advanced quantum field theory example sheet 1 · may wish to consult the rst few sections of d....

Part III Maths Lent Term 2017

David Skinner [email protected]

Advanced Quantum Field Theory

Example Sheet 1

Please email me with any comments about these problems, particularly if you

spot an error. Problems with an asterisk (∗) may be more difficult.

1. Integration of a fermionic (i.e. anticommuting) variable θ is defined by the rules∫dθ θ = 1 and

∫dθ 1 = 0 ,

often called Berezinian integration. If B is an invertible n× n matrix and θi, θi, ηi and

ηi are independent fermionic variables (i = 1, . . . , n), show that

Z(η, η) :=

∫dnθ dnθ exp

(θiBijθ

j + ηiθi + θiηi

)= detB exp(ηi(B

−1)ijηj) .

Use this result to obtain an expression for normalized expectation value

〈θi1 · · · θir θj1 · · · θjs〉 :=1

Z(0, 0)

∫dnθ dnθ exp

(−θiBijθj

)θi1 · · · θir θj1 · · · θjs

and show that it vanishes whenever r 6= s and whenever r, s > n. Interpret your answer

in terms of Feynman graphs. [The result is Wick’s theorem for this fermionic theory.]

2. Consider the partition function

Z(λ) =1√2π

∫R

dx e−12x2− λ

4!x4 (†)

for a zero–dimensional QFT with a quartic interaction with λ > 0.

(a) By expanding the integral in λ obtain the nth order perturbative expansion

Zn(λ) =

n∑`=0

(− λ

4!

)` (4`)!

4` (2`)! `!

and show for ` ≤ 3 that the coefficients a` of λ` in this expression are the sums of

automorphism factors of the relevant loop Feynman graphs. (At two loops there

is only one graph, at three loops there are two graphs and at four loops there are

four.)

1

(b) Optional but instructive: Using any computer package, plot Zn(λ = 110) against n

to see that there is a region in n where Zn appears to converge, before blowing up

as n is increased.

(c) Show that the minimum value of a`λ` occurs when ` ≈ 3

2λ . Hence show that the

Borel transform BZ(λ) =∑∞

`=01`! a` λ

` converges provided |λ| < 32 and that in this

case

Z(λ) =

∫ ∞0

dz e−z BZ(zλ)

so that Z(λ) may be recovered from its Borel transform.

(d) By expanding e−12x2 in the integral (†) obtain the strong coupling expansion

Z(λ) =1

2√π

∞∑L=0

(−1)L

L!Γ

(L

2+

1

4

)(6

λ

)L2

+ 14

for Z(λ) as a series in 1/√λ. For λ = 1

10 how many terms does one need to obtain

the value at which the weak coupling expansion appeared to converge?

3. Consider the action S(φ, ψ, ψ) = m2

2 φ2+M

2 ψψ+λφ2ψψ describing a real bosonic variable

φ coupled to two independent fermionic (Grassmann) variables ψ, ψ.

(a) Treating this as a zero–dimensional QFT, write down the Feynman rules for the

propagators and the interaction.

(b) Integrate out the fermions to show that the effective action for φ has an infinite

expansion

Seff(φ) =∞∑n=0

λnn!φn

in terms of an infinite series of couplings λn whose values you should find. Which

correlation functions can be computed using Seff?

(c) Working to order λ2, compute 12〈φ

2〉 first by using the original action and then by

using the effective action Seff . Check your results agree.

4. Let M be an N ×N Hermitian matrix and consider the integral

Z(a;N) =

∫d2NM exp

(−1

2tr(M2)− a

Ntr(M4)

)where a is a coupling constant. The measure d2NM represents an integral over the real

and imaginary parts of each entry of M.

(a) Represent the propagator as a “double line” where one line edge represents the

rows and the other edge represents the columns of M. What are the Feynman rules

for this action?

2

(b) Show that Z(a;N)/Z(0;N) can be reduced to an integral over the eigenvalues

{λi} of M. [You may use without proof that the measure d2NM is invariant under

M→ U−1MU for any unitary matrix U .]

(c) Show that Z(a;N)/Z(0;N) admits a perturbative expansion of the form

lnZ(a;N)

Z(0;N)=

∞∑g=0

N2−2g

( ∞∑n=0

(−a)nFg,n

)

where Fg,n is a combinatoric number, independent of N and a. (You are not

required to find an explicit expression for Fg,n.)

(d) (*) Show that Fg,n may be interpreted as the number of ways to cover a genus

g Riemann surface with n squares. [For help with this part of the question, you

may wish to consult the first few sections of D. Bessis, C. Itzykson & B. Zuber,

Quantum Field Theory Techniques in Graphical Enumeration, Adv. Applied Maths

1, 109-157, (1980).]

5. Consider the Quantum Mechanics of a particle moving on Rn with Hilbert space H =

L2(Rn, dnx). Obtain (Euclidean time) path integral expressions for the following Heisen-

berg picture transition functions:

(a) TrH(P e−TH

), where P is the parity operator P : xa → −xa.

(b) 〈ψf |e−TH |ψi〉, where ψi,f (x) = 〈x|ψi,f 〉 are arbitrary states in the Hilbert space.

where T is the proper time on the wordline.

Suppose n = 1 and the worldline action includes the potential term 12m

2ω2x2. Given

that the heat kernel for the (Euclidean) harmonic oscillator is

〈x|e−TH |y〉 =

√mω

2π sinωtexp

(−mω (x2 + y2) cosωt− 2xy

2 sinωt

),

evaluate your expressions for (a) and (b) explicitly in the case that |ψi,f 〉 are the ground

state of the harmonic oscillator. Check that they agree with what you expect from QM,

working directly in the energy basis.

6. In a 1d QFT, let x be a real bosonic scalar field and let ψ and ψ be fermionic fields.

Consider the action

S =

∫dτ

[1

2

(∂x

∂τ

)2

+ ψ∂ψ

∂τ+

1

2λ2h′(x)2 + λh′′(x) ψψ

]

where λ is a coupling constant and h(x) is a smooth function of x(τ) (and h′ the

derivative of this function).

3

http://www.sciencedirect.com/science/article/pii/0196885880900081

(a) Show that this action is invariant under the transformations

δx = εψ − εψδψ = ε(−x+ λh′(x))

δψ = ε(x+ λh′(x))

where ε and ε are constant fermionic parameters and x = ∂x/∂τ . Find the con-

served charges Q and Q associated to these two symmetries.

(b) Show that QQ + QQ = 2H where H is the Hamiltonian associated to the above

Lagrangian. Hence or otherwise show that the energies of the system are non–

negative, and that the ground state |Ψ〉 obeys Q|Ψ〉 = Q|Ψ〉 = 0.

(c) Put this theory on a circle and take x(τ), ψ(τ) and ψ(τ) each to be periodic. Show

that the partition function Z =∫DxDψDψ e−S with these boundary conditions

is independent both of the radius of the circle and of λ, provided λ 6= 0.

(d) What is the operator expression to which this partition function corresponds?

[Hint: Think carefully about the fermionic boundary conditions.]

(e) (∗) Argue that the only field configurations that contribute to Z have x = h′(x) =

0. By expanding the action in a neighbourhood of these configurations, evaluate

Z in the case that h(x) is a polynomial of degree n with isolated roots.

4




Example Sheet 2



1. Furry’s theorem states that 〈Aµ1(k1) · · · Aµn(kn)〉 = 0 when n is odd, where A(k) is the

photon field in momentum space. It is a consequence of charge conjugation invariance.

(a) In scalar QED, charge conjugation swaps φ and φ. How must the photon field Aµtransform if the action is to be invariant?

(b) Prove Furry’s theorem in scalar QED using the path integral.

(c) Does Furry’s theorem hold for off–shell photons with kµkµ 6= 0?

2. In the Lorenz gauge ∂µAµ = 0, the classical equation of motion for the photon that

follows from the QED action is �Aµ = ejµ = eψγµψ.

(a) Obtain the Dyson–Schwinger equation

�(x)〈Aµ(x)Aν(y)ψ(x1)ψ(x2)〉 = e〈jµ(x)Aν(y)ψ(x1)ψ(x2)〉−δ4(x−y) δµν〈ψ(x1)ψ(x2)〉

for the photon in the quantum theory. What is the corresponding equation for the

electron?

(b) Use the Dyson–Schwinger equation to related the QED correlation function 〈jµ(x)ψ(x1)ψ(x2)〉to the exact electron–photon vertex function.

(c) Assuming the path integral measure is invariant, show that the Ward identity

(k1 − k2)µΓµ(k1, k2) = iS−1(k2)− iS−1(k1) (0.1)

obtained in lectures implies that the momentum–space three-point function 〈Aµ(p)ψ(k1)ψ(k2)〉obeys

pµ〈Aµ(p)ψ(k1)ψ(k2)〉 = 0

when the electrons are on–shell, whether or not p2 = 0. What is the significance

of this result?

1

3. Suppose that an action involving N real scalar fields φi is invariant under global SO(N)

rotations of the different scalars. Show that the corresponding charges obey the so(N)

algebra [Qa, Qb] = if cabQc.

4. Show that under the redefinition gi → g′i(gj) of the couplings of a theory at scale Λ, the

β-functions transform as

βi → β′i =∂g′i∂gj

βj .

Show that in a theory with a single coupling g, the first two terms in the β–function

β(g) = ag3 + cg5 +O(g7) are invariant under any coupling constant redefinition of the

form g → g′ = g + O(g3). Show that it is possible to choose this redefinition so as to

remove all terms except these first two.

5. A certain theory has a single coupling g, with β-function β(g) = −ag3 − bg5 + O(g7).

Solve the Callan–Symanzik equation for the running coupling to find

1

g2= a ln

Λ2

µ2+b

aln

(ln

Λ2

µ2

)+O(1/ ln(Λ2/µ2)) ,

or equivalently

µ2 = Λ2 e− 1

ag2 (ag2)−ba(1 +O(g2)

).

Now suppose that g′ = g + cg3 + O(g5). Show that g′2 can be expresssed in terms of

µ′2 as above, where ln(µ′2/µ2) = c/b. What is the significance of this calculation?

6. Let ψi denote a (fermionic) Dirac spinor field transforming in the fundamental represen-

tation of a U(N) gauge group, and let ψj denote the Dirac conjugate spinor transforming

in the antifundamental. Let (Aµ)ij denote the gauge field for this interaction. Write

down all possible U(N) gauge invariant local operators involving these fields that are

relevant or marginal near the Gaussian critical point, in the cases that the space–time

has dimension d = 4, d = 3 and d = 2.

7. Consider a four dimensional theory whose only couplings are a mass parameter m2 and

a marginally relevant coupling g.

(a) Write down generic expressions for the β-functions in such a theory to lowest

non–trivial order. (You should be able to identify the values of the classical contri-

butions to the β-functions, and the sign of the leading–order quantum correction

to β(g).)

(b) Sketch the RG flows for this theory.

(c) Suppose that g(Λ′) = 0.1 when the cut–off Λ′ is fixed at 105 GeV. If m2(Λ′) is

measured to be 100 GeV, what value of m2(Λ) would be needed at the higher scale

Λ = 1019 GeV?

2

(d) Suppose you changed your value of m2(Λ) by one part in 1020. What would be

the change in m2(Λ′)?

3




Example Sheet 3



1. Consider the theory given by the action

S[φ] =

∫ddx

1

2∂µφ∂

µφ+1

2m2φ2 +

g

3!φ3 +

λ

4!φ4 .

where φ is a real scalar field.

(a) Determine all connected one loop graphs, complete with their appropriate symme-

try factors, which contribute to

〈φ(x)φ(y)〉 , 〈φ(x)φ(y)φ(z)〉 and 〈φ(x)φ(y)φ(z)φ(w)〉 ,

expressing your answer in terms of integrals over d-dimensional loop momenta.

[You are not required to evaluate the integrals.]

(b) Now set λ = 0 so that just the cubic interaction remains. Determine the momen-

tum space correlation function∫ ∏3

i=1 ddxi eipi·xi 〈φ(x1)φ(x2)φ(x3)〉 to one loop

accuracy.

2. Consider a (neutral) scalar field φ with potential V (φ) = 12m

2φ2 + 16µ

ε/2g(µ)φ3 in

dimension d = 6− ε. Here µ is an arbitrary mass scale introduced so that the coupling

g(µ) is dimensionless.

(a) Draw the one-loop one particle irreducible graph which contributes to the propa-

gator at order g2.

(b) Using dimensional regularisation, show that the divergent part of the corresponding

integral for the six dimensional theory is

−1

ε

g2

(4π)3

(m2 +

1

6p2),

where p is the external momentum. Also compute the divergence corresponding

to the one particle irreducible one-loop graph that gives a g3 correction to three

point function, and find the one loop divergence for the one point function.

1

(c) Show that in six dimensions all these divergences may be cancelled by introducing

the counterterm action

Sct[φ] =

∫ddxLct :=

1

ε

1

6(4π)3

[∫ddx

1

2g2(∂φ)2 + µ−εV ′′(φ)3

].

Check that Lct has dimension d.

(d) Determine the β-function for the coupling g and show that β(g) < 0 at small g.

Does the theory have a continuum limit in perturbation theory? Do you expect

this to survive non–perturbatively?

3. Scalar QED describes the interactions of a photon with a complex scalar field. In d

dimensions it is defined by the action

S[A, φ] =

∫ddx

1

4FµνFµν +

1

2(Dµφ)∗Dµφ+

m2

2φ∗φ

where Fµν = ∂µAν − ∂νAµ and Dµφ = ∂µφ+ ieAµφ.

(a) Show that, not including counterterms, there are two distinct 1-loop Feynman

graphs that contribute to vacuum polarization in scalar QED. One of these dia-

grams leads to an integral that is independent of the external momentum. What

is its role?

(b) By considering vacuum polarization, show that when d = 4, the 1-loop β-function

for the dimensionless coupling g corresponding to the charge e is

β(g) =g3

48π2

in the MS scheme. How does the theory behave at scales far below the mass of the

scalar?

4. Consider the theory of a scalar field φ(x) and fermionic Dirac field ψ, with action

S[φ, ψ] =

∫d4x

1

2∂µφ∂

µφ+1

2m2φ2 + ψ(i/∂ +M)ψ + g φψγ5ψ +

λ

4!φ4

in four dimensional Euclidean space, where γ5 = +γ1γ2γ3γ4 is the product of Dirac

matrices obeying {γµ, γν} = 2 δµν in Euclidean signature.

(a) Show that (γ5)2 = +1 and that {γ5, γµ} = 0. Hence show that the action is

invariant under the global transformation

φ→ −φ ψ → e−iπγ5/2ψ .

Assuming that the path integral measure is also invariant under this transforma-

tion, show that renormalization cannot generate any vertices involving odd powers

of the scalar field unless they are accompanied by an odd power of ψγ5ψ, as in the

original action.

2

(b) What counterterms are necessary when studying the continuum limit of this the-

ory? Using dimensional regularization and the on-shell renormalization scheme,

evaluate these counterterms to 1-loop accuracy, and show that the physical ampli-

tudes are finite.

5. Consider Yang–Mills theory with a semi–simple gauge group G, minimally coupled to

a Dirac spinor transforming in the fundamental representation.

(a) How do the gauge field and fermion transform under a constant gauge transforma-

tion h ∈ G?

(b) Obtain an expression for the Noether current JNoetherµ for this transformation, and

show that it obeys the ‘naive’ conservation equation ∂µJNoetherµ = 0.

(c) Using the equations of motion, show that the corresponding Noether charge reduces

to an integral over a (d−2)-dimensional surface at infinity. What is the significance

of this fact?

6. (*) In General Relativity, the analogue of gauge transformations are diffeomorphisms

of a space–time M . Show that there are no local observables in Quantum Gravity. So

what do experimentalists measure?

3




Example Sheet 4


spot an error. Questions marked with an asterisk may be more challenging.

1. Let tA be the generators of a Lie algebra g, [tA, tB] = ifCABtC , and let cA be anticom-

muting variables. Show that

Q := cAtA −1

2fABCc

BcC∂

∂cA

satisfies Q2 = 0. Suppose tA = 0 and also that fABC = kCDfDAB is completely antisym-

metric, where kCD is the Killing form on g. If X = fABC cAcBcC show that QX = 0

but that X 6= QY .

2. Consider a gauge-fixed action for a free (Abelian) gauge field Aµ of the form

S =

∫dDx

(−1

4FµνFµν + h ∂µAµ +

ξ

2h2 + c ∂2c

)where h is an auxiliary bosonic field and (c, c) are anticommuting ghost and antighost

fields.

(a) Verify that this action is invariant under the BRST transformations δAµ = ε∂µc, δc =

0, δc = −εh, δh = 0 and that δ is nilpotent.

(b) Show that the action can be written in the form

S = −∫

dDx

(1

2ΦT∆Φ + c(−∂2)c

)

where Φ =

(Aµh

), ΦT is its transpose and where

∆ =

(−∂2δµν + ∂µ∂ν ∂ν

−∂µ −ξ

).

1

(c) Obtain equations for the normalized correlation functions 〈Φ(x) Φ(0)T〉 and 〈c(x) c(0)〉and show that∫

dDx e−ip·x〈Φ(x) Φ(0)T〉 = − i

p2

(δ νµ − (1− ξ)pµp

ν

p2ipµ

−ipν 0

)∫

dDx e−ip·x〈c(x) c(0)〉 = − i

p2.

(d) Assuming that 〈δY 〉 = 0 for any Y , consider 〈δ (Φ(x) c(0))〉 and show that we must

have 〈h(x)h(0)〉 = 0. Obtain also a relation between 〈c(x) c(0)〉 and 〈Aµ(x)h(0)〉which should be verified.

3. For a gauge theory coupled to scalars the single particle states are

|AAµ (p)〉 , |φi(p)〉 , |cA(p)〉 , |cA(p)〉 ,

where A runs over a basis of the adjoint representation and i similarly indexes the R

representation. These states have non-zero scalar products

〈AAµ (p)|ABν (p′)〉 = ηµνδABδpp′ 〈φi(p)|φj(p′)〉 = δijδpp′

〈cA(p)|cB(p′)〉 = 〈cA(p)|cB(p′)〉 = δABδpp′

where δpp′ := (2π)D−1 2p0 δ(D−1)(p−p′). The non-zero action of the BRST charge Q is

given by

Q|AAµ (p)〉 = αpµ|cA(p)〉 , Q|φi(p)〉 =∑A

viA|cA(p)〉

Q|cA(p)〉 = βpµ|AAµ (p)〉+∑i

vAi|φi(p)〉

while the ghost charge Qgh acts non-trivially as

Qgh|cA(p)〉 = i|cA(p)〉 , Qgh|cA(p)〉 = −i|cA(p)〉 .

Verify that this is compatible with Q and Qgh being Hermitian if α, β, viA and vAi are

related appropriately. Assume a basis has been chosen so that∑

i vAiviB = δABρA and

so is diagonal. Find the conditions under which the BRST charge Q2 = 0. Use this to

determine the possible physical single particle states.

4. Consider a gauge invariant Lagrangian density of the form

L(A) = −1

4tr(FµνX(D2)Fµν

),

where DλFµν = ∂λFµν + [Aλ, Fµν ] and where X(D2) = 1 + (−D2)r/Λ2r for some scale

Λ. The full quantum Lagrangian with gauge fixing and ghost fields is

Lq(A, c, c) = L(A)− 1

2ξtr(∂µAµX(∂2) ∂νAν

)+ tr

(c X(∂2) ∂µDµc

).

2

(a) Show that the Feynman rules require that the gauge and ghost propagators are

each proportional to p−2−2r.

(b) Show that there must be vertices with n gauge field legs n = 3, . . . 2r+ 4 and with

2r + 4 − n powers of momentum, but that there is just a single vertex involving

both ghost and gauge fields with 2r + 1 momentum factors.

(c) Hence show that, in four dimensions, the superficial degree of divergence of an

`-loop Feynman graph with EA external gauge field lines and Egh ghost field lines

is

δ = 4− EA − Egh − 2r(`− 1) .

5. ∗Consider pure (= no charged matter) electrodynamics with Lagrangian L = − 14e2FµνF

µν .

Let Wγ [A] := exp(

i∮γ Aµ dxµ

)be a Wilson loop around a closed curve γ.

(a) Show that

〈Wγ [A]〉 = exp

[− e2

8π2

∮γ

dxµ∮γ

dyµ1

(x− y)2

].

(b) Now suppose γ is a large rectangle with space-like width L and time-like length

T . Compute 〈Wγ [A]〉 in the limit T � L. By comparing your result to the

usual expression for time evolution, show that the potential between two point-

like charges at fixed separation L in electrodynamics is V (L) = −e2/4πL.

(c) In Feynman gauge, the propagator for a non-Abelian gauge field is

〈ABµ (x)ACν (y)〉 = iηµν δBC

∫d4p

(2π)4e−ip·(x−y)

p2.

Compute the expectation value of a Wilson loop in pure SU(N) Yang-Mills theory

to lowest non-trivial order in the coupling g2. [Your result should depend on a

choice of the representation R of su(N).]

(d) Show that, to this order, the Coulomb potential of non-Abelian gauge theory is

V (L) = −g2C2(R)/4πL2, where C2(R) is the quadratic Casimir of the R represen-

tation.

3

advanced quantum field theory example sheet 1 · may wish to consult the rst few sections of d....

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