Global Fits in the Neutrino Sector Mona Dentler Workshop ”Fit(s) for the LHC run-2“ 11 October 2016
General Remark LHC is not a neutrino experiment → neutrinos are not detected directly @ LHC But neutrinos are interesting for the physics (SM @ high energies, BSM) we want to study @ LHC: • neutrino mass • sterile neutrino • CP-violation • non standard interactions ⇒ understanding neutrino physics can ”make fit for the LHC run-2 “! 2 / 25
Introduction “ We are entering the precision era of neutrino physics ” 3 / 25
Introduction “ We are entering the precision era of neutrino physics ” Experiments can agree – or disagree – more “subtly” 3 / 25
Introduction “ We are entering the precision era of neutrino physics ” Experiments can agree – or disagree – more “subtly” • experiments have rarely exactly the same layout • ⇒ different systematics , parameter spaces , etc... 3 / 25
Introduction “ We are entering the precision era of neutrino physics ” Experiments can agree – or disagree – more “subtly” • experiments have rarely exactly the same layout • ⇒ different systematics , parameter spaces , etc... starting point for global fits ! 3 / 25
Neutrino Fit Parameters 4 / 25
Neutrino Fit Parameters In the SM, there are seven free parameters for the neutrino, e.g.: • Three masses m i , i = e , µ, τ • Three mixing angles θ ij , i , j = e , µ, τ • One CP phase δ CP 5 / 25
Neutrino Fit Parameters In the SM, there are seven free parameters for the neutrino, e.g.: • Three masses m i , i = e , µ, τ • Three mixing angles θ ij , i , j = e , µ, τ • One CP phase δ CP Most precise parameter values (not bounds) come from oscillation experiments Oscillation experiments are only sensitive to ∆ m 2 ij ≡ m 2 i − m 2 j ⇒ one parameter principally not accessible 5 / 25
Neutrino Fit Parameters In the SM, there are seven free parameters for the neutrino, e.g.: • Three masses m i , i = e , µ, τ • Three mixing angles θ ij , i , j = e , µ, τ • One CP phase δ CP Most precise parameter values (not bounds) come from oscillation experiments Oscillation experiments are only sensitive to ∆ m 2 ij ≡ m 2 i − m 2 j ⇒ one parameter principally not accessible Nevertheless, in this talk concentrate on oscillation experiments 5 / 25
Neutrino Oscillation Parameters • Neutrino flavor eigenstate | ν α � and mass eigenstate | ν k � not aligned: N � | ν α � = U ∗ α k | ν k � k =1 • Neutrino propagates as mass eigenstate. Time evolution : | ν k ( t ) � = exp( − iE k t ) | ν k � • Transition probability P αβ = | � ν β | ν α � | 2 N � P αβ = U ∗ α k U β k U α j U ∗ β j exp( − i ( E k − E j ) t ) k , j =1 • In the ultrarelativistic limit , using J αβ kj ≡ U ∗ α k U β k U α j U ∗ β j : kj ) sin 2 (∆ ij )+2 Re ( J αβ Im ( J αβ � � P αβ = δ αβ − 4 kj ) sin (∆ ij ) k > j k > j 6 / 25
Neutrino Oscillation Parameters • Neutrino flavor eigenstate | ν α � and mass eigenstate | ν k � not aligned: N � | ν α � = U ∗ α k | ν k � k =1 • Neutrino propagates as mass eigenstate. Time evolution : | ν k ( t ) � = exp( − iE k t ) | ν k � • Transition probability P αβ = | � ν β | ν α � | 2 N � P αβ = U ∗ α k U β k U α j U ∗ β j exp( − i ( E k − E j ) t ) k , j =1 • In the ultrarelativistic limit , using J αβ kj ≡ U ∗ α k U β k U α j U ∗ β j : kj ) sin 2 (∆ ij )+2 Re ( J αβ Im ( J αβ � � P αβ = δ αβ − 4 kj ) sin (∆ ij ) k > j k > j ∼ Jarlskog invariant: measures CP violation 6 / 25
Neutrino Oscillation Parameters • Neutrino flavor eigenstate | ν α � and mass eigenstate | ν k � not aligned: N � | ν α � = U ∗ α k | ν k � k =1 • Neutrino propagates as mass eigenstate. Time evolution : | ν k ( t ) � = exp( − iE k t ) | ν k � • Transition probability P αβ = | � ν β | ν α � | 2 N � P αβ = U ∗ α k U β k U α j U ∗ β j exp( − i ( E k − E j ) t ) k , j =1 • In the ultrarelativistic limit , using J αβ kj ≡ U ∗ α k U β k U α j U ∗ β j : kj ) sin 2 (∆ ij )+2 Re ( J αβ Im ( J αβ � � P αβ = δ αβ − 4 kj ) sin (∆ ij ) k > j k > j with frequency parameter ∆ ij ≡ ∆ m 2 ij L / (4 E ) 6 / 25
Neutrino Oscillation Parameters Not all experiments are equally sensitive to all six parameters. ⇒ derive effective transition probability For example: “atmospheric frequency” • θ 31 ≈ 0 ⇒ Im ( P αβ ) = 0 ( δ CP attached to θ 31 ) • ∆ m 2 31 ≈ ∆ m 2 | ∆ m 2 32 | ≫ | ∆ m 2 32 , 21 | 32 ) sin 2 (∆ 31 ) − 4 J αβ ⇒ P αβ ≈ δ αβ − 4( J αβ 31 + J αβ 21 sin (∆ 21 ) • frequency such that ∆ 31 ≈ 1 , ⇒ ∆ 21 ≪ 1 32 ) sin 2 (∆ 31 ) ⇒ P αβ ≈ δ αβ − 4( J αβ 31 + J αβ ⇒ P ee ≈ 1 , P µ e ≈ P e µ ≈ 0 P µµ ≈ 1 − sin 2 2 θ 23 sin 2 (∆ 31 ) 7 / 25
Neutrino Oscillation Parameters Not all experiments are equally sensitive to all six parameters. ⇒ derive effective transition probability For example: “atmospheric frequency” • θ 31 ≈ 0 ⇒ Im ( P αβ ) = 0 ( δ CP attached to θ 31 ) • ∆ m 2 31 ≈ ∆ m 2 | ∆ m 2 32 | ≫ | ∆ m 2 32 , 21 | 32 ) sin 2 (∆ 31 ) − 4 J αβ ⇒ P αβ ≈ δ αβ − 4( J αβ 31 + J αβ 21 sin (∆ 21 ) • frequency such that ∆ 31 ≈ 1 , ⇒ ∆ 21 ≪ 1 32 ) sin 2 (∆ 31 ) ⇒ P αβ ≈ δ αβ − 4( J αβ 31 + J αβ ⇒ P ee ≈ 1 , P µ e ≈ P e µ ≈ 0 P µµ ≈ 1 − sin 2 2 θ 23 sin 2 (∆ 31 ) cannot distinguish sin 2 θ 23 from 1 − “Octant Degeneracy”: sin 2 θ 23 7 / 25
Neutrino Oscillation Parameters Not all experiments are equally sensitive to all six parameters. ⇒ derive effective transition probability “Eight-fold” Degeneracy octant degeneracy, ( δ CP , θ 13 ),( δ CP , mass-hierarchy ( sgn ∆ m 2 31 )) Use global fits to resolve degeneracies • combining information from detectors at different baselines • using additional oscillation channels • spectral information (wide band beam) • adding information on θ 13 from a reactor experiment • adding information from ( Mt scale) atmospheric neutrino experiments 7 / 25
Global Fit to 3-Flavor Oscillations Gonzalez-Garcia, Maltoni, Schwetz, 1409.5439 8 / 25
Global Fit to 3-Flavor Oscillations Gonzalez-Garcia, Maltoni, Salvado, Schwetz, 2012 Gonzalez-Garcia, Maltoni, Schwetz, 1409.5439 8 / 25
How to perform a global fit on ν -oscillations 9 / 25
How to perform a global fit on ν -oscillations The goal • build a χ 2 -function including all relevant systematics for each experiment • find the global minimum of this χ 2 -function The challenges • high-dimensional parameter space (6 + systematics) • event spectra depend non-trivially on pull parameters (systematics) • calculating expected spectra is computationally costly cf. full 3-flavor probability P αβ 10 / 25
How to perform a global fit on ν -oscillations The strategy I: simple + fast algorithm Start from Powell’s algorithm to find minimum http://mathfaculty.fullerton.edu 11 / 25
How to perform a global fit on ν -oscillations The strategy I: simple + fast algorithm Start from Powell’s algorithm to find minimum Modify search strategy: • divide search direction into true parameters: computationally expensive pull parameters (systematics): not so expensive • in each step, go exclusively in one direction (true/ pull) 11 / 25
How to perform a global fit on ν -oscillations The strategy I: simple + fast algorithm Start from Powell’s algorithm to find minimum Modify search strategy: • divide search direction into true parameters: computationally expensive pull parameters (systematics): not so expensive • in each step, go exclusively in one direction (true/ pull) The Drawback Powell’s algorithm will only find local minimum 11 / 25
How to perform a global fit on ν -oscillations The strategy I: simple + fast algorithm Start from Powell’s algorithm to find minimum Modify search strategy: • divide search direction into true parameters: computationally expensive pull parameters (systematics): not so expensive • in each step, go exclusively in one direction (true/ pull) The strategy II: optimal starting position • use pre-scans in some plane, e.g. systematics turned off ⇒ use outcome as start value for proper fit • use knowledge about degeneracies (e.g. octant) to find the respective degenerate solution • when doing a parameter-scan: use outcome @ previous grid point as starting position 11 / 25
How to perform a global fit on ν -oscillations The strategy III: even faster Use highly optimized algorithms for manipulating 3 × 3 matrices, e.g. Cardano’s (analytical) formula to calculate eigenvalues J. Kopp, arXiv:physics/0610206 12 / 25
How to perform a global fit on ν -oscillations The strategy III: even faster Use highly optimized algorithms for manipulating 3 × 3 matrices, e.g. Cardano’s (analytical) formula to calculate eigenvalues J. Kopp, arXiv:physics/0610206 The software GLoBES General Long Baseline Experiment Simulator https://www.mpi-hd.mpg.de/personalhomes/globes/ Huber, Kopp, Lindner, Rolinec, Winter, hep-ph/0701187 12 / 25
Global Fits on Sterile Neutrinos 13 / 25
Global Fits on Sterile Neutrinos Sterile neutrino 101 14 / 25
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