SIMCal-W: Supersimple EW Loop Analysis

• SIMCal-W is a supersimple, high-energy one-loop electroweak analysis for e⁻e⁺→W⁻W⁺ that isolates the four dominant helicity-conserving amplitudes with compact analytic formulas in both the SM and MSSM.
• It employs augmented Sudakov and box-induced logarithmic functions to capture leading corrections and achieves errors below 1% at center-of-mass energies above 0.7 TeV.
• The method streamlines computational implementation for collider phenomenology and enhances the capability for new physics searches by enabling efficient model discrimination.

The term SIMCal-W refers to the "supersimple" (sim) high-energy one-loop electroweak (EW) analysis of the four leading helicity-conserving (HC) amplitudes in the process $e^- e^+ \to W^- W^+$. This method provides compact, transparent, and accurate analytic formulas—both in the Standard Model (SM) and Minimal Supersymmetric Standard Model (MSSM)—for the dominant amplitude components at large center-of-mass energies. The core utility of the SIMCal-W approach is to capture the essential dynamical content and leading logarithmic enhancements in one-loop EW corrections, enabling efficient theoretical predictions for high-energy $e^- e^+ \to W^- W^+$, as well as for model discrimination and new physics searches (Gounaris et al., 2013).

1. Theoretical Foundation and Motivation

SIMCal-W is rooted in the observation that, at high energies ($\sqrt{s} \gg m_W$) and fixed angle, the $e^- e^+ \to W^- W^+$ process is overwhelmingly dominated by HC amplitudes. Helicity-violating components, scaling as $m_W/\sqrt{s}$ or $m_W^2/s$, are negligible and omitted. The method leverages the on-shell-renormalized amplitude structure at the one-loop level—initially expressed in Passarino–Veltman (PV) reductions—and isolates the leading logarithmic contributions using asymptotic expansions as in Beccaria et al. This isolates the four HC amplitudes: two transverse–transverse (TT; $F_{--+}, F_{-+-}$) and two longitudinal–longitudinal (LL; $F_{-00}, F_{+00}$).

2. Analytical Structure and Formulation

The SIMCal-W expressions factorize each amplitude as the Born-level result multiplied by a universal one-loop correction, itself rendered as a sum of compact augmented Sudakov and box forms. The principal analytic building blocks are as follows:

• Augmented Sudakov forms:
  • $$\overline{\ln^2 x_{Vi}} \equiv \ln^2 \left( \frac{ -x - i\epsilon }{ m_V^2 } \right) + 4 L_{aVi}$$
  • $$\overline{\ln x_{ij}} \equiv \ln \left( \frac{-x - i\epsilon}{m_i m_j} \right) + b_0^{ij}(m_a^2) - 2$$
• Box-induced forms:
  • $$\overline{\ln^2 r_{xy}} \equiv \ln^2 \left( \frac{ -x - i\epsilon }{ -y - i\epsilon } \right) + \pi^2$$
  • $\ln r_{xy}$

In practice, infrared (IR) regularization is achieved by setting $m_\gamma = m_Z$, with corrections for arbitrary $m_\gamma$ given by straightforward additive terms.

The generic “supersimple” amplitude for TT and LL channels in the SM is:

• For TT:

$$F^{sim}_{\lambda\mu\nu} = F^{Born}_{\lambda\mu\nu} \cdot \frac{\alpha}{16\pi s_W^2} \cdot \{ \text{linear combination of logs} \}$$

• For LL:

$$F^{sim}_{\lambda 00} = F^{Born}_{\lambda 00} \cdot \frac{\alpha}{4\pi} \cdot \{ \text{linear combination of logs} \}$$

Explicit formulas for both TT and LL amplitudes comprise $\sim$20 terms each, detailed in Appendix A of the reference (Gounaris et al., 2013).

3. Modifications for MSSM and New Physics

In the MSSM, the sim amplitudes acquire additional terms accounting for SUSY loop contributions:

• TT amplitudes are modified by chargino and neutralino loops:

$$+ \sum_j |Z^N_{1j} s_W + Z^N_{2j} c_W|^2 \, \overline{\ln t_{\chi^0_j \tilde{e}_L}} + 2c_W^2 \sum_j |Z^+_{1j}|^2 \, \overline{\ln t_{\chi^+_j \tilde{\nu}}}$$

• LL amplitudes are complemented by sfermion, extra-Higgs, and mixed boxes; their explicit structure is available in eqs. (A.17–A.18).

No angle-independent constant terms arise in the MSSM; the gauge-Higgs and gaugino contributions jointly guarantee Sudakov logarithm cancellations consistent with gauge invariance.

For potential new physics, one includes anomalous gauge coupling (AGC) corrections and $Z^\prime$-exchange effects as additive terms to the sim SM amplitudes. AGC enhancements, parameterized as in eqs. (B.1–B.4), predominantly amplify the LL channels as $s/m_W^2$, bifurcating their behavior from the TT sector and thus supporting new physics discrimination via polarized cross sections. $Z^\prime$ mixing effects manifest as resonances near $s \approx m_{Z^\prime}^2$ and can preserve SM cancellation patterns at high energy (Appendix B.2).

4. Numerical Validation and Accuracy

A detailed numerical benchmark employing $\sqrt{s}$ from 0.5 TeV up to 5 TeV and SUSY parameters $(\tilde{m} \sim 0.5$–$2$ TeV, $\tan\beta=20$, $\mu=400$ GeV, $M_1=250$ GeV, $M_2=500$ GeV$)$ reveals maximum relative errors between full one-loop and sim amplitudes as follows:

Energy (TeV)	SM Error	MSSM Error
0.5	~2%	~3%
1.0	≤0.5%	≤1%
2.0	≤0.2%	≤0.5%

Above $\sqrt{s} \gtrsim 0.7$ TeV, the sim approximation matches the full one-loop calculation to better than 1% for both TT and LL amplitudes and all scattering angles [(Gounaris et al., 2013), Figs. 3–4].

5. Implementation and Phenomenological Relevance

SIMCal-W amplitudes are purely algebraic functions of $(s, t)$ or equivalently the scattering angle $\theta$, gauge/Higgs/matter masses, and mixing matrices. Their analytic transparency allows immediate identification of the origin of large logarithmic terms (gauge, Yukawa, gaugino).

The absence of numerical PV integration enables implementation in any computational language and seamless interfacing with event generators. For collider phenomenology (e.g., ILC/CLIC), HC Born amplitudes can be directly substituted by their one-loop sim-corrected forms (plus standard soft-bremsstrahlung subtraction) to achieve $\mathcal{O}(1\%)$ EW precision without high computational cost.

6. Applications in Searches for New Physics

Applications of SIMCal-W include:

• Rapid evaluation of EW one-loop effects for $e^- e^+ \to W^- W^+$ at future linear colliders, especially for SM/MSSM discrimination or as a baseline in new physics searches.
• Isolation and characterization of AGC or $Z^\prime$ contributions, exploiting the distinctive high-energy behavior of the LL amplitudes.
• Construction of total HC amplitudes for direct phenomenological analysis:

$$F^{total}_{HC} = F^{sim\,SM}_{HC} + F^{AGC}_{HC} + F^{Z^\prime}_{HC}$$

Differential cross sections $d\sigma/d\cos\theta$ are computed from only these sim amplitudes, with HV contributions disregarded.

• Streamlining theoretical predictions for EW processes at multi-TeV scales, providing a robust, model-independent framework for interpreting collider data.

7. Outlook and Further Directions

SIMCal-W offers a baseline for precision EW predictions in $e^- e^+ \to W^- W^+$, essential for future lepton collider programs. Extensions to higher-loop accuracy, inclusion of subleading power corrections, or adaptation to other multi-boson channels constitute plausible theoretical developments prompted by the success and compactness of the supersimple formulas (Gounaris et al., 2013).