REVIEW
Deep Inelastic Structure Functions at two Loops
High Energy Physics - Phenomenology · Submitted 15 Dec 1999 · 48 references
Pith's verdict: A careful, method-driven recomputation of the two-loop DIS coefficient functions; the lasting contribution is the Mellin-space recursion machinery, not the numbers.
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Referee report
The paper
The paper claims
A second, independent two-loop calculation of the deep inelastic structure functions $F_2$, $F_3$ and $F_L$ reproduces the known coefficient functions and anomalous dimensions, and validates a Mellin-space recursion method aimed at three-lo
The reading
What carries the argument
The engine is the projection operator $P_N = [q^{[mu_1}...q^{mu_N]}/N!] (d^N/dp_{mu_1}...dp_{mu_N})$ evaluated at $p=0$, applied to the partonic invariants $T_{i,p}$. It collapses each four-point Feynman diagram to a two-point integral with symbolic powers of the momenta, which is then reduced by integration-by-parts and scaling identities into recursion relations in $N$ — first-order difference equations whose solutions are products of Gamma functions and nested sums. Those sums are harmonic sums $S_{m_1,...,m_k}(N)$, and the paper leans on the fact that the sums it encounters close into that same space, so the whole answer stays expressible in harmonic sums. The inverse Mellin transform then maps harmonic sums to harmonic polylogarithms $H_{m_1,...,m_w}(x)$, the natural basis in $x$-space. Singlet mixing anomalous dimensions $gamma^{(1)}_{gq}$ and $gamma^{(1)}_{gg}$ are reached by a second set of Green's functions in which the photon is replaced by a fictitious scalar $phi$ coupling through $phi F^a_{mu nu} F^{a mu nu}$, whose coefficient functions start at order $alpha_s^0$ for the gluon and therefore expose the gluon-to-gluon and gluon-to-quark entries of the renormalization matrix.
Extended reading notes
Core claim
The paper's claim is that the complete set of two-loop ingredients for unpolarized deep inelastic scattering — the singlet and non-singlet anomalous dimensions $P^{(1)}_{qq}$, $P^{(1)}_{qg}$, $P^{(1)}_{gq}$, $P^{(1)}_{gg}$ and the coefficient functions $c^{(2)}_{2,q}$, $c^{(2)}_{3,q}$, $c^{(2)}_{2,g}$, $c^{(2)}_{L,q}$, $c^{(2)}_{L,g}$ — can be obtained as analytic functions of the Mellin moment $N$ and, after inverse Mellin transformation, as functions of $x$ written in harmonic polylogarithms, and that these agree with the published Zijlstra–van Neerven results. Agreement is the load-bearing content: two calculations sharing almost no intermediate machinery landing on the same rational-and-zeta-valued functions is strong evidence that neither is wrong. The authors also report agreement of the anomalous dimensions with the older literature and partial agreement for $F_L$ with earlier longitudinal-structure-function calculations, with the residual discrepancies attributed in part to a factor of $(1-2x)$ from the gluon polarization sum in $D = 4-2x$ dimensions that had been omitted elsewhere.
Load-bearing premise
The singlet mixing anomalous dimensions are extracted by assuming that operators which are BRST variations or vanish by the equations of motion contribute nothing to on-shell matrix elements, so the renormalization matrix can be truncated to its quark and gluon entries.
Editorial extensions
If this is right
- Every two-loop coefficient function and splitting function used in NNLO analyses of deep inelastic data now has a second, structurally independent derivation behind it.
- The longitudinal structure function $F_L$ at two loops is pinned down, removing an ambiguity that had propagated into determinations of $R = sigma_L/sigma_T$ and into fits of the gluon density.
- The recursion-plus-harmonic-sum method is demonstrated to be practical for two-loop DIS and is put forward as the route to the three-loop anomalous dimensions, the missing ingredient for full NNLO evolution.
- All order-$epsilon^2$ one-loop and order-$epsilon$ two-loop terms are obtained as a byproduct, which is exactly what mass factorization needs to extract three-loop coefficient functions.
- The same machinery is stated to be applicable to the polarized structure function $g_1$ at two loops.
Reading between the lines
- Because the two calculations share essentially no intermediate steps, the agreement functions as an error bar of zero on the two-loop DIS coefficient functions; the natural next check is whether the same recursion machinery reproduces the known three-loop non-singlet splitting functions, which would be a genuine test rather than a replication.
- The claim that unphysical operators decouple from on-shell matrix elements is what makes the singlet truncation legitimate; if a future three-loop computation finds a leftover contribution from such operators, the singlet entries would need revision while the non-singlet sector and coefficient functions would be untouched.
- The harmonic-sum and harmonic-polylogarithm bases used here are the same ones that later organize multi-loop QCD amplitudes generally, so the paper's practical contribution may outlast the specific two-loop numbers it reports.
Original abstract
We present the analytic calculation of the Mellin moments of the structure functions F_2, F_3 and F_L in perturbative QCD up to second order corrections and in leading twist approximation. We calculate the 2-loop contributions to the anomalous dimensions of the singlet and non-singlet operator matrix elements and the 2-loop coefficient functions of F_2, F_3 and F_L. We perform the inverse Mellin transformation analytically and find our results in agreement with earlier calculations in the literature by Zijlstra and van Neerven.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We present the analytic calculation of the Mellin moments of the structure functions F_2, F_3 and F_L in perturbative QCD up to second order corrections and in leading twist approximation. We calculate the 2-loop contributions to the anomalous dimensions of the singlet and non-singlet operator matrix elements and the 2-loop coefficient functions of F_2, F_3 and F_L. We perform the inverse Mellin transformation analytically and find our results in agreement with earlier calculations in the literature by Zijlstra and van Neerven.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present the analytic calculation of the Mellin moments of the structure functions F_2, F_3 and F_L in perturbative QCD up to second order corrections and in leading twist approximation. | contribution 1.00 | 0.95 | 0.39 | asserted 1.00 |
| s1 | We calculate the 2-loop contributions to the anomalous dimensions of the singlet and non-singlet operator matrix elements and the 2-loop coefficient functions of F_2, F_3 and F_L. | contribution 0.93 | 0.97 | 0.17 | asserted 1.00 |
| s2 | We perform the inverse Mellin transformation analytically and find our results in agreement with earlier calculations in the literature by Zijlstra and van Neerven. | result 0.71 | 0.93 | 0.37 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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- P abstract
- S sections
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Figures
Figures reproduced from arXiv:hep-ph/9912355 by the authors.
References
How the literature uses it
48 arXiv papers cite this in 81 sentences
Papers citing it · 47 sentences without a confident label
Uses its method or tool 6 papers
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Therefore the representation of the heavy flavor Wilson coefficients in the region $Q^2 ≫m^2$ can be obtained using the factorization [ref] into massive operator matrix elements (OMEs) and the massless Wilson coefficients [ref] .
The $O(\alpha_s^2)$ Heavy Quark Corrections to Charged Current Deep-Inelastic Scattering at large Virtualities · Introduction · 0.89
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([ref]) can be obtained by an algebraic procedure [ref] based on the fact that harmonic sums occur as coefficients of the Taylor expansion of harmonic polylogarithms.
The double fermionic contribution to the four-loop quark-to-gluon splitting function · Introduction · 0.98
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This transformation can be performed by an algebraic procedure [ref] based on the fact that harmonic sums occur as coefficients of the Taylor expansion of HPLs.
Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond · Results in $x$-$$space · 1.00
Uses its data or values 3 papers
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[equation] where the $A_i$ are known [ref] up to $i=3$, together with the the two [ref] and three loop [ref] results for the longitudinal coefficient function.
On threshold resummation of the longitudinal structure function · Checking the ansatz · 0.96
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Our implementation of the generic NNLO corrections in Eq. [ref] is based on the two-loop DIS structure functions available in the literature [ref] .
Hadronic Final States in DIS at NNLO QCD with Parton Showers · Computational setup · 0.80
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([ref]) to two loops since there are results for $γ_G$ [ref] , $B_δ$ [ref] and $Γ_∧$ [ref] .
Wilson-line geometries and the relation between IR singularities of form factors and the large-x limit of DGLAP splitting functions · The relation · 0.86
Compares with it 4 papers
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The resulting evolution (KPS-CTT) is in full agreement with finite-order calculations if one compares its prediction for the small-$x$, large-$N_c$ $GG$ polarized splitting function of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [ref] to that calculated in [ref] .
Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$ · Introduction · 0.97
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The resulting equations have been cross-checked against the small-$x$ and large-$N_c$ part of the $Δγ_GG$ anomalous dimension to the three known loops [ref] , indicating a complete agreement with the existing fixed-order calculations (see also [ref] for other relevant calculations which, apart from presenting interesting and important results, may also be used for further cross checks of the small-$x$ resummation).
Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept · Introduction · 0.92
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Next, fixing the simple initial conditions $G^(0)_2(^2,zs) = 1$ and $G^(0)(^2,zs) = 0$, the gluon hPDF in Eq. [ref] becomes [equation] from which we can see that our prediction for the resummed all-order in $$ $GG$ polarized anomalous dimension at small $x$ and large-$N_c$ (whose subsequent expansion in powers of $$ we also show) is [equation] Reassuringly, this prediction for the anomalous dimension agrees with the fixed-order calculations out to the existing three-loop level (see [ref] ).
Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept · Intercept, Anomalous Dimension, and Comparison to BER · 0.83
Cites it as background 13 papers
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The massless corrections at NNLO to the deep-inelastic structure functions are available [ref] , while for the corresponding massive corrections in the asymptotic limit, a series of moments has been calculated in the single heavy mass case [ref] for all contributing terms in neutral current deep-inelastic scattering.
Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses · Introduction · 0.83
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In the region $Q^2 ≫m^2$ the heavy flavor Wilson coefficients for deep–inelastic scattering factorize into massive operator matrix elements $A_ij(μ^2/m^2)$ and the massless Wilson coefficients $C_k(Q^2/μ^2)$ [ref] for all but the power suppressed contributions.
Two--Loop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to $O(\epsilon) · Introduction · 0.83
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Here, the first three terms define the NNLO predictions for DIS, which is currently the standard approximation, see [ref] for the splitting functions and [ref] for the DIS coefficient functions at this accuracy.
The strong coupling constant: State of the art and the decade ahead · Strong coupling alphas in fits of parton distributions [Authors: S. Moch (Hamburg Univ.)] · 0.83
Discussion
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