bachelor_thesis/talk/vortrag.org

• Intro
  • Importance of MC Methods
  • Diphoton Process
• Calculation of the XS :TOO_LONG:
  • Approach
  • Result + Sherpa
• Monte Carlo Methods
  • Integration
    • Naive Integration
      • compare to other numeric
    • Change of Variables
    • VEGAS
      • research the drawbacks that led to VEGAS
      • nice visualization of vegas working
      • look at original vegas
  • Sampling
    • Hit or Miss
    • Change of Variables
    • Hit or Miss VEGAS
      • add pic that i've sent Frank
    • Stratified Sampling
    • Observables
  • Outlook
    • Other modern Stuff
• Toy Event Generator
  • Basics
    • Parton Density Functions
      • check s/2
  • Implementation
  • Results
    • Integration with VEGAS
    • Sampling and Observables
• Pheno Stuff
  • Short review of HO Effects
    • LO
    • LO+PS
    • LO+PS+pT
    • LO+PS+pT+Hadronization
    • LO+PS+pT+Hadronization+MI
  • Presentation and Discussion of selected Histograms
    • pT of γγ system
    • pT of leading and sub-leading photon
    • Invariant Mass
    • Angular Observables
    • Effects of Hadronization and MI
    • Summary
• Wrap-Up
  • Summary
  • Lessons Learned (if any)
  • Outlook

What the heck should be in there. Let's draft up an outline. 20 minutes: bloody short, so just results

Intro   1_30m

Importance of MC Methods   SHORT

• important tool in particle physics

  • not just numerical
• also applications in stat. phys and lattice QCD
• somewhat romantic: distilling information with entropy
• interface with exp
• precision predictions within, beyond sm

• validation of new theories

  • some predictions are often more subtle than just the existense of new particles
  • backgrounds have to be substracted

Diphoton Process

• feynman diags and reaction formula
• higgs decay channel
• dihiggs decay
• pure QED

Calculation of the XS :TOO_LONG:   5m

Approach

• formalism well separated from underlying theory

  • but can fool intuition (spin arguments)
  • in the course of semester: learned more about the theory :)
• translating feynman diagrams to abstract matrix elements straight forward

• first try: casimir's trick

  • error in calculation + one identity unknown

• second try: evaluating the matrices directly

  • discovered a lot of tricks
  • error prone

• back to studying the formalism: completeness relation for real photons

  • a matter of algebraic gymnastics
  • boils down to some trace and dirac matrix gymnastics
  • mixing terms cancel out, not zero in themselves
• resulting expression for ME essentially t/u channel propagator (1/(t*u)) and spin correlation 1 + cos(x)^2
• only angular dependencies, no kinematics, "nice" factors
• symmetric in θ

Result + Sherpa

• apply the golden rule for 2->2 processes
• show plots and total xs
• shape verified later -> we need sampling techniques first

Monte Carlo Methods   8m

• one simple idea, can be exploited and refined

• how to extract information from a totally unknown function

  • look at it -> random points are the most "symmetric" choice
  • statistics to the rescue
• what does this have to do with minecraft
• theory deals with truly random (uncorrelated) so that statistics apply, prng's cater to that: deterministic, efficient (we don't do crypto)

Integration

• integration as mean value

• convergence due to law of large numbers

  • independent of dimension
  • trivially parallelism
• result normal distributed with σ due to central limit theorem

• goal: speeding up convergence

  1. modify distribution
  2. integration variable
  3. subdivide integration volume
• all those methods can be somewhat intertwined
• focus on some simple methods

Naive Integration

• why mix in that distribution: we choose it uniform
• integral is mean
• variance is variance of function: stddev linear in Volume!
• include result
• rediculous sample size

TODO compare to other numeric

Change of Variables

• drastic improvement by transf. to η

• only works by chance (more or less)

  • pseudo rapidity eats up angular divergence
• can be shown: same effect as propability density
• implementation is different

VEGAS

• a simple ρ: step function on hypercubes, can be trivially generated
• effectively subdividing the integration volume
• optimal: same variance in every cube
• easier to optimize: approximate optimal rho by step function
• clarify: use rectangular grid and blank out unwated edges with θ function
• nice feature: integrand does not have to be smooth :)

• similar efficiency as the travo case

  • but a lot of room for parameter adjustment and tuning

TODO research the drawbacks that led to VEGAS

TODO nice visualization of vegas working

TODO look at original vegas

• in 70s/80s memory a constraint

Sampling

• why: generate events

  • same as exp. measurements
  • (includes statistical effects)
  • events can be "dressed" with more effects
• usual case: we have access to uniformly distributed random values
• task: convert this sample into a sample of another distribution
• short: solve equation

Hit or Miss

• we don't always know f, may have complicated (inexplicit) form
• solve "by proxy": generate sample of g and accept with propability f/g
• the closer g to f, the better the efficiency
• simplest choice: flat upper bound
• show results etc
• one can optimize upper bound with VEGAS

Change of Variables

• reduction of variance similar to integration
• simplify or reduce variance
• one removes the step of generating g-samples
• show results etc

• hard to automate, but intuition and 'general rules' may serve well

  • see later case with PDFs -> choose eta right away

Hit or Miss VEGAS

• use scaled vegas distribution as g and to hit or miss
• samples for g are trivial to generate
• vegas again approximates optimal distribution
• results etc
• advantage: no function specific input

• problem: isolated parts of the distribution can drag down efficiency

  • where the hypercube approx does not work well
  • especially at discontinuities

TODO add pic that i've sent Frank

Stratified Sampling

• avoid global effects: subdivide integration interval and sample independently
• first generate coarse samples and distribute them in the respective grid points
• optimizing: make cubes with low efficiency small! -> VEGAS
• this approach was used for the self-made event generator and improved the efficiency greatly (< 1% to 30%)

• disadvantage: accuracies of upper bounds and grid weights has to be good

  • will come back to this

Observables

• particle identities and kinematics determine final state

• other observables can be calculated on a per-event base

  • as can be shown, this results in the correct distributions without knowledge of the Jacobian

Outlook

• of course more methods
• Sherpa exploits form propagators etc

• multichannel uses multiple distributions for importance sampling and can be optimized "live"

  • https://www.sciencedirect.com/science/article/pii/0010465594900434

TODO Other modern Stuff

Toy Event Generator   3m

Basics   SHORT

• just sampling the hard xs not realistic

  1. free quarks do not occur in nature
  2. hadron interaction more complicated in general
• we address the first problem here
• quarks in protons: no analytical bound state solution known so-far

Parton Density Functions

• in leading order, high momentum limit: propability to encounter parton at some energy scale with some momentum fraction

• can not be calcualated from first principles

  • have to be fitted from exp. data
  • can be evolved to other Q^2 with DGLAP
  • calculated with lattice QCQ: very recently https://arxiv.org/abs/2005.02102

• scale has to be chosen appropriately: in deep inelastic scattering -> momentum transfer

  • p_T good choice
  • here s/2 (mean of t and u in this case)
• xs formula
• here LO fit and evolution of PDFs

TODO check s/2

Implementation

• find xs in lab frame

• impose more cuts

  • guarantee applicability of massless limit
  • satisfy experimental requirements
• used vegas to integrate
• cuts now more complicated because photons not back to back

• apply stratified sampling variant along with VEGAS

  • 3 dimensions: x1, x2 (symmetric), η
  • use VEGAS to find grid, grid-weights and maxima
  • improve maxima by gradient ascend (usually very fast)
  • improve performance by cythonizing the xs and cut computation
  • sampling routines JIT compiled with numba, especially performant for loops and very easy
  • trivial parallelism through python multiprocessing
  • overestimating the maxima corrects for numerical maximization error
  • assumptions: mc found maximum and VEGAS weights are precise enough
• most time consuming part: multidimensional implementation + debugging
• along the way: validation of kinematics and PDF values through sherpa

Results

Integration with VEGAS

• Python Tax: very slow, parallelism implemented, but omitted due to complications with the PDF library

  • also very inefficient memory management :P
• result compatible with sherpa
• that was the easy part

Sampling and Observables

• observables:

  • usual: η and cosθ
• p_t of one photon and invariant mass are more interesting

• influence of PDF:

  • more weight to the central angles (see eta)
  • p_t cutoff due to cuts, very steep falloff due to pdf
  • same picture in inv mass

• compatibilty problematic: just within acceptable limits

  • for p_t and inv mass: low statistic and very steep falloff
  • very sensitive to uncertainties of weights (can be improved by improving accuracy of VEGAS)
  • prompts a more rigorous study of uncertainties in the vegas step!

Pheno Stuff   2m

• non LO effects completely neglected

• sherpa generator allows to model some of them

  • always approximations

Short review of HO Effects

• always introduce stage and effects along with the nice event picture

LO

• same as toy generator

LO+PS

• parton shower CSS (dipole) activated
• radiation of gluons, and splitting into quarks -> shower like cascades QCD
• as there are no QCD particles in FS: initial state radiation
• due to 4-mom conservation: recoil momenta (and energies)

LO+PS+pT

• beam remnants and primordial transverse momenta simulated
• additinal radiation and parton showers

• primordial p_T due to localization of quarks, modeled like gaussian distribution

  • mean, sigma: .8 GeV, standard values in sherpa
  • consistent with the notion of "fermi motion"

LO+PS+pT+Hadronization

• AHADIC activated (cluster hadr)
• jets of parton cluster into hadrons: non perturbative
• models inspired by qcd but still just models
• mainly affects isolation of photons (come back to that)
• in sherpa, unstable are being decayed (using lookup tables) with correct kinematics

LO+PS+pT+Hadronization+MI

• Multiple Interactions (AMISIC) turned on
• no reason for just one single scattering in event
• based on overlap of hadrons and the most important QCD scattering processes
• in sherpa: shower corrections
• generally more particles in FS, affects isolation

Presentation and Discussion of selected Histograms

pT of γγ system

• Parton showers enhance at higher pT
• intrinsic pT at lower pT (around 1GeV)
• some isolation impact

• but highest in phase space cuts

  • increase is almost one percent
  • pT recoils to the diphoton system usually substract pT from one photon -> harder to pass cuts -> amplified through big probability of low pT events!

pT of leading and sub-leading photon

• shape similar to LO
• first photon slight pT boost

• second almost untouched

  • cut bias to select events that have little effect on sub-lead photon

Invariant Mass

• events with lower m are allowed throgh cuts
• events with very high recoil suppressed: colinear limit…

Angular Observables

• mostly untouched
• biggest difference: total xs and details
• but LO gives good qualitative picture
• reasonable, because LO should be dominating

Effects of Hadronization and MI

• fiducial XS differs because of isolation and cuts in the phase space
• we've seen: parton shower affect kinematics and thus the shape of observables and phase space cuts

• isolation critera:

  • photon has to be isolated in detector
  • allow only certain amount of energy in cone around photon
  • force moinimum separation of photons to prevent cone overlap
• Hadronization spreads out FS particles (decay kinematics) and produces particles like muons and neutrinos that aren't detectable or easily filtered out -> decrase in isolation toll
• MI increases hadr activity in FS -> more events filtered out

Summary

• LO gives qualitative picture
• NLO affect observables shape, create new interesting observables
• some NLO effects affect mainly the isolation
• caveat: non-exhaustive, no QED radiation enabled

Wrap-Up

Summary

• calculated XS
• studied and applied simple MC methods
• built a basic working event generator
• looked at what lies beyond that simple generator

Lessons Learned (if any)

• calculations have to be done verbose and explicit
• spending time on tooling is OK
• have to put more time into detailed diagnosis
• event generators are marvelously complex
• should have introduced the term importance sampling properly

Outlook

• more effects
• multi channel mc
• better validation of vegas