ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 8, с. 1054-1059
ЯДРА
SOME RELATIVISTIC ASPECTS OF NUCLEAR DYNAMICS AT THE ELECTRODISINTEGRATION OF NUCLEI
©2014 V. D. Efros*
National Research Centre "Kurchatov Institute", Moscow, Russia Received June 26, 2013
An approach aimed to extend the applicability range of the nonrelativistic microscopic calculations of electronuclear response functions is reviewed. In the quasielastic peak region these calculations agree with experiment at momentum transfers up to about 0.4 GeV/c, while at higher momentum transfers being beyond 1 GeV/c a disagreement is seen. In view of this, a reference frame where dynamic relativistic corrections are small was employed to calculate the response functions and the results were transformed exactly to the laboratory reference frame. This proved to remove the major part of the disagreement with experiment. All leading-order relativistic corrections to the transition charge operator and to the one-body part of the transition current operator were taken into account in the calculations. Furthermore, a particular model to determine the kinematical inputs of the nonrelativistic calculations was introduced. This model provides the correct relativistic relationship between the reaction final-state energy and the momenta of the knocked-out nucleon and the residual system. The above-mentioned choice of a reference frame in conjunction with this model has led to an even better agreement with experiment.
DOI: 10.7868/S004400271407006X
1. INTRODUCTION
At present the nonrelativistic dynamics framework is the only practical one at performing microscopic few- or many-nucleon calculations. Last years the predictive power of such calculations increased due to the progress in the effective field theory approach to nuclear forces providing a three-nucleon force along with a two-nucleon force. Naturally, it is desirable to test the theory in a wider range of momentum and energy transfers. However, if one speaks of electrodis-integration reactions in the quasielastic kinematics, final states of a system may be described nonrela-tivistically only in a rather limited range of transferred momenta. Here, an approach is reviewed allowing for the considerable extension of the applicability range of the nonrelativistic calculations. The presentation is based on the work done by W. Leidemann, G. Orlan-dini, E.L. Tomusiak, and the author.
2. RESPONSE FUNCTIONS
In the one-photon-exchange approximation the inclusive electron scattering cross section in the laboratory (LAB) frame is given by
d2
a
dQdw
— aMott
-
22
+
q2 - w2 2 в , „ , 2q2 +tan 2 ) T(<?'W)
■Rb(q,co)+ (1)
E-mail: v.efros@mererand.com
where RL and RT are the longitudinal and transverse response functions, respectively. The electron variables are denoted by u (energy transfer), q (momentum transfer), and 6 (scattering angle). All these quantities pertain to the LAB frame.
Let and be the eigenstates of a nuclear Hamiltonian with energies Ei and Ef and total momenta Pi and Pf, where i stands for the initial state and f stands for final states of a reaction. Besides P f, the set f includes additional asymptotic quantum numbers that will be denoted by pf. Let us use the volume element df = dPf dpf. The states ^i and ^f are assumed to be normalized as follows,
<*i|*i/) = ¿(Pi - Pi/), (*f ) = S(f - f') = 5(Pf - Pf/)S(pf - pf/).
Let Q(q,u) and Jt(q,u) be the nuclear charge operator and the transverse component of the nuclear current operator J, Jt = J — q(q • J). Their matrix elements between the states and ^f are proportional to 5(Pf - Pi - q),
<^f |Q(q,u)|*i) = 6(Pf - Pi - q)Qfi, (2) <^f |Jt(q,u)|^) = 6(Pf - Pi - q)(Jt)fi. (Pi = 0 in the LAB frame case considered.)
2
4
The response functions entering Eq. (1) are expressed in terms of the on-shell matrix elements from Eqs. (2),
RL(q, to) = fypfQifQiAEf -Ei-u), (3)
dpf (Jl)if ■ (Jt)fi x (4)
Rt (q,u) =
x 5(Ef - Ei - u).
Here, averagings over the projections of the initialstate spin are performed and summations plus integrations over pf are present.
In the nonrelativistic approximation one has Eij = Pf /(2M0) + eif, where M0 is the mass of a nucleus and ei j are the energies of internal motion. One also has a representation j) = |Pij)\^ij), where the states , f being internal motion states are independent of the quantum numbers Pi j. Let us
define the operators Q and Jt acting in the space of internal motion states,
QS(Pf - Pi - q) = (Pf\Q\Pi),
JtS(Pf - Pi - q) = (Pf\Jt\Pi).
Then in accordance with Eqs. (2) the response functions become
RL(q,u) =
dpf \Q(q,uMi)\2 x (5)
x S(6f - €i + Pj /(2Mq) - u),
dpf №Vt(q,u)№f )■ (6)
Rt (q,u) =
tyf \Jt(q,uM)S(6f - 6i + Pj/(2Mo) - u).
3. NONRELATIVISTIC DYNAMICS RESULTS
Response functions of trinucleons were studied. The nuclear Hamiltonians employed in the calculations included two-body and three-body realistic nuclear interactions plus the Coulomb force. Dependence on the version of a realistic nuclear force proved to be weak. Most of the results were obtained with the Argonne V18 two-nucleon interaction [1] and the Urbana IX three-nucleon interaction [2]. These results will be presented below.
In the longitudinal response case the customary one-body charge transition operator was used in the calculations. It consists of the nonrelativistic operator and the relativistic corrections of the leading M-2 order, MN being the nucleon mass. These
are the Darwin—Foldy and spin—orbit corrections. These corrections were accounted for at obtaining the nonrelativistic dynamics results presented below.
The transition operator used in the transverse response case included the nonrelativistic one-body current consisting of the spin current and convection current plus the usual pion and rho-meson-exchange two-body currents. The calculations are described in [3] and in [4] in the longitudinal case and in the transverse case, respectively, where details of the formulation may also be found.
At not low energies it would be very difficult to obtain reaction final states entering Eqs. (5) and (6) as well as to perform the summations and integrations over infinite sets of states there. All this was avoided with the help of the method of integral transforms. The Lorentz transform was employed. The approach is reviewed, e.g., in [5]. A concise review of the formalism can be found, e.g., in [6]. The approach is applicable directly when the operators in Eqs. (5) are u-independent. Actually they include only numerical factors depending on u which may be divided out. A more general case is considered below.
The numerical results obtained may be considered as being accurate for the present purposes. This conclusion follows from the studies of convergence trends of the calculations. Some of these and similar our results were also confirmed with other methods [7, 8] or other ways to solve arising dynamics equations [9].
A review on nonrelativistic studies by other authors of A = 3 electronuclear reactions is provided in [7].
Let us consider some of the results obtained. In Fig. 1 the calculated transverse response functions [4] along with experimental data [ 10—12] are shown. The agreement with experiment that is observed at q = 250 MeV/c subsequently deteriorates as q increases. In Fig. 2 the calculated longitudinal response functions [13] at higher q values are presented along with the data [10—12]. The nonrelativistic results we discuss here are shown with the dotted curve in the figure. A sharp disagreement is seen also in this case.
4. PRIVILEGED REFERENCE FRAME
In addition to RL and RT of Eqs. (3) and (4), one may define related responses Ri[ and R^. They are given by the expressions of Eqs. (3) and (4) form with the replacement of all the quantities entering there with these quantities but pertaining to another reference frame. Here, a class of reference frames is considered which are obtained via boosting the LAB frame along q. The laboratory responses RL and RT
an,EPHA^ OH3HKA tom 77 № 8 2014
q = 250 MeV/c
RT, 10-3 MeV-1 12
9 6 3 0 12 9 6 3
0 12
8 -
20 40 60 80 100 120 140
400 MeV/c
о Bates, 1988 a Saclay, 1985 ■ World
50
100
150
200
500 MeV/c
100
200
ю, MeV
Fig. 1. Comparison of theoretical and experimental Rt at q = 250, 400, and 500 MeV/c. Theoretical Rt with contributions of one-body (dotted curve) and one-body + + two-body transition operators (solid curve). Experimental data are from [10] (triangles), [11] (circles), and [12] (squares).
can be expressed in terms of such RL and Rff with the help of the relationships
q2 Efr
Rl (q,w)
(qfr)2 Mo Efr
RL (qfr ,w r ),
(7)
Rr(q^) =
Here qfr, f and Ef are the corresponding quantities pertaining to a reference frame considered. One has
qfr = Y (q - ßw), Ufr = Y (w - ßq), Pif = -ßYMo, Ef = yMo ,
(8)
where 3 is the velocity of a reference frame, 7 = (1 — — 32)-1/2, and Pf is the initial-state momentum in this reference frame. The origin of the factor q2/(qfr)2 in (7) is shown in [14], see also, e.g., [15]. The factor
Rl, 10-3 MeV 8 г q = 500 MeV/c
01-,-L
50 100 150 200 250 300 350
Q, MeV
Fig. 2. Longitudinal response functions calculated in various reference frames. Experimental data are from [10] (squares), [11] (triangles), and [12] (circles).
Ef /M0 arises since we adopt the usual normalization of a state of a nucleus to unity instead of its covariant normalization.
In a genuine relativistic theory any and RfT from Eqs. (7) would lead to the same RL and RT.
This, of course, is not the case if RL and RT are calculated in the nonrelativistic approximation. The responses RL obtained from Rf[ that are calculated nonrelativistically are shown in Fig. 2. The anti-lab (AL) frame, PfL = -qAL, PAL = 0, the Breit (B) frame, PB = —qB/2, PB = qB/2, and one more reference frame (ANB) described below were employed.
The nonrelativistic expressions for the responses RL and RfT are of Eqs. (5) and (6) form with the replacement P2/(2M0 ) by P2/(2M
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