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(* $Id: coupling.mli 4663 2013-09-23 13:51:54Z msekulla $
Copyright (C) 1999-2013 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
with contributions from
Christian Speckner <cnspeckn@googlemail.com>
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
(* The enumeration types used for communication from [Models]
to [Targets]. On the physics side, the modules in [Models]
must implement the Feynman rules according to the conventions
set up here. On the numerics side, the modules in [Targets]
must handle all cases according to the same conventions. *)
(* \thocwmodulesection{Propagators}
The Lorentz representation of the particle. NB: O'Mega
treats all lines as \emph{outgoing} and particles are therefore
transforming as [ConjSpinor] and antiparticles as [Spinor]. *)
type lorentz =
| Scalar
| Spinor (* $\psi$ *)
| ConjSpinor (* $\bar\psi$ *)
| Majorana (* $\chi$ *)
| Maj_Ghost (* SUSY ghosts *)
| Vector
(*i | Ward_Vector i*)
| Massive_Vector
| Vectorspinor (* supersymmetric currents and gravitinos *)
| Tensor_1
| Tensor_2 (* massive gravitons (large extra dimensions) *)
| BRS of lorentz
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{2.2}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
[Prop_Scalar]
& \multicolumn{2}{l|}{%
$\displaystyle\phi(p)\leftarrow
\frac{\ii}{p^2-m^2+\ii m\Gamma}\phi(p)$} \\\hline
[Prop_Spinor]
& $\displaystyle\psi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\psi(p)$
& $\displaystyle\psi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\psi(p)$ \\\hline
[Prop_ConjSpinor]
& $\displaystyle\bar\psi(p)\leftarrow
\bar\psi(p)\frac{\ii(\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}$
& $\displaystyle\psi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\psi(p)$ \\\hline
[Prop_Majorana]
& \multicolumn{1}{c|}{N/A}
& $\displaystyle\chi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\chi(p)$ \\\hline
[Prop_Unitarity]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon_\mu(p)\leftarrow
\frac{\ii}{p^2-m^2+\ii m\Gamma}
\left(-g_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}\right)\epsilon^\nu(p)$} \\\hline
[Prop_Feynman]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon^\nu(p)\leftarrow
\frac{-\ii}{p^2-m^2+\ii m\Gamma}\epsilon^\nu(p)$} \\\hline
[Prop_Gauge]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon_\mu(p)\leftarrow
\frac{\ii}{p^2}
\left(-g_{\mu\nu}+(1-\xi)\frac{p_\mu p_\nu}{p^2}\right)\epsilon^\nu(p)$} \\\hline
[Prop_Rxi]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon_\mu(p)\leftarrow
\frac{\ii}{p^2-m^2+\ii m\Gamma}
\left(-g_{\mu\nu}+(1-\xi)\frac{p_\mu p_\nu}{p^2-\xi m^2}\right)
\epsilon^\nu(p)$} \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:propagators} Propagators. NB: The sign of the
momenta in the spinor propagators comes about because O'Mega
treats all momenta as \emph{outgoing} and the charge flow for
[Spinor] is therefore opposite to the momentum, while the charge
flow for [ConjSpinor] is parallel to the momentum.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|r|l|}\hline
[Aux_Scalar]
& $\displaystyle\phi(p)\leftarrow\ii\phi(p)$ \\\hline
[Aux_Spinor]
& $\displaystyle\psi(p)\leftarrow\ii\psi(p)$ \\\hline
[Aux_ConjSpinor]
& $\displaystyle\bar\psi(p)\leftarrow\ii\bar\psi(p)$ \\\hline
[Aux_Vector]
& $\displaystyle\epsilon^\mu(p)\leftarrow\ii\epsilon^\mu(p)$ \\\hline
[Aux_Tensor_1]
& $\displaystyle T^{\mu\nu}(p)\leftarrow\ii T^{\mu\nu}(p)$ \\\hline
[Only_Insertion]
& \multicolumn{1}{c|}{N/A} \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:aux-propagators} Auxiliary and non propagating fields}
\end{table}
If there were no vectors or auxiliary fields, we could deduce the propagator from
the Lorentz representation. While we're at it, we can introduce
``propagators'' for the contact interactions of auxiliary fields
as well. [Prop_Gauge] and [Prop_Feynman] are redundant as special
cases of [Prop_Rxi].
The special case [Only_Insertion] corresponds to operator insertions
that do not correspond to a propagating field all. These are used
for checking Slavnov-Taylor identities
\begin{equation}
\partial_\mu\Braket{\text{out}|W^\mu(x)|\text{in}}
= m_W\Braket{\text{out}|\phi(x)|\text{in}}
\end{equation}
of gauge theories in unitarity gauge where the Goldstone bosons are
not propagating. Numerically, it would suffice to use a vanishing
propagator, but then superflous fusions would be calculated in
production code in which the Slavnov-Taylor identities are not tested. *)
type 'a propagator =
| Prop_Scalar | Prop_Ghost
| Prop_Spinor | Prop_ConjSpinor | Prop_Majorana
| Prop_Unitarity | Prop_Feynman | Prop_Gauge of 'a | Prop_Rxi of 'a
| Prop_Tensor_2 | Prop_Vectorspinor
| Prop_Col_Scalar | Prop_Col_Feynman | Prop_Col_Majorana
| Prop_Col_Unitarity
| Aux_Scalar | Aux_Vector | Aux_Tensor_1
| Aux_Col_Scalar | Aux_Col_Vector | Aux_Col_Tensor_1
| Aux_Spinor | Aux_ConjSpinor | Aux_Majorana
| Only_Insertion
(* \begin{JR}
We don't need different fermionic propagators as supposed by the variable
names [Prop_Spinor], [Prop_ConjSpinor] or [Prop_Majorana]. The
propagator in all cases has to be multiplied on the left hand side of the
spinor out of which a new one should be built. All momenta are treated as
\emph{outgoing}, so for the propagation of the different fermions the
following table arises, in which the momentum direction is always downwards
and the arrows show whether the momentum and the fermion line,
respectively are parallel or antiparallel to the direction of calculation:
\begin{center}
\begin{tabular}{|l|c|c|c|c|}\hline
Fermion type & fermion arrow & mom. & calc. & sign \\\hline\hline
Dirac fermion & $\uparrow$ & $\uparrow~\downarrow$ &
$\uparrow~\uparrow$ & negative \\\hline
Dirac antifermion & $\downarrow$ & $\downarrow~\downarrow$ &
$\uparrow~\downarrow$ & negative \\\hline
Majorana fermion & - & $\uparrow~\downarrow$ & - & negative \\\hline
\end{tabular}
\end{center}
So the sign of the momentum is always negative and no further distinction
is needed.
\end{JR} *)
type width =
| Vanishing
| Constant
| Timelike
| Running
| Fudged
| Custom of string
(* \thocwmodulesection{Vertices}
The combined $S-P$ and $V-A$ couplings (see
tables~\ref{tab:dim4-fermions-SP}, \ref{tab:dim4-fermions-VA},
\ref{tab:dim4-fermions-SPVA-maj} and~\ref{tab:dim4-fermions-SPVA-maj2})
are redundant, of course, but they allow some targets to create
more efficient numerical code.\footnote{An additional benefit
is that the counting of Feynman diagrams is not upset by a splitting
of the vectorial and axial pieces of gauge bosons.} Choosing VA2 over
VA will cause the FORTRAN backend to pass the coupling as a whole array *)
type fermion = Psi | Chi | Grav
type fermionbar = Psibar | Chibar | Gravbar
type boson =
| SP | SPM | S | P | SL | SR | SLR | VA | V | A | VL | VR | VLR | VLRM
| TVA | TLR | TRL | TVAM | TLRM | TRLM
| POT | MOM | MOM5 | MOML | MOMR | LMOM | RMOM | VMOM | VA2
type boson2 = S2 | P2 | S2P | S2L | S2R | S2LR
| SV | PV | SLV | SRV | SLRV | V2 | V2LR
(* The integer is an additional coefficient that multiplies the respective
coupling constant. This allows to reduce the number of required coupling
constants in manifestly symmetrc cases. Most of times it will be equal
unity, though. *)
(* The two vertex types [PBP] and [BBB] for the couplings of two fermions or
two antifermions ("clashing arrows") is unavoidable in supersymmetric
theories.
\begin{dubious}
\ldots{} tho doesn't like the names and has promised to find a better
mnemonics!
\end{dubious} *)
type 'a vertex3 =
| FBF of int * fermionbar * boson * fermion
| PBP of int * fermion * boson * fermion
| BBB of int * fermionbar * boson * fermionbar
| GBG of int * fermionbar * boson * fermion (* gravitino-boson-fermion *)
| Gauge_Gauge_Gauge of int | Aux_Gauge_Gauge of int
| Scalar_Vector_Vector of int
| Aux_Vector_Vector of int | Aux_Scalar_Vector of int
| Scalar_Scalar_Scalar of int | Aux_Scalar_Scalar of int
| Vector_Scalar_Scalar of int
| Graviton_Scalar_Scalar of int
| Graviton_Vector_Vector of int
| Graviton_Spinor_Spinor of int
| Dim4_Vector_Vector_Vector_T of int
| Dim4_Vector_Vector_Vector_L of int
| Dim4_Vector_Vector_Vector_T5 of int
| Dim4_Vector_Vector_Vector_L5 of int
| Dim6_Gauge_Gauge_Gauge of int
| Dim6_Gauge_Gauge_Gauge_5 of int
| Aux_DScalar_DScalar of int | Aux_Vector_DScalar of int
| Dim5_Scalar_Gauge2 of int (* %
$\frac12 \phi F_{1,\mu\nu} F_2^{\mu\nu} = - \frac12
\phi (\ii \partial_{[\mu,} V_{1,\nu]})(\ii \partial^{[\mu,} V_2^{\nu]})$ *)
| Dim5_Scalar_Gauge2_Skew of int
(* %
$\frac12 \phi F_{1,\mu\nu} \tilde{F}_2^{\mu\nu} = -
\phi (\ii \partial_\mu V_{1,\nu})(\ii \partial_\rho V_{2,\sigma})\epsilon^{\mu\nu\rho\sigma}$ *)
| Dim5_Scalar_Vector_Vector_T of int (* %
$\phi(\ii\partial_\mu V_1^\nu)(\ii\partial_\nu V_2^\mu)$ *)
| Dim5_Scalar_Vector_Vector_TU of int (* %
$(\ii\partial_\nu\phi) (\ii\partial_\mu V_1^\nu) V_2^\mu$ *)
| Dim5_Scalar_Vector_Vector_U of int (* %
$(\ii\partial_\nu\phi) (\ii\partial_\mu V^\nu) V^\mu$ *)
| Dim6_Vector_Vector_Vector_T of int (* %
$V_1^\mu ((\ii\partial_\nu V_2^\rho)%
\ii\overleftrightarrow{\partial_\mu}(\ii\partial_\rho V_3^\nu))$ *)
| Tensor_2_Vector_Vector of int (* %
$T^{\mu\nu} (V_{1,\mu}V_{2,\nu} + V_{1,\nu}V_{2,\mu})$ *)
| Tensor_2_Vector_Vector_1 of int (* %
$T^{\mu\nu} (V_{1,\mu}V_{2,\nu} + V_{1,\nu}V_{2,\mu} - g_{\mu,\nu}V_1^\rho V_{2,\rho} )$ *)
| Dim5_Tensor_2_Vector_Vector_1 of int (* %
$T^{\alpha\beta} (V_1^\mu
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta V_{2,\mu}$ *)
| Dim5_Tensor_2_Vector_Vector_2 of int
(* %
$T^{\alpha\beta}
( V_1^\mu \ii\overleftrightarrow\partial_\beta (\ii\partial_\mu V_{2,\alpha})
+ V_1^\mu \ii\overleftrightarrow\partial_\alpha (\ii\partial_\mu V_{2,\beta}))$ *)
| Dim7_Tensor_2_Vector_Vector_T of int (* %
$T^{\alpha\beta} ((\ii\partial^\mu V_1^\nu)
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta
(\ii\partial_\nu V_{2,\mu})) $ *)
(* As long as we stick to renormalizable couplings, there are only
three types of quartic couplings: [Scalar4], [Scalar2_Vector2]
and [Vector4]. However, there are three inequivalent contractions
for the latter and the general vertex will be a linear combination
with integer coefficients:
\begin{subequations}
\begin{align}
\ocwupperid{Scalar4}\,1 :&\;\;\;\;\;
\phi_1 \phi_2 \phi_3 \phi_4 \\
\ocwupperid{Scalar2\_Vector2}\,1 :&\;\;\;\;\;
\phi_1^{\vphantom{\mu}} \phi_2^{\vphantom{\mu}}
V_3^\mu V_{4,\mu}^{\vphantom{\mu}} \\
\ocwupperid{Vector4}\,\lbrack 1, \ocwupperid{C\_12\_34} \rbrack :&\;\;\;\;\;
V_1^\mu V_{2,\mu}^{\vphantom{\mu}}
V_3^\nu V_{4,\nu}^{\vphantom{\mu}} \\
\ocwupperid{Vector4}\,\lbrack 1, \ocwupperid{C\_13\_42} \rbrack :&\;\;\;\;\;
V_1^\mu V_2^\nu
V_{3,\mu}^{\vphantom{\mu}} V_{4,\nu}^{\vphantom{\mu}} \\
\ocwupperid{Vector4}\,\lbrack 1, \ocwupperid{C\_14\_23} \rbrack :&\;\;\;\;\;
V_1^\mu V_2^\nu
V_{3,\nu}^{\vphantom{\mu}} V_{4,\mu}^{\vphantom{\mu}}
\end{align}
\end{subequations} *)
type contract4 = C_12_34 | C_13_42 | C_14_23
(*i\begin{dubious}
CS objected to the polymorphic [type 'a vertex4], since it broke the
implementation of some of his extensions. Is there another way of
getting coupling constants into [Vector4_K_Matrix], besides the brute
force solution of declaring the possible coupling constants here?
\textit{I'd like to put the blame on CS for two reasons: it's not clear
that the brute force solution will actually work and everytime a new
vertex that depends non-linearly on coupling contanst pops up, the
problem will make another appearance.}
\end{dubious}i*)
type 'a vertex4 =
| Scalar4 of int
| Scalar2_Vector2 of int
| Vector4 of (int * contract4) list
| DScalar4 of (int * contract4) list
| DScalar2_Vector2 of (int * contract4) list
| GBBG of int * fermionbar * boson2 * fermion
(* In some applications, we have to allow for contributions outside of
perturbation theory. The most prominent example is heavy gauge boson
scattering at very high energies, where the perturbative expression
violates unitarity. *)
(* One solution is the `$K$-matrix' ansatz. Such unitarizations typically
introduce effective propagators and/or vertices that violate crossing
symmetry and vanish in the $t$-channel. This can be taken care of in
[Fusion] by filtering out vertices that have the wrong momenta. *)
(* In this case the ordering of the fields in a vertex of the Feynman
rules becomes significant. In particular, we assume that $(V_1,V_2,V_3,V_4)$
implies
\begin{equation}
\parbox{25mm}{\fmfframe(2,3)(2,3){\begin{fmfgraph*}(20,20)
\fmfleft{v1,v2}
\fmfright{v4,v3}
\fmflabel{$V_1$}{v1}
\fmflabel{$V_2$}{v2}
\fmflabel{$V_3$}{v3}
\fmflabel{$V_4$}{v4}
\fmf{plain}{v,v1}
\fmf{plain}{v,v2}
\fmf{plain}{v,v3}
\fmf{plain}{v,v4}
\fmfblob{.2w}{v}
\end{fmfgraph*}}}
\qquad\Longrightarrow\qquad
\parbox{45mm}{\fmfframe(2,3)(2,3){\begin{fmfgraph*}(40,20)
\fmfleft{v1,v2}
\fmfright{v4,v3}
\fmflabel{$V_1$}{v1}
\fmflabel{$V_2$}{v2}
\fmflabel{$V_3$}{v3}
\fmflabel{$V_4$}{v4}
\fmf{plain}{v1,v12,v2}
\fmf{plain}{v3,v34,v4}
\fmf{dots,label=$\Theta((p_1+p_2)^2)$,tension=0.7}{v12,v34}
\fmfdot{v12,v34}
\end{fmfgraph*}}}
\end{equation}
The list of pairs of parameters denotes the location and strengths
of the poles in the $K$-matrix ansatz:
\begin{equation}
(c_1,a_1,c_2,a_2,\ldots,c_n,a_n) \Longrightarrow
f(s) = \sum_{i=1}^{n} \frac{c_i}{s-a_i}
\end{equation} *)
| Vector4_K_Matrix_tho of int * ('a * 'a) list
| Vector4_K_Matrix_jr of int * (int * contract4) list
type 'a vertexn = unit
(* An obvious candidate for addition to [boson] is [T], of course. *)
(* \begin{dubious}
This list is sufficient for the minimal standard model, but not comprehensive
enough for most of its extensions, supersymmetric or otherwise.
In particular, we need a \emph{general} parameterization for all trilinear
vertices. One straightforward possibility are polynomials in the momenta for
each combination of fields.
\end{dubious}
\begin{JR}
Here we use the rules which can be found in~\cite{Denner:Majorana}
and are more properly described in [Targets] where the performing of the fusion
rules in analytical expressions is encoded.
\end{JR}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, S, Psi)]:
$\mathcal{L}_I=g_S\bar\psi_1 S\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_S\bar\psi_1 S$
& $\psi_2\leftarrow\ii\cdot g_S\psi_1 S$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_S S \bar\psi_1$
& $\psi_2\leftarrow\ii\cdot g_SS\psi_1$ \\\hline
[F13] & $S\leftarrow\ii\cdot g_S\bar\psi_1\psi_2$
& $S\leftarrow\ii\cdot g_S\psi_1^T{\mathrm{C}}\psi_2$ \\\hline
[F31] & $S\leftarrow\ii\cdot g_S\psi_{2,\alpha}\bar\psi_{1,\alpha}$
& $S\leftarrow\ii\cdot g_S\psi_2^T{\mathrm{C}} \psi_1$\\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_SS\psi_2$
& $\psi_1\leftarrow\ii\cdot g_SS\psi_2$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_S\psi_2 S$
& $\psi_1\leftarrow\ii\cdot g_S\psi_2 S$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, P, Psi)]:
$\mathcal{L}_I=g_P\bar\psi_1 P\gamma_5\psi_2$} \\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_P\bar\psi_1\gamma_5 P$
& $\psi_2\leftarrow\ii\cdot g_P \gamma_5\psi_1 P$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_P P\bar\psi_1\gamma_5$
& $\psi_2\leftarrow\ii\cdot g_P P\gamma_5\psi_1$ \\\hline
[F13] & $P\leftarrow\ii\cdot g_P\bar\psi_1\gamma_5\psi_2$
& $P\leftarrow\ii\cdot g_P\psi_1^T {\mathrm{C}}\gamma_5\psi_2$ \\\hline
[F31] & $P\leftarrow\ii\cdot g_P[\gamma_5\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $P\leftarrow\ii\cdot g_P\psi_2^T {\mathrm{C}}\gamma_5\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_P P\gamma_5\psi_2$
& $\psi_1\leftarrow\ii\cdot g_P P\gamma_5\psi_2$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_P \gamma_5\psi_2 P$
& $\psi_1\leftarrow\ii\cdot g_P \gamma_5\psi_2 P$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, V, Psi)]:
$\mathcal{L}_I=g_V\bar\psi_1\fmslash{V}\psi_2$} \\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_V\bar\psi_1\fmslash{V}$
& $\psi_{2,\alpha}\leftarrow\ii\cdot
(-g_V)\psi_{1,\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
g_V\fmslash{V}_{\alpha\beta} \bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot (-g_V)\fmslash{V}\psi_1$ \\\hline
[F13] & $V_\mu\leftarrow\ii\cdot g_V\bar\psi_1\gamma_\mu\psi_2$
& $V_\mu\leftarrow\ii\cdot
g_V (\psi_1)^T {\mathrm{C}}\gamma_{\mu}\psi_2$ \\\hline
[F31] & $V_\mu\leftarrow\ii\cdot g_V[\gamma_\mu\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $V_\mu\leftarrow\ii\cdot
(-g_V)(\psi_2)^T {\mathrm{C}}\gamma_{\mu}\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_V\fmslash{V}\psi_2$
& $\psi_1\leftarrow\ii\cdot g_V\fmslash{V}\psi_2$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
g_V\psi_{2,\beta}\fmslash{V}_{\alpha\beta}$
& $\psi_{1,\alpha}\leftarrow\ii\cdot
g_V\psi_{2,\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, A, Psi)]:
$\mathcal{L}_I=g_A\bar\psi_1\gamma_5\fmslash{A}\psi_2$} \\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_A\bar\psi_1\gamma_5\fmslash{A}$
& $\psi_{2,\alpha}\leftarrow\ii\cdot
g_A\psi_{\beta}[\gamma_5\fmslash{A}]_{\alpha\beta}$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot g_A
[\gamma_5\fmslash{A}]_{\alpha\beta} \bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot g_A \gamma_5\fmslash{A}\psi$ \\\hline
[F13] & $A_\mu\leftarrow\ii\cdot g_A\bar\psi_1\gamma_5\gamma_\mu\psi_2$
& $A_\mu\leftarrow\ii\cdot
g_A \psi_1^T {\textrm{C}}\gamma_5\gamma_{\mu}\psi_2$ \\\hline
[F31] & $A_\mu\leftarrow\ii\cdot
g_A[\gamma_5\gamma_\mu\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $A_\mu\leftarrow\ii\cdot
g_A \psi_2^T {\textrm{C}}\gamma_5\gamma_{\mu}\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_A\gamma_5\fmslash{A}\psi_2$
& $\psi_1\leftarrow\ii\cdot g_A\gamma_5\fmslash{A}\psi_2$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot g_A
\psi_{2,\beta}[\gamma_5\fmslash{A}]_{\alpha\beta}$
& $\psi_{1,\alpha}\leftarrow\ii\cdot
g_A\psi_{2,\beta}[\gamma_5\fmslash{A}]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions} Dimension-4 trilinear fermionic couplings.
The momenta are unambiguous, because there are no derivative couplings
and all participating fields are different.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, T, Psi)]:
$\mathcal{L}_I=g_TT_{\mu\nu}\bar\psi_1
[\gamma^\mu,\gamma^\nu]_-\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_T
\bar\psi_1[\gamma^\mu,\gamma^\nu]_-T_{\mu\nu}$
& $\bar\psi_2\leftarrow\ii\cdot g_T \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_T T_{\mu\nu}
\bar\psi_1[\gamma^\mu,\gamma^\nu]_-$
& $\bar\psi_2\leftarrow\ii\cdot g_T \cdots$ \\\hline
[F13] & $T_{\mu\nu}\leftarrow\ii\cdot g_T\bar\psi_1[\gamma_\mu,\gamma_\nu]_-\psi_2$
& $T_{\mu\nu}\leftarrow\ii\cdot g_T \cdots $ \\\hline
[F31] & $T_{\mu\nu}\leftarrow\ii\cdot g_T
[[\gamma_\mu,\gamma_\nu]_-\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $T_{\mu\nu}\leftarrow\ii\cdot g_T \cdots $ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_T T_{\mu\nu}[\gamma^\mu,\gamma^\nu]_-\psi_2$
& $\psi_1\leftarrow\ii\cdot g_T \cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_T [\gamma^\mu,\gamma^\nu]_-\psi_2 T_{\mu\nu}$
& $\psi_1\leftarrow\ii\cdot g_T \cdots$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions} Dimension-5 trilinear fermionic couplings
(NB: the coefficients and signs are not fixed yet).
The momenta are unambiguous, because there are no derivative couplings
and all participating fields are different.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SP, Psi)]:
$\mathcal{L}_I=\bar\psi_1\phi(g_S+g_P\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot\bar\psi_1(g_S+g_P\gamma_5)\phi$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot\phi\bar\psi_1(g_S+g_P\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $\phi\leftarrow\ii\cdot\bar\psi_1(g_S+g_P\gamma_5)\psi_2$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F31] & $\phi\leftarrow\ii\cdot[(g_S+g_P\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot \phi(g_S+g_P\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot(g_S+g_P\gamma_5)\psi_2\phi$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SL, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\phi(1-\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_L\bar\psi_1(1-\gamma_5)\phi$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_L\phi\bar\psi_1(1-\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $\phi\leftarrow\ii\cdot g_L\bar\psi_1(1-\gamma_5)\psi_2$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F31] & $\phi\leftarrow\ii\cdot g_L[(1-\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_L\phi(1-\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_L(1-\gamma_5)\psi_2\phi$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SR, Psi)]:
$\mathcal{L}_I=g_R\bar\psi_1\phi(1+\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_R\bar\psi_1(1+\gamma_5)\phi$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_R\phi\bar\psi_1(1+\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $\phi\leftarrow\ii\cdot g_R\bar\psi_1(1+\gamma_5)\psi_2$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F31] & $\phi\leftarrow\ii\cdot g_R[(1+\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_R\phi(1+\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_R(1+\gamma_5)\psi_2\phi$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SLR, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\phi(1-\gamma_5)\psi_2
+g_R\bar\psi_1\phi(1+\gamma_5)\psi_2$}\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SP} Combined dimension-4 trilinear fermionic couplings.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VA, Psi)]:
$\mathcal{L}_I=\bar\psi_1\fmslash{Z}(g_V-g_A\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot\bar\psi_1\fmslash{Z}(g_V-g_A\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}\bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $Z_\mu\leftarrow\ii\cdot\bar\psi_1\gamma_\mu(g_V-g_A\gamma_5)\psi_2$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F31] & $Z_\mu\leftarrow\ii\cdot
[\gamma_\mu(g_V-g_A\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot\fmslash{Z}(g_V-g_A\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
\psi_{2,\beta}[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VL, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\fmslash{Z}(1-\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_L\bar\psi_1\fmslash{Z}(1-\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
g_L[\fmslash{Z}(1-\gamma_5)]_{\alpha\beta}\bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $Z_\mu\leftarrow\ii\cdot g_L\bar\psi_1\gamma_\mu(1-\gamma_5)\psi_2$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F31] & $Z_\mu\leftarrow\ii\cdot
g_L[\gamma_\mu(1-\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_L\fmslash{Z}(1-\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
g_L\psi_{2,\beta}[\fmslash{Z}(1-\gamma_5)]_{\alpha\beta}$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VR, Psi)]:
$\mathcal{L}_I=g_R\bar\psi_1\fmslash{Z}(1+\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_R\bar\psi_1\fmslash{Z}(1+\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
g_R[\fmslash{Z}(1+\gamma_5)]_{\alpha\beta}\bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $Z_\mu\leftarrow\ii\cdot g_R\bar\psi_1\gamma_\mu(1+\gamma_5)\psi_2$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F31] & $Z_\mu\leftarrow\ii\cdot
g_R[\gamma_\mu(1+\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_R\fmslash{Z}(1+\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
g_R\psi_{2,\beta}[\fmslash{Z}(1+\gamma_5)]_{\alpha\beta}$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VLR, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\fmslash{Z}(1-\gamma_5)\psi_2
+g_R\bar\psi_1\fmslash{Z}(1+\gamma_5)\psi_2$}\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-VA} Combined dimension-4 trilinear
fermionic couplings continued.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Psibar, S, Chi)]: $\bar\psi S\chi$}\\\hline
[F12] & $\chi\leftarrow\psi S$
& [F21] & $\chi\leftarrow S \psi$ \\\hline
[F13] & $S\leftarrow \psi^T{\rm C}\chi$
& [F31] & $S\leftarrow \chi^T {\rm C}\psi$ \\\hline
[F23] & $\psi\leftarrow S\chi$
& [F32] & $\psi\leftarrow\chi S$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, P, Chi)]: $\bar\psi P\gamma_5\chi$}\\\hline
[F12] & $\chi\leftarrow \gamma_5 \psi P$
& [F21] & $\chi\leftarrow P \gamma_5 \psi$ \\\hline
[F13] & $P\leftarrow \psi^T {\rm C}\gamma_5\chi$
& [F31] & $P\leftarrow \chi^T {\rm C}\gamma_5\psi$ \\\hline
[F23] & $\psi\leftarrow P\gamma_5\chi$
& [F32] & $\psi\leftarrow\gamma_5\chi P$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, V, Chi)]: $\bar\psi\fmslash{V}\chi$}\\\hline
[F12] & $\chi_{\alpha}\leftarrow-\psi_{\beta}\fmslash{V}_{\alpha\beta}$
& [F21] & $\chi\leftarrow-\fmslash{V}\psi$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T {\rm C}\gamma_{\mu}\chi$
& [F31] & $V_{\mu}\leftarrow \chi^T {\rm C}(-\gamma_{\mu}\psi)$ \\\hline
[F23] & $\psi\leftarrow\fmslash{V}\chi$
& [F32] & $\psi_\alpha\leftarrow\chi_\beta\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, A, Chi)]: $\bar\psi\gamma^5\fmslash{A}\chi$}\\\hline
[F12] & $\chi_{\alpha}\leftarrow\psi_{\beta}\lbrack \gamma^5 \fmslash{A} \rbrack_{\alpha\beta}$
& [F21] & $\chi\leftarrow\gamma^5\fmslash{A}\psi$ \\\hline
[F13] & $A_{\mu}\leftarrow \psi^T {\rm C}\gamma^5\gamma_{\mu}\chi$
& [F31] & $A_{\mu}\leftarrow \chi^T {\rm C}(\gamma^5 \gamma_{\mu}\psi)$ \\\hline
[F23] & $\psi\leftarrow\gamma^5\fmslash{A}\chi$
& [F32] & $\psi_\alpha\leftarrow\chi_\beta\lbrack \gamma^5 \fmslash{A} \rbrack_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-maj} Dimension-4 trilinear couplings
including one Dirac and one Majorana fermion}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Psibar, SP, Chi)]:
$\bar\psi\phi(g_S+g_P\gamma_5)\chi$}\\\hline
[F12] & $\chi \leftarrow (g_S+g_P\gamma_5)\psi \phi$
& [F21] & $\chi\leftarrow\phi(g_S+g_P\gamma_5)\psi$ \\\hline
[F13] & $\phi\leftarrow \psi^T {\rm C}(g_S+g_P\gamma_5)\chi$
& [F31] & $\phi\leftarrow \chi^T {\rm C}(g_S+g_P\gamma_5) \chi$ \\\hline
[F23] & $\psi\leftarrow \phi(g_S+g_P\gamma_5)\chi$
& [F32] & $\psi\leftarrow(g_S+g_P\gamma_5)\chi\phi$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, VA, Chi)]:
$\bar\psi\fmslash{Z}(g_V - g_A\gamma_5)\chi$}\\\hline
[F12] & $\chi_\alpha\leftarrow
\psi_\beta[\fmslash{Z}(-g_V-g_A\gamma_5)]_{\alpha\beta}$
& [F21] & $\chi\leftarrow\fmslash{Z}(-g_V-g_A\gamma_5)]
\psi$ \\\hline
[F13] & $Z_\mu\leftarrow \psi^T {\rm C}\gamma_\mu(g_V-g_A\gamma_5)\chi$
& [F31] & $Z_\mu\leftarrow \chi^T {\rm C}\gamma_\mu(-g_V-g_A\gamma_5)\psi$ \\\hline
[F23] & $\psi\leftarrow\fmslash{Z}(g_V-g_A\gamma_5)\chi$
& [F32] & $\psi_\alpha\leftarrow
\chi_\beta[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SPVA-maj} Combined dimension-4 trilinear
fermionic couplings including one Dirac and one Majorana fermion.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, S, Psi)]: $\bar\chi S\psi$}\\\hline
[F12] & $\psi\leftarrow\chi S$
& [F21] & $\psi\leftarrow S\chi$ \\\hline
[F13] & $S\leftarrow \chi^T {\rm C}\psi$
& [F31] & $S\leftarrow \psi^T {\rm C}\chi$ \\\hline
[F23] & $\chi\leftarrow S \psi$
& [F32] & $\chi\leftarrow\psi S$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, P, Psi)]: $\bar\chi P\gamma_5\psi$}\\\hline
[F12] & $\psi\leftarrow\gamma_5\chi P$
& [F21] & $\psi\leftarrow P\gamma_5\chi$ \\\hline
[F13] & $P\leftarrow \chi^T {\rm C}\gamma_5\psi$
& [F31] & $P\leftarrow \psi^T {\rm C}\gamma_5\chi$ \\\hline
[F23] & $\chi\leftarrow P \gamma_5 \psi$
& [F32] & $\chi\leftarrow \gamma_5 \psi P$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, V, Psi)]: $\bar\chi\fmslash{V}\psi$}\\\hline
[F12] & $\psi_\alpha\leftarrow-\chi_\beta\fmslash{V}_{\alpha\beta}$
& [F21] & $\psi\leftarrow-\fmslash{V}\chi$ \\\hline
[F13] & $V_{\mu}\leftarrow \chi^T {\rm C}\gamma_{\mu}\psi$
& [F31] & $V_{\mu}\leftarrow \psi^T {\rm C}(-\gamma_{\mu}\chi)$ \\\hline
[F23] & $\chi\leftarrow\fmslash{V}\psi$
& [F32] & $\chi_{\alpha}\leftarrow\psi_{\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, A, Psi)]: $\bar\chi\gamma^5\fmslash{A}\psi$}\\\hline
[F12] & $\psi_\alpha\leftarrow\chi_\beta\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$
& [F21] & $\psi\leftarrow\gamma^5\fmslash{A}\chi$ \\\hline
[F13] & $A_{\mu}\leftarrow \chi^T {\rm C}(\gamma^5\gamma_{\mu}\psi)$
& [F31] & $A_{\mu}\leftarrow \psi^T {\rm C}\gamma^5\gamma_{\mu}\chi$ \\\hline
[F23] & $\chi\leftarrow\gamma^5\fmslash{A}\psi$
& [F32] & $\chi_{\alpha}\leftarrow\psi_{\beta}\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-maj'} Dimension-4 trilinear couplings
including one Dirac and one Majorana fermion}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, SP, Psi)]: $\bar\chi\phi(g_S+g_P\gamma_5)\psi$}\\\hline
[F12] & $\psi\leftarrow(g_S+g_P\gamma_5)\chi\phi$
& [F21] & $\psi\leftarrow \phi(g_S+g_P\gamma_5)\chi$ \\\hline
[F13] & $\phi\leftarrow \chi^T {\rm C}(g_S+g_P\gamma_5) \psi$
& [F31] & $\phi\leftarrow \psi^T {\rm C}(g_S+g_P\gamma_5)\chi$ \\\hline
[F23] & $\chi\leftarrow\phi(g_S+g_P\gamma_5)\psi$
& [F32] & $\chi \leftarrow (g_S+g_P\gamma_5)\psi \phi$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, VA, Psi)]:
$\bar\chi\fmslash{Z}(g_V - g_A\gamma_5)\psi$}\\\hline
[F12] & $\psi_\alpha\leftarrow
\chi_\beta[\fmslash{Z}(-g_V-g_A\gamma_5)]_{\alpha\beta}$
& [F21] & $\psi\leftarrow\fmslash{Z}(-g_V-g_A\gamma_5)\chi$ \\\hline
[F13] & $Z_\mu\leftarrow \chi^T {\rm C}\gamma_\mu(g_V-g_A\gamma_5)\psi$
& [F31] & $Z_\mu\leftarrow \psi^T {\rm C}\gamma_\mu(-g_V-g_A\gamma_5)\chi$ \\\hline
[F23] & $\chi\leftarrow\fmslash{Z}(g_V-g_A\gamma_5)]
\psi$
& [F32] & $\chi_\alpha\leftarrow\psi_\beta[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SPVA-maj'} Combined dimension-4 trilinear
fermionic couplings including one Dirac and one Majorana fermion.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, S, Chi)]: $\bar\chi_a S\chi_b$}\\\hline
[F12] & $\chi_b\leftarrow\chi_a S$
& [F21] & $\chi_b\leftarrow S \chi_a$ \\\hline
[F13] & $S\leftarrow \chi^T_a {\rm C}\chi_b$
& [F31] & $S\leftarrow \chi^T_b {\rm C}\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow S\chi_b$
& [F32] & $\chi_a\leftarrow\chi S_b$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, P, Chi)]: $\bar\chi_a P\gamma_5\psi_b$}\\\hline
[F12] & $\chi_b\leftarrow \gamma_5 \chi_a P$
& [F21] & $\chi_b\leftarrow P \gamma_5 \chi_a$ \\\hline
[F13] & $P\leftarrow \chi^T_a {\rm C}\gamma_5\chi_b$
& [F31] & $P\leftarrow \chi^T_b {\rm C}\gamma_5\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow P\gamma_5\chi_b$
& [F32] & $\chi_a\leftarrow\gamma_5\chi_b P$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, V, Chi)]: $\bar\chi_a\fmslash{V}\chi_b$}\\\hline
[F12] & $\chi_{b,\alpha}\leftarrow-\chi_{a,\beta}\fmslash{V}_{\alpha\beta}$
& [F21] & $\chi_b\leftarrow-\fmslash{V}\chi_a$ \\\hline
[F13] & $V_{\mu}\leftarrow \chi^T_a {\rm C}\gamma_{\mu}\chi_b$
& [F31] & $V_{\mu}\leftarrow - \chi^T_b {\rm C}\gamma_{\mu}\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow\fmslash{V}\chi_b$
& [F32] & $\chi_{a,\alpha}\leftarrow\chi_{b,\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, A, Chi)]: $\bar\chi_a\gamma^5\fmslash{A}\chi_b$}\\\hline
[F12] & $\chi_{b,\alpha}\leftarrow\chi_{a,\beta}\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$
& [F21] & $\chi_b\leftarrow\gamma^5\fmslash{A}\chi_a$ \\\hline
[F13] & $A_{\mu}\leftarrow \chi^T_a {\rm C}\gamma^5\gamma_{\mu}\chi_b$
& [F31] & $A_{\mu}\leftarrow \chi^T_b {\rm C}(\gamma^5\gamma_{\mu}\chi_a)$ \\\hline
[F23] & $\chi_a\leftarrow\gamma^5\fmslash{A}\chi_b$
& [F32] & $\chi_{a,\alpha}\leftarrow\chi_{b,\beta}\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-maj2} Dimension-4 trilinear couplings
of two Majorana fermions}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, SP, Chi)]:
$\bar\chi\phi_a(g_S+g_P\gamma_5)\chi_b$}\\\hline
[F12] & $\chi_b \leftarrow (g_S+g_P\gamma_5)\chi_a \phi$
& [F21] & $\chi_b\leftarrow\phi(g_S+g_P\gamma_5)\chi_a$ \\\hline
[F13] & $\phi\leftarrow \chi^T_a {\rm C}(g_S+g_P\gamma_5)\chi_b$
& [F31] & $\phi\leftarrow \chi^T_b {\rm C}(g_S+g_P\gamma_5) \chi_a$ \\\hline
[F23] & $\chi_a\leftarrow \phi(g_S+g_P\gamma_5)\chi_b$
& [F32] & $\chi_a\leftarrow(g_S+g_P\gamma_5)\chi_b\phi$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, VA, Chi)]:
$\bar\chi_a\fmslash{Z}(g_V-g_A\gamma_5)\chi_b$}\\\hline
[F12] & $\chi_{b,\alpha}\leftarrow\chi_{a,\beta}[\fmslash{Z}(-g_V-g_A\gamma_5)]_{\alpha\beta}$
& [F21] & $\chi_b\leftarrow\fmslash{Z}(-g_V-g_A\gamma_5)]\chi_a$ \\\hline
[F13] & $Z_\mu\leftarrow \chi^T_a {\rm C}\gamma_\mu(g_V-g_A\gamma_5)\chi_b$
& [F31] & $Z_\mu\leftarrow \chi^T_b {\rm C}\gamma_\mu(-g_V-g_A\gamma_5)\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow\fmslash{Z}(g_V-g_A\gamma_5)\chi_b$
& [F32] & $\chi_{a,\alpha}\leftarrow
\chi_{b,\beta}[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SPVA-maj2} Combined dimension-4 trilinear
fermionic couplings of two Majorana fermions.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Gauge_Gauge_Gauge]:
$\mathcal{L}_I=gf_{abc}
A_a^\mu A_b^\nu\partial_\mu A_{c,\nu}$}\\\hline
[_] & $A_a^\mu\leftarrow\ii\cdot
(-\ii g/2)\cdot C_{abc}^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A^b_\rho A^c_\sigma$\\\hline
\multicolumn{2}{|l|}{[Aux_Gauge_Gauge]:
$\mathcal{L}_I=gf_{abc}X_{a,\mu\nu}(k_1)
( A_b^{\mu}(k_2)A_c^{\nu}(k_3)
-A_b^{\nu}(k_2)A_c^{\mu}(k_3))$}\\\hline
[F23]$\lor$[F32] & $X_a^{\mu\nu}(k_2+k_3)\leftarrow\ii\cdot
gf_{abc}( A_b^\mu(k_2)A_c^\nu(k_3)
-A_b^\nu(k_2)A_c^\mu(k_3))$ \\\hline
[F12]$\lor$[F13] & $A_{a,\mu}(k_1+k_{2/3})\leftarrow\ii\cdot
gf_{abc}X_{b,\nu\mu}(k_1)A_c^\nu(k_{2/3})$ \\\hline
[F21]$\lor$[F31] & $A_{a,\mu}(k_{2/3}+k_1)\leftarrow\ii\cdot
gf_{abc}A_b^\nu(k_{2/3}) X_{c,\mu\nu}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-bosons} Dimension-4 Vector Boson couplings with
\emph{outgoing} momenta.
See~(\ref{eq:C123}) and~(\ref{eq:C123'}) for the definition of the
antisymmetric tensor $C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)$.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[Scalar_Vector_Vector]:
$\mathcal{L}_I=g\phi V_1^\mu V_{2,\mu}$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $\phi\leftarrow\ii\cdot g V_1^\mu V_{2,\mu}$
& [F32] & $\phi\leftarrow\ii\cdot g V_{2,\mu} V_1^\mu$ \\\hline
\multicolumn{4}{|l|}{[Aux_Vector_Vector]:
$\mathcal{L}_I=gX V_1^\mu V_{2,\mu}$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $X\leftarrow\ii\cdot g V_1^\mu V_{2,\mu}$
& [F32] & $X\leftarrow\ii\cdot g V_{2,\mu} V_1^\mu$ \\\hline
\multicolumn{4}{|l|}{[Aux_Scalar_Vector]:
$\mathcal{L}_I=gX^\mu \phi V_\mu$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $\leftarrow\ii\cdot g\cdots$
& [F32] & $\leftarrow\ii\cdot g\cdots$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:scalar-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[Scalar_Scalar_Scalar]:
$\mathcal{L}_I=g\phi_1\phi_2\phi_3$}\\\hline
[F13] & $\phi_2\leftarrow\ii\cdot g \phi_1\phi_3$
& [F31] & $\phi_2\leftarrow\ii\cdot g \phi_3\phi_1$ \\\hline
[F12] & $\phi_3\leftarrow\ii\cdot g \phi_1\phi_2$
& [F21] & $\phi_3\leftarrow\ii\cdot g \phi_2\phi_1$ \\\hline
[F23] & $\phi_1\leftarrow\ii\cdot g \phi_2\phi_3$
& [F32] & $\phi_1\leftarrow\ii\cdot g \phi_3\phi_2$ \\\hline
\multicolumn{4}{|l|}{[Aux_Scalar_Scalar]:
$\mathcal{L}_I=gX\phi_1\phi_2$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $X\leftarrow\ii\cdot g \phi_1\phi_2$
& [F32] & $X\leftarrow\ii\cdot g \phi_2\phi_1$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:scalars}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Vector_Scalar_Scalar]:
$\mathcal{L}_I=gV^\mu\phi_1
\ii\overleftrightarrow{\partial_\mu}\phi_2$}\\\hline
[F23] & $V^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)\phi_1(k_2)\phi_2(k_3)$ \\\hline
[F32] & $V^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)\phi_2(k_3)\phi_1(k_2)$ \\\hline
[F12] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g(k_1^\mu+2k_2^\mu)V_\mu(k_1)\phi_1(k_2)$ \\\hline
[F21] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g(k_1^\mu+2k_2^\mu)\phi_1(k_2)V_\mu(k_1)$ \\\hline
[F13] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\mu-2k_3^\mu)V_\mu(k_1)\phi_2(k_3)$ \\\hline
[F31] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\mu-2k_3^\mu)\phi_2(k_3)V_\mu(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:scalar-current}
\ldots}
\end{table} *)
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Aux_DScalar_DScalar]:
$\mathcal{L}_I=g\chi
(\ii\partial_\mu\phi_1)(\ii\partial^\mu\phi_2)$}\\\hline
[F23] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g (k_2\cdot k_3) \phi_1(k_2) \phi_2(k_3) $ \\\hline
[F32] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g (k_3\cdot k_2) \phi_2(k_3) \phi_1(k_2) $ \\\hline
[F12] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g ((-k_1-k_2) \cdot k_2) \chi(k_1) \phi_1(k_2) $ \\\hline
[F21] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g (k_2 \cdot (-k_1-k_2)) \phi_1(k_2) \chi(k_1) $ \\\hline
[F13] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g ((-k_1-k_3) \cdot k_3) \chi(k_1) \phi_2(k_3) $ \\\hline
[F31] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g (k_3 \cdot (-k_1-k_3)) \phi_2(k_3) \chi(k_1) $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dscalar-dscalar}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Aux_Vector_DScalar]:
$\mathcal{L}_I=g\chi V_\mu (\ii\partial^\mu\phi)$}\\\hline
[F23] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g k_3^\mu V_\mu(k_2) \phi(k_3) $ \\\hline
[F32] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g \phi(k_3) k_3^\mu V_\mu(k_2) $ \\\hline
[F12] & $\phi(k_1+k_2)\leftarrow\ii\cdot
g \chi(k_1) (-k_1-k_2)^\mu V_\mu(k_2) $ \\\hline
[F21] & $\phi(k_1+k_2)\leftarrow\ii\cdot
g (-k_1-k_2)^\mu V_\mu(k_2) \chi(k_1) $ \\\hline
[F13] & $V_\mu(k_1+k_3)\leftarrow\ii\cdot
g (-k_1-k_3)_\mu \chi(k_1) \phi(k_3) $ \\\hline
[F31] & $V_\mu(k_1+k_3)\leftarrow\ii\cdot
g (-k_1-k_3)_\mu \phi(k_3) \chi(k_1) $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:vector-dscalar}
\ldots}
\end{table}
*)
(* Signify which two of three fields are fused: *)
type fuse2 = F23 | F32 | F31 | F13 | F12 | F21
(* Signify which three of four fields are fused: *)
type fuse3 =
| F123 | F231 | F312 | F132 | F321 | F213
| F124 | F241 | F412 | F142 | F421 | F214
| F134 | F341 | F413 | F143 | F431 | F314
| F234 | F342 | F423 | F243 | F432 | F324
(* Explicit enumeration types make no sense for higher degrees. *)
type fusen = int list
(* The third member of the triplet will contain the coupling constant: *)
type 'a t =
| V3 of 'a vertex3 * fuse2 * 'a
| V4 of 'a vertex4 * fuse3 * 'a
| Vn of 'a vertexn * fusen * 'a
(* \thocwmodulesection{Gauge Couplings}
Dimension-4 trilinear vector boson couplings
\begin{subequations}
\begin{multline}
f_{abc}\partial^{\mu}A^{a,\nu}A^b_{\mu}A^c_{\nu} \rightarrow
\ii f_{abc}k_1^\mu A^{a,\nu}(k_1)A^b_{\mu}(k_2)A^c_{\nu}(k_3) \\
= -\frac{\ii}{3!} f_{a_1a_2a_3} C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)
A^{a_1}_{\mu_1}(k_1)A^{a_2}_{\mu_2}(k_2)A^{a_3}_{\mu_3}(k_3)
\end{multline}
with the totally antisymmetric tensor (under simultaneous permutations
of all quantum numbers $\mu_i$ and $k_i$) and all momenta \emph{outgoing}
\begin{equation}
\label{eq:C123}
C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3) =
( g^{\mu_1\mu_2} (k_1^{\mu_3}-k_2^{\mu_3})
+ g^{\mu_2\mu_3} (k_2^{\mu_1}-k_3^{\mu_1})
+ g^{\mu_3\mu_1} (k_3^{\mu_2}-k_1^{\mu_2}) )
\end{equation}
\end{subequations}
Since~$f_{a_1a_2a_3}C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)$ is totally symmetric
(under simultaneous permutations of all quantum numbers $a_i$, $\mu_i$ and $k_i$),
it is easy to take the partial derivative
\begin{subequations}
\label{eq:AofAA}
\begin{equation}
A^{a,\mu}(k_2+k_3) =
- \frac{\ii}{2!} f_{abc}C^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3) A^b_\rho(k_2)A^c_\sigma(k_3)
\end{equation}
with
\begin{equation}
\label{eq:C123'}
C^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3) =
( g^{\rho\sigma} ( k_2^{\mu} -k_3^{\mu} )
+ g^{\mu\sigma} (2k_3^{\rho} +k_2^{\rho} )
- g^{\mu\rho} (2k_2^{\sigma}+k_3^{\sigma}) )
\end{equation}
i.\,e.
\begin{multline}
\label{eq:fuse-gauge}
A^{a,\mu}(k_2+k_3) = - \frac{\ii}{2!} f_{abc}
\bigl( (k_2^{\mu}-k_3^{\mu})A^b(k_2) \cdot A^c(k_3) \\
+ (2k_3+k_2)\cdot A^b(k_2)A^{c,\mu}(k_3)
- A^{b,\mu}(k_2)A^c(k_3)\cdot(2k_2+k_3) \bigr)
\end{multline}
\end{subequations}
\begin{dubious}
Investigate the rearrangements proposed in~\cite{HELAS} for improved
numerical stability.
\end{dubious} *)
(* \thocwmodulesubsection{Non-Gauge Vector Couplings}
As a basis for the dimension-4 couplings of three vector bosons, we
choose ``transversal'' and ``longitudinal'' (with respect to the first
vector field) tensors that are odd and even under permutation of the
second and third argument
\begin{subequations}
\begin{align}
\mathcal{L}_T(V_1,V_2,V_3)
&= V_1^\mu (V_{2,\nu}\ii\overleftrightarrow{\partial_\mu}V_3^\nu)
= - \mathcal{L}_T(V_1,V_3,V_2) \\
\mathcal{L}_L(V_1,V_2,V_3)
&= (\ii\partial_\mu V_1^\mu) V_{2,\nu}V_3^\nu
= \mathcal{L}_L(V_1,V_3,V_2)
\end{align}
\end{subequations}
Using partial integration in~$\mathcal{L}_L$, we find the
convenient combinations
\begin{subequations}
\begin{align}
\mathcal{L}_T(V_1,V_2,V_3) + \mathcal{L}_L(V_1,V_2,V_3)
&= - 2 V_1^\mu \ii\partial_\mu V_{2,\nu} V_3^\nu \\
\mathcal{L}_T(V_1,V_2,V_3) - \mathcal{L}_L(V_1,V_2,V_3)
&= 2 V_1^\mu V_{2,\nu} \ii\partial_\mu V_3^\nu
\end{align}
\end{subequations}
As an important example, we can rewrite the dimension-4 ``anomalous'' triple
gauge couplings
\begin{multline}
\ii\mathcal{L}_{\textrm{TGC}}(g_1,\kappa,g_4)/g_{VWW}
= g_1 V^\mu (W^-_{\mu\nu} W^{+,\nu} - W^+_{\mu\nu} W^{-,\nu}) \\
+ \kappa W^+_\mu W^-_\nu V^{\mu\nu}
+ g_4 W^+_\mu W^-_\nu (\partial^\mu V^\nu + \partial^\nu V^\mu)
\end{multline}
as
\begin{multline}
\mathcal{L}_{\textrm{TGC}}(g_1,\kappa,g_4)
= g_1 \mathcal{L}_T(V,W^-,W^+) \\
- \frac{\kappa+g_1-g_4}{2} \mathcal{L}_T(W^-,V,W^+)
+ \frac{\kappa+g_1+g_4}{2} \mathcal{L}_T(W^+,V,W^-) \\
- \frac{\kappa-g_1-g_4}{2} \mathcal{L}_L(W^-,V,W^+)
+ \frac{\kappa-g_1+g_4}{2} \mathcal{L}_L(W^+,V,W^-)
\end{multline}
\thocwmodulesubsection{$CP$ Violation}
\begin{subequations}
\begin{align}
\mathcal{L}_{\tilde T}(V_1,V_2,V_3)
&= V_{1,\mu}(V_{2,\rho}\ii\overleftrightarrow{\partial_\nu}
V_{3,\sigma})\epsilon^{\mu\nu\rho\sigma}
= + \mathcal{L}_T(V_1,V_3,V_2) \\
\mathcal{L}_{\tilde L}(V_1,V_2,V_3)
&= (\ii\partial_\mu V_{1,\nu})
V_{2,\rho}V_{3,\sigma}\epsilon^{\mu\nu\rho\sigma}
= - \mathcal{L}_L(V_1,V_3,V_2)
\end{align}
\end{subequations}
Here the notations~$\tilde T$ and~$\tilde L$ are clearly
\textit{abuse de langage}, because
$\mathcal{L}_{\tilde L}(V_1,V_2,V_3)$ is actually the
transversal combination, due to the antisymmetry of~$\epsilon$.
Using partial integration in~$\mathcal{L}_{\tilde L}$, we could again find
combinations
\begin{subequations}
\begin{align}
\mathcal{L}_{\tilde T}(V_1,V_2,V_3) + \mathcal{L}_{\tilde L}(V_1,V_2,V_3)
&= - 2 V_{1,\mu} V_{2,\nu} \ii\partial_\rho V_{3,\sigma}
\epsilon^{\mu\nu\rho\sigma} \\
\mathcal{L}_{\tilde T}(V_1,V_2,V_3) - \mathcal{L}_{\tilde L}(V_1,V_2,V_3)
&= - 2 V_{1,\mu} \ii\partial_\nu V_{2,\rho} V_{3,\sigma}
\epsilon^{\mu\nu\rho\sigma}
\end{align}
\end{subequations}
but we don't need them, since
\begin{multline}
\ii\mathcal{L}_{\textrm{TGC}}(g_5,\tilde\kappa)/g_{VWW}
= g_5 \epsilon_{\mu\nu\rho\sigma}
(W^{+,\mu} \ii\overleftrightarrow{\partial^\rho} W^{-,\nu}) V^\sigma \\
- \frac{\tilde\kappa_V}{2} W^-_\mu W^+_\nu \epsilon^{\mu\nu\rho\sigma}
V_{\rho\sigma}
\end{multline}
is immediately recognizable as
\begin{equation}
\mathcal{L}_{\textrm{TGC}}(g_5,\tilde\kappa) / g_{VWW}
= - \ii g_5 \mathcal{L}_{\tilde L}(V,W^-,W^+)
+ \tilde\kappa \mathcal{L}_{\tilde T}(V,W^-,W^+)
\end{equation}
%%% #procedure decl
%%% symbol g1, kappa;
%%% vector V, Wp, Wm, k0, kp, km;
%%% vector v, V1, V2, V3, k1, k2, k3;
%%% index mu, nu;
%%% #endprocedure
%%%
%%% #call decl
%%%
%%% global L_T(k1,V1,k2,V2,k3,V3)
%%% = (V1.k2 - V1.k3) * V2.V3;
%%%
%%% global L_L(k1,V1,k2,V2,k3,V3)
%%% = - V1.k1 * V2.V3;
%%%
%%% global L_g1(k1,V1,k2,V2,k3,V3)
%%% = - V1(mu) * ( (k2(mu)*V2(nu) - k2(nu)*V2(mu)) * V3(nu)
%%% - (k3(mu)*V3(nu) - k3(nu)*V3(mu)) * V2(nu) );
%%%
%%% global L_kappa(k1,V1,k2,V2,k3,V3)
%%% = (k1(mu)*V1(nu) - k1(nu)*V1(mu)) * V2(mu) * V3(nu);
%%%
%%% print;
%%% .sort
%%% .store
%%%
%%% #call decl
%%%
%%% local lp = L_T(k1,V1,k2,V2,k3,V3) + L_L(k1,V1,k2,V2,k3,V3);
%%% local lm = L_T(k1,V1,k2,V2,k3,V3) - L_L(k1,V1,k2,V2,k3,V3);
%%% print;
%%% .sort
%%% id k1.v? = - k2.v - k3.v;
%%% print;
%%% .sort
%%% .store
%%%
%%% #call decl
%%%
%%% local [sum(TL)-g1] = - L_g1(k0,V,km,Wm,kp,Wp)
%%% + L_T(k0,V,kp,Wp,km,Wm)
%%% + (L_T(km,Wm,k0,V,kp,Wp) - L_T(kp,Wp,k0,V,km,Wm)) / 2
%%% - (L_L(km,Wm,k0,V,kp,Wp) - L_L(kp,Wp,k0,V,km,Wm)) / 2;
%%%
%%% local [sum(TL)-kappa] = - L_kappa(k0,V,km,Wm,kp,Wp)
%%% + (L_T(km,Wm,k0,V,kp,Wp) - L_T(kp,Wp,k0,V,km,Wm)) / 2
%%% + (L_L(km,Wm,k0,V,kp,Wp) - L_L(kp,Wp,k0,V,km,Wm)) / 2;
%%%
%%% local delta =
%%% - (g1 * L_g1(k0,V,km,Wm,kp,Wp) + kappa * L_kappa(k0,V,km,Wm,kp,Wp))
%%% + g1 * L_T(k0,V,kp,Wp,km,Wm)
%%% + ( g1 + kappa) / 2 * (L_T(km,Wm,k0,V,kp,Wp) - L_T(kp,Wp,k0,V,km,Wm))
%%% + (- g1 + kappa) / 2 * (L_L(km,Wm,k0,V,kp,Wp) - L_L(kp,Wp,k0,V,km,Wm));
%%%
%%% print;
%%% .sort
%%%
%%% id k0.v? = - kp.v - km.v;
%%% print;
%%% .sort
%%% .store
%%%
%%% .end *)
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_T]:
$\mathcal{L}_I=gV_1^\mu
V_{2,\nu}\ii\overleftrightarrow{\partial_\mu}V_3^\nu$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)V_{2,\nu}(k_2)V_3^\nu(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)V_3^\nu(k_3)V_{2,\nu}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(2k_2^\nu+k_1^\nu)V_{1,\nu}(k_1)V_2^\mu(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(2k_2^\nu+k_1^\nu)V_2^\mu(k_2)V_{1,\nu}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu-2k_3^\nu)V_1^\nu(k_1)V_3^\mu(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu-2k_3^\nu)V_3^\mu(k_3)V_1^\nu(k_1)$ \\\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_L]:
$\mathcal{L}_I=g\ii\partial_\mu V_1^\mu
V_{2,\nu}V_3^\nu$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu+k_3^\mu)V_{2,\nu}(k_2)V_3^\nu(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu+k_3^\mu)V_3^\nu(k_3)V_{2,\nu}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(-k_1^\nu)V_{1,\nu}(k_1)V_2^\mu(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(-k_1^\nu)V_2^\mu(k_2)V_{1,\nu}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu)V_1^\nu(k_1)V_3^\mu(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu)V_3^\mu(k_3)V_1^\nu(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-TGC}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_T5]:
$\mathcal{L}_I=gV_{1,\mu}
V_{2,\rho}\ii\overleftrightarrow{\partial_\nu}
V_{3,\sigma}\epsilon^{\mu\nu\rho\sigma}$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}-k_{3,\nu})
V_{2,\rho}(k_2)V_{3,\sigma}(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}-k_{3,\nu})
V_{3,\sigma}(k_3)V_{2,\rho}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(2k_{2,\nu}+k_{1,\nu})
V_{1,\rho}(k_1)V_{2,\sigma}(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(2k_{2,\nu}+k_{1,\nu})
V_{2,\sigma}(k_2)V_{1,\rho}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu}-2k_{3,\nu})
V_{1,\rho}(k_1)V_{3,\sigma}(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu}-2k_{3,\nu})
V_{3,\sigma}(k_3)V_{1,\rho}(k_1)$ \\\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_L5]:
$\mathcal{L}_I=g\ii\partial_\mu V_{1,\nu}
V_{2,\nu}V_{3,\sigma}\epsilon^{\mu\nu\rho\sigma}$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}+k_{3,\nu})
V_{2,\rho}(k_2)V_{3,\sigma}(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}+k_{3,\nu})
V_{2,\rho}(k_2)V_{3,\sigma}(k_3)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{1,\rho}(k_1)V_{2,\sigma}(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{2,\sigma}(k_2)V_{1,\rho}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{1,\rho}(k_1)V_{3,\sigma}(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{3,\sigma}(k_3)V_{1,\rho}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-TGC5}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim6_Gauge_Gauge_Gauge]:
$\mathcal{L}_I=gF_1^{\mu\nu}F_{2,\nu\rho}
F_{3,\hphantom{\rho}\mu}^{\hphantom{3,}\rho}$}\\\hline
[_] & $A_1^\mu(k_2+k_3)\leftarrow-\ii\cdot
\Lambda^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A_{2,\rho} A_{c,\sigma}$\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim6-TGC}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim6_Gauge_Gauge_Gauge_5]:
$\mathcal{L}_I=g/2\cdot\epsilon^{\mu\nu\lambda\tau}
F_{1,\mu\nu}F_{2,\tau\rho}
F_{3,\hphantom{\rho}\lambda}^{\hphantom{3,}\rho}$}\\\hline
[F23] & $A_1^\mu(k_2+k_3)\leftarrow-\ii\cdot
\Lambda_5^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A_{2,\rho} A_{3,\sigma}$\\\hline
[F32] & $A_1^\mu(k_2+k_3)\leftarrow-\ii\cdot
\Lambda_5^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A_{3,\sigma} A_{2,\rho}$\\\hline
[F12] & $A_3^\mu(k_1+k_2)\leftarrow-\ii\cdot$\\\hline
[F21] & $A_3^\mu(k_1+k_2)\leftarrow-\ii\cdot$\\\hline
[F13] & $A_2^\mu(k_1+k_3)\leftarrow-\ii\cdot$\\\hline
[F31] & $A_2^\mu(k_1+k_3)\leftarrow-\ii\cdot$\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim6-TGC5}
\ldots}
\end{table} *)
(* \thocwmodulesection{$\textrm{SU}(2)$ Gauge Bosons}
An important special case for table~\ref{tab:dim4-bosons} are the two
usual coordinates of~$\textrm{SU}(2)$
\begin{equation}
W_\pm = \frac{1}{\sqrt2} \left(W_1 \mp \ii W_2\right)
\end{equation}
i.\,e.
\begin{subequations}
\begin{align}
W_1 &= \frac{1}{\sqrt2} \left(W_+ + W_-\right) \\
W_2 &= \frac{\ii}{\sqrt2} \left(W_+ - W_-\right)
\end{align}
\end{subequations}
and
\begin{equation}
W_1^\mu W_2^\nu - W_2^\mu W_1^\nu
= \ii\left(W_-^\mu W_+^\nu - W_+^\mu W_-^\nu\right)
\end{equation}
Thus the symmtry remains after the change of basis:
\begin{multline}
\epsilon^{abc} W_a^{\mu_1}W_b^{\mu_2}W_c^{\mu_3}
= \ii W_-^{\mu_1} (W_+^{\mu_2}W_3^{\mu_3} - W_3^{\mu_2}W_+^{\mu_3}) \\
+ \ii W_+^{\mu_1} (W_3^{\mu_2}W_-^{\mu_3} - W_-^{\mu_2}W_3^{\mu_3})
+ \ii W_3^{\mu_1} (W_-^{\mu_2}W_+^{\mu_3} - W_+^{\mu_2}W_-^{\mu_3})
\end{multline} *)
(* \thocwmodulesection{Quartic Couplings and Auxiliary Fields}
Quartic couplings can be replaced by cubic couplings to a non-propagating
auxiliary field. The quartic term should get a negative sign so that it the
energy is bounded from below for identical fields. In the language of
functional integrals
\begin{subequations}
\label{eq:quartic-aux}
\begin{multline}
\mathcal{L}_{\phi^4} = - g^2\phi_1\phi_2\phi_3\phi_4
\Longrightarrow \\
\mathcal{L}_{X\phi^2}
= X^*X \pm gX\phi_1\phi_2 \pm gX^*\phi_3\phi_4
= (X^* \pm g\phi_1\phi_2)(X \pm g\phi_3\phi_4)
- g^2\phi_1\phi_2\phi_3\phi_4
\end{multline}
and in the language of Feynman diagrams
\begin{equation}
\parbox{21mm}{\begin{fmfgraph*}(20,20)
\fmfleft{e1,e2}
\fmfright{e3,e4}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{plain}{v,e4}
\fmfv{d.sh=circle,d.si=dot_size,label=$-\ii g^2$}{v}
\end{fmfgraph*}}
\qquad\Longrightarrow\qquad
\parbox{21mm}{\begin{fmfgraph*}(20,20)
\fmfleft{e1,e2}
\fmfright{e3,e4}
\fmf{plain}{v12,e1}
\fmf{plain}{v12,e2}
\fmf{plain}{v34,e3}
\fmf{plain}{v34,e4}
\fmf{dashes,label=$+\ii$}{v12,v34}
\fmfv{d.sh=circle,d.si=dot_size,label=$\pm\ii g$}{v12}
\fmfv{d.sh=circle,d.si=dot_size,label=$\pm\ii g$}{v34}
\end{fmfgraph*}}
\end{equation}
\end{subequations}
The other choice of signs
\begin{equation}
\mathcal{L}_{X\phi^2}'
= - X^*X \pm gX\phi_1\phi_2 \mp gX^*\phi_3\phi_4
= - (X^* \pm g\phi_1\phi_2)(X \mp g\phi_3\phi_4)
- g^2\phi_1\phi_2\phi_3\phi_4
\end{equation}
can not be extended easily to identical particles and is therefore
not used. For identical particles we have
\begin{multline}
\mathcal{L}_{\phi^4} = - \frac{g^2}{4!}\phi^4
\Longrightarrow \\
\mathcal{L}_{X\phi^2}
= \frac{1}{2}X^2 \pm \frac{g}{2}X\phi^2 \pm \frac{g}{2}X\phi^2
= \frac{1}{2}\left(X \pm \frac{g}{2}\phi^2\right)
\left(X \pm \frac{g}{2}\phi^2\right)
- \frac{g^2}{4!}\phi^4
\end{multline}
\begin{dubious}
Explain the factor~$1/3$ in the functional setting and its
relation to the three diagrams in the graphical setting?
\end{dubious}
\thocwmodulesubsection{Quartic Gauge Couplings}
\begin{figure}
\begin{subequations}
\label{eq:Feynman-QCD}
\begin{align}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\threeexternal{k,,\mu,,a}{p}{p'}
\fmf{gluon}{v,e1}
\fmf{fermion}{e2,v,e3}
\fmfdot{v} \end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} + & \ii g\gamma_\mu T_a
\end{split} \\
\label{eq:TGV}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\threeexternal{1}{2}{3}
\fmf{gluon}{v,e1}
\fmf{gluon}{v,e2}
\fmf{gluon}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
& g f_{a_1a_2a_3} C^{\mu_1\mu_2\mu_3} (k_1,k_2,k_3)
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3,d4,e4}
\fmf{gluon}{v,e1}
\fmf{gluon}{v,e2}
\fmf{gluon}{v,e3}
\fmf{gluon}{v,e4}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmflabel{4}{e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e1}
\fmf{warrow_right}{v,e2}
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - & \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3}) \\
\mbox{} - & \ii g^2 f_{a_1a_3b}f_{a_4a_2b}
(g_{\mu_1\mu_4} g_{\mu_2\mu_3} - g_{\mu_1\mu_2} g_{\mu_3\mu_4}) \\
\mbox{} - & \ii g^2 f_{a_1a_4b}f_{a_2a_3b}
(g_{\mu_1\mu_2} g_{\mu_3\mu_4} - g_{\mu_1\mu_3} g_{\mu_4\mu_2})
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:gauge-feynman-rules} Gauge couplings.
See~(\ref{eq:C123}) for the definition of the antisymmetric
tensor $C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)$.}
\end{figure}
\begin{figure}
\begin{equation}
\label{eq:Feynman-QCD'}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3,d4,e4}
\fmf{gluon}{v12,e1}
\fmf{gluon}{v12,e2}
\fmf{gluon}{v34,e3}
\fmf{gluon}{v34,e4}
\fmf{dashes}{v12,v34}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmflabel{4}{e4}
\fmfdot{v12,v34}
\fmffreeze
\fmf{warrow_right}{v12,e1}
\fmf{warrow_right}{v12,e2}
\fmf{warrow_right}{v34,e3}
\fmf{warrow_right}{v34,e4}
\end{fmfgraph*}}} \,=
\mbox{} - \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3})
\end{equation}
\caption{\label{fig:gauge-feynman-rules'} Gauge couplings.}
\end{figure}
The three crossed versions of
figure~\ref{fig:gauge-feynman-rules'} reproduces the quartic coupling in
figure~\ref{fig:gauge-feynman-rules}, because
\begin{multline}
- \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3}) \\
= (\ii g f_{a_1a_2b} T_{\mu_1\mu_2,\nu_1\nu_2})
\left(\frac{\ii g^{\nu_1\nu_3} g^{\nu_2\nu_4}}{2}\right)
(\ii g f_{a_3a_4b} T_{\mu_3\mu_4,\nu_3\nu_4})
\end{multline}
with $T_{\mu_1\mu_2,\mu_3\mu_4} =
g_{\mu_1\mu_3}g_{\mu_4\mu_2}-g_{\mu_1\mu_4}g_{\mu_2\mu_3}$. *)
(* \thocwmodulesection{Gravitinos and supersymmetric currents}
In supergravity theories there is a fermionic partner of the graviton, the
gravitino. Therefore we have introduced the Lorentz type [Vectorspinor].
*)
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, MOM, Ferm)]:
$\bar\psi_1(\ii\fmslash{\partial}\pm m)\phi\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-(\fmslash{k}\mp m)\psi_1S$
& [F21] & $\psi_2\leftarrow-S(\fmslash{k}\mp m)\psi_1$ \\\hline
[F13] & $S\leftarrow \psi^T_1 {\rm C}(\fmslash{k}\pm m)\psi_2$
& [F31] & $S\leftarrow \psi^T_2 {\rm C}(-(\fmslash{k}\mp m)\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow S(\fmslash{k}\pm m)\psi_2$
& [F32] & $\psi_1\leftarrow(\fmslash{k}\pm m)\psi_2 S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, MOM5, Ferm)]:
$\bar\psi_1(\ii\fmslash{\partial}\pm m)\phi\gamma^5\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow(\fmslash{k}\pm m)\gamma^5\psi_1P$
& [F21] & $\psi_2\leftarrow P(\fmslash{k}\pm m)\gamma^5\psi_1$ \\\hline
[F13] & $P\leftarrow \psi^T_1 {\rm C}(\fmslash{k}\pm m)\gamma^5\psi_2$
& [F31] & $P\leftarrow \psi^T_2 {\rm C}(\fmslash{k}\pm m)\gamma^5\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow P(\fmslash{k}\pm m)\gamma^5\psi_2$
& [F32] & $\psi_1\leftarrow(\fmslash{k}\pm m)\gamma^5\psi_2 P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, MOML, Ferm)]:
$\bar\psi_1 (\ii\fmslash{\partial}\pm m)\phi(1-\gamma^5)\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-(1-\gamma^5)(\fmslash{k}\mp m)\psi_1\phi$
& [F21] & $\psi_2\leftarrow-\phi(1-\gamma^5)(\fmslash{k}\mp m)\psi_1$ \\\hline
[F13] & $\phi\leftarrow \psi^T_1 {\rm C}(\fmslash{k}\pm m)(1-\gamma^5)\psi_2$
& [F31] & $\phi\leftarrow \psi^T_2 {\rm C}(1-\gamma^5)(-(\fmslash{k}\mp m)\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow\phi(\fmslash{k}\pm m)(1-\gamma^5)\psi_2$
& [F32] & $\psi_1\leftarrow(\fmslash{k}\pm m)(1-\gamma^5)\psi_2 \phi$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, LMOM, Ferm)]:
$\bar\psi_1 \phi(1-\gamma^5)(\ii\fmslash{\partial}\pm m)\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-(\fmslash{k}\mp m)\psi_1(1-\gamma^5)\phi$
& [F21] & $\psi_2\leftarrow-\phi(\fmslash{k}\mp m)(1-\gamma^5)\psi_1$ \\\hline
[F13] & $\phi\leftarrow \psi^T_1 {\rm C}(1-\gamma^5)(\fmslash{k}\pm m)\psi_2$
& [F31] & $\phi\leftarrow \psi^T_2 {\rm C}(-(\fmslash{k}\mp m)(1-\gamma^5)\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow\phi(1-\gamma^5)(\fmslash{k}\pm m)\psi_2$
& [F32] & $\psi_1\leftarrow(1-\gamma^5)(\fmslash{k}\pm m)\psi_2 \phi$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, VMOM, Ferm)]:
$\bar\psi_1 \ii\fmslash{\partial}_\alpha V_\beta \lbrack \gamma^\alpha, \gamma^\beta \rbrack \psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-\lbrack\fmslash{k},\gamma^\alpha\rbrack\psi_1 V_\alpha$
& [F21] & $\psi_2\leftarrow-\lbrack\fmslash{k},\fmslash{V}\rbrack\psi_1$ \\\hline
[F13] & $V_\alpha\leftarrow \psi^T_1 {\rm C}\lbrack\fmslash{k},\gamma_\alpha\rbrack\psi_2$
& [F31] & $V_\alpha\leftarrow \psi^T_2 {\rm C}(-\lbrack\fmslash{k}, \gamma_\alpha\rbrack\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow\rbrack\fmslash{k},\fmslash{V}\rbrack\psi_2$
& [F32] & $\psi_1\leftarrow\lbrack\fmslash{k},\gamma^\alpha\rbrack\psi_2 V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-MOM} Combined dimension-4 trilinear
fermionic couplings including a momentum. $Ferm$ stands for $Psi$ and
$Chi$. The case of $MOMR$ is identical to $MOML$ if one substitutes
$1+\gamma^5$ for $1-\gamma^5$, as well as for $LMOM$ and $RMOM$. The
mass term forces us to keep the chiral projector always on the left
after "inverting the line" for $MOML$ while on the right for $LMOM$.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, S2LR, Ferm)]: $\bar\psi_1 S_1 S_2
(g_L P_L + g_R P_R) \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow S_1 S_2 (g_R P_L + g_L P_R) \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow S_1 S_2 (g_L P_L + g_R P_R) \psi_2$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T_1 C S_2 (g_L P_L + g_R P_R) \psi_2$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T_1 C S_1 (g_L P_L + g_R P_R) \psi_2$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow \psi^T_2 C S_2 (g_R P_L + g_L P_R) \psi_1$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow \psi^T_2 C S_1 (g_R P_L + g_L P_R) \psi_1$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, S2, Ferm)]: $\bar\psi_1 S_1 S_2
\gamma^5 \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow S_1 S_2 \gamma^5 \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow S_1 S_2 \gamma^5 \psi_2$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T_1 C S_2 \gamma^5 \psi_2$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T_1 C S_1 \gamma^5 \psi_2$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow \psi^T_2 C S_2 \gamma^5 \psi_1$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow \psi^T_2 C S_1 \gamma^5 \psi_1$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, V2, Ferm)]: $\bar\psi_1 \lbrack \fmslash{V}_1 , \fmslash{V}_2 \rbrack \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow - \lbrack \fmslash{V}_1 , \fmslash{V}_2 \rbrack \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \lbrack \fmslash{V}_1 , \fmslash{V}_2 \rbrack \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_{1\:\alpha} \leftarrow \psi^T_1 C \lbrack \gamma_\alpha , \fmslash{V}_2 \rbrack \psi_2$ \\ \hline
[F124] [F142] [F214] & $V_{2\:\alpha} \leftarrow \psi^T_1 C (-\lbrack \gamma_\alpha , \fmslash{V}_1 \rbrack) \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_{1\:\alpha} \leftarrow \psi^T_2 C (-\lbrack \gamma_\alpha , \fmslash{V}_2 \rbrack) \psi_1$ \\ \hline
[F412] [F421] [F241] & $V_{2\:\alpha} \leftarrow \psi^T_2 C \lbrack \gamma_\alpha , \fmslash{V}_1 \rbrack \psi_1$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-mom2} Vertices with two fermions ($Ferm$ stands
for $Psi$ and $Chi$, but not for $Grav$) and two bosons (two scalars,
scalar/vector, two vectors) for the BRST transformations. Part I}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, SV, Ferm)]: $\bar\psi_1 \fmslash{V} S \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow - \fmslash{V} S \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \fmslash{V} S \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_\alpha \leftarrow \psi^T_1 C \gamma_\alpha S \psi_2$ \\ \hline
[F124] [F142] [F214] & $S \leftarrow \psi^T_1 C \fmslash{V} \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_\alpha \leftarrow \psi^T_2 C (- \gamma_\alpha S \psi_1)$ \\ \hline
[F412] [F421] [F241] & $S \leftarrow \psi^T_2 C (- \fmslash{V} \psi_1)$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, PV, Ferm)]: $\bar\psi_1 \fmslash{V} \gamma^5 P \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow \fmslash{V} \gamma^5 P \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \fmslash{V} \gamma^5 P \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_\alpha \leftarrow \psi^T_1 C \gamma_\alpha \gamma^5 P \psi_2$ \\ \hline
[F124] [F142] [F214] & $P \leftarrow \psi^T_1 C \fmslash{V} \gamma^5 \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_\alpha \leftarrow \psi^T_2 C \gamma_\alpha \gamma^5 P \psi_1$ \\ \hline
[F412] [F421] [F241] & $P \leftarrow \psi^T_2 C \fmslash{V} \gamma^5 \psi_1$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, S(L/R)V, Ferm)]: $\bar\psi_1 \fmslash{V} (1 \mp\gamma^5) \phi \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow - \fmslash{V} (1\pm\gamma^5) \phi \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \fmslash{V} (1\mp\gamma^5) \phi \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_\alpha \leftarrow \psi^T_1 C \gamma_\alpha (1\mp\gamma^5) \phi \psi_2$ \\ \hline
[F124] [F142] [F214] & $\phi \leftarrow \psi^T_1 C \fmslash{V} (1\mp\gamma^5) \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_\alpha \leftarrow \psi^T_2 C \gamma_\alpha (-(1\pm\gamma^5) \phi \psi_1)$ \\ \hline
[F412] [F421] [F241] & $\phi \leftarrow \psi^T_2 C \fmslash{V} (-(1\pm\gamma^5) \psi_1)$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-mom2} Vertices with two fermions ($Ferm$ stands
for $Psi$ and $Chi$, but not for $Grav$) and two bosons (two scalars,
scalar/vector, two vectors) for the BRST transformations. Part II}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, POT, Psi)]: $\bar\psi_\mu S \gamma^\mu \psi$}\\\hline
[F12] & $\psi\leftarrow - \gamma^\mu \psi_\mu S$
& [F21] & $\psi\leftarrow - S\gamma^\mu \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \gamma^\mu \psi$
& [F31] & $S\leftarrow \psi^T{\rm C} (-\gamma^\mu)\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\gamma_\mu\psi$
& [F32] & $\psi_\mu\leftarrow \gamma_\mu \psi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, S, Psi)]: $\bar\psi_\mu \fmslash{k}_S S \gamma^\mu \psi$}\\\hline
[F12] & $\psi\leftarrow \gamma^\mu \fmslash{k}_S \psi_\mu S$
& [F21] & $\psi\leftarrow S\gamma^\mu \fmslash{k}_S \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \psi$
& [F31] & $S\leftarrow \psi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\fmslash{k}_S\gamma_\mu\psi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \psi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, P, Psi)]: $\bar\psi_\mu \fmslash{k}_P P \gamma^\mu \gamma_5 \psi$}\\\hline
[F12] & $\psi\leftarrow \gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu P$
& [F21] & $\psi\leftarrow P\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F13] & $P\leftarrow \psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\psi$
& [F31] & $P\leftarrow \psi^T {\rm C}\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow P\fmslash{k}_P \gamma_\mu \gamma_5 \psi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_P \gamma_\mu \gamma_5 \psi P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, V, Psi)]: $\bar\psi_\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\gamma^\mu\gamma^5\psi$}\\\hline
[F12] & $\psi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \psi_\mu V_\alpha$
& [F21] & $\psi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \psi$
& [F31] & $V_{\mu}\leftarrow \psi^T {\rm C} \gamma^5 \gamma^{\rho} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$ \\\hline
[F23] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \psi $
& [F32] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \psi V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-gravdirac} Dimension-5 trilinear
couplings including one Dirac, one Gravitino fermion and one additional particle.The option [POT] is for the coupling of the supersymmetric current to the derivative of the quadratic terms in the superpotential.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Psibar, POT, Grav)]: $\bar\psi \gamma^\mu S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow - \gamma_\mu \psi S$
& [F21] & $\psi_\mu\leftarrow - S \gamma_\mu\psi$ \\\hline
[F13] & $S\leftarrow \psi^T{\rm C}\gamma^\mu\psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} (-\gamma^\mu) \psi$ \\\hline
[F23] & $\psi\leftarrow S\gamma^\mu\psi_\mu$
& [F32] & $\psi\leftarrow \gamma^\mu\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Psibar, S, Grav)]: $\bar\psi \gamma^\mu \fmslash{k}_S S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \psi S$
& [F21] & $\psi_\mu\leftarrow S \fmslash{k}_S \gamma_\mu\psi$ \\\hline
[F13] & $S\leftarrow \psi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \psi$ \\\hline
[F23] & $\psi\leftarrow S\gamma^\mu\fmslash{k}_S\psi_\mu$
& [F32] & $\psi\leftarrow \gamma^\mu\fmslash{k}_S\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Psibar, P, Grav)]: $\bar\psi \gamma^\mu\gamma^5 P\fmslash{k}_P \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow -\fmslash{k}_P \gamma_\mu \gamma^5 \psi P$
& [F21] & $\psi_\mu\leftarrow -P\fmslash{k}_P \gamma_\mu \gamma^5 \psi$ \\\hline
[F13] & $P\leftarrow \psi^T {\rm C}\gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F31] & $P\leftarrow -\psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\psi$ \\\hline
[F23] & $\psi\leftarrow P \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F32] & $\psi\leftarrow \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Psibar, V, Grav)]: $\bar\psi\gamma^5\gamma^\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \psi V_\alpha$
& [F21] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \psi$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T {\rm C} \gamma^5 \gamma^\rho \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$
& [F31] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \psi$ \\\hline
[F23] & $\psi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$
& [F32] & $\psi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack\psi_\mu V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-diracgrav} Dimension-5 trilinear
couplings including one conjugated Dirac, one Gravitino fermion and one additional particle.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, POT, Chi)]: $\bar\psi_\mu S \gamma^\mu \chi$}\\\hline
[F12] & $\chi\leftarrow - \gamma^\mu \psi_\mu S$
& [F21] & $\chi\leftarrow - S\gamma^\mu \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \gamma^\mu \chi$
& [F31] & $S\leftarrow \chi^T{\rm C} (-\gamma^\mu)\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\gamma_\mu\chi$
& [F32] & $\psi_\mu\leftarrow \gamma_\mu \chi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, S, Chi)]: $\bar\psi_\mu \fmslash{k}_S S \gamma^\mu \chi$}\\\hline
[F12] & $\chi\leftarrow \gamma^\mu \fmslash{k}_S \psi_\mu S$
& [F21] & $\chi\leftarrow S\gamma^\mu \fmslash{k}_S \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \chi$
& [F31] & $S\leftarrow \chi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\fmslash{k}_S\gamma_\mu\chi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \chi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, P, Chi)]: $\bar\psi_\mu \fmslash{k}_P P \gamma^\mu \gamma_5 \chi$}\\\hline
[F12] & $\chi\leftarrow \gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu P$
& [F21] & $\chi\leftarrow P\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F13] & $P\leftarrow \psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\chi$
& [F31] & $P\leftarrow \chi^T {\rm C}\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow P\fmslash{k}_P \gamma_\mu \gamma_5 \chi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_P \gamma_\mu \gamma_5 \chi P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, V, Chi)]: $\bar\psi_\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\gamma^\mu\gamma^5\chi$}\\\hline
[F12] & $\chi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \psi_\mu V_\alpha$
& [F21] & $\chi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \chi$
& [F31] & $V_{\mu}\leftarrow \chi^T {\rm C} \gamma^5 \gamma^{\rho} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$ \\\hline
[F23] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \chi $
& [F32] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \chi V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-gravmajo} Dimension-5 trilinear
couplings including one Majorana, one Gravitino fermion and one
additional particle. The table is essentially the same as the one
with the Dirac fermion and only written for the sake of completeness.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Chibar, POT, Grav)]: $\bar\chi \gamma^\mu S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow - \gamma_\mu \chi S$
& [F21] & $\psi_\mu\leftarrow - S \gamma_\mu\chi$ \\\hline
[F13] & $S\leftarrow \chi^T{\rm C}\gamma^\mu\psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} (-\gamma^\mu) \chi$ \\\hline
[F23] & $\chi\leftarrow S\gamma^\mu\psi_\mu$
& [F32] & $\chi\leftarrow \gamma^\mu\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Chibar, S, Grav)]: $\bar\chi \gamma^\mu \fmslash{k}_S S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \chi S$
& [F21] & $\psi_\mu\leftarrow S \fmslash{k}_S \gamma_\mu\chi$ \\\hline
[F13] & $S\leftarrow \chi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \chi$ \\\hline
[F23] & $\chi\leftarrow S\gamma^\mu\fmslash{k}_S\psi_\mu$
& [F32] & $\chi\leftarrow \gamma^\mu\fmslash{k}_S\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Chibar, P, Grav)]: $\bar\chi \gamma^\mu\gamma^5 P\fmslash{k}_P \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow -\fmslash{k}_P \gamma_\mu \gamma^5 \chi P$
& [F21] & $\psi_\mu\leftarrow -P\fmslash{k}_P \gamma_\mu \gamma^5 \chi$ \\\hline
[F13] & $P\leftarrow \chi^T {\rm C}\gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F31] & $P\leftarrow -\psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\chi$ \\\hline
[F23] & $\chi\leftarrow P \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F32] & $\chi\leftarrow \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Chibar, V, Grav)]: $\bar\chi\gamma^5\gamma^\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \chi V_\alpha$
& [F21] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \chi$ \\\hline
[F13] & $V_{\mu}\leftarrow \chi^T {\rm C} \gamma^5 \gamma^\rho \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$
& [F31] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \chi$ \\\hline
[F23] & $\chi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$
& [F32] & $\chi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack\psi_\mu V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-majograv} Dimension-5 trilinear
couplings including one conjugated Majorana, one Gravitino fermion and
one additional particle. This table is not only the same as the one
with the conjugated Dirac fermion but also the same part of the
Lagrangian density as the one with the Majorana particle on the right
of the gravitino.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, S2, Psi)]: $\bar\psi_\mu S_1 S_2
\gamma^\mu \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow - \gamma^\mu S_1 S_2 \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow \gamma_\mu S_1 S_2 \psi$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T_\mu C S_2 \gamma^\mu \psi$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T_\mu C S_1 \gamma^\mu \psi$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow - \psi^T C S_2 \gamma^\mu \psi_\mu$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow - \psi^T C S_1 \gamma^\mu \psi_\mu$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, SV, Psi)]: $\bar\psi_\mu S \fmslash{V} \gamma^\mu \gamma^5 \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow \gamma^5 \gamma^\mu S \fmslash{V} \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow \fmslash{V} S \gamma_\mu \gamma^5 \psi$ \\ \hline
[F134] [F143] [F314] & $S \leftarrow \psi^T_\mu C \fmslash{V} \gamma^\mu \gamma^5 \psi$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T_\rho C S \gamma_\mu \gamma^\rho \gamma^5 \psi$ \\ \hline
[F413] [F431] [F341] & $S \leftarrow \psi^T C \gamma^5 \gamma^\mu \fmslash{V} \psi_\mu$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T C S \gamma^5 \gamma^\rho \gamma_\mu \psi_\rho$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, PV, Psi)]: $\bar\psi_\mu P \fmslash{V} \gamma^\mu \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow \gamma^\mu P \fmslash{V} \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow \fmslash{V} P \gamma_\mu \psi$ \\ \hline
[F134] [F143] [F314] & $P \leftarrow \psi^T_\mu C \fmslash{V} \gamma^\mu \psi$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T_\rho C P \gamma_\mu \gamma^\rho \psi$ \\ \hline
[F413] [F431] [F341] & $P \leftarrow \psi^T C \gamma^\mu \fmslash{V} \psi_\mu$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T C P \gamma^\rho \gamma_\mu \psi_\rho$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, V2, Psi)]: $\bar\psi_\mu f_{abc} \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack\gamma^\mu \gamma^5 \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow f_{abc} \gamma^5 \gamma^\mu \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow f_{abc} \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack \gamma_\mu \gamma^5 \psi$ \\ \hline
[F134] [F143] [F314] [F124] [F142] [F214] & $V_\mu^a \leftarrow\psi^T_\rho C f_{abc} \lbrack \gamma_\mu , \fmslash{V}^b \rbrack \gamma^\rho \gamma^5 \psi$ \\ \hline
[F413] [F431] [F341] [F412] [F421] [F241] & $V_\mu^a \leftarrow\psi^T C f_{abc} \gamma^5 \gamma^\rho\lbrack \gamma_\mu , \fmslash{V}^b \rbrack \psi_\rho$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-gravferm2boson} Dimension-5 trilinear
couplings including one Dirac, one Gravitino fermion and two additional bosons. In each lines we list the fusion possibilities with the same order of the fermions, but the order of the bosons is arbitrary (of course, one has to take care of this order in the mapping of the wave functions in [fusion]).}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Psibar, S2, Grav)]: $\bar\psi S_1 S_2
\gamma^\mu \psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow - \gamma_\mu S_1 S_2 \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi \leftarrow \gamma^\mu S_1 S_2 \psi_\mu$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T C S_2 \gamma^\mu \psi_\mu$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T C S_1 \gamma^\mu \psi_\mu$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow - \psi^T_\mu C S_2 \gamma^\mu \psi$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow - \psi^T_\mu C S_1 \gamma^\mu \psi$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Psibar, SV, Grav)]: $\bar\psi S \gamma^\mu \gamma^5 \fmslash{V} \psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow \fmslash{V} S \gamma^5 \gamma^\mu \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi\leftarrow \gamma^\mu\gamma^5 S\fmslash{V}\psi_\mu$ \\ \hline
[F134] [F143] [F314] & $S \leftarrow \psi^T C \gamma^\mu \gamma^5 \fmslash{V}\psi$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T C \gamma^\rho \gamma^5 S \gamma_\mu \psi_\rho$ \\ \hline
[F413] [F431] [F341] & $S \leftarrow \psi^T_\mu C \fmslash{V} \gamma^5 \gamma^\mu \psi$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T_\rho C S \gamma_\mu \gamma^5 \gamma^\rho \psi$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Psibar, PV, Grav)]: $\bar\psi P \gamma^\mu \fmslash{V} \psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow \fmslash{V}\gamma_\mu P \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi\leftarrow \gamma^\mu\fmslash{V} P\psi_\mu$ \\ \hline
[F134] [F143] [F314] & $P \leftarrow \psi^T C \gamma^\mu\fmslash{V}\psi_\mu$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T C P \gamma^\rho \gamma_\mu \psi_\rho$ \\ \hline
[F413] [F431] [F341] & $P \leftarrow \psi^T_\mu C \fmslash{V}\gamma^\mu \psi$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T_\rho C P \gamma_\mu \gamma^\rho \psi$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Psibar, V2, Grav)]: $\bar\psi f_{abc} \gamma^5 \gamma^\mu \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack\psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow f_{abc} \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack \gamma_\mu \gamma^5 \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi\leftarrow f_{abc} \gamma^5\gamma^\mu\lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack\psi_\mu$ \\ \hline
[F134] [F143] [F314] [F124] [F142] [F214] & $V_\mu^a \leftarrow\psi^T C f_{abc} \gamma^5\gamma^\rho\lbrack \gamma_\mu , \fmslash{V}^b \rbrack\psi_\rho$ \\ \hline
[F413] [F431] [F341] [F412] [F421] [F241] & $V_\mu^a \leftarrow\psi^T_\rho C f_{abc}\lbrack \gamma_\mu , \fmslash{V}^b \rbrack\gamma^\rho\gamma^5 \psi$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-gravferm2boson2} Dimension-5 trilinear
couplings including one conjugated Dirac, one Gravitino fermion and two additional bosons. The couplings of Majorana fermions to the gravitino and two bosons are essentially the same as for Dirac fermions and they are omitted here.}
\end{table}
*)
(* \thocwmodulesection{Perturbative Quantum Gravity and Kaluza-Klein Interactions}
The gravitational coupling constant and the relative strength of
the dilaton coupling are abbreviated as
\begin{subequations}
\begin{align}
\kappa &= \sqrt{16\pi G_N} \\
\omega &= \sqrt{\frac{2}{3(n+2)}} = \sqrt{\frac{2}{3(d-2)}}\,,
\end{align}
\end{subequations}
where~$n=d-4$ is the number of extra space dimensions. *)
(* In~(\ref{eq:graviton-feynman-rules3}-\ref{eq:dilaton-feynman-rules5}),
we use the notation of~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}:
\begin{subequations}
\begin{equation}
C_{\mu\nu,\rho\sigma} =
g_{\mu\rho} g_{\nu\sigma} + g_{\mu\sigma} g_{\nu\rho}
- g_{\mu\nu} g_{\rho\sigma}
\end{equation}
\begin{multline}
D_{\mu\nu,\rho\sigma}(k_1,k_2) =
g_{\mu\nu} k_{1,\sigma} k_{2,\rho} \\
\mbox{}
- ( g_{\mu\sigma} k_{1,\nu} k_{2,\rho}
+ g_{\mu\rho} k_{1,\sigma} k_{2,\nu}
- g_{\rho\sigma} k_{1,\mu} k_{2,\nu}
+ (\mu\leftrightarrow\nu))
\end{multline}
\begin{multline}
E_{\mu\nu,\rho\sigma}(k_1,k_2) =
g_{\mu\nu} (k_{1,\rho} k_{1,\sigma}
+ k_{2,\rho} k_{2,\sigma} + k_{1,\rho} k_{2,\sigma}) \\
\mbox{}
- ( g_{\nu\sigma} k_{1,\mu} k_{1,\rho}
+ g_{\nu\rho} k_{2,\mu} k_{2,\sigma}
+ (\mu\leftrightarrow\nu))
\end{multline}
\begin{multline}
F_{\mu\nu,\rho\sigma\lambda}(k_1,k_2,k_3) = \\
g_{\mu\rho} g_{\sigma\lambda} (k_2 - k_3)_{\nu}
+ g_{\mu\sigma} g_{\lambda\rho} (k_3 - k_1)_{\nu}
+ g_{\mu\lambda} g_{\rho\sigma} (k_1 - k_2)_{\nu}
+ (\mu\leftrightarrow\nu)
\end{multline}
\begin{multline}
G_{\mu\nu,\rho\sigma\lambda\delta} =
g_{\mu\nu} (g_{\rho\sigma}g_{\lambda\delta} - g_{\rho\delta}g_{\lambda\sigma})
\\ \mbox{}
+ ( g_{\mu\rho}g_{\nu\delta}g_{\lambda\sigma}
+ g_{\mu\lambda}g_{\nu\sigma}g_{\rho\delta}
- g_{\mu\rho}g_{\nu\sigma}g_{\lambda\delta}
- g_{\mu\lambda}g_{\nu\delta}g_{\rho\sigma}
+ (\mu\leftrightarrow\nu) )
\end{multline}
\end{subequations} *)
(* \begin{figure}
\begin{subequations}
\label{eq:graviton-feynman-rules3}
\begin{align}
\label{eq:graviton-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{dbl_dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & - \ii \frac{\kappa}{2} g_{\mu\nu} m^2
+ \ii \frac{\kappa}{2} C_{\mu\nu,\mu_1\mu_2}k^{\mu_1}_1k^{\mu_2}_2
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{h_{\mu\nu}}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{dbl_dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - \ii \frac{\kappa}{2} m^2 C_{\mu\nu,\mu_1\mu_2}
- \ii \frac{\kappa}{2}
(& k_1k_2 C_{\mu\nu,\mu_1\mu_2} \\
&\mbox{} + D_{\mu\nu,\mu_1\mu_2}(k_1,k_2) \\
&\mbox{} + \xi^{-1} E_{\mu\nu,\mu_1\mu_2}(k_1,k_2))
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{p}{p'}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{dbl_dots}{v,e3}
\fmfdot{v}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - \ii \frac{\kappa}{2} m g_{\mu\nu}
- \ii \frac{\kappa}{8}
(& \gamma_{\mu}(p+p')_{\nu} + \gamma_{\nu}(p+p')_{\mu} \\
& \mbox{} - 2 g_{\mu\nu} (\fmslash{p}+\fmslash{p}') )
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:graviton-feynman-rules3} Three-point graviton couplings.}
\end{figure}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Graviton_Scalar_Scalar]:
$h_{\mu\nu} C^{\mu\nu}_{0}(k_1,k_2)\phi_1\phi_2$}\\\hline
[F12|F21]
& $\phi_2 \leftarrow \ii\cdot
h_{\mu\nu} C^{\mu\nu}_{0} (k_1, -k-k_1)\phi_1 $ \\\hline
[F13|F31]
& $\phi_1 \leftarrow \ii\cdot
h_{\mu\nu} C^{\mu\nu}_{0} (-k-k_2, k_2)\phi_2 $ \\\hline
[F23|F32]
& $h^{\mu\nu} \leftarrow \ii\cdot
C^{\mu\nu}_0 (k_1,k_2)\phi_1\phi_2 $ \\\hline
\multicolumn{2}{|l|}{[Graviton_Vector_Vector]:
$h_{\mu\nu} C^{\mu\nu,\mu_1\mu_2}_1(k_1,k_2,\xi)
V_{\mu_1}V_{\mu_2} $}\\\hline
[F12|F21] & $ V^\mu_2 \leftarrow \ii\cdot h_{\kappa\lambda}
C^{\kappa\lambda,\mu\nu}_1(-k-k_1,k_1\xi) V_{1,\nu}$ \\\hline
[F13|F31] & $ V^\mu_1 \leftarrow \ii\cdot h_{\kappa\lambda}
C^{\kappa\lambda,\mu\nu}_1(-k-k_2,k_2,\xi) V_{2,\nu}$ \\\hline
[F23|F32]
& $h^{\mu\nu} \leftarrow \ii\cdot
C^{\mu\nu,\mu_1\mu_2}_1(k_1,k_2,\xi)
V_{1,\mu_1}V_{2,\mu_2} $ \\\hline
\multicolumn{2}{|l|}{[Graviton_Spinor_Spinor]:
$h_{\mu\nu} \bar\psi_1
C^{\mu\nu}_{\frac{1}{2}}(k_1,k_2)\psi_2 $}\\\hline
[F12] & $ \bar\psi_2 \leftarrow \ii\cdot
h_{\mu\nu} \bar\psi_1 C^{\mu\nu}_{\frac{1}{2}}(k_1,-k-k_1) $ \\\hline
[F21] & $ \bar\psi_2 \leftarrow \ii\cdot\ldots $ \\\hline
[F13] & $ \psi_1 \leftarrow \ii\cdot
h_{\mu\nu}C^{\mu\nu}_{\frac{1}{2}}(-k-k_2,k_2)\psi_2$ \\\hline
[F31] & $ \psi_1 \leftarrow \ii\cdot\ldots $ \\\hline
[F23] & $ h^{\mu\nu} \leftarrow \ii\cdot
\bar\psi_1 C^{\mu\nu}_{\frac{1}{2}}(k_1,k_2)\psi_2 $ \\\hline
[F32] & $ h^{\mu\nu} \leftarrow \ii\cdot\ldots $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:graviton-three-point} \ldots}
\end{table}
Derivation of~(\ref{eq:graviton-scalar-scalar})
\begin{subequations}
\begin{align}
L &= \frac{1}{2} (\partial_\mu \phi) (\partial^\mu \phi) - \frac{m^2}{2} \phi^2 \\
(\partial_\mu\phi) \frac{\partial L}{\partial(\partial^\nu\phi)}
&= (\partial_\mu\phi)(\partial_\nu\phi) \\
T_{\mu\nu} &= -g_{\mu\nu} L +
(\partial_\mu\phi) \frac{\partial L}{\partial(\partial^\nu\phi)}
+
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
C^{\mu\nu}_{0}(k_1,k_2)
&= C^{\mu\nu,\mu_1\mu_2} k_{1,\mu_1} k_{2,\mu_2} \\
C^{\mu\nu,\mu_1\mu_2}_1(k_1,k_2,\xi)
&= k_1k_2 C^{\mu\nu,\mu_1\mu_2}
+ D^{\mu\nu,\mu_1\mu_2}(k_1,k_2)
+ \xi^{-1} E^{\mu\nu,\mu_1\mu_2}(k_1,k_2) \\
C^{\mu\nu}_{\frac{1}{2},\alpha\beta}(p,p')
&= \gamma^{\mu}_{\alpha\beta}(p+p')^{\nu}
+ \gamma^{\nu}_{\alpha\beta}(p+p')^{\mu}
- 2 g^{\mu\nu} (\fmslash{p}+\fmslash{p}')_{\alpha\beta}
\end{align}
\end{subequations} *)
(* \begin{figure}
\begin{subequations}
\label{eq:dilaton-feynman-rules3}
\begin{align}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa 2m^2 - \ii \omega \kappa k_1k_2 \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{\phi(k)}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa g_{\mu_1\mu_2}m^2
- \ii \omega \kappa
\xi^{-1} (k_{1,\mu_1}k_{\mu_2} + k_{2,\mu_2}k_{\mu_1}) \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{p}{p'}{\phi(k)}
\fmf{fermion}{e1,v,e2}
\fmf{dots}{v,e3}
\fmfdot{v}
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa 2m
+ \ii \omega \kappa \frac{3}{4}(\fmslash{p}+\fmslash{p}')
\end{align}
\end{subequations}
\caption{\label{fig:dilaton-feynman-rules3} Three-point dilaton couplings.}
\end{figure}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dilaton_Scalar_Scalar]:
$\phi \ldots k_1k_2\phi_1\phi_2 $}\\\hline
[F12|F21] & $ \phi_2 \leftarrow \ii\cdot k_1(-k-k_1)\phi\phi_1 $ \\\hline
[F13|F31] & $ \phi_1 \leftarrow \ii\cdot (-k-k_2)k_2\phi\phi_2 $ \\\hline
[F23|F32] & $ \phi \leftarrow \ii\cdot k_1k_2\phi_1\phi_2 $ \\\hline
\multicolumn{2}{|l|}{[Dilaton_Vector_Vector]:
$\phi \ldots $}\\\hline
[F12] & $ V_{2,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F21] & $ V_{2,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F13] & $ V_{1,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F31] & $ V_{1,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F23] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
[F32] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
\multicolumn{2}{|l|}{[Dilaton_Spinor_Spinor]:
$\phi \ldots $}\\\hline
[F12] & $ \bar\psi_2 \leftarrow \ii\cdot\ldots $ \\\hline
[F21] & $ \bar\psi_2 \leftarrow \ii\cdot\ldots $ \\\hline
[F13] & $ \psi_1 \leftarrow \ii\cdot\ldots $ \\\hline
[F31] & $ \psi_1 \leftarrow \ii\cdot\ldots $ \\\hline
[F23] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
[F32] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dilaton-three-point} \ldots}
\end{table} *)
(* \begin{figure}
\begin{subequations}
\label{eq:graviton-feynman-rules4}
\begin{align}
\label{eq:graviton-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} &
- \ii g\frac{\kappa}{2} C_{\mu\nu,\mu_3\rho}(k_1-k_2)^{\rho} T^{a_3}_{n_2n_1}
\end{split} \\
\label{eq:graviton-scalar-vector-vector}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - g \frac{\kappa}{2} f^{a_1a_2a_3}
(& C_{\mu\nu,\mu_1\mu_2} (k_1-k_2)_{\mu_3} \\
& \mbox{} + C_{\mu\nu,\mu_2\mu_3} (k_2-k_3)_{\mu_1} \\
& \mbox{} + C_{\mu\nu,\mu_3\mu_1} (k_3-k_1)_{\mu_2} \\
& \mbox{} + F_{\mu\nu,\mu_1\mu_2\mu_3}(k_1,k_2,k_3) )
\end{split} \\
\label{eq:graviton-yukawa}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{plain}{v,e3}
\fmf{dbl_dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & \ii g\frac{\kappa}{4}
(C_{\mu\nu,\mu_3\rho} - g_{\mu\nu}g_{\mu_3\rho})
\gamma^{\rho} T^{a_3}_{n_2n_1}
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:graviton-feynman-rules4} Four-point graviton couplings.
(\ref{eq:graviton-scalar-scalar-scalar}),
(\ref{eq:graviton-scalar-vector-vector}),
and~(\ref{eq:graviton-yukawa)} are missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but should be generated
by standard model Higgs selfcouplings, Higgs-gaugeboson couplings, and
Yukawa couplings.}
\end{figure} *)
(* \begin{figure}
\begin{subequations}
\label{eq:dilaton-feynman-rules4}
\begin{align}
\label{eq:dilaton-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa (k_1 + k_2)_{\mu_3} T^{a_3}_{n_1,n_2} \\
\label{eq:dilaton-scalar-vector-vector}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&= 0 \\
\label{eq:dilaton-yukawa}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{plain}{v,e3}
\fmf{dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{fermion}{e1,v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
- \ii \frac{3}{2} \omega g \kappa \gamma_{\mu_3} T^{a_3}_{n_1n_2}
\end{align}
\end{subequations}
\caption{\label{fig:dilaton-feynman-rules4} Four-point dilaton couplings.
(\ref{eq:dilaton-scalar-scalar-scalar}),
(\ref{eq:dilaton-scalar-vector-vector})
and~(\ref{eq:dilaton-yukawa}) are missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but could be generated
by standard model Higgs selfcouplings, Higgs-gaugeboson couplings,
and Yukawa couplings.}
\end{figure} *)
(* \begin{figure}
\begin{subequations}
\label{eq:graviton-feynman-rules5}
\begin{align}
\label{eq:graviton-scalar-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{plain}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & - \ii g^2 \frac{\kappa}{2} C_{\mu\nu,\mu_3\mu_4}
(T^{a_3}T^{a_4} + T^{a_4}T^{a_3})_{n_2n_1}
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{h_{\mu\nu}}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - \ii g^2 \frac{\kappa}{2}
(& f^{ba_1a_3} f^{ba_2a_4} G_{\mu\nu,\mu_1\mu_2\mu_3\mu_4} \\
& \mbox + f^{ba_1a_2} f^{ba_3a_4} G_{\mu\nu,\mu_1\mu_3\mu_2\mu_4} \\
& \mbox + f^{ba_1a_4} f^{ba_2a_3} G_{\mu\nu,\mu_1\mu_2\mu_4\mu_3} )
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:graviton-feynman-rules5} Five-point graviton couplings.
(\ref{eq:graviton-scalar-scalar-scalar-scalar}) is missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but should be generated
by standard model Higgs selfcouplings.}
\end{figure} *)
(* \begin{figure}
\begin{subequations}
\label{eq:dilaton-feynman-rules5}
\begin{align}
\label{eq:dilaton-scalar-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{plain}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\ii \omega g^2 \kappa g_{\mu_3\mu_4}
(T^{a_3}T^{a_4} + T^{a_4}T^{a_3})_{n_2n_1} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{\phi(k)}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&= 0
\end{align}
\end{subequations}
\caption{\label{fig:dilaton-feynman-rules5} Five-point dilaton couplings.
(\ref{eq:dilaton-scalar-scalar-scalar-scalar}) is missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but could be generated
by standard model Higgs selfcouplings.}
\end{figure} *)
(* \thocwmodulesection{Dependent Parameters}
This is a simple abstract syntax for parameter dependencies.
Later, there will be a parser for a convenient concrete syntax
as a part of a concrete syntax for models. There is no intention
to do \emph{any} symbolic manipulation with this. The expressions
will be translated directly by [Targets] to the target language. *)
type 'a expr =
| I | Const of int
| Atom of 'a
| Sum of 'a expr list
| Diff of 'a expr * 'a expr
| Neg of 'a expr
| Prod of 'a expr list
| Quot of 'a expr * 'a expr
| Rec of 'a expr
| Pow of 'a expr * int
| Sqrt of 'a expr
| Sin of 'a expr
| Cos of 'a expr
| Tan of 'a expr
| Cot of 'a expr
| Atan2 of 'a expr * 'a expr
| Conj of 'a expr
type 'a variable = Real of 'a | Complex of 'a
type 'a variable_array = Real_Array of 'a | Complex_Array of 'a
type 'a parameters =
{ input : ('a * float) list;
derived : ('a variable * 'a expr) list;
derived_arrays : ('a variable_array * 'a expr list) list }
(* \thocwmodulesection{More Exotic Couplings}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim5_Scalar_Vector_Vector_T]:
$\mathcal{L}_I=g\phi
(\ii\partial_\mu V_1^\nu)(\ii\partial_\nu V_2^\mu)$}\\\hline
[F23] & $\phi(k_2+k_3)\leftarrow\ii\cdot g
k_3^\mu V_{1,\mu}(k_2) k_2^\nu V_{2,\nu}(k_3)$ \\\hline
[F32] & $\phi(k_2+k_3)\leftarrow\ii\cdot g
k_2^\mu V_{2,\mu}(k_3) k_3^\nu V_{1,\nu}(k_2)$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu \phi(k_1) (-k_1^\nu-k_2^\nu) V_{1,\nu}(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (-k_1^\nu-k_2^\nu)V_{1,\nu}(k_2) \phi(k_1)$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu \phi(k_1) (-k_1^\nu-k_3^\nu)V_{2,\nu}(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (-k_1^\nu-k_3^\nu)V_{2,\nu}(k_3) \phi(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-scalar-vector-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim6_Vector_Vector_Vector_T]:
$\mathcal{L}_I=gV_1^\mu
((\ii\partial_\nu V_2^\rho)%
\ii\overleftrightarrow{\partial_\mu}
(\ii\partial_\rho V_3^\nu))$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\mu - k_3^\mu) k_3^\nu V_{2,\nu} (k_2)
k_2^\rho V_{3,\rho}(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\mu - k_3^\mu) k_2^\nu V_{3,\nu} (k_3)
k_3^\rho V_{2,\rho}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (k_1^\nu+2k_2^\nu) V_{1,\nu} (k_1)
(-k_1^\rho-k_2^\rho) V_{2,\rho}(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (-k_1^\rho-k_2^\rho) V_{2,\rho}(k_2)
(k_1^\nu+2k_2^\nu) V_{1,\nu} (k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (k_1^\nu+2k_3^\nu) V_{1,\nu} (k_1)
(-k_1^\rho-k_3^\rho) V_{3,\rho}(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (-k_1^\rho-k_3^\rho) V_{3,\rho}(k_3)
(k_1^\nu+2k_3^\nu) V_{1,\nu} (k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim6-vector-vector-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Tensor_2_Vector_Vector]:
$\mathcal{L}_I=gT^{\mu\nu}
(V_{1,\mu}V_{2,\nu} + V_{1,\nu}V_{2,\mu})$}\\\hline
[F23] & $T^{\mu\nu}(k_2+k_3)\leftarrow\ii\cdot g
(V_{1,\mu}(k_2) V_{2,\nu}(k_3) + V_{1,\nu}(k_2) V_{2,\mu}(k_3))$ \\\hline
[F32] & $T^{\mu\nu}(k_2+k_3)\leftarrow\ii\cdot g
(V_{2,\nu}(k_3) V_{1,\mu}(k_2) + V_{2,\mu}(k_3) V_{1,\nu}(k_2))$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
(T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1)) V_{1,\nu}(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
V_{1,\nu}(k_2)(T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1))$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
(T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1)) V_{2,\nu}(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
V_{2,\nu}(k_3) (T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1))$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:tensor2-vector-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim5_Tensor_2_Vector_Vector_1]:
$\mathcal{L}_I=gT^{\alpha\beta}
(V_1^\mu
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta V_{2,\mu})$}\\\hline
[F23] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
V_1^\mu(k_2)V_{2,\mu}(k_3)$ \\\hline
[F32] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
V_{2,\mu}(k_3)V_1^\mu(k_2)$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
T_{\alpha\beta}(k_1) V_1^\mu(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
V_1^\mu(k_2) T_{\alpha\beta}(k_1)$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
T_{\alpha\beta}(k_1) V_2^\mu(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
V_2^\mu(k_3) T_{\alpha\beta}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-tensor2-vector-vector-1}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim5_Tensor_2_Vector_Vector_2]:
$\mathcal{L}_I=gT^{\alpha\beta}
( V_1^\mu \ii\overleftrightarrow\partial_\beta (\ii\partial_\mu V_{2,\alpha})
+ V_1^\mu \ii\overleftrightarrow\partial_\alpha (\ii\partial_\mu V_{2,\beta}))
$}\\\hline
[F23] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_3^\beta-k_2^\beta) k_3^\mu V_{1,\mu}(k_2)V_2^\alpha(k_3)
+ (\alpha\leftrightarrow\beta)$ \\\hline
[F32] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_3^\beta-k_2^\beta) V_2^\alpha(k_3) k_3^\mu V_{1,\mu}(k_2)
+ (\alpha\leftrightarrow\beta)$ \\\hline
[F12] & $V_2^\alpha(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\beta+2k_2^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))
(k_1^\mu+k_2^\mu) V_{1,\mu}(k_2)$ \\\hline
[F21] & $V_2^\alpha(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\mu+k_2^\mu) V_{1,\mu}(k_2)
(k_1^\beta+2k_2^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))$ \\\hline
[F13] & $V_1^\alpha(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\beta+2k_3^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))
(k_1^\mu+k_3^\mu) V_{2,\mu}(k_3)$ \\\hline
[F31] & $V_1^\alpha(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\mu+k_3^\mu) V_{2,\mu}(k_3)
(k_1^\beta+2k_3^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-tensor2-vector-vector-1'}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim7_Tensor_2_Vector_Vector_T]:
$\mathcal{L}_I=gT^{\alpha\beta}
((\ii\partial^\mu V_1^\nu)
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta
(\ii\partial_\nu V_{2,\mu}))$}\\\hline
[F23] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
k_3^\mu V_{1,\mu}(k_2) k_2^\nu V_{2,\nu}(k_3)$ \\\hline
[F32] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
k_2^\nu V_{2,\nu}(k_3) k_3^\mu V_{1,\mu}(k_2)$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
T_{\alpha\beta}(k_1) (-k_1^\nu-k_2^\nu)V_{1,\nu}(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (-k_1^\nu-k_2^\nu)V_{1,\nu}(k_2)
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
T_{\alpha\beta}(k_1)$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
T_{\alpha\beta}(k_1) (-k_1^\nu-k_3^\nu) V_{2,\nu}(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (-k_1^\nu-k_3^\nu) V_{2,\nu}(k_3)
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
T_{\alpha\beta}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim7-tensor2-vector-vector-T}
\ldots}
\end{table} *)
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)

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