SLAC-PUB-607 June 1969 (TH) and (EXP)

Ciintcr Wolf

Stanford Linear Accelerator Center Stanford University, Stanford, California 94305

\Vor1< sll~q’:‘I‘tcYl by Ihe u. S. f21oinic i<:nc~~gy Cloinlnis:;ir)l:.

Presented at the conference on m and K?T interactions, ANL, May 14-16, 1969.

1. IN’1’I101:1 JC’I’ION

The Chew-Low cxtraI)olat.ionl to find on-mass-shell scattering cross sections

from rcnctions cxpcctcd to invol\rc scnticring of a virtual particle dcpci~ls criti-

tally on what tJrpc of cstr:tpoI:llion I’unction is used with a limjtcd numl~cr of ovcnts

(several thousand). The work of Baton ct al. , -- 2(-o 7r 71 scattering), Ma et al. , 3a

--

(T-+p scattering), and Marateck et al. , 4

(7r-77’ and 7r+7fiT- scattering), showed that --

linear or quadratic extrapolation functions5 fit the data equally well but i%e extrap-

olated cross sections differ significantly: In general the higher order extrapola-

tion leads to larger cross sections. However, neither of these simple function::

may give the correct answer as was shown by Schlein and coworkers 3a-h who

used data on the reaction pp -p;~+n to determine the ~‘p elastic scattering cross

section in the A region (A = A(123C)) via the Chew-Low extrapolation. Since the

n+p elastic scattering cross section is well measured, this is a place where the

extrapc;Ilition procedure can be CllCciiCd. Both linerirly and quadratically extra-

polated cross section values did not reproduce the on-shell measurements. IIOW-

ever, good agreement wa s found using the Diirr-Pilkuhn 6 (DP) paramcferization

whi.ch relates off-, ‘1~11 and on-shell scattering cross sections.

Recently a slightly diffcrcnt paramcterization for off-shell scattering ampli-

tudes has been proposed by IXnccke and Dlirr 7 (l3D). As it: turns out, t.heir treat-

ment has certain advantages over that of Dtirr and Pilkuhn in that (a) it describes

the data up to larger momcnl um transfers (up to -1 GeV2 as compared to -0.5

(I t,l _<,O. 3 GcV2) Ihe 13D and DP 1);11’:1.Int’t(‘l’i%3tiolls l.cacl to identical results. 8a “’

-2-

- i- np-7r 7rn . (1)

The contribution of this diagram to the differential cross section for rencti.on (1)

is given by

where

S

P”

P

t

in

qt

+&L 1 2 4np*2s

m Cit(Jn-r+(m 2 t) --& g2 I’.iNT@)

square of total cm energy

cm momentum in the initial state

pion mass

square of the four momentum transfer bctwecn into, Ang and

outgoing nucleon

7rN7r+ rest mass

momentum of the eschangcd pion in the T-T-+ rest fral,le:

q: = (t-(m -f v)‘) (t-(n~-p)2)/4m2

(2)

(3)

For t ~1-1~ qt Ixxomcs the on-shell momentum (I, q 2 m2 = 4 - p2

g NNn coupling constant; g2= 2 X14. G in the case of reaction (1).

-3-

quantity ZT-T.,.( 2 m, t) to the pion pole, 1 =/A :

According to r<q. (2), C?,-,+(m) t) in the OPE model is given by the fol.lowing

expression:

c ‘IT-T,~(nl) t) = i

Yt cr,-,dmt) 2 -y- CJ n-ti(ln) Fm~(t) u~-~iTf(ln)

i (4)

which when cvaluatcd at the pion pole, reduces to u .-,+tm)*

III. THE BENECKE DiinR PARAIKli:TERI%ATION

In order to evaluate the OPE formulae (Z), (4), we have to know FNNn(t) and

the relationship lwtwccn off-shell and on-shell scattering c ross sections. Let us

assume that the scattering at the TT vcrtes proceeds through a given angular

momentum state a. Then one can replace u T-TT+(ni, t) in Eqs. (2)) and (4) by the

partial cross section ~$-~+(ni, 1).

Benecke and Diirr have studied the connection bctwccn on-shell and off-shell

scattering assuming that the elasiic scattering of particles a, and h via

can 1X approximated by the exch~Ingc of a scal:,r particle s wi.tll mass in x. (SW?

Fig. 2. ) Going off the mass shell, e. g., with particle a, the relation I,etwccn the

-4-

The functions uQ(x) have the following general propertics

1. u,(x) -X 28 for x << 1.

Hence, near threshold one gets back the Born result:

2,C! clmr-+.(m) (crR, qtR<<l)

1 for example,

(7)

(8)

2. up, - + Pn (4x2) for x>>l. X

7 i.e., for large t the off-&cl.1 cross section decreases like Itjmu in co;;?rast to

the Diirr-Pilkuhn paramctcriznt~ion which lcacls to a fall off proportional to 1 t 1-l.

An estimate of the parameter R may be found in the following way: The form

factor F aid) associated u7ith a vertex ahe is defined as the ratio of the actual

vertex function to the vertex function given by the Born result. In the I3D parnm-

eterization this leads for the p7r vertex to

-5-

R. It will be assumctl that there is one parameter R for each partial wave con-

tributing to the off-shell scattering cross section; e. g., a lt P

for the p-wave

amplitude of the OTT system, h RA for the (3/Z, 3/Z) partial wave of the nN system,

Ck.

For bound state scattering, as in the case of the NN7i vcrtes, the off-shell

behavior will be calculattid a la Diirr-Pilkuhn since the BD paramcterization

leads to complex expressions. Therefore,

with

Q2 E -p2(4mZ,, - p21

4mZ,

and

Qp x -t(lmi - t) --

4m;

where m N nucl.con mass; RN is a prametcr.

i% ----A+“ (12)

1’11 *A+% (13)

r-+1, -A ++‘-P” (14)

0 7r J, - np (15)

-6-

for reactions (12-15): ltn, RN, and It . 0

They were dctcrminecl by fitting the

OPE cxprcssions 10 the dilfercntinl cross sections d~/dlt I for the reactions (IZ-

15) as mcasurcd Ior bc~~m momcntn lztwccn 1. G and 10 GeV/c. 10-13

Figures

3 through 6 show tl:c c~pcrimc::tal data together with the OPE CWVCS. The OPE

curves give an excellent description of the differential cross sections up to

Itl 51 c;,v2 f or all four reactions, and at all momenta. In a separate fit the OPE

cross sections were multiplictl with the square of a function G(t), which may be

thought of as a correction to the pion propagator. The following ansntz was made

for G(t):

G(t) =+ C-P

In addition to Rh , RN, RP the parameter c was varied. The fit gave c = 140 GeV2

which shows tllat the data do not require an additional form factor.

Knowing the values of RA, RN, and R one can go on to fit the R parameters P

of other partial waves of the 7~7i and the TN systems. For instance with the as-

sumption that the reactions

n-p -t nf” (17) and

0 7r p”‘ng (18)

2 1/a ‘L’hc pionic radius of tlic> i~uclcon, <L-~> :- 1. 06 -k 0. 04 f, is l:wgcr tlxtn

2 l/2 the electric charge raclius of the proton (Tp) = 0.8:: f 0. 02f lG by ;~hllt 20

2 l/2 percent, but its val.uc is close to the runs radius for nN jntcrxtions, <rNn> =-

1. If, olkinccl from an optical lnoclel analysis of TN diffraction scattering.

l/2 The rms radius for the ANi-i vcrtcx, <rkNr> L 0,86 k 0. 02 agrees rather well

with Iho value of 0. 84f determined for the ANY vcrtcx from the Ml transition

form factor measured in ep scattering, 17

Table I suggests that for E 2 1, the rms radii for the baryonic vertices arc

of the order of 0. 8 -- 1. Of and for the mcsonic vertices 0. 6 - 0. ‘7f. Therefore,

a good guess for the value of R Rand hence for the form factor can be obtained -

from Eq. (10) choosing a proper <r 2 l/2 > value, e. g., 0. 9f for the baryonic ver-

t,ices and 0.65f for pionic vertices. This can be a useful procedure in the case

of small partial waves for which it is difficult to determine the R parameter

from the t distributions. For s-wave states the value of R is practically zero,

implying u (q .R)/uf(qR) N 1 for these vertices. P t

V. IMPLICATIONS I?O!t THE EXTRAPOLATION FUNCTION

We are now in a position to calculate form factors and off-shell. cross scc-

tions for various vertices in the I3D p:lr:lmctel~i~ation. Figure 8 shows the

quare of the form factor as a function of t for the NNr, ANT, o’ir~ and fm ver-

tices, The fall-off with increasing -t is roughly the same in all. four cases.

The behavior of the off+hcll sc:Lttering cross sections cm lx rent1 off from

-8-

and ANr vertices a rapid incrcnsc with increasing -t is olxscrvcd. The explan:t-

tion for the phenomenon is that in t.hc case of the: NNn and ANT vertices one is

close to thrcsllold ((1, qt =” 0), hence e. g. )

(19)

For the pm and fin vertices, one is far from threshold (m =z 2/1)), thcreforc (for

-t 5 0.3 GeV2)

qtOpn7i(1n,t)-qqa~“(m) i, f 0 f

The analysis indicates that also for other vertices, this type of correspondence

holds when one is far from threshold and I tl is smal.1.

Let us now turn to the function C T-.iT+(m, t) defined in Eqs. (3), and (4).

From the above discussion it is clear that in the p region and above the t depend-

ence of C TiT-7T.,.(n~, t) is mainly that of F2 NNT(t). (Note: This is also true if the

7r7r s-waves are taken into account since qt 0 ‘=O(m, t) = q (j=‘(m). ) Since ITiNT

drops with increasing momentum transfer the function ETVn+(m) t) will rise when

going to the pole. 18

In Fig. 9, the \alucs of ET-,, (m, t) as obtained by Marateck et al. ,4 for --

events scl.ected from a 20 McV band around the p” mass are shown. The: data

-9-

(b) the cxtrapolatcd cross section values differ by 30-50 percent clepcnding

on which extrapolation function is usccl : the liiieflr extrapol:~tion lads

to u - T7r +(m P

) r: S5 f 15 mb, the 13D parnmctcrization yields 0 _ ~ .+m P

) =

124 f 4 mb. From unitarity WC expect the cross section to be either

11G mb or 132 mb dcpcntling on whether only the rcsonnnt p wave is con-

sidered or a resonating T = 0 s-wave is also taken into account. IIence

the cross section value obtained using the BD parametcrization agrees

with the expected vsluc whereas the linear extrapolation underestimates

the cross section by -4 0 percent.

VI. CONCLUSION

The number of events presently available does not allow a model independent

determination of the ~7r scattering cross sections via a Chew-Low extrapolation.

Reliable results can be expected when the Bcnecke-Diirr (TX)) parameterization

is used for the pole extrapolation as was shown for reaction (1) in the p region.

‘Ihis conclusion is strcngthcned

(a) by the sucess of the OPE fits to the reactions (12-T5);8b

(1)) by the fact that in the cast of the reaction pp-- A++n the Chew-Low ex-

trapolation lcad to lhc correct values for the elastic r+p scattering cross

section in the A rc:;ion v:hcn the TXirr-Pilkuhn (DT?) p:trnll7et.crization

was :qqdictl; 32 ) 1,

. (c) by the good results obtainc~tl \iritll the I)!‘ l)nr;lmctcrization for reactions

Of IllC tJ’J)c I<N -NNl<TTi. m, 1 9 ‘l’lic~ I:tsl two poini:; :l]q)lJ’ :1s v;cll for lhc

131) i):lr::ltl(‘tc~J.iz;ttjoli ;;iwc’ 21 sill:111 1. v;iluc:; (:;cc 1171I’,~,~jil(‘1i(~II) T)(,th

~)rO(‘C’tiliL+t’!5 :I I’(! cciu i I ;!]<>jJf .

- 10 -

ACKNOWLEDGEMENTS

I wish to thank my colleagues from Group

B at SLAC for comments.

- 11 -

‘I’Al<I,I;: I:

fits to the monwntun~ tr:tnsfctl* distril,utions.

--

12 R q2 y2 Vcrtcx (Gd) (f ) (9

RN NNn 2.8G k 0. 08 0.57 A= 0.02 1.06 * 0.04

12A ANT 1.76 * 0.03 0.35 YJL 0.01 0.86 * 0. 02

R P v* 2.31 f 0.19 0.46 .t- 0. 04 0. 65 5 0. 05

RSll Sll NIT 0. 03 0. 0.

RPll I’ll Nn 0.34 fixed 0. 1 0. 14

RD13 D13 4.5 fixed 0. 9 0.9

I~D15 D15 N;- 5 . 5

RF15 F15 Nn 4.5 fixed 0.9 0.8

RF17 F17 Nn 4.5 fixed 0.9

RF37 F37 NT 4. 5 fixed 0. 9 0.7

Roo (mqT’O(7T7r) 0.0’ 0. 0.

Rf f 7l7i 3.23 * 1.4G ,654: .29 0.58 If 0.2G

R g g lm 4.5 fixed 0. 9 0.5

R20 (TTT) T=-2 71’ Otazr) o. o 0.01 0. 0.

R22 (7x) TX-2 ,MtTli) 3.59 i 1,19 0” 72 i 0.24 0. 72 3~ 0.24

lI24. (W 1’L2,+l(““) 4.5 fixcd 0.9 0.6

- 12 -

1. G. F. Chew and I;‘. II:. Low, Phys. Rev. 113, lG40 (1959).

2. J. I?. Baton, G. Laurcns, and J. Reignier, Phys. Letters 2513, 419 (19G7).

3a. %. Ming Ma, C. A. Smith, R. J. Sprafka, 13. Colton, and P. E. Schlein,

Report UCLA-1030 Rev., University of California, Los Angeles, (1968) (to

be published);

b. P. E. Schlein, Report UCLA-1029 (19G8), and review talk at The Informal

Meeting on Experimental nks0ll Spectroscopy, University of Philadelphia,

(April 26-27, 1968). Published in Meson Spectroscopy, Eds. C. Balmy

and A. H. Rosenfcld (W. A. Benjamin, Inc., New York, 1968).

4. S. Marateck et al. , Phys. Rev. Letters 21, 1613 (196s). -- -

5. The quadratic and cubic extrapolations given in Ref. 3a should be called linear

and quadratic extrapolatic:ns. For a discussion of the general extrapolation

function see J. Naisse, and J. Rcignier, Fortschritte der Physik 12, 523 -

(1964).

6. H. P. Diirr alld II. Pilkuhn, Nuovo Cimento 2, 899 (19G5).

7. J. Benecke and II. P. Diirr, Nuovo Cimento 56 269 (1968). -’

8a. G. Wolf, Phys. Rev. Letters 2, 925 (19G7);

b. G. Wolf, Report No. SLAC-PUB-544 (1969) and Phys. Rev. , to be publislmd.

9. E. Clemcntel a:~~ C. Villi, Nuovo Cimento 4, 1207 (1956).

+f -1--I- 10. a--A A :

(a) 11. C. Dehnc, E. Rnubold, 1’. ,%ding, M. \i7. Toucher, G. \Trolf and

11. Lollrrll:lllll, Phys. Loticti~s 2, 185 (19!?i), and II. C. 11ehnct, Universil~~

- 13 -

c . v. All c~:~-lk~~~c:lli, I:. l~‘l~c~nch, A. l*‘risli, :~nd I. Mi:?hcjd:t , Nuovo (‘inlcnto

48, 360 (1957), (5.7 Gc\yc). -

11. 1’1) -*++n:

(a) S. Coletti, J. Kidd, L. Mandelli, V. Pclosi, S. Ratti, V. Russo,

L. Tallonc, I?. %:I mpicbri , C. C:ISO , F. Conk, hl. Damcri, C. CIWSSO ,

and G. Tomassini, Nuovo Cimento G, 479 (1967), (4. 0 GcV/c).

(b) II. C. Delme, J. Diaz, I;. Strijmcr, A. Schmitt, W. I?. Swanson, and

G. Wolf, Nuovo Cimento 53 232 (19(X), (10. 0 GeV/c). -’

12. n+p -A++p’:

(a) N. Gclfand, Columbia University, NEVIS Report 13’7, (June 1%X),

(unpublished) (2.35 GeV/c); the data at 2-4 GeV/c and 6.95 GeV/c have

been taken from

(Is) D. Brown, G. Gidal , R. W. Birge, R. Bacastow , S. Yiu Fung,

W. Jackson and R. Pu, Phys. Rev. Letters 19, 664 (19G7); -

(C) P. Slntterly, II. Kraybill, B. Froman, and T. Ferbel, Report UR-875-153,

University of Rochester (July 1966)) (unpublished) (6,95 GeV/c);

(d) Anchcn-Berlin-Birmi~~~lham-inn-l-[anlburgr- London (I. C. )-hliinchen col-

laboration, Nuovo Cimento 35, 659 (1965); and -

(e) Phys. Rev. 138, 6897 (1965), (4. 0 GcV/c);

(f) J. Ballam, A. Brotly, G. Cktdwick, Z. G. ‘I. Guirngossi&, TV. B,

Johnson, R. R. Larsen, D.\V. G. S. Lcith, and 1:. Pickup, privntc

communication, (16” 0 CcV/c).

13 - , 7r p-n/F:

(a) S:lcl:ly-C3rs~:y-T3nri-I~~lol~~~:~ coll:ll~oi-~ltion, Nuovo Cimcnto 29 515 (19G3), --.-’

- 14 -

(c) J. II. l’oirict‘, N. N. I:i:<\v:is, N. n1. cason; 1. I)cJxtIo, v. P. I<cwlcJ,

W. 1). Shcph:rrd , 15. II. Lynn, TI. Yule , W. ~clovc , R. L:hrlich, and

A. L. 13&c-r, Phys. Rev. 1(;::, 1462 (1967), (S. 0 CeV/c).

14. Aachen-.~irr~~in~l~~~~~~-I~o~~~~-II~~i~~l~i~r~-L~~n~lon (I. C. )-Miinchcn collalior~~tion

Nuovo Cimrtilto 31 -’ 7LI!) (131; i j.

15. J. A. Poirier, N. N. Uiswas, N. M. Cason, I. Dcrado, V. P. Kenncy,

w. D. Slqml, 13. II. S~v\‘)l:i~ TI. Yuta, W. Selove, R. Ehrkh, A. L. Baker,

Phys . Rev. 163, 1462 (1967).

2 l/2 16. This value of <rI,> was obtained from a straight line fit to the values of

the electric form factor of the proton as mcssured at momentum transfers

below lfB2 by 13. Dudclzalc, G. Sauvage, and I?. Lehmann, Nuovo Cimcnto

28, 18 (1962), and T. Jansscns: R. Hofstadter, E. B. Hughes, and M. R. -

Ytariail, Phys. Rev. 142, 922 (1966).

17. W. Bartel, B. Dudelzak, II. Krehbicl, J. McElroy, U. Meyer-Gerkhout,

W. Schmidt, V. Walther, and G. Weber , Phys. Letters 27B, GGO (1968).

18. The behavior of the extrapolation function near the pole depends on the t.ype

of vertices invol;,cd. For example, for a determination of

from the reaction

+ + - +--t nr.,--n71A

the correspondin:; estr::Ix~lniion function in the 01’11: motlel is

W. I<. Slatcar, D. 11. Stork, :mtl II. I<. Ticho, Phys. Letters 28B, 202 (1968).

- 16 -

FICURI!! CAP’I’JONS

1. OPIS diagram for the reaction n’-p - 7fIT-7r+n.

2. Approximation of the scattering process a i- b -a’ t- b’ by the ctchangc of a

sc:tlnr pnrticlc x with nl:k:;r. In x’

3. TT Diffcrcntial cross sections dcr/dltl for events of reaction (12) in the A ,

++ A mass region . The curvc?s give the result of the OPE fit.

(a) at 3. G GeV/cl’a (1.13 GcV < MAA < 1.33 GeV)

(b) at 5.7 CBV/C ‘Oc (1.1.5 GeV < MzA < 1.35 GeV).

4. Differential cross sections da/d ItI for events of reaction (13) in the A*

region. The curves give the result of the OPE fit.

(a) at 4 GeV/c”a (1.08 GeV < M < 1.40 GcV)

(b) at 10 GeV/cllb (1,125 < M < 1.325 GeV).

5. Differential cross sections do/d It 1 for events of reaction (14) in the A ++ ,

PO region. The curves in (a) - (d) give the result of the OPE fit. The curve

for the 1G GeV/c case, (e), is a prediction of the OPE model.

(a) at 2.35 GcV/C~~” (0.675 GeV < m < 0.825 GeV, 1.185 GeV < IQ < 1.285

GeV).

(b) at 3 -4 GeV/clBb (0. G8 GeV < m < 0.8G Gel’, 1. 1.2 GeV < iv1 < 1.32 GeV).

(c) at 4 GeV/clZd (0. GG GcV < m < 0.8G GeV, 1.12 GeV < M < 1.32 GeV).

w at 6.95 GeV/c 12’ (0. G4 GcV < m < 0.88 GcV, 1012 GcV < bl < 1.42 GcV).

(e) at 16 GcV/c12f (01. G8 &V < m < 0. 8G GeV, 1.12 GeV < >I < 1. 32 CkV).

6;. The differential cross sections dn/d ItI for cvcnts of reaction (15) in the /lo

region. ‘I’ho CUI*\~CS gi;rc t1:c rosulis of the 01’12 fit.

(a) at 1. 59 Gn\‘/clnn (0. 616 COV < 111 < 0. ST, @I’)

(b) :Lt 2. 75 CkxV/c 13’) (0, 65 (k!\~ < 111 < 0. 85 (k\‘)

(c) at a, 0 mv/,13c (0. 675 Gt>\i < 111 < 0. 8’75 (;<I\‘).

- 17 -

7. ‘I’hc tljfl’elx:nli:d cmss s(~i:lion:i tl~r/ti (t 1 for the cvonts of that x:!ction

0 n-p -nf . The curves give the rcsu1.t of the 01% fit.

(a) at 4 GCv/G14 (1. l.G GcV < m < 1, 38 GcV)

(b) at 8 GcV/C~~ (1.17 GcV < m < 1.37 C;eV)

The diffcrcntial cross: scxtion tier/d it I for the cvcnts of the reaction

0 n-p+ng . The curve givccs the result of the OPE fit.

(c) at 8 GeV/c? (1. 60 GtdV < m < 1. 75 GeV).

8. The t depcndencc of the form factor and the off-shell cross section (see test)

for the vertices

(a) NN7r

(b) ANn

(c) PTT

(d) fTn

9. The fi-nction Z,-,+( m, t) as measured for events of the reaction

with an effective mass of the 7~-7r+ system between 0. 76 GeV and 0. 78 GeV40

Also shown are two different extrapolations of Cn-?r+(m, t) to the pion pole

(t =p2): one which assumes a linear t dependence for C,-,+(m, t:), another

which util izcs the Beneckc -Diirr (ED) para meterization.

- 18 -

I t= \ \ \ \ \ \ \

/ I I I I I I

;;; 1.0 I L’

pp-n++n++

5.7 GeV/c

1 I I I I I I I I

0 0.5

Itl EeV*) 105785

Fig. 3

IO

PP---A”” I I I I I I I

-

-

0 0.5

0.5

Itl (GeV2)

t

1057138

Fig. 4

0 P

a> E(g 10

b-t- - u-O

I.0

1.59 GeVk

IO

I.0

0.1

- 77- p-n PO

-,---l-l- - (cl

8.0 GeV/c

0 0.5

It I (GeV2)

Fig. 6

1

1 10578’7

n E

b -0

IO

G- > g 1.0

- 5 - a

0.1 - 0

7-r-p -nf 0

+ L-r lT-- I I I I I

- - - (aI - -

1.0

0.1

0.01 _ 0 0.5

- -7~ P--g

0

L-T+ lT--

I t I GeV2)

Fig. 7

105789

- a rc

- *

mm

-- /

- \

F ‘b

Lfl <U

! / 2

- R

” .

0 - ul

-0

I

0.76 GeV< M ,-T-+<0.7b GeV C(mb)

a

00 - a*

50-

0 4 8 12 -t f i2) 1332 Al

Fig. 9