[Paper Review] Using SCET to calculate electroweak corrections in gauge boson production

Jui-yu Chiu, Andreas Fuhrer+3 more|arXiv (Cornell University)|May 7, 2009

Particle physics theoretical and experimental studies4 references4 citations

TL;DR

This paper extends Soft-Collinear Effective Theory (SCET) to resum electroweak Sudakov logarithms in gauge boson production at high energies, introducing the $\Delta$ regulator to handle collinear singularities while preserving soft-collinear factorization. It computes resummed form factors for both transverse and longitudinally polarized W bosons, using the Goldstone boson equivalence theorem, and derives all-order logarithmic resummation for massive gauge bosons in a spontaneously broken $SU(2)$ theory.

ABSTRACT

We extend an effective theory framework developped in Refs. [1,2] to sum electroweak Sudakov logarithms in high energy processes to also include massive gauge bosons in the final state. The calculations require an additional regulator on top of dimensional regularization to tame the collinear singularities. We propose to use the Delta regulator, which respects soft-collinear factorization.

Motivation & Objective

• To extend the SCET framework to include massive gauge bosons in the final state, beyond previous treatments of form factors and four-fermi processes.
• To address collinear singularities in effective field theory calculations involving massive gauge bosons, which require a regulator beyond dimensional regularization.
• To preserve soft-collinear factorization in the presence of massive vector bosons by introducing a new regulator, the $\Delta$ regulator.
• To compute resummed electroweak corrections for both transverse and longitudinally polarized W bosons in high-energy scattering processes.
• To generalize existing SCET-based resummation techniques to processes with gauge bosons in the final state, relevant for LHC phenomenology.

Proposed method

• The $\Delta$ regulator is introduced as a new regularization scheme that respects soft-collinear factorization and tames collinear singularities in massive gauge boson amplitudes.
• The effective theory is constructed via matching from the full electroweak theory to SCET, with separate matching at high and low scales to resum large logarithms.
• The Sudakov form factor is computed using the renormalization group evolution of matching coefficients $C_T(Q)$ and $D_T(M)$, with anomalous dimensions $\gamma_T(\mu)$ derived from the effective theory.
• For longitudinally polarized W bosons, the Goldstone boson equivalence theorem is applied, mapping the physical amplitude to a scalar field scattering amplitude involving Higgs and Goldstone fields.
• The matching coefficients $D_L^{(\phi\phi)}$ and $D_L^{(hh)}$ are computed at the low scale, incorporating finite corrections and the non-trivial factor $\mathcal{E}$ that accounts for truncation effects.
• The full resummed form factor is obtained as $F_{O_T}(Q^2) = C_T(Q) \exp\left[\int_Q^M \frac{d\mu}{\mu} \gamma_T(\mu)\right] D_T(M)$, with all logarithms summed to all orders.

Experimental results

Research questions

• RQ1 How can electroweak Sudakov logarithms be resummed in processes with massive gauge bosons in the final state using effective field theory?
• RQ2 What regulator preserves soft-collinear factorization in SCET when massive gauge bosons are present, and how does it compare to dimensional regularization?
• RQ3 How does the Goldstone boson equivalence theorem facilitate the computation of amplitudes for longitudinally polarized W bosons in the effective theory?
• RQ4 What are the structure and resummation properties of the matching coefficients $D_T(M)$ and $D_L(M)$ for massive vector bosons in SCET?
• RQ5 Can the framework developed for the $SU(2)$ Higgs model be generalized to the full Standard Model gauge group?

Key findings

• The $\Delta$ regulator successfully tames collinear singularities in massive gauge boson amplitudes while preserving soft-collinear factorization, enabling consistent effective field theory calculations.
• The resummed form factor for transverse W bosons is given by $F_{O_T}(Q^2) = C_T(Q) \exp\left[\int_Q^M \frac{d\mu}{\mu} \gamma_T(\mu)\right] D_T(M)$, with $D_T(M)$ containing finite corrections and logarithmic terms up to $\mathsf{L_M}^2$.
• For longitudinally polarized W bosons, the effective theory computes the amplitude via scalar Higgs and Goldstone fields, with matching coefficients $D_L^{(\phi\phi)}$ and $D_L^{(hh)}$ including $f_s(1,1)$, $f_s(1,z^2)$, and $f_s(z^2,1)$ integrals.
• The correction factor $\mathcal{E}$ in Eq. (16) is non-running and accounts for truncation effects in the Goldstone boson matrix elements, ensuring gauge invariance at high energies.
• The result confirms that at high energies, the leading logarithmic corrections are dominated by $\alpha \mathsf{L}^2$ terms, and the resummation framework can be extended to the full Standard Model gauge group.