Download Unified Description of Neutrino-Nucleus Interactions within Nuclear

Unified Description of Neutrino-Nucleus Interactions
within Nuclear Many-Body Theory
Omar Benhar
INFN and Department of Physics, “Sapienza” University
I-00185 Roma, Italy
Based on work done in collaboration with
A. Ankowski, A. Cipollone, N. Farina, C. Losa, A. Loreti, A. Lovato, D. Meloni, and E. Vagnoni
Terzo Incontro Nazionale di Fisica Nucleare INFN2016
Laboratori Nazionali di Frascati, November 15, 2016
O UTLINE
?
Motivation
I
I
?
The electroweak nuclear response
I
I
?
Low-energy regime: supernova neutrinos
High-energy regime: oscillation signal of accelerator-based
experiment
Low-energy regime: mean field dynamics, short- and long-range
correlations
High-energy regime: the impulse approximation. Single particle
motion in interacting many-body systems
Summary & Outlook
1 / 25
S UPERNOVA N EUTRINOS
Supernova Neutrinos
1500 km
Core collapse
tcollapse ~100 ms
? Neutrino diffuse out of
neutrinos diffuse
dense newly born
out of the dense
neutron stars, carrying
newly born
away ∼ 1053 erg neutron star
carry away
~ 3 x 1053 ergs
3X107 km
10 km
Shock wave
Eshock~1051ergs
Text
Quasi-static
~1s
100 km
• The time structure of the neutrino signal depends on how
? The time structure ofheat
the isneutrino
signal
oncore
heat
transported
in thedepends
neutron star
(1013-1015 g/cm3 ). transport in the neutron
star
core,
the
density
of
which
is
The spectrum is set by scattering in a hot (T=5-10
MeV) and
•
−3so dense (1011-1013 g/cm3 ) neutrino-sphere.
not
nB ∼ 1013 − 1015 g cm
? The spectrum is determined by scattering in the hot outer region
called neutrino-sphere, with typical temperature and density
T ∼ 5 − 10 MeV and nB ∼ 1011 − 1013 g cm−3
? Energy scale Eν
<
∼
10 MeV
2 / 25
N EUTRINO O SCILLATIONS
? Neutrinos produced in weak interaction processes, e.g. β-decay,
are not in mass-eigenstates ⇒ a neutrino created with a specific
flavor (νe , νµ or ντ ) can be detected at a later time with different
flavor
? Probability of appearance of a new flavor after travelling a
distance L (consider two flavors, for simplicity)
Observable'Oscilla9on'Parameters'
2
2
Pα→β = sin 2θ sin
∆m2 L
4Eν
∆m2 = m2ν1 − m2ν2
? The energy scale of long-baseline experiments (L ∼ few 100s km)
is Eν ∼ few 100s MeV to few GeV
ElectronNNucleus'ScaPering'XIII'
Elba,'June'26,'2014'
4'
3 / 25
N EUTRINO I NTERACTIONS ARE W EAK
. Total cross section of the process
3.2 Decomposition
of the Cross Section
4 (Quasi)Elastic Scattering
. Neutrino-nucleon
scattering
νµ + n → µ− + p
2.25
• Non-resonant background / Deep inelastic scattering (DIS):
Barish et al, Phys. Rev. D16, 1977
Allasia et al, Nucl. Phys. B343, 1990
Mann et al, Phys. Rev. Lett. 31, 1973
Baker et al, Phys. Rev. D23, 1981
2
l, ν
ν
1.75
CC: νN →1.5
lX
N
σ [10-38 cm2]
W, Z
(3.37)
1.25
1
NC: νN → νX
X
(3.38)
0.75
0.5
ν stands here for every kind of neutrino flavor as well as for its
antiparticle. The term
0.25
”quasielastic” refers to the fact that the neutrino changes its identity to a charged lepton. If
0
the outgoing lepton is still a neutrino, the reaction is denoted as ”elastic”.
The term ”deep 1
0.1
Eν [GeV]
inelastic” refers to the kinematical regime where both Q2 and the mass of
the hadronic
−38
2
σνN ∼ Figure
10 4.7: Total
cmcross
,section
σνN
/σeN ∼
final state are large compared to typical hadron masses.
for ν n → µ− p.
10
−6
10
µ
Having those three basic processes at hand we can describe all relevant physical reactions. The cross section is then a sum of all single contribution, namely
the production of
1.25
nucleons, of pions, etas and kaons, etc:
νe
σ CC,N C =
σ(N)
! "# $
+
σ(π)
!"#$
+
σ(η)
!
+ σ(K)1 +νµ . . .
"#
$
ντ
(3.39)
4 / 25
D ETECTING N EUTRINOS R EQUIRES B IG D ETECTORS
. The SuperKamiokande detector, in
Japan, is filled with 12.5 million
gallon of ultra-clean water
. The MiniBooNE detector,at FNAL, is
filled with 800 tons of mineral oil
. The detected signal results from
neutrino interactions with Oxygen
and Carbon nuclei
. A quantitative understanding of the
their response to neutrino
interactions is required for the
interpretation of the measured cross
sections
5 / 25
N EUTRINO -N UCLEON I NTERACTIONS
? Neutrino interactions are mediated by the gauge bosons W ± and
Z0 , whose masses are in the range ≈ 80 − 90 GeV
? In all processes to be discussed in this talk the momentum
2
transfer q is such that q 2 << MW,Z
∼ 80 − 90 GeV ⇒ Fermi
theory of weak interaction works just fine
W, Z0
J` µ
G
LF = √ JN µ J` µ
2
ū`− γ µ (1 − γ5 )uν
=
ūν 0 γ µ (1 − γ5 )uν
(CC)
(N C)
? The nucleon current can be cast in the non relativistic limit
(
ūp γµ (1 − gA γ5 )un → χ†sp (gµ0 + gA giµ σi )χsn (CC)
JN µ =
ūn0 γµ (1 − cA γ5 )un → χ†s0n (gµ0 + cA giµ σi )χsn (N C)
6 / 25
N EUTRINO -N UCLEON C ROSS S ECTION
? x-section of the charged-current process ν` + n → `− + X
dσ ∝ Lλµ W λµ
. Lλµ is determined by the lepton kinematical variables (more on
this later)
. under general assumptions W λµ can be written in terms of five
structure functions Wi (q 2 , (p · q)) (p is the nucleon four
momentum)
W2
W3
W4
+ ελµαβ qα pβ + 2 + q λ q µ 2
m2N
mN
mN
W5
+(pλ q µ + pµ q λ ) 2
mN
W λµ = −g λµ W1 + pλ pµ
? In principle, the structure functions Wi can be extracted from the
measured cross sections
7 / 25
N EUTRINO -N UCLEUS I NTERACTION
? Replace the nucleon tensor witht the nuclear response tensor
X
Wλµ =
h0|Jλ† |nihn|Jµ |0iδ (4) (p + k − pn − k 0 )
n6=0
? Consider, for example, a neutral current process
ν + A → ν0 + X
? In non relativistic regime
W (q, ω) ∝
GF
GF
cA 2
λµ
ρ
σ
L
W
=
(1
+
cos
θ)S
+
(3
−
cos
θ)S
λµ
4π 2
4π 2
3
where cos θ = (k · k0 )/(|k||k0 |), while S ρ and S ρ are the nuclear
responses in the density and spin-density channels, respectively.
8 / 25
? density response
Sρ =
1 X
|h0|J0 |nihn|J0 |0iδ (4) (P0 + q − Pn )
N n
? spin-density response (α, β = 1, . . . 3)
X
ρ
Sρ =
Sαα
α
ρ
Sαβ
1 X
=
|h0|Jα |nihn|Jβ |0iδ (4) (P0 + q − Pn )
N n
? Neutral weak current
X
X
J0 =
ji0 =
eiq·xi
i
? Outstanding issues
i
,
Jα =
X
jiµ =
i
X
eiq·xi σα
i
. Model nuclear dynamics
. Determine the nuclear initial and final states
9 / 25
M ODELING N UCLEAR D YNAMICS
? ab initio (bottom-up) approach
X p2
X
X
i
H=
+
vij +
vijk
2m j>i
i
k>j>i
. vij provides a very accurate descritpion of the two-nucleon system,
and reduces to Yukawa’s one-pion-exchange potential at large
distances
. inclusion of vijk needed to explain the ground-state energies of the
three-nucleon systems
. vij is spin and isospin dependent, and strongly repulsive at short
distance
. note: nuclear interactions can not be treated in perturbation theory
in the basis of eigenstates of the non interacting system. Either the
interaction or the basis states need to be “renormalized” to
incorporate non perturbative effetcs
? Mean field (independent particle) approximation
nX
o
X
X
vij +
vijk →
Ui
j>i
k>j>i
i
10 / 25
E FFECTIVE I NTERACTION
? The effective interaction in nuclear matter at density ρ is defined
through the relation [kF = (3π 2 ρ/2)1/3 ]
h0|H|0i =
3 kF2
+ h0F G |Veff |0F G i
5 2m
? unlike the bare NN potential, Veff is well behaved, and can be
used to perform perturbative calculations in the basis of
eigenstates of the non interacting system
? the response can be also computed using the Fermi gas states
and the corresponding effective operators, defined through
µ
|0F G i
hn|J µ |0i = hnF G |Jeff
11 / 25
E FFECTIVE I NTERACTION
? Comparison between bare and CBF effective interaction at
nuclear matter equilibrium density
1000
(a)
1000
eff
vS=0,T
=1 (r)
v(r) [MeV]
600
400
200
200
0
-200
1
1.5
r [fm−1 ]
80
2
2.5
bare (r)
vS=1,T
=1
400
-200
0.5
0
(c)
60
eff
vS=1,T
=1 (r)
600
0
0
(b)
800
bare (r)
vS=0,T
=1
v(r) [MeV]
v(r) [MeV]
800
0.5
1
1.5
r [fm−1 ]
2
2.5
eff (r)
vt,T
=1
bare (r)
vt,T
=1
40
20
0
-20
-40
0
0.5
1
1.5
r [fm−1 ]
2
2.5
12 / 25
I NTERACTION E FFECTS
? Mean field effects
. Change of nucleon energy spectrum
ek =
X
k2
hkk0 |Veff |kk0 ia
+
2m
0
k
. Effective mass
1
1 dek
=
m?k
|k| d|k|
? Correlation effects
. Effective operators couple the ground state to
two-particle–two-hole (2p2h) final states, thus removing strength
from the 1p1h sector
µ
µ
M2p2h = h2p2h|Jeff
|0i 6= 0 → M1p1h = h1p1h|Jeff
|0i < h1p1h|J µ |0i
13 / 25
? Nucleon energy spectrum and
Effective mass in
isospin-symmetric matter at
equilibrium density
? Quenching of Fermi transition
strength in isospin-symmetric
matter at equilibrium density
14 / 25
q-E VOLUTION OF I NTERACTION E FFECTS
? Density response of isospin-symmetric matter at equilibrium
density
|q| = 3.0 fm−1
|q| = 1.8 fm−1
|q| = 0.3 fm−1
15 / 25
L ONG -R ANGE C ORRELATIONS
? At low momentum transfer the space resolusion of the neutrino
becomes much larger than the average NN separation distance
(∼ 1.5 fm), and the interaction involves many nucleons
← λ ∼ q −1 →
d
? Write the nuclear final state as
a superposition of 1p1h states
(RPA scheme)
|ni =
N
X
i=1
+
+
+...
Ci |pi hi )
16 / 25
E FFECTS OF LONG - RANGE CORRELATIONS
|q|-evolution of the density-response of isospsin-symmetric
O. Benhar, N.
Farina / Physics Letters Bcarried
680 (2009) 305–309
nuclear matter.
Calculation
out within CBF using a
realistic
nuclear
hamiltonian.
ph states, while being eigenstates of the HF Hamiltonian
I
|q| ≈ 480 MeV
(12)
ek ,
ven by Eq. (10), are not eigenstates of the full nuclear
n. As a consequence, there is a residual interaction V res
duce transitions between different ph states, as long as
momentum, q, spin and isospin are conserved.
e included the effects of these transitions, using the
coff (TD) approximation, which amounts to expanding
ate in the basis of one 1p1h states according to [27]
S M) =
!
i
|q| ≈ 300 MeV
c iT S M | p i h i , T S M ),
(13)
hi + q, S and T denote the total spin and isospin of the
le pair and M is the spin projection along the quantiza-
q, the excitation energy of the state | f ), ω f , as well as
ents c iT S M , are determined solving the eigenvalue equa-
|q| ≈ 60 MeV
HF
+ V res )| f ) = ( E 0 + ω f )| f ),
(14)
is the ground state energy. Within our approach this
diagonalizing a N h × N h matrix whose elements are
+ e − e )δ + (h p , T S M | V |h p , T S M ). (15)
OB
Farina,
PLB 680
Fig. 3. and
Nuclear N.
matter
response calculated
within305,
the TD (2009)
(squares) and correlated
HF (diamonds) approximations, for the case of Fermi transitions. Panels (A), (B) and 17 / 25
−1
N EUTRINO M EAN F REE PATH IN N EUTRON M ATTER
? The mean free path of non degenerate neutrinos at zero
temperature is obtained from
Z
G2
d3 q 1
= Fρ
(1 + cos θ)S(q, ω) + C2A (3 − cos θ)S(q, ω)
3
λ
4
(2π)
where S and S are the density (Fermi) and spin (Gamow Teller)
response, respectively
2.6
CTD full expression
CTD simplified expression
CTD without collective mode
λ/λF G
2.4
2.2
2
1.8
1.6
5
10
15
20
25
Eν [MeV]
30
35
40
? Both short and long range correlations important
18 / 25
H IGH -E NERGY R EGIME :
I
THE I MPULSE
A PPROXIMATION
The IA amounts to replacing
2
2
q,ω
q,ω
Σ
x
i
i
Jµ =
X
i
I
jµ (i) , |Xi → |x, px i ⊗ |R, pR i .
Nuclear dynamics and electromagnetic interactions are
decoupled. Relativistic effect can be properly accounted for
Z
dσA = d3 kdE dσN Ph (k, E)
. The electron-nucleon cross section dσN can be written in terms of
stucture functions extracted from electron-proton and
electron-deuteron scattering data
. The hole spectral function Ph (k, E), momentum and energy
distribution of the knocked out nucleon, can be obtained from ab
initio many-body calculations
19 / 25
O XYGEN S PECTRAL F UNCTION
• shell model states account for ∼ 80% of the strenght
• the remaining ∼ 20%, arising from NN correlations, is located at
high momentum and large removal energy
20 / 25
Q UASI E LASTIC E LECTRON S CATTERING
I
Elementary interaction vertex described in terms of the vector
(p,n)
(p,n)
, precisely measured over a broad
and F2
form factors, F1
range of Q2
I
Position and width of the peak are determined by Ph (k, E)
The tail extending to the region of high energy loss is due to
nucleon-nucleon correlations in the initial state, leading to the
occurrence of two particle-two hole (2p2h) final states
I
21 / 25
A NALYSIS OF M INI B OO NE CCQE DATA
.
.
MiniBooNe CCQE data
MiniBooNE flux
22 / 25
C ONTRIBUTION OF D IFFERENT R EACTION M ECHANISMS
I
In neutrino interactions the lepton kinematics is not determined.
The flux-averaged cross sections at fixed Tµ and cos θµ picks up
contributions at different beam energies, corresponding to a
variety of kinematical regimes in which different reaction
mechanisms dominate
. x = 1 → Eν 0.788 GeV , x = 0.5 → Eν 0.975 GeV
. Φ(0.975)/Φ(0.788) = 0.83
23 / 25
“F LUX AVERAGED ” E LECTRON -N UCLEUS X -S ECTION
I
The electron scattering x-section off Carbon at θe = 37◦ has been
measured for a number of beam energies
I
Theoretical calculations of the flux-averaged cross sections must
include all relevant reaction mechanism—one and two -nucleon
emission, excitation of nucleon resonances and deep inelastic
scattering—within a consistent approach
24 / 25
S UMMARY & O UTLOOK
? The study of neutrino interactions with nuclear matter and
nuclei is a strongly cross-disciplinary field, with applications in a
variety of areas, ranging from the astrophysics of compact stars
to the search of neutrino oscillations
? Recent studies carried out within nuclear many-body theory
using realistic hamiltonians have revealed a variety of dynamical
effects, whose effects on the nuclear response are large
? The emerging picture suggests that a unified description of the
nuclear response in a broad kinematical range cortresponding to
neutrino energies ranging from few MeV to few GeV
25 / 25
Backup slides
26 / 25
N EUTRINO -N UCLEON I NTERACTIONS
? In the regime of momentum transfer (q) discussed in this talk
Fermi theory of weak interaction works just fine
W, Z0
? x-section of the charged-current process ν` + n → `− + X
dσ ∝ Lλµ W λµ
. Lλµ is determined by the lepton kinematical variables (more on
this later)
. under general assumptions W λµ can be written in terms of five
structure functions Wi (q 2 , (p · q)) (p is the nucleon four
momentum)
W2
W3
W4
+ ελµαβ qα pβ + 2 + q λ q µ 2
m2N
mN
mN
W5
+(pλ q µ + pµ q λ ) 2
mN
W λµ = −g λµ W1 + pλ pµ
27 / 25
? In principle, the structure functions Wi can be extracted from the
measured cross sections
? In the charged-current elastic sector ν` + n → `− + p they can be
expressed in terms of vector (F1 (q 2 ) and F2 (q 2 )), axial-vector
(FA (q 2 )) and pseudoscalar (FP (q 2 )) form factors
2
q2
q
2
(F1 + F2 ) + 2 m2N −
FA 2
W1 = 2 −
2
2
2 q
W2 = 4 F1 2 −
F2 2 + FA 2 = 2W5
4 m2N
W3 = −4 (F1 + F2 ) FA
q2
F2 2
q2
2
W4 = −2 F1 F2 + 2 m2N +
+
F
−
2
m
F
F
P
N P A
2
4 m2N
2
? according to the CVC hypothesis, F1 and F2 can be related to the
electromagnetic form factors, measured by electron-nucleon
scattering, while PCAC allows one to express FP in terms of the
axial form factor (more on this later)
28 / 25
Figure 4 shows Rosenbluth separation results performed in the 1970’s as the ratio GEp /GD , where GD is the
dipole FF given below by Eq. 14; it is noteworthy that these results strongly suggest a decrease of GEp with
increasing Q2 , a fact noted in all four references [Ber71, Pri71, Bar73, Han73]. As will be seen in section
3.4, the slope of this decrease is about half the one found in recent recoil polarization experiments. Left
out of this figure are the data of Litt et al. [Lit70], the first of a series of SLAC experiments which were
going to lead to the concept of “scaling” based on Rosenbluth separation results, namely the empirical relation
µp GEp /GM p ∼ 1. Predictions of the proton FF GEp made in the same period and shown in Fig. 4 are from
Refs. [Iac73, Hoh76, Gar85], all three are based on a dispersion relation description of the FFs, and related to
the vector meson dominance model (VMD).
V ECTOR F ORM FACTORS
? Proton data
? Neutron
(deuteron) data
Figure 5: Data base for GEp obtained by the Rosenbluth method; the references are [Han63, Lit70, Pri71,
Ber71, Bar73, Han73, Bor75, Sim80, And94, Wal94,
Chr04, Qat05].
Figure 6:
Data base for GM p obtained by the
Rosenbluth method; the references are [Han63, Jan66,
Cow68, Lit70, Pri71, Ber71, Han73, Bar73, Bor75,
Sil93, And94, Wal94, Chr04, Qat05].
A compilation of all GEp and GM p data obtained by the the Rosenbluth separation technique is shown in
Figs. 5 and 6; in these two figures both GEp and GM p have been divided by the dipole FF GD given by:
GD =
1
with GEp = GD , GM p = µp GD , and GM n = µn GD .
(1 + Q2 /0.71GeV 2 )2
(14)
It is apparent from Fig. 5 that the cross section data have lost track of GEp above Q2 ∼ 1 GeV2 . It is difficult
to obtain G2E for large Q2 values by Rosenbluth separation from ep cross section data for several reasons; first,
Figure 20: GEn data as in Fig. 18, compared to the
fits [Kel04] (thick line) and [Gal71] (thin solid line).
Platchkov’s fits [Pla90] with 3 different N N potentials
10
Figure 21: The complete data base for GM n , from
cross section and polarization measurements. Shown
as a solid curve is the polynomial fit by Kelly [Kel04];
29 / 25
Axial structure of the nucleon
A XIAL F ORM FACTOR
(anti)neutrino scattering off protons [8, 9, 10], off deuterons [1
Fe) [17, 18] or composite targets like freon [19]-[22] and propa
of figure 1 we show the available values for the axial mass
scattering experiments. As pointed out in [24], reference
Frascati (1970)
Frascati (1970) GEn=0
Frascati (1972)
DESY (1973)
Daresbury (1975) SP
Daresbury (1975) DR
Daresbury (1975) FPV
Daresbury (1975) BNR
Daresbury (1976) SP
Daresbury (1976) DR
Daresbury (1976) BNR
DESY (1976)
Kharkov (1978)
Olsson (1978)
Saclay (1993)
MAMI (1999)
Average
Argonne (1969)
Argonne (1973)
CERN (1977)
? Dipole
parametrization
Argonne (1977)
CERN (1979)
BNL (1980)
BNL (1981)
2
FA (Q ) =
Argonne (1982)
gA
Fermilab (1983)
2
[1 + (Q2 /MA2 )]
BNL (1986)
BNL (1987)
BNL (1990)
Average
0.85
0.95
1.05
1.15
1.25
MA [GeV]
. gA from neutron β-decay
Figure 1.
Axial mass MA extractions. Left pane
The weigh
0.021) GeV. Right panel: From charged pion electr
weighted average is MA = (1.069 ± 0.016) GeV. N
experiment contains both the statistical and systemat
the systematical errors were not explicitly given. Th
refer to different methods evaluating the correction
explained in the text.
antineutrino scattering
experiments.
. axial mass MA from (quasi) elastic ν- and and
ν̄-deuteron
experiment
severe uncertainties in either knowledge of the incident
30 /neut
25
TAMM -D ANCOFF ( RING ) A PPROXIMATION
? Propagation of the particle-hole pair produced at the interaction
vertex gives rise to a collective excitation. Replace
|phi → |ni =
N
X
i=1
Ci |pi hi )
? The energy of the state |ni and the coefficients Ci are obtained
diagonalizing the hamiltonian matrix
Hij = (E0 + epi − ehi )δij + (hi pi |Veff |hj pj )
X
k2
ek =
+
hkk0 |Veff |kk0 ia
2m
0
k
? The appearance of an eigenvalue, ωn , lying outside the
particle-hole continuum signals the excitation of a collective
mode
31 / 25
E XCITATION OF C OLLECTIVE M ODES
? Density (a) and spin-density (b) responses of isospin-symmetric
nuclear
atA 901
equilibrium
density 45
A. Lovato et al.matter
/ Nuclear Physics
(2013) 22–50
−3
Fig. 13. Fermi (a) and Gamow–Teller (b) response−1
functions of SNM at ρ = 0.16 fm , evaluated at q = 0
? |q| = 0.1, 0.15, 0.20, 0.25,0.20,
0.30,
0.40
and
0.50
fm
using the
v6 + UIX potential and correlation functions.
0.25, 0.30,
0.40, and
0.50 fm−1
"
distribution entering the VGS calculation at three-body cluster level, the result of whic
noted VGS3b .
The static structure functions corresponding to the Fermi and Gamow–Teller transit
displayed in panels (a) and (b) of Fig. 14, respectively. The CTD results have been
"
32 / 25
? Mean free path of a non degenerate neutrino in neutron matter.
Left: density-dependence at k0 = 1 MeV and T = 0 ; Right:
energy dependence at ρ = 0.16 fm−3 and T = 0, 2 MeV
33 / 25
? Density and temperature dependence of the mean free path of a
non degenerate neutrino at k0 = 1 MeV and ρ = 0.16 fm−3
34 / 25