Generic Dalitz model - A measurement of the CKM angle from studies of DK∏ Dalitz plots

Chapter 9 Summary

B.2 Generic Dalitz model

The generic Dalitz model allows for spinless three-body decays to be generated according to a DP model defined at run time, while previous models only defined a hardcoded amplitude model at compile time. This development allows the user to define and modify a DP model through the XML interface described in Sec. B.1. Each DP decay model is defined in XML by adalitzDecaytag containing a number of resonance tags, which correspond to the resonant and nonresonant components of the model. The dalitzDecay tag has two mandatory attributes, name and daughters, which take the name of the decaying particle and a space separated list of the daughters, respectively. Additionally a third attribute,probMax, may be defined. This attribute defines the maximum allowed value of the squared amplitude across the phase space, which is used to convert amplitudes into probabilities when generating decays using the accept/reject method. This value must be larger than the square of the total amplitude obtained at any point in the phase space to avoid probabilities larger than unity but should otherwise be as small as possible to speed up generation. IfprobMaxis not defined then a special run is performed to determine a safe value based on a million random points in the phase space. This value is determined as the square of the maximum amplitude found plus 10 %. If a value larger thanprobMaxis found during generation then a warning message is produced to inform the user that the generated events are invalid.

Each component of the model has an associated complex coefficient expressed as either real and imaginary parts or as a magnitude and phase. These are stored in the real, imag, magand phaseattributes, respectively. In addition, the phasespace distribution of each resonance is defined by theresDaughtersandshape attributes. The resDaughters attribute must be given the names of two of the daughter particles separated by a space, while theshapeattribute describes the lineshape of the resonance and takes one of the values listed in Table B.6. Also listed, in Table B.7 are the additional attributes corresponding to each lineshape, which are used to define any shape parameters. The lineshape of each component is also dependent on spin and, in the case of resonant contributions, the mass and width of the resonance. For many particles, these quantities are available using the particle attribute, which obtains values from the main EvtGenparticle list. Where a particle is not known toEvtGenor different values are needed, the attributes mass,width and spinmay be defined. Finally, the radii of the production and decay Blatt–Weisskopf barrier factors may be specified using the attributesBlattWeisskopfFactorParent andBlattWeisskopfFactorResonance.

Table B.6: Summary of lineshapes available for resonances.

Lineshape Description

RBW CLEO ZEMACH Relativistic Breit–Wigner (Zemach tensor spin formalism) RBW Relativistic Breit–Wigner (helicity spin formalism)

GS CLEO ZEMACH Gounaris–Sakurai (Zemach tensor spin formalism)

GS Gounaris–Sakurai (helicity spin formalism)

GAUSS CLEO ZEMACH Gaussian (Zemach tensor spin formalism) GAUSS Gaussian (helicity spin formalism)

Flatte Flatt´e lineshape

LASS LASS parameterisation for Kπ S-wave

NonRes Flat nonresonant term

NonRes Lin Nonresonant term with a linear form factor NonRes Exp Nonresonant term with an exponetial form factor

Table B.7: Summary of attributes available for resonance lineshapes.

Attribute Description LASS

a LASS scattering length

r LASS effective range

R Resonant-part magnitude

phiR Resonant-part phase

B Nonresonant-part magnitude

phiB Nonresonant-part phase

cutoff Cutoff for the nonresonant part NonRes Exp

An example DP generated using this generic model is shown in Fig. B.1, while the XML used to define this DP is given below.

<resonance mag="1.0" phase="0.0" particle="K*0"

resDaughters="pi+ K-" shape="RBW_CLEO_ZEMACH" BlattWeisskopfFactorParent="5.0"/>

<resonance mag="3.0" phase="-130.3"

width="0.164" mass="1.463" spin="0" resDaughters="K- pi+" shape="RBW" BlattWeisskopfFactorParent="5.0"/>

<resonance mag="6.5" phase="29" particle="K’’*0"

resDaughters="pi+ K-" shape="RBW_CLEO_ZEMACH" BlattWeisskopfFactorParent="5.0"/>

<resonance mag="5.01" phase="16.3"

width="0.470" mass="0.809" spin="0" resDaughters="K- pi+" shape="RBW" BlattWeisskopfFactorParent="5.0"/> </dalitzDecay>

]

) [GeV

π

−

K

(

m

1

2

3 ]

) [GeV

π

−

K(

m

1

2

3

Figure B.1: The Dalitz plot distribution of D+ → K−π+π+ decays as generated by

Appendix C

Laura++

developments

This appendix details my main contributions to the Laura++ package — these are

• cubic spline interpolation of 2D histograms; • new isobar coefficients for the determination ofγ; • a model-independent description of partial waves;

• a parameteric description of narrow peaking backgrounds

C.1 2D spline interpolation

The signal efficiency across the DP and background contributions are both described using 2D histograms in the SDP co-ordinates m0 and θ0. Interpolation is used to determine the value of the efficiency term between bin centres of the input histogram. While it is simple to perform a linear interpolation of the input histogram in the “cell” between four bin centres (as offered by the pre-existingLau2DHistDPclass), this results in discontinuities in the derivatives of the efficiency function between cells. As described in Sec. 6.3.3, a cubic function,

f(x, y) = 3 X i=0 3 X j=0 aijxiyj, (C.1)

may be defined in each cell, wherex andy run from 0 to 1 and the coefficients aij

are chosen to ensure that the efficiency,, and its derivatives _dmd0, _dθd0 and d

dm0_dθ0 are continuous between cells.

This functionality is provided by theLau2DCubicSplineclass, which receives a 2D histogram in its constructor and precalculates the coefficients in each cell

and provides an evaluate method to evaluate the spline at a given point. The Lau2DSplineDPclass acts as a wrapper for a Lau2DCubicSpline object and imple- ments theLau2DAbsHistDP abstract base class, which is also implemented by the Lau2DHistDPclass allowing the two models to be used interchangeably.

A further development is the ability to separate the efficiency model out into separate components each described by an implementation of theLau2DAbsHistDP class. The full efficiency model is provided by theLauEffModel class, which has been updated to contain an array of pointers toLau2DAbsHistDPobjects. Existing functionality is preserved by the methodssetEffHistoandsetEffSpline, which initialise an efficiency model based on a single histogram or spline, while subsequent calls to the new methodsaddEffHistoandaddEffSpline allow the efficiency to be determined as the product of an arbitrary number of splines and histograms.

Furthermore, an alternative efficiency model,LauWeightedSumEffModel, has been developed to allow multiple efficiency models to be summed together. This functionality is useful, e.g., to combine multiple trigger pathways with different efficiency maps.

In document A measurement of the CKM angle from studies of DK∏ Dalitz plots (Page 191-196)