Running of the Coupling in the Φ -Theory and in the Standard Model

Home , Beta function (physics), Coupling constant, Landau pole

Running of the coupling in the φ4-theory and in the Standard Model

Olli Koskivaara

Research training thesis Supervisor: Kimmo Kainulainen October 9, 2014 Abstract

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

The second example of the running of the coupling is more of a phenomenological one. The three gauge couplings of the Standard Model and their scale dependence is studied. A plot of the inverses of the couplings as a function of energy is produced. It is noted that the couplings get near each other at an energy scale of approximately 1016 GeV but yet do not meet each other. A search for a theory predicting the uniﬁcation of the gauge couplings is mentioned.

Tiivistelmä

Tässä erikoistyössä kytkentävakion jouksemista tutkitaan ilmiönä kahden eri esimerkin kautta. Ensin määritetään φ4-teorian kytkinvakion juokse- misen 1-silmukkatasolla määräävä yhtälö. Tämä tehdään yksityiskohtai- sesti laskemalla ensin teorian ensimäisen kertaluvun korjaukset ja sitten renormalisoimalla 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 tehdään lyhyt katsaus beetafunktioiden ominaisuuksiin.

Toinen esimerkki on luonteeltaan fenomenologisempi. Standardimallin kolmea mittakenttäkytkentää ja niiden riippuvuutta energiaskaalasta tar- kastellaan tuottamalla kuva käänteisistä kytkentävakioista energian funk- tioina. Kytkentävakioiden havaitaan lähestyvän toisiaan energiaskaalan 1016 GeV lähettyvillä kohtaamatta kuitenkaan toisiaan. Lopuksi maini- taan kiinnostus mittakenttäkytkentöjen yhdistymisen ennustavaa teoriaa kohtaan.

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 coupling constant. This running is a feature of many interacting theories and plays a great role in modern physics. To put it brieﬂy, the phenomenon is about the energy scale dependence of the coupling constant describing the strength of the interaction of a theory.

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

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

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

1Not to be confused with the beta function familiar from mathematics, which we will actually 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-theory consists 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 theory supplemented with the interaction term,

1 2 1 λ L = ∂ φ − m2φ2 − φ4. (1) 2 µ 2 4! This toy model is often used as an example in quantum ﬁeld theoretic calculations since it is quite simple and exhibits many features present in more complicated theories. The Feynman rules for the theory are given in Appendix A.

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

(n) G ≡ hΩ |T{φ(x1)φ(x2) ··· φ(xn)}| Ωi . (2)

The simplest nontrivial connected Feynman diagrams, from which all other 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., they cannot be split into two separate diagrams by cutting one of the internal lines.

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

2 can hence be written as

" n # n ! (n) (2) D (D) (n) G (p1,..., pn) = ∏ G (pi) (2π) δ ∑ pi Γ (p1,..., pn). (3) i=1 i=1

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

= + 1PI

+ 1PI 1PI + ··· ,

where

1PI ≡ + (4) + + ··· = Γ(2)

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

Γ(4) = 1PI

  (5) ≡ +  + perm. + ··· , where perm. includes the two other topologically nonequivalent one-loop diagrams which are obtained by permuting the incoming and outgoing momenta.

3 2.2 One-loop corrections

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

2.2.1 Propagator

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

iλ Z d4 p i = − , 2 (2π)4 p2 − m2 + iε

1 where the factor 2 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 integral above is quadratically divergent. This is why we shall use dimensional regularization, i.e., we calculate the integral in a general dimension D. Our integral becomes iλ Z dD p i − µ4−D , 2 (2π)D p2 − m2 + iε where µ is an arbitrary parameter with dimensions of mass introduced to keep the dimensions of the integral unchanged. This step is quite crucial because this scale, entering our theory via the parameter µ, will play a key role in the behaviour of the coupling constant.

In order to calculate the regularized integral we shall use a trick known as the Wick rotation. We start by integration over p0. Due to the Feynman prescription iε the integrand has poles above and below the real axis, as shown in Figure 1. The Wick rotation corresponds to a change of variables D D 2 2 p0M −→ ip0E ; d pM −→ id pE, pM −→ −pE, (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 along the real axis to integration along the imaginary axis. The ﬁgure shows the whole integration contour together with the two poles above and below the real axis.

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

If f (z) is an analytic function, and f 0(z) is continuous at each point within and on a closed contour C, then I f (z) dz = 0. C

Since the contour in Figure 1 encloses no poles, the theorem above tells us that our integral over the contour vanishes. Furthermore, the integrand vanishes as |p0| → ∞, and thus the arcs do not contribute at inﬁnity. Hence the integrals over real and imaginary axis have to be equal and opposite. Using this fact and the change of variables (6) our integral reduces to a Euclidean one: iλ Z dD p i iλ Z dD p 1 − µ4−D = − µ4−D E . π D 2 − 2 π D 2 2 2 (2 ) p m + iε 2 (2 ) pE + m Here we have abandoned the prescription iε since it no longer plays any role after the Wick rotation.

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

5 Euclidean space:

∞ D ∞ Z Z Z 2π 2 Z dDx = dDΩ xD−1 dx = xD−1 dx, Γ D 0 2 0 where Γ is the gamma function [3]. This gives us

∞ D D D−1 iλ Z d p 1 iλ 2π 2 Z dp p − µ4−D E = − µ4−D E E (7) 2 (2π)D p2 + m2 2 Γ D (2π)D p2 + m2 E 2 0 E and we are only left to calculate the integral over pE. This can be done using the following result derived in Appendix B.1 using the Euler beta function: ∞ Z xα−1 Γ(α)Γ(β − α) dx = aα−β . (x + a)β Γ(β) 0 First we write

D − 1 1 2 1 pD−1 dp = pD−2 dp2 = p2 dp2 E E 2 E E 2 E E in our integral. Then starting from (7) we get

∞ ∞ D D D−1 D 2 2 −1 iλ 2π 2 Z dp p iλ π 2 Z p − µ4−D E E = − µ4−D dp2 E 2 Γ D (2π)D p2 + m2 2 (2π)DΓ D E p2 + m2 2 0 E 2 0 E D H H D D D − iλ π 2 Γ H Γ 1 − 2 1 = − µ4−D 2H 2 m2 DH D 2 (2π) Γ HH Γ(1) 2H |{z} = 1 4−D iλm2 µ2 2 D = − 4π Γ 1 − . 32π2 m2 2

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

e iλm2 µ2 2 e − 4π Γ − 1 . (8) 32π2 m2 2

6 Next we shall carry out some small-e approximations for this result. For the ﬁrst part involving e we write

e 2 µ2 2 e ln 4π µ e µ2 4π = e 2 m2 = 1 + ln 4π + O(e2). m2 2 m2

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

(−1)n 1 Γ(e − n) = + Ψ(n + 1) + O(e) , n ∈ N, n! e where Ψ is the dilogarithm function deﬁned as the logarithmic derivative of the gamma function,

d n 1 Ψ(x) = ln [Γ(x)] , Ψ(n + 1) = −γ + ∑ , dx i=1 i with γ = −Ψ(1) = 0.5772... being the Euler–Mascheroni constant [4]. Applying this to our gamma function yields

e 1 e e 2 Γ − 1 = e Γ = − 1 + − γ + O(e) 2 2 − 1 2 2 e where we ﬁrst used the recursive property of the gamma function, Γ(x + 1) = xΓ(x), and then expanded in terms of e.

Using these approximations in (8) results in

iλm2 2 γe e γe µ2 + 1 − γ − + 1 + − ln 4π + O(e). 32π2 e 2 2 2 m2 Taking the limit e → 0 we can conclude that at one-loop order the correction to the propagator is

iλm2 1 1 γ 1 µ2 Γ(2)(p2) = + − + ln 4π . (9) 16π2 e 2 2 2 m2

The correction thus consists of a divergent part ∼ e−1 and a finite part which will be insignificant for our purposes. By analytically continuing our original integral to D dimensions we extracted the infinities to one term which is easy to handle. Another observation to be made here is that, 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 diagrams, corresponding to s-, t- and u-channel scatterings. The three diagrams are related to each other by crossing symmetry, which means that they can be transformed to each other just by interchanging the Man- delstam variables suitably. It will therefore sufﬁce to calculate explicitly only one of them, e.g. the s-channel contribution represented by the loop diagram in equation (5).

Using the Feynman rules of our theory we have that the integral to be calculated is (−iλ)2 Z d4k i i , 2 (2π)4 k2 − m2 [(k − p)2 − m2] 1 where p is the sum of the incoming momenta, the factor 2 is a symmetry factor and we have dropped the prescriptions iε. The procedure to calculate this integral is very similar to what we already did with the propagator. First we note that the integral is divergent, this time logarithmically. This leads us to dimensional regularization and calculation of the integral λ2 Z dDk 1 µ4−D . 2 (2π)D (k2 − m2) [(k − p)2 − m2]

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

1 1 Z dx = , ab [xa + (1 − x)b]2 0 which after some algebra reduces our denominator as

1 1 Z dx = . 2 2 2 2 n o2 (k − m ) [(k − p) − m ] 2 2 2 0 [k − (1 − x)p] − m + p x(1 − x) We are then left to calculate 1 λ2 Z dDk Z dx µ4−D . D n o2 2 (2π) 2 2 2 0 [k − (1 − x)p] − m + p x(1 − x)

8 Since our integrals are now convergent after the regularization, we can switch the order of integration and shift the integration parameter as k −→ ek = k − p(1 − x), which leaves the measure dk unchanged. This gives us

1 λ2 Z Z dDek µ4−D dx . h i2 2 D 2 2 2 0 (2π) ek − m + p x(1 − x)

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

1 λ2 Z Z dDek µ4−D dx h i2 2 D 2 2 2 0 (2π) ek − m + p x(1 − x) 1 iλ2 Z Z dDek = µ4−D dx E h i2 2 D 2 2 2 0 (2π) −ekE − m + p x(1 − x)

D 1 ∞ D−1 iλ2 2π 2 Z Z dek ek = µ4−D dx E E 2 D (2π)D h i2 Γ 2 2 2 2 0 0 ekE + m − p x(1 − x) D −1 D 1 ∞ 2 2 iλ2 π 2 Z Z ekE = µ4−D dx dek2 2 D D E h i2 (2π) Γ 2 2 2 2 0 0 ekE + m − p x(1 − x)

D H 1 2 H D D Z D − iλ π 2 Γ H Γ 2 − h i 2 2 = µ4−D 2H 2 dx m2 − p2x(1 − x) DH D 2 (2π) Γ HH Γ(2) 2H |{z} 0 = 1 1 4−D iλ2 Z 4πµ2 2 D = dx Γ 2 − 32π2 [m2 − p2x(1 − x)] 2 0 1 e iλ2 Z 4πµ2 2 e = dx Γ . 32π2 [m2 − p2x(1 − x)] 2 0 Using the same small-e expansions as before and taking the limit e → 0 we get that the correction to the four-point function from the s-channel

9 loop is

 1  iλ2 1 γ 1 4πµ2 1 Z p2  − − − − 2 + ln 2 dx ln 1 2 x(1 x) . 16π e 2 2 m 2 m  0

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 integral. These ﬁnite parts of the correction are not relevant for our purposes, and therefore we merely stress that the overall correction to the four-point function at one-loop order is of the form

3iλ2 1 + FINITE . (10) 16π2 e

2.3 Renormalization

If we were studying the non-interacting Klein–Gordon theory, the parameters of the Lagrangian would correspond to physical parameters, e.g. m appearing in the Lagrangian would indeed be the physical mass observed in experiments. However, as seen in the previous section, adding the interaction to the theory causes divergences as we calculate corrections to the parameters. Because in experiments we measure ﬁnite quantities, this suggests that the parameters of our Lagrangian do not correspond to the physical ones.

It is useful to rewrite our Lagrangian (1) as

1 2 1 λ L = ∂ φ − m2φ2 − 0 φ4 , 2 µ 0 2 0 0 4! 0 where φ0, m0 and λ0 are the so-called bare parameters, which may be inﬁnite and cannot in general be measured in experiments. The next step is to introduce the renormalized (physical, measurable, ﬁnite) parameters φR, mR and λR related to the bare ones by

1/2 φ0 ≡ Zφ φR , (11) −2 λ0 ≡ Zφ ZλλR , (12) 2 −1 2 m0 ≡ Zφ ZmmR , (13)

10 where the factors Zφ, Zλ and Zm contain all the possible infinities. The point of this redefinition is that it allows us to split the Lagrangian into a renormalized one and a part involving the infinities called the counterterm Lagrangian.

Instead of dwelling upon the usefulness of the counterterms, we focus on the relation between the renormalized and bare coupling constant. For simplicity we will from now on consider a massless theory. Furthermore, since regularization will be necessary at some point, it is useful to deal from the beginning with a Lagrangian appropriate for D = 4 − e dimensions e 1 2 µ λ L = ∂ φ − R φ4 + L , e 2 µ R 4! R c.t. e where µ adjusts λR to be dimensionless and Lc.t. is the counterterm La- grangian. Now the relation between the bare and renormalized coupling reads

e −2 λ0 = µ Zφ ZλλR. (14)

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

(n) G0 ≡ hΩ |T{φ0(x1)φ0(x2) ··· φ0(xn)}| Ωi 1/2 1/2 1/2 = hΩ|T {Zφ φR(x1)Zφ φR(x2) ··· Zφ φR(xn)}|Ωi n/2 = Zφ hΩ |T{φR(x1)φR(x2) ··· φR(xn)}| Ωi n/2 (n) = Zφ GR . (15)

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

(n) (n) G Γ ∼ 0 , 0 h (2)in G0 which together with equation (15) yields

(n) −n/2 (n) Γ0 = Zφ ΓR . (16)

11 Now that we have renormalized our theory, we can deﬁne the physical coupling constant as the renormalized proper vertex function evaluated at some arbitrary momenta:

(n) 0 0 0 λR = iΓR (p1, p2,..., pn). It is important to realize that the reference point can indeed be freely chosen; we could have just as well deﬁned

(n) 0 0 0 bλR = iΓR (pb1, pb2,..., pbn),

(n) where ΓR is evaluated at some different momenta which can be related to a measured value of the coupling constant. It can be shown that the freedom in the choice of this subtraction point does not practically affect the values of physical quantities, such as the cross section σ, depending on the coupling constant. The numerical difference caused by different choices would always occur at higher orders in the coupling constant expansion than in the original calculation, i.e., there is no difference at the order we decide to calculate the values. This ensures that our perturbation theory is well deﬁned.

2.4 The Callan–Symanzik equation and the renormalization group

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

All in all we can rewrite equation (16) as

(n) n/2 −e (n) ΓR (p, λR, µ, e) = Zφ λ0µ , e Γ0 (p, λ0, e) .

12 n/2 Dividing both sides by Zφ and taking the derivative with respect to µ yields

∂ h n (n) i Z− /2 λ µ−e, e Γ (p, λ , µ, e) = 0. (17) ∂µ φ 0 R R

−e Keeping in mind that there is a µ-dependence in λR = λR (λ0µ ) too, opening equation (17) gives ∂ n (n) n ∂ (n) Z− /2 Γ + Z− /2 Γ = 0 ∂µ φ R φ ∂µ R n/2 ∂ −n/2 ∂λR ∂ ∂ (n) ⇔ µZφ Zφ + µ + µ ΓR = 0. (18) ∂µ ∂µ ∂λR ∂µ Deﬁning

∂ β(λ ) ≡ µ λ , (19) R ∂µ R 1 ∂ γ(λ ) ≡ µ ln Z (20) R 2 ∂µ φ and observing that

n ∂ ∂ ∂ −nγ = − µ ln Z = µ ln Z−n/2 = µZn/2 Z−n/2, 2 ∂µ φ ∂µ φ φ ∂µ φ we can write equation (18) as ∂ ∂ (n) µ + β (λR) − nγ (λR) ΓR (λR, µ) = 0. (21) ∂µ ∂λR This equation is known as the Callan–Symanzik equation and the renormalization group equation (RGE)3. It tells how changing the scale is related to changes 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 order to 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 our theory at different scales. This set of transformations is not however a group in the mathematical 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 in this work. Basically it explains our theory’s deviations from the scaling behaviour a similar classical theory would have [1].

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

1 4 " 2 # 2 ( ) p Ω (p, λ , µ, e) ≡ Γ 4 (p, λ , µ, e) i , R R R ∏ (2) i=1 ΓR (pi, λR, µ, e) which satisﬁes the homogeneous renormalization group equation (HRGE) ∂ ∂ µ + β (λR) Ω (λR, µ) = 0. (22) ∂µ ∂λR

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 it should not depend on our choice of λR and µ. This means that we can ﬁnd a parametrization which satisﬁes Ω (λR, µ) = Ω λ¯ R(t), µ¯(t) ¯ ∂ ∂λR ∂ ∂µ¯R ∂ ⇔ 0 = Ω λ¯ R(t), µ¯(t) = + Ω λ¯ R, µ¯ , (23) ∂t ∂t ∂λ¯ R ∂t ∂µ¯R where λ¯ R corresponds to the coupling constant at scale µ¯. Comparing equation (23) with the HRGE (22) one sees that ∂λ¯ (t) β λ¯ (t) = R , (24) R ∂t where λ¯ R(0) = λR corresponds to no scaling at all (µ¯(0) = µ).

As mentioned, the invariant charge is physical. We can thus, similarly as with the proper vertex functions, deﬁne the renormalized coupling constant as Ω evaluated at some reference momenta p0: 0 λR ≡ iΩ(p , λR, µ, e)

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. This observation together with the scaling properties discussed above yields t 0 0 −t 0 iΩ(e p , λR, µ, e) = iΩ(p , λR, µe , e) = iΩ(p , λ¯ R(t), µ, e) ≡ λ¯ R(t). So by scaling the momenta we obtained another value for the coupling constant; the coupling “runs” with the scale. What is the equation governing 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 has actually 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(λ ) = µ λ µ−eZ2 Z−1 . (25) R ∂µ 0 φ λ

The renormalization factors Zφ and Zλ are to be constructed in such a way that they cancel the divergences in the renormalized Lagrangian. From equation (9) we see that the one-loop correction to the propagator has no p2-dependence. This means that there is no wave function renormaliza- 4 tion at one-loop level, i.e. Zφ = 1.

From equation (10) we have that the total divergence in the corrections to the four-point function is 3iλ2µ−e 1 o , 16π2 e where we have adjusted the result to ﬁt our regularized Lagrangian Le. In order to have a counterterm canceling the divergence we need a renormalization factor 3λ µ−e 1 Z = 1 + o λ 16π2 e 4This can be justiﬁed by expanding Γ2(p2) around the chosen subtraction point and noticing that any counterterm would be p2-dependent. The absence of wave function renormalization at one-loop level is a special feature of the φ4-theory; e.g. in the Yukawa theory a wave function renormalization is required at one-loop level [3].

15 for the coupling. At one-loop order the renormalized coupling constant is then

2 −e ! ( ) i3λ µ 1 3λ 1 λ ≡ iΓ 4 = i −iλ µ−e + 0 = λ µ−e 1 − 0 R R 0 16π2 e 0 16π2 e −e −1 = λ0µ Zλ . Substituting this into equation (25) yields

∂ 3λ 1 3λ2µ−e β = µ λ µ−e 1 − 0 = −eλ µ−e + 0 , ∂µ 0 16π2 e 0 16π2 where we can ﬁnally take the limit e → 0 to obtain

3λ2 β(λ) = . (26) 16π2

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

∂λ 3λ2 = . ∂t 16π2 This simple differential equation is easily solved, the solution being

λ(0) λ(t) = . (27) − 3tλ(0) 1 16π2 We can further write this result without the scaling parameter t. Let λ(0) = λ(Q0) be the coupling constant at some scale Q0. Then at another scale Q we have for some value of the parameter t t Q Q = Q0e ⇔ t = ln . Q0 Using this in equation (27) yields

λ(Q0) λ(Q) = . (28) 1 − 3λ(Q0) ln Q 16π2 Q0

This result, equation (28), is what we have been aiming for from the beginning. 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 equation (28) to determine the coupling at other scales. A notable feature is that the coupling constant increases with increasing scale Q.

Furthermore, we can characterize the behaviour of the coupling constant by just one dimensionful parameter, namely the scale at which the coupling diverges. From equation (28) we see that the coupling constant becomes inﬁnite at some scale Qe at which the denominator vanishes:

! 16π2 3λ(Q0) Qe − ⇔ 3λ(Q0) 1 2 ln = 0 Qe = Q0 e (29) 16π Q0

This scale is known as the Landau pole, and it can be used to remove the dependence on a reference scale Q0 in equation (28). This procedure is known as dimensional transmutation since it replaces the dimensionless parameter λ characterizing the interaction by a dimensionful one. It should be noted that the scale in equation (29) may be far beyond the reach of our 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 is 4 very large if λ(Q0) is small. Indeed, the φ -theory allows a perturbative coupling constant expansion only at small mass scales (large distances). If we start with a small λ for which the perturbation is sensible and begin to increase the scale, we will need to add more and more terms to the expansion 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 property of e.g. quantum chromodynamics (QCD).5 Perturbation is valid at large scales but breaks down at small scales.

For quantum electrodynamics (QED) the situation is similar to that of the φ4-theory: the coupling becomes stronger at larger scales. The Landau 34 pole of QED corresponds to an energy scale ΛLandau ≈ 10 GeV. The pole causes no problems, because QED is expected to be uniﬁed with other interactions below this scale. Furthermore, quantum gravity steps in at scale 1019 GeV. [4]

5In 2004 David J. Gross, H. David Politzer and Frank Wilczek received the Nobel Price in 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 the Callan–Symanzik beta-function of the theory in hand. Indeed, looking at 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 theory increases with the scale. This is exactly the case for the φ4-theory; from equation (26) we see that β > 0. Similarly a negative beta-function implies an inverse relationship between the coupling and the scale, which leads to 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 properties are controlled by the fixed point structure of the theory. Fixed points are those values of the coupling constant for which the beta-function vanishes. Let λF be such a point of a certain theory, i.e. β(λF) = 0. Expanding 0 β around its zero to first order gives β(λ) ≈ (λ − λF)β (λF). Then using the definition of the beta-function we get

∂ µ λ = (λ − λ )β0(λ ) ∂µ F F β0(λ ) eλ − λ µ F ⇒ F = e , (30) λ − λF µ where we integrated from µ to µe. Looking at equation (30) we see that the sign of the derivative of the beta- 0 function β (λF) tells about the behaviour of the coupling constant close 0 to the fixed point. If β (λF) < 0, then λ → λF independently of the initial value eλ as the scale µ increases, and λF is said to be an UV stable fixed 0 point. If on the other hand β (λF) > 0, then λ tends to λF as µ decreases and goes away from λF for increasing µ. In this case λF is called an IR stable 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; we may have an interacting theory where the coupling constant actually is a constant. Such theories are called conformal ﬁeld theories. One interesting feature of these theories is that due to the vanishing beta-function they are scale 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, occurring at 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 only this so called Gaussian fixed point corresponding to zero coupling. There is however no reason why a theory could not have a beta-function with multiple zeros, corresponding to fixed points at non-zero couplings. The study 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 Standard Model and examine how they run with the energy scale. Mathematical derivations are left aside, and emphasis will be on the phenomenology related to the couplings.

The Standard Model is a gauge field theory based on the symmetry group SU(3) ⊗ SU(2) ⊗ U(1), and the interactions are specified by three cou- 0 pling constants usually denoted by gs, g and g corresponding to the gauge symmetries SU(3), SU(2) and U(1) respectively. These gauge couplings define three of the fundamental forces: strong nuclear, weak nuclear and electromagnetism. In order to compare these interactions, it is useful to define the set of couplings as

5 (g0)2 5 α ≡ αem = 2 , (31a) 3 4π 3 cos θW g2 α ≡ αw = 2 , (31b) 4π sin θW g α ≡ s , (31c) s 4π where the subscripts in the couplings stand for electromagnetic, weak and strong, and θW is the Weinberg angle and α the ﬁne-structure constant. 5 The factor 3 in the deﬁnition of αem is related to normalizing U(1) in order to get a possible SU(5) structure. [8, 9]

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

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

αi (Q0) αi(Q) = , (32) 1 − b α (Q ) ln Q i i 0 Q0 where the coefﬁcients bi are given by  2πb = 41 ,  1 10 19 2πb2 = − 6 ,  2πb3 = −7.

One can use experimentally measured values for the ﬁne-structure constant, the Weinberg angle and the strong coupling constant to determine the couplings in equation (31) at a certain scale. Choosing this scale as the mass of the Z-boson MZ = 91.1876 ± 0.0021 GeV [11], the values of the 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 function 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; the asymptotic freedom of QCD and the increase of the QED coupling with energy. It also gives us a hint about something new: a possible unifi- 16 cation of all the three couplings at some large energy ΓU ∼ 10 GeV, where new physics may have been emerged. Even if we were to take into account the errors in the coupling constant measurements, the couplings would not meet at one point within the scope of the Standard Model. This has led physicists to search for a theory that would predict the unification of the three gauge interactions in the same way as electromagnetism

20 60

55 1 50 αem− 45

30 1 αw− 25

Inverse coupling 20

15 1 10 αs− 5

5 10 15 20 10 10 10 10 Energy (GeV)

Figure 2: The inverses of the three couplings αi as functions of energy. The energy axis is logarithmically scaled. This ﬁgure was produced using MATLAB interface; see Appendix C for details. and weak interaction are already combined into electroweak theory in the Standard Model. Supersymmetry and its extensions are probably the 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. Gravity starts much weaker than the three other forces and is extrapolated to join them at about 1019 GeV. The most ambitious theories try to add this to the unification, resulting in a theory combining all known fundamental forces to a single one. This Theory of Everything (ToE) would have to take care of both quantum field theory and general relativity. Many models have been constructed, string theory being the most famous candidate, but none of them has been agreed on generally. The main problem lies in the difficulty to experimentally verify these theories; no man-made particle 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-theory at one-loop order. This was done by ﬁrst calculating the one-loop corrections to the theory, then renormalizing it and ﬁnally using the thus obtained beta-function to study the scale dependence of the coupling. The coupling was found to increase with increasing momentum scale. A short introduction to beta-functions was also given, mainly underlining their importance in studying coupling constants.

Utilizing the insight gained from the φ4-calculations we then studied the running of the three gauge couplings of the Standard Model. A plot of their inverses against energy was produced. The plot shows that the couplings get near each other at an energy scale of approximately 1016 GeV but do not meet each other. A theory which predicts the joining of the couplings could be used to describe all the three interactions by a single coupling constant. A quest for such a general theory and the absence of one was noted.

The gauge couplings and their non-unification in Standard Model is of course just one example of the phenomenological aspects of the running of the coupling. Another interesting example of the phenomenon is the electroweak 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 the vacuum is unstable, indicating that our universe might exist in a “false” vacuum and could spontaneously fall to a lower state at the same time ceasing to exist as we know it. This is however mere speculation, and more precise calculations and experiments need to be done before mak- ing any definite conclusions. Although intriguing, the vacuum stability and many other phenomena related to the running of the coupling are beyond 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 for Physics and Engineering, 3rd Edition, Cambridge University Press, New York, 2007.

[3] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, 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, Retrieved August 19, 2014, from http://www.nobelprize.org/.

[7] T. J. Hollowood, Renormalization Group and Fixed Points in Quantum Field 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, Renormalization constants and beta functions for the gauge couplings of the Standard Model 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 Extra Dimensions, 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:

i For each internal propagator: = . p p2 − m2 + iε

For each vertex: = −iλ.

Z d4k Integrate over undetermined loop momenta: . (2π)4

External lines do not need to be considered since we are only interested in amputated 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 from the calculations.

B.1 Beta function identity

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

1 Z B(m, n) ≡ tm−1(1 − t)n−1dt, m, n > 0, 0

24 and it is related to the gamma function by [2] Γ(m)Γ(n) B(m, n) = . Γ(m + n) Let us use this to calculate an integral of the form ∞ Z xα−1 dx. (x + a)β 0 A change of variables 1 a x = a − 1 , dx = − du, u u2 results in ∞ 1 Z xα−1 Z dx = aα−β uβ−α−1(1 − u)α−1 du = aα−β B(β − α, α) (x + a)β 0 0 Γ(β − α)Γ(α) = aα−β . Γ(β)

B.2 Feynman parametrization

The Feynman parametrization 1 1 Z dx = ab [xa + (1 − x)b]2 0 is rather easy to prove. First it is noted that b 1 1 1 1 1 Z dz = − = . ab b − a a b b − a z2 a Then one makes a change of variables z = ax + b(1 − x), dz = (a − b)dx, and the desired result follows immediately: 0 1 1 1 Z a − b Z dx = dx = . ab b − a [ax + b(1 − x)]2 [xa + (1 − x)b]2 1 0

25 Appendix C MATLAB code

The plot of the three couplings was generated with MATLAB using the following 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,'\(\alpha_{\mathrm{em}}^{-1}\)','FontSize',13) text(1e4,28,'\(\alpha_{\mathrm{w}}^{-1}\)','FontSize',13) text(1e4,10,'\(\alpha_{\mathrm{s}}^{-1}\)','FontSize',13)