# Quelle:Mac/Niegel 2009

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 Autor Martin Florian Niegel Titel Search for Supersymmetry in Trimuon Final States with the CMS Detector Ort Karlsruhe Datum 18. Dezember 2009 Anmerkung Dissertation, Fakultät für Physik (PHYSIK), Institut für Experimentelle Kernphysik (IEKP), Karlsruher Institut für Technologie; Referent: Prof. Dr. W. de Boer (Institut fur Experimentelle Kernphysik), Korreferent: Prof. Dr. G. Quast (Institut fur Experimentelle Kernphysik) URN http://nbn-resolving.org/urn:nbn:de:swb:90-189499 URL http://digbib.ubka.uni-karlsruhe.de/volltexte/1000018949 Literaturverz. ja Fußnoten ja Fragmente 4

Fragmente der Quelle:
 Zuletzt bearbeitet: 2015-02-23 01:19:46 Graf Isolan

 Typus KomplettPlagiat Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 16, Zeilen: 7-12
Quelle: Niegel 2009
Seite(n): 25, Zeilen: 23-28
2.2.3 SUSY Mass Spectrum

In the MSSM, the masses of the SUSY particles can be calculated via the renormalization group equations (RGE), which are derived from the Lagrangian. With a given initial condition at the GUT scale, the solution of the RGE link the values at the GUT scale with the electroweak scale and thus determine the mass matrices of gauginos, squarks and leptons.

2.3.3 SUSY Mass Spectrum

In the MSSM, the masses of the SUSY particles can be calculated via the renormalization group equations (RGE), which are derived from the Lagrangian. With a given initial condition at the GUT scale, the solution of the RGE link the values at the GUT scale with the electroweak scales and thus determine the mass matrices of gauginos, squarks and leptons.

 Anmerkungen Identisch; ohne Hinweis auf eine Übernahme. Sichter (Graf Isolan)

 Zuletzt bearbeitet: 2015-02-25 13:37:15 Graf Isolan

 Typus KomplettPlagiat Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 17, Zeilen: 1ff. (komplett)
Quelle: Niegel 2009
Seite(n): 25-26, Zeilen: 25:29-32 - 26:1-22
Neutralinos and Charginos

Neutralinos and Charginos are the mass eigenstates of the neutral and charged fields, respectively. Their mass eigenstates are mixed states of gauginos and higgsinos:

$\chi = \left(\begin{array}{c}\tilde B\\ \tilde W^3\\ \tilde H^0_d\\ \tilde H^0_u\end{array} \right)\quad , \quad \psi =\left(\begin{array}{c}\tilde W^+\\ \tilde H^+\end{array} \right)\quad , \quad\quad (2.35)$

where χ and Ψ are the Majorana neutralino and Dirac chargino fields, respectively.

The corresponding neutralino mass matrix can be written as:

$\begin{array}{c} M^{(0)} \end{array} = \left( \begin{array}{cccc} M_1 & 0 & -M_Z \cos \beta \sin \theta_W& M_Z \sin \beta \sin \theta_W\\ 0 & M_2 & M_Z \cos \beta \cos \theta_W& -M_Z \sin \beta \cos \theta_W\\ -M_Z \cos \beta \sin \theta_W& M_Z \cos \beta \cos \theta_W & 0 & -\mu\\ M_Z \sin \beta \sin \theta_W& -M_Z \sin \beta \cos \theta_W & -\mu & 0 \end{array} \right) \quad (2.36)$

with the gaugino masses M1, M2, the weak mixing angle θW and tan β, the ratio of two Higgs vacuum expectation values. The physical masses of the neutralinos are given by the eigenvalues of this matrix. The neutralino mass eigenstates are denoted as $\begin{array}{c}\chi^0_1\end{array}$, $\begin{array}{c}\chi^0_2\end{array}$, $\begin{array}{c}\chi^0_3\end{array}$, $\begin{array}{c}\chi^0_4\end{array}$ with $\begin{array}{c}m_{\chi_1^0}\end{array}\le \begin{array}{c}m_{\chi_2^0}\end{array}\le \begin{array}{c}m_{\chi_3^0}\le \begin{array}{c}m_{\chi_4^0}.\end{array}\end{array}$

The mass matrix for the charginos is given by:

$\begin{array}{c} M^{(c)} \end{array} = \left( \begin{array}{cc} M_2 & \sqrt{2} M_W \sin \beta \\ \sqrt{2} M_W \cos \beta & \mu \end{array} \right ), \quad (2.37)$

which leads to two chargino eigenstates $\begin{array}{c}m_{\chi^\pm_{1,2}}\end{array}$ with the mass eigenvalues

$\begin{array}{rl} m_{\chi^\pm_{1,2}}^2 = & \frac{1}{2} M_2^2 + \frac{1}{2} \mu^2 + \frac{1}{2} M_W^2 \\ & \mp \frac{1}{2} \sqrt{(M_2^2 - \mu^2)^2 + 4 M_W^4 \cos^2 2\beta + 4 M_W^2 (M_2^2 + \mu^2 + 2 M_2 \mu \sin 2\beta)} \end{array}. \quad (2.38)$

The gluino is the only color octet fermion. Since SU(3) is unbroken, the gluino does not mix with other MSSM particles. The mass of the physical particle is defined by the gaugino mass parameter $\begin{array}{c} m_{\tilde g} \equiv M_3. \end{array}$

Approximately, the gaugino mass parameters at the electroweak scale are:

$\begin{array}{lcll} M_3 & \simeq & 2.7 m_{1/2} & \quad\quad (2.39) \\ M_2 & \simeq & 0.8 m_{1/2} & \quad\quad (2.40) \\ M_1 & \simeq & 0.4 m_{1/2} & \quad\quad (2.41) \end{array}$

The physical masses of the neutralinos are obtained by diagonalizing the mass matrix Eq. 2.36. In the mSUGRA model, the lightest neutralino is dominantly bino-like [and the next-to-lightest neutralino is mostly wino-like, with masses close to M1 and M2, respectively.]

[Seite 25]

Neutralinos and Charginos

Neutralinos and Charginos are the mass eigenstates of the neutral and charged fields, respectively. Their mass eigenstates are mixed states of gauginos and higgsinos:

$\chi = \left(\begin{array}{c}\tilde B\\ \tilde W^3\\ \tilde H^0_1\\ \tilde H^0_2\end{array} \right)\quad , \psi =\left(\begin{array}{c}\tilde W^+\\ \tilde H^+\end{array} \right)\quad , \quad\quad (2.64)$

[Seite 26]

where χ and Ψ are the Majorana neutralino and Dirac chargino fields, respectively.

The corresponding neutralino mass matrix reads as:

$\begin{array}{c} M^{(0)} \end{array} = \left( \begin{array}{cccc} M_1 & 0 & -M_Z \cos \beta \sin \theta_W& M_Z \sin \beta \sin \theta_W\\ 0 & M_2 & M_Z \cos \beta \cos \theta_W& -M_Z \sin \beta \cos \theta_W\\ -M_Z \cos \beta \sin \theta_W& M_Z \cos \beta \cos \theta_W & 0 & -\mu\\ M_Z \sin \beta \sin \theta_W& -M_Z \sin \beta \cos \theta_W & -\mu & 0 \end{array} \right) \quad (2.65)$

with the gaugino masses M1, M2, the weak mixing angle θW and tan β, the ratio of two Higgs vacuum expectation values. The physical masses of the neutralinos are given by the eigenvalues of this matrix. The neutralino mass eigenstates are denoted as $\begin{array}{c}\chi^0_1\end{array}$, $\begin{array}{c}\chi^0_2\end{array}$, $\begin{array}{c}\chi^0_3\end{array}$, $\begin{array}{c}\chi^0_4\end{array}$ with $\begin{array}{c}m_{\chi_1^0}\end{array}\le \begin{array}{c}m_{\chi_2^0}\end{array}\le \begin{array}{c}m_{\chi_3^0}\le \begin{array}{c}m_{\chi_4^0}.\end{array}\end{array}$

The mass matrix for the charginos given by:

$\begin{array}{c} M^{(c)} \end{array} = \left( \begin{array}{cc} M_2 & \sqrt{2} M_W \sin \beta \\ \sqrt{2} M_W \cos \beta & \mu \end{array} \right ) \quad (2.66)$

leads to two chargino eigenstates $\begin{array}{c}m_{\chi^\pm_{1,2}}\end{array}$ with the mass eigenvalues

$\begin{array}{rl} m_{\chi^\pm_{1,2}}^2 = & \frac{1}{2} M_2^2 + \frac{1}{2} \mu^2 + \frac{1}{2} M_W^2 \\ & \mp \frac{1}{2} \sqrt{(M_2^2 - \mu^2)^2 + 4 M_W^4 \cos^2 2\beta + 4 M_W^2 (M_2^2 + \mu^2 + 2 M_2 \mu \sin 2\beta)} \end{array}. \quad (2.67)$

The gluino is the only color octet fermion. Since the SU(3) is unbroken, the gluino does not mix with other MSSM particles. The mass of the physical particle is defined by the gaugino mass parameter $\begin{array}{c} m_{\tilde g} \equiv M_3. \end{array}$

Approximately, the gaugino mass parameters at the electroweak scale are:

$\begin{array}{lcll} M_3 & \simeq & 2.7 m_{1/2} & \quad\quad (2.68) \\ M_2 & \simeq & 0.8 m_{1/2} & \quad\quad (2.69) \\ M_1 & \simeq & 0.4 m_{1/2} & \quad\quad (2.70) \end{array}$

The physical masses of the neutralinos are obtained by diagonalizing the mass matrix Eq. 2.65. In the mSUGRA model, the lightest neutralino is dominantly bino-like and the next-to-lightest neutralino is mostly wino-like, with masses close to M1 and M2, respectively.

 Anmerkungen Identisch; ohne Hinweis auf eine Übernahme. Sichter (Graf Isolan)

 Zuletzt bearbeitet: 2015-02-26 12:04:01 Graf Isolan

 Typus KomplettPlagiat Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
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Seite(n): 26-27, Zeilen: 26:20-24 - 27:1-20
[In the mSUGRA model, the lightest neutralino is dominantly bino-like] and the next-to-lightest neutralino is mostly wino-like, with masses close to M1 and M2, respectively. The mass of the lightest chargino is approximately given by M2. Hence the masses of the next-to-lightest neutralino and the lightest chargino are similar, and approximately two times the mass of the lightest neutralino.

Sleptons and Squarks

The masses of left-handed and right-handed fermions are equal. But their superpartners are bosons and the masses of left-handed and right-handed sfermions can be different:

$\begin{array}{lcll} \tilde m_{e_L}^2 & = & \tilde m^2_{L_i} + m^2_{E_i} + M_Z^2 \cos (2\beta)\left(-{\displaystyle \frac{1}{2}} + \sin^2 \theta_W \right) &\quad \quad (2.42) \\ \\ \tilde m_{\nu_L}^2 & = & \tilde m^2_{L_i} + M_Z^2 \cos (2\beta)\left(\displaystyle{\frac{1}{2}}\right) &\quad \quad (2.43) \\ \\ \tilde m_{e_R}^2 & = & \tilde m^2_{E_i} + m^2_{E_i} - M_Z^2 \cos (2\beta)\left(\sin^2 \theta_W \right) &\quad \quad (2.44) \\ \\ \tilde m_{u_L}^2 & = & \tilde m^2_{Q_i} + m^2_{U_i} + M_Z^2 \cos (2\beta)\left(+ \displaystyle{\frac{1}{2}} + \sin^2 \theta_W \right) &\quad \quad (2.45) \\ \\ \tilde m_{d_L}^2 & = & \tilde m^2_{Q_i} + m^2_{D_i} + M_Z^2 \cos (2\beta)\left(-\displaystyle{\frac{1}{2} + \frac{1}{3}} \sin^2 \theta_W \right) &\quad \quad (2.46) \\ \\ \tilde m_{u_R}^2 & = & \tilde m^2_{U_i} + m^2_{U_i} + M_Z^2 \cos (2\beta)\left(\displaystyle {\frac{2}{3}} \sin^2 \theta_W \right) &\quad \quad (2.47) \\ \\ \tilde m_{d_R}^2 & = & \tilde m^2_{D_i} + m^2_{D_i} - M_Z^2 \cos (2\beta)\left({\displaystyle \frac{1}{3}}\sin^2 \theta_W \right) \quad .&\quad \quad (2.48) \end{array}$

On the right side of the equations, the terms denoted as $\tilde m$ are calculated with the RGE, the mass terms m are the fermion masses. The index i denotes the three generations.

Furthermore non-negligible Yukawa couplings lead to a mixing between the electroweak eigenstates and the mass eigenstates of the third generation sleptons and squarks. Due to small Yukawa couplings the mixing is negligible for the first and second generation. Therefore the mass eigenstates corresponds to the interaction eigenstates, which have been introduced above. The mass matrices for the third generation can be written as:

$\begin{array}{lcll} {\mathcal M}^{\tilde t} & = & \left( \begin{array}{cc} \tilde m_{t_L}^2 & m_t (A_t - \mu \cot \beta)\\ m_t (A_t - \mu \cot \beta) & \tilde m_{t_R}^2 \end{array} \right) &\quad\quad (2.49)\\ \\ {\mathcal M}^{\tilde b} & = & \left( \begin{array}{cc} \tilde m_{b_L}^2 & m_b (A_b - \mu \tan \beta)\\ m_b (A_b - \mu \tan \beta) & \tilde m_{b_R}^2 \end{array} \right) &\quad\quad (2.50)\\ \\ {\mathcal M}^{\tilde \tau} & = & \left( \begin{array}{cc} \tilde m_{\tau_L}^2 & m_\tau (A_\tau - \mu \tan \beta)\\ m_\tau (A_\tau - \mu \tan \beta) & \tilde m_{\tau_R}^2 \end{array} \right)\quad . &\quad\quad (2.51) \end{array}$

[Seite 26]

In the mSUGRA model, the lightest neutralino is dominantly bino-like and the next-to-lightest neutralino is mostly wino-like, with masses close to M1 and M2, respectively. The mass of the lightest chargino is approximately given by M2. Hence the masses of the next-to-lightest neutralino and the lightest chargino are similar, and approximately two times the mass of the lightest neutralino.

[Seite 27]

Sleptons and Squarks

The masses of left-handed and right-handed fermions are equal. But their superpartners are bosons and the masses of left-handed and right-handed sfermions can be different:

$\begin{array}{lcll} \tilde m_{e_L}^2 & = & \tilde m^2_{L_i} + m^2_{E_i} + M_Z^2 \cos (2\beta)\left(-{\displaystyle \frac{1}{2}} + \sin^2 \theta_W \right) &\quad \quad (2.71) \\ \\ \tilde m_{\nu_L}^2 & = & \tilde m^2_{L_i} + M_Z^2 \cos (2\beta)\left(\displaystyle{\frac{1}{2}}\right) &\quad \quad (2.72) \\ \\ \tilde m_{e_R}^2 & = & \tilde m^2_{E_i} + m^2_{E_i} - M_Z^2 \cos (2\beta)\left(\sin^2 \theta_W \right) &\quad \quad (2.73) \\ \\ \tilde m_{u_L}^2 & = & \tilde m^2_{Q_i} + m^2_{U_i} + M_Z^2 \cos (2\beta)\left(+ \displaystyle{\frac{1}{2}} + \sin^2 \theta_W \right) &\quad \quad (2.74) \\ \\ \tilde m_{d_L}^2 & = & \tilde m^2_{Q_i} + m^2_{D_i} + M_Z^2 \cos (2\beta)\left(-\displaystyle{\frac{1}{2} + \frac{1}{3}} \sin^2 \theta_W \right) &\quad \quad (2.75) \\ \\ \tilde m_{u_R}^2 & = & \tilde m^2_{U_i} + m^2_{U_i} + M_Z^2 \cos (2\beta)\left(\displaystyle {\frac{2}{3}} \sin^2 \theta_W \right) &\quad \quad (2.76) \\ \\ \tilde m_{d_R}^2 & = & \tilde m^2_{D_i} + m^2_{D_i} - M_Z^2 \cos (2\beta)\left({\displaystyle \frac{1}{3}}\sin^2 \theta_W \right)\quad . &\quad \quad (2.77) \end{array}$

On the right side of the equations, the terms denoted as $\tilde m$ are calculated with the RGE, the mass terms m are the fermion masses. The index i denotes the three generations.

Furthermore non-negligible Yukawa couplings lead to a mixing between the electroweak eigenstates and the mass eigenstates of the third generation sleptons and squarks. Due to small Yukawa couplings the mixing is negligible for the first and second generation. Therefore the mass eigenstates corresponds to the interaction eigenstates, which have been introduced above. The mass matrices for the third generation reads as:

$\begin{array}{lcll} {\mathcal M}^{\tilde t} & = & \left( \begin{array}{cc} \tilde m_{t_L}^2 & m_t (A_t - \mu \cot \beta)\\ m_t (A_t - \mu \cot \beta) & \tilde m_{t_R}^2 \end{array} \right) &\quad\quad (2.78)\\ \\ {\mathcal M}^{\tilde b} & = & \left( \begin{array}{cc} \tilde m_{b_L}^2 & m_b (A_b - \mu \tan \beta)\\ m_b (A_b - \mu \tan \beta) & \tilde m_{b_R}^2 \end{array} \right) &\quad\quad (2.79)\\ \\ {\mathcal M}^{\tilde \tau} & = & \left( \begin{array}{cc} \tilde m_{\tau_L}^2 & m_\tau (A_\tau - \mu \tan \beta)\\ m_\tau (A_\tau - \mu \tan \beta) & \tilde m_{\tau_R}^2 \end{array} \right)\quad . &\quad\quad (2.80) \end{array}$

 Anmerkungen Identisch; ohne Hinweis auf eine Übernahme. Sichter (Graf Isolan)

 Zuletzt bearbeitet: 2015-02-27 13:06:45 Graf Isolan

 Typus KomplettPlagiat Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
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Quelle: Niegel 2009
Seite(n): 27, Zeilen: 21-24
The mass eigenstates of the third generation are:

[...]

The mass eigenstates of the third generation are:
 Anmerkungen Schließt die auf den Vorderseiten begonnene identische Übernahme ab. Sichter (Graf Isolan)