End of NBO
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
230
sciencestuff.sty
230
sciencestuff.sty
@@ -47,6 +47,7 @@
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\numberwithin{table}{section}
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%---- abbreviations
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\providecommand{\sm}{\textsc{sm}\xspace}
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\providecommand{\eom}{\textsc{e.o.m.}\xspace}
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\providecommand{\cft}{\textsc{CFT}\xspace}
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@@ -54,10 +55,12 @@
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\providecommand{\qed}{\textsc{QED}\xspace}
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\providecommand{\qcd}{\textsc{QCD}\xspace}
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\providecommand{\ope}{\textsc{o.p.e.}\xspace}
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\providecommand{\dof}{\textsc{d.o.f.}\xspace}
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\providecommand{\cy}{\textsc{CY}\xspace}
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\providecommand{\lhs}{\textsc{lhs}\xspace}
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\providecommand{\rhs}{\textsc{rhs}\xspace}
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\providecommand{\ap}{\ensuremath{\alpha'}\xspace}
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\providecommand{\sgn}{\ensuremath{\mathrm{sign}}}
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%---- remap greek letters
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@@ -256,6 +259,45 @@
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\providecommand{\hPsi}{\ensuremath{\widehat{\Uppsi}}\xspace}
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\providecommand{\hOmega}{\ensuremath{\widehat{\Upomega}}\xspace}
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\providecommand{\ualpha}{\ensuremath{\underline{\upalpha}}\xspace}
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\providecommand{\ubeta}{\ensuremath{\underline{\upbeta}}\xspace}
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\providecommand{\ugamma}{\ensuremath{\underline{\upgamma}}\xspace}
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\providecommand{\udelta}{\ensuremath{\underline{\updelta}}\xspace}
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\providecommand{\uepsilon}{\ensuremath{\underline{\upepsilon}}\xspace}
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\providecommand{\uzeta}{\ensuremath{\underline{\upzeta}}\xspace}
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\providecommand{\ueta}{\ensuremath{\underline{\upeta}}\xspace}
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\providecommand{\utheta}{\ensuremath{\underline{\uptheta}}\xspace}
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\providecommand{\uiota}{\ensuremath{\underline{\upiota}}\xspace}
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\providecommand{\ukappa}{\ensuremath{\underline{\upkappa}}\xspace}
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\providecommand{\ulambda}{\ensuremath{\underline{\uplambda}}\xspace}
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\providecommand{\umu}{\ensuremath{\underline{\upmu}}\xspace}
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\providecommand{\unu}{\ensuremath{\underline{\upnu}}\xspace}
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\providecommand{\uxi}{\ensuremath{\underline{\upxi}}\xspace}
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\providecommand{\upi}{\ensuremath{\underline{\uppi}}\xspace}
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\providecommand{\urho}{\ensuremath{\underline{\uprho}}\xspace}
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\providecommand{\usigma}{\ensuremath{\underline{\upsigma}}\xspace}
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\providecommand{\utau}{\ensuremath{\underline{\uptau}}\xspace}
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\providecommand{\uupsilon}{\ensuremath{\underline{\upupsilon}}\xspace}
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\providecommand{\uphi}{\ensuremath{\underline{\upphi}}\xspace}
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\providecommand{\uchi}{\ensuremath{\underline{\upchi}}\xspace}
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\providecommand{\upsi}{\ensuremath{\underline{\uppsi}}\xspace}
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\providecommand{\uomega}{\ensuremath{\underline{\upomega}}\xspace}
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\providecommand{\uvarepsilon}{\ensuremath{\underline{\upvarepsilon}}\xspace}
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\providecommand{\uvartheta}{\ensuremath{\underline{\upvartheta}}\xspace}
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\providecommand{\uvarpi}{\ensuremath{\underline{\upvarpi}}\xspace}
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\providecommand{\uvarphi}{\ensuremath{\underline{\upvarphi}}\xspace}
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\providecommand{\uGamma}{\ensuremath{\underline{\Upgamma}}\xspace}
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\providecommand{\uDelta}{\ensuremath{\underline{\Updelta}}\xspace}
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\providecommand{\uTheta}{\ensuremath{\underline{\Uptheta}}\xspace}
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\providecommand{\uLambda}{\ensuremath{\underline{\Uplambda}}\xspace}
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\providecommand{\uXi}{\ensuremath{\underline{\Upxi}}\xspace}
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\providecommand{\uPi}{\ensuremath{\underline{\Uppi}}\xspace}
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\providecommand{\uSigma}{\ensuremath{\underline{\Upsigma}}\xspace}
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\providecommand{\uUpsilon}{\ensuremath{\underline{\Upupsilon}}\xspace}
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\providecommand{\uPhi}{\ensuremath{\underline{\Upphi}}\xspace}
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\providecommand{\uPsi}{\ensuremath{\underline{\Uppsi}}\xspace}
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\providecommand{\uOmega}{\ensuremath{\underline{\Upomega}}\xspace}
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%---- numerical sets
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\providecommand{\1}{\ensuremath{\mathds{1}}\xspace}
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@@ -479,6 +521,59 @@
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\providecommand{\hatY}{\ensuremath{\widehat{Y}}\xspace}
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\providecommand{\hatZ}{\ensuremath{\widehat{Z}}\xspace}
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\providecommand{\undera}{\ensuremath{\underline{a}}\xspace}
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\providecommand{\underb}{\ensuremath{\underline{b}}\xspace}
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\providecommand{\underc}{\ensuremath{\underline{c}}\xspace}
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\providecommand{\underd}{\ensuremath{\underline{d}}\xspace}
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\providecommand{\undere}{\ensuremath{\underline{e}}\xspace}
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\providecommand{\underf}{\ensuremath{\underline{f}}\xspace}
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\providecommand{\underg}{\ensuremath{\underline{g}}\xspace}
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\providecommand{\underh}{\ensuremath{\underline{h}}\xspace}
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\providecommand{\underi}{\ensuremath{\underline{i}}\xspace}
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\providecommand{\underj}{\ensuremath{\underline{j}}\xspace}
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\providecommand{\underk}{\ensuremath{\underline{k}}\xspace}
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\providecommand{\underl}{\ensuremath{\underline{l}}\xspace}
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\providecommand{\underm}{\ensuremath{\underline{m}}\xspace}
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\providecommand{\undern}{\ensuremath{\underline{n}}\xspace}
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\providecommand{\undero}{\ensuremath{\underline{o}}\xspace}
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\providecommand{\underp}{\ensuremath{\underline{p}}\xspace}
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\providecommand{\underq}{\ensuremath{\underline{q}}\xspace}
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\providecommand{\underr}{\ensuremath{\underline{r}}\xspace}
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\providecommand{\unders}{\ensuremath{\underline{s}}\xspace}
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\providecommand{\undert}{\ensuremath{\underline{t}}\xspace}
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\providecommand{\underu}{\ensuremath{\underline{u}}\xspace}
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\providecommand{\underv}{\ensuremath{\underline{v}}\xspace}
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\providecommand{\underw}{\ensuremath{\underline{w}}\xspace}
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\providecommand{\underx}{\ensuremath{\underline{x}}\xspace}
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\providecommand{\undery}{\ensuremath{\underline{y}}\xspace}
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\providecommand{\underz}{\ensuremath{\underline{z}}\xspace}
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\providecommand{\underA}{\ensuremath{\underline{A}}\xspace}
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\providecommand{\underB}{\ensuremath{\underline{B}}\xspace}
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\providecommand{\underC}{\ensuremath{\underline{C}}\xspace}
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\providecommand{\underD}{\ensuremath{\underline{D}}\xspace}
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\providecommand{\underE}{\ensuremath{\underline{E}}\xspace}
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\providecommand{\underF}{\ensuremath{\underline{F}}\xspace}
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\providecommand{\underG}{\ensuremath{\underline{G}}\xspace}
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\providecommand{\underH}{\ensuremath{\underline{H}}\xspace}
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\providecommand{\underI}{\ensuremath{\underline{I}}\xspace}
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\providecommand{\underJ}{\ensuremath{\underline{J}}\xspace}
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\providecommand{\underK}{\ensuremath{\underline{K}}\xspace}
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\providecommand{\underL}{\ensuremath{\underline{L}}\xspace}
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\providecommand{\underM}{\ensuremath{\underline{M}}\xspace}
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\providecommand{\underN}{\ensuremath{\underline{N}}\xspace}
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\providecommand{\underO}{\ensuremath{\underline{O}}\xspace}
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\providecommand{\underP}{\ensuremath{\underline{P}}\xspace}
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\providecommand{\underQ}{\ensuremath{\underline{Q}}\xspace}
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\providecommand{\underR}{\ensuremath{\underline{R}}\xspace}
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\providecommand{\underS}{\ensuremath{\underline{S}}\xspace}
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\providecommand{\underT}{\ensuremath{\underline{T}}\xspace}
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\providecommand{\underU}{\ensuremath{\underline{U}}\xspace}
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\providecommand{\underV}{\ensuremath{\underline{V}}\xspace}
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\providecommand{\underW}{\ensuremath{\underline{W}}\xspace}
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\providecommand{\underX}{\ensuremath{\underline{X}}\xspace}
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\providecommand{\underY}{\ensuremath{\underline{Y}}\xspace}
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\providecommand{\underZ}{\ensuremath{\underline{Z}}\xspace}
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%---- calligraphic letters
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\providecommand{\cA}{\ensuremath{\mathcal{A}}\xspace}
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@@ -751,6 +846,60 @@
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\providecommand{\bccY}{\ensuremath{\overline{\mathscr{Y}}}\xspace}
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\providecommand{\bccZ}{\ensuremath{\overline{\mathscr{Z}}}\xspace}
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\providecommand{\ucA}{\ensuremath{\underline{\mathcal{A}}}\xspace}
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\providecommand{\ucB}{\ensuremath{\underline{\mathcal{B}}}\xspace}
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\providecommand{\ucC}{\ensuremath{\underline{\mathcal{C}}}\xspace}
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\providecommand{\ucD}{\ensuremath{\underline{\mathcal{D}}}\xspace}
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\providecommand{\ucE}{\ensuremath{\underline{\mathcal{E}}}\xspace}
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\providecommand{\ucF}{\ensuremath{\underline{\mathcal{F}}}\xspace}
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\providecommand{\ucG}{\ensuremath{\underline{\mathcal{G}}}\xspace}
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\providecommand{\ucH}{\ensuremath{\underline{\mathcal{H}}}\xspace}
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\providecommand{\ucI}{\ensuremath{\underline{\mathcal{I}}}\xspace}
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\providecommand{\ucJ}{\ensuremath{\underline{\mathcal{J}}}\xspace}
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\providecommand{\ucK}{\ensuremath{\underline{\mathcal{K}}}\xspace}
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\providecommand{\ucL}{\ensuremath{\underline{\mathcal{L}}}\xspace}
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\providecommand{\ucM}{\ensuremath{\underline{\mathcal{M}}}\xspace}
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\providecommand{\ucN}{\ensuremath{\underline{\mathcal{N}}}\xspace}
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\providecommand{\ucO}{\ensuremath{\underline{\mathcal{O}}}\xspace}
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\providecommand{\ucP}{\ensuremath{\underline{\mathcal{P}}}\xspace}
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\providecommand{\ucQ}{\ensuremath{\underline{\mathcal{Q}}}\xspace}
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\providecommand{\ucR}{\ensuremath{\underline{\mathcal{R}}}\xspace}
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\providecommand{\ucS}{\ensuremath{\underline{\mathcal{S}}}\xspace}
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\providecommand{\ucT}{\ensuremath{\underline{\mathcal{T}}}\xspace}
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\providecommand{\ucU}{\ensuremath{\underline{\mathcal{U}}}\xspace}
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\providecommand{\ucV}{\ensuremath{\underline{\mathcal{V}}}\xspace}
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\providecommand{\ucW}{\ensuremath{\underline{\mathcal{W}}}\xspace}
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\providecommand{\ucX}{\ensuremath{\underline{\mathcal{X}}}\xspace}
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\providecommand{\ucY}{\ensuremath{\underline{\mathcal{Y}}}\xspace}
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\providecommand{\ucZ}{\ensuremath{\underline{\mathcal{Z}}}\xspace}
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\providecommand{\uccA}{\ensuremath{\underline{\mathscr{A}}}\xspace}
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\providecommand{\uccB}{\ensuremath{\underline{\mathscr{B}}}\xspace}
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\providecommand{\uccC}{\ensuremath{\underline{\mathscr{C}}}\xspace}
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\providecommand{\uccD}{\ensuremath{\underline{\mathscr{D}}}\xspace}
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\providecommand{\uccE}{\ensuremath{\underline{\mathscr{E}}}\xspace}
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\providecommand{\uccF}{\ensuremath{\underline{\mathscr{F}}}\xspace}
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\providecommand{\uccG}{\ensuremath{\underline{\mathscr{G}}}\xspace}
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\providecommand{\uccH}{\ensuremath{\underline{\mathscr{H}}}\xspace}
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\providecommand{\uccI}{\ensuremath{\underline{\mathscr{I}}}\xspace}
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\providecommand{\uccJ}{\ensuremath{\underline{\mathscr{J}}}\xspace}
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\providecommand{\uccK}{\ensuremath{\underline{\mathscr{K}}}\xspace}
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\providecommand{\uccL}{\ensuremath{\underline{\mathscr{L}}}\xspace}
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\providecommand{\uccM}{\ensuremath{\underline{\mathscr{M}}}\xspace}
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\providecommand{\uccN}{\ensuremath{\underline{\mathscr{N}}}\xspace}
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\providecommand{\uccO}{\ensuremath{\underline{\mathscr{O}}}\xspace}
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\providecommand{\uccP}{\ensuremath{\underline{\mathscr{P}}}\xspace}
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\providecommand{\uccQ}{\ensuremath{\underline{\mathscr{Q}}}\xspace}
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\providecommand{\uccR}{\ensuremath{\underline{\mathscr{R}}}\xspace}
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\providecommand{\uccS}{\ensuremath{\underline{\mathscr{S}}}\xspace}
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\providecommand{\uccT}{\ensuremath{\underline{\mathscr{T}}}\xspace}
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\providecommand{\uccU}{\ensuremath{\underline{\mathscr{U}}}\xspace}
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\providecommand{\uccV}{\ensuremath{\underline{\mathscr{V}}}\xspace}
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\providecommand{\uccW}{\ensuremath{\underline{\mathscr{W}}}\xspace}
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\providecommand{\uccX}{\ensuremath{\underline{\mathscr{X}}}\xspace}
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\providecommand{\uccY}{\ensuremath{\underline{\mathscr{Y}}}\xspace}
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\providecommand{\uccZ}{\ensuremath{\underline{\mathscr{Z}}}\xspace}
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%---- roman letters
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\providecommand{\rA}{\ensuremath{\mathrm{A}}\xspace}
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@@ -834,6 +983,33 @@
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\providecommand{\trY}{\ensuremath{\widetilde{\mathrm{Y}}}\xspace}
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\providecommand{\trZ}{\ensuremath{\widetilde{\mathrm{Z}}}\xspace}
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\providecommand{\urA}{\ensuremath{\underline{\mathrm{A}}}\xspace}
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\providecommand{\urB}{\ensuremath{\underline{\mathrm{B}}}\xspace}
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\providecommand{\urC}{\ensuremath{\underline{\mathrm{C}}}\xspace}
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\providecommand{\urD}{\ensuremath{\underline{\mathrm{D}}}\xspace}
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\providecommand{\urE}{\ensuremath{\underline{\mathrm{E}}}\xspace}
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\providecommand{\urF}{\ensuremath{\underline{\mathrm{F}}}\xspace}
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\providecommand{\urG}{\ensuremath{\underline{\mathrm{G}}}\xspace}
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\providecommand{\urH}{\ensuremath{\underline{\mathrm{H}}}\xspace}
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\providecommand{\urI}{\ensuremath{\underline{\mathrm{I}}}\xspace}
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\providecommand{\urJ}{\ensuremath{\underline{\mathrm{J}}}\xspace}
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\providecommand{\urK}{\ensuremath{\underline{\mathrm{K}}}\xspace}
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\providecommand{\urL}{\ensuremath{\underline{\mathrm{L}}}\xspace}
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\providecommand{\urM}{\ensuremath{\underline{\mathrm{M}}}\xspace}
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\providecommand{\urN}{\ensuremath{\underline{\mathrm{N}}}\xspace}
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\providecommand{\urO}{\ensuremath{\underline{\mathrm{O}}}\xspace}
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\providecommand{\urP}{\ensuremath{\underline{\mathrm{P}}}\xspace}
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\providecommand{\urQ}{\ensuremath{\underline{\mathrm{Q}}}\xspace}
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\providecommand{\urR}{\ensuremath{\underline{\mathrm{R}}}\xspace}
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\providecommand{\urS}{\ensuremath{\underline{\mathrm{S}}}\xspace}
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\providecommand{\urT}{\ensuremath{\underline{\mathrm{T}}}\xspace}
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\providecommand{\urU}{\ensuremath{\underline{\mathrm{U}}}\xspace}
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\providecommand{\urV}{\ensuremath{\underline{\mathrm{V}}}\xspace}
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\providecommand{\urW}{\ensuremath{\underline{\mathrm{W}}}\xspace}
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\providecommand{\urX}{\ensuremath{\underline{\mathrm{X}}}\xspace}
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\providecommand{\urY}{\ensuremath{\underline{\mathrm{Y}}}\xspace}
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\providecommand{\urZ}{\ensuremath{\underline{\mathrm{Z}}}\xspace}
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\providecommand{\hrA}{\ensuremath{\widehat{\mathrm{A}}}\xspace}
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\providecommand{\hrB}{\ensuremath{\widehat{\mathrm{B}}}\xspace}
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\providecommand{\hrC}{\ensuremath{\widehat{\mathrm{C}}}\xspace}
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@@ -1155,6 +1331,59 @@
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\providecommand{\bffY}{\ensuremath{\overline{\mathfrak{Y}}}\xspace}
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\providecommand{\bffZ}{\ensuremath{\overline{\mathfrak{Z}}}\xspace}
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\providecommand{\uffa}{\ensuremath{\underline{\mathfrak{a}}}\xspace}
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\providecommand{\uffb}{\ensuremath{\underline{\mathfrak{b}}}\xspace}
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\providecommand{\uffc}{\ensuremath{\underline{\mathfrak{c}}}\xspace}
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\providecommand{\uffd}{\ensuremath{\underline{\mathfrak{d}}}\xspace}
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\providecommand{\uffe}{\ensuremath{\underline{\mathfrak{e}}}\xspace}
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\providecommand{\ufff}{\ensuremath{\underline{\mathfrak{f}}}\xspace}
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\providecommand{\uffg}{\ensuremath{\underline{\mathfrak{g}}}\xspace}
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\providecommand{\uffh}{\ensuremath{\underline{\mathfrak{h}}}\xspace}
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\providecommand{\uffi}{\ensuremath{\underline{\mathfrak{i}}}\xspace}
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\providecommand{\uffj}{\ensuremath{\underline{\mathfrak{j}}}\xspace}
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\providecommand{\uffk}{\ensuremath{\underline{\mathfrak{k}}}\xspace}
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\providecommand{\uffl}{\ensuremath{\underline{\mathfrak{l}}}\xspace}
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\providecommand{\uffm}{\ensuremath{\underline{\mathfrak{m}}}\xspace}
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\providecommand{\uffn}{\ensuremath{\underline{\mathfrak{n}}}\xspace}
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\providecommand{\uffo}{\ensuremath{\underline{\mathfrak{o}}}\xspace}
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\providecommand{\uffp}{\ensuremath{\underline{\mathfrak{p}}}\xspace}
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\providecommand{\uffq}{\ensuremath{\underline{\mathfrak{q}}}\xspace}
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\providecommand{\uffr}{\ensuremath{\underline{\mathfrak{r}}}\xspace}
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\providecommand{\uffs}{\ensuremath{\underline{\mathfrak{s}}}\xspace}
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\providecommand{\ufft}{\ensuremath{\underline{\mathfrak{t}}}\xspace}
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\providecommand{\uffu}{\ensuremath{\underline{\mathfrak{u}}}\xspace}
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\providecommand{\uffv}{\ensuremath{\underline{\mathfrak{v}}}\xspace}
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\providecommand{\uffw}{\ensuremath{\underline{\mathfrak{w}}}\xspace}
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||||
\providecommand{\uffx}{\ensuremath{\underline{\mathfrak{x}}}\xspace}
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\providecommand{\uffy}{\ensuremath{\underline{\mathfrak{y}}}\xspace}
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\providecommand{\uffz}{\ensuremath{\underline{\mathfrak{z}}}\xspace}
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\providecommand{\uffA}{\ensuremath{\underline{\mathfrak{A}}}\xspace}
|
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\providecommand{\uffB}{\ensuremath{\underline{\mathfrak{B}}}\xspace}
|
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\providecommand{\uffC}{\ensuremath{\underline{\mathfrak{C}}}\xspace}
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\providecommand{\uffD}{\ensuremath{\underline{\mathfrak{D}}}\xspace}
|
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\providecommand{\uffE}{\ensuremath{\underline{\mathfrak{E}}}\xspace}
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\providecommand{\uffF}{\ensuremath{\underline{\mathfrak{F}}}\xspace}
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\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
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||||
|
||||
\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}
|
||||
|
||||
|
||||
349
sec/app/massive.tex
Normal file
349
sec/app/massive.tex
Normal file
@@ -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}
|
||||
|
||||
|
||||
292
sec/app/tensor_wave.tex
Normal file
292
sec/app/tensor_wave.tex
Normal file
@@ -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}
|
||||
@@ -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.
|
||||
|
||||
|
||||
|
||||
@@ -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 $N$.
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
22
thesis.tex
22
thesis.tex
@@ -70,10 +70,16 @@
|
||||
|
||||
%---- coordinates
|
||||
\newcommand{\pX}{\ensuremath{X'}\xspace}
|
||||
\newcommand{\kmkr}{\ensuremath{\qty{k_+,\, l,\, \vb{k},\, r}}}
|
||||
\newcommand{\kmkrN}[1]{\ensuremath{\qty{k_{\qty(#1)\, +},\, l_{\qty(#1)},\, \vb{k}_{\qty(#1)},\, r_{\qty(#1)}}}}
|
||||
\newcommand{\mkmkr}{\ensuremath{\qty{-k_+,\, -l,\, -\vb{k},\, r}}}
|
||||
\newcommand{\mkmkrN}[1]{\ensuremath{\qty{-k_{\qty(#1)\, +},\, -l_{\qty(#1)},\, -\vb{k}_{\qty(#1)},\, r_{\qty(#1)}}}}
|
||||
\newcommand{\kmkr}{\ensuremath{\qty{k_+,\, l,\, \vec{k},\, r}}}
|
||||
\newcommand{\kmr}{\ensuremath{\qty{k_+,\, k_-,\, l,\, r}}}
|
||||
\newcommand{\kmkrN}[1]{\ensuremath{\qty{k_{\qty(#1)\, +},\, l_{\qty(#1)},\, \vec{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{}
|
||||
|
||||
Reference in New Issue
Block a user