diff --git a/sciencestuff.sty b/sciencestuff.sty index d21faa9..42051b6 100644 --- a/sciencestuff.sty +++ b/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} diff --git a/sec/app/massive.tex b/sec/app/massive.tex new file mode 100644 index 0000000..6f62258 --- /dev/null +++ b/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} + + diff --git a/sec/app/tensor_wave.tex b/sec/app/tensor_wave.tex new file mode 100644 index 0000000..d7c1af0 --- /dev/null +++ b/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} diff --git a/sec/part1/dbranes.tex b/sec/part1/dbranes.tex index 8e37c12..95dc3b9 100644 --- a/sec/part1/dbranes.tex +++ b/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. diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex index 954533f..2c7995e 100644 --- a/sec/part1/introduction.tex +++ b/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 $N$. diff --git a/sec/part2/divergences.tex b/sec/part2/divergences.tex index 79540fa..f0de0a3 100644 --- a/sec/part2/divergences.tex +++ b/sec/part2/divergences.tex @@ -7,7 +7,7 @@ This claim was already questioned in the literature where the $O$-plane orbifold This orbifold should in fact be stable against the gravitational collapse but it exhibits divergences in the amplitudes (see the discussion in \cite{Cornalba:2004:TimedependentOrbifoldsString}). In what follows we show a direct computation showing that the presence of the divergence is not related to a gravitational response. -What has gone unnoticed is that in the Null Boost Orbifold (\nbo) \cite{Liu:2002:StringsTimeDependentOrbifold} even the four \emph{open string} tachyons amplitude is divergent. +Unnoticed in the Null Boost Orbifold (\nbo) \cite{Liu:2002:StringsTimeDependentOrbifold}, even the four \emph{open string} tachyons amplitude is divergent. Since we are working at tree level gravity is not an issue. In fact in Equation (6.16) of \cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplitude in the divergent region reads \begin{equation} @@ -47,7 +47,7 @@ Since these terms arise from string theory also through the exchange of massive A deeper understanding of the subject requires the study of the polarisations of the massive state on the orbifold as seen from the covering Minkowski space before the computation of the overlap of the wave functions. We then go back to string theory and we verify that in the \nbo the open string three points amplitude with two tachyons and one first level massive string state does indeed diverge when some physical polarisation are chosen. -We then introduce the generalised Null Boost Orbifold (\gnbo) as a generalization of the \nbo which still has a light-like singularity and is generated by one Killing vector. +We then introduce the generalised Null Boost Orbifold (\gnbo) as a generalisation of the \nbo which still has a light-like singularity and is generated by one Killing vector. However in this model there are two directions associated with $\cA$, one compact and one non compact. We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation~\cite{Estrada:2012:GeneralIntegral}. However if a second Killing vector is used to compactify the formerly non compact direction, the theory has again the same problems as in the \nbo. @@ -62,7 +62,7 @@ In particular the scalar \qed on the \bo can be defined and the first term which Again three points open string amplitudes with one massive state diverge. \subsection{Scalar QED on NBO and Divergences} -\label{sect:NOscalarQED} +\label{sec:NOscalarQED} As discussed the four open string tachyons amplitude diverges in the \nbo. The literature on the subject (see for instance~\cite{Cornalba:2004:TimedependentOrbifoldsString} and references therein) suggests that this can be cured by the eikonal resummation. @@ -288,14 +288,14 @@ We therefore have \end{equation} Using Fourier transforms it follows that the eigenmodes are \begin{equation} - \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) = e^{i k_+ v + i l z + i \vec{k} \cdot \vec{x}}\, - \tphi_{\kmkr}(u), + \tphi_{k_-kr}(u), \end{equation} with \begin{equation} - \tphi_{\kmkr}(u) + \tphi_{k_-kr}(u) = \frac{1}{\sqrt{\qty( 2 \pi )^D~ \abs{2 \Delta k_+\, u}}} e^{ @@ -305,7 +305,7 @@ with \end{equation} and \begin{equation} - \phi^*_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) + \phi^*_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) = \phi_{\mkmkr}\qty(u,\, v,\, z,\, \vec{x}). \end{equation} @@ -313,18 +313,18 @@ We chose the numeric factor in order to get a canonical normalisation: \begin{equation} \begin{split} & - \qty( \phi_{\kmkrN{1}},\, \phi_{\kmkrN{2}} ) + \qty( \phi_{k_-krN{1}},\, \phi_{k_-krN{2}} ) \\ = & \int \dd[D-1]{\vec{x}}\, \int \dd{u}\, \int \dd{v}\, \finiteint{z}{0}{2\pi} \abs{\Delta u}\, - \phi_{\kmkrN{1}}\, \phi_{\kmkrN{2}} + \phi_{k_-krN{1}}\, \phi_{k_-krN{2}} \\ = & \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\, - \delta( r_{(1)} - r_{(2)})\, + \delta( r_{\qty(1)} - r_{\qty(2)})\, \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\, \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}. \end{split} @@ -337,8 +337,8 @@ We can then perform the off-shell expansion \int \dd{k_+} \int \dd{r} \infinfsum{l} - \cA_{\kmkr}\, - \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}), + \cA_{k_-kr}\, + \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}), \end{equation} such that the scalar kinetic term becomes \begin{equation} @@ -349,8 +349,8 @@ such that the scalar kinetic term becomes \int \dd{r} \infinfsum{l} \qty(r - M^2)\, - \cA_{\kmkr}\, - \cA_{\kmkr}^*. + \cA_{k_-kr}\, + \cA_{k_-kr}^*. \end{equation} @@ -461,7 +461,7 @@ We proceed hierarchically: first we solve for $a_v$ and $a_i$ whose equations ar We get the solutions: \begin{equation} \begin{split} - \norm{\tildea_{\kmkr\, \alpha}(u)} + \norm{\tildea_{k_-kr\, \alpha}(u)} \,= \mqty(% \tildea_u @@ -478,11 +478,11 @@ We get the solutions: \in \qty{ \underline{u}, \underline{v}, \underline{z},\underline{i} } } - \cE_{\kmkr\, \underline{\alpha}} - \norm{\tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)} + \pol{\alpha} + \norm{\tildea^{\underline{\alpha}}_{k_-kr\, \alpha}(u)} \\ & = - \cE_{\kmkr\, \underline{u}} + \pol{u} \mqty( 1 \\ @@ -492,10 +492,10 @@ We get the solutions: \\ 0 )\, - \tphi_{\kmkr}(u) + \tphi_{k_-kr}(u) \\ & + - \cE_{\kmkr\, \underline{v}} + \pol{v} \mqty( \frac{i}{2 k_+ u} + @@ -507,10 +507,10 @@ We get the solutions: \\ 0 )\, - \tphi_{\kmkr}(u) + \tphi_{k_-kr}(u) \\ & + - \cE_{\kmkr\, \underline{z}} + \pol{z} \mqty( \frac{l}{\Delta k_+ \abs{u}} \\ @@ -520,10 +520,10 @@ We get the solutions: \\ 0 )\, - \tphi_{\kmkr}(u) + \tphi_{k_-kr}(u) \\ & + - \cE_{\kmkr\, \underline{j}} + \pol{j} \mqty( 0 \\ @@ -533,7 +533,7 @@ We get the solutions: \\ \delta_{\underline{ij}} )\, - \tphi_{\kmkr}(u), + \tphi_{k_-kr}(u), \label{eq:Orbifold_spin1_pol} \end{split} \end{equation} @@ -548,15 +548,22 @@ then we can expand the off-shell fields as \qty{ \underline{u}, \underline{v}, \underline{z},\underline{i} } } \infinfsum{l} - \cE_{\kmkr\, \underline{\alpha}}\, - {a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ), + \pol{\alpha}\, + {a}^{\underline{\alpha}}_{k_-kr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ), \end{equation} -where ${a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x}) = \tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)\, e^{i\, \qty( k_+ v + l z + \vec{k} \cdot \vec{x})}$ and $\int \ccD k = \int \dd[D-3]{\vec{k}} \int \dd{k_+} \int \dd{r}$. +where +\begin{equation} + a^{\underline{\alpha}}_{k_-kr\, \alpha}\qty(u,\, v,\, z,\, \vec{x}) + = + \tildea^{\underline{\alpha}}_{k_-kr\, \alpha}(u)\, + e^{i\, \qty( k_+ v + l z + \vec{k} \cdot \vec{x})} +\end{equation} +and $\int \ccD k = \int \dd[D-3]{\vec{k}} \int \dd{k_+} \int \dd{r}$. We can also compute the normalisation as \begin{equation} \begin{split} - \qty(a_{(1)},\, a_{(2)}) + \qty(a_{\qty(1)},\, a_{\qty(2)}) & = \int \dd[D-3]{\vec{x}} \int \dd{u} @@ -566,48 +573,48 @@ We can also compute the normalisation as \\ & \times g^{\alpha\beta}\, - a_{\kmkrN{1}\, \alpha}\, a_{\kmkrN{2}\, \beta} + a_{k_-krN{1}\, \alpha}\, a_{k_-krN{2}\, \beta} \\ & = - \cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}} + \genpolN{1} \circ \genpolN{2} \\ & \times \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\, - \delta( r_{(1)} - r_{(2)})\, + \delta( r_{\qty(1)} - r_{\qty(2)})\, \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\, \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}, \end{split} \end{equation} where:\footnotemark{} \footnotetext{% - We use a shortened version of the polarizations $\cE$ for the sake of readability. - We write $\cE_{(n)\, \underline{\alpha}} = \cE_{\kmkrN{n}\, \underline{\alpha}}$ thus hiding the understood dependence of the components of $\cE_{(n)}$ on the momenta. + We use a shortened version of the polarisations $\cE$ for the sake of readability. + We write $\polabbrN{\alpha}{n} = \polN{\alpha}{n}$ thus hiding the understood dependence of the components of $\cE_{(n)}$ on the momenta. } \begin{equation} \begin{split} - \cE_{(1)} \circ \cE_{(2)} + \cE_{\qty(1)} \circ \cE_{\qty(2)} = - - \cE_{(1)\, \underline{u}}\, \cE_{(2)\, \underline{v}} - - \cE_{(1)\, \underline{v}}\, \cE_{(2)\, \underline{u}} - + \cE_{(1)\, \underline{z}}\, \cE_{(2)\, \underline{z}} + - \polabbrN{u}{1}\, \polabbrN{v}{2} + - \polabbrN{v}{1}\, \polabbrN{u}{2} + + \polabbrN{z}{1}\, \polabbrN{z}{2} + \eta^{\underline{ij}}\, - \cE_{(1)\, \underline{i}}\, \cE_{(2)\, \underline{j}}. + \polabbrN{i}{1}\, \polabbrN{j}{2}. \end{split} \end{equation} Finally the Lorenz gauge reads \begin{equation} - \eta^{i \underline{j}}\, k_i\, \cE_{\kmkr\, \underline{j}} + \eta^{i \underline{j}}\, k_i\, \pol{j} - - k_+\, \cE_{\kmkr\, \underline{u}} + k_+\, \pol{u} - - \frac{\vec{k}^2 + r}{2\, k_+} \cE_{\kmkr\, \underline{v}} + \frac{\norm{\vec{k}}^2 + r}{2\, k_+} \pol{v} = 0, \label{eq:explicit_orbifold_Lorenz} \end{equation} -which does not impose any constraint on the transverse polarization -$\cE_{\kmkr\, \underline{z}}$. +which does not impose any constraint on the transverse polarisation +$\pol{z}$. The photon kinetic term becomes \begin{equation} \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cE ] @@ -617,9 +624,9 @@ The photon kinetic term becomes \int \dd{r} \infinfsum{l}\, \frac{r}{2}\, - \cE_{\kmkr}\, + \cE_{k_-kr}\, \circ - \cE_{\kmkr}^*. + \cE_{k_-kr}^*. \end{equation} @@ -647,9 +654,9 @@ Its computation involves integrals such as \int \dd{u}\, \abs{\Delta u}\, \qty(\frac{l}{u})^2 - \finiteprod{i}{1}{3} \tphi_{\kmkrN{i}} + \finiteprod{i}{1}{3} \tphi_{k_-krN{i}} \sim - \int_{u \sim 0} \dd{u}\, + \int\limits_{u \sim 0} \dd{u}\, \qty(\frac{l^2}{\abs{u}^{\frac{5}{2}}}) e^{% -i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +)}} @@ -661,9 +668,9 @@ and \int \dd{u}\, \abs{\Delta u}\, \qty(\frac{1}{u}) - \finiteprod{i}{1}{3} \tphi_{\kmkrN{i}} + \finiteprod{i}{1}{3} \tphi_{k_-krN{i}} \sim - \int_{u \sim 0} \dd{u}\, + \int\limits_{u \sim 0} \dd{u}\, \qty(\frac{1}{u\, \abs{u}^{\frac{1}{2}}}) e^{% -i \finitesum{i}{1}{3} \frac{l_{\qty(i)}^2}{2\, \Delta^2 k_{\qty(i)\, +}} @@ -678,10 +685,10 @@ In this case however the integral vanishes if we set $l_{\qty(i)} = 0$ before it This however suggests that when all $l_{\qty(i)} = 0$, i.e.\ when the eigenfunctions are constant along the compact direction $z$, something suspicious is happening. On the other side when at least one $l$ is different from zero we have an integral such as: \begin{equation} - \int_{u \sim 0} \dd{u}\, + \int\limits_{u \sim 0} \dd{u}\, \abs{u}^{-\nu}\, e^{i \frac{\cA}{u}} \sim - \int_{t \sim \infty} \dd{t}\, + \int\limits_{t \sim \infty} \dd{t}\, t^{\nu-2}\, e^{i \cA t}. \end{equation} All $l_{\qty(i)}$ are discrete but $k_{\qty(i)\, +}$ are not thus $\cA$ has an isolated zero. @@ -691,7 +698,7 @@ The second integral has the same issues when all $l_{\qty(*)} = 0$ but, since it We can give in any case meaning to the cubic terms and we get:\footnotemark{} \footnotetext{% - The notation $(2) \rightarrow (3)$ meaning is that all previous terms inside the curly brackets appear again in exactly the same structure but with momenta of particle $(3)$ in place of those of particle $(2)$. + The notation $\qty(2) \rightarrow \qty(3)$ meaning is that all previous terms inside the curly brackets appear again in exactly the same structure but with momenta of particle $\qty(3)$ in place of those of particle $\qty(2)$. } \begin{equation} \begin{split} @@ -711,37 +718,35 @@ and we get:\footnotemark{} & \times e~ \delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)},\, 0}\, - \qty(\cA_{\mkmkrN{2}})^*\, \cA_{\kmkrN{3}} + \qty(\cA_{\mkmkrN{2}})^*\, \cA_{k_-krN{3}} \\ & \times - \left\lbrace - \cE_{\kmkrN{1}\, \underline{u}}\, + \Biggl\lbrace + \polN{u}{1}\, k_{\qty(2)\, +}\, \cI_{\qty{3}}^{\qty[0]} - \right. - \\ - & + - \cE_{\kmkrN{1}\, \underline{z}}\, - \frac{% - k_{\qty(2)\, +} l_{\qty(1)} - - - l_{\qty(2)} k_{\qty(1)\, +} - }{\Delta k_{\qty(1)\, +}}\, - \cJ_{\qty{3}}^{\qty[-1]} - \\ - & + - \cE_{\kmkrN{1}\, \underline{v}}\, - \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)}) - \\ - & - - \left. + \\ + & + + \polN{z}{1}\, + \frac{% + k_{\qty(2)\, +} l_{\qty(1)} + - + l_{\qty(2)} k_{\qty(1)\, +} + }{\Delta k_{\qty(1)\, +}}\, + \cJ_{\qty{3}}^{\qty[-1]} + \\ + & + + \polN{v}{1}\, + \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)}) + \\ + & - \eta^{\underline{i}\, j}\, - \cE_{\kmkrN{1}\, \underline{i}}\, - k_{(2)_j}\, + \polN{i}{1}\, + k_{\qty(2)_j}\, \cI_{\qty{3}}^{\qty[0]}\, - - \qty( (2) \rightarrow (3) ) - \right\rbrace, + \qty( \qty(2) \rightarrow \qty(3) ) + \Biggr\rbrace, \label{eq:sQED_cubic_final} \end{split} \end{equation} @@ -768,26 +773,26 @@ where \end{equation} In the previous expressions we also defined for future use: \begin{eqnarray} - \cI_{(1) \dots (N)}^{\qty[\nu]} + \cI_{\qty(1) \dots (N)}^{\qty[\nu]} = \cI_{\qty{N}}^{\qty[\nu]} & = & \infinfint{u}\, \abs{\Delta u}\, u^{\nu}\, \finiteprod{i}{1}{N} - \tphi_{\kmkrN{i}} + \tphi_{k_-krN{i}} \\ \cJ_{\qty{N}}^{\qty[\nu]} & = & \infinfint{u}\, \abs{\Delta}\, \abs{u}^{1 + \nu} - \finiteprod{i}{1}{N} \tphi_{\kmkrN{i}}. + \finiteprod{i}{1}{N} \tphi_{k_-krN{i}}. \end{eqnarray} For the sake of brevity from now on we use \begin{eqnarray} - \tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}}, + \tphi_{\qty(i)} & = & \tphi_{k_-krN{i}}, \\ - \tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}} + \tphi_{\qty(i)} & = & \tphi_{k_-krN{i}} \end{eqnarray} when not causing confusion. @@ -826,45 +831,41 @@ which can be expressed using the modes as: \delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)} + l_{\qty(4)},\, 0} \\ & \times - \left\lbrace + \Biggl\lbrace e^2\, - \qty(\cA_{\mkmkrN{3}})^* \cA_{\kmkrN{4}} - \right\rbrace - \\ - & \times - \left[ - \qty(\cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}})\, - \cI_{\qty{4}}^{\qty[0]} - \right. - \\ - & - - \frac{i}{2} - \cE_{\kmkrN{1}\, \underline{v}}\, \cE_{\kmkrN{2}\, \underline{v}} - \qty(% - \frac{1 }{k_{\qty(2)\, +}} - + - \frac{1}{k_{\qty(1)\, +}} - )\, - \cI_{\qty{4}}^{\qty[-1]} - \\ - & + - \left. - \frac{1}{2}\, - \frac{\cE_{\kmkrN{1}\, \underline{v}} \cE_{\kmkrN{2}\, \underline{v}} }{\Delta^2} - \qty(% - \frac{l_{\qty(1)}}{k_{\qty(1)\, +}} - - - \frac{l_{\qty(2)}}{k_{\qty(2)\, +}} - )^2\, - \cI_{\qty{4}}^{\qty[-2]} - \right] - \\ - & - - \left. + \qty(\cA_{\mkmkrN{3}})^* \cA_{k_-krN{4}} + \\ + & \times + \Biggl[ + \qty(\genpolN{1} \circ \genpolN{2})\, + \cI_{\qty{4}}^{\qty[0]} + \\ + & - + \frac{i}{2} + \polN{v}{1}\, \polN{v}{2} + \qty(% + \frac{1 }{k_{\qty(2)\, +}} + + + \frac{1}{k_{\qty(1)\, +}} + )\, + \cI_{\qty{4}}^{\qty[-1]} + \\ + & + + \frac{1}{2}\, + \frac{\polN{v}{1} \polN{v}{2} }{\Delta^2} + \qty(% + \frac{l_{\qty(1)}}{k_{\qty(1)\, +}} + - + \frac{l_{\qty(2)}}{k_{\qty(2)\, +}} + )^2\, + \cI_{\qty{4}}^{\qty[-2]} + \Biggr] + \\ + & - \frac{g_4}{4}\, \ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r})\, \cI_{\qty{4}}^{\qty[0]} - \right\rbrace, + \Biggr\rbrace, \end{split} \end{equation} where @@ -876,8 +877,8 @@ where \qty(\cA_{\mkmkrN{2}})^*\, \\ & \times - \cA_{\kmkrN{3}}\, - \cA_{\kmkrN{4}}. + \cA_{k_-krN{3}}\, + \cA_{k_-krN{4}}. \end{split} \end{equation} When setting $l_{\qty(*)} = 0$ all the surviving terms are divergent. @@ -893,12 +894,12 @@ From the discussion in the previous section the origin of the divergences is the When $l = 0$ the highest order singularity of the Fourier transformed d'Alembertian equation vanishes. Explicitly we have: \begin{equation} - A\, \ipd{u} \tphi_{\kmkr} + A\, \ipd{u} \tphi_{k_-kr} + - B(u)\, \tphi_{\kmkr} + B(u)\, \tphi_{k_-kr} = A\, e^{-\int^u \frac{B(u)}{A} du}\, - \ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{\kmkr} ] + \ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{k_-kr} ] = 0, \end{equation} @@ -908,7 +909,7 @@ with \qquad B(u) = - -\qty(\vec{k}^2 + r) + -\qty(\norm{\vec{k}}^2 + r) - i\, k_+\, \frac{1}{u} - @@ -999,7 +1000,7 @@ We can perform the usual Fourier transform and the function $B(u)$ becomes B(u) & = - - (\vec{k}^2 + r) + \qty(\norm{\vec{k}}^2 + r) - i\, k_+\, \frac{u}{u^2 + \epsilon^2} - @@ -1103,7 +1104,7 @@ Performing the same steps as before we get \begin{equation} B(u) = - - (\vec{k}^2 + r) + - \qty(\norm{\vec{k}}^2 + r) - i\, k_+\, \frac{1}{u} + @@ -1115,9 +1116,10 @@ Though appealing, the study of the string in the presence of this non trivial ba \subsection{NBO Eigenfunction from the Covering Space} +\label{sec:Eigenmodes_from_Covering} -We recover the eigenfunctions from the covering Minkowski space in order to elucidate the connection between the polarizations in \nbo and in Minkowski. -Moreover we generalise the result to a symmetric two index tensor which is the polarization of the first massive state to compute the two-tachyons--one-massive-state amplitude in the next section and to show that it diverges. +We recover the eigenfunctions from the covering Minkowski space in order to elucidate the connection between the polarisations in \nbo and in Minkowski. +Moreover we generalise the result to a symmetric two index tensor which is the polarisation of the first massive state to compute the two-tachyons--one-massive-state amplitude in the next section and to show that it diverges. \subsubsection{Spin 0 Wave Function from Minkowski space} @@ -1233,11 +1235,11 @@ We can now use the Poisson resummation to finally get:\footnotemark{} \footnotetext{% In the expression we insert the variables $\vec{k}$ and $\vec{x}$ for completeness. - We also set $r = 2\, k_+ k_- -k_2^2 - \vec{k}^2$. + We also set $r = 2\, k_+ k_- -k_2^2 - \norm{\vec{k}}^2$. } \begin{equation} \begin{split} - \Psi_{[k_+\, k_-\, k_2\, \vec{k}]}\qty(u,\, v,\, z,\, \vec{x}) + \Psi_{\qty[k_+\, k_-\, k_2\, \vec{k}]}\qty(u,\, v,\, z,\, \vec{x}) & = \sqrt{2\pi}~ \frac{2 e^{-i \frac{\pi}{4}}}{\Delta} @@ -1254,7 +1256,7 @@ to finally get:\footnotemark{} - \frac{l^2}{2 \Delta^2 k_+}\, \frac{1}{u} + - \frac{r + \vec{k}^2}{2 k_+} u + \frac{r + \norm{\vec{k}}^2}{2 k_+} u + \vec{k} \cdot \vec{x} ] @@ -1265,7 +1267,7 @@ to finally get:\footnotemark{} & = \cN\, \infinfsum{l} - \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) e^{i\, l\, \frac{k_2}{\Delta k_+}}, \end{split} \label{eq:Psi_phi} @@ -1280,7 +1282,7 @@ when $k_+ \neq 0$ and where The fact that $\Psi$ depends only on the equivalence class $\qty[k_+\, k_-\, k_2\, k]$ allows us to restrict $0 \le \frac{k_2}{\Delta\, \abs{k_+}} < 2 \pi$ so that we can invert the previous expression and get: \begin{equation} \begin{split} - \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) & = \frac{1}{\cN}\, \frac{1}{2 \pi \Delta \abs{k_+}} @@ -1353,31 +1355,29 @@ with \end{equation} Notice that we are not imposing any gauge condition. Moreover if $(\epsilon_+,\, \epsilon_-,\, \epsilon_2)$ are constant then $(\epsilon_u,\, \epsilon_v,\, \epsilon_z)$ are generic functions. -It is worth stressing that they are not the polarizations in the orbifold which are in any case constant: the fact that they depend on the coordinates is simply the statement that not all eigenfunctions of the vector d'Alembertian are equal. +It is worth stressing that they are not the polarisations in the orbifold which are in any case constant: the fact that they depend on the coordinates is simply the statement that not all eigenfunctions of the vector d'Alembertian are equal. Building the corresponding function on the orbifold amounts to summing the images created by the orbifold group: \begin{equation} \begin{split} - \cN + \cN\, \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) = \infinfsum{n} - \vec{\epsilon} \cdot \qty( \cK^{-n} \dd{x})~ - \psi_{k}\qty( \cK^{-n} x) + \vec{\epsilon} \cdot \qty( \cK^n \dd{x})~ + \psi_{k}\qty( \cK^n x) = \infinfsum{n} - \cK^n + \cK^{-n} \vec{\epsilon} \cdot \dd{x}~ - \psi_{\cK^n k}\qty(x). + \psi_{\cK^{-n} k}\qty(x). \end{split} \end{equation} Under the action of the Killing vector $\epsilon$ transforms exactly as the $k$ since it is induced by $\epsilon \cdot \cK^n \dd{x} = \cK^{-n} \epsilon \cdot \dd{x}$, that is: \begin{equation} \epsilon = - \mqty(% - \epsilon_+ \\ \epsilon_2 \\ \epsilon_- - ) + \mqty( \epsilon_+ \\ \epsilon_2 \\ \epsilon_- ) \equiv \cK^{-n} \epsilon = @@ -1393,676 +1393,1280 @@ However the pair $\qty(\vec{k},\, \vec{\epsilon})$ transforms with the same $n$ There is therefore only one equivalence class $\qty[\vec{k},\, \vec{\epsilon}]$ and not two separate classes $\qty[\vec{k}]$, $\qty[\vec{\epsilon}]$. In other words, a representative of the combined equivalence class is the one with $0 \le k_2 < 2 \pi \Delta \abs{k_+}$ when $k_+ \neq 0$. -%%% TODO %%% - -We now proceed to find the eigenfunctions on the orbifold -in orbifold coordinates. -We notice that $\dd{u}, \dd{v}$ and $\dd{z}$ are -invariant and therefore their coefficients in $a$ are as well. -So we write -\begin{align} -\cN - \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) - =& - \sum_n \epsilon \cdot ( \cK^{ n } \dd x)\, - \psi_{k}( \cK^{ n } x) -\nonumber\\ - =& - \dd{v}\, - \left[ \epsilon_+ \sum_n \psi_{ k}( \cK^{ n } x) \right] - + - \dd{z}\, (\Delta u) - \left[ - \epsilon_2 \sum_n \psi_{ k}( \cK^{ n } x) - + - \epsilon_+ \Delta \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x) - \right] - \nonumber\\ - &+ - \dd{u} - \left[ - \epsilon_- \sum_n \psi_{ k}(\cK^{ n }x) - + - \epsilon_2 \Delta \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x) - + - \frac{1}{2} - \epsilon_+ \Delta^2 - \sum_n (z + 2\pi n)^2 \psi_{ k}(\cK^{ n }x) - \right] -. -\end{align} -From direct computation we get\footnote{Notice that these expressions may be written using Hermite polynomials.} -\begin{align} - & - \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x) - = - \qty( - \frac{1}{i \Delta\, u} - \frac{\partial}{\partial k_2} - - - \frac{k_2}{\Delta k_+} - ) \Psi_{[k]}\qty(\qty[x]) - \nonumber\\ -& - \sum_n (z + 2\pi n)^2 \psi_{ k}(\cK^{ n }x) - = - \qty( - \frac{1}{i \Delta\, u} - \frac{\partial}{\partial k_2} - - - \frac{k_2}{\Delta k_+} - )^2 \Psi_{[k]}\qty(\qty[x]) - . -\label{eq:sum_z_psi} -\end{align} - Then it follows that -\begin{align} -\cN - \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) - = - & - \dd{v}\, - \left[ \epsilon_+\, \Psi_{[k]}\qty(\qty[x]) - \vphantom{\frac{\partial}{\partial k_2}} - \right] - \nonumber\\ - & - + - \dd{z}\, (\Delta u) - \left[ - \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} - \Psi_{[k]}\qty(\qty[x]) - + - \epsilon_+ \, \frac{-i}{u} \frac{\partial}{\partial k_2} \Psi_{[k]}\qty(\qty[x]) - \right] - \nonumber\\ - &+ - \dd{u} - \Biggl[ - \qty( - \epsilon_- - \epsilon_2 \frac{k_2}{k_+} - + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 - ) - \, \Psi_{[k]}\qty(\qty[x]) - + - \frac{i}{2 u} \frac{\epsilon_+}{k_+ } \, \Psi_{[k]}\qty(\qty[x]) - \nonumber\\ - &\phantom{d u [} - + - \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} - \frac{-i}{ u} \frac{\partial}{\partial k_2} \Psi_{[k]}\qty(\qty[x]) - + - \frac{1}{2} - \epsilon_+ - \frac{-1}{ u^2} \frac{\partial^2}{\partial k_2{}^2} \Psi_{[k]}\qty(\qty[x]) - \Biggr] - , - \label{eq:a_uvz_from_covering} -\end{align} -where many coefficients of $\Psi$ or its derivatives contain -$k_2$. -They cannot be expressed using the quantum -numbers of the orbifold $\kmkr$ but are invariant on the orbifold and -therefore are new orbifold quantities which we can interpret as -orbifold polarizations. -Using \eqref{eq:Psi_phi} we can finally write -\begin{align} -% \cN - \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) - = -% \cN - \sum_l - & -% \frac{1}{\sqrt{|k_+|}} -\phi_{\kmkr}(u,v,z,\vec{x}) - e^{ i l \frac{k_2}{ \Delta k_+} } - \Bigg\{ - \dd{v} \Bigl[ \epsilon_+ \Bigr] - \nonumber\\ - &+ - \dd{z}\, (\Delta u) - \Biggl[ - \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} - + - \epsilon_+ \frac{1}{\Delta u} \frac{l}{k_+} - \Biggr ] - \nonumber\\ - &+ - \dd{u} - \Biggl[ -% \epsilon_- - \qty( - \epsilon_- - \epsilon_2 \frac{k_2}{k_+} - + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 - ) - + - \frac{i}{2 u} \frac{\epsilon_+}{k_+ } - \nonumber\\ - & - \phantom{ \dd{u} [ } - + - \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} - \frac{1}{u} \frac{l}{\Delta k_+} - + - \epsilon_+ \frac{1}{2 u^2} - \qty(\frac{l}{ \Delta k_+} )^2 - \Biggr] - \Bigg\} - . - \label{eq:spin1_from_covering} -\end{align} - -If we compare with \eqref{eq:Orbifold_spin1_pol} we find -\begin{align} - \cE_{\kmkr\, \underline{v}} &= \epsilon_+ - \nonumber\\ - \cE_{\kmkr\, \underline{z}} &= \mathrm{sign}(u) - \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} - \nonumber\\ - \cE_{\kmkr\, \underline{u}} &= - \epsilon_- - \epsilon_2 \frac{k_2}{k_+} - + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 - , - \label{eq:eps_calE} -\end{align} -which implies that the true polarizations $(\epsilon_+, \epsilon_-, \epsilon_2)$ -and -$\cE_{\kmkr\, \underline{*}}$ are constant as it turns out from direct computation. - -A different way of reading the previous result is that the -polarizations on the orbifold are the coefficients of the highest power -of $u$. - -We can also invert the previous relations to get -\begin{align} - \epsilon_+ - &= - \cE_{\kmkr\, \underline{v}} - \nonumber\\ - \epsilon_2 - &= - \cE_{\kmkr\, \underline{z}} \mathrm{sign}(u) + \frac{k_2}{k_+} \cE_{\kmkr\, \underline{v}} - \nonumber\\ - \epsilon_- &= - \cE_{\kmkr\, \underline{u}} - + \frac{k_2}{k_+} \cE_{\kmkr\, \underline{z}} \mathrm{sign}(u) - + \frac{1}{2} \qty( \frac{k_2}{k_+} )^2 \cE_{\kmkr\, \underline{v}} - , - \label{eq:calE_eps} -\end{align} -and use them in Lorenz gauge $k \cdot \epsilon=0$ in order to get the -expression of Lorenz gauge with orbifold polarizations. -If the definition of orbifold polarizations is right the result cannot -depend on $k_2$ since $k_2$ is not a quantum number of orbifold -eigenfunctions. -Taking in account $k_- = \frac{\vec{k}^2+ k_2^2 + r}{2 k_+}$ in $k \cdot -\epsilon=0$ -we get -exactly the expression for the Lorenz gauge for orbifold polarizations -\eqref{eq:Lorenz_gauge}. - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection{Tensor Wave Function from Minkowski space} -Once again, we can use the analysis of the previous section in the case of a -second order symmetric tensor wave function. -Again we suppress the dependence on $\vec{x}$ and $\vec{k}$ with a caveat: the Minkowskian polarizations $S_{+\, i}$, $S_{-\, i}$ and $S_{2\, i}$ do transform non trivially, therefore we give the full expressions in Appendix~\ref{app:NO_tensor_wave} even if these components contribute in a somewhat trivial way since they behave effectively as a vector of the orbifold. - -We start with the usual wave in flat space and we express either in -the Minkowskian coordinates -\begin{alignat}{4} -\cN - \psi^{[2]}_{k\, S}(x^+,x^-,x^2) - &= S_{\mu \nu}\, \psi_k(x)\, \dd x^{\mu}\, \dd x^{\nu} - \nonumber\\ - &= - % (\epsilon_+ d x^+ + \epsilon_- dx^- + \epsilon_2 d x^2) - % \otimes - % (\etp d x^+ + \etm dx^- + \etd d x^2) -% \qty( - ( - S_{+\, +}\, \dd x^+\, \dd x^+ - & - + - 2 S_{+\, 2}\, \dd x^+\, \dd x^2 - & - + - 2 S_{+\, -}\, \dd x^+\, \dd x^- - \nonumber\\ - & - & - + - 2 S_{2\, 2}\, \dd x^2\, \dd x^2 - & - + - 2 S_{2\, -}\, \dd x^2\, \dd x^- - \nonumber\\ - & - & - & - + - 2 S_{-\, -}\, \dd x^-\, \dd x^- - ) - e^{i\qty( k_+ x^+ + k_- x^- + k_2 x^2 )} - \nonumber\\ - , -\end{alignat} -or in orbifold coordinates -\begin{align} -\cN - \psi^{[2]}_{k\, S}(x) - =& S_{\alpha \beta}\, \psi_k(x)\, \dd x^\alpha\, \dd x^\beta - \nonumber\\ - =& - \Bigl\{ - (\dd{v})^2\, - [S_{+\, +}] - \nonumber\\ - & - + - \dd{v}\, \dd{z}\,\Delta u - [ 2 S_{+\, 2} - + S_{+\, +} \Delta z ] - \nonumber\\ - & - + - \dd{v}\, \dd{u}\, - [ 2 S_{+\, -} - + 2 S_{+\, 2} \Delta z - + S_{+\, +} \Delta^2 z^2 ] - \nonumber\\ - & - + - \dd{z}^2\,\Delta^2 u^2\, - [ S_{2\, 2} - + 2 S_{+\, 2} \Delta z - + S_{+\, +} \Delta^2 z^2 ] - % \nonumber\\ - % & - % + - % d z\, d v\, - % [ S_{2\, 2} \Delta^2 u^2 + S_{+\, 2} \Delta^3 u^2 z - % + \frac{1}{4} S_{+\, +} \Delta^4 u^2 z^2 ] - \nonumber\\ - & - + - \dd{z}\, \dd{v}\, \Delta u\, - [ 2 S_{-\, 2} - + 2 (S_{2\, 2} + S_{+\, -} ) \Delta z - + 3 S_{+\, 2} \Delta^2 z^2 - + S_{+\, +} \Delta^3 z^3 - ] - \nonumber\\ - & - + - \dd{u}^2\, - [ - S_{-\, -} - + 2 S_{-\, 2} \Delta z - + (S_{2\, 2} + S_{+\, -}) \Delta^2 z^2 - + S_{+\, 2} \Delta^3 z^3 - + \frac{1}{4} S_{+\, +} \Delta^4 z^4 - ] - \Bigr\} - \nonumber\\ - &\times - e^{i\left[ k_+ v - + - \frac{2 k_+ k_- - k_2^2}{2 k_+} u - + \frac{1}{2} \Delta^2 k_+ u \qty( z+ \frac{k_2}{\Delta k_+})^2 - \right]} -. -\end{align} -Now we define the tensor on the orbifold as a sum over all images as -\begin{align} -\cN - \Psi^{[2]}_{[k\, S]}\qty(\qty[x]) - &= - \sum_n ( \cK^{ n }d x) \cdot S \cdot ( \cK^{ n } \dd x)~ - \psi_{k}( \cK^{ n } x) - \nonumber\\ - &= - \sum_n \dd x \cdot ( \cK^{ - n }S ) \cdot \dd x~ - \psi_{ \cK^{ -n } k}( x) - . - \end{align} - In the last line we have defined the induced action of the Killing vector on - $(k, S)$ which can be explicitely written as - \begin{align} - \cK^{-n} - \qty( - \begin{array}{c} - S_{ + + }\\ - S_{ + 2 }\\ - S_{ + - }\\ - S_{ 2 2 }\\ - S_{ 2 - }\\ - S_{ - - } - \end{array} - ) - = - % - \qty( - \begin{array}{c} - S_{ + + }\\ - S_{ + 2 } + n \Delta S_{ + + }\\ - S_{ + - } + n \Delta S_{ + 2 } + \frac{1}{2} n^2 \Delta^2 S_{ + + }\\ - S_{ 2 2 } + 2 n \Delta S_{ + 2 } + n^2 \Delta^2 S_{ + + }\\ - S_{ 2 - } + n \Delta (S_{ 2 2 } +S_{ + - }) - + \frac{3}{2} n^2 \Delta^2 S_{ + 2 } + \frac{1}{2} n^3 \Delta^3 S_{ + + }\\ - S_{ - - } + 2 n \Delta S_{ - 2 } - + n^2 \Delta^2 (S_{ 2 2 } + S_{ + - } ) + n^3 \Delta^3 S_{ + 2 } - + \frac{1}{4} n^4 \Delta^4 S_{ + + } - \end{array} - ) -. - \end{align} - - In orbifold coordinates - to compute the tensor on the orbifold simply - amounts to sum over all the shifts - $z \rightarrow (z+2\pi n)$ and the use of the generalization of - \eqref{eq:sum_z_psi}, i.e. to substitute - $(\Delta\,z)^j \psi_k \rightarrow - \qty( - \frac{1}{i u} \frac{\partial}{\partial k_2} -- \frac{ k_2}{ \Delta k_+} - )^j - \Psi_{[k]}\qty(\qty[x])$. - When expressing all in the $\phi$ basis - this last step is equivalent to - $(\Delta\, z)^j \psi_k \rightarrow - \qty( \frac{l}{\Delta\, u\, k_+} )^j - + \dots - $. - We identify the basic polaritazions on the orbifold by - considering the highest power in $u$ and get - \begin{align} - \cS_{u\,u} - &= - \frac{1}{4}{{K^4\,S_{+\,+}}} - +K^2\,S_{+\,-} - -K^3\,S_{+\,2} - +S_{-\,-} - -2\,K\,S_{-\,2} - +S_{2\,2}\,K^2 -\nonumber\\ - \cS_{u\,v} - &= - \frac{1}{2} {{K^2\,S_{+\,+}}} - +S_{+\,-} - -K\,S_{+\,2} - \nonumber\\ - \cS_{u\,z} - &= - - \frac{1}{2} {{K^3\,S_{+\,+}}} - -K\,S_{+\,-} - +\frac{3}{2} {{K^2\,S_{+\,2}}} - +S_{-\,2} - -K\, S_{2\,2} -\nonumber\\ - \cS_{v\,v} - &= - S_{+\,+} -\nonumber\\ - \cS_{v\,z} - &= - S_{+\,2}-K\,S_{+\,+} -\nonumber\\ - \cS_{z\,z} - &= - K^2\,S_{+\,+}-2\,K\,S_{+\,2}+S_{2\,2} -. - \end{align} -with $K= \frac{k_2}{ k_+}$. -The previous equations can be inverted into -\begin{align} -S_{-\,-} -&= -% {{K^2\,\qty(4\,\cS_{z\,z}+4\,\cS_{u\,v})+4\,K^3\,\cS_{v\,z}+K^4\,\cS_{v\,v}+8\,K\,\cS_{u\,z}+4\,\cS_{u\,u}}\over{4}} - K^2\,\qty(\cS_{z\,z}+\cS_{u\,v}) - +K^3\,\cS_{v\,z} - +\frac{1}{4} K^4\,\cS_{v\,v} - +2\,K\,\cS_{u\,z} - +\cS_{u\,u} -\nonumber\\ -S_{+\,-} -&= -%{{2\,K\,\cS_{v\,z}+K^2\,\cS_{v\,v}+2\,\cS_{u\,v}}\over{2}} - K\,\cS_{v\,z} - +\frac{1}{2} K^2\,\cS_{v\,v} - +\cS_{u\,v} -\nonumber\\ -S_{-\,2} -&= -% {{K\,\qty(2\,\cS_{z\,z}+2\,\cS_{u\,v})+3\,K^2\,\cS_{v\,z}+K^3\,\cS_{v\,v}+2\,\cS_{u\,z}}\over{2}} - K\,\qty(\cS_{z\,z}+\cS_{u\,v}) - +\frac{3}{2}\,K^2\,\cS_{v\,z} - +\frac{1}{2} K^3\,\cS_{v\,v} - +\cS_{u\,z} -\nonumber\\ -S_{+\,+} -&= -\cS_{v\,v} -\nonumber\\ -S_{+\,2} -&= -\cS_{v\,z}+K\,\cS_{v\,v} -\nonumber\\ -S_{2\,2} -&= -\cS_{z\,z}+2\,K\,\cS_{v\,z}+K^2\,\cS_{v\,v} -. -\end{align} -Since we plan to use the previous quantities in the case of the first -massive string state we compute the relevant quantities: the trace +In order to write the eigenfunctions on the orbifold in orbifold coordinates we notice that $\dd{u}, \dd{v}$ and $\dd{z}$ are invariant. +We write \begin{equation} - \tr(S)=\cS_{z\,z}-2\,\cS_{u\,v} + \begin{split} + \cN\, + \Psi^{[1]}_{\qty[\vec{k},\, \vec{\epsilon}]}\qty(\qty[x]) + & = + \infinfsum{n} + \epsilon \cdot \qty( \cK^n \dd{x})\, + \psi_{k}( \cK^n x) + \\ + & = + \dd{v}\, + \qty[ \epsilon_+\, \infinfsum{n} \psi_k\qty( \cK^n x) ] + \\ + & + + \dd{z}\, + \qty(\Delta u)\, + \qty[% + \epsilon_2\, \infinfsum{n} \psi_k\qty( \cK^n x) + + + \epsilon_+\, \Delta\, \infinfsum{n} \qty(z + 2\pi n) \psi_k\qty(\cK^n x) + ] + \\ + & + + \dd{u}\, + \Biggl[ + \epsilon_-\, \infinfsum{n} \psi_k\qty(\cK^n x) + + + \epsilon_2\, \Delta\, \infinfsum{n} \qty(z + 2 \pi n) \psi_k\qty(\cK^n x) + \\ + & + + \frac{1}{2} \epsilon_+\, \Delta^2\, \infinfsum{n} \qty(z + 2 \pi n)^2 \psi_k(\cK^n x) + \Biggr]. + \end{split} +\end{equation} +From a direct computation we get:\footnotemark{} +\footnotetext{% + These expressions may be written using Hermite polynomials. +} +\begin{equation} + \begin{split} + \infinfsum{n} \qty(z + 2\pi n)\, \psi_k(\cK^n x) + & = + \qty(% + \frac{1}{i\, \Delta u} + \pdv{}{k_2} + - + \frac{k_2}{\Delta k_+} + ) + \Psi_{\qty[k]}\qty(\qty[x]) + \\ + \infinfsum{n} \qty(z + 2\pi n)^2\, \psi_k(\cK^n x) + & = + \qty(% + \frac{1}{i\, \Delta\, u} + \pdv{}{k_2} + - + \frac{k_2}{\Delta k_+} + )^2 + \Psi_{\qty[k]}\qty(\qty[x]). + \end{split} + \label{eq:sum_z_psi} +\end{equation} +Then it follows that +\begin{equation} + \begin{split} + \cN\, + \Psi^{[1]}_{\qty[\vec{k},\, \vec{\epsilon}]}\qty(\qty[x]) + & = + \dd{v}\, + \qty[% + \epsilon_+\, \Psi_{\qty[k]}\qty(\qty[x]) + ] + \\ + & + + \dd{z}\, + \qty(\Delta u)\, + \qty[% + \frac{\epsilon_2 k_+ - \epsilon_+ k_2}{k_+}\, + \Psi_{\qty[k]}\qty(\qty[x]) + - + \epsilon_+\, \frac{i}{u} \pdv{}{k_2} \Psi_{\qty[k]}\qty(\qty[x]) + ] + \\ + & + + \dd{u}\, + \Biggl[ + \qty(% + \epsilon_- + - + \epsilon_2 \frac{k_2}{k_+} + + + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 + )\, + \Psi_{\qty[k]}\qty(\qty[x]) + + + \frac{i}{2 u} \frac{\epsilon_+}{k_+}\, + \Psi_{\qty[k]}\qty(\qty[x]) + \\ + & - + \frac{\epsilon_2 k_+ - \epsilon_+ k_2}{k_+} + \frac{i}{u} \pdv{}{k_2} \Psi_{\qty[k]}\qty(\qty[x]) + - + \frac{1}{2} \epsilon_+\, + \frac{1}{u^2}\, + \pdv[2]{}{k_2} + \Psi_{\qty[k]}\qty(\qty[x]) + \Biggr]. + \end{split} + \label{eq:a_uvz_from_covering} +\end{equation} +Many coefficients of $\Psi$ or its derivatives contain $k_2$. +They cannot be expressed using the quantum numbers $k_-kr$ of the orbifold but are invariant on it. +They are new orbifold quantities we interpret as orbifold polarisations. +Using~\eqref{eq:Psi_phi} we can finally write +\begin{equation} + \begin{split} + \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) + & = + \infinfsum{l} + \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + e^{i\, l \frac{k_2}{\Delta k_+}} + \\ + & \times + \Biggl\lbrace + \dd{v}\, \epsilon_+ + \\ + & + + \dd{z}\, \qty(\Delta u) + \qty[ % + \frac{\epsilon_2 k_+ - \epsilon_+ k_2}{k_+} + + + \epsilon_+ \frac{1}{\Delta u}\, \frac{l}{k_+} + ] + \\ + & + + \dd{u}\, + \Biggl[ + \qty(% + \epsilon_- + - + \epsilon_2\, \frac{k_2}{k_+} + + + \frac{1}{2} \epsilon_+\, \qty( \frac{k_2}{k_+} )^2 + ) + + + \frac{i}{2 u}\, \frac{\epsilon_+}{k_+} + \\ + & + + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + \frac{1}{u}\, \frac{l}{\Delta k_+} + + + \epsilon_+\, \frac{1}{2 u^2}\, \qty(\frac{l}{ \Delta k_+})^2 + \Biggr] + \Biggr\rbrace. + \end{split} + \label{eq:spin1_from_covering} +\end{equation} + +If we compare the last expression with~\eqref{eq:Orbifold_spin1_pol} we find: +\begin{equation} + \begin{split} + \pol{v} + & = + \epsilon_+ + \\ + \pol{z} + & = + \sgn(u) + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + \\ + \pol{u} + & = + \epsilon_- + - + \epsilon_2\, \frac{k_2}{k_+} + + + \frac{1}{2} \epsilon_+\, \qty( \frac{k_2}{k_+} )^2, + \end{split} + \label{eq:eps_calE} +\end{equation} +which implies that the true polarisations $\qty(\epsilon_+,\, \epsilon_-,\, \epsilon_2)$ and $\pol{*}$ are constant as it turns out from direct computation. +A different way of reading the previous result is that the polarisations on the orbifold are the coefficients of the highest power of $u$. + +We can also invert the previous relations to get: +\begin{equation} + \begin{split} + \epsilon_+ + & = + \pol{v} + \\ + \epsilon_2 + & = + \pol{z}\, \sgn(u) + + + \pol{v}\, \frac{k_2}{k_+} + \\ + \epsilon_- + & = + \pol{u} + + + \pol{z}\, \sgn(u)\, \frac{k_2}{k_+} + + + \pol{v}\, \frac{1}{2} \qty( \frac{k_2}{k_+} )^2, + \end{split} + \label{eq:calE_eps} +\end{equation} +and use them in Lorenz gauge $\vec{k} \cdot \vec{\epsilon} = 0$ in order to get the gauge conditions expressed with the orbifold polarisations. +If the definition of orbifold polarisations is right the result cannot depend on $k_2$ since it is not a quantum number of orbifold eigenfunctions. +Taking into account $k_- = \frac{\norm{\vec{k}}^2+ k_2^2 + r}{2 k_+}$ in $\vec{k} \cdot \vec{\epsilon} = 0$ we get exactly the expression for the Lorenz gauge for orbifold polarisations~\eqref{eq:Lorenz_gauge}. + + +\subsubsection{Tensor Wave Function from Minkowski space} + +We can use the analysis of the previous section in the case of a second order symmetric tensor wave function. +Again we suppress the dependence on $\vec{x}$ and $\vec{k}$ with a caveat: the Minkowskian polarisations $S_{+\, i}$, $S_{-\, i}$ and $S_{2\, i}$ transform non trivially, therefore we give the full expressions in \Cref{sec:NO_tensor_wave} even if these components behave effectively as a vector of the orbifold. + +We start with the usual wave in flat space and we express either in the Minkowskian coordinates +\begin{equation} + \begin{split} + \cN\, \psi^{[2]}_{k\, S}\qty( x^+,\, x^-,\, x^2 ) + & = + S_{\mu\nu}\, + \psi_k( x )\, \dd{x}^{\mu}\, \dd{x}^{\nu} + \\ + & = + \Bigl(% + S_{++}\, \dd{x}^+\, \dd{x}^+ + + + 2\, S_{+\, x}\, \dd{x}^+\, \dd{x}^2 + + + 2\, S_{+\, -}\, \dd{x}^+\, \dd{x}^- + \\ + & + + 2\, S_{2\, 2}\, \dd{x}^2\, \dd{x}^2 + + + 2\, S_{2\, -}\, \dd{x}^2\, \dd{x}^- + \\ + & + + 2\, S_{-\, -}\, \dd{x}^-\, \dd{x}^- + \Bigr) + e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )}, + \end{split} +\end{equation} +or in orbifold coordinates +\begin{equation} + \begin{split} + \cN\, + \psi^{[2]}_{k\, S}(x) + & = + S_{\alpha\, \beta}\, \psi_k(x)\, \dd{x}^{\alpha}\, \dd{x}^{\beta} + \\ + & = + \Biggl\lbrace + \dss[2]{v}\, + S_{+\, +} + \\ + & + + \dd{v}\, \dd{z}\, \Delta u + \qty[% + 2\, S_{+\, 2} + + + S_{+\, +} \Delta z + ] + \\ + & + + \dd{v}\, \dd{u}\, + \qty[% + 2\, S_{+\, -} + + + 2\, S_{+\, 2} \Delta z + + + S_{+\, +} \Delta^2 z^2 + ] + \\ + & + + \dss[2]{z}\, \Delta^2 u^2\, + \qty[% + S_{2\, 2} + + + 2\, S_{+\, 2} \Delta z + + + S_{+\, +} \Delta^2 z^2 + ] + \\ + & + + \dd{z}\, \dd{v}\, \Delta u\, + \qty[% + 2\, S_{-\, 2} + + + 2\, \qty(S_{2\, 2} + + + S_{+\, -})\, \Delta z + + + 3\, S_{+\, 2} \Delta^2 z^2 + + + S_{+\, +}\, \Delta^3 z^3 + ] + \\ + & + + \dd{u}^2\, + \qty[% + S_{-\, -} + + + 2\, S_{-\, 2} \Delta z + + + \qty(S_{2\, 2} + S_{+\, -})\, \Delta^2 z^2 + + + S_{+\, 2} \Delta^3 z^3 + + + \frac{1}{4} S_{+\, +} \Delta^4 z^4 + ] + \Biggr\rbrace + \\ + & \times + e^{% + i\, \qty[% + k_+ v + + + \frac{2\, k_+ k_- - k_2^2}{2 k_+} u + + + \frac{1}{2} \Delta^2 k_+ u \qty( z + \frac{k_2}{\Delta k_+})^2 + ] + }. + \end{split} +\end{equation} +Now we define the tensor on the orbifold as a sum over all images as +\begin{equation} + \begin{split} + \cN\, + \Psi^{[2]}_{[k\, S]}\qty(\qty[x]) + & = + \infinfsum{n} + \qty( \cK^n \dd{x}) \cdot S \cdot ( \cK^n \dd{x})~ + \psi_{k}( \cK^n x) + \\ + & = + \infinfsum{n} + \dd{x} \cdot ( \cK^{-n}\, S ) \cdot \dd{x}~ + \psi_{\cK^{-n} k}\qty(x). + \end{split} +\end{equation} +In the last line we have defined the induced action of the Killing vector on $\qty(\vec{k}, S)$ which can be explicitely written as: +\begin{equation} + \cK^{-n} + \mqty(% + S_{ +\, + } + \\ + S_{ +\, 2 } + \\ + S_{ +\, - } + \\ + S_{ 2\, 2 } + \\ + S_{ 2\, - } + \\ + S_{ -\, - } + ) + = + \mqty(% + S_{ +\, + } + \\ + S_{ +\, 2 } + n \Delta S_{ +\, + } + \\ + S_{ +\, - } + n \Delta S_{ +\, 2 } + \frac{1}{2} n^2 \Delta^2 S_{ +\, + } + \\ + S_{ 2\, 2 } + 2 n \Delta S_{ +\, 2 } + n^2 \Delta^2 S_{ +\, + } + \\ + S_{ 2\, - } + n \Delta \qty(S_{ 2\, 2 } + S_{ +\, - }) + \frac{3}{2} n^2 \Delta^2 S_{ +\, 2 } + \frac{1}{2} n^3 \Delta^3 S_{ +\, + } + \\ + S_{ -\, - } + 2 n \Delta S_{ -\, 2 } + n^2 \Delta^2 \qty(S_{ 2\, 2 } + S_{ +\, - } ) + n^3 \Delta^3 S_{ +\, 2 } + \frac{1}{4} n^4 \Delta^4 S_{ +\, + } + ). +\end{equation} + +Computing the tensor on the orbifold in its own coordinates is equivalent to summing over all the shifts $z \rightarrow \qty(z + 2 \pi n)$ and the use of a generalisation of~\eqref{eq:sum_z_psi}, i.e.\, to substitute $\qty(\Delta\, z)^j \psi_k \rightarrow \qty( \frac{1}{i\, u} \pdv{}{k_2} - \frac{k_2}{\Delta\, k_+} )^j \Psi_{\qty[k]}\qty(\qty[x])$. +When expressing all in the $\phi$ basis, the last step is equivalent to $\qty(\Delta\, z)^j \psi_k \rightarrow \qty(\frac{l}{\Delta\, u\, k_+})^j + \dots$. +We identify the basic polarisations on the orbifold by considering the highest power in $u$: +\begin{equation} + \begin{split} + \cS_{u\,u} + & = + \frac{1}{4} K^4\, S_{+\,+} + + + K^2\, S_{+\,-} + - + K^3\, S_{+\,2} + + + S_{-\,-} + - + 2\, K\, S_{-\,2} + + + S_{2\,2}\, K^2 + \\ + \cS_{u\,v} + & = + \frac{1}{2} K^2\, S_{+\,+} + + + S_{+\,-} + - + K\, S_{+\,2} + \\ + \cS_{u\,z} + & = + -\frac{1}{2} K^3\,S_{+\,+} + - + K\, S_{+\,-} + + + \frac{3}{2} K^2\, S_{+\,2} + + + S_{-\,2} + - + K\, S_{2\,2} + \\ + \cS_{v\,v} + & = + S_{+\,+} + \\ + \cS_{v\,z} + & = + S_{+\,2} - K\, S_{+\,+} + \\ + \cS_{z\,z} + & = + K^2\, S_{+\,+} - 2\, K\, S_{+\,2} + S_{2\,2}. + \end{split} +\end{equation} +where $K = \frac{k_2}{k_+}$. +The previous equations can be inverted to get: +\begin{equation} + \begin{split} + S_{-\,-} + & = + K^2\, \qty(\cS_{z\,z} + \cS_{u\,v}) + + + K^3\, \cS_{v\,z} + + + \frac{1}{4} K^4\, \cS_{v\,v} + + + 2\, + K\, + \cS_{u\,z} + + + \cS_{u\,u} + \\ + S_{+\,-} + & = + K\, \cS_{v\,z} + + + \frac{1}{2} K^2\, \cS_{v\,v} + + + \cS_{u\,v} + \\ + S_{-\,2} + & = + K\, \qty(\cS_{z\,z} + \cS_{u\,v}) + + + \frac{3}{2}\, K^2\, \cS_{v\,z} + + + \frac{1}{2} K^3\, \cS_{v\,v} + + + \cS_{u\,z} + \\ + S_{+\,+} + & = + \cS_{v\,v} + \\ + S_{+\,2} + & = + \cS_{v\,z} + + + K\, \cS_{v\,v} + \\ + S_{2\,2} + & = + \cS_{z\,z} + + + 2\, K\, \cS_{v\,z} + + + K^2\, \cS_{v\,v}. + \end{split} +\end{equation} +Since we plan to use the previous quantities in the case of the first massive string state we compute the relevant quantities. +In particular we have the trace: +\begin{equation} + \tr(S) = \cS_{z\,z} - 2\, \cS_{u\,v} \end{equation} and the transversality conditions -\begin{align} - % - % v - trans ~\cS_{v} - =& - (k\cdot S)_{+} - = - -\frac{\qty(r + \vec{k}^2)}{2\, k_+}\, +\begin{equation} + \begin{split} + \text{trans}~ + \cS_{v} + & = + \qty(\vec{k} \cdot S)_{+} + = + -\frac{\qty(r + \norm{\vec{k}}^2)}{2\, k_+}\, \cS_{v\,v} - -k_{+}\,\cS_{u\,v}, - \nonumber\\ - % - % z - trans ~\cS_{z} - =& - (k\cdot S)_{2} - -K (k\cdot S)_{+} -= - -\frac{\qty(r + \vec{k}^2)}{2\, k_+}\, - \cS_{v\,z} - -k_{+}\,\cS_{u\,z}, -\nonumber\\ - trans ~\cS_{u} - =& - (k\cdot S)_{-} - -K (k\cdot S)_{2} - %+ \frac{1}{2} K^2 (k\cdot S)_{+} - + \frac{1}{2} K^2 (k\cdot S)_{+} - = - -\frac{\qty(r + \vec{k}^2)}{2\, k_+}\, - \cS_{u\,v} - -k_{+}\,\cS_{u\,u} - . -\end{align} -where we used $k_-= (r+\vec{k}^2+k_2^2)/(2 k_+)$. -These conditions correctly do no depend on $K$ since $k_2$ is -not an orbifold quantum number. - + - + k_{+}\, \cS_{u\,v}, + \\ + \text{trans}~ + \cS_{z} + & = + \qty(\vec{k} \cdot S)_{2} + - + K~ \qty(vec{k} \cdot S)_{+} + = + -\frac{\qty(r + \norm{\vec{k}}^2)}{2\, k_+}\, + \cS_{v\,z} + - + k_{+}\, \cS_{u\,z}, + \\ + \text{trans}~ + \cS_{u} + & = + \qty(\vec{k} \cdot S)_{-} + - + K~ \qty(\vec{k} \cdot S)_{2} + + + \frac{1}{2} K^2 \qty(vec{k} \cdot S)_{+} + = + -\frac{\qty(r + \norm{\vec{k}}^2)}{2\, k_+}\, + \cS_{u\,v} + - + k_{+}\, \cS_{u\,u} + \end{split} +\end{equation} +where we used $k_- = \frac{\qty(r + \norm{\vec{k}}^2 + k_2^2)}{\qty(2 k_+)}$. +These conditions do not depend on $K$ since $k_2$ is not an orbifold quantum number. The final expression for the orbifold symmetric tensor is - \begin{align} - % \cN - \Psi^{[2]}_{[k,\, S]}\qty([x]) - & - = -% \cN - \sum_l -\phi_{\kmkr}(u,v,z,\vec{x}) - e^{ i l \frac{k_2}{ \Delta k_+} } -\nonumber\\ -% -% v v - & - \Big\{ - (\dd{v})^2\, - [\cS_{v v} ] - \nonumber\\ -% -% v z - & - + -2 - \Delta\, u\, - \dd{v}\, \dd{z}\, -% [ 2 S_{+\, 2} -% + S_{+\, +} \frac{l}{ \Delta\, u\,k_+} ] - \Bigl[ - \cS_{v\,z} - + - \qty( - \frac{L \cS_{v\,v}}{\Delta} - ) - \frac{1}{u} +\begin{equation} + \begin{split} + \Psi^{[2]}_{\qty[\vec{k},\, S]}\qty(\qty[x]) + & = + \infinfsum{l} + \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + e^{i\, l \frac{k_2}{\Delta k_+}} + \\ + & \times + \Biggl\lbrace + \dss[2]{v}\, \cS_{v\, v} + \\ + & + + 2\, \Delta\, u\, \dd{v}\, \dd{z}\, + \qty[% + \cS_{v\,z} + + + \qty(\frac{L \cS_{v\,v}}{\Delta}) + \frac{1}{u} + ] + \\ + & + + 2\, \dd{v}\, \dd{u}\, + \qty[% + \cS_{u\,v} + + + \qty(\frac{L\, \cS_{v\,z}}{\Delta} + \frac{i\, \cS_{v\,v}}{2\, k_{+}}) + \frac{1}{u} + + + \qty(\frac{L^2\, \cS_{v\,v}}{2\, \Delta^2}) + \frac{1}{u^2} + ] + \\ + & + + \qty(\Delta\, u)^2 \dd{z}^2\, + \qty[% + \cS_{z\,z} + + + \qty(\frac{2\, L\, \cS_{v\,z}}{\Delta} + \frac{i\, \cS_{v\,v}}{k_{+}}) + \frac{1}{u} + + + \qty(\frac{L^2\, \cS_{v\,v}}{\Delta^2}) + \frac{1}{u^2} + ] + \\ + & + + 2\, \Delta\, u\, \dd{z}\, \dd{u}\, + \Bigl[% + \cS_{u\,z} + + + \qty(% + \frac{L\, \cS_{z\,z}}{\Delta} + + + \frac{3\, i\, \cS_{v\,z}}{2\, k_{+}} + + + \frac{L\, \cS_{u\,v}}{\Delta} + ) + \frac{1}{u} + + + \qty(% + \frac{3\, L^2\, \cS_{v\,z}}{2\, \Delta^2} + + + \frac{3\, i\, L\, \cS_{v\,v}}{2\, \Delta\, k_{+}} + ) + \frac{1}{u^2} + \\ + & + + \qty(\frac{L^3\,\cS_{v\,v}}{2\,\Delta^3}) + \frac{1}{u^3} \Bigr] - \nonumber\\ -% -% v u - & - + - 2 - \dd{v}\, \dd{u}\, -% [ 2 S_{+\, -} -% + 2 S_{+\, 2} \frac{l}{\Delta\, u\, k_+} -% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^2 ] -\Bigl[ -\cS_{u\,v} -+ -\qty( -\frac{L\,\cS_{v\,z}}{\Delta}+\frac{i\,\cS_{v\,v}}{2\,k_{+}} -) -\frac{1}{u} -+ -\qty( -\frac{L^2\,\cS_{v\,v}}{2\,\Delta^2} -) -\frac{1}{u^2} -\Bigr] - \nonumber\\ - % -% z z - & - + - (\Delta\, u)^2 - \dd{z}^2\, -% [ S_{2\, 2} -% + 2 S_{+\, 2} \frac{l}{\Delta\, u\, k_+} -% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^2 ]\, - \Bigl[ - \cS_{z\,z} -+ -\qty( -\frac{2\,L\,\cS_{v\,z}}{\Delta}+\frac{i\,\cS_{v\,v}}{k_{+}} -) - \frac{1}{u} - + -\qty( -\frac{L^2\,\cS_{v\,v}}{\Delta^2} -) -\frac{1}{u^2} - \Bigr] - \nonumber\\ - % - % z v - & - + - 2 - \Delta u\, - \dd{z}\, \dd{u}\, - % [ 2 S_{-\, 2} - % + 2 ( S_{2\, 2} + S_{+\, -} ) \frac{l}{\Delta\, u\, k_+} - % + 3 S_{+\, 2} \qty(\frac{l}{\Delta\, u\,k_+})^2 - % + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^3 - % ]\, - \Bigl[ - \cS_{u\,z} -+ -\qty( - \frac{L\,\cS_{z\,z}}{\Delta} - +\frac{3\,i\,\cS_{v\,z}}{2\,k_{+}}+\frac{L\,\cS_{u\,v}}{\Delta} -) - \frac{1}{u} - + -\qty( - \frac{3\,L^2\,\cS_{v\,z}}{2\,\Delta^2} - +\frac{3\,i\,L\,\cS_{v\,v}}{2\,\Delta\,k_{+}} -) - \frac{1}{u^2} - \nonumber\\ - & - \phantom{+\Delta u\,d z\, d v\,} - + -\qty( -\frac{L^3\,\cS_{v\,v}}{2\,\Delta^3} -) -\frac{1}{u^3} - \Bigr] - \nonumber\\ - % - % u u - & - + - \dd{u}^2\, - % [ - % S_{-\, -} - % + 2 S_{-\, 2} \frac{l}{\Delta\, u\,k_+} - % + (S_{2\, 2} + S_{+\, -}) \qty(\frac{l}{\Delta\, u\,k_+})^2 - % + S_{+\, 2} \qty(\frac{l}{\Delta\, u\,k_+})^3 - % + \frac{1}{4} S_{+\, +} \qty(\frac{l}{\Delta\, u\,k_+})^4 - % ] - % - \Bigl[ -\cS_{u\,u} -+ -\qty( -\frac{i\,\cS_{z\,z}}{k_{+}}+\frac{2\,L\,\cS_{u\,z}}{\Delta}+\frac{i\,\cS_{u\,v}}{k_{+}} -) - \frac{1}{u} - + -\qty( - \frac{L^2\,\cS_{z\,z}}{\Delta^2}+\frac{3\,i\,L\,\cS_{v\,z}}{\Delta\,k_{+}} - -\frac{3\,\cS_{v\,v}}{4\,k_{+}^2}+\frac{L^2\,\cS_{u\,v}}{\Delta^2} -) - \frac{1}{u^2} - \nonumber\\ - & - \phantom{ + d u^2\, } - + -\qty( -\frac{L^3\,\cS_{v\,z}}{\Delta^3}+\frac{3\,i\,L^2\,\cS_{v\,v}}{2\,\Delta^2\,k_{+}} -) - \frac{1}{u^3} + \\ + & + + \dd{u}^2\, + \Bigl[ + \cS_{u\,u} + + + \qty(% + \frac{i\, \cS_{z\,z}}{k_{+}} + + + \frac{2\, L\, \cS_{u\,z}}{\Delta} + + + \frac{i\, \cS_{u\,v}}{k_{+}} + ) + \frac{1}{u} + \\ + & + + \qty(% + \frac{L^2\, \cS_{z\,z}}{\Delta^2} + + + \frac{3\, i\, L\, \cS_{v\,z}}{\Delta\, k_{+}} + - + \frac{3\, \cS_{v\,v}}{4\, k_{+}^2} + + + \frac{L^2\, \cS_{u\,v}}{\Delta^2} + ) + \frac{1}{u^2} + \\ + & + + \qty(% + \frac{L^3\, \cS_{v\,z}}{\Delta^3} + + + \frac{3\, i\, L^2\, \cS_{v\,v}}{2\, \Delta^2\, k_{+}} + ) + \frac{1}{u^3} + + + \qty(\frac{L^4 \cS_{v\,v}}{4 \Delta^4}) + \frac{1}{u^4} + \Bigr] + \Biggr\rbrace, + \end{split} +\end{equation} +where $L = \frac{l}{k_+}$. + + +\subsection{Overlaps of Wave Functions and Their Derivatives} + +In this section we compute overlaps of wave functions. +We give their expressions using both integrals over the eigenfunctions and sums of products of delta functions. +The latter is the expression which is naturally obtained by computing tree level string amplitudes on the orbifold when one starts with Minkowski amplitudes and adds the images. +This is equivalent to computing emission vertices on the orbifold and then their correlation functions since this amounts to transfer the sum over the spacetime images to the sum of the polarisations images. + + +\subsubsection{Overlaps Without Derivatives} + +We start from the simplest case of the overlap of $N$ scalar wave functions. +We compute the overlap of orbifold wave functions and then we express it as sum of images of the corresponding Minkowski overlap thus establishing a dictionary between Minkowski and orbifold spaces. +Explicitly we consider the following overlap where all the polarisations $\cA_{(i)}$ have been set to one +\begin{equation} + \begin{split} + I^{(N)} + & = + \int\limits_{\Omega} \dd[3]{x}\, + \sqrt{-\det g}~ + \finiteprod{i}{1}{N} \Psi_{\qty[k_{\qty(i)\, +}\, k_{\qty(i)\, -}\, k_{\qty(i)\, 2}]}\qty(\qty[x^+,\, x^-,\, x^2])) + \\ + & = + \int\limits_{\ccM^{1,2}} \dd[3]{x}\, + \sqrt{-\det g}~ + \psi_{k_{\qty(1)\, +}\, k_{\qty(1)\, -}\, k_{\qty(1)\, 2}}\qty(x^+,\, x^-,\, x^2))\, + \\ + & \times + \finiteprod{i}{2}{N} \infinfsum{m_{(i)}} + \psi_{k_{\qty(i)\, +}\, k_{\qty(i)\, -}\, k_{\qty(i)\, 2}}( \cK^{m_{(i)} }\qty(x^+,\, x^-,\, x^2) ) + \\ + & = + \int\limits_{\ccM^{1,2}} \dd[3]{x}\, + \sqrt{-\det g}~ + \psi_{k_{\qty(1)\, +}\, k_{\qty(1)\, -}\, k_{\qty(1)\, 2}}\qty(x^+,\, x^-,\, x^2)) + \\ + & \times + \finiteprod{i}{2}{N} \infinfsum{m_{(i)}} + \psi_{\cK^{m_{(i)}}\qty(k_{\qty(i)\, +}\, k_{\qty(i)\, -}\, k_{\qty(i)\, 2}) }\qty(x^+,\, x^-,\, x^2) + \\ + & = + \qty(2\pi)^3 + \delta\qty(\infinfsum{i} k_{\qty(i)\, +})\, + \\ + & \times + \eval{% + \finiteprod{i}{2}{N} \infinfsum{m_{(i)}} + \delta\qty(\infinfsum{i} \cK^{m_{(i)}}~ k_{\qty(i)\, 2} )\, + \delta\qty(\infinfsum{i} \cK^{m_{(i)}}~ k_{\qty(i)\, -} )\, + }_{m_{\qty(1)} = 0}, + \end{split} +\end{equation} +where $\Omega = \ccM^{1,2} / \Gamma$ is the fundamental region identifying the orbifold. +We used the unfolding trick to rewrite the integral as an integral over $\ccM^{1,2}$ thus dropping the sum over the images of particle $\qty(1)$. +We then moved the action of the Killing vector from $x$ to $k$ and finally we used the usual $\delta$ definition. +The previous integral can be expressed as: +\begin{equation} + \begin{split} + I^{(N)} + & = + \cN^N + \sum_{ \qty{ l_{\qty(i)} } \in \Z^N } + e^{% + i \finitesum{i}{1}{N} l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}} + } + \int\limits_{\Omega} \dd[3]{x}\, + \sqrt{-\det g}\, + \finiteprod{i}{1}{N}\phi_{k_-rN{i}}\qty(\qty[x]) + \\ + & = + \cN^N + \sum_{ \qty{ l_{\qty(i)} } \in \Z^N } + e^{% + i \finitesum{i}{1}{N} l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}} + }\, + \qty(2\pi)^2 + \delta\qty( \finitesum{i}{1}{N} k_{\qty(i)\, +} ) + \delta_{\finitesum{i}{1}{N} l_{\qty(i)},\, 0}\, + \cI_{\qty{N}}^{\qty[0]}. + \end{split} +\end{equation} +From the last expression we recover the overlap of the wave functions as: +\begin{equation} + \begin{split} + & \int\limits_{\Omega} \dd[3]{x}\, + \finiteprod{i}{1}{N} + \phi_{k_-rN{i}}\qty( \qty[x] ) + \\ + = & + \frac{1}{\cN^N}\, + \finiteprod{i}{1}{N} + \int\limits_0^{2\pi \Delta \abs{k_{\qty(i)\, +}}} + \frac{\dd{k_{\qty(i)\, 2}}}{2 \pi \Delta \abs{k_{\qty(i)\, +}}}\, + e^{% + -i\, l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, i}} + }\, + I^{(N)} + \\ + = & + \qty(2\pi)^3\, + \delta\qty(\infinfsum{i} k_{\qty(i)\, +} )\, + \frac{1}{\cN^N}\, + \finiteprod{i}{1}{N} + \int\limits_0^{2\pi \Delta \abs{k_{\qty(i)\, +}}} + \frac{\dd{k_{\qty(i)\, 2}}}{2 \pi \Delta \abs{k_{\qty(i)\, +}}}\, + e^{% + -i\, l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}} + }\, + \\ + \times & + \finiteprod{j}{2}{N} \infinfsum{m_{(j)}} + \delta\qty( \finitesum{j}{2}{N} \cK^{m_{(j)} } k_{\qty(j)\, 2} )\, + \delta\qty( \finitesum{j}{2}{N} \cK^{m_{(j)} } k_{\qty(j)\, -} ). + \end{split} +\end{equation} +It follows from the explicit expression of $\cI_{\qty{n}}^{\qty[0]}$ that all overlaps $I^{(N)}$ for $N \ge 4$ diverge. + +Intuitively we are in fact summing over infinite distributions with accumulation points of their support. +Nevertheless the existence of the accumulation point is not sufficient since the three scalars overlap, i.e.\ the three tachyons amplitude, converges. + + +\subsubsection{An Overlap With One Derivative} + +Since we will also compute the amplitude involving two tachyons and one photon, as a preliminary step we consider the overlap in Minkowski space: +\begin{equation} + J_{Mink} + = + i\, + \qty(2\pi)^3\, + \qty(\epsilon_{\qty(1)} \cdot k_{\qty(2)\, 2})\, + \delta\qty( \infinfsum{i} k_{\qty(i)\, +} )\, + \delta\qty( \infinfsum{i} k_{\qty(i)\, 2} )\, + \delta\qty( \infinfsum{i} k_{\qty(i)\, -} ). +\end{equation} +Summing over momenta and polarisations we then get to an expression which depends on equivalence classes as: +\begin{equation} + \begin{split} + J\qty(% + \qty[k_{\qty(1)}, \epsilon_{\qty(1)}],\, + \qty[k_{\qty(2)}],\, + \qty[k_{\qty(3)}] + ) + & = + i\, \qty(2\pi)^3 \delta\qty( \infinfsum{i} k_{\qty(i)\, +} )\, + \\ + & \times + \sum_{\qty{ m_{(i)} } \in \Z^3} + \delta_{m_{\qty(1)},\, 1}\, + \qty(% + \cK^{m_{\qty(1)}} \epsilon_{\qty(1)} + \cdot + \cK^{m_{\qty(2)}} k_{\qty(2)\, 2} + ) + \\ + & \times + \delta\qty( \infinfsum{i} \cK^{m_{(i)} } k_{\qty(i)\, 2} )\, + \delta\qty( \infinfsum{i} \cK^{m_{(i)} } k_{\qty(i)\, -} ). + \end{split} + \label{eq:Spin_001_overlap_from_covering} +\end{equation} +The expression depends only on equivalence classes.\footnotemark{} +\footnotetext{% + In order to prove it, under $\qty(k_{\qty(1)},\, \epsilon_{\qty(1)}) \rightarrow \cK^{s}\qty(k_{\qty(1)},\, \epsilon_{\qty(1)})$ we can use $\cK^{s} a \cdot b = a \cdot \cK^{-s} b$ and the invariance of deltas $\delta^3(\cK^{s}a) = \delta^3(a)$. +} + +The previous expression can be written as +\begin{equation} + J + = + \int\limits_{\Omega} \dd[3]{x}\, + \eta^{\mu\nu}\, + \Psi^{[1]}_{\qty[k_{\qty(1)}, \epsilon_{\qty(1)}]\, \mu}\qty(\qty[x])\, + \ipd{\nu} \Psi_{ \qty[k_{\qty(2)}] }\qty(\qty[x])\, + \Psi_{ \qty[k_{\qty(3)}] }\qty(\qty[x]) +\end{equation} +where we performed the unfolding using $a_{\qty[k_{\qty(1)},\, \epsilon_{\qty(1)}]\, \mu}\qty(\qty[x])$.\footnotemark{} +\footnotetext{% + Clearly we can perform the unfolding using whichever other field and this amount to keep the corresponding $m_{(i)}$ fixed in place of $m_{\qty(1)}$. +} +Notice that the previous expression is invariant despite the fact that the derivatives $\ipd{\mu}$ are not well defined on the orbifold. +The fact that $\Psi^{[1]}_{\mu}$ is not invariant in turns helps in recovering the required invariance. + +We can then evaluate the previous expression with Minkowskian polarisations using~\eqref{eq:spin1_from_covering} which is nothing else but a rearrangement of terms of~\eqref{eq:Spin_001_overlap_from_covering}. +We have: +\begin{equation} + \begin{split} + J + & = + i\, \cN^2 + \sum_{ \qty{ l_{\qty(i)} } \in \Z^3 } + e^{% + i\, \finitesum{i}{1}{3} l_{\qty(i)} \frac{k_{\qty(i)\, 2}}{\Delta k_{\qty(i)\, +}} + }\, + \qty(2\pi)^2\, + \delta\qty( \finitesum{i}{1}{3} k_{\qty(i)\, +} ) + \delta_{\finitesum{i}{1}{3} l_{\qty(i)}} + \\ + & \times + \int\limits_{\Omega} \dd[3]{x}\, + \finiteprod{i}{1}{3} + \phi_{k_-rN{i}}\qty(\qty[x])) + \Biggl\lbrace + \epsilon_{\qty(1)\, +}\, + \qty[ + \frac{i}{2 u} + + + \frac{l_{\qty(2)}^2}{k_{\qty(2)\, +}} \frac{1}{2 \Delta^2\, u^2} + + + \frac{\rN{2}}{2 k_{\qty(2)\, +}} + ] + \\ + & + + \frac{1}{\Delta\, u} + \qty[% + \epsilon_{\qty(1)\, 2}\, + + + \frac{1}{\Delta u} \epsilon_{\qty(1)\, +} \frac{l_{\qty(1)}}{k_{\qty(1)\, +}} + ]\, + l_{\qty(2)} + \\ + & + + \qty[% + \epsilon_{\qty(1)\, -} + + + \epsilon_{\qty(1)\, 2} \frac{1}{\Delta u} \frac{l_{\qty(1)}}{k_{\qty(1)\, +}} + + + \epsilon_{\qty(1)\, +} \frac{1}{2 (\Delta u)^2} \frac{l_{\qty(1)}^2}{k_{\qty(1)\, +}^2} + ]\, + k_{\qty(2)\, +} + \Biggr\rbrace. + \end{split} + \label{eq:divergence_overlap_spin1} +\end{equation} +Divergences occur when $l = 0$ because of the absence of the factor $e^{i \frac{A}{u}}$. +However all explicit factors $\frac{1}{u}$ come always with $l$: when $l = 0$ they do not give any contribution. +The divergence in this case comes actually only from the contribution of the first line $\eval{\ipd{u}\phi}_{l = 0} = -\frac{1}{2u} \eval{\phi}_{l = 0}$. +Since we still have to subtract the contribution of the exchange $\qty(2) \leftrightarrow \qty(3)$ the contribution is cancelled in scalar \qed or with Abelian tachyons. +It does not cancel when considering the non Abelian case and the related colour factors unless one uses a kind of principal part regularisation since replacing $\finiteint{u}{-\abs{a}}{\abs{b}}\, \frac{\sgn(u)}{\abs{u}^{\frac{3}{2}}}$ with $\lim\limits_{\delta \rightarrow 0} \qty[\finiteint{u}{-\abs{a}}{-\abs{\delta}} + \finiteint{u}{-\abs{\delta}}{\abs{b}}]\, \frac{\sgn(u)}{\abs{u}^{\frac{3}{2}}}$ gives a finite result. + + +\subsubsection{An Overlap With Two Derivatives} +\label{sec:overlap} + +We can generalise the previous expressions to more general cases. +Since we use the results from~\Cref{sec:Eigenmodes_from_Covering} we miss some non trivial contributions from polarisations like $\cS_{v\, i}$. +These contributions do not alter the final result. +However for completeness we give the lengthy full expression in~\Cref{sec:NO_full_TTS}. + +We consider:\footnotemark{} +\footnotetext{% + The underlying idea is to compute the amplitudes involving two tachyons and one massive state. +} +\begin{equation} + K + = + \int\limits_{\Omega} \dd[3]{x}\, + \sqrt{-\det g}~ + \eta^{\mu\nu}\, \eta^{\rho\sigma}\, + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, \mu\rho}\qty(\qty[x])\, + \ipd{\nu}\ipd{\sigma} \Psi_{ \qty[k_{\qty(2)}] }\qty(\qty[x])\, + \Psi_{ \qty[k_{\qty(1)}] }\qty(\qty[x]), +\end{equation} +in Minkowskian coordinates or +\begin{equation} + K + = + \int\limits_{\Omega} \dd[3]{x}\, + \sqrt{-\det g}~ + g^{\alpha\beta}\, g^{\gamma\delta}\, + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, \alpha\gamma}\qty(\qty[x])\, + D_{\beta} \ipd{\delta} \Psi_{ \qty[k_{\qty(2)}] }\qty(\qty[x])\, + \Psi_{ \qty[k_{\qty(1)}] }\qty(\qty[x]) +\end{equation} +in orbifold coordinates where we need to use covariant derivatives. +Using the unfolding trick over $\qty(3)$ we get +\begin{equation} + \begin{split} + K + & = + \qty(2\pi)^3\, + \delta\qty( \infinfsum{i} k_{\qty(i)\, +} )\, + \finiteprod{i}{2}{N} \infinfsum{m_{(i)}} + S_{\qty(3) \mu \rho}\, + \qty(\cK^{m_{\qty(2)}} k_{\qty(2)\, 2})^{\mu} + \qty(\cK^{m_{\qty(2)}} k_{\qty(2)\, 2})^{\rho}\, + \\ + & \times + \delta\qty( \infinfsum{i} \cK^{m_{(i)}} k_{\qty(i)\, 2} )\, + \delta\qty( \infinfsum{i} \cK^{m_{(i)}} k_{\qty(i)\, -} ). + \end{split} + \label{eq:Spin_002_overlap_from_covering} +\end{equation} +Explicitly in orbifold coordinates we can write +\begin{equation} + \begin{split} + K + & = + \int\limits_{\Omega} \dd[3]{x}\, + \sqrt{-\det g}~ + \Biggl[ + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, u u }\, + \ipd{v}^2 \Psi_{ \qty[k_{\qty(2)}] } + - + \frac{2}{ \qty(\Delta u)^2 } + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, u z }\, + \ipd{v} \ipd{z} \Psi_{ \qty[k_{\qty(2)}] } + \\ + & + + 2\, \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, u v }\, + \ipd{v} \ipd{u} \Psi_{ \qty[k_{\qty(2)}] } + -\qty( -\frac{L^4 \cS_{v\,v}}{4 \Delta^4} -) -\frac{1}{u^4} - \Bigr] - \Bigr\} - , - \end{align} - with $L=\frac{l}{k_+}$. + \frac{1}{ \qty(\Delta u)^4 } + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, z z }\, + \qty(% + \ipd{z}^2 \Psi_{ \qty[k_{\qty(2)}] } + - + \Delta^2 u\, \ipd{v} \Psi_{ \qty[k_{\qty(2)}] } + ) + \\ + & - + \frac{2}{ \qty(\Delta u)^2 } + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, z v }\, + \qty(% + \ipd{z} \ipd{u} \Psi_{ \qty[k_{\qty(2)}] } + - + \frac{1}{u}\, \ipd{z} \Psi_{\qty[k_{\qty(2)}] } + ) + + + \Psi^{[2]}_{\qty[k_{\qty(3)}, S_{\qty(3)}]\, v v }\, + \ipd{u}^2 \Psi_{ \qty[k_{\qty(2)}] } + \Biggr] + \Psi_{ \qty[k_{\qty(1)}] }. + \end{split} +\end{equation} +Keeping the terms which do not vanish when all $l = 0$ and considering only the leading order in $\frac{1}{u}$ we get +\begin{equation} + K + \sim + \int \dd{u}\, \abs{u}\, + \frac{3}{4} + \frac{\qty(k_{\qty(2)\, +} + k_{\qty(3)\, +})^2}{k_{\qty(3)\, +}^2}\, + \cS_{\qty(3)\, v v}~ + \frac{1}{u^2} + \eval{\finiteprod{i}{1}{3} \phi_{\qty(i)}}_{l_{\qty(*)} = 0}, + \label{eq:divergence_overlap_spin2} +\end{equation} +which is divergent as $\abs{u}^{-\frac{5}{2}}$. + + +\subsection{Three Points Amplitudes with One Massive State in String Theory} +\label{sec:NO3ptsMassive} + +We consider string amplitudes including massive states. +They are obtained using the inheritance principle and therefore they are connected to the integrals and relations derived in~\Cref{sec:overlap}. +In particular we want to use the inheritance principle on the momenta and polarisations, i.e.\ we start form amplitudes in Minkowski expressed with momenta and polarisations and then we implement on them the projection on the orbifold. +In particular it is worth stressing that, as there is one Killing vector acting on the spacetime coordinates, there is only one common Killing vector action on all the momenta and polarisations of each field as discussed for spin-1 and spin-2 cases. +Moreover this approach gives the complete answer only for tree level amplitudes since inside the loops twisted states may be created in pairs. +The final result is that the open string amplitude with two tachyons and the first massive (level 2) state diverges and there is no obvious way of curing it since the divergence is also present in the Abelian sector. +The open string expansion we use is +\begin{equation} + X\qty(u,\, \baru) + = + x_0 + - + i\, 2\ap\, p\, \ln(\abs{u}) + + + i\, \sqrt{\frac{\ap}{2}} + \sum_{n \in \Z \setminus \qty{0}} \frac{\alpha_n}{n} + \qty( u^{-n} + \baru^{-n} ). +\end{equation} + + +\subsubsection{First Massive State in String Theory} + +Before computing the amplitude we would like to review the possible polarisations of the first massive state in open string. +The first massive vertex is: +\begin{equation} + \begin{split} + V_M\qty(x;\, k,\, S,\, \xi) + & = + \colon + \qty(% + \frac{i}{\sqrt{2 \ap}} + \xi \cdot \ipd{x}^2 X\qty(x,\, x) + + + \qty( \frac{i}{\sqrt{2 \ap}} )^2 + S_{\mu\nu}\, \ipd{x} X^{\mu}\qty(x,\, x)\, \ipd{x} X^{\nu}\qty(x,\, x) + ) + \\ + & \times + e^{i\, k \cdot X\qty(x,\, x)} + \colon. + \end{split} +\end{equation} +The corresponding state is: +\begin{equation} + \lim\limits_{x \rightarrow 0} + V_M\qty(x;\, k,\, S,\, \xi) \ket{0} + = + \ket{k,\, S,\, \xi} + = + \qty( \xi \cdot \alpha_{-2} + \alpha_{-1} \cdot S \cdot \alpha_{-1} ) + \ket{k}. +\end{equation} +For the state to be physical we require: +\begin{align} + \begin{split} + \qty( L_0 - 1) + \ket{k,\, S,\, \xi} + & = + 0 + \qquad \Rightarrow \qquad + \ap k^2 = -1 + \\ + L_1 + \ket{k,\, S,\, \xi} + & = + 0 + \qquad \Rightarrow \qquad + S \cdot k + \xi = 0 + \\ + L_2 + \ket{k,\, S,\, \xi} + & = + 0 + \qquad \Rightarrow \qquad + k \cdot \xi + \tr S = 0. + \end{split} +\end{align} +String gauge invariance allows us to add: +\begin{equation} + L_{-1} + \qty( \chi \cdot \alpha_{-1} \ket{k} ) + = + \qty( \chi \cdot \alpha_{-2} + \chi \cdot \alpha_{-1}\, k \cdot \alpha_{-1} ) + \ket{k}, +\end{equation} +subject to the physical constraints $\ap k^2 + 1 = 0$ and $\chi \cdot k = 0$. +In critical string theory there is another gauge invariance generated by $L_{-2} + \frac{3}{2} L_{-1}^2$. +We can add a multiple of +\begin{equation} + \qty( L_{-2} + \frac{3}{2} L_{-1}^2 ) + \ket{k} + = + \qty(% + \frac{5}{2} k \cdot \alpha_{-2} + + + \frac{3}{2} \qty( k \cdot \alpha_{-1} )^2 + + + \frac{1}{2} \alpha_{-1}^2 + )\, + \ket{k}, +\end{equation} +to set $a = 0$. +Therefore the only non trivial \dof refer to $S^{TT}$, that is: +\begin{equation} + \tr S^{TT} = k \cdot S^{TT} = \xi = 0. +\end{equation} + +We check that, given $k = \qty(k_+,\, k_-,\, k_2,\, \vec{k})$ such that $-2\, k_+ k_- + k_2^2 + \norm{\vec{k}}^2 = -1$, we can find a non trivial $S^{TT}$ with non vanishing components in the directions $\pm,\, 2$ only. +In fact we find a two parameters family of solutions. +The parameters may be taken to be $S_{+\, +}$ and $S_{+\, 2}$. +Explicitly we have +\begin{equation} + \mqty(% + S_{+\, +} \\ + S_{+\, -} \\ + S_{+\, 2} \\ + S_{-\, -} \\ + S_{-\, 2} \\ + S_{2\, 2} + ) + = + \mqty(% + 1 \\ + -\frac{k_-}{k_+} \\ + 0 \\ + \frac{k_- \qty(k_- k_+ -2 k_2^2)}{k_+^3} \\ + -2 \frac{k_- k_2}{k_+^2} \\ + -2 \frac{k_-}{k_+} + )\, + S_{+\, +} + + + \mqty(% + 0 \\ + \frac{k_2}{k_+} \\ + 1\\ + \frac{2 k_2 \qty(-k_- k_+ + k_2^2)} {k_+^3} \\ + \frac{k_- k_+ -2 k_2^2} {k_+^2} \\ + 2 \frac{k_2}{k_+} + )\, + S_{+\, 2} +\end{equation} +There is even a non trivial solution for the special case $k = \qty(k_+,\, k_- = \frac{1}{k_+},\, k_2 = 0, \vec{k} = \vec{0})$. + +Using the expressions for $S^{T T}$ in orbifold coordinates, we check that there are two possible indepdendent polarisations $\cS_{v\, v}$ and $\cS_{v\, z}$ which correspond to the those used above. +The non trivial solution is: +\begin{equation} + \mqty(% + \cS_{v\, v} \\ + \cS_{u\, v} \\ + \cS_{v\, z} \\ + \cS_{u\, u} \\ + \cS_{u\, z} \\ + \cS_{z\, z} + ) + = + \mqty(% + 1 \\ + - \frac{r + \norm{\vec{k}}^2}{2 k_+^2} \\ + 0 \\ + \qty( \frac{r + \norm{\vec{k}}^2}{2 k_+^2} )^2 \\ + 0 \\ + -2 \frac{r + \norm{\vec{k}}^2}{2 k_+^2} + )\, + \cS_{v\, v} + + + \mqty(% + 0 \\ + -\frac{r + \norm{\vec{k}}^2}{2 k_+^2} \\ + 1 \\ + 0 \\ 0\\ 0\\ + )\, + \cS_{v\, z}. +\end{equation} + + +\subsubsection{Two Tachyons and the First Massive State} + +This Minkowskian amplitude is given by the sum of two colour ordered sub-parts as: +\begin{equation} + \cA_{TTM} + = + A_{T_{\qty(1)} T_{\qty(2)} M_{\qty(3)}}\, + \tr(T_{\qty(1)} T_{\qty(2)} T_{\qty(3)}) + + + A_{T_{\qty(2)} T_{\qty(1)} M_{\qty(3)}}\, + \tr(T_{\qty(2)} T_{\qty(1)} T_{\qty(3)}). +\end{equation} +We find: +\begin{equation} + \begin{split} + A_{T_{\qty(1)} T_{\qty(2)} M_{3)}} + & = + \left\langle\left\langle k_{\qty(1)} \right.\right|\, + V_T\qty(1;\, k_{\qty(2)})\, + \qty(% + \alpha_{-1} \cdot S_{\qty(3)}^{TT} \cdot \alpha_{-1} \ket{k_{\qty(3)}} + ) + \\ + & = + \left\langle\left\langle k_{\qty(1)} \right.\right|\, + e^{i\, k_{\qty(2)} \cdot x_0}\, + e^{-\sqrt{2 \ap} k_{\qty(2)} \cdot \alpha_{1}}\, + \qty(% + \alpha_{-1} \cdot S_{\qty(3)}^{TT} \cdot \alpha_{-1} \ket{k_{\qty(3)}} + ) + \\ + & = + \qty(2\pi)^D\, + \qty(\sqrt{2 \ap})^2\, + \delta^D\qty(\finitesum{i}{1}{3} k_{\qty(i)} )\, + k_{\qty(2)} \cdot S_{\qty(3)}^{TT} \cdot k_{\qty(2)}. + \end{split} +\end{equation} +The transversality of $S_{\qty(3)}^{TT}$ finally leads to: +\begin{equation} + \cA_{TTM} + = + 2\, + \qty(2\pi)^D\, + \qty(\sqrt{2 \ap})^2\, + \delta^D\qty( \finitesum{i}{1}{3} k_{\qty(i)} )\, + k_{\qty(2)} \cdot S_{\qty(3)}^{TT} \cdot k_{\qty(2)}\, + \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)} ). +\end{equation} +Then we can compute the orbifold amplitude as: +\begin{equation} + \begin{split} + \cA_{TTM} + & = + \qty(2\pi)^{D-2} + \delta^{D-3}\qty( \finitesum{i}{1}{3} \vec{k}_{\qty(i)} ) + \delta\qty( \finitesum{i}{1}{3} k_{\qty(i)\, +} ) + \\ + & \times + 2 \qty(\sqrt{2 \ap})^2\, + \sum_{ \qty{m_{\qty(1)},\, m_{\qty(2)},\, m_{\qty(3)}} \in \Z^3 }\, + \delta_{m_{\qty(3)},\, 1}\, + \qty(\cK^{m_{\qty(2)}} k_{\qty(2)}) + \cdot + S_{\qty(3)}^{TT} + \cdot + \qty(\cK^{m_{\qty(2)}} k_{\qty(2)}) + \\ + & \times + \delta\qty( \finitesum{i}{1}{3} \qty(\cK^{m_{\qty(i)}} k_{\qty(i)\, 2}) )\, + \delta\qty( \finitesum{i}{1}{3} \qty(\cK^{m_{\qty(i)}} k_{\qty(i)\, -}) )\, + \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)} ). + \end{split} +\end{equation} +Such amplitude can then be expressed using an overlap: +\begin{equation} + \begin{split} + \cA_{TTM} + & = + 2\, \qty(-i \sqrt{2 \ap} )^2\, + \int\limits_{\Omega} \dd^3x\, + g^{\mu\nu}\, g^{\rho\sigma}\, + \Psi^{[2]}_{\qty[k_{\qty(3)},\, S_{\qty(3)}]\, \mu\rho}\qty(\qty[x])\, + \ipd{\nu}\ipd{\sigma} \Psi_{\qty[k_{\qty(2)}] }\qty(\qty[x])\, + \Psi_{\qty[k_{\qty(1)}] }\qty(\qty[x]) + \\ + & \times + \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)}), + \\ + & = + 2\, \qty(-i \sqrt{2 \ap} )^2\, + \int\limits_{\Omega} \dd^3x\, + g^{\alpha\beta}\, g^{\gamma\delta}\, + \Psi^{[2]}_{\qty[k_{\qty(3)},\, S_{\qty(3)}]\, \alpha\gamma}\qty(\qty[x])\, + D_{\beta} \partial_{\delta} \Psi_{\qty[k_{\qty(2)}] }\qty(\qty[x])\, + \Psi_{\qty[k_{\qty(1)}] }\qty(\qty[x]) + \\ + & \times + \tr\qty( \liebraket{T_{\qty(1)}}{T_{\qty(2)}}_+\, T_{\qty(3)} ). + \end{split} +\end{equation} +As discussed in~\Cref{sec:overlap} the integral is divergent when $S_{+\, +} =\cS_{v\, v} \neq 0$ and the divergence cannot be avoided even introducing a Wilson line around $z$ since the amplitude involves an anticommutator which does not vanish in the Abelian sector. + \subsection{Summary and Conclusions} diff --git a/thesis.tex b/thesis.tex index d15e228..40e0410 100644 --- a/thesis.tex +++ b/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{}