Gauge-Invariant Longitudinal Modes in the Herwig 7 Electroweak Parton Shower

Abstract

Longitudinal electroweak gauge bosons are the most technically delicate ingredient of electroweak parton showers: in the broken Standard Model, the gauge component of a longitudinal polarisation does not cancel diagram by diagram, but is related by Ward identities to amplitudes with an insertion of the associated would-be Goldstone field. Building on the default Herwig 7 treatment based on the Dawson-subtracted longitudinal current, we construct a gauge-invariant longitudinal scheme in which the Dawson remainder is retained as a well-defined contribution and completed by a Ward-identity-fixed Goldstone-matching term. We derive helicity-resolved building blocks and compact quasi-collinear splitting kernels for $q\to q'V$ and $V\to V'V''$ branchings, and implement the scheme in Herwig 7 as a switchable alternative to the default longitudinal basis. The completion leaves the transverse sector unchanged and modifies only longitudinal entries through controlled symmetry-breaking terms, including Yukawa-sensitive contributions in massive-fermion channels. In shower-level studies, we find that the Dawson and gauge-invariant prescriptions coincide at high evolution scales, while differing at lower scales precisely in channels where symmetry breaking is active. Exclusive single-emission observables can moreover display non-monotonic scheme dependence once Sudakov suppression and kinematic constraints are accounted for. A first multi-emission LHC-like study confirms numerical stability and yields controlled, interpretable shifts in observables that probe propagating electroweak shower currents, whereas quantities dominated by promptly contracted, near on-shell vector-boson matrix elements remain largely unchanged.

Figures (8)

• Figure 1: Spin-averaged quasi-collinear splitting kernels $P(z)$ for representative $q\to q'V$ branchings at fixed transverse scale $\tilde{q}=500~\mathrm{GeV}$, comparing the original Dawson longitudinal prescription (red) with the GI completion (blue). Left: $b\to t\,W^-$, for which the Ward-identity (Goldstone) matching terms are Yukawa-enhanced through the heavy-quark mass and induce a visible deformation of the longitudinal contribution over a broad range of $z$. Right: $b\to b\,Z^0$, where the Goldstone couplings are parametrically suppressed, and the two prescriptions are practically indistinguishable at the level of the inclusive kernel.
• Figure 2: Comparison of the analytic $V_0\to V_1V_2$ splitting kernels in the Dawson and GI prescriptions for three representative channels: $W\to WZ$ (left), $Z\to W^+W^-$ (middle) and $\gamma\to W^+W^-$ (right). In each case we show the spin-summed kernel $F^{(\mathrm{X})}(z,\tilde{q}^2)$ as a function of the lightcone momentum fraction $z$ at fixed evolution scale $\tilde{q}$ (with pole masses for the external vectors).
• Figure 3: Single-resummed final-state radiation (SRS) comparison between the Dawson and GI longitudinal prescriptions for $q\to q'V$ branchings in a clean $e^+e^-\to q\bar{q}$ setup at $\sqrt{s}=10~\mathrm{TeV}$ (no hadronisation and no decays; exactly one electroweak gauge boson in the event record). Left column: charged-current emission ($q\to q'W^+$); right column: neutral-current emission ($q\to qZ^0$). Shown are the boson transverse-momentum spectrum $p_{\perp,V}$ (top) and the lightcone momentum fraction proxy $z$ used in the shower mapping (bottom); ratio panels display GI/Dawson.
• Figure 4: SRS shower comparison between the Dawson and GI longitudinal prescriptions for boson-line splittings in a $pp$ sample at $\sqrt{s}=13~{\rm TeV}$. Left column: splitting-tagged $W^\pm\to W^\pm Z^0$; right column: splitting-tagged $W^\pm\to W^\pm\gamma$. Upper row: transverse-momentum spectrum of the parent $W^\pm$, $p_{\perp,W}$; lower row: reconstructed lightcone momentum fraction $z$ of the splitting as used in the Herwig kinematics map.
• Figure 5: As Fig. \ref{['fig:vvv_srs_wj']}, but for splitting-tagged $Z^0\to W^+W^-$ branchings at $\sqrt{s}=13~{\rm TeV}$. Left: parent-$Z^0$ transverse-momentum spectrum; right: reconstructed splitting variable $z$.