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Gauge_Symmetries_non SUSY

Gauge Symmetries non-SUSY

Definition of Gauge groups

The Gauge groups are defined by the array Gauge. An entry reads

Gauge[[/i|i]]={Name, Dimension, Name of Gauge Group, Coupling, Expand, Global};

The different parts have the following meaning:

  1. Name: This is the name for the gauge groups which also fixed the name for the vector bosons and ghosts
  2. Dimension: This defines the dimension of the S**U(N) gauge group: U[1] for an Abelian gauge group or SU[N] with integer N for a non-Abelian gauge group.
  3. Name of Gauge Group: This is the name of the gauge group, e.g. hypercharge, color or left. This choice is import because all matter particles charged under a non-Abelian gauge group carry an corresponding index. The name of the index consists of the first three letter of the name plus a number. Hence, it must be taken care that the first three letters of different gauge group names are not identical. Also the name for the indices in the adjoint representation are derived from this entry.
  4. Coupling: The name of the coupling constant, e.g. g1
  5. Expand: Values can be True or False. If it is set to True, all sums over the corresponding indices are evaluated during the calculation of the Lagrangian. This is normally done non-Abelian gauge groups which get broken like the S**U(2)L in the MSSM.
  6. Global: Transformation under global symmetries
Example: Standard model color group
Gauge[[/3|3]] = {G,  SU[3],  color,  g3, False};

The consequence of this entry is

  1. Gluon and gluon ghost are named VG respectively gG
  2. The S**U(3) generators, the Gell-Mann matrices, are used
  3. The color index is abbreviated colX (for X = 1,2, ...)
  4. The strong coupling constant is named g3
  5. The sums over the color indices are not evaluated

Models with several U(1) gauge groups

In the case of several Abelian gauge groups, there is an additional particulariyt: Gauge kinetic mixing.

SARAH uses

Dμ = ∂μ − i(ḡaQa + ḡb**a)Āμa − i(ḡa**bQa + ḡbQb)Āμb

for the covariant derivatives to write the Lagrangian in that case. For that purposes, it generates new gauge couplings

g<>A<>B

for the off-diagonal couplings. Here gA and gB are the names for the diagonal gauge couplings defined in Gauge, i.e the first letter is always dropped.

Example

In the case of a gauge sector containing

Gauge[[/1|1]] = {R,  U[1],  right,  gR, False};
Gauge[[/2|2]] = {BL,  U[1],  bminusl,  gBL, False};

the off-diagonal gauge couplings are called

gRBL
gBLR

See also