Inclusive electron scattering in a relativistic Green function approach

12 Pages·0.44 MB·English

by  Andrea Meucci

Inclusive electron scattering in a relativistic Green function approach
Andrea Meucci, Franco Capuzzi, Carlotta Giusti, and Franco Davide Pacati
Dipartimento di Fisica Nucleare e Teorica, Universit`a di Pavia and
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy
(Dated: February 8, 2008)
A relativistic Green function approach to theinclusive quasielastic (e,e′) scattering is presented.
The single particle Green function is expanded in terms of the eigenfunctions of the nonhermitian
optical potential. This allows one to treat final state interactions consistently in the inclusive and
in the exclusive reactions. Numerical results for the response functions and the cross sections for
differenttarget nucleiandin awiderangeof kinematicsarepresentedand discussed incomparison
with experimentaldata.
3 PACSnumbers: 25.30.Fj,24.10.Jv,24.10.Cn
0
0
2 I. INTRODUCTION approach considered here, where the components of the
nuclearresponsearewrittenintermsofthesingleparticle
n
a Theinclusiveelectronscatteringinthequasielasticre- optical model Green function. This result was originally
J gion addresses to the one-body mechanism as a natu- derived by arguments based on the multiple scattering

7 ralinterpretation. However,whentheexperimentaldata theory [12] and successively by means of the Feshbach
2 of the separation of the longitudinal and transverse re- projectionoperatorformalism[13–16]. The spectralrep-
resentation of the single particle Green function, based
sponses were available, it became clear that the expla-
1 on a biorthogonal expansion in terms of the eigenfunc-
nation of both responses necessitated a more compli-
v
tionsofthenonhermitianopticalpotential,allowsoneto
4 cated frame than the single particle model coupled to
performexplicitcalculationsandtotreatfinalstateinter-
8 one-nucleon knockout.
0 A review till 1995 of the experimental data and their actions consistently in the inclusive and in the exclusive
1 possible explanations is given in Ref. [1]. Thereafter, reactions. Important and peculiar effects are given, in
0 the inclusive reactions, by the imaginary part of the op-
only few experimental papers were published [2, 3].
3 ticalpotential,whichisresponsiblefortheredistribution
New experiments with high experimental resolution are
0
of the strength among different channels.
plannedatJLab[4]inordertoextracttheresponsefunc-
/
h tions.
t
- From the theoretical side, a wide literature was pro-
cl ducedinordertoexplainthemainproblemsraisedbythe Inapreviouspaperofours[15]the approachwasused
u the separation,i.e., the lack of strength in the longitudi- in a nonrelativistic frame to perform explicit calcula-
n nal response and the excess of strengthin the transverse tionsofthelongitudinalandtransverseinclusiveresponse
v: one. The more recent papers are mainly concerned with functions. The main goal of this paper is to extend the
i the contribution to the inclusive cross section of meson method to a relativistic frame and produce similar re-
X
exchange currents and isobar excitations [5–7], with the sults. Although some differences and complications are
r effect of correlations [8, 9], and the use of a relativistic due to the Dirac matrix structure, the formalism follows
a
frame in the calculations [7]. thesamestepsandapproximationsasthosedevelopedin
At present, however, the experimental data are not thenonrelativisticframeofRefs. [15,16]. Thenumerical
yet completely understood. A possible solution could results obtained in the relativistic approach allow us to
be the combined effect of two-body currents and tensor check the relevance of relativistic effects in the kinemat-
correlations [8, 10, 11]. ics already considered in Ref. [15] and can be applied
In this paper we want to discuss the effects of final to a wider range of kinematics where the nonrelativistic
state interactions in a relativistic frame. Final state in- calculations are not reliable.
teractions are an important ingredient of the inclusive
electron scattering, since they are essential to explain
the exclusiveone-nucleonemission,whichgivesthemain In Sec. II the hadron tensor of the inclusive process
contribution to the inclusive reaction in the quasielastic is expressed in term of the relativistic Green function,
region. The absorptionin agivenfinalstate due, e.g., to which is reduced in Sec. III to a single particle expres-
the imaginary part of the optical potential, produces a sion. The problem of antisymmetrization is discussed in
loss of flux that is appropriate for the exclusive process, Sec. IV. In Sec. V the Green function is calculated in
but inconsistent for the inclusive one, where all the al- terms of the spectral representation related to the opti-
lowed final channels contribute and the total flux must cal potential. In Sec. VI the results of the calculation
be conserved. are reported and compared with the experimental data.
This conservation is preserved in the Green function Summary and conclusions are drawn in Sec. VII.
2
II. THE GREEN FUNCTION APPROACH Inthispapertheinterestisfocusedonrelativisticwave
functions for initial and finalstates. Therefore,the (A+
A. Definitions and main properties 1)-body Hamiltonian H is the sum of one nucleon free
Dirac Hamiltonians and two nucleons interactions Vjj′,
i.e.,
In the one photon exchange approximation the inclu-
sive cross sectionfor the quasielastic (e,e′) scattering on A+1 A+1
1
a nucleus is given by [1] H = αj ·pj +βjM + 2 Vjj′ , (7)
j=1 j,j′=1
σ =K(2ε R +R ) , (1) X(cid:0) (cid:1) X
inc L L T
where the 4 4 Dirac matrices, α and β , act on the
j j
×
where K is a kinematical factor and bispinor variables of the nucleon j. No particular as-
ε = Q2 1+2q2 tan2(ϑ /2) −1 (2) sumInpotirodneristomeaxdperoesnstthhee4h×ad4romnatternixsosrtrinuctteurrmesofofVjsji′n.-
L q2 Q2 e gleparticlequantities,thesameapproximationsasinthe
(cid:18) (cid:19)
nonrelativistic case [15] are required. The first one con-
measures the polarization of the virtual photon. In Eq.
sists in retaining only the one-body part of the charge-
(2) ϑe is the scattering angle of the electron, Q2 = current operator Jµ. Thus, we set
q2 ω2, and qµ =(ω,q) is the four momentum transfer.
−
Allnuclearstructureinformationiscontainedinthe lon- A+1
gitudinaland transverseresponsefunctions, RL and RT, Jµ(q)= jiµ(q) , (8)
defined by i=1
X
where jµ acts only on the variables of the nucleon i. By
R (ω,q) = W00(ω,q) , i
L tot Eq. (8),one canexpressthe hadrontensorasthe sumof
RT(ω,q) = Wt1o1t(ω,q)+Wt2o2t(ω,q) , (3) two terms, i.e.,
intermsofthediagonalcomponentsofthehadrontensor Wµµ(ω,q)=Wµµ(ω,q)+Wµµ(ω,q) , (9)
tot coh
where Wµµ(ω,q) is the incoherent hadron tensor [17],
Wµµ(ω,q) = Ψ Jµ(q) Ψ 2
tot |h f | | 0i| which contains only the diagonal contributions jµ†Gjµ,
ZXf whereas the coherent hadron tensor Wµµ(ω,q) giatheirs
δ(E +ω E) . (4) coh
× 0 − f the residual terms of interference between different nu-
HereJµisthenuclearcharge-currentoperatorwhichcon- cleons. As the incoherent hadron tensor, also Wcµoµh(ω,q)
can be expressed in terms of single particle quantities
nects the initial state Ψ of the nucleus, of energy
0
| i (see Sect. 9 of Ref. [16]), but for the transferred mo-
E , with the final states Ψ , of energy E, both eigen-
0 f f
| i menta consideredin this paper we cantakeadvantageof
states of the (A + 1)-body Hamiltonian H. The sum
thehigh-qapproximation[18]andretainonlyWµµ(ω,q).
runsoverthescatteringstatescorrespondingtoallofthe
Thistermcanbefurthersimplifiedusingthesymmetryof
allowed asymptotic configurations and includes possible
G for the exchange of nucleons and the antisymmetriza-
discretestates. Asmadefor Ψ ,inthefollowingthede-
f
| i tion of Ψ . Therefore we write
generacy indexes will be omitted whenever unnecessary. | 0i
aTthee. gInroournddersttaoteav|oiΨd0ciomispaliscsautmioends toof lbitetlenoinndteergeesnteirn- Wtµoµt(ω,q) ≃ Wµµ(ω,q)=−Aπ+1
thepresentcontext,weneglectrecoileffectsandconsider
Im Ψ jµ†(q)G(E )jµ(q) Ψ ,(10)
only point-like nucleons, without distinguishing between × h 0 | f | 0i
protonsandneutrons. Unless statedotherwise,the wave where jµ(q) is the component of Jµ(q) related to an ar-
functions are properly antisymmetrized. bitrarily selected nucleon. Due to the well-known com-
The hadron tensor of Eq. (4) can equivalently be ex- pleteness property, i.e.,
pressed as
1
dE G†(E) G(E) =1 , (11)
Wtµoµt(ω,q)=−π1ImhΨ0 |Jµ†(q)G(Ef)Jµ(q)|Ψ0i , (5) 2πiZ (cid:0) − (cid:1)
the incoherent hadron tensor fulfills the energy sum rule
where E = E + ω and G(E ) is the Green function
f 0 f
related to H, i.e., dωWµµ(ω,q)=(A+1) Ψ0 jµ†(q)jµ(q) Ψ0 . (12)
h | | i
Z
1
G(E )= . (6)
f
E H +iη
f− B. Projection operator formalism
Here and in all the equations involving G the limit for
η +0isunderstood. Itmustbeperformedaftercalcu- This formalism yields an expression of the incoherent
→
lating the matrix elements between normalizable states. hadrontensor ofEq. (10)in terms ofeigenfunctions and
3
Green functions of the optical potentials related to the the calculations, is correct for sufficiently high values of
variousreactionchannels. Apartfromcomplicationsdue the transferred momentum q. Thus, the hadron tensor
to the Dirac matrix structure, we follow the same steps of Eq. (10) can be expressed as the sum
and approximations as in the nonrelativistic treatment
[12, 13, 15, 16]. Wµµ(ω,q)=Wµµ(ω,q)+Wµµ(ω,q) , (18)
d int
Let us decompose H as
of a direct term
H =α p+βM +U +H , (13)
R
· Wµµ(ω,q) = Wµµ(ω,q) ,
d n
whereα p+βM isthekineticenergyofanarbitrarilyse- n
· X
lected nucleon,U is the interactionbetweenthis nucleon A+1
Wµµ(ω,q) = Im Ψ jµ†(q)P G(E )P
and the other ones, and HR is the residual Hamiltonian n − π h 0 | n f n
of A interacting nucleons. Such a decomposition cannot jµ(q) Ψ , (19)
0
beperformedinthephysicalspaceofthetotallyantisym- × | i
metrized (A+1) nucleon wave functions. Therefore, we and of a term
must operate in the Hilbert space of the wave func-
tions which are antisymmetrized onHly for exchanges of Wµµ(ω,q) = Wµµ(ω,q) ,
int n
the nucleons of HR. This treatment is presented here Xn
only for sake of simplicity. In Sect. IV we shall discuss Ac+1
Wµµ(ω,q) = Im Ψ jµ†(q)P G(E)Q
itsphysicaldrawbacksandoutlinethenecessarychanges. n − π h 0 | n f n
Let n and ε denote the antisymmetrizedeigenvec- jµ(q) Ψ , (20)
tors of|Hi rela|tedi to the discrete and continuous eigen- c × | 0i
R
valuesεnandε,respectively. Weintroducetheoperators which gathers the contributions due to the interference
Pn,projectingontothen-channelsubspaceof ,andQn, between the intermediate states ra;n related to differ-
H | i
projectingontotheorthogonalcomplementarysubspace, ent channels.
i.e., Wenotethattheinterferencetermdoesnotcontribute
to the energy sum rule. In fact Eq. (11) yields
P = dr ra;n n;ra ,
n
| ih |
a Z dω Wµµ(ω,q) = (A+1) Ψ jµ†(q)P Q
X n h 0 | n n
Q = 1 P . (14)
n n Z
− jµ(q) Ψ =0 . (21)
Here ra;n is the unsymmetrized vector obtained from c × | 0i
| i
the tensor product between the discrete eigenstate n Thus, the full contribution to the sum rule of the inco-
of H , and the orthonormalized eigenvectors ra (|a =i herent hadron tensor is given only by the direct term,
R
| i
1,2,3,4) of the position and the spin of the selected nu- i.e.,
cleon. The eigenvectors ra have been chosen only for
| i
sake of definiteness, as every complete orthonormalized dω Wµµ(ω,q)= dω Wµµ(ω,q)
d
setofsinglenucleonvectorswoulddefinethesameopera-
Z Z
torsPn. Apartfromminordifferencesduetothe present = (A+1) Ψ jµ†(q) P jµ(q) Ψ , (22)
0 n 0
relativistic context, P and Q are the projection op- h | | i
n n n
erators of the Feshbach unsymmetrized formalism [19]. X
Note, for later use, the relations which, as a pure consequence of the omission of the con-
tinuous channels described by P , is smaller than the
c
[Pn,α p+βM] = 0 value of Eq. (12).
·
H P = ε P . (15)
R n n n
Moreover,we introduce the projectionoperatoronto the III. SINGLE PARTICLE EXPRESSION OF THE
continuous channel subspace, i.e., HADRON TENSOR
P = dε dr ra;ε ε;ra . (16) A. Single particle Green functions
c
| ih |
Z a Z
X
For the time being, we disregard the effects of inter-
Due to the completenessofthe set ra;n , ra;ε , one
{| i | i} ference between different channels and consider only the
has
directcontributiontothehadrontensorofEq. (19). The
matrix elements of P G(E)P in the basis ra;n define
P +P =1 . (17) n n
n c | i
a single particle Green function (E) having a 4 4
Xn matrix structure, i.e., Gn ×
Then, we insert Eq. (17) into Eq. (10) disregarding the
contributionof P . This approximation,whichsimplifies ra (E) r′a′ n;ra G(E+ε ) r′a′;n . (23)
c n n
h |G | i≡h | | i
4
NotethatheretheenergyscaleisinaccordancewithRef. with
[16] and differs from Ref. [15].
The self-energy of Gn(E) is determined following the λn ≃A+1 , (33)
same steps used by Feshbach to determine the optical n
X
potential from the Schr¨odingerequation [19]. One starts
and the symbol f g denotes the scalar product
from the relation h | i
(E α p βM H U +ε +iη)(P +Q ) f g = drf⋆(ra)g(ra) . (34)
− · − − R− n n n h | i
a Z
G(E+εn)Pn =Pn ,(24) X
×
In Eq. (30) the hadron tensor is expressed in terms of
projectsbothsidesbyP andthenbyQ ,usesEqs. (15),
n n single particlesquantities. As in the nonrelativistic case,
resolves Q GP in terms of P GP , and finally obtains
n n n n ϕ are the eigenstates of the optical potential, i.e.,
n
| i
(E−α·p−βM −Vn(E)+iη) (α·p+βM +Vn(E0−εn))|ϕni
×PnG(E+εn)Pn =Pn , (25) =(E0−εn)|ϕni . (35)
with If Ψ is the eigenstate of H corresponding asymptoti-
E
| i
cally to a nucleon, of momentum k, colliding with a tar-
Vn(E) = PnUPn get nucleus in the bound state εn , the single particle
| i
1 vectors χ (E ε ) representing the elastic scattering
+ PnUQnE−QnHQn+εn+iηQnUPn .(26) wave fun|ctnions−hn;nrai| ΨEi are eigenstates of the same
optical potential, i.e.,
Using Eq. (14) for P and considering the matrix ele-
n
ments in the basis ra;n of both sides of Eq. (25), one (α p+βM + n(E εn)) χn(E εn)
| i · V − | − i
has =(E ε ) χ (E ε ) . (36)
n n n
− | − i
1
n(E)= , (27) SinceE isthetotalenergy√k2+M2+εn,theargument
G E h (E)+iη
− n of n is the kinetic energy (including the rest mass) of
V
the emitted nucleon.
where
h (E)=α p+βM + (E) , (28)
n n
· V B. Interference hadron tensor
and (E) has the 4 4 matrix structure defined by
n
V ×
Theproblemofexpressingtheinterferencehadronten-
ra n(E) r′a′ n;ra Vn(E) r′a′;n . (29) sor Wµµ in a one-body form is treated in Ref. [15] in
h |V | i≡h | | i n
the nonrelativistic context. It is argued that the contri-
Thus, h (E) is the self-energy of the Green function
n buticon of Wµµ can be included into the direct hadron
(E) and (E) is the related mean field. Using the n
Gn Vn tensor Wµµ by the simple replacement
same arguments as in the nonrelativistic case, one finds n
c
that (E) is the unsymmetrized Feshbach optical po-
Vn (E) eff(E) 1 ′ (E) (E)
tential [19], related to the channel n, for the relativistic Gn →Gn ≡ −Vn Gn
Hamiltonian H. q
1 ′ (E) , (37)
UsingthefirstEq. (14)andEq. (23),thedirecthadron × −Vn
tensor Wµµ(ω,q) of Eq. (19) becomes q
n where ′ (E) is the energy derivative of the Feshbach
Vn
1 optical potential.
Wµµ(ω,q) = λ Im ϕ jµ†(q) (E ε )
n −π n h n | Gn f− n In Ref. [21] the problem is considered anew from a
jµ(q) ϕ , (30) rigorouspointofview. The interference hadrontensoris
n
× | i expressedexactly asa seriesinvolvingenergyderivatives
where the initial state | ϕni, normalized to 1, is repre- ofVn(E), ofincreasingorder,plus aresidualtermwhich
sented by the bispinor defined by the matrix elements cannotbereducedtoasingleparticleform. Theseriesis
expectedtofastlyconvergenearthequasielasticpeakand
A+1 at intermediate energies. It is argued that in this region
ra ϕ n;ra Ψ , (31)
h | ni≡ λ h | 0i of momenta and energies the residual term is negligible.
r n
Thus, one recovers the result of Eq. (37) and second
λn is the related spectral strength [20] order corrections which do not seem to give a sizable
contribution.
λ =(A+1) dr n;ra Ψ 2 , (32) Neither the treatment nor the conclusions change if
n 0
|h | i| one considers the relativistic Hamiltonian H. Thus, for
a Z
X
5
the hadron tensor of Eq. (18) we use the approximated IV. ANTISYMMETRIZATION
expression obtained from Eq. (30) with the replacement
(37): For sake of simplicity the treatment of Sects. II and
III is based on the unsymmetrized projection operator
1
Wµµ(ω,q) = −π λnImhϕn |jµ†(q)Genff(Ef−εn) Pn defined in Eq. (14), leading to the Green function
of Eq. (23). In this Section we examine the draw-
n n
X G
jµ(q) ϕ . (38) backs of this formulation and the possible alternatives.
n
× | i On the mathematical ground, deserves the name of
n
G
Since the interference term Wµµ of Eq. (20) has no in- Greenfunctionsinceitfulfillsthesumrule(39),whichis
n
fluenceontheenergysumruleofthetotalhadrontensor a qualifying property. Moreover,and intimately related,
opfroExqim. a(t1io8n),laeandaintgurtaol qEuqe.scti(o3n8)armisaeys wchhaenthgeerththeesuapm- GL2n(Ris3)a.nHinevnecret,ibitlsesoeplfe-reanteorrgyonistnhoetwahffoelceteHdilbbyeratnsypuance-
rule. Actually,onecanobservethatinEq. (37) (E)is due restriction of domain and by the related mathemat-
n
modified by factors which change neither its prGoperties ical troubles.
of analyticity in the energy complex plane nor its high Notwithstanding, the optical potential n related to
V
energy behavior. This fact is used in Ref. [15] to prove n suffers from the drawback of having spurious eigen-
G
the relation functions. In fact H has both antisymmetrized and un-
symmetrized eigenvectors Ψ and the latter ones gen-
E
−π1 dE Imhra|Genff(E)|r′a′i ehraavteeneoigpehnyfusniccatliomnesahnnin;rga.|N|oΨitEoioloefxVisnts(Eto−mεank)ewahdiicsh-
Z
1 tinction inside this unphysical degeneracy. Besides, it is
= dE Im ra (E) r′a′
n apparentthat cannotbecomparedwithanyempirical
−π h |G | i n
= δ(r−Z r′)δaa′ . (39) opTtihcaelrmemodeedlyVpoisteanttiarel.atment based on projection op-
erators onto antisymmetrized states, although their in-
Therefore, the energy sum rule obtained from Eq. (38)
clusion into the hadron tensor is more laborious. Two
is exactly the same as in Eq. (22), i.e., the correct sum
approaches are available.
ruleofthe incoherenthadrontensor,apartfromthecon-
a) The first one (see Subsect. 3.2 of Ref. [16] and
tribution of the continuous channels.
Appendix B of Ref. [15]) uses the Feshbach projection
operator onto antisymmetrized states, i.e.,
C. Excited states of the residual nucleus
PnF = drdr′a†ra |ni
As neither microscopic nor empirical calculations are a,a′Z
X
eaxvaciitlaebdlestfaotretsheεopt,icaalcopmotmenotniaplrVanctaicsesorceilaatteedswthitehmthtoe × hra|(1−Kn)−1 |r′a′ihn|ar′a′ , (43)
n
thegroundstat|epoitentialV0 bymeansofanappropriate where a†ra creates a nucleon in the state |rai and Kn is
energy shift. Here, as in Ref. [15], we use the kinetic en- the one-body density matrix defined by
ergy prescription for the shifts (see Sect. 5 of Ref. [16]),
naturally suggested by the plane wave impulse approx- hra|Kn |r′a′i≡hn|a†r′a′ara |ni . (44)
imation. Such a prescription preserves the value of the
The results of the previous Section remain true with the
kinetic energy (including the rest mass), directly related
replacement
tothevalueoftheopticalpotentialvariableintheenergy
scale adopted here. Therefore we set
ra (E) r′a′ ra F(E) r′a′
h |Gn | i→h |Gn | i
Vn(E)≃V0(E) , (40) ≡ dshn|araE H +1ε +iηa†sb |ni
n
which implies Xb Z −
sb (1 K )−1 r′a′ , (45)
n
× h | − | i
(E) (E) . (41)
n 0
G ≃G where F is the Green function related to the “sym-
Gn
Using these approximations in Eq. (38), we write metrized” Feshbach optical potential F [22]. In this
Vn
approach the spurious degeneracy disappears, since one
1
Wµµ(ω,q) = λ Im ϕ jµ†(q) eff(E ε ) operates in a Hilbert space of antisymmetrized states,
−π n h n | G0 f− n but at the price of new drawbacks [16, 23], namely: (1)
n
× jµ(qX)|ϕni . (42) GriFnseitsoniontcofurrlleyctinDvyerstoinbleeqaunadtiosnos;in(2s)omboethcaseFs aitndgiveFs
Theseapproximationsdonotchangetheenergysumrule are not symmetric for exchange ra r′a′ anGdntherefoVrne
of Wµµ(ω,q). F is nonhermitian below the thres↔hold of the inelastic
Vn
6
processes; (3) the usual nonlocal models of potential are potential related to by the equations
G
nporonbloacbalyl sitnraudcteuqruea.te for VFn, which shows a complicate (E) = 1 , (49)
G E h(E)+iη
Inshort,theapproachbasedon F disguisesnontrivial −
Vn h(E) = α p+βM + (E) . (50)
mathematical problems and it is not really useful, since · V
this potential bears no close relation with the empirical The use of this Green function does not change the ex-
optical model potential. pressionsofthe normalizedinitialstates ϕ andofthe
n
| i
b) The second approach, where the above drawbacks relatedspectroscopicfactorsλn,definedinEqs. (31)and
disappear, is the one of Ref. [16]. It is based on the (32), respectively. Equivalently, they can be written as
extended projection operator of Ref. [24]
1
ra ϕn = λn−2 n ara Ψ0 , (51)
h | i h | | i
Pn(p+h) = dr a†ra+ara |ni λn = dr n ara Ψ0 2 . (52)
a Z |h | | i|
Xn a† +(cid:0)a , (cid:1) (46) Xa Z
× h | ra ra Duetothecomplexnatureof (E)thespectralrepresen-
V
(cid:0) (cid:1) tationof (E)involvesabiorthogonalexpansioninterms
which leads to of the eigGenfunctions of h(E) and h†(E). We consider
the incoming wave scattering solutions of the eigenvalue
ra (E) r′a′ ra (p+h)(E) r′a′
h |Gn | i→h |Gn | i equations, i.e.,
1
≡ hn|araE H +εn+iηa†r′a′ |ni E −h†(E) |χE(−)(E)i = 0 , (53)
−
+ hn|a†r′a′E+H 1ε iηara |ni . (47) (cid:0)(E −h(E)(cid:1))|χ˜E(−)(E)i = 0 . (54)
− n− Thechoiceofincomingwavesolutionsisnotstrictlynec-
essary,butitis convenientinordertohaveaclosercom-
(p+h)(E) is the full Green function, including particle
Gn parison with the treatment of the exclusive reactions,
and hole contributions. It fulfills the sum rule of Eq.
wherethefinalstatesfulfillthisasymptoticconditionand
(39),isfullyinvertibleandproducesmathematicallycor- are the eigenfunctions χ(−)(E) of h†(E).
rect Dyson equations. The related mean field (p+h)(E) | E i
Vn The eigenfunctions of Eqs. (53) and (54) satisfy the
has no spurious eigenfunctions corresponding to unsym-
biorthogonality condition
metrizedstatesanditspropertiesofnonlocalityandsym-
metrymakeitmoreeasilycomparablewiththeempirical χ(−)(E) χ˜(−)(E) =δ( ′) , (55)
h E | E′ i E −E
optical model potentials. Therefore, we understand that
and, in absence of bound eigenstates, the completeness
in the following equations will denote the full Green
Gn relation
function (p+h) of the relativistic Hamiltonian H. The
Gn ∞
associated mean field n is nonlocal as in the nonrel- d χ˜(−)(E) χ(−)(E) =1 , (56)
ativistic case and doesVnot conserve the primarily 4×4 ZM E | E ih E |
matrix structure of Vjj′. where the nucleon mass M is the threshold of the con-
tinuum of h(E).
Eqs. (55) and (56) are the mathematical basis for the
V. SPECTRAL REPRESENTATION OF THE biorthogonal expansions. The contribution of possible
HADRON TENSOR boundstatesolutionsofEqs. (53)and(54)canbe disre-
gardedinEq. (56)sincetheireffectonthehadrontensor
is negligible at the energy and momentum transfers con-
In this Section we consider the spectralrepresentation
sidered in this paper.
of the single particle Green function which allows prac-
Inserting Eq. (56) into Eq. (49) and using Eq. (54),
tical calculations of the hadron tensor of Eq. (42). In
one obtains the spectral representation
expanded form, it reads
∞
1 (E) = d χ˜(−)(E)
Wµµ(ω,q) = −π λnImhϕn |jµ†(q) 1−V′(E) G ZM E | E i
Xn q 1 χ(−)(E) . (57)
(E) 1 ′(E)jµ(q) ϕn , (48) × E−E +iηh E |
× G −V | i
Therefore, Eq. (48) can be written as
q
where E = Ef εn. Here and below, the lower index 1 ∞ 1
0 is omitted in−the Green functions and in the related Wµµ(ω,q) = Im d
−π EE ε +iη
quantities. According to the discussion of the previous n (cid:20)ZM f− n−E
X
Section, we understand that is the full particle-hole
Green function of Eq. (47) anGd that is the mean field × Tnµµ(E,Ef−εn) , (58)
V (cid:21)
7
where to experimental data, are local and involve scalar and
vector components only. The necessary replacement of
Tµµ( ,E) = λ ϕ jµ†(q) 1 ′(E) χ˜(−)(E) the mean field by the empirical optical model potential
n E nh n | −V | E i is, however, a delicate step.
q
χ(−)(E) 1 ′(E)jµ(q) ϕ . (59) In the nonrelativistic treatment of Refs. [15, 21] this
×h E | −V | ni
replacementis justified onthe basisofthe approximated
q
The limit for η +0, understood before the integral equation (holding for every state ψ )
→ | i
of Eq. (58), can be calculated exploiting the standard
symbolic relation Im ψ 1 ′(E) (E) 1 ′(E) ψ
h | −V G −V | i
1 1 q q
lim = iπδ(E ) , (60) Im ψ 1 ′ (E) (E) 1 ′ (E) ψ ,(62)
η→0E +iη P E − −E ≃ h | −VL GL −VL | i
−E (cid:18) −E(cid:19) q q
where denotes the principal value of the integral. where L(E)isthelocalphase-equivalentpotentialiden-
TherefoPre, Eq. (58) reads tified wVith the phenomenologicalopticalmodelpotential
and (E)istherelatedGreenfunction. InRef. [21]the
L
G
proof of Eq. (62) is based on two reasons: (1) a model
Wµµ(ω,q)= ReTµµ(E ε ,E ε ) of (E) commonly used in dispersion relation analyses;
n " n f− n f− n (2)Vthe combined effect of the factor 1 ′(E) and
X
−V
1 ∞ 1 of the Perey factor, which connects the eigenfunctions
−πPZM dEEf−εn−EImTnµµ(E,Ef−εn)# , (61) otofrV(E1) and′(EV)L,(Ein)t.rodWuecesdtrteossatchcaotunittpfiosrjuinsttetrhfeerefnacce-
−V
effects, which allows the replacement of (E) by (E).
whichseparatelyinvolvestherealandimaginarypartsof p V VL
Tµµ. Although the Perey effect is not sufficiently known for
n
Some remarks on Eqs. (59) and (61) are in order. Let theDiracequation,wehaveareasonableconfidencethat
us examine the expression of Tµµ( ,E) at = E = Eq. (62) holds also in the present relativistic context.
E ε for a fixed n. This is thne mEost impoErtant case Therefore, we insert Eq. (62) into Eq. (48). Then, all
f n
sinc−e it appears in the first term in the right hand side the developments of this Section can be repeated with
of Eq. (61), which gives the main contribution. Disre- the simple replacement of (E) by L(E).
V V
garding the square root correction, due to interference
effects, one observes that in Eq. (59) the second matrix
element(withtheinclusionof√λn)isthetransitionam- VI. RESULTS AND DISCUSSION
plitude forthe singlenucleonknockoutfromanucleusin
the state Ψ leaving the residual nucleus in the state
| 0i The cross sections and the response functions of the
n . The attenuation of its strength, mathematically
| i inclusive quasielastic electron scattering are calculated
due to the imaginary part of †, is related to the flux fromthesingleparticleexpressionofthecoherenthadron
V
lost towards the channels different from n. In the inclu-
tensor in Eq. (61). After the replacement of the
sive response this attenuation must be compensated by meanfield (E)bytheempiricalopticalmodelpotential
a corresponding gain due to the flux lost, towards the V
(E), the matrix elements of the nuclear current oper-
L
channel n, by the other final states asymptotically origi- Vator in Eq. (59), which represent the main ingredients
natedbythechannelsdifferentfromn. Inthedescription
of the calculation, are of the same kind as those giv-
provided by the spectral representation of Eq. (61), the ing the transitionamplitudes ofthe electroninduced nu-
compensation is performed by the first matrix element
cleonknockoutreactioninthe relativisticdistortedwave
in the right hand side of Eq. (59), where the imaginary impulse approximation (RDWIA) [25]. Thus, the same
partof hastheeffectofincreasingthestrength. Similar
treatment can be used which was successfully applied to
V
considerationscanbe made,onthe purely mathematical describe exclusive (e,e′p) and (γ,p) data [25, 26].
ground, for the integral of Eq. (61), where the ampli-
Thefinalwavefunctioniswrittenintermsofitsupper
tudes involved in Tµµ have no evident physical meaning
n component following the direct Pauli reduction scheme,
as = E ε . We want to stress here that in the
f n i.e.,
E 6 −
Green function approach it is just the imaginary part of
which accounts for the redistribution of the strength
Ψ
V f+
among different channels. χ(−)(E)= 1 , (63)
ThematrixelementsinEq. (59)containthemeanfield E M + +S†(E) V†(E)σ·pΨf+ !
(E) andits hermitianconjugate †(E), whichare non- E −
V V
local operator with a possibly complicated matrix struc- where S(E) and V(E) are the scalar and vector energy-
ture. Neither microscopic nor empirical calculations of dependent components of the relativistic optical poten-
(E) are available. In contrast, phenomenological opti- tial for a nucleon with energy E [27]. The upper com-
V
cal potentials are available. They are obtained from fits ponent, Ψ , is related to a Schr¨odinger equivalent wave
f+
8
function, Φ, by the Darwin factor, i.e.,
f
Ψ = D†(E)Φ , (64) 12C(e,e')
f+ E f
q S(E) V(E)
D (E) = 1+ − . (65)
E
M +
E
Φ isatwo-componentwavefunctionwhichissolutionof
f
aSchr¨odingerequationcontainingequivalentcentraland
spin-orbitpotentials obtainedfromthe scalarandvector
potentials [28, 29].
As no relativistic optical potentials are available for
the bound states, then the wave function ϕ is taken
n
asthe Dirac-HartreesolutionofarelativisticLagrangian
containing scalar and vector potentials [30, 31].
Concerning the nuclear current operator, no unam-
biguous approach exists for dealing with off-shell nucle-
ons. Inthepresentcalculationsweusethecc2expression
of the 1-body current [32–34]
κ
jµ = F (Q2)γµ+i F (Q2)σµνq , (66)
cc2 1 2M 2 ν
where κ is the anomalous part of the magnetic moment,
F and F are the Dirac and Pauli nucleon form factors,
1 2
which are taken from Ref. [35], and σµν =(i/2)[γµ,γν].
Current conservation is restored by replacing the lon-
gitudinal current by [33] FIG. 1: Longitudinal (upper panel) and transverse (lower
panel)responsefunctionsforthe12C(e,e′)reactionatq=400
ω
JL = Jz = J0 . (67) MeV/c. Solid and dotted lines represent the NLSH results
q with and without the inclusion of the factor in Eq. (68), re-
| |
spectively. Dashedlinesgivetheresultwithouttheintegralin
The calculations have been performed with the same
Eq. (61). Dot-dashedlines arethecontributionof integrated
bound state wave functions and optical potentials as in
single nucleon knockout only. The dataare from Ref. [36].
Refs. [25, 26], where the RDWIA was able to reproduce
(e,e′p) and (γ,p) data.
The relativistic bound state wave functions have been
A. The RL and RT response functions
obtained from the code of Ref. [30], where relativistic
Hartree-Bogoliubov equations are solved in the context
Thelongitudinalandtransverseresponsefunctions for
of a relativistic mean field theory to reproduce single-
12C at q = 400 MeV/c are displayed in Fig. 1 and com-
particle properties of severalspherical and deformed nu-
paredwiththe Saclaydata[36]. The lowenergytransfer
clei [31]. The scattering state is calculated by means of
values are not given because the relativistic optical po-
the energy- and mass number-dependent EDAD1 com-
tentials are not available at low energies.
plex phenomenological optical potential of Ref. [27],
The agreement with the data is generally satisfactory
whichisfittedtoprotonelasticscatteringdataonseveral
for the longitudinalresponse. Incontrast,the transverse
nuclei in an energy range up to 1040 MeV.
responseisunderestimated. Thisisasystematicresultof
In the calculations the residual nucleus states n are
| i the calculations. It may be attributed to physicaleffects
restricted to be single particle one-hole states in the tar-
which are not considered in the present approach, e.g.,
get. Apureshellmodelisassumedforthe nuclearstruc-
meson exchange currents.
ture, i.e., we take a unitary spectral strength for each
The effect of the integralin Eq. (61) is also displayed.
single particle state and the sum runs over all the occu-
Atvariancewiththe nonrelativisticresult,herethis con-
pied states.
tribution is important andessentialto reproduce the ex-
Theresultspresentedinthefollowingcontainthe con-
perimental longitudinal response.
tributions of both terms in Eq. (61). The calculation of
AsexplainedinSect. III,thecontributionarisingfrom
the second term, which requires integration over all the
interferencebetweendifferentchannels,seeEqs. (37)and
eigenfunctions of the continuum spectrum of the optical
(62), gives rise to the factor
potential, is a rather complicate task. This termwas ne-
glected in the nonrelativistic investigation of Ref. [15],
where its contribution was estimated to be very small. 1 ′ (E)= 1 βS′(E) V′(E) . (68)
−VL − −
In the present relativistic calculations the effect of this q
p
termcanbe significantanditisthereforeincludedinthe Wesee,however,thathereitgivesonlyaslightcontribu-
results. tion, due to a compensation between the energy deriva-
9
12C(e,e') 12C(e,e')
FIG. 2: Longitudinal (upper panel) and transverse (lower FIG. 3: The same as in Fig. 1, but for q =500 MeV/c. The
panel)responsefunctionsforthe12C(e,e′)reactionatq=400 data are from Ref. [36].
MeV/c. Solid lines represent the NLSH results, dashed lines
theNL3 results. Data as in Fig. 1.
channels. The role of the integral in Eq. (61) decreases
increasing the momentum transfer. At q = 500 MeV/c
tives S′(E) and V′(E), while in the nonrelativistic cal- its contribution is smaller than at q = 400 MeV/c, but
culation an overall reduction was observed, which was still important to reproduce the experimental longitudi-
necessary to reproduce the data [15]. nal response, while at q = 600 MeV/c the effect is neg-
Thecontributionfromalltheintegratedsinglenucleon ligible and the two curves with and without the integral
knockout channels is also drawn in Fig. 1. It is signif- overlap. The effect of the inelastic channels, indicated
icantly smaller than the complete calculation. The re- in the figures by the difference between the complete re-
duction, which is larger at lower values of ω, gives an sults and the contribution from all the integrated single
indication of the relevance of inelastic channels. nucleonknockoutchannels,isalwaysvisibleandevensiz-
For the calculations in Fig. 1 the Hartree-Bogoliubov able,butitdecreasesincreasingthemomentumtransfer.
equations for the single particle bound states have been The response functions for 40Ca at q = 450 MeV/c
solvedusing NLSH choices for the parametersof the rel- are shown in Fig. 5 and compared with the Saclay [37]
ativistic mean field theory Lagrangian. In Fig. 2 a com- and the MIT-Bates [3] data. The results obtained with
parison is shown between the results obtained with this the NLSH set ofparametershavebeenplotted, since the
choice of parameters and a different choice, i.e., NL3. results with other sets are almost equivalent. The calcu-
The shapes of the responses calculated with the differ- lated response functions are of the same order of mag-
ent bound states show small differences. Their integrals nitude as the MIT-Bates data, while for the Saclay data
must be unchanged, according to the fact that the sum thelongitudinalresponseisoverestimatedandthetrans-
rule has to be preserved. verse response underestimated. The factor in Eq. (68)
Thelongitudinalandtransverseresponsefunctions for produces and enhancement which is minimal but visible
12Catq =500andq =600MeV/caredisplayedinFigs. in the figure.
3and4,respectively,andcomparedwiththeSaclaydata
[36]. The bound state wave function have been obtained
withtheNLSHparametrization. AsalreadyfoundinFig. B. The inclusive cross section
1 at q = 400 MeV/c, a good agreement with the data
is obtained in both cases for the longitudinal response, Investigation of inclusive electron scattering in the re-
while the transverse response is underestimated. Also in gion of large q is of great interest to provide informa-
Figs. 3 and4only a slighteffect is givenby the factorin tion on the nuclear wave functions and excitation and
Eq. (68) arising from the interference between different decay of nucleon resonances. Several experiments have
10
12C(e,e') 40Ca(e,e')
FIG. 4: The same as in Fig. 1, but for q =600 MeV/c. The FIG. 5: Longitudinal (upper panel) and transverse (lower
data are from Ref. [36]. panel) response functions for the 40Ca(e,e′) reaction at q =
450 MeV/c. The Saclay data (open circles) are from Ref.
[37], the MIT-Bates (black circles) are from Ref. [3]. Line
convention as in Fig. 1.
been carried out to explore this region. The separa-
tion of the longitudinal and transverse components of
the nuclear response would be very interesting, but it is
a scattering angle of 32o. The NLSH wave functions
very difficult to performbecause of the decreasingof the
≃
have been used in the calculations. The agreement with
longitudinal-transverse ratio with increasing q. Precise
dataisgoodinalltheconsideredsituations. Theintegral
measurements over a kinematical range that would al-
in Eq. (61) produces a reduction which is now essential
low longitudinal-transverse separation for several nuclei
to reproduce the data at 700 MeV, which correspond to
are howeverplanned in the future at JLab, where the E-
a momentum transfer of . 400 MeV/c. Its contribution
01-016approvedexperiment[4]will make a precise mea-
can be neglected in the other kinematics, where q 600
surementinthemomentumtransferrange0.55 q 1.0
≃
≤ ≤ MeV/c. TheeffectofthefactorinEq. (68)isverysmall.
GeV/c in order to extract the response functions.
In this Subsection we focus our attention on experi-
mentalcrosssectionswithω .300MeV,sinceourmodel
doesnotinclude mesonexchangecurrentsandisobarex- VII. SUMMARY AND CONCLUSIONS
citation contributions.
The calculated inclusive 12C(e,e′) cross section is dis- A relativistic approachto inclusive electron scattering
played in Fig. 6 in comparison with the SLAC data [38] in the quasielastic region has been presented. This work
in a kinematics with a beam energy Ee = 2020 MeV can be considered as an extension of the nonrelativistic
and a scattering angle of 15o. The bound state wave many-body approach of Ref. [15]. The components of
≃
function has been obtained with the NLSH set. A visi- thehadrontensorarewrittenintermsofGreenfunctions
ble enhancement is produced by the factor in Eq. (68). of the optical potentials related to the various reaction
The effect of the integral in Eq. (61) gives a significant channels. The projection operator formalism is used to
reduction whichunderestimates the data. As in the case derive this result. An explicit calculation of the single
of the transverseresponse of Figs. 1 - 5, the discrepancy particle Green function can be avoided by means of its
might be due to two-body mechanisms which are here spectral representation, based on a biorthogonal expan-
neglected. sion in terms of the eigenfunctions of the nonhermitian
In order to extend our analysis to different kinematics optical potential (E) and of its hermitian conjugate.
and target nuclei, we consider in Fig. 7 the 16O(e,e′) The interference bVetween different channels is takeninto
inclusive cross section data taken at ADONE-Frascati accountby the factor 1 ′(E), which also allowsthe
−V
[2]withbeamenergyrangingfrom700to1200MeVand replacement of the mean field (E) by the phenomeno-
p V