LPOTT - PION and KAON elastic scattering from spin 12 nuclei in momentum space

Q0= 1,

—

I~F—pl>pr’ otherwise,

150MeV/c), (123)

R. H. Landau

126

/

Pion and kaon elastic scattering

Since P depends on the irA off-shell momenta and scattering angle: p2 = ( Po + k)2 ~(k2 + k’2 + 2k.k’),

(124)

the integrations in (122) are rather lengthy since they are done for all k and k’. N. B. To perserve consistency, only the Thomas potentials can be used in (122)

error may result for other

—

g ‘5. 2.6.5. Fermi averaging “folding” In principle, LPOTT could easily be used to evaluate the ground state expectation value of the optical potential operator [6], ‘-

(k’(Ujk) + A- 1 ~fd3p~*(

(125)

p-q)(f~t~(co)Ii)~~(p).

The order-of-magnitude or more increase in running time has not permitted this, however, and various approximations are made. The most extreme is the factorization approximation of eqs. (56) and (57). In practice we usually use a “folding” approximation to (125) [2,4,6]: (k’jU(k)

in which we integrate only the energy, the irN corn, =

c3( p)2

s( p)

=

(fIt~(co)li),

(A - l)p(q)fd3p[ ~~(P)I2/A}

—

to,

fd~p~[l~( p)12/AI

(K(

(flt(w)pi) when the t matrices

are first calculated in

p)It.,( ~( p))~(p)),

(127)

(128)

m~— m~ + E,~(PIab)EN( p) —2 P’abP’

K(p)

=

P~b

(icO/mN)[E,,(sco)+EN(ico)].

[s(p)

dependence of

(126)

—

(ma + mN)2][s(p)

—

(ma —

mN)2]

(129)

/4s(p),

(130)

The formulas used for [~~I~~(p)I2]/A are from the harmonic oscillator shell model: AP 1~(p), 2

A

Ei~~

=

4P1~(p)+(A—4)P1~(p), ~ + 12P1~+ (A 16)[Pid + p2~]/2,

4A16,

4P1~+12PIP+24(Pld+P2S)/2+(A—40)PIf,

40A48,

—

2),

P1~(p)

=

A4,

N exp(

—

(pa)

4 exp(

P

—

2exp(

P 2~= 2N( pa) (pa)2)/l5, N

=

—

16

A

40,

(131)

(pa)2)/3,

a3/ir3”2,

1~(p)= 4N(pa)

2exp(_ (pa)2)/2, 4(pa)2) 6exp( (pa)2)/l05.

P 2~(p)= 3N(1

—

P

—

2~(p)

=

8N( pa)

(132)

R.H. Landau

/ Pion and kaon elastic scattering

127

2.7. Annihilation potential Annihilation (true absorption) of pions can be included with a second order potential [6,8]: (k’lUt2~(E0)Ik)=

4irA(A

—

—

~) {~0~0~ ~

2~.ç~(2iT)

g~(sc0)

k~cos okk.}P(k

c0(k0) g~~)

+

k),

(133)

0,a1)=(648,289)MeV/c,

(134)

—

g~ (‘ce) 2,(a g~(Sc)=S~/(ct~+Sc2), gp(Sc)=SpSc/(a~-~-K2)

2(q) =fe4~~p2(r)d3r.

(135)

~

For the harmonic oscillator density of (66), + a(12

—

q2a2)/8 + a2(15

xexp(_q2a2/8)/[ir3/2a3(l

—

4q2a2 + q~a~/l6)/l6]

(136)

+~a)221/~],

whereas for the Wood—Saxon shapes, LPOTT calculates momentum and is calculated from (95) with

?( q) numerically.

iç~ in

(133) is the irNN corn

2. (137) 0) +ED(2k0/A)] 2_ k~(l 2/A) The partial wave decomposition of U~2~ is computed numerically via (42) and (43). At present, LPOTF calculates the isoscalar annihilation strengths B 0 and C0 from ir + d pp total cross sections [6], and prints out these values. Future modifications will employ the more microscopic (isobar) model 3He. of Yoo and Landau [27] which will also provide a spin and isospin dependence as needed e.g. for S,,D

=

[E~(k

—

—~

—

3. Nwnerical techniques 3.1. Solution of Lippmann—Schwinger equation by matrix inversion

We reduce eq. (20) to a set of linear equations. First, we execute the principal value prescription by subtracting an integral of zero value (the Haftel—Tabakin [28] procedure): 2

ç~O

R(k Ik)U(k Ik)+—J

p2U(k’Ip)R(plk) dp Ea(kO)+EA(kO)~Ea(P)~EA(P)

—k~U(k’Ik

2)/2~t(k 0)R(k0Ik) 0)

(k~—p

(138) (for clarity we drop the L ±).Next, the nonsingular integral is approximated as a W1-weighted grid of N appropriately chosen points k1 (see GAUSS, section 4): R(k’ 1k)

U(k’ 1k)

+~ ~

wj[ k~U(k’ik~)R(k~ik’) ko2U(k~iko)R(koik)J —

sum

over a

(139)

R.H. Landau

128

/

Pion and kaon elastic scattering

Now we define the (N + l)th (NI) grid point as k0, kN±l=kNI=ko,

(140)

and combine the Green’s function and weight into a D0: 2 2 W,k Do(k/)=_E(k)E(k)~ 1=l,N,

(l4la)

Do(kNl)= —~k~2~t(k 0)~ k~—k~

(l4lb)

An equivalent form of (139) for k’ and k on the grid of points is thus R(k1Ik1)=U(k,Ik1)—~U(k1Ik,)D0(k,)R(k,Ik1)

(142)

R(k,Ik,)+~U(k1Ik1)D0(k,)R(k1Ik1)=U(k,Ik1).

(143)

or

This is the desired set of linear equations which can now be solved [F][R]

=

by

matrix inversion, (144)

[U],

(145) F(k1,k1)=~1~+U(k,,k~)D0(k1),

(146a)

[F]=[1]—[UG].

(146b)

To avoid dealing with complex matrices, the real and imaginary parts of R and U are separated and the number of linear equations is doubled from N 1 to N2 2( N + 1), =

R=R’+iR’,

U=U”+iU’,

(147)

R~ (N2\ U~ (N2\_ UrD0 —U’D0 R~ /N2\ 2 R’ R’ U’ U’D0 UrD0 (N2) The matrices U and F are stored as a linear array with the algorithm,

(148)

Ut(k,Ik~)= U~=U~=U[(j— l)N2+i], i,j= l,Nl,

(149a)

~

=

UNI+ 1] = U[(j—

l)N2

+ i+

Nh,

(l49b)

l)Nl +:].

(l49c) 2 (the 2 for J L ±4).The present U and are dimensioned U(M, and F(M)ofwhere (2N1) version of F LPOTT has M 4356, so 2) a maximum 32 gridMpoints is possible. A change in the dimension statement would permit more grid points. At any one time, the U and F matrices only for one L value are stored in the memory. The values for all L values are stored on tape 7 and read in as needed. k—F[(k

=

=

=

R.H. Landau / Pion and kaon elastic scattering

129

3.2. Coordinate space wavefunction The basis of our procedure is similar to that described in PIPIT [17]. LPOTT will produce a deck (file) which can be used to calculate the irA wavefunction (possible with spin too) separately from running LPOTF if the subroutine WAVFN, COULFN and SPBESL are used. Our notation is that of Rodberg and Thaler [20], ~~r)=~U,(r)it(2l+ e’~

(J,(r)

—

.

-j~-— sin(.&r

—

l)P,(cos9),

y ln(2kr)

—

l~-+

(150)

(151)

~L + &L).

After converting our standing wave conventions to outgoing waves, rearranging our F matrix to give the conventional Moller operators, renormalizing the A /( A 1) factors present in the KMT procedure, and using Coulomb waves (if one of the “exact” Coulomb procedures is chosen), the Moller wave operator evaluated on our grid of points is —

QT(k

1, k1)

=

A

1

—

~

+AQR(

k1, k1)

T’(kNI, kNI)1 (

NI’

(152)

NI

F,~,”is determined by the matrix inversion in (145) and T’/R’ is the ratio before Coulomb matching. The wavefunction for 1 ‘Store (number of t ‘s calculated) is =

—~-—{cos~LFL(kr)+sin8LGL(kr)),r>Rcut U,(r)=

(153a)

NI

(t
~ Jt(1Cm~~)12T(1CN+I, km)ZL,

r < Rcut,

(l53b)

m= I NI

ZL

=

kR

(cos s~LFL(kRCUl)+ sin ~LGL(kRcut))/ ~ J/(kmRCUt)QT(kN+ I’ km). cut

For I>

‘Store

(153c)

m=1

the Born approximation should be used, (154)

U,(r)=FL(kr)/kr.

The Coulomb wavefunction routine RCWFN by Barnett et al. [30] is used to calculate the FL’s and GL’s.

4. Description of subroutines In section 4.1 we tabulate and identify all subroutines in the order of their listing. In Figs. 1—8 we give flowcharts with equation number references for the MAIN program and the most important subroutines, OPTP, TNUC, TNUCTH, TNOFF, TPICM, RHOKFF and FFACT. In section 4.2 we describe the storage in COMMON’s and of the potential matrix on TAPE 7. 4.1. Subroutine tabulation

1. MAIN 2. BESLIA

R- and T-matrix via matrix inversion in momentum space. exp( z) times spherical Bessel function of argument iz, (68). —

)>

RH. Landau / Pion and kaon elastic scattering

130

MAIN LPQ~T Pie

LiOTT

~ t~~4-3~_~.( ~NP?INPUT ~

1515(t

I

~)

~

-~

(5)-(~Il1)

~

L~OPlN

L~e~co~*bI j(s~ ~4~(r), D(E)

I

0’)

LPUMI

~l~ I I L~

çLPLM )~LPJ’MX>~

I

I

N5f’INI

H~0TI

~‘~1

_____

_____

_-~~--~H’fr)J >(N6FIN-?,-~-~---~ ~ 4.

1—

__ 1L tL”

~-~
?

fr’~-~~I~—-—-N 4iaç(~ (~) ITi~-iP’4~I Y _______

C.~JC[1]L~L),

~

I

H

____

~)

I

~66x-?)-~ L~

~

~

u

I~’t~r~I ~

j~J

~

.~ti.c*ecTLt~~1t1 ~

N5flN—N5~tNtI

Fig. I. Flowchart of main program LPOTT.

3. 4. 5 6. 7. 8. 9. 10.

CGC2 FFMHE3 FFHE3 FHE3Gs FMMM FFPN FFCHE3 LEGPOL LAGRNG

2,(65). Clebsch—Gordon coefficient squared, Analytic fit of McCarthy et al. to the (C0I~’2”3) 3He magnetic form factor, (84a). Assembly of charge and magnetic form factors for 3He and 3He. Interpolation on McMillan’s tn-nucleon form factors. Dipole fit to the proton’s charge form factor (79). Analytic fit of McCarthy et al. to the 3He charge form factor, (84a). Legendre polynomials up to order 50 (15). Simultaneous Lagrange interpolation on an entire matrix (R. Landau’s modification of IBM SSP routine).

/

RH. Landau

Pion and kaon elastic scattering

131

oPrP 1~J5

(~aII ~

MAIN 7~J

I

I

~

I

P~ TA?~~

N

~PUM.I

I

ii~izI ~

I ________

~~I4~I4~(I2)

,L.

(

~)J

(l~5)

C.~ft~vrnOPrP

__________

L

I

~ fixed)

Call p~Hor~rFI /(WP~)Nsv~aIILj

J

4

_____

I

VCo~IL(Wft~)I

I

VArWP.)I

(4~)

TN Li C.

j

_ _______

~ ~

~

I

___________ ___________

I

ii

S~ipfiva~h 1~PE

I

T~wfbl1e4’T

(Ø~)_________

L.PLJt~I

____________

4

C~I~I Ca~ ,,~,

I

L~LJM41

L~f~~M

___

LG~) Ui.ai~(k/i~’)

~)

I.~~(~~/k) j—1~— ~

____

Y~p~~N ___

[~&r I ___________

CaIITNUCTH

________

_____ 1211 1’ l{r2.Iz~1I

jz-NPr~

~L1

-11 • 1

Fig. 2. Flowchart of OPTP.

11. RHOOFR 12. RHOK 13. SPBESL 14. TNCM 15. TNUC

~NLiC.$~

______

(iio~)

________

1

~ri~

Fig. 3. Flowchart of TNUC.

Unnormalized coordinate space nuclear densities of Fermi or HO type, (75), (66). Numerical Fourier transformation of a coordinate space density into a form factor, (75b). Spherical Bessel function of the first kind, (75b). Calls appropriate subroutine for projectile—nucleon t matrix, i.e. TPICM or TKCM. Kinematically maps the two body t matrix from the 7rN to irA corn and does a numerical partial wave decomposition to find BNUC, (89)—(90), (110).

R. H. Landau / Pion and kaon elastic scattering

132

TI’JU CT H

W ~o

(~Th I

~

I

TNOFF-

1~ (~)

~

4~r ~)

NW~Y~>

/‘~Call~

~NU?J

N

I~rh~~fr

~

I

~

__________

I~

KI~’fy5wS

________________ __________

~

N

I C”)(~)

~

~

—< NIrTY(5) _________

~

I

I I

I

9 bNbNr

20. FFACT 21. MINV

I

4

I______

I

~ (Ill)

~

I

‘~)1~) JI

(Ill)

I ~.<

(“i)

____________

I

°(~)

E~b(~J1

NlrY(~n>~L,_ I LOCAL. (jl’~)

T~sI

~~olO9 ,J,o

I ~

,~< I ~~/~0~I

.

jTN1JC(~~~~ ~ I~4

TFOLD VCOUL VC OPTP

wirrr&~>~J~N~

____________

~

l~N~Ift)\~

I

~

~

~

~

16. 17. 18. 19.

I

______________

IE-~I

Fig. 4. Flowchart of TNUCTH.

A4~h1NsIQM4

________________

I?>±~_~

[

<____________

(~)(9~

NIFTYC5)2?~

________

I

(102)

______

2~.~CL4)

L Call MIXUP

Fig. 5. Flowchart of TNOFF.

Folds (Fermi averages) 2 body I matrices, (127). Partial wave decomposition of cut-off Coulomb potential, (42). Coulomb potential in momentum space, (44), (47). Assembly and storage of the optical potential from t matrices and form factors, (64)—(65). Form factor as a function of momentum transfer, (77). Matrix inversion.

)

R.H. Landau / Pion and kaon elastic scattering

133

TPIC..W~ ~ZG(6~ilar4w TKCM) (~ll

f~

I~~NGM (TNO~

‘Jr

~)

(_I~CaII Jr ~

________

b~4~al~~’ ~

___________

N

(ICC.)

~

N

________

I~ti:~] J.(~b))] (toG) ~ ( CakuLa~To’. 1 t~ ____

Ta

_______

1

‘~

~.

____

~C~allMIXUP ________

f

Pr~tT~.

~~A)(Ne~

‘~_____

_______

j

I

I

I

LNI~rY~ Cø1IiWXUP

I 9~~)

Fig. 6. Flowchart of TPLCM.

22. 23. 24. 25. 26. 27. 28.

PIMIX MIXUP PLPRME RHOKFF TPICM TPIRSL TNOFF

29. LOGPLT 30. TNUCTH 31. VABS 32. WAVFN 33. PAUL! INTGL QAV 34. G

Combine eigenchannel amplitudes into ir ±p and ir ±n amplitudes (107)—( 108). Calls pion, kaon or nucleon mixup. Derivative of Legendre polynomial, (14). Partial wave decomposition of form factor via projection, (76). Calculates and stores on-shell irN eigenchannel amplitude from phase shifts, (106). irN phases from Rowe, Salomon and Landau fit. Off-shell two body amplitudes from on-shell amplitudes in the two body corn; fixed K, K’, ~ all L, (lll)—(l20). Sernilogarithmic printer plotting routine; (adapted by RHL). Angle-dependent 2-body t matrix in the meson—nucleus corn. The code is given in table 3. Partial wave decomposition of the annihilation (“ true absorption”) potential, (42). Coordinate space wave function from F’ matrix, (153). Modification terms for irN denominator arising from Pauli blocking (121 )—( 124) (A.W. Thomas, author). Thomas’s separable potential for PAUL!.

R. H. Landau / Pion and kaon elastic scattering

134

FfACT ~ ~HO~WN VCOUL ~

P~HQKFf

.~.

4,2 C~fl

F-FH~~ (~).(s~)

A 20

y

~JIifl

--.--*.-

____

4 ~(I~CALL~yN~

I

F-rMHr~ I I

I

N

_____

I F~MMM

I

~

I

~

~

I

I-

~:

~

(17b)

I )o
RCWFN TKCM TKPMAR KMIXUP GAUSS2 GAUSS 40. BLOCK DATA

I ~ll r?AGT

~)tMoK(~l~,)

I

~

____

~

2 3

4 5

6 7

8

(VA.)

4~(~)

I

I

____

(m)

__________

(ia’)

Fig. 8. Flowchart of RHOKFF.

Coulomb wavefunctions routine of ref. [19]. Calculation and storage of K + N amplitudes (M. Paez, author). The Martin analysis of K~’N phase shifts (M. Paez, author). Combine eigenchannel amplitudes into K~p and K~n amplitudes. Scaled Gaussian—Legendre integration points (S. Pieper, programmer). Three nucleon wave function data of McMillan.

Table 3 Code for T~~N(9): TOFTH(I) I

L~ I

T

_________

Fig. 7. Flowchart of FFACT.

35. 36. 37. 38. 39.

~ll I

I

FF~PN

Re/Im

N

Re Im Re Im Re Im Re Im

p p n n p p n n

n, p

Spin flip X X X X

I VT

R.H. Landau / Pion and kaon elastic scattering

135

4.2. Storage LPOTT uses labelled COMMON blocks and EQUIVALENCE’s to transmit variables among subroutines and to save memory space. Consequently, care is required before changes are made to variable storage.

4.2.1. COMMON blocks 1) /NLSPFL/ (length = 8712) MEMBERS - BIAS NAME (LENGTH) 0 1850 3276 4777

GIN ECMDEN AIMW NECMS

(1400) (25) (1400) (1)

1400 1875 4676 4778

KAPG (100) FMN (1) AKAPD (100) DUMMY(3934)

1500 DENOM (350) 1876 REW (1400) 4776 NKAPGS (1)

This is the largest COMMON block (length 8712 words in MAIN) and contains variables for the nonlocal separable potential, g(p) and folding. GIN (14, 100) and KAPG (100) are the inputted g,,(sc) and K of eq. (111), a the eigenchannel label (1—14) of table 2, and 100 values for K are possible (actual number = NKAPGS). DENOM (14, 25) and ECMDEN (25) are the tabulated Da(~) and 25tZ”s of (lll)—(1l2), whereas FMN is used in the subtraction, (117) of the P11 pole from the Coronis—Landau potential. REW(14, 100), AIMW (14, 100) and AKAPD(100) are used to store the real and imaginary parts of the 2-body t matrices (SCøItaIIC0), (106a), for the 14 eigenchannels a, and a possible 100 values for AKAPD (actual number = NECMS). The rest of /NLSPFL/ in MAIN is filled with a DUMMY variable so that there will be room for the EQUIVALENCE of GIN to U (see section 4.2.2). =

2) /SEC2/ (128) MEMBERS - BIAS NAME (LENGTH) 0 56 100 103 106

BNUC BNF XBARC XN NWAVES

(20) 20 BN (16) 72 RETIJ (1) 101 XI (1) 104 NZ (1) 107 NIFTY

(16) (14) (1) (1) (20)

36 86 102 105 127

BNUCF (20) IMTIJ(l4) XPI(l) NES(l) NA(1)

This is the key information-transmitting block. [BNUC(20), BN(l6)] and [BNUCF(20), BNF(16)] are the physical, meson—nucleon, off-shell [nonflip] and [spin-flip] I matrices, B~’~’, in the [ir-nucleus, ir-nucleon] reference frames, respectively, eqs. (62), (63) and (108). The 16 and 20 correspond to the NL index of table 1. RETIJ( 14) and IMTIJ( 14) are the real and imaginary parts of the corresponding eigenchannel amplitudes, (sc’~çIK), (106) and (Ill). The 14 corresponds to the a index. XBARC = ,tlc = 197.3289, XI ir = 3.141593, XPI = projectile mass = (139.578, 493.668), XN is the average nucleon mass XN

=

~-[Nzx938.2s6+

(NA-NZ)x939.55].

NZ, NES, NWAVES, NIFTY and NA are input parameters (see fig. 2). 3) /SEC4/ The input parameters for the nuclear matter and spin distributions (see section 4.2).

(155)

R. H. Landau

136

/

Pion and kaon elastic scattering

4) /SEC6/ Input sizes for “exact” Coulomb procedures used in VCOUL and VC. 5) /ABSP/ S and P wave strengths of absorptive potential for pions as used in VABS. 6) /ENPAUL/, /ENQAV/ Internal transmission of data among PAUL!, INTGL, QAV for Pauli blocking computation. 7) /MMM/

Transmission of 3N wavefunction information in BLOCK DATA to FMMM. 8) /SEC5/ RHO(4, 50) (and DRHO) are the IMAX (~ 50) amplitudes present in the partial wave decompositioin of the nuclear form factor, eq. (61), for a fixed value of KP = k’ and K = k. The “4” indicates the densities:

3

—

proton matter p(q),

2 — neutron matter p(q),

—

proton matter ~( q),

4

—

156

neutron matter ~( q).

RHO contains the final amplitudes used in OPTP, DRHO is used for intermediate éalculations. 4.2.2. EQUIVALENCE in MAIN Class — GIN, length = 8712, MEMBER: BIAS 0 0 0 12 43 60 84 108 144 180 1500 1876 3849 4776

NAME DSIG U REUL X2 X5 X6 X8 X10 Xl3 X16 DENCM REW DFLIP NKAPGS

(LENGTH) (95) (8712) (50) (12) (12) (12) (12) (12) (12) (12) (350) (1400) (95) (0)

0 0 24 50 72 89 120 156 192 1850 3276 3949

Y Xl X3 YY X7 X86 Xll Xl4 Xl7 ECMDEN AIMW POLAR 4777 NECMS

(50) 0 CHART (12) 0 H (12) 36 X4 (50) 50 AIMUL (12) 77 X76 (1) 96 X9 (12) 132 Xl2 (12) 168 Xl5 (12) 1400 KAPG (25) 1875 FMN (1400) 3749 DNOF (61) 4676 AKAPD (0) 4778 DUMMY

(3721) (240) (12) (50) (1) (12) (12) (12) (100)

(0) (95) (100) (3934)

The EQUIVALENCE statement in MAIN is a built-in overlaying device. First, the memory allocated to COMMON/NLSPFL/ is used to store the Hollerith information which explains the NIFTY codes (H(12,20), Xl, X2, X3,...). Then, after OPTP calculates and stores on TAPE 7 the potential matrix elements for all L, k’ and k, /NLSPFL/ is used to store the (complex) potential matrix UJL±I/2(k’,k), eq. (149), (U(4356,2)) for all (k’, k) but only one L. /NLSPFL/ is then reused for each L value. After all the T’s have been calculated, /NLSPFL/ is used to store variables needed to calculate differential cross sections and polarizations (Y, YY, DSIG, DNOF, DFLIP, POLAR). Then it is used in producing the printer plots by LOGPLT (CHART) and finally in setting up the coordinate space wave functions (REUL, AIMUL).

R.H. Landau

/

Pion and kaon elastic scattering

Table 4 Changes for IBM OLD no.

Change

NEW no.

NEW LINE

1, 2

comments

2C 2c

PROGRAM LPOYF (OUTPUT = 64,TAPEI = 64,TAPE2 = 64,TAPE3 = 64, 1= OUTPUT,TAPE4 = 64,TAPE8 = 64,TAPE7,DEBUG = OUTPUT)

75

insert

76 77

110 120

remove

142-3

COMPLEX * 16 H, X1,X2,X3,X4,X5,X6,X7,X8,X9,Xl0,X1 l,X12, 1 X13,X14,X15,Xl6,X17,X86,X76

112 122

7(,(X16(1),H(l,16)),(X17(1),H(1,17))_______ 7 IOHSIN,KAPO ,IOHSO,NO ANG ,2*1OHUNDEF —, IOHE3B,NO AAY

replace

144

READ(5,730,END = 650) NR,LXMAX

261-2

comments

262c

263c

CALL SECOND (CP) WRITE (6,800)CP

348—9

replace

349

READ (2,860,END

360-1

replace

360

READ (4,870,END__201)ECMDEN(NK),(DENOM(NW,NK),NW =1,14)

378—9

comments

377ç 378C 381C 382ç 412ç

CALL SECOND (CP) WRITE (6,800) CP CALL SECOND (CP) WRITE (6,800) CP IF (LDUM.LE.1) CALL TIME (3,3)

856

AlO

855

710 FORMAT (7H NIFTY(,12,2H)= ,13,lOX,A8,A2)

867

comment

866

ç800 FORMAT (IHO,12X,15H TIME(SECONDS)= ,Fl0.2)

1959

()-~ *

1958

DATA VCOULL,REVABS,IMVABS/150*0.OE+00/

2950

()-.o

2949

DATA SIGL/2*l.,3*-l.,l.,

3028

ABS

3027

IF (CABS(ZSIG(N)).NE.O.) ZSIG(N) = l/ZSIG(N)/TWOPI2

3032

ABS

3031

IF (CABS(ZSIG(N)).NE.O.) ZSIG(N)=l/ZSIG(N)/TWOPI2

4045

comment

4044C

REAL DLOG

4143

DLOG

4142

Y=SUM+ALOG(DUM)*CONST

4388

300

4387 4388ç

IF (ABS(GP).GT.1.OE+70) GO TO 110 NB CHANGED EXPONENT TO 70 FOR IBM

4800

REAL -, REAL*8

4800

IMPLICIT REAL * 8 (A-H, O-Z) REAL*8 XS(NFTS), WTS(NPTS) REAL*8 HH(2) REAL*8 GX(l 18), WT( 118) REAL*8 Xl(52), Wl(59), X2(59), W2(59)

()—

*

382-3 413

4801 4802 4804 4805

*

4897

remove

5037

add after C -

4801 4802 4804 4805

=

180)KAPG(NK),

(GIN(NWAVE,NK),NWAVE = 1,14)

1,1,5*— l.,1./, IX/O/,

SUBROUTINE GAUSS(NPTS,KODE,A,B,XS,WTS) SPECIAL SUB FOR IBM SO LPOTF CAN RUN IN REALs4 WITH GAUSS REAL * 8 DA,DB,DXS(48), DWTS(40) DIMENSION XS(48),WTS(48) DA = A DB = B CALL GAUSS2(NPTS,KODE,DA,DB,DXS,DWTS) DO ION = 1,NPTS XS(N) = DXS(N) 10 WTS(N)= DWTS(N) RETURN END

137

138

RH. Landau

/

Pion and kaon elastic scattering

While it may be fairly simple to remove this EQUIVALENCE statement if problems arise, problems will (ultimately) arise if this statement or some of the variables are changed without proper care. 4.2.3. TAPE 7

As indicated in fig. 2, for each (fixed) value of k and k’ OPTP calculates the potential matrix UL±(k’ k), (17), for a//values of L, and then writes these out on TAPE 7 (NSPIN = 1,2 for J = L ±1/2). This is then repeated for a different set of k and k’ with the matrix U stored in the linear form given by the algorithm (149). Of course when the integral equation (138) is solved we need know U

÷(k’1k) for a fixed L ±and all values of k and k’. A good portion of OPTP concerns itself with skipping records on TAPE 7 keeping L ±fixed and varying k and k’ (except if L> LBorfl, since then ony the on-shell point is needed). In the present version of LPOTF we save considerable time by unFORMATTED READ and WRITE statements for TAPE 7. The user with more (or virtual) memory available may choose to save even more time by storing U( k’ I k) in core. 1

4.3. Running on IBM The present version of LPOTT has been converted to run on the IBM 360/195 at the Rutherford High Energy Physics Laboratory, Didcot, Berks. We find that the single precision IBM version reproduces the phases and cross sections of the test case up to four decimal places. This is sufficient since it matches (or exceeds) the precision of the elementary two body input. The changes made are given in table 4.

5. Running LPOTI’ 5.1. Files

LPOTT uses as many as 8 files for input, output and temporary storage. The run parameters are on TAPE 5 and are changed for each specific calculation desired (the input parameters are described in detail in section 4.2). The other files are: TAPE 1 Tables of irN or KN phase shifts and absorption parameters (see READ statement, card no. 2751 in TPICM and FORMAT 170). The RSL [24] irN and Martin K* N phases [30] have analytic form and are thus built in; the Almehed—Lovelace [31] (DELAL) and CERN theory [32] (DCERN) irN phases are supplied on cards, as are the Watt’s [33] (KNWATI’) K~N phases. Note, if a meson—nucleon amplitude is requested for an energy w less than those tabulated, LPOTT will automatically use one of the built in forms; in many instances the user may choose to “cut off” some of the initial tabulated phases so LPOTT is forced to use the analytic form up to the chosen energy and the tabulated ones beyond that. Note, too, that if the user requests to “fold” the two-body t matrices, LPOTT will insert 25 more entries into the table for energies lower than the first tabulated one in order to guarantee a smooth approach to lower energy. On latter application the user may care to increase NES (see section 5.2.2) to incorporate all the tabulated 1 ‘s. TAPE 2 Tables of irN or KN separable potentials, g,( p )‘s, (111) (see READ statement nos. 346,348 and FORMATS 850,860 in MAIN). The Coronis—Landau [12] potentials GCL and Thomas potentials [13] GTH are supplied.

RH. Landau / Pion and kaon elastic scattering

139

TAPE 3 Tables of meson—nucleon t matrices, (106), produced and then interpolated on by LPOTT (see WRITE statement nos. 2780—2782 and FORMATS 205,200 in TPICM or TKCM). Since for a given set of input phase shifts the two-body t matrices will be the same from run-to-run and different nuclei, computer time (and printer paper) will be saved if this file is permanently stored and then used again on later runs (such as our file TALNF). If the user requests the t matrices to be folded, (127), the folded t matrices will also be written onto TAPE 3 and should be saved for later application (see section 5.2.2). In this case, however, there will be a slight dependence on the nucleus since the momentum distribution, (131 )—( 132), is A-dependent. Consequently, we store these folded I matrices as files TFOLDHE, TFOLDC, etc. The exception to this storage on TAPE 3 occurs when NIFTY(3) 2,3 (see section 5.2.2) in which case off-shell t matrices also are stored on TAPE 3 (section 5.4). =

TAPE 4 Tables of the meson—nucleon Fredholm determinant Da(W) of (111) (see READ statements nos. 357—362 and FORMATS 850,870 in MAIN). As the beam energy falls below 50 MeV, the subthreshold extrapolation required by the three-body energy, (101), is required more often and these D,~’sare then used. We supply D functions for the Thomas potentials [131(DENOMT) and for the Coronis-Landau potentials [12] (DENCL and DENCLP); DENCL has a P11 pole at m,,, (116), whereas DENCLP has it removed via the procedure (117). Meaningful results demand that the separable potential function g,,,(p) on TAPE 3 and the Fredhoim determinant on TAPE 4 be consistent. TAPE 5 The input/run parameters (see section 5.2). TAPE 6 The OUTPUT file for LPOTT, normally routed to a printer with 6 lines to the inch (for standard size 5 cycle semilog printer plots). TAPE 7 The scratch file used by OPTP to store potential matrix during execution. Since this matrix changes with each choice of input parameters, it should not be saved and reused. TAPE 8 When the user requests that coordinate space wavefunctions be calculated, LPOTT will place F~’( k’ I k) (the wave matrix ~L) and other data on TAPE 8. As explained in section 5.3, if these data are saved, they can be used independently to reconstruct the meson—nucleus wavefunction.

5.2. Run parameters (TAPE 5) 5.2.1. FORMATS

Card I NR,LXMAX (4X,14) Card2 E (F10.4) Card 3 NGP, KODE, B, NANG, YMIN1, YMIN2 (14, 13, ElO.2, 14, 2F5.2)

R. H. Landau / Pion and kaon elastic scattering

140

Card 4 ACHP, ACMP, WSP, ACHN, ACMN, WSN, RCOUL, RCUT (8F10.4) Card 5 NZ, NA, (NIFTY(N), N 1, 20) (215, 1X, 212, 811, 1013) Card 6 NES, NWAVES, BOR, BOI, COR, COt (215, 4F10.5) =

5.2.2. Definitions NR switch for meson—nucleus wave equation (18) or Green’s function (4). 1: Schrodinger equation with relativistic kinematics (7), 0: nonrelativistic Schrodinger equation (6), nonrelativistic velocity in Coulomb parameter, <0: nonrelativistic Schrodinger equation (6), relativistic velocity in Coulomb parameter, ~ 4: “approximate” Klein—Gordon equation (9). LXMAX number of ir—A partial waves (L values) to be calculated in (15). Restriction: 25 due to DIMENSION statement in MAIN, yet this can be increased to 45. It is better to overestimate LXMAX somewhat since for L> LBORN (calculated by LPOTT) the Born approximation is used and only the on-shell potential matrix calculated. E meson lab kinetic energy in MeV. Restriction: upper energy of input 2-body phase shifts (2013 MeV for the Almehed—Lovelace irN phases, 2022 MeV for the Martin K~N phases). NGP number of grid points used in solution of integral equation (20), (138) via (139). The Gauss—Legendre integration point routine GAUSS was programmed by S. Pieper. 16 is usually good for a light nucleus — unless iI’(r) is wanted when 24 or 32 may be needed; 24—32 for heavy nuclei. Restriction: The DIMENSION N of U and F in MAIN depends on NGP (N 4 X (NGP + 1)2) with 32 the present maximum. For more grid points, only the DIMENSION in MAIN needs to be changed. KODE,B: KODE (standard 022) is a 3-digit decimal number written as HTU with the digits having the following functions: H 1: a square root singularity is removed from the choice of points at the lower limit of the integral. H 0: no special action. T: (1 to 5) T, A and B determine the interval the points are on. Normally LPOTT sets A k0 and reads =

=

=

=

=

=

=

=

=

in B. However, if B is read in as 0, LPOTT sets A 200 MeV. T 1, the interval is A to B (B may be less than A). T 2 (standard) the interval is 0 to ~ with 50% of the points on (0, A) and 50% on (A, ciz). B is not used. T 3, the interval is — ~ to ~ with 50% of the points on (— A, to A). B is not used. T 4, the interval is B to infinity with 50% of the points on (B, A + 2B). As B 0, the scaling approaches case 2. T 5, the interval is 0 to B with 50% of the points on (0, AB/(A + B)). As B — ~, this scaling approaches case 2. U 0,1,2: Gauss—Legendre points. U 0 requires NGP 2—16 (by 2), 20, 24, 32, or 40 or 48. U means GAUSS will use the next smaller value of NGP contained in GAUSS, if NGP is not available and U 2, changes NGP to the next larger number. U 5 returns equal-spaced poits, Simpson’s rule if NGP odd, rectangular elsewise. NANG Angular step size in degrees for which the differential cross sections over the interval (0°, 180°)are to be computed. <0: use INANGI and plot cross sections on printer plot (— 3 is standard), 0: then NANG set to 3. YMIN1, YMIN2: controls for minimum ordinates (cross sections) on semilog printer plots. 10~’M~I mb/sr: lower limit on 1st plot, 10~YM!N2: for the next two. =

= =

= =

—+

=

=

=

=

=

=

=

=

=

R.H. Landau

/

Pion and kaon elastic scattering

141

ACHP, ACMP, WSP: Proton distribution size parameters. NIFTY(16) = 0 : (66) is used, WSP ignored, for A = 3, (85)—(87) is used, NIFTY(16) = 1,2: (75a) is used with w = WSP, NIFTY(16) = 3: (78) is used for 4He, (84) for 3He or 3H (i.e. this is the magnetic sizes), (88) for 2H. The “standard” values are given in the text. ACHN, ACMN, WSN: Neutron distribution size parameters. The same coding as for protons above (future versions will use these parameters for the p2 potential in some cases). RCOUL: The size R in fm of the charged sphere for the Coulomb potential, (47), R~= ~J5/3 Rrms, e.g. RC(C, 0, Ca, Pb)=(2.6, 3.1, 4.65, 8.1) fm. RCUT: Coordinate space cut off radius at which the inner logarithmic derivative (calculated in momentum space) is matched onto ~~-shifted Coulomb waves, (49). NZ: Nuclear charge number. NA: Nuclear mass number. NIFTY: The switches to control the calculation. NIFTY (1) =0: ir~scattering, 1: ir — scattering, 2: ir + scattering and then ir — with same nuclear potential (approximate Coulomb), 3: irk, ir scattering and then charge exchange (CEX) (ir, ir°) or (irk, ir°) (approximate Coulomb), 4: ir0 scattering, 5: proton scattering (not yet available), 6: irk, ir0 scattering and then CEX, 7: K + scattering, 8: K°scattering, 9: neutron scattering (not yet available), — 1: K~,K°scattering and then CEX, —2: p, n scattering and then CEX (not yet available). NIFTY (2) =NIFTY (2) x 5 MeV is the value of the binding energy IEBI in the three body energy (101). NIFTY (2) <0 is possible. NIFTY (3) = 0: no coordinate space wavefunction, I: calculate coordinate space wavefunction and write parameterization on TAPE 8 (see 5.3), 2: calculate wavefunction parametrization and off-shell 2-body matrices; write onto TAPE 8 and TAPE 3, 3: calculate off-shell 2-body t matrices and write onto TAPE 3. NIFTY (4) = 1: separable potential model, (111), for off shell 2-body t matrices, 0: “constant” off shell behavior, (118), 2: local, Laplacian model generalized to D and F waves, (119). NIFTY (5) = 0: scattering angle-dependent subenergy with on-shell 2 body kinematics, W 2B0, (99); 2 body momenta (angle transformation) based on Madelstam invariants, (95)—(96), I: subenergy defined as 2 body “incoming” energy, (91); 2 body momenta from Mandelstam invariants, (95)—(96), 2: subenergy defined as “outgoing” energy, (92); 2 body momenta from Mandeistam invariants, (95)—(96), 5: subenergy defined as 3-body energy, (101); 2 body momenta from Mandelstam invariants, (95)—(96), 7: subenergy defined as 3-body energy (101):

R.H. Landau

142

/

Pion and kaon elastic scattering

2 body momenta from magic vector (AAY) prescription, (93)—(94), 9: subenergy defined from on shell 2 body kinematics ~a2BO (99); 2 body momenta from magic vector (AAY) prescription, (93)—(94). NIFTY (6) =0: no spin flip scattering; full multiple scattering series (single scattering term always calculated for R Born TBorn), 1: no spin flip scattering; calculate R matrix up to and including double scattering terms (SS term still calculated), 3: include spin flip scattering; full multiple scattering series (separate SS), 4: Set U 0 as check, 5: No spin flip; use SS term as full R matrix (Rms RBorfl) then (21) for unitarized Tms. NIFTY (7) 0: No pion annihilation (true absorption); all 2 body partial wave amplitudes, 1: No annihilation; only S wave 2 body amplitudes, 2: No annihilation; only S and P wave 2 body amplitudes, 3: Include annihilation: all 2 body amplitudes, 4: Annihilation via Yoo—Landau isobar model: all 2 body amplitudes (not available), 5: Annihilation potential printed but not added to U; only S and P wave amplitudes. NIFTY (8) =0: Do not read in phase shifts (use stored t’s); do not fold t matrices, 1: Do not read in phase shifts (use stored I ‘s); fold t matrices, 2: Read in phase shifts and store t’s on TAPE 3; for setup runs. do not fold t matrices, 3: Read in phase shifts; fold t matrices and store I’s onTAPE 3, 4: Use Martin’s analytic form for K~N phase shifts, do not store ‘s on TAPE 3; do not fold, 5: Use Martin’s analytic form for K~Nphase shifts; fold t matrices and store I’s on TAPE 3. NIFTY (9) =0: No Pauli blocking, 1: Pauli blocking of irN t ‘s (must use Thoma’s potentials). NIFTY (lO)= 0 No Coulomb force, 1: Approximate inclusion of the Coulomb force with amplitude phase modifications (55), 2: Approximate inclusion of the Coulomb force without phase modification, (54), 3: Exact Coulomb via matching with uniform charge sphere, 4: Exact Coulomb via matching with realistic charge distribution. NIFTY (11) Upward shift of ir - (or K ± )—nucleon subenergy in MeV. (12) Upward shift of ,r~(or K°)—nucleonsubenergy in MeV. (13)= Upward shift of .~r (or K°)—nucleuschannel E. (14)= Upward shift of ir~ (or K~)—nucleus channel E. (Negative NIFTY’s are OK). =

=

=

=

= =

R. H. Landau

/

Pion and kaon elastic scattering

143

NIFTY (16)=0: Harmonic oscillator p(q), (67), Harmonic oscillator p2(q), (136), 1: Wood—Saxon p(q), (75), Harmonic oscillator ?( q), (136), 2: Wood—Saxon p(q), (75), Wood—Saxon p2(q), (numerical), 3: Special form factors for 3He, 3H, 4He, 2H. NIFTY (17)=0: KMT optical potential with (A — 1)/A scaling. 1: Watson optical potential, no (A — 1)/A scaling. NES Number of energies for table of two body I matrices. If phase shifts are to be read in, NES should be set equal to the number of energies for which the phases have been tabulated. If t matrices stored on TAPE 3 are to be used, NES should be less than or equal to the number of energies there. NWAVES Number of two body eigen channels; normally 14 for pions and 12 for kaons (see table 2). (BOR, BOI), (COR, COl): The (real, imaginary) parts of the S and P wave strengths respectively of the pion annihilation potential, (133). The units are m~4and m6. If zero (or no) values are read in, LPOTT sets (BOR, BOl) (—0.04, 0.04) m~4,(COR, COI) (0.0, 0.08) m;6. =

=

=

=

5.3. Coordinate space wavefunctions The use of the subroutines WAVFN, COULFN and SPBESL with the data on file TAPE 8 permit coordinate space wavefunctions to be calculated even without the other subroutines of LPOTT. The formulas are given in section 3.2. A different data deck/file on TAPE 8 is needed for each energy and/or potential model and the user should supply his own MAIN (with a COMMON block /NLSPFL/ to transfer data). USAGE: Call WAVFN (NSPWF, LMAXWF, RMEV). NSPWF Number of spin terms, 1 for 0 ® 0, 2 for 0 ® LMAXWF Maximum number of partial waves (1 values) for which wavefunction at this RMEV is needed. If LMAXWF is greater than the number of I values for which 8 have been calculated (LSTORE), WAVFN will return F,(kr)/kr. If the wave matrix was calculated with a Coulomb potential included (“exact”), WAVFN will automatically match and return Coulomb-distorted waves; the user has no choice other than to use a different set of data cards. RMEV r (in fm)/hc. REUL(NSPIN, I) real part of U 1_ 1(r). AIMUL (NSPIN, 1) imaginary part of U, 1(r). =

~.

=

=

=

=

Structure of data deck Card 1: NZ, NA, NIFTY(l-20), E, XGAM (Coulomb parameter) Card 2: ACHP, ACMP, WSP, ACHN, ACMN, WSN, RCOUL, RCUT Card 3: NGP, LSTORE Cards 4—8: KK(1,,..., NGP + I) (the grid points) then for I 1, 2 LSTORE Card 5: 1, NSPIN Card 6: Re RL, Im RL, Re TL, Im TL 12R(i), Re ~R(’) Card 7: Re ~R(D’ Im ~R(l)’ —Im Re ~R(2)’ Im ~R(2), — Im ~R(2), Re ~R(2) =

R. H. Landau / Pion and kaon elastic scattering

144

where RL = on-shell R’ matrix before matching, — I)/2i. TL = on-shell T’ matrix after matching, (i~e2’~

5.4. Output of off-shell 2 body

t

matrices

If NIFTY (3) = 2 or 3, LPOTT will write the off-shell two body t matrices onto TAPE 3. These matrices are in the ~‘rAcom and may be useful in DWIA calculations. This output follows statement label 170 in OPTP and contains identifiers (as in section 5.2) describing the run.

6. Test case and set up The test case if for ir~—‘2C scattering at 50 MeV with the three body subenergy, the AAY angle transformation, the annihilation potential, the exact inclusion of the Coulomb force, the CL irN potential model, the RSL irN phase shifts and unfolded t matrices. The ~rN t matrices were computed and tabulated in a former run (see section 6.2). 6.1. Description of output

First the run parameters of section 5.2 are printed out and explained. Then Tab E,~, ~ and ~ are printed. The do ioop over L(7r — A) then begins with L = 0 and identifies itself; since OPTP is called for the first time, all terms in VL(k’ I k) are calculated and stored, and the half-off-shell (k’ = k 0, k = k1,..., k N,) terms for L = 1 are printed: Re V(total), V(Coulomb), Re V(annihilation) - nonflip + flip, Im V(total), Im V(annihilation) — nonflip + flip. Then the strengths and partial wave decomposition of the annihilation potential are tabulated. At this point the output takes a standard form repeated for all the partial waves. The determination of the FL matrix (146) is printed (N.B. a very large determinant, D>> 1, often occurs when a term in the potential is undefined, check array storage and dimensions). Next are printed the unnormalized, on-shell RL matrix (148), normalized RL and TL matrices (22), and the irA complex phase shift. Note that the symmetry present in the F matrix of (146) and (148) should be present in the four RL matrix elements printed as a check. The amplitudes and phase calculated up to this point may include a cut-off Coulomb potential, but have not been matched to external Coulomb waves. If an “exact” Coulomb procedure is chosen, the amplitudes will be printed out again after the matching (49) (the effect of the Coulomb phase aL on TL is also indicated). Finally, the single scattering or Born approximation to R L and T~,(35) and (36), are printed both before and after Coulomb matching. The above format continues until L = LBORN is reached. After that only the Born approximation results are computed. After all the TL’s and RL’s are calcualted, the differential cross sections2,tare Firstare given in the hentabulated. the first four tabulated irA cornlab, are:and, 0cm, finally, cos °Cm’the da/dQ (full MS), da/dQ (Born), t = (k~— k~,) in the Coulomb modified nuclear amplitude fNc of (53) is tabulated: RefNc(OCm), ImfNc(Ocm), IfNC(°cm)12 The total cross sections (28)—(30) then follow. The printout ends with semilogarithmic printer plots. The first is of da/dQ, the second is of do/dQ (“ +“ points) and da/dQ (Born) as “0” points. k~oni(1T— A)

R. H. Landau / Pion and kaon elastic scattering

145

6.2. Output for spin ~ nucleus and charge exchange The general format is the same as above except that for each L value there are two calculations for J = L ± For L = 0 this is provided as a check since the two should be identical. For spin there is also an extra line added to the differential cross section tabulation at each angle. It contains the spin—nonflip and flip cross sections, first and second terms of (26), for both multiple scattering and single scattering, and then the polarization (27). The printer plotting also gets extended: the first plot contains da/dQ +“ and da/dQ (nonflip without f~1) = “0”, the second da/dQ +“ and da/dQ (spin flip) = “0”, and a ~.

-~

=“

=“

third linear plot containing the polarization is added (+ ‘s for positive P(0), — ‘s for negative). For charge exchange scattering (CEX), first ir then ir (or ~0) and, finally, (ir ir°)results are presented with an approximate Coulomb procedure. After the first two calculations, the subtraction indicated in (38)—(40) is performed, the resulting t matrices are printed, and the differential cross sections tabulated. The total “elastic” cross section should be identified with the total CEX cross section. The plots have the same meaning as before. —‘

—,

6.3. Set up We have written LPOTT such that a good amount of information is printed in tables when the user requests the two body phase shifts to be read in (NIFTY(8) = 2, 3). Although we have found this information to be invaluable when setting up and checking LPOTI’ at a new installation, it would be a waste of time and paper to have it on each run. Consequently, we usually have two, short, set up runs to read in the two body phase shifts and tabulate and store the 1) nonfolded and 2) folded two body t matrices (the t ‘s are tabulated and latter read from STREAM/TAPE 3 see section 5.1): —

1) Input (TAPE 5) for nonfolded t ‘s 1

1

48.6000 2 22 0.20E+05

1.5900 6 12

3—2.00—4.00

1.6600 0101000200

0~0 0

1.5900

0

0

0

0

1.6600

0

0

0

0.0

0

3.0000

7.0000

0

The t matrices on TAPE 3 are then saved as, e.g., the file TALNF, and can be used for any nucleus or model (except “folded”). Input (TAPE 5) for folded t ‘s

2)

1

1

48.60000 2 22

1.5900 6 12

.20E+05

3—2.00—4.00

1.6600 0.0000 0101000300 0 0

0

1.5900 0 0 0

1.6600 0

0

0

0.0000

3.0000

7.0000

0

The t matrices on TAPE 3 are then saved as, e.g., the file TFOLDC. Since the folded t matrices depend somewhat on the nucleus via (126) (131), we store a different file for each nucleus. In later runs, such as the “test case”, these previously computed files are used as input. If needed, the output from these set-up runs will be supplied.

146

R.H. Landau / Pion and kaon elastic scattering

Acknowledgements This program is the offspring of theoretical physics research conducted over a number of years with a number of collaborators. The author is proud to acknowledge their contributions to both the physics and computing. In particular, Dr. S.C. Phatak wrote the original program for solving the momentum space Lippmann—Schwinger equation and its vestiges are still present in MAIN and OPTP; Mr. M. Paez has programmed the kaon subroutines; and Dr. A.W. Thomas wrote the Pauli subroutines. The support and hospitality of TRIUMF/UBC and University of Surrey are also acknowledged, as is continued support from the National Science Foundation and the Research Council of Oregon State University. Finally, Drs. R.C. Barrett, P. Bird and J. Tostevin has been most helpful and cooperative in converting LPOTT to the Rutherford IBM system.

References [I] K.M. Watson, Phys. Rev. 89 (1953) 575. N.C. Francis and K.M. Watson, Phys. Rev. 92 (1953) 291. A.K. Kerman, H. McManus and R.M. Thaler, Ann. Phys. 8 (1959) 551. [2] R.H. Landau, S.C. Phatak and F. Tabakin, Ann. Phys. (N.Y.) 78 (1973) 299. S.C. Phatak, F. Tabakin and R.H. Landau, Phys. Rev. C7 (1973) 1804. [3] R.H. Landau, Ann. Phys. (N.Y.) 92 (1975) 205. [4] RH. Landau, Phys. Rev. C15 (1977) 2127. [5] RH. Landau, LAMPF Workshop on Pion Single Charge Exchange, Los Alamos, NM (Jan. 1979) LASL LA-7892C. [6] R.H. Landau and A.W. Thomas, NucI. Phys. A302 (1978) 461; Phys. Lett. 41B (1976) 361. [7] A.W. Thomas and RH. Landau, Phys. Lett. 77B (1978) 155. [8] A.W. Thomas and R.H. Landau, Phys. Rep. 58 (1980) 121. [9] Y. Alexander and RH. Landau, Phys. Lett. 84B (1979) 292. [10] M. Paez and R.H. Landau, Phys. Rev. C24 (1981) 1120. [11] M. Paez and R.H. Landau, Phys. Rev. C24 (1981) 2689. [12] C. Coronis and RH. Landau, Phys. Rev. C24 (1981) 605. [13] A.W. Thomas, Nucl. Phys. A258 (1976) 417. [14] R.H. Landau and F. Tabakin, Phys. Rev. D5 (1972) 2746. J. Londergan, K. McVoy and E. Moniz, Ann Phys. (N.Y.) 86 (1974)147. [15] C.M. Vincent and S.C. Phatak, Phys. Rev. ClO (1974) 391. [16] T.-S.H. Lee and F. Tabakin, Nucl. Phys. A226 (1974) 253. [17] K.A. Eisenstein and F. Tabakin, Comput. Phys. Commun. 12 (1976) 237. [18] ML. Goldberger and KM. Watson, Collision Theory (Wiley, New York. 1964). [19] A.R. Barnett, D.H. Feng, J.W. Steed and L.J.B. Goldfarb, Comput. Phys. Commun. 8 (1974) 377. [20] H.F. Ehrenberg, R. Hofstadter, U. Meyer-Berkhoat, D.G. Ravenhall and S.E. Sobottka, Phys. Rev. 113 (1959) 666. R.C. Barrett and D.F. Jackson, Nuclear Sizes and Structure (Clarendon, Oxford, 1977). [21] R.F. Frosch, iS. McCarthy, RE. Rand and MR. Yearian, Phys. Rev. 160 (1967) 874. [221 B. Bartoli, F. Felicetti and V.S. Silvestrini, Rev. Nuovo Cimento 2 (1972) 241. [23] J.S. McCarthy, I. Sick, R.R. Whitney and M.R. Yearian, Phys. Rev. Lett. 25 (1970) 884. H. Collard et al., Phys. Rev. 138 (1965) B57. M. McMillan, Phys. Rev. C3 (1971) 1702. [24] G. Rowe, M. Salomon and R.H. Landau, Phys. Rev. C18 (1978) 584. M. Salomon, TRIUMF report TRI-74-2 (original version of RSL fit). [25] R.H. Landau, to be published. [26] RH. Landau and F. Tabakin, Nucl. Phys. A231 (1974) 445. [27] K.B. Yoo and RH. Landau, Phys. Rev. C25 (1982) 489. [28] MI. Haftel and F. Tabakin, NucI. Phys. 158 (1970) 1. [29] L.S. Rodberg and R.M. Thaler, Introduction to Quantum Theory of Scattering (Academic Press, New York & London, 1967). [30] B.R. Martin, Nucl. Phys. B94 (1975) 413. [31] S. Almehed and C. Lovelace, NucI. Phys. B40 (1972) 157. [32] D.J. Herdon, A. Barbaro.Galtieri and A.H. Rosenfeld, UCRL-20030 ~rN(1970), CERN Theoretical Fit. [33] S.J. Watts, DV. Bugg, A.A. Carter, M. Coupland, E. Eisenhandler, A. Astburg, G.H. Grayer, A.W. Robertson, T.P. Shah and C. Sutton, Queen Mary and Rutherford Report (1981).

RH. Landau

/

Pion and kaon elastic scattering

TEST RUN OUTPUT 16 22 8. 2SE+95 3-2.98—3.88 NO OF GRID POINTS. 16 1.5598 1.5508 8.8 1.5590 1.5508 0.8 3.9988 7.8808 6 12 9 181795893 8 8 0 0 0 9 0 8 e a HIFTY( 1)— 9 P1+ HIFTYC 2)— 1 s58E SHIFT HIFTY( 3)— 5 NO ~.F. NIFTY( 4)— 1 NLSP—G(P N!FTY( 5>— 7 E38~ AAV NIFTY( 6>. 8 HO SPIW MS NIFTYC 7>— 5 AOS—OLLSP NIFTYC 8>— 8 HO DELIFLD NIFTYC 9>. S NO PAULI HIFTYCIO)— 3 EXACT COUL NtFTY~11). S PIG~PI—~k. OHIFI HIFTY(12>— I PI+~KO SHIFT MIFTYC1]>— S PI-CHANH -. SHI MIFTY~14s I P1+CMANN&L SHIFT HIFTY(15>— 8 NOT USED SHIFT NIFTY(1G) I RNQG.RHO2C NIFTY(17)= I KHT MES.NUAYE8.SIR.0II~CSR~C0I— 59 14 5.9 0.9 8.5 5.9 HES,NUAVES~89R.0SI,C8R~C0I 59 14 —8.14199 0.94898 0.8 0.08008 ,csii,~,* AMASS— l2.0095s*,**s*,I* TPI EPILA~ ECNCSORTS) ICCN PLAO 55.5959

189.5785

11447.9297

RELATIVISTIC CALCULATION ENERGY MOMENTUM 58.5095 126.1715

126.1715

128.2898

147

148 -~ S.,

S.~

U) U)

Cl) U)

•

‘.4 S.

U)

U.

U)

a

a •

0.

U)~U)U)U)U)

-~

I~

o z 5-

Cl) U)

~ •U)~.U) IIiiIiuIiIIIIII WWWWWWWWWWWWWWW t..U)~C’)

U.

.U)F,PJv,r-U): a —.pj~oU)-.—~~ v,’c U) ..0~~

U. U) X

It) U) I Ut

0) U)

—

a

U) —

Cl. E

—.

IIIIIiII!III

II

U) I

Z +

)DU) U)U)UI ‘UWF.. U)

~

F..

~

U)

*~

U) U)

•U)U)

5.5

UJWW WW WUIW L’.IUiUI LU F. U)IS.LU’D(J~0’FCF.Ui~

LI.

UI U)

-‘

U) IU

. I~)

U)S.~

U) U)

~ ~ U) U) U) U)

•Co U) U)U) a

U)

Ut . U) f_ U)

-

-‘

0.

—

U. 5-

•U)U)U)U)U)U)U)U)U)U)U)U)U)•

U) U)

UI x

— .51

~.

UI

‘.

Cl) U)

U)

U)U)U)~U)U)U)U)U)U)~”.J

a

0. —

.J Ii. ~ Z

I

a’

U. —.

..i I’..

C.) >

Ut U)

U) ~ U)

p.4 U) It)

U)

U)

N a

aN

~..

~

UI

UI

0) U) a UC

U) U) U)

II a

N a

U)

Ui

UI

~

It) U) I U

U) + Ut a’ It)

U)

—

U) I Ut a’

I + UIUJUI U)

U)

UI

I

U) U)

—

.

.

.

‘U

F..-.0C CaOtfs. .a~ V)U)-.

~

— —

U) 01

N

N •

Z ~ I...

N 0. ~

UI

a~ U) II.

C Z

~ a

_j CO Z

a

• a.

0~ U)

•

‘~

,-.

p.r-s

5))

F.1

—

—

a

U)

a’ I Ut

U) I

U)

~‘

Ut

a’~

l X 0

+ + UI UIUI U)U)kI 1)105)5. ~U)U)

CO a UI

SInU) U)U) 04..5

It) U) I LII

x

U)U)UI I I

-. CS.

—

-.

a

.4

a) a’ F.) PS. U) •

~iU)O•04a’ IIIIIIIIIIIIIIIII WUI1.UUJWIUWWWWWUIUIUIUIWU.C U)f_C )IU)U)-.S.)It-.5U)f_U)).CWU)U)U)U)U)U)U)a’U)a’ •~U)It~*U)—.U)U)lfl*U)U)~U) .0)$ U) U)U)—It~C.ab)Tr’*..5U)’U.aPU5-PsU)U)a’U)U)a’U)U)U)a’ ~N.AIF.) U)s.I(JU)U)U)..5U)U)...* 0J~U) . . F) a’a’a’a’a’~ a’a’a’a’a’a’a’a’ U).

U) Z 04 .

.-‘UJ ..x U)

q U)

—

*

111111111 WWWWUJWWWLII a ~U)U)U)U) U)U)F)U)Inl~It~f_’U

~

CO Z

5..

_.

“

a 1 a

.

CO X

I

5

U) a U.

—

a

UI a

~

Ut

l U.. ~ .

X.5C U) LI. 5)) UI ~ .4

a ~.

Z Ut p.

C

CU

.*~~F-.U))Dp.i**•*

U)

I • I

0.

U.

t

S.

I

U)

U) UINa’U)U)U)U)U)U)a’U)U) I I I a I-

a U) + W~ 0+ U) U) — •

U)

~

UI pUI U)

XII ...

a)~ I.. a .

~

..

a’ ~ U) ~

—

a

UI ~

z—. a’ U): IIILII ~ 1.1.0,

X I

Z.-I.

X..

a ~

m

I.1~ . 0)01 a

a

0.010

a

15.5

~ CJCIa

~

za

U) ~ U) ~ C.) p. 14

N 001 U)UI N 5-. 1515-. .

~ ~

a XII

Ui..

4

a~

p. U. U)

. .

.a.).i

-

~I-

LI. a

OFIX

~

~

UI ‘.-

~

aa,..~ II-)~ 05-

I —

I

5W I ..IU).

II

~

C C.) I.. I.)

U) ~ a’ U) IS..

U) 05)1

5-

U)0C UI P.101

a’ i UI U)

+ + UIUJ U)U) ~ f_F.. —p., U)~ ~ 0404

a

&‘. a

~_

a4U)U)U)U).~5U)*M

a’ I

X U I-. a

..

UI a

U) I

N a’

0. I U)0)U)*U)U)U)U)~5 U)U)U)U)U)U)a.~U)U)U)U)U)~~U)ZU) +++++++++++++++ +afS.,U).U)U)U)U)U)U)U) UtUtUtWUtL1lL11WWIllUtWWI~IU,I&t—U)1 I I) I I I U)’UU)U)04C)IV~C$f_ U)N-F)

p..,>

U) cga)wInU)w,a)u)w,c.IU)ar. UI P-V)U)Ps U)U)0)U))’)MU)F..~ P01 . U)P0U)..504 0405 U)5..a’InWf_0)a’0101PSWII~) .J . .X •U)OU)5*’*.5 a U)a’U)U)U)U)U)a’U)U)U)U)U)U). U)s.. I I I I I I I I I I ~. Z LUWWWWWUIWUIUI f_U) )U)U)a’F)~ ~ ac51wp..1v—U)r-a’U)soU) UI f_U)0)a’U)U)U)~U)r..r.. IZ~-I)U)U)ClU)U)P..r’.

~

UI U)

a’~ ~

i~ U)

‘5

U)

WUI

—

—

U)

.

~0 U) U)U) U)U)U)U)U)U) U) a. U)

_J

S.~. ..IW .~

~

U) II’

—

U) X a U)

——

—

— •

C

a

Ps

—

U)

—

.

f_U)U)W0)U~IflInU)U)fS.U)*5U)O U)s..~

i

..

~ UI U)

a

U)

U a’ F..

a.

-

I I I I i i I UIUIWWLIJUJUJLUUJWUJUIWWWW U) In~5IF.~t.1 0)~JF,+ U) •U)U)PsCsICUU)’P U’.l) ‘UInU)U)U) U) 04

j

UI

1~

-‘

5-

0) U) a ~

~

Ui

F.)

U) U) U)

)III)I~I~I~II IIIUIWUIWWIUIUWWLIILUUILIJW, 5.5 •U)U)U) f_SCSCSILDl)I LO~.... U)l~’UPsO-.b)U)U)NU)O4*U) U) U)—.-F.J* -~l)I-..5Ca In...F.. ~‘W a’a’a’a’U)a’U)••

~• ..j

a

UI

5-..

N

U) I U U)

Ill a

In

II a

N~ — U) . F.. U) 0) U)U)U)

Ut a 0.

. ~.

I

U) U)

.

.5

U) It) p~

E

-‘

U) U)

—

~.

C

~

f_

X

~ -.

-

a.,~ a.’.. ~

~p. U)

/

R. H. Landau

N) 01 ND V

CD U) U)

—

CM

—

-..

—

CII

01 U) U) V

*

—

U) C) N.

P..

01

ND

U) CM I’)

—

—

ND U) I UI

a N.

a N.

a N.

a N.

U) V

UI U)

.1 UI U)

UI U)

I

...I

II

II

—

II

U)

ND U)

U IC

U) If) (II U)

— N) C) C)

CU U) 0) 0)

If) U) I UI P.. V

U)

~

U)

Ia’

U) V U) 1’)

N a

N. U

aN

aII

II a

U)

•

U) N IC 0

— U) U) UI 0. I IC I— I I IC I S..’

I.. U

U) U) + UI — V CM V 0) — —

•-I01

VVV ..U)) V VU)CU WU)V CO VP..

N. UI

.

.

.

a

..J ~ CD SC

a I I I I I

U 10 ~ It— ,..I U.

.

U) a’ U) + U It) PC SI C) — NJ) CM

a

U)C)U) I —, IC

SI

IaU)

CD IC

—

S..~

a ut r U)-

CU) XLII N-. UI U) U)

II a N.

3

~J

‘C

C) V 01 0)

U) P.. 01 C)

N. N. 01 C)

U)

U)

U)

U)

N a

N a

a

a

N. UI

N. UI

I)

a

01

.J

..I

UI Ca

UI Ca

— U) U) 0)

N.

W 10

U) V — -

01 ND U) I UI U) C’) M C’) P.. U) I’)

N. UI

N. UI

UI IC

N

N. U

U) V C) N. C) C)

S.’

a

XII —.41Sf N. 01 . IC Sf

U) IC 0 .1

•

41Sf

U.N)

-

wa’ U)U) ++ 111111 ...V “’0) —V *0) (MU) NDP.

SI UI + Lii U) 01 U) V ‘C NO 01

..l*

U)U)

—

U) (.1 IC — m UI N. 01

101’) 0... Zr’)

a

XCII IC 0.SI C +— U)UIN. ZN) C~ 0U)Z _IU)..” N ~ND)C.’. ON)U)Sf UIIf)41 . (‘10.. - 0..Sf 0101 • Sf

C C.)

N. SI

UI . (001

U) U N. Ii.

..

UI 41

LU .1 (.3 ~ ZU) U) 10+

UI

.

U)C)

.“

a

—

a

Ia

C.) N.

IC

U)

N. U 01 IC II

UI~

U)S..’Sf

SC

IC...I-. U).U)SC CX X41Sf XC’..’ ~ZN.

IC.. i-sc —..

asc

ZO) 0. II CCI OUt

N NI”

SC -

3.1

-

SC

‘a

‘a

aN. U) IC

N.

..J015.1 IC. 41Sf C)’..’ ZN.

—

U) U) U) I IC I Ia I I IC I I’.. (MU) N.

IC 0 N 41 U. N. ~SX

~-a

a’ IC 0

XX UJ”

rr

.j ‘

U) (.3 N. I.) 01 XI

UI~. SC

IC.. N.Sf 5-

N IC 0

IC

41 UI I— U. 01

..~Sf

-

U)

a

‘a

U) U) Ut Na’ ‘-‘UI N .101”

— U) + U ND N. ND UI V If) —

NC U) I UI U) P.. N) U) — 01 V

‘a

0.N. U) IC

U) I UI C) C’. 0) U) N) NC —

II UIW U)U) ICV —— SIT CMV P.01 $)U)

U) I UI U) N) C) )“I C) I”) —

.

-

010101

-

U)

I

ND.....,

— 0 I uj I.. N. ‘C I’. V PC C)

010110)

III UIUUI If)...I.1) 0) O’.P.. I’) VP.. 0110W) P..U)C’I C. 0)01 01-.—’ .

.

.

SITU) I

—.

C)

IC

CD IC

Ia

Ia

I

IC (.3 N.

-

UI CC IC

—

II) P. CM SI

I

N a N.

N a N.

II a N.

II a N.

..I

..I

-J

.1

.J

Lii U)

U U)

...

U) C. U) CI

UI U) U) C)

— SI SI U)

—

0

0

—

—

U Ca

UI 10

U) C) U)

N

N. UI

UI Ca

u

N

a

..)

0)0. -

0 (.3

U)01

U) U) UI CIa’

N. U

_)

S.UJ N

IC N

CD

01,-Sf

IC a

IC...).. 4101Sf CX X0.Sf XC’.’ ~)XN.

.

-

-

Sf

-

01

N

N a N. J

SI SI

i

N

a

a

a

a

a

N. UI

N. Ut

N. UI

N. t&i

N. Ut

ICOC

a 41 oa’ “

Sf IC.. N.SC

Sf

-N.

N N” ‘.‘Sf

41W

UI.”.

.

SC -

asc

SC

‘a

‘a

ZN-. 0 SC

N.

CISIP. V 01)’.. C) C)-’ P..U)U) C)IDC’) CIVC.

— -

U) U IC U) — U) 41 Lii 0.

01Sf IC. 41Sf 0’a ZN.

-

.

U) U) I UI C) N. 01 U) *

C) II)

.

-

01010 I

0

U)U)SI III UIUIUI U)U)U) ND..I* UlU))’.. N)U)V —‘NOV 010’.—’ V...Ot .

.

U) I UI ND CM IC U) U) 01 —

.

.

I IC — I I SC

U) IC

(0 IC

.‘.l

I —

‘.

—

IC C.) ~

0141 N.

I

IC U N.

a )~.

U) U I’—

0 N 41 1.1.

U,

UI ~

U) ‘a ~‘.

XX %QIa

41Sf N. .

041

ZSC

zr

ZI.Ii N-III C) U)

N. LU

a

a

SC N U1a~. SC

~. P-SC — ~SC

...t ~ CD IC

a I

‘a

I

0.1” 0 IC

I I I

SC N

a

.

-

U) SC

U)aSC X0’a ~ICN.

P-lI.

N.

~,01l

‘CC’) U)—. SIC) SI.’. 0301

— V ‘V C) —

CUll) ‘CC) SIN. VU) P-U)

0) U) I UI a’ SI N) NJ) N) N. V

a

U i UI ~ XCII U) U) I (11111 ~‘.~fl U,V

.

-

.

U)C) I I

41 CD IC — IC C.) N. .U)

.

-V.0.. .0.3.1

-

3.1

I I I I I V

-

‘a

aN. 0 IC

ZU) 41 N 010 C)UI II “ N” 3.1 IaSC . ,JSC aX IC SC 41Sf ‘a C’..’ ~ ZN.

-

U) (0 IC — IC UI N.

a IC

01 ZIC ~1- N N.5C. ZOX 11141 ICC. CO-Sf Z01 Sf

01 SC 0 ..I ~ U) .1.)

‘a

a

.

U)U) I I

IC 0 I —

IX UI N. U. ‘U)

(101

IC N UI” SC X N.SC — aX

IC..J..

C) I UI I’)

C C.)

—u N

OIC. Z41IC

U)U) II 111111.‘.P.

U)

-J

I-. CI

01.-ISC

SI

(‘4CM SIT II 111111 ‘CU)

UIU)U)

U)

U)U) UI ,JN~

001

-

0141

COlIC IN....’ II ~P-)C.-

41 IC

‘a

Na’

0

U Ca ~ I— .,,t 0. IC

rota

a

p~

V U) I UI — — 0~ U) 0) V C’)

IC 0.U)0 I — 011111—

U.

ZN. ~X

‘NV SI C) II UIW -‘II 10* ICU) —C’. SIC’) ‘CU) 0101

IC-’.

U) I.~3 N.

-

UI

PC 01 I CU — ‘C U) U) U) II) —

0101 II UILIJ •.,C) NISI SIll) VU) C’) I’.. 0)01 VU)

10(I) CCII IC’ 0) UI. CCCII 01

IC

—

U)rC

I-Ill

a

IC

01

U) UI

U)

I I

“.

41 IC U) .11

‘a

a

..lU)fI)

SI + LII — If) — C’) ND ND

a

.

‘a

.,I

01 I LII C’. 01 U) 01 V P. U)

CD IC

41 IC C

.JN~

-.

.

11.01

a

U) Sf

—— 0101 II UIUJ P-U) P-CU MI’) 0)1’UlU) (‘ICC —— 001

IaN.

‘—

a

I I I I I

U) a UI 1 (.3 ~ IC—’ U) 10 I UIU1

41

X”Z 0.U)C I — Till It. Z0t01 omr ..JN’~—’I ~01X~ 0’..’U)Sf C.).’.U) -

IC Sf

~

a

IC II —.41Sf N. -

a

0) 1)1 ‘C V

N a N.

011CC’) U)U)U) III LNJUIUI

.

SIC)

a

41 UI N. U.

-

LII 41

.

U. IC

U)V OW) IC U) I’i UI CISU)

a

-

UI U) ~ N. —

IaN.

00) ICUI ItLII C) (IC

-

0.Sf

—

W...If) V 0101 — OSU) 01-~ND V0)CC -‘CDI..

-

V U) C’. It’)

— I

C) C’. 0’.

U)

SI U) N) CU

U) 0101

..l.~Cl1

.

CU V C’) ND

UI U)

-

N

—

NCCM0J U)U)U) III UJWUI

— U) I UI U) CM U) N) I’.

U)U) II UIUJ ‘Cr’. VU) 0101 -~0J U)U) CMV 1CC’)

U)C)C) I ~)SIa’ 014101 I++ U1WUI VU).~’ V (041 U)NDN. U)U).’. VVU) CIU)01 C’)—.—.

I

ZN~IC N.01 XX Ut’-.

-‘U) I

U)

CD.~ •01U) III UIUIUI (‘11CC)

N.

SC O • 41 II.

U)

II a N.

II

a N.

If) V

SI

*

V U) I’) CII

CO U) I UI

UI

149

U) U) U) C)

It)

II

ND V 01 C)

-

+ UI U) C) If) NJ) SI ND

P.. V

-

V ‘C 01

* •

Pion and kaon elastic scattering

DC UI N. U. 01 C) IC C -J ~ 0 CC

‘a

N. UI ~U)

ZOl 4141

CU. SC IC-

J01W N SC N N.’~’ UI” 0) SC IC .1IC-

SC N i-Sf ‘

a

-

0.Sf ‘a

0.N. U) IC

I I I I I

N. CC

a

01Sf

N.W

IC. 01Sf C’a ZN.

— ~Sf

-

‘a

0.P.. 0 IC

150

R. H. Landau f) ~ z t, -. (I)

II. z

/

Pion

and kaon elastic scattering

NNN~NN-.~~ ~ ++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++•.+ wwww~.jwww~ WWWWWWWWWWWWWWWWWWWIIJWWWWWWWWWI.~IIJUIWWWWWWWWUJIaWWWWWWUJ wp,.wCo ~ p~-r~w ~ ~VIN0~ tOt~Co~~ ~*~CoCoCo ~ ~

~~tatwa~.

~ ~

r~)—mp-~(g ~ (.3

~

TCoCoCoCoCoS~• 1111111

11111

till

I

1111111

Ii

x I‘-

l~ Iii

~

cg~a~s, *Co*l~-~ v~cor~wv,s,

~

~ Co Co

a -

Co

S..

a ~ \ Cl) Co

~ r’~c.j ~ ~CoCoCoO~ ++++++++++ WWWWWWWIMWW Co c~or..CoS ~C.1lS.~CSCoCoCo ~

Co Co

Co

~Co Co Co

Co

Co Co

Co

~ CoCoCoCoCo~CoCoCo +++++++++++++++++++++++++++++++++++++++++++++++++++ WWWWWWWWUJUILIJUIWWWWWWIUWWWWWUJWWWWWWWWWWWWWWWIUWWLUUJWWWWWW Co*Co IS. CoCoCoCo*~..WCoCo*t~Wl~.WWW NJISJ.-..-. PIS-I~.

I... 1• .

~ .D

CoCoCoCoCo~~CoCoCo

Co I W * CD

a Co \ Cd) Co .Co a ..J S.’

CoCoIS. It~**1’)IS~Id)F~M id) WWWWC~WCDWWCoWW C,

Co

CoC.IDWC.r’-IS’CD%O* ~sCOisCDtCIMCdI

Co*W)*.IW)CoC,C,WCIJIS.

CoUICo CdIDICoP..’.Olfl *

*~Id)CoC~d)ICDSW)CoU~CoU~Co*W”td)CCdP.. Co -~*I~U~W

CoCoSCoCo

~

WCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCo ~

N

tYsOCC,W CoCoCo

o

CoCoCoCoCoCoCoCo

CoCo~CDW Co a

Co

Co CoCo SCoCo CoCo •CoSSCoCoSCoCoCoCoCoCo +++++~++++++++++++,+++#~+++.+,.+,++++4’++++4++++++++I WC~IWI&IWWWtijWWWWWWWWI~IWWWWWWWWWb.lWWWWWWWIUWWWWWWWWWWW

Co (.3

r.. Co (II

*CoCoCoCoCoCoCoCoCo ++++++~+++ WWWWWUIWWIIJW Co CoNCo Iv,cgr-..Cocgr,—r.....Co —** Co CoCoCol~CoCoCoCo

•m*w~’ClC~CoCoO~C~CoC,Co

CoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoSCoTCoCoWCoCoCoCoCoCoCoCoCoCoCoCoCoCoCoSCoCoCoCoCoTCo—. llllliIlIlliIllilllillIlllIlll

*N,d)~CooCCo

~d)id)CU0JC.I

Co~IflCo~*iS.Co1d)CD~ ..‘.C’lPjN

01*~N~***

Z 0

*V~I)WSCDWIS.P..P..CiP

W)lf)NI

*****

I— (.3 UI ($1

~ ~~~CoCoCoCo~Co 11111111

-. a

a Co UI UI

*CoCoCoCoCoCoCoCoCo •#+++++4.++ UIWUiLUWWWIIIIIJW V~CoCo...iS.. IS~C’JC’iCoWF-V) *w,0ird)l’-—.-.—pd)w, I-a CoCoCoCoCoCoCoSCoCo

*U~CNIS.Co III

CoCoCoCoCoCo 1111111111,111111111111

Ill

ill

lit

CoCoCoCoCoCoCoCCoCoCoCoCoCoCo lIIIIilIl~I 11111

a

_J Lu X

I-— U103

I

a Co \ (3 I

Cl) Co

SCoSCoSCoSCoCoCoCoCoCoCoCoCoCoCoSCoCoCoCoCoSCoCoCoWCoSCoCoCoCoCoSCoCoCoCoCoSCoCoWCoCoSCoCo

P—Wi.eI0C1

S.’ (I)

~.CoCoCoCoCoCoCoCoCo CoCoSCoCoCoCoCoCoCoCoCoCoCoCoCo5(CiSCoCiICoCoCoSCoWCoWCoSCoCoCoOiCoCoCoCoCoCoCoSCoWCoCoCoCoCo—

Co~*Co CoCo

CoWIS*IS.CoCo01Co*CoC,***1CoCoW)a.Co SCoW)CoCoW*CoCo*IS.S~Pd)W)i~i.CoC,C,Co **CoCoCo**WCoCoC,Co0lC,C,C,O)CtCo 111111111111111

111111

111111111

* if) WI

Z 0 — i.(.3 Lii Cfl I CI)

• I

CoSCoCoCoCoCoCoSS CoCoSCoSCoSCoSCoSSCoCoSCoSSSCoCoCoCoCoSCoSCoCoCoOCoTCoSCoCoCoCoCoSCoSCoCoCoSCoCoCoCo

• (.3 I =

CoVd)’.bC,011f)Co *P.. •I~)’JOCOIV)Co.*IS.S ~CaC,(~~Co ,tCoJ01 ~fd)MI’)***

*P..CoI’)CDqtOdN~0I,*iS..Std)WC,i)IW)Co.~* IS.SI~)CC,i)lV)Co~*IS.S Co5CoC,CoS ***~WIWICD~OCDIS~IS.FS.Co

—

II

cit

)~ S.’ ,-

(~ ix UI Z Lii I I

CoCoSCoCoCoSCoCoOSOSCoCoSSCoSCoCoCoSCoSCoSCoCoCoCoCoCoSCoCoSCoSCoCoCoeCoCoOSCoCoCoS IS. ++++++++++.P++4++++++++++++++++,+++++++++++++4++++++ WI LULUWUILUW WWLUWWWWWWWWWWWWWWWIIIWIiiWWWWWUIWWWUIWUJWW — Co~ ***Po-.C,Co* CoSPJflIC*i..S~SI)StD4d)OJ.. ~ OJOICoCoCoCo**Co..dICIi)1CoCor,.l~CDV)WtU~WI *Co*W I.-~IS.C3~DI.ICbW*DCOCDWCDLU U CoCoCoCoCoCoCoSCoS CoCoSCoSCoCoCoCoCoSCoCoSCoCoCoCo•CoCoCoCoCoOCoWCoWCoCoCoCoCoSCoSSCoCoCoCoSCoCoCoCoCoCoCoCo CoC,U~CoCoCD** ~ CoC,OCQIC,CICICo

UI

I (.3 I I I— I Cl)

.

•SCoCoCoCoCo•CoS f+++++++++ WWWWWUIWUIWW Dt)N*r’-*..C’t IS~Co O’tCOCCIS. *V)01’0N)*CCiCo

r (.3 0 (.3

CO ...i UI

...... CoCo CoCoCoCoCoCo CoCoCoCoCoCo~CoCoIPCoCoCoCoCoCoCoCo +++++,+++++++#++#++4+++++++++++++++++,+++i.+++++++++ LUWUIWW WWIIJUJWWWWWUIINILIIWWWW WUIWLiIUJWWWIIIUIUJWWUI&IJWWWWIIJ co CD*CDCo CoCo*Co.iWOl*MCoCoP..W)IS)ISt CoCoIfl—CoN*WCoCoCoCo1** ~

(.1

(.3

Ci) UI 0 ix ci

CoCoCoCoCoCoCoCoCoCoCoTTCo~CoCoCoCo — ++ +++++++ ++++++++ ++++++++++++++++++4•+++++++++++++,++++++++,+ U$WUIWUIWUIWUI LUWW WWIIJWWUIWICILCIWL.IIii&.JWWWUJWWWWMIIIIWWWUIWWWWUIWWWWWWW CoCD-IV~PCoCoCo 01CoV)01CoN~ONDCC-.*WWCord)I~CDP CoU~...IS.* CoM* CoCo CDP~CiiCo~...0Id)JN)i’~id)IS’rd) I-

I’-

OJ 01 I CQ .J.

CI) 0 ix (.3

I

UI ar.. ZCD

a

CoiI~ .-iI’) ..3-. ....I

1

CoO ri— 0 3C i~0 CJix uia. COD. I

..J a I-. 0 i...

..J

-a

U) ups OCo CJCo

RH. Landau SIG(TH) -

19Es03

59.9

•

-

32E+92

-

1aE+92

6

.

+

• •

1.559

•

-

++

•

99

9.9

1.559

.

-

.

.

.

.

.

•

•

• .

•

. •. •

. • •

+

o.. ae 9~ +

9+

9 + . 9• 4. 9

. • • .

•

.

• •

• • • .

•

. .

a

a

1.559 •

I

151

a a

a.S

0++++. 0+4 -

++

.4

+ 4.

•

4,

+

-

a a

•

•

9

• •

a

.

.9

0.

•

a

.

+++++ +

a

191795993

1.559 +

•

.32E+01

a

12

+++s

/ Pion and kaon elastic scattering

9..

+

.

+0

•

IOE+91 +

-

32E+9G

-

LOE+99

32E—9t

+9

94+ .

-a •

•

:

, . •

0

-

. .

9

• .

• •

:

•

:

.

•

.

as

.19E-91

-

32E—$2

-

IOE-02 ~o.

.

•

•

•

•

300

•

.

.

•

•

.

-

-

•