The left-cut for partial waves in terms of physical amplitudes

Technical Summary: The Left-Cut for Partial Waves in Terms of Physical Amplitudes

Problem Statement
In the study of $2 \to 2$ scattering within Effective Field Theories (EFTs), such as Chiral Perturbation Theory (ChPT) for light hadrons or the Electroweak Symmetry Breaking Sector (EWSBS), perturbative unitarity breaks down at high energies. To extend predictive power into the resonance region, physicists rely on unitarization methods like the Inverse Amplitude Method (IAM), the $N/D$ approach, or improved $K$-matrix methods. These techniques require a rigorous treatment of the partial wave amplitude's analytic structure, specifically the "left-hand cut" (unphysical region where $s \le 0$).

While exact $S$-matrix methods (e.g., Roy equations, Froissart-Gribov representation) establish that the left-hand cut is dictated by crossing symmetry from physical channels, practical implementations often struggle with numerical stability. Specifically, the known Mandelstam-Chew representation of partial waves over the left cut has been shown to exhibit numerical instability. Consequently, there is a need for a robust, explicit algebraic representation that connects the left-hand cut directly to physical amplitudes to systematically quantify uncertainties in unitarization schemes.

Methodology
The author develops a novel algebraic series representation for the partial wave amplitude $t^I_J(s)$ over the left-hand cut. The derivation proceeds through the following steps:

1. Partial Wave Projection: The partial wave is defined via a projection of the full scattering amplitude $T(s, t, u)$ over the unphysical integration domain traversing the left cut ($s \le 0$). The integration path connects the physical $t$- and $u$-channel regions.
2. Expansion around Physical Endpoints: The integration domain is partitioned into segments $x \in [-1, 0]$ and $x \in [0, 1]$. The reduced amplitude $A(s, x)$ is expanded in a Taylor series around the physical endpoints ($x = \pm 1$), corresponding to the physical $t$- and $u$-channel cuts. This relies on the assumption that the amplitude and its derivatives remain analytic in a neighborhood just below the physical branch cuts.
3. Derivative Identification: The derivatives of the amplitude with respect to the integration variable $x$ at the endpoints are identified with derivatives of the amplitudes in the $t$- and $u$-channels.
4. Dispersion Relations: Using a fixed-$t$ dispersion relation (derived in [24]), the derivatives of the scattering amplitude over the right cut are expressed as integrals of the imaginary parts of the physical amplitudes.
5. Crossing Symmetry: The crossing matrices ($C_t$ and $C_u$) are employed to relate the isospin channels of the crossed processes to the original channel.
6. Series Summation: The resulting infinite series of derivatives is summed analytically (utilizing symbolic computation tools) to yield closed-form integral representations for specific angular momenta $J$.

Key Contributions and Results
The paper derives an explicit formula expressing the left-hand cut of arbitrary isospin ($I$) and angular momentum ($J$) partial waves as an integral of the imaginary parts of physical amplitudes over the right-hand cut.

• General Representation: The partial wave $t^I_J(s)$ is expressed as a sum of integrals involving $\text{Im } T^{I'}(s' + i\epsilon, s, 0)$, weighted by logarithmic and polynomial kernels dependent on $s$, $s'$, and the crossing matrices.
• Explicit Forms for Low $J$: The author provides explicit integral expressions for $J=0, 1, 2$. For instance, for $J=0$, the result involves a logarithmic kernel $\log(\frac{s'}{s' + s - 4m^2})$.
• SU(2) Isospin Specifics: The imaginary parts of the partial waves over the left cut are explicitly computed for the $IJ = 00, 11, 20, 22$ channels in an SU(2) isospin-symmetric theory (relevant for pion-pion scattering).
• Numerical Stability: Unlike the Mandelstam-Chew representation, which suffers from numerical instability, this new representation is claimed to be robust numerically for Next-to-Leading Order (NLO) SU(2) isospin amplitudes.
• Channel Dependence: A quantitative analysis reveals a distinct difference in the magnitude of the left-cut imaginary parts across channels. The vector channel ($IJ=11$) exhibits the smallest imaginary part on the left cut, while the scalar ($IJ=00$) and tensor channels exhibit significantly larger values.

Significance and Claims
The paper posits that this formulation offers a valuable tool for systematically quantifying left-hand cut uncertainties in unitarization methods.

• Explaining IAM Performance: The author suggests that the derived results provide a theoretical justification for the phenomenological success and failure of the Inverse Amplitude Method (IAM). The IAM approximates the left-hand cut using perturbative unitarity conditions. The paper argues that the IAM works remarkably well for the $IJ=11$ channel (predicting the $\rho$ meson) because the left-cut contribution is small. Conversely, it faces systematic challenges in the $IJ=00$ channel (predicting the $f_0(500)$ or $\sigma$ meson) because the left-cut imaginary part is large, making the perturbative approximation less accurate.
• Uncertainty Quantification: The formalism allows for the translation of systematic uncertainties from the right-cut (where unitarization techniques are applied) directly to the left-cut approximation, offering a path to rigorous error control in EFT predictions for new physics scales.

The author notes that while the representation holds for $s$ in the range $-28m^2 \le s < 0$, future work is required to compare this representation directly with the Mandelstam-Chew representation and to fully implement the uncertainty quantification framework in practical unitarization applications.