Splitting the vertices with 4 colored particles

The CompHEP Lagrangian tables do not describe explicitly the color structure of a vertex. If color particles are present in the vertex, the following implicit contractions are assumed (supposing $p,q,r$ are color indices of particles in the vertex):

• $\delta_{pq}$ for two color particles $A^1_p$, $A^2_q$;
• $\lambda_{pq}^r$ for three particles, which are color triplet, antitriplet and octet;
• $f^{pqr}$ for three color octets.

Other color structures are forbidden in CompHEP.

So, to introduce the 4-gluon vertex $f^{pqr}G_\mu^qG_\nu^rf^{pst}G_\mu^sG_\nu^t$ one should split this 4-legs vertex into 3-legs vertices $f^{pqr}G_\mu^qG_\nu^rX_{\mu\nu}^p$:

Here the field $X_{\mu\nu}^p$ is a Lorenz tensor and color octet, and this field has constant propagator. If gluon name in CompHEP is 'G', the name 'G.t' is used for this tensor particle; its indices denoted as 'm_' and 'M_' ('_' is the number of the particle in table item).

The described transformation is performed by LanHEP automatically and transparently for the user. Each vertex containing 4 color particles is split to 2 vertices which are joined by an automatically generated auxiliary field.

The same technique is applied in the MSSM where more vertices with 4 color particles appear: vertices with 2 gluons and 2 squarks and vertices with 4 squarks. However, the large amount of vertices with 4 squarks requires many auxiliary fields, which can easily break CompHEP limitations on the particles number. It is possible however to reduce significantly the number of vertices and auxiliary fields if one introduce auxiliary fields at the level of multiplets.

The vertices with 4 squarks come from $DD$ and $F^*F$ terms. For example, there is the term $\frac{1}{2}D^a_G D^a_G$ in the Lagrangian,

$$D^a_G = g_s(Q_i^*\lambda^a Q_i + D^*\lambda^aD + D^*\lambda^aD),$$

where $Q,D,U$ are squarks multiplets, and $\lambda$ is Gell-Mann matrix. Instead of evaluating this expression and that splitting all vertices independently, one can introduce one color octet auxiliary field $\xi^a$ and write this Lagrangian term as $D^a_G \xi^a$.

Other $DD$ terms contain both color and colorless particles. Thus, the term $D_A^iD_A^i$ with

$$D_A^i= g_1(Q^* T^i Q + L^* T^i L + H_1^* T^i H_1 + H_2^* T^i H_2),$$

can be represented as

$$g_1(Q^* T^i Q)\xi^i + g_1^2(Q^* T^i Q)(L^* T^i L + H_1^* T^i... ...* T^i H_2) + g_1^2(L^* T^i L + H_1^* T^i H_1 + H_2^* T^i H_2)^2$$

where $\xi^i$ is the triplet of auxiliary fields. This terms can also be written in the another form:

$$g_1(Q^* T^i Q + L^* T^i L)\xi^i + g_1^2(Q^* T^i Q + L^* T^i L... ... H_1 + H_2^* T^i H_2) + g_1^2(H_1^* T^i H_1 + H_2^* T^i H_2)^2,$$

where all vertices with 4 scalars (except vertices with Higgs particles) are splitted. Although the latter splitting is not obligatory, it can reduce significantly the amount of vertices.

The similar technique is applicable to the $F^*F$ terms, with the transformation $F_i^*F_i\rightarrow F_i^*\xi_i^* + F_i\xi_i$.

Thus, we distinguish two types of vertices splitting: splitting at multiplet level and splitting at vertices level. Note that splitting the vertices with two gluons and two squarks must be done at vertices level after combining the similar terms, otherwise they would contain the elements of squark mixing matrices.

The vertices splitting at multiplet level is implemented in LanHEP mainly for MSSM needs. The first case refers to $DD$ terms. The user should declare several let-substitutions and then put in lterm statement the squared sum:

let a1=g*Q*tau*q/2,
a2=g*L*tau*l/2,
a3=g*H1*tau*h1/2,
a4=g*H2*tau*h2/2.
lterm - ( a1 + a2 + a3 + a4 ) ** 2 / 2.

In this case LanHEP looks for the square of the sum of several let-substitution symbols, each containing two color or merely scalar particles. If such expression is found, it is replaced as in the previous formulas.

The vertices splitting in $F^*F$ terms is performed by dfdfc function (see previous section). After taking the variational derivative the monomials with two color or scalar particles (except Higgs ones) are multiplied by auxiliary fields, thus mediating the vertices with 4 color (scalar) particles.

The multiplet level vertices splitting is controlled by the statement

option SplitCol1=N.

where N is a number:

-1

remove all vertices with 4 color particles from Lagrangian;

0

turn off multiplet level vertices splitting;

1

allows vertices splitting with 4 color multiplets;

2

allows vertices splitting with any 4 scalar multiplets except Higgs ones (more generally, any multiplets containing vev's).

The value of this option can be set to different values before executing different lterm statements.

The vertices level splitting is performed after combining similar terms of the Lagrangian. This splitting can be controlled by the statement

option SplitCol2=N.

where N is a number:

0

disable vertex level splitting;

1

enable vertex level splitting (only for vertices with 4 color particles).

For CompHEP output, the default value is 2 for SplitCol1 and 1 for SplitCol2. For LaTeX output, default value is 0 for both options.