Download Report

QFT II
2
Contents
1 Path Integral Quantisation
1.1 Path Integral in Quantum Mechanics . . . . . . . . . .
1.2 The path integral for scalar fields . . . . . . . . . . . .
1.3 Generating functional for correlators . . . . . . . . . .
1.4 Free scalar field theory . . . . . . . . . . . . . . . . . .
1.5 Perturbative expansion in interacting theory . . . . . .
1.6 Connected diagrams . . . . . . . . . . . . . . . . . . .
1.7 The 1PI effective action . . . . . . . . . . . . . . . . .
1.8 Γ[ϕ] as a quantum effective action and background field
1.8.1 Euclidean QFT & statistical field theory . . . .
1.9 Grassmann algebra calculus . . . . . . . . . . . . . . .
1.10 The fermionic path integral . . . . . . . . . . . . . . .
1.11 The Schwinger-Dyson equations . . . . . . . . . . . . .
2 Renormalisation of Quantum Field Theory
2.1 Superficial devergence and power counting . . .
2.2 Renormalisability and BPHZ theorem . . . . . .
2.3 Renormalisation of φ4 theory up to 2-loops . . .
2.3.1 1-loop renormalisation . . . . . . . . . .
2.4 Renormalisation of QED revisited . . . . . . . .
2.5 The renormalisation scale . . . . . . . . . . . .
2.6 The Callan-Symanzyk (CS) equation . . . . . .
2.7 Computation of β-functions in massless theories
2.8 The running coupling . . . . . . . . . . . . . . .
2.9 RG flow of dimensional operators . . . . . . . .
2.10 Wilson effective action & renormalisation group
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
method
. . . . .
. . . . .
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
7
8
9
9
11
11
12
13
15
16
18
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
21
21
22
23
24
24
26
26
27
28
29
30
3 Quantisation of Yang-Mills-Theory
3.1 Recap of classical YM-Theory . . . . . . . . . . . .
3.2 Gauge fixing the path integral . . . . . . . . . . . .
3.3 Feddeev-Popov ghosts . . . . . . . . . . . . . . . .
3.4 Canonical quantisation and asymptotic Fock space .
3.5 BRST symmetry and the physical Hilbert space . .
3.6 Feynman rules for scattering . . . . . . . . . . . . .
3.7 1-loop renormalisation of YM theory & β-function .
3.8 A very brief look at QCD . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
31
31
32
34
36
36
39
41
42
3
.
.
.
.
.
.
.
.
.
.
.
4
4 Symmetries in QFT
4.1 The chiral anomaly . . . . . . . . . . . .
4.2 Relation to Aliyah-Singer-Index-Theorem
4.3 Chiral gauge theories & anomalies . . . .
4.4 Spontaneous Symmetry Breaking . . . .
4.5 The Higgs mechanism . . . . . . . . . .
CONTENTS
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
45
45
47
48
50
53
5 Important formulas and proofs
55
5.1 Proof to 1PI effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Applications to Grassmann calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 1
Path Integral Quantisation
1.1
Path Integral in Quantum Mechanics
Aim: Compute the transition amplitude
ˆ
hqF , tF |qI , tI i = hqF |e−iH(tF −tI ) |qI i
Therefore, divide T = tF − tI into (N+1) intervals of δt =
Z Y
N
hqF , tF |qI , tI i = lim
N →∞
T
N +1
ˆ
and insert 1 =
ˆ
(1.1)
R
dqk |qk ihqk |:
ˆ
dqk hqF |e−iHδt |qN ihqN |e−iHδt . . . |q1 ihq1 |e−iHδt |qI i
k=1
Suppose H(p, q) = f (p) + V (q), then:1
ˆ
−iHδt
hqk+1 |e
Z
|qk i =
dpk −iH(pk ,qk )δt ipk (qk+1 −qk )
e
e
2π
Then:
N
dp0 Y dpk dqk i PNk=0 (pk qk+1 −qk −H(pk ,qk ))δt
δt
e
.
2π k=1 2π
Z
hqF , tF |qI , tI i = lim
N →∞
(1.2)
This is written as the Path integral:
Z
q(tf )=qf
hqF , tF |qI , tI i =
Dq(t)Dp(t)ei
R tF
tI
dt[pq−H(p,q)]
˙
q(ti )=qi
Where we often write:
Z
tF
Z
tF
dt [pq˙ − H(p, q)] =
dt L(p, q) =: S(p, q)
tI
1
tI
In the limit δt → 0
5
(1.3)
6
CHAPTER 1. PATH INTEGRAL QUANTISATION
Special case H(p, q) =
p2
2m
One can now evaluate the
+ V (q):
R
Dp(t) integral, since it appears2
dpk i[pk (qk+1 −qk )−δt p2k ]
i m (q
−qk )2
2m = e δt(1−i) k+1
e
2π
q
−im
Then as δt → 0 with C := 2πδt(1−i)
:
Z
hqF , tF |qI , tI i = lim C
N +1
Z Y
N
N →∞
Z
dqk ei
s
−im
2πδt(1 − i)
R tF
tI
dtL(q,q)
˙
k=1
qF
=
Dq(t) eS(q,q)˙
(1.4)
qI
Correlation functions
A similar procedure leads to
Z
hqF , tF |ˆ
qH (t1 )|qI , tI i =
Dq(t)Dp(t) q(t1 ) ei
R tF
tI
L(p,q)
(1.5)
for tF > t1 > tI . In general one gets
hqF , tF |T
n
Y
Z
qˆH (ti )|qI , tI i =
Dq(t)Dp(t)
i=1
n
Y
q(ti ) ei
R tF
tI
i=1
for tF > ti > tI .
We’ll need the initial/final states replaced by |Ωi. Therefore write
ˆ
ˆ
|qI , tI i = eiHtI |qI i = eiHtI
X
|nihn|qI i
n
= |ΩihΩ|qI i +
X
e
iEn tI
|nihn|qI i
n6=Ω
For tI → tI (1 − i), in the limit tI → −∞ and equivalently for tF one gets:
2
One has to use the Gaussian integral (5.1)
L(p,q)
(1.6)
1.2. THE PATH INTEGRAL FOR SCALAR FIELDS
hΩ|Ωi = (hqF |ΩihΩ|qI i)−1
lim
7
T →∞(1−i)
(hqF , T |qI , −T i)
(1.7)
Thus
hΩ|T
n
Y
R
qˆH (ti )|Ωi =
i=1
1.2
lim
T →∞(1−i)
RT
Q
Dq(t)Dp(t) ni=1 q(ti ) ei −T dtL(p,q)
RT
R
Dq(t)Dp(t) ei −T dtL(p,q)
(1.8)
The path integral for scalar fields
• Aim: Compute
ˆ
hφF , tF |φI , tI i = hφF |e−iH(tF −tI ) |φI i
with
Z
H=
Z
4
d4 x
d x H(φ, π) =
1
1 2
π (x) + (∇φ(x))2 + V (φ(x))
2
2
Procedure
• IR-cutoff ↔ finite 3D volume V
UV cutoff ↔ discreting V
⇒ 3D QM problem for a finite # of d.o.f.
• In the end remove the IR- & UV- cutoff in 3D space → divergences to be taken care of via
renormalisation.
Abbreviated form of this procedure:
Z
φF (x)
Dφ(x)Dπ(x) ei
hφF , tF |φI , tI i =
R tF
tI
˙
d4 x[π(x)φ(x)−H]
.
(1.9)
φI (x)
Since the π-integral is Gaussian:
Z
φF (x)
hφF , tF |φI , tI i =
Dφ(x) ei
R tF
tI
˙
d4 xL(φ,φ)
.
(1.10)
φI (x)
The same way as in QM we get the ’master formula’ for time ordered correlators:
hΩ|T
n
Y
i=1
ˆ i )|Ωi =
φ(x
R
lim
T →∞(1−i)
RT 4
Q
˙
Dφ(x) ni=1 φ(xi ) ei −T d xL(φ,φ)
.
RT
R
4
˙
Dφ(x) ei −T d xL(φ,φ)
(1.11)
8
CHAPTER 1. PATH INTEGRAL QUANTISATION
Euclidean QFT
Write fcts in Euclidean coordinates:
xE = (x1E , x2E , x3E , x3E ) := (x1 , x2 , x3 , ix0 ).
This replaces ei
R
d4 xL(φ(x))
7−→ e−
R
d4 xE L(φ(xE ))
(1.12)
.
The Greens function (= correlation functions) in Euclidean coordinates G(xE1 , . . . , xE2 ) are called
’Schwinger functions’. In ’typical’ QFTs these can be analytically rotated back to Minkowski time.
The Osterwald-Schrader theorem gives precise condition for when this is possible.
Conclusion
2 ways to ’define’ a QFT starting from classical theory:
1. Canonical formalism
ˆ
Classical φ(x, t) 7−→ ’quantum operator’ φ(x).
→ Define Hilbert space of states and derive the correlation function in the interaction picture.
insert formula
2. Path integral formalism
State = wave functional of the classical field.
Define a ’quantum operator’ φˆH (x) by giving its matrix elements:
hφ1 , t1 |φˆH (x)|φ2 , t2 i =
Z
φ2 (x)
Dφ(x)Dπ(x) φ(x) ei
R t2
t1
d4 xL(φ,π)
.
(1.13)
φ1 (x)
For a detailed treatment of wavefunctionals see Hatfield, QFT of Print particles and Strings.
1.3
Generating functional for correlators
Def: Given V a K vector space with K ∈ {R, C}. A functional T is a mapping T : V → K.
R
E.g. φ(x) 7→ S[φ] = d4 xL(φ(x)).
Consider the functional
Z
Z[J] :=
Dφ eiS[φ]+i
R
d4 xJ(x)φ(x)
(1.14)
for J(x) a classical source term.
Taylor expansion shows that
Z
∞
Z[J] X in
=
d4 x1 . . . d4 xn J(x1 ) . . . J(xn ) G(x1 , . . . , xn ),
Z[0]
n!
n=0
Qn ˆ
with G(x1 , . . . , xn ) = hΩ|T i=1 φ(xi )|Ωi.
To solve this we need the tools of functional calculus. Some of them are given in chapter 5.
E.g. we get the formula
(1.15)
1.4. FREE SCALAR FIELD THEORY
9
δ
eiφJ = iJ(x)eiφJ .
δφ(x)
Back to Z[J] this gives:
G(x1 , . . . , xn ) =
1
δ
δ
...
Z[J]|J=0
Z[0] iδJ(x1 )
iδJ(xn )
(1.16)
Thus Z[J] is called generating functional for G(x1 , . . . , xn ).
1.4
Free scalar field theory
R
Consider S0 [φ] = d4 x(− 21 φ(x)(∂x2 + m20 )φ(x)). In the path integral the contour is along t(1 − i).
For the free theory one can proof, that equivalently one can shift m0 7→ m0 − i. Thus
Z
i
Z0 [J] = Dφe 2 φ(Kφ)+iφJ ,
(1.17)
where K(x, y) = −δ (4) (x − y)(∂x2 + m20 − i), that is the inverse of the Feynman propagator
Z
i
d4 p
e−ip(x−y) .
DF (x − y) =
2
4
2
(2π) p − m0 + i
Thus one gets for the free theory
Z
Z0 [J] =
1
−1
Dφe− 2 φDF
φ+iφJ
(1.18)
Completing the square and with φ0 = φ + K −1 J (Dφ = Dφ0 ) one gets:
1
Z0 [J] = Z0 [0]e 2 (iJ)DF (iJ)
(1.19)
Z0 [0] does not play a role for the correlators. (Maybe later sketch how to compute it)
From G0 (x1 , . . . , xn ) =
δ
iδJ(x1 )
1
δ
. . . iδJ(x
e 2 iJDF J |J=0 we get the n-point function of the free theory.
n)
E.g. G0 (x1 , x2 ) = DF (x1 − x2 ). The Feynman diagrams are a graphical way to organize the combinatorics of taking the derivatives.
1.5
Perturbative expansion in interacting theory
Now: S[φ] = S0 [φ] +
R
d4 xLi nt(x).
A few dirty tricks then lead to:
10
CHAPTER 1. PATH INTEGRAL QUANTISATION
Z[J] = ei
R
δ
d4 xLint ( iδJ(x)
)
= Z0 [0] ei
= Z0 [0] e
R
d4 xL
Z0 [J]
δ
int ( iδJ(x) )
1 δ
D δ
2 δφ F δφ
ei
R
1
e 2 iJDF iJ
d4 xL
int (φ)+iJφ
(1.20)
|φ=0
(1.21)
Proof of (1.21): see elsewhere ;).
Two ways to set up perturbation theory
1. From (1.20) with G(x1 , . . . , xn ) =
scription for taking the derivative.
δ
iδJ(x1 )
Z[J]
δ
. . . iδJ(x
|J=0 where the Feynman rules give pren ) Z[0]
2. From (1.21) with the definition:
1δ
δ
DF δφ
2
δφ
Def: hF [φ]i0 := e
F [φ]
Then we get:
R
4
hφ(x1 ) . . . φ(xn ) ei d xLint (φ)+iJφ i0
R
G(x1 , . . . , xn ) =
4
hei d xLint (φ) i0
(1.22)
h. . . i0 leads to a simple proof of Wick’s theorem.
In particular the φ(x) in (1.22) are the external points in FD. Therefore
hei
R
d4 xLinz (φ)
P
i0 = e
Vi
,
(1.23)
where Vi are the values of the possible vacuum bubbles.
On the counting of loops and factors of ~
Consider a fully connected FD in momentum space with
E = #of
I = #of
V = #of
L = #of
external points
internal lines
vertices
loops
A little bit of combinatorics leads to
L=I −V +1
(1.24)
1.6. CONNECTED DIAGRAMS
11
If we resemble ~ we see that the number of additional factors of ~ to a scattering amplitude with
fixed E is equal to
I −V =L−1
(1.25)
Thus a loop expansion is an expansion in ~. Thus the ’classical results’ are the tree level diagrams.
1.6
Connected diagrams
Denote by C the sum over all fully connected diagrams.
Aim: Find generating functional for all fully connected Green’s functions G(C) (x1 , . . . , xn ):
X 1 Z
iW [J] :=
d4 x1 . . . d4 xn iJ(x1 ) . . . iJ(xn )G(C) (x1 , . . . , xn )
n!
(1.26)
W [J] : effective action.
One can convince oneself that:
Z[J]
Z[J]
= eiW [J] ⇔ iW [J] = ln
Z[0]
Z[0]
(1.27)
We define now:
τ (x1 , . . . , xn ) =
δ
δ
...
iW [J]
iδJ(x1 )
iδJ(xn )
(1.28)
Then:
G(C) (x1 , . . . , xn ) = τ (x1 , . . . , xn )|J=0
1.7
The 1PI effective action
The generating functional for 1PI connected diagrams is the object defined as follows:
First step
δW [J]
δJ(x)
Z[J]
where
= eiW [J]
Z[0]
Def: ϕJ (x) :=
(1.29)
12
CHAPTER 1. PATH INTEGRAL QUANTISATION
ϕJ is the 1-point function of φ(x):
J=0
ˆ
ˆ
ϕJ = hΩ|φ(x)|Ωi
J −→ hΩ|φ(x)|Ωi ≡ 0
Second step
Assume ϕJ is invertible (in J). Then we define the 1PI effective action as the Legendre transformation
of W [J]:
Γ[ϕ] := W [Jϕ ] − ϕJϕ
(1.30)
δΓ
= −Jϕ (x).
δϕ(x)
(1.31)
It follows that
Claim: iΓ[ϕ] is the generating functional for the 1PI connected, amputated Green’s function Γn (x1 , . . . , xn ):
Z
∞
X
1
iΓ[φ] =
d4 x1 . . . d4 xn ϕ(x1 ) . . . ϕ(xn )Γn (x1 . . . xn )
n!
n=0
I.e.
˜ n (x1 . . . xn ) :=
Γ
δ
δ
...
iΓ[ϕ]
δϕ(x1 )
δϕ(xn )
then
˜ n (x1 . . . xn )|ϕ=0
Γn (x1 , . . . , xn ) = Γ
See the proof in 5.1.
1.8
Γ[ϕ] as a quantum effective action and background field
method
Replacing S[ϕ] by Γ[ϕ] in the path integral and computing just at tree-level gives the full quantum
theory.
We will now see that we can write:
e
i
W [J]
~
Z
i
Dφ e ~ (S[φ]+Jφ)
=
i
= e ~ (Γ[ϕ)+Jϕ) (evaluated at ϕ s.t.
δΓ
= −J)
δϕ
1.8. Γ[ϕ] AS A QUANTUM EFFECTIVE ACTION AND BACKGROUND FIELD METHOD 13
Interpretation
• The path integral is responsible for ’quantum fluctuations’.
δΓ
δϕ
• Replacing S[φ] by Γ[ϕ] (with
the result in full beauty.
= −J) and working at tree level (i.e. no
R
Dφ any more) gives
⇒ Γ[ϕ] = S[ϕ] + ~K[ϕ] for some object K[ϕ].
δΓ
• The equation δϕ
= −J, i.e.
the classical e.o.m. δS
= 0.
δφ
δΓ
|
δϕ J=0
= 0 is the quantum effective equation of motion replacing
Background field method
Consider
3
e
i
W [J]
~
W [J]=Γ[ϕ]+Jϕ
δΓ
=−J
δϕ
−→
e
Z
˜ e ~i (S[φ]+Jφ)
Dφ
=
i
Γ[ϕ]
~
Z
=
δΓ
(φ−ϕ))
˜ e ~i (S[φ]− δϕ
Dφ
ˆ
Now define f (x) := φ(x) − ϕ(x). Since ϕ(x) is the vacuum expectation of φ(x),
we view f (x) as the
quantum fluctuation around the background ϕ(x). We can write
e
i
Γ[ϕ]
~
Z
δΓ
f)
˜ e ~i (S[ϕ−f ]− δϕ
Df
.
=
(1.32)
A Taylor expansion of S arround ϕ and Γ[ϕ] = S[ϕ] + ~K[ϕ] leads to the formula:
Z
Γ[ϕ] = S[ϕ] − i~ ln
˜ e
Df
i
~
1 δ2 S
f
f −~ δK
f +O(f 3 )
2 δϕ2
δϕ
= S[ϕ] + ~K[ϕ]
(1.33)
This has to be solved perturbatively.R In particular the 1-loop correction to Γ is due to the lowest
˜ . Since K is by itself already 1-loop, the relevant piece is
fluctuation terms in the integrand of Df
only the quadratic one and
Z
2
− 21 f − ~i δ S2 f
(1−loop)
˜
δϕ
K
= −i ln Df e
1.8.1
Euclidean QFT & statistical field theory
Executive summary of what QFT is about
• Classical field φ ' continuum limit description of a system of N harmonic oscillators with
classical action SE (φ).
3
R
˜ =
Dφ
1
Z[0]
R
Dφ, s.t. the integral is normalized to 1
14
CHAPTER 1. PATH INTEGRAL QUANTISATION
• In cl. limit (~ → 0) φ takes a definite value given by the cl. e.o.m.
δSE
=0
δφ
(1.34)
• For ~ finite, quantum fluctuations arise. These are encoded in ZE [J] via the path integral.
• We compute expectation values of correlation functions. In particular
1
ϕ(x)J := hφ(x)iJ =
Z[0]
Z
1
Dφ φ(x)e− ~ (SE [φ]−φ·J) .
(1.35)
δWE
.
δJ(x)
(1.36)
• This can be computed as
ϕ(x)J = −
• In terms of the Legendre trafo we have
δΓE
=J
δϕ
(1.37)
Then γE gives a quantum effective action after integrating out the quantum fluctuations.
This is formally the same structure of classical statistical field theory.
•
1.9. GRASSMANN ALGEBRA CALCULUS
1.9
15
Grassmann algebra calculus
To get the right result for (Dirac) fermion fields as quantum operators with anti-commutation relation
the classical fields (plugged in the path integral) have to anti-commute, too. So we need the notion
of anti-commuting numbers.First consider a finite # of d.o.f. by
ψ A (t, x) → ψi (t)
These ψi (t) take values in a so-called Grassmann-Algebra A. First
Let θi , i = 1, . . . , n be a basis of a n-dimensional complex vector space. In addition define a bilinear
anti-commuting relation
θi θj = −θj θi .
Then the Grassmann-Algebra is defined as the space
A=
n
M
Λk V
(1.38)
k=0
with Λk V the complete anti-symmetric n-fold tensor product of V. A typical elements of A is of the
form
1
1
aij θi θj + · · · + ai1 ...in θi1 . . . θin
2!
n!
The θi are called Grassmann numbers. A is a graded algebra where we define:
a + ai θ i +
(1.39)
Def: An element of Λ2k V has grade 0 (even).
An element of Λ2k+1 V has grade 1 (odd).
Differentiation an integration
One defines functions:
1
1
f (θ) = a + ai θi + aij θi θj + · · · + ai1 ...in θi1 . . . θin
2
n!
.
Differentiation is given as follows:
•
∂
θ
∂θi j
•
∂
(a1 f1 (θ)
∂θi
= δij ,
∂
a
∂θi
= 0 ∀a∈C
+ a2 f2 (θ)) = a1 ∂θ∂ i f1 (θ) + a2 ∂θ∂ i f2 (θ) (linearity)
• If f1 has defined grade s, then the graded Leibniz rule holds:
∂
∂
∂
(f1 (θ)f2 (θ)) = (
f1 (θ))f2 (θ) + (−1)s f1 (θ)
f2 (θ)
∂θi
∂θi
∂θi
For the integral I[f (θ)] as a functional of f (θ) we claim
linearity and translation invariance in the
R
integration variable. It follows that -if we normalise dθ θ = 1 -
16
CHAPTER 1. PATH INTEGRAL QUANTISATION
Z
dθ(a + bθ) = b =
∂
(a + bθ)
∂θ
(1.40)
The generalized form is
Z
∂
θj .
(1.41)
∂θi
Then the Grassmann measure is defined as dn θ = dθn dθn−1 . . . dθi and because of dθi dθj = −dθj dθi
we have
dθi θj = δij =
Z
dn θ θi1 . . . θin = εi1 ...in
(1.42)
Some applications are written down in chapter 5.2.
1.10
The fermionic path integral
First, consider single fermion by
ψˆA (t, x) → ψˆi (t)
(1.43)
ignoring the spinor components A and taking i = 1. We seek the states, s.t.
ˆ
ψ(t)|ψ,
ti = ψ(t)|ψ, ti
(1.44)
with ψ(t) a complex Grassmann number. Suppressing also t-dependence we must find the Hilbert
space acted upon by ψˆ and ψˆ† with
ˆ ψ}
ˆ = 0 = {ψˆ† , ψˆ† }, {ψ,
ˆ ψˆ† } = 1
{ψ,
(1.45)
This is called a fermionic harmonic oscillator. We assume the existence of a ground state |0i s.t.
ˆ = 0 and define |1i = ψˆ† |0i. By means of anti-commutation relations
ψ|0i
ψˆ† |1i = 0
ˆ = |0i
ψ|1i
ˆ
Claim: The eigenstate |ψi with ψ|ψi
= ψ|ψi takes the form
ˆ†
|ψi = |0i − ψ|1i = e−ψψ |0i.
(1.46)
ˆ
ˆ
ˆ = ψ|0i = ψ|0i − ψ 2 |1i = ψ|ψi
ψ(|0i
− ψ|1i) = −ψψ|1i
= ψ ψ|1i
(1.47)
Indeed,
This is the fermionic analogue of a bosonic coherent state. The complex conjugate state is:
ˆ
hψ| = h0| − h1|ψ ∗ = h0|e−ψψ
∗
(1.48)
1.10. THE FERMIONIC PATH INTEGRAL
Norm: hψ|ψ 0 i = 1 + ψ ∗ ψ 0 = eψ
17
∗ ψ0
R
∗
Completeness: dψ ∗ dψ |ψie−ψ ψ hψ| = 1
Let’s now compute the qm transition amplitude:
ˆ
hψF , tF |ψI , tI i = hψF |e−iH(tF −tI ) |ψI i
(1.49)
ˆ generally given by H
ˆ = ψˆ† M ψˆ for some M.
with H
We proceed as in the bosonic case4 and use some simple algebra to get to
hψF , tF |ψI , tI i = lim
N →∞
Z
=
Z Y
N
∗
dψj∗ dψj eψN +1 ψN +1 e
PN
∗
∗
j=0 (−ψj+1 (ψj+1 −ψj )−iδtH(ψj+1 ,ψj ))
j=1
Dψ ∗ Dψ e−
R tF
tI
(ψ ∗ i ∂ψ
−H)
∂t
(1.50)
Generalization to Field Theory
Consider the classical action
¯ =
S(ψ, ψ)
Z
µ
¯
d x ψ(x)(iγ
∂µ − m0 )ψ(x) + Lint ≡
4
Z
d4 x L
(1.51)
With this we generalize the above result to field theory as
Z
hψF (xF ), tF |ψI (xI ), tI i =
¯ x)Dψ(t, x) ei
Dψ(t,
R tF
tI
¯
d4 x L(ψ,ψ)
.
(1.52)
To project initial and final state to the vacuum |Ωi, we use the same tricks as in the bosonic case.
The generating functional for fermionic correlation functions is then defined via
Z
¯ η ψ+iψη
¯
iS[ψ,ψ]+i¯
¯
Z[η, η¯] = DψDψe
(1.53)
Then for instance
¯ 2 )|Ωi =
hΩ|T ψ(x1 )ψ(x
1
δ
−δ
Z[η, η¯]
Z[0] iδ η¯(x1 ) iδη(x2 )
(1.54)
Attention to the minus!
The same methods as in the bosonic theory can be applied to deduce from this the fermionic Feynman
rules.
4
Split the time interval and inserting 1
18
1.11
CHAPTER 1. PATH INTEGRAL QUANTISATION
The Schwinger-Dyson equations
If the measure is invariant, path integral formalism shows very smart how symmetries carry over to
the quantum theory.
Consider a general theory defined by
Z
Z[J] =
Dφ ei(S[φ]+φ·J) .
(1.55)
Assume the trafo
φ → φ0 = φ + ε∆φ + O(ε2 )
(1.56)
leaves the measure Dφ invariant. Then
Z
Z[J] =
Dφ ei(S[φ]+φ·J)
Z
=
i(S[φ]+φ·J)
Dφ e
δS
2
1+i
+ J ε∆φ + O(ε )
δφ
(1.57)
Thus
Z
0=
i(S[φ]+φ·J)
Dφ e
δS
+ J ∆φ.
δφ
(1.58)
Now: For ∆φ(y) = δ (4) (y − x) the assumption holds and we get
Z
Dφ
δS
+ J(x) ei(S[φ]+φ·J) = 0
δφ(x)
This is the celebrated Schwinger-Dyson equation.
Note: The cl. e.o.m. in presence of a source J are
δS
+J =0
δφ
(1.59)
(1.60)
Thus the cl. e.o.m. holds asQan operator equation.
δ
Now acting on (1.57) with ni=1 iδJ(x
and taking J = 0 leads to
i)
n
n
X
δS Y
hΩ|T
φ(xi )|Ωi = i
hΩ|T φ(x1 ) . . . φ(xi−1 )δ(x − xi )φ(xi+1 ) . . . φ(xn )|Ωi
δφ(x) i=1
i=1
(1.61)
In this form the Schwinger-Dyson equation shows that the e.o.m. hold inside a correlation function
up to contact terms.
1.11. THE SCHWINGER-DYSON EQUATIONS
19
Ward-Takahashi identity
Now consider the trafo with ∆φ(y) = δ (4) (y − x)φ(x), for a global continuous classical symmetry δφ
s.t.
δS
δφ(x) = −∂µ j µ (x).
δφ(x)
(1.62)
Here j µ is the Noether current. If the measure is invariant this leads to
Z
0 = Dφei(S[φ]+φJ) (−∂µ j µ (x) + J(x)δφ(x)) .
Again taking
Qn
δ
i=1 iδJ(xi )
∂µ hΩ|T j
= −i
µ
(1.63)
and J = 0 gives the Ward-Takahashi identity:
n
Y
φ(xi )|Ωi
i=1
n
X
hΩ|T φ(x1 ) . . . φ(xi−1 )δφ(x)δ (4) (x − xi )φ(xi+1 ) . . . φ(xn )|Ωi
(1.64)
i=1
Back to (1.57). We can write it as
δS
| δ + J Z[J] = 0
δφ φ= iδJ
(1.65)
With the abbreviations φ(xi ) = φi and J(xi ) = Ji ,
X 1
1
S[φ] = − φi DF−1ij φj +
Yim
φ . . . φi1
1 ...im i1
2
m!
m
and a little bit of algebra the Schwinger-Dyson equation can be written as
X
1
δ
δ
δ
Z[J] = (DF )ki Ji +
Yiim1 ...im−1
...
iδJk
(m − 1)!
iδJi1
iδJim−1
m
Graphical interpretation later.
!
Z[J]
(1.66)
20
CHAPTER 1. PATH INTEGRAL QUANTISATION
Chapter 2
Renormalisation of Quantum Field
Theory
2.1
Superficial devergence and power counting
Which are the relevant diagrams concerning the UV divergence in QFT? For simplicity: Scalar theory
in d dimensions with the bare Lagrangian
(n)
1
1
λ
L0 = (∂φ)2 − m20 φ2 − 0 φn
2
2
n!
Consider for a connected diagram:
(2.1)
E: # of external lines
L: # of loops
V : # of vertices
I: # of internal propagators
We had seen:
L=I −V +1
(2.2)
nV = 2I + E
(2.3)
In addition it’s clear that
Naively the momentum integrals structure in the UV for such a diagram is
Z
dd k1 . . . dd kL
k12 . . . kI2
(2.4)
Def: D = power of mom. in nominator - power of mom. in denominator.
For above theory:
D = dL − 2I
21
(2.5)
22
CHAPTER 2. RENORMALISATION OF QUANTUM FIELD THEORY
If we introduce a momentum cutoff Λ in the UV, then the amplitude M naively scales as follows
with Λ:
D > 0 : M ∼ ΛD
superficially divergent
−|D|
D<0:M∼Λ
D = 0 : M ∼ log(Λ)
superficially finite
superficially log-divergent
According to this diagrams with D ≥ 0 are divergent. Therefore D is called superficial degree of
divergence. D does not always reflect the actual divergence or finiteness properties of the diagram.
But except for issues of divergent subdiagrams, all divergences come at most from superficially divergent diagrams.
For φn theory in d dimensions we get with (2.3) and (2.4) for (2.5):
d−2
d−2
V −
E
D =d− d−n
2
2
2.2
(2.6)
Renormalisability and BPHZ theorem
Definition: A theory is called (power-counting)
• renormalisable: # of superficial divergent amplitudes is finite, but divergences appear at every
order in perturbation theory.
• super-renormalisable: # of superficial divergent Feynman diagrams is finite
• non-renormalisable: # of superficial divergent amplitudes is infinite.
By dimensional considerations one verifies, that (2.6) is also given by
(n)
D = d − [λ0 ]V −
d−2
E.
2
(2.7)
Thus, a theory is
(n)
• renormalisable ↔ [λ0 ] = 0
(n)
• super-renormalisable ↔ [λ0 ] > 0
(n)
• non-renormalisable ↔ [λ0 ] < 0
The BPHZ theorem summarizes the concept of (power-counting) renormalisability:
• For the moment ignore the issue of divergent subgraphs. If a theory is renormalisable, at each
order of perturbation theory only a finite number of divergent diagrams, parametrized by a
finite number of divergent constants, appear. One can absorb these divergences order by order
in the counterterms of the renormalized Lagrangian such that all physical ampitudes are finite.
• The counterterms create new Feynman diagrams relevant at the next order. These will cancel
the divergent subdiagrams (if present) at the next order.
• All of this leads to a perturbative adjustment of a finite number of counterterms in the renormalized Lagrangian, and thus predictivity is maintained.
The proof of the complete cancellation of divergent subdiagrams by lower order counter terms can
be found in Zimmerman’s 1970 article. Here: Demonstration in φ4 theory up to 2-loops.
2.3. RENORMALISATION OF φ4 THEORY UP TO 2-LOOPS
2.3
23
Renormalisation of φ4 theory up to 2-loops
Start with the Lagrangian
1
1
λ0
L = (∂µ φ)2 − m20 φ2 − φ4
(2.8)
2
2
4!
in d = 4 and with [λ0 ] = 0. We then have D = 4 − E. Since the action is invariant under φ → −φ,
any diagram with a odd number of external lines vanishes. Thus the only superficially divergent
amplitudes are
• the vacuum energy with E = 0, D = 4, wich is trivially absorbed into V0
(2.9)
• the propagator with E = 2, D = 2:
(2.10)
• the 4-point ampitude with E = 0, D = 0:
(2.11)
In a theory with dimensionless coupling constant the dimension of the amplitude agrees with D.
Thus we parametrise the second amplitude
p
p
= aλ2 + bpλ + cp2 log(λ) + finite terms
Due to the symmetry p → −p of L in momentum space b ≡ 0. By the same logic (with d dimensionless)
= d log(λ) + finite terms
Definition If the divergence is multiplied by a simple monomial in p as above, it is called local
divergence.
Absorb the 3 divergent constants a, b, d into 3 counterterms in the renormalised Lagrangian, order
by order in renormalised perturbation theory.
Define the renormalised field
φr =
1
1
Z2
with Z the wavefunction renormalisation and rewrite
φ
1
1
λ
1
1
δλ
L = (∂µ φr )2 − m2 φ2r − φ4r + δZ (∂µ φr )2 − δm φ2r − φ4r
2
2
4!
2
2
4!
(2.12)
(2.13)
24
CHAPTER 2. RENORMALISATION OF QUANTUM FIELD THEORY
where δZ = Z − 1, δm = m20 Z − m2 and δλ = λ0 − λ. Thus our new Feynman rules:
p
∼
p p
i
p2 −m2 +i
∼ i(p2 δZ − δm )
∼ iδλ
∼ iλ
Our renormalisation condition:
p
p
p1
!
=
i
p2 −m2
+ terms regular at p2 = m2
p3
amp.
= −iλ at s = 4m2 , t = u = 0
p4
p2
s,t,u are the Mandelstam variables
s = (p1 + p2 )2 , t = (p1 − p3 )2 , u = (p1 − p4 )2
p
p
=
(2.14)
i
p2 −m2 −M 2 (p2 )
Now
Thus the first renormalisation condition implies:
!
M 2 (p2 )|p2 =m2 = 0,
2.3.1
1PI
d
!
M 2 (p2 ) = 0.
2
dp
(2.15)
1-loop renormalisation
=
•
+
that is
−iλ
2
Z
d4 k
i
(1)
(1)
) ≡ −iM 2 (p2 )|1-loop .
+ i(p2 δZ − δm
4
2
2
(2π) k − m + iε
(2.16)
We compute the integral by dimensional regularisation. With d = 4 − ε we get
•
=
i λ
m2 ( 2ε
2 16π 2
1
+ 1 − log(m2 ))(e−γ 4π) 2 ε
... More lengthy calculations and stuff, that I will do when I look through the exercises.
2.4
Renormalisation of QED revisited
¯ µ Aµ ψ. With
QED interaction ∼ ψγ
(2.17)
2.4. RENORMALISATION OF QED REVISITED
25
Eγ = number of external photons
Ee = number of external electrons
(2.18)
the superficial degree of divergence of an amplitude is
3
D = 4 − Eγ − Ee
2
(2.19)
Since [e] = 0 the theory is renormalisable. Amplitudes with D ≥ 0:
1. Vacuum energy with D=4.
2. The photon propagator with D = 2, the electron propagator with D = 1 and the eeγ vertex
with D = 0.
3. The 1-photon amplitude with D = 3:
4. The 3-photon amplitude with D = 1:
5. The 4-photon amplitude with D = 0:
Due to discrete and continuous symmetries of the theory only the diagrams from 1. and 2. are
actually divergent.
• Amplitudes 3. and 4. vanish to all orders due to the discrete symmetry of charge: LQED
is invariant under
j µ → −j µ and Aµ → −Aµ
(2.20)
¯ µ ψ. Then a diagram with only odd number of external photons vanishes since
with j µ ∼ ψγ
e.g.
=
∼ hΩ|T j µ |Ωi.
(2.21)
26
CHAPTER 2. RENORMALISATION OF QUANTUM FIELD THEORY
• Diagram 5. is finite as a consequence of gauge symmetry. Can be shown by exploiting the
Ward identities.
The remaining diagrams of 2. are divergent, but again by symmetry the actual divergence of the
first two diagrams of 2. are just logarithmic ones. (Maybe insert argumentation)
Thus all diagrams are logarithmically divergent.
2.5
The renormalisation scale
µ: mass or renormalisation scale introduced by the arbitrary renormalisation condition.
Illustration by considering a massless theory
1
L = (∂φ0 )2 − λ0 φ20
2
in d = 4. Impose the renormalisation conditions
1
(2.22)
1. M 2 (p2 ) = 0 at p2 = −µ2 ,
2.
d
M 2 (p2 )
dp2
= 0 at p2 = −µ2 ,
3. and for the vertex:
= −iλ at (p21 + p22 )2 = (p21 + p23 )2 = (p21 + p24 )2 = −µ2
1
Now φ = Z − 2 φ0 . Condition 1. and 2 imply
Z
i
d4 x eipx hΩ|T φ(x)φ(0)|Ωi|p2 =−µ2 = 2 |p2 =−µ2
(2.23)
p
i.e.
Z
iZ
(2.24)
d4 x eipx hΩ|T φ0 (x)φ0 (0)|Ωi|p2 =−µ2 = 2 |p2 =−µ2 .
p
I.e. he wavefunction renormalisation factor Z is not the residue of the propagator at the physical
mass p2 = 0, but at p2 = −µ2 .
If we the renormalised 4-correlator, we see that its Fourier transformation is finite, but explicitly and
implicitly depends on µ at 1-loop, because λ is defined via renormalisation 3. at the scale µ.
2.6
The Callan-Symanzyk (CS) equation
Start with the bare cuttoff dependent correlator
Y
G(0)
(x
,
.
.
.
,
x
)
=
hΩ|T
φ0 (xi )|Ωi|conn. = G(0)
1
n
n
n (x; λ0 , m0 )
i
satisfying
1
In renormalisation of φ4 we identified µ with the mass m of φ
(2.25)
2.7. COMPUTATION OF β-FUNCTIONS IN MASSLESS THEORIES
27
d (0)
G (x; λ0 , m0 )|λ0 ,m0 fixed = 0.
dµ n
(2.26)
Compare with the cuttoff independent renormalised correlator
Y
Gn (x1 , . . . , xn ) = hΩ|T
φ(xi )|Ωi|conn. = Gn (x; λ, m, µ)
(2.27)
i
1
Since going to the renormalised Lagrangian is only a reformulation with φ = Z − 2 φ0 :
n
Gn (x1 , . . . , xn ) = Z − 2 G(0)
n (x1 , . . . , xn ).
(2.28)
Together with (2.26) we conclude
µ
d n
Z 2 Gn (x; λ, m; µ)|λ0 ,m0 fixed = 0.
dµ
(2.29)
By the chain rule and the definitions
dλ
|λ ,m
dµ 0 0
dm2
|λ ,m
βm2 := µ
dµ 0 0
µ dZ
1 d
|λ0 ,m0 =
log Z|λ0 ,m0
γφ :=
2Z dµ
2 dµ
βλ := µ
(2.30)
we arrive at the celebrated Callan-Symanzik or renormalisation group equation
∂
∂
∂
µ
+ βλ
+ βm2
+ n · γφ Gn (x; λ, m; µ) = 0.
∂µ
∂λ
∂m2
(2.31)
• By first computing Gn (x; λ, m; µ) and plugging it into the CS equ. allows us to compute
βλ , βm2 , γφ explicitly in perturbation theory.
• To leading order βλ , βm2 , γφ will not depend on the renormalisation scheme.
2.7
Computation of β-functions in massless theories
Consider a massless theory with dimenionless coupling and all divergences are log.
G2 (p) =
2
i
i p2
Λ2
i
i
i
+ 2 ( A log
+
finite)
+
(ip2δ
)
+ ...
Z
p p i
−p2
p2 p2
p2
with δZ = A log Λµ2 + finite s.t. G2 (p) is finite. At leading order the CS equation gives
(2.32)
28
CHAPTER 2. RENORMALISATION OF QUANTUM FIELD THEORY
1 ∂
γφ = µ δZ
2 ∂µ
(2.33)
i.e. γπ = −A.
More generally:
GN = tree level + 1-loop PI + vertex counter-term + external legs
"
#
X
Y i
Λ2
Λ2
−iλ − iB log
,
− iδλ + (−iλ)
Ai log
=
p2i
−p2
−p2i − δZi
i
i
(2.34)
• All terms except −iλ are of order λ2 .
• only δλ , δZi are µ-dependent.
If we rearrange the CS equation with this we get
∂
β(λ) = µ
∂µ
1 X
−δλ + λ
δZi
2 i
!
= −2B − λ
X
Ai .
(2.35)
i
To leading order the finite terms do not enter the expression for β or γ. Thus the 1-loop results are
independent of the renormalisation scheme.
1-loop β-function of QED
Insert later.
2.8
The running coupling
By the renormalisation condition the defined coupling is dependent of the renormalisation scale:
λ ≡ λ(µ).
λ(µ) gives the strength of the interaction at energy scale µ.
The evolution of this coupling λ(µ) is encoded in the β-function. The following is also called renormalisation group flow equation:
µ
d
d
λ(µ) =
λ(µ) = β(λ)
dµ
d log µ
(2.36)
2.9. RG FLOW OF DIMENSIONAL OPERATORS
29
With an expicit λ at reference scale µ∗ , this can be solved by
Z
λ(µ)
λ∗
dλ0
=
β(λ0 )
Z
µ
d log µ0 = log
µ∗
µ
µ∗
(2.37)
E.g. the 1-loop β-function in QED
β(e) =
gives for the fine structure constant α ≡
1 3
e
12π 2
(2.38)
e2
:
4π
α(µ) =
α∗
2 ∗
1 − 3π
α log µµ∗
(2.39)
Alternative RG behaviours
1. β(λ) > 0: At some scale, λ(µ) ceases to be perturbative. If β(λ) > 0 even non-perturbative,
then λ increases indefinitely. Thus the theory is ill-defined in the UV since the coupling diverges
(Landa pole).
But perturbative well defined in the IR, since λ(µ) → 0 as µ → 0. The fixed point λ = 0 is
called Gaussian fixed point.
2. β(λ) < 0: Perturbative in the UV but ceases to be perturbative in the IR. For λ(µ) → 0 as
µ → ∞, the theory is called asymptotically free in the UV.
3. β ≡ 0: The theory is scale-independent or conformal. Thus the theory is finite!
It is possible, that β interpolates between 1.,2. and 3..
2.9
RG flow of dimensional operators
Suppose we add the a renormalisable action a term
Z
dd xCi Odi
Odi is somme local operator of mass dimension di . Then define γi via
∂
∂
∂
0= µ
+β
+ nγφ + γi Ci
Gn (x; λ, Ci ; µ).
∂µ
∂λ
∂Ci
Now define a dimensionless coupling gi via
Ci = gi µd−di
(2.40)
(2.41)
(2.42)
With this the CS equation becames
∂
∂
∂
+β
+ nγφ + βi
Gn (x; λ, gi ; µ),
0= µ
∂µ
∂λ
∂γi
(2.43)
30
CHAPTER 2. RENORMALISATION OF QUANTUM FIELD THEORY
where βi := (γi + di − d)gi . Now gi satisfies the RG flow equation
d
gi = βi
d log µ
(2.44)
For λ 1 and gi 1 an aprox. solution is
∗
gi (µ) = gi (µ )
µ
µ∗
di −d
(2.45)
Crucial conclusion:
Non-renormalisable coupling (d − di < 0) become irrelevant in the IR. Superrenormalisable couplings are relevant in the IR.
2.10
Wilson effective action & renormalisation group
Chapter 3
Quantisation of Yang-Mills-Theory
3.1
Recap of classical YM-Theory
• H : Lie group.
• Lie(H) : corresponding Lie algebra.
• H 3 h = exp(igα),
α ∈ Lie(H), g ∈ R.
• T a : Basis of Lie(H) = generators of H
• T a , T b = ifcab T c , fcab : structure constants.
• Jacobi identity:
a b c c a
T , T , T + [T , T ] , T b + T b , T c , T a = 0.
(3.1)
• Killing form (invariant bilinear form)
T a ◦ T b ∈ R, s.t. ∀h∈H hT a h−1 ◦ hT b h−1 = T a ◦ T b
(3.2)
[T a , T c ] ◦ T b = −T a [T c , T b ]
(3.3)
that also satisfies
• H compact: Killing form is positive definite, and one can normalise
1
κab = T a ◦ T b = δ ab
2
(3.4)
• Typically T a are given in the defining matrix representation. In many such peresentations:
T a ◦ T b ≡ trT a T b .
31
(3.5)
32
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
• Aµ ≡ Aaµ T a : Lie(H)-valued gauge potential transforming in the adjoint representation of
H:
i
Aµ 7→ U Aµ U −1 + (∂µ U )U −1 , where U (x) = eigα(x)
g
(3.6)
To linear order this is
Aµ 7→ Aµ − Dµ α(x),
(3.7)
where the adjoint covariant derivative acts on Lie(H)-valued fields as
Dµ α(x) := ∂µ α(X) + ig[Aµ , α(x)].
(3.8)
• Associated curvature (=field strength) with the connection Aµ :
1
a
Ta
[Dµ , Dν ] ≡ Fµν
ig
= ∂µ Aν − ∂ν Aµ + ig[Aµ , Aν ],
Fµν =
(3.9)
that transforms as
Fµν 7→ U Fµν U −1
(3.10)
Dµ Fαβ + Dβ Fµα + Dα Fβµ = 0
(3.11)
and satisfies the Bianchi identity
• The pure Yang-Mills Lagrangian
1
1 a µν
L = − trFµν F µν = − Fµν
F
2
4
a
(3.12)
is gauge invariant. The e.o.m. for Aµ :
Dµ F µν = 0 ≡ ∂µ F µν + ig[Aµ , F µν ]
3.2
(3.13)
Gauge fixing the path integral
• One finds
Π0 =
∂L
= 0,
∂ A˙ 0
(3.14)
a non-dynamical field. Variation of the action with respect to A0 (Lagrange multiplier) implies
the non-dynamical constraint
Di F 0i = 0.
(3.15)
3.2. GAUGE FIXING THE PATH INTEGRAL
33
• Requires the technology of quantisation of constrained systems.
• In path integral formalism we would write
1 aν
a
3
d x − A (KA )ν + O(A )
2
Z
4
S=
(3.16)
with iDF = K −1 . But K is not invertable.
• First perform path integral quantisation and fix the redundency later.
• A := space of all gauge fields Aµ (x).
• Gauge trafo on Aµ with h ∈ H:
i
Ahµ := hAµ h−1 + (∂µ h)h−1
g
• Quotient space of physically equivalent fields.
A/H := Aµ ∼ Ahµ : Aµ ∈ A, h ∈ H
Aim: Define path integral on the physical space A/H as
Z
dµ[A]eiS[A] .
(3.17)
(3.18)
(3.19)
A/H
I.e. we must construct suitable measure s.t.
Z
Z
Z
dµ(h)
DA =
A
dµ(A).
(3.20)
A/H
H
Therefore we need a gauge fixing condition
!
F (A) = 0
(3.21)
for some unique h ∈ H.
(3.22)
for some F : A → Lie(H).
Ideally, for given A ∈ A
F (Ah ) = 0
On the lefthand side of (3.20) we can insert 1 given by
Z
1=
dµ[h]δ[F (Ah )]M (Ah )
(3.23)
H
h
(A )
with M (Ah ) = det ∂F∂h
. Then
Z
Z
DA =
A
Z
dµ[h]
H
DAδ[F (Ah )]M (Ah )
(3.24)
A
With the non-trivial assumption
DA = DAh
(3.25)
34
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
we get
Z
Z
Z
DA =
dµ[h]
A
DAδ[F (A)]M (A)
|
{z
}
R
≡
Since S[A] = S[Ah ] and
R
(3.26)
A
H
dµ[A]
A/H
dµ[h] is just an overall factor Vol(H), we define
Z
Z
iS(A)
DAδ[F (A)]M (A)e
≡ dµ[A]eiS[A] .
Z=
H
(3.27)
A
˜ = 0, (3.23) receives contributions only near h = id. With this we
• With A˜ = Ah0 such that F [A]
can rewrite
Z
DA δ[F (A)] det∆FP eiS[A]
Z'
(3.28)
A
with the Faddeev-Popov matrix
∆FP = −
∂F (A)
Dµ .
∂Aµ
(3.29)
With this for any gauge invariant operator:
RT
hΩ|T O(A)|Ωi =
3.3
4 xL
DA δ[F (A)] det∆FP O(A) ei −T d
A
RT
R
i −T
d4 xL
DA
δ[F
(A)]
det∆
e
FP
A
R
lim
T →∞(1−i)
(3.30)
Feddeev-Popov ghosts
Technical steps
• Rewrite the delta function
Z
δ[F (A)] =
DB a (x) ei
R
d4 xB a (x)F a (x)
(3.31)
with B(x) = B a (x)T a : auxiliary field called Nakahashi-Lautrup field.
• Similarly to (??) with c¯(x), c(x) Grassmann-valued, Lie(H)-valued fields, called FeddeevPopov ghosts and anti-ghosts:
Z
det∆FP =
Thus altogether
D¯
cDc ei
R
d4 x¯
ca (x)(∆FP c(x))a
(3.32)
3.3. FEDDEEV-POPOV GHOSTS
35
Z
DA δ[F (A)]det∆FP e
iS[A]
Z
=
Z
DA
Z
DB
Z
D¯
c
Dc eiS[A,B,¯c,c]
(3.33)
with
Z
S[A, B, c¯, c] =
One could also require
1 a 2
a
a
a
a
d x − (Fµν ) + B (x)F (x) + c¯ (x)(∆FP c(x)) .
4
4
(3.34)
1
F (A(x)) = f (x)
(3.35)
Rewriting the delta-function as
Z
i
Df δ[F (A) − f ] e− 2ξ
δ[F (A)] →
R
d4 xf a (x)f a (x)
(3.36)
and
1. performing the delta function gives
Z
Z=
Z
DA
Z
D¯
c
Dc ei
R
a )2 − 1 F a F a +¯
ca (∆FP c)a )
d4 x(− 41 (Fµν
2ξ
(3.37)
2. as before introducing the auxiliary field B(x) and performing some algebra gives
Z
Z
DA
Z
DB
Z
D¯
c
Dc eiS[A,B,¯c,c]
(3.38)
with
Z
S[A, B, c¯, c] =
1 a 2
ξ
) + B a F a + B a B a + c¯(∆FP c)a
d x − (Fµν
4
2
4
(3.39)
E.o.m. of the non-kinetic field B a (x): B a (x) = − 1ξ F a (A(x)).
We often work with the gauge-fixing condition
∆Fp
1
instead of F (A) = 0
F (A) = ∂µ Aµ
and
∂F
=−
Dµ = −∂ µ Dµ
∂Aµ
(3.40)
(3.41)
36
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
3.4
Canonical quantisation and asymptotic Fock space
Canonical quantisation is required to establish the Hilbert space of states. Faddeev-Popov treatment
led to the gauge fixed Lagrangian
1 a 2
ξ
L = − (Fµν
) − ∂ µ B a Aaµ + B a B a + ∂ µ c¯Dµ ca
4
2
(3.42)
that gives the canonical conjugate fields
a µ0
Πaµ
Πac = −c¯˙a
A = (F ) ,
ΠaB = −Aa0 ,
Πac¯ = c˙ − gf abc Ab0 cc
The problem of Πa0
A = 0 is solved by the auxiliary field.
We get the (anti-)commutation relations:
[Aaj (t, x), A˙ bk (t, y)] = iδ ab δkj δ (3) (x − y)
(3.43)
[Aa0 (t, x), B b (t, y)] = iδ ab δ (3) (x − y)
{c
a
(t, x), Πbc (t, y)}
a
= {¯
c
(t, x), Πbc¯(t, y)}
(3.44)
ab (3)
= iδ δ (x − y)
(3.45)
(3.46)
Note that for unitarity of the S-matrix, we need
ca† = ca , c¯a† = −¯
ca
(3.47)
According to LSZ logic and in Feynman gauge ξ = 1 all asymptotic in- and out-fields enjoy a free
mode expansion. In particular
A˜aµ (x)
Z
=
3
d3 k 1 X a
ikx
√
(k, λ)[aaλ (k)e−ikx + aa†
]
λ (k)e
(2π)3 Ek λ=0 µ
(3.48)
with the polarisation vector satisfying
(k, λ)(k,
˙ λ0 ) = ηλλ0
(3.49)
˜ a via its e.o.m. B a = −∂ µ A˜a (for ξ˜ = 1) we get the normal commutation
By integrating out B
µ
relations for A. As in QED the resulting Fock-space contains negative and zero-norm states. Also
the ghost sector contains zero-norm states.
3.5
BRST symmetry and the physical Hilbert space
Aims
1. Define Hphys by a criterion that guaranties a positive definite norm.
2. Time evolution (S-matrix) does not lead out of Hphys .
3.5. BRST SYMMETRY AND THE PHYSICAL HILBERT SPACE
37
Idea
Criterion for Hphys is related to a symmetry of the full interacting theory.
2
The gauge-fixed Lagrangian is no longer gauge gauge invariant but possesses a residual global,
fermionic symmetry - the BRST symmetry:
• Define the Grassmann odd operator S:
SAµ = −Dµ c = −(∂µ c + ig[Aµ , c])
i
Sc = g{c, c}
2
S¯
c = −B
SB = 0
(3.50)
• S has the decisive properties
1. S is nilpotent:
S2 = 0
(3.51)
1
L = − Fµνa F µνa − Sψ(x)
4
(3.52)
2. The Lagrangian can be written as
for ψ = c¯a ∂ µ Aaµ + 2ξ c¯a B a
3. The action is invariant under S.
Finally, the BRST symmetry transformation is defined as
δ Φ = SΦ,
Φ ∈ {A, c, c¯B}
(3.53)
Clearly δ2 = 0 and δ L = 0.
µ
By Noether’s theorem: Noether current JBRST
with
µ
∂µ JBRST
=0
(3.54)
and its charge
Z
QBRST =
0
d4 x JBRST
with Q˙ BRST = 0.
ˆ BRST that generates the BRST transformation:
Now, QBRST → Q
2
Then 2. follows automatically.
(3.55)
38
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
ˆ BRST , Φ]
ˆ = iδ Φ.
ˆ
[Q
(3.56)
2
ˆ2
Crucially: Q
BRST = 0 since S = 0. (Proof?)
Q˙ BRST = 0 implies
ˆ Q
ˆ BRST ] = 0.
[H,
Finally,
ˆ†
Q
BRST
(3.57)
ˆ BRST .
=Q
Some mathematical stuff
H: Vectorspace, Q : H → H : nilpotent operator: Q2 = 0.
• Φ ∈ kerQ is called Q-closed.
• ψ ∈ ImQ is called Q-exact.
• Since Q2 = 0: ImQ ∈ kerQ.
• If H has an inner product · and Q† = Q w.r.t. it:
ψ · ψ = 0 for ψ ∈ ImQ
(3.58)
ψ · Φ = 0 for ψ ∈ ImQ, φ ∈ kerQ.
(3.59)
and
Thus Q-exact states are null and orthogonal to Q-closed states. We say
ψ1 ψ˜2 ⇔ ψ1 − ψ2 ∈ ImQ
(3.60)
and define the Q-cohomology:
Cohom(Q) := kerQ/ImQ ≡ {closed Φ}/{exact ψ}
(3.61)
Now back to physics
ˆ BRST : space of states on which the time evolution operator is independent of the specific
1. kerQ
choice of gauge-fixing. (Proof?)
ˆ BRST : space of states which have zero overlap with all other states.
2. ImQ
With this we define
ˆ BRST ≡ kerQ
ˆ BRST /ImQ
ˆ BRST
Hphys = CohomQ
ˆ BRST lead to the following properties of Hphys :
The properties of Q
(3.62)
3.6. FEYNMAN RULES FOR SCATTERING
39
ˆ BRST ) is of form
Φ ∈ Hphys = Cohom(Q
Φ = ξ µ A˜µ (k) with k 2 = 0
for ξ µ · kµ = 0 and where we identify
ξ µ ξ˜µ + kµ (longitudinal null states are trivial)
Unitarity of the S-matrix
Since L = L† the S-matrix is unitary on the entire Fock space, i.e.
X
hα|S † |γihγ|S|βi = hα|1|βi.
(3.63)
γ
ˆ and all states of Im(Q)
ˆ are null, it is also unitary on Hphys ,
But since S does not lead out of Ker(Q)
i.e. for |φT i, |ψT i ∈ Hphys transverse polarised states:
hφT |1|ψT i =
X
hφT |S † |χT ihχT |S|ψT i
(3.64)
χT ∈Hphys
3.6
Feynman rules for scattering
Start from the gauge-fixed action
S[A, c¯, c] = S[A] + Sghost
(3.65)
with
Z
S[A] =
Z
Sghost = −
1
1
d4 x(− (Fµν )2 − (∂µ Aµa )2 )
4
2ξ
d4 x¯
ca ∂µ (Dµ c)a
Now we can rewrite
S[A] = S0 [A] + Sint [A]
Z
Z
1 a
4
b
with S0 [A] = d x Aµ (x) d4 y iDF−1µν
ab (x − y)Aν (y)
2
1 µ ν
(4)
µν 2
with iDF−1µν
ab (x − y) = δ (x − y)(η ∂x − (1 − )∂x ∂y )δab
ξ
With this we get as solution for the propagator (in momentum space):
−i
pµ pν
ab
˜
DF µν = 2
ηµν − (1 − ξ) 2
δ ab
p + i
p
Thus
(3.66)
(3.67)
(3.68)
40
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
DFabµν (x)
d4 p −i
(2π)4 p2 + i
Z
=
pµ pν
ηµν − (1 − ξ) 2
δ ab e−ipx
p
(3.69)
• ξ = 0 : Landau gauge.
• ξ = 1 : Feynman-t’Hooft gauge.
Next include A3 and A4 interactions from
Z
g 2 abc ade µb νc
4
abc µ νa b c
Sint [A] = d x gf ∂ A Aµ Aν − f f A A Aµd Aνe
4
(3.70)
Cubic vertex
If we work out all 3! = 6 contractions we get
(b, ν)
p
k
= −gf abc [η µν (k − p)γ + η νγ (p − q)µ + η µγ (q − k)ν ]
q
(a, µ)
(c, γ)
Quartic vertex
4! possible contractions, but every 4. is equivalent → 6 different terms for
(a, µ)
(b, ν)
(c, γ)
(d, σ)
Result:
−ig 2 [fabe fcde (η µγ η νσ − η µσ η νγ )
+face fbde (η µν η γσ − η µσ η νγ )
+fade fbce (η µν η γσ − η µγ η νσ )
Next the ghost sector:
Z
Lghost =
Z
=
d4 x c¯a (−∂ µ Dµ c)a
d4 x(¯
ca (−∂ 2 δ ab )cb + g¯
ca ∂ µ f abc Abµ cc )
(3.71)
Ghost propagator
DFab = δ ab
i
p2
(3.72)
3.7. 1-LOOP RENORMALISATION OF YM THEORY & β-FUNCTION
41
Cubic vertex
(µ, b)
p
←
a
c
This is: gpµ f abc . ((ipµ ) is for the outgoing momentum)
Minimal coupling of YM theory to charged matter
E.g. a fermion in fundamental representation
ψi (x) 7→ Uij (x)ψj (x) = (eigα(x) )ij ψj (x)
(3.73)
with αij = αa (Tfa )ij . We add the interaction term
Lψ = ψi (iγ µ Dµ − m)ij ψj
(3.74)
with Dµ ij = δij ∂µ + igAaµ (x)Tfa ij
Then for the fermion propagator we get
S˜F
ij (p)
=
iδij
γ·p−m
(3.75)
and for the vertex Af f :
−igγ µ Tfa ij .
(3.76)
In addition, add for in/outgoing states:
aµ (k, λ)
for ingoing gauge boson ”a”: colour index.
∗a
µ (k, λ)
for outgoing gauge boson ”a”
ui
u¯i
vi
v¯i
for
for
for
for
ingoing fermion ”i”: flavour index
outgoing fermion ”i”
outgoing anti-fermion ”i”
ingoing anti-fermion ”i”
Important: Ghost field cancel loop contributions due to unphysical vector bosons and thus serve
as ”negative d.o.f.”.
Optical theorem
Blabla
3.7
Blabla
1-loop renormalisation of YM theory & β-function
42
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
Preliminaries on group theory
The second Casimir C2 (R) of the generators of a Lie algebra in representation R is
TRa TRa = C2 (R)1
(3.77)
trTRa TRb = C(R)δ ab
(3.78)
dim(R)C2 (R) = C(R)dim(H)
(3.79)
if we normalise the TRa s.t.
then
For the computation of δ1 , δ2 , δ3 have a look at the script or the exercise sheets.
As a result we can compute the β-function YM theory coupled to nf fermion (in d = 4):
β(g) = −
4
g 3 11
( C2 (H) − nf C(Rf ).
2
(4π) 3
3
(3.80)
Of all known renormalisable QFTs in 4D, YM teory is the only one which is proven to be
asymptotically free!
3.8
A very brief look at QCD
QCD is a SU(3) YM coupled to 6 quarks in fundamental representation.
g2
Thus the coupling αs = − 4π
runs as
αs (µ) =
αs (µ0 )
(µ0 )
log µµ0
1 − b0 αs2π
From experiment : For µ0 ' 91.2 GeV (' mass of Z-boson): αs (µ0 ) ' 0.118.
Strong scale: ΛQCD where
αs → ∞ as µ → ΛQCD
according to perturbative analysis! Then ΛQCD ' 200 MeV.
Phase structure of QCD
1. Deconfined phase
Energies ΛQCD : Weakly coupled free quarks and gluons.
(3.81)
3.8. A VERY BRIEF LOOK AT QCD
43
2. Confined phase (=hadronic phase)
Observable d.o.f. are bounded states of given by colour-singlets. From
• qi fundamental, i.e. qi 7→ Uij qj
U ∈ SU (3),
• q i anti-fundamental, i.e. q i 7→ Uji q j
one can form 2 types of SU (3) singlets:
(a) mesons: qi q i .
(b) barions: ijk qi qj qk .
Useful concept for hard scattering involving hadrons is given by the parton-distribution-function
(PDF):
Consider hadron with momentum P . Then the PDF Pf (ξ) is given by
Pf (ξ)dξ := probability to find constituent f ((anti-)quark or gluon) inside the hadron with momentum p = ξP
To leading order in αs , the PDF is independent of the energy scale.
44
CHAPTER 3. QUANTISATION OF YANG-MILLS-THEORY
Chapter 4
Symmetries in QFT
4.1
The chiral anomaly
• Classically the Noether current j µ of a global, continuous symmetry of L is conserved: ∂µ j µ =
0
• Ward identity:
∂µ hj µ (x)O(y)i = 0
up to contact terms as x → y
(4.1)
if the measure respects the symmetry.
• If the measure is not invariant then (4.1) does not hold and the symmetry is called anomalous.
• Anomalies of global symmetries: acceptable.
Anomalies of gauge symmetries: No go, inconsistency!
Example: Global axial current anomaly in QED.
¯ µ Dµ − m)ψ − 1 Fµν F µν with Dµ = ∂µ − ieAµ
L = ψ(iγ
4
(4.2)
• Gauge symmetry:
ψ(x) 7→ eieα(x) ψ(x)
i
Aµ 7→ Aµ − ∂µ α(x)
e
For α(x) ≡ α ∈ R constant: Global U (1)V or vector symmetry with conserved current
¯ µ ψ : ∂µ j µ = 0 classically
jVµ = ψγ
V
• If m = 0: Global chiral (axial) U (1)A symmetry:
ψ(x) 7→ eiαγ5 ψ(x)
iαγ5
¯
¯
ψ(x)
7→ ψ(x)e
45
(4.3)
46
CHAPTER 4. SYMMETRIES IN QFT
with the classically conserved axial Noether current
¯ µ γ5 ψ
j5µ = ψγ
• If in Z =
R
(4.4)
DψDψ¯ eiS also Dψ 0 Dψ¯0 = DψDψ¯ then
Z
0=
DψDψ¯ ∂µ j5µ . . . eiS = h∂µ j5µ . . . i up to contact terms
(4.5)
• We must define the measure by a gauge-invariant regularisation procedure. To this we go
to Euklidean space. Then DE = γEµ Deµ = γEµ (∂Eµ − ieAEµ ) is hermitian and
Z
R
¯
DE ψ
¯ d4 xE (ψi
Z = DψDψe
.
(4.6)
Now drop the E.
• Since D is hermitian: ∃ orthogonal eigenbasis ψm :
Dψm = λm ψm
with λm real.
(4.7)
We then expand
ψ(x) =
X
¯
ψ(x)
=
X
am ψm
†
am = hψm
, ψi
(4.8)
¯bm ψ¯m
¯ ψm i
bm = hψ,
(4.9)
m
m
Q
• Naively: DψDψ¯ = m dam d¯bm . But we need a gauge-invariant UV regularisation procedure:
1
D/Λ)2
ψm (x)
Define: ψ˜m (x) := e− 2 (
Then we define:
Z = lim
Λ→∞
with SΛ = −
P
m,n
Y
dan d¯bm e−SΛ
(4.10)
n,m
¯ Dψ
am¯bn hψ˜¯n , (−iD)ψ˜m i = −ψi
• Analysis of the invariance of the measure (first finite Λ):
From ψ 0 (x) = (1 + iα(x)γ5 )ψ(x) ≡
P
a0m =
X
n
a0n :
Mnm am
(4.11)
n
Z
Mnm = δnm + i
Likewise: ¯b0n =
X
m
2
D/λ)
ψm (x).
d4 xα(x)ψn† (x)γ5 e−(
¯bm Mmn
(4.12)
(4.13)
4.2. RELATION TO ALIYAH-SINGER-INDEX-THEOREM
⇒
Y
da0n d¯b0n = (det M )−2
Y
dan d¯bn
47
(detM )−2 ≡ Jacobian.
(4.14)
n
n
Since det(1 + εC) = exp(εtrC + O(2 )):
Z
−2
= exp(−2i d4 xα(x)A(x))
X
D/Λ)2
†
(x)γ5 e−(
ψm (x)
ψm
A(x) = lim
(det M )
Λ→∞
(4.15)
(4.16)
m
• If Aµ ≡ 0 an eigenbasis of D is given by
A
ψm
≡ uA eikm x , uA : basis of spinors
(4.17)
• For Aµ 6= 0, this is still the asymptotic form for high energies (km ).
Crucial: Only UV-region can be responsible for non-zero A.
• From this point some ”simple” algebra leads to the result
−2
(det M )
= exp
i
16π 2
Z
4
µναβ
d xα(x)
2
Fµν Fαβ e
(4.18)
in Euklidean space.
• This implies (in Euklidean space)
∂µ hj5µ . . . i = −
e2 µναβ
hFµν Fαβ . . . i
16π 2
(4.19)
The method is called Fujikawa-method.
• Generalized to d=2n dimensions:
∂µ hj5µ . . . i = (−1)n
4.2
2en µ1 ...µ2n
hFµ1 µ2 . . . Fµ2n−1 µ2n . . . i
n!(4π)n
(4.20)
Relation to Aliyah-Singer-Index-Theorem
We had seen:
∂µ hµ5 i = 2iA
Z
Z
X
4
D/Λ
d xA(x) = d4 x
ψn† γ5 e−
ψn
n
(4.21)
(4.22)
48
CHAPTER 4. SYMMETRIES IN QFT
• If Dψm = λm ψm then
Dγ5 ψm = −λm γ5 ψm
(4.23)
because {D, γ5 } = 0.
• All eigenvalues λn 6= 0 come in pairs (λn , −λn ) and the corresponding eigenfunctions ψn and
γ5 ψn are orthogonal.
Thus only the zero modes ψ0 of D contribute to
R
d4 xA(x).
• Since γ5 = γ5† and γ52 = 1, the zero modes group into
γ5 ψ0i+ = +ψ0i+
i = 1, . . . , n+
(4.24)
γ5 ψ0i−
i = 1, . . . , n−
(4.25)
=
−ψ0i−
and we get
Z
d4 xA(x) = n+ − n− ≡ IndD
≡ index of the Dirac operator.
(4.26)
(4.27)
Thus
1
1
IndD = h∂µ j5µ i = −
2i
32π 2
Z
d4 x µναβ Fµν Fαβ
(4.28)
In form language
1
IndD = −
8π
4.3
Z
F ∧F
Chiral gauge theories & anomalies
Generalisation to non-abelian YM theory:
¯ µ γ5 T a ψ and we need to compute h(p, b), (k, c)|∂ µ j a |0i.
• j5µa = ψγ
µ5
R
The triangle diagrams show, that
(4.29)
4.3. CHIRAL GAUGE THEORIES & ANOMALIES
49
∂µ j5µa = tr(TRa {TRb , TRc })(−
e2 µναβ
Fµν Fαβ
16π 2
(4.30)
• A non-zero chiral anomaly becomes problem only if the axial symmetry is gauged:
¯ µ Dµ ψ + LYM
L = ψiγ
Dµ = ∂µ − igAaµ TRa
(4.31)
Aay couples to the vector current and j5µ is harmless.
Rewrite the first part:
Lferm = ψ¯L iγ m uDµ ψL + ψ¯R iγ m uDµ ψR
(4.32)
with
1
ψL = (1 − γ5 )ψ
2
1
ψR = (1 + γ5 )ψ
2
in representation R1 = R
(4.33)
in representation R2 = R
(4.34)
• In a more general theory:
If R1 6= R” , the theory is called chiral.
• New left handed field:
ψL0 := CψR C † = −iγ 2 ψR∗
(4.35)
¯ 2 , the conjugate representation of R2 , defined by
It transforms in R
TRa¯ = −(TRa )∗ = −(TRa )T
(for T a† = T a )
(4.36)
Thus we can write:
Lferm = ψ¯L iDψL
(4.37)
¯ µ 1 (1 − γ5 )T a ψ
j µa := ψγ
R
2
(4.38)
¯2
with ψL transforms in R = R1 ⊕ R
• In this formulation, Aaµ couples to
¯ this contains an intrinsically chiral part.
and if R 6= R,
50
CHAPTER 4. SYMMETRIES IN QFT
• The anomaly is then
∂µ j µa =
e2
b
c
µναβ Fµν
Fαβ
Aabc
2 · 16π 2
(4.39)
with Aabc = tr(TRa {TRb , TRc }).
Important conclusion
¯ an Aabc 6= 0, then the anomaly of j5µa implies that the gauge invariance of the theory is
If R 6= R
anomalous because the anomalous current j µ = jVµ + jAµ is the Noether current that follows from the
global version of the gauge transformation
ψL 7→ eiα
aT a
R
ψL .
(4.40)
A chiral gauge theory is consistent only if
Aabc = tr(TRa {TRb , TRc }) = 0
4.4
(4.41)
Spontaneous Symmetry Breaking
The physical theory has some symmetry, but the ground state is degenerate and the vacuum does
not respect this symmetry.
Discrete Symmetries
1
L = ∂ µ φ∂µ φ − V (φ)
2
with φ real scalar and assume V (φ) = V (−φ), s.t. we have Z2 -symmetry.
Then (take g > 0)
1
1
V (φ) = m2 φ2 + gφ4 + non-renormalisable terms.
2
4!
(4.42)
(4.43)
4.4. SPONTANEOUS SYMMETRY BREAKING
51
1. m2 > 0:
V (φ)
φ
Figure 4.1:
The classical minimum at φ0 = 0 respects the Z2 -symmetry.
2. m2 < 0: Up to an constant we can write V (φ) = C(φ2 − v 2 )2
V (φ)
φ
Figure 4.2:
φ = 0 is not a ground state. The classical ground state at φ0 = ±v is degenerate and the
symmetry there is broken.
Excitation around the vacuum (e.g. +v):
φ = v + f,
(4.44)
1
1
1
L = ∂µ f ∂ µ f − g(v 2 f 2 + vf 3 + f 4 )
2
6
4
(4.45)
then
This looks like a theory of a massive scalar f with
g
m2f = v 3
3
ˆ = v.
Define particles as excitations of fˆ arround hφi
Note: f(x) has no Z2 -symmetry left- the theory is spontaneously broken.
(4.46)
52
CHAPTER 4. SYMMETRIES IN QFT
Continuous symmetries
SSB of a continuous, global symmetry → massless modes = Goldstone modes.
Example: real scalar fields φ = (φ1 , . . . , φn ). With φ · φ =
1
L = ∂ µ φ · ∂µ φ − V (φ),
2
P
r
φ2r and
1
V (φ) = g(φ2 − v 2 )2 , g > 0
8
(4.47)
• We get a ”Maxican hat potential” with full symmetry group G = O(n) with a classical ground
state at φ20 = 0.
here
→ vacuum manofold M0 = {φ0 : V (φ0 ) = Vm in} = S n−1
• Even after SSB there is a residual symmetry corresponding to the remaining flat directions.
Take φ0 = (0, . . . , 0, v) ⇒ φ0 invariant under H = O(n − 1).
• φ = (φ⊥ , v + f ) then
f is massive m2f = gv 2 .
φ⊥ has no mass term. (n-1)
The massless fields are called Goldstone modes or Goldstone bosons.
More generally
• Starting with V and symmetry G:
V (U (g)φ) = V (φ) ∀g ∈ G.
(4.48)
M0 = {φ0 : V (φ0 ) = Vm in}
(4.49)
• SSB to vacuum manifold
stability group H ⊂ G U (h)φ0 = φ0 ∀h ∈ H.
• Assume: ∀φ0 , φ00 ∈ M0 ∃g ∈ G : φ00 = U (g)φ0 i.e. Hφ0 ' Hφ00 then:
M0 = G/H
(4.50)
Goldstone theorem-classical
Given a quantum field theory with SSB from G to H as above, then there exist (dimG− dimH)
zero mass scalars = Goldstone bosons.
Proof: ...
4.5. THE HIGGS MECHANISM
53
Goldstone’s theorem in QFT
Meep...
4.5
The Higgs mechanism
Consider a gauge theory with SSB from a
gauge G −→ gauge H.
The Higgs effect is the phenomenon that the (dim G- dim H) Goldstone bosons are ”eaten”, i.e.
absorbed by (dim G- dim H) vector bosons, which thereby become massive.
54
CHAPTER 4. SYMMETRIES IN QFT
Chapter 5
Important formulas and proofs
Gaussian integral
Z
∞
dx e
1
ax2 +bx
2
r
=
∞
5.1
2π b2
e 2a
a
(5.1)
Proof to 1PI effective action
˜ n that justifies this claim.
We will establish a relation between τn an Γ
i) n=2
δ
(−iJ(x2 ))
δϕ(x1 )
δ
τ2 (x1 , x2 ) =
ϕ(x2 )
iδJ(x1 )
˜ 2 (x1 , x2 ) =
Γ
Z
⇒
˜ 2 (x2 , x3 ) = −δ(x1 − x3 )
d4 x2 τ2 (x1 , x2 )Γ
˜ −1
⇒ τ 2 = −Γ
2
We have ϕJ |J=0 = 0. Therefore should
(c)
G2 (x1 , x2 ) = −Γ−1
2 (x1 , x2 ).
(5.2)
Let’s see if this can be justified:
First , recall the fully connected Feynman propagator:
(c)
G2 (x1 , x2 )
Z
=
=
(c)
d4 p (c)
G (p)e−ip(x1 −x2
(2π)4 2
(5.3)
i
2 2
with G2 (p) = p2 −m2 −M
2 (p2 ) by Dyson resummation, and thus −iM (p ) is the value of all amputated
0
1PI diagrams at 1-loop and higher.
Second, what is the 1PI 2-point fct?
55
56
CHAPTER 5. IMPORTANT FORMULAS AND PROOFS
Z
Γ2 (x1 , x2 ) =
d4 p
Γ2 (p)eip(x1 −x2 )
(2π)4
(5.4)
Later with the help of (5.2) we will see, that this means
Γ2 (p) = i(p2 − m20 − M 2 (p2 )
(5.5)
Thus (5.2) is justified.
ii) n=3
By chain rule one can verify
δ
=
iδJ(x1 )
Z
d4 yτ (x, y)
δ
δϕ(y)
(5.6)
x2
Graphical representation
..
˜ 1 , . . . , xn ) ≡
Γ(x
..
x1
Γ
xn
x2
(5.7)
..
τ (x1 , . . . , xn ) ≡
..
x1
W
xn
(5.8)
˜ 2 (x, y)−1 is graphically represented as
So the result τ2 (x, y) = −Γ
x
y
W
= −(
and
δ
iδJ(x)
=
x
W
Finally, by definition
begins:
y
x
y
Γ
)−1
δ
.
δϕ(y)
δ
iδJ(x)
adds a further leg to ever τ and
τ3 (x, y, z) =
=
δ
δϕ(x)
˜ The final countdown
to every Γ.
δ
τ2 (y, z)
iδJ(x)
x
W
w
δ
δϕ(w)
−(
y
Γ
z
)−1
5.2. APPLICATIONS TO GRASSMANN CALCULUS
57
Now we have to calculate the derivative:
δ ˜ −1
Γ (y, z) = −
δϕ(w) 2
=−
Z
˜ −1 (y, u)
d4 ud4 v Γ
Z
˜ 3 (w, u, v)τ2 (v, z).
d4 ud4 v τ2 (y, u)Γ
δ ˜
˜ −1 (v, z)
Γ(u, v)Γ
δϕ(w)
With this we can now write
W
=
W
Γ
W
W
.
Thus Γ3 gives indeed the amputated 1PI connected 3-point functions, as external legs merely carry
correction due to the fully resummed propagator G2 .
This procedure can be generalized (graphically) to higher n and thus the claim is proven inductively
for all Γn with n > 3.
Now let us shortly come back to the interpretation of Γ2 . We have (easiest to prove if doing the
integrals and check it!)
y
y
x
x
=−
W
W
Γ
W
This also holds at tree level, and if you work it out you will see, that you get the right result. ;)
5.2
Applications to Grassmann calculus
Fermionic Gauss integral
Consider n = 2m:
Z
1
dn θ e 2 θi Aij θj , Aij = −Aji
Z
1
dn θ Ai1 i2 . . . Ain−1 in θi1 . . . θin
= m
2 m!
√
1
= m Ai1 i2 . . . Ain−1 in i1 ...in := P f (A) = det A
2 m!
I=
In the complex case:
Z
∗
1
i ...i j ...j Bi j . . . Bin jn
n! 1 n 1 n 1 1
≡ det B
dn θdn θ∗ eθi Bij θj =