Download F34TPP Particle Physics 1 Lecture one

F34TPP Particle Physics
1
Lecture one
1. reinstate ~ and c into Dirac’s equation
γ µ ∂µ ψ + mψ = 0,
(1.1)
where the γ µ are dimensionless.
2. reinstate ~ and c into the Klein-Gordon equation.
3. the magnetic moment of an electron is given by µ = g 2me e s, where s is the spin, reinstate
~ and c.
4. the quantum corrected Coulomb potential for an electron is
α
α e−2mr
,
V (r) = −
1+ √
r
4 π (mr)3/2
(1.2)
reinstate ~ and c.
5. show that V µ Vµ = Vν V ν .
6. show ηµν η νρ = δµρ , where δµρ is the Kronecker delta.
7. show that if Vµ = ηµν V ν , then we are forced to have V µ = η µν Vν
8. show that η µν ηµν = 4.
2
Lecture two
1. show, using the Dirac algebra, that γ 5 γ µ = −γ µ γ 5 .
2. show, using the Dirac algebra, that (γ 5 )2 = I.
3. using the properties of γ 5 , and the cyclicity of the matrix-trace, show that T r(γ µ ) = 0.
NB, cyclicity of the trace is just that T r(A1 A2 ...An−1 An ) = T r(An A1 A2 ...An−1 ).
4. Using properties of the matrix-trace, and the Dirac algebra, show that T r(γ µ γ ν ) = 4η µν
5. show that T r(γ 5 ) = 0
6. show that γ µ γµ = 4I
1
3
Lecture three
1. confirm that T a , T b = iabc T c , where abc is the totally antisymmetric Levi-Civita symbol,
with 123 = 1.
2. confirm that S a , S b = iabc S c . Do this using the properties if the Dirac algebra, rather
than in any specific representation of te algebra.
3. calculate (σ 3 )2 , (σ 3 )3 , (σ 3 )4 then, confirm that
cos(θ/2) − i sin(θ/2)
0
−iθ/2 σ 3
e
=
0
cos(θ/2) + i sin(θ/2)
= I cos(θ/2) − iσ 3 sin(θ/2)
(3.3)
(3.4)
This is the matrix analogue of eiθ = cos θ + i sin θ.
4. show that
PL PL
PR PR
PL PR
γ µ PL
γ µ PR
4
=
=
=
=
=
PL
PR
0 = PR PL
PR γ µ
PL γ µ
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
Lecture four
1. show explicitly that Dµ Dν ψ transforms covariantly, i.e. Dµ Dν ψ → eiqα Dµ Dν ψ
2. evaluate the field strength Fµν defined in the lectures
3. using your knowledge of how the four-vector potential Aµ relates to the electric and magnetic fields, show that the field strength is just a re-packaging of electric and magnetic
fields.
4. the complex Klein-Gordon equation, ∂µ ∂ µ φ − m2 φ = 0, has a global symmetry φ0 = e−iΛ φ.
Write down the local, gauged version of the complex Klein-Gordon equation.
5
Lecture five
1. Take the non-relatistic limit of the gauged, complex Klein-Gordon equation (by writing
φ(t, x) = e−imt ϕ(t, x) and dropping second order time derivatives of ϕ) to show that particles described by such fields have zero magnetic moment. (Assume that the background
field is purely magnetic, A0 = φ = 0.)
2
6
Lecture six
1. Starting with x = 1 + bx, use the technique that solved the propogator integral equation
to show that
1 + b + b2 + ... =
1
1−b
(6.10)
2. The Schrödinger propogator is actually given by
Z
dE dp e−iE(tf −t)+ip(xf −x)
G0 (tf , xf , t, x) =
(2π)2 E − p2 /2m + i
where is a small parameter. Integrate this, using
Z
1
eiτ t
Θ(t) =
dτ
2πi
τ − i
(6.11)
(6.12)
to get an explicit (non-integral) expression for the Schrödinger propogator.
3. show that
r
ψ(t, x) =
m
exp
2πit
imx2
2t
(6.13)
1
satisfies the free Schrödinger equation iψ̇ = − 2m
ψ 00 .
4. find an expression for the momentum-space propogator of the Dirac equation.
7
Lecture seven
1. by varying the following action functionals, evaluate their equations of motion
Z
1
1
λ
S1 =
d4 x[− ∂µ φ∂ µ φ − m2 φ2 − φ4 ]
2
2
4
Z
λ
S2 =
d4 x[−∂µ φ∂ µ φ? − m2 φφ? − (φφ? )2 ]
2
Z
1
g
d4 x[ψ̄γ µ ∂µ ψ − ∂µ φ∂ µ φ + λφψ̄ψ − φ4 ]
S3 =
2
4
Z
1
1
S4 =
d4 x[−∂µ φ∂ µ φ? − ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − m2 φφ? + λφ? φϕ]
2
2
(7.14)
(7.15)
(7.16)
(7.17)
2. write down the Feynman rules for the theories given above.
3. draw the second-order and fourth order Feynman diagrams for the scattering of ψψ → ψψ
in theory with action S3 .
3
8
Lecture eight
1. confirm that GL, SL, O, SO, Sp do indeed all form groups, under matrix multiplication.
p
2. taking the standard definition for the length of a vector as V T V , show that V 0 = OV
has the same length as V , where O is a matrix in O(N).
3. write an O(n) matrix as a set of column vectors,
O = (V 1 , V 2 , ...V n )
(8.18)
and see what the condition OT O imposes on these vectors.
4. By introducing the symplectic vector η, where the components of η are the position and
momentum, (η1 = q, η2 = p) rewrite Hamilton’s equations as a symplectic equation.
9
Lecture nine
1. show that
U = e−iθ
a σ a /2
= cos(θ/2) − in.σ sin(θ/2)
(9.19)
and that U † U = I, det(U ) = 1, hence U ∈SU(2)
2. show that the group U(1) is, topologically, a circle
3. show that the group SU(2) is, topologically, a three-sphere
4. the field strength for a non-Abelian gauge field is defined analogously to the Abelian one,
[Dµ , Dν ]φ = −iqFµν φ
(9.20)
0
• how does Fµν
relate to Fµν ?
• In the Abelian case we had Fµν = ∂µ Aν − ∂ν Aµ , what is the corresponding relation
for non-Abelian gauge fields?
10
1.
Lecture ten
• We have seen that σ/2 generate a j = 12 representation of SU(2), write down the
explicit matrices corresponding to j + , j − , j 3 for this rep’.
• calculate the eigenvalues and eigenvectors of j 3 . Label the eigenvectors | 21 , ± 12 i according to their j 3 eigenvalue.
• evaluate j + | 21 , 12 i, j + | 21 , − 21 i, j − | 12 , 12 i, j − | 12 , − 21 i, and confirm that they agree with
the general result.
4
2. Now forget about the explicit rep used above and consider the components of a matrix in
the j = 12 rep defined by
Xmm0 = hj, m|X̂|j, m0 i
(10.21)
where X̂ is some operator, and m labels the rows of the matrix, with m’ the columns.
Use the general formulae
j 3 |j, mi = m|j, mi
p
j ± |j, mi =
j(j + 1) − m(m ± 1)|j, m ± 1i
(10.22)
(10.23)
to calculate the matrices associated with the operators j 1 , j 2 , j 3 . Note that you should
take the m-values to run from +j to −j, so that the first row, first column corresponds
to m = m0 = +j.
11
Lecture eleven
1. confirm that if isospin is conserved, then h0, 0|Ĥint |1, 0i = 0.
2. show that if isospin is conserved
I3
hI, I 3 |Ĥint |I, I 03 i
I 03
(11.24)
I(I + 1) 0 3
hI , I |Ĥint |I, I 3 i
I 0 (I 0 + 1)
(11.25)
hI, I 3 |Ĥint |I, I 03 i =
and hence that it vanishes unless I 3 = I 03 .
3. show that if isospin is conserved
hI 0 , I 3 |Ĥint |I, I 3 i =
and hence that it vanishes unless I = I 0 .
4. given that
|π + i|pi = |3/2, 3/2i
r
r
2
1
−
|π i|pi =
|3/2, 1/2i −
|1/2, 1/2i
3
3
M1 := h1/2, 1/2|Ĥint |1/2, 1/2i
M3 := h3/2, 3/2|Ĥint |3/2, 3/2i
(11.26)
hπ + p|Ĥint |π + pi = M3
2
1
hπ − p|Ĥint |π − pi =
M1 + M3
3
3
(11.30)
(11.27)
(11.28)
(11.29)
show that
5
(11.31)
5. In the lectures we combined two j = 1/2 reps to make one j = 1 and a j = 0, writing
this as 2 ⊗ 2 = 3 ⊕ 1. Now combine three j = 1/2 reps and see what you get. To do
this, denote |1/2, 1/2i by ↑ and |1/2, −1/2i by ↓, then write down all combinations that
are: completely anti-symmetric under interchange of any pair; anti-symmetric under the
interchange ofo the first pair; anti-symmetric under the interchange ofo the second pair;
totally anti-symmetric. So then write down what 2 ⊗ 2 ⊗ 2 is. We shall need this result
when we look at baryons.
6. If you are feeling brave, try the following. Start with two j = 1/2 reps consisting of
|1/2, 1/2i and |1/2, −1/2i, then construct the singlet and triplet states. i.e. starting from
|1/2, 1/2i|1/2, 1/2i, |1/2, 1/2i|1/2, −1/2i,... To do this, use the raising/lowering operators
on the product space.
12
Lecture twelve
the following question set is about mesons, colourless particles formed out of quark-antiquark
pairs.
1. We know that under an SU(2) isospin transformation
u
u
→U
d
d
where U is an SU(2) matrix, that can in general be described by
a
b
U =
,
|a|2 + |b|2 = 1
−b? a?
(12.32)
(12.33)
Given that anti-particles transform under isospin as
¯ → ((ū, d)U
¯ †
(12.34)
(ū, d)
−d¯
u
¯ = −|1/2, 1/2i,
show that
transforms in the same way as
. In other words, |di
ū
d
|ūi = |1/2, −1/2i.
−d¯
u
2. As
transforms in the same way as
. then we may combine them in the
ū
d
same way as before, to form the product 2 ⊗ 2, where the first 2 is composed of |ui and
¯ and |ūi. Using your knowledge of how products
|di, while the second 2 is composed of |di
of |1/2, 1/2i and |1/2, −1/2i combine, write down the triplet and singlet states in terms
¯ and |ūi.
of |ui, |di, |di
3. Just as for baryons, the total wave function of a meson is composed of several parts
Ψtotal = ψspace ψspin ψf lavour ψcolour
• for the lowest energy mesons, what symmetry property should ψspace have?
6
(12.35)
• what are the possible choices for ψspin , given that we have two quarks, each with spin
↑ or ↓.
• how many possible combinations of qi and q̄i are there for three quark flavours?
• of these possible combinations, which is the SU(3) singet, i.e. invariant under u → d,
or u → s etc.?
• with this singlet removed, arrange the remaining states into an octet, where the top
row of the hexagon has S=1, the middle has S=0, and the bottom has S=-1. The
states should be arranged such that the left-most states have lowest isospin, with
isospin increasing to the right. You will probably need linear combinations of your
initial guess-states. The two centre states should be constructed such that one of
them is part of an isospin triplet, and the other is orthogonal to te triplet.
13
Lecture thirteen
1. under parity, how does the magnetic dipole moment change, is it a vector or pseudo vector?
Why does a parity conserving theory allow for fundamental particles to have a magnetic
dipole moment.
2. Last week there was a signal from an alien race, and we have been in constant communication since then, deciding to build spacecraft and meet halfway. To avoid diplomatic
embarrassment it is decided that the ambassadors should shake by the right hand when
they meet - fortunately they do have hands. Suppose that an electrically neutral particle
exists with an electric dipole moment, how would you communicate to them what we
mean by left and right?
14
Lecture fourteen
1. Now that we know parity is violated in the decay of cobalt, how should we use this to
communicate to our alien friends what we mean by left and right?
2. are the following quantities even under T (T : Φ → Φ), or odd (T : Φ → −Φ).
• x
• t
• p
• the electric field E
• the magnetic field B
• electric charge
• angular momentum
• spin
• electromagnetic vector potential A
7
• magnetic dipole moment
• electric dipole moment
3. would the presence of a magnetic dipole moment violate T ?
15
Lecture fifteen
1. We have been in contact with our alien friends, and decided that in order to make sure we
agreed on left and right (in order to shake hands when we meet) we would both perform
the cobalt decy to nickel experiment as performed by Wu. Having agreed to shake right
hands at the first meeting you are surprised to note, when you arrive at the rendezvous,
that the alien offers their left hand. Assuming the aliens are competent at experiments,
and you value you life, what should you do?
2. As previously, assume there exists an electrically charged, neutral particle with an electric
dipole moment. Use the same capacitor experiment to tell our alien friends the difference
between left and right. Now suppose they live in an anti-matter galaxy, would they agree
with you about what is left and right when you meet with them?
16
Lecture sixteen
1. which of the following reactions are allowed, and which are forbidden, by conservation
laws in te standard model
νµ p → µ+ n
νe p → e− π + n
K + → π 0 µ+ νµ
(16.36)
(16.37)
(16.38)
2. assuming there are only three quark flavours (uds), draw the lowest order diagram for the
decay
K 0 → µ− µ+
(16.39)
If we include the charm quark, show that there is another diagram that contributes.
3. Draw Feynman diagrams for the following decays of the D0 (= cū)
D0 → K − π + π 0
D0 → π − π + π 0
(16.40)
(16.41)
4. if the Cabbibo angle is 13◦ , by what factor should the rates for the two decays
K − → π 0 e− ν̄e
π − → π 0 e− ν̄e
differ?
8
(16.42)
(16.43)
5. estimate the Cabbibo angle from the decays of a n(= udd) and Λ0 (= uds)
n → pe− ν̄e , with time constant = 900s, branching ratio = 1
energy scale = 1.3MeV
0
Λ → pe− ν̄e , with time constant = 2.63 × 1 −−10 s, branching ratio = 8.4 × 10−4
energy scale = 177.3MeV
where you may assume that the phase-space factor varies with energy as E 5 .
17
Lecture seventeen
1. the typical cross section for a necleon-neutrino interaction is σ ∼ 10−47 m2 . Given that
the mean-free-path (the average distnce travelled by a particle) is
l = (σn)−1
(17.44)
estimate the mean-free-path of a neutrino in lead. Express your answer in light years!
2. Calculate the MNS matrix, UM N S , explicitly, by multiplying together the three matrices
defining it (denote cos θ12 by c12 , sin θ23 by c23 etc.).
3. given that the mass eigenstates are related to the



νe
 νµ  = UM N S 
ντ
flavour eigenstates by

ν1
ν2 
ν3
(17.45)
express the mass eigenstates as
ν1 = a1 νe + b1 νµ + c1 νµ
ν2 = a2 νe + b2 νµ + c2 νµ
ν3 = a3 νe + b3 νµ + c3 νµ
(17.46)
(17.47)
(17.48)
where the coeficients are known in terms of the mixing angles from the previous question
(You will have to invert U to do this, fortunately it is a unitary matrix - actually special
orthogonal as I’ve switched off the phase - which make this easy to do). Now use the
experimental data to calculate these coefficients, confirming that a2i + b2i + c2i = 1 for each
vaue of i.
18
Lecture eighteen
1. Consider the Lagrangian
λ
L = −∂µ φ∂ µ φ? − (φφ? − η 2 /2)2
4
(18.49)
This describes a field theory with a global U(1) symmetry φ → e−iα φ - no gauge fields.
9
• calculate the equation of motion for this Lagrangian
• by looking at the Lagrangian, at what value of |φ| is the potential minimized?
• if the field takes the constant value that minimizes the potential, does it satisfy the
field equations?
• take the field to be a small perturbation about the vacuum
1
φ = √ eiξ (η + χ)
2
(18.50)
where η is the constant that appears in te Lagrangian, with ξ and χ spacetimedependent fields. Substitute this into the Lagrangian to find a Lagrangian for ξ and
χ. Note, you only need to consider terms that are at most quadratic in ξ and χ.
• what are the masses of ξ and χ?
10