Implementation of K-matrix formalism in the

amplitude modelBy Rudin Petrossian-Byrne

Supervisors: Dr. Marco Gersabeck & Stefanie Reichert

CERN Summer Student Programme 2014

Before The Physics…. Hi!

My name is Rudin

from Dublin

studying Physics (Undergraduate)

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The big pictureMixing and CPV

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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 – First Evidence from BaBar and Belle for mixing in charmed neutral mesons

𝒙=𝑀 1−𝑀 2

Γ

Charm Quark

∝ (10− 2 )𝒚=Γ 1− Γ2

2 Γ

No significant evidence for CPV in charm to date

• Mixing parameters very small

2012 – LHCb find evidence for mixing in a single measurement

• SM ‘predicts’ small CPV

↪Observation at current sensitivity would imply New Physics

R

Phys. Rev. Lett. 110 (2013) 101802

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𝑫𝟎{𝒄𝒏𝒖𝑫𝟎→𝑲 𝒔 𝝅+¿𝝅−¿

(𝑚𝑜𝑑𝑒𝑟𝑛𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛)

Allows measuring x and y parameters directly

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𝑫𝟎{𝒄𝒏𝒖𝑫𝟎→𝑲 𝒔 𝝅+¿𝝅−¿

(𝑚𝑜𝑑𝑒𝑟𝑛𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛)

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

𝑫𝟎

𝝅+¿ ¿

𝝅−

𝑲 𝒔

{𝑞𝑎𝑎Detected by TT tracker.

Bend through magnet. Detected by T1, T2, T3.

𝝅+¿ ¿

𝝅−

{𝑎 decay in VELO as well decay outside VELO harder to reconstruct𝝅+¿ ¿

𝝅−

𝝅−

𝝅+¿ ¿

𝑲 𝒔

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𝑫𝟎→𝑲 𝒔 𝝅+¿𝝅−¿

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.

(𝑚𝑜𝑑𝑒𝑟𝑛𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛)

𝑫𝟎

𝑲 𝒔

𝝅−

𝝅+¿ ¿

↪

𝑫𝟎{𝒄𝒏𝒖

LHCb Simulation

𝒎𝑲𝒔 𝝅¿ ¿

𝒎𝑲𝒔𝝅

−

𝟐

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What we observe:Dalitz plot is characterised by Resonances + non-resonant background

Resonances (bands of high )Corresponds to

↑Intermediate particle, e.g.

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Not homogenously populated with events Some daughter energies more probable

P

𝑫𝟎→𝑲 𝒔 𝝅+¿𝝅−¿

LHCb Simulation

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Resonant decay treated as a superposition of 2-body decays e.g. and

Approximate by a relativistic Breit–Wigner probability distribution function

is clearly affected by resonances

Each resonance has a probability amplitude

𝑫𝟎→𝑲 𝒔 𝝅+¿𝝅−¿

THE ISOBAR MODEL

𝑀=∑𝑟

𝑎𝑟𝑀𝑟

𝑴𝒓= ⟨𝜋+¿𝜋−¿𝑟 ⟩ 𝜟𝒓 ¿

Great for isolated Resonances!

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PROBLEMIf Resonances overlapTotal probability 1

↪Not good

↪Theorists are unhappy

↪Should probably do something about that…

Great for isolated Resonances!

BAD for overlappingresonances

𝒎𝝅𝝅𝟐

𝒎𝝅𝝅𝟐

Curtis A. Meyer, Carnegie Mellon University, ‘A k-matrix tutorial’ (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

𝑫𝟎

𝝅+¿ ¿

𝝅−

𝝅+¿ ¿

𝝅−

𝑫𝟎

𝑲 𝒔

𝑲 𝒔

𝒓

𝒓

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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, ‘A k-matrix tutorial’ (2008)

𝒎𝝅𝝅𝟐

𝒎𝝅𝝅𝟐

─ Breit-Wigner─ K-matrix

─ Breit-Wigner─ K-matrix

𝑴𝒓= ⟨𝜋+¿𝜋−¿𝑟 ⟩ 𝑭 ⟨𝐾 𝑠𝑟|𝐷0 ⟩

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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

𝑃 𝑗=∑𝛼

𝛽𝛼0 𝑔𝛼 𝑗

0

𝑚𝛼2 −𝑚𝜋𝜋

2 + 𝑓 𝜋𝜋 𝑗𝑝𝑟 1 −𝑠0

𝑝𝑟

𝑚𝜋𝜋2 −𝑠0

𝑝𝑟

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!

• Positive cross-checks (e.g. decay time )

Implementation complete

• Observing things (printing to screen) changes variables.

memory issues probably to blame

GPU memory issues currently being addressed

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THANK YOU

Questions?

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Spare Slides

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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

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Outline Background: Mixing and CP violation in the charm sector

Parametrisations:

Implementation & Results

Onto the Physics….

• Isobar model • K-matrix

• Detection• Dalitz Plot & Resonances