Threshold resummation of rapidity distributions at fixed partonic rapidity

This paper derives a general all-order expression for threshold resummation of rapidity distributions in colorless final-state processes at fixed partonic rapidity, determines the coefficients up to NNLL accuracy for the Drell-Yan process by matching with NNLO results, and demonstrates agreement between their direct QCD approach and previous SCET-based findings.

The Gist

Imagine you are trying to predict the weather for a specific city, but you only have data from a massive, chaotic storm system. In the world of particle physics, scientists try to predict what happens when tiny particles smash into each other at nearly the speed of light. One of the most famous "storms" is the Drell-Yan process, where two protons collide and produce a heavy, invisible particle (like a Z boson or a Higgs boson) that flies off.

To make accurate predictions, physicists use a tool called Quantum Chromodynamics (QCD). But QCD is incredibly messy. When you try to calculate the odds of these collisions, your math explodes with infinite numbers unless you use a technique called resummation. Think of resummation as a way to "tame" the infinite chaos by grouping similar, tiny errors together until they make sense.

This paper, written by a team of physicists, tackles a very specific, tricky version of this problem. Here is the breakdown in everyday language:

1. The Problem: The "Slow-Down" vs. The "Sideways"

Imagine two cars driving toward each other on a highway.

• The Old Way (Double-Soft Limit): In previous studies, physicists looked at the scenario where the cars slow down almost to a complete stop right before they crash. The resulting explosion (the new particle) barely moves. It's a "dead stop" scenario.
• The New Way (Single-Soft Limit): This paper asks: "What if the cars slow down, but the explosion still has a specific speed and direction?" Imagine the cars slow down, but the explosion shoots out sideways at a specific angle. This is the "fixed rapidity" scenario.

The authors realized that while the "dead stop" scenario was well understood, the "sideways explosion" scenario was much harder to calculate because the math gets tangled. The particle isn't just sitting still; it's moving, and that movement changes how the "messy" quantum effects behave.

2. The Solution: A New Map for the Chaos

The authors developed a new mathematical "map" to navigate this specific type of chaos.

• The Analogy: Imagine you are trying to predict the path of a leaf falling in a wind tunnel.
  • If the wind is blowing straight down (the old way), you can predict the leaf's path easily.
  • If the wind is blowing sideways and the leaf is spinning (the new way), the path is complex.
  • The authors created a new set of rules (a Renormalization Group approach) that acts like a GPS for this spinning leaf. It allows them to predict the leaf's path accurately, even when the wind is blowing in a weird, specific direction.

3. The "Double-Check" (SCET vs. dQCD)

In physics, there are two different languages used to describe these storms:

1. dQCD (Direct QCD): The traditional, heavy-duty language used by most experimentalists.
2. SCET (Soft-Collinear Effective Theory): A newer, more specialized language that breaks the problem down into smaller, easier pieces.

For a long time, it was like trying to translate a poem from French to German; the meaning was there, but the words didn't always match up perfectly.

• The Paper's Achievement: The authors took the result they found using their new "Direct QCD" map and translated it into the "SCET" language. They then compared it to a result that other scientists had previously found using SCET.
• The Result: The two maps matched perfectly! It's like two different hikers climbing a mountain from opposite sides, meeting at the summit, and realizing they both drew the exact same map of the peak. This proves their new math is correct.

4. Why Does This Matter?

You might ask, "Why do we care about a particle moving sideways?"

• Precision is Key: The Large Hadron Collider (LHC) is a machine that smashes particles to find new physics. To know if they found something new (like a new type of particle), they need to know exactly what the old stuff looks like.
• The "Background Noise": The Drell-Yan process is like the background noise in a concert. If you want to hear a new instrument (new physics), you need to know the background noise perfectly.
• The Impact: By understanding the "sideways" movement better, the authors have reduced the "fuzziness" in their predictions. This means when the LHC sees a weird signal, scientists can be more confident: "Is this a glitch in our math, or is it a discovery?"

Summary

Think of this paper as a team of cartographers who realized that while they had a perfect map for a flat, calm ocean, they didn't have a good map for a ocean with a specific, strong current. They drew a new map for that current, checked it against a different type of map used by sailors, and confirmed they matched. This new map helps future explorers (physicists at the LHC) navigate the stormy seas of particle collisions with much greater confidence.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of resumming large logarithmic corrections in the rapidity distributions of colorless final states (such as Drell-Yan lepton pairs or Higgs bosons) produced in hadron collisions.

• Context: Standard threshold resummation typically considers the limit where the partonic center-of-mass energy $\sqrt{s}$ approaches the mass of the final state $M$ ($s \to M^2$). In this "double-soft" limit, the final state is produced at rest ($y=0$), and both parton momentum fractions $x_1, x_2 \to 1$.
• The Gap: Previous results often assumed the rapidity $y$ was zero or treated it as a subleading effect. However, in collider phenomenology, one often needs the distribution at fixed, non-zero rapidity (or fixed longitudinal momentum $p_z$).
• The Specific Limit: The authors investigate the single-soft limit, defined as $s \to s_{\min}(p_z)$ with $p_z$ (and thus rapidity $y$) fixed. In terms of scaling variables, this corresponds to $x_1 \to 1$ while $x_2$ remains fixed (or vice versa). This limit is distinct from the double-soft limit and requires a different resummation framework because the kinematic constraint is not fully decoupled from the partonic kinematics.

2. Methodology

The authors employ a Direct QCD (dQCD) approach based on renormalization-group (RG) improvement, generalizing methods previously developed for inclusive cross-sections and transverse momentum distributions.

• Kinematic Analysis:
  • They analyze the phase space in the single-soft limit, demonstrating that the dynamics are governed by a single soft scale $\Lambda^2_{ss} \propto M^2(1-x_1)$.
  • Unlike transverse momentum resummation (which involves two soft scales), the fixed-rapidity limit is purely collinear in nature. The resummation dynamics become entangled with the evolution of Parton Distribution Functions (PDFs).
• Renormalization Group Improvement:
  • The coefficient function is factorized into a hard part and a soft/collinear part.
  • They derive a physical anomalous dimension $\gamma$ that depends on the Mellin variables $N_1$ and $N_2$. In the single-soft limit, the dependence on the soft variable $N_1$ is resummed, while $N_2$ acts as a fixed parameter.
  • The resummed coefficient function is expressed in Mellin space as an exponential of integrals involving the cusp anomalous dimension and a non-cusp function $D$.
• Matching to Fixed Order:
  • To determine the unknown coefficients in the resummation formula (specifically the constants and the $D$-function), the authors perform a matching procedure against the fixed-order Next-to-Next-to-Leading Order (NNLO) results for the Drell-Yan process.
  • A significant technical hurdle was transforming the NNLO results, originally provided in variables $(\tau, u)$ (where $u$ is related to the scattering angle), into the scaling variables $(x_1, x_2)$ required for Mellin resummation. This involved deriving new distributional identities to handle the singular Jacobian of the transformation near the threshold.

3. Key Contributions

1. General Resummation Formula: Derivation of a general expression for the resummation of rapidity distributions valid to all logarithmic orders in the single-soft limit ($x_1 \to 1$, fixed $x_2$).
2. NNLL Accuracy for Drell-Yan: Explicit determination of resummation coefficients up to Next-to-Next-to-Leading Logarithmic (NNLL) accuracy for the quark-antiquark nonsinglet channel in the Drell-Yan process.
3. Distributional Identities: Development of a rigorous method to transform fixed-order distributions from $(\tau, u)$ space to $(x_1, x_2)$ space, providing explicit identities up to NNLO. This is crucial for matching fixed-order calculations with resummation in physical variables.
4. SCET vs. dQCD Equivalence: A detailed translation and comparison of the authors' dQCD results with previous results obtained using Soft-Collinear Effective Theory (SCET) (specifically Refs. [14] and [22]). They prove that the two approaches yield identical results when scales are matched appropriately.

4. Key Results

• Resummation Structure: The resummed coefficient function $C(N_1, N_2)$ in the single-soft limit factorizes into a hard constant $C^{(c)}$ and an exponential containing the resummed logarithms. The exponent involves the cusp anomalous dimension $A(\alpha_s)$ and a process-dependent function $D(\alpha_s, N_2)$.
• Coefficients Determination:
  • The authors determined the NNLL coefficients $D_1, D_2$ and the constant $g_0$ by matching to the NNLO fixed-order result.
  • They found that the single-soft $D$-function contains contributions from the evolution of the PDFs (collinear evolution) up to the soft scale, in addition to the standard soft radiation terms.
  • Specifically, $D_{ss}^{(1)}(N_2)$ is identified with the leading-order nonsinglet quark anomalous dimension $\hat{\gamma}^{(0)}_{qq}(N_2)$, and $D_{ss}^{(2)}(N_2)$ includes the NLO anomalous dimension and specific constant terms.
• SCET Agreement: By translating the SCET evolution equations (hard, beam, and soft functions) into the dQCD language, the authors demonstrated exact agreement between their dQCD derivation and the SCET results of Ref. [14] (and extended to N3LL in Ref. [22]). This validates the dQCD approach for rapidity distributions and clarifies the mapping between SCET scales and dQCD resummation scales.

5. Significance

• Phenomenological Precision: This work provides the necessary theoretical tools to improve the precision of rapidity distribution predictions for the LHC and future colliders, particularly in regions where the final state is boosted (non-zero rapidity).
• Theoretical Consistency: It resolves the ambiguity regarding the equivalence of dQCD and SCET approaches for this specific observable, showing that the "collinear" nature of the single-soft limit in dQCD corresponds correctly to the beam function evolution in SCET.
• Methodological Extension: The techniques developed for transforming distributions between variable sets and handling the single-soft limit can be applied to other semi-inclusive processes, such as Semi-Inclusive Deep Inelastic Scattering (SIDIS), which the authors identify as a natural direction for future work.
• Conceptual Clarity: The paper clarifies that while double-soft resummation is driven by soft gluon emission, single-soft resummation at fixed rapidity is driven by collinear evolution from the hard scale down to the soft scale, effectively resumming the logarithms of the momentum fraction of the radiating parton.

In summary, this paper successfully extends threshold resummation to the physically relevant case of fixed rapidity, providing explicit NNLL results for Drell-Yan and establishing a rigorous bridge between direct QCD calculations and SCET formalisms.