R-Parity Violation

R-parity, defined as R= $(-1)^{2S+3B+L}$, is a discrete multiplicative symmetry where $S$ is the particle spin, $B$ is the baryon number, and $L$ is the lepton number. All SM particles have R=1, while all superpartners have R=$-1$, so a single SUSY particle cannot decay into just SM particles if R-parity is conserved. In this case, the lightest superpartner (LSP) is absolutely stable. Astrophysical considerations imply that a stable LSP should be electrically neutral. Viable candidates are the lightest neutralino, the lightest sneutrino, or alternatively the gravitino. Since the LSP can carry away energy without interacting in a detector, the apparent violation of momentum conservation is an important part of SUSY phenomenology. Also, when R-parity is conserved, superpartners must be produced in pairs from a SM initial state. The breaking of the R-parity symmetry would result in lepton and/or baryon number violating processes. While there are strong experimental constraints on some classes of R-parity violating interactions, others are hardly constrained at all.

One simple extension of the MSSM is to break the multiplicative R-parity symmetry. Presently, neither experiment nor any theoretical argument demand R-parity conservation, so it is natural to consider the most general case of R-parity breaking. It is convenient to introduce a function of superfields called the superpotential, from which the Feynman rules for R-parity violating processes can be derived. The R-parity violating (RPV) terms which can contribute to the superpotential are:

$$W_{RPV} = \lambda_{ijk} L^i L^j \bar{E}^k + \lambda^{'}_{ij... ...{''}_{ijk} \bar{U}^i \bar{D}^j \bar{D}^k + \epsilon_i L_iH_2$$ (152)

where $i,j,k$ are generation indices (1,2,3), $L^i_1 \equiv \nu^i_L$, $L^i_2=\ell^i_L$ and $Q^i_1=u^i_{L}$, $Q^i_2=d^i_{L}$ are lepton and quark components of ${\bf SU(2)_L}$ doublet superfields, and $E^i=e^i_{R}$, $D^i=d^i_{R}$ and $U^i=u^i_R$ are lepton, down and up- quark ${\bf SU(2)_L}$ singlet superfields, respectively. The unwritten ${\bf SU(2)_L}$ and ${\bf SU(3)_C}$ indices imply that the first term is antisymmetric under $i \leftrightarrow j$, and the third term is antisymmetric under $j \leftrightarrow k$. Therefore, $i \neq j$ in $L^i L^j \bar{E}^k$ and $j \neq k$ in $\bar{U}^i \bar{D}^j \bar{D}^k$. The coefficients $\lambda_{ijk}$, $\lambda^{'}_{ijk}$, $\lambda^{''}_{ijk}$, and $\epsilon_i$ are Yukawa couplings, and there is no a priori generic prediction for their values. In principle, $W_{RPV}$ contains 48 extra parameters over the $R$-parity-conserving MSSM case. In PYTHIA the effects of the last term in eq. () are not included.

Expanding eq. () as a function of the superfield components, the interaction Lagrangian derived from the first term is

$${\cal{L}}_{LLE} = \lambda_{ijk} \left\{ \tilde{\nu}_L^i e_L^... ...bar{e}^k_R + (\tilde{e}_R^k)^* \nu_L^i e^j_L + h.c. \right\}$$ (153)

and from the second term,

$${\cal{L}}_{LQD} = \lambda^{'}_{ijk} \left\{ \tilde{\nu}_L^i d_L^j... ...+ \tilde{d}_L^j \nu_L^i \bar{d}^k_R - \tilde{u}_L^j e_L^i \bar{d}^k_R + \right.$$

$$\left. (\tilde{d}_R^k)^* \nu_L^i d^j_L - (\tilde{d}_R^k)^* e_L^i u^j_L + h.c. \right\}$$ (154)

Both of these sets of interactions violate lepton number. The $\bar U\bar D\bar D$ term, instead, violates baryon number. In principle, all types of R-parity violating terms may co-exist, but this can lead to a proton with a lifetime shorter than the present experimental limits. The simplest way to avoid this is to allow only operators which conserve baryon-number but violate lepton-number or vice versa.

There are several effects on the SUSY phenomenology due to these new couplings: (1) lepton or baryon number violating processes are allowed, including the production of single sparticles (instead of pair production), (2) the LSP is no longer stable, but can decay to SM particles within a collider detector, and (3) because it is unstable, the LSP need not be the neutralino or sneutrino, but can be charged and/or colored.

In the current version of PYTHIA, decays of supersymmetric particles to SM particles via two different types of lepton number violating couplings and one type of baryon number violating couplings can be invoked (Details about the $L$-violation implementation and tests can be found in [Ska01]).

Complete matrix elements (including $L-R$ mixing for all sfermion generations) for all two-body sfermion and three-body neutralino, chargino, and gluino decays are included (as given in [Dre00]). The final state fermions are treated as massive in the phase space integrations and in the matrix elements for $\b$, $\t$, and $\tau$.

The existence of $R$-odd couplings also allows for single sparticle production, i.e. there is no requirement that SUSY particles should be produced in pairs. Single sparticle production cross sections are not yet included in the program, and it may require some rethinking of the parton shower to do so. For low-mass sparticles, the associated error is estimated to be negligible, as long as the $R$-violating couplings are smaller than the gauge couplings. For higher mass sparticles, the reduction of the phase space for pair production becomes an important factor, and single sparticle production could dominate even for very small values of the $R$-violating couplings. The total SUSY production cross sections, as calculated by PYTHIA in its current form are thus underestimated, possibly quite severely for heavy-mass sparticles.

Three possibilities exist for the initializations of the couplings, representing a fair but not exhaustive range of models. The first, selected by setting IMSS(51)=1 for LLE, IMSS(52)=1 for LQD, and/or IMSS(53)=1 for UDD type couplings, sets all the couplings, independent of generation, to a common value of $10^{-\mbox{\scriptsize\texttt{RMSS(51)}}}$, $10^{-\mbox{\scriptsize\texttt{RMSS(52)}}}$, and/or $10^{-\mbox{\scriptsize\texttt{RMSS(53)}}}$, depending on which couplings are activated.

Taking now LLE couplings as an example, setting IMSS(51)=2 causes the LLE couplings to be initialized (in PYINIT) to so-called `natural' generation-hierarchical values, as proposed in [Hin93]. These values, inspired by the structure of the Yukawa couplings in the SM, are defined by:

$$\begin{array}{rcl} \vert\lambda_{ijk}\vert^2 & = & (\mbox{\... ...{0.7cm} \hat{m}\equiv \frac{m}{v} = \frac{m}{126\mathrm{GeV}}$$ (155)

where $m_{q_i}$ is the arithmetic mean of $m_{u_i}$ and $m_{d_i}$.

The third option available is to set IMSS(51)=3, IMSS(52)=3, and/or IMSS(53)=3, in which case all the relevant couplings are zero by default (but the corresponding lepton or baryon number violating processes are turned on) and the user is expected to enter the non-zero coupling values by hand. (Where antisymmetry is required, half of the entries are automatically derived from the other half, see IMSS(51)=3 and IMSS(53)=3.) RVLAM($i$,$j$,$k$) contains the $\lambda_{ijk}$, RVLAMP($i$,$j$,$k$) contains the $\lambda'_{ijk}$ couplings, and RVLAMB($i$,$j$,$k$) contains the $\lambda''_{ijk}$ couplings.