Numerical analytic continuationor

a nice interpolation and extrapolation method

Ralf-Arno Tripolt

Palaver

Frankfurt, October 29, 2018

Analytic continuation - the idea

[commons.wikimedia.org]

The interpolation method

The interpolation method

The interpolation method

For details...

[courtesy L. Holicki]

arXiv: 1610.03252

arXiv: 1801.10348

Simple Example: f(x) = 1/(x+ 100)

I we use 2 input points for theSPM

I can we reconstruct thefunction?

-300 -200 -100 0 100 200 300

-0.2

-0.1

0.0

0.1

0.2

x

f(x)

f(x)Input

f(x)=1

x + 100

Simple Example: f(x) = 1/(x+ 100)

I Only 2 input points areneeded to reconstructf(x) = 1

x+100

I it is the “first guess” of themethod

-300 -200 -100 0 100 200 300

-0.2

-0.1

0.0

0.1

0.2

x

f(x)

f(x)InputSPM

f(x)=1

x + 100

Another simple example: f(x) = x

I we use 3 input points for theSPM

I can we reconstruct thefunction?

-10 -5 0 5 10

-10

-5

0

5

10

x

f(x)

f(x)=xInput

Another simple example: f(x) = x

I 3 input points are needed toreconstruct f(x) = x

I with 15 digits precision oneobtains for example

f(x) =22 + 1.8 · 1015x

1.8 · 1015 − x ≈ x

-10 -5 0 5 10

-10

-5

0

5

10

x

f(x)

f(x)=xInputSPM

Another example: f(x) = ex

I we use 11 input points

I can we reconstruct thefunction?

-10 -5 0 5 10

0

5000

10000

15000

20000

x

f(x)

f(x)=exp(x)Input

Another example: f(x) = ex

-10 -5 0 5 10

0

5000

10000

15000

20000

x

f(x)

f(x)=exp(x)InputSPM

I for 11 input points we obtain

f(x) =263504 + 170536x+ 46451x2 + 10389x3 + 756x4 + 148x5

265568− 98809x+ 15473x2 − 1274x3 + 55x4 − x5

Another example: f(x) = ex

-10 -5 0 5 10 15 20-1×106

-500000

0

500000

1×106

x

f(x)

f(x)=exp(x)InputSPM

I for 11 input points we obtain

CN (x) =263504 + 170536x+ 46451x2 + 10389x3 + 756x4 + 148x5

265568− 98809x+ 15473x2 − 1274x3 + 55x4 − x5

What is a spectral function?

Ω

ΡHΩL

What is a spectral function?

Ω

ΡHΩLΠ

Ω = mΠ

What is a spectral function?

Ω

ΡHΩLΠ

Ω = mΠ

2 Γ

What is a spectral function?

Ω

ΡHΩLΠ

Ω = mΠ

2 Γ

Π Ψ Ψ

Ω ³ 2mΨ

Model for a spectral function

We use

ρ(ω2) = − 1

πIm

(1

ω2 −M2 −Π(ω2)

)with the self energy

Π(ω2) = S1 log(T 21−ω2)+S2 log(T 2

2−ω2)

and the parametersM = 50 MeV,S1 = 2000 MeV2,T1 = 0 MeV,S2 = 3000 MeV2,T2 = 300 MeV,

ω → ω + iε with ε→ 0

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]ρ[MeV

-2]

ρ(ω)

Model for a spectral function

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]

ρ[MeV

-2]

ρ(ω)Input

Model for a spectral function

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]

ρ[MeV

-2]

ρ(ω)InputSPM

Model for a spectral function

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]

ρ[MeV

-2]

ρ(ω)Input

Model for a spectral function

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]

ρ[MeV

-2]

ρ(ω)InputSPM

Model for a spectral function

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]

ρ[MeV

-2]

ρ(ω)Input

Model for a spectral function

0 100 200 300 400 5001.×10-7

5.×10-71.×10-6

5.×10-61.×10-5

5.×10-5

ω [MeV]

ρ[MeV

-2]

ρ(ω)InputSPM

Plot f(x) in the complex plane

We find poles in the complexplane at:ωP ≈ (236.43± i12.64) MeV

This pole characterizes the

particle which is associated to

the peak in the spectral

function

Model for overlapping resonances - 3 particles

The form factor could be measuredby experiment:

F (s) =

3∑i=1

M2i

M2i − s− iΓi

M2i√s

(k(s)

k(M2i )

)3with

k(s) =

√s

2

√1− 4m2

π/s

and the parametersM1 = 0.5 GeV,Γ1 = 0.2 GeV,M2 = 0.6 GeV,Γ2 = 0.1 GeV,M3 = 0.7 GeV,Γ3 = 0.15 GeV,mπ = 0.137 GeV

0.1 0.2 0.3 0.4 0.5 0.6

1.5

2.0

2.5

3.0

s @GeV2D

»FHsL»

Model for overlapping resonances - 3 particles

0.1 0.2 0.3 0.4 0.5 0.6

1.5

2.0

2.5

3.0

s [GeV2]

|F(s)|

|F(s)|

Input

SPM

Plot F(s) in the complex plane

exact result, F (s): reconstruction, SPM:

Complex pole of the charged ρ(770) meson

[ALEPH collaboration, Phys. Rept. 421, 191-284, 2005, arXiv:hep-ex/0506072]

Complex pole of the charged ρ(770) meson

[R.-A. T., I. Haritan, J. Wambach and N. Moiseyev, arXiv:1610.03252]

Complex pole of the charged ρ(770) meson

We find the complex pole ofthe charged ρ(770) meson tobe at

√sρ = Mρ − iΓρ/2 with

Mρ = 761.8± 1.9 MeV,

Γρ = 139.8± 3.6 MeV.

[R.-A. T., I. Haritan, J. Wambach and N. Moiseyev, arXiv:1610.03252]

The analytic continuation problem

Calculations at finite temperature are often performed using imaginary energies:

Ω

ip0

Ω

ip0

→

The analytic continuation problem

Analytic continuation problem: How to get back to real energies?

Ω

ip0

?

?

Analytic Continuation - Free Particle

We want to reconstruct thespectral function of a freeparticle

ρ(ω) = sgn(ω)δ(ω2 −m2)

starting from the freepropagator

DE(p0) =1

p20 +m2

with p0 = 2nπT ,m = 200 MeV, andT = 1/β = 10 MeV.

-1500 -1000 -500 0 500 1000 15000

5

10

15

20

25

p0 @MeVD

DE

Hp 0L@G

eV

-2

D

Ω

ip0

?

?

Analytic Continuation - Free Particle

We choose 51 input pointsfrom the Euclidean propagatorDE(p0) and apply the SPM toobtain the real-timepropagator DR(ω = ip0) andthe spectral function:

ρ(ω) = − 1

πImDR(ω)

-2000 -1000 0 1000 2000

0.5

1.0

2.0

5.0

10.0

20.0

p0 @MeVD

DE

Hp 0L@G

eV

-2

D

0 200 400 600 800 100010-20

10-15

10-10

10-5

1

105

1010

w @MeVD

rHwL

@Me

V-

2D

Summary

The Schlessinger point method can be used to obtain the analytic continuationof a function that is given in the form of numerical data.

I one can reconstruct the underlying function not only along the real axisbut also in the complex plane

I can be used to identify resonance poles and to “predict” decay thresholds(branch cuts)

I can be used to perform an analytic continuation based on Euclidean data

Schlessinger Point Method (SPM)

Given a finite set of N data points (xi, fi) we construct the rational interpolantp(x)/q(x) with polynomials p(x) and q(x) that is given by the continued fraction

p(x)/q(x) = CN (x) =f1

1 +a1(x− x1)

1 +a2(x− x2)

... aN−1(x− xN−1)

,

where the coefficients ai are given recursively by a1 = f1/f2−1x2−x1

and

ai =1

xi − xi+1

(1 +

ai−1(xi+1 − xi−1)

1+

ai−2(xi+1 − xi−2)

1+· · · a1(xi+1 − x1)

1− f1/fi+1

)The polynomials are of order (N/2− 1, N/2) for an even number of input points and

((N − 1)/2, (N − 1)/2) for an odd number of input points

[L. Schlessinger, Physical Review, Volume 167, Number 5 (1968)]

[R.W. Haymaker and L. Schlesinger, Mathematics in Science and Engineering, Volume 71, Chapter 11 (1970)]

[H.J. Vidberg and J.W. Serene, Journal of Low Temperature Physics, Vol. 29, Nos. 3/4 (1977)]

[A. Pilaftsis and D. Teresi, Nucl. Phys. B 874 (2013) 594-619]

[G. Marko, U. Reinosa and Z. Szep, arXiv: 1706.08726]

Analytic Continuation and Radius of Convergence

I an analytic continuation into thecomplex plane can be performed bychoosing x in CN (x) to be complex,i.e. x = αeiθ

I rational interpolants can exactlyreproduce polar singularities, thusextending the ‘radius of convergence’to the first non-polar singularity, e.g. abranch point

I even non-polar singularities may bewell approximated by poles and zerosof the rational fraction

I a rational fraction can have only onesheet in the complex plane - amany-sheeted function can only bereconstructed on a single sheet

BP