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Multiplicity distribution of produced gluons in deep inelastic scattering: main equations and their homotopy solutions for heavy nuclei
High Energy Physics - Phenomenology · Submitted 23 Mar 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text.
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In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. We obtained three results. First, we get the new derivation of the equations for the cross sections of productions of $n$-cut Pomerons in the final states ($\sigma_n$). These equations coincide with the equations that have been derived using the Abramovsky, Gribov and Kancheli (AGK) cutting rules but based on the dipole approach to QCD. Second, we developed the homotopy approach for finding the solutions to these equations. It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. Third, we found the analytical solution for $\sigma_n$ at large $n\,\gtrsim\,N(z) = 2 N_0 \,z\,\exp( z^2/(2 \kappa))$ with $z = \ln( r^2\,Q^2_s )$. Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. | background 0.84 | 0.32 | 0.05 | asserted 0.68 |
| s1 | We obtained three results. | contribution 0.72 | 0.76 | 0.04 | asserted 0.98 |
| s2 | First, we get the new derivation of the equations for the cross sections of productions of $n$-cut Pomerons in the final states ($\sigma_n$). | result 0.70 | 0.96 | 0.17 | asserted 1.00 |
| s3 | These equations coincide with the equations that have been derived using the Abramovsky, Gribov and Kancheli (AGK) cutting rules but based on the dipole approach to QCD. | result 0.81 | 0.60 | 0.10 | asserted 1.00 |
| s4 | Second, we developed the homotopy approach for finding the solutions to these equations. | contribution 0.56 | 0.96 | 0.08 | asserted 1.00 |
| s5 | It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. | method 0.98 | 0.88 | 0.05 | asserted 0.97 |
| s6 | Third, we found the analytical solution for $\sigma_n$ at large $n\,\gtrsim\,N(z) = 2 N_0 \,z\,\exp( z^2/(2 \kappa))$ with $z = \ln( r^2\,Q^2_s )$. | result 0.99 | 0.96 | 0.29 | asserted 1.00 |
| s7 | Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text. | result 0.99 | 0.93 | 0.31 | asserted 1.00 |
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Original abstract
In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. We obtained three results. First, we get the new derivation of the equations for the cross sections of productions of $n$-cut Pomerons in the final states ($\sigma_n$). These equations coincide with the equations that have been derived using the Abramovsky, Gribov and Kancheli (AGK) cutting rules but based on the dipole approach to QCD. Second, we developed the homotopy approach for finding the solutions to these equations. It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. Third, we found the analytical solution for $\sigma_n$ at large $n\,\gtrsim\,N(z) = 2 N_0 \,z\,\exp( z^2/(2 \kappa))$ with $z = \ln( r^2\,Q^2_s )$. Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text.
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References
How the literature uses it
2 arXiv papers cite this in 4 sentences
Papers citing it · 3 sentences without a confident label
Uses its method or tool 1 paper
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In this talk and these proceedings, based mainly on [ref] , we use this equivalence to replace the AGK cutting rules by the evolution equations.
Multiplicity distributions in DIS for heavy nucleus · Introduction · 0.90
What it cites · 39 sentences without a confident label
Builds on 2 papers
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We continue [ref] to develop the approach for the solution of the non-linear evolution equations that govern the deep inelastic processes (DIS) in QCD.
Homotopy solution to non-linear evolution for heavy nuclei · Introduction · 0.99
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We continue [ref] to develop the approach for the solution of the non-linear evolution equations that govern the deep inelastic processes (DIS) in QCD.
Modified homotopy approach for diffractive production in the saturation region · Introduction · 0.99
Uses the method or tool of 5 papers
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Introducing $ΔY,r⃗_ik,b⃗= 1 - 𝒩Y,r⃗_ik,b⃗$ we rewrite ME7 as follows: ∂/∂Y σ_nY,r⃗_01,b⃗ = /2 π∫d^2 r_2 Kr⃗_01| r⃗_12, r⃗_02 × [ - σ_nY,r⃗_01,b⃗ + σ_nY,r⃗_12,b⃗ ΔY,r⃗_02,b⃗ + ΔY,r⃗_12,b⃗ σ_nY,r⃗_02,b⃗ + ∑_k=1^n-1 σ_n - kY,r⃗_02,b⃗ σ_kY,r⃗_12,b⃗ ] The homotopy approach for finding $ΔY,r⃗_ik,b⃗$ has been developed in our previous paper (see Ref. [ref] ).
Modified homotopy approach for diffractive production in the saturation region · Leading twist BFKL kernel: homotopy approach for the solutions · 0.91
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To find solutions of EQXS00 we use the simplified BFKL kernel suggested in Ref. [ref] .
Solution to the evolution equation for high parton density QCD · Leading twist BFKL kernel: homotopy approach for the solutions · 0.96
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The equations $σ_n$ of EQXS00 for the leading twist BFKL kernel K2 take the form: ∂/∂Y σ_nY,r⃗_01,b⃗ = - 2 lnr_01 Q_s(Y,b⃗) σ_nY,r⃗_01,b⃗ + σ_nY,r⃗_01,b⃗ _lnQ_s^-2(Y,b⃗)^lnr_01^2dlnr_02^2 ΔY,r⃗_02,b⃗ + ΔY,r⃗_01,b⃗ _lnQ_s^-2(Y,b⃗)^lnr_01^2dlnr_02^2σ_nY,r⃗_02,b⃗ + ∑_k=1^n-1 _lnQ_s^-2(Y,b⃗)^lnr_01^2dlnr_02^2σ_n - kY,r⃗_02,b⃗ σ_kY,r⃗_01,b⃗ Introducing $z = lnr^2_01 Q^2_sY,b⃗$ ($z' = lnr^2_02 Q^2_sY,b⃗$ and $= Y - Y_A$ we rewrite EQXS01 as follows: ∂/∂ σ_n,z + κ ∂/∂z σ_n,z = - z σ_n,z + σ_n,z _ξ_0^A^z dz' Δ,z' + Δ,z _ξ_0^A^zdz' σ_n,z' + ∑_k=1^n-1 _ξ_0^A^zdz' σ_n - k,z' σ_k,zIn the following subsections we develop the homotopy approach [ref] for solving EQXS02 for small $n$.
Homotopy solution to non-linear evolution for heavy nuclei · Leading twist BFKL kernel: homotopy approach for the solutions · 0.95
Compares with 1 paper
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This asymptotic behaviour coincides with the solution with the general BFKL kernel [ref] .
Solution to the evolution equation for high parton density QCD · Region I · 0.98
Cites as support 3 papers
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Second, in the simple but instructive one dimensional models for $σ_n$, the evolution equation can be written for $σ_n$ which do not address the AGK cutting rules in an explicit way [ref] .
Particle production in the toy world: multiplicity distribution and entropy · AGK equations without AGK cutting rules · 0.97
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We calculate the entropy ($S_E$) of the produced gluons and show that $S_E = lnN_DIS$ where $N_DIS$ is the multiplicity in the deep inelastic scattering process, confirming the main ideas of Ref. [ref] .
Deep inelastic scattering as a probe of entanglement · Introduction · 0.99
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It is important to note, that this behaviour has been proven on general theoretical grounds [ref] and has been seen in the experimental data on DIS [ref] .
Geometric scaling for the total gamma^* p cross section in the low x region · Introduction · 0.99
Cites as background 21 papers
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The scattering amplitude of the colourless dipole with the size $x_01$ which determines the DIS cross section, satisfies the Balitsky-Kovchegov (BK) non-linear equation [ref] : ∂/∂Y N_01Y = /2 π∫d^2x_2 x^2_01/x^2_02 x^2_12 [ N_02Y + N_12Y - N_01Y - N_02YN_12Y ] where $N_ikY=NY, x⃗_ik,b⃗$ is the scattering amplitude of the dipoles with size $x_ik = |x⃗_ik| = |x⃗_i - x⃗_k|$ and with rapidity $Y$ at the impact parameter $b⃗ = x⃗_0+x⃗_1/2$.
Factorization and high-energy effective action · Introduction · 0.89
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This distribution has become a hot subject during the past several years [ref] especially because of the new view on the entropy in DIS.
1211.69111 · Introduction · 0.86
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This distribution has become a hot subject during the past several years [ref] especially because of the new view on the entropy in DIS.
Entanglement entropy and entropy production in the Color Glass Condensate framework · Introduction · 0.86
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