From 6f2efba172506f55afd193892cae6703c33150b8 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Fri, 30 Oct 2020 12:25:29 +0100 Subject: [PATCH] Correct typo with regex Signed-off-by: Riccardo Finotello --- sec/part2/divergences.tex | 207 +++++++++++++++++++------------------- thesis.pdf | Bin 3104605 -> 3105797 bytes 2 files changed, 106 insertions(+), 101 deletions(-) diff --git a/sec/part2/divergences.tex b/sec/part2/divergences.tex index c3e777c..ff2b41a 100644 --- a/sec/part2/divergences.tex +++ b/sec/part2/divergences.tex @@ -287,14 +287,14 @@ We therefore have \end{equation} Using Fourier transforms it follows that the eigenmodes are \begin{equation} - \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) = e^{i k_+ v + i l z + i \vec{k} \cdot \vec{x}}\, - \tphi_{k_-kr}(u), + \tphi_{\kmkr}(u), \end{equation} with \begin{equation} - \tphi_{k_-kr}(u) + \tphi_{\kmkr}(u) = \frac{1}{\sqrt{\qty( 2 \pi )^D~ \abs{2 \Delta k_+\, u}}} e^{ @@ -304,7 +304,7 @@ with \end{equation} and \begin{equation} - \phi^*_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + \phi^*_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) = \phi_{\mkmkr}\qty(u,\, v,\, z,\, \vec{x}). \end{equation} @@ -312,7 +312,7 @@ We chose the numeric factor in order to get a canonical normalisation: \begin{equation} \begin{split} & - \qty( \phi_{k_-krN{1}},\, \phi_{k_-krN{2}} ) + \qty( \phi_{\kmkrN{1}},\, \phi_{\kmkrN{2}} ) \\ = & \int\limits_{\R^{D-3}} \dd[D-3]{\vec{x}}\, @@ -320,7 +320,7 @@ We chose the numeric factor in order to get a canonical normalisation: \infinfint{v}\, \finiteint{z}{0}{2\pi} \abs{\Delta u}\, - \phi_{k_-krN{1}}\, \phi_{k_-krN{2}} + \phi_{\kmkrN{1}}\, \phi_{\kmkrN{2}} \\ = & \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\, @@ -337,8 +337,8 @@ We can then perform the off-shell expansion \infinfint{k_+} \infinfint{r} \infinfsum{l} - \cA_{k_-kr}\, - \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}), + \cA_{\kmkr}\, + \phi_{\kmkr}\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 \infinfint{r} \infinfsum{l} \qty(r - M^2)\, - \cA_{k_-kr}\, - \cA_{k_-kr}^*. + \cA_{\kmkr}\, + \cA_{\kmkr}^*. \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_{k_-kr\, \alpha}(u)} + \norm{\tildea_{\kmkr\, \alpha}(u)} \,= \mqty(% \tildea_u @@ -479,7 +479,7 @@ We get the solutions: \qty{ \underu, \underv, \underz,\underi } } \pol{\alpha} - \norm{\tildea^{\underline{\alpha}}_{k_-kr\, \alpha}(u)} + \norm{\tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)} \\ & = \pol{u} @@ -492,7 +492,7 @@ We get the solutions: \\ 0 )\, - \tphi_{k_-kr}(u) + \tphi_{\kmkr}(u) \\ & + \pol{v} @@ -507,7 +507,7 @@ We get the solutions: \\ 0 )\, - \tphi_{k_-kr}(u) + \tphi_{\kmkr}(u) \\ & + \pol{z} @@ -520,7 +520,7 @@ We get the solutions: \\ 0 )\, - \tphi_{k_-kr}(u) + \tphi_{\kmkr}(u) \\ & + \pol{j} @@ -533,7 +533,7 @@ We get the solutions: \\ \delta_{\underline{ij}} )\, - \tphi_{k_-kr}(u), + \tphi_{\kmkr}(u), \label{eq:Orbifold_spin1_pol} \end{split} \end{equation} @@ -549,13 +549,13 @@ then we can expand the off-shell fields as } \infinfsum{l} \pol{\alpha}\, - {a}^{\underline{\alpha}}_{k_-kr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ), + {a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ), \end{equation} where \begin{equation} - a^{\underline{\alpha}}_{k_-kr\, \alpha}\qty(u,\, v,\, z,\, \vec{x}) + a^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x}) = - \tildea^{\underline{\alpha}}_{k_-kr\, \alpha}(u)\, + \tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)\, e^{i\, \qty( k_+ v + l z + \vec{k} \cdot \vec{x})} \end{equation} and $\int \ccD k = \int\limits_{\R^{D-3}} \dd[D-3]{\vec{k}} \infinfint{k_+} \infinfint{r}$. @@ -573,7 +573,7 @@ We can also compute the normalisation as \\ & \times g^{\alpha\beta}\, - a_{k_-krN{1}\, \alpha}\, a_{k_-krN{2}\, \beta} + a_{\kmkrN{1}\, \alpha}\, a_{\kmkrN{2}\, \beta} \\ & = \genpolN{1} \circ \genpolN{2} @@ -624,9 +624,9 @@ The photon kinetic term becomes \infinfint{r} \infinfsum{l}\, \frac{r}{2}\, - \cE_{k_-kr}\, + \cE_{\kmkr}\, \circ - \cE_{k_-kr}^*. + \cE_{\kmkr}^*. \end{equation} @@ -654,7 +654,7 @@ Its computation involves integrals such as \int \dd{u}\, \abs{\Delta u}\, \qty(\frac{l}{u})^2 - \finiteprod{i}{1}{3} \tphi_{k_-krN{i}} + \finiteprod{i}{1}{3} \tphi_{\kmkrN{i}} \sim \int\limits_{u \sim 0} \dd{u}\, \qty(\frac{l^2}{\abs{u}^{\frac{5}{2}}}) @@ -668,7 +668,7 @@ and \int \dd{u}\, \abs{\Delta u}\, \qty(\frac{1}{u}) - \finiteprod{i}{1}{3} \tphi_{k_-krN{i}} + \finiteprod{i}{1}{3} \tphi_{\kmkrN{i}} \sim \int\limits_{u \sim 0} \dd{u}\, \qty(\frac{1}{u\, \abs{u}^{\frac{1}{2}}}) @@ -720,7 +720,7 @@ and we get:\footnotemark{} \delta_{\finitesum{i}{1}{3} l_{\qty(i)},\, 0}\, \\ & \times - \qty(\cA_{\mkmkrN{2}})^*\, \cA_{k_-krN{3}} + \qty(\cA_{\mkmkrN{2}})^*\, \cA_{\kmkrN{3}} \\ & \times \Biggl\lbrace @@ -782,19 +782,19 @@ In the previous expressions we also defined for future use: \infinfint{u}\, \abs{\Delta u}\, u^{\nu}\, \finiteprod{i}{1}{N} - \tphi_{k_-krN{i}} + \tphi_{\kmkrN{i}} \\ \cJ_{\qty{N}}^{\qty[\nu]} & = & \infinfint{u}\, \abs{\Delta}\, \abs{u}^{1 + \nu} - \finiteprod{i}{1}{N} \tphi_{k_-krN{i}}. + \finiteprod{i}{1}{N} \tphi_{\kmkrN{i}}. \end{eqnarray} For the sake of brevity from now on we use \begin{eqnarray} - \tphi_{\qty(i)} & = & \tphi_{k_-krN{i}}, + \tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}}, \\ - \tphi_{\qty(i)} & = & \tphi_{k_-krN{i}} + \tphi_{\qty(i)} & = & \tphi_{\kmkrN{i}} \end{eqnarray} when not causing confusion. @@ -835,7 +835,7 @@ which can be expressed using the modes as: & \times \Biggl\lbrace e^2\, - \qty(\cA_{\mkmkrN{3}})^* \cA_{k_-krN{4}} + \qty(\cA_{\mkmkrN{3}})^* \cA_{\kmkrN{4}} \\ & \times \Biggl[ @@ -879,8 +879,8 @@ where \qty(\cA_{\mkmkrN{2}})^*\, \\ & \times - \cA_{k_-krN{3}}\, - \cA_{k_-krN{4}}. + \cA_{\kmkrN{3}}\, + \cA_{\kmkrN{4}}. \end{split} \end{equation} When setting $l_{\qty(*)} = 0$ all the surviving terms are divergent. @@ -896,12 +896,12 @@ 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_{k_-kr} + A\, \ipd{u} \tphi_{\kmkr} + - B(u)\, \tphi_{k_-kr} + B(u)\, \tphi_{\kmkr} = A\, e^{-\int^u \frac{B(u)}{A} du}\, - \ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{k_-kr} ] + \ipd{u} \qty[ e^{+\int^u \frac{B(u)}{A} du} \tphi_{\kmkr} ] = 0, \end{equation} @@ -1269,7 +1269,7 @@ to finally get:\footnotemark{} & = \cN\, \infinfsum{l} - \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) e^{i\, l\, \frac{k_2}{\Delta k_+}}, \end{split} \label{eq:Psi_phi} @@ -1284,7 +1284,7 @@ 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_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) & = \frac{1}{\cN}\, \frac{1}{2 \pi \Delta \abs{k_+}} @@ -1336,7 +1336,7 @@ The explicit expression for the eigenfunction with constant $\epsilon_+$, $\epsi } \\ & = - \cN + \cN\, \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(u,\, v,\, z), \end{split} \end{equation} @@ -1370,7 +1370,7 @@ Building the corresponding function on the orbifold amounts to summing the image \psi_{k}\qty( \cK^n x) = \infinfsum{n} - \cK^{-n} + \cK^{-n}\, \vec{\epsilon} \cdot \dd{x}~ \psi_{\cK^{-n} k}\qty(x). \end{split} @@ -1508,7 +1508,7 @@ Then it follows that \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 cannot be expressed using the quantum numbers $\kmkr$ 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} @@ -1516,7 +1516,7 @@ Using~\eqref{eq:Psi_phi} we can finally write \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) & = \infinfsum{l} - \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) e^{i\, l \frac{k_2}{\Delta k_+}} \\ & \times @@ -1931,7 +1931,7 @@ The final expression for the orbifold symmetric tensor is \Psi^{[2]}_{\qty[\vec{k},\, S]}\qty(\qty[x]) & = \infinfsum{l} - \phi_{k_-kr}\qty(u,\, v,\, z,\, \vec{x}) + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) e^{i\, l \frac{k_2}{\Delta k_+}} \\ & \times @@ -2870,26 +2870,29 @@ We therefore need solve: \end{equation} To this purpose, we introduce a Fourier transformation over $v,\, w,\, z,\, \vec{x}$: \begin{equation} - \phi_r\qty( u,\, v,\, w,\, z,\, \vec{x}) - = - \infinfsum{l}\, - \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}} - \infinfint{k_+} - \infinfint{p}\, - e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )} - \tphi_{k_-krgen}(u), + \begin{split} + & \phi_r\qty( u,\, v,\, w,\, z,\, \vec{x}) + \\ + & = + \infinfsum{l}\, + \int\limits_{\R^{D-4}} \dd[D-4]{\vec{k}} + \infinfint{k_+} + \infinfint{p}\, + e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )} + \tphi_{\kmkrgen}(u), + \end{split} \end{equation} where we defined $k_+,\, p,\, l,\, \vec{k}$ as associated momenta to $v,\, w,\, z,\, \vec{x}$ respectively. We find: \begin{equation} - \phi_{k_-krgen}\qty( u,\, v,\, w,\, z,\, \vec{x} ) + \phi_{\kmkrgen}\qty( u,\, v,\, w,\, z,\, \vec{x} ) = e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )} - \tphi_{k_-krgen}( u ). + \tphi_{\kmkrgen}( u ). \end{equation} where \begin{equation} - \tphi_{k_-krgen}( u ) + \tphi_{\kmkrgen}( u ) = \frac{1}{2 \sqrt{\qty(2 \pi)^D \abs{\Delta_2 \Delta_3 k_+}}}\, \frac{1}{\abs{u}} @@ -2906,7 +2909,7 @@ where These solutions present the right normalisation, as we can verify through the product: \begin{equation} \begin{split} - & \left( \phi_{k_-krgenN{1}},\, \phi_{k_-krgenN{2}} \right) + & \left( \phi_{\kmkrgenN{1}},\, \phi_{\kmkrgenN{2}} \right) \\ & = \int\limits_{\R^{D-4}} \dd[D-4]{\vec{x}} @@ -2917,8 +2920,8 @@ These solutions present the right normalisation, as we can verify through the pr 2 \abs{\Delta_2 \Delta_3} u^2 \\ & \times - \phi_{k_-krgenN{1}}~ - \phi_{k_-krgenN{2}} + \phi_{\kmkrgenN{1}}~ + \phi_{\kmkrgenN{2}} \\ & = \delta^{D - 4}\qty( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)} )\, @@ -2942,7 +2945,7 @@ Then we have the off-shell expansion: \infinfint{r} \\ & \times - \frac{\cA_{k_-krgen}}{\abs{u}} + \frac{\cA_{\kmkrgen}}{\abs{u}} e^{% i\, \qty(% k_+ v + p w + l z + \vec{k} \cdot \vec{x} @@ -3115,15 +3118,15 @@ These equations can be solved using standard techniques through a Fourier transf \\ & \times e^{i\, \qty( k_+ v + p w + l z + \vec{k} \cdot \vec{x} )} - \tildea_{k_-krgen\, \alpha}(u). + \tildea_{\kmkrgen\, \alpha}(u). \end{split} \end{equation} -We first solve the equations for $\tildea_{k_-krgen\, v}$ and $\tildea_{k_-krgen\, i}$ since they are identical to the scalar equation~\eqref{eq:scalar_eom}. -We then insert their solutions as sources for the equations for $\tildea_{k_-krgen\, u}$, $\tildea_{k_-krgen\, w}$ and $\tildea_{k_-krgen\, z}$. +We first solve the equations for $\tildea_{\kmkrgen\, v}$ and $\tildea_{\kmkrgen\, i}$ since they are identical to the scalar equation~\eqref{eq:scalar_eom}. +We then insert their solutions as sources for the equations for $\tildea_{\kmkrgen\, u}$, $\tildea_{\kmkrgen\, w}$ and $\tildea_{\kmkrgen\, z}$. The solutions can be written as the expansion: \begin{equation} \begin{split} - \norm{\tildea_{k_-krgen\, \alpha}(u)} + \norm{\tildea_{\kmkrgen\, \alpha}(u)} & = \mqty(% \tildea_u @@ -3139,16 +3142,16 @@ The solutions can be written as the expansion: \\ & = \sum\limits_{\ualpha \in \qty{\underu, \underv, \underw, \underz, \underi}} - \cE_{k_-krgen\, \ualpha}\, - \norm{\tildea^{\ualpha}_{k_-krgen\, \alpha}(u)} + \cE_{\kmkrgen\, \ualpha}\, + \norm{\tildea^{\ualpha}_{\kmkrgen\, \alpha}(u)} \\ & = - \cE_{k_-krgen\, \underu}\, + \cE_{\kmkrgen\, \underu}\, \mqty( 1 \\ 0 \\ 0 \\ 0 \\ 0 )\, - \tphi_{k_-krgen} + \tphi_{\kmkrgen} \\ & + - \cE_{k_-krgen\, \underv}\, + \cE_{\kmkrgen\, \underv}\, \mqty(% \frac{i}{2 k_+ u} + @@ -3163,10 +3166,10 @@ The solutions can be written as the expansion: \\ 0 )\, - \tphi_{k_-krgen} + \tphi_{\kmkrgen} \\ & + - \cE_{k_-krgen\, \underw}\, + \cE_{\kmkrgen\, \underw}\, \mqty( \frac{1}{4 k_+ \abs{u}} \qty( \frac{l + p}{\Delta_2^2} - \frac{l - p}{\Delta_3^2} ) @@ -3179,10 +3182,10 @@ The solutions can be written as the expansion: \\ 0 )\, - \tphi_{k_-krgen} + \tphi_{\kmkrgen} \\ & + - \cE_{k_-krgen\, \underz}\, + \cE_{\kmkrgen\, \underz}\, \mqty( \frac{1}{4 k_+ \abs{u}} \qty( \frac{l + p}{\Delta_2^2} + \frac{l - p}{\Delta_3^2} ) @@ -3195,20 +3198,20 @@ The solutions can be written as the expansion: \\ 0 )\, - \tphi_{k_-krgen} + \tphi_{\kmkrgen} \\ & + - \cE_{k_-krgen\, \underj}\, + \cE_{\kmkrgen\, \underj}\, \mqty( 0 \\ 0 \\ 0 \\ 0 \\ \delta_{\underline{i j}} )\, - \tphi_{k_-krgen} + \tphi_{\kmkrgen} \end{split} \end{equation} Consider the Fourier transformed functions: \begin{equation} - a^{\ualpha}_{k_-krgen\, \alpha}\qty( u,\, v,\, w,\, z,\, \vec{x} ) + a^{\ualpha}_{\kmkrgen\, \alpha}\qty( u,\, v,\, w,\, z,\, \vec{x} ) = e^{i\, \qty(k_+ v + p w + l z + \vec{k} \cdot \vec{x})} - \tildea^{\ualpha}_{k_-krgen\, \alpha}( u ), + \tildea^{\ualpha}_{\kmkrgen\, \alpha}( u ), \end{equation} then we can expand the off shell fields as \begin{equation} @@ -3223,8 +3226,8 @@ then we can expand the off shell fields as \\ & \times \sum_{\ualpha \in \qty{\underu, \underv, \underw, \underz, \underi}} - \cE_{k_-krgen\, \alpha}\, - a^{\ualpha}_{k_-krgen\, \alpha}(x). + \cE_{\kmkrgen\, \alpha}\, + a^{\ualpha}_{\kmkrgen\, \alpha}(x). \end{split} \end{equation} @@ -3241,7 +3244,7 @@ We can compute the normalisation as: 2 \abs{\Delta_2 \Delta_3} u^2 \\ & \times - \qty(g^{\alpha\beta}\, a_{k_-krgenN{1}\, \alpha}\, a_{k_-krgenN{2}\, \beta}) + \qty(g^{\alpha\beta}\, a_{\kmkrgenN{1}\, \alpha}\, a_{\kmkrgenN{2}\, \beta}) \\ & = \delta^{D-4}\qty( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)} )\, @@ -3251,7 +3254,7 @@ We can compute the normalisation as: \delta\qty( r_1 - r_2 ) \\ & \times - \cE_{k_-krgenN{1}} \circ \cE_{k_-krgenN{2}}, + \cE_{\kmkrgenN{1}} \circ \cE_{\kmkrgenN{2}}, \end{split} \end{equation} where @@ -3285,16 +3288,16 @@ where is independent of the coordinates. The Lorenz gauge now reads: \begin{equation} - \eta^{i\underj}\, k_i \, \cE_{{k_-krgen} \underj} + \eta^{i\underj}\, k_i \, \cE_{{\kmkrgen} \underj} - k_+ - \cE_{k_-krgen\, \underu} + \cE_{\kmkrgen\, \underu} - \frac{\vec{k}^2 + r}{2 k_+} - \cE_{k_-krgen\, \underv} + \cE_{\kmkrgen\, \underv} = 0. \end{equation} -As in the previous case, the constraint equation does not pose any condition on the transverse polarisations $\cE_{k_-krgen\, \underw}$ and $\cE_{k_-krgen\, \underz}$. +As in the previous case, the constraint equation does not pose any condition on the transverse polarisations $\cE_{\kmkrgen\, \underw}$ and $\cE_{\kmkrgen\, \underz}$. \subsubsection{Cubic Interaction} @@ -3331,16 +3334,16 @@ In the case of the \gnbo we find: & \times e~ \cA^*_{\mkmkrgenN{2}} - \cA_{k_-krgenN{3}} + \cA_{\kmkrgenN{3}} \\ & \times \Biggl\lbrace - \cE_{k_-krgenN{1}\, \underu}~ + \cE_{\kmkrgenN{1}\, \underu}~ k_{\qty(2)\, +}~ \cI_{\qty{3}}^{\qty[0]} \\ & + - \cE_{k_-krgenN{1}\, \underv}~ + \cE_{\kmkrgenN{1}\, \underv}~ \Biggl[ \qty( \frac{\vec{k}_{\qty(2)}^2 + r_{(2)}}{2 k_{\qty(2)\, +}} )\, \cI_{\qty{3}}^{\qty[0]} @@ -3370,7 +3373,7 @@ In the case of the \gnbo we find: \Biggr] \\ & + - \qty( \cE_{k_-krgenN{1}\, \underw} - \cE_{k_-krgenN{1}\, \underz} ) + \qty( \cE_{\kmkrgenN{1}\, \underw} - \cE_{\kmkrgenN{1}\, \underz} ) \\ & \times \Biggl[ @@ -3402,18 +3405,18 @@ where we defined: \cI_{\qty{N}}^{\qty[\nu]} & = \infinfint{u} - 2\, \abs{\Delta_2 \Delta_3} u^2\, u^{\nu}\, \finiteprod{i}{1}{N} \tphi_{k_-krgenN{i}}, + 2\, \abs{\Delta_2 \Delta_3} u^2\, u^{\nu}\, \finiteprod{i}{1}{N} \tphi_{\kmkrgenN{i}}, \\ \cJ_{\qty{N}}^{\qty[\nu]} & = \infinfint{u} - 2\, \abs{\Delta_2 \Delta_3} u^2\, \abs{u}^{\nu}\, \finiteprod{i}{1}{N} \tphi_{k_-krgenN{i}}. + 2\, \abs{\Delta_2 \Delta_3} u^2\, \abs{u}^{\nu}\, \finiteprod{i}{1}{N} \tphi_{\kmkrgenN{i}}. \end{split} \end{equation} While in the \nbo case we need to regularise the integrals at least taking their principal part when all $l_{(*)} = 0$ in~\eqref{eq:nbo_div_integral}, the \gnbo does not need any specific manipulation. -In fact the form of $\tphi_{k_-krgenN{i}}$ in~\eqref{eq:GNBO_reg_wave_functions} prevents the formation of isolated zeros in the phase factor proportional to $u^{-1}$: the presence of the continuous momentum $p$, contrary to the \nbo where all momenta are discrete, gives the integrals a distributional interpretation, similar to a derivative of a Dirac $\delta$ function. +In fact the form of $\tphi_{\kmkrgenN{i}}$ in~\eqref{eq:GNBO_reg_wave_functions} prevents the formation of isolated zeros in the phase factor proportional to $u^{-1}$: the presence of the continuous momentum $p$, contrary to the \nbo where all momenta are discrete, gives the integrals a distributional interpretation, similar to a derivative of a Dirac $\delta$ function. \subsubsection{Quartic Interactions} @@ -3454,16 +3457,16 @@ As for the \nbo, we consider the quartic interaction for the scalar \qed action: \Biggl\lbrace e^2 \cA^*_{\mkmkrgenN{3}} - \cA_{k_-krgenN{4}} + \cA_{\kmkrgenN{4}} \\ & \times \Biggl[ - \cE_{k_-krgenN{1}} \circ \cE_{k_-krgenN{2}}\, + \cE_{\kmkrgenN{1}} \circ \cE_{\kmkrgenN{2}}\, \cI_{\qty{4}}^{\qty[0]} \\ & - i - \cE_{k_-krgenN{1}\, \underv}\, - \cE_{k_-krgenN{2}\, \underv} + \cE_{\kmkrgenN{1}\, \underv}\, + \cE_{\kmkrgenN{2}\, \underv} \\ & \times \Biggl( @@ -3497,8 +3500,8 @@ As for the \nbo, we consider the quartic interaction for the scalar \qed action: \cA^*_{\mkmkrgenN{2}} \\ & \times - \cA_{k_-krgenN{3}} - \cA_{k_-krgenN{4}} + \cA_{\kmkrgenN{3}} + \cA_{\kmkrgenN{4}} \cI_{\qty{4}}^{\qty[0]} \Biggr\rbrace, \end{split} @@ -3514,16 +3517,16 @@ where we defined: \\ \tilde{\cE}_{\pm\,\left( a, b \right)} & = - \cE_{k_-krgenN{a}\, \underv} + \cE_{\kmkrgenN{a}\, \underv} \\ & \times - \qty( \cE_{k_-krgenN{b}\, \underw} \pm \cE_{k_-krgenN{b}\, \underz} ) + \qty( \cE_{\kmkrgenN{b}\, \underw} \pm \cE_{\kmkrgenN{b}\, \underz} ) \\ & - - \cE_{k_-krgenN{b}\, \underv} + \cE_{\kmkrgenN{b}\, \underv} \\ & \times - \qty( \cE_{k_-krgenN{a}\, \underw} \pm \cE_{k_-krgenN{a}\, \underz} ) + \qty( \cE_{\kmkrgenN{a}\, \underw} \pm \cE_{\kmkrgenN{a}\, \underz} ) \end{split} \end{equation} for simplicity. @@ -3931,6 +3934,8 @@ Now we use the basic trick used in Poisson resummation \abs{\frac{k_+ x^+}{k_- x^-}}^{-i \frac{l}{2 \Delta} }\, \infinfint{s} e^{i\, 2 \pi\, l\, s} + \\ + & \times e^{i\, \sgn(k_+\, x^+) \sqrt{\abs{k_+ k_- x^+ x^-}} \qty{ \Lambda^s @@ -3973,7 +3978,7 @@ Now we use the basic trick used in Poisson resummation \end{equation} where the last line represents the change of quantum number from $m\, \beta$ to $m\, l$ and allows us to identify \begin{equation} - \cN_{\text{BO}} + \cN_{\text{BO}}\, \tphi_{\lsi}(\tau) = \frac{1}{2\pi}\, diff --git a/thesis.pdf b/thesis.pdf index c35453e5e5959810d63332e5268edec360c4491b..29d73b83d534c0e28df7a4bdb3e5f3046e79f135 100644 GIT binary patch delta 138741 zcmV(&K;ggLzP$m3$h`r8gaL#Cgad>Ggaxz)@5%x+HJ9p7oY*30vHGk^Q~yR+x7&pw=S 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