Electroweak Sudakov Logarithms and Real Gauge-Boson Radiation in the TeV Region

TL;DR

The paper analyzes how electroweak Sudakov logarithms generate large negative corrections at TeV energies and how real W/Z radiation can counterbalance them under various phase-space restrictions. Using a progression from a massive abelian theory to a spontaneously broken SU(2) model and the SM, it shows that the degree of cancellation between virtual and real effects is highly sensitive to which soft and collinear emissions are included in the observable. Bloch-Nordsieck violations in non-abelian contexts further complicate the cancellation, depending on external state charges and helicities. The findings highlight the importance of modeling real radiation and phase-space cuts for accurate high-energy predictions, with implications for both lepton and hadron colliders and planned extensions to hadronic processes.

Abstract

Electroweak radiative corrections give rise to large negative, double-logarithmically enhanced corrections in the TeV region. These are partly compensated by real radiation and, moreover, affected by selecting isospin-noninvariant external states. We investigate the impact of real gauge boson radiation more quantitatively by considering different restricted final state configurations. We consider successively a massive abelian gauge theory, a spontaneously broken SU(2) theory and the electroweak Standard Model. We find that details of the choice of the phase space cuts, in particular whether a fraction of collinear and soft radiation is included, have a strong impact on the relative amount of real and virtual corrections.

Figures (4)

• Figure 1: Different restrictions on the real emission process. The momentum of the undetected gauge boson (wavy line) is assumed to lie within the shaded area. In Scenario A (collinear and soft) we require the final state fermions to be almost back-to-back or the emitted boson to lie within a cone around the initial state fermions. In Scenario B (collinear) the emitted boson has to be within any of the cones around the fermions.
• Figure 2: Relative NLO corrections to the four-fermion process in the abelian toy theory as a function of the center of mass energy $\sqrt{s}$ in TeV. In each plot the lower solid line represents the virtual correction (with $M=80$ GeV, $\alpha=0.03$, $n_f=6$, $n_s=1$) and the dashed lines refer to the sum $\Delta\sigma = \sigma^{(V)} + \sigma^{(R)}$ with different restrictions on the real emission process. The individual dashed lines (green/red/blue, from bottom to top in each plot) refer to $z_c=0.75/0.5/0.25$ and no angular cut (top), $\theta_{f\bar{f}}^c = 175^\circ/170^\circ/165^\circ$ (middle) and $\theta_{F b}^c=15^\circ/30^\circ/45^\circ$ (bottom). In the lower two plots we fixed $z_c=0.5$ and $\theta_{I b}^c=10^\circ$. The dotted curves indicate the contribution from initial state radiation (corresponding to $\theta_{f\bar{f}}^c =180^\circ$ and $\theta_{F b}^c=0^\circ$, respectively).
• Figure 3: Relative NLO corrections to the four-fermion process in a spontaneously broken $SU(2)$ theory with neutral (upper plots) and charged (lower plots) initial states (notation and numerical input values from Figure \ref{['fig:U1:fourferm:cuts']}).
• Figure 4: Relative NLO electroweak corrections to $e^+e^-\to q\bar{q}$ as a function of the center of mass energy in TeV. In each plot the lower solid line represents the virtual correction (with $\alpha=1/128$ and $s_w^2=0.231$) and the dashed lines refer to the sum $\Delta\sigma = \sigma^{(V)} + \sigma^{(R)}$ with different restrictions on the real emission process (according to the scenarios from Figure \ref{['fig:scenarios']}). The individual dashed lines (green/red/blue, from bottom to top in each plot) refer to $\theta_{f\bar{f}}^c = 175^\circ/170^\circ/165^\circ$ (left) and $\theta_{F b}^c=15^\circ/30^\circ/45^\circ$ (right). We further set $z_c=0.5/0.7$ in the upper/lower plots and $\theta_{I b}^c=10^\circ$. The dotted curves indicate the contribution from initial state radiation (corresponding to $\theta_{f\bar{f}}^c =180^\circ$ and $\theta_{F b}^c=0^\circ$, respectively).