Methods of Experimental Particle Physics

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!

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)

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

“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

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!

Dimensional Regularization • Need to calculate the phase space in d dimensions in • Use: • Then: • Table shows results for several discrete values of d

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

Standard Integrals • Summary of the integrals we will need to calculate P in d dimensions:

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

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

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

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

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 ?!!

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”

Running Couplings • 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

Unstable Particles • 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 Z Z g e e 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

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

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

Methods of Experimental Particle Physics