- Email: [email protected]
Neutrino and anti-neutrino scattering off Zn isotopes with quasi-particle RPA calculations
Progress in Particle and Nuclear Physics 66 (2011) 430–435
Contents lists available at ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
Neutrino and anti-neutrino scattering off Zn isotopes with quasi-particle RPA calculations V. Tsakstara a,c,∗ , T.S. Kosmas a,b,c , J. Sinatkas d a
Theoretical Physics Section, University of Ioannina, GR-45110 Ioannina, Greece
b
GSI Theoretical Physics Devision, D-64291 Darmstadt, Germany
c
Institute of Nuclear Physics, TU Darmstadt, D-64289, Germany
d
Informatics & Computer Technology Department, TEI of W. Macedonia, GR-52100, Greece
article
info
Keywords: Neutrino nucleus cross sections Supernova neutrino detection Nuclear detector response Quasi-particle random phase approximation
abstract We performed detailed calculations for the cross-sections of (anti)neutrinos with Zn isotopes, contents of the COBRA double beta decay detector at Gran Sasso. We utilized a quasi-particle random phase approximation (QRPA) relying on realistic two-body forces. We present and discuss original double and single differential cross-sections of the neutral current reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ . The response of these isotopes to the energy-spectra of supernova neutrinos is studied by convoluting the original cross-sections with the two-parameter Fermi–Dirac neutrino energy distribution for various mean neutrino energies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the detection of low-energy astrophysical neutrinos is searched in conjunction with double beta and neutrinoless double beta decays by employing appropriate nuclear detectors in terrestrial experiments [1–3]. Among the various promising double beta decay detectors the Cd-based semiconductors CdTe and CdZnTe, which are currently used in several areas of physics, have been proposed to be used in the COBRA double beta decay experiment [1]. This experiment has the great advantage that the double beta emitters are part of the detector itself and, in contrast to scintillators, it has good energy resolution [1]. In the present work (see also Refs. [4,5]) our goal is to study the nuclear response to supernova (SN) neutrino energy spectra of all stable nuclear isotopes of the COBRA detector [4,5]. In the semiconductor CdZnTe the stable Zn isotopes play an important role [1] and a systematic investigation of their response as low-energy neutrino detectors has not yet been addressed. In the present paper, we firstly focus on the reactions of (anti)neutrinos with 64,66 Zn, the most abundant isotopes in the natural Zn, and calculate their reaction cross-sections within the context of the quasi-particle random phase approximation (QRPA). We present original results for the double differential, d2 σ /dΩ dω and single differential, dσ /dω cross-sections. The folding procedure is subsequently carried out with known neutrino-energy distributions in order to obtain convoluted single differential cross-sections of the type [dσ (ω)/dω]fold [4,5]. 2. Brief description of the formalism In a terrestrial nuclear detector, but also in the stellar interior, the low energy neutrinos (εν ≤ 100 MeV) interact with nuclei via neutral current reactions described by
νx (˜νx ) + (A, Z ) −→ νx (˜νx ) + (A, Z )∗ , ∗
Corresponding author at: Theoretical Physics Section, University of Ioannina, GR-45110 Ioannina, Greece. E-mail address: [email protected] (V. Tsakstara).
0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.01.046
(1)
V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
431
where x = e, µ, τ . νe and ν˜ e interact also through the charged current reactions as
νe (˜νe ) + (A, Z ) −→ e− (e+ ) + (A, Z ± 1)∗ ,
(2)
while others, νµ , ν˜ µ , ντ , ν˜ τ do not participate in charged current reactions since they do not have sufficient energies to produce the heavy leptons µ± and τ ± . This is why from a collapsing star νµ , ν˜ µ , ντ and ν˜ τ neutrinos are emitted with higher average energies [6,7]. On the other hand, due to the fact that νe interact more than the ν˜ e (because of the neutronrich matter of the stellar environment), they have lower average energy as compared to that of the ν˜ e . Neutrino interactions in nuclei are mediated by boson exchange (W ± for the charged and Z for the neutral current reactions) between the neutrino (lepton sector) and the nucleon (baryon sector). The lepton sector is relevant to fundamental properties of neutrinos and weak interactions, while the nucleon sector is relevant to the nuclear weak responses we are interested in in the present work. In the incoming-neutrino energy range considered in the present paper, the weak interaction neutrino–nucleus Hamiltonian is written according to the standard model in the usual effective current–current form as [8]
∫
ˆ W = − √G H 2
ˆµ d3 xjlept µ (x)J (x).
(3)
µ (G = 1.1664 × 10−5 GeV−2 is the weak coupling constant) where jlept µ and J denote the leptonic and hadronic currents, respectively. According to V-A theory, the leptonic current takes the form
¯ jlept µ = ψνℓ (x)γµ (1 − γ5 )ψνℓ (x),
(4)
where ψνℓ are the (anti)neutrino spinors. From a nuclear physics point of view, only the hadronic current of Eq. (3) is of interest. Focusing on the neutral current processes, the structure of both vector and axial–vector components (neglecting the pseudo-scalar contributions) is written as (isospin representation)
Jλ = ψ N
F1Z
(qµ )γλ + 2
F2Z
(qµ ) 2
iσλν qν 2M
+
FAZ
(qµ )γλ γ5 τ0 ψN , 2
(5)
(τ0 = +1 for protons and τ0 = −1 for neutrons). The superscript Z denotes Z -exchange (neutral current) processes, ψN represent the nucleon (proton or neutron) spinors with mass M and γλ , γ5 , σλν are the Dirac matrices. In Eq. (5), FiZ , i = 1, 2, represent the weak nucleon form factors given in terms of the well known charge and p electromagnetic form factors for protons (F1,2 ) and neutrons (F1n,2 ) by the expressions (CVC-theory) Z (p,n)
F1,2
=
1 2
− sin2 θW F1p,,2n τ0 − sin2 θW F1p,,2n
(6)
(θW is the Weinberg angle, sin2 θW = 0.2325). Also FAZ stands for the neutral current axial–vector form factor, FAZ = 1 F (q2 )τ , where for FA (q2µ ) we employ the dipole ansatz and use as static value (at q = 0) gA = −1.258. We note that 2 A µ 0 in the present work, no-quenching effect was taken into consideration for the axial vector coupling constant. In the considered neutral-current processes, the low energy neutrinos (or anti-neutrinos) with initial four-momentum π ki = (εi , ki ) are elastically or inelastically scattered from Zn isotopes. The initial nuclear state (|i⟩ ≡ |Ji i Mi ⟩) is supposed πf
π
to be the ground state, |Ji i ⟩ = |0+ ⟩ state. After the reaction, the nucleus is generally left in an excited state |f ⟩ ≡ |Jf Mf ⟩ (Mf and Mi are the magnetic quantum numbers). The final and initial states are assumed to have well defined spin J and parity π . The double differential cross-section of the reactions (1) are written as d2 σi→f
dΩ dω
ν/¯ν
−
1
sf ,si ,Mf ,Mi
(2Ji + 1)
= (2π )4 kf εf
−
|⟨f |Hˆ W |i⟩|2 ,
(7)
where εf (kf ) represents the energy (momentum) of the outgoing lepton. After applying a multipole analysis of the weak hadronic current as in Donnelly–Walecka method [8–10], the double differential cross section of Eq. (7) reads d2 σi→f
dΩ dω
ν/˜ν
= δ(Ef − Ei − ω)
2G2 εf2
cos2
∞ θ− Jf 2
2 ω M J (q) + L J (q) Ji q
π (2Ji + 1) J =0 2 ∞ 1 qµ θ − + − 2 + tan2 |⟨Jf ‖TJ mag (q)‖Ji ⟩|2 + |⟨Jf ‖TJ el (q)‖Ji ⟩|2
2q
∓ 2 tan
θ 2
2
−
q2µ q2
+ tan
J =1
2
θ 2
1/2
∞ − J =1
ℜe⟨Jf ‖TJ mag (q)‖Ji ⟩⟨Jf ‖TJ el (q)‖Ji ⟩∗ .
(8)
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V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
The δ -function in the right hand side of the latter equation denotes the energy conservation, as
ω = E f − E i = εi − εf , where ω is the excitation energy of the nucleus, Ei and Ef represent the energy of the initial (ground) and final state of the J , L J , TJ el and TJ mag are given in Refs. [8–11]. studied nucleus, respectively. The definitions of the multipole operators M Eq. (8) is also written in the compact way d2 σi→f
dΩ dω
2G2 εf2 cos2 (θ /2)
=
π (2Ji + 1)
ν/˜ν
[CV + CA + CVA ] ,
(9)
where the term σV is a summation over the contributions of the vector multipole operators as CV =
∞ − Jf J =0
+
∞ −
2 M Ji J (q) + ω L ( q ) J q
−
J =1
q2µ
2
2q2
+ tan
θ
2
mag Tjel (q)‖Ji ⟩|2 . |⟨Jf ‖ Tj (q)‖Ji ⟩|2 + |⟨Jf ‖
(10)
The term σA is a summation over the contributions of the axial vector multipole operators as
σA =
∞ − Jf J =0
+
∞ −
2 5 5 M Ji J (q) + ω L ( q ) J q
−
J =1
q2µ
2
2q2
+ tan
θ
2
mag5 |⟨Jf ‖ Tj (q)‖Ji ⟩|2 + |⟨Jf ‖ Tjel5 (q)‖Ji ⟩|2 .
(11)
The term CVA contains the product of transverse vector and transverse axial vector contributions as CVA = ∓2 tan
θ 2
−
q2µ q2
2
+ tan
1/2
θ
∞ −
2
ℜe⟨Jf ‖TJ mag (q)‖Ji ⟩⟨Jf ‖TJ el (q)‖Ji ⟩∗ .
(12)
J =1 mag5
For normal parity transitions, CVA contains contributions of TJel and TJ
operators and for abnormal parity transitions it
mag
contains matrix elements of TJ and TJel5 . In the last expression as well as in Eq. (8) the (−) sign corresponds to scattering of neutrinos and the (+) to scattering of anti-neutrinos. The magnitude of the 3-momentum transfer q and the square of the 4-momentum transfer q2µ appearing in Eqs. (5), (8) and (10)–(12) are written in terms of the kinematical parameters (the laboratory scattering angle θ and the lepton energies εi and εf ) as
1/2
q = |q| = ω2 + 4εi εf sin2 (θ /2)
,
q2µ = −4εi εf sin2 (θ /2).
(our convention here is that of Ref. [11]). 3. Results and discussion In our neutral-current neutrino scattering processes 64,66
Zn(ν, ν ′ )64,66 Zn∗ , π
64,66
Zn( ν, ν ′ )64,66 Zn∗ ,
(13)
the ground states (|J ⟩ = |0 ⟩) of the 64,66 Zn isotopes are constructed by solving iteratively the BCS equations while the final (excited) states are obtained within the context of a pp–nn type quasi-particle random phase approximation (QRPA) as described in Ref. [11]. In the first step of the calculational procedure, the original results for the double differential crosssection d2 σ /dΩ dω of Eq. (8) are obtained for both nuclear isotopes, 64,66 Zn. Fig. 1, shows the variation of double differential cross-section as a function of the excitation energy of the nucleus ω and the scattering angle θ of the outgoing lepton, for 64 Zn (left) and 66 Zn (right). For all excitation energies ω in the range 0 < ω < 25 − 30 MeV, the cross-section is clearly backward peaked (θ ≈ 180), a result that comes from the contribution of the transverse terms of the neutrino–64,66 Zn interaction. In the next step, single differential cross-sections, dσ /dω, were obtained by integrating numerically over angles Eq. (9). J ) and As can be seen from Eqs. (9)–(12), for J π = 0± transitions, the interference term vanishes, i.e., only the Coulomb (M π ± longitudinal (LJ ) components of the hadronic current contribute to the cross-sections (for J = 0 CVA = 0) and, therefore, they are the same both for neutrino and anti-neutrino reactions. Table 1 illustrates our results for the J π = 0− multipole states which come from the axial vector component of the hadronic current (abnormal parity transitions). In Table 2, we tabulate the total contribution to dσ /dω of each of the multi-polarities 0− , 0+ , 1− , 1+ , 2− , 2+ , 3− , 3+ for the reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ (incoming neutrino energy εν = 40 MeV). The important +
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433
Fig. 1. The double differential cross section d2 σ /dΩ dω as function of the excitation energy ω of the nucleus and the laboratory scattering angle θ : for Zn (left) and 66 Zn (right). The incoming neutrino energy is εν = 40 MeV. The surfaces correspond to the J π = 2+ multi-polarity but similar pictures come out of the results of other pronounced multi-polarities (1+ , 1− , etc.) 64
Table 1 Individual contributions of the 0− multipole states into the differential cross-section dσ /dω for the reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ . Apparently, only the axial vector component of the hadronic current contributes to the 0− multi-polarity. The initial neutrino energy used is εν = 40 MeV. 64
66
Zn
ω (MeV)
dσ /dω (10
4.572 6.293 6.842 7.541 8.937 9.017 9.897 10.320 11.218 12.380 12.849 13.203 13.395 14.354 15.621 20.018
172.130 1.603 30.590 5.064 3.524 3.885 4.586 0.576 1.226 0.064 0.283 0.088 0.531 2.303 0.024 2.441
−42
2
cm MeV
−1
)
Zn
ω(MeV)
dσ /dω(10−42 cm2 MeV−1 )
6.242 7.117 7.921 8.438 8.802 9.496 9.799 9.918 11.343 12.186 13.021 13.158 13.616 13.903 15.972 19.185
58.323 0.056 19.665 1.051 21.186 3.732 0.144 9.075 0.429 0.281 1.878 1.275 2.261 0.207 0.299 3.012
Table 2 − + − + − + − + Comparison between the total contributions to dσ /dω|ν and dσ /dω| ν of the multi-polarities 0 , 0 , 1 , 1 , 2 , 2 , 3 and 3 , for the reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ (εν = 40 MeV).
ν/ ν
dσ /dω(×10−40 ) cm2 MeV−1 Dominant multi-polarity
64
Zn
66
Zn
ν ν ν ν
Total
0−
0+
1−
1+
2−
2+
3−
3+
2.289 2.289 1.229 1.229
4.773 4.773 5.176 5.176
36.522 34.002 39.290 36.041
22.064 24.442 17.800 20.336
1.974 2.378 1.871 2.260
19.629 19.388 21.075 20.900
11.900 11.190 13.988 13.957
0.161 0.186 0.155 0.179
99.312 98.648 100.584 100.078
conclusion which comes out of the results of this table, is the fact that for normal parity transitions, π = (−)J , the crosssections of neutrinos are greater, but for abnormal parity transitions, π = (−)J +1 , the cross-sections of anti-neutrinos exceed those of neutrinos. By summing the rows of Table 2, we obtain the relation between the total cross-sections of neutrino reactions and anti-neutrino reactions for the two isotopes. By including in the summations all multi-polarities (practically those with J π ≤ 8± ) the obtained result is tot dσ
dω
ν
≃ 0.98
tot dσ
. dω ν
We note that, in the present work the spuriousity of the 1− multipolarity has not been studied in detail. In Fig. 2, we compare the differential cross-sections dσ /dω(ω) of neutrino reactions with those of the anti-neutrino ones (see Eq. (13)) obtained for the 2+ multi-polarities (the incoming neutrino energy is εν = 40 MeV). It is clear that the overall cross-sections for neutrinos are larger although for some individual states it may happen the opposite. This is also illustrated in Fig. 3, where
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V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
Fig. 2. Single differential cross-section dσ /dω as a function of the excitation energy ω of the nucleus for neutrinos and anti-neutrinos in 64 Zn (left) and 66 Zn (right), for the 2+ multipole states.
Fig. 3. Difference between dσ /dω|ν of the reaction Zn(ν, ν ′ )Zn∗ and dσ /dω| ν, ν ′ )Zn∗ for the 64 Zn (left) and 66 Zn (right), respectively. ν of the reaction Zn( These results refer to the dominant multipole states 2+ .
we plot the difference dσ /dω|ν − dσ /dω| ν = −2[dσ /dω]VA , i.e. the differential cross-section of neutrinos minus that of anti-neutrinos which is twice the interference term of vector and axial vector contribution, versus the excitation energy ω for all the 2+ multipole states, for 64 Zn (left) and 66 Zn (right) isotopes. We note that we have paid special attention on the 2+ multi-polarity because most of the low-lying excitations of the isotopes 64,66 Zn are 2+ states [12]. In order to estimate the response of a nuclear isotope to a specific source of neutrinos, the original differential crosssections of neutrino-nucleus reactions must be folded with the neutrino energy distribution of the source in question. In the case of the single differential cross-section dσ (εν , ω)/dω, the folding is defined by the expression [5]
[
dσ (ω) dω
]
∫
∞
= fold
ω
dσ (εν , ω) dω
η(εν )dεν ,
(14)
where the η(εν ) represent the neutrino-energy distribution of the assumed neutrino source. For SN-neutrinos we are interested in the present work, a two-parameter Fermi–Dirac or Power-Law distribution is appropriate [6]. In the case of 66 Zn, such results obtained by folding the cross-section dσ /dω with a Fermi–Dirac distribution are shown in Fig. 4. More specifically, in this figure we show the mean energy ⟨εν ⟩ dependence of [dσ (ω)/dω]fold in a set of excitation energies ω in which [dσ (ω)/dω]fold presents pronounced peaks. The value of the degeneracy parameter was ηdg = 2.7 and the mean energies were ⟨εν ⟩ = 12, 16, 20 and 24 MeV. We conclude that, the folded differential cross-sections increase with the mean energy (or the temperature) ⟨εν ⟩. This increase is larger at higher excitation energies, region ω = 25–30 MeV, which means that signals of supernova neutrinos of type νx and νx , x = µ, τ (high mean energies), cause stronger responses at high excitations of the detector. Before closing, it is worth mentioning that, in extracting information through experiments based on neutral current neutrino–nucleus scattering, the most important problem is not only the very small interaction cross-sections, but also the
V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
435
Fig. 4. Folded differential cross-section for the reaction 66 Zn(ν, ν ′ )66 Zn∗ , averaged over a Fermi–Dirac distribution with mean energies ⟨εν ⟩ = 12, 16, 20 and 24 MeV.
difficulty to have a visible signal (e.g. for a detection of a γ de-excitation below the nucleon emission threshold a huge-mass detector is needed) [1–3]. 4. Conclusions We calculated state-by-state the contributions to the double and single differential cross-sections of the reactions Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ . We found that the dominant multi-polarities are the J π = 1+ , 1− and 2+ . We paid special attention on the individual cross-sections of the 2+ multipole states. The convolution showed that the folded cross-sections for a specific neutrino detector are strongly dependent on the mean energy ⟨εν ⟩ of the incoming neutrino, especially for high lying nuclear excitations. Also there is a clear temperature (T ) increase of the folded cross sections for a fixed value of the width w of the neutrino energy distribution. 64,66
Acknowledgements This research was supported by the Π ENE∆ No. 03E∆807 project of the General Secretariat for Research and Technology of the Hellenic Ministry of Development and the Helmholtz International Center for Facility for Antiproton and Ion Research (HIC for FAIR) within the framework of the LOEWE program. References [1] K. Zuber, Phys. Lett. B 519 (2001) 1; K. Zuber, Prog. Part. Nucl. Phys. 57 (2006) 235. [2] H. Ejiri, Phys. Rep. 338 (2000) 265. [3] T.S. Kosmas, V. Tsakstara, J. Phys. Conf. Ser. 203 (2010) 012090. [4] V. Tsakstara, T.S. Kosmas, P.C. Divari, J. Phys. Conf. Ser. 203 (2010) 012093. [5] V. Tsakstara, T.S. Kosmas, P.C. Divari, J. Sinatkas, Prog. Part. Nucl. Phys. 64 (2010) 411. [6] H.-T. Janka, B. Muller, Phys. Rep. 256 (1995) 135. [7] K. Langanke, Acta Phys. Polon. B 39 (2008) 265. [8] T.W. Donnelly, R.D. Peccei, Phys. Rep. 50 (1979) 1. [9] T.W. Donnelly, J.D. Walecka, Nuclear Phys. A 274 (1976) 368. [10] C. Haxton, Phys. Rev. D 36 (1987) 2283. [11] V.C. Chasioti, T.S. Kosmas, Nuclear Phys. A 829 (2009) 234. [12] T.S. Kosmas, V. Tsakstara, P.C. Divari, J. Sinatkas, AIP Conf. Proc. 1180 (2009) 140.
Contents lists available at ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
Neutrino and anti-neutrino scattering off Zn isotopes with quasi-particle RPA calculations V. Tsakstara a,c,∗ , T.S. Kosmas a,b,c , J. Sinatkas d a
Theoretical Physics Section, University of Ioannina, GR-45110 Ioannina, Greece
b
GSI Theoretical Physics Devision, D-64291 Darmstadt, Germany
c
Institute of Nuclear Physics, TU Darmstadt, D-64289, Germany
d
Informatics & Computer Technology Department, TEI of W. Macedonia, GR-52100, Greece
article
info
Keywords: Neutrino nucleus cross sections Supernova neutrino detection Nuclear detector response Quasi-particle random phase approximation
abstract We performed detailed calculations for the cross-sections of (anti)neutrinos with Zn isotopes, contents of the COBRA double beta decay detector at Gran Sasso. We utilized a quasi-particle random phase approximation (QRPA) relying on realistic two-body forces. We present and discuss original double and single differential cross-sections of the neutral current reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ . The response of these isotopes to the energy-spectra of supernova neutrinos is studied by convoluting the original cross-sections with the two-parameter Fermi–Dirac neutrino energy distribution for various mean neutrino energies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the detection of low-energy astrophysical neutrinos is searched in conjunction with double beta and neutrinoless double beta decays by employing appropriate nuclear detectors in terrestrial experiments [1–3]. Among the various promising double beta decay detectors the Cd-based semiconductors CdTe and CdZnTe, which are currently used in several areas of physics, have been proposed to be used in the COBRA double beta decay experiment [1]. This experiment has the great advantage that the double beta emitters are part of the detector itself and, in contrast to scintillators, it has good energy resolution [1]. In the present work (see also Refs. [4,5]) our goal is to study the nuclear response to supernova (SN) neutrino energy spectra of all stable nuclear isotopes of the COBRA detector [4,5]. In the semiconductor CdZnTe the stable Zn isotopes play an important role [1] and a systematic investigation of their response as low-energy neutrino detectors has not yet been addressed. In the present paper, we firstly focus on the reactions of (anti)neutrinos with 64,66 Zn, the most abundant isotopes in the natural Zn, and calculate their reaction cross-sections within the context of the quasi-particle random phase approximation (QRPA). We present original results for the double differential, d2 σ /dΩ dω and single differential, dσ /dω cross-sections. The folding procedure is subsequently carried out with known neutrino-energy distributions in order to obtain convoluted single differential cross-sections of the type [dσ (ω)/dω]fold [4,5]. 2. Brief description of the formalism In a terrestrial nuclear detector, but also in the stellar interior, the low energy neutrinos (εν ≤ 100 MeV) interact with nuclei via neutral current reactions described by
νx (˜νx ) + (A, Z ) −→ νx (˜νx ) + (A, Z )∗ , ∗
Corresponding author at: Theoretical Physics Section, University of Ioannina, GR-45110 Ioannina, Greece. E-mail address: [email protected] (V. Tsakstara).
0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.01.046
(1)
V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
431
where x = e, µ, τ . νe and ν˜ e interact also through the charged current reactions as
νe (˜νe ) + (A, Z ) −→ e− (e+ ) + (A, Z ± 1)∗ ,
(2)
while others, νµ , ν˜ µ , ντ , ν˜ τ do not participate in charged current reactions since they do not have sufficient energies to produce the heavy leptons µ± and τ ± . This is why from a collapsing star νµ , ν˜ µ , ντ and ν˜ τ neutrinos are emitted with higher average energies [6,7]. On the other hand, due to the fact that νe interact more than the ν˜ e (because of the neutronrich matter of the stellar environment), they have lower average energy as compared to that of the ν˜ e . Neutrino interactions in nuclei are mediated by boson exchange (W ± for the charged and Z for the neutral current reactions) between the neutrino (lepton sector) and the nucleon (baryon sector). The lepton sector is relevant to fundamental properties of neutrinos and weak interactions, while the nucleon sector is relevant to the nuclear weak responses we are interested in in the present work. In the incoming-neutrino energy range considered in the present paper, the weak interaction neutrino–nucleus Hamiltonian is written according to the standard model in the usual effective current–current form as [8]
∫
ˆ W = − √G H 2
ˆµ d3 xjlept µ (x)J (x).
(3)
µ (G = 1.1664 × 10−5 GeV−2 is the weak coupling constant) where jlept µ and J denote the leptonic and hadronic currents, respectively. According to V-A theory, the leptonic current takes the form
¯ jlept µ = ψνℓ (x)γµ (1 − γ5 )ψνℓ (x),
(4)
where ψνℓ are the (anti)neutrino spinors. From a nuclear physics point of view, only the hadronic current of Eq. (3) is of interest. Focusing on the neutral current processes, the structure of both vector and axial–vector components (neglecting the pseudo-scalar contributions) is written as (isospin representation)
Jλ = ψ N
F1Z
(qµ )γλ + 2
F2Z
(qµ ) 2
iσλν qν 2M
+
FAZ
(qµ )γλ γ5 τ0 ψN , 2
(5)
(τ0 = +1 for protons and τ0 = −1 for neutrons). The superscript Z denotes Z -exchange (neutral current) processes, ψN represent the nucleon (proton or neutron) spinors with mass M and γλ , γ5 , σλν are the Dirac matrices. In Eq. (5), FiZ , i = 1, 2, represent the weak nucleon form factors given in terms of the well known charge and p electromagnetic form factors for protons (F1,2 ) and neutrons (F1n,2 ) by the expressions (CVC-theory) Z (p,n)
F1,2
=
1 2
− sin2 θW F1p,,2n τ0 − sin2 θW F1p,,2n
(6)
(θW is the Weinberg angle, sin2 θW = 0.2325). Also FAZ stands for the neutral current axial–vector form factor, FAZ = 1 F (q2 )τ , where for FA (q2µ ) we employ the dipole ansatz and use as static value (at q = 0) gA = −1.258. We note that 2 A µ 0 in the present work, no-quenching effect was taken into consideration for the axial vector coupling constant. In the considered neutral-current processes, the low energy neutrinos (or anti-neutrinos) with initial four-momentum π ki = (εi , ki ) are elastically or inelastically scattered from Zn isotopes. The initial nuclear state (|i⟩ ≡ |Ji i Mi ⟩) is supposed πf
π
to be the ground state, |Ji i ⟩ = |0+ ⟩ state. After the reaction, the nucleus is generally left in an excited state |f ⟩ ≡ |Jf Mf ⟩ (Mf and Mi are the magnetic quantum numbers). The final and initial states are assumed to have well defined spin J and parity π . The double differential cross-section of the reactions (1) are written as d2 σi→f
dΩ dω
ν/¯ν
−
1
sf ,si ,Mf ,Mi
(2Ji + 1)
= (2π )4 kf εf
−
|⟨f |Hˆ W |i⟩|2 ,
(7)
where εf (kf ) represents the energy (momentum) of the outgoing lepton. After applying a multipole analysis of the weak hadronic current as in Donnelly–Walecka method [8–10], the double differential cross section of Eq. (7) reads d2 σi→f
dΩ dω
ν/˜ν
= δ(Ef − Ei − ω)
2G2 εf2
cos2
∞ θ− Jf 2
2 ω M J (q) + L J (q) Ji q
π (2Ji + 1) J =0 2 ∞ 1 qµ θ − + − 2 + tan2 |⟨Jf ‖TJ mag (q)‖Ji ⟩|2 + |⟨Jf ‖TJ el (q)‖Ji ⟩|2
2q
∓ 2 tan
θ 2
2
−
q2µ q2
+ tan
J =1
2
θ 2
1/2
∞ − J =1
ℜe⟨Jf ‖TJ mag (q)‖Ji ⟩⟨Jf ‖TJ el (q)‖Ji ⟩∗ .
(8)
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The δ -function in the right hand side of the latter equation denotes the energy conservation, as
ω = E f − E i = εi − εf , where ω is the excitation energy of the nucleus, Ei and Ef represent the energy of the initial (ground) and final state of the J , L J , TJ el and TJ mag are given in Refs. [8–11]. studied nucleus, respectively. The definitions of the multipole operators M Eq. (8) is also written in the compact way d2 σi→f
dΩ dω
2G2 εf2 cos2 (θ /2)
=
π (2Ji + 1)
ν/˜ν
[CV + CA + CVA ] ,
(9)
where the term σV is a summation over the contributions of the vector multipole operators as CV =
∞ − Jf J =0
+
∞ −
2 M Ji J (q) + ω L ( q ) J q
−
J =1
q2µ
2
2q2
+ tan
θ
2
mag Tjel (q)‖Ji ⟩|2 . |⟨Jf ‖ Tj (q)‖Ji ⟩|2 + |⟨Jf ‖
(10)
The term σA is a summation over the contributions of the axial vector multipole operators as
σA =
∞ − Jf J =0
+
∞ −
2 5 5 M Ji J (q) + ω L ( q ) J q
−
J =1
q2µ
2
2q2
+ tan
θ
2
mag5 |⟨Jf ‖ Tj (q)‖Ji ⟩|2 + |⟨Jf ‖ Tjel5 (q)‖Ji ⟩|2 .
(11)
The term CVA contains the product of transverse vector and transverse axial vector contributions as CVA = ∓2 tan
θ 2
−
q2µ q2
2
+ tan
1/2
θ
∞ −
2
ℜe⟨Jf ‖TJ mag (q)‖Ji ⟩⟨Jf ‖TJ el (q)‖Ji ⟩∗ .
(12)
J =1 mag5
For normal parity transitions, CVA contains contributions of TJel and TJ
operators and for abnormal parity transitions it
mag
contains matrix elements of TJ and TJel5 . In the last expression as well as in Eq. (8) the (−) sign corresponds to scattering of neutrinos and the (+) to scattering of anti-neutrinos. The magnitude of the 3-momentum transfer q and the square of the 4-momentum transfer q2µ appearing in Eqs. (5), (8) and (10)–(12) are written in terms of the kinematical parameters (the laboratory scattering angle θ and the lepton energies εi and εf ) as
1/2
q = |q| = ω2 + 4εi εf sin2 (θ /2)
,
q2µ = −4εi εf sin2 (θ /2).
(our convention here is that of Ref. [11]). 3. Results and discussion In our neutral-current neutrino scattering processes 64,66
Zn(ν, ν ′ )64,66 Zn∗ , π
64,66
Zn( ν, ν ′ )64,66 Zn∗ ,
(13)
the ground states (|J ⟩ = |0 ⟩) of the 64,66 Zn isotopes are constructed by solving iteratively the BCS equations while the final (excited) states are obtained within the context of a pp–nn type quasi-particle random phase approximation (QRPA) as described in Ref. [11]. In the first step of the calculational procedure, the original results for the double differential crosssection d2 σ /dΩ dω of Eq. (8) are obtained for both nuclear isotopes, 64,66 Zn. Fig. 1, shows the variation of double differential cross-section as a function of the excitation energy of the nucleus ω and the scattering angle θ of the outgoing lepton, for 64 Zn (left) and 66 Zn (right). For all excitation energies ω in the range 0 < ω < 25 − 30 MeV, the cross-section is clearly backward peaked (θ ≈ 180), a result that comes from the contribution of the transverse terms of the neutrino–64,66 Zn interaction. In the next step, single differential cross-sections, dσ /dω, were obtained by integrating numerically over angles Eq. (9). J ) and As can be seen from Eqs. (9)–(12), for J π = 0± transitions, the interference term vanishes, i.e., only the Coulomb (M π ± longitudinal (LJ ) components of the hadronic current contribute to the cross-sections (for J = 0 CVA = 0) and, therefore, they are the same both for neutrino and anti-neutrino reactions. Table 1 illustrates our results for the J π = 0− multipole states which come from the axial vector component of the hadronic current (abnormal parity transitions). In Table 2, we tabulate the total contribution to dσ /dω of each of the multi-polarities 0− , 0+ , 1− , 1+ , 2− , 2+ , 3− , 3+ for the reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ (incoming neutrino energy εν = 40 MeV). The important +
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433
Fig. 1. The double differential cross section d2 σ /dΩ dω as function of the excitation energy ω of the nucleus and the laboratory scattering angle θ : for Zn (left) and 66 Zn (right). The incoming neutrino energy is εν = 40 MeV. The surfaces correspond to the J π = 2+ multi-polarity but similar pictures come out of the results of other pronounced multi-polarities (1+ , 1− , etc.) 64
Table 1 Individual contributions of the 0− multipole states into the differential cross-section dσ /dω for the reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ . Apparently, only the axial vector component of the hadronic current contributes to the 0− multi-polarity. The initial neutrino energy used is εν = 40 MeV. 64
66
Zn
ω (MeV)
dσ /dω (10
4.572 6.293 6.842 7.541 8.937 9.017 9.897 10.320 11.218 12.380 12.849 13.203 13.395 14.354 15.621 20.018
172.130 1.603 30.590 5.064 3.524 3.885 4.586 0.576 1.226 0.064 0.283 0.088 0.531 2.303 0.024 2.441
−42
2
cm MeV
−1
)
Zn
ω(MeV)
dσ /dω(10−42 cm2 MeV−1 )
6.242 7.117 7.921 8.438 8.802 9.496 9.799 9.918 11.343 12.186 13.021 13.158 13.616 13.903 15.972 19.185
58.323 0.056 19.665 1.051 21.186 3.732 0.144 9.075 0.429 0.281 1.878 1.275 2.261 0.207 0.299 3.012
Table 2 − + − + − + − + Comparison between the total contributions to dσ /dω|ν and dσ /dω| ν of the multi-polarities 0 , 0 , 1 , 1 , 2 , 2 , 3 and 3 , for the reactions 64,66 Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ (εν = 40 MeV).
ν/ ν
dσ /dω(×10−40 ) cm2 MeV−1 Dominant multi-polarity
64
Zn
66
Zn
ν ν ν ν
Total
0−
0+
1−
1+
2−
2+
3−
3+
2.289 2.289 1.229 1.229
4.773 4.773 5.176 5.176
36.522 34.002 39.290 36.041
22.064 24.442 17.800 20.336
1.974 2.378 1.871 2.260
19.629 19.388 21.075 20.900
11.900 11.190 13.988 13.957
0.161 0.186 0.155 0.179
99.312 98.648 100.584 100.078
conclusion which comes out of the results of this table, is the fact that for normal parity transitions, π = (−)J , the crosssections of neutrinos are greater, but for abnormal parity transitions, π = (−)J +1 , the cross-sections of anti-neutrinos exceed those of neutrinos. By summing the rows of Table 2, we obtain the relation between the total cross-sections of neutrino reactions and anti-neutrino reactions for the two isotopes. By including in the summations all multi-polarities (practically those with J π ≤ 8± ) the obtained result is tot dσ
dω
ν
≃ 0.98
tot dσ
. dω ν
We note that, in the present work the spuriousity of the 1− multipolarity has not been studied in detail. In Fig. 2, we compare the differential cross-sections dσ /dω(ω) of neutrino reactions with those of the anti-neutrino ones (see Eq. (13)) obtained for the 2+ multi-polarities (the incoming neutrino energy is εν = 40 MeV). It is clear that the overall cross-sections for neutrinos are larger although for some individual states it may happen the opposite. This is also illustrated in Fig. 3, where
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V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
Fig. 2. Single differential cross-section dσ /dω as a function of the excitation energy ω of the nucleus for neutrinos and anti-neutrinos in 64 Zn (left) and 66 Zn (right), for the 2+ multipole states.
Fig. 3. Difference between dσ /dω|ν of the reaction Zn(ν, ν ′ )Zn∗ and dσ /dω| ν, ν ′ )Zn∗ for the 64 Zn (left) and 66 Zn (right), respectively. ν of the reaction Zn( These results refer to the dominant multipole states 2+ .
we plot the difference dσ /dω|ν − dσ /dω| ν = −2[dσ /dω]VA , i.e. the differential cross-section of neutrinos minus that of anti-neutrinos which is twice the interference term of vector and axial vector contribution, versus the excitation energy ω for all the 2+ multipole states, for 64 Zn (left) and 66 Zn (right) isotopes. We note that we have paid special attention on the 2+ multi-polarity because most of the low-lying excitations of the isotopes 64,66 Zn are 2+ states [12]. In order to estimate the response of a nuclear isotope to a specific source of neutrinos, the original differential crosssections of neutrino-nucleus reactions must be folded with the neutrino energy distribution of the source in question. In the case of the single differential cross-section dσ (εν , ω)/dω, the folding is defined by the expression [5]
[
dσ (ω) dω
]
∫
∞
= fold
ω
dσ (εν , ω) dω
η(εν )dεν ,
(14)
where the η(εν ) represent the neutrino-energy distribution of the assumed neutrino source. For SN-neutrinos we are interested in the present work, a two-parameter Fermi–Dirac or Power-Law distribution is appropriate [6]. In the case of 66 Zn, such results obtained by folding the cross-section dσ /dω with a Fermi–Dirac distribution are shown in Fig. 4. More specifically, in this figure we show the mean energy ⟨εν ⟩ dependence of [dσ (ω)/dω]fold in a set of excitation energies ω in which [dσ (ω)/dω]fold presents pronounced peaks. The value of the degeneracy parameter was ηdg = 2.7 and the mean energies were ⟨εν ⟩ = 12, 16, 20 and 24 MeV. We conclude that, the folded differential cross-sections increase with the mean energy (or the temperature) ⟨εν ⟩. This increase is larger at higher excitation energies, region ω = 25–30 MeV, which means that signals of supernova neutrinos of type νx and νx , x = µ, τ (high mean energies), cause stronger responses at high excitations of the detector. Before closing, it is worth mentioning that, in extracting information through experiments based on neutral current neutrino–nucleus scattering, the most important problem is not only the very small interaction cross-sections, but also the
V. Tsakstara et al. / Progress in Particle and Nuclear Physics 66 (2011) 430–435
435
Fig. 4. Folded differential cross-section for the reaction 66 Zn(ν, ν ′ )66 Zn∗ , averaged over a Fermi–Dirac distribution with mean energies ⟨εν ⟩ = 12, 16, 20 and 24 MeV.
difficulty to have a visible signal (e.g. for a detection of a γ de-excitation below the nucleon emission threshold a huge-mass detector is needed) [1–3]. 4. Conclusions We calculated state-by-state the contributions to the double and single differential cross-sections of the reactions Zn(ν, ν ′ )64,66 Zn∗ and 64,66 Zn( ν, ν ′ )64,66 Zn∗ . We found that the dominant multi-polarities are the J π = 1+ , 1− and 2+ . We paid special attention on the individual cross-sections of the 2+ multipole states. The convolution showed that the folded cross-sections for a specific neutrino detector are strongly dependent on the mean energy ⟨εν ⟩ of the incoming neutrino, especially for high lying nuclear excitations. Also there is a clear temperature (T ) increase of the folded cross sections for a fixed value of the width w of the neutrino energy distribution. 64,66
Acknowledgements This research was supported by the Π ENE∆ No. 03E∆807 project of the General Secretariat for Research and Technology of the Hellenic Ministry of Development and the Helmholtz International Center for Facility for Antiproton and Ion Research (HIC for FAIR) within the framework of the LOEWE program. References [1] K. Zuber, Phys. Lett. B 519 (2001) 1; K. Zuber, Prog. Part. Nucl. Phys. 57 (2006) 235. [2] H. Ejiri, Phys. Rep. 338 (2000) 265. [3] T.S. Kosmas, V. Tsakstara, J. Phys. Conf. Ser. 203 (2010) 012090. [4] V. Tsakstara, T.S. Kosmas, P.C. Divari, J. Phys. Conf. Ser. 203 (2010) 012093. [5] V. Tsakstara, T.S. Kosmas, P.C. Divari, J. Sinatkas, Prog. Part. Nucl. Phys. 64 (2010) 411. [6] H.-T. Janka, B. Muller, Phys. Rep. 256 (1995) 135. [7] K. Langanke, Acta Phys. Polon. B 39 (2008) 265. [8] T.W. Donnelly, R.D. Peccei, Phys. Rep. 50 (1979) 1. [9] T.W. Donnelly, J.D. Walecka, Nuclear Phys. A 274 (1976) 368. [10] C. Haxton, Phys. Rev. D 36 (1987) 2283. [11] V.C. Chasioti, T.S. Kosmas, Nuclear Phys. A 829 (2009) 234. [12] T.S. Kosmas, V. Tsakstara, P.C. Divari, J. Sinatkas, AIP Conf. Proc. 1180 (2009) 140.