\documentclass[ ,final % use final for the camera ready runs %% ,draft % use draft while you are working on the paper %% ,numberedheadings % uncomment this option for numbered sections %% , % add further options here if necessary ] {aipproc} \usepackage{amsmath,amssymb} \layoutstyle{8x11double} %\documentclass[11pt,a4paper]{article} \usepackage{amsmath,amssymb} %\usepackage{epsfig,graphicx} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqn{\begin{eqnarray}} \def\eeqn{\end{eqnarray}} %\topmargin -0.4cm %\headsep=0.0cm %\headheight=0.0cm %\textheight 24.6cm %\oddsidemargin -0.3cm %\evensidemargin -0.3cm %\textwidth 15.9cm \begin{document} \title{ BFKL Pomeron and production amplitudes in $N=4$ SUSY} %\footnote{Talk at the International Conference "Landau-100"} \classification{11.30.Pb} \keywords{Pomeron, supersymmetry} \author{L.N.Lipatov}{address={Petersburg Nuclear Physics Institute, Russia}} \begin{abstract} Theoretical approaches to the problem of the high energy hadron-hadron scattering in the Regge kinematics are reviewed. It is shown, that the gluon in QCD is reggeized and the Pomeron is a two gluon composite state. Further, the equation for the multi-gluon composite states is integrable at $N_c\rightarrow \infty$. Due to the AdS/CFT correspondence in $N=4$ SUSY the BFKL Pomeron is equivalent to the reggeized graviton. The important properties of the maximal transcendentality and integrability are realized in this model. Multi-gluon scattering amplitudes are investigated in the Regge limit. The BDS ansatz for them is not valid beyond one loop due to the presence of the Mandelstam cuts. The hamiltonian for the corresonding reggeon states coincides with the hamiltonian of an integrable open Heisenberg spin chain. \end{abstract} %\date{} \maketitle \section{Regge approach to high energy interactions} Hadron-hadron scattering in the Regge kinematics \beq s=(p_A+p_B)^2 =(2E)^2 >> \vec{q}^2=-(p_{A'}-p_A)^2 \sim m^2 \eeq is usually described in terms of a $t$-channel exchange of the Reggeon \begin{eqnarray} &&A_p(s,t)=\xi _p(t)\,g(t)\,s^{j_p(t)}\,g(t)\,,\,\,j_p(t)\nonumber \\ &&=j_0+\alpha 't\,,\,\,\xi_p(t)=\frac{e^{-i\pi j_p(t)}+p}{\sin (\pi j_p)}\,, \end{eqnarray} where $j_p(t)$ is the Regge trajectory which is assumed to be linear, $j_0$ and $\alpha '$ are its itercept and slope, respectively. The signature factor $\xi_p$ is a complex quantity depending on the Reggeon signature $p=\pm 1$. Around 40 years ago V.N. Gribov constructed the phenomenological field theory for all possible Reggeon interactions. A special Reggeon - Pomeron with the vacuum quantum numbers and positive signature was introduced to explain an approximately constant behavior of total cross-sections at high energies and a fullfillment of the Pomeranchuck theorem $\sigma _{h\bar{h}}/\sigma _{hh}\rightarrow 1$. %\[ %s=4E^2\gg -t=|q|^2 \approx E^2\,\theta ^2\, %\] In the Born approximation of QCD the elastic amplitude for two colored particle scattering is factorized \begin{eqnarray} &&M_{AB}^{A^{\prime }B^{\prime }}(s,t)|_{Born}=\Gamma _{A^{\prime }A}^{c}\,\frac{2s}{t}\,\Gamma _{B^{\prime }B}^{c}\,,\,\, \Gamma _{A^{\prime }A}^{c}\nonumber \\ &&=g\,T_{A^{\prime }A}^{c}\,\delta _{\lambda _{A^{\prime }}\lambda _{A}}\,, \end{eqnarray} where $T^c$ are the generators of the color group $SU(N_c)$ in the corresponding representation and $\lambda _r$ are helicities of the colliding and final state particles. In the leading logarithmic approximation (LLA) this amplitude has the Regge form~\cite{BFKL} \beq M_{AB}^{A^{\prime }B^{\prime }}(s,t)=M_{AB}^{A^{\prime }B^{\prime }}(s,t)|_{Born}\,s^{\omega (t)},\,\,\alpha _{s}\ln s \sim 1 \,, \eeq where the gluon Regge trajectory with the use of the dimensional regularization can be written as follows \begin{eqnarray} &&\omega (-|q|^2)=-\frac{\alpha _s N_c}{(2\pi ) ^{2-2\epsilon}}\,|q^2|\int \frac{\mu^{2\epsilon}\,d^{2-2\epsilon}k}{|k|^2|q-k|^2}\nonumber \\ &&\approx -a\,\left(\ln \frac{|q^2|}{\mu ^2}-\frac{1}{\epsilon}\right)\,,\,\, a=\frac{\alpha _s\,N_c}{2\pi}\,\left(4\pi\,e^{-\gamma}\right)^\epsilon \end{eqnarray} and $\gamma$ is the Euler constant $\gamma =-\psi (1)$. This Regge trajectory was calculated also in two-loop approximation in QCD~\cite{trajQCD} and in supersymmetric gauge theories~\cite{trajN4}. Further, the scattering amplitude in the multi-Regge kinematics for produced gluons with momenta $k_r$ \begin{eqnarray} &&s\gg s_1\,,\,\,s_2\,,...\,,\,\,s_{n+1}\gg t_1\,,\,\,t_2\,,...\,,\,\,t_{n+1}\,;\,\nonumber \\ &&\,s_r=(k_{r-1}+k_r)^2\,,\,\, t_r=-|q_r|^2 \end{eqnarray} has the form~\cite{BFKL} \begin{eqnarray} &&M_{2\rightarrow 1+n} =2s\,\Gamma _{A^{\prime }A}^{c_1}\, \frac{s_1^{\omega (-|q_1|^2)}}{|q_1|^2} \,gT_{c_2c_1}^{d_1}C(q_2,q_1)\, \nonumber \\ &&\times\frac{s_2^{\omega (-|q_2|^2)}}{|q_2|^2}...C(q_{n},q_{n-1})\, \frac{s_{n}^{\omega (-|q_n|^2)}}{|q_{n}|^2}\,\Gamma _{B^{\prime }B}^{c_n}\,. \end{eqnarray} Here $C$ are the Reggeon-Reggeon-gluon vertices \beq C(q_2,q_1)=\frac{q_2\,q_1^*}{q_2^*-q_1^*} \eeq and we used the complex notations $k=k_x+ik_y$ for the transverse components of momenta. \section{BFKL equation} Because the production amplitudes in QCD are factorized, one can write a Bethe-Salpeter-type equation for the total cross-section $\sigma _t$. It is governed by the Pomeron exchange. The Pomeron wave function satisfies the equation of Balitsky, Fadin, Kuraev and Lipatov (BFKL)~\cite{BFKL} \beq E\,\Psi (\vec{\rho}_{1},\vec{\rho}_{2})=H_{12}\,\Psi (\vec{\rho}_{1},\vec{% \rho}_{2})\;,\,\,\Delta =-\frac{\alpha _{s}N_{c}}{2\pi }\,E\,, \eeq where $\sigma _t \sim s^{\Delta _{max}}$. The BFKL Hamiltonian in the coordinate representation $\rho$ is \begin{eqnarray} &&H_{12}=\ln \,|p_1p_2|^{2}+ \frac{1}{p_1p_2^*}(\ln |\rho _{12}|^{2})p_1p_2^*\nonumber \\ &&+\frac{1}{p_1^*p_2}(\ln |\rho _{12}|^{2})p_1^*p_2 -4\psi (1)\,, \end{eqnarray} where $\rho _{12}=\rho _1-\rho _2$. It is invariant under the M\"obius transformations~\cite{int1, moeb} \beq \rho _k \rightarrow \frac{a\rho _k+b}{c\rho _k+d} \eeq and has the property of the holomorphic separability \begin{eqnarray} &&H_{12}=h_{12}+h^*_{12}\,,\,\,h_{12}=\ln (p_1p_2)+ \frac{1}{p_1}\ln (\rho _{12}) \,p_1\nonumber \\ &&+\frac{1}{p_2}\ln (\rho _{12}) \,p_2 -2\psi (1)\,. \label{pairh} \end{eqnarray} Here we used the complex notations $\rho _r =x_r+iy_r, \,p_r=i\partial _r$ for two-dimensional transverse coordinates and their canonically conjugated momenta. For the principal series of unitary representations of the M\"{o}bius group the conformal weights are \beq m=\gamma +n/2\,,\,\,\widetilde{m}=\gamma -n/2\,,\,\,\gamma =1/2+i\nu \,, \eeq where $\gamma$ is the anomalous dimension of the twist-2 operators and $n=0,\pm 1, \pm 2,...$ is the conformal spin. The Bartels-Kwiecinski-Praszalowicz (BKP) equation for colorless composite states of several reggeized gluons has the form~\cite{BKP} \begin{eqnarray} &&E\,\Psi (\vec{\rho}_{1},...\vec{\rho}_n)= H\,\Psi (\vec{\rho}_{1},...\vec{\rho}_n )\;,\,\,H\nonumber \\ &&=\sum _{k0$ but $s_1,s_3<0$ the Regge factorization for the BDS amplitude is broken~\cite{BLS} \begin{eqnarray} &&\frac{M_{2\rightarrow 4}}{\Gamma (t_1)\Gamma (t_3)} = C \left(\frac{-s_1}{\mu ^2}\right)^{\omega (t_1)} \Gamma (t_2,t_1,\ln -\kappa _{12} )\nonumber \\ &&\times\left(\frac{-s_2}{\mu ^2}\right)^{\omega (t_2)} \Gamma (t_3,t_2, \ln -\kappa _{23} ) \left(\frac{-s_3}{\mu^2}\right)^{\omega (t_3)}, \label{24mixedregion} \end{eqnarray} where the coefficient $C$ is given below \begin{equation} C=\exp \left[\frac{\gamma _K(a)}{4} \,i\pi \,\left( \ln \frac{\vec{q}_1^2\vec{q}_3^2}{(\vec{k}_1+\vec{k}_2)^2\mu ^2} -\frac{1}{\epsilon}\right)\right]. \label{coeffC} \end{equation} Similarly for the BDS amplitude describing the transition $3\rightarrow 3$ in the physical region, where $s,s_2=t'_2>0$ but $s_1,s_3<0$ we obtain the result \begin{eqnarray} &&\frac{M_{3\rightarrow 3}}{\Gamma (t_1)\Gamma (t_3)} = C' \, \left(\frac{-s_1}{\mu ^2}\right)^{\omega (t_1)} \Gamma (t_2,t_1,\ln -\kappa _{12} )\nonumber \\ &&\times\left(\frac{-s_2}{\mu ^2}\right)^{\omega (t_2)} \Gamma (t_2,t_1, -\ln \kappa _{23} ) \left(\frac{-s_3}{\mu^2}\right)^{\omega (t_3)}, \end{eqnarray} where the phase factor $C'$ is \begin{equation} C'=\exp \left[\frac{\gamma _K(a)}{4} \,(-i\pi ) \, \ln \frac{(\vec{q}_1-\vec{q}_2)^2\, (\vec{q}_2-\vec{q}_3)^2}{(\vec{q}_1+\vec{q}_3-\vec{q}_2)^2\,\vec{q}_2^2} \right], \label{coeffC'} \end{equation} which also contradicts the Regge factorization. The reason for these drawbacks is that just in these kinematical regions the amplitudes $A_{2\rightarrow 4}$ and $A_{3\rightarrow 3}$ should contain the Mandelstam cuts in the $j$-pane of the $t_2$-channel~\cite{BLS}. Therefore the BDS amplitudes for these processes are not correct beyond 1 loop. \section{Mandelstam cuts in the adjoint representation at LLA} The Mandelstam cuts in the elastic amplitude appear only in the non-planar diagrams because the integrals over the Sudakov variables $\alpha =2kP_A/s$ and $\beta =2kp_B$ for the reggeon momenta $k$ and $q-k$ should have the singularities above and below the corresponding integration contours. For the case of planar diagrams this Mandelstam condition is fulfilled only for inelastic amplitudes starting from six external particles in the kinematical region where $s,s_2>0$ and $s_1,s_3<0$. Two reggeons in the $t_2$-channel with an adjoint representation of the gauge group $SU(N_c)$ can also scatter each from another. The corresponding contribution to the imaginary part in the $s_2$-channel for the amplitude $A_{2\rightarrow 4}$ can be written as follows~\cite{BLS} \begin{eqnarray} &&\frac{1}{\pi}\Im _{s_2} M_{2\rightarrow 4}=s_2^{\omega (t_2)}\, \int _{\sigma -i\infty}^{\sigma +i\infty}\nonumber \\ &&\times\frac{d\omega}{2\pi i} \left(\frac{s_2}{\mu ^2}\right)^\omega \, \widetilde{f}_2(\omega )\, \end{eqnarray} The reduced partial wave $\widetilde{f}_2(\omega)$ is given by \begin{eqnarray} \label{f-reduced} &&\widetilde{f}_2(\omega )=\hat{\alpha}_\epsilon \, |q_2|^2\int d^{2-2\epsilon}k\,d^{2-2\epsilon}k'\,\nonumber \\ &&\times\Phi _1 (k,q_2,q_1)\, G_{\omega}(k,k',q_2)\, \Phi_3 (k',q_2,q_3)\, \end{eqnarray} where $\Phi _{1,3}$ are the impact factors \begin{eqnarray} \label{phi3} &&\Phi_1(k,q_2,q_1)= \frac{k_1^* (q_2-k)^*}{q_2^*(k+k_1)^*}\,,\,\,\nonumber \\ &&\Phi_3(k',q_2,q_3)=\frac{k_2(k'-q_2)}{q_2(k'-k_2)}\,. \end{eqnarray} The Green's function $G_{\omega}(k,k',q_2)$ satisfies the BFKL-type equation \begin{eqnarray} &&\omega G_{\omega}^{(8_A)}(k,k',q_2) = \frac{(2\pi)^3 \delta^{(2)}(k-k')}{|k|^2 |k+q_2|^2}\nonumber \\ &&+ \frac{1}{|k|^2 |k+q_2|^2} \left(K^{(8_A)} \otimes G_{\omega}^{(8_A)} \right) (k,k',q_2)\,, \label{BFKLoctet} \end{eqnarray} where \begin{eqnarray} &&K^{(8_A)}(k,k';q_2)\nonumber \\ &&= \delta^{(2)}(k-k') \left( \omega(-|k|^2) + \omega(-|q_2-k|^2)-2\omega (-|q|^2) \right) \nonumber \\ &&+ \frac{a}{2} \frac{k^* (q_2-k) k'(q_2-k')^* + c.c.}{|k-k|^2}\,. \end{eqnarray} The infrared divergencies are extracted from $M_{2\rightarrow 4}$ in the form of the Regge factor $s_2^{\omega (t_2)}$ and coincide with those of the BDS amplitude, as it should be. The partial wave $\widetilde{f}_2(\omega)$ contains the infrared divergency only in one loop \begin{eqnarray} \label{C-phase} &&\hat{\alpha}_\epsilon \, |q_2|^2\int d^{2-2\epsilon}k\, \frac{k^*q_1^*}{q_2^* (k+k_1)^*}\,\,\, \frac{1}{|k|^2 |q_2 -k|^2}\,\,\, \frac{k q_3}{q_2 (k-k_2)} \nonumber \\ &&= \frac{a}{2} \left( \ln \frac{|q_1|^2 |q_3|^2}{|k_1+k_2|^2 \mu^2} - \frac{1}{\epsilon}\right)\,, \end{eqnarray} which is also compatible with the BDS result. But in upper loops the iteration of the above equation leads to terms which are absent in the BDS amplitude. For example, in two loops we obtain for the imaginary part of $A_{2\rightarrow 4}$ in the $s_2$-channel the more complicated expression~\cite{BLS2} \beq A_{s_2}=\frac{a^2}{2}\,\ln s_2 \,\ln \frac{|q_1-q_3|^2|q_2|^2}{|q_1|^2|k_2|^2}\, \ln \frac{|q_1-q_3|^2|q_2|^2}{|q_3|^2|k_1|^2}\,. \eeq It is symmetric with respect to the simultaneous transmutation of momenta \beq k_1 \leftrightarrow k_2\,,\,\, q_1 \leftrightarrow -q_3\,. \eeq The same expression is valid also for the imaginary part in the $s$-channel. In a similar way we can calculate the $s$-channel imaginary part of the amplitude for the transition $3\rightarrow 3$ \beq A_{s}^{3\rightarrow 3}=\frac{a^2}{2}\,\ln t'_2\,\ln \frac{|q_2-q_1-q_3|^2|q_2|^2}{|k_1|^2|k_2|^2}\, \ln \frac{|q_2-q_1-q_3|^2|q_2|^2}{|q_3|^2|q_1|^2}\,. \eeq Moreover, the BFKL equation for the state with adjoint quantum numbers can be solved exactly and we obtain for the imaginary part in $s_2$-channel~\cite{BLS2} \begin{eqnarray} &&\Im M_{2\rightarrow 4}\sim s_2^{\omega (t_2)}\sum _{n=-\infty}^\infty \int _{-\infty}^\infty \frac{d\nu }{\nu ^2+\frac{n^2}{4}}\,\left(\frac{q_3^*k^*_1}{k^*_2q_1^*}\right)^{i\nu -\frac{n}{2}}\, \nonumber \\ && \times\left(\frac{q_3k_1}{k_2q_1}\right)^{i\nu +\frac{n}{2}}\, \exp \left(\omega (\nu , n)\,\ln s_2\right)\,, \end{eqnarray} where the eigenvalue of the reduced BFKL kernel for the adjoint representation is \beq \omega (\nu , n) =-a\left( \psi (i\nu +\frac{|n|}{2})+ \psi (-i\nu +\frac{|n|}{2})-2\psi (1)\right)\,. \eeq It turns out, that the leading singularity of the $t_2$-partial wave corresponds to $n=1$ and is situated at \[ j-1=\omega (t_2)+a(4\ln 2-2). \] \section{Multi-reggeon Mandelstam cuts} Let us consider now the Mandelstam cuts constructed from several reggeons~\cite{intBDS}. The non-vanishing contribution from the exchange of $n+1$ reggeons appears in the planar diagrams only if the number of external lines is $r\ge 2n+4$. In the case of production of $2n$ gluons with the same helicity the amplitude in $N=4$ SUSY is proportional to the Born expression. In the leading logarithmic approximation for the $n+1$-reggeon contribution to the $s_{n+1}$-channel the proportionality factor has the form~\cite{intBDS} \begin{eqnarray} &&f_{LLA}^{2\rightarrow 2+2n}=\left(i\,\frac{g^2\,N_c}{4\pi}\right)^{n}\, Q^*\widetilde{Q}\nonumber \\ &&\int \prod _{l=1}^n\frac{\mu ^{2\epsilon}d^{2-2\epsilon } p_l}{(2\pi)^{1-2\epsilon}}\,\frac{\mu ^{2\epsilon}d^{2-2\epsilon } p'_l}{(2\pi)^{1-2\epsilon}}\,\nonumber \\ &&\times\prod _{l=1}^{n} \frac{k^*_lk_{2r-l}}{|p_l|^2}\,\frac{G(p,p';s_{n+1})}{|p_{n+1}|^2}\Phi _1\, \Phi_2\,, \end{eqnarray} where $Q$ and $\widetilde{Q}$ are momentum transfers between momenta $p_A,p_{A'}$ and $p_B,p_{B'}$. The impact factors are \begin{equation*} \Phi _1(\vec{p_1},...,\vec{p}_{n+1})= \end{equation*} \begin{eqnarray*} \prod_{l=1}^n \frac{p^{*}_{l+1}}{(Q^*-\sum _{s=1}^lp^{ *}_s-\sum _{s=1}^{l-1}k^*_s)(Q^*-\sum _{t=1}^{l+1}p^{*}_t -\sum _{t=1}^{l-1}k_t^*)}, \end{eqnarray*} \begin{equation*} \Phi _2(\vec{p}'_1,...,\vec{p}'_{n+1})= \end{equation*} \begin{eqnarray*} \prod _{l=1}^n\frac{p'_{l+1}}{(\widetilde{Q}+\sum _{s=1}^lp'_s- \sum _{s=1}^{l-1}k_{2n -s+1})(\widetilde{Q}+\sum _{t=1}^{l+1}p^{\prime }_t -\sum _{t=1}^{l-1}k_{2n-t+1})}. \end{eqnarray*} The multi-reggeon Green function satisfies the equation~\cite{intBDS} \begin{eqnarray} &&\frac{\partial}{\partial \ln s_{n+1}}\, G(\vec{p},\vec{p}';s_{n+1})\nonumber \\ &&= K\,G(\vec{p},\vec{p}';s_{n+1})\,,\,\,\nonumber \\ &&G(\vec{p},\vec{p}';0) = \prod _{l=1}^n\frac{(2\pi )^{1-2\epsilon}}{\mu ^{2\epsilon}}\, \delta ^{2-2\epsilon}(p_{l}-p'_{l})\,. \end{eqnarray} Here the kernel $K$ in LLA can be expressed in terms of the infraredly stable Hamiltonian $H$ \beq K=\omega (t) -\frac{g^2N_c}{16 \pi ^2}\,H\,,\,\, \omega (t)=a\left(\frac{1}{\epsilon}-\ln \frac{-t}{\mu _2}\right)\,, \,\,t=-|q|^2\,, \eeq \beq H=\ln \frac{|p_1|^2|p_{n+1}|^2}{|q|^4}+\sum _{l=1}^{n}H_{l,l+1}\,, \eeq where the pair Hamiltonians are \begin{eqnarray} &&H_{l,l+1}=\ln |p_l|^2+\ln |p_{l+1}|^2\nonumber \\ &&+ p_l\,p_{l+1}^*\,\ln |\rho _{l,l+1}|^2\,\frac{1}{p_l\,p_{l+1}^*}\nonumber \\ &&+ p_l^*\,p_{l+1}\,\ln |\rho _{l,l+1}|^2\,\frac{1}{p_l^*\,p_{l+1}}\,. \end{eqnarray} \section{Integrable open Heisenberg spin chain} The Hamiltonian for the $n+1$-gluon composite state in the ajoint representation has the property of the holomorphic separability~\cite{intBDS} \beq H=h+h^*\,,\,\,h=\ln \frac{p_1\,p_{n+1}}{q^2}+ \sum _{l=1}^{n}h_{l,l+1}\,, \eeq where \beq h_{l,l+1}=\ln p_l+\ln p_{l+1}+ p_l\,\ln \rho _{l,l+1}\,\frac{1}{p_l}+ p_{l+1}\,\ln \rho _{l,l+1}\,\frac{1}{p_{l+1}}\,. \eeq Using the duality transformations (cf.~\cite{dual}) \begin{equation} p_1=z_{0,1}\,,\,\,p_r=z_{r-1,r}\,,\,\,q=z_{0,n}\,,\,\,\rho _{r,r+1}= i\frac{\partial}{\partial z_r}=i\partial _r\,, \end{equation} the holomorphic hamiltonian can be rewritten in the form invariant under the M\"{o}bius transformations \begin{equation} z_k\rightarrow \frac{az_k+b}{cz_k+d}\,. \end{equation} Therefore we can put \begin{equation} z_0=0\,,\,\,z_n=\infty \,, \end{equation} Further, by regrouping the terms one can present it in another form~\cite{intBDS} \begin{equation} \ln (z_{1}^2\partial _1)+ \ln (\partial _{n-1})+2\gamma + \sum_{r=1}^{n-2}\,h'_{r,r+1}\,, \end{equation} where \[ h'_{r,r+1}=\ln (z_{r,r+1}^2\partial _r)+ \ln (z_{r,r+1}^2\partial _{r+1})-2\ln z_{r,r+1}+ 2\gamma \] \begin{equation} =\ln (\partial _r)+\ln (\partial _{r+1}) +\frac{1}{\partial _r}\,\ln z_{r,r+1}\,\partial _r+ \frac{1}{\partial _{r+1}}\,\ln z_{r,r+1}\,\partial _{r+1} +2\gamma \,. \end{equation} The pair hamiltonian $h'_{r,r+1}$ coincides in fact with the expression (\ref{pairh}) in the coordinate representation acting on the wave function with non-amputated propagators. The remarkable property of $h$ is its commutativity with the matrix element $D(u)$ of the monodromy matrix (\ref{ABCD}) introduced above for the description of integrability of the BKP equations in the multi-color QCD~\cite{intBDS} \beq [D(u),h]=0\,. \eeq Therefore if we write $D(u)$ as a polynomial in $u$ \begin{equation} D(u)=\sum _{k=0}^{n-1}u^{n-1-k}\,q'_k\,, \end{equation} then the differential operators \begin{equation} q'_k=-\sum _{0