QFT Generalizations via Taylor Expansion Cognitive Framework

I. Mathematical Framework and Established Physics

1.1 The Effective Field Theory Framework (Established)

The effective field theory (EFT) approach is a well-established framework in particle physics. The Standard Model Lagrangian can be systematically extended with higher-dimension operators:

\[\mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \sum_{d=5}^{\infty} \sum_i \frac{c_i^{(d)}}{\Lambda^{d-4}} \mathcal{O}_i^{(d)}\]

where:

$\Lambda$ is the scale of new physics (experimentally constrained)
$c_i^{(d)}$ are Wilson coefficients (dimensionless)
$\mathcal{O}_i^{(d)}$ are operators of dimension $d$
The expansion is valid for $E \ll \Lambda$

Domain of Validity: This expansion converges for energies $E \ll \Lambda$. Current LHC constraints place $\Lambda \gtrsim 1-10$ TeV for most operators.

1.2 Types of Expansions in QFT

1. Perturbative Expansions (Established)

Parameter: Coupling constant $g$ or $\alpha = g^2/4\pi$
Convergence: Asymptotic series, requires $\alpha \ll 1$
Example: QED with $\alpha \approx 1/137$

2. Derivative Expansions (Established)

Parameter: $p/\Lambda$ where $p$ is momentum
Convergence: Convergent for $p < \Lambda$
Example: Chiral perturbation theory with $\Lambda_\chi \approx 1$ GeV

3. Large-N Expansions (Established)

Parameter: $1/N$ where $N$ is number of colors/flavors
Convergence: Typically asymptotic
Example: QCD with $N_c = 3$

II. Concrete Example 1: Higgs Effective Field Theory (Established)

2.1 Mathematical Framework

After integrating out heavy new physics at scale $\Lambda$, the Higgs sector is described by:

\[\mathcal{L}_{\text{Higgs EFT}} = \mathcal{L}_{\text{SM}} + \frac{c_H}{\Lambda^2} \mathcal{O}_H + \frac{c_{HW}}{\Lambda^2} \mathcal{O}_{HW} + \frac{c_{HB}}{\Lambda^2} \mathcal{O}_{HB} + \cdots\]

Key dimension-6 operators:

$\mathcal{O}H = \partial\mu (H^\dagger H) \partial^\mu (H^\dagger H)$
$\mathcal{O}{HW} = H^\dagger H W{\mu\nu}^I W^{I\mu\nu}$
$\mathcal{O}{HB} = H^\dagger H B{\mu\nu} B^{\mu\nu}$

2.2 Experimental Constraints

Current LHC Bounds (Run 2, 139 fb⁻¹):

$ c_H/\Lambda^2 < 0.15$ TeV⁻²
$ c_{HW}/\Lambda^2 < 0.05$ TeV⁻²
$ c_{HB}/\Lambda^2 < 0.08$ TeV⁻²

This translates to $\Lambda > 0.5-1.5$ TeV assuming $c_i \sim 1$.

2.3 Future Experimental Sensitivity

HL-LHC Projections (3000 fb⁻¹):

Expected sensitivity: $\delta(c_i/\Lambda^2) \sim 0.01-0.02$ TeV⁻²
Probe scales up to $\Lambda \sim 3-5$ TeV
Key channels: $h \to ZZ^*$, $h \to \gamma\gamma$, $Vh$ production

FCC-ee Projections:

Precision on Higgs couplings: 0.1-0.5%
Indirect reach: $\Lambda \sim 10-30$ TeV
Model-independent global fit possible

2.4 Observable Effects

The operator $\mathcal{O}_H$ modifies the Higgs self-coupling: $\lambda_{hhh} = \lambda_{\text{SM}} \left(1 + \frac{3 c_H v^2}{\Lambda^2}\right)$

For $c_H = 1$ and $\Lambda = 1$ TeV:

Deviation: $\delta\lambda/\lambda \approx 17\%$
HL-LHC sensitivity: $\delta\lambda/\lambda \sim 50\%$
FCC-hh sensitivity: $\delta\lambda/\lambda \sim 5\%$

III. Concrete Example 2: Chiral Perturbation Theory (Established)

3.1 Mathematical Framework

Low-energy QCD is described by an expansion in $p/\Lambda_\chi$ and $m_q/\Lambda_\chi$:

\[\mathcal{L}_{\chi PT} = \mathcal{L}^{(2)} + \mathcal{L}^{(4)} + \mathcal{L}^{(6)} + \cdots\]

Leading order ($\mathcal{O}(p^2)$): $\mathcal{L}^{(2)} = \frac{f_\pi^2}{4} \text{Tr}[\partial_\mu U \partial^\mu U^\dagger] + \frac{f_\pi^2 B_0}{2} \text{Tr}[M_q (U + U^\dagger)]$

where $U = \exp(2i\pi^a T^a/f_\pi)$, $f_\pi \approx 93$ MeV.

Next-to-leading order ($\mathcal{O}(p^4)$): 10 low-energy constants (LECs) $L_1, \ldots, L_{10}$.

3.2 Convergence Properties

The expansion parameter is $p/(4\pi f_\pi) \approx p/(1.2\text{ GeV})$.

Pion-pion scattering amplitude: $A(s,t,u) = \frac{s}{f_\pi^2} + \frac{1}{96\pi^2 f_\pi^4}[s^2 \log(s/\mu^2) + t^2 \log(t/\mu^2) + u^2 \log(u/\mu^2)] + \mathcal{O}(p^6)$

Convergence breakdown at $\sqrt{s} \sim 4\pi f_\pi \approx 1.2$ GeV (ρ meson mass).

3.3 Experimental Tests

Pion decay constant: $f_\pi = 92.21(14)$ MeV (0.15% precision)

Pion-pion scattering lengths (in units of $m_\pi^{-1}$):

$a_0^0 = 0.220(5)$ (experiment)
$a_0^0 = 0.219(3)$ (χPT at NNLO)

Kaon decays: $K \to \pi\pi$ amplitudes test χPT to 1% level.

IV. Concrete Example 3: Non-Commutative QFT (Plausible/Speculative)

4.1 Mathematical Framework

Status: Theoretically motivated by string theory, but no experimental evidence.

Spacetime coordinates satisfy: $[\hat{x}^\mu, \hat{x}^\nu] = i\theta^{\mu\nu}$

The Moyal product replaces ordinary multiplication: $f \star g = f \exp\left(\frac{i}{2} \overleftarrow{\partial_\mu} \theta^{\mu\nu} \overrightarrow{\partial_\nu}\right) g$

Expansion in powers of $\theta$: $f \star g = fg + \frac{i}{2}\theta^{\mu\nu} \partial_\mu f \partial_\nu g - \frac{1}{8}\theta^{\mu\nu}\theta^{\rho\sigma} \partial_\mu \partial_\rho f \partial_\nu \partial_\sigma g + \mathcal{O}(\theta^3)$

4.2 QED on Non-Commutative Spacetime

The action becomes: $S_{NC-QED} = \int d^4x \left[-\frac{1}{4}F_{\mu\nu} \star F^{\mu\nu} + \bar{\psi} \star (i\gamma^\mu D_\mu - m) \psi\right]$

Leading correction to electron-photon vertex: $\delta\Gamma^\mu = -\frac{e^3}{48\pi^2} \theta^{\alpha\beta} k_{1\alpha} k_{2\beta} \gamma^\mu$

4.3 Experimental Constraints

Current bounds (assuming $\theta^{0i} = 0$):

From Lamb shift: $\sqrt{ \theta } > 10^{-20}$ m
From synchrotron radiation: $\sqrt{ \theta } > 10^{-19}$ m
From CMB: $\sqrt{ \theta } > 10^{-19}$ m

Future sensitivity:

Next-gen atomic interferometry: $\sqrt{ \theta } \sim 10^{-22}$ m
Space-based gravitational wave detectors: indirect constraints

4.4 Quantitative Predictions

For $\sqrt{|\theta|} = 10^{-19}$ m:

Modification to g-2: $\delta a_\mu \sim 10^{-24}$ (unobservable)
Energy dependence: Effects grow as $E^2/M_{NC}^2$ where $M_{NC} = 1/\sqrt{ \theta }$

V. Systematic Classification of QFT Extensions

5.1 Established Extensions

Effective Field Theory: Well-defined, experimentally tested
Chiral Perturbation Theory: Convergent below 1 GeV
Heavy Quark Effective Theory: Systematic $1/m_Q$ expansion
NRQCD/NRQED: Non-relativistic expansions

5.2 Plausible Extensions

Higher-dimension operators: Constrained by precision tests
Modified dispersion relations: Constrained by astrophysics
Lorentz violation: Severely constrained but not ruled out

5.3 Highly Speculative Extensions

Non-commutative geometry: No experimental evidence
Fractional derivatives: Mathematical curiosity
Emergent spacetime: Highly speculative

VI. Conclusions

The Taylor expansion framework provides a systematic way to understand QFT generalizations:

Established Success: EFT and χPT demonstrate the power of systematic expansions with clear convergence properties and experimental validation.
Future Prospects: Next-generation experiments will probe higher-order operators with unprecedented precision, potentially revealing new physics at the TeV scale.
Theoretical Guidance: The framework helps organize our thinking about beyond-Standard Model physics, but must be grounded in experimental reality.

Clear Limitations: Not all physics can be captured by Taylor expansions, and convergence must be carefully analyzed in each case.

VII. Comparative Analysis and Summary

7.1 Summary Table: QFT Expansion Frameworks

| Framework | Expansion Parameter | Convergence Radius | Current Experimental Status | Key Observable | |———–|——————-|——————-|—————————|—————-| | QED Perturbation Theory | $\alpha \approx 1/137$ | Asymptotic series | Tested to $10^{-12}$ precision | $g-2$ of electron | | QCD Perturbation Theory | $\alpha_s(M_Z) \approx 0.12$ | Asymptotic for $E > 2$ GeV | Tested to 1% at LHC | Jet cross sections | | Chiral Perturbation Theory | $p/(4\pi f_\pi) \approx p/1.2$ GeV | $p < 1$ GeV | Tested to 0.1% precision | $\pi\pi$ scattering | | Heavy Quark EFT | $\Lambda_{QCD}/m_Q$ | $m_Q > 1$ GeV | Tested to 1% for $b$ quarks | $B$ meson decays | | Higgs EFT | $v^2/\Lambda^2$ | $\Lambda > 1$ TeV | Constraints: $\Lambda > 0.5-1.5$ TeV | Higgs couplings | | SMEFT (dim-6) | $E^2/\Lambda^2$ | $E < \Lambda$ | Bounds: $\Lambda > 1-10$ TeV | Triple gauge couplings | | Non-commutative QFT | $p^2\theta$ | $p < 1/\sqrt{|\theta|}$ | Bounds: $\sqrt{|\theta|} > 10^{-19}$ m | Modified dispersion | | Large-N QCD | $1/N_c$ | Asymptotic | $N_c = 3$ in nature | Meson spectrum |

7.2 Flowchart: Systematic EFT Construction

┌─────────────────────────────────┐
│   1. Identify Degrees of Freedom │
│      and Symmetries              │
└────────────┬────────────────────┘
          │
          ▼
┌─────────────────────────────────┐
│   2. Choose Power Counting       │
│      Scheme                      │
│   • Identify small parameters    │
│   • Assign scaling dimensions    │
└────────────┬────────────────────┘
          │
          ▼
┌─────────────────────────────────┐
│   3. Construct Operator Basis    │
│   • List all allowed operators   │
│   • Remove redundancies via EOM  │
│   • Apply integration by parts   │
└────────────┬────────────────────┘
          │
          ▼
┌─────────────────────────────────┐
│   4. Match to UV Theory          │
│   • Calculate Wilson coefficients│
│   • Run to low energy scale      │
└────────────┬────────────────────┘
          │
          ▼
┌─────────────────────────────────┐
│   5. Compute Observables         │
│   • Organize by power counting   │
│   • Include loop corrections     │
└────────────┬────────────────────┘
          │
          ▼
┌─────────────────────────────────┐
│   6. Fit to Experimental Data    │
│   • Extract LECs                 │
│   • Assess convergence           │
└─────────────────────────────────┘

7.3 Wilson Coefficient Correlations

Correlation Matrix Structure In SMEFT, Wilson coefficients are not independent due to:

SU(2)×U(1) gauge invariance: Links operators in the same gauge multiplet
Flavor symmetries: Relates coefficients across generations
Experimental observables: Multiple operators contribute to same process Example: Higgs-Gauge Sector Correlations The operators $\mathcal{O}{HW}$, $\mathcal{O}{HB}$, and $\mathcal{O}_{HWB}$ are correlated through: $\begin{pmatrix} \delta g_{hZZ} \\ \delta g_{hWW} \\ \delta g_{h\gamma\gamma} \end{pmatrix} = \begin{pmatrix} c_W^2 & s_W^2 & -2s_Wc_W \\ 1 & 0 & 0 \\ 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} c_{HW}/\Lambda^2 \\ c_{HB}/\Lambda^2 \\ c_{HWB}/\Lambda^2 \end{pmatrix}$ where $s_W = \sin\theta_W$, $c_W = \cos\theta_W$. Global Fit Results (Current LHC data):
- Strong correlation ($\rho > 0.8$) between $c_{HW}$ and $c_{HWB}$
- Measuring $h\to\gamma\gamma$ constrains linear combination
- Individual extraction requires multiple channels Correlation Coefficients (Typical values):
```
1
2
3
4
5
    c_HW    c_HB    c_HWB   c_H
c_HW     1.00   -0.15    0.82   0.05
c_HB    -0.15    1.00   -0.73   0.02
c_HWB    0.82   -0.73    1.00   0.08
c_H      0.05    0.02    0.08   1.00
```
  7.4 Machine Learning in EFT Analysis
  
  Current Applications:
Neural Network Amplitude Regression
- Train on Monte Carlo samples with different Wilson coefficients
- Interpolate amplitudes for arbitrary coefficient values
- Speed up: 10³-10⁴× faster than full simulation
Optimal Observable Construction
- Use ML to find observables maximally sensitive to specific operators
- Example: Neural networks identify angular distributions sensitive to anomalous triple gauge couplings

Global Fit Optimization

Gaussian processes for likelihood surface mapping
Efficient exploration of high-dimensional Wilson coefficient space

Handles non-Gaussian posteriors naturally Concrete Example: SMEFT Analysis with ML

# Pseudocode for ML-enhanced EFT fitting
class SMEFTAnalyzer:
 def __init__(self):
   self.amplitude_network = TrainedNeuralNetwork()
   self.likelihood_gp = GaussianProcess()
 def predict_observable(self, wilson_coeffs, kinematics):
   # Fast amplitude evaluation
   return self.amplitude_network(wilson_coeffs, kinematics)
 def global_fit(self, data):
   # Bayesian optimization using GP
   return self.likelihood_gp.maximize(data)

Performance Metrics: - Traditional χ² fit: ~10⁴ likelihood evaluations - ML-enhanced fit: ~10² likelihood evaluations - Uncertainty quantification: Comparable to traditional methods Future Directions:

Symbolic Regression for EFT
- Discover optimal operator bases
- Identify hidden correlations
- Automate redundancy removal
Anomaly Detection
- Identify deviations from SM without assuming specific EFT
- Model-independent searches for new physics
- Real-time analysis at LHC
Quantum Machine Learning
- Quantum circuits for amplitude calculation
- Potential exponential speedup for loop calculations
- Currently limited to toy models Challenges and Limitations: - Training data quality crucial - Interpretability vs. performance trade-off - Systematic uncertainty quantification - Extrapolation beyond training region dangerous

Choose Theme