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					      Mixing and CPV in the D System
   Mixing in the D system
   Time-integrated CPV and new physics
   Mixing in neutral D system
      Time-dependent
      Time-integrated
   Summary




Charm 2007, Ithaca, NY, 8/06/2007   Brian Meadows, U. Cincinnati
                         Mixing Parameters
   Mixing in the D system arises from the existence of two
    mass eigenstates D1 and D2


     with eigenvalues


   It is usually defined by four parameters:


     where M = (m1+m2) / 2 and  = (1+2) / 2




Charm 2007, Ithaca, NY, 8/06/2007         Brian Meadows, U. Cincinnati
                    Decay Rates and CPV
   Decay amplitudes                 and                  evolve at time t to be




     where

   With no CPV in either mixing or decay
     we expect that                 and                where
      is the strong phase difference between decays                and

   With CPV in direct decay                      or in mixing
     we expect that                    where

Charm 2007, Ithaca, NY, 8/06/2007              Brian Meadows, U. Cincinnati
CPV and Mixing in Charm Meson Decays


   Time-integrated CPV and new physics
   Mixing in neutral D system
      Time-dependent
      Time-integrated
   Summary




Charm 2007, Ithaca, NY, 8/06/2007   Brian Meadows, U. Cincinnati
                Time-integrated CPV results


   Recent results:
      D0 → K+K−, π+π−
      D0 → K+K−π0, π+π−π0
   Older result (not covered here):
      D0 → K+K−π+ -- PRD 71, 091101 (2005)




Charm 2007, Ithaca, NY, 8/06/2007      Brian Meadows, U. Cincinnati
                         CPV in D0 → K+K−/π+π−
     CP asymmetry:
Experimentally tricky to measure with per-mille
systematics:
                            Tagging efficiency asymmetry for soft pion in D*+
                            → D0 π+ studied with control sample of D0 → K−π+
                            events.
                            •Crucial to get this from data, not MC!
                            •Control sample corrected for K+/K− and π+/π− efficiency
                            asymmetry as function of polar angle and momentum.


            MC simulation
                             Forward-backward production asymmetry
                             •From Z/γ interference & higher-order QED diagrams
                             •These effects are odd in cos(θ*)
                             •CP asymmetry is even in cos(θ*)
                             •... so measure aCP in bins of |cos(θ*)| & odd terms vanish
                                                                           385/fb, PRL 100,061803
                                                                           (2008)
 Charm 2007, Ithaca, NY, 8/06/2007                    Brian Meadows, U. Cincinnati
                         CPV in D0 → K+K−/π+π−
Plotting CP asymmetry in bins of |cos(θ*)|:
                                                           Systematics




                                               Last bin excluded
                                                    (due to
                                                 acceptance)




Results are consistent with zero CP asymmetry:


                                                          385/fb, PRL 100,061803
                                                          (2008)
 Charm 2007, Ithaca, NY, 8/06/2007   Brian Meadows, U. Cincinnati
                  CPV in D0 → K+K−π0/π+π−π0
•Move to three-body mode -- we now have more tools:
•Look for rate asymmetry in bins of |cos(θ*)| as before
•Look for asymmetry in distribution.
•Second point is crucial -- CP asymmetry may pop up in
one corner of phase space or in one intermediate
resonance.
•Remember: Direct CPV is not universal.
•Localized asymmetry may be washed out -- or even cancelled --
when looking at integral over whole phase space.
•Several ways used to check for distribution asymmetry:
•Bin-by-bin difference in normalized Dalitz plot (model-
independent)
•Difference in angular moments (model-independent)
•Differences in amplitudes & phases of components in Dalitz plot fit
 Charm 2007, Ithaca, NY, 8/06/2007     Brian Meadows, U. Cincinnati
                 CPV in D0 → K+K−π0/π+π−π0
    Look for distribution                         Angular distribution asymmetry
                                                    (first three Legendre polynomial moments only
    asymmetry in normalized                                          shown here):
    Dalitz plots:




       Efficiency-corrected Dalitz plots


         P(χ2) = 32.8%




                              P(χ2) = 16.6%

                                              385/fb, arXiv:0802.4035
             Normalized residuals              accepted by PRD-RC

  No evidence of CP violation found
Charm 2007, Ithaca, NY, 8/06/2007
                                                     No evidence of CP violation fo
                                                     Brian Meadows, U. Cincinnati
                  CPV in D0 → K+K−π0/π+π−π0
Asymmetries in Dalitz plot           385/fb, arXiv:0802.4035   Asymmetries in phase-
fits?  0     + − 0
                                      accepted by PRD-RC
                                                               space-integrated rates?
           D →π π π




           D0 → K+K−π0
                                                 No evidence of CP violation found
                                            c.f. Belle:
                                           [arXiv:0801.2439, 532/fb]

                                                   Thus, no evidence for CP
                                                   violation found in any of the
No evidence of CP violation found                  four tests. U. Cincinnati
 Charm 2007, Ithaca, NY, 8/06/2007                    Brian Meadows,
     Low MassS-wave K and  Systems
                                    Brian Meadows
                                    University of Cincinnati


       S- waves in heavy flavour physics ?
       What is known about S- wave -+ and K -+
        scattering and how this should apply to D decays
       Measurements of S- wave component
           D  K -++
           Other modes

       Summary

Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
       S-waves in Heavy Flavour physics ?
   Low mass K and  S- wave systems are of intrinsic interest and important for
    understanding the spectroscopy of scalar mesons – existence of low mass  or 
    states in particular
      This is not covered in this talk, though a review of recent theoretical and experimental
       efforts focussing on pole parameters for  (476–628)− i (226–346) and of  (694-
       841)-i(300-400) MeV/c2 cites many of the relevant references:
         D. V. Bugg, J. Phys. G 34, 151 (2007).


   The S- wave is also both ubiquitous and “useful”
      Interference in hadronic final states through Dalitz plot analyses plays a major role in
       studying much that is new in flavour physics:
           CKM 
           D0-D0 mixing
           Sign of cos2, etc….

   General belief is that P- and D- waves are well described by resonance
    contributions, but that better ways to parameterize the S- wave systems are
    required as our targets become more precise.
      This talk focusses on recent attempts to improve on this situation.


Charm 2007, Ithaca, NY, 8/06/2007                      Brian Meadows, U. Cincinnati
                 What is Known about K Scattering ?
                  SLAC/LASS experiment E135: K -p  K -+n (11 GeV/c)          NPB 296, 493 (1988)


                                                                               +++   Total S-wave
                                                                               +++   I = 1/2
                    L=0                             L=0
 Phase degrees




                                                                               +++   I = 3/2
   Phase 0




                                             |T |
                                                                                      I =3/2 Phase 0



                                          |T |
                                                                                           K +p  K ++n




                     M (K -+) (GeV/c2)             M (K -+) (GeV/c2)
                                                                                           K -p  K –-D++


                   I- spins are separated using I=3/2 phases from
                     K +p  K ++n and K -p  K –-D++ (13 GeV/c)
                                                                                      M (K§ §) (GeV/c2)

            No evidence for (800) – yet ~no data below 825 MeV/c2 either.     Estabrooks, et al, NP B133, 490 (1978)


Charm 2007, Ithaca, NY, 8/06/2007                            Brian Meadows, U. Cincinnati
       Effective Range Parametrization (LASS)
                                                                NPB 296, 493 (1988)


   Scattering amplitude is unitary (elastic) up to K’ threshold
    (for even L):
                                                  where:

      S-wave (I = 1/2):                S-wave (I = 3/2):



                                           No resonances:
       One resonance:
                                       a   “scattering lengths”
         M0 ~1435 ; 0 ~275   MeV/c2   b   “effective ranges”




   Strictly, only valid below ~1460 MeV/c2.

Charm 2007, Ithaca, NY, 8/06/2007          Brian Meadows, U. Cincinnati
                            S-wave Scattering (I = 0)
                  Excellent Data from - p  - + n
                       G. Greyer, et al, NP B75, 189-245 (1975)
                      (several analyses - including other reactions)

                                                                                      B. Hyams, et al, NP B64, 134 (1973)

                                         I=0                                                           Im T
00 (degrees)




                                                         Au, Morgan,
                                                     Pennington, PR D35,
                                                      1633-1664 (1987)




                                                        PT
                                                                                    KK                                Re T
                                                                                 Threshold
                                   KK
                                Threshold

                                                    M() (MeV/c2

                No evidence for (500) – essentially no data below 500 MeV/c2 either.

Charm 2007, Ithaca, NY, 8/06/2007                                          Brian Meadows, U. Cincinnati
                    S-wave Scattering (I = 2)
                 from N. Achasov and G. Shestakov, PRD 67, 243 (2005)

                                                       Data included in fit:
                                                        + p  + + n (12.5 GeV/c)
                               02                         W. Hoogland, et al, NP B69,
     (degrees)




                                                           266-278 (1974)

                                                          + d  - - ppspec
     0 2




                                                                                 (9 GeV/c)
                                                           N. Durusoy, et al, PL B45, 517-
                                                           520 (1973)

                                                       NOTE - 02 is negative.


    Fit assumes amplitude to be unitary:
                                               Reasonable assumption
                                                up to §§ threshold




Charm 2007, Ithaca, NY, 8/06/2007               Brian Meadows, U. Cincinnati
    How This Should Apply to 3-body D Decays
   Decays have amplitudes F(s) related to scattering amplitude
    T(s) by:
           Ff (s) = Tfk (s) Qk (s) Intermediate states
    Weak decay/fragmentation:
                                     D+                              +
    • I-spin not conserved                      Q    k T
    • k scattering on +during                                            Scattering:
                                                                 f          kf
      fragmentation can impart
      an overall phase                                               K-
                                           +

              Watson theorem:        Up to elastic limit (for each L and I )
             K -+ phase has same dependence on s as elastic scattering
                        but there can be an from overall phase shift.


                              Behaviour of Q(s) is unknown.

Charm 2007, Ithaca, NY, 8/06/2007                       Brian Meadows, U. Cincinnati
                         Conventional Approach –
                        Breit-Wigner Model “BWM”
   The “isobar model” ignores all this, and problems of double-counting:
                             “NR”              {12}               {13}              {23}            2
                                    1                     1                  1                  2
                                        2
                                    3                     2                  3                  3
                                                      3                  2                  1
   Amplitude for channel {i j} with angular momentum L:

             NR - constant                  R form                  D form          spin
                (L=0)                       factor                  factor         factor



   In the BWM each resonance “R” (mass mR, width R) described as:




   Lots of problems with this theoretically – especially in S- wave

Charm 2007, Ithaca, NY, 8/06/2007                         Brian Meadows, U. Cincinnati
                  Study D Decay Channels with
                   Large S-wave Component
   D +  K -++ (shown to right)                               E791




                                                                           Asymmetry
         Prominent feature:
          Strong asymmetry in K*(892) bands
          F-B asymmetry vs. K*(892) Breit-




                                                   M 2(K -+)
            Wigner phase (inset) is zero at 560.                                        BW
          (Differs from LASS where this is
            zero at 135.50
    Interference with large S– wave
     component.
    Shift in S–P relative phase wrt
     elastic scattering by -79.50                                          M 2(K -+)


                                                                   0
    Another channel with similar features w.r.t. the  (770) is D+  -++


Charm 2007, Ithaca, NY, 8/06/2007                       Brian Meadows, U. Cincinnati
          (800) in BWM Fit to D+  K-++
                           E791: E. Aitala, et al, PRL 89 121801 (2002)

Without (800):                                                                Fraction %     Phase 0



     NR ~ 90%
     Sum of fractions 130%
     Very Poor fit (10-5 %)
BUT                                                                              S~89 %
     Inclusion of  makes
      K0*(1430) parameters                                          M1430 = 1459 § 7 § 12 MeV/c2
      differ greatly from PDG                                       1430 = 175 § 12 § 12 MeV/c2
      or LASS values.
                                                                      M = 797 § 19 § 42 MeV/c2
Similarly, (500) is required in D+  -++                           = 410 § 43 § 85 MeV/c2

E791: E. Aitala, et al, PRL 86:770-774 (2001)
                                                                               2/d.o.f. = 0.73 (95 %)

       Can no longer describe S- wave by a single BW resonance
       and constant NR term for either K -+ or for -+ systems.
        Search for more sophisticated ways to describe S- waves
Charm 2007, Ithaca, NY, 8/06/2007                       Brian Meadows, U. Cincinnati
                             New BWM Fits Agree

                                                       NEW RESULTS from both FOCUS and
                                                       CLEO c support similar conclusions:
                                                       •  required (destructively interferes
                                                       with NR) to obtain acceptable fit.
                                                       • K0*(1430) parameters significantly
                                                       different from LASS.




   These BW parameters are not physically meaningful
   ways to describe true poles in the T- matrix.




    FOCUS - arXiv:0705.2248v1 [hep-ex] 2007
    CLEO c - arXiv:0707.3060v1 [hep-ex] 2007


Charm 2007, Ithaca, NY, 8/06/2007                       Brian Meadows, U. Cincinnati
               E791 Quasi-Model-Independent
               Partial Wave Analysis (QMIPWA)
                                    E791 Phys.Rev. D 73, 032004 (2006)

   Partial Wave expansion in angular momentum L of K -+
    channels from D+  K-++ decays




    Decay amplitude                      :
      S- wave (L = 0): Replace BWM by discrete points cne in
      P- or D- wave: Define as in BWM

       Parameters (cn, n) provide quasi-model independent estimate
       of total S- wave (sum of both I- spins).
       (S- wave values do depend on P- and D- wave models).

Charm 2007, Ithaca, NY, 8/06/2007                         Brian Meadows, U. Cincinnati
Compare QMIPWA with LASS for S-wave
                          arg{F0(s)}            |F0 (s) |
                                                                     E791
                                                                     LASS




       S-wave phase for E791 is shifted by –750 wrt LASS.
       Energy dependence compatible above ~1100 MeV/c2.
           Parameters for K*0(1430) are very similar – unlike the BWM
            Complex form-factor for the D+  1.0 at ~1100 MeV/c2 ?

   Not obvious if Watson theorem is broken in these decays ?

Charm 2007, Ithaca, NY, 8/06/2007               Brian Meadows, U. Cincinnati
                  Watson Theorem Breaking vs. I = 3/2 ?
                   FOCUS / Pennington: D  K-++   arXiv:0705.2248v1 [hep-ex] 2007
                                                                                      K-matrix fit using LASS Data
                                                                                        For I=1/2 production vector:
S- wave phase (deg.)




                                                           LASS I=1/2
                                                             phase




                                                                                         Includes separate I=3/2 wave
                                                                                          Big improvement in 2.
                         Total K-+                     I =1/2 K-+
                          S- wave                         S- wave                              Large Data sample:
                                                                                       52,460 § 245 events (96.4% purity)

                                         s 1/2 (GeV/c2)
                                                                                      Observations:
                                                                                      I=½ phase does agree well with LASS as
                                                                                           required by Watson theorem except near
      S- wave fractions (%): I=1/2:             207.25 § 24.45 § 1.81 § 12.23              pole (1.408 GeV/c2)
                                         I=3/2: 40.50 § 9.63 § 0.55 § 3.15            This possibility is built in to the fit model
                                                            stat.   syst.   Model
                                                                                      Huge fractions of each I- spin interfere
      P- and D- wave fractions & phases ~same as BWM fit.                                 destructively.
                                                                                      What about P- wave ?


Charm 2007, Ithaca, NY, 8/06/2007                                              Brian Meadows, U. Cincinnati
                          CLEO c: D  K-++
                            arXiv:0707.3060v1 [hep-ex] Jul 20, 2007



    Very clean sample from
     (3770) data:
         67,086 events with 98.9 %
         purity.


    BWM fit similar to E791
        (800) in S- wave is required
         (as a Breit-Wigner) with NR.
        K* (1410) in P- wave not
         required



Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
                          CLEO c: D  K-++
                            arXiv:0707.3060v1 [hep-ex] Jul 20, 2007


   BWM fit is also significantly improved by adding I=2 ++
    amplitude – repairs poor fit to ++ inv. mass spectrum.

   Best fit uses a modification of E791 QMIPWA method …




                BWM fit                                          QIMPWA fit
Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
  Total S- wave from D+  K-++ Decays

 • General agreement
   is good
 • All differ from LASS
   (blue curves, 2nd row)



     CLEO c (Solid line)
      arXiv:0707.3060v1, 2007



     E791 (Error bars)
     Phys.Rev.D73:032004, 2006



       FOCUS (Range)
       arXiv:0705.2248v1, 2007



                                    M(K- +) (GeV/c2)

Charm 2007, Ithaca, NY, 8/06/2007   Brian Meadows, U. Cincinnati
                             CLEO c: D  K-++
                              arXiv:0707.3060v1 [hep-ex] Jul 20, 2007


   QMIPWA (E791 method applied to all waves and channels!)
    Define wave in each channel as:
         F(s) = C(s) + ae             i   R(s)
                                    Breit-Wigner type
       Interpolation table          of propagator:
      (26 complex values)           K-+ S- wave – K0*(1430)
                                    K-+ P- wave – K*(890)
                                    D- wave        – K2*(1420)
                                    ++ S- wave – R = 0



   Total of ~ 170 parameters:                                   • Is final fit converged. (Errors?)
                                                                 • Is solution unique?
      BUT – only float C(s) for one wave at a time.              • Is I=2 wave over-constraint?


Charm 2007, Ithaca, NY, 8/06/2007                           Brian Meadows, U. Cincinnati
         New Data from CLEO c: D  -++
                            arXiv:0704.3965v2 [hep-ex] Jul 20, 2007
                                                                                   BWM fits
    Use 281 pb-1 sample (3770):
     •    ~4,086 events including
          background.
         Had to remove large slice in
          m+- invariant mass
          corresponding to
                                                               FOCUS: Phys.Lett.B585:200-212,2004
           D+  Ks+                                           E. Aitala, et al, PRL 89 121801 (2002)


    General morpholgy similar to
     E791 and FOCUS
         Standard BWM fit requires a 
          amplitude much the same

    Introduced several variations in                                          CLEO c
     S- wave parametrization:
     …………………..
Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
                          Complex Pole for :
                                    J. Oller: PRD 71, 054030 (2005)




   Replace S- wave Breit-Wigner
    for  by complex pole:




                                                        arXiv:0704.3965v2 [hep-ex] Jul 20, 2007




   Best fit:



Charm 2007, Ithaca, NY, 8/06/2007                        Brian Meadows, U. Cincinnati
Linear  Model inspired Production Model
    Black, et al. PRD 64, 014031 (2001), J. Schecter et al., Int.J.Mod.Phys. A20, 6149 (2005)

                                                                        arXiv:0704.3965v2, 2007
Replace S- wave  and f0 (980) by weakly mixed
  complex poles:
                                 Weakly mixed
               Unitary        Poles  and f0(980)




           . . . + usual BW terms for f0 (1350) and f0 (1500)                                   %
                                                                                                %
    Full recipe includes both weak and strong
     mixing between  and f0(980)                                                               %

     – 7 parameters in all

                                                                               Excellent fit:

Charm 2007, Ithaca, NY, 8/06/2007                      Brian Meadows, U. Cincinnati
                          CLEO c: D  -++
                            arXiv:0704.3965v2 [hep-ex] Jul 20, 2007




   A fourth, “custom model” for S-
    wave (Achasov, et. Al., priv. comm.)
    also gave excellent fit


   All models tried (including
    BWM):
      Give essentially the same non
       S- wave parameters
      Provide excellent descriptions of
       the data



Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
          Moments Analysis in D+  K-K++
                                                           Focus: hep-ex/0612032v1 (2007)
    K++ channel has no resonances
                                                           6400 Events before  cut.
    Remove  meson in          K+K+   channel
    Allows Legendre polynomial
     moments analysis in K-+ channel
     free from cross-channel:
      |S| similar to LASS
                                                        where                 (in K –   +   CMS)
      Phase was not computed, but appears
     to be shifted ~900 wrt LASS.


                                                 |S|2                  S*P                   |P|2




Charm 2007, Ithaca, NY, 8/06/2007                   Brian Meadows, U. Cincinnati
                    S- Wave in B  J/ K+-
   Similar analysis (more complex due to
    vector nature of J/) on K- + system
   Mass dependence of S- and P-wave
    relative phase in K-+ system was used to
    determine sign: cos 2 > 0
   A clear choice agrees with the LASS data
    with overall shift + radians.




                                                         Clearly an interesting
                                                           way to probe the
                                                             K- + S- wave
              89 fb-1     PRD 71: 032005 (2005)

Charm 2007, Ithaca, NY, 8/06/2007                 Brian Meadows, U. Cincinnati
                                                        +
                   S- Wave in            D+       K+-
   FB asymmetry in K- + system in these
    decays observed by FOCUS to follow
    closely the LASS behaviour.

                          Phys.Lett.B621:72-80,2005




                                                        Clearly an interesting
                                                          way to probe the
                                                            K- + S- wave

Charm 2007, Ithaca, NY, 8/06/2007                Brian Meadows, U. Cincinnati
               Some K S-wave Measurements
                Compared to LASS Amplitude
                                                               S – P        | Amplitude |     | Amplitude |
                                     Decay Process          Meas. – LASS
                                                                            m(K ) < 1 GeV     m(K ) > 1 GeV
  Use of LASS S- wave
                                                              ( deg. )
                                                                               Unknown;
  parametrization or                B+      K+   -   +       ~0           (M/p) | ALASS |   Similar to LASS

  determination of
                                                                               used in fit


  relative S-P phase in
                                                                            Poorly defined ;
                                    B0  J/ K+ -           ~ + 180                           Similar to LASS
                                                                             to be updated

  various Dalitz plot               B+  K+ - +            ~ ± 180           Unknown            Unknown
  analyses leads to a
  confusing picture.                D0  K- K+ 0             ~ - 90        Similar to LASS    Similar to LASS


                                                                            Very different ;   Similar to LASS
                                                                                                 get ~ same
                                    D+  K- + +             ~ - 75        significant rise
                                                                                                  K0*(1430)
  More channels are                                                         toward threshold
                                                                                               mass and width

  needed to understand              D+  K- K+ +             ~ - 90        Similar to LASS    Similar to LASS

  any pattern.
                                    D+  K- + l               ~0          Similar to LASS    Similar to LASS


                                                            Adapted from W.M. Dunwoodie, Workshop on 3-Body
                                                            Charmless B Decays, LPHNE, Paris, Feb. 1-3, 2006


Charm 2007, Ithaca, NY, 8/06/2007                               Brian Meadows, U. Cincinnati
                                    Conclusions
   The most reliable data on S- wave scattering are still from
    LASS or CERN-Munich data.
   More information on very low mass data may be accessible
    through study of
      semi-leptonic D decays
      larger samples of B  J/ K-(-)+ decays
   New techniques seem to yield information on the S- wave in
    various decay modes, BUT it is not yet obvious how to carry
    that over information from one decay to another.
      Understanding this will require a systematic study of many more D
       and B decays
      This should remain a goal before it becomes a limiting systematic
       uncertainty in other heavy flavour analyses.


Charm 2007, Ithaca, NY, 8/06/2007           Brian Meadows, U. Cincinnati
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Charm 2007, Ithaca, NY, 8/06/2007      Brian Meadows, U. Cincinnati
                               Charged (800) ?
Babar: D0  K-K+0
                                                                                    ?
          11,278 § 110 events (98% purity)


Tried three recipes for K§0 S-wave:
     1.    LASS parametrization
     2.    E791 fit
                                   *
     3.    NR and BW’s for  and K0 (1430)                  ?
    Best fit from #1 rotated by ~-900.
    No need for + nor -, though not
     excluded:
     Fitted with:
           M = (870§ 30) MeV/c2,    Not consistent
            = (150§ 20) MeV/c2       With “”
                                                         385 fb-1: PRC-RC 76, 011102 (2007)


Charm 2007, Ithaca, NY, 8/06/2007                    Brian Meadows, U. Cincinnati
Partial Wave Analysis in D0  K-K+0
    Region under  meson is ~free                                                               p- s

     from cross channel signals:
     allows Legendre polynomial
     moments analysis in K-K+ channel:
     (Cannot do this is K channels)


                                                                       |S|                            |P|




        where                       (in K –K   +   CMS)


                |S| consistent with either
            a0(980) or f0(980) lineshapes.                Babar: 385 fb-1: PRC-RC 76, 011102 (2007)


Charm 2007, Ithaca, NY, 8/06/2007                         Brian Meadows, U. Cincinnati
          Compare QMIPWA with BWM Fit
                                                      arg{F(s)}

   Red curves are §1
    bounds on BWM fit.                     S

   Black curves are §1
    bounds on QMIPWA fit.
                                           P
   Completely flexible S-
    wave changes P- & D-
    waves.                                 D


                                                       E791 Phys.Rev. D 73, 032004 (2006)

                  (S- wave values do depend on P- and D- wave models).
Charm 2007, Ithaca, NY, 8/06/2007                   Brian Meadows, U. Cincinnati
       E791 Require (500) in D+  -++
                              E. Aitala, et al, PRL 86:770-774 (2001)

                                                                                  Fraction %     Phase 0
Without (500):
                                                 With 
    NR ~ 40% dominates
     (1400) >  (770) !!
    Very Poor fit (10-5 %)

                                                                                    S~116 %
Observations:

    NR and  phases differ by ~
     1800                                         No 

    Inclusion of  makes
     K0*(1430) parameters differ
     greatly from PDG or LASS
                                                                            2/d.o.f. = 0.90 (76 %)
     values.


                  This caught the attention of our theorist friends !

Charm 2007, Ithaca, NY, 8/06/2007                         Brian Meadows, U. Cincinnati
          FOCUS / Pennington: D  K-++
                            arXiv:0705.2248v1 [hep-ex] May 15, 2007


   Use K-matrix formalism to separate I- spins in S-wave.

   The K-matrix comes from their fit to scattering data T(s)
    from LASS and Estabrooks, et al:
        Extend T(s) to K threshold using PT
        I= 1/2: 2-channels (K and K’ ) one pole (K                *
                                                                     1430)
        I= 3/2: 1-channel (K only) no poles

   This defines the D+ decay amplitudes for each I- spin:

         where
                                                       T- pole is at: 1.408 – i 0.011 GeV/c2

Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
          FOCUS / Pennington: D  K-++
                            arXiv:0705.2248v1 [hep-ex] May 15, 2007

   Amplitude used in fit:


                   I- spin 1/2 and 3/2                     Usual BWM model for
                       K-+ S-wave                           P- and D- waves

   P- vectors are of form:
                                                                k=1 K ; k=2 K’

                      Same as pole
                       in K-matrix

    that can have s-dependent phase except far from pole.


Charm 2007, Ithaca, NY, 8/06/2007                     Brian Meadows, U. Cincinnati
           … Is Watson Theorem Broken ?

   E791 concludes:
         “If the data are mostly I= 1/2 , this observation indicates that
        the Watson theorem, which requires these phases to have the same
        dependence on invariant mass, may not apply to these decays
        without allowing for some interaction with the other pion.”
      Point out that their measurement can include an I =3/2 contribution
       that may influence any conclusion.

   Note:
      They also make a perfectly satisfactory fit (2 /  = 0.99) in which the
       S-wave phase variation is constrained to follow the LASS shape up
       to K’ threshold.


Charm 2007, Ithaca, NY, 8/06/2007             Brian Meadows, U. Cincinnati

				
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