Superscaling analysis of quasielastic electron scattering with relativistic effective mass

0.57 MB·

by  J. E. Amaro

Superscaling analysis of quasielastic electron scattering with relativistic effective mass
J.E. Amaro,1,∗ E. Ruiz Arriola,1,† and I. Ruiz Simo1,‡
1Departamento de F´ısica At´omica, Molecular y Nuclear
and Instituto Carlos I de F´ısica Te´orica y Computacional
Universidad de Granada, E-18071 Granada, Spain.
(Dated: January 20, 2017)
We provide a parametrization of a new phenomenological scaling function obtained from a chi-
squarefittoaselectedsetof(e,e’)crosssectiondataexpandingabandcenteredaroundthequasielas-
tic peak. We start from a re-analysis of quasielastic electron scattering from nuclear matter within
therelativistic mean fieldmodel. Thecrosssection dependsontherelativistic effectivemassof the
7 nucleon, m∗N, and it scales with respect to a new scaling variable, ψ∗. This suggests a new super-
1 scaling approach with effective mass (SuSAM*) for predicting quasielastic cross sections within an
0 uncertainty band. The model reproduces previously established results on the enhancement of the
2 transverse response function as compared to thetraditional relativistic Fermi gas.

n
a
J INTRODUCTION (e,e′) cross section is that it is the result of contribu-

9 tions from many unseen processes and interferences that
1 Thedescriptionofquasielasticelectronscatteringcross cannotbe disentangledeasily. Fromsimplistic viewpoint
section is still an open problem in theoretical nuclear the cross section can be regarded as the sum of nucleon
]
h physics. The recent neutrino experiments with accelera- knockout plus multinucleon knockout plus pion-nucleon
t tors have emphasized the importance of a global precise emission plus additional inelastic processes in the deep
-
l descriptionofleptonscatteringfromnucleiatintermedi- region. Both1p-1hand1π1p-1harecontaminatedby2p-
c
u ate energies [1–5]. Although a great deal of progress has 2h and cannot be unambiguously separated from it [15–
n been achieved with models based on first principles de- 17]. From the theoretical point of view the one-nucleon
[ scriptionofthenuclearsystem[6],alternativeapproaches knockout process generates the quasielastic peak. But

1 basedonthespectralfunction[7,8]ortheshellmodel[9], experimentally this peak can only be isolated from the
v to mention some recent studies, have been put forward. data of the longitudinal response function for moderate
7 As a general rule, the models cannot provide yet a com- momentum transfer where meson-exchangecurrents and
1 pletedescriptionofthewholesetof(e,e′)dataatthefull pion emission are predominantly transverse [18, 19].
4
rangeofkinematicsneeded,speciallyforhighmomentum Scalingideashavepromptedstudieswhicharepromis-
5
0 and energy transfers, where a relativistic description be- ing phenomenologicalalternativescomplementary to the
. comesmandatory[10,11]. Nottomentionthatthereare theoretical microscopic models [20]. The superscaling
1
still basic issues like gauge invariance that are not easy approach (SuSA) [18, 19, 21, 22] exploited the scaling
0
7 to control and generate well known ambiguities. properties of the reduced longitudinal response function
1 In addition to the relativistic corrections in the kine- (divided by the corresponding single-nucleon structure
v: matics and in the current operator, the importance of function) when plotted against an appropriate scaling
i relativistic corrections stemming from the dynamics has variableψ′. Thisallowedtoimplementaphenomenologi-
X been emphasized from a fully relativistic mean field cal- callongitudinalscalingfunction fL(ψ′)fitted to electron
r culation [12]. In those studies the combination of scalar data. This scaling function embodies implicitly all the
a
and vector relativistic potentials naturally produce an genuinely quasielastic nuclear physics processes. Thus
enhancement of the lower components of relativistic nu- any model aiming to describe the quasielastic reaction
cleon wave functions [13] and correspondingly an en- should be able to describe fL.
hancement of the transverse response function. This is Alltheprocessesviolatingscalingcontributemainlyto
a genuine relativistic dynamical effect, and it goes away, the transverse response. Within the SuSA approach the
forexample,afterasemirelativisticapproximationwhere “quasielastic” part of the transverse response function
thelowercomponentsareneglectedorprojectedout[11]. was computed by assuming the same scaling properties
It is known that the transverse cross section is larger as the longitudinal response, and that a transverse (un-
than the predictions of the independent nucleon model. known) scaling function fT exists. In the original SuSA
Whilethetransverseenhancementhasprimarilybeenat- approach it was simply assumed that fT = fL [22] and
tributed to multi-nucleon processes via meson-exchange this allowed to construct a manageable model to predict
currents and ∆ excitation [14], it should be noted that neutrino cross sections from the (e,e′) data. Although
this enhancement can also be regarded partly as due to such an assumption was not based on data, semirela-
relativity. tivistic models like that of ref. [11] give in fact fT ≃fL.
A major difficulty in the description of the inclusive The relativistic mean field (RMF) framework to finite
2
nuclei reproduces the experimental f function rather the vector and scalar potentials generate an effective
L
well and therefore this model can be used to test the mass m∗ for the nucleon in the medium. Our present
N
validity of the f =f assumptionunder relativistic dy- approach, called SuSAM* (super scaling approach with
T L
namics. This model is based on the Dirac-Hartree ap- M∗), enjoys the good features of the RMF in nuclear
proximation of ref. [23]. In [12] it was actually found matter. It keeps gauge invariance (that SuSA violates
that in the RMF model f > f ; the precise value of becauseitintroducesanenergyshifttoaccountforsepa-
T L
f depends on the treatment of the off-shell ambiguities rationenergy)anddescribesthedynamicalenhancement
T
of the current operator. Under the CC2 prescription of of both the lower components of the relativistic spinors
de Forest [24], f is about 20% larger than f , while it and the transverse response function.
T L
is almost twice as large for CC1 [12]. The CC2 results In ref. [26] the effective mass was randomly modified
for the (e,e′) cross section seem more reasonable, and aroundthe meanvalue 0.8±0.1,approximatelysimulat-
so this was the prescription used in the recent upgrade ing the dispersion band of the real data. In the present
SuSA-v2 [25]. This new model includes nuclear effects workweinsteadfitaselectionofexperimentaldatawhich
which are theoretically-inspired by the RMF by using a are considered “true” quasielastic based on a data den-
transversescalingfunctionf whichisdifferentfromf . sity criterium. Our main goal is to provide a simple fit
T L
and that also has an additional dependence on the mo- ofthenewphenomenologicalψ∗-scalingfunction, f∗(ψ∗)
mentumtransferq. ThereforetheSuSA-v2resultsdonot as the sum of two Gaussian functions, and similar fits
scaleanymore,althoughthemodelconservesthe’scaling’ for the dispersion band as well. This simple formula al-
name by tradition. lowstopredictthequasielasticcrosssectionforarbitrary
In this work we explore a new scaling approach to kinematics together with a uncertainty band, providing
describe, in a wide kinematical range, the (e,e′) data the maximum information from the scaling properties of
by minimizing undesirable contaminationsfrominelastic the available (e,e′) data, with few parameters. The un-
scattering or other effects beyond the quasielastic con- certainty bands describe about 1000 “quasielastic” data
ditions. We proceed by exploiting the scaling formal- outofthe∼2500existingdatafor12C.Thedatathatlie
ism and, at the same time, the proved good properties outside the uncertainty band, are those generated by in-
of the relativistic mean field, which already includes by elastic processes or low energy nuclear effects that break
construction the transverse response enhancement. Our scaling explicitly.
goal is to return to the description of the quasielastic
peak with only one phenomenological scaling function,
f∗(ψ∗), by fitting a subset of conveniently selected data. FORMALISM
Thus, we expect that the double differential cross sec-
tion improves over the SuSA model [18, 25]. We remind We follow the notation introduced in Ref. [26]. We
that the description of the SuSA model is not fully sat- assume that an incident electron scatters off a nucleus
isfactorybecauseonlythelongitudinalresponsefunction transferring momentum q and energy ω
was fitted and not the cross section. The SuSAv2 was The quasielastic cross section is
animprovementbyusing atheoreticaltransversescaling
dσ
function coming fromthe RMF but atthe costofviolat- =σ {v R +v R } (3)
dΩ′dǫ′ Mott L L T T
ing scaling [25]. In our approach, however, we maintain
the scaling with respect to a new scaling variable ψ∗ to where σ is the Mott cross section, and θ the scatter-
Mott
be defined shortly. ing angle. R (q,ω) and R (q,ω) are the nuclear longi-
L T
In a previous work [26] we started exploring the ψ∗ tudinal and transverse response functions, respectively.
scalingideainthecontextoftheRMFfornuclearmatter. The four-momentum transfer is Q2 = ω2−q2 < 0. Fi-
In that study we obtained the best value of the effective nally the kinematical factors v ,v are defined by
L T
mass
Q4
m∗ v = (4)
M∗ = N =0.8. (1) L q4
m
N θ Q2
v = tan2 − . (5)
This value providesthe bestscalingbehaviorofthe data T 2 2q2
with a large fraction of data concentrated around the
universal scaling function of the relativistic Fermi gas
RMF in nuclear matter
3
f (ψ∗)= (1−ψ∗2)θ(1−ψ∗2) (2)
RFG
4
The SuSAM* model is inspired by the RMF in nu-
The ψ∗ variable was inspired by the mean field theory, clearmatterwhichwesummarizehere. Inthis modelwe
that provides a reasonable description of the quasielas- consider single-nucleon excitations with initial nucleon
tic response function [27, 28]. In the interacting RFG energy E = p2+m∗ 2 in the mean field. The final
N
p
3
momentumofthe nucleonis p′ =p+qandits energyis in the medium due to the effective mass according to
E′ = p′2+m∗ 2. Note that initial and final nucleons
khFav=et2ph2e5sMameeVe/ffcNefcotriv1e2Cm.aTsshemn∗Nu.clTeahrerFeesprmonismeofumnecnttiounms G∗E = F1−τmmN∗NF2 (19)
can be written in the factorized form for K =L,T m∗
G∗ = F + NF . (20)
M 1 m 2
N
R =r f∗(ψ∗), (6)
K K
We use here the successful CC2 prescription of the elec-
whererL andrT arethesingle-nucleoncontribution,tak- tromagnetic nucleon current, that reproduces the exper-
ing into account the Fermi motion imental superscaling function [12]. Using the CC1 op-
erator obtained through the Gordon reduction produces
ξ
r = F (ZUp +NUn) (7) the same effects as in the RMF of ref. [12]. The same
K m∗ η3κ K K
N F modificationofformfactorsinthe mediumwasexplored
and f∗(ψ∗) is the scaling function, given by Eq. (2). It in ref. [29], but the ψ∗-scaling was not investigated in
depends only on the new scaling variable ψ∗, that is the that context. For the free Dirac and Pauli form factors,
F and F , we use the Galster parametrization.
minimumkinetic energyofthe initialnucleondividedby 1 2
Note that our formalism generalizes the conventional
the kinetic Fermi energy. The minimum energy allowed
SuSA formulae. In fact for M∗ =1 we recoverthe SuSA
for a nucleon inside the nucleus to absorb the virtual
photon (in units of m∗ ) is results and definitions.
N
The factorization of Eq. (6) implies that the cross
1 section in the RMF for nuclear matter also factorizes as
ǫ0 =Max κ 1+ −λ,ǫF −2λ , (8) theproductofthescalingfunctiontimesasinglenucleon
τ
( r )
cross section. In this approach the scaling function is a
where we have introduced the dimensionless variables parabola.
λ = ω/2m∗ , (9)
N
The SuSAM*
κ = q/2m∗ , (10)
N
τ = κ2−λ2, (11)
IntheSuSAM*approachwecomputethecrosssection
η = k /m∗ , (12)
F F N by using the above formulas, but replacing the scaling
ξ = 1+η2 −1, (13) function(2)byaphenomenologicalonewhichweextract
F F
from the experimental data.
q
ǫ = 1+η2, (14) We start with the more than 2500 experimental (e,e′)
F F
cross section data for 12C [30]. For every kinematical
q
Usually [20] these variables are defined dividing by the pointwecomputethecorrespondingexperimentalscaling
nucleon mass m instead of m∗ . The definition of the function f∗
N N exp
scaling variable is
dσ
ǫ −1 f∗ = dΩ′dǫ′ exp (21)
ψ∗ = ǫ0 −1sgn(λ−τ) (15) exp σMott(cid:0)(vLrL(cid:1)+vTrT)
r F
ψ∗ is negativeto the left ofthe quasielasticpeak (λ<τ) Inref. [26]weperformedananalysisofthe experimental
scaling function for the bulk of data [30, 31] by plotting
and positive on the right side.
them against the scaling variable ψ∗. A large fraction of
The single nucleon response functions are
the data then collapses into a cloud with an asymmet-
κ2 (G∗)2+τ(G∗ )2 rical shape as seen in Fig. 1. The cloud of data forms
U = (G∗)2+ E M ∆ (16)
L τ E 1+τ a thick band. The selection of data was made in ref.
(cid:20) (cid:21)
[26]by measuringthe densityofpoints clusteredabovea
(G∗)2+τ(G∗ )2
U = 2τ(G∗ )2+ E M ∆ (17) given threshold n, inside a circle of radius r = 0.1. The
T M 1+τ
selection of data depends on the chosen value of n. In
where the quantity ∆ has been introduced this work we use n = 25, meaning that we neglect all
the points with less than 25 neighbours inside a circle of
τ 1 ξ radius r = 0.1. The number of surviving data is around
∆= ξ (1−ψ∗2) κ 1+ + F(1−ψ∗2) . (18)
κ2 F τ 3 1000. Thethicknessofthedatacloudmeasuresthesmall
" r #
degree of scaling violation around the quasielastic peak.
Oneofthe consequencesofthepresentRMFapproachis The neglected data correspond to inelastic excitations
thatthe electricandmagneticformfactors,aremodified and low energy processes that highly violate scaling and
4
1 16
exp SuSAM
fit 14 q = 500 MeV/c M∗ = 0.8
0.8 min RFG
max 1 12
−
]
0.6 V 10
∗) e
ψ G 8
∗( [
f 0.4 )
ω 6
(
L
0.2 R 4
2
0
0
-3 -2 -1 0 1 2 3
100 200 300 400
ψ∗
6
q = 700 MeV/c
2 5
1
−
] 4
1.5 V
e
G
3
[
)
∗ )
ψ 1 ω
∗f( LR( 2
1
0.5
0
200 400 600
0
-3 -2 -1 0 1 2 3 2
ψ∗ q = 1 GeV/c
1 1.5
−
FIG. 1: Top panel: Phenomenological M∗-scaling function V]
f∗(ψ∗) and itsuncertaintyband,fm∗in <f∗ <fm∗ax, for 12C, Ge
1
obtainedfrom afittotheexperimentaldata. Onlydatawith [
density n ≥ 25 inside a circle with radius r = 0.1 have been ω)
included in the fit. Bottom panel: comparison of the present L(
fit to thebulk set of world data. Dataare from ref. [30–32] R 0.5
a1 a2 a3 b1 b2 b3
0
central -0.0465 0.469 0.633 0.707 1.073 0.202
200 400 600 800
min -0.0270 0.442 0.598 0.967 0.705 0.149
0.8
max -0.0779 0.561 0.760 0.965 1.279 0.200
q = 1.3 GeV/c
TABLEI:Parametersofourfitofthephenomenologicalscal-
ingfunctioncentralvalue,f∗(ψ∗),andofthelowerandupper 1 0.6
−
boundaries (min and max, respectively). V]
e
G
0.4
[
cannot be considered quasielastic processes, as can be ω)
seen in the lower panel of Fig. 1. L(
R 0.2
In Fig. 1 we showalso our new parametrizationofthe
scaling function after a fit to the selected (quasielastic)
experimental data. They are well described as a sum of
0
two Gaussian functions 200 400 600 800 1000 1200
f∗(ψ∗)=a e−(ψ∗−a1)2/(2a22)+b e−(ψ∗−b1)2/(2b22) (22) ω [MeV]
3 3
The coefficients are given in table 1. FIG.2: Longitudinalresponsefunctionof12CintheSuSAM
The lower and upper limits of the experimental data model, for several values of the momentum transfer. The
relativisticFermigasresultsforeffectivemassM∗ =1and0.8
band have also been parametrized as sum of two Gaus-
are also shown. The Fermi momentum is kF =225 MeV/c.
5
1
SuSAM SuSAM
20 q = 500 MeV/c ∗ q = 0.5 GeV/c ∗
M = 0.8 M = 0.8
RFG 0.8 RFG
1
− 15
]
V
0.6
e )
G ψ
ω)[ 10 f(L 0.4
(
T
R 5
0.2
0 0
100 200 300 400 -2 -1 0 1 2 3
12 1
q = 700 MeV/c
10
0.8 q = 700 MeV/c
1
−
] 8
V
0.6
e )
G ψ
6
[ (
ω) fL 0.4
( 4
T
R
0.2
2
0 0
200 400 600 -2 -1 0 1 2 3
1
q = 1 GeV/c
6
0.8 q = 1 GeV/c
−1 5
]
V
0.6
e 4 )
G ψ
[ (
ω) 3 fL 0.4
(
T 2
R
0.2
1
0 0
200 400 600 800 -2 -1 0 1 2 3
3 1
q = 1.3 GeV/c
0.8 q = 1.3 GeV/c
1
−
] 2
V
0.6
e )
G ψ
[ (
ω) fL 0.4
( 1
T
R
0.2
0 0
200 400 600 800 1000 1200 -2 -1 0 1 2 3
ω [MeV] ψ
FIG. 3: Transverse response function of 12C in the SuSAM FIG. 4: Traditional longitudinal scaling function fL(ψ) of
model, for several values of the momentum transfer. The 12CintheSuSAMmodel,forseveralvaluesofthemomentum
relativisticFermigasresultsforeffectivemassM∗ =1and0.8 transfer. The relativistic Fermi gas results for effective mass
are also shown. The Fermi momentum is kF =225 MeV/c. M∗ = 1 and 0.8 are also shown. The Fermi momentum is
kF =225 MeV/c.
6
1 0.8
SuSAM RFG
q = 0.5 GeV/c ∗ q=500MeV/c
M = 0.8 0.7
q=700MeV/c
0.8 RFG q=1000MeV/c
0.6 q=1300MeV/c
0.6 0.5
)
ψ )
( ψ 0.4
T (
f 0.4 fL
0.3
0.2 0.2
0.1
0
-2 -1 0 1 2 3 0
1 -2 -1 0 1 2 3
0.8
0.8 q = 700 MeV/c 0.7
0.6
0.6
)
ψ 0.5
(
fT 0.4 ψ() 0.4
T
f
0.3
0.2
0.2
0 0.1
-2 -1 0 1 2 3
1 0
-2 -1 0 1 2 3
ψ
0.8 q = 1 GeV/c
FIG. 6: Traditional scaling analysis of the longitudinal and
0.6 transverse scaling functions of 12C in the SuSAM model, for
)
ψ severalvaluesofthemomentumtransfer. Theuniversalrela-
(
T tivistic Fermi gas scaling function is also shown for compari-
f 0.4
son
0.2
sians,withcoefficientsamin/max,bmin/max,givenintable
i i
1 as well.
0
-2 -1 0 1 2 3 Our phenomenological scaling function describes the
1 center values of the selected data cloud and the aver-
age thickness. The thickness can be interpreted as a
0.8 q = 1.3 GeV/c fluctuation produced by nuclear effects beyond the im-
pulse approximation (finite size effects, short-range NN-
correlations, long-range RPA, meson-exchange currents,
0.6
ψ) virtual∆excitation,two-particleemission,finalstatein-
( teraction) [26].
T
f 0.4
0.2 RESULTS
0 In this section we use the new phenomenological scal-
-2 -1 0 1 2 3
ing function ofthe SuSAM* model, givenby eq. (22), to
ψ compute the response functions and cross section of 12C
andthecorrespondinguncertaintybandderivedfromthe
FIG.5: TraditionaltransversescalingfunctionfT(ψ)of12C thickness of the fitted data set.
in the SuSAM model, for several values of the momentum Infigs2and3weshowthelongitudinalandtransverse
transfer. The relativistic Fermi gas results for effective mass
response functions for several values of the momentum
M∗ = 1 and 0.8 are also shown. The Fermi momentum is
kF =225 MeV/c.
7
0.7 ground state, which are thought to be caused by finite
size effects and nuclear correlations [34, 35]. Another
0.6
smaller tail appears for low energy also produced by nu-
0.5 clear effects.
The structureofthe highenergytailisdifferent inthe
) 0.4 longitudinal and transverse responses. The longitudinal
ψ
f(L 0.3 response presents a prominent shoulder, which is not so
prominentinthetransverseone,althoughbothresponses
0.2 have been calculated with the same phenomenological
scalingfunction. Giventhatthescalingfunctiondoesnot
0.1
present a similar shoulder, one can trace back its origin
tothe energydependence ofthe singlenucleonresponses
0
-2 -1 0 1 2 3 inthemedium,thathavebeencalculatedwithM∗ =0.8,
0.8 see Eqs. (7–20).
In comparing the ratio between R and R from figs.
0.7 T L
2and3onecanalsoobserveanenhancementforthecase
0.6 M∗ = 0.8. This is related with the known enhancement
0.5 of the lower components in the relativistic spinor in the
) nuclear medium.
ψ 0.4
( InFigs4and5weshowthelongitudinalandtransverse
T
f
0.3 scalingfunctionsf (ψ)andf (ψ),obtainedformthere-
L T
sponse of Fig.s 2, 3, by dividing by the single nucleon
0.2
responses with effective mass M∗ = 1. These can be
0.1 directly compared to the traditional SuSA scaling func-
tions. They are plotted against the original scaling vari-
0
-2 -1 0 1 2 3 able ψ for different values of q. It is apparent that they
ψ′ are almost independent on q. They are compared to the
scalingfunctionofthefreeRFG.Thef (ψ)scalingfunc-
L
tionissmallerthanthefreeoneasinthecaseofthephe-
FIG. 7: Traditional scaling analysis of the longitudinal and
transverse scaling functions of 12C in the SuSAM model, for nomenologicalSuSA scalingfunction. The fT(ψ) scaling
severalvaluesofthemomentumtransfer. Theyareplottedas function is larger and almost of the same size as the free
a function of the shifted variable ψ′ so that the maximum in one, being this again a consequence of the enhancement
each curveis reached at ψ′ =0. Forcomparison we also plot of the transverse response function.
the data of the longitudinal scaling function obtained from
Theψ-scalingpropertiesofthe SuSAM* modelcanbe
theexperimental RL(q,ω) datain ref. [21].
better observed in fig. 6. The scaling is only approx-
imate as it happens with the RMF model. There is a
shift with respect to the RFG parabola, which increases
transfer. They are comparedto the free RFG andto the slightlywiththemomentumtransfer. Thisincreaseisap-
interacting RFG with M∗ =0.8. The effect of the effec- proximately linear. The enhancement of the transverse
tive mass is a shift of the responses to higher energies, response also increases with q (bottom panel of Fig. 6).
because the position of the quasielastic peak is given by In fig. 7 we compare the ψ-scaling function of our
ω = q2+m∗N2 −m∗N. Note that this shift gives the model with the experimental data of the longitudinal
correct position of the quasielastic peak without need of scalingfunctionf (ψ)obtainedinref. [21]. Thelongitu-
p L
introducing a separation energy parameter [27, 33]. In dinal scaling function is well reproduced, except a slight
fact the position of the peak with the SuSAM* model disagreement for low ψ′, because we fit the cross sec-
almost coincides with the RFG with M∗ =0.8. tion and not R . The transverse scaling function in our
L
Theintroductionofthephenomenologicalscalingfunc- model is evidently larger than the experimental f (ψ′)
L
tion produces a prominent tail for high energy transfer, in an amount about 20 %, similar to the RMF results of
whichextendsmuchhighthatthe upperendoftheRFG [12].
responses, which is more in accordance with the experi- In figs. 8–13 we show the predictions of our model
mentaldata. Thishighendisaconsequenceofthemaxi- forthe(e,e′)crosssectioncomparedtotheexperimental
mummomentumforthenucleonsintheFermigas,bound data. Ourglobaldescriptionisquiteacceptablegiventhe
bytheFermimomentum,correspondingtoψ∗ =1. Since few parameters of the SuSAM* model. A large fraction
the data in Fig.1 extends above ψ∗ = 1, this indicates ofthedatafallinsideouruncertaintyband. Infact,most
the presence of finalstate interactionsand alsohigh mo- ofthe data usedto performthe fit, anddisplayedinFig.
mentum components in the nuclear wave function in the 1 (top) are inside our prediction bands by construction.
8
ǫ=160MeV, θ=36◦ ǫ=161MeV, θ=60◦ ǫ=200MeV, θ=36◦
2500 250 1400
1200
2000 200
1000
1500 150 800
1000 100 600
400
500 50
200
0 0 0
10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
ǫ=200MeV, θ=60◦ ǫ=240MeV, θ=36◦ ǫ=240MeV, θ=60◦
180 900 100
V] 160 800
e 140 700 80
M
120 600
/
nb 100 500 60
[
Ω 80 400 40
d 60 300
ω
/d 40 200 20
dσ 20 100
0 0 0
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 0 20 40 60 80 100120
ǫ=280MeV, θ=60◦ ǫ=320MeV, θ=36◦ ǫ=320MeV, θ=60◦
60 450 40
400 35
50
350 30
40 300
25
250
30 20
200
15
20 150
100 10
10
50 5
0 0 0
0 50 100 0 25 50 75 100 0 50 100 150 200
ω [MeV] ω [MeV] ω [MeV]
FIG.8: SuSAMpredictionsanduncertaintybandsforthequasielastic (e,e′)cross section for severalkinematicscompared to
theexperimental data.
The data that lie outside our prediction bands are those assigned to the phenomenological scaling function from
clearly in the inelastic or deep region and those corre- the dispersion of the data set around the central value.
sponding to low excitation energy, and therefore break
With this scaling function we have calculated the lon-
ψ∗-scaling because they fall outside the quasielastic re-
gitudinaland transverseresponsefunctions and the con-
gion defined in fig. 1 (top).
ventional scaling functions of the SuSA model. Our
model contains the enhancement of the transverse com-
ponents of the electromagnetic current by construction.
CONCLUSIONS This confirms the RMF interpretation of the enhance-
ment of the transverseresponse in terms ofthe relativis-
tic modification of the lower components of the nucleon
In this paper we have investigated a novel scaling ap-
proachbasedonanewscalingvariableψ∗ extractedfrom spinorsinthe medium, thatwe encodewith the effective
nucleon mass reduction of M∗ =0.8.
the scalingpropertiesofthe RMFmodelin nuclearmat-
ter. Within this model we have obtained a phenomeno- Finally we have computed the differential quasielas-
logical scaling function f∗(ψ∗) from the inclusive (e,e’) tic cross section with an uncertainty band generated by
reaction data off 12C, after a selection procedure of the thescalingfunctionthicknessandcomparedtotheworld
quasielastic subset of data. This new phenomenological 12C(e,e′) data, with a reasonable description of around
scalingfunctionhasbeenparametrizedasthesumoftwo one thousand data, which thus can be tagged as truly
Gaussians. Additionally, an uncertainty band has been quasielastic.
9
ǫ=361MeV, θ=60◦ ǫ=400MeV, θ=36◦ ǫ=401MeV, θ=60◦
25 250 18
16
20 200 14
12
15 150
10
8
10 100
6
5 50 4
2
0 0 0
0 50 100 150 200 250 0 50 100 150 200 0 50 100150200250300
ǫ=440MeV, θ=60◦ ǫ=480MeV, θ=36◦ ǫ=480MeV, θ=60◦
14 140 10
9
V] 12 120
8
e
M 10 100 7
/
b 8 80 6
n
[ 5
Ω 6 60 4
d
ω 4 40 3
d
/ 2
σ 2 20
d 1
0 0 0
0 50 100150200250300 0 100 200 300 0 100 200 300
ǫ=500MeV, θ=60◦ ǫ=519MeV, θ=60◦ ǫ=560MeV, θ=36◦
9 8 80
8 7 70
7 6 60
6
5 50
5
4 40
4
3 30
3
2 2 20
1 1 10
0 0 0
0 100 200 300 400 0 100 200 300 400 0 100 200 300 400
ω [MeV] ω [MeV] ω [MeV]
FIG.9: SuSAMpredictionsanduncertaintybandsforthequasielastic (e,e′)cross section for severalkinematicscompared to
theexperimental data.
Thismodelpredictsaquasielasticcrosssectiondirectly ACKNOWLEDGEMENTS
from the data without any theoretical assumption, be-
sidestherequirementsofgaugeinvariance,relativityand
scaling, which determines the values of the relativistic This work is supported by Spanish DGI (grant
effective mass and the Fermi momentum. The rest of FIS2014-59386-P) and Junta de Andalucia (grant
nuclear effects contributing to the quasielastic reaction, FQM225). I.R.S. acknowledges support from the Min-
havebeenencodedintotheparametrizedscalingfunction isterio de Economia y Competitividad (grant Juan de la
f∗(ψ∗) within an uncertainty band. Any model aiming Cierva-Incorporacion).
to describe the quasielastic cross section at intermediate
energiesshould lie inside the SuSAM* uncertaintyband.
Therefore,our scaling function parametrizationprovides
a novel test for theoretical scaling studies. This imposes
∗ Electronic address: [email protected]
constraints over the transverse enhancement, additional
† Electronic address: [email protected]
to those imposed by the longitudinal scaling function in
‡ Electronic address: [email protected]
the SuSA model. In line with the currentrevival of elec-
[1] V Lyubushkin et al. (NOMAD Collaboration), Eur.
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[3] A. Aguilar-Arevalo et al. (MiniBooNE Collaboration),
10
ǫ=560MeV, θ=60◦ ǫ=560MeV, θ=145◦ ǫ=620MeV, θ=36◦
6 60
5 0.4 50
4 0.3 40
3 30
0.2
2 20
0.1
1 10
0 0 0
0 100 200 300 400 0 100 200 300 400 500 0 100 200 300 400 500
ǫ=620MeV, θ=60◦ ǫ=680MeV, θ=36◦ ǫ=680MeV, θ=60◦
4 40 3
V] 3.5 35 2.5
Me 3 30
2
/ 2.5 25
b
n
[ 2 20 1.5
Ω
d 1.5 15 1
ω
d 1 10
σ/ 0.5 5 0.5
d
0 0 0
0 100 200 300 400 500 0 100200300400500600 0 100200300400500600
ǫ=730MeV, θ=37◦ ǫ=961MeV, θ=37◦ ǫ=1108MeV, θ=37.5◦
25 9 4.5
8 4
20 7 3.5
6 3
15
5 2.5
4 2
10
3 1.5
5 2 1
1 0.5
0 0 0
0 200 400 600 0 200 400 600 800 0 200 400 600 800
ω [MeV] ω [MeV] ω [MeV]
FIG. 10: SuSAM predictions and uncertainty bands for the quasielastic (e,e′) cross section for several kinematics compared
to theexperimental data.
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