Running of the coupling in the

φ4-theory and in the Standard Model

Olli Koskivaara

Research training thesisSupervisor: Kimmo Kainulainen

October 9, 2014

Abstract

In this research training thesis the running of the coupling constant as aphenomenon is studied within two different frameworks. First the equa-tion governing the running of the φ4-theory coupling at one-loop order isderived. This is done in detail by first calculating the one-loop correctionsto the theory and then renormalizing it. The renormalization procedureis treated more generally by e.g. deriving the Callan–Symanzik equation.Using the properties of the renormalization group and the one-loop cor-rections calculated earlier the running of the coupling is obtained. Thecoupling is found to increase with increasing energy scale. A short dis-cussion on beta-functions is also given.

The second example of the running of the coupling is more of a phe-nomenological one. The three gauge couplings of the Standard Modeland their scale dependence is studied. A plot of the inverses of the cou-plings as a function of energy is produced. It is noted that the couplingsget near each other at an energy scale of approximately 1016 GeV but yetdo not meet each other. A search for a theory predicting the unificationof the gauge couplings is mentioned.

Tiivistelmä

Tässä erikoistyössä kytkentävakion jouksemista tutkitaan ilmiönä kahdeneri esimerkin kautta. Ensin määritetään φ4-teorian kytkinvakion juokse-misen 1-silmukkatasolla määräävä yhtälö. Tämä tehdään yksityiskohtai-sesti laskemalla ensin teorian ensimäisen kertaluvun korjaukset ja sittenrenormalisoimalla se. Renormalisaatioprosessi käsitellään hiukan yleisem-mällä tasolla esimerkiksi johtamalla Callanin–Symanzikin yhtälö. Kyt-kentävakion juokseminen saadaan käyttämällä renormalisaatioryhmän o-minaisuuksia sekä aiemmin laskettuja silmukkakorjauksia. Kytkentäva-kion havaitaan voimistuvan energiaskaalan kasvaessa. Lopuksi tehdäänlyhyt katsaus beetafunktioiden ominaisuuksiin.

Toinen esimerkki on luonteeltaan fenomenologisempi. Standardimallinkolmea mittakenttäkytkentää ja niiden riippuvuutta energiaskaalasta tar-kastellaan tuottamalla kuva käänteisistä kytkentävakioista energian funk-tioina. Kytkentävakioiden havaitaan lähestyvän toisiaan energiaskaalan1016 GeV lähettyvillä kohtaamatta kuitenkaan toisiaan. Lopuksi maini-taan kiinnostus mittakenttäkytkentöjen yhdistymisen ennustavaa teoriaakohtaan.

i

Contents

1 Introduction 1

2 Running of the coupling in the φ4-theory 2

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 One-loop corrections . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 2→ 2 scattering . . . . . . . . . . . . . . . . . . . . . 8

2.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 The Callan–Symanzik equation and the renormalization group 12

2.5 Running of the coupling . . . . . . . . . . . . . . . . . . . . . 15

3 Beta-functions 18

4 Running couplings in the Standard Model 19

5 Conclusions 22

Appendix A Feynman rules 24

Appendix B Mathematical details 24

B.1 Beta function identity . . . . . . . . . . . . . . . . . . . . . . . 24

B.2 Feynman parametrization . . . . . . . . . . . . . . . . . . . . 25

Appendix C MATLAB code 26

ii

1 Introduction

This work concerns the phenomenon known as the running of the cou-pling constant. This running is a feature of many interacting theories andplays a great role in modern physics. To put it briefly, the phenomenonis about the energy scale dependence of the coupling constant describingthe strength of the interaction of a theory.

We start by introducing the φ4-theory and some basic tools used in quan-tum field theory. We then proceed to calculate in detail the one-loopcorrections to the theory, main goal being the equation governing therunning of the coupling of the φ4-theory at this level. In order to get tothis result, we need to visit the realm of renormalization. Renormaliza-tion is a powerful tool used to extract meaningful and finite results fromseemingly infinite outcomes. It deals with such fundamental questionsas what is really meant by the very notions of electric charge, mass, etc. Italso plays a major role in the theory of running couplings.

Once we have arrived at our final result concerning the φ4-theory, we saya few words about the role of beta-functions1, which we will meet duringthe φ4-calculations, in physics. It turns out that once a beta-function of agiven theory is known, the running of the coupling is completely dictatedby it.

After becoming familiar with the mathematics related to the running ofthe coupling with φ4-theory, we turn to some phenomenology. The en-ergy scale dependence of the Standard Model gauge couplings is studiedby producing a plot of their inverses as a function of energy. We finishour discussion by pondering on the possible implications of this plot.

1Not to be confused with the beta function familiar from mathematics, which we willactually also encounter in this work.

1

2 Running of the coupling in the φ4-theory

2.1 Preliminaries

Let us start by defining the theory we are dealing with. The φ4-theoryconsists of a massive real scalar field φ(x) with a quartic self-interaction.The Lagrangian of the theory is thus that of the Klein–Gordon theorysupplemented with the interaction term,

L =12(∂µφ

)2 − 12

m2φ2 − λ

4!φ4. (1)

This toy model is often used as an example in quantum field theoreticcalculations since it is quite simple and exhibits many features present inmore complicated theories. The Feynman rules for the theory are givenin Appendix A.

Before rushing into calculations some important concepts should be in-troduced. The information about scattering events is given by correlationfunctions also known as Green’s functions and n-point functions. The n-particle propagator is a time-ordered vacuum expectation value of thefields φ(xi),

G(n) ≡ 〈Ω |T φ(x1)φ(x2) · · · φ(xn)|Ω〉 . (2)

The simplest nontrivial connected Feynman diagrams, from which allother diagrams can be constructed, are the so called one-particle irreducible(1PI) diagrams. These diagrams are determined by two requirements.They are amputated, which means that all external lines have been re-moved. Additionally, they are not allowed to have any cutlines, i.e., theycannot be split into two separate diagrams by cutting one of the internallines.

Since only the 1PI diagrams are significant for our calculations, it is use-ful to define for such diagrams the proper vertex function Γ(n), which isobtained from the n-particle momentum space correlation function by re-moving all the propagators on the external legs and omitting the deltafunction indicating the conservation of momentum. The relation be-tween the correlation function G(n) and the proper vertex function Γ(n)

2

can hence be written as

G(n)(p1, . . . , pn) =

[n

∏i=1

G(2)(pi)

](2π)Dδ(D)

(n

∑i=1

pi

)Γ(n)(p1, . . . , pn). (3)

The fact that 1PI diagrams can be regarded as the basic building blocksof Feynman diagrams simplifies calculations tremendously. For instancein the φ4-theory the exact propagator can be written diagrammatically asa geometric series of the form

= + 1PI

+ 1PI 1PI + · · · ,

where

1PI ≡ +

+ + · · · = Γ(2)

(4)

consists of all 1PI diagrams. The exact four-point can be represented asa similar series where each term is constructed from the collection of 1PIdiagrams

Γ(4) = 1PI

≡ +

+ perm.

+ · · · ,

(5)

where perm. includes the two other topologically nonequivalent one-loopdiagrams which are obtained by permuting the incoming and outgoingmomenta.

3

2.2 One-loop corrections

In this section we will calculate the one-loop corrections to the propagatorand to the four-point function. These results will be needed later whenwe start to renormalize our theory.

2.2.1 Propagator

The one-loop correction to the propagator is given by the first 1PI diagramin equation (4), this is the so-called tadpole diagram. Using the Feynmanrules listed in Appendix A we have2

= − iλ2

∫ d4p(2π)4

ip2 −m2 + iε

,

where the factor 12 in front of the integral is a symmetry factor originating

from all possible contractions when forming the tadpole diagram.

For example by introducing a cutoff one readily sees that the integralabove is quadratically divergent. This is why we shall use dimensionalregularization, i.e., we calculate the integral in a general dimension D.Our integral becomes

− iλ2

µ4−D∫ dD p

(2π)Di

p2 −m2 + iε,

where µ is an arbitrary parameter with dimensions of mass introduced tokeep the dimensions of the integral unchanged. This step is quite crucialbecause this scale, entering our theory via the parameter µ, will play akey role in the behaviour of the coupling constant.

In order to calculate the regularized integral we shall use a trick knownas the Wick rotation. We start by integration over p0. Due to the Feynmanprescription iε the integrand has poles above and below the real axis, asshown in Figure 1. The Wick rotation corresponds to a change of variables

p0M −→ ip0E ; dD pM −→ idD pE, p2M −→ −p2

E, (6)

where in the subscripts M stands for Minkowskian and E for Euclidean.2The factors 2π appearing in integrals like this are mere conventions essentially aris-

ing from Fourier transformations between momentum and position space integrals [1].

4

Figure 1: The Wick rotation accounts for changing the integration alongthe real axis to integration along the imaginary axis. The figure shows thewhole integration contour together with the two poles above and belowthe real axis.

We will further need an important result from complex analysis calledthe Cauchy’s integral theorem, which essentially states the following [2]:

If f (z) is an analytic function, and f ′(z) is continuous at each pointwithin and on a closed contour C, then∮

C

f (z)dz = 0.

Since the contour in Figure 1 encloses no poles, the theorem above tells usthat our integral over the contour vanishes. Furthermore, the integrandvanishes as |p0| → ∞, and thus the arcs do not contribute at infinity.Hence the integrals over real and imaginary axis have to be equal andopposite. Using this fact and the change of variables (6) our integralreduces to a Euclidean one:

− iλ2

µ4−D∫ dD p

(2π)Di

p2 −m2 + iε= − iλ

2µ4−D

∫ dD pE

(2π)D1

p2E + m2

.

Here we have abandoned the prescription iε since it no longer plays anyrole after the Wick rotation.

Now we can use the result for D-dimensional angular integral valid for

5

Euclidean space:

∫dDx =

∫dDΩ

∞∫0

xD−1 dx =2π

D2

Γ(D

2

) ∞∫0

xD−1 dx,

where Γ is the gamma function [3]. This gives us

− iλ2

µ4−D∫ dD pE

(2π)D1

p2E + m2

= − iλ2

µ4−D 2πD2

Γ(D

2

) ∞∫0

dpE

(2π)DpD−1

Ep2

E + m2(7)

and we are only left to calculate the integral over pE. This can be doneusing the following result derived in Appendix B.1 using the Euler betafunction:

∞∫0

xα−1

(x + a)βdx = aα−β Γ(α)Γ(β− α)

Γ(β).

First we write

pD−1E dpE =

12

pD−2E dp2

E =12

(p2

E

)D2 −1

dp2E

in our integral. Then starting from (7) we get

− iλ2

µ4−D 2πD2

Γ(D

2

) ∞∫0

dpE

(2π)DpD−1

Ep2

E + m2= − iλ

2µ4−D π

D2

(2π)DΓ(D

2

) ∞∫0

dp2E

(p2

E)D

2 −1

p2E + m2

= − iλ2

µ4−D πD2

(2π)DHHHHΓ(D

2

)HHHHΓ

(D2

)Γ(1− D

2

)Γ(1)︸︷︷︸= 1

(m2)D

2 −1

= − iλm2

32π2

(4π

µ2

m2

) 4−D2

Γ(

1− D2

).

Since we are interested in phenomena close to four dimensions, it is sen-sible to write our results in terms of a parameter ε ≡ 4 − D which isassumed to be small. The loop integral becomes

− iλm2

32π2

(4π

µ2

m2

) ε2

Γ(ε

2− 1)

. (8)

6

Next we shall carry out some small-ε approximations for this result. Forthe first part involving ε we write(

4πµ2

m2

) ε2

= eε2 ln(

4π µ2

m2

)= 1 +

ε

2ln(

4πµ2

m2

)+O(ε2).

For the gamma function we use its Laurent expansion around its poles:

Γ(ε− n) =(−1)n

n!

[1ε+ Ψ(n + 1) +O(ε)

], n ∈ N,

where Ψ is the dilogarithm function defined as the logarithmic derivativeof the gamma function,

Ψ(x) =d

dxln [Γ(x)] , Ψ(n + 1) = −γ +

n

∑i=1

1i

,

with γ = −Ψ(1) = 0.5772... being the Euler–Mascheroni constant [4].Applying this to our gamma function yields

Γ(ε

2− 1)=

1ε2 − 1

Γ(ε

2

)= −

(1 +

ε

2

)(2ε− γ

)+O(ε)

where we first used the recursive property of the gamma function,Γ(x + 1) = xΓ(x), and then expanded in terms of ε.

Using these approximations in (8) results in

iλm2

32π2

[2ε+ 1− γ− γε

2+(

1 +ε

2− γε

2

)ln(

4πµ2

m2

)]+O(ε).

Taking the limit ε→ 0 we can conclude that at one-loop order the correc-tion to the propagator is

Γ(2)(p2) =iλm2

16π2

[1ε+

12− γ

2+

12

ln(

4πµ2

m2

)]. (9)

The correction thus consists of a divergent part ∼ ε−1 and a finite partwhich will be insignificant for our purposes. By analytically continuingour original integral to D dimensions we extracted the infinities to oneterm which is easy to handle. Another observation to be made here isthat, although written as the argument of the two-point function Γ(2), p2

does not appear in the correction.

7

2.2.2 2→ 2 scattering

For the four-point function the one-loop correction consists of three dia-grams, corresponding to s-, t- and u-channel scatterings. The three dia-grams are related to each other by crossing symmetry, which means thatthey can be transformed to each other just by interchanging the Man-delstam variables suitably. It will therefore suffice to calculate explicitlyonly one of them, e.g. the s-channel contribution represented by the loopdiagram in equation (5).

Using the Feynman rules of our theory we have that the integral to becalculated is

(−iλ)2

2

∫ d4k(2π)4

ik2 −m2

i[(k− p)2 −m2]

,

where p is the sum of the incoming momenta, the factor 12 is a symme-

try factor and we have dropped the prescriptions iε. The procedure tocalculate this integral is very similar to what we already did with thepropagator. First we note that the integral is divergent, this time loga-rithmically. This leads us to dimensional regularization and calculationof the integral

λ2

2µ4−D

∫ dDk(2π)D

1(k2 −m2) [(k− p)2 −m2]

.

The next thing to do is to introduce a Feynman parametrization (see Ap-pendix B.2 for proof)

1ab

=

1∫0

dx

[xa + (1− x)b]2,

which after some algebra reduces our denominator as

1(k2 −m2) [(k− p)2 −m2]

=

1∫0

dx[k− (1− x)p]2 −m2 + p2x(1− x)

2 .

We are then left to calculate

λ2

2µ4−D

∫ dDk(2π)D

1∫0

dx[k− (1− x)p]2 −m2 + p2x(1− x)

2 .

8

Since our integrals are now convergent after the regularization, we canswitch the order of integration and shift the integration parameter ask −→ k = k − p(1− x), which leaves the measure dk unchanged. Thisgives us

λ2

2µ4−D

1∫0

dx∫ dD k

(2π)D[k2 −m2 + p2x(1− x)

]2 .

The momentum integral is calculated exactly in the same way as for thepropagator, first performing the Wick rotation and then carrying out theangular integrals and using the beta function identity:

λ2

2µ4−D

1∫0

dx∫ dD k

(2π)D[k2 −m2 + p2x(1− x)

]2

=iλ2

2µ4−D

1∫0

dx∫ dD kE

(2π)D[−k2

E −m2 + p2x(1− x)]2

=iλ2

2µ4−D 2π

D2

Γ(D

2

) 1∫0

dx∞∫

0

dkE

(2π)DkD−1

E[k2

E + m2 − p2x(1− x)]2

=iλ2

2µ4−D π

D2

(2π)DΓ(D

2

) 1∫0

dx∞∫

0

dk2E

(k2

E

)D2 −1

[k2

E + m2 − p2x(1− x)]2

=iλ2

2µ4−D π

D2

(2π)DHHHHΓ(D

2

)HHHHΓ

(D2

)Γ(2− D

2

)Γ(2)︸︷︷︸= 1

1∫0

dx[m2 − p2x(1− x)

]D2 −2

=iλ2

32π2

1∫0

dx

4πµ2

[m2 − p2x(1− x)]

4−D2

Γ(

2− D2

)

=iλ2

32π2

1∫0

dx

4πµ2

[m2 − p2x(1− x)]

ε2

Γ(ε

2

).

Using the same small-ε expansions as before and taking the limit ε → 0we get that the correction to the four-point function from the s-channel

9

loop is

iλ2

16π2

1ε− γ

2+

12

ln(

4πµ2

m2

)− 1

2

1∫0

dx ln[

1− p2

m2 x(1− x)] .

The other two one-loop diagrams give a contribution of the same form,the only difference being in the total momentum p2 appearing in the inte-gral. These finite parts of the correction are not relevant for our purposes,and therefore we merely stress that the overall correction to the four-pointfunction at one-loop order is of the form

3iλ2

16π2

(1ε

+ FINITE)

. (10)

2.3 Renormalization

If we were studying the non-interacting Klein–Gordon theory, the param-eters of the Lagrangian would correspond to physical parameters, e.g. mappearing in the Lagrangian would indeed be the physical mass observedin experiments. However, as seen in the previous section, adding the in-teraction to the theory causes divergences as we calculate corrections tothe parameters. Because in experiments we measure finite quantities, thissuggests that the parameters of our Lagrangian do not correspond to thephysical ones.

It is useful to rewrite our Lagrangian (1) as

L =12(∂µφ0

)2 − 12

m20φ2

0 −λ0

4!φ4

0 ,

where φ0, m0 and λ0 are the so-called bare parameters, which may beinfinite and cannot in general be measured in experiments. The next stepis to introduce the renormalized (physical, measurable, finite) parametersφR, mR and λR related to the bare ones by

φ0 ≡ Z1/2φ φR , (11)

λ0 ≡ Z−2φ ZλλR , (12)

m20 ≡ Z−1

φ Zmm2R , (13)

10

where the factors Zφ, Zλ and Zm contain all the possible infinities. Thepoint of this redefinition is that it allows us to split the Lagrangian into arenormalized one and a part involving the infinities called the countertermLagrangian.

Instead of dwelling upon the usefulness of the counterterms, we focus onthe relation between the renormalized and bare coupling constant. Forsimplicity we will from now on consider a massless theory. Furthermore,since regularization will be necessary at some point, it is useful to dealfrom the beginning with a Lagrangian appropriate for D = 4− ε dimen-sions

Lε =12(∂µφR

)2 − µελR

4!φ4

R + Lc.t. ,

where µε adjusts λR to be dimensionless and Lc.t. is the counterterm La-grangian. Now the relation between the bare and renormalized couplingreads

λ0 = µε Z−2φ ZλλR. (14)

From equations (2) and (11) we see how the n-particle propagator scaleswhen the fields are renormalized:

G(n)0 ≡ 〈Ω |T φ0(x1)φ0(x2) · · · φ0(xn)|Ω〉

= 〈Ω|T Z1/2φ φR(x1)Z1/2

φ φR(x2) · · · Z1/2φ φR(xn)|Ω〉

= Zn/2φ 〈Ω |T φR(x1)φR(x2) · · · φR(xn)|Ω〉

= Zn/2φ G(n)

R . (15)

How does the proper vertex function Γ(n) scale under renormalization?From equation (3) we see that

Γ(n)0 ∼

G(n)0[

G(2)0

]n ,

which together with equation (15) yields

Γ(n)0 = Z−n/2

φ Γ(n)R . (16)

11

Now that we have renormalized our theory, we can define the physicalcoupling constant as the renormalized proper vertex function evaluatedat some arbitrary momenta:

λR = iΓ(n)R (p0

1, p02, . . . , p0

n).

It is important to realize that the reference point can indeed be freelychosen; we could have just as well defined

λR = iΓ(n)R ( p0

1, p02, . . . , p0

n),

where Γ(n)R is evaluated at some different momenta which can be related

to a measured value of the coupling constant. It can be shown that thefreedom in the choice of this subtraction point does not practically affectthe values of physical quantities, such as the cross section σ, dependingon the coupling constant. The numerical difference caused by differentchoices would always occur at higher orders in the coupling constant ex-pansion than in the original calculation, i.e., there is no difference at theorder we decide to calculate the values. This ensures that our perturba-tion theory is well defined.

2.4 The Callan–Symanzik equation and the renormaliza-tion group

Let us look back at equation (16) relating the renormalized and bareproper vertex functions to each other. Since bare parameters are inde-pendent of the so-called “hidden” scale µ, the bare proper vertex func-tion does not depend on µ. It may only depend on momenta which wewill collectively call p, parameters of the bare Lagrangian and ε. Thefield renormalization factor Zφ on the other hand depends on µ throughthe dimensionless combination λ0µ−ε. The renormalized proper vertexfunction also depends on µ, both explicitly and implicitly, because therenormalized parameters themselves depend on µ. Obviously it also de-pends on ε, but stays finite in the limit ε→ 0.

All in all we can rewrite equation (16) as

Γ(n)R (p, λR, µ, ε) = Zn/2

φ

(λ0µ−ε, ε

)Γ(n)

0 (p, λ0, ε) .

12

Dividing both sides by Zn/2φ and taking the derivative with respect to µ

yields

∂

∂µ

[Z−n/2

φ

(λ0µ−ε, ε

)Γ(n)

R (p, λR, µ, ε)]= 0. (17)

Keeping in mind that there is a µ-dependence in λR = λR (λ0µ−ε) too,opening equation (17) gives(

∂

∂µZ−n/2

φ

)Γ(n)

R + Z−n/2φ

∂

∂µΓ(n)

R = 0

⇔(

µZn/2φ

∂

∂µZ−n/2

φ + µ∂λR

∂µ

∂

∂λR+ µ

∂

∂µ

)Γ(n)

R = 0. (18)

Defining

β(λR) ≡ µ∂

∂µλR, (19)

γ(λR) ≡12

µ∂

∂µln Zφ (20)

and observing that

−nγ = −n2

µ∂

∂µln Zφ = µ

∂

∂µln(

Z−n/2φ

)= µZn/2

φ

∂

∂µZ−n/2

φ ,

we can write equation (18) as[µ

∂

∂µ+ β (λR)

∂

∂λR− nγ (λR)

]Γ(n)

R (λR, µ) = 0. (21)

This equation is known as the Callan–Symanzik equation and the renormal-ization group equation (RGE)3. It tells how changing the scale is related tochanges in the coupling constant and field strength. To be more precise,it tells us how to shift the coupling λR when we shift the scale µ in orderto keep physical quantities invariant.

The functions β and γ appearing in the RGE are known as the Callan–Symanzik beta-function and the anomalous dimension, respectively. β (λR)will be of great importance to us, because it governs the running of the

3Here the word group refers to transformations between different versions of ourtheory at different scales. This set of transformations is not however a group in themathematical sense of the word, since there is no inverse for every element [3].

13

coupling constant. The anomalous dimension γ on the other hand re-lates to shifts in the field strength, and we will not be dealing with it inthis work. Basically it explains our theory’s deviations from the scalingbehaviour a similar classical theory would have [1].

Since we are not really interested in the anomalous dimension γ, it wouldbe useful to get rid of it one way or another. This can be done by defininga dimensionless ratio of Green’s functions sometimes called an invariantcharge [5]

Ω (p, λR, µ, ε) ≡ Γ(4)R (p, λR, µ, ε)

4

∏i=1

[p2

i

Γ(2)R (pi, λR, µ, ε)

] 12

,

which satisfies the homogeneous renormalization group equation (HRGE)[µ

∂

∂µ+ β (λR)

∂

∂λR

]Ω (λR, µ) = 0. (22)

Let us now look at a renormalization group transformation

T : µ −→ etµ ≡ µ(t),

where t is just a parameter characterizing the transformation. Consist-ing of renormalized proper vertex functions Ω is physical, and hence itshould not depend on our choice of λR and µ. This means that we canfind a parametrization which satisfies

Ω (λR, µ) = Ω(λR(t), µ(t)

)⇔ 0 =

∂

∂tΩ(λR(t), µ(t)

)=

(∂λR

∂t∂

∂λR+

∂µR

∂t∂

∂µR

)Ω(λR, µ

), (23)

where λR corresponds to the coupling constant at scale µ. Comparingequation (23) with the HRGE (22) one sees that

β(λR(t)

)=

∂λR(t)∂t

, (24)

where λR(0) = λR corresponds to no scaling at all (µ(0) = µ).

As mentioned, the invariant charge is physical. We can thus, similarlyas with the proper vertex functions, define the renormalized couplingconstant as Ω evaluated at some reference momenta p0:

λR ≡ iΩ(p0, λR, µ, ε)

14

What if we now scale the momenta as p0 → et p0 and evaluate Ω again?Since Ω is dimensionless, it can only depend on the ratio of p0 and µ. Thisobservation together with the scaling properties discussed above yields

iΩ(et p0, λR, µ, ε) = iΩ(p0, λR, µe−t, ε) = iΩ(p0, λR(t), µ, ε) ≡ λR(t).

So by scaling the momenta we obtained another value for the couplingconstant; the coupling “runs” with the scale. What is the equation gov-erning this running?

2.5 Running of the coupling

In order to answer the previous question, we need to calculate the Callan–Symanzik beta-function β for our theory. Fortunately the dirty work hasactually already been done in Section 2.2 where we calculated the one-loop corrections for the φ4-theory. Using the definition of β and equation(14) we can write

β(λR) = µ∂

∂µ

(λ0µ−εZ2

φZ−1λ

). (25)

The renormalization factors Zφ and Zλ are to be constructed in such a waythat they cancel the divergences in the renormalized Lagrangian. Fromequation (9) we see that the one-loop correction to the propagator has nop2-dependence. This means that there is no wave function renormaliza-tion at one-loop level, i.e. Zφ = 1. 4

From equation (10) we have that the total divergence in the corrections tothe four-point function is

3iλ2oµ−ε

16π21ε

,

where we have adjusted the result to fit our regularized Lagrangian Lε.In order to have a counterterm canceling the divergence we need a renor-malization factor

Zλ = 1 +3λoµ−ε

16π21ε

4This can be justified by expanding Γ2(p2) around the chosen subtraction point andnoticing that any counterterm would be p2-dependent. The absence of wave functionrenormalization at one-loop level is a special feature of the φ4-theory; e.g. in the Yukawatheory a wave function renormalization is required at one-loop level [3].

15

for the coupling. At one-loop order the renormalized coupling constantis then

λR ≡ iΓ(4)R = i

(−iλ0µ−ε +

i3λ20µ−ε

16π21ε

)= λ0µ−ε

(1− 3λ0

16π21ε

)= λ0µ−εZ−1

λ .

Substituting this into equation (25) yields

β = µ∂

∂µ

[λ0µ−ε

(1− 3λ0

16π21ε

)]= −ελ0µ−ε +

3λ20µ−ε

16π2 ,

where we can finally take the limit ε→ 0 to obtain

β(λ) =3λ2

16π2 . (26)

Now we can combine equations (24) and (26) to get

∂λ

∂t=

3λ2

16π2 .

This simple differential equation is easily solved, the solution being

λ(t) =λ(0)

1− 3tλ(0)16π2

. (27)

We can further write this result without the scaling parameter t. Letλ(0) = λ(Q0) be the coupling constant at some scale Q0. Then at anotherscale Q we have for some value of the parameter t

Q = Q0et ⇔ t = ln(

QQ0

).

Using this in equation (27) yields

λ(Q) =λ(Q0)

1− 3λ(Q0)16π2 ln

(QQ0

) . (28)

This result, equation (28), is what we have been aiming for from the be-ginning. It governs the running of the coupling constant of the (massless)φ4-theory at one-loop order. Indeed; if the coupling constant is known,

16

for example from experiments, at some scale Q0, then one can use equa-tion (28) to determine the coupling at other scales. A notable feature isthat the coupling constant increases with increasing scale Q.

Furthermore, we can characterize the behaviour of the coupling constantby just one dimensionful parameter, namely the scale at which the cou-pling diverges. From equation (28) we see that the coupling constantbecomes infinite at some scale Q at which the denominator vanishes:

1− 3λ(Q0)

16π2 ln

(QQ0

)= 0 ⇔ Q = Q0 e

16π2

3λ(Q0) (29)

This scale is known as the Landau pole, and it can be used to remove thedependence on a reference scale Q0 in equation (28). This procedure isknown as dimensional transmutation since it replaces the dimensionless pa-rameter λ characterizing the interaction by a dimensionful one. It shouldbe noted that the scale in equation (29) may be far beyond the reach ofour treatment, and an accurate expression would probably require a non-perturbative analysis.

Equation (29) tells us that the scale at which the Landau pole occurs isvery large if λ(Q0) is small. Indeed, the φ4-theory allows a perturbativecoupling constant expansion only at small mass scales (large distances).If we start with a small λ for which the perturbation is sensible and beginto increase the scale, we will need to add more and more terms to theexpansion due to the increase in λ according to equation (28).

The case could be the opposite: if the sign in the denominator in equation(28) was positive, the coupling constant would vanish at large small scales(small distances). This is known as asymptotic freedom, and it is a propertyof e.g. quantum chromodynamics (QCD).5 Perturbation is valid at largescales but breaks down at small scales.

For quantum electrodynamics (QED) the situation is similar to that of theφ4-theory: the coupling becomes stronger at larger scales. The Landaupole of QED corresponds to an energy scale ΛLandau ≈ 1034 GeV. Thepole causes no problems, because QED is expected to be unified withother interactions below this scale. Furthermore, quantum gravity stepsin at scale 1019 GeV. [4]

5In 2004 David J. Gross, H. David Politzer and Frank Wilczek received the Nobel Pricein Physics “for the discovery of asymptotic freedom in the theory of the strong interaction” [6].

17

3 Beta-functions

All these behaviours related to the coupling constant are depicted by theCallan–Symanzik beta-function of the theory in hand. Indeed, lookingat the definition of the beta-function (19) one readily sees that if a beta-function of a theory is positive, then the coupling constant of the theoryincreases with the scale. This is exactly the case for the φ4-theory; fromequation (26) we see that β > 0. Similarly a negative beta-function impliesan inverse relationship between the coupling and the scale, which leadsto asymptotic freedom.

Beta-functions can further be used to examine the high- and low-energy,i.e. ultraviolet (UV) and infrared (IR), properties of a theory. These prop-erties are controlled by the fixed point structure of the theory. Fixed pointsare those values of the coupling constant for which the beta-function van-ishes. Let λF be such a point of a certain theory, i.e. β(λF) = 0. Expandingβ around its zero to first order gives β(λ) ≈ (λ− λF)β′(λF). Then usingthe definition of the beta-function we get

µ∂

∂µλ = (λ− λF)β′(λF)

⇒ λ− λF

λ− λF=

(µ

µ

)β′(λF)

, (30)

where we integrated from µ to µ.

Looking at equation (30) we see that the sign of the derivative of the beta-function β′(λF) tells about the behaviour of the coupling constant closeto the fixed point. If β′(λF) < 0, then λ→ λF independently of the initialvalue λ as the scale µ increases, and λF is said to be an UV stable fixedpoint. If on the other hand β′(λF) > 0, then λ tends to λF as µ decreasesand goes away from λF for increasing µ. In this case λF is called an IRstable fixed point.

The word “stable” refers to the fact that, once evolved to its stable value,the coupling constant does not run anymore. This is quite interesting; wemay have an interacting theory where the coupling constant actually is aconstant. Such theories are called conformal field theories. One interestingfeature of these theories is that due to the vanishing beta-function they arescale invariant. Usually the scale invariance, often present in the classical

18

version of a theory, is broken in the quantum theory by the interaction. 6

Equation (26) tells us that the φ4-theory has one fixed point, occurringat λ = 0. In this case the φ4-theory of course just reduces to the Klein–Gordon theory describing a free particle. QED and QCD also have onlythis so called Gaussian fixed point corresponding to zero coupling. Thereis however no reason why a theory could not have a beta-function withmultiple zeros, corresponding to fixed points at non-zero couplings. Thestudy of fixed point structures essentially leads to the concept of univer-sality used to classify different theories related to each other [7].

4 Running couplings in the Standard Model

In this chapter we take a look at the coupling constants of the StandardModel and examine how they run with the energy scale. Mathematicalderivations are left aside, and emphasis will be on the phenomenologyrelated to the couplings.

The Standard Model is a gauge field theory based on the symmetry groupSU(3) ⊗ SU(2) ⊗ U(1), and the interactions are specified by three cou-pling constants usually denoted by gs, g and g′ corresponding to thegauge symmetries SU(3), SU(2) and U(1) respectively. These gauge cou-plings define three of the fundamental forces: strong nuclear, weak nu-clear and electromagnetism. In order to compare these interactions, it isuseful to define the set of couplings as

αem ≡53(g′)2

4π=

53

α

cos2 θW, (31a)

αw ≡g2

4π=

α

sin2 θW, (31b)

αs ≡gs

4π, (31c)

where the subscripts in the couplings stand for electromagnetic, weak andstrong, and θW is the Weinberg angle and α the fine-structure constant.The factor 5

3 in the definition of αem is related to normalizing U(1) inorder to get a possible SU(5) structure. [8, 9]

6This phenomenon is known as anomalous symmetry breaking: a classical symmetrydoes not survive to the quantum theory.

19

Each of these three couplings has a specific equation governing the run-ning, just as with the coupling of the φ4-theory studied earlier. Theseequations are derived from the corresponding beta-functions, which areagain obtained by investigating the basic interactions of the theory at de-sired order, renormalizing, using the renormalization group etc. After allthese not-so-trivial endeavors the equations describing the running of thecouplings in equation (31) at one-loop order turn out to be [8, 10]

αi(Q) =αi (Q0)

1− biαi (Q0) ln(

QQ0

) , (32)

where the coefficients bi are given by2πb1 = 41

10 ,2πb2 = −19

6 ,2πb3 = −7.

One can use experimentally measured values for the fine-structure con-stant, the Weinberg angle and the strong coupling constant to determinethe couplings in equation (31) at a certain scale. Choosing this scale asthe mass of the Z-boson MZ = 91.1876± 0.0021 GeV [11], the values ofthe coupling constants become [8]

αem(MZ) = 0.017,αw(MZ) = 0.034,αs(MZ) = 0.118.

Using these values and equation (32) we can plot the couplings as a func-tion of energy. Figure 2 shows the graphs of the inverses of the couplings,starting from MZ.

This figure exhibits the kind of features we were expecting to see; theasymptotic freedom of QCD and the increase of the QED coupling withenergy. It also gives us a hint about something new: a possible unifi-cation of all the three couplings at some large energy ΓU ∼ 1016 GeV,where new physics may have been emerged. Even if we were to take intoaccount the errors in the coupling constant measurements, the couplingswould not meet at one point within the scope of the Standard Model. Thishas led physicists to search for a theory that would predict the unifica-tion of the three gauge interactions in the same way as electromagnetism

20

105

1010

1015

1020

5

10

15

20

25

30

35

40

45

50

55

60

Energy (GeV)

Inverse

coupling

α−1em

α−1w

α−1s

Figure 2: The inverses of the three couplings αi as functions of energy.The energy axis is logarithmically scaled. This figure was produced usingMATLAB interface; see Appendix C for details.

and weak interaction are already combined into electroweak theory inthe Standard Model. Supersymmetry and its extensions are probablythe most well-known efforts towards this so-called Grand Unified Theory(GUT). Currently no model is generally accepted as a valid GUT.

Of course there is also the fourth one of the fundamental forces. Gravitystarts much weaker than the three other forces and is extrapolated to jointhem at about 1019 GeV. The most ambitious theories try to add this tothe unification, resulting in a theory combining all known fundamentalforces to a single one. This Theory of Everything (ToE) would have to takecare of both quantum field theory and general relativity. Many modelshave been constructed, string theory being the most famous candidate,but none of them has been agreed on generally. The main problem liesin the difficulty to experimentally verify these theories; no man-madeparticle accelerator can reach the energies of the Planck scale ∼ 1019 GeV.

21

5 Conclusions

In this work we calculated the running of the coupling in the φ4-theoryat one-loop order. This was done by first calculating the one-loop cor-rections to the theory, then renormalizing it and finally using the thusobtained beta-function to study the scale dependence of the coupling.The coupling was found to increase with increasing momentum scale. Ashort introduction to beta-functions was also given, mainly underliningtheir importance in studying coupling constants.

Utilizing the insight gained from the φ4-calculations we then studied therunning of the three gauge couplings of the Standard Model. A plotof their inverses against energy was produced. The plot shows that thecouplings get near each other at an energy scale of approximately 1016

GeV but do not meet each other. A theory which predicts the joiningof the couplings could be used to describe all the three interactions bya single coupling constant. A quest for such a general theory and theabsence of one was noted.

The gauge couplings and their non-unification in Standard Model is ofcourse just one example of the phenomenological aspects of the runningof the coupling. Another interesting example of the phenomenon is theelectroweak vacuum stability in the Standard Model. The quartic self-coupling of the Higgs boson decreases with increasing energy and at suf-ficiently large energies it would turn negative. This would mean that thevacuum is unstable, indicating that our universe might exist in a “false”vacuum and could spontaneously fall to a lower state at the same timeceasing to exist as we know it. This is however mere speculation, andmore precise calculations and experiments need to be done before mak-ing any definite conclusions. Although intriguing, the vacuum stabilityand many other phenomena related to the running of the coupling arebeyond the scope of this work and are left for future study.

22

References

[1] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cam-bridge University Press, New York, 2014.

[2] S. J. Bence, M. P. Hobson and K. F. Riley, Mathematical Methods forPhysics and Engineering, 3rd Edition, Cambridge University Press, NewYork, 2007.

[3] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum FieldTheory, Westview Press, 1995.

[4] L. Álvarez-Gaumé and M. Á. Vázquez-Mozo, An Invitation to Quan-tum Field Theory, Springer, Berlin, 2012.

[5] R. Balian and J. Zinn-Justin, Methods in Field Theory, North-Holland &World Scientific, Singapore, 1981.

[6] Nobelprize.org, All Nobel Prizes in Physics, Nobelprize.org, RetrievedAugust 19, 2014, from http://www.nobelprize.org/.

[7] T. J. Hollowood, Renormalization Group and Fixed Points in QuantumField Theory, Springer, New York, 2013.

[8] D. I. Kazakov, Beyond the Standard Model (In Search of Supersymmetry),Lectures given at the European School for High Energy Physics, Cara-mulo, 2000, arXiv:hep-ph/0012288.

[9] L. N. Mihaila, J. Salomon and M. Steinhauser, Renormalizationconstants and beta functions for the gauge couplings of the StandardModel to three-loop order, Physical Review D 86 (2012) 096008, doi:10.1103/PhysRevD.86.096008.

[10] M. L. Alciati et al., Proton Lifetime from SU(5) Unification in ExtraDimensions, Journal of High Energy Physics 0503 (2005) 054, doi:10.1088/1126-6708/2005/03/054.

[11] J. Beringer et al. (Particle Data Group), Physical Review D 86 (2012)010001, doi: 10.1103/PhysRevD.86.010001.

23

Appendix A Feynman rules

For the φ4-theory the Feynman rules in momentum space are:

For each internal propagator:p

=i

p2 −m2 + iε.

For each vertex: = −iλ.

Integrate over undetermined loop momenta:∫ d4k

(2π)4 .

External lines do not need to be considered since we are only interested inamputated 1PI diagrams. Momentum has to be conserved in all vertices.In addition, possible symmetry factors are to be taken into account.

Appendix B Mathematical details

This Appendix contains some mathematical technicalities left out fromthe calculations.

B.1 Beta function identity

The beta function, also called the Euler beta function and the Euler integralof the first kind is defined by

B(m, n) ≡1∫

0

tm−1(1− t)n−1dt, m, n > 0,

24

and it is related to the gamma function by [2]

B(m, n) =Γ(m)Γ(n)Γ(m + n)

.

Let us use this to calculate an integral of the form∞∫

0

xα−1

(x + a)βdx.

A change of variables

x = a(

1u− 1)

, dx = − au2 du,

results in∞∫

0

xα−1

(x + a)βdx = aα−β

1∫0

uβ−α−1(1− u)α−1 du = aα−β B(β− α, α)

= aα−β Γ(β− α)Γ(α)Γ(β)

.

B.2 Feynman parametrization

The Feynman parametrization

1ab

=

1∫0

dx

[xa + (1− x)b]2

is rather easy to prove. First it is noted that

1ab

=1

b− a

(1a− 1

b

)=

1b− a

b∫a

dzz2 .

Then one makes a change of variables

z = ax + b(1− x), dz = (a− b)dx,

and the desired result follows immediately:

1ab

=1

b− a

0∫1

a− b

[ax + b(1− x)]2dx =

1∫0

dx

[xa + (1− x)b]2.

25

Appendix C MATLAB code

The plot of the three couplings was generated with MATLAB using thefollowing code:

a=91.1876

f=@(t) (1/0.017)-(41/(20*pi))*log(t/a)

g=@(t) (1/0.034)+(19/(12*pi))*log(t/a)

h=@(t) (1/0.118)+(7/(2*pi))*log(t/a)

fplot(f,[a 1e22],'-r')

hold on

fplot(g,[a 1e22],'-g')

fplot(h,[a 1e22],'-b')

hold off

set(0,'defaulttextinterpreter','latex')

set(gca,'XScale','log')

xlabel('Energy (GeV)','FontSize',11)

ylabel('Inverse coupling','FontSize',11)

text(1e4,50,'\(\alpha_\mathrmem^-1\)','FontSize',13)

text(1e4,28,'\(\alpha_\mathrmw^-1\)','FontSize',13)

text(1e4,10,'\(\alpha_\mathrms^-1\)','FontSize',13)

26