Fermion vertices in 1+1 and 3+1 space-time dimensions

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Master Thesis

Fermion vertices in 1+1 and 3+1

space-time dimensions

by

Jelmer Doornenbal

10595759

December 2020

MSc Physics

Theoretical Physics

At the Dutch National Institute

for Subatomic Physics, NIKHEF

Supervisor/Examiner:

Examiner:

Em. Prof. Dr. Piet Mulders

Dr. Juan Rojo

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Abstract

In this thesis, Feynman diagrams with fermion vertices will be calculated in two and four dimensions in order to compare both dimensions. Specifically, the anomalous magnetic moment and mass renormalization will be considered in multiple theories. At first glance, one would expect there not to be an equivalent of the anomalous magnetic moment in two dimensions, because the magnetic field is absent in this dimension.

However, it is still possible to distinguish the vertex contributions between parts that are equivalent to the spin-dependent part and the spin-independent part. There-fore, two-point and three-point function diagrams have been calculated in other to compare the contribution of the fermion vertices in two and four dimensions. These results have been applied to several two- and four-dimensional models in order to find a possible connection between the one-loop structure in the two space-times.

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Chapter 1 Introduction

One of the biggest achievements in physics in the latter half of the 20th century, is the formulation of the Standard Model of Particle Physics. It is a set of mathemat-ical formulae that describe elementary particles and their interactions. Specifmathemat-ically, the theory encapsulates three of the four fundamental forces in nature. These forces are the electromagnetic, weak and strong interactions. The Standard Model is able to predict experimental results and explain phenomena involving all these forces.

However, in the search for a theory of everything, one must find a theory that describes the three forces of the Standard Model combined with the gravitational force. This unification has yet to be completed. Other questions that go beyond the Standard Model are issues on naturalness and the absence of supersymmetry. Therefore, while the Standard Model is generally believed to be self consistent, it is not the perfect theory.

In the light of this ascertainment, other fundamental questions about the Stan-dard Model itself can be raised. Primarily, the origin and the role of symmetries in the Standard Model are unknown. The theory has a built-in gauge symmetry of SU (3) ⊗ [SU (2) × U (1)]. Only when these symmetries are assumed to be present, the Standard Model can be properly formulated. Hence, investigating the origin and possible simplifications of the symmetries of this theory is compelling.

One way to simplify the symmetry structure of the Standard Model has been pro-posed by my supervisor in [12]. His theory therefore inspired the research described in this thesis. This theory uses multipartite entanglement in order to explain the symmetries of the Standard Model. The theory is constructed from a supersym-metric qubit basis with right states and left states. These states form the basis of the Hilbert space. In this Hilbert space, two fermionic and two bosonic states can be formed through using fermionic creation operators. Moreover, excitations are created through the application of bosonic creation operator on these states.

Essentially, this theory consists of left- and right-movers in a (1+1)-dimensional space. In order to gain in a complexity to the same degree as the Standard Model, a three-fold symmetry has to be introduced. In particular, this theory uses the tripar-tite space H(3) = H ⊗ H ⊗ H combined with SO(3) symmetry, time inversion and parity reversal symmetry to show how the symmetries of the Standard Model could

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emerge. This extension from a one-dimensional Hilbert space to a three-dimensional Hilbert space is comparable with the common example of the three-dimensional har-monic oscillators that can be built up from a direct product of three one-dimensional harmonic oscillators.

The dynamics of this theory are built up from the supersymmetric Wess-Zumino La-grangian in (1+1)-dimension.[21] The notion that one can build up a four-dimensional theory from a theory in two dimensions by essentially adding rotations raises cer-tain questions. Specifically, one might wonder whether it is possible to have a two-dimensional equivalent of an otherwise known as purely four-dimensional phe-nomena. One example of an aforementioned effect could be the anomalous magnetic moment, which is given by the higher order contributions of quantum effects to a particle’s magnetic moment. [10] These effects have been calculated up to tenth order in perturbation theory for Quantum Electrodynamics (QED) and agree with experiments to more than 10 significant digits.

The possibility of a two-dimensional equivalent of the anomalous magnetic moment will be examined through studying different fermion vertices. At first glance, one would expect this not to exist, since there clearly is no such thing as a magnetic moment in two dimensions. The electromagnetic tensor only has an electric field component. Whereas the magnetic field, that would have been perpendicular to this field, is absent. However, it is still possible to distinguish momenta in, often referred to in four dimensions as, a spin-dependent part and a spin-independent part through the Gordon decomposition and split them up into form factors.

Another goal of this thesis is to determine whether a link between this four-dimensional constant and the value found in two dimensions exists. In order to perform these calculations, the structure generated by 2d gamma matrices will be explored. This basis is less complicated than in four dimensions, since the basis is spanned by a combination of only contains scalars, pseudoscalars and vectors.

In order to get a complete view of the terms contributing to a possible anoma-lous magnetic moment, the full structure of contributions caused by all vertex in-teractions that span the Dirac basis will be examined. These calculations will be performed in general n dimensions with the goal to compare two dimensional and four dimensional results. Through this approach, the known result for four dimen-sional QED can be retraced. Moreover, by calculating the fermion self energy and applying the Ward identity all these terms can be independently checked. These calculations will also be performed on the supersymmetric extensions of QED.

Apart from QED, the anomalous magnetic moment will also be calculated in a particular case of the (1+1)-dimensional Wess-Zumino model. Not only was this theory a building block for that what inspired this thesis, the supersymmetric Wess-Zumino model in 2d has the interesting property that it contains a massive vector field and Yukawa interactions. The implications of these properties on the anoma-lous magnetic moment in this theory will be examined. Moreover, a more detailed look on other characteristics of this model, such as cancellation of mass corrections, will be given. These cancellations are a solution to the naturalness problem.

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Chapter 2 The anomalous magnetic moment

In this chapter, the basic concepts involved in the determination of the anomalous magnetic moment will be explained. This explanation will lead to an example calcu-lation of the commonly known four dimensional anomaly to the magnetic moment first calculated by J. Schwinger in 1948. However, the chapter will first start by explaining what a magnetic dipole moment is. Afterwards, Feynman diagrams will be introduced as our method of calculating scattering amplitudes for the relevant processes in a quantum field theory. The outcome of these calculations can be split up into form factors that allow us to identify the contributing part to the magnetic moment.

2.1 Magnetic dipole moment

As stated in the introduction, the aim of this thesis is to investigate a possible two-dimensional anomalous magnetic moment. In order to properly understand the question, we first examine what a magnetic dipole moment is. The magnetic dipole moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. It is defined as a vector ~µ that, with an external magnetic field ~B, creates an aligning torque ~τ , as

~

τ = ~µ × ~B. (2.1)

A classical current moving through a wire produces a magnetic field. The magnetic dipole moment, or magnetic moment, in this configuration is given by,

~ µ = I

Z

d~a = I~a, (2.2)

where I is the current flowing though the wire, and ~a is the vector area. Similarly, if we would examine a current flowing through a circular wire in the xy-plane, it would create a magnetic field in the z-direction through the Lorentz force. This magnetic field would be aligned with the magnetic moment ~µ, which is given by

~ µ = I

Z

d~a = Iπr2 z.ˆ (2.3) In this formula, r is the radius of the circular wire and I is the current. [8] Since the current is defined as the charge per unit time that passes a given point, the magnetic

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moment can also be linked to the angular momentum ~L of the particles. Therefore, in the usual four-dimensional setting, we have particles with charge q and angular velocity ω moving through the loop. This leads to the following expression for ~µ,

~ µ = q 2π/ωπr 2 _{z =}_ˆ q 2m ~ L. (2.4)

Even though quantum particles can’t be described as orbiting distributions of uni-form charge-to-mass ratio, the magnetic moment of these Dirac particles, that are without internal structure, has a very similar form. For instance, electrons have a magnetic dipole moment of,

~

µ = g e 2me

~

S. (2.5)

In the formula, g is the so-called g-factor or the dimensionless magnetic moment, e is the charge of the electron, me is the mass of the electron and ~S is the spin angular

momentum. The g-factor is a dimensionless constant that links the spin angular momentum of a particle with its magnetic moment. At tree level, we will later see that g = 2 for fermions. However, the g-factor is susceptible to quantum effects at higher orders. 99% of these corrections come from QED interactions, where fermions interact with the electromagnetic field through photons. This anomaly is called the anomalous magnetic moment. [10] In this thesis, we will focus on the tree level contribution and the first order correction originating from this theory.

2.2 Feynman diagrams

In order to calculate the magnetic moment of the electron, one must find its predic-tion given by Quantum Electrodynamics. This can be found by calculating Feynman diagrams. A Feynman diagram is a pictorial representation of a perturbative con-tribution to the amplitude from some initial state to a final state. The lines in these diagrams represent particles and the points where they interact are called vertices. [14]

For each different quantum field theory, one can deduce the set of corresponding Feynman rules, such that the scattering amplitudes of the diagrams can be calcu-lated. They are related to each other via the following formula,

iM˙(2π)nδ(n)(pA+ pB−

X pF) =

(sum of all connected, amputated Feynman diagrams) , (2.6)

where pAand pBare the incoming momenta, pF is the outgoing momenta and n is the

number of dimensions. Moreover, connected and amputated diagrams are diagrams with only one part from which one can not truncate or amputate its external legs. In particular, the scattering process corresponding to our calculation of the magnetic moment is given by the class of Feynman diagrams located on the next page. In this figure, the gray circle implies a summation over all amputated diagrams. These diagrams represent electron scattering with one or multiple virtual photons.

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Figure 2.1: The class of Feynman diagrams used to calculate the magnetic moment of the electron. [14]

In the known case for the anomalous magnetic moment, the set of Feynman rules of Quantum Electrodynamics was used. The Lagrangian of this theory is,

L = −1 4FµνF

µν _{+ ¯}_{Ψi( /}_{∂ − m)Ψ + eA}

µ× ¯ΨγµΨ, (2.7)

where Fµν is the electromagnetic tensor, Ψ is the electron field and Aµis the

electro-magnetic field. The first two terms describe free photons and the last term describes the interaction between photons and the electromagnetic field. The complete set of the Feynman rules of QED can be found in the appendix.

2.3 Form factors

In the previous section, our method of performing field theoretical calculations on interactions between particles through Feynman diagrams and their corresponding rules has been introduced. The next step in the calculation of the anomalous mag-netic moment, is the splitting of the outcome of these calculations in two parts by introducing form factors. By using the Feynman rules given in the appendix, the calculation of the magnetic dipole moment can be performed. The analysis of the next three sections has been based on information given in ”An Introduction to Quantum Field Theory” by Peskin and Schroeder(1995). Applying these rules, the diagram on the left-hand side of Figure 2.1 has the following amplitude,

iM = ie2(¯u(p0)Γµ(p0, p)u(p)) 1 q2 ((¯u(k

0

)γµu(k)) , (2.8)

where the sum of vertex functions is given by −ieΓµ(p0, p). In this formula, u(p), ¯u(p0) are the spinors or anti-spinors with associated momentum p, p0, k or k0 and q is the momentum of the massless propagator. To lowest order, Γµ _{= γ}µ_{, which means that}

the lowest order diagram simply has one vertex interaction between two fermions coupling through a gamma matrix γµ_{. However, in general the vertex function Γ}µ

can be written as a linear combination of γµ _{and momenta p}µ_{, p}0µ_,

Γµ= Aγµ+ B(p0µ+ pµ) + C(p0µ− pµ_), _(2.9)

in this formula, A, B and, C have to be functions of the only non-trivial scalar q2. Moreover, in the expression above C has to be 0 due to the Ward identity: qµΓµ= 0.

This identity states that the longitudinal polarization of a photon is unphysical and hence has to be equal to 0. [14]

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Therefore, the most simplified form of the vertex function Γµ is,

Γµ = Aγµ+ B(p0µ+ pµ). (2.10) However, this expression can be rewritten in a convenient way through the Gordon decomposition. This formula splits the Dirac current into a part associated with the motion of the center of mass of the particles and a part associated with spin.

¯

u(p0)γµu(p) = ¯u(p0) p

0µ_{+ p}µ 2m + iσµν 2m u(p). (2.11) In this formula σµν ₌ i 2[γ

µ_{, γ}ν_{]. This formula allows us to write the general sum of}

vertex diagrams Γµ _{as an expression containing two form factors,}

Γµ(p0, p) = γµF1(q2) +

iσµν 2m F2(q

2_), _(2.12)

where F1(q2) and F2(q2) are the so-called electric form factor and the magnetic form

factor respectively. Hence, F1 is helicity-preserving, while F2 is helicity-flipping.

Since the form factor F2 is the so-called magnetic form factor, our calculations will

be mainly focused on finding this factor. These form factors are defined as the Fourier transform of the electric charge density ρ(~r),

F (~q) = Z

d3r exp((i~q · ~r)ρ(~r), (2.13) Another name for these form factors is structure functions, since it is used to describe the structure of a nucleus. In this light, one can view q2 as a measure for the radius. In order to relate the form factors to physically measurable quantities, we can also express Γµ _{in terms of the Sachs form factors instead,}

GE(q2) = F1(q2) −

q2

4m2F2(q

2_), _(2.14)

GM(q2) = F1(q2) + F2(q2). (2.15)

From these two formulas, we can deduce a charge Q and a static magnetic dipole moment µ [24],

Q = eGE(0), (2.16)

µ = eGM(0)

2m . (2.17)

2.4 Anomalous magnetic moment

These form factors all contain complete information about the effect of the elec-tromagnetic field on the fermions. This allows us to investigate the contribution of quantum mechanics on the particles. Particularly, the contribution of quantum mechanics on the magnetic moment is our interest. This is often referred to as the anomalous magnetic moment. In the upcoming paragraphs the factors that con-tribute to these quantum effects will be derived and explained.

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Following the Feynman rules and the QED Lagrangian, the scattering amplitude of the diagram in Figure 2.1 is,

iM = ie ¯ u(p0)(γµF1(q2) + iσµν 2m F2(q 2 ))u(p) ˜ A(~q). (2.18)

This equation can be simplified by realizing that spin-coupling can be isolated by setting ~p = 0. Moreover, since the momenta are on-shell, only the term proportional to ~q are important to keep track of. This allows the following expansion of the non-relativistic Dirac spinors in the non-non-relativistic limit,

u(k) = √ k · σζ √ k · ¯σζ ≈√m(1 + 1 2~q · (~σ/2m) ζ) √ m(1 − 1₂~q · (~σ/2m) ζ) , (2.19)

where, k is the momentum of the spinor, σ is a Pauli matrix and ζ is a two component up- or down-spinor. This expansion can be used to simplify equation (2.18). Without necessarily showing all the algebraic steps involved, the simplification leads to,

iM = 2meζ0† 1 2m ijk_qj_σk_[F 1(0) + F2(0)]) ζ ˜Ak(~q). (2.20) In this formula, ˜Ak(~q) is the electromagnetic four-potential. In order to relate this expression to magnetism and spin, we substitute the Fourier transform of the mag-netic field and the normalization of relativistic states into (2.20),

˜

B(~q) = −iijkqiA˜k(~q), (2.21) hSk_{i = 2mζ}0†

Skζ. (2.22)

This leads to the following expression for the scattering amplitude,

iM = −e

m(F1(0) + F2(0))hS

k_{i ˜}_B(~_q). _(2.23)

This expression can be related to the magnetic moment through interpreting the scattering amplitude M as the Born approximation to electron scattering in a po-tential well. This popo-tential would then be of the following form,

V (~(x)) = −hµ(~x)i · ~B(~x), (2.24) where hµ(~x)i is the expected value of the magnetic moment. This allows us to derive an equation for this magnetic moment by comparing it to (2.5),

~ µ = 2[F1(0) + F2(0)] e 2m ~S = g e 2m ~ S. (2.25)

Hence, the earlier introduced g-factor is given by [14],

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It is now a simple task to calculate the value of these two form factors, F1(q2) and

F2(q2) by examining the lowest order Feynman diagram. This diagram is shown

below in Figure 2.2.

Figure 2.2: The lowest order Feynman diagram.

At lowest order, Γµ _{= γ}µ_{, hence the scattering amplitude is given by,}

iM = −ie¯u(p0)γµu(p). (2.27) Therefore, by comparing the result with the general formula for the scattering am-plitude in equation (2.18). We can conclude that at the lowest order F1(0) = 1

and F2(0) = 0. In particular, F1 = 1 at all orders, since it gives the coefficient in

the eAµΨγ¯ µΨ-coupling in the Dirac equation. It implicates that the electric charge

corresponds to what can be measured with Coulomb’s Law at large distances. This would not get any radiative corrections. [16] Therefore, all corrections from quan-tum mechanics is encapsulated in the factor F2, which is zero at lowest order, but

non-zero at higher orders. Therefore, the contribution to the magnetic moment of these quantum effects or the anomalous magnetic moment is,

F2(0) = a =

g − 2

2 . (2.28)

2.5 The anomalous magnetic moment in 4d

In previous sections, we have built up an understanding on what the anomalous magnetic moment is and how to calculate it. Although, it was not the goal of this thesis, it is instructive to show the known calculation of the anomalous mag-netic moment in four dimensional QED. Specifically in this section, the first order contribution of quantum effects to the magnetic dipole moment will be calculated. This contribution comes from the amputated one-loop diagrams, since the tree-level contribution is absent on F2(0) as shown earlier. The diagram is shown below.

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The diagram on the last page was shown with assigned momenta. [14] The incoming momentum is labeled with p0, the outgoing momentum is labeled with p, the third momenta is labeled with q and the internal loop momentum k. By applying the Feynman rules to first order, we find the following expression,

iM = −ie¯u(p0)Γµu(p) = ¯u(p0)[ Z d4_k (2π)4 −i k2_{+ i}(−ieγ ν₎ i(/p 0_{+ /}_{k + m)} (p0 _{+ k)}2_{− m}2_{+ i}(−ieγ µ₎ _(2.29) i(/p + /k + m)

(p + k)2_{− m}2_{+ i}(−ieγν)]u(p) = ¯u(p 0 )e3 Z d4k (2π)4 Nµ D u(p).

In these expressions, the abbreviation /p = γµpµ is used. The numerator and the

denominator are given by,

Nµ _{= γ}ν

(/p0+ /k + m)γµ(/p + /k + m)γν, (2.30)

D = (k2_{+ i) [(p}0

+ k)2− m2_{+ i] [(p + k)}2_{− m}2 _{+ i].} _(2.31)

The denominator can be simplified by using Feynman paramaters. This technique helps to rewrite these three factors in the denominator to a single term, raised to the third power. In general, this formula is given by,

1 A1A2...An = Z 1 0 dx1....dxnδ(Σxi− 1) (n − 1)! []x1A1+ x2A2+ ...xnAn]n (2.32)

In this case a product of three denominators has to be simplified. The formula used to rewrite the denominator is,

1 D = Z 1 0 dx1dx2dz δ(x1+ x2+ z − 1) 2 D3. (2.33)

Using this formula, the new denominator D with an extra term ∆ is,

D = (k + x1p + x2p0)2− ((1 − z)2 m2− x1x2 q2) = (k + x1p + x2p0)2− ∆, (2.34)

∆ = ((1 − z)2 m2 − x1x2 q2). (2.35)

Now, spherical symmetry of the integral can be used in order to further simplify our equations. By defining a shift for the loop momentum to complete the square l = k + x1p + x2p0, the denominator can be rewritten as,

1 D = Z 1 0 dx1dx2dz δ(x1+ x2+ z − 1) 2 (l2_{− ∆ + i)}3. (2.36)

This leads to a new formula for the scattering amplitude with a yet to be simplified numerator. Γµ(p0, p) = −2ie2 Z 1 0 dx1dx2dz δ(x1+ x2+ z − 1) Z d4l (2π)4 Nµ (l2 _{− ∆ + i)}3. (2.37)

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Substituting this momentum shift into the numerator gives,

Nµ_{= γ}µ_(/_{k − x}

1/p + (1 − x2)/p0+ m)γν(/k + (1 − x)1/p − x2/p0+ m)γµ

= γµ(/k + z/p + (1 − x2)/q + m)γν(/k + z/p − x2/q + m)γµ, (2.38)

where /q = /p0 − /p.

The numerator can be simplified using the following properties of gamma matrices in d = 4,

γµγµ = 4, (2.39)

γµγνγµ= −2γν, (2.40)

γµγργσγµ= 4 gρσ, (2.41)

γµγνγργσγµ= −2 γσγργν. (2.42)

A full list of gamma matrix identities in n dimensions can be found in the appendix. Hence, the numerator then is,

Nµ_{= −2m}2_γµ_{+ 4m(2zp}µ_{+ (1 − 2x} 2)qµ)

− 2(/l + z/p − x2/q)γν(/l + z/p + (1 − x2)/q). (2.43)

It is important to note that integrating over terms linear in l will not give any contribution due to spherical symmetry. Using the anti-commutation relation: {γµ

, /p} = 2pµ, the Dirac equation:/pu(p) = mu(p), ¯u(p0)/p0 = ¯u(p0)m and lµlν = gµν l

2

D

the numerator becomes,

Nµ_{= γ}µ_{l2_{− 2(z + x}

1x2)q2− 2(1 − 2z − z2)m2}

− 2mz(z − 1)(p0µ+ pµ) − 2m(z − 2)(x1− x2)qµ. (2.44)

where 1 − x − y − z = 0 has been used. Moreover, terms that are anti-symmetric under the exchange of x and y can be removed, since the integrand is symmetrical. The last simplification can be done through substituting the Gordon decomposition:

¯

u(p0)γµu(p) = ¯u(p0)[(p

0

+ p)µ 2m +

iσµνqν

2m ]u(p). (2.45) Altogether, the simplified form of the numerator is now,

Nµ_{= γ}µ_l2_{− q}2_γµ_{(2(z + x}

1x2) + 2m2γµ(4z − z2− 1) − imσµνqν(1 − z)(2z). (2.46)

Similarly, to the general form of Γµ in the previous section, one can write down a general form for the numerator

Nµ _{= N}

1γµ+ N2

iσµν_q ν

2m . (2.47)

Specifically, since we are interested in the anomalous magnetic moment, we are interested in calculated F2(0). This is given by the following to be calculated integral

F2(q2 = 0) = −2ie2 Z dx1dx2dz δ(x1+ x2+ z − 1) Z ddl (2π)d m2(1 − z)4z (l2_{− ∆ + i)}3. (2.48)

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This contour integral can be calculated by first applying a Wick rotation. [14] Introducing an Euclidian 4-momentum variable lE: l0 = ilE0, l = lE. This helps us

evaluate the integral in (3+1)d as,

Z _d4_l (2π)4 1 [l2_{− ∆]}m = i (−1)m 1 (2π)4 Z d4lE 1 [l2 E + ∆]m = i(−1) m (2π)4 Z dΩ4 Z ∞ 0 dlE l3_E [l2 E+ ∆]m = i(−1) m (4π)2 1 (m − 1)(m − 2) 1 ∆m−2. (2.49)

A complete list of these integral expressions can be found in the appendix. Therefore, in this case with m = 3 and q2 _{= 0,}

Z _d4_l (2π)4 N2 [l2_{− ∆]}3 = −i 2(4π)2 N2 ∆ = −i 32π2 4z (1 − z). (2.50) The usage of this expression leads to an integral coming from the Feynman param-eters. Calculating this integral leads to the final result for the anomalous magnetic moment, a = g − 2 2 = F2(q 2 _{= 0) =} e2 16π2 Z dx1dx2dz δ(x1+ x2+ z − 1) 4z 1 − z = e 2 16π2 Z 1 0 dz (1 − z) 4z 1 − z = e2 8π2 = α 2π. (2.51)

This is the known result for four-dimensional Quantum Electrodynamics. It was first found by Julian Schwinger in 1948. [18] In this formula, α is the fine structure constant, which is defined as α ≡ _4πe2. Nowadays, the anomalous magnetic moment of the electron has been calculated analytically up to eighth order and are even calculated through algorithms up to tenth order in perturbation theory. [10] The calculated values for the anomalous magnetic moment agree with experiments to more than 10 digits, making it the most accurately verified prediction in all of physics. These three values are given below,

• First order: ae = _2πα = 0.0011614,

• Theoretical value: ae = 0.001159652181643(764),

• Experimental value: ae= 0.00115965218073(28).

As mentioned previously, the electromagnetic force is not the only force that causes this anomaly. The weak force and the strong force also contribute to the anomalous magnetic moment of the electron. However, due to the low mass of the electron these contributions are very small of the order of 1%. For the more massive fermion, the muon, these contributions are already much larger. More interestingly, when adding these corrections obtained from calculations, one finds that they differ from experimental data with 3.5σ. This indicates a process that goes beyond the Standard Model. [3]

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Chapter 3 General dimension n diagram

calculations

In the previous chapter, the anomalous magnetic moment has been introduced and the known contribution of Quantum Electrodynamics in four dimensions has been calculated. By building upon this knowledge, we can try to perform a similar cal-culation in two dimensions in order to answer our main question on the existence of a two dimensional equivalent of the anomalous magnetic moment. In order to properly perform this calculation, we must focus on the specific differences between the two spacetimes. That is the first topic of this chapter. Mainly, the differences will be examined through defining gamma matrices. These were the matrices that coupled the fermions to the massless photon field in QED causing higher order cor-rections. Later on, these gamma matrices will be used to build up a basis that span all the possible interactions in any theory. This will help us show the existence of terms related to spin even in two dimensions. Finally, the vertex function and the self-energy diagram will be calculated in general n dimensions with all possible interactions.

3.1 Gamma matrices conventions

The aim of this section is to introduce the conventions regarding gamma matrices. In this thesis, from on here out the chosen metric has the following signature: ηµν =

ηµν _{= diag(+1, −1, ...., −1). Since, our systems live in flat Minkowski space the}

metric tensor gµν = ηµν. In d dimensions, Gamma matrices are a set of d 2d/2× 2d/2

-matrices γµ, such that their anti-commutation relations generate a Clifford Algebra Cl1,d−1(R): {γµ, γν} = 2ηµνI2d/2. [11] In order to define the gamma matrices, we

use the fact that a known set of matrices, specifically the set of Pauli matrices, also generate an algebra that is isomorphic to the Clifford Algebra in (3+1)-d: {σi_{, σ}j_{} = 2δ}ij_{I. This will help us relate the gamma matrices to Pauli matrices.}

These matrices are the following,

σ0 _{= 1 =}1 0 0 1 , σ1 =0 1 1 0 , σ2 =0 −i i 0 , σ3 =1 0 0 −1 . (3.1)

The indices of these matrices can be lowered by multiplying with the metric tensor gµν and then contracting: gµνσµ = σν. Therefore, the sigma matrices with lower

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raised again, this can be done via gµνσµ= σν.

In particular, our interest lies in the representations of this algebra in (1+1)d and (3+1)d. There are multiple representations that generate the Clifford Algebra. The chiral representation has been chosen, because in the chiral representation the prod-uct of all gamma matrices is diagonal. In two dimensions, in order to have the gamma matrices generate the Clifford Algebra and for the product of the two gamma matrices to be diagonal, the following representation comes up:

γ0 = ρ1 =0 1 1 0 , γ1 = iρ2 =0 −1 1 0 . (3.2)

Lowering the indices of these gamma matrices can be done through using a sim-ilar formula: gµνγµ = γν. Hence, the gamma matrices with lower indices are

γ0 = g00γ0 = γ0 and γ1 = g11γ1 = −γ1. The product of both matrices is

di-agonal: γ5 = γ0γ1 = σ3 = −γ0γ1. Alternatively, γ5 = _2!1µνγµγnu and µνγ5 = 1

2![γ

µ_{, γ}ν_{] = −iσ}µν_.

In order to extend the two dimensional representation to a four dimensional one, a tensorproduct can be taken between the time-component of γν in (1+1)d: γ0 = σ1

and the time component σ0 to get,

γ0 = ρ1⊗ σ0 =     0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0     . (3.3)

Moreover, a tensorproduct between the space-component of γν _{in (1+1)d: iρ}2 _and

each individual space-component σi: γi = iρ2⊗ σi _gives,

γ1 = iρ2⊗ σ1 =     0 0 0 −1 0 0 −1 0 0 1 0 0 1 0 0 0     , γ2 = iρ2⊗ σ2 =     0 0 0 i 0 0 −i 0 0 −i 0 0 i 0 0 0     , γ3 = iρ2⊗ σ3 ₌     0 0 −1 0 0 0 0 1 1 0 0 0 0 −1 0 0     .

In a similar way as before, the indices of the gamma matrices can be lowered with gµνγµ = γν in order to find a similar result: γ0 = γ0 and γi = −γi. The four gamma

matrices can be summarized into one as,

γµ= 0 σ µ ¯ σµ ₀ , γ5 = iγ0γ1γ2γ3 = −σ0 0 0 σ0 , (3.4)

where σµ = (σ0, σi) and ¯σµ _{= (σ}0_{, −σ}i_{). Similarly to the (1+1)-d, γ}

5 can be

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3.2 Fierz identity

Now that the gamma matrices have been properly defined in 2d and 4d, we can form a basis spanned by these matrices. Specifically, by taking appropriate products between the gamma matrices γµ, one can form a set of d × d matrices that span the

d × d dimensional space of all matrices. This exact expansion is given by the Fierz Identity, which allows us to rewrite a product of two spinors as a linear combination of the elements of products of the Clifford Algebra and the individual spinors. [11]

ψ ¯χ = 1 2d/2 d X n=0 1 n!(−) n(n−1)/2_γ µ1...µn( ¯χγ µ1...µn_ψ). _(3.5)

For d = 2, the basis exists of three different elements: ψ ¯χ = 1₂(( ¯χψ) + γµ( ¯χγµψ) − 1

2γµν( ¯χγ

µν_{ψ)). This expression can be rewritten using the following relations: γ} µν = 1 2[γµ, γν] = µνγ5, µν µν _{= 2 to,} ψ ¯χ = 1 2(( ¯χψ) + γµ( ¯χγ µ_{ψ) − γ} 5( ¯χγ5ψ)) ≡ 1 2(cS1 + cV µγ µ_{+ c} Pγ5). (3.6)

The three different elements can be identified as one scalar S, one pseudoscalar P and one vector V. Hence, the three matrices that span the space of 2 × 2 matrices are 1, γµ and γ5. These matrices also form the Lorentz structure of vertex:

Γ = CS1 + CV µγµ+ CPγ5. (3.7)

Apart from this Lorentz structure, the Fierz identity can help us with the simplifi-cation of products of gamma matrices. We define the following matrices: Γ1_S = 1, Γ1

V and Γ2V = γµ and Γ1P = iγ5. Using these definitions, there is a formula that helps

us reduce a product of three gamma matrices to one gamma matrix. [13]

Γq_IΓs_JΓIq = CIJΓsJ. (3.8)

with CIJ given by the following matrix, where scalar corresponds to index 1, vector

to index 2 and pseudoscalar to index 3.

CIJ = 1 2   1 1 −1 2 0 2 −1 1 1  . (3.9)

For which individual entry of CIJ has been calculated using the following formula:

ΓAΓB = 1 2dTr[Γ

A_Γ

CΓBΓD]ΓDΓC. (3.10)

This multiplication table will be commonly used throughout this thesis. Similarly, there is a matrix CIJ for the sigma matrices. The four matrices that span the space

of 2 × 2 hermitian matrices are Γ1

S = 1, Γ1V = iσ1, ΓV2 = iσ2 and Γ1P = iσ3. Hence,

CIJ = 1 2   1 −1 −1 −2 0 −2 −1 −1 1  . (3.11)

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For d = 4, expanding the Fierz identity gives, ψ ¯χ = 1₄(( ¯χψ)+γµ( ¯χγµψ)−1₂γµν( ¯χγµνψ)− 1 3!γµνρ( ¯χγ µνρ_{ψ) +} 1 4!γµνρσ( ¯χγ µνρσ_{ψ)). Using γ} µν = ₂i[γµ, γν] = σµν, γµνρ = iµνργ5γρ,

µνρµνρ = 6, γµνρσ = iµνρσγ5 and µνρσµνρσ = 24, this can be rewritten to,

ψ ¯χ =1 4(( ¯χψ) + γµ( ¯χγ µ_{ψ) − σ} µν( ¯χσµνψ) − γ5γµ( ¯χγ5γµψ) + γ5( ¯χγ5ψ)) ≡1 4(cs1 + c µ vγµ+ cµνT σµν + cAµγµγ5+ cPγ5). (3.12)

Therefore, the 16 matrices that span the space of 4 × 4 matrices are 1, γµ_{, σ}µν_{, γ}µ_γ 5

and γ5. Once more, these matrices also form the Lorentz structure:

Γ = CS1 + CV µγµ+ CT µνγµ[γν] + CA µγµγ5+ CPγ5. (3.13)

Moreover, defining the following 16 matrices: Γ1_S = 1, Γ1_V through Γ4_v = γµ, Γ1_T through Γ6

T = σµν(µ < ν), Γ1Athrough Γ4A= γµγ5 and Γ1P = γ5. These five groups of

matrices are either scalars S, pseudoscalars P, vectors V, axial-vectors A or tensors T. A product of three gamma matrices can be reduced to one gamma matrix using (3.8), with CIJ in this case being,

CIJ = 1 8       2 2 −1 −2 2 8 −4 0 −4 −8 −24 0 −4 0 −24 −8 −4 0 −4 8 2 −2 −1 2 2       . (3.14)

In this matrix, the indices 1, 2, 3, 4 and 5 correspond to S, V, T, A and P respectively.

Comparing the structures between two- and four-dimensional gamma matrices, two differences immediately come to mind. The first characteristic is the striking sim-plicity of the two-dimensional structure. Its basis only contains three elements and four matrices in total, compared to five elements with 16 matrices in four dimen-sions. The number of possible interactions in two dimensional theories therefore would also be much smaller and easier to calculate. The second characteristic stems from the first one. In two dimensions one can get every single element through com-bining γ0 and γ1. It is this property that will have interesting consequences later for

two dimensional QED and the Wess-Zumino model.

3.3 The vertex function in n dimensions

In this section, the vertex function will be calculated in general n dimensions. In order to acquire a result, the gamma matrix conventions and identities as discussed earlier will be used. This to be calculated three-point function will also contain all possible couplings that span the Dirac basis in that dimension defined by the Fierz identity. This will give us insight in what possible coupling does contribute to an anomalous magnetic moment. In four dimensions, we have seen that we need at least one coupling that contains a gamma matrix in order to use the Gordon decomposition.

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By using the two dimensional gamma matrices as defined before, we can deduce a two dimensional equivalent of the Gordon decomposition,

¯

u(p0)γµu(p) = ¯u(p0)[(p

0_{+ p)}µ

2m − µν_q

νγ5

2m ]u(p). (3.15) Again, we need at least one coupling that contains a gamma matrix in order to have a term that would have a non-zero F2. Following from this Gordon decomposition,

the splitting of the vertex function in two form factors in two dimensions is,

Γµ(p0, p) = γµF1(q2) −

µν_q νγ5

2m F2(q

2_), _(3.16)

Recalling the Fierz identity up to two dimensions and four dimensions, we can write out the total expansion of possible coupling combinations. In two dimensions this Dirac basis is given by,

Γ2d= CS1 + CPγ5+ CV µγµ, (3.17)

where CS, CP and CV are the coupling constants associated with scalar, pseudoscalar

or vector coupling respectively. In four dimensions, the Dirac basis contains more components,

Γ4d = CS1 + CPγ5+ CV µγµ+ CA µγµγ5+ CT µνγµ[γν]. (3.18)

In this equation, the first three couplings are the same as in two dimensions. How-ever, two couplings have been added: the axial vector coupling with constant CA

and the tensor coupling with constant CT. These two expansions can be used to

write down the full contribution of the first order loop diagram as shown in Figure 2.3. This formula is the following,

iM = −ie¯u(p0)Γ(3)(p0, p)u(p) = ¯u(p0)[C_a2 Z _dn_l (2π)n −i l2_{+ µ}2_{+ i}(−iΓ ν₎ i(/p 0_{+ /l + m)} (p0_{+ l)}2 _{− m}2_{+ i}(−iγ µ₎ _(3.19) i(/p + /l + m)

(p + l)2_{− m}2_{+ i}(−iΓν)]u(p) = ¯u(p 0 ) − i Z dn_l (2π)n Nµ D u(p),

where now the mass of the propagator is not necessarily 0, but given by µ. The next step, as previously showed, is to simplify the numerator and the denominator. These are given by,

Nµ = Γν(/p0+ /l + m)γµ(/p + /l + m)Γν (3.20)

D = (l2+ µ2+ i) [(p0+ l)2− m2+ i] [(p + l)2− m2+ i] (3.21) The denominator can be simplified in the same way as in the previous calculation using Feynman parameters in order to unify the three factors as,

D = (l + x1p + x2p0)2− ((1 − z)2 m2+ zµ2− x1x2 q2)

= (l + x1p + x2p0)2− ∆, (3.22)

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Similarly, to the example of the four-dimensional calculation, the spherical symmetry of the integral helps us simplify our equations. By applying the substitution: l = l + x1p + x2p0, the numerator can be simplified to,

N = Γν(/l − x1/p + (1 − x2)/p0+ m)γµ(/l + (1 − x)1/p − x2/p0 + m)Γν

= Γν(/l + z/p + (1 − x2)/q + m)γµ(/l + z/p − x2/q + m)Γν. (3.24)

Further simplification of the numerator is a bit more involved than for the four di-mensional case in QED. In this case, the five different elements that span the Dirac basis in 4d create five different numerators to be computed in general n dimen-sions. In order to fully simplify these five equations, we use the following general n dimensional gamma matrix identities [14],

{γµ, γ5} = 0 (3.25) {/p, γµ_{} = 2p}µ _(3.26) γµγµ= n, (3.27) γµγνγµ = (2 − n)γν, (3.28) γµγργσγµ = 4 gρσ− (4 − n)γργσ, (3.29) γµγνγργσγµ = −2 γσγργν + (4 − n)γνγργσ (3.30)

By applying these gamma matrix identities, combined with the same simplification process as in the earlier example calculation, the five numerators can be calculated. However, in order to have a unified notation between both dimensions, we need to define one common Gordon decomposition to use through: iσµν = −µνγ5. This

equality is true in two dimensions, therefore from now on we only use the Gordon decomposition of 4d: (2.45). The simplified numerator are the following,

NSV S = CS2[ 2 − n n γ µ_l2_{+ γ}µ_(m2_{(z + 1)}2_{) + x} 1x2q2) − imσµνqν(z2− 1)], (3.31) NP V P = CP2[ n − 2 n γ µ l2 + γµ(m2(−(1 − z)2) − x1x2q2) + imσµνqν(1 − z)2], (3.32) NV V V = CV2[ (2 − n)2 n γ µ_l2_{+ γ}µ_(m2_(−2z2_{− 2 + 8z + (4 − n)(1 − z)}2_{) + q}2_{((4 − n)x} 1x2 − 2(1 − x1)(1 − x2))) − imσµνqν((1 − z)(2z + (4 − n)(1 − z)))], (3.33) NAV A = CA2[ (2 − n)2 n γ µ l2+ γµ(m2(−2z2− 2 − 8z + (4 − n)(z + 1)2) + q2((4 − n)x1x2 − 2(1 − x1)(1 − x2))) − imσµνqν(−2z2− 6z + (4 − n)(z2− 1))], (3.34) NT V T = CT2[ 2 − n 2 (n − (2 − n) 2_)γµ_l2_{+ γ}µ_(m2_(z2_{(9(4 − n) − (4 − n)}2₎ + 2z(3(4 − n)2) + 9n − 16) + γµ(q2(3(4 − n)x1x2+ (9n − 16)(1 − x1)(1 − x2))) − imσµν_q ν(z2(3(4 − n) − (4 − n)2) + z(3(4 − n) + 2(4 − n)2 + 2(4 − n)2+ 9n − 16) + 1(−n − (4 − n)2))]. (3.35)

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At first, we were only looking for a two-dimensional equivalent of the anomalous magnetic moment. Therefore, in this thesis our analysis will only continue on the first three numerator in n = 2. Analysis on the other numerators can be found in the appendix. Using formula (2.47), we can write down the expressions for F2(0) in

all three cases as,

F₂iV i = −ie Z dx1dx2dz δ(x1+ x2+ z − 1) Z _dn_l (2π)n NiV i 2 D . (3.36) Specifically for each case F2(0) is given by,

F₂SV S(0) = −iC_S2 Z dx1dx2dz δ(x1+ x2+ z − 1) Z _dn_l (2π)n 2m2_(z2_{− 1)} (l2_{+ ∆ − i)}3, F₂P V P(0) = −iC_P2 Z dx1dx2dz δ(x1+ x2+ z − 1) Z _dn_l (2π)n −2m2_{(1 − z)}2 (l2_{+ ∆ − i)}3, F₂V V V(0) = −iC_V2 Z dx1dx2dz δ(x1+ x2+ z − 1) Z _dn_l (2π)n 2m2_{((1 − z)(2z + (4 − n)(1 − z)))} (l2_{+ ∆ − i)}3

By first integrating over l and then over x1, x2 and z in n = 2, we can calculate F2,

F₂SV S(0) = −C_S2 1 2πm 2 Z 1 0 dz (1 + z)(1 − z) 2 ((1 − z)2_m2_{+ zµ}2₎2 (3.37) = −C_S2 1 2π 1 2[ 1 m2 log( µ2 m2) + 2(µ2_{− 2m}2₎ µm2pµ2_{− 4m}2(log( µ(µ −pµ2_{− 4m}2₎ 2m2 − 1)] F₂P V P(0) = −C_P2 1 2πm 2 Z 1 0 dz (1 − z) 3 ((1 − z)2_m2_{+ zµ}2₎2 (3.38) = −C_P2 1 2π 1 2[ 4 µ2 _{− 4m}2 − 1 m2log( µ2 m2) − 2µ(µ2− 6m2₎ m2_(µ2_{− 4m)}3₂(log( µ(µ −pµ2_{− 4m}2₎ 2m2 − 1)] F₂V V V(0) = C_V2 1 2π2m 2 Z 1 0 dz (1 − z) 2 ((1 − z)2_m2_{+ zµ}2₎2 (3.39) = C_V2 1 2π 1 2[ 4 µ2_{− 4m}2 + 16m2 µ(µ2_{− 4m}2₎3₂(log( µ(µ −pµ2_{− 4m}2₎ 2m2 − 1)]

As you can tell, in the limit of µ → 0, where the propagator would be massless, as in 4d QED, only F₂P V P and F₂V V V contain one finite term, the other terms are logarithmically infrared-divergent. Therefore, more information on the mass ratio between photons and fermions in 2 dimensional QED is necessary in order to make a definitive conclusion on what the anomalous magnetic moment in that case is. This shall be the main topic in chapter 4.

One interesting feature that can be noticed from these results is that the addition of these three different terms leads to,

ΣF2(0) = F2SV S(0) + F P V P 2 (0) + F V V V 2 (0) = 1 2π 1 2[ 1 m2 log( µ2 m2(−C 2 S+ C 2 P) + 1 µ2_{− 4m}2(−4C 2 P + 4C 2 V) + log( µ(µ −pµ2_{− 4m}2₎ 2m2 − 1) (− 2(µ 2_{− 2m}2₎ µm2_pµ2_{− 4m}2C 2 S+ 2µ(µ2_{− 6m}2₎ m2_(µ2_{− 4m)}32 C_P2 + 16m 2 µ(µ2_{− 4m}2₎32 C_V2)] (3.40)

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Choosing C_S2 = C_P2 = C_V2 leads to all terms cancelling and the result would be F2 = 0. Concretely, this means that if there were a theory in two dimensions

con-taining a scalar field, a pseudoscalar field and a vector field couple to fermions with equal masses, there would not be a contribution of quantum effects to the magnetic moment. This creates the suggestion that investigating more complex theories than QED could show us something profound about the anomalous magnetic moment.

3.4 Field strength renormalization

In the previous section, the anomalous magnetic moment was calculated in two di-mensional QED as a function of electron charge e, electron mass m and the photon mass µ. The anomalous magnetic moment, however, was only one part of the vertex function that was split up into the two form factors. The form factors F1 can also

be calculated from the numerators. In chapter 2, it was mentioned that this factor F1 = 1 at all orders. Therefore, this section will show how this is exactly the case

for all couplings in n dimensions and use this fact to independently check our results.

The form factors F1 can be calculated starting from the numerators given in section

3.4. By applying the Gordon decomposition in a similar way as in (2.47), the integrals for F1 can be written down. These integrals can be found in the appendix.

The integral for n dimensional QED is the following,

F₁V V V(0) = C_V2 1 (4π)n/2( (2 − n)2 2 Γ(2 − n 2) Z 1 0 dz (1 − z) 1 (m2_{(1 − z)}2_{+ zµ}2₎2−n₂ + Γ(3 − n 2) Z 1 0 dz m2(1 − z)−2z 2_{+ 8z − 2 + (4 − n)(1 − z)}2 (m2_{(1 − z)}2_{+ zµ}2₎3−n₂ ). (3.41)

This integral contains a ultraviolet divergence in d = 4 due to the Γ(2 −n₂)-term. In two dimensions, this integral also is not equal to zero as the higher order contribution of this term should have been. Therefore, these terms have to be renormalized such that the anomalous magnetic moment nor the electric charge grow infinitely with these high order loop corrections. Quantum Electrodynamics is renormalized through defining counterterms that absorb these divergences. The renormalized QED Lagrangian is defined as follows,

L = −1 4(F µν r ) 2 + ¯ψr(i /∂ + m)ψr− e ¯ψrγµψrArµ− 1 4δ3(F µν r ) 2 + ¯ψr(iδ2∂ − δ/ m)ψr− eδ1ψ¯rγµψrArµ. (3.42)

In this formula, ψr is the renormalized wavefunction, δ1, δ2, δ3 and δm are the added

counterterms that cancel the divergences that come up in amplitude calculations. [14] δ1 is linked to the vertex function, δ2 and δm to the electron-self energy and δ3

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These counterterms fix the electron mass at m, the residues of the photon and electron propagators at 1, and the electron charge at e through the following four renormalization conditions, Σ(/p = m) = 0, (3.43) d dpΣ(/p)|/p=m= 0, (3.44) Π(q2 = 0) = 0, (3.45) ieΓµ(q = 0) = −ieγµ (3.46) From these normalization conditions, one has to set δ1 = δF1(0), which is the sum of

F1(0) at each order. After this renormalization, F1(0) = 1 at all orders as previously

noted. This counterterm is related to the counterterm of the electron self-energy through the Ward identity: δ1 = δ2. This identity shows that the coefficients of

/

∂ and eAµ _{do not get affected by radiative corrections. [16] By way of explanation,}

this means that the relative charge of the protons and electrons does not change in this quantum field theory.

This consequence of the Ward identity can be used as a separate way to check our calculation of F1 and indirectly F2 by calculating δ2 = _dpdΣ(/p)|/p=m related to the

electron self-energy. The electron self-energy is associated with the process given by the following diagram,

Figure 3.1: One-loop correction of the electron self-energy.

This diagram contains an incoming fermion with momentum p and a propagator loop with momentum l. Similarly to the calculation of the vertex function, all possible couplings will be examined that together span the Dirac basis (3.13). The scattering amplitude is therefore given by,

− iM = i( /P + M ) P2_{− M}2 − iΣ2(P ) i( /P + M ) P2_{− M}2 , where Σ2(P ) = Z dnl (2π)n −i l2_{− µ}2_{+ i}(−iΓ ν₎ i( /P + /l + M ) (P + l)2_{− M}2_{+ i}(−iΓν) Σ2(P ) = − Z _dn_l (2π)n N D (3.47)

where the numerator N and the denominator D are given by,

N = Γν_{(/l + /}_{P + M )Γ}

ν (3.48)

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The denominator can again be simplified with Feynman parameters to,

D = (l + zP )2− ((1 − z)2_µ2_{+ zM}2_{− z(1 − z)P}2_{) = (l − zP )}2_{− ∆,} _(3.50)

∆ = (1 − z)2µ2+ zM2− z(1 − z)P2 (3.51) Similarly to the previous calculations, the numerator can be simplified by first shift-ing the loop momentum l = l + zP and usshift-ing spherical symmetry to remove terms linear in l. Afterwards, the gamma matrix identities will create the last simplifica-tion, which gives the following five numerators,

ΣSS₂ = C_S2 1 (4π)n/2Γ(2 − n 2) Z 1 0 dz z/p + m (−z(1 − z)p2_{+ zµ}2_{+ (1 − z)m}2₎2−n 2 , (3.52) ΣP P₂ = C_P2 1 (4π)n/2Γ(2 − n 2) Z 1 0 dz −z/p + m (−z(1 − z)p2_{+ zµ}2_{+ (1 − z)m}2₎2−n₂ , (3.53) ΣV V₂ = C_V2 1 (4π)n/2Γ(2 − n 2) Z 1 0 dz (2 − n)z/p + nm (−z(1 − z)p2_{+ zµ}2_{+ (1 − z)m}2₎2−n₂ , (3.54) ΣAA₂ = C_A2 1 (4π)n/2Γ(2 − n 2) Z 1 0 dz (2 − n)z/p − nm (−z(1 − z)p2_{+ zµ}2_{+ (1 − z)m}2₎2−n₂ , (3.55) ΣT T₂ = C_T2 1 (4π)n/2Γ(2 − n 2) Z 1 0 dz ((2 − n) 2_{− n)z/p − n(n + 1)m} (−z(1 − z)p2_{+ zµ}2_{+ (1 − z)m}2₎2−n₂ . (3.56)

It is important to remember that the last two amplitudes only appear in 4d, since the 2d Lorentz structure does not contain axial vectors or tensors. From these expressions, the counterterm δ2 can be computed through δ2 = _dpdΣ(/p)|/p=m. As an

example, the derivative of ΣV V₂ will be shown and compared to F₁V V(0), δ2 = d dpΣ(/p)|/p=m= C 2 V 1 (4π)n/2Γ(2 − n 2)[ Z 1 0 dz (2 − n)z (m2_{(1 − z)}2_{+ zµ}2₎2−n₂ (3.57) +m 2_{((2 − n)z + n)(n − 4)z(z − 1)} (m2_{(1 − z)}2_{+ zµ}2₎3−n₂ ] (3.58)

It can be shown through integration by parts that this is indeed equal to −F1(0)

and hence δ1 = δ2. The radiative mass corrections therefore cancels with the F1

contribution of the vertex function, even though the diagrams seem unrelated. All other terms can be found in the appendix.

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Chapter 4 2d Quantum Electrodynamics

In the previous chapter, we calculated the anomalous magnetic moment in 2 dimen-sions from all three couplings. The result, however, was inconclusive, since the result depended on the mass of the photon. Therefore, in this chapter we will dive deeper into the properties of two-dimensional electromagnetism and two-dimensional Quan-tum Electrodynamics. These properties will then be used to calculate the anomalous magnetic moment in two dimensions with massive and massless fermions.

4.1 Electromagnetism in two dimensions

First, we will describe two-dimensional electromagnetism. As previously mentioned, the electromagnetic tensor only has one component F01 while the gauge field Aµ has

two components for µ = 0, 1. These facts lead to a number of interesting properties compared to the four dimensional theory. [19] The Lagrangian that describes two dimensional electromagnetic systems is given by the following formula,

L = − 1 2e2F01F

01_{+ A}

µjµ. (4.1)

In this formula, it is the gauge field Aµ that couples to the current jµ. From this

Lagrangian a couple of interesting properties can be deduced. Firstly, the equation of motion of this theory predicts that this theory only allows for a constant electric field: ∂0F01 = ∂1F01 = 0. This can be explained by considering that, while the

gauge field Aµ_{in two dimensions has two components, only one of them is physical.}

Since, one can use gauge symmetry to fix A0 = 0, only the spatial component of

the gauge field is left over. The electric field is then time-independent. After im-posing the equation of motion, the longitudinal fluctuations of A1 _{are filtered out.}

This implies that there are no transverse waves in two dimensions, and hence the electromagnetic field has no degrees of freedom to propagate.

Moreover, after adding matter to the system, one can deduce that the electric field emitted by a point charge is constant. Moreover, the energy required to have such a point charge is infinite. Finally, the energy stored in the system grows linearly with distance. Therefore, one can say that charges in two dimensions are confined. [19]

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While, as mentioned in the previous paragraphs, electromagnetism in 2d has no propagating solutions, its quantum properties are still worth exploring. Specifically, the Lagrangian that describes the quantum levels of electromagnetism is given by,

L = − 1 2e2F01F

01₊ θ

2πF01. (4.2)

In this Lagrangian, the added θ is the so called theta angle. This theta angle can be viewed as a total derivative and therefore it does not affect the equation of motion. It is a gauge that can be added to our system. In this theory, it can be interpreted as a background electric field that for example adds first order phase transitions.

4.2 Quantum Electrodynamics in two dimensions

However, we are not only interested in the properties of the quantum levels of electromagnetism in two dimensions. Specifically, this thesis aims to examine models that could give rise to terms comparable to what is perceived as an anomalous magnetic moment. A model that contains some of the properties of two dimensional electromagnetism as discussed, while possibly having a non-zero anomalous magnetic moment is the Schwinger model. [17] This model describes two dimensional QED. Its Lagrangian is the following,

L = − 1 2e2F01F 01₊ θ 2πF01+ i ¯ψ(∂µγ µ_{− eA} µγµ)ψ − im ¯ψψ. (4.3)

In this theory with massive fermions, it is possible to have a background electric field F in two dimensions. This background electric field can be seen as if it is created by two external charges at plus and minus infinity. It has a set of stable vacua associated with it: θ = 2πF/e. When the background electric field is strong enough, electron-positron pairs can be produced. Moreover, the external charges become screened. The particles are still influenced by the same confining potential as in the previous two models.

When fermions are massless, m = 0, equation (4.3) is the Lagrangian of the fully soluble massless Schwinger model. In this case, an electron and a positron are not simply connected and interacting through an electromagnetic field. Since due to the fact that these particles are massless, they now move with the speed of light. This causes the destruction of an electric field by the vacuum polarization. This is, in fact, means that the particles are no longer influenced by a linear confining potential. They are now screened as a Yukawa potential. Moreover, the background field is in this case not present. Therefore, the stable vacua, which were created by the background electric field, now have become degenerate. This indicates a sponta-neous breaking of a symmetry. Specifically, it is the chiral symmetry that has been broken due to the fact that axial vector current is not conserved in two dimensions.

∂µψγ¯ µγ5ψ =

e π

µν_∂

µAν (4.4)

Through this breaking of chiral symmetry, the photon obtains a mass. [4] This is a direct product from the fact that in 2d axial vectors and vectors are not linear independent: γµγ5 = −µνγν. It can be calculated through examining the vacuum

polarization diagram. The diagram that corresponding to this interaction is given on the next page. [14]

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Figure 4.1: Vacuum polarization diagram.

The scattering amplitude belonging to this diagram, with a fermion loop with loop momentum k and incoming momentum q, can be calculated as follows,

iΠµν(q) = (−ie)2(−1 2) Z _d2_k (2π)2 Tr[γ µi(/k + m) k2_{− m}2 )γ ν i(/k + /q + m) (q + k)2 _{− m}2)] = i(q2gµν − qµqν)e 2 π, (4.5)

where −ie is the coupling constant and (−1₂) comes for the fermion loop. The struc-ture of this expression is one of a photon mass term. These massive excitations of this theory are that of neutral massive particles with mass: µ2₀ = e_π2. [17] This mass of the photon arose from the vacuum polarization amplitude through a pole at q2 _{= 0.}

This pole emerged from an intermediate state between a fermion and anti-fermion through their at low energy scales. The appearance of his pole in two dimensions again shows that the theory is not reguralizable in a way such that it is both gauge invariant and the axial vector current is conserved. This non-conservation shows the loss of right-moving fermions and the sudden appearance of left-moving fermions.

Concretely, through the properties discussed above, fermions are absent in the phys-ical spectrum of both cases. The physphys-ical state with the lowest energy is a bound state of a massive fermion-antifermion. This boson, the Schwinger boson, is free if the fermion are massless. Moreover, excitations of higher states are also free n-particle states. If the fermions are massive, however the Schwinger boson is now interacting with other particles. This creates a spectrum full of n-bosonic bound states. Due to these interations, the Schwinger boson, or the photon field, now gets extra mass corrections. To first order, these mass corrections are now given by,

µ2₁ = µ2₀+ µ0m exp γ. (4.6)

In this formula, µ1 is the mass of the Schwinger boson in the massive model, mu0

the mass of the Schwinger boson in the massless case, m the fermion mass and γ the Euler-Mascheroni constant. [2]

4.3 Anomalous magnetic moment in 2d QED

In the previous two sections, we examined the properties of two dimensional electro-magnetism and Quantum Electrodynamics in order to attempt to find an expression for the photon mass. This attempt succeeded and it showed us that in 2d QED with massless fermions the photons have the following mass: µ2₀ = e_π2. In this mass-less limit, the anomalous magnetic moment can now be calculated using previously calculated expression (3.39) . The result is the following,

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FQED2 2 (m 2 = 0, q2 = 0) = e 2 π 1 µ2 = 1. (4.7)

Hence, the anomalous magnetic moment to first order is equal to the electric form factor F1. In total, the g-factor relating the spin angular momentum of a particle

to its magnetic moment is now to first order: g = 2(1 + 1) = 4. This g-factor is relatively large. Its size can be compared to the g-factor of the neutron, another neutral massive particle, which has gn= 3.82608545(90). [7]

The anomalous magnetic moment can also be calculated in massive two dimensional QED using the photon mass of (4.6). In order to be able to rewrite the to be calculated integral (3.39), we examine the ratio of the photon mass and the fermion mass. This ratio is given by,

µ2 1 m2 = e2 πm2 + e m√πexp(γ). (4.8) Using this ratio, the anomalous magnetic moment in two dimensional QED is the following, FQED2 2 (q 2 _{= 0, m}2_{) =} (e + exp(γ) √ πm)

((e + exp(γ)√πm) (e2_{+ exp(γ)}√_{πem − 4πm}2₎₎3/2 (4.9)

[e2 q

e + exp(γ)√πm e2_{+ exp(γ)}√_{πem − 4πm}2 + 4πe3/2m2(tanh−1

√

e (e + exp(γ)√πm) p

(e + exp(γ)√πm) (e2_{+ exp(γ)}√_{πem − 4πm}2₎

!

− tanh−1 e

2_{+ exp(γ)}√_{πem − 2πm}2

√

ep(e + exp(γ)√πm) (e2 _{+ exp(γ)}√_{πem − 4πm}2₎

! )].

Therefore, an equivalent of the anomalous magnetic moment does indeed exist in two dimensions. This equation is, however. quite complicated due to the extra correction term in the mass of the photon and the arc-tangents. Therefore, we are unable to find a connection between the four dimensional value of the anomalous magnetic moment and the result above. Especially since both e and m are free parameters of this theory. However, looking back at equation (3.40), we remember that it might still be possible to find an insightful answer in a massive model when we turn to more complicated systems. Therefore, we will turn to supersymmetric systems in the next chapters in hope of finding a less complicated mass ratio between the photon and the fermion and cancellation between arc-tangent terms.

A less complicated relation between the mass of the photon and the mass of the fermion might already give us some insight. Since, if we were to calculate the anomalous magnetic moment in massive two dimensional in the limit where the photon mass is twice the mass of the fermion: µ = 2m, we find

FQED2 2 (q 2 _{= 0, µ}2 _{= 4m}2_{) =} e2 2πm2 Z 1 0 dz 2(1 − z) 2 ((1 − z)2_{+ 4z)}2 = 2α 3m2. (4.10)

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Chapter 5 2d Wess-Zumino Model

In the previous chapter, we calculated the anomalous magnetic moment in two di-mensional QED. In the massless limit, we found that F2 = 1, while in the massive

limit we found a very complicated equation depending on the free parameters e and m. We also considered the limit in the massive case where the mass of the photon was twice the mass of the fermion. In that case, the anomalous magnetic moment to first order was given by _3m2α2. Although, these results itself are quite interesting,

it does not give us any insight into how it could be related to the four dimensional quantity. Therefore, theories with different interactions or a specific ratio between the mass of the fermion and the photon will have to be considered. These different interactions could cause cancellations as the example in chapter 3.

Therefore, in this thesis a second two dimensional model will be examined: the 2d Wess-Zumino model. This is the model upon which the emergence theory of the Standard Model from my supervisor P.J. Mulders was built. [12] This model has some interesting properties that will be introduced in the next section that might help us gain some insight.

5.1 Wess-Zumino Model

The Wess-Zumino model is an interacting field theory that contains supersymme-try. Supersymmetry is a concept that links the two groups of elementary particles: bosons and fermions. This connection between the two particles, in for example the Standard Model, has never been found. In an extended version of the Standard Model each particle would have an supersymmetric partner. [23] The introduction of these particles could solve numerous problems, such as the hierachy problem, already preluded in the introduction. In the next section, it will be shown that the introduction of supersymmetry in the Wess-Zumino model does indeed solve the hierachy problem for that theory.

First, the Wess-Zumino model shall be properly introduced. It has a Lagrangian that contains a scalar field, pseudoscalar field and a spinor field. It is invariant under supergauge transformations, which in 4d means that it is invariant under the transformation between spinors and tensors. [21] We will follow most of the derivation done in the article ”The roots of the Standard Model of Particle Physics” by Piet Mulders (2016).

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The 2d Wess-Zumino Lagrangian is the sum of free Lagrangian and the Wess-Zumino supersymmetric potential [12], L = 1 2∂ µ_φ S∂µφS+ 1 2∂ µ_φ P∂µφP + i ¯ψ /∂ψ − V (φ, ζ), (5.1)

where φS is an CP-even scalar field and φP is an CP-odd pseudoscalar field. These

two fields can be recombined to right- and left-moving field as: φR/L= (φS±φP)/

√ 2. The real fermion field ζ is recombined to a spinor ψ = (ζR, −iζL)/

√

2. The Wess-Zumino potential V (φ, ζ) contains a pair of auxillary bosonic fields F and G in the following supersymmetric form,

V (φ, ζ) = 1 2(F

2_{+ G}2 _{+ λ}2_{) + M (F φ}

S+ GφP + ¯ψψ) + λF

+ g0(F (ψS2 − ψ2P) + 2GψSψp − ¯ψ(φS+ φPγ1)ψ). (5.2)

From this formula, F and G can be eliminated through the equations of motion,

F + λ = −M φS − g0(φ2S− φ2P) (5.3)

G = −M ψP − 2g0φSφP (5.4)

in order to cast the potential in a convenient form. This gives us the following Lagrangian, L = 1 2∂ µ φS∂µφS+ 1 2∂ µ φP∂µφP + ¯ψ(i /∂ − M − g0φS− g0φPγ1)ψ − 1 2M 2 (φ2_S+ φ2_P) − M g0φS(φ2S + φ 2 P) − 1 2g 2 0(φ 2 S+ φ 2 P) 2 + λ(g0(φ2S− φ 2 P) + M φS). (5.5)

This Lagrangian shows that this theory has Yukawa interactions between bosons and fermions and between bosons themselves. It can be rewritten into a form that could possibly explain the emergence of the symmetries of the Standard Model. Through first defining v0 = M/2g0, we notice that it can be rewritten in a form

that represents the non-zero vacuum expectation value of one of the bosonic fields: ΦS = φS/v0 + 1 = cosh η and ΦP = φp/v0 = sinh η with hΦSi = 1, hΦpi = 0.

Therefore, the Wess-Zumino is now the following,

L = v 2 0 2∂ µ ΦS∂µΦS+ v2 0 2 ∂ µ ΦP∂µΦP + ¯ψ(i /∂ − M 2 − g0v0ΦS− g0v0ΦPγ 1 )ψ −1 2M 2_v2 0(Φ 2 SΦ 2 P) − 1 2g 2 0v 4_{(1 − Φ}2 S+ Φ 2 P) + λ(g0v02(Φ 2 S− Φ 2 P − 1). (5.6)

Moreover, this suggests that ΦP is a vector field in d = 2: Aµ = φP(nµ− ¯nµ)/2

with AµAµ = −φ2P. This allows us to rewrite the pseudoscalar coupling in terms

of a coupling with a vector field Aµ_{: ¯}_{ψg /}_{Aψ Therefore, the fermionic part of the}

Lagragian can now be written as,

LF = i ¯ψ( /∂ − M + g0φS+ gγµAµ)ψ. (5.7)

In this equation, we notice that the covariant derivative arose: iDµ = i∂µ+ gAµ.

Moreover, the bosonic part of the Lagrangian can also be rewritten,

1 2∂ µ_φ P∂µφP = − 1 4FµνF µν₋ 1 2(∂µA µ₎2_. _(5.8)

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The combining of the fields φL/Rcomes with a dynamical gauge field corresponding

to its complex phase freedom. This results into the absorption of a part of the potential through the covariant derivative. This covariant derivative can be used to rewrite this Lagrangian. Therefore, in d = 2 with a gauge field Aµ generated by U(1), the covariant derivative absorbs a part of the potential as,

DµΦSDµΦS = ∂µΦS∂µΦS+ 2g2v02Φ2SAµAµ. (5.9)

Combining these relations, the Lagrangian now is given by the following,

L = v 2 0 2D µ_Φ SDµΦS− 1 4 g2 4g2 0 FµνFµν+ ¯ψ(i /∂ − M 2 − g0v0ΦS− g /A)ψ − 1 2g 2 0v 4 (1 − Φ2_S+ Φ2_P) + λ(g0v02(Φ 2 S− Φ 2 P − 1). (5.10)

Setting v0 = 1 for normalization and λ = 0, we can expand the bosonic fields around

their minimum ΨS = (1 + H), the following mass terms appear,

LM = 1 2M 2 H2− 1 2 g2 g2 0 M2AµAµ− M ¯ψψ. (5.11)

This spontaneous symmetry breaking has caused the non-dynamical gauge field Aµ

to gain a mass. It is the Goldstone mode of the broken U(1) gauge originating from the opposite phases of the fields. In order for this theory to be supersymmetric, g = g0, such that all fields have a mass M corresponding to them. Therefore,

the (1+1)-d Wess-Zumino model only has one coupling constant g and one mass M , which are even related through M = 2g. This theory involves interactions among massive scalar fields and massive vector fields with fermions. It will be these interactions that contribute to a non-zero anomalous magnetic moment. The Feynman rules of this theory are shown in the appendix.

5.2 Non-renormalization

In the previous section, the (1+1)-d Wess-Zumino model has been introduced. This model is supersymmetric, meaning that half-integer spin fermions are linked with integer spin bosons. In this model, the supersymmetry is unbroken and these par-ticles have the same mass. In the Standard Model however, this symmetry is not present. The Higgs mass only gets radiative corrections through interaction with fermions. The first order loop diagram is given in the figure below.

Fermion vertices in 1+1 and 3+1 space-time dimensions

Master Thesis