The Fate of Ultra-Collinear Modes in On-Shell Massive Sudakov Form Factors

Abstract

Individual multi-loop diagrams for the massive Sudakov form factor contain an infinite tower of ultra-collinear momentum regions. We show that, for the on-shell form factor in QCD, these contributions cancel to all orders as a consequence of gauge invariance, so the leading-power SCET$_{\rm II}$ factorization formula is unchanged. Using the $η$ rapidity regulator, we compute the soft function and the massive jet function of the quark and gluon Sudakov form factors through two loops and resum logarithms at NNLL accuracy, including hierarchies of fermion masses. We also show that with a gauge-boson mass regulator, the infinite tower of modes is truncated and ultra-collinear and ultra-soft modes become manifest and factorize explicitly, providing a direct EFT derivation of the regulated infrared dependence.

Figures (8)

• Figure 1: Power-counting diagram for a momentum $k$ in light-cone coordinates, with $n_+\!\cdot k/Q\sim\lambda^{a}$ and $n_-\!\cdot k/Q\sim\lambda^{b}$ (hence $k_\perp/Q\sim\lambda^{(a+b)/2}$), where $\lambda\equiv m/Q$. Diagonal lines $a+b=\text{const}$ (and the shaded bands between them) correspond to the massless virtuality $k^2\sim Q^2\lambda^{a+b}$. For an on-shell massive collinear line, the induced off-shellness $\Delta_c\equiv (p_c+k)^2-m^2$ scales as $\Delta_c\sim Q^2\lambda^{\min(b,\,a+2)}$; the contours highlight $\Delta_c\sim m^2$ and $\Delta_c\sim m^4/Q^2$. Marked points indicate the hard, hard-collinear, (anti-)collinear, soft, ultra-soft, and ultra-collinear modes used in the EFT description.
• Figure 2: Chain of EFTs for massive energetic particles/jets, organized by the relevant virtuality scale: QCD (hard) at $\mu^2\!\sim\!Q^2$, matched onto SCET$_{\mathrm{I}}$ (hard-collinear) at $\mu^2\!\sim\!mQ$, then SCET$_{\mathrm{II}}$ (collinear/soft) at $\mu^2\!\sim\!m^2$, and finally onto bHQET (ultra-collinear) for scales below $m^2$.
• Figure 3: Example of Ward identity cancellation at two loops. The two leftmost diagrams denote specific regions of QCD diagrams. The second diagram from the right is a contribution to $\sqrt{Z}$ in SCET$_{\rm II}$. The rightmost diagram is a contribution to $\sqrt{\mathfrak Z_1}$ in bHQET$_1$. Heavy fermion lines in bHQET$_1$ are denoted by dashed lines. Wilson lines are denoted by double lines.
• Figure 4: Another two-loop example of Ward-identity cancellation. The two leftmost diagrams denote the ultra-collinear region of the corresponding QCD graphs. The second diagram from the right is the SCET$_{\rm II}$ contribution to $\sqrt{Z}$, and the rightmost diagram is the bHQET$_1$ contribution to $\sqrt{\mathfrak Z_1}$. Heavy-fermion lines in bHQET$_1$ are dashed and Wilson lines are drawn as double lines.
• Figure 5: Two-loop example of Ward-identity cancellation in a non-abelian gauge theory. The three leftmost diagrams denote the ultra-collinear regions of the corresponding QCD graphs. The second diagram from the right is the SCET$_{\rm II}$ contribution to $\sqrt{Z}$, and the rightmost diagram is the bHQET$_1$ contribution to $\sqrt{\mathfrak Z_1}$.