An Essential Toolkit for Particle Physics · 2021. 5. 17. · Patrick Komiske – Machine Learning...
Transcript of An Essential Toolkit for Particle Physics · 2021. 5. 17. · Patrick Komiske – Machine Learning...
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
T
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T0
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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
Ubiquity of ML in HEP
Future Directions
Patrick Komiske – The Hidden Geometry of Particle Collisions 3
T T0
ΣMD(T , T0)
...
...
...
Particles Observable
Per-Particle Representation Event Representation
F
Energy/Particle Flow Network
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!
Patrick Komiske – Machine Learning in Particle Physics
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
Patrick Komiske – Machine Learning in Particle Physics
My Perspective on ML in HEP
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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
Ubiquity of ML in HEP
Future Directions
Patrick Komiske – The Hidden Geometry of Particle Collisions 7
T T0
ΣMD(T , T0)
...
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Particles Observable
Per-Particle Representation Event Representation
F
Energy/Particle Flow Network
Latent Space
Disclaimers
‣ Examples biased towards collider physics
‣ References are not exhaustive
Patrick Komiske – Machine Learning in Particle Physics
“Uncontroversial” Applications of Machine Learning in HEP
ML can provide a uniformly improved, drop-in replacement for some mathematical tasks
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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!
Patrick Komiske – Machine Learning in Particle Physics
Improvement to Computational Speed/Efficiency with ML
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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
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Local Metropolis
6 8 10 12 14
0.5
1
2
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50% ML models
70% ML models
Flow-based MCMC
Patrick Komiske – Machine Learning in Particle Physics
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
Patrick Komiske – Machine Learning in Particle Physics
Deep Sets for Particle Jets
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Particles Observable
Per-Particle Representation Event Representation
F
Energy/Particle Flow Network
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)
!
<latexit sha1_base64="LJYIOSy+p3EJ0LfNjUDkHZjl3q4=">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</latexit><latexit sha1_base64="LJYIOSy+p3EJ0LfNjUDkHZjl3q4=">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</latexit><latexit sha1_base64="LJYIOSy+p3EJ0LfNjUDkHZjl3q4=">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</latexit>
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!
Patrick Komiske – Machine Learning in Particle Physics
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]
Patrick Komiske – Machine Learning in Particle Physics
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 …
Patrick Komiske – Machine Learning in Particle Physics
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)
<latexit sha1_base64="Y9jnBAHs2CS9VmAdmFCxThiBfTo=">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</latexit>
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=">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</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
Pythia 8.235,√s = 14 TeV
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
Patrick Komiske – The Hidden Geometry of Particle Collisions
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
Patrick Komiske – Machine Learning in Particle Physics
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]
Patrick Komiske – Machine Learning in Particle Physics
… 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
Ubiquity of ML in HEP
Future Directions
Patrick Komiske – The Hidden Geometry of Particle Collisions 19
T
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T0
<latexit sha1_base64="UYzj50ehFJVZ4zFLWtoVED1TcCs=">AAACB3icdVDLSsNAFL3xWeurPnZuBovoKiQitO4KblxW6AvaUCbTSTt0MgkzEyGEfoA/4Fb/wJ249TP8Ab/DSapgfRwYOJxzL/fM8WPOlHacN2tpeWV1bb20Ud7c2t7Zreztd1SUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT69yv3tLpWKRaOk0pl6Ix4IFjGBtpMEgxHpCMM9as9NhperYlzXHAP0mru0UqDYOoUBzWHkfjCKShFRowrFSfdeJtZdhqRnhdFYeJIrGmEzxmPYNFTikysuKzDN0YpQRCiJpntCoUL9vZDhUKg19M5lnVD+9XPzL6yc6qHsZE3GiqSDzQ0HCkY5QXgAaMUmJ5qkhmEhmsiIywRITbWpauBLQVITxrGyK+fo9+p90zm3Xsd2bi2qjPm8ISnAEx3AGLtSgAdfQhDYQiOEeHuDRurOerGfrZT66ZH3uHMACrNcPJ++akw==</latexit><latexit sha1_base64="UYzj50ehFJVZ4zFLWtoVED1TcCs=">AAACB3icdVDLSsNAFL3xWeurPnZuBovoKiQitO4KblxW6AvaUCbTSTt0MgkzEyGEfoA/4Fb/wJ249TP8Ab/DSapgfRwYOJxzL/fM8WPOlHacN2tpeWV1bb20Ud7c2t7Zreztd1SUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT69yv3tLpWKRaOk0pl6Ix4IFjGBtpMEgxHpCMM9as9NhperYlzXHAP0mru0UqDYOoUBzWHkfjCKShFRowrFSfdeJtZdhqRnhdFYeJIrGmEzxmPYNFTikysuKzDN0YpQRCiJpntCoUL9vZDhUKg19M5lnVD+9XPzL6yc6qHsZE3GiqSDzQ0HCkY5QXgAaMUmJ5qkhmEhmsiIywRITbWpauBLQVITxrGyK+fo9+p90zm3Xsd2bi2qjPm8ISnAEx3AGLtSgAdfQhDYQiOEeHuDRurOerGfrZT66ZH3uHMACrNcPJ++akw==</latexit><latexit sha1_base64="UYzj50ehFJVZ4zFLWtoVED1TcCs=">AAACB3icdVDLSsNAFL3xWeurPnZuBovoKiQitO4KblxW6AvaUCbTSTt0MgkzEyGEfoA/4Fb/wJ249TP8Ab/DSapgfRwYOJxzL/fM8WPOlHacN2tpeWV1bb20Ud7c2t7Zreztd1SUSELbJOKR7PlYUc4EbWumOe3FkuLQ57TrT69yv3tLpWKRaOk0pl6Ix4IFjGBtpMEgxHpCMM9as9NhperYlzXHAP0mru0UqDYOoUBzWHkfjCKShFRowrFSfdeJtZdhqRnhdFYeJIrGmEzxmPYNFTikysuKzDN0YpQRCiJpntCoUL9vZDhUKg19M5lnVD+9XPzL6yc6qHsZE3GiqSDzQ0HCkY5QXgAaMUmJ5qkhmEhmsiIywRITbWpauBLQVITxrGyK+fo9+p90zm3Xsd2bi2qjPm8ISnAEx3AGLtSgAdfQhDYQiOEeHuDRurOerGfrZT66ZH3uHMACrNcPJ++akw==</latexit><latexit sha1_base64="eE4yOUN4thEvzVLQ7QphRH0EVIo=">AAACB3icdVDLSgMxFM34rPVVdekmWERXw4wIrbuCG5cV+oLOUDJppg1NMiHJCMPQD/AH3OofuBO3foY/4HeYaStYHwcCh3Pu5Z6cSDKqjee9Oyura+sbm6Wt8vbO7t5+5eCwo5NUYdLGCUtUL0KaMCpI21DDSE8qgnjESDeaXBd+944oTRPRMpkkIUcjQWOKkbFSEHBkxhixvDU9G1SqnntV8yzgb+K73gxVsEBzUPkIhglOOREGM6R13/ekCXOkDMWMTMtBqolEeIJGpG+pQJzoMJ9lnsJTqwxhnCj7hIEz9ftGjrjWGY/sZJFR//QK8S+vn5q4HuZUyNQQgeeH4pRBk8CiADikimDDMksQVtRmhXiMFMLG1rR0JSaZ4HJatsV8/R7+TzoXru+5/u1ltVFfVFQCx+AEnAMf1EAD3IAmaAMMJHgAj+DJuXeenRfndT664ix2jsASnLdPrzSaPw==</latexit>
ΣMD(T , T0)
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...
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Particles Observable
Per-Particle Representation Event Representation
F
Energy/Particle Flow Network
Latent Space
Patrick Komiske – Machine Learning in Particle Physics
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
Patrick Komiske – Machine Learning in Particle Physics
Thank You!
21
Patrick Komiske – The Hidden Geometry of Particle Collisions
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
Patrick Komiske – Machine Learning in Particle Physics
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)
<latexit sha1_base64="UyQwCnKpcfCTpWH6V6nRvy120lk=">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</latexit>
t(0)j = Pr(truth is j)
<latexit sha1_base64="fh4lBDacBDgmo9XTnetGi80Hqpo=">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</latexit>
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
Patrick Komiske – The Hidden Geometry of Particle Collisions
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]
Patrick Komiske – The Hidden Geometry of Particle Collisions
101 102 103
Energy Scale Q (GeV)
0
1
2
3
4
5
6
7
8
CorrelationDim
ension
EMD: Intrinsic Dimension
Pythia 8.235,√s = 14 TeV
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)
<latexit sha1_base64="XKuBaJ7fsvZeohVRYf7hETWOHbM=">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</latexit>
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
Patrick Komiske – The Hidden Geometry of Particle Collisions
101 102 103
Energy Scale Q (GeV)
0
1
2
3
4
5
6
7
8
CorrelationDim
ension
EMD: Intrinsic Dimension
Pythia 8.235,√s = 14 TeV
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)
<latexit sha1_base64="TwWtGcO0QQbG7yKasfTozXEthQo=">AAAEfnicdZPdbtMwFMe9LsAoXx1cchNRgYY0StJ2/bhAmuCGK7RKdJvUlMpxnNRq4kSOM1ZFeUQegOfgFgQnbpaqSTlJ5KNzfj4++du2I5/F0jB+HjQOtXv3Hxw9bD56/OTps9bx88s4TAShUxL6obi2cUx9xulUMunT60hQHNg+vbJXn/L81Q0VMQv5V7mO6DzAHmcuI1hCaNHyviwsSW9lyinzlp3sZPLWikQYyVCffNtkHBZk1ik8LICOaKz8MgMTPkwsV2CSOlnqTDLL53q96KLVNjqGMr3umIXTRoVdLI4bfy0nJElAuSQ+juOZaURynmIhGfFp1rSSmEaYrLBH01slRKa/hpiju6GAj0tdRXdAHMTxOrCBDLBcxtVcHtybswVeUbmz6CyR7mieMh4lknKyWdxNfB2ky5XWHSYokf4aHEwEg7Z1ssQglIT9aDYth7qwaarHNEpEBD+VCs/OUqNzNhycgjpmz8yHs2E/28VDgbm3xUfGSHGGwrsDs4IL6pTsoDdWpbtdxRrV0p7A6xLuDY2c2g5ZrW26LQ67CBi8eyjbT8p+FQQwYJx+J2EQYO6kVqHBzJyn6vTsKtM2swpfiFDh76Sp83mjFVj1XieVBhV0o0udvZOg3jb9T/VSjj1TNjKpOU24Mub2pozBhv3CGZvllbnswjHpdCf99vnH4vIcoZfoFTpBJhqic/QZXaApIugH+oV+oz8a0t5o77T3G7RxUMx5gXZMG/0D5AGApA==</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
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
Patrick Komiske – The Hidden Geometry of Particle Collisions
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
<latexit sha1_base64="Fvp1cxlsrBf2/ymk5NqFqTxXRXM=">AAAFlHicdZRdT9swFIZTRjbWja1s0m52E62aBBJCSRtokYYElHZsGggmvqSmVI7jhKiOkznORmX5Z+5iv2U3c0ILrcOsRDrxeezz5ryW3QSHKTPNP5WFJ4v602dLz6svXi6/el1beXORxhmF6BzGOKZXLkgRDgk6ZyHD6CqhCEQuRpfuqJPnL38imoYxOWPjBA0iEJDQDyFgcmpYix0XUON0SFadCLAbCLDxZW3HSX9Q1icD7vgUQG4JfiycNIuGfLRjietjByOfrToM3TLuJBlNePfoQIiZPdbvw6/D0ZpDw+CGrV0TMazVzQ2zGEY5sCZBXZuMk+HK4rLjxTCLEGEQgzTtW2bCBhxQFkKMRNXJUpQAOAIB6suQgAilA150Rhgf5Yxn+DGVL2FGMTu7goMoTceRK8lcb6rm8snHcv2M+e0BD0mSMUTgXSE/wwaLjbzNhhdSBBkeywBAGkqtBrwBsplMmlGdK3N7p7XqeMiXLhZfPKaABEjww7Ojb4J37JbdNBUE5z11cXZPbVntVmdboXJ38ANiWuaeilDkTfPdbqNhdpT8bI3cnV5PAeSZku6AKdPrmWZ7S2ECihCZEk27tb2pyohCNqNkr2n1mraCeICOZtU0GpuN9kGpEhgL/v3zvuDb9nrxCNlxgn7BOIoA8bgTUxKIviXPd36E5ztet4SYp7GrsjOtL+N5x9UFUxfKdP7LClx0oUwW9RT0fxqigDCVvXepjAeUqPTEr0e2ZrSkeGpdmfZyw1T8wcbHpEj3SlpyRwu2Ki8QS70uysFFY8OyN+xTu767P7lKlrT32gdtVbO0lrarHWon2rkGtd/a38piRdff6Z/0jt69QxcqkzVvtbmhH/8DHSnybQ==</latexit>
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)}
<latexit sha1_base64="Zeez8oe+uMocMvmbtKCgpjVI1No=">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</latexit>
jet from dataset medoids
Patrick Komiske – The Hidden Geometry of Particle Collisions
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)
�
�
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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
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: 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
[PTK, Metodiev, Thaler, 2004.04159]
Patrick Komiske – The Hidden Geometry of Particle Collisions
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)
�
�
�
�
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Ei ≥ 0
)
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dPi→ig '
2αs
πCa
dθ
θ
dz
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sha1_base64="subA+vmJKmyZTD6XsMW4hQeuWIs=">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</latexit><latexit 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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=">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
<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=">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</latexit> λ
<latexit sha1_base64="wXk7AmmngymkoNcky3vNULvpq4c=">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</latexit>
1− λ<latexit sha1_base64="feDmrmhe5e0qNCrjBVSp9Fat0fI=">AAAFG3icdZRNb9MwGMe9LcAoL+vGkUtEhcQBqjhNm/Y2wYXjkOg2qakqx3maRc2bHAdaRfkocIXvwQ1x5cDH4BvgOFmzJsVKZMv+PXn+/j+O7dj3Eq5pfw4Oj5R79x8cP+w8evzk6Un39OwyiVJGYUojP2LXNknA90KYco/7cB0zIIHtw5W9elesX30ClnhR+JFvYpgHxA29pUcJF1OL7pkVpyz2IcNvLF+EOSRfdHtaX5NNbQ9wNeihql0sTo/+Wk5E0wBCTn2SJDOsxXyeEcY96kPesdIEYkJXxIVsLTXn6ksx56jLiIk35Kqc3QFJkCSbwBZkQPhN0lwrJveu2YysgO8knaV8OZ5nXhinHEJaJl+mvsojtTBFdTwGlPsbMSCUeUK2Sm8II5QL6zody4Gl8FdqzEq/8oy5dp5p/aE5ei3cwQNcdEPTyHfxiJHQrfGxNpacJnF9hBs4J7bLAMJtADaHBTnSJzKNqbcDbD+tE2DdKEBjJGWZmtnmGTi1HkMKwUOj7Jr4XXY0KDXoutSuNbfqMrLZwgNTK6i6y1s2Qv1xcaoEJp491N3tSUjAAgvhM42CgIROVp3hfIbnmcVhzXcr1cN5g6+K0uBvS9XmC6ENWGpvk2XxGmxV0Ta9rXYjoD4F+zIIl1sJCufb7K3JbWPgP/q3hu8JKQshYzriksD13TARzTSqwQRvL4lLXfwYff2D0Tt/W10Xx+g5eoFeIYxMdI7eows0RRSt0Vf0DX1Xvig/lJ/KrxI9PKhinqGdpvz+B3OjuAQ=</latexit>
[PTK, Metodiev, Thaler, 2004.04159]
Patrick Komiske – The Hidden Geometry of Particle Collisions
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 )
<latexit sha1_base64="F9gCX1fxuBDAQh3AIhXiVu/Lf4M=">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</latexit>
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=">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</latexit>
Smooth IRC invariance
Eliminates common observables with hard boundaries
[Sterman, Weinberg, PRL 1997; Sterman, PRD 1978; Banfi, Salam, Zanderighi, JHEP 2005]
Patrick Komiske – The Hidden Geometry of Particle Collisions
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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✏<latexit sha1_base64="jgUrJhLGALjMeIyRRVBYIdUwEyk=">AAACLnicbZDLSsNAFIZP6q3WW73s3ASL4EJKIoJdFty4rGBboQ1lMj1ph04mYWYilJDn8DV8Abf6BoILcSc+htO0gm39YeDjP+fMOfx+zJnSjvNuFVZW19Y3ipulre2d3b3y/kFLRYmk2KQRj+S9TxRyJrCpmeZ4H0skoc+x7Y+uJ/X2A0rFInGnxzF6IRkIFjBKtLF6ZTft5p905MD3Uqfq5DpfgqyLsWI8ElmvXPk17WVwZ1CpH0GuRq/81e1HNAlRaMqJUh3XibWXEqkZ5ZiVuonCmNARGWDHoCAhKi/Nr8rsU+P07SCS5glt5+7fiZSESo1D33SGRA/VYm1i/lfrJDqoeSkTcaJR0OmiIOG2juxJTnafSaSajw0QKpm51aZDIgnVJs25LQGORRhnJROMuxjDMrQuqq5TdW8vK/XaNCEowjGcwBm4cAV1uIEGNIHCIzzDC7xaT9ab9WF9TlsL1mzmEOZkff8AvcOl4w==</latexit><latexit sha1_base64="jgUrJhLGALjMeIyRRVBYIdUwEyk=">AAACLnicbZDLSsNAFIZP6q3WW73s3ASL4EJKIoJdFty4rGBboQ1lMj1ph04mYWYilJDn8DV8Abf6BoILcSc+htO0gm39YeDjP+fMOfx+zJnSjvNuFVZW19Y3ipulre2d3b3y/kFLRYmk2KQRj+S9TxRyJrCpmeZ4H0skoc+x7Y+uJ/X2A0rFInGnxzF6IRkIFjBKtLF6ZTft5p905MD3Uqfq5DpfgqyLsWI8ElmvXPk17WVwZ1CpH0GuRq/81e1HNAlRaMqJUh3XibWXEqkZ5ZiVuonCmNARGWDHoCAhKi/Nr8rsU+P07SCS5glt5+7fiZSESo1D33SGRA/VYm1i/lfrJDqoeSkTcaJR0OmiIOG2juxJTnafSaSajw0QKpm51aZDIgnVJs25LQGORRhnJROMuxjDMrQuqq5TdW8vK/XaNCEowjGcwBm4cAV1uIEGNIHCIzzDC7xaT9ab9WF9TlsL1mzmEOZkff8AvcOl4w==</latexit><latexit sha1_base64="jgUrJhLGALjMeIyRRVBYIdUwEyk=">AAACLnicbZDLSsNAFIZP6q3WW73s3ASL4EJKIoJdFty4rGBboQ1lMj1ph04mYWYilJDn8DV8Abf6BoILcSc+htO0gm39YeDjP+fMOfx+zJnSjvNuFVZW19Y3ipulre2d3b3y/kFLRYmk2KQRj+S9TxRyJrCpmeZ4H0skoc+x7Y+uJ/X2A0rFInGnxzF6IRkIFjBKtLF6ZTft5p905MD3Uqfq5DpfgqyLsWI8ElmvXPk17WVwZ1CpH0GuRq/81e1HNAlRaMqJUh3XibWXEqkZ5ZiVuonCmNARGWDHoCAhKi/Nr8rsU+P07SCS5glt5+7fiZSESo1D33SGRA/VYm1i/lfrJDqoeSkTcaJR0OmiIOG2juxJTnafSaSajw0QKpm51aZDIgnVJs25LQGORRhnJROMuxjDMrQuqq5TdW8vK/XaNCEowjGcwBm4cAV1uIEGNIHCIzzDC7xaT9ab9WF9TlsL1mzmEOZkff8AvcOl4w==</latexit><latexit sha1_base64="nY45iuKraCLDFewaTX5iwTSikYs=">AAACLnicbZDLSsNAFIYn9VbrLerSTbAILqQkIthlwY3LCvYCTSiT6Uk7dDIJMxMhhDyHr+ELuNU3EFyIO/ExnKYRbOsPAx//OWfO4fdjRqWy7Xejsra+sblV3a7t7O7tH5iHR10ZJYJAh0QsEn0fS2CUQ0dRxaAfC8Chz6DnT29m9d4DCEkjfq/SGLwQjzkNKMFKW0PTydzik4EY+15mN+xCFyuQuxBLyiKeD836r2mtglNCHZVqD80vdxSRJASuCMNSDhw7Vl6GhaKEQV5zEwkxJlM8hoFGjkOQXlZclVtn2hlZQST048oq3L8TGQ6lTENfd4ZYTeRybWb+VxskKmh6GeVxooCT+aIgYZaKrFlO1ogKIIqlGjARVN9qkQkWmCid5sKWAFIexnlNB+Msx7AK3cuGYzecu6t6q1lGVEUn6BSdIwddoxa6RW3UQQQ9omf0gl6NJ+PN+DA+560Vo5w5Rgsyvn8ARReljw==</latexit>
δ
<latexit sha1_base64="CTit7owv6Rbo1QgImtmrLEJTECw=">AAACLHicbZDLSsNAFIZPvFtv8bJzEyyCC6mJCHZZcONSwVqhCWUyPWkHJ5MwMxFCyGP4Gr6AW30DNyJu+xxO0wpq/WHg4z/nzDn8YcqZ0q77bs3NLywuLa+s1tbWNza37O2dW5VkkmKbJjyRdyFRyJnAtmaa410qkcQhx054fzGudx5QKpaIG52nGMRkIFjEKNHG6tknhV990pWDMCjchlvpeAZKv49ck7Jn178tZxa8KdRbe1DpqmeP/H5CsxiFppwo1fXcVAcFkZpRjmXNzxSmhN6TAXYNChKjCorqptI5NE7fiRJpntBO5f6cKEisVB6HpjMmeqj+1sbmf7VupqNmUDCRZhoFnSyKMu7oxBmn5PSZRKp5boBQycytDh0SSag2Wf7aEmEu4rSsmWC8vzHMwu1pw3Mb3vVZvdWcJAQrsA8HcAQenEMLLuEK2kDhEZ7hBV6tJ+vN+rA+J61z1nRmF37JGn0B2F6k3w==</latexit><latexit sha1_base64="CTit7owv6Rbo1QgImtmrLEJTECw=">AAACLHicbZDLSsNAFIZPvFtv8bJzEyyCC6mJCHZZcONSwVqhCWUyPWkHJ5MwMxFCyGP4Gr6AW30DNyJu+xxO0wpq/WHg4z/nzDn8YcqZ0q77bs3NLywuLa+s1tbWNza37O2dW5VkkmKbJjyRdyFRyJnAtmaa410qkcQhx054fzGudx5QKpaIG52nGMRkIFjEKNHG6tknhV990pWDMCjchlvpeAZKv49ck7Jn178tZxa8KdRbe1DpqmeP/H5CsxiFppwo1fXcVAcFkZpRjmXNzxSmhN6TAXYNChKjCorqptI5NE7fiRJpntBO5f6cKEisVB6HpjMmeqj+1sbmf7VupqNmUDCRZhoFnSyKMu7oxBmn5PSZRKp5boBQycytDh0SSag2Wf7aEmEu4rSsmWC8vzHMwu1pw3Mb3vVZvdWcJAQrsA8HcAQenEMLLuEK2kDhEZ7hBV6tJ+vN+rA+J61z1nRmF37JGn0B2F6k3w==</latexit><latexit sha1_base64="CTit7owv6Rbo1QgImtmrLEJTECw=">AAACLHicbZDLSsNAFIZPvFtv8bJzEyyCC6mJCHZZcONSwVqhCWUyPWkHJ5MwMxFCyGP4Gr6AW30DNyJu+xxO0wpq/WHg4z/nzDn8YcqZ0q77bs3NLywuLa+s1tbWNza37O2dW5VkkmKbJjyRdyFRyJnAtmaa410qkcQhx054fzGudx5QKpaIG52nGMRkIFjEKNHG6tknhV990pWDMCjchlvpeAZKv49ck7Jn178tZxa8KdRbe1DpqmeP/H5CsxiFppwo1fXcVAcFkZpRjmXNzxSmhN6TAXYNChKjCorqptI5NE7fiRJpntBO5f6cKEisVB6HpjMmeqj+1sbmf7VupqNmUDCRZhoFnSyKMu7oxBmn5PSZRKp5boBQycytDh0SSag2Wf7aEmEu4rSsmWC8vzHMwu1pw3Mb3vVZvdWcJAQrsA8HcAQenEMLLuEK2kDhEZ7hBV6tJ+vN+rA+J61z1nRmF37JGn0B2F6k3w==</latexit><latexit sha1_base64="3pF/Jb7/cAnqPIgLpv26uMSvwlE=">AAACLHicbZDNSsNAFIUn9a/Wv6pLN4NFcCE1EcEuC25cVrCtkIQymd60QyeTMDMRSshj+Bq+gFt9AzcibvscTtsItvXAwMe59869nCDhTGnb/rRKa+sbm1vl7crO7t7+QfXwqKPiVFJo05jH8jEgCjgT0NZMc3hMJJAo4NANRrfTevcJpGKxeNDjBPyIDAQLGSXaWL3qZebNPnHlIPAzu27PdLECudcHrkneq9Z+LbwKTgE1VKjVq068fkzTCISmnCjlOnai/YxIzSiHvOKlChJCR2QArkFBIlB+Nrspx2fG6eMwluYJjWfu34mMREqNo8B0RkQP1XJtav5Xc1MdNvyMiSTVIOh8UZhyrGM8TQn3mQSq+dgAoZKZWzEdEkmoNlkubAlhLKIkr5hgnOUYVqFzVXfsunN/XWs2iojK6ASdonPkoBvURHeohdqIomf0it7Qu/VifVhf1ve8tWQVM8doQdbkB1+ypIs=</latexit>
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
[PTK, Metodiev, Thaler, 2004.04159]
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
…
Patrick Komiske – The Hidden Geometry of Particle Collisions
Perturbation Theory in the Space of Events
32
Infrared singularities of massless gauge theories appear on each PN
✏ → 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>
Soft Collinear
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
<latexit sha1_base64="1fXZyS78TMKvsIeI3ezYebd9gz0=">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</latexit><latexit sha1_base64="1fXZyS78TMKvsIeI3ezYebd9gz0=">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</latexit><latexit sha1_base64="1fXZyS78TMKvsIeI3ezYebd9gz0=">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</latexit><latexit sha1_base64="qKorqtoDZgNc12Sexz2smIaqQ9g=">AAACDnicdVDLSgMxFM3UV62vVpdugkVwY8m09rUruHElFewD2lIyaaYNzWSGJKOUYf7BH3Crf+BO3PoL/oDfYaatYEUPBA7n3Ms9OU7AmdIIfViptfWNza30dmZnd2//IJs7bCs/lIS2iM992XWwopwJ2tJMc9oNJMWew2nHmV4mfueOSsV8catnAR14eCyYywjWRhpmc30P6wnBPGrGw+j63I6H2TwqoEq5XkIQFcrIrtbrhiBUqZWK0DYkQR4s0RxmP/sjn4QeFZpwrFTPRoEeRFhqRjiNM/1Q0QCTKR7TnqECe1QNonn0GJ4aZQRdX5onNJyrPzci7Ck18xwzmQRVv71E/MvrhdqtDSImglBTQRaH3JBD7cOkBzhikhLNZ4ZgIpnJCskES0y0aWvliktnwgvijCnm+/fwf9IuFmxUsG8u8o3asqI0OAYn4AzYoAoa4Ao0QQsQcA8ewRN4th6sF+vVeluMpqzlzhFYgfX+BRTjnJM=</latexit>
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.)
<latexit sha1_base64="W6sDhi0jd4X5vbTHYOvFDyiMti8=">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</latexit>
[Larkoski, Thaler, JHEP 2014; Larkoski, Marzani, Thaler, PRD 2015]
[PTK, Metodiev, Thaler, 2004.04159]
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
<latexit sha1_base64="OJ26IaP8BcK3QyL1R+qvb41yMn0=">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</latexit>
[Sterman, PRD 1979; Banfi, Salam, Zanderighi, JHEP 2005]
V (E) = T2(E)
✓
1 +1
lnE(E)/T3(E)
◆
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Patrick Komiske – The Hidden Geometry of Particle Collisions
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
[PTK, Metodiev, Thaler, 2004.04159]