End of NBO

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

sciencestuff.sty

@@ -47,6 +47,7 @@
 \numberwithin{table}{section}
 
 %---- abbreviations
+
 \providecommand{\sm}{\textsc{sm}\xspace}
 \providecommand{\eom}{\textsc{e.o.m.}\xspace}
 \providecommand{\cft}{\textsc{CFT}\xspace}
@@ -54,10 +55,12 @@
 \providecommand{\qed}{\textsc{QED}\xspace}
 \providecommand{\qcd}{\textsc{QCD}\xspace}
 \providecommand{\ope}{\textsc{o.p.e.}\xspace}
+\providecommand{\dof}{\textsc{d.o.f.}\xspace}
 \providecommand{\cy}{\textsc{CY}\xspace}
 \providecommand{\lhs}{\textsc{lhs}\xspace}
 \providecommand{\rhs}{\textsc{rhs}\xspace}
 \providecommand{\ap}{\ensuremath{\alpha'}\xspace}
+\providecommand{\sgn}{\ensuremath{\mathrm{sign}}}
 
 %---- remap greek letters
 
@@ -256,6 +259,45 @@
 \providecommand{\hPsi}{\ensuremath{\widehat{\Uppsi}}\xspace}
 \providecommand{\hOmega}{\ensuremath{\widehat{\Upomega}}\xspace}
 
+\providecommand{\ualpha}{\ensuremath{\underline{\upalpha}}\xspace}
+\providecommand{\ubeta}{\ensuremath{\underline{\upbeta}}\xspace}
+\providecommand{\ugamma}{\ensuremath{\underline{\upgamma}}\xspace}
+\providecommand{\udelta}{\ensuremath{\underline{\updelta}}\xspace}
+\providecommand{\uepsilon}{\ensuremath{\underline{\upepsilon}}\xspace}
+\providecommand{\uzeta}{\ensuremath{\underline{\upzeta}}\xspace}
+\providecommand{\ueta}{\ensuremath{\underline{\upeta}}\xspace}
+\providecommand{\utheta}{\ensuremath{\underline{\uptheta}}\xspace}
+\providecommand{\uiota}{\ensuremath{\underline{\upiota}}\xspace}
+\providecommand{\ukappa}{\ensuremath{\underline{\upkappa}}\xspace}
+\providecommand{\ulambda}{\ensuremath{\underline{\uplambda}}\xspace}
+\providecommand{\umu}{\ensuremath{\underline{\upmu}}\xspace}
+\providecommand{\unu}{\ensuremath{\underline{\upnu}}\xspace}
+\providecommand{\uxi}{\ensuremath{\underline{\upxi}}\xspace}
+\providecommand{\upi}{\ensuremath{\underline{\uppi}}\xspace}
+\providecommand{\urho}{\ensuremath{\underline{\uprho}}\xspace}
+\providecommand{\usigma}{\ensuremath{\underline{\upsigma}}\xspace}
+\providecommand{\utau}{\ensuremath{\underline{\uptau}}\xspace}
+\providecommand{\uupsilon}{\ensuremath{\underline{\upupsilon}}\xspace}
+\providecommand{\uphi}{\ensuremath{\underline{\upphi}}\xspace}
+\providecommand{\uchi}{\ensuremath{\underline{\upchi}}\xspace}
+\providecommand{\upsi}{\ensuremath{\underline{\uppsi}}\xspace}
+\providecommand{\uomega}{\ensuremath{\underline{\upomega}}\xspace}
+\providecommand{\uvarepsilon}{\ensuremath{\underline{\upvarepsilon}}\xspace}
+\providecommand{\uvartheta}{\ensuremath{\underline{\upvartheta}}\xspace}
+\providecommand{\uvarpi}{\ensuremath{\underline{\upvarpi}}\xspace}
+\providecommand{\uvarphi}{\ensuremath{\underline{\upvarphi}}\xspace}
+\providecommand{\uGamma}{\ensuremath{\underline{\Upgamma}}\xspace}
+\providecommand{\uDelta}{\ensuremath{\underline{\Updelta}}\xspace}
+\providecommand{\uTheta}{\ensuremath{\underline{\Uptheta}}\xspace}
+\providecommand{\uLambda}{\ensuremath{\underline{\Uplambda}}\xspace}
+\providecommand{\uXi}{\ensuremath{\underline{\Upxi}}\xspace}
+\providecommand{\uPi}{\ensuremath{\underline{\Uppi}}\xspace}
+\providecommand{\uSigma}{\ensuremath{\underline{\Upsigma}}\xspace}
+\providecommand{\uUpsilon}{\ensuremath{\underline{\Upupsilon}}\xspace}
+\providecommand{\uPhi}{\ensuremath{\underline{\Upphi}}\xspace}
+\providecommand{\uPsi}{\ensuremath{\underline{\Uppsi}}\xspace}
+\providecommand{\uOmega}{\ensuremath{\underline{\Upomega}}\xspace}
+
 %---- numerical sets
 
 \providecommand{\1}{\ensuremath{\mathds{1}}\xspace}
@@ -479,6 +521,59 @@
 \providecommand{\hatY}{\ensuremath{\widehat{Y}}\xspace}
 \providecommand{\hatZ}{\ensuremath{\widehat{Z}}\xspace}
 
+\providecommand{\undera}{\ensuremath{\underline{a}}\xspace}
+\providecommand{\underb}{\ensuremath{\underline{b}}\xspace}
+\providecommand{\underc}{\ensuremath{\underline{c}}\xspace}
+\providecommand{\underd}{\ensuremath{\underline{d}}\xspace}
+\providecommand{\undere}{\ensuremath{\underline{e}}\xspace}
+\providecommand{\underf}{\ensuremath{\underline{f}}\xspace}
+\providecommand{\underg}{\ensuremath{\underline{g}}\xspace}
+\providecommand{\underh}{\ensuremath{\underline{h}}\xspace}
+\providecommand{\underi}{\ensuremath{\underline{i}}\xspace}
+\providecommand{\underj}{\ensuremath{\underline{j}}\xspace}
+\providecommand{\underk}{\ensuremath{\underline{k}}\xspace}
+\providecommand{\underl}{\ensuremath{\underline{l}}\xspace}
+\providecommand{\underm}{\ensuremath{\underline{m}}\xspace}
+\providecommand{\undern}{\ensuremath{\underline{n}}\xspace}
+\providecommand{\undero}{\ensuremath{\underline{o}}\xspace}
+\providecommand{\underp}{\ensuremath{\underline{p}}\xspace}
+\providecommand{\underq}{\ensuremath{\underline{q}}\xspace}
+\providecommand{\underr}{\ensuremath{\underline{r}}\xspace}
+\providecommand{\unders}{\ensuremath{\underline{s}}\xspace}
+\providecommand{\undert}{\ensuremath{\underline{t}}\xspace}
+\providecommand{\underu}{\ensuremath{\underline{u}}\xspace}
+\providecommand{\underv}{\ensuremath{\underline{v}}\xspace}
+\providecommand{\underw}{\ensuremath{\underline{w}}\xspace}
+\providecommand{\underx}{\ensuremath{\underline{x}}\xspace}
+\providecommand{\undery}{\ensuremath{\underline{y}}\xspace}
+\providecommand{\underz}{\ensuremath{\underline{z}}\xspace}
+\providecommand{\underA}{\ensuremath{\underline{A}}\xspace}
+\providecommand{\underB}{\ensuremath{\underline{B}}\xspace}
+\providecommand{\underC}{\ensuremath{\underline{C}}\xspace}
+\providecommand{\underD}{\ensuremath{\underline{D}}\xspace}
+\providecommand{\underE}{\ensuremath{\underline{E}}\xspace}
+\providecommand{\underF}{\ensuremath{\underline{F}}\xspace}
+\providecommand{\underG}{\ensuremath{\underline{G}}\xspace}
+\providecommand{\underH}{\ensuremath{\underline{H}}\xspace}
+\providecommand{\underI}{\ensuremath{\underline{I}}\xspace}
+\providecommand{\underJ}{\ensuremath{\underline{J}}\xspace}
+\providecommand{\underK}{\ensuremath{\underline{K}}\xspace}
+\providecommand{\underL}{\ensuremath{\underline{L}}\xspace}
+\providecommand{\underM}{\ensuremath{\underline{M}}\xspace}
+\providecommand{\underN}{\ensuremath{\underline{N}}\xspace}
+\providecommand{\underO}{\ensuremath{\underline{O}}\xspace}
+\providecommand{\underP}{\ensuremath{\underline{P}}\xspace}
+\providecommand{\underQ}{\ensuremath{\underline{Q}}\xspace}
+\providecommand{\underR}{\ensuremath{\underline{R}}\xspace}
+\providecommand{\underS}{\ensuremath{\underline{S}}\xspace}
+\providecommand{\underT}{\ensuremath{\underline{T}}\xspace}
+\providecommand{\underU}{\ensuremath{\underline{U}}\xspace}
+\providecommand{\underV}{\ensuremath{\underline{V}}\xspace}
+\providecommand{\underW}{\ensuremath{\underline{W}}\xspace}
+\providecommand{\underX}{\ensuremath{\underline{X}}\xspace}
+\providecommand{\underY}{\ensuremath{\underline{Y}}\xspace}
+\providecommand{\underZ}{\ensuremath{\underline{Z}}\xspace}
+
 %---- calligraphic letters
 
 \providecommand{\cA}{\ensuremath{\mathcal{A}}\xspace}
@@ -751,6 +846,60 @@
 \providecommand{\bccY}{\ensuremath{\overline{\mathscr{Y}}}\xspace}
 \providecommand{\bccZ}{\ensuremath{\overline{\mathscr{Z}}}\xspace}
 
+\providecommand{\ucA}{\ensuremath{\underline{\mathcal{A}}}\xspace}
+\providecommand{\ucB}{\ensuremath{\underline{\mathcal{B}}}\xspace}
+\providecommand{\ucC}{\ensuremath{\underline{\mathcal{C}}}\xspace}
+\providecommand{\ucD}{\ensuremath{\underline{\mathcal{D}}}\xspace}
+\providecommand{\ucE}{\ensuremath{\underline{\mathcal{E}}}\xspace}
+\providecommand{\ucF}{\ensuremath{\underline{\mathcal{F}}}\xspace}
+\providecommand{\ucG}{\ensuremath{\underline{\mathcal{G}}}\xspace}
+\providecommand{\ucH}{\ensuremath{\underline{\mathcal{H}}}\xspace}
+\providecommand{\ucI}{\ensuremath{\underline{\mathcal{I}}}\xspace}
+\providecommand{\ucJ}{\ensuremath{\underline{\mathcal{J}}}\xspace}
+\providecommand{\ucK}{\ensuremath{\underline{\mathcal{K}}}\xspace}
+\providecommand{\ucL}{\ensuremath{\underline{\mathcal{L}}}\xspace}
+\providecommand{\ucM}{\ensuremath{\underline{\mathcal{M}}}\xspace}
+\providecommand{\ucN}{\ensuremath{\underline{\mathcal{N}}}\xspace}
+\providecommand{\ucO}{\ensuremath{\underline{\mathcal{O}}}\xspace}
+\providecommand{\ucP}{\ensuremath{\underline{\mathcal{P}}}\xspace}
+\providecommand{\ucQ}{\ensuremath{\underline{\mathcal{Q}}}\xspace}
+\providecommand{\ucR}{\ensuremath{\underline{\mathcal{R}}}\xspace}
+\providecommand{\ucS}{\ensuremath{\underline{\mathcal{S}}}\xspace}
+\providecommand{\ucT}{\ensuremath{\underline{\mathcal{T}}}\xspace}
+\providecommand{\ucU}{\ensuremath{\underline{\mathcal{U}}}\xspace}
+\providecommand{\ucV}{\ensuremath{\underline{\mathcal{V}}}\xspace}
+\providecommand{\ucW}{\ensuremath{\underline{\mathcal{W}}}\xspace}
+\providecommand{\ucX}{\ensuremath{\underline{\mathcal{X}}}\xspace}
+\providecommand{\ucY}{\ensuremath{\underline{\mathcal{Y}}}\xspace}
+\providecommand{\ucZ}{\ensuremath{\underline{\mathcal{Z}}}\xspace}
+
+\providecommand{\uccA}{\ensuremath{\underline{\mathscr{A}}}\xspace}
+\providecommand{\uccB}{\ensuremath{\underline{\mathscr{B}}}\xspace}
+\providecommand{\uccC}{\ensuremath{\underline{\mathscr{C}}}\xspace}
+\providecommand{\uccD}{\ensuremath{\underline{\mathscr{D}}}\xspace}
+\providecommand{\uccE}{\ensuremath{\underline{\mathscr{E}}}\xspace}
+\providecommand{\uccF}{\ensuremath{\underline{\mathscr{F}}}\xspace}
+\providecommand{\uccG}{\ensuremath{\underline{\mathscr{G}}}\xspace}
+\providecommand{\uccH}{\ensuremath{\underline{\mathscr{H}}}\xspace}
+\providecommand{\uccI}{\ensuremath{\underline{\mathscr{I}}}\xspace}
+\providecommand{\uccJ}{\ensuremath{\underline{\mathscr{J}}}\xspace}
+\providecommand{\uccK}{\ensuremath{\underline{\mathscr{K}}}\xspace}
+\providecommand{\uccL}{\ensuremath{\underline{\mathscr{L}}}\xspace}
+\providecommand{\uccM}{\ensuremath{\underline{\mathscr{M}}}\xspace}
+\providecommand{\uccN}{\ensuremath{\underline{\mathscr{N}}}\xspace}
+\providecommand{\uccO}{\ensuremath{\underline{\mathscr{O}}}\xspace}
+\providecommand{\uccP}{\ensuremath{\underline{\mathscr{P}}}\xspace}
+\providecommand{\uccQ}{\ensuremath{\underline{\mathscr{Q}}}\xspace}
+\providecommand{\uccR}{\ensuremath{\underline{\mathscr{R}}}\xspace}
+\providecommand{\uccS}{\ensuremath{\underline{\mathscr{S}}}\xspace}
+\providecommand{\uccT}{\ensuremath{\underline{\mathscr{T}}}\xspace}
+\providecommand{\uccU}{\ensuremath{\underline{\mathscr{U}}}\xspace}
+\providecommand{\uccV}{\ensuremath{\underline{\mathscr{V}}}\xspace}
+\providecommand{\uccW}{\ensuremath{\underline{\mathscr{W}}}\xspace}
+\providecommand{\uccX}{\ensuremath{\underline{\mathscr{X}}}\xspace}
+\providecommand{\uccY}{\ensuremath{\underline{\mathscr{Y}}}\xspace}
+\providecommand{\uccZ}{\ensuremath{\underline{\mathscr{Z}}}\xspace}
+
 %---- roman letters
 
 \providecommand{\rA}{\ensuremath{\mathrm{A}}\xspace}
@@ -834,6 +983,33 @@
 \providecommand{\trY}{\ensuremath{\widetilde{\mathrm{Y}}}\xspace}
 \providecommand{\trZ}{\ensuremath{\widetilde{\mathrm{Z}}}\xspace}
 
+\providecommand{\urA}{\ensuremath{\underline{\mathrm{A}}}\xspace}
+\providecommand{\urB}{\ensuremath{\underline{\mathrm{B}}}\xspace}
+\providecommand{\urC}{\ensuremath{\underline{\mathrm{C}}}\xspace}
+\providecommand{\urD}{\ensuremath{\underline{\mathrm{D}}}\xspace}
+\providecommand{\urE}{\ensuremath{\underline{\mathrm{E}}}\xspace}
+\providecommand{\urF}{\ensuremath{\underline{\mathrm{F}}}\xspace}
+\providecommand{\urG}{\ensuremath{\underline{\mathrm{G}}}\xspace}
+\providecommand{\urH}{\ensuremath{\underline{\mathrm{H}}}\xspace}
+\providecommand{\urI}{\ensuremath{\underline{\mathrm{I}}}\xspace}
+\providecommand{\urJ}{\ensuremath{\underline{\mathrm{J}}}\xspace}
+\providecommand{\urK}{\ensuremath{\underline{\mathrm{K}}}\xspace}
+\providecommand{\urL}{\ensuremath{\underline{\mathrm{L}}}\xspace}
+\providecommand{\urM}{\ensuremath{\underline{\mathrm{M}}}\xspace}
+\providecommand{\urN}{\ensuremath{\underline{\mathrm{N}}}\xspace}
+\providecommand{\urO}{\ensuremath{\underline{\mathrm{O}}}\xspace}
+\providecommand{\urP}{\ensuremath{\underline{\mathrm{P}}}\xspace}
+\providecommand{\urQ}{\ensuremath{\underline{\mathrm{Q}}}\xspace}
+\providecommand{\urR}{\ensuremath{\underline{\mathrm{R}}}\xspace}
+\providecommand{\urS}{\ensuremath{\underline{\mathrm{S}}}\xspace}
+\providecommand{\urT}{\ensuremath{\underline{\mathrm{T}}}\xspace}
+\providecommand{\urU}{\ensuremath{\underline{\mathrm{U}}}\xspace}
+\providecommand{\urV}{\ensuremath{\underline{\mathrm{V}}}\xspace}
+\providecommand{\urW}{\ensuremath{\underline{\mathrm{W}}}\xspace}
+\providecommand{\urX}{\ensuremath{\underline{\mathrm{X}}}\xspace}
+\providecommand{\urY}{\ensuremath{\underline{\mathrm{Y}}}\xspace}
+\providecommand{\urZ}{\ensuremath{\underline{\mathrm{Z}}}\xspace}
+
 \providecommand{\hrA}{\ensuremath{\widehat{\mathrm{A}}}\xspace}
 \providecommand{\hrB}{\ensuremath{\widehat{\mathrm{B}}}\xspace}
 \providecommand{\hrC}{\ensuremath{\widehat{\mathrm{C}}}\xspace}
@@ -1155,6 +1331,59 @@
 \providecommand{\bffY}{\ensuremath{\overline{\mathfrak{Y}}}\xspace}
 \providecommand{\bffZ}{\ensuremath{\overline{\mathfrak{Z}}}\xspace}
 
+\providecommand{\uffa}{\ensuremath{\underline{\mathfrak{a}}}\xspace}
+\providecommand{\uffb}{\ensuremath{\underline{\mathfrak{b}}}\xspace}
+\providecommand{\uffc}{\ensuremath{\underline{\mathfrak{c}}}\xspace}
+\providecommand{\uffd}{\ensuremath{\underline{\mathfrak{d}}}\xspace}
+\providecommand{\uffe}{\ensuremath{\underline{\mathfrak{e}}}\xspace}
+\providecommand{\ufff}{\ensuremath{\underline{\mathfrak{f}}}\xspace}
+\providecommand{\uffg}{\ensuremath{\underline{\mathfrak{g}}}\xspace}
+\providecommand{\uffh}{\ensuremath{\underline{\mathfrak{h}}}\xspace}
+\providecommand{\uffi}{\ensuremath{\underline{\mathfrak{i}}}\xspace}
+\providecommand{\uffj}{\ensuremath{\underline{\mathfrak{j}}}\xspace}
+\providecommand{\uffk}{\ensuremath{\underline{\mathfrak{k}}}\xspace}
+\providecommand{\uffl}{\ensuremath{\underline{\mathfrak{l}}}\xspace}
+\providecommand{\uffm}{\ensuremath{\underline{\mathfrak{m}}}\xspace}
+\providecommand{\uffn}{\ensuremath{\underline{\mathfrak{n}}}\xspace}
+\providecommand{\uffo}{\ensuremath{\underline{\mathfrak{o}}}\xspace}
+\providecommand{\uffp}{\ensuremath{\underline{\mathfrak{p}}}\xspace}
+\providecommand{\uffq}{\ensuremath{\underline{\mathfrak{q}}}\xspace}
+\providecommand{\uffr}{\ensuremath{\underline{\mathfrak{r}}}\xspace}
+\providecommand{\uffs}{\ensuremath{\underline{\mathfrak{s}}}\xspace}
+\providecommand{\ufft}{\ensuremath{\underline{\mathfrak{t}}}\xspace}
+\providecommand{\uffu}{\ensuremath{\underline{\mathfrak{u}}}\xspace}
+\providecommand{\uffv}{\ensuremath{\underline{\mathfrak{v}}}\xspace}
+\providecommand{\uffw}{\ensuremath{\underline{\mathfrak{w}}}\xspace}
+\providecommand{\uffx}{\ensuremath{\underline{\mathfrak{x}}}\xspace}
+\providecommand{\uffy}{\ensuremath{\underline{\mathfrak{y}}}\xspace}
+\providecommand{\uffz}{\ensuremath{\underline{\mathfrak{z}}}\xspace}
+\providecommand{\uffA}{\ensuremath{\underline{\mathfrak{A}}}\xspace}
+\providecommand{\uffB}{\ensuremath{\underline{\mathfrak{B}}}\xspace}
+\providecommand{\uffC}{\ensuremath{\underline{\mathfrak{C}}}\xspace}
+\providecommand{\uffD}{\ensuremath{\underline{\mathfrak{D}}}\xspace}
+\providecommand{\uffE}{\ensuremath{\underline{\mathfrak{E}}}\xspace}
+\providecommand{\uffF}{\ensuremath{\underline{\mathfrak{F}}}\xspace}
+\providecommand{\uffG}{\ensuremath{\underline{\mathfrak{G}}}\xspace}
+\providecommand{\uffH}{\ensuremath{\underline{\mathfrak{H}}}\xspace}
+\providecommand{\uffI}{\ensuremath{\underline{\mathfrak{I}}}\xspace}
+\providecommand{\uffJ}{\ensuremath{\underline{\mathfrak{J}}}\xspace}
+\providecommand{\uffK}{\ensuremath{\underline{\mathfrak{K}}}\xspace}
+\providecommand{\uffL}{\ensuremath{\underline{\mathfrak{L}}}\xspace}
+\providecommand{\uffM}{\ensuremath{\underline{\mathfrak{M}}}\xspace}
+\providecommand{\uffN}{\ensuremath{\underline{\mathfrak{N}}}\xspace}
+\providecommand{\uffO}{\ensuremath{\underline{\mathfrak{O}}}\xspace}
+\providecommand{\uffP}{\ensuremath{\underline{\mathfrak{P}}}\xspace}
+\providecommand{\uffQ}{\ensuremath{\underline{\mathfrak{Q}}}\xspace}
+\providecommand{\uffR}{\ensuremath{\underline{\mathfrak{R}}}\xspace}
+\providecommand{\uffS}{\ensuremath{\underline{\mathfrak{S}}}\xspace}
+\providecommand{\uffT}{\ensuremath{\underline{\mathfrak{T}}}\xspace}
+\providecommand{\uffU}{\ensuremath{\underline{\mathfrak{U}}}\xspace}
+\providecommand{\uffV}{\ensuremath{\underline{\mathfrak{V}}}\xspace}
+\providecommand{\uffW}{\ensuremath{\underline{\mathfrak{W}}}\xspace}
+\providecommand{\uffX}{\ensuremath{\underline{\mathfrak{X}}}\xspace}
+\providecommand{\uffY}{\ensuremath{\underline{\mathfrak{Y}}}\xspace}
+\providecommand{\uffZ}{\ensuremath{\underline{\mathfrak{Z}}}\xspace}
+
 %---- groups
 
 \providecommand{\OO}[1]{\ensuremath{\mathrm{O}(#1)}\xspace}
@@ -1165,6 +1394,7 @@
 \providecommand{\GL}[2]{\ensuremath{\mathrm{GL}_{#1}(#2)}\xspace}
 
 %---- algebras
+
 \providecommand{\liebraket}[2]{\ensuremath{\left[ #1,\, #2 \right]}}
 \providecommand{\no}[1]{\ensuremath{\colon #1 \colon}\xspace}
 

sec/app/massive.tex

@@ -0,0 +1,349 @@
+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}
+
+

sec/app/tensor_wave.tex

@@ -0,0 +1,292 @@
+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}

sec/part1/dbranes.tex

@@ -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.
 
 

sec/part1/introduction.tex

@@ -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 $N$.

thesis.tex

@@ -70,10 +70,16 @@
 
 %---- coordinates
 \newcommand{\pX}{\ensuremath{X'}\xspace}
-\newcommand{\kmkr}{\ensuremath{\qty{k_+,\, l,\, \vb{k},\, r}}}
+\newcommand{\kmkr}{\ensuremath{\qty{k_+,\, l,\, \vec{k},\, r}}}
-\newcommand{\kmkrN}[1]{\ensuremath{\qty{k_{\qty(#1)\, +},\, l_{\qty(#1)},\, \vb{k}_{\qty(#1)},\, r_{\qty(#1)}}}}
+\newcommand{\kmr}{\ensuremath{\qty{k_+,\, k_-,\, l,\, r}}}
-\newcommand{\mkmkr}{\ensuremath{\qty{-k_+,\, -l,\, -\vb{k},\, r}}}
+\newcommand{\kmkrN}[1]{\ensuremath{\qty{k_{\qty(#1)\, +},\, l_{\qty(#1)},\, \vec{k}_{\qty(#1)},\, r_{\qty(#1)}}}}
-\newcommand{\mkmkrN}[1]{\ensuremath{\qty{-k_{\qty(#1)\, +},\, -l_{\qty(#1)},\, -\vb{k}_{\qty(#1)},\, r_{\qty(#1)}}}}
+\newcommand{\kmrN}[1]{\ensuremath{\qty{k_{\qty(#1)\, +},\, k_{\qty(#1)\, -},\,l_{\qty(#1)},\, r_{\qty(#1)}}}}
+\newcommand{\mkmkr}{\ensuremath{\qty{-k_+,\, -l,\, -\vec{k},\, r}}}
+\newcommand{\mkmkrN}[1]{\ensuremath{\qty{-k_{\qty(#1)\, +},\, -l_{\qty(#1)},\, -\vec{k}_{\qty(#1)},\, r_{\qty(#1)}}}}
+\newcommand{\pol}[1]{\ensuremath{\mathcal{E}_{\kmkr\, \underline{#1}}}}
+\newcommand{\polN}[2]{\ensuremath{\mathcal{E}_{\kmkrN{#2}\, \underline{#1}}}}
+\newcommand{\polabbrN}[2]{\ensuremath{\mathcal{E}_{\qty(#2)\, \underline{#1}}}}
+\newcommand{\genpolN}[1]{\ensuremath{\mathcal{E}_{\kmkrN{#1}}}}
 
 %---- BEGIN DOCUMENT
 
@@ -141,6 +147,14 @@
 \label{sec:details_reflection}
 \input{sec/app/reflection.tex}
 
+\section{Tensor Wave Functions on NBO}
+\label{sec:NO_tensor_wave}
+\input{sec/app/tensor_wave.tex}
+
+\section{Overlap of Second Level Massive States on NBO}
+\label{sec:NO_full_TTS}
+\input{sec/app/massive.tex}
+
 
 %---- BIBLIOGRAPHY
 \cleardoubleplainpage{}