Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball...

64
Non-perturbative methods and Chiral Perturbation Theory applied to Meson–Meson and Nucleon–Nucleon interactions Seminar for JLab & Indiana U. for Amplitude Analysis Miguel Albaladejo IFIC, U. Valencia January 18, 2013

Transcript of Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball...

Page 1: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Non-perturbative methodsand

Chiral Perturbation Theoryapplied to

Meson–Meson and Nucleon–Nucleoninteractions

Seminar for JLab & Indiana U. for Amplitude Analysis

Miguel Albaladejo

IFIC, U. Valencia

January 18, 2013

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Short biographical presentation

Name Miguel Albaladejo SerranoBorn August 6, 1981, Murcia (Spain)

Undergrad. U. of MurciaGrad. U. of Valencia

Master Th. Meson-Meson scattering up to√

s = 2 GeV and its spectroscopyDate June 15, 2007Advisors E. Oset (Valencia), J. A. Oller (Murcia)

Ph.D. U. of MurciaPh.D. Th. Non-perturbative methods and Chiral Perturbation Theory

applied to Meson-Meson and Nucleon-Nucleon interactionsDate December 4, 2012Advisor J. A. Oller (Murcia)

Postdoc IFIC, Valencia U.Date January 1, 2013 - July 31, 2013

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Main publications

I authors: M. Albaladejo and J.A. Ollertitle: Identification of a Scalar Glueballjournal: Phys. Rev. Lett. 101, 252002 (2008)

I authors: M. Albaladejo, J.A. Oller and L. Rocatitle: Dynamical generation of pseudoscalar resonancesjournal: Phys. Rev. D 82 094019 (2010)

I authors: M. Albaladejo and J.A. Ollertitle: Nucleon-Nucleon Interactions from Dispersion Relations: Elastic Partial Wavesjournal: Phys. Rev. C 84, 054009 (2011)

I authors: M. Albaladejo and J.A. Ollertitle: Size of the σ meson and its naturejournal: Phys. Rev. D 86, 034003 (2012)

I authors: M. Albaladejo, J.A. Oller, E. Oset, G. Rios and L. Rocatitle: Finite volume treatment of ππ scattering and limits to phase shifts extractionfrom lattice QCDjournal: JHEP 1208, 071 (2012)

I authors: M. Albaladejo and J.A. Ollertitle: Nucleon-Nucleon Interactions from Dispersion Relations: Coupled Partial Wavesjournal: Phys. Rev. C 86, 034005 (2012)

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Outline of the seminar

Part I: The Scalar Sector and the Scalar Glueball

Part II: Scalar-Pseudoscalar scattering andheavy pseudoscalar resonances

Part III: On the size and the nature of the σ meson

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Part IThe Scalar sector and the scalar glueball

[M. Albaladejo and J. A. Oller, Phys. Rev. Lett. 101, 252002 (2008)]

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Introduction

Objective and tools:Problem: Scalar mesons identification. How many? Where?σ, f0(980), f0(1370), f0(1500), f0(1710)...

Very broad resonances, strongly coupled channels open up in the nearby ofresonances, which have very different natures: dynamically generated, qq,glueballs. . .Our objective: study of strongly interacting channels with quantum numbersI = 0, I = 1/2 JPC = 0++ for

√s 6 2 GeV.

We use Chiral Lagrangians, implementing Unitarity through N/D (UChPT).Coupled channel dynamics. Channels to be considered:

I = 0: ππ, KK , ηη, σσ, ηη′, ρρ, ωω, η′η′, ωφ, φφ, K∗K∗, a1(1260)π, π?πI = 1/2, I = 3/2: Kπ, Kη and Kη′[Jamin,Oller,Pich,NPB,587,331(’00)][NPB,622,279(’02)]

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Lagrangians and AmplitudesU(3) O(p2) Lagrangian

Gasser and Leutwyler

L = L2 + LS8 + LS1

N

Coupled channel partial wave: N/D[Oller,Oset,PR,D60,074023(’99)]

T = (I + N(s)g(s))−1 N(s)

T N N T= +

Large Nc limit implies η′ becomes the ninth Goldstone boson: SU(3)→ U(3).

L2 =f 2

4〈DµU†DµU〉+

f 2

4〈χ†U + χU†〉 −

12

M21η

21

DµU = ∂µU − irµU + iUlµ = ∂µU − ig [vµ,U]

In our case, rµ = lµ = gvµ: vectorial resonances nonet. Yang-Mills fields,minimal coupling, local gauge symmetry.Mixing: η1, η8 → η, η′. Mixing angle is θ ≈ −20.

N(s): Nij amplitudes matrix, obtained from Lagrangian, i , j = 1, . . . 13g(s): Unitarization loops (diagonal) matrix (including all intermediate states).

gi (s) = gi (s0)− s − s08π2

∫ ∞sth,i

ds ′ pi (s ′)/√

s ′(s ′ − s0)(s ′ − s + iε)

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Lagrangians (II)

Vector field vµ is given by:

vµ =

ρ0√2

+ ω8√6

ρ+ K∗+

ρ− − ρ0√2

+ ω8√6

K∗0

K∗− K∗0 − 2ω8√6

+1√3ω1

We assume ideal mixing between ω8, ω1 to generate physical φ, ω.Derivative piece of L2 has interactions of: ΦΦ, V ΦΦ y VV ΦΦ:

LΦΦ2 = f 2

4 〈∂µU∂µU†〉 LVV ΦΦ2 = g2

⟨Φ2vµvµ − vµΦvµΦ

⟩LV ΦΦ2 = − igf 2

4 〈∂µU[vµ,U†] + [vµ,U]∂µU†〉g is determined through decay width ρ→ ππ, from LV ΦΦ

2 , being g = 4.23

We introduce explicit resonances (JPC = 0++) from RChPT[Ecker et al.,NPB,321,311(’88)][PLB,223,425(’89)].

LS8 = cd 〈S8uµuµ〉+ cm 〈S8χ+〉 , LS1 = cd S1 〈uµuµ〉+ cmS1 〈χ+〉

χ+ = u†χu† + uχ†u, U(x) = u(x)2,uµ = iu†DµUu† = u†µ

S(i)1 singlet and S(i)

8 octet scalar resonance, with M, cd and cm fitted to data.

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σσ channel amplitudes. Pion rescatteringWe want to obtain σσ amplitudes starting from our lagrangian. σ is S-wave ππinteraction, |σ〉 = |ππ〉0, [Oller,Oset,NPA,620,438(’97)]

Ti

=⇒Ti

D−1

D−1

=⇒

1s−sσ

1s−sσ

Ti

gσPion rescattering, given by factor D−1(s) = (1 + t2G(s))−1, with:

t2 =s −m2

π/2f 2π

(4π)2G(s) = α+ logm2π

µ2− σ(s) log

σ(s)− 1σ(s) + 1

To isolate transition amplitude Ni→σσ :

limsi→sσ

Ti→(ππ)0(ππ)0

DII (s1)DII (s2)=

Ni→σσg2σππ

(s1 − sσ)(s2 − sσ)=⇒ Ni =

g2σππ

t2(sσ)2Ti→(ππ)0(ππ)0

Now, close to the σ pole:

t2(s1)

DII (s1)=

t2(s1)

1 + t2(s1)GII (s1)' −

g2σππ

s1 − sσ+ regular terms

Further, gσππ can be calculated through:

1g2σππ

= −d

ds1

(t2(s1)−1 + GII (s1)

)=

1t22(s1)f 2

−dGII (s1)

ds1

∣∣∣sσ

=⇒g2σ

t2(sσ)2' f 2

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Results: free parametersM (MeV) cd (MeV) cm (MeV)

S(1)8 1290± 5 25.8± 0.5 25.8± 1.1

S(2)8 1905± 13 20.3± 1.4 −13.9± 2.0

S(1)1 894± 13 14.4± 0.3 46.6± 1.1

First octet (M(1)8 , c(1)

d , c(1)m ) fixed to:

[Jamin,Oller,Pich,NPB,587,331(’00)][NPB,622,279(’02)].

Second octet has only fixed its mass, M(2)8 , but not its couplings.

Singlet mass M(2)1 is more difficult to fix, but for lower values, we can obtain

the same physics with higher couplings.

Since SU(3) breaking is milder in the vector sector, we take just onesubstraction constant for the whole set of vector channels,

aρρ = aωω = aK?K? = aωφ = aφφ

Together, we have aππ, aKK , aηη, aηη′ , aη′η′ and aσσ.

Total: just 12 free parameters, for about 370 data points.

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Results: Comparison with experimental data

0

90

180

270

360

450

500 750 1000 1250 1500 1750

δ0 0

(deg

) 0102030

275 350 425

00.5

11.5

500 750 1000 1250 1500 1750

η0 0

√s (MeV)

180

270

3601000 1250 1500 1750

δ1,

2(d

eg)

Etkin et al.Cohen et al.

0

0.5

1

1000 1250 1500 1750

| S1,

2|

√s (MeV)

0

0.1

0.2

1000 1250 1500 1750 2000

| S1,

3|2

√s (MeV)

0

0.04

0.08

0.12

1500 1600 1700 1800 1900

∣ ∣ S 1,5∣ ∣2

√s (MeV)

0306090

120150180210240

750 1000 1250 1500 1750

φ(d

eg)

√s (MeV)

50

100

150

200

250

750 1000 1250 1500 1750

A

√s (MeV)

ππ → ππ: Although reaching energies of 2GeV, description of low energy data is stillrather goodππ → KK : Near threshold, Cohen data arefavored.ππ → ηη,ηη′: In good agreement for a lowweight on χ2. In addition, the data areunnormalized.I = 1/2 K−π+ → K−π+ amplitude andphase from LASS.

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Spectroscopy. Pole content: Summary

I = 0√

s (MeV) Resonance M (PDG) Γ (PDG)456± 6− i 241± 7 σ ≡ f0(500)983± 4− i 25± 4 f0(980) 980± 10 40− 1001466± 15− i 158± 12 ← f L

0 f0(1370) 1200− 1500 200− 5001602± 15− i 44± 15 ← f R

0 f0(1500) 1505± 6 109± 71690± 20− i 110± 20 f0(1710) 1724± 7 137± 81810± 15− i 190± 20 f0(1790) 1790+40

−30 270+30−60

I = 1/2√

s (MeV) Resonance M (PDG) Γ (PDG)708± 6− i 313± 10 κ ≡ K∗0 (800) − −1435± 6− i 142± 8 K∗0 (1430) 1414± 6 290± 211750± 20− i 150± 20 K∗0 (1950) − −

About the σ meson, we will discuss laterRemarkable agreement in f0(980) width with Madrid Group[García-Martín et al.,PRL,107,072001(’11)].We discuss now in detail each resonance/pole in detail

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Spectroscopy. Pole content: f0(1370)

f0(1370)

f L0 :√

s = 1466± 15− i 158± 12 MeV

Shift in mass peak:Mf0(1370) ' 1370 MeV.Strong coupling to σσ, ππΓ(f0(1370)→4π)Γ(f0(1370)→ππ)

= 0.30± 0.12, in goodagreement with the interval0.10− 0.25 of [Bugg,EPJC,52,55(’07)]

M (MeV) Γ (MeV)PDG 1200− 1500 200− 500Our 1370± 15 316± 24

1000 1100 1200 1300 1400 15000

50

100

150

200

250

Re√s (MeV)

Im√s (MeV)

0

500

1000

1500

2000

2500

3000

1000 1100 1200 1300 1400 1500

|Tσσ→σσ|2

Re√s (MeV)

√s ' 1.37 GeV

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Spectroscopy. Pole content: f0(1500)

f0(1500)

f R0 :√

s = 1602± 15− i 44± 15 MeVThe mass peak is at 1.5 GeV, due toηη′ threshold. Effect similar toa0(980) with the KK threshold[Oller,Oset,PR,D60,074023(’99)].The width is Γ ≈ 1.2×88 ' 105 MeV,because a BW at (1.6− i0.04) GeV iscut by ηη′ thresold.Complicated region. Importantcoupled channel dynamics. Threeinterfering effects give raise tof0(1500):

1 f R0 : Pole at (1.6− i0.04) GeV

2 f0(1370): Pole at (1.47− i0.16) GeV3 Nearby thresholds: ηη′, ωω

M (MeV) Γ (MeV)PDG 1505± 6 109± 7Our 1502 105± 36

1300 1400 1500 1600 17000

50

100

150

200

250

fR0

fL0

Re√s (MeV)

Im√s (MeV)

0

1000

2000

3000

4000

5000

6000

1300 1400 1500 1600 1700

|Tηη′ →

ηη′ |2

Re√s (MeV )

ηη′ threshold

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Spectroscopy. Pole content: f0(1710)

f0(1710)

√s = 1690± 20− i 110± 20 MeV

Shifted in mass, bump at√

s ≈ 1.7GeV.The width “measured” on the real axisis Γeff ' 160 MeV.The effective width can depend on theconcrete process.

M (MeV) Γ (MeV)PDG 1724± 7 137± 8Our 1700± 20 160± 40BR Our value PDG

Γ(KK)Γ

0.36± 0.12 0.38+0.09−0.19

Γ(ηη)Γ

0.22± 0.12 0.18+0.03−0.13

Γ(ππ)

Γ(KK)0.32± 0.14 0.41+0.11

−0.17Γ(ηη)

Γ(KK)0.64± 0.38 0.48± 0.15

1500 1550 1600 1650 1700 1750 18000

50

100

150

200

f0(1710)

Re√s (MeV)

Im√s (MeV)

0

6000

12000

18000

24000

1500 1600 1700 1800

|Tη′ η

′ →η′ η

′ |2

Re√s (MeV)

Γ ' 160 MeV

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Spectroscopy. Pole content: f0(1790)

f0(1790)

√s = 1810± 15− i 190± 20 MeV

Weak signal on the real axisIt couples weakly to KK , a majordifference with respect to f0(1710), asalso observed by BESII.It is the partner of the pole at1.75− i0.15 GeV in I = 1/2.These poles originate from the higherbare octet, S(2)

8 , with M(2)8 = 1905,

c(2)d = 20.3, c(2)

m = −13.9 MeV.

M (MeV) Γ (MeV)BESII 1790+40

−30 270+30−60

Our 1810± 15 380± 40

1700 1750 1800 1850 19000

100

200

300

400

Re√s (MeV)

Im√s (MeV)

0

400

800

1200

1600

2000

1700 1750 1800 1850 1900

|Tηη′ →

ηη′ |2

Re√s (MeV)

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WA102 and CBC data

0

2

4

6

8

500 1000 1500

Eve

nts/

0.02

GeV

√s (MeV)

×104

WA102 pp→ ppπ+π−

0

1

2

3

1000 1250 1500 1750

Eve

nts/

0.04

GeV

√s (MeV)

×103

WA102 pp→ ppK+K−

0

1

2

3

4

1250 1500 1750

Eve

nts/

0.08

GeV

√s (MeV)

×102

WA102 pp→ ppηη

0

3

6

9

12

1250 1500 1750

Eve

nts/

9.2

MeV

√s (MeV)

×103

CBC pp→ π0ηη

2

4

6

8

10

12

1500 1550 1600 1650 1700 1750E

vent

s/12

MeV

√s (MeV)

×101

CBC pp→ π0ηη ′

Similarly as done by WA102 Collaboration, we put a coherent sum of Breit–Wigners:Ai (√

s) = NRi (√

s) +∑

jaj eiθj gj;i

M2j −s−iMj Γj

, j =

σ, f0(980), f0(1370), f R

0√

s 6 mηη′σ, f0(980), f0(1710), f0(1790)

√s > mηη′

Note that Mj , Γj and gj;i are fixed from the previous study on scattering (the poles onprevious slide), without changing them.A constant is included to ensure continuity.

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Couplings of f R0 and f0(1710): the lightest scalar glueball

[Chanowitz,PRL,95,172001(’05)]: Chiral suppression mechanism of the coupling of theglueball to qq. Coupling ∝ mq. This implies |gss | |gnn|The coupling pattern of f0(1710) suggests an enhancement in ss production.

Coupling (GeV) f R0 f0(1710)

gπ+π− 1.31± 0.22 1.24± 0.16gK0K0 2.06± 0.17 2.00± 0.30gηη 3.78± 0.26 3.30± 0.80gηη′ 4.99± 0.24 5.10± 0.80gη′η′ 8.30± 0.60 11.70± 1.60

Coupling (GeV) f R0 f0(1710)

gss 11.5± 0.5 13.0± 1.0gns −0.2 2.1gnn −1.4 1.2

gss/6 1.9± 0.1 2.1± 0.2

We can obtain gss , gns and gnn from gηη, gηη′ and gη′η′ (mixing anglesinβ = −1/3):

3gη′η′ = 2gss + gnn + 2√2gns

3gηη′ = −√2gss +

√2gnn + gns

3gηη = gss + 2gnn − 2√2gns

√3η = −ηs +

√2ηn√

3η′ =√2ηs + ηn

ηs = ss ηn = (uu + dd)/√2

Together with the OZI rule requires |gss | |gns |.This is precisely what we obtain from the previous couplings and equations.A similar reasoning on KK leads to expect gK0K0 = gss/6.

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Lattice support to the Chiral Supression Mechanism

Couplings of the lightest pseudoscalar glueball from quenched latticecalculations [Sexton et al.,PRL,75,4563(’95)]:

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8

λ mρ

(mps

)2

0++ glueballChiral suppression

This result is compatible with the chiral suppression mechanism[Chanowitz,PRL,95,172001(’05)]

Mass [Chen et al.,PRD,73,014516(’06)]:

Mgb0++ = 1.66± 0.05 GeV

Mf0(1710) = 1.69± 0.02 GeV

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Summary

We have performed the first coupled channel study of the meson-mesonS-waves for I = 0 and I = 1/2 up to 2 GeV with 13 coupled channels.Interaction kernels are determined from Chiral Lagrangians and implemented inN/D–type equations, avoiding ad-hoc parameterizationsWe have fewer free parameters, compared with previous works in the literature,for a vast quantity of data.All scalar resonances below 2 GeV are generated:

I = 0: σ, f0(980), f0(1370), f0(1500), f0(1710) and f0(1790)I = 1/2: κ, K∗0 (1430) and K∗0 (1950)

The structure of the couplings to pairs of pseudoscalars implies that:f0(1370), K∗0 (1430) (and a0(1450)) remain as pure octet.f R0 and f0(1710), which are the same pole reflected on different sheets, have adecay pattern showing an enhanced ss production, in agreement with the chiralsuppression mechanism of glueball decay, soIt should be considered as the lightest unmixed 0++ glueball.

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Part IIDynamical generation of pseudoscalar resonances

[M. Albaladejo, J. A. Oller and L. Roca, Phys. Rev. D 82, 094019 (2010)]

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Introduction – MotivationExcited Pseudoscalars above 1 GeV

I = 1 π(1300), π(1800)

I = 1/2 K(1460), K(1630)

I = 0

η(1295), η(1405), η(1475)︸ ︷︷ ︸η(1440)?

[Masoni et al.,JPG,32(’06)][Klempt,Zaitsev,PR,454(’07)][Wu et al.,PRL,108(’11)]

I = 0 X(1835)?Exotic quantum numbers? Glueballs?[Chanowitz,PRL,46,981(’81)][Ishikawa,PRL,46,978(’81)][Close,PRD,55,5749(’97)]

π+

K+K 0

π−

K− K 0

π0

η1η8

General Idea

K

K

K

f0(980)

K

K(1460)

Scalars dynamically generatedin PS–PS interactions.

Is it possible to generatedynamically heavier PS withS–PS interactions?

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

FormalismOur approach

S1(p1) S2(p2)

P1(k1) P2(k2)

g1 g2

T (sin)

q

p1 − q p2 − q

S–PS interaction kernels are given by thetriangle diagram. Internal lines are K[Alarcon et al.,PR,D80;PR,D82(’10)]

Basic point: this diagram is enhancedbecause MSi ' 2mK (f0(980), a0(980))

Kernel

LK = 2ig1 g2

∫d4q

(2π)4T ((P − q)2)

(q2 − m2K + iε)((p1 − q)2 − m2

K + iε)((p2 − q)2 − m2K + iε)

Dependence of the vertices is known:sin = (P − q)2 ⇔ Inserting a dispersion relationg1, g2 ⇔ couplings of S1,2 to KK . Residues. No (new) free parameter.

Dispersion Relation

T (sin) = T (sA) +∑

i

sin − sA

sin − si

Resi

si − sA+

sin − sA

π

∫ ∞sth

ds′ImT (s′)

(s′ − sin)(s′ − sA)

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Loop functions from the DR and S-wave projection

The insertion of the Dispersion Relation induces the appearence of three and fourpoint one loop functions, that we project onto S partial wave

p1 p2

p4 p3

` + p1

` ` + p1 + p2

p1 p2

p4 p3

` + p1

`− p3 − p4

`− p4

`

C3 =i2

∫ +1

−1d cos θ

∫d4`

(2π)41

((`+ p1)2 −m21)(`2 −m2

2)((`+ p1 + p2)2 −m23)

C4 =i2

∫ +1

−1d cos θ

∫d4`

(2π)41

((`+ p1)2 −m21)(`2 −m2

2)((`− p3 − p4)2 −m23)((`− p4)2 −m2

4)

These loop functions are worked out analytically and numerically, and checked,when possible, against OneLOop [van Hameren, Papadopoulos, Pittau,JHEP0909,106(’09)]

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Coupling the channels

IG Channels0+ (η) a0π, f0η, f0η′1/2 (K) f0K , a0K1− (π) f0π, a0η, a0η′1+ a0π3/2 a0K

T =TL

1 + TLG

As an example, the I = 1/2 amplitudes areconstructed as:

TL(f0K → f0K) =g2

f02

LK [3T I=1KK→KK + T I=0

KK→KK ]

TL(f0K → a0K) =

√3gf0ga02

LK [T I=1KK→KK − T I=0

KK→KK ]

TL(a0K → a0K) =g2

a02

LK [3T I=0KK→KK + T I=1

KK→KK ] .

Page 26: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ResultsI = 1/2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1.2 1.4 1.6 1.8

|T(f

0K→f 0K

)|2/1

05

√s (GeV)

K(1460)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1.2 1.4 1.6 1.8

|T(a

0K→a0K

)|2/1

05

√s (GeV)

K(1460)

f0(980)K and a0(980)K channelsClear resonant peak M ' 1460 MeVΓ ' 100 MeV (Exp. Γ ' 250 MeV)Generated in three-body interactionsalso [Martinez-Torres et al.,PR,D84(’11)]

asub = −0.5 (Natural sized)|gf0K/ga0K | ' ( 20.2

5.7 )1/4 ' 1.4K?(892)π not included (but couplesto K(1460))

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ResultsI = 1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 1.2 1.4 1.6 1.8 2

|T(a

0η′→a0η′ )|2/105

√s (GeV)

π(1800)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1 1.2 1.4 1.6 1.8

|T(a

0η→a0η)|2/1

05

√s (GeV)

π(1300)

f0(980)π, a0(980)η and a0(980)η′

a0η′ almost elastic (higher threshold)aa0η′ = −1.3⇒ π(1800)

Γ ' 200 MeV (Exp. Γ = 208± 12 MeV)

af0π,a0η = −2 ⇒ π(1300)

M ' 1.2 GeV, Γ ' 200 MeV.Generated in three-body interactionsalso [Martinez-Torres et al.,PR,C83(’11)]

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ResultsI = 0

0

2

4

6

8

10

12

14

16

1.2 1.4 1.6 1.8 2

|T(f

0η′→f 0η′ )|2/1

05

√s (GeV)

X(1835)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1.2 1.4 1.6 1.8

|T(f

0η→f 0η)|2/105

√s (GeV)

η(1475)

a0(980)π, f0(980)η and f0(980)η′

a0η′ almost elastic (higher threshold)aa0η′ = −1.2⇒ X(1835)

Γ ' 70 MeV (Exp.Γ = 67.7± 20.3± 7.7 MeV)

aa0π,f0η ' −1 ⇒ η(1475) ?M ' 1.45 GeVΓ ' 150 MeV (Exp. 85± 9 MeV)K?(892)K not included (but couplesto η(1475))

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ResultsI = 0

0

2

4

6

8

10

12

14

16

1.2 1.4 1.6 1.8 2

|T(f

0η′→f 0η′ )|2/1

05

√s (GeV)

X(1835)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1.2 1.4 1.6 1.8

|T(f

0η→f 0η)|2/105

√s (GeV)

η(1475)

Regarding η(1475):Resonant peak at M ' 1.45 GeVCouples mostly to a0π, as η(1475) ⇒ identify it as η(1475)

Low mass and absence of η(1405) ⇒ possible interpretation as single η(1440)[Wu et al.,PRL,108(’11)]

We do not find any signal for η(1295)

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ResultsI = 3/2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

1 1.2 1.4 1.6 1.8 2

∣ ∣ T(a± 0K

±→

a± 0K

±)∣ ∣2 /1

05

√s (GeV)

Single channel: a0(980)±K± (pure I = 3/2).Exotica1 = −0.5 (as in I = 1/2)Resonant behavior, in accordance with aprediction of Longacre [Longacre,PRD,42,874(’90)]

This channel was not isolated in I = 1/2(K(1460)) experiments[Daum et al.,NPB,187,1(’81)][Brandenburg et al.,PRL,36,1239(’76)]

The other exotic channel IG = 1+ shows noresonant behavior

Page 31: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Summary

We have studied the S-PS interactions in the region 1− 2 GeVScalars: f0(980), a0(980) dynamically generated in PS-PS interactionsWe have studied the π,K ,η-alike channels, as well as the exotic channelsThese interactions dynamically generate some resonances experimentally seenWhence, a large contribution to these physical signals has a dynamical origin

Resonance IG Width (MeV) PropertiesK(1460) I = 1

2 Γ & 100 |gf0K/ga0K | ' 1.4π(1800) IG = 1− Γ ' 200 a0η′ elasticπ(1300) IG = 1− Γ & 200 a0π, f0η coupled channelsX(1835) IG = 0+ Γ ' 70 f0η′ elasticη(1475) IG = 0+ Γ ' 150 f0η elasticExotic I = 3

2 Γ ' 200 a0K threshold

Page 32: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Part IIIOn the size and the nature of the σ meson

[M. Albaladejo and J. A. Oller, Phys. Rev. D 86, 034003 (2012)]

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Introduction

We now make a specific study of the σ meson nature and sizeElastic, low energy ππ interactions are considered (SU(2) Chiral Lagrangians)O(p4) ππ amplitude (T4) is calculated. It involves LECs (li) as well as loopfunctions:

T (s) =V (s)

1 + V (s)G(s)

' V2(s) + V4(s)− V 22 (s)G(s) + · · ·

≡ T2(s) + T4(s) + · · ·V2(s) = T2(s)

V4(s) = T4(s) + T 22 (s)G(s)

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Comparison with other determinations

Γσ/2

(MeV

)

Mσ (MeV)

ABCDEFGH

220

240

260

280

300

420 440 460 480 500

Ref. Mσ (MeV) Γ/2 (MeV) a00 b00M2

π

A 441+16−8 272+9

−13 - -B 484± 17 255± 10 0.233± 0.013 0.285± 0.012C 457+14

−13 279+11−7 - -

D 463± 6+31−17 254± 6+33

−34 0.218± 0.014 0.276± 0.013E 452± 12 260± 15 - -F 470± 50 285± 25 - -G 456± 12 241± 14 - -This work 440± 10 238± 10 0.219± 0.005 0.281± 0.006Average 453± 5 258± 5 0.220± 0.003 0.279± 0.003Mean 458± 14 261± 17 0.223± 0.007 0.280± 0.004New PDG 400− 550 200− 350 − −

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Dependence of sσ with mπ

400

500

600

700

800

900

1000

1100

1200

0 100 200 300 400 500 600

(MeV

)

mπ (MeV)

FuIAM

LONLO

Dyn. n = 1Dyn. n = 2Que. n = 1Que. n = 2

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

Γσ/2

(MeV

)

mπ (MeV)

IAMLO

NLO

0

2

4

6

8

10

12

0 100 200 300 400 500 600 700

s σ/m

2 π

mπ (MeV)

Re(sσ ) LO

Re(sσ ) NLO

−Im(sσ ) LO

−Im(sσ ) NLO

3.7

3.8

3.9

4

300 400 500 600 700

Re(

s σ/m

2 π)

mπ (MeV)

fπ, li depend on mπ (ChPT).Agreement with IAM and Lattice.Evolution of the σ pole:

1 Physical pion mass: pole at II RS(resonance).

2 Mσ < 2mπ , Γσ → 0 (II RS still)mπ ' 310–330 MeV.

3 Mσ . 2mπ (threshold) and jumpsto I RS (bound state).

References:[Prelovsek et al.,PRD,82,094507(’10)](Lattice)[Pelaez,Rios,PRD,82,114002(’10)] (IAM)[Fu,CPL,28,081202(’11)] (Lattice)

Page 36: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Scalar form factor and quadratic scalar radius of the σ

Now, we study the scalar form factor of the σ meson, FσFrom this form factor we can extract the quadratic scalar radiusWe need the amplitude for ππ scattering in the presence of a scalar source,H(q)

=⇒H(q)

=⇒H(q)

Page 37: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ππH → ππ amplitudeKinematics

π(p1)

π(p2)

π(p3)

π(p4)

H(q)

Mandelstam-like variables:

s = (p3 + p4)2 s ′ = (p1 + p2)2

t = (p1 − p3)2 t ′ = (p2 − p4)2

u = (p1 − p4)2 u′ = (p2 − p3)2

s + t + u + s ′ + t ′ + u′ = q2 + 8m2π

Perturbative Amplitude

Calculated from O(p4) chiral lagrangians:

li , i = 1, . . . , 4.Loops: A0, B0(p2), C0(s, s ′, q2)

Amplitude An(s, s ′, t, t ′, u, u′, q2)

Page 38: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ππH → ππ amplitudeResummed Amplitude

TS (s, s′, q2) =W (s, s′, q2)(

1 + V (s)G(s))(

1 + V (s′)G(s′))

= W2 + W4

−W2V (s)G(s)−W2V (s′)G(s′) + · · ·= Φ2 + Φ4 + · · ·

• Resummed amplitude• Kernel• Perturbative amplitude

Matching both series, we have:W2 = Φ2

W4 = Φ4 + W2V (s)G(s) + W2V (s′)G(s′)Φn is the angular projection of the amplitude An:

Φn =

∫d2ΩAn(s, s′, t, t′, u, u′, q2)

Perturbative Amplitude

Calculated from O(p4) chiral lagrangians:

li , i = 1, . . . , 4.Loops: A0, B0(p2), C0(s, s ′, q2)

Amplitude An(s, s ′, t, t ′, u, u′, q2)

Page 39: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Scalar form factor of the σ meson

H(q)

11+V (s′)G(s′)

11+V (s)G(s)

W (s, s′, q2)

=⇒ gσ gσ

Fσ(q2)

1s′−sσ

1s−sσ

H(q)

Fσ(q2) g2σ

(s − sσ)(s ′ − sσ)= lim

s,s′→sσ

W (s, s ′, q2)

(1 + V (s)G(s)) (1 + V (s ′)G(s ′))

Close to the σ pole, sσ, we have:

V (s)

1 + V (s)G(s)

s'sσ= − g2σ

s − sσ+ · · ·

Scalar Form Factor of the σ meson

Fσ(q2) =g2σ

V (sσ)2W (sσ, sσ, q2)

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Radius of the σ meson

From Fσ(q2) we can get 〈r 2〉σS :

Fσ(q2) = Fσ(0)

(1 +

q2

6 〈r2〉σS + · · ·

)⇐⇒ 〈r 2〉σS =

6Fσ(0)

∂Fσ(q2)

∂q2

∣∣∣∣q2=0

0

0.5

1

1.5

2

2.5

3

50 100 150 200 250 300 350 400 450 500 550 600

√ 〈r2〉σ s

(fm

)

mπ (MeV)

Re√〈r2〉σs

−Im√〈r2〉σs

[1] [Jamin et al.,PRD,74(’06)]

[2] [Oller,Roca,PL,B651(’07)]

[3] [Amendolia et al.,NPB,277(’86)]

[4] [Dally et al.,PRL,48(’82)]

[5] [de Forcrand,Liu,PRL,69(’92)][Hou et al..PRD,64(’01)]

Diverges:in the chiral limit as logmπ (as〈r2〉πS,V ): infinite pion cloudwhere mσ = 2mπ : zero binding energybound state

Imaginary part is small, despite large σwidthAround 0.5 fm in a wide range of mπ

Page 41: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Radius of the σ meson

From Fσ(q2) we can get 〈r 2〉σS :

Fσ(q2) = Fσ(0)

(1 +

q2

6 〈r2〉σS + · · ·

)⇐⇒ 〈r 2〉σS =

6Fσ(0)

∂Fσ(q2)

∂q2

∣∣∣∣q2=0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

50 100 150 200 250 300 350

〈r2 〉

σ s(f

m2 )

mπ (MeV)

Re 〈r2〉σs−Im 〈r2〉σs

[1] [Jamin et al.,PRD,74(’06)]

[2] [Oller,Roca,PL,B651(’07)]

[3] [Amendolia et al.,NPB,277(’86)]

[4] [Dally et al.,PRL,48(’82)]

[5] [de Forcrand,Liu,PRL,69(’92)][Hou et al..PRD,64(’01)]

At physical mπ :

〈r2〉σS = (0.19± 0.02)− i (0.06± 0.02) fm2√〈r2〉σS = (0.44± 0.03)− i (0.07± 0.03) fm

For comparison:

〈r2〉Kπs = 0.1806± 0.0049 fm2[1]

〈r2〉πs = 0.65± 0.05 fm2[2]

〈r2〉π±

V = 0.439± 0.008 fm2[3]

〈r2〉K±

V = 0.28± 0.07 fm2[4]

Scalar glueballs (expected): 0.1− 0.2 fm[5]

Page 42: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Naive view of the 4q vs. ππ molecule question

Molecule

π π

q

q q

q

4q

σq

q q

q

Intermediate

π π

q

q q

q

〈r 2〉σS is an estimation of σ’s size.Pion would overlap so much thatthey are tightly packed. σ is acompact state.Hence, a 4q nature seems moreadequate than a molecule picture.For large mπ values, mσ . 2mπ, andσ becomes a 2π molecule.

Page 43: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Naive view of the 4q vs. ππ molecule question

Molecule

π π

q

q q

q

4q

σq

q q

q

0

0.5

1

1.5

2

2.5

3

50 100 150 200 250 300 350 400 450 500 550 600

√ 〈r2〉σ s

(fm

)

mπ (MeV)

Re√〈r2〉σs

−Im√〈r2〉σs

〈r 2〉σS is an estimation of σ’s size.Pion would overlap so much thatthey are tightly packed. σ is acompact state.Hence, a 4q nature seems moreadequate than a molecule picture.For large mπ values, mσ . 2mπ, andσ becomes a 2π molecule.

Page 44: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Summary

We have performed a detailed calculation of ππ scattering at low energyThe σ meson pole (440± 10− i238± 10 MeV) is well reproduced andcompared with other valuesWe have studied the dependence of the pole position with the pion mass,finding remarkable agreement with lattice and IAM resultsThe σ meson quadratic scalar radius is calculated, finding an small value for it(4q vs. molecule question)The Feynman–Hellmann theorem has been checked for the σ meson

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Other works I have done. . .

NN interactionsfrom Dispersion Relations

[Albaladejo, Oller, PR,C84(’11)][Albaladejo, Oller, PR,C86(’12)]

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300

deg

rees

|p| (MeV)

1S0-25

-20

-15

-10

-5

0

0 50 100 150 200 250 300

deg

rees

|p| (MeV)

1P1

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

|p| (MeV)

3P0

-25

-20

-15

-10

-5

0

0 50 100 150 200 250 300

|p| (MeV)

3P1

Finite volume effects in ππ interactions[M. Albaladejo et al., JHEP 1208,071(’12)]

200

300

400

500

600

700

800

900

1.5 2 2.5 3 3.5 4 4.5 5

E(M

eV)

Lmπ

I = 0

IAMN/D

BS

1

2

3

4

0

10

20

30

40

50

60

70

80

90

100

300 400 500 600 700 800

δ(d

eg)

√s (MeV)

I = 0

L = ∞ IAMn = 2 IAML = ∞ N/Dn = 2 N/D

Now working, among other things, at heavy-meson molecules (DD,. . . ) in a FiniteVolume and possibilities of finding them in Lattice QCD

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Non-perturbative methodsand

Chiral Perturbation Theoryapplied to

Meson–Meson and Nucleon–Nucleoninteractions

Seminar for JLab & Indiana U. for Amplitude Analysis

Miguel Albaladejo

IFIC, U. Valencia

January 18, 2013

Page 47: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

References I

I R. Machleidt and D. R. Entem, Phys. Rep., 503 (2011) 1.

I E. Epelbaum, H.-W. Hammer and U.-G. Meißner, Rev. Mod. Phys. 81 (2009)1773.

I S. Weinberg, Phys. Lett. B 251 (1990) 288.

I S. Weinberg, Nucl. Phys. B 363 (1991) 3.

I S. Weinberg, Phys. Lett. B 295 (1992) 114.

I C. Ordóñez, L. Ray and U. van Kolck, Phys. Rev. Lett. 72 (1994) 1982; Phys.Rev. C 53 (1996) 2086.

I D. R. Entem and R. Machleidt, Phys. Lett. B524 (2002) 93; Phys. Rev. C66(2002) 014002; Phys. Rev. C 68 (2003) 041001.

I D. R. Entem and R. Machleidt, Phys. Rev. C68 (2003) 041001.

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I M. Albaladejo and J. A. Oller, in preparation.

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References II

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I G. P. Lepage, arXiv:nucl-th/9706029.

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References III

I G. P. Lepage, arXiv:hep-ph/0506330.

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References IV

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References V

I L. Alvarez-Ruso, J. A. Oller and J. M. Alarcon, Phys. Rev. D 80, 054011 (2009).

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Page 52: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

References VI

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References VII

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

σ width effects on g(s)With sσ complex, Ni→σσ would be complex, thus violating unitarity. So we take sireal, but varying according to a mass distribution.

Lehman propagator representation (dispersion relation):

P(s) = − 1π

∫ ∞√

sth

ds ′ =P(s ′)s − s ′ + iε ,

In a first approach:

=P(s ′) ∝ =(

1s ′ −m2

σ + i mσΓσ(s ′)

)Γσ(s ′) = Γσ

√1− 4m2

π/s ′1− 4m2

π/m2σ∫ ∞

√sth

ds ′=P(s ′) = 1

g(s) (unitarity loop) for σ channel can be written as:∫ ∞√

sth

ds1∫ ∞√

sth

ds2=P(s1)=P(s2) g4(s; s1, s2) ,

Page 55: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Spectroscopy, poles and Riemann sheets

When parameters are fitted to data, we extrapolate our amplitudes to thes-complex plane, finding poles, and we identify them with resonances through√s0 ≈ M − iΓ/2.Analytical extrapolations are needed in the Tij to the different Riemann sheets,which appear because of the cuts in G(s) at opening thresholds:

p(s) =

√s − (ma + mb)2

√s − (ma −mb)2

2√

s

Using continuity, for√

s real, > ma + mb, we have:

G II (s + iε) = G(s − iε) = G(s + iε)− 2i=G(s + iε) = G(s + iε) +i4π

p(s)√s

Page 56: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Spectroscopy. Couplings of f0(1370)

f0(1370) (I = 0)Coupling bare (MeV) final (MeV)gπ+π− 3.9 3.59± 0.18gK0K0 2.3 2.23± 0.18gηη 1.4 1.70± 0.30gηη′ 3.7 4.00± 0.30gη′η′ 3.8 3.70± 0.40

K∗0 (1430) (I = 1/2)

Coupling bare (MeV) final (MeV)gKπ 5.0 4.8gKη 0.7 0.9gKη′ 3.4 3.8

Bare coupling are very similar to the physical ones.Bare: those of S(1)

8 , with M(1)8 = 1.29 GeV, c(1)

d = c(1)m = 25.8 MeV.

The first scalar octet is a pure one, not mixed with the nearby f0(1500) norf0(1710)

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ππ scattering at NLO in SU(2)

δ0 0

(deg

)

√s (MeV)

LONLO

AverageFroggatt

S118E865

NA48/20

20

40

60

80

100

120

250 300 350 400 450 500 550 600 650 700 750 800

-505

10152025

275 300 325 350 375

δ2 0

(deg

)

√s (MeV)

LONLO

Hoogland et al.Losty et al.

-30

-25

-20

-15

-10

-5

0

300 400 500 600 700 800 900 1000

We consider I = 0, 2 S-wave. Freeparameters: li (i = 1, . . . , 4) and a(subtraction constant)

Fit √sσ (MeV)LO 465± 2− i 231± 7

NLO 440± 10− i 238± 10Fit a00 b0

0M2π

LO 0.209± 0.002 0.278± 0.005NLO 0.219± 0.005 0.281± 0.006

Fit a l1 l2 l3 l4 χ2

LO −1.36± 0.12 - - - - 1.6NLO −1.2± 0.4 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5 0.7

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

ππ scattering at NLO in SU(2)

f π(M

eV)

mπ (MeV)

NLOBeane et al.

JLQCD

85

90

95

100

105

110

115

120

0 100 200 300 400 500 600

a2 0

mπ (MeV)

NLOBeane et al.Feng et al.

Fu

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 100 200 300 400 500 600

We consider I = 0, 2 S-wave. Freeparameters: li (i = 1, . . . , 4) and a(subtraction constant)

Fit √sσ (MeV)LO 465± 2− i 231± 7

NLO 440± 10− i 238± 10Fit a00 b0

0M2π

LO 0.209± 0.002 0.278± 0.005NLO 0.219± 0.005 0.281± 0.006

Fit a l1 l2 l3 l4 χ2

LO −1.36± 0.12 - - - - 1.6NLO −1.2± 0.4 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5 0.7

Page 59: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Comparing with other determinations of the LECsRef. l1 l2 l3 l4GL −2.3± 3.7 6.0± 1.3 2.9± 2.4 4.6± 0.9CGL −0.4± 0.6 4.31± 0.11 − 4.4± 0.2ABT 0.4± 2.4 4.9± 1.0 2.5+1.9

−2.4 4.20± 0.18PP −0.3± 1.1 4.5± 0.5 − −GKMS 0.37± 1.96 4.17± 0.47 − −BCT − − − 4.4± 0.3OR − − − 4.5± 0.3DFGS − − −15± 16 4.2± 1.0NA48/2 − − 2.6± 3.2 −RBC-UKQCD − − 2.57± 0.18 3.83± 0.9PACS-CS − − 3.14± 0.23 4.04± 0.19ETM − − 3.70± 0.07± 0.26 4.67± 0.03± 0.1JLQCD/TWQCD − − 3.38± 0.40± 0.24+0.31

−0 4.09± 0.50± 0.52MILC − − 2.85± 0.81+0.37

−0.92 3.98± 0.32+0.51−0.28

This work 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5

l1

GL

CGL

ABT

PP

GKMS

This work

-6 -5 -4 -3 -2 -1 0 1 2 3

l2

GL

CGL

ABT

PP

GKMS

This work

3.5 4 4.5 5 5.5 6 6.5 7 7.5

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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Comparing with other determinations of the LECsRef. l1 l2 l3 l4GL −2.3± 3.7 6.0± 1.3 2.9± 2.4 4.6± 0.9CGL −0.4± 0.6 4.31± 0.11 − 4.4± 0.2ABT 0.4± 2.4 4.9± 1.0 2.5+1.9

−2.4 4.20± 0.18PP −0.3± 1.1 4.5± 0.5 − −GKMS 0.37± 1.96 4.17± 0.47 − −BCT − − − 4.4± 0.3OR − − − 4.5± 0.3DFGS − − −15± 16 4.2± 1.0NA48/2 − − 2.6± 3.2 −RBC-UKQCD − − 2.57± 0.18 3.83± 0.9PACS-CS − − 3.14± 0.23 4.04± 0.19ETM − − 3.70± 0.07± 0.26 4.67± 0.03± 0.1JLQCD/TWQCD − − 3.38± 0.40± 0.24+0.31

−0 4.09± 0.50± 0.52MILC − − 2.85± 0.81+0.37

−0.92 3.98± 0.32+0.51−0.28

This work 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5

l3

GL

ABT

RBC/UKQCD

MILC

PACS-CS

ETM

JLQCD/TWQCD

NA48/2

This work

-2 -1 0 1 2 3 4 5 6

l4

GLCGLABTBCTDFGSORRBC/UKQCDMILCPACS-CSETMJLQCD/TWQCDThis work

3 3.5 4 4.5 5 5.5

Page 61: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

An issue with the amplitude. . .

→ 1D1

= 1(p1+q)2−m2

π= 1

q22 −2|~q||~p| cos θ

. Diverges on integration

Upon resummation, one should have: → +

Because on-shell factorization, one really has: → 2×

W(1 + VG)2

' W − 2WVG

Indeed, the perturbative calculation of this diagram gives half this result.This means that this kind of terms should be resummed with a factor(1 + VG)−1 and not (1 + VG)−2. In that case, they do not contribute to thedouble σ-pole necessary for the form factor.Diagrammatically:

=⇒

Page 62: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

1/Γ vs. r

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700

r(f

m)

mπ (MeV)

√r2

1/Γ1/Γ

Page 63: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Feynman–Hellmann theorem for the σ meson

1

2

3

4

5

6

7

8

100 200 300 400 500 600

Re

s σ

mπ (MeV)

Re(sσ ) LO

Re(sσ ) NLO

−Re(

Fσ (0)

2B

)NLO

0

1

2

3

4

50 100 150 200 250 300

Ims σ

mπ (MeV)

Im(sσ ) LO

Im(sσ ) NLO

−Im(

Fσ (0)

2B

)NLO

sσ =dsσdm2 =

dsσdm2

π

dm2π

dm2︸︷︷︸ChPT

Feynman–Hellmann theorem

dsσdm2 = −Fσ(0)

2B

It is worth noting that:

These two quantities are obtained fromthe expansion of two differentinteracting kernelssσ : the derivative does not act onexternal legs in ππ diagrams Spares

Page 64: Non-perturbative methods and Chiral Perturbation Theory ... · Scalarsectorandglueball Pseudoscalarresonances Sizeandnatureofσ Spares Spectroscopy. Polecontent: Summary I = 0 s (MeV)

Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares

Feynman–Hellmann theorem for the σ meson

1.4

1.6

1.8

2

2.2

2.4

2.6

50 100 150 200 250 300

Re

s σ

mπ (MeV)

Re(sσ ) LO

Re(sσ ) NLO

−Re(

Fσ (0)

2B

)NLO

0

1

2

3

4

50 100 150 200 250 300

Ims σ

mπ (MeV)

Im(sσ ) LO

Im(sσ ) NLO

−Im(

Fσ (0)

2B

)NLO

sσ =dsσdm2 =

dsσdm2

π

dm2π

dm2︸︷︷︸ChPT

Feynman–Hellmann theorem

dsσdm2 = −Fσ(0)

2BIt is worth noting that:

These two quantities are obtained fromthe expansion of two differentinteracting kernelssσ : the derivative does not act onexternal legs in ππ diagrams Spares