Dispersive analysis for pion-kaon scattering€¦ · K B 0 s + K J= Crossed channel ˇˇ!KK :...

Dispersive analysis for pion-kaon scattering

J. Ruiz de Elvira

Institute for Theoretical Physics, University of Bern

TRR110 workshop: amplitudes for three-body final states, Garching, July 12th, 2018

Motivation: Why πK scattering?

Low energies: test chiral dynamics in the strange-quark sector

B Scattering lengths lowest energy observables

→ Spontaneous and explicit chiral symmetry breaking

Higher energies: resonances, hadron spectrum

→ κ(800) non-ordinary meson, PDG “needs confirmation”

Input for Heavy-meson decays: CP-violation and New Physics searches

D+

π+

π+

K−

B0s

π+

K−

J/Ψ

Crossed channel ππ → K K : first inelastic contribution to ππ scattering

→ Γ(f0(500)→ K K ) nature of the σ meson

→ Nucleon form factors, g − 2 . . .

J. Ruiz de Elvira (ITP) Pion-kaon scattering TRR110 workshop 2

Pion-kaon interaction

Simplest scattering process involving strangeness

Two independent amplitudes Is = 1/2, 3/2 or I± = +,−

T 1/2 = T + + 2T−, T 3/2 = T + − T−

Leading-order O(p2) = O(m2i ) predictions for πK

a− =mπmK

8π(mπ + mK )f 2π

+O(m4i ) a+ = O(m4

i )

Next-to-leading order: low-energy constants L1−8 [Bernard, Kaiser, Meißner 91]

B Loop contribution suppressed at threshold

a−LECs =2mK m3

π

π(mπ + mK )f 4π

L5 +O(m6i ) [Weinberg 1966]

a+LECs =

2m2K m2

π

π(mπ + mK )f 4π

(4(L1 + L2 − L4) + 2L3 − L5 + 2(2L6 + 2L8)) +O(m6i )

Next-to-next to leading order: C1−32, 10 for a− and 23 for a+

[Bijnens,Dhonte,Talavera 2004], [Bijnens, Ecker 2014]

KK

π π

KK

π π

1/f 2π

(a)

KK

π π

Li

(b)


πK scattering lengths

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA

FLAG 16



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA

FLAG 16BE14 p4 14



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA

FLAG 16BE14 p4 14BE14 p6 Ci =0



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA FLAG 16BE14 p4 14BE14 p6 Ci =0

BE14 p6



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6


Current status

Experimental values: DIRAC collaboration

B lifetime of πK atoms at CERN⇒ isovector scattering length

Γ1S ∝∣∣∣T(π+K−→π0K 0)

∣∣∣2 ∝ |a−|2[Deser, Goldberger, Baumann, Thirring 1954]

B Current result:

a− = (−0.072+0.031−0.020)m−1

π

[DIRAC 2017]

→ huge uncertainties

B Room for improvement: near future increase statistics by 10

3s 3p 3d

2s 2p

1s

Γ1s

ǫ1s



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17


Current status

Lattice analysis: unquenched results only

B Constraint from semileptonic Kl3 decays [Flynn, Nieves 2007]

a1/2 = 0.179(17)(14)m−1π

B NPLQCD: dynamical Nf = 2 + 1 calculation, mπ = 290− 600 MeV [NPLQCD 2006]

a1/2 = (0.173+0.003−0.016)m−1

π , a3/2 = (−0.057+0.003−0.006)m−1

π

B Z. Fu: dynamical Nf = 2 + 1 fermions, mπ = 330− 466 MeV [Fu 2012]

a1/2 = 0.182(4)m−1π , a3/2 = −0.051(2)m−1

π

B PACs: dynamical Nf = 2 + 1 fermions, mπ = 170− 710 MeV [PACs-Cs 2014]

a1/2 = 0.142(14)(27)m−1π , a3/2 = −0.047(2)(2)m−1

π


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07


Current status



a1/2 = 0.179(17)(14)m−1π


a1/2 = (0.173+0.003−0.016)m−1

π , a3/2 = (−0.057+0.003−0.006)m−1

π


a1/2 = 0.182(4)m−1π , a3/2 = −0.051(2)m−1

π


a1/2 = 0.142(14)(27)m−1π , a3/2 = −0.047(2)(2)m−1

π


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06


Current status



a1/2 = 0.179(17)(14)m−1π


a1/2 = (0.173+0.003−0.016)m−1

π , a3/2 = (−0.057+0.003−0.006)m−1

π


a1/2 = 0.182(4)m−1π , a3/2 = −0.051(2)m−1

π


a1/2 = 0.142(14)(27)m−1π , a3/2 = −0.047(2)(2)m−1

π


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06Fu 12


Current status



a1/2 = 0.179(17)(14)m−1π


a1/2 = (0.173+0.003−0.016)m−1

π , a3/2 = (−0.057+0.003−0.006)m−1

π


a1/2 = 0.182(4)m−1π , a3/2 = −0.051(2)m−1

π


a1/2 = 0.142(14)(27)m−1π , a3/2 = −0.047(2)(2)m−1

π


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06Fu 12


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06Fu 12PACS 14


Current status

Dispersive determination: Roy-Steiner equations analysis

B Most precise results up to date

a1/2 = 0.224(22)m−1π , a3/2 = −0.045(8)m−1

π

[Büttiker, Descontes-Genon, Moussallam 2003]

→ More than 3.5σ from O(p4) ChPT results

This talk: where does this discrepancy come from?

→ Overview of Roy-Steiner equation for πK scattering


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06Fu 12PACS 14


Current status

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06Fu 12PACS 14Buettiker et al. 04


Current status

Dispersive determination: Roy-Steiner equations analysis

B Most precise results up to date

a1/2 = 0.224(22)m−1π , a3/2 = −0.045(8)m−1

π

[Büttiker, Descontes-Genon, Moussallam 2003]

→ More than 3.5σ from O(p4) ChPT results

This talk: where does this discrepancy come from?

→ Overview of Roy-Steiner equation for πK scattering


Why Roy-Steiner equations?

Roy(-Steiner) eqs. = Partial-Wave (Hyberbolic) Dispersion Relationscoupled by unitarity and crossing symmetry

Respect all symmetries: analyticity, unitarity, crossing

Model independent⇒ the actual parametrization of the data is irrelevant once it is used inthe integral.

Framework allows for systematic improvements (subtractions, higher partial waves, ...)

PW(H)DRs help to study processes with high precision:

ππ-scattering: [Ananthanarayan et al. (2001), García-Martín et al. (2011)]

πK -scattering: [Büttiker et al. (2004)]

γγ → ππ scattering: [Hoferichter et al. (2011)]

πN scattering: [Hoferichter et al. (2015)]


Roy equations for ππ scattering

ππ → ππ⇒ fully crossing symmetric in Mandelstam variables s, t , and u = 4Mπ − s − t

Start from twice-subtracted fixed-t DRs

T I(s, t) = c(t) +1π

∞∫4m2π

ds′

s′2

[s2

(s′ − s)−

u2

(s′ − u)

]ImT I(s′, t)

Subtraction functions c(t) are determined via crossing symmetry

→ functions of the I=0,2 scattering lengths: a00 and a2

0

PW-projection and expansion yields the Roy-equations [Roy (1971)]

t IJ (s) = ST I

J (s) +∞∑

J′=0

(2J′ + 1)∑

I′=0,1,2

∞∫4m2π

ds′K II′JJ′ (s′, s) Im t I′

J′ (s′)

K II′JJ′ (s′, s) ≡ kernels⇒ analytically known


Solving Roy-equations: flow information

Roy-equations rigorously valid for a finite energy range

⇒ introduce a matching point sm

only partial waves with J ≤ Jmax are solved

Assume isospin limit

InputHigh-energy region: Imt I

J (s) for s ≥ sm and for all J

Higher partial waves: Imt IJ (s) for J > Jmax and for all s

Inelasticities η(s)

OutputSelf-consistent solution for δIJ (s) for J ≤ Jmax and sth ≤ s ≤ sm

Subtraction constants


Roy-equations: existence and uniqueness

Solution characterized by subtraction constants and high-energy input (a, A)

Existence and uniqueness depends on δi dynamically at sm

m =∑

i

mi , mi =

⌊

2δi (sm)π

⌋if δi (sm) > 0,

−1 if δi (sm) < 0,

bxc ⇒ largest integer ≤ x .[Gasser, Wanders 1999, Wanders 2000]

m = 0, a unique solution exists for any (a, A)

m > 0, m-parameter family of solutions for any (a, A)

m < 0, only for a specific choice of the input constrained by |m| conditions

Physical solution characterized by smooth matching


Roy-equations: ππ results

Solution for the ππ S0-wave

400 600 800 1000 1200 1400

s1/2

(MeV)

0

50

100

150

200

250

300

CFDOld K decay data

Na48/2K->2 π decay

Kaminski et al.Grayer et al. Sol.B

Grayer et al. Sol. C

Grayer et al. Sol. D

Hyams et al. 73

δ0

(0)

1000 1100 1200 1300 1400

s1/2

(MeV)

0

0.5

1

η0

0(s)

Cohen et al.Etkin et al.Wetzel et al.Hyams et al. 75

Kaminski et al.Hyams et al. 73

Protopopescu et al.

CFD .

ππ KK

ππ ππ

Garcia-Martin, Kaminski, Pelaez, JRE (2011)


Roy–Steiner equations for πK : differences to ππ Roy equations

Key differences compared to ππ Roy equations

Crossing: coupling between πK → πK (s-channel) and ππ → K K (t -channel)

⇒ need a different kind of dispersion relations [Hite, Steiner 1973, Büttiker et al. 2004]

Unitarity in t-channel, e.g. single-channel approximation

Imf J±(t) = σπt f J

±(t)t lJ (t)∗ tIJ f I

J

⇒Watson’s theorem: phase of f J±(t) equals δIJ [Watson 1954]

B solution in terms of Omnès function [Muskhelishvili 1953, Omnès 1958]

→ ππ scattering δIJ need as input


Roy–Steiner equations for πK : flow of information

Limited range of validity: hyperbolic dispersion relations

√s ≤√

sm = 0.985 GeV t ≤ tm = 2 GeV2

Input/Constraints

S- and P-waves above matching points > sm (t > tm)

Inelasticities

Higher waves (D-, F-, · · · )ππ phase shifts below the K Kthreshold

Output

S- and P-wave phase-shifts at lowenergies s < sm (t < tm)

Subtraction constants

Subthreshold parameters: c+00, c+

01, c−00

→ πK scattering lengths


Solving Roy-Steiner equations for πK : t-channel

General solution in terms of Omnes function [Muskhelishvili 1953, Omnes 1958]

Ω(t) = exp

(tπ

tm∫tπ

δ(t′)dt′

t′(t′−t)

), with Ω(t ∼ tm) ∼ |t ′ − tm|

δ(tm)π

δIJ (tm) > π⇒ one free parameter αIJ

gIJ (t) = ∆IJ (t) +tnΩIJ (t)

tm−t

(αIJ + t

π

tm∫tπ

dt′

t′ n+1∆IJ (t′)sin δIJ (t′)

t′−ttm−t|Ω(t′)| + t

π

∞∫tm

dt′

t′ n+1Im gIJ (t′)

t′−ttm−t|Ω(t′)|

)∆IJ (t) ≡ subtraction constants, s-channel pw, J>1 t-channel pw

gIJ (tm) is continuous but gIJ ′(tm) has a cusp (divergent)

→ αIJ analytically determined from a non-cusp condition

Ω(t ′)−1 obeys a dispersion relation⇒ Ω−1IJ (t) = Pn(t)− 1

π

tm∫tπ

dt′

t′ nsin δIJ (t′)

|ΩIJ (t′)|(t′−t)

→ ∆IJ (t)|ST polynomial in t ⇒ gIJ (tm)|ST can be computed analytically

→ ∆IJ (t)|s−channel expanded as polynomial using splines

Exact solution of the MO problem while solving the s-channel RS equations


Solving s-channel: strategy

Parameterize S and P waves up to s < sm

B Imposing a continuous and differentiable matching point

Introduce as many subtractions as necessary to match d.o.f [Gasser, Wanders 1999]

B m = 0, three non-cusp conditions: S1/2, S3/2 and P1/2

→ three free parameters: c+00, c+

01, c−00

Minimize difference between LHS and the RHS on a grid of points Wj

χ2phys =

∑l,Is

N∑j=1

(Re f Is

l±(Wj )− F [cIsnm, f

Isl±](Wj )

)2

Re f Isl±(Wj )2

B F [cIsnm, f

Isl±](Wj ) ≡ right hand side of RS-equations

Parametrization and subthreshold parameters are the fitting parameters


Roy-Steiner solution: preliminary results

0.5 0.6 0.7 0.8 0.9s GeV2

−0.4−0.20.00.20.40.60.81.0 f

1/20

flfr

0.5 0.6 0.7 0.8 0.9s GeV2

−1.0

−0.5

0.0

0.5

1.0 f1/21

0.5 0.6 0.7 0.8 0.9s GeV2

−0.5−0.4−0.3−0.2−0.10.00.1 f

3/20

0.5 0.6 0.7 0.8 0.9s GeV2

−0.10−0.08−0.06−0.04−0.020.000.02 f

3/21

f tot


Preliminary results: s-channel phase shifts

0.65 0.70 0.75 0.80 0.85 0.90 0.95√s GeV2

0

10

20

30

40

50

60

70

80 δ1/20 (s)

LRSLACLASS

0.65 0.70 0.75 0.80 0.85 0.90 0.95√s GeV2

0

50

100

150

200 δ1/21 (s)

LRSLACLASS

0.65 0.70 0.75 0.80 0.85 0.90 0.95√s GeV2

−35

−30

−25

−20

−15

−10

−5

0 δ3/20 (s)

LRSLAC


Preliminary results: t-channel solution

0.0 0.5 1.0 1.5 2.0t GeV2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

| g0 0( t

) |

LongacreCohen

0.0 0.5 1.0 1.5 2.0t GeV2

0

1

2

3

4

5

6

7

| g1 1( t

) |

Cohen


Preliminary results: kappa pole

600 650 700 750 800 850 900 950M (MeV)

-400

-350

-300

-250

-200

-150

-100 −

Γ/2

(M

eV)

Breit-Wigner-like parameterizations

Zhou et al.PelaezBugg

Bonvicini et al.Descotes-Genon et al.Padé result, Pelaez, Rodas, JREPelaez and RodasColangelo, Maurizio, JRE (preliminary)


πK Scattering length puzzle: BDM solution

Fixed-t dispersion relation

B cI(t) extracted from a hyperbolic dispersion relation at threshold

s · u = m2K −m2

π

Twice/once subtracted version for I = +,−

c+(t) = 8π(mπ + mK )a+ + b+t

c−(t) = 8π(mπ+mK )mπmK

a−pπ(t)qK (t)

b+ extracted from a sum rule involving a−

a+ and a− extracted from RS self-consistent solution

χ2phys =

∑l,Is

N∑j=1

(Re f Is

l (sj )− F [f Isl ](sj )

)2

Re f Isl (sj )2

B F [f Isl ](sj ) ≡ right hand side of RS-equations


Roy-Steiner solution for πK

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.4 0.5 0.6 0.7 0.8 0.9 1

Re fIs

l(s)

s (GeV2)

f01/2

f11/2

f03/2


Dispersive determination of πK scattering lengths

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA




Stability of the solution

How stable is the solution?

→ ππ scattering case⇒ universal band in the a00, a2

0 plane [Ananthanarayan et al. (2001)]

0.15 0.2 0.25 0.3

a0

0

−0.06

−0.05

−0.04

−0.03

−0.02

a02

S0

S1

S2

S3

B similar for πN scattering [Hoferichter et al. (2015)]

→ look for πK RS solutions in a grid


Universal band on the scattering length plane

0.16 0.18 0.20 0.22 0.24 0.26 0.28

- 0.08

- 0.07

- 0.06

- 0.05

- 0.04

- 0.03

a01/ 2 M π

a03/2M

π

0

20

40

60

80


Subthreshold parameter plane

0.5 1.0 1.5 2.0 2.5 3.0

8.0

8.2

8.4

8.6

8.8

9.0

9.2

9.4

C +00

C-00

0

5

10

15


πK scattering lengths: preliminary results

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA





0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA


BE14 p6RχPT p6Dirac 17Flynn et al. 07NPLQCD 06Fu 12PACS 14Buettiker et al. 04Universal Band


Looking for a unique solution

Unique solution: as many subtractions as necessary to match d.o.f [Gasser, Wanders 1999]

→ two non-cusp conditions⇒ two free scattering lengths [Büttiker, Descontes-Genon, Moussallam 2003]

Are the two scattering lengths independent?

8π(mπ + mK )a−SR

mπmK=

1π

∫ ∞sth

ds′

λ(s′)Im F−(s′, t ′) +

1π

∫ ∞tπ

dt ′

t ′Im

G1(t ′, s′)8pπ(t ′)pK (t ′)

Ks Kt Ds Dt As At Total

a−i,SR (mπ102) 0.86 6.10 0.60 0.13 0.00 0.37 8.05a− (mπ102) – – – – – – 8.96

→ once a+ is fixed, a− is obtained from a convergent sum rule

New approach: HDR subtracted at subthreshold

B three non-cusp conditions⇒ three subthreshold parameters constrained by SR

χ2tot = χ2

phys +

(c−00 − c−00,SR

)2

∆c−002

+

(c10

+ − c+10,SR

)2

∆c+10

2


Unique solution for πK scattering

- 2 - 1 0 1 2

7.0

7.5

8.0

8.5

C +00

C-00

5

10

15

20


πK scattering without χ2c−00

- 2 - 1 0 1

7.0

7.5

8.0

8.5

9.0

C +00

C-00

0

10

20

30



0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA





0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30a

120Mπ

−0.080

−0.075

−0.070

−0.065

−0.060

−0.055

−0.050

−0.045

−0.040

a3 2 0Mπ

CA



HDR


Thank you


Spare slides


πK subthreshold parameter plane

−3 −2 −1 0 1 2 3C +00

7.0

7.5

8.0

8.5

9.0

9.5

10.0C

− 00

C-RdE-M

FLAG

Bijnens 13

GL∗

Ci=0

BE14

BE14

FIT 10

Lang

B-M-DG


Dispersion relations in a nutshell

Effective field theories⇒ systematically improvable but

B number of LECs increase rapidly

B convergence problems: low-lying resonances, strong rescattering effects

Dispersion relations: analyticity, crossing, unitarity

B analitycity constrains the energy depedence of scattering amplitude

B crossing symmetry connects different physical regions

B unitarity constrains imaginary part

Roy(-Steiner) eqs. =Partial-Wave (Hyberbolic) Dispersion Relationscoupled by unitarity and crossing symmetry

→ model independent aproach

→ analytic continuation for the complex plane⇒ resonances, unphysical regions


From Cauchy’s theorem to dispersion relations

Cauchy’s Theorem

f (s) =1

2πi

∫∂Ω

ds′f (s′)s′ − s

→ analyticity

Subtractions

f (s) = f (0) +sπ

∫cuts

ds′Im f (s′)s′(s′ − s)

Imaginary part from Cutkosky rules→ forward direction: optical theorem

Unitarity for partial waves

Im f (s) = σ(s)|f (s)|2, f (s) =η(s)e2iδIJ (s) − 1

2iσ(s), σ(s) =

√1−

4m2π

s



Dispersion relation

f (s) =1π

∫cuts

ds′Im f (s′)s′ − s

→ analyticity

Subtractions

f (s) = f (0) +sπ

∫cuts





2iσ(s), σ(s) =

√1−

4m2π

s



Dispersion relation

f (s) =1π

∫cuts

ds′Im f (s′)s′ − s

→ analyticity

Subtractions

f (s) = f (0) +sπ

∫cuts





2iσ(s), σ(s) =

√1−

4m2π

s

t(s) t(s)


Roy equations: range of convergence

Convergence for T I(s, t) guaranteed for t < 4m2

Where does the partial wave expansion converge?

Assumption: Mandelstam analyticity [Mandelstam (1958,1959)]

T (s, t) =1π2

∫∫ds′dt ′

ρst (s′, t ′)(s′ − s)(t ′ − t)

+ (t ↔ u) + (s ↔ u)

→ integration on the support of the double spectral densities ρ

Boundaries of ρ

(I) (II)

Lehmann ellipses

→ largest ellipses, which do not enter any ρ

-40 -20 0 20 40

-40

-20

0

20

40

s [M2π ]

u[M

2 π]

ρtuρsu

ρst

[Lehmann (1958)]


Outline

Why is pion-kaon scattering important?

B Relevance of pion-kaon scattering lengths

Chiral perturbation in SU(3)

B Convergence of the chiral series

Dispersive techniques in meson-meson scattering

B Roy-Steiner equation for pion-kaon scattering

On a dispersive determination of πK scatteringin collaboration with Gilberto Colangelo and Stefano Maurizio


Roy equations and resonance pole parameters

tIJ (s) known in the Lehmann ellipsis

Resonances

→ poles on unphysical Riemann sheets

SII(s − iε, t) = SI(s + iε, t)

→ t IIIJ (s) = tIJ (s) · (1 + 2i Σ(s)tIJ (s))−1

Elastic scattering: II RS is known exactly

Coupled channels

tIJ (s) =

t(11)IJ (s) t(12)

IJ (s)

t(12)IJ (s) t(22)

IJ (s)

Σ(s) =

(σ1(s) 0

0 σ2(s)

)

→ III and IV RS require crossed channels

t IVIJ (s) = t(11)

IJ (s)−2iσ2(s)t(12)

IJ (s)2

1 + 2iσ2(s)t(22)IJ (s)

0 20 40 60

Res

-40

-20

0

20

40

Ims


Dip vs no-dip solutions

The dip vs no-dip⇒ long-standing controversy [Pennington, Bugg, Zou, Achasov] · · ·

→ no clear preference for any of the two scenarios in previous works

Is it possible to satisfy Roy Equations with a non-dip scenario?

1000 1100 1200 1300 1400

s1/2(MeV)

0

0.25

0.5

0.75

1

η00(s)

Cohen et al.Etkin et al.Wetzel et al.ndUFDCFD 2010ndCFD enlarged errors

→ the non-dip scenario is rejected by DR


f0(500) pole 2010

400 500 600 700 800 900 1000 1100 1200

Re sσ1/2

(MeV)

-500

-400

-300

-200

-100

0

Im s

σ1/2 (

MeV

)

PDT estimate (2006)PDT citations (2006)Dobado, Pelaez 1996Colangelo, Gasser, Leutwyler 2001Pelaez 2004Caprini, Colangelo, Leutwyler 2006Garcia-Martin, Pelaez, Yndurain 2007Zhou 2004CFD + disersive relations


f0(500) pole 2012

400 500 600 700 800 900 1000 1100 1200

Re sσ1/2

(MeV)

-500

-400

-300

-200

-100

0

Im s

σ1/2 (

MeV

)

PDT estimate (2012)PDT citations (2006)Dobado, Pelaez 1996Colangelo, Gasser, Leutwyler 2001Pelaez 2004Caprini, Colangelo, Leutwyler 2006Garcia-Martin, Pelaez, Yndurain 2007Zhou 2004CFD + disersive relations


f0(980) pole 2010

960 980 1000 1020 1040 1060 1080 1100

Re sσ1/2

(MeV)

-100

-50

0

Ims σ1/

2 (M

eV)

PGD citations (2006)PGD estimate (2006)Caprini et alCFD + dispersion relations


f0(980) pole 2012

960 980 1000 1020 1040 1060 1080 1100

Re sσ1/2

(MeV)

-100

-50

0

Ims σ1/

2 (M

eV)

PGD citations (2006)PGD estimate (2012)Caprini et alCFD + dispersion relations


Dispersive analysis for pion-kaon scattering€¦ · K B 0 s + K J= Crossed channel ˇˇ!KK :...

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Transcript of Dispersive analysis for pion-kaon scattering€¦ · K B 0 s + K J= Crossed channel ˇˇ!KK :...