1. Automated Search for Conserved Quantities in Particle Reactions Oliver Schulte School of Computing Science Simon Fraser University oschulte@cs.sfu.ca

2. SFU Particle Physics Group2 Outline What’s this about? Finding conserved quantities in particle reactions Algorithm Data Findings Introducing extra particles to fit the data better

3. SFU Particle Physics Group3 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.

4. SFU Particle Physics Group4 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.

5. SFU Particle Physics Group5 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.”

6. SFU Particle Physics Group6 How much can we rule out?  -   - + n  -   - +    -  e - +  + e n  e - + e + p p + p  p + p +  observed reactionsnot yet observed reactions n  e - + e p + p  p + p +  +  can’t rule out Hypothetical Scenario

7. SFU Particle Physics Group7 The Vector Representation for Reactions Fix n particles. Reaction  n-vector: list net occurrence of each particle.

8. SFU Particle Physics Group8 Conserved Quantities in Vector Space

9. SFU Particle Physics Group9 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 r 1, …, r k are allowed, so is any linear combination.

10. SFU Particle Physics Group10 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 .

11. SFU Particle Physics Group11 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

12. SFU Particle Physics Group12 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)

13. SFU Particle Physics Group13 The Data: Reactions At least one decay for each particle with a decay mode. Particle utility converts to vector representation.

14. SFU Particle Physics Group14 Why Decays? Wanted: linearly independent reactions. Proposition: Decays of distinct particles are linearly independent.

15. SFU Particle Physics Group15 # independent quantities  # unstable particles Defn. A particle is stable if it has no decay mode, e.g., Proposition 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

16. SFU Particle Physics Group16 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#

17. SFU Particle Physics Group17 Finding #2 Program matches particle-antiparticle pairings. There is an analytic explanation.

18. SFU Particle Physics Group18 Finding #3 Different runs seem to produce version of the lepton family laws e.g., {- Muon#, - Electron#, -Tau#}. No analytic explanation.

19. SFU Particle Physics Group19 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).

20. SFU Particle Physics Group20 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.

21. SFU Particle Physics Group21 Hidden Particles, Finding #1 The standard selection rules are maximally strict with respect to transitions among nonneutrinos.

22. SFU Particle Physics Group22 Critical Reaction for  e   e Discovered by Computer Finding if  e =  e, then the process Υ + Λ 0  p + e - cannot be ruled out with selection rules.

23. SFU Particle Physics Group23 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.

24. SFU Particle Physics Group24 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.