1

USEFUL RELATIONS IN QUANTUM FIELDTHEORY

LECTURE NOTES

In this set of notes I summarize many useful relations in Quantum FieldTheory that I was sick of deriving or looking up in the “correct”

conventions (see below for conventions)!

Notes Written by: JEFF ASAF DROR

2014

Contents

1 Introduction 4

2 Classical Field Theory 5

2.0.1 Important Relations . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.0.2 Free Real Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . 5

2.0.3 Free Complex Scalar Field . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Solutions of the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Massless Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Feynman Rules 10

3.1 Deriving the Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Symmetry Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Standard Model 12

4.1 φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Scalar QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 Spinor QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.4 Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.5 CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Polarized Calculations 15

5.1 Polarization and Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 Calculational Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Renormalization 18

6.1 On-Shell Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Quantum Mechanics 20

7.1 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.2 Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Special Relativity 21

2

CONTENTS 3

9 Mathematics 229.1 Anticommuting Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9.1.1 Sigma Matrices and Levi Civita Tensors . . . . . . . . . . . . . . 229.1.2 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.2 Complete the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.3 Degrees of Freedom in a Matrix . . . . . . . . . . . . . . . . . . . . . . . 259.4 Feynman Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.5 n - sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.6 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.7 Sample Loop Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9.7.1 d-dimensional integrals in Minkowski space . . . . . . . . . . . . . 289.7.2 Dimensional Regularization and the Gamma Function . . . . . . . 28

Chapter 1

Introduction

In this note I summarize many important relations I constantly look up throughout mytime working in Particle Theory and in particular calculating Feynman diagrams. I tryto derive some of these relationships if the derivations are straightforward but many arejust quoted. One of the most frustrating events for me is to find some formula and notknow what conventions they are using. In this report I follow Peskin and Schroeder andI use the +−−− metric convention as well as the standard Weyl basis with a positiveγ0. Natural units are used throughout.

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 ajd268@cornell.edu.

4

Chapter 2

Classical Field Theory

2.0.1 Important Relations

The Euler-Lagrangian equations of motion are

∂L∂φ

+ ∂µ∂L

∂(∂µφ)(2.1)

The conjugate momenta of the field is given by

π =∂L

∂(∂0φ)(2.2)

The Hamiltonian is given by

H =

∫d3xπ∂0φ− L (2.3)

The formula for the current is

jµ(x) =∂L

∂(∂µφ)δφ− J µ (2.4)

where J µ found by finding the change in L through a Taylor expansion.

The energy-momentum tensor is given by

T µν ≡∂L

∂(∂µφ)∂νφ− Lδµν (2.5)

2.0.2 Free Real Scalar Field

The Klien Gordan Lagrangian for a real scalar field is

L =1

2∂µφ∂

µφ−m2φ2 (2.6)

5

6 CHAPTER 2. CLASSICAL FIELD THEORY

Quantizing the fields gives

φ(x) =

∫d3p

(2π)3

1√2ωp

(ape

ip·x + a†pe−ip·x) (2.7)

π(x) =

∫d3p

(2π)3(−i)√ωp2

(ape

ip·x − a†pe−ip·x)

(2.8)

or in an equivalent but more convenient form,

φ(x) =

∫d3p

(2π)3

1√2ωp

(ap + a†−p

)eip·x (2.9)

π(x) =

∫d3p

(2π)3(−i)√ωp2

(ap − a†p

)eip·x (2.10)

and the commutation relations are[ap, a

†p′

]= (2π)3δ(3)(p− p′) (2.11)

as well as

[φ(x), π(x′)] = iδ(3)(p− p′) (2.12)

[φ(x), φ(x′)] = [π(x), π(x′)] = 0 (2.13)

2.0.3 Free Complex Scalar Field

φ =

∫d3p

(2π)3

1√2Ep

(ape

ip·x + b†pe−ip·x) (2.14)

φ† =

∫d3p

(2π)3

1√2Ep

(a†pe

−ip·x + bpeip·x) (2.15)

T µν = ∂µφ∗∂νφ+ ∂µφ∂νφ∗ − δµνL (2.16)

H =

∫d3x

(πµπ + ∂iφ

∗∂iφ+m2φ∗φ)

(2.17)

2.1 Solutions of the Dirac Equation

In field theory we often use u(p) and v(p) as solutions to the Dirac equation (with theexponentials factored out). These obey(

/p−m)u(p) = 0 (2.18)(

/p+m)v(p) = 0 (2.19)

(2.20)

2.1. SOLUTIONS OF THE DIRAC EQUATION 7

which in turn imply

u†(p)(γ0/pγ

0 −m)

= 0 (2.21)

v†(p)(γ0/pγ

0 +m)

= 0 (2.22)

(2.23)

u(p)(/p−m

)= 0 (2.24)

v(p)(/p+m

)= 0 (2.25)

(2.26)

The zero momentum solutions take the form

us(0) =√m

(χs

χs

)(2.27)

vs(0) =√m

(χs

−χs)

(2.28)

These can be boosted to an arbitrary momentum through

e−12ηp·K (2.29)

where η = sinh−1(|p|m

)is the rapidity, p is the unit vector of the boost, and Kj ≡ − i

2γjγ0

is the boost matrix.It is straightforward to calculate the boost matrix explicitly:

Kj = − i2

(0 σj

−σj 0

)(0 11 0

)= − i

2

(σj 00 −σj

)(2.30)

which gives the boost:

e−12ηp·K = exp

(−η

2p ·(

σ 00 −σ

))(2.31)

=

(e−

12ηp·σ 0

0 e12ηp·σ

)(2.32)

=

(cosh η

2− p · σ sinh η

20

0 cosh η2

+ p · σ sinh η2

)(2.33)

2.1.1 Massless Limit

Deriving the form of the equations in the massless limit is straightfoward. We have theequation

γµ∂µψ = 0 (2.34)

8 CHAPTER 2. CLASSICAL FIELD THEORY

Take solutions of the form ψ = eip·xu:

pµγµu = 0 (2.35)

pµ

(0 σµ

σµ 0

)u = 0 (2.36)

This gives two sets of equations that are completely decoupled for the left and right

handed part of u. We consider the two solutions indepedently. Consider u =

(u+

0

):

pµσµu+ = 0 (2.37)

We now define p± ≡ p0 ± p3, z ≡ p1 + ip2, and u+ ≡(αβ

). This gives

(p+ zz p−

)(αβ

)= 0 (2.38)

There are two linearly indepdent solutions to this equation.

1. If p− 6= 0 then we can write β = − zp−α (including β = 0)

2. If p− = 0⇒ z = 0, then β can equal anything and α = 0.

but

z

p−=z√p+/√p−√

p+p−=√p+/p−e

iφ (2.39)

where we have defined

eiφ ≡ p1 + ip2√p+p−

(2.40)

With this we can write our solutions as√p−

−√p+eiφ

00

,

0√2p0

00

(2.41)

where we have normalized our spinors to the condition u†u = 2p0. For the other twolinearly independent solutions we have the equation,

pµσµu− = 0 (2.42)(

p− −z−z p+

)(αβ

)= 0 (2.43)

Our two linearly independent solutions are

2.2. SUM RULES 9

1. If p− 6= 0 then we can write α = (z/p−)β (including α = 0)

2. If p− = 0⇒ z = 0, then α can equal anything and β = 0.

From before we can writez

p−=√p+/p−e

−iφ (2.44)

and 00√

p+e−iφ

√p−

,

00√2p0

0

(2.45)

2.2 Sum Rules

The spin sum formula for the Dirac spinors are given by∑s

usausb = (/p+m)ab (2.46)∑

s

vsavsb = (/p−m)ab (2.47)

The polarization sum rule for external vector bosons is given by

∑λ

εµ(k;λ)εν(k;λ) =

−ηµν (massless boson)

−ηµν + kµkνm2 (massive boson)

(2.48)

Chapter 3

Feynman Rules

3.1 Deriving the Feynman Rules

To properly derive the Feynman rules can be difficult. However determining the interac-tions 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 mustbe an accomadating i somewhere else. Consider an arbitrary interaction:

Lint = g (φn1φm2 ...) (3.1)

where the particular fields in the interaction are irrelevant. Then the Feynman rule forthe interaction will just be

x −→ ig (3.2)

Note that the sign of the terms are conserved. Positive Lagrangian terms give positiveinteraction 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 forthese 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 calcula-tion is to account for identical particles in the calculation. There can be many diagramscorresponding 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 whichis called the Symmetry factor.

1. Each vertex contributes a suppression factor. For example in φ4 theory we typicallyhave a 4! suppression factor for each vertex. Of course the value of these is depen-dent on the definition of the couplings but we define our couplings on purpose sowe 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 youswap 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 ColinMorningstar[Bourjaily(2013)]. The idea is as follows. Let n be the number of verteicesof a diagram, η be the coupling constant suppression factor, and r the multiplicity of adiagram. Then the Symmetry factor is given by

S =n! (η)n

r(3.4)

and the amplitude is given by

M =1

SM1 diagram (3.5)

The multiplicity of a diagram is the number of different contractions in the Wick expan-sion, or the number of ways to connect all the external lines to the vertices. This canbe found by first drawing out the edges of each external line and points coming out of avertex. 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 waysto connect the second initial line. Continuing on and counting the number of ways eachline can be connected we have

r = (8)(3)(4)(3)(2)(1) (3.6)

This gives a symmetry factor of

S =2! (4!)2

(8)(3)(4)(3)(2)(1)= 2 (3.7)

Chapter 4

Standard Model

The Standard Model charges are summarized below:

doublet T3 = σ3/2 Y Q = T3 + Y2(

νee−

)L

12

−12

−1

−1

0

−1(ud

)L

12

−12

1313

23

−13

singletseR 0 −2 −1uR 0 4

323

dR 0 −23

−13

Higg’s sector(φ+

φ0

) 12

−12

1

1

1

0

4.1 φ4 Theory

The scalar propagator is

ff → i

p2 −m2 + iε(4.1)

4.2 Scalar QED

The Lagrangian for scalar QED is given by

L = (Dµφ)†Dµφ−m2φ†φ− λ

4(φ†φ)2 − 1

4FµνF

µν (4.2)

where Dµ = ∂µ − ieAµ. The vertices are given by

Lint = −ieAµ((∂µφ†)φ− φ(∂µφ)

)= e2AµA

µφ†φ− λ

4(φ†φ)2 (4.3)

12

4.3. SPINOR QED 13

Depending on the relative directions of the lines going into vertex and the type of particlewe get a different vertex factor. This gives the following rules

e−FEe−

v→ −ie(pµ1 + pµ2) (4.4)

e+e+

v→ −ie(−pµ1 − p

µ2) (4.5)

e−

Dge+

→ −ie(pµ1 − pµ2) (4.6)

e−

Eg e+

→ −ie(pµ1 − p

µ2) (4.7)

and

Duv

→ 2ie2gµν (4.8)

4.3 Spinor QED

af p→Ffb =i(/p+m)ab

p2 −m2 + iε=

(i

/p−m+ iε

)ab

(4.9)

af ←pFfb =i(−/p+m)ab

p2 −m2 + iε=

(i

−/p−m+ iε

)ab

(4.10)

af p→fb =i(−/p+m)ab

p2 −m2 + iε=

(i

−/p−m+ iε

)ab

(4.11)

af ←pfb =i(/p+m)ab

p2 −m2 + iε=

(i

/p−m+ iε

)ab

(4.12)

µg k→ggν = − i

k2 + iε

(gµν − (1− ξ)kµkν

k2

)(4.13)

bgaE

= −ie (γµ)ab (+ momentum conservation) (4.14)

(4.15)

14 CHAPTER 4. STANDARD MODEL

4.4 Weak Interactions

`−FEν`

vW−→ − ig

2√

2γµ(1− γ5) (4.16)

4.5 CKM Matrix

The CKM matrix in the Wolfenstein parametrization is [Gibbons(2013)]

VCKM =

1− λ2

2λ Aλ3(ρ− iη)

−λ 1− λ2

2Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

(4.17)

Chapter 5

Polarized Calculations

5.1 Polarization and Spin

For some reason a thorough discussion of polarization calculations is missing from thepopular Quantum Field Theory books. The discussion here is an amalgam of what I’vefound 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 inthe rest from of the particle is what we often call its spin. We denote this polarization assome 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 bysµr . Now what should the first component of the four-vector be? In the rest frame thereis no other degree of freedom for the spin. We set s0

r to zero.

To boost back to the lab frame we apply a Lorentz Transformation in the −p directiononto the four-vector. Recall in matrix form the transformation matrices applied onto avector (t, r):

t′ = γ (t− r · v) (5.1)

r′ = r +

(γ − 1

v2r · v − γt

)v (5.2)

15

16 CHAPTER 5. POLARIZED CALCULATIONS

The spatial component of the spin in the lab frame is

s = λ +

(−γ − 1

v2λ · v − 0

)(−v)

= λ +

(γλ · (γmv)

mγ− λ · γmv

γm

)vγm

= λ +

((γ − 1)λ · pE2(γ2 − 1)

γ2

)p

= λ +p(λ · p)

E2m(E +m)E2

= λ +p(λ · p)

m(E +m)(5.3)

and the time component is

s0 = γλ · vm/m

=p · λm

(5.4)

So we have

sµ =

(p · λm

,λ +p(p · λ)

m(m+ E)

)(5.5)

In the case that the spin is measured along the direction of motion (i.e. p ‖ λ) we have

sµ =1

m(|p| , pE) (5.6)

Note that

s2 =(p · λ)2

m2− λ2 − 2

(p · λ)2

m(E +m)− p2(p · λ)2

m(m+ E)

= (p · λ)2

(1

m2− 2

m(E +m)− E2 −m2

m2(m+ E)2

)− 1

= (p · λ)2

(E2 + 2mE +m2 − 2Em− 2m2 − E2 +m2

m2(E +m)2

)− 1

= −1 (5.7)

and

sµpµ =

p · λEm

− p · λ− (E2 −m2)(p · λ)

m(m+ E)

= (p · λ)

(E

m− 1− E −m

m

)= (p · λ)

E −m− E +m

m= 0 (5.8)

5.2. CALCULATIONAL TRICKS 17

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) (5.10)

Σ(s)v(p, s) = v(p, s) (5.11)

Σ(−s)u(p, s) = Σ(−s)v(p, s) = 0 (5.12)

5.2 Calculational Tricks

When doing a calculation with some polarized particles there are some useful tricks thatcan be implemented to simplify the math. The key result that can be used to derive allthe following relations is

u(p,λ)u(p,λ) =(/p+m

)Spin Projection Operator︷ ︸︸ ︷1 + γ5/s

2(5.13)

v(p,λ)v(p,λ) =(/p−m

) 1 + γ5/s

2(5.14)

With this we can now find a variety of important relations (we suppress the polariza-tion and momentum dependence in u:

uγµu = tr (uγµu)

= tr (γµuu)

=1

2tr(γµ(/p+m

) (1 + γ5/s

))=

1

2tr(γµ/p)

= 2pµ (5.15)

uγµγ5u = tr(uγµγ5u

)= tr

(γµγ5uu

)=

1

2tr(γµγ5

(/p+m

) (1 + γ5/s

))=

1

2tr (+mγµ/s)

= 2msµ (5.16)

Chapter 6

Renormalization

Renormalization schemes is subtle topic with a lot of depth. Here we just present thebare-bones needed to do calculations

6.1 On-Shell Renormalization

There are two renormalization conditions to consider. First corresponds to the massrenormalization. Consider the full particle propagator (with the external lines amputated,i.e. ignore the external legs contribution):

p→FFp→+

p→FpFp→+

p→FxxxxxxxxxxxxxFp→(6.1)

whereFpF represents a sum of loop corrections andFxxxxxxxxxxxxxF is the counterterm.It can be shown (see [Perelstein(2013)], pg. 36 for a detailed example) that the Greenfunction corresponding the above computation are (in the case of φ− 4 theory)

〈Ω|φ(x)φ(y)|Ω〉 =i

p2 −m2p + Σ(p2) + iε

(6.2)

where Σ(p2) is the sum of the amputated diagrams above. There are two shift factorshere. There is a shift in the pole and there is a also a shift in the amplitude of thisfactor. The first renormalization condition is that when the incoming particle is on-shell(p2 = m2

p), the loop contribution (Σ) is zero. In other words(p→FpFp→

+p→FxxxxxxxxxxxxxFp→

) ∣∣∣∣p2=m2

p

= 0 (6.3)

This makes sence sinceFF should give the pical propagator for an on-shell particle.

The second renormalization condition is for the normalization of the Green’s function

18

6.1. ON-SHELL RENORMALIZATION 19

not to change. We can rewrite equation as

〈Ω|φ(x)φ(y)|Ω〉 =i

p2 −m2p + Σ(m2) + (p2 −m2) dΣ

dp2

∣∣∣∣p2=mp

+ iε

(6.4)

=i

(p2 −m2p)

(1 + dΣ

dp2

∣∣∣∣p2=mp

)+ Σ(m2) + iε

(6.5)

(6.6)

where we only keep terms of order p2 −m2 since we are considering the conditions on Σnear the pole (these are on-shell conditions after all!). Thus we require (in φ− 4 theory):

dΣ

dp2

∣∣∣∣p2=m2

= 0 (6.7)

The third renormalization condition is for the coupling. The sum of the 4 vertexdiagrams (this can of course be done for any types of couplings but we consider 4 externallegs for concreteness).

DE

+ DpE

+DxxxxxxxxxxxxxE

(6.8)

The renormalization condition is DpE

+DxxxxxxxxxxxxxE

∣∣∣∣s+u+t=4m2

= 0 (6.9)

where λ is the renormalized couplings.

Chapter 7

Quantum Mechanics

7.1 Commutation Relations

[x, p] = i~ (7.1)

7.2 Quantum Harmonic Oscillator

The creation annhillation operators are defined by

a ≡ mω

2~

(x+

i

mωp

)(7.2)

a† =√mω2~

(x− i

mωp

)(7.3)

or

x =

√~

2mω(a+ a†) (7.4)

p = i√mω~2(a† − a) (7.5)

which give

a† |n〉 =√n+ 1 |n+ 1〉 (7.6)

a |n〉 =√n |n− 1〉 (7.7)

We also have,〈0|ana†n|0〉 = n! (7.8)

This is easiest to see by starting with n = 1 and going on recursively to higher n.

20

Chapter 8

Special Relativity

In special relativity we have covariant (xµ) and contravariant (xµ) vectors. Contravariantvectors 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,∇) (8.3)

∂µ ≡ (∂0,−∇) (8.4)

21

Chapter 9

Mathematics

9.1 Anticommuting Matrices

9.1.1 Sigma Matrices and Levi Civita Tensors

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)(9.1)

They obey the commutation relations,

[σa, σb] = 2iεabcσc (9.2)

and the anticommutation relations,

σa, σb = 2δab (9.3)

Furthermore any product of Pauli matrices can be written as

σaσb = iεabcσc + δab (9.4)

It is a common practice to exponentiate a linear combination of these matrices. We derivethe general formula below,

eiθiσi = 1 + iθiσi −1

2θiθjσiσj − ... (9.5)

=

(1− 1

2(θiσi)

2 + ...

)+ i

(θiσi −

1

3!(θiσi)

3 + ...

)(9.6)

but

θiθj(σiσj) = θiθj(iεijkσk + δij) (9.7)

= θ2 (9.8)

and

(θiσi)3 = θ2θiσi (9.9)

22

9.1. ANTICOMMUTING MATRICES 23

so

eiθiσi = cos θ + iθiσiθ

sin θ (9.10)

σµσν + σν σµ = 2ηµν (9.11)

We also have

tr σµσν = 2ηµν (9.12)

(σµ)αα(σµ)ββ = 2δ βα δ β

α (9.13)

Furthremore,

σµν =1

2iεµνρσσρσ (9.14)

We haveε0123 = −ε0123 = +1 (9.15)

and

εijkεimn = δmj δ

nk − δnj δmk (9.16)

εjmnεimn = 2δij (9.17)

εijεin = δnh (9.18)

Furthermore, we have

εσµνρεσαβγ = 6δ[µα δ

νβδ

ρ]γ (9.19)

εµνρσερσαβ = −4δ[µα δ

ν]β ≡ −4

(δµαδ

νβ − δναδνβ

)(9.20)

where the square brackets denote all possible permutations of µ, ν, ρ. Each permutationhas a negative sign if it is an odd permutation.

The three dimensional extension of the Pauli matrices are:

1√2

0 1 01 0 10 1 0

,1√2

0 −i 0i 0 −i0 i 0

,

1 0 00 0 00 0 −1

(9.21)

9.1.2 Gamma Matrices

In the Weyl basis the Gamma matrices can be written

γµ =

(0 σµ

σµ 0

)(9.22)

where σµ = 1,σ and σµ = 1,−σ. They obey the defining commutation relation

γµ, γν = 2gµν (9.23)

24 CHAPTER 9. MATHEMATICS

This leads to many important relations. For example

γµγµ =1

2γµγµ + γµγµ (9.24)

=1

2

2gµµ

(9.25)

= 4 (9.26)

γµγαγµ = γµ(2gαµ − γµγα

)(9.27)

= 2γα − 4γα (9.28)

= −2γα (9.29)

In the Weyl basis the Gamma-5 matrix takes the form

γ5 =

(−1 00 1

)(9.30)

and has the properties that

(γ5)† = γ5 (9.31)

(γ5)2 = 1 (9.32)γ5, γµ

= 0 (9.33)

The γ5 matrix can be used to form projection operators:

PL =1− γ5

2(9.34)

PR =1 + γ5

2(9.35)

Trace Technology

(uγµv)∗ = v†γµ† (9.36)

= vγ0㵆γ0u (9.37)

= vγ0(γ0γµγ0)γ0u (9.38)

(uγµv)∗ = vγµu (9.39)

We have

tr (γµγν) =1

2tr (γµγν + γµγν) (9.40)

=1

2tr (γµγν + 2ηµν − γνγµ) (9.41)

=1

2tr (γµγν + 2ηµν − γµγν) (9.42)

=1

2tr (2ηµν) (9.43)

= 4ηµν (9.44)

9.2. COMPLETE THE SQUARE 25

where we have used the cyclic property of traces.Furthermore we have

tr(γ5)

= 0 (9.45)

tr(γµγνγ5

)= 0 (9.46)

tr(γµγνγργσγ5

)= −4iεµνρσ (9.47)

9.2 Complete the Square

To complete the square we want to take an expression of the form ax2 + bx + c tod(x+ e)2 + f . Expanding the second form gives

dx2 + 2dex+ de2 + f (9.48)

first off, clearly a = d. So we make the identifications, b = 2ae, ae2 + f = c. However, wetypically want the inverted form of these equations. So we have

d = a (9.49)

e =b

2a(9.50)

f = c− b2

4a(9.51)

9.3 Degrees of Freedom in a Matrix

The number of degrees of freedom in a unitary matrix are found below. The number offree parameters in a general complex matrix is 2N2. Unitarity implies that U †U = 1. Wedefine the elements of U as aij + ibij. Then unitarity implies

(aij + ibij)(aji − ibji) = δij (9.52)

which gives two equations,

aijaji + bijbji = δij (9.53)

bijaji − aijbji = 0 (9.54)

The first equation is symmetric under interchanging i↔ j and gives

1 + 2 + ...+N =N(N + 1)

2(9.55)

conditions (just think of it as a matrix equation and the independent equations makingup a top right triangle of the matrix).

26 CHAPTER 9. MATHEMATICS

The second equation is antisymmetric under changing i↔ j (the component of i = jvanishes trivially as doesn’t offer an extra constraint). This gives

1 + 2 + ...+N − 1 =N(N − 1)

2(9.56)

conditions.Thus the number of free parameters in a Unitary matrix is

2N2 −N2 = N2 (9.57)

9.4 Feynman Parameters

Two denometers can be combined as

1

AB=

∫ 1

0

dx1

[xA+ (1− x)B]2(9.58)

n denomenators can be combined as

1

A1A2...An=

∫ 1

0

dx1 dx2... dxnδ(∑i

xi − 1)(n− 1)!

[x1A1 + ...+ xnAn]n(9.59)

This is done below for a common two denomenators:

1

(`− p)2 −m21 + iε

1

`2 −m22 + iε

=

∫dx

1

[x((`− p)2 −m21 + iε) + (1− x)(`2 −m2

2 + iε)]2

=

∫dx

1

[`2 − 2`px+ p2x+ (m22 −m2

1)x−m22 + iε]

2

=

∫dx

1

[(`− px)2 + p2x(1− x) + (m22 −m2

1)x−m22 + iε]

2

=

∫dx

1

[(`− px)2 −∆ + iε]2(9.60)

where we have defined ∆ ≡ −p2x(1− x) + (m21 −m2

2)x+m22.

9.5 n - sphere

The surface area of a unit n− sphere is

Sn = (n+ 1)π(n+1)/2

Γ(n+1

2+ 1) (9.61)

Note in this notation a sphere in our 3D world corresponds to S2 = 4π.

9.6. INTEGRALS 27

9.6 Integrals

9.7 Sample Loop Integral

One common integral (usually in φ− 4 theory) is the following

∫d4`

1

`2 −∆ + iε(9.62)

We perform this integral here to refer to this result in the future. We assume φ− 4 so wecan work with the cutoff. The integral can be split as follows

=

∫d3`d`0

1

`20 − `2 −∆ + iε

The poles are at

`20 − `2 −∆ + iε = 0

`0 = ±(√

`2 + ∆− iε)

The positions of the poles are shown below

Re(`0)

Im(`0)

××

We can apply a Wick rotation such that `0 → −i`0. This gives

= i

∫d`d`0

1

−`20 − `2 −∆ + iε

= −i∫d4`E

1

`2E + ∆

28 CHAPTER 9. MATHEMATICS

where we define a Eulidean four vector, `E = (`0, `) with `2E ≡ `2

0 + `2. Since we are nowfar from the poles we also omit the iε factors.

= −i2π2

∫dkE

k3E

k2E + ∆

= −i2π2

∆

∫ Λ

0

∆1/2 dx∆3/2x3

x2 + 1

= −i2π2∆

∫ Λ/m

0

dxx3

x2 + 1

= −iπ2∆

(Λ2

∆− log

(1 + Λ2/∆

))= −iπ2

(Λ2 −∆ log

(∆ + Λ2

∆

))(9.63)

9.7.1 d-dimensional integrals in Minkowski space

9.7.2 Dimensional Regularization and the Gamma Function

There are two particularly useful integrals when using dim-reg([Peskin and Schroeder(1995)],pg. 251): ∫

dd`E(2π)d

1

(`2E + ∆)n

=1

(4π)d/2Γ(n− d

2

)Γ(n)

∆d/2−n (9.64)∫dd`E(2π)d

`2E

(`2E + ∆)2

=1

(4π)d/2d

2

Γ(n− d

2− 1)

Γ(n)∆d/2−n+1 (9.65)

The Gamma function is the generalization of the factorial. It obeys the relationships

Γ(n) = (n− 1)! , (9.66)

Γ(n+ 1) = nΓ(n) , (9.67)

and

Γ

(1

2

)=√π (9.68)

When using “dim reg” to regular your integrals it is often useful to expand Γ(ε) to firstorder:

Γ(ε) =1

ε− γ +

1

12(6γ2 + π2)ε+O(ε2) (9.69)

Γ(−1 + ε) = −1

ε+ (γ − 1) (9.70)

where γ ≈ 0.577.

9.7. SAMPLE LOOP INTEGRAL 29

To expand any other terms of the form ∆ε one can use

∆ε = elog ∆ε

(9.71)

= eε log ∆

= 1 + ε log ∆ (9.72)

∫dd`

(2π)d1

(`2 −∆)n=

(−1)ni

(4π)d/2Γ(n− d

2

)Γ(n)

(1

∆

)n− d2

(9.73)∫dd`

(2π)d`2

(`2 −∆)n=

(−1)n−1i

(4π)d/2d

2

Γ(n− d

2− 1)

Γ(n)

(1

∆

)n− d2−1

(9.74)∫dd`

(2π)d`µ`ν

(`2 −∆)n=

(−1)n−1i

(4π)d/2gµν

2

Γ(n− d

2− 1)

Γ(n)

(1

∆

)n− d2−1

(9.75)∫dd`

(2π)d(`2)2

(`2 −∆)n=

(−1)ni

(4π)d/2d/(d+ 2)

4

Γ(n− d

2− 2)

Γ(n)

(1

∆

)n− d2−2

(9.76)∫dd`

(2π)d`µ`ν`ρ`σ

(`2 −∆)n=

(−1)ni

(4π)d/2Γ(n− d

2− 2)

Γ(n)

(1

∆

)n− d2−2

× 1

4(gµνgρσ + gµρgνσ + gµσgνρ)

(9.77)

Bibliography

[Bourjaily(2013)] J. Bourjaily. Physics 513, quantum field theory - homework 7.http://www-personal.umich.edu/ jbourj/qft.htm, 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.

[Gibbons(2013)] L. Gibbons. Introduction to the standard model lecture notes.pages.physics.cornell.edu/ ajd268/Notes/IntroSM-Notes.pdf, 2013.

[Peskin and Schroeder(1995)] M. Peskin and D. Schroeder. An Introduction to QuantumField Theory. West View Press, 1995.

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