[Paper Review] Are bottom PDFs needed at the LHC?

Maria Ubiali|arXiv (Cornell University)|Jul 21, 2014

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This paper investigates whether bottom quark parton distribution functions (PDFs) are necessary in LHC phenomenology by comparing the 4-flavor and 5-flavor schemes for bottom-quark-initiated processes. It proposes a physically motivated factorization scale based on kinematic and dynamical considerations, showing that matching predictions between schemes at this scale yields strong agreement, thus justifying the use of both schemes complementarily in precision LHC physics.

ABSTRACT

Processes involving bottom quarks play a crucial role in the LHC phenomenology, from flavour physics to Higgs characterisation and as a window to new physics, appearing both as signals and irreducible background in BSM searches. These processes can be described in QCD either in a 4-flavor or 5-flavor scheme. In the former, $b$ quarks appear only in the final state and are considered massive. In 5-flavor schemes, calculations include $b$ quarks in the initial state. Possibly large logarithms originating from the collinear splitting of gluons into bottom pairs are resummed into the $b$ parton distribution function (PDF). In this contribution, I describe a simple method to assess the size of the logarithms in processes initiated by bottom quarks and show how a substantial and justified agreement between calculations in the two schemes can be achieved. As a consequence both calculations can be used in different context. To conclude, an overview of the current studies aiming to generalise the current appraisal is given and some preliminary results are discussed.

Motivation & Objective

• To determine whether bottom PDFs are necessary in LHC phenomenology for processes initiated by bottom quarks.
• To resolve the long-standing discrepancy between 4-flavor and 5-flavor scheme predictions, particularly in Higgs and top quark production.
• To establish a physically motivated criterion for choosing the factorization scale in the 5-flavor scheme that reconciles results with the 4-flavor scheme.
• To assess the size and impact of initial-state collinear logarithms in phenomenologically relevant LHC processes.
• To enable reliable, high-order QCD predictions by unifying insights from both schemes for future LHC analyses.

Proposed method

• Derives a factorization scale in the 5-flavor scheme based on the kinematic suppression of collinear logarithms, using a weighting function derived from the partonic phase space.
• Applies a dynamical criterion based on the evolution of the bottom PDF, truncating DGLAP evolution at NLO to assess the impact of resummed logarithms.
• Compares total cross sections in the 4-flavor and 5-flavor schemes at the derived scale, showing improved agreement without ad hoc matching.
• Uses analytic expressions for leading-order cross sections in the 4-flavor scheme, particularly in the collinear limit, to study the distribution of logarithmic factors.
• Extends the analysis to processes with two initial-state bottom quarks (e.g., $pp \to Hbb$, $pp \to Zbb$), analyzing the suppression of logarithmic scales by phase space factors.
• Proposes a generalization of the method to differential distributions and final-state collinear logarithms via time-like DGLAP evolution, enabling a unified treatment of initial- and final-state resummation.

Experimental results

Research questions

• RQ1 What is the typical size of initial-state collinear logarithms $\log(\mathcal{Q}^2/m_b^2)$ in LHC processes involving bottom quarks?
• RQ2 By what physical criterion can the factorization scale in the 5-flavor scheme be consistently chosen to match 4-flavor scheme predictions?
• RQ3 Why does the discrepancy between 4- and 5-flavor schemes decrease at the LHC compared to the Tevatron, despite higher energy?
• RQ4 How do kinematic phase space factors suppress the effective scale of collinear logarithms in $b$-initiated processes?
• RQ5 To what extent can 4-flavor scheme calculations with massive $b$ quarks provide accurate predictions when compared to 5-flavor schemes with resummed logarithms?

Key findings

• The resummation of initial-state collinear logarithms $\log(\mu_f^2/m_b^2)$ via DGLAP evolution in the 5-flavor scheme has a small effect at NLO, especially at small $x$, justifying the use of truncated expansions.
• A physically motivated factorization scale for the 5-flavor scheme is derived as $m_t/4$ for single top production and $m_W/3$ for $Wb$ associated production, matching phenomenological choices with theoretical justification.
• At this scale, the total cross sections in the 4-flavor and 5-flavor schemes agree within expected uncertainties, resolving long-standing discrepancies.
• For $pp \to Hbb$ and $pp \to Zbb$, the logarithmic factors are suppressed by phase space, reducing the effective scale of $\log(\mathcal{Q}^2/m_H^2)$ or $\log(\mathcal{Q}^2/m_Z^2)$, which explains the mild impact of resummation.
• The improved 5-flavor scheme with the derived scale yields a reliable NLO QCD prediction for heavy charged Higgs boson production, suitable for LHC Run 2 searches.
• The study establishes that both 4-flavor and 5-flavor schemes provide complementary and reliable information, with 4-flavor schemes better suited for exclusive observables and 5-flavor schemes for inclusive rates at higher orders.