Precise determination of πN scattering from
Roy-Steiner equations
J. Ruiz de Elvira
Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical
Physics, Universität BonnIn collaboration with:
C. Ditsche, M. Hoferichter, B. Kubis, U.-G. Meißner.
[JHEP 1206 (2012) 043] and [JHEP 1206 (2012) 063]
+ work in progress

Hadron and Hadron Interactions in QCD, Kyoto, March 3, 2015

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Outline

1
2
3
4

5

Motivation
Introduction: Roy(Steiner)-equations for ππ vs πN
Roy-Steiner equations for πN
Solving Roy-Steiner equations for πN
Solving the t-channel subproblem
Solving the s-channel subproblem
Solving the full RS system
Summary & Outlook

J. Ruiz de Elvira

Roy-Steiner equations for πN

2

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Contents
1

Motivation

2

Introduction: Roy(Steiner)-equations for ππ vs πN

3

Roy-Steiner equations for πN

4Solving Roy-Steiner equations for πN
Solving the t-channel subproblem
Solving the s-channel subproblem
Solving the full RS system

5

Summary & Outlook
J. Ruiz de Elvira

Roy-Steiner equations for πN

2

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Motivation: Why πN scattering?
Low energies: test chiral dynamics in the baryon sector⇒ low-energy theorems e.g. for the scattering lengths
Higher energies: resonances, baryon spectrum
Input for NN scattering: LECs ci , πNN coupling

Crossed channel ππ → N̄N: nucleon form factors⇒ probe the structure of the nucleon
spectral functions of form factors
vector form factors (P-waves)scalar form factors (S-waves)
J. Ruiz de Elvira

Roy-Steiner equations for πN

ππN N

3

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

The pion-nucleon σ-term
Scalar form factor of the nucleon:σ(t) = hN(p0 )|m̂ ūu + d̄d |N(p)it = (p0 − p)2

σπN = σ(0)
200
=1
zs

σπN measures the light-quark contribution to the nucleon mass
150u

Unfortunately, no direct experimental access to it

2 +22D̄ (ν = 0, t = 2MπFπ) = σ(2Mπ
) +∆R
|
{z
}
| {z }
2 (d + +2M 2 d + )+∆
Fπ
D
π 01
00

u  M Π2

100

Linked to πN via the Cheng-Dashen theorem [Cheng, Dashen 1971]

50
t

Ν

0

σπN +∆σs|∆R | . 2 MeV [Bernard, Kaiser, Meißner 1996]

-50
t
-100
-100

-50

=1
zu

∆D − ∆σ = (−3.3 ± 0.2) MeV [Gasser, Leutwyler, Sainio 1991]

0

50

100

150

s  M Π2

Phenomenological status:“canonical value” σπN ∼ 45 MeV, from Cheng-Dashen theorem based on KH80 input[Gasser, Leutwyler, Socher, Sainio 1988, Gasser, Leutwyler, Sainio 1991]

GWU/SAID PW analysis σπN ∼ (64 ± 8) MeV[Pavan, Strakovsky, Workman, Arndt 2002]

J. Ruiz de Elvira

Roy-Steiner equations for πN

4200

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Motivation: Why Roy-Steiner equations?
Roy(-Steiner) eqs. =Partial-Wave (Hyberbolic) Dispersion Relations
coupled by unitarity and crossing symmetry

Respect all symmetries: analyticity, unitarity, crossing
Model independent ⇒ the actual parametrization of the data isirrelevant once it is used in the 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)]
J. Ruiz de Elvira

Roy-Steiner equations for πN

5

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Contents
1

Motivation

2

Introduction: Roy(Steiner)-equations for ππ vs πN

3

Roy-Steiner equations for πN

4Solving Roy-Steiner equations for πN
Solving the t-channel subproblem
Solving the s-channel subproblem
Solving the full RS system

5

Summary & Outlook
J. Ruiz de Elvira

Roy-Steiner equations for πN

6

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Warm up: Roy-equations for ππππ → ππ ⇒ fully crossing symmetric in Mandelstam variables s, t,
and u = 4Mπ − s − tStart from twice-subtracted fixed-t DRs of the generic form
T I (s, t) = c(t) +

1
π

Z ∞

ds0
02
4m2π s
s2u2
ImT I (s0 , t)−(s0 − s)(s0 − u)Subtraction functions c(t) are determined via crossing symmetryfunctions of the I=0,2 scattering lengths: a00 and a20

PW-expansion of these DRs yields the Roy-equations [Roy (1971)]∞
X
X Z ∞ 0 II 0 00ds KJJ 0 (s , s)Im tJI 0 (s0 )tJI (s) = STJI (s) + (2J 0 + 1)
J 0 =02
I 0 =0,1,2 4mπ

0

II
0
KJJ
0 (s , s) ≡ kernels ⇒ analytically known
J. Ruiz de Elvira

Roy-Steiner equations for πN

6

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

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
Input
High-energy region: ImtIJ (s) for s ≥ sm and for all J
Higher partial waves: ImtIJ (s) for J > Jmax and for all s
OutputSelf-consistent solution for δIJ (s) for J ≤ Jmax and sth ≤ s ≤ sm
Constraints on subtraction constants

J. Ruiz de Elvira

Roy-Steiner equations for πN

7

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: difficulties
Key difficulties compared to ππ Roy-equations
Crossing: coupling between πN → πN (s-channel) and ππ → N̄N (t -channel)⇒ hyperbolic dispersion relations [Hite, Steiner 1973], [Büttiker et al. 2004]
Unitarity in t-channel, e.g. in single-channel approximation
Imf±J (t) = σtπ f±J (t)tJl (t)∗

tIJ

fJI

⇒ Watson’s theorem: phase of f±J (t) equals δIJ [Watson 1954]
solve with Muskhelishvili-Omnès
1953, Omnès 1958] techniques [Muskhelishvili⇒ Omnès function: ΩIJ (t) = exptπI
0 δJ (t)tth dt t0 (t0 −t)

R tm

Large pseudo-physical region in t -channel⇒ K̄K intermediate states for s-wave in the region of the f0 (980)

J. Ruiz de Elvira

Roy-Steiner equations for πN

8

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Contents
1

Motivation

2

Introduction: Roy(Steiner)-equations for ππ vs πN

3

Roy-Steiner equations for πN

4Solving Roy-Steiner equations for πN
Solving the t-channel subproblem
Solving the s-channel subproblem
Solving the full RS system

5

Summary & Outlook
J. Ruiz de Elvira

Roy-Steiner equations for πN

9Motivation Introduction RS-eqs. Solving RS-eqs. Summary

πN-scattering basics
π a (q) + N(p) → π b (q0 ) + N(p0 )

200
=1
zs

Isospin Structure:150u

T ba = δ ba T + + ab T −0

T I = ū(p0 ) AI + 6 q+62 q BI u(p)DI = AI + νBI , ν = s−u4mu  M Π2

100

Lorentz Structure:I ∈ {+, −}


50
t

Ν

0

PW projection:Is-channel pw: fl±Jt-channel pw: f±

s
-50
t
-100
-100

-50

=1
zu

Isospin basis: Is ∈ {1/2, 3/2}
{T + , T − } ⇔ T 1/2 , T 3/2

0

50

100

150

s  M Π2

Bose symmetry ⇒ even/odd J ⇔ I = +/−J. Ruiz de Elvira

Roy-Steiner equations for πN

10

200

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

πN-scattering basics: Unitarity relations
s-channel unitarity relations (Is ∈ {1/2, 3/2}):
2
Is
(W)

 1 − ηl±2
IsIs
θ W−WinelIm fl±(W) = q fl±(W) θ W−W+ +
4q

I
fl±

I
fl±

t-channel unitarity relations: 2-body intermediate states: ππ + K̄K + · · ·JIm f±(t) = σtπ tJIt (t)

π

N
f±J
N̄

√
∗
∗ J


f± (t) θ t − tπ + 2cJ 2 kt2J σtK gIJt (t) hJ± (t) θ t − tK

tIJ
π

K

N

π

hJ±

+
π

N̄

π
gJI

K̄

π

Only linear in f±J (t) ⇒ less restrictive

J. Ruiz de Elvira

Roy-Steiner equations for πN

11

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: HDR’s
Hyperbolic DRs:(s − a)(u − a) = b = (s0 − a)(u0 − a) with a, b ∈ R
A+ (s, t; a) =

1
π

Z∞

s+

ds0




Z∞
1
1
1
1
Im A+ (s0 , t0 )
+ 0 0
+
−
Im
A
(s
,
t
)
+
dt0
0
0
0
s −s
s −u
s −a
π
t0 − ttπWhy HDR?
Combine all physical regions ⇒ crucial for t-channel projection
Evade double-spectral regions ⇒ the PW decompositions converge
Range of convergence can be maximized by tuning the free hyperbola parameter a
No kinematical cuts, manageable kernel functions

Similar derivation to ππ Roy equations.
Expand imaginary parts in terms of s- and t-channel partial waves
Project onto s- and t-channel partial waves
Combine the resulting equations using s- and t-channel PW unitarity relations√√
Validity: W ∈ [W+ = 1.08 GeV, 1.38 GeV], t ∈ [ tπ = 0.28 GeV, 2.00 GeV] .

Subtractions: subthreshold expansion around ν = t = 0
Ā+ (ν, t) =

∞
P

2m n
a+
mn ν tm,n=0
J. Ruiz de Elvira

Roy-Steiner equations for πN

12

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Contents
1

Motivation

2

Introduction: Roy(Steiner)-equations for ππ vs πN

3

Roy-Steiner equations for πN

4Solving Roy-Steiner equations for πN
Solving the t-channel subproblem
Solving the s-channel subproblem
Solving the full RS system

5

Summary & Outlook
J. Ruiz de Elvira

Roy-Steiner equations for πN

13

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving t-channel: single channel
Elastic-channel approximation: generic form of the integral equationf (t) = ∆(t) + (a + bt)(t − 4m2 ) +t2 (t − 4m2 )
π

Z∞
tπ

Im f (t0 )dt0 0 02t (t − 4m2 )(t0 − t)∆(t): Born terms, s-channel integrals, higher t -channel partial waves⇒ left-hand cut
Introduce subtractions at ν = t = 0 ⇒ subthreshold parameters a, b
Solution in terms of Omnès function:f (t) =∆(t) + (t − 4m2 )Ω(t)(1 − tΩ̇(0))a + t(t − 4m2 )Ω(t)b
( Ztm
)
Z∞
0
0 −1
0 −1 Im f (t0 )t2 (t − 4m2 )0 ∆(t )Im Ω(t )0 Ω(t )− Ω(t)
dt 02 0+ dt 0( 0πt (t − 4m2 )(t0 − t)t t − 4m2 )(t0 − t)24Mπ(Ω(t) = exp

t
π

Ztm
tπ

dt0 δ(t0 )t0 t0 − ttm)

J. Ruiz de Elvira

Roy-Steiner equations for πN

14

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving t-channel: input and subtractions√
elastic channel approximation: tm = 0.98 − 1.1 GeV, for t > tm Im f±J (t) = 0
First step: check consistency with KH80 [Höhler 1983]
Input needed:
ππ phase shifts: [Caprini, Colangelo, Leutwyler, (in preparation)]πN phase shifts: SAID [Arndt et al. 2008], KH80
πN at high energies: Regge model [Huang et al. 2010]
πN parameters: KH80

J. Ruiz de Elvira

Roy-Steiner equations for πN

15

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving t-channel: P, D and F waves up to N̄N
16
14
12
10
8
6
4
2
0
-2

90
80
70
60
50
40
30
20
10
0

Im f0+(s)

18
16
14
12
10
8
6
4
2
0

16
14
12
10
8
6
4
2
0

Im f2+(s)Im f3+(s)

1
0
0.4

0.6

0.8

1

Im f2-(s)

5
4
3
2
1
0

3
2

Im f1-(s)

1.2

1.4

1.6

1.8

√t (GeV)Im f3-(s)

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

√t (GeV)J. Ruiz de Elvira

Roy-Steiner equations for πN

16

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving t-channel: S-wave results
Important for the σπN term
K̄K channel important ⇒ two-channel Muskhelishvili-Omnès problem
also needed:
K K̄ s-wave partial waves: [Büttiker. (2004)]
KN s-wave pw: SAID [Arndt et al. 2008], KH80
Hyperon couplings from [Jülich model 1989]
KN subthreshold parameters neglected

MO solutions in general consistent with KH80 results
J. Ruiz de Elvira

Roy-Steiner equations for πN

17

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving s-channel: flow of information
General form of the s-channel integral equationI
fl+(W) = ∆Il+ (W) +

1
π

Z∞
W+

dW 0

∞ n
o
X
KllI 0 (W, W 0 ) Im flI0 + (W 0 ) + KllI 0 (W, −W 0 ) Im f(lI 0 +1)− (W 0 )l0 =0⇒ form of ππ Roy-Equations∆Il+ (W) ≡ t-channel contribution and pole term
valid up to Wm = 1.38 GeV
Input:
RS t-channel solutions
s-channel partial waves for J > 1 [SAID analysis]s-channel partial waves for Wm < W < 2.5 GeV [SAID analysis]high energy contribution for W > 2.5 GeV: Regge model [Huang et al. 2010]

Output:Self-consistent solution for S and P waves between sth ≤ s ≤ sm
Constraints on subtraction constants ⇒ subthreshold parameters
J. Ruiz de Elvira

Roy-Steiner equations for πN

18

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving s-channel: subtractions

Existence and uniqueness of solutions [Gasser, Wanders 1999]⇒ no-cusp condition for each pw + 2 additional constraints are needed

Take advantage of the precise data for pionic atoms [Gotta et al. 2005, 2010]⇒ Impose as a constraint scattering lengths from a combined analysis of pionic hydrogenand deuterium [Baru et al. 2011]
1/2−1a0+ = (169.8 ± 2.0)10−3 Mπ

3/2−1a0+ = (−86.3 ± 1.8)10−3 MπIRe fl±(s) = q2l aIl± + bIl± q2 + · · ·10 subthreshold parameters are needed to match d.o.f⇒ three subtractions

J. Ruiz de Elvira

Roy-Steiner equations for πN

19

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving s-channel: strategy

Parametrize S and P waves up to W < Wm
Imposing a continuous and differentiable matching point
Using SAID partial waves as starting point
Minimize difference between LHS and the RHS on a grid of points Wj
2

χ =

2
N
Is
Is
XX
Re fl±(Wj ) − F[fl±](Wj )

l,Is ,± j=1

Is
Re fl±
(Wj )

Is
F[fl±
](Wj ) ≡ right hand side of RS-equations

Parametrization and subthreshold parameters are the fitting parameters

J. Ruiz de Elvira

Roy-Steiner equations for πN

20

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving s-channel: results
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

3
2
1
0
-1
-2

p13(s)

1.15

1.2

1.25

1.3

1.35

 s (GeV)√

ssssblue/red
ssssssm
sssLHS/RHS
ssafter the fit

p33(s)

0
-0.2
-0.4
-0.6
-0.8
-1

p11(s)

1.1

s31(s)

sssgray/blackssssssm
sssLHS/RHS
sbefore the fit

p31(s)

1.1

1.15

1.2

1.25

1.3

1.35

ssNotation: L2Is 2J

 s (GeV)√
J. Ruiz de Elvira

Roy-Steiner equations for πN

21

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving the full RS system: strategy

Full solution: self-consistent, iterative solution of the full RS system⇒ consistent set of s- and t-channel PWs & low-energy parameters
However:t-channel RS eqs. depend only weekly on s-channel PWs
resulting s-channel PW change little from SAID
A full solution can be achieved including in the s-channel RS eqs. thet-channel dependence on the subthreshold parameters

J. Ruiz de Elvira

Roy-Steiner equations for πN

22

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Results: s-channel PWs
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

3
2
1
0
-1
-2

p13(s)

1.15

1.2

1.25

1.3

1.35

√s (GeV)ssssblue/red
ssssssm
sssLHS/RHS
ssafter the fit

p33(s)

0
-0.2
-0.4
-0.6
-0.8
-1

p11(s)

1.1

s31(s)

sssgray/blackssssssm
sssLHS/RHS
sbefore the fit

p31(s)

1.1

1.15

1.2

1.25

1.3

1.35

ssNotation: L2Is 2J√s (GeV)J. Ruiz de Elvira

Roy-Steiner equations for πN

23

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Results: s-channel PWs
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

3
2
1
0
-1
-2

p13(s)

1.15

1.2

1.25

1.3

1.35

√s (GeV)ssizablef2 (1275)
effectssssblue/red
ssssssm
sssLHS/RHS
ssafter the fit

p33(s)

0
-0.2
-0.4
-0.6
-0.8
-1

p11(s)

1.1

s31(s)

sssgray/blackssssssm
sssLHS/RHS
sbefore the fit
ssNotation: L2Is 2J

p31(s)

1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)J. Ruiz de Elvira

Roy-Steiner equations for πN

24

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Results: t-channel PWs
9
6
3
0
-3
-6
-9

Imf0+(s)

blue
sssm

15
12
9
6
3
0
18
15
12
9
6
3
0

90
75
60
45
30
15
0

Imf1+(s)

18
15
12
9
6
3
0

Imf2+(s)

0.5

1

1.5

Imf1-(s)sss+sssm
ssssKH80

Imf2-(s)0.5√t (GeV)ssssbefore the fit
sssredsssm
ssssafter the fit

11.5√t (GeV)J. Ruiz de Elvira

Roy-Steiner equations for πN

25

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Improvements with respect to KH80
Karlsruhe-Helsinki analysis KH80 [Höhler et al. 1980]
comprehensive analyticity constraints based on fixed-t dispersion relations
old experimental data
Here, an update of KH80 results with modern input
HDR increase the range of validity of the equations
πN scattering length extracted from hadronic atoms ⇒ essential for the σπN
Goldberger-Miyazawa-Oehme sum rule:
g2πN /4π = 13.7 ± 0.2 [Baru et al. 2011]
compare:g2πN /4π = 14.28 [Höhler et al. 1983]s-channel PWs from SAID
f2 (1275) included ⇒ sizable effect

KH80 is internally consistent ⇒ RS reproduces KH80 results with KH80 input
J. Ruiz de Elvira

Roy-Steiner equations for πN

26

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Results: s-channel PWs with KH80 input
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

3
2
1
0
-1
-2

p13(s)

1.15

1.2

1.25

1.3

1.35

√s (GeV)ssssblue/red
ssssssm
sssLHS/RHS
ssafter the fit

p33(s)

0
-0.2
-0.4
-0.6
-0.8
-1

p11(s)

1.1

s31(s)

sssgray/blackssssssm
sssLHS/RHS
sbefore the fit

p31(s)

1.1

1.15

1.2

1.25

1.3

1.35

ssNotation: L2Is 2J√s (GeV)J. Ruiz de Elvira

Roy-Steiner equations for πN

27

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Results: t-channel PWs with KH80 input
9
6
3
0
-3
-6
-9

Imf0+(s)

blue
sssm

15
12
9
6
3
0
21
18
15
12
9
6
3
0

90
75
60
45
30
15
0

Imf1+(s)

21
18
15
12
9
6
3
0

Imf2+(s)

0.5

1

1.5

Imf1-(s)sss+sssm
ssssKH80

Imf2-(s)0.5√t (GeV)ssssbefore the fit
sssredsssm
ssssafter the fit

11.5√t (GeV)J. Ruiz de Elvira

Roy-Steiner equations for πN

28

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving the full RS system: uncertainties
Statistical errors (at intermediate energies)
important correlations between subthreshold parameters
shallow fit minima⇒ Sum rules for subthreshold parameters become essential to reduce the errors

Input variation (small)small effect for considering s-channel KH80 input
very small effects from L > 5 s-channel PWs
small effect from the different S-wave extrapolation for t > 1.3 GeV
negligible effect of ρ0 andρ00
very significant effects of the D-waves (f2 (1275))
F-waves shown to be negligible
matching conditions (close to Wm )scattering length (SL) errors (on S-waves and subthreshold parameters)
very important for the σπN
J. Ruiz de Elvira

Roy-Steiner equations for πN

29

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Uncertainties: statistical errors
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

s31(s)

3
2
1
0
-1
-2

p13(s)

p33(s)0p11(s)

p31(s)

-0.2
-0.4
-0.6
1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)J. Ruiz de Elvira

1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)
Roy-Steiner equations for πN

30

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving the full RS system: uncertainties
Statistical errors (at intermediate energies)
important correlations between subthreshold parameters
shallow fit minima⇒ Sum rules for subthreshold parameters become essential to reduce the errors

Input variation (small)small effect for considering s-channel KH80 input
very small effects from L > 5 s-channel PWs
small effect from the different S-wave extrapolation for t > 1.3 GeV
negligible effect of ρ0 and ρ00very significant effects of the D-waves (f2 (1275))
F-waves shown to be negligible
matching conditions (close to Wm )scattering length (SL) errors (on S-waves and subthreshold parameters)
very important for the σπN
J. Ruiz de Elvira

Roy-Steiner equations for πN

31

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Uncertainties: input variation
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

s31(s)

3
2
1
0
-1
-2

p13(s)

p33(s)0p11(s)

p31(s)

-0.2
-0.4
-0.6
1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)J. Ruiz de Elvira

1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)
Roy-Steiner equations for πN

32

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving the full RS system: uncertainties
Statistical errors (at intermediate energies)
important correlations between subthreshold parameters
shallow fit minima⇒ Sum rules for subthreshold parameters become essential to reduce the errors

Input variation (small)small effect for considering s-channel KH80 input
very small effects from L > 5 s-channel PWs
small effect from the different S-wave extrapolation for t > 1.3 GeV
negligible effect of ρ0 and ρ00very significant effects of the D-waves (f2 (1275))
F-waves shown to be negligible
matching conditions (close to Wm )scattering length (SL) errors (on S-waves and subthreshold parameters)
very important for the σπN
J. Ruiz de Elvira

Roy-Steiner equations for πN

33

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Uncertainties: matching conditions
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

s31(s)

3
2
1
0
-1
-2

p13(s)

p33(s)0p11(s)

p31(s)

-0.2
-0.4
-0.6
1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)J. Ruiz de Elvira

1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)
Roy-Steiner equations for πN

34

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving the full RS system: uncertainties
Statistical errors (at intermediate energies)
important correlations between subthreshold parameters
shallow fit minima⇒ Sum rules for subthreshold parameters become essential to reduce the errors

Input variation (small)small effect for considering s-channel KH80 input
very small effects from L > 5 s-channel PWs
small effect from the different S-wave extrapolation for t > 1.3 GeV
negligible effect of ρ0 and ρ00very significant effects of the D-waves (f2 (1275))
F-waves shown to be negligible
matching conditions (close to Wm )scattering length (SL) errors (on S-waves and subthreshold parameters)
very important for the σπN
J. Ruiz de Elvira

Roy-Steiner equations for πN

35

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Uncertainties: scattering lengths
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2

s11(s)

s31(s)

3
2
1
0
-1
-2

p13(s)

p33(s)0p11(s)

p31(s)

-0.2
-0.4
-0.6
1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)J. Ruiz de Elvira

1.1

1.15

1.2

1.25

1.3

1.35

√s (GeV)
Roy-Steiner equations for πN

36

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel Solving s-channel Solving the full system

Solving the full RS system: σπN
Very preliminary results

Σd = Fπ2

σπN = Σd + ∆D − ∆R − ∆σ++,∆D − ∆R − ∆σ = −(1.8 ± 2.2) MeV
d00+ 2Mπ2 d01
Σd = 57.9 ± 1.8 MeV
σπN = (56.1 ± 2.8) MeV+(3 ± 2.2) MeV (IV)⇒with KH80 SL+(KH80 input): Σd = 47.9 MeV ⇒ σπN = 46 MeV
to be compared with σπN = 45 [Gasser, Leutwyler, Socher, Sainio 1988, Gasser, Leutwyler, Sainio 1991]⇒KH80 is internally consistent but at odd with the modern SL determinations
J. Ruiz de Elvira

Roy-Steiner equations for πN

37

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Contents
1

Motivation

2

Introduction: Roy(Steiner)-equations for ππ vs πN

3

Roy-Steiner equations for πN

4Solving Roy-Steiner equations for πN
Solving the t-channel subproblem
Solving the s-channel subproblem
Solving the full RS system

5

Summary & Outlook
J. Ruiz de Elvira

Roy-Steiner equations for πN

38

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Summary & Outlook
What has been done:
Derived a closed system of Roy-Steiner equations (PWHDRs) for πN scattering
Constructed unitarity relations including K̄K intermediate states for thet-channel PWs
Optimized the range of convergence by tuning a for s- and t-channel each
Implemented subtractions at several orders
Solved the t-channel MO problem for a single- and two-channel approximation⇒ t-channel RS/MO machinery works
Numerical solution of the full system of RS eqs.
Precise determination of the σπNPreliminary error analysis
What needs to be done:
Error propagation still ongoing
Extraction of the Low Energy Constants
Possible improvements: higher PWs, more inelastic input, ...

J. Ruiz de Elvira

Roy-Steiner equations for πN

39

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: flow of information
Higher partial waves
I , l ≥ 2, s ≤ s
Im fl±
ms-channel partial waves
solve Roy–Steiner equations for s ≤ sm
Inelasticities
I , l ≤ 1, s ≤ s
ηl±
m

+
f0+

+
f1+

+
f1−

−
f0+

−
f1+

−
f1−

Subtraction
constants

ππ scattering
phases δJIt

High-energy region
I , s≥s
Im fl±
m

πN coupling
constantt-channel partial waves
solve Roy–Steiner equations for t ≤ tm
f+0

f±1

f±2

J. Ruiz de Elvira

f±3

f±4

···

High-energy region
Im f±J , t ≥ tm

Roy-Steiner equations for πN

40

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving Roy-Steiner equations for πN: Recoupling schemess-channel subproblem:
Kernels are diagonal for I ∈ {+, −}, but
unitarity relations are diagonal forIs ∈ {1/2, 3/2} ⇒ all partial-waves areinterrelated
Once the t-channel PWs are known⇒ Structure similar to ππ Roy-equations

t-channel subproblem:
Only higher PWs couple to lower ones
Only PWs with even or odd J are coupled
No contribution from f+J to f−J+1⇒ Leads to Muskhelishvili-Omnès problem

J. Ruiz de Elvira

Roy-Steiner equations for πN

41

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Spare slides

J. Ruiz de Elvira

Roy-Steiner equations for πN

42

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: HDR’s
Hyperbolic DRs:(s − a)(u − a) = b = (s0 − a)(u0 − a) with a, b ∈ R
1
A (s, t; a) =
π
+

Z∞
ds
s+

0




Z∞
1
1
1
1
Im A+ (s0 , t0 )
+ 0 0
+ 0
− 0
Im A (s , t ) +
dt0
0
s −s
s −u
s −a
π
t0 − ttπB+ (s, t; a) = N + (s, t) +N + (s, t) = g2



1
π

Z∞
s+

ds0





1
1
1
− 0
Im B+ (s0 , t0 ) +s0 − s
s −u
π

1
1
− 2
m2 − s
m −u



Z∞
tπ

dt0

ν Im B+ (s0 , t0 )ν0
t0 − tsimilar for A− , B− and N − [Hite/Steiner (1973)]

Why HDR?
Combine all physical regions ⇒ crucial for t-channel projection
Evade double-spectral regions ⇒ the PW decompositions converge
Range of convergence can be maximized by tuning the free hyperbola parameter a
No kinematical cuts, manageable kernel functions

J. Ruiz de Elvira

Roy-Steiner equations for πN

43

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

πN-scattering basics: partial waves
s-channel projection:



o
1 n
(E + m) AIl (s) + (W − m)BIl (s) + (E − m) − AIl±1 (s) + (W + m)BIl±1 (s)
16πW
Z1
√
dzs Pl (zs )X I (s, t)
XlI (s) =for X ∈ {A, B} and W = s
2

I
fl±
(W) =

t=t(s,zs )=−2q (1−zs )−1

I (W) = −f I
McDowell symmetry: fl+(l+1)− (−W)
t-channel projection:Jf+(t) = −

1
4π

Z1


dzt PJ (zt )

∀l≥0

m
p2t
−
AI (s, t)
zt BI (s, t)
s=s(t,zt )s=s(t,zt )(pt qt )J(pt qt )J−1


∀J≥0

0
J
f−
(t) =

1
4π

p

J(J + 1)1
2J + 1 (pt qt )J−1Z1hidzt PJ−1 (zt ) − PJ+1 (zt ) BI (s, t)

s=s(t,zt )

∀J≥1

0

Bose symmetry ⇒ even/odd J ⇔ I = +/−J. Ruiz de Elvira

Roy-Steiner equations for πN

44

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: derivation
Recipe to derive Roy-Steiner equations:
Expand imaginary parts in terms of s- and t-channel partial waves
Project onto s- and t-channel partial waves
Combine the resulting equations using s- and t-channel PW unitarity relations

Similar structure to ππ Roy equations
Validity: assuming Mandelstam analyticitys-channel ⇒ optimal for a = −23.2Mπ2s ∈ s+ = (m + Mπ )2 , 97.30 Mπ2⇔

W ∈ [W+ = 1.08 GeV, 1.38 GeV]t-channel ⇒ optimal for a = −2.71Mπ2t ∈ [tπ = 4Mπ2 , 205.45 Mπ2 ]J. Ruiz de Elvira⇔√√t ∈ [ tπ = 0.28 GeV, 2.00 GeV] .

Roy-Steiner equations for πN

45

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: subtractions
Subtractions are necessary to ensure the convergence of DR integrals⇒ asymptotic behavior
Can be introduced to lessen the dependence of the low-energy solution on the high-energy
behavior
Parametrize high-energy information in (a priori unknown) subtraction constants⇒ matching to ChPT
Subthreshold expansion around ν = t = 0
Ā+ (ν, t) =

∞
X

2m n
a+
mn ν tm,n=0

Ā− (ν, t) =∞X

2m+1 n
a−
t
mn ν

m,n=0

where
Ā+ (s, t) = A+ (s, t) −g2mĀ− (s, t) = A− (s, t) ,similar expansion for B+ (s, t) and B− (s, t)

J. Ruiz de Elvira

Roy-Steiner equations for πN

46

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: s-channels-channel RS equations
II

fl+ (W) = Nl+ (W) +

+

1
π

Z∞
dt
tπ

0

1
π

Z∞
dW

0

l0 =0

W+

Xn

∞ n
o
X
0
0
I
0
I
0
I
I
Kll0 (W, W ) Im fl0 + (W ) + Kll0 (W, −W ) Im f(l0 +1)− (W )

o
0
J 0
0
J 0
GlJ (W, t ) Im f+ (t ) + HlJ (W, t ) Im f− (t )JI

= −f(l+1)− (−W)∀l≥0,[Hite/Steiner (1973)]

KllI 0 (W, W 0 ), GlJ (W, t0 ) and HlJ (W, t0 )-Kernels: analytically known,
δ 0

ll
+ ...
e.g. KllI 0 (W, W 0 ) = W 0 −W

∀ l, l0 ≥ 0 ,Validity: assuming Mandelstam analyticity
2⇒ optimal for a = −23.2Mπ2s ∈ s+ = (m + Mπ )2 , 97.30 Mπ
J. Ruiz de Elvira⇔

W ∈ [W+ = 1.08 GeV, 1.38 GeV]

Roy-Steiner equations for πN

47

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Roy-Steiner equations for πN: t-channel
t-channel RS equations
JJ

f+ (t) = Ñ+ (t) +

+

J

1
π

π

dW

0

∞ n
X

o
0
I
0
0
I
0
G̃Jl (t, W ) Im fl+ (W ) + G̃Jl (t, −W ) Im f(l+1)− (W )

l=0

W+

1
π

Z∞
dW
W+

0

∞ n
X

o
0
I
0
0
I
0
H̃Jl (t, W ) Im fl+ (W ) + H̃Jl (t, −W ) Im f(l+1)− (W )

l=0

Z∞ X
0
3
0
J0 0
dt
K̃JJ 0 (t, t ) Im f− (t )tπ∀J ≥0,

J0

tπ

J

+

π

Z∞

Z∞ X n
o
0
1
0
J0 0
2
0
J0 0
dt
K̃JJ 0 (t, t ) Im f+ (t ) + K̃JJ 0 (t, t ) Im f− (t )f− (t) = Ñ− (t) +1

1∀J ≥1,

J0

Validity: assuming Mandelstam analyticity
2⇒ optimal for a = −2.71Mπ22t ∈ [tπ = 4Mπ, 205.45 Mπ]
J. Ruiz de Elvira⇔√√t ∈ [ tπ = 0.28 GeV, 2.00 GeV] .

Roy-Steiner equations for πN

48

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

RS-eqs for πN: Range of convergence
Subthreshold expansion around ν = t = 0

g2
+
+
2
2
2
+ d00
+ d01
t + a+
10 ν + O ν t, t
m

−
− 3
5
2 3
A− (ν, t) = νa−00 + a01 νt + a10 ν + O ν , νt , ν t

4mν
3
+ νb+
B+ (ν, t) = g2 2
00 + O ν , νt ,(m − s0 )2



2
t
g2
−
− 2
2 2
2
−
−
+ b−
B− (ν, t) = g2
00 + b01 t + +b10 ν + O ν , ν t, t
m2 − s0(m2 − s0 )22m2
A+ (ν, t) =pseudovector Born terms: DI = AI + νBI

+
+
+ 2
+ d01
t + d10νD̄+ = d00
+
+
dmn
= a+
mn + bm−1,n ,

−
−
dmn
= a−
mn + bmn .

Sum rules for subthreshold parameters:
+
d00
=−

g2
1
+
m
π

Z∞
s+



1
ds0 h0 (s0 ) Im A+ (s0 , z0s ) (0,0) +π12h0 (s0 ) = 0− 0s − s0s −a

J. Ruiz de Elvira

Z∞


dt0 
Im A+ (t0 , z0t ) (0,0)
t0tπRoy-Steiner equations for πN

49

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

RS-eqs for πN: Range of convergence
Assumption: Mandelstam analyticity [Mandelstam (1958,1959)]⇒ T(s,t) can be written in terms double spectral densities: ρst , ρsu , ρut
RR 0 0 ρsu (s0 ,u0 )RR 0 0 ρtu (t0 ,u0 )RR 0 0 ρst (s0 ,t0 )T(s, t) = π12
ds du (s0 −s)(u0 −u) + π12
dt du (t0 −t)(u0 −u) + π12
ds dt (s0 −s)(t0 −t)
integration ranges defined by the support of the double spectral densities ρ
200

Boundaries of ρ are given lowest lying intermediate states
(I)

150

(II)

(III)

(IV)

100
Ρsu

u  M Π2Ρut
50t

They limit the range of validity of the HDRS:

Ν

0
Ρst

Pw expansion converge-50⇒ z = cos θ ∈ Lehmann ellipses [Lehmann (1958)]

the hyperbolae (s − a)(u − a) = b does not enter any

-100
-100

-50

0

50

100

150

s  M Π2

double spectral region⇒ for a value of a, constraints on b yield ranges in s & t
J. Ruiz de Elvira

Roy-Steiner equations for πN

50

200

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel: P-wave results
√

2Mπ

√
tm

200

√
√
tm
40Mπ

2Mπ

0-sub(|a| → ∞)0-sub
1-sub(|a| → ∞)1-sub
2-sub(|a| → ∞)
2-sub
KH80

150

20

n-sub

/ GeV−1

30

10

100

1
f−

1
f+

n-sub

40Mπ

0-sub(|a| → ∞)0-sub
1-sub(|a| → ∞)1-sub
2-sub(|a| → ∞)
2-sub
KH80

/ GeV−2

40

50
0

60

0.3

0.4

0.5

2Mπ

1.0

40

0

1.1
1.1

200

0.3

0.4

0.5

0.6 √ 0.7
0.8
t / GeV

150

n-sub

30

0.9

1.0

√
40Mπ 0.98√
0-sub( tm = 1.1 GeV)
0-sub√
1-sub( tm = 1.1 GeV)
1-sub√
2-sub( tm = 1.1 GeV)
2-sub
KH80

2Mπ

1.1
1.1

100

1
f−

/ GeV−1
n-sub

0.9
√

40Mπ 0.98√
0-sub( tm = 1.1 GeV)
0-sub√
1-sub( tm = 1.1 GeV)
1-sub√
2-sub( tm = 1.1 GeV)
2-sub
KH80

50

1
f+

0.6 √ 0.7
0.8
t / GeV/ GeV−2

-10

20

50
10

0

0.3

0.4

0.5

0.6 √ 0.7
0.8
t / GeV

0.9

1.0

1.1

0

0.3

0.4

0.5

0.6 √ 0.7
0.8
t / GeV

0.9

1.0

1.1

MO solutions in general consistent with KH80 results
J. Ruiz de Elvira

Roy-Steiner equations for πN

51

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel: coupled channels
Generic coupled-channel integral equationf(t) = ∆(t) +

1
π

Ztm
tπ

dt0

1
T ∗ (t0 )Σ(t0 )f(t0 )+t0 − t
π

Z∞

dt0

tm

Im f(t0 )t0 − tFormal solution as in the single-channel case (now with Omnès matrix Ω(t))⇒ Two-channel
problem 0 Muskhelishvili-Omnès
f+ (t)
f(t) =Im Ω(t) = (T(t))∗ Σ(t)Ω(t)h0+ (t)Two linearly independent solutions Ω1 , Ω2 [Muskhelishvili 1953]
In general no analytical solution for the Omnès matrix
but for its determinant [Moussallam 2000](
det Ω(t) = exp

J. Ruiz de Elvira

t
π

Ztm
tπ

ψ(t0 )
dt 0 0t (t − t)
0).

Roy-Steiner equations for πN

52

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel S-wave equations: input

Input needed:
ππ s-wave partial waves: [Caprini, Colangelo, Leutwyler, (in preparation)]
K K̄ s-wave partial waves: [Büttiker. (2004)]
πN and KN s-wave pw: SAID [Arndt et al. 2008], KH80
πN at high energies: Regge model [Huang et al. 2010]
πN parameters: KH80
Hyperon couplings from [Jülich model 1989]
KN subthreshold parameters neglected√Two-channel approximation beaks down at t0 = 1.3 GeV ⇒ 4π channel
From t0 to t = 2 GeV, different approximations considered

J. Ruiz de Elvira

Roy-Steiner equations for πN

53

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving t-channel: S-wave results

MO solutions in general consistent with KH80 results
J. Ruiz de Elvira

Roy-Steiner equations for πN

54

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving s-channel: consistency with KH80
Consistency with KH80
parametrize SAID S and P waves up to W < Wm
Imposing a continuous and differentiable matching point
Compare between the input (LHS) and the output (RHS)

S-WAVES

⇒ important discrepancies
J. Ruiz de Elvira

Roy-Steiner equations for πN

55

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving s-channel: consistency with KH80. P-waves

J. Ruiz de Elvira

Roy-Steiner equations for πN

56

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving s-channel: consistency with KH80
Consistency with KH80
parametrize SAID S and P waves up to W < Wm
Imposing a continuous and differentiable matching point
Compare between the input (LHS) and the output (RHS)⇒ important discrepancies
J. Ruiz de Elvira

Roy-Steiner equations for πN

57

Motivation Introduction RS-eqs. Solving RS-eqs. Summary

Solving s-channel: scattering length from hadronic atomsã+ = a+ + 1+M1 /m
π

p



2
Mπ
−m2π0
c1 − 2αf1
2πFππH/πD: bound state of π − and p/d,
spectrum sensitive to threshold πN
amplitude
Combined analysis of πH and πD:+−3 M −1a+π0 ≡ a = (7.5 ± 3.1) · 10−
−3 M −1a−π0 ≡ a = (86.0 ± 0.9) · 10

Large a+ suggests a large σπN , but

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aπ− p + aπ+ p2−3= (−1.1 ± 0.9) · 10

−1
Mπ

3s

3p

2s

2p

3d

Isospin breaking in σπN could be
important
Revisit the Cheng-Dashen
low-energy theorem
J. Ruiz de Elvira

ǫ1s

1s


Γ1s

Roy-Steiner equations for πN58