Fitting assumptions

\(\chi^2\) definition

The overall fit quality is quantified by the log-likelihood, or \(\chi^2\) function, defined as

\[\chi^2 \left ( {\textbf{c}} \right ) \equiv \frac{1}{n_{\rm dat}}\sum_{i,j=1}^{n_{\rm dat}} \left (\sigma^{(\rm th)}_i \left( {\textbf{c}} \right) -\sigma^{(\rm exp)}_i\right) ({\rm cov}^{-1})_{ij} \left ( \sigma^{(\rm th)}_j \left( {\textbf{c}}\right) -\sigma^{(\rm exp)}_j\right)\]

where \(\sigma_i^{\rm (exp)}\) and \(\sigma^{\rm (th)}_i \left(\textbf{c} \right )\) are the central experimental data and corresponding theoretical predictions for the i-th cross-section, respectively.

Individual fits

Individual (one-parameter) fits correspond to varying a single EFT coefficient while keeping the rest fixed to their SM values. While such fits neglect the correlations between the different coefficients, they provide a useful baseline for the global analysis, since there the CL intervals will be by construction looser (or at best, similar) as compared to those of the one-parameters fits.

They are also computationally inexpensive, as they can be carried out analytically from a scan of the \(\chi^2\) profile without resorting to numerical methods.

Another benefit is that they facilitate the comparison between different EFT analyses, which may adopt different fitting bases but whose individual bounds should be similar provided they are based on comparable data sets and theoretical calculations.

In the scenario where a single EFT coefficient, \(c_j\), is allowed to vary while the rest are set to zero, the theoretical cross-section (for \(\Lambda=1\) TeV) given by simplifies to

\[\sigma_m^{\rm (th)}(c_j)= \sigma_m^{\rm (sm)} + c_j\kappa_{m,j} + c_j^2 \tilde{\kappa}_{m,jj}\]

which results in a quartic polynomial form for the \(\chi^2\) when inserted into the \(\chi2\) definition, namely:

\[\chi^2(c_j) = \sum_{k=0}^4 a_k \left(c_j\right)^k\]

Restricting the analysis to the linear order in the EFT expansion further simplifies to a parabolic form:

\[\chi^2(c_j) = \sum_{k=0}^2 a_k \left(c_j\right)^k = \chi^2_0 + b\left( c_j-c_{j,0} \right)^2\]

where \(c_{j,0}\) is the value of \(c_j\) at the minimum of the parabola, and in this case linear error propagation (Gaussian statistics) is applicable.

To determine the values of the quartic polynomial coefficients \(a_k\), it is sufficient to fit this functional form to a scan of the \(\chi^2\) profile obtained by varying the EFT coefficient \(c_j\) when all other coefficients are set to their SM value.

The associated 95 % CL interval to the coefficient \(c_j\) can then be determined by imposing the condition

\[\chi^2(c_j)-\chi^2(c_{j,0}) \equiv \Delta \chi^2 \le 5.91\]

We note that if the size of the quadratic \(\mathcal{O}\left(\Lambda^{-4}\right)\) corrections is sizable, there will be more than one solution for \(c_{j,0}\) and one might end up with pairwise disjoint CL intervals.