Physics Letters B 788 (2019) 380–387
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Structure of parton quasi-distributions and their moments
A.V. Radyushkin
a,b,∗
a
Physics Department, Old Dominion University, Norfolk, VA 23529, USA
b
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
25 July 2018
Received
in revised form 22 November 2018
Accepted
26 November 2018
Available
online 28 November 2018
Editor:
B. Grinstein
We discuss the structure of the parton quasi-distributions (quasi-PDFs) Q (y, P
3
) outside the “canonical”
−1 ≤ y ≤ 1 support region of the usual parton distribution functions (PDFs). Writing the y
n
moments of
Q (y, P
3
) in terms of the combined x
n−2l
k
2l
⊥
-moments of the transverse momentum distribution (TMD)
F (x, k
2
⊥
), we establish a connection between the large-|y| behavior of Q (y, P
3
) and large-k
2
⊥
behavior of
F (x, k
2
⊥
). In particular, we show that the 1/k
2
⊥
hard tail of TMDs in QCD results in a slowly decreasing
∼ 1/|y| behavior of quasi-PDFs for large |y| that produces infinite y
n
moments of Q (y, P
3
). We also
relate the ∼ 1/|y| terms with the ln z
2
3
-singularities of the Ioffe-time pseudo-distributions M(ν, z
2
3
).
Converting the operator product expansion for M(ν, z
2
3
) into a matching relation between the quasi-
PDF
Q (y, P
3
) and the light-cone PDF f (x, μ
2
), we demonstrate that there is no contradiction between
the infinite values of the y
n
moments of Q (y, P
3
) and finite values of the x
n
moments of f (x, μ
2
).
© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
In the original Feynman approach [1], the parton distribution
functions (PDFs) f (x) were introduced as the infinite momentum
P
3
→∞limit of distributions in the longitudinal k
3
= yP
3
mo-
mentum
of partons. These distributions basically coincide with the
quasi-PDFs Q (y, P
3
) introduced more recently by X. Ji [2].
As
is well-known, “x” of the parton model corresponds to the
ratio x = k
+
/P
+
of the light-cone-plus components of the parton
and hadron momenta, rather than the ratio y = k
3
/P
3
of their
third Cartesian components. However, in the P
3
→∞limit, the
difference between y and x disappears.
In
the parton model, f (x)’s were treated as k
⊥
-integrals of
more detailed f (x, k
⊥
) distributions that involve also the trans-
verse
momentum k
⊥
. From the start, it was understood by Feyn-
man
that the P
3
→∞ limit exists only if f (x, k
⊥
) rapidly de-
creases
with k
⊥
, so that the integral over k
⊥
does not diverge. This
happens, in particular, in the theories/models with transverse mo-
mentum
cut-off k
⊥
, e.g., in super-renormalizable models, but
not in QED and other renormalizable field theories.
One
may ask two natural questions. First, why the shape of
Q (y, P
3
) for a finite P
3
differs from that of f (x)? Second, how
does the shape of Q (y, P
3
) convert into that of f (x) when
*
Correspondence to: Thomas Jefferson National Accelerator Facility, Newport
News, VA 23606, USA.
E-mail
address: radyush@jlab.org.
P
3
→∞? A qualitative answer is that the parton’s longitudinal
momentum k
3
= yP
3
comes from two sources: from the motion of
the hadron as a whole (xP
3
) and from a Fermi motion of quarks in-
side
the hadron, so that (y − x)P
3
∼ 1/R
hadr
. As P
3
→∞,the role
of the y − x ∼ 1/P
3
R
hadr
fraction decreases and Q ( y, P
3
) → f (x).
In
this picture, the ( y − x)P
3
part has the same physical ori-
gin
as the parton’s transverse momentum. Hence, one should be
able to relate quasi-PDFs to the transverse momentum distribu-
tions
(TMDs) and quantify the difference between Q ( y, P
3
) and
f (x) in terms of TMDs f (x, k
⊥
).
An
important point is that the components of k
⊥
may take any
values from −∞ to ∞, even when the distribution in k
⊥
is mostly
restricted to a limited range, like in a Gaussian e
−k
2
⊥
/
2
. Similarly,
the (y − x)P
3
part of the k
3
-distribution may take any values.
As a result, Q (y, P
3
) formally has the −∞ < y < ∞ support re-
gion,
though possibly with a rapid decrease (say, like e
−y
2
P
2
3
/
2
)
for large y.
In
other words, for a finite P
3
, there is no requirement that
the fraction y is smaller than 1 or positive. Even in a fast-moving
hadron, there is some probability that a parton moves in the op-
posite
direction, and hence, that some other parton has the mo-
mentum
k
3
larger than P
3
. Still, with increasing P
3
, the chances
for fractions outside the [0, 1] segment decrease rapidly, reflecting
the large-k
⊥
dependence of the relevant TMD f (x, k
⊥
).
When
Q (y, P
3
) ∼ e
−y
2
P
2
3
/
2
, one may consider y
n
moments of
quasi-PDFs Q (y, P
3
) calculated over the whole −∞ < y < ∞ axis
https://doi.org/10.1016/j.physletb.2018.11.047
0370-2693/
© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
下载后可阅读完整内容，剩余7页未读，立即下载
weixin_38703277
- 粉丝: 6
- 资源: 915
我的内容管理
展开
最新资源
- JavaScript DOM事件处理实战示例
- 全新JDK 1.8.122版本安装包下载指南
- Python实现《点燃你温暖我》爱心代码指南
- 创新后轮驱动技术的电动三轮车介绍
- GPT系列：AI算法模型发展的终极方向？
- 3dsmax批量渲染技巧与VR5插件兼容性
- 3DsMAX破碎效果插件：打造逼真碎片动画
- 掌握最简GPT模型：Andrej Karpathy带你走进AI新时代
- 深入解析XGBOOST在回归预测中的应用
- 深度解析机器学习：原理、算法与应用
- 360智脑企业内测开启，探索人工智能新场景应用
- 3dsmax墙砖地砖插件应用与特性解析
- 微软GPT-4助力大模型指令微调与性能提升
- OpenSARUrban-1200：平衡类别数据集助力算法评估
- SQLAlchemy 1.4.39 版本特性分析与应用
- 高颜值简约个人简历模版分享
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
