Topicsin U(3) χPTdynamicsandrelated spectroscopy.is invariant as well as any function of X Witten...

Introduction Analytical calculation Numerical analysis SS , PP correlators. Average duality NC pole trajectories Conclusions

Topics in U(3) χPT dynamics and relatedspectroscopy.

J. A. Oller

Departamento de FısicaUniversidad de Murcia

in collaboration with Zhi-Hui GuoPhys. Rev. D84(2011)034005, others forthcoming

J. A. Oller Universidad de Murcia

Topics in U(3) χPT dynamics and related spectroscopy.


Outline

1 Introduction2 Analytical Calculation3 Numerical results4 SS , PP correlators5 Average duality6 NC pole trajectories7 Conclusions




IntroductionAlong the years we have been very interested in the scalardynamics and spectrum combining ChPT with non-perturbativetechniques from S-matrix theory:

Unitarization techniques based on the N/D method,Bethe-Salpeter equation, Omnes functions, solving theMuskhelishvili-Omnes equation in coupled channels, etc.Oset,JAO NPA620’97;NPA629’98;PRD’99

JAO, PLB’98;PLB’00;NPA’03;NPA’03;PRD’05

Oset, Pelaez, JAO PRL’98; PRD’99

Meißner,JAO NPA’01

Jamin,Pich,JAO NPB’00;NPB’02;EPJC’02;JHEP’04;PRD’06

Roca,JAO EPJA’07;PLB’07;EPJA’08; Roca,Schat,JAO PLB’08

Albaladejo,JAO PRL’08; Albaladejo,Roca,JAO PRD’10

Guo,JAO PRD’11

Thanks! to all my collaboratorsJ. A. Oller Universidad de Murcia



Now we want to study simultaneously important QCD results:

i) Large NC QCD. Pole trajectories with varying NC

qq resonances: M ∼ O(N0C ), Γ ∼ O(1/Nc)

Glueball resonances: M ∼ O(N0C ), Γ ∼ O(1/N2

c )Tetraquarks dissolve in the continuumOller, Oset PRD’99 M2

S ∝ f 2 ∝ Nc

Pelaez et al PRL’04,PRL’06,. . . ,PRD’11

ii) Second (Weinberg) spectral function sum rules for SS − SSand SS − PP correlatorsBijnens, Gamiz, Prades JHEP’01

Sanz-Cillero, Trnka PRD’10

iii) Average or semi-local dualityPelaez, Pennington, Ruiz de Elvira, Wilson PRD’11




The large NC dependence of ii) and iii) are of great interest:

In the physical world the σ or f0(600) dominates the isoscalarscalar channel at low energiesThe f0(600) disappears with growing NC (e.g. for a tetraquark, ameson-meson molecule or resonance)

iii) (and also ii)) requires of a strong scalar contribution to achievethe cancellations needed for NC large.

Which is its origin?

Interesting puzzle




Previous studies of the NC trajectory for the f0(600) pole werebased on SU(2) or SU(3) ChPT

However, in the large NC limit:

Neither SU(2) nor SU(3) ChPT have the right degrees offreedom

The η1 becomes the ninth Goldstone boson in the chiral limit

The η becomes similarly light as the π for NC → ∞Weinberg PRD’75




In the chiral limit mu = md = ms = 0 the QCD Lagrangian isinvariant under UL(3)⊗ UR(3) symmetry.

SUL(3)⊗ SUR(3) → SUV (3) is Spontaneously Broken. Goldstonebosons appear π, K , η

UV (1) ≡ UL+R Conserved Baryon Number.

UA(1) ≡ UL−R Neither Conserved nor Goldstone Boson.

Puzzle:Goldstone mode: There should be an η1 with a mass <

√3mπ

Weinberg PRD’75 but η is much heavier.




The ninth axial singlet current has an anomalous divergence Adler

PR’69 Fujikawa PRD’80

Jµ (0)5 = qγµγ5q

∂µJµ (0)5 =

g2

16π2

1

NcTrc(GµνG

µν)

Large Nc QCD Nc → ∞, g2Nc → constant’t Hooft NPB’74, Witten NPB’79

UL(NF )⊗ UR(NF ) → UL+R(NF )Entire Nonet of Goldstone bosonsresults.

Coleman, Witten PRL’80

Knecht, de Rafael PLB’98

The explicit breaking of chiral symmetry due to quark masses andthe UA(1) anomaly is treated perturbatively.




Combined power expansion in the light quark masses and 1/Nc

This formalism is set up inDi Vecchia, Veneziano NPB’80Rosenzweig, Schechter, Trahern PRD’80

Witten Ann.Phys.’80

The Leading Order in 1/Nc and the Derivative Expansion wasworked out.

Herrera-Siklody, Latorre, Pascual, Taron NPB’97

Generalization of Gasser, Leutwyler Ann.Phys.’84, NPB’85 fromSUL(3)⊗ SUR(3) to UL(3)⊗ UR(3)

Generating functional in the presence of external sourcesZ[l , r , s, p, θ]. Chiral Lagrangian to O(p4) and all orders in 1/Nc




Bilinear Quark operators (currents) and the Topological Chargeoperator coupled to external sources:

L = LQCD + qLγµℓµ(x)qL + qRγµr

µ(x)qR − qR(s(x) + ip(x))qL

− qL(s(x)− ip(x))qR − g2

16π2

θ(x)

NcTrc(GµνG

µν)

gL = I + i(β − α) , gR = I + i(β + α)

θ(x) → θ(x)− 2〈α(x)〉The extra term in the fermionic determinant due to the anomaly iscompensated.

The non-Abelian anomaly cannot be compensated. TheWess Zumino Witten term has to be added by handWess and Zumino PLB’71 , Witten NPB’83

Adler, PR’69, Bardeen, PR’69 , Adler and Bardeen, PR’69J. A. Oller Universidad de Murcia



DµU → gR(DµU)g †L

U = exp

(i

√2

fΦ

),

Φ =

1√2π0 + 1√

6η8 +

1√3η1 π+ K+

π− −1√2π0 + 1√

6η8 +

1√3η1 K 0

K− K 0 −2√6η8 +

1√3η1

This reflects the pseudo-Goldstone nature of the η1

Nc → ∞ enforces a nonet symmetry, with the π, K , η and η′ allhaving identical properties.

There is then only one decay constant f for the octet and singletof pseudo-Goldstone bosons.




In addition one has the specific gluonic content of the η1 due toUA(1) anomaly, which explicitly breaks chiral symmetry.

Both (pseudo-Goldstone boson+ gluonic content) are related bythe combination

X (x) = 〈logU(x)〉+ iθ(x) ≡ i

√2NF

fη1 + θ(x)

is invariant as well as any function of X Witten NPB’79, Leutwyler

PLB’96

θ(x) couples in the QCD Lagrangian through the topologicalcharge

− g2

16π2

Tr(GµνG

µν)

Nc.

Extra suppression factor 1/Nc .

Vertex suppression 1/N3/2c per η1




Leading Order Lagrangian in the Chiral Expansion:

L0+2 = −W0(X ) +W1(X )〈DµU†DµU〉+W2(X )〈U†χ+ χ†U〉

+ iW3(X )〈U†χ− χ†U〉+W4(X )〈U†DµU〉〈U†DµU〉+W5(X )〈U†(DµU)〉Dµθ +W6(X )DµθD

µθ

W4(0) = 0 , W1(0) = W2(0) =f 2

4 from quadratic fields

M2η1

∣∣UA(1)

= −2NF

f 2W ′′

0 (0) ∝1

Nc

Herrera-Siklody et al. NPB’97

L4 =57∑

i=0

ci (µ)Oi

The βi functions are also calculated.J. A. Oller Universidad de Murcia



These Lagrangians were employed by Borasoy et al

S-wave meson-meson scattering from unitarized U(3) chiralLagrangians PRD’03The interaction kernel is calculated only at the tree-level

Photoproduction off nucleons of η and η′ PRD’01

One-loop calculation in Infrared Regularization for:η-η′ mixing EPJA’01η′ → ηππ decay NPA’02

It is stressed that Mη′ is large.

But Mη′ also appears from vertices and there is nothing like‘Baryon number conservation’ that acts in the meson-baryonsector.

Proliferation of free parameters. Poor predictive power.




1/Nc counting for counterterms Leutwyler PLB’96

G (X ) = g(X

Nc)N

2−N(TrF )−N(θ)c

for Arbitrary Number of Flavors.

The expansion of g in powers of X/Nc has only leadingcoefficients of order 1.

f ∼√Nc → Each additional Meson Loop is suppressed by N−1

c

δ-expansion: p2 ∼ mq ∼ 1/Nc ∼ δ

Loops are suppressed by p2/f 2 ∼ O(δ2)

For low energy implications of U(3) theory Leutwyler PLB’96; Kaiser

and Leutwyler EPJC’00




Lagrangians up to NLO, O(δ), have been thoroughly studiedHerrera-Siklody et al. NPB’97; Kaiser, Leutwyler, EPJC’00

Lδ0 : B , F , M20

Lδ: 11 operators.

η − η′ mixing:NLO O(δ2) Herrera-Siklody et al. PLB’98One loop calculation, plus the pseudoscalar decay constantsKaiser, Leutwyler hep-ph/9806336

The η-η′ mixing does not follow the naive one-mixing-anglescheme.

η′ → ηππ: Tree level input at O(δ2) with ππ Final StateInteractions resummed employing Unitary χPT based on theN/D method Escribano, Masjuan, Sanz-Cillero, JHEP’11However, they take the mass and mixing angle for η, η′ from other

works. Inconsistency between input and output.J. A. Oller Universidad de Murcia



Spectral function Sum Rules for SS − SS and SS − PP correlators

S(a)s (s) = q(x)

λa

√2q(x) , P(a)(x) = q(x)iγ5

λa

√2q(x)

ΠS ,P(Q) = −i

∫d4x e iqxT 〈S ,P(x)S ,P(0)〉

Constraint: Non-trivial relation between the scalar spectrum(resonances) and the pseudoscalar one (Goldstone bosons andresonances) in the chiral limit.

Bernard, Duncan, LoSecco, Weinberg PRD’75∫ ∞

0ds[ImΠ

(0)S (s)− ImΠ

(3)P (s)

]= 0 =

∫ ∞

0ds[ImΠ

(3)S (s)− ImΠ

(0)P (s)

]

Moussallam EPJC’99, HEP’00; Leutwyler, NPB’90∫ ∞

0ImΠ

(0−3)SS (s)ds = 0 =

∫ ∞

0ImΠ

(0−3)PP (s)ds




• We perform the first complete calculation of the meson-mesonscattering within U(3) χPT at one loop level, with monomials ofO(δ2) and O(δ3).

• We include explicit exchange of tree level resonances instead oflocal counterterms: V (1−−), S (0++) and P (0−+) resonances.

• These amplitudes are then unitarized so that we can studyresonances and their properties.

Are the results stable under the inclusion of the η1?

η1 becomes in large Nc the ninth pseudo-Goldstone boson.SU(3) χPT results are ill-defined for Nc → ∞.

In large Nc there should be a light isosinglet pseudoscalar withmass ∼ mπ

Influence on the running of the pole positions with Large Nc .




Jamin,Oller,Pich, NPB’00 I=1/2 S-wave meson-meson scatteringIt was not a full one-loop calculation for the kernel.

Beisert and Borasoy, PRD’03 studied S-wave meson-meson scatteringfrom Lδ0 and Lδ

The interaction kernel is calculated at tree level.Local terms instead of resonance exchanges.

Another framework is non-relativistic effective field theory Kubis

and Schneider EPJC’09 studied cusp effects in η′ → ηππ similarly toK → 3π Colangelo,Gasser,Kubis,Rusetksy PLB’06




Relevant Chiral Lagrangian




L(0) =F 2

4〈uµuµ〉 +

F 2

4〈χ+〉+

F 2

3M2

0 (ln det u)2

where

u = ei Φ√

2F , U = u2 ,

uµ = iu†DµUu† = u†µ , χ± = u†χu† ± uχ†u

Φ =

√3π0+η8+

√2η1√

6π+ K+

π− −√3π0+η8+

√2η1√

6K 0

K− K 0 −2η8+√2η1√

6




Li ’s generically correspond to the higher order local operators.

At O(δ) one has O(Ncp4) and O(p2) operators:

L(δ) = L2〈uµuνuµuν〉 + (2L2 + L3)〈uµuµuνuν〉+ L5〈DµU

†DµU(U†χ+ χU)〉+ L8〈U†χU†χ+ χ†Uχ†U〉+ . . .

+ F 2Λ1〈uµ〉〈uµ〉+ F 2Λ2 ln(detU)〈U†χ− χ†U〉+ . . .

Λ2 is relevant for η-η′ mixing. (Not for meson-mesonscattering).

Λ1 only gives rise to η1 field renormalization. If included inthe fit its resulting value vanishes.




O(p4)

L(δ2) = (L1 − L2/2)〈uµuν〉2 + L4〈uµuµ〉〈χ+〉+ . . .

+ L18iDµX 〈DµU†χ− DµUχ†〉+ L25iX 〈U†χU†χ− χ†Uχ†U〉+ . . .

[Kaiser and Leutwyler, ’00] [Herrera-Siklody, et al., ’97]

In the current discussion, we assume the resonance saturationof ChPT counterterms and exploit resonance exchanges tocalculate the meson-meson scattering.

The monomials proportional to Λ1 and Λ2 are not generatedthrough resonance exchange. No double counting.




Vector, Scalar and Pseudoscalar resonance operators:

LV =FV

2√2〈Vµν f

µν+ 〉+ iGV

2√2〈Vµν [u

µ, uν ]〉

LS = cd〈 S8uµuµ 〉+ cm〈 S8χ+ 〉+ cdS1〈 uµuµ 〉+ cmS1〈χ+ 〉+ cd〈 S9uµ 〉〈 uµ 〉

LP = idm〈P8χ−〉+ i dmP1〈χ−〉Ecker, Gasser, Pich, de Rafael NPB’89

Elimination of the mixing terms between P resonances andpseudo-Goldstone bosons:

P8 → P8 + idmM2

P8

(χ− − 1

3〈χ− 〉I3×3

),

P1 → P1 + idmM2

P1

〈χ− 〉 ,




LP =1

2〈∇µP8∇µP8 −M2

P8P28 〉+

1

2

(∂µP1∂µP1 −M2

P1P21

)

+ idmM2

P8

〈∇µP8∇µχ− 〉+ idmM2

P1

∇µP1∇µ〈χ− 〉

− d2m

2M2P8

〈χ−χ− 〉+( d2

m

6M2P8

− d2m

2M2P1

)〈χ− 〉〈χ− 〉+ · · · ,

However, the properties of the pseudoscalar resonances are not wellsettled:

δL82

〈χ+χ+ + χ−χ−〉

No δL7 because we assume the large NC relations

dm =1√3dm , MP1 = MP8




dm = 30 MeV , MP8 = 1350 MeV

Ecker et al. NPB’89; Golterman, Peris PRD’00; Sanz-Cillero, Trnka

PRD’10

For the scalar resonances we always assume the large NC

constraint:

cd =1√3cd , cm =

1√3cm

We include two scalar nonetsThe second one with a bare mass & 2 GeV is taken from Jamin,

Pich, JAO NPB’00 since they were more sensitive to the moremassive second scalar nonet:

c ′d = c ′m =√3 c ′d =

√3 c ′m ≃ 40 MeV

MS ′8= MS ′

1≃ 2.5MeV




Perturbative calculation of the scattering amplitudes




S

P

Figure: Relevant Feynman diagrams for the pseudoscalar self-energy.

The leading order η-η′ mixing has to be solved exactly

η = cos θ η8 − sin θ η1 ,

η′ = sin θ η8 + cos θ η1 ,

sin θ = −1/

√1 +

(3M2

0 − 2∆2 +√9M4

0 − 12M20∆

2 + 36∆4)2/32∆4

∆2 = m2K −m2

π , sin θ → 0 for ∆2 → 0 or M20 → ∞.




m2η =

M20

2+m2

K −

√M4

0 − 4M20∆

2

3 + 4∆2

2,

m2η′ =

M20

2+m2

K +

√M4

0 − 4M20∆

2

3 + 4∆2

2,

Figure: The dot denotes the mixing of η8 and η1 at leading order, whichis proportional to m2

K −m2π.

θ = −16.2o+2.8o

−2.9o .




The NLO η-η′ mixing can be treated perturbatively

L =1 + δη

2∂µη∂

µη +1 + δη′

2∂µη

′∂µη′ + δk ∂µη∂µη′

−m2

η + δm2η

2η η −

m2η′ + δm2

η′

2η′η′ − δm2 η η′ .

(ηη′

)=

(cos θδ − sin θδsin θδ cos θδ

)(1 +

δη2

δk2

δk2 1 +

δη′2

)(ηη′

).

Because of δk the one-mixing angle scheme for η-η′ is notappropriate.




S

Figure: Relevant Feynman diagrams for the pseudoscalar decay constants.

We expressed all the amplitudes in terms of physical masses andFπ = 92.4 MeV

Reshuffling of the LO contributions.

Fπ = F

{1 +

1

16π2F 2π

[A0(m

2π) +

1

2A0(m

2K )

]

+

[4cd cm(m

2π + 2m2

K )

F 2πM

2S1

− 8cd cm (m2K −m2

π)

3F 2πM

2S8

]}

This allows to know the NC -dependence of Fπ at the one-looplevel in U(3) ChPT including 1/NC subleading terms




NLO Meson-meson scattering O(δ3):

+ + crossed

S

+

+ + crossed

S , V

+ crossed+

We keep all contributions up to O(δ3) and calculate all theone-loop diagrams O(δ4)In addition, resonance exchanges generate an infinite tower ofhigher order contributions from resonance propagators

1

M2 − t=

1

M2

(1 +

t

M2+

t2

M4+ . . .

)




Not depicted

Wave Function Renormalization

η-η′ Mixing:

For NLO scattering only the LO mixing is required

For LO scattering one needs the mixing up to NLO

Rewriting of F in terms of Fπ for the LO scattering

Fπ = F

{1 +

1

16π2F 2π

[A0(m

2π) +

1

2A0(m

2K )

]

+

[4cd cm(m

2π + 2m2

K )

F 2πM

2S1

− 8cd cm (m2K −m2

π)

3F 2πM

2S8

]}.

http://www.um.es/oller/u3FullAmp16.nb




Partial wave amplitude and its unitarization

Why unitarizing?

1 In U(3) ChPT one has the large s quark mass and anomalymass term.In many kinematical regions the two-body pseudoscalarthresholds are much larger than the typical three-momenta.This enhances the unitarity two-pseudoscalar reducible loopcontributions Weinberg NPB91

2 We also go over the resonance region where the unitarityupper bound in partial waves is easily reached

Unitarity must be treated in a non-perturbative way




Partial wave projection:

T IJ (s) =

1

2(√2)N

∫ 1

−1dx PJ(x)T

I [s, t(x), u(x)] ,

where PJ(x) denote the Legendre polynomials and (√2)N is a

symmetry factor to account for the identical particles, such asππ, ηη, η′η′.

This defines the Unitary Normalization Oller,Oset NPA’97

ImT IJmn =

∑

k

θ(s − skth)ρkTIJ ikT

IJ

∗kj (1)

Identical and non-identical particle states are treated in the sameway.




The N/D method Chew, Mandelstam PR’60 is employedapproximately for unitarizing TJ :

TJ =N

D,

where

ImD = N ImT−1J = −ρN , for s > 4m2 ,

ImD = 0 , for s < 4m2 ,

ImN = D ImTJ , for s < 0 ,

ImN = 0 , for s > 0 ,

due to the fact that the unitarity condition for the elastic channels > 4m2 is

ImT−1IJ = −ρθ(s − sth) ,

with ρ =√1− 4m2/s/16π = q/8π

√s .




One can now write the dispersion relations for N and D:

D(s) = aSL(s0)−s − s0π

∫ ∞

4m2

N(s ′) ρ(s ′)(s ′ − s)(s ′ − s0)

ds ′

N(s) =

∫ 0

−∞

D(s ′) ImTJ(s′)

s ′ − sds ′ .

It can be greatly simplified if one imposes the perturbative solutionfor N(s) in terms of the left hand cut (LHC) discontinuity Oller,

Oset PRD’99, Oller PLB’00

No LHC:N(s) = 1

D(s) = aL +∑

i

Ri

s − si+ g(s) = Q(s)−1 + g(s)

g(s) =aSL(s0)

16π2+

s − s0π

∫ ∞

4m2

ρ(s ′)(s ′ − s)(s ′ − s0)

ds ′ .




TIJ(s) =Q(s)

1 + g(s)Q(s),

Including LHC (e.g. in our perturbative calculation there is LHCfrom crossed exchange of resonances and crossed loops)

Q(s) → N(s)

N(s) only has LHC

TIJ(s) =N(s)

1 + g(s) N(s),

Matching with TJ(s)|χPT = T2 + TR + TL up to one-loop at

O(p4):

T2 + TR + TL = N(s)− N(s)g(s)N(s) +O(~2)

N(s) = T2 + TR + TL + N(s)g(s)N(s)




The generalization to the inelastic case is straightforward:

TIJ(s) = [1 + g(s) · NIJ(s)]−1 · NIJ(s) ,

For IJ = 00 channel we have 5 channels: ππ, KK , ηη, ηη′ and η′η′

N00 (s) =

Nππ→ππ Nππ→KK Nππ→ηη Nππ→ηη′ Nππ→η′η′

Nππ→KK NKK→KK NKK→ηη NKK→ηη′ NKK→η′η′

Nππ→ηη NKK→ηη Nηη→ηη Nηη→ηη′ Nηη→η′η′

Nππ→ηη′ NKK→ηη′ Nηη→ηη′ Nηη′→ηη′ Nηη′→η′η′

Nππ→η′η′ NKKη′η′ Nηη→η′η′ Nηη′→η′η′ Nη′η′→η′η′




g00 (s) =

gππ 0 0 0 00 gKK 0 0 00 0 gηη 0 00 0 0 gηη′ 00 0 0 0 gη′η′




For IJ = 10 there are 3 channels: π0η, KK and π0η′

N(s)10 =

Nπη→πη Nπη→KK Nπη→πη′

Nπη→KK NKK→KK NKK→πη′

Nπη→πη′ NKK→πη′ Nπη′→πη′

g(s)10 =

gπη 0 00 gKK 00 0 gπη′




For IJ = 1/2 0 and 1/2 1 there are 3 channels: Kπ, Kη and Kη′

N(s)1/20 =

NKπ→Kπ NKπ→Kη NKπ→Kη′

NKπ→Kη NKη→Kη NKη→Kη′

NKπ→Kη′ NKη→Kη′ NKη′→Kη′

g(s)1/20 =

gKπ 0 00 gKη 00 0 gKη′




For the IJ = 11 there are 2 channels

N11 (s) =

(Nππ→ππ Nππ→KK

Nππ→KK NKK→KK

),

g11 (s) =

(gππ 00 gKK

).

IJ = 3/2 0:

N(s)3/20 = NKπ→Kπ

g(s)3/20 = gKπ

IJ = 2 0:

N(s)20 = Nππ→ππ ,

g(s)20 = gππ




IJ = 01

N01 (s) = NKK→KK ,

g01 (s) = gKK .




Numerical results




Observables fitted:

I = J = 0: δ00ππ→ππ, |S00ππ→ππ|, 1

2 |S00ππ→KK

|, δ00ππ→KK

I = J = 1: δ11ππ→ππ

I = 1/2 J = 0 , 1: δ120

πK→πK , δ121

πK→πK

I = 2 J = 0 : δ20ππ→ππ

I = 3/2 J = 0 : δ320

πK→πK

I = 1 J = 0 : πη event distribution around a0(980)

dNπη

dEπη= qπη N

∣∣TKK→πη(s) + c Tπη→πη(s)∣∣2 .

mη, mη′




Subtraction Constants: The number of free ones can be reducedby applying Isospin and SU(3) symmetry. We also assume nonetsymmetry in some cases. Jido,Oller,Oset,Ramos,Meißner, NPA’03

• Isospin Symmetry requires that all the aIJSL are the sameseparately for ππ, KK and Kπ

• Nonet Symmetry requires that all aIJSL are the same for a given J

a120 ,Kπ

SL = a320 ,Kπ

SL

a00 ,KKSL = a1 0 ,KK

SL = a00 , ηηSL = a00 , ηη′

SL = a00 , η′η′

SL = a1 0 , πη′

SL

a00 , ππSL = a20 , ππSL

All the subtraction constants in the vector channels are set equalto a00 , ππSL (play a little role).Unitarization is much less important for these channels.




Large NC relation for the singlet scalar couplings:

cd =1√3cd , cm =

1√3cm

14 relevant free parameters:

cd = (20+2−5)MeV , cm = (42+4

−9)MeV ,

MS8= (1400+70

−60)MeV , MS1= (1100+30

−60)MeV ,

GV = (62.1+1.9−2.1)MeV , a1 0 ,πηSL = 2.0+3.3

−4.5 ,

a00 ,ππSL = −1.27+0.12−0.12 , a00 ,KK

SL = −0.95+0.33−0.16 ,

a12 0 ,Kπ

SL = −1.12+0.12−0.17 , a

12 0 ,Kη

SL = −0.08+0.38−1.04 ,

a12 0 ,Kη

′

SL = −1.25+1.11−1.23 , δL8 = (0.23+0.29

−0.19)× 10−3 ,

M0 = (950+50−50)MeV , Λ2 = −0.37+0.19

−0.19 ,

with χ2/d.o.f = 714/(348− 16) ≃ 2.15.

nσ = ∆χ2/√

2χ2 ≤ 2 to get the errors, nσ = 2 Etkin et al. PRD’82

GV = 55 MeV Gasser and Leutwyler ’85J. A. Oller Universidad de Murcia



• Large Nc U(3) nonet symmetry:

a00SL ≃ a120

SL

M8 = M1 (∼ 30% discrepancy)

M8 and M1 in good agreement with previous determinations[I,II]

cd = (20+2−5) and cm = (42+4

−9) MeV are compatible with

previous determinations [I] cd = 19.1+2.4−2.1 ; [II] cd ≃ 24 ; [III]

cd = 26± 7. cm always with large errors.

[I] Oller, Oset PRD’99;[II] Jamin, Oller, Pich NPB’00; [III] Guo,

Sanz-Cillero PRD’09

L4 = 0.09± 0.04× 10−3

L5 = 0.67+0.03−0.19 × 10−3

Our results

L4 = 0.75± 0.75× 10−3

L5 = 0.58± 0.13× 10−3

Bijnens, Jemos NPB’12

We can obtain good fits with 6 free parameters less.




0

50

100

150

200

250

300

200 400 600 800 1000 1200

Energy (MeV)

New-FitOld-Fit

0

0.2

0.4

0.6

0.8

1

1.2

1000 1100 1200 1300

Energy (MeV)

New-FitOld-Fit

0.2

0.25

0.3

0.35

0.4

0.45

1000 1100 1200 1300

Energy (MeV)

New-FitOld-Fit

120

160

200

240

280

1000 1100 1200 1300

Energy (MeV)

New-FitOld-Fit

δ00

ππ→ππ(D

egrees)

|S00

ππ→ππ|

|S00

ππ→KK|/2

δ00

ππ→KK(D

egrees)




0

50

100

150

200

800 1000 1200 1400 1600

Energy (MeV)

100

200

300

400

800 850 900 950 1000 1050

Eve

nts/

25M

eV

Energy(MeV)

-40

-30

-20

-10

0

200 400 600 800 1000 1200

Energy (MeV)

-40

-30

-20

-10

0

600 800 1000 1200 1400

Energy (MeV)

δ1 20

Kπ→Kπ(D

egrees)

δ20

ππ→ππ(D

egrees)

δ3 20

Kπ→Kπ(D

egrees)




0

30

60

90

120

150

180

500 600 700 800 900 1000 1100 1200

Energy (MeV)

0

30

60

90

120

150

180

700 800 900 1000 1100 1200

Energy (MeV)

δ11

ππ→ππ(D

egrees)

δ1 21

Kπ→Kπ(D

egrees)

• We need to distinguish between the ρ(770) and K ∗(892) baremasses in order to fit the phase shifts accurately:

Mρ = (801.0+7.0−7.5)MeV , MK∗ = (909.0+7.5

−6.9)MeV




• Unitarity (vector self-energies due to two-pseudoscalar loops)does not give the necessary shift to the vector octet bare mass.

• Unitarity provides the right width for every vector resonance.




Poles from the unitarized amplitudesBy crossing the real axis between the thresholds Tn < s < Tn+1

one directly reaches the Riemann sheet

(−,−, ..−︸︷︷︸n

,+,+, . . .)

Tn Tn+1

Resonance

Asymmetric shape. Asymmetric shape.Resonance Rise Right. Cusp Fall Right.Cusp Rise Left. Resonance Fall Right.




R M (MeV) Γ/2 (MeV) |Residues|1/2 (GeV) Ratios

σ 441 247 3.02 (ππ) 0.50 (KK/ππ) 0.17 (ηη/ππ)

0.33(ηη′/ππ) 0.11(η′η′/ππ)

f0(980) 977 29 1.8(ππ) 2.6(KK/ππ) 1.6(ηη/ππ)

1.0(ηη′/ππ) 0.7(η′η′/ππ)

f0(1370)(iii) 1359 169 3.2(ππ) 1.0(KK/ππ) 1.2(ηη/ππ)

1.5(ηη′/ππ) 0.7(η′η′/ππ)

f0(1370)(v) 1496 152 2.2(ππ) 1.4(KK/ππ) 1.2(ηη/ππ)

2.9(ηη′/ππ) 1.7(η′η′/ππ)

κ 641 304 4.8(Kπ) 0.86(Kη/Kπ) 0.747(Kη′/Kπ)

K∗

0 (1430)(iii) 1483 132 4.4(Kπ) 0.33 (Kη/Kπ) 1.22(Kη′/Kπ)

a0(980) 1006 22 2.45(πη) 1.9 (KK/πη) 0.03(πη′/πη)

a0(1450) 1460 174 4.5(πη) 0.36(KK/πη) 0.97+0.2−0.1(πη′/πη)

ρ(770) 760 71 2.44(ππ) 0.64(KK/ππ)

K∗(892) 892 25 1.85(Kπ) 0.91(Kη/Kπ) 0.41(Kη′/Kπ)

φ(1020) 1019 1.9 0.85(KK)




σ or f0(600) , IJ = 00 (−,+,+,+,+)

Mσ = 440+3−3MeV , Γσ/2 = 258± 12MeV ,

|gσππ| = 3.02+0.03−0.03GeV ,

|gσKK |/|gσππ| = 0.51+0.03−0.02 , |gσηη|/|gσππ| = 0.06+0.03

−0.01

|gσηη′ |/|gσππ| = 0.16+0.03−0.02 , |gση′η′ |/|gσππ| = 0.05+0.03

−0.03

T → − gi gjs −M2

, g2σππ =

−1

2πi

∮

|s−sσ |=R

T II(s) ds .

Mσ = 441+16−8 , Γσ/2 = 272+9

−13 Caprini et al. PRL’06

Mσ = 484± 17 , Γσ/2 = 255± 10 Garcıa-Martın et al. PRD’07

Mσ = 456± 6 , Γσ/2 = 241± 17 Albaladejo, Oller PRL’08

Narison et al. PLB’10 interpret the large KK coupling of the σas an indication for a glueball nature.We also have such a large coupling. In our case it correspondsto a dynamically generated resonance.




f0(980) , IJ = 00 (−,+,+,+,+)

Mf0 = 981+9−7MeV , Γf0/2 = 22+5

−7MeV ,

|gf0ππ| = 1.7+0.3−0.3GeV

|gf0KK |/|gf0ππ| = 2.3+0.3−0.2 , |gf0ηη|/|gf0ππ| = 1.6+0.3

−0.3

|gf0ηη′ |/|gf0ππ| = 1.2+0.1−0.2 , |gf0η′η′ |/|gf0ππ| = 0.7+0.4

−0.5

f0(1370) , IJ = 00 (−,−,−,+,+)

Mf0 = 1401+58−37MeV , Γf0/2 = 106+36

−23MeV ,

|gf0ππ| = 2.4+0.2−0.1GeV

|gf0KK |/|gf0ππ| = 0.62+0.04−0.05 , |gf0ηη|/|gf0ππ| = 0.9+0.1

−0.1

|gf0ηη′ |/|gf0ππ| = 1.7+0.4−0.6 , |gf0η′η′ |/|gf0ππ| = 1.1+0.4

−0.5

Both resonances have strong couplings to states with η, η′




a0(1450) , IJ = 10 , (−,−,−)

Ma0 = 1368+68−68MeV , Γa0/2 = 71+48

−23MeV ,

|ga0πη| = 2.3+0.4−0.5GeV

|ga0KK |/|ga0πη| = 0.6+0.7−0.2 , |ga0πη′ |/|ga0πη| = 0.6+0.2

−0.1

PDG: Γ = 265± 13 MeV. We are lacking important decaychannels: ωππ, a0(980)ππρ(770) , IJ = 11 (−,+)

Mρ = 762+4−4MeV , Γρ/2 = 72+2

−2MeV ,

|gρ ππ| = 2.48+0.03−0.05GeV , |gρKK |/|gρππ| = 0.64+0.01

−0.01

K ∗(892) , IJ = 1/2 1 (−,+,+)

MK∗ = 891+3−4MeV , ΓK∗/2 = 25+2

−1MeV ,

|gK∗ πK | = 1.86+0.05−0.05GeV

|gK∗Kη|/|gK∗Kπ| = 0.91+0.03−0.02 , |gK∗Kη′ |/|gK∗Kπ| = 0.45+0.08

−0.08




φ(1020) , IJ = 01 (−,+,+)

Mφ = 1014.7+0.30.3 MeV , Γφ/2 = 1.50+0.04

−0.08MeV ,

|gφKK | = 1.50+0.04−0.08GeV

ΓKK = 3 MeV, close to 3.5 MeV PDG. We are lacking theπππ state.




SS , PP correlators. Sum rules

δabΠR(p2) = i

∫d4x e ip·x < 0|T [Ra(x)Rb(0)]|0 > ,

In the chiral limit we have the second spectral function sum rules:

∫ ∞

0

[ImΠR(s)− ImΠR′(s)

]ds = 0 ,

Ra refers to Sa or Pa with a = 0, 1, 3 or 8.Splitting between hadronic and high energy regions:

∫ s0

0


]ds +

∫ ∞

s0


]ds = 0

s0 = 2.5, 3, 3.5 GeV2




For s > s0 we employ the OPE expression (or theoretical spectralfunction) from Jamin,Munz ZPC’93 calculated up to O(αs) andincluding up to dimension 5 operators.

The theoretical spectral functions cancel each other in the chirallimit. The second integral for s > s0 is 0

Wi = 16π

∫ s0

0ImΠi (s) ds , i = S8, S0, S3,P0,P8,P3 ,

W =

∑i=(S8,S0,S3,P0,P8,P3)Wi

3× 6,

σ =∑

i=(S8,S0,S3,P0,P8,P3)

√(Wi −W )2

17,




WS0 WS8 WS3 WP0 WP8 WP3 W σ σ/W

Physical masses

8.6 9.0 9.6 7.4 7.5 7.7 7.0 7.2 7.4 8.9 11.3 10.1 9.0 1.5 0.16

mq = 0

5.8 5.9 6.1 9.6 9.7 9.7 5.4 5.6 5.7 5.5 7.5 7.5 6.9 1.5 0.22

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3

a=0a=8a=3

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5 3

a=0a=8a=3

16πIm

ΠSa×3/N

C

16πIm

ΠSa×3/N

C

s(GeV2)s(GeV2)

NC = 3 NC = 30




The equality of the integrals for the spectral functions issatisfied at the level of 20%

The lightest peak for a = 0 is the f0(600)

It is interesting the movement of the peaks with NC

For NC → ∞ only the bare poles remains

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3

a=0a=8a=3

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5 3

a=0a=8a=3

16πIm

ΠSa×3/N

C

16πIm

ΠSa×3/N

C

s(GeV2)s(GeV2)

NC = 3 NC = 30




0

2

4

6

8

10

12

0 3 6 9 12 15 18 21 24 27 30 0

2

4

6

8

10

12

0 3 6 9 12 15 18 21 24 27 30

WSa×3/N

C

WSa

WP

a×3/N

C

WP a

a = 0 (Chiral Limit)a = 0 (Chiral Limit)

a = 0 (Physical Point)a = 0 (Physical Point)





NCNC

In the large NC limit the SS , PPcorrelators reduce to

S, P

c2m = F 2

8 + d2m

perfectly fulfilled bycm = 42 MeV from our fit




Calculation of the spectral functions

Scalar Form Factors

ImΠSa(s) =∑

i

ρi (s)Fai (s)F

a∗i (s) θ(s − sthi )

ρi (s) =

√[s − (mP +mQ)2][s − (mP −mQ)2]

16π s=

qi8π

√s

F ai (s) =

1

B〈 0| q λa√

2q | (PQ)i 〉

ImF Ij (s) =

Z∑

k=1

T IJjk (s)

∗ρk(s)F

Ik(s)




Resumming of the right hand cut

TIJ(s) =[1 + NIJ(s) gIJ(s)

]−1NIJ(s)

FI (s) =[1 + NIJ(s) gIJ(s)

]−1RI (s)

=[1− NIJ(s) gIJ(s)

]RI (s) +O(g2)

with

NIJ(s) = TIJ(s)(2)+Res+Loop + TIJ(s)

(2) gIJ(s)TIJ(s)(2) ,

RI (s) = F I (s)(2)+Res+Loop

+ NIJ(s)(2) gIJ(s)FI (s)

(2) .

RI (s) has no cutsMeißner, JAO NPA’01

It is determined perturbatively in Resonance U(3) ChPT




FI (s): One-loop Resonance U(3) ChPT calculation:Guo, JAO forthcoming

(a) (b) (c)

SS

(d) (e) (f)




Scalar Radius of the pion

F uu+ddππ (s) = F uu+dd

ππ (0)

[1 +

1

6〈 r2 〉πS s + ...

]

〈 r2 〉πS = 0.5 fm2

Dispersive studies:〈 r2 〉πS ≃ 0.6 fm2

Colangelo, Gasser, Leutwyler

NPB’01; Roca, JAO PLB’07




Pseudoscalar Form Factors: ImΠPa(s) =∑

i π δ(s −m2Pi) |Ha

i (s)|2

(a) (b)

S

(c)

P

(e)(d)




Semi-local duality

Relation between of s-channel dynamics (including sum ofresonances) and t-channel Regge pole exchanges

For energies not high the duality works at the average level

Pelaez, Pennington, Ruiz de Elvira, Wilson PRD’11 [I] applied averageduality for studying the f0(600) and its non-standard nature

They applied the Inverse Amplitude Method (IAM) for unitarizingChPT at the two-loop level and paid attention to the ρ(770) andf0(600) resonances




Fixed-t scattering, ν = (s − u)/2

∫ ν2

ν1

ν−n ImA(I )t,Regge(ν, t) =

∫ ν2

ν1

ν−n ImA(I )t,Hadrons(ν, t)

ν2 − ν1 = O(GeV2).The main point is that for the isotensor t-channel caseπ+π− → π−π+ there is no qq Regge exchange and it is suppressed

A(2)t (s, t) =

1

3A(0)s (s, t)− 1

2A(1)s (s, t) +

1

6A(2)s (s, t)

The s-channel I = 0 (f0(600)) and I = 1 (ρ(770)) contributionsshould cancel in the integration for It = 2.




We use the same parameterizations as Pelaez et al. PRD’11 for

ImA(I )t,Regge(ν, t) that contain the ρ trajectory for It = 1 and the

Pomeron for It = 0. No isotensor Regge contribution.

ImA(I )s,Hadrons(ν, t) =

∑

J

(2J + 1) ImA(I )J (s)PJ(zs) ,

ImA(I )J (s) are the partial wave that we already calculated and

studied.

R In =

∫ ν2ν1

ν−n ImA(I )t (ν, t)

∫ ν3ν1

ν−n ImA(I )t (ν, t)

,

it is calculated with Regge theory and in terms of hadronicdegrees of freedom. The results are compared.ν1 =threshold, ν2 = 1 GeV2 and ν3 = 2 GeV2.

We take n = 0, 1, 2, 3, as n increases the integrals are more andmore sensitive to the low energy region.




As n increases one is increasingly sensitive to the splitting in massbetween the lightest resonances

n R0n R0

n R1n R1

n F 21n F 21

n

t = tth t = 0 t = tth t = 0 t = tth t = 0

νmax = 2 GeV2

Regge 0 0.225 0.233 0.325 0.353 ∼ 0 ∼ 0

1 0.425 0.452 0.578 0.642 ∼ 0 ∼ 0

2 0.705 0.765 0.839 0.908 ∼ 0 ∼ 0

3 0.916 0.958 0.966 0.990 ∼ 0 ∼ 0

Ours 0 0.669 0.628 0.836 0.817 -0.113 0.040

S + P 1 0.837 0.812 0.919 0.908 -0.230 -0.087

Waves 2 0.934 0.924 0.966 0.962 -0.129 0.028

3 0.979 0.976 0.989 0.988 0.169 0.345

Ours 0 0.410 0.400 0.453 0.468 0.531 0.587

S + P + D 1 0.653 0.643 0.694 0.706 0.154 0.236

Waves 2 0.850 0.844 0.875 0.882 0.027 0.155

3 0.954 0.953 0.965 0.968 0.225 0.388




D-waveWe consider here the exchange of a nonet of tensor resonancesemploying the Lagrangians of Ecker, Zaune, EPJC’07 extended toU(3)These exchanges contribute to the NIJ function

LT = −1

2〈TµνD

µν,ρσT Tρσ 〉+gT 〈Tµν{uµ, uν} 〉+β〈Tµ

µ uνuν 〉+γ〈Tµ

µχ+ 〉 ,

Tµν =

a02√2+

f 82√6+

f 12√3

a+2 K ∗+2

a−2 − a02√2+

f 82√6+

f 12√3

K ∗02

K ∗−2 K ∗0

2 −2f 82√6+

f 12√3

µν

.

Ideal mixing:

f2(1270) =

√2

3f 12 +

√1

3f 82 , f ′2 =

√1

3f 12 −

√2

3f 82 .




Short distance constraints: forward elastic partial waves satisfyonce subtracted dispersion relation. We generalize them to theU(3) case Guo,JAO forthcoming.

They are important to obtain meaningful results once tensorresonances are includedSeveral O(p4) monomials are needed. (The results are thenindependent of β but β = −gT avoids including O(p6)monomials.)




F II ′n =

∫ νmax

ν1ν−n ImA

(I )t (ν, t)

∫ νmax

ν1ν−n ImA

(I ′)t (ν, t)

νmax = 1 or 2 GeV2.Most interesting I = 2 and I = 1

A(1)t (s, t) =

1

3A(0)s (s, t) +

1

2A(1)s (s, t)− 5

6A(2)s (s, t)

A(2)t (s, t) =

1

3A(0)s (s, t)− 1

2A(1)s (s, t) +

1

6A(2)s (s, t)

F 21n → −1 if scalar contribution is dropped

F 21n → +1 if vector contribution is dropped




n R0n R0

n R1n R1

n F21n F21nt = tth t = 0 t = tth t = 0 t = tth t = 0

νmax = 2 GeV2

Regge 0 0.225 0.233 0.325 0.353 ∼ 0 ∼ 0

1 0.425 0.452 0.578 0.642 ∼ 0 ∼ 0

2 0.705 0.765 0.839 0.908 ∼ 0 ∼ 0

3 0.916 0.958 0.966 0.990 ∼ 0 ∼ 0

Ours 0 0.669 0.628 0.836 0.817 -0.113 0.040

S + P 1 0.837 0.812 0.919 0.908 -0.230 -0.087

Waves 2 0.934 0.924 0.966 0.962 -0.129 0.028

3 0.979 0.976 0.989 0.988 0.169 0.345

Ours 0 0.410 0.400 0.453 0.468 0.531 0.587

S + P + D 1 0.653 0.643 0.694 0.706 0.154 0.236

Waves 2 0.850 0.844 0.875 0.882 0.027 0.155

3 0.954 0.953 0.965 0.968 0.225 0.388

The numbers in absolute value are typically much smaller thanunity. Average duality is fulfilled quite accurately




n R0n R0

n R1n R1

n F21n F21nt = tth t = 0 t = tth t = 0 t = tth t = 0

νmax = 2 GeV2

Regge 0 0.225 0.233 0.325 0.353 ∼ 0 ∼ 0

1 0.425 0.452 0.578 0.642 ∼ 0 ∼ 0

2 0.705 0.765 0.839 0.908 ∼ 0 ∼ 0

3 0.916 0.958 0.966 0.990 ∼ 0 ∼ 0

Ours 0 0.669 0.628 0.836 0.817 -0.113 0.040

S + P 1 0.837 0.812 0.919 0.908 -0.230 -0.087

Waves 2 0.934 0.924 0.966 0.962 -0.129 0.028

3 0.979 0.976 0.989 0.988 0.169 0.345

Ours 0 0.410 0.400 0.453 0.468 0.531 0.587

S + P + D 1 0.653 0.643 0.694 0.706 0.154 0.236

Waves 2 0.850 0.844 0.875 0.882 0.027 0.155

3 0.954 0.953 0.965 0.968 0.225 0.388

For n = 0, the D waves make a too large contribution. They overbalance

the ρ(770). The ρ(1450) gives a negligible contribution because its BR

to ππ (≃ 6%) is too small [PDG]. Thus, heavier vector resonances are

needed. For larger n they have no impact due to their large mass.




Extrapolating in NC

Due to the uncertainties in the large NC values of the resonanceparameters we consider several options:

Scenario 1: Leading running with NC

{cd(NC ), cm(NC ), cd(NC ), cm(NC ),GV (NC )}

= {cd(3), cm(3), cd(3), cm(3),GV (3)} ×√

NC

3,

{MS1(NC ),MS8(NC ),Mρ(NC ),MK∗(NC ),Mω(NC ),Mφ(NC ), aSL(NC )}= {MS1(3),MS8(3),Mρ(3),MK∗(3),Mω(3),Mφ(3), aSL(3)}For the other cases subleading 1/NC corrections are introduced as:

A(NC ) = α+ β1

NC

α = A(∞) , β = 3 [A(3)− A(∞)]




Scenario 2: Large NC expression for GV

GV (NC ) =Fπ(NC )√

3

High energy constraint of pion vector f.f. at one-loop level Pich,Rosell,Sanz-Cillero JHEP’109, radiative τ decays

Guo, Roig PRD’113016, ππ scattering Guo,JAO PRD’11; Guo, Sanz-Cillero, Zheng JHEP’07, model for ππ

scattering in extra dimensions Chivukula et al. PRD’07

GV (NC ) = GV (NC = 3)

√NC

3

×[1 +

GV (NC = 3)− GNorV (NC → ∞)

GV (NC = 3)

(3

NC

− 1

)]

GNorV (NC → ∞) = GV (NC → ∞)

√3NC




Scenario 3: Large NC degenerate masses for ρ(770) and S1(1000)M(∞) = 930 MeVMean between the bare ρ and S1 massesVery close to the NC → ∞ ρ mass of [I]

M2(NC ) = M2(NC = 3)

[1 +

M2(NC = 3)−M2(NC → ∞)

M2(NC = 3)

(3

NC

− 1

)]

Scenario 4: D-waves are includedThe couplings are proportional to

√NC




-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=0

S-1S-2S-3S-4

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=3

NCNC

NCNC

F21

0

F21

1

F21

2

F21

3




Thus, heavier vector resonances are needed. For larger n they haveno impact due to their large mass. S-5 includes a ρ′.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=0

S-1S-2S-3S-4S-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20 25 30

n=3

NCNC

NCNC

F21

0

F21

1

F21

2

F21

3

A 40% BR to ππ is needed for this other resonance ρ′




IAM: Leading large NC behavior

Li (Nc) = Li (3)

(NC

3

)α

Average duality is fulfilled for two-loop IAM but not for one-loopIAM

400 600 800 1000 1200Mρ(MeV)

-200

-160

-120

-80

-40

0

-Γρ/2

(MeV

) Nc=3

Nc=6

Nc=9Nc=12

two-loop ρ pole trajectory case A

Nc

Mρ in one-loop IAM at large NC is ≃ 750 MeV

Mρ in two-loop IAM at large NC is ≃ 940 MeV

400 600 800 1000 1200Mσ (MeV)

-500

-400

-300

-200

-100

0

-Γσ/2

(MeV

)

Nc=3

Nc=6

Nc=9

Nc=12

two-loop σ pole trajectory case A

Nc

qq-like structure for large NC

√sσ = 1.2 − i 0.23 GeV at NC = 12




For NC large

ImA(I=1)J=1 (s) =

πMρΓρσ(M2

ρ )δ(s −M2

ρ ) , σ(s) =√

1− 4m2π/s

∫ νmax

ν1

ν−n ImAI=2t,ρ (ν, t) = −3

2

π ΓρM1−2nρ

σ(M2ρ )

Ratio between two- and one-loop contributions

Γρ,two−loop

Γρ,one−loop

(Mρ,two−loop

Mρ,one−loop

)1−2n σ(M2ρ,one−loop)

σ(M2ρ,two−loop)

For n = 3 one has a reduction of 0.43.ρ(770) contribution is strongly suppressed in two-loop IAM.Why?

To which resonance does the f0(600) pole correspond at large NC?If it were a qq [I] (think of the ρ) its mass movement will be muchsmaller




Our study answers these two questions:First: The short distance constraints imply, when including crossedρ-exchanges, a transition of GV from

GV (3) ≃Fπ(3)√

2→ GV (∞)Nor ≃ Fπ(3)√

3

Second: This qq resonance around 1 GeV corresponds to S1 thatcontributes for NC = 3 to the f0(980) resonance pole (anothercontribution is a KK bound state Oset, JAO NPA’97)




Running of pole positions with Nc

For the first time the Nc dependence of the pseudo-Goldstonemasses and mixing angle are taken into account for determiningresonance properties with increasing Nc .

There is not large Nc limit of SU(3) χPT.

The η1 becomes the ninth pseudo-Goldstone boson.

M20 ∼ Λ2 ∼ 1/Nc

cd ∼ cm ∼ cd ∼ cm ∼ GV ∼ F ∼√Nc

M2V ∼ M2

S8∼ M2

S1∼ B ∼ aSL ∼ O(N0

c )

We take full Scenario 3 (includes subleading 1/NC corrections)

We solve for M2π, M

2K , M

2η , M

2η′ and the LO mixing angle θ as

functions of NcJ. A. Oller Universidad de Murcia



Pseudoscalar masses with varying Nc

0

100

200

300

400

500

600

700

800

900

1000

1100

0 5 10 15 20 25 30

Mass(M

eV)

NC

π

Kη

η′




Leading order 1/Nc → ∞ prediction (M0 → 0):

M2η = Mπ = (139.5+4.4

−4.6)2 MeV2 ,

M2η′ = 2M2

K −M2π = (721.5+17.4

−11.1)2 MeV2 .

Ideal Mixing (OZI rule is exact): θ = −57.7o

-60

-50

-40

-30

-20

-10

0 5 10 15 20 25 30

θ(D

egrees)

NC




σ-Pole Trajectory

200

300

400

500

600

700

800

900

1000

1100

1200

100 200 300 400 500 600 700

mimic SU(3)vector reduced

full result

Γ/2

(MeV

)

Mass (MeV)

NC = 3

σ

Oller, Oset PRD’99 M2S ∝ f 2 ∝ Nc

Pelaez et al ’04,’06,. . . ,’10 σ Mass always increases with Nc

At the two-loop order it moves to a pole with zero width at 1 GeV.

We also obtain such a pole but it comes from the bare scalarsinglet MS1 ≃ 1 GeV (At Nc = 3 it contributes to the f0(980).)




1

M2V − t

→ 1

M2V

,

NLO local terms in χPT. Vector Reduced case (Blue Triangles)

The σ-pole trajectory is then more similar to that of the IAMone-loop study Pelaez PRL’04

Sensitivity to higher order local terms.

When full vector propagators are kept their crossed exchangecontributions cancel mutually with those from crossed loops alongthe RHC (

√s . 1 GeV) Oller, Oset PRD’99

For increasing Nc loops are further suppressed and this cancellationis spoilt: More sensitivity to the LHC contributions.

The σ pole blows-up in the complex plane (not qq). Dynamicallygenerated resonance.




Mimic SU(3) case: Mixing is set to zero and η1 is kept in the loops. η8,

η1 masses are frozen. Differences highlight the role of η′. Differences for

the σ case are negligible.

0

10

20

30

40

50

960 980 1000 1020 1040 1060 0

40

80

120

160

200

1320 1340 1360 1380 1400

mimic SU(3)vector reduced

full result

0

20

40

60

80

100

120

140

1320 1340 1360 1380 1400 1420 1440 1460 0

20

40

60

80

100

120

140

1340 1360 1380 1400 1420 1440

4567

Γ/2

(MeV

)

Γ/2

(MeV

)

Γ/2

(MeV

)

Γ/2

(MeV

)

Mass (MeV)Mass (MeV)


NC = 3

NC = 3

NC = 3NC = 3

NC = 3

NC = 3

NC = 3NC = 3

NC = 3f0(980)

f0(1370)

K∗0(1430)

a0(1450)




f0(980)The Pole moves for Nc → ∞ to the bare scalar singlet on the realaxis with mass MS1 ≃ 1 GeV.

The f0(980) has an important KK bound state contribution to itsnature Weinstein, Isgur PRL’82; PRD’83; PRD’90; Weinstein, PRD’93;

Janssen et al. (Julich model) PRD’95; Oller, Oset NPA’97; PRD’99.It disappears in the large Nc limit.

f0(1370), K∗0 (1430), a0(1450)

All of them evolve to the octet of bare scalar resonancesMS8 = 1370+172

−57 MeV.

Large differences between our full results and mimic SU(3) case,indicating large couplings to η′ channels.




κ: Track of (+,−,+) pole

0

300

600

900

1200

1500

1800

300 400 500 600 700

0

200

400

600

800

1000

1200

1000 1200 1400 1600 1800 2000

Γ/2

(MeV

)

Γ/2

(MeV

)


NC = 3NC = 3

a0(980)κ

Similar trajectory as for the σ. The mass decreases with Nc andthe width increases very fast.

In the vector reduced case the mass keeps increasing with Nc .a0(980)It disappears deep in the complex plane, not qq. Dynamicallygenerated resonance.




ρ(770), K ∗(892)

0

10

20

30

40

50

60

70

80

760 770 780 790 800

mimic SU(3)full result

0

5

10

15

20

25

30

890 895 900 905 910

mimic SU(3)full result

Γ/2

(MeV

)

Γ/2

(MeV

)


NC = 3NC = 3

ρ(770) K∗(892)

qq trajectories: Mass O(N0c ) and Width O(1/Nc) Pelaez PRL’04

Above Nc = 13 the Kη threshold becomes lighter than theK ∗(892) mass (kink in the residue to Kη at Nc = 14).For the φ(1020) the situation is the same. There is sensitivity tothe slight movement of the nearby KK threshold with Nc .




Conclusions

A complete one loop calculation of all meson-mesonscattering amplitudes within U(3) χPT was worked out.

It is included the explicit exchange of tree level scalar andvector resonances in the s-,t- and u-channels.

The perturbative amplitudes are unitarized using an approachbased on the N/D method, while treating perturbatively thecrossed channel dynamics.

More resonances are generated than included. Dynamicalgeneration of resonances from the self-interactions betweenthe lightest pseudoscalars.




Nc dependence considered for various resonance quantities,such as pole positions and residues.

Peculiar trajectories indicating dynamical generation of thelightest scalar resonances σ, κ, a0(980).

Bare singlet scalar pole S1 present in the f0(980) with mass∼ 1 GeV.

It does not evolve from the σ and it stands both at Nc = 3and Nc → ∞.

f0(1370), K∗0 (1430) and a0(1450) move to the bare octet of

scalar resonances S8 with mass ∼ 1.37 GeV.

Vector resonances ρ(770), K ∗(892) and φ(1020) follow atextbook qq large Nc trajectory.




Non-trivial QCD constraints are satisfied quite satisfactorily byour partial waves

They imply interesting relations between the scalar spectrumand the pseudoscalar one and also with the vector resonancespectrum

Interesting transition between NC = 3 to NC → ∞. Thestrength of the f0(600) (dynamical-)resonance progressivelyevolves to the S1 (bare-)resonance

For semi-local duality this is accompanied by a weakening ofthe ρ contribution (GV → Fπ/

√3 for large NC ). This also

sheds light to large NC QCD.

For the PP correlators this is accompanied by the change ofthe couplings to the pseudoscalar sources




Our study of all these facets simultaneoulsy: chiral symmetry,unitarity, analyticity, QCD second spectral function sum rules,QCD semi-local duality and large NC QCD strengthens ourunderstanding of the scalar spectrum and its nature.

f0(600) non-standard qq resonance.

Existence of a bare isosinglet scalar at 1 GeV.

First qq octet at around 1.4 GeV.



Topicsin U(3) χPTdynamicsandrelated spectroscopy.is invariant as well as any function of X Witten...

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