Automated Search for Conserved Quantities in Particle Reactions Oliver Schulte School of Computing Science Simon Fraser University [email protected]

Automated Search for Conserved Quantities in Particle Reactions

Oliver SchulteSchool of Computing ScienceSimon Fraser [email protected]

SFU Particle Physics Group 2

Outline

What’s this about?Finding conserved quantities in particle reactions Algorithm Data Findings

Introducing extra particles to fit the data better


CS Goals

• Basic research (good enough): Write programs that match the theories from physics.

• Previous work:

• Kobacas, Valdes-Perez on discovering selection rules.

• Valdes-Perez, Zytkow on (re)discovering particle substructure (Physical Review E, 1996)

• Practical use (icing on the cake): analyze data to help with new discoveries.


The Goal: Find Absolutely Conserved Quantities

Omnes (1971), Introduction to Particle Physics.

“The method [of assigning quantum numbers] is rather lengthy … so that we give the procedure in detail, once and for all.”

Want a program for assigning quantum numbers.


Basic Principle: Disallow as much as you can

• 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.”

• Kenneth Ford (1965). “Everything that can happen without violating a conservation law does happen.”


How much can we rule out?

- - + n

- - +

- e- + + e

n e- + e + p

p + p p + p +

observed reactions not yet observed reactions

n e- + e

p + p p + p + +

can’t rule out

Hypothetical Scenario


The Vector Representation for Reactions

• Fix n particles.

• Reaction n-vector: list net occurrence of each particle.

1 2 3 4 5 6 7

Process p 0 - e+ e- e - e- + + e 0 0 1 0 -1 -1 -1

p e+ + 0 1 -1 0 -1 0 0 0

p + p p + p + 0 0 -1 0 0 0 0 0


Conserved Quantities in Vector Space

1 2 3 4 5 6 7

Process p 0 - e+ e- e - e- + + e 0 0 1 0 -1 -1 -1

p e+ + 0 1 -1 0 -1 0 0 0

p + p p + p + 0 0 -1 0 0 0 0 0

Baryon Number 1 0 0 0 0 0 0

Electric Charge 1 0 -1 1 -1 0 0


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 . i 1

k akrk


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.


System for Finding a Maximally Strict Set of Selection Rules

Read in Observed Reactions

Convert to list of vectors R

Compute basis Q for nullspace R

from database

using conversion utility

Maple function nullspace


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

• One entry for same type, different masses

• e.g., just one entry for Σ(1385), Σ(1670)


The Data: Reactions

• At least one decay for each particle with a decay mode.

• Particle utility converts to vector representation.


Why Decays?

Wanted: linearly independent reactions.Proposition: Decays of distinct particles are linearly independent.

1 2 3 4 5 6 7 8

Process + 0

e-

e γ

+ + + 1 0 -1 0

0 γ + γ 0 1 0 0 - e- + + e 0 0 -1 0 -1 -1 -1 0


# independent quantities # unstable particles

Defn. A particle is stable if it has no decay mode, e.g., , e , e ,p, p ,ve, ve, v ,v ,v ,vProposition Fix n particles, allowed reactions R.

1. dim(R) n - # unstable particles

2. # independent conserved quantities # stable particles

• e.g. n = 193 particles, 11 stable 11 conserved quantities

• because of antiparticles, can be improved to 6 conserved quantities


Finding #1

{Baryon#, E. Charge, Muon#, Electron#, Tau#} is basis for nullspace of possible 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#


Finding #2

Program matches particle-antiparticle pairings.There is an analytic explanation.


Finding #3

Different runs seem to produce version of the lepton family lawse.g., {- Muon#, - Electron#, -Tau#}.No analytic explanation.


More Particles can lead to stricter Conservation Principles

Well-known example: if e = e, then n + n p + p + e- + e- should be possible.(Williams Ch.12.2).


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

1. r is a linear combination of R

2. but only with fractional coefficients.

h id d e n p a rt ic le s

N oh id d e n p a rt ic le s

o b s e rv e d tra n s itio n s

lin e a r c o m b in a tio n sw ith in te g e r c o e ff ic ie n ts

lin e a r c o m b in a tio n sw ith f ra c tio n a l c o e ff ic ie n ts


Hidden Particles, Finding #1

The standard selection rules are maximally strict with respect to transitions among nonneutrinos.


Critical Reaction for e e Discovered by Computer

Finding if e = e , then the process Υ + Λ0 p + e- cannot be ruled out with selection rules.

coefficient observed process ½ Y + + - ½ Y e+ + e- ½ Λ0 p + e + e- ½ Λ0 p + - ½ - - + -½ + e+ + e +

unobserved Υ + Λ0 p + + + - + e-


Conclusions

Program computes maximally strict set of selection rules.Good match with {Baryon#, E. Charge, Muon#, Electron#, Tau#} Classifies reactions as possible or

impossible in exact agreement. Reproduces particle-antiparticle pairings

Extra particle: Computes a critical experiment to test if e = e .


Further Work

Search for partially conserved quantities like strangeness.Historical Analysis of Data (R. Coleman).Pitch: looking for coauthor/proofreader for interdisciplinary or physics publication.

Documents

Automated Search for Conserved Quantities in Particle Reactions Oliver Schulte School of Computing Science Simon Fraser University [email protected]