POPxf

Array of $M$ names identifying each observable $O_m$. Must be an array of unique, non-empty strings, with at least one entry.

Array of $S$ names identifying each model parameter $C_s$ (e.g., Wilson coefficient names). Must be an array of unique, non-empty strings, with at least one entry. In general, this includes $S_\mathbb{R}$ real-valued and $S_\mathbb{C}$ complex-valued parameters with $S = S_\mathbb{R} + S_\mathbb{C}$. The real-valued parameters and the real and imaginary parts of the complex-valued parameters are used as the $R=S_\mathbb{R} + 2\ S_\mathbb{C}$ independent variables of all polynomial terms and can be grouped together in a real-valued parameter vector $\vec{C}$ of length $R$.

Defines the parameter basis (e.g. an operator basis in an EFT). At least one of the two subfields wcxf and custom has to be present. If both subfields are present, any element of parameters (see above) not belonging to the wcxf basis is interpreted as belonging to the custom basis. The subfields are defined as follows:

The renormalisation scale in GeV at which the parameter vector $\vec{C}$, the polynomial coefficients ${\vec{p}_k \supset \vec{b}_k, \vec{c}_k, ...}$, and the observable coefficients ${\vec{o}_m \supset \vec{b}_m, \vec{c}_m, ...}$ and their uncertainties $\vec{\sigma}_m$ are defined. The parameter vector $\vec{C}$ that enters a given polynomial $P_k$ or observable $O_m$ has to be given at the same scale at which the polynomial coefficients $\vec{p}_k$ or observable coefficients $\vec{o}_m$ are defined, such that the polynomial or observable itself is scale-independent up to higher-order corrections in perturbation theory.

This field can take one of two forms:

• single number: A common scale $\mu$ at which all polynomial coefficients $\vec p_k$ or observable coefficients $\vec o_m$ are defined.

  • If the observables $O_m$ are expressed in terms of polynomials $P_k$, the polynomials are functions of the parameters evolved to the common scale $\mu$:

    $$P_k = a_{k} + \vec{C}(\mu) \cdot \vec{b}_{k}(\mu) + \dots\ $$

  • If the observables $O_m$ are themselves polynomials, they are themselves functions of the parameters evolved to the common scale $\mu$:

    $$O_m = a_m + \vec{C}(\mu) \cdot \vec{b}_m(\mu) + \dots\ $$

• array of numbers: An array defining separate scales $\mu_k$ of polynomial coefficients $\vec p_k$ if metadata.polynomial_names is present, or separate scales $\mu_m$ of observable coefficients $\vec o_m$ if metadata.polynomial_names is absent.

  • If metadata.polynomial_names is present, the observables $O_m$ are expressed in terms of polynomials $P_k$ and each polynomial is a function of the parameters evolved to its corresponding scale $\mu_k$:

    $$P_k = a_{k} + \vec{C}(\mu_k) \cdot \vec{b}_{k}(\mu_k) + \dots\ $$

    The length and order of the array defining the scales $\mu_k$ must match those of the field metadata.polynomial_names. To avoid ambiguities, the following restrictions apply to this case:

    • data.observable_central must be absent;
    • data.observable_uncertainties must be absent or only define uncertainties for the parameter-independent terms (i.e. only the SM uncertainties in EFT applications).

  • If metadata.polynomial_names is absent, the observables $O_m$ are themselves polynomials and each observable is a function of the parameters evolved to its corresponding scale $\mu_m$:

    $$O_m = a_m + \vec{C}(\mu_m) \cdot \vec{b}_m(\mu_m) + \dots\ $$

    The length and order of the array defining the scales $\mu_m$ must match those of the field metadata.observable_names.

This field is required to express observables as functions of polynomials. It requires the simultaneous presence of metadata.observable_expressions and data.polynomial_central.

Array of $K$ names identifying the individual polynomials $P_k$ that enter the observable predictions through the functions defined in metadata.observable_expressions (see below). Must contain unique, non-empty strings.

This field is required to express observables as functions of polynomials. It requires the simultaneous presence of metadata.polynomial_names and data.polynomial_central.

Defines how each observable is constructed from the named polynomials. Must be an array of $M$ objects, one per observable. The length and order of the array must match those of the observable_names field. Each object must contain:

Specifies the maximum degree of polynomial terms included in the expansion. If omitted, the default value is 2 (i.e., quadratic polynomial). Values higher than 2 may be used to represent observables involving higher-order terms in the model parameters. The current implementation of the JSON schema defining the data format supports values up to 5. Higher degrees are not prohibited in principle but are currently unsupported to avoid excessively large data structures.

Collects relevant data that may be required by a third party to reproduce the prediction. Each element of the array should be an object that corresponds to a step in the workflow and has three predefined fields: description, tool and inputs, specified below. In addition, any additional fields containing data deemed useful in this context can be included.

Schematic example:

  "reproducibility": [
    {
      "description": "Description of the first step",
      "tool": { ... },
      "inputs": { ... }
    },
    {
      "description": "Description of the second step",
      "tool": { ... },
      "inputs": { ... }
    },
    ...
  ]

The predefined fields are as follows:

Optional free-form metadata for documentation purposes. May include fields such as authorship, contact information, date, description of the observable, information identifying the associated correlation file (e.g. hash value or filename), or external references. The format is unrestricted, allowing any JSON-encodable content.

This field is required to express observables as functions of polynomials. It requires the simultaneous presence of metadata.polynomial_names and metadata.observable_expressions.

An object representing the central values of the polynomial coefficients $\vec{p}_k$ for each named polynomial $P_k$. Each key must be a monomial key as defined above. Each value must be an array of $K$ numbers whose order matches metadata.polynomial_names.

An object representing the central values of the observable coefficients $\vec{o}_m$ for each observable $O_m$. In case the observables are not themselves polynomials, the observable coefficients correspond to the polynomial approximation of the observables obtained from a Taylor expansion of the observable expressions defined in metadata.observable_expressions. Each key must be a monomial key as defined above. Each value must be an array of $M$ numbers whose order matches metadata.observable_names.

An object representing the uncertainties on the observable coefficients $\vec{\sigma}_m$ for each observable $O_m$. In case the observables are not themselves polynomials, the observable coefficients correspond to the polynomial approximation of the observables obtained from a Taylor expansion of the observable expressions defined in metadata.observable_expressions. The fields specify the nature of quoted uncertainty. In many cases there may only be a single top-level field, "total", but multiple fields can be used to specify a breakdown into several sources of uncertainty (e.g., statistical, scale, PDF, ...). To avoid mistakes, the names of the top-level fields must not have the format of a monomial key (i.e., stringified tuples as defined above). The value of each top-level field can either be an object or an array of floats. Objects must have the same structure as observable_central, arrays must have length $M$. If instead of an object, an array of floats is specified, it is assumed to correspond to the parameter independent uncertainty only (e.g. the uncertainty on the SM prediction). This would be equivalent to specifying an object containing a single element with the monomial key of the constant term (e.g. "('','')" for a quadratic polynomial).