Isoscalar ππ scattering and the σ/f0(500) resonance

Raúl Briceño rbriceno@jlab.org

[ with Jozef Dudek, Robert Edwards & David Wilson]

Lattice 2016Southampton, UK

July, 2016

HadSpec Collaboration

Motivation

K

⇡

⇡

b

d

u

d

d

u

d

d

⇡

⇡

u

d

u

d

d

⇡

⇡

u

d

u

d

d

dd

u unn dd

ud

ud

dd nn

⇡⇡

f0(500)/� f0(500)/�

scattering/spectroscopy precision tests of SM

long range nuclear forcesSample of previous lattice efforts:

Alford & Jaffe (2000)Prelovsek, et al. (2010)Fu (2013)Wakayama, et al. (2015)Howarth & Giedt, (2015)Z. Bai et al. (RBC, UKQCD) (2015)

The experimental situation

summary of various experiments

E� = 449(2216) MeV

�� = 550(24) MeV

“f0(500)/�”

The experimental situation

Peláez (2015)

bound state

threshold

Im[s] Infinite volume

s = E2cm

first Riemann sheet

Re[s]

branch cut - where scattering takes place

Finite vs. infinite volume spectrum

s = E2cm

Im[s]

Re[s]

second Riemann sheet

Infinite volume

narrow resonance

broad resonance

Re[s]

Finite vs. infinite volume spectrum

sR = (ER � i2�R)

2

Finite vs. infinite volume spectrum

finite volume eigenstates

finite volume

“only a discrete number of modes can exist in a finite volume”

no continuum of states:no cutsno sheet structureno resonances

no asymptotic states:no scattering

Lüscher formalismspectrum satisfy:

scattering amplitude

an exact mapping

finite volume spectrum

det[F�1(EL, L) +M(EL)] = 0

L

EL = finite volume spectrum

L = finite volume

F = known function

M = scattering amplitude

Lüscher formalism

Lüscher (1986, 1991) [elastic scalar bosons]

Rummukainen & Gottlieb (1995) [moving elastic scalar bosons]

Kim, Sachrajda, & Sharpe/Christ, Kim & Yamazaki (2005) [QFT derivation]

 Bernard, Lage, Meissner & Rusetsky (2008) [Nπ systems]

RB, Davoudi, Luu & Savage (2013) [generic spinning systems]

Feng, Li, & Liu (2004) [inelastic scalar bosons]

Hansen & Sharpe / RB & Davoudi (2012) [moving inelastic scalar bosons]

RB (2014) / RB & Hansen (2015) [moving inelastic spinning particles]

spectrum satisfy: det[F�1(EL, L) +M(EL)] = 0

Extracting the spectrumTwo-point correlation functions:

Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)]

e.g. ⇡[000]⇡[110]

m⇡ = 236 MeV

C2pt.ab (t,P) ⌘ h0|Ob(t,P)O†

a(0,P)|0i =X

n

Zb,nZ†a,ne

�Ent

-0.004

0

0.004

0.008

0.012

0.016

0 4 8 12 16 20 24 28 32 36

-0.002

0

0.002

0 4 8 12 16 20 24

eE0tC(t, 0)

Extracting the spectrumTwo-point correlation functions:

Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)]

e.g. ⇡[000]⇡[110]

m⇡ = 236 MeV

C2pt.ab (t,P) ⌘ h0|Ob(t,P)O†

a(0,P)|0i =X

n

Zb,nZ†a,ne

�Ent

-0.004

0

0.004

0.008

0.012

0.016

0 4 8 12 16 20 24 28 32 36

-0.002

0

0.002

0 4 8 12 16 20 24

(d)

(b)(a)

(c)(c)

close up

all ⌘⌘KK⇡⇡mm �

KK|thr.

⌘⌘|thr.…

Extracting the spectrumTwo-point correlation functions:

‘Diagonalize’ correlation function variationally Use a large basis of operators with the same quantum numbers

e.g.

Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)]

C2pt.ab (t,P) ⌘ h0|Ob(t,P)O†

a(0,P)|0i =X

n

Zb,nZ†a,ne

�Ent

a tE

cm

0.1

0.15

0.20

0.25

⌘⌘KK⇡⇡

�

~d =~PL

2⇡= [110]

m⇡ = 391MeV

L/as = 24

all ⌘⌘KK⇡⇡mm �

KK|thr.

⌘⌘|thr.…

Extracting the spectrumTwo-point correlation functions:

‘Diagonalize’ correlation function variationally Use a large basis of operators with the same quantum numbers

e.g.

Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)]

C2pt.ab (t,P) ⌘ h0|Ob(t,P)O†

a(0,P)|0i =X

n

Zb,nZ†a,ne

�Ent

a tE

cm

0.1

0.15

0.20

0.25

⌘⌘KK⇡⇡

�

~d =~PL

2⇡= [110]

m⇡ = 391MeV

L/as = 24

correct spectrum

Finite volume spectra

mπ=236 MeV mπ=391 MeV

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24 32 400.08

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24 32 400.08

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24 32 400.08

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24 32 400.08

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24 32 40

Finite volume spectra

600

700

800

900

1000

1100

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

500

600

700

800

900

1000

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

mπ=391 MeVmπ=236 MeV

det[F�1(EL, L) +M(EL)] = 0 Spectrum satisfies:

Use a various parametrizations

One channel, ignoring partial wave mixing:cot �0(Ecm) + cot�(P,L) = 0

e.g. [unitarity]M�1 = K�1 + I, Im(I) = �⇢

K =g2

s0 � s+ c

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24 32 400.08

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24 32 400.08

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24 32 40

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Scattering amplitude vs mπ

HadSpec Collaboration

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Scattering amplitude vs mπ

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Scattering amplitude vs mπ

[bound state]

HadSpec Collaboration

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Scattering amplitude vs mπ

HadSpec Collaboration

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Scattering amplitude vs mπ

[no narrow resonance]

HadSpec Collaboration

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0

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150 200 250 300 350 400

The σ/f0(500) vs mπs0 = (E� � i

2��)2, g2�⇡⇡ = lim

s!s0(s0 � s) t(s)

�Im

ps 0

=1 2��/M

eV

-300

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0 300 500 700 900

0

200

400

600

800

150 200 250 300 350 400

The σ/f0(500) vs mπs0 = (E� � i

2��)2, g2�⇡⇡ = lim

s!s0(s0 � s) t(s)

�Im

ps 0

=1 2��/M

eV

The σ/f0(500) vs mπs0 = (E� � i

2��)2, g2�⇡⇡ = lim

s!s0(s0 � s) t(s)

0

200

400

600

800

150 200 250 300 350 400

-300

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0 300 500 700 900

�Im

ps 0

=1 2��/M

eV

-300

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-100

0 300 500 700 900

The σ/f0(500) vs mπs0 = (E� � i

2��)2, g2�⇡⇡ = lim

s!s0(s0 � s) t(s)

400 500 600 700 800 900 1000 1100 1200Re s

σ

1/2 (MeV)

-500

-400

-300

-200

-100

Im s σ

1/2 (M

eV)

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

Historical perspectiveJ. R. Peláez (2015)Review of Particle Physics (RPP)

�Im

ps 0

=1 2��/M

eV

The σ/f0(500) vs mπs0 = (E� � i

2��)2, g2�⇡⇡ = lim

s!s0(s0 � s) t(s)

-300

-200

-100

0 300 500 700 900

UχPT - Nebreda & Peláez (2015)

mπ~350 MeV

The σ/f0(500) vs mπs0 = (E� � i

2��)2, g2�⇡⇡ = lim

s!s0(s0 � s) t(s)

0

200

400

600

800

150 200 250 300 350 400

UχPT - Nebreda & Peláez (2015)

!!-KK / f0(980)

dispersive analysis

chiral extrapolation

Elastic form factors of composite particles

Outlook

!!-KK / f0(980)

dispersive analysis

chiral extrapolation

Elastic form factors of composite particles

Outlook

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!!-KK / f0(980)

dispersive analysis

chiral extrapolation, more quark masses(?)

Elastic form factors of composite particles

Outlook� 1/�

E?⇡⇡/MeVEcm/MeV

m⇡ = 140 MeV

Bolton, RB & Wilson Phys.Lett. B757 (2016) 50-56.

!!-KK / f0(980)

dispersive analysis

chiral extrapolation, more quark masses(?)

Elastic form factors of composite particles

Outlook

2.0 2.1 2.32.2 2.4 2.5

2.0 2.1 2.32.2 2.4 2.5

2.0

0

050100

4.0

6.0

RB, Hansen - Phys.Rev. D94 (2016) no.1, 013008.RB, Hansen - Phys.Rev. D92 (2015) no.7, 074509.RB, Hansen, Walker-Loud - Phys.Rev. D91 (2015) no.3, 034501. Bernard, D. Hoja, U. G. Meissner, and A. Rusetsky (2012)

formalism understood:

RB, Dudek, Edwards, Thomas, Shultz, Wilson - Phys.Rev. D93 (2016) 114508.RB, Dudek, Edwards, Thomas, Shultz, Wilson - Phys.Rev.Lett. 115 (2015) 242001

first implementation: πγ*-to-ππ/πγ*-to-ρ

Take-home message

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Dudek EdwardsWilson

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150 200 250 300 350 400 arXiv:1607.05900 [hep-ph]HadSpec Collaboration

HadSpec talksResonance in coupled channels- David Wilson, Monday 10:30

Searches for charmed tetraquarks- Gavin Cheung, Monday 13:55

Radiative transitions in charmonium - Cian O'Hara, Monday 17:25

Optimised operators and distillation - Antoni Woss, Tuesday 14:40

a0 resonance in πη, KK - Jozef Dudek, Tuesday 15:50

Charmed meson spectroscopy - David Tims, Thursday 14:20

Dπ, Dη and DsK scattering - Graham Moir, Thursday 15:00

DK scattering - Christopher Thomas, Thursday 15:20

Charmed-bottom mesons - Nilmani Mathur, Friday 15:40

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Scattering amplitude vs mπ

[Levinson’s theorem]

HadSpec Collaboration