The hadronic vacuum polarisation contribution to the muon g 2

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1 The hadronic vacuum polarisation contribution to the muon g 2 Alex Keshavarzi (in collaboration with Daisuke Nomura & Thomas Teubner [KNT17]) LFC17: Old and New Strong Interactions from LHC to Future Colliders ECT, Trento, Italy 14 th September 2017 a µ update 14 th September / 28

Introduction Setting the scene... The Standard Model (SM) contributions to a µ a SM µ = a QED µ + a EW had, LbL µ + aµ + aµ + a BSM?? µ QED Known to five-loop (12,672 diagrams) 99.99% of a SM µ 0.

2 Introduction Setting the scene... The Standard Model (SM) contributions to a µ a SM µ = a QED µ + a EW had, LbL µ + aµ + aµ + a BSM?? µ QED Known to five-loop (12,672 diagrams) 99.99% of a SM µ 0.001% of δa SM µ EW a) b) c) γ W W µ νµ Z H Known to two-loop (with m H known) % of a SM µ 0.2% of δa SM µ HVP Non-perturbative (experimental input) 0.006% of a SM µ 73% of δa SM µ HLbL BSM Non-perturbative (data input + model/lattice) % of asm µ 27% of δa SM µ??????? % of a exp µ aµ update 14 th September / 28

BNL (new from shift in λ) had, LOVP aµ still dominated uncertainty Potential for improvement from experimental data and data combination New

3 Introduction Setting the scene... The previous analysis... [HLMNT(11), J. Phys. G38 (2011), ] HLMNT(11): HMNT (06) JN (09) Davier et al, τ (10) Davier et al, e + e (10) JS (11) HLMNT (10) HLMNT (11) experiment BNL BNL (new from shift in λ) had, LOVP aµ still dominated uncertainty Potential for improvement from experimental data and data combination New x4 accuracy measurements planned from Fermilab and J-PARC = If a SM µ & a EXP µ improve as planned... g-2 discrepancy > 7σ! a µ aµ update 14 th September / 28

Introduction Basics of the hadronic vacuum polarisation The vacuum polarisation (VP) id µν γ (q 2 ) = Undressed (tree-level QED) q 2 Free propagator id γ µν (q 2 ) = igµν q 2 (sum of all 1PI hadronic

4 Introduction Basics of the hadronic vacuum polarisation The vacuum polarisation (VP) id µν γ (q 2 ) = Undressed (tree-level QED) q 2 Free propagator id γ µν (q 2 ) = igµν q 2 (sum of all 1PI hadronic blobs...) Quantum fluctuations q 2 q 2 Dressed propagator = ig µν q 2( 1 Π γ(q 2 ) ) Virtual loop(s) = VP = charge screening = Running QED coupling: α α(q 2 α ) = 1 + 4παReΠ γ(q 2 ) = α 1 α(q 2 ) where α(q 2 ) = α lep (q 2 ) + α (5) had (q2 ) + α(q 2 ) top q 2 q 2 aµ update 14 th September / 28

5 Introduction Basics of the hadronic vacuum polarisation The hadronic vacuum polarisation (HVP) Hadronic blob contains sum of all hadronic states q 2 q 2 γ π 0 γ γ γ π + π γ γ K + K γ.... aµ update 14 th September / 28

6 Introduction Basics of the hadronic vacuum polarisation The hadronic vacuum polarisation (HVP) Hadronic blob contains sum of all hadronic states J/ψ Υ(1s 6s) 1000 ψ(2s) q 2 q 2 R(s) ρ/ω φ s = γ π 0 γ γ γ π + π γ γ K + K γ Requires knowing entire hadronic spectrum (s th s ) Then, how do we calculate a had,vp µ? aµ update 14 th September / 28

Calculating a had,vp µ The set-up Dispersion relation (analyticity) and the optical theorem So far, we know we need to relate The hadronic vacuum polarisation tensor, Π had (q 2 ) The hadronic

7 Calculating a had,vp µ The set-up Dispersion relation (analyticity) and the optical theorem So far, we know we need to relate The hadronic vacuum polarisation tensor, Π had (q 2 ) The hadronic spectrum We do this in two steps... 1) Analyticity Relate real part of Π(s) to its imaginary part 2) Optical theorem Relate imaginary part of Π(s) to total e + e hadrons cross section Π had (q 2 ) = q2 π s th ds ImΠ had(s) s(s q 2 iε) ( s ) ImΠ had (s) = σ had (s) 4πα...convolute with one loop contribution from coupling of virtual photon to muon... a had,lovp µ = 1 4π 3 s th ds σ 0 had,(γ) (s)k(s) aµ update 14 th September / 28

8 Calculating a had,vp µ The set-up HVP dispersion integral input Important points to note: a had,lovp µ = 1 ds σ 4π had,(γ)(s)k(s) 0 3 s th Can also define dispersive integral for NLOHVP with identical data input ( Integral has 1/s dependence K(s) = m2 µ 3s K(s) ) Must undress cross section of VP effects: σ had σ 0 had Cross section must include FSR effects: σ 0 had σ 0 had,(γ) Can conventionally define hadronic R-ratio: R(s) = σ0 had,γ(s) σ pt(s) a had,lovp µ = α2 ds 3π 2 s th s R(s)K(s) σ0 had,γ(s) 4πα 2 /3s aµ update 14 th September / 28

9 Calculating a had,vp µ The set-up Hadronic cross section input a had,lovp µ = α2 ds 3π 2 s th s R(s)K(s), where K(s) = m2 µ 3s K(s) J/ψ Υ(1s 6s) 1000 ψ(2s) R(s) ρ/ω φ aµ update 14 th September / 28

10 Calculating a had,vp µ The set-up Hadronic cross section input a had,lovp µ = α2 ds 3π 2 s th s R(s)K(s), where K(s) = m2 µ 3s K(s) Non-perturbative (Experimental data, isopsin, ChPT...) J/ψ ψ(2s) Υ(1s 6s) Perturbative (pqcd) R(s) ρ/ω φ Non -perturbative/ perturbative (Experimental data, pqcd, Breit-Wigner...) Must build full hadronic cross section/r-ratio... aµ update 14 th September / 28

11 Calculating a had,vp µ The set-up Building the hadronic R-ratio m π s 2 GeV Input experimental hadronic cross section data* Combine all available data in exclusive hadronic final states (π + π, K + K,...) Sum 35 exclusive channels Robust treatment of experimental errors Estimate missing data input (isospin relations, ChPT...) 2 s 11.2 GeV Can use experimental inclusive R data* or pqcd Must use data at quark flavour thresholds Combine all available R data Robust treatment of experimental errors Include narrow resonances 11.2 s < GeV Calculate R using pqcd (rhad) *σ had experiments KLOE BaBar SND CMD-(2/3) KEDR BESIII More old and new... aµ update 14 th September / 28

12 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? a µ update 14 th September / 28

13 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? Radiative corrections of data and the corresponding error estimate a µ update 14 th September / 28

14 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? Radiative corrections of data and the corresponding error estimate When combining data......how to best amalgamate large amounts of data from different experiments a µ update 14 th September / 28

15 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? Radiative corrections of data and the corresponding error estimate When combining data......how to best amalgamate large amounts of data from different experiments...the correct implementation of correlated uncertainties (statistical and systematic) a µ update 14 th September / 28

16 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? Radiative corrections of data and the corresponding error estimate When combining data......how to best amalgamate large amounts of data from different experiments...the correct implementation of correlated uncertainties (statistical and systematic)...finding a solution that is free from bias a µ update 14 th September / 28

17 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? Radiative corrections of data and the corresponding error estimate When combining data......how to best amalgamate large amounts of data from different experiments...the correct implementation of correlated uncertainties (statistical and systematic)...finding a solution that is free from bias The reliability of the integral and error estimate a µ update 14 th September / 28

18 Data combination Data treatment Combining experimental data Question: What are the main points of concern when correcting, combining and integrating experimental data to evaluate a µ? Radiative corrections of data and the corresponding error estimate When combining data......how to best amalgamate large amounts of data from different experiments...the correct implementation of correlated uncertainties (statistical and systematic)...finding a solution that is free from bias The reliability of the integral and error estimate The choices when estimating unmeasured hadronic final states a µ update 14 th September / 28

19 Data combination Data treatment Radiative corrections (for data that is not σ 0 had,(γ) ) Vacuum polarisation corrections Fully updated, self-consistent VP routine: [vp knt v3 0] Cross sections undressed with full photon propagator (must include imaginary part), σ 0 had(s) = σ had (s) 1 Π(s) 2 If correcting data, apply corresponding radiative correction uncertainty Take 1 3 of total correction per channel as conservative extra uncertainty Final state radiation corrections For π + π, include through sqed approximation [Eur. Phys. J. C 24 (2002) 51, Eur. Phys. J. C 28 (2003) 261] For higher multiplicity states, difficult to estimate correction Apply conservative uncertainty Need new, more developed tools to increase precision here (e.g. - CARLOMAT 3.1 [Eur.Phys.J. C77 (2017) no.4, 254 ]?) a µ update 14 th September / 28

20 Data combination Data treatment Clustering data Re-bin data into clusters Better representation of data combination through adaptive clustering algorithm More and more data risk of over clustering σ 0 (s) [nb] δ = 5 MeV a µ = (1.73 ± 0.21) loss of information on resonance Scan cluster sizes for optimum solution (error, χ 2, check by sight...) Scanning/sampling by varying bin widths Clustering algorithm now adaptive to points at cluster boundaries Fit of all data Imprecise, dense data Precise, sparse data σ 0 (s) [nb] δ = 55 MeV a µ = (1.48 ± 0.08) Fit of all data Imprecise, dense data Precise, sparse data aµ update 14 th September / 28

21 Data combination Covariance matrices Correlation and covariance matrices Correlated data beginning to dominate full data compilation... Non-trivial, energy dependent influence on both mean value and error estimate KNT17 prescription Construct full covariance matrices for each channel & entire compilation Framework available for inclusion of any and all inter-experimental correlations If experiment does not provide matrices... Statistics occupy diagonal elements only Systematics are 100% correlated If experiment does provide matrices π KLOE Fit: a + π µ = ± 2.89 σ 0.2 Matrices must satisfy properties of a 0 (e + e π + π ) π KLOE(08): a + π µ = ± 3.31 π 1200 KLOE(10): a + π µ = ± 3.19 π KLOE(12): a µ covariance matrix + π = ± e.g. - KLOE π + π γ(γ) combination covariance matrices update = Originally, NOT a positive semi-definite matrix: (Corrected in separate work to be published) σ 0 /fit σ 0 (e + e π + π ) [nb] aµ update 14 th September / 28

22 Data combination Data combination Systematic bias and use of the data/covariance matrix Data is re-binned using an adaptive clustering algorithm Iterative fit of covariance matrix as defined by data D Agostini bias HLMNT11 Non-linear χ 2 minimisation fitting nuisance parameters Penalty trick bias R m = [Nucl.Instrum.Meth. A346 (1994) ] KNT17 Fix the covariance matrix in an iterative χ 2 minimisation Free from bias R m Unbiased Result: R m = R m = 1 Non linear χ 2 Minimisation 0.9 Unbiased Result: R m = R m = 1 Linear (Fixed) χ 2 Minimisation Log 10 [df k /df l ] Log 10 [df k /df l ] Allows for increased fit flexibility and full use of energy dependent, correlated uncertainties aµ update 14 th September / 28

23 Data combination Data combination Linear χ 2 minimisation Redefine clusters to have linear cross section Fix covariance matrix with linear interpolants at each iteration (extrapolate at boundary) N tot N tot χ 2 = i=1 j=1 ( R (m) i R i ) m C 1 ( i (m), j (n))( R (n) j R j ) n Through correlations and linearisation, result is the minimised solution of all neighbouring clusters...and solution is the product of the influence of all correlated uncertainties The flexibly of the fit to vary due to the energy dependent, correlated uncertainties benefits the combination...and any data tensions are reflected in a local χ 2 error inflation Local error inflation σ 0 (e + e π + π ) δa µ π + π (Local χ 2 Inflation)/δa µ π + π (No χ 2 Inflation) Global χ 2 min /d.o.f. = 1.26 aµ update 14 th September / σ 0 (e + e π + π ) [nb]

24 Data combination Data combination Integration Trapezoidal rule integral Consistency with linear cluster definition High data population Accurate estimate from linear integral 1400 Quadratic Linear Cubic Quadratic Linear Cubic σ 0 (e + e π + π ) [nb] σ 0 (e + e Κ + Κ ) [nb] Higher order polynomial integrals give (at maximum) differences of 10% of error Estimates of error non-trivial at integral borders Extrapolate/interpolate covariance matrices aµ update 14 th September / 28

25 Results Results from individual channels π + π channel [preliminary] Large improvement for 2π estimate BESIII [Phys.Lett. B753 (2016) ] and KLOE combination provide downward influence to mean value 1400 (σ 0 - σ 0 Fit )/σ0 Fit σ 0 (e + e - π + π - ) BaBar (09) Fit of all π + π - data CMD-2 (03) CMD-2 (06) SND (04) KLOE combination BESIII (15) a µ π + π - (0.6 s 0.9 GeV) = ( ± 1.32) x χ 2 min /d.o.f. = σ 0 (e + e - π + π - ) [nb] σ 0 (e + e - π + π - ) [nb] BaBar (09) Fit of all π + π - data KLOE combination CMD-2 (07) SND (06) CMD-2 (04) BESIII (15) Fit of all π + π data: ± 1.32 Direct scan only: ± Correlated & experimentally corrected σ 0 ππ(γ) data now entirely dominant π a + π µ (0.6 s 0.9 GeV) x KLOE combination: ± 2.00 BaBar (09): ± 2.72 BESIII (15): ± 4.22 a π+ π µ (0.305 s 2.00 GeV): HLMNT11: ± 3.09 KNT17: ± 1.93 (no radiative correction uncertainties) aµ update 14 th September / 28

26 Fit of all π + π - π 0 data SND (15) CMD-2 (07) Scans BaBar (04) SND (02,03) CMD-2 (95,98,00) DM2 (92) ND (91) CMD (89) DM1 (80) Results Fit of all π + π - π + π - data BaBar (12) CMD-2 (04) SND (03) CMD-2 (00,00) ND (91) DM2 (90) CMD (88) OLYA (88) DM1 (79,82) GG2 (81) M3N (79) ORSAY (76) Fit of all K + K - data CMD-2 (08) Scans SND (07) BCF (86) DM1 (81) DM2 (87) DM2 (83) OLYA (81) CMD (91) CMD-2 (95) SND (00) Scans Babar (13) SND (16) Scans Results from individual channels Fit of all π + π - π 0 π 0 data SND (03) CMD-2 (99) DM2 (90) OLYA (86) MEA (81) GG2 (80) M3N (79) Other notable exclusive channels 1000 π + π π 0 50 π + π π + π 50 π + π π 0 π 0 σ 0 (e + e - π + π - π 0 ) [nb] σ 0 (e + e - π + π - π + π - ) [nb] σ 0 (e + e - π + π - π 0 π 0 ) [nb] HLMNT11: ± 0.99 KNT17: ± 0.70 σ 0 (e + e - K + K - ) [nb] K + K HLMNT11: ± 0.47 KNT17: ± 0.14 σ 0 (e + e - K 0 S K0 L ) [nb] K 0 SK 0 L HLMNT11: ± 1.26 KNT17: ± 1.19 Fit of all K 0 S K0 L data CMD-3 (16) Scans BaBar (14) SND (06) CMD-2 (03) SND (00) - Charged Modes SND (00) - Neutral Modes CMD (95) Scans DM1 (81) HLMNT11: ± 0.46 KNT17: ± HLMNT11: ± 0.16 KNT17: ± 0.12 aµ update 14 th September / 28

27 Results Results from individual channels KKπ, KKππ and isospin New BaBar data for KKπ and KKππ removes reliance on isopsin (only KS 0 = KL) 0 KKπ KSK 0 Lπ 0 0 [Phys.Rev. D95 (2017), ] σ 0 (KKπ) [HLMNT(11) isospin estimate] σ 0 (KKπ) [Data] KKππ K 0 S K0 L π+ π [Phys.Rev. D80 (2014), ] K 0 S K0 S π+ π [Phys.Rev. D80 (2014), ], K 0 S K0 L π0 π 0 [Phys.Rev. D95 (2017), ] K 0 S K± π π 0 [arxiv: ] σ 0 (KKππ) [HLMNT(11) isospin estimate] σ 0 (KKππ) [Data] σ 0 [nb] 8 6 σ 0 [nb] HLMNT11: 2.77 ± 0.15 KNT17: 2.82 ± 0.14 But, still reliant on isospin estimates for π + π 3π 0, π + π 4π 0, KK3π HLMNT11: 3.31 ± 0.58 KNT17: 2.42 ± 0.09 aµ update 14 th September / 28

28 Results Results from individual channels Inclusive New KEDR inclusive R data ranging 1.84 s 3.05 GeV [Phys.Lett. B770 (2017) ] and 3.12 s 3.72 GeV [Phys.Lett. B753 (2016) ] Exclusive Data Combination Inclusive Data pqcd KEDR (2016) Fit of all inclusive R data KEDR (16) BaBar R b data (09) BESII (09) BES (06) CLEO (07) BES (02) Crystal Ball (88) R(s) 2.3 R(s) LENA (82) MD-1 (96) BES (99) pqcd had, LOVP aµ (1.84 s 2.00 GeV): pqcd : 6.42 ± 0.03 Data : 6.88 ± had, LOVP aµ (2.60 s 3.73 GeV): pqcd (inflated errors) : ± 0.38 Data : ± 0.14 = Choose to adopt entirely data driven estimate from threshold to 11.2 GeV aµ update 14 th September / 28

29 Results KNT17 update Contributions to mean value below 2GeV σ 0 [nb] PRELIMINARY A.KESHAVARZI σ 0 (e + e hadrons) π + π π + π π 0 K + K π + π π 0 π 0 π + π π + π K 0 S K0 L π 0 γ KKππ KKπ (π + π π + π π 0 π 0 ) no η ηπ + π (π + π π + π π 0 ) no η ωπ 0 ηγ (π + π π 0 π 0 π 0 ) no η ηφ ωηπ 0 All remaining ηω π + π π + π π + π (π + π π 0 π 0 π 0 π 0 ) no η aµ update 14 th September / 28

30 Results KNT17 update Contributions to uncertainty below 2GeV dr(s) PRELIMINARY A.KESHAVARZI σ 0 (e + e hadrons) π + π π + π π 0 K + K π + π π 0 π 0 π + π π + π K 0 S K0 L π 0 γ KKππ KKπ (π + π π + π π 0 π 0 ) no η ηπ + π (π + π π + π π 0 ) no η ωπ 0 ηγ ωηπ 0 ηω π + π π + π π + π aµ update 14 th September / 28

31 Results KNT17 update R(s) for m π s < J/ψ Υ(1s 6s) 1000 ψ(2s) R(s) ρ/ω φ Full compilation data set for hadronic R-ratio to be made available soon =...complete with full covariance matrix aµ update 14 th September / 28

32 Results KNT17 update KNT17 aµ and α (5) had (M Z) 2 update [preliminary] (g 2) µ α(m 2 Z) HLMNT(11): ± 4.27 had, LOVP This work: aµ = ± 1.26 stat ± 2.02 sys ± 0.31 vp ± 0.70 fsr = ± 2.42 exp ± 0.77 rad value (error) 2 = ± 2.54 tot had, NLOVP aµ = 9.83 ± 0.04 tot HLMNT11: ( ± 1.38 tot) 10 4 This work: α (5) had (M Z) 2 = ( ± 0.39 stat ± 0.64 sys ± 0.08 vp ± 0.82 fsr ) 10 4 = ( ± 0.76 exp ± 0.83 rad ) 10 4 = ( ± 1.13 tot) 10 4 a µ had,lo VP 0.9 m π 2 rad. m 1.4 π Accuracy better then 0.4% (uncertainties include all available correlations) Full KNT17 VP package [vp knt v3 0.f] available for use aµ update 14 th September / 28

33 Results KNT17 update KNT17 vs. DHMZ17 vs. FJ17 [preliminary] Different data treatment/methods produce very different results Channel s 1.8 GeV KNT17 DHMZ17 FJ17 π + π ± ± 2.58 π + π 2π ± ± π + 2π ± ± 0.31 K + K ± ± 0.41 KSK 0 L ± ± 0.24 Total HVP s < GeV ± ± ± 3.25 Between 1.8 s 2 GeV, KNT use data, DHMZ use pqcd BUT, pqcd = 8.30 ± 0.09, KNT data = 8.42 ± 0.29, DHMZ data = 7.71 ± 0.32 DHMZ17 use correlated systematics differently in determination of the mean value Determining π + π using only local weighted average gives ± 2.84 Much better agreement when neglecting the effect of correlated uncertainties on the mean value a µ update 14 th September / 28

34 Results KNT17 update KNT17 a SM µ update [preliminary] QED (0.02) (0.01) [Phys. Rev. Lett. 109 (2012) ] EW (0.20) (0.10) [Phys. Rev. D 88 (2013) ] LO HLbL (2.60) 9.80 (2.60) [EPJ Web Conf. 118 (2016) 01016] NLO HLbL 0.30 (0.20) [Phys. Lett. B 735 (2014) 90] HLMNT11 KNT17 LO HVP (4.27) (2.54) this work NLO HVP (0.07) (0.04) this work NNLO HVP 1.24 (0.01) [Phys. Lett. B 734 (2014) 144] Theory total (4.94) (3.62) this work Experiment (6.33) world avg Exp - Theory 26.1 (8.0) 28.1 (7.3) this work a µ 3.3σ 3.9σ this work a µ update 14 th September / 28

35 Results KNT17 update KNT17 a SM µ update [preliminary] DHMZ (10) JS (11) HLMNT (11) Jegerlehner (17) DHMZ (17) This work [KNT(17)] 3.9σ BNL 7.1σ BNL (x4 accuracy) PRELIMINARY (a µ SM x ) aµ update 14 th September / 28

36 Conclusions Conclusions Can use dispersive techniques to calculate HVP and HLbL contributions to a µ Many necessary changes made in order to improve data combination When combining data......all covariance matrices are correctly constructed with a framework that can accommodate any available correlations...employ a linear χ 2 minimisation that has been shown to be free from bias New method shows improvements in all channels due to increased fit flexibility Less reliance on isospin for estimated states with more measured final states a had,lovp µ accuracy better than 0.4% (g 2) µ theory initiative to collaborate to produce single estimate for a SM µ Eagerly awaiting new FNAL result! a µ update 14 th September / 28

37 Conclusions Extra Slides a µ update 14 th September / 28

38 Conclusions Fixing the covariance matrix [JHEP 1005 (2010) 075, Eur.Phys.J. C75 (2015), 613] Apply a procedure to fix the covariance matrix ( C I i (m), j (n)) = C stat( i (m), j (n)) + Csys( i (m), j,n)) R mr R (m) i R (n) n, j in an iterative χ 2 minimisation method that, to our best knowledge, is free R from bias m Fixing with theory value regulates influence Can be shown from toy models to be free from bias Swift convergence Comparison with past results shows HLMNT11 estimates are largely unaffected Allows for increased fit flexibility and full use of energy dependent, correlated uncertainties Unbiased Result: R m = R m = 1 Linear (Fixed) χ 2 Minimisation Log 10 [df k /df l ] aµ update 14 th September / 28

39 Conclusions Kaon FSR study σ 0 (e + e - K + K - ) [nb] PRELIMINARY BUT K + K cross section is totally dominated by φ resonance No phase space for creation of hard real photons at φ Inclusive FSR correction is large over-correction No longer apply FSR correction New Fit CMD-2 (08) Scans SND (07) BCF (86) DM1 (81) DM2 (87) DM2 (83) OLYA (81) CMD (91) CMD-2 (95) SND (00) Scans Babar (13) SND (16) Scans Inclusive FSR correction was previously applied to K + K cross section KLN theorem requires all virtual and soft corrections necessarily included in given cross section Only hard real radiation is left to be corrected for η cut (s)/η(s) CMD-2 (95) SND (00) σ 0 (e + e - K + K - ) σ 0 (e + e - K + K - ) [nb] aµ update 14 th September / 28

40 Conclusions Properties of a covariance matrix Any covariance matrix, C ij, of dimension n n must satisfy the following requirements: As the diagonal elements of any covariance matrix are populated by the corresponding variances, all the diagonal elements of the matrix are positive. Therefore, the trace of the covariance matrix must also be positive n n Trace(C ij) = σ ii = Var i > 0 i=1 It is a symmetric matrix, C ij = C ji, and is, therefore, equal to its transpose, C ij = C T ij The covariance matrix is a positive, semi-definite matrix, i=1 a T C a 0 ; a R n, where a is an eigenvector of the covariance matrix C Therefore, the corresponding eigenvalues λ a of the covariance matrix must be real and positive and the distinct eigenvectors are orthogonal b C a = λ a(b a) = a C b = λ b (a b) if λ a λ b (a b) = 0 The determinant of the covariance matrix is positive: Det(C ij) 0 a µ update 14 th September / 28

41 Conclusions Tests of reliability of f k method Did the f k method incur a bias? Compare f k method and fixed matrix method with only multiplicative normalisation uncertainties. If we see differences in mean value, then bias previously influenced the fit. Previous results unreliable If we see no differences in mean value, then bias did not influence fit (any change comes from improved treatment of systematics) Previous results reliable Example - π + π Set 1 - CMD-2(06) (0.7% Systematic Uncertainty), Set 2 - CMD-2(06) (0.8% Systematic Uncertainty), Set 3 - SND(04) (1.3% Systematic Uncertainty) From GeV Fit Method: f k method Fixed matrix method Channel a µ χ 2 min /d.o.f. aµ χ2 min /d.o.f. Difference π + π ± ± a µ update 14 th September / 28

Physics from 1-2 GeV

Physics from 1-2 GeV Caterina Bloise INFN, Frascati Laboratory, Italy What next LNF: Perspectives of fundamental physics at the Frascati Laboratory Frascati, November 11, 2014 1 / 19 1 Introduction 2 Kaon