1 Introduction and main results↩︎

1.1 The anomalous gyromagnetic factor↩︎

A spin-\(1/2\) fermion with mass \(m\) and charge \(e\) has an intrinsic magnetic dipole moment \(\boldsymbol{\mu} = -g (e/2m) \mathbf{S}\), where \(\mathbf{S}\) is the spin three-vector and \(g\) a dimensionless number known as gyromagnetic factor. The quantity \(\mathfrak{a}\equiv (g-2)/2\) is called anomalous gyromagnetic factor; it receives nonzero contributions from all the fundamental interactions \(\mathfrak{a}= \mathfrak{a}_{\mathrm{\small qed}} + \mathfrak{a}_{\mathrm{weak}} + \mathfrak{a}_{\mathrm{\small qcd}}\) and it can be computed in the Standard Model, the theory providing the more fundamental description of elementary particles. The current theoretical value of \(\mathfrak{a}\) for the electron, obtained as result of decades of computations, is \(\mathfrak{a}_e = 0.0011596521817877\dots\), while its experimental value is \(\mathfrak{a}_e^{\mathrm{\small exp}} = 0.00115965218059(13)\) Aoyama_2012?. In the case of the muon, the agreement is also impressive but a bit less precise Aguillard_2023?. Such results provide a very stringent test for the validity of the Standard Model, and possible differences could give an insight on new physical phenomena.

The first theoretical prediction about the \(g\)-factor is due to Dirac, who obtained the result \(g = 2, \mathfrak{a}= 0\) in 1928 Dirac_1930?. Experimental deviations from this value were attributed to quantum radiative contributions due to the interaction with the electromagnetic field. Their evaluation was problematic due to the appearence of diverging integrals (“infinities”) in the computations due to the large energy behavior, and it was finally achieved using the theory of renormalization. Such a procedure is based on the fact that the parameters appearing in the model (called bare parameters), are not the observed ones (called dressed parameters), as they also aquire radiative corrections. It turns out that one can absorb all the divergences in the bare parameters, thus expressing the anomalous gyromagnetic factor as a series in terms of the dressed ones. The corrections to \(\mathfrak{a}\) due to the electroweak forces (there are also corrections due to strong forces which are expected to be much smaller) can be therefore written as a series expansion in \(e\) and \(\mathsf{g}_{\mathrm{w}}\) with \[\label{eq:perturbative95expansion95g} \mathfrak{a}_{\mathrm{\small ew}} \equiv \mathfrak{a}_{\mathrm{\small qed}} + \mathfrak{a}_{\mathrm{\small weak}} =\sum_{n,m} A_{n,m} e^n \mathsf{g}_{\mathrm{w}}^m\tag{1}\] where \(e\) is the electric charge and \(\mathsf{g}_{\mathrm{w}}\) the weak coupling; they can be expressed in terms of the fine structure constant \(\alpha\), that is \(\alpha=e^2/(\hbar c) \approx 1/137\) and \(\mathsf{g}_{\mathrm{w}}^2=4\pi \alpha/\sin^2 \theta_W\) with \(\sin^2\theta_W=0,232 \dots\) and \(\theta_W\) is the Weinberg angle. The smallness of the couplings motivates the perturbative approach. Due to perturbative renormalizablity, \(A_{n,m}\) is finite at all orders. The lowest order electromagnetic contribution \(A_{2,0} e^2=\alpha/\pi\) was obtained by Schwinger in 1948 Schwinger_1948?, while \(A_{4,0} e^4=-(\alpha/\pi)^2 0.328 \dots\) was initially computed by Karplus and Kroll in 1950 Karplus_Kroll_1950? with an error corrected by Petermann and Sommerfeld in 1958 Petermann_1958?, Sommerfeld_1958?. \(A_{6,0}\) has been computed analytically Laporta_1996? and \(A_{8,0}\), \(A_{10,0}\) are known numerically Aoyama_2012?. The lowest order contribution coming from Weak forces is \(A_{0,2}\mathsf{g}_{\mathrm{w}}^2=\mathfrak{a}_{\mathrm{\small z}, 1} + \mathfrak{a}_{\mathrm{\small w}, 1}\), where \(\mathfrak{a}_{\mathrm{\small z}, 1}\) and \(\mathfrak{a}_{\mathrm{\small w}, 1}\) are the neutral and the charged contribution, respectively mediated by the \(Z^0, W^\pm\) bosons Weinberg_1964?. This contribution was computed in 1972 by Jackiw and Weinberg Jackiw_Weinberg_1972? as well as Altarelli, Cabibbo and Maiani Altarelli_1972? within the first perturbative order. In particular, \[\label{eq:JW95result} \mathfrak{a}_{\mathrm{\small z}, 1} = \frac{m^2}{M^2} \frac{\lambda^2}{4\pi^2} \frac{1- 5 \kappa^2}{3} \left[ 1 + \mathcal{O}\left( \frac{m^2}{M^2} \right) \right] =\bar \mathfrak{a}_{\mathrm{\small z}, 1}\left[ 1 + \mathcal{O}\left( \frac{m^2}{M^2} \right) \right]\tag{2}\] where \(\lambda = \mathsf{g}_{\mathrm{w}} (4 \sin^2 \theta_W - 1)/(4 \cos \theta_W), \kappa = (4\sin^2 \theta_W - 1)^{-1}\) and \(M\) is the \(Z^0\) mass. This prediction was historically relevant, because it anticipated the experimental discovery of weak neutral currents Rousset_1994?. Note that \(\mathfrak{a}_{\mathrm{\small z}, 1}\) is proportional to the square of the ratio between the fermion mass and the \(Z^0\) mass, which is a very small quantity (\(\approx 10^{-6}\)) for the electron; it is still small but larger (\(\approx 10^{-3}\)) for the muon. By summing up the first few terms of 1 and adding the QCD contribution, which is computed numerically, one gets the theoretical value mentioned above.

Despite the accuracy of the result obtained truncating the series expansions, the procedure appears mathematically unclear. The series are not convergent and even not asymptotic to any QFT, see e.g. Fredenhagen_2007?; the Higgs sector, necessary to make the Standard Model renormalizable, is trivial, that is, there is no way to define the theory in the removed cutoff limit, unless the limiting theory is non-interacting (see Frohlich_1982?, Aizenman_Copin_2021?).

A possible rigorous approach consists in considering the Standard Model as an effective theory valid up to certain energy scales and therefore requiring an energy cutoff \(\Lambda\) to be defined. One expects that \(\mathfrak{a}\) (and all the physical quantities) are independent on the regularization \(\Lambda\) and agree with the perturbative computations at lowest order, up to an error which is small in a range of values of the couplings including their physical values. There are however no results putting such expectations on a quantitative, rigorous footing. The main difficulty is that one has to quantitatively fuflill two opposite requirements: from one side, \(\Lambda\) must large enough so that no effects are seen, as no sign of cutoff is experimentally observed; on the other hand, the theory is well-defined only up to an energy scale that decreases if the coupling size increases. In the present paper we present the first rigorous result on the anomalous gyromagnetic factor, focusing on the \(Z^0\) contributions of Weak forces and proving upper and lower bounds for it. An announcement of a weaker version of this result with no techical details is in Mastropietro_2024?.

1.2 The regularized neutral Electroweak model↩︎

The neutral sector of the Electroweak model (see e.g. Jackiw_Weinberg_1972? with \(W=A=0\)) describes a fermion \(\psi\) with mass \(m\) interacting with a boson \(\mathcal{Z}_\mu\) with mass \(M\). The theory is chiral, in the sense that the \(\mathcal{Z}_\mu\) boson couples in different ways with the left-handed and the right-handed Weyl components of \(\psi\). We consider an Euclidean regularization of the neutral sector of the Electroweak model obtained via an ultraviolet cutoff \(\Lambda\), to be kept fixed, and an infrared finite-volume cutoff \(L\), to be removed at the end. The model is defined by its expectations, expressed by the functional integral \[\require{physics} \expval{F} = \frac{\int \mathbb{P}_{{g}^{\le N}}(\dd \psi) \int \mathbb{P}_{v}(\dd \mathcal{Z}) \exp[\sum_{s = \pm} \int_{\mathcal{M}_L} \!\! \dd[4]{x} \, \psi^+_s \sigma^\mu_s \psi^-_s i\lambda_s \mathcal{Z}_\mu] F(A,\psi)}{\int \mathbb{P}_{{g}^{\le N}}(\dd \psi) \int \mathbb{P}_{v}(\dd \mathcal{Z}) \exp[\sum_{s = \pm} \int_{\mathcal{M}_L} \!\! \dd[4]{x} \psi^+_s \sigma^\mu_s \psi^-_s i\lambda_s \mathcal{Z}_\mu]},\] where

• Spacetime is represented by the four-dimensional torus \(\mathcal{M}_L \equiv [-L/2, L/2]^4\) with periodic boundary conditions and a finite volume \(L^4\). The spacetime metric is Euclidean, so the metric tensor is simply \(\mathrm{diag}(1, 1, 1, 1)\). A natural norm on \(\mathcal{M}_L\) that is compatible with periodic boundary conditions is given by \(\abs{a}_L \equiv (L/\pi)(\sum_{j=1}^4 \sin^2(\pi a_j/L))^{1/2}\). For every \(a \in \mathbb{R}^4\), we also define the vector \((a)_{\scriptstyle{\mathbb{T}}}= [L/(2\pi)](\sin(2\pi a_1/L), \dots, \sin(2\pi a_4/L))\).

• The ultraviolet regularization is realized through the function \(\chi_N(k) \equiv \chi_0(2^{-N}k)\), where \(\chi_0\) is a rotationally invariant smooth function on \(\mathbb{R}^4\) such that \(\chi_0(k)=0\) for \(\abs{k} \ge 2\) and \(\chi_0(k)=1\) for \(\abs{k} \le 1\). We assume that \(\chi_0\) has Gevrey class \(2\). The cutoff scale associated with \(\chi_N\) is \(\Lambda\equiv 2^N\).

• Given the set \(\mathcal{D}_{L, N} = (2\pi \mathbb{Z}/L)^4 \cap \mathop{\mathrm{supp}}(\chi_N)\) and a function \(\hat{\mathcal{Z}}_{k \mu} \colon \mathcal{D}_{L, N} \to \mathbb{R}^{4 \abs*{\mathcal{D}_{L, N}}}\), we define the Gaussian measure \[\require{physics} \label{eq:bosonic95gaussian95measure} \mathbb{P}_{v}(\dd \mathcal{Z}) \equiv \mathcal{N}_\mathcal{Z} \prod_{k \in \mathcal{D}_{L, N}} \prod_{\mu = 0}^3 \dd{\hat{\mathcal{Z}}_{k \mu}} \, \exp \left( - \frac{1}{2} \frac{1}{L^4} \sum_{k \in \mathcal{D}_{L, N}} \!\!\! \hat{\mathcal{Z}}_{k \mu} \left[\hat{v}^{-1}(k) \right]^{\mu \nu} \! \hat{\mathcal{Z}}_{-k, \nu} \right),\tag{3}\] where \[\label{eq:free95boson95propagator} v^{\mu \nu}(x-y) =\frac{1}{L^4} \sum_{k \in \mathcal{D}_{L}} \frac{\delta_{\mu \nu} \chi_N(k)}{\abs*{k}^2 + M^2} \, e^{i k \cdot (x-y)} \equiv \frac{\delta_{\mu \nu}}{L^4} \sum_{k \in \mathcal{D}_{L}} e^{i k \cdot (x-y)} \, \hat{v}(k),\tag{4}\] see e.g. Bauerschmidt_2019?. The constant \(\mathcal{N}_\mathcal{Z}\) is chosen so that \(\require{physics} \int \mathbb{P}_{v}(\dd \mathcal{Z}) = 1\).

• Given the finite-dimensional Grassmann algebra generated by the anticommuting variables \(\lbrace \hat{\psi}^\pm_{k \alpha} \rbrace_{\alpha=1, \dots 4, k \in \mathcal{D}_{L, N}}\), we define the Grassmann integral as \(\require{physics} \int \dd{\hat{\psi}^\pm_{k \alpha}} = 0, \int \dd{\hat{\psi}^\pm_{k \alpha}} \, \psi^{\pm}_{k \alpha} = 1\), see e.g. Feldman_2002?, and we let \[\require{physics} \label{eq:fermionic95gaussian95measure} \mathbb{P}_{{g}^{\le N}}(\dd \psi) \equiv \mathcal{N}_\psi \prod_{k \in \mathcal{D}_{L, N}} \prod_{\alpha = 1}^4 \dd{\hat{\psi}^+_{k \alpha}} \dd{\hat{\psi}^-_{k \alpha}} \, \exp \left( - \frac{1}{L^4} \sum_{k \in \mathcal{D}_{L, N}} \!\!\! \hat{\psi}^+_{k \alpha} \left[({\hat{g}}^{\le N})^{-1}(k) \right]^{\alpha \beta} \! \hat{\psi}^-_{k \beta} \right)\tag{5}\] with \[\label{eq:free95fermion95propagator} {g}^{\le N}(x) \equiv \frac{1}{L^4} \sum_{k \in \mathcal{D}_{L}} \frac{\chi_N(k)}{i k_\mu \gamma^\mu_N + \mathsf{m}_N} \, e^{i k \cdot x} \equiv \frac{1}{L^4} \sum_{k \in \mathcal{D}_{L}} e^{i k \cdot x} \, {\hat{g}}^{\le N}(k).\tag{6}\] As before, the constant \(\mathcal{N}_\psi\) is chosen so that \(\require{physics} \int \mathbb{P}_{{g}^{\le N}}(\dd \psi) = 1\); the \(4 \times 4\) complex matrices \(\lbrace \gamma^\mu_N \rbrace_{\mu = 0}^3\), \(\mathsf{m}_N\) take the form \[\label{eq:gamma95matrices95and95mass95matrix} \gamma^\mu_N \equiv \begin{pmatrix} 0 & Z^+_N \sigma_+^\mu \\ Z^-_N \sigma_-^\mu & 0 \end{pmatrix}, \qquad \mathsf{m}_N \equiv \begin{pmatrix} m^+_N & 0 \\ 0 & m^-_N \end{pmatrix},\tag{7}\] where \(\sigma_{\pm} \equiv (\mathbb{1}_{2 \times 2}, \mp i \sigma_1, \mp i \sigma_2, \mp i \sigma_3)\) and \(\sigma_1, \sigma_2, \sigma_3\) are the standard Pauli matrices. In the following sections, we will often use the standard Euclidean gamma matrices \[\label{eq:standard95gamma95matrices} \gamma^\mu \equiv \begin{pmatrix} 0 & \sigma_+^\mu \\ \sigma_-^\mu & 0 \end{pmatrix}, \qquad \gamma^5 \equiv \begin{pmatrix} \mathbb{1}_{2 \times 2} & 0 \\ 0 & -\mathbb{1}_{2 \times 2} \end{pmatrix}.\tag{8}\] together with the Feynman slash notation \(\cancel{k} \equiv \gamma^\mu k_\mu\). The gamma matrices satisfy the anticommutation relations \(\lbrace \gamma^\mu, \gamma^\nu \rbrace = 2 \delta^{\mu \nu}\).

• We define \(\psi_\alpha^\epsilon(x) \equiv L^{-4} \sum_{k \in \mathcal{D}_{L, N}} \hat{\psi}_{k \alpha}^\epsilon \, e^{i \epsilon k \cdot x}\), thought as a Grassmann-valued smooth periodic function on \(\mathcal{M}_L\). We will often use the notation \[\psi^-(x) \equiv \begin{pmatrix} \psi^-_1 \\ \psi^-_2 \\ \psi^-_3 \\ \psi^-_4 \end{pmatrix}(x) \equiv \begin{pmatrix} \psi^-_- \\[2pt] \psi^-_+ \end{pmatrix}(x), \qquad \psi^+(x) \equiv \begin{pmatrix} \psi_1^+ \\ \psi^+_2 \\ \psi^+_3 \\ \psi^+_4 \end{pmatrix}^{\!\!\mathsf{T}}(x) \equiv \begin{pmatrix} \psi^+_+ \\ \psi^+_- \end{pmatrix}^{\!\!\mathsf{T}}(x),\] where the \(\mathsf{T}\) superscript denotes transposition (so \(\psi^+(x)\) is thought as a row vector) and the two-dimensional Grassmann fields \(\lbrace \psi^-_s(x) \rbrace_{s = \pm}\) are equal to the left-handed (\(s = -\)) and right-handed (\(s = +\)) Weyl components of the Dirac field \(\psi^-(x)\). Throughout the rest of the paper, we use Greek indices to denote the components of the Dirac fields \(\psi^+, \psi^-\). Given a pair of components \(\psi^+_\alpha, \psi^-_\beta\), the indices \(\alpha, \beta\) are said to have opposite chiralities if \(\alpha, \beta \in \lbrace 1, 2 \rbrace\) or \(\alpha, \beta \in \lbrace 3, 4 \rbrace\). Conversely, \(\alpha, \beta\) are said to have similar chiralities if \(\alpha \in \lbrace 1, 2 \rbrace, \beta \in \lbrace 3, 4 \rbrace\) or \(\alpha \in \lbrace 3, 4 \rbrace, \beta \in \lbrace 1, 2 \rbrace\).

• Finally, the constants \(\lambda_+, \lambda_-\) are related with the physical parameters as \(2\lambda \equiv \lambda_+ + \lambda_-, 2\lambda \kappa \equiv \lambda_- - \lambda_+\).

The connected correlation functions of our theory are obtained from the generating functional \[\require{physics} \begin{gather} \label{eq:generating95functional} W[J, \eta] \equiv \log \int \mathbb{P}_{{g}^{\le N}}(\dd \psi) \, \mathbb{P}_{v}(\dd \mathcal{Z}) \, \times \\ \times \exp \left[\sum_{s = \pm} \int_{\mathcal{M}_L} \!\! \dd[4]{x} \, [\psi^+_s \sigma^\mu_s \psi^-_s (i\lambda_s \mathcal{Z}_\mu + Z^{J, s}_N J_\mu) + \eta^+_s \psi^-_{-s} + \psi^+_s \eta^-_{-s}](x) \right]. \end{gather}\tag{9}\] By letting \[\require{physics} \label{eq:schwinger95functions} G_{\underline{\alpha}, \underline{\beta}, \underline{\mu}}^{r,s}(\underline{x}, \underline{y}, \underline{z}) \equiv \frac{\partial}{\partial \eta^-_{\underline{\beta}}(\underline{y})} \frac{\partial}{\partial \eta^+_{\underline{\alpha}}(\underline{x})} \frac{\partial}{\partial J_{\underline{\mu}}(\underline{z})} W[J, \eta] \eval_{J, \eta = 0},\tag{10}\] the two-point function and the vertex function are given by \[S_{\alpha \beta}(x, y) \equiv G_{\alpha, \beta}^{1, 0}(x, y), \qquad (\mathfrak{I}^\mu)_{\alpha \beta}(x, y, z) \equiv G_{\alpha, \beta, \mu}^{2, 1}(x, y, z).\] Since the system is translationally invariant, we define \[\require{physics} \label{ct} \hat{S}(k) \equiv \int_{\mathcal{M}_L} \dd[4]{x} e^{-i k \cdot x} \, S(x, 0), \qquad \hat{\mathfrak{I}}^\mu(p', p) \equiv \int_{\mathcal{M}_L^2} \dd[4]{x} \dd[4]{y} e^{-i p' \cdot x + i p \cdot y} \, \mathfrak{I}^\mu(x, y, 0).\tag{11}\] Finally, the amputated vertex function in Fourier space is \[\hat{\Gamma}^\mu(p', p) \equiv [\hat{S}(p')]^{-1} \cdot \hat{\mathfrak{I}}^\mu(p', p) \cdot [\hat{S}(p)]^{-1}(p).\] After integrating out the boson fields, the integral 9 can be written as \[\require{physics} \label{eq:generating95functional95fermionic} W[J, \eta] = \log \int \mathbb{P}_{{g}^{\le N}}(\dd \psi) \, \exp( {V}^{N}[\psi, J, \eta] )\tag{12}\] with \[\require{physics} \begin{gather} \label{eq:ultraviolet95fermionic95potential} {V}^{N}[\psi, J, \eta] \equiv - \frac{\lambda^2}{2} \int_{\mathcal{M}_L^2} \dd[4]{x} \dd[4]{y} \, (\psi^+ \Upsilon^\mu \psi^-)(x) \, v(x - y) \, (\psi^+ \Upsilon^{\mu} \psi^-)(y) \, + \\ + \sum_{s = \pm} \int_{\mathcal{M}_L} \dd[4]{x} (Z^{J, s}_N \psi^+_s \sigma^\mu_s J_\mu \psi^-_s + \eta^+_s \psi^-_{-s} + \psi^+_s \eta^-_{-s})(x) \end{gather}\tag{13}\] and \(\lambda \equiv (\lambda_+ + \lambda_-)/2, \Upsilon^\mu \equiv \gamma^\mu(1 - \kappa \gamma^5),\kappa \equiv (\lambda_+ - \lambda_-)/(2\lambda_+ + 2\lambda_-)\). Whenever there is no risk of ambiguity, we adopt the condensed notation \((J, \eta) \equiv \omega\), it being understood that \(\omega= (\omega_1, \dots, \omega_{12}) = (J_0, \dots, \eta^-_4)\).

If possible, the bare parameters \(Z^s_N, Z^{J, s}_N, m^s_N\) have to be chosen so that the dominant parts of the correlation functions describe a fermion whose charge and field strength renormalization are equal to \(1\) and whose mass is equal to \(m\). In particular, the two-point function and the vertex function must satisfy, in the limit \(L\to\infty\), the renormalization conditions \[\begin{align} \label{eq:conditions95correlations} \begin{aligned} \hat{S}(k) & =\frac{1}{i \cancel{k}+m} \cdot \left( 1 + \mathcal{O}\left[ \left( \frac{\max \lbrace m, \abs*{k}\rbrace}{\Lambda} \right)^\theta \right] \right), \\ \hat{\Gamma}^\mu(p', p) & = \gamma^\mu + \mathcal{O}\left[ \left( \frac{\max \lbrace m, \abs{p}, \abs*{p'} \rbrace}{\Lambda} \right)^\theta \right] \end{aligned} \end{align}\tag{14}\] for some \(\theta>0\).

1.3 The regularized gyromagnetic factor↩︎

The regularized gyromagnetic factor is given by \[\label{eq:g-2:def} \mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\equiv \sum_{\ell = 0}^K \frac{(im)^\ell}{\ell!} \mathcal{A}^{(\ell)}(0)\tag{15}\] with \[\label{eq:noncovariant95gfunction} \mathcal{A}(z) \equiv \frac{z \epsilon_{abc} \tr[\gamma^5 (1 + \gamma^0) (\partial_{p'} - \partial_p)^a \hat{\Gamma}^b(p_z, p_z) \, \gamma^c] + 2 \tr[\gamma_a \hat{\Gamma}^a(p_z, p_z)]}{6 \tr[(1 + \gamma^0) \hat{\Gamma}^0(p_z, p_z)]} - 1\tag{16}\] where \(a, b, c \in \lbrace 1, 2, 3 \rbrace\) and \(p_z \equiv (z, 0, 0, 0)\). If the formal \(L, N, K \to\infty\) limit is taken, this definition formally coincides with the standard one (see e.g. Peskin_1995?). More details about this definition are provided in Appendix 5. Our main result is the following

The result rigorously shows that the anomalous gyromagnetic factor in the effective regularized theory coincides with the truncation of the perturbative expansion with cutoff removed (whose sum does not exist), up to regularizaton-dependent corrections which are small in a suitable region of the parameter space. Therefore, this result allows to estimate the error due to truncation in the formal expansion, even if it is non convergent. The physical regime corresponds to \(m/M\) and \(\lambda\) small with \(m/M \ll \lambda\) (in reality, \(\lambda\sim 10^{-2}\) and \(m/M\sim 10^{-6}\) in the case of electrons). Hence, the first term appearing in the remainder \(R_\lambda\) is surely small; further corrections due to higher values of \(K\) would be dominated by this one. The second term comes from the difference between the lowest order contribution of the regularized theory and the one without cutoff, and it is small if \(\Lambda\) is chosen to be much higher than the boson mass \(M\). The third term takes into account higher orders in \(\lambda\) and it is small provided that \(\lambda\) is sufficiently small, that is \(\lambda \ll \mathcal{O}( (M/\Lambda)\log^{-1}(M/m))\). A crucial aspect of our result is the fact that this term depends logarithmically on \(M/m\): a power law dependence would imply a completely unrealistic smallness condition such as \(\lambda \ll \mathcal{O}(m/\Lambda)\). This is achieved by proving suitable cancellations in the expansion, implying that all the higher orders contributions are \(\mathcal{O}(m^2/M^2)\). The choice \(K=4\) is enough to get upper and lower bounds in terms of the Jackiw-Weimberg formula 2 .

1.4 Sketch of the proof↩︎

After integrating out the bosons, the amputated vertex function appearing in 16 is expressed by Grassmann integrals which can be analyzed by means of rigorous Renormalization Group methods. The amputated vertex function is written as a series in the coupling \(\lambda\) and the running coupling constants, that is \[\label{eq:vertex95splitting} \hat{\Gamma}^\mu(p', p)= \hat{\Gamma}_a^\mu(p', p)+\hat{\Gamma}^\mu_b(p', p),\tag{17}\] where the dominant part, \(\hat{\Gamma}_a^\mu(p', p)\), only depends from the running coupling constants; \(\hat{\Gamma}^\mu_b(p', p)\) is subdominant and can be expressed as a series in \(\lambda\), namely \[\label{eq:subdominant95vertex95expansion} \hat{\Gamma}^\mu_b(p', p)=\sum_{n=1}^{+\infty} \lambda^{2 n} \hat{\Gamma}_n^\mu(p', p; \lambda).\tag{18}\] Note that the above is not a power series in \(\lambda\), as the running coupling constants are also functions of \(\lambda\) (that is, \(\hat{\Gamma}_n^\mu(p', p; \lambda)\) depend from \(\lambda\)). It turns out that, by fixing the bare constants so that the renormalization conditions 14 are verified, the contribution to \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) coming from \(\hat{\Gamma}_a^\mu\) is vanishing. The contribution coming from \(\lambda^2 \hat{\Gamma}_1^\mu(p', p; \lambda)\) is expressed by several renormalized graphs whose sum reconstructs the Jackiw-Weinberg result, up to corrections which are included in the first and second term of ?? . In particular, the dominant part of \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) turns out be of order \(\lambda^2 m^2/M^2\). The higher orders (\(\lambda^{2 n} \hat{\Gamma}_n^\mu(p', p; \lambda)\), with \(n\ge 2\)) can also be written as sums over renormalized Feynman graphs. In order to prove convergence, however, it is convenient not to bound each graph separately, but to control their sum using suitable determinant bounds which exploit cancellations caused by the anticommutativity of fermionic variables. A dimensional bound for \(\hat{\Gamma}_n^\mu(p', p; \lambda)\) would be \(\mathcal{O}(C^n (\Lambda^2/M^2)^n)\); this estimate is enough for convergence, but not for ensuring the lowest order dominance, as it does not contain the small \(m^2/M^2\) factor. An improvement of the bound for the contributions coming from \(\hat{\Gamma}^\mu_b(p', p)\) follows from certain cancellations due to exact or approximate symmetries; the last ones are indeed violated by the presence of the fermion mass, which is however small. As a consequence, the renormalized scaling dimension is smaller than or equal to \(-2\) and this produces the desidered \(m^2/M^2\) gain up to some harmless logarithmic corrections.

1.5 Open problems and remarks↩︎

The regularized neutral Electroweak model considered in this paper shares some similarities with Condensed Matter models that admit an emerging Quantum Field Theory description, like graphene, Hall systems or Weyl semimetals. In such systems, the lattice of ions among which the conduction elctrons are hopping provides a physical realization of the ultraviolet cutoff. A basic property of these Condensed Matter models is the universality of certain transport coefficients, that is, the independence of such quantities from the microscopic details. In particular, universality is present in the optical conductivity of graphene (\(\sigma=(e^2/h) \cdot 1/(2\pi)\)) or in the Hall conductance (\(\sigma_n = n e^2/h\), where \(n \in \mathbb{N}\)). These coefficients are strictly related with quantum anomalies of the emerging QFT description.

The transport coefficients, togheter the anomalous gyromagnetic factor of the electron, are measured with very high precision and their theoretical values are used to determine the fine structure constant. Note that, in the case of the transport coefficients, the theoretical prediction coincides with the lowest order computation in perturbation theory, while the anomalous gyromagnetic factor is a series with non vanishing coefficients at any order. The Renormalization Group approach used in this paper provides an explanation of this fact: the crucial point is that transport coefficients (as most of the correlation functions that are actually measured in Condensed Matter experiments) are usually associated to marginal or relevant terms, while the anomalous gyromagnetic factor is associated to an irrelevant one.

The universality of the transport coefficients in presence of a weak short range many-body interaction has been proven in \(3+1\) dimensions Mastropietro_2007?, Giuliani_Mastropietro_Porta_2019?, Mastropietro_2024_Vanishing? or in \(2+1\) dimensions Giuliani_Mastropietro_Porta_2019?, Giuliani_Mastropietro_Porta_2017? in the case of graphene. The transport coefficients are decomposed as sums of two terms as in 17 . In this case, however, only the first term (involving marginal couplings) gives a nonzero contribution; the second one, involving irrelevant terms, is exactly zero. No need of improvment of estimates with respect to power counting is necessary in this case. From this fact follows the universality of transport coefficients or the related phenomenon of the non-renormalization of quantum anomalies in presence of a finite cutoff. The anomalous gyromagnetic factor, on the contrary, only gets contributions from higher orders in the renormalized expansion, which involve irrelevant interactons.

There are several natural extensions of the present work, which could be possibly treated with the methods introduced in this paper and their generalizations. A natural question is to improve the smallness condition on the coupling, \(\lambda \ll \mathcal{O}( (M/\Lambda)\log^{-1}(M/m))\). In the case of renormalizable models, one could hope to replace such condition with \(\lambda \ll \mathcal{O}(\log^{-1}(\Lambda/M)\log^{-1}(M/m))\), which would allow to reach much higher cutoffs. The model considered here is perturbatively non renormalizable due to the presence of anomalies, so this improvment is not expected here. However, after including the quark contributions with realistic values of the charges, the anomalies are expected to cancel and perturbative renormalizablity is recovered, so one can hope to get such a result in this generalized framework. As a starting point, one could consider a vector model like massive QED\(_4\), and even in this case the construction up to exponentially high cutoff is unknown. As a preliminary step, one would like to repeat the analysis presented here with lattice cutoff (which is necessary for the anomaly cancellation), where the symmetries at the basis of the above mentioned cancellations are different. The issue of QED\(_4\) contributions to the gyromagnetic factor with vanishing photon mass is also a natural problem to be considered, see e.g. Kaplan_2021?.

The paper is structured in the following way. In Section 2 we developed a Renormalization Group analysis and a tree expansion for the correlations, incorporating exact and emeging symmetries. In Section 3 we prove bounds for the coefficient of the expansion for the vertex function, proving its convergence using determinant bounds and cluster expansion formulas for truncared expectations. Finally, in Section 4 we prove the decomposition 18 and the relative bounds with improvements due to cancellations, leading to the proof of the main result.

2 Renormalization Group analysis↩︎

2.1 Multiscale decomposition↩︎

The Grassmann integral occurring in 12 is perfomed iteratively using the Gaussian convolution identity; namely, if \(g = g_1 + g_2\), we have \[\require{physics} \label{eq:measure95additivity} \int \mathbb{P}_{g_1}(\dd \psi) \, F(\psi)=\int \mathbb{P}_{g_1}(\dd \psi_1) \int \mathbb{P}_{g_1}(\dd \psi_2) \, F(\psi_1+\psi_2)\tag{19}\] for every functional \(F\). This allows to represent the generating functional 12 as \[\require{physics} \label{eq:generating95functional95on95scale95h} W[\omega] = \log \int \mathbb{P}_{{g}^{\le h}}(\dd \psi) \exp(\mathscr{V}^{h}[\psi, \omega]),\tag{20}\] where \(h \in \lbrace h^\star, \dots, N \rbrace\), \(h^\star = \lfloor \log_2 m \rfloor\), \(\chi_h(k) \equiv \chi_0(2^{-h} k)\) and \[\label{eq:renormalized95propagator} {\hat{g}}^{\le h}(k) \equiv \chi_h(k) \cdot \begin{pmatrix} m_h^+ & i Z_h^+ \sigma^\mu_+ k_\mu \\ i Z_h^- \sigma^\mu_- k_\mu & m_h^- \\ \end{pmatrix}^{\!-1} \equiv \frac{\chi_h(k)}{i k_\mu \gamma^\mu_h + \mathsf{m}_h}.\tag{21}\] The effective potential on scale \(h\) takes the form \[\require{physics} \label{eq:veff95kernels} {\mathscr{V}}^{h}[\psi, \omega] = \sum_{A, B} \int \dd{\underline{x}} \dd{\underline{y}} \, {W}^{h}(A, \underline{x}; B, \underline{y}) \Psi(A, \underline{x}) \Omega(B, \underline{y}),\tag{22}\] where \({W}^{h}(A, \underline{x}; B, \underline{y})\) are smooth, periodic, translationally invariant kernels and the field monomials \(\Psi(A, \underline{x}) \Omega(B, \underline{y})\) contain at most three derivatives on each field \(\psi^\pm\). Here, we adopt the same notation introduced in Giuliani_Mastropietro_Rychkov_2021?: given a pair of index sets \(A^\pm = \lbrace A_1^\pm, \dots, A_s^\pm \rbrace\) with \(A^\pm_\ell = (\alpha_\ell^\pm, \underline{\delta}_\ell^\pm) \subseteq \lbrace 1, \dots, 4 \rbrace \times \mathbb{N}^4\), we let \[\label{eq:field95monomials} \Psi(A, \underline{x}) \equiv \prod_{a = 1}^{\abs*{A^+}} \partial_{\underline{\delta}_a^+} \psi^+_{\alpha_a^+}(x_a^+) \prod_{b = 1}^{\abs*{A^-}} \partial_{\underline{\delta}_b^-} \psi^-_{\alpha_b^-}(x_b^-),\tag{23}\] where \(A \equiv A^+ \sqcup A^-\), \(\underline{x} = (x_1^+, \dots, x_s^-)\) and \(\partial_{\underline{\delta}_\ell^\pm} = \prod_{r = 1}^4 (\partial_r)^{\delta_\ell^{\pm, r}}\) with \(\abs*{\underline{\delta}_\ell^\pm} \equiv \sum_{r = 1}^4 \abs*{\delta_\ell^{\pm, r}} \le 3\). In the same way, we let \[\label{eq:source95field95monomials} \Omega(B, \underline{y}) \equiv \prod_{a = 1}^{\abs*{B_J}} J^{\mu_a}(x_a) \prod_{b = 1}^{\abs*{B_{\eta}^+}} \eta^+_{\alpha^+_b}(y_b^+) \prod_{c = 1}^{\abs*{B_{\eta}^-}} \eta^-_{\alpha^-_c}(y_c^-),\tag{24}\] where the index set \(B = B_J \sqcup B^+_\eta \sqcup B^-_\eta\) is constructed as before.

The representation 20 coincides with 12 when \(h = N\), so it will be fully determined once we show how to pass from an integer \(h\) to \(h-1\).

2.2 Localization↩︎

We write 20 by decomposing \({\mathscr{V}}^{h}[\psi, \omega]\) as \[\require{physics} \label{eq:generating95functional95RL} W[\omega] = \log \int \mathbb{P}_{{g}^{\le h}}(\dd \psi) \exp(\mathcal{L}{\mathscr{V}}^{h}[\psi, \omega] + \mathcal{R}{\mathscr{V}}^{h}[\psi, \omega]),\tag{25}\] where \(\mathcal{R}=\mathbb{1}-\mathcal{L}\). Depending on the structure of the kernel \(W(A, \underline{x}; B, \underline{y})\), the localization operator \(\mathcal{L}\) is defined as \[\require{physics} \label{eq:localization95definition} \mathcal{L}\left[ W(A, \underline{x}; B, \underline{y}) \, \Psi(A, \underline{x})\Omega(B, \underline{y}) \right] \equiv \begin{cases} (\mathcal{L}_{0, x} + \mathcal{L}_{1, x} + \bar{\mathcal{L}}_{2, x})\left[ W_{\alpha \beta}(x, y) \psi^+_\alpha(x) \psi^-_\beta(y) \right] \\[5pt] (\mathcal{L}_{0, z} + \bar{\mathcal{L}}_{1, z}) \left[ W_{\alpha \beta \mu}(x, y, z) \psi^+_\alpha(x) \psi^-_\beta(y) J_\mu(z) \right] \\[5pt] (\mathcal{L}_{0, x} + \bar{\mathcal{L}}_{1, x}) \left[ W_{\alpha \beta \underline{\delta}}(x, y) \partial_{\underline{\delta}_1} \psi^+_\alpha(x) \partial_{\underline{\delta}_2} \psi^-_\beta(y) \eval_{\underline{\delta} = 1} \right] \\[5pt] \bar{\mathcal{L}}_{0, x} \left[ W_{\alpha \beta \underline{\delta}}(x, y) \partial_{\underline{\delta}_1} \psi^+_\alpha(x) \partial_{\underline{\delta}_2} \psi^-_\beta(y) \eval_{\underline{\delta} = 2} \right] \\[5pt] \bar{\mathcal{L}}_{0, z} \left[ W_{\alpha \beta \mu \underline{\delta}}(x, y, z) \partial_{\underline{\delta}_1} \psi^+_\alpha(x) \partial_{\underline{\delta}_2} \psi^-_\beta(y) J_\mu(z) \eval_{\underline{\delta} = 1} \right] \\[5pt] 0 \qquad \text{In any other case} \end{cases}\tag{26}\] where \(\abs*{\underline{\delta}} \equiv \abs*{\underline{\delta}_1} + \abs*{\underline{\delta}_2}\) and

• If \(x_0\) is a spacetime point among \(\underline{x}, \underline{y}\), the operator \(\mathcal{L}_{\ell, x_0}\) acts as \[\require{physics} \label{eq:taylor95localization95definition} \mathcal{L}_{\ell, x_0} \left[ W(A, \underline{x}; B, \underline{y}) \, \Psi(A, \underline{x})\Omega(B, \underline{y}) \right] \equiv W(A, \underline{x}; B, \underline{y}) \, \frac{1}{\ell!} \frac{\dd^\ell}{\dd t^\ell}\Psi(A, \underline{x}_t)\Omega(B, \underline{y}_t) \eval_{t = 0},\tag{27}\] where the spacetime points \(\underline{x}_t, \underline{y}_t\) are obtained by replacing every point \(z\) occurring among \(\underline{x}, \underline{y}\) with the interpolating point \(I_t(x_0, z) \equiv x_0 + t(z - x_0)_{\scriptstyle{\mathbb{T}}}\).

• \(\bar{\mathcal{L}}_{\ell, x_0}\) acts on a kernel with spinor indices \(\alpha, \beta\) as \[\bar{\mathcal{L}}_{\ell, x_0}= \begin{cases} \mathcal{L}_{\ell, x_0} & \mathrm{if } \alpha, \beta \mathrm{ have similar chiralities} \\ \mathcal{P}\mathcal{L}_{\ell, x_0} & \mathrm{if } \alpha, \beta \mathrm{ have opposite chiralities} \end{cases}\] with \(\mathcal{P}(\, \cdot \,) \equiv (\, \cdot \,) \vert_{m_N = 0}\).

The renormalization operator \(\mathcal{R}= \mathbb{1}- \mathcal{L}\) can be decomposed as \(\mathcal{R}\equiv \mathcal{R}^0 + \mathcal{R}^1\), where \(\mathcal{R}^0 = \mathbb{1}- \mathcal{L}_{0, x_0} - \mathcal{L}_{1, x_0} - \cdots - \mathcal{L}_{R - 1, x_0}\) and \(\mathcal{R}^1\) is either equal to \(0\) or to \((\mathbb{1}- \mathcal{P})\mathcal{L}_{R - 1, x_0}\) (both \(R\) and \(x_0\) depend on the kernel on which \(\mathcal{R}\) acts). The \(\mathcal{R}^0\) operator will be called Taylor renormalization.

An important property of the localization operation is that

Note that several terms which should appear in the above expression based on definition 26 are indeed absent due to symmetry properties. In particular, the \(\bar{\mathcal{L}}_{\ell, x_0}\) operators displayed in 26 have no influence on \(\mathcal{L}{\mathscr{V}}^{h}[\psi, \omega]\).

The action of the renormalization operator on the kernel representation 22 can be understood as follows. Consider a smooth periodic function \(F \colon \mathcal{M}_L^2 \to \mathbb{C}\) whose Fourier coefficients take the form \(\hat{F}(k_0, k) = f_1(k_0) f_2(k)\), where \(f_2\) vanishes as long as the norm of its argument is greater than \(2^N\). If \(\mathcal{R}^0_{R, x_0} = \mathbb{1}- \mathcal{L}_{0, x_0} - \cdots - \mathcal{L}_{R-1, x_0}\) and \(W(x_0, x) = W(0, x - x_0)\) is a smooth, periodic, translationally invariant kernel, then \[\require{physics} \label{eq:ren95kernel95eq} \int \dd{x_0} \dd{x} \mathcal{R}^0_{R, x_0} [W(x_0, x) F(x_0, x)] = \int \dd{x_0} \dd{x} (\mathscr{R}^0_{R, x_0} W)^{\underline{\alpha}}(x_0, x) \frac{\partial F}{\partial x^{\underline{\alpha}}}(x_0, x)\tag{28}\] with \[\require{physics} \label{eq:ren95taylor95kernel} (\mathscr{R}^0_{R, x_0} W)^{\underline{\alpha}}(x_0, x) \equiv \int_0^1 \dd{t} \frac{(1-t)^{R-1}}{(R-1)!} \int \dd{y} W(0,y) \, y^{\underline{\alpha}}_{\scriptstyle{\mathbb{T}}}\, \delta_N(x - x_0 - t y_{\scriptstyle{\mathbb{T}}}).\tag{29}\] Here, \(\underline{\alpha} = (\alpha_1, \dots, \alpha_R) \in \lbrace 0, \dots, 3 \rbrace^R\) is a multi-index: given any \(x \in \mathbb{R}^4\), the notations \(x^{\underline{\alpha}}\) and \(\partial/\partial x^{\underline{\alpha}}\) respectively stand for \(\prod_{\ell = 1}^R x^{\alpha_\ell}\) and \(\prod_{\ell = 1}^R \partial/\partial x^{\alpha_\ell}\) and sum over \(\alpha_1, \dots, \alpha_R\) is understood in both 28 and 29 . Finally, the “regularized delta” \(\delta_N(x)\) is a periodic function on \(\mathcal{M}_L\) whose Fourier coefficients are given by \(\hat{\delta}_N(k) \equiv \chi_{N+1}(k)\).

Formula 29 is an immediate consequence of the interpolation identity \[\require{physics} \begin{gather} \label{eq:lm:interpolating95formula} \int \dd{x_0} \dd{x} \mathcal{R}^0_{R, x_0} [W(x_0, x) F(x_0,x)] = \\ = \int \dd{x_0} \dd{x} W(x_0, x) \sum_{\underline{\alpha}} \int_0^1 \dd{t} \frac{(1-t)^{R-1}}{(R-1)!} (x - x_0)^{\underline{\alpha}}_{\scriptstyle{\mathbb{T}}}\, \partial_{\underline{\alpha}} F(x_0, I_t(x_0, x)). \end{gather}\tag{30}\] Note that, according to our initial assumption, \(\hat{F} \hat{\delta}_N = \hat{F}\). In position space, this equality becomes \[\require{physics} F(x_0, z) = \int \dd{y} F(x_0, y) \delta_N(y - z) \qquad \forall z \in \mathcal{M}_L,\] so the right hand side of 30 can be rewritten as \[\require{physics} \label{eq:delta95insertion} \int \dd{x_0} \dd{x} \dd{y} \int_0^1 \dd{t} \delta_N(y - x_0 - t(x - x_0)_{\scriptstyle{\mathbb{T}}}) \, W(0, x-x_0) \frac{(1-t)^{R-1}}{(R-1)!} (x - x_0)^{\underline{\alpha}}_{\scriptstyle{\mathbb{T}}}\, \partial_{\underline{\alpha}} F(x_0, y),\tag{31}\] where we used the fact that \(W(x_0, x) = W(0, x - x_0)\). After performing the change of variables \(x \mapsto x + x_0\) together with the relabelling \(x \leftrightarrow y\), the integral 31 becomes \[\require{physics} \int \dd{x}_0 \dd{x} \dd{y} \int_0^1 \dd{t} \delta_N(x - x_0 - t y_{\scriptstyle{\mathbb{T}}}) \, W(0, y) \frac{(1-t)^{R-1}}{(R-1)!} y^{\underline{\alpha}}_{\scriptstyle{\mathbb{T}}}\, \partial_{\underline{\alpha}} F(x_0, x)\] and by comparing this with 28 we deduce that \((\mathscr{R}_{R, x_0}^0 W)^{\underline{\alpha}}\) is indeed given by 29 .

The above construction can be applied to the \(\mathcal{L}, \mathcal{R}^1\) operators as well and it can be generalized in order to deal with kernels having two external fermionic legs and one \(J\) source field. By adopting the synthetic notation \(\underline{x}\) for either \(x_1\) or \((x_1, x_2)\), we have \[\require{physics} \begin{align} \begin{aligned} \int \dd{x_0} \dd{\underline{x}} \mathcal{L}_{\ell, x_0} [W(x_0, \underline{x}) F(x_0, \underline{x})] = \int \dd{x_0} \dd{\underline{x}} (\mathscr{L}_{\ell, x_0} W)^{\underline{\alpha}}(x_0, \underline{x}) \partial_{\underline{\alpha}} F(x_0, \underline{x}) \end{aligned} \\ \begin{align} \label{eq:renormalization95kernel95action} \int \dd{x_0} \dd{\underline{x}} \mathcal{R}^a_{R, x_0} [W(x_0, \underline{x}) F(x_0, \underline{x})] = \int \dd{x_0} \dd{\underline{x}} (\mathscr{R}^a_{R, x_0} W)^{\underline{\alpha}}(x_0, \underline{x}) \partial_{\underline{\alpha}} F(x_0, \underline{x}) \end{align} \end{align}\tag{32}\] with \(a = 0, 1\) and \[\require{physics} \begin{align} \begin{aligned} \tag{33} (\mathscr{L}_{\ell, x_0} W)^{\underline{\alpha}}(x_0, \underline{x}) & \equiv \frac{1}{\ell!} \binom{\ell}{\underline{\alpha}} (\underline{x} - \underline{x}_0)^{\underline{\alpha}}_{\scriptstyle{\mathbb{T}}}\, \delta_N^{(n)}(\underline{x} - \underline{x}_0) \int \dd{\underline{y}} W(0, \underline{y}), \end{aligned} \\ \begin{align} \tag{34} (\mathscr{R}_{R, x_0}^0 W)^{\underline{\alpha}}(x_0, \underline{x}) & \equiv \binom{R}{\underline{\alpha}} \int_0^1 \dd{t} \frac{(1-t)^{R-1}}{(R-1)!} \int \dd{\underline{y}} W(0, \underline{y}) \, \underline{y}^{\underline{\alpha}}_{\scriptstyle{\mathbb{T}}}\, \delta^{(n)}_N(\underline{x} - \underline{x}_0 - t \underline{y}_{\scriptstyle{\mathbb{T}}}), \end{align} \\ \begin{align} \tag{35} (\mathscr{R}_{R, x_0}^1 W)^{\underline{\alpha}}(x_0, \underline{x}) & \equiv -(\mathscr{S}\mathscr{L}_{R-1, x_0} W)^{\underline{\alpha}}(x_0, \underline{x}), \end{align} \end{align}\] where \(\mathscr{S}(\, \cdot \,) \equiv (\, \cdot \,) - (\, \cdot \,) \vert_{m_N = 0}\), \(\underline{x}_0 \equiv (x_0, \dots, x_0)\) and \(n=1\) or \(n=2\) depending on whether \(W\) carries a \(J\) source field or not. This time, \(\underline{\alpha}\) is a multi-index of the form \(\underline{\alpha} = (\underline{\alpha}_1, \dots, \underline{\alpha}_n)\) with \(\underline{\alpha}_\ell \in \lbrace 0, \dots, 3 \rbrace^{\abs*{\underline{\alpha}_\ell}}\). The constraints \(\sum_{\ell = 1}^n \abs*{\underline{\alpha}_\ell} = \ell, R, R-1\) are respectively required in 33 , 34 and 35 , whereas \[\binom{R}{\underline{\alpha}} \equiv \frac{R!}{\abs*{\alpha_1}! \cdots \abs*{\alpha_n}!}, \qquad \underline{x}^{\underline{\alpha}} \equiv \prod_{\ell=1}^n \prod_{s=1}^{\abs*{\alpha_\ell}} x_\ell^{(\alpha_\ell)_s}, \qquad \partial_{\underline{\alpha}} \equiv \prod_{\ell=1}^n \prod_{s=1}^{\abs*{\alpha_\ell}} \frac{\partial}{\partial x_\ell^{(\alpha_\ell)_s}}.\]

2.3 Single scale integration↩︎

Thanks to the localization operator defined by 26 , we can finally show how to rewrite 20 by lowering the scale from \(h\) to \(h-1\). At first, we reabsorb the quadratic terms contained into \(\mathcal{L}{\mathscr{V}}^{h}[\psi, \omega]\) inside the Gaussian measure, thus getting \[\require{physics} \label{eq:generating95functional95RL95explicit} W[\omega] = \log \int \mathbb{P}_{{(g')}^{\le h}}(\dd \psi) \, \exp( {V}^{h}[\psi, \omega]),\tag{36}\] where the functional \({V}^{h}\) is defined as \[\require{physics} \mathcal{L}{V}^{h}[\psi, \omega] \equiv \sum_{s = \pm} \int_{\mathcal{M}_L} \dd[4]{x} (Z^{J, s}_h \psi^+_s \sigma_s^\mu J_\mu \psi^-_s)(x), \qquad \mathcal{R}{V}^{h}[\psi, \omega] \equiv \mathcal{R}{\mathscr{V}}^{h}[\psi, \omega]\] and the renormalized propagator is \[\label{eq:propagator95splitting} {(\hat{g}')}^{\le h}(k) \equiv \frac{\chi_{h-1}(k)}{i k_\mu \gamma^\mu_{h-1} + \mathsf{m}_{h-1}} + \frac{f_{h}(k)}{i k_\mu \tilde{\gamma}^\mu_{h}(k) + \tilde{\mathsf{m}}_{h}(k)} \equiv {\hat{g}}^{\le h-1}(k) + {\hat{g}}^{h}(k)\tag{37}\] with \(f_h(k) \equiv \chi_h(k) - \chi_{h-1}(k)\) and \(Z^s_{h-1} \equiv Z^s_h + \beta^s_h, \,\, m^s_{h-1} \equiv m^s_h + \beta^{m,s}_h\). The matrices \(\tilde{\gamma}^\mu_h(k), \tilde{\mathsf{m}}_h(k)\) are defined as \(\gamma^\mu_h, \mathsf{m}_h\) (see 21 ) with \(Z^s_h + \beta^s_h \chi_h(k), \, m^s_h + \beta^{m, s}_h \chi_h(k)\) in place of \(Z^s_h, m^s_h\). The renormalized potential takes the form \[\require{physics} \label{eq:veff95kernelsr} \mathcal{R}{V}^{h}[\psi, \omega] = \sum_{A, B} \int \dd{\underline{x}} \dd{\underline{y}} \, (\mathscr{R}{W}^{h})(A, \underline{x}; B, \underline{y}) \Psi(A, \underline{x}) \Omega(B, \underline{y}),\tag{38}\] where \(\mathscr{R}\) is either equal to \(\mathbb{1}\) or to a suitable combination of \(\mathscr{R}^0_{R, x_0}\) and \(\mathscr{R}^1_{R, x_0}\), where both \(R\) and \(x_0\) depend on the structure of the kernel on which they act in accordance with definition 26 . We stress that there are at most three derivatives for each fermion field; indeed, the \(\mathcal{R}\) operator respectively produces three and two extra derivatives when acting on \(W \psi^+ \psi^-\) and \(W \psi^+ \partial \psi^-\) and one extra derivative when acting on \(W \psi^+ \partial^2 \psi^-\). Similarly, \(\mathcal{R}\) respectively produces two extra derivatives and one extra derivative when acting on \(J \psi^+\psi^-\) and \(J \psi^+ \partial \psi^-\). In any other case, \(\mathcal{R}\) behaves as the identity operator.

The running coupling constants \(Z^{J, s}_{h-1}, Z^s_{h-1}, m^s_{h-1}\) are entirely determined by \(Z^{J, s}_h\), \(Z^s_h\), \(m^s_h\), \(\lambda\) by means of the relations \[Z^{J, s}_{h-1} = Z^{J, s}_h + \beta^{J, s}_h, \qquad Z^s_{h-1} = Z^s_h + \beta^s_h, \qquad m^s_{h-1} = m^s_h + \beta^{m, s}_h.\] The quantities \(\beta^{J, s}_h, \beta^s_h, \beta^{m, s}_h\) are known as beta functions on scale \(h\) and they depend on \(\lambda\) and on the set \(\lbrace Z^{J, s}_j, Z^s_j, m^s_j \rbrace_{j = h}^N\).

Thanks to 19 , we can write \[\require{physics} \begin{align} \label{eq:generating95functional95additivity} \begin{aligned} W[\omega] & = N_h + \log \int \mathbb{P}_{{g}^{\le h-1}}(\dd \psi)\int \mathbb{P}_{{g}^{h}}(\dd \psi) \exp ({V}^{h}[\psi, \omega]) \\ & = N_h + \log \int \mathbb{P}_{{g}^{\le h-1}}(\dd \psi) \exp({\mathscr{V}}^{h-1}[\psi, \omega]) \end{aligned} \end{align}\tag{39}\] where \(N_h\) is an inessential constant and \[\require{physics} \label{eq:V95h95integration95definition} {\mathscr{V}}^{h-1}[\psi, \omega] \equiv \log \int \mathbb{P}_{{g}^{h}}(\dd \psi) \, \exp ( {V}^{h}[\psi, \omega] ).\tag{40}\] This construction shows how to unambiguously pass from \(h\) to \(h-1\) inside 20 . In the next section, we shall explicitly prove (see 50 ) that \({\mathscr{V}}^{h}[\psi, \omega]\) has always the same structure for every scale \(h\), that is, it admits an expansion like 22 with no more than three derivatives per fermion field.

2.4 Tree expansion↩︎

The Gallavotti-Nicolò tree expansion (or, in what follows, the tree expansion tout-court) is a convenient method to represent the recursive structure of multiple Renormalization Group steps Gallavotti_1995?, Mastropietro_Nonpert_Renorm?, Gentile_2001?. As a starting point, we express 40 as \[\label{eq:truncated95expval95potential} {\mathscr{V}}^{h-1}[\psi, \omega] = \sum_{n = 1}^{+\infty} \frac{1}{n!} \mathcal{E}^{h}_{\scriptscriptstyle{\mathrm{T}}}(\underbrace{{V}^{h}[\, \cdot + \psi, \omega], \dots, {V}^{h}[\, \cdot + \psi, \omega]}_{n \text{ times}}),\tag{41}\] where the truncated expectation value of any set of \(n\) functionals \(\lbrace \mathscr{O}_1, \dots, \mathscr{O}_n \rbrace\) is \[\require{physics} \mathcal{E}^{h}_{\scriptscriptstyle{\mathrm{T}}}(\mathscr{O}_1, \dots, \mathscr{O}_n) \equiv \frac{\partial^n}{\partial t_1 \cdots \partial t_n} \log \int \mathbb{P}_{{g}^{h}}(\dd \psi) \, \exp(t_1 \mathscr{O}_1 + \cdots + t_n \mathscr{O}_n) \eval_{t_1 = 0, \dots, t_n = 0}\] (in the following sections, we will often write \(\mathcal{E}^{h}_{\scriptscriptstyle{\mathrm{T}}}(\lbrace \mathscr{O}_\ell \rbrace_{\ell=1}^n)\) in place of \(\mathcal{E}^{h}_{\scriptscriptstyle{\mathrm{T}}}(\mathscr{O}_1, \dots, \mathscr{O}_n)\)). If \(h = N\), we depict 41 in the following way, \[\tag{42} {\mathscr{V}}^{N-1} = \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fxhmvceo.png}\tag{43}\end{figure} \,\, + \,\, \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/uqpscjag.png}\tag{44}\end{figure} \,\, + \,\, \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/dqrkmsxo.png}\tag{45}\end{figure} \,\, + \,\, \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/jvbthrak.png}\tag{46}\end{figure} \,\, + \cdots,\] it being understood that each dot with label \(N+1\) represents a \({V}^{N}[\psi, \omega]\) factor. If every \({V}^{h}\) factor appearing at the right hand side of 41 is decomposed as \(\mathcal{L}{V}^{h} + \mathcal{R}{V}^{h} = \mathcal{L}{V}^{h} + \mathcal{R}{\mathscr{V}}^{h}\), this equality can be recast into \[\label{eq:truncated95expval95potential95splitting} {\mathscr{V}}^{h-1} = \sum_{n = 1}^{+\infty} \frac{1}{n!} \sum_{\lbrace \mathscr{O}_1, \dots, \mathscr{O}_n \rbrace} \mathcal{E}^{h}_{\scriptscriptstyle{\mathrm{T}}}(\mathscr{O}_1, \dots, \mathscr{O}_n),\tag{47}\] where each \(\mathscr{O}_\ell\) can be either equal to \(\mathcal{L}{V}^{h}[\, \cdot + \psi, \omega]\) or \(\mathcal{R}{\mathscr{V}}^{h}[\, \cdot + \psi, \omega]\). The renormalized potential \(\mathcal{R}{\mathscr{V}}^{h}[\psi, \omega]\) admits an expansion with the same structure as 47 , namely \[\label{eq:truncated95expval95potential95renormalized} \mathcal{R}{\mathscr{V}}^{h} = \sum_{n = 1}^{+\infty} \frac{1}{n!} \sum_{\lbrace \mathscr{O}_1, \dots, \mathscr{O}_n \rbrace} \mathcal{R}\mathcal{E}^{h+1}_{\scriptscriptstyle{\mathrm{T}}}(\mathscr{O}_1, \dots, \mathscr{O}_n);\tag{48}\] this time, each \(\mathscr{O}_\ell\) can be either equal to \(\mathcal{L}{V}^{h+1}[\, \cdot + \psi, \omega]\) or \(\mathcal{R}{\mathscr{V}}^{h+1}[\, \cdot + \psi, \omega]\). After plugging 48 into 47 , we obtain an expansion for \({\mathscr{V}}^{h-1}[\psi, \omega]\) as a function of \(\mathcal{L}{V}^{h}\), \(\mathcal{L}{V}^{h+1}\), \(\mathcal{R}{\mathscr{V}}^{h+1}\). This construction can be iterated by writing \(\mathcal{R}{\mathscr{V}}^{h+1}\) in terms of \(\mathcal{L}{V}^{h+2}, \mathcal{R}{\mathscr{V}}^{h+2}\), then \(\mathcal{R}{\mathscr{V}}^{h+2}\) in terms of \(\mathcal{L}{V}^{h+3}, \mathcal{R}{\mathscr{V}}^{h+3}\), and so on; ultimately, \({\mathscr{V}}^{h-1}\) will be expressed as a complicated sum of nested truncated expectation values of \(\mathcal{L}{V}^{h}\), \(\mathcal{L}{V}^{h+1}, \dots, \mathcal{L}{V}^{N-1}\), \({V}^{N}\). With a simple generalization of the graphical representation 42 , each term of this expansion can be associated with a Gallavotti-Nicolò tree:

An example of Gallavotti-Nicolò tree is provided in figure 1. The set of all the Gallavotti-Nicolò trees whose root label is equal to \(h\) and whose endpoint labels are less or equal than \(N+1\) is denoted by \(\mathcal{T}_{h|N}\). For every \(j \in \mathbb{Z}\), the elements of \(\mathcal{T}_{j|j}\) are called trivial trees; the set \(\mathcal{T}'_{h|N} \equiv \mathcal{T}_{h|N} \setminus \mathcal{T}_{h|h}\) only consists of nontrivial trees. Given some \(\tau \in \mathcal{T}_{h|N}\), the unique vertex \(w_0 \in \mathrm{V}(\tau)\) with label \(h+1\) will be called first nonroot vertex. For every pair of vertices \(v, w \in \tau\), we write \(v \succ w\) (\(w \prec v\)) if \(v\) is a child of \(w\) and \(v > w\) (\(w < v\)) if \(h_v > h_w\). Furthermore, the notation \(v \triangleleft w\) (\(w \triangleright v\)) is used if there exists a path \(\lbrace a_1, \dots, a_n \rbrace \subseteq \mathrm{V}(\tau)\) such that \(v = a_1 \prec \cdots \prec a_n = w\). For every \(v \in \mathrm{V}(\tau)\), we define \(\tau_v\) as the rooted subtree of \(\tau\) obtained by taking \(v\) itself, its parent vertex and all the vertices \(\lbrace w \colon w \triangleright v \rbrace\), together with the respective edges (by construction, \(\tau_v\) belongs to \(\mathcal{T}_{h_v - 1|N}\)). Finally, \(s_v\) denotes the number of children of \(v\).

Each tree is recursively assigned a value in the following way:

According to this definition, the complicated expansion described above can be synthetically represented as \[\label{eq:sum95over95trees} {\mathscr{V}}^{h}[\psi, \omega] = \sum_{\tau \in \mathcal{T}_{h|N}'} \mathop{\mathrm{Val}}(\tau)[\psi, \omega].\tag{49}\]

The tree expansion can also be defined at the level of the kernels \({W}^{h}\) appearing in 22 . To do this, for every \(\tau \in \mathcal{T}_{h|N}, v \in \mathrm{V}(\tau)\) we define

• A set \(P_v = P_v^+ \cup P_v^-\) whose elements are called external fields of \(v\). If \(v\) is an endpoint, its external fields are equal to the fermion fields occurring into the explicit expression of \(\mathop{\mathrm{Val}}(\tau_v)[\psi, \omega]\); if \(v\) is not an endpoint, then \(P_v \subseteq \cup_{w \succ v} P_w\). A general collection \(\lbrace P_v \rbrace_{v \in \mathrm{V}(\tau)}\) is denoted by \(\underline{P}\).

• A set \(\Delta_v\) that contains the derivative indices falling on the external fields of \(v\). As before, \(\underline{\Delta}\) denotes a general collection \(\lbrace \Delta_v \rbrace_{v \in \mathrm{V}(\tau)}\).

• A set \(I_v \equiv \cup_{w \succ v} P_w \setminus P_v\) whose elements are called internal fields of \(v\). We say that a vertex \(v \in \mathrm{V}(\tau)\) is trivial (with respect to a given \(\underline{P}\)) if \(I_v = \varnothing\). Moreover, if \(v \succ w\), we let \(Q_v \equiv P_v \cap I_w\).

• A set \(U_v = U^J_v \cup U^{\eta, +}_v \cup U^{\eta, -}_v\) that contains all the source fields appearing inside the endpoints that are connected with \(v\) by an increasing path of vertices.

Thanks to 32 , the expansion 49 becomes \[\require{physics} \begin{align} \label{eq:sum95over95trees95P} \begin{aligned} {\mathscr{V}}^{h}[\psi, \omega] & = \sum_{\tau \in \mathcal{T}_{h|N}'} \sum_{\underline{P}, \underline{\Delta}} \int \dd{\underline{x}(P_{w_0})} \dd{\underline{y}(U_{w_0})} \, W_{\tau, \underline{P}, \underline{\Delta}}(\underline{x}(P_{w_0}), \underline{y}(U_{w_0})) \Psi(P_{w_0}, \Delta_{w_0}) \Omega(U_{w_0}), \\ \mathcal{R}{\mathscr{V}}^{h}[\psi, \omega] & = \begin{multlined}[t] \sum_{\tau \in \mathcal{T}_{h|N}'} \sum_{\underline{P}, \underline{\Delta}} \int \dd{\underline{x}(P_{w_0})} \dd{\underline{y}(U_{w_0})} \, (\mathscr{R}W)_{\tau, \underline{P}, \underline{\Delta}}(\underline{x}(P_{w_0}), \underline{y}(U_{w_0})) \, \times \\\times \Psi(P_{w_0}, \Delta'_{w_0}) \Omega(U_{w_0}), \end{multlined} \end{aligned} \end{align}\tag{50}\] where \(w_0\) is the first nonroot vertex of \(\tau\), the monomials \(\Psi(P_{w_0}, \Delta_{w_0}), \Omega(U_{w_0})\) are constructed as in 23 , 24 and \(\underline{x}(P_{w_0}), \underline{y}(U_{w_0})\) are the spacetime points that the field monomials \(\Psi(P_{w_0}, \Delta_{w_0}), \Omega(U_{w_0})\) depend on. The set \(\Delta'_{w_0}\) differs from \(\Delta_{w_0}\) due to the extra derivatives produced the renormalization operator, as showed in 28 . The first line of 50 shows that \({\mathscr{V}}^{h}\) does indeed admit an expansion of the form 22 for every scale \(h\).

The kernel \(W_{\tau, \underline{P}, \underline{\Delta}}\) is recursively defined as \[\require{physics} \label{eq:W95tau95recursive} W_{\tau, \underline{P}, \underline{\Delta}}(\underline{x}, \underline{y}) \equiv \frac{1}{s_{w_0}!} \! \int \! \prod_{v \succ w_0} \!\! \dd{\underline{x}(Q_v)} \mathcal{E}^{h_{w_0}}_{\scriptscriptstyle{\mathrm{T}}}(\lbrace \Psi(Q_v, \Delta_v') \rbrace_{v \succ w_0}) \! \prod_{v \succ w_0} \!\! \mathscr{R}_v W_{\tau_v, \underline{P}, \underline{\Delta}}(\underline{x}(P_v), \underline{y}(U_v)),\tag{51}\] where \(\mathscr{R}_v = \mathscr{R}\) or \(\mathscr{R}_v = \mathbb{1}\) depending on whether a renormalization operator acts on \(v\) or not and \(\Delta'_v\) is constructed in the same way as \(\Delta'_{w_0}\). If \(v\) is an endpoint, \(W_{\tau_v, \underline{P}, \underline{\Delta}}\) is equal to one among \[\begin{align} \begin{aligned} \tag{52} {W}^{h; \alpha \beta \mu}_{(J)}(x, y, z) & \equiv \delta_N(x - y) \delta_N(x - z) (\gamma^\mu_{J,h})_{\alpha \beta}, \end{aligned} \\ \begin{align} \tag{53} {W}^{N; \alpha \beta \sigma \rho}_{(\lambda)}(x_1, x_2, x_3, x_4) & \equiv - \frac{\lambda^2}{2} \delta_N(x_1 - x_2) \delta_N(x_3 - x_4) v(x_1 - x_3) \, (\Upsilon^\mu)_{\alpha \beta} (\Upsilon_\mu)_{\sigma \rho}, \end{align} \\ \begin{align} \tag{54} {W}^{N; \alpha \beta}_{(\eta)}(x, z) & \equiv \delta_{\alpha \beta} \delta_N(x - z), \end{align} \end{align}\] based on the sets \(P_v, \Delta_v, U_v\) and on the structure of \(\tau\) (the matrix \(\gamma^\mu_{J, h}\) is constructed as in 21 with \(Z^J_h\) in place of \(Z_h\) and \(\Upsilon^\mu \equiv \gamma^\mu - \kappa \gamma^\mu\gamma^5\)). This suggests that endpoints can be naturally classified as \(J\), \(\lambda\) or \(\eta\) endpoints. By construction, \(J\) endpoints can occur at every energy scale, while \(\lambda, \eta\) endpoints must lie on scale \(N+1\); moreover, definition 2 implies that \(W_{\tau, \underline{P}, \underline{\Delta}}\) is identically vanishing if \(\tau\) contains a \(J\) endpoint lying on scale \(h\) that is not immediately preceded by a nontrivial vertex with label \(h-1\).

2.5 Feynman diagrams↩︎

The truncated expectation value of a product of fermionic fields admits the representation \[\label{eq:TExp95feynman} \mathcal{E}^{h}_{\scriptscriptstyle{\mathrm{T}}}(\psi^+_{\alpha_1}(x_1) \cdots \psi^+_{\alpha_n}(x_n) \psi^-_{\beta_1}(y_1) \cdots \psi^-_{\beta_n}(y_n)) = \sum_{\varphi\in \mathcal{F}} \epsilon_\varphi\prod_{\ell \in \mathrm{E}(\varphi)} {g}^{h}_{\alpha_{\ell_1} \beta_{\ell_2}}(x_{\ell_1}, x_{\ell_2}),\tag{55}\] where \(\mathcal{F}\) is the set of all the possible connected graphs (called Feynman diagrams) obtained by drawing the spacetime points \(x_1, \dots, x_n, y_1, \dots, y_n\), reorganizing the fields in pairs of the form \(\lbrace \psi^-_{\alpha_a}(x_a), \psi^+_{\beta_b}(y_b) \rbrace\) and drawing an oriented line from \(y_b\) to \(x_a\) for each of such pairs (\(\epsilon_\varphi\) is the sign of the permutation needed to convert the product 55 into the set of ordered pairs associated with \(\varphi\)). By iteratively applying 55 to 51 , we can express \(W_{\tau, \underline{P}, \underline{\Delta}}\) as a sum over Feynman diagrams with a multiscale structure determined by the tree \(\tau\). The procedure to construct a multiscale Feynman diagram associated with a tree \(\tau\) goes as follows:

• Draw an interaction vertex for each endpoint of \(\tau\). All the possible interaction vertices are listed in Figure 2.

• If a family of endpoints has a common nontrivial parent vertex \(v \in \mathrm{V}(\tau)\), connect the corresponding family interaction vertices by contracting some of their external legs, respecting the directions of the arrows. A single scale propagator \({g}^{h_v}\) is associated with each contracted line. As a result, one obtains a cluster for each family of endpoints sharing the same nontrivial parent vertex.

• Iterate the above construction following the vertex structure of \(\tau\). Namely, if the vertices \(v_1, \dots, v_n\) associated with a family of clusters and/or interaction vertices have a common nontrivial parent vertex \(w\), connect these clusters and/or interaction vertices with a set of lines carrying the single scale propagator \({g}^{h_w}\).

• If the \(\mathcal{R}\) operator acts nontrivially on a vertex \(v\), it must act on the corresponding cluster as well.

While computing the value of a diagram, one needs to sum over all the momenta flowing within its internal propagators; in the \(L \to +\infty\) limit, these sums are replaced by integrals and the Kronecker deltas \(\delta_{p - k}, \delta_{p_1 - p_3 - q}, \cdots\) are replaced by Dirac deltas.

To see how renormalization acts at the level of Feynman diagrams, it is convenient to work directly in Fourier space. As an example, let us consider a kernel \(W\) acting on a field monomial of the form \(\psi^+_\alpha \psi^-_\beta J_\mu\). Recalling that the highest order localizations occurring in 26 do not contribute to \(\mathcal{L}{\mathscr{V}}^{h}[\psi, \omega]\) (see ?? ), a straightforward application of 26 yields \[\mathcal{R}\int W^{\alpha \beta \mu} \psi^+_\alpha \psi^-_\beta J_\mu = \frac{1}{L^8} \sum_{p', p \in (2\pi Z/L)^4} [\hat{W}^{\alpha \beta \mu}(p', p) - \hat{W}^{\alpha \beta \mu}(0, 0)] \hat{\psi}^+_{\alpha p'} \hat{\psi}^-_{\beta p} \hat{J}_{\mu, p' - p}.\] Hence, renormalizing a multiscale Feynman diagram with two external fermionic legs and one \(J\) source leg simply amounts to subtract its zero Fourier mode. Similarly, one can see that the renormalization operator acts on a two-legged multiscale Feynman diagram by subtracting its first-order Taylor polynomial centered around zero. Some examples of this will be provided in Section 4.

We will use the Feynman diagram representation to compute \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) at lowest order in \(\lambda^2\). However, to obtain bounds on \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) at all orders it is convenient to represent the truncated expectation values in a different way, as discussed below. This is due to the fact that Feynman diagrams have bad combinatorial properties.

3 Bounds on the renormalized expansion↩︎

In order to bound the kernels \(W_{\tau, \underline{P}, \underline{\Delta}}\) occurring in 50 , it is convenient to introduce a suitable weighted \(L^1\) norm Giuliani_Mastropietro_Rychkov_2021?. For any finite subset \(A \subseteq \mathcal{M}_L\) and for every integer \(h \le N\), we define the weight function as \(\mathsf{u}_h(A) \equiv \exp (c \sqrt{2^h \, \mathrm{diam}(A)})\), where the notion of diameter is defined with respect to the distance \(\abs*{\, \cdot \,}_L\) introduced in Section 1.2 (if \(A = \lbrace x, y \rbrace\), \(\mathrm{diam}(A)\) reduces to \(\abs*{x - y}_L\)) and \(c\) is some positive constant such that \(\require{physics} \int \dd[4]{x} \abs*{{g}^{h}(x)} \mathsf{u}_h(x), \int \dd[4]{x} \abs*{\delta_N(x)} \mathsf{u}_{N+1}(x) \le +\infty\). It will be clear from Lemma 3 that this constant exists and it is determined by the choice of the cutoff function \(\chi_0\).

The effect of the renormalization on kernels is described by the following lemma.

Lemma 2 is the key to prove that the renormalization operator improves the naïve dimensional estimate of the kernel on which it acts. Before getting into the details of this, we need to introduce a few more notations. Given tree \(\tau \in \mathcal{T}_{h|N}\) together with a suitable choice of the sets \(\underline{P}, \underline{\Delta}\), consider any nontrivial vertex \(v \in \mathrm{V}(\tau)\). Then

• \(D_v \equiv 4 - 3(\abs*{P_v} + n^\eta_v)/2 - n^J_v - \mathfrak{d}_v\) and \(\bar{D}_v \equiv D_v + 2 n_v + n^\eta_v\), where are respectively equal to the number of \(\eta, J, \lambda\) endpoints that are connected with \(v\) by an increasing path of vertices and \(\mathfrak{d}_v\) is the number of derivatives falling on the external fields of \(v\) before the action of the renormalization operator.

• \(h'_v\) denotes the scale of the first nontrivial vertex that precedes \(v\). If such vertex does not exist, \(h'_v \equiv h\).

• If \(v\) is either an endpoint or the first nonroot vertex, we let \(R_v \equiv 0\). In any other case, \[R_v \equiv \begin{cases} 3 & \quad \abs*{P_v} = 2, \mathfrak{d}_v = 0, n^J_v = 0, n^\eta_v = 0 \\ 2 & \quad \abs*{P_v} = 2, \mathfrak{d}_v = 1, n^J_v = 0, n^\eta_v = 0 \\ 2 & \quad \abs*{P_v} = 2, \mathfrak{d}_v = 0, n^J_v = 1, n^\eta_v = 0 \\ 0 & \quad \text{otherwise} \end{cases}\]

3.1 The flow of the running coupling constants↩︎

3.2 Integration of the lowest scale↩︎

Thanks to the presence of a nonvanishing fermion mass, the norms of the renormalized propagator \({(g')}^{\le h^\star}\) (defined as in Section 2.3) satisfy the bounds ?? , ?? , so it is possible to interrupt the Renormalization Group flow on scale \(h^\star - 1\). In fact, \({(g')}^{\le h^\star}\) is equal to the periodization of the function displayed in 63 with \(h = h^\star\) and with \(\chi_0\) in place of \(f_0\). Following the same argument used in the proof of Lemma 3, it can be easily seen that the “denominator” \(u(k) = i \tilde{\gamma}^\mu_{h^\star}(2^{h^\star}k) k_\mu + 2^{-h^\star}\tilde{\mathsf{m}}_{h^\star}(2^{h^\star}k)\) satisfies the bound \(\norm*{\partial_{k_\mu}^j u} \le c_2 C^j (j!)^2\) for every \(j \in \mathbb{N}, k \in \mathop{\mathrm{supp}}(\chi_0)\), as it happens for \({g}^{h}\). Moreover, the first factor appearing in 65 is manifestly controlled by some scale-independent constant \(c_3\), while \[\label{eq:mass95dominance95over95kinetic95term} \sup_{k \in \mathop{\mathrm{supp}}(\chi_0)}\abs{k^2 \tilde{Z}^+_{h^\star}(2^{h^\star} k) \cdot \tilde{Z}^-_{h^\star}(2^{h^\star}k) + 2^{-2h^\star}[\tilde{m}^s_{h^\star}(2^{h^\star}k)]^2}^{-1} \le \abs{2^{-2h^\star}[\tilde{m}^s_{h^\star}(2^{h^\star}k)]^2}^{-1} \le c_4,\tag{91}\] where \(c_4\) is another scale-independent constant. We therefore have \(\norm*{u^{-1}(k)} \le c_3 c_4 \equiv c_1\) and the proof of Lemma 3 can be followed with no further variations.

Once the integration with respect to \({(g')}^{\le h^\star}\) is explicitly performed, we are left with the expansion \[W[\omega] = \sum_{\tau \in \mathcal{T}'_{h^\star - 1|N}} \mathop{\mathrm{Val}}(\tau)[\omega],\] it being understood that the propagator associated with the lowest scale is \({g}^{h^\star} \equiv {(g')}^{\le h^\star}\). Note that \(\mathop{\mathrm{Val}}(\tau)\) does not depend on \(\psi\), because every \(\psi\) field has been integrated away during the last Renormalization Group step. Since \({g}^{h^\star}\) behaves as a single scale propagator, the bound ?? still applies to every tree \(\tau \in \mathcal{T}'_{h^\star - 1|N}\).

3.3 Infinite volume limit↩︎

The bounds obtained in Theorem 2 are uniform with respect to the spacetime volume. Indeed, it can be shown that the sequence of finite-volume kernels \(\lbrace W_{\tau, \underline{P}, \underline{\Delta}, L} \rbrace_{L \in \mathbb{N}}\) converges in the \(L \to +\infty\) limit with respect to the uniform norm on every fixed compact subset of \(\mathbb{R}^{4(\abs*{P_{w_0}} + n^J_{w_0} + n^\eta_{w_0})}\), where \(w_0\) is the first nonroot vertex of \(\tau\). A complete proof of this statement is rather long, so we shall not discuss it here; a detailed treatment can be found in Giuliani_Mastropietro_2010?. Although our model differs from the one analyzed in Giuliani_Mastropietro_2010?, the proof can be adapted to the present case with little effort. The main idea consists in noticing that the infinite volume limits of the functions \(\delta_N, W_{(\eta)}^N, W_{(J)}^N, W_{(\lambda)}^N, {g}^{N}\) exist and they are equal to the inverse Fourier transforms of \(\chi_{N+1}(k), \hat{W}_{(\eta)}^N(k), \hat{W}_{(J)}^N(p, p'), \hat{W}_{(\lambda)}^N(k_1, k_2, k_3), {\hat{g}}^{N}(k)\), thought as functions of \(\mathbb{R}^4\) vectors. As a consequence, the kernels \(W_{\tau, \underline{P}, \Delta, L}\) can be defined in the infinite volume limit for every \(\tau \in \mathcal{T}'_{N-1|N}\) (this is realized by taking 51 and replacing every function \(\delta_N, W_{(\eta)}^N, \dots\) with its infinite volume limit). The difference between \(W_{\tau, \underline{P}, \Delta, L}\) and \(W_{\tau, \underline{P}, \Delta}\) can be shown to vanish as \(L \to +\infty\) (see Giuliani_Mastropietro_2010?). The existence of the infinite volume limits of the above kernels implies the existence of the infinite volume limits of the running coupling constants \(m^s_{N-1}, Z^s_{N-1}, Z^{J, s}_{N-1}\) as well as those of the functions \(W_{(\eta)}^{N-1}, W_{(J)}^{N-1}, W_{(\lambda)}^{N-1}, {g}^{N-1}\). Relying on the recursive definition 50 , we can iteratively apply the above construction to every kernel \(W_{\tau, \underline{P}, \underline{\Delta}, L}\) for every \(\tau \in \mathcal{T}'_{j|N}\), with \(j = h^\star-1, \dots, N-1\).

From now on, we will always assume that the infinite volume limit has been taken.

4 Evaluation of the anomalous gyromagnetic factor↩︎

Based on the results discussed in Section 2 and 3, we can finally address the problem of evaluating \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\), which is expressed by (19) in terms of \(\hat{\Gamma}^\mu(p', p)\) and its derivatives computed at \(p=p'=0\). The bounds ?? , ?? yield to the conclusion that the tree expansions for the correlation functions \(\hat{\Gamma}^\mu(p', p)\) are absolutely convergent if \(\lambda \le \mathcal{O}(M/\Lambda)\) and bounded at any order \(n\) by \(\mathcal{O}((\lambda M/\Lambda)^{2n})\). This bound is however not sufficient to prove our main result, which estabilishes that \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) is bounded above and below by a \(\mathcal{O}(m^2/M^2)\) constant. In order to get this result we note that, based on Theorem 2 and Theorem 3, it is \[\begin{align} \begin{aligned} \tag{92} \hat{\Gamma}^\mu(p', p) & = \gamma^\mu + \sum_{n = 1}^{+\infty} \lambda^{2n} \hat{\Gamma}^\mu_n(p', p; \lambda), \end{aligned} \\ \begin{align} \tag{93} \hat{S}(k) & = \sum_{n = 0}^{+\infty} \lambda^{2n} \hat{S}_n(k; \lambda) \equiv \frac{1}{i \cancel{k} + m} \left(1 + \sum_{n = 1}^{+\infty} \lambda^{2n} \hat{G}_n(k; \lambda) \right), \end{align} \end{align}\] where each contribution of order \(n\) inlcudes the sum over all the possible trees with \(n\) endpoints of type \(\lambda\); in particular, the first of the above expressions coincides with 17 . The simple form of the \(n = 0\) terms is due to the fact that we have imposed the renormalization conditions via Theorem 3.

In this section, we shall prove three properties:

• If \(\abs*{p'}, \abs{p} \le m/2\) and \(n \ge 1\), the bound for \(\lambda^{2n} \hat{\Gamma}^\mu_n(p', p; \lambda)\) can be improved by a factor \(m^2/M^2\), provided that the value of \(\lambda\) is sligthly decreased. This is due to the presence of symmetries implemented by our choice of the \(\mathcal{R}\) operator defined in Section 2.2.

• This \(m^2/M^2\) improvement is not present in the \(n = 0\) term of the expansion 92 ; however, the contribution to \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) coming from this term is vanishing.

• We separate the \(n = 1\) term from the rest and we show that it gives the same value of the Jackiw-Weinberg formula 2 up to subdominant corrections.

In what follows, the symbol \(\mathcal{O}(\lambda f)\) is used to denote any function of \(\lambda\) that satisfies \(\abs{\mathcal{O}(\lambda f)} \le C \cdot \abs{\lambda f}\) for some constant \(C > 0\) that does not depend on \(m, M, \Lambda, \lambda\).

4.1 Bounds for \(\hat{\Gamma}^\mu(p', p)\) and \(\hat{S}(k)\)↩︎

By combining Theorem 2 and Theorem 3, we exhibit a bound on the two-point function and the amputated vertex function. In particular, we prove that the subdominant part of \(\hat{\Gamma}^\mu(p', p)\) is suppressed by a factor \(m^2/M^2\).

4.2 Tree expansions at lowest order↩︎

As we are ultimately interested in evaluating the derivatives of \(\mathcal{A}(z)\) at \(z = 0\), it is sufficient to determine the functions \(\hat{S}(k), \hat{\mathfrak{I}}^\mu(p', p)\) within the region \(\abs{k}, \abs*{p'}, \abs{p} \le m/2\). Due to the compact support properties of the single scale propagators, the only relevant trees are

for the interacting two-point function, and

for the full vertex function. The corresponding Feynman diagrams are shown in Figure 4. It is immediate to see that \(\omega_1, \omega_2, \varphi_1\) respectively come from \(\xi_1, \xi_2, \tau_1\). If \(h_1 = h_2 = h^\star\), \(\tau_2\) and \(\tau_3\) degenerate into the same tree which gives rise to \(\varphi_2 \vert_{h_1 = h^\star}\), \(\varphi_3 \vert_{h_1 = h^\star}\), \(\varphi_4^1 \vert_{h_1 = h_2 = h^\star}\), \(\varphi_5^1 \vert_{h_1 = h_2 = h^\star}\); in any other case, \(\tau_2\) produces \(\varphi_2, \varphi_3\) and \(\tau_3\) produces \(\varphi_4^{1, 2}, \varphi_5^{1, 2}\) or \(\varphi_4^0, \varphi_5^0\) depending on whether \(h_1 \ne h_2\) or \(h_1 = h_2\).

In terms of the above Feynman diagrams, the lowest orders of the tree expansions for \(\hat{S}(k)\) and \(\hat{\mathfrak{I}}^\mu(p', p)\) are \[\begin{align}\hat{S}_0(k; \lambda) = \hat{S}(k; \omega_1, \lambda), \qquad \lambda^2\hat{S}_1(k; \lambda) = \hat{S}(k; \omega_2, \lambda), \qquad \hat{\mathfrak{I}}^\mu_0(p', p; \lambda) = \hat{\mathfrak{I}}^\mu(p', p; \varphi_1, \lambda),\\ \begin{align} \lambda^2 \hat{\mathfrak{I}}^\mu_1(p', p; \lambda) = \sum_{j=2}^3 \hat{\mathfrak{I}}^\mu(p', p; \varphi_j, \lambda) + \sum_{a = 0}^2 \sum_{j=4}^5 \hat{\mathfrak{I}}^\mu(p', p; \varphi_j^a, \lambda), \end{align} \end{align}\] where, given any diagram \(\varphi\), the notations \(\hat{S}(k; \varphi, \lambda)\), \(\hat{\mathfrak{I}}^\mu(p', p; \varphi, \lambda)\) stand for the contributions to \(\hat{S}(k), \hat{\mathfrak{I}}^\mu(p', p)\) coming from \(\varphi\). By inverting the series expansion for \(\hat{S}(k)\), it is easy to see that \[\label{eq:S95inverse95coefficients} (\hat{S}^{-1})_0(k; \lambda) = [\hat{S}_0(k; \lambda)]^{-1}, \qquad (\hat{S}^{-1})_1(k; \lambda) = - [\hat{S}_0(k; \lambda)]^{-1} \, \hat{S}_1(k; \lambda) \, [\hat{S}_0(k; \lambda)]^{-1};\tag{99}\] after plugging the explicit expressions of the Feynman diagrams shown in Figure 4 into 99 and 98 , one finds \[\begin{align} \label{eq:amp95vertex95first95order} \begin{aligned} \hat{\Gamma}^\mu_0(p', p; \lambda) & = \gamma^\mu, \\ \lambda^2 \hat{\Gamma}^\mu_1(p', p; \lambda) & = \sum_{a = 0}^2 \sum_{j = 4}^5 \, [{g}^{h^\star}(p')]^{-1} \, \hat{\mathfrak{I}}^\mu(p', p; \varphi_j^a, \lambda) \, [{g}^{h^\star}(p)]^{-1} \\ & \equiv \sum_{a = 0}^2 \sum_{j = 4}^5 \hat{\Gamma}^\mu(p', p; \varphi_j^a, \lambda). \end{aligned} \end{align}\tag{100}\] As it happens in ordinary perturbation theory, diagrams \(\varphi_2, \varphi_3\) have no influence on \(\hat{\Gamma}^\mu_1(p', p; \lambda)\), because \[\label{eq:self95energy95insertion} \sum_{j=2}^3 \hat{\mathfrak{I}}^\mu(p', p; \varphi_j, \lambda) = \hat{S}_1(p'; \lambda) \, \hat{\mathfrak{I}}^\mu_0(p', p; \lambda) \, \hat{S}_0(p; \lambda) + \hat{S}_0(p'; \lambda) \, \hat{\mathfrak{I}}^\mu_0(p', p; \lambda) \, \hat{S}_1(p; \lambda)\tag{101}\] and this sum is exactly cancelled by the terms with \((n_1, n_2, n_3) = (1, 0, 0), (0, 0, 1)\) occurring inside 98 . The particular structure displayed in 101 arises because \(\mathcal{R}\) acts identically on every cluster that can be disconnected by cutting a single fermionic line.

We are now left to explicitly compute the function \(\mathcal{A}(z) \equiv [\mathcal{F}(z) - \mathcal{Q}(z)]/\mathcal{Q}(z)\), from which the anomalous gyromagnetic factor will be recovered. Since \(\hat{\Gamma}^\mu(p', p)\) satisfies 92 , the difference \(\abs*{\mathcal{Q}(z) - 24}\) is controlled by the right hand side of ?? with \(n = 1\). Therefore, if \(\lambda^2 \Lambda^2/M^2\) is sufficiently small, the function \(\mathcal{Q}(z)\) is never vanishing and \(\mathcal{A}(z)\) is well-defined. Relying on 16 , we can write \[\mathcal{A}(z) = \mathcal{A}_0(z; \lambda) + \mathcal{A}_2(z; \lambda) + \mathcal{A}_{> 2}(z; \lambda),\] where, according to Theorem 4, \(\abs*{\mathcal{A}_{> 2}(z; \lambda)}\) is controlled by the left hand side of ?? with \(n = 2\).

4.3 Vanishing of \(\mathcal{A}_0(z; \lambda)\) and of the non-triangular contributions to \(\mathcal{A}_2(z; \lambda)\)↩︎

Before getting into the actual calculation of \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\), we prove that \(\mathcal{A}_0(z; \lambda) = 0\), so \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) does not receive any contribution from the dominant part of \(\hat{\Gamma}^\mu(p', p)\). We also show that \(\mathcal{A}_2(z; \lambda)\) only depends on the triangular diagrams \(\varphi^0_4, \varphi^1_4, \varphi^2_4\).

Due to the anticommutation properties of the \(\gamma^\mu\) matrices, any term proportional to \(\gamma^\mu\) that appears within the expansion of \(\hat{\Gamma}^\mu(p', p)\) annihilates the difference \(\mathcal{F}(z) - \mathcal{Q}(z)\); moreover, every term proportional to \(\gamma^5 \gamma^\mu\) annihilates both \(\mathcal{F}(z)\) and \(\mathcal{Q}(z)\).

It is easy to see that \(\hat{\Gamma}^\mu_0(p', p; \lambda)\) and the local parts of \(\hat{\Gamma}^\mu(p', p; \varphi_5^{0, 1, 2}, \lambda)\) are linear combinations of \(\gamma^\mu\) and \(\gamma^5 \gamma^\mu\), so they do not contribute to \(\mathcal{A}(z)\), nor to \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\). On the other hand, the non local parts of \(\hat{\Gamma}^\mu(p', p; \varphi_5^{0, 1, 2}, \lambda)\) are linear combinations of integrals of the form \[\require{physics} \label{eq:loop95integral} H^\mu(p', p) = \hat{v}(k) \Upsilon_\nu \int \frac{\dd[4]{q}}{(2\pi)^4} \tr[\Upsilon^\nu {g}^{h_1}(k + q) \, (\gamma^\mu_J)_{\max(h_1, h_2)} \, {g}^{h_2}(q)] \equiv \hat{v}(k) \Upsilon_\nu D^{\nu \mu}(k)\tag{102}\] with \(k = p' - p\) and \(h^\star \le h_1, h_2 \le N\). The \(\mathrm{SO}(4)\) covariance of the theory implies that \(D^{\nu \mu}(0)\) is proportional to \(\delta^{\nu \mu}\), so \(H^\mu(p_z, p_z) = \hat{v}(k) \Upsilon_\nu D^{\nu \mu}(0)\) is a linear combination of \(\gamma^\mu\) and \(\gamma^5 \gamma^\mu\). Relying on the parity cancellation \(\partial^\alpha D^{\nu \mu}(0) = 0\), it can be seen that \((\partial_{p'} - \partial_p)^\alpha H^\mu(p_z, p_z) = \partial^\alpha \hat{v}(0) \Upsilon_\nu D^{\nu \mu}(0)\) is a linear combination of \(\gamma^\mu\) and \(\gamma^5 \gamma^\mu\) as well. We conclude that any integral of the form 102 annihilates the function \(\mathcal{F}(z) - \mathcal{Q}(z)\), so the non local parts of diagrams \(\varphi_5^{0, 1, 2}\) have no influence on \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\).

The combined values of \(\varphi_4^{0, 1, 2}\) will be conveniently represented by the function \[\sum_{j=0}^2 \hat{\Gamma}^\mu(p', p; \varphi_4^j, \lambda) \equiv \hat{\Gamma}^\mu_\triangle(p', p; \lambda),\] called triangle integral. Explicitly, \[\require{physics} \begin{gather} \label{eq:triangle95multiscale} \hat{\Gamma}^\mu_\triangle(p', p; \lambda) = - \sum_{h_1, h_2}^{\star h^\star} \int \frac{\dd[4]{q}}{(2\pi)^4} \, \Upsilon_\nu \, {\hat{g}}^{h_1}(p' - q) \, (\gamma^\mu_J)_{\max(h_1, h_2)} \, {\hat{g}}^{h_2}(p - q) \, \Upsilon^\nu \, + \\ - \sum_{h_1, h_2 > h^\star} \int \frac{\dd[4]{q}}{(2\pi)^4} \, \mathcal{R}[ \Upsilon_\nu \, {\hat{g}}^{h_1}(p' - q) \, (\gamma^\mu_J)_{\max(h_1, h_2)} \, {\hat{g}}^{h_2}(p - q) \, \Upsilon^\nu ], \end{gather}\tag{103}\] where \(\sum_{h_1, h_2}^{\star h^\star}\) denotes a sum constrained by the requirement that at least one among \(h_1, h_2\) must be equal to \(h^\star\). According to the above argument, the function \(\mathcal{A}_2(z; \lambda)\) is entirely determined by \(\hat{\Gamma}^\mu_\triangle(p', p; \lambda)\).

4.4 Extraction of the \(\lambda\)-independent part of \(\mathcal{A}_2(z; \lambda)\)↩︎

Since the running coupling constants \(Z^{J, s}_h, Z^s_h\) depend on \(\lambda\) and \(h\), it is not immediate to explicitly perform the sum over \(h_1, h_2\) inside the triangle integral 103 . For this reason, we split the function \(\hat{\Gamma}^\mu_\triangle(p', p; \lambda)\) into the sum of a \(\lambda\)-independent part (for which the sum over \(h_1, h_2\) could be trivially performed) and a small remainder.

4.5 Existence of the \(\Lambda\to +\infty\) limit of \(\hat{\Gamma}^\mu_\triangle(p', p)\)↩︎

Thanks to Lemma 4, from now on we can turn our attention on the \(\lambda\)-independent triangle integral \(\hat{\Gamma}^\mu_\triangle(p', p)\). Since the \({g}^{h}_0\) propagator scales in the same way as \({g}^{h}\), the bound ?? implies that \[\label{eq:non95optimal95estimate} \sup_{\abs*{p'}, \abs{p} \le m/2} \norm*{\hat{\Gamma}^\mu_\triangle(p', p)} \le C \cdot \frac{m^2}{M^2} \cdot \log^2 \left( \frac{\Lambda}{m} \right).\tag{108}\] so \(\hat{\Gamma}^\mu_\triangle(p', p)\) could apparently diverge as \(\Lambda\) goes to \(+\infty\). However, this is not the case, because the estimate 108 is not optimal. To improve it, it is necessary to exploit the decay of \(\hat{v}(q)\) as \(\abs{q} \to +\infty\), as we did in the proof of Lemma 4.

4.6 Evaluation of \(\mathcal{A}_2(z)\)↩︎

Thanks to Lemmas 4 and 5, we can write \[\label{eq:A95295decomposition} \mathcal{A}_2(z; \lambda) = \mathcal{A}_2(z) + R(z; \lambda),\tag{110}\] where the function \(\mathcal{A}_2(z)\) has the same form as \(\mathcal{A}_2(z; \lambda)\) with \(\hat{\Gamma}^\mu_{\triangle, \infty}(p', p)\) in place of \(\hat{\Gamma}^\mu_2(p', p)\) and \[\label{eq:R95estimate95A952} \abs{R(z; \lambda)} \le C_1 \cdot \frac{m^2}{M^2} \cdot \frac{\lambda^4 \Lambda^2}{M^2} \log \left( \frac{M}{m} \right) \frac{M}{\Lambda} + C_2 \cdot \lambda^2 \frac{m^2}{M^2} \frac{M^2}{\Lambda^2}.\tag{111}\] The explicit form of \(\hat{\Gamma}^\mu_{\triangle, \infty}(p', p)\) is \[\require{physics} \begin{align} \label{eq:lambda95independent95triangle95limit} \begin{aligned} \hat{\Gamma}^\mu_{\triangle, \infty}(p', p) & = \lim_{\Lambda\to +\infty} \left(- \int \frac{\dd[4]{q}}{(2\pi)^4} \, \Upsilon_\nu {\hat{g}}^{\le N}_0(p' - q) \, \gamma^\mu \, {\hat{g}}^{\le N}_0(p - q) \Upsilon^\nu - L^\mu \right) \\ & \equiv \lim_{\Lambda\to +\infty} ( \hat{\mathscr{G}}^\mu_{\triangle, \Lambda}(p', p) - L^\mu), \end{aligned} \end{align}\tag{112}\] where \(L^\mu\) denotes the local term that comes from the action of the renormalization operator (see 103 , recalling that \(\mathcal{R}= 1 - \mathcal{L}\)). Moreover, \[\begin{gather} \label{eq:gfunction95first95order} \mathcal{A}_2(z) = \frac{\lambda^2}{24} \biggl( z \epsilon_{a b c} \tr[\gamma^5 (\partial_{p'} - \partial_p)^a \hat{\Gamma}^b_{\triangle, \infty}(p_z, p_z) \gamma^c]\, + \\ + 2 \tr[\gamma_a \hat{\Gamma}^a_{\triangle, \infty}(p_z, p_z)] - 6 \tr[(1 + \gamma^0) \hat{\Gamma}^0_{\triangle, \infty}(p_z, p_z)] \biggr). \end{gather}\tag{113}\] It is easy to see that \(L^\mu\) does not contribute to \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\), because it is equal to a linear combination of \(\gamma^\mu\) and \(\gamma^5 \gamma^\mu\) (the absence of such contributions has already been discussed in Section 4.3). Therefore, we can rewrite 113 in terms of \(\hat{\mathscr{G}}^\mu_{\triangle, \Lambda}(p', p)\) alone, \[\begin{gather} \label{eq:gfunction95first95order95no95local95part} \mathcal{A}_2(z) = \lim_{\Lambda\to +\infty} \frac{\lambda^2}{24} \biggl( z \epsilon_{a b c} \tr[\gamma^5 (\partial_{p'} - \partial_p)^a \hat{\mathscr{G}}^b_{\triangle, \Lambda}(p_z, p_z) \gamma^c]\, + \\ + 2 \tr[\gamma_a \hat{\mathscr{G}}^a_{\triangle, \Lambda}(p_z, p_z)] - 6 \tr[(1 + \gamma^0) \hat{\mathscr{G}}^0_{\triangle, \Lambda}(p_z, p_z)] \biggr), \end{gather}\tag{114}\] and then plug 112 into 114 . A straightforward calculation yields \[\require{physics} \begin{align} \label{eq:A95cutoff} \begin{aligned} \mathcal{A}_2(z) & = 4 \lambda^2 \lim_{\Lambda\to +\infty} \int \frac{\dd[4]{q}}{(2\pi)^4} \frac{T_z(q) \chi(q) \chi^2(q - p_z)}{[q^2 + M^2][(p_z - q)^2 + m^2]^2}, \\ T_z(q) & \equiv (\kappa^2 + 1)[2 z^2 - 3 z q^0 + (q^0)^2 - q_a q^a/3] + 2im(\kappa^2 - 1)(z - q^0), \end{aligned} \end{align}\tag{115}\] where a sum over the spatial index \(a \in \lbrace 1, 2, 3 \rbrace\) is understood. This can be further simplified thanks to the identity \[\require{physics} \label{eq:feynman95parametrization} \frac{1}{ab^2} = 2 \int_0^1 \dd{x} \frac{x}{[a(1 - x) + bx]^3} \quad \forall a, b \in \mathbb{C}\setminus \lbrace 0 \rbrace,\tag{116}\] known as Feynman’s parametrization Schwartz_2013?: by letting \(a \equiv q^2 + M^2, b \equiv (p_z - q)^2 + m^2\) and subsequently performing the variable change \(q \mapsto q + p_z x\), 115 becomes \[\require{physics} \label{eq:limit95shifted95cutoffs} \mathcal{A}_2(z) = 8 \lambda^2 \lim_{\Lambda\to +\infty} \int_0^1 \dd{x} \int \frac{\dd[4]{q}}{(2\pi)^4} \, \frac{x T_z(q + x p_z)}{[q^2 + \Delta^2(x, z)]^3} \, \chi(q + x p_z) \chi^2(q + (x - 1)p_z),\tag{117}\] where \(\Delta^2(x, z) \equiv (1 - x)M^2 + z^2x(1-x) + m^2 x\). Due to the fact that the polynomial \(T_z(q)\) contains the quadratic terms \((q^0)^2 - q_a q^a/3\) (see 115 ), the function \(f_{x, z}(q) \equiv x T_z(q + x p_z)[q^2 + \Delta^2(x, z)]^{-3}\) decays as \(1/q^4\) when \(\abs{q} \to +\infty\); hence, although the limit 117 exists, it cannot be recklessly brought under the integral sign. This problem is circumvented by means of a subtle cancellation arising from rotational symmetry, as discussed below.

Fix an arbitrary \(c \in \mathbb{R}^4\) and consider the difference \(\chi(q + c) - \chi(q)\). Then \[\label{eq:chi95estimate} \abs{\chi(q + c) - \chi(q)} \le \abs{c} \cdot \norm*{\partial \chi}_\infty \, \mathbb{1}_\mathcal{B}(q) \le (\mathrm{const}) \cdot \frac{\mathbb{1}_\mathcal{B}(q)}{\Lambda},\tag{118}\] where \(\mathcal{B} \equiv \lbrace q \in \mathbb{R}^4 \colon \Lambda- \abs{c} \le \abs{q} \le 2 \Lambda+ \abs{c} \rbrace\) and the relation \(\norm{\partial \chi}_\infty = 2^{-N} \norm{\partial \chi_0}_\infty\) has been exploited. Now, since \(\abs{f_{x, z}(q)} \asymp 1/q^4\), we have \[\require{physics} \label{eq:chi95estimate95function} \int \frac{\dd[4]{q}}{(2\pi)^4} \, \abs{f_{x, z}(q)} \, \abs{\chi(q + c) - \chi(q)} \lesssim \frac{(\mathrm{const})}{\Lambda} \int_\Lambda^{2 \Lambda} \frac{q^3\dd{q}}{q^4} \lesssim \frac{1}{\Lambda} \overset{\Lambda\to +\infty}{\longrightarrow} 0.\tag{119}\] Based on 119 , the limit 117 can be equivalently computed with \(\chi^3(q)\) in place of \(\chi(q + x p_z) \chi^2(q + (x - 1)p_z)\), namely \[\require{physics} \label{eq:limit95after95centering} \mathcal{A}_2(z) = 8 \lambda^2 \lim_{\Lambda\to +\infty} \int_0^1 \dd{x} \int \frac{\dd[4]{q}}{(2\pi)^4} \, \frac{x T_z(q + x p_z)}{[q^2 + \Delta^2(x, z)]^3} \, \chi^3(q).\tag{120}\] Thanks to the rotational symmetry of the function \(\chi^3(q)[q^2 + \Delta^2(x, z)]^{-3}\), we have \[\require{physics} \int \frac{\dd[4]{q}}{(2\pi)^4} \frac{(q^0)^2 - q_a q^a/3}{[q^2 + \Delta^2(x, z)]^3} \, \chi^3(q) = 0,\] so the dangerous quadratic terms occurring in the numerator of 120 cancel out exactly. We stress that this argument could not be applied to 117 , because the function that multiplies \(f_{x, z}(q)\) inside 117 is not rotationally invariant.

In absence of the above quadratic terms, Lebesgue’s dominated convergence theorem holds. If the limit is brought under the integral sign, an elementary integration gives \[\require{physics} \label{eq:gfunction95pert} \mathcal{A}_2(z) = \frac{\lambda^2}{4\pi^2} \int_0^1 \dd{x} \, \frac{z^2 (\kappa^2 + 1)x(x^2 - 3x + 2) + 2imz(\kappa^2 - 1) x(1-x)}{(1 - x)M^2 + z^2x(1-x) + m^2 x}.\tag{121}\]

The function \(\mathcal{A}_2(z)\) admits a unique holomorphic extension \(\tilde{\mathcal{A}}_2(z)\) defined inside an open disk of radius \(M/2\) centered around \(z = 0\); in particular, this disk contains the point \(z = im\). This is true because the denominator of 121 cannot vanish as long as \(x \in [0, 1]\) and \(\abs*{z} \le M/2\). According to Cauchy’s integral formula, we have \[\require{physics} \label{eq:cauchy95estimates} \abs*{m^\ell \tilde{\mathcal{A}}_2^{(\ell)}(0)} = \abs{\ell! \, m^\ell \oint_{\abs{z} = M/2} \frac{\dd{z}}{2\pi i} \frac{\tilde{\mathcal{A}}_2(z)}{z^{\ell + 1}}} \le \ell! \left( \frac{2m}{M} \right)^\ell \max_{\theta \in [0, 2\pi]} \abs*{\tilde{\mathcal{A}}_2((M/2)e^{i \theta})},\tag{122}\] so \[\abs*{m^\ell \tilde{\mathcal{A}}_2^{(\ell)}(0)} \le C \lambda^2 \, \ell! \left( \frac{2m}{M} \right)^\ell.\] Since \(K = 4\) and \(M > 10m\), this implies that \[\abs{\sum_{\ell = 0}^K \frac{(im)^\ell}{\ell!} \mathcal{A}^{(\ell)}_2(0) - \tilde{\mathcal{A}}_2(im)} \le C \lambda^2 \frac{m^4}{M^4}.\] Finally, an easy calculation shows that \[\label{eq:A95expansion} \tilde{\mathcal{A}}_2(im) = \frac{m^2}{M^2} \, \frac{\lambda^2}{4\pi^2} \frac{1- 5 \kappa^2}{3} + \mathcal{O}\left( \frac{\lambda^2 m^4}{M^4} \right) = \bar{\mathfrak{a}}_{\mathrm{\small z}, 1}(1 + \mathcal{O}(m^2/M^2));\tag{123}\] consequently, there exists a \((\lambda, M, m, \Lambda)\)-independent constant \(C_0\) such that \[\abs{\sum_{\ell = 0}^K \frac{(im)^\ell}{\ell!} \mathcal{A}^{(\ell)}_2(0) - \bar{\mathfrak{a}}_{\mathrm{\small z}, 1}} \le C_1 \lambda^2 \frac{m^4}{M^4}.\]

4.7 Proof of Theorem 1↩︎

According to definition 15 , we can write \[\label{eq:holomorphic95estimate} \mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}= \sum_{\ell = 0}^K \frac{(im)^\ell}{\ell!} \, [\mathcal{A}^{(\ell)}_0(0; \lambda) + \mathcal{A}^{(\ell)}_2(0; \lambda) + \mathcal{A}^{(\ell)}_{>2}(0; \lambda)].\tag{124}\] We know from Section 4.3 that \(\mathcal{A}_0(z; \lambda)\) is identically zero. Moreover, relying on the fact that \(\mathcal{A}(z) = [\mathcal{F}(z) - \mathcal{Q}(z)]/\mathcal{Q}(z)\) with \(\mathcal{Q}(z) = 24 + \sum_{n \ge 1} \lambda^{2n} \mathcal{Q}_n(z; \lambda)\), Theorem 4 implies that \(m^\ell \abs*{\mathcal{A}^{(\ell)}_{>2}(0; \lambda)}\) is bounded by the right hand side of ?? with \(n = 2\) and a \(\ell\)-dependent overall constant. Finally, \(m^\ell \mathcal{A}^{(\ell)}_2(z; 0)\) can be decomposed as in 110 and the remainder \(m^\ell R^{(\ell)}(z; \lambda)\) is estimated as in 111 with a \(\ell\)-dependent overall constant. Based on 123 , we obtain \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}= \bar{\mathfrak{a}}_{\mathrm{\small z}, 1}(1 + R_\lambda)\), where \[\abs*{R_\lambda} \le C_1 \frac{M^2}{\Lambda^2} + C_2 \frac{m^2}{M^2} + C_3 \frac{\lambda^2\Lambda^2}{M^2} \cdot \log^4 \left(\frac{M}{m}\right) \log^4 \left( \frac{\Lambda}{M} \right) + C'_3 \frac{\lambda^2\Lambda^2}{M^2} \log \left( \frac{M}{m} \right) \frac{M}{\Lambda}.\] Note that \(C'_3 (\lambda^2 \Lambda^2/M^2) \log (M/m) \cdot (M/\Lambda)\) is subdominant with respect to the remainder multiplied by \(C_3\), so it can be reabsorbed inside it. This completes the proof of Theorem 1.

Acknowledgments↩︎

We acknowledge support from the MUR, PRIN 2022 project MaIQuFi cod.20223J85K3. This work has been carried out under the auspices of the GNFM of INdAM.

5 The regularized anomalous gyromagnetic factor↩︎

In this section we provide some details about the definition of regularized anomalous gyromagnetic factor given by 15 . In the context of perturbative Quantum Field Theory, the anomalous gyromagnetic factor is defined in Minkowski spacetime starting from the probability amplitude for a process in which a single fermion is scattered by a weak, external electromagnetic field \(A_\mu(x)\). This amplitude reads \[\require{physics} \label{eq:scattering95amplitude} - \frac{1}{2\sqrt{p_0 \, p'_0}} \bar{u}_{p' \xi'} \, \hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p', p) \, u_{p \xi} \, \hat{A}_\mu(k) \eval_{(p')^2 = p^2 = - m^2}\tag{125}\] within the first order in \(A_\mu\), where \(\hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p', p)\) is the Minkowskian amputated vertex function (see e.g. Peskin_1995?). Here, \(k = p' - p\), \(p^2 \equiv -(p_0)^2 + \abs{\boldsymbol{p}}^2\) and \(\lbrace u_{p \xi} \rbrace_{\xi = \pm}\) are two linearly independent solutions of the equation \((i p_\mu \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}+ \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}) u_{p \xi} = 0\), where \(\gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\) are the Minkowskian gamma matrices, \(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}} \equiv \sqrt{-p^2}\) and \(p^2 \equiv -(p^0)^2 + \abs{\mathbf{p}}^2\). Explicitly, \[\mathbb{C}^4 \ni u_{p \xi} = \begin{pmatrix} \sqrt{-p^0 + i\mathbf{p} \cdot \boldsymbol{\sigma}} \, e_\xi \\ \sqrt{-p^0 - i\mathbf{p} \cdot \boldsymbol{\sigma}} \, e_\xi \end{pmatrix}\] where \(\xi \in \lbrace -1/2, 1/2 \rbrace\) labels the helicity states of the fermion and \(\lbrace e_{-1/2}, e_{1/2} \rbrace \subseteq \mathbb{C}^2\) is the canonical basis of \(\mathbb{C}^2\). The constraint \(p^2=-m^2\) occurring in 125 is called mass shell condition. Lorentz symmetry constraints the vertex matrix element to take the form Nowakowski_2005? \[\label{eq:vertex95parametrization} \bar{u}_{p' \xi'} \, \hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}\, u_{p \xi} = \bar{u}_{p' \xi'} \, \biggl[ \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}[F + \gamma^5 F_5]- \frac{i(p'+p)^\mu}{\abs*{p'}_{\! \scriptscriptstyle \mathrm{M}}+\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}} [G + \gamma^5 G_5] +\frac{(p'-p)^\mu}{\abs*{p'}_{\! \scriptscriptstyle \mathrm{M}}+\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}} [H + \gamma^5 H_5] \biggr] u_{p \xi},\tag{126}\] where \(F, F_5, \dots, H_5\) are called form fators and they depend on \(\abs*{p'}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{k}_{\! \scriptscriptstyle \mathrm{M}}\). In the non interacting case (\(\lambda=0\)), one has \(F=1\), \(F_5, G, G_5, H, H_5 = 0\) and \(\hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p', p) = \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\).

If \(\mathbf{k} = \mathbf{p}' - \mathbf{p}\) is small with respect to \(m\), the parametrization 126 implies that 125 takes the form \[\label{eq:low95energy95scattering} (-i) [F + G](m, m, 0) \cdot \biggl[ \delta_{\xi' \xi} \hat{A}^0(\mathbf{k}) +\frac{2F(m, m, 0)}{[F + G](m, m, 0)} \cdot \left(-\frac{1}{2m} \mathbf{S}_{\xi' \xi} \cdot \hat{\mathbf{B}}(\mathbf{k}) \right) \biggr] + \cdots,\tag{127}\] where \(\mathbf{S} = (\sigma_1, \sigma_2, \sigma_3)/2\) are the standard spin-\(1/2\) matrices and \(\hat{\mathbf{B}}(\mathbf{k}) \equiv i \mathbf{k} \times \hat{\mathbf{A}}(\mathbf{k})\) is the Fourier transform of the external magnetic field. The neglected terms, denoted by \((\, \cdots)\), are either higher orders in \(\mathbf{p}, \mathbf{p}'\) or terms that do not contribute to the gyromagnetic factor Nowakowski_2005?. To derive 127 , one uses the basic relation \[\bar{u}_{p' \xi'} \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, u_{p \xi}= \bar{u}_{p' \xi'} \left[ \frac{(p' + p)^\mu}{2m} - \frac{[\gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}, \gamma^\nu_{\scriptscriptstyle{\mathrm{M}}}]}{4m} \, k_\nu \right] u_{p \xi},\] which, once plugged into 126 , shows that a term proportianal to \(\mathbf{S}_{\xi' \xi} \cdot \hat{\mathbf{B}}\) arises even in the non-interacting case. Recalling that the gyromagnetic factor is the coefficient \(g\) appearing inside the relation \(\boldsymbol{\mu} = -g (e/2m) \mathbf{S}\), the usual perturbative definition of anomalous gyromagnetic factor can be read off from 127 (see Peskin_1995?): \[\label{eq:g95form95factors} \mathfrak{a}= \frac{-G(m, m, 0)}{[F + G](m, m, 0)}.\tag{128}\] The form factors can be expressed in terms of the amputated vertex function as \[\require{physics} \begin{align} \tag{129} F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0) & = \frac{i\epsilon_{\alpha \mu \sigma \beta}}{48 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} p^\sigma \, (\partial_{p'} - \partial_p)^\alpha \tr[\gamma^5 \mathcal{C}_{p'} \, \hat{\Gamma}_{\scriptscriptstyle{\mathrm{M}}}^\mu(p', p) \, \mathcal{C}_{p} \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] \eval_{p, p} \\ \tag{130} (F + G)(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0) & = -\frac{i}{4 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} \tr[\mathcal{C}_{p} \, p_\mu \hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p, p)] \end{align}\] where \(\mathcal{C}_{p} = -i \cancel{p} + \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}\) and \(\epsilon_{\alpha \mu \sigma \beta}\) is the Levi-Civita symbol. A proof of these relations will be given below.

As a consequence of the Lorentz covariance of the formal continuum theory, both \(F\) and \(G\) depend on \(p\) only through \(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}\), so we can evaluate them at \(p = (\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0, 0, 0)\) without loss of generality. With this choice, a combination of 128 and 129 gives a formal characterization of \(\mathfrak{a}\) in terms of the Minkowskian amputated vertex function; namely, a straightforward computation shows that \(\mathfrak{a}= \mathcal{A}_{\scriptscriptstyle{\mathrm{M}}}(m)\), where \[\mathcal{A}_{\scriptscriptstyle{\mathrm{M}}}(s) \equiv \frac{i s \epsilon_{a b c} \tr[\gamma^5 (1 - i \gamma^0_{\scriptscriptstyle{\mathrm{M}}})(\partial_{p'} - \partial_p)^a \hat{\Gamma}^b_{\scriptscriptstyle{\mathrm{M}}}(p_s, p_s) \gamma^c_{\scriptscriptstyle{\mathrm{M}}}] + 2 \tr[ (\gamma_{\scriptscriptstyle{\mathrm{M}}})_a \hat{\Gamma}^a_{\scriptscriptstyle{\mathrm{M}}}(p_s, p_s) ]}{-6i \tr[(1 - i \gamma^0_{\scriptscriptstyle{\mathrm{M}}}) \hat{\Gamma}^0_{\scriptscriptstyle{\mathrm{M}}}(p_s, p_s)]} - 1\] and \(p_s \equiv (s, 0, 0, 0)\). Recalling that \(\gamma^0_{\scriptscriptstyle{\mathrm{M}}}= i \gamma^0\) and \(\gamma^a_{\scriptscriptstyle{\mathrm{M}}}= \gamma^a \,\, \forall a = 1, 2, 3\), the Euclidean function \(\mathcal{A}(z)\) defined in 16 is obtained by applying the formal Wick rotation \[s \mapsto -iz, \quad \hat{\Gamma}^0_{\scriptscriptstyle{\mathrm{M}}}(p_s, p_s) \mapsto i\hat{\Gamma}^0(p_z, p_z), \quad \hat{\Gamma}^a_{\scriptscriptstyle{\mathrm{M}}}(p_s, p_s) \mapsto\hat{\Gamma}^a(p_z, p_z) \,\,\, \forall a = 1, 2, 3\] on \(\mathcal{A}_{\scriptscriptstyle{\mathrm{M}}}(s)\). Our non perturbative definition of \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) consists in the \(K\)-th MacLaurin polynomial of \(\mathcal{A}(z)\) evaluated at \(z = im\), as the rest is subdominant in \(m^2/M^2\).

We finally give a derivation of 129 and 130 . As a preliminary step, it is convenient to multiply both sides of 126 by \(u_{p' \xi'}\) on the left and by \(\bar{u}_{p \xi}\) on the right and subsequently sum over \(\xi', \xi \in \lbrace -1/2, 1/2 \rbrace\). Thanks to the identity \(\sum_{\xi=\pm 1/2} u_{p \xi} \bar{u}_{p \xi} = -i\cancel{p} + \abs*{p}_{\! \scriptscriptstyle \mathrm{M}} \equiv \mathcal{C}_{p}\), this converts 126 into \[\label{eq:form95factors95matrix} \mathcal{C}_{p'} \, \hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p', p) \, \mathcal{C}_{p} = \mathcal{C}_{p'} \, X^\mu_{\scriptscriptstyle{\mathrm{M}}}(p', p) \, \mathcal{C}_{p},\tag{131}\] where \(X^\mu_{\scriptscriptstyle{\mathrm{M}}}(p', p)\) denotes the matrix appearing at the right hand side of 126 .

To prove 130 , we take the trace of both sides of 131 and evaluate them at \(p = p'\), thus getting \[\label{eq:F95G95trace} \tr[\mathcal{C}_{p} \, \hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p, p) \, \mathcal{C}_{p}] = \tr \left[ \, \mathcal{C}_{p} \! \left( \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}} , 0) - \frac{ip^\mu}{\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}} G(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0) \right) \mathcal{C}_{p} \right].\tag{132}\] Here, \(H, H_5\) are absent because they are multiplied by \((p - p)^\mu = 0\) and \(G_5\) and \(F_5\) are wiped away because \(\tr[ \mathcal{C}_{p} \, \gamma^5 \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, \mathcal{C}_{p} ] = \tr[ \mathcal{C}_{p} \, \gamma^5 \, \mathcal{C}_{p} ] = 0\). By exploiting the trivial matrix identity \(\mathcal{C}_{p} \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\mathcal{C}_{p} = - \mathcal{C}_{p} \, i p^\mu \, \mathcal{C}_{p}/\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}\) and the cyclicity of the trace, we can rewrite 132 as \[\label{eq:F95G95normalization95quasi95final} \tr[\hat{\Gamma}^\mu_{\scriptscriptstyle{\mathrm{M}}}(p, p) \, \mathcal{C}_{p}^2] = - \frac{i p^\mu}{\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}} (F + G)(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0) \tr[ \mathcal{C}_{p}^2 ];\tag{133}\] knowing that \(p^2 = -\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2\), \(\mathcal{C}_{p}^2 = 2 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}} \mathcal{C}_{p}\) and \(\tr[\mathcal{C}_{p}^2] = 8 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2\), formula 130 follows after contracting both sides of 133 with \(p_\mu\).

We now prove 129 . At first, we verify that the right hand side of this equality cannot depend on any form factor different from \(F\), nor on the derivatives of \(F\) itself. For instance, we see from 131 that \(G\) contributes with the term \[\require{physics} \frac{i\epsilon_{\alpha \mu \sigma \beta}}{48 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} p^\sigma (\partial_{p'} - \partial_p)^\alpha \tr[\gamma^5 \left( \mathcal{C}_{p'} \, \frac{-i(p' + p)^\mu}{\abs*{p'}_{\! \scriptscriptstyle \mathrm{M}} + \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}} \, \mathcal{C}_{p} \right) \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] \eval_{p, p} G(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0),\] which vanishes because it can be written as a sum of traces of the form \(\tr[\gamma^5 \gamma^{a_1}_{\scriptscriptstyle{\mathrm{M}}}\cdots \gamma^{a_n}_{\scriptscriptstyle{\mathrm{M}}}]\) with \(n < 4\). An immediate extension of this argument leads to the conclusion that the right hand side of 129 does not receive any contribution from \(G, H, \partial F, \partial G, \partial H\). The form factors \(F_5, G_5, H_5\) and their derivatives are instead cancelled due to the complete antisymmetry of the Levi-Civita symbol. To see why this is the case, let us consider the term proportional to \(F_5\), which is \[\require{physics} \label{eq:F595contribution} \frac{i\epsilon_{\alpha \mu \sigma \beta}}{48 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} p^\sigma (\partial_{p'} - \partial_p)^\alpha \tr[\gamma^5 \, \mathcal{C}_{p'} \, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\gamma^5 \, \mathcal{C}_{p} \, \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] \eval_{p, p} F_5(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0).\tag{134}\] Thanks to the properties of the \(\gamma^5\) matrix Weinberg_1995?, we can write \[\require{physics} \begin{align} \label{eq:axial95form95factor95vanishing} \begin{aligned} (\partial_{p'} - \partial_p)^\alpha \tr[\gamma^5 \, \mathcal{C}_{p'} \, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\gamma^5 \, \mathcal{C}_{p} \, \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] \eval_{p, p} & = (\partial_{p'} - \partial_p)^\alpha \tr[(i \cancel{p}' + \abs*{p'}_{\scriptscriptstyle{\mathrm{M}}}) \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}(i \cancel{p} - \abs*{p}_{\scriptscriptstyle{\mathrm{M}}}) \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] \eval_{p, p}\\ & = i\tr[\gamma^\alpha_{\scriptscriptstyle{\mathrm{M}}}\, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, \cancel{p} \, \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] -i\tr[\cancel{p} \, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, \gamma^\alpha_{\scriptscriptstyle{\mathrm{M}}}\, \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}]. \end{aligned} \end{align}\tag{135}\] If \(\eta = \mathrm{diag} (-1, 1, 1, 1)\) denotes the Lorentzian metric tensor, the trace of a product of \(4\) gamma matrices satisfies \(\tr \, [\gamma^\alpha_{\scriptscriptstyle{\mathrm{M}}}\, \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}\, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, \gamma^\nu_{\scriptscriptstyle{\mathrm{M}}}] = 4(\eta^{\alpha \beta} \eta^{\mu \nu} - \eta^{\alpha \mu} \eta^{\beta \nu} + \eta^{\alpha \nu} \eta^{\beta \mu})\), so both the traces appearing in the last line of 135 vanish when they are contracted with the totally antisymmetric tensor \(\epsilon_{\alpha \mu \sigma \beta}\); this causes the whole term 134 to vanish. A similar conclusion can be drawn for \(G_5, H_5, \partial F_5, \partial G_5, \partial H_5\).

According to the above considerations, the right hand side of 129 consists in a single term which is proportional to \(F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0)\), namely \[\require{physics} \label{eq:F95factor95RHS95surviving} \frac{i\epsilon_{\alpha \mu \sigma \beta}}{48 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} p^\sigma \, (\partial_{p'} - \partial_p)^\alpha \tr[\gamma^5 \mathcal{C}_{p'} \, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, \mathcal{C}_{p} \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] \eval_{p, p} F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0);\tag{136}\] since \(\tr[\gamma^5 \mathcal{C}_{p'} \, \gamma^\mu_{\scriptscriptstyle{\mathrm{M}}}\, \mathcal{C}_{p} \gamma^\beta_{\scriptscriptstyle{\mathrm{M}}}] = 4i \epsilon^{\tau \mu \nu \beta} (p')_\tau p_\nu\), this is also equal to \[\require{physics} \frac{i\epsilon_{\alpha \mu \sigma \beta}}{48 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} p^\sigma \cdot 4i \epsilon^{\tau \mu \nu \beta} \left( (\partial_{p'} - \partial_p)^\alpha (p')_\tau p_\nu \eval_{p, p} \right) \, F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0).\] After computing the derivatives and exploiting the identity \(\epsilon_{\alpha \mu \sigma \beta} \epsilon^{\alpha \mu \nu \beta} = 6 \delta^\nu_\sigma\), the right hand side of 129 reduces to \[\frac{i p^2}{48 \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}^2} \cdot (8i \cdot 6) F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0) = F(\abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, \abs*{p}_{\! \scriptscriptstyle \mathrm{M}}, 0),\] so 129 holds.

6 Proof of Lemma 1↩︎

Preliminarily, we show that \(\beta^{m, s}_h \vert_{m_N = 0} = m^s_h \vert_{m_N = 0} = 0\) on all scales. The key observation is that the effective potential \({\mathscr{V}}^{N-1}[\psi, \omega] \vert_{m_N = 0}\) is manifestly invariant under the global chiral \(\mathrm{U}(1)\) transformation \(\hat{\psi}^\epsilon_{k s} \mapsto e^{i\epsilon \alpha_s} \hat{\psi}^\epsilon_{k s}, \, \hat{\eta}^\epsilon_{k s} \mapsto e^{i \epsilon \alpha_s} \hat{\eta}^\epsilon_{k s}\), where \(\alpha_+\) and \(\alpha_-\) are two independent phases. The term \(-\beta^{m, s}_{N-1} \vert_{m_N = 0} (\psi^+_s \psi^-_{-s})(x)\) is not invariant under this transformation, so it must be \(\beta^{m, s}_{N-1} \vert_{m_N = 0} = 0\) and therefore \(m^s_{N-2} \vert_{m_N = 0} = 0\) (recall that \(m^s_{N-1} = m^s_N\)). This implies that \({\mathscr{V}}^{N-2}[\psi, \omega] \vert_{m_N = 0}\) is still invariant under the global chiral \(\mathrm{U}(1)\) transformation introduced above, so both \(\beta^{m, s}_{N-2} \vert_{m_N = 0}\) and \(m^s_{N-3} \vert_{m_N = 0}\) vanish. By iterating this argument, one infers that \(\beta^{m, s}_h \vert_{m_N = 0} = m^s_{h-1} \vert_{m_N = 0} = 0\) for every \(h = h^\star, \dots, N\).

Now we turn our attention on ?? . Since the theory is translationally invariant, the first line of 26 is equal to \[\require{physics} \int_{\mathcal{M}_L^2} \!\! \dd[4]{x} \dd[4]{y} \mathcal{L}\left[ W^{\alpha \beta}(x,y) \, \psi^+_\alpha(x) \psi^-_\beta(y) \right] = \int_{\mathcal{M}_L} \! \dd[4]{x} (\psi^+_\alpha [b^{\alpha \beta} + c^{\mu \alpha \beta} \partial_\mu + d^{\mu \nu \alpha \beta} \partial_\mu \partial_\nu] \psi^-_\beta)(x)\] with \[\require{physics} \begin{align}b^{\alpha \beta} = \int_{\mathcal{M}_L} \dd[4]{z} W^{\alpha \beta}(z), \! \quad c^{\mu \alpha \beta} = \int_{\mathcal{M}_L} \dd[4]{z} (z_{\scriptstyle{\mathbb{T}}})^\mu W^{\alpha \beta}(z),\\ \begin{align} d^{\mu \nu \alpha \beta} = \frac{1}{2} \int_{\mathcal{M}_L} \dd[4]{z} (z_{\scriptstyle{\mathbb{T}}})^\mu (z_{\scriptstyle{\mathbb{T}}})^\nu \bar{W}^{\alpha \beta}(z), \end{align} \end{align}\] where a sum over repeated indices is understood the scale labels are temporarily suppressed, \(W^{\alpha \beta}(z)\) stands for \(W^{\alpha \beta}(z, 0)\) and the kernel \(\bar{W}^{\alpha \beta}(z)\) must be evaluated at \(m_N = 0\) when the spinor indices \(\alpha, \beta\) have different chiralities. If the coefficients \(b^{\alpha \beta}, c^{\mu \alpha \beta}, d^{\mu \nu \alpha \beta}\) are thought as \(4 \times 4\) complex matrices whose entries are labelled by \(\alpha, \beta\), we can decompose them along the standard basis \(\lbrace \mathbb{1}, \gamma^\mu, \gamma^\mu \gamma^5, [\gamma^\mu, \gamma^\nu]/2 \rbrace\): \[\begin{align} \label{eq:gamma95matrices95base95decomposition} \begin{aligned} b & = b_\mathbb{1}\mathbb{1}+ b_5 \gamma^5 + (b_v)_\rho \gamma^\rho + (b_a)_\rho \gamma^\rho \gamma^5 + (b_t)_{\theta \rho} [\gamma^\theta, \gamma^\rho]/2, \\ c^\mu & = (c_\mathbb{1})^\mu \mathbb{1}+ (c_5)^\mu \gamma^5 + (c_v)_\rho^\mu \gamma^\rho + (c_a)_\rho^\mu \gamma^\rho \gamma^5 + (c_t)_{\theta \rho}^\mu [\gamma^\theta, \gamma^\rho]/2, \\ d^{\mu \nu} & = (d_\mathbb{1})^{\mu \nu} \mathbb{1}+ (d_5)^{\mu \nu} \gamma^5 + (d_v)_\rho^{\mu \nu} \gamma^\rho + (d_a)_\rho^{\mu \nu} \gamma^\rho \gamma^5 + (d_t)_{\theta \rho}^{\mu \nu} [\gamma^\theta, \gamma^\rho]/2. \end{aligned} \end{align}\tag{137}\] Even if the spacetime volume is finite, the theory is invariant under the action of a discrete group \(G \subseteq \mathrm{SO}(4)\) generated by all the \(\pi/2\) rotations around each pair of axes. The precise meaning of this sentence is the following: if we consider the maps \[\begin{align} S(\vec{\theta}, \vec{\xi}) & \equiv \begin{pmatrix} \displaystyle \exp \left( \frac{i \vec{\sigma} \cdot (\vec{\theta} + \vec{\xi})}{2} \right) & 0 \\[3pt] \displaystyle 0 & \displaystyle \exp \left( \frac{i \vec{\sigma} \cdot (\vec{\theta} - \vec{\xi})}{2} \right) \end{pmatrix}, \\ U(\vec{\theta}, \vec{\xi}) & \equiv \exp[\vec{\theta} \cdot \vec{L} + \vec{\xi} \cdot \vec{K}], \end{align}\] where \(\vec{\sigma}\) are the Pauli matrices and the \(4 \times 4\) matrices \(\vec{L}, \vec{K}\) are defined as \[\begin{align} L_1 = \begin{pmatrix} 0 & & & \\ & 0 & 0 & 0 \\ & 0 & 0 & -1 \\ & 0 & 1 & 0 \\ \end{pmatrix}, \qquad L_2 = \begin{pmatrix} 0 & & & \\ & 0 & 0 & 1 \\ & 0 & 0 & 0 \\ & -1 & 0 & 0 \\ \end{pmatrix}, \qquad L_3 = \begin{pmatrix} 0 & & & \\ & 0 & -1 & 0 \\ & 1 & 0 & 0 \\ & 0 & 0 & 0 \\ \end{pmatrix}, \\[4pt] K_1 = \begin{pmatrix} 0 & -1 & & \\ 1 & 0 & 0 & 0 \\ & 0 & 0 & 0 \\ & 0 & 0 & 0 \\ \end{pmatrix}, \qquad K_2 = \begin{pmatrix} 0 & & 1 & \\ & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ & 0 & 0 & 0 \\ \end{pmatrix}, \qquad K_3 = \begin{pmatrix} 0 & & & -1 \\ & 0 & 0 & 0 \\ & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix}, \end{align}\] the generating functional 12 is invariant under the transformations \[\begin{align} \hat{\psi}^-_k \mapsto S(\vec{\theta}, \vec{\xi}) \, \hat{\psi}^-_{U(\vec{\theta}, \vec{\xi})k}, \qquad \hat{\psi}^+_k \mapsto \hat{\psi}^+_{U(\vec{\theta}, \vec{\xi})k} \, S^{-1}(\vec{\theta}, \vec{\xi}), \\ \hat{\eta}^-_k \mapsto S(\vec{\theta}, \vec{\xi}) \, \hat{\eta}^-_{U(\vec{\theta}, \vec{\xi})k}, \qquad \hat{\eta}^+_k \mapsto \hat{\eta}^+_{U(\vec{\theta}, \vec{\xi})k} \, S^{-1}(\vec{\theta}, \vec{\xi}), \\\hat{J}_k \mapsto U^{-1}(\vec{\theta}, \vec{\xi}) \hat{J}_{U(\vec{\theta}, \vec{\xi})k}\end{align}\] only provided that \(\vec{\theta}, \vec{\xi} \in (\pi \mathbb{Z}/2)^3\). This constraint on \(\vec{\theta}, \vec{\xi}\) is due to the finiteness of the spacetime volume: if \(L < +\infty\), the generating functional is left invariant by those \(\mathrm{SO}(4)\) transformations that preserve the shape of the cubic spacetime \(\mathcal{M}_L = [-L/2, L/2]^4\) and the structure of the discrete Fourier space \(\mathcal{D}_{N, L} = (2\pi \mathbb{Z}/L)^4 \cap \mathop{\mathrm{supp}}\chi_N\). As \(\vec{\theta}, \vec{\xi}\) span \((\pi \mathbb{Z}/2)^3\), the matrices \(U(\vec{\theta}, \vec{\xi})\) constitute a four-dimensional, complex representation of the finite group \(G\) denoted by \(\mathcal{U}\). This representation is also irreducible: in fact, the whole canonical basis of \(\mathbb{C}^4\) is recovered by acting on the vector \(e_0 \equiv (1, 0, 0, 0)\) with the matrices \(\mathbb{1}_4, \exp(\pi K_1/2), \exp(-\pi K_2/2), \exp(\pi K_3/2) \in \mathcal{U}(G)\), so the only invariant subspaces of \(\mathcal{U}\) are \(\lbrace 0 \rbrace \subset \mathbb{C}^4\) and \(\mathbb{C}^4\) itself.

Depending on the number of their spacetime indices, the coefficients \((b_v)_\rho, (b_a)_\rho\), \((b_t)_{\theta \rho}, (c_\mathbb{1})^\mu, (c_5)^\mu, (c_v)_\rho^\mu, \dots\) can be thought as tensors belonging to the representation spaces of \(\mathcal{U}, \mathcal{U}^{\otimes 2}, \mathcal{U}^{\otimes 3}, \dots\). The \(G\)-symmetry of the theory implies that each of these tensors must belong to a one-dimensional invariant subspace of the corresponding representation. To see a concrete example of this, consider the term \[\require{physics} \int \dd[4]{x} [\psi^+ \, (b_v)_\rho \gamma^\rho \, \psi^-](x).\] If this term is invariant under the above transformations, it must be \[S^{-1}(\vec{\theta}, \vec{\xi}) \, (b_v)_\rho \gamma^\rho \, S(\vec{\theta}, \vec{\xi}) = (b_v)_\rho \gamma^\rho\] for every \(\vec{\theta}, \vec{\xi} \in (\pi\mathbb{Z}/2)^3\). Thanks to the identity \(S^{-1}(\vec{\theta}, \vec{\xi}) \, \gamma^\rho \, S(\vec{\theta}, \vec{\xi}) = U(\vec{\theta}, \vec{\xi})^{\rho}_{\,\, \mu} \, \gamma^\mu\), this can be rewritten as \[(b_v)_\rho = U(\vec{\theta}, \vec{\xi})^{\rho}_{\,\, \mu} \, (b_v)^\mu \qquad \forall \vec{\theta}, \vec{\xi} \in (\pi\mathbb{Z}/2)^3,\] showing that the vector \(((b_v)_\rho)_{\rho = 0, \dots, 3} \in \mathbb{C}^4\) belongs to a one-dimensional invariant subspace of the representation \(\mathcal{U}\). Similar considerations apply to the other coefficients as well.

Now consider the element of \(G\) represented by \(\exp\,\![(\pi, 0, 0) \cdot (\vec{L} + \vec{K})] = - \mathbb{1}_4\). The invariance of \((b_v)_\rho\), \((b_a)_\rho\), \((c_\mathbb{1})^\mu\), \((c_5)^\mu\), \((c_t)_{\theta \rho}^\mu\), \((d_v)_\rho^{\mu \nu}\), \((d_a)_\rho^{\mu \nu}\) under the action of this element implies that \((b_v)_\rho = - (b_v)_\rho, (b_a)_\rho = - (b_a)_\rho\) and so on; therefore, all the coefficients carrying an odd number of spacetime indices are identically vanishing. The fact that \(\mathcal{U}\) is irreducible implies that all the coefficients with two spacetime indices are proportional to the Kronecker delta. Indeed, let us consider \((b_t)_{\theta \rho}\) for concreteness. The invariance argument used above yields \[\label{eq:rank95295tensor95invariance} \mathcal{U}_\mu^{\,\, \theta}(g) \, \mathcal{U}_\nu^{\,\, \rho}(g) (b_t)_{\theta \rho} = (b_t)_{\mu \nu} \qquad \forall g \in G,\tag{138}\] or, in a more compact matrix form, \[\label{eq:rank95295tensor95invariance95matrix} \mathcal{U}(g) \cdot b_t \cdot \mathcal{U}(g)^T = b_t \qquad \forall g \in G.\tag{139}\] Recalling that \(\mathcal{U}(g) \mathcal{U}(g)^T = \mathbb{1}_4\) (\(\mathcal{U}\) is orthogonal, because \(\vec{L}, \vec{K}\) are skew-symmetric), we can multiply both sides of 139 by \(\mathcal{U}(g)\) on the right, thus getting \([b_t, \mathcal{U}(g)] = 0 \, \forall g \in G\). Then, since \(\mathcal{U}\) is irreducible, Schur’s lemma guarantees that \((b_t)_{\theta \rho} = b_t \, \delta_{\theta \rho}\) for some \(b_t \in \mathbb{C}\). The same argument holds for \((c_v)_\rho^\mu, (c_a)_\rho^\mu, (d_\mathbb{1})^{\mu \nu}, (d_5)^{\mu \nu}\) as well.

Based on the above considerations, the decomposition 137 becomes \[\label{eq:lm:coefficients95matrix95form95final} b = b_\mathbb{1}\mathbb{1}+ b_5 \gamma^5, \qquad c^\mu = (c_v + c_a \gamma^5) \gamma^\mu, \qquad d^{\mu \nu} = (d_\mathbb{1}\mathbb{1}+ d_5 \gamma^5) \delta^{\mu \nu} + (d_t)^{\mu \nu}_{\theta \rho} [\gamma^\theta, \gamma^\rho]/2.\tag{140}\] Since \[b_\mathbb{1}\mathbb{1}+ b_5 = \begin{pmatrix} b + b_5 & 0 \\ 0 & b - b_5 \end{pmatrix}, \qquad (c_v + c_a \gamma^5) \gamma^\mu = \begin{pmatrix} 0 & (c_v + c_a) \sigma^\mu_+ \\ (c_v - c_a) \sigma^\mu_- & 0 \end{pmatrix},\] the matrix structure of \(b, c^\mu\) clearly agrees with ?? . According to 140 , we have \[\label{eq:d95structure} d^{\mu \nu} = \begin{pmatrix} \delta^{\mu \nu}(d_\mathbb{1}+ d_5) + (d_t)^{\mu \nu}_{\theta \rho} \, \sigma_+^{[\theta} \sigma_-^{\rho]}/2 & 0 \\ 0 & \delta^{\mu \nu}(d_\mathbb{1}- d_5) + (d_t)^{\mu \nu}_{\theta \rho} \, \sigma_-^{[\theta} \sigma_+^{\rho]}/2 \end{pmatrix},\tag{141}\] where the notation \(\sigma_\pm^{[\theta} \sigma_\mp^{\rho]}\) stands for the antisymmetrization \(\sigma_\pm^{\theta} \sigma_\mp^{\rho} -\sigma_\pm^{\rho} \sigma_\mp^{\theta}\). Formula 141 shows that the only potentially nonvanishing entries of the matrix \(d^{\mu \nu}\) are labelled by spinor indices \(\alpha, \beta\) with opposite chiralities, so these entries must be evaluated at \(m_N = 0\). The same argument used to prove that \(\beta^{m, s}_{h-1} \vert_{m_N = 0} = 0\) then implies that \(d^{\mu \nu} = 0\), so we conclude that the action of \(\mathcal{L}\) on \(W^{\alpha \beta}\) agrees with ?? . An entirely similar procedure can be followed with the remaining lines of 26 .

7 Some estimates at higher orders↩︎

Although the bound provided by Theorem 4 is based on the tree expansion, it clearly holds for each renormalized multiscale Feynman diagram that contributes to \(\hat{S}(k)\) or \(\hat{\Gamma}^\mu(p', p)\), at least when \(\abs{k}, \abs*{p'}, \abs{p}\) are small enough. To concretely appreciate the subtle mechanism that allows to extract the overall \(m^2/M^2\) factor, it is instructive to analyze a few concrete examples and work out the bounds ?? “by hand”. In the standard perturbative QFT framework, it is practically impossible to give an a priori estimate on the size of a Feynman diagram; yet, the non perturbative formalism adopted in this paper allows to exhibit a meaningful bound on any multiscale renormalized Feynman diagram, without even writing down its explicit expression.

7.0.0.1 A fourth order diagram.

Let us consider the diagram displayed in Karplus_Kroll_1950?, which contributes to the fourth order correction to the anomalous gyromagnetic factor in ordinary perturbative QED\(_4\). In our model, there are plenty of renormalized multiscale Feynman diagrams with the same structure as this. An example is given by figure 10. Here, black boxes have been used to evidence the clusters associated with the vertices of the tree: proceeding from the innermost to the outermost cluster, we encounter one propagator on scale \(h_1\), two propagators on scale \(h_2\), one propagator on scale \(h_3\) and two propagators on scale \(h^\star\).

According to definition 26 , the \(\mathcal{R}\) operator acts nontrivially on the penultimate cluster. The \((h^\star-1)\)-weighted norm of the diagram can be estimated by using the same methods discussed in the proof of Theorem 2; namely, after constructing a spanning tree \(T_h\) for each cluster with scale \(h\), we bound the propagators lying along \(T_h\) with their weighted \(L^1\) norm and the other ones with their \(L^\infty\) norm. Moreover, the action of \(\mathcal{R}\) on the cluster with scale \(h_3\) results in an extra \(2^{(h^\star - h_3) \cdot 2}\) factor, so we get \[\norm*{W}_{h^\star-1} \le \underbrace{\left(\frac{C \lambda^2}{M^2}\right)^2}_{\text{Endpoints}} \cdot \,\, \underbrace{2^{-h^\star - h^\star}}_{h^\star \text{ cluster}} \,\, \cdot \underbrace{2^{3h_3}}_{h_3 \text{ cluster}} \cdot \,\, \underbrace{2^{3h_2 - h_2}}_{h_2 \text{ cluster}} \,\, \cdot \underbrace{2^{-h_1}}_{h_1 \text{ cluster}} \cdot \,\, \underbrace{2^{2(h^\star - h_3)}}_{\mathcal{R}}\] and this can be easily rewritten as \[\label{eq:diagram95estimate} \norm*{W}_{h^\star-1} \le \left(\frac{C \lambda^2 \Lambda^2}{M^2}\right)^2 \cdot 2^{2(h_1 - N) + 2(h_2 - N) -2h^\star} \cdot 2^{(h_3 - h^\star) \cdot (0 - 2) + (h_2 - h_3) \cdot (-3 - 0) + (h_1 - h_2) \cdot (-3 - 0)},\tag{142}\] thus reproducing the structure displayed in ?? (for each cluster, we evidenced the difference \(D_v - R_v\), where \(v \in \mathrm{V}(\tau)\) is the corresponding vertex on the tree). The short memory property is recovered by multiplying 142 by \(1 = 2^{2(h^\star - N)} \cdot 2^{2(N - h^\star)}\) and redistributing the factor \(2^{2(N - h^\star)}\) among the various clusters as \(2^{2(N - h^\star)} = 2^{2(N - h_1) + 2(h_1 - h_2) + 2(h_2 - h_3) + 2(h_3 - h^\star)}\); the result is \[\norm*{W}_{h^\star-1} \le \left(\frac{C \lambda^2 \Lambda^2}{M^2}\right)^2 \cdot 2^{2(h^\star - N)} \cdot 2^{2(h_2 - N) -2h^\star} \cdot 2^{(h_3 - h^\star) \cdot 0 + (h_2 - h_3) \cdot (-1) + (h_1 - h_2) \cdot (-1)}.\] Finally, recalling that \(2^{h^\star} \simeq m\), we use the bound \(2^{2(h_2 - N)} \le 1\) and rearrange the remaining factors as \[\norm*{W}_{h^\star-1} \le \frac{C^2 \lambda^4 \Lambda^2}{M^2} \cdot \frac{m^2}{M^2} \cdot 2^{-2h^\star} \cdot 2^{(h_3 - h^\star) \cdot 0 + (h_2 - h_3) \cdot (-1) + (h_1 - h_2) \cdot (-1)}.\] As expected, an overall \(m^2/M^2\) factor appears. Also, the sum over the scale differences \(h_2 - h_3, h_1 - h_2\) can be safely performed due to the damping factors \(2^{-(h_2 - h_3)}, 2^{-(h_1 - h_2)}\), whereas the undamped sum over \(h_3 - h^\star\) is bounded by \(\abs*{N - h^\star + 1} \simeq \log(\Lambda/m) \le \log^4(\Lambda/m)\). Since the amputation procedure cancels the \(2^{-2h^\star}\) factor, we proved that our (amputated) fourth order diagram satisfies the bound ?? . It is worth noting that this estimate is not optimal, because it neglects the gain produced by \(2^{2(h_2 - N)}\) and it overcounts the number of \(\log(\Lambda/m)\) factors. Nevertheless, this analysis makes it evident that some diagrams (or, more generally, some trees) are more relevant than others. In principle, the precise number of \(\log(\Lambda/m)\) factors and the overall damping produced by the \(\lambda\) endpoints can be worked out case by case.

7.0.0.2 A fourth order diagram with two nested renormalizations.

As a second example, we consider the diagram depicted in Figure 11, which has the same structure as the one displayed in Karplus_Kroll_1950?.

Now the renormalization operator acts nontrivially on two clusters, respectively lying on scale \(h_1\) and on scale \(h_2\). Proceeding as before, we obtain \[\norm*{W}_{h^\star-1} \le \underbrace{\left(\frac{C \lambda^2}{M^2}\right)^2}_{\text{Endpoints}} \cdot \,\, \underbrace{2^{-h^\star - h^\star}}_{h^\star \text{ cluster}} \,\, \cdot \,\, \underbrace{2^{3h_2 - h_2}}_{h_2 \text{ cluster}} \,\, \cdot \,\, \underbrace{2^{3h_1 - h_1}}_{h_1 \text{ cluster}} \,\, \cdot \,\, \underbrace{2^{2(h^\star - h_2) + 2(h_2 - h_1)}}_{\mathcal{R}}.\] Again, this can be rearranged as \[\label{eq:second95diagram95clusters95bound} \norm*{W}_{h^\star-1} \le \left(\frac{C \lambda^2 \Lambda^2}{M^2}\right)^2 \cdot 2^{2(h_1 - N) + 2(h_2 - N) -2h^\star} \cdot 2^{(h_2 - h^\star) \cdot (0 - 2) + (h_1 - h_2) \cdot (0 - 2)}\tag{143}\] and the short memory property is evidenced by inserting a \(1 = 2^{2(h^\star - N)} \cdot 2^{2(N - h^\star)}\) factor and subsequently writing \(N - h^\star\) as \((N - h_1) + (h_1 - h_2) + (h_2 - h^\star)\). Since \(2^{2(h_2 - N)} \le 1, 2^{h^\star} \simeq m\), the short memory property transforms 143 into \[\norm*{W}_{h^\star-1} \le \frac{C \lambda^4 \Lambda^2}{M^2} \frac{m^2}{M^2} \cdot 2^{-2h^\star} \cdot 2^{(h_2 - h^\star) \cdot 0 + (h_1 - h_2) \cdot 0}.\] The sum over the scale differences is not damped, so it produces a \(\abs*{N - h^\star + 1}^2 \simeq \log^2(\Lambda/m) \le \log^4(\Lambda/m)\) factor: once the amputation is taken into account, we conclude that this diagram satisfies ?? .

7.0.0.3 A tenth order diagram.

We now consider the diagram represented in Figure 12, that contributes to \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) at tenth order in \(\lambda\). It should be noted that this diagram lies at the current limit of the ordinary perturbative calculations in Minkowskian QED\(_4\): it is extremely demanding (and perhaps impossible) to express its contribution to \(\mathfrak{a}_\mathrm{\small z}^\mathrm{\small r}\) in a closed form Aoyama_2012?.

The same approach followed in the previous cases yields \[\begin{gather} \norm*{W}_{h^\star} \le \underbrace{\left(\frac{C \lambda^2}{M^2}\right)^5}_{\text{Endpoints}} \cdot \,\, \underbrace{2^{-h^\star - h^\star}}_{h^\star \text{ cluster}} \,\, \cdot \underbrace{2^{3h_1 - h_1}}_{h_1 \text{ cluster}} \cdot \,\, \underbrace{2^{3h_2 + 3 \cdot (-h_2)}}_{h_2 \text{ cluster}} \,\, \cdot \,\, \underbrace{2^{3h_3 -h_3}}_{h_3 \text{ cluster}} \,\, \cdot \underbrace{2^{3h_4}}_{h_4 \text{ cluster}} \, \times \\ \times \underbrace{2^{3h_5}}_{h_5 \text{ cluster}} \,\, \cdot \,\, \underbrace{2^{2(h^\star - h_1) + 2(h_1 - h_3) + 3(h_2 - h_4) + 3(h_2 - h_5)}}_{\mathcal{R}} \end{gather}\] which is equivalent to \[\begin{gather} \norm*{W}_{h^\star} \le \left(\frac{C \lambda^2 \Lambda^2}{M^2}\right)^5 \,\, \cdot \,\, 2^{2(h_2 - N) + 2(h_2 - N) + 2(h_3 - N) + 2(h_4 - N) + 2(h_5 - N) - 2h^\star} \, \times \\ \times 2^{(h_1 - h^\star)(0 - 2) + (h_2 - h_1) (-2 - 0) + (h_3 - h_1)(0 - 2) + (h_4 - h_2)(1 - 3) + (h_5 - h_2)(1 - 3)}. \end{gather}\] Once again, we see that the estimate provided by Theorem 2 is correctly reproduced. The short memory factor can be extracted as before, by multiplying the right hand side by \(1 = 2^{2(h^\star - N)} \cdot 2^{2(N - h^\star)}\).