N3LO non singlet anomalous dimesions

doc/source/index.rst

@@ -36,6 +36,7 @@ EKO is ...
     theory/FlavorSpace
     theory/pQCD
     theory/DGLAP
+    theory/N3LO_ad
     theory/Matching
 
     zzz-refs

doc/source/refs.bib

@@ -654,3 +654,56 @@ @article{Luthe:2017ttg
     pages = "166",
     year = "2017"
 }
+
+@article{Davies:2016jie,
+    author = "Davies, J. and Vogt, A. and Ruijl, B. and Ueda, T. and Vermaseren, J. A. M.",
+    title = "{Large-$n_f$ contributions to the four-loop splitting functions in QCD}",
+    eprint = "1610.07477",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "LTH-1106, NIKHEF-2016-049",
+    doi = "10.1016/j.nuclphysb.2016.12.012",
+    journal = "Nucl. Phys. B",
+    volume = "915",
+    pages = "335--362",
+    year = "2017"
+}
+
+@article{Moch:2017uml,
+    author = "Moch, S. and Ruijl, B. and Ueda, T. and Vermaseren, J. A. M. and Vogt, A.",
+    title = "{Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond}",
+    eprint = "1707.08315",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "DESY-17-106, NIKHEF-2017-034, LTH-1139",
+    doi = "10.1007/JHEP10(2017)041",
+    journal = "JHEP",
+    volume = "10",
+    pages = "041",
+    year = "2017"
+}
+
+@article{Davies:2022ofz,
+    author = "Davies, J. and Kom, C. -H. and Moch, S. and Vogt, A.",
+    title = "{Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering}",
+    eprint = "2202.10362",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "LTH 1289, DESY 21-225",
+    month = "2",
+    year = "2022"
+}
+
+@article{Henn:2019swt,
+    author = "Henn, Johannes M. and Korchemsky, Gregory P. and Mistlberger, Bernhard",
+    title = "{The full four-loop cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills and QCD}",
+    eprint = "1911.10174",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-th",
+    reportNumber = "IPhT-T19/155, MIT-CTP/5160, MPP-2019-233",
+    doi = "10.1007/JHEP04(2020)018",
+    journal = "JHEP",
+    volume = "04",
+    pages = "018",
+    year = "2020"
+}

doc/source/theory/N3LO_ad.rst

@@ -0,0 +1,144 @@
+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 in Mellin space it will included harmonics sum up to weight 7, for which an
+analytical contribution is not available.
+
+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
+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.
+
+Non-singlet sector
+------------------
+
+In the non-singlet sector we construct a parametrization for
+:math:`\gamma_{ns,-}^{(3)},\gamma_{ns,-}^{(3)},\gamma_{ns,s}^{(3)}` where:
+
+    .. math ::
+        \gamma_{ns,s}^{(3)} = \gamma_{ns,v}^{(3)} - \gamma_{ns,-}^{(3)}
+
+In particular, making explicitly the dependence on :math:`n_f`, the non-singlet anomalous dimensions include
+the following terms:
+
+    .. list-table:: non-singlet 4-loop Anomalous Dimensions
+        :header-rows: 1
+
+        *   -
+            - :math:`n_{f}^0`
+            - :math:`n_{f}^1`
+            - :math:`n_{f}^2`
+            - :math:`n_{f}^3`
+
+        *   - :math:`\gamma_{ns,-}^{(3)}`
+            - |T|
+            - |T|
+            - |T|
+            - |T|
+
+        *   - :math:`\gamma_{ns,+}^{(3)}`
+            - |T|
+            - |T|
+            - |T|
+            - |T|
+
+        *   - :math:`\gamma_{ns,s}^{(3)}`
+            -
+            - |T|
+            - |T|
+            -
+
+Some of these parts are known analytically exactly (:math:`\propto n_f^2,n_f^3`),
+while others are available only in the large :math`N_c` limit (:math:`\propto n_f^0,n_f^1`).
+In |EKO| they are implemented as follows:
+
+    * the part proportional to :math:`n_f^3` is common for :math:`\gamma_{ns,+}^{(3)},\gamma_{ns,-}^{(3)},\gamma_{ns,v}^{(3)}`
+      and is exact :cite:`Davies:2016jie` (Eq. 3.6).
+
+    * In :math:`\gamma_{ns,s}^{(3)}` the part proportional to :math:`n_f^2`
+      is exact :cite:`Davies:2016jie` (Eq. 3.5).
+
+    * In :math:`\gamma_{ns,s}^{(3)}` the part proportional to :math:`n_f^1` is
+      parametrized in x-space and copied from :cite:`Moch:2017uml` (Eq. 4.19, 4.20).
+
+    * 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. 3.3, 3.8, 3.9, 3.10) and coming
+            from small-x resummation.
+            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)}}
+
+            Note the expressions are evaluated with the exact values of the |QCD|
+            Casimir invariants, to better agree with the :cite:`Moch:2017uml` parametrization.
+
+        -   The large-N limit :cite:`Moch:2017uml`, which reads (Eq. 2.17):
+
+            .. math ::
+                \gamma_{ns} \approx A_4 S_1(N) - B_4 + C_4 \frac{S_1(N)}{N} - (D_4 + \frac{1}{2} A_4) \frac{1}{N} + \mathcal{O}(\frac{\ln^k(N)}{N^2})
+
+            This limit is common for all :math:`\gamma_{ns,+}^{(3)},\gamma_{ns,-}^{(3)},\gamma_{ns,v}^{(3)}`.
+            The coefficient :math:`A_4`, being related to the twist-2 spin-N operators,
+            can be obtained from the |QCD| cusp calculation
+            :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
+            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
+            from quark number conservation we can trivially obtain:
+            :math:`\gamma_{ns,s}(1)=\gamma_{ns,-}(1)=0`.
+
+        -   The difference between the known moments and the known limits is parametrized
+            in Mellin space. The basis includes:
+
+            .. list-table::
+                :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:`(1-x)\ln^2(1-x)`
+                    - :math:`\mathcal{M}[(1-x)\ln^2(1-x)] \approx \frac{S_1^2(N)}{N^2}`
+                *   - :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:`\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}`
+
+
+            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,-}`.
+
+            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.
+            It is checked that this contribution is much more smaller than the values of :math:`B_4`.