Non-perturbative renormalization for lattice massive QED$_2$: the ultraviolet problem

TL;DR

This work presents a non-perturbative construction of lattice Massive QED$_2$ that preserves Ward identities and implements a Wilson term to avoid fermion doubling. By integrating out ultraviolet bosonic degrees of freedom, the authors derive a fermionic effective action with non-local kernels and prove a convergent expansion for sufficiently small coupling $| ext{e}| o ext{e}_0$, with cut-off independent bare parameters. They develop a robust multiscale fermionic analysis, employing Gallavotti–Nicolò trees and a Battle–Brydges–Federbush framework to bound all relevant and marginal terms, including intricate diagrams and higher-degree kernels. The ultraviolet results, combined with their previous infrared analysis, lead to a complete non-perturbative construction of the model and yield current conservation and Adler–Bardeen anomaly non-renormalization at the lattice level, offering insights applicable to 4D gauge theories. The methodology and bounds establish a rigorous template for controlling UV and IR behavior in lattice gauge theories, with potential implications for non-perturbative studies of the Standard Model's electroweak sector.

Abstract

We consider a lattice regularization, preserving Ward Identities (WI) and with a Wilson term, of the Massive QED$_2$, describing a fermion with mass $m$ and charge $\mathsf{e}$ interacting with a vector field with mass $M$, in the regime $m\ll M\ll a^{-1}$ ($a$ being the lattice spacing) which is the suitable one to mimic a realistic 4d massive gauge theory like the Electroweak sector. The presence of the lattice and of the mass $m$ breaks any solvability property. In this paper we prove that the effective action obtained after the integration of the ultraviolet degrees of freedom is expressed by expansions which are convergent for values of the coupling $|\mathsf{e}|\le \mathsf{e}_0$, with $\mathsf{e}_0$ independent on $a$ and $m$, and with cut-off-independent bare parameters. By combining this result with the analysis of the infrared part in previous papers we get a complete construction of the model and a number of properties whose analogous are expected to hold in 4d. The analysis is done by integrating out the bosons and reducing to a fermionic theory; however, with respect to the case with momentum regularizations (which break essential features like the WI), the resulting effective fermionic action has not a simple form and this requires the developments of new methods to get the necessary bounds.

Figures (16)

• Figure 1: (i) graphical representation of the contribution to the term $\mathcal{O}(\psi^{12})$ of the fermionic interaction, obtained by selecting in $w_{6,0}$ the tree with bonds $\{1,2\},\{1,3\},\{1,4\},\{4,5\},\{4,6\}$; (ii) graphical representation for the boson propagator.
• Figure 2: The two possibilities arising from the extraction of a fermionic line (black solid line attached to the vertex "1"), depending on whether the vertex it is attached to, "1", is a leaf (a) or a not-leaf (b) of the spanning tree. The red sub-tree corresponds to a source term of type $G^{(n)}$ (with $n=7$ for (a) and $n=3$ for (b)) and the green one to a source term of type $\dot{G}^{(n)}$ (with $n=5$).
• Figure 3: Example of a Gallavotti-Nicolò tree $\tau\in\mathcal{T}_{h,N}$, where the vertices $v\in V(\tau)$ are outlined as black dots.
• Figure 4: Graphical representation of Eq. \ref{['eq_15']}. The graphs are numbered as in the order they appear in the expansion of Eq. \ref{['eq_15']}: graphs $(a),(b)$ are relative to $\mathcal{A}_1$, $(c)-(e)$ to $\mathcal{A}_2$, $(f)-(l)$ to $\mathcal{A}_3$ and $(m)-(o)$ to $\mathcal{A}_4$. Solid and wiggly lines, with dotted endpoints, represent fermion and boson propagators respectively, while dashed and curly lines, with black dotted endpoints, represent delta functions. Blobs (big North East line patterned circles) represent kernels, associated to external fields distinguished by the type of attached lines to them: each solid, wiggly, dashed and curly line attached to a blob identifies a $\psi,G,\dot{G}$ and $\ddot{G}$ field respectively.
• Figure 5: (a): synthetic representation for $v_n(\underline{b})O^n_{\underline{b}}$, obtained from Fig. \ref{['fig_kernel']} by first collapsing each sub-tree rooted in $b_2,b_3$ and $b_4$ into one dark blob (the label $(n)$ stands for the order of the sub-tree), and then by thinking of the sub-trees rooted at $b_3$ and $b_4$ as one only tree rooted at $b'$, with $b'=b$ (the dashed line stands for a Dirac delta). (b): contribution to the kernel $W^{(h)|1,1}_{\emptyset;\emptyset;\emptyset}$ arising from the bare kernel in (a).

Theorems & Definitions (40)

• Remark 1.1
• Remark 1.2
• Lemma 1.3: $\xi-$independence
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Proposition 2.1
• proof
• Theorem 3.1
• Lemma 3.2