José A. Oller Univ. Murcia, Spain Scalar Dynamics Final State Interactions in Hadronic D decays José A. Oller Univ. Murcia, Spain Introduction FSI in the D +   -  +  + decay FSI in the D s +   -  +  + decay FSI in the D +   -  +  + decay Mixing Angle of the Lightest Scalar Nonet Determination of CHPT Counterterms Bad Honnef, January

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 f 0 (600) or  E791 Collaboration PRL 86, 770 (2001) D +   -  +  +, and the K * 0 (800) or  mesons E719 Collaboration PRL 89, (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 f 0 (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) s 12 =(p 1 +p 2 ) 2, s 13 =(p 1 +p 3 ) 2 a 0 e i  0 : Non-Resonant Term A n : 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), f 0 (980), f 2 (1270), f 0 (1370),  0 (1450), 

BW  (s)=(s-m  2 +i m   (s) ) -1 energy dependent width ! BW 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:  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

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 f 0 (980) resonances appear as poles in the D matrix (they are dynamically generated) Thus, the  and f 0 (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 f 0 (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 D 11 the background accompanying the  pole is negligible (No Adler zero)

1.3 FSI in the D s + !  -  +  + decay E791 Collaboration PRL86,765 (2001) 625 events Isobar Model: f 0 (980),  0 (770), f 2 (1270), f 0 (1370) with a f 0 (980) dominant contribution. M f 0 (980) =(977 § 4) MeV, g K = 0.02 § 0.05, g  =0.09 § 0.01 g K >> g  and g K 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+! -++Ds+! -++ Notice that the f 0 (980) resonant pole position is already fixed from scattering from UCHPT, as given in the D matrix The  and f 0 (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, (2002) events only 6% background Similar situation to the D + !  -  +  + case   Isobar Model: Without   + :  2 /dof=2.7 ! CL= With   + :  2 /dof=0.73 ! CL=95%

  +  WITH   +

BW  (s)=(s-m  2 +i m   (s) ) -1 energy dependent width BW BW does not follow the experimental S-wave I=1/2  phase shifts M  =797 § 47 MeV   =410 § 97 MeV I=1/2 K  Phase shifts (degrees) Absolute Value Square 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 N c 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 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

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 GeV 2 Events/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 D 11 the background accompanying the  pole is negligible (No Adler zero) Points from E791 fit

K * 0 (1430) E M K * 0 (1430) = 1459 § 9 MeV  K * 0 (1430) = 175 § 17 MeV PDG... M K * 0 (1430) = 1412 § 6 MeV  K * 0 (1430) = 294 § 23 MeV Jamin,Pich,JAO ( , ) MeV ' (M,  /2) In their fit (6.10) (1450,142) MeV employed 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 f 0 (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 1 Spectroscopy: I) II)III) 2. THE MIXING ANGLE OF THE LIGHTEST SCALAR NONET Partial Waves from Oset,Oller PRD60,074023(99) UCHPT

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   a 0 (980) f 0 (980) J.A.O. NPA727(03)353; hep-ph/ [0,1]

a 0 (980),  : Pure Octet States From we calculate: 0.70 § § 0.05 Table § 0.25 Jamin,Pich,J.A.O NPB587(00)331 with  -  ´ mixing 1.10 § 0.03 This is an indication that systematic errors are larger than really shown in table 1 from the 3 T-matrices

 f 0 (980) System: Mixing Ideal Mixing: Morgan, PLB51,71(´74); f 0 (980), a 0 (980),  ( v 1200), f 0 ( v 1100) cos 2  = 0.13 Jaffe, PRD15,267(´77); f 0 (980), a 0 (980),  ( v 900), f 0 ( v 700) cos 2  = 0.33 Dual Ideal Mixing Scadron, PRD26,239(´82); f 0 (980), a 0 (980),  (800), f 0 (750) cos 2  = 0.39 Napsuciale (hep-ph/ ),+ Rodriguez Int.J.Mod.Phys.A16,3011(´01); f 0 (980), a 0 (980),  (900), f 0 (500) cos 2  = 0.87 Black, Fariborz, Sannino, Schechter PRD59,074026(´99) (Syracuse group); f 0 (980), a 0 (980),  (800), f 0 (600) cos 2  = 0.63

Standard Unitary Symmetry Model analysis in vector and tensor nonets of coupling constants. Okubo, PL5,165(´63) Glashow,Socolow,PRL15,329(´65) g(  !  8  8 )=0

I:g  , g f 0 KK g 8 =10.5 § 0.5GeV cos 2  =0.93 § 0.06 II: All couplings g 8 =6.8 § 2.3 GeV cos 2  =0.76 § 0.16 III: Fit, multiplying errors g 8 =8.2 § 0.8 GeV cos 2  =0.89 § 0.06 III: Fit, Global error g 8 =7.7 § 0.8 GeV cos 2  =0.74 § 0.10 III: Fit, 20% + systematic error g 8 =7.0 § 0.8 GeV cos 2  =0.69 § 0.10 From a 0 (980),  g 8 =8.7 § 1.3 GeV Paper We average by taking the extrem values for each entry, x+  (x) and x-  (x): 12 numbers g 8 10 numbers cos 2  g 8 = 8.2 § 1.8 GeV cos 2  =0.80 § 0.15 |  |: ( )° g 1 = 4.69 § 2.7GeV

Removing the values from Strategy I (higher than the rest) A)g 8 =7.7 § %, cos 2  =0.76 § %, |  |: (17-39)° Previous result B) g 8 = 8.2 § 1.8 GeV 21% cos 2  =0.80 § % |  |: ( )°  g   = g  KK = g   = g   =4.0 § 1.4 g  KK =1.2 § 0.4 g   =0. g   =3.6 § 1.5 g  KK =1.0 § 0.4 g   =0 f 0 (980) g f 0  = g f 0 KK = g f 0  = g f 0  =2.4 § 1.1 g f 0 KK =3.5 § 0.9 g f 0  =2.81 § 0.6 g f 0  =3.0 § 1.1 g f 0 KK =3.4 § 0.8 g f 0  =2.9 § 0.6 a 0 (980) g a 0  = g a 0 KK = g a 0  =3.4 § 1.1 g a 0 KK =4.22 § 1.1 g a 0  =3.7 § 1.4 g a 0 KK =4.5 § 1.4  g  K  = g  K  = g  K  =5.2 § 1.6 g  K  =1.7 § 0.5 g  K  =5.5 § 1.7 g  K  =1.8 § 0.6

A) g 8 =7.7 § 1.7 GeV 22%, cos 2  =0.76 § %,  : (17-39)° g1=5.5 § The  is mainly the singlet state. The f 0 (980) is mainly the I=0 octet state. 2.Very similar to the mixing in the pseudoscalar nonet but inverted.  Octet !  Singlet ;  ´ Singlet ! f 0 (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) Sign of  At the f 0 (980) peak. Clearly the f 0 (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.cos 2  =0.77 § 0.15;  =28 § 11;  is mainly a singlet and the f 0 (980) is mainly the I=0 octet state. g 8 =7.7 § 1.7, g 1 =5.5 § 2.3 GeV 2.These scalar resonances satisfy a Linear Mass Relation. 3.The value of cos 2  is compatible with: –Ideal mixing –U(3)xU(3) with U A (1) breaking model of Napsuciale from the U(2)xU(2) model of t´Hooft,hep-ph/ –Syracuse group, Black,Fariborz,Sannino,Schechter PRD59(‚99)074026

3. Order p 6 Chiral Couplings from the Scalar K  Form Factor DECAYS UNITARITY OF THE CABIBBO- KOWAYASHI-MASKAWA MATRIX |V ud | 2 +|V us | 2 +|V ub | 2 =1

3.1 K l3 Decays: K   l + ν t=(p K - p π ) 2 ; F + (t) is the vector K π form factor F 0 (t) is the scalar K π form factor F + (0)= F 0 (0)

From Γ K one can obtain |V us |, once F + (0) is known Leutwyler, Roos Z.Phys C25(´84)91 New high precission experiments on K e3 are ongoing or prepared (CMD2, NA48, KTEV, KLOE) to improve Γ K Decay Rate I K results from integrating over phase space the form factor dependence on t

From BT, F + (0) only depends on two new order p 6 chiral counterterms: C r 12, C r 34 The same countererms are also the only ones that appear in the slope and the curvature of F 0 (t), and hence by a knowledge of F 0 (t) one can determine those counterterms. This is our aim So that an order p 6 calculation of F + (0) becomes feassible From Γ K one can obtain |V us |, once F + (0) is known Leutwyler, Roos Z.Phys C25(´84)91 New high precission experiments on K e3 are ongoing or prepared (CMD2, NA48, KTEV, KLOE) to improve Γ K O(p 6 ) in CHPT isospin limit Post, Schilcher, EPJC25(´02)427; Bijnens, Talavera NPB699(´03)341 (BT)

3.2 O(p 6 ) Counterterms ( C r i ), F + (0), F K / F π F + (t), F 0 (t) calculated recently at order p 6 in CHPT. Post,Silcher EPJC25(´02)427, Bijnens, Talavera NPB669(´03)341 (BT03) Some differences were found in both calculations for t  0. We follow BT03 that employs the O(p 6 ) CHPT Lagrangian of Bijnens,Colangelo,Ecker NPB508(´97)263 ? L i r O(p 4 ); No C r i L i r from Amoros,Bijnens,Talavera NPB602(01)87 =  

Momentum dependence Dependence on L r i not on C r i Can be fixed from the t-dependence, slope and curvature, of the scalar form factor F 0 (t)=F K  (t)

3.3 Strangeness Changing Scalar K π, K η´ Form Factors P= π, η´, η F 0 (t)  F Kπ (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 F K  and F K  ´ 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: 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:

m=1: Fits 6.10K1, 6.11K1 Only one vanishing independent solution G 1 (s)={G 11 (s), G 12 (s)} Normalized: G 11 (0) =1  F 0 (s) = F 0 (0) G 11 (s), F 0 (s)´=F 0 (0) G 11 (s)´, F 0 (s)´´=F 0 (0) G 11 (s)´´ But F 0 (0) appears as well as a necessary input... CONSISTENCY We solve for F + (0)=F 0 (0)

GeV -2 GeV -4 F 0 (0) =  C r 12 =(2.8  2.9) C r 12 +C r 34 =(2.4  1.6) F + (0)| C =  The presently (UP TO NOW) estimated value is the one from Leutwyler&Ross ZPC25(´84)91 F + (0)| C LR =   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 

To be improved... We take F K /F  = 1.22  0.01 from Leutwyler, Roos ZPC25(´84)91 (PDG source)  F 0 (0) =  LR84 already employed F + (0) (O(p 4 ) CHPT + F + (0)| C LR ) Our results for F + (0)| C is biassed by the previous F + (0)| C LR F K /F  = 1.20  F + (0)| C =   F 0 (0) =  F K /F  = 1.24  F + (0)| C =  F 0 (0) =  One needs an independent determination of F K /F ,maybe lattice F 0 (0) =  F 0 (0)=0.976  BT03 employing F + (0)| C LR

 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 Unitarity Cut W=  s We perform a dispersion relation for the inverse of the partial wave (the unitarity cut is known) The rest g(s): Single unitarity bubble

g(s)= 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 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:

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.

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 S  SB.

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 S  SB. Let us apply the chiral unitary approach –LEADING ORDER: g is order 1 in CHPT Oset, Oller, NPA620,438(97) a SL  -0.5 only free parameter, equivalently a three-momentum cut-off   0.9 GeV

σ

All these resonances were dynamically generated from the lowest order CHPT amplitudes due to the enhancement of the unitarity loops. Pole positions and couplings

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

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).

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)

–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.