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VEVs

VEVs

The particles responsible for breaking a gauge symmetry receive a VEV. After the symmetry breaking, these particles are parametrized by a scalar ϕ and a pseudo scalar σ part and the VEV v:

S = \frac{1}{\sqrt{2}} \left( \phi_S + i \sigma_S + v_S \right)

Implementation in SARAH 

This is in SARAH done by

DEFINITION[$EIGENSTATES][VEVs] =
{Particle Name, {{VEV, Coefficient 1},
      {Pseudoscalar, Coefficient 2},{Scalar, Coefficient 3},({Phase})};
  1. Name: The name of the particle receiving a VEV
  2. VEV: Name of the VEV
  3. Scalar: Name of the scalar component
  4. Pseudoscalar: Name of the pseudo scalar component
  5. Coefficient 1,2,3: The different (numerical) coefficients.
  6. Phase: Optional phase

All indices carried by the particle receiving the VEV are automatically added to the scalar and pseudo scalar part. The scalar, pseudo scalar and the VEV are handled as real parameters in SARAH . The phase is only an optional argument and can be skipped for Higgs sectors without CP violation.

Example

In the MSSM, the HiggsHd0 andHu0 get VEVsvd andvu:

H_u^0 = \frac{1}{\sqrt{2}} \left(v_u + i \sigma_u +\phi_u \right) \, , \hspace{1cm} H_d^0 = \frac{1}{\sqrt{2}} \left(v_d + i \sigma_d +\phi_d \right)

This is done in SARAH by using

DEFINITION[EWSB][VEVs]=
{{SHd0, {vd, 1/Sqrt[2]}, {sigmad, I/Sqrt[2]},{phid,1/Sqrt[2]}},
 {SHu0, {vu, 1/Sqrt[2]}, {sigmau, I/Sqrt[2]},{phiu,1/Sqrt[2]}},
};

To add a relative phase, use

DEFINITION[EWSB][VEVs]=
{{SHd0, {vd, 1/Sqrt[2]}, {sigmad, I/Sqrt[2]},{phid,1/Sqrt[2]}},
 {SHu0, {vu, 1/Sqrt[2]}, {sigmau, I/Sqrt[2]},{phiu,1/Sqrt[2]},{eta}},
};

This is interpreted as

H_u^0 = \frac{e^{i \eta}}{\sqrt{2}} \left(v_u + i \sigma_u +\phi_u \right) \, , \hspace{1cm} H_d^0 = \frac{1}{\sqrt{2}} \left(v_d + i \sigma_d +\phi_d \right)

Aligned VEVs

The standard definition of a model with broken electric charge due to VEVs charged slepton VEVs looks like

DEFINITION[EWSB][VEVs]=
  {...
   {SeL, {vL, 1/Sqrt[2]}, {sigmaL,I/Sqrt[2]},{phiL,1/Sqrt[2]}},
   {SeR, {vR, 1/Sqrt[2]}, {sigmaR,I/Sqrt[2]},{phiR,1/Sqrt[2]}},
  };

With this definition, all three generations of left and right sleptons would get a VEV. However, usually one is only interested in the case that staus receive VEVs. This can now be defined by

DEFINITION[EWSB][VEVs]=
  {..,
   {SeL, {vL[3], 1/Sqrt[2]}, {sigmaL,I/Sqrt[2]},{phiL,1/Sqrt[2]}},
   {SeR, {vR[3], 1/Sqrt[2]}, {sigmaR,I/Sqrt[2]},{phiR,1/Sqrt[2]}}};

If one wants to consider smuon and stau VEVs, vL[2,3], vR[2,3] can be used.

Complex VEVs

To define complex VEVs, it is possible to give the phase as last argument:

DEFINITION[EWSB][VEVs]=
  {{SHd0, {vd, 1/Sqrt[2]}, {sigmad,I/Sqrt[2]},{phid,1/Sqrt[2]}},
   {SHu0, {vu, 1/Sqrt[2]}, {sigmau,I/Sqrt[2]},{phiu,1/Sqrt[2]},{eta}}};

This is understood as H_u^0 \to \frac{\exp(i \eta)}{\sqrt{2}} \left(v_u + i \sigma_u + \phi_u\right). Another possibility to define complex VEVs is to define

DEFINITION[EWSB][VEVs]=
 {{SHd0, {vdR, 1/Sqrt[2]}, {vdI, I/Sqrt[2]},
                      {sigmad,I/Sqrt[2]},{phid,1/Sqrt[2]}},
  {SHu0, {vuR, 1/Sqrt[2]}, {vuI, I/Sqrt[2]},
                      {sigmau,I/Sqrt[2]},{phiu,1/Sqrt[2]}}
   };

which is understood as

H_d^0 \to \frac{1}{\sqrt{2}}\left(v_d^R + i v_d^I + i \sigma_d + \phi_d \right)\,,\hspace{1cm} H_u^0 \to \frac{1}{\sqrt{2}}\left(v_u^R + i v_u^I + i \sigma_u + \phi_u \right) \, .

This format has the advantage that the tree-level tadpole equations are also in the complex case are purely polynomials and can be used numerically with dedicated codes like HOM4PS2 which is used by Vevacious .

See also