The Standard Model Effective Field Theory Integrating UV Models via Functional Methods FAGNER CINTRA CORREIA A dissertation submitted to the Institute for Theoretical Physics - Sao Paulo State University in accordance with the requirements for the degree of DOCTOR OF PHILOSOPHY. Instituto de Física Teórica UNIVERSIDADE ESTADUAL PAULISTA “JÚLIO DE MESQUITA FILHO” AUGUST 2017 Número: IFT-T.004/17. Word count: 59394 Dedicado à Camilla. ABSTRACT It will be presented the principles behind the use of the Standard Model Effective Field Theory as a consistent method to parametrize New Physics. The concepts of Matching and Power Counting are covered and a Covariant Derivative Expansion introduced to the construction of the operators set coming from the particular integrated UV model. The technique is applied in examples including the SM with a new Scalar Triplet and for different sectors of the 3-3-1 model in the presence of Heavy Leptons. Finally, the Wilson coefficient for a dimension-6 operator generated from the integration of a heavy J-quark is then compared with the measurements of the oblique Y parameter. Keywords: Covariant Derivative Expansion, Standard Model Effective Field Theory, 3-3-1 Models, Triplet Scalar, Oblique Parameters. iii RESUMO O Modelo Padrão Efetivo é apresentado como um método consistente de parametrizar Física Nova. Os conceitos de Matching e Power Counting são tratados, assim como a Expansão em Derivadas Covariantes introduzida como alternativa à construção do conjunto de operadores efetivos resultante de um modelo UV particular. A técnica de integração funcional é aplicada em casos que incluem o MP com Tripleto de Escalares e diferentes setores do modelo 3-3-1 na presença de Leptons pesados. Finalmente, o coeficiente de Wilson de dimensão-6 gerado a partir da integração de um quark-J pesado é limitado pelos valores recentes do parâmetro obliquo Y. Palavras-Chave: Expansão em Derivadas Covariantes, Modelo Padrão Efetivo, Modelos 3-3-1, Tripleto Escalar, Parâmetros Oblíquos. v AUTHOR’S DECLARATION I declare that the work in this dissertation was carried out in accordance with the requirements of the he Institute for Theoretical Physics - UNESP’s Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate’s own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author. SIGNED: .................................................... DATE: .......................................... vii PREFACE Some time ago I lived the circumstance of having to search for a place on a map. I was with Prof. Pleitez. After a few minutes we were not even close to realize where we were or about the point we were trying to reach. Then he told me something that I will probably never forget - “This is the problem with the maps, you never know where you are going”. Well, the purpose of a map should be exactly the opposite, of course. It is true, in fact, that some of them may require a small experience as a scout. However, the point about my supervisor’s conclusion was that it remounted me a memory of my first year in college. At that time I was used to think about Physics exactly as a map, but a map made from a previous knowledge on the treasure. The treasure of Higgs phenomenology, of dark matter, neutrino and flavor physics, etc. After studying the topic of the present work I have realized about our problem that day. The scale of that map was not the approprite one. The place we were looking for was not too close, but not too far. Most of those symbols and lines were indeed not necessary, and we did not need the information about places that we were not able to reach in any case. At the other hand if the map had only our current street and something about the neighbor, it would be still useless. What we needed was a map in an intermediary scale, a simplified one, an effective map. The present work is about maps and is my ackowledgment to Prof. Vicente Pleitez. ix TABLE OF CONTENTS Page List of Tables xv List of Figures xvii I Introduction 1 1 The Method in High Energy Physics 3 1.0.1 Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.0.2 Top-down approach and Precision Observables . . . . . . . . . . . . . . . . . 5 II The Covariant Derivative Expansion Literature Review 9 2 The Covariant Derivative Expansion 11 2.1 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 On the Effective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 On the sign of determinant for real, complex scalars and fermions . . . . . 15 2.3 Evaluating the Functional Determinant - On the Universal Formula . . . . . . . . 17 2.3.1 Evaluation of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 The Universal Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Evaluating the Functional Determinant - On the Mixed Terms . . . . . . . . . . . . 29 2.4.1 The Matching Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.3 On the Meaning of the Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Fundamentals of Effective Field Theories 43 3.1 The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The Weinberg Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 The Decoupling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xi TABLE OF CONTENTS IIIThe Standard Model Effective Field Theory 57 4 The 3-3-1 Model with Heavy Leptons 59 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Gauge Structure and Scalars in the 3-3-1HL . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.1 Particle Content in Different Versions . . . . . . . . . . . . . . . . . . . . . . 66 4.2.2 Self-Interactions of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.3 Vacuum Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.4 Gauge-Fixing Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.5 Gauge-Boson Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.6 Scalar Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.7 The Potential for particular models . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.8 Self-Interactions of Gauge Bosons . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Fermions in the 3-3-1HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.1 Gauge interactions of the Fermions . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.2 Yukawa Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5 Matching in the Functional Formalism 89 5.1 Triplet Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.1.1 Matching at the Log-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.1.2 Wilson Coefficients from Mixed Loops . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Comments on the Triplet Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Theories with Kinetic Mixing of Gauge-Bosons . . . . . . . . . . . . . . . . . . . . . 101 5.4 Integrating the Z′ of the 3-3-1HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.1 At tree-level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4.2 At Loop-Level: Mixed Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 The J-quarks in the 3-3-1HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Precision Observables 109 6.1 Operator Mixing and Anomalous Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Running the Wilson Coefficients - Considerations . . . . . . . . . . . . . . . . . . . . 113 6.3 J-quarks and Electroweak Precision Observables . . . . . . . . . . . . . . . . . . . . 113 6.3.1 Scaled Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3.2 The Y Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.3.3 The Role of the cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7 Conclusions 121 A Formulae 125 xii TABLE OF CONTENTS A.1 On Gamma Functions and MS scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2 Properties for Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.3 Finite Components of I2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.4 On CP-odd Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B Example from a Discrete Picture 135 Bibliography 141 xiii LIST OF TABLES TABLE Page 5.1 The set of dimension-six Bosonic operators . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.1 Bosonic Operators - Running Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2 The column c2B of the anomalous dimension matrix for the dim-6 Bosonic operators. 115 6.3 The 95 % CL measurement to the Precision Observables. . . . . . . . . . . . . . . . . . 118 xv LIST OF FIGURES FIGURE Page 2.1 Effective Action as a Generating Functional - λφ3. . . . . . . . . . . . . . . . . . . . . . 14 2.2 Effective Action as a Generating Functional - λφ4. . . . . . . . . . . . . . . . . . . . . . 14 2.3 Light UV Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Effective Vertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 1LPI graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 The Decoupling Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.1 The graphs present in the mixed-heavy matching procedure. . . . . . . . . . . . . . . . 106 xvii Part I Introduction 1 CHAPTER 1 THE METHOD IN HIGH ENERGY PHYSICS The title of this chapter remounts a complex and diverse area that will not be entirely, or even approximately, covered by the present work. It has been chosen, however, because the set of elements defining a methodology for particle physics will contain (i) a complete quantum field theory (or UV, for short), (ii) how to extract predictions from it and (iii) and how to connect it with the experiment. Here, these three topics are introduced in a specific form. The starting point consists in a technique for simplifying the computation from the UV model, redefining it like an Effective Field Theory (EFT). It will be seen that the use of an EFT must preserve the overall of quantum field theory paradigms and it can be better considered as a tool, a very consistent one, appropriate for many branches of theoretical physics. In High Energy Physics this technique will play an important role in the context where new degrees of freedom do not emerge asymptotically in the experiments, what is the current status of the measurements coming from the Large Hadron Collider since the discovery of the Higgs boson. The construction of an EFT may be performed by two approaches of integration - functional methods or Feynman diagrams - followed by two views of matching - subtraction or integration by regions. The term integration, or to integrate a model, express the procedure of cutting a heavy sector of the model out of the set of external particles. As mentioned, it follows from the phenomenological fact that new physics have not emerged in this form yet. The term matching express the correspondence into the both modes of describing the physical process - At a given scale the EFT implies the same predictions as the UV complete theory. The content of the Chapter 2 covers one combination of the points above. The purpose is to present a technique of integration that rewrites the UV theory directly into its Effective form by preserving the original symmetries. The method is called Covariant Derivative Expansion, a concept revealing its own meaning. The presentation is supported by two recent works of B. Henning et al. ([32] and [33]) and it involves the task of writing a series from the non-local components of the UV theory after the heavy particles have been integrated out. The series must preserve the covariant derivative intact. The functional that must be expanded consists of the first quantum correction to the Effective Action of both theories. These determinants are a more general result than those from a perturba- tive expansion and they will be called Log corrections, in contrast to the term 1-loop corrections. Since the generators of one-particle-irreducible graphs are the fundamental objects for defining the analytical structure of a theory, the matching is performed by equating the Effective Actions, denoted by Γ, and solving to the so-called Wilson coefficients. It will be seen that the equations 3 CHAPTER 1. THE METHOD IN HIGH ENERGY PHYSICS are defined order-by-order, what permits the construction of the EFT in a systematic manner. In addition, the matching equations emerge with an important conceptual meaning - the Wilson coefficients, ci, receives contributions from the difference of ΓUV and ΓEFT Log corrections. Since, by definition, the EFT must agree with the UV in the soft region of momentum scales, the ci is corrected at this level with the information about the heavy line in the hard region. By matching through subtraction is meant to perform the Log computation in both theories and then to identify the difference between them. At the other, during the match through integration by regions, the many loop integrals are computed already considering q2 ∼Λ inside the integrand, with Λ denoting a heavy scale. The methods are equivalent and must lead to the same results. In the literature the combination of Feynman diagrams plus subtractions is present, for example, in the work about QCD corrections to weak interactions of Buchalla et. all in [8]. For integration by regions it can be mentioned the textbook of J.Donoghue et. al, [21]. To the functional approach plus integration by regions it follows the recent work of Fuentes-Martin et.al, [28]. As mentioned before, the second article of B.Henning et al. [33] presents the last combination, i.e. functional methods followed by the subtraction, but does not contain a final expression to the case where both light and heavy particles run into the loops. The Chapter 2 propose a complement to this topic. 1.0.1 Renormalizability The Top-Down approach described above is confirming the statement that the effective the- ory comprises the principles of a quantum field theory and is in fact addressing any possible controversy about non-renormalizability. The EFT composes an alternative on searching for New Physics through the virtual effects of particles that cannot be produced as free states, and explores how they may enhance the Standard Model parameters. The Chapter 3 develops three fundamental results that may support these assertions - the Operator Product Expansion, the Weinberg Theorem and the Decoupling Theorem. The presentation about infinities and renormalizability will not intend to saturate or repeat the original fonts, namely the works of Bogolyubov and Shirkov [5], Peskin and Schroeder [42] and more recently Matthew Schwartz [49]. The aim is to explore the total of components behind the EFT technique, thus composing a consistent chain. The acceptance of the presence of infinities is specially a consequence of a better conceptual comprehension of the theory. If the computation of a divergent loop resulted in a non-analytical piece on the external momentum, this would require an insertion of a non-local term in the Lagrangian, thus violating the locality of the theory. The Weinberg theorem will prove, however, that any divergent piece emerging from the theory must be proportional to polynomials in the external momenta, what is translated into the locality of the counterterms. The presence of divergent terms with a non-analytical structure on the external momenta could be, perhaps, a better definition of non-renormalizability. The fact of having to include terms consistently, 4 although an indefinite number of times, which are suppressed by the completion scale of the theory and that can handle with the divergences is, at the most, a practical issue. If such an infinite, but discrete, set of operators are proposed a priori, this final theory, of no definite form and preserving the symmetries of its complete version, would still be predictive. It is important to remark that the presence of infinities occurs at the very beginning in quantum field theory. By placing a harmonic oscillator to every point in space-time, for example, at the same time an infinity amount of energy is associated to the universe [49]. The result must not be inconvenient since every measurement in nature is only meaningful once it is performed through a comparison, a difference. Every number should correspond a variation. At some cases, to perform a variation (or subtraction) can be something trivial, like estimating the position of a body from some reference point. However, this variation in quantum field theory, although very systematic, is not direct. It must occur for every parameter and fields through the renormalization procedure. The variables performing the subtraction are the counterterms and rely on the above mentioned feature of the theory - the infinities appears as polynomials in the external momenta, a property connected to the local aspect of the quantum theory. During the renormalization procedure the choice for the subtraction scale must be done, and the final result is independent from it. It is important to emphasize that to state that some piece of the Lagrangian is independent of the renormalization scale is not to say that the couplings does not depend on the physical scale. What is being changed, in truth, is the starting point, the reference where the subtraction was chosen to be performed. That is equivalent to say that the first measurement of the fine-structure constant α could have been done at a larger scale than ∼ me - This single measurement would be sufficient to provide exactly the same predictions of QED. In other words, the renormalization group equations does not come from a hypothesis of independence of a physical process on the energy, but from the independence of the object (correlation functions) associated with the given process under the change of the scale that defines its components (couplings and operators). A Green’s function is invariant under translation as a physical process will also be. Finally, the functional form of a running coupling on the energy scale as well as its Landau pole can contribute to define the properties of a given phenomena. 1.0.2 Top-down approach and Precision Observables The previous discussion may clarify the question about when the perturbative analysis is impor- tant. The answer is - when the scale of the process computed is not exactly the scale where the coupling constant were extracted. If it was, only the tree-level computation would be necessary, although this would not mean that there would not be a subtraction present. The subtraction for tree-level exact results was implicit during the fitting of the constants with the data. The renormalization group equations will then perform the sum of corrections at once and will transfer them to the couplings, leaving the last task of computing a tree-level matrix element. The point about this discussion, therefore, is that the matching procedure provides the boundary 5 CHAPTER 1. THE METHOD IN HIGH ENERGY PHYSICS conditions for solving the RGE. The running of the Wilson coefficients will be associated with the many corrections from the light-particles present in the renormalizable sector of the theory, namely the Standard Model. These assertions reveal the advantage of working in a top-down approach. It is clear since the starting point that the aim of making an effective model is nothing more than simplifying the analysis of the original UltraViolet theory. By the correspondence principle one extracts the couplings at the heavy scale that can be run down to the electroweak scale. The final set of operators will then be considered for an arbitrary number of process and through a tree-level analysis. According to the Decoupling Theorem, the error committed on working exclusively in a Stan- dard Model framework must be proportional to 1 ma , being m a heavy-scale and a approximately two. The aim of raising an SMEFT is exactly to reduce this error by including new local inter- actions suppressed by the same power. What is going to be shown in the Chapter 3, therefore, precedes the matching procedure and can be summarized as follows - (i) Consider the initial UV complete theory as a large set of interactions containing those of the Standard Model as a subset, i.e. ΓUV =ΓSM +ΓSM+H (1.1) where H is meant to be a heavy sector. (ii) To explore the consequences of proposing a low-energy version of the UV model consisting in the SM theory with changed couplings and masses, emerged by cutting out ΓH from the graphs generated by ΓUV : ΓUV →ΓSM (1.2) such that the 1LPI graph, i.e. 1PI graphs in the light-fields, generated by Γ(n) UV will be Γ(n) UV =Γ(n) SM × [ 1+O ( 1 ma )] (1.3) The first task is to prove that exists a simplified theory at low energies which is able to provide the same results as the complete theory, at some level of precision. Next, the error is inserted as local operators into the low-energy renormalizable theory, representing the value of the coupling constant in the heavy scale. Since the exchange of light particles involves renormalizable couplings, the coefficients will be run down to the low-energy scale through these light corrections. One comment, the differential equation will also be coupled, which corresponds a dependence of the anomalous dimension for the higher-dimension operators on the SM parameters. Finally, the Decoupling Theorem confirms that the purpose of constructing an EFT is not of defending the use of non-renormalizable theories as a final description of the nature, but to develop a technique for studying it. The SM is, by hypothesis, a renormalizable low-energy sector of a more complex complete theory. 6 The present work develops the principles discussed above. The Part II consist of a literature review about the functional methods of matching and fundamentals of Effective Field Theories. The Part III contains the introduction of gauge theories with spontaneously symmetry breaking, where the 3-3-1 models with Heavy leptons have been chosen as the main example. The choice was motivated by the possibility of exploring features that are not present in the Standard Model. Apart from that, it is in general alleged that these theories present a pattern of symmetry breaking following strictly the path 3⊗3⊗1 → 3⊗2⊗1 → U(1), which includes the Standard Model. Thus, a SMEFT can be extracted from it, leading to a entire connection with the rest of the work. The Chapter 5 apply the results of Part II to a set of models, including some specific sectors of the 3-3-1HL. The defined set of Wilson coefficients must be run down and face some Electroweak Precision Observables in Chapter 6, thus informing about how far the experimental measurements are from being sensible to the theories at the loop level. 7 Part II The Covariant Derivative Expansion Literature Review 9 CHAPTER 2 THE COVARIANT DERIVATIVE EXPANSION Consider an UltraViolet complete theory (UV for short) formulated in order to present two sectors separated by different scales. The light sector can be generically denoted by φ and the heavy sector by Φ. The generating functional is expressed like [53] ZUV [JΦ, Jφ]= ∫ DΦDφei ∫ x(LUV [Φ,φ]+Jφφ+JΦΦ) (2.1) On what follows the Effective Action for the Light-UV theory is given by the following Legendre transform ΓL,UV [φ]=−i log ZUV [JΦ = 0, Jφ]− ∫ x Jφφ (2.2) i.e. the source for heavy fields must be equal to zero. In other words the sector Φ are not allowed to emerge as external particles and the solution for Γ can be obtained through the saddle-point approximation by expanding the integrand around the classical solutions: δSUV [Φ,φ] δΦ ∣∣∣ JΦ=0 = 0, δSUV [Φ,φ] δφ = 0 (2.3) such that ΓL,UV [φ]≡ΓL,UV [Φ[φ],φ] after the replacement of Φ[φ] as an implicit functional of the light fields. The UV Effective Action expressed in terms of background fields at the Log order is given by ΓUV [φ,Φ]= SUV [φ,Φ]+ iα logdet ( δ2SUV δ(φ,Φ)2 ) (2.4) 2.1 Locality The concept of locality will be present in some of the most important steps of the matching procedure and defines, for example, the correct mode of performing the loop counting in the Effective and in the Light-UV theory. Consider a theory for two scalar fields defined by the Lagrangian [49]: L =−1 2 φ ( 2+M2) φ− 1 2 π ( 2+m2) π+ λ 2 φπ2 (2.5) The equations of motion for φ, when applied back to L , may convert it into a theory for the light fields π. In other words, δS δφ = 0 → −( 2+M2) φ+ λ 2 π2 = 0 (2.6) where the solution may be given in terms of the Green functions to the Klein-Gordon operator( 2+M2) x Gφ xy = δxy (2.7) 11 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION and here the continuous index is a shorthand for the variables defining functions and operators or, for example, δxy ≡ δ(x− y). Besides, the upper index denote the dependence on M and will be omitted on what follows. The above expression implies φx = λ 2 〈Gxyπ 2 y〉 ≡ λ 2 ∫ y Gxyπ 2 y (2.8) where it was used a shorthand notation (by [48]) that will frequently be evoked along this work - Repeated space-time indices inside brackets < ...> are being integrated over the total volume. By plugging Eq.(2.8) back into Eq.(2.5), it follows L = −λ 2 8 〈Gxzπ 2 z〉 ( 2+M2) x 〈Gxyπ 2 y〉− 1 2 π ( 2+m2) π+ λ 4 〈Gxyπ 2 y〉π2 x = λ2 8 π2 x〈Gxzπ 2 z〉− 1 2 π ( 2+m2) π (2.9) and, thus, the function L on a particular x is now being simultaneously affected by the total field configuration from the integral inside the brackets. This new quality is what defines the Lagrangian as a non-local object. The Fourier transform of Gxy is given by Gxy = ∫ dqe−iq(x−y)Gq Eq.(2.7)→ Gxy = ∫ dq e−iq(x−y) −q2 +M2 (2.10) by abbreviating dq ≡ dq (2π)4 . In a more symbolic form, Eq.(2.7) can be rewritten as1 Gφ xy = ( 2+M2)−1 x δxy (2.11) It is known from the Feynman rules in coordinate space that, at tree-level, the momentum running in Eq.(2.10) must collapse into the momenta p2 of the external particles. Thus, motivated by the scenario where 2∼ p2 ¿ M2, the r.h.s. may be expanded [49] into local terms through Gφ xy = 1 M2 ( 1− 2 M2 + ( 2 M2 )2 −·· · ) x δxy (2.12) which turns even more spurious when the propagator runs inside loops. As it will repeatedly explored, to force this expansion is in fact what is behind the necessity of a consistent matching procedure. Finally, the replacement of φ by the respective e.o.m. solution encodes its influence to the Green function. The field is out of the set of asymptotic particles, but is certainly present through virtual effects. 2.2 On the Effective Action The Effective Action will always be denoted by the Greek letter Γ. In order to perform the Legendre transformation in the beginning of this chapter it is first needed to solve the set of 1Or O−1O ≡ 1, for a generic operator O . Formally, it is equivalent to O−1 x Ax ≡ ∫ y Gxy Ay, where OxGxy = δxy. 12 2.2. ON THE EFFECTIVE ACTION classical equations in terms of the sources and then replace the solutions in the functional generator formula, commonly obtained in the saddle point approximation. The classical result for the Effective Action (i.e. the first term in the expansion) is composed by the classical action given in terms of the field-sources, here denoted by the sub-index ‘c’, like in φc and ψc [48]. The aim of this section is to provide both a qualitative and formal explanation about the Effective Action as a generating functional and its role during the construction of Effective Field Theories. It may be assumed as known the quality of Γ[φ] as the generator of amputated one-particle irreducible (1PI) diagrams. In a theory composed by a single self-interacting field, this means that the corrections coming from the Log in Eq.(2.4), diagrammatically, are represented by loops without external lines and such that it cannot be decomposed into independent graphs through a single cut of propagator. The saddle-point approximation [48], commonly adopted in order to solve a closed form to the functional generators, consists in a Taylor expansion on the Action made by assuming that the leading contribution to integrals like ∫ x e−a(x) is given by the region around the minimum of a(x). Therefore, the procedure is an independent method compared with a perturbation expansion for a quantum field theory. In other words, the Log may enclose simultaneously the quantum correction in different orders of the small parameter of the theory, i.e. it can associate different n-point functions. This assertion will turn clearer when the Universal Formula for the determinant computation is presented (see Section 2.3). It is still important to distinguish about two concepts already present at this point. Again, from the Effective Action definition it can be seen that it corresponds to an object that can be written in an arbitrarily higher dimension on the classical fields. The term ‘effective’ here is rather clear - its components are in fact what is connected with physical processes, whose effect is weighted by the coefficient of the correspondent n-point function. In summary, the set of higher-dimension operators defines a set of physical processes. A given operator is accompanied by a coefficient informing the importance (or suppression) of the correspondent scattering, for example. In this sense, the Effective Action is an object able to provide physical intuition about the considered phenomena. Along this work there will be rarely found mention to Feynman diagrams, but it might be important to include some comments. The Effective Action generate n-point functions containing exclusively internal lines or, equivalently, amputated Green’s function. Thus, the diagrams are composed by dots and propagators running in loops. By integrate out a field, i.e. by replacing it into the theory through the solution of its classical equation of motion, the field will then be forbidden to represent asymptotic degrees of freedom. This new theory must be called UV-Light and its respective ΓLUV will generate 1-Light Particle Irreducible diagrams (or 1-LPI, for short) - Irreducible graphs only on the light internal lines. Therefore, in this framework ΓLUV may generate diagrams containing heavy internal lines. This may be seen, for example, from the first term of Eq.(2.9). One Effective Field Theory consists on bringing the 1LUV theory into a local description of the interactions. Here, the respective action for the EFT will again generate dots 13 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION (a) (b) FIGURE 2.1. In a λφ3 theory, for example, the effective action can generate the 4-point function (a) but not (b), where • and ◦ denote an insertion of one and two external lines, respectively. (a) (b) FIGURE 2.2. In a λφ4 theory, the determinant can generate a set of 4-point functions containing graphs of different order in perturbation theory. Above, the • and ◦ denote an insertion of one and two external lines, respectively. (a) (b) FIGURE 2.3. The Effective Action for the 1LUV of Eq.(2.9) may generate the 4-point functions above, where the dashed and full lines represent the propagation of the light π and the heavy φ, respectively. (a) (b) FIGURE 2.4. One effective vertex (b) from the local expansion of the LUV theory (a). and loops, although the former is commonly considered sufficient for a well accurate analysis. In an Effective Theory for π’s of the Eq.(2.9) only the classical term, or the linear contribution on the Wilson coefficients, are present. The n-point function in this approximation will be represented only by cn, in reference to the operator On, and a box 2 (or ⊗). 14 2.2. ON THE EFFECTIVE ACTION (a) (b) FIGURE 2.5. In a theory containing self- or mixed interactions of light-fields, the 1LUV (a) and the EFT (b) may present the above graphs. The 2 represents an effective vertex; 2.2.1 On the sign of determinant for real, complex scalars and fermions Along the development of an Universal Formula to the first quantum correction of the Effective Action, the authors in [32] first sought for a general expression2 related to graphs which include solely heavy lines. The first step consisted in defining a generic expression for the Log piece: Γ(1) UV = icsTr log(−P2 +m2 +Ux) (2.13) with Pµ ≡ iDµ = i∂µ+Aµ the covariant derivative and U(x) a function representing constant configurations of light fields. One important point observation, therefore, relies on the extraction of the constant cs, holding the information about the specie of particles being integrated. The following lines are intended to treat this topic and concern fermions, real and complex scalars. The Eq.(2.13) is derived from the first correction to the generating functional on the saddle point approximation. For a theory containing real and spinor fields [48], it implies the expression eiSeff[Φ,Ψ†,Ψ]|c = ∫ DΦDΨ†DΨeiS[Φ,Ψ†,Ψ] (2.14) In fact this expression is already assuming the final result for the Effective Action which will be fully demonstrated in the coming text. The action in the r.h.s. must be rotated to the Euclidean space and then expanded through S = S ∣∣∣ c + 〈 η† δS δΨ† ∣∣∣ c 〉 + 〈 δS δΨ ∣∣∣ c η 〉 + 〈δS δΦ ∣∣∣ c ρ 〉 +1 2 {〈 ρx δS δΦxδΦy ∣∣∣ c ρ y 〉 +2 〈 η† x δS δΨ† xδΨy ∣∣∣ c ηy 〉 +2 〈 ρx δS δΦxδΨy ∣∣∣ c ηy 〉 +2 〈 η† x δS δΨ† xδΦy ∣∣∣ c ρ y 〉} +·· · (2.15) where ρ ≡Φ−Φc and η ≡Ψ−Ψc. The sub-index ‘c’ stands for the fields on the solution of the classical equations of motion, which eliminate the first derivatives. Moreover, the factor of two inside the brackets accounts for the second derivative of mixed terms. As mentioned before, repeated space-time indices inside < ·· · > are being integrated over. 2The notation will be consistently preserved. 15 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION It was left implicit in Eq.(2.15) that the action is defined in Euclidean space. At the end of a functional integration, the fields must be rotated back to Minkowski variables, what in summary will imply an overall multiplication by the factor −i. The extraction of cs concerns only the second line and for real scalars it should come from eiSeff = ∫ DΦ exp [ −i ∫ x 1 2 Φ (−P2 +M2 +Ux ) Φ ] (2.16) When the expansion (2.15) is performed the corrections are in fact being integrated over Euclidean (or ‘bar’) variables, such that SE = ∫ x 1 2 Φ ( −P 2 +M2 +Ux ) Φ (2.17) Note that the minus sign is preserved in the definition just to follow −P2 ⊃−(i∂µ)2 = ∂2 =−∂2 . By replacing the expansion SE = SE|c + 1 2 < ρx δ2SE δΦxδΦy ρ y >+·· · back into Eq.(2.16): e−SE = e−SE |c × ∫ DΦ exp [ −1 2 〈ρx δ2SE δΦxδΦy ρ y〉 ] (2.18) Since ∫ DΦ e− 1 2 〈ΦAΦ〉 = (detA)− 1 2 = exp[−1 2Tr log A], after rotating the expressions back to Minkowski space the Effective Action at loop order must be given by eiSeff = eiS|c × exp [ −1 2 Tr log(−P2 +M2 +Ux) ] (2.19) or Seff = S ∣∣ c + i 2 Tr log(−P2 +M2 +Ux) (2.20) which leads to cs = 1 2 (real scalars). It turns out that this half factor, just in front of the trace, originates from the Taylor expansion instead of the Lagrangian. Thus, although the procedure for complex scalars must follow in the same way, now the Taylor coefficient acquires an additional factor of two from mixed terms, as already recorded in Eq.(2.15). From ∫ DΦ†DΦ e−〈Φ † AΦ〉 = (detA)−1, it follows that cs = 1 (complex scalars). Finally, the constant for fermions must consider the inverse relation for a Gaussian integration of Grassmann fields, i.e. ∫ DΨ†DΨ e−〈Ψ † AΨ〉 = (detA). Moreover, this result would still lead to Seff = S ∣∣ c − iTr log(− /P +M+Fx) (2.21) where Fx is denoting constant light fields. In order to convert the above result into the desired quadratic form of Eq.(2.20) one may resort to the invariance of this trace under flipping the signs of gamma matrices: S(1) eff = − i 2 Tr[log(−i /D+M+Fx)+ log(−i /D+M+Fx)] = − i 2 Tr[log(−i /D+M+Fx)+ log(i /D+M+Fx)] = − i 2 Tr [ log ( /D2 + (M+Fx)2 − i[ /D,Fx] )] (2.22) 16 2.3. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE UNIVERSAL FORMULA Proceeding according to the notation in [33], from the identity /D2 = D2 − i 2σ µνG′ µν and G′ µν ≡ [Dµ,Dν], it follows that S(1) eff =− i 2 Tr [ log ( D2 +M2 +Uferm )] (2.23) where Uferm ≡− i 2 σµνG′ µν− i[ /D,Fx]+2MFx +F2 x (2.24) and cs =−1 2 (fermions). 2.3 Evaluating the Functional Determinant - On the Universal Formula The task of determining the first quantum corrections for the Effective Action involves a solid comprehension to the meaning of a functional trace. One complete treatment on this topic was developed by the authors HLM3 in [32] and, for completeness, it will be summarized in this subsection. In a general case, the loop correction consists of Tr f (x̂, q̂) (2.25) where the hats refer to the operators form. The representation for the arguments will recline on the choice of a basis, either in momentum or position space. To the former case, the trace can be defined as Tr f (x̂, q̂)= ∫ dq tr 〈q| f (x̂, q̂)|q〉 (2.26) with the small ‘tr’ now accounting exclusively for internal indices. The representation q̂ in the position basis is given by q̂ = i∂x, thus corresponding the commutation relation [x̂, q̂]=−i (2.27) Moreover, the plane wave convention follows from the product 〈x|q〉 = e−iq·x. By proceeding with the notation of HLM, the differentials must always hide the 2π4 factor like dq ≡ d4q 2π4 and dx ≡ d4x. Finally, the representation for x̂ and q̂ on the aforementioned basis is such that 〈x| f (x̂, q̂)= f (x, i∂x), 〈q| f (x̂, q̂) (2.27)= f (−i∂q, q) (2.28) Next, a complete set of position eigenstates4, 1= ∫ x |x〉〈x|, is inserted in Eq.(2.26), what implies Tr f (x̂, q̂) = ∫ dxdq tr 〈q|x〉〈x| f (x̂, q̂)|q〉 = ∫ dxdq tr eiq·x f (x, i∂x) e−iq·x (2.29) 3Abbreviation for B. Henning, X. Lu and H. Murayama; 4Remark that, analogously, the identity for the discrete case arises like I=∑ i |ei〉〈ei |, from the set of generators {ei} defining an orthonormal basis of a n-dimensional vectorial space. 17 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION From the Baker-Campbell-Hausdorff formula, the momentum operator is in fact being translated into5 i∂x → i∂x + q. By flipping the momentum sign, the final result may be written as Tr f (x̂, q̂)= ∫ dxdq tr f (x, i∂x − q) (2.30) The Effective Field Theory, under development, is marked by the matching procedure, one equation that must in general reveal a piece like δ2SUV δΦ2 , noted in HLM as the term defining the set of Wilson coefficients coming from the exclusive case of heavy fields running in the loops. This portion can be expressed like S(1) eff ⊃ icsTrlog(−P2 +m2 +Ux) (2.31) whose components were already presented in Section (2.2.1). From Eq.(2.29) it is translated into S(1) eff ⊃ ics ∫ dxdq tr eiq·x log(−P2 +m2 +Ux) e−iq·x = ics ∫ dxdq tr log [−(Pµ− qµ)2 +m2 +Ux ] (2.32) One interesting approach for treating the above expression was proposed by Mary Gaillard in [29] and reproduced in [32] with the intent of reaching a closed formula. The author in [29] proposed a general matrix-valued function g, dependent on the derivatives (∂p,∂x) and under the initial condition g(0,0)= 1 (2.33) such that its expansion around (0,0) would be given by g = 1+ g(1)|0 + g(2)|0 +·· · (2.34) Once g−1 acts in the right, it is in fact acting on the identity. Moreover, since the g(i)’s are at least of first order in derivatives, it follows that it should be equal to the unit. On acting in the left, the situation would not be different. The expansion of g leaves total derivatives of the operator, what should vanish by the boundary conditions of the integrand. Thus, only g(0,0) acts, and the insertion of g in both sides of Eq.(2.32) is trivial. At the end, the function is assumed with the form g ≡ eP·∂q (2.35) 5This follows from the analyticity of f (x̂, q̂), resulting that the BCH formula may in fact be applied on the identity vicinity, being sufficient to verify only eiq·x̂ q̂ e−iq·x̂; 18 2.3. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE UNIVERSAL FORMULA or ∫ eP·∂qAe−P·∂q = ∫ eP·∂qA(e−P·∂q I) = ∫ eP·∂qA = ∫ ( 1+ (P ·∂q)n n! ) A, n ≥ 1 = ∫ A+F∣∣∞−∞ = ∫ A (2.36) By applying the Gaillard procedure to Eq.(2.32), the task converts into simplifying the expression S(1) eff ⊃ ics ∫ dxdq tr eP·∂q log [−(Pµ− qµ)2 +m2 +Ux ] e−P·∂q (2.37) by requesting the BCH formula once more: eB Ae−B = ∞∑ n=0 1 n! Ln B A, where LB A ≡ [B, A] (2.38) The first component, (Pµ− qµ), may be translated after one brief observation:[ P ·∂q, qµ ] = P ·∂qbqµ− qµP ·∂q = Pνδ ν µ+ qµP ·∂q − qµP ·∂q = Pµ (2.39) and, thus eP·∂q (Pµ− qµ)e−P·∂q = ∞∑ n=0 1 n! (LP∂q )nPµ− ∞∑ n=0 1 n! (LP∂q )nqµ = ∞∑ n=0 1 n! (LP∂q )nPµ − ( qµ+ ∞∑ n=1 1 n! (LP∂q )nqµ ) (2.40) where the last sum can be rewritten as ∞∑ n=1 1 n! (LP∂q )nqµ (2.39)= Pµ+ ∞∑ n=2 1 n! (LP∂q )n−1Pµ = Pµ+ ∞∑ n=1 1 (n+1)! (LP∂q )nPµ (2.41) and, in Eq.(2.40), eP·∂q (Pµ− qµ)e−P·∂q = −qµ+ ∞∑ n=1 1 n! (LP∂q )nPµ− ∞∑ n=1 1 (n+1)! (LP∂q )nPµ = −qµ+ ∞∑ n=1 n (n+1)! (LP∂q )nPµ (2.42) 19 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION The sum can still be simplified via ∞∑ n=1 n (n+1)! (LP∂q )nPµ = ∞∑ n=0 n+1 (n+2)! (LP∂q )n+1Pµ = ∞∑ n=0 n+1 (n+2)! (LP∂q )n(LP∂q Pµ) = − ∞∑ n=0 n+1 (n+2)! { Ln P∂q [ Dν,Dµ ]} ∂qν (2.43) where it has been considered LP∂q Pµ = PνPµ∂qν +Pν(∂qνPµ)−PµPν∂qν = [ Pν,Pµ ] ∂qν (2.44) since Pµ is momentum-independent. Finally, by strictly following the HLM notation, all the momentum derivatives inside LP∂q can be placed to the right, what implies eP·∂q (Pµ− qµ)e−P·∂q = −qµ− ∞∑ n=0 n+1 (n+2)! { Ln Pα [ Dν,Dµ ]} ∂n qα∂qν = −qµ− ∞∑ n=0 n+1 (n+2)! [ Pα1 , [ · · · [ Pαn , [ Dν,Dµ ]]]] ∂n ∂qα1 · · ·∂qαn ∂qν ≡ −( qµ+ G̃νµ∂qν ) (2.45) where G̃νµ is thus referring a general form to the field-strength. The same analysis can proceed for the simpler case of the U component inside the determi- nant: eP·∂qUe−P·∂q = ∞∑ n=0 1 n! [ Pα1 , [ · · · [ Pαn ,U ]]] ∂n ∂qα1 · · ·∂qαn ≡ Ũ (2.46) These previous expressions permit to represent S(1) eff in a simplified form S(1) eff = ics ∫ dxdq tr log [ −( qµ+ G̃νµ∂qν )2 +m2 +Ũ ] ≡ ∫ dxL (1) eff (2.47) where the second line remark that the correction can be expressed equivalently in terms of a Lagrangian. For moving to the Log calculation, the prescription first consists in convert it into an integral over m2, i.e. L (1) eff = ics ∫ dqdm2 tr [ −( qµ+ G̃νµ∂qν )2 +m2 +Ũ ]−1 (2.48) and then to expand the squared terms: L (1) eff = −ics ∫ dqdm2 tr [ ∆−1 +{ q,G̃∂q }+ (G̃∂q)2 −Ũ ]−1 = −ics ∫ dqdm2 tr { ∆−1 [ 1+∆[ −(G̃∂q)2 −{ q,G̃∂q }+Ũ ]]}−1 (2.49) 20 2.3. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE UNIVERSAL FORMULA where the omitted Lorentz indices are contracted according to Eq.(2.48). Moreover, ∆≡ (q2−m2)−1 and may be considered as the variable for a matrix expansion like( A−1(1− AB) )−1 = (1− AB)−1 A = ∞∑ n=0 (AB)n A (2.50) which converts the quantum correction to the effective Lagrangian into a sum of integrals, or L (1) eff =−ics ∞∑ n=0 In (2.51) where6 In ≡ ∫ dqdm2 tr [ ∆ [ −(G̃∂q)2 −{ q,G̃ } ∂q +Ũ ]]n ∆ (2.54) Above it was also considered the identity7 { qµ,G̃νµ∂ν }= { qµ,G̃νµ } ∂ν+ G̃νµ [ qµ,∂ν ] (2.55) and, from [ qµ,∂ν ]=−δµν and the antisymmetry of G̃µν, it follows that { qµ,G̃νµ∂ν }= { qµ,G̃νµ } ∂ν. Thus, the effective Lagrangian at Log level is represented by the expression of Eq.(2.51). There, the In consists basically in operations on ∆. As it will be recurrently emphasized, a series expansion is always present during the matching procedure and must be truncated up to the desired order in the fields dimension after a power counting at the level of G̃νµ and Ũ, both containing higher-dimension operators (or HDO’s for short). In other words, the L (1) eff is a series of In, which turns out to be a series in HDO’s. By interrupting the sum at some order in the fields, it is established a number n for Eq.(2.51), implying a set that in fact composes the theory. One example - It is known a priori that G̃νµ and Ũ are at least linear in the light fields. Then, if the expansion is chosen to cease at dim-6 operators, it follows that no In will contribute for n > 6. 2.3.1 Evaluation of Integrals The formula Eq.(2.51) implies the task of calculating seven integrals. In this section the main tools for achieving this are completely developed. The final result will be, as presented by HLM, 6There is one additional information behind the integral after the power series conversion of the log. Being Fx the primitive of fx, i.e. fx = ∂xFx, the transformation is in fact given by F(m2)−F(m2 0)= ∫ m2 m2 0 dm2 f (m2) (2.52) and m0 can be chosen such that F(m2 0)= 0. In the context, Fx = log(a+ x), the primitive of 1 a+x . Since the final result must given in terms of a truncated series, the integrand becomes fx ' gx, or∫ m2 m2 0 dm2 g(m2)=G(m2)−G(m2 0) (2.53) such that Fx ' Gx. In other words, the primitive for the truncation must still be zero in the inferior limit of the integral; 7The lower-upper convention for Lorentz indices will not be followed henceforth; 21 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION one Universal Formula for the matching of quantum corrections from process involving only heavy internal lines. As demonstrated in last section, this formula must not entail the total set of operators behind a loop calculation, since it does not include process with light internal lines. However, from its simplicity and clarity it might be sufficient in some contexts. The next part of this work will turn this argument clear through some examples of application. The reason of only seven integrals has been mentioned before and is justified by the truncation of the series up to dimension-six operators. Since the functions inside In are at least linear in the fields, it follows that it must be interrupted at n = 6. Although apparently n = 6 is a small number, the final amount of integrals may be large and arduous to be computed. In HLM all the fundamental elements to for this task were presented and the result for I1 completely described. Here this result will be complemented with three new integrals, the most complex ones related with I2. The choice for what integrals to consider is motivated by the presence of different techniques during the computation, including, for instance, counting of divergences, subtraction scheme, Wick rotation, Gamma functions and master integrals, etc. In summary, it is important to identify the complete set of conceptual and technical manipulations which compose the method for treating these objects before the task of writing a final expression. As mentioned, the first criteria corresponds the counting of divergences and the integration on m2 can be very informative on this part. For n ≥ 1 it automatically arises like∫ dm2∆p = ∆ p−1 p−1 , p ≥ 2 and ∆= (q2 −m2)−1 (2.56) where p is at least of order two, since the effect of a q derivation is to increase the ∆ degree: ∂qµ∆ k =−2k× qµ∆k+1 (2.57) Next, the counting is performed for the integral in q which, due to momentum derivatives, must arise like ∫ d4q ∆kq2a (2.58) where a ∈N and, in Euclidean space∫ d4q∆kq2a ∝ i(−1)k+a ∫ ∞ 0 dq q3+2a (q2 +m2)k , where q = q2 0 +q2 = i (−1)k+a 2 ∫ ∞ m2 du q2(a+1) uk = i (−1)k+a 2 ∫ ∞ m2 du (u−m2)(a+1) uk = i 2 a+1∑ j=0 ( a+1 j ) (−1)k+a+ j(m2) j ∫ ∞ m2 du u(a+1− j) uk (2.59) which is convergent whenever k+ j− (a+1)> 1 and, since its minimum occurs for j = 0, it follows k > 2+a (2.60) 22 2.3. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE UNIVERSAL FORMULA The above expression is a central criteria and will be constantly required. By looking at the definition of In of Eq.(2.54), the object inside the brackets is operating on the outside ∆ with an arbitrary number of q-derivatives. The counting can be directly made by first excluding one propagator from the integral on m2 effect, as Eq.(2.56). If at some point the integrand for q is in the form of ∆k such that k > 2, from Eq.(2.57), by applying (2a) additional derivatives over ∆ implies the final power k̃ ≡ k+2a > 2+2a > 2+a, and the integrand remains convergent. This last conclusion is therefore sufficient to conclude that no divergence should be found in In for n > 2. Moreover, even for I2 all the pieces with derivatives must imply k with a minimum k = 4 with a = 1, such that only the coefficient for trU2 must be regulated. This and other examples are completely worked in Appendix (A.3). The integral I1 was completed calculated in HLM and here only one of its components were chosen to be registered. Nevertheless, the rest of divergent terms will be calculated in detail, namely I0 and I2, thus supplementing the evaluation section of [32]. The finite part will consist, in general, of systematic application of the master integrals and here the equivalent pieces present in I2 will saturate the examples at this topic. 2.3.1.1 Evaluating I0 From the formula Eq.(2.54) it is clear that I0 is a constant factor and does not play any role in the theory although it can be one important object of illustration. It follows that I0 = ∫ dqdm2∆ (2.61) For this sort of function it is adequate to first perform the integral on the variable q, calculated in Appendix A.1 and given by Eq.(A.21). Thus8 I0 = ∫ dm2 [ i (4π)2 m2 ( log ( m2 µ2 ) −1 )] = i (4π)2 { m4 2 logm2 − m4 2 (logµ2 +1)− m4 4 } = i (4π)2 { −3 4 m4 + m4 2 log m2 µ2 } (2.63) 2.3.1.2 Evaluating I1 To recapitulate, the I1 is given by9 I1 ≡ ∫ dqdm2 tr ∆ [ −(G̃∂q)2 −{ q,G̃ } ∂q +Ũ ] ∆ (2.64) 8From ∫ b a x log x = x2 2 log x ∣∣∣b a − x2 4 (2.62) 9Along the rest of the work the notation A has been chosen for denoting any matrix under internal indices. However, for simplicity this must not be followed in this chapter; 23 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION where G̃νµ ≡ ∞∑ n=0 n+1 (n+2)! [ Pα1 , [ · · · [ Pαn , [ Dν,Dµ ]]]] ∂n ∂qα1 · · ·∂qαn (2.65a) Ũ ≡ ∞∑ n=0 1 n! [ Pα1 , [ · · · [ Pαn ,U ]]] ∂n ∂qα1 · · ·∂qαn (2.65b) and Pµ = iDµ. By assuming that m2 commutes with the fields present both in G and U, the trace can be completely separated from the integrals, thus resulting in just a factor to the operators. The generalization for non-commuting m2 was performed by [22] and, for simplicity, will not be reproduced here. The authors in [32] then start with Ũ by noting that [∂xµ + A,U] = (∂xµb+A) U −U (∂xµ + A) = (∂xµU)+ [A,U] (2.66) such that, whenever the internal indices are retained in a single commutator, the trace operation results a total derivative and therefore must vanish10. Thus, from Eq.(2.65), only n = 0 must remain in the series such that I1 ⊃ ∫ dqdm2 tr ∆Ũ∆ = tr U × ∫ dqdm2∆2 (2.56)= tr U × ∫ dq∆ = tr U × I0 1 (2.67) and, finally, from Eq.(A.21)11: I1 ⊃ i m2 (4π)2 ( log ( m2 µ2 ) −1 ) × tr U (2.68) The authors then move to the anti-commutator piece which, for being a linear function with a commutator, must also be zero, leaving the final term to be I1 ⊃− ∫ dqdm2 tr ∆ G̃µσG̃νσ ∂2 ∂qµ∂qν ∆ (2.69) The above formula may provide the first example of power counting. The evaluation of I1, as well as any other integral, has been truncated to include operators only up to dim-6 in the fields. During the expansion, it is important to emphasize that the covariant derivative must be counted as dim-1 object, a condition for achieving a covariant formula. Thus, Gνµ is dim-2 and, from Eq.(2.65a), both G̃ must cease at n = 2, i.e. G̃µσ = 1 2 Gµσ+ 1 3 [ Pα,Gµσ ] ∂ ∂qα + 1 8 [ Pα2 , [Pα1 ,Gµσ] ] ∂2 ∂qα1∂qα2 +O(dim5) (2.70) 10This will certainly not be the case for the product of traceless terms; 11The following result differs in sign with [32], but is correct in their Universal Formula; 24 2.3. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE UNIVERSAL FORMULA such that the product G̃2, up to dim-6, implies: G̃µσG̃νσ∂ 2 µν = { 1 4 GµσGνσ∂ 2 µν+ 1 16 Gµσ [ Pα2 , [Pα1 ,Gνσ] ] ∂2 α1α2 ∂2 µν+ 1 9 [Pα,Gµσ][Pβ,Gνσ]∂2 αβ∂ 2 µν+ + 1 16 [ Pα2 , [Pα1 ,Gµσ] ] Gνσ∂ 2 α1α2 ∂2 µν } (2.71) where it was made implicit that ∂µ stands for 4-momentum indices. Moreover, the O(P5) vanish since they are odd under the integrated momentum12. From the criteria for convergence of Eq.(2.60) all the terms with four derivatives would correspond at most to k = 5 and a = 2, being therefore convergent. Thus, only the dim-4 operator above must present a regulated coefficient. The next result is similarly presented in HLM and here reproduced for a matter of fixing the notation for the Appendix. Thus,∫ dqdm2∆∂2 µν∆ (2.56)= 2gµν ∫ dq ( −1 2 ∆2 + 1 3 q2∆3 ) (A.15)= 2gµν ( −1 2 (iA2)+ 1 3 (iA2) ) = −i gµν 3 A2 (2.72) It must be noted that the replacement qµqν → q2 4 gµν must only be performed at the stage of integration. From Eq.(A.15) the A2 is given by A2 = Γ ( ε 2 ) (4π)2 ( 4πµ2 m2 ) ε 2 (2.73) which can be expanded into A2 = 1 (4π)2 ( 2 ε +ψ(2)+ log ( 4πµ2 m2 )) (2.74) Since in the MS scheme the pole is subtracted along with log(4π) and the Euler-Mascheroni constant γ present in ψ(2)= 1−γ, it follows, finally, that A2 MS→ 1 (4π)2 ( 1− log ( m2 µ2 )) (2.75) or I1 ⊃ i (4π)2 1 12 ( 1− log ( m2 µ2 )) × tr (GµνGµν) (2.76) After the I1 analysis, the authors of HLM simplify the above results through systematic application of the covariant derivative properties that are partially reproduced in Appendix A.2. 12For Lorentz violating dim-5 operators, see [6]. 25 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION 2.3.1.3 Evaluation of I2 This section intents to extract the last divergent part of the formula for dim-6 operators in an effective Lagrangian at quantum level. From Eq.(2.54) the I2 can be expressed like, I2 ≡ ∫ dqdm2 tr ∆ [ −(G̃∂q)2 −{ q,G̃ } ∂q +Ũ ] ∆ [ −(G̃∂q)2 −{ q,G̃ } ∂q +Ũ ] ∆ (2.77) By power counting under the dim-6 criteria the following items can be selected I2 ⊃ ∫ dqdm2 tr { ∆ (G̃∂q)2 ∆ { q,G̃ } ∂q ∆− (2.78a) −∆ (G̃∂q)2∆ Ũ∆+ (2.78b) +∆ { q,G̃ } ∂qb ∆ (G̃∂q)2 ∆+ (2.78c) +∆ { q,G̃ } ∂qb ∆ { q,G̃ } ∂q ∆− (2.78d) −∆ { q,G̃ } ∂qb ∆ Ũ ∆− (2.78e) −∆ Ũ ∆ { q,G̃ } ∂q ∆− (2.78f) −∆ Ũ ∆ (G̃∂q)2 ∆+ (2.78g) +∆ Ũ ∆ Ũ ∆ } (2.78h) In summary, only one term must be neglected a priori, namely, (G̃∂q)2(G̃∂q)2, for producing operators of, at least, dim-8. On the task of extracting divergences it is worthy to argue the above items in separate: a. The criteria Eq.(2.60) must be applied after the observation of Eq.(2.56), i.e. the mass integration, that is equivalent to exclude one of the ∆’s during the counting. For dim-6 implies that only the n = 0 in the series of both G̃ is being taken into account. From Eq.(2.57), the final power for the propagators must be k = 5, while a = 2. Therefore, the term is convergent; b. Here, by power counting, both the n = 0,1 in the expansion for Ũ must be taken. For n = 1, however, the integral of q has an old integrand and a vanishing result. For n = 0, the counting leads to a = 1 and k = 4 and no divergence will be present here; c. The same conclusions for the first line are made to this case; d. This is a potentially divergent term that will require a more particular treatment over the anti-commutator piece. From power counting, the series for both G̃ must run for m,n ∈ [0,1], simultaneously13. For (m,n)= (0,0), the integrand presents a = 2 and k = 4, thus breaking 13A requirement from the parity on q and disallowance of dim-5 operators; 26 2.3. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE UNIVERSAL FORMULA the convergence criteria. Nevertheless, explicitly it is given by 2.78d = ∆ {q,G}∂qb ∆ [ qµGνµ∂ ν ∆ ] = ∆ {q,G}∂qb ∆ [−2Gνµqµqν ∆ ] = 0 (2.79) as a consequence of the Gνµ antisymmetry. This result might suggest one incorrect conclu- sion that the anti-commutators would always vanish for the same field-strength property. To elucidate it, the complete case for (m,n) = (1,1) is treated in Appendix A.3 and may provide an useful intuition on the application of the series expansion during the matching. For completeness, here a = 1 and k = 4, i.e. the integral is finite; e. This is also a potentially divergent piece. From power counting, the powers to be analyzed where m,n ∈ {0,1} in the G̃ and Ũ series. Here, if m+n is an odd number they should vanish from the parity of q14. For (m,n)= (1,1), the convergence criteria reads with a = 2, k = 5, the coefficient is finite and has been computed in Appendix A.3 . The term for (m,n)= (0,0) vanishes from Gνµ anti-symmetry; f. Equivalent to the previous case; g. Equivalent to item 2.78b; h. Finally, this is the term that could render divergences. Again, from q-parity, each term in the series must run simultaneously. The (m,n)= (0,0) is certainly divergent, since a = 0 and k = 2. For (m,n)= (1,1), it follows a = 1 and k = 4, thus corresponding a finite coefficient. Therefore, to the total I2, only the Wilson coefficient for tr(U2) must be renormalized, what is extracted straightforwardly from the identities of Appendix A.1 and results: I2 ⊃ tr(U2)× ∫ dqdm2 ∆3 (2.56)= tr(U2) 2 × ∫ dq∆2 (A.14a)= i tr(U2) 2 × A2 (2.80) Finally15, I2 ⊃ i 2(4π)2 ( 1− log ( m2 µ2 )) × tr(U2) (2.81) 14If the number of momentum and derivatives is odd then, by simple counting in the power of q, the total integrand emerges like an odd function; 15The result does not agree with the universal formula in HLM, but is corrected in [53]; 27 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION 2.3.2 The Universal Formula The previous section presented the totality of Wilson coefficients whose divergences were regu- larized and then subtracted via the MS scheme. In order to recapitulate, the work on those integrals involves the search for a final expression to the quantum correction of the Effective Action whenever it can be expressed in the form S(1) eff ⊃ icsTrlog(−P2 +m2 +Ux) (2.82) which in general corresponds to non-mixed terms of heavy particles in the original UV theory. It is worthy to note the above choice for the symbol Seff instead of Γ. Such conversion will be clarified in the next section. Along the derivation, a power-counting over the fields was consistently performed and the series truncated at dimension-six operators, one condition that will be made clear at the next chapter. From the definition of a Covariant Derivative Expansion, Pµ = iDµ was preserved intact along the derivation and counted like an one-dimensional field. At the other hand, the piece Ux, an arbitrary function of SM fields, can assume order one or two, depending on the nature of the correspondent vertex. Furthermore, the mass matrix m2 has been presumed to commute with both U and the field-strength. On what follows, the formula is presented including the notation and results registered in the updated version [33]. The finite coefficients are assumed to be correct and have been reproduced in subsequent extensions like [28] and [53]. Finally, L (1) eff ⊃ cs (4π)2 tr { m4 2 [ 3 2 − log ( m2 µ2 )] +m2 [( 1− log ( m2 µ2 )) U ] + +m0 [ 1 2 ( 1− log ( m2 µ2 )) U2 − 1 12 ( 1− log ( m2 µ2 )) GµνGµν ] + + 1 m2 [ 1 12 [ Dµ,U ]2 − 1 12 UGµνGµν− 1 6 U3 + 1 60 [ Dµ,Gµν ]2 − 1 90 Gν µGρ νGρ ν ] + + 1 m4 [ 1 24 U4 − 1 12 U [ Dµ,U ]2 + 1 120 [ Dµ, [ Dµ,U ]]2 + 1 60 [ Dµ,U ] [Dν,U]Gµν+ + 1 40 U2GµνGµν+ 1 60 [ U ,Gµν ][ U ,Gµν ]]+ + 1 m6 [ 1 20 U2 [ Dµ,U ]2 − 1 60 U5 + 1 30 U [ Dµ,U ] U [ Dµ,U ]]+ + 1 m8 [ 1 120 U6 ]} (2.83) where it is important to remark the minus sign in the definition of Eq.(2.51). One important conclusion immediately extracted from the above expression is invariance under the symmetry of the original Lagrangian. As a brief review, for a theory based on arbitrary fields Ψ and symmetric under the transformation Ψ→Ψ′ =VΨ (2.84) 28 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS the covariant derivative is defined such that (DµΨ)→ (DµΨ)′ =V (DµΨ) or Dµ→ D′ µ =V DµV−1 (2.85) The commutator of these objects defines de gauge-field strength [ Dµ,Dν ]∝ Fµν and, from Eq.(2.85), Fµν→ F ′ µν =V FµνV−1 (2.86) At this point, therefore, the presence of a trace over the internal indices is sufficient to demon- strate the invariance of Eq.(2.83) under V : tr[V AV−1]= trA (2.87) Apart from that, the Universal Formula express the real meaning of quantum corrections to the Effective Action. Although the independent powers for the heavy mass exclusively represents a 1-loop computation, it may be noted that the arbitrary function U is present at different levels of the expansion. This function is what in fact contains the small perturbative parameter and, therefore, the formula contain corrections associated with different n-point functions. As emphasized during the power counting, the covariant derivative is necessarily of dimension one and the function U can be of dim-2, for example, in the scalar sector. In this context, the formula breaks in an even more simplified version. The most complete case, at the other hand, will certainly occur with trilinear vertex of scalars, what forces the presence of couplings with positive dimension and these functions to be of order one. Despite of its closed and model-independent form, the UF is limited to the case where the internal lines defining the Effective Action contain exclusively the same field. Nevertheless, a priori there is not a decisive argument that could avoid the additional Wilson coefficients coming from mixed heavy and light internal lines. This scenario motivated the authors of HLM to update their original work into a new version including this topic, namely in [33] and improved in [28] and [53]. As it can be seen, therefore, it has been a very recent and interesting topic of discussion. The next section is centered on the matter of mixed corrections and the attempt of performing a similar matching through a covariant expansion. 2.4 Evaluating the Functional Determinant - On the Mixed Terms The last section has tried to elucidate some technical aspects behind the computation of a functional determinant, but did not clarify the conceptual meaning behind the term Effective Theory. One example is the constant lack of consistency on calling the Log correction as an action. The present section develops a more formal presentation to the logical path before the construction of an EFT and its correct connection with Effective Actions. To achieve this task it is important to consider the general case where corrections from both light and internal lines are equally present. 29 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION It has been shown that the first term defining the Effective Action is just composed by the classical action, i.e. by the functional which defines the theory. Thus, the action is already a generator functional. Although the One-Particle Irreducible graphs are fundamental elements for the analytical properties of a given model, they do not comprise the overall set of n-point functions present in the S-matrix - the classical action can be considered for a tree-level analysis, i.e. for representing any type of process exclusively through tree-level diagrams. The Log correction, therefore, will be used as a source of more accurate predictions, then including in the description the independent elements of the ultraviolet structure of the theory. The 1PI are generated by the Effective Action and the Effective Theory will be constructed by taking advantage on the theoretical strength of this functional generator. The Lagrangian for the EFT is by definition written as a general series over a set of operators. Their coefficients are just weights on that specific process represented by the matrix element. If the series is defined a priori this would be a model independent Effective Theory, a bottom-up proposal emerged from principles that will follow a previous knowledge or a primary assumption about the particular phenomena. In many contexts this effective proposal is considered sufficient to the specific analysis - All the operators, for example, that could be generated through Log cor- rections are then assumed to be out of the current precision. However, the quantum correction to the EFT Effective Action can be computed with no restriction. Analogously, this would correspond to improve the predictions in higher orders for the Wilson coefficients, turn them more accurate. Here, however, the treatment follows a different direction. The Effective Theory will be built from a previous hypothesis on the Ultra-Violet (UV for short) complete model. The purpose is still to raise a series of higher-dimension operators, but now from a different perspective - The hypothesis of a UV complete model, more general than the SM, is necessarily accompanied by the phenomenological fact that any new heavy field, with independent properties, has appeared asymptotically in the experiments. Since these UV versions are in general more complicated, it is important to develop a technique to simplify their analysis. By considering them entirely during the computation of a particular process would represent a large step compared with the experiments achievement, apart from the expected complexity likely to be required. The EFT at this level arises as one attempt to reach new physics information in a more gradual manner. This will define a method and the Effective Actions the most important tool. From the Decoupling Theorem (see Chapter 3) it is known a priori that if a theory intrinsically contains a heavy sector, this can be eliminated of the complete model by just cutting these heavy lines out of any graph, i.e. by considering exclusively its renormalizable low-energy sector. The procedure consists in a redefinition of fields and couplings of the low sector and - the most important feature - is such that the error committed decreases with the heavy scale. Now, assuming one particular UV theory is to assume the Standard Model as this low-energy model. Since the Effective Action is related with the ultraviolet property of the complete theory, it is therefore the adequate object to translate any information about these heavy particles to inside 30 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS the constant couplings of these first corrections. Thus, to propose dim-6 operators is to propose a representation to the error committed in assuming the Standard Model as a final theory for describing process at intermediary energy scales. The composition of LEFT must change as the matching between the Effective Actions ΓEFT and ΓLUV is done in different order from the saddle-point expansion. These previous sentences are then stating a clear separation into an Effective Theory and a Light-UV Theory. The latter, however, will be definitely present during the rise of the former, following a type of recursive construction. In summary, there are present two important and distinct concepts for the method: • ΓLUV: The Effective Action for the Light-UV model, i.e. the theory where heavy degrees of freedom are disallowed to appear as external fields, is constructed by following the steps: 1. Make the Effective Action for the UV theory, including all the fields at the same level and by following, for example, the method of saddle point approximation for extracting corrections, expanding all the fields around their background. 2. Afterwards, replace the heavy fields by their classical, and non-local, solution. At this context, there will not be an expansion into local terms. 3. In this sense, the Effective Action at classical level16 for the theory of Eq.(2.5) will just be given by Eq.(2.9). 4. The final and unique model is still the UV theory, but the Effective Action may be represented exclusively by a particular set of fields. The first correction for ΓLUV will not be constructed from the Lagrangian in Eq.(2.9) but from the correspondent correction for the effective action of the complete Eq.(2.5). Again, after this resolution, the heavy fields are then replaced by their classical and non-local representation. • ΓEFT: The matching is performed at the level of Effective Actions and, therefore, the Effective Field Theory must emerge recursively, according to the desired level of accuracy for the Wilson coefficients. In other words, each level for the matching, either at classical or at quantum level, will imply different and independent theories. 1. The first theory, i.e. the starting point, will certainly be present in any of the improved versions and is defined by equating the Γ(0) EFT[φ] with Γ(0) LUV[Φ̂c[φ],φ] when the classical fields Φc are made local Φc → Φ̂c, employing the notation from [53]. Since Γ(0) is just the classical action, the theory of S(0) eff will be the local version of SLUV ≡ Γ(0) LUV. In the example of Section 2.1, the L (0) eff is given by Eq.(2.9) after the insertion of the expansion Eq.(2.12). This theory may be represented, for instance, as the SM plus n 16i.e. with no determinants; 31 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION higher-dimension operators: L (0) eff =LSM + n∑ i=1 c(0) i Oi (2.88) Regardless its origin, L (0) eff must be phenomenologically constrained and theoretically explored at any order in its coupling constants . 2. The next theory will certainly entail the previous one but now including a new set of operators apart from additional contributions to the zeroth coefficients. Since this will be considered the ultimate effective theory, it can be written without superscript: Leff ≡L (0) eff +L (1) eff =LSM + n∑ i=1 ( c(0) i + c(1) i ) Oi + p∑ j=n+1 c(1) j O j (2.89) As mentioned before, L (1) eff is obtained recursively by equating the quantum correction Γ(1) LUV[Φc[φ],φ], expanded into local terms after the Trace computation and truncated by power counting, with the equivalent Γ(1) EFT[φ], which now includes a determinant of δ2S(0) eff δφ2 , i.e. the quantum correction to the Effective Action of the theory L (0) eff . The formal clarification, the consequences and points on how the matching is performed will be treated in detail along the next section. 2.4.1 The Matching Procedure Before moving to the first application of matching by functional methods, one final procedure must also be developed and, if possible, by involving the same transparent steps as those behind the universal formula. The work [33] was dedicated to address this topic and, although it has been later extended by the authors in [28] and [25], here most of their method must be preserved, apart from one unpretentious attempt to leave the formulas with a simpler aspect. It will be seen that the task of performing a Covariant Derivative Expansion, i.e. of matching while maintaining the covariant derivative intact, is achieved through the expansion of the 1LUV Effective Action into a series of local operators after the Trace computation, being the truncation defined according to a power counting fixed a priori. On what follows, first it will be done a brief but complete presentation of the entire set of Wilson coefficients from the formalism developed in [33]. At this point, the review is general but covers in particular the Chapter 3 of the referred article. Once the coefficients related to the Universal Formula are fully identified, the focus are then directed to the additional terms, there called mixed terms, by manifest reasons. At this point it will be argued that the final form of mixed operators remains in a sense obscure in their work, with a general trace that, although correct, is exhibited without a consistent support. The present section is in fully agreement on the incontestable importance of all the concepts and results presented in [33], apart from their accuracy. Here, however, is aimed to find the 32 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS same universality as in Eq.(2.83), via one additional formula, preserving at the same time the conceptual richness contained in both HLM articles. The Matching procedure may be considered as the Correspondence Principle for Effective Theories - The Ultra-Violet complete model and its Effective variant must agree at some scale µ. In general the matching scale is assumed to be determined by the mass of the particles that were integrated out of the UV theory, since it is being considered excessively heavy to emerge as external degrees of freedom. The procedure consists on equating both theories represented by their respective Effective Actions. If the ϕ is chosen to represent the complete set of Standard Model fields, i.e. those present as asymptotic particles, then Φ will be denoting the counterpart, a set of fields running exclusively through virtual processes. In the context of this Standard Model Effective Field Theory the actions will be represented by SUV [ϕ,Φ] = SSM[ϕ]+SΦ[ϕ,Φ] (2.90a) SEFT [ϕ] = SSM[ϕ]+∑ i ci(µ) Oi[ϕ] (2.90b) At µ= M, the heavy scale, the ‘Light’ version of the UV must define the EFT, order-by-order, at the level of Effective Actions: Γ(0) EFT [ϕ] = Γ(0) L,UV [ϕ], at µ= M (2.91) Γ(1) EFT [ϕ] = Γ(1) L,UV [ϕ], at µ= M (2.92) The UV Effective Action, at the Log order is given by ΓUV [ϕ,Φ]= SUV [ϕ,Φ]+ iα logdet ( δ2SUV δ(ϕ,Φ)2 ) (2.93) and, by definition, is being expressed in terms of background fields. The ΓL,UV is then constructed after ΓUV and through the replacement of Φ as an implicit functional of the light fields, from the solution of the classical equation of motion δSUV [ϕ,Φ] δΦ ∣∣∣∣ Φc[ϕ] = 0 (2.94) Here it will be chosen a representation for Φc motivated by both the notation in [33] and the general aspect of phenomenologically relevant Lagrangians, with a tree-level piece like LUV ⊃ 1 2 Φ OΦ Φ + 1 2 ϕ Oϕ ϕ − ΦBϕ + QΦ (2.95) where QΦ, a ‘quartic’ operator may depend on N-power of heavy fields, with N ∈ [3,4]. The operator OΦ may also be a function of the light fields, OΦ =OΦ(ϕ) and, as one example, in the case of a Scalar Fields it would be OΦ ≡ [∆Φ]−1 − A(ϕ), ∆Φ ≡ [P2 −M2]−1 (2.96) 33 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION As before, Pµ ≡ iDµ. The equation defining the classical Φc, Eq.(2.94), is chosen to be solved linearly, discarding the contribution from Q, which results in Φc[ϕ]= [ OΦ ]−1 Bϕ (2.97) The Effective Action for the LUV , the fundamental object behind the construction of a Effective Theory, will be given by ΓL,UV [ϕ]=ΓUV [ϕ,Φc[ϕ]] (2.98) At the EFT side the Effective Action requires a thorough conceptual analysis. The action in Eq.(2.90b) is the final representation of a theory which is being constructed recursively and, therefore, remains as an symbolic object. At the quantum level , the correspondent Effective Action is given by the familiar formula ΓEFT [ϕ]= SEFT [ϕ]+ iα logdet ( δ2SEFT [ϕ] δϕ2 ) (2.99) Since the Wilson coefficients are determined after the local expansion of ΓL,UV , order-by-order, the inclusion of the Log term in the equation for the still undetermined c(1) i ’s is only meaningful once it contains only the zeroth-order action, S(0) EFT , emerged after the tree-level matching. It is crucial to remark that in terms of the action for the EFT - instead of the Effective Action - the upper-index is denoting the order in which the matching and the Wilson coefficients were extracted. That is to say that the theory is being constructed by steps and following SEFT = S(0) EFT +S(1) EFT +·· · . The logical significance of Eq.(2.99) is acquired by ΓEFT [ϕ] = SEFT [ϕ]+ iα logdet ( δ2S(0) EFT [ϕ] δϕ2 ) (2.90)= SSM[ϕ]+ N∑ i=1 ci(µ) Oi[ϕ]+ iα logdet ( δ2S(0) EFT [ϕ] δϕ2 ) = SSM[ϕ]+ p∑ m=1 ( c(0) m + c(1) m ) Om[ϕ]+ N∑ j=p+1 c(1) j O j[ϕ] +iα logdet ( δ2S(0) EFT [ϕ] δϕ2 ) (2.100) where the split of the Wilson coefficients is just turning explicit that the matching at quantum level can originate a new set of operators. Finally, at zeroth-order the Effective Action for the EFT is given by Γ(0) EFT [ϕ]= S(0) EFT = SSM[ϕ]+ p∑ m=1 c(0) m Om[ϕ] (2.101) and, at the Log order, Γ(1) EFT [ϕ]= N∑ i=1 c(1) i Oi[ϕ]+ iα logdet ( δ2Γ(0) EFT [ϕ] δϕ2 ) (2.102) 34 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS where N ≥ p. Note, therefore, that the presence of S(1) EFT in the above expression provides the equation for the Log correction of the EFT, such that the matching Γ(1) EFT [ϕ]=Γ(1) L,UV [ϕ], is now represented by N∑ i=1 c(1) i Oi[ϕ]+ iα logdet ( δ2Γ(0) EFT [ϕ] δϕ2 ) = iα logdet ( δ2SUV [ϕ,Φ] δ(ϕ,Φ)2 ∣∣∣∣ Φc[ϕ] ) (2.103) which must be solved for c(1) i once the r.h.s is expanded through a covariant series of local terms. Having found the corrections to c(0) i , and possible new operators, the EFT is then redefined and corresponds a new and independent theory, more accurate than that raised by the tree-level matching. The Effective Field Theory is, in general, experimentally constrained only at linear order on the final ci. That is to say that the action defining the final EFT is explored like a classical generator functional. Thus, the n-point function behind the process of interest is given by the correspondent Feynman diagram with a single dot, usually expressed like ⊗, i.e17 δnSEFT (δφ)n ≡ ci ⊗ for Oi =ϕ1 · · ·ϕn (2.106) The reason for not considering higher perturbative corrections on ci, for a specific process, is comprehensible - given the observable, the matching at quantum level is assumed to be sufficient precise, and the loop corrections including light particles can be summarized from the Renormalization Group Equations. The ci(M) provide the initial condition to the RGE, which then provide their appropriate evolution to the scale where the matrix element 〈Oi〉 is in fact evaluated. As clarified in the Section 3.1, the separation into high and low energy regimes for coefficients and operators, respectively, is one of the most important step composing the method for investigate phenomenology via the EFT approach. Here, it corresponds to the fact that the information about the propagation of the heavy particles is all contained in the Wilson coefficients to the local operators. The evaluation of 〈Oi〉 is usually performed by non-perturbative methods and, even for classical processes18, like Kaon decays, still retain large uncertainties. The task of running ci(M) down to the scale µ is motivated by the following hypothesis - The improvement of non-perturbative techniques in the determination of the matrix element 〈Oi〉 will not be sufficient to fully explain the particular anomaly being analyzed. The possible 17The Feynman rule can be more transparent in the momentum space: δnSEFT δφ1 · · ·δφn ∝ ci n∏ i=2 δ(xi − x1) (2.104) which converts into G(n) p ∝ ci ∫ dx e−ip·x n∏ i=2 δ(xi − x1)= ciδ ( n∑ i=1 pi ) (2.105) where a≡ (a1, · · · ,an) and the final delta is therefore representing the momentum conservation; 18Here, by classical it is meant historically relevant; 35 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION disagreement between observable and theory should be explained by a precise determination of the Wilson coefficients. The persistence of a anomaly might be a sign of New Physics and the current ∆I/2 rule from the QCD corrections for meson decays is one of the most important examples of this philosophy [21]. The main topic of this section consists on solving the Eq.(2.103) to the c(1) i , what involves one additional step related to the fact that now δ2SUV [ϕ,Φ] δ(ϕ,Φ)2 is an Hessian matrix including field indices, i.e. H ≡ δ2SUV [ϕ,Φ] δ(ϕ,Φ)2 = ( δ2S δΦδΦ δ2S δΦδϕ δ2S δϕδΦ δ2S δϕδϕ ) (2.107) In the Appendix of [33], however, the authors have shown one important identity that permits H to be written as a sum of two independent logarithms, what finally provides a very logical meaning to the form of Eq.(2.103). It follows: logdet ( δ2SUV [ϕ,Φ] δ(ϕ,Φ)2 ∣∣∣∣ Φc[ϕ] ) = logdet ( δ2SUV [ϕ,Φ] δΦ2 ∣∣∣∣ Φc[ϕ] ) + logdet δ2Γ(0) L,UV [ϕ] δϕ2  (2.108) where Γ(0) L,UV [ϕ]= SUV [ϕ,Φ[ϕ]]. Thus, the matching equation, at Log level, converts into N∑ i=1 c(1) i Oi[ϕ] = iα logdet ( δ2SUV [ϕ,Φ] δΦ2 ∣∣∣∣ Φc[ϕ] ) + +iα logdet δ2Γ(0) L,UV [ϕ] δϕ2 − logdet ( δ2Γ(0) EFT [ϕ] δϕ2 ) (2.109) The above equation implies a strong simplification on the procedure. The first line can be promptly identified with the matter of the previous section and its relative operators will then be extracted from the Universal Formula of Eq.(2.83). The correspondent coefficients was denoted by the authors of [33] like ci,heavy, since they are connected with diagrams including only heavy internal lines. The second term, inside the brackets in Eq.(2.109), is a new object and the correspondent Wilson coefficients were called ci,mixed by HLM. Here, none of these terminology will be adopted, despite of their importance in a conceptual level. Notwithstanding, the main question to be addressed is what is the real difference between these two similar Logs. As has been stated recurrently, the determination of ci is done through a local expansion of Γ(0) L,UV in Eq.(2.109) up to some predetermined power in the fields. The Γ(0) EFT is originated by a power counted truncation performed at the linear level. Notwithstanding, now the expansion, made a posteriori, is in a non-linear context. This justify the fact that the difference inside the brackets does not trivially vanishes - Γ(0) L,UV is intact a priori and only expanded as the argument of a Log function. The correspondent series will certainly contain the Γ(0) EFT piece, but in such a way that the cancellation still have operators on the desired field dimension. In summary, the 36 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS power counting at the quantum level is independent of that performed linearly, what in these terms sounds like a reasonable statement. The precise cancellation of common terms and the counting in this framework can be explicitly achieved by considering a generic representation for the UV Lagrangian as Eq.(2.95). The classical Effective Action for the LUV corresponds simply to the action SUV [ϕ,Φ[ϕ]], or: Γ(0) L,UV = ∫ x ( 1 2 Φc OΦ Φc + 1 2 ϕ Oϕ ϕ − ΦcBϕ + QΦc ) (2.110) and, from Eq.(2.97), it follows that Γ(0) L,UV = ∫ x ( 1 2 ϕ Oϕ ϕ − 1 2 Bϕ [ OΦ ]−1 Bϕ + QΦc ) (2.111) The series to be truncated during the raising of Γ(0) EFT usually comes exclusively from [ OΦ ]−1, since the QΦc is expected to break the adopted power counting up dim-6. Here this truncation will be denoted like: [ OΦ ]−1 =O Φ (0) +OΦ(1) (2.112) where the ‘bar’ in the r.h.s is referring to local operators. The O Φ (0) is the part that saturates the power counting during the matching at tree-level and OΦ(1) is the reminiscent non-local part that might be present to the power counting during the Log matching. Since the importance of ΓL,UV is based entirely on providing one equation for the Effective Theory construction, the inverse operator can be replaced by ‘bar’ operators without loss of generality. In other words [ OΦ ]−1 can always be expanded exactly into local and non-local operators via a recursive formula. For example, if A(ϕ)= 0 in the representation of Eq.(2.96): [OΦ]−1 =∆Φ 1= − [ 1 M2 + P2 M2 1 M2 −P2 ] 2= − [ 1 M2 + P2 M2 ( 1 M2 + P2 M2 1 M2 −P2 )] : n= − [ 1 M2 + P2 (M2)2 +·· ·+ (P2)n−1 (M2)n + (P2)n (M2)n 1 M2 −P2 ] = − 1 M2 ( 1+ P2 M2 +·· ·+ (P2)n−1 (M2)n−1 )∣∣∣∣ local + (P2)n (M2)n 1 P2 −M2 (2.113) where it also has been assumed commutation into D and M components. In order to simplify the analysis, the quartic self-interaction term QΦc will be assumed to not enter at this level of the matching. Thus, from the previous arguments, the Eq.(2.111) can be rewritten as Γ(0) L,UV = ∫ x ( 1 2 ϕ Oϕ ϕ − 1 2 Bϕ [ O Φ (0) +OΦ(1) ] Bϕ ) (2.114) At the other hand, the Γ(0) EFT , by definition, corresponds to the same form, although containing the ‘zeroth-order’ piece, i.e. Γ(0) EFT = ∫ x ( 1 2 ϕ Oϕ ϕ − 1 2 Bϕ O Φ (0) Bϕ ) (2.115) 37 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION Back to the brackets of Eq.(2.109), it follows that N∑ i=1 c(1) i Oi[ϕ] ⊃ iα logdet δ2Γ(0) L,UV [ϕ] δϕ2 − logdet ( δ2Γ(0) EFT [ϕ] δϕ2 ) = iα { Trlog ( Oϕ− [ H(0) +H(1) ])−Trlog ( Oϕ−H(0) )} (2.116) where H(i) is now denoting the Hessian matrix of 1 2 Bϕ OΦ(i) Bϕ over light-fields, i.e. H [.]= δ2 δϕ2 [.]. Note that minus sign has been factorized out of the bracket. Moreover, the sum of the first term is then exploring the fact that H is a linear application19. Next, the trace of the light operator can be subtracted through the identity N∑ i=1 c(1) i Oi[ϕ] ⊃ iα { Trlog ( 1− [ Oϕ ]−1 [ H(0) +H(1) ])−Trlog ( 1− [ Oϕ ]−1 H(0) )} ≡ iα { Trlog ( 1− [ A(0) +A(1) ])−Trlog ( 1−A(0) )} (2.117) and for future references the definition of A(i) must be registered explicitly: A(i) ≡ [ Oϕ ]−1 H [ 1 2 Bϕ OΦ(i) Bϕ ] (2.118) The power counting will finally be made transparent through a series for the logarithm: N∑ i=1 c(1) i Oi[ϕ] ⊃ iαTr ∑ p 1 p { A p (0) − [ A(0) +A(1) ]p } (2.119) The cancellation of A(0), remembering that these are the diagrams coming exclusively from the EFT, can then be directly identified after a binomial expansion of the last bracket: N∑ i=1 c(1) i Oi[ϕ] ⊃ iαTr ∑ p 1 p { A p (0) − p∑ k=0 ( p k ) A p−k (0) A k (1) } = −iαTr ∑ p 1 p { p∑ k=1 ( p k ) A p−k (0) A k (1) } (2.120) The above formula is the final complement to the matching at the log-level and was introduced in [33] in a generic form, without mentioning the dependence of the coefficients. It is important to mention that the binomial may be expressed in its canonical formula once the operators A(i) commute. In a more specific case, the binomial coefficients are informing the number of combinations to the non-commuting objects, like for example: (A+B)4 ⊃ 6A2B2 or 6A2B2 → A2B2 + ABAB+BA2B+ AB2 A+BABA+B2 A2 (2.121) 19Consider f (x) and g(x) two functions of multi-variables x, then: H [ α f (x)+βg(x) ]=αH [ f (x)]+βH [g(x)] for constants α and β. 38 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS Thus, although in a compact form, the result of Eq.(2.120) can be very arduous to compute, even for the simplest dim-6 criteria. In summary, the final formula to the matching at log-level will be given by N∑ i=1 c(1) i Oi[ϕ]= iαTr log ( δ2SUV [ϕ,Φ] δΦ2 ∣∣∣∣ Φc[ϕ] ) + iαTr ∑ p 1 p { p∑ k=1 ( p k ) A p−k (0) A k (1) } (2.122) The first trace will be computed through the Universal Formula Eq.(2.83) and represents the set of process containing heavy internal lines only. The second piece, from its complexity, requires additional comments: • The trace is complementing the set of operators from processes containing both light and heavy internal lines inside the loop; • The only simplification that has been done on the UV theory concerns the omission of quartic self-interactions represented by QΦ in Eq.(2.95). In fact these terms will be more suppressed at this level, since the derivatives are being taken, a posteriori, in the Higgs fields. The Wilson coefficients dependent on the QΦ parameters are then being left to the first part of the matching, with heavy-lines only. Nevertheless, in order to be complete, the dependence in λΦ corresponds to an additional piece in the Hessian H(1) and does not imply any change in the derivation of Eq.(2.122); • As it was mentioned for Eq.(2.96), the generic operator OΦ may still contain a function on the light fields, so-called A(ϕ). Remembering that OΦ ≡ [∆Φ]−1 − A(ϕ), ∆Φ ≡ [P2 −M2]−1 (2.123) In the framework where A(ϕ) 6= 0, the expansion of [OΦ]−1 will require one additional step given by [OΦ]−1 = 1 P2 −M2 − A ' ∆Φ+∆Φ A ∆Φ+·· · (2.124) where the ∆ operator, which includes the covariant derivative, is opened on the right. The two terms in the above series are usually sufficient and, then the local expansion of Eq.(2.113) can be taken. All the steps for the derivation of Eq.(2.120) follows in the same way, although now the O Φ (1) is complemented with terms in the form O Φ (1) ⊃ ∆Φ(0) A ∆Φ(1) + ∆Φ(1) A ∆Φ(0) +∆Φ(1) A ∆Φ(1). • Since the main purpose of the present work is to clarify the light-heavy matching as represented in Eq.(2.122), the A(ϕ) parameters will not be included at this level and their contribution to the Wilson coefficients will be left exclusively to the Universal Formula 39 CHAPTER 2. THE COVARIANT DERIVATIVE EXPANSION application. Notwithstanding, the previous item was completely covered by [33], [28] and [25] for the example of the Electroweak Theory with a Triplet Scalar. In order to compare their results with those coming from the application of the formula Eq.(2.122), the model will also be treated in the Chapter 5. 2.4.2 Summary In their presentation of mixing terms, the authors of HLM chose a subtraction a posteriori, made inside the functional trace, what may render a lack a clarity. Here it was intended to separate the pieces at the beginning, resulting in a series that can then be used in an algorithmic form. In summary, • It was first supposed a generic Lagrangian to the UV theory, in Eq.(2.95), considering both tre-level pieces as quartic self-interactions. Although the aspect resembles the scalar case, at principle no additional complexity should be found in the fermionic case, already discussed in Appendix of [33]; • The formula for subtraction at the Log-level, Eq.(2.109), was then applied for both the generic UV and the EFT action; • The LUV theory was split into local and non-local parts according to the definition of the EFT at tree-level. This marks where the series must be recovered; • The Log is expanded and the linear local piece is extracted. The Eq.(2.120) is the final expression. Comments on the last section: • It is not necessary to carry two calculations, the Eq.(2.120) is already including the subtrac- tion; • The Hessian matrix is also unique, for both local or non-local parts, what will be illustrated in Chapter 5 and can actually be noted in the definition of Eq.(2.118). Thus, even the Hessian is carried just once; • When the operators O contains additional functions of the light fields they may be replaced by the series: [OΦ]−1 = 1 P2 −m2 − A = ∞∑ n=0 [ 1 P2 −m2 A ]n 1 P2 −m2 (2.125) 40 2.4. EVALUATING THE FUNCTIONAL DETERMINANT - ON THE MIXED TERMS 2.4.3 On the Meaning of the Subtraction During these computations some questions must arise in a very natural way: (i) Why is this subtraction needed since the UV theory is known? or in other terms - (ii) Since the real purpose is to integrate out a heavy field, what results in the ΓL,UV theory, why it is not sufficient to just plug this non-local object into the equation for the Effective Action, Eq.(2.93), and perform the local expansion a posteriori? Finally, (iii) what is the importance of the matching? All of these questions are very logical. In fact, one starts with the UV theory and extracts the 1PI generator functional at the Log-level via the Effective Action. Every step can be done once and for all, resulting in a sum of operators informing on the significance for a given physical process. The answer for the above questions relies on a single statement - The Effective Action has the status of a Generating Functional, not of a Theory. This consists in an independent and very different perspective. The Effective Theory will assume the complete set of operators generated by the Effective Action, at leading or Log-level, as the action of a new and independent model, valid and very precise at low-energies according to a definite scale. For being a new model, the associate generator functional of 1PI vertex is given by the usual Log correction, of Eq.(2.99). Thus, if this theory was emerged by the LUV only, its Log correction would result in coefficients for the n-point functions which was in reality counted twice. The subtraction is preventing this double counting and making the process consistent. The previous paragraph can cover the question (i) and leads to a rectification of question (ii) - The procedure of integrating out a field does not resolve the ‘real purpose’ to the construction of an EFT, but the necessity of calculating complex processes, often involving a large number of particles or events. If it is then feasible to propose a theory with a reduced number of degrees of freedom, it is certain that this would be technically significant. Finally, the third question is reached. This new theory is by definition an approximate variant of something more universal. It is therefore required a correspondence principle, some coherent argument that guarantees its limit of validity - the principle is the matching procedure. 41 CHAPTER 3 FUNDAMENTALS OF EFFECTIVE FIELD THEORIES One of the most important tools during the extraction of physical predictions from a quantum field theory is the set of differential equations composing the renormalization group, and not only for practical purposes, but for its conceptual relevance. The running coupling analysis is a fundamental part of the applicability of Effective Field Theories. The EFT Lagrangian is defined after the replacement of the set composed by non-local product of operators for a sum of local terms whose coefficients will carry all the information about the original small distance of a heavy particle propagation. As mentioned before, the renormalization procedure is based in a fundamental principle which can be stated in many equivalent forms, and in this work it will be chosen as - any measurement in physics consist on variations. In other words, a number may not have any meaning unless it is giving through a comparison and such that the final and relevant physical result do not depend on the reference for the subtraction. This invariance on a ‘translation of references’ is what gives origin to the renormalization group equations and results in a strong simplification of calculations in perturbation theory. For illustrating these statements, consider a generic interaction in the form Sint ⊃ eχχσ (3.1) Here, a mention to the physical nature of the fields χ and σ is not necessary. Assume that the coupling constant e has been measured at some reference scale p2 0 from the fitting of a particular set of data to the scattering χχ→ χχ. The S-matrix element is associated with the time-ordered correlation function S(4) ∼ 〈Ω|T{χ(x1)χ(x2)χ(x3)χ(x4)}|Ω〉, with |Ω〉 and χ the vacuum and fields in the interacting scenario, respectively. For perturbative QFT this object is better to be rewritten in terms of correlation functions in the free theory, like S(4) ∼ 〈0|T{χ0(x1)χ0(x2)χ0(x3)χ0(x4) eiSI [χ0,σ0]}|0〉 (3.2) where SI denotes the interacting action and the sub-index zero the free fields operators. The operators on the external points create the free asymptotic particles while, diagrammatically, the fields on SI represent internal lines connecting these points. In other words, the interaction can be represented by a set of free fields allowed to virtually play a role during the short amount of time that the process occurs. By considering the reduced notation S(4) ∼ 〈eiSI 〉 the simplest non-trivial term from the interaction with σ is the tree-level piece S(4) ∼ (ie)2 ∣∣∣ p0 〈(χχσ)x(χχσ)y〉 ≡ (ie)2 ∣∣∣ p0 〈−〉 (3.3) 43 CHAPTER 3. FUNDAMENTALS OF EFFECTIVE FIELD THEORIES where the small line is just denoting a tree-level computation. The higher-order terms in the expansion of SI will correspond to Feynman diagrams, for example, from all the products of 1-loop interactions such that, by factorizing e2, can be represented like: S(4) ∼ (ie)2 ∣∣∣ p0 [〈−〉+ (ie)2〈◦〉+ (ie)4〈◦×◦〉+ ·· ·+ (ie)2n〈◦× · · ·×◦〉]≡ (ie)2 ∣∣∣ p0 〈leading log〉 ∣∣∣p0 p (3.4) Along the summation of the first loops, the fields will get rescaled such that the matrix element may contain a dependence in the reference scale p0. Suppose the process occurs on the physical scale p2 and that the dependence of the brackets on the subtraction scale can be written like 〈leading log〉 ∣∣∣p0 p ∼ 1 1− e2 12π2 log ( p2 p2 0 ) ×〈−〉 ∣∣∣ p (3.5) where again 〈−〉 ∣∣∣ p is just representing a simple matrix element of a tree-level diagram. It is clear that if the first measurement of e had been performed in a different value p̄0, instead of p0, the correspondent change in e(p̄) would be compensated by a change in the log, such that the total S(4), would remain invariant on the particular choice of reference1. The Renormalization Group Equations emerge as technique for summing logs in a direct manner, by taking advantage of this feature - physical quantities will not depend on the choice of subtraction point. In general, for representing this arbitrariness for the renormalization scale, the symbol µ is preferred instead of p0 [49]. By replacing Eq.(3.5) in Eq.(3.4), S(4) can be represented like S(4) ∼ ( ie(p2) )2 〈−〉 ∣∣∣ p (3.6) with the effective coupling constant absorbing the rescaling factor. Thus, the solution of the RGE’s will correspond to an efficient way to perform perturbative corrections, turning the predictions more accurate. It is certain, therefore, that if the physical scale of the process was the same of the first measurement, i.e. p0, there would be no need for calculating corrections. In fact, as it will be treated in more detail, once the renormalization parameters for the fields involved in the desired process are known, there is no necessity for calcu