CoDEx: Wilson coefficient calculator connecting SMEFT to UV theory
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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 1loop diagrams of that BSM theory. It then computes the associated WCs up to 1loop 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 electroweak 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 dimension6 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 beyondstandard 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 electroweak 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 nonobservation 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 nonSM 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 cutoff 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 Smatrix in powers of \(\Lambda ^{1}\) (UVscale), 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, dimension6 operators are more suppressed than the dimension5 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 Smatrix 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 cutoff \(\Lambda \) 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 nonzero offdiagonal 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 1loop 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 1loop results depend on the choice of renormalization scheme; thus the choice of the scheme is very important and crucial. Our code provides the 1loop results under the \(\overline{\text {MS}}\) renormalization scheme. We would like to make a remark on the results obtained at the 1loop level: The present version of this code can integrate out only heavy propagators, thus the 1loop 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 1loop 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 1loop 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 electroweak 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 electroweak \(SU(2)_L\) 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.

It will integrate out the heavy BSM field(s) while respecting the gauge invariance.

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

It will perform RG evolutions of the effective operators generated at some high scale and provide the operators at the electroweak scale.
Running ‘InstallCoDEx’ can be customized using its options
Options  Default  Detail 

None  True \(\rightarrow \) warning messages for notebooks created with  
newer Mathematica version automatically disabled\({}^{\mathrm{a}}\)  
None \(\rightarrow \) user will be asked by a dialog  
False \(\rightarrow \) warnings will not be disabled  
None  True \(\rightarrow \) Previous installations automatically deleted  
None \(\rightarrow \) user will be asked by a dialog  
False \(\rightarrow \) the directory will be overwritten  
False  True \(\rightarrow \) Install latest development version  
False \(\rightarrow \) Install the latest stable version  
Specifies custom full path to installation directory 
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
Main functions provided by CoDEx
Function  Details 

Opens the CoDEx guide, with all help files listed  
Calculates WCs generated from treelevel processes  
Calculates WCs generated from 1loop processes  
Generic function for WCs calculation with choices for level, bases etc. given with OptionValues  
Creates representation of heavy fields  
Use the output to construct BSM Lagrangian  
Given a List, returns the LaTeX output of a tabular environment, displayed and/or copied to clipboard\({}^{\mathrm{a}}\)  
Applied on a list of WCs from a specific operator basis, reformats the output in the specified style  
RG Flow of WCs of dim. 6 operators in Open image in new window basis, from matching scale to a lower (arbitrary) scale  
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 
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).
3.2 How to build the Lagrangian
3.2.1 Building the Lagrangian: an example
Generic form of individual fields
Pattern  Details 

Fieldname, used as arrayhead to construct the Lagrangian  
Number of components in the heavy field  
Dimensionality of the heavy field under \(SU(3)_C\)  
Dimensionality of the heavy field under \(SU(2)_L\)  
Hypercharge of the heavy field under \(U(1)_Y\)  
Spin quantum number of the heavy field  
Variable representing the mass of the heavy field 
3.2.2 Other examples of defining fields
Available functions for calculating Wilson coefficients (WCs) for a given BSM Lagrangian
Function  Details 

Calculates treelevel Wilson coefficients  
Calculates 1 loop level Wilson coefficients  
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).
The Options for codexOutput. Other than these, this function also takes all Options of formPick
Option  Default value  Details 

True  Shows an animation while computing  
Open image in new window (for Version\(\ge \) 11)  Appearence of the animation  
Choice of basis of the Dim.6 operators  
List  The output format  
Takes the same input as Open image in new window  
If left blank, will give the treelevel  
All  The level at which output is evaluated  
‘All’ means the result calculates both tree and loop level results and combines them  
True  Turns the ‘Integration by Parts’ option on  
Detail available on the Open image in new window page 
3.3.1 Tree level
3.3.2 1loop level
3.3.3 One function to find them all
3.4 RGEs of WCs: anomalous dimension matrix and choice of basis
For a given BSM Lagrangian we can compute the effective dimension6 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 electroweak (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).

Open image in new window takes a list as its first argument. List output from Open image in new window , Open image in new window and Open image in new window functions should be used as the first argument.

Open image in new window takes Wilson coefficients of dimension6 operators in the Open image in new window basis only.

The second argument is the scale at which the full BSM theory is matched with the EFT.

Open image in new window can be any energy scale below the matching scale.
 Load the package through
 Say that the following is the output, in the form of a list of WCs, that you had found from an earlier session of Open image in new window and had saved. Let us give it a name: These WCs are evaluated at the high scale. Now, to compute the WCs at the electroweak scale ( Open image in new window ) we need to perform their RGEs. After setting the matching scale (high scale) at the mass of the heavy particle (‘ Open image in new window ’), we have to recall the function Open image in new window as
 One can reformat, save, and/or export all these WCs corresponding to the effective operators at the electroweak scale ( Open image in new window ) to LaTeX, using Open image in new window . Below is an illustrative example: Here we have shown a truncated version of the resulting long table.^{2}
 Remember that the RGE of WCs can only be performed in the Open image in new window basis (as it a complete one) and not in the Open image in new window basis.
3.5 Detailed example: electroweak \(SU(2)_L\) real singlet scalar
Effective operators and Wilson coefficients for real singlet scalar. These results are calculated in the \(\overline{\text {MS}}\) renormalization scheme. The 1loop 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 crosschecked 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 (1loop 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 (1loop 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.First, load the package:
 2.We have to define the field \(\phi \) as
 3.Then we need to build the relevant part of the Lagrangian (involving the heavy field only). As a sidenote, 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.Next, we need to construct the symmetry generators: (See the documentation of Open image in new window for details.)
 5.The last step is the computation of effective operators and associated WCs as
 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 1loop 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 1loop results.
 7.
As is demonstrated in Sect. 3.4, these resulting WCs can then be run down to the electroweak 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 multicores, 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. Treelevel 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 treelevel and/or 1loop 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 electroweak 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 electroweak 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 1loop 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) nondegenerate multiple heavy BSM fields [58].
Footnotes
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.G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, JHEP 06, 045 (2007). arXiv:hepph/0703164 [hepph]ADSCrossRefGoogle Scholar
 2.R. Contino, M. Ghezzi, C. Grojean, M. Muhlleitner, M. Spira, JHEP 07, 035 (2013). arXiv:1303.3876 [hepph]ADSCrossRefGoogle Scholar
 3.B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, JHEP 10, 085 (2010). arXiv:1008.4884 [hepph]ADSCrossRefGoogle Scholar
 4.E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 10, 087 (2013). arXiv:1308.2627 [hepph]ADSCrossRefGoogle Scholar
 5.E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 01, 035 (2014). arXiv:1310.4838 [hepph]ADSCrossRefGoogle Scholar
 6.R. Alonso, E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 04, 159 (2014). arXiv:1312.2014 [hepph]ADSCrossRefGoogle Scholar
 7.R.J. Furnstahl, N. Klco, D.R. Phillips, S. Wesolowski, Phys. Rev. C 92, 024005 (2015). arXiv:1506.01343 [nuclth]ADSCrossRefGoogle Scholar
 8.H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993)ADSCrossRefGoogle Scholar
 9.D.B. Kaplan, in Beyond the standard model 5. Proceedings, 5th Conference, Balholm, Norway, April 29May 4, 1997 1995. arXiv:nuclth/9506035 [nuclth]
 10.A.V. Manohar, Perturbative and nonperturbative aspects of quantum field theory. Proceedings, 35. Internationale Universitätswochen für Kern und Teilchenphysik: Schladming, Austria, March 29, 1996, Lect. Notes Phys. 479, 311 (1997). arXiv:hepph/9606222 [hepph]
 11.C.P. Burgess, Ann. Rev. Nucl. Part. Sci. 57, 329 (2007). arXiv:hepth/0701053 [hepth]ADSCrossRefGoogle Scholar
 12.I.Z. Rothstein, arXiv:hepph/0308266 [hepph] (2003)
 13.B. Gripaios, D. Sutherland, arXiv:1807.07546 [hepph] (2018)
 14.A. Falkowski, B. Fuks, K. Mawatari, K. Mimasu, F. Riva, V. Sanz, Eur. Phys. J. C 75, 583 (2015). arXiv:1508.05895 [hepph]ADSCrossRefGoogle Scholar
 15.A. Celis, J. FuentesMartin, A. Vicente, J. Virto, Eur. Phys. J. C 77, 405 (2017). arXiv:1704.04504 [hepph]ADSCrossRefGoogle Scholar
 16.J.C. Criado, Comput. Phys. Commun. 227, 42 (2018). arXiv:1710.06445 [hepph]ADSCrossRefGoogle Scholar
 17.J. Aebischer, J. Kumar, D.M. Straub, arXiv:1804.05033 [hepph] (2018)
 18.J. Aebischer et al., Comput. Phys. Commun. 232, 71 (2018b). arXiv:1712.05298 [hepph]ADSCrossRefGoogle Scholar
 19.B. Henning, X. Lu, H. Murayama, JHEP 01, 023 (2016). arXiv:1412.1837 [hepph]ADSCrossRefGoogle Scholar
 20.B. Henning, X. Lu, H. Murayama, JHEP 01, 123 (2018). arXiv:1604.01019 [hepph]ADSCrossRefGoogle Scholar
 21.B. Henning, X. Lu, T. Melia, H. Murayama, JHEP 08, 016 (2017). arXiv:1512.03433 [hepph]ADSCrossRefGoogle Scholar
 22.J. EliasMiro, J.R. Espinosa, E. Masso, A. Pomarol, JHEP 11, 066 (2013). arXiv:1308.1879 [hepph]ADSCrossRefGoogle Scholar
 23.J. EliasMiró, C. Grojean, R.S. Gupta, D. Marzocca, JHEP 05, 019 (2014). arXiv:1312.2928 [hepph]ADSCrossRefGoogle Scholar
 24.E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 09, 063 (2013b). arXiv:1305.0017 [hepph]ADSCrossRefGoogle Scholar
 25.A.V. Manohar, Phys. Lett. B 726, 347 (2013). arXiv:1305.3927 [hepph]ADSCrossRefGoogle Scholar
 26.Z. Han, W. Skiba, Phys. Rev. D 71, 075009 (2005). arXiv:hepph/0412166 [hepph]ADSCrossRefGoogle Scholar
 27.G. Cacciapaglia, C. Csaki, G. Marandella, A. Strumia, Phys. Rev. D 74, 033011 (2006). arXiv:hepph/0604111 [hepph]ADSCrossRefGoogle Scholar
 28.F. Bonnet, M.B. Gavela, T. Ota, W. Winter, Phys. Rev. D 85, 035016 (2012a). arXiv:1105.5140 [hepph]ADSCrossRefGoogle Scholar
 29.F. Bonnet, T. Ota, M. Rauch, W. Winter, Phys. Rev. D 86, 093014 (2012b). arXiv:1207.4599 [hepph]ADSCrossRefGoogle Scholar
 30.F. del Aguila, J. de Blas, Elementary particle physics and gravity. Proceedings, 10th Hellenic Schools and Workshops, Corfu 2010, Corfu, August 29September 12, 2010, Fortsch. Phys. 59, 1036 (2011). arXiv:1105.6103 [hepph]
 31.C. Grojean, E.E. Jenkins, A.V. Manohar, M. Trott, JHEP 04, 016 (2013). arXiv:1301.2588 [hepph]ADSCrossRefGoogle Scholar
 32.R. Contino, A. Falkowski, F. Goertz, C. Grojean, F. Riva, JHEP 07, 144 (2016). arXiv:1604.06444 [hepph]ADSCrossRefGoogle Scholar
 33.J. Brehmer, A. Freitas, D. LopezVal, T. Plehn, Phys. Rev. D 93, 075014 (2016). arXiv:1510.03443 [hepph]ADSCrossRefGoogle Scholar
 34.L. Berthier, M. Trott, JHEP 02, 069 (2016). arXiv:1508.05060 [hepph]ADSCrossRefGoogle Scholar
 35.M. Gorbahn, J.M. No, V. Sanz, JHEP 10, 036 (2015). arXiv:1502.07352 [hepph]ADSCrossRefGoogle Scholar
 36.L. Berthier, M. Trott, JHEP 05, 024 (2015). arXiv:1502.02570 [hepph]ADSCrossRefGoogle Scholar
 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, 126 June 2009 pp. 5–70, (2011). arXiv:1006.2142 [hepph]
 38.Z.U. Khandker, D. Li, W. Skiba, Phys. Rev. D 86, 015006 (2012). arXiv:1201.4383 [hepph]ADSCrossRefGoogle Scholar
 39.C. Englert, E. Re, M. Spannowsky, Phys. Rev. D 88, 035024 (2013). arXiv:1306.6228 [hepph]ADSCrossRefGoogle Scholar
 40.S. Banerjee, S. Mukhopadhyay, B. Mukhopadhyaya, JHEP 10, 062 (2012). arXiv:1207.3588 [hepph]ADSCrossRefGoogle Scholar
 41.S. Banerjee, S. Mukhopadhyay, B. Mukhopadhyaya, Phys. Rev. D 89, 053010 (2014). arXiv:1308.4860 [hepph]ADSCrossRefGoogle Scholar
 42.J.D. Wells, Z. Zhang, JHEP 06, 122 (2016a). arXiv:1512.03056 [hepph]ADSCrossRefGoogle Scholar
 43.M.K. Gaillard, Nucl. Phys. B 268, 669 (1986)ADSCrossRefGoogle Scholar
 44.O. Cheyette, Phys. Rev. Lett. 55, 2394 (1985)ADSCrossRefGoogle Scholar
 45.L. Lehman, A. Martin, Phys. Rev. D 91, 105014 (2015). arXiv:1503.07537 [hepph]ADSMathSciNetCrossRefGoogle Scholar
 46.C.W. Chiang, R. Huo, JHEP 09, 152 (2015). arXiv:1505.06334 [hepph]ADSCrossRefGoogle Scholar
 47.R. Huo, JHEP 09, 037 (2015). arXiv:1506.00840 [hepph]CrossRefGoogle Scholar
 48.R. Huo, Phys. Rev. D 97, 075013 (2018). arXiv:1509.05942 [hepph]ADSCrossRefGoogle Scholar
 49.L. Lehman, A. Martin, JHEP 02, 081 (2016). arXiv:1510.00372 [hepph]ADSCrossRefGoogle Scholar
 50.J.D. Wells, Z. Zhang, JHEP 01, 123 (2016b). arXiv:1510.08462 [hepph]ADSCrossRefGoogle Scholar
 51.A. Drozd, J. Ellis, J. Quevillon, T. You, JHEP 03, 180 (2016). arXiv:1512.03003 [hepph]ADSCrossRefGoogle Scholar
 52.F. del Aguila, Z. Kunszt, J. Santiago, Eur. Phys. J. C 76, 244 (2016). arXiv:1602.00126 [hepph]ADSCrossRefGoogle Scholar
 53.J. FuentesMartin, J. Portoles, P. RuizFemenia, JHEP 09, 156 (2016). arXiv:1607.02142 [hepph]ADSCrossRefGoogle Scholar
 54.S.A.R. Ellis, J. Quevillon, T. You, Z. Zhang, Phys. Lett. B 762, 166 (2016). arXiv:1604.02445 [hepph]ADSCrossRefGoogle Scholar
 55.A. Alloul, N.D. Christensen, C. Degrande, C. Duhr, B. Fuks, Comput. Phys. Commun. 185, 2250 (2014). arXiv:1310.1921 [hepph]ADSCrossRefGoogle Scholar
 56.M. Schunter, “Textableform, Version 1.0,” Teichstr, 1 (2018)Google Scholar
 57.W.R. Inc., “Mathematica, Version 8.0,” Champaign, IL (2010)Google Scholar
 58.S. Das Bakshi, J. Chakrabortty, S. Patra, In preparationGoogle Scholar
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