Renormalization of Factorially Divergent Series: Quantum Field Theory in Zero Dimensions

Master’s Thesis

Supervisor: Bren Schaap Prof. dr. Ronald Kleiss Dedicated to my personal group Preliminary remarks This research is based almost entirely on information from the book Quantum Field Theory: A Diagrammatic Approach [1] by Ronald Kleiss, and is a continuation of the bachelor’s theses [2, 3, 4] of Dirk van Buul, Ilija Milutin and Mila Keijer. The subject of this research was revitalized by Michael Borinsky through his work [5, 6, 7], and he was the one who pointed Ronald Kleiss in the right direction. I would therefore like to thank all of them for introducing me to this fascinating subject. Contents

1 Introduction to zero-dimensional QFT 6 1.1 Random numbers ...... 7 1.1.1 Green’s functions ...... 7 1.1.2 The Schwinger-Dyson equations ...... 8 1.1.3 Connected Green’s functions ...... 9 1.2 Feynman diagrams ...... 10 1.2.1 Diagrammatic equations ...... 12

2 Renormalization 14 2.1 Renormalizing ϕ3 theory ...... 15 2.2 Factorially divergent series ...... 18 2.2.1 Retrieving αf and cf ...... 20 2.3 The factorially divergent nature of Green’s functions ...... 21 2.4 Asymptotic behaviour of renormalized ϕ3 theory ...... 22 2.4.1 The improvement factor ...... 22 2.4.2 Asymptotic behaviour of the tadpole ...... 24 2.5 Freedom of choice ...... 26 2.6 Physical interpretation ...... 29

3 Real ﬁelds 30 3.1 ϕ4 theory ...... 31 3.2 ϕQ theory for odd Q ...... 33 3.3 ϕQ theory for even Q ...... 35 3.4 Renormalizing ϕ123 theory ...... 37

4 QED 39 4.1 Bald QED ...... 40 4.2 Counterterm QED ...... 43 4.2.1 Computing the counterterm ...... 43 4.2.2 Everything is in the counterterm ...... 44 4.2.3 The improvement factor of Counterterm QED ...... 45 4.3 Quenched QED ...... 47 4.3.1 The improvement factor of Quenched QED ...... 47 4.4 Furry QED ...... 49 4.4.1 The improvement factor of Furry QED ...... 49

4 5 Combination theories 51 3/4 5.1 ϕz theory ...... 52 5.1.1 Unphysical z and the saddlepoint approximation ...... 53 5.1.2 Physical z ...... 54 5.1.3 The improvement factor ...... 55 5.1.4 Determining z ...... 57 3/6 5.2 ϕz theory ...... 58 5.3 Pure ϕ3/4 theory ...... 61

6 QCD 64 6.1 Asymptotics of H0,0 ...... 65 6.1.1 Unphysical z > 0 ...... 66 6.1.2 Physical z = −1...... 68 6.2 Renormalizing xQCD ...... 69 6.2.1 The improvement factor for xQCD ...... 70

7 Higgs 71 7.1 Asymptotics of H0,0 ...... 72 7.2 Renormalizing Higgs theory ...... 73

Conclusion 75

Appendices 76 A Factorially divergent pullbacks ...... 77 B Computing the counterterm in CQED (alternative method) ...... 79 C Saddlepoint approximation in QCD ...... 81

Bibliography 83

5 Chapter 1

Introduction to zero-dimensional QFT

“Look yonder,” said my Guide, “[...] I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions.” — Edwin A. Abbot, Flatland

It might come as a surprise to us four-dimensional creatures that zero dimensions can be so interesting. Zero-dimensional spacetimes, also known as “points”, contain neither space nor time, and yet hundreds of pages have been written about them. This thesis is an addition to the subcollection of writings about quantum field theory (QFT) in zero dimensions. The sole purpose of this first chapter is to lay the theoretical foundations that are necessary for this research. For those familiar with QFT in zero dimensions, it is merely a demonstration of notation. For others who are familiar with QFT in four dimensions, it is a good reminder of the fundamental basics of QFT. For everyone else, it is a minimal introduction to QFT in zero dimensions. In the first section we introduce the probabilistic interpretation of quantum field theory; this includes random numbers, probability density functions, and expectation values. In the second section we introduce the famous Feynman diagrams, along with their associated Feynman rules.

6 1.1 Random numbers

Zero-dimensional quantum field theory concerns itself with everything there is to know about all possible zero-dimensional spacetimes that are filled with random (complex) numbers. For any given spacetime, these numbers are generated by a set of stochastic variables {ϕi}, called “fields”, which we can group together into one vector ~ϕ. These fields obey a (spacetime-specific1) probability density which we will write as 1 1 P(~ϕ) = exp − S(~ϕ) (1.1) N ~ where the action S(~ϕ) is a function of the fields and N is the normalization constant. Every theory is fully specified by its corresponding action. When we observe the values of the fields in one zero-dimensional spacetime, the resulting numbers look random and nothing can be said about the underlying probability. This is in contrast to a spacetime where these numbers change over time and where we can probe the underlying probability density by repeated measurements.2 To emulate this behaviour in zero dimensions and get a good definition of probability, we can observe a collection of independent and identically distributed spacetimes. In other words: we cannot say anything about a single value of a field but we can say things about expectation values.

1.1.1 Green’s functions

Q ni We will look at the expectation values G~n = h i ϕi i which are called the Green’s functions. Trivially, G~0 = h1i = 1. To make life a little easier we deﬁne Z 1 Y ni ~ H~n ≡ ( ϕi ) exp − S(~ϕ) dϕ (1.2) i ~ with the integral implicitly running from −∞ to +∞ so that

H~n G~n = (1.3) H~0 and H~0 is an explicit way of writing N. In general, the integrals H~n can not be solved analytically. In this research we solve these integrals perturbatively to get a series expansion in ~. To be able to say things about the collection of Green’s functions we deﬁne the path integral

1For comparison: to our best understanding the four-dimensional spacetime we live in is generated by the action of the Standard Model. 2This is completely analogous to how one would test if a die is fair or not.

7 1 Z 1 1 Z(J~) ≡ exp − S(~ϕ) + J~ · ~ϕ dϕ~ N ~ ~ Z ki 1 Y X 1 Jiϕi 1 = ( ) exp − S(~ϕ) dϕ~ N ki! ~ ~ i ki≥0 ki X Y 1 Ji = G~k (1.4) ki! ~ ~k i ~ where bookkeeping device Ji is called a source and the sum runs over all vectors k with integer entries ki ≥ 0. This is our ﬁrst of many encounters where we use a P 1 k series expansion (in this case exp(x) = k≥0 k! x ) to simplify an integral. Every Green’s function G~n can be retrieved from the path integral by applying the correct operator:

" ni # Y ∂ ~ G~n = ~ Z(J) . ∂Ji i J~=~0 1.1.2 The Schwinger-Dyson equations∗ We will now deduce properties of the collection of Green’s functions from the path integral. Let us define ∂ as the vector with ith component ∂ and consider the ∂J~ ∂Ji equality Z ∂S ∂ ~ 1 ∂S ∂ 1 1 ~ ~ ~ − Ji Z(J) = ~ − Ji exp − S(~ϕ) + J · ~ϕ dϕ ∂ϕi ∂J~ N ∂ϕi ∂J~ ~ ~ Z 1 ∂S 1 1 ~ ~ = (~ϕ) − Ji exp − S(~ϕ) + J · ~ϕ dϕ N ∂ϕi ~ ~ 1 Z ∂ 1 1 = −~ exp − S(~ϕ) + J~ · ~ϕ dϕ~ N ∂ϕi ~ ~ = 0 where in the last step the integral over ϕi vanishes because the probability density goes to zero at its end points. From this we find a Schwinger-Dyson equation (SDe) ∂S ∂ ~ ~ ~ Z(J) = JiZ(J) (1.5) ∂ϕi ∂J~ for every field in the theory. From the SDe’s we can obtain a relationship between the individual Green’s functions. For this we need to plug in the path integral as expansion of Green’s functions and group together the terms with the same powers of Ji . This method is very powerful in cases where computing H~n is computationally heavy: instead of having to compute an integral for every Green’s function you consider, it suffices to calculate a few integrals and use the (in general relatively simple) SDe’s to compute the rest of the Green’s functions. ∗This subsection is not essential to understanding the main line of reasoning in this thesis.

8 1.1.3 Connected Green’s functions While the Green’s functions fully describe P(~ϕ), the average quantum ﬁeld theorist will use connected Green’s functions since these are directly related to quantities we can measure, such as cross sections and decay times. The connected Green’s functions are deﬁned by2

ki ~ X Y 1 Ji ln(Z(J)) = C~k (1.6) ki! ~ ~k i where this time normalization requires C~0 = 0. Every Green’s function can be written in terms of connected Green’s functions, and vice versa. For instance, for a theory with only one ﬁeld, C1 = G1 = hϕi corresponds to the mean, C2 = 2 2 2 G2 − G1= hϕ i − hϕi corresponds to the variance, etc. The connected Green’s functions are the main subject of study in this research. Just as the Schwinger-Dyson equations yield a set of equations for the Green’s functions, it is also possible to obtain a set of equations for the connected Green’s functions. We shall do this in section 1.2.1, but ﬁrst we must climb onto the shoulders of Richard P. Feynman.

2Using the series expansion of ln(1 + x).

9 1.2 Feynman diagrams

If we consider only real-valued ﬁelds3 we can write the most general expression of the action as

X Y 1 ki S(~ϕ) = λ~k ϕi ki! ~k i and distinguish the terms we encounter:4

• The constant term λ~0 does not contribute to the theory because it gets ab- sorbed by the normalization constant.

• The constants λeî are called counterterms and we will encounter an example in section 4.2. ~ • The quadratic terms with k = 2êi are called mass terms with µi ≡ λ2êi a positive real number referred to as the “mass” of field ϕi. ~ • The quadratic terms with k =e î +e ˆj (i 6= j) are normally not considered.

• All higher-order terms are called interaction terms for which λ~k is called the coupling constant. Let us now consider a theory described by an action which only contains mass and interaction terms 1 X 1 2 X Y ki P S(~ϕ) = 2 µiϕi + λ~k ϕi θ(3 ≤ i ki) (1.7) ki! i ~k i where the function θ(x) is 1 if x is true and 0 if x is false. What would we do if we wanted to calculate a connected Green’s function by hand? This is where the famous Feynman diagrams come in: every Green’s function can be expressed perturbatively in terms of Feynman diagrams. Instead of working with symbolic equations, it is possible to write diagrammatic equations that make sense visually. A Feynman diagram is a graph — built up out of lines and vertices — which stands for an expression that can be determined using Feynman rules. Every type of line and every type of vertex has a factor associated with it. For the theory of equation 1.7 the building blocks are

↔ ~ , ↔ ~ , etc. µ1 µ2 (1.8) λ λ ↔ − 1,1,1 , ↔ − 3,3,5,8 , etc. ~ ~

~ where λa,b,c is shorthand for λ~k with k =e ˆa +e ˆb +e ˆc . The Feynman rules state that the value of the whole Feynman diagram is equal to the product of its building blocks, multiplied by the sm factor of the diagram.

3We will encounter complex-valued ﬁelds in chapter 4. 4 th eˆi is the “unit vector” with a one in the i place and zeros everywhere else.

10 The sm factor Every Feynman diagram D is weighted by a factor sm(D): the product of the symmetry factor s(D) and the multiplicity m(D). The multiplicity of a diagram is the number of distinct ways in which distinct labels can be assigned to external lines5 of the same type. To get the symmetry factor of a diagram that has at least one external line, multiply:

1 • a factor k! for every set of k internal lines that may be permuted without changing the diagram;

1 • a factor p! for every set of p disjunct connected pieces that may be permuted without changing the diagram.

Computing a connected Green’s function is like computing the matrix element M for a scattering or decay process. To compute a connected Green’s function C~n , 6 take the sum of all connected Feynman diagrams with exactly ni external lines of type i. In general, this sum has an infinite number of terms, but terms containing more building blocks contribute less to the final result. This means that in practice a computation is an approximation for which the error decreases the more “higher-order” terms are computed. To be more specific: the perturbative expansion of a connected Green’s function is equal to the value of the lowest level diagrams (called tree level) multiplied by a power series in terms of dimensionless parameters. This is because higher-order (also called loop-order) terms are the same as tree-level diagrams but with loops added to them (see figure 1.1 for an example) and every loop evaluates to such a dimensionless parameter by the Feynman rules.

Figure 1.1: Examples of a tree diagram and a (four-)loop diagram.

5An internal line is a line that is connected to vertices at both ends. An external line is a line that is not an internal line. 6Also include semi-connected diagrams (all parts in a semi-connected diagrams are connected to at least one external line) to get G~n . To get H~n, include disconnected diagrams as well.

11 1.2.1 Diagrammatic equations∗ Feynman diagrams are a powerful tool to construct equations that relate connected Green’s functions to each other. There are two main types of such equations: 1) equations that look at subdiagrams of a connected Green’s function and 2) equations that morph one connected Green’s function into another. Application of both of these types can be found in section 4.2.1.

Subdiagrams A connected Green’s function is a sum of an infinite number of diagrams. By grouping these diagrams in different ways, different relationships emerge. A peculiar feature of this infinity is that every connected Green’s function can be found as a subdiagram in every other connected Green’s function.2 To make it possible to draw an infinite number of diagrams, we introduce a shaded area in our Feynman diagrams; a shaded area with lines attached to it is shorthand for all possible connected Feynman diagrams with precisely that number of external lines. Look for example at figure 1.2. Here we see that C5 (second diagram on the right) can be found in C3 (diagram on the left). Such a diagrammatic equation leads to the algebraic equation

−λ 2 −λ C = 3~ + 4 C + ... 3 µ3 µ 5 which can then be used in an algebraic derivation.

= + + . . .

Figure 1.2: Example of how C3 can contain C5.

Morphing

−1 As seen in equation 1.8, every line comes with a factor µi and every vertex comes with a factor λ~n. In other words, a diagram D with a lines of type i and b vertices of −a b type ~n is proportional to the factor µi λ~n . This means that there exist operators ∂ ∂ −µi D = a ·D and λ~n D = b ·D ∂µi ∂λ~n that “count” the number of lines and vertices of a certain type. Because these operations are diagram dependent, we can also apply them to a sum of diagrams such as a connected Green’s function. ∗This subsection is not essential to understanding the main line of reasoning in this thesis. 2At least in general. This is not true for theories in which subsets of ﬁelds do not interact with each other.

12 The counting operators can be used to apply a morphing operation (such as adding, removing or replacing a component) in all possible ways. For instance, all possible diagrams D0 where one line of type i in D is replaced by a line of type j can be generated with the operation

µ2 ∂ − i D µj ∂µi and all possibilities of adding an external line of type j to a vertex of type n can be generated with the operation λ ∂ ~ n+1 D . µj ∂λn Of course one can think of a vast number of operations, but one in particular will be of great importance to us later. It is possible to add a vertex to a diagram by placing it in the middle of a line. One can think of this as cutting the line open, thereby splitting the line in two, and then gluing it together again with the added vertex. The corresponding operation

−λ ∂ ∂ −~ D = λ D ~ ∂µi ∂µi can be accompanied by attaching a component C to the new vertex ∂ Cλ D ∂µi such as an external line. As mentioned before, these operations can be applied to connected Green’s functions. The operations in which the number of external lines changes, as in the previous examples, can be used to morph one into another, thus creating an equation that relates two connected Green’s functions.

13 Chapter 2

Renormalization

The main goal of this thesis is to investigate the transformation of connected Green’s functions under renormalization. Renormalization is the process in which the parameters of a theory (e.g. µ, λ) are expressed in terms of measured quantities. The best way to learn about renormalization is via an example; in section 2.1 we will first introduce the concept of renormalization by renormalizing ϕ3 theory. After that we will set up the mathematical framework of factorially divergent series (2.2) that is necessary to understand the behaviour of connected Green’s functions at large loop order (2.3) and the difference between the original and the renormalized ones (2.4). In section 2.5 we again renormalize ϕ3 theory but this time using a different renormalization scheme. Section 2.6 concludes this chapter with a brief physical interpretation of the results.

14 2.1 Renormalizing ϕ3 theory

ϕ3 theory is a good example for learning how renormalization works because it includes many facets also present in other theories. We start with the action 1 1 S(ϕ) = µϕ2 + λ ϕ3 (2.1) 2 3! 3 including one mass term and one three-point interaction term. This action leads to the Feynman rules λ ↔ ~ ↔ − 3 (2.2) µ ~ and these Feynman rules combine into the dimensionless parameter

λ2 u = 3~ ∝ (2.3) µ3 that represents an added loop. The dot indicates a vertex that can be attached to a line.

Computing the connected Green’s functions To calculate the desired connected Green’s functions, we must first compute the 1 integrals Hn perturbatively. We find Z n 1 Hn = ϕ exp(− S(ϕ))dϕ ~ k Z X 1 λ3 µ = − ϕn+3k exp − ϕ2 dϕ k! 6 2 k≥0 ~ ~ p k 2 X λ3 ~ p! ∝ − p p θ(p = n + 3k even) (2.4) 6 µ 2 k≥0 ~ k!( 2 )!2 where we first used the series expansion of the exponential for the interaction term and then the integral

s p 2 Z 1 1 2 2π p! p − 2 µϕ ~ ~ ϕ e ~ dϕ = p p θ(p even) (2.5) µ µ 2 ( 2 )!2 for the mass term. Often it is more convenient to substitute ~ for u and doing so results in the expression n 6µ X k p! Hn ∝ − u p p θ(p = 6k − 2n) (2.6) λ k 2 3 n 36 (2k − n)!( 2 )!2 k≥d 2 e by redeﬁning k → 2k − n in equation 2.4.

1We will ignore the fact that all integrands, including the normalization, diverge for ϕ → −∞.

15 Taking all Hn and using the steps from section 1.1 leads to the connected Green’s functions of the ϕ3 theory −λ C = γ 3~t (u) = + ... 1 1 µ2 1 ( (2.7) λ n−2 n−2 C = γ ~ − 3~ t (u) = γ + ... n≥2 n µ µ2 n n

2 where γn is the sum of the sm factors of the tree-level diagrams and tn(u) = 1+O(u) is the “tail” of the connected Green’s function.

Renormalization We now choose to reparameterize the theory by introducing the parametersµ ˆ and ˆ λ3 in such a way that the two connected Green’s functions

C = ~t (u) ≡ ~ 2 µ 2 µˆ (2.8) λ 2 λˆ 2 C = − 3~ t (u) ≡ − 3~ 3 µ3 3 µˆ3 become free of loop corrections. One could think of this as performing a measure- 3 ment of these two quantities after which C2 and C3 are simply numbers. We would now like to express all connected Green’s functions in terms of the new parameters. For this we define a new dimensionless parameter 2 ˆ2 C3 λ3~ uˆ ≡ 3 ≡ 3 (2.9) C2 µˆ which we can express in terms of the old dimensionless parameter as 2 t3(u) uˆ(u) = u 3 ≡ uρˆ(u) (2.10) t2(u) withρ ˆ(u) = 1 + O(u) another series in u. Equation 2.10 can be inverted to yield an equation for u in terms of the new parameteru ˆ u(û) =uρ ˆ (û) (2.11) by tweaking the coefficients of ρ(û) order by order such thatu ˆ(ûρ(û)) =u ˆ. This leads to the expressions −λ −λˆ C = γ 3~t (u) ≡ γ 3~tˆ (û) 1 1 µ2 1 1 µˆ2 1 !n−2 (2.12) λ n−2 λˆ C = γ ~ − 3~ t (u) ≡ γ ~ − 3~ tˆ (û) n≥2 n µ µ2 n n µˆ µˆ2 n in terms of the new tails tˆn(û).

2 1 Strictly speaking C1 does not have a tree-level contribution so γ1 = 2 is the sm factor of the one-loop diagram. 3This is only a thought experiment since there is no notion of “measuring” in zero dimensions.

16 Comparing the tails

What did this renormalization procedure yield? Of course the expressions of C2 and C3 are much simpler now, but there is more. When we write

X (k) k tn(u) = tn u k≥0 (2.13) ˆ X ˆ(k) k tn(ˆu) = tn uˆ k≥0

(k) (k) we can compare the coeﬃcients tn and tˆn with each other. In ﬁgure 2.1 the ratio (k) (k) 4 tn /tˆn is plotted as function of loop order k. We notice three things:

1. the coeﬃcients of the new tails are, for n ≥ 2, smaller than the coeﬃcients of the original tails;

2. the ratios seem to tend asymptotically toward a constant value;

3. the behaviour for n = 1 seems to be fundamentally diﬀerent than for n ≥ 2.

The ﬁrst property is a nice extra that renormalization gives us. The second and third properties will be discussed in section 2.4 but ﬁrst we will need to construct a mathematical framework in which we can analyse asymptotic behaviour.

Figure 2.1: The ratio of the tail coeﬃcients in ϕ3 theory.

4The computations in this research were performed, and the plots were made, using MapleTM.

17 2.2 Factorially divergent series

P n A formal power series f(x) = n≥0 fnx is called factorially divergent (fd) when the coeﬃcients fn grow as

n 1 fn ≈ af cf Γ(αf n + βf ) · 1 + O( n ) (2.14)

+ for large n and some constants af , cf ∈ R6=0, αf ∈ N and βf ∈ R. As we will see in section 2.3, Green’s functions are, in genera, factorially divergent. Extensive work was done by Michael Borinsky [5, 6, 7] on factorially divergent series with αf = 1. To explicitly distinguish this special case, we will call series with αf > 1 factorially super divergent (fsd). Where Borinsky mainly focuses on 1 computing the O( n ) corrections using rigorous mathematics, we are only interested in (intuitively understanding) the leading behaviour. It is possible to asymptotically compare fd’s to each other when we deﬁne the following relations: f f(x) ≺ g(x) if lim n = 0 (2.15) n→∞ gn f(x) g(x) if g(x) ≺ f(x) (2.16)

fn f(x) ∼ g(x) if lim ∈ R6=0 (2.17) n→∞ gn f f(x) g(x) if lim n = 1 (2.18) n→∞ gn from which it can be veriﬁed that f(x) ≺ g(x) if [αf < αg] or [αf = αg ∧ cf < cg] or [αf = αg ∧ cf = cg ∧ βf < βg]. The corollary f(x) + g(x) f(x) if g(x) ≺ f(x) directly shows the power of these relations. We will call a series t(x) properly factorially divergent (pfd) if t0 = 1 and t1 6= 0. k By deﬁning the leading term of an fd series f(x) as L(f, x) = fkx with fk 6= 0 and ∀j

The fd shift Let g(x) fd then the expression xg(x) has coeﬃcients

n−1 (xg)n = gn−1 ≈ agcg Γ(αgn − αg + βg) ag ⇒ axg = cxg = cg βxg = βg − αg (2.19) cg so that multiplying with x “shifts” βg by αg such that xg(x) ≺ g(x).

5Sometimes called “reversion”.

18 The pfd product Let f(x) and g(x) pfd then their product is asymptotically equal to their sum

n X n X f(x) · g(x) = x fkgn−k n≥0 k=0 X n x (f0gn + fng0) n≥0 = f(x) + g(x) (2.20) because the sum over k is dominated by fn and/or gn. We get the expression

p f(x) p · f(x) ∀p ∈ R (2.21) by generalizing the multiplication.

The pfd pullback After reparameterization we get expressions of the form f(xg(x)) called pullbacks. For pfd’s f(x) and g(x) we have derived the asymptotic behaviour of the factorially divergent pullback as

( g1 f1xg(x) + f(x) exp if αf = 1 f(xg(x)) cf (2.22) f1xg(x) + f(x) if αf > 1

(see appendix A for full derivation). Computing the actual expression for a pullback to large loop order is computationally heavy. Therefore we have used Horner’s method [8] to optimize eﬃciency.

The pfd inversion If we start with the equation y = xf(x) where f(x) is pfd then it is always6 possible to find a pfd series g(y) such that x = yg(y) solves the equation x = xf(x)g(xf(x)). We will be interested in two properties of the series g(y): the first-order term g1 and the asymptotic behaviour. 2 First of all we can find g1 by looking only at terms up to O(x ) in the equation

x = xf(x)g(xf(x)) 2 2 = x(1 + f1x + O(x )) 1 + g1x(1 + O(x)) + O(x ) 2 = x(1 + f1x + g1x + O(x ))

⇒ g1 = −f1 (2.23) which gives us a remarkably simple relationship.

6This is guaranteed by the Lagrange inversion theorem.

19 For the asymptotic behaviour, we can make use of previous results. We start by observing that y = xf(x) = yg(y)f(yg(y)) implies that

1 = g(y)f(yg(y)) g(y) + f(yg(y))

( g1 g(y) + f(y) exp if αf = 1 cf g(y) + f(y) if αf > 1 where we threw away the pullback term f1yg(y) in the last step since we know for sure that f1yg(y) ≺ g(y). Rearranging gives us

( f1 −f(y) exp − if αf = 1 g(y) cf (2.24) −f(y) if αf > 1 the asymptotic behaviour of g(y). Note that this means that g(y) ∼ f(y). Computing the actual expression of the inverse to large loop order is computationally heavy. Therefore we have used Maple’s powseries:-reversion() com- mand to optimize eﬃciency.

2.2.1 Retrieving αf and cf

In order to use equations 2.22 and 2.24 we have to be able to retrieve f1, cf and αf from a given series expansion f(x). Since f1 is a low-order term, it can be computed by hand (see e.g. equation 2.37). Retrieving cf and αf from a series is nothing more than taking the limit of the ratio

f Γ(α n + β + α ) n+1 f f f αf lim = lim cf cf · (αf n) (2.25) n→∞ fn n→∞ Γ(αf n + βf )

(n+k)! k and reading oﬀ cf and αf . The approximation n! ≈ n for large n will turn out to be very useful to compute these ratios quickly.

20 2.3 The factorially divergent nature of Green’s functions

The goal of this section is to demonstrate that Green’s functions in ϕ3 theory are factorially divergent. The steps followed in this section hold for most theories described in this research, but in chapter 5 we will see that the deﬁnition of a factorially divergent series (equation 2.14) does not cover all zero-dimensional theories. Our road map is to ﬁrst prove that H0 is fd and then take this result to show, step by step, that all series in a theory are fd. Taking equation 2.6 with n = 0 and using Stirling’s approximation r √ nn 2π z z n! ≈ 2πn or Γ(z) ≈ e z e for large factorials we get

X (6k)! H ∝ uk 0 36k(2k)!(3k)!23k k≥0 k k k X 1 1 66 k uk √ 36 · 23 2233 e k≥0 2πk k X 1 3 Γ(k) uk (2.26) 2π 2 k≥0

1 3 as proof for H0 being fd with af = 2π , cf = 2 , αf = 1 and βf = 0. To go from H0 P (k) k to Hn we write Hn = k≥dn/2e Hn u and calculate

H(k) 6µn (6k)−2n 2µn lim n = − = − (2.27) k→∞ (k) λ (2k)−n(3k)−n2−n λ H0 3 3 implying that Hn ∼ H0 ∀n which means that all Hn are fd. The next step is to go from Hn to Gn. Writing Hn(u) for the tail of Hn we ﬁnd

Hn Hn(u) Gn = = L(Hn, u) Hn − L(Hn, u) · H0 Hn (2.28) H0 H0

dn/2e because L(Hn, u) ∝ u so that Gn is similar to H0 as well. Going from Gn to Cn is a bit more subtle. Cn is equal to Gn plus a sum of products of Gm’s with m < n. Since L(Gm, u) is at least O(u), we infer that (just like in equation 2.28) Cn Gn which means also Cn ∼ H0. So, indeed, all Green’s functions are fd as we wanted to show. As a bonus we also discovered that all (connected) Green’s functions are asymptotically similar to one another. Because taking out the lead term to get the tails tn(u) only shifts βf , and all series in a theory are essentially a combination of these tn’s, all series in a theory have the same αf and cf .

21 2.4 Asymptotic behaviour of renormalized ϕ3 theory

The goal of this section is to predict the asymptotic behaviour as seen in figure 2.1 and verify this numerically. It is clear that the behaviour of C1 is different than that of Cn≥2. Therefore we will first focus on Cn≥2 (section 2.4.1) and then look at C1 (section 2.4.2). To apply our fd knowledge, we must determine the mutual comparison between all tails tn(u) and u(û). We know that all connected Green’s functions are similar to each other, and with the help of the general expressions for the connected Green’s functions (equation 2.7) we find

∀n, m : Cn ∼ Cm n−1 ⇒ ∀n ≥ 2 : ut1(u) ∼ u tn(u)

⇒ ∀n ≥ 4 : t1(u) ∼ t2(u) ≺ t3(u) ≺ tn(u) where we substituted ~ in favour of u. Immediately we also ﬁnd 2 t3(u) 2 −3 ρˆ(u) = 3 t3(u) + t2(u) ∼ t3(u) t2(u)

⇒ u(û) =uρ ˆ (û) ≺ tn≥3(û) from the pfd inversion. This means that the asymptotic behaviour of an expression will be dominated by the tail with the largest n.

2.4.1 The improvement factor

Our starting point for determining the asymptotic behaviour of Cn≥2 after renormalization is the definition of the renormalized tails from equation 2.12. Substituting ˆ the definitions ofµ ˆ and λ3 gives us n−2 , ˆ ! n−3 ˆ µˆ λ3 λ3 t2(u(û)) tn≥2(û) ≡ 2 2 tn(u(û)) = n−2 tn(u(û)) (2.29) µ µ µˆ t3(u(û)) in which we recognize pfd products and pfd pullbacks. Using our fd knowledge we find

tˆn≥4(û) (n − 3)t2(u(û)) − (n − 2)t3(u(û)) + tn(u(û)) (1) (1) (1) [(n − 3)t2 − (n − 2)t3 + tn ]u(û) ρ1 + [(n − 3)t2(û) − (n − 2)t3(û) + tn(û)] exp cf ρ1 tn(û) exp (2.30) cf as our desired result. Alternatively the derivation7 n−3 ˆ t2(u) ρ1 tn≥4(û) = n−2 tn(u) [tn(u)]u=u(û) tn(û) exp (2.31) t3(u) u=u(û) cf

7 Note that this derivation holds only because tn(u) u(ˆu).

22 is a bit less rigorous but somewhat easier to follow. The fact that tˆn(û) ∼ tn(û) for n ≥ 4 means that we indeed predict that the ratio of the coefficients of the tails tends asymptotically to a constant value as was already suggested by figure 2.1. This brings us to the improvement factor 8 I for ϕ3 theory

t(k) ρ ρˆ I[ϕ3] ≡ lim n = exp − 1 = exp 1 (2.32) k→∞ (k) tˆn cf cf which turns out to be a computable constant that is independent of n. Of course we want to verify that we found the correct asymptotic value. For this we must computeρ ˆ1 and cf .

Determining ρˆ1

ρˆ1 is a low-order quantity which means that it can be calculated by hand. Recapit- ulating the deﬁnition ofu ˆ(u) we have

2 2 C3 t3(u) uˆ(u) = 3 = u 3 (2.33) C2 t2(u) on the one hand and

2 uˆ(u) = uρˆ(u) = u 1 +ρ ˆ1u + O(u ) (2.34) on the other hand. This tells us that, to ﬁndρ ˆ1, we only need to compute C2 and C3 to one-loop order. Computing C2 and C3 by hand is done by drawing all Feynman diagrams with, respectively, two or three external lines. In this case we are only interested in diagrams containing a maximum of one loop. The results can be found in ﬁgures 2.2 and 2.3. Applying the Feynman rules gives the expressions

C = ~ 1 + 1 u + 1 u + O(u2) = ~ 1 + u + O(u2) (2.35) 2 µ 2 2 µ

λ 2 λ 2 C = − 3~ 1 + u + 3 u + 3 u + O(u2) = − 3~ 1 + 4u + O(u2) (2.36) 3 µ3 2 2 µ3 that we can plug into equation 2.33 to get

2 2 2 C3 (1 + 4u + O(u )) 2 uρˆ(u) = 3 = u 2 3 = u 1 + 5u + O(u ) (2.37) C2 (1 + u + O(u )) from which we can read oﬀρ ˆ1 = 5.

8 (k) (k) The notion of “improvement” is based on the fact that tˆn < tn .

23 3 Figure 2.2: All terms to one-loop order for C2 in ϕ theory.

3 Figure 2.3: All terms to one-loop order for C3 in ϕ theory.

Determining cf

We already found cf using Stirling’s formula but normally we would use a faster method. Starting with the expansion X (6k)! H ∝ uk 0 36k(2k)!(3k)!23k k≥0 and using what we learned in section 2.2.1 we ﬁnd H(k+1) (6k)6 3 lim 0 = k (2.38) k→∞ (k) 36(2k)2(3k)323 2 H0 3 so we can read oﬀ αf = 1 and cf = 2 .

Numerical veriﬁcation

3 Plugging in the computed values ofρ ˆ1 = 5 and cf = 2 determines ρˆ 10 I[ϕ3] = exp 1 = exp ≈ 28.03 (2.39) cf 3 as improvement factor for ϕ3 theory. In ﬁgure 2.4 we indeed see that the ratios of the tails tend to this value.

2.4.2 Asymptotic behaviour of the tadpole

C1 is called the tadpole because the lowest-order diagram resembles the larva of an amphibian. Repeating for n = 1 what we did for the( other renormalized tails , ˆ ! ˆ λ3 λ3 t2(u(û)) t1(û) ≡ 2 2 t1(u(û)) = t1(u(û)) (2.40) µ µˆ t3(u(û)) immediately shows us that tn=1(u(û)) is no longer the largest pfd series in the expression for tˆn=1(û). From this it should be no surprise that the asymptotic behaviour is different. Applying fd techniques as before results in9 ˆ t2(u) ρ1 t1(û) = t1(u) [−t3(u)]u=u(û) −t3(û) exp (2.41) t3(u) u=u(û) cf

9 Again note that this derivation holds only because t3(u) u(ˆu).

24 Figure 2.4: The ratio of the tail coefficients in ϕ3 theory together with the predicted asymptotic improvement factor. which shows us that the tadpole becomes asymptotically larger after renormalization. This t1(u) ≺ tˆ1(u) indeed corresponds to the qualitative behaviour we saw (k) ˆ(k) in figure 2.1. Quantitatively we can verify this result by plotting the ratio t3 /t1 together with the predicted asymptotic value (see figure 2.5).

(k) ˆ(k) Figure 2.5: The ratio t3 /t1 (red curve) as a function of loop order k together with 10 the predicted asymptotic value of − exp( 3 ) ≈ −28.03 (black line).

25 2.5 Freedom of choice

Until now we have chosen to renormalize such that C2 and C3 become free of loop corrections, but there is nothing that keeps us from making a diﬀerent choice. In this section we investigate the consequences of choosing a renormalization scheme 10 where C2 and C4 are free of loop corrections. To make a distinction between the two choices, we will denote the renormalized ˜ quantities in the new scheme with tildes instead of carets. Deﬁneµ ˜ and λ3 by

C = ~t (u) ≡ ~ 2 µ 2 µ˜ (2.42) λ2 3 λ˜2 3 C = γ 3~ t (u) ≡ γ 3~ 4 4 µ5 4 4 µ˜5 in an analogous way as before and

λ˜2 u˜ = 3~ (2.43) µ˜3 as the new dimensionless parameter. In the following we will repeat the steps from the previous section for our new renormalization scheme. To expressu ˜ in terms of u, we must calculate C4 to ﬁrst loop order. From ﬁgure 2.6 we determine λ2 3 C = 3~ 3 + 3 u + 3 u + 12 u + 12 u + 6u + 3u + O(u2) 4 µ5 2 2 2 2 λ2 3 = 3 3~ 1 + 8u + O(u2) (2.44) µ5

11 which shows that γ4 = 3. Now we have

C4 t4(u) (1 + 8u + ...) u˜ = 2 = u 2 = u 2 = u(1 + 6u + ...) ≡ uσ˜(u) (2.45) γ4C2 t2(u) (1 + u + ...) which can be inverted to get u =uσ ˜ (˜u) withσ ˜1 = −σ1 = 6. Note that we inter- changed ρ for σ to make a clear distinction between the two schemes.

3 Figure 2.6: All terms to one-loop order for C4 in ϕ theory.

10Again, there is nothing that prevents us from choosing anything else. We could just as well have chosen, say, C37 and C42 but this would make the equations unnecessarily clumsy and would yield no new insights. 11 Note that dividing out γ4 is necessary to be able to meaningfully compare the old and new tails.

26 Similar to before, we have expressions for the renormalized tails in terms of the unrenormalized tails

, ˜ ! s ˜ λ3 λ3 t2(u(˜u)) t1(˜u) ≡ 2 2 t1(u(˜u)) = t1(u(˜u)) µ µ˜ t4(u(˜u)) n−2 (2.46) , ˜ !n−2 s ˜ µ˜ λ3 λ3 1 t2(u(˜u)) tn≥2(˜u) ≡ 2 2 tn(u(˜u)) = tn(u(˜u)) µ µ µ˜ t2(u(˜u)) t4(u(˜u)) but this time t3(u) is no longer the largest tail in the expression for t˜3(˜u). Therefore we will distinguish three cases when determining the asymptotic behaviour of the renormalize tails: n = 1, n = 3 and n ≥ 5. First of all, the asymptotic behaviour of the renormalized tadpole

σ ˜ 1 1 1 t1(˜u) − 2 t4(u) u=u(˜u) − 2 t4(˜u) exp (2.47) cf is not very surprising, given what we found before. The crux lies in the asymptotic behaviour of C3 σ ˜ 1 1 1 t3(˜u) − 2 t4(u) u=u(˜u) − 2 t4(˜u) exp (2.48) cf which now mimics the asymptotic behaviour of the tadpole. All other tails ˜ σ1 tn≥5(˜u) [tn(u)]u=u(˜u) tn(˜u) exp (2.49) cf revert to the old behaviour. All predicted behaviour is verified in figures 2.7 and 2.8. 3 The predicted improvement factor fromσ ˜1 = 6 and cf = 2 σ˜ I˜[ϕ3] ≡ exp 1 = exp(4) ≈ 54.60 (2.50) cf is larger than before. Although the improvement factor increases for n ≥ 5, we now not only have t˜1(˜u) t1(˜u) but also t˜3(˜u) t3(˜u). A larger improvement factor is nice, but its benefits do not outweigh the disadvantage of an asymptotically larger three-point Green’s function. Therefore the most logical choice for a renormalization scheme is to renormalize the asymptotically smallest tails in the theory.

27 3 Figure 2.7: The ratio of the tail coeﬃcients in ϕ theory renormalized on C2 and C4, together with the predicted asymptotic improvement factor.

(k) ˜(k) (k) ˜(k) Figure 2.8: The ratios t4 /t1 and t4 /t3 as a function of loop order k together with the predicted asymptotic value of −2 exp(4) ≈ −109.

28 2.6 Physical interpretation

Perturbation theory is essential for making theoretical predictions in quantum field theory; without it, we would not be able to test theories against reality. Perturbation theory is inseparable from Feynman diagrams and renormalization. Further understanding of factorially divergent series, diagrams and renormalization can therefore play a role in advancing theoretical particle physics. Diagrams can be divided into (overlapping) categories. We have already encoun- tered connected diagrams, but there are many more types, such as disconnected, semi-connected and one-particle irreducible (1PI) diagrams. Asymptotic calculations can tell us what proportion of all diagrams is of a certain type.[7] In this section we will interpret some results. The first remarkable result follows from the factorially divergent nature of Green’s functions. The fact that Hn Cn tells us that, at high loop order, the total weight of disconnected diagrams is negligible with respect to the total weight of connected 12 diagrams. A similar conclusion can be drawn from Hn Gn and Gn Cn for semi-connected diagrams. The interpretation of the improvement factor has to do with subdiagrams. I[ϕ3] ≈ 28 implies that a random connected diagram in ϕ3 theory has a weighted probability 1 of 1− I ≈ 0.96 to contain at least one two- or three-legged subdiagram that contains a loop. The improvement factor in theories with αf = 1 originates from the exponential factor that arises in the pfd pullback. For theories with αf > 1 this factor is unity, which means that they have an improvement factor of one. Therefore we can already conclude that fsd theories contain a negligible number of diagrams that are affected by renormalization. These interpretations on the level of the diagrams are the reason why we bother with ϕ3 theory. While ϕ3 theory is ill-defined over the real axis, one can choose a different curve in the complex plane such that the theory becomes well-defined. (See [1] for an elaborate discussion.) For us the unphysical nature does not matter because our focus is purely on the combinatorial interpretation. The last result that should be interpreted is the fact that tˆ1 t1 . Although outside the scope of this research, renormalization can be reformulated in terms of interaction counterterms. In this formulation – loosely speaking – renormalization removes certain diagrams. For the tadpole, renormalization “removes” more diagrams than there were in the first place. Because of this “overcompensation”, the ˆ(k) (k) terms t1 flip sign and become larger than the terms t1 .

12Note that this does not necessarily mean that the number of connected diagrams is larger than the number of non-connected diagrams. If we look for instance at semi-connected diagrams with one loop and two external lines in ϕ3 theory, there are ﬁve diagrams of which only two are connected, but, because of symmetry, the weight of these two is more than twice the weight of the other three.

29 Chapter 3

Real ﬁelds

In the previous chapter we introduced the subject of factorially divergent series and showed how the framework can be used to understand the behaviour of connected Green’s function at large loop order. Applying the fd knowledge to ϕ3 theory resulted in a (scheme-dependent) improvement factor that quantifies the effect of renormalization. The goal of this chapter is to find the improvement factor for different theories containing real fields and one interaction term. First we will work out ϕ4 theory and after that generalize what we learned to ϕQ theory for all Q ∈ N. Of course there are countless other theories; in the last section we will look at one particular example, namely ϕ123 theory.

30 3.1 ϕ4 theory

The simplest realistic1 theory is the ϕ4 theory. To make life a bit easier, we define the corresponding action to be 1 1 S(ϕ) = µϕ2 − λ ϕ4 (3.1) 2 4! 4 by redefining λ4 → −λ4. The consequence of redefining λ4 is that the Feynman rule for the vertex changes to + λ4 so that we won’t be bothered by minus signs in our ~ calculations. Note that the probability density is only normalizable for λ4 < 0. The associated dimensionless parameter of this theory λ u = 4~ ∝ (3.2) µ2 deviates significantly from the dimensionless parameter in ϕ3 theory; this will be elaborated on in the next two sections. All odd Green’s functions are zero2 while the other integrals read

p k 2 X λ4 ~ p! H2n ∝ p p θ(p = 2n + 4k) (3.3) 24 µ 2 k≥0 ~ k!( 2 )!2 which reduce to the simple form

X (4k)! H ∝ uk (3.4) 0 24kk!(2k)!22k k≥0 for n = 0. Taking the limit

H(k+1) (4k)4 2 lim 0 = k (3.5) k→∞ (k) 24k(2k)222 3 H0

2 4 shows that αf = 1 and cf = 3 for the ϕ theory. Since αf = 1 we predict an improvement factor larger than one. The general expression for the connected Green’s functions is

!n−1 λ 2 n−1 λˆ 2 C = γ ~ 4~ t (u) ≡ γ ~ 4~ tˆ (ˆu) (3.6) 2n 2n µ µ3 2n 2n µˆ µˆ3 2n which we renormalize on C2 and C4

ˆ 3 ˆ ~ λ4~ C4 λ4~ C2 ≡ ,C4 ≡ 4 , uˆ ≡ 2 ≡ 2 (3.7) µˆ µˆ C2 µˆ 1Realistic in the sense that all integrals converge. 2This can be understood on the level of the probability density (the action is symmetric) and on the level of Feynman diagrams (it is not possible to draw a graph with an odd number of external lines using only four-point vertices).

31 as instructed by section 2.5. From ﬁgures 3.1 and 3.2 we get

t (u) (1 + 7 u + ...) uˆ(u) = u 4 = u 2 = u 1 + 5 u + ... (3.8) 2 1 2 2 t2(u) (1 + 2 u + ...)

5 from which we can read oﬀρ ˆ1 = 2 . Same as before we have t2n ≺ t2n+2 meaning that u ≺ t2n≥4 and

n−2 ˆ t2(u) ρ1 t2n≥6(û) n−1 t2n(u) [t2n(u)]u=u(û) t2n(û) exp (3.9) t4(u) u=u(û) cf so we predict

ρˆ 15 I[ϕ4] ≡ exp 1 = exp ≈ 42.52 (3.10) cf 4 which is veriﬁed in ﬁgure 3.3.

1 Figure 3.1: C2 to one-loop order. From left to right the sm factors are 1 and 2 .

4 3 Figure 3.2: C4 to one-loop order. From left to right the sm factors are 1, 2 and 2 .

Figure 3.3: Veriﬁcation of the improvement factor of ϕ4 theory.

32 3.2 ϕQ theory for odd Q

Now that we have determined the asymptotic behaviour of ϕ3 theory and ϕ4 theory, we can generalize the used methods to ϕQ theory for arbitrary Q. As mentioned in the previous section, the expression for the dimensionless parameter u is inherently different for ϕ3 theory with respect to ϕ4 theory. As it turns out, this is a fundamental difference between theories with an odd-point vertex and theories with an even-point vertex. Therefore we will treat each case separately. Like the trick with the minus sign in the previous section, we define the action for odd Q as

Q−3 1 µ 2 λ S(ϕ) = µϕ2 − ϕQ (3.11) 2 ~ Q!

Q−3 µ 2 by redeﬁning the coupling constant as λQ ≡ − λ so that the dimensionless ~ parameter

2 Q−2 2 λQ~ λ u = ≡ ~ (3.12) µQ µ3 is the same for all Q. Computing

n+3k k 2 X λ ~ p! Hn ∝ p p θ(p = n + Q · k even) (3.13) Q! µ 2 k≥0 ~ k!( 2 )!2 which reduces for n = 0 to

k X u (Q · 2k)! H ∝ (3.14) 0 Q!2 (2k)!(Q · k)!2Q·k k≥0 yields the limit

H(k+1) 1 (2Qk)2Q (2Q)Q lim 0 = kQ−2 · (3.15) k→∞ (k) Q!2 (2k)2(Qk)Q2Q 4Q!2 H0 from which we can read oﬀ αf = Q − 2. This remarkable result tells us that for all odd Q > 3 the series in a ϕQ theory are super divergent, meaning that ϕ3 is the only odd-point theory with a non-trivial improvement factor. The improvement factor

I[ϕQ] = 1 ∀ odd Q > 3 (3.16) is veriﬁed in ﬁgure 3.4 for ϕ5 and ϕ7 theory.

33 Figure 3.4: Demonstration of the trivial improvement factor for ϕQ theory with odd Q > 3. Top: ϕ5 theory. Bottom: ϕ7 theory.

34 3.3 ϕQ theory for even Q

We deﬁne the action for even Q as

Q−4 1 µ 2 λ S(ϕ) = µϕ2 − ϕQ (3.17) 2 ~ Q!

Q−4 µ 2 by redeﬁning the coupling constant as λQ ≡ − λ so that the dimensionless ~ parameter

Q −1 λQ~ 2 λ~ u = Q ≡ 2 (3.18) µ 2 µ is the same for all Q. Computing

k n+2k X λ ~ p! H2n ∝ p p θ(p = 2n + Q · k) (3.19) Q! µ 2 k≥0 ~ k!( 2 )!2 which reduces for n = 0 to

k X u (Qk)! H ∝ (3.20) 0 Q Qk Q! 2 k≥0 k!( 2 k)!2 yields the limit

(k+1) Q Q H0 1 (Qk) Q −1 Q 2 lim = k 2 · (3.21) (k) Q Q Q k→∞ Q! 2 2 Q! H0 k( 2 k) 2

Q 4 from which we can read oﬀ αf = 2 −1. Like before, ϕ theory is the only even-point theory with a non-trivial improvement factor. The improvement factor

I[ϕQ] = 1 ∀ even Q > 4 (3.22) is veriﬁed in ﬁgure 3.5 for ϕ6 and ϕ8 theory.

35 Figure 3.5: Demonstration of the trivial improvement factor for ϕQ theory with even Q > 4. Top: ϕ6 theory. Bottom: ϕ8 theory.

36 3.4 Renormalizing ϕ123 theory The fact that ϕQ theory has a trivial improvement factor for Q > 4 does not mean that ϕ3 theory and ϕ4 theory are the only theories with a non-trivial improvement factor. In this section we will work out the ϕ123 theory as an example. The ϕ123 action

1 2 2 2 S(ϕ1, ϕ2, ϕ3) = 2 µ(ϕ1 + ϕ2 + ϕ3) − gϕ1ϕ2ϕ3 (3.23) contains three real fields (labelled by the numbers 1,2,3) with mass µ and an interaction term that couples the three fields to each other with coupling constant g = −λ1,2,3. The masses do not have to be identical, but it is always possible to redefine the fields and the coupling constant such that this action is retrieved. The corresponding Feynman rules

g ↔ ~ ↔ (3.24) µ ~ combine into the familiar g2 u = ~ (3.25) µ3 as dimensionless parameter. Using the shorthand N = n1 + n2 + n3 the triple integrals

ZZZ 1 n1 n2 n3 − S(ϕ1,ϕ2,ϕ3) Hn1,n2,n3 = ϕ1 ϕ2 ϕ3 e ~ dϕ1dϕ2dϕ3

N+3k 3 k 2 X 1 g ~ Y pj! ∝ pj θ(pj = nj + k even) (3.26) pj k! µ 2 k≥0 ~ j=1 ( 2 )!2 show explicitly what the Feynman rules already imply: the Green’s functions are symmetric in their indices and are equal to zero unless the nj are all even or odd. Taking the case N = 0

3 X 1 (2k)! H ∝ uk (3.27) 0,0,0 (2k)! k!2k k≥0 gives the limit

H(k+1) (2k)4 lim 0,0,0 2k (3.28) k→∞ (k) k323 H0,0,0 from which we can read oﬀ αf = 1 and cf = 2. Even though there are only two parameters, we can renormalize four connected Green’s functions because C2,0,0 = C0,2,0 = C0,0,2. Since everything is symmetric in

37 the indices, in the following we will work with n1 ≤ n2 ≤ n3. Now renormalize such that 2 2 2 ~ gˆ~ C1,1,1 gˆ ~ C0,0,2 ≡ ,C1,1,1 ≡ , uˆ ≡ ≡ (3.29) 3 3 3 µˆ µˆ C0,0,2 µ to get

2 2 t1,1,1(u) (1 + 4u + ...) uˆ(u) = u 3 = u 3 = u(1 + 5u + ...) (3.30) t0,0,2(u) (1 + u + ...) from which we can read offρ ˆ1 = 5. Since ϕ123 theory has no tadpole, and all connected Green’s functions with N = 2 and N = 3 are renormalized, we predict that the ratio tends to the improvement factor for all tails. Combiningρ ˆ1 = 5 and cf = 2 gives the improvement factor ρˆ1 5 I[ϕ123] ≡ exp = exp ≈ 12.18 (3.31) cf 2 which is verified in figure 3.6.

Figure 3.6: Veriﬁcation of the improvement factor of ϕ123 theory.

38 Chapter 4

QED

QED (quantum electrodynamics) is the theory that describes electromagnetic interactions between photons and electrically charged particles. The goal of this chapter is to first formulate a naive zero-dimensional QED model containing one massive photon and one charged fermion, and then investigate multiple ways to upgrade it into a more realistic model. In this case “realistic” means that the theory will behave more like full-fledged four-dimensional QED. Every particle that carries charge has its own antiparticle. These two units are not separate identities, but rather both sides of the same particle-antiparticle coin. The two sides of the coin have the same mass and carry the same amount of charge, but the sign of the charge is flipped. For example, the electron carries a negative unit charge while its antiparticle, the positron, carries a positive unit charge. In the first section a complex-valued field will be introduced to implement the notion of charge. What is interesting about having multiple variants of one theory is that we will be able to compare quantities in these models and determine their effect on asymptotic behaviour, renormalization, and the improvement factor.

39 4.1 Bald QED

The simplest action we can write down for a zero-dimensional QED toy model has two fields 1 S(ϕ, B) = mϕϕ¯ + µB2 + eϕBϕ¯ (4.1) 2 in which the complex-valued field ϕ acts as charged fermion with mass m, the real-valued field B acts as photon with mass µ, and the interaction term couples them together with coupling constant e > 0. We will call this theory “Bald QED” (BQED).

e ↔ ~ ↔ ~ ↔ − m µ ~

Figure 4.1: The Feynman rules for Bald QED.

The fact that we now have complex-valued fields changes the Feynman rules a bit. As can be seen in figure 4.1 we introduce fermion lines that have an orienta- tion, and we depict the photon as a wavy line to visually distinguish photons from fermions. Note that the vertex has one arrow pointing inwards and one arrow pointing outwards; this charge conservation results in diagrams always having the same number of inward-pointing as outward-pointing external lines. The dimensionless parameter that follows from these Feynman rules e2 α = ~ = (4.2) m2µ will be called α as a homage to the fine-structure constant. Since we now know what we have to calculate to predict the asymptotic behaviour of a theory, we will focus only on calculating αf , cf andρ ˆ1. The expansion of the integrals ZZZ n n l n n l − 1 S(ϕ,B) Hn,l ≡ Nhϕ¯ ϕ B i = ϕ¯ ϕ B e ~ dϕdϕdB¯

p k n+k 2 X e ~ ~ (n + k)!p! ∝ − p p θ(p = l + k even) (4.3) m µ 2 k≥0 ~ k!( 2 )!2 can be found by writing dϕdϕ¯ = πds with s = |ϕ|2 =ϕϕ ¯ and using the equation Z ∞ q q − m s ~ s e h ds ∝ q! (4.4) 0 m for the natural number q = n + k. For n = l = 0 the integrals reduce to X (2k)! H ∝ αk (4.5) 0,0 k!2k k≥0

40 from which we can compute the limit

H(k+1) (2k)2 lim 0,0 α = 2k (4.6) k→∞ (k) k · 2 H0,0 to read oﬀ αf = 1 and cf = 2 . We can use the mass parameters m and µ to renormalize C1,0 and C0,2. Because the connected Green’s function C0,3 has no tree-level diagram, t1,1(α) ≺ t0,3(α) and we can use e to renormalize C1,1 as the next smallest tail. Now renormalize such that eˆ 2 C ≡ ~ ,C ≡ ~ ,C ≡ − ~ , 1,0 mˆ 0,2 µˆ 1,1 mˆ 2µˆ 2 2 (4.7) C1,1 eˆ ~ αˆ ≡ 2 ≡ 2 C1,0C0,2 mˆ µˆ with (see ﬁgures 4.2 through 4.4)

2 2 t1,1(α) (1 + 6α + ...) αˆ(α) = α 2 = α 2 = α(1 + 7α + ...) (4.8) t1,0(α) t0,2(α) (1 + 2α + ...) (1 + α + ...) from which we read offρ ˆ1 = 7. For (n, l) ∈/ {(0, 1), (0, 2), (1, 0), (1, 1)}, the renormalized tails ρ1 tˆn,l(ˆα) tn,l(α(ˆα)) tn,l(ˆα) exp (4.9) cf and the renormalized tadpole t1,0(α) ρ1 tˆ0,1(ˆα) = t0,1(α) −t1,1(α(ˆα)) −t1,1(ˆα) exp (4.10) t1,1(α) α=α(ˆα) cf behave as seen before. Therefore we predict an improvement factor for Bald QED of ρˆ 7 I[bqed] = exp 1 = exp ≈ 33.12 (4.11) cf 2 which is verified in figure 4.5. (See also [7, 5, 1, 3].) Only the renormalized tadpole tˆ0,1 ≺ t0,1 again behaves differently from the rest.

41 Figure 4.2: C1,0 up to one-loop order for BQED.

Figure 4.3: C0,2 up to one-loop order for BQED.

Figure 4.4: C1,1 up to one-loop order for BQED.

Figure 4.5: The ratio of the tail coeﬃcients in Bald QED together with the predicted asymptotic improvement factor.

42 4.2 Counterterm QED

In four dimensions the QED tadpole evaluates to zero because of momentum conservation. Our ﬁrst approach to making our QED model more realistic will be removing the tadpole from the theory. We will do this by adding a counterterm T to the Bald QED action 1 S(ϕ, B) = mϕϕ¯ + µB2 + eϕBϕ¯ + TB (4.12) 2 where T is tuned in such a way that it exactly cancels the whole tadpole, and nothing else. We will call this theory “Counterterm QED” (CQED). In terms of diagrammatics this entails that we add the vertex T ↔ − (4.13) ~ to the Feynman rules. To get an expression for T , it is possible to take the expansion

p k j n+k 2 X e T ~ ~ (n + k)!p! Hn,l ∝ − − p p θ(p = l + k + j even) m µ 2 k,j≥0 ~ ~ k!j!( 2 )!2 (4.14) and solve the equation H0,1 = 0. The disadvantage of this method is that it is very slow, even at low loop order. Luckily we can do better. In appendix B a scheme is applied in which Bald QED is transformed into Counterterm QED by redeﬁning parameters. This method is already a big improvement, but it contains an inversion which is still suboptimal. In the following we will take a third route.

4.2.1 Computing the counterterm The goals of this part are to derive a diﬀerential equation for T , and solve it. The derivation relies on the following relationships T C = − (4.15) 1,0 e m C = − T (4.16) 0,2 eµ

C1,1 = eC0,2∂mC1,0 (4.17) m = T ∂ T (4.18) eµ m which we will prove ﬁrst using diagrammatic equations.

Diagrammatic equations

T is tuned such that C0,1 = 0 but we can also write C0,1 as

0 = = + (4.19)

43 which gives T = −eC1,0 proving equation 4.15. Comparing the equations e C = = + = ~ − C (4.20) 1,0 m m 1,1 e C = = + = ~ − C (4.21) 0,2 µ µ 1,1

m m gives us C0,2 = µ C1,0 = − eµ T proving equation 4.16. To go from Cn,l to Cn,l+1 in Counterterm QED we must take a fermion line in Cn,l and tag the dressed photon line C0,2 onto it

= (4.22)

e which gives the equation Cn,l+1 = − C0,2(− ∂m)Cn,l. This is in contrast to Bald ~ ~ QED (where we can just add a photon line) because now we do not have tadpoles to build from. Taking (n, l) = (1, 0) proves equation 4.17.

The differential equation We start from equation 4.20 and fill in what we have found T 1 − = ~ − T ∂ T (4.23) e m µ m to find a differential equation for T . It is not immediately clear how to solve this e~ equation but we can make it dimensionless by writing T ≡ − m R(α) with R(α) pfd. Now we find e e ∂α µ ∂ T = ~ R(α) − ~ ∂ R(α) = α[R(α) + 2α∂ R(α)] (4.24) m m2 m ∂m α e α so that equation 4.23 reads

2 2 R(α) = 1 + αR(α) + 2α R(α)∂αR(α) (4.25) which is much nicer. It is possible to ﬁnd R(α) by iterating equation 4.25 but the solution is actually known [9] and it can be generated extremely fast because there exists a simple recursive relation between the coeﬃcients.

4.2.2 Everything is in the counterterm Now that we have computed the tadpole counterterm, what can we do with it? The straightforward option is to plug it into the expansion of the integrals (equation 4.14) and continue as before, but we will again take a diﬀerent route. As seen in the previous part C0,2 , C1,0 and C1,1 can be expressed in terms of T . In this part we will show that actually all Gn,l , and thus all Cn,l , are fully determined by T .

44 The path integral as expansion of Green’s functions reads

X 1 J¯nJ nHl Z(J,¯ J, H) = G (4.26) n!n!l! n+n+l n,l n,l≥0 ~ where J, J¯ and H are the sources. Now the two independent SDe’s read

2 [µ~∂H + e~ ∂J ∂J¯ + T ]Z = HZ (4.27) 2 ¯ [m~∂J + e~ ∂H ∂J ]Z = JZ which entail at the level of the Green’s functions that for n, l ≥ 0

eGn+1,l + µGn,l+1 + TGn,l − l~Gn,l−1 = 0 (4.28) and for n ≥ 1, l ≥ 0

eGn,l+1 + mGn,l − n~Gn−1,l = 0 . (4.29)

Replacing Gn,l+1 in (4.28) by (4.29) we get for n ≥ 1, l ≥ 0 mµ eG + (T − )G + n~G − l G = 0 . (4.30) n+1,l e n,l e n−1,l ~ n,l−1

By inserting G0,0 = 1 and G0,1 = 0 we can generate all Green’s functions from T by calculating T G1,0 = − e from (4.28)|(n = 0, l = 0) mµ T ~µ Gn+1,0 = ( e2 − e )Gn,0 − e2 nGn−1,0 from (4.30)|(n ≥ 1, l = 0) ~ T e G0,l+1 = l µ G0,l−1 − µ G0,l − µ G1,l from (4.28)|(n = 0, l ≥ 1) ~ m Gn,l+1 = n e Gn−1,l − e Gn,l from (4.29)|(n ≥ 1, l ≥ 0) where caution has to be taken in which order the last two lines are computed. In practice this means that we can ﬁrst compute all Gn,0 and then compute {G0,l,Gn≥1,l} by looping over increasing l. Now the connected Green’s functions are retrieved as usual.

4.2.3 The improvement factor of Counterterm QED Repeating the steps for Bald QED, but now with slightly modiﬁed tails (see ﬁgures 4.6 through 4.8) due to the absence of tadpoles, gives

2 2 t1,1(α) (1 + 4α + ...) αˆ(α) = α 2 = α 2 = α(1 + 5α + ...) (4.31) t1,0(α) t0,2(α) (1 + α + ...) (1 + α + ...)

[bqed] from which we findρ ˆ1 = 5. As before, cf = 2 since T ∼ C0,1 changes nothing in that respect. Therefore the improvement factor for Counterterm QED is 5 I[cqed] = exp ≈ 12.18 (4.32) 2 which is verified in figure 4.9. (See also [3].)

45 The CQED improvement factor is slightly smaller than for Bald QED; this was to be expected since CQED has fewer diagrams to begin with. Furthermore it is tempting to look for similarities with ϕ123 theory because they have the same improvement factor. The one-loop diagrams of the here renormalized connected Green’s functions do indeed coincide, but the fact that cf = 2 in both cases is more of a coincidence than an intrinsic property, and further comparisons also break down.

Figure 4.6: C1,0 up to one-loop order for CQED.

Figure 4.7: C0,2 up to one-loop order for CQED.

Figure 4.8: C1,1 up to one-loop order for CQED.

Figure 4.9: The ratio of the tail coeﬃcients in Counterterm QED together with the predicted asymptotic improvement factor.

46 4.3 Quenched QED

Furry’s theorem states that in four-dimensional, “adult” QED all diagrams including fermion loops with an odd number of attached photon lines cancel out. The nice thing about this theorem is that the tadpole diagram automatically vanishes. Before we mimic this behaviour in the next section, we will ﬁrst look at a theory where all fermion loops vanish. We fabricate “Quenched QED” (QQED) by adding a counterterm to the action of Bald QED but, in contrast to the previous section, this time we know in advance what counterterm we should add.[1] Starting with the action 1 e S(ϕ, B) = mϕϕ¯ + µB2 + eϕBϕ¯ − ln 1 + B (4.33) 2 ~ m leads to a sum of two integrals that are both very similar to Bald QED. Carrying out the integrals we get

X (k) (k) Hn,l ∝ En,l θ(l + k even) + On,l θ(l + k odd) (4.34) k≥0 where

k n+k l+k e 2 (n + k)!(l + k)! E(k) = − ~ ~ n,l l+k l+k ~ m µ k!( )!2 2 2 (4.35) k n+k l+k+1 e e 2 (n + k)!(l + k + 1)! O(k) = − ~ ~ n,l l+k+1 l+k+1 m m µ 2 ~ k!( 2 )!2

e arise from respectively the 1 and m B terms. Calculating

" k+1 l+k+1 k+1 l+k+1 # e 2 e 2 (l + k + 1)! O(k) + E(k+1) = (−1)k ~ − ~ 0,l 0,l l+k+1 l+k+1 m µ m µ 2 ( 2 )!2 = 0 makes the cancellations explicit.

4.3.1 The improvement factor of Quenched QED

~ Since all fermion loops are suppressed, C0,2 = µ only consists of the leading term. Therefore there is no need to renormalize the photon mass; i.e.µ ˆ = µ. Combining this with the altered first-order contribution of C1,1 (see figures 4.10 and 4.11) we find

2 2 t1,1(α) (1 + 3α + ...) αˆ(α) = α 2 = α 2 = α(1 + 4α + ...) (4.36) t1,0(α) (1 + α + ...)

47 (k) (k) (k) from which we read oﬀρ ˆ1 = 4. Since En,l and On,l are so similar to H0,0 for Bald QED, it should come as no surprise that again cf = 2. Therefore the improvement factor of Quenched QED is

ρˆ I[qqed] = exp 1 = exp(2) ≈ 7.39 (4.37) cf which is veriﬁed in ﬁgure 4.12. (See also [7, 5, 1].)

Figure 4.10: C1,0 up to one-loop order for Quenched QED.

Figure 4.11: C1,1 up to one-loop order for Quenched QED.

Figure 4.12: The ratio of the tail coeﬃcients in Quenched QED together with the predicted asymptotic improvement factor.

48 4.4 Furry QED

As stated before, we would like to mimic Furry’s theorem by cancelling all fermion loops that have an odd number of photon lines attached to them. We will do this by adding a counterterm (again taken from [1]) to the action of Bald QED

e 1 2 ~ 1 + m B S(ϕ, B) = mϕϕ¯ + µB + eϕBϕ¯ − ln e (4.38) 2 2 1 − m B that is signiﬁcantly more complicated than in the case of Quenched QED. Luckily e we can expand the counterterm in terms of m B. The integrals then read

p k j n+k 2 X e e ~ ~ aj(n + k)!p! Hn,l ∝ − p p θ(p = l + k + j even) m m µ 2 k,j≥0 ~ k!( 2 )!2 (4.39) where aj has no closed form but it comes from the expansion r 1 + x X = a xj (4.40) 1 − x j j≥0 and can therefore be easily calculated.

4.4.1 The improvement factor of Furry QED

The connected Green’s functions C0,2, C1,0 and C1,1 to ﬁrst loop order are identical to those of Counterterm QED. This means that alsoρ ˆ = 5 is the same as for CQED. 1 aj+1 Since aj ∼ √ for large j [10] and thus → 1 it does not change anything to j aj cf = 2. Therefore the improvement factor of Furry QED is

5 I[fqed] = exp ≈ 12.18 (4.41) 2 which is veriﬁed in ﬁgure 4.13. (See also [7, 5, 1].)

We can conclude that altering the Bald QED action does not change the factorially divergent behaviour of αf = 1 and cf = 2. Alterations that eﬀect the one-loop diagrams of the renormalized connected Green’s functions do inﬂuence the improvement factor viaρ ˆ1. An extreme case was seen in Quenched QED where the removal [qqed] ~ of all fermion loops resulted in C0,2 = µ having only one contributing diagram, thus removing the need to renormalize the photon mass µ altogether.

49 Figure 4.13: The ratio of the tail coeﬃcients in Furry QED together with the predicted asymptotic improvement factor.

50 Chapter 5

Combination theories

Until now we have looked only at theories with one interaction term. While this is perfectly fine for the photon, which only interacts with other particles via a three- point coupling, the other bosons of the Standard Model interact with themselves via three-point and four-point couplings. The goal of this section is to investigate theories containing one real field and two interaction terms. 3/4 In the first section we describe the ϕz theory, which is a combination of the ϕ3 theory and the ϕ4 theory. The interaction terms in this theory are related via the parameter z in order to make it possible to apply the fd framework. In the 3/6 second section we make a detour and look at the ϕz theory as an example for other combination theories. In the last section we come back to combining the ϕ3 theory and the ϕ4 into the ϕ3/4 theory, but this time no relation is assumed between the interaction terms.

51 3/4 5.1 ϕz theory

3/4 The ϕz action

λ λ2 1 S(ϕ) = 1 µϕ2 − 3 ϕ3 − z 3 ϕ4 (5.1) 2 3! µ 4! describes a real ﬁeld ϕ that interacts with itself via a three-point and a four-point 2 λ3 interaction. The four-point coupling constant λ4 = z µ with some constant z ∈ R is chosen such that λ λ2 u ≡ 4~ = z 3~ ≡ zu (5.2) 4 µ2 µ3 3 and z = u4 is actually the ratio between the dimensionless parameters of a ϕ3 theory u3 4 and a ϕ theory. In the following, u = u3 will be used as dimensionless parameter. 3/4 Previous work on ϕz theory with z > 0 was done by Keijer and Kleiss.[4, 1] In this section we will recapitulate and expand upon that research. Notably we will investigate the case z < 0 because the normalization integral H0 diverges for positive z, which makes z > 0 unphysical. The expansion of the integrals

l p k 2 2 X λ3 zλ3 ~ p! Hn = p p θ(p = n + 3k + 4l even) (5.3) 6 24µ µ 2 k,l≥0 ~ ~ k!l!( 2 )!2 shows nothing new, but rewriting

X u k zul p! H0 = p p θ(p = 6k + 4l) 36 24 (2k)!l!( )!2 2 k,l≥0 2 (5.4) X s = Fs(z)u s≥0 where s = k + l (or k = s − l) and

s s X X (3z)l (6s − 2l)! F (z) ≡ F (z, s, l) ≡ (5.5) s 288s (2s − 2l)!l!(3s − l)! l=0 l=0

(s) shows that the value H0 is now the outcome of a sum Fs(z). We would guess that the coefficients still grow factorially divergent but this is not guaranteed. When z < 0 the sum is alternating and huge cancellations take place, making it difficult to say anything about the asymptotics. Therefore we will first look at the (unphysical) case z > 0.

52 5.1.1 Unphysical z and the saddlepoint approximation The function F (z, s, l) is “bell-shaped” in l for all positive values of z. By this we mean that for fixed z and s the function F (z, s, l) is peaked around a maximum at l0 and falls off to zero away from its maximum. As a consequence the coefficient is bounded by F (z, s, l0) ≤ Fs(z) ≤ s · F (z, s, l0) which means that determining the asymptotic form of F (z, s, l0) is enough to calculate αf and cf in the normal manner. However, we are also interested in af (for reasons that will become clear later) so we need the exact asymptotic form of Fs(z). The idea of finding the asymptotic form of Fs(z) is to approximate the sum over l by an integral, and make a saddlepoint approximation around its maximum. The saddlepoint approximation is given by

r2π F (z) ≈ F (z, s, l ) (5.6) s 0 L00

00 so we will need to determine the asymptotic expression of l0, F (z, s, l0) and L (which is deﬁned in equation 5.11). First we have to ﬁnd the the value of l0. If F (z, s, l0) is a maximum, then we know that the slope is zero at l0 i.e. F (z, s, l0) ≈ F (z, s, l0 + 1). This means that l0 must satisfy the equality

2 2 F (z, s, l0 + 1) (2s − 2l0) (3s − l0) (s − l0) 1 = 3z 2 = 3z (5.7) F (z, s, l0) (6s − 2l0) l0 (3s − l0)l0 which is solved by √ 3 + 6z − 9 + 24z l = s ≡ y(z) · s (5.8) 0 2 + 6z but we can also write (3 − y)y z(y) = (5.9) 3(1 − y)2 in order to replace z and l0 in favour of y and s. The found value l0 = ys can now be used to determine the two other factors. Using Stirling’s approximation yields

F (z, s, l0) = F (z(y), s, ys) (3z(y))y s ((6 − 2y)s)! = 288 ((2 − 2y)s)!(ys)!((3 − y)s)! y s  (3−y)y  s (1−y)2 1 (6 − 2y)s ≈  288  2π (2 − 2y)s · ys · (3 − y)s

(6 − 2y)6−2y s ss · (2 − 2y)2−2yyy(3 − y)3−y e 1 (3 − y)3 s Γ(s) ≈ √ (5.10) 2πp(1 − y)y 18(1 − y)2 2πs

53 from which we start to see the factorially divergent behaviour. The term

∂2 ln F (z, s, l) L00 ≡ − ∂l2 l=l0 ln F (z, s, l + 1) − 2 ln F (z, s, l ) + ln F (z, s, l − 1) − 0 0 0 12 F (z, s, l0 + 1)F (z, s, l0 − 1) = − ln 2 F (z, s, l0) 3 + y 1 = − ln 1 − + O (3 − y)(1 − y)ys s2 3 + y ≈ (5.11) (3 − y)(1 − y)ys is approximated with Taylor expansions. Inserting what we found into the equation of the saddlepoint approximation (5.6) gives

1 r3 − y (3 − y)3 s F (z) A (z) ≡ Γ(s) (5.12) s s 2π 3 + y 18(1 − y)2 implying H0 is fd with af and cf dependent on z. This is veriﬁed in ﬁgure 5.1 for multiple values of z. Note that this is where af becomes important.

(s) Figure 5.1: Ratio between the coeﬃcients H0 and their predicted asymptotic value, plotted for multiple values of z.

5.1.2 Physical z Even though the steps we followed for positive z do not hold for negative z, we can hypothesize that the asymptotic behaviour found for H0 is continuous at z = 0. We

54 would like to prove that this is mathematically correct, but we do not know how.1 What we can do is test this hypothesis numerically. The asymptotic behaviour is 3 verified in figure 5.1 for multiple values of z ∈ − 8 , 0 and the hypothesis seems to hold.2 3 3 A second complication arises at z = − 8 . For values of z ≥ − 8 the value y(z) 3 is real, but for values of z < − 8 the value y(z) is complex and the asymptotic relation cannot possibly hold as is. Yet we noticed oscillatory behaviour in H0 for 3 z < − 8 which does hint at a complex phase. Plotting the real part of the asymptotic 1 expression magically resembled 2 H0. Therefore our second hypothesis is that 3 F (z) 2 Re[A (z)] ∀z < − (5.13) s s 8 which is based on plots like figure 5.2.

(s) Figure 5.2: Ratio between the coeﬃcients H0 and 2 Re[As(z)] for z = −1.

3 3 So what happens at z = − 8 ? The nondimensionalized action

2 ˜ λ3 µ 1 2 1 3 z 4 S(x) ≡ 3 S x = x − x − x (5.14) µ λ3 2 6 24

3 3 happens to have two local minima for − 8 < z < 0 but only one for z < − 8 . One 3 could therefore call z = − 8 a phase transition point. How this phase transition is linked with the series expansion of H0 is an open question.

5.1.3 The improvement factor The form of the connected Green’s functions and the renormalization scheme are 3 the same as in ϕ theory, with the exception that the prefactors γn and coeﬃcients (s) tn now depend on z. The renormalized dimensionless parameter is equal to

t (u)2 (1 + (4 + 7 z)u + ...)2 uˆ(u) = u 3 = u 2 = u 1 + (5 + 11 z)u + ... (5.15) 3 1 3 2 t2(u) (1 + (1 + 2 z)u + ...) 1Diving into such a proof would be more suitable for a mathematician. 2 3 If z = − 8 then y = −3 and equation 5.11 has to be altered. 3 1 This action corresponds to a one-dimensional Mexican hat potential for z = − 3 ; see also chapter 7.

55 11 so thatρ ˆ1 = 5 + 2 z. With 19y2 − 27y + 30 (3 − y)3 ρˆ = and c = (5.16) 1 6(1 − y)2 f 18(1 − y)2 we predict

2 3/4 ρˆ1 19y − 27y + 30 I[ϕz ] ≡ exp = exp 3 3 (5.17) cf (3 − y) as z-dependent improvement factor. This improvement factor holds for values of 3 3 z ≥ − 8 (see e.g. ﬁgure 5.3) but the oscillatory behaviour for z < − 8 creates a seemingly chaotic ratio between the original and renormalized tails.

3/4 3 Figure 5.3: Demonstration of the improvement factor for ϕz theory with z = − 8 .

As a sanity check it is good to look at two limiting cases of the improvement 3 4 factor, namely the ϕ limit z → 0 and the ϕ limit z → −∞. In these limits u3 and u4, respectively, become dominant and we should recover the improvement factors for theories with only one interaction term. The first limit 3/4 3/4 10 3 lim I[ϕz ] = lim I[ϕz ] = exp = I[ϕ ] z→0 y→0 3 is pretty straightforward. The second limit cannot be compared in this way because of the difference in renormalization scheme (see section 2.5). If we would have 3/4 renormalized our ϕz theory on C2 and C4 instead, we would have gotten a different ρˆ1 and consequently the improvement factor 182y4 − 591y3 + 960y2 − 855y + 324 I˜[ϕ3/4] = exp 3 z (3 − y)3(8y2 − 15y + 9)

56 from which we can compute4 ˜ 3/4 ˜ 3/4 15 4 lim I[ϕz ] = lim I[ϕz ] = exp = I[ϕ ] z→−∞ y→1 4 to complete our sanity check.

5.1.4 Determining z One could ask oneself: “Which value of z should I choose?” For this we do a small 2 C3 thought experiment in which we “measure” certain values. The value ofu ˆ = 3 can C2 be measured (either directly or via measurements of C2 and C3) and the result is independent of z. Now we do an extra measurement of the (dimensionless) quantity C2C4 D = 2 . On the one handu ˆ and D are now simply numbers. On the other hand C3 we have the relationship C C 2 4 ˆ 2 2 D = 2 = (3 + z)t4(û) = (3 + z) + (3 + 3z − 3z )û + O(û ) (5.18) C3 from which we can determine an approximation z ≈ (D−3)+(33−21D+3D2)û+... in terms of measured quantities up to arbitrary order in perturbation theory. So it is not necessary to choose a value for z beforehand; you can simply measure it like you would with any other parameter in a physical model.

4Note that Im(y) → 0 as z → −∞ such that z → −∞ nicely corresponds to z → +∞.

57 3/6 5.2 ϕz theory

It is perhaps not really surprising that combining two theories with αf = 1 leads to a theory with αf = 1. But what happens when we combine a factorially divergent theory with a factorially super divergent theory? 3/6 We will look at the ϕz action

λ λ2 1 S(ϕ) = 1 µϕ2 − 3 ϕ3 − z 3 ϕ6 (5.19) 2 3! ~ 6! where z = u6 is the ratio between the dimensionless parameters of a ϕ3 theory and u3 a ϕ6 theory. The expansions of the corresponding integrals are given by

l p k 2 2 X λ3 zλ3 ~ p! Hn = 2 p p θ(p = n + 3k + 6l even) (5.20) 6 6! µ 2 k,l≥0 ~ ~ k!l!( 2 )!2 resulting in an expansion in u = u3

k l X u zu p! X s H0 = p p θ(p = 6k + 6l) = Fs(z)u (5.21) 36 720 2 k,l≥0 (2k)!l!( 2 )!2 s for n = 0 with the altered

s s X X 1 z l (6s)! F (z) ≡ F (z, s, l) ≡ (5.22) s 288s 20 (2s − 2l)!l!(3s)! l=0 l=0

3/4 again a sum. The function F (z, s, l) has the same general shape as in the ϕz case and we can therefore repeat the steps from the previous section. The equation for maximizing F (z, s, l) given positive z

F (z, s, l + 1) z (s − l )2 1 = 0 0 (5.23) F (z, s, l0) 5 l0 is solved by

5 r5 25 l = s + − s + ≡ s − x (5.24) 0 2z z 4z2 which diﬀers from the y · s we saw before. We will make use of the properties5

5 s s s 5 x2 = (s − x) , 1 , exp x + (5.25) z s − x s − x 2z in the following calculations.

5 x The third property was derived by writing it as exp −s ln 1 − s and using a Taylor expansion for the logarithm up to second order.

58 Plugging in l0 and using Stirling’s approximation

1 s z s−x (6s)! F (z, s, l ) = 0 288 20 (2x)!(s − x)!(3s)! 3 s 5 (s − x)x 1 1 s3s 12s−x ≈ z 10 z x2 2π px(s − x) (s − x)s e 1 3 s 5 (s − x) x r s s s ≈ z Γ(2s) e (5.26) 2π 40 z x2 πx(s − x) s − x and then using the properties of equation 5.25

5 s 2x e 2z 3 e F (z, s, l ) z Γ(2s) · √ (5.27) 0 2π 40 πx reveals the factorially super divergence. Combining this with the term

(2x − 1)(2x)(s − x) 2 L00 = − ln ≈ (5.28) (2x + 1)(2x + 2)(s − x + 1) x results in

r 5 s 2π e 2z 3 F (z) F (z, s, l ) A (z) = z Γ(2s) · e2x (5.29) s 0 L00 s 2π 40 as saddlepoint approximation. This result is made plausible for z > 0 by ﬁgure 5.4, but it is certainly less accurate at low loop order for small z.

(s) Figure 5.4: Ratio between the coeﬃcients H0 and their predicted asymptotic value, plotted for multiple values of z.

3/6 6 So the ϕz theory has the same αf and cf as the ϕ theory, but an extra term e2x pops up which does not ﬁt in the factorially divergent framework as deﬁned in

59 section√ 2.2. The good news is that all machinery stays the same if it is added as n 6 a bf term to equation 2.14. Finding out if more such terms pop up in different combination theories would be an interesting topic for further research. 3/6 Since ϕz theory is factorially super divergent, we predict that the improvement factor tends to one for positive z and this is verified in figure 5.5 (top). While af and cf are real for all z, the factor x becomes complex for all negative values of z. We see this reflected in oscillatory behaviour of H0 and the ratio of the tails (see bottom figure 5.5), but this time the oscillations around the improvement factor seem to dampen at large loop order. 3/Q 4/Q Numerical verification showed that ϕz and ϕz theories with Q ≥ 5 also have an improvement factor of one.

3/6 Figure 5.5: Demonstrations of the trivial improvement factor for ϕz theory. Top: z = 1. Bottom: z = −1.

6 Such a term would sit between βf and cf in importance when it comes to the ≺ comparison.

60 5.3 Pure ϕ3/4 theory

In the previous two sections all coupling constants were written in terms of λ3. Doing so results in less freedom because there is one fewer parameter that can be renormalized. In this section we will revisit the combination theory with a three- and four-point interaction, but this time leaving λ4 free. The ϕ3/4 action λ λ S(ϕ) = 1 µϕ2 − 3 ϕ3 − 4 ϕ4 (5.30) 2 3! 4! is normalizable for λ4 < 0. Because we do not insist on a relation between λ3 and λ4, there are now two independent dimensionless parameters λ2 λ u = 3~ , u = 4~ (5.31) 3 µ3 4 µ2 and all series will be bivariate in these parameters. The integrals

p k l 2 X λ3 λ4 ~ p! Hn = p p θ(p = n + 3k + 4l even) (5.32) 6 24 µ 2 k,l≥0 ~ ~ k!l!( 2 )!2 reduce to the bivariate series expansion

X (k,l) k l H0 = H0 u3u4 k,l≥0 (5.33) 1 k 1 l (6k + 4l)! H(k,l) ≡ 0 36 24 (2k)!l!(3k + 2l)!23k+2l for n = 0. Since we are now dealing with a 2D grid instead of a single axis, it is not a priori clear what a fruitful definition of “asymptotic” is. Therefore we will first look at the two edge cases, namely the asymptotics along the u3 axis (l = 0, k → ∞) and along 3/4 3 the u4 axis (k = 0, l → ∞). For these cases ϕ theory reverts back to ϕ theory and ϕ4 theory respectively, of which we already know what the asymptotics are. The important conclusion of the two edge cases is that the asymptotics look different along the u3 axis than along the u4 axis. Based on this conclusion, analysis of the data, and some trial and error, the most logical approach seems to be to focus k on lines that go radially outward from the origin. On such lines the ratio a = l is fixed. Using Stirling’s approximation for large l a + 1 (6a + 4)3a+2 l H(al,l) ≈ Γ (a + 1)l − 1 0 p(2π)3a 36a · 24 · (2a)2a · (a + 1)a+1 2 and then rewriting in terms of the Manhattan distance from the origin r = k + l

1 r 3a+2 ! ( a r, 1 r) a + 1 (6a + 4) a+1 H a+1 a+1 Γ(r − 1 ) (5.34) 0 p(2π)3a 36a · 24 · (2a)2a · (a + 1)a+1 2 gives a form that resembles a factorially divergent series.

61 Renormalizing ϕ3/4 theory

For C1, C2 and C3 the tree-level diagrams do not contain four-point vertices, so it is possible to take out the leading term to get the tails t1,2,3(u3, u4). For higher Green’s functions the tree-level diagrams consist of both three- and four-point vertices, which complicates matters. Take for instance

λ2 3 λ 3 2 C = 3 3~ + 4~ + O( 4) = ~ (3u + u ) + O(u2) (5.35) 4 µ5 µ4 ~ µ 3 4 with a leading term containing 3u3 + u4 that cannot be divided out without leaving the realm of multivariate formal power series. Therefore we write the connected Green’s functions as n λ n C = ~ s (u , u ) ,C = 3~ ~ s (u , u ) 2n µ 2n 3 4 2n+1 µ2 µ 2n+1 3 4 with sn not starting with 1. The renormalization scheme renormalizes three connected Green’s functions λˆ 2 2 C ≡ ~ ,C ≡ 3~ ,C ≡ ~ (3ˆu +u ˆ ) (5.36) 2 µˆ 3 µˆ3 4 µˆ 3 4 with 2 ˆ2 ˆ C3 λ3~ C4 λ4~ uˆ3 ≡ 3 ≡ 3 , uˆ4 ≡ 2 − 3ˆu3 ≡ 2 (5.37) C2 µˆ C2 µˆ as renormalized dimensionless parameters. These parameters can be expressed in terms of the old parameters by expressions

2 t3(u3, u4) 2 uˆ3(u3, u4) = u3 3 = u3 + O(u ) t2(u3, u4) 2 (5.38) s4(u3, u4) t3(u3, u4) 2 uˆ4(u3, u4) = 2 − 3u3 3 = u4 + O(u ) t2(u3, u4) t2(u3, u4) that meet the requirements to be simultaneously invertible. Calculating the pullbacks s sˆ (û , uˆ ) = 2n (u (û , uˆ ), u (û , uˆ )) 2n 3 4 tn 3 3 4 4 3 4 2 (5.39) s2n+1 sˆ2n+1(û3, uˆ4) = n−1 (u3(û3, uˆ4), u4(û3, uˆ4)) t3t2 and plotting

(k,l) sˆn (k,l) (5.40) sn shows interesting asymptotic behaviour (see e.g. ﬁgure 5.6). For instance, the ratio tends to I[ϕ4] ≈ 42.52 for (k = 0, l → ∞) but not to I[ϕ3] ≈ 28.03 for (k → ∞, l = 0). The challenge of predicting this behaviour is left for future research.

62 Figure 5.6: Ratio of the coeﬃcients of C5 before and after renormalization.

63 Chapter 6

QCD

QCD (quantum chromodynamics), also known as the strong force, is the theory that describes the interactions between quarks and gluons. The goal of this chapter is to look at a zero-dimensional theory that consists of one quark and one gluon. The diﬀerence with the QED models of chapter 4 will be that the gluon also interacts with itself through a three-gluon and a four-gluon vertex. Fundamental to QCD is that there is a single coupling constant g for all interactions. The action we therefore settled on is 1 g2 1 S(q, q,¯ G) = mqq¯ + 1 µG2 + gqGq¯ + g G3 + G4 2 3! µ 4!

g where the extra factor µ of the four-gluon vertex is needed so that all tree diagrams are of the same order. Sadly enough the above action gives rise to two independent dimensionless parameters1

g2 g2 u = ~ , v = ~ µ3 m2µ which makes it impossible to analyse the asymptotics within the current framework. µ Our solution is to ﬁx the ratio x = m between the gluon and quark masses. Anticipating what is to come, we simultaneously include the z-parameter (which will be set to z = −1 in the end) to arrive at an alternative action

2 µ 1 2 1 3 g 1 4 S(q, q,¯ G) = qq¯ + 2 µG − gqGq¯ − g G − z G (6.1) x 3! µ 4! z=−1 for xQCD. The goal of this chapter is to analyse the asymptotics and a renormalization scheme for this xQCD theory.

√ 2 1 g ~ The dimensionless quantity uv = mµ2 also pops up, which makes it even worse than pure ϕ3/4 theory.

64 6.1 Asymptotics of H0,0 The perturbative expansion of the integrals generated by the xQCD action

j p 2 n+k 2 X g k g i zg x~ ~ (n + k)!p! Hn,l = p p 3! 4!µ µ µ 2 k,i,j≥0 ~ ~ ~ k!i!j!( 2 )!2 (6.2) θ(p = l + k + 3i + 4j even) reduces to

j k+i k i 2 2 X xg g zg ~ ~ p! H0,0 = 3 p p µ 3!µ 4!µ µ 2 k,i,j≥0 i!j!( 2 )!2 (6.3) θ(p = k + 3i + 4j even) for n = l = 0. Tricky about H0,0 is that the number of three-point vertices of each kind can be either even or odd. To handle these two cases, we split the triple sum in two. This makes it possible to redeﬁne2 k, i

2i j X k+i+j 2k 1 z p! H0,0 = u x p p θ(p = 2k + 6i + 4j) 6 24 2 k,i,j≥0 (2i)!j!( 2 )!2 2i+1 j X k+i+j+1 2k+1 1 z p! + u x p p θ(p = 2k + 6i + 4j + 4) 6 24 2 k,i,j≥0 (2i + 1)!j!( 2 )!2 X s ≡ [Es(x, z) + Fs(x, z)] u (6.4) s≥0 such that a series in u emerges. For the “even” coeﬃcients Es we rewrite j = r − i and k = s − r such that s r X X Es(x, z) ≡ E(s, r, i) (6.5) r=0 i=0 r i 2s z 2 p! E(s, r, i) ≡ x 2 p p θ(p = 2s + 2r + 2i) 24x 3z 2 (2i)!(r − i)!( 2 )!2 and for the “odd” coeﬃcients Fs we rewrite j = r − i and k = s − r − 1 such that

s−1 r X X Fs(x, z) ≡ F (s, r, i) (6.6) r=0 i=0 x2s z r 2 i p! F (s, r, i) ≡ 2 p p θ(p = 2s + 2r + 2i + 2) 6x 24x 3z 2 (2i + 1)!(r − i)!( 2 )!2 where E(s, r, i) and F (s, r, i) implicitly depend on x and z. As anticipated, a negative value of z produces an alternating sum with huge cancellations. The plan of action is to ﬁnd the asymptotics of xQCD for positive z and then see if we can make an analytical continuation to z = −1. 2(k, i) → (2k, 2i) or (k, i) → (2k + 1, 2i + 1)

65 6.1.1 Unphysical z > 0

For any given values of s, x, z the coefficients Es(x, z) and Fs(x, z) are just numbers. Since the functions E(s, r, i) and F (s, r, i) are so similar, we will focus on Es(x, z) and note the differences with Fs(x, z) afterwards. The goal of this subsection is to find the asymptotic expression for these coefficients in the unphysical case where z is positive. 1 2 Calculating Es(x, z) consists of summing over ∼ 2 s points (r, i). Distributing these points on a Cartesian grid, the points make up a triangle T . We are mainly interested in finding αf and cf , and for those it suffices to find the point (r0, i0) ∈ T for which the function E(s, r, i) is largest. Depending on the values of x and z, the point (r0, i0) lies either in the interior or on the boundary of the triangle. This is in contrast with the previous chapter where the maximum was always in the interior. In the following we will look at both cases separately.

Maximum in interior

If the point (r0, i0) lies in the interior of T , then E(s, r0, i0) is a local maximum and we can repeat the steps from the previous chapter. The point (r0, i0) is a solution to the system of equations

E(s, r , i + 1) 1 (s + r + i )(r − i ) 1 = 0 0 0 0 0 0 E(s, r , i ) 3z i2 0 0 0 (6.7) E(s, r0 + 1, i0) z (s + r0 + i0) 1 = 2 E(s, r0, i0) 12x (r0 − i0) since the slope is zero in all directions. Given x, z > 0 ∧ (r0, i0) ∈ T this system of equation is solved by 2x + z r0 = 2 s ≡ as 12x − 4x − z (6.8) 2x i = s ≡ bs 0 12x2 − 4x − z

1 2 if and only if x > 2 ∧ 0 < z < 6x − 3x . This means that the maximum lies in the 1 2 interior if and only if x > 2 ∧ 0 < z < 6x − 3x . Plugging in the found values of r0 and i0 yields

12x2 − 4x − z 24x4 s Γ(s) E(s, as, bs) √ √ (6.9) 2π 2xz 12x2 − 4x − z 2πs as asymptotic expression for the maximum value. Using the two-dimensional saddlepoint approximation from appendix C leads to

r 4 s 3 x 24x 1 Es(x, z) √ Γ(s + ) (6.10) π 12x2 − 4x − z 12x2 − 4x − z 2 for the asymptotic expression of the actual coeﬃcient.

66 Repeating the above steps for Fs(x, z) results in the same system of equations and therefore the same point (r0, i0). Also F (s, as, bs) E(s, as, bs) because of the 1+a+b (s) equality b = 6x . Therefore H0,0 2Es(x, z) or, explicitly,

r 3 2x 24x4 s H(s) A(s) ≡ √ Γ(s + 1 ) (6.11) 0,0 I π 12x2 − 4x − z 12x2 − 4x − z 2

1 2 if x > 2 ∧ 0 < z < 6x − 3x.

Maximum on boundary If there is no local maximum in the interior of T then the maximum must lie on the boundary. After investigating E(s, r, i), we know that the maximum lies on the edge r0 = s for. On this edge the function reduces to

3/4 E(s, s, i) = F [ϕz ](z, s, s − i) (6.12)

3/4 3 the ϕz -function from equation 5.5. This means that

s X 3/4 E(s, s, i) = Fs[ϕz ](z) (6.13) i=0 and i0 = (1 − y)s with √ 3 + 6z − 9 + 24z (3 − y)y y = , z = (6.14) 2 + 6z 3(1 − y)2

3/4 just as for the ϕz theory. We were not able to find the values of af and βf , but we (3−y)3 can conclude that αf = 1 and cf = 18(1−y)2 . Now it is time to look at Fs(x, z). The maximum of F (s, r, i) lies asymptotically at the same i0 but the r-sum for this coefficient only runs up to s−1 which causes it to deviate from Es(x, z). Knowing that the behaviour of the coefficients is determined by the peaks, we can conclude from the ratio

F (s, s − 1, (1 − y)s) 6x(1 − y) (6.15) E(s, s, (1 − y)s) (3 − y) that the coeﬃcients are proportional to one another

6x(1 − y) F (x, z) E (x, z) (6.16) s (3 − y) s with a proportionality factor that depends on x and z. Even though we cannot compute βf for general x and z, we can look at three 1 2 limiting cases. First of all, for x > 2 ∧ z = 6x − 3x the expansions of the two cases 3One might wonder why we did not redefine i → r − i from the start, but this would have caused difficulties when finding the maximum in the interior.

67 (maximum in the interior4 and on the boundary) must coincide. This means that 1 1 2 βf ≈ 2 for x > 2 ∧ z ≈ 6x − 3x . The other two limits are the heavy quark limit (x → 0) and the dominating four-point vertex limit (z → ∞). In both limits the 3/4 xQCD action approaches that of the ϕz action and so βf ↓ 0 in both limits. This brings us to the conclusion that

(3 − y)3 s H(s) ∼ A(s) ≡ Γ(s + β ) , β ∈ 0, 1 (6.17) 0,0 B 18(1 − y)2 f f 2

1 2 if x ≤ 2 ∨ z ≥ 6x − 3x .

6.1.2 Physical z = −1

(s) It is not a priori clear which asymptotic expansion of H0,0 should be used for an analytical continuation to z = −1. It is diﬃcult to numerically compare H0,0 with the asymptotic expression of the boundary AB because y is complex and we do not have the exact value of af and βf . Therefore our approach was to compare H0,0 with the asymptotic expression of the interior AI . We found that H0,0 AI if and 1 1 only if x > 2 . For x ≤ 2 we found that H0,0 displays oscillatory behaviour. From this we predict that

24x4 1 c = for x > (6.18) f 12x2 − 4x + 1 2

1 and that the ratios of the tails behave chaotically for x ≤ 2 .

4For z = 6x2 − 3x the asymptotic expansion of equation 6.11 is halved because half of the peak lies outside the triangle.

68 6.2 Renormalizing xQCD

In this section we will set up a renormalization scheme for the xQCD theory with z = −1. By ﬁxing x and thus removing m as a free parameter, only µ and g are left to renormalize. This entails that we can only renormalize one two-point and one other connected Green’s function. With the option in mind of adding more quarks, we chose to renormalize such that the gluon functions C0,2 and C0,3 become free of loop corrections.5 Deﬁning

2 2 2 ~ gˆ~ C0,3 gˆ ~ C0,2 ≡ ,C0,3 ≡ 3 , uˆ ≡ 3 = 3 (6.19) µˆ µˆ C0,2 µˆ the connected Green’s functions take the form g gˆ C = γ (x) ~t (u; x) ≡ γ (x) ~tˆ (ˆu; x) 0,1 0,1 µ2 0,1 0,1 µˆ2 0,1 (6.20) g 2n+l−2 gˆ 2n+l−2 C = γ (x)~ ~ t (u; x) ≡ γ (x)~ ~ tˆ (ˆu; x) n,l6=0,1 n,l µ µ2 n,l n,l µˆ µˆ2 n,l before and after renormalization. Because γ0,2(x) = γ0,3(x) = 1 the renormalized dimensionless parameter equals

2 t0,3(u; x) 3 2 1 uˆ(u) = u 3 = u 1 + (4x + 3x + x − 2 )u + ... (6.21) t0,2(u; x)

3 2 1 from which we can read oﬀρ ˆ1 = 4x + 3x + x − 2 .

Figure 6.1: Ratio of the tails for xQCD with x = 2 and improvement factor 3731 exp( 768 ) ≈ 128.78.

5 Choosing C1,0 and/or C1,1 instead showed no fundamentally diﬀerent behaviour.

69 6.2.1 The improvement factor for xQCD 1 (s) ˆ(s) We already predicted for x ≤ 2 that the ratio of the tails t /t will behave chaotically, and this is exactly what we see. We can therefore say nothing meaningful for this regime. 1 For the regime x > 2 we found 1 24x4 ρˆ = 4x3 + 3x2 + x − , c = (6.22) 1 2 f 12x2 − 4x + 1 which combine into

5 4 3 2 ρˆ1 96x + 40x + 8x − 14x + 6x − 1 I[xQCD] ≡ exp = exp 4 (6.23) cf 48x as xQCD improvement factor.6 Equations 6.20 and 6.21 imply

t0,2 ∼ t1,0 ∼ t0,1 ∼ uˆ ≺ t0,3 ∼ t1,1 ≺ tn,l ∀ 2n + l ≥ 4 (6.24) as asymptotic comparisons for the tails. Therefore we predict that the ratio of the tails tends to I[xQCD] at large loop order for all connected Green’s functions with at least four external lines. This predicted behaviour is verified in figure 6.1 for x = 2. In the same figure we can see three outliers. The tadpole C0,1 tends to zero as normal, but C1,0 and C1,1 seem to tend to their own asymptotic constants that are neither zero nor the improvement factor. The relation

µˆ t1,0(ûρ(û); x) tˆ1,0(û; x) ≡ t1,0(u(û); x) ≡ µ t0,2(ûρ(û); x) ρ (1) (1) 1 cf uρˆ (û)[t1,0 − t0,2] + e [t1,0(û; x) − t0,2(û; x)] ρ1 ρ1 cf 2 1 1 cf −e uˆρˆ(û)(x − 2 x − 2 ) + e [t1,0(û; x) − t0,2(û; x)] ρ1 c 2 e f t1,0(û; x) − t0,2(û; x) − (2x − x − 1)ût0,3(û; x) (6.25) proves that tˆ1,0 ∼ t1,0 and the relation

2 2 ˆ g~ gˆ~ t1,1(ûρ(û); x) t1,1(û; x) ≡ 3 3 t1,1(u(û); x) ≡ µ µˆ t0,3(ûρ(û); x) ρ1 c e f [t1,1(û; x) − t0,3(û; x)] (6.26) proves that tˆ1,1 ∼ t1,1. In the interest of time, we did not compute the exact asymptotic relation for these outliers because doing so would probably entail finding the asymptotic expression for H1,0, H0,2 and H0,3 like we did for H0,0. The asymptotic 2 expression for H1,1 is not needed because t1,0(u; x) x ut1,1(u; x) can be proven using Feynman diagrams.

6 5 1 To get a feeling for this factor: I[xQCD] = exp 3 ≈ 5.29 for x = 2 and increases as x increases. Why the improvement factor becomes larger for larger x is still an open question.

70 Chapter 7

Higgs

3/4 As icing on the cake we will look at one final theory. The symmetry of the ϕ 1 − 3 action is spontaneously broken (see equation 5.14), analogous to the Mexican hat potential of the Higgs field in the Standard Model. Therefore we will investigate the action m µ 1 1 µ 1 S(ϕ, ϕ,¯ H) = mϕϕ¯ + 1 µH2 + ϕHϕ¯ + H3 + H4 2 v v 3! 3 v2 4! containing a fermionic field ϕ with mass m, a Higgs field H with mass µ, and µ mass-dependent interactions. By rewriting v = − λ we get an action m 1 1 λ2 1 S(ϕ, ϕ,¯ H) = mϕϕ¯ + 1 µH2 − λϕHϕ¯ − λ H3 + H4 (7.1) 2 µ 3! 3 µ 4! that is quite similar to the one for QCD. Due to the mass-dependent fermion-Higgs coupling, the parameter

λ2 u = ~ µ3 is the only dimensionless parameter in the Higgs theory. As a result it is possible to add an arbitrary number of fermions to this theory without needing to ﬁx mass ratios, as would be necessary for QED or QCD. A nice consequence is that it is possible to renormalize all masses. In this research we will limit ourselves to one fermion.

71 7.1 Asymptotics of H0,0 The perturbative expansion of the integrals generated by the Higgs action

j p k i 2 n+k 2 X mλ λ 1 λ ~ ~ (n + k)!p! Hn,l = − p p µ 3! 3 4!µ m µ 2 k,i,j≥0 ~ ~ ~ k!i!j!( 2 )!2 (7.2) θ(p = l + k + 3i + 4j even) reduces to

j k+i k i 2 2 X λ λ 1 λ ~ ~ p! H0,0 = − 3 p p µ 3!µ 3 4!µ µ 2 k,i,j≥0 i!j!( 2 )!2 (7.3) θ(p = k + 3i + 4j even)

1 for n = l = 0. This form is exactly the same as for QCD with x = 1, z = − 3 and g → λ (see equation 6.3). Therefore we know immediately from equation 6.11 that

6 72s H(s) √ Γ(s + 1 ) (7.4) 0,0 5 π 25 2

72 so αf = 1 and cf = 25 .

72 7.2 Renormalizing Higgs theory

The connected Green’s functions take the form

λ λˆ C = γ ~t (u) ≡ γ ~tˆ (ˆu) 0,1 0,1 µ2 0,1 0,1 µˆ2 0,1 µ n λ 2n+l−2 C = γ ~ ~ t (u) n,l6=0,1 n,l µ m µ2 n,l (7.5) !2n+l−2 µˆ n λˆ ≡ γ ~ ~ tˆ (ˆu) n,l µˆ mˆ µˆ2 n,l before and after renormalization. We chose to renormalize such that

ˆ 2 2 ˆ2 ~ ~ λ~ C0,3 λ ~ C1,0 ≡ ,C0,2 ≡ ,C0,3 ≡ 3 , uˆ ≡ 3 = 3 mˆ µˆ µˆ C0,2 µˆ because of the option of adding more fermions to the theory. The renormalized dimensionless parameter equals

2 t0,3(u) 25 uˆ(u) = u 3 = u 1 + u + ... (7.6) t0,2(u) 2

25 72 soρ ˆ1 = 2 . Combined with cf = 25 this gives ρˆ 625 I[Higgs] ≡ exp 1 = exp ≈ 76.73 (7.7) cf 144 as improvement factor which is verified in figure 7.1. This time the only outlier (apart from the tadpole) is C1,1. For its renormalized tail we get , ! λ 2 λˆ 2 tˆ (û) ≡ ~ ~ t (u(û)) 1,1 mµ2 mˆ µˆ2 1,1

t0,2(u(û)) ≡ t1,1(u(û)) t1,0(u(û))t0,3(u(û))

t1,1(ûρ(û)) − t0,3(ûρ(û)) ρ1 c e f [t1,1(û) − t0,3(û)] (7.8) which is verified in figure 7.2. The qualitative behaviour tˆ1,1 ∼ t1,1 can indeed be seen in figure 7.1.

73 Figure 7.1: Veriﬁcation of the Higgs improvement factor.

Figure 7.2: Veriﬁcation of the tˆ1,1 behaviour.

74 Conclusion

Perturbation expansions in zero-dimensional quantum field theory exhibit factorially divergent behaviour. This asymptotic behaviour can often, but not always, be captured by the given definition of a factorially divergent series. Mathematical understanding of factorially divergent series — especially the pfd pullback and pfd inversion — leads to understanding the large loop order behaviour of connected Green’s functions after renormalization. The difference between unrenormalized and renormalized connected Green’s functions is quantified by the improvement factor. For super divergent theories the improvement factor is always one. For non-super divergent theories the improvement factor is a combination of the exponential divergence and the one-loop terms of the renormalized functions. The latter implies that the improvement factor depends on the renormalization scheme. However, while there are an infinite number of ways to renormalize a theory, there are only a small number of logical scheme choices. Theories containing one interaction term can be easily analysed using the current framework. For instance, ϕQ theory was analysed for all Q. Theories containing multiple interaction terms demand a more involved analysis which makes it difficult to generalize procedures and conclusions. Extending the framework to multivariate formal power series could solve this problem.

There are in this world optimists who feel that any sym- bol that starts oﬀ with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer. — E.J. McShane

75 Appendices

76 Appendix A

Factorially divergent pullbacks

A pullback f(xg(x)) of properly factorially divergent series f(x), g(x) occurs nat- urally in the renormalization scheme, and we would like to know the asymptotic behaviour of these factorially divergent pullbacks. In this appendix we will derive the factorially divergent pullback

( g1 f1xg(x) + f(x) exp if αf = 1 f(xg(x)) cf f1xg(x) + f(x) if αf > 1 as stated in equation 2.22. If f(x) and g(x) are properly factorially divergent series, then

X n f(xg(x)) = fn(xg(x)) n≥0 expresses the factorially divergent pullback of the two. To determine the asymptotic behaviour of this pullback, we ﬁrst split oﬀ the largest term proportional to g(x). We have (xg(x))n nxng(x) ≺ xg(x) for n ≥ 2 so neglecting the higher powers

X n f(xg(x)) = 1 + f1xg(x) + fn(xg(x)) n≥2 X n 2 n f1xg(x) + fnx (1 + g1x + O(x )) n≥2 already reveals the term f1xg(x). Now we look at the terms proportional to f(x) by ﬁrst using the binomial theorem

X n n f(xg(x)) f1xg(x) + fnx (1 + g1x) n≥2 n X X n = f xg(x) + f xn (g x)k 1 n k 1 n≥2 k=0 n X X n − k = f xg(x) + xn f gk 1 n−k k 1 n≥2 k=0

77 and then regrouping xn. The approximation

n − k n − k f ≈ a cn−kΓ(α (n − k) + β ) n−k k f f f f k

n−k (αf n − αf k)! (n − k)! ≈ af cf Γ(αf n + βf ) (αf n)! k!(n − 2k)! 1 1 1 ≈ fn α f k (αf −1)k (cf · αf ) k! n is where αf becomes important. For αf > 1 the sum over k is completely dominated by k = 0 and therefore

f(xg(x)) f1xg(x) + f(x) if αf > 1 proves the ﬁrst result. For αf = 1 the approximation leads to

n k X n X 1 g1 f(xg(x)) f1xg(x) + x fn k! cf n≥2 k=0 which we recognize as the series expansion of the exponential function as n → ∞ and therefore g1 f(xg(x)) f1xg(x) + f(x) exp if αf = 1 cf proves the second result.

78 Appendix B

Computing the counterterm in CQED (alternative method)

In this appendix we describe an alternative way of determining the tadpole counterterm T for Counterterm QED (section 4.2). This is done by taking Bald QED and transforming it into Counterterm QED by redeﬁning parameters. First let us compute the tadpole t ≡ C0,1 of a Bald QED with masses M and µ. The tadpole obeys the diagrammatic equation

= which translates to e e2 e2 t = − ~ + C t + C (B.1) Mµ Mµ 1,0 Mµ 1,1 by following the Feynman rules. We proceed by writing C1,0 and C1,1 in terms of t. µ C1,0 is C0,1 with the incoming photon and its vertex removed, so C1,0 = − e t. C1,1 is e~ C1,0 with an extra external photon on any fermion line, so C1,1 = µ ∂M C1,0 = −~∂M t. e~ Plugging these results into equation B.1 and writing t = − Mµ Q(u) with Q(u) pfd 2 e ~ and u ≡ M 2µ we get a diﬀerential equation for the tail

2 2 Q(u) = 1 + uQ(u) + uQ(u) + 2u ∂uQ(u) (B.2) which can be solved by iterating. The ﬁeld functions for the photon A(H) + t and fermion φ(J) as deﬁned in [1] obey the SDe’s ¯ µA(H) = H − e(φφ + ~∂J φ) − µt (B.3) (M + et)φ(J) = J − e(Aφ + ~∂H φ) where A(H) explicitly does not contain C0,1. Comparing this with the SDe’s for CQED ¯ µA(H) = H − e(φφ + ~∂J φ) − T (B.4) mφ(J) = J − e(Aφ + ~∂H φ)

79 we read oﬀ e T = µt = − ~Q(u) M (B.5) m = M + et = M(1 − uQ(u)) as expressions for the counterterm T and fermion mass m in terms of M and u. Now we can write α in terms of u

M 2 α(u) = u = u(1 − uQ(u))−2 (B.6) m which can be inverted to obtain u(α). Writing the pullback as R(α) ≡ Q(u(α)) brings us to our ﬁnal result for the counterterm e T = − ~ (1 − u(α)R(α)) R(α) (B.7) m in terms of m and α.

80 Appendix C

Saddlepoint approximation in QCD

(s) In this appendix we will derive the exact asymptotic form of H0,0 in QCD when z > 0 and 0 < r0 < s. This is a straightforward generalisation of the saddlepoint approximations in chapter 5, but the tedious calculations are better left out of the main text. The asymptotic expression for the peak value of E(s, r, i) is given by

12x2 − 4x − z 24x4 s Γ(s) E(s, r0, i0) √ √ 2π 2xz 12x2 − 4x − z 2πs as per equation 6.9. The double sum Es(x, z) can be approximated by a two- dimensional saddlepoint approximation 2π Es(x, z) ≈ E(s, r0, i0) (C.1) p 00 00 00 00 LrrLii − LriLir

00 where the terms L are all second derivatives of ln(E(s, r, i)) at the point (r0, i0). First we compute the curvature in the r-direction 00 E(s, r0 + 1, i0)E(s, r0 − 1, i0) Lrr = − ln 2 E(s, r0, i0) (2s + 2r + 2i + 1)(r − i ) = − ln 0 0 0 0 (2s + 2r0 + 2i0 − 1)(r0 − i0 + 1) 1 1 ≈ − − s + r0 + i0 r0 − i0 12x2 − 4x − z = (12x2 − z) (C.2) 12x2zs

81 and the curvature in the i-direction 00 E(s, r0, i0 + 1)E(s, r0, i0 − 1) Lii = − ln 2 E(s, r0, i0) (2s + 2r + 2i + 1)(2i − 1)(i )(r − i ) = − ln 0 0 0 0 0 0 (2s + 2r0 + 2i0 − 1)(2i0 + 1)(i0 + 1)(r0 − i0 + 1) 1 2 1 ≈ − − − ) s + r0 + i0 i0 r0 − i0 12x2 − 4x − z = (12x2 + 12xz − z) (C.3) 12x2zs just like in chapter 5. By choosing a convenient deﬁnition for the derivative with respect to i we get the cross term 00 E(s, r0 + 1, i0)E(s, r0, i0 − 1) Lri = − ln E(s, r0 + 1, i0 − 1)E(s, r0, i0) (2s + 2r + 2i + 1)(r − i + 2) = − ln 0 0 0 0 (2s + 2r0 + 2i0 − 1)(r0 − i0 + 1) 1 1 ≈ − + s + r0 + i0 r0 − i0 12x2 − 4x − z = −(12x2 + z) (C.4) 12x2zs

00 00 and Lir equals Lri since derivatives commute. All calculations are the same for Fs(x, z) meaning that Es(x, z) Fs(x, z) and (s) H0,0 2Es(x, z). Combing everything gives √ (s) 2πx 12xz H0,0 ≈ 2E(s, r0, i0) 3 s (12x2 − 4x − z) 2 r 3 2x 24x4 s √ Γ(s + 1 ) (C.5) π 12x2 − 4x − z 12x2 − 4x − z 2 as the result.

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