Application of the Spectral Function Formalism to Electronand Neutrino-Nucleus Scattering

Transcription

1 Application of the Spectral Function Formalism to Electronand Neutrino-Nucleus Scattering Artur M. Ankowski Center for Neutrino Physics, Virginia Tech ECT* Doctoral Training Program 2017 Microscopic Theories of Nuclear Structure, Dynamics and Electroweak Currents Trento, June 28 29, 2017

2 Outline 1) Introduction Neutrino oscillations in a nutshell Accurate neutrino-energy reconstruction requires an accurate modeling of nuclear effects 2) Impulse approximation Why to test nuclear models using electron scattering data Fermi gas model Shell model Spectral function approach Final-state interactions in the spectral function approach 3) Summary 2

3 Neutrino oscillations in a nutshell να ν1, ν2, ν3 να, νβ, νγ source detector ν's produced in a given flavor α, mixture of mass eigenstates different masses propagate with different phases, detected mixture of mass eigenstates is, in general, different; appearance of another flavors, β and γ 3

4 Neutrino oscillations in a nutshell In the simplest case of 2 flavors Example [K. Abe et al. (T2K Collaboration), PRD 91, (2015)] 4

5 Neutrino oscillations in a nutshell In the simplest case of 2 flavors Example [K. Abe et al. (T2K Collaboration), PRD 91, (2015)] 5

6 Neutrino oscillations in a nutshell In the simplest case of 2 flavors Example [K. Abe et al. (T2K Collaboration), PRD 91, (2015)] 6

7 Neutrino oscillations in a nutshell What we do know: X. Qian & P. Vogel, arxiv: What needs to be determined: mass hierarchy (sign of Δm322), θ23 octant (dominant flavor in ν3), CP violation in the lepton sector 7

8 Matter-antimatter asymmetry APS/Alan Stonebraker 8

9 Matter-antimatter asymmetry mixing matrix PDG 2008 The quark contrib. is too low by several orders of magnitude 9

10 Matter-antimatter asymmetry mixing matrix NuFIT 3.0 (2016) PDG

11 Energy reconstruction

12 Neutrino beams Aguilar-Arevalo et al. (MiniBooNE), D 81, (2010) 12

13 What precision are we reaching? J. J.Hignight Hignight(IceCube), (IceCube),APS APSApril AprilMeeting, Meeting,

14 What precision are we reaching? At the T2K kinematics (~600 MeV), 10% uncertainty (current T2K), ~60 MeV 2% uncertainty (current global fits), ~10 MeV At the NOvA and DUNE kinematics, values x4 5. Effects considered to be small need to be accounted for accurately to avoid biases. 14

15 Kinematic reconstruction In quasielastic scattering off free nucleons, v + p l + n and v + n l + p, we can deduce the neutrino energy from the charged lepton's kinematics. No need to reconstruct the nucleon kinematics. E' E'and andθθknown known ME ' +const E= M E ' + k ' cosθ 15

16 Kinematic reconstruction In nuclei the reconstruction becomes an approximation due to the binding energy, Fermi motion, final-state interactions, two-body interactions etc. E' E'and andθθknown known (M ϵ) E ' +const E M ϵ E ' + k ' cos θ 16

17 Free-proton events For targets containing H, the (ν and ν) pion-production events on free protons could be separated out, based on the balance of the transverse momentum. Lu et al., PRD 92, (2015) 17

18 Unknown monochromatic beam Consider the simplest (unrealistic) case: the beam is monochromatic but its energy is unknown and has to be reconstructed E' E'and andθθknown known EE==?? PoS(NuFact2014)004 18

19 Unknown monochromatic e beam Sealock et al., PRL 62, 1350 (1989) E' E'== MeV MeV θθ== deg deg ΔE' ΔE'==55MeV MeV 19

20 Unknown monochromatic e beam Sealock et al., PRL 62, 1350 (1989) E' E'== MeV MeV θθ== deg deg ΔE' ΔE'==55MeV MeV for forϵϵ==25 25MeV MeV EE== MeV MeV ΔE= ΔE=77MeV MeV 20

21 Unknown monochromatic e beam Sealock et al., PRL 62, 1350 (1989) E' E'== MeV MeV θθ== deg deg ΔE' ΔE'==55MeV MeV for forϵϵ==25 25MeV MeV EE== MeV MeV ΔE= ΔE=77MeV MeV true truevalue value EE== MeV MeV 21

22 Unknown monochromatic e beam θ (deg) E' (MeV) ΔE' (MeV) Assuming ϵ = 25 MeV rec. E true E 1285 ± ± ± ± ±

23 Unknown monochromatic e beam θ (deg) E' (MeV) ΔE' (MeV) ± ± ± 3 Appropriate ϵ value? true E ϵ ± ± 5 Sealock et al., PRL 62, 1350 (1989) O'Connell et al., PRC 35, 1063 (1987) Barreau et al., NPA 402, 515 (1983) 23

24 Unknown monochromatic e beam θ (deg) E' (MeV) ΔE' (MeV) ± ± ± 3 Appropriate ϵ value? true E ϵ ± ± 5 22 different E different Q different E different Q different different θθ different different ϵϵ 24

25 Realistic calculations vs Erec 25

26 Realistic calculations vs Erec Same Same physics physics drives drives the the QE QE peak peak position position and and relates relates the the kinematics kinematics to to neutrino neutrino energy energy 26

27 Polychromatic beam In modern experiments, the neutrino beams are not monochromatic, and the energy must be reconstructed from the observables, typically E' and cos θ under the CCQE event hypothesis. E' E'and andθθknown known EE==?? 27

28 CCQE events In practice, CCQE event candidates are defined as containing no pions observed. + CCQE (any number of nucleons) pion production and followed by absorption undetected pions CCQE with pions from FSI 0π events 28

29 Recall the monochromatic-beam case º QE, 1p1h QE, 2p2h Delta 29

30 CCQE events of given l ± kinematics Omar NuFact11, PRL 105, (2010) 30

31 CCQE events of given l ± kinematics Very different processes and neutrino energies contribute to CCQE-like events of a given E' and cos θ. An undetected pion typically lowers the reconstructed energy by ~ MeV. Note that in the reconstruction formula, MΔ = 1232 MeV would be more suitable than M' = 939 MeV. 31

32 Absorbed or undetected pions undetected π absorbed π, irreducible T. T. Leitner Leitner && U. U. Mosel Mosel PRC PRC 81, 81, (2010) (2010)

33 Calorimetric energy reconstruction Advantage: applicable to any final states Insensitive to nuclear effects when missing energy «neutrino energy Otherwise, requires input from nuclear models A.M.A., arxiv: π neutrino neutron nuclear deexcitation (γ, p, n, d, α) possibly delayed charged lepton 33

34 Impulse approximation 34

35 Impulse approximation Assumption: the dominant process of lepton-nucleus interaction is scattering off a single nucleon, with the remaining nucleons acting as a spectator system. 35

36 Impulse approximation Assumption: the dominant process of lepton-nucleus interaction is scattering off a single nucleon, with the remaining nucleons acting as a spectator system. It is valid when the momentum transfer q is high enough, as the probe's spatial resolution is ~1/ q. 36

37 Impulse approximation Hole spectral function Particle spectral function Elementary cross section 37

38 Impulse approximation Describes the ground-state properties of the target nucleus 38

39 Impulse approximation Characterizes the interaction vertex 39

40 Impulse approximation Ensures the energy conservation and Pauli blocking 40

41 Impulse approximation Hole spectral function Particle spectral function Elementary cross section 41

42 Impulse approximation For scattering in a given angle, neutrinos and electrons differ only due to the elementary cross section. In neutrino scattering, uncertainties come from (i) interaction dynamics and (ii) nuclear effects. It is highly improbable that theoretical approaches unable to reproduce (e,e') data would describe nuclear effects in neutrino interactions at similar kinematics. 42

43 Much more than the vector part... electrons muon neutrinos vector part 43

44 How relevant is the precision? Expected sensitivity of DUNE to CP violation as a function of exposure for a νe signal normalization uncertainties between 5% + 1% and 5% + 3%. Acciari et al., arxiv:

45 Off-shell effects Consider a nucleus stable against emission of nucleons. As in its ground state, EA = MA, the energy cannot be decreased by emission of a nucleon EA = EA-1 + Ep < EA-1 + M so the energy of a nucleon in the nucleus is lower than M. V.R. Pandharipande, Nucl. Phys. B (Proc. Suppl.) 112, 51 (2002) 45

46 Off-shell effects In a nuclear model, the initial nucleon's energy may differ from the on-shell energy by a constant be a function of the momentum lack 1:1 correspondence with momentum increasing sophistication 46

47 Off-shell effects The elementary cross section, contains two tensors with only the hadron one affected by off-shell effects. 47

48 Off-shell effects The current appearing in the hadron tensor is known on the mass shell, or equivalently 48

49 Off-shell effects The prescription of de Forest [NPA 392, 232 (1983)]: to approximate the off-shell hadron tensor, one can use the on-shell expression with the same momentum transfer and a modified energy transfer, with 49

50 Off-shell effects The prescription of de Forest [NPA 392, 232 (1983)]: as the initial nucleon's energy is now E p= M 2 + p2 in our calculations, and the final energy is an observable, the energy transfer has to be ω the difference between the lepton ω and hadron ~ is transferred to the spectator system of (A-1) nucleons. 50

51 Off-shell effects Examples of an oversimplified treatment: ~ elem elem(q22) on-shell σ on-shell σ (Q ) Llewellyn-Smith Llewellyn-Smith elem off-shell σ off-shell σelem ~=ω with ω with ω = ω 51

52 Coulomb effects deeper binding acceleration deceleration deeper potential 52

53 Coulomb effects deeper binding acceleration deceleration deeper potential For 12C, it gives * MeV, the difference relevant for CP measurements 53

54 Fermi gas model

55 Fermi gas model Imagine an infinite space filled uniformly with nucleons 55

56 Fermi gas model Due to the translational invariance, the eigenstates can be labeled using momentum, ψ( x)=c e i p x. 56

57 Fermi gas model L π Due to the boundary conditions, pi = +n π 2 2 every state occupies (2 π / L)3 in the momentum space -L/2 +L/2 57

58 Fermi gas model L π Due to the boundary conditions, pi = +n π 2 2 every state occupies (2 π / L)3 in the momentum space pf Momentum space Coordinate space 58

59 Electron scattering carbon, 500 MeV, 60 deg Fermioffgas model Moniz et al., PRL 26, 445 (1971) 59

60 Electron scattering carbon, 500 MeV, 60 deg Fermioffgas model described by ε driven by pf Moniz et al., PRL 26, 445 (1971) 60

61 Fermi gas model Whitney et al., PRC 9, 2230 (1974) 61

62 Fermi gas model What happens at kinematics other than 500 MeV, 60 deg? Barreau et al., NPA 402, 515 (1983) 62

63 Shell model

64 Shell model In a spherically symmetric potential, the eigenstates can be labeled using the total angular momentum. p1/ MeV p3/ MeV s1/ MeV See e.g. Cohen, Concepts of Nuclear Physics, McGraw-Hill,

65 Example: oxygen nucleus Leuschner et al., PRC 49, 955 (1994) 65

66 Example: oxygen spectral function s1/2 p3/2 p1/2 Fermi gas: d - function 44 66

67 Depletion of the shell-model states De Witt Huberts, JPG 16, 507 (1990) 67

68 Depletion of the shell-model states Spectroscopic strengths/ipsm The observed depletion is ~35% for the valence shells (LRC and SRC) and ~20% when higher missing energy is probed (SRC). NIKHEF: 208Pb(e,e'p)207Tl SRC LRC D. Rohe, NuInt05 Benhar et al, PRC 41, R24 (1990)68

69 Spectral function approach

70 Short-range correlations The main source of the depletion of the shell-model states at high E are short-range nucleon-nucleon correlations. Yielding NN pairs (typically pn pairs) with high relative momentum, they move ~20% of nucleons to the states of high removal energies. 70

71 Short-range correlations The hole spectral function can be expressed as describes the contribution of the shell-model states, vanishes at high p or high E relevant only at high p and E 71

72 Short-range correlations Momentum distributions Benhar&Pandharipande, RMP 65, 817 (1993) 72

73 Short-range correlations Momentum distributions SRC don't depend on the shell structure or finite-size effects, only on the density 73

74 Local-density approximation The correlation component in nuclei can be obtained combining the results for infinite nuclear matter obtained at different densities: Benhar et al., NPA , (1994) 74

75 Comparison to C(e, e') data RFG SF data: Baran et al., PRL 61, 400 (1988) 75

76 Energy conservation 76

77 Energy conservation 77

78 Energy conservation 78

79 Energy conservation 79

80 Energy conservation 80

81 Final-state interactions Their effect on the cross section is easy to understand in terms of the complex optical potential: the real part modifies the struck nucleon's energy spectrum: it differes from M 2 + p ' 2 the imaginary part reduces the single-nucleon final states and produces multinucleon final states Horikawa et al., PRC 22, 1680 (1980) 81

82 Final-state interactions In the convolution approach, with the folding function Nucl. Nucl. transparency transparency 82

83 Nuclear transparency Rohe et al., PRC 72, (2005) 83

84 Nuclear transparency no correlations NN correlations reduce FSI O. NuInt05 84

85 Short-range correlations Pair distribution function of NM Benhar et al., PRC 44, 2328 (1991) 85

86 Real part of the optical potential We account for the spectrum modification by This procedure is similar to that from the Fermi gas model to introduce the binding energy in the argument of δ(...). 86

87 Optical potential by Cooper et al. C( C(pp,, p' p')) MeV MeV Deb et al., PRC 72, (2005) C( C(pp,, p' p')) MeV MeV 87

88 Optical potential by Cooper et al. obtained from Cooper et al., PRC 47, 297 (1993) 88

89 Simple comparison Real part of the OP acts in the final state shifts the QE peak to low ω at low q (to high ω at high q ) Binding energy in RFG acts in the initial state shifts the QE peak to high ω low q high q 89

90 Why to focus on quasielastic? Benhar et al., RMP 80, 189 (2008) 90

91 Comparison to C(e, e') data RFG SF data: Baran et al., PRL 61, 400 (1988) 91

92 Comparison to C(e, e') data RFG SF + FSI data: Baran et al., PRL 61, 400 (1988) 92

93 Compared calculations RFG model ε = 25 MeV pf = 221 MeV SF calculation, LDA treatment of Pauli blocking SF calculation without FSI SF calculation, step function 69 93

94 Compared calculations Elastic scattering and excitation of low-ex levels Calcs. include QE by 1-body current only Giant resonance Ex = 22.6 MeV, Γ = 3.2 MeV 70 94

95 Comparisons to C(e,e') data Barreau et al., NPA 402, 515 (1983) 95

96 Comparisons to C(e,e') data Barreau et al., NPA 402, 515 (1983) 96

97 Comparisons to C(e,e') data Barreau et al., NPA 402, 515 (1983) Baran et al., PRL 61, 400 (1988) Whitney et al., PRC 9, 2230 (1974) 97

98 Comparisons to C(e,e') data The supplemental material of PRD 91, (2015) shows comparisons to the data sets collected at 54 kinematical setups energies from ~160 MeV to ~4 GeV, angles from 12 to 145 degrees, at the QE peak, the values of momentum transfer from ~145 to ~1060 MeV/c and 0.02 Q (GeV/c)2. 98

99 CCQE MINERvA data SF calculations with FSI vs. SF calculation without FSI Fields et al., PRL 111, (2013) A. M. A., PRD 92, (2015) Fiorentini et al., PRL 111, (2013) 99

100 CCQE MINERvA data 100

101 Measurement of the spectral function of argon in JLab

102 What do we know about Ar? deg oxygen x argon Anghinolfi et al., JPG 21, L9 (1995) 102

103 What do we know about Ar? nuclear excitations by up to ~11 MeV Cameron & Singh, Nucl. Data Sheets 102, 293 (2004) angular distributions of 40Ar(p, p') for a few excitation lvls. Fabrici et al., PRC 21, 830 & 844 (1980); De Leo et al., PRC 31, 362 (1985); Blanpied et al., PRC 37, 1304 (1988) angular distributions of 40Ar(p,d)39Ar Tonn et al., PRC 16, 1357 (1977) 103

104 What do we know about Ar? n-ar total cross section form energies < 50 MeV Winters et al., PRC 43, 492 (1991) Ar(νe,e) cross section from the mirror 40Ti 40Sc decay 40 Bhattacharya et al., PRC 58, 3677 (1998) Gammov-Teller strength distrib. for 40Ar 40K from 0 (p, n) Bhattacharya et al., PRC 80, (2009) Ar(n, p)40cl cross section between 9 and 15 MeV 40 Bhattacharya et al., PRC 86, (R) (2012) 104

105 Spectral function of Ca 40 Mougey et al., NPA 262, 461 (1976) 105

106 Approximated SF of Ar 40 AMA and J. Sobczyk, PRC 77, (2008) RFG Anghinolfi et al., NPA 602, 405 (1996) 106

107 Experiment E at JLab We propose a measurement of the coincidence (e,e'p) cross section on argon. This data will provide the experimental input indispensable to construct the argon spectral function, thus paving the way for a reliable estimate of the neutrino cross sections. Benhar et al., arxiv:

108 Experiment E at JLab Primary goal: extraction of the proton shell structure of 40Ar from (e,e'p) scattering spectroscopic factors, energy distributions, momentum distributions. Secondary goal: improved description of final-state interactions in the argon nucleus. 108

109 Relevance for DUNE Neutrino oscillations Reduction of systematic uncertainties from nuclear effects, especially for the 2nd oscillation maximum. Proton decay Probed lifetime affected by the partial depletion of the shell-model states. Supernova neutrinos Information on the valence shells essential for accurate simulations and detector design. 109

110 Impulse approximation p' elementary cross section prec = p in IA k' k spectral function nuclear transparency 110

111 (Anti)parallel kinematics, p' q Energy conservation Momentum conservation Impulse Approximation, prec = p 111

112 Neutron spectral function of Ar

113 Kinematic settings Data collected Feb - Mar

114 Expected energy distributions 114

115 Momentum distributions 115

116 Summary An accurate description of nuclear effects, including finalstate interactions, is crucial for an accurate reconstruction of neutrino energy. Theoretical models must be validated against (e,e') data to estimate their uncertainties. The spectral function formalism can be used in Monte Carlo simulations to improve the accuracy of description of nuclear effects. 116

117 Backup slides

118 Vagnoni et al., PRL 118, (2017) 118

119 Conserved current Conserved energy Not SRC, simple π reabsorption: Weinstein et al., PRC 94, (2016) 119

120 Other NC and CC QE data A. M. A, PRC 86, (2012) 120

121 Side remark: relativistic kinematics A.M.A. & O. Benhar, PRC 83, (2011) Sizable differences between the relativistic and nonrelativistic results at neutrino energies ~500 MeV. 121

122 Side remark: relativistic kinematics A.M.A. & O. Benhar, PRC 83, (2011) At q ~540 MeV, semi-relativistic result is 5% lower than the exact cross section. 122

123 Short-range correlations Acciari et al. (ArgoNeuT), PRD 90, (2014) Not SRC, simple π reabsorption: Weinstein et al., PRC 94, (2016) 123