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Relative Flux, FD/ND , using Low- Technique: Part-I H. Duyang, Sanjib R. Mishra, & Xinchun Tian with contributions from Maxim Gonchar & Roberto Petti 01 Low- Idea

  1. Relative Flux, FD/ND , using Low- ν Technique: Part-I H. ¡Duyang, ¡ ¡Sanjib ¡R. ¡Mishra, ¡& ¡Xinchun ¡Tian ¡ with ¡contributions ¡from ¡ ¡ Maxim Gonchar & Roberto Petti 01

  2. Low- ν Idea ¡ ➾ SRM, Wold.Sci. 84(1990), Ed.Geesman 3 Determination of Relative Neutrino Flux The dynamics of neutrino-nucleon scattering implies that the number of events in a given energy bin with E had < ν 0 is proportional to the neutrino (antineutrino) flux in that energy bin up to corrections O ( ν 0 /E ν ) and O ( ν 0 /E ν ) 2 . The method follows from the general expression of the ν -nucleon di ff erential cross section. By invoking the assumptions of locality, Lorentz invariance, CP-invariance, and the V-A current structure of the lepton vertex, the expression of the di ff erential cross section is: d σ ν ( ν ) dxdy = G 2 + y 2 F ME � (1 − y − Mxy 1 − y � � � 2 E ) F ν ( ν ) 2 2 xF ν ( ν ) xF ν ( ν ) ± y (2) 2 1 3 2 π The symbols have their usual meanings; the structure functions F i are functions of x and Q 2 . It should be noted that the above expression is independent of the specifics of nucleon composition; in particular no assumption about quark/partons as nucleon constituents need be invoked. Using ν = E ν × y , and integrating the ν -N di ff erential cross section with respect to x (from 0 to 1) and ν (from 0 to ν 0 ), we get: � 1 � ν 0 d σ N ( ν < ν 0 ) = Φ ( E ν ) . dxd ν dxd ν 0 0 0 / 2 E ν ) F 2 + ν 3 F 1 ± ( ν 2 − ν 3 � � 0 0 0 ( ν 0 − ν 2 ) F 3 = C . Φ ( E ν ) . 6 E 2 6 E 2 2 E ν ν ν (3) � 1 � ν 0 where F i = 0 F i ( x ) dxd ν , N ( ν < ν 0 ) is the number of events in a given energy bin (E ν ) 0 with hadronic energy less than ν 0 , C is a constant, and the term Mxy 2 E ν has been suppressed for simplicity. It should be noted that the integrals F i contain the appropriate factors of x in the integrand for the structure functions xF 3 and 2 xF 1 . By rearranging terms as coe ffi cients of ( ν 0 /E ν ) and its powers we arrive at the more amenable form: ( F 2 ∓ F 3 ) + ν 2 � � F 2 − ν 0 0 N ( ν < ν 0 ) = C . Φ ( E ν ) . ν 0 ( F 2 ∓ F 3 ) 6 E 2 2 E ν ν � � A + ( ν 0 ) B + ( ν 0 ) 2 C + O ( ν 0 ) 3 = C . Φ ( E ν ) . ν 0 E ν E ν E ν N( ν<ν 0 ) α φ(E ν ) up to ( ν 0 / E ), ( ν 0 / E ) 2 19 02

  3. ⇐ RELATIVE FLUX WITH LOW- ν METHOD ✦ Relative ν µ , ¯ ν µ flux vs. energy from low- ν 0 method: (S. R. Mishra, Wold. Sci. 84 (1990), Ed. Geesm N ( E ν , E Had < ν 0 ) = k Φ ( E ν ) f c ( ν 0 E ν ) the correction factor f c ( ν 0 /E ν ) → 1 for ν 0 → 0 : ➳ � 2 � � � B C f c ( ν 0 E ν ) = 1 + 2 A + ..... ν 0 ν 0 A − E ν E ν � 1 � 1 where A = G 2 0 F 2 ( x ) dx , B = − G 2 F M/ π F M/ π 0 ( F 2 ( x ) ∓ x F 3 ( x )) dx and � 1 C = B − G 2 0 F 2 ( x ) [(1 + 2 Mx/ ν ) / (1 + R ( x, Q 2 )) − Mx/ ν − 1] dx F M/ π ✦ In practice use MC to calculate the correction factor normalized at high E ν : σ ( E ν , E Had < ν 0 ) f c ( E ν ) = σ ( E ν →∞ , E Had < ν 0 ) where the denominator is evaluated at the highest energy accessible in the spectrum. = ⇒ Need precise muon energy scale and good resolution at low ν values = ⇒ Reliable flux predictions for E ν � 2 ν 0 → DUNE spectra require ν 0 ≃ 0 . 25 ÷ 0 . 50 GeV − etti 03

  4. correction ν 0 1.1 correction 1 0.9 0.8 neutrino 0.7 anti − neutrino 0.6 0.5 10 20 30 40 50 60 Enu Figure 1: ν 0 correction for ν 0 = 1.0 GeV as a function of E ν for ν µ and ν µ Empirical Parametrization of π + , K + , π − , and K − 5 * (B/A) , ¡ (C/A) ¡ ➾ ¡ Low-­‑ ν ¡Processes ¡ ¡ using the Low- ν Events in ND _ Our analysis entails an empirical prarametrization (EP) of the secondary π ± and K ± pro- Figure ¡of ¡Merit ¡for ¡ (B/A) : ¡ ¡ ν ¡~ ¡ -0.3 ; ¡ ¡ ν ¡~ ¡ -1.5 ( NOMAD spectrum) ¡ duction in 120 GeV p-NuMI target as a function of x F and p T using the relative flux determined by the low- ν events in the ND. The analysis should be contrasted with the * Error ¡on ¡ (B/A) ¡ ¡ ⇔ ¡Err. ¡in ¡Low-­‑ ν ¡Interactions ‘traditional’ method of using the low- ν events, as in CCFR/NuTEV and in the MINOS- ND: start with data CC events with E Had ≤ ν 0 correct for acceptance and smearing; apply the low- ν correction to obtain the relative ν -flux at ND. (The analysis of the in- clusive ν µ -cross section by Debdatta and Donna [2] essentially use this method.) The advantage of the EP analysis is as follows: • ND and FD Flux: The EP constraints of pions and kaons allows us to accurately predict the FD flux predicated on the ND low- ν events. • The ν e and ν e Flux: Constraining the normalization and energy dependence of π + , and, hence, of µ + , and K + allows us to predict the ν e / ν µ ratio at the ND and the FD [3]. 04 19

  5. MINOS Coll., PRD 81 (2010) 072002 (MINOS)( (MINOS)( A. Bodek et al., EPJC 72 (2012) 1973 ) 4.4 ) 2 4.4 2 cm GENIE MC <0.25 ν cm GENIE MC ν <0.25 4.2 4.2 4 4 -38 -38 3.8 GENIE MC <0.5 GENIE MC <0.5 ν ν 3.8 (10 (10 3.6 3.6 3.4 GENIE MC <1.0 3.4 ν GENIE MC ν <1.0 σ σ 3.2 3.2 3 3 GENIE MC <2.0 ν GENIE MC <2.0 2.8 ν 2.8 2.6 2.6 GENIE MC ν <5.0 2.4 GENIE MC <5.0 ν 2.4 2.2 2.2 MINOS ν <1 2 2 MINOS ν <1 1.8 1.8 MINOS ν <2 1.6 1.6 1.4 MINOS <2 ν 1.4 1.2 MINOS <5 ν 1.2 1 1 MINOS <5 ν 0.8 0.8 0.6 ⟸ 0.6 0.4 0.4 0.2 0.2 0 0 -2 -1 -1 5 10 10 2 10 2 3 4 5 6 78 10 20 30 40 -1 × × 1 2 × 10 1 2 3 4 5 6 7 8 910 20 30 40 50 E (GeV) E (GeV) Fraction of Events with Cut for Neutrino on Carbon * ( B/A ) , ¡ ( C/A ) ¡ ➾ ¡ Low-­‑ ν ¡Processes ¡ ¡ _ Figure ¡of ¡Merit ¡for ¡ ( B/A ) : ¡ ¡ ν ¡ ~ ¡ -0.3 ; ¡ ¡ ν ¡~ ¡ -1.5 ( for ¡NOMAD ) ¡ *Error ¡on ¡ ( B/A ) ¡ ⇔ ¡Err. ¡in ¡Low-­‑ ν ¡Interactions 05

  6. ✦ Low- ν technique only provides RELATIVE BIN-TO-BIN flux as a function of E ν , NOT ABSOLUTE flux = ⇒ Implicit constraint of fixed flux integral (introduces correlation among bins) ✦ Freedom to chose the energy range used to impose the normalization constraint = ⇒ E.g. E ν bins with higher statistics / smaller systematic uncertainties ✦ The correction factor f c ( E ν ) can be a ff ected by model uncertainties on (anti)neutrino-nucleus cross-sections (QE, RES, DIS) ● Typically keep f c ( E ν ) at the level of few percent or below (small ν 0 /E ν ) to minimize model uncertainties (correction to correction); ● For ν 0 = 0 . 25 ÷ 0 . 50 GeV samples almost entirely QE ( 99 ÷ 75% ) and RES; ● Low- ν sensitive only to model uncertainties modifying the total cross-section vs. E ν (integrated over Q 2 and other kinematic variebles) = ⇒ Shape of σ ( E ν ) intrinsically more stable 06

  7. High-­‑Resolu,on ¡Fine ¡Grain ¡Tracker: ¡ ¡ Reference ¡ND ¡of ¡DUNE ¡ ECAL ¡ μ ¡ Detector ¡ Dipole-­‑B ν Transition ¡Radiation ➳ e +/-­‑ ¡ID ¡⇒ γ ¡ ¡ ¡ ¡ ¡ STT ¡& ¡ dE/dx ¡ ➳ Proton, ¡π +/-­‑ , ¡K +/-­‑ ¡ ¡ Ar-­‑Target Magnet/Muon ¡Detector ➳ μ +/- e +/- ( ⇒ Absolute ¡Flux ¡measurement ) 1X0 ¡~ ¡600 ¡cm ¡/ ¡ 1 λ ¡~ ¡1200 ¡cm ☙ ~ 3.5m ¡x ¡3.5m ¡x ¡6.5m ¡STT ¡(ρ≃0.1gm/cm 3 ) ¡ ☙ 4 π -­‑ECAL ¡in ¡a ¡Dipole-­‑B-­‑Field ¡(0.4T) ☙ 4 π -­‑μ-­‑Detector ¡(RPC) ¡in ¡Dipole ¡and ¡ Downstream ☙ Pressurized ¡Ar-­‑target ¡(≃x5 ¡FD-­‑Stat) ➾ LAr- FD 07

  8. Composition ¡of ¡the ¡Neutrino ¡Beam ¡ (1) ν μ ¡ ¡ ⇒ π + ⊕ K + ➾ ID’d by 𝜈 - _ (2) ν μ ¡ ¡ ⇒ π - ⊕ K - ⊕ 𝜈 + ( ⇐ π + ) ➾ ID’d by 𝜈 + (3) ν e ¡ ¡ ⇒ K + ⊕ 𝜈 + ( ⇐ π + ) ⊕ K 0 L ➾ ID’d by e - _ (4) ν e ¡ ¡ ⇒ K 0 L ⊕ K - ⊕ 𝜈 - ( ⇐ π - ) ⊕ Charm ➾ ID’d by e + Need: Accurate identification & measurement of each specie: ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ Required ¡for ¡the ¡need ¡redundancy! ¡ ¡ 08

  9. NOMAD ¡Experience ¡ _ ν μ ¡ ¡ ⇒ π + ⊕ K + ν μ ¡ ¡ ⇒ π - ⊕ K - ⊕ 𝜈 + ( ⇐ π + ) 10 3 10 4 10 2 10 3 10 10 2 1 10 -1 10 100 200 300 100 200 300 E(GeV) E(GeV) _ ν e ¡ ¡ ⇒ K 0 L ⊕ K - ⊕ 𝜈 - ( ⇐ π - ) ν e ¡ ¡ ⇒ K + ⊕ 𝜈 + ( ⇐ π + ) 10 2 ⊕ Charm ⊕ K 0 L 10 10 1 1 -1 10 -2 10 -1 10 100 200 300 50 100 150 200 E(GeV) E(GeV) 09

  10. MINOS ¡Experience ¡ Events/GeV Data signal Total π 5 10 background K 0 K L µ 4 10 3 10 2 10 10 20 30 40 50 60 Evis(GeV) Example of Low- ν EP fit to the MINOS low energy (LE) data (J. Ling and S.R. Mishra) 10

  11. Salient Considerations in Low- ν Flux Analysis: ✴ Measurement and in situ calibration of Leptons: 𝜈 & e ✴ Calibration of ECAL: Response to π +/- , Proton, n, π 0 in a dedicated Test-beam ! ✴ Differential Cross-sections of Low- ν Processes: Measure ➳ QE, ¡ ¡Resonance, ¡& ¡DIS ¡ ✴ Theoretical Errors in estimation of fc(E) : ✴ Constraining Non-Prompt Background ✴ Empirical-Parametrization of π /K Diff-Xsec ✴ Constraints from external Hadro-Production Experiments ✴ Beam Transport Errors in MC: Affects the acceptance ✴ Experimental Errors 11

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