Post on 08-Aug-2020
1
Methods of Experimental
Particle Physics
Alexei Safonov
Lecture #4
Course Web-site
• Our web-site is up and running now
• http://phys689-hepex.physics.tamu.edu/
• Thanks to Aysen!
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Lab Schedule • We will continue with finishing up Lab #1 this
week
• We updated the list of “tasks” for Lab #1 be in the
submitted “lab report”
• We realized it was too vague for people with no past
experience with ROOT, now all exercises are listed explicitly
• If you submitted your report already, you don’t need to re-
submit it
• Will make sure further exercises are more explicitly
listed
• The first homework assignment will be
distributed soon (by email and on the web-site)
• Calculation of the e-e- scattering cross-section
• Format for submissions: PDF file based on Latex (a
template with an example will be provided)
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QED Beyond Leading Order • Feynman diagrams
are just a visual way
to do perturbative
expansion in QED
• The small parameter
is a=e2/4p~1/137
• If we want higher
precision, we must
include higher order
diagrams
• But that’s where
troubles start
showing up
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“Photon Propagator” at Higher Orders • Imagine you are calculating a
diagram where two fermions
exchange a photon
• Instead of just normal photon
propagator, you will have to
write two and in between include
a new piece for the loop:
• Integrate over
all allowed
values of k
• Divergent b/c
of terms
d4k/k4
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Some Math Trickery
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• We want to calculate that integral even if we
know it has a problem
• Introduce Feynman parameter
• Some more trickery and substitutions:
• If integrated to L instead of infinity:
• This is really bad!
Dimensional Regularization • Need to calculate the phase space in d
dimensions in
• Use:
• Then:
• Table shows results for
several discrete values of d 7
How Bad is the Divergence?
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• Need to take an integral:
• But that’s beta function:
• Then:
• A pole at d=4, to understand the magnitude of the
divergence, use and
• The integral diverges as 1/e – logarithmic divergence
Standard Integrals
• Summary of the integrals we will need to
calculate P in d dimensions:
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Final Result • Now we can calculate the original integral:
• And the answer is:
• Where
• Terrific, but it’s really a mess. It’s an infinity
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How to Interpret It? • Let’s step back and think what is it we have
been calculating. The idea was to calculate this:
• We just did the first step in the calculation
• One can write the above as a series
• And drop qmqn terms (they will disappear anyway)
• This looks like kind of like photon propagator
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Interpretation Attempt • As we said, it kind of looks like a photon
propagator but with one tiny problem:
• The photon has non-zero mass!
• To be exact, it now has infinite mass P(q2)*q2
• That’s a dead end and a lousy one
• The QFT would seem like a complete nonsense
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Solution • Maybe what we calculated is not the propagator
• Remember in physics processes the quantity
we calculated enters with e2:
• Why don’t we push this infinity
• …into the “new” electrical charge definition
calculated at q2=0
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Charge Renormalization • Let’s summarize:
• We can hide this infinity, but the new charge is equal
to the old charge plus infinity
• What if the original charge we used was actually a
minus infinity?
• … However strange that may sound, the new charge
is then a finite quantity
• … but not really a constant, it depends on q2:
• Subtracting the 1/e infinity from P2 we get the q2
dependence:
• Is electrical charge dependent on q2 ?!!
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Running Coupling • Well maybe… Coupling becomes stronger at
smaller distances (or higher energies)
• If so, the fine structure constant depends on q2:
• But it depends slowly
• 1/137 at q2=0 and 1/128 at |q2|=m(Z)
• It actually can be not that crazy…
• Leads to “electrical charge screening”
• and “vacuum polarization” 15
Running Couplings
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• You may
have seen
these before
• What’s
plotted is 1/a
• We will talk
about other
forces later
Renormalizability of a Theory • This is not the only divergent diagram
• E.g. this one diverges too:
• A similar mechanism: the “bare” electron mass is
infinite, but after acquiring an infinite correction
becomes finite and equal to the mass of a physical
electron
• It can still depend on q2 so mass is also running
• The trick is to hide all divergences
simultaneously and consistently
• If you can do that, you got a “renormalizable theory”
• QED is renormalizable and so is the Standard Model 17
Z
e e
g
• In some sense in QED there are no “unstable” particles
• Electron can’t decay to two photons
• In QED you can’t do anything except to emit or absorb a
photon, so particles can’t decay via QED interactions
• But in the electroweak model Z boson can decay to
pairs of muons
• Corrections have different behavior because corrections for the
left diagram have a second component with an extra “i”
(something to do with how propagators multiply)
• G comes from
• Corrected propagator becomes: 𝟏
𝒑𝟐−𝒎𝟐 →𝟏
𝒑𝟐−𝒎𝟐+𝒊𝒎𝚪
• Correspondingly, various cross-section diagrams will acquire
dependence 𝟏
(𝒑𝟐−𝒎𝟐)𝟐+𝒎𝟐𝚪𝟐 and have no divergence at the pole
Unstable Particles
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Z
e
Renormalization Group Equations • A consistent schema how to get all running
parameters (masses, charges) dependences on
q2 for a particular theory
• Important as lagrangians are often written at
some high scale where they look simple
• SUSY often uses the GUT scale
• But physical masses (at our energies) can be
different
• In SUSY phenomenology,
masses often taken to be
universal at GUT scale
• Interactions - split and evolve
differently to our scale
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Types of Divergences • What we talked about so far have been ultra-violet
divergences (they appear as we integrate towards
infinite values of momentum in the integral)
• One can also regularize them using cut-off scale Lambda
• You sort of say beyond that theory either doesn’t make sense and
there must be something that will regulate things, like a new heavy
particle(s)
• In condensed matter, ultraviolet divergences often have a natural
cut-off, e.g. the size of the lattice in crystal
• Not all theories suffer from them, e.g. the QCD doesn’t
• Another type is “infrared divergences”:
• The amplitude (and the cross-section) for emitting an infinitely
soft photon is infinite
• In QED the trick is to realize that emitting a single photon is not
physical: you need to sum up single and all sorts of multiple
emissions, then you get a finite answer
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Near Future
• Wednesday lecture – accelerator physics
by Prof. Peter McIntyre
• Originally this topic was planned for about a
week from now but due to my travel we will
schedule it earlier
• Next lectures:
• Weak Interactions and the Electroweak
theory
• Standard Model, particle content, interactions
and Higgs
• Physics at colliders including a short review
of QCD 21