Resummed perturbative series of scalar quantum field theories in two-particle-irreducible formalism

0] ]0

Introduction

1]Introduction Introduction]1

Quantum field theory (QFT) is an essential tool in order to understand phenomena occuring from atomic ( m) to the Planck scale ( m). It provides a theoretical framework to calculate scattering cross sections, particle lifetimes, and other observables of processes of the quantum world. Its most powerful strength is demonstrated by the fact that every elementary interaction known in particle physics can be described by a corresponding relativistic quantum field theory. The main reason of the introduction of the field description is that the processes of particle physics changing the particle number can not be described by the simple application of quantum mechanics (e.g. see the Klein-paradox [1]). Furthermore, QFT combines quantum mechanics and special relativity naturally: it explains the relation between spin and statistics, and solves the causality problem of (relativistic) quantum mechanics. Quantum field theories also play an important role in statistical- and condensed-matter physics, e.g. in the description of critical phenomena and quantum phase transitions. It is extensively used in atomic and nuclear physics as well.

In field theory we have infinitely many degrees of freedom: one assigns dynamical variables (typically scalars, vectors, spinors and/or matrices) to every point of the three (or more generally ) dimensional space and follow for the time evolution of them. The classical dynamics is determined through the Lagrange formalism as a generalization of classical mechanics. In the quantum version of the theory, the variables are considered to be linear operators acting on the Hilbert space of the quantum states in question (in QFT this is the so-called Fock space). The key of the transition to QFT from the classical version is the so-called canonical equal-time commutation relation.

In order to obtain measurable quantities, one has to compute -point functions, which are vacuum expectation values of the time ordered product of the fields. The most successful method for obtaining them is perturbation theory, in which every quantity is expressed through power series (perturbative series) of small parameters of the theory, i.e. the coupling constants. Although general analyses proved that perturbative series are asymptotic [3], therefore the convergence radius is zero, very valuable results can be obtained within this framework, e.g. the perturbatively solved quantum electrodynamics (QED) has extremely accurate predictive power. For example its prediction for the anomalous magnetic moment of the electron agrees with the experimentally measured value to more than 10 significant digits (this is the most accurately verified prediction in the history of physics) [1]. However, for strongly coupled theories perturbation theory completely breaks down; the series do not seem to show even practical convergence at all. At low enough energy, this is the case of the fundamental theory of the strong interaction, the quantum chromodynamics (QCD) [2]. One of the most challenging task of contemporary particle physics is to solve QCD in different setups relevant for phenomena from accelerator experiments to internal structure of compact stellar objects.

The most productive and mathematically well controlled method to attack the problem of large couplings is lattice field theory. It is a fully non-perturbative approach, in which the degrees of freedom are reduced in a way that the quantum equations of motion are solved in a lattice, instead of the continuum formulation [5]. The method typically requires very huge computational resources, nevertheless, it seems to be the most competitive method in order to describe non-perturbative features of the spectra of strongly coupled theories.

A more traditional alternative way to deal with the problem is the application of functional methods. The best known technical framework is offered by the infinite set of Dyson-Schwinger equations. In the first chapter, we will discuss the main properties of this approach, in particular a variant called two-particle-irreducible (2PI) formalism, which is the central topic of this thesis.

Before we proceed to review the literature on 2PI formalism, let us shortly discuss a central problem of every quantum field theory. QFT in its original formulation is ill-defined: divergences are obtained even at the lowest orders of perturbation theory. These divergences come from the ultra-violet regime of the momenta, or in other words from spacetime points which are very close to each other. This implies that QFT at very short distances must break down, which gives no surprise; we should not expect that in an arbitrary small neighborhood of a given point infinitely many physically relevant degrees of freedom can appear. We should think of QFT as an effective model of some more fundamental theory (e.g. string theory), and expect that the details of the dynamics at this scale are not relevant for obtaining results to a much larger scale. This expectation leads us to the idea of renormalization.

In order to obtain well-defined quantities, first we have to regularize QFT. The most natural regularization is to build up equations on a lattice, or the use of a cutoff in the momenta. On the regularized theory, we can perform the so-called renormalization programme. This means that our lack of knowledge of the short range dynamics is hidden into a finite number of physical quantities determined by appropriate measurements or fixed by some conditions. If this can be done in a consistent way, where the cutoff can really be thought as a large quantity, therefore formally(!) can be sent to infinity with obtaining finite results, then the theory is said to be renormalizable.

Renormalization and renormalizability have an extended literature and in perturbation theory it is worked out in great details. Usually the way we treat the theory is to think of the model parameters as cutoff (defined as the highest allowed momentum) dependent objects, which can be separated into renormalized parts and counterterms. In perturbation theory we treat also the counterterms as perturbations, which can be determined from the requirement of the cancellation of the appearing divergences. Pioneers of perturbative renormalizaton with rigorous proofs built upon Feynman’s diagram technique were Bogoliubov, Parasiuk, Hepp and Zimmermann [8, 9], together with Weinberg [10]. A very detailed description of their work and the general theory of perturbative renormalization can be found in [3].

2PI formalism generalizes the idea of the 1PI effective action [4]. The so-called 2PI effective action has two types of variables: mean fields and propagators. Stationary conditions of this action lead to self-consistent equations for these quantities. The solution induces a partial resummation of the perturbative series (i.e. resums certain infinite classes of Feynman diagrams), therefore it is a rather nontrivial question, how the perturbative renormalizability is carried over into certain 2PI approximations. One of the goals of this thesis is to show various methods, which are capable to formulate 2PI renormalizability.

The formalism was developed in the 1960s for non-relativistic field theories of condensed matter physics [11], and was generalized to relativistic theories in the 1970s [15] by Cornwall, Jackiw and Tomboulis. Referring to the latter paper, the method is sometimes called CJT formalism. The -derivable approximation appellation is also used with reference to the paper [14], but these mean the very same formulation. Although the technique is known for a long time, the applications have become an especially active research topic in the last 10 years placing emphasis on non-equilibrium simulations. This is due to the fact, that 2PI formalism is the only known solution to the so-called secular time-evolution problem. Considering scalar fields for instance, the simplest non-local contribution to the 2PI effective action leads to non-secular time evolution showing thermalization at late times [21]. Same observations were made in theories coupling scalars to fermions [22]. Some of the non-perturbative effects of 2PI approximations can be captured by combining the approach with the large-N expansion. In out-of-equilibrium phenomena, the 2PI 1/N expansion at next-to-leading order proved to be very valuable in studying various problems, including thermalization [23, 24, 25] and cosmological preheating [22, 26], non-thermal fixed points [27] and decoherence [28]. The lowest ordered 2PI approximations in equilibrium also show good convergence of thermodynamical quantities, such as the pressure [29, 30]. Discussions of finite temperature gauge theories can be found in [31, 32]. There were attempts to determine transport coefficients in [33, 34, 35]. Phenomenological studies have also appeared using various 2PI approximations, see [36, 37, 38, 39]. Very recently promising applications appeared for the evaluation of critical exponents for continuous phase transition [40].

The renormalization of approximations obtained from the 2PI technique was investigated first for scalar theories at finite temperature in real time formalism [41, 42] and also in imaginary time formalism [43, 44]. The method was then extended to fermions [45] and to QED [46]. Renormalization with non-vanishing field expectation values and therefore the determination of the effective potential was discussed in [44, 47, 48, 49, 50]. In case of theories with more complicated global symmetries renormalization techniques were developed in [51]. Although many papers discussed 2PI renormalization and renormalizability, there are still some open questions, in particular the standardized qualification of its numerical realizations is missing.

The present thesis deals with quantum field theories in equilibrium. Its goal is to map and solve some problems of renormalization both from an analytic and a numerical point of view, together with applications of some newly developed techniques to scalar quantum field theories. It is important to stress that publications discussing the accuracy of numerical realizations of 2PI renormalization in a reproducible manner are virtually missing from the literature. One of the aims of this thesis is to fill the gap caused by this absence of interest.

The greater part of the models examined here are used as effective theories of strong interactions. With this, the thesis would like to contribute to developing the treatments of models of phenomenological importance beyond perturbation theory, which - as already discussed - is crucial when large coupling constants appear. It is important to stress that this thesis has no task to do phenomenology. We wish to search for appropriate methods applicable for phenomenological investigations.

The structure of the thesis is as follows. In Chapter 1, we provide a discussion of the functional methods of QFT. The task of this part is to present pedagogically the usual generator functionals of the Green’s functions in order and end up at 2PI formalism. As an intermediate step, we will also discuss the Dyson-Schwinger equations and establish connections to two-particle-irreducibility. In Chapter 2, we introduce the models investigated in the thesis. At this point we will discuss some fundamental aspects of the large-N technique, which will also be a central subject of our investigations during the next chapters. In Chapter 3, we deal with the O(N) model in 2PI formalism at next-to-leading order of the large-N expansion. We will use the so-called auxiliary field formulation, and demonstrate its renormalizability with constructing appropriate counterterms explicitly. The equivalence between the original and the auxiliary field formulation will also be discussed. The task of Chapter 4 is to present a careful numerical study of 2PI renormalization, in which the one component theory was employed. The first part of this chapter nevertheless deals with a theoretical issue: we show that the minimal subtraction procedure used in Chapter 3 is equivalent to imposing appropriate renormalization conditions on some quantities. This will allow us in one hand to obtain explicitly finite 2PI equations, and on the other hand the opportunity to compare the convergence of the numerical solutions of these equations with the ones containing counterterms. We will discuss the features of various numerical algorithms and produce very accurate results. In Chapter 5 we turn again to a phenomenologically more important theory, the symmetric meson model. We will present an approximate large-N solution in the broken phase, which turns out to be renormalizable and fulfills Goldstone’s theorem. Various symmetry breaking patterns will be investigated, and the renormalized effective potential will be constructed. We will investigate the vacuum structure of the theory which reveals new extrema of the effective potential not observed in previous perturbative computations.

The thesis is based on the following four publications:

• G. Fejős, A. Patkós, Zs. Szép: Renormalized effective actions for the O(N) model at next-to-leading order of the 1/N expansion
Phys. Rev. D80, 025015 (2009), arXiv:0902.0473 [hep-ph]

• G. Fejős, A. Patkós: A renormalized large N solution of the U(N)U(N) linear sigma model in the broken symmetry phase
Phys. Rev. D82, 045011 (2010), arXiv:1005.1382 [hep-ph]

• G. Fejős, A. Patkós: Spontaneously broken ground states of the U(N)U(N) linear sigma model at large N
Phys. Rev. D84, 036001 (2011), arXiv:1103.4799 [hep-ph]

• G. Fejős, Zs. Szép: Broken symmetry phase solution of the model at two-loop level of the -derivable approximation
Phys. Rev. D84, 056001 (2011), arXiv:1105.4124 [hep-ph]

Acknowledgements

First of all I would like express my gratitude to my advisor, Prof. András Patkós for his continuous support and useful advises over the last five years. His painstaking and most valuable guidance has been an indispensable help during my undergraduate and PhD years.

I am very greatful to Zsolt Szép for his precision and endurance. He introduced me to the numerical work, which was an enormous help. Without him, the thesis in its current form could not have been completed.

I thank Urko Reinosa and Julien Serreau for the illuminating discussions during my three months visit in France.

I benefited a lot from the discussions and lectures of Antal Jakovác.

I thank my fellow students Gergely Markó and István Szécsényi for various discussions related to the topics of my thesis.

Travel grants from the Doctoral School of Eötvös University are greatfully acknowledged. My work was supported by the Hungarian Research Fund (OTKA) under contract No. K77534 and No. T068108.

Chapter 1 Functional techniques in quantum field theory

1] ]1 In this chapter we review the standard functional techniques of quantum field theory. We will go through the ordinary generator functionals and show explicitly the properties of these quantities. The motivation for such a summary is that our goal at the end of this chapter is to arrive at the so-called two-particle-irreducible (2PI) formalism, which is a generalization of the standard one-particle-irreducible (1PI) formulation. The best way to achieve this is to build up the usual generator functionals from the beginning to see the correspondence between the different approaches. In this introductory chapter we present the functional techniques in a less formal but nevertheless expressive way. The discussion is built in a way from which the 2PI formalism can be obtained very naturally. Derivations will be presented mainly graphically, using the language of the Feynman-diagram technique. For a greater transparency we illustrate the general line of thinking within the one component theory in the symmetric phase, nevertheless all steps can be performed easily in arbitrary theories.

In this thesis we work in units and this chapter will contain calculations only in Minkowski space. The metric is defined as . In this thesis only scalar quantum fields will appear in space-time dimensions, with Lagrangians parametrised as

 L[∂μϕ,ϕ]=−12ϕ(∂μ∂μ+m2)ϕ−U(ϕ), (1.1)

where is a multicomponent -number scalar field and describes the self-interaction. The -point Green function is defined as a time-ordered vacuum expectation value of Heisenberg field operators

 Gn(x1,x2,...,xn)=⟨0|T(^ϕ(x1)^ϕ(x2)...^ϕ(xn))|0⟩. (1.2)

In functional integral formalism these functions can be obtained as [4]

 Gn(x1,x2,...,xn)=∫Dϕϕ(x1)ϕ(x2)...ϕ(xn)eiS[ϕ]∫DϕeiS[ϕ] (1.3)

where refers to the action of the configuration : .

The Green functions are not measurable quantities, but with the help of the so-called reduction formulae (Lehmann, Symanzik and Zimmermann, 1955), they can be turned into physical transition amplitudes, see details in [1]. Due to this result, one can prove that the Green functions incorporate all information which can be extracted from a quantum field theory. Therefore, it is worthwhile to concentrate on building up equations for these quantities, which we eventually do in the next sections: the corresponding Dyson-Schwinger and 2PI equations will be derived. Before turning to this task, let us discuss the generator functionals of the Green functions. The properties of these functionals summarized here are quite general and can be obtained in arbitrary theories.

1.1 Generator functionals

The generator functional of the scalar Green functions is a functional of a source , which is an arbitrary -number function:

 Z[J]=∞∑n=0inn!∫d4x1...d4xnJ(x1)...J(xn)Gn(x1...xn), (1.4)

therefore the Green functions can be obtained by functional differentiation as

 Gn(x1,x2,...,xn)=(−i)nδnZ[J]δJ(x1)...δJ(xn)∣∣∣J=0≡(−i)nδnZ[J]δJn∣∣∣J=0. (1.5)

We will often not write down explicitly the spacetime arguments and use the short-hand notation as shown by the second equality. can be represented with functional integrals:

 Z[J]=∫Dϕei(S[ϕ]+∫Jϕ)∫DϕeiS[ϕ], (1.6)

as it can be easily seen by expanding in Taylor series. To calculate and eventually the Green functions we split the action into two parts:

 S[ϕ]=S0[ϕ]+SI[ϕ], (1.7)

where refers to the gaussian part (quadratic in ), while to the interaction:

 S0[ϕ]=−12∫d4xϕ(x)(∂μ∂μ+m2)ϕ(x),SI[ϕ]=−∫d4xU[ϕ(x)]. (1.8)

We write as

 Z[J]=∫Dϕei(S[ϕ]+∫Jϕ)/∫DϕeiS0[ϕ]∫DϕeiS[ϕ]/∫DϕeiS0[ϕ]≡, (1.9)

where refers to the operation of the gaussian averaging. The evaluation of the numerator and the denominator can be made by expanding the exponentials in the brackets, since Green functions of the gaussian theory can be easily calculated [1]. Here we implicitly assumed that in order to have a positive definite quadratic form for the gaussian part, therefore the expansion makes sense. For each term in the expansion one can associate a diagram with defining model specific Feynman rules. For of a one component field these are the following [1]:

 \includegraphics[bb=145482334431,scale=0.7]feyn1.pdf=ΔF(x−y)
 \includegraphics[bb=145482334431,scale=0.7]feyn2.pdf=−iλ∫d4z
 \includegraphics[bb=135482334441,scale=0.7]feyn3.pdf=i∫d4zJ(z),

where is the Feynman (or perturbative) propagator:

 ΔF(x−y)=−i(∂2+m2)−1(x,y)=∫d4k(2π)4ieik(x−y)k2−m2+iϵ. (1.10)

Every diagram must be divided by a symmetry factor, which is defined as the dimension of the graph’s symmetry group. Note that the -point vertex appears only in the expansion of the numerator, due to the term proportional to the source. In the averages every possible diagram without external legs appear, these are the so-called vacuum graphs. Using the Feynman rules (keeping also the symmetry factors in mind), in theory the first few terms of the denominator are

 =1+\includegraphics[bb=106540200631,scale=0.21]graf1.pdf+\includegraphics[bb=106540200631,scale=0.21]graf1.pdf\includegraphics[bb=106540200631,scale=0.21]graf1.pdf+\includegraphics[bb=63540234631,scale=0.21]graf2.pdf+\includegraphics[bb=106585200631,scale=0.21]graf4.pdf+\includegraphics[bb=106540200631,scale=0.21]graf1.pdf\includegraphics[bb=63540234631,scale=0.21]graf2.pdf+\includegraphics[bb=63540234631,scale=0.21]graf3.pdf+... (1.11)

We can observe that some terms contain disconnected diagrams, which offers a possibility for a simplification of the expression. Let be the set of the internally connected diagrams:

 V={\includegraphics[bb=106540200631,scale=0.21]graf1.pdf\includegraphics[bb=63540234631,scale=0.21]graf2.pdf\includegraphics[bb=106585200631,scale=0.21]graf4.pdf\includegraphics[bb=63540234631,scale=0.21]graf3.pdf...}.

We can check that every term in (1.11) can be written in the following form:

 ∏i1ni!(Vi)ni,

where is the th element of and equals to the number of occurrences of this connected diagram in the term in question (the factorials are symmetry factors of the interchanging of disconnected diagrams). The denominator of (1.9) is therefore

 =∑{ni}∏i1ni!(Vi)ni, (1.12)

where the summation goes through all ordered sets . It is easy to show that this expression can be factored:

 =(∑n11n1!Vn11)(∑n21n2!Vn22)(∑n31n3!Vn33)...=∏i∑ni1ni!Vnii. (1.13)

We recognize the definition of the exponential function and obtain

 =∏iexpVi=exp∑iVi=exp(\includegraphics[bb=106540200631,scale=0.21]graf1.pdf+\includegraphics[bb=63540234631,scale=0.21]graf2.pdf+\includegraphics[bb=106585200631,scale=0.21]graf4.pdf+\includegraphics[bb=63540234631,scale=0.21]graf3.pdf+...). (1.14)

This is the linked cluster theorem, which states that the sum of all possible vacuum diagrams is equal to the exponential of the sum of all the connected ones. The numerator of (1.9) can be evaluated the very same way. We have to keep in mind that there we also have a -point vertex, therefore in addition to the diagrams of , we have new vacuum graphs. Let be the extended set of the vacuum diagrams:

 ~V={\includegraphics[bb=106540200631,scale=0.21]graf1.pdf\includegraphics[bb=63540234631,scale=0.21]graf2.pdf\includegraphics[bb=106585200631,scale=0.21]graf4.pdf\includegraphics[bb=63540234631,scale=0.21]graf3.pdf...\includegraphics[bb=165482318441,scale=0.35]graf5.pdf\includegraphics[bb=165482318441,scale=0.35]graf6.pdf\includegraphics[bb=165482318441,scale=0.35]graf7.pdf \includegraphics[bb=165482318441,scale=0.35]graf8.pdf\includegraphics[bb=165482318441,scale=0.35]graf9.pdf...\includegraphics[bb=165482318441,scale=0.355]graf10.pdf\includegraphics[bb=195482298441,scale=0.355]graf11.pdf...\includegraphics[bb=180482338441,scale=0.355]graf12.pdf...}.

Note that the crosses do not correspond to external legs, these are -point vertices, therefore every diagram of is indeed a vacuum graph. Following the method described above we arrive at

 =exp(\includegraphics[bb=106540200631,scale=0.21]graf1.pdf+\includegraphics[bb=63540234631,scale=0.21]graf2.pdf+\includegraphics[bb=106585200631,scale=0.21]graf4.pdf+\includegraphics[bb=63540234631,scale=0.21]graf3.pdf+...\includegraphics[bb=165482333441,scale=0.35]graf5.pdf+\includegraphics[bb=165482333441,scale=0.35]graf6.pdf+
 +\includegraphics[bb=165482333441,scale=0.35]graf7.pdf+\includegraphics[bb=174482328441,scale=0.35]graf8.pdf+\includegraphics[bb=175482328441,scale=0.35]graf9.pdf+...\includegraphics[bb=206482298441,scale=0.355]graf10.pdf+\includegraphics[bb=206482298441,scale=0.355]graf11.pdf+\includegraphics[bb=205482338441,scale=0.355]graf12.pdf...). (1.15)

Since is the quotient of (1.1) and (1.14), only diagrams involving the source remain [1]:

 Z[J]=exp(\includegraphics[bb=165482333441,scale=0.35]graf5.pdf+\includegraphics[bb=165482333441,scale=0.35]graf6.pdf+\includegraphics[bb=165482333441,scale=0.35]graf7.pdf+\includegraphics[bb=174482328441,scale=0.35]graf8.pdf+\includegraphics[bb=175482328441,scale=0.35]graf9.pdf...

 +\includegraphics[bb=186482328441,scale=0.355]graf10.pdf+\includegraphics[bb=186482328441,scale=0.355]graf11.pdf+...+\includegraphics[bb=186482348441,scale=0.355]graf12.pdf+...). (1.16)

If now we are interested in the value of the various -point functions, then we should re-expand the exponential function and read off the coefficients of the power series in the source . We immediately recognize that in our concrete example only even number of sources could appear in every term, therefore when we take the source to zero only Green functions of even variables are non-zero. (In the broken phase of theory also Green functions of odd variables are present.) For example the - and -point functions read as

 G2(x1,x2)=\includegraphics[bb=186482328441,scale=0.355]graf5b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf6b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf7b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf8b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf9b.pdf..., (1.17)
 G4(x1,x2,x3,x4)=\includegraphics[bb=186482328441,scale=0.355]graf13.pdf+\includegraphics[bb=186482328441,scale=0.355]graf14.pdf+\includegraphics[bb=186482328441,scale=0.355]graf15.pdf+\includegraphics[bb=186482328441,scale=0.355]graf10b.pdf+

 +\includegraphics[bb=186482328441,scale=0.355]graf16.pdf+\includegraphics[bb=186482328441,scale=0.355]graf17.pdf+...\includegraphics[bb=186482328441,scale=0.355]graf18.pdf+...\includegraphics[bb=186482328441,scale=0.355]graf19.pdf+...\includegraphics[bb=186482328441,scale=0.355]graf20.pdf+... (1.18)

The crosses from the endpoints disappeared, they indicated the presence of the sources (therefore neither multiplicative factors nor integration is needed corresponding to these endpoints). This means that these diagrams are not vacuum graphs anymore but have and external legs, respectively.

We recognize that there are lots of disconnected diagrams appearing in the series of the Green functions. (We note that in the symmetric phase of the theory contains only connected diagrams - see (1.17). This would not be the case in the broken phase, since then there would be a -point vertex in the original Lagrangian, which could also produce nonzero contribution to .) Recalling (1.1) it is possible to build up a generator functional which generates only the connected pieces, which provides great simplifications in further calculations. We introduce as

 iW[J]=logZ[J]. (1.19)

Factoring an is just a convention, which will prove to be useful later. Using (1.1) we have

 iW[J]=\includegraphics[bb=165482333441,scale=0.35]graf5.pdf+\includegraphics[bb=165482333441,scale=0.35]graf6.pdf+\includegraphics[bb=165482333441,scale=0.35]graf7.pdf+\includegraphics[bb=174482328441,scale=0.35]graf8.pdf+\includegraphics[bb=175482328441,scale=0.35]graf9.pdf...

 +\includegraphics[bb=186482328441,scale=0.355]graf10.pdf+\includegraphics[bb=186482328441,scale=0.355]graf11.pdf+...+\includegraphics[bb=186482348441,scale=0.355]graf12.pdf+... (1.20)

If we think of as a power series of the source:

 iW[J]=∑ninn!∫d4x1...d4xnJ(x1)...J(xn)Wn(x1,x2,...,xn), (1.21)

it is obvious that the quantities refer to sums of connected diagrams. These are to be called as connected Green functions or cumulants and can be calculated as

 Wn(x1,...,xn)=(−i)mδn(iW[J])δJn∣∣∣J=0. (1.22)

In our example due to the absence of nonzero , we have and , however and further Green functions are greatly simplified:

 W2(x1,x2)=\includegraphics[bb=186482328441,scale=0.355]graf5b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf6b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf7b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf8b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf9b.pdf..., (1.23)

 W4(x1,x2,x3,x4)=\includegraphics[bb=186482328441,scale=0.355]graf10b.pdf+\includegraphics[bb=186482328441,scale=0.355]graf18.pdf+...\includegraphics[bb=186482328441,scale=0.355]graf19.pdf+\includegraphics[bb=186482328441,scale=0.355]graf21.pdf+... (1.24)

It turns out that even more simplification can be made, which leads us to the idea of the quantum effective action. Let us introduce the so-called semi-classical or mean-field as

 ¯ϕ=δW[J]δJ, (1.25)

which is the vacuum expectation value of in the presence of the source . We assume that this relation can be inverted uniquely: for a given only one exists and vice-versa. The quantum effective action is the functional Legendre-transform of the generator of the connected Green functions [4]:

 Γ[¯ϕ]=W[J]−∫J¯ϕ. (1.26)

Using (1.25) we have the usual relation

 δΓ[¯ϕ]δ¯ϕ=∫δWδJδJδ¯ϕ−∫δJδ¯ϕ¯ϕ−J=−J. (1.27)

Let us try to calculate the diagrams contributing to this quantity. We have

 eiW[J]=∫Dϕei(S[ϕ]+∫Jϕ)∫DϕeiS[ϕ], (1.28)

therefore

 eiΓ[¯ϕ]=∫Dϕei(S[ϕ]+∫J(ϕ−¯ϕ))∫DϕeiS[ϕ]=∫Dϕei(S[¯ϕ+ϕ]+∫Jϕ)∫DϕeiS[ϕ], (1.29)

where in the second equality we performed the following change of the integration variable in the numerator: . The shifted action in the numerator can be expanded around :

 S[¯ϕ+ϕ]=S[¯ϕ]+S′[¯ϕ]ϕ+12S′′[¯ϕ]ϕ2+..., (1.30)

which gives new vertices.

We are now equipped to calculate such a quantity like (1.29). As we have already seen, the denominator is the exponential of all possible connected vacuum diagrams with couplings of the original Lagrangian. The numerator contains the same type of graphs but with vertices of the shifted Lagrangian extended with an additional term for the -point vertex coming from the source. Diagrams of the denominator cancel the diagrams of the numerator which do not contain at least one of the new vertices or the source. This is a consequence of the normalization , which implies .

Before starting to calculate the remaining diagrams by brute force, let us point out an important simplification taking place. The key observation is that the -point function of the shifted action is zero by construction [7]:

 ∫Dϕϕei(S[¯ϕ+ϕ]+∫Jϕ)∫DϕeiS[ϕ] = ∫Dϕ(ϕ−¯ϕ)ei(S[ϕ]+∫J(ϕ−¯ϕ))∫DϕeiS[ϕ]=−¯ϕeiΓ[¯ϕ] (1.31) + ∫Dϕϕei(S[ϕ]+∫J(ϕ−¯ϕ))∫DϕeiS[ϕ]=−¯ϕeiΓ[¯ϕ]+e−i∫J¯ϕ1iδδJeiW[J] = eiΓ[¯ϕ](δW[J]δJ−¯ϕ)=0.

This gives a great simplicity of the structure of . Consider a diagram of the numerator in which two subdiagrams and are connected with one single line. This is to be called -particle-reducible (1PR). Then find every possible diagram where is replaced with some other subdiagram. In the sum of these diagrams, the exact -point function will appear hanging on . Since we saw that the -point function is identically zero, we may forget about diagrams with the previously indicated property (see Figure 1.1). In other words, we may omit all 1PR diagrams and consider only one-particle-irreducible (1PI) ones. These do not fall into two parts if one internal line is cut. The expressions corresponding to these diagrams do not refer to the -point vertices at all, therefore the source does not appear in , as it should. We may write

 eiΓ[¯ϕ]=∫DϕeiS[¯ϕ+ϕ]∫DϕeiS[ϕ]∣∣∣1PI. (1.32)

When we are interested in the physical case (i.e. ), from (1.27) we have

 δΓ[¯ϕ]δ¯ϕ=0, (1.33)

which means that the semi-classical field is an extremal configuration of . This is the reason for calling it quantum effective action. In agreement with the expectations, after expanding the shifted action in (1.32) around and ignoring fluctuations, becomes the classical action: .

In theory the shifted action introduces the following new and -point vertices (the -point vertex is irrelevant due to the previous analysis):

 \includegraphics[bb=127481228512,scale=0.8]feyn4.pdf=(−iλ)12∫d4z¯ϕ2(z),
 \includegraphics[bb=143481228512,scale=0.8]feyn5.pdf=(−iλ)∫d4z¯ϕ(z).

Note, that some sticks appeared at the end of the dashed lines. These refer to the multiplication(s) of and to the integration over spacetime. The first few terms in the series of turns out to be:

 Γ[¯ϕ]=S[¯ϕ]−i[\includegraphics[bb=192492246371,scale=0.65]graf26.pdf+\includegraphics[bb=192492246371,scale=0.65]graf27.pdf+\includegraphics[bb=192482291371,scale=0.65]graf28.pdf (1.34)
 +\includegraphics[bb=177482279371,scale=0.65]graf22.pdf+\includegraphics[bb=172482279331,scale=0.65]graf25.pdf+...\includegraphics[bb=172482259331,scale=0.65]graf24.pdf...]

We may think of as a power series of :

 Γ[¯ϕ]=∑n1n!∫d4x1...d4xnΓn(x1,...,xn)¯ϕ(x1)...¯ϕ(xn), (1.35)

where the coefficients are the so-called proper vertices. From (1.34) we can read for example that

 Γ2(x1,x2)=−(∂2+m2)δ(x1−x2)−i[\includegraphics[bb=192492246371,scale=0.65]graf26b.pdf+\includegraphics[bb=192492246371,scale=0.65]graf27b.pdf+\includegraphics[bb=192482281371,scale=0.65]graf28b.pdf...],
 Γ4(x1,x2,x3,x4)=λδ(x1−x2)δ(x1−x3)δ(x1−x4)−i[\includegraphics[bb=177482269371,scale=0.65]graf22b.pdf
 (1.37)

Proper vertices with odd number of variables are zero in the symmetric phase of the theory. Note that diagrams in (LABEL:1-gamma2_diag) and (1.37) do not have sticks on their “legs”, as the multiplicative factors (their integrals more precisely) were not included in the definition of the proper vertices. One also has to note that the factorials in (1.35) were “eaten up” by the symmetry factors of diagrams still containing these sticks (i.e. diagrams in (1.34)).

is strongly related to the connected -point function, since

 δδJ(y)=∫zδ¯ϕ(z)δJ(y)δδ¯ϕ(z)=∫zδ2W[J]δJ(y)δJ(z)δδ¯ϕ(z), (1.38)

which applied to the function gives

 δ(x−y)=∫zδ2W[J]δJ(y)δJ(z)δJ(x)δ¯ϕ(z)=−∫zδ2W[J]δJ(y)δJ(z)δ2Γ[¯ϕ]δ¯ϕ(z)δ¯ϕ(x). (1.39)

This shows that in functional sense

 (δ2WδJ2)−1=−δ2Γδ¯ϕ2, (1.40)

which implies that

 iW−12=Γ2. (1.41)

The squared bracket of (LABEL:1-gamma2_diag) including the factor is called the self-energy, and is denoted by . With the help of (1.40) we arrive at the Dyson-equation of the propagator

 iW−12(x1,x2)=−(∂2+m2)δ(x1−x2)−Σ(x1,x2), (1.42)

to which we will refer several times in the forthcoming sections.

It is possible to organize the series of in terms of the number of loops. When one works in ordinary units, it turns out to be a series in . Introducing the notations

 Γ[¯ϕ]=S[¯ϕ]+Γℏ[¯ϕ]+Γℏ2[¯ϕ]+..., (1.43)

the leading order piece (beyond the classical value) can be resummed in a rather compact form, in which one has to take into account every 1-loop diagram:

 iΓℏ = \includegraphics[bb=151498211331,scale=0.63]graf31.pdf+\includegraphics[bb=131498211331,scale=0.63]graf32.pdf+\includegraphics[bb=131498221331,scale=0.63]graf33.pdf+... (1.44)

Taking into account the symmetry factors, the sum of these diagrams is

 iΓℏ = 12(−iλ2)∫xΔF(x,x)¯ϕ2(x) (1.45) + 14(−iλ2)2∫x∫yΔF(x,y)ΔF(y,x)¯ϕ2(x)¯ϕ2(y) + 16(−iλ2)3∫x∫y∫zΔF(x,y)ΔF(y,z)ΔF(z,x)¯