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Isoscalar ππ scattering and the σ/f0(500) resonance
Raúl Briceño [email protected]
[ 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
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
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
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 400.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
24 32 40
-1
-0.5
0
0.5
1
-0.06 -0.03 0 0.03 0.06 0.09 0.12
Scattering amplitude vs mπ
HadSpec Collaboration
-1
-0.5
0
0.5
1
-0.06 -0.03 0 0.03 0.06 0.09 0.12
Scattering amplitude vs mπ
[scattering lengths]-1
HadSpec Collaboration
-1
-0.5
0
0.5
1
-0.06 -0.03 0 0.03 0.06 0.09 0.12
Scattering amplitude vs mπ
[bound state]
HadSpec Collaboration
0
30
60
90
120
150
180
0.01 0.03 0.05 0.07 0.09 0.11 0.13
Scattering amplitude vs mπ
HadSpec Collaboration
0
30
60
90
120
150
180
0.01 0.03 0.05 0.07 0.09 0.11 0.13
Scattering amplitude vs mπ
[no narrow resonance]
HadSpec Collaboration
-300
-200
-100
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
-300
-200
-100
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
-200
-100
0 300 500 700 900
�Im
ps 0
=1 2��/M
eV
-300
-200
-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
-300
-200
-100
0 300 500 700 900
!!-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
-1
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0
0.5
1
-0.06 -0.03 0 0.03 0.06 0.09 0.12
Dudek EdwardsWilson
0
30
60
90
120
150
180
0.01 0.03 0.05 0.07 0.09 0.11 0.13
-300
-200
-100
0 300 500 700 900
0
200
400
600
800
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
0
30
60
90
120
150
180
0.01 0.03 0.05 0.07 0.09 0.11 0.13
Scattering amplitude vs mπ
[Levinson’s theorem]
HadSpec Collaboration