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Geometric scaling as traveling waves
High Energy Physics - Phenomenology · Submitted 16 Sep 2003
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Geometric scaling is for the first time shown to result from an exact solution of nonlinear QCD evolution equations.
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The abstract, sentence by sentence
We show the relevance of the nonlinear Fisher and Kolmogorov-Petrovsky- Piscounov (KPP) equation to the problem of high energy evolution of the QCD amplitudes. We explain how the traveling wave solutions of this equation are related to geometric scaling, a phenomenon observed in deep-inelastic scattering experiments. Geometric scaling is for the first time shown to result from an exact solution of nonlinear QCD evolution equations. Using general results on the KPP equation, we compute the velocity of the wave front, which gives the full high energy dependence of the saturation scale.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show the relevance of the nonlinear Fisher and Kolmogorov-Petrovsky- Piscounov (KPP) equation to the problem of high energy evolution of the QCD amplitudes. | contribution 0.88 | 0.81 | 0.16 | asserted 1.00 |
| s1 | We explain how the traveling wave solutions of this equation are related to geometric scaling, a phenomenon observed in deep-inelastic scattering experiments. | contribution 0.51 | 0.77 | 0.16 | asserted 1.00 |
| s2 | Geometric scaling is for the first time shown to result from an exact solution of nonlinear QCD evolution equations. | result 0.72 | 0.91 | 0.62 | asserted 1.00 |
| s3 | Using general results on the KPP equation, we compute the velocity of the wave front, which gives the full high energy dependence of the saturation scale. | result 0.77 | 0.94 | 0.20 | asserted 1.00 |
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The paper
Original abstract
We show the relevance of the nonlinear Fisher and Kolmogorov-Petrovsky- Piscounov (KPP) equation to the problem of high energy evolution of the QCD amplitudes. We explain how the traveling wave solutions of this equation are related to geometric scaling, a phenomenon observed in deep-inelastic scattering experiments. Geometric scaling is for the first time shown to result from an exact solution of nonlinear QCD evolution equations. Using general results on the KPP equation, we compute the velocity of the wave front, which gives the full high energy dependence of the saturation scale.
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References
How later papers used it
55 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 built on it.
Builds on it 1 of 3 papers with this label · 1 by the same authors
By the same authors
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Recalled previously known results on geometric scaling from studies of nonrunning and running BK equations and generalized them for generic n.
Extended Geometric Scaling from Generalized Traveling Waves (2009) (same authors) · 0.97 · paraphrased: the licence does not allow quoting
Uses its method or tool 2 of 4 papers with this label
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Used this paper's expansion together with changes in variables to reexpress an equation as the FKPP equation.
Exclusive $J/\Psi$ and $\rho^0$ vector meson production using BK evolution theory (2024) · 0.88 · paraphrased: the licence does not allow quoting
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Exploited a formal analogy between its problem and traveling wave solutions to reaction-diffusion processes to compute the universal scaling solution and its sub-asymptotic deviations.
Anomalous diffusion in QCD matter (2021) · 0.81 · paraphrased: the licence does not allow quoting
How the literature uses it
53 arXiv papers cite this in 99 sentences
Papers citing it · 79 sentences without a confident label
Builds on it 3 papers
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[ref] for the standard running and nonrunning FKPP cases, the generic solution reads U_n(X,T) ∝T^1n+2 G_n( L-(v_cT)^1n+1+c_n(T)T^1n+2) e^{L-(v_cT)^1n+1+c_n(T)} , Leading where $n=(0,1)$ is the previously defined index for nonrunning and running cases, $τ_n(L,T)≡L-(v_cT)^1n+1+c_n(T)$ is the original scaling variable extrapolated from the values obtained in [ref] , $c_n(T)$ the corresponding subleading universal contributions to the scaling variable and $$ a universal critical slope value characterizing the scaling in the wave interior when $z ≡τ_n(L,T)/T^1n+2→0.$ Note the major constraint in this limit $G_n(z)∼z$ which expresses the “absorptive” property [ref] due to the nonlinear terms.
Extended Geometric Scaling from Generalized Traveling Waves · 4. Extending Geometric Scaling · 0.98
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As shown in [ref] the dispersion in the final saturation scales are caused by fluctuations in the low density tail of the initial condition and can be realized by averaging over the initial saturation momenta drawn from a Gaussian distribution in the logarithm of the saturation scale, [equation] Our treatment extends the work of [ref] to include fluctuations in both nucleons in which case the event averaging can be performed according to [equation] where the two saturations scale $Q_o,1^2$ and $Q_o,2^2$, fluctuate event-by-event around the mean saturation scales $o,1$ and $o,2$ respectively.
Probing proton fluctuations with asymmetric rapidity correlations · Rapidity distribution in proton-proton collisions · 0.95
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Our method is the extension of a strategy originally introduced in relation with analytic studies of the BFKL evolution in the presence of saturation [ref] , whose deeper justification [ref] lies in the correspondence between high–energy evolution in QCD and the reaction–diffusion process in statistical physics [ref] .
CCFM Evolution with Unitarity Corrections · Introduction · 0.80
Uses its method or tool 3 papers
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Relying on the mapping between the evolution equation for $$ and the equations governing the propagation of traveling wave fronts in non-linear physics [ref] , we can compute the large $L$ asymptotic behaviour of the saturation momentum, even without having analytic control of the full non-linear evolution equation [ref] .
Transverse momentum broadening from NLL BFKL to all orders in pQCD · Introduction · 0.89
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As suggested by Munier and Peschanski [ref] , one can only keep the first three terms of the expansion to avoid the mathematical difficulties regarding the infinite-order differential equation.
Analytic solution of Balitsky-Kovchegov equation with running coupling constant using homogeneous balance method · Analytical solution of BK equation with fixed coupling constant · 0.83
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We find in particular that the latter reaches a universal scaling solution at late times (large $L$) that we compute analytically along with its sub-asymptotic deviations exploiting a formal analogy between the present problem and traveling wave solutions to reaction-diffusion processes [ref] .
Anomalous diffusion in QCD matter · Introduction · 0.81
Compares with it 4 papers
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This kinematic region is always near and forward to the identified leading edge regions characterized (see [ref] for $n=0,1$) by $L∼v_c T^1n+1+O(T^1n+2).$ For instance, $L∼(v_c+K)T$ for the nonrunning case and $L∼v_c T^2/3$ for the running case.
Extended Geometric Scaling from Generalized Traveling Waves · 4. Extending Geometric Scaling · 0.98
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These results coincide indeed with the corresponding predictions of the non–linear, BK, equation, as deduced via the correspondence with the FKPP equation [ref] .
CCFM Evolution with Unitarity Corrections · Unitarity and Saturation momentum · 0.83
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The solution in this work is similar to the traveling wave front introduced by Munier and Peschanski [ref] .
An Analytical Solution of the Balitsky-Kovchegov Equation with the Homogeneous Balance Method · Discussions and summary · 1.00
Cites it as background 6 papers
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The idea of intrinsic fluctuations of the saturation momentum goes back to Ref. [ref] , which extended a correspondence between high-energy QCD and the reaction-diffusion problem that was originally identified in Ref. [ref] at the level of mean field approximation.
Intrinsic Fluctuations of the Proton Saturation Momentum Scale in High Multiplicity p+p Collisions · Additional Sources of Fluctuations · 0.95
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The relation between the BK equation and the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation has been found [ref] .
An Analytical Solution of the Balitsky-Kovchegov Equation with the Homogeneous Balance Method · Introduction · 0.82
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In the last years, it has been shown [ref] that the BK equation lies in the same class of universality than the Fisher–Kolmogorov-Petrovsky-Piscunov (F-KPP) equation.
Saturation QCD predictions with heavy quarks at HERA · QCD predictions for the amplitude and heavy quarks · 0.88
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