Resonances in Coupled-Channel Scattering · 2016. 7. 25. · 0(500) resonance. This means that the...

Resonances in Coupled-Channel Scattering

David Wilson

Lattice 2016University of Southampton

24-30 July 2016

David Wilson 2Resonances in coupled-channel scattering

Coupled-channel scattering

This talk: Topical report of recent coupled-channel scattering results from the Hadron Spectrum Collaboration

The method:- Build large correlation matrices with a diverse range of operators

- Extract many energy levels using the variational method

- Use these energies with extensions of Lüscher’s method to obtain infinite volume scattering amps

- Investigate the poles of the scattering amplitudes to obtain resonance information

Topics I won’t cover: The HALQCD method, Finite Volume Hamiltonian, EFTs in a box, etc.



Sounds hard... why bother?

a0(980), f0(980)a1(1260)X(3872), and other XYZ statesN★(1440), 𝝠(1405), ...

all decay into multiple final statesall are resonant enhancements in multiple channelsto understand these rigorously, we need coupled-channel analyses

Combinations of Kþ!"!þ candidates that are consis-tent with originating from a common vertex with"2vtxðKþ!"!þÞ=ndf < 9, with each charged hadron (h)separated from all PVs ["2IPðhÞ> 9] and having pTðhÞ>0:25 GeV, are selected. The quantity "2IPðhÞ is defined asthe difference between the "2 of the PV reconstructed withand without the considered particle. Kaon and pion candi-dates are required to satisfy ln½LðKÞ=Lð!Þ&> 0 and


Extracting resonance properties

π

π

π

π

excited states seen as resonant enhancements in the scattering of lighter stable particles

�/�

Ecm/GeV

⇢(770)



π

π

π

π

excited states seen as resonant enhancements in the scattering of lighter stable particles

} }Bound states Meson-meson continuum

Infinite volume

Finite volume

E

E

Cij(t)vnj = �n(t, t0)Cij(t0)v

nj

0.10

0.15

0.20



0.10

0.15

0.20

0.10

0.15

0.20

build a large basis of operators: O† ⇠ ̄� !D ... !D compute large correlation matrices:

solve GEVP:

Cij(t) = h0| Oi(t)O†j(0) |0i

[000]T�1

m⇡ = 236 MeV

(⇡⇡)† =X

~p1+ ~p2=~P

C(~p1, ~p2)⇡†(~p1)⇡†(~p2)⇡† =

X

i

v⇡i O†i

0.10

0.15

0.20



0.10

0.15

0.20

0.10

0.15

0.20

add in operators using a variationally optimal pion⇡⇡

combine in pairs

[000]T�1

m⇡ = 236 MeV

0.10

0.15

0.20



0.10

0.15

0.20

0.10

0.15

0.20

essential to have operators that overlap onto “meson” and “meson-meson” contributions to the physical spectrum

[000]T�1

m⇡ = 236 MeV


ρ resonance Phase shifts via the Lüscher method:

0.10

0.15

0.20

KK

��

��

��

��

��

��

��

atEcm

tan �1 =⇡3/2q

Z00(1; q2)

Z00(1; q2) =X

n2Z3

1

|~n|2 � q2

[000]T�1

m⇡ = 236 MeV

�1/�


ρ resonance with moving frames

��

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atEcm

�/�

m⇡ = 236 MeV

~P = [000]

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atEcm

�/�

m⇡ = 236 MeV

~P = [000]~P = [001]~P = [011]~P = [111]~P = [002]

��

��

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�=[��]�=[��]

�=[��]�=[��]

�=[��]�=[��]

�=[��]�=[��]

�=[��]�=[��]



atEcm

�/�

one volume, 22 energy levels... lots of constraint m⇡ = 236 MeV

- for more see Antoni Woss Tuesday 26 Jul 2016 at 14:40PRD 92 094502, arXiv:1507.02599

~P = [000]~P = [001]~P = [011]~P = [111]~P = [002]



Many derivations, all in agreement:

He, Feng, Liu 2005 - two channel QM, strong couplingHansen & Sharpe 2012 - field theory, multiple two-body channelsBriceño & Davoudi 2012 - strongly-coupled Bethe-Salpeter amplitudesGuo et al 2012 - Hamiltonian & Lippmann-Schwinger

Also derivations in specific channels, or for a specific parameterization of the interactions like NREFT, chiral PT, Finite Volume Hamiltonian, etc.

Briceño 2014 - Generalised to scattering of particles with non-zero spin, and spin-½.

Significant steps towards a general 3-body quantization condition have been made - see Stephen Sharpe on Tuesday 26 Jul 2016 at 15:40 for the latest

infinite volume scattering t-matrix

known finite-volume functionsphase space

Direct extension of the elastic quantization condition derived by Lüscher

det [1+ i⇢(E) · t(E) · (1+ iM(E,L))] = 0

��

-�-�-�-�-��


Determinantt = (⇡⇡ ! ⇡⇡)

det

[1+

i⇢·t

·(1+

iM(L

))]

�⇡⇡1�/�

�� -��

�

��

��

��

��

��

��

atEcm

E0 E1 E2 E3Real

Imag

det

[1+

i⇢·t

·(1+

iM(L

))]

��

-�-�-�-�-�

��


Determinantt = (⇡⇡ ! ⇡⇡)

KK

�⇡⇡1�/�

�� -��

�

��

��

��

��

��

��

atEcm

E0 E1 E2 E3

��

-�-�-�-�-��

�� -��

�

��

��

��

��

��

��


Determinantt =

✓⇡⇡ ! ⇡⇡ 0

0 KK̄ ! KK̄

◆

det

[1+

i⇢·t

·(1+

iM(L

))]

KK

�⇡⇡1�/�

�KK̄1

��

��

��⌘

atEcm

��

-�-�-�-�-��

�� -��

�

��

��

��

��

��

��


Determinantt =

✓⇡⇡ ! ⇡⇡ ⇡⇡ ! KK̄KK̄ ! ⇡⇡ KK̄ ! KK̄

◆

det

[1+

i⇢·t

·(1+

iM(L

))]

KK

�⇡⇡1�/�

�KK̄1

⌘

��

��

��

atEcm

��

-�-�-�-�-��

�� -��

�

��

��

��

��

��

��


Determinantt =

✓⇡⇡ ! ⇡⇡ ⇡⇡ ! KK̄KK̄ ! ⇡⇡ KK̄ ! KK̄

◆

det

[1+

i⇢·t

·(1+

iM(L

))]

KK

�⇡⇡1�/�

�KK̄1

⌘

��

��

��

atEcm

��

-�

-�

-�

-�

-�

�

�

�

�

�

��

-�-�-�-�-��

�� -��

�

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Determinantt =

✓⇡⇡ ! ⇡⇡ ⇡⇡ ! KK̄KK̄ ! ⇡⇡ KK̄ ! KK̄

◆

det

[1+

i⇢·t

·(1+

iM(L

))]

KK

�⇡⇡1�/�

�KK̄1

⌘

��

��

��

atEcm


Amplitude parameterization

determinant condition:- several unknowns at each value of energy- energy levels typically do not coincide- underconstrained problem for a single energy

one solution: use energy dependent parameterizations- Constrained problem when #(energy levels) > #(parameters)- Essential amplitudes respect unitarity of the S-matrix

t =

✓⇡⇡ ! ⇡⇡ ⇡⇡ ! KK̄KK̄ ! ⇡⇡ KK̄ ! KK̄

◆

S†S = 1 ! Im t�1 = �⇢

t�1 = K�1 � i⇢K-matrix approach:

Kij =gigj

m2 � s + �ije.g.:

⇢ij = �ij2kiEcm

det [1+ i⇢(E) · t(E) · (1+ iM(E,L))] = 0


ρ resonance into the coupled-channel region

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

30 32 340.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

30 32 340.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

30 32 34

KK

m⇡ = 236 MeV


ρ resonance into the coupled-channel region

-30

0

30

60

90

120

150

180

0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

0.7

0.8

0.9

1.00.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

PRD 92 094502, arXiv:1507.02599

m⇡ = 236 MeV


ρ resonance pole

��

��

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��

�=[��]�=[��]

�=[��]�=[��]

�=[��]�=[��]

�=[��]�=[��]

�=[��]�=[��]

atEcm

�/�

m⇡ = 236 MeV

-0.04

-0.03

-0.02

-0.01

0.08 0.10 0.12 0.14 0.16

-0.018

-0.017

-0.016

-0.015

-0.014

-0.013

-0.012

0.128 0.129 0.130 0.131 0.132 0.133

tij ⇠cicjs0 � s

near a pole:

⇡⌘-KK̄-⇡⌘0

I = 1 J = 0


An a0 resonance

m⇡ = 391 MeV

- for more see Jozef Dudek, Tuesday 26 Jul 2016 at 16:50PRD 93 094506, arXiv:1602.05122

Figure 58. Partial wave analysis of the K−π+ → K−π+ amplitudes deduced from theLASS results of Fig. 57, showing the magnitude and phase for the S , P and D-waves [103].

Figure 59. Mass distribution for π0η from the GAMS experiment [11] on π−p → (π0η)n ,where both π0 and η are detected in the γγ decay mode. Partial wave analysis reveals thatthe J = 0 wave has two possible resonances a0(980) and a0(1430) in this mass region.

a0(980) a0(1430)

GAMS, Alde et al PLB 203 397, 1988.

⇡+p ! (⇡0⌘)n

E⇡⌘|thr. KK̄|thr.}

a0(980)

12 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle

-100 -50 0 50 100E [MeV]

0

1e-05

2e-05

3e-05

4e-05

5e-05

6e-05

7e-05

σππ

[1/M

eV2 ]

CMD-2SNDKLOEE791

Fig. 9. Results for the ππ cross section. The curves are resultsbased on Flatté distributions taken from the Refs. [14,15,16,17].

-200 -100 0 100 200E [MeV]

0

50

100

150

δ πη (

deg)

AchasovBuggTeigeKLOE

Fig. 10. πη phase shift δ in the J = 0, I = 1 partial wave.The curves are results based on Flatté distributions taken fromRefs. [8,9,12,13].

0 50 100 150 200 250 300E [MeV]

0,2

0,4

0,6

0,8

1

ηπη


Fig. 11. Inelasticity in the πη J = 0, I = 1 partial wave. Thecurves are results based on Flatté parameters taken from Refs.[8,9,12,13].

-100 -50 0 50 100E [MeV]

0

5e-05

1e-04

σπη

[1/M

eV2 ]


Fig. 12. Total cross section in the πη J = 0, I = 1 partialwave. The curves are results based on Flatté parameters takenfrom Refs. [8,9,12,13].

Baru et al, Eur. Phys. J. A23 523, 2005.

�⇡⌘/M

eV�2

(E � 2mK)/MeV

5⇥ 10�5

10�4


An a0 resonance

0.17

0.19

0.21

0.23

0.25

0.27

0.29

16 20 24

0.17

0.19

0.21

0.23

0.25

0.27

0.29

16 20 24

0.17

0.19

0.21

0.23

0.25

0.27

0.29

16 20 24

0.17

0.19

0.21

0.23

0.25

0.27

0.29

16 20 24

0.17

0.19

0.21

0.23

0.25

0.27

0.29

16 20 24

0.17

0.19

0.21

0.23

0.25

0.27

0.29

16 20 24

m⇡ = 391 MeV

⇡⌘-KK̄-⇡⌘0


0.17

0.19

0.21

0.23

16 20 24

0.17

0.19

0.21

0.23

16 20 24

0.17

0.19

0.21

0.23

16 20 24

0.17

0.19

0.21

0.23

16 20 24

0.17

0.19

0.21

0.23

16 20 24

a0 resonance - two channel region

m⇡ = 391 MeV

⇡⌘-KK̄

Kij =gigj

m2 � s + �ijm = (0.2214± 0.0029± 0.0004) · a�1t

2

6666664

1 0.58 �0.06 �0.51 0.39 0.021 �0.63 �0.87 0.84 �0.49

1 0.52 �0.68 0.831 �0.90 0.53

1 �0.781

3

7777775

g⇡⌘ = ( 0.091± 0.016± 0.009) · a�1tgK¯K = (�0.129± 0.015± 0.002) · a�1t

�⇡⌘,⇡⌘ = �0.16± 0.24± 0.03�⇡⌘, K¯K = �0.56± 0.29± 0.04�K¯K,K¯K = 0.12± 0.38± 0.08

�2/Ndof

= 58.047�6 = 1.41

using 47 energy levels

David Wilson 27Resonances in coupled-channel scatteringm⇡ = 391 MeV

��

30

−30

−60

60

0.18 0.19 0.20 0.21 0.22 0.23

0.18 0.19 0.20 0.21 0.22 0.23

90

120

1.0

242016

0.8

0.6

0.4

0.2

0

150

180

0.18 0.19 0.20 0.21 0.22 0.23242016

0.7

0.6

0.5

0.4

0.2

0.3

0.1

a0 resonance - two channel region⇡⌘-KK̄

from 47 energy levelsS-wave


a0 resonance pole

-0.12

-0.08

-0.04

0 0.16 0.20 0.24

-0.015

-0.010

-0.005

0.005

0.010

0.015

0.195 0.200 0.205 0.210 0.215 0.220

tij ⇠cicjs0 � s

-0.2

-0.1

0.1

0.2

-0.2 -0.10.1

0.2

- for more see Jozef Dudek, Tuesday 26 July 2016 at 16:50


Other calculations

-30

0

30

60

90

120

150

180

0.7

0.8

0.9

1.0

0.16 0.18 0.20 0.22 0.24 0.26 0.28

0.16 0.18 0.20 0.22 0.24 0.26 0.28

242016

Coupled ⇡K � ⌘K

��

��

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��

��

��

��

-��

-��

-��

��

��

atEcm

�/�

⌘

⌘D⇡0

⌘D⌘0

⌘DsK̄0

�D⇡0

�DsK̄0

�D⌘0

��

��

atEcm

�/�

⌘

Coupled D⇡ �D⌘ �Ds ¯K

Combined S & P-wave analysis80 energy levels from 3 volumesarXiv:1406.4158, PRL 113 (2014) no.18, 182001

Combined S & P-wave analysis3 coupled channels in S-wave47 energy levels from 3 volumesarXiv:1607.????

m⇡ = 391 MeV

- Graham Moir, Thursday 28 July 2016 at 15:00


The f0(500)/� resonance - see Raul Briceño, Tuesday 26 July 2016 at 15:20

elastic scattering withvacuum quantum numbers⇡⇡ in I = 0, J = 0

300 400 500 600 700 800 900 1000 1100 1200 1300 1400s1/2 (MeV)

0

30

60

90

120

150

180

210

240

270

300

330

δ0

0(s)

Solution B Solution ASolution C Solution DSolution EProtopopescu et al. (Table VI)Estabrooks & Martin s-channelEstabrooks & Martin t-channelKaminski et al.

Grayer et al.

300 400 500 600 700 800 900 1000s1/2 (MeV)

0

30

60

90

120

150

δ1(s)

Protopopescu et al.

Estabrooks & Martin

Figure 2: Data on ⇡⇡ ! ⇡⇡ scattering phase shifts: Protopopescu et al. from [31], Grayer et al. from [33] (SolutionB also from [32]), Estabrooks and Martin from [35], Kaminski et al. from[36]. Left panel: The scalar-isoscalar phaseshift �(0)0 . Note the huge di↵erences due to systematic uncertainties, which exist even within data sets from the sameexperimental collaboration [33] (Something similar happens with [31], but we only show the most commonly used andconsistent data set). Please note that there is no Breit-Wigner-like sharp increase of 180o on the phase between thresholdand 800 MeV. Such sharp phase increase is seen around 980 MeV, corresponding to the f0(980) meson, although startingover a background phase of about 100o degrees. Right panel: For comparison we also show the vector-isovector �1 phaseshift, where the ⇢(770) resonance can be seen to follow the familiar Breit-Wigner shape [38] to a very good degree ofapproximation.

Sometimes, as in [33], statistical uncertainties were provided for each set of solutions. However,since these data sets are incompatible among themselves within statistical uncertainties, the dif-ferences between sets should be interpreted as an indication of the systematic uncertainty. As anexample, the left panel of Fig.2 displays the data on ⇡⇡! ⇡⇡ scattering phase shifts of the scalarisoscalar wave. Note the large di↵erences even within data sets coming from the same exper-iment [33] (Solution B was published first in [32]) due to systematic uncertainties. Somethingsimilar happens with [31], but we only show the most commonly used data set, since it will beseen later that the others are even more inconsistent with fundamental dispersive constraints.

Another relevant indication of the interest on ⇡⇡ scattering in the early seventies was theappearance of Ke4 experiments [39, 40]. These correspond to the K ! ⇡⇡e⌫ decay and providean indirect measurement of the �00 � �1 phase combination well below 500 MeV, a region thatcould not be reached with ⇡N ! ⇡⇡N experiments. At that time these low energy data were notvery determinant in the � discussion, but we will see that recent Ke4 experiments have actuallybeen decisive to enter the precision era for light scalars.

At this point, and in view of Fig.2 it is important to emphasize that the � is so wide thatright from the very beginning it was clear that the familiar Breit-Wigner description [38], validfor narrow isolated resonances, is not appropriate to describe the S-wave data. Actually, note inFig.2 that there is no isolated Breit-Wigner shape around 500-600 MeV, corresponding to a � orf0(500) resonance. This means that the � resonance does not appear as a peak in the ⇡⇡ ! ⇡⇡cross section nor in many other amplitudes which contain in the final state two pions with thequantum numbers of the f0(500). Of course, a Breit-Wigner-like shape over a background phaseof about 100 degrees is seen around 980 MeV in Fig.2, corresponding to the f0(980), but even

8

π

π

π

π

arXiv:1607.05900


The f0(500)/� resonance

500

600

700

800

900

1000

24 32 40 24 32 40 24 32 40 24 32 40 24 32 40

600

700

800

900

1000

1100

16 20 24 16 20 24 16 20 24 16 20 24 16 20 24

m⇡ = 391 MeV

m⇡ = 236 MeV

elastic scattering withvacuum quantum numbers⇡⇡ in I = 0, J = 0

- see Raul Briceño, Tuesday 26 July 2016 at 15:20



0

30

60

90

120

150

180

0.01 0.03 0.05 0.07 0.09 0.11 0.13

arXiv:1607.05900- see Raul Briceño, Tuesday 26 July 2016 at 15:20

50

0

100

150

200

200

400

02.0 2.1 2.32.2

2.0 2.1 2.32.2


Future directionstwo-body coupled-channel

further operator structures - glueball, tetraquark, ...

formalism for three-body and beyond - needed for higher energies - needed to get closer to the physical mass

f0(980)

DD̄

DD̄?

N⇡

�a ! bc

- see Gavin Cheung, Monday 25 July 2016 at 14:55

- see Stephen Sharpe, Tuesday 26 July 2016 at 15:40

π

γ

π

π

Briceño et al, Phys.Rev.Lett. 115 (2015)

- see also Luka Leskovec on 29 July 2016 at 17:50

π π

ππ


Promising results so far ...... but a lot still to do

-30

30

60

90

120

150

180

1000 1050 1100 1150 1200 1250 1300

0.250.500.75

1000 1050 1100 1150 1200 1250 1300

/ MeV

m⇡ = 391 MeV


Backup

-1

-0.5

0

0.5

1

-0.06 -0.03 0 0.03 0.06 0.09 0.12





-300

-200

-100

0 300 500 700 900

- see Raul Briceño, Tuesday 26 Jul 2016 at 15:20

400 500 600 700 800 900 1000 1100 1200Re s

σ

1/2 (MeV)

-500

-400

-300

-200

-100

Im s σ

1/2 (

MeV

)

PDG estimate 1996-2010poles in RPP 2010poles in RPP 1996

Figure 4: The � or f0(400 � 1400) resonance poles listed in the RPP 1996 edition (Black squares) together with thosealso cited in the 2010 edition [70] (Red circles). Note the much better consistency of the latter and the general absenceof uncertainties in the former. The huge light gray area corresponds to the uncertainty band assigned to the � from 1996to 2010.

before, a very significant part of the apparent disagreement between di↵erent poles in Fig.2 isnot coming from experimental uncertainties when extracting the data, but from the use of modelsin the interpretation of those data and unreliable extrapolations to the complex plane. Actually,di↵erent analyses of the same experiment could provide dramatically di↵erent poles, dependingon the parameterization or model used to describe the data and its later interpretation in terms ofpoles and resonances. Maybe the most radical example are the three poles from the Crystal Barrelcollaboration, lying at (1100� i300) MeV [68], (400� i500) MeV and (1100� i137) MeV [69],corresponding to the highest masses and widths in that plot. These poles were compiled togetherin the RPP although they even lie in di↵erent Riemann sheets. Moreover we will see in Sect.2that all three lie outside the region of analyticity of the partial wave expansion (Lehmann-Martinellipse [71]).

Therefore it should be now clear that in order to extract the parameters of the � pole, whichlies so deep in the complex plane and has no evident fast phase-shift motion, it is not enough tohave a good description of the data. As a matter of fact, many functional forms could fit verywell the data in a given region, but then di↵er widely with each other when extrapolated outsidethe fitting region. For instance, if all data were consistent (which they are not) one can alwaysfind a good data description using polynomials, or splines, which have no poles at all. Hence, tolook for the � pole, the correct analytic extension to the complex plane, or at least a controlledapproximation to it, is needed. Unfortunately that has not always been the case in many analyses,and thus the poles obtained from poor analytic extensions of an otherwise nice experimentalanalysis are at risk of being artifacts or just plain wrong determinations. This, together with thehuge uncertainty attached to the � in the RPP, is what made many people outside the communityto think that no progress was made in the light scalar sector for many decades.

However, progress was being made and the other remarkable feature of Fig.2 is that by 2010most determinations agreed on a light sigma with a mass between 400 and 550 MeV and a half

13


The f0(500)/� resonance from J. R. Pelaez, arXiv:1510.00653


The f0(500)/� resonance from J. R. Pelaez, arXiv:1510.00653

400 500 600 700 800 900 1000 1100 1200Re s

σ

1/2 (MeV)

-500

-400

-300

-200

-100

Im s σ

1/2 (

MeV

)

RPP estimate 2012RPP estimate 1996-2010poles used for estimate in RPP 2012

400 425 450 475 500 525 550Re s

σ

1/2 (MeV)

-350

-325

-300

-275

-250

-225

-200

Im s σ

1/2 (

MeV

)

Conservative dispersive estimateRPP restricted range 2012RPP estimate 2012"most advanced" dispersive analyses according to RPP 2012

Figure 7: Left panel: Compilation of f0(500) or � resonance poles in the 2012 RPP edition. The light gray area standsfor the uncertainty assigned to the � poles from 1996 until 2010. The darker gray rectangle is the new uncertaintyestimated in the summary tables, Eq.(1). Right panel: Detail of the new uncertainty band. We only show the four“most advanced dispersive analyses” according to the 2012 RPP and as a darker area the “more radical” and “restrictedrange of f0(500) parameters”, Eq.(2), if one averages those four analyses. The dashed rectangle corresponds to the“conservative dispersive estimate”

ps� = 449+22�16 � i(275 ± 12)MeV which, as explained in the text, takes into account

that the di↵erences between these four dispersive approaches are of systematic nature.

pole based on data descriptions consistent with dispersive constraints, and whose poles are ob-tained from sound analytic extrapolations to the complex planes, should be taken into account.Of course, this was well known to the RPP authors and that is why in the 2012 RPP “Note onscalar mesons below 2 GeV”, they suggested that “One might also take the more radical pointof view and just average the most advanced dispersive analyses, [97, 111, 116, 117]...”, whichwe show in the right panel of Fig.7, “which provide a determination with minimal bias”. Byaveraging the values obtained in those four references a more restricted range of parameters isestimated at the 2012 RPP:

ps� = (446 ± 6) � i(276 ± 5)MeV (RPP2012 restricted range) (2)

In the left panel of Fig.7, the area covered by this “restricted” uncertainty would be almostimperceptible within the darker rectangle, and hence we show in the right panel an expandedview of the darker rectangle and just the “most advanced dispersive analyses” according to the2012 RPP. Thus the “restricted range of parameters” corresponds to the smallest and even darkerrectangle in the middle of the plot.

At the risk of being annoying, these uncertainties may now be too small, since the di↵er-ences between those four determinations are more of a systematic than statistical nature. Thus,weighting them as if the uncertainties and di↵erences were statistical to obtain an even smalleruncertainty is somewhat optimistic. Moreover, an uncertainty of less than 3% is hard to achievedue to isospin breaking e↵ects, which are not incorporated into these formalisms (except maybein the experimental uncertainties), and to the absence of 4⇡ channels, although we have seenabove that this e↵ect is very small. A suggestion would be to take as a conservative dispersiveestimate the band that covers [111] and [116], since the pole in [97] was not really calculatedwith the analytic extension of Roy equations but from the phenomenological representation andcan be considered superseded by the results of [111]. In addition the result of [117] lies withinthis estimate (it uses results from the [97, 111] group as input). That is:

ps� = 449+22�16 � i(275 ± 12)MeV (Conservative dispersive estimate) (3)

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⇡⌘-KK̄-⇡⌘0 - for more see Jozef Dudek, Tuesday 26 Jul 2016 at 16:50

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An a0 resonance - three channel region

Resonances in Coupled-Channel Scattering · 2016. 7. 25. · 0(500) resonance. This means that the...

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