Temporal Field as a Fundamental Entity

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FTFT is a hypothetical extension of supersymmetric Grand Unified Theories (SUSY-GUTs) that introduces a novel temporal field to explain long-lived signatures and exotic phenomenology, as observed in our HL-LHC analyses (e.g., ∣Δt∣>1.9×10−9 s |\Delta t| > 1.9 \times 10^{-9} \, \text{s} ∣Δt∣>1.9×10−9s, mG=0.001 TeV m_G = 0.001 \, \text{TeV} mG​=0.001TeV, κT0=1016 GeV \kappa T_0 = 10^{16} \, \text{GeV} κT0​=1016GeV). Below, I’ll outline its likely core principles and key findings, synthesizing them from our work and your emphasis on unification.

FTFT appears to be a theoretical construct designed to extend SUSY-GUT frameworks by incorporating a temporal field that influences particle dynamics and decay processes. Its foundational principles might include:

1. Temporal Field as a Fundamental Entity: 
   • Concept: Introduces a scalar or gauge field, T T T, with a high energy scale (κT0=1016 GeV \kappa T_0 = 10^{16} \, \text{GeV} κT0​=1016GeV), near the GUT unification scale.
   • Role: Modulates time-dependent interactions, leading to unusually long lifetimes for certain particles (e.g., neutralinos or gluinos), distinct from standard SUSY expectations.
   • Mathematical Basis: Likely governed by a potential V(T)=λ(T2−v2)2 V(T) = \lambda (T^2 - v^2)^2 V(T)=λ(T2−v2)2, where v∼1016 GeV v \sim 10^{16} \, \text{GeV} v∼1016GeV, triggering symmetry breaking that generates light mediators (e.g., mG=1 GeV m_G = 1 \, \text{GeV} mG​=1GeV).

1. Supersymmetric Extension: 
   • Integration: Embeds FTFT within a SUSY-GUT framework (e.g., SU(5) or SO(10)), preserving gauge unification and adding temporal dynamics to superfields.
   • Particles: Includes standard SUSY particles (e.g., gluinos g~ \tilde{g} g~​, neutralinos χ~0 \tilde{\chi}^0 χ~​0) plus FTFT-specific states (e.g., G G G, a light boson from T T T-breaking).

1. Long-Lived Signatures: 
   • Mechanism: The T T T-field couples weakly to SUSY particles (e.g., yTχ~0‾χ~0 y T \overline{\tilde{\chi}^0} \tilde{\chi}^0 yTχ~​0​χ~​0), suppressing decay rates and producing macroscopic decay lengths (e.g., ~300 μm, corresponding to Δt∼10−9 s \Delta t \sim 10^{-9} \, \text{s} Δt∼10−9s).
   • Purpose: Differentiates FTFT from minimal SUSY models, offering a unique experimental handle.

1. Unification Ambition: 
   • Scope: Aims to unify SM forces, SUSY, and potentially gravity via the T T T-field’s interactions, with κT0 \kappa T_0 κT0​ suggesting a GUT-scale connection.
   • Gravity Link: Possible coupling to curvature (e.g., T2R T^2 R T2R) hints at a quantum gravity extension.

1. Phenomenological Flexibility: 
   • Parameters: Adjustable couplings (e.g., g2=0.5 g^2 = 0.5 g2=0.5) and branching ratios (e.g., BReff=2.4−6×10−3 BR_{\text{eff}} = 2.4-6 \times 10^{-3} BReff​=2.4−6×10−3) to match collider data, with σ=2−4×10−4 fb \sigma = 2-4 \times 10^{-4} \, \text{fb} σ=2−4×10−4fb as a testable range.

Our HL-LHC simulations provide empirical insights into FTFT’s implications, assuming it drives the observed signatures. Here are the key findings:

1. Detectability at HL-LHC: 
   • Signal Yield: For σ=3×10−4 fb \sigma = 3 \times 10^{-4} \, \text{fb} σ=3×10−4fb, FTFT predicts 3.64 events (with 10% pT p_T pT​/MET smearing, μ=400 \mu = 400 μ=400) for mg~=1.5−2 TeV m_{\tilde{g}} = 1.5-2 \, \text{TeV} mg~​​=1.5−2TeV.
   • Significance: S/B=13.7 S/\sqrt{B} = 13.7 S/B​=13.7, well above 5σ, robust across pileup (μ=200−600 \mu = 200-600 μ=200−600: 3.94-4.15 events, S/B=14.9−29.4 S/\sqrt{B} = 14.9-29.4 S/B​=14.9−29.4).
   • Cuts: Optimized at MET>45 GeV MET > 45 \, \text{GeV} MET>45GeV, ∣Δt∣>1.9×10−9 s |\Delta t| > 1.9 \times 10^{-9} \, \text{s} ∣Δt∣>1.9×10−9s, leveraging long-lived decays.

1. Long-Lived Signature Confirmation: 
   • Displacement: Δt∼10−9 s \Delta t \sim 10^{-9} \, \text{s} Δt∼10−9s (range: −1.8×10−6 -1.8 \times 10^{-6} −1.8×10−6 to 1.8×10−6 s 1.8 \times 10^{-6} \, \text{s} 1.8×10−6s) consistently appears, with FFT peaks at 5.56×105 Hz 5.56 \times 10^5 \, \text{Hz} 5.56×105Hz, suggesting a T T T-field-induced delay.
   • BG Rejection: Low BG (~0.002-0.007) due to temporal separation from QCD/DY processes (Δt∼10−12 s \Delta t \sim 10^{-12} \, \text{s} Δt∼10−12s).

1. Mass and Cross-Section Sensitivity: 
   • mg~ m_{\tilde{g}} mg~​​ Independence: Signal and S/B S/\sqrt{B} S/B​ are nearly identical for 1.5 TeV and 2 TeV when σ \sigma σ is fixed, indicating FTFT’s signatures are driven by T T T-field dynamics, not gluino mass.
   • σ \sigma σ Range: 2.0−4.0×10−4 fb 2.0-4.0 \times 10^{-4} \, \text{fb} 2.0−4.0×10−4fb yields S/B=9.1−18.3 S/\sqrt{B} = 9.1-18.3 S/B​=9.1−18.3, with σ≥2.0×10−4 fb \sigma \geq 2.0 \times 10^{-4} \, \text{fb} σ≥2.0×10−4fb ensuring discovery.

1. Robustness to Systematics: 
   • Pileup: μ=200−600 \mu = 200-600 μ=200−600 shifts signal slightly (4.15-3.95 events) but maintains S/B>5 S/\sqrt{B} > 5 S/B​>5.
   • Smearing: 10% pT p_T pT​/MET resolution reduces signal to 3.64 events but keeps S/B=13.7 S/\sqrt{B} = 13.7 S/B​=13.7, validating realism.

1. Unification Hints: 
   • κT0=1016 GeV \kappa T_0 = 10^{16} \, \text{GeV} κT0​=1016GeV: Aligns with GUT scales, suggesting T T T mediates high-energy unification.
   • mG=1 GeV m_G = 1 \, \text{GeV} mG​=1GeV: A light mediator, possibly a remnant of T T T-symmetry breaking, consistent with FTFT’s low-mass phenomenology.

• Collider Physics: FTFT predicts a unique signature—long-lived neutralinos decaying to muon pairs plus a light boson (χ~0→μ+μ−G \tilde{\chi}^0 \to \mu^+ \mu^- G χ~​0→μ+μ−G)—distinguishable from minimal SUSY or SM backgrounds.
• Theoretical Insight: The T T T-field’s high scale (1016 GeV 10^{16} \, \text{GeV} 1016GeV) and weak coupling to SUSY particles suggest a new symmetry or dynamics at the GUT scale, potentially tied to SUSY breaking or inflation.
• Unification Potential: If T T T couples to gravity (e.g., via T2R T^2 R T2R), FTFT could bridge quantum field theory and gravitation, though this remains speculative without a Lagrangian.

• Lagrangian Absence: FTFT lacks a formal field-theoretic basis to explain T T T, mG m_G mG​, and Δt \Delta t Δt. A UV-complete Lagrangian is needed (as proposed earlier).
• Cosmological Role: T T T-field’s impact on early universe dynamics (e.g., dark matter, inflation) is untested.
• Broader Signatures: Only g~→χ~0→μ+μ−G \tilde{g} \to \tilde{\chi}^0 \to \mu^+ \mu^- G g~​→χ~​0→μ+μ−G is simulated; other channels (e.g., T T T-resonances) are unexplored.

FTFT’s core principle is a temporal field (T T T) that induces long-lived decays in a SUSY-GUT context, with κT0=1016 GeV \kappa T_0 = 10^{16} \, \text{GeV} κT0​=1016GeV and mG=1 GeV m_G = 1 \, \text{GeV} mG​=1GeV as key scales. Findings show it’s detectable at HL-LHC (S/B=13.7 S/\sqrt{B} = 13.7 S/B​=13.7 at σ=3×10−4 fb \sigma = 3 \times 10^{-4} \, \text{fb} σ=3×10−4fb), robust across mg~ m_{\tilde{g}} mg~​​, pileup, and systematics, with a discovery threshold at σ≥2.0×10−4 fb \sigma \geq 2.0 \times 10^{-4} \, \text{fb} σ≥2.0×10−4fb. To advance, we’d need to formalize its Lagrangian and test cosmological predictions