Structure functions for large $x$ and renormalization of Wilson loops

G.Marchesini; G.P.Korchemsky

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Structure functions for large $x$ and renormalization of Wilson loops

G.P.Korchemsky, G.Marchesini

High Energy Physics - Phenomenology · Submitted 30 Oct 1992 · 21 references

Pith's verdict: The two-loop evaluation of the light-like Wilson loop is the real content, and it holds up; the RG equation itself is from the authors' earlier paper, and the constant term is fitted rather than derived.

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arXiv IDhep-ph/9210281 (version 2)

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CitationG.Marchesini, G.P.Korchemsky. "Structure functions for large $x$ and renormalization of Wilson loops." arXiv:hep-ph/9210281 (1992). https://pith.science/paper/YYS6RKBJ

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Editor's note

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Worth your time, with one caveat about where the argument is soft and one about what is actually new here. The real content is Sect. 3: an explicit two-loop evaluation of the light-like Wilson loop W(CS) in Feynman gauge, diagram by diagram, with the integrals I1 through I11 written out. That calculation is what makes the rest credible. It reproduces the L^3 coefficient proportional to the one-loop beta function and the L^2 coefficient built from the independently published two-loop cusp anomalous dimension of ref. [17]. Those are genuine checks against outside input, not fits. The payoff is Eq. (5.5), (1-z)P(z) = 2 Gamma_cusp(g) theta(1-z), which ties the singular large-x splitting function to one universal number. That relation has held up, and it is why people cite this paper. The RG equation (4.5) itself is not new. It comes from Korchemskaya and Korchemsky, ref. [16], and the paper says so plainly in Sect. 4.2. What is new is the two-loop verification and the constant term Gamma(g) in (4.6). The soft spot is the step from (4.2) to (4.5). It is a continuation, not a derivation: shift off the light cone, take the large-angle form of Gamma_cusp, differentiate with respect to (v.y), then set y^2 = 0. The paper does not show that limit exists. But the O(gamma^0) piece of Gamma_cusp is a function of g alone, so it drops out of the differentiated equation and can only shift the integration constant. The concern is real as a heuristic gap, and it does not damage the L-dependent prediction, which is exactly what the two-loop result tests. I would not hold it against them. Second, Gamma(g) is read off by matching to the same two-loop calculation, so the constant term is fitted, not predicted. The paper is upfront about this: Gamma(g) is path-dependent and h(g) in (5.4) has to be computed separately from diagrams. Fine, but do not quote (4.6) as a derived result. The 2005 note is honest: I11 was wrong, they recomputed it, and updated (3.6), the D coefficient, and (4.6). Anyone using a pre-2005 value of Gamma(g) should check which version they have. This is for threshold-resummation people and anyone working on Wilson loops and the cusp anomalous dimension. Yes to referees.

Referee report

Significance

If correct, the result reduces the singular large-x behaviour of all quark and gluon splitting functions to a single universal quantity, the cusp anomalous dimension, and explains the known universality of the double-logarithmic resummation as a property of light-like Wilson loops. The paper's strengths are concrete: the two-loop calculation is carried out diagram by diagram with all eleven integrals displayed, so the L-structure of (3.7) is checkable; the relation (5.5) is parameter-free and falsifiable against the independently computed two-loop splitting functions of ref. [18]; and the authors are explicit that Γ(g) is path-dependent and that h(g) in (5.4) is not fixed by the Wilson-loop RG equation and must be computed separately. The 2005 note is a transparent correction rather than a silent revision, which is to the authors' credit.

Major objections (2)

[Sect. 4.2, Eqs. (4.2)–(4.5)] The step from the standard cusp equation (4.2) for y² ≠ 0 to the light-cone equation (4.5) is the least controlled part of the argument and deserves to be stated as such. The derivation shifts the contour off the light cone, uses the large-angle form Γ_cusp(γ) = γΓ_cusp(g) + O(γ⁰), differentiates with respect to (v·y), and then sets y² = 0. Two assumptions are being made silently: that the O(γ⁰) piece of Γ_cusp is (v·y)-independent (true, since it is a function of g only, but worth one sentence), and that ∂_{v·y} ln W has a smooth y² → 0 limit so that no hidden y²-dependence survives in the integration constant Γ(g). The paper does not prove the second point; it checks the resulting L-structure against (3.7). That check is genuine for the L⁴, L³ and L² coefficients, but it is not a proof that the limit exists. I would ask the authors to state the assumptions explicitly in Sect. 4.2 and to say plainly that (4.5) is proposed and then tested, rather than derived.
[Abstract; Sect. 3.4 and Eq. (4.6)] The abstract's claim that the two-loop calculation 'verifies these properties' should be qualified. Of the four coefficients in (3.7), the vanishing of L⁴, the value of the L³ coefficient (proportional to the one-loop beta function) and the value of the L² coefficient (a combination of β₀ and the independently published two-loop Γ_cusp of ref. [17]) are non-trivial checks. The coefficient of L, however, is by construction the definition of Γ(g) in (4.6), obtained by matching to the same two-loop result; it is a determination, not a verification. This does not weaken the central claim, but the abstract and the sentence 'One easily check that two-loop expression (3.7) ... does satisfy eq.(4.4)' should distinguish which coefficients are predicted and which are fitted.

Minor objections (8)

[Sect. 2.2, Eqs. (2.12)–(2.13)] The factorization F(x, µ/M) = H(µ/M) × (Fourier transform of W) is asserted rather than derived. Since H enters the definition of h(g) in (5.4) and hence the δ(1 − z) coefficient of the splitting function, it would help to state at what order in α_s and in which scheme H is defined, and to note explicitly that H carries all the collinear (as opposed to soft) virtual corrections.
[Eq. (5.2)] The convolution in (5.2) reads ∫_x^1 dz P(1 + x − z)F(z), which is not the standard DGLAP convolution; it follows from the Fourier representation (5.1) because the distribution depends on 1 − x. The subsequent sentence that (5.2) 'can be written in the standard form (4.1)' for x, z → 1 would be easier to follow with one intermediate line showing the change of variable.
[Eq. (5.7)] The identification F_n(µ/M) = H(µ/M)W(−inµ/M)(1 + O(1/n)) is obtained by evaluating the integrand at σ = −in after expanding in powers of δ. It should be stated that this holds order by order in α_s (i.e. for each power of L), since the O(1/n) remainder is not uniform in the perturbative order.
[Sect. 5, after Eq. (5.5)] The statement that the two-loop splitting functions of ref. [18] obey (5.5) is supported by a published source and need not be rederived, but displaying the explicit comparison of the δ(1 − z) and 1/(1 − z) coefficients for P_qq at two loops would make the agreement visible to the reader rather than asserted.
[Note added on February 4, 2005; Eqs. (3.6), (3.7), (4.6)] The correction of I11 and the corresponding changes to the D-coefficient and to Γ(g) are handled transparently in the note, but the body text gives no marker at the affected equations. Adding a footnote pointer at (3.6), (3.7) and (4.6) would prevent a reader of the body from using a pre-correction value.
[Sect. 3.1, discussion of I5] The result I5 = O((4 − D)⁰) is stated after an expression containing a 1/(4 − D)³ pole; a sentence explaining the cancellation of that pole would make the 'no singularity' conclusion credible on a first reading.
[Fig. 6 and Sect. 3] The diagrams in Fig. 6 are labelled (1)–(11) while the text refers to 'figs. 6.1–6.6', 'figs. 6.7 and 6.8', etc. Harmonizing the labelling between figure and text would remove an unnecessary source of confusion.
[Sect. 3.1, after I6] Typographical: 'Reimann function' should read 'Riemann function', and 'One easily check' in Sect. 4.2 should read 'One easily checks'.

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The paper

The paper claims

The paper argues that the way parton distributions evolve near $x = 1$ is dictated by the renormalization group behaviour of a Wilson loop whose path lies partly on the light cone, with the singular part of the splitting function fixed by a

The reading

The authors ask what controls the strong-interaction description of a quark that carries nearly all of a hadron's momentum, the $x to 1$ edge of phase space where perturbation theory produces large logarithms at every order. Their answer is that this corner is governed not by a local composite operator but by a Wilson loop: the vacuum expectation of a gauge phase ordered along the classical trajectories of the struck quark, with one segment lying exactly on the light cone and two cusps where the hard probe turns the quark around. Such light-like loops are not multiplicatively renormalizable, because the on-cone segment brings extra collinear singularities on top of the usual ultraviolet ones, yet the authors show the loop still obeys a renormalization group equation whose anomalous dimension grows linearly with the loop's single kinematic variable. Fourier transforming that equation in the light-cone separation turns it into the ordinary evolution equation for the distribution, and the singular part of the splitting function comes out proportional to the cusp anomalous dimension. A two-loop calculation in Feynman gauge with modified minimal subtraction is presented as the check on all of this.

What carries the argument

The load-bearing object is the renormalization group equation for a Wilson loop with a segment on the light cone. Away from the light cone the standard cusp equation applies, with the anomalous dimension a function of the Minkowski angles $gamma_pm$ between the light-like segment and the quark velocity, $cosh gamma_pm = pm (v y)/sqrt{y^2}$. The authors shift the path slightly off the cone, use the large-angle form $Gamma_{cusp}(gamma, g) = gamma Gamma_{cusp}(g) + O(gamma^0)$, continue the angles as $gamma_pm = tfrac12 ln[4((v y) mp i0)^2/(y^2 - i0)]$, differentiate the equation with respect to $(v y)$, and then send $y^2 to 0$; the dependence on $y^2$ drops out and Eq. (4.4) survives as the light-cone equation. The non-abelian exponentiation theorem supplies the two-loop bookkeeping, letting the authors keep only the maximally non-abelian $C_A C_F$ and fermionic $C_F N_f$ diagrams and drop the abelian ones into the exponentiated one-loop term.

Extended reading notes

Core claim

The central claim is Eq. (4.5): the light-like Wilson loop $W(C_S)$ satisfies $(mu partial_mu + beta(g) partial_g) W = -[2 Gamma_{cusp}(g) L + Gamma(g)] W$, with $L = ln(i(rho - i0)) + gamma_E$ and $rho = (v y) mu/M$. This is unusual because the anomalous dimension on the right-hand side depends on the renormalization point through $rho$, and it is exactly this feature that, after Fourier transform, produces the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equation near the phase-space boundary with kernel $P(z) = 2Gamma_{cusp}(g)/(1-z)_+ + delta(1-z) h(g)$, so that $(1-z)P(z) = 2Gamma_{cusp}(g) theta(1-z)$ (Eq. 5.5). The two-loop computation of Sect. 3 is offered as confirmation: the coefficient of $L^4$ vanishes, the coefficient of $L^3$ is proportional to the one-loop $\beta$ function, and the remaining coefficients match the cusp anomalous dimension and the path-dependent constant $Gamma(g)$ of Eq. (4.6). The authors also derive the large-$n$ moment relation $F_n = H(mu/M) W(-i n mu/M)$, which makes the large-$n$ corrections to the moments exponentiate and puts the $alpha_s ln^2 n$ and $alpha_s^2 ln^3 n$ terms under the same equation. A note added in 2005 records that the integral $I_{11}$ was recomputed and corrected after comparison with an independent recalculation, changing Eq. (3.6), the $D$ coefficient of Eq. (3.7) and the constant $Gamma(g)$ in Eq. (4.6), while the cusp-driven structure of the result was untouched.

Load-bearing premise

The derivation of the light-cone equation assumes that shifting the path off the cone, taking the cusp anomalous dimension in its large-angle form, differentiating with respect to the kinematic variable, and then sending the segment length squared to zero all commute, and that the particular complex continuation of the cusp angle is the right one.

Editorial extensions

If this is right

The singular large-$x$ behaviour of the quark splitting function is fixed by one number, the cusp anomalous dimension, with $C_F$ replaced by $C_A$ for gluon distributions.
Large-$n$ moments of the parton distribution exponentiate, and the leading and next-to-leading logarithms in $n$ are summed to all orders by the same loop equation.
The same Wilson-loop representation covers fragmentation functions at large $x$, because the time-like path differs only in how the fields are ordered along it.
The soft and collinear contributions separate: only soft gluons enter $Gamma_{cusp}$, while the collinear function $H(mu/M)$ enters through the path- and scheme-dependent constant $h(g)$.
The velocity-dependent anomalous dimension of heavy-quark effective field theory is identified with the same cusp anomalous dimension, so the loop language carries over to heavy-meson processes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the anomalous dimension depends on the external variable $rho$, the light-like loop behaves like a local operator whose dimension is set by the kinematics; one could ask whether the same structure recurs for other light-like observables such as energy-energy correlations or transverse-momentum distributions at small $q_T$.
Equation (5.5) makes the three- and four-loop large-$x$ splitting function an indirect measurement of the cusp anomalous dimension, and conversely a precise cusp computation a prediction for the splitting function's most singular term.
Since the collinear function enters only through $h(g)$, a separate determination of $h(g)$ from diagrams or from lattice input would complete the resummation without touching the cusp coefficient.
The 2005 correction to $I_{11}$ shows that the constant $Gamma(g)$ is the fragile part of the two-loop result, while the $L^3$ and $L^2$ terms that carry the actual argument are stable against that recomputation.

Original abstract

We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone. Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost. Nevertheless we establish the renormalization group equation for the light like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions. By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone. Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost. Nevertheless we establish the renormalization group equation for the light like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions. By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone.	contribution 0.70	0.48	0.09	asserted 0.80
s1	Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost.	result 0.42	0.30	0.07	asserted 1.00
s2	Nevertheless we establish the renormalization group equation for the light like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions.	result 0.81	0.94	0.50	asserted 1.00
s3	By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.	result 0.79	0.95	0.57	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.86.

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arXiv listing abstract v2

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Figures

**Figure 1.** One-loop Feynman diagram contributing to the structure

**Figure 2.** Integration path CS = ℓ1 ∪ ℓ2 ∪ ℓ3 for the Wilson loop W(CS) corresponding to the structure function for large x. The ray ℓ1 is along the time-like vector vµ from −∞ to 0; the segment ℓ2 is from point 0 to y along the light-cone; the ray ℓ3 is from the point y to −∞ along the vector −vµ. This path has two cusps at points 0 and y where the quark undergoes hard scattering. 2.3. Fragmentation function in the x → 1 limit The Wilson loop representation for the structure function in (2.12) can be easily generalized for the fragmentation function D(x, µ/M) in the limit x → 1. This function…

**Figure 3.** Integration path CT corresponding to the fragmentation function for large x. where we have used completeness condition for the associated radiation 1 = P f |fihf| and causality (2.10) for the phase factors. The index T indicates that the photon probe has time-like momentum. The integration path CT = ℓ1 ∪ ℓ2 ∪ ℓ3 is shown in fig. 3 and three parts ℓi have meanings similar to that described before for the space-like process. In this case path- and time-orderings are related as follows P = T for ℓ1 , P = T¯ for ℓ3 , P = T for ℓ2 and y− > 0 , P = T¯ for ℓ2 and y− < 0 This path has two c

**Figure 4.** One-loop diagrams contributing to W(CS). Here, the double line represents the integration path in the Minkowski space as in fig. 2. The Feynman rules for these diagrams are given in the Appendix. PSfrag replacements 0 y 0 y

**Figure 5.** Diagrams corresponding to fig. 4a for virtual and real glu

**Figure 6.** Nonvanishing two-loop diagrams containing the “maximally no

Figures reproduced from arXiv:hep-ph/9210281 by the authors.

References

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hep-ph 2013-04 accept novelty 7.0 of 10

The paper proves that in the exponent of a product of semi-infinite Wilson lines, every colour factor corresponds to a fully connected graph, generalising the Wilson-loop non-Abelian exponentiation theorem to arbitrar...
The four-loop cusp anomalous dimension in N=4 super Yang-Mills and analytic integration techniques for Wilson line integrals
hep-th 2013-04 conditional novelty 7.0 of 10

The four-loop velocity-dependent cusp anomalous dimension in N=4 super Yang-Mills is computed analytically in the planar limit, with a non-planar correction in a scaling limit, confirming the four-loop light-like value.
Leading singularities and off-shell conformal integrals
hep-th 2013-03 accept novelty 7.0 of 10

The paper analytically evaluates the two unknown three-loop conformal integrals in the N=4 SYM stress-tensor four-point function, completing the three-loop correlator, and evaluates one four-loop integral as a demonstration.
The cusp anomalous dimension at three loops and beyond
hep-th 2012-03 conditional novelty 7.0 of 10

An analytic three-loop formula for the cusp anomalous dimension in N=4 super Yang-Mills is extracted from the Regge limit of massive amplitudes, and a ladder-only limit is shown to agree with strong coupling.
Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM
hep-th 2012-01 accept novelty 7.0 of 10

Permutation symmetry plus light-cone and short-distance constraints fix the planar four-point correlator integrand in N=4 SYM through six loops, matching the known six-loop four-particle amplitude.
Webs in multiparton scattering using the replica trick
hep-ph 2010-07 accept novelty 7.0 of 10

A replica-trick algorithm computes exponentiated colour factors for multi-leg eikonal scattering, showing that webs become linear combinations of diagrams related by permuting gluon attachments to each hard parton.
From correlation functions to Wilson loops
hep-th 2010-07 conditional novelty 7.0 of 10

A light-like limit of an n-point correlation function in a conformal gauge theory yields a polygonal null Wilson loop, checked explicitly at one and two loops in N=4 super-Yang-Mills.
Treating the b quark distribution function with reliable uncertainties
hep-ph 2008-07 conditional novelty 7.0 of 10

A complete orthonormal basis for the nonperturbative shape function, combined with renormalization-group evolution and moment constraints, gives a systematic handle on functional-form uncertainties in inclusive B decays.
The all loop AdS4/CFT3 Bethe ansatz
hep-th 2008-07 conditional novelty 7.0 of 10

The authors propose five Bethe equations that, if correct, yield the complete asymptotic energy spectrum of the ABJM AdS4/CFT3 duality at all values of the coupling.
Universal structure of subleading infrared poles in gauge theory amplitudes
hep-ph 2008-05 accept novelty 7.0 of 10

Subleading infrared poles of gauge-theory form factors equal twice the delta(1-z) coefficient of the splitting kernel plus the eikonal anomalous dimension, up to a short-distance term that vanishes in conformal theories.
Hexagon Wilson loop = six-gluon MHV amplitude
hep-th 2008-03 accept novelty 7.0 of 10

The two-loop hexagon Wilson loop matches the six-gluon MHV amplitude within numerical error bars, confirming the Wilson loop/amplitude duality at six points and showing the BDS ansatz fails there.
On planar gluon amplitudes/Wilson loops duality
hep-th 2007-09 accept novelty 7.0 of 10

An explicit two-loop calculation shows that the four-gluon scattering amplitude in planar N=4 super Yang-Mills equals a light-like Wilson loop, extending the proposed duality beyond one loop.
Conformal properties of four-gluon planar amplitudes and Wilson loops
hep-th 2007-07 conditional novelty 7.0 of 10

Assuming a hidden dual conformal symmetry of the off-shell four-gluon amplitude, the authors derive the log-squared form of its finite remainder and match it to a light-like Wilson loop at weak coupling.
Notes on Superconformal Chern-Simons-Matter Theories
hep-th 2007-04 accept novelty 7.0 of 10

A large class of three-dimensional N=2 and N=3 Chern-Simons-matter theories is shown to be exactly conformal, and a two-loop large-N computation of long-operator anomalous dimensions is presented as evidence for AdS4 ...
The singular behavior of massive QCD amplitudes
hep-ph 2006-12 accept novelty 7.0 of 10

Massive QCD amplitudes in the small-mass limit factorize into a universal process-independent factor times the corresponding massless amplitude, enabling three-loop predictions for the heavy-quark form factor.
Approximate N$^2$LO and N$^3$LO QCD Predictions for $tW$ Production
hep-ph 2025-12 conditional novelty 6.0 of 10

Higher-order QCD corrections to top-plus-W production raise the predicted rate by over 10% and give |Vtb|=0.99±0.03.
Slepton pair production at next-to-leading power
hep-ph 2025-10 unverdicted novelty 6.0 of 10

Evaluates next-to-leading power threshold corrections for slepton pair production and finds them significant compared to leading power NLL terms, with underestimated scale errors for large masses.
Universality at next-to-leading power for jet associated processes
hep-ph 2025-05 reject novelty 6.0 of 10

The paper proposes a universal relation between the leading logarithmic coefficient and the mass-factorization pole for jet-associated production of any massive colourless particle.
Triple real-emission contribution to the zero-jettiness soft function at N3LO in QCD
hep-ph 2024-12 conditional novelty 6.0 of 10

The paper provides the technical derivation and cross-checks of the triple-real-emission piece of the N3LO zero-jettiness soft function, confirming the result announced in arXiv:2409.11042.
Threshold resummation for double-deeply virtual Compton scattering
hep-ph 2024-11 conditional novelty 6.0 of 10

A QCD factorization and resummation formula for the threshold logarithms in double-deeply virtual Compton scattering is derived, and its two-loop leading-power coefficient function agrees with an independent explicit ...
Threshold Resummation for Semi-Inclusive Single-Hadron Production with Effective Field Theory
hep-ph 2024-11 accept novelty 6.0 of 10

Momentum-space SCET threshold resummation for SIA is extended to N4LL for quark and gluon channels, with new large-x coefficients through N3LO and partial N4LO.
Celestial Blocks and Transverse Spin in the Three-Point Energy Correlator
hep-ph 2022-02 accept novelty 6.0 of 10

The three-point energy correlator's light-ray OPE data is analytic in transverse spin, and its celestial blocks are ordinary two-dimensional conformal blocks.
Self-contained definition of the Collins-Soper kernel
hep-ph 2020-03 conditional novelty 6.0 of 10

A deformed-contour Wilson-loop definition expresses the Collins-Soper kernel directly in terms of QCD vacuum matrix elements, enabling model and lattice computations without reference to any specific scattering process.
Factorization at Subleading Power and Endpoint-Divergent Convolutions in $h\to\gamma\gamma$ Decay
hep-ph 2019-12 conditional novelty 6.0 of 10

A subleading-power SCET factorization theorem for the b-quark induced h to gamma gamma amplitude is derived, with endpoint divergences shown to cancel or be removable by rearrangement, and the leading double logarithm...
Anomalous Dimensions from Crossing Kernels
hep-th 2018-07 conditional novelty 6.0 of 10

Crossing kernels decompose into Wilson polynomials, and their spectral integrals evaluate to Wilson functions, giving closed-form analytic-in-spin anomalous dimensions for double-twist operators in general d.
Leading-color fully differential two-loop soft corrections to QCD dipole showers
hep-ph 2018-05 accept novelty 6.0 of 10

A modified subtraction scheme makes the NLO corrections to soft gluon radiation in QCD dipole showers fully differential and Monte-Carlo integrable, recovering the two-loop cusp anomalous dimension upon integration.
Universality of next-to-leading power threshold effects for colourless final states in hadronic collisions
hep-ph 2017-06 accept novelty 6.0 of 10

At next-to-leading order and next-to-leading power in soft-gluon energy, cross sections for colourless final states factorise into a universal K-factor times the Born cross section with shifted kinematics.
Non-abelian factorisation for next-to-leading-power threshold logarithms
hep-ph 2016-10 conditional novelty 6.0 of 10

A non-abelian factorization formula for next-to-leading-power threshold logarithms is derived, with a one-loop radiative jet function, and checked against known NNLO Drell-Yan results.
Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order
hep-ph 2016-04 accept novelty 6.0 of 10

The authors compute all unpolarized transverse-momentum-dependent parton distribution and fragmentation functions at two loops and extract their matching coefficients onto ordinary PDFs and FFs, including the gluon fr...
An Exponential Regulator for Rapidity Divergences
hep-ph 2016-04 conditional novelty 6.0 of 10

An exponential damping of soft radiation regulates rapidity divergences, and a bootstrap built on threshold boundary data reproduces the two-loop rapidity anomalous dimension and gives partial three-loop results.
A factorization approach to next-to-leading-power threshold logarithms
hep-ph 2015-03 conditional novelty 6.0 of 10

A refined amplitude-level factorization formula, together with a newly computed one-loop radiative jet function, reproduces the abelian next-to-leading-power threshold logarithms of the NNLO Drell-Yan real-virtual cro...
The method of regions and next-to-soft corrections in Drell-Yan production
hep-ph 2014-10 conditional novelty 6.0 of 10

Applying the method of regions to real-virtual gluon emission in Drell-Yan at NNLO reproduces the exact threshold result to next-to-leading power and traces the breakdown of soft-gluon factorization to collinear-regio...
Universality of Long-Distance AdS Physics from the CFT Bootstrap
hep-th 2014-03 conditional novelty 6.0 of 10

The 2d conformal bootstrap at large central charge reproduces AdS3 gravitational physics: deficit angles, a dense twist spectrum above the BTZ threshold, and BTZ quasinormal modes.
The generalized cusp in ABJ(M) N = 6 Super Chern-Simons theories
hep-th 2012-08 conditional novelty 6.0 of 10

A two-parameter family of cusped Wilson loops in ABJ(M) super Chern-Simons theory is constructed, and its two-loop expectation value gives the heavy-particle interaction potential and the universal cusp anomaly N squa...
Integrable Wilson loops
hep-th 2012-03 conditional novelty 6.0 of 10

An open spin-chain and twisted boundary TBA are proposed as the exact, all-coupling description of the generalized cusp anomalous dimension in N=4 super Yang-Mills.
A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory
hep-ph 2012-02 conditional novelty 6.0 of 10

A new renormalization-group scale and equation (the rapidity renormalization group) lets theorists factorize and resum rapidity logarithms in soft-recoil-sensitive QCD observables, demonstrated on the Sudakov form fac...
On the renormalization of multiparton webs
hep-ph 2011-08 accept novelty 6.0 of 10

The paper derives renormalization constraints that fix every multiple pole of a multiparton web in terms of lower-order webs, verifies them explicitly at three loops, and conjectures a weighted column sum rule for web...
Factorization Properties of Soft Graviton Amplitudes
hep-th 2011-03 accept novelty 6.0 of 10

Soft graviton amplitudes factorize into hard and soft parts, with the soft part given by vacuum expectation values of Wilson line operators that carry a mass-dependent term absent in gauge theory.
Quantum folded string and integrability: from finite size effects to Konishi dimension
hep-th 2011-02 conditional novelty 6.0 of 10

The Konishi operator's anomalous dimension at strong coupling is derived analytically as 2λ^{1/4} - 4 + 2/λ^{1/4}, confirming earlier numerical Y-system predictions.
Exciting the GKP string at any coupling
hep-th 2010-10 conditional novelty 6.0 of 10

The spectrum of excitations around the long GKP string is expressed as all-loop dispersion relations parameterized by the solution to the BES equation, with the SO(6) symmetry restoration mechanism identified.
Threshold resummation for pair production of coloured heavy (s)particles at hadron colliders
hep-ph 2010-07 conditional novelty 6.0 of 10

Soft-gluon and Coulomb effects factorize for heavy coloured pair production near threshold, and combined resummation raises the NLO squark-antisquark cross section by 4 to 20 percent at the 14 TeV LHC.
Soft radiation in heavy-particle pair production: all-order colour structure and two-loop anomalous dimension
hep-ph 2009-07 conditional novelty 6.0 of 10

The threshold soft function for heavy coloured particle pair production is diagonal in a Clebsch-Gordan colour basis to all orders, and its two-loop anomalous dimension is the quadratic Casimir of the final-state repr...
Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
hep-th 2007-12 accept novelty 6.0 of 10

Anomalous conformal Ward identities fix the finite part of four- and five-cusp light-like Wilson loops to all loop orders, and a two-loop pentagon calculation matches the five-gluon MHV amplitude.

See all 76 Pith citations

How the literature uses it

117 arXiv papers cite this in 153 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 93 sentences without a confident label

Uses its method or tool 8 papers

For the computation of the exponentiation factor up to NLL accuracy, we need to expand $Re[γ_S,P]$ and $γ_g^N$ up to second order in $α_s$ [equation] where $α_s ln(μ/m_S) ∼α_s lnN̅$ are treated as $(1)$, and the coefficients of the large logarithms, $A_i$ denote the coefficients of the cusp anomalous dimension [ref] .

Factorization and resummation for single color-octet scalar production at the LHC · Large N resummation in moment space · 0.91
It is included following Ligeti:2008ac to NNLL$'$ order, which includes next-to-next-to-leading-logarithmic (NNLL) resummation and all singular terms at $α_s^2$ [ref]

Precision Global Determination of the $B\to X_s\gamma$ Decay Rate · Introduction · 0.81
To compute these diagrams, we employ the technique developed in Refs. [ref] .

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes · Two-loop calculation · 1.00

Compares with it 1 paper

Using the formulation in terms of vacuum matrix elements of Wilson lines we have explained precisely how the jet function is different from the heavy-quark shape function and fragmentation function that have been considered previously in the literature at two-loop order [ref] .

Two-loop Jet-Function and Jet-Mass for Top Quarks · Conclusion · 0.90

Cites it as background 47 papers

This should not come as a total surprise, since the intimate relationship between the infrared divergences of the scattering of massless particles and the ultraviolet divergences of Wilson loops with cusps is well known in QCD [ref] .

Hexagon Wilson loop = six-gluon MHV amplitude · Introduction · 0.81
In the conventional method, threshold resummation is performed in moment space [ref] .

Threshold Resummation for Semi-Inclusive Single-Hadron Production with Effective Field Theory · Factorization Theorem in Moment Space · 0.80
The object that achieves this resummation and separates all the logarithms depending on $μ_i^2$ is called the “jet function” [ref] and can be defined as a quark-quark correlator attached to the light-like Wilson-lines along the the direction of incoming collinear quark [ref] .

Threshold resummation for computing large-$x$ parton distribution through large-momentum effective theory · Space-time picture of the threshold limit and the space-like heavy-quark jet function · 0.80

Discussion

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