Download Seminar Quantum Field Theory - Institut für Theoretische Physik III

Seminar
Quantum Field Theory
Institut für Theoretische Physik III
Universität Stuttgart
2013
Contents
Renormalization
1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Superficial Degree of Divergence and Renormalizability . .
2.1
Renormalizability of QED . . . . . . . . . . . . . . . . .
2.2
Renormalizability of the φn -Theory . . . . . . . . . . . .
3
Regularization and Renormalization: Bare Perturbation Theory
3.1
The Four-Point-Function: Pauli-Villars Regularization .
3.2
The Two-Point-Function: Dimensional Regularization . .
4
Dresses Perturbation Theory . . . . . . . . . . . . . . . . . . . .
5
The Renormalization Group . . . . . . . . . . . . . . . . . . . .
5.1
Renormalization Group of the Ising Model . . . . . . . .
5.2
Renormalization-Group in QFT: Wilsons Approach . . .
Bibliography
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Chapter 0
Renormalization
1 Motivation
When we calculate scattering amplitudes at higher orders in perturbation theory it
turns out that the Feynman diagrams contain divergent integrals. In this chapter
the method called Regularization is presented which makes these integrals finit by
introducing a high momentum cutoff without violating gauge and Lorentz invariance. This high momentum cutoff is somehow arbitrary while physical predictions
like scattering amplitudes or correlation functions must be cutoff-independent.
The procedure to make physical predictions cutoff-independent again after Regularization is called Renormalization.
2 The Superficial Degree of Divergence and
Renormalizability
2.1 Renormalizability of QED
When we want to calculate scattering amplitudes at higher order in perturbation theory we have to compute a sum of Feynman diagrams. In a Feynman
diagram each loop corresponds to a potentially divergent integral over the whole
momentum space.
v
Renormalization
∝
Z
d4 k1 d4 k2 · · · d4 kL
(k/i − m) · · · (kj2 ) · · · (kn2 )
(1)
The internal lines of these loops correspond to propagators wich are part of the
integrand. In QED the electron propagator carries the momentum to the power
of one in the denominator, the photon propagator even carries the momentum to
the power of two in the denominator. These propagators weaken the strength or
even compensate the divergence which occurs because of the integration.
Since the divergences occur due to the integration over the high momentum degrees of freedom the divergences are called UV-divergences. The divergent integrals can be cured by introducing a high momentum cutoff as upper integration
boundary. Such a cutoff can be used to measure the strength of the divergence.
Simple examples of one dimensional integrals can illustrate this procedure:
Z
Z
Z Λ
dk
−→
k
Z Λ
dk k −→
dk
∝
k
log Λ
(2)
dk k ∝
Λ2
(3)
We call the first integral logarithmically divergent, the second integral quadratically divergent. The one dimensional integrals can also justify the following
definition of the ”Superficial Degree of Divergence” (SDD):
D ≡ (power of k in numerator) − (power of k in denominator)
= 4L − Pe − 2Pγ
(4)
(5)
As an example we calculate the SDD of an integral occurring as a so called radiative correction in QED, the fermion self-energy: In this Feynman diagram (fig. 1
the loop corresponds to the integral:
2
− iΣ2 (p) = (−ie)
vi
Z
Z 4
d4 k
i
dk
µ −igµν ν
γ 2
γ ∝
4
2π p/ − k/ − m + i k + i
k3
(6)
2 The Superficial Degree of Divergence and Renormalizability
Figure 1: The fermion self-energy
For the SDD we obtain:
D =4−3=1
(7)
In QED the overall SDD of a Feynman diagram can be calculated just by counting
the number of loops (L), the number of internal fermion propagators (Pe ) and the
the number of internal photon propagators (Pγ ). The SDD turns out to be
D = 4L − Pe − 2Pγ .
(8)
Although it seems sensible to assume that the cutoff-dependence of a Feynman
diagam could be calculated in the manner
Z Λ
d4 k1 d4 k2 · · · d4 kL
∝ ΛD
(k/i − m) · · · (kj2 ) · · · (kn2 )
(9)
it turns out that this assumption is often wrong. In fig. 2 some simple QED diagrams illustrate this difference between the SDD and the actual divergent behavior
of the diagrams.
Nevertheless the SDD turns out to be a useful tool in order to estimate whether a
quantum field theory is renormalizable or not or in other words whether we are able
to calculate scattering amplitudes also at higher orders in perturbation theory or
not. In order to determine the renormalizability of a QFT it is necessary to express
the SDD not in dependence of the number of loops and internal propagators like in
eq. 5, but in dependence of the external lines and the number of vertices. Since in
section 3.2 it will be necessary to do calculations not only in our fourdimensional
spacetime, we will do this replacement in an arbitrary dimension of spacetime d.
The following abbreviations will be used to determine the SDD in QED:
vii
Renormalization
Figure 2: QED diagrams which illustrate the difference between the SDD (middle
column) and the actual divergent behavior (cutoff-dependence, right column): In the first diagram is D = 0 but the diagram is finite since there
isn’t an integration to compute. In the third diagram the divergence
is weakened due to the ward identity so it is logarithmically divergent
although it is D = 2. In the forth diagram D is negative which could
naively considered to be a hint that the diagram is finite, but it contains
a divergent sub-diagram making it logarithmically divergent. Just in the
second and the fifth diagram the SDD match with the actual degree of
divergence. Source [1].
viii
2 The Superficial Degree of Divergence and Renormalizability
D = superficial degree of divergence
d = dimension of spacetime
Pe , Pγ = number of electron (photon) propagators
Ne , Nγ = number of external electron (photon) lines
V = number of vertices
In a D-dimensional spacetime we have to take d times the number of loops in eq.
5:
D = dL − Pe − 2Pγ
(10)
In the original Feynman rules each propagator has an integral. Each Vertex
however carrys a delta function for momentum conservation which reduces the
number of integrations. Since one of these delta functions enforces the overall
momentum conservation, there remain a single integration for each loop. This
consideration leads to the expression
L = Pe + Pγ − V + 1.
(11)
On the other hand we know that at each there meet a single photon line and two
electron lines. Therefore we are able to express the number of vertices with the
number of lines:
1
V = 2Pγ + Nγ = (2Pe + Ne )
2
(12)
The number of the propagators counts twice in these equations since they connect
two vertices. Now we can use eq.11 and 12 in order to eliminate L,Pe and Pγ in
eq.10. We find an expression for the SDD:
!
!
!
d−4
d−2
d−1
D =d+
V −
Nγ −
Ne
2
2
2
(13)
Now consider a QED in less than four dimensions. Than the factor in front of the
number of vertices in eq. 13 becomes negative. Remember that the number of
vertices correspond to the order in perturbation theory. So it turns out that in this
ix
Renormalization
case at sufficiently high order in perturbation theory the additional Feynman diagrams of the scattering amplitudes obtain a smaller and smaller (finally negative)
SDD. In this case we can be sure that at sufficiently high order in perturbation
theory there won’t occur any divergent diagrams anymore. Such a QFT we call a
Super-Renormalizable Theory.
In the case of a fourdimensional spacetime the dependence on V vanishes completely in eq. 13 and we obtain:
3
D = 4 − 2Nγ − Ne
2
(14)
Since there is no dependence on the number of vertices in this case a divergent
amplitude stays divergent also at higher orders in perturbation theory. But fortunately due to the dependence on the number of external legs there should be a
finite number of divergent scattering amplitudes. Such a QFT we call a Renormalizable Theory.
Now consider the case of a spacetime with more than four dimensions. Then
there occurs a positive factor in front of V in eq. 13. This means that at higher
orders in perturbation theory the SDD of the additional diagrams of all scattering amplitudes becomes larger and larger. In such a case we can be sure that
finally every scattering amplitude becomes divergent at a sufficiently high order
in perturbation theory. We can’t cope with such a situation with the methods of
regularization and renormalization. Such a theory is called a Non-Renormalizable
Theory.
These definitions of Renormalizability should be summarized:
Super-Renormalizable theory: Only a finite number of Feynman diagrams superficially diverge.
Renormalizable theory: Only a finite number of amplitudes superficially diverge,
but divergence occure in all orders in perturbation theory.
Non-Renormalizable theory: All amplitudes are divergent at a sufficiently high
order in perturbation theory.
As explained in a fourdimensional spacetime QED turns out to be renormalizable.
Since the SDD is independent of the number of vertices, there is a finite number of
x
2 The Superficial Degree of Divergence and Renormalizability
Figure 3: The seven superficially divergent amplitudes in fourdimensional QED.
Diagram (a) describes an unobservable shift in the vacuum energy. This
diagram is irrelevant to scattering processes, diagrams (b) and (d) vanish
because of symmetries, in diagram (e) (photon scattering) the divergent
parts cancel due to the Ward-identity, the diagrams (c), (f) and (g) turn
out to be logarithmically divergent (even though D¿0 for (c) and (f)).
The latter three amplitudes have to be regularized and renormalized in
order to compute QED-scattering processes in arbitrary high order of
perturbation theory. Source [1].
superficially divergent amplitudes. Fig. 3 shows these seven superficially divergent
amplitudes as well as the three amplitudes which turn out to be actually divergent.
2.2 Renormalizability of the φn -Theory
The computations of regularization and renormalization in QED are very extensive. Since this complexity in calculations distract the effort to understand the
basic ideas of the renormalization theory this chapter will just deal with the renormalization of the scalar φ4 -theory (For Renormalization in QED please read [2]).
But at first we investigate the renormalizability of a scalar φn -theory in a ddimensional spacetime. The Lagrangian density of such a QFT is
L=
λ0
1
1
(∂µ φ)2 − m20 φ2 − φn
2
2
n!
(15)
In the scalar φn -theory just two Feynman rules, a propagator corresponding to
the internal lines and a coupling constant (λ0 ) corresponding to the vertices, have
xi
Renormalization
to be taken into account in order to calculate the loop integrals:
F
de
ed
p
=
p2
i
− m20 + i
= iλ0
The vertex with for legs occurs in this manner in the φ4 -theory, in a φn -theory this
vertex would carry accordingly n legs. In order to calculate the SDD we introduce
the following abbreviations:
d = dimension of spacetimeP
= number of internal lines/propagators
(16)
L = number of loops
(17)
N = number of external legs
(18)
V = number of vertices
(19)
Since the propagator contains the momentum squared in the denominator the
SDD is calculated with the equation
D = dL − 2P.
(20)
Again we can replace the number of propagators with the number of vertices and
external legs:
L=P −V +1
nV = N + 2P
(21)
(22)
The former equation follows the considerations we did when we calculated the
SDD in QED. The latter equation is valid because at each vertex of a φn -theory
there meet n lines. In this context the number of propagators count twice since
they connect two vertices. Replacing P and L in eq. 20 with the results of eq. 21
and 22 we obtain
xii
2 The Superficial Degree of Divergence and Renormalizability
"
!
#
!
d−2
d−2
D =d+ n
−d V −
N
2
2
(23)
We are especially interested in the case of a fourdimensional spacetime. It turns
out that in this dimension n < 4 would produce a negative factor in front of
V (number of vertices). With the considerations we discussed in section 2.1, we
identify this case with a super-renormalizable theory. In the case d = 4 and n = 4
the number of vertices vanish completely in eq.23. This theory therefore turns
out to be renormalizable. For d = 4 and n > 4 however the factor in front of
V becomes positive and we end up with a non-renormalizable theory. This is
exactly the reason why we are always talking about the φ4 -theory. This theory
is interesting enough (n ≤ 3-theories describe rather boring interactions) while it
stays renormalizable.
Nevertheless we should keep in mind that the φ4 -theory becomes super renormalizable in less than four dimensions. This means that our loop-integrals could
become finite if we calculate them in less then four dimensions. This fact turns out
to be useful for the certain procedure of regularisation (dimensional regularization
sec. 3.2).
Now I want to discuss an other approach to the question whether a QFT is
renormalizable or not. In the system of unity which is usually used in QFT it
is c = ~ = 1. Therefore every unity can be expressed in the unity of the mass.
For example the energy has the dimension [E] = [mass]. We will count now the
dimension of the unities. Therefore we define the masspower [massd ] ≡ d. For
1
≡ −1 and therefore [dxd ] ≡ −d. Since the
the length x we obtain [x] = [mass]
action has to be dimensionless the dimension of the Lagrangian density is [L] ≡ d.
The kinetic term of the Lagrangian density furthermore tells us that [φ] ≡ d−2
.
2
The coupling term in the Lagrangian density has to have the same dimension as
the Lagrangian density: [λφn ] ≡ d. Now we can calculate the dimension of the
coupling constant:
d−2
[λ] = d −
2
!
(24)
It turns out that the dimension of the coupling constant is exactly the factor
which occurs in eq. 23 in front of the number of vertices. A similar calculation
can be carried out also in other QFTs. Therefore we are able to formulate an
xiii
Renormalization
other possibility to estimate the renormalizability of a QFT:
Coupling constant with positive mass dimension [λ] > 0:
Super-Renormalizabile Quantum Field Theory
Dimensionless coupling constant [λ] = 0: The theory can either be renormalizable or not. In the case of the φ < n-theory we end up with a renormalizable
theory
Coupling constant with negative mass dimension [λ] < 0:
Non-Renormalizabile Quantum Field Theory
In QED the dimension of the coupling constant is the dimension of the electric
charge. In our chosen system of unity it is [e] = 0 and therefore we end up with a
renormalizable QED. Now have a look on a quantum field theory of Gravitation.
The coupling constant of such a theory would be λgrav = Gm. With G the
Gravitation Constant with the dimension [G] = −2. We end up with a negative
dimension of the coupling constant [Gm] = −1. Unfortunately a quantum field
theory of gravitation turns out to be non-renormalizable.
In order to give a more physical argument for the fact that [λ] < 0 leads to a nonrenormalizable theory consider a perturbation series of a scattering amplitude. In
higher orders higher powers of the coupling constant occur: Since we have to keep
the right overall dimension of the sum there has to occur a factor in each addend
which compensates the dimension of the coupling constant. This factor can just
be built with the cutoff of the integration which has the dimension [Λ] = 1. In the
QFT of gravitation a perturbation series of a scattering-amplitude would show
the following behavior.
M ∝ λgrav + λ2grav Λ · · · + λ3grav Λ2 · · · + λ4grav Λ3 · · ·
(25)
In higher orders the cutoff-dependence become worse and worse. Such a cutoffdependence can’t be renormalized.
xiv
3 Regularization and Renormalization: Bare Perturbation Theory
vacuum energy shift
∝ Λ2 + p2 log Λ + finite terms
∝ log Λ + finite terms
Figure 4: The divergent amplitudes in φ4 -theory and there actual dependence on
a high momentum cutoff. Just the two- and four-point-function need
to be regularized and renormalized since the vacuum energy shift is
unobservable
3 Regularization and Renormalization: Bare
Perturbation Theory
In this section the whole procedure of regularization and renormalization will be
carried out in the φ4 -theory. As discussed in the previous section this theory is
renormalizable and therefore there exist a finite number of divergent amplitudes
in this theory and these amplitudes are divergent in all orders of the perturbation
series. The three actually occurring divergent amplitudes are shown in fig. 4.
Since one of these amplitudes describe an unobservable shift in the vacuum energy
we end up with just two amplitudes which need to be regularized and renormalized:
the two-point-function and the four-point function.
There are two equivalent methods of Renormalization, called ”Bare Perturbation
Theory” and ”Dressed Perturbation Theory” This section deals with the bare
perturbation theory.
3.1 The Four-Point-Function: Pauli-Villars Regularization
There are several equivalent methods to regularize a divergent integral which
means an introduction of a cutoff which makes the integral finite. In order
to regularize the four-point-function we will use a method called ”Pauli-VillarsRegularization”.
xv
Renormalization
In second order perturbation theory the four-point-function which describes the
scattering process of two scalar particles is a sum of four Feynman-diagrams.
≈
The three diagrams containing a loop turn out to be divergent. The first diagram
correspond to the following integral:
(−iλ)2 Z d4 k
i
i
≈
4
2
2
2
(2π) k − m (k + p)2 − m2
(26)
But before we start the regularization we introduce a new system of coordinates
which simplifies the calculations: The Mandelstam-Coordinates.
s = p2 = (p1 + p2 )2
(27)
t = (p1 − p3 )2
(28)
u = (p1 − p4 )2
(29)
The Mandelstam coordinate s describes the center of mass energy. The coordinates t and u describe the scattering angles.
In order to calculate eq.26 we do the Wick-rotation in order to end up in a Euclidean space in which we can easily introduce spherical coordinates. After the
Wick-rotation the integral becomes
M2 =
(−iλ)2 2 Z d4 k
1
1
i
4
2
2
2
(2π) k − m (p − k)2 − m2
(30)
Now we use the so called Feynman-trick to transform our product in the denominator of the integrand into a sum:
Z 1
1
1
=
dα
xy
(αx + (1 − α)y)2
0
(31)
Using this equation, shifting our integration variable k → k + pα and introducing
the abbreviation c2 = m2 − α(1 − α)p2 we end up with the integral
xvi
3 Regularization and Renormalization: Bare Perturbation Theory
Z 1
0
dα
Z
d4 k
1
(2π)4 (k 2 − c2 + i)2
(32)
So far we just simplified the integral, we haven’t regularized anything. This integral is still divergent. Now we come to the important step called regularization. The integral will be made finite by introducing a cutoff. In Pauli-VillarsRegularization this cutoff occurs as a high momentum cutoff. Since the integral is
divergent because of the integration over the high momentum degrees of freedom,
the physical interpretation of what we actually do by regularizing an integral is
quite obvious when we use the Pauli-Villars-method: We ignore the high momentum degrees of freedom. Nevertheless in the most QFTs we are not allowed
two introduce the cutoff as an upper integration bound as we have done it in the
simple onedimensional examples eq. 3. This is due to the fact that such a procedure would violate gauge invariance. The method of Pauli-Villars-Regularization
introduces the high-momentum cutoff Λ in an additional addend in the integrand
(we ignore the α-integration for a moment):
Z
"
#
1
1
d4 k
− 2
≡ R(Λ2 , c2 )
4
2
2
2
2
2
(2π) (k − c + i)
(k − Λ + i)
(33)
For Λ going to infinity we obtain again our divergent integral. For small k 2 the
second addend can be neglected since Λ is of a huge value. On the other hand
for k 2 even much huger than Λ the two addends cancel each other. All in all the
second addend make the integrand vanish slowly when we go to high momentum.
Calculating the derivation of eq. 33 one time after c2 and one time after Λ2 we
end up with integrals which are listed in integration tables:
∂R Z d4 k
−2
i
=
=−
2
4
2
2
3
∂c
(2π) (k − c + i)
16π 2 c2
∂R
i
=
2
∂Λ
16π 2 Λ2
!
i
Λ2
R=
log 2
16π 2
c
!
2 Z 1
iλ
Λ2
M2 =
dα log
32π 2 0
m2 − α(1 − α)p2 + i
(34)
(35)
(36)
(37)
xvii
Renormalization
Λ2
= iCλ log
s
2
!
= (−iλ)2 iV (s, Λ)
(38)
In the last step when we introduced the function V (s, Λ) as an abbreviation for the
(now finite) integral we remembered that p2 is nothing else than the Mandelstam
coordinate s. For the other two loop-integrals contributing to the second order
in perturbation series we obtain exactly the same result except that we have to
replace s with the other Mandelstam-coordinates u and t. The whole scattering
amplitude in second order perturbation theory becomes:
iM = −iλ0 + (−iλ0 )2 [iV (s, Λ) + iV (t, Λ) + iV (u, Λ)]
(39)
Renormalization of the four-point-function
By introducing a high momentum cutoff without violating neither the gauge invariance nor the Lorentz invariance we obtained a finite scattering amplitude for
the four-point-function. This amplitude depends on a high but yet arbitrary cutoff Λ. Physical predictions should of course be independent of this cutoff. The
procedure used to make the physical prediction cutoff independent is called renormalization.
Let’s have a look on equation 39. As explained the cutoff Λ is not well defined.
Fortunately there is another constant which is so far not well defined: the constant λ0 is a constant which was introduces in the noninteracting theory. The
coupling constant shall be a measure for the strength of the interaction between
two particles. The value of the coupling constant therefore must be determined
in an experiment. A suitable experiment for the determination would be a scattering process between two particles with certain initial conditions. The coupling
constant can be nothing else than the scattering amplitude we measure with these
initial conditions. In short: before we can calculate anything we have to determine our coupling constant in a scattering experiment. Therefore we have to
choose some arbitrary initial conditions which means a certain initial set of Mandelstam coordinates s0 , u0 and t0 . This set of Mandelstam coordinates is called
the renormalization point. The value of the coupling constant depends on the
renormalization point. With this considerations it turns out, that λ0 isn’t the
physical coupling constant, which is from now on called λp :
xviii
3 Regularization and Renormalization: Bare Perturbation Theory
− iλp = −iλ0 + (−iλ0 )2 [iV (s0 , Λ) + iV (t0 , Λ) + iV (u0 , Λ)]
(40)
This equation we change after λ0 and take just the terms of first and second order
in λP into account (we want to calculate the second order in perturbation series).
− iλ0 = −iλP + (−iλP )2 [iV (s0 , Λ) + iV (t0 , Λ) + iV (u0 , Λ)] = −iλp Zλ (Λ) (41)
Equation 39 isn’t a physical equation because it contains an unphysical constant
Λ. Since λ0 , the so called bare coupling constant, wasn’t well defined it can also
be regarded as an unphysical constant. So far we have outsourced these two
unphysical constants into an unphysical equation 39. Although this equation is
not physical it redefines λ0 . If we use this equation in order to replace λ0 in eq.
39 we end up with a scattering amplitude which contains neither Λ nor λ0 but
the physical coupling constant λP defined in a scattering experiment:
M = −iλP +
iCλ2P
s0
t0
u0
+ log
+ log
log
s
t
u
(42)
Since we are calculating scattering amplitudes in perturbation series the predictions of equation 42 are just valid if s, u and t do not differ too much from
the renormalization point s0 , t0 , u0 we used to define the value of the coupling
constant. For low energy scattering processes a suitable renormalization point is
t0 = u0 = 0 (the four external legs are of equal length) and s0 = 4m2 (center of
mass in in rest).
3.2 The Two-Point-Function: Dimensional Regularization
In first order perturbation theory there occurs a correction to the propagator:
fpf F
=
k
=
+
+
F
k
i
i
2
+
(−iΣ(k
))
+
k 2 − m2 + i
k 2 − m2 + i
(43)
The loop corresponds to a divergent integral. As explained in the previous sections
the φ4 -theory is super-renormalizable in less than four dimensions. That’s why we
xix
Renormalization
can calculate the integral in a spacetime-dimension d < 4 in which it isn’t divergent
at all. This consideration leads to a procedure called dimensional regularization:
λ0 Z dd p
1
− iΣ(p ) = −i
d
2
2
(2π) p − m20 + i
2
(44)
After a Wick-rotation we obtain a Euclidean metric and are able to introduce
spherical coordinates. The integral we want to calculate is of the form:
I=
Z
Z
Z
dd p
pd−1
=
dΩ
dp
d−1
p2 − m2
p2 − m2
(45)
Without going through the calculations (see also [3]) in detail I present the result
for equation 44:
iλ0 m20
−iλ0 m20 Γ(1 − (d/2))
≈
− iΣ(p ) =
32π 2
16π 2
2
1
+ ...
4−d
(46)
So far d was the dimension of spacetime and therefore a natural value. But now
we end up with an equation which is not defined just for natural values but for all
real values d < 4. For d = 4 we obtain again an infinite result. In this procedure
of dimensional regularization we introduced our cutoff by choosing a real-valued
spacetime dimension very close to four.
Since the physical meaning of the Pauli-Villars-cutoff is much more obvious from
now on we consider the two point function also to be Pauli-Villars-regularized.
If we do the two-point-calculation more in detail taking also repeated loops into
account we end up with the expression:
iZφ (Λ)
−
+ Σ(p2 , Λ)) + i
(47)
In this equation Σ(p2 , Λ) and Zφ (Λ) are cutoff dependent constants which diverge
when Λ goes to infinity.
= i∆0 (p) =
p2
(m20
Renormalization of the two-point function
After the regularization we are again confronted with a cutoff-dependent object.
To renormalize this object we have to redefine a constant which was so far not well
defined: the bare mass m0 . In order to introduce the physical mass of the particle
mP in the calculation we have again to choose something like the renormalization
xx
4 Dresses Perturbation Theory
point. In QFT the mass of a particle is defined in its propagator. The pole of the
propagator has to occure at the value of the particle mass. This physical mass
can be determined in an experiment. Therefore we can identify:
m20 = m2p − Σ(m2 , Λ)
(48)
The cutoff-dependent constant Zφ (Λ) is determined by choosing the residue of the
pole to be equal one. In order to achieve that the propagator appears completely
invariant after renormalization Zφ (Λ) is usually referred to the vertices or to the
field functions in the case of the external legs (field functions are not observable).
We end up with renormalized field-functions, a renormalized mass, a renormalized
coupling constant and an invariant propagator:
m20 = m2p − Σ(m2 , Λ)
i
∆0 (p) = 2
p − m2P + i
iZλ (Λ)λp
iλ0 = −
(Zφ (Λ))2
φ0 =
q
Zφ (Λ)φP = ZφP
(49)
(50)
(51)
(52)
If we replace with these equations the bare constants and the bare fields in the
regularized integrals in second order perturbation theory, all scattering amplitudes become finite and cutoff-independent. In many QFTs the cutoff-dependent
constants (Zφ , Zλ , Σ) are not independent from each other. In the φ4 -theory for
example it turns out that Zφ = Zλ = Z
4 Dresses Perturbation Theory
So far we started the calculations with the bare constant λ0 , φ0 and m0 of the
noninteracting theory and redefined them in the procedure of renormalization.
This method of renormalization is called bare perturbation theory. It is also
possible to start with the physical constants λP , mP and φP . For this method
called dressed perturbation theory we have to introduce the physical constants
in the Lagrangian density. We replace φ0 with equation 52 and introduce the so
xxi
Renormalization
called counterterms:
δZ = Z − 1
(53)
δm = m20 Z − m2p
(54)
δλ = λ0 Z 2 − λP
(55)
Now the Lagrangian density depends on physical constants..
L=
δλ
1
λP 4 1
1
1
(∂µ φP )2 − m2P φ2P −
φP + δZ (∂µ φP )2 − δm φ2P − φ4P
2
2
4!
2
2
4!
(56)
...but contains counterterms. These counterterms lead to new elements in the
Feynman diagrams:
Now λ and m stand for the physical constants. Now the two particle scattering
process in second order perturbation theory contains an additional diagram:
≈
+
Of course the loop integrals are still divergent and we have regularize them. Now
we obtain
iM = −iλP + (−iλP )2 [iV (s, Λ) + iV (t, Λ) + iV (u, Λ)] + iδZ .
(57)
Again we choose a renormalization point at which we want the scattering ampli-
xxii
5 The Renormalization Group
tude to be equal the coupling constant (iM = −iλP ). Again a natural choice
for the renormalization point would be s = 4m2 , t = u = 0. In order to achieve
iM = −iλP the counterterm must be
h
i
δZ (Λ) = −λ2P V (4m2 , Λ) + 2V (0, Λ)
(58)
Now the cutoff-dependence of δZ cancels the other cutoff-dependent parts in eq.
57 and we obtain again the finite cutoff-independent scattering amplitudes for
arbitrary values of s, u and t (eq. 42.
5 The Renormalization Group
When we calculate scattering amplitudes in QFT we are working in Fourier space.
A high momentum cutoff in the integration therefore ignores the high momentum
Fourier components. In position space these components cause fluctuations of
the field amplitudes over small distances. Therefore the high momentum cutoff
smoothen the field functions. The small distance fluctuations occur due to the
Heisenberg relation ∆E∆t ∝ ~ which allows a violation of the energy conservation
over small periods of time resulting in a continuous production and destruction
of virtual particles in the vacuum. All in all the cutoff ignores quantum processes
occurring on small length- and timescales at high energy regions. The idea to
ignore the high energy region or the atomic length scale is useful in many physical
theories in order to simplify the calculations (just consider classical mechanics),
however in QFT the situation is slightly different: In QFT we are not just allowed
to ignore the high energy processes we even must ignore them to be able to
calculate any physical prediction. The reason must be that at high energies,
on small length- and timescales the so far developed QFTs loose their validity
and must be replaced by a more comprehensive theory (maybe string theory or
quantum loop theories). Theories in which essential interactions take place on
small length scales therefore must turn out to be non-renormalizable. A deeper
insight into the nature of the cutoff and in what we are doing when we choose our
renormalization point gives the so called Renormalization-Group-Theory.
xxiii
Renormalization
5.1 Renormalization Group of the Ising Model
Renormalization is a procedure not only useful in QFTs but also in other parts
of physics for example in condensed matter physics. In order to illustrate the
idea of the renormalization group I will introduce the concept using the example
of the Ising model. Consider a 2D-lattice of spin- 12 -particles in which the spin
can just choose between two directions: up or down. An interaction between the
spins occurs just between the next neighbors. Nevertheless there will of course
occur a correlation also between spins separated over larger distances. From a
macroscopic point of view the spin system appears as a continuum and also the
correlation function can be treated as depending continuously on the position.
Nevertheless the correlation function contains a natural cutoff in position space:
It does not make sense to study correlations on length scales smaller than the
lattice constant. If we are not interested in spin fluctuations on small distances
but want to analyze the macroscopic behavior of the system we could try to
increase the cutoff by building block-spins (fig.5): We replace nine spins in a
square by just a single spin by just counting the up-spins in the square. If their
are five or more of them the whole block is replaced by an up-spin or in the other
case of four or less up-spins by a down-spin. In a second step we have to reduce
the size of our lattice. This procedure changes the cutoff L (smallest distance on
which we want to take fluctuations into account).
The blockspin-transformation ignores all spin fluctuations within a spin-block.
When we want to study the macroscopic behavior of the 2D-ferromagnet described
by the Ising-model this procedure can simplify the calculations. What we have
changed by doing the transformation is the Hamiltonian describing the lattice.
The original Hamiltonian of the Ising-model is:
H = −J
XX
n
sn sn+e + B
e
X
(59)
sn
n
In this Hamiltonian n counts the spins in the lattice, e counts the next neighbors of
each spin, J is the coupling constant in this model. The blockspin-transformation
changes this Hamiltonian:
H̃ = RL (H) = −J1
XX
n
xxiv
e
sn sn+e + B1
X
n
sn + . . .
(60)
5 The Renormalization Group
Figure 5: Blockspin-transformation in the Ising-model.
RL is an operator leading to a new cutoff-length L. J1 and B1 are now of different
values than J and B. There occur also further terms in the Hamiltonian for example addends proportional to s3 . The blockspin-transformation can be continued
iteratively in order to achieve larger cutoff-length:
RL : H 7−→ H̃
(61)
RL0 : H̃ 7−→ H̄
(62)
(RL0 ◦ RL ) : H 7−→ H̄
(63)
Eq.63 gives a hint that the operation RL representing the blockspin-transformation
builds a group-like structure: RL0 ◦RL = RL00 . Since RL ◦RL = RL there exists also
a neutral element. But there are no inverse elements because with each blockspintransformation we loose informations about fluctuations on small length scales.
These informations can’t be regained. Therefore what we call the renormalization
group is actually a semi-group.
Now consider the blockspin transformation to be a transformation continuous in
L. When you change L you obtain a trajectory of the effective Hamiltonian in
xxv
Renormalization
the space of all Hamiltonians Hef f (L). In QFT we have obviously also a natural
cutoff Λ if we reduce this cutoff Λ0 = bΛ; b < 1 we obtain analogously a different
effective Lagrangian density following a trajectory in the space of all Lagrangian
densities Lef f (bΛ). The equation describing this trajectory (cutoff-dependence of
Lef f ) is called the renormalization-group-flow. Effectively we change the coupling
constant and the mass and also additionally occurring factors when we change
the cutoff. Therefore the renormalization-group-flow can also be considered to
describe a trajectory in the parameter-space of these constants (λ(b),m(b),. . .).
5.2 Renormalization-Group in QFT: Wilsons Approach
In QFT there exist a renormalization-group-transformation similar to the blockspintransformation in the Ising model. The transformation can be analyzed in pathintegral formalism. The basis of the Feynman rules is the equation
Z[J] =
Z
Dφei
R
[L+Jφ]
(64)
We simplify the problem by evaluating the RG-transformation in φ4 -theory without any sources or sinks (J = 0):
Z=
Z
[Dφ]Λ exp −
Z
"
1
1
λ
d x (∂µ φ)2 + m2 φ2 + φ4
2
2
4!
d
#!
(65)
In order to regularize the occurring divergences we already introduced in eq. 65
the necessary natural cutoff Λ by
[Dφ]Λ =
Y
(66)
dφ(k).
|k|<Λ
This means we just take fields into account with Fourier-components smaller than
Λ. Like in the Ising model we can change the cutoff (bΛ, b < 1) but it is crucial that
by doing that we do not change physical predictions so Z should stay unchanged.
Therefore we have to change our Lagrangian density and obtain what I call the
effective Lagrangian density Lef f :
Z=
Z
[Dφ]bΛ exp −
Z
d
d xLef f
(67)
The reduction of the cutoff without changing the result means effectively that we
xxvi
5 The Renormalization Group
integrate out the high momentum degrees of freedom. This step is of course irreversible. The effective Lagrangian density contains an infinite number of addends:
Z
dd xLef f =
1
1
1
dd x [ (1+∆Z)(∂µ φ)2 + (m2 +∆m2 )φ2 + (λ+∆λ)φ4 +∆C(∂µ φ)4 +∆Dφ6 +· · · ]
2
2
4!
(68)
Z
In order to obtain an integral directly comparable to the original integral we
have to rescale the momentum, the position and the field. After rescaling in the
integration the cutoff is again Λ:
k0 =
k
b
h
i1
x0 = xb φ0 = b2−d (1 + ∆Z)
2
φ
(69)
This step of rescaling correspond to the reduction of the lattice size in the last
step of the blockspin-transformation in the RG-theory of the Ising model. We end
up with the effective action:
Z
d
d xLef f =
Z
1
1
1
dd x [ (∂µ φ0 )2 + m02 φ02 + λ0 φ04 + C 0 (∂µ φ0 )4 + D0 φ06 + · · · ] (70)
2
2
4!
The new constants in the effective Lagrangian density are related to the original
constants:
m02 = (m2 + ∆m2 )(1 + ∆Z)−1 b−2
(71)
λ0 = (λ + ∆λ)(1 + ∆Z)−2 bd−4
(72)
C 0 = (C + ∆C)(1 + ∆Z)−2 bd
(73)
D0 = (D + ∆D)(1 + ∆Z)−3 b2d−6
(74)
We see that a reduction of the cutoff causes a shift of the constants. In section
3 we chose a certain renormalization point and therefore certain values for the
coupling constant and the mass. Now we find out that a reduction of the cutoff changes these values. This means that by choosing a certain renormalization
point(set of Mandelstam coordinates which determine values for the coupling constant and the mass) we fix the cutoff which was so far necessary but undefined.
On the other hand by the choice of a certain cutoff we also choose a certain renor-
xxvii
Renormalization
malization point. We end up with a cutoff-dependent coupling constant and a
cutoff-dependent mass.
The value of the cutoff is determined by the factor b. The equations 71-74 determine the RG-flow of the Lagrangian density. Each point on the trajectory in
the space of all Lagrangian densities refer to a certain set of constants, a certain
cutoff and a certain renormalization point. Now consider this flow next to the
fixed point L0 = 12 (∂µ φ)2 . This Lagrangian density is left unchanged under a
RG-transformation (reduction of the cutoff). Next to this point we can linearize
the equations 71-74:
m02 = m2 b−2
(75)
λ0 = λbd−4
(76)
C 0 = Cbd
(77)
D0 = Db2d−6
(78)
The cutoff-dependence of the constants is of the form α0 = bx α. When we choose
a small value of b leading to a cutoff next to the fixed point we realize that those
constants with x > 1 (C, D) become infinitesimally small. Such constants are
called irrelevant. Constants with x < 1 (m) on the other hand increase in there
importance and are therefore called relevant. Constants with x = 0 (λ in the case
of a fourdimensional spacetime) are called marginal. The consideration next to
the fixed point lead to a deeper insight into the nature of renormalizable theories:
Theories are renormalizable if there is just a finite number of relevant and marginal
constants. Just in this case the integration stays calculable.
(The behavior of the renormalization group flow next to a fixed point is especially
interesting in condensed matter physics. The fixed points then remarks the second
order phase transitions. The fact that next to the fixed point several constants
turn out to be irrelevant leads to a similar behavior of different systems next to the
fixed point. This results in connections between the critical exponents describing
the behavior of material constants near the phase transition.)
Now we want to study the cutoff-dependence of the constants away from the
fixed point. The cutoff-dependence of the coupling constant is now of course
more complicated than eq. 76 predicts. The dependence is usually expressed as
xxviii
5 The Renormalization Group
differential equation with a theory-dependent function β:
∂λ
(79)
∂Λ
Since we found out that each cutoff Λ refers to a certain renormalization point
µ (the renormalization point is usually determined by three Mandelstam coordinates, now we assume that u = s = t = µ which is actually impossible but it
simplifies the problem enormously) we can replace the cutoff in 79 by the renormalization point µ:
β(Λ, λ) = Λ
β(µ, λ) = µ
∂λ
∂µ
(80)
The renormalization point determines the energy-region at which we want to
evaluate our scattering experiment. Eq. 76 predicts that the coupling constant
isn’t constant at all but depends on the energy-region. This equation describing
the energy-dependence of coupling constant is called The Running of the Coupling
Constant. In QED for low energy regions eq. 80 can be solved:
e2 (µ) ≈
e2 (µ0 )
1 − (e2 (µ0 )/6π 2 ) log( µµ0 )
(81)
Eq. 81 predicts that the strength of the coupling increases in QED when we go
to higher energies. In fig. 5.2 the running of this coupling constant is depicted.
Going to higher energies the electric charge increases. The RG-theory says that
when we go to higher energies we reduce the cutoff. This is problematic since
the the theory must fail if our renormalization point is in the region of the cutoff.
As explained beyond the cutoff of a field theory a new physic must occur. But
fortunately in the moment when we come in danger that our renormalization point
meets the cutoff and the QED breaks down this new physic occurs in the shape
of the weak interaction. We are able to unify the QED and the theory of weak
interaction to a new physic a new field theory with a new cutoff far beyond the
energy-region of this electroweak unification.
xxix
Renormalization
Figure 6: Running of the coupling constants: Energy dependence of the coupling constants of the QED, the electroweak interaction and the strong
interaction
xxx
Bibliography
[1] M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory.
Reading, Ma [u.a.]: Addison-Wesley, 2. print. ed., 1996. viii, xi
[2] xi
[3] xx
xxxi