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Methods of Experimental Particle Physics

Alexei Safonov

Lecture #4

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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• 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!

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Dimensional Regularization• Need to calculate the phase space in d

dimensions in

• Use:

• Then:

• Table shows results for several discrete values of d

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How Bad is the Divergence?• 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

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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”

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Running Couplings

• You may have seen these before

• What’s plotted is 1/a

• We will talk about other forces later

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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

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Z

ee

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: • Correspondingly, various cross-section diagrams will acquire

dependence and have no divergence at the pole

Unstable Particles

Z

e

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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