Chapter 4
Renormalization
Cut off dependence in a few basic quantities this into redefinition:
1
L = − t̄aµν0 t¯0aµν + ψ¯0 (ip0 m0 )ψ0
4
1
Aren. = √ A0
ZA
1
ψren. = � ψ0
Zψ
ZTψ̄ψA gren. ψ¯ren. γ µ ψren. Aµren. = g0 ψ¯0 γ µ ψ0 Aµ0
(4.1)
(4.2)
(4.3)
(4.4)
1
2
g0 Zψ ZA
ZTψ̄ψA
= ψ¯0 ψ0 m0
m 0 Zψ
=
Zψ̄ψ
gren. =
Zψ̄ψ ψ¯ren. ψren. mren.
mren.
(4.5)
(4.6)
(4.7)
The wave-function renormalizations are absorbed into normalizing the 1–particle
states. The renormalized couplings are physical particles matrix of renormalized
states. Indeed, these are the parameters define the theory, at least perturbatively.
We fix these to experiment. Then, other quantities must be expressed finite terms
using them (and we remove the cut off).
Notice that in a gauge theory, we must use a gauge-invariant regulator to get the
simplicity. Otherwise, there are A2 (no derivatives res is like), AσA, AAA, A4 , etc.
coupling appearing. We may set them to zero and entree conspiracies to record gauge
symmetry (throwing their a gauge invariant regulation).
30
31
3
gren.
ZA2
=
g0 vector 3 − pt.
ΓA 3
ZA
= √
g0 vector 4 − pt. ∝ g 2
ΓA 4
(4.8)
(4.9)
1
ZA2 Zψ
=
Γψ̄ψA
..
.
= Z
(4.10)
(4.11)
Za
= universal (W ard identity)
Zν̄mA
(4.12)
In Abelian theory are equal 1. In N.A. theory, not generally (+ gauge dependent).
Figure 4.1: Abelian Theory.
¯ ψ
To clarify this structure, let’s work an example. Correlation function of Bare ψ,
(inverse propagator).
bare =
+
Figure 4.2: Bare.
bare = −i(p/ − m) +
�
Λ
(−i)(i)γµ (p/ − k/ + m)γ µ
d4 k
2
(ie)
(2π)4
k 2 [(p − k)2 − m2 ]
(4.13)
32
CHAPTER 4. RENORMALIZATION
k0
k0= i kE
Figure 4.3: Bare Diagram.
Renormalized:
bare =
+
Figure 4.4: Bare.
bare = −iZ(p/ − mR ) + e2i (no cut − of f )
(4.14)
k
bare =
+
p
p−k
Figure 4.5: Renormalized.
bare = −i(p/ − m) + ie2
�
Λ
�
S3 = 2π 2
bare = −i(p/ − m) +
ie2
8π 2
d4 k γµ (p/ − k/ + m)γ µ
(2π)4 k 2 [(p − k)2 − m2 ]
�
Λ
��
new Eulidec
dΩdkk 3
�
/ + 4m
−2(p/ − k)
k 2 [(p − k )2 − m2 ]
(4.15)
(4.16)
(4.17)
We are going to keep only the divergent part. We shall subtract by normalizing
the log divergence (only) at µ.
linear + log divergent
(4.18)
33
log
�
:
Λ
dΩdkk 3
−2p/ + 4m
k 2 [(p − k)2 − m2 ]
dΩdkk 3
[−2p/ + 4m]
k4
= ln Λ[−2p/ + 4m]
� Λ
−2k/
drdkk 3 2
linear :
k [(p − k)2 − m2 ]
� Λ
−2k/
dΩdkk 3 2
=
k − 2pk + p2 + m2
� Λ
−2k/
dΩdkk 3
=
2
2pk 2
2
k (1 − ( k ) + kp2 +
=
=
�
dΩk/kµ =
=
�
Λ
dΩdkk 3
�
�
�
Λ
Λ
dΩdkk 3
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
m2
k2
−2k/
2pk
k6
dΩkµ kν γν
1 2
γk
4
(4.24)
(4.25)
(4.26)
(4.27)
� Λ
p/
−2k/
2pk
=
dkk 3 4
6
k
k
= p/ ln Λ
(4.28)
(4.29)
Putting it all together (as discussed):
bare = −i(p/ − m0 ) − i
Λ
e2
ln [−2p/ + 4m0 + p/]
2
µ
γπ
= iZψ (p/ − mr )
(4.30)
(4.31)
2
Zψ = 1 −
e
Λ
ln
2
8π
µ
(4.32)
Finite matrix d� t� at |p| = µ
mR (µ) = m0 [1 − 3
Λ
e2
ln ]
2
8π
µ
(4.33)
Check Ward identify:
(−ie)3 γν (p/ − k/)γµ (p/ − k/)γν (i2 )(−i)
bare = −ieγµ +
k 2 ((k − p)2 )2
�
ie3 Λ
−2k/γµ k/
= −ieγµ − 2
dΩdkk 3
k6
8π
�
Λ
(4.34)
(4.35)
34
CHAPTER 4. RENORMALIZATION
µ
p−k
p−k
k
p
p
Figure 4.6: Ward Identity.
= −ieγµ − ie
3
�
dk −2 k 5
γα γµ γ α
2
6
8π 4 k
ie3
γµ ln Λ
8π 2
Λ
e2
= 1 + 2 ln
8π
µ
2
e
Λ
= 1 − 2 ln
8π
µ
= −ieγµ −
ZT−1
ψ̄ψA
ZTψψA
¯
(4.36)
(4.37)
(4.38)
(4.39)
Check gauge-invariance of mR (µ):
Figure 4.7: Added Stuff.
d4 k αkµ kν γµ (p/ − k/ + m)γν
added stuf f ∝
(2π)4 k 2
k 2 (p − k)2
need ∝ p/ − m
�
Λ
:
2a
Λ
(4.41)
3
dkk 1
1
( (−2)p/ + 4m)
6
k
4
4
� Λ
/
dΩdkkµ kν (−)(γµ k γν (2pk)k 3 )
1
linear :
8π 2
k8
� Λ
1
dΩdk(−)k/(2pk)k 3 )
=
8π 2
k6
� Λ
1
(−2)pk
1
/ 3
4
=
dk
− p/ + m
8π 2
k4
log
1
8π 2
�
(4.40)
Effective mass: m�R (µ1 ) equivalent to m2R (µ2 ) if same m0 , Λ
(4.42)
(4.43)
(4.44)
(4.45)
35
m�R (µ1 )
Λ
e2
1 − 3 8π
2 ln µ
1
=
m2R (µ2 )
Λ
e2
1 − 3 8π
2 ln µ
2
(4.46)
2
Λ
e
1 − 3 8π
2 ln µ
m2R (µ2 )
1
=
Λ
e2
m�R (µ1 )
ln
1 − 3 8π
2
µ2
!
= 1−3
(4.47)
e2
µ2
ln
2
8π
µ1
e2
mR (µ)
= 3 2
∂ ln µ
8π
(4.48)
(4.49)
Renormalization group: N.B. : e2 µ too.
Physical significance:
1. With normalization at µ, no large loop for processes with parameter a ∼ µ.
2. Internally, not rotate for µ to Ω.
Anticipate:
dg(µ)
= −b0 g 3 (b0 > 0)
d ln µ
d g12
= 2t0
dt
1
1
µ
− 2
= 2b0 ln
2
g (µ) g (µ0 )
µ0
1
1
µ
e 2b0 g2 (µ) = e 2b0 g2 (µ0 )
µ0
1
(4.50)
(4.51)
(4.52)
(4.53)
1
µ = [µ0 e 2b0 g2 (µ0 ) ]e 2b0 g2 (µ)
(4.54)
Take µ → ∞, dimensional transmutation perturbative coupling.
Lattice: Given g21(µ) , determine µ ∝ a1 . Measure M
fixed, independent of µ as
µ
g → 0.
Good news: Check approach to continuous limit. Bad news: Decreasing g(µ)
required exponentially bigger lattice.
With opposite sign for b0 , coupling blows up as µ → ∞(a → 0), no limiting theory
guaranteed.
36
CHAPTER 4. RENORMALIZATION
Running coupling and dimensional transmutation:
Figure 4.8: Running Coupling
dg(µ)
= −b0 g 3 (b0 > 0)
d ln µ
d g12
= 2b0
d ln µ
1
1
µ
= 2
+ 2b0 ln
2
g (µ)
g (µ0 )
µ0
(4.55)
(4.56)
(4.57)
Bare charge → 0
1
1
µ
= e 2b0 g2 (µ) e 2b0 g2 (µ0 )
µ0
1
(4.58)
1
µ0 = e 2b0 g2 (µ) µe 2b0 g2 (µ0 )
(4.59)
Identify from µ → ∞ limit [finite].
More accurate:
dg
= −b0 g 3 + b1 g 5
d ln µ
d g12
1
= 2b0 − 2b1 1
d ln µ
g2
du
2b1
= 2b0 −
dt
u
u0 = 2b0 t
du
b1
= 2b0 −
dt
b0 t
b1
u � 2b0 t − ln t
b0
(4.60)
(4.61)
(4.62)
(4.63)
(4.64)
(4.65)
37
1
g 2 (µ)
ln µ0
µ0
µ
µ
b1
− ln ln
µ 0 b0
µ0
1
µ
b1
− ln
� 2b0 ln
µ0 b0 2b0 g 2
1
1
b1
+ 2 ln
= ln µ −
2
2b0 g
2b0 2b0 g 2
b1
1
1
−
2
� µe 2b0 g2 (µ) [ 2 ] 2b0 f inite as µ → ∞
g (µ)
� 2b0 ln
(4.66)
(4.67)
(4.68)
(4.69)
Relation to lattice (fundamental): Hold µ0 fixed (reference mass)
b1
1
− 1 2 1 2b2 1
2b0 g
a(g) ∼ ∼ e
(4.70)
[ 2] 0
µ
g
µ0
Small a corresponds to small g. N.B.: Exponential dependence. Test: Physical
masses stay constant as g → 0. Explicitly: Measures correlation fall-off e−λn , where
n is the lattice sparcing.
This is e−M L with L = an for M = λa .
λ
e−λn = e− a an
→ µ0 e
(4.71)
b1
2b2
1
[ 12 ] 0
2
2b0 g
g
λ
(4.72)
Fixing µ0 , we see λ → 0 as g → 0. Large correlation lengths, variating of lattice
artifacts. Alternatively, aM → 0 without asymptotic freedom we would be lost here,
in tertudation theory. Long correlation lengths can occur at critical points. To nail
things down let’s look at this more closely:
dg
d ln µ
ln(g − gC )
g(µ) − gC
g(µ0 ) − gC
so we have, g(µ) → gC
∼
= −b0 (g − gC )
(4.73)
= −b0 ln µ
µ0
= ( )b0
µ
g(µ) − gC b1
)0
µ = µ0 (
g(µ0 ) − gC
(4.74)
λ
e−nλ = e− a na
λ
M =
a
(4.75)
(4.76)
(4.77)
(4.78)
38
CHAPTER 4. RENORMALIZATION
β (g)
Fixed Point
g
Figure 4.9: Fixed Point
a(g(µ)) ∼
1
1 g(µ) − gC b1
∼ (
)0
µ
µ0 g(µ0 ) − gC
(4.79)
M = λa fixed ⇒ λ → 0, M1 � a. Alternatively, IR fixed points, but these do not
support non-zero masses.
Antiscreening as paramagnetism:
Vacuum polarization:
E.D ∝ �Foi F oi
B.H ∝ µ−1 Fij F ij
(4.80)
(4.81)
�µ = 1
(4.82)
In reality
either is varying g.
Antiscreening (� < 1) = paramagnetism (µ > 1) magnetism is easier to think
about, since particles do not get produced.
Eclass. =
1 2
B
2g 2
(4.83)
But zero point energies, 12 hω (Bose), − 21 hω (Fermi) would like to throw it out,
but it is field-dependent and cut off dependent.
Interpretation using running coupling:
Wilson approach: Contribution of modes between µ0 + µ (say µ > µ0 ) is
39
= −n ln
µ 2
B
µ0
1
E =
2g 2 (µ
�
0)
��
(4.84)
B2
�
(4.85)
cut of f at µ0
= (
1
2g 2 (µ)
�
− n ln
��
cut of f at µ
n = b0
µ
) B2
µ0
�
(4.86)
(4.87)
Result and physical interpretation:
n=
1
1
[3{−2T (R 1 + 8T (R1 ))}]
[−{T (R0 ) − 2T (R 1 + 2T (R1 ))}] +
2
2
2
96π
96π 2
(4.88)
To aid in physical interpretation, pick a particular direction (e.g., τ3 ) in internal
space for B, and pretend it is electromagnetism.
First term from orbital diamagnetism (Lambda). Note 2 for polarizations, − for
fermion.
N.B. This applies to 0–point energy of virtual particles. Real particles do not have
this.
Second term from spin paramagnetism (Pauli). Note g–factor 2 (also for vectors),
− for fermion, 13 ratio as in solid state.
S2
=4
S 21
(4.89)
2
AF comes from gluon paramagnetism. Reference: F. W., RMP 71 S85 (Ceturean
Issue).
Learning from QED: With these insights, we could extract the non-Abelian β
function from the Abelian one.
Numerical value:
QCD : T (R
1 ) = f
2
1
2
T (R1 ) = 3
1
f
f
[+2 − 2 × 3 + 3(−2 + 8 × 3)]
2
96π
2
2
1
1
2
=
[66 − 4f ] =
[11 − f ]
2
2
96π
16π
3
(4.90)
(4.91)
b0 =
(4.92)
40
CHAPTER 4. RENORMALIZATION
b0 > 0 for f ≤ 16. Gluons heavily dominate for f = 3.
Application for unified theories:
SU (3)
−8F + 66
SU (2)
F(
(4.93)
−4
����
f ermion f actor
= −8F + 44
4
����
number of doublets
Higgs
⎛
×
1
) + 44
2
����
Tdoublet
(4.94)
⎞
2
⎜
⎜
⎜
normalized charge : ⎜
⎜
⎜
⎝
U (1)
�
1
2
×
2
2
−3
1
1
q 2 = F ( + (3 × 42 + 6(−1)2 + 1(−6)2 ))
2 60
3
W.S.B.
= 2F −→ −8F −
Higgs
10
−3
⎟
⎟
⎟ 1
⎟√
⎟
⎟ 60
⎠
(4.95)
(4.96)
(4.97)
3 equations: constraint, tu , gu
1
1
= bi tu + 2
2
gu
gi
1
1
− 2 = (b3 − b2 )tu
2
g2
g3
(4.98)
(4.99)
Note: unchanged by complete multiplets (no F )
1
g22
1
g12
−
−
1
g32
1
g22
=
b 3 − b2
b 2 − b1
(4.100)
Fixing scale: note exponential dependence
tu =
1
=
g
u2
1
g22
b3
−
1
g32
b2
−
1
−
b2 g 2
2
1
b2
−
=
1
b3 g32
1
b3
1
=
−
1
=
b3 g32
b3 g12 −
b2 g12
2
3
b 3 − b2
b3 gu2
1
b2 gu2
−
1
b2 g22
(4.101)
1 1
1
1
b3 + b2 1
1
= ( 2 + 2) + 3
( 2 − 2 ) (4.102)
2
2 g2 g 3
2 b − b g 2 g3
4.1. SUSY MODIFICATIONS
4.1
41
SUSY Modifications
1. Gauge Modification
2R 1 − 2R1
(4.103)
2
−6R 1 − 24R1
(4.104)
2
R
1
=
2
11
����
group theory
66 → 54
44 → 36
× ����
2 ×
chialitic
9
1
⇒ f actor
2
11
����
(4.105)
majorna
(4.106)
(4.107)
2. Higgs Modification
2 doublets
f ermion partner
−1 → 2 × (−1 + �2 −
6) like 6 Higgs doublet
�� �
(4.108)
(4.109)
(4.110)
× 21
3. Matter Modification
×
3
by same reasoning
2
(4.111)
Example: N = 4 gauge field. 4 adjoint fermions. 6 Higgs.
diamagnetism : 2 −8
paramagnetism : 8 −8
(4.112)
Numerical aspects:
SUSY:
1
:
b3 −b2
22 + 12
b 3 − b2
=
b 2 − b1
44 − 12 +
3
10
18 + 62
b 3 − b2
=
b 2 − b1
36 + 6(− 12
+
=
3
)
10
75
� 0.514
146
=
35
� 0.604
58
(4.113)
(4.114)
42
CHAPTER 4. RENORMALIZATION
1
1
→
1
18 + 62
22 + 2
0.044 → 0.048
(4.115)
(4.116)
Raising scale looks small but appears in exponential (p–decay).
SUSY:
100 + 16F −
b3 + b2
=
3
2
b −b
22 + 12
1
2 F =3
=
62 −
22 +
1
2
1
2
=
41
� 2.73
15
90 + 24F − 3 F =3 3
b3 + b2
� 0.60
=
=
22 + 3
5
b 3 − b2
(4.117)
(4.118)
b2 : (very little remaining)
44 − 8F −
1
1
= 19 → 36 − 12F − 3 = −3
2
2
(4.119)
Other applications:
mb
→ 3 f rom minimal SU (5)
mτ
Vacuum instability and Colman–Weinberg:
ϕ
ϕ4 λ( )
µ
(4.120)
(4.121)
interpret as energy from extra nodes.
Too many fermions. moment µ, field strength ϕ (as in our magnetic calculation).
Weak Coupling Massless
Figure 4.10: Weak Coupling Massless
4.1. SUSY MODIFICATIONS
43
Table 4.1: Running of coupling
1
g2
2
1
g2
1
SU (3)b33
SU (2)b23
U (1)b10
1
g2
3
1
− 2
g
2
−
=
tu =
1
2
gu
=
b3 −b2
b2 −b1
1
− 12
g2
g
2
3
3
b −b2
1
b2 g 2
2
1
b2
1
b3 g 2
3
1
− 3
b
SM
66 − 8F
44 − 8F − 21
3
−8F − 10
M SSM
54 − 12F
36 − 12F − 3
−12F − 59
nearly flat
0.514
0.604
agrees with expectation
0.044
0.048
larger scale
important for p-decay
close to Planck scale
2.73
0.60
families and extra stuff
−
appear here
bigger coupling desirable?
Running of coupling – summary sheet
Other applications: N = 4 SUSY, glue, 4 Majorna fermions, 6 scalars. Dia: count
DOF. Para: 4 fermion, β = 0
Normal SU(5):
mb
= 1 at unif ication
mτ
(4.122)
mb
�3
mτ
(4.123)
Normalizes to
Vacuum instability
λ(
µ 4
ϕ
)ϕ negates (AF ) contribution → λ( )ϕ4 f rom f ermions
µ0
µ0
(4.124)
Variational interpretation?
< ϕ >← a+
k , ak
(4.125)
mix mode in |k|–dependent manner. C.F. our magnetic field calculation.
Colman-Weinberg:
Electroweak breaking through leavry sTop. t instability ulicrod when t̂ comes in.
PQCD:
Simplest case: e+ e− → hadrons
44
CHAPTER 4. RENORMALIZATION
Figure 4.11: Vacuum Instability
V(φ )
Weak Coupling
m=0
φ
pass through zero
Figure 4.12: Spontanous Symmetry Breaking
σ(p, g (µ) , µ)
= σ(λp, g(λµ), λµ)
σ(λp, g (µ) , µ)
1
=
σ(p, g(λµ), µ)
��
�
λ2 �
(4.126)
(4.127)
expand perturbutively
(
+
)
2
Figure 4.13: Expand Perturbatively.
Potential problem from soft and collisions divergence. They do not occur in in­
clusive qualities.
Jet phenomena: Observation, heuristic explanation, hard scattering, rate. StermanWeinberg comes, energy bite. Stiull no scale – same argument. Extremum pattern.
Hadmic gets over 6 nodes of reguitode.
4.1. SUSY MODIFICATIONS
45
Figure 4.14: W–Exchange.
OPE, DIS: W–exchange.
Nodes up to k 2 ∼ m2ω .
Simplify to
+
Different Q #s
Figure 4.15: Simplify.
Operator versions:
X small x �
) −→
Ci (x)Oi (o)
(4.128)
2
Singularities of C ordered by dimensions of Oi keep low dimensions. Ci (x, g, µ)
includes renormalization of O.
J(x)J(
Ci (x, g, µ) = [µ
−1
∂
∂
+ β(g) − γθ (λ)]Ci = 0
∂g
∂µ
−
Ci (λ x, g(µ0 ), µ0 ) = Ci (x, g(λµ0 ))e
� g1
g0
(4.129)
γθ dg
β
ḡ − c0
= Ci (x, g(λµ0 ))( ) b0
g
(4.130)
One can also have operators mixing. For DIS, light–cone singularities (tracing
quarks and glouns) are important.
x
x x2 →0 � (i) µνα1 ···αn
(0)xα1 ···xαn
J µ ( )J ν (− ) ≈
C(x) O
2
2
but not individual component.
(4.131)
46
CHAPTER 4. RENORMALIZATION
Twist ≡ dimension – spin
< O > → pµ pν · · · (symmetric)
∂
g
x →
or
∂g 2
g2
(4.132)
(4.133)
)2 is what appears.
( pq
q
2
Bj limit:
g2 → ∞
2pq
q2
(4.134)
finite (≡ x1 ).
QCD has twist–tor families of operators (respectively chirality).
↔
↔
g (L) γα1 �α2 · · · �αn q(L)
↔
↔
Gaµα1 �α2 · · · �αn−1 Gaαn µ
D = 3 + n − 1, J = n
(4.135)
D = 4 + n − 2, J = n
(4.136)
First family includes nonsinglets, currents. Tµν also appears (but not by itself).
Twist J governs J th moment of appropriate structure function.
Many testable predictions: Parton model sum rules and g 2 correlations. CG
relation and g 2 correlations. E: Max with oth order correlations. Bj scalar deviations.