Extended Massive Ambitwistor String II

Technical Summary: Extended Massive Ambitwistor String II

Problem Statement
This paper serves as a detailed continuation of the work presented in [1], which introduced an extension to the massive ambitwistor string model of [2, 3]. The primary objective is to rigorously establish the consistency of this extended model, specifically addressing claims made in [1] regarding the vanishing of vacuum partition functions and the cosmological constant, as well as the properties of higher-loop scattering amplitudes. The model aims to provide a modular invariant and unitary formulation of $N=8$ supergravity in twistor space, with an embedded Super-Yang-Mills (SYM) theory, while investigating its Ultraviolet (UV) finiteness.

Methodology and Model Construction
The authors revisit the action of the massive ambitwistor string, introducing an auxiliary action ($S_{aux}$) and extending the worldsheet gauge symmetry.

• Extended Little Group: The little group is extended from $SU(2)$ to $SU(2) \times SU(2) \times SU(2)$ (or its complexification $SL(2,\mathbb{C})^3$). This extension is achieved by adding auxiliary fermionic and bosonic worldsheet spinors ($\rho, \tau, \tau', \tilde{\tau}'$) and corresponding Lagrange multipliers.
• Anomaly Cancellation: The inclusion of these auxiliary fields ensures the vanishing of the Virasoro central charge ($c=0$) and the little group anomaly coefficient, rendering the BRST charge nilpotent and the model anomaly-free.
• Physical Spectrum: While the off-shell description involves three pairs of twistors, the authors note that on-shell, a "tiny group" gauge transformation reduces the description to two twistors, mapping the third pair to zero. Consequently, the physical spectrum and tree-level scattering amplitudes remain unchanged from the original model, but the extended symmetry provides necessary virtual excitations for loop amplitudes.
• Vertex Operators: Fixed and integrated vertex operators are constructed using Picture Changing Operators (PCOs) to handle the extended gauge symmetries. The authors demonstrate that for tree amplitudes, the auxiliary fields decouple, recovering the results of [1].

Key Contributions and Results

1. Partition Function and Cosmological Constant:
   The paper calculates the partition function for Riemann surfaces of genus $g \geq 1$. By utilizing chiral splitting of the measure and analyzing the zero modes of the spinor fields on both even and odd spin structures, the authors demonstrate that the partition function vanishes identically for all $g \geq 1$.

   • On even spin structures, the partition function vanishes because the relevant fields are "orphan" (lacking zero modes) and cannot be saturated by the PCOs.
   • On odd spin structures, while zero modes exist, the specific counting of bosonic and fermionic components (particularly the auxiliary fields) leads to a cancellation of uncanceled zero modes.
   • Result: The vanishing of the partition function implies that the cosmological constant is zero at all orders of perturbation theory.

2. Modular Invariance of Loop Amplitudes:
   The authors derive all-multiplicity loop amplitudes for genus $g \geq 1$ on both even and odd spin structures.

   • A critical technical step involves ensuring that the boundary conditions of all chiral ghosts align with those of the spinors. This alignment is shown to be necessary for the modular invariance of the amplitude.
   • The integration over the moduli space is shown to be well-defined and localized to a set of points in the fundamental domain, determined by the solutions to the polarized scattering equations.
   • The amplitudes are proven to be modular invariant, with the measure and integrand transforming appropriately under the modular group $Sp(2g, \mathbb{Z})$.

3. Factorization and Unitarity:
   The paper investigates the factorization of amplitudes on degenerated Riemann surfaces:

   • Non-separating Degeneration: The amplitude is evaluated on a non-separating divisor at infinity. The authors show that the process of reducing a genus $g$ amplitude to a genus $g-1$ amplitude (with $n+2$ external particles) is exact and reversible. The insertion of a complete set of physical states at the degeneration point confirms unitary factorization.
   • Separating Degeneration: By leveraging the reversibility of the non-separating case, the authors argue that separating degeneration also yields correct factorization, reducing the problem to the known tree-level factorization established in [1].

4. UV Finiteness:
   The paper addresses the UV behavior of the theory:

   • Formal Argument: Modular invariance restricts the integration to a fundamental domain, mapping potential UV divergences (small moduli) to IR divergences (large moduli), suggesting formal UV completeness.
   • Scaling Argument: A detailed scaling analysis is performed for single-loop amplitudes on a non-separating divisor. By analyzing the behavior of the polarized scattering equations and the reduced determinants of the matrices $H$ and $G$ as momentum scales to infinity ($\Lambda \to \infty$), the authors show that the amplitude scales as $\Lambda^{-6}$ or better.
   • Conclusion: This scaling behavior proves that the theory is UV-finite at least at the single-loop level for reduced amplitudes. The authors note that while this does not constitute a rigorous proof for all loop orders, it supports the formal claim of UV finiteness derived from modular invariance.

Significance and Claims
The paper claims to provide a robust, modular invariant, and unitary formulation of $N=8$ supergravity in twistor space that naturally embeds Super-Yang-Mills theory. The primary significance lies in:

• Vanishing Cosmological Constant: Proving that the vacuum energy vanishes to all orders, a crucial feature for a consistent theory of gravity.
• UV Behavior: Offering evidence that $N=8$ supergravity may be UV-finite, challenging the conventional expectation of divergences at high loop orders.
• Consistency: Demonstrating that the extended massive ambitwistor string model is free of anomalies and possesses the correct factorization properties required for a unitary quantum field theory.

The authors conclude that while the model is compelling, further work is needed to sharpen the UV finiteness claims (potentially to $\Lambda^{-8}$ behavior) and to address infrared divergences associated with massless particles, possibly through the development of a twistor space string field theory.