Seminar Quantum Field Theory

Institut f¨ ur Theoretische Physik III Universit¨ at 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

. . . . . . . . . . .

v v v v xi xv xv xix xxi xxiii xxiv xxvi xxxi

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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].

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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

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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.

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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

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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)

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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 :

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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

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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.

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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: ˜ = RL (H) = −J1 H

XX n

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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 →H ˜ 7−→ H ¯ RL0 : H ¯ (RL0 ◦ RL ) : H 7−→ H

(61) (62) (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

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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| 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

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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.

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Renormalization

Figure 6: Running of the coupling constants: Energy dependence of the coupling constants of the QED, the electroweak interaction and the strong interaction

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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

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