Threshold electrodisintegration of the deuteron caused by backward scattering of electrons

Volume 48B, number 2

PHYSICS LETTERS

THRESHOLD

ELECTRODISINTEGRATION

CAUSED BY BACKWARD

SCATTERING

21 January 1974

OF THE DEUTERON OF ELECTRONS

S A. SMIRNOV and S V. TRUBNIKOV Steklov Mathematteal Instttute, USSR Academy of Scwnces, Moscow, USSR

Recewed 18 October 1973 Using a new relativistic integral representation for the matrix element of electromagnetic current in terms of experimental phase shifts alone (representation of Shlrokov RS) the cross section o f the threshold electrodlslntegratlon of the deuteron caused by backward scattered electrons is calculated for the S-wave deuteron in the interval of the four-momentum transfers squared 0.16 ~
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21 January 1974

Feynman diagrams, corresponding to the intermediate states of the lowest masses). In prlnc@e (see dmcusslon In ref. [1 1 ]) the value of the MEC contribution can be estimated as the difference between experimental and calculated cross sections. We neglect the effects of inelastic n-p channels. RS proved to a certain extent its effectiveness m the application to the calculation of the charge form factor of the deuteron [ 14, 15 ]. The general discussion of deuteron electrodasmtegratIon and RS application to this problem are contained ia ref [16]. We have calculated the cross section for q2 ~< 5 fm - 2 , i.e. m the kinematical situation of the experiments [1, 1 7 - 2 0 ] . The contribution of the D-wave in the deuteron is ignored. The results of the interesting papers [ 6 - 8 ] show that for low q2 the D-state contributions to the impulse approximation and to the two-plon exchange current cancel each other and so in this region the S-wave deuteron without MEC Is a reasonable approximation to compare with the experimental data (for high values of q2 the D-state contribution is large [6, 7]). We take into account only the S-wave in the final n-p system because near the threshold the velocities of the outgoing nucleons in their center-of-mass system are small and the centrifugal barrier suppresses all but S-wave. The calculation of the differential cross section is reduced to the calculation of the current matrix element (nplluld) m the one-photon-exchange approximation. We use the multipole parametrxsatlon for in terms of the inelastic mvanant form factors G(s, t), introduced m [21 ] The backward scattering slmphfies the kinematics of the process, and the formula for the cross section appears as d2o dEed~e

71~2MD~2 -

1 2(27r)4(2_M)2 E2X/~

{G (s t)[ 2 + IGt(s, t)12}, s '

(1)

where Gs, t describes, respectively, the magnetic dipole transition to 1So-, 3Sl-States of the final n-p system;M and M D are the nucleon and the deuteron masses respectively, a = 1/137 Is the electromagnetic coupling constant; t ! Ee, E e are the imtlal and final electron energies, t ~ _ q 2 < 0, for 0 e = 180 °, q2 = 4EeEe; s = (kp+kn) 2 is the int variant mass of the final n-p system, s = 4M 2 + 2Mw = 4M 2 + 2M(E e - E e - e ) - ½q2. RS leads to the following formula for G~(s, t) (in lnvanant variables)

GQ(s,t)=_I' F 271ILs,=M2D '

s2(s't)

Res

-- B~- lIr/)2711 (s-

f

] sl(s,t)

Gg(s, s", t)ABt(s" ) S

""D

(2)

~o / '" G:(S'",S",t)ABt(s") I Y ds'" s-sAB~(s'")-"'-Irl 2(s t)ds,, S"~M~DD J' 4M2 Sl (s'",t)

here ~ means s or t, F is the normalization constant. We have calculated P from the condition of normalization of the theoretical value of the cross section to the experimental one at a point (we have chosen the point q2 = 0.1 6 fro-2). The first term In eq. (2) describes the Born approximation, the second one is caused by the FSI. B(s) is the relativxstlc generalization of the Jost function, AB(s) = B(s +1~7)- B ( s - i t / ) is the discontinuity on the cut. G~O (s, s,t t) describes the electromagnetic form factor of the noninteracting n-p system (the exphcit formula for G~O (s, s,t t) see below). The limits of integration m eq. (2), determining the position of anomalous thresholds, are given by (2M2 _ t ) ( s _ 2 M 2) -+1 x/(_t)(4M2 _t)s(s_4M,). s2,1(s, t) = 2M 2 + 1 2M 2 2M z

(3)

The Jost functions m terms of the phase shifts have the form. Bt(s)= (l

M2-4M2~ ~s - 4 M

106

] fit(s);

Bs(S) = ~'s(s)'

[1 j d g 6 ~ ( s " } B'~(s) = exp -. , (71 4M 2

s--S

(4)

Volume 48B, number 2

PHYSICS LETTERS

21 January 1974

5s, t are the 1So- and 3Sl-phase sh]fts of n-p scattering respectwely. The &rect calculation of G~(s, s', t) gwes us (the techmque of such principally simple but cumbersome calculations is contained m ref. [14])

G°(s, s', t) = H l ( H 2 sin (tp+ if))GVN + [H 3 cos ~0-sln l~(H 4 sin ~ - / / 5 ) ] GVN/4M2}, Gt(s , s',

H1

t) = ~ / 2 H I { H

2 sin (~0- ~ ) G EN S -- [H3 cos q0+H 4 sln ~0 Sllt ~]

- 8 ~/ss' X/(s s - t) . 1 V/3,'(s-aM2)(s"-aM 2) lx/~--tf~'

H2 =X/(-1)(M2 +ss't/X)'

n 3 = 2 H 4 cos ~ + H2M [2M(x/~-+xR) + t] sxn t~/n6, H 5 = M ( a M 2 - t) [(X/~-+XR) 2 - t] (x/s-- x/~-)

GSN/4M2},

sin

n 4 = Mt(s+s'-t)/2x/~,

l~/2H6~/r~,

(5)

n 6 = 2 M + V/S+ N/~,

X = S2 + S'2 + t 2 -- 2(SS'+St+s't), cotan

= (M

t] +

_

L

cotan ~ = [M(s+s'-t)H 6 + V / ~ ( 4 M 2 _t)]/H2H6X/~, S,V

2

GE,M(N)( q ) are the ]soscalar and isovector charge and magnetic nucleon form factors.

, The cross section is calculated using eqs. ( 1 - 5 ) . ]'he numerical calculation of Jost funct,ons (4) is made with one subtraction but the form factors (2) are independent of the value o f the subtraction constant. The cut-off of the integrals m (2), (4) for co = 2 GeV leads to enough precision m the calculations. The experimental phase shifts were gwen m ref. [22] 3Sl-phase shifts in the energy region 0 ~< co ~< 460 MeV and 1So-phase shifts m the energy region 0 ~< co ~< 750 MeV. Then the phase shifts were extrapolated to higher energies in accord with the results

a,o

~oo 200 ~00

Ct~odE; , '

.,..

5O 2O 4O

2 0.5 a2

.~,,

]

Fzg 1. The calculated value of the deuteron electrodlsmtegratlon cross section (m umts 10 -34 cm2/sr (MeV/c)) near threshold E t o ( e = 0 98 Ethreshold) at 0 e = 180 wzth the phase 6 (•) = - l r (sohd hne), the same value with the phase 6 (=) = 0 (dashed hne), q2

zs m fm -2. The experimental points are taken from [1] e, [17, 18] A, [19] X and [20] o. 107

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21 January 1974

of the high-energy phase shift analysis [23] (see fig. 1 In ref. [15]). The results are almost independent of the high-energy tall of the phase shifts. Details of such calculations are described in ref. [15]. We have used the dipole fit and scahng law for GEp, GMp, GMn, GEn was set equal to zero. The charge term (term proportional to GEN In Gs°,t, see eq. (5)) has a purely relativistic origin for the S-wave deuteron. The contribution of this term to the cross section is extremely small. The results of the calculations are presented in fig 1. The calculated cross section for q2 ~< 5 fm - 2 coincides within the experimental error with the data [1, 1 7 - 2 0 ] . We think this result to be of some interest as an example of a completely relativistic calculation. To calculate the cross section for large values of q2 we have to include m our formalism [14] the correction due to deuteron D-state and to introduce MEC corrections from the nearest diagrams [ 6 - 8 ] . RS allows one to calculate the deuteron disintegration differential cross section with electrons scattered at small and intermediate angles in the 1.s., and further to solve the interesting question of determination of the neutron charge form factor It is possible to calculate the deuteron photodlslntegratlon, plon electroproduction on the deuteron, the structure of nucleon isobars using RS. Fxnally, RS makes it possible to calculate electromagnetic properties of other composite hadron systems [24], and to calculate the matrix elements of other local operators. The authors are deeply indebted to Dr. V.I. Kukuhn for great support, to Professor Yu.M. Shlrokov and Dr. V.E. Troltskl for useful discussion, and to Mrs. Kuznetsova and Mrs. Pogrebkova for assistance in numerical calculations.

References [1] [2] [3] [4]

D. Gamchot, B Grossetete and D.B. Isabelle, Nucl. Phys. A178 (1972) 545. W. Bartel et al., Phys. Lett 39B (1972) 407 J.S. Levlnger, Conf. on Nuclear three-body problem, Budapest (1971) U. Amaldl, Intern. Conf. on Few partlcle problems m the nuclear interaction, Los Angeles (1972) (preprlnt ISS 72/14, Roma, 1972). [5] W B. Atwood and G B. West, Phys. Rev D7 (1973) 773. 16] M Gari, J. Hockert and D O. Rlska, to be pubhshed. [7] D.O Rlska, Asllomar Conf. (1973). [8] D.O Rlska and G E. Brown, Phys Lett 38B (1972) 193 [9] J.S. Levlnger, Nuclear photo-disintegration (Oxford University Press, 1960). [10] F.M. Renard, J. Tranh Thanh Van and M. Le Bellac, Nuovo Clm. 38 (1965) 565, 1688. [11] B Bosco, P P Delsanto and F Erdas, Nuovo Clm. 33 (1964) 1240 [121 I.J. McGee, Phys. Rev 158 (1967) 1500; 161 (1967) 1640 [13] Yu.M. Shlrokov, Nucl. Phys B6 (1968) 159. [ 14] V.P. Kozhevmkov et al., Theor Math. Phys (USSR) 10 (1972) 47; V E. Troltskl, S.V Trubmkov and Yu.M. Shirokov, Theor. Math. Phys. (USSR) 10 (1972) 209,349. [15] V.I. Kukuhn et al., Phys Lett. 39B (1972) 319; Problems of atomic science and technology (series' High energy and atomic nuclei physics) 2(4) (1973) 15, Kharkov (m Russian). [16] S.V. Trubnlkov, Theor Math Phys. (USSR) 12 (1972) 390; Problems of atomic science and technology (series: High energy and atomxc nuclei physics) 2(4) (1973) 3, Kharkov (m Russian). [17] G.A. Peterson and WC. Barber, Phys. Rev. 128 (1962) 812. [18] W.C. Barber et al, Nucl Phys. 41 (1963) 461. [19] J Goldemberg and C. Schaerf, Phys. Lett. 20 (1966) 193. [20] R.E. Rand et al., Phys Rev. Lett. 18 (1967) 469. [21] A.A Cheshkov and Yu. M. Shlrokov, JETP (Soviet Phys.) 44 (1963) 1982; V.M Dubovik and A A. Cbeshkov, Physics of elementary particles and atomic nuclei, Dubna (m press). [22] M.N McGregor, R A. Arndt and R.M Wright, Phys Rev. 169 (1968) 1149; 173 (1968) 1272; 182 (1969) 1714 [23] N Hoshlzakl, T, Kadota and K. Makl, Prog. Theor. Phys. 45 (1972) 1123. [24] A.I. Kirdlov, Yu.M. Shlrokov and V E. Troxtskl, Phys. Lett. 39B (1972) 249.

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