An Essential Toolkit for Particle Physics · 2021. 5. 17. · Patrick Komiske – Machine Learning in Particle Physics Improvement to Computational Speed/Efﬁciency with ML 9 Fast

Machine Learning An Essential Toolkit for Particle Physics

Massachusetts Institute of Technology

Center for Theoretical Physics

Patrick T. Komiske III

Snowmass Computational Frontier Workshop – ML Subgroup

August 10, 2020

Lightning Review

Ubiquity of ML in HEP

Future Directions

Patrick Komiske – The Hidden Geometry of Particle Collisions 2

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ΣMD(T , T0)

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Particles Observable

Per-Particle Representation Event Representation

F

Energy/Particle Flow Network

Latent Space

Lightning Review


Future Directions


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ΣMD(T , T0)

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Latent Space

Patrick Komiske – Machine Learning in Particle Physics

Machine Learning Permeates High-Energy Physics

4

*2016

*when I entered HEP

ML in HEP is a young, vibrant, growing field with exciting potential!


Machine Learning Fundamentals

5

The Power of ML

‣ Interpolation in high-dimensional spacesCombats the curse of dimensionality

‣Automatic feature extractionEnsures relevant features are not missed

‣Asymptotically optimizes performanceProvides useful/practical statistical power

Comes at a cost

Loses analytic understandability/tractability

Cannot easily convey what features are used

Training is difficult with few global guarantees

Responsible ML Considerations

‣Available dataData source, number of samples, labels, reliability

‣Learning paradigmFully/weakly/un-supervised, classification/regression/generation

‣ Inputs and outputsSize/shape, symmetries, dimensionality

‣Model architectureExpressibility, loss function, hyperparameters, validation/testing

‣Deployment strategyModel implementation, training/evaluation speed, uncertainties


My Perspective on ML in HEP

6

Machine learning is here to stay in high-energy physics

Calculus

‣ Fundamental to quantitative study of functions

‣ Taught to all undergraduate physics students

‣ Essential for understanding modern physics

Machine Learning

‣ Fundamental to statistical data analysis

‣ Increasingly taught in undergrad science programs

‣ Increasingly essential for modern physics/science

Machine learning straddles theory/experiment/computation divide

Close coordination between theory and experiment is essential in the current era of uncertainty in particle physics

Key Questions During the Snowmass Process‣Where has ML already had an

impact in high-energy physics?

Part II of this talk

‣ What is the role of ML in particle physics (and vice-versa) in the future?

Part III of this talk

Lightning Review


Future Directions


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Latent Space

Disclaimers

‣ Examples biased towards collider physics

‣ References are not exhaustive


“Uncontroversial” Applications of Machine Learning in HEP

ML can provide a uniformly improved, drop-in replacement for some mathematical tasks

8

Efficient Monte Carlo Integration[Bendavid, 1707.00028; Klimek, Perelstein, 1810.11509]

“A continuum implementation of the VEGAS algorithm”

DNN has 10-100x smaller error with the same or fewer samples!

Positive MC-Weight Resampling[Andersen, Gutschow, Maier, Prestel, 2005.09375;

Nachman, Thaler, 2007.11586]

Central value preserved … using only positive weights …

while preserving uncertainty … and requiring fewer events!


Improvement to Computational Speed/Efficiency with ML

9

Fast Electromagnetic Calorimeter Simulation[Paganini, de Oliviera, Nachman, 1705.02355 1712.10321;

Erdmann, Glombitza, Quast, 1807.01954]

Generative models can be O(102-105) x quicker than full detector-simulation

Wasserstein GANs have stabler training and good agreement

GAN

Geant4

Improved MCMC for Lattice Field Theory[Albergo, Kanwar, Shanahan, 1904.12072; Talk by G. Kanwar]

Normalizing flows can be used to sample from complicated distributions

Invertible and tractable Jacobian leads to

Power law growth of autocorrelation time avoided with ML

6 8 10 12 14

0.5

1

2

5

Local Metropolis

6 8 10 12 14

0.5

1

2

5

50% ML models

70% ML models

Flow-based MCMC


Neural Network Architectures for Particle Physics

Maximally appropriate ML architectures respect symmetries of the underlying data

Particle physics events are naturally point clouds (alternatively, images e.g. calorimeters)

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Point cloud: "A set of data points in space" –Wikipedia

LIDAR data from self-driving car sensor

An unordered, variable length collection of particles

Due to quantum-mechanical indistinguishability

Due to probabilistic nature of event formation

Multi-jet event at CMS


Deep Sets for Particle Jets

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Latent Space

PFN({pµ1, . . . , p

µM}) = F

MX

i=1

Φ(pµi )

!

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Particle Flow Network (PFN)

Fully general latent space

EFN({pµ1, . . . , p

µM}) = F

MX

i=1

ziΦ(p̂i)

!

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Energy Flow Network (EFN)

IRC-safe latent space

[Zaheer, Kottur, Ravanbhakhsh, Póczos, Salakhutdinov, Smola, 1703.06114;

PTK, Metodiev, Thaler, 1810.05165;

EnergyFlow Python Package]

0.0 0.2 0.4 0.6 0.8 1.0

Quark Jet Efficiency

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Significance

Improvem

ent

Quark vs. Gluon Jets

Pythia 8.230,√s = 14 TeV

R = 0.4, pT ∈ [500, 550] GeV

PFN-ID

RNN-ID

EFPs

DNN

CNN

Multiplicity

nSD

Improved performance (and training) compared to RNN and CNN

−R −R/2 0 R/2 R

Translated Rapidity y

−R

−R/2

0

R/2

R

Translated

Azimuthal

Angle

¡

Energy Flow Network Latent Space (` = 256)

Latent space visualization reveals what the network has learned

Dynamic pixel sizing related to collinear singularity of QCD!


Other Physics-Inspired Architectures

12

Deep Impact Parameter Sets[ATL-PHYS-PUB-2020-014]

Achieves 2x better flavor tagging than an RNN

“Optimised” includes additional features per track

Dynamic Graph CNNs (e.g. Particle Net)[Wang, Sun, Liu, Sarma, Bronstein, Solomon, 1801.07829;

Qu, Gouskos, 1902.08570]

Preserves permutation symmetry while prioritizing relationships between inputs

Lorentz Group Equivariant NN[Bogatskiy, Anderson, Offermann, Roussi, Miller, Kondor, 2006.04780]


Improved Performance on Classic HEP Tasks with ML

ML optimizes performance

13

(Jet) Classification[Kasieczka, Plehn, et al., 1902.09914;

many many other references…]

Community top-tagging comparison

Regression (e.g. to remove pileup)[PTK, Metodiev, Nachman, Schwartz, 1707.08600;

ATL-PHYS-PUB-2019-028;see also Arjona Martínez, Cerri, Spiropulu, Vlimant, Pierini 1810.07988]

PUMML

(Full) Phase-Space Reweighting[Cranmer, Pavez, Louppe, 1506.02169;Andreassen, Nachman, 1907.08209]

Likelihood-free inference via classification and Neyman-Pearson

DCTR uses a single high-dimensional reweighting…

to match ECF distributions … and multiplicity distributions!

ML can enable an unbinned version of a binned method

High-dimensional reweighting is useful for many tasks …


OmniFold – Unbinned, Full Phase-Space Unfolding

14

[Andreassen, PTK, Metodiev, Nachman, Thaler, 1911.09107;PTK talk at ML4Jets 2020]

ωn(m) = νpushn−1 × L[(1,Data), (νpush

n−1 , Sim)](m)

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νn(t) = νn−1(t)× L[(ωpulln

,Gen), (νn−1,Gen)](t)

<latexit sha1_base64="TGZF0tmc+uEZDQhMlm0LXQYEDM8=">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</latexit>

Step 1 –

Step 2 –

OmniFold – i.e. continuous IBU

p(n)unfolded(t) = νn(t)× pGen(t)

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Unfold any* observable using universal weights pGen(t) νn(t)

*Observables should be chosen responsibly

Detector-level Particle-level

Simulation

Syn

theti

cN

atu

ral Data

Generation

Truth

Pull

Step 1:

νn−1

Data−−−→ ωn

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Step 1 – Reweights Simn-1 to data, pulls weights back to particle-level Genn-1

Push

Step 2:

νn−1

ωn

−−→ νn<latexit sha1_base64="USI/aHUkKNmen4gwHqyTXZ7rTGs=">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</latexit><latexit sha1_base64="USI/aHUkKNmen4gwHqyTXZ7rTGs=">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</latexit><latexit sha1_base64="USI/aHUkKNmen4gwHqyTXZ7rTGs=">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</latexit><latexit sha1_base64="aANR4xqa5G8N4A3PGP5I4W3UWFE=">AAACJ3icdVDdShtBGJ21tmpsa6yX3gwGQYQuuxqSeCd446UF8wPZEGYn3yaD87PMzKrLsu/Q1+gL9La+gXeil73xOZxsItTSHhg4nHM+vvlOnHJmbBA8eSvvVt9/WFvfqG1+/PR5q779pWdUpil0qeJKD2JigDMJXcssh0GqgYiYQz++Opv7/WvQhil5afMURoJMJUsYJdZJ4/phJLNxIb+GJY5uNZvOLNFa3RSREjAlzimdUUXKcb0R+Eet43bYwoHfbJ6ErbYj7VZw0uzg0A8qNNASF+P6czRRNBMgLeXEmGEYpHZUEG0Z5VDWosxASugVmcLQUUkEmFFR3VTifadMcKK0e9LiSv1zoiDCmFzELimInZm/vbn4L2+Y2aQzKphMMwuSLhYlGcdW4XlBeMI0UMtzRwjVzP0V0xnRhFpX45stCeRSpGXNFfN6Pf4/6R35YeCH35qN086yonW0i/bQAQpRG52ic3SBuoii7+gn+oXuvB/evffgPS6iK95yZge9gff7BX9hqAI=</latexit>

Step 2 – Reweights Genn-1 to (step 1)-weighted genn-1, pushes weights to detector-level Simn

OmniFold weights particle-level Gen to be consistent with Data once passed through the detector

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Norm

alizedCross

Section

D/T: Herwig 7.1.5 default

S/G: Pythia 8.243 tune 26

Delphes 3.4.2 CMS Detector

Z+jet: pZT

> 200 GeV, R = 0.4

“Data”

“Truth”

Sim.

Gen.

IBU ln ρ

OmniFold

−14 −12 −10 −8 −6 −4 −2

Soft Drop Jet Mass ln ρ

0.85

1.0

1.15

Ratioto

Tru

th

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Norm

alizedCross

Section

D/T: Herwig 7.1.5 default

S/G: Pythia 8.243 tune 26

Delphes 3.4.2 CMS Detector

Z+jet: pZT

> 200 GeV, R = 0.4

“Data”

“Truth”

Sim.

Gen.

IBU τ(β=1)21

OmniFold

0.0 0.2 0.4 0.6 0.8 1.0 1.2

N -subjettiness Ratio τ(β=1)21

0.85

1.0

1.15

Ratioto

Tru

th

IRC safe Sudakov safe

Patrick Komiske – The Hidden Geometry of Particle Collisions

Non-Parametric ML – The Energy Mover’s Distance (EMD)

15

[PTK, Metodiev, Thaler, PRL 2019;

applied on CMS Open Data: PTK, Mastandrea, Metodiev, Naik, Thaler, 1908.08542]

EMD between energy flows defines a metric on the space of events

Triangle inequality

0 ≤ EMD(E , E 0) ≤ EMD(E , E 00) + EMD(E 00, E

0)<latexit sha1_base64="+zmPsKcG0R3U9IVeOZ97W+27Rbw=">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</latexit>

EMD: 125.4 GeV

Top Jet 1 Top Jet 2

101 102 103

Energy Scale Q (GeV)

0

1

2

3

4

5

6

7

8

CorrelationDim

ension

EMD: Intrinsic Dimension


R = 1.0, pT ∈ [500, 550] GeV

Top jets

W jets

QCD jets

Hadrons

Partons

Decays

EMD is the work required to rearrange one event into another

Intrinsic dimensionality of dataset highlights physics at all scales


ML Enables New Theoretical Paradigms – Six Decades of Collider Techniques as Geometry

16[Slide from talk by Eric Metodiev]

[PTK, Metodiev, Thaler, 2004.04159]

*See backup for more on IRC safety and perturbative calculability


Handling Uncertainties

ML can assist in handling uncertainties (but can also create its own difficulties)

17

Bayesian neural networks can estimate some uncertainties[Bollweg, Haußmann, Kasieczka, Luchmann, Plehn, Thompson, 1904.10004;

Kasieczka, Luchmann, Otterpohl, Plehn, 2003.11099]

Adversarial training to reduce dependence on uncertain regions[Englert, Galler, Harris, Spannowsky, 1807.08763]

[Nachman, 1909.03081]

Sources of uncertainty in a statistical analysis

Parametrized models could enable efficient profiling to handle systematic uncertainties

[similar to Baldi, Cranmer, Faucett, Sadowski, Whiteson, 1601.07913]


… And So Much More HEP-Specific ML

18

Unsupervised Learning

e.g. JUNIPR[Andreassen, Feige, Frye, Schwartz, 1804.09720, 1906.10137]

Probabilistic formation of a Pythia jet

Anomaly Detection

tons of recent work[Collins, Howe, Nachman, 1805.02664;

Farina, Nakai, Shih, 1808.08992;Heimel, Kasieczka, Plehn, Thompson, 1808.08979;

Cerri, Nguyen, Pierini, Spiropulu, Vlimant, 1811.10276;Blance, Spannowsky, Waite, 1905.10384;

See LHC Olympics 2020 anomaly detection workshop]

CWoLa hunting to enhance a resonance search

Weakly Supervised Learning

e.g. LLP, CWoLa[Dery, Nachman, Rubbo, Schwartzman, 1702.00414;

Metodiev, Nachman, Thaler, 1708.02949;Cohen, Freytsis, Ostdiek, 1706.09451;

PTK, Metodiev, Nachman, Schwartz, 1801.10158]

Topic Modeling

e.g. Jet Topics[Metodiev, Thaler, 1802.00008;

PTK, Metodiev, Thaler, 1809.01140]

Decorrelation Methods

e.g. DDT, Planing, DisCo[Dolen, Harris, Marzani, Rappoccio, Tran, 1603.00027;

Chang, Cohen, Ostdiek, 1709.10106;Kasieczka, Shih, 2001.05310;

Kasieczka, Nachman, Schwartz, Shih, 2007.14400]

Decorrelating ML taggers from a key observable

Lightning Review


Future Directions


T

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T0

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ΣMD(T , T0)

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...

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F


Latent Space


Thoughts on the Future

20

‣Machine learning will be essential in maximizing HEP potentialWe should capitalize on the opportunity to optimize

ML is both a computational tool and a useful formalism/language

‣Diversity of applications is impressive and will become more soRapidly advancing beyond “hammer and nail” approach for ML in HEP

New observables, architectures, paradigms have been developed

‣Collaboration across traditional lines will enable successLearn from and contribute to the highly-vibrant ML community

ML in HEP needs insight from theory and experiment

‣ML strongly overlaps with other computational frontier areasSoftware workflows, reproducible analyses, public datasets are critical

[Reviews of Modern Physics Cover December 2019from Machine learning and the physical sciences]

HEPML-LivingReview has a thorough and organized list of papers


Thank You!

21


EnergyFlow Python Package

22

Keras/Tensorflow implementations of Energy/Particle Flow Networks

Interfaces with reprocessed CMS 2011A Jet Primary Dataset hosted on Zenodo

Detailed examples, demos, and documentation

pip3 install energyflow


Iterated Bayesian Unfolding (IBU)

Histogram-based unfolding method for a small number of observables

23

[Richardson, 1972; Lucy, 1974; D’Agostini, 1995]

Choose observable(s) and binning at detector-level and particle-level

measured distribution: true distribution: mi = Pr(measure i)

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t(0)j = Pr(truth is j)

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Rij = Pr(measure i | truth is j)

<latexit sha1_base64="ETtI5OCzflED9PlkUZ1h2OWyS3I=">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</latexit>

is in general non-square and non-invertibleR

Calculate response matrix from generated/simulated pairs of events Rij

t(n)j =

X

i

Pr(truthn−1 is j | measure i)× Pr(measure i)

<latexit sha1_base64="ADBnR1bWljaQEBAERUGHp8Bz/gE=">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</latexit>

=

X

i

Rijt(n−1)j

Pk Rikt

(n−1)k

×mi

<latexit sha1_base64="qULT0dK3m1jeMPNqms4zJs2e/qE=">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</latexit>

Calculate new particle-level distribution using Bayes’ theorem

Iterate procedure to remove dependence on prior


Visualizing Geometry in the Space of Events

24

t-Distributed Stochastic Neighbor Embedding (t-SNE)MNIST handwritten digits

[L. van der Maaten, G. Hinton, JMLR 2008 ]

4 7

9

6

5

3

8

0 2

1

74

5

1

89

3

0 2

6

Gray contours represent the density of jets

Each circle is a particular W jet

Geometric space of W jets

"bottom heavy""o

ne

pro

ng"

"balan

ced"

"top heavy"

Constraints: W Mass and

𝜙 = 0 preprocessing

z

1 – z

𝜃

z

1 – z𝜙

W Jet

[PTK, Metodiev, Thaler, PRL 2019]


101 102 103


0

1

2

3

4

5

6

7

8

CorrelationDim

ension



R = 1.0, pT ∈ [500, 550] GeV

Top jets

W jets

QCD jets

Hadrons

Partons

Decays

Quantifying Event-Space Manifolds

25

[Grassberger, Procaccia, PRL 1983; PTK, Metodiev, Thaler, PRL 2019]

Correlation dimension: how does the # of elements within a ball of size Q change?

dim(Q) = Q∂

∂Qln

X

i

X

j

Θ(EMD(E i, E0

j) < Q)

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Nneigh.(Q) ∝ Qdim =⇒ dim(Q) = Qd

dQlnNneigh.(Q)

<latexit sha1_base64="TwWtGcO0QQbG7yKasfTozXEthQo=">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</latexit>

dim ≃ 1 dim ≃ 2

dim ≃ 0dim → 0

Correlation dimension lessons:Decays are "constant" dim. at low QComplexity hierarchy: QCD < W < TopFragmentation increases dim. at smaller scalesHadronization important around 20-30 GeV

*Preliminary LL calculation for QG jets in backup


101 102 103


0

1

2

3

4

5

6

7

8

CorrelationDim

ension



R = 1.0, pT ∈ [500, 550] GeV

Top jets

W jets

QCD jets

Hadrons

Partons

Decays

Quantifying Event-Space Manifolds

26

[Grassberger, Procaccia, PRL 1983; PTK, Metodiev, Thaler, PRL 2019]

Correlation dimension: how does the # of elements within a ball of size Q change?

dim(Q) = Q∂

∂Qln

X

i

X

j

Θ(EMD(E i, E0

j) < Q)

<latexit sha1_base64="XKuBaJ7fsvZeohVRYf7hETWOHbM=">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</latexit>

Nneigh.(Q) ∝ Qdim =⇒ dim(Q) = Qd

dQlnNneigh.(Q)

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dim ≃ 1 dim ≃ 2

dim ≃ 0dim → 0

Correlation dimension lessons:Decays are "constant" dim. at low QComplexity hierarchy: QCD < W < TopFragmentation increases dim. at smaller scalesHadronization important around 20-30 GeV

*Preliminary LL calculation for QG jets in backup

101 102

Energy Scale Q [GeV]

0

1

2

3

4

5

6

CorrelationDim

ension

AK5 Jets, |ηjet| < 1.9

pjetT

∈ [399, 401] GeV

CHS, Tracks, pPFCT

> 1 GeV

Scaled to 400 GeV, Rotated

CMS 2011 Open Data

CMS 2011 Simulation

Pythia 6 Generation

… in CMS Open Data

*More in backup


Visualizing Geometry in CMS Open Data

nth moment of EMD distribution for a dataset

27

50 100 150 200

Mean EMD to Dataset Q1,track [GeV]

10−4

10−3

10−2

10−1

100

101

102DifferentialCross

Section

[pb/G

eV]

CMS 2011 Open Data

AK5 Jets, |ηjet| < 1.9

pjetT

∈ [399, 401] GeVCHS, pPFC

T> 1 GeV, Tracks

Rotated, Scaled to 400 GeV Q̄n(I) =n

v

u

u

t

1

N

NX

k=1

(EMD(I,Jk))n

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EMD for anomaly detection

4 medoids in each bin of anomaliness Q̄1

[PTK, Mastandrea, Metodiev, Naik, Thaler, PRD 2019; code and datasets at energyflow.network]

k-eventiness?

How far does this go?

Vk =1

N

NX

i=1

min {EMD(Ji,K1), . . . ,EMD(Ji,Kk)}

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jet from dataset medoids


N-particle Manifolds in the Space of Events

= set of all N-particle configurations = PN

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28

(

NX

i=1

Ei δ(n̂− n̂i)

�

�

�

�

�

Ei ≥ 0

)

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−R −R/2 0 R/2 R

Rapidity y

−R

−R/2

0

R/2

R

AzimuthalAngle

φ

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P3

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P2<latexit sha1_base64="HCtUNp3JQxizVnLfnOcMYXRPGOE=">AAACDHicdVDLSgMxFL3js9ZXfezcBIvgqmRa7WNXcOOygq1COwyZNKPBTGZIMkIZ5hf8Abf6B+7Erf/gD/gdZloFFT0QOJxzL/fkBIng2mD85szNLywuLZdWyqtr6xubla3tgY5TRVmfxiJWlwHRTHDJ+oYbwS4TxUgUCHYR3JwU/sUtU5rH8txMEuZF5ErykFNirORXNkcRMdeUiKyX+1k99ytVXMPN404DI1w7xm6r07EE42a7UUeuJQWq3V2YoudX3kfjmKYRk4YKovXQxYnxMqIMp4Ll5VGqWULoDbliQ0sliZj2smnwHB1YZYzCWNknDZqq3zcyEmk9iQI7WcTUv71C/MsbpiZsexmXSWqYpLNDYSqQiVHRAhpzxagRE0sIVdxmRfSaKEKN7erHlZBNZJTkZVvM1+/R/2RQr7m45p4dVbvtWUNQgj3Yh0NwoQVdOIUe9IFCCvfwAI/OnfPkPDsvs9E553NnB37Aef0AdLWcWQ==</latexit><latexit sha1_base64="HCtUNp3JQxizVnLfnOcMYXRPGOE=">AAACDHicdVDLSgMxFL3js9ZXfezcBIvgqmRa7WNXcOOygq1COwyZNKPBTGZIMkIZ5hf8Abf6B+7Erf/gD/gdZloFFT0QOJxzL/fkBIng2mD85szNLywuLZdWyqtr6xubla3tgY5TRVmfxiJWlwHRTHDJ+oYbwS4TxUgUCHYR3JwU/sUtU5rH8txMEuZF5ErykFNirORXNkcRMdeUiKyX+1k99ytVXMPN404DI1w7xm6r07EE42a7UUeuJQWq3V2YoudX3kfjmKYRk4YKovXQxYnxMqIMp4Ll5VGqWULoDbliQ0sliZj2smnwHB1YZYzCWNknDZqq3zcyEmk9iQI7WcTUv71C/MsbpiZsexmXSWqYpLNDYSqQiVHRAhpzxagRE0sIVdxmRfSaKEKN7erHlZBNZJTkZVvM1+/R/2RQr7m45p4dVbvtWUNQgj3Yh0NwoQVdOIUe9IFCCvfwAI/OnfPkPDsvs9E553NnB37Aef0AdLWcWQ==</latexit><latexit sha1_base64="HCtUNp3JQxizVnLfnOcMYXRPGOE=">AAACDHicdVDLSgMxFL3js9ZXfezcBIvgqmRa7WNXcOOygq1COwyZNKPBTGZIMkIZ5hf8Abf6B+7Erf/gD/gdZloFFT0QOJxzL/fkBIng2mD85szNLywuLZdWyqtr6xubla3tgY5TRVmfxiJWlwHRTHDJ+oYbwS4TxUgUCHYR3JwU/sUtU5rH8txMEuZF5ErykFNirORXNkcRMdeUiKyX+1k99ytVXMPN404DI1w7xm6r07EE42a7UUeuJQWq3V2YoudX3kfjmKYRk4YKovXQxYnxMqIMp4Ll5VGqWULoDbliQ0sliZj2smnwHB1YZYzCWNknDZqq3zcyEmk9iQI7WcTUv71C/MsbpiZsexmXSWqYpLNDYSqQiVHRAhpzxagRE0sIVdxmRfSaKEKN7erHlZBNZJTkZVvM1+/R/2RQr7m45p4dVbvtWUNQgj3Yh0NwoQVdOIUe9IFCCvfwAI/OnfPkPDsvs9E553NnB37Aef0AdLWcWQ==</latexit><latexit sha1_base64="KOxF7/xLsYdqqJTwGj/l/E7wiKw=">AAACDHicdVDLSgMxFM34rPXRqks3wSK4KpnWvnYFNy4r2Ae0w5BJM21oJjMkGaEM8wv+gFv9A3fi1n/wB/wOM20FK3ogcDjnXu7J8SLOlEbow9rY3Nre2c3t5fcPDo8KxeOTngpjSWiXhDyUAw8rypmgXc00p4NIUhx4nPa92XXm9++pVCwUd3oeUSfAE8F8RrA2klssjAKspwTzpJO6SSV1iyVURvVaq4ogKteQ3Wi1DEGo3qxWoG1IhhJYoeMWP0fjkMQBFZpwrNTQRpF2Eiw1I5ym+VGsaITJDE/o0FCBA6qcZBE8hRdGGUM/lOYJDRfqz40EB0rNA89MZjHVby8T//KGsfabTsJEFGsqyPKQH3OoQ5i1AMdMUqL53BBMJDNZIZliiYk2Xa1d8elcBFGaN8V8/x7+T3qVso3K9u1Vqd1cVZQDZ+AcXAIbNEAb3IAO6AICYvAInsCz9WC9WK/W23J0w1rtnII1WO9f+/qcBQ==</latexit>

P1

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: manifold of events with two particlesP2

<latexit sha1_base64="P56Ieon4CUqn4FN83EWloqcl8tM=">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</latexit>

: manifold of events with one particle P1

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: manifold of events with three particlesP3

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…

PN ⊃ PN−1 ⊃ · · · ⊃ P3 ⊃ P2 ⊃ P1

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by soft and collinear limits

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Uniform event, not contained in any PN



N-particle Manifolds in the Space of Events

= set of all N-particle configurations = PN

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29

(

NX

i=1

Ei δ(n̂− n̂i)

�

�

�

�

�

Ei ≥ 0

)

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dPi→ig '

2αs

πCa

dθ

θ

dz

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sha1_base64="2jWaCQ/7ibZA4T9XCP9LqvIisJQ=">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</latexit><latexit sha1_base64="2jWaCQ/7ibZA4T9XCP9LqvIisJQ=">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</latexit>

PN

<latexit sha1_base64="JsZOYfja2vHFGx/cDtLrSJZ4kCM=">AAACDHicdVDLSgMxFL1T3/XR8bFzEyyCqyFTq627ghtXUsGq0JaSSTNtaCYzJBmhDPML/oBb/QN34tZ/8Af8DtNWQUUPBA7n3Ms9OUEiuDYYvzmFufmFxaXlleLq2vpGyd3cutJxqihr0VjE6iYgmgkuWctwI9hNohiJAsGug9HpxL++ZUrzWF6accK6ERlIHnJKjJV6bqkTETOkRGTNvJed5z23jD18XPUrPsLeEfbrFTwjJ7VD5Ht4inJjB6Zo9tz3Tj+macSkoYJo3fZxYroZUYZTwfJiJ9UsIXREBqxtqSQR091sGjxH+1bpozBW9kmDpur3jYxEWo+jwE5OYurf3kT8y2unJqx3My6T1DBJZ4fCVCATo0kLqM8Vo0aMLSFUcZsV0SFRhBrb1Y8rIRvLKMmLtpiv36P/yVXF87HnX1TLjfqsIViGXdiDA/ChBg04gya0gEIK9/AAj86d8+Q8Oy+z0YLzubMNP+C8fgCFmZxj</latexit><latexit sha1_base64="JsZOYfja2vHFGx/cDtLrSJZ4kCM=">AAACDHicdVDLSgMxFL1T3/XR8bFzEyyCqyFTq627ghtXUsGq0JaSSTNtaCYzJBmhDPML/oBb/QN34tZ/8Af8DtNWQUUPBA7n3Ms9OUEiuDYYvzmFufmFxaXlleLq2vpGyd3cutJxqihr0VjE6iYgmgkuWctwI9hNohiJAsGug9HpxL++ZUrzWF6accK6ERlIHnJKjJV6bqkTETOkRGTNvJed5z23jD18XPUrPsLeEfbrFTwjJ7VD5Ht4inJjB6Zo9tz3Tj+macSkoYJo3fZxYroZUYZTwfJiJ9UsIXREBqxtqSQR091sGjxH+1bpozBW9kmDpur3jYxEWo+jwE5OYurf3kT8y2unJqx3My6T1DBJZ4fCVCATo0kLqM8Vo0aMLSFUcZsV0SFRhBrb1Y8rIRvLKMmLtpiv36P/yVXF87HnX1TLjfqsIViGXdiDA/ChBg04gya0gEIK9/AAj86d8+Q8Oy+z0YLzubMNP+C8fgCFmZxj</latexit><latexit sha1_base64="JsZOYfja2vHFGx/cDtLrSJZ4kCM=">AAACDHicdVDLSgMxFL1T3/XR8bFzEyyCqyFTq627ghtXUsGq0JaSSTNtaCYzJBmhDPML/oBb/QN34tZ/8Af8DtNWQUUPBA7n3Ms9OUEiuDYYvzmFufmFxaXlleLq2vpGyd3cutJxqihr0VjE6iYgmgkuWctwI9hNohiJAsGug9HpxL++ZUrzWF6accK6ERlIHnJKjJV6bqkTETOkRGTNvJed5z23jD18XPUrPsLeEfbrFTwjJ7VD5Ht4inJjB6Zo9tz3Tj+macSkoYJo3fZxYroZUYZTwfJiJ9UsIXREBqxtqSQR091sGjxH+1bpozBW9kmDpur3jYxEWo+jwE5OYurf3kT8y2unJqx3My6T1DBJZ4fCVCATo0kLqM8Vo0aMLSFUcZsV0SFRhBrb1Y8rIRvLKMmLtpiv36P/yVXF87HnX1TLjfqsIViGXdiDA/ChBg04gya0gEIK9/AAj86d8+Q8Oy+z0YLzubMNP+C8fgCFmZxj</latexit><latexit sha1_base64="7cLdUU0n89KEDkB05clWTPEiXwM=">AAACDHicdVDLSgMxFM34rPXRUZdugkVwNWRqtXVXcONKKtgHtEPJpJk2NJMZkoxQhvkFf8Ct/oE7ces/+AN+h5m2ghU9EDiccy/35PgxZ0oj9GGtrK6tb2wWtorbO7t7JXv/oK2iRBLaIhGPZNfHinImaEszzWk3lhSHPqcdf3KV+517KhWLxJ2extQL8UiwgBGsjTSwS/0Q6zHBPG1mg/QmG9hl5KCLqltxIXLOkVuvoDm5rJ1B10EzlMECzYH92R9GJAmp0IRjpXouirWXYqkZ4TQr9hNFY0wmeER7hgocUuWls+AZPDHKEAaRNE9oOFN/bqQ4VGoa+mYyj6l+e7n4l9dLdFD3UibiRFNB5oeChEMdwbwFOGSSEs2nhmAimckKyRhLTLTpaulKQKcijLOiKeb79/B/0q44LnLc22q5UV9UVABH4BicAhfUQANcgyZoAQIS8AiewLP1YL1Yr9bbfHTFWuwcgiVY718M7ZwP</latexit>

PN−1

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– Infrared Divergences

=

✏ → 0<latexit sha1_base64="+7VgSN6bIS94rg923ECXL9Ylmgg=">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</latexit>

=λ

<latexit sha1_base64="wXk7AmmngymkoNcky3vNULvpq4c=">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</latexit>

1− λ<latexit sha1_base64="feDmrmhe5e0qNCrjBVSp9Fat0fI=">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</latexit>

Energy flow is unchanged by exact soft/collinear emissions

PN ⊃ PN−1 ⊃ · · · ⊃ P3 ⊃ P2 ⊃ P1

<latexit sha1_base64="7R/N0re+hPKVdr2pFymV3jerLd4=">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</latexit>

by soft and collinear limits

Functions of energy flow automatically satisfy exact IRC invariance!

Real and virtual divergences appear naturally together

✏ → 0<latexit sha1_base64="+7VgSN6bIS94rg923ECXL9Ylmgg=">AAAFKXicdZTLjtMwFIY9MwGGcuvAkk1EhcQCVXGaNu1uBBuWg0RnRmqqynFOM1Fzk+PAVFG2vAxs4T3YAVuegDfAcdKmTYqVyJb9nZzf/3Fsx76XcE37dXR8oty5e+/0fufBw0ePn3TPnl4mUcooTGnkR+zaJgn4XghT7nEfrmMGJLB9uLJXb4v1q4/AEi8KP/B1DPOAuKG39CjhYmrRVS0Ot1x+J4sYCV3IMwvixPOj0OKRli+6Pa2vyaa2B7ga9FDVLhZnJ38tJ6JpACGnPkmSGdZiPs8I4x71Ie9YaQIxoSviQnYrM+fqSzHnqMuIiTfkqpzdA0mQJOvAFmRA+E3SXCsmD67ZjKyA7yWdpXw5nmdeGKccQlomX6a+yiO1sEh1PAaU+2sxIJR5QrZKbwgjlAsjOx3LgaVwu/QsTlksNpUx184zrT80R6+FO3iAi25oGvk+vrG4wsfaWHKaxPURbuCc2C4DCLcB2BwW5EifyDSm3g6w/bROgHWjAI2RlGVqZptn4NR6DCkED42ya+K77GhQatB1qV1rbtVlZL2FB6ZWUHWXt2yE+uPiVAlMPAeo3e1JSMACC+ETjYKAhE5mVTWZ4Xm2c7o3lerhvMFXRWnwm1K1+UJoA5ba22RZvAZbVbRNb6vdCKhPwaEMwuVWgsL5NrsxuW0M/Ef/1vADIWUhZExHXBK4vhsmoplGNZjg7SVxqYsfo6+/N3rnb6rr4hQ9Ry/QK4SRic7RO3SBpoiiz+gr+oa+K1+UH8pP5XeJHh9VMc/QXlP+/AMEzb76</latexit> λ





Defining IRC Safety Precisely

Infrared and collinear safety is a proxy for perturbative calculability of an observable

30

O(pµ1, . . . , p

µM ) = O(0pµ

0, p

µ1, . . . , p

µM )

O(pµ1, . . . , p

µM ) = O(λpµ

1, (1− λ)pµ

1, . . . , p

µM )

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Exact IRC invariance

Guarantees observable is well-defined on energy flows

Allows for pathological observables, e.g. pseudo-multiplicity

O(pµ1, . . . , p

µM ) = lim

✏→0

O(✏pµ0, p

µ1, . . . , p

µM )

O(pµ1, . . . , p

µM ) = lim

pµ

0→p

µ

1

O(�pµ0, (1− �)pµ

1, . . . , p

µM )

<latexit sha1_base64="mQ0QVF/Ecr7NfTXvr+0eZMnP9FI=">AAAIPnicfZVNj9M4GIBTvgpldxngyCVixGpYdSs7TTvJAQkGKBwY8SEGkCYlchI3E43jRI7DUEW5cYXfxN/gD3Bb7ZUjzteQOB2iRnrz+nm/XduJSZBwAL4Nzp2/cPHS8PKV0dU//vzr2tb1G2+SKGUuPnAjErF3DkowCSg+4AEn+F3MMAodgt86xw+L9bcfMEuCiL7m6xgvQ+TTYBW4iAuVvfXDChE/chFRn+/ENnxvhelYtYgX8WQc2/vF913173tCFYR2ZuE4CUhELR6BvGXZ6NXYBpWLU1+NM7XxZlmj38RU+0FrnyJm47UTmohiPSSWMpBXrnbgv7X2rnqmf3trG0xA+ah9AdbCtlI/L+zrlyaWF7lpiCl3CUqSQwhivswQ44FLcD6y0gTHyD1GPj4UIkUhTpZZOaJcvSM0nrqKmHgpV0tt2yJDYZKsQ0eQRW2JvFYoN60dpnxlLLOAxinH1K0CrVKiim4V81a9gGGXk7UQkMsCkavqHiGGXC52xagTpkyKOyTvqj9WJYwsD6/ELiu/sogh6uM8e/p6/1meGaZhwrmEkMA/Et7SUwpopmbqEhWnLCanyBxA8MCUEIa9Zv3xY00DD6X1TgzxLBYSIPa8GBpqmMUCAEPO1mcY04aAe8YMPJK9BLyVyYMpXEzlYjzEjtvZaNpMM2Q/PkPrPHv1ZC/PTH1c/oqOU3ziRmGIqJdZEaN+fgiXmcXxR97t+DbM8y5NHJlttb6PFx2XDZop9OmiZAkuu9Any3gSelYOoU+5zJ5OqY/7jMp0Pa8NrjnrZdyMrk97xcBk/NcYN6UiptfLpZhoyXZHzZE4h+0yNPMdsTsnMwDNORg3B00pGHAK802Ga0xIdNKxnZnjSpiWtlDfnck7ubKt+1ObgvmsshCmu6ZZhp0bU22jKceItCx1qMHK0tBqF+bu9MxSbcevd7emm2NNn4p3c5yqPJkvbc4uqY3Palz+G/4qo03rNS2Ol9Gd9gHn4yjEnK1zoW7kDOkxijEbF1M+CTx+dG86mQVURHqExfHP8L44ip8LAvGI/ZNZiPmhWBbXgW+NC2kkbhgo3yd94Y02gfpEf6lv39+r75rLyi3ltrKjQGVXua88VV4oB4o7eD/4NPg8+DL8Ovw+/G/4f4WeG9Q2N5XOM/zxE3yJ3O8=</latexit>

Smooth IRC invariance

Eliminates common observables with hard boundaries

[Sterman, Weinberg, PRL 1997; Sterman, PRD 1978; Banfi, Salam, Zanderighi, JHEP 2005]


More EMD Geometry – Continuity in the Space of Events

31

E

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O

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E0

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δ

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An observable O is EMD continuous at an event E if, forany ✏ > 0, there exists a � > 0 such that for all events E 0:

EMD(E , E 0) < � =⇒ |O(E)−O(E 0)| < ✏.

<latexit sha1_base64="i51gici9SArRx4fYDtAj5Q4j0wQ=">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</latexit>

Classic definition of continuity in a metric spaceϵ − δ


IRC Safety = EMD Continuity*

Towards a geometric definition of IRC Safety

*on all but a negligible set‡ of events

‡a negligible set is one that contains no positive-radius EMD-ball

…


Perturbation Theory in the Space of Events

32

Infrared singularities of massless gauge theories appear on each PN

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Soft Collinear

PN

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PN−1

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Sudakov safety

Some observables have discontinuities on PN for some N

A resummed IRC-safe companion can mitigate the divergences

Event geometry suggests N-(sub)jettiness as universal companion

p(OSudakov) =

ZdOComp. p(OSudakov|OComp.) p(OComp.)

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[Larkoski, Thaler, JHEP 2014; Larkoski, Marzani, Thaler, PRD 2015]


Fixed-order calculability

Is a statement of integrability on each PN

EMD continuity must be upgraded to EMD-Hölder continuity on each PN

Example: is EMD continuous but not EMD Hölder continuous (it is Sudakov safe)

limE!E0

O(E)−O(E 0)

EMD(E , E 0)c= 0, c > 0

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[Sterman, PRD 1979; Banfi, Salam, Zanderighi, JHEP 2005]

V (E) = T2(E)

✓

1 +1

lnE(E)/T3(E)

◆

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Hierarchy of IRC Safety Definitions

33

All Observables Measurable at a collider

Defined on Energy Flows Invariant to exact infrared & collinear emissions everywhere except a negligible set of events

Infrared & Collinear Safe EMD continuous everywhere except a negligible set of events

EMD Hölder Continuous Everywhere invariant to infinitesimal

infrared & collinear emissions

Sudakov Safe Discontinuous on some

N-particle manifolds


Documents

An Essential Toolkit for Particle Physics · 2021. 5. 17. · Patrick Komiske – Machine Learning in Particle Physics Improvement to Computational Speed/Efﬁciency with ML 9 Fast