Learning Conservation Principles in Particle Physics

1
Learning Conservation Principles in Particle
Physics

• Oliver Schulte
• School of Computing Science
• Simon Fraser University
• oschulte_at_cs.sfu.ca

2
The Scientific Problem

• 100s of known reactions
• Empirical questions
• What are the laws of particle interaction?
• Are there particles we havent seen?
• What is the relationship between matter and
  antimatter?
• Valdes 1994, Kocabas 1991, Machine Learning

3
Objectives

• Application Support high-level knowledge
  discovery, scientific model construction, data
  analysis.
• Machine Learning
• New Algorithms for Learning in Linear Spaces
• Apply and illustrate Computational Learning Theory

4
Outline

• Finding conserved quantities in particle
  reactions
• Algorithm
• Data
• Findings
• Learning-theoretic analysis
• Introducing extra particles to fit the data
  better
• A New Experiment

5
Additive Conservation Principles Selection
Rules
6
Basic Learning Principle Disallow as much as you
can

• Kenneth Ford (1965).
• Everything that can happen without violating a
  conservation law does happen.
• Nobel Laureate Leon Cooper (1970).
• In the analysis of events among these new
  particles, where the forces are unknown and the
  dynamical analysis, if they were known, is almost
  impossibly difficult, one has tried by observing
  what does not happen to find selection rules,
  quantum numbers, and thus the symmetries of the
  interactions that are relevant.

7
How much can we rule out?
Hypothetical Scenario
observed reactions
not yet observed reactions
?- ? ?- n ?- ? m- nm m- ? e- nm ne n ?
e- ne pp p ? p p ?
n ? e- ne
p p ? p p ? ?
cant rule out
8
The Vector Representation for Reactions

• Fix n particles.
• Reaction ? n-vector list net occurrence of each
  particle.

9
Conserved Quantities in Vector Space
10
Conserved Quantities are in the Null Space of
Observed Reactions

• Let q be the vector for a quantum number, r for a
  reaction.
• Then q is conserved in r ? q ? r 0.
• Let Q be a matrix of quantities. Then Qr 0 ? r
  is allowed by Q.
• So if r1, , rk are allowed, so is any linear
  combination .

k
a
r
F
k
k
i

1
11
Maximally strict selection rules basis for
nullspace of observations

• Defn A list of selection rules Q is maximally
  strict ? nullspace(Q) span(R).
• Proposition Q is maximally strict ? span(Q)
  R?.

12
System for Finding a Maximally Strict Set of
Selection Rules

• Read in Observed Reactions

from database
Convert to list of vectors R
using conversion utility
Compute basis Q for nullspace R?
Maple function nullspace
13
Database Conversion Utility
14
The Data Particles

• Particles from Review of Particle Physics
• Total 193 particles
• Separate entries for particle and anti-particles
• e.g., p, p 2 entries

15
The Data Reactions

• At least one decay for each particle with a
  decay mode.
• 182 out of 193 particles have decay modes.
• Particle utility converts to vector
  representation.

16
Why Decays?

• Wanted linearly independent reactions.
• Proposition Assuming Special Relativity, decays
  of distinct particles are linearly independent.

17
Finding 1 Classifying Reactions

• E. Charge, Baryon, Muon, Electron, Tau is
  basis for nullspace of known reactions.

1. Output of Program is equivalent classifier to
   standard rules.
2. All absolutely conserved quantum numbers are
   linear combinations of Baryon, E. Charge,
   Muon, Electron, Tau e.g., Lepton Muon
   Electron Tau

18
Finding 2 Matter/Antimatter

• Observation
• Physicists rules match particle-antiparticle
  pairings.
• On repeated runs, program always matches
  particle-antiparticle pairings.

Proposition If there is any basis that matches
particle-antiparticle pairings, then all bases
match particle-antiparticle pairings.
19
Physicists Rules Match Matter/Antimatter
Pairings
20
Finding 3 Clustering Simplicity Standard
Quantities

• Observation Different runs often produce version
  of the lepton family laws Baryon, Muon,
  Electron, Tau.
• Is there something special about these laws?
• Williams (1997) these laws have no basis in
  fundamental physical principles.

21
Conservation Principles classify reactions and
cluster particles

• A particle p carries a quantity q if the value
  of q for p ? 0.
• Observation The standard conservation principles
  have disjoint carriers.

Baryon
Electron
Muon
Tau
22
Physicists Quantities Have Disjoint Carriers
23
Clustering by Conservation Principles is Unique

• Theorem. Let q1, q2, q3, q4 be any quantities
  such that
• charge, q1, q2, q3, q4 classify reactions as
  charge, B, E, M, T do, and
• q1, q2, q3, q4 have disjoint carriers.
• Then the carriers of the qi are the same as the
  carriers of B, E, M, T.

24
Clustering by Conservation Principles is Unique
Illustration
Baryon
Electron
Muon
Tau
Carriers
Quantum1
Quantum2
Quantum3
Quantum4
Any alternative set of 4 Qs with disjoint
carriers
25
Computational Search for Clustering Conservation
Principles

• Take electric charge as given.
• Choose suitable objective function to encourage
  clustering.
• Minimizing objective function -gtrediscovers
  standard principles.
• Work with Mark Drew.

26
Learning-Theoretic Analysis

• The maximally strict learner is a PAC-learner.
• Given n particles, e tolerance, 1- d confidence,
  a sample of n/e x ln(n/d) suffices.
• E.g. 90 accuracy, 90 confidence ?14,600
  data points.

Proposition. The maximally strict learner is the
only learner that identifies a correct set of
conservation principles in the limit with at most
n mind changes.
27
More Particles can lead to stricter Conservation
Principles

• Well-known example if ?e ?e, then n n ? p
  p e- e- should be possible.
• Elliott and Engel (May 2004)What aspects of
  still-unknown neutrino physics is it most
  important to explore? it is clear that the
  absolute mass scale and whether the neutrino is a
  Majorana or Dirac particle are crucial issues.

28
When do more particles lead to stricter
Conservation Principles?

• Theorem An extra particle yields stricter
  selection rules for a set of reactions R ?there
  is a reaction r such that
• r is a linear combination of R
• but only with fractional coefficients.

29
Critical Reaction for ?e ? ?e Discovered by
Computer
Finding if ?e ?e , then the process ? ?0 ? p
e- cannot be ruled out with selection rules.
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Conclusions

• Program computes maximally strict set of
  selection rules.
• Good match with Baryon, Charge, Muon,
  Electron, Tau
• Classifies reactions as possible or impossible in
  exact agreement.
• Reproduces particle-antiparticle pairings
• Clustering particles given Charge leads to
  complete agreement.
• Extra particle Computes a novel critical
  experiment to test if ?e ?e .

31
Polynomial Time Algorithm for Deciding if New
Particle is Needed

• Theorem (Smith 1861). Let A be an integer matrix.
  Then there are matrices U,V,S such that
• A USV
• S is diagonal (S Smith Normal Form of A)
• U,V are unimodular.
• Theorem (Giesbrecht 2004). Let R be the matrix
  whose rows are the observed reactions. Then a new
  particle is needed ? Smith Normal Form of RT has
  a diagonal entry outside of 0,1,-1.