anything concern ing the dynamical origin of the resonance. 1 2 Previous works by Parkinson and others have succeeded in building up a relativistic K-matrix formalism which is very similar to the Breit-Wigner formula in form, but does O A not have any of the above objections. ** This work represents an extension of the new parametri- zation, which has been rather successful in describing the rho meson, to the case of a high-spin baryon resonance, the P^ in pion-nucleon scattering. It is intended to be the first in a series of parametrizations of direct channel poles to be used as a rigorous check of duality. It is also hoped that the series of works can shed some light on the signifi cance of the pomeranchon. In order to make a systematic study of various reso nances, the phase space function has been studied in detail. It is found that if the widths of resonances are ignored, i.e., if there is no "spread" in the masses of resonances, it is possible to write down simple general phase space func tions for all channels involving such particles. In this work the zero width assumption has been used for certain channel thresholds. The refinement of non-zero widths will be attempted later. Using recent phase shifts of Carter et al.^ and a com pilation of older results,^ the parametrization has been used to determine the mass and width of the P^ resonance, as well as the coupling constants and CttN *A * General aspects of pion-nucleon scattering and its kin ematics will be studied in Chapter II. In Chapter III the 3 general parametrization formula will be developed. Chapter IV deals principally with the phase space function which plays an important role in the parametrization. Chapter V studies the problem of parametrization of the P^3* ^ a^-so includes a discussion of data handling. The last chapter sums up the findings of this work. The notations of reference 4 will be followed. The is sometimes denoted by A for short. CHAPTER II PION-NUCLEON SCATTERING 1. Introduction The interaction between pions and nucleons is one of the fundamental problems of subatomic physics. For nuclear physics, the determination of the nuclear potential, the long-range component of which is dominated by the interaction between pions and nucleons, is to be a giant step forward. It will bring theoretical nuclear physics to the same level of development as present day atomic and molecular physics, where the many-body problem is the only outstanding obstacle. To particle physics, pion-nucleon scattering is espe cially important because of the availability of high inten sity pion beams at a wide energy range. The large amount of data collected from pion-nucleon scattering not only serves as a check on most theoretical concepts in hadron physics, it is also the source of many new ideas. In particular, phase shift analyses of partial wave am plitudes for this scattering process have produced evidence for a series of new particles or resonances, and this dis covery has been extended to other scattering processes. The 4 5 presence of a large number of resonances, all of which are considered as essential constituents as the pion and the nucleon, is a new feature only found in particle physics. Their presence provides very severe constraints on all dynamical models of strong interactions. Various detailed dynamical models have found different degrees of success in describing experimental results. No table among low energy models are the Chew-Low theory and the effective range approximation. Among high energy models are Regge pole theory and the Veneziano model. The low en ergy theories will be briefly discussed later in this chap ter. 2. Kinematics In the present work only Lorentz-invariant dynamical variables will be used. The relationship between the invar iants and ordinary physical variables such as energy and mo mentum is established here. In addition, only two-body reactions will be considered From a practical point of view, this restriction is un avoidable because three- or more-body kinematics is compli cated and the kinematic singularities of corresponding heli- city amplitudes are not known. A theoretical implication of this restriction will be taken up later. A typical reaction is, therefore, 6 1 + 2 3 + 4 where particles ia,l>..)4 have four-momenta p^, rest masses ek, spins s^, and helicity components X^. If particles 1 and 2 are a pion and a nucleon, corresponding to the initial state of the scattering, the above reaction will be called the direct channel or s channel. By contrast, the cross channels are 1 + 3 -+ 2 + 4 (t channel) 1 + 4 2 + 3* (u channel) The terminologies will be made reasonable in a moment. In the direct channel, particles 3 and 4 may be any one of a number of combinations. They may be the pion and the nucleon in the case of elastic scattering, or the pion and the A, the pion and the Pjj resonance, in the case of in elastic scattering. These different combinations will also be called "channels", or more correctly, the exit channels. The basic Lorentz-invariant dynamical variable in the direct channel is the total center-of-mass energy squared, 2 2 S (Pj + P2) = (p3 + P4) (1) 7 where the are measured in the direct channel. The momen tum transfers t and u are defined by t (Pj P3>2 = (p2 P4)2 (2) u (pj p4)2 = (p2 P3)2. (3) The three invariants are not independent, s + t + u = 2^ m2 (4) where the summation is over the ir.cident and the exit chan nels . If 0 is the scattering angle in the center-of-mass frame of the direct channel, then sin 0 = 2[s4>(s,t)]^/S12(s)S34(s) (S) cos 0 = [s2 + 2st sZ.m| + (T12U12T34U34^2^/si2^s^S34^s^ (6) where - stu B(T,3013T24024)i t(T1212I3434)4 + - (m2 + m2 m2 m2)(m2m2 m2m2) (7) Tij = (mi + mj)2 (8) 8 (9) i Sij(s) = C(s Tid)(s Uid)]2. (10) In the two-dimensional space spanned by s, t and u, not all points are "physical". In other words, there are regions in this space that are inaccessible for realistic scattering processes. This follows from the fact that the four-momenta are constrained by 2 m, . (11) The boundary of the physical region is given by (12) All square roots in the above are chosen to be positive ' in the physical region, so that the sine of 0 is positive in the direct channel. threshold. In the center-of-mass frame of the direct channel, the total energy is JL W s2 (13) 9 and the magnitude of the three-momentum of either particle i or particle j is kid = Sid(s)/2W. (14) In the laboratory system, where the nucleon target is at rest, the total energy of the incident pion is E = (s m2 ^i2)/2m (15) where m is the nucleon mass and p the pion mass. 3. The Scattering Amplitude An experiment in strong interactions consists of ob serving the initial state of two or more particles, allowing them to interact, and then observing the final states of the arbitrary number of particles resulting. The S-matrix ele ment, Pfi = represents the probability of |f> being the final state, given the initial state |i> . It is well established that strong interactions have very short ranges. Since all weak but long-range interac tions such as electromagnetism and gravitation are ignored 10 in theoretical investigation and subtracted out of experi mental results, the finite range of strong interactions guarantees that |f> and |i} are free particle states under ordinary experimental situations. In the following all par ticles in the incident and exit channels are regarded as free, and suitable kinematics is applied. The helicity components of particles in a given channel can be used instead of the spin projection along a fixed axis in space for a complete description of hadron states. When this is done, a helicity amplitude can be defined in terms of the S-matrix. In particular, the helicity amplitude h^ is defined by where SA|i The differential cross section is given by (18) where k is the magnitude of the center-of-mass three momen tum in the direct channel, for the incident particles. 11 The helicity amplitude defined here is related to 7 the helicity amplitude of Jacob and Wick by k f JW and to the usual helicity amplitude such as seen in Wang o and King7 by (19) 8 (20) where q is the magnitude of in the exit channel. Partial wave expansion follows: the center-of-mass three-momentum for S Ap and % are defined as % -k + (2.) % (2J + l)h^d^(9) (22) where df (0) is the d function of the rotation matrix ele- Ap ment dAp(0) = The total cross section, according to equations (l8) and (22), is therefore given by 12 (24) 4 The Phase Shift According to equation (l6), and since there must be unit probability for an initial state to end up in some final state} i = zt = Here the completeness of final states |f> has been assumed. If, in addition, as is customary, the initial state is normalized, then it follows that SfS = i (26) since the initial state \y is arbitrary. This is the unitarity relation for the scattering matrix. Combining equations (21) and (26), it is found that the unitarity relation for the partial wave is SJV = JU (27) In the case of elastic pion-nucleon scattering, the 13 partial wave scattering matrix has only four helicity compo nents: ++ +- -+ where + denotes +5 and denotes -5. By parity invariance, it is found that the first equals the last, and the second equals the third. Using these relations, the unitarity re lation for elastic scattering becomes + J* J J* J S++S+- + S+-b++ 0. (28) Define (29) and the unitarity relation becomes even simpler: (30) Conventionally, one insures that this condition is sat isfied by writing where the phase shift bj is a real function of W. (3D 14 In terms of the helicity amplitudes, 4-jrk 2ik where 1 4ik [e 2i J- 2i + e J+ - 2] - i D + hj] 1 ij(W) , h(W) Â£ e sin J(W). (32) (33) A similar expression can be obtained for h^_. Above the first inelastic threshold, the phase shift can no longer be real. The unitarity relation now reads = l n^. The phase shift must therefore have a positive imaginary part. However, it is conventional to factor out the imagi nary part, in the form of an absorption parameter t|, and make the phase shift real even for inelastic processes. Hence equation (31) becomes (35) T|j is equal to unity when the scattering is purely elastic, and it is smaller than unity when the scattering is partial ly inelastic. 15 5. The Breit-Wigner Formula From the definition of h, equation (33) it can be found that, if inelasticity can be ignored, J = I 1 l k cot 6 j i * (36) A resonance is said to exist at W if cot 6_. goes r j * through zero for this value of W with a negative derivative. If a power series expansion of cot 6j is made, then in the neighborhood of cot 6J (Wr w)/Â£r r> o (37) and r/2 - w ir/2' (38) Inserting this result into equation (24) and neglecting the other partial waves, the resonance cross section is found to be O' (W) res (j + 1) Ja. (Wr w) 2 + r2/4 (39) This is the well-known Breit-Wigner formula. At-resonance energy, the partial-wave cross section is, 16 according to the Breit-Wigner formula, which is commonly called the unitarity limit, since this is the absolute maximum of a partial-wave cross section by equation (33) It should be pointed out that in the derivation of equation (39) the fact that h^ and h*| | (h^_) have different normalization is used. 6. The Chew-Low Model The Chew-Low model is the first model that successfully predicted the gross characteristics of pion-nucleon scatter ing at pion energies below a few hundred MeV.1 In particu lar, it predicted the resonance. A non-relativistic pseudovector interaction between pions and nucleons is assumed in this model, from which, with the additional assumption of a nucleon at rest, the interaction hamiltonian of the following form is obtained: Ht Z (V a + vV) (41) j. m m m mm Vj if( m m m tion operators for single pions, k is the three momentum and 17 G the energy of the pion, and m represents all pion quantum numbers, and lastly, n(k ) is the Fourier transform of the nucleon charge density. In order to ensure the convergence of the necessary integrations n(k ) must be cut off above some k . max By virtue of the pseudovector interaction and the as sumption that the nucleon remains rigidly at rest, the model predicts that all scatterings will be in the P-state. The Low scattering equation is solved with the assump tion that the only two diagrams of importance are 7T \ IT TT ' N N N jir - N Figure 1. Diagrams Finally, the following relation is derived for the J*=|- phase shift k3 Vcot -^2 (1 rE) 4* (43) where f is the renormalized coupling constant and r a con stant approximately given by r 3 f2 Ema* > E is the pion energy corresponding to k max max Relations similar to equation (43) are predicted for the other P-wave phase shifts. However, for these phase 18 shifts r < 0. This means that there are no resonances in these partial waves. Experimentally the other phase shifts are small in the low energy region. The Chew-Low model was a break-through for field theory. It was the first calculation involving a strong interaction which gave any significant agreement with experiment. By its specific assumption it is limited to low energies. It is also a particular case of the effective range approximation, which will be discussed next. 7. The Effective Range Approximation It is convenient for the discussion of effective range approximation to use the partial wave scattering amplitudes h^(w) defined in equation (33)* Here the new variable w, the square of k, is preferred over W the total energy. From the definition, it is found that Im h^(w) = k|h*(w)| (45) if inelasticity can be ignored. From equations (36) and (45) it follows that Im [ ] = -k. (46) h(w) It is possible to seek a power series expansion in the low energy region of w by using analyticity However, the expansion cannot be made simply for h^(w), for they have a 19 branch point at w=0. The function M(w) is defined as M^(w) = l/h^(w) + ik. (47) M(w) is analytic in the neighborhood of w=0 since Im M = 0 for w real and positive, and therefore by the Schwarz re flection principle M(w) has no discontinuity across the pos itive real axis. Furthermore, in the physical region, (48) Thus the power series expansion will take the following form, where the parameters are conventional: (49) a is commonly called the scattering length and r the effec tive range. The effective range approximation is made by keeping only the first two terms in the expansion. Because of threshold behavior of helicity amplitudes, about which a later chapter will be devoted to, equation (49) is not always the most convenient expansion to make. A general form commonly seen is k2J+l coÂ£ c JL -f irw. U tt (SO) CHAPTER III THE K-MATRIX FORMALISM In a parameterization of the dynamics of scattering pro cesses it is most desirable to take care of all kinematic effects explicitly in the formalism. In a dispersion theory these effects are generated by kinematic singularities of the scattering amplitudes. It will be assumed that these kinematic singularities factorize, i.e., for the scattering from channel i to chan nel f, the helicity amplitude can be written as I 1 hJ(f,i) [p(s)]2 [p^s)]2 (51) where s) depends only on dynamical variables and charac teristics of channel x and M^(f,i) contains only dynamical singularities. The function p^(s) will be called the J-th partial-wave phase space function for channel x, or simply the phase space function for channel x if J is understood. The phase space function is real above the channel threshold and it vanishes below the threshold. By their definitions in equations (21) and (22), it is found that 20 21 sf = 1_ + 2ihf . (52) Combining equations (27) and (52), the following relation is obtained: Im [ ] -J,. (53) hJ If it is defined that 9 where T is the threshold in channel m and 6(s-T ) is the m m step function, then L1 2 hJ C/] (55) () 'dir J,fi ApfPi l*W>|2 (56) IbC i1 (57) The last equation is the key to the K-matrix formalism. If, furthermore, we define (s) pj(s') ds' s' s ie + p.s. (58) where p.s. stands for "possible subtraction", then it follows from equation (57) that a K-matrix can be defined as follows: 22 (59) and that it is real for positive real s. In both and V? are analytic in s, so is K**. can also be written as JC 1 KJRJ addition, if Equation (39) (60) Equation (53) has been obtained as a general result of unitarity. It can be obtained more readily from equation (46) and the relation between the two helicity amplitudes. However, the result obtained this way is valid only for the low energy region. By definition is a diagonal matrix. The diagonal elements of this matrix will sometimes be simply denoted by R(s) . At this point the following physical considerations can be made: (1) For the sake of resonance study, it is only necessary to use a single incident channel. Mathematically, this means that both M. and are column matrices. (Note the helicity indices have been suppressed in favor of channel indices.) (2) Near a resonance, elements of the K-matrix are assumed to have the simple form (61) r where s is the position of the resonance, and C is a matrix r ^ 23 of coupling constants that factorizes: CiCj (62) Equation (6l) is the "pole approximation" of the scattering amplitude and it contains the only dynamical singularity as sumed for the formalism. The factorization property of the residue of the pole corresponds to the non-degeneracy of the resonance. (3) In the neighborhood of a resonance, the M-matrix ele ments have a form similar to the expression (6l). That is, the amplitude should indicate a simple pole at the resonance energy. To ensure that this is the case, according to equa tion (59), .R^ should be at least quadratic in (s-s^) in the neighborhood of s^. Hence, equation (58) should be changed to be (s) [ i (s-sr)2 OO /*.')ds' T (s'-sr)2(s,-s-ie) + p.s ] &y (03) Combining the above considerations, the following pa- rametrization formula is found: if'(s) C. (sr-s) XjC^R^s) iZjC2p^(s)0(s-Tj) (64) CHAPTER IV PHASE SPACE FUNCTION In order to have a well-defined parametrization formula, it remains to find a unique expression for the phase space function. The phase space function for a two-body channel is a product of two factors: the regular two-body phase space function for spinless particles, as required by kinematics, and the kinematic singularities of the helicity amplitude, which is caused by the presence of spins. 1. The Regular Two-Body Phase Space Function The regular phase space function for spinless particles is well-known.11. It is the Jacobian of the transformation from the momentum space to the phase space or invariant space. In a two-body channel, the Jacobian depends only on a single variable, s. By definition, in the center-of-mass frame, dpQ = | d4px d4p2 4(Pj+p2-P) (pj-mj) (p2-m2) (65) or Po s [(s ~ T12)(s U12)]2 (66) 24 25 where TJ2 and are respectively the threshold and the pseudothreshold of the channel consisting of particles 1 and 2. The normalization of this function and the normalization of the coupling constants are closely related. In order to avoid confusion, has been defined with no numerical coef ficient here. The coupling constant is going to carry an overall normalization. The presence of spins does not change the Jacobian from the momentum space to the invariant space. But it introduces kinematic singularities in the helicity amplitudes. 2. Kinematic Singularities of Helicity Amplitudes The kinematic singularity structure of helicity ampli- 8 o i2 11 tudes has been studied extensively. J It includes singularities at (a) the boundary of the physical region, (b) the threshold and the pseudothreshold, and (c) the or igin of the s-plane. (a) The Boundary of the Physical Region The d function of the rotation matrix is related to the Jacobi polynomials by <.() E WX+M C- w1^1- (lx-nl,lx+i*l) P(J-M) (cOS 0) (67) 26 where M=max( |X\ y (|| ) and N*=min( |X\, ) From the defini tion of the partial-wave expansion it can be seen that h^ contains the factors (68) By equations (5) and (12), the above expression represents a series of zeros or singularities at the boundary of the phys ical region. These are called t zeros or singularities, be cause for given s, the positions of these zeros or singular ities are determined by the scattering angle through t, the momentum transfer in the direct channel* Since only direct channel kinematic singularities are of interest in the parametrization, these possible singular ities at the boundary of the physical region, or cross chan nel singularities, are ignored in this work. (b) The Threshold and the Pseudothreshold The covariant helicity operator approach of King and o others is employed here. The direct channel covariant helicity operators corres ponding to particles 1 and 2 are defined by the relations (69) (70) 27 where /u is a summation index and is the Casimir operator for the spin of particle i. In the s channel cen- ter-of-mass frame these operators reduce to the ordinary helicity operators. Similarly, the t channel covariant helicity operators for particles 1 and 2 are defined by F* = 2 pÂ£ / S13(t) (71) f\= 2 wj2* pj / S24(t) (72) They also reduce to helicity operators in the t channel cen- ter-of-mass frame. The transformations between the eigenstates of the two sets of operators are given by ^S(pr,X) = Zx, dJxTx(/r)^(pr,X') (73) a where X and X' are eigenvalues of the operators Fr and Fr, respectively, Jn is the intrinsic spin of particle r, and r=l or 2. y*2 is the angle between -pj and -p^ in the rest frame of particle 2, while ^ is the angle between -p2 and p^ in the rest frame of particle 1. The coordinate axes are always chosen such that In terms of the invariants s and t, the angles are given by the relations (s + m2 m2)(t + m2 m2) 2m2M sl2() Sjji) cos (74) 28 cos where and S12(s> S24(t) M = "4 m2 + ml "3 2m1[<))(s,t) ]2 810 = Si2(s^S13(t) 2m2C4(s,t) ]2 Sn ^ = Sl2(s)S24(t) (75) (76) (77) From their definitions the operators and F can be seen to be ill-defined at the threshold and the pseudo- threshold. The covariant helicity amplitudes, f ^eing eigenstates of these operators, are singular at these values of s. Introducing a new set of amplitudes f^(s,t) in which the states for particles 3 and 4 remain unchanged but the states for 1 and 2 are now eigenstates of F^ and F^, it is found that S\W,t) dX*X1(;<:i)dK'^2(^2)f*3X4xV The singularities at the threshold and the pseudothreshold are now isolated in the d functions, because f ,(s,t) A3X4X is regular at these points. After an analysis of the asymptotic behavior of the d s o functions, it is found that the amplitudes f behave as 29 is T34] '^(J3+J4) (79) and (80) near the final-state threshold and pseudothreshold respec tively. Thus the product of the expressions in the above is the singularity to be included in the phase space function. However, it should be noted here that the kinematic singu larity of the helicity amplitude is, by equation (51), the square root of p^(s), but not p^(s) itself. (c) The Origin In the above discussion the origin has not been found to be a singular point of the helicity amplitude, but it is often pointed out that the origin is a singular point of the 12 scattering amplitude. Cohen-Tannoudji et al. concluded that the helicity amplitude does not have singularities at this point. It is only the linear combinations of helicity amplitudes commonly used for Reggeization that are singular here. This is also pointed out by King, since the singular ity of a linear combination can be different from the singu- o larity of a single helicity amplitude. 3. The Phase Space Function The product of the expressions in equations (66), (79) 30 and (80) is not quite the phase space function yet. The phase space function has been defined for partial wave ampli- tudes only. Referring to equation (22), it is seen that the singularity of d^(0) has to be studied, too. According to the asymptotic behavior for the d functions, 9,14 () ~ [cos e]J C(s -T34)(s -U34)] J (81) because of equation (6). Thus, finally, J>j(s) = i [(s T )(s U )] jH-j.-j . 3 4 (82) It must be pointed out that the threshold and pseudo threshold singularities have been obtained for particles with well-defined masses, and the parametrization is dealing with channels that often involve many unstable particles or resonances. In using equation (82) it is assumed that the widths of the contributing channels are sufficiently small to be ignored. This is a limitation of the present formal ism. CHAPTER V PARAMETRIZATION OF THE P^3 RESONANCE 1. Introduction The parameterization formula gives general representa tions of all matrix elements of a given partial-wave heli- city amplitude. The theory does not have the power to pre dict the exact behavior of individual matrix elements. The matrix elements can only be determined by "parametrization" against experimental data. The P^^ phase shifts, here chosen as the only experi mental data input, can be expected to determine one particu lar matrix element. However, because of the way phase shifts are defined in equation (33)* it turns out that a linear combination of two matrix elements is determined instead. 4 The P^3 phase shift is related to h_, which is proportional 1 4- to the linear combination hj j+h* This raises two questions. First, since singularities of certain linear combina tions of helicity amplitudes are known, is it not a better way to start directly with the parametrization of the proper linear combination? Theoretically, this is an attractive alternative. In fact, it is not known a priori that a pa rametrization developed for h^ can produce an adequate ap- 31 32 proximation for because they may have different kinematic singularities. But practically, direct parametri- zation of linear combinations is very unattractive because the kinematic singularities involved are such that a higher number of subtractions is usually needed, in which case the calculation may be obscured by the presence of a large num ber of subtraction constants. Other linear combinations simply do not have known singularities, so that it is impos sible to use such an approach even if the subtraction con stants are not an objection. In pion-nucleon scattering, both difficulties are present. J J * Second, since the linear combination IvT -h^ or h;, A-H* + is related to the phase shift, is it not advisable to make a simultaneous parametrization of the an<* D^3 reso nances, so that individual matrix elements, i.e., h^ and h? might be determined? The answer is definitely yes, it A-H would seem. But the helicity amplitudes themselves are not physically important for the strong interaction, because they are not parity-invariant. The physically important am plitudes are the parity-invariant linear combinations. 2. Basis of Computation In Table I are listed the various channels that con tribute to the resonance, their characteristics, and the appropriate phase space functions. The phase space functions for the channels are ob- 33 Table I. Characteristics of Contributing Channels Meson Baryon T2 [MeV] 1 U2 [MeV] P pion nucleon 1077.9 843.7 1 S C(.-T)(s-0)]3/2 pion P1 33 1373.6* 1094.4* i [(s-T)(s-U)* pion P1 *11 1609.6 1340.4 1 a C(s-X)(s-U)]3/2 varied in actual computation as the P'. mass is changed. tained from equation (82). While for the pion-P^ channel only two subtractions are required to make the dispersion integral convergent, in the other channels three are needed. Thus the dispersion integral for the second channel has the form of equation (63), without the p.s. term, the same inte grals for the other channels are of the following form p^(s) ds* R^(s) Â£ (s s )3 - 3 (s' s )3(s' s iÂ£) Ai r (83) The real part of the denominator in equation (64) is denoted by D(s), and the imaginary part by G( s) They are: D(s) = sr s + z(s sr)2 C2Rj( s) (84) G(s) = Zj C2p(j)0(s T.) (85) where z is an overall subtraction constant that is made nec essary by the extra subtraction, and 6(s) is a step function 34 The phase shift is given by tan 6j(s) = (86) Because of the fact that most coupling constants in volved in the parametrization of the are unknown at the moment, one of the objectives of this work is to make cer tain predictions on their values. The procedure is to use phase shift data and the parametrization formula to find a best fit, with the coupling constants and the subtraction constant, as well as the mass of the resonance, as free pa rameters. The computer routine PARFIT at the Department of Physics and Astronomy of the University of Florida is used for this purpose. 3. Phase Shifts Two sets of phase shifts have been used. Set A is tak en from Carter et al.^ Set B is compiled from older re sults. The Cavendish-Rutherford collaboration of Carter et al. recently reported an extensive series of cross section meas urements at the CERN synchrocyclotron in the energy range around the Pj^ resonance. The reported error bars are much smaller than all previously available results. An interest ing feature of the new data is an apparent shift of the peak of the cross section to a lower mass. The phase shifts re- 35 ported by this group are extracted directly from the total cross sections. Set A is used to yield more accurate information on the resonance: its mass and width, and possibly also its coupling constants. In Set B, values of phase shift reported by Moorhouse, Kirsopp, Johnson, Donnachie, Bareyre et al,.1'* and a set from Berkeley called Path 1^ are taken on equal footings and their average determined. Since there is no available basis to prefer any one set of these over any other, no weight factors have been assigned in the average. Since different listings cover different energy ranges, the number of entries at any particular energy can vary con siderably. In the lower range where entries typically num ber two or three, the standard deviation is determined by the following observation: In the Saclay isovector pion-pion phase shifts,the quoted uncertainty roughly forms a band of constant width about the mean when plotted graphically. In the absence of further information, this observation has been applied to assign standard deviations for set B. In the higher energy range where data entries are more numerous, the standard deviation is taken to be one half of the maxi mum difference among the sets, generally. And in order to achieve some kind of uniformity throughout the energy range, the width of the band in the lower range has been determined at the upper energy range. 36 The results are listed in column 2 of Table IV. 4. Left Half-Plane Singularities In the complex s-plane, the partial-wave helicity am plitude has dynamical singularities at poles and branch cuts corresponding to all "communicating1' channels or possible exit channels as required by unitarity.1^ These are known as the right-hand singularities for they occur in the right half-plane. There are also left-hand singularities, occurr ing in the left half-plane, because of cross channel right- hand singularities. The N/D method of Ball and Parkinson, which is equiva lent to a K-matrix formalism using the R-matrix of equation (63), which in turn approximates the contribution of right- hand singularities, uses one form of the effective range ap- 3 proximation for the left-hand singularities. With an in creasing number of subtractions in the R-matrix, it has been found that more left-hand poles are generated. In particu lar, the R-matrix of equation (83) generates a pair of com plex conjugate left-hand poles. The symmetric poles about the left-hand branch cuts along the negative real axis is a better approximation than a single pole. A wider range of validity can be expected than the effective range approxima tion . This line of reasoning has been used in works on the approximation of a Regge cut by a pair of complex conjugate 37 17 Regge poles in the J-plane. In this respect, it should be noted that if the parame- trization is to extend to higher and higher energies, reso nances of higher and higher J values will have to be in cluded. Then, by equation (82), the phase space function will be so modified that more and more subtractions are nec essary in the R-matrix. And this indeed is going to extend the region of validity of the parametrization to higher and higher energies. Returning to the present problem, it is seen that the energy range is relatively small. Thus a two-pole approxi mation should be sufficient. 5. Contributing Channels In equation (64) the summations are over all contribut ing channels. These are also called communicating channels or possible exit channels. In principle they cannot be ex hausted, unless experimentalists find an upper limit for re sonance production. Practically, the choice is made in the following manner. From the set of well-established particles, i.e., the ones listed by the Particle Data Group with complete quantum l8 number and mass-width specifications, all meson-baryon pairs that have the right quantum number combinations are selected, and their thresholds and pseudothresholds, accord ing to equations (8) and (9), calculated. Most thresholds 38 lie beyond the top of the energy range of phase shift data. Referring to equation (64) it is found that channels whose thresholds are higher than the top of the energy range under consideration contribute only to the function D(s), through the R-matrix. Elements of the R-matrix are rela tively small for those channels with high thresholds, as can be seen from equation (83)* Here a high threshold means a large s'-s in the denominator of the integrand, and conse quently a small matrix element. Thus all channels with very high thresholds are not considered important, and a total of eight channels are finally selected as possible contributing channels. These are listed in Appendix B as a comment in the computer program actually used in the calculation. The maximum energy for the phase shift data is at 172 MeV. Arranged in order of increasing thresholds, the fourth channel threshold is at (1660 MeV)^ and the eighth at (1783 MeV)^. In fitting the phase shift data it is found that all channels higher than the third can be ignored without affect ing the chi-square of the fits. Thus, for the energy range under consideration, there are only three contributing chan nels. These are listed in Table I. The practical aspect of two-body kinematics has been studied in Chapter II. From a theoretical point of view, the exclusive use of two-body channels is an application of the idea of direct channel resonance dominance. That is, three- or more-body channels, though experimentally observed to be dominating at times, are themselves dominated by two- 39 body channels. As an example, in pion-nucleon scattering the pion pro duction process of two pions and a nucleon in the exit chan nel are often observed. Rather than trying to solve a three- body problem, the view is taken that the "extra" pion has come mostly from either a rho meson or any one of the baryon resonances. (The word "come" must be understood in a very loose sense.) Thus, the inclusion of pion-baryon resonance and rho meson-nucleon channels should describe the three- body channel adequately. Direct channel dominance, used in this manner, is quite similar to Feynman diagrams in field theory, though there is a subtle difference. In Feynman diagrams, the scattering is assumed to have gone through a virtual intermediate state. In the S-matrix theory there is merely a statement of domi nance in the amplitude, and a subsequent substitution. 6. Computation The general procedure of actual computation is as fol lows: A set of values for the resonance mass and the cou pling constants in various channels, plus the subtraction constant, is supplied to the program PARFIT, which uses equa tion (86) to determine the chi-square value on the basis of experimental phase shifts and standard deviations. PARFIT also automatically minimizes the chi-square by adjusting the 40 free parameters, that is, the coupling constants and the sub traction constant. At the end of computation PARFIT produces the best fit, with all relevant data. The mass of the is then shifted and a new best fit found. The best overall fit with the mass as a parameter is taken to be the final result. In addition, the number of contributing channels can be varied in each chi-square calculation, by varying the value of PAR. Thus, the K-matrix formalism provides the most general form for the parametrization. The number of contributing channels is determined by the sensitivity to chi-square fit, and the actual parameters determined by best fit against ex perimental data. And a complete parametrization for the P^ is obtained. The mechanics of actual calculation are contained in the Appendices. The routine PARFIT is not included. CHAPTER VI RESULTS AND CONCLUSIONS 1. Comparison with Breit-Wigner Formula The K-matrix can be made relativistic simply by using phase space functions and dynamical variables that are all Lorentz invariant. This has been done in Chapters II, III, and IV. Thus the parametrization is relativistic. And this removes the first objection. It should be remarked here that it is not the aesthetic aspect of non-relatiyistic nature of the Breit-Wigner formu la that prompted the objection. Rather, it is the fact that non-relativistic mechanics restricts all formalisms to low energies, or narrow energy ranges, and this restriction is quite undesirable. The summation in the parametrization formula, equation (64), in the real part of the denominator specifically, in cludes both open and closed channels for any particular en ergy. Hence nearby singularities can influence the dynamics of the scattering. The zeros of the denominator in equation (64) do not appear on all sheets of the scattering amplitude, in gener- 41 42 The zero of the denominator in equation (64) is deter mined, in part, by the expression (87) The relative contribution of each channel toward producing 3 the resonance is then given by'' CjRj(sr). (88) This gives a rough indication of the relative importance of each channel in the dynamical origin of the resonance. 2. General Description of Phase Shift Fits Set A Reasonable fits to phase shift data have been obtained. The best chi-square value for 14 points is 15*3 correspond ing to a resonance mass of 1230.4 MeV. Results of the pa- rametrization fit are listed in Table II. Table III contains the phase shifts and the values of chi-square at all data points, together with the experimental data used. It is of interest to note that the lowest data point carries the worst chi-square. If this data point is deleted, the best chi-square value would be 10.7 for 13 points. Pre viously, the data point at the other end of the spectrum has l8 been considered a bad fluctuation. 43 Table II. Parameters from Phase Shift Fits Set A Set B resonance mass 1230.4 MeV 1235.0 MeV Â£- resonance width 120 MeV 122 MeV SrNA 0.1551*0.0002 0.1543*0.0010 CttAA 0.0830*0.0073 0.1029*0.0073 CrrN'A 0.8048*0.0202 1.018 *0.031 subtraction constant -0.0488*0.0003 -0.0506*0.0016 chi-square 15.31 63.68 mean chi-square 1.09 1.63 * full width at half 4 maximum. With a total of five de facto parameters (three cou pling constants, one subtraction constant, and the resonance mass), the fit to set A has not been very good. But it should be pointed out that different channels contribute differently to the resonance, and not all coupling constants affect the quality of the fit equally significantly. In fact, the introduction of some more channels and their cou pling constants does not change the chi-square value at all. The parametrization used here is not an optimum mathematical approximation. Set B The best chi-square value for the 39-point set B is 44 Table III. Phase Shifts Fitted According to Carter et al 5 W [MeV] Exp. [deg.] Theory 1139 11.87*0.15 11.54 4.75 1160 21.41iO.06 21.42 0.01 1177 33.170.10 33.36 3.50 1178 34.36db0.20 34.26 0.25 1190 45.64iO.ll 45.66 0.04 1193 47.83iO.ll 47.81 0.04 1206 62.82iO.l8 62.61 1.31 1210 67.730.21 67.48 1.37 1215 73.51dbO.30 73.63 0.15 1227 84.940.98 85.97 1.10 1244 102.05i0.51 102.3 0.21 1261 114.41iO.23 114.6 0.45 1280 124.03i0.17 124.2 0.85 1301 13l.960.15 131.8 1.27 63.7* This result is as good as can be expected, because systematic errors among different sources are expected to be important. For this very reason results of this fit should not be taken too seriously, especially those pertaining to the resonance. In Table II are listed main findings of this fit. In Table IV are the phase shifts. Similar to the parametrization of the rho meson,^ in this relatively wide energy range the theoretical values de viate from experimental phase shifts significantly near the 45 Table IV. Phase Shifts Fitted According to Compilation Data W [MeV] Esq). [ deg. ] Theory 1094 1.4 0.9 0.88 0.33 1104 2.6 0.9 2.01 O.42 1109 3.6 0.9 2.76 0.88 1113 4.2 0.9 3.44 0.71 1127 7.6 0.9 6.66 1.32 1160 21.0 1.0 19.79 1.47 1177 31.8 1.1 31.12 0.39 1185 37.9 1.1 37.77 0.00 1195 45.9 1.3 47.28 1.12 1197 48.2 1.4 49.32 O.64 1202 54.2 1.4 54.59 0.08 1213 65.8 1.4 66.71 0.52 1216 69.4 1.4 70.05 M O 1231 85.1 1.4 86.09 0.61 1235 90.1 1.4 90.04 0.00 1247 100.6 1.4 100.7 0.04 1252 105.2 1.3 104.7 0.15 1254 107.5 1.2 106.2 1.20 1258 109.4 1.0 109.0 0.07 1268 117.7 l.P 115.4 5.16 1275 119.8 1.0 119.3 0.23 1291 126.9 0.6 126.8 0.02 1320 136.7 1.3 136.8 0.01 1362 144.9 0.6 146.5 7.33 1390 149.2 1.0 151.2 3.86 1416 153.3 1.1 154.6 1.81 1442 156.3 1.1 157.5 1.28 1470 160.2 1.4 160.2 0.00 1481 l6l.l 1.7 161.1 0.00 1500 164.5 2.5 162.6 0.59 1512 166.0 0.9 163.4 *fr H GO 1524 166.1 1.0 164.2 3.45 46 Table IV. continued W [MeV] Exp. [deg.] Theory 9C2 1543 168.3 + 2.2 165.4 1.69 1572 170.3 + 2.8 167.1 1.31 1601 172.2 + 3.0 168.6 1.41 1617 171.2 + 1.7 169.3 1.30 1629 173.2 + 4.0 169.0 1.11 1658 174.1 + 3.6 167.6 3.24 1672 175.5 + 2.6 166.7 11.56 top of the range. The chi-square for a 38-point fit, for example, would be 52.1, a significant improvement. The introduction of higher contributing channels does not change this top-of-the-range deviation. Since set B covers a wide energy range, in the calcula tion of the chi-square values the number of contributing channels has been varied. The result is that only the low est three channels contribute, the same channels as present for phase shift data set A. Altogether eight channels, with O thresholds up to (1783 MeV) have been tried. It may be conjectured that a certain channel with still higher thresh old contributes significantly by virtue of a huge coupling constant. At least this cannot be ruled out yet. And this very high threshold may solve the difficulty at the higher end of the range. 47 3. The Resonance Mass Set A The mass of 1230.4 MeV for the resonance is lower 18 than most values reported. The lone exception is the "nu clear" result reported by Carter et al.^ themselves. The shift to lower values is in agreement with most recent works. However, it is worth noting that various forms of the Breit- Wigner formula, all based on the same experimental data, 18 consistently give higher masses. The full width at half maximum is 120 MeV. Set B The resonance mass is high compared with the one for set A. This is typical of older results. It is believed that, due to higher systematic and statistical errors, in formation concerning the extracted from this fit is of very limited value. 4. The Subtraction Constant The subtraction constant is very small, both for set A and for set B. It is believed that the magnitude of the subtraction constant is a measure of the goodness of fit, too. For example, in the successful rho meson parametriza- tion, it has been found that not only is the chi-square low, the subtraction constant is practically zero. A physical 48 interpretation is the following: the subtraction contributes to offset any deficiency in the knowledge of kinematic and dynamic effects. Had the last two effects been properly handled* there would be no need for a non-zero subtraction constant. This is reinforced by a mathematical interpreta tion of the subtraction: it is merely a free term whose function is to reduce the chi-square. 5. The Coupling Constants A. The Pion-Nucleon-P^^ Coupling Each term in the expansion for -G(s) can be identified with a certain partial decay width for the resonance; if it is evaluated at the resonance energy. In particular; using the value of obtained in the parametrization; we have r cK2 H + v2)l2 H v2}^ m* 0.81 (89) in pion mass unit. This can be compared with the full width at half maximum of 0.87. They represent two different ways in which the width can be defined. 21 According to Mathews; this partial decay width is also given by 49 *2 SttNA 2 47T 3 p 3 (mA + ->' - a. (90) mA Comparing the last two expressions, it can be seen that 2 / >2 2 ,2 JTNA (mA + "n> ~ "V 'ttNA 4tt 48 m3 (91) g: 'irNA Or that a of 0.1551 corresponds to a ^ of 0.37, in dimensionless unit. 20 Ebel et al. report a g fl-NA 47T of 0.33, which they main tain is inferior to the value of 0.34 obtained from non- relativistic spinless Born approximation. By inference, it would seem that the value of 0.37 is quite acceptable. B. The Pion-Pl^-P^ Coupling 22 According to Rushbrooke, the partial width correspond ing to the decay of a resonance into a pion and a 33 resonance is given by g 'TTAA 47r 3m C3( m2 + m2 m2 a A ir 2m. ^ + 3^ 2p2 m2 + m2 m2 /A a 7r 2mA e om ttAA _1 rr 4-rr m 2m. 2mA) ] (92) since the center-of-mass momentum of the decay product is very small. According to the K-matrix formalism, the same quantity is given by Combining the two we have 2 mi 4ir 4mA * (94) Our value of C,^ =0.083 therefore corresponds to the value ry 0.243 for SirAA/47T in dimensionless unit. 2 3 2 Hori and Kanki reported that is approximately o one ninth of g^^. The relativistic SU(6) model of Sakita 2 a 2 and Wali, ^ on the other hand, predicted that g is about 2 2 5 26 nine times of g Sutherland and Michael predicted 1TNA 2 that g-rrv\a is even larger by a factor of about two. Our re- 2 suit is not in close agreement with any of the above. 2 is shown here to be two-thirds of g^^. C. The Pion-Pj^-P^ Coupling The K-matrix formalism developed here cannot be applied to the decay of the PJj into a pion and a resonance, though the appropriate branching ratio and total width have been measured experimentally. Thus no comparison can be made at this point. It is worth noting that a similar problem occurs in reference 22. The decay of a spin| particle into a spin--| particle and a spin-0 particle warrants special equivalence relations for satisfactory results. 51 6. General Remarks In spite of its many known and suspected deficiencies, the formalism developed here not only points to a way by which some experimental results can be derived through a theoretical model, it also opens up an approach by which the whole problem of hadron dynamics may be attacked. Traditionally, hadron physics has two distinct compo nents, the low energy and the high energy theories. Duality is a way to reconcile the two components. The present ap proach, on the other hand, is to extend the low energy theo ry into higher and higher energies. If the K-matrix parame- trization scheme is successful there, a unified theory of hadron physics is in sight. And it may eventually be able to check the validity of duality, and of many other inter esting concepts popular in high energy physics. Of more practical concern, and related to the quality of these phase shift fits, is the fact that phase shifts must be deduced from "direct" experimental data through some kind of parametrization first. Existing parametrizations commonly used are all based on the Breit-Wigner formula. In this light the chi-square values mentioned in this work can not be too small. The present work offers an alternative to the variations of the Breit-Wigner formula. Because of the many objections of the latter as studied in Chapter I, and their removal in the K-matrix formalism, it is believed that many difficulties associated with present-day description of 52 the phase shift results can be removed. APPENDIX A MECHANICS OF CALCULATION The basic principal integral is defined by^ OO f(s,T,U) P j [(s' T) (s' U) ] 2 (Al) T so that _i _! 1 for s>T, f(s,T,U)=-2[(s-T)(s-U)] 2Coth C (^ 3 (A2) 2 1 || '2 for T)s>U, f(s,T,U)= 2[(T-s)(s-U)] tan C(fE) 3 (A3) 1 1T 2 for U>s, f(s,T,U)= 2[(T-s)(U-s)] tanh C(^Ef) 3 (A4) and the derivative of f(s,T,U) with respect to s is -1 f'(s,T,U)=[(s-T)(s-U)] [-l+|(T+U-2s)f(s,T,U)3 (AS) which may be used iteratively to produce higher derivatives with respect to the same variable. 53 APPENDIX B COMPUTER PROGRAM c channel # meson baryon psf T U c 1 pion 140 ncln 940 3 1080 800 c 2 pion 140 delta 1234 1 1374 1094 c 3 pion 140 N1470 H70 3 1610 1330 c 4 pion 140 N1520 1520 1 1660 1380 c 5 pion 40 N1535 1535 3 1675 1395 c 6 kaon 496 sigma 1190 3 1686 694 c 7 rho 765 ncln 940 1 1705 175 c 8 eta 549 delta 1234 1 1783 685 c 1. energy unit in pion mass for all calculations C 2 input energy unit is the MeV C 3. normalization of energy by variable "unit C 4* A(par) is the subtraction constant C 5* numerical differentiation is used C 6. exp data input: UCRL20030 compilation implicit real*8 (a-h,k-z) real x integer psf common/data/unit,m,T(8),U(8),psf(8)/intg/w(39,8),energy( +39)/stpdf/fx,x(400)/mass/pion,kaon,rho,eta,ncIn,delta, +N1470,N1520,N1535,sigma read (1,1) eta,kaon,pion,rho,delta,N1470,N1520,N1535> 4-ncln, sigma 1 format (I0d8.l) read (1,2) psf 2 format (8i7) do 4 i"l>39 54 55 read (1,3) energy(i) 3 format (2x,d8.l) 4 continue do 5 1=1,400 5 x( i) =0.1 call callit stop end double precision function f(A,L) implicit real*8 (a-h,k-z) real rgm,A(l),duml,dum2 integer psf,L,par cnvr=l80./3.14159265358979dO common/data/unit,m,T(8),U(8),psf(8)/intg/w(39,8),energy( +39)/stpdf/fx/srchit/rgm(50),duml(2554),par,dum2(9),inx +/mass/pion,kaon,rho,eta,ncln,delta,N1470,N1520,N1535, +sigma if (inx.ne.l) go to 15 inx=2 read (1,11) delta 11 format (dl2.4) write (3,20) delta unit=pion T(l)=((ncln+pion)/unit)**2 U( l)=((ncln-pion)/unit)**2 T(2)=((delta+pion)/unit)**2 U( 2) *=( ( delta-pion)/unit)**2 T(3)=((N1470+pion)/unit)**2 U(3)=((N1470-pion)/unit)**2 T(4)((Nl520+pion)/unit)**2 U(4)-((N1520-pion)/unit)**2 T(5)=((N153 5+pion)/unit)**2 U(5)=((N1535~pion)/unit)**2 T(6)=((sigma+kaon)/unit)**2 U( 6) =( ( sigma-kaon) /unit) *#2 56 T(7)=((ncln+rho)/unit)**2 U(7)((ncln-rho)/unit)**2 T( 8)=( ( delta+eta) /unit)#*2 U(8)((delta-eta)/unit)**2 m(delta/unit)**2 isum!=,par-l do 14 il,39 s=(energy(i)/unit)#*2 do 12 jl,isum 12 w(i,j)=v( j,s) write (3,13) (w(i,j), jl,isum) 13 format (8gl6.3) 14 continue 15 i=rgm(l) sm(rgm(2)/unit)**2 g=0 .dO isum^par-l do 16 j^ljisura if (s.lt.T(j)) go to 16 g^g-Aij)**2*fcn(psf(j) ,s,T( j) ,U(J)) 16 continue d=m-s+A(par)*(m-s)#*2 do 17 3=1,isum 17 d=d+A(j)**2*w(i,j) h=dabs( d) if (h.lt.l.d-20) go to 19 god*=g/ d if (god.lt.0) go to 18 fdatan(god)*cnvr f x**f return 18 fl80.0+datan( god)*cnvr fxf return f=90.dO fx~f 19 57 return 20 format ('l'jgSO.) end double precision function fen (dmml,dmm2,dmm3,dmm4) implicit real*8 (a-h,o-z) integer dmml go to (21,22,23), dmml 21 fcn"dsqrt((dmm2-dmm3)*(dmm2-dmm4))/dmm2 return 22 fcn^O.dO return 23 fcn(dsqrt((dmm2dmm3)*(cimm2-clmm4)))**3/din2 return end double precision function v(i,r) implicit real*8 (a-h,m-z) integer psf common/data/unit,m,T(8),U(8),psf(8) d=psf(i) P*T(i) z=U(i) am-p bm-z e*m-r x*=r-p y=r-z Pi3.14159265358979dO cutoff99d25 go to (41*42,43), j 41 g=a*b/m/e gpg*(1.dO/a+1.dO/b-1.dO/m-1.dO/e) v*>( -p*z#e**2*q(0.dO,p,z,cutoff) /r/nr**2+x*y*q( r ,p,z,cutof f +)/r+e##2*gp*q(m,p,z,cutoff)+e**2*g*qp(m,p,z,cutoff))/pi return 58 42 v=0. dO return 43 ha**2*b**2/m/e hp=h*(2.dO/a+2.dO/b-1.dO/m-1.dO/e) hpphp*(2.dO/a+2.dO/b-1.dO/m-1.dO/e)+h*(1,d0/m**2+l.dO/ +e**2-2.dO/a**2-2.dO/b**2) v-(p**2*z**2*e**3*q(0.dO,p,z,cutoff)/r/m**3-x**2*y**2*q( +r,p,z,cutoff)/r+e**3*hpp*q(m,p,z,cutoff)/2.d0+e**3*hp*qp +(m,p,z,cutoff)+e**3*h*qpp(m,p,z,cutoff)/2.dO)/pi return end double precision function q(s,t,u,lmda) implicit real*8 (a-z) f actor(lmda-t)/(lmda-u) if (s.lt.u) go to 32 if (s.eq.u) go to 33 if (s.lt.t) go to 31 q-l./dsqrt((s-t)*(s-u))*dlog((1.+dsqrt((s-t)/(s-u)/ +f actor))/(1.-dsqrt((s-t)/(s-u)/f actor))) return 31 q2./dsqrt((t-s)*(s-u))*datan(dsqrt((s-u)/(t-s)*factor) ) return 32 q=>l ./dsqrt( ( s-u)*( s-t) )*dlog( ( 1 .+dsqrt( ( s-u) /( s-t)* +f actor))/(1.-dsqrt((s-u)/(s-t)*f actor))) return 33 q^ ./(t-u)*dsqrt( factor) return end double precision function qp(s,t,uflmda) implicit real*8 (a-z) if (s.eq.u) go to 34 factor*dsqrt( ( lmda-u) *( lmda-t) ) /( s-lmda) qp(factor+((t+u)/2.-s)*q(s,t,u,lmda))/((s-u)*(s-t)) return 59 34 f actor=dsqrt((lmda-t)/(lrada-u)) qp=2,/3.*factor/(t-u)*(2./(t-u)+l./(lmda-u)) return end double precision function qpp(s,t,u,Irada) implicit real*8 (a-z) if (s.eq.u) go to 35 qpP(((t+u)/2.-s)*qp(s,t,u,lmda)-q(s,t,u,Irada)*(1,+((t+u +)/2.-s)*(1./(s-t)+l./(s-u)))-dsqrt((lmda-t)*(lmda-u))/(s +-lmda)*(1./(s-t)+l./(s-u)+l./(s-lmda)))/((s-t)*(s-u)) return 35 factor=dsqrt((lmda-t)/(lmda-u)) qpp=4,/l5.*factor/(t-u)**2*(8./(t-u)+4/(lmda-u)-3.*(t-u +)/(lmda-u)**2) return end REFERENCES 1. 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BIOGRAPHICAL SKETCH Ji- , Yin-po Tschang, in Chinese was born in Chungking, China on June 20, 1944 He received his early education in Taipei, Singapore, and Hong Kong, before entering Chung Chi College of the Chinese University of Hong Kong on a four-year Hong Kong Government Scholarship. In 1966 he graduated with the degree of B.S. cum laude and since then has been in the Graduate School of the University of Florida. In December, 1972, he received the degree of Doctor of Philosophy with a major in physics. He is married to the former Miss Kwok-lan Chan, of Hong Kong. They have a seven-month-old boy, Yuan Tschang, 63 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Michael T. Parkinson, Chairman Assistant Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Mkrt. Arthur A. Broyles ^ Professor of Physics and Physical Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Zoran R. Pop-Stojattovic Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Samuel B. Assistant Tricke; Professor Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jjahn R. Sabin Associate Professor of Physics & Chemistry This dissertation was submitted to the Department of Physics and Astronomy in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December| 1972 Dean, Graduate School |

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UNIVERSITY OF FLORIDA
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