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f_i(0) = \left\{1-a_s(t_F) t_F +\begin{array}{rcl} + f_i(0) &=&\displaystyle \Bigg\{1-a_s(t_F) t_F P_{ij}^{(0)} + a_s^2(t_F)\left[-t_F P_{ij}^{(1)}+ t_F^2 \frac12\left( P_{il}^{(0)}\otimes P_{lj}^{(0)} - \beta_0 P_{ij}^{(0)} - \right)\right]\right\}\otimes f_j(t_F) + \mathcal{O}(a_s^3)\,. + \right)\right]\\ +\\ +&+&\displaystyle a_s^3(t_F)\Bigg[ \frac12 t_F^2 \left(P_{il} + ^{(0)}\otimes P_{lj} ^{(1)}+P_{il} ^{(1)} \otimes P_{lj} + ^{(0)}\right)-\frac{1}{2} t_F^2 \beta _1 P_{ij} + ^{(0)}+\frac{1}{2} t_F^3\beta _0 P_{il} ^{(0)}\otimes P_{lj} + ^{(0)}\\ +\\ +&-&\displaystyle \frac{1}{3} t_F^3\beta _0^2 P_{ij}^{(0)}-\frac{1}{6}t_F^3 P_{il} ^{(0)}\otimes P_{lk} ^{(0)} \otimes + P_{kj} ^{(0)}-t_F^2\beta _0 P_{ij} ^{(1)}- t_F P_{ij} ^{(2)}\Bigg]\Bigg\}\otimes f_j(t_F) + \mathcal{O}(a_s^4)\,. +\end{array} \label{expDGLAP1} \end{equation} In addition, using Eq.~(\ref{BETAsimp}), one easily finds: \begin{equation} a_s(t_F)= - a_s(t_R)\left[1+a_s(t_R)\beta_{0}(t_R-t_F)+\mathcal{O}(a_s^2)\right]\,, + a_s(t_R)\left[1+a_s(t_R)\beta_{0}(t_R-t_F)+a_s^2(t_R)\left(\beta_{1} (t_R-t_F)+\beta_{0}^2 (t_R-t_F)^2\right)+\mathcal{O}(a_s^3)\right]\,, \label{BETAexp} \end{equation} which can be plugged into Eq.~(\ref{expDGLAP1}) to give: \begin{equation} - f_i(0) = \left\{1-a_s(t_R) t_F +\begin{array}{rcl} + f_i(0) &=&\displaystyle \Bigg\{1-a_s(t_R) t_F P_{ij}^{(0)} + a_s^2(t_R)\left[-t_F P_{ij}^{(1)}+ t_F^2 \frac12\left( P_{il}^{(0)}\otimes P_{lj}^{(0)} + \beta_0 P_{ij}^{(0)} \right)- - t_Ft_R\beta_{0}P_{ij}^{(0)}\right]\right\}\otimes f_j(t_F) + - \mathcal{O}(a_s^3)\,. + t_Ft_R\beta_{0}P_{ij}^{(0)}\right]\\ +\\ +&+&\displaystyle a_s^3(t_R)\Bigg[\frac{1}{2} t_F^2 \left(P_{il} + ^{(0)} P_{lj} ^{(1)}+P_{il} ^{(1)} P_{lj} + ^{(0)}\right)+\frac{1}{2} t_F^2 \beta _1 P_{ij} ^{(0)}-t_F t_R + \beta _1 P_{ij} ^{(0)}-\frac{1}{2} t_F^3\beta _0 P_{il} + ^{(0)}\otimes P_{lj} ^{(0)}\\ +\\ +&-&\displaystyle \frac{1}{3} t_F^3\beta _0^2 P_{ij} ^{(0)}-\frac{1}{6} + t_F^3 P_{il} ^{(0)}\otimes P_{lk} ^{(0)}\otimes P_{kj} ^{(0)}+ + t_F^2 t_R\beta _0 P_{ik} ^{(0)}\otimes P_{kj} ^{(0)}+ t_F^2 t_R + \beta _0^2 P_{ij} ^{(0)}- t_F t_R^2 \beta _0^2 P_{ij} ^{(0)}\\ +\\ +&+&\displaystyle t_F^2\beta _0 P_{ij} ^{(1)}-2 t_F t_R \beta _0 P_{ij} ^{(1)}- t_F P_{ij} ^{(2)}\Bigg]\Bigg\}\otimes f_j(t_F) + + \mathcal{O}(a_s^4)\,. +\end{array} \label{expDGLAP2} \end{equation} +This equality can be conveniently written as: +\begin{equation} +f_i(0) =\sum_{k=0}^3a_s^k(t_R)f_i^{[k]}(t_R,t_F)+ +\mathcal{O}(a_s^4)\,, +\label{expDGLAP3} +\end{equation} +where the coefficients $f_i^{[k]}(t_R,t_F)$ can be read off from +Eq.~(\ref{expDGLAP2}). + It is also useful to consider $t_R=0$ in Eq.~(\ref{BETAexp}), that is equivalent to set $\mu_R=Q$, which gives: \begin{equation} a_s(0)= - a_s(t_R)\left[1+a_s(t_R)\beta_{0}t_R+\mathcal{O}(a_s^2)\right]\,. + a_s(t_R)\left[1+a_s(t_R)\beta_{0}t_R+a_s^2(t_R)\left(\beta_{1} t_R+\beta_{0}^2t_R^2\right) \right]+\mathcal{O}(a_s^4)\,. \label{BETAexp1} \end{equation} - We are now ready to use these equations to derive the scale variation terms to be included in ZM structure functions. Truncating the perturbative series to $\mathcal{O}(\alpha_s^2)$, they are written in terms of PDFs and coefficient functions as: \begin{equation} - F(t_R,t_F) / x = \left[\sum_{k=0}^{2} a_s^k(t_R) - \widetilde{\mathcal{C}}_i^{(k)}(t_R,t_F)\right]\otimes f_i(t_F)\,, + F(t_R,t_F) / x = \left[\sum_{k=0}^{3} a_s^k(t_R) + \widetilde{\mathcal{C}}_i^{(k)}(t_R,t_F)\right]\otimes f_i(t_F) + \mathcal{O}(a_s^4)\,, \label{NonZeroScales} \end{equation} -where the symbol $\otimes$ represents the convolution that is not -necessary to write explicitly here. Since structure functions are -physically observable, they must be renormalisation and factorisation -scale invariant, that is: +where the symbol $\otimes$ represents the convolution. Since structure +functions are physically observable, they must be renormalisation and +factorisation scale invariant, that is: \begin{equation} F(t_R,t_F) = F(0,0)\,, \label{invariance} \end{equation} -up to subleading, \textit{i.e.} $\mathcal{O}(\alpha_s^3)$, -corrections. Since: +order by order in perturbation theory. Since: \begin{equation} - F(0,0) / x = \left[\sum_{k=0}^{2} + F(0,0) / x = \left[\sum_{k=0}^{3} a_s^k(0) \widetilde{C}_i^{(k)}\right]\otimes - f_i(0)\,, + f_i(0) + \mathcal{O}(a_s^4)\,, \label{ZeroScales} \end{equation} where $\widetilde{C}_i^{(k)}$ are the usual perturbative contributions @@ -783,44 +815,88 @@ \section{Renormalisation and factorisation scale variations} identity in Eq.~(\ref{invariance}). By doing so, one finds the explicit expression of the ``generalised'' coefficient functions $\widetilde{\mathcal{C}}_i^{(k)}(t_R,t_F)$ that include also the scale -variation terms. The result is: -\begin{equation} - \begin{array}{rcl} - F(0,0)/x &=&\bigg\{ \widetilde{C}_j^{(0)} \\ \\ - &+&\displaystyle a_s(t_R)\left[\widetilde{C}_j^{(1)}- t_F - \widetilde{C}_i^{(0)} \otimes P_{ij}^{(0)}\right]\\ \\ - &+&\displaystyle a_s^2(t_R)\bigg[\widetilde{C}_j^{(2)} + t_R\beta_0 - \widetilde{C}_j^{(1)} -t_F \left(\widetilde{C}_i^{(0)} \otimes - P_{ij}^{(1)}+\widetilde{C}_i^{(1)} \otimes P_{ij}^{(0)}\right)\\ \\ - &+&\displaystyle \frac{t_F^2}2 \widetilde{C}_i^{(0)} \otimes \left( - P_{il}^{(0)}\otimes P_{lj}^{(0)} + \beta_0 P_{ij}^{(0)} \right)- - t_Ft_R\beta_{0}\widetilde{C}_i^{(0)} \otimes - P_{ij}^{(0)}\bigg]\bigg\}\otimes f_j(t_F)+\mathcal{O}(a_s^3)\,. -\end{array} -\end{equation} -Finally, using the identity in Eq.~(\ref{invariance}), it is easy to -find that: +variation terms. It is convenient to first compute renormalisation +scale variations while leaving $t_F=0$: \begin{equation} \begin{array}{rcl} \displaystyle - \widetilde{\mathcal{C}}_j^{(0)}(t_R,t_F) &=& \displaystyle + \widetilde{\mathcal{C}}_j^{(0)}(t_R,0) &=& \displaystyle \widetilde{C}_j^{(0)} \\ \\ \displaystyle - \widetilde{\mathcal{C}}_j^{(1)}(t_R,t_F) &=& \displaystyle - \widetilde{C}_j^{(1)}-t_F \widetilde{C}_i^{(0)} \otimes P_{ij}^{(0)} - \\ \\ \displaystyle \widetilde{\mathcal{C}}_j^{(2)}(t_R,t_F) &=& - \displaystyle \widetilde{C}_j^{(2)} + t_R\beta_0 \widetilde{C}_j^{(1)} - -t_F \left(\widetilde{C}_i^{(0)} \otimes - P_{ij}^{(1)}+\widetilde{C}_i^{(1)} \otimes P_{ij}^{(0)}\right)\\ \\ - &+&\displaystyle\frac{t_F^2}2 \widetilde{C}_i^{(0)} \otimes \left( - P_{il}^{(0)}\otimes P_{lj}^{(0)} + \beta_0 P_{ij}^{(0)} \right)- - t_Ft_R\beta_{0}\widetilde{C}_i^{(0)} \otimes P_{ij}^{(0)}\,. + \widetilde{\mathcal{C}}_j^{(1)}(t_R,0) &=& \displaystyle + \widetilde{C}_j^{(1)}\\ + \\ +\displaystyle \widetilde{\mathcal{C}}_j^{(2)}(t_R,0) &=& + \displaystyle + \widetilde{C}_j^{(2)} + + + t_R\beta_0 + \widetilde{C}_j^{(1)}\,,\\ +\\ +\displaystyle \widetilde{\mathcal{C}}_j^{(3)}(t_R,0) &=& + \displaystyle + \widetilde{C}_j^{(3)}+2 t_R\beta _0 \widetilde{C}_j^{(2)} + +t_R \left(\beta_1 +\beta _0^2 t_R\right) \widetilde{C}_j^{(1)}\,, \end{array} -\label{generalizedCF} +\label{CFtR} +\end{equation} +so that: +\begin{equation} + F(t_R,0) / x = \left[\sum_{k=0}^{3} a_s^k(t_R) + \widetilde{\mathcal{C}}_i^{(k)}(t_R,0)\right]\otimes f_i(0) + \mathcal{O}(a_s^4)\,. +\label{NonZeroScalestR} +\end{equation} +We can now use Eq.~(\ref{expDGLAP3}) to express the PDF $f(0)$ in +terms of $f(t_F)$ finally obtaining: +\begin{equation} + F(t_R,0) / x = \sum_{k=0}^{3} a_s^k(t_R) + \sum_{j=0}^{k}\widetilde{\mathcal{C}}_i^{(k-j)}(t_R,0)\otimes f_i^{[j]}(t_R,t_F) + \mathcal{O}(a_s^4)\,. +\label{NonZeroScalestRtF} \end{equation} -Unsurprisingly, setting $\mu_F=\mu_R=Q$, that results in $t_F=t_R=0$, -one finds -$\widetilde{\mathcal{C}}_j^{(k)}(0,0) =\widetilde{C}_j^{(k)}$ as -required by construction. +In the absence of factorisation scale variations, only the term with $j=0$ +contributes to the inner sum. + + +% \newpage +% The result is: +% \begin{equation} +% \begin{array}{rcl} +% F(0,0)/x &=&\bigg\{ \widetilde{C}_j^{(0)} \\ \\ +% &+&\displaystyle a_s(t_R)\left[\widetilde{C}_j^{(1)}- t_F +% \widetilde{C}_i^{(0)} \otimes P_{ij}^{(0)}\right]\\ \\ +% &+&\displaystyle a_s^2(t_R)\bigg[\widetilde{C}_j^{(2)} + t_R\beta_0 +% \widetilde{C}_j^{(1)} -t_F \left(\widetilde{C}_i^{(0)} \otimes +% P_{ij}^{(1)}+\widetilde{C}_i^{(1)} \otimes P_{ij}^{(0)}\right)\\ \\ +% &+&\displaystyle \frac{t_F^2}2 \widetilde{C}_i^{(0)} \otimes \left( +% P_{il}^{(0)}\otimes P_{lj}^{(0)} + \beta_0 P_{ij}^{(0)} \right)- +% t_Ft_R\beta_{0}\widetilde{C}_i^{(0)} \otimes +% P_{ij}^{(0)}\bigg]\\ +% \\ +% &+&\displaystyle a_s^3(t_R)\bigg[\widetilde{C}_j^{(3)} \bigg]\bigg\}\otimes f_j(t_F)+\mathcal{O}(a_s^4)\,. +% \end{array} +% \end{equation} +% Finally, using the identity in Eq.~(\ref{invariance}), it is easy to +% find that: +% \begin{equation} +% \begin{array}{rcl} +% \displaystyle +% \widetilde{\mathcal{C}}_j^{(0)}(t_R,t_F) &=& \displaystyle +% \widetilde{C}_j^{(0)} \\ \\ \displaystyle +% \widetilde{\mathcal{C}}_j^{(1)}(t_R,t_F) &=& \displaystyle +% \widetilde{C}_j^{(1)}-t_F \widetilde{C}_i^{(0)} \otimes P_{ij}^{(0)} +% \\ \\ \displaystyle \widetilde{\mathcal{C}}_j^{(2)}(t_R,t_F) &=& +% \displaystyle \widetilde{C}_j^{(2)} + t_R\beta_0 \widetilde{C}_j^{(1)} +% -t_F \left(\widetilde{C}_i^{(0)} \otimes +% P_{ij}^{(1)}+\widetilde{C}_i^{(1)} \otimes P_{ij}^{(0)}\right)\\ \\ +% &+&\displaystyle\frac{t_F^2}2 \widetilde{C}_i^{(0)} \otimes \left( +% P_{il}^{(0)}\otimes P_{lj}^{(0)} + \beta_0 P_{ij}^{(0)} \right)- +% t_Ft_R\beta_{0}\widetilde{C}_i^{(0)} \otimes P_{ij}^{(0)}\,. +% \end{array} +% \label{generalizedCF} +% \end{equation} +% Unsurprisingly, setting $\mu_F=\mu_R=Q$, that results in $t_F=t_R=0$, +% one finds +% $\widetilde{\mathcal{C}}_j^{(k)}(0,0) =\widetilde{C}_j^{(k)}$ as +% required by construction. In order to provide an operative formulation of scale variations, it is necessary to specify the basis in which PDFs are expressed. The diff --git a/docs/latex/src/codes/gfunctions.nb b/docs/latex/src/codes/gfunctions.nb index a9c083f2f..af630fa65 100644 --- a/docs/latex/src/codes/gfunctions.nb +++ b/docs/latex/src/codes/gfunctions.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 173359, 4514] -NotebookOptionsPosition[ 162382, 4315] -NotebookOutlinePosition[ 162820, 4332] -CellTagsIndexPosition[ 162777, 4329] +NotebookDataLength[ 233943, 6095] +NotebookOptionsPosition[ 222208, 5882] +NotebookOutlinePosition[ 222653, 5899] +CellTagsIndexPosition[ 222610, 5896] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -22,7 +22,7 @@ Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["g - functions for alpha_s", "Title"]], "Input", CellChangeTimes->{{3.874026907670014*^9, 3.874026942447508*^9}}, - CellLabel->"In[5]:=",ExpressionUUID->"18a01ae8-6ae1-4e34-801f-22213a11bca6"], + 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7*Power(\[Lambda],3)*Subscript(b,3) - - 6*Log(1 - \[Lambda])*Subscript(b,3) + 18*\[Lambda]*Log(1 - \ -\[Lambda])*Subscript(b,3) - 18*Power(\[Lambda],2)*Log(1 - \ -\[Lambda])*Subscript(b,3) + 6*Power(\[Lambda],3)*Log(1 - \ -\[Lambda])*Subscript(b,3))/ - (3.*Power(-1 + \[Lambda],3)*Power(Subscript(\[Beta],0),2)) + (Log(\ -\[Kappa])*(6*Power(\[Lambda],2)*Power(Subscript(b,1),2)*Subscript(\[Beta],0) + - 6*Power(\[Lambda],3)*Power(Subscript(b,1),2)*Subscript(\[Beta],0) \ -+ 12*\[Lambda]*Log(1 - \ -\[Lambda])*Power(Subscript(b,1),2)*Subscript(\[Beta],0) + - 6*Power(Log(1 - \[Lambda]),2)*Power(Subscript(b,1),2)*Subscript(\ -\[Beta],0) - 18*\[Lambda]*Power(Log(1 - \ -\[Lambda]),2)*Power(Subscript(b,1),2)*Subscript(\[Beta],0) - - 6*Power(\[Lambda],2)*Subscript(b,2)*Subscript(\[Beta],0) - \ -6*Power(\[Lambda],3)*Subscript(b,2)*Subscript(\[Beta],0)))/(3.*Power(-1 + \ +2*Power(Log(1 - \[Lambda]),3)*Power(Subscript(b,1),3) - + 6*\[Lambda]*Power(Log(1 - \[Lambda]),3)*Power(Subscript(b,1),3) + 6*\ +\[Lambda]*Subscript(b,1)*Subscript(b,2) - \ +15*Power(\[Lambda],2)*Subscript(b,1)*Subscript(b,2) + + 5*Power(\[Lambda],3)*Subscript(b,1)*Subscript(b,2) + 6*Log(1 - \ +\[Lambda])*Subscript(b,1)*Subscript(b,2) - 18*\[Lambda]*Log(1 - \ +\[Lambda])*Subscript(b,1)*Subscript(b,2) + + 12*Power(\[Lambda],2)*Log(1 - \ +\[Lambda])*Subscript(b,1)*Subscript(b,2) - 12*Power(\[Lambda],3)*Log(1 - \ +\[Lambda])*Subscript(b,1)*Subscript(b,2) - 6*\[Lambda]*Subscript(b,3) + + 15*Power(\[Lambda],2)*Subscript(b,3) - \ +7*Power(\[Lambda],3)*Subscript(b,3) - 6*Log(1 - \[Lambda])*Subscript(b,3) + \ +18*\[Lambda]*Log(1 - \[Lambda])*Subscript(b,3) - + 18*Power(\[Lambda],2)*Log(1 - \[Lambda])*Subscript(b,3) + 6*Power(\ +\[Lambda],3)*Log(1 - \[Lambda])*Subscript(b,3))/(3.*Power(-1 + \ \[Lambda],3)*Power(Subscript(\[Beta],0),2)) + + (Log(\[Kappa])*(6*Power(\[Lambda],2)*Power(Subscript(b,1),2)*Subscript(\ +\[Beta],0) + \ +6*Power(\[Lambda],3)*Power(Subscript(b,1),2)*Subscript(\[Beta],0) + + 12*\[Lambda]*Log(1 - \[Lambda])*Power(Subscript(b,1),2)*Subscript(\ +\[Beta],0) + 6*Power(Log(1 - \[Lambda]),2)*Power(Subscript(b,1),2)*Subscript(\ +\[Beta],0) - + 18*\[Lambda]*Power(Log(1 - \ +\[Lambda]),2)*Power(Subscript(b,1),2)*Subscript(\[Beta],0) - \ +6*Power(\[Lambda],2)*Subscript(b,2)*Subscript(\[Beta],0) - + 6*Power(\[Lambda],3)*Subscript(b,2)*Subscript(\[Beta],0)))/(3.*\ +Power(-1 + \[Lambda],3)*Power(Subscript(\[Beta],0),2)) + (Power(Log(\[Kappa]),2)*(6*\[Lambda]*Subscript(b,1)*Power(Subscript(\ \[Beta],0),2) + 6*Log(1 - \ \[Lambda])*Subscript(b,1)*Power(Subscript(\[Beta],0),2) - @@ -4125,9 +5688,10 @@ Subscript(B,1)*((3*Power(\[Lambda],2)*Power(Subscript(b,1),2)*Subscript(\ 0),3) + 2*Power(\[Lambda],3)*Power(Subscript(\[Beta],0),3)))/(3.*Power(-1 + \ \[Lambda],3)*Power(Subscript(\[Beta],0),2)))\ \>", "Output", - CellChangeTimes->{3.899905323772543*^9, 3.9067772856945953`*^9}, + CellChangeTimes->{3.899905323772543*^9, 3.9067772856945953`*^9, + 3.908854907178638*^9, 3.908855432177671*^9, 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6*LQ*Log(\[Kappa])*Subscript(b,1)*Subscript(B,1)*Subscript(\[Beta],0) \ -- 6*Power(LQ,2)*Subscript(B,2)*Subscript(\[Beta],0) + \ -12*LQ*Log(\[Kappa])*Subscript(B,2)*Subscript(\[Beta],0) - - Power(LQ,4)*Subscript(A,1)*Power(Subscript(\[Beta],0),2) + \ -4*Power(LQ,3)*Log(\[Kappa])*Subscript(A,1)*Power(Subscript(\[Beta],0),2) - - 6*Power(LQ,2)*Power(Log(\[Kappa]),2)*Subscript(A,1)*Power(Subscript(\ -\[Beta],0),2) - 2*Power(LQ,3)*Subscript(B,1)*Power(Subscript(\[Beta],0),2) + - 6*Power(LQ,2)*Log(\[Kappa])*Subscript(B,1)*Power(Subscript(\[Beta],0),\ -2) - 6*LQ*Power(Log(\[Kappa]),2)*Subscript(B,1)*Power(Subscript(\[Beta],0),2) \ -- +- 6*Power(LQ,2)*Subscript(B,2)*Subscript(\[Beta],0) + + 12*LQ*Log(\[Kappa])*Subscript(B,2)*Subscript(\[Beta],0) - \ +Power(LQ,4)*Subscript(A,1)*Power(Subscript(\[Beta],0),2) + + 4*Power(LQ,3)*Log(\[Kappa])*Subscript(A,1)*Power(Subscript(\[Beta],0),\ +2) - 6*Power(LQ,2)*Power(Log(\[Kappa]),2)*Subscript(A,1)*Power(Subscript(\ +\[Beta],0),2) - + 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Alphas(Q) / FourPi; - const double cp2 = cp * cp; - const double cp3 = cp * cp2; - Set CoefFuncs = FObjQ.C0.at(k); - if (PerturbativeOrder > 0) - CoefFuncs += cp * FObjQ.C1.at(k); - if (PerturbativeOrder > 1) - CoefFuncs += cp2 * FObjQ.C2.at(k); - if (PerturbativeOrder > 2) - CoefFuncs += cp3 * FObjQ.C3.at(k); - return CoefFuncs; - }; - else - Cf = [=] (double const& Q) -> Set - { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const double cp = Alphas(xiR * Q) / FourPi; - const double cp2 = cp * cp; - const double cp3 = cp * cp2; - const double b0 = beta0qcd(FObjQ.nf); - const double b1 = beta1qcd(FObjQ.nf); - Set CoefFuncs = FObjQ.C0.at(k); - if (PerturbativeOrder > 0) - CoefFuncs += cp * FObjQ.C1.at(k); - if (PerturbativeOrder > 1) - CoefFuncs += cp2 * ( FObjQ.C2.at(k) + ( tR * b0 ) * FObjQ.C1.at(k) ); - if (PerturbativeOrder > 2) - CoefFuncs += cp3 * ( FObjQ.C3.at(k) + ( 2 * tR * b0 ) * FObjQ.C2.at(k) + ( tR * ( b1 + pow(b0, 2) * tR ) ) * FObjQ.C1.at(k) ); - return CoefFuncs; - }; - - // Define distribution-function functions - const auto DistF = [=, &g] (double const& Q) -> Set - { - return Set{FObj(Q, Couplings(Q)).ConvBasis.at(k), DistributionMap(g, InDistFunc, xiF * Q, skip)}; - }; - - // Add pair to the observable - Obs.AddConvolutionPair(Cf, DistF); - } - else - { - // Declare vectors of coefficient functions and PDFs, each - // one having as many entried a perturbative orders - // needed. - std::vector(double const&)>> Cfv(PerturbativeOrder + 1); - std::vector(double const&)>> DistFv(PerturbativeOrder + 1); - - // LO - Cfv[0] = [=] (double const& Q) -> Set { return FObj(Q, Couplings(Q)).C0.at(k); }; - DistFv[0] = [=, &g] (double const& Q) -> Set - { - return Set{FObj(Q, Couplings(Q)).ConvBasis.at(k), DistributionMap(g, InDistFunc, xiF * Q, skip)}; - }; - - // NLO - if (PerturbativeOrder > 0) - { - Cfv[1] = [=] (double const& Q) -> Set { return Alphas(xiR * Q) / FourPi * FObj(Q, Couplings(Q)).C1.at(k); }; - DistFv[1] = [=, &g] (double const& Q) -> Set - { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const double cp = Alphas(xiR * Q) / FourPi; - const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); - const Set F{CMap, DistributionMap(g, InDistFunc, xiF * Q)}; - const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; - const Set D1 = ( - cp * tF ) * P0F; - return Set{FObjQ.ConvBasis.at(k), D1.GetObjects()}; - }; - } - - // NNLO - if (PerturbativeOrder > 1) - { - Cfv[2] = [=] (double const& Q) -> Set - { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - return pow(Alphas(xiR * Q) / FourPi, 2) * ( FObjQ.C2.at(k) + ( tR * beta0qcd(FObjQ.nf) ) * FObjQ.C1.at(k) ); - }; - DistFv[2] = [=, &g] (double const& Q) -> Set - { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const double cp2 = pow(Alphas(xiR * Q) / FourPi, 2); - const double b0 = beta0qcd(FObjQ.nf); - const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); - const Set F{CMap, DistributionMap(g, InDistFunc, xiF * Q)}; - const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; - const Set P1F = FObjQ.P.SplittingFunctions.at(1) * F; - const Set P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0F; - const Set D2 = cp2 * ( ( - tF ) * P1F + ( pow(tF, 2) / 2 ) * P0P0F + ( - tF * tR * b0 + pow(tF, 2) * b0 / 2 ) * P0F ); - return Set{FObjQ.ConvBasis.at(k), D2.GetObjects()}; - }; - } - - // NNNLO - if (PerturbativeOrder > 2) - { - Cfv[3] = [=] (double const& Q) -> Set - { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - return pow(Alphas(xiR * Q) / FourPi, 3) * ( FObjQ.C3.at(k) + ( 2 * tR * beta0qcd(FObjQ.nf) ) * FObjQ.C2.at(k) - + ( tR * ( beta1qcd(FObjQ.nf) + pow(beta0qcd(FObjQ.nf), 2) * tR ) ) * FObjQ.C1.at(k) ); - }; - DistFv[3] = [=, &g] (double const& Q) -> Set - { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const double cp3 = pow(Alphas(xiR * Q) / FourPi, 3); - const double b0 = beta0qcd(FObjQ.nf); - const double b1 = beta1qcd(FObjQ.nf); - const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); - const Set F{CMap, DistributionMap(g, InDistFunc, xiF * Q)}; - const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; - const Set P1F = FObjQ.P.SplittingFunctions.at(1) * F; - const Set P2F = FObjQ.P.SplittingFunctions.at(2) * F; - const Set P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0F; - const Set P0P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0P0F; - const Set P0P1F = 0.5 * ( FObjQ.P.SplittingFunctions.at(0) * P1F + FObjQ.P.SplittingFunctions.at(1) * P0F ); - const Set D3 = cp3 * ( ( pow(tF, 2) * b1 / 2 - tF * tR * b1 - pow(tF, 3) * pow(b0, 2) / 3 + pow(tF, 2) * tR * pow(b0, 2) - tF * pow(tR, 2) * pow(b0, 2) ) * P0F - + ( pow(tF, 2) * b0 - 2 * tF * tR * b0 ) * P1F - + ( - tF ) * P2F - + ( - pow(tF, 3) * b0 / 2 + pow(tF, 2) * tR * b0 ) * P0P0F - + ( - pow(tF, 3) / 6 ) * P0P0P0F - + pow(tF, 2) * P0P1F ); - return Set{FObjQ.ConvBasis.at(k), D3.GetObjects()}; - }; - } - - // Combine coefficient functions and PDFs - for (int i = 0; i <= PerturbativeOrder; i++) - for (int j = 0; j <= PerturbativeOrder - i; j++) - Obs.AddConvolutionPair(Cfv[i], DistFv[j]); - } - - /* - // Define coefficient function functions that multiply F - const auto Cf = [=] (double const& Q) -> Set - { - const double cp = Alphas(xiR * Q) / FourPi; - const double cp2 = cp * cp; - const double cp3 = cp * cp2; - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - Set CoefFuncs = FObjQ.C0.at(k); - if (PerturbativeOrder > 0) - CoefFuncs += cp * FObjQ.C1.at(k); - if (PerturbativeOrder > 1) - CoefFuncs += cp2 * ( FObjQ.C2.at(k) + tR * beta0qcd(FObjQ.nf) * FObjQ.C1.at(k) ); - if (PerturbativeOrder > 2) - CoefFuncs += cp3 * FObjQ.C3.at(k); - return CoefFuncs; - }; - - // Define distribution-function functions - const auto DistF = [=, &g] (double const& Q) -> Set - { - return Set{FObj(Q, Couplings(Q)).ConvBasis.at(k), DistributionMap(g, InDistFunc, xiF * Q, skip)}; - }; - - // Create Observable - Obs.AddConvolutionPair(Cf, DistF); - - // Include scale variation terms if necessary, that is when - // the perturbative order is higher than zero. In addition, - // since the only pure tR-dependent term is already included - // above in the C2 term, we also require tF be different from - // zero. - // !!! Scale variations at N3LO are not implemented yet. !!! - if (PerturbativeOrder > 0 && tF != 0) + // Split the computation into different cases to avoid + // computing more convolutions than necessary. + if (xiF == 1) { - // Define coefficient function functions that multiply P0 * F - const auto CfP0 = [=] (double const& Q) -> Set + // Declare and then allocate coefficient function + // according to whether renormalisation-scale-variation + // terms have to be included or not. + std::function(double const&)> Cf; + + if (xiR == 1) + Cf = [=] (double const& Q) -> Set + { + const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); + const double cp = Alphas(Q) / FourPi; + const double cp2 = cp * cp; + const double cp3 = cp * cp2; + Set CoefFuncs = FObjQ.C0.at(k); + if (PerturbativeOrder > 0) + CoefFuncs += cp * FObjQ.C1.at(k); + if (PerturbativeOrder > 1) + CoefFuncs += cp2 * FObjQ.C2.at(k); + if (PerturbativeOrder > 2) + CoefFuncs += cp3 * FObjQ.C3.at(k); + return CoefFuncs; + }; + else + Cf = [=] (double const& Q) -> Set + { + const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); + const double cp = Alphas(xiR * Q) / FourPi; + const double cp2 = cp * cp; + const double cp3 = cp * cp2; + const double b0 = beta0qcd(FObjQ.nf); + const double b1 = beta1qcd(FObjQ.nf); + Set CoefFuncs = FObjQ.C0.at(k); + if (PerturbativeOrder > 0) + CoefFuncs += cp * FObjQ.C1.at(k); + if (PerturbativeOrder > 1) + CoefFuncs += cp2 * ( FObjQ.C2.at(k) + ( tR * b0 ) * FObjQ.C1.at(k) ); + if (PerturbativeOrder > 2) + CoefFuncs += cp3 * ( FObjQ.C3.at(k) + ( 2 * tR * b0 ) * FObjQ.C2.at(k) + ( tR * ( b1 + pow(b0, 2) * tR ) ) * FObjQ.C1.at(k) ); + return CoefFuncs; + }; + + // Define distribution-function functions + const auto DistF = [=, &g] (double const& Q) -> Set { - const double cp = Alphas(xiR * Q) / FourPi; - const double cp2 = cp * cp; - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - Set CoefFuncs = ( - cp * tF ) * FObjQ.C0.at(k); - if (PerturbativeOrder > 1) - CoefFuncs += ( - cp2 * tF ) * ( FObjQ.C1.at(k) + ( tR - tF / 2 ) * beta0qcd(FObjQ.nf) * FObjQ.C0.at(k) ); - return CoefFuncs; + return Set{FObj(Q, Couplings(Q)).ConvBasis.at(k), DistributionMap(g, InDistFunc, xiF * Q, skip)}; }; - // Define distribution-function functions obtained as P0 * F - const auto DistP0F = [=, &g] (double const& Q) -> Set + // Add pair to the observable + Obs.AddConvolutionPair(Cf, DistF); + } + else + { + // Declare vectors of coefficient functions and PDFs, each + // one having as many entried a perturbative orders + // needed. + std::vector(double const&)>> Cfv(PerturbativeOrder + 1); + std::vector(double const&)>> DistFv(PerturbativeOrder + 1); + + // LO + Cfv[0] = [=] (double const& Q) -> Set { return FObj(Q, Couplings(Q)).C0.at(k); }; + DistFv[0] = [=, &g] (double const& Q) -> Set { - const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const Set P0 = FObjQ.P.SplittingFunctions.at(0); - return Set{FObjQ.ConvBasis.at(k), (P0 * Set{P0.GetMap(), DistributionMap(g, InDistFunc, xiF * Q)}).GetObjects()}; + return Set{FObj(Q, Couplings(Q)).ConvBasis.at(k), DistributionMap(g, InDistFunc, xiF * Q, skip)}; }; - Obs.AddConvolutionPair(CfP0, DistP0F); + // NLO + if (PerturbativeOrder > 0) + { + Cfv[1] = [=] (double const& Q) -> Set { return Alphas(xiR * Q) / FourPi * FObj(Q, Couplings(Q)).C1.at(k); }; + DistFv[1] = [=, &g] (double const& Q) -> Set + { + const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); + const double cp = Alphas(xiR * Q) / FourPi; + const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); + const Set F{CMap, DistributionMap(g, InDistFunc, xiF * Q)}; + const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; + const Set D1 = ( - cp * tF ) * P0F; + return Set{FObjQ.ConvBasis.at(k), D1.GetObjects()}; + }; + } - // The remaining terms are O(as^2) + // NNLO if (PerturbativeOrder > 1) { - // Define coefficient function functions that multiply P1 * F - const auto CfP1 = [=] (double const& Q) -> Set + Cfv[2] = [=] (double const& Q) -> Set { - return ( - pow(Alphas(xiR * Q) / FourPi, 2) * tF ) * FObj(Q, Couplings(Q)).C0.at(k); + const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); + return pow(Alphas(xiR * Q) / FourPi, 2) * ( FObjQ.C2.at(k) + ( tR * beta0qcd(FObjQ.nf) ) * FObjQ.C1.at(k) ); }; - - // Define distribution-function functions obtained as P0 * F - const auto DistP1F = [=, &g] (double const& Q) -> Set + DistFv[2] = [=, &g] (double const& Q) -> Set { const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const Set P1 = FObjQ.P.SplittingFunctions.at(1); - return Set{FObjQ.ConvBasis.at(k), (P1 * Set{P1.GetMap(), DistributionMap(g, InDistFunc, xiF * Q)}).GetObjects()}; + const double cp2 = pow(Alphas(xiR * Q) / FourPi, 2); + const double b0 = beta0qcd(FObjQ.nf); + const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); + const Set F{CMap, DistributionMap(g, InDistFunc, xiF * Q)}; + const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; + const Set P1F = FObjQ.P.SplittingFunctions.at(1) * F; + const Set P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0F; + const Set D2 = cp2 * ( ( - tF ) * P1F + ( pow(tF, 2) / 2 ) * P0P0F + ( - tF * tR * b0 + pow(tF, 2) * b0 / 2 ) * P0F ); + return Set{FObjQ.ConvBasis.at(k), D2.GetObjects()}; }; + } - Obs.AddConvolutionPair(CfP1, DistP1F); - - // Define coefficient function functions that multiply P0 * P0 * F - const auto CfP0P0 = [=] (double const& Q) -> Set + // NNNLO + if (PerturbativeOrder > 2) + { + Cfv[3] = [=] (double const& Q) -> Set { - return ( pow(Alphas(xiR * Q) * tF / FourPi, 2) / 2 ) * FObj(Q, Couplings(Q)).C0.at(k); + const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); + return pow(Alphas(xiR * Q) / FourPi, 3) * ( FObjQ.C3.at(k) + ( 2 * tR * beta0qcd(FObjQ.nf) ) * FObjQ.C2.at(k) + + ( tR * ( beta1qcd(FObjQ.nf) + pow(beta0qcd(FObjQ.nf), 2) * tR ) ) * FObjQ.C1.at(k) ); }; - - // Define distribution-function functions obtained as P0 * P0 * F - const auto DistP0P0F = [=, &g] (double const& Q) -> Set + DistFv[3] = [=, &g] (double const& Q) -> Set { const StructureFunctionObjects FObjQ = FObj(Q, Couplings(Q)); - const Set P0 = FObjQ.P.SplittingFunctions.at(0); - return Set{FObjQ.ConvBasis.at(k), (P0 * ( P0 * Set{P0.GetMap(), DistributionMap(g, InDistFunc, xiF * Q)} )).GetObjects()}; + const double cp3 = pow(Alphas(xiR * Q) / FourPi, 3); + const double b0 = beta0qcd(FObjQ.nf); + const double b1 = beta1qcd(FObjQ.nf); + const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); + const Set F{CMap, DistributionMap(g, InDistFunc, xiF * Q)}; + const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; + const Set P1F = FObjQ.P.SplittingFunctions.at(1) * F; + const Set P2F = FObjQ.P.SplittingFunctions.at(2) * F; + const Set P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0F; + const Set P0P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0P0F; + const Set P0P1F = 0.5 * ( FObjQ.P.SplittingFunctions.at(0) * P1F + FObjQ.P.SplittingFunctions.at(1) * P0F ); + const Set D3 = cp3 * ( ( pow(tF, 2) * b1 / 2 - tF * tR * b1 - pow(tF, 3) * pow(b0, 2) / 3 + pow(tF, 2) * tR * pow(b0, 2) - tF * pow(tR, 2) * pow(b0, 2) ) * P0F + + ( pow(tF, 2) * b0 - 2 * tF * tR * b0 ) * P1F + + ( - tF ) * P2F + + ( - pow(tF, 3) * b0 / 2 + pow(tF, 2) * tR * b0 ) * P0P0F + + ( - pow(tF, 3) / 6 ) * P0P0P0F + + pow(tF, 2) * P0P1F ); + return Set{FObjQ.ConvBasis.at(k), D3.GetObjects()}; }; - - Obs.AddConvolutionPair(CfP0P0, DistP0P0F); } + + // Combine coefficient functions and PDFs + for (int i = 0; i <= PerturbativeOrder; i++) + for (int j = 0; j <= PerturbativeOrder - i; j++) + Obs.AddConvolutionPair(Cfv[i], DistFv[j]); } - */ + // Finally insert Observable F.insert({k, Obs}); } @@ -2375,54 +2281,67 @@ namespace apfel const double tR = 2 * log(xiR); const double tF = 2 * log(xiF); + // Gather coupling constant and its powers, beta function + // coefficients, and splitting functions convoluted with the input + // PDFs. const double cp = AlphasQ / FourPi; const double cp2 = cp * cp; const double cp3 = cp * cp2; - Set Cf = FObjQ.C0.at(k); - if (PerturbativeOrder > 0) - Cf += cp * FObjQ.C1.at(k); - if (PerturbativeOrder > 1) - Cf += cp2 * ( FObjQ.C2.at(k) + tR * beta0qcd(FObjQ.nf) * FObjQ.C1.at(k) ); - if (PerturbativeOrder > 2) - Cf += cp3 * FObjQ.C3.at(k); - - // Convolute coefficient function with set of distributions - Set SF = Cf * Set {FObjQ.ConvBasis.at(k), InDistFuncQ}; - - // Include scale variation terms if necessary, that is when the - // perturbative order is higher than zero. In addition, since the - // only pure tR-dependent term is already included above in the C2 - // term, we also require tF be different from zero. - // !!! Scale variations at N3LO are not implemented yet. !!! - if (PerturbativeOrder > 0 && tF != 0) - { - // Get splitting functions P0 - const Set P0 = FObjQ.P.SplittingFunctions.at(0); - const Set SetInDistFuncQ{P0.GetMap(), InDistFuncQ}; + const double b0 = beta0qcd(FObjQ.nf); + const double b1 = beta1qcd(FObjQ.nf); + + const ConvolutionMap CMap = FObjQ.P.SplittingFunctions.at(0).GetMap(); + const Set F{CMap, InDistFuncQ}; + const Set P0F = FObjQ.P.SplittingFunctions.at(0) * F; + const Set P1F = FObjQ.P.SplittingFunctions.at(1) * F; + const Set P2F = FObjQ.P.SplittingFunctions.at(2) * F; + const Set P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0F; + const Set P0P1F = 0.5 * ( FObjQ.P.SplittingFunctions.at(0) * P1F + FObjQ.P.SplittingFunctions.at(1) * P0F ); + const Set P0P0P0F = FObjQ.P.SplittingFunctions.at(0) * P0P0F; + + // Allocate vectors of coeffient functions and PDFs + std::vector> Cfv(PerturbativeOrder + 1); + std::vector> DistFv(PerturbativeOrder + 1); - // Define coefficient function functions that multiply P0 * F - Set CfP0 = ( - cp * tF ) * FObjQ.C0.at(k); - if (PerturbativeOrder > 1) - CfP0 += ( - cp2 * tF ) * ( FObjQ.C1.at(k) + ( tR - tF / 2 ) * beta0qcd(FObjQ.nf) * FObjQ.C0.at(k) ); - - // Include scale variation term - SF += CfP0 * Set {FObjQ.ConvBasis.at(k), (P0 * SetInDistFuncQ).GetObjects()}; + // LO + Cfv[0] = FObjQ.C0.at(k); + DistFv[0] = Set {FObjQ.ConvBasis.at(k), InDistFuncQ}; - // The remaining terms are O(as^2) - if (PerturbativeOrder > 1) - { - // Get splitting functions P1 - const Set P1 = FObjQ.P.SplittingFunctions.at(1); + // NLO + if (PerturbativeOrder > 0) + { + Cfv[1] = cp * FObjQ.C1.at(k); + DistFv[1] = Set {FObjQ.ConvBasis.at(k), (( - cp * tF ) * P0F).GetObjects()}; + } - const Set CfP1 = ( - pow(AlphasQ / FourPi, 2) * tF ) * FObjQ.C0.at(k); - SF += CfP1 * Set {FObjQ.ConvBasis.at(k), (P1 * SetInDistFuncQ).GetObjects()}; + // NNLO + if (PerturbativeOrder > 1) + { + Cfv[2] = cp2 * ( FObjQ.C2.at(k) + ( tR * b0 ) * FObjQ.C1.at(k) ); + DistFv[2] = Set {FObjQ.ConvBasis.at(k), (cp2 * ( ( - tF ) * P1F + ( pow(tF, 2) / 2 ) * P0P0F + ( - tF * tR * b0 + pow(tF, 2) * b0 / 2 ) * P0F )).GetObjects()}; + } - const Set CfP0P0 = ( pow(AlphasQ * tF / FourPi, 2) / 2 ) * FObjQ.C0.at(k); - SF += CfP0P0 * Set {FObjQ.ConvBasis.at(k), (P0 * ( P0 * SetInDistFuncQ )).GetObjects()}; - } + // NNNLO + if (PerturbativeOrder > 2) + { + Cfv[3] = cp3 * ( FObjQ.C3.at(k) + ( 2 * tR * b0 ) * FObjQ.C2.at(k) + ( tR * ( b1 + pow(b0, 2) * tR ) ) * FObjQ.C1.at(k) ); + DistFv[3] = Set {FObjQ.ConvBasis.at(k), (cp3 * ( ( pow(tF, 2) * b1 / 2 - tF * tR * b1 - pow(tF, 3) * pow(b0, 2) / 3 + pow(tF, 2) * tR * pow(b0, 2) - tF * pow(tR, 2) * pow(b0, 2) ) * P0F + + ( pow(tF, 2) * b0 - 2 * tF * tR * b0 ) * P1F + + ( - tF ) * P2F + + ( - pow(tF, 3) * b0 / 2 + pow(tF, 2) * tR * b0 ) * P0P0F + + ( - pow(tF, 3) / 6 ) * P0P0P0F + + pow(tF, 2) * P0P1F )).GetObjects() + }; } - // Combine set and return + // Convolute coefficient function with set of distributions + Set SF = Cfv[0] * DistFv[0]; + for (int i = 0; i <= PerturbativeOrder; i++) + for (int j = 0; j <= PerturbativeOrder - i; j++) + if (i != 0 && j != 0) + SF += Cfv[i] * DistFv[j]; + + // Combine set of distributions and return return SF.Combine(); }