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Improving the Weizs\"acker-Williams Approximation in Electron-Proton Collisions

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S. Frixione, M. L. Mangano, P. Nason, G. Ridolfi

High Energy Physics - Phenomenology · Submitted 26 Oct 1993 · 5 references

Pith's verdict: The angular-cut Weizsäcker-Williams spectrum (Eq. 27) is a solid, durable fix; the untagged-case improvement is a one-parameter fit with a disclosed free ξ, and the abstract's framing slightly outruns it.

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arXiv IDhep-ph/9310350 (version 1)
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CitationG. Ridolfi, M. L. Mangano, P. Nason, S. Frixione. "Improving the Weizs\"acker-Williams Approximation in Electron-Proton Collisions." arXiv:hep-ph/9310350 (1993). https://pith.science/paper/UZDGEGUQ
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The paper

The paper claims

The paper argues that the standard Weizsäcker–Williams photon distribution, routinely used to approximate electron-proton collisions, can be wrong by tens of percent, and derives a corrected version that keeps a universal non-logarithmic, $

The reading

Electron-proton experiments are usually analysed by pretending the electron beam is really a beam of photons, with a probability distribution supplied by the Weizsäcker–Williams formula. That formula keeps only the large logarithm and discards everything of order one, which the authors say is unsafe: at HERA energies the logarithm is only about 15 to 20, so the neglected pieces can be worth tens of percent. They redo the derivation without dropping the electron-mass terms, keeping the dependence on the angular cut applied to the scattered electron, and arrive at a closed expression containing one extra, process-independent piece that grows as the photon energy fraction $y$ goes to zero. Comparing against exact leading-order heavy-quark electroproduction, they find that adding this single term moves the approximation from roughly 6–7% off to about 1% for charm and 0.1% for bottom at HERA. They then argue that when no angular cut is applied, the effective scale inside the logarithm is fixed by the mass of the heavy object produced rather than by the kinematics of the electron.

What carries the argument

The load-bearing object is the improved distribution of eq. (27), obtained by keeping the full electron-mass dependence in the electron tensor $T^{��ν}$ and in the Jacobian of the change of variables $(p',��) �� (q^2,y)$, which turns out to be simply $d^3p'/E' = ��dq^2dy$. Integrating over $q^2$ between the two kinematic bounds produces a logarithm plus a term proportional to $m_e^2 y /(E^2(1-y)^2��_c^2 + m_e^2y^2)$; the second piece is the correction the paper is arguing for. A secondary mechanism is the step-function model of eq. (34), in which the virtual-photon cross section is taken equal to the real one below $|q^2| = ��^2M^2$ and zero above, converting the no-cut case into a single effective scale $Q^2_{WW} = ��min(ys-4M^2, ��^2M^2)$.

Extended reading notes

Core claim

The central claim is that the commonly used Weizsäcker–Williams distribution, $f^{(e)}_��(y) = (��_{em}/2��)[(1+(1-y)^2)/y]��log[(Sy-S_{min})(1-y)/(m_e^2y^2)]$ plus terms of order one, is not good enough for photoproduction analyses at HERA, and that a better distribution follows from keeping the electron mass exactly through the phase-space integration. Carrying out that integration between $q^2_{max} = -m_e^2y^2/(1-y)$ and $q^2_{min}$ set by the anti-tag angle $��_c$, the authors obtain a distribution whose non-logarithmic part is singular in $y$ and depends only on $m_e$, $E$ and $��_c$ — not on the hard process. Because this term is universal, it can be added once and used for any electroproduction calculation, and the paper reports that doing so collapses the discrepancy with the exact heavy-quark cross sections from 6–7% to about 1% (charm) and 0.1% (bottom) for cuts of $5��10^{-2}$ and $5��10^{-3}$ radians. For the complementary case with no angular cut, the authors argue that the upper limit of the $q^2$ integration is instead controlled by the mass $M$ of the produced heavy system, giving an effective scale $Q^2_{WW} = ��^2M^2$ with $��$ of order one, and that the residual non-logarithmic term can be absorbed into a rescaling $��' �� 0.6��$.

Load-bearing premise

The no-angular-cut analysis rests on replacing the virtual-photon cross section by a sharp step at a scale $��M$, with $��$ a number of order one that the authors themselves call essentially unconstrained; the good agreement at $��=1$ is therefore a one-parameter match to the exact answer rather than an independent prediction.

Editorial extensions

If this is right

  • Any photoproduction cross section computed with the leading-logarithmic photon distribution alone carries a systematic error of order 6–30% at HERA energies, depending on the angular cut and the hard scale.
  • Adding the universal non-logarithmic term of eq. (27) makes the approximate heavy-quark cross sections agree with the exact leading-order electroproduction calculation at the 1% level for charm and 0.1% for bottom.
  • The discrepancy grows with beam energy at fixed angular cut, so the approximation degrades as one moves from HERA kinematics to higher-energy electron-hadron collisions.
  • When no electron anti-tag cut is applied, the correct effective scale in the logarithm is dynamical, of order the heavy-quark mass, rather than the kinematic bound $ys-4M^2$.
  • Non-factorisable corrections from the $q^2$ dependence of the hadronic form factors are negligible in the presence of a small-angle cut but can be of order $1/y$ when the full phase space is integrated over.

Reading between the lines

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

  • The universal character of the non-logarithmic term suggests it should be checkable in any electroproduction channel, not just heavy quarks; a natural test is dijet or jet-plus-photon photoproduction, where the hard scale is set by the jet transverse energy rather than a quark mass.
  • Because the correction is singular in $y$, the largest improvements should appear in the low-$y$ region of the photon spectrum, which is exactly where the gluon density is largest — so the effect on predicted event rates may be more pronounced in the $y$ distribution than in the total cross section.
  • The step-function treatment of the no-cut case effectively introduces a hand-tuned parameter $��$; a more principled version would replace it with a measured or modelled vector-meson-dominance-type transition form factor, which would make the no-cut prediction parameter-free.
  • If the same electron-mass bookkeeping is applied to positron beams or to polarised beams, the correction term should be unchanged at leading order, which offers a cheap consistency check of the claimed universality.
Original abstract

We critically examine the validity of the Weizs\"acker-Williams approximation in electron-hadron collisions. We show that in its commonly used form it can lead to large errors, and we show how to improve it in order to get accurate results. In particular, we present an improved form that is valid beyond the leading logarithmic approximation in the case when a small-angle cut is applied to the scattered electron. Furthermore we include comparisons of the approximate expressions with the exact electroproduction calculation in the case of heavy-quark production.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,573 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.27 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We critically examine the validity of the Weizs\"acker-Williams approximation in electron-hadron collisions. We show that in its commonly used form it can lead to large errors, and we show how to improve it in order to get accurate results. In particular, we present an improved form that is valid beyond the leading logarithmic approximation in the case when a small-angle cut is applied to the scattered electron. Furthermore we include comparisons of the approximate expressions with the exact electroproduction calculation in the case of heavy-quark production.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We critically examine the validity of the Weizs\"acker-Williams approximation in electron-hadron collisions. contribution 0.71 0.80 0.06 unclear 0.75
s1 We show that in its commonly used form it can lead to large errors, and we show how to improve it in order to get accurate results. contribution 0.62 0.95 0.79 asserted 0.92
s2 In particular, we present an improved form that is valid beyond the leading logarithmic approximation in the case when a small-angle cut is applied to the scattered electron. contribution 0.91 0.94 0.30 asserted 1.00
s3 Furthermore we include comparisons of the approximate expressions with the exact electroproduction calculation in the case of heavy-quark production. contribution 0.76 0.92 0.07 asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.53 · evidence from proof or derivation 0.84.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

58 later papers cite it

Evidence: substantial. 56 of the citing papers share no author with it, and 34 of those carry a confident label from Pith's citation classifier.

Pith read 26 of the 65 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 11 of 34 papers with this label
Compares with it 1 paper · 1 by the same authors

By the same authors

How the literature uses it

57 arXiv papers cite this in 65 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 · 26 sentences without a confident label

Uses its method or tool 33 papers
  • Electron-positron bremsstrahlung is the source of photons, which can be well formulated in Weizäcker-Williams approximation(WWA) [ref] , [equation] where $α_em=1/137$, $Q^2_min = m_e^2 x^2/(1-x)$ and $Q^2_max = (Eθ_c)^2(1-x) + Q^2_min$ with $x = E_γ/E_e$, $θ_c=32 mrad$ the maximum scattered angular cut in order to ensure the photon to be real, and $E=E_e=√(s)/2$, which is $√(s)=240$ at the CEPC.

    Prompt $J/\psi$ photoproduction within the non-relativistic QCD framework at the CEPC · Prompt $J/ψ$ Photoproduction in NRQCD Framework · 0.99

  • Under the condition of applying a small-angle cut to the scattered muon, the photon density function, characterized by the Weizsäcker-Williams approximation (WWA), is given by [ref] [equation] where $x=E_γ/E_μ$, $E_γ$ and $E_μ$ are photon and muon energies. $α$ represents the fine structure constant, and $m_μ$ denotes the muon mass, we do not neglect the higher-order terms of the muon mass. $Q^2_min$ and $Q^2_max$ are given by [equation] where $β=√(1-m^2_μ/E^2_μ)$, $A=E_μ(1+β)(1-x)$, the scattering angle cut $θ_c$ is determined by experiment [ref] .

    Production of doubly heavy baryon at the Muon-Ion Collider · Calculation technology · 0.99

  • Initial photons can be generated through the bremsstrahlung effect which is described by the Weizsacker-Williams approximation (WWA) [ref] , [equation] where $Q^2_min=m^2_ex^2/(1-x)$, $Q^2_max=(θ_c√(s)/2)^2(1-x)+Q^2_min$, $m_e$ is the mass of electron, $s$ is the squared center-of-mass (CM) energy, and $θ_c=32$ mrad is the maximum scattering angle of the electron or positron [ref] .

    Production of doubly charmed tetraquark $T_{cc}$ via photon-photon fusion at electron-positron colliders · FORMULATION · 0.98

Compares with it 1 paper
  • [ref] , provided that in the latter: a) one identifies $E^2θ_c^2$ with $s$ (which can be understood from the physical viewpoint by noting that $E^2θ_c^2$ is the “large” scale of the problem); b) one neglects all terms which are suppressed by powers of $m^2$ (some of these have been kept in ref.

    Initial conditions for electron and photon structure and fragmentation functions · Final result · 1.00

Cites it as background 3 papers

Discussion

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