claims depot shelf
The Strong CP Problem
Formal claims (Lean)
Stated claims
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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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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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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 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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In all 2HDMs of the class where the flavor symmetry forces arg det(yu yd) = 0 at tree level, there are no one-loop contributions to θbar, independent of the mass of the second Higgs doublet, and all neutral scalar interactions conserve CP. This appears in Sec. 5.1 ("there are no one-loop contributions to θbar due to the exchange of a charged boson, regardless of the Abelian flavor symmetry imposed as long as it ensures arg det(yu yd) = 0 at tree-level") and App. C. If correct, the strong CP problem is solved radiatively in this class and the extra scalars can sit near 1 TeV, with FCNC bounds close to a TeV rather than 20 TeV.
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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 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 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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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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Theoretical constructions extend the window for the axion mass and couplings beyond conventional regions, and bounds from cosmology, astrophysics and experimental searches are updated to account for these models.
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The gravitational emergence of the local axion field associated with the superselection theta parameter is worked out using as basic ingredients the quantum representation of classical geometries in terms of coherent states and the interpretation of the theta QCD superselection rule in quantum information terms as the lack of a quantum reference frame for topological charge.