A Renewed Numerical Exploration of the Triviality of the Quartic Scalar Field Theories

A Renewed Numerical Exploration of the Triviality of the Quartic Scalar Field Theories Jorge L. deLyra∗ Universidade de S˜ao Paulo Instituto de F´ısica Rua do Mat˜ao, 1371, 05508-090 S˜ao Paulo, SP, Brazil November 11 2025 Abstract We explore the triviality properties of the quantum field theory of a scalar field with a single real component and a quartic potential term in the action. The scalar quantum field theories are important because they constitute the scalar sector of the standard model of particle physics. The single scalar field left over after spontaneous symmetry breaking has been implemented in that model is known as the Higgs field. The triviality of the theory means that, although its ensemble is not Gaussian, all its expectation values are identical to those that can be obtained from a Gaussian theory, with appropriately chosen parameters. We present here numerical results with high-quality statistics, that strongly support the conclusion that the theory is in fact trivial. We measure the dimensionless renormalized mass parameter of the scalar field theory in several different ways, some of which would be valid only if the theory was in fact trivial. One of these measurements has a global character over the whole lattice, since it is done in momentum space, and is in fact the usual way to measure this parameter. The other five measurements are done in position space and have a local character, being valid on single lattice sites, single lattice links or single diagonals of plaquettes. The expressions for the local observables used for these measurements are derived with the use of the Gaussian ensemble of a free theory, and therefore can be valid only if the quartic theory is in fact trivial. We will show that not only there is strong evidence of triviality, but that in some cases this triviality seems to be in force already on finite lattices, instead of only in the continuum limit. ∗Email: [email protected] ORCID: 0000-0002-8551-9711 1 Keywords: Quantum Field Theory, Euclidean Lattice, Triviality, Higgs Field, Quartic Scalar Theories, Vacuum Expectation Value, Propagator. DOI: 10.5281/zenodo.17582908 2 1 Introduction The problem of the triviality of the polynomial scalar field theories is a longstanding one. The basic numerical facts associated to this have been known for a long time [1]. There is still no analytic proof of triviality in the particularly important case of four-dimensional spacetime, but a rather involved proof for five or more dimensions was published more than four decades ago, in 1981 [2, 3]. Since that time there has constantly been some amount of confusion about this issue, so that an effort to analyze and describe the situation in simpler and clearer terms, even in the absence of a complete proof in the most relevant case, seems plainly justified. In this paper the scalar field theory will be defined in a mathematically precise fashion using a discrete representation on what is known as an Euclidean lattice. The continuum theory in Euclidean space is to be obtained as a limit from this discrete Euclidean construction, after which a Wick rotation is executed, leading to the corresponding continuum theory in Minkowski spacetime. In this paper we will examine under a new light some aspects of a well-known family of scalar theories in quantum field theory, known as the λϕ4polynomial scalar theories. This is a family of quartic non-linear theories of scalar fields that has great fundamental importance due to the fact that it plays a vital role in the structure of the so-called standard model of particle physics. The scalar sector of that standard model contains a scalar field theory of this family, for a doublet of complex scalar fields, that is essential to implement the process of spontaneous symmetry breaking which is at the very core of the structure of that standard model. Once the process of symmetry breaking has been completed, a single real scalar field component is left over, with its dynamics given by an action that contains a quartic term. This remaining uncharged massive scalar field is known as the Higgs field. Here we will examine the quantum dynamics of this single-component scalar field, which has a non-quadratic action that contains a potential term given by a quartic polynomial on the field. By construction this theory displays the phenomenon of spontaneous symmetry breaking. We will show some very interesting facts about the theory, facts that are related to the well-known issue that is described as the triviality of the quartic scalar theories, in particular in the case of four spacetime dimensions. This property is better analyzed in terms of the dimensionless lattice fields, rather than in terms of the corresponding dimensionfull fields in the continuum, as is sometimes done. This is so because the two-point function of the dimensionless lattice fields can be properly normalized as a standard statistical correlation function, equal to one at the origin, while the two-point function of the corresponding dimensionfull fields in the continuum is a Green’s function, which is singular at the origin. A more detailed discussion of this can be found in Appendix A. It is important to note that this triviality property has no direct bearing on the role that the scalar theories play as part of the standard model. Sometimes opinions are expressed to the effect that this scalar theory cannot be trivial because this would prevent it from having the effect that it must have within the structure of the standard model. However, this is simply not true, since the trivial scalar theory is perfectly capable of generating a finite and non-zero vacuum expectation value of the field and a finite and non-zero renormalized mass, which are the requirements for the role that the scalar theory plays in the standard model. Such opinions are usually associated with what can be characterized as perturbative thinking about the scalar theory, but one should not attempt to draw fundamental structural conclusions from the perturbative series of this theory, since it is a completely divergent series. A simple proof of this fact, based on the well-known facts about power series established in the theory of analytic functions, can be found in Appendix B. This paper is organized as follows: in Section 2 we review the quartic scalar theory; 3 in Section 3 we introduce the local observables for the renormalized mass parameter; in Section 4 we describe the computer program used in the numerical exploration; in Section 5 we present the numerical results; and in Section 6 we present our conclusions. 2 Review of the Quartic Scalar Theory on the Lattice Here we will give the definition and describe the main structural properties of the theory, in order to establish the notation and the basic mathematical facts about it. If we are to make any meaningful and precise statements about any theory, it is imperative that we start from a precise mathematical definition of that theory. We will use as the very definition of any given quantum field theory its realization on a discrete Euclidean lattice. In this constructive definition a discrete version of the theory is set up on finite lattices in Euclidean space, in a way which is valid for arbitrarily large values of the size of the lattice. The corresponding theory in the Minkowski continuum is obtained when we make the lattice infinitely large and its mesh infinitely fine, and execute a Wick rotation from the resulting Euclidean space to the corresponding Minkowski spacetime. 2.1 Definition of the Theory In order to define the theory and establish the notation, we will consider a discrete Cartesian lattice with Nsites in each one of dspacetime directions, hence with a total of Nd sites and dNdlinks between neighboring sites. Roughly speaking, the lattice is a discrete representation of spacetime. If the components of the d-vector ~n are the ddiscrete integer coordinates of the lattice sites in ddimensions, then the lattice field ϕ=ϕ(~n) is defined as a real function of these discrete coordinates, where ~n = (n1,...,nd),(1) nµ= 1,...,N, (2) µ= 1,...,d, (3) where ϕ(~n) is a single-component real scalar field, which on the lattice formalism is always a dimensionless quantity. The relevant geometrical elements of the lattice itself are its sites ~n, its links ℓµbetween neighboring sites along the direction µ, and its plaquettes. By definition the lattice consists of Ndsites, and of dNdlinks between neighboring sites. We assume that periodic boundary conditions are used on the lattice, meaning that along any given direction µthe site nµ=N+ 1 is identified with the site nµ= 1, and that the site nµ= 0 is identified with the site nµ=N. Given a site ~n and a direction µ, the link ℓµ simply connects the site ~n to its neighboring site ~n+1µin that direction, and is the natural locus for the finite difference of the two values of the field at either site. Here 1µrepresents the link from the site ~n to its next neighbor in the positive direction µ, going from a given value of nµto the value nµ+ 1. The plaquettes are two-dimensional objects constructed as follows. One starts at a given site ~n and considers two positive directions departing from it, say µand ν. The plaquette is the square generated by the two links ℓµand ℓν. Since there are d!/[2!(d−2)!] ways to choose the two positive directions from a given site, it is clear that there are d(d−1)Nd/2 plaquettes in an N-lattice in ddimensions. If we consider the diagonals of plaquettes, since there are two diagonals per plaquette, it is clear that the total number of such diagonals on the N-lattice in ddimensions is given by d(d−1)Nd. Note that these diagonals are necessary to help upgrade the Cartesian lattice to a simplicial complex, over which a definite metric 4 geometry can be unequivocally defined. Therefore it is important to be able to determine the lengths of these diagonals, as well as the lengths of the links. The Euclidean action functional defining the scalar field theory on the lattice is given in this d-dimensional discrete position space by SN[ϕ] = Nd X ~n    1 2 d X µ=1 [∆µϕ(~n)]2+α 2ϕ2(~n) + λ 4ϕ4(~n)  ,(4) where the sum over ~n runs over all Ndlattice sites, so that it can be expressed as a ddimensional multiple sum, Nd X ~n ≡ N X n1=1 ... N X nd=1 ,(5) and where the lattice squared mass parameter αis a dimensionless real parameter, while the lattice coupling constant λis a dimensionless positive real parameter. The operator ∆µ represents the finite difference of the fields across the lattice link starting at the site ~n in the positive direction µ. Note that αcan be negative so long as λis strictly positive, thus ensuring the energetic stability of the theory or, in quantum terminology, the existence of its ground state. These stability constraints will be discussed in more detail in the subsequent Subsection 2.2. If λis zero then αhas to be positive, and in this case our theory reduces to the free quantum field theory of the single real scalar field ϕ(~n). If we introduce a lattice spacing awith physical dimensions of length it is not difficult to show that for α≥0 the expression shown in Equation (4) tends to the usual classical expression for the action of this theory in the continuum limit, within a finite cubic box in ddimensions, S[φ] = IV ddx   1 2 d X µ=1 [∂µφ(~x )]2+m2 2φ2(~x ) + Λ 4φ4(~x )  ,(6) where we introduced, still on finite lattices, the dimensionfull quantities given by the squared mass m2=α/a2, the dimensionfull coupling constant Λ = ad−4λ, and the dimensionfull field φ(~x ) = a(2−d)/2ϕ(~n), where ~x =a~n are now dimensionfull coordinates on the lattice, so that the minimum possible variation of a coordinate is dxµ=a, the d-dimensional integration element is ddx=ad, the lattice length is L=Na and the lattice volume is V=Ld. After that we take the continuum limit, in which N→ ∞ and a→0, in such a way that the lattice length L=Na remains constant, under some auxiliary conditions involving the dimensionless parameters αand λof the theory. The observable quantities of the quantum field theory are the average values of functionals of the field on the statistical distribution of field configurations, also known as an ensemble, defined by the exponential of minus the action. These are called expectation values, and given an arbitrary observable O[ϕ], which in general is some functional of the field, its expectation value is defined as the ratio of multiple integrals ON=Z[dϕ]Ne−SN[ϕ]O[ϕ] Z[dϕ]Ne−SN[ϕ] ,(7) where the integrals indicated are functional integrals, that is, sums over the set of all possible field configurations ϕ(~n) on the lattice. On a finite N-lattice in ddimensions they can be written as multiple integrals in a real space of dimension Nd, with the measure 5 [dϕ]N= Nd Y ~n dϕ(~n),(8) where the product over ~n runs over all Ndlattice sites, so that it can be expressed as a d-dimensional multiple product, Nd Y ~n ≡ N Y n1=1 ... N Y nd=1 .(9) The integrals over ϕ(~n) at each site ~n, shown in Equation (7), are taken from ϕ(~n)→ −∞ to ϕ(~n)→ ∞. In the N→ ∞ continuum limit the integrals in Equation (7) become infinite-dimensional integrals, also known as functional integrals in the continuum. Some of the most important expectation values of the theory are defined in momentum space, so let us introduce the representation of the theory in that space. The field ϕ(~n), defined at first in position space, can also be represented in momentum space, in the case of the use of periodic boundary conditions on the lattice, by its Fourier components eϕ, also known as the coefficients of the expansion of the field in lattice modes. If ~κ are the discrete integer momentum coordinates of these modes on the conjugate lattice in momentum space, then the Fourier components eϕ=eϕ(~κ) are defined as complex functions of these modes, where ~κ = (κ1,...,κd),(10) κµ=κmin,...,κmax,(11) µ= 1,...,d, (12) and where the extremes of the discrete momentum coordinates κµare given by κmin =         −N−1 2for odd N, −N 2+ 1 for even N, (13) κmax =         N−1 2for odd N, N 2for even N. (14) The conjugate lattice has the same geometrical structure of the original lattice, being Cartesian with Npoints along each one of dindependent directions, but which represent now modes rather than sites. In this paper we will use only even values of N, because for algorithmic reasons this is the most important case for the numerical simulations, due to the use of periodic boundary conditions. The finite Fourier transformation and its inverse, between these two representations of the field, are exact relations given by eϕ(~κ) = 1 Nd Nd X ~n eı2π N~κ·~nϕ(~n),(15) ϕ(~n) = Nd X ~κ e−ı2π N~κ·~n eϕ(~κ),(16) 6 where the sum over ~κ runs over all Ndlattice modes, so that it can be expressed as a d-dimensional multiple sum, Nd X ~κ ≡ κmax X κ1=κmin ... κmax X κd=κmin ,(17) and where the set of complex exponentials shown in these transformation formulas constitutes a complete and orthogonal basis for the representation of arbitrary functions on the lattice, known as the Fourier basis. The first expression, in Equation (15) above, defines the Fourier components of the field, and the second expression, in Equation (16) above, can be interpreted as the decomposition of the field in lattice modes. Note that, since ϕ(~n) is real, from Equation (15) it follows that the Fourier components satisfy the constraints eϕ∗(~κ) = eϕ(−~κ).(18) Modes ~κ with all components equal to either 0 or N/2 are real modes, since in these cases the complex exponential functions representing the modes are real numbers, and hence the Fourier components are also real. For even Nthere are 2dreal modes, which are at the vertices of a d-dimensional cube within the conjugate lattice in momentum space, of extent N/2 in each direction, going from the zero mode (0,...,0) to the maximal mode (N/2,...,N/2). For odd Nthere is a single real mode, the zero mode ~κ =~ 0. Note that from Equation (15) it follows that the coefficient of the zero mode eϕ0=eϕ(~ 0) is a full-lattice block variable, that is, the average of the field over the whole lattice, eϕ0=1 Nd Nd X ~n ϕ(~n).(19) This real quantity will be instrumental in the determination of the state of symmetry breaking of the quartic polynomial theory. The observable eϕ2 0or, more precisely, the corresponding observable formed by the difference of expectation values σ2 0=eϕ2 0−heϕ0i2,(20) which is the squared width of the distribution of values of eϕ0, can be used as an order parameter for the two phases of the theory, since for large Nit tends to zero in the symmetric phase, and remains different from zero in the broken-symmetric phase. Although the symmetry breaking mechanism only becomes exact in the N→ ∞ limit, this observable can be used to determine approximately the location of the phase transition in the (α, λ) parameter plane, even on finite lattices. It is noteworthy that the dimensionless renormalized mass parameter αR, to be discussed in what follows, can also be used for this purpose. On finite lattices the direction of symmetry breaking tends to drift (in our case here to flip sign), in such a way that the expectation value heϕ0iis averaged out to zero during the drift. This drift only ceases completely in the N→ ∞ limit. In the same way, due to this drift and consequent averaging out, the expectation value vR=hϕ(~n)iof the field, which in the continuum limit is different from zero only in the broken-symmetric phase, is always zero on finite lattices, in either phase, and hence cannot be used as the order parameter. In order to have vR6= 0 on finite lattices it is necessary to break the symmetry by hand, by either using fixed boundary conditions oriented in a predetermined way, or including in the theory an external source j(~n) with a predetermined orientation, which can be done by adding the term j(~n)ϕ(~n) to the summand of the action 7 shown in Equation (4). Note that the introduction of such an external source would cause vR(~n) to depend on position, unless j(~n) was actually constant over the lattice. The action of the λϕ4theory shown in Equation (4) can be written in terms of eϕ(~κ), as SN[eϕ], but its form in this representation is neither simple nor particularly useful. By transforming the fields to momentum space we may write the expectation value, in terms of the Fourier transforms of the fields, as ON=Z[deϕ]Ne−SN[eϕ]O[eϕ] Z[deϕ]Ne−SN[eϕ] ,(21) where the measure is now given symbolically by [deϕ]N= Nd Y ~κ deϕ(~κ),(22) where for a more precise description deϕ(~κ) should be replaced by dℜ[eϕ(~κ)]dℑ[eϕ(~κ)], and where the values of ~κ and the real and imaginary parts of eϕ(~κ) have to be selected so as to neither under-count nor over-count the Ndindependent variables, that is, they must take into account the constraints given in Equation (18) and the fact that some modes are real and thus have no imaginary parts. The most fundamental expectation value in the theory is that of the two-point function in momentum space, also known as the propagator, which is defined as D|eϕ(~κ)|2EN=Z[deϕ]Ne−SN[eϕ]|eϕ(~κ)|2 Z[deϕ]Ne−SN[eϕ] .(23) This observable defines the important physical quantity that is the renormalized dimensionless squared mass parameter αRof the theory. This parameter is related to the correlation length of the field on the lattice. In terms of the dimensionfull renormalized mass mRit is given by αR=m2 Ra2, so that finite values of mRin the N→ ∞,a→0 continuum limit correspond to αR→0. This condition implies that all continuum limits converge to the critical curve of the theory, as will be discussed in detail in the subsequent Subsection 2.3. The observable parameter αRcan be defined as the position of the single pole of this momentum-space two-point function, when it is written in the Minkowski spacetime by means of a Wick rotation. The ratios of high-dimensional integrals that define the expectation values can be calculated efficiently and with good precision by stochastic methods known as Monte Carlo simulations, or stochastic simulations. Since there are no known exact solutions of these non-linear theories, we will have to use these numerical methods in order to calculate physical quantities. 2.2 Stability Constraints Examining the action SN[ϕ] given in Equation (4) we now observe that, for any finite value of N, in order for the integrals over each degree of freedom ϕ(~n) to exist and be finite, and hence for the Euclidean ensemble defined in Equations (7) and (21) above to be welldefined, we need to have one of the following two conditions hold: either λ > 0 with any real value of α, or λ= 0 with α > 0. This corresponds to the condition that SN[ϕ] must have a lower bound. 8 In any case, the condition λ < 0 implies that the integrals do not exist, because in this case SN[ϕ] has no lower bound, as one can see by considering arbitrarily large constant field configurations, and therefore the factor exp(−SN[ϕ]) in the integrands diverges very fast to +∞at the two ends of the integration interval, that is, both for ϕ(~n)→ −∞ and for ϕ(~n)→ ∞, so that the integrals do not exist. This is true on each finite N-lattice, and therefore it is also true in any N→ ∞ continuum limit. In terms of the Minkowski theory, this condition is translated as the fact that the Hamiltonian of the theory has no lower bound, and that therefore there is no ground state of the quantum theory. Note that this argument implies that for λ < 0 neither the classical theory nor the quantum theory exist, for any and all finite values of N, so that this limitation on the possible values of λis not only a consequence of the N→ ∞ continuum limit. If the limitation is violated, then the theory fails to exist on each and every one of the finite-N lattices, all the way to the N→ ∞ continuum limit. This fundamental limitation on the values of λalso determines the behavior of the perturbative expansion in this theory. The definition of the observables of the theory given in Equation (7) implies that, given any observable ON, and so long as the theory exists at all, its expectation value is a function of the real parameters αand λof the theory. If there is a perturbative series for this observable, developed around the exactly solvable case of the free theory for λ= 0, then it must be given by hONi(α, λ) = ∞ X n=0 an(α)λn,(24) which is a real power series on λ, developed around the reference point λ= 0. The possible usefulness of this series comes from the fact that in principle the coefficients an(α) can be calculated exactly by analytical means, since they are to be calculated for λ= 0, and therefore are within the confines of the free theory, where all observables can be calculated exactly, and where every such calculation reduces to the calculation of multiple Gaussian integrals. However, ultimately this series turns out not to be useful, because it can be shown that the series is divergent at all points, except for its reference point λ= 0. A simple proof of this fact can be found in Appendix B. 2.3 Fundamental Properties We must now describe the phenomenology of the lattice theory, in particular with respect to the mechanism of the continuum limit and to the mechanism of symmetry breaking, which are related to each other. Here we must point out that the actions shown in Equations (4) and (6) are invariant by an O(1) discrete reflection symmetry, in which we flip the signs of the fields. Both mechanisms are also closely related to the critical behavior of the lattice theory in the N→ ∞ limit, if we interpret it as a statistical model of a set of random variables ϕ(~n). The other crucially important property of the theory is that of the so-called triviality, which we will also describe in the subsequent Subsection 2.4. The critical diagram of the λϕ4theory can be found in Figure 1. This is a qualitative diagram in the space of the parameters of the theory, in our case here the (α, λ) parameter plane. The qualitative aspects of the diagram are the same for any dimension d≥3, including the important case d= 4. Some aspects of the diagram are over-emphasized for visualization purposes. The alternative parameters rand θ, that will be used instead of αand λin the numerical work, are defined to be such that α=−rcos(θ),(25) λ=rsin(θ).(26) 9 4 Description of the Program Here we will give a general description of the numerical programs used to generate the data shown in the graphs presented in the subsequent Section 5. There are several programs involved, and the whole set is freely available on the Web, where it is being distributed under the GNU GPL. A compressed tar file with all the programs, auxiliary files and documentation can be easily and freely downloaded from Zenodo [4]. The programs have been developed and tested only of the Linux operating system, but it should not be too difficult to port then to other Unix-like operating systems. Overall there are about 7000 lines of Fortran code in a little more than 150 modules, and a few C modules for interface with the Linux system calls. The contents of the tar file should work right out of the box in any Linux system that has the necessary development software installed, such as the GNU Fortran compiler gfortran, the GNU C compiler gcc and the make utility. The programs run with standard double-precision (64-bit) real variables. The source code is extensively documented internally by comment lines, and there is a README file with the basic instructions for compiling and running the programs. The stochastic simulations are worked out in two phases, in the first of which a randomly chosen initial configuration of the field is relaxed to a typical one for the set of free parameters in use, and in the second of which statistical samplings are collected. There are two main Monte Carlo programs, one called Main relaxer for the relaxation phase and one called Main sampler for the sampling phase. In the relaxation phase we also run a couple of feedback mechanisms in order to adjust two technical parameters, the range of the random walk in the Metropolis algorithm and the number of clusters in the Wolff cluster algorithm, in order to ensure that the acceptance rates of both the Metropolis updates and the Wolff cluster updates are kept as close as possible to 50%, in order to maximize the efficiency of the process of diffusion of the Markovian chain through the space of configurations. This works well almost always, the exception being deep in the broken-symmetric phase, where the Wolff clusters span essentially the whole lattice, so that the Wolff cluster acceptance rate approaches 100% even with the construction of only a single cluster. There are a few additional processing phases after the statistical sampling phase is finished. The additional programs are meant to do basic statistical analysis of the results, such as the calculation of the final averages and of their standard statistical errors, which is what the program Main aversig does, and then to do the data analysis, with a program called Main propres for the results relating to the momentum-space propagator, including the fits in momentum space, and another program called Main lobsres for the results relating to the local observables. In these last two phases equivalent data is consolidated, in order to improve the statistical quality of the results. In the program Main propres data for different values of ~κ that result in the same value of the eigenvalue −ρ2(~κ) of the Laplacian operator are consolidated. In the program Main lobsres geometrically equivalent data are consolidated over sites, links and plaquette diagonals, as allowed by the intrinsic symmetries of the problem. This type of consolidation is possible, and further decreases the statistical errors in the data, because these measurements are related by symmetry transformations under which the theory is invariant, and because they are, by and large, statistically independent from one another. In the case of the measurements in momentum space the various measurements which are consolidated are equivalent because the propagator can depend on the modes ~κ only through the eigenvalues −ρ2(~κ) of the Laplacian, and not directly on the values of the individual integer coordinates κµ. In this case the statistical independence of the various measurements is related to the very fact that the theory is trivial, and hence does not couple 16 different lattice modes, as we can verify a posteriori. In the case of the measurements associated with the local observables, the various measurements which are consolidated are equivalent because they are related to each other by the discrete translation and rotation invariances of the lattice. In this case they are almost completely statistically independent because there is very little correlation between the fields at different sites. In fact, it is possible to show that in the continuum limit the fields at two different sites become completely uncorrelated from one another. A simple proof of this fact, based on the physical requirements on the two-point Green’s function that are necessary for any well-behaved quantum field theory in the continuum limit, can be found in Appendix A. The program that runs the longest is Main sampler, specially if the lattice is large and high-precision statistics are required. The numerical work presented in this paper involved approximately 400,000 hours of CPU time on a single 4 GHz processor. This program is prepared to run in parallel on many CPUs or cores of a multiprocessing machine, as well as on many nodes of a parallel machine, such as a cluster of processing nodes. Due to the parallelization the actual wall-clock time for producing the data was reduced to approximately 1500 hours, or about 2 months. Each instance of the program on a given processor writes out sampling data for each one of the observables, with one sample per line in the output data files. All this is done within a well-structured standardized sub-directory tree. There are control files that record the number of processors involved in a particular run and the number of sub-averages already produced for that run, as well as various log files. Once a particular run is done, all the output files, for each one of the various observables, one from each processor, can be concatenated together in a single consolidated output data file, including the contributions from all processors, for that observable. Included in the source code tar file there is a shell script called collect parallel data that does this. The Monte Carlo program uses two different algorithms in order to update the field configurations and create a Markovian chain of configurations. The direction of the field values, which in our case here is just their orientations, as given by their signs, is updated by the Wolff cluster algorithm [5], which is the most effective one for this purpose, since it is numerically efficient and essentially eliminates any troublesome auto-correlations along the Markovian chain. In order to update the magnitude of the fields we use the Metropolis algorithm [6], with the lattice set in what is called a checkerboarding configuration, in which we separate the sites into two groups of mutually independent sites, the odd sites and the even sites, as defined by the parity of the sum of all their dinteger coordinates nµ. In this way we have that every next-neighbor of an even site is an odd site, and vice-versa. This allows us to update one group all at once, while keeping the other group fixed, and hence to avoid giving origin to unwanted correlations within that group, along the Markovian chain. Due to this, and given that we use periodic boundary conditions on the lattice, we are able to use only even values of N, starting at N= 2, because otherwise the separation of the sites into two mutually non-interacting groups would be impossible. Monte Carlo programs can be parallelized in a trivial way, since it is enough to have each independent processor generate independent sub-averages towards a global total average. Each processor can do this in a completely independent fashion, without depending on other processors at all, and will produce statistically independent results so long as the process at each processor uses a different value of the random seed. The seed for the random number generator will be chosen and periodically changed, from each sub-average to the next, using the true random number generator /dev/urandom of the Linux operating system, that does this by collecting entropy during the operation of the hardware, and thus keeping and using a cache of entropy. The use of this native Linux random number generator is probably the 17 greatest difficulty in porting the programs to other operating systems, in which this system facility may not be available. The implementation in the program of the Fourier transform and its inverse presents particular challenges. This transform is represented by a very large double-precision real matrix, of dimensions Nd×Nd, with elements involving either exponential functions or sines and cosines. Since the Fourier transformation is executed many times, the repeated calculation of these matrix elements becomes very expensive in terms of processing time. On the other hand, only a relatively small collection of different real numbers appears in these matrix elements, being repeated many times along them. The solution adopted was to calculate these real numbers only once, then store the small number of different values of these elements, and use them in the transformation matrix via an indexation system, which maps the integer coordinates of each matrix element to the corresponding double-precision real value. In this way a very large number of dynamical real numbers was traded for a much smaller number of static real numbers, plus the corresponding integers that are the addresses of those real numbers in the transformation matrix. These indexation integers can be defined as short (16-bit) integers, further reducing the memory size requirements. The idea of integer indexation is also used for several other ends in the program, such as making contiguous lists of even and odd sites, contiguous lists of the neighbors of even and odd sites, contiguous lists of the neighbors of arbitrary sites, and contiguous lists of sites, links and plaquette diagonals. The use of such indexation matrices makes the program both more efficient and easier to read, so long as the names of the variables involved are judiciously chosen. In this program the definition of a single sweep of the lattice, for the acquisition of a single sample, consists of three parts, first a single trial at the Metropolis update of every single even site, then a single trial at the Metropolis update of every single odd site, and finally the construction of a certain number of Wolff clusters, to ensure that every site was tested once, on average, to be flipped as part of a cluster. The samplings were organized in sub-averages of a global average, to enable us to calculate the dispersion of the sub-averages around the total average, and therefore to evaluate the standard statistical errors of the final averages. We used large sub-averages, of 25000 sweeps each, with a total of 1000 such sub-averages. This corresponds to 25 ×106sweeps in total, and to that same number of samples, thus corresponding to a nominal statistical error level of 1 in 5000, or 0.02%. It should be noted here that the Main sampler program makes rather extensive use of disk space. Even moderately large lattices can easily produce more than 1 TB of raw data on large statistics runs. When concatenating the various output data files from individual processors into a single global output data file, this disk space requirement will momentarily double. After the calculation of the global averages and of their standard statistical errors by the program Main aversig, in the statistical analysis phase, the raw data can be deleted, which will radically reduce the disk space requirements for permanent storage of the final results and their standard errors. 5 Numerical Results The numerical results presented here can be classified in two basic types. First we have the results in momentum space, as presented in the development of the theory that was given in Section 2. Then we have the results in position space, based on the local observables defined and discussed in Section 3. The results in momentum space are all based on the fits to the propagator, from which we obtain the residue R(α, λ) of the pole of the propagator and the renormalized mass parameter αR(α, λ), according on the expression 18 presented in Equation (27). The results obtained in momentum space for the residue R(α, λ) are subsequently used in the expressions of the local observables in position space. Therefore our first task here is to document the quality of these momentum-space fits. After that, we can report on the behavior of the results in momentum space, and finally on the behavior of the results in position space. 5.1 Fits to the Propagator Given the momentum-space propagator as calculated by the stochastic programs, we need to obtain from it the two essential observables, the residue R(α, λ) of the pole and the renormalized mass parameter αR(α, λ). In the general case the propagator eg2(~κ) is defined as the subtracted two-point function in momentum space, eg2(~κ) = D|eϕ(~κ)|2E−|heϕ(~κ)i|2,(44) however, since in our case here, given that we are working on finite lattices without external sources or fixed boundary conditions, we always have for the expectation value of the field at sites hϕ(~n)i= 0 for all ~n, which implies at once that heϕ(~κ)i= 0 for all ~κ, we can in fact simply use eg2(~κ) = D|eϕ(~κ)|2E.(45) In all cases of interest here this function can be well fitted by the expression of the propagator of the free theory, for all strictly positive values of ρ2(~κ), with modified values of R(α, λ), which is just 1 in the free theory, and αR(α, λ), which is just the parameter αin the free theory. Since our objective here is to obtain the values of R(α, λ) and αR(α, λ) from the numerical data, we consider this function normalized in the form Ndeg2(~κ), and then invert it, thus obtaining the function f(~κ) in momentum space, representing the inverted and appropriately rescaled propagator, f(~κ) = 1 Ndeg2(~κ) =αR(α, λ) R(α, λ)+1 R(α, λ)ρ2(~κ) =a0(α, λ) + a1(α, λ)ρ2(~κ),(46) where a0(α, λ) and a1(α, λ) are now the two parameters of a linear fit in terms of ρ2(~κ), and where f(~κ) may also be written as f(ρ2). Once a0(α, λ) and a1(α, λ) are determined, the residue of the pole is given by R(α, λ) = 1/a1(α, λ) and the mass parameter is given by αR(α, λ) = a0(α, λ)/a1(α, λ). In addition to this, since the functions above depend only on the eigenvalues of the Laplacian ρ2(~κ), and there are several different modes ~κ that result in the same value of ρ2(~κ), we can consolidate the values of the function f(~κ) = f(ρ2) in terms of the values of ρ2(~κ), taking averages and propagating the errors appropriately. This results in a certain number nof independent values of ρ2(~κ), which we will denote by xi, for i= 1,...,n. Note that we start the counter iat one, which means that we exclude the zero-mode ~κ =~ 0, with ρ2(~κ) = 0. We therefore have the function in this notation, leaving aside the dependencies on αand λfor the time being, f(xi) = a0+a1xi.(47) The problem of finding R,αRand their standard errors is thus reduced to the problem of finding a0,a1and their standard errors. This can be done by the procedure of fitting the 19 numerical data to the linear expression given in Equation (47) above. We therefore have aχ2-fit minimization problem, with the ndata points yi= 1/[Ndeg2,i] to be compared to the fitting function fi=f(xi) at the positions xi=ρ2 i. Given that the standard errors σi associated to the data points yican vary significantly with the values of i, we will use the weighted version of the χ2function, which is defined as χ2(a0, a1) = n X i=1 a0+a1xi−yi σi2 .(48) It is important to emphasize here that we will use only the range i= 1,...,n for the data points, avoiding the value i= 0, which is the zero mode, because it gets severely entangled with the symmetry breaking process. The use of these weights can be interpreted in several ways. First, this is a way to give more influence over the overall result of the fitting process to those terms that have smaller errors and therefore are more precise. Second, this makes the sum defining χ2(a0, a1) intrinsically dimensionless, regardless of the physical units or nature of the quantities involved. Third, when we consolidate physically equivalent terms and propagate errors, this automatically uses the fact that a particular value of irefers to the average of several independent measurements, and increases its influence accordingly, since under Gaussian error propagation σ2 idecreases linearly with the number of consolidated observations, so that 1/σ2 ieffectively counts the number of measurements that were consolidated in a particular value of i. The function χ2(a0, a1) is a positive function that goes to infinity in every limit to infinity along the (a0, a1) plane, in any direction. It is a continuous and differentiable function of both a0and a1, and therefore has a single point of minimum. The position of this minimum can be obtained analytically, if we consider the two partial derivatives of χ2(a0, a1) with respect to a0and a1, ∂χ2 ∂a0 = 2 n X i=1 a0+a1xi−yi σ2 i ,(49) ∂χ2 ∂a1 = 2 n X i=1 a0+a1xi−yi σ2 i xi.(50) The condition determining the location of the extremal point is that both partial derivatives be zero at the point of minimum, which can be used to determine a0and a1, resulting in the pair of linear equations a0 n X i=1 1 σ2 i!+a1 n X i=1 xi σ2 i!= n X i=1 yi σ2 i!,(51) a0 n X i=1 xi σ2 i!+a1 n X i=1 x2 i σ2 i!= n X i=1 xiyi σ2 i!,(52) which forms quite a simple linear system of equations. This can easily be solved for a0and a1. Multiplying Equations (51) and (52) by appropriate factors and subtracting these two equations in order to eliminate a1we get for a0, a0= n X i=1 x2 i σ2 i! n X i=1 yi σ2 i!− n X i=1 xi σ2 i! n X i=1 xiyi σ2 i! n X i=1 1 σ2 i! n X i=1 x2 i σ2 i!− n X i=1 xi σ2 i!2.(53) 20 All these sums can be calculated numerically in a straightforward manner. Multiplying now the same Equations (51) and (52) by other appropriate factors and subtracting these two equations in order to eliminate a0we get for a1, a1= n X i=1 1 σ2 i! n X i=1 xiyi σ2 i!− n X i=1 xi σ2 i! n X i=1 yi σ2 i! n X i=1 1 σ2 i! n X i=1 x2 i σ2 i!−a n X i=1 xi σ2 i!2.(54) Again all these sums can be calculated in a straightforward manner. Note that the denominator is the same as in Equation (53), and that it does not depend on the data points yi. Note also that all the sums are quite simple and follow a regular pattern. There is one sum without xior yi, there are two sums with powers of xionly, one sum with only yiand one sum with the product xiyi, which we denote as Σx0y0= n X i=1 1 σ2 i ,(55) Σx1y0= n X i=1 xi σ2 i ,(56) Σx2y0= n X i=1 x2 i σ2 i ,(57) Σx0y1= n X i=1 yi σ2 i ,(58) Σx1y1= n X i=1 xiyi σ2 i .(59) In terms of these quantities our results can be written as a0=Σx2y0Σx0y1−Σx1y0Σx1y1 Σx0y0Σx2y0−Σ2 x1y0 ,(60) a1=Σx0y0Σx1y1−Σx1y0Σx0y1 Σx0y0Σx2y0−Σ2 x1y0 .(61) There are two types of errors involved in the fitting process. On the one hand there is the overall fitting error, which describes how well the analytical expression used fits the data. On the other hand there is the uncertainty in the values of a0and a1due to the numerical errors σiof yi. These last ones can be calculated by straightforward propagation of errors. In order to do this we need to calculate the partial derivatives of a0and a1with respect to yj. Since yidoes not appear in the denominator of our results in Equations (53) and (54), this calculation is quite simple, and we get ∂a0 ∂yj =1 σ2 j Σx2y0−xjΣx1y0 Σx0y0Σx2y0−Σ2 x1y0 ,(62) ∂a1 ∂yj =1 σ2 j xjΣx0y0−Σx1y0 Σx0y0Σx2y0−Σ2 x1y0 .(63) Using the standard formula for the propagation of errors we now get, after some rather remarkable cancellations and simplifications 21 Global Fitting Errors ∆S/S d = 3 d= 4 d= 5 N= 2 3.1389 ×10−43.0615 ×10−42.2366 ×10−4 N= 4 2.1905 ×10−40.9212 ×10−40.4239 ×10−4 N= 6 2.0070 ×10−40.6554 ×10−4 N= 8 2.0195 ×10−40.6779 ×10−4 Table 1: Average global fitting errors. σa0=sΣx2y0 Σx0y0Σx2y0−Σ2 x1y0 ,(64) σa1=sΣx0y0 Σx0y0Σx2y0−Σ2 x1y0 .(65) Therefore, after some lengthy algebraic work we get simple formulas for both the coefficients and their errors, always with the same denominator. The other type of error, relating to the fit as a whole, can be estimated by comparing areas. If we interpolate the discrete functions fiand yiby continuous functions over a range of values of ρ2, we can then consider comparing the two areas under the curves of the graphs of these functions, one for the fitting function and another for the data. The area between these two curves, compared to the area under one of them, will tell us how good the fit is. In our discrete environment here what we can do is to look at the differences between the fitting function fiand the data yi, over the range i= 1,...,n, ∆i=fi−yi,(66) where the fitting function fiis given in Equation (47). Since we do not want variations with opposite signs to cancel each other, we consider the sum of the squares of these differences, which has the dimensions of squared area, (∆S)2= n X i=1 (fi−yi)2.(67) Note that this is essentially the definition of the χ2function, so that we will actually use the usual weighted definition of this function, for the same reasons as before, (∆S)2= n X i=1 fi−yi σi2 ,(68) which in principle should be divided by the normalization factor n X i=1 1 σ2 i .(69) However, since we are going to compare this quantity to another average of the same type, with the same normalization, we may ignore this normalization factor for now. As a reference value for this quantity we compare it to the weighted squared area S2under the fitting function, defined in the same way, 22 0 4 8 12 16 0 4 8 12 16 0030 deg. 0060 deg. 0120 deg. 0030 deg. 0060 deg. 0120 deg. Fits of the Inverted Propagator f ρ2 θ= 30o θ= 60o θ= 120o θ= 30o θ= 60o θ= 120o Figure 2: Examples of the fits to the inverted propagator for d= 4, N= 8, r= 1.0, and the angles θ= 30o,θ= 60oand θ= 120o. The data are represented by the symbols, and the fits are represented by the lines. S2= n X i=1 fi σi2 ,(70) which should also be divided by the normalization factor in Equation (69). We thus get the ratio representing the quality of the fit, ∆S S= v u u t n X i=1 ∆i σi2 v u u t n X i=1 fi σi2.(71) This number, on a scale given by the interval [0,1], will give us an idea of how good the fit is, that is, of how well the data is represented by the function used for the fit. The value zero indicates a perfect fit. It is to be expected that this quantity cannot be significantly smaller than the relative statistical errors which are intrinsic to the number of samplings used in the statistics, in our case here 2 ×10−4, possibly divided by the square root of the number of consolidated entries in each case. In Table 1 one can see the average value of ∆S/S, calculated over a significant part of the (C, πe) parameter plane, for each dimension dand for each lattice size N. The global fitting errors are smaller for the larger lattices because in these cases we have a larger number of equivalent data that were consolidated 23 0 30 60 90 120 150 180 0.0 1.0 2.0 3.0 4.0 5.0 r=0000.1 r=0000.2 r=0000.5 r=0001.0 r=0002.0 r=0005.0 Measurements of αRin Momentum Space αR θ r= 0.1 r= 0.2 r= 0.5 r= 1.0 r= 2.0 r= 5.0 Figure 3: An example of the momentum-space propagator, for the case d= 4 and N= 8, measured over a significant part of the parameter plane. together. As one can see, these overall fitting errors are essentially compatible with the expected statistical fluctuations of the runs, so that the fits are essentially perfect within the statistical expectations. In Figure 2 we can see three examples of the fit to the inverted propagator in a few typical cases, with d= 4, N= 8, r= 1.0 and three values of θ, with θ= 30oin the brokensymmetric phase, θ= 60onear the phase transition, and θ= 120oin the symmetric phase. In this graph the data points are represented by the symbols and the linear fits are represented by the lines. The behavior of the fitting function and of the data, as well as the quality of the fit, are essentially the same in all other cases. This graph is meant only at giving an illustrative visual idea of the quality of the fits. The fits shown were calculated numerically by the graphics package, instead of by the exact analytical approach described above, and used to construct Table 1. The correlation coefficient of the fit returned by the graphics package was always essentially equal to 1.0, indicating an excellent fit. As one can see, the data error bars are quite small, barely visible in the scale of the graph. 5.2 Results in Momentum Space The results in momentum space were obtained by scanning the parameter space of the theory, with the use of the coordinates (r, θ), and then using the techniques described in Subsection 5.1 in order to calculate Rand αR. In Figure 3 we can see the results for the dimensionless renormalized mass parameter αRobtained from the momentum-space propagator, over a significant part of the parameter plane, for the case d= 4, N= 8, with r= 0.1,...,5.0. For each curve shown the θ→0 limit to the left side is exactly equal to twice the value of the θ→πlimit to the right side, which corresponds to the free theory, 24 0 30 60 90 120 150 180 0.0 0.5 1.0 1.5 2.0 N=002 N=004 N=006 N=008 Illustration of the Continuum Limit αR θ N= 2 N= 4 N= 6 N= 8 Figure 4: An example of the initial part of the sequence of lattices that constitute the continuum limit, for the case d= 4 and r= 1.0. and is itself given by the value of r. All the curves emanate from a point close to the θaxis, which is an approximation of the position of the critical point for these values of dand N. This illustrates how we have αR 0 when we increase Nand approach the critical curve in the parameter plane. In Figure 4 we can see an illustration of the first few steps towards the continuum limit, for the case d= 4 and r= 1.0, using the range N= 2,...,8. Again we see that we have αR 0 when we approach the critical point, including when we take the continuum limit. Note that the value of αRincreases to both sides from the position of the critical point. Therefore, although the position of the minimum of αRcan be used as the best evaluation of the position of the critical point, for given values of dand N,αRis not really an order parameter in the usual sense. As one can see in Figure 4, the continuum limit proceeds very fast with N. This is due to the fact that the proximity to the continuum limit, in terms of the critical behavior of the lattice interpreted as a statistical system, is approximately proportional, not to N, but to the total number Ndof sites in the lattice. The continuum limit is indeed tied up with the critical behavior of the lattice system. The behavior of the order parameter σ2 0defined in Equation (20) can be seen in Figure 5, for the case d= 4 and r= 1.0. In the continuum limit the order parameter tends to zero in the symmetric phase, to the right of the critical point, and remains different from zero in the broken-symmetric phase, to the left of the critical point. The curves σ2 0(θ) shown diverge to infinity as −α/λ = cot(θ) when we make θ→0, independently of the values of Nand r. The value of these curves at the smallest value of θthat was used, which was 5o, is approximately given by cot(π/36) ≃11.43, also independently of the values of Nand r. As one can see, the finite-lattice curves approach a limiting curve very fast. Despite 25 Interestingly, these are exactly the two momenta that are relevant for the role that the scalar field theories play in the standard model of particle physics. It is very likely that the results shown here can be extended do the family of scalar field theories with non-linear terms given by λϕ2p, with integer p > 2, and possibly to any stable potential V(ϕ). This exploration led us to introduce the various local observables that can be used for the determination of the renormalized mass parameter αR, as given in Equations (38) through (42). These observables were revealed to be precise tools for the determination of this parameter, and ultimately for the determination of the renormalized mass. It is clear that these local observables for the renormalized mass parameter αRcan be very useful instruments. In fact, in any case in which we have any reason to introduce into the theory an external source that is dependent on position, and which is more than just an infinitesimal one meant at the taking of derivatives, they constitute the only reasonable way to measure the renormalized mass. While only two of the five local observables defined here work with complete precision, this can be attributed to the symmetry breaking mechanism, and to the fact that in our simulations here the random drift of the direction of symmetry breaking makes unavailable to us the observables needed for the necessary subtractions of the other three local observables. It is to be expected that when an external source is present and the subtracted observables are used, at least four of the five observables defined here, those given in Equations (39) through (42), would be usable with the same high level of accuracy and precision. This would, of course, require the realization of whole new set of simulations, for example with the inclusion in the action of a constant external source. Data Availability Statement Data sharing not applicable to this article as no experimental datasets were generated or analyzed during the current study. Nevertheless, since the numerical simulation data used in this paper is rather extensive, and took almost half a million hours of CPU time to produce, the whole set of statistically processed data, as well as a fairly complete set of vector-formatted graphs, are available for download from Zenodo [7]. Conflict of Interest Statement The authors hereby certify that there are no actual or potential conflicts of interest of any of the authors in relation to this article. 32 A No Correlations in the Continuum Limit Let us show that the dimensionless lattice fields at any two different lattice sites become completely uncorrelated from each other in the continuum limit. This will be based on the general properties that the dimensionfull two-point function is expected to have in the continuum limit, in any well-defined quantum field theory. Let us first consider the two-point function of the dimensionfull field in the continuum limit. Given our current circumstances here, in which we have hφiN= 0, for simplicity we will use the non-subtracted expectation values of the fields, but the same argument can be repeated for the subtracted ones when it is the case to do so. In momentum space it is given by an expression such as e φ(~p ) 2=R(~p ) |~p |2+m2 R ,(76) where ~p is the four-momentum, while in position space we have Dφ(~x1)φ(~x2)E=G2(|~x1−~x2|),(77) where G2is a two-point Green’s function. This position-space two-point Green’s function is related to the two-point correlation function between the corresponding dimensionless lattice fields, as we will discuss momentarily. The function G2above has the general property that in the continuum limit it diverges to infinity when ~x1→~x2, and has finite and non-zero values for all ~x16=~x2. It is to be expected that this would be the case for any well-defined quantum field theory, since G2is a Green’s function. We can provide some examples of this using the free theory, where everything can be calculated in detail by analytic means. On finite lattices the squared dispersion of values σ2(N) of the field at any arbitrarily given site, which is defined in Equations (73) and (74), is a finite number, of course. In addition to this, in the cases d≥3 the N→ ∞ limit of this quantity exists and is finite, of the order of 1, in fact. The values in the N→ ∞ limit can be obtained with good precision by adding up progressively larger finite sums and extrapolating to N→ ∞ limit, and are given approximately by the numbers shown in Equation (75). The Green’s functions G2(|~x1−~x2|) in the continuum limit are finite and non-zero away from the origin |~x1−~x2|= 0. If r=|~x1−~x2|is the distance between the points ~x1and ~x2we have the exact results of the free theory, within a very large lattice box, d= 3 : G2(r) = 1 2 1 (2πr)e−mRr, d= 4 : G2(r) = 1 2π mR 2πr K1(mRr), d= 5 : G2(r) = π1 + mRr (2πr)3e−mRr, (78) where Knis the modified cylindrical Bessel function of the second kind and order n. The limits of these functions in the case in which the mass mRgoes to zero are given by an inverse-power decay with the distance, which depends on the spacetime dimension d, 33 d= 3 : lim mR→0G2(r) = 1 2 1 (2πr), d= 4 : lim mR→0G2(r) = 1 (2πr)2, d= 5 : lim mR→0G2(r) = π1 (2πr)3. (79) For finite and non-zero values of mRand large values of r, with r≫1/mR, the asymptotic behavior of these functions is given by an exponential decay with the distance, d= 3 : G2(r)≃1 2 1 (2πr)e−mRr, d= 4 : G2(r)≃1 2 √mR (2πr)3/2e−mRr, d= 5 : G2(r)≃1 2 mR (2πr)2e−mRr. (80) As one can see, all these functions diverge as an inverse power of rfor r→0. Back to the general case, so long as the Green’s function diverges at the origin, and since the relation between the dimensionless lattice field ϕand the dimensionfull field φis given by ϕ=a(d−2)/2φ, it follows that the corresponding relation between the lattice correlation function and the two-point function is given by Dϕ(~n1)ϕ(~n2)E=Dφ(~x1)φ(~x2)Ead−2 =G2(r)ad−2.(81) Since we know that for d≥3 the lattice correlation function on the left-hand side has a finite and non-zero value at ~n1=~n2in the continuum limit, which is the result of the product of the factor ad−2→0 with the divergence of the dimensionfull two-point Green’s function G2(r) at the origin, it follows that at any other point, where the two-point function G2(r) is finite in the continuum limit, we must have that the lattice correlation function goes to zero as ad−2, where d≥3. Therefore the dimensionless lattice fields at two different sites become completely uncorrelated to each other in the continuum limit. B Divergence of the Perturbative Series Here we will prove, by “reductio ad absurdum”, that the perturbative series of the λφ2p d scalar polynomial theory defined on the Euclidean lattice, with any even integer power 2p with p≥2, is necessarily divergent for all values of λexcept for the reference point λ0= 0 of the series, for all spacetime dimensions d≥3, and for all lattice sizes N≥2, all the way to the N→ ∞ continuum limit. What is meant here by divergence is simply the lack of convergence, in the usual sense used in the theory of analytic functions. B.1 Definition of the Theory The definition that we will adopt here for the scalar field theories is the one discussed in detail in Subsection 2.1, obtained from their realization on the Euclidean lattice, in d≥3 spacetime dimensions, for all even integer powers 2pwith p≥2, and any finite value of the lattice size N≥2. The action of the theory in terms of these parameters is given, for 34 the case p= 2, in Equation (4), and can be simply generalized to 2pinstead of 4. The expectation values of the observables ON[ϕ] are defined by the ratios of Nd-dimensional multiple integrals shown in Equation (7), again generalized to the power 2p. The continuum theory in Minkowski space is obtained in the N→ ∞ limit, when we make a Wick rotation from the resulting Euclidean space to the corresponding Minkowski space. If there is a perturbative series for this observable, developed around the exactly solvable case of the free theory for λ= 0, then it must be given by the expression shown in Equation (24), which is a real power series on λ, developed around the reference point λ= 0. It is important to recall that, as was shown before in Subsection 2.2, the theory does not exist at all for λ < 0, a fact that can be easily generalized to the case of the power 2p. B.2 Analytic Functions and Power Series Since the perturbative series shown in Equation (24) is a power series on λ, we may extend it to complex values of λ, and then use the power-series theorems that can be established in the theory of complex analytic functions in order to establish some facts about it. After this is done we may reduce the series back to the real case, in which all the facts established in the complex case continue to be equally valid in their corresponding real forms. We will use the following mathematical facts from the theory of analytic functions, which are all rather basic theorems that can be found in any text on that theory [8], and that we will quickly review here. Given a complex power series around a point z0of the complex plane, which we denote here as f(z) = ∞ X n=0 an(z−z0)n,(82) which presumably is some function of z, so long as the series is convergent, and where {an, n = 0,...,∞} is some arbitrary set of finite complex coefficients, which for the time being remain unspecified, the following set of mathematical facts holds. 1. The series is trivially convergent at the point z=z0, since in this case it reduces to a single term, f(z0) = a0. 2. If the coefficients anare such that a second point z16=z0exists in the complex plane, such that the series is convergent at z1, then it follows that the series is absolutely convergent in the strict interior of a disc in the complex plane, which is centered at z0and has radius given by ρ1=|z1−z0|. 3. Since absolute convergence implies, in particular, plain point-wise convergence, it then follows that the series is convergent in that open disc. 4. The series is also uniformly convergent within any closed set contained strictly within that open disc, and therefore can be term-wise integrated and differentiated. 5. Within that open disk the series converges to an analytic function f(z), that satisfies the Cauchy-Riemann conditions. 6. Where it converges the power series is identical to the Taylor series of the analytic function f(z), written around the reference point z0. 7. If that analytic function f(z), when maximally extended to the complex plane, has a singularity of any type at some point outside the open disk of radius ρ1, then there 35 is a maximal disc of convergence of the power series. The radius of the maximal disc of convergence is known as the convergence radius ρc, and is given by the distance from the reference point z0to the singular point of the analytic function f(z) which is closer to z0. That singularity of f(z) is therefore located at the rim of the maximal disc of convergence. 8. With the use of the facts given in items 2, 3 and 5, it can now be shown by “reductio ad absurdum” that at any point strictly outside the maximal disc of convergence the series is necessarily divergent. This is simply done, since the analytic function f(z) has a singularity at some point on the rim of the maximal disc of convergence, so that if there is a point outside that disc where the series converges, then it follows from items 2, 3 and 5 above that the series converges to an analytic function at that singular point, which is absurd. Therefore the series diverges everywhere strictly outside the maximal disc of convergence. A completely divergent series is one for which the convergence radius ρcis zero. Such a series diverges everywhere except at the reference point z0. It should be noted here that strictly outside its maximal convergence disk a power series diverges very fast. B.3 Application of the Analytic Results Let us now apply these results to our case here, with λ0=z0. Given an observable ON[ϕ], the series shown in Equation (24) is clearly convergent at λ0= 0, where the theory reduces to the free theory. Let us now assume that there is a point λ1>0 for which the series converges. Extending the series to the complex λplane we conclude that there must be a disc around λ0= 0, with non-zero radius, within which the series converges. This disc includes the segment (−λ1, λ1) of the real λaxis. Since this disc includes negative real values of λ, we therefore conclude that the series converges at points where the theory does not exist at all, which is absurd. Therefore, we conclude that λ1must be outside the maximal disc of convergence of the series, and hence the radius of convergence ρcmust be smaller than λ1, ρc< λ1.(83) Since this is true for all strictly positive real values of λ1, no matter how small, we conclude that we must have ρc= 0.(84) In other words, the perturbative series must be completely divergent, that is, divergent everywhere but at the reference point λ0= 0, for any observable ON[ϕ], for all finite values of the lattice size N, and for any spacetime dimension d≥3. Of course, this then implies that the same is true in any continuum limit, in which we necessarily have that N→ ∞. 36 References [1] I. A. Fox and I. G. Halliday, “A monte carlo study of φ4in four-dimensions,” Phys. Lett. B, vol. 159, pp. 148–150, 1985. [2] M. Aizenman, “Proof of the triviality of ϕ4 dfield theory and some mean-field features of ising models for d > 4,” Physical Review Letters, vol. 47, no. 1, 1981. [3] M. Aizenman, “Geometric analysis of φ4fields and ising models. parts i and ii,” Communications in Mathematical Physics, vol. 86, pp. 1–48, 1982. [4] J. L. deLyra, “Program for the triviality of the quartic scalar model.” DOI/Zenodo. Freely downloadable compressed tar file with Fortran source code, available at the URL https://zenodo.org/records/XXXXXXXX or at the URL https://doi.org/10.5281/zenodo.XXXXXXXX. [5] U. Wolff Phys. Rev. Lett., vol. 62, p. 361, 1989. [6] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” The Journal of Chemical Physics, vol. 21, no. 6, pp. 1087–1092, 1953. [7] J. L. deLyra, “Data and graphs for the triviality of the quartic scalar model.” DOI/Zenodo. Freely downloadable compressed tar file with all the processed data in readable text format and all the graph data in readable and plotable formats, available at the URL https://zenodo.org/records/XXXXXXXX or at the URL. [8] R. V. Churchill, Complex Variables and Applications. McGraw-Hill, second ed., 1960. 37