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Quantum Interference in Showering:
The LPM effect, what is is, and whyits theoretical development is still
interesting 70 years later
Peter ArnoldUniversity of Virginia
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[Oversimplification: Only electromagnetic shower shown.]
High energy particles traveling through matter lose energy viasuccessive bremsstrahlung and pair production:
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High energy particles traveling through matter lose energy viasuccessive bremsstrahlung and pair production:
[Oversimplification: Only electromagnetic shower shown.]
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High energy particles traveling through matter lose energy viasuccessive bremsstrahlung and pair production:
[Oversimplification: Only electromagnetic shower shown.]
radiationlength X0
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LandauPomeranchukMigdal (LPM) effect
What is the LPM Effect?
A coherence effect that complicates calculations of bremsstrahlung or pairproduction when a very high energy particle scatters from a medium.
Places it comes up in QED
• Very high energy cosmic rays showering in the atmosphere.• Certain beam dump experiments designed to measure the LPM effect.
• Energy loss of high energy jets in a quarkgluon plasma.
• Complete leadingorder calculations of the viscosity and other transport coefficients of a weaklycoupled quarkgluon plasma.
Places it comes up in QCD
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• Coherence in highenergy, smallx scattering from a large nucleus: color glass condensate.
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The LPM Effect
brem from several successive (small angle) collisions not very different frombrem from one collision.
Result: a reduction of the naive brem rate.
Naively
brem rate ~ n v ~ (density of scatterers) c
At very high energy,
probabilities of brem from successive scatterings no longer independent;
Problem
Result: highenergy brem rate independent of E.
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Example: stopping distance
If LPM effect ignored: stopping distance
Actual result (weak coupling): stopping distance
(up to logs)
(up to logs)
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The LPM Effect (QED)
Warmup: Recall that light cannot resolve details smaller than its wavelength.
[Photon emission from different scatterings have same phase coherent.]
Now: Just Lorentz boost above picture by a lot!
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The LPM Effect (QED)
Note: (1) bigger E requires bigger boost more time dilation longer formation length (2) big boost this process is very collinear.
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versus
Are these two possibilities in phase? Or does the interference average to zero?
IN PHASE if (i) everything is nearly collinear ✓ (ii) particle and photon have nearly same velocity ✓ (speed of light)
An alternative picture 7/37
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A real calculation involves...
soft photons: Can do a classical EM calculation
hard photons: Much trickier!
(Landau+Pomeranchuk 1953)
(Migdal 1955)
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Experimental Measurement of LPM (QED)
SLAC E146
Phys. Rev. Lett. 75 (1995) 2949.
independent collisions
LPM effect
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An application to QCD
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For this talk, simplify discussion by focusing on ...
QGP
(hadronization not shown)
(initial vacuum-like radiation not shown)
In-medium evolution of a jetConsider cartoon of
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QGP
Cascades that stop in-medium
QGP
• Qualitative points we'll discuss generalize.
• Formalism generalizeable as well.
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The LPM Effect (QCD)There is a qualitative difference for soft bremsstrahlung.:
QED
Softer brem photon longer wavelength less resolution more LPM suppression
QCD
Unlike a brem photon, a brem gluon can easily scatter from the medium.
vs.
Upshot: Soft brem more important in QCD than in QED (for highE particles in a medium)
Softer brem gluon easier for brem gluon to scatter less collinearity less LPM suppression
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Stopping distance still
If LPM effect ignored: stopping distance
Actual result (weak coupling): stopping distance
(up to logs)
(up to logs)
QCDlike theories w/ gravity duals(infinitely strong coupling):
maxstopping distance
And for later comparison...
[Chesler, Jensen, Karch, Yaffe; Gubser, Gulotta, Pufu, Rocha; Hatta, Iancu, Mueller (2008)
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An idealized Monte Carlo pictureof in-medium evolution
As time passes,
roll classical dice for probability of each splitting
weighted by the quantum calculation of the single splitting rate
(including LPM effect) for each vertex above.
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Built-in assumption:
Consecutive splittings are quantum-mechanically independent.
Heuristic attempts to improve on this in real Monte Carlos:
JEWELrecent versions of MARTINI
(But are they?)
Here, I want to talk about
What's known from first-principles(-ish) QCD calculations?
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Single splitting rate
Collisions with the medium
generate chances for bremsstrahlung
with
prob of emission ~ per collision
prob of emission ~ per formation length
indistinguishable from
So, actually,
Naive picture (ignoring LPM)
LPM effect
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Rate calculated for QCD by
BDMPS-Z for QCD (1990s)
and investigated/generalized in numerous ways by many since,with a particular shout-out to ASW [Armesto, Salgado, and Wiedemann].
Units of made up as
Where medium parameter defined by
average net transversemomentum kick from medium times distance traveled( )
Dependence on medium:
2
[Baier, Dokshitzer, Mueller, Peigne, Schiff; B. Zakharov]
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So chance of overlap (i.e. “rolling dice separately” breaking down) is
∝ a
Chance of brem ~ a per formation time
means that two consecutive splittings will typically look like
Consecutive emissions
How big is “a” ??
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How big is s?Nothing to do with whether medium is
sQGP / perfect liquidvs.
weakly-coupled QGP
[ s(T ) big ] [ s(T ) small ]
s on previous slide associated with emission vertex:
costs roughly s(Q) with Q
a few GeV
panic and/or fool aroundwith AdS/CFT energy loss vs.
LPM-based analysis
[ s(Q) big ] [ s(Q
) small ]
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Does the wisdom of the ages tell usif s(few GeV) is small?
Particle physics in vacuum:
Small for some things, like matching lattice calculationsto continuum MS-bar s
High-temperature physics:
Bad news (except possibly if one does sophisticatedresummations of perturbation series)
Overlapping formation times effects on cascade:
∝ a
We should calculate it and see.
effect on
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Soft emissionSoft emissions are generally enhanced by logs.Path-breaking authors found small-x-like double logs in this case,
Blaizot & MehtarTani; Iancu; Bin Wu (2014)
This is a BIG effect for large E.
following Liou, Mueller, Bin Wu (2013)
Also, leading logs can be resummed to all orders! ...
But they found soft emission effects could be absorbed into themedium parameter
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Effect turns out to be
stopping distance or equivalently
(for large enough E)
Result for sample values of s(Q):
stopping distance
QCDlike theoriesw/ gravity duals max
stopping distance
Compare to ...
stopping distance
something involving a Bessel fn
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Beyond the soft limit
What about overlap effects that can't be absorbed into ?
Back to small(ish) ...
Soft emission corrections summary
• Absorbable into medium parameter
• Size of effect controlled by (after resummation)
∝ as
At leading log order
And how big are those effects for relevant sizes of ?
Also, what about sub-leading logs? Are they also resummable into medium parameters?
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What we've doneComputed the effect of the overlap for hard emissions
∝ as
In broad brush: interesting and fun field theory problem.In calculational detail: a pain in the ass.
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Formalism for LPM: single bremShorthand henceforth: Draw
as simply
But will be even more convenient to draw as
Can (formally) interpret this as 3 particles moving forward in time [Zakharov 1990's]:
2 particles from the amplitude (evolving with e iHt)1 particle from the conjugate amplitude (evolving with e +iHt)
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Will show that evolution in can be described by
3particle nonrelativistic Quantum Mechanics in 2 dimensions
with weird properties:
●
● (i.e. H is nonHermitian)
interference vanishes as Dt ∞, as it must!
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Kinetic terms:
Energy of a highpz particle:
Evolution of is e iHt with1
2
3
conjugate evolveswith e +iHt
This is 2dimensional nonrelativistic QM with
As promised,
Potential term:Potential term:
V (b1,b2,b3) incorporates (statistically averaged) effect of collisions with the medium.
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Potential terms:
To motivate form, think of something else...
A classical Boltzman analysis of scattering:
gain term loss term
Fourier transform:
with
This looks like a Schrodingerish equation:
with
In our problem, this physics gives V :
1
2
3
(QED)
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How to put the calculation together:
(1) Solve for propagation in 3particle QM in shaded region.
(2) Tie together with QFT matrix elements for vertices
How to put the calculation together:
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Simplifcation: 3particle QM 1particle QM
Method 1. Can solve numerically.
Method 2. High energies very collinear b 's small.
So make small B approximation to V (B ) a harmonic oscillator problem
( a nonHermitian one: )
[ Zakharov (2004+); CaronHuot & Gale (2010) ]
[ Baier et al . (1998) ]
[ BDMPSZ (1990's) ]
Can use various symmetries of problem to get rid of 2 d.o.f.
Simplifcation: Harmonic Oscillator
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Formalism for LPM: double bremExample of an interference contribution:
To compute : Sew together QFT matrix elementfor vertices with QM evolution in between.
Simplify : Using symmetries, as before.
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ugliest bit = 2particle QM evolution
Can imagine
● numerics [ have not done ]
● harmonic osc. approximation [ have done! ]
Harmonic osc. sounds very straightforward, but in fact quite complicated.
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What do we do?
● For 4particle (effectively 2particle) evolution, find eigenmodes and frequencies of
● Construct corresponding propagator for 4particle (2particle) evolution. [Also do the same for 3particle (1particle) evolution.]
● Combine with QFT matrix elements for splitting vertices.
● Analytically integrate over all vertex times except Dt :
● Analytically integrate over all vertex transverse positions.
Result:
● Final Dt integral easy to do numerically.
nonHermitiansprings
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Complications
Color: During 4particle evolution, T1+T2+T3+T4 = 0 is not enough to fix color factors Ti∙ ∙Tj .
Color dynamics is nontrivial!
For now : Work in large Nc limit. [Not necessary if the brems are soft.]
Formalism: Getting straight the formalism for 4particles effectively 2 particles.
Helicities: Helicities of highenergy particles contract nontrivially in
Must use helicitydependent DGLAP splitting functions at vertices.
Divergences: Each timeordered diagram diverges as Dt 0.
Must handle carefully (and nontrivially), even though the amplitude (blue) is just a tree diagram!
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Published Work
+permutations of
(x, y, 1xy)
All diagrams for overlap of two real gluon emissions
+permutations of
(x, y, 1xy)
Still in progress
[ all for g gg ggg ]
virtual corrections, e.g.
Diagrams with 4gluon vertices:
correct single brem rate
Crossed diagrams:
Sequential diagrams: [subtle to separate from consecutive splittings calculated with leadingorder formalism!]
e.g.
with Shahin Iqbal and Han-Chih Chang34/37
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Results for real double brem
correction to double brem dueto overlapping formation times
where f(x,y) varies from 1.05 to -0.90 and isshown on the right.
Qualitative Point
Effect of overlapping formation times enhancesthe rate except when one gluon is very soft.
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Virtual diagrams: what’s the holdup?
UV divergent vertex correction:renormalization of charge, etc.
But can’t I look up in my favorite textbook how to compute UV part of the amplitude
Example:
in QFT?
Yes, but I earlier treated these particles in the approximation
2-dim QM with “mass” pz
So I need to match UV renormalization of underlying gauge theory to calculationsIn the effective QM theory used for
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SummarySubtle problems in the field theory description of veryhigh energy showering
e.g.
can be reduced to problems in
2dimensional nonrelativistic nonHermitian quantum mechanics
and even
2dimensional nonrelativistic nonHermitian harmonic oscillators!
(Just when you thought you couldn't learn anything more from the harmonic oscillator...)
Coming in the future
Are the O(s) corrections to physical, infraredsafe quantities characterizing showerdevelopment small (after accounting for the known running of q(E) due to soft brem)?
To wit, is the basic physical assumption behind inmedium Monte Carlo simulationson firm ground?
^
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EXTRA
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+permutations of
(x, y, 1xy)
Sequential diagrams:
What was subtle about
A: Have to avoid double counting with Monte Carlo based on singlesplitting rates:
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vs
Monte Carlo (MC)
How to account for correction from
?
Add a g ® ggg Monte Carlo possibility to account for correction:
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where
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RESERVE
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[Aside: A personal gauge-gravity frustration]
What’s the 1/(Nc) correction to the infinite-coupling result E1/3 ?
QCDlike theoriesw/ gravity duals max
stopping distance
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1
2
3
QED:
(bar over G means charge e 2 factored out)
QCD:
Color factors Ti∙ ∙Tj are fixed (not dynamical) because
e.g.