The ALLM parameterization of sigma_{tot}(gamma* p) - an update

Aharon Levy; Halina Abramowicz

REVIEW

The ALLM parameterization of sigma_{tot}(gamma* p) - an update

Halina Abramowicz, Aharon Levy

High Energy Physics - Phenomenology · Submitted 17 Dec 1997 · 23 references

Pith's verdict: A useful, honestly documented refit that became a standard tool — but it is an interpolation, not a determination, and the paper never quantifies its own parameters.

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arXiv IDhep-ph/9712415 (version 2)

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CitationAharon Levy, Halina Abramowicz. "The ALLM parameterization of sigma_{tot}(gamma* p) - an update." arXiv:hep-ph/9712415 (1997). https://pith.science/paper/P7Y3UA4W

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Editor's note

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Bottom line: this is a refit, not a new idea, and it is worth having. The functional form of Eqs. 1–6 is untouched from ALLM91; what is new is the 23-number parameter set in Table 2, fitted to 1356 points spanning photoproduction up to Q2 = 5000 GeV2, with chi2/ndf = 0.97. The physically interesting bit is the Pomeron scale mP^2 going from 10.7 to 49.5 GeV2, which pulls the soft-to-hard turnover down to a few GeV2 (Fig. 4). That is the reason people cite this paper, and it has held up in use for corrections and extrapolation. The paper also does several things right that this genre usually doesn't: the full parameter table is published, Table 1 breaks chi2 down by dataset, and the text says plainly that the Q2 = 0 intercept was fixed to the Donnachie–Landshoff value rather than fitted. Everything is reproducible from the equations plus Table 2. Figures 9–12, where the parameterization is compared against GRV, MRSR1 and a Regge fit to the BPC data, are the most substantive part of the paper and are not just curve-on-data plots; the demonstration in Fig. 12 that a slope extracted the same way from data and from the parameterization reproduces the same x-cut artefacts is a real consistency check, not a tautology. Credit for that. Where it is soft: no parameter errors, no correlation matrix, no error band anywhere. So the headline shift in mP^2 is unquantified in significance, and a reader can't tell which of the 23 numbers the data actually pin. Second, the stress-test concern lands. Table 1 shows per-point residuals from 0.51 (ZEUS BPC) to 1.35 (ZEUS-94), with photoproduction at 1.15, and the paper never mentions it, never quotes relative normalizations, and never discusses correlated systematics. A single global fit with no per-experiment scale freedom has to absorb that spread into the shape parameters. So 'good description of all the available data' should be read as 'good compromise fit across datasets of differing internal consistency.' That doesn't break the claim — the claim is modest by construction — but the text should say it. Minor bookkeeping: '23 parameters determined from a fit' is loose, since Lambda^2 is identical in both columns of Table 2 and aP1 is fixed externally; about 21 are actually free. Who should read it: anyone doing DIS radiative corrections, acceptance, or extrapolation into unmeasured regions, and anyone wanting a smooth analytic stand-in for sigma_tot(gamma*p) across the transition. It is a tool paper, and judged as a tool it is solid. Send it to referees — mainly to force the error analysis and the normalization discussion into the record.

Referee report

Significance

If the result holds, it is a genuinely useful piece of phenomenology: a closed-form, publicly specifiable expression for σ_tot(γ*p) valid across the whole (x, Q²) plane, with an explicit parameter list (Table 2) that others can implement for acceptance, radiative-correction, and extrapolation purposes. The paper deserves credit for three things that are not universal in this genre. First, Table 1 gives the per-data-set χ² breakdown, which lets the reader see where the fit is good and where it is not, rather than quoting only a global number. Second, the comparisons in §5.1–5.2 are not merely in-sample overlays: GRV, MRSR1, and the Regge fit to the BPC data are independent constructions, and the paper uses them to locate the transition region at Q² ≈ 1–3 GeV², which is a falsifiable, checkable statement. Third, the authors state explicitly (§4) that the Q² = 0 Pomeron intercept was fixed to the Donnachie–Landshoff value because the HERA photoproduction data do not constrain it, which is the right kind of disclosure. The claim is modest by construction — that a smooth interpolant exists and fits well — and the evidence presented is commensurate with that claim.

Major objections (2)

[§3 and Table 1] Table 1 shows that the fit quality is not uniform across data sets: ZEUS-94 contributes 253.8 to χ² for 188 points (1.35 per point) and the combined photoproduction set 262.3 for 228 points (1.15 per point), while H1-94 gives 127.1/193 (0.66 per point) and ZEUS-BPC 17.3/34 (0.51 per point). A spread of this size across ten data sets is the expected signature of unaccounted relative normalization offsets and correlated systematic uncertainties, and the paper neither introduces per-experiment scale factors nor discusses the spread. This matters for the headline sentence of the abstract, which claims a good description of all the data: a data set at 1.35 per point is being described less well than the global number suggests. The authors should either add relative normalization parameters to the fit or, at minimum, state in the text that the quoted χ²/ndf is a compromise across data sets of differing internal consistency, and identify which sets drive the residual.
[§4 and Table 2] No parameter uncertainties, no correlation matrix, and no error band on any curve are given anywhere in the paper. For a parameterization whose stated practical purpose (§1) is to be used by others for acceptance and radiative corrections, this is a real gap: a user cannot propagate the fit uncertainty into a correction, and a reader cannot judge whether the headline structural change between ALLM91 and ALLM97 — the roughly fivefold increase in m_P² from 10.676 to 49.457 GeV², which the text itself singles out as the biggest difference — is a significant, data-driven shift or a shallow direction in a 20-plus-parameter fit. At minimum the authors should quote errors on the Table 2 parameters and state whether the m_P² shift is many standard deviations. A residual plot (pull vs Q² and vs x, aggregated over data sets) would also substantiate the claim that the description is good everywhere rather than on average.

Minor objections (7)

[§2 and Table 2] The text states that 'there are altogether 23 parameters to be determined from a fit to the data', but §4 says the Q² = 0 Pomeron intercept was fixed to the Donnachie–Landshoff value, and Table 2 lists a_P1 = −0.0808 and Λ² = 0.06527 as identical in ALLM91 and ALLM97. The number of parameters actually floated in the fit, and hence the ndf used in χ²/ndf = 0.97, should be stated explicitly.
[Abstract vs §4.3] The abstract quotes the fitted range as 0 ≤ Q² < 5000 GeV², while §4.3 refers to figure 8 as covering 0 ≤ Q² ≤ 2000 GeV². These should be reconciled, or the distinction between the range of data used and the range displayed should be made explicit.
[§1] The sentence claiming that the parameterization 'can reproduce features like the slope in Q² and in x, where all other parameterizations fail' is stronger than the paper's own §5 supports: the text there states that GRV agrees with the data for Q² > 5 GeV² and MRSR1 for Q² ≳ 3–4 GeV². The claim should be softened to describe the specific region (Q² ≲ 3 GeV²) where the other parameterizations deviate.
[Eq. (3)] The typeset definition of t is garbled in the manuscript as 'ln Q2+Q20Λ2 / ln Q20Λ2'. The intended expression should be checked and written unambiguously, since t enters every Q²-dependent parameter through Eqs. (5)–(6).
[Fig. 3 caption] The caption states that the dotted line shows the uncertainty of the fit, but no other uncertainty is quantified in the paper. The reader should be told how that band was obtained (parameter errors propagated, or a fixed fractional variation).
[§5.2] The text says a minimum of four data points at different x values 'were required' to extract λ at fixed Q², but does not state the x window used for each point. Since the section's own point is that λ depends strongly on the x cut, the window should be given, at least for the curves shown in Fig. 12.
[Reference [4]] ALLM-N, used for the comparison curves in Fig. 2 and Fig. 6, is cited as a thesis (TAUP-2350, 1996) and is not publicly available in a form the reader can check. Either a published reference should be substituted or the relevant parameter values should be reproduced in the paper.

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The paper

The paper claims

A single 23-parameter Regge-inspired formula can describe the total virtual-photon–proton cross section across five orders of magnitude in $x$ and from $Q^2=0$ up to $5000$ GeV$^2$, with $1{,}356$ data points fitted at $1.03$ per point.

The reading

The authors want to know whether one analytic function, built out of the two exchange amplitudes that Regge theory uses for soft hadron scattering, can describe the proton structure function and the total photon–proton cross section over the entire kinematic range that experiments had reached by 1997 — from real photoproduction at $Q^2=0$ through the transition region at a few GeV$^2$ to deeply inelastic scattering at $x$ as small as $3.5$ parts per million. Their earlier version of this parameterization had been fitted only to pre-HERA data and, while it predicted the first HERA photoproduction and high-$Q^2$ measurements well, it overshot the data in the intermediate region $5 < Q^2 < 25$ GeV$^2$. The update refits all 23 parameters to 1356 published points from ten experiments and reaches $1.03$ per point. If the claim holds, no separate soft and hard machinery is needed: one expression covers photoproduction, the transition, and the scaling-violation region, and can be used for acceptance and radiative corrections and for extrapolating into corners of the kinematic plane where no data exist.

What carries the argument

The machinery is a two-component ansatz: $F_2(x,Q^2) = (Q^2/(Q^2+m_0^2))[F_2^{P} + F_2^{R}]$, where $F_2^{P} = c_P(t) x_P^{a_P(t)}(1-x_P)^{b_P(t)}$ and similarly for the Reggeon term, with $t = 1.03$ encoding the $Q^2$ dependence through $t = 1.03$. The scaled variables $x_{P,R}$ replace Bjorken $x$ by $1/(1+(W^2-M^2)/(Q^2+m_{P,R}^2))$, so that the effective Pomeron and Reggeon masses $m_P^2$, $m_R^2$ govern how the curve bends as $Q^2$ falls toward zero and the expression reduces to a total photoproduction cross section. Four of the six coefficient functions vary with $Q^2$ as $f_1 + f_2 t^{f_3}$ and the remaining two as $g_1 + (g_1-g_2)[1/(1+t^{g_3}) - 1]$, which is what makes the Pomeron intercept rise with $Q^2$ while the Pomeron coupling falls. The $Q^2/(Q^2+m_0^2)$ prefactor is the effective-photon-mass factor that enforces the correct behaviour at $Q^2 = 0$. All 23 parameters are fixed by a single global fit; the mechanism carrying the physics is that the $Q^2$ dependence of the intercept, not the overall normalization, is what distinguishes a working fit from a failing one.

Extended reading notes

Core claim

The paper's claim is that the updated ALLM97 parameterization gives a good description of all available data over $3.5$ parts in $10^6 < x < 0.85$ and $0 < Q^2 < 5000$ GeV$^2$, at $1.03$ per data point. The decisive change from the 1991 fit is in the effective Pomeron mass scale $m_P^2$, which rose from $10.7$ to $49.5$ GeV$^2$, roughly a factor of five. That single shift makes the Pomeron intercept $a_P(Q^2)$ depart from its real-photon value early, so the transition from soft to hard behaviour sets in at lower $Q^2$ than before, which is exactly where the old fit failed. The intercept at $Q^2=0$ was fixed to the Donnachie–Landshoff value, since the HERA photoproduction data alone do not pin it down. Table 1 reports the residuals set by set: the ZEUS-94 sample contributes $253.8$ to $1.03$ per point's worth of total for 188 points, and the combined photoproduction sample $262.3$ for 228 points, so the agreement is good but not uniform across experiments. The authors also use the fitted function to locate the soft-to-hard transition by comparing the measured slope $dF_2/dQ^2$ against a QCD-evolution parameterization and a Regge fit, and conclude the crossover sits around $1$ to $3$ GeV$^2$.

Load-bearing premise

The fit treats 1356 points from ten independent experiments as if each carried only its own quoted uncertainty, with no relative normalization factors between datasets and no correlated systematic errors, so the fitted parameters may be an effective interpolation rather than a determination of the Pomeron and Reggeon couplings.

Editorial extensions

If this is right

Acceptance and radiative corrections for $ep$ experiments can be computed from one closed-form expression valid from $Q^2=0$ to $5000$ GeV$^2$, instead of stitching together separate soft and hard parameterizations at a matching scale.
The soft-to-hard transition can be located empirically at roughly $1$ to $3$ GeV$^2$ by watching where the measured slope $dF_2/d\ln Q^2$ stops rising as $x$ falls, since ALLM97 reproduces that turnover while GRV and MRSR1 do not.
The measured low-$x$ slope $\lambda = d\ln F_2/d\ln x$ can be read as the Pomeron intercept $\alpha_P - 1$ only when the fit uses data below $x < 10^{-3}$; wider $x$ windows bias the extracted intercept, and the parameterization reproduces that bias.
The parameterization can be extrapolated into kinematic corners with no data, which is what makes it usable as a generator-level input rather than only as a fit summary.

Reading between the lines

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

The claim is a statement about interpolation quality, not about the reality of Pomeron and Reggeon exchange; a 23-parameter fit that lands at $1.03$ per point over ten heterogeneous datasets is consistent with the functional form being a good effective description even if the individual fitted couplings carry little independent meaning.
Because the fit does not appear to include relative normalization factors between the ten datasets, the residual pattern in Table 1 (ZEUS-94 at $1.35$ per point, photoproduction at $1.15$) is the place to look first: a refit with per-experiment normalizations and correlated systematics would show whether the parameters are stable or merely absorbing inter-experiment offsets.
The prediction that the slope turnover sits at $1$ to $3$ GeV$^2$ is the sharpest testable statement in the paper, since it is a shape prediction in a region that was then only sparsely measured; a denser scan there would either confirm the turnover location or force the Pomeron mass scale to move again.
If the same two-component form continues to hold at much higher $Q^2$ and lower $x$ than the fitted range, that would suggest the Regge-inspired $t$-dependence is capturing something robust about the $Q^2$ evolution rather than an accident of the fitted window.

Original abstract

The ALLM parameterization of sigma_{tot}(gamma* p) has been updated by using all published F_2 data to determine its parameters. The fit yields a chi^2/ndf of 0.97 for the 1356 data points. The updated ALLM parameterization, ALLM97, gives a good description of all the available data in the whole x and Q^2 range studied so far (3\times10^{-6} < x < 0.85, 0 \le Q^2 < 5000 GeV^2).

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.21 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

The ALLM parameterization of sigma_{tot}(gamma* p) has been updated by using all published F_2 data to determine its parameters. The fit yields a chi^2/ndf of 0.97 for the 1356 data points. The updated ALLM parameterization, ALLM97, gives a good description of all the available data in the whole x and Q^2 range studied so far (3\times10^{-6} < x < 0.85, 0 \le Q^2 < 5000 GeV^2).

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	The ALLM parameterization of sigma_{tot}(gamma* p) has been updated by using all published F_2 data to determine its parameters.	contribution 0.80	0.85	0.14	asserted 1.00
s1	The fit yields a chi^2/ndf of 0.97 for the 1356 data points.	result 1.00	0.93	0.07	asserted 1.00
s2	The updated ALLM parameterization, ALLM97, gives a good description of all the available data in the whole x and Q^2 range studied so far (3\times10^{-6} < x < 0.85, 0 \le Q^2 < 5000 GeV^2).	result 0.73	0.89	0.85	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.69 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
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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

Figures

**Figure 1.** The total γ ∗p cross section as function of W2 , for different Q2 values. The curves are the expectations of the ALLM parameterization. 9

**Figure 2.** The total γ ∗p cross section as function of W2 , for different Q2 values. The curves are the results of the ALLM-N parameterization. 10

**Figure 3.** The intercept of the Pomeron trajectory as functio

**Figure 4.** The intercept of the Pomeron trajectory as functio

**Figure 5.** The dependence of the proton structure function,

**Figure 6.** The effective virtual photon-proton cross section

**Figure 7.** The total virtual photon-proton cross section as fu

**Figure 8.** The total γ ∗p cross section as function of W2 , for different Q2 values. The curves are the expectations of the ALLM97 parameterization (full line) and those of the MRSR1 one (dotted line). 15

**Figure 9.** The slope of F2 with respect to ln x as function of x for Q2 values as indicated in the figure. The round dots are the HERA data and the triangle symbols are the results of the GRV94HO parameterization. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 -5 10 -4 10 -3 10 -2 10 -1

**Figure 10.** The slope of F2 with respect to ln x as function of x for Q2 values as indicated in the figure. The round dots are the HERA data, the triangle symbols are the results of the MRSR1 parameterization and the full squares come from a Regge fit to the BPC data. 16

**Figure 11.** The slope of F2 with respect to ln x as function of x for Q2 values as indicated in the figure. The full dots are the HERA data and the open symbols are the results of ALLM97. 17

**Figure 12.** The slope λ as function of Q2 for different x cuts, as indicated in the figure. The dashed line is αP - 1 as calculated from the ALLM97 parameterization, while the full line is the value of λ as obtained from the parameterization in the same way as from the data. 18

Figures reproduced from arXiv:hep-ph/9712415 by the authors.

References

Works cited (23)

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B. Surrow, private communication. 8 1 10 100 1000 10000 10000010 -2 10 -1 1 10 10 2 Figure 1: The total γ∗p cross section as function of W 2, for diﬀerent Q2 values. The curves are the expectations of the ALLM parameterization. 9 1 10 100 1000 10000 100000 10 -2 10 -1 1 10 10 2 Figure 2: The total γ∗p cross section as function of W 2, for diﬀerent Q2 valu...

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How the literature uses it

36 arXiv papers cite this in 43 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 · 23 sentences without a confident label

Uses its method or tool 10 papers

In our Monte Carlo, tau energy losses are simulated with [ref] using the ALLM97 parametrization to describe the photo-nuclear interactions [ref] .

Tau Appearance from High-Energy Neutrino Interactions · Introduction · 0.98
Three of them, the ALLM [ref] , FFJLM (Fiore et al.) [ref] and LUX-like [ref] were already used in our previous publications and are described in more detail in Ref. [ref] .

Production of $e^{+} e^{-}$ in proton-lead collision: photon-photon fusion · Structure function parametrizations · 0.90
Throughout this paper we evaluate the neutrino-nucleon cross section using the nCTEQ15 parton distribution functions [ref] and adopt the Abramowicz-Levin-Levy-Maor (ALLM) parameterization of the proton structure function [ref] for photonuclear energy loss, as discussed in more detail in Ref. [ref] .

POEMMA's Target of Opportunity Sensitivity to Cosmic Neutrino Transient Sources · POEMMA's Effective Area, Exposure, and Sensitivity · 0.83

Uses its data or values 7 papers

Concretely, we use the ALLM97 fit to $F_2$ [ref] which represents the measured data points of [ref] - [ref] within their errors, except maybe for the region of very low $Q^2$ where the fit appears to be slightly worse.

On the Range of Validity of the Dipole Picture · Bounds on Ratios of $F_2$ at Different Values of $Q^2$ · 0.94
In this region, we directly take the structure functions measured by the HERMES experiment with the GD-11P and GD-11D fits [ref] with the ALLM parametrization [ref] .

The Photon Content of the Neutron · The LUXqed photon PDF formalism · 1.00
We assume the mean cross sections from Ref. [ref] and the energy losses from Ref. [ref] in constructing the lookup tables.

Prospects for High-Elevation Radio Detection of >100 PeV Tau Neutrinos · Tau Neutrino Propagation & Tau Lepton Decay · 0.92

Compares with it 1 paper

[ref] fitted $F_2$ data prior to 1997 above the resonance region, and is provided with and without $Q^2$ dependence.

Extraction of the neutron F2 structure function from inclusive proton and deuteron deep-inelastic scattering data · Isoscalar corrections with structure functions · 0.87

Cites it as background 2 papers

The dominant energy loss processes of high-energy muons are electron pair production [ref] , bremsstrahlung [ref] , and inelastic nuclear interaction [ref] .

Coulomb corrections to the bremsstrahlung and electron pair production cross section of high-energy muons on extended nuclei · Introduction · 0.97
Together with the production of electron-positron pairs [ref] and the inelastic nuclear interaction [ref] it describes the dominant contribution to the energy loss of high-energy muons.

Radiative corrections to the average bremsstrahlung energy loss of high-energy muons · Introduction · 0.90

Discussion

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B. Surrow, private communication. 8 1 10 100 1000 10000 10000010 -2 10 -1 1 10 10 2 Figure 1: The total γ∗p cross section as function of W 2, for diﬀerent Q2 values. The curves are the expectations of the ALLM parameterization. 9 1 10 100 1000 10000 100000 10 -2 10 -1 1 10 10 2 Figure 2: The total γ∗p cross section as function of W 2, for diﬀerent Q2 valu...

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