]>NUPHB115537115537S0550-3213(21)00234-010.1016/j.nuclphysb.2021.115537The Author(s)High Energy Physics – PhenomenologyFig. 1The Feynman rules for the sigma model. The solid line and dashed line represents the π and σ propagators, respectively.Fig. 1Fig. 2The tree level diagram for the Hc and Ha operator insertions. The crossed circle represents a generic π-bilinear operator.Fig. 2Fig. 3The one-loop tadpole diagram for the Hc and Ha operator insertions. This diagram is leading in N because the two factors 1N from the π-σ coupling cancels the factor N from the loop.Fig. 3Fig. 4The Feynman diagram for the one-loop electron self energy. The double lines are dressed electron propagators and the crosses are counter terms. They contribute to the energy shift δEN. Notice that the fourth diagram corresponds to the counter term Z3−12FμνFμν that couples the background and radiative photon fields.Fig. 4Fig. 5Dynamical response of the scalar Hamiltonian HS in the presence of the fermion Ψ, generating a contribution to the fermion mass The dotted line represents the dynamical Higgs particles h and the crossed circle denotes the scalar Hamiltonian linear in h. The coupling g between the Higgs field and the fermion is proportional to fermion mass.Fig. 5Fig. 6Quantum anomalous energy contribution to the nucleon mass seen as a dynamical response of the anomalous scalar field in the presence of the nucleon. The dotted line represents the intermediate scalar particles with couplings gNNs proportional to the nucleon mass, which is dominated by a single Higgs particle in the Higgs mechanism.Fig. 6Fig. 7The mψ¯ψ insertion at internal line of the first diagram in Fig. 4. It corresponds to the mass derivative of the internal electron propagator. Notice that all the electron lines are dressed.Fig. 7Fig. 8The mψ¯ψ insertion at external line of the first diagram in Fig. 4. It corresponds to the mass derivative of the external Dirac wave function. For simplicity we only draw the case in which the internal electron is forward moving and the derivative is taken at the out-going wave function. The other cases are similar. The first diagram corresponds to the first term at right hand side of equality in Eq. (B.6), while the other two diagrams with back-moving lines combine to produce the second term in Eq. (B.6).Fig. 8Table 1Mass decomposition in various schemes: While the naive Hamiltonian Hc gives scheme-dependent results, the anomaly contribution is always present and scheme-independent. The total mass is of course fixed.Table 1SchemeHcHa ≡ HSH

k2≤ΛUV2m2m2m

latticem2m2m

|k1| ≤ ΛUVmm2m

DRmm2m

k42λ2+k12≤ΛUV2λm1+λm2m

Scale symmetry breaking, quantum anomalous energy and proton mass decompositionXiangdongJiabxji@umd.eduYizhuangLiucd⁎yizhuang.liu@uj.edu.plAndreasSchäferdAndreas.Schaefer@physik.uni-regensburg.deaCenter for Nuclear Femtography, SURA, 1201 New York Ave. NW, Washington, DC 20005, USACenter for Nuclear FemtographySURA1201 New York Ave. NWWashingtonDC20005USACenter for Nuclear Femtography, SURA, 1201 New York Ave. NW, Washington, DC 20005, USAbDepartment of Physics, University of Maryland, College Park, MD 20742, United States of AmericaDepartment of PhysicsUniversity of MarylandCollege ParkMD20742United States of AmericaDepartment of Physics, University of Maryland, College Park, MD 20742cInstitut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, GermanyInstitut für Theoretische PhysikUniversität RegensburgRegensburgD-93040GermanyInstitut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, GermanydInstitute of Theoretical Physics, Jagiellonian University, 30-348 Kraków, PolandInstitute of Theoretical PhysicsJagiellonian UniversityKraków30-348PolandInstitute of Theoretical Physics, Jagiellonian University, 30-348 Kraków, Poland⁎Corresponding author.Editor: Hong-Jian HeAbstractWe study the anomalous scale symmetry breaking effects on the proton mass in QCD due to quantum fluctuations at ultraviolet scales. We confirm that a novel contribution naturally arises as a part of the proton mass, which we call the quantum anomalous energy (QAE). We discuss the QAE origins in both lattice and dimensional regularizations and demonstrate its role as a scheme-and-scale independent component in the mass decomposition. We further argue that QAE role in the proton mass resembles a dynamical Higgs mechanism, in which the anomalous scale symmetry breaking field generates mass scales through its vacuum condensate, as well as its static and dynamical responses to the valence quarks. We demonstrate some of our points in two simpler but closely related quantum field theories, namely the 1+1 dimensional non-linear sigma model in which QAE is non-perturbative and scheme-independent, and QED where the anomalous energy effect is perturbative calculable.1IntroductionIn relativistic quantum field theories, a bound state is an irreducible representation of Poincaré group characterized by its mass and spin, making it a natural question to ask if and how these two quantities can meaningfully be decomposed into different contributions. In non-relativistic quantum mechanics the mass is simply the sum of the masses of the individual components minus the binding energy which can be unambiguously decomposed into familiar kinematic energy and potential energy. Furthermore, in the macroscopic world, the binding energy effects on mass are entirely negligible and mass is just the sum of individual parts. In the opposite limit showcased in the electroweak theory, the masses of elementary particles arise entirely from their interactions with the Higgs potential, which acquires a vacuum condensate after the well-known spontaneous gauge symmetry breaking, or the Higgs mechanism [1].In quantum chromodynamics (QCD), the fundamental theory of strong interactions, the decomposition of proton mass and spin has been quite interesting for many years. While consensus seems finally to be reached with respect to spin, some controversy seems still ongoing for its mass. The original work by one of us [2,3] studied the decomposition of proton mass based on the QCD energy-momentum tensor (EMT) Tμν which was split into trace and traceless parts. It was found in Refs. [2,3] that 14 of the proton mass can be attributed to the trace, a statement which reminds of the virial theorem for non-relativistic systems. Moreover, a mass sum rule that contains quark and gluon kinematic energy terms, a quark mass term and an anomalous term has been proposed. These and related results inspired new variants of the mass sum rule [4–7] and interesting discussions on the subject in the literature [8–10]. In our view, the controversy is not about mathematical consistency of the different suggestions but on their phenomenological relevance. The latter debate could be ended if, e.g. the quantum anomalous energy (QAE) contribution could be unambiguously extracted from heavy-quarkonium electroproduction off a nucleon at threshold as is tried at Jefferson Lab and is on the physics program for the future Electron-Ion Collider [11–13]. While these efforts are ongoing an alternative is provided by lattice QCD calculations. The masses for nucleon and other hadrons have been numerically calculated to such high accuracy [14–17] that the configurations have to contain all major effects contributing to it. Because lattice QCD evidence has thus reached comparable reliability as direct experiments it also suffices to determine the value for QAE in the nucleon on the lattice, similar to what was done in [18–21].Despite this progress in understanding the mass structure of a proton, open questions remains, in particular concerning the anomalous term, see e.g. Ref. [7]. Let us mention that the connection between the trace anomaly and the nucleon mass as well as the related low-energy theorems/effective theories have already been studied in the 70's [22,23] and that this discussion thus approaches its 50th anniversary. The identification of a piece of QCD Hamiltonian as anomalous contribution in [2] was based on these pioneering studies. In a recent paper [24], the authors have not only investigated the anomaly contribution to the QCD energy and argued that the quantum anomalous energy (QAE) is a meaningful part of the proton mass decomposition but it has been suggested that the QAE mechanisms in QCD are very similar to those of Spontaneous Symmetry Breaking (SSB). Results for QED (quantum electrodynamics), QCD and large-N 1+1 non-linear sigma model were presented in support of this analogy. In the current paper, we provide a more detailed arguments and derivations of [24].In section 2, we will first provide a general review to mass generation and dimensional transmutation in QCD-like theories. We emphasize that the bare coupling constant that leads to the continuum limit is a function of the ratio between the ultraviolet (UV) cutoff and the physical scales. We show how renormalization group equation (RGE) and trace anomaly naturally appear as result of dimensional transmutation.In section 3, based on the principles given in section 2 we derive the mass sum rule by studying the corresponding Ward identities in detail, paying special attention to regularization and renormalization. We first work in lattice QCD, in which the derivation was first made in Refs. [4,25], papers which, unfortunately, seem to have been largely forgotten for many years. We then generalize the derivation to continuum regularization such as dimensional regularization. We show that while the “naive” Hamiltonian 12(E→2+B→2) is scheme dependent, the decomposition into traceless and trace part is scheme independent. We identity the operators corresponding to the traceless and trace parts in different schemes and argue that it is the traceless part that can be naturally interpreted as the quark-gluon kinematic energy. We also comment on the renormalization of QCD EMT and compare the results in [6,7] to the more familiar renormalization properties of twist-2 quark and gluon operators [1].In section 4, we explicitly demonstrate the results of the previous section in 1+1 dimensional O(N) non-linear sigma model in the solvable large N limit that exhibits asymptotically freedom, dimensional transmutation and dynamical mass generation. We show that the quantum anomalous energy contributes half of the mass of the pion-like bosons in such theories, consistent with the “virial theorem” stated above. We study the mass-sum rule in different schemes and show that a proper mass-sum rule does requires a scheme and scale independent anomalous contribution.In section 5, we study the anomalous energy contribution in QED. We show that although there is no dynamical scale generation, there does exists a QAE contribution to the electron pole mass, as well as to the binding energy of a hydrogen atom in the presence of a background field. In particular, the famous O(α5) Lamb-shift receives a trace anomaly contribution which we will calculate.In section 6, we relate the QAE contribution to the proton mass to a dynamical Higgs effect. We compare the more standard Higgs mechanism for the fermion mass generation to the mass generation in the 1+1 non-linear sigma model. We show that in both cases, the scalar part of the Hamiltonian is proportional to the Higgs field, and that mass generation can be measured either through the mass term due to the scalar vacuum condensate or through the response of the Higgs field in the presence of other fields. We then generalize to QCD and relate the QAE contribution to the Higgs-like coupling of scalar resonances to the proton. In the chiral limit and assuming the dominance of the lowest glueball, the glueball-proton coupling is proportional to the proton mass, similar to the Higgs coupling which has been tested at the LHC [26–28]. We also discuss the pion case. The results are consistent with an effective theory [29] for the lowest glueball and its coupling to pions, which is also presented here.Finally, we draw conclusions and give an outlook. Some technical details are presented in Appendices.2Review on scale generation, RGE and trace anomalyIn this section we review the mass generation and dimensional transmutation in QCD-like theories. We emphasize that in order to take the continuum limit, the coupling constant of a cutoff theory must depends on the UV cutoff and the physical mass scales of the theory in continuum non-trivially. We show how the trace-anomaly naturally arises as a consequence of this scale-dependency by providing a path-integral based derivation of the trace anomaly Tμμ and the renormalization group equation (RGE) for two-point functions. The advantage of this derivation is that it does not require the Lorentz invariance in prior and can be applied to lattice-like regularization as well. In section 3 the same method will be used to investigate the mass-sum rules and to derive the QAE.2.1Pure SU(3)To simplify the discussion let us first consider pure SU(3) Yang-Mills (YM) theory. At classical level the theory is massless and has conformal symmetry. As such, it has no mass scale. Any bound state mass must be either zero or infinity, making this theory not very interesting.However, the conformal symmetry of SU(3) YM theory is broken at the quantum level, where the theory must be defined as limit of a theory with an UV cutoff, such as lattice gauge theory in which the cutoff is given by lattice spacing a. In the cutoff theory, the correlation length ζ (assumed to be finite) for a gauge-invariant correlation function 〈O(x)O(0)〉c=〈O(x)O(0)〉−〈O〉2∼e−x/ζ is a dimensionless function of the bare coupling constant g0, ζ/a=f(g0). The operator O can be chosen to be F2 or some other gauge-invariant operator. One expects ζ to be the inverse of the glueball mass M, ζ=1/M in the continuum limit a→0. Therefore, one has to tune g0 such that f(g0)=1/Ma goes to infinity as a→0. This implicitly introduces a physical scale M into the problem and fixes g0=g0(1/Ma) as a function of 1/Ma. This process has been called “dimensional transmutation” [30]. In fact, one expects from perturbation theory that as a→0, g0(1/Ma) approaches 0 logarithmically (asymptotic freedom). The existence of a finite correlation length at small g0 that approaches ∞ in unit of a as g0→0 is a genuinely non-perturbative effect and one of the most important properties of the QCD vacuum.For pure YM theory, since there is only one free dimensional parameter, the mass scale M, which can be extracted from any two-point function, actually determines the full theory. All other physical scales of the theory are proportional to M. Among them there are not only masses, but also the string tension σ=cM2 that characterizes the linear q¯q confinement potential. The number c only depends on the SU(3) group. Despite this relation, we should point out that in general the confinement and spontaneous mass generation is not necessarily related. The confining phase is characterized by non-vanishing string tension or area law for Wilson-loops. But there is also the standard Higgs phase in which color magnetic charges are confined while color electric charges only got screened, characterized by area law for 't Hooft loops [31]. And both of the two phases are gapped with finite correlation lengths. Nevertheless, it is widely believed that the Higgs and confining phase are smoothly connected [32] and we will explore the similarity between dimensional transmutation and Higgs mechanism in Sec 3.1.2.2SU(3) plus fermionsWhen one adds fermions with Nf flavors, which we assume to be massless, the classical theory has the UA(1) symmetry that is known to be broken by the famous UA(1) anomaly, as well as the SU(Nf)L×SU(Nf)R chiral symmetry for Nf>1 that gets spontaneously broken down to SU(Nf)V by the chiral condensate. In the strictly massless case, the chiral condensate is proportional to the cubic power of the scale M∼ΛQCD introduced before. The same mass scale M is thus related to three different phenomena: mass scale generation, color-confinement and spontaneous chiral symmetry breaking. The contribution from instantons explains the chiral symmetry breaking quite well in the instanton-liquid model [33,34], and has been supported by lattice results [35]. While instantons might account for a large portion of the hadron mass [10], it is known [36] that they can not explain confinement. On the other hand, the confinement has been used the main physical mechanism to generate mass for hadrons in the MIT bag model [37].When one includes non-zero fermion masses, say for degenerate u and d quarks, there are two parameters of the theory: the bare coupling constant g0 and the bare quark mass m0. One must fix them by two physical mass scales, namely M1 and M2=mπ. One then has the dimensional transmutation relations m0=mπf(M1a,mπa) and g0=g0(M1a,mπa).Due to the presence of mass scales, the naive scaling invariance of the classical theory is broken in the quantum version of the theory. Nevertheless, one still has the extended scale invariance under simultaneous rescaling of space-time and physical mass scales. The Ward identity of this invariance is the RGE or the trace anomaly, as we will discuss next.2.3Trace anomaly from anomalous scale symmetry breakingAfter introducing the dimensional transmutation, in this section, using the Euclidean path-integral formalism, we provide a derivation of the trace anomaly and the RGE for two-point functions. For simplicity we only consider pure-YM like theories. The methods here will be used later to derive QCD Hamiltonian responsible for the mass sum rule and the QAE. Our convention for Euclidean coordinates is (x4,x→) where x4 is the imaginary time (also in 2D case discussed later). We consider a general interpolating operator O(x4,x→) for a given hadron, which for simplicity we assume to be renormalization group (RG) invariant. We study the two-point function(2.1)G(T,p→)=〈O(T,p→)O(0,−p→)〉=∫Dϕe−1g02S[ϕ]O(T,p→)O(0,−p→)∫Dϕe−1g02S[ϕ] with some given time T. In this expression, ϕ is a spin zero field and 1g02S(ϕ) is the Euclidean action. It can be written as(2.2)S[ϕ]=∫d4xS(ϕ(x))|ΛUV, where we use the symbol |ΛUV to denote usage of an UV regulator with value ΛUV. As we have emphasized, we have g0=g0(ΛUV/M) where M is the physical scale of the theory, which we choose to be the mass M of the hadron created by O to guarantee the finiteness of the results as we take the limit ΛUV→∞. At large T, the two-point function G(T,p→)∼e−MT is controlled by the mass of the hadron. Thus we can extract the hadron mass using:(2.3)M=−limT→∞1TlnG(T,p→). In the following we only consider the rest frame, i.e. p→=0. With these definitions we can derive the RGE and trace anomaly. Let us consider the scale transformation x→x′=λx and ϕ′(x′)=λ−dϕ(x) where d is the naive mass dimension of the theory. In terms of ϕ′ and x′ the two point function becomes:(2.4)G′(T,0→)=〈O′(T,0→)O′(0,0→)〉=λ−2dO+3G(λ−1T,0), where dO is the naive mass dimension of the operator O and +3 comes from the Fourier transformation to momentum space. On the other hand, the action for ϕ′(x′) becomes(2.5)S[ϕ′]=∫d4x′S(ϕ′(x′))|λ−1ΛUV, which leads to the identity(2.6)λ−2dO+3G(λ−1T,0→)=∫Dϕ′e−g0−2(ΛUVM)∫d4x′S(ϕ′(x′))|λ−1ΛUVO′(T,0→)O′(0,0→)∫Dϕ′e−g0−2(ΛUVM)∫d4x′S(ϕ′(x′))|λ−1ΛUV. If it were not for the Λ dependence of the UV regulator, the right hand side would be the same as G(T,0→). The presence of ΛUV results in a λ dependence from the mismatch between g0(ΛUV/M) and λ−1ΛUV. In fact, in terms of ΛUV′=λ−1ΛUV, one simply has g0(ΛUV′/(λ−1M)). Thus, the equation is equivalent to(2.7)λ−2dO+3G(λ−1T,0→)=∫Dϕ′e−g0−2(ΛUV′λ−1M)∫d4x′S(ϕ′(x′))|ΛUV′O′(T,0→)O′(0,0→)∫Dϕ′e−g0−2(ΛUV′λ−1M)∫d4x′S(ϕ′(x′))|ΛUV′. Now, all λ dependence has been absorbed into g0. By taking one derivative with respect to λ and evaluating it at λ=1, the equation reads(2.8)(−2dO+3−TddT)G(T,0→)=〈2β(g0)g03∫d4xS(ϕ(x))O(T,0→)O(0,0→)〉c. Here(2.9)β(g0)=ΛUVdg0(ΛUV/M)dΛUV is the bare beta function and the 〈〉c denotes the connected part. By comparing with the Ward-identity for scale-transformation (Eq. (42) in Ref. [3])(2.10)(−2dO+3−TddT)G(T,0→)=〈∫d4xTμμ(x)O(T,0→)O(0,0→)〉c, one can identify the trace part Tμμ of the EMT as(2.11)Tμμ(x)=2β(g0)g03S(ϕ(x)). The derivation here also shows that the operator 2β(g0)g03S(ϕ(x)) must be RG invariant and scheme independent.One further notice that at large T≫M, Eq. (2.8) is dominated by linear terms in T(2.12)MTG(T,0)=T〈2β(g0)g03∫d3x→O(T,0→)S(ϕ(0,x→))O(0,0→)〉c. Thus one obtains the scale-setting relation(2.13)M=limT→∞〈∫d3x→O(T,0→)Tμμ(0,x→)O(0,0→)〉c〈O(T,0→)O(0,0→)〉=〈P→=0|∫d3x→Tμμ(x→)|P→=0〉〈P→=0|P→=0〉. Here we use |P→=0〉 to denote the lightest hadron state created by the interpolating field O.To summarize, in this section we have provided a review of the far-reaching consequences of dimensional transmutation in QCD-like theories. Starting from the path-integral representation of the theory with a generic UV cutoff, it has been shown that as a result of the scale-dependency of the bare coupling constant, a non-vanishing scalar field naturally emerges in the continuum limit and can be identified as the trace of the energy-momentum tensor. We should mention that although all the results in this section are well-known, our derivation of the trace-anomaly has the advantage that it does not require Lorentz-invariance in prior and can be applied to lattice-like regularizations as well. In the next section the same method will be adopted to study the Hamiltonian-based mass sum rules.3Quantum anomalous energy and mass sum ruleTo derive a mass sum rule, we need to examine the traditional derivation of the EMT, in particular, the Hamiltonian density T00 more carefully in the presence of a regulator. The standard way to obtain the EMT is to study the change of the Lagrangian under an infinitely small space-time transformation xμ→xμ+δxμ(x). In the presence of the cutoff, this shift could potentially change the UV cutoff and extra attention must be paid. For example, the scale transformation changes a→λa and as we have seen in the previous section, the λ dependence of a can be absorbed into g0, resulting in the trace anomaly. In this section we show that individual components of Tμν such as T00 in the cutoff scheme can also suffer from such effects, leading to an anomalous contrition to the mass sum rule.As in the previous section, let's consider the two point function G(T,0→). We would like to derive the mass sum rule as a Ward identity for the following transformation:(3.1)x4′=λx4,(3.2)ϕ′(x4′,x→)=ϕ(x4,x→). The two point function in terms of the ϕ′ field then becomes G(λ−1T,0→). On the other hand, the same two point function can be obtained from the functional integral over the action for ϕ′, which can be written in the generic form:(3.3)S[ϕ′]=1g02∫d4x′(λSτ(ϕ′(x′))+λ−1Ss(ϕ′(x′)))λ−1ΛUV,ΛUV. Here Sτ(ϕ′(x′)), Ss(ϕ′(x′)) can be roughly identified with 12F4iF4i=−12E2, 14FijFij=12B2, respectively. The symbol λ−1ΛUV,ΛUV indicates that the UV cutoff in the x4 direction has been modified to λ−1ΛUV, while in the spatial direction it remained as before. If there were no λ dependence of the cutoff, following the same logic as before, evaluating one derivative with respect to λ and taking the large T limit, one would end up with the following sum rule:(3.4)M=〈P→=0|1g02∫d3x→[Ss(ϕ(0,x→))−Sτ(ϕ(0,x→))]|P→=0〉〈P→=0|P→=0〉. This would amount to the “naive” Hamiltonian(3.5)Hc=1g02∫d3x→[Ss(ϕ(0,x→))−Sτ(ϕ(0,x→))]. However, due to the possible λ dependence of the cutoff, there can be an anomalous contribution Ha to H. In the following subsections we investigate this Ha for different regulators.3.1Lattice and cutoff regularizationIn this subsection we investigate the lattice regularization in more detail. We restrict ourselves to the Wilson action for pure YM theory and stay in the infinite volume limit. In terms of the link variables Uμ(x), the standard Wilson action reads:(3.6)S[U]=−1g02∑x∑μνTrPμν(U,x), in which we sum over the trace of the Wilson-loop Pμν(U,x) for the elementary plaquettes in the μν plane at position x. Pμν(U,x) is defined as the product of the link variables along the boundary of the plaquette. Neglecting the effects of scale transformations, on which we will focus in this section, in the limit of an infinitesimal a one gets Uμ(x)=e−iaAμ(x) and Pμν(U,x)=e−ia2Fμν(x) and thus −∫d4x14F2 for the action in the continuum.Given this information, let us construct the rescaled version of the theory, equivalent to(3.7)S[ϕ′]=1g02∫d4x′(λSτ(ϕ′(x′))+λ−1Ss(ϕ′(x′)))λ−1ΛUV,ΛUV, where the correlation length in the imaginary time direction is rescaled by a factor λ. To avoid taking a derivative with respect to the cutoff, one absorbs the λ dependence into the parameters of the action which gives(3.8)S[ϕ′]=1g02(ΛUV/M,λ)∫d4x′(λSτ(ϕ′(x′))+λ−1Ss(ϕ′(x′)))ΛUV. This expression is equivalent to expression (3.7). Our task is thus reduced to determining the λ dependence of g0.In lattice regulation, the above suggests to investigate the action Sλ(3.9)Sλ[U]=−1g02∑x(λPτ(x)+1λPs(x)), where Pτ contains the sum over the temporal plaquettes in the (x4,xi) plane and Ps over all purely spatial plaquettes. This action has already been investigated in Ref. [38]. Below we provide a brief introduction to its properties. Naively, expanding to leading order in a, it seems that Eq. (3.9) is already sufficient to produce the required rescaling in x4 direction. However, due to the presence of a hard cutoff, the loop integral of the λ dependent propagators can not be simply rescaled back by x4→λ−1x4, e.g.,(3.10)∫p2≤Λ2d4p(2π)41λp42+λ−1p→2+λ−1m2≠∫p2≤Λ2d4p(2π)41p2+m2. Therefore, additional λ dependencies get introduced by loop integrals that must be compensated by a change in g0.Non-perturbatively, this point can be argued in the following way. One notices that since all the field variables and coupling constants are dimensionless, what really characterizes the theory are the correlation lengths ζτ in temporal and ζs in spatial directions, measured in natural lattice units. Both of them are functions of g0 and λ. As g0→0, one expects that both approaches ∞, but that their ratio approaches λ:(3.11)ζτ=λf(g0,λ),(3.12)ζs=f(g0,λ). In order for the continuum limit to be just a rescaled version of the original theory, one must identify f(g0,λ)=1Ma. As a result, we conclude that g0 in Eq. (3.9) must be λ dependent,(3.13)g0=g0(Ma,λ), for the continuum limit of the theory defined by Eq. (3.9) to be given simply by rescaling. This is equivalent to stating that the physical scale remains the same, whereas the cutoff in x4 direction is rescaled, which is the viewpoint of Ref. [38]. It has been furthermore proven in that paper that the λ derivative of g0 is 14 of the beta function(3.14)dg0(aM,λ)dλ|λ=1=−14adg0(Ma)da=14β(g0). We will provide a derivation of Eq. (3.14) in Appendix A. As one can see from the derivation, the result relies crucially on the lattice symmetry and that the space-time dimension is 4. In d dimensions the above relation can be generalized with 14 replaced by 1d.With help of Eq. (3.14), by comparing the λ derivatives of two point functions in a way similar to Sec. 2.3, the mass-sum rule can be obtained by combining Eq. (A.3) and Eq. (3.14):(3.15)M=〈P→=0|Hc+Ha|P→=0〉〈P→=0|P→=0〉, where Hc is given by the lattice QCD version of Eq. (3.5)(3.16)Hc=1g02∑x→(Pτ(0,x→)−Ps(0,x→)), and the quantum anomalous energy (QAE) contribution reads(3.17)Ha=β(g0)2g03∑x→(Pτ(0,x→)+Ps(0,x→)). Comparing with the tensor decomposition of the Hamiltonian [2,3],(3.18)H=HT+HS=∫d3x→TT00(x→)+14∫d3x→Tμμ(x→), where TTμν is the traceless (tensor) part of the EMT, one found that in lattice regularization the Hc equals to the tensor part HT of the Hamiltonian, while Ha equals to the scalar part HS of the Hamiltonian(3.19)HT=Hc,HS=Ha. The tensor part of the Hamiltonian HT contributes 34 of the hadron mass, while the scalar part contributes 14, in consistency with the virial theorem [2]. It is not difficult to see that the derivation can be adopted to generic cutoff schemes that preserve the lattice symmetry.We shall also mention that if one maintain a generic λ in the derivation above, then the Hamiltonian has the form(3.20)H=1g02∑x→(−λ2Pτ(0,x→)+Ps(0,x→))+2λg03dg0dλ∑x→(λ2Pτ(0,x→)+Ps(0,x→)). Eq. (3.20) will be used later in Sec. 4.3.2Dimensional regularization and renormalization of EMTIn dimensional regularization (DR), the spatial dimension has been changed to D−1=3−2ϵ, while the temporal direction remains one-dimensional. Therefore, the rescaling in temporal direction (3.1) will encounter no conflict with the UV-cutoff and we expect that rescaled action agrees with the naive one(3.21)S=14g02(ϵ,gr(Mμ))μ2ϵ∫dx4d3−2ϵx→(2λF4iF4i+λ−1FijFij), without any λ dependence in g0. As a result, the naive Hc=12(E→2+B→2) is the full-Hamiltonian and contains both the scalar and tensor parts HT and HS. A general lesson that we will learn from the non-linear sigma model in Sec. 4 is that the finner the cutoff in the temporal direction, the larger the proportional in the full Hamiltonian carried by the “naive” one.The classical-looking Hamiltonian mixes the scalar and tensor contributions can also be explained by investigating the Lorentz symmetry of the theory. The dimensional-regularized theory has a SO(1,3−2ϵ) symmetry group instead of the SO(1,3) in the continuum. In 4-D, 12(E→2+B→2) is the 00 component of a traceless rank two tensor −FμρFρν+gμν4F2, which remains true in d-dimension with a factor of 1/4 in the second term. However, in 4−2ϵ dimensions the traceless tensor is −FμρFρν+gμν4−2ϵF2 and differs from the EMT in d-dimension by an “evanescent” operator proportional to ϵF2 [22,39]. After taking the ϵ→0 limit, it is −FμρFρν+gμν4−2ϵF2 becomes the tensor part in 4-D, while ϵF2 remains finite despite the ϵ in front due to the presence of 1ϵ ultraviolet poles and becomes the trace-anomaly. This is a good demonstration of the fact that the difference in SO(1,3) and SO(1,3−2ϵ) combined with the presence of UV divergence has far reaching consequences for renormalization of tensorial operators and their traces [39], in particular, the EMT for QCD as we will review now.Notice that although the discussions up to here are only for pure-YM, one can generalize them to the case of full QCD similarly. In the notation of [6,7], the EMT for QCD reads(3.22)Tμν=O1μν+O2μν4+O3μν, with the operators:(3.23)O1μν=−FμρFρν,(3.24)O2μν=gμνF2,(3.25)O3μν=ψ¯iγ(μDν)ψ,(3.26)O4μν=gμνmψ¯ψ. Although Tμν is UV finite after summing over all the terms, none of the individual operators above have simple renormalization property, due to the fact that they contain both scalar and tensor representations of the Lorentz group SO(1,3−2ϵ). To simplify the renormalization property and fully utilize the Lorentz structure, a standard way to proceed [1–3] is to decompose them into the tracefull and traceless parts according to the Lorentz group SO(1,3−2ϵ) and renormalize separately(3.27)Tμν=TSμν+TTμν, where(3.28)TTμν=(O1μν+O2μν4−2ϵ)+(O3μν−O4μν4−2ϵ),(3.29)TSμν=gμν4−2ϵ(mψ¯ψ−2ϵ4−2ϵF2), are tensor and scalar parts of the EMT in 4−2ϵ dimensions. Under renormalization, operators belonging to tensor and scalar representations do not mix with each other and become the tensor and scalar operators for the renormalized theory in 4-D after taking ϵ→0 in the end. For more details regarding the standard way of renormalizing the energy-momentum tensor, see Ref. [3].Instead of renormalizing the trace and traceless parts of EMT separately, in Ref. [6,7] the renormalization is performed for the operators O1, O2, O3 by directly subtracting the 1ϵ poles [39] without separating the tensor and scalar contributions. As far as renormalization is concerned, this is perfectly fine. However, this renormalization procedure does not respect the Lorentz symmetry and the resulting finite operators O1,R, O2,R, O3,R mixes different Lorentz representations. In particular, the renormalized operator 12(E→2+B→2)R in this renormalization procedure mixes the tensor and scalar representations of the Lorentz group and is physically less useful. In contrary, the notation 12(E→2+B→2)R has commonly been reserved for the 00-component of the renormalized traceless tensor [2,40,41] and can be measured directly through deep-inelastic scattering as the momentum fraction carried by gluons. More importantly, the non-standard renormalization scheme advocated in Refs. [6,7] hides the scheme and scale-independent QAE contribution in the mass.To summarize, in the subsection we have shown that the “naive” Hamiltonian in dimensional regularization mixes the scalar and tensor contribution from two viewpoints: one is based on time-rescaling property and another is based on investigating the Lorentz invariance. In the next subsection we will argue that a maximally scheme-independent mass sum rule must preserve the Lorentz structure and a clear separation between the tensor and scalar energy contributions.3.3Scheme-independent mass decompositionIn the previous two subsections, we argued that the operator form of the mass sum rule is sensitive to the regularization scheme. In a cutoff scheme such as the symmetric lattice scheme for pure YM, we obtain the classical-looking term Hc=∫d3x12g02(E→2+B→2) as well as the anomalous term Ha, while in DR we only get the “naive” one. In other regularization schemes such as the asymmetric lattice scheme, we get another linear combination.Thus, the contribution of the “naive” Hamiltonian Hc to the nucleon mass is scheme dependent and has not the clear physical interpretation one would usually expect. However, what we are interested is a maximally scheme independent decomposition of the nucleon mass. As was already discussed in [2,3], the decomposition of the Hamiltonian into a traceless (scalar) and trace (tensor) parts is such, H=HT+HS. A crucial point is that the trace and trace-less parts of EMT correspond to two different irreducible representations of the Lorentz group. Therefore, this separation is unique and independent of any interpretation issue. In both dimensional and lattice regularization, the trace part is(3.30)HS=Ha=β(g)2g3∫d3x→S(0,x→), while the expressions for the traceless part HT look different in different schemes but give identical contributions to the hadron mass(3.31)HT(lattice)=∫d3x→2g02(E→2+B→2),HT(DR)=∫d3−2ϵx→g02μ2ϵ(2−2ϵ4−2ϵE→2+24−2ϵB→2),〈M|HT(lattice)|M〉=〈M|HT(DR)|M〉=34M. It is the traceless part (3.31), but not the naive Hamiltonian that naturally corresponds to the gluon kinematic energy. The trace part of the energy momentum tensor corresponds to higher twist operators and thus cannot describe the energy contribution of the twist-two gluon parton distribution function, i.e. its second Mellin moment ∫dxxfg(x) which can be obtained by extrapolating experimental data and direct lattice calculations.Generalizing to the case of full QCD, one has the following decomposition in DR:(3.32)H=HT+HS≡(Hg+Hq)+Hm+Ha,(3.33)HS=Ha+14Hm,(3.34)HT=(Hg+Hq)+34Hm, where(3.35)Ha=14∫d3x→(β(g)2gF2+γmmψ¯ψ)R,(3.36)Hm=∫d3x→(mψ¯ψ)R,(3.37)Hg+Hq=∫d3x→(2−2ϵ4−2ϵE→2+24−2ϵB→2+ψ¯(−iα→⋅D)ψ)R, in which the lower-script R denotes renormalized version. Of these Ha, Hg+Hq and Hm are separately scale invariant, while Hg and Hq are not. The decomposition preserves the Lorentz symmetry and can be obtained by looking at the 00 components of the EMT renormalized using the standard methods in Refs. [1–3] as discussed in Sec. 3.2.Let's now consider the extraction of all the matrix elements in our decomposition. We first denote the nucleon sigma term as(3.38)σN=〈P|mψ¯ψ|P〉2MN. We then notice the relation(3.39)〈P|TTμν|P〉=2(PμPμ−gμν4)(xq(μ2)+xg(μ2)) which relates the matrix element of a twist-2 operator to moments of gluon and quark PDFs. The momentum fractions simply satisfy xq(μ2)+xg(μ2)=1. Given these, we define quantities Mg, Mq, Mm and Ma by(3.40)Mg+Mq=〈Hg+Hq〉=3MN4(xq(μ2)+xg(μ2))−3σN4,(3.41)Mm=〈Hm〉=σN,(3.42)Ma=〈Ha〉=MN−σN4, where MN is the nucleon mass. Therefore, the anomalous energy for the nucleon equals 14 of the nucleon mass minus the nucleon sigma term σN. Essentially, two scale invariant quantities, MN and σN are required. It is interesting to see that one has Mg+Mq=3Ma, a result of the virial theorem.To summarize, in this section we have reviewed the mass sum-rule in various regularization schemes. We have shown that due to the presence of UV cutoff in the temporal direction, the Hamiltonian of QCD-like theories in lattice cutoff requires a term equivalent to the QAE in addition to the “classical” one. We have shown that in dimensional regularization, the anomalous energy is hidden in the naive classical-looking Hamiltonian and a scheme independent mass decomposition must preserve the Lorentz structure and treat the scalar and tensor contributions at different footing. We also commented on the renormalization of EMT in QCD. In the next section we use the non-linear sigma model in 1+1 dimension as an example to further demonstrate these points.4Mass generation and trace-anomaly in the 1+1 dimensional non-linear sigma modelAs illustration of the results in Sec. 3, we investigate in this section the mass decomposition of the 2-dimensional non-linear sigma model in the large N limit in detail [24]. We work exclusively in Euclidean formulation of the theory. The model [42,43] consists of an N component scalar field π=(π1,...πN) normalized by ∑a=1Nπaπa=1. The action reads(4.1)S=12g02∫d2x(∂μπa)(∂μπa). Here, g0 is the dimensionless coupling constant. This model is O(N) rotational invariant πa→Oabπb. A perturbative analysis of the model can be performed by using the parametrization π=(g0π1,...g0πN−1,1−g02∑i=1N−1πi2) near the north pole which is identified with the perturbative vacuum. In this parametrization, the action for π1,..πN−1 reads:(4.2)S=∫d2x12∑a=1N−1(∂μπa)(∂μπa)+g02∑a=1,b=1N−1(πa∂μπa)(πb∂μπb)21−g02∑i=1N−1πi2. The O(N) symmetry is broken to O(N−1) in the perturbative vacuum and the remaining N−1 πas are massless Goldstone bosons. One then expands the square root and treats the resulting terms as perturbations. One can show that the resulting theory is renormalizable to all orders in g0, and that the theory is asymptotically free [42,43]. However, due to the infrared divergences in 2d, the perturbative analysis fails to capture the vacuum structure of the theory. Instead of having N−1 massless modes, one expects that the theory is gapped and dimensional transmutation occurs in a similar way as for QCD in 4d. The πa fields are all massive and the SO(N) invariance should be unbroken. But unlike QCD, there is no color charges in the theory that got confined. Here, we should notice that while there is quite convincing theoretical and numerical evidence that these statements are true, a formal proof is still missing, as far as we know.In the large N limit of the theory, defined by taking N→∞ with λ0=g02N being fixed, the theory is exactly savable and one can use it as a tool to investigate the mass structure in asymptotically free theory with mass generation. We will first provide a self-contained review of the model in large N limit in Sec. 4.1, then investigate the mass structure of the model in Sec. 4.2-4.4. From the discussion in Sec. 2, the trace anomaly of the theory is given by(4.3)Tμμ=β(g0)g0∑a=1N(∂μπa∂μπa), where the β(g0) is the beta function of the theory to be given later, which implies that the anomalous energy term reads:(4.4)Ha=β(g0)2g0∫dx1∑a=1N((∂1πa)2+(∂4πa)2). We will show that regardless of the regularization scheme, the operator form of Ha remains the same and contributes to half of the πa mass. On the other hand, we will show that the naive kinematic plus potential energy contribution T+V or Hc, similar to the 12(E→2+B→2) in gauge theory is sensitive to the regularization scheme and mixes the traceless and trace contributions in the case of DR. According to the discussions in Sec. 3.1 and 3.2, the explicit operator forms that reduce to the Hamiltonian H in the continuum limit is regulator dependent. However, the total contribution of H remains the same in all the regularization schemes.4.1Review of the large N limit of the modelFor the convenience for our discussion here we present here a self-contained introduction to the solution in the large N limit defined similar to that of QCD as λ=g02N fixed while N→∞, for more details see Ref. [42,43].One first introduces an auxiliary field σˆ and rewrites the action as:(4.5)S=12g02∫d2x(∂μπa)(∂μπa)+i∫d2xσˆ2g02(∑aπaπa−1). (By integrating out σˆ, one recover the constraint ∑aπaπa=1.) One then integrates out πa instead and ends up with the following effective action for σˆ(4.6)S[σ]=N2Trln(−∂2+iσˆ)−i2g02∫d2xσˆ=N(12Trln(−∂2+iσˆ)−i2λ0∫d2xσˆ). At large N, one expects the action to be dominated by the saddle point at σˆ=−im2 where m is the fundamental mass scale of the model that will become the mass of the πa fields as we will see. The saddle point satisfies the gap equation:(4.7)∫d2k(2π)21k2+m2=1λ0, which determines the bare λ0 as a function of the mass m and the UV cutoff. To check the dominance of the saddle-point, one can expand the effective action around it σˆ=−im2+σ. The linear term vanishes due to the gap equation, and the quadratic term for σ reads(4.8)S2[σ]=N4Tr(1−∂2+m2σ1−∂2+m2σ)=N2∫d2p(2π)2σ†(p)Σ−1(p)σ(p) where the Tr denotes the trace in coordinate or momentum space and the inverse propagator is:(4.9)Σ−1(p)=∫d2k2(2π)21((p−k)2+m2)(k2+m2) which is convergent and positive definite. At zero momentum, we have Σ−1(0)=18πm2. This guarantees the stability of the saddle point. One further notices that all the higher order terms for σ are proportional to N, therefore, after rescaling σ→1Nσ, the action reads:(4.10)S=NS0+14Tr(1−∂2+m2σ1−∂2+m2σ)+∑i≥31Ni2−1Si, and we obtain a systematic expansion in 1/N. This shows the dominance of the large N saddle-point.Given the large N solution based on the auxiliary field, we move to the original fields πa in Eq. (4.5). To leading order in N, m2πaπa is the mass term for πa. Therefore, the large N saddle point indicates that the πa form a massive SO(N) vector multiplet in contrast to the SO(N−1) multiplet obtained in the perturbative approach. All the higher order contributions can be generated from the following Feynman rules.•The massive field πa→g0πa represented by a solid line has the propagator δabk2+m2.•The σ field represented by a dashed line has the propagator Σ(p). At zero momentum it reads Σ(0)=8πm2.•The interaction between two πa and one σ is represented by the vertex −iN.•The one-loop self-energy diagram for the σ propagator, as well as the one-loop tadpole diagrams for σ have to be discarded. One can show that the resulting theory is renormalizable to all orders in 1N. There are three types of divergent diagrams. The two point function for π is quadratically divergent, the tadpole-diagram for σ is logarithmically divergent and the σ−π vertex is also logarithmically divergent. These divergencies can be removed by corresponding charge and field renormalization, and the resulting theory is equivalent to the original nonlinear sigma model with a running 1g02 beyond the leading order result. The Feynman rules for the theory are shown in Fig. 1.After introducing the large N solution, we should mention that beyond the large N limit, the O(N) non-linear sigma model is widely believed to be integrable and has been investigated in Refs. [44,45]. Moreover, in the O(3) case, there are instanton solutions which has been suggested to be responsible for the mass generation [46]. It is interesting to combine the various viewpoints to produce a deeper understanding of the mass structure for the model. These are left for a future work.4.2Mass decomposition in hard cutoff k2≤ΛUV2 and lattice regularizationIn this subsection we study the mass-decomposition in the hard cutoff k2≤ΛUV2 and lattice regularization. We first consider the hard-cutoff. With this cutoff, λ0=Ng02 is fixed by the gap equation Eq. (4.7)(4.11)1g02N=∫k2≤ΛUV2d2k(2π)21k2+m2=14πlnΛUV2m2. As a result, the β function reads:(4.12)β(g0)=ΛUVdg0dΛUV=−Ng034π. Therefore, the anomalous term Ha is:(4.13)Ha=−Ng028π∫dx1∑a=1N((∂1πa)2+(∂4πa)2). According to the discussion in Sec. 3.1, in the symmetric cutoff the Hamiltonian has the form(4.14)H=Hc+Ha,(4.15)Hc=12∫dx1∑a=1N((∂1πa)2−(∂4πa)2). We now evaluate the matrix element of Hc and Ha in the massive πa state |πa,P→=0〉 at rest.We first study Ha, more precisely, we calculate the forward matrix-element of Ha in the πa state at rest. In the leading order of N there are two diagrams. The first one, shown in Fig. 2, is of tree level and the second one shown in Fig. 3 is of one-loop level. It is leading because the factor N from the loop cancels the factor (1N)2 from the vertices. The diagram gives (note that k2=(k4)2+(k1)2):(4.16)Ea=12m(Ng024πm2+Ng028πΣ(0)∫k2≤ΛUV2d2k(2π)2k2(k2+m2)2). Using Σ(0)=8πm2 one obtains(4.17)Ea=〈πa,P→=0|Ha|πa,P→=0〉〈πa,P→=0|πa,P→=0〉=12mNg024πm2(1−Σ(0)2∫k2≤ΛUV2d2k(2π)21(k2+m2)2)+12mNg02m2∫k2≤ΛUV2d2k(2π)21(k2+m2). The first line vanishes due to the identity 12Σ(0)∫d2k(2π)21(k2+m2)2=1, even in the presence of the regulator, while the second line equals m22m=m2 after using the gap equation Eq. (4.7). Therefore, we conclude that(4.18)Ea=m2, in consistency with the virial theorem.We then study the contribution Ec of Hc in the πa state. At leading order in N there are again two contributions given by the tree-level diagram in Fig. 2 and the one-loop diagram in Fig. 3. The tree level diagram can be easily calculated and contributes to m2, while the tadpole diagram contribution reads(4.19)Ec|Fig. 3=12mN(−iN)2Σ(0)12∫k2≤ΛUV2d2k(2π)2(k4)2−(k1)2(k2+m2)2=0. Therefore, we conclude that in the cutoff scheme k2≤ΛUV2 the kinematic energy, i.e. the expectation value of Hc, contributes half of the mass. Therefore the average of H equals to m, consistent with the fact that the πa field has mass m.Similarly, the lattice cutoff can be treated using the same methods. The only difference is that ∂2 becomes the lattice version of the Laplacian. The gap equation reads in lattice regularization(4.20)1g02N=∫−πaπadk4dk1(2π)21m2+2a2(2−cosak4−cosak1). With the notation A(k,a)=2a2(2−cosak4−cosak1), Σ(p) is given by:(4.21)Σ−1(p,a)=12∫−πaπadk4dk1(2π)21m2+A(p−k,a)1m2+A(k,a). It is straightforward to show that Σ(p,a)→Σ(p) as a→0, therefore the propagator for the sigma field remains the same in lattice regularization.These equations allow us to check whether all sum rules for the lattice cutoff are the same as for the hard cutoff k2≤ΛUV in the continuum limit, which is the case. For example, the contribution Ha reads:(4.22)Ea=12mNg024πm2(1−Σ(0)2∫−πaπad2k(2π)21(A(k,a)+m2)2)+12mNg02m2∫−πaπad2k(2π)21A(k,a)+m2 which is m2 thanks to the gap equation.4.3Mass decomposition in hard cutoff |k1|≤ΛUVWe then investigate the scheme in which we only impose a hard cutoff |k1|≤ΛUV in the spatial direction while the k4 is unbounded. This regulator breaks the rotational invariance explicitly. Since the k4 integral can be rescaled freely, we expect that in this case the naive Hamiltonian Hc term generates the complete hadron mass and(4.23)H=Hc. Most of above calculations remain valid. The gap equation reads(4.24)1g02N=∫|k1|≤ΛUVd2k(2π)21k2+m2=12πlnΛUVm+ln22π. Also the beta function remains the same. We first study the contribution of Ha. The diagrams for Ha are again given by Fig. 2 and Fig. 3. The total result reads(4.25)Ea=12mNg024πm2(1−Σ(0)2∫|k1|≤ΛUVd2k(2π)21(k2+m2)2)+12mNg02m2∫|k1|≤ΛUVd2k(2π)21(k2+m2), which can be shown to become m2 in the limit ΛUV→∞ by using the gap equation.We then study the contribution for H=Hc, the tree diagram remains the same and equals m2, while the tadpole diagram becomes(4.26)Ec|Fig. 3=12mN(−iN)2Σ(0)12∫|k1|≤ΛUVd2k(2π)2k42−k12(k2+m2)2. Unlike in the case of a symmetric cutoff, the integral does not vanish(4.27)∫|k1|≤ΛUVd2k(2π)2k42−k12(k2+m2)2=14π in the ΛUV→∞ limit. The fact that a naively vanishing integral is non-vanishing due to the presence of cutoff signals the anomalous nature of this one-loop contribution. Using Σ(0)=8πm2 we obtain again m2. Thus in the |p1|≤ΛUV scheme one found that the average of H equals to m.Again, the contribution of the total Hamiltonian equals to the mass of the πa field. We should emphasize that although the contribution of Hc in this cutoff equals to the total mass of πa, the one-loop part is in fact of anomalous nature and equals to that of Ha. The part that can be identified as “classical” T+V contribution remains to be m2.More generally, we can show that in the regularization scheme k42λ2+k12≤ΛUV2 where λ>0 is positive, corresponding to the time-rescaled theory x4′=λx4 discussed in Sec. 3.1. In this case, the running of g0 is λ dependent:(4.28)1g02N=12πlnΛUVm−12πln(λ+12λ). From which one has(4.29)λdg0dλ=β(g0)1+λ. At λ=1, corresponding to the symmetric cutoff case, one has λdg0dλ=β(g0)2. At generic λ, from Eq. (3.20), the total Hamiltonian reads(4.30)H=Hc+λg0dg0dλ∫dx1∑a=1N((∂1πa)2+(∂4πa)2), where we have used Eq. (4.29) in the last equality. The contribution of Ha, which is contained in the one-loop diagram of Hc as well as the last term in Eq. (4.30) can be calculated as(4.31)Ea=12mNg024πm2(1−Σ(0)2∫k42λ2+k12≤ΛUV2d2k(2π)21(k2+m2)2)+12mNg02m2∫k42λ2+k12≤ΛUV2d2k(2π)21(k2+m2)=m2, and remains to the m2, while the average of the Hc part in this case can be calculated as mλ1+λ, thus using Eq. (4.30) one has(4.32)〈πa,P→=0|H|πa,P→=0〉〈πa,P→=0|πa,P→=0〉=mλ1+λ+m1+λ=m, which equals to the mass of the πa.4.4Mass decomposition in dimensional regularizationFinally, we study DR. For DR with space-time dimension d=2−2ϵ one should change g02→g02μ2ϵ. The gap equation reads:(4.33)1g02N=μ2ϵ∫d2−2ϵk(2π)2−2ϵ1k2+m2=(μm)2ϵΓ(ϵ)(4π)1−ϵ=14πϵ+2ln(μm)−γE+ln(4π)4π. The beta function is again given by −g034π. In DR, the total Hamiltonian reads(4.34)H=Hc, similar to the |k1|≤ΛUV case. Let's first study the contribution of Ha. The diagrams are again given in Fig. 2 and Fig. 3. The result reads(4.35)Ea=12mNg024πm2(1−12Σ(0)μ2ϵ∫d2−2ϵk(2π)2−2ϵ1(k2+m2)2)+12mNg02μ2ϵm2∫d2−2ϵk(2π)2−2ϵ1(k2+m2), which again equals m2 as ϵ→0 thanks to the gap equation.We then discuss the Hc. The tree level diagram remains the same as before and equals m2. The tadpole diagram now reads:(4.36)Ec|Fig. 3=12mN(−iN)2Σ(0)12μ2ϵ∫dk4d1−2ϵk1(2π)2−2ϵk42−k12(k2+m2)2. Since we have split the space-time of dimension 2−2ϵ into a 1-dimensional time and a 1−2ϵ dimensional space, the integral reads:(4.37)∫dk4d1−2ϵk1(2π)2−2ϵk42−k12(k2+m2)2=ϵ1−ϵ∫d2−2ϵk(2π)2−2ϵk2(k2+m2)2→14π, where the factor ϵ1−ϵ comes from 12−2ϵ−1−2ϵ(2−2ϵ). As ϵ→0, there is a 14πϵ pole from the integral which cancels the ϵ in front, leading to a finite result. This is identical to the mechanism of trace-anomaly in DR and we again see that the non-vanishing one-loop diagram for Hc is of anomalous nature. By using Σ(0)=8πm2 one obtains m2. Thus we found that the contribution of H again equals to the mass of the πa field.In dimensional regularization, one can check that the traceless part of the EMT produces indeed m2. HT in 2−2ϵ dimensions reads:(4.38)HT=1μ2ϵ∫d1−2ϵx∑a(−1−2ϵ2−2ϵ(∂4πa)2+12−2ϵ(∂1πa)2) with which the Hamiltonian can be equivalently written as(4.39)H=HT+Ha. We now verify that the contribution of HT is indeed m2. The tree level diagram remains m2. The tadpole diagram is now proportional to(4.40)ET|Fig. 3=∫dk4d1−2ϵk1(2π)2−2ϵ1(k2+m2)2(−1−2ϵ2−2ϵk42+12−2ϵk12), which in turn is proportional to(4.41)(−1−2ϵ2−2ϵ12−2ϵ+12−2ϵ1−2ϵ2−2ϵ)≡0. The vanishing of the one-loop contribution for HT is identical to that of Hc case in symmetric cutoff. Again, only the tree-level contribution to Hc can be identified as the “classical” T+V contribution.To summarize, in this section we have investigated the mass structure of 1+1 non-linear sigma model in detail. In all the schemes, the anomalous term Ha has the same operator form and contributes half of the πa mass. The total Hamiltonian, although differs in operator forms in different regularization schemes, contribute to the total πa mass. On the other hand, although the naive Hamiltonian Hc has the same form as the classical one, the contribution is actually regulator dependent and has no universal physical meaning. In regularization schemes that treat both directions equally, Hc can be identified as HT while in schemes in which the k4 integral can be re-scaled back and forth, such as the |k1|≤ΛUV scheme and the dimensional regularization scheme, Hc can be identified as the full Hamiltonian. However, even in regulators where Hc equals to H formally, we still found that the one-loop contribution of Hc is of anomalous nature and can be identified as Ha. We summarize the various contributions in Table 1. The example of the 1+1 dimensional sigma model thus illustrates that the QAE is part of the total mass of πa regardless regularization schemes. This suggests that its contribution to the total mass might be related to some very general property of the theory under study.5Virial theorem and perturbative anomalous energy contribution in QEDIn the previous sections, it has been shown that due to the cutoff dependency of the coupling constants, a trace anomaly is generated and contributes to the energy. In this section we study the case of QED. In QED, although there is no dynamical scale generation, there are still UV divergences that lead to the beta function β(e)=e312π2 and mass anomalous dimension γm=3e28π2. As a consequence, there is an anomalous contribution to the energy(5.1)Ha=14∫d3x→(β(e)2eF2+γmmψ¯ψ), in addition to the mass term(5.2)Hm=∫d3x→mψ¯ψ, where Fμν is the electromagnetic field strength and ψ is the electron field. In this section we study Ha for free electrons and for bond-states in a background field.5.1Virial theorem and non-relativistic reductionBefore coming to the QAE, let's consider the virial theorem for QED in the non-relativistic limit and show that it reduces to the corresponding virial theorem in quantum mechanics. One first notices that the scalar part of the Hamiltonian reads(5.3)HS=14Hm+Ha, where Ha and Hm are given in Eqs. (5.1), (5.2). In the mean time, the full Hamiltonian reads in symmetric regularization scheme:(5.4)H=∫d3x→(12(E→2+B→2)+ψ†(−iα→⋅D→)ψ)+Hm+Ha. Therefore, the virial theorem, ES=E/4, indicates that(5.5)〈P→=0|∫d3x→(12(E→2+B→2)+ψ†(−iα→⋅D→)ψ)|P→=0〉=3〈P→=0|Ha|P→=0〉, where |P→=0〉 is a state in the rest frame. One then notices that in the non-relativistic limit, the anomalous contributions are unimportant. Neglecting Ha in Eq. (5.5), one obtains the relation:(5.6)〈P→=0|(E→2+B→2)/2+(−iα→⋅D→)|P→=0〉=0. We now show that Eq. (5.6) reduces to the virial theorem for the hydrogen-like systems in non-relativistic quantum mechanics.We chose the Coulomb gauge ∇⋅A→=0 where A→ contains only transverse part A→=A→T. The quantization of QED in this gauge is explained in many textbooks [47]. The temporal component A0 decouples from the transverse part of the gauge field and is expressed as,(5.7)A0=−e∇2ψ¯γ0ψ. By using E→=−∂tA→T−∇A0 and the explicit solution of A0 in Eq. (5.7), the transverse and longitudinal parts of the electric field decouple from each other and one has the relation(5.8)∫d3x→12(E→2+B→2)=∫d3x→12(E→T2+B→T2)+e22∫d3x→d3y→ψ†ψ(x→)ψ†ψ(y→)4π|x→−y→|, where E→T=−∂tA→T and B→T=∇×A→T are the radiative part of the photon field. For a positronium state, to leading order in a non-relativistic expansion the contribution of the transverse radiation fields A→T is negligible, thus the virial theorem Eq. (5.6) reduces to(5.9)〈P→=0|ψ†(−iα→⋅∇→)ψ|P→=0〉+〈P→=0|e22∫d3x→d3y→ψ†ψ(x→)ψ†ψ(y→)4π|x→−y→||P→=0〉=0, by using Eq. (5.8).We now investigate the consequence of Eq. (5.9) on the leading component of the positronium state(5.10)|P→=0〉=∫d3k→2(2π)3Ekψss′(k→)as†(k→)bs′†(−k→)|0〉 where the creation and annihilation operators are normalized as [as(k→),as′†(k→′)]+=(2π)32Ekδs,s′δ3(k→−k′→), and similarly for b and b†. The non-relativistic wave function is normalized as(5.11)∫d3k→(2π)3ψss′†(k→)ψss′(k→)=1. Using the free field(5.12)ψ(x)=∫d3k→(2π)32Ek∑s(as(k→)us(k)e−ik⋅x+bs†(k→)vs(k)eik⋅x), in the non-relativistic limit the matrix elements can be calculated as(5.13)〈P→=0|ψ†(−iα→⋅∇→)ψ|P→=0〉=2∫d3k→(2π)3ψss′†(k→)ψss′(k→)k2m, and(5.14)〈P→=0|e22∫d3x→d3y→ψ†ψ(x→)ψ†ψ(y→)4π|x→−y→||P→=0〉=−e2∫d3k→d3q→(2π)3ψss′†(k→)ψss′(q→)|k→−q→|2. Therefore, the relation Eq. (5.9) simply reduces to the non-relativistic virial theorem(5.15)〈V〉=−2〈T〉, where(5.16)〈V〉=−e2∫d3k→d3q→(2π)3ψss′†(k→)ψss′(q→)|k→−q→|2,(5.17)〈T〉=∫d3k→(2π)3ψss′†(k→)ψss′(k→)k2m, are the kinematic and Coulomb energy. In conclusion, for non-relativistic hydrogen-like systems the virial theorem simply reduces to the classical relation 〈V〉=−2〈T〉.5.2The anomalous contribution to the electron pole massAfter discussing the virial theorem, we now return to the perturbative QAE in QED. The simplest quantity is the electron pole mass me, defined as the pole of the inverse electron propagator [48]. Here we show that although small, the perturbative QAE does contributes to the electron pole mass.We first consider the MS‾ scheme at renormalization scale μ, where the minimally renormalized electron mass m=m(μ) that appears in the Lagrangian is not the electron pole mass me, instead, at one-loop order one has the relation [49](5.18)me=m(μ)(1+α(μ)π). The one-loop beta function reads β(e)=e312π2 and γm=3e28π2. At one-loop order, the F2 term does not contribute and only the γmmψ¯ψ term contributes to the anomalous energy Ha(5.19)Ea=3α8πm(μ). On the other hand, one can show that at one loop order the mass term contributes as(5.20)Em=〈P→=0|Hm|P→=0〉〈P→=0|P→=0〉=(1−α2π)m(μ), which gives the correct result Ea+14Em=me4.We then consider the on-shell renormalization scheme [1,47], where the mass parameter of the renormalized Lagrangian equals to the pole mass. At one-loop order the mass anomalous dimension equals to γm=3e28π2 which is the same as in MS‾. Therefore the contribution of the QAE is(5.21)Ea=3α8πme. On the other hand, the contribution of the mass terms now reads(5.22)Em=(1−3α2π)me, which is again consistent with the relation Ea+14Em=me4.5.3The anomalous contribution in a background field (Lamb shift)In this subsection we investigate the contribution of the anomalous term to the QED radiative correction of hydrogen atom binding energies. The naive Lagrangian density of the system with a background Coulomb field reads:(5.23)L=ψ¯(iγ⋅∂−m)ψ−eψ¯γμψ(Aμ+Aμ)−14FμνFμν, where Aμ=Ze4π|x→|δ0μ is the Coulomb field of the heavy nucleus and Aμ is the QED photon field. We first study the trace anomaly of such a system, which is related to its renormalization properties. It is easy to see that all the UV divergences of the system can be taken care of using the standard wave function renormalization Z1−1=Z2−1 for electron and Z3−1 for photon, as well as the mass counter-term mδm order by order in perturbation theory. The counter-terms in terms of the renormalized fields AR,ψ¯R are(5.24)δL=(Z1−1)ψ¯R(iγ⋅∂−m)ψR−e(Z1−1)ψ¯Rγμψ(Aμ,R+Aμ)−δmmZ1ψ¯RψR−Z3−14FRμνFμν,R−Z3−12FRμνFμν, where the last term is needed to cancel the UV divergence of vacuum polarization diagrams with one Aμ and one Aμ insertion. Again, Z2−1=Z1−1 follows from the Ward identity and the bare charge is related to the renormalized charge by eRAR=e0A0. Thus e0=Z3−12eR and the beta function is purely determined by Z3. By re-scaling with e0A0→A0 and eA→A, in terms of the bare fields the Lagrangian reads:(5.25)L=ψ0iγ⋅(∂+A0+A)ψ0−Z34e2F02−Z3−12e2F0μνFμν−(1+δm)mψ¯0ψ0, where all the cutoff dependencies are absorbed into the renormalization constants Z3 and δm, which are precisely those which determine the QED beta function and the electron mass anomalous dimension. From now on until the end of this section, without mention all the field operators are renormalized and we will omit the lower-script R for all the renormalized quantities. Thus, one has for the trace anomaly in the background field:(5.26)Tμμ=β(e)2eFμνFμν+β(e)eFμνFμν+m(1+γm)ψ¯ψ, which implies that(5.27)Ha=14∫d3x→(β(e)2eFμνFμν+β(e)eFμνFμν+mγmψ¯ψ). Let's investigate the contribution of Ha, 〈N|Ha|N〉 to the energy shift for a given bound state |N〉 in the background field A when radiative corrections are included.We first review the field theoretical calculation of the energy-shift in the presence of a background, following the notation and approach in Weinberg's book [47]. One first quantize the theory in the background field A without QED corrections. The energy levels are labeled as |N〉, while the positive and negative energy solutions to the Dirac equation are denoted by uN(x→) and vN(x→). This theory is regarded as the “free” theory. We then add QED interactions and treat them as perturbations. The resulting perturbation theory in background field differs from the standard QED by that all the electron propagators are “dressed” [47] in the external field, but otherwise remains very similar. One can show [47] that the one-loop correction δEN to the energy level N can be calculated efficiently using covariant perturbation theory as(5.28)δEN=∫d3p→d3p→′(2π)6u¯N(p→′)Σ(EN,p→′;EN,p→)uN(p→), where Σ(EN,p→′;EN,p→) is the one-loop electron self energy diagram with dressed propagator in the background field. See Fig. 4 for the one-loop self energy diagrams that contributes to δEN.Similarly, when interactions are added, the state |N〉 in the background field will also receive radiative corrections. We need to calculate the QAE contribution(5.29)δENa=〈N|∫d3x→(β(e)8eFμνFμν+β(e)4eFμνFμν+14mγmψ¯ψ)|N〉, in the perturbed wave function |N〉. It can be calculated in covariant perturbation theory using standard Feynman rules for operator insertions, paying attention to insertion of quark-bilinear operators at external legs. More conveniently, using the fact that the trace anomaly can be obtained as mass-derivatives of counter-terms in the Hamiltonian as explained in Appendix B, δENa can also be obtained from the Feynman diagrams for δEN by taking mass derivatives in the counter-terms (Z3−1) and δm. To lowest order in radiative correction, the β(e)8eFμνFμν term in Eq. (5.29) does not contribute, therefore one only needs to consider the β(e)4eFμνFμν and γmψ¯ψ terms.We first calculate 〈N|∫d3x→β(e)4eFμνFμν|N〉. To lowest order in radiative corrections, the Feynman diagram is identical to the last diagram in Fig. 4 with (Z3−1) replaced by the mass derivative or the beta function(5.30)β(e)2e=m4ddm(Z3−1)=α6π, and can be directly calculated as(5.31)〈N|∫d3x→β(e)4eFμνFμν|N〉=α6πie∫d3p→d3p→′(2π)6u¯N(p→′)Γμ(EN,p→′;EN,p→)uN(p→)Aμ(p→′−p→), where(5.32)Γμ(p′,p)=i(p′−p)2+i0(−(p−p′)2gμν+(p−p′)μ(p−p′)ν)γν. For Aμ that only has temporal component, the last term is proportional to EN−EN and vanishes. Therefore, one can simplify the result to(5.33)〈N|∫d3x→β(e)4eFμνFμν|N〉=α26π∫d3x→uN†(x→)uN(x→)|x→|, which is of order mα3. Our results Eq. (5.33) for the photonic contribution to QAE differs has an additional −2 factor compared with Eq. (7) in Ref. [50]. However, the total results in Ref. [50] for the energy shift is still correct possibly due to another discrepancy when evaluating the contribution of mψ¯ψ.We then calculate the mass anomalous dimension term. The contribution is identical to the second diagram in Fig. 4 with δm replaced by mγm. It can be evaluated simply as(5.34)〈N|∫d3x→14mγmψ¯ψ|N〉=14γmm∫d3x→u¯N(x→)uN(x→)=14γmEN. Therefore, by summing the above contributions one gets for the total anomalous part:(5.35)δEn,ja=α26π∫d3x→un,j†(x→)un,j(x→)|x→|+3α8πEn,j, for the energy level N≡n,j, where n is the radial quantum number and j is the total spin. The first term is the photonic contribution while the second term is the fermionic contribution. Here un,j(x→) is the quantum-mechanical wave function that solves the Dirac equation in a static Coulumb field, and En,j is the bound state energy. In the non-relativistic limit, Eq. (5.35) can be further expanded in α and contains contributions at O(α), O(α3) and O(α5). The O(α) and O(α3) contributions will be canceled by other terms, while the contribution at O(α5) reads(5.36)δEn,ja,(5)=−7meα524πn4(38−12j+1). This contributes to the famous Lamb shift at O(α5).We shall also mention that the electron mass Hm gives a non-trivial contribution to the bound-state energy as well. In appendix B, we show that after adding the electron mass contribution, the total scalar energy contribution is 14 of the bond-state energy, consistent with the virial theorem.6Anomalous energy contribution as Higgs mechanismIn both QCD and the non-linear sigma-model, the QAE generates a non-perturbative contribution characterized by a new mass scale (dimensional transmutation [30]). It is then natural to consider the QAE itself responsible for the scale generation. In this section we take this view seriously and show that the anomalous scalar field can be considered as a dynamical Higgs field [24], and the QAE contribution to the mass comes from its dynamical response to the matter, in analogy to the Higgs mechanism [1] for the fermion masses in the standard model.To see this analogy, let's first review the Higgs mechanism for fermion mass generation in a simplified context without gauge symmetry. Introduce a complex scalar ϕ=12(σ+iπ) with action(6.1)L=ϕ†(−∂μ∂μ+μ2)ϕ−λ4!(ϕ†ϕ)2, where μ2>0. The saddle point of the potential V(ϕ)=−μ2|ϕ|2+λ4!|ϕ|4 satisfies(6.2)2μ2|ϕc|=λ3!|ϕc|3. Let's expand the field around the saddle point ϕ=(|ϕc|+12h+i12π). To leading order, the field h is massive with Mh2=2μ2, while π is massless. In the classical theory, the canonical EMT reads:(6.3)Tμν=2ϕ†∂μ∂νϕ−gμνL. In quantum theory, Collins and others have shown that one needs to add the term −16(∂μ∂ν−gμν∂2)ϕ2 to make all the matrix elements finite. To our desired order in λ, up to total derivative terms, the trace of the EMT reads(6.4)Tμμ=−2μ2|ϕ2|, which can be easily derived using the equation of motion. Therefore, the scalar part of the Hamiltonian reads:(6.5)HS=−∫d3x→(12|ϕc|h+14h2).In the presence of the massless fermion Ψ with Yukawa type coupling(6.6)Lint=−Ψ¯Ψgϕ+ϕ†2, which generated a fermion mass term with mΨ=2gϕc. The above also yield a dynamical coupling between the fermion and the Higgs particle, (−g)hΨ¯Ψ, which is proportional to the fermion mass mΨ. This dynamical coupling generates a response of the Higgs field in the presence of the fermion, which contributes to the fermion mass,(6.7)〈Ψ|HS|Ψ〉=(−g)fsmh2=14gϕc=14mΨ, where fs=−12μ2ϕc is a scalar decay constant. The 1mh2 is due to the zero-momentum propagator of the Higgs field. Therefore, the scalar part of the Hamiltonian contributes 1/4 of the fermion mass through the dynamical Higgs. See Fig. 5 for a depiction of the mechanism. This simple example demonstrates that the mass of the fermions can also be measured by the response of the fluctuating part of the scalar field in the presence of the matter [24].Similarly, for the non-linear sigma model, the QAE contribution to the meson mass can be explained in term of a dynamical Higgs mechanism as follows. One first notices that using the equation of motion, the anomalous Hamiltonian can also be re-written in terms of the auxiliary scalar(6.8)Ha=−iNm28π∫dx1σ, where the dimensionless scalar σ=(Σ−〈Σ〉)/m2 contains the quantum fluctuation part. This is similar to the Higgs example above, in that the scalar part of the Hamiltonian is linear in the sigma field. Its contribution to the pion mass is determined by 〈πi|σ|πi〉. By using the ππσ vertices in Eq. (4.5), and the dominance of the zero-momentum σ propagator 〈σ(0)σ(0)〉=8π/(Nm2) in the intermediate state, the response of the scalar σ to πi state exactly makes Ha contributing 12 of the πi mass. We shall mention that the propagator of σ [43] contains only a cut starting at the two-π threshold p2=4m2 but no poles, unlike the Higgs field h in the previous example. Nevertheless, the zero-momentum propagator of σ contributes to the average of the anomalous Hamiltonian exactly the same way as the zero-momentum propagator of the Higgs field h.6.1Dynamical scalar and QAE contribution to the nucleon mass and pressureThe idea that the mass is generated from the response of the scalar field in the presence of the external source can be generalized to QCD. For simplicity, we consider the limiting case of massless up and down quarks. The anomalous Hamiltonian comes entirely from the gluon composite scalar, Ha=∫d3x→Φ(x), where Φ(x)=β(g)/(8g)FμνFμν(x). As in the non-linear sigma model, its contribution to the nucleon mass can be seen as a form of dynamical Higgs-mechanism, which is consistent with that the Higgs and confining phases of matter-coupled gauge theory are smoothly connected [32,51].It is interesting to recall that for the infinite-heavy Q¯Q state separated by r in pure gauge theory, it has been shown [4,25] that the non-perturbative contribution of Ha to the static potential is 14(V(r)+rV′(r)). At large r where the confinement potential dominate V(r)∼σr, the anomalous contribution is exactly one half of the confinement potential.The scalar field Φ(x) has a vacuum condensate Φ0=〈0|Φ|0〉 [52,53]. However, in the presence of the nucleon, the quantum response is measured by(6.9)ϕ(x)=Φ(x)−Φ0, which is a dynamical version of the MIT bag-model constant B [37]. Its contribution to the nucleon mass can be seen as the response of the scalar field to the nucleon source,(6.10)Ea=〈ϕ〉N=〈N|ϕ(x)|N〉, where the nucleon state is normalized as 〈N|N〉=(2π)3δ3(0). If ϕ(x) is a static constant B inside the nucleon, Ea will be of order BV, where V is the effective volume in which the valence quarks are present. In the MIT bag model, the nucleon mass is entirely determined by the bag constant, in line with the view that the QAE determines the mass scale.We should point out that it has been proposed that static response of the composite gluon scalar ϕ in the nucleon state can be measured in the electro-production of heavy quarkonium on the proton [11,41,54–58] or leptoproduction of heavy quarkonium at large photon virtuality [59]. The color dipole from the quarkonium will be an effective probe of the F2. This also provides a direct determination of the QAE contribution to the mass. Nevertheless, it has also been found recently that the near threshold production of heavy meson is dominated by the twist-two tensor contribution instead of the 0++ scalar contribution based on holographic QCD [60,61] or perturbative analysis [62,63]. To further clarify the role of trace anomaly in the heavy-meson production requires more elaborate QCD analysis and is left for future work.Similar to fermion masses in elementary particle physics, we can also consider a dynamical response of the ϕ in the presence of the nucleon through a tower of scalar 0++ spectral states, as in the Higgs model. Assume an effective coupling between the nucleon and scalar gNNϕN¯Nϕ, the QAE contribution to the mass can be related to the scalar field response function,(6.11)〈N|ϕ|N〉=igNNϕ〈ϕ(0)ϕ(0)〉 where 〈ϕ(0)ϕ(0)〉 is the zero-momentum propagator of the scalar field ϕ. If the propagator is dominated by a series of scalar resonances, or 〈ϕ(0)ϕ(0)〉=∑sifs2−ms2, one has [24](6.12)〈N|ϕ|N〉=∑sgNNsfsms2. Here ms is the mass of the scalar resonances, fs=〈s|ϕ|0〉 is the decay constant and gNNs≡gNNϕfs is the coupling of the nucleon to the scalars. See Fig. 6 for a depiction.One might assume the dominance of the lowest mass scalar glueball-like state, generically called σ, for the above equation. If the coupling constant gNNs can be extracted through experiment, one can perform a consistency check on the σ dominance picture by combining the glueball masses and the decay constants extracted from lattice QCD calculations [64,65]. In fact, for the lowest glueball state σ, one can say more. In [29], an effective action for σ that is consistent at tree level with the Ward-identity for Tμμ has been constructed. Introduce a dimension-1 scalar field Σ, which replaces the scalar field Φ through the relation(6.13)Φ≡−mσ464|Φ0|Σ4, where mσ is a mass parameter the meaning of which will be explained later. The effective action for Σ is(6.14)L=12∂μΣ∂μΣ−V(Σ), where the effective potential V(Σ) is(6.15)V(Σ)=mσ4256|Φ0|Σ4lnΣC. The potential has a minimum at Σ=σ0 constrained by the relation 4lnσ0C=−1. In terms of σ0 and mσ, the vacuum condensate can be expressed as(6.16)|Φ0|2=mσ4256σ04, which determines C as a function of mσ and |Φ0|, the two independent parameters of the theory. The dilatation symmetry that was broken at quantum level in the original theory is broken in the effective theory at classical level by the dynamically generated potential V(Σ). This can be viewed as the realization of dimensional transmutation in an effective Higgs phase.By expanding Σ=σ0+σ, one can show that the σ is a massive scalar with mass equals to mσ. Furthermore, one expand Φ to leading order in σ(6.17)ϕ=Φ−Φ0=mσ4σ0364|Φ0|σ+O(σ2), from which the decay constant fσ can be extracted as [29](6.18)fσ=mσ|Φ0|. Assume that the coupling between the nucleon N and the scalar Σ is given by the Yukawa coupling(6.19)LYukawa=−gNNσN¯NΣ. The nucleon mass can be measured exactly through two ways. One way is through the mass term generated by the vacuum expectation value σ0 or mN=gNNσσ0, from which one can extract [24](6.20)gNNσ=mNmσ4|Φ0| in the chiral limit.The second way is through the response of Σ in the presence of the nucleon. By assuming the σ dominance in the intermediate state, one has(6.21)〈N|ϕ|N〉=fσgNNσmσ2=mN4, which we have used Eq. (6.20) and the formula for fσ in Eq. (6.18). Thus we confirm that the scalar response is proportional to the nucleon mass, exactly as 1/4 predicted by virial theorem. This also exactly corresponds to the Higgs model mentioned earlier.Of course, the above simple picture is modified strongly by the presence of light quarks. We also comment that beside the chiral effective theory, the interaction between the scalar glueballs and the nucleons can also be investigated in the framework of holographic QCD [60]. The scalar spectrum in QCD is more complicated, and is in between the simple Higgs and the 1+1 sigma models. However, the coupling between the nucleon (or any other hadrons) and with the scalars must be proportional to its mass, same as in the Higgs case which has been tested recently at LHC [26–28].6.2Anomalous energy contribution to pion massFinally, we show that the anomalous contribution also plays an important role to the pion mass. For simplicity we only consider the two-flavor case with equal quark masses m for up and down quarks u and d. The theory has two mass scales, the quark mass m and ΛQCD. The scalar part of the Hamiltonian reads(6.22)HS=∫d3x→14m(u¯u+d¯d)+Ha, where Ha is the anomalous part of the Hamiltonian. In the m≪ΛQCD limit, it is well known [1] that the mass Mπ for the Golstone boson or the pion relates to the quark mass and the chiral-condensate through the Gell-Mann-Oakes-Renner (GMOR) [66] relation:(6.23)Mπ2=−1fπ2〈0|m(u¯u+d¯d)|0〉. Therefore, the contribution of the mass-term to the pion mass can be calculated as(6.24)〈π|∫d3x→14m(u¯u+d¯d)|π〉=14m∂Mπ∂m=18Mπ, where the pion state is normalized as 〈π|π〉=(2π)3δ3(0) and we have applied the Feynman-Hellman theorem [67] in the first equality and Eq. (6.23) in the last equality. One can also derive this relation by inserting pion intermediate state in the correlation function 〈0|∂μjμ5(x)m(u¯u+d¯d)(y)∂νjν5(z)|0〉 and using the Ward-identity for the axial current. This results, in the context of QCD mass decomposition, is noticed originally in Ref. [3]. Since the scalar part of the Hamiltonian contributes to 14 of the pion mass in total due to the virial-theorem, we found that the anomalous contribution to the pion mass equals to(6.25)〈π|Ha|π〉=18Mπ. Although the pion being the lowest-energy state, the anomalous part contributes to a significant amount of its mass, equally to the quark mass term. The quark mass term fails to dominate in the scalar part. This is however understandable since even in the chiral limit the theory still has dimensional transmutation which gives rise to the ΛQCD. The pion mass in Eq. (6.23) involves a mixing between the quark mass m and the chiral-condensate measured by ΛQCD. Therefore, it is natural that both the quark mass term and the anomalous term contributes equally to the pion mass.Assuming the sigma dominance, an effective action for the coupling between the pion and the lightest glueball field has been constructed in Ref. [68] and reads(6.26)L=fπ24(Σσ0)2Tr∂μU†∂μU−(Σσ0)3m〈0|u¯u|0〉Tr(U†+U)−13m〈0|u¯u|0〉(Σσ0)4, where U=exp(ifπ∑a=13πaτa) is the standard SU(2) matrix for the pion and Σ, σ0 are given in previous subsection. The last term is required to maintain σ0 as the minimal of the potential for Σ. By expanding the Lagrangian above, the coupling between the pion and the σ is proportional to the mass square Mπ2 of the pion.More specifically, the ππσ coupling term Lππσ reads(6.27)Lππσ=∂μπa∂μπaσσ0−32Mπ2πaπaσσ0=−12Mπ2πaπaσσ0, where we have used the equation of motion in the first term. Therefore, in the massive pion state the expectation value of ϕ can be calculated as(6.28)〈π|ϕ|π〉=12Mπ×Mπ2σ0fσmσ2=Mπ8, where we have used the relation fσmσ2σ0=14 which follows from Eq. (6.16) and Eq. (6.18). The factor 12Mπ is due to overall normalization of the state. This is consistent with Eq. (6.25) that the anomalous contribution is responsible for 18 of the pion mass. Furthermore, it indicates that the scalar field response inside the pion vanishes in the chiral limit, consistent with the expectation that the Goldstone boson arises from the chiral rotation of the QCD vacua.7ConclusionIn this paper we expand our previous study on the implications of anomalous scale symmetry breaking effect on the nucleon mass structure in QCD and other relativistic quantum field theories [24]. The scale symmetry breaking generates a non-perturbative anomalous contribution to the QCD energy called QAE and therefore to all hadron masses. The QAE also sets the scale for the contributions of more familiar quark and gluon kinetic energies.We start by explaining the role of UV divergences in generating the mass scale of QCD through the so-called dimensional transmutation. We demonstrate through a path integral formulation of a two-point function that the trace anomaly naturally arises as consequences of the UV cut-off dependence in QCD-like theories. Furthermore, the QAE contribution to the hadron mass can be derived as resulting from UV cut-off dependence of couplings and quark masses by investigating the time-rescaling property of the theory. Contrary to some mis-understandings in the literature, the naive expression for the Hamiltonian is scheme dependent, but the QAE contrition is scheme independent and is a key physical part of the nucleon mass. We emphasize the importance of Lorentz invariance when renormalizing tensor operators in DR and show that the maximally-scheme-independent decomposition of the hadron mass is facilitated by separating the trace and traceless contributions.We then study the scale symmetry breaking effect and the mass structure in the large N non-linear sigma model in 1+1 dimensions. We demonstrate explicitly in different UV regularization schemes that the QAE is indeed scheme independent and is a crucial part of the mass of the πa particles. On the other hand, the naive Hamiltonian is regulator sensitive and lacks universal physical meaning. Furthermore, the fundamental mass scale in the model is generated through a scalar field that develops a non-vanishing vacuum condensate, resembling the Higgs mechanism of generating quark and electron masses. We also show that in QED, although there is no scale generation, the effect of QAE is perturbative and non-zero, and contributes to the electron pole mass and the famous Lamb-shift.Finally, inspired by the non-linear sigma model, we explore the similarity between the QAE contribution to proton mass generation and the Higgs effect. Similar to the sigma model and standard Higgs mechanism, the nucleon mass can be measured through either static or dynamical response of the Higgs field in the presence of the nucleon. By interpreting the Higgs particles as scalar glueballs, one can determine that the nucleon-glueballs coupling is proportional to nucleon mass, similar to the case of standard Higgs mechanism. We show how these ideas work in an effective theory in which the dimensional transmutation is realized in an effective Higgs phase and the anomalous scalar field acquires a dynamical generated potential. In the chiral limit the QAE contributes to 18 of the pion mass and the pion-glueball coupling is proportional to pion mass square.However, the connection between the anomalous scalar field F2 and the laws of fundamental QCD is not clear to us at present time. In particular, F2 may play crucial role in color confinement as well as spontaneous chiral symmetry breaking. In the MIT bag model, the response of the F2 is related to the bag constant B which plays a role of negative pressure to confine quarks. A more microscopic model for the F2-assisted confinement is provided by 't Hooft [69] in which the Lagrangian of F2 is modified by scalar coupling and generates the flux tubes between colored sources. In the instanton liquid model [10], the F2 contribution is related to the average density of instantons that sets up the fundamental mass scale of the theory. However, a full picture for the role of F2 in the mass generation of the proton and confinement awaits further studies and understanding of lattice QCD simulations [70].CRediT authorship contribution statementAll authors contributed equally to the paper.Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsWe thank K.-F. Liu, Z.-E. Meziani, F. Yuan, and I. Zahed for discussions related to the proton mass. This material is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-SC0020682. A.S. acknowledges support from the German Research Foundation (DFG) through the Trans Regional Collaborative Research Center number 55 (SFB/TRR-55).Appendix ADerivation of Eq. (3.14)In this appendix we provide a derivation of Eq. (3.14). The general idea, similar to that in Ref. [38], is to explore the lattice symmetry. Let's consider two two-point functions in the ensemble given by the action Sλ in Eq. (3.9). One of them extends in temporal direction as before, while the other extends in the spacial direction ei where i=1,2,3(A.1)Gτ(T,0)=∫DUO(Te4,p→=0)O(0,p→=0)e−1g02(λ)Sλ[U]∫DUe−1g02(λ)Sλ[U],(A.2)Gi(T,0)=∫DUO(Tei,p→=0)O(0,p→=0)e−1g02(λ)Sλ[U]∫DUe−1g02(λ)Sλ[U]. Here, e4 and ei are unit vectors along the (imaginary) time and the i-th spacial directions. The scalar operator O is assumed to be local and scale invariant. The notation (Tei,x→) is meant to indicate that the i-th coordinate is T, and the remaining three directions are labeled by x→ with Fourier conjugating variable p→.One notice that, in the continuum limit, Gτ(T,0) becomes the time rescaled version of the two-point function and behaves as e−MT/λ at large T, while Gi(T,0) is not rescaled and behaves as e−MT. By comparing the λ derivatives as before, one obtains the relations:(A.3)−λ2〈1g02∑x→Pτ(x→)〉τ+λ〈1g02∑x→Ps(x→)〉τ+2λ2g03dg0dλ〈∑x→S(x→)〉τ=M,(A.4)−λ2〈1g02∑x→Pτ(x→)〉i+λ〈1g02∑x→Ps(x→)〉i+2λ2g03dg0dλ〈∑x→S(x→)〉i=0. Here the averages 〈〉i and 〈〉τ are defined as(A.5)〈A(x→)〉i=limT→∞〈O(Tei,p→=0)A(04,x→)O(0,p→=0)〉c〈O(Tei,p→=0)O(0,p→=0)〉,(A.6)〈A(x→)〉τ=limT→∞〈O(Te4,p→=0)A(0i,x→)O(0,p→=0)〉c〈O(Te4,p→=0)O(0,p→=0)〉, where 0i≡0ei and 04≡0e4. At λ=1 the symmetry of the hypercubic lattice implies the following relations between the averages(A.7)〈∑x→P4i(x→)〉τ=〈∑x→Pkj(x→)〉k|k≠j,(A.8)〈∑x→Pij(x→)〉τ=〈∑x→Plm(x→)〉k|k≠l,m,(A.9)〈∑x→S(x→)〉τ=〈∑x→S(x→)〉i, where P4i and Pij denotes plaquettes in 4i and ij planes. One now adds Eq. (A.3) to the sum of Eqs. (A.4) over i=1,2,3 and take λ=1. All the averages over Ps and Pτ in the first two terms of Eq. (A.3) and Eq. (A.4) cancel out thanks to Eqs. (A.7) and (A.8), left only with(A.10)M=4×2g03dg0dλ|λ=1〈∫d3x→S(0,x→)〉τ. Comparing with equation Eq. (2.13) one finds(A.11)dg0dλ|λ=1=β(g0)4, which finishes the derivation of Eq. (3.14).Appendix BQAE from mass derivative of HamiltonianIn this appendix we show that the trace anomaly in QED can be obtained from mass derivative of the Hamiltonian in on-shell renormalization scheme where UV cut-off can be traded with the electron mass as the infrared cut-off. This idea can be generalized to other renormalization scheme which we will not consider here.More specifically, the mass-derivative of the QED Hamiltonian gives the trace anomaly(B.1)m∂H∂m=∫d3x→Tμμ(x→)≡4HS, where m is on-shell electron mass. In terms of bare fields and the conjugate momenta, the Hamiltonian in Coulomb gauge reads(B.2)H=∫d3x→(e022(Π→)2+12e02B→2)+∫d3x→ψ¯(−iγ→⋅D→+mZm)ψ+e02∫d3x→d3y→ψ†ψ(x→)ψ†ψ(y→)4π|x→−y→|, where A→ and Π→ are gauge vector potential and the conjugating fields, satisfying the transverse commutation relation [Πi(k→),Aj(x→)]=−i(δij−kikik2)e−ik→⋅x→. Notice that the non-standard appearances of e0 after re-scaling the gauge potential by this factor. The electron pole mass m relates to the bare mass through m0=mZm. Treating e0 and Zm as dimensionless parameters, the naive mass derivative of the Hamiltonian looks like ∫d3x→mZmψ¯ψ. However, to make the theory UV finite, the bare coupling constants e0 and the mass renormalization constant Zm must be function of the m and the UV cutoff Λ:(B.3)e02=e2(1+e212π2lnΛ2m2+..),(B.4)Zm=1−3e216π2lnΛ2m2+... Therefore, the mass derivative of the Hamiltonian also depends on the beta function β(e0)=−m∂e0∂m and the mass anomalous dimension γm=m∂lnZm∂m. Using the relation F2=2(B→2−E→2) and E→=g02Π→, one has the mass derivative of the Hamiltonian(B.5)m∂H∂m=∫d3x→(mZm(1+γm)ψ¯ψ+β(e0)2e03F2), where we have combined the E→2 from the radiation field and the Coulomb field into the total F2, which is just Eq. (B.1).Beside the operator proof, here we also provide a diagrammatic argument of the above derivation, using the QED in background field in Sec. 5 as an example. We show that: taking mass derivatives in one-loop Feynman diagrams Fig. 4 for δEN will exactly produce the one-loop Feynman diagrams for insertion of 4HS. The mass derivative has four origins: the explicit mass dependency of the electron propagator, the implicit mass dependency in the energy level EN, the mass dependencies in renormalization constants δm and Z3−1, and the implicit mass dependency in the wave function uN.The mass derivative of the fermion propagator 1iγ⋅D−m simply reduces to mψ¯ψ operator insertion in the internal electron line as shown in Fig. 7. The mass dependency in EN will lead to the wave function renormalization in external legs. The mass dependencies in renormalization constants δm and Z3−1 will exactly lead to the anomalous energy contribution. Finally, the mass derivative of the external wave function uN is more complicated, which is shown the remaining diagrams where the mψ¯ψ are inserted at external legs. One can see this by using the relation(B.6)m∂∂muN(x→)=∑N′≠NuN′(x→)∫d3y→mu¯N′(y→)uN(y→)EN−EN′+∑N′≠NvN′(x→)∫d3y→mv¯N′(y→)uN(y→)EN+EN′. 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