Polarization interference in exclusive $V+$jets at all orders in $\alpha_s$

Technical Summary: Polarization Interference in Exclusive V +jets at All Orders in $\alpha_s$

Problem Statement
The paper addresses the theoretical description of transverse-longitudinal polarization interference in the production of electroweak (EW) gauge bosons ($V = \gamma^*, Z^{(*)}, W^{(*)}$) in association with jets ($pp \to V + \text{jets}$). While the "polarized propagator" formalism has been successfully used to model scattering rates and describe LHC polarization data, previous studies often relied on inclusive approximations (e.g., structure functions) or narrow-width/pole approximations. A rigorous understanding of how polarization interference behaves in exclusive processes—where jet kinematics are fixed—at arbitrary orders of the strong coupling constant $\alpha_s$ has been lacking. Specifically, the conditions under which the interference between transverse ($\lambda = \pm 1$) and longitudinal ($\lambda = 0$) helicity states vanishes or remains significant, particularly in the presence of fiducial cuts, require clarification.

Methodology
The authors employ a novel approach based on newly identified analytical decompositions of polarized gauge boson propagators in covariant gauges ($R_\xi$). Key methodological elements include:

• Formalism: The work operates entirely within the polarized propagator formalism, decomposing the unpolarized propagator $\Pi^V_{\mu\nu}$ into outer products of polarization vectors for transverse ($\lambda=T$), longitudinal ($\lambda=0$), and scalar ($\lambda=S$) states.
• Current Conservation: The derivation relies on the conservation of the incoming QCD current ($G^\nu_{\text{in}}$) and the outgoing lepton current ($J^\mu_{h,h'}$). Crucially, the authors assume massless leptons and work at the lowest order in EW theory, ensuring these currents are conserved ($q \cdot J = 0$ and $q \cdot G_{\text{in}} = 0$).
• Tensor Decomposition: The authors utilize a specific decomposition of the propagator into transverse ($\Phi_{\mu\nu}$) and forward-backward ($\Theta_{\mu\nu}$) tensor structures. This allows for a manifest power counting of mass-over-energy factors in covariant gauges, a feature previously more common in axial gauges.
• Exclusivity: Unlike prior works that integrated over hadronic activity, this analysis remains fully exclusive with respect to the $n$-jet system kinematics while integrating over the decay products of the boson.

Key Results
The paper derives the polarization interference term $I^{\text{V+jets}}_{\text{pol}}$ at the fully differential level and integrates over the decay phase space of the boson into massless leptons. The primary findings are:

1. Vanishing Interference for $\gamma^*$: For the virtual photon ($\gamma^* \to \ell^+\ell^-$), the transverse-longitudinal polarization interference vanishes exactly after integrating over the lepton kinematics (specifically the polar angle), provided parity is conserved ($g_L = g_R$). This recovers the well-known inclusive results of Lam & Tung [41] but extends them to the exclusive regime.
2. Parity Violation and $W/Z$ Bosons: For $W$ and $Z$ bosons, parity violation ($g_L \neq g_R$) softens the cancellations.
   • For the $Z$ boson, the interference is partially suppressed but non-zero.
   • For the $W$ boson, where parity violation is maximal, the interference remains non-vanishing after integration.
3. Scaling Behavior: The magnitude of the polarization interference relative to the unpolarized cross section scales linearly with $1/\gamma_V$ (where $\gamma_V = E_V/\sqrt{q^2}$). In the limit where the boson is highly boosted ($E_V \gg \sqrt{q^2}$), the longitudinal component and the interference term vanish, consistent with the behavior of massless spin-1 states.
4. Impact of Fiducial Cuts: The authors provide a simple formula to estimate the impact of experimental acceptance cuts. While ideal phase-space integration leads to strong cancellations, realistic cuts (e.g., $p_T > 25$ GeV, $|\eta| < 2.5$) disrupt these cancellations.
   • For $Z + \text{jets}$ at $\sqrt{s}=13$ TeV with realistic kinematics ($p_T^Z \sim 50-250$ GeV), the interference is estimated to be at the percent level (e.g., $\lesssim 5\%$ for $p_T^Z \sim 50$ GeV, dropping to $\sim 2\%$ for higher $p_T$).
   • The interference is generally bounded to be at most $\mathcal{O}(10\%)$ in $V + \text{jets}$ processes.

Significance and Claims
The paper claims to establish a rigorous, all-orders-in-$\alpha_s$ framework for understanding polarization interference in exclusive processes without relying on narrow-width approximations. The significance of the work lies in:

• Theoretical Clarity: It demonstrates that the vanishing or suppression of polarization interference is intrinsically tied to parity conservation and current conservation, rather than specific diagrammatic cancellations.
• Methodological Utility: The derived tensor identities (Eq. 11) simplify the organization of polarized amplitudes and make power counting transparent in covariant gauges.
• Applicability: The results are asserted to hold for:
  • Multiboson processes (under narrow-width approximations where non-resonant contributions are suppressed).
  • Heavy ion collisions ($pA$ and $AA$), assuming factorization holds.
  • New physics scenarios involving massive, colorless, chiral gauge bosons (e.g., $Z'$ or $W'$), provided the couplings preserve current conservation.

The authors conclude that the polarized propagator formalism is not merely a computational tool for Monte Carlo generators but a rigorous approach capable of pushing the state-of-the-art in understanding EW boson production at ultra-high energies.