Phys 253a
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Problem Set 10 Solutions December 2, 2010
1. (a) The phase space integral for 1 2 is 1 2m d4 q d3 p (2)4 4 (P - p - q) 3 2E (2)3 E (2) p q
This is easiest to evaluate in the center of mass frame, P = (m , 0). If we first do the integral over q
Phys 253a
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Problem Set 9 Solutions December 2, 2010 1. (a) There are lots of diagrams. Most of them are just vertex and propagator corrections. The other diagram we'll deal with is the one that's the same, but with the photon lines crossed in the middle.
253a/Schwartz Due September 14, 2010
Problem Set 1
1. Special Relativity Review a) If two protons are approaching each other with 14 TeV center of mass energy, how close are they to the speed of light (in miles/hour)? b) How fast is one proton moving with
253a/Schwartz Due October 26, 2010
Problem Set 7
1. Majorana representation a) Write out the form of the Lorentz generators in the Majorana representation. b) Calculate 5 = i0 1 2 3 in the Majorana representation. 2. In practice, we only rarely use explic
Physics 253a
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Problem Set 5 Solutions October 26, 2010 1. (a) We need a W ee coupling and a W coupling in order to get a decay of the form e - W- e- An appropriate Lagrangian is L=
f =e,e
1 1 - f ( + m2 )f - W ( + m2 )W + gW + gW ee W f 2 2
Here, we've
253a/Schwartz Due November 30, 2010
Problem Set 12
1. * (This problem involves path integrals, so I made it a star problem. But it is a rather easy path integral problem, so everyone should be able to do it. The real track two parts are marked *.) Furry's
Phys 253a
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Problem Set 2 Solutions September 30, 2010 1. (a) Remember that dD x has mass dimension -D, while derivatives have mass dimension 1. We'll denote the mass dimension of a field by []. Then since the action has mass dimension 0,
2 0 = [dD x F ]
Physics 253a
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Problem Set 1 Solutions September 21, 2010
cm 1. (a) In the COM frame, one of the protons has energy E2 = m = m(1 - v 2 )- 2 2m2 (where c = 1). Solving for v gives v 1 - with = E 2 . The mass of the cm 2 proton is about 1 GeV, so 140002 10-
Physics 253a
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Problem Set 4 Solutions October 7, 2010 1. (a) The tree level diagram has a single vertex ig
Recall that the LSZ formula amputates the external legs, so we just have iM = ig. It turns out this process is actually kinematically disallowed.
253a/Schwartz Due November 16, 2010
Problem Set 10
1. Particle decays. Recall that the decay rate is given by the general formula d = d 3 p2 1 1 |M|2 (2)3 2E2 2E1 d 3 pn 1 (2)4 4(p1 - p2 - (2)3 2En - pn) (1)
a) Evaluate the phase space integrals for 1 2 d
Phys 253a
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Problem Set 11 Solutions December 1, 2010
1. (a) On the previous problem set, we calculated the rate for a scalar decay to two particles i i and found = where :=
mh v .
2 mh i (1 - 4x2 )3/2 = i 8
2 mh 2 x (1 - 4x2 )3/2 i 8 i
(b) The important
Phys 253a
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Problem Set 12 Solutions December 4, 2010 1. (a) The kinetic term for is (D ) D = ( - ieA ) ( + ieA ) If we switch and , then clearly we should take A -A to leave the Lagrangian invariant. (b) We have an expression for the correlation function
Physics 253a
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Problem Set 3 Solutions October 6, 2010
1. (a) In the center of mass frame, the e+ and e- approach each other with opposite cm cm momenta and total energy Ecm : pe+ = E2 (1, 1, 0, 0) and pe- = E2 (1, -1, 0, 0). The muon and anti-muon look t
253a/Schwartz Due October 5, 2010
Problem Set 4
1. Consider the Lagrangian for 3 theory 1 g L = - ( + m2) + 3 2 3! (1)
a) Draw a tree level feynman diagram for the decay . Write down the corresponding amplitude using the Feynman rules. b) Now consider the
Phys 253a
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Problem Set 6 solutions November 7, 2010
1. (a) There are many diagrams one can draw (I count 23 at order e4 ). Many of these are corrections to the two and three point functions for the scalars or the photon and will cancel in small groups. W
253a/Schwartz Due October 12, 2010
Problem Set 5
1. Radioactive decay. The muon decays to an electron and two neutrinos through an intermediate massive particle called the W - boson. The muon, electron and W - all have charge - 1. a) Write down a Lagrangi
253a/Schwartz Due September 21, 2010
Problem Set 2
Problems marked with a * are optional, but strongly recommended. 1. Dimensional analysis. a) A photon coupled to a complex scalar field in D-dimensions has action S= 1 2 dDx - F - + gA + 3 + 4 (1) , D - 1
UNIVERSITY OF CALIFORNIA, SANTA BARBARA Department of Physics Physics 221A Prof: Joe Polchinski http:/www.kitp.ucsb.edu/~joep/Web221A/221A.html FINAL EXAM Open notes, homework, solutions, Srednicki (the text, not the person). Please do not discuss the tes
UNIVERSITY OF CALIFORNIA, SANTA BARBARA Department of Physics Physics 221A Quantum Field Theory Fall 2007
http:/www.kitp.ucsb.edu/~joep/Web221A/221A.html FINAL EXAM SOLUTIONS 1. a) The propagator for eld i = A, B, C is i/(k 2 + m2 i ). There is an ABB i v
253a/Schwartz Due September 28, 2010
Problem Set 3
Problems marked with a * are optional, but strongly recommended. 1. We found for e+e- + - that the cross section was proportional to 1 + cos2. a) Work out the Lorentz invariant quantities s = (pe+ + pe-)2
253a/Schwartz Due October 19, 2010
Problem Set 6
1. Consider the following diagram for scalar alchemy, Pb- + Pb + Au+ + Au -:
a) what other diagrams contribute at the same order in perturbation theory? b) show explicitly that the sum of the diagrams is ga
253a/Schwartz Due November 2, 2010
Problem Set 8
1. Moller scattering is the process e-e- e-e-. Of the tree-level processes in QED in is especially interesting because it involves identical particles. a) Calculate the spin-averaged differential cross sect
Phys 253a
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Problem Set 8 Solutions November 22, 2010
1. (a) The t and u channels both contribute to Moller scattering. In Feynman gauge ( = 1), we have iM = (-ie)2 u3 u1 u4 u2 M = -ig -ig - (-ie)2 u4 u1 u3 u2 t u
e2 e2 u3 u1 u4 u2 - u4 u1 u3 u2 . t u
The
253a/Schwartz Due November 9, 2010
Problem Set 9
1. Consider the following diagram for e+e- + - in QED
a) What other diagrams contribute at the same order in perturbation theory? b) Show explicitly that the sum of this diagram and the diagram with the pho
253a/Schwartz Due November 23, 2010
Problem Set 11
The Higgs is a (hypothetical) real scalar particle which couples to the standard model as 1 1 2 Lhiggs = - ( + m2 ) + iii + i(i - mi + gzZ)i + Z ( + m2 )Z + Z Z + Z h 2 2 (1)
i is called a Yukawa Coupling