An Extra-Dimensional Axion in a 5D Warped Orbifold GUT

TL;DR

This work embeds a QCD axion into a 5D warped orbifold GUT, realizing the axion as the zero mode of the fifth component of a bulk U(1)_C gauge field. Owing to the warped geometry, gauge couplings run logarithmically above the KK scale, allowing the axion decay constant $f_a$ to lie in the conventional window $10^9$–$10^{12}$ GeV while maintaining perturbative unification. The analysis shows that bulk and boundary effects, including a CFT contribution $b_{ m CFT}$, can be accommodated without spoiling unification, and that axion quality is protected by warping and SUSY, even with bulk charged matter. A complementary 4D holographic dual with a baryonic PQ symmetry provides a consistent field-theoretic interpretation, suggesting a robust connection between geometry, holography, and axion phenomenology in warped flux compactifications.

Abstract

We study the QCD axion arising from the 5th component of a bulk $U(1)$ gauge field in a five-dimensional warped grand unified theory, and determine the viable range of the axion decay constant $f_a$. Unlike flat extra dimensions, where gauge couplings run quickly above the Kaluza--Klein (KK) scale, the logarithmic running in warped geometries permits substantially smaller $f_a$ while preserving perturbative gauge coupling unification. However, bulk tree-level contributions to the gauge coupling -- interpreted holographically as CFT renormalization -- place a lower bound on $f_a$. We find that the conventional QCD axion window $10^{9}\,\mathrm{GeV} \lesssim f_a \lesssim 10^{12}\,\mathrm{GeV}$ is readily compatible without losing perturbativity, provided the AdS curvature is near the Planck scale. Thus, the 5D warped orbifold GUT naturally accommodates a high-quality QCD axion in a grand unified theory that provides an effective description of string-theoretic warped flux compactifications, admitting complementary geometric and holographic descriptions of the axion.

Figures (4)

• Figure 1: Schematic plot showing the field content of the extra-dimensional one-form QCD axion in a 5D warped orbifold GUT. The purple and red lines denote the UV and IR branes located at $y=0$ and $y=\pi R$ respectively. The bulk contains the $SU(5)_{\rm GUT}$ and axion $U(1)_{C}$ vector supermultiplets, while the MSSM chiral multiplets $\{10_i,\;\bar{5}_i\}$ and Higgs fields $\{24_{\Phi},\;5_H,\;\bar{5}_H\}$ are localized on the UV brane.
• Figure 2: The Standard Model couplings $\alpha^{-1}_{i}=(4\pi)/g_{i}^{2} \,\,(i=1,2,3)$ as a function of $q$ for several values of $f_a$ and $b_{\rm CFT}$. The purple, orange and green solid lines correspond to $\alpha_i^{-1}(q)$ for $(kR,b_{\rm CFT},f_{a})= {\color{violet}(6,6,5\times10^{9}~{\rm GeV})},{\color{orange}(4,6,2\times10^{12}~{\rm GeV})}$ and ${\color{teal}(4,3,4\times10^{12}~{\rm GeV})}$ respectively, while the black dashed lines show the MSSM running.
• Figure 3: The value of $\alpha^{-1}_{\rm GUT}(M_{\rm GUT})$ as a function of $b_{\rm CFT}$. Each orange, purple, green and black solid line corresponds to $\alpha_{\rm GUT}(M_{\rm GUT})$ for $kR={\color{orange}9}, {\color{violet}6}$${\color{teal}4}$ and $2$ respectively. Next to $kR$, we also specify the order of magnitude of the corresponding axion decay constant $f_{a}$. The purple and gray shaded regions indicate, respectively, where $\alpha_{\rm GUT}^{-1}(M_{\rm GUT}) < 4\pi$ (signaling loss of perturbativity) and where $b_{\rm CFT} < 3$. Parameter values lying outside these shaded domains remain viable.
• Figure 4: An $SU(2)$ instanton diagram assuming small instantons of size $\rho\simeq M_{\rm GUT}^{-1}$. The zero mode lines of the $SU(2)$ doublet quarks and leptons are closed up by the $c_{i}\frac{QQQL}{M_{P}^{2}}$ operators with $i=1,2,3$ the generation index. The two Higgsino zero mode solid lines and the four $SU(2)$ gaugino zero mode wavy lines are closed up by the mass insertions with the gravitino mass, $m_{3/2}$, parametrizing the supersymmetry-breaking soft masses. The red lines correspond to the preon zero mode legs and there are $N_{\rm CFT}$ pairs of closed legs.