Embed Size (px)
Citation preview
Neutrino-floor with nuclear structure calculations
Dimitrios K. Papoulias
IFIC (CSIC-Valencia U.)
15th MultiDark Consolider Workshop, 3–5 April, Zaragoza
1 / 26
Outline
1 IntroductionPhysics Motivations of neutrino-nucleus studiesTheoretical background
2 CEνNS and WIMP-nucleus ratesElectromagnetic neutrino propertiesNew Z ′ and scalar mediatorsImpact on the neutrino floor
3 Nuclear PhysicsNuclear radiiNuclear structure methods (BCS, QRPA)Conventional methods for calculating the nuclear form factor
4 Summary and Outlook
2 / 26
Physics Motivations of CEνNS
SM CEνNS reaction (conventional)
να + (A, Z)→ να + (A, Z), α = (e, µ, τ)
Conventional, well-studied ν-process theoretically
Finally observed by COHERENT in August 2017, CONUS (hints)(other: MINER, TEXONO, CONNIE, Ricochet, νGEN, ν-cleus etc.)
Very high experimental sensitivity (low detector threshold) is required
Z
(A,Z)
να
(A,Z)
να
irreducible background for direct dark matter experiments: neutrino-floor "
can probe nuclear form factors "
any deviation from the SM would indicate a glimpse on new physics (NSIs, EM
properties, novel mediators) "
competitive determination of sin2 θW at low-energy
valuable tool for sterile oscillation searches
important in supernova dynamics (investigate deep sky)
study gA quenching of electroweak interactions
3 / 26
SM Cross sections and Nuclear Transition Matrix Elements
The SM CEνNS diff. cross section with respect to the scattering angle θ takes the form
dσSM,να
d cos θ=
G 2F
2πE 2ν (1 + cos θ)
∣∣∣〈g .s.||MSMV ,να ||g .s.〉
∣∣∣2
Eν : incident neutrino energy
Q2 = 4E 2ν sin2 θ
2: 4-momentum transfer (from kinematics: −q2 ≡ Q2 = −ω2 + q2 > 0)
|g.s.〉 = |Jπ〉 ≡ |0+〉: the nuclear ground state(for even-even nuclei is explicitly constructed by solving the BCS Eqs.)
gp(n)V
: polar-vector coupling of proton (neutron) to the Z boson
The SM nuclear matrix element is given in terms of the electromagneticproton(neutron) nuclear form factors FZ(N)(Q
2) (CVC theory)
For SM g .s.→ g .s. transitions (i.e. |0+〉 → |0+〉) only the Coulomb operatorcontributes∣∣∣MSM
V ,να
∣∣∣2 ≡ ∣∣∣〈g .s.||M00||g .s.〉∣∣∣2 =
[gp
VZFZ (Q2) + gnVNFN (Q2)
]2
D.K. Papoulias and T.S. Kosmas, Phys.Lett. B728 (2014) 482
4 / 26
Analysis of the COHERENT data: SMSM diff. cross section
dσSM
dTN
(Eν ,TN ) =G 2
F M
π
[(QV
W )2
(1−
MTN
2E 2ν
)
+ (QAW )2
(1 +
MTN
2E 2ν
)]F 2(TN ) ,
SM vector and axial vectorcouplings
QVW =
[gV
p Z + gVn N],
QAW =
[gA
p (Z+ − Z−) + gAn (N+ − N−)
],
single-bin counting problem (flux, quenchingfactor, and acceptance uncertainties areincorporated)
χ2(s2
W ) = minξ,ζ
[(Nmeas − NSM
να(s2
W )[1 + ξ]− B0n [1 + ζ])2
σ2stat
+
(ξ
σξ
)2
+
(ζ
σζ
)2 ],
search between 6 ≤ nPE ≤ 30
0 10 20 30 40 50
−10
0
10
20
30
40
COHERENT:14.57 kg CsI
Number of Photoelectrons (nPE)
Counts/2PE
COHERENT data
this work
0.1 0.2 0.3 0.4 0.50
2
4
6
8COHERENT:
14.57 kg CsI
90% C.L.
sin2 θW
∆χ2
simple syst.
full syst.
D.K. Papoulias and T.S. Kosmas, Phys.Rev. D97 (2018) 033003
for future perspectives see Canas et al., Phys.Lett. B784 (2018) 159-162 5 / 26
Analysis of the COHERENT data: EM properties
Neutrino magnetic momentcontrbution
(dσ
dTN
)SM+EM
= GEM(Eν ,TN )dσSM
dTN
,
GEM = 1 +1
G 2F
M
(QEM
QVW
)2 1−TN/EνTN
1− MTN2E2ν
.
EM charge: QEM =πaEMµνα
meZ
Vogel et al. Phys.Rev. D39 (1989) 3378
Neutrino charge radiusredefinition of the weak mixing angle
sin2θW → sin2
θW +
√2πaEM
3GF
〈r2να〉 .
see also Cadeddu et al., arXiv:1810.05606
10−10 10−9 10−8 10−7 10−60
5
10
15
COHERENT:
14.57 kg CsI
90% C.L.
µνα/µB
∆χ2
µνα = µν
µνα = µνeµνα = µνµ
−80 −60 −40 −20 0 20 40 60 800
5
10
15
20
COHERENT:14.57 kg CsI
90% C.L.
〈r2να〉 [10−32cm2]
∆χ2
rνα = rνrνα = rνerνα = rνµ
D.K. Papoulias and T.S. Kosmas, Phys.Rev. D97 (2018) 033003
6 / 26
Analysis of the COHERENT data: combined constraints
0.1 0.2 0.3 0.4 0.5−12
−10
−8
−6
?
COHERENT:
14.57 kg CsI90% C.L.
sin2(θW )
log(µν/µB)
−10 −5 0 5 10−12
−10
−8
−6
?
COHERENT:
14.57 kg CsI90% C.L.
〈r2ν〉 [10−32cm2]
log(µν/µB)
−12 −11 −10 −9 −8 −7 −6−12
−10
−8
−6
COHERENT:
14.57 kg CsI90% C.L.
log (µνe/µB)
log( µνµ/µB
)
−60 −40 −20 0 20 40−60
−40
−20
0
20
40
?
COHERENT:14.57 kg CsI
90% C.L.
〈r2νe〉 [10−32cm2]
〈r2 νµ〉[
10−32cm
2]
D.K. Papoulias and T.S. Kosmas, Phys.Rev. D97 (2018) 033003 7 / 26
Analysis of the COHERENT data: novel light mediatorsvector Z ′ mediator Cerdeno,JHEP 1605 (2016) 118
Lvec = Z ′µ(
gqV
Z′ qγµq + gνVZ′ νLγ
µνL
)+ 1
2M2
Z′Z′µZ ′µ
Z ′ contribution to CEνNS cross section
(dσ
dTN
)SM+Z′
= G2Z′ (TN )
dσSM
dTN
,
GZ′ = 1−1
2√
2GF
QZ′QV
W
gνVZ′
2MTN + M2Z′,
Z ′ charge: QZ′ =(
2guVZ′ + gdV
Z′)
Z +(
guVZ′ + 2gdV
Z′)
N
Scalar φ mediator Dent et al. PRD 96 (2017) 095007
Lsc = φ(
gqSφ
qq + gνSφ νRνL + H.c.
)− 1
2M2φφ
2
φ contribution to CEνNS cross section
(dσ
dTN
)scalar
=G 2
F M2
4π
G2φM4
φTN
E 2ν
(2MTN + M2
φ
)2F 2(TN )
Gφ =gνSφ QφGF M2
φ
scalar charge: Qφ =∑N ,q g
qSφ
mNmq
f(N )
T,q
for NSI in CEνNS see: Liao et al. PLB 775 (2017)
Abdullah et al. PRD98 (2018) 015005
Dutta et al. PRD 93 (2016) 013015
−4 −2 0 2 4 6−10
−8
−6
−4
−2
0
2
degeneracy area
COHERENT:14.57 kg CsI
90%C.L.
log (MZ′)
log( g
2 Z′)
−4 −2 0 2 4 6−10
−8
−6
−4
−2
0
2
COHERENT:14.57 kg CsI
90%C.L.
log (Mφ)
log( g
2 φ
)
D.K. Papoulias and T.S. Kosmas, Phys.Rev. D97 (2018) 033003 8 / 26
WIMP-nucleus cross sectionCross section in lab. frame
dσ(u, υ)
du=
1
2σ0
(1
mpb
)2 c2
υ2
dσA(u)
du,
spin (axial current) & scalarcontributions
dσA
du=[
f 0A Ω0(0)
]2F00(u)
+ 2f 0A f 1
A Ω0(0)Ω1(0)F01(u)
+[
f 1A Ω1(0)
]2F11(u) +M2(u) .
coherent contribution
M2(u) =(
f 0S [ZFZ (u) + NFN (u)]
+ f 1S [ZFZ (u)− NFN (u)]
)2.
model dependent parametersf 0A , f 1
A for the isoscalar and isovector parts of theaxial-vector current
f 0S , f 1
S for the isoscalar and isovector parts of the
scalar current
Spin structure coefficients
Fρρ′ (u) =∑λ,κ
Ω(λ,κ)ρ (u)Ω
(λ,κ)
ρ′ (u)
Ωρ(0)Ωρ′ (0)
with ρ, ρ′ = 0, 1 for the isoscalar and isovectorcontributions
Ω(λ,κ)ρ (u) =
√4π
2Ji + 1
× 〈Jf ||A∑
j=1
[Yλ(Ωj )⊗ σ(j)
]κ
jλ(√
u rj )ωρ(j)||Ji 〉 .
ω0(j) = 1 and ω1(j) = τ(j) with τ = +1(−1)for protons (neutrons)Ωj : solid angle for the position vector of the j-thnucleon.
evaluation of the reduced nuclear matrix element(first calculate the single particle matrixelements)
〈ni li ji ||t(l,s)J ||nk lk jk〉 =√(2jk + 1)(2ji + 1)(2J + 1)(s + 1)(s + 2)
×
li 1/2 jilk 1/2 jkl s J
〈li ||√4π Y l ||lk〉 〈ni li |jl (kr)|nl lk〉 ,
D.K. Papoulias et al., Adv. Adv.High Energy Phys. 2018
(2018) 6031362 9 / 26
WIMP-nucleus cross sectionCross section in lab. frame
dσ(u, υ)
du=
1
2σ0
(1
mpb
)2 c2
υ2
dσA(u)
du,
spin (axial current) & scalarcontributions
dσA
du=[
f 0A Ω0(0)
]2F00(u)
+ 2f 0A f 1
A Ω0(0)Ω1(0)F01(u)
+[
f 1A Ω1(0)
]2F11(u) +M2(u) .
coherent contribution
M2(u) =(
f 0S [ZFZ (u) + NFN (u)]
+ f 1S [ZFZ (u)− NFN (u)]
)2.
model dependent parametersf 0A , f 1
A for the isoscalar and isovector parts of theaxial-vector current
f 0S , f 1
S for the isoscalar and isovector parts of the
scalar current
Spin structure coefficients
Fρρ′ (u) =∑λ,κ
Ω(λ,κ)ρ (u)Ω
(λ,κ)
ρ′ (u)
Ωρ(0)Ωρ′ (0)
with ρ, ρ′ = 0, 1 for the isoscalar and isovectorcontributions
Ω(λ,κ)ρ (u) =
√4π
2Ji + 1
× 〈Jf ||A∑
j=1
[Yλ(Ωj )⊗ σ(j)
]κ
jλ(√
u rj )ωρ(j)||Ji 〉 .
ω0(j) = 1 and ω1(j) = τ(j) with τ = +1(−1)for protons (neutrons)
Ωj : solid angle for the position vector of the j-th
nucleon.
0 1 2 3 40
0.2
0.4
0.6
0.8
1
71Ga
u
Fρρ′
F00
F01
F11
D.K. Papoulias et al., Adv. Adv.High Energy Phys. 2018
(2018) 603136210 / 26
WIMP-nucleus ratesdifferential WIMP-nucleus event rate
dR(u, υ)
dq2= Ntφ
dσ
dq2f (υ) d3
υ, φ = ρ0υ/mχ
with the dimensional parameter u = q2b2/2
ρ0 is the local WIMP density
0 20 40 60 80 100 120 140 1600
50
100
150
71Ga
D1
Tthres = 0 keV
Tthres = 5 keV
Tthres = 10 keV
0 20 40 60 80 100 120 140 1600
50
100
150
71Ga
D2
Tthres = 0 keV
Tthres = 5 keV
Tthres = 10 keV
0 20 40 60 80 100 120 140 1600
50
100
150
71Ga
D3
mχ [GeV]
Tthres = 0 keV
Tthres = 5 keV
Tthres = 10 keV
0 20 40 60 80 100 120 140 1600
100
200
300
71Ga
D4
mχ [GeV]
Tthres = 0 keV
Tthres = 5 keV
Tthres = 10 keV
f (υ): distribution of WIMP velocity(Maxwell-Boltzmann)for consistency with the LSP
WIMP-nucleus rate
〈R〉 =(f 0A )2D1 + 2f 0
A f 1A D2 + (f 1
A )2D3
+ A2(
f 0S − f 1
S
A− 2Z
A
)2
|F (u)|2D4 .
with
Di =
∫ 1
−1dξ
∫ ψmax
ψmin
dψ
∫ umax
umin
G(ψ, ξ)Xi du ,
andX1 = [Ω0(0)]2 F00(u) ,
X2 =Ω0(0)Ω1(0)F01(u) ,
X3 = [Ω1(0)]2 F11(u) ,
X4 =|F (u)|2 .
D.K. Papoulias et al., Adv. Adv.High Energy
Phys. 2018 (2018) 603136211 / 26
Neutrino Backgrounds to Dark Matter SearchesSolar neutrinosW. C. Haxton, R. G. Hamish Robertson, and A.M. Serenelli, Ann. Rev. Astron. Astrophys. 51(2013), 21
Low-energy Atmospheric neutrinos(FLUKA simulations)G. Battistoni, A. Ferrari, T. Montaruli, and P. R.
Sala, Astropart. Phys. 23 (2005) 526
Diffuse Supernova neutrinosS. Horiuchi, J. F. Beacom, and E. Dwek, Phys.
Rev. D79 (2009) 083013
type Eνmax [MeV] flux [cm−2s−1]
pp 0.423 (5.98± 0.006)× 1010
pep 1.440 (1.44± 0.012)× 108
hep 18.784 (8.04± 1.30)× 103
7Below 0.3843 (4.84± 0.48)× 108
7Behigh 0.8613 (4.35± 0.35)× 109
8B 16.360 (5.58± 0.14)× 106
13N 1.199 (2.97± 0.14)× 108
15O 1.732 (2.23± 0.15)× 108
17F 1.740 (5.52± 0.17)× 106
Solar neutrino fluxes and uncertainties in theframework of the high metallicity SSM
10−1 100 101 102 10310−4
100
104
108
1012
127Xe
NeutrinoEnergy [MeV]
NeutrinoFlux[M
eV
−1cm
−2s−
1]pp 7Be861.3
17F atm : νµ
hep 8B atm : νe DSN : νe
pep 13N atm : νe DSN : νe
7Be384.315O atm : νµ DSN : νx
D.K. Papoulias et al., Adv. Adv.High Energy Phys. 2018
(2018) 6031362
12 / 26
Differential Event CEνNS rates
10−3 10−2 10−1 100 101 102 10310−6
10−3
100
103
10671Ga
Recoil Energy [keV]
Diff
.E
vent
Rate
[ton
keV
year]
−1
pp pep 7Be861.313N 17F DSN
hep 7Be384.38B 15O atm tot
10−3 10−2 10−1 100 101 102 10310−6
10−3
100
103
10673Ge
Recoil Energy [keV]
10−3 10−2 10−1 100 101 102 10310−6
10−3
100
103
10675As
Recoil Energy [keV]
Diff
.E
vent
Rate
[ton
keV
year]
−1
pp pep 7Be861.313N 17F DSN
hep 7Be384.38B 15O atm tot
10−3 10−2 10−1 100 101 102 10310−6
10−3
100
103
106127I
Recoil Energy [keV]
13 / 26
Exploring the neutrino-floor through CEνNS constraints
use of Deformed Shell Model calculations
oblate-prolate structure of nuclei is considered
R =DSMevents
Helmevents
0 200 400 600 800 1,00010−1
100
101
102
Tthres [keV]
R
71Ga73Ge75As127I
D.K. Papoulias et al., Adv. Adv.High Energy Phys. 2018
(2018) 6031362
Dark Matter Events
0 20 40 60 80 100 1200
0.5
1
1.5
71Ga
73Ge
75As
127I
Tthres [keV]
Events
[kg−1year−
1]
Neutrino floor
10−3 10−2 10−1 100 101 102 103
10−2
100
102
104
SM
µν = 4.3 × 10−9 µB
MZ′ = 1 GeV
g2Z′ = 10−6
µνe = 2.9 × 10−11 µB
MZ′ = 10 MeV
g2Z′ = 10−6
Tthres [keV]
Events
[ton−1year−
1]
71Ga73Ge75As127I
related works: Boehm et al. JCAP 1901 (2019), Cerdeno et al. JHEP 1605 (2016), Billard et al. JCAP 1811 (2018)14 / 26
slide from: M. Cadeddu @ NuFact 2018
15 / 26
Probing the neutron radii: COHERENT exp.
COHERENT F ′/F stat. uncertainty syst. uncertainty b (fm−1) 〈R2n 〉1/2 (fm)
phase I 1 current current 2.30+0.36−0.54 5.64+0.99
−1.2
scenario I 10 σstat = 0.2 σsys = 0.14 2.10+0.16−0.21 5.56+0.97
−0.49
scenario II 100 σstat = 0.1 σsys = 0.07 2.10+0.08−0.08 5.56+0.19
−0.23
Model independent form factor expansion Patton et al. Phys.Rev. C86 (2012)
Fp,n(Q2) ≈ 1− Q2
3!〈R2
p,n〉+Q4
5!〈R4
p,n〉 −Q6
7!〈R6
p,n〉+ · · · ,
0 2 4 6 80
2
4
6
8
⟨Rn
2⟩1/2 (fm)
⟨Rn4⟩1/4(f
m)
Unphysical
current
0 2 4 6 80
2
4
6
8
⟨Rn
2⟩1/2 (fm)
⟨Rn4⟩1/4(f
m)
Unphysical
scenario I
0 2 4 6 80
2
4
6
8
⟨Rn
2⟩1/2 (fm)⟨R
n4⟩1/4(f
m)
Unphysical
scenario II
Papoulias et al. arXiv:1903.0372216 / 26
Evaluation of the form factors (Klein-Nystrand)
Follows from the convolution of a Yukawapotential with range ak = 0.7 fm over aWoods-Saxon distribution, approximated asa hard sphere with radius RA.
FKN = 3j1(QRA)
qRA
[1 + (Qak )2
]−1
The rms radius is: 〈R2〉KN = 3/5R2A + 6a2
kS. Klein and J. Nystrand, Phys.Rev. C60 (1999) 014903
current
scenario I
scenario II
2 3 4 5 6 7 8 90
2
4
6
8
10
⟨Rn
2⟩1/2 (fm)
Δχ
2
Papoulias et al. arXiv:1903.03722
17 / 26
Evaluation of the form factors (Helm)
Convolution of two nucleonic densities, one being a uniform density with cut-off radius R0,(namely box or diffraction radius) characterizing the interior density and a second one that isassociated with a Gaussian falloff in terms of the surface thickness s.
FHelm(Q2) = FB FG = 3j1(QR0)
qR0e−(Qs)2/2
The first three moments
⟨R2
n
⟩=
3
5R2
0 + 3s2
⟨R4
n
⟩=
3
7R4
0 + 6R20 s2 + 15s4
⟨R6
n
⟩=
1
3R6
0 + 9R40 s2 + 63R2
0 s4 + 105s6 .
j1(x) is the known first-order Spherical-Bessel function
box or diffraction radius R0 (interior density)
s = 0.9 fm: surface thickness of the nucleus from spectroscopy data (Gaussian fallof).
J. Engel, Phys.Lett. B 264 (1991) 114
18 / 26
Evaluation of the form factors (Symmetrized Fermi)
Adopting a conventional Fermi (Woods-Saxon) charge density distribution, the SF form factor iswritten in terms of two parameters (c, a)
FSF
(Q2)
=3
Qc [(Qc)2 + (πQa)2]
[πQa
sinh(πQa)
] [πQa sin(Qc)
tanh(πQa)− Qc cos(Qc)
],
The first three moments
⟨R2
n
⟩=
3
5c2 +
7
5(πa)2
⟨R4
n
⟩=
3
7c4 +
18
7(πa)2c2 +
31
7(πa)4
⟨R6
n
⟩=
1
3c6 +
11
3(πa)2c4 +
239
15(πa)4c2 +
127
5(πa)6 .
c: half-density radius
a fm: diffuseness
surface thickness: t = 4a ln 3
J.D. Lewin and P.F. Smith, Astropart.Phys. 6 (1996) 87-112
19 / 26
Probing nuclear form factors: COHERENT exp.
0.5 1.0 1.5 2.0 2.5 3.0
2
4
6
8
10
r0 (fm)
s(f
m)
current
0.5 1.0 1.5 2.0 2.5 3.0
2
4
6
8
10
r0 (fm)
s(f
m)
scenario I
0.5 1.0 1.5 2.0 2.5 3.0
2
4
6
8
10
r0 (fm)
s(f
m)
scenario II
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
2
4
6
8
10
a (fm)
c(f
m)
current
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
2
4
6
8
10
a (fm)
c(f
m)
scenario I
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
2
4
6
8
10
a (fm)
c(f
m)
scenario II
2 4 6 8 10 120
1
2
3
4
5
6
RA (fm)
ak(f
m)
current
2 4 6 8 10 120
1
2
3
4
5
6
RA (fm)
ak(f
m)
scenario I
2 4 6 8 10 120
1
2
3
4
5
6
RA (fm)
ak(f
m)
scenario II
Papoulias et al. arXiv:1903.03722 20 / 26
Impact of form factor on CEνNS: COHERENT exp.
Fp
Fn
Helm
FSM
KN
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.610-5
10-4
10-3
10-2
10-1
100
Q (fm-1)
F2(Q
2)
Fp
Fn
Helm
FSM
KN
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.610-5
10-4
10-3
10-2
10-1
100
Q (fm-1)
F2(Q
2)
Papoulias et al. arXiv:1903.03722
5 10 15 20 25 30 35-20
-10
0
10
20
30
40
Photoelectrons (PE)
Ev
en
ts/2
PE
χmin2 (DSM) = 2.73
χmin2 (Helm) = 3.18 χmin
2 (SF) = 3.14
χmin2 (KN) = 2.88
5 10 15 20 25 30 350.0
0.5
1.0
1.5
2.0
2.5
3.0
Photoelectrons (PE)
Ev
en
ts/2
PE
NHelm-NDSM
NSF-NDSM
NKN-NDSM
Bon
21 / 26
Nuclear ground state (BCS)Within the context of the quasi-particle random phase approximation (QRPA) method the formfactors for protons (neutrons) are obtained as
FNn =1
Nn
∑j
j〈g .s.||j0(|q|r)||g .s.〉(
vp(n)j
)2
where j =√
2j + 1, Nn = Z (or N), vp(n)j are the BCS probability amplitudes, determined by
solving iteratively the BCS equations.
T.S. Kosmas, J.D. Vergados, O. Civitarese and A. Faessler, NPA 570 (1994) 637
After choosing the active model space the following important parameters must be
properly adjustedthe harmonic oscillator (h.o.) size parameter b
the two pairing parameters gp (n)pair for proton (neutron) pairs that renormalise the monopole (pairing) residual
interaction of the Bonn C-D two-body potential (describing the strong two-nucleon forces)
η, π, ρ, ω, σ, φ
p(n) p(n)
p(n) p(n)
Realistic proton and neutron form factors
The Bonn C-D residual interaction is mediatedvia one-meson exchangeR. Machleidt, Phys.Rev. C63 (2001) 024001
22 / 26
In-COHERENT calculationsInteraction Hamiltonian for neutral-current (NC) neutrino-nucleus scattering
〈f |Heff |i〉 =GF√
2
∫d3x 〈`f |j lept
µ (x)|`i 〉〈Jf |J µ(x)|Ji 〉
with 〈`f |j leptµ |`i 〉 = ναγµ(1− γ5)να e−iq·x, q : 3−momentum transfer
In the Donnelly-Walecka multipole decomposition method, the NC, double diff. SM crosssection from an initial |Ji 〉 to a final |Jf 〉 nuclear state (constructed explicitly throughQRPA realistic nuclear structure calculations), reads
d2σi→f
dΩ dω=
G 2F
π
εiεf
(2Ji + 1)
( ∞∑J=0
σJCL +
∞∑J=1
σJT
),
εi (εf ) is the initial (final) neutrino energy and ω is the nucleus excitation energy.Contributions to σJ
CL (Coulomb-longitudinal) and σJT (transverse electric-magnetic)
components T. W. Donnelly and R. D. Peccei, Phys. Rept. 50 (1979) 1
σJCL =(1 + a cos θ)|〈Jf ||MJ (κ)||Ji 〉|2 + (1 + a cos θ − 2b sin2 θ)|〈Jf ||LJ (κ)||Ji 〉|2
+[ωκ
(1 + a cos θ) + d]
2<e|〈Jf ||LJ (κ)||Ji 〉||〈Jf ||MJ (κ)||Ji 〉|∗ ,
σJT =(1− a cos θ + b sin2 θ)
[|〈Jf ||T mag
J (κ)||Ji 〉|2 + |〈Jf ||T elJ (κ)||Ji 〉|2
]∓[
(εi + εf )
κ(1− a cos θ)− d
]2<e|〈Jf ||T mag
J (κ)||Ji 〉||〈Jf ||T elJ (κ)||Ji 〉|∗
where the parameters a = 1, b = εiεf /κ2, d = 0 are obtained from the kinematics and κ = |q|23 / 26
Evaluation of the Nuclear Matrix Elements
Seven new operators are defined (proton-neutron representation) as
T JM1 ≡ MJ
M (κr) = δLJ jL(κr)Y LM (r),
T JM2 ≡ ΣJ
M (κr) = MJJM · σ,
T JM3 ≡ Σ′JM (κr) = −i
[1
κ∇×MJJ
M (κr)
]· σ,
T JM4 ≡ Σ′′JM (κr) =
[ 1
κ∇MJ
M (κr)]· σ,
T JM5 ≡ ∆J
M (κr) = MJJM (κr) ·
1
κ∇,
T JM6 ≡ ∆′JM (κr) = −i
[ 1
κ∇×MJJ
M (κr)]·
1
κ∇,
T JM7 ≡ ΩJ
M (κr) = MJM (κr)σ ·
1
κ∇ .
Closed compact analytic formulae for the single-particle reduced ME (upper) and many-body reduced ME (lower) for QRPAcalculations, are deduced.
〈(n1`1)j1||T Ji ||(n2`2)j2〉 = e−y yβ/2
nmax∑µ=0
P i, Jµ yµ, y = (κb/2)2
, nmax = (N1 + N2 − β) /2, Ni = 2ni + `i
〈f ‖T J‖0+gs〉 =
∑j2≥j1
〈j2‖T J‖j1〉J
[Xj2 j1
up(n)j2
vp(n)j1
+ Yj2 j1v
p(n)j2
up(n)j1
]
V.Ch. Chasioti and T.S. Kosmas, Nucl. Phys. A 829 (2009) 234P.G. Giannaka, D.K. Papoulias, T.S. Kosmas, unpublished (for any configuration (j1, j2)J)
24 / 26
Summary and Outlook
The neutrino floor constitutes an irreducible background in directdetection dark matter experiments
sensitivity era of direct dark matter detection experiments in CEνNS-induced backgroundevents from several astrophysical sources (Solar, Atmospheric, DSNB)
advanced nuclear physics methods enable a better determination of the neutrino floor indark matter searches
nuclear structure (neutron form factor, rms radius etc)
Study of CEνNS contributions to the neutrino floor due to new physicsinteractions
neutrino magnetic moments
new light mediators e.g. Z ′, φ
25 / 26
Thank you for your attention !
26 / 26
Extras
1 / 25
Impact of the nuclear form factor
The dσ/dTN , dσ/d cos θ and σtot cross sections for 48Ti.
0 0.1 0.2 0.3 0.4 0.5 0.610−6
10−5
10−4
10−3
10−2
10−1
100
48Ti
20 MeV
50 MeV
80 MeV
100 MeV
120 MeV
TN (MeV)
dσ/dTN
(10−37cm
2M
eV
−1)
FOPF=1
0 20 40 60 80 100 120 140 16010−6
10−5
10−4
10−3
10−2
10−1
100
101
10248Ti
Eν (MeV)
dσ/dcosθ(1
0−39cm
2)
15o
60o
90o
165o
0 20 40 60 80 100 120 140 16010−2
10−1
100
101
102
48Ti
Eν (MeV)
σto
t(1
0−39cm
2)
FOPF=1
Differences can be up to 30%
The effect is more important forheavier nuclear isotopes
D.K. Papoulias, PhD Thesis
2 / 25
SM Event rates for supernova neutrinos
Various nuclear methods tested D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
10−2
10−1 20Ne
dN
dTN
(ton−1keV
−1)
F = 1Shell-Model
F (q2)BCSexp.
1
2
3
4
5
20Ne
counts
(ton−1year−
1)
10−1 100 101 10210−3
10−2
10−1
100
40Ar
TN (keV)
dN
dTN
(ton−1keV
−1)
F = 1Shell-Model
F (q2)BCSexp.FOP
10−1 100 101 102
2
4
6
8
10
12
40Ar
T thres.N (keV)
counts
(ton−1year−
1)
Dark-matter detectors are also sensitive to Supernova neutrinos3 / 25
SM Event rates for supernova neutrinos (continued)
Various nuclear methods tested D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
10−3
10−2
10−1
100
101 76Ge
dN
dTN
(ton−1keV
−1)
F = 1Shell-Model
F (q2)BCSexp.
0
5
10
15
20
25
30
76Ge
counts
(ton−1year−
1)
10−1 100 101 10210−3
10−2
10−1
100
101 132Xe
TN (keV)
dN
dTN
(ton−1keV
−1)
F = 1Shell-Model
F (q2)BCS
10−1 100 101 1020
10
20
30
40 132Xe
T thres.N (keV)
counts
(ton−1year−
1)
Dark-matter detectors are also sensitive to Supernova neutrinos4 / 25
Study of the neutrino-floor at Dark Matter detectors
10−3 10−2 10−1 100 101 102 103
10−3
10−1
101
103
10571Ga
Tthres [keV]
Events
[ton−1year−
1]
pp pep 7Be861.313N 17F DSN
hep 7Be384.38B 15O atm tot
10−3 10−2 10−1 100 101 102 103
10−3
10−1
101
103
10573Ge
Tthres [keV]
10−3 10−2 10−1 100 101 102 103
10−3
10−1
101
103
10575As
Tthres [keV]
Events
[ton−1year−
1]
pp pep 7Be861.313N 17F DSN
hep 7Be384.38B 15O atm tot
10−3 10−2 10−1 100 101 102 103
10−3
10−1
101
103
105127I
Tthres [keV]
5 / 25
Multipole expansion of the hadronic current
At low and intermediate energies, any semi-leptonic process is described by an effective
interaction Hamiltonian, written in terms of the leptonic j leptµ and hadronic J µ currents as
Heff =G√
2
∫d3x j lept
µ (x)J µ(x) ,
Leptonic current ME, between an initial |`i 〉 and a final state |`f 〉〈`f |j leptµ |`i 〉 = `µ e−iq·x .
Define a complete orthonormal set of spatial unit vectors: l =∑λ=0,±1 lλe†λ
Expand the plane wave as:
e iq·x =∑
l
i l√
4π(2l + 1)jl (ρ)Yl0(Ωx ) , ρ = κ|x|, κ = |q|
The Clebsch-Gordan coefficient limits the sum on l to three terms, l = J and J ± 1.Evaluating for λ = ±1, one finds
eqλe iq·x = −∞∑
J≥1
√2π(2J + 1)iJ
λjJ (ρ)YλJJ1 +
1
κ∇×
[jJ (ρ)YλJJ1
],
and for λ = 0
eq0e iq·x =−i
κ
∞∑J≥0
√4π(2J + 1)iJ∇ [jJ (ρ)YJ0] .
T.W. Donnelly and J.D.Walecka, Nucl. Phys. A 274 (1976) 368 6 / 25
Tensor multipole operators
Substituting one finds
〈f |Heff |i〉 = − G√2〈f |∑
J≥0
√4π(2J + 1)(−i)J
(l3LJ0(κ)− l0MJ0(κ)
)
+∑λ=±1
∑J≥1
√2π(2J + 1)(−i)J lλ
(λT mag
J−λ(κ) + T elJ−λ(κ)
)|i〉 .
The multipole expansion procedure gives 8 independent irreducible tensor multipoleoperators, acting on the nuclear Hilbert space and having rank Jfour operators are defined for the polar vector component Jλ = (ρ, J) and
four for the the axial vector component J5λ = (ρ5, J5) of the hadronic current
MJM (κ) = McoulJM − Mcoul5
JM =
∫drMJ
M (κr)J0(r),
LJM (κ) = LJM − L5JM = i
∫dr
(1
κ∇MJ
M (κr)
)· J (r),
T elJM (κ) = T el
JM − T el5JM =
∫dr
(1
q∇×MJJ
M (κr)
)· J (r),
T magJM (κ) = T mag
JM − T mag5JM =
∫drMJJ
M (κr) · J (r) ,
the V-A structure of the weak interaction is adopted: Jµ = Jµ − J5µ = (ρ, J)− (ρ5, J5) .
7 / 25
Required Nuclear Matrix Elements
We proceed by defining
MJM (κr) = McoulJM + Mcoul5
JM
= F V1 MJ
M (κr)− iκ
MN[FAΩJ
M (κr) +1
2(FA + q0FP )Σ
′′JM (κr)] ,
LJM (κr) = LJM + L5JM
=q0
κF V
1 MJM (κr) + iFAΣ
′′JM (κr)] ,
T elJM (κr) = T el
JM + T el5JM
=κ
MN[F V
1 ∆′JM (κr) +
1
2µV ΣJ
M (κr)] + iFAΣ′JM (κr)] ,
T magJM (κr) = T mag
JM + T magn5JM
= − q
MN[F V
1 ∆JM (κr)− 1
2µV Σ
′JM (κr)] + iFAΣJ
M (κr)] ,
with FX (Q2), X=1,A,P and µV (Q2) being the free nucleon form factors
CVC Theory: only seven operators are linearly independentPolar-vector: Coulomb Mcoul
JM , longitudinal LJM , transverse electric T elJM [with normal
parity π = (−)J ] and transverse magnetic T magJM [with abnormal parity π = (−)J+1].
Axial-vector: Mcoul5JM , L5
JM , T el5JM (with abnormal parity) and T mag5
JM (with normal parity).
J. D. Walecka, Theoretical Nuclear And Subnuclear Physics, World Scientific, Imperial College Press
8 / 25
Evaluation of the Nuclear Matrix Elements
Seven new operators are defined (proton-neutron representation) as
T JM1 ≡ MJ
M (κr) = δLJ jL(κr)Y LM (r),
T JM2 ≡ ΣJ
M (κr) = MJJM · σ,
T JM3 ≡ Σ′JM (κr) = −i
[1
κ∇×MJJ
M (κr)
]· σ,
T JM4 ≡ Σ′′JM (κr) =
[ 1
κ∇MJ
M (κr)]· σ,
T JM5 ≡ ∆J
M (κr) = MJJM (κr) ·
1
κ∇,
T JM6 ≡ ∆′JM (κr) = −i
[ 1
κ∇×MJJ
M (κr)]·
1
κ∇,
T JM7 ≡ ΩJ
M (κr) = MJM (κr)σ ·
1
κ∇ .
Closed compact analytic formulae for the single-particle reduced ME (upper) and many-body reduced ME (lower) for QRPAcalculations, are deduced.
〈(n1`1)j1||T Ji ||(n2`2)j2〉 = e−y yβ/2
nmax∑µ=0
P i, Jµ yµ, y = (κb/2)2
, nmax = (N1 + N2 − β) /2, Ni = 2ni + `i
〈f ‖T J‖0+gs〉 =
∑j2≥j1
〈j2‖T J‖j1〉J
[Xj2 j1
up(n)j2
vp(n)j1
+ Yj2 j1v
p(n)j2
up(n)j1
]
V.Ch. Chasioti and T.S. Kosmas, Nucl. Phys. A 829 (2009) 234P.G. Giannaka, D.K. Papoulias, T.S. Kosmas, unpublished (for any configuration (j1, j2)J)
9 / 25
Concrete Example
(n1l1)j1 − (n2l2)j2 J µ = 0 µ = 1 µ = 2 µ = 3 µ = 4
0p1/2 − 0s1/2 1√
16
0p3/2 − 0s1/2 1 −√
13
0
0d5/2 − 0s1/2 2 −√
35
0
0f5/2 − 0p1/2 2 −√
75
√2063
0
0d7/2 − 0p1/2 4√
863
0
1p3/2 − 1s1/2 1 −√
59
√4
45−√
445
0
0d3/2 − 0f7/2 3 −√
16175
√4
45−√
161575
0
1f5/2 − 2p3/2 2 0√
87
−√
288343
√2888
27783−√
3227783
The Geometrical coefficients P6, Jµ for the the 〈j1||∆′JM ||j2〉.
PJµ are simple rational numbers for the diagonal elements (not shown here), or square
roots of rational numbers for the non-diagonal elements.
Calculable for any configuration
Advantages: the method is faster, the coefficients can be computed at once and stored,there is no-need to evaluate the Talmi integrals
D.K. Papoulias and T.S. Kosmas, J.Phys.Conf.Ser. 410 (2013) 012123
10 / 25
The nuclear BCS wavefunction
The concept of quasi-particles, made out of particle and hole components with certainoccupation amplitudes, is introduced.
The nucleons are assumed to couple in pairs with vanishing angular momentum formingbosonic states and behave as being in a superconducting state.
The ground state (g .s.) wavefunction of an even-even nucleus can be represented by theansatz
|BCS〉 =∏α>0
(ua − va c†α c†α
)|core〉 .
c†αc†α: creates an identical nucleonic pair and c†α (cα) is the creation (annihilation)operator for physical particles.
Baranger notation: α ≡ (a,mα), where a ≡ (na, la, ja). For instance, in the case where agiven Hamiltonian is invariant under time reversal, such as the Harmonic Oscillator basis(spherical basis) one would readily write|α〉 = |nalaja; mα〉 |α ≡ −α〉 = |nalaja;−mα〉, mα > 0 .
for the assumed spherical nuclei the ua and va parameters are independent of theprojection quantum number mα.
ua and va, are considered as variational parameters representing the probabilityamplitudes, (v2
a and u2a constitute the probability that a conjugate pair state (α, α) is
occupied or unoccupied respectively).
The latter are not independent and obey the condition
u2a + v2
a = 1 for all a .
J. Suhonen, From Nucleons to Nucleus Concepts of Microscopic Nuclear Theory, Springer 2007
11 / 25
BCS quasi-particles I
We define the quasi-particle creation (annihilation) operator a† (a) as linear combinationsof particle operators via the Bogoliubov-Valatin (BV) transformation
a†α =uac†α + vacα, c†α = uaa†α − vaaα ,
a†α =uac†α − vacα, c†α = uaa†α + vaaα ,
aα =uacα + vac†α, cα = uaaα − vaa†α ,
aα =uacα − vac†α, cα = uaaα + vaa†α .
The corresponding anticommutation relations preserve the basic commutation relations(i.e. BV is a quantum mechanical canonical transformation), as
a†α, a†β = 0, aα, aβ = 0, aα, a†β = δαβ ,
emphasising that the quasi-particles are (generalised) fermions like the physical particlesthey are built from.
This discussion mostly follows the notation of:
J. Suhonen, From Nucleons to Nucleus Concepts of Microscopic Nuclear Theory, Springer 2007
Other notable books:P. Ring and P. Schuck, The Nuclear Many-body Problem, Springer, Heidelberg, Germany, 1980
A. de Shalit and H. Feshbach, Theoretical Nuclear Physics: Nuclear structure, John Wiley & Sons Inc, New York, USA, 1974
12 / 25
BCS quasi-particles II
the quasi-particle is to be viewed as partly particle and simultaneously partly hole, ratherthan bare particle or bare hole.
In other words, the operator a†α creates a quasi-particle, in an orbital α, that is a particlewith probability amplitude ua and a hole with probability amplitude va.
above the Fermi surface (v2a small) it is nearly particle, while below the Fermi surface (u2
asmall) it is nearly hole.
In this framework, a j-orbital, is occupied with (2j + 1)v2j particles and (2j + 1)u2
j holes.In addition, through the BV transformation, it is achieved a representation of the groundstate of pairwise interacting particles in terms of a gas of non-interacting quasi-particles.
Disadvantage: the BV transformation does not conserve the particle number due to themixing of creation and annihilation operators.
aα and aα operations on the BCS vacuum: aα|BCS〉 = 0, aα|BCS〉 = 0
contraction properties (normal ordered operators):
aαa†β ≡ 〈BCS |aαa†β |BCS〉 = δαβ , all other contractions = 0 .
J. Suhonen, From Nucleons to Nucleus Concepts of Microscopic Nuclear Theory, Springer 2007
13 / 25
Quasi-particle representation of the nuclear Hamiltonianwithin the context of BCS
In second quantisation the many-body nuclear Hamiltonian, H = T + V , is expressed as asum of a one-quasi-particle term that describes the quasi-particle mean-field, and aresidual interaction, as
H =∑α
εαc†αcα +1
4
∑αβγδ
vαβγδc†αc†βcδcγ ,
where the first sum represents the one-body kinetic energy, T , of the Hamiltonian, whilethe second sum represents the two-body potential V .The antisymmetrised two-nucleon interaction matrix elements are denoted by
vαβγδ = 〈αβ|V |γδ〉 − 〈αβ|V |δγ〉 ,The normalised and antisymmetrised two-nucleon states with respect to the appropriateparticle vacuum |0〉 (e.g. |0〉 ≡ |core〉 in our case) are given by
|αβ〉 = c†αc†β |0〉, |γδ〉 = c†γc†δ |0〉 .In angular-momentum-coupled representation, the normalised two-nucleon state takes theform
|ab; JM〉 = Nab(J)[c†a c†b
]JM|0〉 ,
with
Nab(J) =
√1 + δab(−1)J
1 + δab.
J. Suhonen, From Nucleons to Nucleus Concepts of Microscopic Nuclear Theory, Springer 2007 14 / 25
BCS equations IThe BCS equations can be derived by minimising the BCS ground-state expectation value
E = 〈BCS |H|BCS〉 ,which can be viewed as a constrained variational problem with respect to the occupationamplitudes uα and vα.Fixing the weak point of the method (e.g. the non-conservation of the particle number):the variational problem is forced to reproduce the correct average particle number,n ≡ (Z ,N), subject to the constraint
〈BCS |H|BCS〉 = n , n =∑
a
j2a v2
a .
J. Suhonen, From Nucleons to Nucleus Concepts of Microscopic Nuclear Theory, Springer 2007
In our numerical calculations, protons and neutrons are treated separately (proton-neutronpairing is ignored). The variational problem is then solved by employing the method of theLagrange multipliers (in order to become unconstrained) which enter via the definition ofthe auxiliary Hamiltonian
H ≡ H − λn .
In this context, the parameter λ, namely the chemical potential (or the Fermi energy), isdetermined through the variational problem
δ〈BCS |H|BCS〉 = 0 ,
or equivalently
δ
(H0 − λ
∑a
j2a v2
a
)≡ δH0 = 0 .
15 / 25
BCS equations II
The auxiliary Hamiltonian becomes
H0 =∑
a
(εa − λ) j2a v2
a +1
2
∑abJ
v2a v2
b J2 [Nab(J)]−2 〈ab; J|V |ab; J〉
+1
2
∑ab
ja jbuavaubvb〈aa; 0|V |bb; 0〉 ,
thus, the single-particle energies εa are sifted as εa → εa − λ.
It is convenient to introduce the following abbreviations
∆b ≡−[jb
]−1∑a
jauava〈aa; 0|V |bb; 0〉 ,
µb ≡−[jb
]−2∑aJ
v2a J2 [Nab(J)]−2 〈ab; J|V |ab; J〉 ,
ηb ≡εb − λ− µb .
∆b: is called the pairing-gap
µa: is the self-energy, describes a renormalisation of the single-particle energy, εa, due tothe fact that the energy of a nucleon in the orbital, a, gets additional contributions fromits interactions with the other nucleons.
J. Suhonen, From Nucleons to Nucleus Concepts of Microscopic Nuclear Theory, Springer 2007
16 / 25
Evaluation of the form factors (Shell-Model)
The form factor FZ (q2), for h.o. wavefunctions reads
FZ (q2) =1
Ze−(|q| b)2/4Φ (|q| b,Z) , Φ (q b,Z) =
Nmax∑λ=0
θλ(|q| b)2λ .
The radial nuclear charge density distribution ρp(r) is written in compact form, as
ρp(r) =1
π3/2b3e−(r/b)2
Π( rb,Z), Π (χ,Z) =
Nmax∑λ=0
fλχ2λ
The coefficients fλ are expressed as
fλ =∑
(n,`)j
π1/2(2j + 1)n! Cλ−`n`
2Γ(
n + ` + 32
)The coefficients θλ are expressed as
θλ =
√π
4λ
Nmax∑(n,`)j
(2n+`>λ)
2n∑m=s
(2j + 1)n!C mn`Λλ(m + `, 0)(` + m)!
2Γ(n + ` + 32
)
with
Λk (n, `) =(−)k
k!
(n + ` + 1/2
n − k
), C m
n` =m∑
k=0
Λm−k (n, `)Λk (n, `), s =
0, if λ− ` ≤ 0
λ− ` if λ− ` > 0
T.S. Kosmas and J.D. Vergados, Nucl.Phys. A 536 (1992) 72
17 / 25
Evaluation of the form factors (Fractional occupationprobabilities, FOP)
Shell model assumes that the proton occupation probabilities equal to unity (zero) for thestates below (above) the Fermi surface.Introduce depletion and occupation numbers, to parametrise the partially occupied levelsof the states that satisfy the relation T.S. Kosmas and J.D. Vergados, NPA 536 (1992) 72∑
(n,`)jall
αn`j (2j + 1) = Nn, Nn = Z or N
In this approximation we have a number of active surface nucleons
Π(χ,Z , αi ) =Π(χ,Z2)α1
Z1 − Z2+ Π(χ,Z1)
[α2
Zc − Z1− α1
Z1 − Z2
]+ Π(χ,Zc )
[Z ′ − Z
Z ′ − Zc− α2
Zc − Z1− α3
Z ′ − Zc
]+ Π(χ,Z ′)
[Z − Zc
Z ′ − Zc+
α3
Z ′ − Zc− α4
Z ′′ − Z ′
]+ Π(χ,Z ′′)
[α4
Z ′′ − Z ′− λ
Z ′′′ − Z ′′
]+ Π(χ,Z ′′′)
λ
Z ′′′ − Z ′′,
with λ = α1 + α2 − α3 − α4 D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
fit the parameters to reproduce the proton charge density distribution
18 / 25
Evaluation of the form factors (experimental data)
The proton nuclear form factor is evaluated through the Fourier transformof the proton charge density distribution, as
FZ (q2) =4π
Z
∫ρp(r)j0(|q|r) r2 dr ,
with j0 being the zero order spherical Bessel function.D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
The proton charge density distribution ρp(r) is obtained from a modelindependent Fourier-Bessel analysis of electron scattering experiments
ρ(r) =
∑Nv=1 av j0(vπr/R) for r ≤ R ,
0 for r > R ,
where R is the cutoff radius and N = R|q|max/π.H. De Vries, C.W. De Jager and C. De Vries, At. Data and Nucl. Data Tables 36 (1987) 495536
this method assumes FN(q2) ' FZ (q2) (not always a goodapproximation)
19 / 25
Bonn C-D potential
The Bonn C-D potential is based upon meson exchange
All mesons with masses below the nucleon mass are included, i.e. π, η, ρ(770), ω(782)and two scalar-isoscalar σ bosons
Lagrangians describing the couplings of themesons of interest to nucleons
Lπ0NN =− gπ0 ψiγ5τ3ψφ(π0)
Lπ±NN =−√
2gπ± ψiγ5τ±ψφ(π±)
LσNN =− gσψψφ(σ)
LωNN =− gωψγµψφ
(ω)µ
LρNN =− gρψγµτψ · φ(ρ)
µ −fρ
4Mpψσµντψ ·
(∂µφ
(ρ)ν − ∂νφ(ρ)
µ
)
η, π, ρ, ω, σ, φ
p(n) p(n)
p(n) p(n)
ψ is the nucleon field, φ denotes a meson field, Mp is the proton mass and τ3,± are thePauli matrices (a vanishing coupling of η to the nucleon is assumed).
Bonn C-D fits the world proton-proton data below 350 MeV
R. Machleidt, Phys.Rev. C63 (2001) 024001
20 / 25
Construction of the nuclear ground state |i〉The gap-equation: consider the monopole part of the Bonn C-D potential (pairing interaction)
∆b = −g
p (n)pair
2jb
∑a
ja∆a√η2
a + ∆2a
〈aa; 0|V |bb; 0〉 ,
pairing gaps are obtained through the three-point formula as follows
∆expn =−
1
4[Sn(A− 1, Z)− 2Sn(A.Z) + Sn(A + 1, Z)] ,
∆expp =−
1
4
[Sp (A− 1, Z − 1)− 2Sp (A.Z) + Sp (A + 1, Z + 1)
],
where, Sn (Sp ) denotes the experimental separation energy for neutrons (protons), respectively, of the target nucleus(A, Z) and the neighbouring nuclei (A± 1, Z ± 1) and (A± 1, Z).D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
Nucleus model-space b ∆p ∆n gppair gn
pair12C 8 (no core) 1.522 4.68536 4.84431 1.12890 1.1964816O 8 (no core) 1.675 3.36181 3.49040 1.06981 1.13636
20Ne 10 (no core) 1.727 3.81516 3.83313 1.15397 1.2760028Si 10 (no core) 1.809 3.03777 3.14277 1.15568 1.2313532S 15 (no core) 1.843 2.03865 2.09807 0.8837 0.95968
40Ar 15 (no core) 1.902 1.75518 1.76002 0.94388 1.0134848Ti 15 (no core) 1.952 1.91109 1.55733 1.05640 0.9989076Ge 15 (no core) 2.086 1.52130 1.56935 0.95166 1.17774
114Cd 18 (core 16O) 2.214 1.41232 1.35155 1.03122 0.98703132Xe 15 (core 40Ca) 2.262 1.19766 1.20823 0.98207 1.13370
The values of proton gppair and neutron gn
pair pairs that renormalise the residual interaction and reproduce the respective
empirical pairing gaps ∆p and ∆n . The active model space and the harmonic oscillator parameter, for each isotope, are also
presented. 21 / 25
The excited nuclear states-Diagonalisation of QRPA Eqs.
The excited states |f 〉 are derived by solving the QRPA equations:(A B−B −A
)(Xν
Y ν
)= ΩνJπ
(Xν
Y ν
),
First we define new amplitudes Pm, Rm
(A− B)Pm = Rm , (A+ B)Rm = Ω2mPm
which are related to the X ,Y through
Xm =
√1
2(Ω1/2
m Pm + Ω−1/2m Rm) , Y m =
√1
2(−Ω1/2
m Pm + Ω−1/2m Rm)
Finally, the diagonalisation of the QRPA equations gives:
X and Y forwards and backwards going amplitudes
QRPA excitation spectrum
V. Tsakstara and T.S. Kosmas, Phys.Rev. C83 (2011) 054612
22 / 25
Low-lying QRPA Excitation Spectrum
Reproducibility of the excitation spectrum is achievedQRPA spectra fit well the experimental data for low lying excitations
P.G. Giannaka and T.S. Kosmas, Phys.Rev. C92 (2015) 014606
23 / 25
Comparison of the nuclear methods
D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
0 1 2 3 4 5 6 7
0.02
0.04
0.06
0.08
0.1 40Ar
r(fm)
ρ(r
)
Shell-ModelFOPexp.
0 1 2 3 4 5 6 7
48Ti
r(fm)
0 0.5 1 1.5 2 2.5
10−4
10−3
10−2
10−1
100
40Ar
q(fm−1)
∣ ∣ F(q
2)∣ ∣
Shell-Model
F (q2)BCSFOPexp.
0 0.5 1 1.5 2 2.5
48Ti
q(fm−1)
D.K. Papoulias and T.S. Kosmas, Adv.High Energy Phys. 2015 (2015) 763648
24 / 25
Deformed Shell ModelThe construction of the many-body wave functions for the initial |Jπi 〉 and final |Jπf 〉 nuclear states in the framework of DSMinvolves performance of the following steps
(i) selection of model space consisting of a given set of spherical single-particle (sp) orbits, sp energies and theappropriate two-body effective interaction matrix elements.
(ii) Assuming axial symmetry and solving the HF single particle equations self-consistently, the lowest-energy prolate (oroblate) intrinsic state for the nucleus in question is obtained
(iii) The various excited intrinsic states then are obtained by making particle-hole (p-h) excitations over thelowest-energy intrinsic state (lowest configuration).
(iv) Then, because the HF intrinsic nuclear states |χK (η)〉 (K is azimuthal quantum number and η distinguishes states
with same K) do not have definite angular momentum, angular momentum projected states |φJMK (µ)〉 are constructed
as,
|φJMK (η)〉 =
2J + 1
8π2√
NJK
∫dΩ DJ∗
MK (Ω)R(Ω)|χK (η)〉 .
In the previous expression, Ω = (α, β, γ) represents the Euler angles, R(Ω) denotes the known general rotation
operator and the Wigner D-matrices are defined as DJMK (Ω) = 〈JM|R(Ω)|JK〉. Here, NJK is the normalisation
constant which by assuming axial symmetry is defined as
NJK =2J + 1
2
∫ π0
dβ sin β dJKK (β)〈χK (η)|e−iβJy |χK (η)〉 , (1)
where the functions dJKK (β) are the diagonal elements of the matrix dJ
MK (β) = 〈JM|e−iβJy |JK〉.(v) Finally, the good angular momentum states φJ
MK are orthonormalised by band mixing calculations and then, interms of the index η, it is possible to distinguish between different states having the same angular momentum J,
|ΦJM (η)〉 =
∑K,α
SJKη(α)|φJ
MK (α)〉 . (2)
25 / 25
