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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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commit f97b9f87a1
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349
sec/app/massive.tex Normal file
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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}

292
sec/app/tensor_wave.tex Normal file
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For the sake of completeness we report the expression of the full \nbo tensor wave function.
In what follows $L = \frac{l}{k_+}$.
We have
\begin{equation}
\begin{split}
\mqty(
S_{u\, u}
\\
S_{u\, v}
\\
S_{u\, z}
\\
S_{u\, i}
\\
S_{v\, v}
\\
S_{v\, z}
\\
S_{v\, i}
\\
S_{z\, z}
\\
S_{z\, i}
\\
S_{i\, i}
)
& =
\Biggl\lbrace
\cS_{u\, u}
\mqty(
1
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
+
\cS_{u\, v}
\mqty(
\frac{i}{k_+\, u} + \frac{L^2}{\Delta^2\, u^2}
\\
1
\\
L
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
+
\cS_{u\, z}
\mqty(
\frac{2\, L}{\Delta\, u}
\\
0
\\
\Delta\, u
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
+
\cS_{u\, i}
\mqty(
0
\\
0
\\
0
\\
1
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
\\
& +
\cS_{v\, v}
\mqty(
-\frac{3}{4\, k_+^2\, u^2}
+
\frac{3\, i\, L^2}{2\, \Delta^2\, k_+\, u^3}
+
\frac{L^4}{4\, \Delta^4\, u^4}
\\
\frac{i}{2\, k_+\, u}
+
\frac{L^2}{2\, \Delta^2\, u^2}
\\
\frac{3\, i\, L}{2\, k_+\, u}
+
\frac{L^3}{2\, \Delta^2\, u^2}
\\
0
\\
1
\\
L
\\
0
\\
\frac{i\, \Delta^2\, u}{k_+}
+
L^2
\\
0
\\
0
\\
)\,
+
\cS_{v\, z}
\mqty(
\frac{3\, i\, L}{\Delta\, k_+\, u^2}
+
\frac{L^3}{\Delta^3\, u^3}
\\
\frac{L}{\Delta\, u}
\\
\frac{3\, L^2}{2\, \Delta\, u}
+
\frac{3\, i\, \Delta}{2\, k_+}
\\
0
\\
0
\\
\Delta\, u
\\
0
\\
2\, \Delta\, L\, u
\\
0
\\
0
\\
)\,
\\
& +
\cS_{v\, i}
\mqty(
0
\\
0
\\
0
\\
\frac{i}{2\, k_+\, u}
+
\frac{L^2}{2\, \Delta^2\, u^2}
\\
0
\\
0
\\
1
\\
0
\\
L
\\
0
\\
)\,
+
\cS_{z\, z}
\mqty(
\frac{i}{k_+\, u}
+
\frac{L^2}{\Delta^2\, u^2}
\\
0
\\
L
\\
0
\\
0
\\
0
\\
0
\\
\Delta^2\, u^2
\\
0
\\
0
\\
)\,
+
\cS_{z\, i}
\mqty(
0
\\
0
\\
0
\\
\frac{L}{\Delta\, u}
\\
0
\\
0
\\
0
\\
0
\\
\Delta\, u
\\
0
\\
)\,
\\
& +
\cS_{i\, j}
\mqty(
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
\delta_{i j}
\\
)\,
\Biggr\rbrace
\phi_{\kmkr}.
\end{split}
\end{equation}

2
sec/part1/dbranes.tex
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@@ -2049,7 +2049,7 @@ Explicitly we impose the four real equations in spinorial formalism
f_{{\bart+1}\, (s)} - f_{{\bart-1}\, (s)},
\end{equation}
where we used the mapping~\eqref{eq:def_omega} to write the integrals in the $\omega$ variables.
This equation has enough degrees of freedom to fix completely the two complex parameters $C_1$ and $C_2$.
This equation has enough \dof to fix completely the two complex parameters $C_1$ and $C_2$.
The final generic solution is thus uniquely determined.

2
sec/part1/introduction.tex
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@@ -1280,7 +1280,7 @@ The field $\cA^a$ forms a vector representation of the group \SO{D-1-p} and from
\label{fig:dbranes:chanpaton}
\end{figure}
It is also possible to add non dynamical degrees of freedom to the open string endpoints.
It is also possible to add non dynamical degrees of freedom (\dof) to the open string endpoints.
They are known as \emph{Chan-Paton factors}~\cite{Paton:1969:GeneralizedVenezianoModel}.
They have no dynamics and do not spoil Poincaré or conformal invariance in the action of the string.
Each state can then be labelled by $i$ and $j$ running from $1$ to $N$.

2210
sec/part2/divergences.tex
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