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End of NBO

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-05 17:49:06 +02:00
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We report the full expression of the overlap with two derivatives considered in the main text.
It corresponds to the colour ordered amplitude of two tachyons and one level-2 massive state:
\begin{equation}
\begin{split}
K
& =
\cN^2
\int \dd[D]{x}\,
\sqrt{-\det g}
\\
& \times
\Biggl[
u^{-3}\, \ffs^{(-3)}_{\qty{\cS};\, \kmkrN{i}}
+
u^{-2}\, \ffs^{(-2)}_{\qty{\cS};\, \kmkrN{i}}
\\
& +
u^{-1}\, \ffs^{(-1)}_{\qty{\cS};\, \kmkrN{i}}
+
\ffs^{(0)}_{\qty{\cS};\, \kmkrN{i}}
\\
& +
u\, \ffs^{(1)}_{\qty{\cS};\, \kmkrN{i}}
\Biggr]~
\prod_{j = 1}^3 \phi_{\kmkrN{j}}
\end{split}
\end{equation}
where $i = 1,\, 2,\, 3$ and:
\begin{equation}
\begin{split}
\ffs^{(-3)}_{\qty{\cS},\, \kmkrN{i}}
& =
\Biggl(
-
\frac{%
k_{\qty(2)\, +}^4\, l_{\qty(3)}^4
-
4\, k_{\qty(2)\, +}^3\, k_{\qty(3)\, +}\, l_{\qty(2)}\, l_{\qty(3)}^3
}{%
4\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^4\, \Delta^3
}
\\
& -
\frac{%
6\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^2\, l_{\qty(2)}^2\,l_{\qty(3)}^2 + k_{\qty(3)\, +}^4\, l_{\qty(2)}^4
}{%
4\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^4\, \Delta^3
}
\Biggr)\,
\cS_{v\, v},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\ffs^{(-2)}_{\qty{\cS},\, \kmkrN{i}}
& =
\Biggl(
-
\frac{%
3 i\, \qty(%
k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}\, l_{\qty(3)}^2
+
k_{\qty(2)\, +}^3\, l_{\qty(3)}^2
)
}{%
2\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}^3\, \Delta
}
\\
& +
\frac{%
i\, \qty(%
2\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}^2\, l_{\qty(2)}\, l_{\qty(3)}
+
3\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}\, l_{\qty(2)}\, l_{\qty(3)}
)
}{%
k_{\qty(2)\, +}\, k_{\qty(3)\, +}^3\, \Delta
}
\\
& -
\frac{%
3 i\, \qty(%
k_{\qty(3)\, +}^3\, l_{\qty(2)}^2
+
k_{\qty(2)\, +}\, k_{\qty(3)\, +}^2\, l_{\qty(2)}^2
)
}{%
2\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}^3\, \Delta
}
\Biggr)\,
\cS_{v\, v}
\\
& -
\qty(%
\frac{%
l_{\qty(3)}\,
\qty(%
k_{\qty(2)\, +}^2\, l_{\qty(3)}^2-3\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}\, l_{\qty(2)}\, l_{\qty(3)}
+
3\, k_{\qty(3)\, +}^2\, l_{\qty(2)}^2
)
}{%
k_{\qty(3)\, +}^3\, \Delta^2
}
)\,
\cS_{v\, z},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\ffs^{(-1)}_{\qty{\cS},\, \kmkrN{i}}
& =
\qty(%
-
\frac{%
\qty(%
k_{\qty(2)\, +}\, l_{\qty(3)}
-
k_{\qty(3)\, +}\, l_{\qty(2)}
)^2
}{%
k_{\qty(3)\, +}^2\, \Delta
}
)\,
\cS_{u\, v}
\\
& +
\Biggl(
-
\frac{%
k_{\qty(2)\, +}^2\, l_{\qty(3)}^2\, \qty(r_{(2)} + \norm{\vec{k}_{(2)}}^2)\,
+
k_{\qty(3)\, +}^2\, l_{\qty(2)}^2\, \qty(r_{(2)} + \norm{\vec{k}_{(2)}}^2)\,
}{%
2\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^2\, \Delta
}
\\
& +
\frac{%
2\, k_{\qty(2)\, +}^3\, k_{\qty(3)\, +}\, l_{\qty(2)}\, l_{\qty(3)}
}{%
k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^2\, \Delta
}
\\
& +
\frac{%
3\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^2\, \Delta
6\, k_{\qty(2)\, +}^3\, k_{\qty(3)\, +}\, \Delta
3\, k_{\qty(2)\, +}^4\, \Delta
}{%
4\, k_{\qty(2)\, +}^2\, k_{\qty(3)\, +}^2
}
\Biggr)\,
\cS_{v\, v}
\\
& -
\Biggl(%
\frac{%
i\, \qty(%
3\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}\, l_{\qty(3)}
+
3\, k_{\qty(2)\, +}^2\, l_{\qty(3)}
)
}{%
k_{\qty(3)\, +}^2
}
\\
& +
\frac{%
i\, \qty(%
2\, k_{\qty(3)\, +}^2\, l_{\qty(2)}
+
3\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}\, l_{\qty(2)}
)
}{%
k_{\qty(3)\, +}^2
}
\Biggr)\,
\cS_{v\, z}
\\
& +
\qty(%
\frac{%
k_{\qty(2)\, i}\, l_{\qty(3)}\,
\qty(%
k_{\qty(2)\, +}\, l_{\qty(3)}
-
2\, k_{\qty(3)\, +}\, l_{\qty(2)}
)
}{%
k_{\qty(3)\, +}^2\, \Delta
}
)\,
\cS_{v\,{i}}
\\
& +
\qty(%
-
\frac{%
\qty(%
k_{\qty(2)\, +}\, l_{\qty(3)}
-
k_{\qty(3)\, +}\, l_{\qty(2)}
)^2
}{%
k_{\qty(3)\, +}^2\, \Delta
}
)\,
\cS_{z\, z},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\ffs^{(0)}_{\qty{\cS},\, \kmkrN{i}}
& =
\qty(%
-\frac{%
i\, k_{\qty(2)\, +}\, \qty(k_{\qty(3)\, +} + k_{\qty(2)\, +})\, \Delta
}{%
k_{\qty(3)\, +}
}
)\,
\cS_{u\, v}
\\
& +
\qty(%
-
\frac{%
2\, k_{\qty(2)\, +}\,
\qty(%
k_{\qty(2)\, +}\, l_{\qty(3)}
-
k_{\qty(3)\, +}\, l_{\qty(2)}
)
}{%
k_{\qty(3)\, +}
}
)\,
\cS_{u\, z}
\\
& +
\qty(%
-
\frac{%
i\, \qty(%
k_{\qty(3)\, +}
+
k_{\qty(2)\, +})\, \Delta\, \qty(r_{(2)} + \norm{\vec{k}_{(2)}}^2)
}{%
2\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}
}
)\,
\cS_{v\, v}
\\
& +
\qty(%
-
\frac{%
l_{\qty(3)}\, \qty(r_{(2)} + \norm{\vec{k}_{(2)}}^2)
-
2\, k_{\qty(2)\, +}\, k_{\qty(3)\, +}\, l_{\qty(2)}
}{%
k_{\qty(3)\, +}
}
)\,
\cS_{v\, z}
\\
& +
\qty(%
\frac{%
i\, k_{\qty(2)\, i}\, k_{\qty(2)\, +}\, \Delta
}{%
k_{\qty(3)\, +}
}
)\,
\cS_{v\,{i}}
\\
& +
\qty(%
-
\frac{%
i\, k_{\qty(2)\, +}\, \qty(k_{\qty(3)\, +} + k_{\qty(2)\, +})\, \Delta
}{%
k_{\qty(3)\, +}
}
)\,
\cS_{z\, z}
\\
& +
\qty(%
\frac{%
2\, k_{\qty(2)\, i}\,
\qty(%
k_{\qty(2)\, +}\, l_{\qty(3)}
-
k_{\qty(3)\, +}\, l_{\qty(2)}
)
}{%
k_{\qty(3)\, +}
}
)\,
\cS_{z\,{i}},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\ffs^{(1)}_{\qty{\cS},\, \kmkrN{i}}
& =
\qty(%
-k_{\qty(2)\, +}^2\, \Delta
)\,
\cS_{u\, u}
\\
& +
\qty(%
-\Delta\, \qty(r_{(2)} + \norm{\vec{k}_{(2)}}^2)\,
)\,
\cS_{u\, v}
\\
& +
\qty(%
2\, k_{\qty(2)\, i}\, k_{\qty(2)\, +}\, \Delta
)\,
\cS_{u\,{i}}
\\
& +
\qty(%
-
\frac{%
\Delta\, \qty(r_{(2)} + \norm{\vec{k}_{(2)}}^2)^2
}{%
4\, k_{\qty(2)\, +}^2
}
)\,
\cS_{v\, v}
\\
& +
\qty(%
2\, k_{\qty(2)\, i}\, k_{\qty(2)\, +}\, \Delta
)\,
\cS_{v\,{i}}
\\
& +
\qty(- k_{\qty(2)\, i} k_{\qty(2)\, j}\, \Delta)\,
\cS_{{i}\,{j}}.
\end{split}
\end{equation}