Virtual photons in pion electroproduction

Nuclear Physis A379 (1982) 407-414 Q North-Holland Publishing Company

VIRTUAL PHOTONS IN PION ELEGTROPRODUCTION L. TIATOR and L.E . WRIGHT* Institut für Kernphysik, Icharmes Gutenberg Universität, D-6S00 Mainz, Federal Republic of Germany

Received 2 November 1981 (Revised 16 December 1981) Abstrsxt: We examine the applicability of the virtual photon spectrum to pion production experiments by using an exact calculation of pion electroproduction from the proton and from 3He . After deriving a new virtual spectrum formula which takes into account the recoil of light target nuclei and the proper electron kinematics near the endpoint, we find that the concept of a virtual photon spectrum in such experiments is very good .

1. Introduction The concept of a virtual photon spectrum originated by Weizsäcker t) and Williams 2) has been used extensively to relate nuclear excitations induced by electron scattering to the same excitations induced by real photons. In such experiments the final electron is not observed and the target is bathed in a spectrum of virtual photons with energies from zero up to the initial electron's kinetic energy, called the endpoint . The total electron-induced cross section to some final state ~e(e ;) is related to the photon induced cross section vY to the same final state by writing -m° dt~ ~e(Ei) - ~ 0

Ne(Eif

~)~Y(~) - r

where e, is the incident electron energy and Ne(E ;, m) is the virtual photon spectrum . Most users of the virtual photon spectrum as given in eq . (1) implicitly assume an infinitely massive target, i.e. no recoil effects in the virtual spectrum itself . In recent years there has been considerable interest in the use of the virtual photon concept in pion electroproduction from nuclei 3). In this paper we will examine the applicability of the virtual photon spectrum to pion electroproduction from the proton and 3He. In sect. 2, we give an alternative derivation of the virtual photon spectrum for the case of pion production that is particularly appropriate for the case of differential cross sections and permits the inclusion of recoil effects. In evaluating the resulting integrals over electron angles analytically we avoid a number of approximations that are normally made. In sect . 3, we compare our results to the virtual spectrum of Dalitz-Yennie `) to an exact numerical integration over electron angles, and to a full calculation of pion electroproduction . ~ Alexander-von-Humboldt Fellow on leave from Ohio University, Athens, Ohio, USA. 407

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L. Bator and L,E. Wright / Virtual photons

2. Virtual photon spectrum for pion production In the notation of Bjorken and Dre11 5), the differential cross sections for electron (e) and photon (y) induced pion production in the lab frame are 6): derre

_ tr 2 Mf(VLW`L +

VTWT + VpW`p cos 2~  + ViWi cos ~,.) 3 s(E`) d kn d 3kt , 2or Z k;efEEi 4w (2)

where the kinematics are shown in fig. 1 and we have used three-momentum conservation to integrate over the final momentum Pf of the recoiling nucleus. The

Ei . Fig . 1, Kinematics of pion electroproduction .

pion azimuthal angle ~ ~ is measured about the virtual photon axis . The arguments of the energy delta functions are given by

where the final recoil energies are EÉ =JMr +(qo - k~) ~

(7)

The nuclear structure functions Wf ( j = L, T, P, I) depend on the pion kinematics and on the virtual photon energy and momentum (tv, q) while the Vi ( j =L, T, P, I) only depend on the electron kinematia. The function VT is k?kf,sin Z B~ t z Vr =- z9w + -q-+ and the remaining functions are given explicitly in ref. 6). In order to make a virtual photon spectrum interpretation of pion electroproduction, we assume that VTWT dominates eq . (2) after integrating over electron

L. Tiator and L.E. Wüght / Virtual photons

409

scattering angles, and furthermore that WT can be evaluated at 9~ = 0°, and pulled out of any integral over electron angles . With this forward peaking approximation (FPA), W°T~a,_o = WT, we can then relate the electron and photon induced cross sections by integrating eqs. (2) and (3) over dk.~ to obtain d 2tr~ -_ ~ Ei Idï' Y/dkn~ aVTwki dv,, I dkf I d~ (9) dE d,(lÂb Ei ~dï"/dk,~~ -tr ;qw k~ r dn;°b dE,~

where we have differentiated with respect to E to assist in comparing the double differential cross section for electroproduction to the single differential cross section for photoproduction. We will use eq . (9) as the basic relation between electron and photon induced pion production for the case of a bound final nuclear system . The factor dkt/dE can be evaluated simply from energy and momentum conservation and can be written as _dkf __ - (EClkr)(Ei +E.~-(E,~lk,.)4' k,.) dE4 D

(10)

where the denominator factor D is given by Et

D =M;+e;-E,~+kt [(kA cos B~-k ;) cos 6e +k sin B sin 6° cos ¢°] .

(11)

All angles in eq . (11) are measured in the lab frame. The factors kt, st, dkt/dE~, Ei and dE`/dk in eq. (9) depend on the electron scattering angle, but do not vary too much for the values of the angle which contribute significantly to the integral and are usually evaluated at their FPA values to obtain the Dalitz-Yennie virtual spectrum . However, the angular dependence in the factor D is not negligible for light target nuclei, and as stressed by Furui'), the variation of kf with electron scattering angle is rather important near the endpoint . In our evaluation of the virtual spectrum we will include the electron scattering angle dependence of these factors but will evaluate Ei, dE°/dkA and the numerator of dkt/dEA at their FPA values . This allows us to write dZ tr~ N~ dtry ~ (12) R~6, dE d,(l -d,(l  mo where the recoil factor R is given by (13) and the virtual photon spectrum for a bound recoiling nucleus is a typo ~ ktVT N~ -- ~ d,fl~ , k. 4

.

(14)

and the label 0 designates the FPA values . The factor Do = M; + s; -E + s f (kA ços B - k;)/ki was introduced so that in the limitof an infinitely

410

L. T"uttor and L .E. Wüghtl Virtual picotons

massive target our virtual spectrum will reduce to the conventional Dalitz-Yennie one. In performing the integral over electron angles in eq. (14), we make two approximations in order to obtain simple ânalytic results. The first one is to neglect the ~° variation in the denominator factor D by setting ¢° = 90°. For the case when the pion is detected this is certainly a good approximation since the coefficient of cos ¢~ in eq. (11) is small as compared to the other terms in D. For the case when the recoiling nucleus is detected (interchange of nuclear and pion coordinates everywhere), this approximation also seems to be good. In addition, we evaluate the kinematic variables in D at their FPA values, and find it convenient to write D=a r +b~cos B~,

(15)

where

and the subscript r denotes recoil since the term Do /D differs from unity only because of the finite mass of the target. The second approximation is to parametrize the dependence of kf (and ef) on the electron scattering angle with a simple form . We find that the parametrizations kf-kf Ef=Ef

ar+6±cos9~-kf(1+ a + b' ~,

a~+b~ cos 9~ = Et

~

âbrsin2z6°),

(17)

2b' ar

(18)

1+ r slnZ Zee

~

where b~ _ (ki/si ) Z b* agrees quite well with the exact values determined by numerically solving the kinematic equations. Note that our parametrization satisfies of -kf = m~ to order (br/a~) Z which is important near the endpoint. In evaluating kf, one should solve the energy conservation equation [eq. (4)] for 8~=0° exactly, particularly for large pion angles . After substituting eqs . (15)-(18) into eq . (14) we perform the angular integrals after neglecting terms of order (b~/a~) Z to obtain : 1V`

__a mô _b ar +b r rr _2al ln ( (a'-b')(a,+b~) _4bl ~ 2~r b' ar-(a'/b')br L\ 1 ôJ \(a~+b')(a~-b~)~ ~ôJ '

( 19)

where a=mé-s ;e°, b=k;k°, a'=a-S, b'=b+S, S= - meb~(Ei/Et - 1)/a*r and a, and b~ are defined in eq . (16). The second and third terms in eq . (19) come from the small second term in VT and in evaluating them we have kept q constant at 40 = wo~ and have neglected terms of order (br/a,)(kf/k;) in order to obtain a simpler result . For energies more than 50 MeV from the endpoint, the second term in VT may require a more exact integration. The conventional Dalitz-Yennie spectrum

L. 7"rator and L.E. Wright / Virtual photons

411

is obtained by letting brja~-.0 in eq . (19) . Finally, eqs. (12), (13) and (19) also apply to the case when the bound recoiling nucleus is detected rather than the pion if the pion and nuclear coordinates are interchanged everywhere . 3. Results and condasions In table 1 we show our calculations for p(e, ~+)e'n for different pion energies and lab angles. The values of the electroproduction cross section given in (a) and (b) are obtained by using the virtual photon spectra of Dalitz-Yennie and our new analytic result in eq . (12). The deviations between ourspectra and the Dalitz-Yennie spectra are quite large near the endpoint and rise with increasing B4 and k. As a check on our approximate analytic result and as a check on the validity of the virtual photon concept as applied to pion production we calculate in (c) the cross section by using the FPA value for WT and numerically integrating eq . (14) over electron angles, and in (d) the cross section with all structure functions including their implicit dependence on electron angles by numerical integration. Details of this complete calculation can be found in ref. e) for the analogous case of pion production from 3He. In cases (a}-(c), the pion photoproduction cross section has been calculated with the Blomgvist and Luget amplitudes 8). Close examination of the results under (c) and (d) in table 1 show that the concept of a virtual spectrum for pion production from the proton is very good. Even at 50 MeV above the endpoint the deviation between the exact calculation and the virtual spectrum result is only 3.5% in the worst case shown, with about 1% due to the longitudinal, polarization and interference terms in eq . (2) and the remainder from the variation in WT. We also note that the conventional Dalitz-Yennie spectrum can be in error by as much as a factor of 2 very near the endpoint, and is generally off by about 5-20% for the proton case. Our new analytic result (b) which includes recoil and the variation of kr with electron angle is in quite good agreement with the exact numerical integration over electron angles (c) with the largest disagreement coming within 100-200 keV of the endpoint . We also observe that the recoil factor R can differ greatly from one and affects not only the magnitude of the cross section but the shape as a function of incident electron energy above threshold as well. As a further check on the concept of the virtuâl photon spectrum as applied to pion production we have reexamined the reaction 3He(e, 3H)e'Tr + analysed in detail in ref. 6). In this case all pion and nuclear coordinates in the kinematics and in eq. (19) must be interchanged as mentioned previously in the text . In fig. 2 we show the cross sections obtained with a Dalitz-Yennie spectrum along with our virtual spectrum and the complete calculation where the nuclear structure functions Wi ( j = L, T, P, I) have been calculated with numerical solutions of the Faddeev equation 9) as discussed in ref. 6) . The agreement between our analytic virtual photon spectrum (dotted curve) and the full calculation (solid line) is excellent.

L. Tiator and L.E. Wüght / Virtual photons

41 2

Tnst .E 1 Values of dZQ°/dE d114b in nb/MeV ~ sr calculated with (a) Dalitz-Yennie, (b) Tiator-Wright, (c) numerical integration with WT = constant, and (d) full calculation at various pion moments and angles as a function of the incident electron energy de; above threshold e ;"'. (T'he values of the recoil factor ./dn~n R and the pion production cross section dQ, are also given.) k = 100 MeV/c

B = 30°

de ; (MeV)

R

0.1 0.3 0.5 1.0 3.0 5.0 10. 50.

1 .078 1 .026 1 .014 1 .004 0.999 0.998 0.998 0.998

0.1029 0.1556 0.1902 0.2483 0.3645 0.4257 0.5130 0.7481

k =100 MeV/c

9~ = 90°

s;h` =199.7 MeV

de; (MeV)

R

a)

b)

c)

d)

0.1 0.3 0.5 1.0 3.0 5.0 10 . 50 .

1 .733 1 .575 1 .536 1 .505 1 .488 1 .485

0.1442 0.1968 0 .2332 0 .2963 0.4263 0.4961

0.0977 0.1557 0.1942 0.2595 0.3913 0.4612 0.5610 0.8208

0.0917 0.1532 0 .1930 0 .2597 0 .3927 0 .4628 0.5626 0.8115

0.0915 0.1529 0.1926 0.2593 0.3924 0.4628 0.5633 0.8196

1 .484 1.484

e; h' =179 .4 MeV a)

0.5965 0.8630

b) 0.0835 0.1385 0.1744 0.2332 0.3501 0.4112 0.4979 0.7270

c) 0.0820 0.1379 0.1737 0.2332 0.3502 0.4112 0.4974 0.7134

d) 0.0820 0.1379 0.1737 0.2332 0.3504 0.4117 0.4986 0.7189

do,./d11~,°,b = 9.34 wb/sr

./dnAb do, =7 .26 Wb/sr

kR =100 MeV/c

B =150°

de ; (MeV)

R

a)

b)

c)

d)

0.1 0.3 0.5 1.0 3.0 5.0 10 . 50 .

2.766 2.414 2.324 2.253 2.210 2.205 2.202 2.201

0.1765 0.2202 0 .2532 0.3129 0.4407 0.5108 0.6125 0.8795

0.1031 0.1562 0.1924 0.2553 0.3855 0.4559 0.5571 0.8175

0.0875 0.1477 0.1868 0.2527 0.3857 0.4566 0.5580 0.8102

0.0873 0.1474 0.1864 0.2521 0.3849 0.4558 0.5575 0.8129

k = 200 MeV/c

B4 = 30°

e;h` = 254 .5 MeV

de ; (MeV)

R

a)

b)

c)

d)

1.140 1.092 1 .080 1.071 1.066

0.0900 0.1379 0.1691 0.2215 0.3260

0.0748 0.1244 0.1564 0.2096 0.3146

0.0748 0.1248 0 .1569 0.2102 0.3153

0.0748 0.1249 0.1571 0.2107 0.3167

0.1 0.3 0.5 1.0 3.0

e~h' =225 .1 MeV

, dv,./dl1;',b =10.76 ~,b/sr

dQ,./dl1;;b =12.79 wb/sr

L. Tiator and L.E. Wüght/ Virtualphotons

de; (MeV)

T.AHLE 1 (cont)

,

b)

c)

a)

R

41 3

5.0 10 . 50 .

1.066 1.065 1.065

k = 200 MeV/c

B~ = 90°

de ; (MeV)

R

a)

b)

c)

d)

0.1 03

2.273 1 .983 1 .909 1.851 1.816 1.811 1.809 1.808

0.3100 0.3867 0.4448 0.5497 0.7747 0.8983 1 .0774 1 .5302

0.1812 0.2745

0.1654 0.2673 0.3349 0.4497 0.6829 0.8077 0.9865 1 .4222

0.1653 0.2574 0.3350 0.4500 0.6836 0.8090 0.9892 1.4315

0.5 1 .0 3.0 5.0 10. 50.

k~ = 200 MeV/c

B~ =150°

e ;'"

0.3810 0.4594 0.6586

0.3695 0.4474 0.6417

= 318.0 MeV

b do,./df1 : = 21 .94 wb/ar

e;~` = 423.E MeV

0.3382 0.4487 0.6779 0.8019 0.9804 1.4232

0.3702 0.4478 0.6357

d) 0.3725 0.4524 0.6583

do,./df1~ = 2.83 Wb/ar

de; (MeV)

R

a)

b)

c)

d)

0.1 0.3 0.5 1.0 3.0 5.0 10 .

5.065 4.104 3.849 3.638 3.503 3.483 3 .472 3.468

0.0829 0.0878 0.0951 0.1105 0.1477 0.1694 0.2016 0.2837

0.0370 0.0500 0.0594 0.0767 0.1151 0.1369 0.1691 0.2496

0.0235 0.0427 0.0540 0:9736 0.1143 0.1367 0.1692 0.2491

0.0231 0.0426 0.0540 0.0734 0.1138 0.1360 0.1680 0.2447

50 .

~

Fig. 2. Inclusive differential cross section for the reaction e(3He, 3H)e'+r + as a function of triton energy at incident electron energy e; = 200 MeV and fixed triton angle B~ = 28.3'. The solid line is obtained by an explicit integration of the coincidence cross section, and the dashed and dotted lines are obtained by using the virtual photon spectra of Dalitz-Yennie and Tiator-Wright respectively . The experimental data are from Skopic et al.'s.

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L. 7lator artd L.E. Wrlght / Virtua! photons

In summary, we have derived a new virtual photon spectrum which includes recoil effects which is appropriate for pion production experiments . We have found that the recoil of the undetected final state gives large effects which depend on the angle and momentum of the detected particle for light nuclei as well as for the case of a detected nucleus and a recoiling pion. Furthermore, these effects are not just an overall factor times the virtual photon spectrum, but also change the shape considerably, particularly near the endpoint where the spectrum has a large gradient . We conclude that our new analytic virtual photon spectrum works quite well for pion production from the proton and 3 He, and believe that it will also work well for heavier nuclei until Coulomb distortion coupled with finite-size corrections' l'lz) begin to play an important role. References 1) C.F. Weizsâcker, Z. Phys. 88 (1934) 612 2) E.J. Williams, Mat. Fys. Medd . Dan. Vid. Selsk . 13 (1935) no . 4 3) P. Stoler, E.J . Winhold, F. O'Brien, P.F. Yergin, D. Rowley, K. Min, J. LeRose, A.M. Bernstein, K.I . Blomgvist, G. Franklin, N. Paras and M. Pauli, Phys . Rev. C22 (1980) 911 4) R.H . Dalitz and D.R. Yennie, Phys . Rev. 105 (1957) 1598 S) J.D. Bjorken and S.D . Drell, Relativistic quantum mechanics (McGraw-Hill, NY, 1964) app. A, B 6) L. Tiator, Ph.D . dissertation, Mainz 1980 ; L. Tiator and D. Drechsel, Nucl. Phys. A360 (1981) 208 7) S. Furui, Prog . Theor. Phys. 58 (1977) 864; Nucl . Phys . A300 (1978) 385 8) I. Blomgvist and J.M. Laget, Nucl . Phys . A280 (1977) 405 9) R.A . Brandenburg, Y.E . Kim and A. Tubis, Phys. Rev. C12 (1975) 1368 10) D.M . Skopic, E.L . Tomusiak, J. Asai and J.J . Murphy II, Phys . Rev. C18 (1978) 2219 11) L.E . Wright and C.W. Soto Vargas, Comp . Phys . Comet. 20 (1980) 337 12) D.S . Onley, to be published