N3LO singlet ad

.pre-commit-config.yaml

@@ -2,7 +2,7 @@
 # See https://pre-commit.com/hooks.html for more hooks
 repos:
   - repo: https://github.com/pre-commit/pre-commit-hooks
-    rev: v4.2.0
+    rev: v4.3.0
     hooks:
       - id: trailing-whitespace
       - id: end-of-file-fixer
@@ -12,7 +12,7 @@ repos:
       - id: debug-statements
       - id: fix-encoding-pragma
   - repo: https://github.com/psf/black
-    rev: 22.3.0
+    rev: 22.6.0
     hooks:
       - id: black
   - repo: https://github.com/asottile/blacken-docs
@@ -25,6 +25,13 @@ repos:
       - id: isort
         args: ["--profile", "black"]
   - repo: https://github.com/asottile/pyupgrade
-    rev: v2.32.1
+    rev: v2.37.1
     hooks:
       - id: pyupgrade
+  - repo: https://github.com/pycqa/pydocstyle
+    rev: 6.1.1
+    hooks:
+    -   id: pydocstyle
+        files: ^src/
+        additional_dependencies:
+          - toml

doc/source/refs.bib

@@ -722,3 +722,57 @@ @misc{Rottoli
   author        = "Rottoli, Luca",
   howpublished  = "private communication"
 }
+
+@article{Bonvini:2018xvt,
+    author = "Bonvini, Marco and Marzani, Simone",
+    title = "{Four-loop splitting functions at small $x$}",
+    eprint = "1805.06460",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    doi = "10.1007/JHEP06(2018)145",
+    journal = "JHEP",
+    volume = "06",
+    pages = "145",
+    year = "2018"
+}
+
+@article{Moch:2021qrk,
+    author = "Moch, S. and Ruijl, B. and Ueda, T. and Vermaseren, J. A. M. and Vogt, A.",
+    title = "{Low moments of the four-loop splitting functions in QCD}",
+    eprint = "2111.15561",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "DESY 21-203, NIKHEF 21-030, LTH 1282",
+    doi = "10.1016/j.physletb.2021.136853",
+    journal = "Phys. Lett. B",
+    volume = "825",
+    pages = "136853",
+    year = "2022"
+}
+
+@article{Soar:2009yh,
+    author = "Soar, G. and Moch, S. and Vermaseren, J. A. M. and Vogt, A.",
+    title = "{On Higgs-exchange DIS, physical evolution kernels and fourth-order splitting functions at large x}",
+    eprint = "0912.0369",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "LTH-857, DESY-09-211, SFB-CPP-09-119, NIKHEF-09-031",
+    doi = "10.1016/j.nuclphysb.2010.02.003",
+    journal = "Nucl. Phys. B",
+    volume = "832",
+    pages = "152--227",
+    year = "2010"
+}
+
+@article{Albino:2000cp,
+    author = "Albino, Simon and Ball, Richard D.",
+    title = "{Soft resummation of quark anomalous dimensions and coefficient functions in MS-bar factorization}",
+    eprint = "hep-ph/0011133",
+    archivePrefix = "arXiv",
+    reportNumber = "CERN-TH-2000-332, EDINBURGH-2000-23",
+    doi = "10.1016/S0370-2693(01)00742-0",
+    journal = "Phys. Lett. B",
+    volume = "513",
+    pages = "93--102",
+    year = "2001"
+}

doc/source/shared/abbreviations.rst

@@ -30,6 +30,15 @@
 .. |N3LO| replace::
    :abbr:`N3LO (Next-to-Next-to-Next-to-Leading Order)`
 
+.. |LL| replace::
+   :abbr:`LL (Leading Log)`
+
+.. |NLL| replace::
+   :abbr:`NLL (Next-to-Leading Log)`
+
+.. |NNLL| replace::
+   :abbr:`NNLL (Next-to-Next-to-Leading Log)`
+
 .. Names
 
 .. |DGLAP| replace::
@@ -60,6 +69,11 @@
 .. |QED| replace::
    :abbr:`QED (Quantum Electrodynamics)`
 
+.. |DIS| replace::
+   :abbr:`DIS (Deep Inelastic Scattering)`
+
+.. |BFKL| replace::
+   :abbr:`BFKL (Balitsky-Fadin-Kuraev-Lipatov)`
 .. external
 
 .. |yadism| replace::

doc/source/theory/N3LO_ad.rst

@@ -2,23 +2,23 @@ N3LO Anomalous Dimensions
 =========================
 
 The |N3LO| |QCD| anomalous dimensions :math:`\gamma^{(3)}` are not yet fully known,
-since they rely on the calculation of 4-loop DIS integrals.
-Moreover the analytical structure of these function is already known to be complicated
+since they rely on the calculation of 4-loop |DIS| integrals.
+Moreover, the analytical structure of these function is already known to be complicated
 since in Mellin space it will included harmonics sum up to weight 7, for which an
-analytical contribution is not available.
+analytical expression is not available.
 
-Here we describe the various assumptions and limits used in order to reconstruct a parametrization
+Here, we describe the various assumptions and limits used in order to reconstruct a parametrization
 that can approximate their contribution.
 In particular we will take advantage of some known physical constrain,
 such as large-x limit, small-x limit, and sum rules, in order to make our reconstruction reasonable.
 
-Generally we remark that the large-x limit correspond to large-N in Mellin space
+Generally, we remark that the large-x limit correspond to large-N in Mellin space
 where the leading contribution comes from the harmonics :math:`S_1(N)`,
 while the small-x region corresponds to poles at :math:`N=0,1` depending on the type of
 divergence.
 
 In any case |N3LO| |DGLAP| evolution at small-x, especially for singlet-like PDFs, will not be reliable
-until the splitting function resummation will not be available up to NNLL.
+until the splitting function resummation will not be available up to |NNLL|.
 
 Non-singlet sector
 ------------------
@@ -33,6 +33,7 @@ In particular, making explicitly the dependence on :math:`n_f`, the non-singlet
 the following terms:
 
     .. list-table:: non-singlet 4-loop Anomalous Dimensions
+        :align: center
         :header-rows: 1
 
         *   -
@@ -80,7 +81,7 @@ In |EKO| they are implemented as follows:
             This part contains the so called double logarithms:
 
             .. math ::
-                \ln(x)^k \quad k=1,..,6, \quad \mathcal{M}[\ln^k(x)] = \frac{1}{N^{(k+1)}}
+                \ln^k(x), \quad \mathcal{M}[\ln^k(x)] = \frac{1}{N^{k+1}}, \quad k=1,..,6
 
             Note the expressions are evaluated with the exact values of the |QCD|
             Casimir invariants, to better agree with the :cite:`Moch:2017uml` parametrization.
@@ -96,7 +97,7 @@ In |EKO| they are implemented as follows:
             :cite:`Henn:2019swt`, while the :math:`B_4` is fixed by the integral of the 4-loop splitting function.
             :math:`C_4,D_4` instead can be computed directly from lower order splitting functions.
             From large-x resummation :cite:`Davies:2016jie`, it is possible to infer further constrains
-            on sub-leading terms :math:`\frac{\ln(N)^k}{N^2}`, since the non-singlet splitting
+            on sub-leading terms :math:`\frac{\ln^k(N)}{N^2}`, since the non-singlet splitting
             functions contain only terms :math:`(1-x)^a\ln^k(1-x)` with :math:`a \ge 1`.
 
         -   The 8 lowest odd or even N moments provided in :cite:`Moch:2017uml`, where
@@ -106,38 +107,36 @@ In |EKO| they are implemented as follows:
         -   The difference between the known moments and the known limits is parametrized
             in Mellin space. The basis includes:
 
-            .. list-table::
+            .. list-table:: :math:`\gamma_{ns,\pm}^{(3)}` parametrization basis
+                :align: center
                 :header-rows: 1
 
                 *   - x-space
                     - N-space
                 *   - :math:`\delta(1-x)`
                     - 1
                 *   - :math:`(1-x)\ln(1-x)`
-                    - :math:`\mathcal{M}[(1-x)\ln(1-x)] \approx \frac{S_1(N)}{N^2}`
+                    - :math:`\mathcal{M}[(1-x)\ln(1-x)]`
                 *   - :math:`(1-x)\ln^2(1-x)`
-                    - :math:`\mathcal{M}[(1-x)\ln^2(1-x)] \approx \frac{S_1^2(N)}{N^2}`
+                    - :math:`\mathcal{M}[(1-x)\ln^2(1-x)]`
                 *   - :math:`(1-x)\ln^3(1-x)`
-                    - :math:`\mathcal{M}[(1-x)\ln^3(1-x)] \approx \frac{S_1^3(N)}{N^2}`
-                *   - :math:`- Li_2(x) + \zeta_2`
+                    - :math:`\mathcal{M}[(1-x)\ln^3(1-x)]`
+                *   - :math:`- \rm{Li_2}(x) + \zeta_2`
                     - :math:`\frac{S_1(N)}{N^2}`
-
-            which model the sub-leading differences in the :math:`N\to \infty` limit, and:
-
-            .. list-table::
-                :header-rows: 1
-
-                *   - x-space
-                    - N-space
                 *   - :math:`x\ln(x)`
                     - :math:`\frac{1}{(N+1)^2}`
                 *   - :math:`\frac{x}{2}\ln^2(x)`
                     - :math:`\frac{1}{(N+1)^3}`
+                *   - :math:`x^{2}, x^{3}`
+                    - :math:`\frac{1}{(N-2)},\frac{1}{(N-3)}`
 
+            The first five functions model the sub-leading differences in the :math:`N\to \infty` limit,
+            while the last three help the convergence in the small-N region. Finally, we add a polynomial part
+            :math:`x^{2}` or :math:`x^{3}` respectively for :math:`\gamma_{ns,+},\gamma_{ns,-}`.
+            For large-N we have the limit:
 
-            to help the convergence in the small-N region. Finally we add a polynomial part
-            :math:`x^{2(3)}` which corresponds to simple poles at :math:`N=-2,-3`
-            respectively for :math:`\gamma_{ns,+},\gamma_{ns,-}`.
+                .. math ::
+                    \mathcal{M}[(1-x)\ln^k(1-x)] \approx \frac{S_1^k(N)}{N^2}
 
             Note that the constant coefficient is included in the fit, following the procedure done
             in :cite:`Moch:2017uml` (section 4), to achieve a better accuracy.
@@ -147,7 +146,7 @@ Singlet sector
 --------------
 
 In the singlet sector we construct a parametrization for
-:math:`\gamma_{gg}^{(3)},\gamma_{gq}^{(3)},\gamma_{qq}^{(3)},\gamma_{qg}^{(3)}` where:
+:math:`\gamma_{gg}^{(3)},\gamma_{gq}^{(3)},\gamma_{qg}^{(3)},\gamma_{qq}^{(3)}` where:
 
     .. math ::
         \gamma_{qq}^{(3)} = \gamma_{ns,+}^{(3)} + \gamma_{qq,ps}^{(3)}
@@ -156,6 +155,7 @@ In particular, making explicitly the dependence on :math:`n_f`, the singlet anom
 the following terms:
 
     .. list-table:: singlet 4-loop Anomalous Dimensions
+        :align: center
         :header-rows: 1
 
         *   -
@@ -172,23 +172,123 @@ the following terms:
             - |T|
 
         *   - :math:`\gamma_{gq}^{(3)}`
-            -
+            - |T|
             - |T|
             - |T|
             - |T|
 
-        *   - :math:`\gamma_{qq,ps}^{(3)}`
+        *   - :math:`\gamma_{qg}^{(3)}`
             -
             - |T|
             - |T|
             - |T|
 
-        *   - :math:`\gamma_{qg}^{(3)}`
-            - |T|
+        *   - :math:`\gamma_{qq,ps}^{(3)}`
+            -
             - |T|
             - |T|
             - |T|
 
 Only the parts proportional to :math:`n_f^3` are known analytically
 :cite:`Davies:2016jie` and have been included so far.
-The rest will be approximated using some known limits.
+The other parts are approximated using some known limits:
+
+* The remaining contributions include the following constrains.
+
+    *   The small-x limit, given in the large :math:`N_c` approximation by
+        :cite:`Davies:2022ofz` (see Eq. 5.9, 5.10, 5.11, 5.12) and coming
+        from small-x resummation of double-logarithms which fix the leading terms
+        for the pole at :math:`N=0`:
+
+            .. math ::
+                \ln^k(x), \quad \mathcal{M}[\ln^k(x)] = \frac{1}{N^{k+1}}, \quad k=4,5,6
+
+    *   The small-x limit, coming from |BFKL| resummation
+        :cite:`Bonvini:2018xvt` (see Eq. 2.32, 2.20b, 2.21a, 2.21b)
+        which fix the leading terms (|LL|, |NLL|) for the pole at :math:`N=1`:
+
+            .. math ::
+                \frac{\ln^k(x)}{x}, \quad \mathcal{M}[\frac{\ln^k(x)}{x}] = \frac{1}{(N-1)^{k+1}}, \quad k=4,5
+
+        Note that in principle also the term :math:`\frac{\ln^6(x)}{x}` could be present at |N3LO|,
+        but they are vanishing.
+        These terms are way larger than the previous ones in the small-x limit and
+        are effectively determining the raise of the splitting functions at small-x.
+        In particular only the expansion for :math:`\gamma_{gg}^{(3)}` is known at |NLL|.
+        |LL| terms respect the representation symmetry :
+
+            .. math ::
+                \gamma_{gq} & \approx \frac{C_F}{C_A} \gamma_{gg}  \\
+                \gamma_{qq,ps} & \approx \frac{C_F}{C_A} \gamma_{qg} \\
+
+
+    *   The large-x limit of the singlet splitting function is different for the diagonal part
+        and the off-diagonal.
+        It is known that :cite:`Albino:2000cp,Moch:2021qrk` the diagonal terms diverge in N-space as:
+
+            .. math ::
+                \gamma_{kk} \approx A_4 S_1(N)  + \mathcal{O}(1)
+
+        Where again the coefficient :math:`A_4` is the |QCD| cusp anomalous dimension. However, :math:`\gamma_{qq,ps}^{(3)}`
+        do not constrain any divergence at large-x or constant term so its expansion will start as
+        :math:`\mathcal{O}(\frac{1}{N^2})`.
+        The off-diagonal do not contain any +-distributions or delta distributions but can include divergent logarithms
+        of the type :cite:`Soar:2009yh`:
+
+            .. math ::
+                \ln^k(1-x) \quad k=1,..,6
+
+        where also in this case the term :math:`k=6` vanish. The values of the coefficient for :math:`k=4,5`
+        can be guessed from the lower order splitting functions. These logarithms are not present in the diagonal
+        splitting function, which can include at most term :math:`(1-x)\ln^4(1-x)`.
+
+
+    *   The 4 lowest even N moments provided in :cite:`Moch:2021qrk`, where we can use momentum conservation
+        to fix:
+
+            .. math ::
+                & \gamma_{qg}(2) + \gamma_{gg}(2) = 0 \\
+                & \gamma_{qq}(2) + \gamma_{gq}(2) = 0 \\
+
+    *   Finally difference between the known moments and the known limits is parametrized
+        in Mellin space. The basis used in this approximation is different for each splitting
+        function as listed in the following tables.
+
+
+        .. list-table::  :math:`\gamma_{gg}^{(3)}` parametrization basis
+            :align: center
+
+            *   - :math:`\frac{1}{(N-1)^2}`
+                - :math:`\frac{1}{(N-1)}`
+                - :math:`1`
+                - :math:`\mathcal{M}[\ln(1-x)](N)`
+
+        .. list-table::  :math:`\gamma_{gq}^{(3)}` parametrization basis
+            :align: center
+
+            *   - :math:`\frac{1}{(N-1)^3}`
+                - :math:`\frac{1}{(N-1)^2}`
+                - :math:`\mathcal{M}[\ln^3(1-x)](N)`
+                - :math:`\mathcal{M}[(1-x)\ln^3(1-x)](N)`
+
+        .. list-table::  :math:`\gamma_{qg}^{(3)}` parametrization basis
+            :align: center
+
+            *   - :math:`\frac{1}{(N-1)^2}`
+                - :math:`\frac{1}{(N-1)}`
+                - :math:`\mathcal{M}[\ln^3(1-x)](N)`
+                - :math:`\mathcal{M}[(1-x)\ln^3(1-x)](N)`
+
+        .. list-table::  :math:`\gamma_{qq,ps}^{(3)}` parametrization basis
+            :align: center
+
+            *   - :math:`\frac{1}{(N-1)^2} - \frac{1}{N^2}`
+                - :math:`\frac{1}{(N-1)} - \frac{1}{N}`
+                - :math:`\frac{S_1^3(N)}{N^2}`
+                - :math:`\frac{S_1^2(N)}{N^2}`
+
+        Note that for :math:`\gamma_{qq,ps},\gamma_{qg}` the parts proportional
+        to :math:`n_f^0` are not present.
+        Furthermore for the part :math:`\propto n_f^2` in :math:`\gamma_{gq}^{(3)}`
+        we adopt a slightly different basis to account fot the fact that the leading
+        contribution for the pole at :math:`N=1` is :math:`\frac{1}{(N-1)^2}`.