<p>In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> gauge-Higgs unification in the Randall-Sundrum warped space the mixing in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>W</mi></math> couplings in the quark sector is induced by masses of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> singlet fermions which are responsible for splitting masses of down-type quarks from those of up-type quarks in each generation. We show that the observed Cabibbo-Kobayashi-Maskawa matrix in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>W</mi></math> couplings is reproduced within experimental errors. Further flavor-changing neutral currents in the quark sector are shown to be naturally suppressed.</p>

Japan

Yutaka Hosotani

Shuichiro Funatsu

Hisaki Hatanaka

Czechia

Yuta Orikasa

Naoki Yamatsu

Origin of CKM matrix and natural FCNC suppression in gauge-Higgs unification

We propose a grand unified theory (GUT) pseudo-Nambu-Goldstone boson (pNGB) dark matter (DM) model based on <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> gauge symmetry. In the GUT model, the Standard Model (SM) gauge symmetry <math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>SM</mi></mrow></msub><mo>≔</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>C</mi></mrow></msub><mo>×</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math> and the “dark” gauge symmetry <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> are unified, where the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> symmetry plays an important role in the stability of DM. The unification of SM fermions and dark sector fermions is partially realized. The gauge symmetry <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> is spontaneously broken to <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gauge symmetry at the GUT scale by the nonvanishing vacuum expectation values of an <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> adjoint scalar field, where the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> gauge symmetry is not usual <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> GUT gauge symmetry. The symmetry is further broken to <math><msub><mi>G</mi><mrow><mi>SM</mi></mrow></msub><mo>×</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> at an intermediate scale. Furthermore, the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> symmetry is broken by the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> doublet and triplet scalar fields at the TeV scale. In the pNGB DM model based on <math><msub><mi>G</mi><mrow><mi>SM</mi></mrow></msub><mo>×</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math>, the residual global <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>V</mi></msub></math> dark custodial symmetry guarantees DM stability. On the other hand, in the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> pNGB DM model, this global symmetry is explicitly broken by the Yukawa interaction and the effective Majorana mass terms. In the scalar sector, the cubic coupling constants of the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> doublet and triplet scalar fields are the order parameters of the <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>V</mi></msub></math> symmetry breaking. To maintain <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>V</mi></msub></math> symmetry and thus the DM stability, we need to tune Yukawa coupling constants and cubic scalar couplings at high accuracy. We find that the allowed DM mass region is quite restricted as the gauge coupling constant of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> is determined by the condition of the gauge coupling unification. To satisfy gauge coupling unification and the current experimental constraint on proton lifetime, we find that three generations of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><msub><mo>)</mo><mi>C</mi></msub></math> adjoint fermions and another three generations of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>L</mi></msub></math> adjoint fermions with the intermediate mass scale are required. We also find that there is no other solution to satisfy simultaneously the gauge coupling unification and the proton decay constraint if one assumes the other symmetry breaking schemes.

China

Cheng-Wei Chiang

Koji Tsumura

Yoshiki Uchida

Pseudo-Nambu-Goldstone dark matter in <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> grand unification

The <math><mi>W</mi></math> boson mass <math><msub><mi>m</mi><mi>W</mi></msub></math> in the grand unified theory inspired <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math> gauge-Higgs unification in the Randall-Sundrum (RS) warped space is evaluated. The muon decay <math><msup><mi>μ</mi><mo>−</mo></msup><mo>→</mo><msup><mi>e</mi><mo>−</mo></msup><msub><mover><mi>ν</mi><mo>¯</mo></mover><mi>e</mi></msub><msub><mi>ν</mi><mi>μ</mi></msub></math> proceeds by the exchange of not only the zero mode of the <math><mi>W</mi></math> boson <math><mrow><mo>(</mo><msup><mrow><mi>W</mi></mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></msup><mo>)</mo></mrow></math> but also Kaluza-Klein (KK) excited modes <math><msup><mi>W</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math> and <math><msubsup><mi>W</mi><mi>R</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup></math> (<math><mi>n</mi><mo>≥</mo><mn>1</mn></math>) at the tree level. The anti–de Sitter curvature of the RS space also affects the relationship among the gauge couplings and the ratio of <math><msub><mi>m</mi><mi>W</mi></msub></math> to the <math><mi>Z</mi></math> boson mass <math><msub><mi>m</mi><mi>Z</mi></msub></math>. The <math><mi>W</mi></math> couplings of leptons and quarks also change. With the given KK mass scale <math><msub><mi>m</mi><mrow><mi>KK</mi></mrow></msub></math> the range of the Aharonov-Bohm phase <math><msub><mi>θ</mi><mi>H</mi></msub></math> in the fifth dimension is constrained. For <math><msub><mi>m</mi><mrow><mi>KK</mi></mrow></msub><mo>=</mo><mn>13</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math>, <math><mn>0.085</mn><mo>≲</mo><msub><mi>θ</mi><mi>H</mi></msub><mo>≲</mo><mn>0.11</mn></math> and <math><mn>80.381</mn><mo>≲</mo><msub><mi>m</mi><mi>W</mi></msub><mo>≲</mo><mn>80.407</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. The predicted value of <math><msub><mi>m</mi><mi>W</mi></msub></math> for <math><mn>13</mn><mo>≤</mo><msub><mi>m</mi><mrow><mi>KK</mi></mrow></msub><mo>≤</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math> lies between <math><msubsup><mi>m</mi><mi>W</mi><mrow><mi>SM</mi></mrow></msubsup><mo>=</mo><mn>80.354</mn><mo>±</mo><mn>0.007</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> in the standard model and <math><msubsup><mi>m</mi><mi>W</mi><mrow><mi>CDF</mi></mrow></msubsup><mo>=</mo><mn>80.4335</mn><mo>±</mo><mn>0.0094</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, the value reported by the CDF Collaboration in 2022.

<math><mrow><mi>W</mi></mrow></math> boson mass in gauge-Higgs unification

We examine <math><mi>W</mi></math> and <math><mi>Z</mi></math> boson pair production processes at electron-positron collider experiments in the <math><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>C</mi></mrow></msub><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>W</mi></mrow></msub><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>X</mi></mrow></msub></mrow></math> gauge-Higgs unification (GHU) model. We find that the deviation of the total cross section for the <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><msup><mi>W</mi><mo>−</mo></msup><msup><mi>W</mi><mo>+</mo></msup></math> process from the Standard Model (SM) in the GHU model with parameter sets, which are consistent with the current experiments, is about 0.5% to 1.5% and 0.6% to 2.2% for <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and 500 GeV, respectively, depending on the initial electron and positron polarization. We find that for the <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>Z</mi><mi>Z</mi></math> process the deviation from the SM in the GHU model is at most 1%. We find that unitarity bound for the <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><msup><mi>W</mi><mo>−</mo></msup><msup><mi>W</mi><mo>+</mo></msup></math> process is satisfied in the GHU model as in the SM, as a consequence of the relationship among coupling constants.

<math><mrow><mi>W</mi></mrow></math> and <math><mi>Z</mi></math> boson pair production at electron-positron colliders in gauge-Higgs unification

Abstract In grand-unified-theory-inspired SO(5) × U(1)$_{X}$ × SU(3)$_{C}$ gauge-Higgs unification (GHU) in the Randall–Sundrum warped spacetime, the W- and Z-couplings of all 4D fermion modes become nontrivial. The W- and Z-couplings of zero-mode quarks and leptons slightly deviate from those in the SM, and the couplings take the matrix form in the space of Kaluza–Klein (KK) states. In particular, the 4D couplings and mass spectra in the KK states depend on the Aharonov–Bohm phase θ$_{H}$ in the fifth dimension. Nevertheless there emerge three astonishing sum rules among those coupling matrices, which guarantees the finiteness of certain combinations of corrections to vacuum polarization tensors. We confirm by numerical evaluation that the equality in the sum rules holds with 5- to 7-digit accuracy. Based on the sum rules we propose improved oblique parameters in GHU. Oblique corrections due to fermion 1-loop diagrams are found to be small.

Taiwan

Coupling sum rules and oblique corrections in gauge-Higgs unification

We examine contributions to single Higgs boson production processes via <math><msup><mi>Z</mi><mo>′</mo></msup></math> and <math><msup><mi>W</mi><mo>′</mo></msup></math> bosons in the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><msub><mo>)</mo><mi>C</mi></msub><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><msub><mo>)</mo><mi>W</mi></msub><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>X</mi></msub></math> gauge-Higgs unification (GHU) model. In particular, we analyze the cross sections of three single Higgs boson production processes <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>Z</mi><mi>h</mi></math>, <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>ν</mi><mover><mi>ν</mi><mo>¯</mo></mover><mi>h</mi></math>, and <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mi>h</mi></math> in the SM and the GHU model. For the Higgs strahlung process <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>Z</mi><mi>h</mi></math>, we show that for a parameter region satisfying the current experimental constraints, a maximum deviation from the SM is about up to 6% for the center-of-mass energy of the initial electron and positron <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and that from the SM is about up to 20% for <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>500</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, depending on the initial polarization of electron and positron. The deviation from the SM is monotonically increasing with respect to <math><msqrt><mi>s</mi></msqrt></math> for <math><msqrt><mi>s</mi></msqrt><mo>≲</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math>. By using the Higgs strahlung process, it is possible to explore up to the region of tens of TeV in terms of the Kaluza-Klein (KK) mass. We also show that the sign of the bulk mass of a lepton multiplet in the GHU model can be determined by examining the deviation of the left-right symmetry of the <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>Z</mi><mi>h</mi></math> process from the SM. The contributions to the cross sections of the <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>ν</mi><mover><mi>ν</mi><mo>¯</mo></mover><mi>h</mi></math> and <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mi>h</mi></math> processes via the <math><msup><mi>Z</mi><mo>′</mo></msup></math> and <math><msup><mi>W</mi><mo>′</mo></msup></math> bosons are relatively small compared to that for <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>Z</mi><mi>h</mi></math> at least for <math><msqrt><mi>s</mi></msqrt><mo>≤</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math>. As is the same as in the SM, the <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><mi>ν</mi><mover><mi>ν</mi><mo>¯</mo></mover><mi>h</mi></math> process gives a main contribution to the single Higgs boson production processes for <math><msqrt><mi>s</mi></msqrt><mo>≳</mo><mn>500</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, but large deviations from the SM are only observed on energy scales close to the masses of the first KK gauge bosons or higher.

Single Higgs boson production at electron-positron colliders in gauge-Higgs unification

We propose a pseudo-Nambu-Goldstone boson (pNGB) dark matter (DM) model based on an additional non-Abelian gauge symmetry <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math>. The gauge symmetry <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> is spontaneously broken to a global custodial symmetry <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>V</mi></msub></math> via the nonvanishing vacuum expectation values of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><msub><mo>)</mo><mi>D</mi></msub></math> doublet and triplet scalar fields. Due to the exact global symmetry <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>V</mi></msub></math>, the lightest <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>V</mi></msub></math> charged particle becomes stable. We assume that the lightest charged particle in the model is the charged complex pNGB, which we regard as DM. It avoids the strong constraints from current DM direct detection experiments due to the property of NGB. We find that the measured energy density of DM can be reproduced when the DM mass is larger than the half of the Higgs mass, where the lower limit generally comes from the constraint of DM invisible decay and the upper limit from DM direct detection experiments depends on the model parameters.

Hajime Otsuka

Takashi Shimomura

Pseudo-Nambu-Goldstone dark matter from non-Abelian gauge symmetry

We examine effects of <math><msup><mi>Z</mi><mo>′</mo></msup></math> bosons in gauge-Higgs unification (GHU) models at <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> Bhabha scatterings. We evaluate differential cross sections in Bhabha scatterings including <math><msup><mi>Z</mi><mo>′</mo></msup></math> bosons in two types of <math><mi>SO</mi><mo>(</mo><mn>5</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>SU</mi><mo>(</mo><mn>3</mn><mo>)</mo></math> GHU models. We find that deviations of differential cross sections in the GHU models from those in the SM can be seen at <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. With 80%-longitudinally polarized electron and 30%-longitudinally polarized positron beams, the left-right asymmetries in the GHU A and B models are resolved at more than <math><mn>3</mn><mi>σ</mi></math> at <math><msub><mi>L</mi><mrow><mi>int</mi></mrow></msub><mo>=</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>. We also show that Bhabha scattering with scattering angle less than 100 mrad can be safely used as luminosity measurements at <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> colliders since the effects of <math><msup><mi>Z</mi><mo>′</mo></msup></math> bosons are well suppressed for small scattering angle. We propose a new observable which can be measured at future TeV-scale <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> linear colliders.

Bhabha scattering in the gauge-Higgs unification

The <math><mi>p</mi><mi>p</mi><mo>→</mo><mo>{</mo><mi>W</mi><mo>,</mo><msup><mi>W</mi><mo>′</mo></msup><mo>}</mo><mo>→</mo><mi>l</mi><mi>ν</mi></math> and <math><mi>p</mi><mi>p</mi><mo>→</mo><mo>{</mo><mi>γ</mi><mo>,</mo><mi>Z</mi><mo>,</mo><msup><mi>Z</mi><mo>′</mo></msup><mo>}</mo><mo>→</mo><msup><mi>l</mi><mo>+</mo></msup><msup><mi>l</mi><mo>−</mo></msup></math> (<math><mrow><mi>l</mi><mo>=</mo><mi>e</mi></mrow></math>, <math><mrow><mi>μ</mi></mrow></math>) processes in the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><msub><mo>)</mo><mi>C</mi></msub><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>X</mi></msub></math> gauge-Higgs unification (GHU) models are studied, where <math><msup><mi>W</mi><mo>′</mo></msup></math> and <math><msup><mi>Z</mi><mo>′</mo></msup></math> bosons are Kaluza-Klein (KK) exited states of the electroweak gauge bosons. From the experimental data collected at the Large Hadron Collider, constraints on the KK mass scale and the Aharonov-Bohm phase are obtained. One can explore the KK mass scale in the grand unified theory inspired GHU model up to 18 TeV for the luminosity <math><mrow><mn>300</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fb</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> and 22 TeV for the luminosity <math><mrow><mn>3000</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fb</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> at <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>14</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math>.

Signals of <math><msup><mi>W</mi><mo>′</mo></msup></math> and <math><msup><mi>Z</mi><mo>′</mo></msup></math> bosons at the LHC in the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gauge-Higgs unification

Abstract In gauge-Higgs unification (GHU), gauge symmetry is dynamically broken by an Aharonov–Bohm (AB) phase, θ$_{H}$, in the fifth dimension. We analyze SU(2) GHU with an SU(2) doublet fermion in the flat M$^{4}$ × (S$^{1}$/Z$_{2}$) spacetime and in the Randall–Sundrum (RS) warped space. With orbifold boundary conditions the U(1) part of gauge symmetry remains unbroken at θ$_{H}$ = 0 and π. The fermion multiplet has chiral zero modes at θ$_{H}$ = 0, which become massive at θ$_{H}$ = π. In other words, chiral fermions are transformed to vector-like fermions by the AB phase θ$_{H}$. The chiral anomaly at θ$_{H}$ = 0 continuously varies as θ$_{H}$, and vanishes at θ$_{H}$ = π. We demonstrate this intriguing phenomenon in the RS space in which no level crossing occurs in the mass spectrum and everything varies smoothly. The flat spacetime limit is singular as the anti-de Sitter curvature of the RS space diminishes, and reproduces the result in the flat spacetime. Anomalies appear for various combinations of Kaluza–Klein excitation modes of gauge fields as well. Although the magnitude of the anomalies depends on θ$_{H}$ and the warp factor of the RS space, it does not depend on the bulk mass parameter of the fermion field controlling its mass and wave function at general θ$_{H}$.