|
|
---
|
|
|
title: VEVs
|
|
|
permalink: /VEVs/
|
|
|
---
|
|
|
|
|
|
[Category:Model](/Category:Model "wikilink") 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 Higgs*H*<sub>*d*</sub><sup>0</sup> and*H*<sub>*u*</sub><sup>0</sup> get VEVs*v*<sub>*d*</sub> and*v*<sub>*u*</sub>:
|
|
|
|
|
|
$H_u^0 = \\frac{1}{\\sqrt{2}} \\left(v_u + i \\sigma_u +\\phi_u \\right) \\, , \\hspace{1cm}
|
|
|
H_d^0 = \\frac{1}{\\sqrt{2}} \\left(v_d + i \\sigma_d +\\phi_d \\right)$
|
|
|
|
|
|
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, <span>vL\[2,3\]</span>, <span>vR\[2,3\]</span> 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 <span>HOM4PS2</span> which is used by <span>Vevacious</span> .
|
|
|
|
|
|
See also
|
|
|
-------- |
|
|
\ No newline at end of file |