introduction to anomalies in qft - usaldiarium.usal.es/vazquez/files/2015/03/uam_slides1.pdfm.Á....

Introduction to Anomalies in QFT

Miguel Á. Vázquez-Mozo Universidad de Salamanca

Universidad Autónoma de Madrid, PhD Course.

Plan of the course

Anomalies: general aspects

The axial anomaly: a case study.

Gauge anomalies

Gravitational anomalies

Anomalies and phenomenology:

Pion decay

Anomaly cancellation and model building

Nonperturbative physics from anomalies. Anomaly matching

Functional methods

Anomalies and topology

Advanced topics (Green-Schwarz mechanism, anomaly inflow…)

M.Á. Vázquez-Mozo Introduction to Anomalies in QFT PhD Course, Universidad Autónoma de Madrid


Bibliography (a sample)

R.A. Bertlmann, “Anomalies in Quantum Field Theory”, Oxford 1996

K. Fujikawa & H. Suzuki, “Path integrals and Quantum Anomalies”, Oxford 2004

L. Álvarez-Gaumé & M.A. Vázquez-Mozo, “Introduction to Anomalies”, Springer (to appear)

M.E. Peskin & D.V. Schroeder, “An Introduction to Quantum Field Theory”, Perseus Books 1995 (Chapter 19)

L. Álvarez-Gaumé & M.A. Vázquez-Mozo, “An Invitation to Quantum Field Theory”, Springer 2012 (Chapter 9)

M.D. Schwartz, “Quantum Field Theory and the Standard Model”, Cambridge 2014

Monographs:

QFT books:

Online Reviews:

J.A. Harvey, “TASI Lectures on Anomalies”, hep-th/0509097



What is an anomaly?


M.Á. Vázquez-Mozo Introduction to Anomalies in QFT PhD Course, Universidad Autónoma de MadridM.Á. Vázquez-Mozo Introduction to Anomalies in QFT PhD Course, Universidad Autónoma de Madrid

In certain situations, some symmetries/invariances of the classical theory can be incompatible with the quantization procedure

In those cases we say the theory has an ANOMALY, or that the symmetry/invariance is ANOMALOUS.

The obvious example is scale invariance. E.g.

E Mathematical Appendix V: ⇣-functions, ⌘-functions and the heat kernel 67

F Mathematical Appendix VI: Computing functional determinants 76

1 Generalities about anomalies

1.1 What is an anomaly?

Symmetries are at the cornerstone of modern physics. It can be said that the success of particlephysics is mostly based on the implementation of the notion of symmetry1.

Despite of this, there are situations in Quantum Field Theory (QFT) in which classicalsymmetries clash with the quantization procedure. When this happens, the correspondingsymmetry of the classical Lagrangian cannot be implemented in the quantum theory. Onesays then that the symmetry is anomalous or that there is an anomaly associated with thissymmetry.

To get a flavor of what we mean by this, we look at what maybe is the most obvious exampleof an anomaly in QFT. We consider a massless �4 theory in four dimensions,

S =

Z

d4x

✓

1

2@µ�@

µ� � �

4!�4

◆

. (1.1)

This classical action is invariant under scale transformations acting as

xµ ! ⇠xµ, �(x) ! ⇠�1�(⇠�1x). (1.2)

This simply reflects the fact that the theory does not have any dimensionfull parameter. Amass term, for example, would break the invariance.

Quantizing the theory we find that some diagrams are divergent and need to be regularized.After renormalization the coupling constant runs, i.e. its value depends on the scale of theprocess under consideration. This is reflected in a nonvanishing beta function at one loop

�(�) =3�2

16⇡2

, (1.3)

that leads to the running

�(µ) =�(µ

0

)

1 � 3

16⇡3�(µ0

) log⇣

µµ0

⌘ . (1.4)

Here µ0

is some reference scale at which the coupling constant takes the value �(µ0

).

1For the time being we include under the label of symmetry both standard symmetries and gauge invariance.Later we will make a crucial distintion between them.

2

Is classically invariant under scale transformations








S =

Z

d4x

✓

1

2@µ�@

µ� � �

4!�4

◆

. (1.1)


xµ ! ⇠xµ, �(x) ! ⇠�1�(⇠�1x). (1.2)



�(�) =3�2

16⇡2

, (1.3)


�(µ) =�(µ

0

)

1 � 3

16⇡3�(µ0

) log⇣

µµ0

⌘ . (1.4)

Here µ0


).


2








S =

Z

d4x

✓

1

2@µ�@

µ� � �

4!�4

◆

. (1.1)


xµ ! ⇠xµ, �(x) ! ⇠�1�(⇠�1x). (1.2)



�(�) =3�2

16⇡2

, (1.3)


�(µ) =�(µ

0

)

1 � 3

16⇡3�(µ0

) log⇣

µµ0

⌘ . (1.4)

Here µ0


).


2

The physics is the same at all scales.


Upon quantization, however, we have divergences to deal with. For example,

12.3 The φ4 Theory: A Case Study 239

(12.31)

The factor of 12 is a symmetry factor. We can take advantage of the calculations made

in the previous section to isolate the divergent part of the diagram as d → 4

(12.32)

To cancel this divergence we add a counterterm − 12δm2φ2 to the Lagrangian density

where δm2 is given by

δm2 = − λm2

16π2

1d − 4

. (12.33)

Adding this counterterm means to include in the Feynman rules a new vertex withtwo external legs

(12.34)

Its contribution to the two point function to order λ exactly cancels the divergent partof the one loop diagram (12.31). There is of course an ambiguity in the definitionof the counterterm because in addition to the pole we could also have subtracted afinite part. For the time being, however, we choose not to do so.

The next divergent diagram in the φ4 theory comes from the one-loop calculationof the four-point function. In fact there are three diagrams contributing at order λ2

(12.35)

The last two diagrams differ in a permutation of the momenta p3 and p4. Sincethe corresponding legs are attached to different vertices the two diagrams are topo-logically nonequivalent. Applying the Feynman rules listed above, we find that thecontribution of these three diagrams can be written as

(12.36)

The regularization of the integrals introduces an energy scale that leads to a running of the coupling:








S =

Z

d4x

✓

1

2@µ�@

µ� � �

4!�4

◆

. (1.1)


xµ ! ⇠xµ, �(x) ! ⇠�1�(⇠�1x). (1.2)



�(�) =3�2

16⇡2

, (1.3)


�(µ) =�(µ

0

)

1 � 3

16⇡3�(µ0

) log⇣

µµ0

⌘ . (1.4)

Here µ0


).


2

This quantum breaking of scale invariance is encoded in the beta function

�(�) =3~�2

16⇡2


Scale anomaly: a quantum mechanical toy model


We illustrate scale anomaly with a quantum mechanical example:

Let us take the Hamiltonian

where the potential is a homogeneous function of degree −2

invariance for certain type of potentials. Let us consider the classical Hamiltonian of a particlewith mass M propagating in a potential V (r)

H =p2

2M+ V (r), (1.5)

and let us require the potential to be a homogeneous function of degree �2

V (�r) = ��2V (r). (1.6)

When this is the case, it is straightforward to see that the Hamilton equations

r =@H

@p, p = �@H

@r, (1.7)

are invariant under the scale transformation

t �! �2t,

r �! �r, (1.8)

p �! ��1p.

So far, our discussion has been valid for arbitrary dimensions. Let us consider first a two-dimensional system. An obvious potential satisfying condition (1.6) is the two-dimensionalDirac delta potential

V (r) = ↵ �(2)(r), (1.9)

with ↵ a real constant. At the classical level, there is no nontrivial scattering by this potential.This is not the case in QM, where there is scattering due to the fact that the particle wave-function extends to the origin and “feels” the potential (but only the s-wave). The Schodingerequation for this problem reads

� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)

where ~2k2 = 2ME, with E the energy of the particle. In momentum space is rewritten as

1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)

whose general solution is

(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4

the EOM

are invariant under


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4



H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4

In particular we consider the 2D potential:

The Schrödinger equation reads


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4

( )

that we solve in momentum space:


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4


H =p2

2M+ V (r), (1.5)


V (�r) = ��2V (r). (1.6)


r =@H

@p, p = �@H

@r, (1.7)


t �! �2t,

r �! �r, (1.8)

p �! ��1p.


V (r) = ↵ �(2)(r), (1.9)


� ~2

2Mr2 (r) + ↵ �(2)(r) (0) =

~2k2

2M (r), (1.10)


1

2M(p2 � k2) (p) = �↵ (r = 0), (1.11)


(±)(p) = (2⇡)2�(2)(p ⌥ k) � 2M↵ (±)(r = 0)1

p2 � k2 ⌥ i✏, (1.12)

4


We transform back to position spacewhere the superscript indicates respectively the outgoing and incoming solutions. We areinterested in the outgoing solution, which transforming back to position space reads

(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)

Setting r = 0 in this equation leads to the consistency condition

(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)

The integral that appears on the right-hand side of Eq. (1.14) is logarithmicaly divergentand need to be regularized. One way to proceed is by introducing a hard momentum cuto↵ ⇤

Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)

In terms of this, the value of the wave function at the origin is given by

(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)

where we have written the result in terms of the particle energy. Substituting this into thesolution to the Schrodinger equation (1.13) we arrive at

(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)

The integral on the right-hand side of this equation can be computed in terms of Hankelfunctions to give

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)

Using the asymptotic expansion of the Hankel function for large values of the argument

H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)

we have the asymptotic expansion of the outgoing wave function for large values of |r|

(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5

To obtain the spectrum of the theory, we set r = 0 to obtain the consistency condition

where the superscript indicates respectively the outgoing and incoming solutions. We areinterested in the outgoing solution, which transforming back to position space reads

(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5

However, the integral is divergent. Using a hard momentum cutoff


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5

we have


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5


With this we have a solution of the Schrödinger equation


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5

The integral can be solved in terms of Hankel functions as


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5

Using its asymptotic expansion for large arguments

we have (as |r| → ∞)


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5

H(1)0 (z) ⇠

r2

⇡zeiz+

i⇡4


Let us now compare our result

with the asymptotic for of the wave function for 2D scattering


(+)(r) = e�ik·r � 2M↵ (±)(0)

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.13)


(+)(0) = 1 � 2M↵ (±)(0)

Z

d2p

(2⇡)21

p2 � k2 � i✏. (1.14)


Z

d2p

(2⇡)21

p2 � k2 � i✏=

1

2⇡

Z

⇤

0

pdp

p2 � k2 � i✏

=1

4⇡log

✓

�⇤2

k2

◆

. (1.15)


(+)(0) =1

1 + 2M↵4⇡

log�

� ⇤

2~2

2ME

� , (1.16)


(+)(r) = e�ik·r/~ � 11

2M↵+ 1

4⇡log

�

� ⇤

2~2

2ME

�

Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏. (1.17)


Z

d2p

(2⇡)2e�ip·r/~

p2 � k2 � i✏=

i

4H

(1)

0

✓

kr

~

◆

. (1.18)


H(1)

0

(z) ⇠r

2

⇡zeiz�

i⇡4 , (1.19)


(+)(r) ⇠ e�ik·r/~ � 1p2⇡kr

eikr~ +

i⇡4

1

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.20)

5Taking into account now the general asymptotic form of the outgoing wave function at largedistances in two dimensions

(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)

we identify the scattering function as

f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)

First of all, the function is independent of the angle, so scattering only takes place in the s-waveas argued above (higher waves do not interact with the delta function potential because thecorresponding wave functions vanish at the origin). The second important point to take intoaccount concerns the dependence of the scattering function on the cuto↵. As we will see now,all dependence on ⇤ can be eliminated in terms of physical (i.e., measurable) quantities.

Looking at Eq. (1.22) we see that the denominator vanish for the following value of theenergy

E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)

The existence of this pole in the scattering amplitude at negatives values of the energy indicatesthe existence of a bound state, whose energy is nonperturbative in the coupling constant ↵.Moreover, that its energy is proportional to ~ indicates that the bound state is a purely quantume↵ect. Now, since the energy of the bound state is a measurable quantity, we can trade ⇤ byE

0

and thus eliminate all dependence of the scattering amplitude in the unphysical cuto↵

f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)

where again we have written the energy in terms of the mometum of the particle. Thus, wefind the di↵erential cross section to be

d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)

If scale invariance would have been preserved in the process of quantization, the result wouldhave been a scale invariant spectrum. However, we have found that the system has a singlebound state with energy E

0

which obviously breaks conformal invariance. Alternatively, thisresult can be read by looking at the phase shifts �n(k), which in two dimensions are defined by

f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

We identify the scattering function as

Taking into account now the general asymptotic form of the outgoing wave function at largedistances in two dimensions

(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

Two important features:

• It is independent of the angle (only s-wave scattering).

• It depends on the (unphysical) cutoff.


To deal with the second problem, we notice that the scattering function


(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

has a pole for negative energy


(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

Thus, the theory has a single bound state and we can trade the cutoff Λ by the observable quantity E0


(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

We have renormalized the theory!

[at this point, sending requires ]

⇤ ! 1↵ ! 0�


We define the scattering function in terms of the phase shifts


(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6Using our result for f (θ ),

The phase shift depends on the particle energy, hence…

Scale invariance is broken!


Comparing with Eq. (1.24) we find that the only nonvanishing shift corresponds to n = 0 and

e2i�0(k) =

1

⇡log

⇣

~2k2

2M |E0|

⌘

+ i

1

⇡log

⇣

~2k2

2M |E0|

⌘

� i=) �n(k) = �n,0 cot

�1

1

⇡log

✓

~2k2

2M |E0

|

◆�

. (1.27)

In a scale invariant theory, the phase shifts should be independent of the energy given the factthat there is no scale that we can use to construct a dimensionless quantity. In our case, thisscale is provided by the energy of the bound state.

What we have just analyzed is an example of a quantum mechanical system in which scaleinvariant is anomalous due to the dynamical generation of an energy scale, a bound state thatdoes not exist in the classical theory. In this respect, the situation is very similar to that foundin QCD, where a scale is dynamically generated that breaks the scale invariance of the classicalaction.

Another example of a classical scale invariant system, this time in three dimensions, is thatof a particle interacting with a inverse square potential

V (r) =↵

r2

. (1.28)

We will not analyze this case in detail, but only summarize its main features. For ↵ > 0 itcan be seen that scale invariance is preserved by quantization. For example, the phase shiftsfor the scattering of a particle o↵ this potential are independent of the particle momentum.In the case of an attractive potential, on the other hand, the situation changes because thequantization shows that the spectrum of the system is continuous and unbounded below [2].There are several ways to handle the problem. One of them is to notice that the correspondingHamiltonian is not self-adjoint, due to the singular character of the wave function at the origin.Alternatively, one can regularize the potential around r = 0 and, as in the case of the deltafunction potential, re-express the cuto↵ scale in terms of the energy of a (single) bound state.Again, the final result shows how scale invariance is broken by quantum e↵ects, showing itsanomalous character.

7


What is going on here?

Classically, the spectrum is scale invariant:

E

0



What is going on here?

Classically, the spectrum is scale invariant:

E E

0 0

E �! E

�2



Quantum mechanically, scale invariance is broken by the presence of the bound state:

E

E0

0




E E

E0

�2

E0

00

E �! E

�2



(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

We have an energy scale that is quantum-mechanically generated. (e.g. as in QCD)



E E

E0

�2

E0

00

E �! E

�2


this bound state is a quantum mechanical affair!


(+)(r) ⇠ eik·r/~ +ei

kr~ +

i⇡4

pr

f(✓), (1.21)


f(✓) = � 1p2⇡k

11

M↵+ 1

2⇡log

�

� ⇤

2~2

2ME

� . (1.22)



E0

= �⇤2~2

2Me

2⇡M↵ < 0. (1.23)


0


f(✓) =

r

2⇡

k

1

log⇣

~2k2

2M |E0|

⌘

� i⇡, (1.24)


d�

d⌦=

2

⇡k

1

1 + 1

⇡log2

⇣

~2k2

2M |E0|

⌘ . (1.25)


0


f(✓) = �i1

X

n=�1

e2i�n(k) � 1p2⇡k

ein✓. (1.26)

6

We have an energy scale that is quantum-mechanically generated. (e.g. as in QCD)


A second example is the 3D Hamiltonian

For the attractive case (α < 0) the potential overcomes the centrifugal barrier for

✓`+

1

2

◆2

< 3M |↵|

and the spectrum becomes continuous and unbounded from below

The Hamiltonian is not self-adjoint!

H =p2

2M+

↵

r2


To define the theory we regularize the Hamiltonian near r = 0, e.g.

V (r) =

(↵r2 |r| > a

1 |r| < a

Renormalizing the parameters of the solution, leads in the a → 0 again to a bound state and the breaking of scale invariance.

The physics of these toy models is similar to dimensional transmutation in QCD

SQCD =

Zd

4x

0

@�1

4F

aµ⌫F

aµ⌫ +

NfX

f=1

Q

fiD/ Q

f

1

A

⇤QCD

quantization

[see e.g. Coon & Holstein, Am. J. Phys. 70 (2002) 513]


Anomalies: the good and the bad


Whether anomalies are bad or good depends on what symmetries/invariances they affect:

• They are harmless and even useful when they affect global (non gauge) symmetries

Scale invariance

Chiral symmetry

asymptotic freedom


23

µ jµk x ihJk x (0.265)

S d4x12 µ! µ!

g4!!4 (0.266)

" g3g2

16#2(0.267)

g µg µ0

1 316#3 g µ0 log µ

µ0

(0.268)

µ JµV x ? 0 (0.269)

µ JµA x ? 0 (0.270)

S d4x i$ $ eJµV A µ (0.271)

0 (0.272)

µ JµA x Ae2

2d4y1d4y2

xµ Cµ%& x,y A x y1 y2 A& x y2

p q µ i' µ(" p,q ? (0.273)

#0 2) (0.274)

Their presence can be also used to extract nonpeturbative information about the theory (anomaly matching)


• They are potentially disastrous when they affect gauge symmetries

Gauge anomalies

Gravitational anomalies

These types of anomalies should be cancelled at all cost, otherwise the theory becomes sick (e.g. nonunitary)

The conditions for anomaly cancellations can be useful for phenomenology (e.g. constraints on the spectrum)

introduction to anomalies in qft - usaldiarium.usal.es/vazquez/files/2015/03/uam_slides1.pdfm.Á....

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