Numerical Study of MRW-Type Unintegrated Double Parton Distribution Functions from Non-Factorized DPDFs

This paper presents a numerical study of MRW-type unintegrated double parton distribution functions constructed from non-factorized GS09 collinear DPDFs, comparing the performance and characteristics of double modified KMRW, double virtuality ordered MRW, and a normalization-matched variant to analyze their transverse momentum dependence, normalization properties, and sensitivity to longitudinal correlations.

The Gist

Imagine a proton not as a solid marble, but as a bustling, crowded dance floor filled with tiny, energetic dancers called partons (quarks and gluons). Usually, when two protons smash into each other in a particle collider, we assume only one pair of dancers from each side bumps into each other. This is called "Single Parton Scattering."

However, at very high energies, it's possible that two separate pairs of dancers collide simultaneously in the same crash. This is Double Parton Scattering (DPS). To understand this chaotic dance, physicists need a map showing not just where the dancers are, but also how fast they are moving sideways (transverse momentum) and how they are correlated with each other.

This paper is a numerical study by the CHROMA Collaboration that creates and tests three different ways to draw this map. Here is the breakdown in simple terms:

1. The Problem: The "Pocket Formula" is Too Simple

For a long time, physicists used a "pocket formula" to estimate these double collisions. It was like assuming the dance floor is empty and the dancers are completely independent of each other. You just multiply the probability of one dancer being there by the probability of another.

• The Flaw: In reality, the dancers are crowded. If one dancer is in a specific spot, it changes the odds of where another dancer can be. Also, the "pocket formula" ignores how fast the dancers are moving sideways. The paper argues we need a more detailed map that accounts for these correlations and sideways movements.

2. The Ingredients: The "GS09" Map

The authors start with a pre-existing, high-quality map of the proton called GS09. This map already knows about the "crowdedness" (correlations) of the dancers. However, this map is "collinear," meaning it only tells you where the dancers are moving forward, not how much they are wiggling sideways.

• The Task: They needed to take this forward-moving map and add the "wiggling" (transverse momentum) to it, creating what they call Unintegrated Double Parton Distribution Functions (UDPDFs).

3. The Three Methods: Three Ways to Add the Wiggle

The paper tests three different "recipes" (prescriptions) to add this sideways movement to the map. Think of these as three different chefs trying to add spice to a stew:

• Recipe A: The "Virtuality Ordered" Chef (DVO-MRW)

  • How it works: This chef adds spice based on a strict rule: "The bigger the wiggle, the more the recipe changes." It looks at the history of the dancers to decide how much they wiggle.
  • The Catch: This chef is a bit messy. Sometimes, after adding the spice, the total amount of stew (the total probability) doesn't match the original recipe exactly. It creates a "normalization mismatch."
  • The Fix: The authors created a Matched Version (MDVO-MRW). This is the same chef, but they add a final "taste test" step to adjust the amount of stew so the total volume is perfect, without changing the flavor profile (the shape of the wiggle).

• Recipe B: The "Normalized Kernel" Chef (DMKMRW)

  • How it works: This chef is very precise. They take the original map and attach a pre-made, perfectly measured "wiggle sticker" to every dancer.
  • The Benefit: Because the stickers are pre-measured, the total amount of stew is guaranteed to be correct from the start. No messy adjustments needed.
  • The Difference: Unlike the first chef, this one doesn't let the wiggle change the underlying map of the dancers; it just adds the wiggle on top.

• Recipe C: The "Old School" Chef (Direct LO-MRW)

  • Why they didn't use it: The paper mentions an older method that requires cutting the map into pieces (like a puzzle) to handle different speeds. The authors found this too complicated and clunky for their needs, so they stuck to the two newer, cleaner recipes above.

4. The Findings: What the Maps Showed

The authors ran simulations to see how these three recipes compared. Here is what they found:

• The "Wiggle" Matters: The way you add the sideways movement changes the final picture significantly, especially when the dancers are moving fast or are near the edge of the dance floor (high energy).
• Correlations are Real: The "crowdedness" of the dance floor matters.
  • If you look for two dancers of the same type (e.g., two "up" quarks), the map shows they are less likely to be found together than the simple "pocket formula" predicts. It's like two people of the same size trying to squeeze into a small corner; they push each other away.
  • If you look for a pair of opposites (e.g., a quark and an anti-quark), they are more likely to be found together. It's like a magnet pair sticking together.
• The Recipe Choice Changes the Result:
  • The Normalized Kernel (DMKMRW) recipe keeps the "wiggle" separate from the "crowdedness." The sideways movement looks the same regardless of where the dancers are.
  • The Virtuality Ordered (DVO-MRW) recipe mixes them together. The "wiggle" changes depending on how crowded the area is.
  • Crucially: Even after fixing the "messy chef's" volume issue (the Matched version), the two recipes still produced different shapes for the sideways movement. This means the choice of recipe is a major source of uncertainty in predicting these collisions.

5. The Conclusion

The paper concludes that to accurately predict what happens when protons smash together at high energies, we cannot use the simple "pocket formula." We must use these detailed maps that account for how partons are correlated.

However, there is a catch: Which recipe you use to add the sideways movement matters. The "Normalized Kernel" and "Virtuality Ordered" methods give different results, especially for high-speed collisions. The authors suggest that future experiments need to be careful about which mathematical "recipe" they use, as it could change the final answer.

In short: They built a better, more detailed map of the proton's interior, tested three different ways to draw the "sideways motion" on that map, and found that the choice of drawing method significantly changes the picture, especially in the most energetic parts of the collision.

Technical Summary

Technical Summary: Numerical Study of MRW-Type Unintegrated Double Parton Distribution Functions from Non-Factorized DPDFs

Problem Statement
Double parton scattering (DPS) is a critical probe of multi-parton correlations within hadrons. While the standard phenomenological approach relies on the "pocket formula," which assumes a factorized form for double parton distribution functions (DPDFs) and integrates out transverse momenta, this approximation fails to satisfy the double DGLAP evolution equations and associated momentum and number sum rules. Furthermore, at high energies, initial-state QCD radiation generates significant transverse momenta ($k_t$), necessitating the use of unintegrated double parton distribution functions (UDPDFs) within the $k_t$-factorization framework.

A specific challenge arises in constructing UDPDFs from non-factorized collinear DPDFs. Conventional Leading Order (LO) MRW (Martin-Ryskin-Watt) prescriptions require a piecewise treatment: the homogeneous (independent evolution) and non-homogeneous (perturbative splitting) components of the DPDF must be treated separately, particularly in the region where transverse momenta are below the input scale ($k_t < \mu_0$). This separation is cumbersome for phenomenological applications, especially when utilizing full, non-factorized DPDF inputs like the GS09 set, where the components are not stored separately. Additionally, existing models often struggle with normalization ambiguities and the generation of unphysical tails above the hard scale.

Methodology
The authors present a numerical study constructing UDPDFs directly from non-factorized collinear DPDFs using three MRW-inspired prescriptions that avoid the piecewise separation of homogeneous and non-homogeneous terms:

1. Input and Evolution: The study utilizes the GS09 DPDFs as input, evolved to unequal scales ($\mu_1 \neq \mu_2$) using a custom numerical double DGLAP evolution framework named ChromaPDFEvolver. This framework solves the double DGLAP equations (including the inhomogeneous "feed" term) in $x$-space using a grid in the variable $u = \ln(x/(1-x))$ and a Runge-Kutta-Fehlberg algorithm. The unequal scale evolution is validated against the momentum sum rule.
2. Prescription 1: Double Virtuality Ordered MRW (DVO-MRW): This model applies a virtuality-ordered last-step evolution to the full collinear DPDF. The virtuality scale $K^2 = k_t^2/(1-z)$ is used for the Sudakov form factor and coupling, ensuring the distribution is zero beyond the kinematically allowed virtuality region. It generates transverse momentum dependence dynamically without requiring a separate low-$k_t$ extrapolation.
3. Prescription 2: Normalization-Matched DVO-MRW (MDVO-MRW): To address normalization deviations inherent in the DVO-MRW model, a multiplicative matching factor is introduced. This preserves the virtuality-ordered transverse momentum shape while forcing the integral over $k_t$ to exactly reproduce the input collinear DPDF.
4. Prescription 3: Double Modified KMRW (DMKMRW): Based on the Modified KMRW (MKMRW) approach, this model assigns normalized transverse kernels to each external parton leg. The transverse dependence is factorized from the collinear DPDF, and the model is constructed to be exactly normalization-preserving by design.

The study compares these models across various partonic channels ($gg, ug, uu, u\bar{u}$) and analyzes the ratio of non-factorized to factorized distributions to isolate the impact of longitudinal correlations.

Key Results

• Validation: The ChromaPDFEvolver successfully preserves the momentum sum rule for unequal scale DPDFs to high numerical accuracy, confirming the validity of the input for UDPDF construction.
• Normalization: The DMKMRW model is inherently normalization-preserving. The DVO-MRW model exhibits non-trivial normalization deviations, particularly in the gluon-gluon channel, which are successfully corrected in the MDVO-MRW variant.
• Transverse Momentum Dependence:
  • In the DMKMRW model, the ratio of non-factorized to factorized distributions is independent of $k_t$. The non-factorized effects manifest purely as a longitudinal correlation factor ($D_{ij}/f_i f_j$) that modifies normalization and flavor dependence but not the $k_t$ shape.
  • In the DVO-MRW/MDVO-MRW models, the non-factorized effects are coupled with the transverse momentum. As $k_t$ increases, the virtuality ordering constraint restricts the integration region, sampling the DPDF closer to the kinematic boundary ($x_1+x_2=1$). This leads to a $k_t$-dependent suppression of non-factorized distributions at large $x$ and large $k_t$.
• Channel Dependence:
  • The $uu$ channel shows strong suppression in non-factorized distributions relative to the factorized approximation, driven by valence number sum rules (finding a second valence quark is constrained).
  • The $u\bar{u}$ channel can show enhancement, reflecting quark-antiquark correlations present in the input GS09 DPDFs.
• Model Comparison: Even when both models are normalization-matched (DMKMRW vs. MDVO-MRW), they yield different transverse momentum shapes. The MDVO-MRW distribution is suppressed at large $k_t$ due to the virtuality ordering constraint ($K^2 < \mu^2$), whereas the DMKMRW distribution retains a tail determined by the normalized kernel.

Significance and Claims
The paper claims that the construction of UDPDFs from non-factorized DPDFs is essential for a consistent $k_t$-factorization description of DPS. The authors demonstrate that:

1. Non-factorized correlations are not universal: They cannot be represented by a simple multiplicative factor independent of kinematics; they depend on flavor, longitudinal momentum fractions, and the specific prescription used to generate transverse momenta.
2. Prescription dependence is a source of uncertainty: The choice between a normalized-kernel approach (DMKMRW) and a virtuality-ordered convolution approach (DVO/MDVO-MRW) leads to significant differences in the predicted $k_t$ shapes, particularly at large transverse momenta and near the kinematic boundary.
3. Practical utility: The proposed prescriptions (specifically MDVO-MRW and DMKMRW) provide compact, single-expression models that can be applied directly to full non-factorized DPDFs, avoiding the cumbersome piecewise treatments of conventional LO-MRW constructions.

The authors conclude that while factorized approximations may suffice for small $x$ and moderate $k_t$, the use of non-factorized UDPDFs is critical for processes sensitive to large $x$, large $k_t$, or specific valence/quark-antiquark correlations. They note that a full assessment of the phenomenological impact requires implementing these distributions in process-level DPS calculations, which is left for future work.