Determining source cumulants in femtoscopy with Gram-Charlier and Edgeworth series
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1 Determining source cumulants in femtoscopy with Gram-Charlier and Edgeworth series M.B. de Kock a H.C. Eggers a J. Schmiegel b a University of Stellenbosch, South Africa b Aarhus University, Denmark VI Workshop on Particle Correlations and Femtoscopy, September 010, Kiev
2 Outline 1 Objective 3 4
3 Nomenclature Measured correlation function C(q) = ρ(q) ρ ref (q) Interpret as a probability density function f (q) = C(q) 1 [C(q) 1 dq Assume noninteracting, identical particles C(q) 1 = d 3 xs(x)e iq x C( q) = C(q)
4 Objective: Describe the shape of the source S(x) in terms of the shape of the correlator C(q) using cumulants. C(q) 1 = d 3 xs(x)e iq x
5 Objective: Describe the shape of the source S(x) in terms of the shape of the correlator C(q) using cumulants. C(q) 1 = d 3 xs(x)e iq x κ (q) r Known κ (x) r Unknown
6 Objective: Describe the shape of the source S(x) in terms of the shape of the correlator C(q) using cumulants. C(q) 1 = d 3 xs(x)e iq x κ (q) r Known Gram-Charlier Edgeworth κ (x) r Unknown
7 How cumulants describe shape Mean κ 1 < 0 < κ 1 Variance κ < κ Skewness κ 3 < 0 < κ 3 Kurtosis κ 4 < κ 4
8 Useful properties of cumulants Translation invariant Additive under sums of random variables Simpler than moments, Gaussian cumulants: κ 1 = µ, κ = σ, κ r = 0 for all r 3 Shift and squeeze, q q µ σ Yields standardised cumulants: γ 1 = 0, γ = 1 γ r = κ r σ r Guassian as a baseline, with γ 4 0 the distance from it
9 Useful properties of cumulants Translation invariant Additive under sums of random variables Simpler than moments, Gaussian cumulants: κ 1 = µ, κ = σ, κ r = 0 for all r 3 Shift and squeeze, q q µ σ Yields standardised cumulants: γ 1 = 0, γ = 1 γ r = κ r σ r Guassian as a baseline, with γ 4 0 the distance from it
10 Useful properties of cumulants Translation invariant Additive under sums of random variables Simpler than moments, Gaussian cumulants: κ 1 = µ, κ = σ, κ r = 0 for all r 3 Shift and squeeze, q q µ σ Yields standardised cumulants: γ 1 = 0, γ = 1 γ r = κ r σ r Guassian as a baseline, with γ 4 0 the distance from it
11 Useful properties of cumulants Translation invariant Additive under sums of random variables Simpler than moments, Gaussian cumulants: κ 1 = µ, κ = σ, κ r = 0 for all r 3 Shift and squeeze, q q µ σ Yields standardised cumulants: γ 1 = 0, γ = 1 γ r = κ r σ r Guassian as a baseline, with γ 4 0 the distance from it
12 Measuring cumulants Defining moments µ (q) r = q r f (q)dq Cumulants in moments: κ (q) r κ (q) = µ(q) κ (q) 4 = µ(q) 4 3(µ(q) ) κ (q) 6 = µ(q) 6 15µ(q) 4 µ(q) + 30(µ(q) ) q Π q q Π q q 4 Π
13 Symmetry of cumulants and generating functions Generating function: q-cumulants κ (q) r = ( i) r d r log S(x) dx r Generating function: Source cumulants κ (x) r = ( i) r d r f (q) log dqr f (0) x=0 q= Easy to measure Hard to measure
14 Gram-Charlier expansion Series expansion: Gram-Charlier series f (q) = f 0 (q) c 1 f (1) 0 (q) + c! f () 0 (q) c 3 3! f (3) 0 (q) + Reference function: Gaussian Distribution f 0 (q) = 1 e q π Derivatives: Hermite functions ( ) d r 1 e q 1 = H r (q) e q dq π π
15 Gram-Charlier in pictures Gaussian + Second Hermite + Fourth Hermite + Sixth Hermite
16 Relating coefficients to cumulants Gram-Charlier with Gaussian reference function: f (q) = f 0 (q) c 4 4! H 4(q)f 0 (q) + c 6 6! H 6(q)f 0 (q) + Fourier Transform S(x) f (0) = x [ e 1 + c 4 4! (ix)4 + c 6 6! (ix)6 + c 8 8! (ix)8 + Expand in cumulants [ S(x) f (0) = exp κ (q)! (ix) + κ(q) 4 4! (ix)4 + κ(q) 6 6! (ix)6 +
17 Relating coefficients to cumulants Gram-Charlier with Gaussian reference function: f (q) = f 0 (q) c 4 4! H 4(q)f 0 (q) + c 6 6! H 6(q)f 0 (q) + Fourier Transform S(x) f (0) = x [ e 1 + c 4 4! (ix)4 + c 6 6! (ix)6 + c 8 8! (ix)8 + Expand in cumulants [ S(x) f (0) = exp κ (q)! (ix) + κ(q) 4 4! (ix)4 + κ(q) 6 6! (ix)6 +
18 Relating coefficients to cumulants Gram-Charlier with Gaussian reference function: f (q) = f 0 (q) c 4 4! H 4(q)f 0 (q) + c 6 6! H 6(q)f 0 (q) + Fourier Transform S(x) f (0) = x [ e 1 + c 4 4! (ix)4 + c 6 6! (ix)6 + c 8 8! (ix)8 + Expand in cumulants [ S(x) f (0) = exp κ (q)! (ix) + κ(q) 4 4! (ix)4 + κ(q) 6 6! (ix)6 +
19 Coefficients in terms of cumulants Equate series and compare powers in x m e x = e x [ 1 + c 4 4! (ix)4 + c 6 6! (ix)6 + c 8 8! (ix)8 + [ 1 + κ(q) 4 4! (ix)4 + κ(q) 6 6! (ix)6 + κ(q) 8 +35(κ(q) 4 ) 8! (ix) 8 For symmetrical distributions c 4 = κ (q) 4 c 6 = κ (q) 6 c 8 = κ (q) (κ(q) 4 ) c 10 = κ (q) κ(q) 6 κ(q) 4 c 1 = κ (q) κ(q) 4 κ(q) (κ(q) 6 ) (κ (q) 4 )3
20 Source cumulants in terms of q cumulants Connecting cumulants { κ (x) = 1 f () (q) f (q) κ (x) = ( i)r d r q=0 κ (q) r = 1 κ (q) dq log f (q) r f (0) [ 1+ 5!! 4! γ(q) 4 7!! 6! γ(q) 6 + 9!! 8! (γ(q) 8 +35(γ(q) 4 ) ) !! 4! γ(q) 4 5!! 6! γ(q) 6 + 7!! 8! (γ(q) 8 +35(γ(q) 4 ) )+... } q=0 Ratio of two infinite series [ !! 4! γ(q) 4 κ (q) 1 + 3!! 4! γ(q) 4 Where do we truncate x m, m = 4?
21 Source cumulants in terms of q cumulants Connecting cumulants { κ (x) = 1 f () (q) f (q) κ (x) = ( i)r d r q=0 κ (q) r = 1 κ (q) dq log f (q) r f (0) [ 1+ 5!! 4! γ(q) 4 7!! 6! γ(q) 6 + 9!! 8! (γ(q) 8 +35(γ(q) 4 ) ) !! 4! γ(q) 4 5!! 6! γ(q) 6 + 7!! 8! (γ(q) 8 +35(γ(q) 4 ) )+... } q=0 Ratio of two infinite series [ !! 4! γ(q) 4 7!! 6! γ(q) 6 κ (q) 1 + 3!! 4! γ(q) 4 5!! 6! γ(q) 6 Where do we truncate x m, m = 6?
22 Source cumulants in terms of q cumulants Connecting cumulants { κ (x) = 1 f () (q) f (q) κ (x) = ( i)r d r q=0 κ (q) r = 1 κ (q) dq log f (q) r f (0) [ 1+ 5!! 4! γ(q) 4 7!! 6! γ(q) 6 + 9!! 8! (γ(q) 8 +35(γ(q) 4 ) ) !! 4! γ(q) 4 5!! 6! γ(q) 6 + 7!! 8! (γ(q) 8 +35(γ(q) 4 ) )+... } q=0 Ratio of two infinite series 1 κ (q) [ 1 + 5!! 4! γ(q) 4 7!! 6! γ(q) 6 + 9!! 8! (γ(q) (γ(q) 4 ) ) 1 + 3!! 4! γ(q) 4 5!! 6! γ(q) 6 + 7!! 8! (γ(q) (γ(q) 4 ) ) Where do we truncate x m, m = 8?
23 Test with a Toy Model Test with Toy Model (Symmetrical Normal Inverse Gaussian): ) (q) 3e3/γ 4 K 1 (3γ (q) q f (q γ (q), γ(q) 4 ) = 3γ (q) 4 γ(q) (γ ) (q) π 4 q + 3γ (q) 4 γ(q) Adjustable kurtosis. Exact κ (q) 0.6, κ(q) 4, κ(q) 6, κ(x), κ(x) 4, κ(x)
24 Compare Toy model and Gram-Charlier Keep x m terms in series. Percentage Deviation κ (x) j (κ (x) j ) SNIG 1 Against q-kurtosis Gaussian limit Asymptotic series becomes worse Unexpected disaster 50%-100% error
25 Compare Toy model and Gram-Charlier Keep x m terms in series. Percentage Deviation κ (x) j (κ (x) j ) SNIG 1 Against q-kurtosis Gaussian limit Asymptotic series becomes worse Unexpected disaster 50%-100% error
26 How do we fix this? Cumulants of Toy model γ (q) r ( ) γ r (q) 1 4 Suggests truncation in γ4 r. Assume f (q) is n-divisible q = q 1 + q + + q n Product of n independent components ( x i S(x) = S σ n Resulting cumulants γ (q) r ( ) r 1 1 n ) n
27 Edgeworth series Gram-Charlier series Edgeworth series S(x) = exp S(x) = exp [ [ x + r=3 γ r (q) (ix) r r! x + γ r (q) (ix) r n r/ 1 r! r=3 Central Limit Theorem lim S(x) = exp [ x n
28 Edgeworth and Gram-Charlier S(x) f (0) = [ exp exp x + [ x + r=3 γ (q) r (ix) r r=3 r! γ (q) r (ix) r n r/ 1 r! = x m F m Gram-Charlier = n w C w Edgeworth Gram-Charlier Powers in x m γ 4 H 4 (q) 4! H γ 6 (q) 6 6! [ γ8 + 35γ 4 H8 (q) 8! Edgeworth Powers in n w [ 1 n γ 4 H 4 (q) 4! 1 [35γ H 8 (q) H n 4 8! + γ 6 (q) 6 6! [ 1 H γ 8 (q) H n 3 8 8! + 10γ 6 γ 10 (q) 4 10! γ4 3 H 1 (q) 1!
29 Compare Edgeworth series and Toy model Keep terms up to order n w Percentage Deviation κ (x) j (κ (x) j ) SNIG 1 Against q-kurtosis Edgeworth accuracy 1% Improving approximation Independent of the value of n
30 Summary Edgeworth series can relate correlation cumulants to source cumulants. Gram-Charlier cannot. Gram-Charlier is a orthogonal polynomial expansion. Simple problem and text book method is surprisingly complex. Reordering gives a large decrease in error. Edgeworth groups terms according to their approach to normality. Why this works is still unclear.
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