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CoDEx: Wilson coefficient calculator connecting SMEFT to UV theory

  • Supratim Das Bakshi
  • Joydeep Chakrabortty
  • Sunando Kumar PatraEmail author
Open Access
Regular Article - Theoretical Physics
  • 87 Downloads

Abstract

CoDEx is a Mathematica\(^{\textregistered }\) package that calculates the Wilson coefficients (WCs) corresponding to effective operators up to mass dimension 6. Once the part of the Lagrangian involving single and multiple degenerate heavy fields, belonging to some beyond standard model (BSM) theory, is given, the package can then integrate out propagators from the tree and 1-loop diagrams of that BSM theory. It then computes the associated WCs up to 1-loop level, for two different bases: Open image in new window and Open image in new window . "CoDEx" requires only very basic information as regards the heavy field(s), e.g., color, isospin, hypercharge, mass, and spin. The package first calculates the WCs at the high scale (mass of the heavy field(s)). We then have an option to perform the renormalization group evolutions (RGEs) of these operators in Open image in new window basis, a complete one (unlike Open image in new window ), using the anomalous dimension matrix. Thus, one can get all effective operators at the electro-weak scale, generated from any such BSM theory, containing heavy fields of spin 0, 1/2, and 1. We provide many example models (both here and in the package documentation) that more or less encompass different choices of heavy fields and interactions. Relying on the status of the present day precision data, we restrict ourselves up to dimension-6 effective operators. This will be generalized for any dimensional operators in a later version.

1 Introduction

It is a perplexing time for particle physics. On one side we are cherishing the discovery of the standard model (SM)-Higgs like particle, considered to be the pinnacle of success of the SM; on the other hand we have enough reason to believe the existence of theories beyond it. To address the shortcomings of the SM, many beyond-standard model (BSM) scenarios are proposed at very different scales. It is believed that any such theory, which contains the SM as a part of it, will affect the electro-weak and the Higgs sector. Thus the precision observables are expected to carry the footprints of the new physics, unless it is in the decoupling limit.

The ongoing and proposed future experiments are expected to improve the sensitivity of these precision observables at per mille level. Thus we can indirectly estimate the allowed room left for some BSM physics, even in the case of non-observation of new resonances. This motivates us to look into the BSM scenario through the tinted glass of standard model effective field theory (SMEFT). The basic idea of SMEFT is quite straightforward: integrate out heavy non-SM degrees of freedom and capture their impact through the higher mass dimensional operators—\(\sum _{i}(1/\Lambda ^{d_i - 4}) C_i \mathcal {O}_i\). Here \(d_i\) is the mass dimensionality of the operator \(\mathcal {O}_i\) (starting from 5), and \(C_i\) is the corresponding Wilson coefficient, a function of the BSM parameters. It is important to note that the choice of operator basis, i.e., the explicit structure of the \(\mathcal {O}_i\), is not unique. Among different choices we restrict ourselves to the Open image in new window [1, 2] and Open image in new window [3, 4, 5, 6] bases. These bases can be transformed from one to another. \(\Lambda \) is the cut-off scale at which all WCs are computed (\(C_i(\Lambda )\)) and usually identified as the mass of the heavy field being integrated out. This EFT approach relies on the validity of the perturbative expansion of the S-matrix in powers of \(\Lambda ^{-1}\) (UV-scale), and the resultant series is expected to pass the convergence test. As this scale is higher than the scale \(M_Z\), where the precision test is performed, dimension-6 operators are more suppressed than the dimension-5 ones and so on. Now we ask where to truncate the \(1/\Lambda \) series. This decision is made case by case, based on the achieved (expected) precision level of the observables at present (future) experiments [7]. One can consult the lectures in [8, 9, 10, 11, 12] where effective field theory has been introduced and discussed in great detail. Several other packages and libraries are available in the literature, which address various issues regarding SMEFT operators and the corresponding Wilson coefficients, from basis transformation to running of the coefficients [13, 14, 15, 16, 17]. A universal data exchange format for BSM Wilson coefficients has also been developed (WCxf) [18].

Now the nagging questions are: (a) Why use SMEFT instead of doing the full calculation, using the supposedly more accurate BSM Lagrangian? (b) How can one ensure that the difference between the results, computed in the SMEFT approach using a truncated S-matrix and those obtained using the full BSM theory, is imperceptible (in the precision tests)?

The computation with the full BSM is involved and tedious, and that too at loop level. The cut-off \(\Lambda \) is chosen in such a way that the \(M_Z/\Lambda \) series is converging, which ensures that the truncation of this series at some finite order is safe and sufficient. Even then, the question remains: How do we connect the physics of two different scales, namely UV and the \(M_Z\)? The WCs, which we are computing using SMEFT, are at the scale \(\Lambda \), but the observables are measured at the \(M_Z\) scale. Hence, we need to evolve the \(C_i(\Lambda )\) to obtain \(C_i(M_Z)\), using the anomalous dimension matrix \((\gamma )\). While performing the renormalization group evolutions (RGEs) of the \(C_i'\), we need to choose \(\gamma \) carefully, as it is basis dependent. Thus we need to choose only those bases in which the precision observables are defined, and it is important to ensure that the basis we are working with is a complete one. As the matrix \(\gamma \) contains non-zero off-diagonal elements, it is indeed possible to generate, through RGEs, some new effective operators, which were absent at the \(\Lambda \) scale. These effective field theory approach has been successfully used in the context of precision data and Higgs phenomenology; for details see [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

With this backdrop, we introduce Open image in new window , a Mathematica\(^{\textregistered }\) package, which can integrate out the heavy field propagator(s) from tree and 1-loop processes and can generate SM effective operators up to dimension 6. It also provides the Wilson coefficients as a function of the BSM parameters.

We know that the 1-loop results depend on the choice of renormalization scheme; thus the choice of the scheme is very important and crucial. Our code provides the 1-loop results under the \(\overline{\text {MS}}\) renormalization scheme. We would like to make a remark on the results obtained at the 1-loop level: The present version of this code can integrate out only heavy propagators, thus the 1-loop diagrams that involve both heavy and light types of propagators are ignored which will have less impact on the presence of other contributions; see Ref. [20] (Sect. 1.2). In the case of operators of dimension \(< 5\), the inclusion of a 1-loop correction is related to the renormalization of the full theory. There are standard tools to compute such loop corrections. In this paper our focus is on the evaluation of higher dimensional operators after integrating out the heavy BSM fields. That is why we have not implemented the loop correction to the operators of mass dimension less than 5. From this perspective, the 1-loop results are incomplete.

The article is organized in the following manner: in Sect. 2, we briefly discuss the underlying principle of Open image in new window from a theoretical perspective. Details as regards downloading and installation are in Sect. 3.1. In the remainder of Sect. 3, we provide a guideline to define the heavy field(s) and build the working Lagrangian (Sect. 3.2), provide a list of all the functions that are necessary to run Open image in new window in detail (Sect. 3.3), and explain the way Open image in new window takes care of the RGEs of the WCs down to the electro-weak scale (Sect. 3.4). In the next section, we provide the user with one detailed work flow to use the package for a model with a single heavy electro-weak \(SU(2)_L\) real singlet scalar in it (Sect. 3.5). Sequentially following these steps should enable one to find the effective operators up to mass dimension 6 and the respective WCs at the high scale for that model. In the appendix we provide various example models to encapsulate different types of fields that are used frequently to build BSM scenarios. One may consult Refs. [4, 5, 6, 42] regarding the running of the SM effective operators.

2 The underlying principle

In this section, we will briefly discuss the adopted method, based on the idea of a Covariant Derivative Expansion (CDE), to integrate out the heavy fields to compute the Wilson coefficients (WCs). This was introduced in Ref. [43] and then extended in Ref. [44]. As we are performing this “integrating out of the field(s)" order by order in perturbation theory, we need to respect the gauge invariance at each and every step. The perturbative expansion thus is required to be done in terms of some gauge covariant quantities. Thus, the covariant derivative is the ‘chosen one’. CDE is not only restricted to quantify the integrating out of heavy fields, rather it has a wider impact; see [19, 20, 21] for details. The method of integrating out different types of heavy fields using functional methods and the basis dependency is discussed in many places in the literature; see [14, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] for details.

Considering the status of present and prospect of future experiments, we can adjudge ourselves safe, when we restrict ourselves to only dimension-6 operators including the tree and 1-loop parts of the effective action. The modus operandi of Open image in new window is based on the method of CDE discussed in [19, 20, 21]. Briefly, this is what Open image in new window can give you:
  • It will integrate out the heavy BSM field(s) while respecting the gauge invariance.

  • It will generate Wilson coefficients at tree and/or 1-loop level.

  • It will perform RG evolutions of the effective operators generated at some high scale and provide the operators at the electro-weak scale.

Let us consider a BSM Lagrangian in the following form:
$$\begin{aligned} \mathcal {L}^{(\mathrm{{BSM}})} \equiv \mathcal {L}^{(\mathrm{{BSM}})} ( \phi , \Phi )=\mathcal {L}^{(\Phi )} + \mathcal {L}^{(\phi )} + \mathcal {L}^{(\phi , \Phi )}_{int}\,, \end{aligned}$$
(1)
where \(\phi \) and \(\Phi \) represent light (SM) and heavy (BSM) fields, respectively. Here \(\mathcal {L}^{(\phi )}, \mathcal {L}^{(\Phi )}\), and \(\mathcal {L}^{(\phi , \Phi )}_{int}\) are three different sectors of the BSM Lagrangian containing, respectively, only heavy fields, only light fields, and both heavy and light fields.
To proceed, we have to first solve the following Euler–Lagrange (EL) equation to compute the solution for the heavy field:
$$\begin{aligned} \frac{\partial }{\partial \Phi } \mathcal {L}^{(\mathrm{{BSM}})}(\phi , \Phi ) = \mathcal {D}_{\mu } \ \frac{\partial }{\partial (\mathcal {D}_{\mu } \Phi )} \mathcal {L}^{(\mathrm{{BSM}})}(\phi , \Phi ) , \end{aligned}$$
(2)
where \(\mathcal {D}_{\mu }\) is the covariant derivative corresponding to the heavy field \(\Phi \). To find the EL-equations, we will concentrate on the \(\mathcal {L}^{(\mathrm{{BSM}})}( \phi , \Phi ) \) part only. If we express this part of the Lagrangian as a polynomial in \(\Phi \), e.g., \(\mathcal {L}^{(\phi )}_{I} \cdot \Phi + \mathcal {L}^{(\phi )}_{II} \cdot (\Phi )^{2} + \ldots \), the coefficients can then be written as
$$\begin{aligned} \mathcal {L}^{(\phi )}_{I} = \ \mathcal {O}_{\mathcal {D}} \cdot \hat{\Phi } , \end{aligned}$$
(3)
where \(\mathcal {O}_{\mathcal {D}}\) contains the information regarding the covariant derivative of the heavy field and the functional of the light (SM) fields. In general, \(\mathcal {O}_{\mathcal {D}}\) is in the form of an elliptic operator, e.g. \(\mathcal {O}_{\mathcal {D}} = \mathcal {D}^{2} + M^{2} + \mathcal {L}_{II}^{\phi } \), where M is the mass of the heavy field to be integrated out. Thus the heavy field solution can be rewritten as
$$\begin{aligned} \hat{\Phi } = [\mathcal {O}_{\mathcal {D}}]^{-1} \cdot \mathcal {L}^{(\phi )}_{I} . \end{aligned}$$
(4)
The operator \([\mathcal {O}_{\mathcal {D}}]^{-1}\) can be Taylor-expanded, where terms are suppressed by \(M^{2n}\) (n takes integer values starting from 1). This series is convergent as M is much larger than the allowed maximum momentum transfer in the low energy theory. Thus we can truncate this series based on our requirement of the mass dimension of the effective operator. It is important to note that in the theories where the \(\mathcal {L}^{(\phi )}_{I}\) term is absent, any tree-level effective operator will not be generated after integrating out \(\Phi \).
The next task is to compute these effective operators at loop level. We will restrict our computation to the 1-loop level, relying on the precision of present data. The effective Lagrangian at 1-loop level is given by [19, 20, 21]
$$\begin{aligned}&\mathcal {L}^{({\text {dim-6}})}_{1{\text {-loop}}}[\phi ] \nonumber \\&\quad =\frac{c_s}{(4\pi )^2} \, \text {tr}\, \Bigg \{ \frac{1}{m^2}\Bigg [ -\frac{1}{60} \, \big (P_{\mu }G_{\mu \nu }'\big )^2 - \frac{1}{90} \, G_{\mu \nu }'G_{\nu \sigma }'G_{\sigma \mu }' \nonumber \\&\qquad -\frac{1}{12} \, (P_{\mu }U)^2 - \frac{1}{6}\, U^3 - \frac{1}{12}\, U G_{\mu \nu }'G_{\mu \nu }' \Bigg ] \nonumber \\&\qquad + \frac{1}{m^4} \Bigg [\frac{1}{24} \, U^4 + \frac{1}{12}\, U \big (P_{\mu }U\big )^2 + \frac{1}{120}\, \big (P^2U\big )^2 \nonumber \\&\qquad +\frac{1}{24} \, \Big ( U^2 G'_{\mu \nu }G'_{\mu \nu } \Big )- \frac{1}{120} \, \big [(P_{\mu }U),(P_{\nu }U)\big ] G'_{\mu \nu } \nonumber \\&\qquad - \frac{1}{120}\, \big [U[U,G'_{\mu \nu }]\big ] G'_{\mu \nu } \Bigg ] + \frac{1}{m^6} \Bigg [-\frac{1}{60} \, U^5 - \frac{1}{20} \, U^2\big (P_{\mu }U\big )^2 \nonumber \\&\qquad - \frac{1}{30} \, \big (UP_{\mu }U\big )^2 \Bigg ] + \frac{1}{m^8} \bigg [ \frac{1}{120} \, U^6 \bigg ]\Bigg \}, \end{aligned}$$
(5)
where \(c_{s} = \frac{1}{2}, 1, - \frac{1}{2}\), and \(\frac{1}{2}\) for real scalar, complex scalar, fermion, and gauge boson, respectively. Here \(P_{\mu } = i \ \mathcal {D}_{\mu } \ \text {and} \ G'_{\mu \nu }=[\mathcal {D}_{\mu } , \mathcal {D}_{\nu }]\). U is a collection of coefficients of the terms which are bi-linear in the heavy field. U can be written in matrix form of \(\delta \Big [ \mathcal {L}^{(\mathrm{{BSM}})}(\phi , \Phi )\Big ]/\delta \Phi _i \delta \Phi _j\), evaluated at \(\hat{\Phi }_i\) [19, 20, 21]. It is important to note that ‘\(\text {tr}\)’ in the above equation is the trace performed over the internal symmetry indices.
Table 1

Running ‘InstallCoDEx’ can be customized using its options

Options

Default

Detail

Open image in new window

None

True \(\rightarrow \) warning messages for notebooks created with

Open image in new window

newer Mathematica version automatically disabled\({}^{\mathrm{a}}\)

None \(\rightarrow \) user will be asked by a dialog

False \(\rightarrow \) warnings will not be disabled

Open image in new window

None

True \(\rightarrow \) Previous installations automatically deleted

Open image in new window

None \(\rightarrow \) user will be asked by a dialog

False \(\rightarrow \) the directory will be overwritten

Open image in new window

False

True \(\rightarrow \) Install latest development version

Open image in new window

False \(\rightarrow \) Install the latest stable version

Open image in new window

Open image in new window

Specifies custom full path to installation directory

\({}^{\mathrm{a}}\)Needed to generate documentation in older versions

Once we find the effective operators and their respective Wilson coefficients at high scale, i.e., the scale of new physics, we can run them down to the electro-weak scale. These operators are evolved with the energy scale as
$$\begin{aligned} \frac{\mathrm{{d}} \mathcal {O}_i}{\mathrm{{d}}\ln \mu }=\gamma _{ij} \mathcal {O}_j, \end{aligned}$$
(6)
where \(\gamma _{ij}\) is the anomalous dimension matrix. This is also implemented in Open image in new window , but only for the Open image in new window basis, as it is the complete one.

3 The package, in detail

3.1 Installing and loading CoDEx

Installing Open image in new window is quite straightforward. This can be done in one of the following ways:

3.1.1 Automatic installation

Open image in new window can be installed in a Mathematica environment by downloading and importing the installer file ‘install.m’ available at this link. The installer can also be automatically loaded inside the Mathematica environment by using the command1
This loads two functions in the working kernel: ‘ Open image in new window ’ and ‘ Open image in new window ’. A typical way of running them is
One can use Open image in new window instead of Open image in new window , which is equivalent to running
Table 2

Main functions provided by CoDEx

Function

Details

Open image in new window

Opens the CoDEx guide, with all help files listed

Open image in new window

Calculates WCs generated from tree-level processes

Open image in new window

Calculates WCs generated from 1-loop processes

Open image in new window

Generic function for WCs calculation with choices for level, bases etc. given with OptionValues

Open image in new window

Creates representation of heavy fields

Use the output to construct BSM Lagrangian

Open image in new window

Given a List, returns the LaTeX output of a tabular environment, displayed and/or copied to clipboard\({}^{\mathrm{a}}\)

Open image in new window

Applied on a list of WCs from a specific operator basis, reformats the output in the specified style

Open image in new window

RG Flow of WCs of dim. 6 operators in Open image in new window basis, from matching scale to a lower (arbitrary) scale

Open image in new window

Prepares the Isospin and Color symmetry generators for a specific model with a specific heavy field content Open image in new window can only be run after this step is done

\({}^{\mathrm{a}}\)This is a simplified version of the package titled TeXTableForm [56]

3.1.2 Download archive file

There is a provision to download and install the program locally (Table 1).

The Open image in new window package is available in both .zip and .tar format in its ‘Github’ repository. While using the ‘ Open image in new window ’ option from the Notebook menu inside Mathematica or manually extracting the downloaded archive file in the ‘Applications’ folder inside the Open image in new window works perfectly, installing CoDEx is made a lot easier by downloading and importing the installer file ‘install.m’ available here.

If for some reason you choose to download the archive files and install CoDEx from them, this can still be done, using Open image in new window . You just need to run it in a slightly different way.

You need to copy the path of the downloaded archive (e.g. if you are working in Windows and have downloaded the .zip file to the ‘Downloads’ folder, your path will be Open image in new window . Let us call it Open image in new window from now).

Then run this command in your notebook after importing the ‘install.m’ file:
This goes through exactly the same steps as in the previous section, but instead of downloading the archive from the server, one uses the local file. After this, the package can always be loaded in Mathematica using
The installer file is not our creation (Table 2). We have edited and simplified the installer for FeynRules [55].

3.2 How to build the Lagrangian

3.2.1 Building the Lagrangian: an example

Let us demonstrate this with a toy example where the Lagrangian is given in its traditional form (Fig. 1):
$$\begin{aligned} L(\Phi ,H)\supset -\eta ~ \Phi ^a \Phi ^a \left| H\right| ^2 + 2 \kappa ~ H \tau ^a \Phi ^a H^{\dagger }-\frac{\lambda _{\Phi }}{4} \left( \Phi ^a \Phi ^a\right) ^2. \end{aligned}$$
(7)
Fig. 1

Flow-chart for CoDEx

Here \(\Phi \) is the heavy field which is going to be integrated out. From the user-end, the information for this heavy field is fed into the code as
which contains the required information (within {...}) about the field to compute the WCs. The properties necessary to define the heavy field are listed in Table 3. Now, this code is equally applicable for multiple heavy BSM fields. In this case, the field definitions will be listed sequentially under the first set of curly braces (see Sect. 3.2.2). In the case of our example model, we have only one heavy field, a real triplet scalar (i.e. Open image in new window \(\rightarrow \) 3, Open image in new window \(\rightarrow \) 1, Open image in new window \(\rightarrow \) 3, Open image in new window \(\rightarrow \) 0, Open image in new window \(\rightarrow \) 0). Let us denote Open image in new window \(\rightarrow \) ‘ph’ and Open image in new window \(\rightarrow \) ‘m’. This represents the field content of our model in the correct way:
Table 3

Generic form of individual fields

Pattern

Details

Open image in new window

Field-name, used as array-head to construct the Lagrangian

Open image in new window

Number of components in the heavy field

Open image in new window

Dimensionality of the heavy field under \(SU(3)_C\)

Open image in new window

Dimensionality of the heavy field under \(SU(2)_L\)

Open image in new window

Hypercharge of the heavy field under \(U(1)_Y\)

Open image in new window

Spin quantum number of the heavy field

Open image in new window

Variable representing the mass of the heavy field

Now that our field definitions are ready, it is time to write the Lagrangian in a form that the code understands. For this purpose, first we have to create the representations of these fields in their component form. This is done by a specific function:
To write the Lagrangian in a compact form one can define the heavy field as
With this definition, the working Lagrangian, given in regular form in Eq. (7), can be written in compact form like

3.2.2 Other examples of defining fields

So far we have mentioned how to integrate out a single heavy field. In the case of multiple fields, the required format of the ‘fieldList’ would look like
Now we can create the heavy fields’ list:
In general, if we have n heavy fields; then Open image in new window is a list, whose ith element is the representation of the ith heavy field. Let us describe different cases in detail, where we have fields of different characteristics. A case with a single field is already shown in Sect. 3.2.1.
Following the same proposal, we can define multiple heavy fields. A possible example is
with which we define
Here the representation for the first heavy field is
and the second field is represented as
Now, these two field representations ( Open image in new window and Open image in new window ) can be used to build the required Lagrangian.
For a spin-1 field, the field definition will be
We do not count the Lorentz components of a heavy field while writing the total number of field components (the second entry in Open image in new window ).
In a similar manner, the spin-1/2 field is represented as
As before, we do not count the Lorentz components of a heavy field.
Use Open image in new window and Open image in new window as the field representation of the fermion (say, \(\psi \)) and its Lorentz conjugate (\(\overline{\psi }\)).
Table 4

Available functions for calculating Wilson coefficients (WCs) for a given BSM Lagrangian

Function

Details

Open image in new window

Calculates tree-level Wilson coefficients

Open image in new window

Calculates 1 loop level Wilson coefficients

Open image in new window

Generic function for WCs calculation up to 1 loop

3.3 How to run the code

We have demonstrated how to build the Lagrangian in the previous section. Here we will discuss the necessary steps that need to be followed to compute the Wilson coefficients (WCs). Of the various options that Open image in new window provides to the users, the first is the choice of operator bases ( Open image in new window ) between (i) Open image in new window , and (ii) Open image in new window . Here Open image in new window is the default one. The next option is the level at which the user wants the WCs (Table 4).

One can compute WCs up to 1 loop using this code.
Table 5

The Options for codexOutput. Other than these, this function also takes all Options of formPick

Option

Default value

Details

Open image in new window

True

Shows an animation while computing

Open image in new window

Open image in new window (for Version\(\ge \) 11)

Appearence of the animation

Open image in new window

Open image in new window

Choice of basis of the Dim.-6 operators

Open image in new window

List

The output format

Open image in new window

Open image in new window

Takes the same input as Open image in new window

If left blank, will give the tree-level

Open image in new window

All

The level at which output is evaluated

All’ means the result calculates both tree and loop level results and combines them

Open image in new window

True

Turns the ‘Integration by Parts’ option on

Detail available on the Open image in new window page

3.3.1 Tree level

To obtain the tree-level Wilson coefficients, one needs to use the function Open image in new window , used as
This will generate the WCs in the Open image in new window basis. Now, to compute the same in Open image in new window basis, one has to simply provide an explicit choice of the operator basis as

3.3.2 1-loop level

To compute the WCs at 1-loop level only, we have to use another function, Open image in new window . This can be used as
Unlike the tree-level case, here we need the transformation property of the heavy fields under the given gauge symmetry. More precisely, the structure of the generators determined by the dimensionality of the heavy field’s representations must be provided explicitly. We have provided their structure up to fundamental and quadruplet cases for the \(SU(3)_C\) and \(SU(2)_L\) gauge groups, respectively. For more exotic BSM particles, we have kept provision for the user to define it. To do so, one has to run a function:
If the dimensionality of the heavy fields are within the mentioned ranges, then one does not need to provide the explicit structures of the generators. Otherwise, one has to provide all the generators explicitly for each and every heavy field as Open image in new window and Open image in new window where ‘ Open image in new window ’ denotes the number of heavy fields, and ‘ Open image in new window ’ and ‘ Open image in new window ’ run from 1 to 3 and 1 to 8, respectively.

3.3.3 One function to find them all

Now, one can wish to get all the WCs, i.e., tree and 1-loop levels together. For this purpose, one can simply use the following function:
Essentially, this function is all one needs to calculate the WCs and even format them in the correct way. With careful choice of Open image in new window , one can obtain results at different levels, with different operator bases, and in different formats. We have enlisted the main options which can be found using Open image in new window .

3.4 RGEs of WCs: anomalous dimension matrix and choice of basis

For a given BSM Lagrangian we can compute the effective dimension-6 operators and the associated Wilson coefficients by integrating out the heavy fields. Open image in new window can provide these results in two different bases: Open image in new window and Open image in new window . In the Open image in new window basis, operators form a complete basis unlike the Open image in new window one. Thus we prefer to perform the running of the Wilson coefficients in Open image in new window basis only. Once the WCs are computed at the high scale, one can run those effective operators using the Anomalous dimension matrix and can compute the operator structures at the electro-weak (EW) scale by using the Open image in new window function. Results of this module will help to connect the EW observables and the BSM physics through the effective operators and the WCs (Table 5).

To perform the RG evolution of the WCs, we need to use the function
This function works in the following way:
Here both the Open image in new window and the Open image in new window can be symbolic inputs. The working principle of this function is as follows:

3.5 Detailed example: electro-weak \(SU(2)_L\) real singlet scalar

Here we demonstrate the work flow of Open image in new window with the help of a complete analysis of a representative model. This and many others are listed in the package documentation. We also list the results of the other models in Appendix A of this article. Say the Lagrangian is
$$\begin{aligned} \mathcal {L}_{\mathrm{{BSM}}}= & {} \mathcal {L}_{\mathrm{{SM}}} \ + \ \frac{1}{2} \ (\mathcal {\partial }_{\mu } \phi )^{2} \ - \ \frac{1}{2} \ m_{\phi }^{2} \ \phi ^{2} - c_{a} |H|^{2} \phi \nonumber \\&- \frac{1}{2} \kappa |H|^{2} \phi ^{2} - \frac{1}{3!} \ \mu \phi ^{3} - \frac{1}{4!} \ \lambda _{\phi } \phi ^{4}. \end{aligned}$$
(8)
Here \(\phi \) is the real singlet scalar. Once this field is integrated out, few effective operators will emerge. To obtain those effective dimension-6 operators and their respective Wilson coefficients using Open image in new window , we need to perform the following steps:
Table 6

Effective operators and Wilson coefficients for real singlet scalar. These results are calculated in the \(\overline{\text {MS}}\) renormalization scheme. The 1-loop result may be altered based on the choice of renormalization scheme, e.g. in this particular case we have noted mismatches with results given in Ref. [19] where a different renormalization scheme has been considered. Here we have highlighted the extra terms obtained in our calculation in the \(\overline{\text {MS}}\) scheme in ‘Red’ color. We have further cross-checked these results in the other scheme, adopted in Ref. [19], explicitly. For results in “Warsaw" basis, no such comparison has been done

(a) “SILH” basis (tree level)

   \(O_H\)

\(\frac{c_a^2}{m^4}\)

   \(O_6\)

\(\frac{\mu c_a^3}{6 m^6}-\frac{\kappa c_a^2}{2 m^4}\)

(b) “SILH” basis (1-loop level)

   \(O_H\)

\({\frac{\mu ^2 c_a^2}{192 \pi ^2 m^6}-\frac{\kappa \mu c_a}{96 \pi ^2 m^4}+\frac{\lambda c_a^2}{16 \pi ^2 m^4}}+\frac{\kappa ^2}{192 \pi ^2 m^2}\)

   \(O_6\)

\({-\frac{\kappa \lambda c_a^2}{32 \pi ^2 m^4}}-\frac{\kappa ^3}{192 \pi ^2 m^2}\)

(c) “Warsaw” basis (tree level)

   \(Q_H\)

\(\frac{\mu c_a^3}{6 m^6}-\frac{\kappa c_a^2}{2 m^4}\)

   \(Q_{\text {HD}}\)

\(-\frac{2 c_a^2}{m^4}\)

(d) “Warsaw” basis (1-loop level)

   \(Q_H\)

\(-\frac{\kappa \lambda c_a^2}{32 \pi ^2 m^4}-\frac{\kappa ^3}{192 \pi ^2 m^2}\)

   \(Q_{\text {HD}}\)

\(\frac{\mu ^2 c_a^2}{96 \pi ^2 m^6}-\frac{\kappa \mu c_a}{48 \pi ^2 m^4}-\frac{\lambda c_a^2}{8 \pi ^2 m^4}+\frac{\kappa ^2}{96 \pi ^2 m^2}\)

  1. 1.
    First, load the package:
     
  2. 2.
     
  3. 3.
    Then we need to build the relevant part of the Lagrangian (involving the heavy field only). As a side-note, we should mention that for Open image in new window to function, it does not need the heavy field kinetic term (the covariant derivative and the mass terms). Thus, the only part of the Lagrangian we need here is
     
  4. 4.
    Next, we need to construct the symmetry generators:
    (See the documentation of Open image in new window for details.)
     
  5. 5.
    The last step is the computation of effective operators and associated WCs as
     
  6. 6.
    The output is obtained in Open image in new window basis and is formatted as a detailed table in Open image in new window . There is provision to export the result in LaTeX format. Table 6c is actually obtained from the output of the code above. We can compute the same in Open image in new window basis as well and for that we have to use
    Output of this can be found in Table 6a. Similarly, results for a 1-loop calculation can be obtained by changing the option value of ‘ Open image in new window ’ to Open image in new window . The default value of Open image in new window is Open image in new window , which combines both tree and 1-loop results.
     
  7. 7.

    As is demonstrated in Sect. 3.4, these resulting WCs can then be run down to the electro-weak scale, using Open image in new window .

     

3.6 Miscellaneous

Open image in new window is written in Wolfram Language\(^{\textregistered }\) [57]. Careful steps have been taken to speed up the code using parallelization over multi-cores, when available, while keeping the customizability for the user. All the example models listed in this article and in the documentation have been run on different processors, with different operating systems and versions of Mathematica.

When run on a 1.6 GHz Intel\(^{\textregistered }\) Core i5 processor, the models take \(\sim \) 20–2000 s to run. Tree-level runs never take more than a minute. The only exception is 2HDM, which we have run in a 16 core Xeon processor, with 32 GB RAM.

How much time it takes to get the WCs for the user’s model depends on its structure and complexity. We hope a user can have a clear idea about run time if she runs all the examples given in documentation as trials.

We hope to enable the output of future versions of Open image in new window in ‘WCxf’ format in addition to the present ones.

4 Summary

Open image in new window allows one to integrate out single and/or multiple degenerate heavy field(s) in a gauge covariant way. The user needs to provide the part of the Lagrangian that involves heavy BSM fields only. One needs to identify that BSM field by providing its number of component fields, spin, mass and quantum numbers under standard model gauge symmetry in a certain way. Open image in new window then integrates out the heavy field propagators from all tree-level and/or 1-loop processes, and generates the Wilson coefficients for an exhaustive set of effective operators in both Open image in new window and Open image in new window bases. It allows the user to run the operators in Open image in new window basis down to the electro-weak scale. As the precision observables can be recast in terms of the effective operators, it will be really helpful to test the BSM physics in the light of electro-weak precision data. A list of example models are provided along with the package. These include a variety of field representations usually used by the BSM model builders.

In the present version of Open image in new window , the user can compute operators up to mass dimension 6 by integrating out particles up to spin 1. This package can integrate out only heavy field propagators at the tree and 1-loop levels. In a future version, we will include a few other aspects, such that it can deal with (i) loops containing light (SM) and heavy (BSM) mixed field propagators, and (ii) non-degenerate multiple heavy BSM fields [58].

Footnotes

  1. 1.

    This requires a live internet connection and Mathematica should be able to connect to the internet.

  2. 2.

    Fun-fact: This table in LaTeX format, and other similar results used in this article are all created using formPick as well.

Notes

Acknowledgements

The authors acknowledge the useful discussions with Soumitra Nandi. This work is supported by the Department of Science and Technology, Government of India, under the Grant IFA12/PH/34 (INSPIRE Faculty Award); the Science and Engineering Research Board, Government of India, under the agreement SERB/PHY/2016348 (Early Career Research Award), and the Initiation Research Grant, agreement no. IITK/PHY/2015077, by IIT Kanpur.

References

  1. 1.
    G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, JHEP 06, 045 (2007). arXiv:hep-ph/0703164 [hep-ph]ADSCrossRefGoogle Scholar
  2. 2.
    R. Contino, M. Ghezzi, C. Grojean, M. Muhlleitner, M. Spira, JHEP 07, 035 (2013). arXiv:1303.3876 [hep-ph]ADSCrossRefGoogle Scholar
  3. 3.
    B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, JHEP 10, 085 (2010). arXiv:1008.4884 [hep-ph]ADSCrossRefGoogle Scholar
  4. 4.
    E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 10, 087 (2013). arXiv:1308.2627 [hep-ph]ADSCrossRefGoogle Scholar
  5. 5.
    E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 01, 035 (2014). arXiv:1310.4838 [hep-ph]ADSCrossRefGoogle Scholar
  6. 6.
    R. Alonso, E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 04, 159 (2014). arXiv:1312.2014 [hep-ph]ADSCrossRefGoogle Scholar
  7. 7.
    R.J. Furnstahl, N. Klco, D.R. Phillips, S. Wesolowski, Phys. Rev. C 92, 024005 (2015). arXiv:1506.01343 [nucl-th]ADSCrossRefGoogle Scholar
  8. 8.
    H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993)ADSCrossRefGoogle Scholar
  9. 9.
    D.B. Kaplan, in Beyond the standard model 5. Proceedings, 5th Conference, Balholm, Norway, April 29-May 4, 1997 1995. arXiv:nucl-th/9506035 [nucl-th]
  10. 10.
    A.V. Manohar, Perturbative and nonperturbative aspects of quantum field theory. Proceedings, 35. Internationale Universitätswochen für Kern- und Teilchenphysik: Schladming, Austria, March 2-9, 1996, Lect. Notes Phys. 479, 311 (1997). arXiv:hep-ph/9606222 [hep-ph]
  11. 11.
    C.P. Burgess, Ann. Rev. Nucl. Part. Sci. 57, 329 (2007). arXiv:hep-th/0701053 [hep-th]ADSCrossRefGoogle Scholar
  12. 12.
    I.Z. Rothstein, arXiv:hep-ph/0308266 [hep-ph] (2003)
  13. 13.
    B. Gripaios, D. Sutherland, arXiv:1807.07546 [hep-ph] (2018)
  14. 14.
    A. Falkowski, B. Fuks, K. Mawatari, K. Mimasu, F. Riva, V. Sanz, Eur. Phys. J. C 75, 583 (2015). arXiv:1508.05895 [hep-ph]ADSCrossRefGoogle Scholar
  15. 15.
    A. Celis, J. Fuentes-Martin, A. Vicente, J. Virto, Eur. Phys. J. C 77, 405 (2017). arXiv:1704.04504 [hep-ph]ADSCrossRefGoogle Scholar
  16. 16.
    J.C. Criado, Comput. Phys. Commun. 227, 42 (2018). arXiv:1710.06445 [hep-ph]ADSCrossRefGoogle Scholar
  17. 17.
    J. Aebischer, J. Kumar, D.M. Straub, arXiv:1804.05033 [hep-ph] (2018)
  18. 18.
    J. Aebischer et al., Comput. Phys. Commun. 232, 71 (2018b). arXiv:1712.05298 [hep-ph]ADSCrossRefGoogle Scholar
  19. 19.
    B. Henning, X. Lu, H. Murayama, JHEP 01, 023 (2016). arXiv:1412.1837 [hep-ph]ADSCrossRefGoogle Scholar
  20. 20.
    B. Henning, X. Lu, H. Murayama, JHEP 01, 123 (2018). arXiv:1604.01019 [hep-ph]ADSCrossRefGoogle Scholar
  21. 21.
    B. Henning, X. Lu, T. Melia, H. Murayama, JHEP 08, 016 (2017). arXiv:1512.03433 [hep-ph]ADSCrossRefGoogle Scholar
  22. 22.
    J. Elias-Miro, J.R. Espinosa, E. Masso, A. Pomarol, JHEP 11, 066 (2013). arXiv:1308.1879 [hep-ph]ADSCrossRefGoogle Scholar
  23. 23.
    J. Elias-Miró, C. Grojean, R.S. Gupta, D. Marzocca, JHEP 05, 019 (2014). arXiv:1312.2928 [hep-ph]ADSCrossRefGoogle Scholar
  24. 24.
    E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 09, 063 (2013b). arXiv:1305.0017 [hep-ph]ADSCrossRefGoogle Scholar
  25. 25.
    A.V. Manohar, Phys. Lett. B 726, 347 (2013). arXiv:1305.3927 [hep-ph]ADSCrossRefGoogle Scholar
  26. 26.
    Z. Han, W. Skiba, Phys. Rev. D 71, 075009 (2005). arXiv:hep-ph/0412166 [hep-ph]ADSCrossRefGoogle Scholar
  27. 27.
    G. Cacciapaglia, C. Csaki, G. Marandella, A. Strumia, Phys. Rev. D 74, 033011 (2006). arXiv:hep-ph/0604111 [hep-ph]ADSCrossRefGoogle Scholar
  28. 28.
    F. Bonnet, M.B. Gavela, T. Ota, W. Winter, Phys. Rev. D 85, 035016 (2012a). arXiv:1105.5140 [hep-ph]ADSCrossRefGoogle Scholar
  29. 29.
    F. Bonnet, T. Ota, M. Rauch, W. Winter, Phys. Rev. D 86, 093014 (2012b). arXiv:1207.4599 [hep-ph]ADSCrossRefGoogle Scholar
  30. 30.
    F. del Aguila, J. de Blas, Elementary particle physics and gravity. Proceedings, 10th Hellenic Schools and Workshops, Corfu 2010, Corfu, August 29-September 12, 2010, Fortsch. Phys. 59, 1036 (2011). arXiv:1105.6103 [hep-ph]
  31. 31.
    C. Grojean, E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 04, 016 (2013). arXiv:1301.2588 [hep-ph]ADSCrossRefGoogle Scholar
  32. 32.
    R. Contino, A. Falkowski, F. Goertz, C. Grojean, F. Riva, JHEP 07, 144 (2016). arXiv:1604.06444 [hep-ph]ADSCrossRefGoogle Scholar
  33. 33.
    J. Brehmer, A. Freitas, D. Lopez-Val, T. Plehn, Phys. Rev. D 93, 075014 (2016). arXiv:1510.03443 [hep-ph]ADSCrossRefGoogle Scholar
  34. 34.
    L. Berthier, M. Trott, JHEP 02, 069 (2016). arXiv:1508.05060 [hep-ph]ADSCrossRefGoogle Scholar
  35. 35.
    M. Gorbahn, J.M. No, V. Sanz, JHEP 10, 036 (2015). arXiv:1502.07352 [hep-ph]ADSCrossRefGoogle Scholar
  36. 36.
    L. Berthier, M. Trott, JHEP 05, 024 (2015). arXiv:1502.02570 [hep-ph]ADSCrossRefGoogle Scholar
  37. 37.
    W. Skiba, in Physics of the large and the small, TASI 09, Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, USA, 1-26 June 2009 pp. 5–70, (2011). arXiv:1006.2142 [hep-ph]
  38. 38.
    Z.U. Khandker, D. Li, W. Skiba, Phys. Rev. D 86, 015006 (2012). arXiv:1201.4383 [hep-ph]ADSCrossRefGoogle Scholar
  39. 39.
    C. Englert, E. Re, M. Spannowsky, Phys. Rev. D 88, 035024 (2013). arXiv:1306.6228 [hep-ph]ADSCrossRefGoogle Scholar
  40. 40.
    S. Banerjee, S. Mukhopadhyay, B. Mukhopadhyaya, JHEP 10, 062 (2012). arXiv:1207.3588 [hep-ph]ADSCrossRefGoogle Scholar
  41. 41.
    S. Banerjee, S. Mukhopadhyay, B. Mukhopadhyaya, Phys. Rev. D 89, 053010 (2014). arXiv:1308.4860 [hep-ph]ADSCrossRefGoogle Scholar
  42. 42.
    J.D. Wells, Z. Zhang, JHEP 06, 122 (2016a). arXiv:1512.03056 [hep-ph]ADSCrossRefGoogle Scholar
  43. 43.
    M.K. Gaillard, Nucl. Phys. B 268, 669 (1986)ADSCrossRefGoogle Scholar
  44. 44.
    O. Cheyette, Phys. Rev. Lett. 55, 2394 (1985)ADSCrossRefGoogle Scholar
  45. 45.
    L. Lehman, A. Martin, Phys. Rev. D 91, 105014 (2015). arXiv:1503.07537 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    C.-W. Chiang, R. Huo, JHEP 09, 152 (2015). arXiv:1505.06334 [hep-ph]ADSCrossRefGoogle Scholar
  47. 47.
    R. Huo, JHEP 09, 037 (2015). arXiv:1506.00840 [hep-ph]CrossRefGoogle Scholar
  48. 48.
    R. Huo, Phys. Rev. D 97, 075013 (2018). arXiv:1509.05942 [hep-ph]ADSCrossRefGoogle Scholar
  49. 49.
    L. Lehman, A. Martin, JHEP 02, 081 (2016). arXiv:1510.00372 [hep-ph]ADSCrossRefGoogle Scholar
  50. 50.
    J.D. Wells, Z. Zhang, JHEP 01, 123 (2016b). arXiv:1510.08462 [hep-ph]ADSCrossRefGoogle Scholar
  51. 51.
    A. Drozd, J. Ellis, J. Quevillon, T. You, JHEP 03, 180 (2016). arXiv:1512.03003 [hep-ph]ADSCrossRefGoogle Scholar
  52. 52.
    F. del Aguila, Z. Kunszt, J. Santiago, Eur. Phys. J. C 76, 244 (2016). arXiv:1602.00126 [hep-ph]ADSCrossRefGoogle Scholar
  53. 53.
    J. Fuentes-Martin, J. Portoles, P. Ruiz-Femenia, JHEP 09, 156 (2016). arXiv:1607.02142 [hep-ph]ADSCrossRefGoogle Scholar
  54. 54.
    S.A.R. Ellis, J. Quevillon, T. You, Z. Zhang, Phys. Lett. B 762, 166 (2016). arXiv:1604.02445 [hep-ph]ADSCrossRefGoogle Scholar
  55. 55.
    A. Alloul, N.D. Christensen, C. Degrande, C. Duhr, B. Fuks, Comput. Phys. Commun. 185, 2250 (2014). arXiv:1310.1921 [hep-ph]ADSCrossRefGoogle Scholar
  56. 56.
    M. Schunter, “Textableform, Version 1.0,” Teichstr, 1 (2018)Google Scholar
  57. 57.
    W.R. Inc., “Mathematica, Version 8.0,” Champaign, IL (2010)Google Scholar
  58. 58.
    S. Das Bakshi, J. Chakrabortty, S. Patra, In preparationGoogle Scholar

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Authors and Affiliations

  • Supratim Das Bakshi
    • 1
  • Joydeep Chakrabortty
    • 1
  • Sunando Kumar Patra
    • 1
    Email author
  1. 1.Department of PhysicsIndian Institute of TechnologyKanpurIndia

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