NuWro - neutrino MC event generator Tomasz Golan 28.04.2017, UW HEP Seminar Outline Neutrino source νe Detector ′ α 1. Motivation να νµ ντ 2. NuWro event generator 3. Final State Interactions 3500 MB data Total 3000 EL on carbon np-nh 2500 EL on hydrogen Irreducible background 2000 Other backgrounds 1500 4. MB NCEL data analysis No. of events 1000 500 0 0 100 200 300 400 500 600 700 Reconstructed kinetic energy (MeV) Tomasz Golan NuWro @ HEP UW 2 / 61 Introduction Basic properties of neutrinos ■ 1 Three generations 2 -spin, no electric charge, small mass Introduction of fermions H Neutrino properties (extremely hard to detect) I II III PMNS matrix Probability of oscillation u c t γ Neutrino oscillation ■ Measurement idea Interactions with elementary particles Example: T2K BOSONS QUARKS s g Energy reconstruction → electroweak theory (Standard Model) d b MC generators ■ ν νµ ν 0 NuWro Interactions with nucleons e τ Z Final state interactions → form factors e µ τ ± MB NCEL analysis → parton distribution functions LEPTONS W Backup slides ■ Interactions with nuclei → nuclear effects ■ The neutrino flavor state is a superposition of the mass states: |ναi = i Uαi |νii νe The neutrino producedP in α state να νβ νµ ντ can be measured in β state. The phenomenon is called neutrino oscillation. Tomasz Golan NuWro @ HEP UW 4 / 61 Pontecorvo-Maki-Nakagawa-Sakata matrix The PMNS matrix defines the mass mixing in the lepton sector: −iδ 1 0 0 c13 0 s13e c12 s12 0 U = 0 c23 s23 0 1 0 −s12 c12 0 iδ 0 −s23 c23 −s13e 0 c13 0 0 1 cij = cos θij sij = sin θij θij - mixing angels δ - CP phase factor Measurements: 2 ■ Solar, reactor and accelerator: sin (2θ12) = 0.846 ± 0.021 2 ■ Atmospheric: sin (2θ23) > 0.92 2 ■ Reactor: sin (2θ13) = 0.093 ± 0.008 δ can be measured if and only if all mixing angles are nonzero. If δ 6= 0 CP is violated. Tomasz Golan NuWro @ HEP UW 5 / 61 Probability of oscillation Introduction 2 Neutrino properties P (να → νβ) = |hνβ(x)|να(y)i| = δαβ PMNS matrix 2 Probability of oscillation ∗ ∗ 2 ∆mij L Neutrino oscillation − 4 Re U UiβUα U sin Measurement idea αi j jβ 4E Example: T2K i>j Energy reconstruction 2 2 2 X ∆mij ≡ mi − mj 2 MC generators ∆m L + 2 Im U ∗ U U U ∗ sin2 ij NuWro αi iβ αj jβ 2E Final state interactions i>j X MB NCEL analysis Backup slides L - traveled distance (fixed), E - neutrino energy (will be discussed) Measurements: ■ 2 −5 2 Solar neutrinos: ∆m21 = (7.53 ± 0.18) · 10 eV ■ 2 −3 2 Atmospheric neutrinos: |∆m32| = (2.44 ± 0.06) · 10 eV Tomasz Golan NuWro @ HEP UW 6 / 61 Neutrino oscillation The probability of neutrino oscillation depends on: Normal hierarchy Inverted hierarchy 2 2 ■ m3 m2 What we want to measure: 2 ∆m21 2 2 ◆ three mixing angles - already measured ∆m32 m1 2 2 ◆ the difference between squared mass of the mass m2 ∆m31 2 2 states - the sign of ∆m is still unknown, ∆m21 32 2 2 what is the mass hierarchy? m1 m3 ? ? ◆ the CP phase factor - is the CP symmetry broken? 0 0 ? P (¯να → ν¯β) = P (νβ → να) = P (να → νβ) C + − ■ What we need to know: charge conjugation P ◆ the distance traveled by a neutrino + + parity transformation ◆ neutrino energy - this is a problem T + + time reversal Tomasz Golan NuWro @ HEP UW 7 / 61 How to measure neutrino oscillation? Neutrino source ν Detector Introduction e Neutrino properties ′ PMNS matrix Probability of oscillation Neutrino oscillation α/α Measurement idea να Example: T2K Energy reconstruction νµ ντ MC generators NuWro Final state interactions MB NCEL analysis Backup slides Disappearance method Number of α-flavor Number of α-flavor neutrinos charged leptons Appearance method Number of α-flavor Number of α′-flavor neutrinos charged leptons Tomasz Golan NuWro @ HEP UW 8 / 61 Example: T2K Target Proton beam Decay volume π + α νµ beam 295 km να 41.4 m µ + 2.5o 39.3 m ■ Off-axis beam ■ Cherenkov detector (Super-Kamiokande) ■ 50 000 tons of ultra-pure water ■ Only charged leptons and final state charged hadrons are visible ■ The neutrino energy is unknown Tomasz Golan NuWro @ HEP UW 9 / 61 The problem with neutrino energy reconstruction ′ ′ Introduction l(k ) p(p ) Neutrino properties PMNS matrix Probability of oscillation In the case of neutrino scattering off Neutrino oscillation Measurement idea nucleon at rest neutrino energy can be Example: T2K W Energy reconstruction calculated from lepton kinematics: MC generators ν(k) n(p) NuWro Final state interactions 2 ′2 ′ 2 Mp = p =(p + k − k ) Quasi-elastic scattering MB NCEL analysis Backup slides 2 2 2 Mp = ml + Mn − 2MnEl k = (Eν, 0, 0,Eν) + 2Eν(Mn − El + |~pl| cos θl) ′ k = (El, ~pl) 2 2 2 p = (Mp, 0, 0, 0) Mp − Mn − ml + 2MnEl ′ Eν = p = p + q 2(Mn − El + |~pl| cos θl) q = k − k′ Tomasz Golan NuWro @ HEP UW 10 / 61 The problem with neutrino energy reconstruction Introduction ■ Usually, the energy reconstruction procedure is based on Neutrino properties PMNS matrix quasi-elastic neutrino-nucleon scattering: Probability of oscillation Neutrino oscillation Measurement idea Example: T2K Energy reconstruction 2 2 2 REC Mp − (Mn − EB) − ml + 2(Mn − EB)El MC generators Eν = 2(Mn − EB − El + |~pl| cos θl) NuWro Final state interactions MB NCEL analysis Backup slides Nucleus Impulse Approximation l ν θl No Fermi motion π nucleon(s) Binding Potential Tomasz Golan NuWro @ HEP UW 10 / 61 The problem with neutrino energy reconstruction Introduction ■ Usually, the energy reconstruction procedure is based on Neutrino properties PMNS matrix quasi-elastic neutrino-nucleon scattering: Probability of oscillation Neutrino oscillation Measurement idea Example: T2K Energy reconstruction 2 2 2 REC Mp − (Mn − EB) − ml + 2(Mn − EB)El MC generators Eν = 2(Mn − EB − El + |~pl| cos θl) NuWro Final state interactions MB NCEL analysis Backup slides Nucleus l Never judge an event ν θl by its final state particles! π nucleon(s) Tomasz Golan NuWro @ HEP UW 10 / 61 Monte Carlo generators Buffon’s needle problem Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips? Georges-Louis Leclerc, blue are good Comte de Buffon 18th century red are bad Monte Carlo without computers If needle length (l) < lines width (t): MC experiment was performed by Mario Lazzarini in 1901 by throwing 3408 2l P = needles: tπ which can be used to estimate π: 2l · 3408 355 π = = = 3.14159292 2l t · #red 113 π = tP Tomasz Golan NuWro @ HEP UW 12 / 61 From Solitaire to Monte Carlo method ■ Stanis law Ulam was a Polish mathematician ■ He invented the Monte Carlo method while playing solitaire ■ The method was used in Los Alamos, performed by ENIAC computer ■ What is a probability of success in solitaire? ◆ Too complex for an analytical calculations ◆ Lets try N = 100 times and count wins ◆ With N → ∞ we are getting closer to correct result Tomasz Golan NuWro @ HEP UW 13 / 61 MC integration (hit-or-miss method) Lets do the following integration using MC method: Introduction MC generators 1 1 2 1 Buffon’s needle problem 1 1 x 1 From Solitaire to MC f(x)dx = x dx = = Hit-or-miss method 2 2 2 4 Crude method Z0 Z0 0 Methods comparison Accept-reject y MC generators Why do we need them? ■ take a random point from The main problem Cooking generator the [0, 1] × [0, 1] square NuWro 1 1 Final state interactions ■ compare it to your f(x) f(x) = 2 x MB NCEL analysis Backup slides ■ repeat N times ■ count n points below the function ■ you results is given by 1 x 1 n n f(x)dx = P · = N N Z0 Tomasz Golan NuWro @ HEP UW 14 / 61 MC integration (crude method) Lets do the following integration using MC method once again: Introduction MC generators 1 1 2 1 Buffon’s needle problem 1 1 x 1 From Solitaire to MC f(x)dx = x dx = = Hit-or-miss method 2 2 2 4 Crude method Z0 Z0 0 Methods comparison Accept-reject ■ One can approximate y MC generators Why do we need them? integral The main problem Cooking generator b N NuWro b − a f(x)dx ≈ f(xi) 1 Final state interactions N f(x) = x Za i=1 2 MB NCEL analysis X Backup slides where xi is a random number from [a, b] ■ It can be shown that crude method is more accurate x than hit-or-miss ■ We will skip the math and look at some comparisons Tomasz Golan NuWro @ HEP UW 15 / 61 N = 100 (hit-or-miss) N = 100 (crude) 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 MC result MC result 0.1 0.1 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 Run number Run number N = 1000 (hit-or-miss) N = 1000 (crude) 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 MC result MC result 0.1 0.1 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 Run number Run number Acceptance-rejection method fmax Introduction For generating events 0.6 MC generators according to a distribution. Buffon’s needle problem 2 From Solitaire to MC 3 −x y 0.4 Hit-or-miss method f(x) = x · e Crude method Methods comparison Accept-reject 0.2 MC generators ■ Evaluate fmax ≥ max(f) Why do we need them? The main problem Cooking generator 0 Note: fmax > max(f) will 0 0.5 1 NuWro affect performance, but the x Final state interactions result will be still correct MB NCEL analysis Backup slides ■ Generate random x 0.4 ■ Accept x with P = f(x) f y max 0.2 ◆ generate a random u from [0, fmax] ◆ 0 accept if u < f(x) 0 0.5 1 x Tomasz Golan NuWro @ HEP UW 17 / 61 Monte Carlo event generators ■ Monte Carlo generators Introduction basically do two things: MC generators Buffon’s needle problem From Solitaire to MC ◆ integrate cross Hit-or-miss method Crude method section formulas Methods comparison Accept-reject MC generators ◆ Why do we need them? generate events The main problem Cooking generator using