Matchete`
Matchete`

DefineGaugeGroup

DefineGaugeGroup[groupName,LieGroup,coupling,fieldName]

defines a gauge group labelled groupName from a simple or Abelian LieGroup, together with its corresponding coupling and gauge bosons labelled fieldName.

Details and Options

  • The LieGroup can either be U1 or a simple group. The simple group are more accurately defined by their Lie algebra (Alg) in this case, although the functions SU, SO, and Sp provide convenient shortcuts to get the Algebra of the group in question.
  • A scalar coupling with the name coupling and EFTOrder 0 is initialized with the gauge group (as with DefineCoupling).
  • DefineGaugeGroup also defines the fundamental and adjoint representations of the corresponding non-Abelian group. They are referred to as groupName[fund] and groupName[adj], respectively.
  • The initialization of a simple gauge group also defines some Clebsch-Gordan (CG) coefficients among the representations: the generators of gen[groupName[fund]] and gen[groupName[adj]] of the fundamental and adjoint representations; Kronecker deltas del[groupName[fund]] and del[groupName[adj]] of the representations; and the structure constant of the group fStruct[groupName].
  • SU(N) groups also initialize the Levi-Civita symbol eps[groupName].
  • Sp(N) groups also initialize the 2-index antisymmetric invariant eps[groupName].
  • The following options can be given:
  • AdjAlphabet Nonespecifies a printing alphabet for indices of the adjoint representation under NiceForm.
    FundAlphabet Nonespecifies a canonical printing alphabet for indices of the fundamental representation under NiceForm.
    NiceForm Defaultprovides the display of the gauge coupling and field under NiceForm formatting.
  • The allowed options for AdjAlphabet and FundAlphabet are
  • Nonedefault index alphabet is uses
    {index1, index2, … }repeated indices of the representation are canonically displayed with as these strings under NiceForm
  • The allowed options for NiceForm are
  • Defaultuses Default format for both coupling and gauge field
    {format, Default}formats the gauge coupling as the string format
    {Default, format}formats the gauge field as the string format
    {format1, format2}formats the coupling as format1 and the field as format2

Examples

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Basic Examples  (1)

An SU(3) gauge group referred to as G, with a gauge field A and a coupling g, is initialized with

Afterwards a the coupling is available with

and a field strength tensor (shortcut FS) with 2 Lorentz and 1 adjoint index:

Scope  (2)

An Abelian gauge group is initialized by using U1 for the LieGroup. For instance the SM hypercharge group is

For Abelian groups the field-strength tensor only has two Lorentz indices, since the Lie algebra is one-dimensional (trivial):

To initialize a field that is charged under this group, one can use the Charges option for DefineField. For, e.g., a charge 1/2 scalar, we may use

where the group name is used a Head for the charge value.

Initialization of non-Abelian gauge groups also initializes fundamental and adjoint representations and a several Clebsch-Gordan coefficients (CG) between them. For instance the SM color group

Initializes the following CGs involving the fundamental and adjoint representations:

More representations and CGs can be defined with DefineRepresentation and DefineCG.

Generalizations & Extensions  (1)

In principle more exotic gauge groups can be initialized by passing a Lie algebra (Alg) for the LieGroup argument. For instance, the special group G2 is initialized by

We see that the trace of the Kronecker delta associated with the fundamental representation is

as is the dimension of the fundamental representation.

Options  (4)

FundAlphabet  (1)

The FundAlphabet option allows the user to specify the canonical labeling (under NiceForm) of repeated indices that belong to the fundamental representation of the gauge group. To use lower case Greek letter starting from α we may specify

The Kronecker delta of the fundamental representation comes with fundamental and anti-fundamental indices. RelabelIndices changes the repeated index labels to the canonical form

AdjAlphabet  (1)

Similarly to FundAlphabet, the AdjAlphabet option allows the user to specify the canonical labeling (under NiceForm) of repeated indices that belong to the adjoint representation of the gauge group. To use lower case Greek letter starting from α we may specify

The structure constant of the group comes with indices in the adjoint representation. RelabelIndices changes the repeated index labels to the canonical form

NiceForm  (2)

The display of the gauge coupling under NiceForm can be changed with this option. For instance,

changes the coupling display to

while using default formatting for the gauge field.

The display of the gauge field under NiceForm can be changed with this option. For instance,

changes the display of the field-strength tensor accordingly: