Signal Modelling with Overdetermined Morphing Technique

Precise modelling of a signal in processes with multiple observables, exhibiting a complex dependency on the underlying parameters, is often a difficult and challenging task. Predicting the results of experimental measurements in high-energy physics reactions serves a good example. The reaction rates and distributions of momenta of the final state particles, depend on the parameters of the underlying physics model in a non-trivial way. The conventional way to predict the experimental of observables is to use the Monte Carlo morphing technique on a finite discrete set of simulated samples. In this article we extend this technique by using the overdetermined morphing basis. We show that our approach yields a better result than the traditional technique in terms of the statistical power and uniformity of the resulting prediction, and delivers better precision. We further demonstrate that the overdetermined morphing is better suited for description of extended regions of phase space than the conventional morphing. We use the modelling of kinematic distributions of the final state particles produced in decays of non-Standard Model Higgs bosons as an example.

💡 Research Summary

The paper addresses the challenge of accurately modeling signals in high‑energy physics processes where multiple observables depend non‑trivially on underlying model parameters. Traditional Monte‑Carlo (MC) morphing predicts experimental observables by linearly combining a finite set of simulated samples. This approach assumes that the squared matrix element, which determines the observable, can be expressed as a polynomial in the model couplings. By generating separate MC samples for each pure‑coupling term (|Oi|²) and each mixed term (Re(Oi* Oj)), one can construct a linear system whose solution yields the morphing weights for any target parameter point.

However, when the number of input samples equals the number of independent polynomial terms, the resulting Λ matrix is square and often poorly conditioned. A large condition number (κ) leads to numerical instability, and the statistical power of the morphed distribution varies dramatically across the parameter space. The authors quantify this effect using two metrics: the effective event ratio N₁ (the fraction of events effectively retained after morphing) and the Kish effective sample size N₂ (which measures the statistical power of the weighted sample). In conventional morphing, N₁ and N₂ show pronounced peaks at the locations of the base samples and drop to near zero in regions far from them, as illustrated in Fig. 1.

To overcome these limitations, the authors propose an “overdetermined morphing” technique. The key idea is to use more MC samples than the number of independent polynomial terms, turning the Λ matrix into an M × N (M > N) rectangular system. They then solve the resulting overdetermined linear equations in a least‑squares sense, minimizing the residual ‖y − Bx‖₂. Rather than using the normal equations (which square the condition number), they employ QR factorisation (B = QR) to obtain a numerically stable solution x = R⁻¹Qᵀy. This approach preserves the condition number of the original matrix and avoids the amplification of rounding errors.

The method is demonstrated on a realistic physics example: vector‑boson‑fusion (VBF) production of a heavy scalar Higgs boson (mass = 9 GeV) followed by decay into a pair of vector bosons (VV). The model includes the Standard Model coupling (fixed to unity) and a beyond‑Standard‑Model (BSM) CP‑even operator parameterized by k_HZZ, which appears both in production and decay vertices. According to the polynomial counting formula (Eq. 6), the minimal number of base samples required to span the BSM coupling space is five. The authors generate these five samples using MadGraph within the Higgs Characterisation framework and compare the standard morphing results with those obtained from an overdetermined basis consisting of fifteen samples covering a broader range of k_HZZ values (−14 to +10).

Figures 2 and 3 present the performance comparison. In the standard morphing case, the effective weight ratio N₁ exhibits strong non‑uniformity, with peaks at the five base points, and the Kish ratio N₂ falls sharply outside the interval spanned by the base samples. Overdetermined morphing, by contrast, yields an almost flat N₁ across the entire k_HZZ range and maintains N₂ ≈ 1 throughout, indicating a uniformly high statistical power. Within the interval −8 ≲ k_HZZ ≲ −1, the overdetermined approach achieves N₂ values roughly three times larger than the standard method, although it also uses three times more MC events, so the Kish efficiency (N₂/N_gen) remains comparable. Outside this window, the standard morphing’s N₂ collapses (N₂ ≪ 1), whereas the overdetermined scheme retains N₂ ≳ O(1), delivering a clear advantage in effective statistics across most of the parameter space.

The paper’s main insights are: (1) Expanding the morphing basis beyond the minimal set and solving the resulting overdetermined system with QR‑based least‑squares dramatically improves the conditioning of the problem and stabilizes the numerical solution. (2) The resulting morphing weights are more uniformly distributed, providing consistent statistical sensitivity throughout the multidimensional parameter space. (3) Because the technique relies only on a polynomial dependence of the observable on the parameters, it is applicable to any physics model with such a structure, not just Higgs‑related processes. (4) The method allows incremental improvement: additional MC samples can be added to the basis without discarding existing ones, enabling systematic refinement of the model’s accuracy.

In summary, overdetermined morphing offers a robust, statistically powerful, and flexible alternative to conventional MC morphing for signal modeling in high‑energy physics. By leveraging QR factorisation and least‑squares fitting, it mitigates the pitfalls of ill‑conditioned bases, ensures uniform coverage of the parameter space, and can be readily generalized to other complex, multi‑observable analyses. This advancement promises to enhance the precision of theoretical predictions and the sensitivity of experimental searches for new physics.