Transverse Momentum Resummation and Analytic Continuation into the Deep Infrared
This presentation explores a breakthrough approach to understanding hadron internal structure through transverse-momentum data. The authors introduce a novel analytic method that performs resummation directly in transverse-momentum space rather than the traditional impact-parameter space, achieving next-to-leading logarithmic accuracy. By analytically continuing calculations into the deep infrared regime, this work identifies fundamental non-perturbative effects rooted in QCD dynamics and opens a new paradigm for transverse-momentum-dependent phenomenology.Script
How do you extract the three-dimensional structure of a proton from the momentum of particles flying out of a collision? The answer lies hidden in a mathematical transition from what we can calculate to what we must measure, and the authors of this work have found a new path through that boundary.
Transverse-momentum-dependent observables are exquisitely sensitive to how quarks and gluons move inside hadrons, but translating theory into predictions has required a detour.
For decades, theorists have worked in impact-parameter space because resummation is manageable there, then used Fourier transforms to connect to the transverse momentum experimentalists measure. But this indirect route buries the physics. Non-perturbative effects get encoded in models of the transformation itself, disconnected from the underlying dynamics of quarks and gluons.
What if you could perform the resummation directly in transverse-momentum space and peer into the non-perturbative regime analytically?
The authors introduce an analytic technique that sidesteps the traditional detour entirely. By carefully handling the edge effects and singularities, they achieve next-to-leading logarithmic resummation directly in momentum space. Then, through analytic continuation into the deep infrared, they identify where perturbative QCD breaks down and non-perturbative physics takes over, all without invoking ad hoc models of the Fourier transform.
This figure shows the transverse-momentum distribution of an up quark inside a proton at two different energy scales. The green line represents fixed-order perturbative calculations, which diverge at low momentum. The resummed curves tame that divergence and extend smoothly into the infrared. Notice how the shape evolves with energy: at 4 giga-electron-volts on the left, the distribution is broader, while at 9 giga-electron-volts on the right, it narrows. This evolution reflects how partons probe different momentum regimes as collision energy changes, and the resummation captures that physics faithfully.
The contrast is stark. Impact-parameter space is mathematically convenient for resummation, but it sits one Fourier transform away from what experiments measure. Every prediction requires modeling that transform, which smuggles in assumptions. The new momentum-space method works directly in the observable frame. Analytic continuation then pinpoints exactly where strong coupling and parton distributions inject non-perturbative corrections, anchoring those effects in fundamental QCD rather than auxiliary parametrizations.
Here we see how different resummation prescriptions compare at low transverse momentum, both at a moderate energy scale of 4 giga-electron-volts and at the much higher Z boson mass. The blue curves represent the next-to-leading logarithmic resummed distributions computed with the new method. The agreement across energy scales and the smooth behavior even as momentum drops toward zero demonstrate that the analytic approach handles infrared physics robustly, without the instabilities that have stymied earlier momentum-space attempts.
This work opens experimental doors. By validating analytic continuation against data from low to high energies, physicists can test whether the non-perturbative sources identified here truly govern hadron structure. Flavor dependence, previously buried in parametrizations, becomes a direct question about quark dynamics. And the entire framework shifts TMD phenomenology from fitting transforms to extracting physics, a genuine paradigm change.
Resummation in momentum space with analytic continuation transforms the boundary between calculation and measurement into a window, revealing the infrared heart of QCD. Visit EmergentMind.com to explore this paper in depth and create your own research video.
