claims depot shelf
The Strong CP Problem
Formal claims (Lean)
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The authors claim that small-instanton contributions to the axion potential in KSVZ models are governed by a counting rule: a vector-like quark (VQ) representation with $2|N|$ conjugate zero-mode pairs is closed by $k$ scalar-Yukawa loops and $r$ mass insertions with $2|N|=2k+r$, and the instanton-size integral is ultraviolet-dominated when $\Delta=b_3+r-4<0$. Because each Yukawa loop lowers $\Del
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The paper's central claim is a quantitative quality-coupling relation: high axion quality implies a large anomalous coupling between the axion and the $U(1)_{B-L}$ gauge boson, and that coupling induces an axion-fermion interaction with a calculable lower bound. Under the charge assignment $q_1=m=l$, $-q_2=n=k$ with equal scalar VEVs, the cubic anomaly-cancellation condition reduces to $3mA-3nB =
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On the paper's own account, the central claim is that the Nelson-Barr scalar $S=(s+ia)/\sqrt{2}$ can be the inflaton. Its imaginary component $a$ has a CP-symmetric double-well potential, so the scalar has two CP-conjugate vacua at $\langle a\rangle=\pm f$; these are the two inflationary valleys, and inflation is quartic hilltop inflation along one of them. The same structure removes the domain-wa
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Fitting all 9-link textures — full-rank Yu and Yd with nine non-zero entries and a single rephasing-invariant phase φ — to the ten flavor observables, the paper finds that viable fits force φ to cluster at multiples of π/8, predominantly π/2, π/8, and 3π/8, the measured values of the unitarity-triangle angles (α, β, γ). The mechanism is a leading-order identity: in most textures the angle carried
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The central discovery is that the charm-coupled PQ sector maps into HMChPT without creating a leading-order symmetry-breaking spurion. The scalar $\sigma$ matches to the spin-symmetric operator $\mathrm{tr}[\bar{H}_a H_a]$, so it preserves heavy-quark spin symmetry and is not excluded by any heavy-meson mass or splitting measurement; its strongest effect is a $\sim 0.5 B_S$ enhancement of $B_s$ mi
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Axion quality is compatible with minimality: the Peccei–Quinn symmetry can be realized as a gauge-origin boundary remnant of five-dimensional U(1) gauge invariance on a flat interval, the axion is the surviving open Wilson-line phase, and a familiar four-dimensional KSVZ/DFSZ anomaly sector still generates the QCD potential, without bulk QCD, warping, or a color Chern–Simons term, while exponentia
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In the unbroken E6^L × E6^R theory the left-right exchange is realized on the Dirac field as spacetime parity and acts trivially on colour and electric charge. Consequently the two would-be colour groups coincide as one vector-like SU(3)_c with a colour-singlet electron, hypercharge is the consistency relation Y = Q - T_L^3 with Q = N/3 and no right-sector generator, and the spontaneous parity for
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For a four-dimensional theory with gauge group G and an automorphism σ that implements charge conjugation, gauging CP means summing over Pin+(4)⋉G fiber bundles, not ordinary CP×G bundles. The paper's main result is that when π0(G)=π1(G)=0, the five-dimensional bordism group that classifies global anomalies satisfies Ω_5^{Pin+⋉G}(pt) ≃ Ω_5^{Spin}(BG). This group is Z2 for G=Sp(N) and trivial other
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The paper establishes that the Murayama QCD-scale axion and its natural extensions are excluded by low-energy mesonic observables. Integrating out the heavy PQ scalar yields an effective chiral Lagrangian for mesons that contains a new spurion I_PQ, which transforms like a quark mass matrix under chiral symmetry but breaks the accidental SU(2) isospin of the leading-order QCD chiral Lagrangian. Th
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According to the paper, in N=1 supersymmetric QCD with N_c=N_f, confinement spontaneously breaks the global baryon-number symmetry U(1)_B, and the resulting Nambu-Goldstone boson is a QCD axion. The axion's Peccei-Quinn shift symmetry is not put in by hand: the gauge-invariant baryon operator is a product of N_c quark superfields, so any operator that breaks baryon number and respects the gauge sy
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For perturbative heterotic E8×E8 compactifications on Calabi-Yau threefolds (six-dimensional internal spaces), the paper's central claim is that the axion mass spectrum has a much stronger lower bound than in type IIB: almost all model-dependent two-form axions are heavy, lifted by worldsheet instantons or gaugino condensation, and the QCD axion, when it solves Strong CP, is the lightest state wit
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The paper argues that a non-invertible selection rule, arising from gauging a $\mathbb{Z}_2$ subgroup of global $\mathbb{Z}_N$ symmetries, enforces a three-zero texture on the down-quark mass matrix. With CP assumed to be an exact symmetry at high energies, the determinant of that matrix is forced to stay real by the non-invertible constraint. CP is then spontaneously broken by complex vacuum expe
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The paper's central claim is that a canonical scalaron emerges from an Einstein–Cartan action whose dimension-four geometric sector is a single complete square, and that matter currents coupled to torsion become, in the Einstein frame, interactions of that scalaron with the currents. For gauge-invariant currents the interaction is a derivative coupling $\nabla_\mu \phi \, j^\mu$ plus a current sel
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The paper's claim is that the dichotomy $\bar{\theta}=0$ versus $\bar{\theta}=\pi$ is not an unavoidable consequence of CP or P invariance but an artifact of assuming the usual $2\pi$ periodicity of the QCD $\theta$ angle. In a gauge theory of the form (4), fractional instantons of the different factors are correlated: the topological charges take the form $\vec Q = \vec n + m \vec q$, and a $2\pi
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The central claim, stated in the author's own terms, is that only third-generation fermions acquire tree-level masses, while the first and second generations gain their masses via quantum corrections induced by new gauge bosons. The mechanism works because each chiral fermion sector couples to a vector-like pair through mass terms $\mu_L$ and $\mu_R$, producing a seesaw-like tree-level mass matrix
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The central discovery is a pair of theorems about two-Higgs-doublet models with a softly broken CP symmetry and an Abelian flavor symmetry. First, when the flavor charges are chosen so that arg det(yu yd) = 0 at tree level for all couplings, because each Yukawa determinant is a single monomial in the two Higgs fields, the one-loop correction to $\theta$-bar also vanishes, independent of the scale
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The central discovery claim is that the strong CP, hierarchy, and cosmological constant problems can be solved together in type IIA flux compactification on $T^6/(\mathbb{Z}_2\times\mathbb{Z}_2)$. The strong CP part works by identifying the axion potential of the four-form-coupled mechanism with the flux-induced potential of the orientifold: the flux constraint $b_{ij} b_j = h_i$ forces $X=0$ in t
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On the paper's own terms, the central claim is that replacing metric gravity by Einstein-Cartan gravity changes the status of Weyl symmetry. In metric gravity a local scale (Weyl) invariance is destroyed by the Weyl anomaly, and a local Weyl-invariant action is essentially the Weyl-squared theory with ghosts. In the EC formulation the vector part of the torsion, $v_\mu$, transforms as a Weyl gauge
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The central claim is that a would-be axion arising from Einstein-Cartan gravity cannot solve the strong CP problem if its only matter interaction is the derivative coupling to the axial fermionic current, because this operator is exactly invariant under $\varphi \to \varphi + c$. The apparent topological coupling $\frac{\alpha}{4\pi} \varphi \operatorname{Tr} G_{\mu\nu} \tilde{G}^{\mu\nu}$ is obta
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The central claim is that the $\theta$-term of QCD is a direct consequence of a global chiral transformation, provided the fermion kinetic term is modified by a non-Abelian Berry connection $A_\mu$ associated with slow variations of the gauge background. With $\alpha(x)=\theta$, Eq. (6) of the paper gives the effective action of QCD together with the topological term $-\frac{g^2}{16\pi^2}\theta\,\
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The paper claims that the modular-invariance mechanism for solving the strong CP problem can be extended to the effective field theories suggested by string compactifications, where light quarks carry mostly positive modular weights and the gauge kinetic functions $f_a$ are non-trivial functions of moduli. The physical combination is $A(S,\tau) = e^{-8\pi^2 f_3(S,\tau)} \det Y_u(\tau) \det Y_d(\ta
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The paper's central claim is that the 'WIMP quality problem' — the puzzle of why R-parity violation is so small — can be solved by the same discrete R-symmetries that solve the mu problem and the strong CP problem. With anomaly-free discrete R-symmetries $Z^R_n$ for $n=4,6,8,12,24$ and two gauge-singlet fields $X,Y$ whose VEVs sit near $f_a \sim 10^{11}$ GeV, the dangerous renormalizable RPV terms
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On the paper's own terms, the discovery is that $\theta$ is a property of the state, not of the laws. The QCD Hamiltonian in temporal gauge ($A_0=0$) already preserves CP, yet it possesses energy eigenstates—the $\theta$ vacua—that violate CP, just as a rotationally invariant hydrogen Hamiltonian has non-spherical excited states. The $\theta$ term in the Lagrangian appears only when one constructs
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The central claim is that a non-invertible symmetry, not an axion, is what makes the QCD vacuum angle unobservably small. In a two-Higgs-doublet extension of the Standard Model with CP assumed exact at high energies and spontaneously broken at an intermediate scale, the authors assign $\widetilde{Z}_5\times\widetilde{Z}_3\times\widetilde{Z}_3$ charges to the three left-handed quark doublets, the r
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The central claim is that the chiral quark condensate of QCD, whose phase is the dynamical degree of freedom for the η′ meson, supports 2π-winding domain walls bounded by cosmic strings in two flavor directions. In the η′ direction, the phases of light-quark condensates wind with the same sign; in the π0 direction, they wind oppositely ($n_{\theta_u} = -n_{\theta_d}$). These defects arise in pure
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The central claim is that the discrete R-symmetry $Z_n^R$ controls both the axion and the fate of the lightest supersymmetric particle. When the MSSM is augmented by two Peccei-Quinn charged fields $X$ and $Y$, the lowest-order R-parity-violating superpotential operator receives a suppression $(f_a/m_P)^N$. The paper finds $N=1$ for $Z_4^R$ and $Z_8^R$, giving induced RPV couplings $\lambda \sim 1
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On the paper's own terms, the discovery is that $\theta$-dependence in hadronic correlators is carried entirely by localized zero modes of the Dirac operator. Because the density of these modes falls like $1/\sqrt{V}$, the chance of finding one inside a nucleon's interaction volume tends to zero as the volume grows, so the CP-odd part of the nucleon correlator vanishes and the neutron electric dip
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The central discovery is that in the BχF construction the exact global U(1) symmetries of the five-dimensional theory do not force a massless boson or forbid quark condensation. Rather, the instanton sum in $N_f=1$ QCD acquires extra factors from bulk zero modes $Q$ that are localized far from the physical boundary, making the full sum invariant under the exact $U(1)\times U(1)$ symmetry while the
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The central claim is that a single bulk axion in flat or Randall-Sundrum extra dimensions, with QCD localized on the infrared brane and QCD as the only source of Peccei-Quinn breaking, can realize only one viable pattern: one zero-mode axion on the canonical QCD line plus a plateau of heavier KK modes. The non-canonical patterns, in which several KK modes share the solution to the strong CP proble
Stated claims
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The paper's central claim is that the conventional wisdom that small instantons only make the QCD axion heavier rests on the assumption that the Standard Model flavor sector is unaltered in the UV. In the general low-energy Lagrangian of Sec. 2, small instantons contribute both to the axion potential and, additively, to the up-quark Yukawa: $\delta y_u^{\rm SI}$ joins the tree-level coupling $\hat{y}_u$, and the up-quark mass is the sum of the two. The axion mass formula (eq. 2.24) has an infrared QCD term and a UV small-instanton term, and both are proportional to $\hat{y}_u$, because in the limit $\hat{y}_u \to 0$ an exact non-anomalous shift symmetry (eq. 2.2) protects the axion. The observed ratio $y_u/y_d = 0.46$ then forces $\delta y_u^{\rm SI}/y_d \simeq 0.46$ in the light limit, and imposing this condition on the small-instanton estimate fixes $\alpha_{s1}(M) \simeq 0.7$ in the product-$SU(3)$ model. With that condition met, the axion mass scales as $m_a \propto \sqrt{\hat{y}_u}$ (eqs. 3.13 and 4.23), so the axion is lighter than standard for $\hat{y}_u \lesssim 10^{-22}$ (product-$SU(3)$) or $\hat{y}_{uL} \lesssim 10^{-25}$ (composite Higgs) and heavier otherwise.
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At tree level the parity symmetry forbids the bare QCD $\theta$ term and makes the determinants of the quark mass matrices real, so $\bar{\theta}=0$; the discovery is that soft parity breaking -- the dimension-two operator that makes $\langle H'\rangle = v' \neq v = \langle H\rangle$ and splits $m_{H'}$ from $m_H$ -- switches on two-loop corrections to the light-quark mass matrices that regenerate $\bar{\theta}$. The corrections organize into three CP-odd rephasing invariants, Eqs. (3.16), (3.18), and (3.19), which feed Eq. (3.30): a fully antisymmetric combination of the three up-type Yukawa outer products, and two invariants built from mixed up- and down-type couplings that contain the inverse Yukawa matrix $B_q^a = x_q^a (x_q^{-1})^a$ and are therefore non-decoupling. The dominant term, proportional to $\operatorname{Im Tr}(A_u^a A_u^b B_d^c)$, is nonzero even if the only CP violation is the CKM phase, is enhanced by the square-root mass ratios $\sqrt{m_b/m_d}$ and $\sqrt{m_b/m_s}$, and is controlled by the splittings of the down-type vector-like quark masses. For the benchmark hierarchies $(m_1/m_3, m_2/m_3) = (10, 3)$ and $(100, 10)$, the probability distribution of $|\bar{\theta}|$ with generic mixings peaks near $10^{-5}$--$10^{-6}$, well above the neutron EDM bound; holding $\bar{\theta}$ below the bound requires the down-type vector-like partners to match to about $10^{-4}$ and the up-type to about $10^{-2}$ through a GIM-like cancellation that suppresses the correction as the heavy masses approach degeneracy, or the new mixing angles to be strongly suppressed. The authors conclude that the inverted vector-like mass hierarchy motivated by the universal seesaw is severely restricted, and they state that the required tuning stays at roughly the 1% and 0.01% level even when higher-order terms in the $v'/M$ expansion are included.
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In the paper's own terms, the central claim is that imposing parity together with leptonic reality ($CP_\ell$) in the three-generation minimal Dirac theory is exactly equivalent to the vanishing of a single rephasing-invariant, CP-odd obstruction. On the physical parity image the compatibility condition $V_\ell^R = K_e^\dagger V_L^* K_\nu$ holds if and only if $I_H = \Im(H_{e\mu}H_{\mu\tau}H_{\tau e}) = 0$, where $H$ is the mass-weighted Hermitian core of the reconstruction. The same invariant controls the two observable projections: the right-left modulus mismatch has rank one, $M = \Lambda\,E\,N^T$, with the exact relation $\Lambda D_\nu = 2\epsilon I_H$, and the Jarlskog orientation obeys $(J_R+J_L)D_\nu = 2 I_H \Xi$, from which the exact identity $J_L = -\frac{1}{2}\Delta J_P$ follows. The paper also derives the leading response near the two CP-conserving origins: on branches with opposite neutrino signs, $\delta^\ell_{CP} - \delta_0 \simeq \epsilon\,\hat{m}_\tau\,C_{\tau 12}\big/\bigl(2(m_{\nu_1}-m_{\nu_2})\bigr)$, so the small solar splitting amplifies a tiny deformation, with a separate resummed quasi-degenerate response that the paper warns cannot be extrapolated to $\epsilon = 0$; in the quark sector the same deformation induces $\bar\theta = C_q\,\epsilon$. The paper is explicit about the limits: near zeros of $\Xi$ and for degenerate spectra the Jarlskog condition is only locally equivalent to full compatibility, and the full invariant condition takes over. The invariant reduction and the exact identities are proven in the text and Appendix A, while the quantitative branch-by-branch solutions and the Figure 1 regions are delegated to the companion Ref. [50].
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The central discovery is that a single gauged chiral U(1)_χ symmetry, with charges chosen to forbid dimension-5 operators in both the clockwork and Nelson-Barr sectors while cancelling anomalies with additional vector-like quarks, resolves the quality problems that plagued earlier Nelson-Barr relaxion constructions. This raises the Nelson-Barr quality ceiling from f ≲ $10^{8}$ GeV to f ≲ $10^{13}$ GeV, and for the benchmark couplings studied the phenomenological ceiling for the scale of CP violation becomes f_pheno_max ~ 4×$10^{11}$ GeV, about eight times larger than in the original NBR model. The charge assignment (Table 1) also allows a type-I seesaw for neutrino masses and, provided the right-handed-neutrino masses and reheating temperature satisfy the Davidson–Ibarra bound, thermal leptogenesis can explain the baryon asymmetry.
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The central claim is that a residual CP-violating angle θ_eff, generated by explicit PQ-breaking or by operators like the chromo-electric dipole moment, produces a linear term in the clock frequency variation: δν_Th/ν_Th = K (2 x_* θ_eff a/f_a + x_π (a/f_a)^2). The linear term oscillates at m_a, scales as 1/f_a, and overtakes the quadratic term after a crossover time t_cross ~ 80 days × (K x_*)^{-4} for θ_eff ~ $10^{{-9}}$. The paper demonstrates that this linear signal is within reach of current and projected nuclear clock experiments, and it also yields a static axion field sourced by Earth, turning nuclear clocks into fifth-force detectors.
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The paper establishes a classification and a bound. On the classification side, the $\mathbb{Z}_3$ and $S_3$ gaugings of $\mathbb{Z}_N\times\mathbb{Z}'_N$ for $N=2,3,4,5$ each generate a definite menu of full-rank quark Yukawa textures; the menus agree for $N=2,4$, while for $N=3,5$ the $S_3$ gauging adds extra one- and two-zero textures, because each $S_3$ class merges two $\mathbb{Z}_3$ classes and its fusion rules are less restrictive, allowing more couplings and hence more full-rank patterns. On the bound side, Appendix A shows that ten zeros in total, whether split 5+5 or 6+4, force either a vanishing mixing angle or vanishing CP violation, so nine is the largest viable total, split as 6+3 or 5+4 between the sectors. For the 5+4 distribution, right-handed weak-basis transformations collapse all four-zero textures into one equivalence class, so the phenomenology reduces to the representative pair $Y_u=T_5$, $Y_d=T_1^{(1)}$ of Eq. (5.25); a $\chi^2$ fit to the $M_Z$-scale inputs returns all six quark masses, $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, and $\delta_{CP}$ within $1\sigma$ (Eqs. (5.28)–(5.29)). The representative parameterization has $\det(Y_uY_d)=acf\,a'c'd'$ real and positive, with the only physical phase sitting in $Y_u$, so the texture contributes nothing to the QCD vacuum angle $\bar\theta$ at tree level while retaining CKM CP violation; a subset of the 6+3 textures shares this property. The paper offers this as a new route toward an axionless strong-CP solution, while stating that a dynamical ultraviolet realization and an analysis of radiative corrections are required to complete the argument, and that the zeros alone do not explain the hierarchy of the nonvanishing couplings.
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Solving only the spatial components of the bulk Yang-Mills or Maxwell equations, $\mathcal{D}_i \bar{F}^{iA}=0$ with the gauge condition $\Delta\,\partial_i \bar{A}^i=0$, extends boundary gauge fields into the disk independently at every time slice, so interior fields depend on boundary fields only at that same instant. For Abelian fields in $n=1$ and $n=2$ this yields closed-form interior solutions built from modified Bessel functions, powers $(r/R)^{|m|}$, and a Green's function for $\bar{A}_0$; these solutions are non-singular throughout the interior for boundary data with zero net topological charge, and when the boundary carries net charge the required singularity lands on the temporal boundary rather than in the bulk. The paper argues the same behavior holds for non-Abelian SU(N) fields in the weak-coupling regime where the equations Abelianize, with no topological obstruction to smooth extension otherwise, so the zero-mode counting of the original construction survives: $n_i+\bar{n}_i$ boundary chiral modes and $|n_i-\bar{n}_i|$ interior mirror modes at the temporal boundary. Consequently the $\eta'$ is gapped at all nonzero momenta while its zero-momentum mode stays gapless, so it is not a Goldstone boson and the $\theta$ angle can be rotated away, preserving the absence of the strong CP problem.
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The central claim is that the strong-coupling dynamics used to explain the scale hierarchy is itself the source of a CP phase, so the reality of all UV couplings and the vanishing of all bare $\theta$-angles are insufficient for a safe SCPV construction. In pure $SU(N_c)$ super Yang-Mills, the instanton computation fixes only the real, $Z_{2N_c}$-invariant product $\langle\lambda\lambda\rangle^{N_c}$, which survives one instanton, real coefficient, per the direct computation; extracting the condensate of a single field requires an $N_c$-th root, producing $N_c$ degenerate vacua of one and the same real Lagrangian, with the realized branch undetermined by the dynamics. The constant term inherits this phase, $C=N_c\Lambda^3 e^{2\pi i k/N_c}$, and it flows into $\bar\theta$ through anomaly mediation unless diluted by gauge mediation with $m_{3/2}\lesssim(3.7\text{--}37)$ keV; the same structure applies to the ADS family, which is safe only for $N'\equiv N_c-N_f\in\{1,2\}$. For the DSB sector, the paper shows that the $N_f=N_c$ quantum moduli constraint is branch-free, yet IYIT still admits purely imaginary meson vacua for roughly half of real couplings; ISS is real for non-degenerate masses but the degenerate limit permits a complex vacuum orientation; and the 3-2 model at $N'=1$ is safe once the real constant term fixes the R-axion at one of two CP-conserving points.
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The discovery claim is that exempting the SM Higgs doublet from the charge convention $Y' = Y - (5/4)(B-L)$ while all SM fermions and the three right-handed neutrinos follow it forbids every dimension-4 Yukawa coupling to the Higgs by gauge invariance. The fermion masses are instead generated by higher-dimensional effective operators, realized renormalizably as chains of vector-like fermions, with the observed hierarchy absorbed into powers of $\langle\xi\rangle^k/\Lambda^k$. Because the Higgs singlets $\xi_d$ and $\xi_u$ carry different $U(1)_{Y'}$ charges, the lowest-order gravity-induced Peccei-Quinn-breaking term is necessarily the operator $O = (1/m!\,n!)\,\xi_d^m \tilde{\xi}_u^n / M_{\rm Pl}^{m+n-4}$ of Eq. (10), whose dimension $m+n$ grows with the chain-link counts, so the axion is automatically high quality.
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This review establishes that the strong CP problem is an unresolved tension between the Standard Model's natural expectation and experimental bounds, and that none of the conventional solutions is without theoretical difficulties. The Peccori-Quinn mechanism remains the most studied solution but suffers from the axion quality problem. Recent critiques question the very existence of a physical theta parameter by arguing that the theta-vacuum construction relies on an unproven hypothesis about gauge transformations at infinity and that the order of sum over topological sectors and the infinite-volume limit may not commute. If these critiques hold, the strong CP problem may be qualitatively different or even absent.
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Under minor assumptions about the 1PI effective action, the paper proves that the only way to have CP conservation for small nonzero θ̅ is if the anomaly-induced pseudoscalar mass term is completely subdominant relative to the mass term from chiral symmetry breaking. Because the U(1) problem is resolved precisely by that anomaly-induced mass, this means the strong-CP problem cannot be solved within QCD without forgoing the solution to the U(1) problem. The topological susceptibility of pure Yang-Mills theory and its QCD generalization play a central role in establishing this tension and in computing corrections to leading-order θ̅ results.
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The central claim is that the $\theta$-induced nEDM can be computed as a Feynman-Hellmann response: $d_n/\bar{\theta} = (1/E_z)\langle N^L| q_{\mathrm{top}}(0) |N^R\rangle_{E_z}$. The uniform Euclidean background electric field makes the dipole interaction anti-Hermitian, so the nucleon ground state is a parity-mixed state of a non-Hermitian Hamiltonian; the paper constructs its left and right eigenstates with a generalized eigenvalue problem in a basis of positive- and negative-parity nucleon interpolating operators. The topological charge operator is sampled at a single time slice and regulated by gradient flow, which replaces the global volume sum of earlier approaches. On three 2+1-flavor domain-wall-fermion ensembles with pion masses 340, 420, and 576 MeV and lattice spacing about 0.11 fm, the signal is nonzero, stable against electric-field strength, and the chiral extrapolation gives $d_n/\bar{\theta} = -0.0050(4)(8)\,e\cdot\mathrm{fm}$.
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The paper argues that in the minimal Higgs parity model, with gauge group $SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)_X$ broken to the Standard Model at scale $v_R$, the baryon asymmetry can be generated by out-of-equilibrium decays of the heavy singlet fermions $S_i$ after $SU(2)_R$ breaking. Each $S_i$ decays through two competing channels, $S_i\to \bar e_a W_R^{(\ast)}$ and $S_i\to \ell_b H_L$, and the coexistence of these channels violates $B-L$ while one-loop interference between them produces a CP asymmetry $\epsilon_{ia}$ that can be parametrically large despite small neutrino masses, because the masses are generated only at five-loop order. The asymmetry exists only because the right-handed charged leptons are non-trivial mixtures of $SU(2)_R$ doublet partners and a heavier vector-like state; if the heavy state were kinematically open, unitarity would cancel the inclusive asymmetry, so the model requires $M_H\gtrsim M_{S_i}$. Feeding $\epsilon_{ia}$ into a Boltzmann treatment with the $28/79$ sphaleron conversion factor, thermal production gives an upper bound on the baryon yield that demands $v_R\gtrsim 6\times 10^{12}$ GeV, whereas non-thermal production via inflaton decay, after washout constraints from inverse decays and a dimension-seven scattering operator are imposed, leaves viable $v_R\gtrsim 10^{4}$ GeV — a window that current LHC bounds and future colliders can directly probe.
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The central claim is that the effective CP-violating angle $\bar{\theta}_{\rm eff}$ can be dynamically relaxed to exactly zero, not by the usual QCD-axion potential but by four-form fluxes. In the Type IIA compactification, the low-energy potential for axions $b^j$ and four-form fields $H^i_4$ takes the form $V \sim \sum_i (b_{ij}b^j - h_i)^2$; integrating out the four-form fields and minimizing the potential over the axions yields $b_{ij}b^j = h_i$. Through the dictionary $H^i_4 \leftrightarrow X$, $b_{ij} \leftrightarrow \mu_{ij}$, $b^j \leftrightarrow a^j$, and $h_i \leftrightarrow \bar{\theta}\tilde\Lambda_{\rm QCD}^2$, this is exactly the field-theory condition $\mu_a a - \bar{\theta}\tilde\Lambda_{\rm QCD}^2 = 0$. The paper argues that this is a discrete vacuum selection, not a tuning: any integer flux pair satisfying the relation automatically gives $\bar{\theta}_{\rm eff}=0$, and Planck-scale corrections are absorbed into the flux parameters rather than shifting the minimum. The paper then verifies the mechanism in Model A, a globally consistent intersecting D6-brane configuration, checks its compatibility with STU and KL-style moduli stabilization, and estimates the anti-D6 uplift backreaction as $|\delta\bar{\theta}|<10^{-11}$.
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The central claim is that modular invariance in N=1 supergravity, together with holomorphy and the requirement that $\tau$ is the only source of CP violation, solves the strong CP problem without an axion and produces quark mass hierarchies from the same consistency condition. The new ingredient is a second modular-anomaly condition, Eq. (13b), which matches the determinant character of the finite representation matrices $\rho_I(\gamma)$ to the threshold multiplier of the gauge kinetic function; earlier analyses contained only the continuous-weight condition (13a). With the Higgs weights obeying $k_{H_u}+k_{H_d}=0$ and $\chi_{H_u}\chi_{H_d}=1$, the quantity $A_{\mathrm{SG}} = e^{-8\pi^2 f_3(S,\tau)}\det(Y_uY_d)(\tau)W_0(S,\tau)^{-C_3}$ is modular invariant. Regularity of $A_{\mathrm{SG}}$ means its divisor is zero, which, in the minimal cusp-only threshold class, translates into $\det(Y_uY_d)\propto\eta^{2K_Y}$ (Eq. (23)). For $K_Y>0$ this forces the determinant to vanish at the cusp $\tau=i\infty$, giving hierarchically separated quark masses; the explicit $\Delta(384)$ model realizes this with $\det(Y_uY_d)\propto\eta^{24}$, $e_{B_3}=12$, $\delta_{\mathrm{GS}}=-3$, and a hidden pure super-Yang-Mills sector with $C_h=3$ whose condensate cancels the remaining dilaton dependence, leaving $A_{\mathrm{SG}}$ a real constant and hence $\bar\theta=0$.
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The paper's claim is that the post-inflation axion window from $10$ to $20$ GHz ($m_a\simeq 40$–$80~\mu$eV) can be scanned with sensitivity approaching the KSVZ benchmark using a plasma haloscope. The axion is a proposed particle that resolves the strong-CP problem and is a dark-matter candidate; the KSVZ benchmark is a standard reference coupling strength for QCD axions. The decisive object is a wire-array metamaterial resonator: an array of metal rods whose collective plasma frequency acts as a bulk property of the unit cell, so the resonant frequency can be set by rod spacing rather than by the confinement volume. ALPHA Phase I plans two tunable implementations—the Rinnegan (ring nested group rotation), which rotates nested rings of spiral-packed rods about one axis, and a symmetric triangular lattice of pivoting rods—both enclosed in a photonic bandgap structure that suppresses unwanted transverse-electric modes. With target parameters $Q=1.3\times10^4$, form factor $C=0.4$, volume $V=0.012$ m$^3$, giving a figure of merit $QC^2V^2=0.3$ m$^6$, the projected figure of merit is two to three orders of magnitude above a single cylindrical cavity scaled to the same frequency, and the system noise is projected at $N_{\rm sys}=1.32$ quanta with near-quantum-limited readout. In 3.7 years of integration the projected discovery potential reaches KSVZ-level couplings across the band.
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The authors claim that small-instanton contributions to the axion potential in KSVZ models are governed by a counting rule: a vector-like quark (VQ) representation with $2|N|$ conjugate zero-mode pairs is closed by $k$ scalar-Yukawa loops and $r$ mass insertions with $2|N|=2k+r$, and the instanton-size integral is ultraviolet-dominated when $\Delta=b_3+r-4<0$. Because each Yukawa loop lowers $\Delta$ by two, maximally Yukawa-closed contractions give the largest UV enhancement: $(M_{\rm Pl}/\Lambda)^{31/3}$ for a single color-15 VQ, $(M_{\rm Pl}/\Lambda)^{13}$ for two copies of the octet doublet $R_{13}$, and the same power 13 for the distinct sets $S_3$-$S_7$. Under the stated assumption that the QCD and small-instanton potentials have equal periodicity and aligned minima, the total potential is $V(a)=-(\chi_{\rm QCD}+\chi_{\rm SI})\cos(\bar{\theta}+a/f_a)$, so the axion mass at fixed $f_a$ is enhanced: the largest values quoted are 3.43 eV for a single VQ, 71.2 MeV for two copies of $R_{13}$, and 65.3 MeV for the $N_{\rm DW}=22$ assignment of the distinct set $S_4$. The axion-photon coupling, by contrast, stays determined by $f_a$ and $E/N$, and the paper exhibits contours in the $(m_a,g_{a\gamma\gamma})$ plane that leave the conventional QCD-axion band.
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The central claim is that the present-day spin-down power of PSR J0437-4715, after standard losses are subtracted with a field normalization that does not come from the torque law being tested, leaves a positive residual that any nonstandard CP-odd radiation channel must fit inside. An unscreened assignment to coherent electric-dipole radiation gives the benchmark $|d_n| < 5.783\times 10^{-25}\,e\,\mathrm{cm}$ at the 90th percentile of the positive branch. Because crustal electrons and magnetospheric plasma screen static electric dipoles, the paper instead promotes a screening-aware channel: an effective neutron magnetic quadrupole moment radiating at the observed orthogonal-polarization-mode transition frequency. With the aligned inner-crust neutron reservoir and the quadrupolar surface-field fraction, the residual gives $|M_n^0| < 7.47\times 10^{-38}\,e\,\mathrm{cm}^2$; assuming a pure-$\bar{\theta}$ origin, this becomes $|\bar{\theta}| < 2.99\times 10^{-9}$ and an equivalent $|d_n| < 4.42\times 10^{-25}\,e\,\mathrm{cm}$. The paper is explicit that these are conditional, effective bounds, not screening-independent measurements.
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The paper's central claim is a quantitative quality-coupling relation: high axion quality implies a large anomalous coupling between the axion and the $U(1)_{B-L}$ gauge boson, and that coupling induces an axion-fermion interaction with a calculable lower bound. Under the charge assignment $q_1=m=l$, $-q_2=n=k$ with equal scalar VEVs, the cubic anomaly-cancellation condition reduces to $3mA-3nB = nm(m^2-n^2)$, and non-negativity of the sums $A,B$ of squared heavy-quark charges gives $C_{B-L}\ge \max(n^2/4,(m^2+3)/4)$ (or the $n\leftrightarrow m$ variant). The loop computation yields $g_{aee}^{(B-L)}\simeq (3\alpha_{B-L}^2/(4\pi^2))\,q_f^2 (m_f/F_a) C_{B-L}\log(v/m_B)$. Because the quality requirement $\delta\bar{\theta}<10^{-10}$ forces $n$ large, this induced coupling has a floor that can exceed the model-independent QCD contribution, extending the observable parameter space of invisible QCD axions and tying low-energy couplings to the UV charge assignments of the heavy chiral quarks.
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On the paper's own account, the central claim is that the Nelson-Barr scalar $S=(s+ia)/\sqrt{2}$ can be the inflaton. Its imaginary component $a$ has a CP-symmetric double-well potential, so the scalar has two CP-conjugate vacua at $\langle a\rangle=\pm f$; these are the two inflationary valleys, and inflation is quartic hilltop inflation along one of them. The same structure removes the domain-wall problem: the CP-invariant ridge at $a=0$ has minimum height $V_{\rm CP}=\lambda_a f^4/4$, while CMB normalization fixes the inflationary plateau energy $V_0\simeq \lambda_s v^4/12$ to be many orders of magnitude smaller for $f\sim v$ and $\lambda_a=O(1)$, so post-inflationary scalar energy cannot cross the ridge and the CP-branch is never repopulated. A small CP-preserving linear deformation $-Cs$ from a real spurion raises the spectral index out of the quartic-hilltop value $n_s\simeq 0.928$--$0.942$ up to the observed $n_s=0.974\pm0.003$ in a finite parameter region. The same vector-like-quark interactions that transmit spontaneous CP violation to the CKM phase provide loop-level reheating through gluons, and a coupling to right-handed neutrinos transmits the phase to the lepton sector, with nonthermal leptogenesis possible if the CP-breaking scale is raised to about $10^{16}$ GeV.
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Fitting all 9-link textures — full-rank Yu and Yd with nine non-zero entries and a single rephasing-invariant phase φ — to the ten flavor observables, the paper finds that viable fits force φ to cluster at multiples of π/8, predominantly π/2, π/8, and 3π/8, the measured values of the unitarity-triangle angles (α, β, γ). The mechanism is a leading-order identity: in most textures the angle carried by φ is exactly one of α, β, γ, because the ratio of CKM elements defining that angle equals a ratio of Yukawa entries. That ratio, together with its argument, defines a 'Yukawa triangle' which coincides with the unitarity triangle at leading order in the small flavor parameters. By fixing the Yukaw
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The paper's central claim is that in the minimal Higgs parity model, an SU(2)_L × SU(2)_R bi-triplet Weyl or Dirac fermion—charge assignments (3,3,0) or (3,3,1)—is accidentally stable dark matter. Scanning all SU(2) multiplets up to dimension three, the authors find that every smaller representation decays through operators of dimension three, four, or five, while the bi-triplet's leading decay operators are dimension six; with a Planck-scale cutoff this gives lifetime τ_X ≳ 10^28 s, comfortably above cosmological bounds. The same gauge structure fixes the freeze-out dynamics: electroweak and new W_R/Z_R mediated annihilations, including coannihilation among multiplet members and the Sommerf
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The central discovery is that the charm-coupled PQ sector maps into HMChPT without creating a leading-order symmetry-breaking spurion. The scalar σ matches to the spin-symmetric operator tr[\bar H_a H_a], so it preserves heavy-quark spin symmetry and is not excluded by any heavy-meson mass or splitting measurement; its strongest effect is a ~0.5 B_S enhancement of B_s mixing. The pseudoscalar a, by contrast, vanishes between ground-state heavy mesons in the static limit, so its direct coupling is suppressed by 1/m_c and its observable footprint is dominated by axion–pion mixing; that mixing produces rare decay rates (D*^0→D^0 a at ~9×10^-4, K+→π+a at ~4×10^-4, B+→K+a at ~2×10^-8) that are ei
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The central claim: assigning the PQ charge to the right-handed charm quark, not to light quarks, zeroes the light-quark spurion and thereby removes the isospin obstruction that otherwise produces a 15% pion mass splitting. With m_φ ≈ 3–4 MeV the charm Yukawa κ_c ≈ 0.44–0.59 is perturbative, f_a ≈ 2.1–2.9 GeV, and the axion mass is 2–2.7 MeV. At this f_a, even the d=6 Planck-suppressed operator gives m_PQ/m_a ~ 10^-14, solving the quality problem without extra symmetries. The model predicts BR(B→Kσ) ≈ 2×10^-5 and ΔN_eff ≈ -0.1, and claims all ten constraint classes pass.
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No established physical observable requires the additional assumptions that globally classify gauge-field configurations into integer topological sectors. All the standard local nonperturbative consequences of the topological charge density follow from minimal QCD (local gauge invariance, causal locality, and the functional integral). The conventional strong CP problem appears only after those extra global assumptions promote a source parameter into a physical vacuum angle.
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Axion quality is compatible with minimality: the Peccei–Quinn symmetry can be realized as a gauge-origin boundary remnant of five-dimensional U(1) gauge invariance on a flat interval, the axion is the surviving open Wilson-line phase, and a familiar four-dimensional KSVZ/DFSZ anomaly sector still generates the QCD potential, without bulk QCD, warping, or a color Chern–Simons term, while exponential suppression protects against dangerous operators.
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Once an anomaly-free discrete Z_n^R symmetry forbids the mu term, the MSSM plus any of the four Kim–Nilles base models develops an accidental global U(1)_PQ. Soft supersymmetry breaking then drives intermediate-scale vacuum expectation values for the PQ-charged singlets, simultaneously regenerating a weak-scale mu, breaking the discrete R-symmetry, and producing a supersymmetric DFSZ axion that solves the strong CP problem.
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A carefully chosen set of U(1)_R charges, together with vector-like fermions and two singlets that break the new symmetry, forces every light fermion mass (except the top) through a universal-seesaw block whose upper-left entry vanishes. The same block form implements a Nelson-Barr mechanism: spontaneous CP violation generates an order-one CKM phase while the determinant of each full quark mass matrix remains real, so the tree-level strong-CP angle vanishes.
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The paper identifies a specific mechanism — the condition D_T W = 0 at a metastable minimum in a mixed F/D-term uplift — that locally disables the DFK bound's inference from the superpotential inequality 2|⟨W⟩| ≤ f_R F to the Planckian conclusion f_R ≳ M_Pl. When the modulus T containing the R-axion is arranged to be supersymmetric at the minimum (D_T W = 0), the R-Goldstone direction has zero projection onto the SUSY-breaking F-term, so the cosmological constant cancellation is handled by the uplift sector rather than forcing f_R to be large. The resulting vacuum is metastable, and demanding its lifetime exceed the age of the universe yields the relaxed bound f_R ≳ 400^{1/4} √(m_{3/2} M_Pl)
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When the PQ symmetry also has a mixed anomaly with a pure Yang-Mills dark SU(Nc), the resulting dark-instanton potential explicitly breaks the residual Z_NDW symmetry. For a dark critical temperature Tc inside the narrow window 0.1–3 MeV (and a temperature ratio ξ ≲ 0.5), the vacuum bias is large enough to drive domain-wall collapse before BBN while the induced shift in the effective theta angle stays below the neutron-EDM bound, thereby solving the domain-wall problem without reintroducing the strong CP problem.
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In the unbroken E6^L × E6^R theory the left-right exchange is realized on the Dirac field as spacetime parity and acts trivially on colour and electric charge. Consequently the two would-be colour groups coincide as one vector-like SU(3)_c with a colour-singlet electron, hypercharge is the consistency relation Y = Q - T_L^3 with Q = N/3 and no right-sector generator, and the spontaneous parity forbids θ_QCD while real-determinant flavour rotors give arg det M = 0, yielding θ-bar = 0 at tree level that coexists with a nonzero CKM phase.
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We promote the accidental B+L symmetry of the Standard Model to a Peccei-Quinn symmetry while realizing spontaneous proton decay radiatively. The PQ anomaly sector consists of vector-like quarks providing a KSVZ-type axion solution to the strong CP problem. After spontaneous PQ breaking a residual Z2 symmetry remains which forbids tree-level proton decay. The VLQs required to generate the QCD anomaly, together with scalar mediators odd under the residual Z2, induce one-loop proton decay through the effective operator u_R u_R d_R e_R. The resulting models lead to distinct predictions for the axion-to-photon coupling.
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The paper establishes that recent computational and theoretical progress now permits the quantification of hadron-level CP violation effects that feed into experimental observables such as electric dipole moments, thereby tightening constraints on physics beyond the Standard Model and offering a path to address the strong CP problem through existing fields alone.
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Spontaneous CP violation after inflation produces stable domain walls unless the reheat temperature lies below the CP-breaking scale. The authors introduce an auxiliary scalar that acquires a large early-universe value with a complex phase; a higher-dimensional interaction then splits the energies of the degenerate CP vacua by an amount that shrinks as the auxiliary field rolls to zero. The resulting bias is large enough to trigger wall decay while the auxiliary field is still displaced, yet disappears at late times so that the low-energy CP-odd phase remains exactly as in the original model. The auxiliary field's residual oscillations automatically constitute a viable dark-matter candidate
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Interactions mediated by novel lightweight particles such as ALPs arise from the Peccei-Quinn mechanism, string theory, and supersymmetry breaking, offer solutions to several challenges in modern physics including the strong CP problem, and can serve as cold dark matter; many predicted interactions are spin-dependent and the review compiles theoretical expressions along with current experimental constraints on them.
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Constructions with gauged U(1)_F flavor symmetries shield the axion from Planck-suppressed operators, yielding an accidental high-quality flavored axion with unit domain wall number. These predict flavor-changing neutral currents at high flavor scales and stochastic gravitational waves from the evolution and decay of gauged flavonic and axionic cosmic-string networks, while global axionic strings can radiate axions to match the observed dark matter abundance. The resulting plateau-valley structure in the GW spectrum provides a distinctive probe of high-quality flavored axion dark matter models complementary to low-energy flavor experiments.
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Big axions are defined as axion models in which a Nambu-Goldstone mode emerges from the collective spontaneous breaking of a network of U(1) symmetries delocalized in theory space. They naturally realize high-quality accidental global symmetries, admit both pre- and post-inflationary cosmological histories, and exhibit rich topological structures that interpolate between ordinary Peccei-Quinn axions and axions which descend from extra-dimensional gauge fields. Little big axions, identified as the minimal phenomenologically viable subclass, provide a robust solution to the strong charge-parity problem in quantum chromodynamics while potentially accounting for some or all of the dark matter of
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The central claim is that an SU(12) × SU(2)_L × U(1)_R theory unifies SU(9) quark color-flavor with SU(3) lepton flavor starting from a single Yukawa shared by up-type quarks and neutrinos; gauged instantons dynamically generate the bottom and tau Yukawas, only two new scalar representations are required for the breaking chain that includes color-flavor deconstruction followed by infrared reunification, and the low-energy gauge group emerges as G_SM equal to the quotient of SU(3)_C × SU(2)_L × U(1)_Y × Z^X_18 by Z_3 × Γ × Z_3 where the discrete gauge symmetry X = B − 3(L_i + L_j − L_k) absolutely stabilizes the proton.
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The central discovery is a framework where mirror symmetry relates the visible sector to an unbroken GUT sector whose confinement dynamically produces a heavy axion mass scale without fine tuning, thereby solving the strong CP problem in a way robust to high-scale effects.
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Allowing for the running gauge coupling, the CP-violating four-fermion interaction becomes relevant in the chirally broken phase. In the presence of a finite quark mass, the RG running of the θ-parameter is shown to be strongly suppressed toward the infrared. The present work clarifies how strong-CP effects generated at UV can non-trivially be transferred to the infrared physics in QCD-like theories.
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Multi-axion theories modify the canonical mass-photon-coupling relation of single-axion solutions to the strong CP problem. The location of the axions in parameter space is fixed by the structure of Peccei-Quinn symmetry breaking together with the relative alignment between the QCD and electromagnetic anomalies. These ingredients yield a general sum rule for N-axion systems that incorporates arbitrary PQ breaking and non-universal anomaly coefficients, thereby generating the full set of qualitative mass-coupling patterns.
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Spontaneous CP violation is realized along flat directions and stabilized through supersymmetry-breaking effects and a non-perturbative dynamics. This addresses the naturalness of the SCPV scale and the presence of problematic higher dimensional operators and radiative corrections spoiling the mechanism. It is explicitly shown that SCPV is realized along flat directions and stabilized through supersymmetry-breaking effects and a non-perturbative dynamics, predicting light SCPV sector particles feebly coupled to the Standard Model particles. Furthermore, the Affleck-Dine mechanism can successfully generate the observed baryon asymmetry with a low reheating temperature compatible with the grav
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The theta dependence of the QCD vacuum energy is described by a set of analytic approaches whose regimes of validity are known, while lattice Monte Carlo simulations of the discretized theory supply direct numerical access to the topological susceptibility and its theta dependence.
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By considering a special embedding of the confining gauge group responsible for the composite axion as well as QCD into a larger product gauge group, the domain wall number is essentially set to unity in the ultraviolet theory. Small instanton effects associated with the UV gauge dynamics induce a controlled explicit breaking of the residual discrete symmetry, providing a bias term in the axion potential. As a result, the domain walls become unstable and decay sufficiently quickly, while the axion solution to the strong CP problem remains intact.
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The authors establish that the chiral U(1) gauge symmetry induces an accidental Peccei-Quinn symmetry whose explicit and spontaneous breaking arises entirely from mirror QCD dynamics. This yields a viable solution to the strong CP problem without massless fermions or a light QCD axion. The same dynamics ensure no stable domain walls or colored relics form, while allowing a reheating temperature high enough for leptogenesis, metastable domain walls that source gravitational waves, and a pseudo-Nambu-Goldstone boson that serves as WIMP dark matter connected to the Standard Model via a kinetically mixed gauge boson.
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Kinetic mixing between QCD and an additional hidden U(1) three-form gauge field shifts the effective theta parameter of QCD and induces a nonzero vacuum expectation value for the operator G tilde G. The paper demonstrates the shift first in a controlled two-dimensional model and then in the four-dimensional three-form formulation, showing that hidden-sector fluxes propagate into observable CP-odd effects without requiring explicit quark mass phases or instanton contributions.
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In the multi-axion framework the domain wall number is determined by the full anomaly structure of the theory rather than by a single axion; this relaxes the E/N >= 8/3 theoretical bound on the QCD axion photon coupling that holds in the single-axion case under the assumption of minimal global structure for the Standard Model gauge group. Combined with the possibility that the QCD axion constitutes only a subdominant fraction of dark matter, the scenario makes the QCD axion harder to detect directly, but a scan of the parameter space shows that in most regions where the QCD axion evades detection an axion-like particle remains visible to next-generation experiments, with the most promising 2
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In frameworks equipped with non-invertible selection rules, a CP-like symmetry restricts tree-level mass matrices; once loop corrections are included, these rules are broken radiatively, introducing flavor-dependent phases that produce leptonic CP violation. The same corrections generate mass terms, as illustrated in the inverse seesaw model where the Majorana mass of the light sterile neutrino N_L appears dynamically together with CP violation. The mechanism is presented as having wider applicability to other CP-related issues such as the strong CP problem, leptogenesis, and baryogenesis.
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In these Nelson-Barr constructions, the transmission of spontaneous CP violation occurs through the mixing of Standard Model quark doublets with vector-like quark doublets. An accidental symmetry of the renormalizable Lagrangian ensures that contributions to the effective theta-bar parameter are absent at one and two loops, appearing first at three loops. This provides a natural explanation for the observed smallness of hadronic CP violation without additional fine-tuning.
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In this Type IIA orientifold model with intersecting D6-branes, the STU model combined with the KL mechanism stabilizes the moduli and breaks supersymmetry, producing an N=1 supersymmetric three-generation MSSM-like spectrum. The model embeds a four-form flux mechanism that solves the strong CP problem and predicts dark matter consisting of both string axions and the lightest neutralino, with their relic abundances calculated to be consistent with observations.
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For a four-dimensional theory with gauge group G and an automorphism σ that implements charge conjugation, gauging CP means summing over Pin+(4)⋉G fiber bundles, not ordinary CP×G bundles. The paper's main result is that when π0(G)=π1(G)=0, the five-dimensional bordism group that classifies global anomalies satisfies Ω_5^{Pin+⋉G}(pt) ≃ Ω_5^{Spin}(BG). This group is Z2 for G=Sp(N) and trivial otherwise. The Z2 for Sp(N) is the familiar Witten anomaly, so if the theory was anomaly-free before gauging CP, it remains anomaly-free after. Since the SU(5) and Spin(10) embeddings of the standard model are anomaly-free before gauging, their gauged-CP versions are anomaly-free as well.
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The paper establishes that the Murayama QCD-scale axion and its natural extensions are excluded by low-energy mesonic observables. Integrating out the heavy PQ scalar yields an effective chiral Lagrangian for mesons that contains a new spurion I_PQ, which transforms like a quark mass matrix under chiral symmetry but breaks the accidental SU(2) isospin of the leading-order QCD chiral Lagrangian. The dominant new operator, O_PQ^1 = tr[I_PQ† U] tr[I_PQ U†], replaces the standard quark-mass operator in its effect on pions. At quadratic order it produces a neutral-charged pion mass splitting of order unity, and at quartic order it changes pion scattering amplitudes from the Weinberg form by facto
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In both the quantum rotor and the quantum pendulum the exact energy levels are known and depend on the vacuum angle θ through standard Bloch-wave or Mathieu-function expressions. When the ACGT prescription is followed—first sending the circle length to infinity and only afterward summing over winding sectors—the computed energies deviate from these exact expressions. The discrepancy demonstrates that the order of limits does not preserve the physical spectrum even in these controlled models.
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The paper claims that in its minimal universal seesaw Pati-Salam model, θ̄ = 0 at tree level because parity forces the quark mass determinants to be real, and the color-sextet and color-octet mass matrices to be hermitian. At one loop, the paper shows that the vast majority of new diagrams—those with charged leptons, scalar and gauge leptoquarks, and corrections to sextet/octet masses—vanish individually, leaving only the neutral-lepton correction to the down-quark mass as a nonzero contribution. That contribution is parametrically suppressed by either M10/κR or M15/κR, so the model can keep θ̄ < 10^-10 by choosing the color sextet or the color octet fermion mass to lie well below the parity
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The central claim is that the universal seesaw framework, combined with a generalized parity that solves the strong CP problem, offers a new and viable leptogenesis channel that avoids the usual high-scale obstruction. Instead of using the ν_R inside the SU(2)_R doublet—whose gauge interactions thermalize and wash out the asymmetry unless the left-right breaking scale is very high—the model uses the gauge-singlet fermions N_L and N_R required for Type-I seesaw neutrino masses. The parity doubling of these singlets means that one generation already contains two heavy states with complex couplings, so the interference of tree and one-loop diagrams yields a nonzero CP asymmetry once the masses
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Supersymmetry provides a natural framework to accommodate spontaneous CP violation by protecting the scale of SCPV from radiative corrections and suppressing problematic higher-dimensional operators generating a strong CP phase. In the exact SUSY limit, the spurion formalism is extended to identify the necessary condition for stabilizing CP-violating phases, and radial vacuum expectation values are stabilized through R-symmetry constraints on the superpotential. In a second construction, CP is spontaneously broken at an intermediate scale along pseudo-flat directions stabilized by soft SUSY breaking and non-perturbative effects of a gauge theory, predicting light scalars in the SCPV sector
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According to the paper, in N=1 supersymmetric QCD with N_c=N_f, confinement spontaneously breaks the global baryon-number symmetry U(1)_B, and the resulting Nambu-Goldstone boson is a QCD axion. The axion's Peccei-Quinn shift symmetry is not put in by hand: the gauge-invariant baryon operator is a product of N_c quark superfields, so any operator that breaks baryon number and respects the gauge symmetry has dimension N_c+2 or higher, strongly suppressing its effect on the axion potential. The Standard Model color group is embedded as a subgroup of the flavor symmetry, so the mixed anomaly between that flavor group and U(1)_B becomes the axion's coupling to gluons. After supersymmetry breakin
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The parity solution to the strong CP problem extends the Standard Model with an SU(2)_R gauge sector that restricts Yukawa interactions. In an appealing neutrino sector structure, small neutrino masses arise naturally while lepton number symmetry is violated substantially at the TeV scale. This permits observable lepton number violating collider signals not suppressed by small neutrino masses. The process μ⁺ μ⁺ → W⁺ W'⁺ at a 10 TeV μ⁺ μ⁺ collider can probe the W' mass up to 10 TeV on-shell and 16 TeV off-shell.
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The central claim is that heavy QCD axions, whose masses are raised well above the usual sub-60 meV range by new small-instanton contributions, would be produced at a muon collider through electroweak vector-boson fusion and would decay predominantly to gluons, so they appear as a narrow resonance in the dijet invariant mass spectrum. The paper argues that a 10 TeV muon collider with 10 ab$^{-1}$ can probe axion decay constants $f_a\sim$ TeV and masses up to $m_a\sim 10$ TeV, a region that existing LHC, LEP, and beam-dump experiments cannot cover. The same logic applies to four classes of ultraviolet completions: product groups, extra dimensions with a bulk scalar axion, mirror worlds, and color unification, all of which predict axions inside the projected sensitivity, while the version with an axion from a bulk gauge field does not yield viable muon-collider parameter space.
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For perturbative heterotic E8×E8 compactifications on Calabi-Yau threefolds (six-dimensional internal spaces), the paper's central claim is that the axion mass spectrum has a much stronger lower bound than in type IIB: almost all model-dependent two-form axions are heavy, lifted by worldsheet instantons or gaugino condensation, and the QCD axion, when it solves Strong CP, is the lightest state with m ~ Λ_QCD²/f. The argument combines the heterotic volume bound (V ≲ 20–30 in string units, from perturbativity and gauge-coupling unification) with the relative strengths of QCD instantons, hidden gaugino condensation, and worldsheet instantons. The only exception is a fibred Calabi-Yau where the
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The central discovery is that a supersymmetric SU(10) chiral gauge theory with one antisymmetric and six antifundamental chiral superfields, perturbed by anomaly-mediated supersymmetry breaking, has a stable nonsupersymmetric vacuum that can be solved exactly in the limit where the supersymmetry-breaking scale m is much smaller than the dynamical scale Λ. The exact non-perturbative superpotential W = (Λ^{23}_{10}(Pf A)(Pf A \bar F \bar F))^{1/3} normally drives a runaway; the anomaly-mediated term V = m(φ_i ∂W/∂φ_i − 3W) + c.c. stabilizes it at b = c = a/√2 = Λ(17Λ/138m)^{3/20}. Around this vacuum, the global SU(6) × U(1)PQ symmetry is broken to Sp(6) — which contains QCD color — and the U(1
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In a supersymmetric Twin Higgs model, mirror-symmetry breaking in the Yukawa couplings plus light sfermions drives a negative thermal mass for the Higgs, so electroweak symmetry stays broken below the twin scale instead of being restored by the plasma; this is electroweak symmetry non-restoration. The same light scalars stabilise the electroweak scale and, in new parameter regions, make the transition first order. With right-handed neutrinos and unbroken B′−L′ in the twin sector, the extra dark relativistic degrees of freedom fall to a level compatible with CMB data. The setup also accommodates minimal axiogenesis, jointly addressing baryon asymmetry, dark matter, and the strong CP problem.
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The paper argues that a non-invertible selection rule, arising from gauging a $\mathbb{Z}_2$ subgroup of global $\mathbb{Z}_N$ symmetries, enforces a three-zero texture on the down-quark mass matrix. With CP assumed to be an exact symmetry at high energies, the determinant of that matrix is forced to stay real by the non-invertible constraint. CP is then spontaneously broken by complex vacuum expectation values of new scalars $\eta_i$ at an intermediate scale, producing the CKM phase; the QCD $\theta$ term stays zero because the determinant of the down-quark mass matrix remains real. The full matter and Higgs sector is taken to transform uniformly under the non-invertible symmetry, so the construction can be embedded into an $SO(10)$ grand unified theory.
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The central claim is that the requirement of reproducing the observed dark matter abundance places an upper bound on the Parity and $SU(2)_R\times U(1)_X$ breaking scale $v_R$ in the doublet-pair WIMP model. The dark matter candidate is the neutral component $\psi^0_r$ of a Parity partner doublet; its charged partner and its $SU(2)_L$ counterparts are heavier by calculable one-loop gauge corrections. For freeze-out to yield the right abundance, the dark matter must annihilate resonantly, through an intermediate $W_R$ or $Z'$ whose mass is about twice the dark matter mass, so its mass is tied to $m_{W_R}/2$ or $m_{Z'}/2$ (a separate branch at about 260 GeV is excluded by searches for long-lived charged particles). Along these resonance branches, the annihilation cross section is fixed by the gauge coupling and $v_R$, and once $v_R$ is too large the cross section is too small, so the relic abundance exceeds the observed value. The paper finds the resulting upper bound is $v_R \simeq 25$–$60$ TeV, with the precise value depending on whether the dark matter and its Parity partner coannihilate; direct detection and collider searches place complementary lower bounds.
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The paper's central claim is that a canonical scalaron emerges from an Einstein–Cartan action whose dimension-four geometric sector is a single complete square, and that matter currents coupled to torsion become, in the Einstein frame, interactions of that scalaron with the currents. For gauge-invariant currents the interaction is a derivative coupling $\nabla_\mu \phi \, j^\mu$ plus a current self-coupling, and the paper defines the decay constants $f_1(\phi)$ and $f_2(\phi)$ in Eq. (3.16). For gauge-dependent currents, the paper's construction uses the shifted torsion fields $S'_\mu = S_\mu + \zeta j_\mu/M_{\rm Pl}^2$ and $T'_\mu = T_\mu + \xi j_\mu/M_{\rm Pl}^2$; integrating by parts converts the resulting $\chi \nabla_\mu j^\mu$ terms into $\phi F\tilde F$ couplings for Chern–Simons currents. The paper also shows that in the minimal fermion kinetic coupling, the scalaron–current interaction alone cannot generate a QCD $\theta$-term potential, but the Chern–Simons coupling can, so the scalaron is a possible but generically not a viable QCD axion because it also couples to dimensionful parameters.
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The paper's claim is that the dichotomy $\bar{\theta}=0$ versus $\bar{\theta}=\pi$ is not an unavoidable consequence of CP or P invariance but an artifact of assuming the usual $2\pi$ periodicity of the QCD $\theta$ angle. In a gauge theory of the form (4), fractional instantons of the different factors are correlated: the topological charges take the form $\vec Q = \vec n + m \vec q$, and a $2\pi$ shift of the $\theta$ angle of one factor can be compensated by shifts of the others precisely when $\vec k \cdot \vec q$ is an integer. The paper shows that if the group is chosen so that condition (7) holds, the $\theta$ angle of the factor containing QCD has period $4\pi$ rather than $2\pi$; imposing CP then forces $\bar{\theta}_1=0$ or $2\pi$, and matching to the Standard Model gives $\bar{\theta} \approx \bar{\theta}_1$, effectively zero. It further shows that the Standard Model's own quotient structure does not enlarge the QCD period, and that the unification groups proposed so far — SU(5), Spin(10), Pati-Salam, trinification, and $SU(6) \times SU(2)$ — fail condition (7), while the illustrative $[SU(6) \times USp(4) \times USp(4)]/\mathbb{Z}_2$ model satisfies it.
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The central claim, stated in the author's own terms, is that only third-generation fermions acquire tree-level masses, while the first and second generations gain their masses via quantum corrections induced by new gauge bosons. The mechanism works because each chiral fermion sector couples to a vector-like pair through mass terms $\mu_L$ and $\mu_R$, producing a seesaw-like tree-level mass matrix of rank one; the sole massive state is the third-generation fermion. Non-universal gauge charges then break the accidental $U(2)^5$ symmetry of the mass Lagrangian, and gauge-boson self-energy diagrams generate the remaining masses: both lighter generations at one loop in the $U(1)_1 \times U(1)_2$ model, with the hierarchy set by the ratio of the two gauge-boson masses; the first generation at two loops in the optimised single-$U(1)_F$ model, where the loop-corrected mass matrix stays rank two at one loop and becomes rank three only through the two-loop formula; and sequential first- and second-generation masses in the $SU(3)_F$ model from the two-step breaking $SU(3)_F \to SU(2)_F \to$ nothing. Numerical fits to the charged fermion masses and CKM mixing are presented for each construction, and the left-right version is claimed to keep the strong CP phase at $\bar{\theta} \lesssim 10^{-14}$.
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In the Nelson-Barr model, spontaneous CP breaking produces a pseudo-Nambu-Goldstone boson $\phi$; when this field is ultralight and forms the local dark matter, it oscillates at frequency $m_\phi$ and, through the field-dependent CKM matrix, induces periodic modulations of weak-interaction parameters and quark masses. The paper's central calculation follows these modulations into nuclear physics: the neutron-proton mass difference shifts through the QCD term $4c_5B_0(m_u-m_d)$, and the tritium binding energy shifts through a one-pion-exchange potential evaluated with a square-well deuteron wavefunction and SU(4) scaling. The combined effect is an oscillating residual in the tritium $\beta$-decay rate, $I(\phi)=2.44\times 10^{-5}\,\phi/f$, with $\phi/f = \sqrt{2\rho_{DM}}/(f m_\phi)\cos(m_\phi t+\delta)$. Applying a frequentist hypothesis test to the null tritium data yields the paper's main result: at 95% CL, decay constants below $7.0\times 10^9$ to $1.4\times 10^7$ GeV are excluded for $m_\phi$ between $3.4\times 10^{-23}$ and $1.7\times 10^{-20}$ eV.
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The central discovery is a pair of theorems about two-Higgs-doublet models with a softly broken CP symmetry and an Abelian flavor symmetry. First, when the flavor charges are chosen so that arg det(yu yd) = 0 at tree level for all couplings, because each Yukawa determinant is a single monomial in the two Higgs fields, the one-loop correction to $\theta$-bar also vanishes, independent of the scale of the second doublet. The reason is that the radiative corrections to the Yukawa matrices share the same zero-texture pattern and real entries as the tree-level matrices, so the phase of each determinant is unchanged. Second, in all such models there exists a phase basis in which all neutral scalar couplings to quarks are real: neutral flavor-changing interactions conserve CP. Consequently, the flavor-changing neutral current bound on the second doublet drops from the usual 20 TeV to about 1 TeV, and the two Z3 sample models have negligible direct neutron-EDM contributions, with two-loop contributions to $\theta$-bar of order $10^{-12}$ and up to $10^{-10}$ respectively.
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The central discovery claim is that the strong CP, hierarchy, and cosmological constant problems can be solved together in type IIA flux compactification on $T^6/(\mathbb{Z}_2\times\mathbb{Z}_2)$. The strong CP part works by identifying the axion potential of the four-form-coupled mechanism with the flux-induced potential of the orientifold: the flux constraint $b_{ij} b_j = h_i$ forces $X=0$ in the vacuum, so the CP-violating combination vanishes dynamically. The cosmological constant part adopts the KL downshift: a supersymmetric Minkowski vacuum is displaced to AdS by a small correction $\Delta W$, then uplifted by an anti-D6-brane contribution. After checking several power-law forms for $\Delta W$, the paper argues that the only viable form is $\Delta W = f_0 U^3$, because it avoids runaway directions, keeps the vacuum near Minkowski, and does not conflict with the swampland distance conjecture. In this case the gravitino mass is $m_{3/2} = |f_0| u^{3/2}/(2^{7/2} s^{1/2} t^{3/2})$, which is below 100 TeV when $|f_0| \lesssim 10^{-6}$ and $s \sim t \sim 100$, $u \sim 0.1$.
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On the paper's own terms, the central claim is that replacing metric gravity by Einstein-Cartan gravity changes the status of Weyl symmetry. In metric gravity a local scale (Weyl) invariance is destroyed by the Weyl anomaly, and a local Weyl-invariant action is essentially the Weyl-squared theory with ghosts. In the EC formulation the vector part of the torsion, $v_\mu$, transforms as a Weyl gauge field, making curvature invariants Weyl-covariant and allowing a Weyl-invariant, ghost-free quadratic action $S_{EC,4} = \int \det e\,(\frac{1}{f^2}F^2 + \frac{1}{\tilde f^2}\tilde F^2 + \frac{2}{f_m^2}F\tilde F)$. Coupled to the Standard Model, after gauge-fixing the Weyl symmetry, for instance by setting $\chi=M_P$, one is left with the graviton, the Standard Model fields, and one extra scalar, the field $a$, an axion-like particle. The paper's conclusions state that this ALP has all properties needed to solve the strong CP problem, and that the smallness of the cosmological constant and of the ALP, Higgs, and heavy neutral lepton masses results from tiny values of the dimensionless Lorentz gauge couplings $f,\tilde f,f_m$. The paper also claims that the EC formulation permits anomaly-free quantum Weyl-invariant theories, while acknowledging that the full quantum theory remains to be constructed.
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The central claim is that a would-be axion arising from Einstein-Cartan gravity cannot solve the strong CP problem if its only matter interaction is the derivative coupling to the axial fermionic current, because this operator is exactly invariant under $\varphi \to \varphi + c$. The apparent topological coupling $\frac{\alpha}{4\pi} \varphi \operatorname{Tr} G_{\mu\nu} \tilde{G}^{\mu\nu}$ is obtained by integrating the derivative coupling by parts and using the chiral anomaly, but the same divergence also contains the fermion-mass term $2i \bar{\Psi} m_\Psi \gamma_5 \Psi$, and these two pieces cancel in the effective potential, as required by the unbroken shift symmetry. Existing gravitational-axion constructions rely on this invalid step, identifying the would-be topological coupling with the true axion coupling while neglecting the fermion-mass contribution. The no-go is evaded if the coupling to fermions explicitly breaks the shift symmetry, as in the general Einstein-Cartan Lagrangian with $\varphi$-dependent $Z(\varphi)$ functions; then loop corrections can generate Yukawa-type couplings, but naive power counting requires severe fine-tuning to keep gravitational contributions below $\Lambda_{\mathrm{QCD}}^4$, while a scale-invariant regularization renders the couplings too weak. Weyl-invariant Einstein-Cartan gravity is the remaining viable route: quantum Weyl invariance forces evanescent $(D-4)$-suppressed operators that, combined with loop poles, generate the needed axion couplings with uncorrelated coefficients.
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The central claim is that $\theta$ is not a quantum number that can be coherently superposed but a superselection sector label of QCD. The relation $\langle \theta' | O | \theta \rangle = 0$ for every gauge-invariant operator $O$ and $\theta' \neq \theta$ makes different $\theta$ vacua dynamically disconnected and unobservably distinct, which is the step the recent superposition argument omitted. Because parity sends $|n\rangle$ to $|-n\rangle$ in the winding-number expansion $|\theta\rangle = \sum_n e^{in\theta} |n\rangle$, only $\theta = 0$ and $\theta = \pi$ are parity eigenstates. A parity-symmetric universe must therefore occupy one of those sectors, and once parity and CP break spontaneously, the strong CP phase $\bar{\theta}$ is generated only by $\mathrm{Arg}\,\mathrm{Det}\,M$. In the minimal left-right symmetric model the radiative contribution to $\bar{\theta}$ is tied to leptonic Yukawa phases, giving the sharp prediction that $\sin(\delta_{CP})$ must be near zero.
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The central claim is that the $\theta$-term of QCD is a direct consequence of a global chiral transformation, provided the fermion kinetic term is modified by a non-Abelian Berry connection $A_\mu$ associated with slow variations of the gauge background. With $\alpha(x)=\theta$, Eq. (6) of the paper gives the effective action of QCD together with the topological term $-\frac{g^2}{16\pi^2}\theta\,\mathrm{Tr}(F_{\mu\nu}\tilde F^{\mu\nu})$, so the vacuum angle appears without being inserted by hand. The functional Berry phase $\Delta\alpha=i\int dx^\mu\,\gamma_5 A_\mu$ accumulated from the adiabatic gauge condition is then not part of the $\theta$-term calculation; instead, it tells how the physical state space must be defined. States are sections parallel transported by $A_\mu$, and the Hilbert space acquires a fibration that carries topological information about the trajectory in configuration space.
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The paper claims that the modular-invariance mechanism for solving the strong CP problem can be extended to the effective field theories suggested by string compactifications, where light quarks carry mostly positive modular weights and the gauge kinetic functions $f_a$ are non-trivial functions of moduli. The physical combination is $A(S,\tau) = e^{-8\pi^2 f_3(S,\tau)} \det Y_u(\tau) \det Y_d(\tau)$, which transforms with modular weight $k_A = 3(k_{H_u}+k_{H_d})$. If the Higgs weights satisfy $k_{H_u}+k_{H_d}=0$ and $A$ is holomorphic in the closure of the fundamental domain, including the cusp $\tau=i\infty$, then $A$ is independent of $\tau$; assuming the constant is real and positive gives $\bar\theta = 0$. The paper argues that the required holomorphicity is consistent with string expectations if the only singularity is the decompactification limit $\tau=i\infty$, so the quark determinant must vanish there like $\Delta^m$ and the gauge factor must have the matching pole $\Delta^{-m}$, with $\Delta$ the modular discriminant. With this structure, spontaneous CP breaking by the vacuum value of $\tau$ produces the CKM phase while $\bar\theta$ remains zero, and the paper demonstrates the mechanism in an explicit model with quark modular weights $(2,4,6)$ whose Yukawa matrices are built from Eisenstein series and have determinant proportional to $\Delta^2$ for each of $Y_u$ and $Y_d$.
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The paper's central claim is that the 'WIMP quality problem' — the puzzle of why R-parity violation is so small — can be solved by the same discrete R-symmetries that solve the mu problem and the strong CP problem. With anomaly-free discrete R-symmetries $Z^R_n$ for $n=4,6,8,12,24$ and two gauge-singlet fields $X,Y$ whose VEVs sit near $f_a \sim 10^{11}$ GeV, the dangerous renormalizable RPV terms are forbidden at tree level but regenerated by Planck-suppressed operators, so the LSP decays with couplings $\lambda \sim (f_a/m_P)^k$. For a subset of models $k=1$, giving $\lambda \sim 10^{-7}$ and lifetimes $10^{-3}$-$10$ s, the LSP is gone before BBN and the dark matter is all axions. For other models $k\ge 3$, the LSP is stable on cosmological timescales. The paper then turns the LHC's null results into an R-parity-quality bound: if higgsinos are light as naturalness requires, the absence of prompt or displaced RPV signals implies $\lambda \lesssim 10^{-4}$-$10^{-2}$ for nearly all trilinear couplings.
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The paper's central claim is that a single confining SU(N_c) theory can supply both the QCD axion and, through its flavor dynamics, the Standard Model gauge group, with a Peccei–Quinn symmetry that is accidentally protected to very high order. Before confinement, the fermion content is vector-like under SU(N_c) but chiral under the weakly gauged SO(6)×SO(4)×Sp(10), which forbids dimension-three mass terms. After condensation the quark bilinears break SU(10)_L×SU(10)_R to SU(10)_V, break the weakly gauged group to U(3)×U(2) ⊃ SU(3)_c×SU(2)_L×U(1)_Y, and spontaneously break the accidental U(1)_PQ, producing the QCD axion among the Nambu–Goldstone bosons. The key quantitative assertion is that the only PQ-violating operator that can obtain a vacuum expectation value has dimension 12; Eq. (10) then gives a residual θ_eff that remains below the neutron EDM bound $10^{-10}$ as long as f_a ≲ $10^{11}$ GeV, so axion dark matter from misalignment is allowed. The same construction fixes the electromagnetic charges of the exotic fermions and predicts the anomaly ratio E/N = −7/3, giving a relatively large axion–photon coupling, and can unify into SO(10) near 2×$10^{16}$ GeV.
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On the paper's own terms, the discovery is that $\theta$ is a property of the state, not of the laws. The QCD Hamiltonian in temporal gauge ($A_0=0$) already preserves CP, yet it possesses energy eigenstates—the $\theta$ vacua—that violate CP, just as a rotationally invariant hydrogen Hamiltonian has non-spherical excited states. The $\theta$ term in the Lagrangian appears only when one constructs a path integral that evolves one particular $\theta$ sector, and that path integral is a solution to the Hamiltonian time evolution, not the definition of the theory. Consequently, imposing parity or CP as a symmetry adds nothing, and fine-tuning a Lagrangian parameter cannot cancel a random measurement outcome. The paper concludes that dynamical cancellation, by the axion or by an $\eta'$ acting as an axion when a quark is massless, is the only genuine solution.
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On the paper's own terms, the central discovery is that the coefficient combination $2L_6-L_4$, which controls the strange-mass-dependent shift of the up-quark mass, is unsuppressed in the $F=N=3$ AMSB supersymmetric QCD theory. Through Eq. (11), $\mu_u/\mu_d \simeq (16B_0 \mu_s/F_0^2)(2L_6-L_4)$, the computed low-energy constants give Eq. (27): the ratio scales as $(7\alpha/18)(\mu_s/m)$, an order-one quantity for $\alpha\sim 1$ and $\mu_s/m$ not too small. This contrasts with the $F<N$ case, where the ratio is parametrically suppressed in $1/N$ and vanishes at large $N$. The authors emphasize that numerical values obtained by extrapolating to $m\sim\Lambda$ lie outside the rigorous validity of their calculation, but with $\alpha\sim 2$ and $\mu_s/m\sim 0.5$ the extrapolated ratio is $\simeq 0.4$, large enough to account for the entire physical up-quark mass and render the QCD $\theta$ angle unphysical.
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The paper's central claim is that the Peccei-Quinn symmetry does not have to be imposed or protected by ad hoc discrete symmetries: in a simplified Pati-Salam model with gauge group $G_{\mathrm{PS}} \times SU(3)_{f_R}$ and the field content of Table 1, the renormalizable Lagrangian has an exact accidental $U(1)_{\mathrm{PQ}}$, and the same gauge structure forbids the higher-dimensional operators that would break it through the large VEVs of $\chi$, $\Phi$, $\Sigma$ and $\Delta$. The model is reported to reproduce the SM flavour structure while predicting three axion mass windows, the high-quality post-inflationary case requiring $m_a \gtrsim 0.01\ \mathrm{eV}$. The anomalon fermions that cancel the $SU(3)_{f_R}$ anomaly are massless at the renormalizable level; their masses are generated by non-renormalizable operators, making them parametrically light---sub-eV dark radiation in the high-quality regime, or keV-scale dark matter for intermediate axion decay constants.
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The central claim is that a non-invertible symmetry, not an axion, is what makes the QCD vacuum angle unobservably small. In a two-Higgs-doublet extension of the Standard Model with CP assumed exact at high energies and spontaneously broken at an intermediate scale, the authors assign $\widetilde{Z}_5\times\widetilde{Z}_3\times\widetilde{Z}_3$ charges to the three left-handed quark doublets, the right-handed up and down singlets, and the two Higgs doublets. The resulting Yukawa structure has an effectively diagonal up-type matrix—off-diagonal entries are suppressed by the mass hierarchy—and a down-type matrix with three zero entries: the (1,1), (1,3) and (3,1) elements are forbidden, while the (2,2) entry is the only complex one. The determinant of $M_u M_d$ is then real at tree level, giving $\bar{\theta}=0$, and the paper estimates that two-loop vertex corrections induce $\Delta\bar{\theta}\sim 10^{-15}$. The same down-type texture is shown to reproduce the observed quark masses and the full quark mixing matrix, including its CP-violating phase, and the light Standard-Model-like Higgs is a combination of the two doublets with the orthogonal heavy doublet at the ten-TeV scale.
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The central claim is that a gauge theory CP violating phase can be relaxed in discrete steps to very small values because it includes the magnetic dual of a 4-form flux, which discharges by the nucleation of membranes. Inside the bubbles surrounded by the membranes, the total CP violating phase is reduced. When the bubbles are produced rapidly during radiation domination in the early universe, near the chiral symmetry breaking scale, they will collide and percolate, melting away into gauge theory radiation and dramatically relaxing CP violation.
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The central claim is that the chiral quark condensate of QCD, whose phase is the dynamical degree of freedom for the η′ meson, supports 2π-winding domain walls bounded by cosmic strings in two flavor directions. In the η′ direction, the phases of light-quark condensates wind with the same sign; in the π0 direction, they wind oppositely ($n_{\theta_u} = -n_{\theta_d}$). These defects arise in pure QCD, without a hidden axion, because the η′ meson is an axion-like field that cancels the θ-angle in the massless-quark limit, albeit with poor quality once quark masses are included. In hidden-axion theories, the paper shows that the minimal string winds the quark-condensate phase in the opposite sense to the Peccei-Quinn phase, so the localized fermion zero modes have opposite chiralities and the string is anomaly-free. The paper also argues that early-universe scenarios with an enhanced QCD scale can make η′ and pion string-wall systems cosmologically dominant, even when a Peccei-Quinn axion is present.
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The paper's central claim is that the hybrid-symmetric frozen-spin storage ring, using radial electric bending at about 4.4 MV/m and alternating-gradient magnetic focusing in a 24-period, roughly 800-meter lattice, can reach a proton EDM sensitivity of $10^{-29}\,e\cdot\mathrm{cm}$ in Phase I using currently available technology. It argues that this design eliminates the dominant systematic error, the average vertical electric field, through simultaneous clockwise and counterclockwise beam storage, and suppresses the radial magnetic field error through spin-based harmonic alignment up to harmonic 24. If correct, this would improve indirect proton EDM limits by about $10^4$ and give the experiment a clear path, in a later phase, toward testing the Standard Model prediction of a proton EDM.
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The central claim is that the discrete R-symmetry $Z_n^R$ controls both the axion and the fate of the lightest supersymmetric particle. When the MSSM is augmented by two Peccei-Quinn charged fields $X$ and $Y$, the lowest-order R-parity-violating superpotential operator receives a suppression $(f_a/m_P)^N$. The paper finds $N=1$ for $Z_4^R$ and $Z_8^R$, giving induced RPV couplings $\lambda \sim 10^{-7}$, and $N=3$ for $Z_6^R$, $Z_{12}^R$ and $Z_{24}^R$, giving $\lambda \sim 10^{-21}$. With $N=1$, the thermally produced higgsino-like LSP decays with a lifetime of order $10^{-3}$ to $10$ seconds, before big bang nucleosynthesis concludes. The authors therefore conclude that the $Z_4^R$ and $Z_8^R$ versions of the model leave a universe with all axion cold dark matter and no WIMPs, in agreement with LZ-2024.
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On the paper's own terms, the discovery is that $\theta$-dependence in hadronic correlators is carried entirely by localized zero modes of the Dirac operator. Because the density of these modes falls like $1/\sqrt{V}$, the chance of finding one inside a nucleon's interaction volume tends to zero as the volume grows, so the CP-odd part of the nucleon correlator vanishes and the neutron electric dipole moment goes as $\sqrt{\chi_t/V}\,|\theta| \to 0$. Strong CP is therefore conserved by QCD dynamics alone, and the experimental upper bound on $|d_n|$ does not force $\theta$ to be small. In the axion extension, integrating out the constant mode of the axion field restricts the path integral to topological charge $Q=0$, giving $\chi_t=0$ and removing the anomaly and the mass mechanism for the $\eta'$ and for the axion itself.
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The central claim is that the order of limits proposed in recent work—infinite volume taken before the sum over topological sectors, Eq. (2)—when applied to the quantum rotor produces a zero topological susceptibility, while the exact spectrum $E_n = \frac{1}{2I}\left(n - \frac{\theta}{2\pi}\right)^2$ gives $\chi_t = d^2E_0/d\theta^2|_{\theta=0} = 1/(4\pi^2 I)$, and lattice simulations agree with the conventional order of limits. Because the sector distribution $p_T(Q)$ is Gaussian with width proportional to $\sqrt{T}$, the proposed double limit is dominated by the $Q = 0$ sector and yields zero for any finite sector cutoff $N$. The simulations, run very close to the continuum thanks to the winding algorithm, extrapolate to the exact value, and they also confirm the linear θ-dependence of the first excited level, $\Delta E_1 = \frac{1}{2I}\left(1 - \frac{\theta}{\pi}\right)$. The paper therefore asserts that the proposal, which would make θ disappear from all observables and remove the strong CP problem without new physics, fails already in the simplest theory that shares the essential features of topology and a θ term.
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The central claim is that the strong CP problem disappears if CP is identified with a modular symmetry that is spontaneously broken only by the real part of the modulus tau. The quark mass matrix M_q then has entries built from real coefficients times modular forms of positive weight; supersymmetry guarantees that no conjugated fields or negative powers appear. The determinant det M_q transforms as a modular form whose weight is the QCD modular anomaly coefficient A = sum_i (2k_Q_i + k_U_i + k_D_i) + k_Hu + k_Hd. Requiring the Higgs doublets to have k_Hu + k_Hd = 0 makes this weight zero, so det M_q is a modular form of weight zero and therefore a constant; CP then forces it real, giving theta_QCD = 0. The CKM phase, by contrast, is controlled by the relative phase of the Eisenstein series E_4 and E_6 and is generically of order unity. The paper presents a concrete three-generation model with modular weights {-6, 0, +6} that reproduces the observed quark mass hierarchies and mixing angles.
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On the paper's own terms, the central discovery is that a global $U(1)_{\rm PQ}$ under which the vector-like quarks $\Psi$ are chirally charged does double duty: it is the symmetry whose spontaneous breaking yields the axion, and it forbids Majorana neutrino mass terms so neutrinos are forced to be Dirac. The charge assignments leave a residual $\mathbb{Z}_3$ under which leptons and scalar leptoquarks rotate by different powers, which selects the dimension-five Dirac operator $(\bar\ell_L \tilde\Phi \nu_R)\sigma^*$ while blocking all Majorana operators. The one-loop diagram with $\Psi$, $\eta$, and $\chi$ then gives the neutrino mass matrix of Eq. (9), and the paper shows that the anomaly factors $N=1,2,3$ and the electromagnetic anomaly ratio $E/N$ vary across the seven viable vector-like quark representations, producing distinct, experimentally distinguishable axion-to-photon couplings. In three of the models, heavy-light quark mixing is large enough to induce flavor-violating axion-quark couplings, which are constrained by rare decays, meson mixing, and top decays. The axion can also account for the dark-matter abundance, with the post-inflationary case selecting $f_a$ in the range preferred by string-network simulations for the $N_{\rm DW}=1$ models.
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The paper's central claim is that a gauged, anomaly-free axial U(1)_a symmetry can turn the Peccei-Quinn symmetry into an accidental global symmetry of two decoupled sectors, so that quantum gravity respects the gauge symmetry while breaking the PQ symmetry only through high-dimensional operators. For the vector-like quark models I and II, the leading dangerous operator is $S^{{3n}}$T^* (or $S^{{mn}}$T^*) suppressed by $M_Pl^{{3n-3}}$; choosing n at least 4 (or m > 3) leaves the induced $\theta$ shift many orders of magnitude below $10^{-10}$. In the SO(10) model the leading operator is $T^{{12}}$S^*/$M_Pl^{9}$, which keeps $\theta$ at or below $10^{-10}$ for f_a up to 6.96 x $10^{11}$ GeV; the axion couplings to electron and nucleon there interpolate between KSVZ and DFSZ values, with E/N = 8/3 and a positive electron coupling that can distinguish the model. All three models have domain wall number one, which avoids the cosmological domain-wall problem.
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The paper's central claim is that in a concrete realization of the modified BBP model with a complex scalar $\varphi$ carrying a $Z_{4n}$ charge, the phase $a \equiv f_a \theta_{\varphi}$ is a light Nelson-Barr axion whose total potential is $V_a = \chi(T)(1-\cos(k a/f_a)) + (m_a^2 f_a^2/(16 n^2))(1-\cos(4n a/f_a))$. The tree-level $Z_{4n}$ term produces the $4n$ vacua and the associated string-wall network, while the QCD-induced term is a potential bias that makes the CP-conserving vacuum $a=0$ the unique lowest-energy state when $k$ and $4n$ are coprime. The paper derives the collapse temperature $T_{\rm dec}$ by balancing the wall tension against this QCD pressure and then evaluates the axion and gravitational wave yields from the collapse. Depending on the cutoff scale, the collapse can account for the observed dark matter abundance as Nelson-Barr axions, or it can generate a gravitational wave background whose peak falls in the nHz band and matches the 15-year pulsar timing array data. The strong CP angle stays protected because the quark mass matrix has real determinant at tree level and the QCD potential, even when subdominant, still selects the CP-conserving minimum.
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The central claim is that the generating functional (5), in which all integrated fields obey the perturbative in/out (emission) boundary conditions, contains only perturbative contributions. Since the incoming fields contain only positive frequencies and outgoing fields only negative frequencies, the fields oscillate at time infinities in Minkowski space and decrease in the Euclidean continuation, so total derivatives in the Lagrangian integrate to zero. Expanding the functional integral at the points $g=-0$, $g=i0$, and $g=i^{2/x}0$ gives finite coefficients, which the paper takes as excluding non-perturbative terms of the form $\exp(-1/g)$, $\exp(-1/g^2)$, and $\exp(-1/g^x)$. The paper concludes that instantons are absent from the integration region, the strong CP problem is resolved without axions, and non-perturbative condensates used in QCD sum rules can in principle be generated by summing the full perturbative series.
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The review's central claim, on its own terms, is that the axion search program is about to make a qualitative leap: the diversity of experimental strategies-resonant cavities, solar helioscopes, laboratory light-shining-through-wall setups, and astrophysical probes-now covers complementary patches of the $(m_a, g_{a\gamma})$ plane, and the projects now under construction or in an advanced planning stage are expected to close much of the gap between current laboratory sensitivity and the QCD axion band. The paper summarizes the current best limits: ADMX has reached the standard QCD band at masses around a few $\mu$eV, a separate cavity run has probed the DFSZ line near 4.55 $\mu$eV, CAST has set the most restrictive direct limit on the solar axion-photon coupling, and ALPS II has completed its first run with full optics expected in 2025. It argues that if these instruments perform as designed, axion or ALP discovery, or at least the exclusion of large parts of the theoretically motivated parameter space, becomes plausible within this decade.
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The central discovery is that in the BχF construction the exact global U(1) symmetries of the five-dimensional theory do not force a massless boson or forbid quark condensation. Rather, the instanton sum in $N_f=1$ QCD acquires extra factors from bulk zero modes $Q$ that are localized far from the physical boundary, making the full sum invariant under the exact $U(1)\times U(1)$ symmetry while the sub-sum contributing to boundary-only observables reduces to a Bessel function that gives the $\eta'$ a mass. The theory's exact axial $U(1)$ is spontaneously broken and realized nonlinearly as a shift symmetry of the $\eta'$, but because the $\eta'$'s zero-momentum mode is an isolated zeromode of a nonlocal effective action, no Goldstone boson appears. The same $Q$ modes dynamically set the topological charge of all accessible gauge configurations to zero, so the strong CP phase can be rotated away and no strong CP violation occurs on the boundary.
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The central claim is that a single bulk axion in flat or Randall-Sundrum extra dimensions, with QCD localized on the infrared brane and QCD as the only source of Peccei-Quinn breaking, can realize only one viable pattern: one zero-mode axion on the canonical QCD line plus a plateau of heavier KK modes. The non-canonical patterns, in which several KK modes share the solution to the strong CP problem and sit at $g_i \gg 1$, are in severe tension with the combined constraints from fifth-force searches, astrophysics, and perturbative unitarity. The proof relies on the KK axion wavefunctions being universal and matching the graviton wavefunctions, so that bounds on the lightest massive graviton directly constrain the axion tower; it holds in both flat and warped geometries and, as the paper shows, also when the number of extra dimensions is increased or the bulk VEV profile is changed.
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The central claim is that at $\theta=\pi$, where CP violation is maximal, the quark matter equation of state in the three-flavor NJL model with vector coupling $G_v=0.1\,G_s$ supports hybrid neutron stars with a maximum mass of $2.05\,M_\odot$, in agreement with the NICER measurement of PSR J0740+6620 and the GW170817 tidal constraints. The star has a pure quark matter core of radius $\sim 1.6$ km, quark matter in a mixed phase out to $\sim 6.5$ km, and a hyperonic outer core; without the vector term the same $\theta=\pi$ equation of state gives only $1.84\,M_\odot$ and fails the two-solar-mass constraint. The $\theta=\pi$ case also lowers the onset of the mixed phase from about $0.44$ fm$^{-3}$ (at $\theta=0$) to $0.35$ fm$^{-3}$ for $G_v=0.1\,G_s$. For the maximum-mass hybrid star, the paper computes an $f$-mode frequency near $2.1$ kHz, about $110$ Hz above a nucleonic star of the same mass, with hyperons and the enlarged quark core contributing separately to the shift.
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The paper's central claim is that the $\theta$-term nEDM is now a lattice-QCD-determined quantity rather than an unknown: $d_n = -0.00148(14)(31) \bar{\theta} \, e\cdot\mathrm{fm}$ and $d_p = 0.0038(11)(8) \bar{\theta} \, e\cdot\mathrm{fm}$, with the neutron value at roughly 10 statistical $\sigma$. This is attributed to three improvements working together: overlap fermions preserve the anomalous Ward identity at finite lattice spacing so the EDM vanishes in the chiral limit; cluster-decomposition error reduction (CDER) suppresses the topological-charge noise that plagued earlier attempts; and partially quenched data supply a controlled chiral interpolation with valence and sea pion-mass dependence of opposite sign, as predicted by partially quenched chiral perturbation theory. The review further reports the CP-violating pion-nucleon coupling $\bar{g}_0 = -0.01713(62) \bar{\theta}$ from the nucleon pseudoscalar form factor, which agrees with the baryon-spectrum value $-15.5(2.0)(1.6)\times 10^{-3} \bar{\theta}$. The consequence is that the strong-CP contribution to the neutron EDM is small compared with current experimental limits, so the experimental search remains a discovery channel for beyond-Standard-Model CP violation.
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The central claim is that the Peccei-Quinn symmetry should be viewed as a flavor symmetry that can be merged with baryon and lepton number, not as an isolated accidental symmetry. In the derivative representation of the axion Lagrangian, the apparent local anomalous couplings to chiral gauge bosons are spurious: they cancel exactly against triangle-graph anomalies, so the physical axion couplings are the same as in the linear or polar representations. Once this equivalence is accepted, one can add couplings such as the seesaw operator φ ν_R^c ν_R to the KSVZ model, and the axion becomes entangled with lepton number. The paper states a counting rule, from the literature, that at most two independent B and/or L combinations can be broken before the axion becomes massive, and it exhibits explicit leptoquark/diquark constructions where the axion is tied to neutrino masses, spontaneous proton decay, and neutron-antineutron oscillations.
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Within the Nelson-Barr framework, a scalar field remains naturally light and, when it constitutes the dark matter, produces periodic time variation in the CKM matrix elements; this variation supplies new experimental signatures accessible to quantum sensors, including nuclear clocks, that go beyond standard axion phenomenology.
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The load-bearing claim is Eq. (122): the coupling of a compact scalar to gauge fields takes the form $S = (k_F/8π^2)∫θ F∧F + (k_G/8π^2)∫θ tr(G∧G)$ with $k_F, k_G$ integers. This is not an assumption about a UV model but a consistency condition on any theory in which $θ$ is compact, because $θ ↦ θ + 2π$ must leave the path integral invariant and the topological charge density integrates to an integer in units of $8π^2$ (the instanton number for $SU(N)$, the squared-flux integral for $U(1)$). Two corollaries follow and are used throughout. First, apparent exceptions must be mass-suppressed: the axion–photon coupling generated by axion–pion mixing is really a $(□θ)F∧F$ operator, allowed precisely because it scales with $m_a^2$. Second, the coefficient $N$ in front of $θ tr(G∧G)$ is what QCD dynamics minimizes against, so the axion potential is $V_{QCD}(θ) = -m_π^2 f_π^2√(1 - 4m_u m_d/(m_u+m_d)^2 sin^2(Nθ/2))$, minimized at the CP-conserving value, with $m_a ≈ 6 μeV × (10^{12} GeV/(f/N))$.