Electroweak radiative corrections to e+ e- ->t anti-t H

We have calculated the complete electroweak O(alpha) radiative corrections to the Higgs-boson production process e+ e- ->t anti-t H in the electroweak Standard Model. Initial-state radiation beyond O(alpha) is included in the structure-function approach. The calculation of the corrections is briefly described, and numerical results are presented for the total cross section. Both the photonic and the genuine weak corrections reach the order of about 10% or even more and show a non-trivial dependence on the Higgs-boson mass and on the scattering energy. We compare our results with two previous calculations that obtained differing results at high energies.

only the main ingredients. Moreover, we compare our results with those of Refs. [ 14] and [ 15]. While we find good agreement with the results of the recent calculation of Ref. [ 15] at all considered energies, our results differ from those of Ref. [ 14] at high energies and close to threshold. Moreover, we could reproduce the QCD corrections of Ref. [ 8] from our results on photonic final-state radiation within statistical integration errors.

Method of calculation
We have calculated the complete O(α) electroweak virtual and real photonic corrections to the process e + e − → ttH, following the same strategy as used in our previous calculation of the corrections to e + e − → ννH [ 16].
The calculation of the one-loop diagrams has been performed in the 't Hooft-Feynman gauge both in the conventional and in the background-field formalism using the conventions of Refs. [ 17] and [ 18], respectively. The renormalization is carried out in the on-shell renormalization scheme, as described there. The electron mass m e is neglected whenever possible.
The calculation of the Feynman diagrams has been performed in two completely independent ways, leading to two independent computer codes for the numerical evaluation. Both calculations are based on the methods described in Ref. [ 17]. Apart from the 5-point functions the tensor coefficients of the one-loop integrals are recursively reduced to scalar integrals with the Passarino-Veltman algorithm [ 19] at the numerical level. The scalar integrals are evaluated using the methods and results of Refs. [ 17,20], where ultraviolet divergences are regulated dimensionally and IR divergences with an infinitesimal photon mass. The 5-point functions are reduced to 4-point functions following Ref. [ 21], where a method for a direct reduction is described that avoids leading inverse Gram determinants which potentially cause numerical instabilities. The two calculations differ in the following points. In the first calculation, the Feynman graphs are generated with FeynArts version 1.0 [ 22]. Using Mathematica the amplitudes are expressed in terms of standard matrix elements and coefficients of tensor integrals. The whole calculation has been carried out in the conventional and in the background-field formalism. The second calculation has been done with the help of FeynArts version 3 [ 23], and the analytical expressions have been generated by FormCalc [ 24] and translated into a C code. The scalar and tensor coefficients, in particular those for the 5-point functions, have been evaluated by own routines, as described above.
The results of the two different codes, and also those obtained within the conventional and background-field formalism, are in good numerical agreement (typically within at least 12 digits for non-exceptional phase-space points and double precision). The agreement of the results in the conventional and background-field formalism, in particular, checks the gauge independence of our results.
The matrix elements for the real photonic corrections are evaluated using the Weylvan der Waerden spinor technique as formulated in Ref. [ 25] and have been successfully checked against the result obtained with the package Madgraph [ 26]. The soft and collinear singularities are treated in the dipole subtraction method [ 27]. Beyond O(α) initial-state-radiation (ISR) corrections are included at the leading-logarithmic level using the structure functions given in Ref. [ 28] (for the original papers see references therein).
The phase-space integration is performed with Monte Carlo techniques in both computer codes. The first code employs a multi-channel Monte Carlo generator similar to the one implemented in RacoonWW [ 29,30] and Lusifer [ 31], the second one uses the adaptive multi-dimensional integration program VEGAS [ 32].

Numerical results
For the numerical evaluation we use the following set of SM parameters [ 33], (3.1) We do not calculate the W-boson mass from G µ but use its experimental value as input. The masses of the light quarks are adjusted to reproduce the hadronic contribution to the photonic vacuum polarization of Ref. [ 34]. We parametrize the lowest-order cross section with the Fermi constant G µ (G µ -scheme), i.e. we derive the electromagnetic coupling α according to α Gµ = √ 2G µ M 2 W s 2 w /π. This, in particular, absorbs the running of the electromagnetic coupling α(Q 2 ) from Q = 0 to the electroweak scale (Q ∼ M Z ) into the lowest-order cross section so that the results are practically independent of the masses of the light quarks. In the relative radiative corrections, we use, however, α(0) as coupling parameter, which is the correct effective coupling for real photon emission.
In the following we separate the photonic corrections, which comprise loop diagrams with virtual photon exchange in the loop and the corresponding parts of the counter terms as well as real photon emission, from the full electroweak O(α) corrections; the remaining non-photonic electroweak corrections are called weak. Since the lowest-order diagrams involve only neutral-current couplings, but no W-boson exchange, this splitting is gauge invariant. Moreover, we separately discuss the higher-order [i.e. beyond O(α)] ISR corrections that are obtained from the convolution of the lowest-order cross section with the leading-logarithmic structure functions, but with the O(α) contribution subtracted. The higher-order ISR is included in the electroweak corrections shown in the following plots.
In addition to the electroweak corrections, we also include results on the QCD corrections which can be deduced from the part of the photonic corrections that is proportional to Q 2 t , where Q t = 2/3 is the relative electric charge of the top quark, i.e. from the final-state radiation (FSR). The QCD correction is obtained from these corrections upon replacing the factor Q 2 t α by C F α s = 4α s (µ 2 )/3. Following Ref. [ 8], the QCD renormalization scale µ is set to the CM energy in the following, and the running of the strong coupling is evaluated at the two-loop level (MS scheme) with five active flavours, normalized by α s (M 2 Z ) as given in Eq. (3.1). For √ s = 500, 800, and 1000 GeV the resulting values for the strong coupling are given by α s (M 2 Z ) = 0.09349, 0.08857, and 0.08642, respectively. In this letter, we consider merely total cross sections without any cuts; distributions will be discussed elsewhere. For reference we give some numbers for the total cross section in lowest order, σ tree , the cross section including electroweak and QCD corrections, σ corr , Figure 1: Lowest-order and corrected cross sections (l.h.s.) as well as relative corrections (r.h.s.) in the G µ -scheme for a CM energy √ s = 500 GeV and the various contributions to the relative corrections defined as δ = σ/σ tree − 1 in Table 1. The last numbers in parentheses correspond to the Monte Carlo integration error of the last given digits. In Away from the kinematic threshold at √ s = 2m t + M H the total cross section is typically of the order of 0.5-3 fb and becomes maximal in the energy range between 700 GeV and 1 TeV for small Higgs-boson masses. As already discussed in Refs. [ 8,9] in detail, the QCD corrections are positive and rather large in the threshold region ( √ s > ∼ 2m t + M H ), where soft-gluon exchange between in the tt system leads to a Coulomb-like singularity. Away from threshold this singularity is diluted by the larger phase space, since the singularity demands a low relative velocity of the quarks. Averaging the singular factor over the phase space leads to the following threshold behaviour of the QCD correction [ 8], where β t is the maximal quark velocity in the tt rest frame. The FSR corrections show the same behaviour 3 , however, suppressed by the factor Q 2 t α/(4α s /3) ∼ 0.02. Although the relative QCD correction becomes rather huge close to threshold, it should be realized that the total cross section decreases rapidly there. In the region above threshold, where the cross section is largest, the QCD correction is only of the order of a per cent. For   (7) Figure 2: Lowest-order and corrected cross sections (l.h.s.) as well as relative corrections (r.h.s.) in the G µ -scheme for a CM energy √ s = 800 GeV CM energies and Higgs-boson masses far above threshold, the QCD corrections even turn negative and reduce the cross section by about 5% at √ s = 1 TeV. This behaviour is expected from the effective Higgs-boson approximation [ 7], as also argued in Ref. [ 8].
The photonic corrections are negative for CM energies below 1 TeV and not too small Higgs-boson masses, i.e. not too far away from threshold. This is due to large virtual photonic corrections which are not cancelled by the corresponding real radiation owing to the decreasing phase-space volume for real-photon emission with increasing M H or decreasing √ s. Therefore, the photonic corrections are large and negative, in particular, at the relatively small scattering energy of √ s = 500 GeV, where they reach −25% to −65% for M H = 100-150 GeV. In this situation resummation of the large photonic corrections is mandatory. The bulk of these contributions is included in our calculation (higher-order ISR) and for √ s = 500 GeV amounts to 3-20%. Away from threshold, the energies. Note that we use the G µ -scheme, and that the weak corrections are shifted by ∼ ±10% when transformed to other schemes like the α(0) or the α(s) schemes (see also Ref. [ 16]). The spikes at M H = 2M W , 2M Z result from thresholds and are well-known from the process e + e − → ZH [ 35]. Photonic and weak corrections partially cancel each other, and the resulting electroweak corrections increase very weakly with √ s apart from the region close to threshold, where the electroweak corrections decrease fast with decreasing √ s and reach of the order of −20% at threshold. Away from threshold, they are at the level of −5%, −8%, and −11% for M H = 115 GeV, 150 GeV, and 200 GeV, respectively, and thus become increasingly negative with increasing Higgs-boson mass.

Comparison with other calculations
The results on the QCD corrections given in Table 1 have been reproduced with the (publically available) computer code based on the calculation of Ref. [ 8]. We found agreement within the statistical integration errors.
For a comparison of the electroweak O(α) corrections with the results of Ref. [ 14] we changed our input parameters to the ones quoted there and switched to the α(0)scheme, where G µ is ignored in the input and all couplings are deduced from α(0). In Table 2 we compare some representative numbers 4 from the calculation of Ref. [ 14] with the corresponding results from our Monte Carlo generator. The numbers in parentheses give the errors in the last digits of our calculation. The tree-level cross sections coincide within 0.03%. Most of the numbers for the one-loop corrected cross sections agree within 1-2%, i.e. roughly within the estimated error of Ref. [ 14]. However, for the corrected cross sections at √ s = 2 TeV, i.e. at high energies, and the one very close to threshold, i.e. for √ s = 500 GeV and M H = 150 GeV, we find differences of 4% and 7%, respectively.
The same holds for the relative corrections. Ours are larger by about 4% at √ s = 2 TeV and smaller by about 5% for the selected cross section close to threshold. Finally, we have also compared the electroweak O(α) corrections with Ref. [ 15], where also the α(0)-scheme has been used. In Table 3 we list the results of Table 2 of Ref. [ 15] and the separate relative photonic and weak corrections of Table 3 of that paper together with the corresponding results from our Monte Carlo generator. Again the numbers in parentheses give the errors in the last digits. We reproduce the results for the lowest-order cross section within the integration errors, which are about 2-3 × 10 −4 . The results for the cross section including electroweak corrections coincide to better than 0.1% which is of the order of the integration error of the results of Ref. [ 15]. The relative electroweak, weak, and photonic corrections agree also within 0.1%. This holds as well for the QCD corrections (not shown in Table 3).

Summary
We have presented results from a calculation of electroweak radiative corrections to the process e + e − → ttH, which is important for a precise determination of the top-Higgs   (2) −5.00(1) this work Table 3: Total cross section in lowest order and including the full electroweak O(α) corrections as well as the relative electroweak, weak, and photonic corrections for various CM energies and Higgs-boson masses for the input-parameter scheme of Ref. [ 15] Yukawa coupling. In detail, we have discussed the impact of photonic corrections at and beyond O(α), the genuine weak O(α) corrections, and the O(α s ) QCD corrections. The photonic and weak corrections both reach the order of ∼ 10% and show characteristic dependences on the Higgs-boson mass and on the scattering energy. Owing to a phase-space effect the (negative) photonic corrections reduce the cross section more and more when the threshold is approached, i.e. with increasing Higgs-boson masses. Close to threshold the resummation of the higher-order ISR corrections becomes mandatory. The weak corrections, which depend on the Higgs-boson masses only moderately, range from about +15% at a CM energy of 500 GeV to about −10% at 1.5 TeV. There are large cancellations between photonic and weak corrections, and the final size of the corrections depends strongly on the input-parameter scheme. These results clearly demonstrate the necessity to include the electroweak corrections in predictions adequate for a future high-luminosity e + e − collider.
We have compared our results with those of other groups. The QCD corrections have been successfully checked against previous calculations. The electroweak corrections agree well with the results of the recent calculation [ 15] but are at variance with the results of Ref. [ 14] at high energies and close to threshold.
the Swiss Bundesamt für Bildung und Wissenschaft and by the European Union under contract HPRN-CT-2000-00149.