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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