1 Methods of Experimental Particle Physics Alexei Safonov Lecture #4

Course Web-site Our web-site is up and running now Thanks to Aysen! 2

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 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) 3

QED Beyond Leading Order Feynman diagrams are just a visual way to do perturbative expansion in QED The small parameter is  =e 2 /4  ~1/137 If we want higher precision, we must include higher order diagrams But that’s where troubles start showing up 4

“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 d 4 k/k 4 5

Some Math Trickery 6 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  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? 8 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/  – logarithmic divergence

Standard Integrals Summary of the integrals we will need to calculate  in d dimensions: 9

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 10

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 11

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  (q 2 )*q 2 That’s a dead end and a lousy one The QFT would seem like a complete nonsense 12

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 q 2 =0 13

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 q 2 : Subtracting the 1/  infinity from  2 we get the q2 dependence: Is electrical charge dependent on q 2 ?!! 14

Running Coupling Well maybe… Coupling becomes stronger at smaller distances (or higher energies) If so, the fine structure constant depends on q 2 : But it depends slowly 1/137 at q 2 =0 and 1/128 at |q 2 |=m(Z) It actually can be not that crazy… Leads to “electrical charge screening” and “vacuum polarization” 15

Running Couplings 16 You may have seen these before What’s plotted is 1/  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 q 2 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  Unstable Particles 18 Z e

Renormalization Group Equations A consistent schema how to get all running parameters (masses, charges) dependences on q 2 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 19

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 20

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