****Abstract:**** The behavior of higher-order radiative corrections due to initial state radiation in processes of electron-positron annihilation is analyzed. Numerical results for energies of future colliders are presented. Uncertainties of the known results on these corrections are estimated. A modified scheme for simultaneous exponentiation of pure photonic and non-singlet pair corrections is presented. Matching of the exponentiated results with the existing analytic higher-order calculations is constructed. A new DIS-like subtraction scheme is discussed.

Keywords: radiative corrections, exponentiation, parton distribution functions, QED, electron-positron annihilation

1 Introduction↩︎

Precise theoretical predictions of radiative corrections to electron-positron annihilation are important for experiments at future electron-positron colliders like FCC-ee [1] and CEPC [2]. Despite the great progress in the development of multi-loop calculation methods, computing higher-order radiative corrections to realistic differential observables in high-energy physics remains a complex task. Meanwhile, the accuracy of the analysis of many observables, first of all related to precision tests of the Standard Model and extraction of its parameters, suffers from uncertainties in the theoretical description of effects due to initial state radiation [3]. One of the methods that allows evaluation of higher-order corrections without their direct calculations is exponentiation of known lowest-order results, which performes a reorganization of the perturbative expansion and resummation of certain contributions to all orders. Another powerful method is the so-called QED structure function approach [4], [5], which allows one to calculate systematically the part of higher-order corrections enhanced by powers of large logarithms. In this article, we discuss matching of these approaches and estimate the corresponding theoretical uncertainties.

Exponentiation in particle physics has been studied since 1960s. In the work of D.R. Yennie, S.C. Frautschi, and H. Suura [6] infrared divergences were resummed into a universal exponential factor. In the Yennie-Frautschi-Suura (YFS) exponentiation formalism, all photons emitted in processes like electron scattering by a potential, bremsstrahlung off nucleons, etc., are separated into soft and hard parts based on their energy, and radiation of soft and virtual photons is associated with infrared singularities, which cancel each other. The remaining contribution can be re-summed in an exponential form. There are also some Monte-Carlo (MC) simulation programs based on the YFS method: PHOTOS [7], KKMC [8], BHLUMI [9], [10], Sherpa [11], and others. Exponentiation of the pure photonic part of the electron structure function (parton distribution function) was derived by V.N. Gribov and L.N. Lipatov [12]. E.A. Kuraev and V.S. Fadin [4] suggested an ad hoc exponentiation procedure which allows to improve accuracy of the finite order perturbative solution and to take into account a part of pair corrections. In the work by G. Passarino [13], connection between exponentiation and structure functions is discussed in detail. Amplitude-based resummation in collinear and IR limits and its YFS MC realization was discussed in the work [14], and CEEX (Coherent Exclusive Exponentiation) [15] and EEX (Exclusive Exponentiation) [16] realization was discussed in [17].

The method of structure functions was introduced first for QED in the work of V.N. Gribov and L.N. Lipatov [18], where the electron structure functions were obtained. The Dokshitzer–Gribov–Lipatov–Altarelli–Parisi evolution equations [19], [20] were derived for QCD based on the latter approach, and they were reduced back to the QED case by E.A. Kuraev and V.S. Fadin [4]. There are numerous applications and further developments of the method within the leading logarithmic approximation, see e.g. refs. [21]–[27]. Application of the method in the next-to-leading order (NLO) approximation was demonstrated for the first time in [28] for derivation of QED radiative corrections due to initial state radiation (ISR) in electron-positron annihilation. Then it was applied for calculations of \(\mathcal{O}(\alpha^2L)\) corrections to a few other processes including muon decay [29], deep inelastic scattering [30], and Bhabha scattering [31], and charge and helicity asymmetries [32]. The structure function approach allows avoiding direct loop calculations. But only the most numerically significant corrections are taken into account. These are corrections enhanced by the so-called large logarithm \[L = \ln \frac{\mu_F^2}{\mu_R^2},\] where \(\mu_F\) is factorization scale and \(\mu_R\) is renormalization scale. The natural choice of \(\mu_R\) in QED is the electron mass, and \(\mu_F\) should correspond to a characteristic (high) energy of the process [33].

The article is organized as follows. In the next Section, ISR radiative corrections to electron-positron annihilation are discussed. In the third Section, we describe exponentiation procedures and suggest an expression for exponentiation of parton distribution function corresponding to both photonic and non-singlet pair corrections. The fourth Section is dedicated to the scheme dependence of the corrections. In the fifth Section, theoretical uncertainties are estimated and discussed.

2 ISR radiative corrections↩︎

Here we apply the parton distribution function (structure function) approach [4]. The cross section of electron-positron annihilation into a virtual photon or \(Z\) boson, decaying then, say, into muon pair \[e^+e^- \to \gamma^{*}(Z^{*}) \to \mu^+\mu^-\] with ISR corrections can be represented in the form of the convolution of two electron parton distribution functions (PDFs) and the partonic cross section [28] \[\begin{align} \label{master} && \frac{d\sigma^{\mathrm{NNLL}}_{\bar{e} e}(s')}{ds'} = \sum \limits_{i,j= e\!, \bar{e}\!, \gamma} \int \limits^{1}_{\bar{z}_1} \int \limits^{1}_{\bar{z}_2} d z_1 d z_2 D_{i e} \left(z_1,\frac{\mu_R^2}{\mu^2_F}\right) D_{j \bar{e}} \! \left( \! z_2,\frac{\mu_R^2}{\mu^2_F} \! \right) \! \nonumber \\ && \times \left( \sigma^{(0)}_{ij} (s z_1 z_2) + \overline{\sigma}^{(1)}_{ij} (s z_1 z_2) + \overline{\sigma}^{(2)}_{ij} (s z_1 z_2) \right) \delta(s' \! - \! sz_1z_2) + \mathcal{O}\left(\frac{m_e^2}{s}\right), \end{align}\tag{1}\] where \(\bar{e}\equiv e^+\) is positron and \(e\equiv e^-\) is electron, \(D_{i e}\) and \(D_{j \bar{e}}\) are PDFs, \(\sigma^{(0,1,2)}_{ij}\) are the Born \((0)\), one-loop \((1)\), and two-loop \((2)\) contributions to cross-sections of annihilation into \(\gamma^*(Z^*)\) at the partonic level, \(s\) is the initial center-of-mass energy squared, \(s'\) is the invariant mass of the produced virtual photon (or \(Z\)-boson) squared, and \(z\) is energy fraction: \(s'=sz\), \(z\equiv z_1z_2\). Bars over \(\overline{\sigma}^{(1,2)}_{ij}\) mean that these contributions are computed for massless partons within the \(\overline{\mathrm{MS}}\) scheme for subtraction of electron mass singularities.

The expansion of the master equation for cross-section takes into account the QED radiative corrections enhanced by the large logarithms and reads \[\begin{align} \label{cij} && \frac{d\sigma^{\mathrm{NNLL}}_{\bar{e} e}(s')}{ds'} = \frac{\sigma^{(0)}_{\bar{e} e}(s')}{s} \biggl\{ \delta(1-z) + \sum \limits^{\infty}_{\substack{k=1 \\ k\geq l \geq k-2}} c_{kl}(z) \left(\frac{\alpha}{2\pi}\right)^k L^{l} + \mathcal{O}(\alpha^kL^{k-3}) \biggr\}, \end{align}\tag{2}\] where \(c_{kl}(z)\) are the coefficients to be computed. Here the sum can be divided into tree parts that consist of the leading (LL), next-to-leading (NLL), and next-to-next-to-leading (NNLL) logarithmic contributions (\(l=k\), \(l=k-1\), and \(l=k-2\), respectively). Higher-order coefficients \(c_{kl}(z)\) in the leading and next-to-leading logarithmic approximations up to \(c_{65}\) were first calculated in [34], and most of those coefficients were independently recalculated and corrected in [35]. Corrections to the results for \(c_{20}\) given in [28] can be found in [36] and [37].

To get the total cross section we have to integrate the differential cross section over \(z\) from \(z_{min}\) corresponding to a certain minimal value of the invariant mass of the muon pair defined by experimental conditions, \[\sigma_{e^+ e^-} = \sigma^{(0)} (s) + \sum\limits_{\substack{k=1 \\ l \leq k}} \left(\frac{\alpha}{2\pi}\right)^k L^l \left[ \;\int \limits_{z_{min}}^{1 - \Delta} dz \sigma^{(0)}(sz) c^{\Theta}_{kl} (z) + c^{\Delta}_{kl}\sigma^{(0)}(s) \right].\] Here \(c^{\Theta}_{kl}\) and \(c^{\Delta}_{kl}\) are the so-called \(\Theta\) and \(\Delta\) parts of the coefficients \(c_{kl}\), see details in [38]. The first corresponds to contributions from hard emission, and the second one corresponds to soft and virtual corrections. \(\Delta\) is a small parameter dividing hard and soft contributions and is taken equal to \(10^{-7}\) or \(10^{-8}\). The final result doesn’t depend on \(\Delta\) for \(\Delta\to 0\). We present numerical results for relative corrections \(h_{kl}\) in percent separately in different orders in \(\alpha\) and \(L\): \[h_{kl} = \left(\frac{\alpha}{2\pi}\right)^k L^l \left( \int \limits_{z_{min}}^{1 - \Delta} dz \sigma^{(0)}(sz) c^{\Theta}_{kl} (z) + c^{\Delta}_{kl} \sigma^{(0)}(s) \right) / \sigma^{(0)} (s) \cdot 100 \%.\]

We estimated numerical values of the leading and next-to-leading logarithmic corrections to initial state radiation in \(e^+e^-\)-annihilation at different energies \(\bigl(\frac{M_Z - 1}{2}\), \(\frac{M_Z}{2}\), \(\frac{M_Z + 1}{2}\), \(160\) GeV, \(240\) GeV and \(3\) TeV\(\bigr)\) and different values of \(z_{min}\) (see Tables 2-4). At \(160\) GeV and \(240\) GeV a radiative return to \(Z\)-resonance can occur due to the effective reduction of the collision energy because of radiation from the initial state, which makes the values of the corrections at \(z_{min}=0.1\) up to ten times larger than the same corrections at \(\sqrt{s} = M_Z\). At \(\sqrt{s}=240\) GeV the return to the resonance is at \(z \approx 0.14\), where \(sz=M_Z^2\), and at \(\sqrt{s}=160\) GeV it is at \(z \approx 0.32\).

3 Exponentiation↩︎

Exponentiation procedure allows resummation of a part of corrections that are most important in the domain of collinear and soft (or virtual) radiation. Exponentiation and resummation of QED corrections is discussed in details in ref. [13].

Exponentiated pure photonic part of the electron PDF \(D_{ee}\) is given by the exact solution of the QED DGLAP evolution equation found by V.N. Gribov and L.N. Lipatov in [12]: \[\label{GL} D_{ee}^{(exp, \gamma)}(z;\beta) = \frac{\beta}{2} \frac{(1-z)^{\frac{\beta}{2}-1}}{\Gamma(\frac{\beta}{2}+1)} \exp \biggl(\frac{\beta}{2} \left( \frac{3}{4} - \gamma_E \right)\biggr),\tag{3}\] for \(\Theta-\)part, where \(\beta = \frac{2 \alpha}{\pi} (L -1)\), \(\Gamma\) is the Euler Gamma-function, \(\gamma_E \approx 0.577\) is the Euler-Mascheroni constant. We also need the \(\Delta-\)part for numerical applications: \[\label{dexp} D_{ee}^{(exp, \gamma) \Delta} \equiv \int\limits_{1-\Delta}^{1}d z D_{ee}^{(exp,\gamma)}(z;\beta)= \exp \left( \frac{\beta}{2} \ln \Delta +\frac{3 \beta}{8}\right) \frac{\exp (-\gamma_E \beta /2)}{\Gamma(1+\beta/2)}.\tag{4}\]

As soon as the cross section has the structure \(\sigma \otimes D \otimes D\), see Eq. (1 ), we need to calculate the convolution of \(D_{ee}^{(exp,\gamma) \otimes 2} = D_{ee}^{(exp,\gamma)} \otimes D_{ee}^{(exp,\gamma)}\) analytically. Technically it is very complicated, but, see review [5], we can present the \(D_{ee}^{(exp,\gamma) \otimes 2}\) as \[D_{ee}^{(exp,\gamma) \otimes 2} (z; \beta)\approx D_{ee}^{(exp,\gamma)} (z; 2 \beta)\label{D2beta}.\tag{5}\] This approximation becomes exact at \(z \rightarrow 1\).

We suggest matching of the known perturbative higher-order corrections with the exponentiated result in the following form of an improved cross-section: \[\sigma^{imp}(z) = \sigma^{pert}(z) - D^{(exp,\gamma) \otimes 2}_{ser}(z) \;\sigma^{(0)}(z) + D^{(exp,\gamma) \otimes 2}_{\infty}(z) \;\sigma^{(0)}(z), \label{exp}\tag{6}\] where \(D^{(exp,\gamma) \otimes 2}_{\infty}\) is the full exponentiated \(D_{ee}^{(exp, \gamma)} \otimes D_{ee}^{(exp,\gamma)}\) given by Eq. 5 . Function \(D^{(exp,\gamma) \otimes 2}_{ser}\) is obtained from \(D^{(exp,\gamma) \otimes 2}_{\infty}\) by its expansion in series in \(\alpha\) and \(L\) and keeping only contributions of those orders which are present in \(\sigma^{pert}\), namely the leading and next-to-leading logarithmic contributions up to \(\mathcal{O}(\alpha^5L^5)\) and \(\mathcal{O}(\alpha^4L^3)\), respectively, together with \(\mathcal{O}(\alpha^2L^0)\) terms. Remind that \(\sigma^{pert}\) is the result of our perturbative calculation [35]: \[\begin{align} \label{sigpert} && \sigma^{pert} (z) = D_{e e} \otimes D_{\bar{e} \bar{e} } \otimes \left(\sigma_{e \bar{e} }^{(0)}(z) + \frac{\alpha}{2 \pi} \overline{\sigma}_{e \bar{e} }^{(1)} (z) + \left( \frac{\alpha}{2 \pi} \right)^2\overline{\sigma}_{e \bar{e} }^{(2)} (z) \right) \nonumber \\ && + 2 D_{\gamma e} \! \otimes \! D_{\bar{e} \bar{e}} \! \otimes \! \left( \! \frac{\alpha}{2 \pi}\sigma_{e \gamma}^{(0)}(z) + \left( \! \frac{\alpha}{2 \pi} \! \right)^2 \overline{\sigma}_{e \gamma}^{(1)} (z) \! \right) + D_{\bar{e} e} \! \otimes \! D_{e \bar{e}} \! \otimes \! \left( \! \frac{\alpha}{2 \pi} \sigma^{(0)}_{e \bar{e} }(z) + \left( \! \frac{\alpha}{2 \pi} \! \right)^2 \overline{\sigma}_{e \bar{e} }^{(1)} (z) \! \right). \end{align}\tag{7}\] Here \(D_{ab}\) are structure functions obtained by iterative solution of the DGLAP equation. In this way, we perform matching of the perturbative results with the exponentiated ones and avoid double counting. Despite the fact that we have explicit perturbative results only in the LL and NLL approximations and the one \(\mathcal{O}(\alpha^2L^0)\) next-to-next-to-leading correction (\(h_{20}\)), we take into account in \(D^{(exp,\gamma) \otimes 2}_{ser}\) all terms up to \(\mathcal{O}(\alpha^5L^0)\) appearing in the expansion. All these higher-order corrections are present in \(D^{(exp,\gamma) \otimes 2}_{\infty}\), and then cancel each other when we calculate an improved cross section according to Eq. (6 ). Note that in \(D^{(exp,\gamma) \otimes 2}_{ser}(z) \;\sigma^{(0)}(z)\) and \(D^{(exp) \otimes 2}_{\infty}(z) \;\sigma^{(0)}(z)\) there is no \(\overline{\sigma}^{(1)}\) because all soft and virtual contributions are included into \(D^{(exp) \otimes 2}\) and because the exponentiated expressions are applicable at \(z \rightarrow 1\) but \(\overline{\sigma}^{(1)}\) tends to zero in this area. To obtain numerical value of the improved cross-section we have to integrate it from some \(z_{min}\), defined by experimental conditions, to \(1\).

We can estimate the \(h_{55}\) pure photonic correction the following way: \[\label{h55} h_{55, \gamma}^{exp}= \left( \;\int \limits_{z_{min}}^{1} dz D^{(exp, \gamma) \otimes 2}_{ser,55} (z) \;\sigma^{(0)}(sz) \right)/ \sigma^{(0)} (s) \cdot 100 \%,\tag{8}\] where \(D^{(exp, \gamma) \otimes 2}_{ser,55} (z)\) is the part of the expansion that consists of only \(\mathcal{O} (\alpha^5 L^5)\) terms. The values of \(h_{55, \gamma}^{exp}\) are presented in percent with respect to Born cross section. Numerical values of the full pure photonic \(h_{55}\) and approximated \(h_{55, \gamma}^{exp}\) corrections are close to each other, as can be seen from Tables 2-4, which justifies application of the exponentiation for estimates of further higher-order contributions.

We can also estimate the residual "tail" consisting of higher-order contributions provided by the exponentiated result beyond the perturbative one: \[\label{tail} \Delta^{D \! D\sigma}_{\infty, \gamma} = \left( \;\int \limits_{z_{min}}^{1} dz D^{(exp, \gamma) \otimes 2}_{\infty} (z) \;\sigma^{(0)}(sz) - \int \limits_{z_{min}}^{1} dz D^{(exp, \gamma) \otimes 2}_{ser} (z) \;\sigma^{(0)}(sz) \right)/ \sigma^{(0)} (s) \cdot 100 \%.\tag{9}\] We can also estimate pair corrections. Only non-singlet (NS) pair corrections can be exponentiated since the singlet ones are not enhanced at \(z\to 1\). In the work [39] exponentiation of QCD corrections for Drell-Yan and DIS processes was discussed taking into account pair corrections. In ref. [40], an exponentiation of non-singlet pair and photonic corrections was constructed as a sum of two "exponents". Here we suggest to perform a joint exponentiation of pure photonic and non-singlet pair corrections in electron PDF. We use the same structure as the expression for pure photonic corrections but modify the parameter \(\beta\) as follows: \[\begin{align} D^{(exp,\gamma + NS)}_{ee}(z;\tilde{\beta}) &=& \frac{\tilde{\beta}}{2} (1-z)^{\frac{\tilde{\beta}}{2} - 1} \exp \left( \frac{3}{4} \frac{\tilde{\beta}}{2} \right) \frac{\exp (-\gamma_E \tilde{\beta} /2)}{\Gamma(1+ \tilde{\beta}/2)}, \nonumber \\ \label{DgammaNS} \tilde{\beta} &\equiv& \frac{2\alpha}{\pi}(L - 1)\left(1+ \frac{\alpha}{6 \pi} \left( L - \frac{7}{3}\right)\right). \end{align}\tag{10}\] The number \(\frac{7}{3}\) was obtained from the comparison of the expanded exponent with the perturbative expression. This coefficient was adjusted to make the terms proportional to \(1/(1-z)\) in the contribution of the order \(\alpha^2 L^1\) being equal. The \(\Delta\)-part of this expression is constructed analogously to Eq. 4 : \[\label{dexpNS} D_{ee}^{(exp, \gamma+NS) \Delta} = \exp \left( \frac{\tilde{\beta}}{2} \ln \Delta +\frac{3 \tilde{\beta}}{8}\right) \frac{\exp (-\gamma_E \tilde{\beta} /2)}{\Gamma(1+\tilde{\beta}/2)}.\tag{11}\] Note that Eq. (10 ) takes into account pure photonic contributions, pure non-singlet pair ones, and mixed (photonic + non-singlet pair) corrections. We can estimate photonic plus non-singlet pair corrections in the same way as in Eq. (8 ), and \(\Delta^{D \! D\sigma}_{\infty, \gamma+NS}\) the same way as in Eq. 9 . In Tables 2-4, they are presented in the columns \(h^{exp}_{55}\) and \(\Delta^{D \! D\sigma}_{\infty}\) correspondingly, pure photonic case is shown in the rows marked "\(\gamma\)", and photonic plus non-singlet pair case is shown in the rows marked "Full".

4 Scheme dependence↩︎

Despite the fact that the sum of all corrections of particular order in \(\alpha\) doesn’t depend neither on the factorization scale value nor on the subtraction scheme choice, the expressions for individual contributions are scheme-dependent. In the \(\overline{\mathrm{MS}}\) scheme we use the following initial conditions for the evolution equation [38]: \[\begin{align} \label{ddd1} && d_{ee}^{(1)} (x)= \left[\frac{1 +x^2}{1-x}(- 1 - 2 \ln(1-x) )\right]_+, \\ && d_{\gamma e}^{(1)}(x) = -\frac{1 + (1-x)^2}{ x}(2 \ln x + 1), \\ && d_{e \gamma}^{(1)}(x) = 0, \end{align}\tag{12}\] and the expression for the relevant NLO splitting function reads \[\begin{align} && P_{ee}^{(1)\Theta}(x) = -\frac{56 x^2}{9}+\left(\frac{8 x^2}{3}+\frac{11 x}{3}-\frac{13}{3 (1-x)}+\frac{5}{3}\right) \ln (x) +\frac{37 x}{3}+\frac{20}{9 x} -\left(\frac{3 x}{2}+\frac{3}{2}\right) \ln ^2(x) \nonumber \\ && \quad + \left((2 x+2) \ln (1-x) -\frac{4 \ln (1-x)}{1-x}\right) \ln (x)-\frac{25}{3}, \nonumber \\ && P_{ee}^{(1) \Delta} =\frac{15}{8} - \frac{13}{3} \zeta_2 + 6 \zeta_3. \end{align}\] In Eq. (12 ) the subscript "\(+\)" means the standard plus prescription. If we compare an expression for \(D_{ee}(x)\) obtained from iterative solution of the evolution equation in the \(\overline{\mathrm{MS}}\) scheme with the \(D^{(exp, \gamma)}_{ser}\), we can see that the \(\overline{\mathrm{MS}}\) scheme generates higher powers of \(\ln(1-z)\) than exponentiation due to the presence of \(\frac{1 +x^2}{1-x} \ln(1-x)\) in \(d_{ee}^{(1)}\). In order-by-order calculations those singular at \(z\to1\) extra terms (they do not correspond to any soft or collinear singularities) cancel out with the corresponding terms from convolution of the leading-order structure function with the next-to-leading order partonic cross section \(\overline{\sigma}^{(1)}\). But in the exponentiated \(D^{(exp \otimes 2)}\) multiplied by \(\sigma^{(0)}\) they don’t cancel out.

We suggest another factorization scheme, where \[\tilde{d}^{(1)}_{ee} (x)= - \left[\frac{1+x^2}{1-x}\right]_+.\] So, we change \(d_{ee}^{(1)}\) and derive the corresponding expressions for \(\widetilde{\sigma}^{(1)}\) and \(P_{ee}^{(1)}\) from matching equalities at \(\mathcal{O}(\alpha^1)\) [35] and \(\mathcal{O}(\alpha^2)\): \[\label{matchingOa1} \delta^{(1)}_{\bar{e} e}(sx) = \bar{\delta}^{(1)}_{\bar{e} e}(sx) + 2 \frac{\alpha}{2\pi}\left[ P^{(0)}_{ee}(y) L + d_{ee}^{(1)}(y)\right] + \mathcal{O}\left(\frac{m_e^2}{s}\right),\tag{13}\] where \(\bar{\delta}_{\bar{e} e}^{(1)} (sx)\equiv \frac{\overline{\sigma}_{\bar{e} e}^{(1)} (sx)}{\sigma_{\bar{e} e}^{(0)} (sx)}\) and \(\delta_{\bar{e} e}^{(1)} (sx)\equiv \frac{\sigma_{\bar{e} e}^{(1)} (sx)}{\sigma_{\bar{e} e}^{(0)} (sx)}\), and \[\begin{align} && \delta^{(2)}_{\bar{e} e}(sx) = \left( \frac{\alpha}{2\pi}\right)^2 L^2 \left[ P_{e \gamma}^{(0)} \otimes P_{\gamma e}^{(0)} + \frac{2}{3} P_{ee}^{(0)} + 2 P_{ee}^{(0)\otimes 2}\right] + \left( \frac{\alpha}{2\pi}\right)^2 L \left[ 2 \bar{\delta}_{e \gamma}^{(0)} \otimes P_{\gamma e}^{(0)} + \frac{2}{3}\bar{\delta}_{\bar{e} e}^{(1)} \right. \nonumber \\ && \quad \left.+ 2 \bar{\delta}_{\bar{e} e}^{(1)} \otimes P_{ee}^{(0)} + 2 d_{\gamma e}^{(1)} \otimes P_{e \gamma}^{(0)} +2 P_{ee}^{(1)} - \frac{20}{9} P_{ee}^{(0)} + 4 P_{ee}^{(0)} \otimes d_{ee}^{(1)}\right] + \left(\frac{\alpha}{2\pi}\right)^2 c_{20}. \end{align}\] We omit the arguments on the right-hand side for brevity. Functions \(\overline{\sigma}^{(2)}\) and \(d_{ee}^{(2)}\), which will appear in \(c_{20}\), must change due to change of \(d_{ee}^{(1)}\), \(P_{ee}^{(1)}\) and \(\overline{\sigma}^{(1)}\). We assume that \(P_{e \gamma}^{(1)}\), \(P_{\gamma e}^{(1)}\), and \(d_{\gamma e}^{(0)}\) don’t change because they contain no unphysical logarithms. From this we can conclude that it is enough to consider the sum \[\frac{2}{3}\bar{\delta}_{\bar{e} e}^{(1)} + 2 \bar{\delta}_{\bar{e} e}^{(1)} \otimes P_{ee}^{(0)} +2 P_{ee}^{(1)} + 4 P_{ee}^{(0)} \otimes d_{ee}^{(1)},\] which must not change for matching. In our new scheme, we have \[\begin{align} \widetilde{P}_{ee}^{(1)}(x) = {P}_{ee}^{(1)}(x) + \frac{4}{3} (1+x^2) \frac{\ln(1-x)}{1-x}, \nonumber \\ \widetilde{P}_{ee}^{(1) \Delta} = - \frac{11}{12} + \frac{8}{3} \ln^2 \Delta - \frac{26}{3} \zeta_2, \nonumber \\ \widetilde{\overline{\sigma}}_{ee}^{(1)} (x) = 2 \frac{1+x^2}{1-x} (\ln z - \ln x), \\ \widetilde{\overline{\sigma}}_{ee}^{(1) \Delta} = 4 \zeta_2 - 1. \end{align}\] In our new scheme \(\widetilde{\overline{\sigma}}_{ee}^{(1)}\) has its theta part equal to zero at \(x=z\), and convolutions become significantly easier.

5 Theoretical uncertainty↩︎

We can estimate the following three different contributions to theoretical uncertainty: \[\begin{align} && \Delta_{LL,exp} = \frac{h_{55}}{h_{44}} (h^{(exp)}_{55} - h_{55}), \tag{14} \\ && \Delta_{NLL} = \frac{h_{43}}{h_{32}} h_{43}, \tag{15} \\ && \Delta_{NNLL} = \frac{h_{20}}{h_{21}} h_{32}. \tag{16} \end{align}\] Eq.@eq:erh66 estimates the uncertainty in \(h_{66}\); Eq.@eq:h54 estimates the unaccounted NLL contribution \(h_{54}\), and Eq.@eq:h31 estimates the unaccounted NNLL contribution \(h_{31}\). We present the results for calculating uncertainties at the Z-peak where the highest experimental precision of the order of \(10^{-3}\%\) can be achieved at future colliders in the so-called Tera-Z mode [41].

Here all the values are in percent with respect to Born cross section, so the current theoretical uncertainty in description of ISR in the total cross section at \(Z\)-peak can be estimated to be about \(4\cdot 10^{-4}\%\). Obviously, calculation of the \(h_{31}\) contribution should be the next step to improve to the accuracy.

At the energy \(\sqrt{s}=240\) GeV the uncertainty is bigger for \(z_{min}=0.1\) because of the radiative return to the resonance, but the experimental requirements are not so strict. Uncertainties in this energy domain will be considered elsewhere.

6 Conclusion↩︎

So, we constructed matching of the existing perturbative results for the leading and next-to-leading higher-order ISR corrections to processes of \(e^+e^-\) annihilation with simulataneous exponentiation of certain photonic and pair corrections. The matching allows one to combine the advantages of the two approaches and thus to improve the resulting precision. Note that any future perturbative result can be easily incorporated within the proposed scheme.

Numerical values of LL and NLL ISR corrections to the cross section of electron-positron annihilation in \(\mathrm{\overline{MS}}\) scheme were calculated. The calculations will be implemented in the ZFITTER program [42], [43]. The suggested new subtraction scheme is useful for evaluation of radiative corrections within the QED structure approach, especially for \(z\to 1\). The theoretical uncertainty in the description of radiative corrections in \(e^+e^-\) annihilation processes should still be improved by inclusion of new perturbative calculations, including effects due final state radiation (FSR) and ISR-FSR interference.

The presented numerical results are obtained for inclusive treatment of the ISR effects. Alternatively, one can apply QED parton showers, which are a kind of exponentiated description of ISR in a differential form. If the latter is relevant for a given observable, one should perform matching of the completely differential perturbative results with parton showers. Our work on this task is in progress within the ReneSANCe Monte Carlo event generator [44].

Acknowledgments↩︎

We are grateful to A.V. Kotikov for fruitful discussions.

7 Appendix↩︎

Here we present Tables with numerical results for particular contributions.

References↩︎