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### Towards a Common Methodology?

Brian Meadows

University of Cincinnati

- Towards Uniformity
- Discussions with CLEO and BELLE
- Methodology
- No recommendations by this group
- Further recommendations

Brian Meadows, U. Cincinnati.

Towards Uniformity(with CLEO and BELLE anyway)

- Discussions were held earlier this year between
- CLEO - David Asner
- Belle - Alex Bondar
- BaBaR - Alexei Dvoretskii, Brian Meadows

We met by telephone (occasionally all of us!)

- Asked by the HFAG for recommendations to make comparison of results of fits to three body decays ("Dalitz Plot fits") from the three experiments less ambiguous.
- As a focus for discussions, 4 important questions were asked.
- Ideas were exchanged – and some guidelines were recommended

http://babar-hn.slac.stanford.edu:5090/hn/aux/brian/recommendations4.ps

Brian Meadows, U. Cincinnati

Question 1

Is it useful to establish common phase (or other) conventions which would simplify comparison and averaging of results from CLEO, BABAR and Belle?

Brian Meadows, U. Cincinnati

Question 1

- An overall phase is not important for comparisons.
- There is no obvious reference phase.
- Usually the “strongest” resonance is chosen (~900 at pole)
- Everyone else is likely to see it too!
- Occasionally “strongest” resonance not the best choice however:
- Consider D0! K0p+p-
- Do not know strangeness of K0! choose r0?
- You may want to compare with D0! K0p0p0
- No r0 there, so use f0(980).
- Neither r0 nor f0(980) is the strongest resonance.

Brian Meadows, U. Cincinnati

Ordering Matters

CM system for “bc channel”:

H!Rk + a

b + c

c

pa=q

a

pc=-p

cos bc = p¢ q

Cyclic permutations for

Other two channels

bc

l

L=l

b

pb=p

Spin function:

Flips sign under a $ b for odd (but not even)L.

Brian Meadows, U. Cincinnati

Ordering Matters …

- Suggested ordering scheme:
- For particle decays:

a) Take particles in descending mass order;

b) Within particles of equal mass, order them by descending charge;

c) Order neutral K and D in order of decreasing Cabbibo favoured status:

K0 before K0 in D decays (converse in B decay);

D0 before D0 in B decay.

- For anti-particle decays same rules,
- reverse charge or particle anti-particle convention of all decay products but NOT their masses

Brian Meadows, U. Cincinnati

Question 2

DP analyses will complicate the problem of providing branching fractions for the various quasi-2-body components. We believe it is necessary to provide branching fractions where interference effects are manageable. What common conventions might be established to facilitate this and criteria established for when it simply makes no sense to quote a branching fraction?

Brian Meadows, U. Cincinnati

Resonant Fractions

- In either the isobar model, or the K matrix formulation, the DP density is effectively sum of terms – one per resonance

/ a0ei0 + k akeikMk

non-resonant resonances k

- Suggest we define fraction fk (k = 0, 1, 2, …) as
- This has intuitive meaning in most B decays - less so in D decays.

Brian Meadows, U. Cincinnati

B0! K

Dalitz BoundariesInterference effects

More important in

D than in B decay

(usually!)

Brian Meadows, U. Cincinnati

Resonant Fractions

- This definition does guarantee that fk/ |ak|2
- BUT, recalling that resonances interfere, and that their central mass can lie outside the Dalitz plot
- Often implies that fk 1
- There will be cases where ak is large, yet fk is small
- Resonances that show up as dips still have fk > 0.
- Also considered (and not recommended) was:
- Does guarantee that fk = 1
- BUT fk can be < 0.

Brian Meadows, U. Cincinnati

Implications for the PDG Averages

- The PDG has acknowledged the difficulties in averaging resonant branching fractions. They propose a modest change in their procedure.
- Basically, they propose dropping the “correction for unseen modes” factor.
- E.g. D+! K-++ has large K*(890)0+ fraction
- So does D+! K00+
- We cannot expect that the ratio between these is ~2:1 as expected from I -spin conservation in the K*(890) decay because
- p+p+ scattering is different from p+p0.
- In particular, r+ is a possibility in the former, not the latter
- Bose symmetrization is required in the former, not the latter.
- Perhaps we should average “couplings” instead?

Brian Meadows, U. Cincinnati

Question 3

It is already clear that and K spectroscopy below 2 GeV is not sufficiently well known to determine the important resonance components to include in a Dalitz Plot. Can some guidance be given on this choice? If ad hoc structures are needed in some mass regions to properly describe the rest of the DP, how should the results of such ad hoc structures be quoted by HFAG or PDG?

Brian Meadows, U. Cincinnati

Question 3

- Understanding the scalar spectrum is one clear and important goal of Dalitz plot analyses
- Another goal is to obtain empirical knowledge of magnitude and phase of amplitude at each point on the Dalitz plot
- Required for certain CP parameter studies
- Necessary for simulations of three body decays

We may have to expect that it is too hard to meet both goals in a single analysis. The first may require input from several experiments.

Brian Meadows, U. Cincinnati

Question 3

- So we acknowledge the notion of an “empirical fit.”
- These already exist – e.g. BaBaR and Belle use two isobars (500) and (1000)!+- to improve their fits to D0! K0+-
- E791 introduced (800) in D+! K-++ and (500) in D+! K-++ to obtain acceptable fits.

It is fair to say that the HEP community does not yet universally accept these isobars as well defined states in the scalar spectrum.

BUT they do provide good, empirical descriptions of the data.

- Properly incorporating features of the scalar spectrum may require more input from the theoretical / phenomenological community, but we have a duty to try

Brian Meadows, U. Cincinnati

Question 4

What final results from DP analyses would be tabulated?

Easy – all the parameters needed to describe the Dalitz plot!

(Experimental issues such as efficiency removed)

Brian Meadows, U. Cincinnati

Question 4

- Quantities reported should include
- Resonant fractions, amplitudes and phases
- Masses and widths of resonances (floated or not)
- If “PDG” Is quoted, then the year should be too!
- Definition of form factors
- Variations on Breit-Wigner forms
- PWA parameters
- … any other parameters.
- BUT
- Resonances introduced for purposes of obtaining an “empirical” fit need a separate specification. These will probably not be included in the PDG averages, nor will other mass or widths determined in their presence.

Brian Meadows, U. Cincinnati

A Common Methodology?

- Several approaches to Dalitz fits:
- Isobar model
- K matrix / P-vector approach
- Partial wave analysis (energy independent)
- Each has its own particular value – depends on channel
- The CLEO / BELLE / BABAR group did not make any recommendation
- But our group …?

Brian Meadows, U. Cincinnati

Further Thoughts

- Couplings rather than resonant fractions?
- Which is most important?
- Is there some way to combine empirical fits from various experiments so that progress with scalar, and other spectroscopy, issues can be made?
- Remove experimental dependencies
- Provide error matrices.
- A database?

Brian Meadows, U. Cincinnati

Obvious Experimental Details

- No recommendations were made on experimental details. However, some of our (my) preferences
- Three body mass be constrained where reasonable - B, D decay, but not (?S), etc.
- Narrow resonances be convoluted with resolution
- Backgrounds be minimized or carefully modeled
- PDF normalization sensitive to narrow resonance

(Importance of importance sampling!)

- Seriously consider making binned and / or c2 fits for channels with highest statistical significance
- Many possible benefits

Brian Meadows, U. Cincinnati

Unconstrained 3-Body Mass

D0!p+p0p-

- Each point defined by 3 coordinates rather than two
- Efficiency is function of 3 rather than 2 coordinates. To obtain it:
- Either smear MC truth table coordinates to match OR
- Make un-binned likelihood fit
- Fit itself must be un-binned likelihood with 3-d normalization.
- No option to make a binned c2 fit (no normalization required).

Brian Meadows, U. Cincinnati

Summary

- CLEO, BaBaR and Belle have agreed to some guidelines for simplifying the job of the HFAG, and perhaps the PDG in comparing results of Dalitz plot fits.
- People need some freedom in their choice of methodology to suit the problem they seek to solve.
- It is acknowledged that “empirical fits” may sometimes be made, and may not always include “real” physics.

Brian Meadows, U. Cincinnati

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