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A Python program to generate the Lagrangians for dark matter models

Project description


This is a Python program which allows one to specify the field content of an extension of the Standard Model of particle physics (SM) and, using this information, generates the most general renormalizable Lagrangian that describes such a model. As the program was written for the study of minimal dark matter models with radiative neutrino masses, it can handle additional fields with the following properties:

  • scalar or Weyl fermion fields
  • SU(3) singlets
  • SU(2) singlets, doublets or triplets
  • arbitrary hypercharge
  • charged under a global ℤ₂ symmetry
  • charged under an arbitrary number of global U(1) symmetries


The program requires Python 3 (tested with Python ≥ 3.4). No external libraries are necessary.


minimal-lagrangians is available on pip, so it can simply be installed by running

pip install minimal-lagrangians


minimal-lagrangians only prints the potential involving at least one new (i. e. non-SM) field, i. e. the kinetic terms and the Standard Model Lagrangian are omitted. The models are not checked for anomalies (tools like SARAH can be used for this purpose).

The new models are currently defined in the file Models can be added in a user-defined file (to be used with the --model-file option, see below) in the following form:

	BSMModel('<model_name>', (
		    # list of fields
		    # type       name   SU(2) rep.  hypercharge
		    # for a scalar field, e.g. a scalar doublet with hypercharge 1:
		    ScalarField ('S',   2,          Y=1),
		    # for a fermion field, e.g. a fermion singlet with hypercharge 0:
		    FermionField('F',   1,          Y=0),
		    # ℤ₂-even scalar field:
		    ScalarField ('S',   1,          Y=0, z2=1),
		    # …
		# optional: parameter values for different hypercharge assignments (offsets), e.g.
		(0, 2, )
	# …

To add fields with global U(1) charges, use the optional parameter u1:

        # type       name   SU(2) rep.  hypercharge  U(1) charges
        # for a scalar field, e.g. a scalar doublet with hypercharge 1:
        ScalarField ('S',   2,          Y=1,         u1=[ 1, 3, ]),
        # for a fermion field, e.g. a fermion singlet with hypercharge 0:
        FermionField('F',   1,          Y=0,         u1=[-1, 2, ]),

The Standard Model fields are assumed to be neutral (z2=1, U(1) charges zero) under the new global symmetries.

Information on how to run the program on the command line can be obtained with minimal-lagrangians -h:

usage: minimal-lagrangians [-h] [--format {LaTeX,SARAH,plain}]
                           [--model-file [path/to/]]
                           [--omit-self-interaction] [--list-discarded-terms]
                           [--sarah-no-scalar-cpv] [--version]
                           model [parameter α]

A Python program to generate the Lagrangians for dark matter models

positional arguments:
  model                 name of the model whose Lagrangian is to be generated
                        (specify “list” in order to list all available models)
  parameter α           value of the model parameter α (determines
                        hypercharges of the fields)

optional arguments:
  -h, --help            show this help message and exit
  --format {LaTeX,SARAH,plain}
                        output format for the generated Lagrangian (default:
  --model-file [path/to/]
                        file containing user-defined models; a file is only
                        read if this option is present (default: ./
                        keep only scalar fields from the model which have
                        unique quantum numbers and absolute hypercharge values
                        (omit duplicates)
                        omit pure self-interactions of the new fields in the
                        Lagrangian, that is, output only interaction terms
                        which involve both SM and new fields (default: output
                        all terms)
                        list redundant terms which were discarded from the
                        Lagrangian due to identities
                        assume that there is no CP violation causing mixing
                        between scalar and pseudoscalar fields for SARAH
  --version             show program’s version number and exit

Test cases can be run using


Among other checks, this currently tests whether the program produces the correct Lagrangian for the following models:


For example, running

minimal-lagrangians --omit-equivalent-scalars T1-1-A 0

prints the Lagrangian for the model T1-1-A with α = 0 from (Restrepo, Zapata, Yaguna; arXiv:1308.3655 [hep-ph]):

 - M_ϕ'² ϕ'^† ϕ' - ½ M_φ² φ²
 - (λ₁ (H ϕ') φ + H.c.)
 - λ₂ (H^† H) (ϕ'^† ϕ') - λ₃ (H^† ϕ') (ϕ'^† H) - λ₄ (ϕ'^† ϕ')² - λ₅ (H^† H) φ² - λ₆ (ϕ'^† ϕ') φ² - (λ₇ (H ϕ')² + H.c.) - λ₈ φ⁴
 - (½ M_ψ ψ ψ + H.c.)
 - (y₁ (ϕ'^† L) ψ + H.c.)


minimal-lagrangians STS

prints the Lagrangian for model C (singlet–triplet scalar) from (Cheung, Sanford; arXiv:1311.5896 [hep-ph]):

 - ½ M_T² Tr(T²) - ½ M_S² S²
 - λ₁ H^† T² H - λ₂ (H^† T H) S - λ₃ (H^† H) S² - λ₄ Tr(T²)² - λ₅ Tr(T²) S² - λ₆ S⁴

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