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Дата изменения: Tue Aug 15 14:53:38 2000
Дата индексирования: Mon Oct 1 22:47:37 2012
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Smoothing

The general idea of the integrand smoothing is trivial. Let us need to evaluate

$$\int_a^b F(x)dx \;\;\;,$$ (7)

and let $F(x)$ have a peak like $f(x)$, where $f(x)$ is a simple symbolically integrable function in contrast to $F(x)$:

$$g(x)=\int_a^x f(x') dx'\;.$$ (8)

Now we may represent the integral (7) as

$$\int_a^b F(x)dx = \int_0^{g(b)}dy\frac{F(g^{-1}(y))}{f(g^{-1}(y))},$$ (9)

where $g^{-1}(y)$ is the inverse function for $g(x)$. The integrand is a smooth function now.

We face very often squared matrix elements which have several poles in one of variables. For example, the $\gamma \rightarrow b,\bar{b}$, $Z \rightarrow b,\bar{b}$ and $H \rightarrow b,\bar{b}$ virtual subprocesses may contribute just to the same amplitude. Although in this case we can evaluate the integral function $g(x)$ symbolically, the inverse function $g^{-1}(y)$ can be computed only as a numerical solution of the corresponding equation. To bypass the calculation of inverse function CompHEP  uses the multi-channel Monte Carlo (branching) method to smooth a sum of peaks.

The idea of the branching method is the following. Let $F(x)$ have two peaks, one is similar to $f_1(x)$ and another to $f_2(x)$. $f_1(x)$ and $f_2(x)$ are singular but elementary functions. Then, instead of one integration (7), we could perform two ones:

$$\int F(x)dx= \int \frac{F(x)f_1(x)}{f_1(x)+f_2(x)} dx + \int \frac{F(x)f_2(x)}{f_1(x)+f_2(x)} dx\;,$$ (10)

but now each integration has only a single peak! It is easy to extend this method for an arbitrary number of peaks.

The branching method was used in [Berends-1995] to separate peaks which came from various diagrams. In that paper there was also proposed to use the expression (10) where $f_i(x)$ is replaced by $\alpha_i f_i(x)$ with a subsequent search for optimal coefficients $\alpha_i$. CompHEP  passes on this weight optimization to Vegas, combining two integrals in one Vegas  hypercube.

As was mentioned above, CompHEP  automatically searches for a polar vector for each angle integration in order to reach a linear relation between $cos\Theta$ and one of the squared sum of momenta which is responsible for the peak. It could happen that various peaks need different polar vectors for the same decay. In this case CompHEP  uses the branching method again, but now for the whole two-dimension sphere integration. In other words, we use the branching equation (10) where $x$ is the two dimensional sphere angle [Ilyin-1996,Kovalenko-1997].