Download useful relations in quantum field theory

1
USEFUL RELATIONS IN QUANTUM FIELD
THEORY
In this set of notes I summarize many useful relations in Quantum Field
Theory that I was sick of deriving or looking up in the “correct”
conventions (see below for conventions)!
Notes Written by: JEFF ASAF DROR
2016
Contents
1 Introduction
1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
2 Classical Field Theory
2.0.1 Important Relations . . .
2.0.2 Free Real Scalar Field . .
2.0.3 Free Complex Scalar Field
2.0.4 Free Dirac Field . . . . . .
2.1 Solutions of the Dirac Equation .
2.1.1 Massless Limit . . . . . .
2.2 Sum Rules . . . . . . . . . . . . .
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5
5
5
6
6
7
8
9
3 Feynman Rules
3.1 Deriving the Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Symmetry Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
10
10
4 Standard Model
4.1 φ4 Theory . . . . . . . .
4.2 Spinor QED . . . . . . .
4.3 Scalar QED . . . . . . .
4.4 Electroweak Interactions
4.5 CKM Matrix . . . . . .
4.6 Supersymmetry . . . . .
4.6.1 Triplet . . . . . .
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18
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5 Polarized Calculations
5.1 Polarization and Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Calculational Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Renormalization
6.1 On-Shell Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
3
7 Quantum Mechanics
7.1 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . .
25
25
25
8 Special Relativity
26
9 Phase space
27
10 Mathematics
10.1 Anticommuting Matrices . . . . . . . . . . . . . . . . . . . .
10.1.1 Sigma Matrices and Levi Civita Tensors . . . . . . .
10.1.2 Gamma Matrices . . . . . . . . . . . . . . . . . . . .
10.2 Complete the Square . . . . . . . . . . . . . . . . . . . . . .
10.3 Degrees of Freedom in a Matrix . . . . . . . . . . . . . . . .
10.4 Feynman Parameters . . . . . . . . . . . . . . . . . . . . . .
10.5 n - sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Sample Loop Integral . . . . . . . . . . . . . . . . . . . . . .
10.7.1 d-dimensional integrals in Minkowski space . . . . . .
10.7.2 Dimensional Regularization and the Gamma Function
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35
35
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Chapter 1
Introduction
In this note I summarize many important relations I constantly look up throughout my
time working in Particle Theory and in particular calculating Feynman diagrams. I try
to derive some of these relationships if the derivations are straightforward but many are
just quoted. One of the most frustrating events for me is to find some formula and not
know what conventions they are using. In this report I follow the Peskin and Schroeder
conventions which I detail in the next sections.
1.1
Conventions
gµν = diag {+ − −−}
(1.1)
The gamma matrices are,
γ0 =
0 1
1 0
,
γi =
0 σi
−σi 0
(1.2)
Natural units are used throughout. The covariant derivatives are given by,
Dµ ≡ ∂µ − igT a Aaµ
(1.3)
and we include a 1/2 in the hypercharge definitions such that,
Q = T3 +
Y
2
(1.4)
The higgs VEV is v ≈ 246 GeV. We also take e < 0 throughout as used in Peskin and
Schroeder.
At this point the notes are awfully disorganized. I hope to fix that in the future.
However, if you find any errors please let me know at [email protected].
4
Chapter 2
Classical Field Theory
2.0.1
Important Relations
The Euler-Lagrangian equations of motion are
∂L
∂L
+ ∂µ
=0
∂φ
∂(∂µ φ)
(2.1)
The conjugate momenta of the field is given by
π=
∂L
∂(∂0 φ)
(2.2)
The Hamiltonian is given by
Z
H=
d3 xπ∂0 φ − L
(2.3)
j µ (x) =
∂L
δφ − J µ
∂(∂µ φ)
(2.4)
The formula for the current is
where J µ found by finding the change in L through a Taylor expansion.
The energy-momentum tensor is given by
T µν ≡
2.0.2
∂L
∂ν φ − Lδ µν
∂(∂µ φ)
(2.5)
Free Real Scalar Field
The Klien Gordan Lagrangian for a real scalar field is
1
L = ∂µ φ∂ µ φ − m2 φ2
2
5
(2.6)
6
CHAPTER 2. CLASSICAL FIELD THEORY
Quantizing the fields gives
Z
φ(x) =
Z
π(x) =
d3 p
1
ip·x
† −ip·x
p
e
a
e
+
a
p
p
(2π)3 2ωp
(2.7)
d3 p
√
(−i) ωp 2 ap eip·x − a†p e−ip·x
3
(2π)
(2.8)
or in an equivalent but more convenient form,
d3 p
1 †
ip·x
p
a
+
a
p
−p e
(2π)3 2ωp
Z
ip·x
d3 p
√
†
π(x) =
(−i)
ω
2
a
−
a
e
p
p
p
(2π)3
Z
φ(x) =
and the commutation relations are
h
i
ap , a†p0 = (2π)3 δ (3) (p − p0 )
(2.9)
(2.10)
(2.11)
as well as
[φ(x), π(x0 )] = iδ (3) (p − p0 )
[φ(x), φ(x0 )] = [π(x), π(x0 )] = 0
2.0.3
(2.12)
(2.13)
Free Complex Scalar Field
d3 p
1
ip·x
† −ip·x
p
a
e
+
b
e
φ=
p
p
(2π)3 2Ep
Z
d3 p
1
† −ip·x
ip·x
p
a
e
+
b
e
φ† =
p
p
(2π)3 2Ep
Z
T µν = ∂ µ φ∗ ∂ν φ + ∂ µ φ∂ν φ∗ − δ µν L
Z
H = d3 x π µ π + ∂i φ∗ ∂i φ + m2 φ∗ φ
2.0.4
(2.14)
(2.15)
(2.16)
(2.17)
Free Dirac Field
Z
ψ=
d3 p
1 X s −ip·x s
s † ip·x s
p
a
e
u
+
b
e
v
p
p
p
p
(2π)3 2Ep s
(2.18)
2.1. SOLUTIONS OF THE DIRAC EQUATION
2.1
7
Solutions of the Dirac Equation
In field theory we often use u(p) and v(p) as solutions to the Dirac equation (with the
exponentials factored out). These obey
p/ − m u(p) = 0
(2.19)
p/ + m v(p) = 0
(2.20)
(2.21)
which in turn imply
u† (p) γ 0 p/γ 0 − m = 0
v † (p) γ 0 p/γ 0 + m = 0
(2.22)
(2.23)
(2.24)
ū(p) p/ − m = 0
v̄(p) p/ + m = 0
(2.25)
(2.26)
(2.27)
The zero momentum solutions take the form
s √
χ
s
u (0) = m
χs
s √
χ
v s (0) = m
−χs
(2.28)
(2.29)
These can be boosted to an arbitrary momentum through
1
e− 2 ηp̂·K
(2.30)
where η = sinh−1 |p|
is the rapidity, p̂ is the unit vector of the boost, and K j ≡ − 2i γ j γ 0
m
is the boost matrix.
It is straightforward to calculate the boost matrix explicitly:
i
i σj
0 σj
0 1
0
Kj = −
=−
(2.31)
1 0
0 −σ j
2 −σ j 0
2
which gives the boost:
− 21 η p̂·K
e
η
σ 0
= exp − p̂ ·
0 −σ
2
− 1 ηp̂·σ
e 2
0
=
1
0
e 2 ηp̂·σ
cosh η2 − p̂ · σ sinh η2
0
=
0
cosh η2 + p̂ · σ sinh η2
(2.32)
(2.33)
(2.34)
8
CHAPTER 2. CLASSICAL FIELD THEORY
2.1.1
Massless Limit
Deriving the form of the equations in the massless limit is straightfoward. We have the
equation
γ µ ∂µ ψ = 0
(2.35)
Take solutions of the form ψ = eip·x u:
pµ
pµ γ µ u = 0
σµ
u=0
0
0
σ̄ µ
(2.36)
(2.37)
This gives two sets of equations that are completely decoupled for the leftand right
u+
handed part of u. We consider the two solutions indepedently. Consider u =
:
0
pµ σ̄ µ u+ = 0
(2.38)
We now define p± ≡ p0 ± p3 , z ≡ p1 + ip2 , and u+ ≡
p+ z̄
z p−
α
β
α
β
. This gives
=0
(2.39)
There are two linearly indepdent solutions to this equation.
1. If p− 6= 0 then we can write β = − pz− α (including β = 0)
2. If p− = 0 ⇒ z = 0, then β can equal anything and α = 0.
but
where we have defined
√
√
z p+ / p − p
z
= √
= p+ /p− eiφ
p−
p+ p−
(2.40)
p1 + ip2
eiφ ≡ √
p+ p−
(2.41)
With this we can write our solutions as
 √
p−
 −√p+ eiφ


0
0





,

0
 √2p0 


 0 
0
(2.42)
2.2. SUM RULES
9
where we have normalized our spinors to the condition u† u = 2p0 . For the other two
linearly independent solutions we have the equation,
pµ σ µ u− = 0
p− −z̄
α
=0
−z p+
β
(2.43)
(2.44)
Our two linearly independent solutions are
1. If p− 6= 0 then we can write α = (z̄/p− )β (including α = 0)
2. If p− = 0 ⇒ z = 0, then α can equal anything and β = 0.
From before we can write
and
2.2
p
z̄
= p+ /p− e−iφ
p−

0


 √ 0 −iφ 

 p+ e
√
p−

0
 0 

 √
 2p0 
0
(2.45)


,
(2.46)
Sum Rules
The spin sum formula for the Dirac spinors are given by
X
usa ūsb = (p/ + m)ab
(2.47)
s
X
vas v̄bs = (p/ − m)ab
(2.48)
s
The polarization sum rule for external vector bosons is given by
(
X
−ηµν (massless boson)
µ (k; λ)¯ν (k; λ) =
µ kν
−ηµν + km
(massive boson)
2
λ
(2.49)
Chapter 3
Feynman Rules
3.1
Deriving the Feynman Rules
To properly derive the Feynman rules can be difficult. However determining the interactions is easy. The important point is to remember that the Lagrangian is a real scalar.
Thus there should generally not be any i’s in it. If there is a complex i then there must
be an accomadating i somewhere else. Consider an arbitrary interaction:
Lint = g (φn1 φm
2 ...)
(3.1)
where the particular fields in the interaction are irrelevant. Then the Feynman rule for
the interaction will just be
−→ ig
(3.2)
x
Note that the sign of the terms are conserved. Positive Lagrangian terms give positive
interaction vertices. Furthermore, there is an i that comes with the term.
Now one subtely is if there is a partial derivative. The proper “replacement” rule for
these is
∂µ → −ipµ
(3.3)
where pµ is the momentum of the particle that ∂µ is acting on.
3.2
Symmetry Factors
When using Feynman diagrams to calculate amplitudes a major difficulty in the calculation is to account for identical particles in the calculation. There can be many diagrams
corresponding to the exact same process so in general we have to account for all of these.
There are 3 contributing factors that result in one factor in front of the amplitude which
is called the Symmetry factor.
1. Each vertex contributes a suppression factor. For example in φ4 theory we typically
have a 4! suppression factor for each vertex. Of course the value of these is dependent on the definition of the couplings but we define our couplings on purpose so
we end up with symmetry factors on the order of unity.
10
3.2. SYMMETRY FACTORS
11
2. There are different ways external particles can be arranged with each vertex. If you
swap all the vertices you get the same diagram.
3. There are equivalent ways to contract the fields in the Wick expansion.
A technique to account for all of these is given by Jacob Bourjaily which credits Colin
Morningstar[Bourjaily(2013)]. The idea is as follows. Let n be the number of verteices
of a diagram, η be the coupling constant suppression factor, and r the multiplicity of a
diagram. Then the Symmetry factor is given by
S=
n! (η)n
r
(3.4)
and the amplitude is given by
M=
1
M
S 1 diagram
(3.5)
The multiplicity of a diagram is the number of different contractions in the Wick expansion, or the number of ways to connect all the external lines to the vertices. This can
be found by first drawing out the edges of each external line and points coming out of a
vertex. Then count the number of ways the lines can be connected.
As an example we consider the “fish” diagram in φ4 theory,
First we draw the edges,
Start with the initial lines. There are eight ways to connect the first line to a vertex.
Then since the initial lines and final lines need to be kept as such, there are only 3 ways
to connect the second initial line. Continuing on and counting the number of ways each
line can be connected we have
r = (8)(3)(4)(3)(2)(1)
(3.6)
2! (4!)2
=2
S=
(8)(3)(4)(3)(2)(1)
(3.7)
This gives a symmetry factor of
Chapter 4
Standard Model
The Standard Model charges are summarized below:
doublet
νe
e− L
u
d L
singlets
eR
uR
dR
Higg’s sector
+ φ
φ0
4.1
T(
3 = σ3 /2 ( Y
1
−1
2
1
−1
−
( 2
(
1
2
1
3
1
3
− 12
0
0
0
(
Q(
= T3 +
0
−1
(
− 12
− 13
−1
4
3
2
3
(
1
1
1
2
2
3
−2
− 23
Y
2
− 13
(
1
0
φ4 Theory
The scalar propagator is
ff
→
12
p2
i
− m2 + i
(4.1)
4.2. SPINOR QED
4.2
13
Spinor QED
fFf
fFf
ff
ff
ggg
g
E
a
a
p→
←p
b
b
=
=
a
p→
b
=
a
←p
b
=
µ
k→
ν
=
b
=
i(p/ + m)ab
=
p2 − m2 + i
i(−p/ + m)ab
=
p2 − m2 + i
i(−p/ + m)ab
=
p2 − m2 + i
i
p/ − m + i
i
−p/ − m + i
(4.2)
ab
(4.3)
ab
i
−p/ − m + i ab
i(p/ + m)ab
i
=
p/ − m + i ab
p2 − m2 + i
i
kµ kν
− 2
gµν − (1 − ξ) 2
k + i
k
−ieQ (γ µ )ab
(+ momentum conservation)
a
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
where e > 0 and Q = −1 for the electron.
Furthermore, incoming and outgoing photons gives,
µ
µ
µ ∗µ
4.3
Scalar QED
The Lagrangian for scalar QED is given by
λ
1
L = (Dµ φ)† Dµ φ − m2 φ† φ − (φ† φ)2 − Fµν F µν
4
4
(4.9)
where Dµ = ∂µ − ieAµ . The vertices are given by
λ
Lint = −ieAµ (∂ µ φ† )φ − φ(∂ µ φ) = e2 Aµ Aµ φ† φ − (φ† φ)2
4
(4.10)
14
CHAPTER 4. STANDARD MODEL
Depending on the relative directions of the lines going into vertex and the type of particle
we get a different vertex factor. This gives the following rules
FE
v
v
Dg
gE
e−
e−
e+
e+
→
−ie(pµ1 + pµ2 )
(4.11)
→
−ie(−pµ1 − pµ2 )
(4.12)
→
−ie(pµ1 − pµ2 )
(4.13)
→
−ie(pµ1 − pµ2 )
(4.14)
2ie2 gµν
(4.15)
e−
e+
e−
e+
and
4.4
Du
v
→
Electroweak Interactions
√
The W ± ≡ (W1 ∓ iW2 )/ 2 boson interactions are given by,
∂µ −i 21 σ a W a
√
z}|{
+
1
g
.
2W
µ
√
ψ̄i Dµ γ ψ = g ψ̄γµ
ψ + ... = √ ψ̄γ µ PL Wµ+ ψ
−
2W
.
2
2
The triple gauge vertices are,
Wν−
%
→
Wµ+ → −ie(g µν (k− − k+ )λ − g νλ (q + k− )µ + g λµ (q + k+ )ν )
→
Wµ+ → −igcW (g µν (k− − k+ )λ − g νλ (q + k− )µ + g λµ (q + k+ )ν )
γλ
Wν−
%
Zλ
(4.16)
4.4. ELECTROWEAK INTERACTIONS
15
The higgs interactions are found through,
0
0
g
g
†
a
a
µ
b
bµ
φ
φ
Y Bµ + gT Wµ
Y B + gT W
2
2
√ † .
1
.√
.
=
. φ0 / 2
. 2g 2 W + W − + (g 0 Y B − gW3 )2
φ0 / 2
4
2
2
g
e
1
= φ20 W + W − + 2 2 φ20
Zµ Z µ
4
4cW sW
2
(4.17)
(4.18)
(4.19)
which gives a higgs-W + − W − Lagrangian (there is an additional factor of 2 since each
higgs in φ0 can get a VEV)1 ,
W
Z
h
→
ig 2 v
2
W
h
2
e v
→ i 2cW
sW
Z
The masses of the vector bosons are obtained by taking φ0 → v:
m2W =
g2v2
,
4
m2Z =
e2 v 2
4cW sW
(4.20)
The left fermion-Z interactions are derived from,
0
g µ
1 g 0 Y B + gW3
.
a
aµ
ψ̄γµ
B Y + gT W
PL ψ = ψ̄γµ
PL ψ
.
g 0 Y B − gW3
2
2
(4.21)
The weinberg angle rotates into the A, Z basis. An angle of 0 corresponds to hypercharge
being fulled aligned with charge (i.e., Bµ = Aµ ):
Bµ
Aµ
cW −sW
=
(4.22)
Wµ3
Zµ
s W cW
and the couplings are related to the electroweak couplings by,
p
|e| = g 0 cW = gsW |e| = g 02 + g 2
(4.23)
and in our conventions,
Q = T3 +
Y
2
(4.24)
This corresponds to,
0
g Y Bµ ±
1
gWµ3
We assume φ = h + v
1
= e (Y ± 1)Aµ +
±1 + s2W (−Y ∓ 1) Zµ
s W cW
(4.25)
16
CHAPTER 4. STANDARD MODEL
The up-type and down-type fermion interactions are,
e
1
up−type
2
LN C
= eQAµ +
− QsW Zµ ūγ µ PL u
cW sW 2
e
1
down−type
2
¯ µ PL d
LN C
+ QsW Zµ dγ
= eQAµ −
cW s W 2
(4.26)
(4.27)
The right handed couplings are
LψR ψR Z + ... = ψ̄γµ g 0 B µ QPR ψ = −
eQsW
ψ̄γµ PR ψZ µ + ...
cW
(4.28)
We now compute the interactions of the goldstones with the vector bosons. These
arise from the higgs kinetic term:
† g
g0
† µ
a
a
a
a
(Dµ H) D H = (∂µ − igT Wµ − i Bµ )H
(∂µ − igT Wµ − i Bµ )H
(4.29)
2
2
g0
2
†
a
a
01
†
a
a
= |∂µ H| + ∂µ H −igT Wµ − ig Bµ H + H igT Wµ + i Bµ ∂ µ H
2
2
2
0
g
+ gT a Wµa + Bµ H (4.30)
2
√
µ
+
0
i
2W
gW
+
g
B
g
2
†
µ
†
√
H − H (...)µ ∂ H
∂µ H
= |∂µ H| −
2gW − −gW 3 + g 0 B
2
1
+ H † (...)µ (...)µ H
(4.31)
4
The first term is just a kinetic term. We first simply the term in square brackets.
√
µ gW + g 0 B
+
π+
g
2W
− √1
0
√
(h − iπ )
[...] = ∂µ π
− h.c.
2
√1 (h + iπ 0 )
2gW − −gW 3 + g 0 B
2
(4.32)
• π 0 π 0 Z:
− gW 3 + g 0 B = −
⇒
e
Z
s w cw
(4.33)
1 e ∂µ π 0 π 0 Z µ − ∂µ π 0 π 0 Z µ = 0
(4.34)
2 s w cw
Thus there is not π 0 π 0 Zµ interaction. This is comforting as the SM doesn’t have a
triple-Z interaction. This can be traced back to the fact that the Goldstones form a
triplet of SU (2)L and so the interactions are parameterized by, abc Wa Wb Wc which
doesn’t have a ZZZ interaction from the structure of the Levi-Cevita tensor.
∆L = −
• hπ 0 Z:
∆L = −
e 1 e i (∂µ h)π 0 Z µ − h ∂µ π 0 Z µ − h.c. = −i
∂µ (hπ 0 )Z µ (4.35)
2 s w cw
s w cw
4.4. ELECTROWEAK INTERACTIONS
17
• hhZ:
∆L = −
1 e
(∂µ h) hZ µ − h.c. = 0
2 sw cw
(4.36)
• π − π + Z, π − π + A:
µ
∆L = (∂µ π − )π + gW 3 + g 0 B − h.c.
µ
1
µ
−
+
2
2A +
= e (∂µ π )π
1 − 2sw Z − h.c.
s w cw
(4.37)
(4.38)
• π ± W ∓ h:
∆L = g ∂µ π − W + µ h + g (∂µ h) π + W − µ − h.c.
= gh ∂µ π + W + µ − ∂µ π + W − µ + g (∂µ h) π − W + µ − π + W − µ
= g ∂µ (hπ − )W + µ − h.c.
(4.39)
(4.40)
(4.41)
2
Now lets consider the terms arising from the H † (...)2 H term. We have,
√
2
1 †
a√+ 2g 2 Wµ− W ± g 2(a + b) · W +
2
|...| = H
H
(4.42)
g 2(a + b) · W + b2 + 2g 2 W + · W −
4
1
2
where aµ ≡ e 2Aµ + sw cw (1 − 2sw ) Zµ and bµ = − cwesw Zµ . The terms arising from this
contribution are given below:
• π + π − , A, Z, W :
e2
∆L = π + π −
4
"
1
2Aµ +
1 − 2s2w Zµ
s w cw
2
#
+ 2g 2 Wµ+ W − µ
(4.43)
• (π 0 )2 , h2 , W 2 , Z 2 :
11 2
h + π0 2
∆L =
42
"
e
s w cw
2
#
µ
Zµ Z + 2g
2
Wµ+ W − µ
(4.44)
• π − (h, π 0 )Z, W + : Top right:
1
∆L = π − h + iπ 0 g(a + b) · W + µ
4
e2 =
h + iπ 0 π − (Aµ − tw Zµ ) W + µ
2
(4.45)
(4.46)
The bottom left contribution is similar. The sum gives,
e2 (Aµ − tw Zµ ) (h + iπ 0 )π − W + µ + h − iπ 0 π + W − µ
2
(4.47)
18
4.5
CHAPTER 4. STANDARD MODEL
CKM Matrix
The CKM matrix in the Wolfenstein parametrization is [Gibbons(2013)]


2
1 − λ2
λ
Aλ3 (ρ − iη)
2

VCKM = 
−λ
1 − λ2
Aλ2
Aλ3 (1 − ρ − iη) −Aλ2
1
4.6
(4.48)
Supersymmetry
In this section we summarize the supersymmetric Feynman rules for a gauge theory.
We continue to take the Peskin and Schroeder convention for the sign in the covariant
derivatives. This comes at the come of deviating from the convention used in Martin’s
notes by a minus sign for the gauge charge. For a field charged under a gauge group the
D term is given by,
Z
√
(4.49)
d4 θΦ† eV Φ = iψ † σ̄ µ Dµ ψ + 2g(φ† T a ψ)λa + ...
It is often useful to have the form of this expression in the case of a field charged under
SU (2) × U (1). We denote the field by uL , dL but they refer to up-type and down-type
fields respectfully. We have,
Z
√ Y
√
(4.50)
d4 θΦ† eV Φ = 2g φ† T a ψ W̃ a + 2g 0 φ† B̃ψ + h.c. + ...
2
√
+
1
2g
W̃
u
g
W̃
+
Y
g
B̃
L
0
†
†
√
= ũL d˜L √
+ h.c.
(4.51)
dL
2g W̃ −
g 0 Y B̃ − g W̃0
2
† 1
†
†
† 1
+
−
˜
˜
= ũL √ g W̃0 + Y g B̃ uL + dL √ gY B̃ − g W̃0 dL + g ũL W̃ dL + dL W̃ uL
2
2
(4.52)
4.6.1
Triplet
Above we considered the additional EW interactions for a field in the fundamental representation. Now we extend this to fields in the adjoint representation. For an adjoint
field we have the new interactions,
√
2g φ† T a ψ W̃ a + gψ † Ta σ̄ µ ψWµa
(4.53)
There are no additional B̃ terms since fields in the adjoint representation of a U (1) aren’t
charged under the group. We start by considering the first term. Using Tbca = ifabc = iabc
we have,
√
√
(4.54)
2g φ† T a ψ W̃ a = 2igφ†b bac ψc W̃ a




0
−W̃ 3 W̃ 2
ψ1
√
†
†
†
3
1  ψ 

= 2g φ1 φ2 φ3
(4.55)
W̃
0
−W̃
2
2
1
ψ3
−W̃
W̃
0
4.6. SUPERSYMMETRY
19
We want to write the fields according to their charged basis. For this we use the transformations,
1
1
ψ−
1 +i
ψ1
φ−
1 +i
φ1
=√
=√
(4.56)
ψ+
ψ2
φ+
φ2
2 1 −i
2 1 −i
We define,


1 i 0
1
U ≡ √  1 −i 0 
2
0 0 1
(4.57)
and so,
√
i 2g

φ†− φ†+

ψ
−
φ†0 U AU †  ψ+ 
ψ0
(4.58)
where,

− √i2 (W̃ − − W̃ + )


1
0
−
+


W̃
0
−
W̃
+
W̃
A≡
2

1
−
+
−
+
√i
√
0
W̃ − W̃
W̃ + W̃
2
2

−W̃ 0
0
(4.59)
Multiplying through gives,
√
− 2g


W̃ 0
0
−W̃ −
ψ−
 0
−W̃ 0 W̃ +   ψ+ 
ψ0
0
−W̃ + W̃ −

φ†− φ†+ φ†0
(4.60)
√
The simpilfication for the second term is identical with g → g/ 2, φ → ψ, and
W̃ → W ,

 

0
−
W
0
−W
ψ
−
µ
µ
†
†
−Wµ0 Wµ+  σ̄µ  ψ+ 
(4.61)
gψ † Ta σ̄ µ ψWµa = −g ψ−
ψ+
ψ0†  0
+
−
ψ0
0
−Wµ Wµ
To rewrite this in terms on the physical gauge bosons we use, Wµ0 = sW Aµ + cW Zµ .
Chapter 5
Polarized Calculations
5.1
Polarization and Spin
For some reason a thorough discussion of polarization calculations is missing from the
popular Quantum Field Theory books. The discussion here is an amalgam of what I’ve
found from Peskin, Srednicki, as well as Bjorken and Drell.
Consider a particle with a spinor u(p, s) or v(p, s) which is at rest. It’s polarization in
the rest from of the particle is what we often call its spin. We denote this polarization as
some 3-vector λ. For example if the particle is polarized along the z axis then λ = (0, 0, 1).
We form a four-vector to represent its “spin”. We denote the rest frame spin vector by
sµr . Now what should the first component of the four-vector be? In the rest frame there
is no other degree of freedom for the spin. We set s0r to zero.
To boost back to the lab frame we apply a Lorentz Transformation in the −p direction
onto the four-vector. Recall in matrix form the transformation matrices applied onto a
vector (t, r):
t0 = γ (t − r · v)
γ−1
0
r · v − γt v
r =r+
v2
20
(5.1)
(5.2)
5.1. POLARIZATION AND SPIN
21
The spatial component of the spin in the lab frame is
γ−1
s = λ + − 2 λ · v − 0 (−v)
v
γλ · (γmv) λ · γmv
−
vγm
=λ+
mγ
γm
(γ − 1)λ · p 2
γ p
=λ+
E 2 (γ 2 − 1)
p(λ · p)
=λ+ 2
E2
E m(E + m)
p(λ · p)
=λ+
m(E + m)
(5.3)
and the time component is
s0 = γλ · vm/m
p·λ
=
m
So we have
µ
s =
p·λ
p(p · λ)
,λ +
m
m(m + E)
(5.4)
(5.5)
In the case that the spin is measured along the direction of motion (i.e. p̂ k λ̂) we have
sµ =
1
(|p| , p̂E)
m
(5.6)
Note that
(p · λ)2
(p · λ)2
p2 (p · λ)2
2
−
λ
−
2
−
m2
m(E + m) m(m + E)
2
E 2 − m2
1
2
−
−
−1
= (p · λ)
m2 m(E + m) m2 (m + E)2
2
E + 2mE + m2 − 2Em − 2m2 − E 2 + m2
2
= (p · λ)
−1
m2 (E + m)2
= −1
s2 =
(5.7)
and
p · λE
(E 2 − m2 )(p · λ)
−p·λ−
m
m(m + E)
E
E−m
= (p · λ)
−1−
m
m
E−m−E+m
= (p · λ)
m
=0
sµ p µ =
(5.8)
22
CHAPTER 5. POLARIZED CALCULATIONS
For a general spin vector the spin projection operator is (see for example [Bjorken and Drell(1964)])
Σ(s) =
1 + γ/s
2
(5.9)
This operator obeys
Σ(s)u(p, s) = u(p, s)
Σ(s)v(p, s) = v(p, s)
Σ(−s)u(p, s) = Σ(−s)v(p, s) = 0
5.2
(5.10)
(5.11)
(5.12)
Calculational Tricks
When doing a calculation with some polarized particles there are some useful tricks that
can be implemented to simplify the math. The key result that can be used to derive all
the following relations is
Spin Projection Operator
z }| {
1 + γ 5 /s
u(p, λ)ū(p, λ) = p/ + m
2
1 + γ 5 /s
v(p, λ)v̄(p, λ) = p/ − m
2
(5.13)
(5.14)
With this we can now find a variety of important relations (we suppress the polarization and momentum dependence in u:
ūγ µ u = tr (ūγ µ u)
= tr (γ µ uū)
1
= tr γ µ p/ + m 1 + γ 5 /s
2
1
= tr γ µ p/
2
= 2pµ
(5.15)
ūγ µ γ 5 u = tr ūγ µ γ 5 u
= tr γ µ γ 5 uū
1
= tr γ µ γ 5 p/ + m 1 + γ 5 /s
2
1
= tr (+mγ µ /s)
2
= 2msµ
(5.16)
Chapter 6
Renormalization
Renormalization schemes is subtle topic with a lot of depth. Here we just present the
bare-bones needed to do calculations
6.1
On-Shell Renormalization
There are two renormalization conditions to consider. First corresponds to the mass
renormalization. Consider the full particle propagator (with the external lines amputated,
i.e. ignore the external legs contribution):
FF
p→
FpF
p→
+
FpF
p→
p→
+
FxxF
FxxF
p→
p→
(6.1)
where
represents a sum of loop corrections and
is the counterterm.
It can be shown (see [Perelstein(2013)], pg. 36 for a detailed example) that the Green
function corresponding the above computation are (in the case of φ − 4 theory)
hΩ|φ(x)φ(y)|Ωi =
p2
−
m2p
i
+ Σ(p2 ) + i
(6.2)
where Σ(p2 ) is the sum of the amputated diagrams above. There are two shift factors
here. There is a shift in the pole and there is a also a shift in the amplitude of this
factor. The first renormalization condition is that when the incoming particle is on-shell
(p2 = m2p ), the loop contribution (Σ) is zero. In other words
FpF
FF
p→
p→
+
FxxF
p→
p→
=0
(6.3)
p2 =m2p
This makes sence since
should give the pical propagator for an on-shell particle.
The second renormalization condition is for the normalization of the Green’s function
23
24
CHAPTER 6. RENORMALIZATION
not to change. We can rewrite equation as
i
hΩ|φ(x)φ(y)|Ωi =
p2
=
−
m2p
+
Σ(m2 )
(p2
+
−
(6.4)
dΣ m2 ) dp
2
+ i
p2 =mp
i
dΣ (p2 − m2p ) 1 + dp
2
(6.5)
!
+ Σ(m2 ) + i
p2 =m
p
(6.6)
where we only keep terms of order p2 − m2 since we are considering the conditions on Σ
near the pole (these are on-shell conditions after all!). Thus we require (in φ − 4 theory):
dΣ =0
(6.7)
dp2 p2 =m2
The third renormalization condition is for the coupling. The sum of the 4 vertex
diagrams (this can of course be done for any types of couplings but we consider 4 external
legs for concreteness).
DE
+
The renormalization condition is


DpE DxxE
DpE DxxE
+
where λ is the renormalized couplings.
+


(6.8)
=0
s+u+t=4m2
(6.9)
Chapter 7
Quantum Mechanics
7.1
Commutation Relations
[x, p] = i~
7.2
(7.1)
Quantum Harmonic Oscillator
The creation annhillation operators are defined by
mω
i
a≡
p̂
x̂ +
2~
mω
√
i
†
p
a = mω2~ x −
mω
(7.2)
(7.3)
or
r
~
(a + a† )
2mω
√
p̂ = i mω~2(a† − a)
x̂ =
(7.4)
(7.5)
which give
√
a† |ni = n + 1 |n + 1i
√
a |ni = n |n − 1i
(7.6)
(7.7)
We also have,
n
h0|an a† |0i = n!
This is easiest to see by starting with n = 1 and going on recursively to higher n.
25
(7.8)
Chapter 8
Special Relativity
In special relativity we have covariant (xµ ) and contravariant (xµ ) vectors. Contravariant
vectors have positive spatial indices:
xµ = (x0 , x)
(8.1)
while covariant vectors have negative spatial indices:
xµ = (x0 , −x)
(8.2)
The derivatives have the opposite sign convention,
∂µ ≡ (∂0 , ∇)
∂ µ ≡ (∂0 , −∇)
26
(8.3)
(8.4)
Chapter 9
Phase space
The differential cross section of a two-body collision is given by,
Z
1
dσ =
dΠ |M|2
2EA 2EB |vA − vB |
(9.1)
where Ei and vi are the incoming particle energies and velocities, while we define,
!
Y d3 pf 1
X
X
4 (4)
dΠ ≡
pf )
(9.2)
(2π)
δ
(
p
−
i
(2π)3 2Ef
i
f
f
If the incoming particles are identical particles we have, EA = EB = Ecm /2, and vA −
vB = 2 |pi | /Ecm , which gives,
Z
1
dσ =
dΠ |M|2
(9.3)
8Ecm |pi |
where pi is the momenta of the incoming particles.
The differential decay rate is,
Z
1
dΓ =
dΠ |M|2
2mA
For a two particle final state the phase space is just,
Z
Z
dΩcm 1 2 |p|
dΠ2 =
4π 8π Ecm
(9.4)
(9.5)
where |p| is the magnitude of the
√ momentum of either outgoing particle (they are equal
in the cm frame), and Ecm = s is the cm energy. In the massless limit this is just, [Q
1: There is factor of 2 problem floating around....]
Z
Z
1
dΩcm
dΠ2 =
(9.6)
8π
4π
27
28
CHAPTER 9. PHASE SPACE
Finally for 2 → 2 process with all massless particles we have,
Z
1
dΩcm
dσ =
|M|2
8πs
4π
(9.7)
If we have a 2 → 2 process with the outgoing particles having equal mass then we can
rewrite the phase space as,
Z − s (1−β)2
4
1
dt
(9.8)
4πs − 4s (1+β)2
q
2
where β ≡ 1 − 4ms and t is the Mandelstam variable. In the massless limit this takes
the form,
Z 0
1
dt
(9.9)
4πs −s
The three particle phase space for unpolarized particles with two massless particles
(i = 1, 2) and one massive (i = 3) is,
Z 1−α2
Z 1−(α2 /(1−x1 ))
s
dx1
dx2
(9.10)
dΠ3 =
128π 3 0
1−α2 −x1
P
√
√
where α ≡ m3 / s, s ≡ ( pi )2 , and xi ≡ 2Ei / s is the momentum fraction of particle
i. For m3 → 0 we have the better known result,
Z
Z 1
Z 1
s
dΠ3 =
dx1
dx2
(9.11)
128π 3 0
1−x1
Z
Chapter 10
Mathematics
10.1
Anticommuting Matrices
10.1.1
Sigma Matrices and Levi Civita Tensors
σ1 =
0 1
1 0
σ2 =
0 −i
i 0
σ3 =
1 0
0 −1
(10.1)
They obey the commutation relations,
[σa , σb ] = 2iabc σc
(10.2)
{σa , σb } = 2δab
(10.3)
and the anticommutation relations,
Furthermore any product of Pauli matrices can be written as
σa σb = iabc σc + δab
(10.4)
σi2 = 1
(σ a )∗ = iσ2 σ a iσ2
Trσi = 0
det σi = −1
(10.5)
(10.6)
(10.7)
(10.8)
They also obey,
It is a common practice to exponentiate a linear combination of these matrices. We
derive the general formula below,
1
eiθi σi = 1 + iθi σi − θi θj σi σj − ...
2
1
1
2
3
= 1 − (θi σi ) + ... + i θi σi − (θi σi ) + ...
2
3!
29
(10.9)
(10.10)
30
CHAPTER 10. MATHEMATICS
but
θi θj (σi σj ) = θi θj (iijk σk + δij )
= θ2
(10.11)
(10.12)
(θi σi )3 = θ2 θi σi
(10.13)
and
so
eiθi σi = cos θ + i
θi σi
sin θ
θ
(10.14)
σ µ σ̄ ν + σ ν σ̄ µ = 2η µν
(10.15)
We also have
tr {σ µ σ̄ ν } = 2η µν
(10.16)
(σ µ )αα̇ (σ̄µ )β̇β = 2δα β δα̇ β̇
(10.17)
Furthremore,
σ µν =
1 µνρσ
σρσ
2i
(10.18)
We have
0123 = −0123 = +1
(10.19)
ijk imn = δjm δkn − δjn δkm
(10.20)
and
imn
jmn in
ij =
=
2δji
(10.21)
δhn
(10.22)
Furthermore, we have
σµνρ σαβγ = 6δα[µ δβν δγρ]
ν]
(10.23)
ν
µνρσ ρσαβ = −4δα[µ δβ ≡ −4 δαµ δβν − δαν δβ
(10.24)
where the square brackets denote all possible permutations of µ, ν, ρ. Each permutation
has a negative sign if it is an odd permutation.
The three dimensional extension of the Pauli matrices are:






0 1 0
0 −i 0
1 0 0
1
1
√  1 0 1  , √  i 0 −i  ,  0 0 0 
(10.25)
2
2
0 1 0
0 i
0
0 0 −1
10.1. ANTICOMMUTING MATRICES
10.1.2
31
Gamma Matrices
In the Weyl basis the Gamma matrices can be written
0 σµ
µ
γ =
σ̄ µ 0
(10.26)
where σ µ = {1, σ} and σ̄ µ = {1, −σ}. They obey the defining commutation relation
{γ µ , γ ν } = 2g µν
(10.27)
This leads to many important relations. For example
1 µ
{γ γµ + γ µ γµ }
2
1 µ
=
2gµ
2
=4
(10.30)
γ µ γ α γµ = γ µ 2gµα − γµ γ α
(10.31)
γ µ γµ =
= 2γ α − 4γ α
= −2γ α
In the Weyl basis the Gamma-5 matrix takes the form
−1 0
5
γ =
0 1
(10.28)
(10.29)
(10.32)
(10.33)
(10.34)
and has the properties that
(γ 5 )† = γ 5
(γ 5 )2 = 1
5 µ
γ ,γ = 0
(10.35)
(10.36)
(10.37)
The γ 5 matrix can be used to form projection operators:
1 − γ5
2
1 + γ5
PR =
2
PL =
(10.38)
(10.39)
Furthermore we have,
a
//b = a · b − iaµ σ µν bν
(10.40)
a
/a
/ = a2
(10.41)
and in particular,
32
CHAPTER 10. MATHEMATICS
Trace Technology
(ūγ µ v)∗ = v † γ µ†
(10.42)
= v̄γ0 γ µ† γ0 u
= v̄γ0 (γ0 γ µ γ0 )γ0 u
(ūγ µ v)∗ = v̄γ µ u
(10.43)
(10.44)
(10.45)
We have
1
tr (γ µ γ ν ) = tr (γ µ γ ν + γ µ γ ν )
2
1
= tr (γ µ γ ν + 2η µν − γ ν γ µ )
2
1
= tr (γ µ γ ν + 2η µν − γ µ γ ν )
2
1
= tr (2η µν )
2
= 4η µν
(10.46)
(10.47)
(10.48)
(10.49)
(10.50)
where we have used the cyclic property of traces.
Furthermore we have
tr γ 5 = 0
tr γ µ γ ν γ 5 = 0
tr γ µ γ ν γ ρ γ σ γ 5 = −4iµνρσ
10.2
(10.51)
(10.52)
(10.53)
Complete the Square
To complete the square we want to take an expression of the form ax2 + bx + c to
d(x + e)2 + f . Expanding the second form gives
dx2 + 2dex + de2 + f
(10.54)
first off, clearly a = d. So we make the identifications, b = 2ae, ae2 + f = c. However, we
typically want the inverted form of these equations. So we have
d=a
b
e=
2a
f =c−
(10.55)
(10.56)
b2
4a
(10.57)
10.3. DEGREES OF FREEDOM IN A MATRIX
10.3
33
Degrees of Freedom in a Matrix
The number of degrees of freedom in a unitary matrix are found below. The number of
free parameters in a general complex matrix is 2N 2 . Unitarity implies that U † U = 1. We
define the elements of U as aij + ibij . Then unitarity implies
(aij + ibij )(aji − ibji ) = δij
(10.58)
aij aji + bij bji = δij
bij aji − aij bji = 0
(10.59)
(10.60)
which gives two equations,
The first equation is symmetric under interchanging i ↔ j and gives
1 + 2 + ... + N =
N (N + 1)
2
(10.61)
conditions (just think of it as a matrix equation and the independent equations making
up a top right triangle of the matrix).
The second equation is antisymmetric under changing i ↔ j (the component of i = j
vanishes trivially as doesn’t offer an extra constraint). This gives
1 + 2 + ... + N − 1 =
N (N − 1)
2
(10.62)
conditions.
Thus the number of free parameters in a Unitary matrix is
2N 2 − N 2 = N 2
10.4
(10.63)
Feynman Parameters
Two denometers can be combined as
1
=
AB
Z
1
dx
0
1
[xA + (1 − x)B]2
(10.64)
n denomenators can be combined as
1
=
A1 A2 ...An
Z
0
1
X
dx1 dx2 ... dxn δ(
xi − 1)
i
(n − 1)!
[x1 A1 + ... + xn An ]n
(10.65)
34
CHAPTER 10. MATHEMATICS
This is done below for a common two denomenators:
Z
1
1
1
= dx
2
2
2
2
2
2
2
(` − p) − m1 + i ` − m2 + i
[x((` − p) − m1 + i) + (1 − x)(`2 − m22 + i)]
Z
1
= dx
2
[`2 − 2`px + p2 x + (m22 − m21 )x − m22 + i]
Z
1
= dx
2
[(` − px)2 + p2 x(1 − x) + (m22 − m21 )x − m22 + i]
Z
1
= dx
(10.66)
2
[(` − px) − ∆ + i]2
where we have defined ∆ ≡ −p2 x(1 − x) + (m21 − m22 )x + m22 .
10.5
n - sphere
The surface area of a unit n − sphere is
Sn = (n + 1)
π (n+1)/2
Γ n+1
+1
2
(10.67)
Note in this notation a sphere in our 3D world corresponds to S2 = 4π.
10.6
Integrals
10.7
Sample Loop Integral
One common integral (usually in φ − 4 theory) is the following
Z
1
d4 ` 2
` − ∆ + i
(10.68)
We perform this integral here to refer to this result in the future. We assume φ − 4 so we
can work with the cutoff. The integral can be split as follows
Z
1
= d3 `d`0 2
2
`0 − ` − ∆ + i
The poles are at
`20 − `2 − ∆ + i = 0
`0 = ±
The positions of the poles are shown below
√
`2
+ ∆ − i
10.7. SAMPLE LOOP INTEGRAL
35
Im(`0 )
×
× Re(` )
0
We can apply a Wick rotation such that `0 → −i`0 . This gives
Z
1
= i d`d`0 2
2
−`0 − ` − ∆ + i
Z
1
= −i d4 `E 2
`E + ∆
where we define a Eulidean four vector, `E = (`0 , `) with `2E ≡ `20 + `2 . Since we are now
far from the poles we also omit the i factors.
Z
k3
2
= −i2π
dkE 2 E
kE + ∆
2 Z Λ
∆3/2 x3
2π
∆1/2 dx 2
= −i
∆ 0
x +1
Z Λ/m
x3
= −i2π 2 ∆
dx 2
x +1
02
Λ
2
2
= −iπ ∆
− log 1 + Λ /∆
∆
∆ + Λ2
2
2
= −iπ Λ − ∆ log
(10.69)
∆
10.7.1
d-dimensional integrals in Minkowski space
10.7.2
Dimensional Regularization and the Gamma Function
There are two particularly useful integrals when using dim-reg([Peskin and Schroeder(1995)],
pg. 251):
Z d
d `E
1
1 Γ n − d2 d/2−n
=
∆
(10.70)
(2π)d (`2E + ∆)n
(4π)d/2 Γ(n)
Z d
d `E
`2E
1 d Γ n − d2 − 1 d/2−n+1
=
∆
(10.71)
(2π)d (`2E + ∆)2
(4π)d/2 2
Γ(n)
36
CHAPTER 10. MATHEMATICS
The Gamma function is the generalization of the factorial. It obeys the relationships
and
Γ(n) = (n − 1)! ,
(10.72)
Γ(n + 1) = nΓ(n) ,
(10.73)
√
1
Γ
= π
2
(10.74)
When using “dim reg” to regular your integrals it is often useful to expand Γ() to first
order:
1
1
− γ + (6γ 2 + π 2 ) + O(2 )
12
1
Γ(−1 + ) = − + (γ − 1)
Γ() =
(10.75)
(10.76)
where γ ≈ 0.577.
To expand any other terms of the form ∆ one can use
∆ = elog ∆
= e log ∆
= 1 + log ∆
Z
(−1)n i Γ n − d2
1
dd `
=
(2π)d (`2 − ∆)n
(4π)d/2 Γ(n)
1
∆
(10.77)
(10.78)
n− d2
(10.79)
n− d −1
2
dd `
`2
(−1)n−1 i d Γ n − d2 − 1
1
=
(10.80)
(2π)d (`2 − ∆)n
(4π)d/2 2
Γ(n)
∆
n− d −1
Z
2
dd `
`µ `ν
(−1)n−1 i g µν Γ n − d2 − 1
1
=
(10.81)
(2π)d (`2 − ∆)n
(4π)d/2 2
Γ(n)
∆
n− d −2
Z
2
dd `
(`2 )2
(−1)n i d/(d + 2) Γ n − d2 − 2
1
=
(10.82)
(2π)d (`2 − ∆)n
(4π)d/2
4
Γ(n)
∆
n− d −2
Z
2
dd ` `µ `ν `ρ `σ
(−1)n i Γ n − d2 − 2
1
1
=
× (g µν g ρσ + g µρ g νσ + g µσ g νρ )
d
2
n
d/2
(2π) (` − ∆)
(4π)
Γ(n)
∆
4
(10.83)
Z
Bibliography
[Bourjaily(2013)] J. Bourjaily. Physics 513, quantum field theory - homework 7.
http://www-personal.umich.edu/ jbourj/qft.htm, 2013.
[Gibbons(2013)] L. Gibbons.
Introduction to the standard model lecture notes.
pages.physics.cornell.edu/ ajd268/Notes/IntroSM-Notes.pdf, 2013.
[Bjorken and Drell(1964)] J.D. Bjorken and S.D. Drell. Relativistic quantum mechanics.
McGraw-Hill, 1964.
[Perelstein(2013)] M. Perelstein. Quantum field theory ii. 2013.
[Peskin and Schroeder(1995)] M. Peskin and D. Schroeder. An Introduction to Quantum
Field Theory. West View Press, 1995.
37