Implementation of K-matrix formalism in the
amplitude modelBy Rudin Petrossian-Byrne
Supervisors: Dr. Marco Gersabeck & Stefanie Reichert
CERN Summer Student Programme 2014
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The big pictureMixing and CPV
Mixing
strange quark
bottom quark
mixing
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The big pictureMixing and CPV
Mixing
me
mixing
strange quark
bottom quark
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The big pictureMixing and CPV with Charm
2007 β First Evidence from BaBar and Belle for mixing in charmed neutral mesons
π=π 1βπ 2
Ξ
Charm Quark
β (10β 2 )π=Ξ 1β Ξ2
2 Ξ
No significant evidence for CPV in charm to date
β’ Mixing parameters very small
2012 β LHCb find evidence for mixing in a single measurement
β’ SM βpredictsβ small CPV
βͺObservation at current sensitivity would imply New Physics
R
Phys. Rev. Lett. 110 (2013) 101802
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π«π{ππππ«πβπ² π π +ΒΏπ βΒΏ
(πππππππππ‘ππ‘πππ)
Allows measuring x and y parameters directly
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π«π{ππππ«πβπ² π π +ΒΏπ βΒΏ
(πππππππππ‘ππ‘πππ)
Allows measuring x and y parameters directly
P P
Protons collide in the VELO
Very short life-time of charm quark travels mm, then decays
π«π
π +ΒΏ ΒΏ
π β
π² π
{πππDetected by TT tracker.
Bend through magnet. Detected by T1, T2, T3.
π +ΒΏ ΒΏ
π β
{π decay in VELO as well decay outside VELO harder to reconstructπ +ΒΏ ΒΏ
π β
π β
π +ΒΏ ΒΏ
π² π
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π«πβπ² π π +ΒΏπ βΒΏ
Allows measuring x and y parameters directly
3-Body Decay Unlike 2-body decays, energy of daughter particles not well-defined Range of possible values.
(πππππππππ‘ππ‘πππ)
π«π
π² π
π β
π +ΒΏ ΒΏ
βͺ
π«π{πππ
LHCb Simulation
ππ²π π ΒΏ ΒΏ
ππ²ππ
β
π
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What we observe:Dalitz plot is characterised by Resonances + non-resonant background
Resonances (bands of high )Corresponds to
βIntermediate particle, e.g.
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Not homogenously populated with events Some daughter energies more probable
P
π«πβπ² π π +ΒΏπ βΒΏ
LHCb Simulation
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Resonant decay treated as a superposition of 2-body decays e.g. and
Approximate by a relativistic BreitβWigner probability distribution function
is clearly affected by resonances
Each resonance has a probability amplitude
π«πβπ² π π +ΒΏπ βΒΏ
THE ISOBAR MODEL
π=βπ
ππππ
π΄π= β¨π+ΒΏπβΒΏπ β© ππ ΒΏ
Great for isolated Resonances!
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PROBLEMIf Resonances overlapTotal probability 1
βͺNot good
βͺTheorists are unhappy
βͺShould probably do something about thatβ¦
Great for isolated Resonances!
BAD for overlappingresonances
ππ π π
ππ π π
Curtis A. Meyer, Carnegie Mellon University, βA k-matrix tutorialβ (2008)
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ALTERNATIVE MODEL
Makes use of K-Matrix formalism to describe resonances with
Assumption: equivalent dynamics to scattering off
Need to consider all possible decays of r in the analysis
π«π
π +ΒΏ ΒΏ
π β
π +ΒΏ ΒΏ
π β
π«π
π² π
π² π
π
π
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P
K-Matrix Formulation
Pseudo-propagatorComes from scatteringdata.
K - Matrix phase-space factor
Production VectorDescribes Couplings of resonances to
Upholds unitarity by construction!Curtis A. Meyer, Carnegie Mellon University, βA k-matrix tutorialβ (2008)
ππ π π
ππ π π
β Breit-Wignerβ K-matrix
β Breit-Wignerβ K-matrix
π΄π= β¨π+ΒΏπβΒΏπ β© π β¨πΎ π π|π·0 β©
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My Job: Implement K-matrix description in fitting code for
Implementation: CODING
Programming language: CUDA ( C++
Running on a GPU (Graphical Processing Unit)
implement P fit to data
extremely fast parallelisation
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What are we fitting to data?
21 floating variables
and are complex
π π=βπΌ
π½πΌ0 ππΌ π
0
ππΌ2 βπππ
2 + π ππ πππ 1 βπ 0
ππ
πππ2 βπ 0
ππ
P
K-matrix: comes entirely from scattering data
phase space matrix: depends only on masses
production vector: describes coupling to resonances
LHCb Simulation
Need to fit
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Wrestling with errors
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STATUS
This Week: Code Working on a CPU!
β’ Positive cross-checks (e.g. decay time )
Implementation complete
β’ Observing things (printing to screen) changes variables.
memory issues probably to blame
GPU memory issues currently being addressed
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What are we fitting to data?
21 floating variables and are complex both real and imaginary parts need to be fitted
K-Matrix
Production vector
Sum over poles
Non resonant term
Correction term