Open image in new window : a Python package for the running and matching of Wilson coefficients above and below the electroweak scale
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Abstract
wilson is a Python library for matching and running Wilson coefficients of higherdimensional operators beyond the Standard Model. Provided with the numerical values of the Wilson coefficients at a high new physics scale, it automatically performs the renormalization group evolution within the Standard Model effective field theory (SMEFT), matching onto the weak effective theory (WET) at the electroweak scale, and QCD/QED renormalization group evolution below the electroweak scale down to hadronic scales relevant for lowenergy precision tests. The matching and running encompasses the complete set of dimensionsix operators in both SMEFT and WET. The program builds on the Wilson coefficient exchange format (WCxf) and can thus be easily combined with a number of existing public codes.
1 Introduction
The Standard Model (SM) [1, 2, 3] is considered to be an effective theory valid only up to a new physics scale \(\varLambda \), which negative searches for new particles at the LHC likely relegate to well above the electroweak (EW) scale. If no light degrees of freedom beyond the SM are assumed, any new physics effect in processes proceeding at energies well below \(\varLambda \) can be described by local interactions among SM fields invariant under the SM gauge symmetry [4, 5]. This effective field theory (EFT) approach [6, 7] to new physics not only allows to resum large logarithms that might invalidate calculations in perturbation theory for vastly different scales relevant in a given process, but also serves as a convenient intermediate step between “model building” in the UV and lowenergy phenomenology. If new physics predictions for experimental observables are expressed in terms of Wilson coefficients of an EFT beyond the SM, the investigation of the lowenergy implications of a concrete new physics model becomes much simpler since only the Wilson coefficients need to be calculated at the appropriate scale.
While the EFT approach to new physics has been ubiquitous in quark flavour physics – dealing with processes at energies of few GeV – for a long time already, the experimental indications that \(\varLambda \) lies well above the electroweak scale have led to the realization that this approach is also valuable for processes of electroweak scale energies like Higgs physics or electroweak precision tests (see [8] and references therein). In contrast to the EFT below the electroweak scale, that is conventionally called the weak effective theory (WET) [9, 10, 11] and only contains QED and QCD gauge interactions, the EFT above the electroweak scale, conventionally called SMEFT^{1} [13, 14, 15], contains \(SU(2)_L\) interactions that do not conserve flavour. Consequently, quantum effects lead to an interesting interplay between processes with and without flavour change and call for a global approach.
 1.
Compute the SMEFT Wilson coefficients at \(\varLambda \).
 2.
Perform the renormalization group (RG) evolution of the SMEFT Wilson coefficients down to the electroweak scale.
 3.
Match the complete set of SMEFT Wilson coefficients onto the WET.
 4.
Perform the RG evolution of WET Wilson coefficients.
 5.
If the process proceeds at energies below the b quark mass, repeat the last two steps for the WET with reduced numbers of quark and lepton flavours as appropriate.
 6.
Compute the process of interest as a function of the lowenergy Wilson coefficients.

The complete basis of SMEFT operators first derived in [4] and for a nonredundant set of operators in [5].

Analytical solutions to the oneloop RG evolution of all flavour violating operators in WET [10].

The complete RG evolution of WET operators [11].

The complete treelevel matching of SMEFT onto the WET [9, 20].

The definition of a Wilson coefficient exchange format (WCxf) that allows to define EFTs, bases of Wilson coefficients, and facilitates exchanging numerical values of Wilson coefficients between different codes [21].

The SMEFT RG evolution was ported from (and is tested against) the DsixTools Mathematica package [22].

The QCD evolution of quark masses and the strong coupling constant is computed with the pythonrundec package that wraps the CRunDec module [23].

The SM \(\overline{\text {MS}}\) parameters at the electroweak scale have been obtained with the mr package [24].
2 Description
2.1 Extraction of standard model parameters in SMEFT
 1.
We determine all the SM parameters in the \(\overline{\text {MS}}\) scheme [25] at the scale \(M_Z\).
 2.
We invert the relations between the effective \(\overline{\text {MS}}\) SM parameters and their counterparts in SMEFT, that are given e.g. in [26].
 3.
We iteratively determine the SM parameters at the UV scale by running up and down with the SM boundary conditions imposed at the scale \(M_Z\) and the Wilson coefficient boundary conditions at the UV scale \(\varLambda \).

For the running of the quark masses to the scale \(M_Z\), we have used the pythonrundec package [23].

For the determination of the running top, W, Z, and Higgs masses, we have used the mr package [24].

For the lepton masses, we have neglected the \(O(\alpha _e)\) shift from the conversion to the \(\overline{\text {MS}}\) scheme.

We do not display uncertainties as fixed values are used in the code. We expect the parametric errors to be subdominant to other uncertainties in the calculation, e.g. from the iterative determination of highscale SM parameters.^{3}
SM \(\overline{\text {MS}}\) parameters at the scale \(M_Z\). Masses are given in units of GeV
Par.  Value  Par.  Value 

\(\alpha _e\)  1 / 127.9  \(m_u\)  0.00127 
\(\alpha _s\)  0.1185  \(m_d\)  0.00270 
\(V_{us}\)  0.2243  \(m_s\)  0.0551 
\(V_{cb}\)  0.04221  \(m_c\)  0.635 
\(V_{ub}\)  0.00362  \(m_b\)  2.85 
\(\gamma \)  1.27  \(m_t\)  169.0 
\(m_e\)  0.000511  \(m_W\)  80.20 
\(m_{\mu }\)  0.1057  \(m_Z\)  91.46 
\(m_{\tau }\)  1.777  \(m_h\)  130.6 
We note that we treat the CKM elements as elements of a unitary \(3\times 3\) matrix. Dimensionsix contributions to the W coupling to quarks are thus not absorbed in effective CKM elements, as done e.g. in [26]. We find this procedure more convenient for our purposes; in particular, it allows to continue to use unitarity relations in lowenergy calculations in flavour physics. While this blurs the connection between these CKM elements and the semileptonic decays that are used to measure them, we note that this connection is anyway blurred in SMEFT due to direct dimensionsix fourfermion contributions to these decays that can lead to a processdependent shift of the apparent CKM element (see e.g. [27] for a discussion of \(s\rightarrow u\) transitions and [28] for \(b\rightarrow c\) transitions).
2.2 RG evolution in SMEFT
As an important caveat, we caution the reader that the numerical inputs and outputs, using the nonredundant basis defined by the WCxf convention, differ from the conventions used in [16, 18, 19], where a redundant basis of flavour indices is employed, by symmetry factors in some cases. We refer to appendix A of [29], where this issue is discussed in detail.
2.3 Matching from SMEFT to WET
We implement the complete treelevel matching from SMEFT to WET as derived in [9]. It includes the full set of nonredundant gaugeinvariant dimension six operators in both theories. The matching is performed at the EW scale.
2.4 RG evolution in WET
 1.
We take the beta functions from [9], discarding terms that are quadratic in dipole operator coefficients (these terms correspond to dimension eight contributions when matching from the SMEFT with linearly realized electroweak symmetry breaking).
 2.We rescale dipole operators and threegluon operators in the following way:$$\begin{aligned}&{\bar{f}}^i_{L} \sigma ^{\mu \nu } f^j_{R}\, F_{\mu \nu } \rightarrow \frac{e}{g_s^2}m_f{\bar{f}}^i_{L} \sigma ^{\mu \nu } f^j_{R}\, F_{\mu \nu } \,, \end{aligned}$$(2)$$\begin{aligned}&{\bar{f}}^i_{L} \sigma ^{\mu \nu } T^A f^j_{R}\, G^A_{\mu \nu } \rightarrow \frac{1}{g_s} m_f {\bar{f}}^i_{L} \sigma ^{\mu \nu } T^A f^j_{R}\, G^A_{\mu \nu } \,, \end{aligned}$$(3)where \(m_f=\text {max}(m_{f_i}, m_{f_j})\). This allows us to write the RGEs in the simple form$$\begin{aligned}&G_\mu ^{A\nu } G_\nu ^{B\rho } G_\rho ^{C\mu } \rightarrow \frac{1}{g_s}G_\mu ^{A\nu } G_\nu ^{B\rho } G_\rho ^{C\mu } \,, \end{aligned}$$(4)Note in particular that there are no linear or mixed terms in \(g_s\) or e. Thanks to the rescalings, the anomalous dimension matrices \(\gamma ^{s,e}\) only contain numbers and ratios of fermion masses, which are RG invariant to \(O(\alpha _s)\) and thus can be treated as constants to good approximation.$$\begin{aligned} \frac{d C_i}{d\ln \mu } = \frac{g_s^2}{16\pi ^2} \sum _j \gamma ^s_{ji} C_i+ \frac{e^2}{16\pi ^2} \sum _j \gamma ^e_{ji} C_i \,. \end{aligned}$$(5)
 3.
3 Installation
Installing wilson only requires a system with Python version 3.5 or above. It works on Linux, Mac OS, and Windows. The most recent version can be installed directly from the Python package index by issuing the command^{6}
in the terminal, without root privileges. This will automatically install the wcxf package and command line interface as well, if not already available on the system. When a new version is available, the package can be upgraded with
4 Usage
4.1 Initializiation
Using the wilson package in a Python script or interactive session starts by creating a Wilson object that represents a point in EFT parameter space. On creating the instance, initial values of the Wilson coefficients have to be specified at some scale, e.g. the new physics scale \(\varLambda \), in a given EFT and basis. For example, the commands

wcxf.WC represents a set of numerical Wilson coefficients at a fixed scale in a fixed EFT and basis;

wilson.Wilson represents a point in the parameter space of the EFT beyond the SM, that can be evolved to different scales and translated to different bases within the same EFT without loss of generality, or matched to EFTs valid at lower energies.
4.2 Matching and running
Running, i.e. performing the RG evolution in SMEFT and WET, as well as matching from SMEFT to WET (and from WET with five active quark flavours to the variants of WET valid below the bottom and charm mass scales) is the main purpose of the wilson package. Having initialized a Wilson object as described in Sect. 4.1 – we will continue to call this instance mywilson – the user can obtain Wilson coefficient values (in the form of wcxf.WC instances) in different EFTs, at different scales, in different bases, through the method match_run:^{7}
The names of admissible EFTs and bases can be found on the WCxf website [32].
We note that the output scale can also be higher than the input scale, but only if the output EFT is the same as the input EFT. In this case, the RG evolution (in WET or SMEFT) will be performed from the low input scale to the high output scale. Since the matching is not bijective, this cannot be done across EFT thresholds.
The default behaviour of the Wilson class can be modified with a few user options that can be modified either on a single instance or globally for all future instances of the class (e.g. when importing the package),

’smeft_accuracy’ – set accuracy of the SMEFT RG evolution to numerical integration (value ’integrate’, default) or leadinglogarithmic approximation (’leadinglog’), which is less accurate but much faster.

’qcd_order’, ’qed_order’ – set the order of QED and QCD anomalous dimensions to be taken into account in the WET RG running. Currently both values are restricted to 1 (default, leading order) or 0 (off).

’smeft_matchingscale’ – set the scale (in GeV) where SMEFT is matched onto WET. Defaults to 91.1876 (the central value of the \(Z^0\) mass).

’mb_matchingscale’, ’mc_matchingscale’ – set the scales (in GeV) where WET is matched onto WET4 and WET4 onto WET3. Default to 4.2 and 1.3, respectively.
4.3 Interfacing with other codes
Since wilson builds on the Wilson coefficient exchange format WCxf, it is straightforward to import and export from and to programs supporting this standard. While the import has already been discussed above, the export can simply leverage the methods provided by the wcxf Python package, e.g.
An even simpler data exchange is possible for codes written in Python themselves. In particular, the flavio package [17], that can compute predictions for a plethora of observables in quark and lepton flavour physics, directly makes use of the wilson package for the RG evolution, matching, and translation, starting from version v0.28. Functions that accept new physics Wilson coefficient values can be directly provided with a Wilson instance. This also allows to compute observables in terms of SMEFT Wilson coefficients. For example,
5 Example
where the parameter variables have been set to numerical values.^{8} This Wilson instance can now be used to compute predictions for the relevant constraints using flavio, as discussed in Sect. 4.3:
Using this procedure, in Fig. 2 we have reproduced the result of Refs. [33, 34], where the four free parameters are scanned as: \(\lambda _{23}^q \in [0.05,0]\), \(\lambda _{23}^{\ell } \in [0.5,0.5], C_{1,3} \in [4,0]\), and the scale \(\varLambda \) is set at 1 TeV. This shows that a simultaneous explanation of the charged and neutral current anomalies is disfavoured in this simplified scenario.
6 Summary
We have presented wilson, a Python package for the RG evolution, matching, and basis translation of Wilson coefficients beyond the SM. Starting from numerical values of Wilson coefficients at a high scale \(\varLambda \), it automatically performs the necessary steps to return the Wilson coefficients at low energies relevant for precision measurements probing physics beyond the SM. Built on the Wilson coefficient exchange format (WCxf), wilson can be easily linked with a number of public codes, e.g. to directly compute the predictions for lowenergy observables, as demonstrated in Sect. 5.
While wilson is currently limited to oneloop RG evolution in SMEFT and WET and to treelevel matching, the structure of the code is general enough to be generalized to higher loop orders in the running and to looplevel matching (which is partially known, see e.g. [20]) in the future. It has already been used in several NP analyses in the context of B anomalies [49] and \(\varepsilon /'\varepsilon \) [50, 51, 52]. Being an open source project with a permissive license,^{9} contributions from the community are welcome via the public code repository [53]. Further information to wilson can be found on the wilson web page https://wilsoneft.github.io/.
Footnotes
 1.
Throughout, we work with the EFT above the electroweak scale with linearly realized electroweak symmetry breaking (see [12] and references therein for a discussion of the nonlinear case).
 2.
Steps 3.–5. can be omitted for observables at electroweak scale energies.
 3.
To check the accuracy of the iterative determination of SM parameters, the class wilson.run.smeft.SMEFT, that is initialized by a wcxf.WC instance, provides a method get_smpar, that computes the predicted values for the SM \(\overline{\text {MS}}\) parameters at the electroweak scale, which should correspond to the values in Table 1.
 4.
 5.
In the WET RG evolution, we restrict ourselves to baryon and lepton number conserving operators for the time being.
 6.
The name of the Python 3 executable might differ depending on the system.
 7.
Tiny nonzero entries of Wilson coefficients can be traced back to a finite numerical precision used for the solution of the RGEs and can safely be neglected.
 8.
Note: wilson does not accept symbolic inputs.
 9.
wilson is released under the MIT license.
Notes
Acknowledgements
The work of D. S. and J. A. is supported by the DFG cluster of excellence “Origin and Structure of the Universe”. We thank Xuanyou Pan and Matthias Schöffel for important technical support in the development phase. J. K. thanks Michael Paraskevas for discussions.
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