Learning the open string spectrum with neural networks

Technical Summary: Learning the Open String Spectrum with Neural Networks

Problem Statement
The tree-level open superstring amplitude possesses a unique mass-squared spectrum consisting of positive integers located on linear Regge trajectories. While this spectrum is well-known from the explicit Veneziano amplitude, the extent to which it is fixed by fundamental physical principles—specifically locality, analyticity, crossing symmetry, and unitarity—remains a subject of investigation. Furthermore, recent work has suggested that the open string spectrum minimizes the entanglement generated in $2 \to 2$ scattering compared to general non-integer spectra. This paper addresses two primary questions: (1) Can the integer mass-squared spectrum and partial-wave coefficients be simultaneously learned using machine learning, and what are the requirements for convergence? (2) Does the integer spectrum indeed correspond to a minimum in entanglement generation compared to non-integer alternatives? Additionally, the author investigates bounds on three-channel crossing symmetric amplitudes and the behavior of non-dual amplitudes.

Methodology
The author employs a "bootstrap" approach utilizing Crossing Symmetric Dispersion Relations (CSDR) with a one-parameter ambiguity ($\lambda$-CSDR). The amplitude is expressed as an integral over its imaginary part, decomposed into partial waves involving Gegenbauer polynomials.

1. Machine Learning Setup (Physics-Informed Neural Networks - PINN):

   • Objective: To learn the mass-squared spectrum ($R_n$) and the partial-wave coefficients ($c^{(R_n)}_\ell$) simultaneously.
   • Architecture: A fully-connected feed-forward neural network with 3 hidden layers (size 128) takes the spin $\ell$ and mass level $R_n$ as inputs and outputs the coefficient $c^{(R_n)}_\ell$.
   • Constraints: The loss function incorporates:
     • Unitarity: Enforced via a Softplus activation function to ensure $c^{(R_n)}_\ell \geq 0$.
     • Wilson Coefficients: Constraints forcing the low-energy expansion coefficients ($W_{10}, W_{01}$) to match known open string values ($-\zeta(3)$ and $7\zeta(4)/4$).
     • $\lambda$-Independence: Constraints ensuring the amplitude is independent of the ambiguity parameter $\lambda$ around a seed value (e.g., $\lambda = 14.6$), a condition satisfied by the full infinite spectrum.
     • Entanglement Minimization: The loss function includes the "Entanglement Power" ($\Phi$), defined as a sum over the spectrum weighted by residues, which is minimized.
   • Optimization: The author uses the Adam optimizer for the neural network weights and gradient descent for the $R_n$ values. A critical technical challenge was ensuring the hypergeometric functions (used in Gegenbauer polynomials) were compatible with PyTorch's autograd for differentiating through the $R_n$ values; this was achieved by using a truncated series representation of the hypergeometric function.

2. Linear Programming (SDPB):

   • To verify the entanglement minimization claim, the author used the Semi-definite Programming Bootstrap (SDPB) package.
   • They generated 100 sets of random non-integer mass spectra ($R_n$) and minimized $\Phi$ subject to the same Wilson coefficient and $\lambda$-independence constraints.

3. Three-Channel Analysis:

   • The author extended the analysis to three-channel crossing symmetric amplitudes (relevant for closed strings/Virasoro-Shapiro) using a local three-channel $\lambda$-CSDR to carve out the allowed space of Wilson coefficients ($W_{10}, W_{01}$).

Key Results

• Learning the Spectrum:

  • The machine learning algorithm successfully learned partial-wave coefficients and mass levels that approximate the open string spectrum, but only when the first mass level ($R_1$) was fixed to its integer value ($R_1=1$).
  • Crucial Finding: The algorithm did not converge to the open string solution when $R_1$ was left free. When $R_1$ (and sometimes $R_2, R_3$) were fixed, the remaining unfixed levels ($R_n$ for $n>1$) optimized to values approximately close to integers but did not converge to exact integers uniformly. Some levels moved closer to integers, while others moved further away. This suggests that within the finite truncation ($N_{\text{max}}$) and the specific ansatz, the integer spectrum is favored but not uniquely recovered from first principles without fixing the mass scale.

• Entanglement Minimization:

  • Linear Programming Results: Out of 300 random non-integer spectra tested, 187 converged to a $\Phi$ value strictly higher than the open string value ($\Phi_{\text{string}} \approx 1.5997$). The remaining 113 failed to satisfy constraints within the required tolerance.
  • Conclusion: The open string integer spectrum generates lower entanglement than general non-integer spectra, providing strong evidence for the entanglement minimization principle in this context.

• Three-Channel Bounds:

  • Using the three-channel $\lambda$-CSDR, the author derived analytic bounds on the Wilson coefficients: $W_{10} \leq 0$ and $0 < W_{01} \leq -\frac{3}{2}W_{10}$.
  • The Virasoro-Shapiro amplitude was successfully located within the allowed region of the $(W_{10}, W_{01})$ space.
  • Spin-0 Dominance: The author demonstrated that the lower boundary of the allowed region is determined by "spin-0 dominance," where the contribution of spin-0 states to the dispersion relations outweighs higher-spin contributions. This necessitates the existence of infinite higher masses and spins to satisfy crossing symmetry.

• Non-Dual Amplitudes:

  • The paper analyzes a "Non-dual" amplitude (the "black star" amplitude from previous work) which violates the dual resonance model.
  • The Regge trajectory of this amplitude exhibits a transition from linear Regge behavior ($\alpha(s) \sim s$) to power-law behavior ($s^0$) for $s_1 \gg 1$ and $s_2 < 0$, resembling a Polchinski-Strassler type transition observed in certain QFT amplitudes.

Significance and Claims
The paper claims to advance the "S-matrix bootstrap" program by demonstrating the feasibility of using Physics-Informed Neural Networks to learn spectral data (masses and couplings) simultaneously, rather than just coefficients for a fixed spectrum. The author highlights that their approach, which utilizes PyTorch-compatible hypergeometric functions to handle non-linear dependencies on mass levels, is a novel technical contribution.

The work provides numerical evidence that the integer mass-squared spectrum of the open string is special because it minimizes entanglement generation in scattering, a property not shared by generic non-integer spectra. However, the author is modest regarding the "learning from first principles" aspect: they explicitly state that their current neural network architecture could not recover the exact integer spectrum without fixing the first pole ($R_1$) to set the mass scale. They suggest that recovering the full integer spectrum without such fixing would require more sophisticated architectures.

Finally, the paper establishes rigorous bounds for three-channel crossing symmetric amplitudes and locates the Virasoro-Shapiro amplitude within this space, linking the boundaries of the allowed region to the physical principle of spin-0 dominance.