Lecture 26
8.324 Relativistic Quantum Field Theory II
Fall 2010
8.324 Relativistic Quantum Field Theory II
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2010
Lecture 26
5.3.3: Beta-Functions of Quantum Electrodynamics
In the case of quantum electrodynamics, we have the Lagrangian
( (
)
)
1 B µν
L = − Fµν
FB − iψ̄B γ µ ∂µ − ieB AB
µ − m B ψB ,
4
1
−1
1
−1
ϵ
2
2
2
with ψB = Z22 ψ, AB
µ = Z3 Aµ , mB = m + δm, and eB = Z3 eµ or αB = Z3 αµ , where α =
2α 1
earlier result that Z3 = 1 − 3π ϵ , we have
[
]
2α2 1
αB = µϵ α +
+ ... ,
3π ϵ
and
βα = −
ϵ
2α2
4α2
2α2
+
=
+ O(α3 ).
3π
3π
3π
(1)
e2
4π .
From our
(2)
(3)
Hence, the running of α(µ) is given by
1
1
2
µ
=
−
log ,
α(µ)
α(µ0 ) 3π
µ0
or, equivalently,
α(µ) =
α0
1−
log
2α0
3π
µ
µ0
(4)
.
(5)
In quantum electrodynamics, α(µ) increases as µ is increased, and α(µ) decreases as µ decreases. In particular,
1
α(μ)
log(μ)
Figure 1: We find a linear relationship between α−1 (µ) and log(µ) with a negative gradient for quantum electrody­
namics.
the Landau pole at α(µ) −→ ∞ is given by
3π
µ = Λ ≡ µ0 e 2α0 ,
(6)
independently of the choice of µ0 . Quantum electrodynamics becomes strongly coupled near the scale Λ. It is
1
convenient to express α(µ) in terms of the physical αphys ≈ 137
which we measure. Consider
e(µ)
e(µ)
∝
k
1 e2 (µ)
.
k 2 1 − Π(k 2 )
(7)
At the one-loop level,
Π(k 2 ) = −
e2 (µ)
π2
ˆ
(
1
dx x(1 − x)
0
1
1 1
− log
2
ϵ
(
D
µ̃2
))
− (Z3 − 1) ,
(8)
Lecture 26
8.324 Relativistic Quantum Field Theory II
Fall 2010
2
2
2
2
2
where µ̃2 ≡ 4πµ
eγ , D ≡ m + x(1 − x)k and m = me + O(α), where me is the physical electron mass. It is
convenient to introduce
α(µ)
αB
α̂(k) ≡
=
.
(9)
2
1 − ΠB (k 2 )
1 − Π(k )
Note that this quantity is finite, although the numerator and denominator of the last term are divergent. α̂(k) is
an effective k-dependent coupling. In particular,
1
.
137
αe = α̂(k = 0) ≈
Now,
ΠM S (k 2 ) =
2α(m)
π
and so
ΠM S (k 2 = 0) =
ˆ
(10)
1
dx x(1 − x) log
0
D
,
µ2
(11)
e2 (µ)
me
2α(µ)
me
.
log
=
log
2
6π
3π
µ
µ
We therefore have
αe =
α(µ)
1−
2α(µ)
3π
log
me
µ
(12)
.
(13)
We can compare this to
α(µ′ ) =
α(µ)
1−
2α(µ)
3π
log
µ′
µ
,
(14)
and we find
αe = α(µ = me ).
Therefore, in the MS scheme,
α(µ) =
1−
αe
log
2αe
3π
(15)
µ
me
,
(16)
3π
and the Landau pole occurs at Λ = me e 2αe ≈ me e5×137 . We now consider α̂(k) =
1.
α(µ0 )
1−ΠM S (k2 )
:
For k 2 ≫ m2e , D ∼ x(1 − x)k 2 , and so
ΠM S (k 2 ) =
α(µ0 )
k2
log 2 + . . . .
3π
µ0
(17)
α̂(k) =
α(µ0 )
2α(µ0 )
log µk0
3π
(18)
Therefore,
1−
+ ...,
and so α̂(k) ≈ α(µ) for µ ≈ k. Note that this is scheme-independent: in any scheme for µ ≫ me ,
α(µ) ≈ α̂(µ).
2.
When k 2 ≪ m2e , we have α̂(k) ≈ αe , but α(µ) −→ 0 as mµe −→ 0. For µ < me , α(µ) differs qualitatively
from the physical coupling. Physically, me becomes important, but that is not tracked by the MS or
^
α(k)
1
137
-log(me)
-log(k)
Figure 2: α̂(k) as a function of log k1 . At the scale of k ∼ me , we have α̂(me ) ∼
2
1
137 .
Lecture 26
8.324 Relativistic Quantum Field Theory II
Fall 2010
MS schemes: α(µ) is no different for a theory with me = 0. It is more transparent to understand the
behaviour of α̂(k) using the Wilsonian approach. The coupling in the Wilsonian action, by definition,
should track α̂(k) closely.
Below the scale of me , the electron becomes heavy, and we can then integrate it out, leaving a pure
Maxwell theory, in which the coupling constant does not run.
For massless quantum electrodynamics, with me = 0, α(µ) is qualitatively correct for µ −→ 0. We then
find that αef f (k) −→ 0 as k −→ 0: The theory is marginally irrelevant.
3.
5.3.4: Beta-Function of Quantum Chromodynamics
The Lagrangian of quantum chromodynamics, with an SU (Nc ) gauge group and Nf quarks, where Nc = 3 and
Nf = 6, is given by
Nf
∑
1
L = − TrFµν F µν − i
ψ̄j (γ µ Dµ − mj ) ψj ,
(19)
4
j=1
with
Fµν
=
∂µ Aν − ∂ν Aµ − ig [Aµ , Aν ] ,
a a
Aµ = Aaµ ta , Fµν = Fµν
t , where ta are the generators of the fundamental representation of SU (Nc ). The covariant
derivative is given by
Dµ ψj = ∂µ ψj − igAµ ψj .
(20)
We redefine Aµ −→ g1 Aµ , and so the Lagrangian becomes
Nf
∑
1
µν
L = − 2 TrFµν F − i
ψ̄j (γ µ Dµ − mj ) ψj ,
4g
j=1
(21)
Dµ ψj = ∂µ ψj − iAµ ψj .
(22)
with
We will outline the computation of the β−function for the pure gauge theory using the language of the Wilsonian
approach, in the Euclidean case. Consider that the theory is provided with a cut off Λ, and bare coupling g = g(Λ).
We now write
′
Aµ = Aµ(Λ ) +A˜µ ,
(23)
(Λ′ )
where Aµ
and
is the part below the scale Λ′ , and A˜µ is the high-energy part, above the scale Λ′ . Then we have
[
]
[
]
′
)
(Λ′ )
SY M [Aµ ] = SY M A(Λ
+
S
A
,
Ã
,
(24)
1
µ
µ
µ
ˆ
DAµ e−S[Aµ ]
ˆ
=
ˆ
=
′
)
DA(Λ
µ e
′
)
DA(Λ
µ e
[
]
′)
−S A(�
µ
ˆ
Dõ e
[
]
′)
−S1 A(�
,õ
µ
[
]
[
]
′)
′)
−S A(�
−∆S A(�
µ
µ
.
We take the derivative expansion of ∆S,
∆S ≈ c log
Λ
Λ′
ˆ
d4 x FΛ2′ + . . . ,
(25)
giving
1
1
Λ
= 2
+ c′ log ′ ,
g 2 (Λ′ )
g (Λ)
Λ
3
(26)
Lecture 26
8.324 Relativistic Quantum Field Theory II
Fall 2010
1 22
where c is a pure g−independent number. A precise calculation gives c = − (4π)
3 NC . So, for the β−function, we
find
(
)
11
2
g3
βg = −
N
−
N
.
(27)
C
F
2
3
3
(4π)
Considering the fermionic sector, we have
ˆ
´ d
Dψ̄Dψ e− d x ψ̄(D/�′ −m)ψ+...
(
)
/ Λ′ − m
∝ det D
elog det(D/�′ −m) .
=
If we define αs =
g2
4π ,
we find
βα = −
α2
2π
(
11
2
NC − NF
3
3
)
= −bα2 ,
(28)
as we found in the gϕ3 theory. So, we have
1
1
µ
−
= b log ′ ,
′
αs (µ) αs (µ )
µ
and hence
αs (µ) =
αs (µ0 )
1 + αs (µ0 )b log
µ
µ0
.
(29)
(30)
The Landau pole occurs at
1
α(μ)
QED
QCD
log(μ)
Figure 3: α−1 scales linearly with log(µ) with a positive gradient in quantum chromodynamics, as with the gϕ3
theory.
αs (µ0 )b log
and so
ΛQCD = µ0 e
ΛQCD
= −1,
µ0
− bα
1
s (µ0 )
≈ 250MeV,
(31)
(32)
independently of our choice of µ0 . Finally, we put our coupling constant in the form
αs (µ) =
1
b log
µ
ΛQCD
.
(33)
Near ΛQCD , quantum chromodynamics becomes strongly coupled. The form
⟨ of this
⟩ coupling leads to many inter­
¯L UR ̸= 0.
esting phenomena, including confinement, and chiral symmetry breaking: U
4
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