Logarithmic Unification From Symmetries Enhanced in the Sub-Millimeter Infrared

TL;DR

This work demonstrates that in theories with a TeV-scale string scale and large extra dimensions, gauge coupling unification can arise from infrared logarithmic variation of bulk fields in two transverse dimensions, rather than from short-distance gauge symmetries. Through explicit $N=2$ and $N=1$ D-brane constructions, the authors show how bulk gravity calculations reproduce the holomorphic running of gauge couplings on brane probes and how a geometric, bulk-based symmetry can induce unification at a scale far above the string scale. They discuss threshold corrections and the role of orbifolds in maintaining the bulk-to-field theory correspondence, and propose that unification may be rooted in bulk geometry rather than traditional UV completions. The framework also suggests possible mechanisms for generating large hierarchies via bulk dynamics and offers a novel reinterpretation of RG flow in higher-dimensional contexts.

Abstract

In theories with TeV string scale and sub-millimeter extra dimensions the attractive picture of logarithmic gauge coupling unification at $10^{16}$ GeV is seemingly destroyed. In this paper we argue to the contrary that logarithmic unification {\it can} occur in such theories. The rationale for unification is no longer that a gauge symmetry is restored at short distances, but rather that a geometric symmetry is restored at large distances in the bulk away from our 3-brane. The apparent `running' of the gauge couplings to energies far above the string scale actually arises from the logarithmic variation of classical fields in (sets of) two large transverse dimensions. We present a number of N=2 and N=1 supersymmetric D-brane constructions illustrating this picture for unification.

Figures (6)

• Figure 1: The picture for 'running' from the infrared. The gauge coupling on our brane is determined by the value of a bulk field $\phi$ evaluated at the position of our brane. Other branes in the bulk a distance $R >> M_s^{-1}$ away act as sources for $\phi$, and if the transverse co-dimension relative to the source brane is 2, then the value of $\phi$ on our brane can be logarithmically sensitive to $R$. In principle, this logarithmic profile of $\phi$ can mimic field theoretic 'running' to an energy scale $R M_s^2$ far above the string scale.
• Figure 2: An $N=2$ SUSY example where the bulk SUGRA equations reproduce the (3+1)-d QFT holomorphic gauge coupling running on a probe D3 brane located at $w$. Source D7 branes are located at positions $w_i$ in the $w=x^8 +i x^9$ plane, and an O7$^-$ orientifold plane is located at $w =0$. At long distances $\gg R$ the total RR charge and tension of these D7/O7 branes cancel and there is no variation in the bulk fields. In this figure we have moved $f=3$ of the D7 branes far away from the D3 and O7 brane, and so from the D3-brane gauge theory perspective 3 $N=2$ hypermultiplets gain a large mass $m=R M_s^2$.
• Figure 3: The general picture describing the duality between 1-loop open string and tree-level closed string diagrams. The worldsheet coordinates for the string are $\xi_1$ and $\xi_2$. If $\xi_1$ is taken to be the worldsheet 'time' coordinate then this diagram represents the exchange of a closed string state between our brane and a 'source' brane in the bulk. If the separation $R$ of the branes is large $R\ell_{\rm s}\gg 1$, then this amplitude is well approximated by the (classical) bulk supergravity solution in the presence of the brane source. On the other hand, if $\xi_2$ is taken to be the worldsheet 'time' coordinate then this diagram represents the 1-loop contribution of a massive stretched open-string to the 2-point function of our brane-localized theory. In special cases (described in the text) only the zero-mode stretched string state contributes, the higher string oscillator states canceling, and the usual QFT beta-function is reproduced.
• Figure 4: A toy $N=2$ theory with unification far above the string scale based on the Hanany-Witten set-up. The thick lines are NS5 branes and the thin ones are D4 branes. The $6$ direction is compactified on a circle, which we indicate by periodically repeating the configuration. The gauge group is $SU(3)^3$ with hypermultiplets transforming as $(3,\bar{3},1) + (1,3,\bar{3}) + (\bar{3},1,3)$. The NS5's are equally spaced so the three gauge couplings are identical. The forces on the NS5's due to the D4's cancel locally so there is no bending of the NS5's.
• Figure 5: Moving some of the D4's a distance $R$ away leaves an $SU(3) \times SU(2) \times U(1)$ gauge group living on the remaining branes. The forces on the NS5's no longer cancel locally and they bend. We see that $l_3 < l_2 < l_1$, so $g_3 > g_2 > g_1$.