diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzndza" "b/data_all_eng_slimpj/shuffled/split2/finalzzndza" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzndza" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\nBhabha scattering is measured with remarkable precision at LEP1\/SLC \nenergies in two kinematical regions identified by the scattering angle: \nat small-angle for the luminosity monitoring and \nat large-angle for the measurements of the $Z$ properties. \n\\par\nSuch results require a continuous effort on the theoretical side to maintain \nthe theoretical error on the prediction within a comparable precision and \nto provide a tool for realistic comparisons with the experimental data \n(namely an event generator).\n\\par\nWe have already given in ~\\cite{NC92} the general description of our approach\nto the problem, \nwhile the features of the first version of our event generator BHAGEN are \ndescribed in ~\\cite{IJMP93}, with instructions on how to get the program \nand how to use it. In ~\\cite{PL94} we have considered the higher order \nQED corrections, including the most relevant contributions and estimating the \nomitted terms; in ~\\cite{PL94} a new version of our event generator, BHAGEN94, \nis also announced. \n\\par\nSeveral improvements have been made subsequently to BHAGEN94, in \norder to increase and control its precision, and \nin ~\\cite{CERN95-03} some results for small-angle \nBhabha scattering cross-section are presented with the label BG94-NEW \nand compared with the results from few other available \ncomputer programs: OLDBIS ~\\cite{OLDBIS}, LUMLOG ~\\cite{LUMLOG}, \nBHLUMI 2.01~\\cite{BHLUMI2} and 40THIEVES ~\\cite{40THIEVES}.\nThe conclusion from the comparisons with the above programs, and also from the\nuse of the new event generator BHAGEN-1PH, which contains the complete hard \nphoton matrix element ~\\cite{1PH}, is that the hard photon part (even in the \nimproved version of BHAGEN94) deserves a more accurate treatment to achieve \nthe 0.1\\% precision, presently aimed at small-angle for luminosity monitoring. \nTo obtain such a result a new procedure and the code BHAGEN95 are developed. \nBHAGEN95 (available on request from the authors) is actually a collection of \nthree programs and in this paper we discuss\nin detail two of them, as the third one is already discussed separately \n~\\cite{1PH}.\nThe current precision for the cross-section is heavily dependent on the chosen \nevent selection and for our code can be estimated to be about 0.1\\%-0.2\\% for \ntypical experimental event selection at small-angle for LEP1\/SLC and LEP2 \nenergies.\nIn references ~\\cite{CERN96} and ~\\cite{BWG_LETT} our results are presented \nin a variety of different event selections and conditions \n(some of those preliminary results are updated in this discussion),\nand compared with the other presently available programs of similar\nprecision:\nBHLUMI 4.03 ~\\cite{BHLUMI4},\nOLDBIS+LUMLOG ~\\cite{OBI+LUMG}, \nSABSPV ~\\cite{SABSPV},\nNLLBHA ~\\cite{NLLBHA}.\n\nThe precision at large-angle depends on the beam energy and \nalso on the event selection used; in the worst case (beam energy a few GeV \nabove the $Z$ peak) can be safely estimated to be about 1\\%, \nmainly due to the raise of the incertitude coming from the second order leading \nlogarithmic correction in the $s-t$-interference terms, which are still \nunknown and usually are assumed to be the same as in the $s$-channel.\nFor the other energy ranges at LEP1\/SLC, and at LEP2,\none can expect with relatively loose cuts a typical accuracy of 0.5\\%,\nwhich can be usefully compared with the other presently available programs, \nin the conditions they are suited for: \nALIBABA ~\\cite{ALIBABA}, \nZFITTER ~\\cite{ZFITTER}, \nTOPAZ0 ~\\cite{TOPAZ0}, \nBHAGENE3 ~\\cite{BHAGENE}, \nUNIBAB ~\\cite{UNIBAB},\nBHWIDE ~\\cite{BHWIDE},\nSABSPV ~\\cite{SABSPV}.\n\n\\par\nThis paper goes through the following sections:\nimplementation of an improved energy distribution for the final photon \nemission in the resummed program BHAGEN94, in section 2;\nconstruction of the BG94-FO code, the \\Ord{\\alpha} version of the event \ngenerator BHAGEN94, in section 3;\nimplementation of necessary weak and QCD corrections in BHAGEN94 up\nto two loops, in section 4;\nconstruction of the BHAGEN95 code, in section 5;\ncomparisons at small-angle, in section 6; \ncomparisons at large-angle, in section 7; \nconclusions, in section 8.\n\\bigskip\n\n\\section{ Improved energy distribution for the final photon \nemission in the resummed program BHAGEN94.}\n\nIn the same notation as in \n~\\cite{NC92}, ~\\cite{IJMP93}, ~\\cite{PL94} and ~\\cite{CERN89},\nthe integrated cross-section is \n\\begin{equation}\n\\sigma(e^{+}e^{-} \\rightarrow e^{+} e^{-} ) = \\int\\limits_{\\Delta\\Omega_-} d\\Omega_- \\quad\n{d\\tilde\\sigma(E_b,\\theta_-)\\over{d\\Omega_-}} \\ ,\n\\label{eq:s1}\n\\end{equation}\nwhere the dressed differential cross-section in our approach is \n\\begin{eqnarray}\n{d\\tilde\\sigma(E_b,\\theta_-)\\over{d\\Omega_-}} = \n&&\\int\\limits_{E_{min}}^{E_b} {d E^0_- \\over{E_b}} F^0(E_b,E^0_-,\\theta_-)\n \\int\\limits_{E_{min}}^{E_b} {d E^0_+ \\over{E_b}} F^0(E_b,E^0_+,\\theta_+) \n\\nonumber \\\\\n&&\\int\\limits_{E_{min}}^{E_-} {d E'_- \\over{E_-}} F'(E_-,E'_-,\\theta_-)\n \\int\\limits_{E_{min}}^{E_+} {d E'_+ \\over{E_+}} F'(E_+,E'_+,\\theta_+) \n\\label{eq:s2} \\\\\n&&{d\\sigma(E^0_+,E^0_-,\\theta_-)\\over{d\\Omega_-}} \\quad \\Theta(cuts) \\ , \\nonumber \n\\end{eqnarray} \n$E_b$ is the beam energy, $\\theta_{\\pm}$ the scattering-angle of the electron \n(positron), $s=4E_b^2$ and $t_{\\pm}=-s \\sin^2{\\theta_{\\pm}\\over2}$.\nIn the laboratory system the electron (positron) energy after \ninitial radiation is $E^0_{\\pm}$, after the scattering is $E_{\\pm}$, \nafter the final radiation is $E'_{\\pm}$, and $E_{min}$ is the minimum energy \nfor the leptons.\n\\par\nIn terms of the variables in the LM-c.m.s. (the c.m.s. after initial radiation \nemitted collinearly to the initial direction, as introduced \nin ~\\cite{NC92}), the incoming fermion energy \n$E^*=\\sqrt{E^0_- E^0_+}$, the scattering angle $\\theta^*(E^0_-,E^0_+,\\theta_-)$ \nand the jacobian $\\left( {d\\Omega^* \\over{d\\Omega_-}} \\right)$,\nthe differential cross-section from Feynman diagrams, \\Ord{\\alpha} complete \nand up to \\Ord{\\alpha^2} leading logarithm, is written as\n\\begin{eqnarray}\n{d\\sigma(E^0_+,E^0_-,\\theta_-)\\over{d\\Omega_-}} = \n&& \\left( {d\\Omega^* \\over{d\\Omega_-}} \\right)\n \\sum_{i=1}^{10} {{d\\bar\\sigma_0^{(i)}(E^*,\\theta^*)}\\over{d\\Omega^*}} \n\\label{eq:s3} \\\\\n&& \\left[ 1+\\delta_N^{(i)}(E_b,\\theta^*) \\right]\n \\left[ 1 + \\tilde C^{(i)}(E^*,\\theta^*) \\right] \n \\left[ 1+\\delta_N^{(i)}(E^*,\\theta^*) \\right] \\nonumber \\ , \n\\end{eqnarray}\nwhere ${{d\\bar\\sigma_0^{(i)}(E^*,\\theta^*)}\\over{d\\Omega^*}}$ are the Born \ndifferential cross-sections for the various channels with vacuum polarization \ninsertions (Dyson resummed as in Eq.(2.6) of ~\\cite{NC92}),\n$\\delta^{(i)}_{N}(E, \\theta)$ (defined in Eq.(16) of ~\\cite{PL94}) \nare the leading logarithmic corrections up to \\Ord{\\alpha^2}, \nand $\\tilde C^{(i)}(E_b,\\theta_-)$ are the \\Ord{\\alpha} complete corrections \nafter subtraction of the terms already included in the previous corrections.\n\\par \nThe function $\\Theta(cuts)$ accounts for the rejection procedure, according to \nthe requested cuts on energies and angles.\n\\par \nAs usual,\n\\begin{eqnarray}\n\\beta(E,\\theta) &=& \\beta_e(E) +\\beta_{int}(\\theta) \\ , \\nonumber \\\\ \n\\beta_e(E) &=& 2 \\left({\\alpha\\over\\pi}\\right) \\left(\\ln {{4E^2}\\over{m_e^2}} -1\\right) \\ , \\\\\n\\beta_{int}(\\theta) &=& 4 \\left({\\alpha\\over\\pi}\\right) \\ln \\left( \\tan {\\theta \\over2} \\right) \\ ,\n\\nonumber \n\\end{eqnarray} \nwith $\\beta(E_b,\\theta_{\\pm}) \\simeq \\beta_e(\\sqrt{s}\/2)$ at large-angle and \n$\\beta(E_b,\\theta_{\\pm}) \\simeq \\beta_e(\\sqrt{-t_{\\pm}}\/2)$ at small-angle.\n\\par\nThe emission functions for initial $(F^0)$ and final $(F')$ emission \nin the notation of ~\\cite{PL94}, and using the results of ~\\cite{S+J}, are\n\\begin{eqnarray}\nF^0(E_b,E^0_{\\pm},\\theta_{\\pm}) &&= D^0(E_b,E^0_{\\pm},\\theta_{\\pm})\n\\Biggl\\{\n {1\\over2} \\left( 1 + \\left( {E^0_{\\pm}\\over{E_b}} \\right)^2 \\right)\n + A^0_m(E_b,E^0_{\\pm}) \n\\label{eq:s4} \\\\\n &&+ {{\\beta(E_b,\\theta_{\\pm})}\\over 8}\n \\left[ -{1\\over2}\n \\left( 1 +3 \\left( {E^0_{\\pm}\\over{E_b}} \\right)^2 \\right)\n \\ln \\left( {E^0_{\\pm}\\over{E_b}} \\right)\n - \\left(1-{E^0_{\\pm}\\over{E_b}}\\right)^2\n \\right] \\Biggr\\} \\ , \\nonumber \n\\end{eqnarray}\nand\n\\begin{eqnarray}\nF'(E_{\\pm},E'_{\\pm},\\theta_{\\pm}) &&= D'(E_{\\pm},E'_{\\pm},\\theta_{\\pm})\n\\Biggl\\{\n {1\\over2} \\left( 1 + \\left( {E'_{\\pm}\\over{E_{\\pm}}} \\right)^2 \\right)\n + A'_m(E_{\\pm},E'_{\\pm}) \n\\label{eq:s5} \\\\\n &&+ {{\\beta(E_{\\pm},\\theta_{\\pm})}\\over 8}\n \\left[ -{1\\over2}\n \\left( 1 +3 \\left( {E'_{\\pm}\\over{E_{\\pm}}} \\right)^2 \\right)\n \\ln \\left( {E'_{\\pm}\\over{E_{\\pm}}} \\right)\n - \\left(1-{E'_{\\pm}\\over{E_{\\pm}}}\\right)^2\n \\right] \\Biggr\\} \\ , \\nonumber \n\\end{eqnarray}\nwith\n\\begin{equation}\nA^0_m(E_b,E^0_{\\pm}) = {1\\over2} {{\\left( 1 - {{E^0_{\\pm}}\\over{E_b}} \\right)}^2\n \\over {\\ln \\left({{2E_b}\\over{m_e}}\\right)^2 - 1 } } \\ , \\quad\nA'_m(E_{\\pm},E'_{\\pm}) ={1\\over2}{{\\left(1-{{E'_{\\pm}}\\over{E_{\\pm}}} \\right)}^2\n \\over {\\ln \\left({{2E'_{\\pm}}\\over{m_e}}\\right)^2 - 1 } } \\ .\n\\end{equation}\nThe radiator functions for initial $(D^0)$ and final $(D')$ emission are\n\\begin{equation}\nD^0(E_b,E^0_{\\pm},\\theta_{\\pm}) = \\div12 \\beta(E_b,\\theta_{\\pm})\n\\left( 1-{E^0_{\\pm}\\over{E_b}} \\right)^{\\div12 \\beta(E_b,\\theta_{\\pm}) -1} \n\\label{eq:s6}\n\\end{equation}\nand\n\\begin{equation}\nD'(E_{\\pm},E'_{\\pm},\\theta_{\\pm}) = \\div12 \\beta(E'_{\\pm},\\theta_{\\pm})\n\\left( 1-{E'_{\\pm}\\over{E_{\\pm}}} \\right)^{\\div12 \\beta(E'_{\\pm},\\theta_{\\pm}) \n-1} \\ ,\n\\label{eq:s7}\n\\end{equation}\nnote that the leading logarithmic corrections, up to \\Ord{\\alpha^2}, \nhave been included in the factors $(1+\\delta_N^{(i)}(E,\\theta))$ of the expression \nin Eq.~(\\ref{eq:s3}). \n\nAs the differential cross-section in Eq.~(\\ref{eq:s3}) does not depend on \nfinal lepton energies $E'_{\\pm}$, in our previous formulations the \nintegrations on these variables were performed analytically, assuming the \nfurther simplification that in the expression for $D'$ in Eq.~(\\ref{eq:s7}) \nthe values for $\\beta(E'_{\\pm},\\theta_{\\pm})$ were taken constant at energy $E^*$,\ni.e. $\\beta(E^*,\\theta_{\\pm})$. \n\\par\nThe simplification allows the analytical integration \n\\begin{eqnarray}\n\\int\\limits_{E_{min}}^{E_-} {d E'_- \\over{E_-}} F'(E_-,E'_-,\\theta_-) &=&\n\\Delta_-^{\\div12 \\beta(E^*,\\theta^*)} G(E^*,\\theta^*,\\Delta_-) \\ , \n\\label{eq:s8} \\\\\n\\int\\limits_{E_{min}}^{E_+} {d E'_+ \\over{E_+}} F'(E_+,E'_+,\\theta_+) &=&\n\\Delta_+^{\\div12 \\beta(E^*,\\theta^*)} G(E^*,\\theta^*,\\Delta_+) \\ ,\n\\label{eq:s9}\n\\end{eqnarray}\nwhere $\\Delta_{\\pm} = 1-E_{min}\/E_{\\pm}$ and the function \n$G(E^*,\\theta^*,\\Delta_{\\pm})$ is explicitly given in Eq.~(23) of ~\\cite{PL94}. \n\\par \nThe procedure is rather good for small values of $\\Delta_{\\pm}$, but the \nextension to larger and more realistic values, corresponding to harder photon \nemission, causes a loss in precision, which justifies the difference with\nBHLUMI 2.01~\\cite{BHLUMI2} and 40THIEVES ~\\cite{40THIEVES} amounting up to \n0.8\\%, as reported in ~\\cite{CERN95-03} for results under label BG94-OLD.\n\\par\nIn the new version of BHAGEN94 the $\\beta$ dependence on $E'_{\\pm}$ is kept\nand the integration (or generation) is done numerically without further \napproximations.\nThe new values (reported in ~\\cite{CERN95-03} under label BG94-NEW) \nreduce the difference with the above mentioned programs to about 0.25\\%.\n\\par\nThe new approach imply the generation of two more variables for the final \nfermion energies $E'_{\\pm}$, for the proper description of the peaks in \nEq.~(\\ref{eq:s7}) the variables $y'_{\\pm}$ are introduced\n\\begin{equation}\ny'_{\\pm} = \n\\left( 1-{E'_{\\pm}\\over{E_{\\pm}}} \\right)^{\\div12 \\beta(E_{\\pm},\\theta_{\\pm})}\\ .\n\\end{equation}\nTo generate within a unit volume, the variables are changed to \n$r'_{\\pm} \\in (0,1)$\n\\begin{equation}\ny'_{\\pm} = r'_{\\pm} \n\\left( 1-{E_{min}\\over{E_{\\pm}}} \\right)^{\\div12 \\beta(E_{\\pm},\\theta_{\\pm})}\\ .\n\\end{equation}\nNote that, even if in the generation procedure the function $\\beta$ is still \ntaken constant in energy, in the actual calculation the exact value is \ncomputed and used to determine the cross-section.\n\\bigskip\n\n\\section{ Construction of the BG94-FO generator.}\n\nWe have extracted from BHAGEN94 a first-order event generator \n\\ ---\\ i.e. including only corrections up to \\Ord{\\alpha} \\ ---\\ \nthat we call BG94-FO, \nkeeping as much as possible unchanged the structure of the program.\n\\par \nThat is achieved by analytically expanding in $\\alpha$ all the formulae \nand then keeping the \\Ord{\\alpha} terms only; the separation between soft and \nhard radiation is reintroduced through the small emitted energy fraction cut \n$\\epsilon$, whose size has to be of the order of the accepted error, \ntypically $\\epsilon \\le 10^{-4}$. \nTo allow the expansion in $\\alpha$, the integrations in Eq.~(\\ref{eq:s2}) are \nsplit into \n\\begin{equation}\n\\int\\limits_{E_{min}}^{E_b} {d E^0_{\\pm}} = \n\\int\\limits_{E_{min}}^{E_b(1-\\epsilon)} {d E^0_{\\pm}} \n +\\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E^0_{\\pm}} \\ , \n\\end{equation}\nand\n\\begin{equation}\n\\int\\limits_{E_{min}}^{E_{\\pm}} {d E'_{\\pm}} =\n\\int\\limits_{E_{min}}^{E_{\\pm}(1-\\epsilon)} {d E'_{\\pm}} \n +\\int\\limits_{E_{\\pm}(1-\\epsilon)}^{E_{\\pm}} {d E'_{\\pm}} \\ .\n\\end{equation}\nAll the resulting terms, which involve more than one integration in the hard \nphoton regions $(E_{min},E_b(1-\\epsilon))$ and $(E_{min},E_{\\pm}(1-\\epsilon))$, \nare omitted, being of higher order.\n\\par\nIn the soft photon regions \n$(E_b(1-\\epsilon),E_b)$ and $(E_{\\pm}(1-\\epsilon),E_{\\pm})$ \nand for smooth functions, \n$E^0_{\\pm}$ can be approximated by $E_b$ and $E'_{\\pm}$ by $E_{\\pm}$, \nwhile special care has to be devoted to the peaking behaviour of\nthe functions $D^0$ and $D'$.\nMoreover from the definition of $E^0_{\\pm}$ (Eq.~(3.6) of ~\\cite{NC92}) \none has that for \n$E^0_{\\pm} \\simeq E_b$ also $E_{\\pm} \\simeq E_b$, and $\\theta_+ = \\theta_-$, \nso Eq.~(\\ref{eq:s2}) becomes\n\\begin{eqnarray}\n{d\\tilde\\sigma(E_b,\\theta_-)\\over{d\\Omega_-}} &\\simeq& \n\\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E^0_- \\over{E_b}} D^0(E_b,E^0_-,\\theta_-)\n\\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E^0_+ \\over{E_b}} D^0(E_b,E^0_+,\\theta_-) \n\\nonumber \\\\\n& &\\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E'_- \\over{E_b}} D'(E_b,E'_-,\\theta_-) \n \\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E'_+ \\over{E_b}} D'(E_b,E'_+,\\theta_-) \n {d\\sigma(E_b,E_b,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\\\\n&+&\\int\\limits_{E_b(1-\\epsilon)}^{E_b}{d E^0_- \\over{E_b}}D^0(E_b,E^0_-,\\theta_-)\n \\int\\limits_{E_b(1-\\epsilon)}^{E_b}{d E^0_+ \\over{E_b}}D^0(E_b,E^0_+,\\theta_-) \n\\nonumber \\\\\n& &\\int\\limits_{E_{min}}^{E_b(1-\\epsilon)} \n {d E'_- \\over{E_b}} F'(E_b,E'_-,\\theta_-) \n \\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E'_+ \\over{E_b}} D'(E_b,E'_+,\\theta_-) \n {d\\sigma(E_b,E_b,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\\\\n&+&\\int\\limits_{E_b(1-\\epsilon)}^{E_b}{d E^0_- \\over{E_b}}D^0(E_b,E^0_-,\\theta_-)\n \\int\\limits_{E_b(1-\\epsilon)}^{E_b}{d E^0_+ \\over{E_b}}D^0(E_b,E^0_+,\\theta_-) \n\\\\\n& &\\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E'_- \\over{E_b}} D'(E_b,E'_-,\\theta_-) \n \\int\\limits_{E_{min}}^{E_b(1-\\epsilon)} \n {d E'_+ \\over{E_b}} F'(E_b,E'_+,\\theta_-) \n {d\\sigma(E_b,E_b,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\\\\n&+&\\int\\limits_{E_{min}}^{E_b(1-\\epsilon)} \n {d E^0_- \\over{E_b}} F^0(E_b,E^0_-,\\theta_-)\n\\int\\limits_{E_b(1-\\epsilon)}^{E_b} {d E^0_+ \\over{E_b}} D^0(E_b,E^0_+,\\theta_+) \n\\nonumber \\\\\n& &\\int\\limits_{E_-(1-\\epsilon)}^{E_-} {d E'_- \\over{E_-}} D'(E_-,E'_-,\\theta_-) \n \\int\\limits_{E_+(1-\\epsilon)}^{E_+} {d E'_+ \\over{E_+}} D'(E_+,E'_+,\\theta_+) \n {d\\sigma(E_b,E^0_-,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\\\\n&+&\\int\\limits_{E_b(1-\\epsilon)}^{E_b}{d E^0_- \\over{E_b}}D^0(E_b,E^0_-,\\theta_-)\n \\int\\limits_{E_{min}}^{E_b(1-\\epsilon)} \n {d E^0_+ \\over{E_b}} F^0(E_b,E^0_+,\\theta_+) \n\\nonumber \\\\\n& &\\int\\limits_{E_-(1-\\epsilon)}^{E_-} {d E'_- \\over{E_-}} D'(E_-,E'_-,\\theta_-) \n \\int\\limits_{E_+(1-\\epsilon)}^{E_+} {d E'_+ \\over{E_+}} D'(E_+,E'_+,\\theta_+) \n {d\\sigma(E^0_+,E_b,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\ ,\n\\end{eqnarray}\nwhere the first term accounts for soft emission only, \nthe second and third account for hard emission from final electron or positron, \nthe fourth and fifth for hard emission from initial electron or positron.\nThe functions $F^0$ and $F'$ are now intended at \\Ord{\\alpha}, that is \nomitting the $\\beta$ term in curly braces of \nEq.~(\\ref{eq:s4}) and Eq.~(\\ref{eq:s5}).\n\\par\nThe analytical integration and subsequent expansion in $\\alpha$ can be easily \ndone for the initial radiation\n\\begin{equation}\n\\int\\limits_{E_b(1-\\epsilon)}^{E_b} \n{d E^0_{\\pm} \\over{E_b}} D^0(E_b,E^0_{\\pm},\\theta_{\\pm}) = \n\\epsilon^{\\div12 \\beta(E_b,\\theta_{\\pm})} = \n1 + {\\div12 \\beta(E_b,\\theta_{\\pm})} \\ln \\epsilon +{\\cal O}(\\alpha^2) \\ ,\n\\end{equation}\nand in the case of final radiation the result is similar, apart from \ncorrections of order $\\epsilon$ \n\\begin{equation}\n\\int\\limits_{E_b(1-\\epsilon)}^{E_b} \n{d E'_{\\pm} \\over{E_b}} D'(E_b,E'_{\\pm},\\theta_{\\pm}) = \n\\epsilon^{\\div12 \\beta(E_b,\\theta_{\\pm})} +{\\cal O}(\\epsilon) = \n1 + {\\div12 \\beta(E_b,\\theta_{\\pm})} \\ln \\epsilon +{\\cal O}(\\alpha^2) \n+{\\cal O}(\\epsilon) \\ .\n\\end{equation}\nIn Eq.~(\\ref{eq:s3}) the expansion in $\\alpha$ is performed and only \n\\Ord{\\alpha} terms are retained: the vacuum polarization contribution \n$\\delta^{(i)}_{VP}$ is defined in Eq.(2.3) of ~\\cite{NC92},\n$\\delta^{(i)}_{N}(E_b)$ is taken up to \\Ord{\\alpha} from Eq.(16) of \n~\\cite{PL94}, and $\\tilde C^{(i)}(E_b,\\theta_-)$, as indicated in\n~\\cite{PL94}, is the nonleading part of the \\Ord{\\alpha} correction\n$C^{(i)}(E_b,\\theta_-)$, given explicitly in ~\\cite{CERN89}. \nWhen an \\Ord{\\alpha} correction from above integrals occurs only the Born \ncross-section is retained, and if no longer necessary the transformation \nto LM-c.m.s. is released. \n\\par \nWith all that the cross-section at \\Ord{\\alpha} becomes \n\\begin{eqnarray}\n&&{d\\tilde\\sigma(E_b,\\theta_-)\\over{d\\Omega_-}} \\simeq\n \\sum_{i=1}^{10} {d\\sigma_0^{(i)}(E_b,\\theta_-) \\over{d\\Omega_-}}\n \\Biggl\\{ 1 +2 \\beta(E_b,\\theta_-) \\ln\\epsilon\n +\\delta^{(i)}_{VP} +2 \\delta^{(i)}_{N}(E_b) + \\tilde C^{(i)}(E_b,\\theta_-) \n\\nonumber \\\\\n&&+\\int_{E_{min}}^{E_b(1-\\epsilon)} {d E'_- \\over{E_b -E'_-}}\n{{\\beta(E'_-,\\theta_-)}\\over{4}} \n\\left[ 1 +\\left({E'_-\\over{E_b}}\\right)^2 \n+{\\left(1-{E'_-\\over{E_b}}\\right)^2 \\over{\\ln (2E'_-\/m_e)^2 -1}} \\right] \n\\nonumber \\\\\n&&+\\int_{E_{min}}^{E_b(1-\\epsilon)} {d E'_+ \\over{E_b -E'_+}}\n{{\\beta(E'_+,\\theta_-)}\\over{4}} \n\\left[ 1 +\\left({E'_+\\over{E_b}}\\right)^2 \n+{\\left(1-{E'_+\\over{E_b}}\\right)^2 \\over{\\ln (2E'_+\/m_e)^2 -1}} \\right] \n\\Biggr\\} \\label{eq:fo1} \\\\\n&&+\\int_{E_{min}}^{E_b(1-\\epsilon)} {d E^0_- \\over{E_b -E^0_-}} \n{{\\beta(E_b,\\theta_-)}\\over{4}} \n\\left[ 1 +\\left({E^0_-\\over{E_b}}\\right)^2 \n+{\\left(1-{E^0_-\\over{E_b}}\\right)^2 \\over{\\ln ({2E_b}\/m_e)^2 -1}} \\right] \n {d\\sigma_0(E_b,E^0_-,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\\\\n&&+\\int_{E_{min}}^{E_b(1-\\epsilon)} {d E^0_+ \\over{E_b -E^0_+}} \n{{\\beta(E_b,\\theta_+)}\\over{4}} \n\\left[ 1 +\\left({E^0_+\\over{E_b}}\\right)^2 \n+{\\left(1-{E^0_+\\over{E_b}}\\right)^2 \\over{\\ln ({2E_b}\/m_e)^2 -1}} \\right] \n {d\\sigma_0(E_b,E^0_+,\\theta_-)\\over{d\\Omega_-}} \n\\nonumber \\ ,\n\\end{eqnarray}\nwhere \n\\begin{equation}\n{d\\sigma_0(E_b,E^0_{\\pm},\\theta_-)\\over{d\\Omega_-}} =\n\\left( {d\\Omega^* \\over{d\\Omega_-}} \\right)\n \\sum_{i=1}^{10} {d\\sigma_0^{(i)}(E^*,\\theta^*) \\over{d\\Omega^*}}\n\\end{equation}\nwith accordingly $E^*=\\sqrt{E_b E^0_{\\pm}}$ and \n$\\theta^*=\\theta^*(E_b,E^0_{\\pm},\\theta_-)$. \n\\par\nIn Eq.~(\\ref{eq:fo1}) the last two lines are due to hard initial emission, \nthe second and third lines are due to hard final emission; in the approximation \nof constant $\\beta$ and symmetrical cuts, the hard final emission contributions \ncan be integrated analytically giving \n\\begin{equation}\n + \\beta(E_b,\\theta_-) \\left[ \\ln{\\Delta\\over{\\epsilon}} -\\Delta\n +{\\Delta^2 \\over{4}} {\\ln(s\/m_e^2) \\over{(\\ln(s\/m_e^2) -1)}} \\right] \n \\sum_{i=1}^{10} {d\\sigma_0^{(i)}(E_b,\\theta_-) \\over{d\\Omega_-}} \\ ,\n\\label{eq:fo2} \\end{equation}\nwith $\\Delta = 1 -E_{min}\/E_b$.\nIn Table 1 of ~\\cite{CERN95-03} Eq.~(\\ref{eq:fo2}) is used to produce the \nresults presented under the label BG94-FO-OLD, which \ndiffer from the exact ones up to 0.87\\%, while the direct generation from \nEq.~(\\ref{eq:fo1}) gives the results under the label BG94-FO-NEW, which have \nonly up to 0.28\\% difference from the exact ones; these are under label OLDBIS \n~\\cite{OLDBIS} and BG94-FO-EXACT (obtained using BHAGEN-1PH ~\\cite{1PH} for\nthe hard photon emission part) and are well in agreement inside the OLDBIS \ntechnical precision of 0.02\\%. \n\\bigskip\n\n\\section{ Implementation of weak and QCD corrections.}\n\n\\def(v_e^2 + a_e^2){(v_e^2 + a_e^2)}\n\\def\\delta^{tZ}_{1}(+){\\delta^{tZ}_{1}(+)}\n\\def\\delta^{tZ}_{1}(-){\\delta^{tZ}_{1}(-)}\n\\def\\[(2s^2-1)]{\\left(2 \\ s_W^2 -1 \\right)}\n\\def{\\left(1+z\\right)}^2{{\\left(1+z\\right)}^2}\n\\def\\delta^{tZ}_{3}(+){\\delta^{tZ}_{3}(+)}\n\\def\\hbox{Re}\\left(\\chi(s)\\right){\\hbox{Re}\\left(\\chi(s)\\right)}\n\\def\\hbox{Im}\\left(\\chi(s)\\right){\\hbox{Im}\\left(\\chi(s)\\right)}\n\\def\\delta^{sZ}_{2}(+){\\delta^{sZ}_{2}(+)}\n\\def\\delta^{sZ}_{1}(+){\\delta^{sZ}_{1}(+)}\n\\def\\delta^{sZ}_{1}(-){\\delta^{sZ}_{1}(-)}\n\nThe previous versions of the program ~\\cite{NC92}, ~\\cite{IJMP93}, including \nthe latest BHAGEN94 ~\\cite{PL94}, did not contain the $Zee$ weak vertex \ncorrection, while the self energy corrections were included in an \napproximate form.\nFor an accuracy better than 1\\% at large-angle those complete corrections \n~\\cite{HOLLIK}, and also some higher order corrections ~\\cite{CERN95-03_2}\nare necessary. \nAfter their inclusion the error due to the neglected weak one loop corrections \nis about 0.06\\% around the $Z$ boson resonance (LEP1) and 0.1\\% at LEP2 \nenergies, so smaller than the other left over photonic corrections.\nThe use of the program at much higher energies (say 1 TeV) requires an update \nof the weak library for having a precision better than 2\\%.\n\nFor the one loop $Zee$ vertex corrections and self-energy corrections \nwe use the results of ~\\cite{HOLLIK}, but we introduce the \nDyson resummation of the self energy, as described for $s$-channel in\n~\\cite{CERN89_1} (we do actually the Dyson resummation in all channels).\nThe higher orders corrections are included as outlined in ~\\cite{CERN95-03_2},\nusing the available results for weak ~\\cite{FLEISCHER} \nand QCD corrections ~\\cite{Kniehl}.\n\nFor completeness we report here the formulae of these corrections as they come\nusing our notation as in ~\\cite{NC92}, ~\\cite{IJMP93}, ~\\cite{PL94} and \n~\\cite{CERN89}.\n\nThe $\\tilde C^{(i)}(E^*,\\theta^*)$ for $(i=4,...,10)$ should be changed in \nthe following way\n\n\\begin{equation}\n \\tilde C^{(i)}(E^*,\\theta^*) \\rightarrow \\tilde C^{(i)}(E^*,\\theta^*)\n + C^{(i)}_{Z}(E^*,\\theta^*) \\ . \n\\end{equation}\n\nThe $C^{(i)}_{Z}(E^*,\\theta^*)$ are \n\n\\begin{equation}\n C^{(4)}_Z ={\n { 8 \\delta^{tZ}_{1}(+) s_W^4 + 2 \\delta^{tZ}_{1}(-) \\[(2s^2-1)]^2 }\n \\over {(v_e^2 + a_e^2)}} \\ , \n\\label{eq:c4}\n\\end{equation}\n\n\\begin{eqnarray}\n C^{(5)}_Z &=& {1 \\over {(v_e^2 + a_e^2) \\left( {\\left(1+z\\right)}^2 + 4(r_V-r_A)\\right) }}\n {\\Bigl[ 8 \\delta^{tZ}_{1}(+) {\\left(1+z\\right)}^2 s_W^4} \\label{eq:c5} \\\\\n & &{ + 2 \\delta^{tZ}_{1}(-) {\\left(1+z\\right)}^2 \\[(2s^2-1)]^2 + 32 \\delta^{tZ}_{3}(+) s_W^2\n \\[(2s^2-1)] \\Bigr]} \\ , \\nonumber \n\\end{eqnarray}\n\n\\begin{eqnarray}\n C^{(6)}_Z &=& { 1 \\over\n {(v_e^2 + a_e^2)^2 \\left[{\\left(1+z\\right)}^2 (1+4r_V r_A) + 4 (1-4r_V r_A) \\right]}}\n { \\Bigl[ 256 \\delta^{tZ}_{1}(+) {\\left(1+z\\right)}^2 s_W^8} \\nonumber\\\\ \n & &{+ 16 \\delta^{tZ}_{1}(-) {\\left(1+z\\right)}^2 \\[(2s^2-1)]^4 \n + 512 \\delta^{tZ}_{3}(+) s_W^4\n \\[(2s^2-1)]^2 \\Bigr] } \\ , \n\\label{eq:c6}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n C^{(7)}_Z &=&{ {1} \\over { 2 (v_e^2 + a_e^2) \\Bigl( (1+z^2)r_V + 2 z r_A \\Bigr) } }\n { \\Bigl[ \\Bigl( 8 (1-z^2) \\hbox{Re}(\\delta^{sZ}_{2}(+)) s_W^2 \\[(2s^2-1)]} \\nonumber\\\\ \n & &{ + 8{\\left(1+z\\right)}^2 \\hbox{Re}(\\delta^{sZ}_{1}(+)) s_W^4 \n + 2{\\left(1+z\\right)}^2 \\hbox{Re}(\\delta^{sZ}_{1}(-)) \\[(2s^2-1)]^2 \\Bigr) } \\label{eq:c7} \\\\ \n & &{ + {{\\hbox{Im}\\left(\\chi(s)\\right)}\\over{\\hbox{Re}\\left(\\chi(s)\\right)}}\n \\Bigl( - 8 (1-z^2) \\hbox{Im}(\\delta^{sZ}_{2}(+)) s_W^2 \\[(2s^2-1)]\n - 8 {\\left(1+z\\right)}^2 \\hbox{Im}(\\delta^{sZ}_{1}(+)) s_W^4 } \\nonumber\\\\ \n & &{ - 2{\\left(1+z\\right)}^2 \\hbox{Im}(\\delta^{sZ}_{1}(-)) \\[(2s^2-1)]^2 \\Bigr) \\Bigr] } \\ , \\nonumber \n\\end{eqnarray}\n\n\\begin{eqnarray}\n C^{(8)}_Z &=& { 1 \\over {(v_e^2 + a_e^2)} }\n { \\Bigl[ 8 \\hbox{Re}(\\delta^{sZ}_{1}(+)) s_W^4\n + 2 \\hbox{Re}(\\delta^{sZ}_{1}(-)) \\[(2s^2-1)]^2 } \\label{eq:c8} \\\\ \n & &{+ {{\\hbox{Im}\\left(\\chi(s)\\right)}\\over{\\hbox{Re}\\left(\\chi(s)\\right)}} \\Bigl( - 8 \\hbox{Im}(\\delta^{sZ}_{1}(+)) s_W^4\n - 2 \\hbox{Im}(\\delta^{sZ}_{1}(-))\\[(2s^2-1)]^2 \\Bigr) \\Bigr] } \\ , \\nonumber \n\\end{eqnarray}\n\n\\begin{eqnarray}\n C^{(9)}_Z &=& {1 \\over { {(v_e^2 + a_e^2)^2 } \\left(1+4 r_V r_A \\right)} }\n { \\Bigl[ 128 \\delta^{tZ}_{1}(+) s_W^8\n + 8 \\delta^{tZ}_{1}(-) \\[(2s^2-1)]^4} \\nonumber\\\\ \n & &{ + 128 \\hbox{Re}(\\delta^{sZ}_{1}(+)) s_W^8\n + 8 \\hbox{Re}(\\delta^{sZ}_{1}(-)) \\[(2s^2-1)]^4 } \\label{eq:c9} \\\\ \n & &{ -{{\\hbox{Im}\\left(\\chi(s)\\right)}\\over{\\hbox{Re}\\left(\\chi(s)\\right)}} \\Bigl( 128 \\hbox{Im}(\\delta^{sZ}_{1}(+)) s_W^8 \n + 8 \\hbox{Im}(\\delta^{sZ}_{1}(-)) \\[(2s^2-1)]^4 \\Bigr) \\Bigr] } \\ ,\\nonumber \n\\end{eqnarray}\n\n\\begin{eqnarray}\n C^{(10)}_Z &=& { 1 \\over { (1+z^2) (v_e^2 + a_e^2)^2 + 8 z v_{e}^{2} a_{e}^{2} } }\n {\\Bigl[ 128 {\\left(1+z\\right)}^2 \\hbox{Re}(\\delta^{sZ}_{1}(+)) s_W^8 } \\label{eq:c10} \\\\ \n & &{+ 8 {\\left(1+z\\right)}^2 \\hbox{Re}(\\delta^{sZ}_{1}(-)) \\[(2s^2-1)]^4 \n + 64 (1-z)^2 \\hbox{Re}(\\delta^{sZ}_{2}(+)) s_W^4 \\[(2s^2-1)]^2 \\Bigr] } \\ .\\nonumber \n\\end{eqnarray}\n\nThe symbols used are \n\\begin{eqnarray}\n&&\\delta_{1}^{tZ}(\\pm)=\\delta_{1,w,V}^{tZ}(\\pm) \n + {{2}\\over{g^Z_{\\pm}}} \\Pi^{\\gamma Z}(t) \\ ,\n \\delta_{1}^{sZ}(\\pm)=\\delta_{1,w,V}^{sZ}(\\pm) \n + {{2}\\over{g^Z_{\\pm}}} \\Pi^{\\gamma Z}(s) \\ , \\label{eq:cdef} \\\\\n&&\\delta_{2}^{sZ}(\\pm)=\\delta_{2,w,V}^{sZ}(\\pm) \n+ \\left( {{1}\\over{g^Z_{+}}} + {{1}\\over{g^Z_{-}}}\\right) \\Pi^{\\gamma Z}(s) \\ ,\n \\delta_{3}^{tZ}(\\pm)=\\delta_{3,w,V}^{tZ}(\\pm) \n+ \\left({{1}\\over{g^Z_{+}}} + {{1}\\over{g^Z_{-}}}\\right) \\Pi^{\\gamma Z}(t) \\ ,\n\\nonumber \\end{eqnarray}\nwhile functions $\\delta_{i,w,V}^{tZ}(\\pm)$, $\\delta_{i,w,V}^{sZ}(\\pm)$\n ($i=1,2,3$), $\\Pi^{\\gamma Z}(t)$ and constants $g^Z_{\\pm}$\nare defined in the second reference of ~\\cite{HOLLIK} in the same notation. \nAs usual we take \n\\begin{equation}\n {\\sin^2(\\theta_W) \\equiv s_W^2 = 1 - c_W^2 = 1-{{M_W^2}\\over{M_Z^2}} } \\ .\n\\label{eq:c11} \n\\end{equation}\n\nThe above formulae take into account the $Zee$ weak vertex corrections and \n$\\gamma Z$ mixing self energy contribution. \nTo include completely the $Z$ boson self energy corrections it is sufficient \nto change in our approach the two functions $\\chi(s)$ and $\\chi'(t)$. \nThe new ones are \n\\begin{eqnarray}\n \\chi(s) &=& { {{1} \\over {16 s_W^2 c_W^2}} {{1}\\over{1+\\Pi^Z(s)}}\n (v_e^2 + a_e^2) {{s}\\over{s-M_Z^2 +is\\Gamma_Z\/M_Z}} }\\ ,\n \\label{eq:c12} \\\\ \n \\chi'(t) &=& { {{1} \\over {32 s_W^2 c_W^2}} {{1}\\over{1+\\Pi^Z(t)}}\n (v_e^2 + a_e^2) {{s}\\over{M_Z^2 -t}} } \\ ,\n \\label{eq:c13} \n\\end{eqnarray}\nwhere \n$\\Pi^Z(s) = \\hbox{Re}[\\hat \\Sigma^Z(s)]\/(s-M_Z^2)$ and $\\hat \\Sigma^Z(s)$ \nis again defined in ~\\cite{HOLLIK}.\nAll the higher order corrections are included into the \n$\\Pi^Z$ and $\\Pi^{\\gamma Z}$ functions, and through them into the formulae\n~(\\ref{eq:c4}-\\ref{eq:cdef}) and ~(\\ref{eq:c12}-\\ref{eq:c13}), which remain\nunchanged.\n\nThe value of $s_W^2$ is calculated using Eq.~(\\ref{eq:c11}) with $M_Z$\nas an input parameter. In the code the value of $M_W$ can be supplied by \nthe user, or is calculated using the Fermi constant $G_F$ as an independent \nparameter.\nIn the latter case we implement the approach outlined in ~\\cite{CERN95-03_2},\nwhich allows to get the value of the $W$ boson mass with higher accuracy\nthan from its direct measurement. \nFor this better precision it is necessary to include the already mentioned \nweak corrections up to two loops to the $\\Delta\\rho$ parameter \n~\\cite{FLEISCHER} and also the QCD corrections to the self energy functions of \nthe bosons (which are needed to properly calculate $\\Delta r$ ) ~\\cite{Kniehl}. \nChecking with the values of the $W$ boson mass presented in ~\\cite{CERN95-03_2} \nas a function of the mass of the top quark $m_t$ (at fixed values of \n$m_H=300$ GeV and $\\alpha_S(M_Z)=0.125$), we obtain a very good agreement \naround $m_t=175$ GeV (only a small difference (0.03\\%) is appreciable \nabove the unrealistic value $m_t$ = 225 GeV). \n\nActually the inclusion in BHAGEN94 (and as a consequence also in BHAGEN95) of \nthe higher order QCD corrections is done in an approximate form. \nIn particular for the first two fermion generations (u,d,s and c quarks, \nfor which the limit $s,-t \\gg m^2$ is valid, with $m$ the quark mass) \nwe use the multiplicative factor $(1+{{\\alpha_S}\\over \\pi}+...)$ \nin front of the contribution due to one quark loop in the self energy \nfunctions of the bosons. \nFor the $t,b$ doublet we use the approximated formulae, \npresented in ~\\cite{Kniehl}, \nfor the doublet contribution to $\\Delta r$ in the $M_W$ calculation.\nOn the contrary in the Bhabha cross-section calculations is necessary the \nknowledge of the separate contributions $\\Pi^Z$ and $\\Pi^{\\gamma Z}$ to the \nboson self energies, we use them in the form described in ~\\cite{CERN95-03_2} \n(BHM\/WOH approach), i.e. we include the leading $\\Delta\\rho^{HO}$ corrections. \nIn this way the corrections are\ncalculated in the point $s=M_Z^2$ and their $s,t$ dependance \n(which is known however only in the QCD two loop contribution) is not taken\ninto account. Also some known nonleading corrections are not included. \nAll this considered the approximation is sufficient to reach at large-angle \nthe projected accuracy of 0.5\\% in the cross-section calculations. \nMoreover other not included QED two loop corrections are expected to be \nmuch bigger than the left over QCD corrections, like the next to leading \ncorrection, which is not yet calculated, or the inferred leading term in \n$s-t$ channel interference, assumed to be identical to the $s$-channel.\n\\bigskip\n\n\\section{ The BHAGEN95 code. }\n\nBHAGEN95 is a collection of three programs to calculate the cross-section\nfor Bhabha scattering from small to large scattering angles at LEP1 and\nLEP2 energies.\nIn its present form the integrated cross-section $\\sigma({\\hbox{BHAGEN95}})$ \nfor a given selection of cuts is calculated as\n\\begin{eqnarray}\n \\sigma({\\hbox{BHAGEN95}})\n = \\sigma({\\hbox{BHAGEN94}})\n -\\sigma^H({\\hbox{BH94-FO}})\n +\\sigma^H({\\hbox{BHAGEN-1PH}}) \\ .\n \\label{eq:cs}\n\\end{eqnarray}\n\n$\\sigma({\\hbox{BHAGEN94}})$ is the integrated cross-section, \nbased on leading \\Ord{\\alpha^2} exponentiated formulae, \nobtained with the Monte Carlo event generator BHAGEN94, \nwith the implementations described in the previous sections 2 and 4.\nThe use of collinear kinematics of initial and final radiation leaves \nsome approximation in the angular distribution, which limits the accuracy\nparticularly in the region of hard photon emission, as remarked at the end of\nsection 2.\n\\par\n$\\sigma^H({\\hbox{BH94-FO}})$ is the integrated cross-section of\n\\Ord{\\alpha} for one hard photon emission, obtained selecting the appropriate \npart in the Monte Carlo event generator BH94-FO, the \\Ord{\\alpha} expansion \nof BHAGEN94 described in section 3.\n\\par\n$\\sigma^H({\\hbox{BHAGEN-1PH}})$ is the integrated cross-section obtained\nwith the one hard photon complete matrix element and exact kinematics,\nimplemented in the Monte Carlo event generator BHAGEN-1PH ~\\cite{1PH}.\n\\par\nThe subtraction of $\\sigma^H({\\hbox{BH94-FO}})$ and its substitution\nwith $\\sigma^H({\\hbox{BHAGEN-1PH}})$ is performed to reduce the error\nin the cross-section, coming from the approximation in the contribution \ndue to the one hard photon emission in BHAGEN94, as discussed in \n~\\cite{CERN95-03}.\nIn this way the one hard photon emission is treated exactly in BHAGEN95.\n\nThe choice of the energy threshold $\\epsilon$ (in units of the beam energy) \nfor a photon to be considered hard should not affect the difference between \n$\\sigma^H({\\hbox{BHAGEN-1PH}})$ and $\\sigma^H({\\hbox{BH94-FO}})$. \nHowever the approximate hard photon treatment in BH94-FO implies that a \ncorrection of the order of $\\epsilon$ is missing, so it is necessary to \nchoose $\\epsilon$ well below the desired accuracy, typically in the range \nof $10^{-5}-10^{-4}$.\nBut even so the difference is really independent on $\\epsilon$ only if \nthe soft photon treatment in BH94-FO is the same as in the exact program \nBHAGEN-1PH. This is indeed the case.\nIn fact changing $\\epsilon$ from $10^{-4}$ to $10^{-5}$ the separate \ncross-sections are growing about 20-30\\% for the event selections described \nin ~\\cite{CERN96}, while the difference is stable within statistical errors.\n\nThe three programs provide cross-sections, which are summed as in \nEq.~(\\ref{eq:cs}) or used to obtain other quantities, such as \nforward-backward asymmetry.\nAlthough the constituent programs are separately genuine event generators, \nthe discussed combination can be used for Monte Carlo integration only.\n\nAt small-angle we estimate the accuracy in the cross-section evaluation,\ndue to the uncontrolled higher orders terms \\Ord{\\alpha^2 L} and\n\\Ord{\\alpha^3 L^3} and to the incertitude in \\Ord{\\alpha^2 L^2} $s-t$\ninterference, to amount comprehensively to about 0.1\\%.\nThe further error, due to the approximate two hard photon contribution \n(strongly dependent on the imposed cuts) is estimated on the basis of the \ncalculated correction for the one hard photon contribution times \n$\\beta_e(s)\\simeq 0.1$, to account for the increase in perturbative order.\n\nIn Table \\ref{Tab1} are presented the values of the ratio $R$, defined as \n \\begin{eqnarray}\n R = 100 {{\\sigma^H({\\hbox{BHAGEN-1PH}})\n -\\sigma^H({\\hbox{BH94-FO}}) } \\over\n {\\sigma({\\hbox{BHAGEN95}})}}\n \\ ,\n \\label{eq:cs1}\n\\end{eqnarray}\nwhich is the correction (in \\%) introduced in BHAGEN95, to account for the \napproximation in the one hard photon contribution in BHAGEN94. \nThe correction is given for the event selections used in ~\\cite{CERN96}, \nin the same notation as for the $t$-channel comparison. \nAs already said, for $\\beta_e(s)\\simeq 0.1$ the same numbers in Table \n\\ref{Tab1} are an estimation (in per mil) of the inaccuracy, due to the \napproximate treatment of two hard photon emission.\nThe values in Table \\ref{Tab1} are acceptably small for cuts closer to the \nones in experiments, which are of the angular asymmetric WN type, with \n$ 0.3 \\le z_{min} \\le 0.7 $, where $z_{min}$ is the energy threshold for the\nfinal clusters for accepting or rejecting the events. \nIndeed in such cases the (included) one hard photon \ncorrections are found to be below 1.4 \\% at LEP1, so that the corrections \nto two hard photon emissions are expected to be below 1.4 per mil. \nFor the values at $z_{min}=0.9$ and calorimetric event selection, \nthese corrections are expected to be much bigger (at a few per mil level).\n\nAll included we estimate at small-angle an accuracy of the order of\n0.1\\%-0.2\\% for typical experimental cuts for both LEP1 and LEP2 energies.\n\\par\nAt large-angle we estimate the accuracy of the \\Ord{\\alpha^2 L^2} $s-t$ \ninterference contribution up to 1\\% (depending on energy and cuts) at LEP1, \nbut much smaller at LEP2.\nThe error, coming from the approximate treatment of two hard photon emission,\nis estimated as explained above for the small-angle case, and is smaller for \nmore stringent acollinearity cut.\nAll included we estimate an accuracy of the order of 1\\% in the worst case\nat LEP1, when the beam energy is a few GeV above the $Z$ boson peak, while\ntypical accuracy for relatively loose cuts is 0.5\\%.\n\nThe three programs run separately.\nThey provide initialization and fiducial volume definition according to\ninput parameters, then starts the generation of events according to \nappropriate variables, which smooth the cross-section behavior.\nRejection is performed through the routine {\\tt TRIGGER}, where the special\ncuts can be implemented.\nThe programs stop when the requested number of accepted events is reached\nor alternatively when the requested accuracy is obtained.\n\nThe following data have to be provided in input:\nmass of the $Z$, mass of the top quark, mass of the Higgs, value of\n$\\alpha_S(M_Z)$, value of $\\Gamma_Z$, the beam energy $E_{beam}$, the\nminimum energy for final leptons $E_{min}$ (larger than 1 GeV), \nminimum and maximum angle for the scattered electron (positron) with \nthe initial electron (positron) direction, maximum acollinearity allowed \nbetween final electron and positron, number of accepted events to be produced, \nnumbers to initialize the random number generator.\nThe following possibilities are also provided: i) to switch on or off the \nleading contribution from virtual and soft emitted pairs ~\\cite{PL94},\nii) to calculate separately the different channel contributions (useful for \ntests), iii) the recording of the generated events of each component program \nin a separate file.\n\nFor \\Ord{\\alpha} programs the minimum and maximum energy allowed for the photon\nhas to be specified.\nThe input of BHAGEN-1PH requires also the maximum acoplanarity, \nand minimum angles of the emitted photon with initial \nand final fermion directions, if the contributions with the collinear photons\nare to be excluded.\n\nEach program returns the input parameters and the values of the cross-section\nobtained with weighted and unweighted events, with the relative statistical\nvariance (one standard deviation).\nOf course, due to the efficiency, the weighted cross-section is usually much\nmore precise than the unweighted one.\nThe total integrated cross-section is then calculated according to Eq.\n(\\ref{eq:cs}).\n\n\n\\begin{table}[!ht]\n\\centering\n\\begin{tabular} {|c|c|c|c|c|c|}\n\\hline\n$ z_{min} $\n& 0.1\n& 0.3\n& 0.5\n& 0.7\n& 0.9\n\\\\\n\\hline\n\\hline\n BARE1(WW) \n& \n& -0.24\n& -0.29\n& -0.23\n& -0.11\n\\\\\n\\hline\n CALO1(WW)\n& \n& -0.46\n& -0.75\n& -1.15\n& -2.29\n\\\\\n\\hline\n CALO2(WW)\n& -0.38\n& -0.60\n& -1.01\n& -1.66\n& -3.57\n\\\\\n\\hline\n SICAL2(WW)\n& +0.35\n& +0.26\n& +0.005\n& -1.02\n& -3.34\n\\\\\n\\hline\n CALO2(NN)\n& -0.54\n& -0.73\n& -1.06\n& -1.76\n& -3.66\n\\\\\n\\hline\n CALO2(NW)\n& +0.01\n& -0.22\n& -0.64\n& -1.39\n& -3.40\n\\\\\n\\hline\n CALO2(WW-all incl.)\n& -0.36\n& -0.57\n& -0.96\n& -1.59\n& -3.42\n\\\\\n\\hline\n CALO2(NN-all incl.)\n& -0.52\n& -0.70\n& -1.02\n& -1.69\n& -3.51\n\\\\\n\\hline\n CALO2(WN-all incl.)\n& +0.00\n& -0.21\n& -0.61\n& -1.33\n& -3.26\n\\\\\n\\hline\n\\hline\n CALO3(NN)\n& -1.14\n& -1.29\n& -1.56\n& -2.19\n& -4.38\n\\\\\n\\hline\n CALO3(NW)\n& -0.21\n& -0.41\n& -0.77\n& -1.61\n& -3.95\n\\\\\n\\hline\n CALO2(WW)\n& \n& -0.61\n& -1.02\n& -1.68\n& -3.62\n\\\\\n\\hline\n SICAL2(WW)\n& \n& +0.25\n& +0.04\n& -1.03\n& -3.40\n\\\\\n\\hline\n\\end{tabular}\n\\caption{\\small \nCorrection R, defined in Eq.~({\\protect\\ref{eq:cs1}}), due to the approximate \ntreatment of the one hard photon emission in BHAGEN94, in \\% of the BHAGEN95 \ncross-section. \nAll event selections are defined in ~\\protect\\cite{CERN96} in the same \nnotation. \nFirst 9 rows are for LEP1 and last 4 rows are for LEP2 energies.\nThe option 'all incl.' means that all possible contributions are included, \nwhile the remaining tests are done only for $t$-channel contributions with \nvacuum polarization switched off. }\n\\label{Tab1}\n\\end{table}\n\\bigskip\n\n\\section{ Comparisons at small-angle. }\n\nThe \\Ord{\\alpha} results are tested with other programs in ~\\cite{CERN95-03} \nfor bare event selection and in ~\\cite{CERN96} also for calorimetric event \nselection, where the precision of 0.03 \\% is confirmed, as a consequence \nof the agreement of different programs within statistical errors. \n\nIn ~\\cite{CERN95-03} the results of BHAGEN94, including the improvement \noutlined here in section 2, are presented for a comparison of the \ncross-section with the final electron and positron scattering \nangles $\\theta_\\pm$ in the range $3^{\\circ} \\leq \\theta_\\pm \\leq 8^{\\circ}$.\nFrom this comparison it is clear that we cannot reach a precision better than\n0.3\\% using only the structure function approach. \n\nTo have a better precision the code BHAGEN95 is settled, with the features \ndescribed in the previous section. \nSeveral preliminary results are already presented and compared in \n~\\cite{CERN96}, so we correct the few which are revised, but \nwe do not repeat here the unchanged ones. \n\nIn Fig. 16 of ~\\cite{CERN96}, in the comparison at small-angle \n($t$-channel only) the bare event selection case (BARE1, WW) shows \ngood agreement (almost inside the 0.1\\% box) \nup to $z_{min} = 0.9$ included among all the programs (BHAGEN95, \nBHLUMI 4.03 ~\\cite{BHLUMI4},\nOLDBIS+LUMLOG ~\\cite{OBI+LUMG}, \nSABSPV ~\\cite{SABSPV},\nNLLBHA ~\\cite{NLLBHA}).\n\nFor calorimetric event selections BHAGEN95 and OLDBIS+LUMLOG \nhave very close results for every event selection cut, but they \nagree (within 0.1\\% at LEP1 and 0.2\\% at LEP2) with BHLUMI and SABSPV \nonly for values of the parameters close to the real experimental ones \n(NW or WW, $0.3\\le z_{min} \\le 0.7$).\nThe difference is larger for more severe cuts (as in the NN case) and \nfor $z_{min} = 0.9$. \n\nIt is proposed in ~\\cite{CERN96} that for OLDBIS+LUMLOG the difference is due \nto the so called 'classical limit' (i.e. zero radiation emission limit)\nwhich could be different in higher orders.\n\nFor BHAGEN95 we believe that the difference is due to the approximate \ntreatment of two hard photons illustrated in previous section, while the\n'classical limit' is sufficiently accurate.\nIn fact for calorimetric event selection, even for $z_{min} \\rightarrow 1.0$, \nthat limit is not approached, as almost collinear hard photons can join the\nfinal lepton in the cluster.\nThe 'classical limit' is approached only for BARE1 event selection, where\nall the programs are in substantial agreement even at $z_{min} =0.9$, \nand remarkably there the estimated error in BHAGEN95 on the basis of \nTable \\ref{Tab1} is only about 0.01 \\%.\nOn the contrary for calorimetric event selection and $z_{min} = 0.9$ \nhard photons are included within very stringent cuts on phase space, \nas the cluster opening is very narrow. \nThe structure function approach (used in the part of BHAGEN95 called BHAGEN94) \nfor this configuration is expected to be inaccurate, as it is not able to mimic \nvery sophisticated cuts.\n\nIn conclusion, at small-angle for the test cross-section called \n'$t$-channel only', the results of BHAGEN95 are in very good agreement \nwith those of OLDBIS+LUMLOG, and inside the errors also with those of \nBHLUMI and SABSPV. \nThe accuracy of BHAGEN95 is very much dependent on the event selection used, \nand is of the order of 0.1\\% - 0.2\\% for typical experimental cuts, \nwhile for more stringent cuts it can amount up to say 0.5\\%, with the \nsource of the error well understood.\n\nThe results for the complete cross-section of BHAGEN95, with all the other \ncontributions included, are presented and compared in Table 18 and in Fig. 17 \nof ~\\cite{CERN96}, repeated here in Fig. \\ref{fig:sical92asy-FG} for \ncompleteness\n\\footnote{The values of BHAGEN95 in Fig. 2 of ~\\cite{BWG_LETT} differ \n slightly from the ones in ~\\cite{CERN96}, due to a second bug \n introduced in the rush-fixing of a previous (actually inactive) \n bug in the routine {\\tt TRIGGER}. \n The corrected results confirm those already presented in \n ~\\cite{CERN96}. }.\n\nIt is interesting to compare the difference $\\Delta\\sigma$ between the \ncomplete results (all included) of Table 18 and the '$t$-channel only' \nresults of Tables 14 and 16 of ~\\cite{CERN96} for BHLUMI and BHAGEN95 \n(for SABSPV values the comparison has no significance due to larger \nstatistical errors). \nIn Table \\ref{Tab2} are presented the values for different angular ranges \n(WW, NN, WN) and energy threshold $z_{min}$, taken from Table 18 of \n~\\cite{CERN96} for BHLUMI and calculated as indicated above for BHAGEN95.\n\nThe difference $\\Delta\\sigma$ is mainly due to vacuum polarization correction, \npresumably implemented in the same way in both programs, using also the \nparameterization in ~\\cite{VPH}, and to the $Z-\\gamma$ interference term.\nThis latter contribution at small-angle is in the range of 1 per mil of \nthe total cross-section, depending mostly on the angular range allowed and \nmuch less on the other details of the selection.\n\nIn the last column of Table \\ref{Tab2} is presented in per mil the variation \n \\begin{eqnarray}\n V = 1000 {{\\Delta\\sigma({\\hbox{BHAGEN95}})\n -\\Delta\\sigma({\\hbox{BHLUMI}}) } \\over\n {\\sigma({\\hbox{BHLUMI,all incl.}})}}\n \\ ,\n \\label{eq:cs2}\n\\end{eqnarray}\nbetween the complete cross-section obtained with BHLUMI and the one obtained \nwith BHAGEN95, due to the difference in vacuum polarization implementation and \n$Z-\\gamma$ interference correction. \nIt can be seen that in the experimentally interesting region the variation \nis statistically significant, always positive and increasing with $z_{min}$, \nand it is almost up to 0.1\\%.\n\\begin{table}[!ht]\n\\centering\n\\begin{tabular} {|c|c|c|c|}\n\\hline\n$ z_{min} $\n& $\\Delta\\sigma$(BHLUMI,WW)\n& $\\Delta\\sigma$(BHAGEN95,WW)\n& V(WW)\n\\\\\n\\hline\n0.1\n&5.140(8) \n&5.203(11)\n&0.46(10)\n\\\\\n\\hline\n0.3\n&5.126(8)\n&5.197(14)\n&0.52(12)\n\\\\\n\\hline\n0.5\n&5.100(8)\n&5.195(14)\n&0.70(12)\n\\\\\n\\hline\n0.7\n&4.994(8)\n&5.125(16)\n&0.99(14)\n\\\\\n\\hline\n0.9\n&4.627(8)\n&4.821(16)\n&1.57(15)\n\\\\\n\\hline\n \n& $\\Delta\\sigma$(BHLUMI,NN)\n& $\\Delta\\sigma$(BHAGEN95,NN)\n& V(NN)\n\\\\\n\\hline\n0.1\n&3.751(7)\n&3.802(14)\n&0.51(16)\n\\\\\n\\hline\n0.3\n&3.742(7)\n&3.801(14)\n&0.60(16)\n\\\\\n\\hline\n0.5\n&3.728(7)\n&3.799(14)\n&0.72(16)\n\\\\\n\\hline\n0.7\n&3.678(7)\n&3.774(14)\n&0.99(16)\n\\\\\n\\hline\n0.9\n&3.430(7)\n&3.579(14)\n&1.64(18)\n\\\\\n\\hline\n\n& $\\Delta\\sigma$(BHLUMI,WN)\n& $\\Delta\\sigma$(BHAGEN95,WN)\n& V(WN)\n\\\\\n\\hline\n0.1\n&3.883(4)\n&3.920(13)\n&0.36(14)\n\\\\\n\\hline\n0.3\n&3.873(4)\n&3.919(13)\n&0.45(14)\n\\\\\n\\hline\n0.5\n&3.854(4)\n&3.916(13)\n&0.61(14)\n\\\\\n\\hline\n0.7\n&3.779(4)\n&3.869(13)\n&0.90(14)\n\\\\\n\\hline\n0.9\n&3.478(4)\n&3.624(14)\n&1.59(15)\n\\\\\n\\hline\n\\end{tabular}\n\\caption{\\small \nThe difference $\\Delta\\sigma$, in nb, between the complete cross-section \nand the '$t$-channel only' contribution, for BHLUMI and BHAGEN95, for several\nevent selections (WW, NN, and WN angular range), for $2E_b = 92.3$ GeV.\nIn the last column is the variation V, defined in \nEq.~({\\protect\\ref{eq:cs2}}), \ngiving in per mil the difference due to vacuum polarization implementation and \n$Z-\\gamma$ interference correction of BHAGEN95 respect to BHLUMI. \nIn brackets is the statistical error on the last digits. } \n\\label{Tab2}\n\\end{table}\n\nIt seems difficult that the difference comes from vacuum polarization \nimplementation, so we have investigated the $Z-\\gamma$ interference \ncontribution. \n\nIn the next section are reported comparisons at large-angle: BHAGEN95 is in \na good agreement (about 0.5\\%) with the other programs (notably with \nALIBABA for BARE event selection), where the contribution of the $Z-\\gamma$ \ninterference term amounts up to half of the cross-section.\n\nThe accuracy of the $Z-\\gamma$ interference term included in BHLUMI is tested \nin ~\\cite{J+P+W}, relaying mostly on the comparison with the ALIBABA code, \nso we have obtained results in the same conditions for BHAGEN95 for comparison.\nThe results of ALIBABA and the Born cross-sections, to which the\nresults are normalized, were taken from ~\\cite{B+P92} for old generation \nluminometers (LCAL: larger angular range between $3.3^{\\circ}$ and \n$6.3^{\\circ}$) and from ~\\cite{B+P93} for new generation luminometers \n(SCAL: smaller angular range between $1.5^{\\circ}$ and $3.15^{\\circ}$, more\nsimilar to the ones used in ~\\cite{CERN96}), \nwhile the results of BHLUMI are taken from ~\\cite{J+P+W} in the same \nconditions. \nIn Table \\ref{Tab3} the results of ALIBABA, BHLUMI and BHAGEN95 \nin the SCAL angular range are inside the 0.01\\% difference, and in the LCAL \nangular range are inside the 0.05\\%.\nThis test is done for very loose cuts: no cut on photon energy and angles\nis applied and only the angular ranges of leptons are restricted.\nIn the results of Table \\ref{Tab2} the agreement between BHLUMI\nand BHAGEN95 is much better for the looser cuts, than for the more severe ones,\nand this could be the reason for the better agreement in Table \\ref{Tab2} .\n\nIt is however difficult to identify with certainty the source of the difference \nwithout a more detailed analysis, particularly if it comes from a different\nuse of the same electroweak parameters, but it would be necessary in case the \naccuracy has to be better than 0.1\\%.\n\\begin{table}[!ht]\n\\centering\n\\begin{tabular} {|c|c|c|c|c|c|c|}\n\\hline\n &\n\\multicolumn{3}{|c|}{LCAL angular range}\n&\\multicolumn{3}{c|}{SCAL angular range} \\\\\n\\hline\n$ 2 E_b $ \n&BHAGEN95 \n&ALIBABA\n&BHLUMI\n&BHAGEN95\n&ALIBABA\n&BHLUMI\n\\\\\n\\hline\n89.661 \n&0.757\n&0.778\n&0.794\n&0.165\n&0.172\n&0.175\n\\\\\n\\hline\n90.036 \n&0.779\n&0.799\n&0.816\n&0.169\n&0.177\n&0.180\n\\\\\n\\hline\n90.411 \n&0.724\n&0.747\n&0.754\n&0.156\n&0.164\n&0.165\n\\\\\n\\hline\n90.786 \n&0.527\n&0.545\n&0.538\n&0.112\n&0.117\n&0.116\n\\\\\n\\hline\n91.161 \n&0.175\n&0.187\n&0.158\n&0.035\n&0.036\n&0.032\n\\\\\n\\hline\n91.563 \n&-0.206\n&-0.206\n&-0.247\n&-0.048\n&-0.050\n&-0.057\n\\\\\n\\hline\n91.911 \n&-0.473\n&-0.479\n&-0.521\n&-0.105\n&-0.110\n&-0.116\n\\\\\n\\hline\n92.286 \n&-0.602\n&-0.609\n&-0.647\n&-0.132\n&-0.136\n&-0.143\n\\\\\n\\hline\n92.661 \n&-0.642\n&-0.650\n&-0.678\n&-0.141\n&-0.145\n&-0.150\n\\\\\n\\hline\n\\end{tabular}\n\\caption{\\small \n$Z-\\gamma$ interference term (in \\% of Born cross-section)\n for BHAGEN95, ALIBABA and BHLUMI for two\ndifferent angular ranges (LCAL and SCAL) as a function of the beam energy. }\n\\label{Tab3}\n\\end{table}\n\\bigskip\n\n\\section{ Comparisons at large-angle. }\n\nAt large-angle some comparisons were presented for LEP1 and LEP2 energies \nin ~\\cite{CERN96}.\nHowever, as mentioned also there, the higher orders weak and QCD corrections \nwere just implemented in BHAGEN95 and the results were very preliminary. \nUnfortunately some bugs were present in the program and, after their \ncorrection and test, the new results of BHAGEN95 can now be compared with \nthe ones of the other programs \nALIBABA ~\\cite{ALIBABA}, \nTOPAZ0 ~\\cite{TOPAZ0}, \nBHAGENE3 ~\\cite{BHAGENE}, \nUNIBAB ~\\cite{UNIBAB},\nBHWIDE ~\\cite{BHWIDE},\nSABSPV ~\\cite{SABSPV}.\n\nWe report here the new results of BHAGEN95 in Fig. \\ref{fig:lab-lep1-bare} \nand Fig. \\ref{fig:lab-lep1-calo} for LEP1 and in Fig. \\ref{fig:lab-lep2-calo} \nfor LEP2 energies (corresponding respectively to Fig. 19, 20 and \nFig. 21 in ~\\cite{CERN96} in the same notation), for scattering angles \n$40^\\circ < \\theta_- < 140^\\circ$ and $0^\\circ < \\theta_+ < 180^\\circ$. \n\nWe estimate that at large-angle the accuracy of the program BHAGEN95 is 0.5\\% \neverywhere, except in the region of a few GeV above the $Z$ boson peak, \nwhere, due to the absence of the calculation of the leading \\Ord{\\alpha^2 L^2} \nterm in $s-t$ channel interference, it can be assumed to be up to 1\\% \n~\\cite{PL94}.\n\nIn Fig. \\ref{fig:lab-lep1-bare}, for LEP1 energy and BARE type event selection \nthe results of BHAGEN95 are in agreement with most of the programs inside 0.5\\% \nfor energies under and on top of the $Z$ boson peak, while above the resonance \nand with more stringent acollinearity cuts ($10^\\circ$) there is \na noticeable agreement with ALIBABA, but some difference from other programs, \nwhich however is still less than 1\\%. \nThe same pattern is kept for the calorimetric event selection CALO, \nshown in Fig. \\ref{fig:lab-lep1-calo}, where unfortunately results from \nALIBABA are no longer available. \n\nIn Fig. \\ref{fig:lab-lep2-calo}, for LEP2 energies and CALO event selection, \nthere is a remarkable clustering in a 2\\% interval of most of the programs.\n\\bigskip\n\n\\section{ Conclusions.} \n\nThe Monte Carlo integrator BHAGEN95 (available on request from the authors) \ncalculates the cross-section of the Bhabha scattering with continuity from \nsmall to large-angle and from LEP1 to LEP2 energies. \n\nThe program contains all the necessary corrections to provide results, \nfor typical experimental event selections, with the precision required \nto usefully compare with experimental measurements and other theoretical \nresults. \nWe estimate a precision of 0.1-0.2\\% at small-angle, in the angular range \nof the new generation of LEP luminometers, mainly due to the inaccuracy in\nin the two hard photon emission contribution. \nAt large-angle for both LEP1 and LEP2 energies, we estimate a precision of \n0.5\\% with the exception of the region of the beam energies a few GeV above \nthe $Z$ boson resonance, where the relevance of the still missing calculation \nof the leading \\Ord{\\alpha^2 L^2} radiative corrections to the $Z-\\gamma$ \ninterference term spoils the accuracy up to 1\\%, as in every other present \ncalculation.\n\nWe have performed a detailed comparison of the photonic and weak corrections \nfor both small-angle (angular range of the new generation of LEP luminometers) \nand large-angle Bhabha scattering. \n\nAt small-angle we obtain an agreement better than 0.02\\% at \\Ord{\\alpha} \nwith the program OLDBIS for every type of event selection. \nIn the comparison beyond the \\Ord{\\alpha}, but limited to the $t$-channel only, \nwith vacuum polarization and $Z-\\gamma$ interference switched off, the results \nof BHAGEN95 agree with the ones of OLDBIS+LUMLOG better than 0.03\\%, \nfor typical experimental event selection, and have a difference less than 0.1\\%\nwith BHLUMI and SABSPV in these conditions. \nFor event selections less realistic for experiments the results of BHAGEN95\n(and also of OLDBIS+LUMLOG) can differ from the ones of BHLUMI and SABSPV\nof a few per mil. We believe to understand the difference (at least in the \nBHAGEN95 case) as due to the inaccuracy in the two hard photon emission \ncontribution, which depends on the details of the event selection used and is \nsmaller for looser cuts. \nWhen all possible contributions are switched on the agreement between \nBHLUMI, SABSPV and BHAGEN95 is inside 0.1\\% for realistic experimental event\nselection, as presented in Fig. \\ref{fig:sical92asy-FG} (same as in Fig. 17 of \n~\\cite{CERN96}). \nThe better agreement in this case is due to a slightly larger (but less \nthan 0.1\\%) contribution of BHAGEN95, coming mainly from the implementation \nof the vacuum polarization and\/or $Z-\\gamma$ interference correction. \nWe have investigated this small difference, but it is impossible to \ndisentangle the contributions without ad hoc runs of all the compared programs.\n\nAt large-angle and in the LEP1 energy range the corrected results of BHAGEN95 \nare shown in Fig. \\ref{fig:lab-lep1-bare} and in Fig. \\ref{fig:lab-lep1-calo}; \nthe difference with the other results is \nwithin 0.5\\% around the $Z$ boson resonance, being bigger \nfor energies a few GeV above the peak, where BHAGEN95 is in agreement with\nALIBABA within 0.5\\% and within 1\\% with other codes.\n\nFor LEP2 energies although the differences between the codes are larger \n(up to 2\\% for the cluster of the results of TOPAZ0, BHWIDE, SABSPV and \nBHAGEN95), as shown in Fig. \\ref{fig:lab-lep2-calo}, the precision is \ncomparable with the foreseen experimental accuracy.\n\\bigskip\n\n\\noindent{\\bf Acknowledgments}\\par\nUseful conversations with the conveners (S.~Jadach and O.~Nicrosini) and \nall the members of the CERN Working Group of the Event Generators for Bhabha \nScattering are gratefully acknowledged. \nWe thank W.~Hollik for useful informations about the BHM\/WOH approach and \nW.~Beenakker about the ALIBABA parameters.\nOne of us (HC) is grateful to the Bologna Section of INFN and to the Department \nof Physics of Bologna University for support and kind hospitality.\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{S1.\\quad Symmetry constraints on the S-matrix}\n\nFirst, we consider $C_3$ rotation symmetry which preserves the valley. We find that\n\\begin{equation}\n\\mathcal S = C_3 \\mathcal S C_3^{-1},\n\\end{equation}\nwhere $C_3$ is a cyclic permutation $(a_1,a_1') \\rightarrow (a_2,a_2') \\rightarrow (a_3,a_3') \\rightarrow (a_1,a_1')$ of the incoming modes which are defined in Fig.\\ {\\color{red} 1}(b) of the main text, and similar for outgoing modes. Next, we discuss the effect of $C_2$ rotation symmetry and time-reversal symmetry $T$. As these symmetries do not conserve the valley, we need to consider both valleys:\n\\begin{equation}\n\\begin{pmatrix} b_K \\\\ b_{K'} \\end{pmatrix} = \\begin{pmatrix} \\mathcal S_K & 0 \\\\ 0 & \\mathcal S_{K'} \\end{pmatrix} \\begin{pmatrix} a_K \\\\ a_{K'} \\end{pmatrix}.\n\\end{equation}\nUnder $C_2$ rotation symmetry, we have\n\\begin{equation}\n\\begin{pmatrix} b_{K'} \\\\ b_K \\end{pmatrix} = \\begin{pmatrix} \\mathcal S_K & 0 \\\\ 0 & \\mathcal S_{K'} \\end{pmatrix} \\begin{pmatrix} a_{K'} \\\\ a_K \\end{pmatrix},\n\\end{equation}\nsuch that $\\mathcal S_{K'} = \\mathcal S_K$. On the other hand, under time-reversal symmetry we have\n\\begin{equation}\n\\begin{pmatrix} a_{K'}^* \\\\ a_K^* \\end{pmatrix} = \\begin{pmatrix} \\mathcal S_K & 0 \\\\ 0 & \\mathcal S_{K'} \\end{pmatrix} \\begin{pmatrix} b_{K'}^* \\\\ b_K^* \\end{pmatrix},\n\\end{equation}\nsuch that $\\mathcal S_{K'} = (\\mathcal S_K)^t$. Hence, the combination $C_2T$ enforces $\\mathcal S_K = (\\mathcal S_K)^t$.\n\n\\section{S2.\\quad S-matrix without forward scattering}\n\nThe $S$-matrix relates valley Hall states that propagate along AB\/BA domain walls at the scattering nodes (AA regions) such that $(b,b')^t = \\mathcal S (a, a')^t$ with $a,a'$ six incoming modes and $b,b'$ six outgoing modes where the prime distinguishes the two valley Hall states as illustrated in Fig.\\ {\\color{red} 1}(b) of the main text. In the absence of forward scattering, we find that the most general $S$-matrix consistent with unitarity and $C_3$ and $C_2T$ symmetry is given by\n\\begin{equation} \\label{eq:Smat1}\n\\mathcal S = e^{i\\varphi} \\begin{pmatrix}\n0 & e^{i \\phi} \\sqrt{P_{d1}} & e^{i \\phi} \\sqrt{P_{d1}} & 0 & \\sqrt{P_{d2R}} & -\\sqrt{P_{d2L}} \\\\\ne^{i \\phi} \\sqrt{P_{d1}} & 0 & e^{i \\phi} \\sqrt{P_{d1}} & -\\sqrt{P_{d2L}} & 0 & \\sqrt{P_{d2R}} \\\\\ne^{i \\phi} \\sqrt{P_{d1}} & e^{i \\phi} \\sqrt{P_{d1}} & 0 & \\sqrt{P_{d2R}} & -\\sqrt{P_{d2L}} & 0 \\\\\n0 & -\\sqrt{P_{d2L}} & \\sqrt{P_{d2R}} & 0 & -e^{-i\\phi} \\sqrt{P_{d1}} & -e^{-i\\phi} \\sqrt{P_{d1}} \\\\\n\\sqrt{P_{d2R}} & 0 & -\\sqrt{P_{d2L}} & -e^{-i\\phi} \\sqrt{P_{d1}} & 0 & -e^{-i\\phi} \\sqrt{P_{d1}} \\\\\n-\\sqrt{P_{d2L}} & \\sqrt{P_{d2R}} & 0 & -e^{-i\\phi} \\sqrt{P_{d1}} &-e^{-i\\phi} \\sqrt{P_{d1}} & 0\n\\end{pmatrix},\n\\end{equation}\nwith $\\varphi$ and $\\phi$ real phases, and with the conditions $2P_{d1}+P_{d2R}+P_{d2L}=1$ and $P_{d1}=\\sqrt{P_{d2R} P_{d2L}}$ which has two solutions. Either all probabilities are nonzero with $00$ can be represented for convenience by the simple parametrization shown as solid line in Fig.~\\ref{figure1}: $\\alpha(x_{F}) =0.9503e^{-ln2(\\frac{x_{F}}{1.3846})^{1.8067}} $ . If an experiment has published cross sections of $J\/\\psi$ in proton nucleus collisions, Eq.~\\ref{eq1} and the solid curve in Fig.~\\ref{figure1} is applied to obtain the corresponding $J\/\\psi$ cross section in proton nucleon collisions. Some of the experimental measurements are only quoted for a limited phase-space. To obtain the total cross sections, the functional forms of $x_{F}$ and $p_{T}$ spectrum shapes utilized for extrapolation are: $d\\sigma\/dx_{F} = a \\times e^{-ln2(\\frac{x_{F}}{b})^{c}}$,and $d\\sigma\/dp_{T} = a \\times \\frac{p_{T}}{(1+b^{2}p_{T}^{2})^{c}}$ respectively, where a, b, and c are free parameters. As illustrated in Fig.~\\ref{E331_Fig}, these two functional forms describe the $x_{F}$ and $p_{T}$ spectra very well. All the measurements are updated with the latest branching fractions ($5.961 \\pm 0.032\\%$ for $J\/\\psi \\rightarrow \\mu^{+} + \\mu^{-}$, $5.971 \\pm 0.032\\%$ for $J\/\\psi \\rightarrow e^{+}+e^{-}$) ~\\cite{PDG}. The treated results on $J\/\\psi$ cross sections are listed in Tab.~\\ref{table1}.They show a good overall consistency, even though some of them contradict with each other. For example, the two measurements (E331 and E444) at 20.6 GeV deviate from each other by roughly 2$\\sigma$. The E705 measurement at 23.8 GeV is higher than the UA6 one at 24.3 GeV by more than 2$\\sigma$.\n\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{drawE331pCpT.pdf}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{drawE331pCxF.pdf}\n\\caption{(color online) Distributions of $Ed\\sigma\/dp_{T}$ (left panel) and $Ed\\sigma\/dx_{F}$ (right panel) in $p+C$ collisions at $\\sqrt{s} = 20.6$ GeV measured by E331 collaboration ~\\cite{E331_1}. The solid lines are fit curves with the functional forms described in the text.}\n\\label{E331_Fig}\n\\end{center}\n\\end{figure*}\n\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{table*}[htbp]\n\\newcommand{\\tabincell}\n\\centering\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\nExperiment&Reaction&$\\sqrt{s}$ (GeV)&$\\sigma_{J\/\\psi}$ (nb\/nucleon)\\\\\n\\hline\n\\hline\nCERN-PS ~\\cite{CERN-PS}&p+A &6.8&0.732$\\pm$0.13\\\\\n\\hline\nWA39 ~\\cite{WA39}&p+p&8.7&2.35$\\pm$1.18\\\\\n\\hline\nIHEP ~\\cite{IHEP}&p+Be&11.5&21.63$\\pm$5.64\\\\\n\\hline\nE331 ~\\cite{E331}&p+Be&16.8&85.15$\\pm$21.30\\\\\n\\hline\nNA3 ~\\cite{alpha_na3}&p+Pt&16.8&95.0$\\pm$17.0\\\\\n\\hline\nNA3 ~\\cite{alpha_na3}&p+Pt&19.4&122.6$\\pm$21\\\\\n\\hline\nNA3 ~\\cite{alpha_na3}&p+p&19.4&120$\\pm$22\\\\\n\\hline\nE331 ~\\cite{E331_1}&p+C&20.6&278$\\pm$32.8\\\\\n\\hline\nE444 ~\\cite{E444}&p+C&20.6&176.5$\\pm$23.3\\\\\n\\hline\nE705 ~\\cite{E705}&p+Li&23.8&271.51$\\pm$29.84\\\\\n\\hline\nUA6 ~\\cite{UA6}&p+p&24.3&171.42$\\pm$22.21\\\\\n\\hline\nE288 ~\\cite{E288}&p+Be&27.4&294.12$\\pm$73.53\\\\\n\\hline\nE595 ~\\cite{E595}&p+Fe&27.4&264$\\pm$56\\\\\n\\hline\nNA38\/51 ~\\cite{NA38, NA51}&p+A&29.1&229.5$\\pm$34.4\\\\\n\\hline\nNA50 ~\\cite{alpha_na50}&p+A&29.1&250.7$\\pm$37.6\\\\\n\\hline\nE672\/706 ~\\cite{E672}&pBe&31.6&343.07$\\pm$75.12\\\\\n\\hline\nE771 ~\\cite{E771}&p+Si&38.8&359.1$\\pm$34.2\\\\\n\\hline\nE789 ~\\cite{E789}&p+Au&38.8&415.04$\\pm$100\\\\\n\\hline\nISR ~\\cite{ISR52}&p+p&52&716$\\pm$303\\\\\n\\hline\nPHENIX ~\\cite{PHENIX200}&p+p&200&4000$\\pm$938\\\\\n\\hline\nCDF ~\\cite{CDF1960}&p+$\\bar{p}$&1960&22560$\\pm$3384\\\\\n\\hline\nALICE ~\\cite{ALICE2760}&p+p&2760&29912.6$\\pm$5384.3\\\\\n\\hline\nALICE ~\\cite{ALICE7000}&p+p&7000&54449.4$\\pm$8494\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{(color online) Updated total ($\\sigma_{J\/\\psi}$) production cross sections in proton-induced interactions.}\n\\label{table1}\n\\end{table*} \n\\section{Results}\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{crosssectionfit.pdf}\n\\caption{(color online) Energy dependence of inclusive $J\/\\psi$ prodcution cross section. The open circle is the fit from CEM shape.The solid line is a function fit as discuss in the text.}\n\\label{figure2}\n\\end{center}\n\\end{figure}\nThe energy evolution of the total inclusive $J\/\\psi$ production cross section in proton induced interactions is shown in Fig.~\\ref{figure2}. The first approach is to use the predicted shape in the Colour Evaporation Model at Next to Leading Order (NLO) ~\\cite{CEM} to describe the energy dependence of $J\/\\psi$ cross section. The central CT10 parton density set ~\\cite{PDF_CT10} and $\\{m,\\mu_{F}\/m,\\mu_{R}\/m\\}=\\{1.27 (GeV), 2.10, 1.60\\}$ set is utilized in the predicted shape, where m is the charm quark mass, $\\mu_{F}$ is the factorization scale, $\\mu_{R}$ is the renormalization scale. The fit is defined such that the normalization of the NLO CEM calculation is left as a free parameter ($\\alpha$): $\\sigma=\\alpha \\times \\sigma_{CEM}$. The second approach is to use a functional form to describe the cross section energy evolution: $f(\\sqrt{s})=a \\times y_{max}^{d}\\times e^{\\frac{-b}{y_{max}+c}}$, where $y_{max} = ln(\\frac{\\sqrt{s}}{m_{J\/\\psi}})$, a, b, c and d are free parameters. As shown in Fig.~\\ref{figure2}, both approaches can describe the energy evolution trend of $J\/\\psi$ cross section. The $\\chi^{2}\/NDF$ for CEM and functional fit are 90.5\/22 and 76.7\/20, respectively. The large $\\chi^{2}$ mainly comes from three experimental points which contradict with the common trend (E331 and E444 measurements at 20.6 GeV, E705 measurement at 23.8 GeV). If we exclude these three data points and refit the results, the $\\chi^{2}\/NDF$ for CEM and functional fit are 41.1\/19 and 16.7\/16, respectively. The values extrapolated (without the three bad experimental points) for the $J\/\\psi$ cross sections at $\\sqrt{s} = $ 39 and 62.4 GeV, utilizing the functional form and the NLO CEM based fit are listed in Table~\\ref{table2}.\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{table}[htbp]\n\\newcommand{\\tabincell}\n\\centering\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multirow{2}{*}{Fit} & \\multicolumn{2}{|c|}{cross section (nb\/nucleon)}\\\\\\cline{2-3}&$\\sqrt{s} = $39 GeV &$\\sqrt{s} = $62.4 GeV\\\\\n\\hline\nNLO CEM& 425$\\pm$20&941$\\pm$42\\\\\n\\hline\nfunction&445$\\pm$30&995$\\pm$66\\\\\n\\hline\nevaluated results&445$\\pm$30$\\pm$20&995$\\pm$66$\\pm$54\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Extrapolated values of the $J\/\\psi$ production cross section at $\\sqrt{s} =$ 39 and 62.4 GeV. The difference between CEM and function fit has been taken as the systematic uncertainties of the extrapolation.}\n\\label{table2}\n\\end{table} \n\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{drawfity.pdf}\n\\caption{(color online) Normalized $J\/\\psi$ production cross section as a function of $y\/y_{max}$. Two function fits are shown: one is Gaussian function, the other one is $\\frac{a}{1-(y\/y_{max})^{2}}e^{-b(ln(\\frac{1+y\/y_{max}}{1-y\/y_{max}}))^{2}}$. The difference between these two curves has been considered as systematic errors.}\n\\label{figure3}\n\\end{center}\n\\end{figure}\n\nThe knowledge of the rapidity dependence of $J\/\\psi$ production at different cms-energies is crucial to obtain a reference for the measurements at mid-rapidity from RHIC. Based on a universal energy scaling behavior in the rapidity distribution obtained at different cms-energies, we explore approaches to the extrapolation of the rapidity distribution. As shown in Fig.~\\ref{figure3}, the y-differential cross sections at different cms-energies have been normalized by the total cross section, and the normalized values are plotted verse $y\/y_{max}$, where $y_{max}$ has been previously defined. Despite of huge cms-energy difference, the treated RHIC ~\\cite{PHENIX200} and LHC ~\\cite{ALICE2760, ALICE7000, LHCb7000} experimental distributions fall into a universal trend, which allows us to perform global fits to all the experimental results with suitable functions. Two functional forms are chosen to do the fits: one is Gaussian function, the other one is $\\frac{a}{1-(y\/y_{max})^{2}}e^{-b(ln(\\frac{1+y\/y_{max}}{1-y\/y_{max}}))^{2}}$, where a and b are free parameters. Both of them can describe the global distribution very well($\\chi^{2}\/NDF = 10.1\/27$ for gaussian fit, $\\chi^{2}\/NDF = 11.2\/27$ for the other fit). With the extrapolated $J\/\\psi$ cross sections and rapidity distributions, the predicted $J\/\\psi$ cross section times branching ratio at $\\sqrt{s} =$ 39 and 62.4 GeV in mid-rapidity are $Br(e^{+}e^{-})d\\sigma\/dy|_{|y|<1.0} = 9.04 \\pm 0.69$ and $17.74 \\pm 1.06$ nb, respectively. The uncertainties include statistical and systematic uncertainties. These values are highly consistent with the estimations from CEM model ($8.7 \\pm 4.5$ nb for 39 GeV, $17.4 \\pm 8.0$ for 62.4 GeV).\n\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{1.pdf}\n\\caption{(color online) $J\/\\psi$ $z_{T}$ distributions for available experimental results from $\\sqrt{s} =$ 10 to 7000 GeV. The solid line is a function fit as discussed in the text.}\n\\label{figure4}\n\\end{center}\n\\end{figure}\n\nThe energy evolution of $J\/\\psi$ transverse momentum distribution are also studied via available experimental measurements from $\\sqrt{s} = $ 10 - 7000 GeV ~\\cite{alpha_na3, E288, E672, E331_1,PHENIX200, ALICE7000, CDF1960, ISR53}. We use part of the world-wide fixed-target data (with only p, Be, Li, and C respectively) measured with incident protons. In this way, we avoid uncertainties due to ignoring any cold nuclear matter effects on the $J\/\\psi$ transverse momentum distributions. To compare the different experimental measurements at different energies and rapidity domains, as shown in Fig.~\\ref{figure4}, the transverse momentum distributions are normalized by their $p_{T}$-integrated cross sections and plotted verse the $z_{T}$ variable, which is defined as $z_{T} = p_{T}\/$. The treated distributions follow a universal trend despite of the different cms-energies and rapidity domains. We can describe the global distributions very well by the following function: $\\frac{1}{d\\sigma\/dy}\\frac{d^{2}\\sigma}{z_{T}dz_{T}dy} =a \\times \\frac{1}{(1+b^{2}z_{T}^{2})^{n}}$ ~\\cite{function}, where $a=2b^{2}(n-1)$, $b=\\Gamma(3\/2)\\Gamma(n-3\/2)\/\\Gamma(n-1)$, and n is the only free parameter. From the fit, we obtain $n=3.93 \\pm 0.03$ with $\\chi^{2}\/NDF = 143.9\/162$. \n\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{ptaverage.pdf}\n\\caption{(color online) $J\/\\psi$ $$ at mid-rapidity as a function of cms-energy from $\\sqrt{s} =$ 10 to 7000 GeV. The solid line is a function fit as discussed in the text.}\n\\label{figure5}\n\\end{center}\n\\end{figure}\nWith the universal shape and $$ information at certain energy and rapidity domain (we focus on mid-rapidity), we can extrapolate the transverse momentum distribution at any cms-energy. Thus the next step is to evaluate the energy evolution of $$. The $$ at mid-rapidity as a function of cms-energy from world-wide experiments ~\\cite{alpha_na3, E288, E672, E331_1,PHENIX200, ALICE7000, CDF1960, ISR53, ISR30, ISR52} is shown in Fig.~\\ref{figure5}. Also, only part of the world-wide fixed-target data (with p, Be, Li, and C respectively) are used to reduce the cold nuclear matter effects. The $$ versus energy can be fitted by the function form: $f(\\sqrt{s}) = p + q ln\\sqrt{s}$, where p, q are free parameters. The fit parameters are $p=0.0023 \\pm 0.0182$, $q = 0.329 \\pm 0.031$ with $\\chi^{2}\/NDF = 41.1\/15$. The estimated $$ from the fit function at $\\sqrt{s} =$ 39 and 62.4 GeV are $1.21 \\pm 0.04$ and $1.36 \\pm 0.04$ GeV\/c, respectively. With these inputs, the transverse momentum distribution at these two cms-energies can be completely determined. \n\n\\renewcommand{\\floatpagefraction}{0.75}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[keepaspectratio,width=0.45\\textwidth]{crosscheck1.pdf}\n\\caption{(color online) The ratios of $J\/\\psi$ $\\sigma|_{|y|<1.0}$ to $\\sigma_{total}$ as a function of cms-energy. The open points are the estimations using the two fit functions in Fig.~\\ref{figure3}.}\n\\label{figure6}\n\\end{center}\n\\end{figure}\n\nThere are rare rapidity distribution measurements in p$+$A collisions at $\\sqrt{s} < 200$. Therefore, the universal energy scaling parameters of rapidity distributions are determined by the measurements at $\\sqrt{s} \\geq 200$ GeV. Its validity at low energy ($<$200 GeV) range still need to be further investigated. But we do have various $x_{F}$ distribution measurements of $J\/\\psi$ in fix-target experiments ~\\cite{IHEP, E331_1, E444, E595, E705, E672, E771, E789}. \nCooperated with the $\\alpha$ verse $x_{F}$ curve in Fig.~\\ref{figure1} and the transverse momentum distributions obtained using the strategy described in the above section, we can evaluate the rapidity distributions via the $x_{F}$ distributions measurements in the fix-target experiments to check the validity of the rapidity interpolation method. The ratios of $J\/\\psi$ $\\sigma|_{|y|<1.0}$ to $\\sigma_{total}$, which are calculated utilizing the evaluated rapidity distributions in fix-target experiments, verse cms-energy are shown in Fig.~\\ref{figure6}. The open points plotted in the figure are the estimations using the two fit functions in Fig.~\\ref{figure3}. In this figure, we can see that our extrapolation strategy also works at low cms-energy range. \n\\section{summary} \nWe study the world-wide data of $J\/\\psi$ production and kinematics at $\\sqrt{s} = 6.8-7000$ GeV. We have developed a strategy to interpolate the $J\/\\psi$ cross section, rapidity distribution, and transverse momentum distribution at any cms-energy in $\\sqrt{s} = 6.8-7000$ GeV. The rapidity and transverse momentum distributions measured in different energies have a universal energy scaling behavior. With this strategy, we predicted that the $J\/\\psi$ cross section times branching ratio at $\\sqrt{s} =$ 39 and 62.4 GeV in mid-rapidity are $Br(e^{+}e^{-})d\\sigma\/dy|_{|y|<1.0} = 9.04 \\pm 0.69$, $17.74 \\pm 1.06$ nb, respectively.\n\n\\section{Acknowledgments}\nWe express our gratitude to the STAR Collaboration and the RCF at BNL for their support. This work was supported in part by the U.S. DOE Office of Science under the contract No. DE-SC0012704; authors Wangmei Zha and Chi Yang are supported in part by the National Natural Science Foundation of China under Grant Nos 11005103 and 11005104, China Postdoctoral Science Foundation funded project, and the Fundamental Research Funds for the Central Universities.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}