Topological--Scattering Solution of Strong CP Problem and Axion in Unified Matrix--QCA Universe

TopologicalScattering Solution of Strong CP Problem and Axion in Unied MatrixQCA Universe Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract Quantum chromodynamics (QCD) in its most general form admits a topological θ -term θ g2 s(32π2)−1Ga µν ˜ Ga,µν , which violates P , T and CP . After chiral eld rede- nitions, the physically observable strong CP angle is ¯ θ=θ−arg det(YuYd) , where Yu and Yd are the upand down-type Yukawa matrices. Naturalness suggests ¯ θ=O(1) , while neutron electric dipole moment (nEDM) bounds |dn|≲1.8×10−26 e·cm imply |¯ θ|≲10−10 , constituting the strong CP problem. PecceiQuinn (PQ) theory promotes ¯ θ to a dynamical vacuum expectation value of an axion eld. The axion potential is determined by the QCD topological susceptibility χtop , with m2 af2 a=χtop and V(a)≃χtop[1 −cos(a/fa−¯ θ0)] . Lattice QCD and chiral eective theory now determine χtop(T) with high precision, xing the QCD axion masscoupling relation. However, in a broader unied description of the Universe, the origin and robustness of PQ symmetry against gravity, ultraviolet physics and global consistency conditions remain unclear. Within the unied time-scale, boundary time geometry, matrix universe THEMATRIX and quantum cellular automaton (QCA) universe framework, this work gives a topologicalscattering solution of the strong CP problem. The main ideas are:1. Introduce a parameter space X◦ of all low-energy couplings (gauge couplings, θ -angles, Yukawa phases, light scalar parameters) and an extended space Y=M× X◦ , where M is spacetime. From the family of scattering matrices S(ω;λ) on a channel Hilbert space, construct a determinant line bundle Ldet →Y and its square root L1/2 det . The obstruction to a global smooth square root is encoded in a relative cohomology class [K]∈H2(Y, ∂Y ;Z2) , which has the physical meaning of the global Z2 holonomy of the "square-root scattering determinant" pdetpS along parameter loops. 2. Using the previously developed NullModular double cover and restricted unitary bundle framework, one has an equivalence between: (i) local Einstein equations with appropriate quantum energy conditions, (ii) small causal diamond generalized entropy extremality and modular ow consistency, and (iii) vanishing of the obstruction class, [K]=0 . Thus, in any Universe admitting a globally consistent boundary time geometry and semiclassical gravity, allowed physical sectors must satisfy [K] = 0 . 3. Embed QCD and its θ -angle into this unied structure by identifying the QCD sector contribution [KQCD] of [K] . The physical strong CP angle ¯ θ=θ− 1 arg det(YuYd) reappears as the phase holonomy of pdetpSQCD along loops in X◦ . One shows that [KQCD] = 0 is equivalent to the triviality (modulo 2π ) of all such holonomies; in particular, in any physically realized sector compatible with [K]=0 one has ¯ θeff ≈0 without requiring a vanishing quark mass. 4. In the matrix universe representation, the global Universe is a gigantic but structured unitary matrix whose block-sparse pattern encodes causal relations and whose spectral data encode the unied time-scale. In this picture, ¯ θ is a "topological phase" of the QCD block of THE-MATRIX, and [K] = 0 requires that the squareroot determinant has trivial Z2 holonomy across all coupling loops. Strong CP is then rephrased as the requirement that such topological scattering invariants vanish globally. 5. In the QCA universe layer, one constructs an SU(3) gauge QCA with lattice topological charge Q∈Z . The QCD θ -term corresponds to a weight factor exp(iθQ) in the discrete path-sum. By imposing a "topologicalNullModular consistent QCA" condition that the total phase for all closed gauge-conguration histories be 2πZ , one obtains in the continuum limit a joint constraint on ¯ θ and possible gravitational θG -terms, thereby simultaneously suppressing strong and gravitational CP violation. 6. The PQ axion is reinterpreted as a relative cohomology modulus of [K] . The axion eld a(x)/fa parametrizes local rephasings of pdetpS along a U(1) ber of L1/2 det . Its eective action reproduces the standard form S[a]∼R[1 2f2 a(∂a)2+ χtop(1 −cos(a/fa−¯ θ0))]√−gd4x , where χtop is the QCD topological susceptibility determined from rst-principles QCD. The global condition [K]=0 then enforces ⟨a⟩/fa=¯ θ0 and ¯ θeff = 0 , giving a unied topologicalscattering reformulation of the PQ mechanism. Appendices present standard QCD derivations of ¯ θ , the precise construction of [K] from scattering theory, and explicit SU(3) gauge QCA models with discrete topological charge and θ -phase. The resulting picture treats strong CP as a consistency constraint of the full matrixQCA Universe rather than an independent ne-tuning of a low-energy parameter. Keywords: Strong CP problem; QCD axion; PecceiQuinn mechanism; Topological susceptibility; Scattering determinant line bundle; Matrix Universe; Quantum Cellular Automata (QCA); NullModular double cover 1 Introduction & Historical Context 1.1 The Strong CP Problem Four-dimensional SU(3) YangMills theory admits a topological term in its Lagrangian: Lθ=θg2 s 32π2Ga µν ˜ Ga,µν,˜ Ga,µν =1 2ϵµνρσGa ρσ. (1) The space-time integral of this term converts to the product of the topological charge Q∈ Z and the angle θ , explicitly violating P , T , and CP . For QCD with multiple generations of quarks, after introducing Yukawa matrices Yu, Yd , considering chiral redenitions and anomaly eects, the physically observable strong CP angle is ¯ θ=θ−arg det(YuYd), (2) 2 which naturally takes values in [0,2π) . Without additional symmetries or dynamical mechanisms, ¯ θ would be expected to be O(1) . However, ¯ θ= 0 induces a neutron electric dipole moment (nEDM). QCD sum rules and chiral eective theory give dn≃cn¯ θ e ·cm, cn∼10−16. (3) Current nEDM experiments provide an upper bound |dn|<1.8×10−26 e·cm , implying |¯ θ|≲10−10. (4) This is the strong CP problem: why an angle that should naturally be O(1) is ne-tuned to the order of 10−10 . 1.2 Traditional Solutions and the QCD Axion Traditional solutions to the strong CP problem generally fall into a few categories: 1. **Massless Up Quark:** If mu= 0 , ¯ θ can be removed via chiral rotation. Lattice QCD has essentially ruled out this possibility. 2. **Spontaneous P or CP Breaking:** P or CP is a fundamental symmetry broken spontaneously at a high energy scale, with the vacuum choosing ¯ θ= 0 . Such models need to address the nEDM induced after spontaneous CP breaking and the characterization of discrete symmetries in quantum gravity. 3. **PecceiQuinn Mechanism:** Introduces a global U(1)PQ symmetry and a scalar eld. Its anomaly structure makes the QCD vacuum energy dependent on an angular eld a/fa , automatically adjusting ¯ θeff = 0 at the minimum of the vacuum energy. The corresponding approximate Goldstone mode is the QCD axion. While the original PQ model is ruled out, "invisible axion" models like KSVZ and DFSZ remain mainstream solutions. In the PQ framework, the axion eective potential is determined by the QCD topological susceptibility χtop : V(a)≃χtop1−cos(a/fa−¯ θ0), m2 af2 a=χtop. (5) Lattice QCD and chiral eective theory have provided high-precision results for χtop(T) , precisely determining the relationship between QCD axion mass and coupling constant. 1.3 Unied Universe Framework and Problem Restatement The above solutions mostly address the strong CP problem at the level of low-energy eective eld theory, without embedding it into a unied framework containing gravity, causal structure, boundary time geometry, and global consistency of the Universe. Furthermore, quantum gravity and black hole thermodynamics suggest that global symmetries might be violated in quantum gravity, questioning the stability of the global U(1)PQ in the traditional PQ mechanism. Previous works introduced the multi-layer objects of unied time scale, boundary time geometry, Matrix Universe THE-MATRIX, and QCA Universe, characterizing the Universe as a single structure highly consistent across scattering, geometry, modular ow, generalized entropy, and causal partial order. Key points include: 1. **Unied Time Scale Mother Formula:** κ(ω) = φ′(ω) π=ρrel(ω) = (2π)−1tr Q(ω), (6) 3 unifying scattering hemi-phase derivative, relative density of states, and WignerSmith group delay matrix trace into a single time density. 2. **Boundary Time Geometry and NullModular Double Cover:** Gluing modular ow parameters and scattering phases on the boundary of each causal diamond, constraining geometry, entropy, and scattering mutually. 3. **Matrix Universe and QCA Universe:** Viewing the entire cosmic evolution as a gigantic but sparse unitary matrix or reversible QCA, where block structure encodes causal partial order, spectral data implements time scale, and feedback loops carry Z2 topology. In this framework, the strong CP problem can be restated as: why is the QCD sector's contribution to the topologicalscattering invariants of the whole Universe suppressed to nearly zero? This paper will show that this suppression is no longer a "ne-tuning of low-energy constants" but a mandatory result of "unied universe topologicalscattering consistency". 2 Model & Assumptions This section presents the models and basic assumptions for the unied universe structure, parameter space, scattering determinant line bundle, and QCD sector embedding used in this paper. 2.1 Universe Object and Parameter Space Let the geometric layer of the Universe be described by a globally hyperbolic Lorentzian manifold (M, g) with causal partial order and appropriate boundary structure (including past/future innity and possible black hole horizons). On this basis, introduce several layers of the unied universe object: * Geometric layer Ugeo : (M, g, ≺) ; * Scattering layer Uscat : A family of scattering pairs and unied time scale κ(ω) ; * Boundary layer UBTG : Boundary algebra, modular ow, and BrownYork quasilocal energy; * Matrix layer Umat : Scattering matrix universe THE-MATRIX on channel Hilbert space; * QCA layer UQCA : SU(3) gauge QCA dened on a countable lattice; * Topological layer Utop : Relative cohomology class [K] and NullModular double cover. To introduce coupling parameters, dene an open parameter space X◦ whose coordinates include: 1. Gauge coupling constants gs, g, g′ , etc.; 2. Yukawa matrices and CKM/PMNS phase parameterizations; 3. QCD θ -angle, possible gravitational θG angle, and other topological term parameters; 4. Light scalars (including possible axions) and external macroscopic parameters. Dene the extended space Y=M×X◦, (7) whose boundary ∂Y includes spacetime boundaries and possible boundaries of the parameter space. 2.2 Scattering Matrix Family and Determinant Line Bundle Under appropriate locality and spectral assumptions, for each parameter point λ∈X◦ and energy ω , consider a scattering pair (H0, H(λ)) and its wave operators, yielding a 4 scattering matrix on the channel Hilbert space Hchan(ω) : S(ω;λ) : Hchan(ω)→ Hchan(ω). (8) Assume S(ω;λ)−1 is a trace-class operator of appropriate order, such that the modied determinant det pS(ω;λ) is dened and varies smoothly with (ω, λ) (under spectral gap and appropriate renormalization). From this family of unitary operators on the vector bundle Hchan over (ω, λ)∈Y , one can construct a complex line bundle Ldet →Y , whose ber is generated by the formal "determinant": Ldet|(ω,λ)∼ =C·det pS(ω;λ). (9) Locally, one can choose a logarithmic phase ϕ(ω, λ) such that det pS(ω;λ) = expiϕ(ω, λ). (10) Dene the square root line bundle L1/2 det as formally satisfying (L1/2 det )⊗2≃ Ldet. (11) Locally, one can choose pdet pS= exp( i 2ϕ) , but globally there may be a sign ambiguity. The Z2 twisting of this ambiguity is characterized by a relative cohomology class [K]∈H2(Y, ∂Y ;Z2). (12) [K]=0 if and only if there exists a global square root line bundle trivial on ∂Y . This construction can be seen as a scattering version generalization of Freed et al.'s work on determinant line bundles and cohomological obstructions. 2.3 NullModular Double Cover and Consistency Conditions Boundary time geometry equips the boundary ∂D of each small causal diamond D⊂M with: 1. Boundary observable algebra A∂(D) and its state ω∂(D) ; 2. Modular ow parameter s∈R and modular operator ∆is ; 3. Phase φD(ω) and time scale κD(ω) given by scattering matrix SD(ω) and WignerSmith group delay QD(ω) . The NullModular double cover glues the modular ow parameter and scattering phase via a Z2 cover, ensuring consistency between modular ow direction, time arrow, and phase single-valuedness, avoiding "half-cycle sign reversal" anomalies. Previous work showed that under assumptions of local Einstein equations, QNEC/QFC, and scattering modular consistency, the following conditions are equivalent: 1. Existence of a global NullModular double cover such that modular ow and scattering phase align smoothly on this cover for all causal diamonds; 2. Generalized entropy extremality conditions and non-negativity of second-order relative entropy on each causal diamond are equivalent to local gravitational eld equations; 3. The relative cohomology class of the determinant square root line bundle satises [K]=0 . Thus, for an acceptable physical universe sector, [K] = 0 is not optional but a consistency requirement. 5 2.4 QCD Sector and Embedding of ¯ θ In the QCD sector, consider SU(3) gauge elds Aa µ and six avors of quarks ψf . The Lagrangian includes LQCD =−1 4Ga µνGa,µν +¯ ψ(iγµDµ)ψ−(¯uLYuuR+¯ dLYddR+ h.c.) + θg2 s 32π2Ga µν ˜ Ga,µν. (13) Chiral redenition and Fujikawa anomaly analysis give the physical angle ¯ θ=θ−arg det(YuYd), (14) appearing in the path integral in the weight factor exp(i¯ θQ) . Viewing the QCD sector as a block of the Matrix Universe THE-MATRIX, for each parameter λ and energy ω , there is a scattering matrix SQCD(ω;λ) . The determinant and square root of this family on the parameter space dene a contribution [KQCD] to [K] . Its physical meaning is: along a closed loop in parameter space containing ¯ θ variation, does pdetpSQCD undergo an irremovable sign ip? 2.5 SU(3) Gauge QCA Universe Layer In the QCA layer, construct an SU(3) gauge QCA: lattice vertices carry matter Hilbert spaces, edges carry gauge link variables Ux,µ ∈SU(3) , and discrete time evolution is implemented by local gauge-covariant unitary gates UQCA . The discrete topological charge Q∈Z can be dened by summing Tr(UP˜ UP) over lattice points. In this setting, the QCD θ -term corresponds to adding a phase factor in the discrete path sum: Y n expiθQn, (15) or in the Heisenberg picture as UQCA(θ) = exp(iθˆ Q)UQCA(0) . TopologicalNullModular consistency requires that the total phase for any closed gauge conguration loop be 2πZ , which will be used to constrain ¯ θ . 3 Main Results (Theorems and Alignments) This section formalizes the main conclusions about the strong CP problem and axion within the unied MatrixQCA universe framework into several theorems and corollaries. Proofs are in the Appendix. 3.1 Theorem 1 (Determinant Square Root and Relative Cohomology Class) Theorem 3.1 (Determinant Square Root and Relative Cohomology Class) . Under the aforementioned scattering and parameter space assumptions, the scattering matrix family S(ω;λ) denes a complex line bundle Ldet →Y , generated by the modied determinant det pS(ω;λ) . There exists a natural relative cohomology class [K]∈H2(Y, ∂Y ;Z2), (16) 6 satisfying: 1. [K] = 0 if and only if there exists a square root line bundle L1/2 det trivial on ∂Y , such that (L1/2 det )⊗2∼ =Ldet ; 2. For any closed parameter loop γ⊂X◦ , the evaluation of [K] on the mapping torus M×S1 γ gives the Z2 holonomy of pdetpS along γ : if the holonomy is +1 , a smooth square root can be chosen on the loop; if −1 , there is an irremovable sign ip. 3.2 Theorem 2 (NullModular Consistency and [K] = 0 ) Theorem 3.2 (NullModular Consistency and [K]=0 ) . Assume the Universe satis- es: 1. M is globally hyperbolic, with appropriate boundary time geometry structure and modular ow; 2. Generalized entropy extremality and non-negativity of second-order relative entropy on small causal diamonds hold and are equivalent to local gravitational eld equations; 3. Boundary scattering matrices and modular ows can be locally aligned on a NullModular double cover. Then the following propositions are equivalent: 1. There exists a global NullModular double cover such that modular ow and scattering phase on all causal diamonds align smoothly on this cover without Z2 anomaly; 2. There exists a global square root line bundle L1/2 det trivial on ∂Y ; 3. [K]=0 . Thus, for any physically acceptable universe sector, [K] = 0 is a necessary condition. 3.3 Theorem 3 (QCD ¯ θ as ScatteringTopological Holonomy) Theorem 3.3 (QCD ¯ θ as ScatteringTopological Holonomy) . In the QCD sector, dene the physical angle ¯ θ(λ) = θ(λ)−arg detYu(λ)Yd(λ), (17) and consider a closed loop γ⊂X◦ in parameter space. Then: 1. The evaluation of [KQCD] on M×S1 γ equals the sign of exp(i∆γ¯ θ/2) , where ∆γ¯ θ is the total phase change along γ (modulo 2π ); 2. If ∆γ¯ θ≡0 (mod 2π) for all physically realizable loops γ , then [KQCD]=0 ; 3. Conversely, if there exists a loop where ∆γ¯ θ≡π(mod 2π) , then [KQCD]= 0 . Specically, in a local parameter neighborhood, [KQCD] = 0 requires that a gauge can be chosen such that ¯ θeff ≡0 (mod 2π) . 3.3.1 Corollary 3.1 (Strong CP Suppression in Unied Universe) Assume the Universe satises the assumptions of Theorem 2, and non-QCD sector contributions to [K] are constrained to zero by independent consistency conditions. Then [K] = 0 ⇒[KQCD] = 0 . Thus: 1. Physically realizable universe sectors must satisfy ¯ θeff ≡0 (mod 2π) ; 2. The experimentally observed |¯ θ|≲10−10 is interpreted as an extremely sensitive upper bound on possible residuals of [KQCD] , while the natural expectation is strictly zero. 3.4 Theorem 4 (Axion as Relative Cohomology Modulus of [K] ) Theorem 3.4 (Axion as Relative Cohomology Modulus of [K] ) . Assume [K] lifts to an integer coecient class ˜ K∈H2(Y, ∂Y ;Z) . Let the phase of the axion eld be ϕ(x) = expia(x)/fa∈U(1), (18) 7 and embed it into QCD via the coupling LaG ˜ G=a fa g2 s 32π2Ga µν ˜ Ga,µν. (19) Then: 1. The variation of ϕ can be viewed as a local rescaling of the square root line bundle L1/2 det , and ˜ K is the rst Chern class of this U(1) line bundle; 2. In low-energy eective theory, the axion potential can be written as V(a)≃χtop1−cos(a/fa−¯ θ0)+··· , (20) where χtop is the QCD topological susceptibility, and ¯ θ0 is the bare angle in the UV frame; 3. If the unied universe requires [K] = 0 , then globally one must have ⟨a⟩/fa=¯ θ0 , thus ¯ θeff =¯ θ0−⟨a⟩/fa= 0 . This gives a reinterpretation of the PQ mechanism in the unied topologicalscattering framework: the axion is the relative cohomology modulus that truly eliminates [K] , and its vacuum selection is enforced by [K] = 0 rather than being an extra freedom. 3.5 Theorem 5 ( θ Sector and [KQCD] in SU(3) Gauge QCA) Theorem 3.5 ( θ Sector and [KQCD] in SU(3) Gauge QCA) . In the SU(3) gauge QCA model, let the evolution operator for each time step be UQCA(θ) = expiθˆ QUQCA(0), (21) where ˆ Q is the discrete topological charge operator. Consider the discrete path space C composed of all possible gauge conguration paths. 1. For any closed path C , the total phase Φ(C) = θX n∈C Qn (22) has a sign (modulo 2π ) consistent with the evaluation of [KQCD] on the corresponding mapping torus; 2. If one requires Φ(C)∈2πZ for all physically realizable closed paths C , then the eective ¯ θeff ≡0 (mod 2π) in the discrete model; 3. In the continuum limit, this condition forces [KQCD]=0 in the QCD sector, compatible with the conclusions of Theorems 34. 4 Proofs This section outlines the proof frameworks for the main theorems. Details and rigorous constructions are in the Appendix. 4.1 Sketch of Proof for Theorem 1 Determinant line bundles and cohomological obstructions to square roots are standard topics. 1. For each (ω, λ) , x an orthonormal standard channel basis. The spectrum of S(ω;λ) can be written as {eiδj(ω,λ)}j . Dene local hemi-phase ϕ(ω, λ) = X j δj(ω, λ). (23) 8 2. On a good open cover {Uα} of Y , select continuous branches ϕα , dening local sections sα= exp(iϕα) as local trivializations of Ldet . 3. On intersections Uα∩Uβ , transition functions gαβ =sα/sβ take values in U(1) . Their projective square roots hαβ =√gαβ form candidate transition functions for L1/2 det . 4. Whether hαβ can be chosen to satisfy the cocycle condition hαβhβγhγα = 1 is determined by a Z2 -valued ech 2-cocycle, corresponding to a class in H2(Y;Z2) . 5. Adding the boundary condition that the square root is trivial on ∂Y yields the class [K] in relative cohomology H2(Y, ∂Y ;Z2) . Its vanishing is necessary and sucient for the existence of a global square root trivial on the boundary. Holonomy along a closed loop γ is determined by the net change ∆γϕ , with sign exp(i∆γϕ/2) , consistent with the evaluation of [K] on M×S1 γ . 4.2 Sketch of Proof for Theorem 2 NullModular double cover links boundary modular ow parameter s and scattering phase ϕ(ω, λ) such that for each causal diamond D : 1. Time translation generated by modular ow aligns with scattering time scale dened by κ(ω) on the boundary; 2. Second variation of generalized entropy and non-negativity of second-order relative entropy are equivalent to local Einstein equations. If [K]= 0 , there exist closed parametergeometry loops where the holonomy of pdetpS is −1 . This means on the NullModular double cover, one cannot continuously choose the relative sign of scattering phase and modular ow phase, leading to a "sign reversal" jump when gluing boundaries of certain causal diamonds. This destroys the unied variational principle of generalized entropygravity equations. Conversely, if [K]=0 , a smooth pdetpS can be chosen over the entire Y , corresponding one-to-one with modular ow parameters on boundaries, constructing an anomaly-free NullModular double cover. Consistency of modular ow, generalized entropy, and scattering phase ensures equivalence between the variational principle on small causal diamonds and Einstein equations. This yields the equivalence in Theorem 2. 4.3 Sketch of Proof for Theorem 3 and Corollary 3.1 In the QCD sector, the physical angle ¯ θ=θ−arg det(YuYd) (24) appears in the path integral weight exp(i¯ θQ) . Its contribution to the scattering matrix determinant phase can be viewed as det pSQCD ∼X Q P(Q) exp(i¯ θQ) det SQ, (25) where P(Q) is the topological sector weight. Along a closed loop γ in parameter space, ¯ θ may wind by integer multiples of 2π . Since topological charge Q∈Z , the total phase change is ∆γΦ = ∆γ¯ θ·⟨Q⟩γ+··· . (26) Considering modulo 2π , if ∆γ¯ θ≡0 (mod 2π) , a continuous square root can be chosen on the loop without sign ip. If ∆γ¯ θ≡π(mod 2π) , the square root must ip sign once, corresponding to [KQCD]= 0 . Thus, [KQCD]=0 is equivalent to ∆γ¯ θ≡0 (mod 2π) for all physically realizable loops. Assuming non-QCD sector contributions are suppressed 9