Fortschritte der Physik / Progress of Physics: Band 26, Heft 10 1978 [Reprint 2021 ed.] 9783112519189, 9783112519172

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MHMS©©I!©S©S1KÌ ÌM ATWiHiysyBia ¡MI® IPÜH®n!£Mi!§®iliIM ©HgilMJ^ÌMiFE1 IDI® ©UGHÌSGSniSS ®SSfi®[SM'ÌHÌg(SIlI!llR! ffiBOTUBiUIK WIM IF. l&MKSüOMJBOlFSio

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HEFT 10 • 1978 • BAND 26

A K A D E M I E

- V E R L A G EVP 10,- M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie -Verlag, D D R - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36221 und 2236229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t IS,— M (Preis lür die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/26/10. @ 1978 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 26, 5 0 9 - 5 6 4 (1978)

The 7tK Scattering and Related Processes C. B . LANG

Institut für Theoretische Physik, Universität Graz, Graz, Austria Contents 1. Introduction

509

2. Notation and kinematics 2.1. General 2.2. Special 2.3. Singularity structure

511 511 ..514 515

3. Status of experimental evidence on TTK —> 7TK 3.1. Experiments 3.2. Notation for the production processes 3.3. Determination of on mass shell TTK scattering 3.4. Main features of nK scattering

518 518 520 523 529

4. Status of experimental evidence on mr -> K K 5. Theory of 71K: real approximations 5.1. Unitary symmetry 5.2. PCAC and current algebra 5.3. Reggeology and duality

534 '

536 536 538 542

6. Theory of 7rK: dispersion techniques 6.1. Continuation to KK -> K K 6.2. Selfconsistent determination of TTK amplitudes 6.2.1. Modified K-matrix approach 6.2.2. Partial wave equations

543 544 550 551 555

7. Conclusion

560

References

561 I. Introduction

Through the last twenty years it was clear that an understanding and adequate description of meson-meson scattering is highly desirable, if not necessary, to understand the forces between the baryons. The simplest mesonic scattering system is the mt system which is of utmost importance to meson-baryon, baryon-baryon, and even electron-baryon scattering since it carries the quantum numbers of the exchanged states. It turned out however, that only in the very low energy domain it_is sufficient to treat TZTZ scattering without any other coupled states. As soon as the K K state is kinematically allowed it does have a drastic effect on the two-pion system. Through this 3 9 Zeitschrift „Fortschiitte der Physik", Heft 10

510

C. B . LANG

the 7r- -> K K and -via crossing- the 7TK scattering processes enter. The latter constitute the simplest mesonic systems with unequal mass kinematics and carrying hypercharge and thus deserve interest of their own. A comprehension of TZTZ and TTK therefore should be obtained simultaneously and preferably by similar methods. Most of the reviews of the last years concentrate on mc scattering ( P E T E R S E N (74), MARTIN (76), P E T E R S E N (77a); the very profound review of P E T E R S E N (71) includes - K scattering — in the meantime however there has been further progress. We therefore discuss the TCK scattering system and the crossed process TT- —> K K with the emphasis on the results obtained in the last few years since 1971. Earlier work is only briefly mentioned — if nothing new has emerged from it in the last years; for a more complete appreciation of this work we then refer to P E T E R S E N (71) and to the quoted original texts. We introduce in ch. 2 the notation which is common for all meson-meson scattering amplitudes, and discuss the singularity structure specific to the TTK kinematic. In ch. 3 we review the methods to isolate from the 7TK production amplitudes the one-pionexchange contribution in order to extract TTK scattering data. These methods are almost identical to those for TCTT scattering; thus we concentrate on the results to get a closed view of the present situattion. In ch. 4 we discuss recent results on mz -> K K amplitudes. Through the experiments of the last years the meson-meson scattering amplitudes have been determined reasonably good between threshold and 1—2 GeV CMS-energy. Of course there are still open problems even in this energy interval — like e.g. the - K scattering lengths — but it may be just a question of time until we have obtained a clarification. There is however no single theoretical model that can explain all the observed structure. An essential step to this aim were the low energy theorems derived by Weinberg for scattering processes with one or two idealized massless ("soft") mesons as incoming and/or outgoing particles. Another step were the dual scattering amplitudes by Veneziano and successors where the sum over all the exchanged resonances is identified with the sum over direct channel resonances. The soft meson theory gives a reasonable first order approximation to the subthreshold region and the Veneziano model is a reasonable first order approximation to the crossing and resonance structure. But it has not become clearer in the last years how the unitarity- and other correction terms should be produced. In ch. 5 we sketch these approaches only for completeness and refer for a further treatment to the quoted references. Chew's intriguing idea of an elementary particle democracy namely that the complete set of elementary particles is the only one consistent with the basic ^-matrix principles: unitarity, analyticity, and crossing, has .initiated numerous attempts in the sixties to "bootstrap" a subset of these particles. Naturally much work was concentrated on the TCTT system with its dominant resonance, the p meson. The idea was to show that the p meson with correct mass and width is the only ^p-wave resonance compatible with the basic principles and the information on free pions alone. None of the attempts in the framework of $-matrix theory — that was where the interest was focussed — led to convincing results. Much more work of the sixties, e.g. models for 7rK scattering based on N/D equations and on Mandelstam's double dispersion relations, is not discussed here because the results did not agree with experiments satisfactorily and nothing new has resulted from that work in the last years. The main source for a better understanding has evolved to be dispersion relations (ch. 6) applied on different curved manifolds and combined such as to give amplitudes that are crossing symmetric. They are a valuable tool to incorporate crossing and analyticity in a feasible way. The central question became what input is necessary and sufficient to assure unique results governed by the basic principles unitarity, analyticity, and crossing. The answer may provide an interesting connection to the "classical"

The 7tK Scattering and Related Processes

511

approach of Lagrangian field theory. In most of the models formulated in this dispersion theory language it turns out that the required input parameters correspond to those provided by an effective Lagrangian of simple form, namely the masses and coupling constants of the stable particles and resonances involved. The list of references is certainly not complete. In particular for the experiments we always try to give at least some of the newest references so that the interested reader may trace them back to get a complete set. 2. Notation and kinematics

2.1. General The scattering matrix element for the transition between di-meson states \i) —> |/) is Sfi = d/i + i{

o 39*

(2.7a)

512

C. B . LANG

and the slope parameter fa1 defined through the threshold expansion Re

«,'(«)J =

Im

+ / ? , V + 0(2>4)

(on1)2 p2l+1

afts)^ =

+

0(p2l+3).

(2.7 b)

(2.7 c)

Invariance under Poincaré group transformation and the mass shell condition for the external particles lead to the total number of two independent invariant kinematic variables; in order to demonstrate the symmetry between the three possible reactions one chooses the three independent variables « =

with

(Pa + Pbf

=

ijPc+

Pdf

t=(Pb~

Pi?

=

(Pa ~ Pcf

U =

Pc?

=

(Pa -

{pb-

s + t + u = 2} Pi2,

(2-8)

Pdf

Pi2 = mi2.

i

The physically possible values of the variables define three physical regions, the socalled $, t, and w-channel ; the boundaries are given through Kibble's function (KIBBLE (60)). The s, t, and «-channel processes are.described by the scattering amplitudes s: A (ab

ed)

(2.9)

t: A (db -> ca) u: A (cb

ad)

where the letters a, b, c and d stand for the type of the particle as well as for its isospin and hypercharge values. The amplitudes are supposed to be meromorphic, real analytic functions in the whole s, t, u plane with cuts and poles consistent with unitarity and crossing. The amplitudes in the physical regions are given as the boundary values of this analytic function in the "+ie-convention" (GUNSON (65), EDEN(66)) A(s) = lim A(s + is).

(2.10)

£—

If we cross one of the final particles (a«| to give an initial antiparticle | a*oc) this state does have simple crossing properties; it has the same transformation properties as the original state because it is obtained by analytic continuation. {aot\

'-a

|a*oi) =

(C-1)««»

|aoc)

|aoc) ^ ( a * « | = (e*- 1 )..» ( a « ' | .

(2.11)

Here a represents the isospin and the hypercharge (Ia, Ya) and tx the 3rd component of the isospin I3. The states a* however do not transform conveniently under isospin transformation — one would need a different set of Clebsch-Gordan coefficients to couple two states a and a * . The newly defined states a however do have the correct isospin transformation properties ( N E V I L L E (67)). The unitary matrices G are defined C„>

with

\Vs | = 1

(2.12)

513

The 7tK Scattering and Related Processes

and obey

(2.13)

The phase factor r\a is arbitrary but has to be fixed before calculation. I f we follow

PETERSEN (71) and choose

Va

(2.14)

=

we find for self-conjugate particles like the tt (and tq) | ax)

which corresponds to

=

|tc± 1) = T[Tt±> K

4)

K

0) = ]7r0)

\K+)

t ) =

(2.15)

| a Krr

7=1/2,3/2;

f-channel : rnz —> K K

1° = 0+, 1+;

7 = 0.

«-channel : KTÏ —> K -

7=1/2,3/2;

7 = 1 .

7 = 1 .

We abbreviate mK = ¡x and mK = M and whenever units are omitted we have vin2 = 1. There are two isospin amplitudes A1 in each channel connected through crossing matrices

C

« =

\

\

1

I

=

I

O

O I

°su

=

C„

=

I •

^

^

I

(2.23)

\|/6 t-channel: The cosine of the scattering angle is « + p2 + q2 ^Vtlt

s— u 4ptqt

(2.24)

-i jt — 4M2 n t — 4«2 where qt = y is the kaon momentum in the c.m. frame, and pt = 1/ — is the pion momentum in the c.m. frame. The values of qt, pt below the corresponding thresholds are defined by the description ( E D E N ( 6 6 ) , GUNSON ( 6 5 ) ) and therefore qt(t < 4 M z ) = i\qt\, pt(t < 4//2) = i\pt\. Thus zt becomes imaginary for real s, u values and 4/î2 < t < 4J7 2 . The partial wave expansion is = 1/2

E

(2l + ViqtPtYaS^Pfa).

even I for even I t odd I for odd I t

(2.25)

TheTCKScattering and Related Processes

515

Many authors define amplitudes A+ = ^L^.= 0 (7t7r -> KK) = i (A1'-1* J/6 3

+ 24'.=3/») (2.26)

.4- = I r A 1 ' - 1 ^ -> KK) = - i (A / .= 1 ' 2 - ^'.=3/2) 2t O and

gf, = - 16?r j/2 «/'(tttt

KK).

(2.27)

s-channel: The cosine of the scattering angle is given through zs = 1 + ^

= (t - u + (M2 - ¡u2)2ls)liq*

(2.28)

with the momentum in the c.m. frame

The sum of partial waves is A'.(s, t)=£

i

(21 + 1) q2W K+TT-P

K°-+n

K - n - > K-7T-P

FIRESTONE ( 7 2 ) , F o x ( 7 4 ) , B A K E R ( 7 5 ) BAKER (75) B A K K E R ( 7 0 ) , CHO ( 7 0 ) , ANTICH ( 7 1 )

(3.1)

In order to obtain from the production cross section K N - > KTCB (here B denotes the outgoing baryon system, e.g. N o r A ^ - rcN) data for on mass shell - K • > - K scattering and eventually — as a test — 7TN B -> TTN (for B = A) scattering amplitudes one tries to extrapolate the experimental results with t < 0 to the value t = mj 1. OPE is the only contribution with a pole at £ = m„2 and therefore no other competing process like e.g. q or A,, exchange should contribute to the residuum. This extrapolation would be no problem for arbitrarily dense data without errors; the average number of events per experiment however is around 10—20 K. This leads to results depending strongly on the type of extrapolation used. If one can afford it one introduces cuts in the data to select the "better" OPE events. In fig. 4 the distribution in the Treiman-Yang angles (i.e. the angle between the plane formed by the initial and final nucleon momenta and the plane formed by the initial K momentum and the momentum of the outgoing K system (TREIMAN (62)) for selected events with ra(-K) SS TO(K*) and »»(TIP) m M(A++) as obtained by MATISON (74) is shown; for small values of |I| one finds the isotropy that is expected from pure OPE mechanism. Further investigations for higher »¿(-K) leads to the conclusion that OPE dominates for m(-p) sy m{A++), small \t\ < 0.1—0.2 GeV2, and not too large W(TTK) < 1 . 2 GeV. Out of the total data without cuts it is obvious that there are competing reactions of comparable strength- this can be clearly seen from the data for K~p K~-+n and K+n K+7i~p which look very different. If OPE were the only mechanism at work they should be equal since K+nr scattering is equivalent to K~tt+ scattering. That the data (cf. F o x (74)) differ strongly just exhibits the different s-channel absorptive structure and the additional ¿-channel structure they are imbedded in. We do not want to go into the details of the experiments and the analyses, for this we would need much more space, we sketch only briefly the main paths that have been used so far to obtain - K phase shifts. More details can be found in the quoted literature and in reviews like e.g. that of SCHLEIN ( 7 1 ) or MARTIN ( 7 6 ) . We do not discuss results from Kt 3 decay at all and refer for these topics to previous review articles (CHOUNET ( 7 2 ) , NATH ( 7 4 ) , LEMONNE ( 7 6 ) ) .

520

C. B. LANG

-180 -90

0

90 180 -180 -90 4>np Idegj

90

0

90

180

90

180 -180 -90 4>ktt tdeg)

180

F i g . 4. T h e T r e i m a n - Y a n g angles 4>nv> ®7ck in t h e jx + p a n d K+TT_ c.m. systems for t h e events K + p —> A + + K * ( 8 9 0 ) ; (a) events with |(| < 0 . 1 GeV 2 , (b) 0.1 GeV 2 S |(| < 0 . 2 GeV 2 , (c) 0 . 2 GeV 2 S |f| < 0 . 3 G e V ! , < 0.4 GeV 2 . T h e results and figures are from MATISOST (74)

(d) 0 . 3 GeV 3 S Id

3.2. Notation for the production processes Before we discuss the different methods to obtain the uK scattering amplitudes let us introduce the notation for the production process KN ->• KTCB. The measured cross section can be written dmnK

ot dU„K

2

where X and X are the initial nucléon and final baryon helicities. The target nucléon is unpolarized and the polarization of the final baryon is undetected. Then we may write i ^

= E

E

; = 0 m = —l

.

V2Ï +

1 LiXm

dUO)

^

(3-3)

with L = 8, P, D, F ... for I = 0, 1, 2, 3 ... ; I andTOare the angular momentum and helicity of the producedTCKsystem, 6 and

K-rrN has an additional factor (—t) in the numerator. Therefore the OPE signal vanishes just between the physical accessible region and the on shell point of the exchanged pion. The background contributions will be important there and the result of the extrapolation to the pion pole doubtful.

The 7TK Scattering and Related Processes

525

ad l b . An attempt to cure some of these flaws is to extrapolate relative quantities like the normalized moments ( F ( m ) . In the definition of these moments (eq. 3.9) one divides by N and therefore one removes at least the structure due to the pion pole and the form factors. Pure OPE mechanism means isotropy in tzK scattering is extended off the mass shell of a pion or a kaon according to the LSZ reduction formalism (LEHMANN (55)). The interpolating pion or kaon field is defined through the divergence of the axial vector current because of the generalized PCAC hypothesis tYL d^{x)=-^ [

K K ) Q - ^ ô ^ n n

rot))

(6.6)

where the Omnes function is defined through

Q(ô(t))

=

exp

Y' ò(t') dt' H J

t'(t' -

t)

y ith

imax =

\M\

(6.7)

Ìthr

Fig. 16. As an example for the application of the finite contour d.r. the figure shows the real part of the function (T° — K°) the contribution from a dispersion integral, and the discrepancy function (determined at points s > Sthr) as obtained in a model calculation of LANG (77). The singularity structure of (T° — K°) is similar to that of T", therefore the figure for T" would not look very different

T o avoid the logarithmic singularity at 4fi2. W e know from ch. 2.3 that in order to project ¿-channel partial waves for t € (4m2, 4Ji 2 ) we need to know the amplitude at complex s values. To get for instance the value at the point f 0 , zto we have s = ^ +

M*-

i0/2 + 2izu\qtJ i v

(6.8)

Dispersing on fixed complex s lines however leads to a breakdown of convergence of the sum of partial waves imaginary parts in the w-channel for u > 68.7fi2 due to the influence of the double spectral region (2.32b). The sum of the real parts has even worse convergence properties. This does not allow to apply discrepancy methods like the finite contour method discussed above.

546

C. B. LANG

A possible way out of these troubles is to choose as dispersion curves hyperbolae of the type (cf. fig. 1) (s — « ) (w — a) = j8. (6.9) Since the curves approach fixed-s or u lines asymptotically < we have no problems concerning the existence of d.r. The curves have a close relationship with the kinematic Kibble function (KIBBLE (60)). 0

4tqt2p(2

=

sin2

=

t(us

-

( M

2

—

(6.10)

/li2)2).

0 is strictly positive in thè physical regions and vanishes at the boundaries. Fixing 0\t2 =

with

« < 0

(6.11a)

as dispersion curve, that is P = «2 - oì[2M2 + 2« 2 ) + (Jf 2 - fi 2 ) 2

(6.11b)

gives hyperbolae with the desired properties, i.e. lying inside the physical s, u and ¿-region above physical threshold. For rcN scattering this was suggested by HITE (72) and has been applied by many authors in the meantime (HITE (75, 76), BORIE (77)). This hyperbola however does not allow a normal ¿-channel partial wave projection in the extended unitarity region t 6 (4jm2, 4M 2 ) because by definition no negative value of oc (this covers the hyperbolae that are asymptotically inside the physical region) gives values of zt £ (— 1, + 1 ) in this interval. A hyperbola running through zt = z0 at t = t0 £ (4M2, 4:M2) has for given oc z0) = 4 ^ o ( l -

P(t0,

z02)

+

( M

(~~2—"j2

~

-

2

ti2)2

+

*

-

0 +

2

-

oc(Z

-

(6.12)

t0)

and the branches E =

P)

with

E

=

2fi2

+

2 M

2

.

-

J

For

1/2

t -

±

t € (ta,

«2 -

(6.13)

j8(«ò, Zo)

we have complex s values, where

tb)

taAk, zo) = ^ - 2 « T 2

zo)

(6.14)

and (t zt

(t -

-

ta)

4Jii2)

(t

-

tb)

1/2

(6.15)

4M 2 )

{t -

For

+

2

2 2

(6.16)

tL

with tL the value corresponding to .s = .smax and zs = —1; the discrepancy function A(t) has the same meaning as before. The cut from extended unitarity t € (4/i2,4M2) is removed with the help of an Omnes function as described above.

T h e 7TK S c a t t e r i n g a n d R e l a t e d P r o c e s s e s

547

A better approximation for ¿-channel partial waves can be obtained by choosing the class of hyperbolae running for a given t0 through zt = z0 with e.g. P3(z0 = ]/3/5) = 0. This implies that for odd amplitudes the deviation from waves at t == t0 is of the order of partial waves with I — 5 and higher. In general for 2 A partial waves (Lmix = 2k — 1) considered one would evaluate the d.r. at all values of zt = z-t where P 2 t+i( z i) = 0. This approximation was proposed by Burkhardt (65) and is related to the method of Gaussian integration; it has been applied for tzk TTTT scattering ( L A N G (73)). B O N N I E R (75) consider such hyperbolae with z0 = ]/l/3 (P 2 (± z o) =_0) and use mapping techniques to derive upper bounds for the modulus of the titz - » K K s-wave. We discuss the results in the context of other determinations below. Choosing a hyperbola running through ZQ = |/3/5 at t = m e 2 allows a simple determination of the value of the TTTT —> K K p-wave near the p pole; this gives the coupling pKK ( B L A T N I K (78)) because the coupling p - - is quite well-known. In order to obtain partial wave relation ( S T E I N E B (70, 7 1 ) for tcN, J O H A N N E S S O N (74) and H E D E G A A B D - J E N S E N (74) for T T K ) we write down once subtracted d.r. for _

glfiven £1,01id

j^Jfiven — u),

A^oM/is

=

s - u

=

2s

+

t -

Z

on curves with « = 0, /? = ji(t, z0). Actually for odd /< no subtraction would be necessary. t

r i m B ' i t ^ s U ' J ) )

t

itnr

rlmBhlt^slt',^))

— 00

(6.17)

Here i = E — sthr — /?/.Sthr denotes the t value where the hyperbola enters the region •s > st)ir- If asymptotic K * exchange dominance is assumed, one subtraction should be enough for the even amplitude. We now project partial' waves +i a/it)

1

=

=

.

1

. pt'qti

,

f J

Ai>(t,

s(t,

z0))

PJzt{t,

s(t,

0)))

dzt

-l o

,-

f

A'>(t,

s(t,

z0))

P,(zo)

(6.18)

dz0

-l where we use the identity z0 =

(6.19)

zt(t,s{t,P(t,z0)j).

We have changed the integration interval to (—1, 0); this is possible because the integrand has to be symmetric in z0. The sum of partial waves converges everywhere on the hyperbola provided t £ ( — 2 8 . 2 j u , 2 , 82.2/n2); these limits are due to the values of s where the hyperbola touches the "mirror image" (u > 0, s > 0) of the s — t double spectral function boundary (2.32b) (the values given by J O H A N N E S S O N (74) and H E D E G A A R D J E N S E N (74) have been corrected). Ai'(t',

s(t',

¡3)) =

]/2

E

pj)

j r

(21

(21

' A i \ t ' , s(f,

=

+

l J p ^ o j ' K O

Pl{zi{t',s(t',

(9)))

• +

(6.20) 1) ?52V*(s(r,

/?)) Pi(zs(s(t',

P),

t')).

548

C. B. LANG

From this we obtain o

«*) =

J *' °>fti'(0's(°> P)) PAzo) dz» -1

t_ F C {21 + l)g{.g{.

l ™J J t thr — 1 t f

t',p) Im a i H n P ^ j t ' ; s(t', /?))) Pf{z0) ^ ^

PtW

t'{t' ~ t)

Z°

f(2l+l)q^x'it,t',p)

t' n J J

pfal ft

— 00 —1

(-Cii1-)

Im o«J.(«(i', P)) Pt(zJ(a(t',

P),

t'(t' — t)

with ? 0,p) A' j we find El}(t,

t' = t) = 0 .

(6.26)

The obvious deviations from the standard partial wave d.r. come from the choice of the hyperbola as dispersion curve. In normal partial wave d.r. one uses the manifold of curves with zt constant for all t. Partial wave integration a t a given t then automatically gives no contribution from other partial waves. As discussed above the problem then comes from the double spectral region where values cannot be determined up to now. In practical applications the contributions from higher partial waves and distant parts of the cuts (so-called "driving terms") are approximated through resonances and Regge models. H e d e g a a b d - J e n s e n (74) determines d and /-waves too and finds the high energy contributions small if he works with hyperbolas of the type su = ft. In all the calculations that used the above discussed dispersion methods the authors had a common set of input with only small differences. For the tzK input parametrizations of the phase shifts up to 1.1—1.2 GeV were used; the s 1/2 -wave close to the shape obtained by F o x (74), with K K i-channel physical region. For t £ (tL — —32/u,2, 0) the sum of s-channel partial waves of the imaginary part of Cts(Ts — Ks) converges, therefore projection into i-channel partial waves is directly possible. For the real part we write a finite contour d.r. (NIELSEN (71, 72)) for fixed t Cts(Ts(s) - Ks(s) =A(s)

1 r"lm 1[CjTis') - Ka(s'))\ M + , ^ ds' 7t I S — S Sthr OO

71 J s«

S'

'

S — S

The last integral can be evaluated explicitely giving _ I ^L i m [C,J[T.(8S) - *.(«*))] In 71 S

Sji

(6.37)

and removes the first order logarithmic singularity at sR introduced by the first integral. We find that the discrepancy function A (s) has a cut only for .s > sR; it therefore will have only little structure below threshold. We can determine its real values in the physical s channel region (z( < —1) for s < sH; we then extrapolate zl(s) to lower s values (zs £ (—1, + 1 ) into the unphysical region. This extrapolation is one from interior points to interior points of the analyticity domain and should therefore be stable (cf. fig. 16). Solving the d.r. for unphysical s values (zt £ ( — 1, +1)) gives the real part of Cts(Ts(s) — Ks(s)) which then may be projected into ¿-channel partial waves. This allows to apply finite contour partial wave d.r.

71 J ti.

t —t

71 J

t

The TTK Scattering and Related Processes

553

where the second integral is explicitely —n

t

Im h{tL) In

(6.39)

tL

and is added for the same purpose as before. Similar reasoning allows to determine A{t) for t £ (tL, 0) and by extrapolation also for positive t values. Solving the above d.r. in turn gives k(t) for positive t values in the physical ¿-channel region. For unequal mass (like in the TTK —> 7IK channel) the cut structure of the partial waves is more complicated (cf. sect. 2) and the partial wave d.r. are somewhat more difficult to handle ( L A N G ( 7 7 ) ) . The analytic continuation procedure is numerically reliable only for not too large distances, i.e. not too large energy values (say, below 1 0 0 / I 2 ) . Exotic partial waves are determined by the i£"-matrix elements exclusively and there we can see the difficulties a t higher energies. In non-exotic partial waves the main structure at higher energies is due to coupled channels and resonances, a deviation of the if-matrix element from its correct value has only small effects on the amplitude. In a model calculation we obtain solutions of_the discussed system of equations for s, p, and d-waves of the 7TK - > TTK, nu TCTT, K K amplitudes below-1.2 GeV. These amplitudes are related to each other by unitarity, crossing and analyticity. In each of the non-exotic partial waves we allow for one coupled single particle state, t h a t becomes a resonance after solving the ii-matrix equation. The coupling of these states of given angular momentum to the corresponding 7TK, TCTZ and K K channels enters through the relevant r-matrix element and is assumed to be constant. We discussed already that these real coupling constants as well as the real positions of the single particle poles are the sort of bare coupling and mass parameters as we find them in a phenomenological Lagrangian ( L A N G (75a, 77)). These parameters — two for each non-exetic partial wave — are adjusted such as to give the observed resonance position and width. Where no experimental evidence is available (e.g. for the couplings of e, p, f to KK) we apply SU(3) relations with a mixing angle of 14° for the «-waves and ideal mixing for p and d-waves. This choice is consistent with the G - M O ( G E L L - M A N N ( 6 1 ) , OKUBO ( 6 7 ) ) relation between the bare masses of the resonances. _ _ There is one further important point; in KK — KK there is a cut below the KK threshold due to the nn intermediate state (extended unitarity). This cut is overlapped b y a left-hand cut (extended unitarity in the crossed channel); in order to avoid complex IF-matrix elements this would require modifications of the above equations ( L A N G (75 a)) and in related approaches ( Y N D U R A I N (75)). We checked the importance of^the contribution due to this cut by calculating the approximate discontinuity of tt (KK KK) in this region 1£ (4/I 2 , 4M2, — ^¡j?); we find a smooth function leading to smooth real partcontributions, small for g_and d-waves, but not negligible for the s-wave. We therefore neglect the kt (KK — KK)-matrix elements for the p and ¿-waves and introduce a one parameter function for the s-wave kf:°{KK

^

KK) =

yt

(6.40)

where y is chosen to give close above K K threshold a phase shift and inelasticity that agrees with experiments. We discuss only some of the results of this coupled channel selfconsistent model calculation, details can be found in L A N G (75a, 77). The parameters for the coupled single particle poles are given together with the resulting resonance and low energy parameters in table 2.

554

C. B . LANG

Table 2 The input parameters and the results of the coupled channel model calculation of LANG (77) Input I

0

I

SR

1

70.0

2 3 2

0

-

1

1 1

2

-

1

0.720

Resonance

TTK

-

5.5 x 10"4

Input I

0 0 1

2 2

I

Scattering Length

R

0

1

0 2

30.0 —

28.3 76.1 —

I

I

0 1 2

0 1 0

0.950 —

0.217 0.047

RKK

T

0.850 0.153 0.027

Resonance

0.323 -0.014 0.030 16.7 X 10"4 2.4 x 10-4 Results KK

Input

-21.26

0.893

0.048

-0.129 -0.673

-

1.421

0.100

-0.066

Results 7T7T —> VTK

S

2

7.40

-

-

K*

, (Sthr) ds X 103

out of ddjds*) r (GeV)

0.902

0.813

—

P f —

0.774 1.268 —

KK

Scattering Length Re AI1 Im AI i —0.816 -0.009 42.6 X 10-6

(5(a) = K/2 yi(GeV)

1.275 0.006 13.2 x 10"6

, (s(«thr)) ds X 10= dk

-11.83 -16.87 -0.475 -0.108 -0.112

—

0.151 0.166 —

Results KK

KK

a,1 (s = 4)

Re AI1 {s = 50)

Im AI1 (s = 50)

"0.416 0.010 7.7 x 10-4

-2.167 -0.036 6.0 x 10"4

0.072 0.024 0.8 x 10"4

*) Using the definition ddjds (s = m2) = 1 jmr The results of all the involved scattering amplitudes agree with the available data in the relevant energy region, they obey two channel or elastic unitarity for TIT-, K K or - K respectively and they obey crossing symmetry in good approximation. We try both assumptions for the 7 = 1 / 2 TCK s-wave: with a x resonance near 1.2 GeV and without it. Comparison of the two solutions (fig. 18) clearly shows that the y. hypothesis is preferable. Widths and masses of the non-exotic p and d-wave resonances have been used to adjust the corresponding input parameters, therefore the obtained agreement with the experiments (cf. fig. 11) is no surprise; the scattering lengths (table 2) however are really predictions of the model. The sign and shape of the exotic ( I = 3/2) phase shifts are predicted in agreement with experiments (fig. 13) below 1 GeV. Deviations at higher

555

The 7tK Scattering and Related Processes

energies indicate that either the method of analytic continuation for the ¿-matrix element is no longer suited or that we have to add driving terms to r due to distant singularities and higher partial waves. We remark also that the determination of phase shifts for the exotic channel from experiments is problematic (cf. sect. 3). In the scattering processes nn K K and nn - > mz we find that one coupled single particle is enough for each partial wave. This is surprising for the s-wave where we find

»/fV? + pole

f

3-UGeV/c

•f

K*p — KO +

(Bingham et ail

2

GeV/c K'p — K'n'nlFox

^

2

GeV/c K'n—K'n'p

+

12

GeV/c

et ail

(Fox et al.l

K*p~K+n-A** ÎHathon

e

t

_

_

LHC contribution {linear threshold 2Ç

modelÌ

L_ 30

40

Kïï)

50

ml

F i g . 1 8 . T h e & 0 1 / 2 - m a t r i x element for the irK «-wave determined from experiments is compared with the prediction of a model calculation with (full line) and without (dashed line) x pole (LANG (77))

that an e, (bare mass 30,u2) is responsible for the phase shift running through n/2 near 40,«2. This state leads to poles of the tztz - > nu amplitude on sheets I I at s£l = (8.78 ± 12.61z)ltt2 (mEl = 415 MeV, T El == 596 MeV) and on sheet I I I at s £l = (31.98 ± 26.37i) p? — a behaviour that is expected from a normal resonance. The drastic increase of the phase at K K threshold with a distinct dip of the inelasticity is due to a S* pole at ss* — (50.82 ± l . 3 7 i ) , u 2 ("ra s . = 998 MeV, r s * = 27 MeV") on sheet I I with no companion pole on sheet I I I . This pole comes out without allowing for an additional single particle pole and therefore it does not have much in common with a resonance like e.g. the p meson. We discussed these aspects already in ch. 5. 6.2.2. Partial wave equations Partial wave equations ( S T E I N E R (70, 71), R O Y (71), Y N D U R A I N (75), J O H A N N E S S O N (78)) are derived by combining d.r. on two sets of curves (algebraic manifolds) — connecting the physical regions of s and ¿-channel or s and «-channel respectively. For 71K scattering the amplitudes A I t { s , t ) are for even I t even against s — u exchange, i.e. ëA1'even(s, 8(s

and odd for odd I t

—

^j«odd(s

t)

= 0

u) =

M) t

)

=

o.

(6.41a) (6.41b)

556

C. B .

Lang

We introduce these symmetry properties in the d.r. explicitely. To connect the s and «-channel physical region we write twice-subtracted d.r. on the curve t = g(s, y) valid for t < 2 resonance formulae and asymptotic Regge behaviour. Once this input is given the partial wave equations restrict the allowed low energy partial waves severely due to crossing and analyticity. Like in the modifiedK-matrix approachin sect. 6.2.1. we expect that the solutions of the system of equations obey the general requirements of crossing, unitarity and analyticity approximately, that is within the restricted energy domain and the required accuracy. Like it is the case for the Roy equations for tztz - > iz~ (BASDEVANT ( 7 4 ) , POOL ( 7 7 ) ) the class of possible solutions for the low partial waves is very large. One has to add a set of restrictions that are obtained from experiments and it is sometimes hard to find the minimal number of such restrictions that are necessary to lead to a single solution. JOHANNESSON (78) treat the 7tK amplitudes below 1.1 GeV; the driving terms are constructed from the I = 2 mesons / and K * (1421) and standard high energy Regge amplitudes. The low energy region is characterized by the input phase