Local density and the RPA corrections in charge current quasielastic neutrino on Oxygen, Argon and Iron scattering

0.24 MB·English

by  Krzysztof M. Graczyk

Lo
al density and the RPA
orre
tions in
4
harge
urrent quasielasti
neutrino on
0
0
Oxygen, Argon and Iron s
attering
2

n ∗
a
Krzysztof M. Gra
zyk
J

0
3
Institute of Theoreti
al Physi
s, University of Wro
ªaw

2
pl. M. Borna 9, 50 - 204 Wro
law
v
3
Poland
5
0
1 February 8, 2008
0
4
0
/
h PACS: 25.30.Pt
t
-
l
c Abstra
t
u
n Numeri
al
omputations of
ross se
tions for quasielasti

harge
urrent
:
v s
atteringof neutrinoon Oxygen, Argonand Ironin Lo
alDensity Approx-
i
X imation(LDA) arepresented. We
onsiderpro
essesforafew GeVneutrino
energy. We in
lude
orre
tions from nu
leon re-intera
tion in nu
leus de-
r
a
s
ribed by relativisti
Random Phase Approximation (RPA). We adopt the
relativisti
Fermigasmodelofnu
leuswithandwithouttakingintoa

ount
the e(cid:27)e
tive mass of nu
leons.
1 Introdu
tion
The study of os
illations phenomena of neutrinos be
ame one of the
most interesting topi
s of parti
le physi
s. Experiments su
h as K2K,
MINOS, MiniBoone, ICARUS
olle
t or will
olle
t data whi
h allow
to establish parameters of os
illations. In the analysis of experimental
∗
e-mail address: kgra
zykift.uni.wro
.pl
1
resultsa goodknowledge of theoreti
al predi
tions of neutrino-nu
leus

ross se
tions is
ru
ial. In long baseline and atmospheri
neutrino
experiments one of the most important be
ome analysis of the data
for a few GeV neutrinos[1℄. In this energy domain the quasielasti

ontribution is important.
The
al
ulations of
ross se
tions require a study of in(cid:29)uen
e of
nu
lear e(cid:27)e
ts. The s
attering amplitude
an be
ombined from two
parts[2℄. Oneisdes
ribedbyweakintera
tion vertexofs
atteringneu-
trinoonfreenu
leonandtheotherwhi
hisevaluatedintheframework
of many body theory and des
ribes the model of nu
leus. The weak
intera
tion vertex is des
ribed very a

urately by Fermi theory a few
GeV neutrino. A
onsideration of the model of nu
leus des
ribed by
Fermi gas (FG) of nu
leons[3℄ is one of the most popular and simple
starting point in dis
ussions of nu
lear e(cid:27)e
ts in that energy regime.
IntheFermigasmodel momenta ofnu
leons areuniformlydistributed
k
F
inside the Fermi sphere whose radius is
alled Fermi momentum - .
It is
onne
ted with baryon density inside nu
leus. For the (cid:28)rst time
this simple approa
h was applied in the s
attering of ele
trons on nu-

lei[4℄-[7℄. Experimental resultsobtained fromthesepro
essesallowed
to establish average Fermi momenta for given nu
lear targets. Su
-

ess of this approa
h
aused for appli
ation of it in neutrino-nu
leus
intera
tions[8, 9, 10, 11℄.
As was mentioned above the Fermi gas model is the good starting
point for further nu
lear e(cid:27)e
ts dis
ussions. One
an evaluate this by

onsideration of some additional e(cid:27)e
ts su
h as binding energy, e(cid:27)e
-
tivemassofnu
leons,(cid:28)nalstateintera
tionandmanyothers. However
the most important seems to be taking into a

ount the
olle
tive be-
havior of nu
leons inside the nu
leus. It
an be done in several ways
e.g. by usingstati
potential, whi
hgetsthequiteeasiestway toapply
it or by appli
ation of the methodology of quantum many body (cid:28)eld
theory.
Oneofthemost
onsistentwaysofdes
ribingofthe
olle
tiveprop-
erties of nu
leons inside nu
leus is the relativisti
Mean Field Theory
(MFT)[12℄. The ground state of nu
leus is des
ribed by relativisti
Fermi gas of nu
leons whi
h has e(cid:27)e
tive mass introdu
ed by self-

onsisten
y equation for given target. The strong intera
tion between
nu
leons is introdu
ed into the model by
onsideration of virtual par-
ti
les whi
h are ex
hanged between nu
leons. They are des
ribed by
the e(cid:27)e
tive intera
tion lagrangian whi
h
ontain of ve
tor and s
alar
(cid:28)elds. The MFT allows to solve this theory. In the ele
tron-nu
leus
2
σ ω
s
attering,neutral andve
tor are
onsidered[13℄. The
ontribution
σ
from (cid:28)eld is fully manifested in the e(cid:27)e
tive mass (the appearing of
this (cid:28)eld lead to the above mentioned self-
onsisten
y equation). In
the
ase of the
harge
urrent neutrino-nu
leus intera
tion, pions and
mesons rho give
ontribution to s
attering amplitude[14, 15℄. Corre
-
tionsdes
ribedbye(cid:27)e
tivelagrangianare
al
ulatedbyusingstandard
tools of quantum perturbation (cid:28)eld theory. It gives rise to in(cid:28)nite
number of diagrams. It is impossible to take into a

ount
ontribu-
tions from all diagrams. One may
hose spe
ial
lass of them whi
h

orrespond to one parti
le - one hole ex
itations. This approa
h is

alled relativisti
Random Phase Approximation (RPA) [16℄.
Experimentalmeasurementsoftheele
tron-nu
leuss
atteringshow
that
harge distribution innu
leus isnot uniform[17, 18℄, whi
h means
that baryon density is not uniform either. The
harge distribution
an
be
al
ulated in the framework of the relativisti
MFT by solving the
Dira
-Hartree self-
onsisten
y equations[19℄. Solution of it is
onsis-
tent with experimental predi
tions. It allows us to use the experimen-
tal results and apply them to the formalism under
onsideration.
The measurements of the s
attering ele
tron on nu
leus allowed to
(cid:28)t
harge density distributions inside nu
leus. The (cid:28)tting is done by
assumption that the simple models of nu
leus su
h os
illator model
in the
ase of light nu
lei e.g. Oxygen and Fermi model in the
ase
of heavier nu
lei e.g. Argon, Iron. The in
lusion into the model of
nu
leus of the lo
al pro(cid:28)le of baryon density (Lo
al Density Approx-
imation - LDA) is the treatment whi
h makes the model of nu
leus
more realisti
. The e(cid:27)e
t of the lo
al density were taken into a

ount
not only in dis
ussions of ele
tron nu
leus data and in neutrino matter
intera
tions[20℄-[23℄.
Inthispaperwewanttodevelop
onsiderationofpreviouspaper[24℄,
where we adopted relativisti
MFT. We presented the analyti
al solu-
tion of the RPA Dayson equation and dis
ussed the RPA
orre
tions
in s
attering of neutrino on nu
leus. Here we are going to extend our

onsideration by appli
ation of Lo
al Density Approa
h (LDA) in the

asesof thesefew targets. We will
ompare resultsfor the
aseof LDA
with
al
ulations whi
h were done by assumption the
onstant baryon
density. The se
ond
ase is des
ribed by average values of Fermi mo-
menta and e(cid:27)e
tive masses for given targets. They were
al
ulated
from experimental
harge density distributions[18℄.
O16 Ar40 Fe56
8 18 26
We
hoose nu
lei , and , whi
h
hara
terize tar-
gets of neutrino experiments: Super Kamionkande, I
arus and Minos.
3
As mentioned above, we are going to
ompare di(cid:27)erential and total

ross se
tions for the nu
lei with
onstant and lo
al density. The
om-
parison be done in various levels of the model (Fermi gas with or
without RPA and with or without e(cid:27)e
tive mass).
The paper is organized as follows. A short des
ription of the for-
malism is presented in se
tion 2. In se
tion 3 numeri
al
omputations
of
ross se
tions are presented and the results are dis
ussed. In this
se
tion we also summarized.
2 The formalism
We
onsider the pro
ess:
−
ν +N(A,Z) → µ +p+N(A−1,Z)
µ
The
rossse
tionperonenu
leoninthelaboratoryframefortheabove
s
attering is the following:
d2σ G2cos2θ |~q|
= − F c Im(L νΠµ ).
d|~q|dq 16π2ρ E2 µ ν (1)
0 F
ρ = k3/3π2 q = k−k′ k
kw′here F F , is the tranLsfµeνr of four-momentum, ,
are neutrino and lepton four momenta. is a leptoni
tensor:
L = 8 k k′ +k′ k −g k k′α±iǫ k′αkβ .
µν µ ν µ ν µν α µναβ
(cid:16) (cid:17)
±
The signdependsonthe
onsideredpro
ess(neutrino/antineutrino).
Nu
lear properties of nu
leus are in
luded in the polarization tensor
Πµν
. This tensor is evaluated by the QFT te
hniques [16℄ and
an be
written asthe sum of two
ontributions. One of them is the free Fermi
gas and se
ond the RPA
orre
tion:
Πµν(q) = Πµν (q)+Πµν (q),
free RPA
d4p
Πµν (q)= −i Tr(∆ (p+q)Γµ(q)∆ (p)Γν(−q)).
free (2π)4 FG FG (2)
Z
Γµ
By elementary weak
harge
urrent nu
leon-nu
leon is des
ribed
and expressed by means of the form fa
tors [25℄:
iσανq
Γα(q ) = F (q2)γα+F (q2) ν +G (q2)γαγ5
µ 1 µ 2 µ 2M A µ (3)
4
G (q ) ν ν
p µ e µ
We omit term sin
e its
ontribution to and
ross se
tions
E ∼ 1GeV
ν
at is negligible. The value of parameters whi
h
hara
ter-
ize the form fa
tors are presented in the appendix.
∆ (p)
FG
By the in (2) we denote the propagators of nu
leon in the
free Fermi sea:
1 iπ
∗
∆ (p) = (p/+M ) + δ(p −E )θ(k −p) .
FG p2α−M∗2+iǫ Ep 0 p F !
∗ (4)
M
is the e(cid:27)e
tive mass,
al
ulated from the self-
onsisten
y equation
k
F
for given Fermi momentum [12℄:
g2 M∗ k +E∗
M∗ = M − s k E∗ −M∗ln F F ,
m2 π2 F F M∗ (5)
s (cid:18) (cid:18) (cid:19)(cid:19)
E = k2 +M∗2 1/2.
F F
(cid:16) (cid:17)
In∗our
al
ulationswewillseparatethe
asewiththefreenu
leonmass
M = M = 939
MeV and with the e(cid:27)e
tive mass.
The RPA part of the polarization tensor is
al
ulated by
onsider-
ation of
ontributions from one parti
le - one hole ex
itations. It leads
Π
RPA
to the
orrespond Dyson equation whose solution gives the (for
more details see [24℄).
2.1 Lo
al Density Approximation
We are going to
ompare
ross se
tions for the
ase of the model of
nu
leus with
onstant baryondensity with the
aseofthe model of nu-

leus with lo
al baryon density. The
onstant value of baryon density
for given nu
leus is evaluated by
al
ulation of average value of Fermi
momentum for given density distribution
d3rk (r)ρ(r)
F
< k > = ,
F
A (6)
R
Z
k (r) = 3 3π2 ρ(r),
Fp
s A (7)
N
k (r) = 3 3π2 ρ(r).
Fn
s A (8)
A= d3rρ(r) ρ(r)
is a atomi
number, -
harge density pro(cid:28)le[18℄.
R
5
The results for the lo
al baryon density are obtained by the in-
tegration of the
ross se
tion per one nu
leon with weight given by
density.
cutoff
dσ = 4π drr2ρ (r)dσ(k (r),M∗(r))| ,
localdensity n,p F pernucleon
Z0
(9)
ρ
wheretheindexofnorpin
orrespondstoneutronorprotondensity
distribution.
< k > < M∗ > [MeV]
F
Nu
leus [MeV℄
Oxygen 199.21 690.78
Argon 216.88 631.37
Iron 216.91 634.84
Table 1: The average values of Fermi momenta and
orresponding e(cid:27)e
tive masses
al
ulated from

hargedistributionsfornu
lei.
r
In presented approa
h, for ea
h values of int∗egrating variable the
k (r) M (r)
F
Fermi momenta and e(cid:27)e
tive masses are
al
ulated. It
is done numeri
ally. The re
onstru
ted distributions of the e(cid:27)e
tive
massinsidenu
leiarepresentedin(cid:28)g. 1. Wealso
al
ulatetheaverage
FermimomentaforOxygen,ArgonandIronaswellase(cid:27)e
tivemasses.
The results are presented in table 1.
3 Numeri
al results and dis
ussion
We show a set of plots whi
h present the di(cid:27)erential
ross se
tions for
1 GeV neutrino energy. We
al
ulated also the ratio of total
ross for
the model of nu
leus with
onstant density to
ross se
tions
al
ulate
in LDA at 1 GeV neutrino energy (table.2). In the end we
omputed
the total
ross se
tions for the three investigated targets. It was made
by taking into a

ount all
onsidered nu
lear e(cid:27)e
ts (LDA, RPA
or-
re
tions and the e(cid:27)e
tive mass).
In the table 2 we made separation of several
ases. We
omputed
ratioof
rossse
tionsforthefreeFermigas,theFermigaswithe(cid:27)e
tive
mass, the Fermi gas with RPA without and with the e(cid:27)e
tive mass.
In the
ase of the model of nu
leus with
onstant density the average
Fermi momenta and the e(cid:27)e
tive masses from table 1 were applied.
As
an be seen the di(cid:27)eren
es between approa
hes are in order of
6
(M∗) (M∗)
Nu
leus FG FG FG + RPA FG + RPA
Oxygen 0.997 0.994 0,997 0.993
Argon 0.994 0.986 0,994 0.989
Iron 0.994 0.987 0,995 0.990
Table 2: Theratio oftotal
ross se
tions
al
ulated fornu
leus withlo
aldensityto
ross se
tionfor
thenu
leuswith
onstant density. The
Mal
∗ulationswere done for1GeV neutrinoenergy. FGmeansthe
freeFermigasmodel. Theappearingof meansofappli
ationofthee(cid:27)e
tivemass.
0÷1.5%
. It does not depend on how sophisti
ated
ompli
ations the
modelofnu
leusis. One
analsonoti
ethatthedi(cid:27)eren
esareslightly
bigger for argon and iron. These two targets are heavier and
harge
distributions for them were extra
ted in the di(cid:27)erent way.
Make our dis
ussion more pre
ise we present details
omparison of
di(cid:27)erential
ross se
tions. In the (cid:28)gure 2 and 3 the di(cid:27)erential
ross
se
tions for s
attering of 1 GeV neutrino on Oxygen, Argon and Iron
are presented. It was
al
ulated for the Fermi gas model of nu
leus
with and without adoption of the e(cid:27)e
tive mass. The di(cid:27)eren
es be-
tween approa
hes with
onstant and lo
al baryon density of nu
leus
are minor. One
an noti
e that only the maximum of the
urve in the
LDA is little
ut. The appli
ation of e(cid:27)e
tive mass lowers the pi
k of
%

ross se
tions by about 30 but also does not
hange the di(cid:27)erentiae
between results for LDA and
onstant baryon density.
In last three (cid:28)gures we
onsider the Fermi gas with
orre
tions
from random phase approximation. RPA
orre
tions are
al
ulated by

ounting in(cid:28)nite number of one loop digrams[14℄. We
onsider
harge
ρ

urrent rea
tions with mesons and pions
ontributions into RPA
polarization tensor. We applied standard values of
oupling
onstans
forthese(cid:28)elds. Theshortrange
orrelat′ionbetweennu
leonsisapplied
g
by using′ Landau-Migdal parameter [14℄. In our
al
ulations we
g = 0.7
adopt .
∗In (cid:28)gure 4 we present
al
ulations with the RPA
orre
tions and
M = M = 939
MeV. Similarly, as was noti
ed in the previous
ase
((cid:28)g.2), only small di(cid:27)eren
es in the maximum are observed. The in-

lusion of the e(cid:27)e
tive mass in the RPA
al
ulations dose not
hange
the LDA e(cid:27)e
t ((cid:28)g. 5). When we
ompare the (cid:28)gure 3 with (cid:28)g. 4 and
(cid:28)gure 5 with (cid:28)gre 6 one
an see that the RPA e(cid:27)e
t de
rease the
ross
se
tions, espe
ially for small transfers of the energy.
We
on
lude that lo
al density e(cid:27)e
t
ould be repla
ed by
al
u-
lation of equivalent average parameters of the model whi
h make the
7
numeri
al
al
ulations faster. The appli
ation of the approa
h does
not depend on
ompli
ation introdu
ed dynami
al model of rea
tion.
A Appendix
•
Form Fa
tors
M =
A
In our
omputations we used dipol form fa
tors[25℄ with:
1.03 GeV M2 = 0.71GeV2 µ = 4.71 g = G (0) = −1.26
, V , and A A .
•
Charge density distributions
We adopt the
harge density pro(cid:28)les from ref. [18℄.
O16
8
Charge density pro(cid:28)le for Oxygen is determined by harmoni
os
illator model
r2 r 2 r 4
ρ(r)= ρ(0)exp − 1+C +C
R2! (cid:18)R(cid:19) 1(cid:18)R(cid:19) ! (10)
where:
ρ(0) = 0.141fm−3 R = 1.833fm C = 1.544. C = 0.
1
Ar40 Fe56
18 26
Inthe
aseof and thetwoparametersFermimodelgives
the following pro(cid:28)les:
ρ(0)
ρ(r)= .
(1+exp((r−c)/C )) (11)
1
in the
ase of Argon:
ρ(0) = 0.176fm−3, C = 3.530fm, C = 0.542fm.
1
in the
ase of Iron:
ρ(0) = 0.163fm−3, C = 4.111fm, C = 0.558fm.
1
A
knowledgments
ThisworkwassupportedbyKBNgrant105/E-344/SPB/ICARUS/P-
03/DZ 211/2003-2005. I would like to thank Prof. Jan Sob
zyk for
useful dis
ussions, remarks and suggestions whi
h resulted in improve-
ment of this paper. I thank also to C. Jusz
zak and J. Nowak for
several
omments.
8
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