删除或更新信息,请邮件至freekaoyan#163.com(#换成@)

Analytic continuation and reciprocity relation for collinear splitting in QCD

本站小编 Free考研考试/2022-01-01

Hao Chen ,
, Tong-Zhi Yang ,
, Hua-Xing Zhu ,
, Yu-Jiao Zhu ,
,
Corresponding author: Hao Chen, chenhao201224@zju.edu.cn
Corresponding author: Tong-Zhi Yang, yangtz@zju.edu.cn
Corresponding author: Hua-Xing Zhu, zhuhx@zju.edu.cn
Corresponding author: Yu-Jiao Zhu, zhuyujiao@zju.edu.cn
Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
Received Date:2020-08-07
Available Online:2021-04-15
Abstract:It is well-known that direct analytic continuation of the DGLAP evolution kernel (splitting functions) from space-like to time-like kinematics breaks down at three loops. We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta. However, splitting functions can be constructed from the squares of (generalized) splitting amplitudes. We establish the rules of analytic continuation for splitting amplitudes, and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions. In this way we derive the time-like splitting functions at three loops without ambiguity. We also propose a reciprocity relation for singlet splitting functions, and provide non-trivial evidence that it holds in QCD at least through three loops.

HTML

--> --> -->
I.INTRODUCTION
Parton distribution functions (PDFs) and fragmentation functions (FFs) provide essential input for accurate determination of various quantities of QCD and the Standard Model [1-3] within the framework of QCD factorization [4]. While PDFs and FFs are intrinsically non-perturbative objects, their scale evolution obeys the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [5-7]. The corresponding evolution kernels are space-like ($ q^2 < 0 $, Fig. 1(a)) splitting functions for PDFs and time-like ($ q'^2 > 0 $, Fig. 1(b)) splitting functions for FFs, both of which can be calculated in QCD perturbation theory. Determining the splitting functions to higher orders is one of the most important tasks of perturbative QCD.
Figure1. (color online) Typical processes used for the determination of PDFs (a) and FFs (b).

Space-like splitting functions were obtained to next-to-next-to-leading order (NNLO) some time ago [8,9], and have more recently been obtained to N3LO for non-singlet functions [10]. However, knowledge for time-like splitting functions is less precise. Direct calculation of time-like splitting functions has been done at NLO in Ref. [11]. At NNLO and beyond, results by direct calculation are not yet available (see Refs. [12-15] for recent progress). However, it has long been noted that space-like deep inelastic scattering (DIS) and $ e^+e^- $ annihilation are kinematically related [16,17]. The easiest way to see this is from the definition of the Bjorken variable $ x_B $ in DIS and the Feynman variable $ x_F $ in $ e^+e^- $,
$x_B = \frac{-q^2}{2 P \cdot q} \,, \qquad x_F = \frac{2 P' \cdot q'}{q'^2} \,,$
(1)
where $ P $ is the incoming hadron momentum in DIS, $ P' $ is the detected hadron momentum in $ e^+e^- $, and $ q $ and $ q' $ are the space-like and time-like momentum transfer, respectively. After crossing, $ P = - P' $, $ q = q' $, one finds the analytic continuation relation $ x_B = 1/x_F $. However, beyond LO, the analytic continuation relation cannot be applied directly to the splitting functions, but to the appropriate bare quantities [11,18]. Analytic continuation of exclusive amplitudes has also been understood at NLO accuracy [19]. Further subtleties arise at NNLO, where additional momentum sum rules, supersymmetry relations, and reciprocity considerations at large $ x $ [20] are needed in order to obtain NNLO non-singlet and singlet time-like splitting functions [21-23]. However, as has been explicitly pointed out in Ref. [23], the third-order corrections to off-diagonal quark-gluon splitting, $ P_{qg}^{{\rm{T}},(2)} $, have only been determined up to an uncertainty proportional to the QCD beta function. Fixing this remaining uncertainty is not only crucial for achieving complete NNLO analysis of parton-to-hadron fragmentation, but is also important for precision jet substructure studies, see e.g. Refs. [24-31].
In this Letter we study the analytic continuation of splitting functions using soft-collinear effective theory [32-35]. We point out that splitting functions, both space-like and time-like, can be extracted from bare transverse-momentum-dependent (TMD) distributions. We identify the origin of the breakdown of direct analytic continuation for splitting functions and TMD distributions, as they are computed from the squares of splitting amplitudes, and are therefore not analytic. Nevertheless, we identify certain holomorphic and anti-holomorphic contributions to TMD distributions, for which a correct rule of analytic continuation can be established. We use this to obtain time-like splitting functions at NNLO from the space-like ones. Our results are in full agreement with those obtained in Refs. [21-23], except for a minor discrepancy in $ P_{qg}^{{\rm{T}},(2)} $. Finally, we propose an all-order generalization of the Gribov-Lipatov reciprocity relation [36] for singlet splitting functions in QCD. Using the time-like splitting functions obtained in this work, we verify this relation to NNLO, where the discrepancy in $ P_{qg}^{{\rm{T}},(2)} $ mentioned above plays an important role.
II.SPLITTING FUNCTIONS FROM TMD DISTRIBUTIONS
TMD distributions are central ingredients in the TMD factorization approach to hard scattering [37-48]. In SCET, they can be conveniently defined as matrix elements of collinear fields integrated over light-cone coordinates. Since for the purpose of analytic continuation, there is no intrinsic difference between quark and gluon TMD distributions, we shall focus on quark TMD distributions in the discussion below. The operator definition for quark TMD PDFs is given by
$\begin{aligned}[b] {\cal{B}}_{q/N}(x_B,b_\perp) =& \sum\limits\int\limits_{X_n} \int \frac{{\rm d}b^-}{2\pi} \, {\rm e}^{-{\rm i} x_B b^- P^+} \\ &\cdot \langle N(P) | \bar{\chi}_n(0,b^-,b_\perp) | X_n \rangle \frac{\not{\bar{n}}}{2} \langle X_n| \chi_n(0) | N(P) \rangle \,, \end{aligned}$
(2)
where $ N(P) $ is a hadron state with momentum $P^\mu = (\bar{n} \cdotP) n^\mu/2 = P^+ n^\mu/2$, with $ n^\mu = (1, 0, 0, 1) $ and $\bar{n}^\mu = (1, 0, 0, -1)$. $ \chi_n(x) = W^\dagger_n(x) \xi_n(x) $ is the gauge-invariant collinear quark field [49], and
$W^\dagger_n(x) = {\cal{P}} \exp{\left({\rm i} g_s \int_0^{\infty} {\rm d}s \, \bar{n} \cdot{{A}}_n(x + s \bar{n} ) {\rm e}^{-\varepsilon s}\right)} $
(3)
is the path-ordered $ n $-collinear Wilson lines in the fundamental representation. Although not necessary, we have inserted a complete set of $ n $-collinear states $ \mathbb{1} = {\sum\nolimits_{X_n}\int}\quad | X_n \rangle \langle X_n | $ into the definition of $ {\cal B}_{q/N} $. Similarly, for an anti-quark $ \bar{q} $ fragmenting into an anti-hadron $ \overline{N} $, the TMD FF can be written as
$\begin{aligned}[b]& {\cal{F}}_{\overline N/\bar q}(x_F,b_\perp) ={\sum\limits\int\limits_{X_n}} x_F^{1-2 \epsilon} \int \frac{{\rm d}b^-}{2\pi} {\rm e}^{{\rm i} b^- P'^+ / x_F} \\& \quad \times \langle 0 | \bar \chi_{n}(0,b^-,b_\perp) | \overline N(P'),X_n \rangle \frac{\not{\bar{n}}}{2} \langle \overline N(P'),X_n | \chi_{n}(0) | 0 \rangle \,,\end{aligned}$
(4)
where $ P'^\mu = (\bar n \cdotP') n^\mu/2 = P'^+ n^\mu/2 $ is the momentum of the final-state detected hadron. At high energy and small $ |\vec{b}_\perp| $, TMD PDFs and FFs admit light-cone operator product expansion onto collinear PDFs and FFs, with perturbative calculable Wilson coefficients, which have been calculated to NNLO [50-58], and very recently to N3LO [59,60]. The Wilson coefficients can be directly calculated by replacing the non-perturbative hadronic state $ N\,(\overline{N}) $ by the perturbative partonic state $ i\,(\bar \imath) $, namely $ {\cal B}_{q/i} $ and $ {\cal F}_{\bar \imath/\bar q} $. The operator definitions in Eqs. (2) and (4) make it clear that they can be computed from squared amplitudes integrated over collinear phase space [61],
$\begin{aligned}[b]{\cal B}_{q/i} =& \sum\limits_{X_n} \int{\rm d} {\rm{PS}}_{X_n} {\rm e}^{- {\rm i} K_\perp\cdot b_\perp} \delta(K^+ - (1 - x_B) P^+) \\ &\times {\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}} \frac{\not{\bar{n}}}{2} {\bf{Sp}}_{X_n q^* \leftarrow i}^{{\rm{S}},*} \,, \\ {\cal F}_{\bar \imath/\bar q} =& \sum\limits_{X_n} x_F^{1 - 2 {\epsilon}} \int{\rm d}{\rm{PS}}_{X_n} {\rm e}^{- {\rm i} K_\perp\cdot b_\perp} \delta\left(K^+ - \left(\frac{1}{x_F} - 1 \right) P'^+\right) \\ &\times {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q^*}^{\rm{T}} \frac{\not{\bar{n}}}{2} {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q^*}^{{\rm{T}},*} \,,\end{aligned}$
(5)
where $ K^\mu $ is the total momentum of $ |X_n \rangle $, and $ {\rm d}{\rm{PS}}_{X_n} $ is the collinear phase space measure. We also define the (generalized) space-like and time-like splitting amplitudes [62,63],
$\begin{aligned}[b]{\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}} \left( k_a^+/P^+, \dots \right) =& \ \langle X_n| \chi_n(0) | V_{P_l}^i(P_r) \rangle \,, \\ {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q^*}^{\rm{T}} (k_a^+/P'^+, \dots) =& \ \langle X_n, V_{P'_l}^{\bar \imath}(P'_r) | \chi_{n}(0) | 0 \rangle \,, \end{aligned}$
(6)
where $ {\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}} $ denotes the amplitudes for parton $ i $ splitting into an off-shell quark $ q^* $ and $ X_n $, and similarly for $ {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q^*}^{\rm{T}} $. $ |V_{P_l}^i(P_r) \rangle $ denotes the partonic state $ i $ with momentum $ P $, decomposed into label momentum and residual momentum $ P^\mu = P^\mu_l + P^\mu_r $, and similarly for $ V_{P'_l}^{\bar \imath}(P'_r) $. Label momentum in SCET is Euclidean-like and does not require causal prescription, while residual momentum does and will be discussed in the next section. When $ X_n $ consists of a single parton, Eq. (6) reduces to the usual $ 1{\rightarrow} 2 $ splitting amplitudes, which are known to two-loop accuracy [64-66]. Results are also available for $ 1{\rightarrow} 3 $ and $ 1{\rightarrow} 4 $ splitting [67-70]. In Eq. (6) we have made explicit the possible functional dependence on $ P^+ $ and $ P'^+ $, where $ k_a $ is any combination of momenta in $ |X_n\rangle $. This is due to reparameterization III invariance in SCET [71], namely the SCET matrix element should be invariant under $ n^\mu {\rightarrow} e^{\lambda} n^\mu $ and $ \bar{n}^\mu {\rightarrow} e^{-\lambda} \bar{n}^\mu $. We have also made implicit in Eq. (5) the averaging over initial-state spin and color, as well as the sum over final-state spin and color.
After proper renormalization and zero-bin subtraction [72], the TMD PDFs and FFs still contain collinear divergence due to the tagged hadron in the initial state or final state. Schematically, at $ n $-th order in perturbation theory, the single pole of the remaining collinear divergences has the following convolution form:
${\cal B}_{q/i}^{(n)} \sim \sum\limits_j \frac{P_{qj}^{{\rm{S}}, (n)}}{n {\epsilon}} \otimes \phi_{ji}^{\rm{bare}} \,, \quad {\cal F}_{\bar \imath /\bar q}^{(n)} \sim \sum\limits_j d_{\bar \imath j}^{\rm{bare}} \otimes \frac{P_{j \bar q}^{{\rm{T}},(n)}}{n {\epsilon}} \,,$
where $ \phi_{ij}^{\rm{bare}} = d_{ij}^{\rm{bare}} = \delta_{ij} $ are the bare partonic PDFs and FFs. Therefore, one can extract the space-like and time-like splitting functions directly from the partonic TMD PDFs and FFs.
III.ANALYTIC CONTINUATION OF SPLITTING AMPLITUDES
In order to understand the analytic continuation for TMD PDFs and FFs, we start with LSZ reduction on the space-like splitting amplitudes:
${\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}}= \int{\rm d}^dx \, {\rm e}^{-{\rm i} P_r \cdot x} \langle X_n |{\rm{T}} \{ \chi_n (0) J_{P_l}^i(x) \} | 0 \rangle \,,$
(7)
where the current $ J^{i}_{P_l}(x)=i(i{\cal{P}}_l+\partial_x)^2 V_{P_l}(x) $ creates a parton state $ i $ from vacuum. Using the fact that the SCET operator $ \chi_n(x) $ is local in residual space, the time-ordering product can be replaced by a (anti-)commutator if $ i $ is a boson (fermion),
${\rm{T}}\{\chi_n(0)J^{i}_{P_l}(x)\} = \theta(-x^0)\left[\chi_n(0),J^{i}_{P_l}(x)\right]_{\mp} \pm J^{i}_{P_l}(x)\chi_n(0) \,. $
(8)
The second term in Eq. (8) does not contribute to the correlator since $ \chi_n(0) $ effectively carries negative energy in the physical process and thus annihilates vacuum $ | 0 \rangle $. Note that since $ \chi_n $ is local, the (anti-)commutator in Eq. (8) vanishes in the space-like region $ \Omega_0 $ of Fig. 2. Thus, we can rewrite the space-like splitting amplitudes as:
Figure2. (color online) Penrose diagram of Minkowski space.

${\bf{Sp}}_{X_n q \leftarrow i}^{\rm{S}}=\int\limits_{x \in \Omega_-}{\rm d}^dx \, {\rm e}^{-{\rm i} P_r \cdot x} \langle X_n | [ \chi(0), J_{P_l}^i(x) ]_{\mp} | 0 \rangle \,,$
(9)
where the $ x $ integral is now restricted to inside the past light-cone, $ \Omega_- $. Demanding analyticity for the splitting amplitudes imposes a unique causal prescription for residual momenta, $ P_r{\rightarrow} P_r+i q_I $ where $ q_I $ is any positive-energy time-like vector.
Similarly, we can write the time-like splitting amplitudes as
${\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q}^{\rm{T}} = \int\limits_{x \in \Omega_+}{\rm d}^d x \, {\rm e}^{{\rm i} P'_r \cdot x} \langle X_n | [ J_{P'_l}^{\bar \imath}(x) , \chi(0) ]_{\mp} | 0 \rangle \,,$
(10)
and again the causal prescription must be $ P^\prime_r {\rightarrow} P^\prime_r + i q_I $. With the causal prescription properly defined, we can now discuss the analytic continuation between space-like and time-like splitting amplitudes.
Since splitting amplitudes are analytic functions of external momentum, we can continue $ P $ and $ P' $ to a common space-like infinity region, where space-like and time-like splitting amplitudes can be shown to equal. Therefore, by the edge-of-the-wedge theorem [73], space-like and time-like splitting amplitudes are actually analytic continuations of each other, although the causal prescription described above tells us their analytic regions are disjoint.
For concreteness and later convenience, we choose a particular path displayed in Fig. 3 as the blue lines (solid and dashed), where we analytically continue the momentum of a time-like splitting amplitude from $ P' $ (red) to $ -P' $ (green). The orange segment of the path, sitting at space-like infinity relative to $ O $, lies inside the region where $ {\bf{Sp}}^{\rm{S}}(-P^\prime)={\bf{Sp}}^{\rm{T}}(P^\prime) $ and does not require a causal prescription. Along the red segment, $ P^\prime $ in $ {\bf{Sp}}^{\rm{T}}(P^\prime) $ should have positive imaginary part $ {\rm{Im}} P^\prime\in \Omega_{+}^{p} $, while along the green segment, $ P^\prime $ in $ {\bf{Sp}}^{\rm{S}}(-P^\prime) $ should have a negative imaginary part $ {\rm{Im}}P^\prime\in\Omega_{-}^{p} $. In principle, every path allowed by analytic continuation should serve the same purpose.
Figure3. (color online) Penrose diagram of real momentum space. We have shown an extra spatial momentum dimension to visualize the path of analytic continuation, the blue lines.

The corresponding contour in the complex $ 1/{P^\prime}^{+} $ plane is depicted schematically in Fig. 4. Note that the orange segment in Fig. 3 cannot be simply shown in this plane of a single variable, so we abstractly use an orange dot at the origin to represent it, which allows us to cross the real line analytically. The physical region of the time-like process sits just below the positive real line with an infinitesimal imaginary part, while the physical region of the space-like process is just above the negative real line. As illustrated in Fig. 4, the correct path connects ${\rm e}^{-{\rm i}0_+}/{P^\prime}^{+}$ and ${\rm e}^{-{\rm i}\pi-{\rm i}0_+}/{{P^\prime}^+}$ for positive $ {P^\prime}^+ $.
Figure4. (color online) Path of analytic continuation from a time-like point T to a space-like point S or vice versa in the $ 1/{P^\prime}^{+} $ plane.

The discussion above determines a unique prescription for the analytic continuation of splitting amplitudes. We define an operator $ \underset{{\rm{T}} {\rightarrow} {\rm{S}}}{\cal{AC}} $ which continues a time-like splitting amplitude from its physical region to a space-like splitting amplitude as
$\begin{aligned}[b]\underset{{\rm{T}} {\rightarrow} {\rm{S}}}{\cal{AC}} \circ {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q}^{\rm{T}} \left(\frac{k_a^+}{P'^+ {\rm e}^{{\rm i}0_+}}, \cdots \right) \equiv & {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q}^{\rm{T}} \left(\frac{k_a^+}{P'^+ {\rm e}^{{\rm i} (\pi +0_+)}}, \cdots \right) \\ =& {\bf{Sp}}_{X_n q \leftarrow i}^{\rm{S}} \left(\frac{k_a^+}{P^+ {\rm e}^{{\rm i}0_+}}, \cdots\right) \,.\end{aligned}$
(11)
Similarly for a space-like to time-like continuation,
$\begin{aligned}[b]\underset{{\rm{S}} {\rightarrow} {\rm{T}}} {\cal{AC}} \circ {\bf{Sp}}_{X_n q \leftarrow i}^{\rm{S}} \left(\frac{k_a^+}{P^+ {\rm e}^{{\rm i}0_+}}, \cdots \right) \equiv & {\bf{Sp}}_{X_n q \leftarrow i}^{\rm{S}} \left( \frac{k_a^+}{P^+ {\rm e}^{-{\rm i} \pi+i0_+}}, \cdots\right) \\ =& {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar q}^{\rm{T}} \left(\frac{k_a^+}{P'^+ {\rm e}^{{\rm i}0_+}}, \cdots \right) \,.\end{aligned}$
(12)
One can also define an analytic continuation operator for complex conjugate amplitudes, $ \overline{\underset{{\rm{T}} {\rightarrow} {\rm{S}}}{\cal{AC}}} $, which amounts to first performing analytic continuation to amplitudes, and then taking complex conjugates. For a tree-level amplitude, $ \underset{{\rm{T}} {\rightarrow} {\rm{S}}}{\cal{AC}} $ and $ \overline{\underset{{\rm{T}} {\rightarrow} {\rm{S}}}{\cal{AC}}} $ become identical.
IV.ANALYTIC CONTINUATION OF TMD DISTRIBUTIONS
Since TMD distributions are obtained from squared amplitudes, analyticity in external momentum is lost. However, for a subset of contributions to TMD distributions at each perturbative order, it is possible to restore analyticity. We define the holomorphic part of TMD PDFs (the anti-holomorphic part is simply the conjugate of the holomorphic part) as:
$\begin{aligned}[b]{\cal B}_{q/i}^{h} =& \int{\rm d} {\rm{PS}}_{X_n} {\rm e}^{-{\rm i} K_\perp\cdot b_\perp} \delta(K^+ - (1 - x_B) P^+) \\ &\times {\rm e}^{- \frac{b_0 \tau}{2} |K^-|} \, {\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}} \left(k_a^+/P^+, \dots \right) \frac{\not{\bar n}}{2} {\bf{Sp}}_{X_n q^* \leftarrow i}^{{\rm{S}}, (0), *} \,,\end{aligned}$
(13)
where $ {\bf{Sp}}^{{\rm{S}}, (0) ,*} $ is the complex conjugate of the tree-level splitting amplitude. We have also inserted a rapidity regulator into the definition of the TMD PDFs, which we choose to be an exponential regulator ${\rm e}^{- b_0 \tau |K^-|/2}$ [55,74]. The advantage of this regulator is that all the end-point $ \delta(1-x) $ terms are absorbed into the soft function [75,76], which can be shown to be the same for Drell-Yan, DIS, or $ e^+e^- $ processes [75,77,78]. We emphasize that the results for splitting functions are independent of rapidity regularization. In the following, we shall restrict our discussion to $ 0<x<1 $, and show that $ {\cal B}_{q/i}^h $ can be analytically continued to $ {\cal F}_{\bar \imath/\bar{q}}^h $, and vice versa.
We introduce a dimensionless light-cone momentum fraction $ y_a = k_a^+/((1-x_B)P^+) $. For $ X_n $ consisting of $ m $ massless partons, the holomorphic part is
${\cal B}_{q/i}^{h,m}(x_B, |P^+|, b_\perp) = \int \prod\limits_{a = 1}^m \frac{{\rm d}^{d-2} \vec{k}_{a,\perp}}{2 (2 \pi)^3} \frac{{\rm d}y_a}{y_a} {\rm e}^{-{\rm i} K_\perp\cdot b_\perp - \frac{b_0\tau}{2} |K^-|} \frac{\delta\left(\displaystyle\sum\nolimits_{l = 1}^m y_l - 1\right) }{|1-x_B| |P^+| } {\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}} \left(\frac{y_a(1-x_B)}{{\rm e}^{{\rm i}0+}}, \dots \right) \frac{\not{\bar n}}{2} {\bf{Sp}}_{X_n q^* \leftarrow i}^{({\rm{S}}, (0) ,*} \,,$
(14)
where in terms of dimensionless light-cone momentum fraction $ |K^-| = |\sum\nolimits_b \vec{k}_{b,\perp}^2 y_b^{-1}|/(|1-x_B| |P^+|) $. The additional $ |P^+| $ dependence in the argument of $ {\cal B} $ results from rapidity regularization. The analytic continuation reads
$\underset{{\rm{S}} {\rightarrow} {\rm{T}}}{\cal{AC}} \circ {\cal B}_{q/i}^{h,m} = \ \int \prod\limits_{a = 1}^m \frac{{\rm d}^{d-2} \vec{k}_{a,\perp}}{2 (2 \pi)^3} \frac{{\rm d}y_a}{y_a} {\rm e}^{-{\rm i} K_\perp\cdot b_\perp - \frac{b_0 \tau}{2} |K^-| } \frac{\delta\left(\displaystyle\sum\nolimits_{l = 1}^m y_l - 1\right) }{|1-x_B| |P^+| } {\bf{Sp}}_{X_n q^* \leftarrow i}^{\rm{S}} \left(\frac{y_a(1-x_B)}{{\rm e}^{{\rm i}(\pi+0_+)}}, \dots \right) \frac{\not{\bar n}}{2} {\bf{Sp}}_{X_n q^* \leftarrow i}^{{\rm{S}}, (0), *}$
(15)
$\quad\quad\quad\quad = \ \int \prod\limits_{a = 1}^m \frac{{\rm d}^{d-2} \vec{k}_{a,\perp}}{2 (2 \pi)^3} \frac{{\rm d}y_a}{y_a} {\rm e}^{-{\rm i} K_\perp\cdot b_\perp - \frac{b_0 \tau}{2} |K^-| } \frac{\delta\left(\displaystyle\sum\nolimits_{l = 1}^m y_l - 1\right) }{|1-x_B| |P^+| } {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar{q}^*}^{\rm{T}} \left(\frac{y_a(1-x_B) }{{\rm e}^{{\rm i}0_+}}, \dots \right) \frac{\not{\bar n}}{2} {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar{q}^*}^{{\rm{T}}, (0), *} \,. $
(16)
Note that the analytic continuation operator only acts on the all-order splitting amplitude, as well as the conjugate of the tree-level splitting amplitude. We can also write down the holomorphic part of the TMD FFs,
${\cal F}_{\bar \imath/\bar{q}}^{h,m}(x_F, |P'^+|,b_\perp) = \ x_F^{1-2 {\epsilon}} \int \prod\limits_{a = 1}^m \frac{{\rm d}^{d-2} \vec{k}_{a,\perp}}{2 (2 \pi)^3} \frac{{\rm d}y'_a}{y'_a} {\rm e}^{-{\rm i} K_\perp\cdot b_\perp - \frac{b_0 \tau}{2} |K^-|} \frac{\delta\left(\displaystyle\sum\nolimits_{l = 1}^m y'_l - 1\right) }{|1/x_F - 1| |P'^+| } {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar{q}^*}^{\rm{T}} \left(\frac{y'_a\left(\dfrac{1}{x_F}-1\right)}{{\rm e}^{{\rm i}0+}}, \dots \right) \frac{\not{\bar n}}{2} {\bf{Sp}}_{X_n \bar \imath \leftarrow \bar{q}^*}^{{\rm{T}}, (0), *} \,,$
(17)
where the lightcone momentum fraction is $ y'_a = k_a^+/ ((1/x_F - 1)P'^+) $ ,
$ |K^-| = |\sum\nolimits_b \vec{k}_{b,\perp}^2 y_b'^{-1}|/(|1/x_F-1| |P'^+| ) $
and we identify the path for analytic continuation
$(1 - x_B) {\rightarrow} \left( \frac{1}{x_F} - 1\right) {\rm e}^{{\rm i} \pi} \,. $
(18)
The analytic continuation between $ {\cal B}^h $ and $ {\cal F}^h $ then reads
${\cal F}_{\bar \imath/\bar{q}}^{h,m}(x_F, |P'^+|, b_\perp) = (-1)^{i_F} x_F^{1-2 {\epsilon}} {\cal B}_{q/i}^{h,m} \left( - \frac{{\rm e}^{{\rm i}\pi}}{x_F}, |P'^+|, b_\perp \right) \,,$
(19)
where $ i_F = 1 $ if $ i $ is a fermion, and 0 if a boson. The minus sign is due to crossing a fermion from initial state to final state. Similarly for gluon TMD distributions, the analytic continuation reads
${\cal F}_{\bar \imath/g}^{h,m}(x_F, |P'^+|, b_\perp) = \ (-1)^{ 1 + i_F} x_F^{1-2 {\epsilon}} \\ \ \cdot {\cal B}_{g/i}^{h,m} \left( - \frac{{\rm e}^{{\rm i}\pi}}{x_F}, |P'^+|, b_\perp \right) \,,$
(20)
where the additional minus sign originates from the operator definition, and we have suppressed the irrelevant Lorentz indices. We stress that the analytic continuation is for bare quantities before PDF or FF renormalization.
We can now apply the analytic continuation rules in Eqs. (19) and (20) to the TMD PDFs. At NLO and NNLO, there are only holomorphic and anti-holomorphic contributions. Therefore the analytic continuation rules determine the TMD FFs completely. At N3LO, the partonic contributions can be decomposed into triple real (RRR), double-real virtual (VRR), double-virtual real (VVR), and virtual-squared real (VV*R) components, as depicted in Fig. 5. The first three contributions are either holomorphic or anti-holomorphic. But the last contribution, VV*R, mixes holomorphic and anti-holomorphic terms, and therefore cannot be determined from analytic continuation. Since this is a relatively simple contribution, we can calculate it directly using the defining equation in Eq. (5). In this way we obtain the bare TMD FFs at N3LO. The results for N3LO TMD FFs will be presented elsewhere. Here we focus on splitting functions. From the single-pole terms of the bare TMD FFs we extract all the time-like splitting functions through NNLO. Comparing the results with those in the literature, we find full agreement except for the non-diagonal quark-gluon splitting. The discrepancy between our results and those presented in Ref. [23] can be written as
Figure5. Contributions from different partonic channels to TMD PDFs at N3LO.

$\begin{aligned}[b]\Delta P_{qg}^{{\rm{T}},(2)}(x) =&P_{qg}^{{\rm{T}},(2)}\Big|_{\rm{this \;work}} - P_{qg}^{{\rm{T}},(2)}\Big|_{[23]} \\=& \frac{\pi^2}{3} (C_F - C_A) \beta_0 \left[ -4 + 8 x + x^2\right. \\&\left.+ 6 (1 - 2 x + 2 x^2) \ln x \right] \,, \end{aligned}$
(21)
where $ P_{qg}^{{\rm{T}},(2)} $ is the coefficient of $ \alpha_s^3/(4 \pi)^3 $ in the off-diagonal singlet splitting matrix, and $ \beta_0 = 11 C_A/3 - 2 n_f/3 $ is the one-loop QCD beta function. In Mellin moment space the discrepancy reads
$\begin{aligned}[b]&- \int_0^1 {\rm d}x\, x^{N-1} \Delta P_{qg}^{{\rm{T}},(2)} (x) = (C_A - C_F) \beta_0 \frac{\pi^2}{3} \left( \frac{12}{(N+1)^2} \right. \\ &\left. \quad - \frac{6}{N^2} - \frac{12}{(N+2)^2} - \frac{4}{N} + \frac{8}{N+1} + \frac{1}{N+2} \right) .\end{aligned}$
(22)
Note that the discrepancy vanishes for $ N = 2 $, as it is completely fixed by the momentum sum rule [22]. In the Appendix we point out potential sources of the discrepancy. For the convenience of the reader we provide the full time-like splitting functions through NNLO as an ancillary file along with the arXiv submission.
V.RECIPROCITY RELATIONS IN QCD
With the full space-like and time-like splitting functions, it is interesting to explore yet another relation between them, the so-called reciprocity relation. Reciprocity for tree-level splitting functions was first proposed by Gribov and Lipatov [36], and says that $ P_{ab}^{{\rm{S}},(0)}(x) = P_{ab}^{{\rm{T}},(0)}(x) $. While the Gribov-Lipatov reciprocity breaks down beyond LO [79,80], consideration from small $ x $ [81,82] and large $ x $ [20,83], as well as from conformal field theory [84,85], suggests that a modified form of the reciprocity relation exists, at least for the non-singlet case.
Our new results are for the singlet case, which for both space-like and time-like splitting can be written as
$\widehat{P}(x,\alpha_s) = \begin{pmatrix} \widetilde{P}_{qq} & 2 n_f P_{qg} \\ P_{gq} & P_{gg} \end{pmatrix} \,,$
(23)
where
$\widetilde{P}_{qq} = P_{qq} + P_{\bar{q} q} + (n_f-1)(P_{q'q}+P_{\bar{q}'q}) \,.$
(24)
For time-like splitting, the $ P_{ij}^{{\rm{T}}} $ can be found in the ancillary file through NNLO. It is also convenient to introduce the Mellin moment of singlet splitting,
$\widehat{\gamma}(N,\alpha_s) = - \int_0^1 {\rm d}x \, x^{N-1} \widehat{P}(x,\alpha_s) \,,$
(25)
and the associated eigenvalues,
$\gamma_\pm = \frac{1}{2}( \pm \sqrt{({\rm{tr}}\widehat{\gamma}\,)^2 - 4 {\rm{det}} \widehat{\gamma} } + {\rm{tr}}\widehat{\gamma}\, ) \,.$
(26)
An important motivation for the reciprocity relation in the singlet case comes from the evolution equation for jet functions in energy correlators [30,31],
$\frac{{\rm d} \vec{J} (\ln \frac{x_L Q^2}{\mu^2})}{{\rm d} \ln\mu^2} = \int_0^1{\rm d}y\, y^N \vec{J}\, \left(\ln \dfrac{x_L y^2 Q^2}{\mu^2}\right) \cdot \widehat{P}^{\,\rm{T}} (y,\alpha_s) \,, $
(27)
where $ x_L $ measures the size of $ N $ tagged particles in a jet. Note that this is a non-local evolution equation. For a fixed coupling, one can write down for Eq. (27) a completely local, power-law solution for $ \vec{J} $, with the power-law exponent given by $ \gamma_{\pm}^{{\rm{T}}} $ evaluated at a shift $ N $. Based on this consideration, we propose the following reciprocity relations for the singlet splitting with running coupling,
$2 \gamma_\pm^{\rm{S}}(N, \alpha_s) = \ 2 \gamma_\pm^{\rm{T}} (N + 2 \gamma_\pm^{\rm{S}} (N, \alpha_s), \alpha_s) \,,$
(28)
$2 \gamma_\pm^{\rm{T}}(N, \alpha_s) = \ 2 \gamma_\pm^{\rm{S}} (N - 2 \gamma_\pm^{\rm{T}} (N, \alpha_s), \alpha_s) \,. $
(29)
The two relations (28) and (29) are not independent. We have verified Eqs. (28) and (29) through NNLO ($ \alpha_s^3 $) using the newly determined time-like singlet splitting functions. However, this relation is violated should we use the $ P_{qg}^{{\rm{T}},(2)} $ from Ref. [23]. We stress that the reciprocity relation is for arbitrary $ N $, and therefore hints at hidden relation between space-like and time-like processes beyond small $ x $ and large $ x $.
VI.CONCLUSION
We have provided a clean theoretical understanding of analytic continuation for TMD distributions and splitting functions using SCET. Employing the analytic continuation rules for holomorphic and anti-holomorphic contributions to TMD distributions, we have determined the time-like splitting functions in QCD through NNLO. For the eigenvalues of the singlet splitting matrix, we propose an all-order reciprocity relation, valid for arbitrary $ N $. We verified this relation through NNLO using the newly determined time-like singlet splitting functions. We leave a deeper understanding of the reciprocity relation to future work.
ACKNOWLEDGEMENTS
We thank Lance Dixon, Yi-Bei Li, Ming-xing Luo, Ian Moult, and Hua-Sheng Shao for helpful discussion.
APPENDIX A
In this Appendix we point out potential sources of discrepancy between our results and those of Ref. [23].
The operations of analytic continuation in Refs. [21-23] are done at cross-section level. At N3LO, this amounts to analytic continuation of the sum of the VVR and its conjugate, the VV*R, the VRR and its conjugate, and the RRR. As explained in the main text, the VV*R itself is neither holomorphic nor anti-holomorphic, and therefore does not obey the simple analytic continuation rule. This constitutes the first source of discrepancy. The second source of discrepancy comes from the VVR contribution and its conjugate, for which a discussion in the $ x_B {\rightarrow} 1 $ limit is sufficient to illustrate the origin of the discrepancy.
Specifically, the VVR contribution of TMD beam functions in the $ x_B {\rightarrow} 1 $ limit is
$\tag{A1}{\cal{B}}_{\rm{VVR}} : C_1 {(1-x_B)^{2 \epsilon}}+ {C_2 {{\rm e}^{{\rm i} \pi \epsilon} }{(1-x_B)^{\epsilon}}}+{C_3 {{\rm e}^{{\rm 2i} \pi \epsilon}}} \,,$
where $ C_1, C_2, C_3 $ are all real constants. Since it only contains a holomorphic part, we can analytically continue the VVR safely. Using the analytic continuation rule given in Eq. (18), we obtain the VVR contribution of TMD fragmentation functions in the limit of $ x_F {\rightarrow} 1 $,
$\tag{A2}{\cal{F}}_{\rm{VVR}} : {\rm e}^{{\rm 2i} \pi \epsilon} \bigg( C_1 {(1-x_F)^{2 \epsilon}}+ {C_2 {(1-x_F)^{\epsilon}}}+{C_3 } \bigg) \,.$
Adding VVR and (VVR)*, we get
$\tag{A3}{\cal{F}}_{\rm{VVR}} + {\rm{c.c.}}: 2 \cos( 2 \pi \epsilon) \bigg( C_1 {(1-x_F)^{2 \epsilon}}+ {C_2 {(1-x_F)^{\epsilon}}}+{C_3 } \bigg).$
However, to the best of our knowledge, the authors of Refs. [21-23] do not have enough information to separate the VVR part of the space-like structure functions from the full results. What they know is the sum of VVR and its complex conjugate, that is
$\tag{A4}\begin{aligned}[b]&{{\rm{VVR}}}+ {\rm{c.c.}} : \\ & 2 C_1 {(1-x_B)^{2 \epsilon}}+ 2 {C_2 \cos(\pi \epsilon) {(1-x_B)^{\epsilon}}}+ 2 {C_3 \cos(2 \pi \epsilon)} \,.\end{aligned}$
Cross-section level analytic continuation of Refs. [21-23] amounts to applying the analytic continuation rule of Eq. (18) to the quantity in Eq. (A4) and then taking the real part. The result reads
$\tag{A5}\begin{aligned}[b] 2 C_1 {(1-x_F)^{2 \epsilon}} \cos(2 \pi \epsilon) +& 2 {C_2 \cos^2(\pi \epsilon) {(1-x_F)^{\epsilon}}} \\ +&2 {C_3 \cos(2 \pi \epsilon)}\,. \end{aligned}$
It is clear that Eq. (A3) (our results) and Eq. (A5) are different, and this therefore constitutes another source of discrepancy.
A similar problem exists for the VRR+c.c. part. However, in that case the difference between th correct analytic continuation and analytic continuation at the cross-section level differs only by imaginary terms, and therefore does not contribute to the final discrepancy.
相关话题/Analytic continuation reciprocity

闂傚倸鍊峰ù鍥敋瑜忛埀顒佺▓閺呯娀銆佸▎鎾冲唨妞ゆ挾鍋熼悰銉╂⒑閸︻厼鍔嬫い銊ユ噽婢规洘绻濆顓犲幍闂佸憡鎸嗛崨顓狀偧闂備焦濞婇弨閬嶅垂閸洖桅闁告洦鍨扮粻娑㈡煕閹捐尙鍔嶉柛瀣斿洦鈷戠痪顓炴噺椤ュ鏌i埥鍡樼弸闂傚倸鍊搁崐鎼佸磹閻戣姤鍊块柨鏃堟暜閸嬫挾绮☉妯诲櫧闁活厽鐟╅弻鐔兼倻濮楀棙鐣烽梺绋垮椤ㄥ棝濡甸崟顖氭闁割煈鍠掗幐鍐⒑閸涘⿴娈曠€光偓閹间礁钃熸繛鎴欏灩缁犵粯淇婇妶鍌氫壕闂佺娅曢崝娆撳箖閿涘嫧鍋撻敐搴℃灍闁稿﹤鐏氶〃銉╂倷閼碱兛铏庨梺鍛婃⒐瀹€鎼佸蓟閿濆憘鏃堝焵椤掑嫬鐤柡澶嬪灩閺嗭箓鏌熸潏鍓х暠缂佺姴顭烽幃褰掑炊椤忓嫮姣㈠銈忚礋閸斿矂鈥旈崘顔嘉ч柛娑卞灣椤斿洤鈹戦埥鍡椾簼妞ゃ劌锕獮鍐灳閺傘儲顫嶉梺闈涚箚濡狙囧箯濞差亝鈷戠紒瀣硶缁犱即鏌涘顒夊剳婵炲棎鍨藉濠氬Ψ閿旀儳骞堥梻浣虹帛濮婄粯鐏欓梺宕囩帛濞茬喖寮婚悢纰辨晬婵炲棙鍨垫俊浠嬫倵閸偅绶查悗姘煎枟缁傛帡鏁冮崒姘辩暰閻熸粌閰i幃妤咁敇閵忊檧鎷绘繛杈剧秬濞咃綁濡存繝鍥ㄧ厱闁规儳顕粻鏍磼濡ゅ啫鏋庨摶鏍煕濞戝崬骞楅弶鍫濈墦濮婃椽妫冨☉杈ㄐ㈤梺鍝勬噺缁捇宕哄☉銏犵婵°倓鑳堕崢鍗炩攽閻愭潙鐏ョ€规洦鍓熼悰顔嘉熼懖鈺冿紲闂佺粯鐟ラ幊鎰板箖閼测晝纾奸弶鍫涘妼缁椦兠归悪鍛暤闁诡喖澧芥禒锕傛偩鐏炶棄绠洪梻鍌氬€峰ù鍥敋閺嶎厼鍨傞柡澶嬶紩濞差亜惟闁宠桨鑳堕崝锕€顪冮妶鍡楃瑨闁挎洩绠撳畷姘跺级濞嗗墽鍞甸悷婊冪Т闇夊瀣閸ㄦ繈鏌涢銈呮瀻缂佸墎鍋ら幃妤呮晲鎼粹€茬盎閻庢鍠栧ḿ鈥愁潖濞差亜绠伴幖杈剧悼閻g敻姊虹涵鍛彧闁告梹顨堥崚鎺撶節濮橆剛鐓戞繝銏f硾閻ジ鎯侀崼銉︹拺闂侇偆鍋涢懟顖涙櫠閹绢喗鐓欐い鏂诲妼濞层倝鏌嬮崶顒佺厓闁搞劍绋掑▍鍡涙煟韫囨挸鏆i柡宀嬬稻閹棃鏁嶉崟顓熸闂備胶枪闁帮絾绂嶉崼鏇犲祦濠电姴鍟崕鐔兼煏婵犲繒鐣卞ù婊勵殜濮婅櫣绮欓幐搴㈡嫳闂佺硶鏅涢崯顐㈠祫闂佸壊鍋侀崕鏌ユ偂濞嗘垹纾藉ù锝夋涧閻忊晠鏌h箛鏇炐㈤棁澶嬬節婵犲倻澧㈤柣锝囧劋閹便劍绻濋崘鈹夸虎閻庤娲忛崝宥囨崲濠靛纾兼繝濠傛噺閸n垰鈹戦悩娈挎毌婵℃彃鎳樺畷鎴﹀川椤栨稑搴婇梺鍓插亖閸庡啿鐣垫笟鈧弻鐔煎箥椤旂⒈鏆梺鍝勬噺閹倿寮婚妸鈺傚亞闁稿本绋戦锟�
547闂傚倸鍊搁崐鎼佸磹妞嬪海鐭嗗ù锝夋交閼板潡姊洪鈧粔鏌ュ焵椤掆偓閸婂湱绮嬮幒鏂哄亾閿濆簼绨介柨娑欑洴濮婃椽鎮烽弶搴撴寖缂備緡鍣崹鍫曞春濞戙垹绠虫俊銈勮兌閸橀亶姊洪崫鍕妞ゃ劌妫楅埢宥夊川鐎涙ḿ鍘介棅顐㈡祫缁插ジ鏌囬鐐寸厸鐎光偓鐎n剙鍩岄柧缁樼墵閺屽秷顧侀柛鎾跺枛瀵粯绻濋崶銊︽珳婵犮垼娉涢敃锕傛偪閸ヮ剚鈷戦悷娆忓缁€鍐┿亜閺囧棗鎳愰惌鍡涙煕閹般劍鏉哄ù婊勭矒閻擃偊宕堕妸锕€闉嶅銈冨劜缁捇寮婚敐澶婄閻庨潧鎲¢崚娑樷攽椤旂》鏀绘俊鐐舵閻e嘲螖閸涱厾顦ч梺鍏肩ゴ閺呮盯宕甸幒妤佲拻濞达絽鎲¢幉鎼佹煕閿濆啫鍔︾€规洘鍨垮畷鐔碱敍濞戞ü鎮i梻浣虹帛閸ㄥ吋鎱ㄩ妶澶婄柧闁归棿鐒﹂悡銉╂煟閺囩偛鈧湱鈧熬鎷�1130缂傚倸鍊搁崐鎼佸磹閹间礁纾瑰瀣捣閻棗銆掑锝呬壕闁芥ɑ绻冮妵鍕冀閵娧呯厒闂佹椿鍘介幑鍥蓟閿濆顫呴柕蹇婃櫆濮e矂姊虹粙娆惧剱闁圭懓娲ら悾鐤亹閹烘繃鏅濋梺鎸庣箓濞诧箓顢樻繝姘拻濞撴埃鍋撻柍褜鍓涢崑娑㈡嚐椤栨稒娅犻柛娆忣槶娴滄粍銇勯幇鈺佺労婵″弶妞介弻娑㈡偐鐠囇冧紣濡炪倖鎸搁崥瀣嚗閸曨剛绡€闁告劦鍘鹃崣鎴︽⒒閸屾瑧绐旈柍褜鍓涢崑娑㈡嚐椤栨稒娅犻柟缁㈠枟閻撴盯鎮橀悙鐧昏鏅堕懠顑藉亾閸偅绶查悗姘煎櫍閸┾偓妞ゆ帒锕︾粔闈浢瑰⿰鍕煉闁挎繄鍋為幆鏃堝煢閳ь剟寮ㄦ禒瀣厽闁归偊鍨伴惃鍝勵熆瑜庨惄顖炲蓟濞戙垹惟闁靛/鍌濇闂備椒绱徊鍧楀礂濮椻偓瀵偊骞樼紒妯轰汗闂佽偐鈷堥崜锕€危娴煎瓨鐓熼柣鏂挎憸閻﹦绱掔紒妯虹闁告帗甯掗埢搴ㄥ箻瀹曞洤鈧偤姊洪崘鍙夋儓闁哥喍鍗抽弫宥呪堪閸曨厾鐦堥梺闈涢獜缁插墽娑垫ィ鍐╃叆闁哄浂浜顕€鏌¢崨顐㈠姦婵﹦绮幏鍛村川婵犲倹娈橀梺鐓庣仌閸ャ劎鍘辨繝鐢靛Т閸熺増鏅舵潏鈺冪=闁稿本绋掑畷宀勬煙缁嬪尅鏀荤紒鏃傚枛閸╋繝宕掑☉杈棃闁诲氦顫夊ú锔界濠靛绠柛娑卞灡閸犲棝鏌涢弴銊ュ箺鐞氭瑩姊婚崒姘偓椋庣矆娴i潻鑰块梺顒€绉撮崒銊ф喐閺冨牆绠栨繛宸簻鎯熼梺瀹犳〃閼冲爼顢欓崶顒佲拺闁告挻褰冩禍婵囩箾閸欏澧甸柟顔惧仱瀹曞綊顢曢悩杈╃泿闂備胶鎳撻顓㈠磻濞戙埄鏁嬫繝濠傛噽绾剧厧霉閿濆懏鎯堟い锝呫偢閺屾洟宕惰椤忣厽銇勯姀鈩冪濠殿喒鍋撻梺瀹犳〃缁€浣圭珶婢舵劖鈷掑ù锝囨嚀椤曟粎绱掔€n偄娴€规洘绻傞埢搴ㄥ箻鐠鸿櫣銈﹂梺璇插嚱缂嶅棝宕抽鈧顐㈩吋閸℃瑧鐦堟繝鐢靛Т閸婅鍒婇崗闂寸箚闁哄被鍎查弫杈╃磼缂佹ḿ绠為柟顔荤矙濡啫鈽夊Δ浣稿闂傚倷鐒﹂幃鍫曞礉瀹€鈧槐鐐寸節閸屻倕娈ㄥ銈嗗姂閸婃鎯屽▎鎰箚妞ゆ劑鍊栭弳鈺呮煕鎼存稑鈧骞戦姀鐘斀閻庯綆浜為崐鐐烘⒑闂堟胆褰掑磿閺屻儺鏁囨繛宸簼閳锋垿鏌涘┑鍡楊伌婵″弶鎮傞弻锝呂旀担铏圭厜閻庤娲橀崹鍧楃嵁閹烘嚦鏃堝焵椤掑嫬瑙︾憸鐗堝笚閻撴盯鏌涢幇鈺佸濠⒀勭洴閺岋綁骞樺畷鍥╊啋闂佸搫鏈惄顖炲春閸曨垰绀冮柍鍝勫枤濡茬兘姊绘担鍛靛湱鎹㈤幇鐗堝剶闁兼祴鏅滈~鏇㈡煙閻戞﹩娈㈤柡浣革躬閺屾稖绠涢幙鍐┬︽繛瀛樼矒缁犳牠骞冨ú顏勭鐎广儱妫涢妶鏉款渻閵堝骸浜滄い锔炬暬閻涱噣宕卞☉妯活棟闁圭厧鐡ㄩ幐濠氾綖瀹ュ鈷戦柛锔诲幖閸斿鏌涢妸銊︾彧缂佹梻鍠栧鎾偄閾忚鍟庨梺鍝勵槸閻楀棙鏅舵禒瀣畺濠靛倸鎲¢悡娑㈡煕濠娾偓缁€浣圭濠婂牆纭€闂侇剙绉甸悡鏇熴亜閹邦喖孝闁告梹绮撻弻锝夊箻鐎涙ḿ顦伴梺鍝勭灱閸犳牠骞冨⿰鍏剧喓鎷犻弻銉р偓娲⒒娴e懙褰掝敄閸ャ劎绠鹃柍褜鍓熼弻锛勪沪閻e睗銉︺亜瑜岀欢姘跺蓟濞戞粎鐤€闁哄啫鍊堕埀顒佸笚缁绘盯宕遍幇顒備患濡炪値鍋呯换鍕箲閸曨個娲敂閸滃啰鑸瑰┑鐘茬棄閺夊簱鍋撹瀵板﹥绂掔€n亞鏌堝銈嗙墱閸嬫稓绮婚悩铏弿婵☆垵顕ч。鎶芥煕鐎n偅宕岄柣娑卞櫍瀹曞綊顢欓悡搴經闂傚倷绀侀幗婊堝窗閹惧绠鹃柍褜鍓涢埀顒冾潐濞叉﹢宕归崸妤冨祦婵☆垰鐨烽崑鎾斥槈濞咁収浜、鎾诲箻缂佹ǚ鎷虹紓鍌欑劍閿氶柣蹇ョ畵閺屻劌顫濋懜鐢靛帗閻熸粍绮撳畷婊冣槈閵忕姷锛涢梺缁樻⒒閸樠囨倿閸偁浜滈柟鐑樺灥閺嬨倖绻涢崗鐓庡闁哄瞼鍠栭、娆撴嚃閳轰胶鍘介柣搴ゎ潐濞叉ê煤閻旂鈧礁鈽夐姀鈥斥偓鐑芥煠绾板崬澧┑顕嗛檮娣囧﹪鎮欓鍕ㄥ亾閺嶎厼鍨傚┑鍌溓圭壕鍨攽閻樺疇澹樼紒鈧崒鐐村€堕柣鎰緲鐎氬骸霉濠婂嫮鐭掗柡宀€鍠栭獮鍡氼槾闁圭晫濞€閺屾稒绻濋崘銊ヮ潚闂佸搫鐬奸崰鏍€佸▎鎾村殐闁宠桨鑳堕崢浠嬫煟鎼淬値娼愭繛鑼枑缁傚秹宕奸弴鐘茬ウ闂佹悶鍎洪崜娆愬劔闂備線娼чˇ顓㈠磹閺団懞澶婎潩椤戣姤鏂€闂佺粯鍔橀崺鏍亹瑜忕槐鎺楁嚑椤掆偓娴滃墽绱掗崒姘毙ч柟宕囧仱婵$柉顧佹繛鏉戝濮婃椽骞愭惔銏紩闂佺ǹ顑嗛幑鍥涙担鐟扮窞闁归偊鍘鹃崢閬嶆椤愩垺澶勬繛鍙夌墱閺侇噣宕奸弴鐔哄幍闂佺ǹ绻愰崥瀣磹閹扮増鐓涢悘鐐垫櫕鍟稿銇卞倻绐旈柡灞剧缁犳盯寮崒妤侇潔闂傚倸娲らˇ鐢稿蓟濞戙垹唯妞ゆ梻鍘ч~鈺冪磼閻愵剙鍔ら柕鍫熸倐瀵寮撮悢铏圭槇闂婎偄娲﹀ú婊堝汲閻樺樊娓婚柕鍫濇缁€澶婎渻鐎涙ɑ鍊愭鐐茬墦婵℃悂濡锋惔锝呮灁缂侇喗鐟╁畷褰掝敊绾拌鲸缍嶉梻鍌氬€烽懗鑸电仚濡炪倖鍨靛Λ婵嬬嵁閹邦厾绡€婵﹩鍓涢鍡涙⒑閸涘﹣绶遍柛銊╀憾瀹曚即宕卞☉娆戝幈闂佸搫娲㈤崝灞炬櫠娴煎瓨鐓涢柛鈩兠崫鐑樻叏婵犲嫮甯涢柟宄版嚇瀹曨偊宕熼锛勫笡闂佽瀛╅鏍窗濡ゅ懎纾垮┑鍌溓规闂佸湱澧楀妯肩矆閸愨斂浜滈煫鍥ㄦ尰椤ョ姴顭跨捄鍝勵仾濞e洤锕俊鎯扮疀閺囩偛鐓傞梻浣告憸閸c儵宕圭捄铏规殾闁硅揪闄勯崑鎰磽娴h疮缂氶柛姗€浜跺娲棘閵夛附鐝旈梺鍝ュ櫏閸嬪懘骞堥妸鈺佺劦妞ゆ帒瀚埛鎴犵磼鐎n偒鍎ラ柛搴㈠姍閺岀喓绮欏▎鍓у悑濡ょ姷鍋涚换妯虹暦閵娧€鍋撳☉娅亝绂掗幆褜娓婚柕鍫濇婢ь剟鏌ら悷鏉库挃缂侇喖顭烽獮瀣晜鐟欙絾瀚藉┑鐐舵彧缁蹭粙骞夐敓鐘茬畾闁割偁鍎查悡鏇炩攽閻樻彃顎愰柛锔诲幖瀵煡姊绘笟鈧ḿ褏鎹㈤崼銉ョ9闁哄洢鍨洪崐鍧楁煕椤垵浜栧ù婊勭矒閺岀喓鈧數枪娴犳粍銇勯弴鐔虹煂缂佽鲸甯楅幏鍛喆閸曨厼鍤掓俊鐐€ら崣鈧繛澶嬫礋楠炲骞橀鑲╊槹濡炪倖宸婚崑鎾剁棯閻愵剙鈻曢柟顔筋殔閳绘捇宕归鐣屼壕闂備浇妗ㄧ粈渚€鈥﹂悜钘壩ュù锝囩《濡插牊淇婇娑氱煂闁哥姴閰i幃楣冨焺閸愯法鐭楁繛杈剧到婢瑰﹤螞濠婂嫮绡€闁汇垽娼ф禒鈺呮煙濞茶绨界紒杈╁仱閸┾偓妞ゆ帊闄嶆禍婊勩亜閹扳晛鐒烘俊顖楀亾闂備浇顕栭崳顖滄崲濠靛鏄ラ柍褜鍓氶妵鍕箳閹存繍浠鹃梺鎶芥敱鐢繝寮诲☉姘勃闁硅鍔曢ˉ婵嬫⒑闁偛鑻崢鍝ョ磼椤旂晫鎳囬柕鍡曠閳诲酣骞囬鍓ф闂備礁鎲″ú锕傚礈閿曗偓宀e潡鎮㈤崗灏栨嫼闂佸憡鎸昏ぐ鍐╃濠靛洨绠鹃柛娆忣槺婢ц京绱掗鍨惞缂佽鲸甯掕灒闂傗偓閹邦喚娉块梻鍌欑濠€閬嶅磻閹剧繀缂氭繛鍡樻嫴婢跺⿴娼╅柤鍝ユ暩閸橀亶鏌f惔顖滅У闁稿鎳愭禍鍛婂鐎涙ḿ鍘甸悗鐟板婢ф宕甸崶鈹惧亾鐟欏嫭绀堥柛蹇旓耿閵嗕礁螣鐞涒剝鏁犻梺璇″瀻閸屾凹妫滄繝鐢靛Х閺佸憡鎱ㄩ弶鎳ㄦ椽鏁冮崒姘憋紮闂佸壊鐓堥崑鍡欑不妤e啯鐓欓悗娑欋缚缁犳﹢鏌$€n亜鏆熺紒杈ㄥ浮閸┾偓妞ゆ帒鍊甸崑鎾绘晲鎼粹剝鐏嶉梺缁樻尭閸熶即骞夌粙搴撳牚闁割偅绻勯ˇ褍鈹戦悙鏉戠仸婵ǜ鍔戦幆宀勫幢濡炴洖缍婇弫鎰板醇閻旂补鍋撻崘顔界厽闁圭儤鍩婇煬顒勬煛瀹€鈧崰搴ㄥ煝閹捐鍨傛い鏃傛櫕娴滄劙姊绘担鍛靛綊顢栭崱娑樼闁归棿绀侀悡鈥愁熆鐠哄搫顦柛瀣崌瀹曠兘顢橀悙鎰╁劜閵囧嫰鏁傞崹顔肩ギ濠殿喖锕ュ浠嬪蓟閸涘瓨鍊烽柤鑹版硾椤忣參姊洪崨濞掝亪骞夐敍鍕床婵炴垯鍨圭痪褔鏌熺€电ǹ浠滈柡瀣Т椤啴濡堕崘銊т痪闂佹寧娲忛崹褰掓偩閻戠瓔鏁冮柨鏇楀亾閸烆垶鎮峰⿰鍐伇缂侇噮鍘藉鍕箾閻愵剚鏉搁梺鍦劋婵炲﹤鐣烽幇鏉跨缂備焦锚閳ь剙娼¢弻銊╁籍閳ь剙鐣峰Ο缁樺弿闁惧浚鍋呴崣蹇斾繆椤栨氨浠㈤柣鎾村姍閺岋綁骞樺畷鍥╊啋闂佸搫鏈惄顖炲春閸曨垰绀冮柍鍝勫枤濡茶埖淇婇悙顏勨偓褏鎷嬮敐鍡曠箚闁搞儺鍓欓悞鍨亜閹哄棗浜惧┑鐘亾閺夊牄鍔庢禒姘繆閻愵亜鈧倝宕㈡總绋垮簥闁哄被鍎查崑鈺呮煟閹达絽袚闁哄懏鐓¢弻娑㈠Ψ椤栫偞顎嶉梺鍛婃礀閸熸潙顫忛搹鍦煓闁圭ǹ瀛╅幏鍗烆渻閵堝啫濡奸柟鍐茬箳缁顓兼径濠勭暰濡炪値鍏橀埀顒€纾粔娲煛娴g懓濮嶇€规洏鍔戦、娆撳礂閸忚偐鏆梻鍌氬€风粈渚€骞夐垾瓒佹椽鎮㈤搹閫涚瑝闂佸搫绋侀崢濂告嫅閻斿吋鐓ユ繝闈涙-濡插綊鏌涙繝鍕幋闁哄本绋戦埢搴ょ疀閿濆棌鏋旀繝纰樻閸嬪懘宕归崹顕呮綎婵炲樊浜濋悞濠氭煟閹邦垰钄奸悗姘嵆閺屾稑螣缂佹ê鈧劙鏌″畝瀣М妤犵偞甯¢幃娆撴偨閸偅顔撻梺璇插椤旀牠宕抽鈧畷婊堟偄妞嬪孩娈鹃梺鍦劋閸╁牆岣块埡鍛叆婵犻潧妫欓ˉ鐘绘煕濞嗗繐鏆炵紒缁樼箓閳绘捇宕归鐣屼壕闂備胶顢婂▍鏇㈠箰閸濄儱寮查梻浣虹帛鏋い鏇嗗懎顥氬┑鐘崇閻撴瑩鏌熼鍡楁噺閹插吋绻濆▓鍨仭闁瑰憡濞婂璇测槈濡攱顫嶅┑顔筋殔閻楀﹪寮ィ鍐┾拺闂傚牃鏅濈粙濠氭煙椤旂厧鈧灝顕f繝姘櫜闁糕剝锚閸斿懘姊洪棃娑氱濠殿喗鎸冲绋库枎閹惧鍘介梺缁樏崯鎸庢叏婢舵劖鐓曢柣妯虹-婢х數鈧娲樺浠嬪春閳ь剚銇勯幒宥夋濞存粍绮撻弻鐔衡偓鐢登规禒婊勩亜閺囩喓鐭嬮柕鍥у閺佸啴宕掗妶鍡╂缂傚倷娴囨ご鎼佸箰閹间緡鏁囧┑鍌溓瑰钘壝归敐鍤借绔熸惔銊︹拻濞达絼璀﹂弨鐗堢箾閸涱喗绀嬮柟顔ㄥ洦鍋愰悹鍥皺閻ゅ洭姊虹紒妯曟垵顪冮崸妤€鏋侀柛鈩冪⊕閻撴洟鏌熼柇锕€鏋涘ù婊堢畺閺岋箓骞嬪┑鎰ㄧ紓浣介哺閹瑰洤鐣烽幒鎴旀瀻闁瑰瓨绻傞‖澶愭⒒娴e憡鍟為柛鏃€娲熼垾锕傛倻閻e苯绁﹂棅顐㈡处缁嬫帡寮查幖浣圭叆闁绘洖鍊圭€氾拷28缂傚倸鍊搁崐鎼佸磹閹间礁纾瑰瀣椤愪粙鏌ㄩ悢鍝勑㈢痪鎹愵嚙椤潡鎳滈棃娑樞曢梺杞扮椤戝洭骞夐幖浣哥睄闁割偅绋堥崑鎾存媴閼叉繃妫冨畷銊╊敊闂傚鐩庨梻鍌欑劍閸庡磭鎹㈤幇顒婅€块梺顒€绉甸崑鍌炴倵閿濆骸鏋熼柍閿嬪灴閹嘲鈻庤箛鎿冧痪闂佺ǹ瀛╅〃濠囧蓟濞戙垹惟闁靛/宥囩缂傚倷绶¢崰鏍偋婵犲嫭宕查柛娑卞幐閺嬪海绱掗娑氼暡缂佺粯绻堥幃浠嬫濞戞鍕冩繝鐢靛仜閸氬鎮烽埡浣烘殾闁靛/鈧崑鎾绘晲鎼粹剝鐏嶉梺鎼炲€曢懟顖濈亙闂佹寧绻傞幊搴ㄥ汲濞嗘垹纾奸柣姗€娼ч弸娑㈡煛鐏炵喎妫涢悿鈧梺鎸庣箓閹冲秶鑺遍崗鑲╃瘈婵炲牆鐏濋悘锟犳煙閸涘﹤鈻曢柣娑卞櫍瀵粙鈥栭妷銉╁弰妞ゃ垺顨婇崺鈧い鎺戝閻掑灚銇勯幒宥囧妽缂佲偓閳ь剟姊哄畷鍥╁笡闁圭懓娲顐﹀箻缂佹ɑ娅㈤梺璺ㄥ櫐閹凤拷128.00闂傚倸鍊搁崐鎼佸磹閻戣姤鍤勯柤鍝ユ暩娴犳氨绱撻崒娆掑厡缂侇噮鍨跺畷婵嗏枎韫囨洘娈鹃梺鍝勵槸缁ㄥ崬危閸喐鍙忔慨妤€妫楀鐐箾閼测晛鏋涙慨濠呮閹瑰嫰濡搁妷锔惧綒闂備胶鎳撻崵鏍箯閿燂拷