PDFs与小x恢复：HELL 2.0的理论精度提升

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"这篇论文详细介绍了在parton分布函数（PDFs）的全球拟合中，如何处理小x（x接近0）区域的logarithmic enhancements问题，这是由HERA对撞机的数据所驱动的。文章提出了名为HELL 2.0的新方法，该方法包含了small-x求和，以更准确地描述深度非弹性散射（DIS）数据。" 在高能物理中，parton分布函数描述了在高能量碰撞中，如质子或核子内部的夸克和胶子的分布情况。PDFs对于理解和预测粒子对撞实验的结果至关重要。当x值非常小时，即质子内部的相对动量份额非常低，小x增强的logarithmic corrections变得显著。这些效应不能简单地通过固定顺序的量子色动力学（QCD）计算来处理，因此需要进行small-x resummation。本文的工作重点在于发展了一种新的技术，以包含无质量结构函数的小x恢复，同时考虑了大量最终状态，如迷克夸克的生成，以及质量共线对数的恢复。作者还引入了一种可变风味数的方法，以在小x方案中保持一致性。这些改进使得PDF的演化能够在重口味阈值下进行x倍恢复，并且恢复后的结果具有次对数（NLL）精度，这是与次次对数（NNLO）订单相匹配的第一次尝试。此外，为了提高理论精度，研究者考虑了运行耦合的全阶处理，这进一步改进了他们之前的工作。所有这些进展都将集成到HELL 2.0的新版本中，使得在NNLO + NLL的理论精度下，对DIS数据中的PDF进行拟合成为可能。这不仅是对PDF精确确定的重要步骤，也将对大型强子对撞机（LHC）和其他未来高能物理实验的精确现象学预测产生深远影响。这篇论文发表在JHEP期刊上，属于开放访问，可以供广泛的科学社区查阅和使用。作者们来自罗马、热那亚和米兰的大学和研究所，展示了他们在高能物理领域的专业性和创新性。通过他们的工作，小x区域的理论描述得到了显著提升，对于理解高能碰撞过程中的基本物理现象具有重要意义。

JHEP12(2017)117

2.1.2 Computation of matching functions

In the derivation above the matching functions K

]

(or ∆K

) are assumed to be given

as an input to the computation of coeﬃcient functions in the n

+ 1 scheme. However, the

very same derivation also allows us to construct the matching functions themselves. This

is true in general (see e.g. ref. [75]), but we focus on the small-x limit for simplicity.

The key observation is that, after scheme change, the coeﬃcient functions in the n

scheme must not contain anymore the collinear logarithms associated to the (n

+ 1)-th

ﬂavour. This is possible only if the matching functions subtract such collinear logarithms

from the massive coeﬃcients ∆ec

a,i

), such that the massless limit m

→ 0 of the

coeﬃcient functions in the new scheme ∆c

a,i

) is ﬁnite. If we further require that

the n

and n

+ 1 scheme are both of the same type, e.g. MS-like, we also need to impose

that the massless limit of the massive coeﬃcient ∆c

a,i

) is just what we would have

computed if the (n

+ 1)-th ﬂavour were massless:

lim

Qm

∆c

a,i

) = ∆c

a,i

(0), i = g, q. (2.25)

(Note that, at the logarithmic accuracy we are interested in, ∆c

a,i

(0) are the same in

the n

and in the n

+ 1 schemes.

) The massless limit eq. (2.25) ensures that in the

+1 scheme the collinear logarithms are properly factorized into the PDFs and resummed

through DGLAP. It also ﬁxes the “constant” (i.e., non mass dependent) part of the function

∆c

a,i

). It does not tell us anything about the power corrections in m

/Q in

the n

+ 1 scheme, which have to be determined by the matching procedure.

The results eq. (2.24) show that, since F

has no collinear singularities at NLL, the

massive coeﬃcient in the n

+ 1 scheme smoothly approaches the massless one at large Q,

without any scheme change to be applied. On the other hand, in the case of F

, eq. (2.23),

which contains collinear singularities in the massless limit, the scheme-change eﬀectively

subtracts the matching function, and the diﬀerence will smoothly tend to the massless

coeﬃcient for Q  m

We can exploit the last consideration to derive the desired resummed expressions for

the matching functions ∆K

, i = g, q. Indeed, eq. (2.23) can be inverted to give ∆K

terms of the massive coeﬃcient functions ∆ec

2,i

(m), which are known, and of the coeﬃcient

functions ∆c

2,i

(m) in the n

+ 1 scheme, the massless limits of which, ∆c

2,i

(0), are known.

We can therefore consider the massless limit m

→ 0 of this expression, keeping in mind

that ∆K

and ∆ec

2,i

(m) are separately logarithmically divergent, and therefore the massless

limit has to be intended as setting to zero power-behaving contributions while keeping the

logarithms ﬁnite. Assuming, or better choosing, that ∆K

contain only logarithms of the

mass m

or mass independent terms, we then ﬁnd

∆K

) = lim

Qm

∆ec

2,i

) − ∆c

2,i

(0), i = g, q, (2.26)

In the actual resummed expressions we compute, there is a non-zero n

dependence in the coeﬃcient

functions ∆c

a,i

(0), which is however subleading and can therefore be ignored.

– 11 –

JHEP12(2017)117

where the limit has to be intended as a formal expression as described above. Note that

including power-behaved mass dependent terms is possible and would simply change the

form of ∆c

2,i

(m) through eq. (2.23), as well as the PDFs in the n

+ 1 scheme according

to eq. (2.12). However, this “factorization” of power behaved contributions is beyond the

control of the collinear factorization framework, so the simplest choice eq. (2.26) is perfectly

acceptable and therefore universally adopted.

The deﬁnition eq. (2.26) also shows that matching functions at NLL satisfy color-

charge relations. Indeed, it is known that both massless and massive coeﬃcient functions

in the quark and gluon channels are related at NLL by

∆c

a,q

(0) =

∆c

a,g

(0), ∆ec

a,q

(m) =

∆ec

a,g

(m). (2.27)

From eq. (2.26) it then immediately follows

∆K

(m) =

∆K

(m). (2.28)

Together, eqs. (2.27) and (2.28) imply through eqs. (2.23) and (2.24) a color-charge relation

∆c

a,q

(m) =

∆c

a,g

(m). (2.29)

for the coeﬃcient functions in the n

+ 1 scheme.

2.1.3 Generalization to neutral and charged currents

The previous results have been derived in the case of photon-exchange DIS. We now com-

ment on their generalization to the more general case of neutral current, where also the Z

boson contributes, and the case of charged current, where the exchanged boson is a W .

Including a Z boson in the discussion is rather trivial. What changes is just the

coupling to the quark, which is diﬀerent to that of the photon both in the Z-only exchange

and in the photon-Z interference. In Z-only exchange both vector and axial couplings

contribute, which we denote generically as g

and g

. If we consider only massless quarks,

the sum of the squares of each coupling, g

+ g

, factorizes. The only subtlety regarding

Z exchange is that in the massive case there is a contribution which is not proportional to

the sum of the squares of the vector and axial couplings. However, in this case, one can

still factor out g

+ g

, but a contribution proportional to g

/(g

+ g

) appears, which is

not present in the photon (or photon-Z interference) case. This term is not problematic,

as it vanishes in the limit of massless quarks. Therefore, the whole construction of the

previous sections, and in particular section 2.1.1, remains unchanged, provided the photon

couplings e

are replaced with the more general coupling (g

+ g

)

for each quark q

The charged-current case is less trivial. In this case, the quark ﬂavour changes after

interacting with the W and this results in two additional complications. Firstly, the quark

These colour-charge relations are strictly speaking only valid when the resummed contributions refer to

the pure-singlet. If they refer to the singlet, in the massless case there is a ﬁxed-order contribution which

needs to be subtracted [61]. Alternatively, we can say that these relations are valid for ∆

a,i

with k ≥ 1.

– 12 –

JHEP12(2017)117

mass changes. In fact, for all practical applications, one of the quarks interacting with the

W can be considered as massless. Indeed, ignoring the top quark which is too heavy to give

a contribution at typical DIS energies, the only two massive quarks are the charm and the

bottom, but their interaction is suppressed by the V

CKM entry and can thus be neglected.

Hence, the only remaining combinations involve either a massless and a massive quarks, or

two massless quarks. The second complication arises due to the fact that the ﬁnal state,

composed by a quark q and an anti-quark ¯q

, is not self conjugate (as it is in the neutral-

current case, where the ﬁnal state contains q ¯q). This means that the non-singlet coeﬃcient

in eq. (2.13) is diﬀerent for q and ¯q, depending on the process. In particular, this implies

that when using eq. (2.14) to derive eq. (2.18), only either the heavy quark q

or the

heavy anti-quark ¯q

contributes: thus, the collinear subtraction, where present, contains

only half of the matching function. Additionally, for the computation of the parity-violating

structure function (which contains a singlet contribution in the massive charged-current

case), the non-singlet LO coeﬃcient has opposite sign depending on whether the initial

state parton is a quark or an anti-quark, speciﬁcally C

CC,NS,(0)

3,q

= 1 and C

CC,NS,(0)

3,¯q

= −1.

Putting everything together, we have that the analogous of eqs. (2.23) and (2.24) are

∆c

2,i

) = ∆ec

2,i

) − ∆K

)/2, i = g, q, (2.30a)

∆c

L,i

) = ∆ec

L,i

), (2.30b)

∆c

3,i

) =







∆ec

3,i

) − ∆K

)/2 if ﬁnal state is q

∆ec

3,i

) + ∆K

)/2 if ﬁnal state is Q¯q,

(2.30c)

where we are denoting with Q the heavy massive quark and with q the companion massless

quark appearing in the ﬁnal state. The massless limit of the latter is just zero, ∆c

3,i

(0) = 0,

since in the massless limit F

is non-singlet and therefore it does not contain logarithmic

enhancement at small x (see appendix A for further details).

2.2 Small-x resummation of coeﬃcient functions and matching functions

We are now ready to discuss the actual resummed expressions for coeﬃcient functions both

in the massless and massive case. Additionally, using eq. (2.26), we also determine the all-

order behaviour of the matching functions. The all-order behaviour of partonic coeﬃcient

functions is obtained using the k

-factorization theorem. In this framework one computes

the gluon-initiated contribution to the process of interest, keeping the incoming gluon oﬀ

its mass-shell. To the best of our knowledge, only the oﬀ-shell coeﬃcient functions that

are necessary to perform the resummation of the photon-induced DIS structure functions

have been presented in the literature, both in the case of massless and massive quarks, see

e.g. [44, 45, 90]. However, in this study we want to resum all DIS coeﬃcient functions, both

the neutral-current and charged-current contributions. Therefore, we perform a general

Note that the opposite sign in these two contribution makes the collinear subtractions in the fully

massless case ineﬀective. Indeed, the collinear singularities in this case cancel automatically. In fact, after

cancellation, the non-singular part vanishes, thus making the massless singlet contribution to F

zero (which

remains true at higher orders, since the underlying reason is the antisymmetry of the F

contribution for

the exchange of the two quark masses). The same mechanism holds in the neutral-current massive case.

– 13 –

JHEP12(2017)117

calculations of DIS oﬀ-shell coeﬃcient functions considering the coupling to the electro-

weak bosons and, where relevant, the interference with the photon-induced contributions.

The calculation is detailed in appendix A, where we collect all the relevant results for the

oﬀ-shell DIS coeﬃcient functions. These results have been implemented in the public code

HELL, version 2.0, and are also accessible through the public code APFEL [91], to which

HELL has been interfaced, which directly constructs resummed structure functions.

2.2.1 Resummed coeﬃcient functions

The resummation of the massless coeﬃcient functions was originally performed to NLL

in ref. [49]. Following ref. [38], in our recent work [61] we have also included formally

subleading, but important, contributions such as the ones related to the running of the

strong coupling. In our approach, the partonic massless coeﬃcient function C

(N, ξ, 0, 0),

which is calculated with an incoming oﬀ-shell gluon (see appendix A for details about our

notation), is convoluted with an evolution operator U(N, ξ),

∆

2,g

(0) = −

dξ

(0, ξ, 0, 0) U (N, ξ) −

, (2.31a)

∆

L,g

(0) = −

dξ

(0, ξ, 0, 0) U (N, ξ), (2.31b)

where

U(N, ξ) = exp



dζ



N, α

(ζQ

)





, (2.32)

and γ

is the small-x resummed anomalous dimension. Note that the above expressions

hold in a factorization scheme denoted Q

MS [47, 49, 92, 93], which diﬀers from MS at

relative O(α

). In the context of small-x resummation this scheme is preferred because it

leads to more stable results [38]. Furthermore, since the explicit N dependence of the oﬀ-

shell partonic coeﬃcient function is subleading, we ﬁnd advantageous to work at NLL with

its N = 0 moment. In the resummed expression of the F

contribution, the subtraction

term U

appears. Its role is to cancel the collinear singularity of C

(0, ξ, 0, 0). Its expression

reads [61]

dξ

(N, α

(ξQ

)) θ(1 − ξ) U(N, ξ) (2.33)

where γ

is the resummed qg anomalous dimension. All ξ integrals extend to ∞ and start

from the position of the Landau pole, ξ

= exp

−1

. In ξ = ξ

the evolution function

U is supposed to vanish (this was e.g. a condition for neglecting a boundary term when

integrating by parts in ref. [61]). However, due to subleading contributions, this is not

always true in our practical construction. While the induced eﬀect is subleading, this fact

is undesirable: we discuss in appendix B.1 how we now deal with this issue.

We can apply the same procedure to the case of a heavy (non-active) ﬂavour. We

start considering neutral currents. Note that in this case the mass of the quark acts as a

regulator and no subtraction term U

appears,

∆

a,g

(m) = −

dξ

(0, ξ, ξ

, ξ

) U(N, ξ), a = 2, L, (2.34)

– 14 –

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