Scalar Dynamics

Final State Interactions in Hadronic D decaysJosé A. OllerJosé A. Oller

Univ. Murcia, SpainUniv. Murcia, Spain

• Introduction

• FSI in the D+ -++ decay

• FSI in the Ds+ -++ decay

• FSI in the D+ -++ decay

•Mixing Angle of the Lightest Scalar Nonet

• Determination of CHPT Counterterms

Bad Honnef, January 21 2004

1. Some decays of D mesons offer experiments with high statistics where the meson-meson S-waves are dominant. This is very intersting and new.

2. This has given rise to observe experimentally with large statistical significance the f0(600) or E791 Collaboration PRL 86, 770 (2001) D+ -++ , and the K*

0(800) or mesons E719 Collaboration PRL 89, 121801 (2002) D+ -++ .

3. Clear observation of the resonance has been also reported by the Collaborations CLEO, Belle and BaBar.

4. There are no Adler zeroes that destroy the bumps of the and , contrary to scattering.

1.1 Introduction

However in the E791 Analyses:• The phases of the Breit-Wigner‘s used for the and do

not follow the I=0 S-wave and I=1/2 K S-wave phase shifts, respectively. Despite that at low two-body energies one expects that the spectator hypothesis should work and then it should occur by Watson‘s theorem.

• Furthermore, the f0(980) resonance has non standard

couplings, e.g. it couples just to pions while having a coupling to kaons compatible with zero.

• The width of the K*0(1430) is a factor of 2 smaller than

PDG value, typical from scattering studies.

1.2 FSI in the D+! -++ decay

D+! -++ E791 Col. PRL 86, 770 (2002) 1686 candidates, Signal:background 2:1 1124 events.

E791 Analysis follows the Isobar Model to study the Dalitz plot

It is based on two assumption: Third pion is an spectator, and one sums over intermmediate two body resonances

D R

INTERMEDIATE RESONANCES R ARE SUMMED

D+! -++ E791 Col. PRL 86, 770 (2002) 1686 candidates, Signal:background 2:1 1124 events.

E791 Analysis follows the Isobar Model to study the Dalitz plot

-(1), +(2), +(3) s12=(p1+p2)2, s13=(p1+p3)2

a0 ei0 : Non-Resonant Term An : Breit-Wigner (BW) term corresponding to

a (12) or (13) resonance

Then is Bose-symmetrized because of the two identical +

It is based on two assumption: Third pion is an spectator, and one sums over intermmediate two body resonances

1.2 FSI in the D+! -++ decay

When in the sum over resonances the + state was not included, then

2/dof=1.5 CL=10-5 WHEN included 2/dof=0.9 CL=76 %

NO

WITH

Exchanged resonances:

0(770) , f0(980), f2(1270), f0(1370), 0(1450),

BW(s)=(s-m2+i m (s) )-1 energy dependent width !

BWBW

BW does not follow the experimental S-wave I=0 phase shifts

M=478§24 MeV

= 324§42 MeV

Laurent series around the pole position at the second Riemann sheet:

From the T-matrix of Oset,J.A.O.,NPA620(1997)435

Laurent series around the pole position at the second Riemann sheet:

PolePole

The movement of the phase of the pole follows the experimental phase shifts

We substitue the BW by

Adler zero: the reason why the background is so large in order to cancel the pole for low energies

background

BW

BW

2

Dashed line, pole instead of BW 2/dof=3/152 s is fixed

At the same time the phase motion of the contribution follows the experimental phase shifts.

Full Final State Interactions (FSI) from the T-matrix of Oset,J.A.O.,NPA620(1997)435 Unitary CHPT

SCATTERING:

In Unitarized Chiral Perturbation Theory the matrix of scattering amplitudes N is fixed by matching at a given chiral order with the full amplitude T calculated in CHPT

is the scalar loop function or unitarity bubble

D-decays:

We now have meson-meson intermediate states

The and f0(980) resonances appear as poles in the D matrix (they are dynamically generated)

Thus, the and f0(980) BW’s are removed and substituted by the previuos expression.

2/dof=2/152 , solid line

2

Dashed line, pole instead of BW 2/dof=3/152

Solid line, full results. The BW’s of the and f0(980) are removed, 2/dof=2/152

FSI are driven by the fixed scattering amplitudes from UCHPT in agreement with scattering experimental data

No background, in the Laurent expansion of D11 the background accompanying the pole is negligible (No Adler zero)

1.3 FSI in the Ds+! -++ decay

E791 Collaboration PRL86,765 (2001) 625 events

Isobar Model: f0(980), 0(770), f2(1270), f0(1370) with a f0(980) dominant contribution.

Mf0(980)=(977 § 4) MeV, gK= 0.02 § 0.05, g=0.09 § 0.01

gK>> g and gK compatible with zero !! In constrast with its proved

affinity to couple with strangeness sources, SU(3) analysis, etc

We employ our formalism to take into account FSI from UCHPT

2

Ds+! -++

Notice that the f0(980) resonant pole position is already fixed from scattering from UCHPT, as given in the D matrix

The and f0(980) poles were fixed in Oset,JAO, NPA620,435 (1997) in terms of just one free parameters plus CHPT at leading order.

1.4 FSI in the D+! K-+- decay

E791 Collaboration PRL 89,121801 (2002) 28400 events only 6% background

Similar situation to the D+ ! -++ case Isobar Model:

Without + : 2/dof=2.7 ! CL=10-11

With + : 2/dof=0.73 ! CL=95%

+ WITH +

BW(s)=(s-m2+i m (s) )-1 energy dependent width

BWBW does not follow the experimental S-wave I=1/2 phase shifts

M=797§47 MeV

=410 § 97 MeV

I=1/

2 K

Phas

e sh

ifts

(de

gree

s)

Abs

olut

e V

alue

Squ

are

BW

Laurent series around the pole position at the second Riemann sheet:

From the T-matrix of M.Jamin, A. Pich, J.A.O., NPB587,331 (2000)

UCHPT matching with U(3) CHPT +Resonances+large Nc constraints (vanishing of scalar form factors for s! 1)

K,K,K’ channels are included

Laurent series around the pole position at the second Riemann sheet:

Pole Pole

The movement of the phase of the pole follows the experimental phase shifts

We substitue the BW by

Adler zero: the reason why the background is so large in order to cancel the pole for low energies

background

BW BW

Dashed line, pole instead of BW 2/dof=6.5/132 s is fixed

At the same time the phase motion of the pole contribution follows the experimental phase shifts.

Points from E791 fit

GeV2

Eve

nts/

0.04

GeV

2

Full Final State Interactions (FSI) from the T-matrix of Jamin, Pich, J.A.O. NPB587,331 (2000)

We now have meson-meson intermediate states

The and K*0(1430) resonances appear as poles in the D matrix

Thus, the and K*0(1430) BW’s are removed and substituted by the

previuos expression.

2/dof=127/128 , solid and dashed lines

Solid line, full results. The BW’s of the and K*

0(1430) are removed

Dashed line: the K channel is removed. Stability.

FSI are driven by the fixed scattering amplitudes from UCHPT in agreement with scattering experimental data

No background, in the Laurent expansion of D11 the background accompanying the pole is negligible (No Adler zero)

Points from E791 fit

K*0(1430) E791... MK*

0(1430)= 1459 § 9 MeV

K*0(1430) = 175 § 17 MeV

PDG... MK*0(1430)= 1412 § 6 MeV

K*0(1430) = 294 § 23 MeV

Jamin,Pich,JAO (1430-1450,140-160) MeV' (M,/2)In their fit (6.10) (1450,142) MeVemployed here

1.5 Summary We have reproduced the E791 Collaboration signal distribution functions in terms of new parameterizations.

In E791 analyses the disagreement between the phase motions of the and and the elastic S-wave I=0,1/2 phase shifts is due to the employment of BW‘s.

Once these BW‘s are substituted by the pole contributions of these resonances the agreement is restored.

We have also reproduced the results of E791 making use of the full results of I=0,1/2 S-wave T-matrices in agreement with scattering and from Uunitarized CHPT.

The reason why the and pole contributions are not distorted in contrast with scattering is the absence of Adler zeroes.

The f0(980) from D decays turns out also with standard properties regarding its

coupling to kaon.

The width of the K*0(1430) from D decays is then in agreement with that from

scattering data and reported in the PDG. That from E791 analysis was a factor 2 smaller.

Dynamically generated resonances.

TABLE 1Spectroscopy:

I) II) III)

2. THE MIXING ANGLE OF THE LIGHTEST SCALAR NONET

Partial Waves from Oset,Oller PRD60,074023(99) UCHPT

mπ( )λ =mπ λ( )m0 mπ ; mK( )λ + =mK λ( )m0 mK ; mη( )λ + =mη λ( )m0 mη + m0=300 MeV

2.1 SU(3) Analyses

Continuous movement from a SU(3) symmetric point: from a SU(3) symmetric point with equal masses, which implies equal subtraction constants to the physical limit

octet

Singlet

a0(980)f0(980)

J.A.O. NPA727(03)353; hep-ph/0306031

2[0,1]

a0(980), : Pure Octet States

g( )a0KŪUK 1 , g( )a0

πη 8 , g( )κKπ12

From we calculate:

GeV 3.17.8|| 8 g

0.70§ 0.041.64 § 0.05 Table 1

2.5 § 0.25 Jamin,Pich,J.A.O NPB587(00)331

with -´ mixing1.10§ 0.03

This is an indication that systematic errors are larger than really shown in table 1 from the 3 T-matrices

f0(980) System: Mixing

Ideal Mixing: cos 2θ =2

3= 0.67

Morgan, PLB51,71(´74); f0(980), a0(980), (v1200), f0(v1100) cos2 = 0.13

Jaffe, PRD15,267(´77); f0(980), a0(980), (v900), f0(v700) cos2 = 0.33

Dual Ideal Mixing

Scadron, PRD26,239(´82); f0(980), a0(980), (800), f0(750) cos2 = 0.39

Napsuciale (hep-ph/9803396),+ Rodriguez Int.J.Mod.Phys.A16,3011(´01);

f0(980), a0(980), (900), f0(500) cos2 = 0.87

Black, Fariborz, Sannino, Schechter PRD59,074026(´99) (Syracuse group);

f0(980), a0(980), (800), f0(600) cos2 = 0.63

σ =ūuu u ŪUd d +

2; f0( )980 = ūussqq If

Standard Unitary Symmetry Model analysis in vector and tensor nonets of coupling constants.Okubo, PL5,165(´63)

Glashow,Socolow,PRL15,329(´65)

g( ! 88)=0

g1

g8

=8

5tan θ

I:g , gf0 KK g8=10.5§0.5GeV cos2=0.93§ 0.06

II: All couplings g8=6.8§2.3 GeV cos2=0.76§ 0.16

III: Fit,

multiplying errors

g8=8.2§0.8 GeV cos2=0.89§ 0.06

III: Fit, Global

error

g8=7.7§0.8 GeV cos2=0.74§ 0.10

III: Fit, 20% +

systematic error

g8=7.0§0.8 GeV cos2=0.69§ 0.10

From a0(980), g8=8.7§1.3 GeV

Paper

We average by taking the extrem values for each entry, x+(x) and x-(x):

12 numbers g8

10 numbers cos2

g8 = 8.2 § 1.8 GeV

cos2=0.80§0.15

||: (13-34.4)°

g1 = 4.69§2.7GeV

Removing the values from Strategy I (higher

than the rest) A) g8=7.7§ 1.7 22% , cos2=0.76§ 0.15 20%, ||: (17-39)°

Previous result

B) g8 = 8.2 § 1.8 GeV 21% cos2=0.80 § 0.15 18% ||: (13-34.4)°

g =2.94-3.01g KK =1.09-1.30g =0.04-0.09

g =4.0 § 1.4g KK =1.2§ 0.4g =0.

g =3.6§1.5g KK =1.0§0.4g =0

f0(980) gf0 =0.89-1.33

gf0 KK =3.59-3.83gf0 =2.61-2.85

gf0 =2.4§1.1

gf0 KK =3.5§0.9gf0 =2.81§0.6

gf0 =3.0§1.1

gf0 KK =3.4§0.8gf0 =2.9§0.6

a0(980) ga0 =3.67-4.08

ga0 KK =5.39-5.60ga0

=3.4§1.1ga0 KK =4.22§1.1

ga0 =3.7§1.4

ga0 KK =4.5§1.4

g K =4.7-5.02g K =2.96-3.10

g K =5.2§1.6g K =1.7§0.5

g K =5.5§1.7g K =1.8§0.6

A) g8=7.7§1.7 GeV 22% , cos2=0.76§0.15 20%, : (17-39)°

g1=5.5§2.3

1. The is mainly the singlet state. The f0(980) is mainly the I=0 octet state.

2. Very similar to the mixing in the pseudoscalar nonet but inverted.

Octet ! Singlet ; ´ Singlet ! f0(980) Octet. In this model is positive.

3. Scalar QCD Sum rules, Bijnens, Gamiz, Prades, JHEP 0110 (01) 009.

LINEAR MASS RELATION

Scalar Form Factors I=0

U.-G. Meissner, J.A.O NPA679(00)671

2.3 Sign of

At the f0(980) peak. Clearly the f0(980) should be mainly strange.

A) >0 ratio= 5.0§3 B) <0 ratio= 0.3§0.2

In U(3) symmetry

2.4 Conclusions

1. cos2=0.77§ 0.15; =28 § 11; is mainly a singlet and the f0(980) is mainly the I=0 octet state. g8=7.7§ 1.7, g1=5.5§2.3 GeV

2. These scalar resonances satisfy a Linear Mass

Relation.

3. The value of cos2 is compatible with:– Ideal mixing

– U(3)xU(3) with UA(1) breaking model of Napsuciale from the U(2)xU(2) model of t´Hooft,hep-ph/9803396

– Syracuse group, Black,Fariborz,Sannino,Schechter PRD59(‚99)074026

3. Order p6 Chiral Couplings from the Scalar K Form Factor

• DECAYS• UNITARITY OF THE CABIBBO-

KOWAYASHI-MASKAWA MATRIX|Vud|2+|Vus|2+|Vub|2=1

M=GF

2Vus

* Lμ CK [ ]F+( )t ( )pK pπ + μ

F_( )t ( )p k pπ μ +

F0( )t = F+( )tt

MK2 Mπ

2F_( )t +

3.1 Kl3 Decays: K l+ ν

t=(pK - pπ )2 ; Lμ = ūuu( )p ν γμ ( )1 γ5 ν( )p l

CK=1

21, for ( )K ,K0 +

F+(t) is the vector K π form factor

F0(t) is the scalar K π form factor

< π( )pπ |ūus γμu |K( )pK >

F+(0)= F0(0)

K+

•From ΓK one can obtain |Vus| , once F+(0) is known Leutwyler, Roos

Z.Phys C25(´84)91

•New high precission experiments on Ke3 are ongoing or prepared (CMD2,

NA48, KTEV, KLOE) to improve ΓK

Decay Rate

F0( )t = F+( )0 1 λ0

t

Mπ2 + c0

t2

Mπ4 +

F+( )t = F+( )0 1 λ+

t

Mπ2 + c+

t2

Mπ4 + m l

2 t ( )MK Mπ2

IK results from integrating over

phase space the form factor dependence on t

ΓK = CK2

GF2 MK

5

192π3S ew |Vus |

2 |F+(0)|2 IK

•From BT, F+(0) only depends on two new order p6 chiral counterterms: Cr

12, Cr34

•The same countererms are also the only ones that appear in the slope and the curvature of F0(t), and hence by a knowledge of F0(t) one can determine those counterterms.

This is our aimSo that an order p6 calculation of F+(0) becomes feassible

•From ΓK one can obtain |Vus| , once F+(0) is known Leutwyler, Roos

Z.Phys C25(´84)91

•New high precission experiments on Ke3 are ongoing or prepared (CMD2,

NA48, KTEV, KLOE) to improve ΓK

•O(p6) in CHPT isospin limit Post, Schilcher, EPJC25(´02)427;

Bijnens, Talavera NPB699(´03)341 (BT)

3.2 O(p6) Counterterms ( Cri ), F+(0) , FK / Fπ

F+(t) , F0(t) calculated recently at order p6 in CHPT .

Post,Silcher EPJC25(´02)427, Bijnens, Talavera NPB669(´03)341 (BT03)

Some differences were found in both calculations for t0.

We follow BT03 that employs the O(p6) CHPT Lagrangian of Bijnens,Colangelo,Ecker NPB508(´97)263

F+( )0 = 1 Δ( )0 + 8

Fπ4

C12r C34

r + ΔKπ2

?

Lir O(p4);

No Cri

Lir from

Amoros,Bijnens,Talavera NPB602(01)87

Δ( )0 = 0.00800.0057 [ ]loops 0.0028 L ir +

= 0.0080 0.0064

C12r C34

r + = F0´( )0 ŪUΔ 1

FK Fπ 1

ΔKπ

8ΣKπ

Fπ4

C12r

Fπ4

8ΣKπ

C12r = [ ]2ŪUΔ 2 F0´´( )0

Fπ4

16

F0( )t = F+( )0FK Fπ 1

ΔKπ

t + 8

Fπ4

2C12r C34

r + ΣKπ t + 8

Fπ4C12

r t2 ŪUΔ ( )t +

Momentum dependence ΔKπ = MK2 Mπ

2 ΣKπ = MK2 Mπ

2 +

ŪUΔ ( )t = ŪUΔ 1( )t t ŪUΔ 2( )t t2 + ŪUΔ 3( )t t3 + O( )t4 +

= 0.259( )9 t 0.840( )31 t2 + 1.291( )170 t2 + Dependence on Lr

i not on Cr

i

rr CC 3412 and Can be fixed from the t-dependence, slope and curvature, of the scalar form factor F0(t)=FK(t)

3

3.3 Strangeness Changing Scalar K π , K η´ Form Factors

<P|μ ūus γμu |K> = i( )ms mu <P|ūus u|K> = iΔKπ

2CKPFKP( )t

P= π, η´, η

F0(t)FKπ(t)

Jamin, Pich, J.A.O. NPB587(´00)331 (JOP00): I=1/2 S-wave T-matrix coupling toghether Kπ, K , K ´

Jamin, Pich, J.A.O. NPB622(´02)279 (JOP02): The scalar form factors FK and FK´

ρKQ( )t =pKQ

8π sθ t ( )MK MQ + 2ImFKP( )t =

Q

FKQ ρKQ TKP,KQ*

Unitarity relates the scalar form factors with the I=1/2 meson-meson scalar amplitudes

The K turns out to be negligible, both experimentally as well as theoretically, and we neglect it in the following.

The form factors satisfy the dispersion

relations:

F1( )s =1

πs th

ds´ρ1( )s´ F1( )s´ T11

* ( )s´

s´ s iε1

πs th

ds´ρ2( )s´ F2( )s´ T12

* ( )s´

s´ s iε +

F2( )s =1

πs th

ds´ρ1( )s´ F1( )s´ T12

* ( )s´

s´ s iε1

πs th

ds´ρ2( )s´ F2( )s´ T22

* ( )s´

s´ s iε +

This is the so called Muskhelishvili-Omnès problem in coupled channels solved in Jamin, Pich, J.A.O. NPB622(´02)279 (JOP02)

For only one channel (elastic case), namely K, one has the Omnès solution:

F1( )s = F1( )0 exps

πs th

ds´δ1( )s´

s´( )s´ s iε

F+( )0 = 1 Δ( )0 + 8

Fπ4

C12r C34

r + ΔKπ2

C12r C34

r + = F0( )0 G11(0)´ ŪUΔ 1 ΣKπŪUΔ 2

FK Fπ 1

ΔKπ

ΣKπF0( )0 G11(0)´´ 2 + Fπ

4

8ΣKπ

m=1: Fits 6.10K1, 6.11K1

Only one vanishing independent solution G1(s)={G11(s), G12(s)}

Normalized: G11(0) =1 F0(s) = F0(0) G11(s) , F0(s)´=F0(0) G11(s)´ , F0(s)´´=F0(0) G11(s)´´

C12r = [ ]2ŪUΔ 2 F0( )0 G11(0)´´

Fπ4

16

But F0(0) appears as well as a necessary input... CONSISTENCY

We solve for F+(0)=F0(0)

F+(0)|C =8

Fπ4

C12r C34

r + ΔKπ2

G11(0)´ = 0.803 , G11(0)´´= 1.661

1ΔKπ

2

ΣKπ

G11(0)´ G11( )0 ´´ΣKπ

2 + +

1

F+( )0 = 1 Δ( )0 + ΔKπ

2

ΣKπ

ŪUΔ 1 ΣKπŪUΔ 2 + FK Fπ 1

ΔKπ

+ +

GeV-2 GeV-4

F0(0) = 0.979 0.009

Cr12=(2.8 2.9) 10-7

Cr12+Cr

34=(2.4 1.6) 10-6

F+(0)|C= - 0.0130.009 The presently (UP TO NOW) estimated value is the one from Leutwyler&Ross ZPC25(´84)91

F+(0)|CLR= - 0.0160.008

Model depedent calculation based on the overlap between the Kaon and Pion quark wave functions in the light cone. Only analytic piece, no loops. Strong Scale dependence in F+(0)|C unknown in LR value, now it is known and we give our values at =M

1ΔKπ

2

ΣKπ

G11(0)´ G11( )0 ´´ΣKπ

2 + +

1

F+( )0 = 1 Δ( )0 + ΔKπ

2

ΣKπ

ŪUΔ 1 ΣKπŪUΔ 2 + FK Fπ 1

ΔKπ

+ +

To be improved...

We take FK/F = 1.22 0.01 from Leutwyler, Roos ZPC25(´84)91

(PDG source) F0(0) = 0.979 0.009.

LR84 already employed F+(0) (O(p4) CHPT + F+(0)|CLR )

Our results for F+(0)|C is biassed by the previous F+(0)|CLR

FK/F = 1.20 F+(0)|C= 0.0280.009 F0(0) = 0.965 0.009

FK/F = 1.24 F+(0)|C= + 0.0010.009 F0(0) = 0.993 0.009One needs an independent determination of FK/F ,maybe lattice

F0(0) = 0.979 0.009

F0(0)=0.976 0.010 BT03 employing F+(0)|CLR

However, our result for the counterterms agree with that of Leutwyler,Roos: CONSISTENCY CHECK – DIFFERENT METHODS.

WE HAVE FIXED THE PROBLEM REGARDING THE STRONG SCALE DEPENDENCE OF SUCH COUNTERTERM

3.4 CONCLUSIONS

• Above threshold and on the real axis (physical region), a partial wave amplitude must fulfill because of unitarity:

General Expression for a Partial Wave Amplitude

ImT ij =k T ik ρk Tkj* ImT 1

ij= ρ i δ ijUnitarity Cut

W=s

We perform a dispersion relation for the inverse of the partial wave (the unitarity cut is known)

T ij1=R ij

1 δ ij g( )s0 i

s s0

π ρ( )s´ ids´

( )s´ s i0+ ( )s´ s0

+

The restg(s): Single unitarity bubble

g(s)= g( )s =1

4π 2aSL σ( )s log

σ( )s 1

σ( )s 1 + +

σ( )s =2 q

sT= ( )R 1 g( )s + 1

T obeys a CHPT/alike expansion

R is fixed by matching algebraically with the CHPT/alike CHPT/alike+Resonances

expressions of T

In doing that, one makes use of the CHPT/alike counting for g(s)

The counting/expressions of R(s) are consequences of the known ones of g(s) and T(s)

The CHPT/alike expansion is done to R(s). Crossed channel dynamics is included perturbatively.

The final expressions fulfill unitarity to all orders since R is real in the physical region (T from CHPT fulfills unitarity pertubatively as employed in the matching).

The re-scattering is due to the strong „final“ state interactions from some „weak“ production mechanism.

Production Processes

ImFi =k Fk ρk Tki*

We first consider the case with only the right hand cut for the strong interacting amplitude, is then a sum of poles (CDD) and a constant. It can be easily shown then:

1R

F=( )I Rg( )s + 1ξ

Finally, ξ is also expanded pertubatively (in the same way as R) by the matching process with CHPT/alike expressions for F, order by order. The crossed dynamics, as well for the production mechanism, are then included pertubatively.

Finally, ξ is also expanded pertubatively (in the same way as R) by the matching process with CHPT/alike expressions for F, order by order. The crossed dynamics, as well for the production mechanism, are then included pertubatively.

LET US SEE SOME APPLICATIONS

Meson-Meson Scalar Sector

1) The mesonic scalar sector has the vacuum quantum

numbers . Essencial for the study of Chiral Symmetry

Breaking: Spontaneous and Explicit .

2) In this sector the mesons really interact strongly.

1) Large unitarity loops.

2) Channels coupled very strongly, e.g. π π- , π η- ...

3) Dynamically generated resonances, Breit-Wigner formulae, VMD, ...

3) OZI rule has large corrections.

1) No ideal mixing multiplets.

2) Simple quark model.

Points 2) and 3) imply large deviations with respect to Large Nc QCD.

0

mmm sdu,,

KK KK,

4) A precise knowledge of the scalar interactions of the

lightest hadronic thresholds, π π and so on, is often

required.– Final State Interactions (FSI) in ´/ , Pich, Palante,

Scimemi, Buras, Martinelli,...

– Quark Masses (Scalar sum rules, Cabbibo suppressed Tau decays.)

– Fluctuations in order parameters of SSB.

4) A precise knowledge of the scalar interactions of the

lightest hadronic thresholds, π π and so on, is often

required.– Final State Interactions (FSI) in ´/ , Pich, Palante,

Scimemi, Buras, Martinelli,...

– Quark Masses (Scalar sum rules, Cabbibo suppressed Tau decays.)

– Fluctuations in order parameters of SSB.

Let us apply the chiral unitary approach– LEADING ORDER:

T= ( )R 1 g( )s + 1

T=T2=R2 R2gR2 .. + g is order 1 in CHPT R=R2=T2

Oset, Oller, NPA620,438(97) aSL-0.5 only free parameter,

equivalently a three-momentum cut-off 0.9 GeV

a0( )980

f0( )980

σ

All these resonances were dynamically generated from the lowest order

CHPT amplitudes due to the enhancement of the unitarity loops.

(GeV) )980(0a(GeV) )980(

0f

90.1|| g

f

3.80 || gf

KK

54.3|| g

a

20.5|| ga

KK

012.0993.0 i 056.0009.1 i

Pole positions and couplings

Br( )a0( )980 πη =0.63Br( )f0( )980 ππ =0.70

In Oset,Oller PRD60,074023(99) we studied the I=0,1,1/2 S-waves.The input included next-to-leading order CHPT plus resonances:1. Cancellation between the crossed channel loops and crossed channel resonance exchanges. (Large Nc violation).2. Dynamically generated renances. The tree level or preexisting resonances move higher in energy (octet around 1.4 GeV). Pole positions were very stable under the improvement of the kernel R (convergence).3. In the SU(3) limit we have a degenerate octet plus a singlet of dynamically generated resonances σ, f0(980), a0(980), κ( )700

Using these T-matrices we also corrected by Final State

Interactions the processes Where the input comes from CHPT at one loop, plus resonances. There were some couplings and counterterms but were taken from the literature. No fit parameters.Oset, Oller NPA629,739(98).

γγ π+π - ,π0π0 ,πη,K+K- ,K0ŪUK 0

1

M2 q2 =1

q2

q2

M4 + q4

M6 + ... q2 + < < M2

Resonances give rise to a resummation of the chiral series at the

tree level (local counterterms beyond O( ).

The counting used to perform the matching is a simultaneous one in the number of loops calculated at a given order in CHPT (that increases order by

order). E.g:– Meissner, J.A.O, NPA673,311 (´00) the πN scattering was

studied up to one loop calculated at O( ) in HBCHPT+Resonances.

CHPT+Resonances

Ecker, Gasser, Pich and de Rafael, NPB321, 311 (´98)

4p

3p

LS ππ =S ūuc mTr( )Χ+ ūuc dTr uμuμ +

– Jamin, Pich, J.A.O, NPB587, 331 (´00), scattering.

• The inclusion of the resonances require the knowlodge of

their bare masses and couplings, that were fitted to experiment

A theoretical input for their values would be very welcome: – The CHUA would reduce its freedom and would increase its

predictive power.

– For the microscopic models, one can then include the so important

final state interactions that appear in some channels, particularly in the

scalar ones. Also it would be possible to identify the final physical poles

originated by such bare resonances and to work simultaneously with

those resonances dynamically generated.

Kπ, Kη, Kη´

LSNN= g sNNŪUΨΨS