阈值恢复的Parton分布：改善QCD计算与NNPDF3.0数据的兼容性

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本文主要探讨了具有阈值恢复的Parton分布函数（PDFs）在量子色动力学（QCD）计算中的应用。Parton分布是描述强子内部基本粒子（如夸克和胶子）的概率密度，它们在高能物理实验中起着关键作用，尤其是在理解和预测深度非弹性散射、Drell-Yan过程以及顶夸克对生产等现象中的硬散射截面。研究者构建了一组新的PDF集合，该集合在传统的Next-to-Leading Order (NLO) 和 Next-to-Next-to-Leading Order (NNLO) 固定阶数计算基础上，增加了软胶子（阈值）效应的Next-to-Next-to-Leading Logarithmic (NNLL) 和 Next-to-Leading Logarithmic (NLL) 求和。这样的改进旨在提供更精确的近似，特别是在接近物理过程的阈值区域，即粒子对的能量接近产生新粒子对所需的最小能量时，这种效应尤为显著。通过与具有阈值的QCD计算相结合，这些PDFs能够修正固定阶次计算中因阈值效应引起的增强，并在一定程度上使恢复的强子横截面更接近实际的物理结果。然而，作者强调了在进行恢复计算时，必须始终使用具有阈值恢复的PDFs，以确保一致性。这意味着在理论预测中，不仅矩阵元素需要阈值恢复，连输入的PDF也应该适应这一处理方式，以获得更准确的预测误差范围。论文JHEP09(2015)191由Springer出版，于2015年9月发布，涵盖了来自深度非弹性散射、Drell-Yan过程和顶夸克对生产数据的NNPDF3.0分析基础。这是一项重要的成果，因为它提供了对阈值效应在PDFs和强相互作用计算中的理解，有助于提高未来在高能物理实验中对标准模型和新物理搜索的精度。本文的核心内容包括阈值恢复技术的应用、其对PDFs的影响，以及如何通过这种方法改善强子横截面的理论预测。这对于理解粒子物理实验中的细节和误差来源具有重要意义，同时也为未来的理论发展和实验设计提供了宝贵的知识基础。

JHEP09(2015)191

we adopt the Minimal Prescription, but we stress that from a practical point of view, dif-

ferences between these prescriptions become only relevant at extremely large values x [120],

a region where no experimental data is available.

2.2 Numerical implementation

In this section we discuss the numerical implementation of the N-soft threshold resum-

mation described above. For the PDF ﬁts performed in this work, the processes that

we are interested in are DIS (both neutral and charged currents), lepton-pair invariant

mass and rapidity distribution for Drell-Yan production, and inclusive top pair production

cross section. For DIS and Drell-Yan, we use a new version of the public code ResHiggs,

written originally [36] to perform threshold resummation of Higgs inclusive cross section,

including several improvements with respect to standard N-soft resummation, and later

extended [37] to also perform (improved) resummation in the Soft-Collinear Eﬀective The-

ory formalism. Because of the inclusion of additional processes, the new version of this

code changes name from ResHiggs to TROLL, standing for TROLL Resums Only Large-x

Logarithms, publicly available at the webpage [121]. To give continuity with the original

code ResHiggs, the ﬁrst version of TROLL is v3.0. For top pair production we use the

public code Top++ [7].

The code TROLL is designed to provide only the contribution of the resummation,

while the ﬁxed-order calculation is obtained from a separate code (in our case, the same

FKgenerator code used for the NNPDF3.0 ﬁts). More speciﬁcally, the output of TROLL is

∆

, deﬁned as the diﬀerence between a resummed K-factor at N

LO+N

LL and a

ﬁxed-order K-factor at N

LO, such that

LO+N

= σ

+ σ

× ∆

, (2.11)

where all the cross sections appearing in the above equation are evaluated with a common

LO+N

LL PDF set. Internally, ∆

is computed by subtracting oﬀ the expansion

of the resummed coeﬃcient up to O(α

) from the coeﬃcient itself, multiplying this by

the parton luminosity, computing the inverse Mellin transform and ﬁnally dividing by the

parton luminosity.

There are several advantages in using the ∆K-factors deﬁned in eq. (2.11): the ﬁxed-

order normalisation is irrelevant, the K-factor is much less sensitive to the input PDFs than

the cross section itself, and ﬁnally, since the resummed contribution has the same kinematic

structure as the Born cross section (soft radiation does not change the kinematics), the

eﬀect of phase space constraints like kinematic cuts are correctly taken into account if they

are applied to the LO cross section in eq. (2.11).

We note that in DIS, Target Mass Corrections (TMCs) at next-to-leading twist are

included in the resummation according to the same prescription used in the NNPDF ﬁtting

code [88], which amounts to multiplying the Mellin transforms of the partonic coeﬃcient

functions by an N-dependent factor. No TMCs are included for the ﬁxed-target DY data.

In ﬁgure 1 we show the ∆K-factors for the neutral current DIS structure function

(x, Q), as a function of x, for Q = 2 GeV and Q = 30 GeV. The plot on the left

– 7 –

JHEP09(2015)191

-0.1

-0.05

0.05

0.1

0.15

0.04 0.3 0.8 0.1

K-factor

Neutral current DIS

solid: including TMC

Q= 2GeV, NLL

Q=30GeV, NLL

-0.1

-0.05

0.05

0.1

0.15

0.04 0.3 0.8 0.1

K-factor

Neutral current DIS

solid: including TMC

Q= 2GeV, NNLL

Q=30GeV, NNLL

Figure 1. ∆K-factors eq. (2.11) for the neutral current DIS structure function F

(x, Q), as a

function of x, for Q = 2 GeV and Q = 30 GeV. The plot on the left corresponds to j = 1, k = 1 in

eq. (2.11), i.e. NLO and NLL, while the one on the right to j = 2, k = 2, i.e. NNLO and NNLL.

The eﬀect of adding TMCs is shown as a thin solid line.

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Figure 2. Same as ﬁgure 1 for the NLL (left plot) and NNLL (right plot) resummation of neutral-

current Drell-Yan invariant mass distribution at the Tevatron and at the LHC.

corresponds to j = 1, k = 1 in eq. (2.11), i.e. NLL to be matched to NLO, while the

one on the right to j = 2, k = 2, i.e. NNLL to be matched to NNLO. We note that the

resummation enhances the cross section at large x, while it gives a very small contribution

at small x, as it should. We also note a dip in the region of intermediate x, which is also

present in ﬁxed-order calculations [122].

TMC eﬀects are also shown as light shadows to the actual curves: as expected, at large

scales they are negligible, while at smaller scales their eﬀect is non-negligible, in particular

at large x, where they reduce the eﬀect of the resummation. Note that in the deﬁnition of

∆K we use the fact that the same TMCs are already included in the LO cross section, so

much of their eﬀect cancels out.

In ﬁgure 2 we show the corresponding ∆K-factors this time for Drell-Yan invariant

mass distributions, for LHC at

√

s = 7 TeV and for the Tevatron at

√

s = 1.96 TeV. In

ﬁgure 3 we also present the results for the lepton pair rapidity distribution in DY processes

– 8 –

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