Infrared Subtraction with Artificial Intelligence

Wenjie He; Xiaohui Liu; Yandong Liu; Zhan Wang

hep-phSep 28, 2026

Infrared Subtraction with Artificial Intelligence

Authors: Wenjie He, Xiaohui Liu, Yandong Liu, Zhan Wang

Organizations: School of Physics and Astronomy, Beijing Normal University, and Key Laboratory of Multiscale Spin Physics (Beijing Normal University), Ministry of Education, Beijing 100875, China · Southern Center for Nuclear Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Science, Huizhou 516000, China

Abstract

We present AI-developed local infrared subtraction, building on projection to Born and EFT matching. The framework separates an integrable radiation term from a finite contribution at Born kinematics, referred to as the Born contact. The contact is determined using the EFT singular distribution in a resolution observable such as N-jettiness $τ_N$ . Under human physics guidance, an LLM develops two implementations. One uses a neural network for phase space projection and fits the contact by matching to EFT cumulants. The other uses an analytic construction that keeps the Born momenta fixed while integrating over radiation. It combines the EFT $δ(τ_N)$ coefficient with finite 4-dimensional radiation integrals to calculate the contact term directly. This gives a local subtraction formula without a slicing parameter, while reusing existing lower-order radiation calculations and EFT singular predictions. As a demonstration, we reconstruct the full NLO correction for massless 3- and 4-jet production in electron-positron annihilation. The attempt to the NNLO dijet production is also made by recursively using the NLO P2B construction with the LLM designing machine-learning controls to reduce the variance of the contact integral. The tested predictions are in good agreement with EERAD3. The numerical calculation and projection-network training use a 2020 Apple M1 MacBook, without GPU acceleration, illustrating the feasibility of the construction with modest computing resources. The appendices develop an extension of the local subtraction to 3-jet NNLO, giving explicit radiation maps and a proposed contact formula. We also show how to integrate over NNLO radiation while keeping the Born momenta fixed, for any number of massless final-state jets. Our results demonstrate how AI can help higher-order calculations by constructing infrared subtraction and improving its numerical integration.

Figures & tables

Panel of Figure 1 — Figure 1: Roles of human physics guidance, the LLM, machine learning and EFT input in the subtraction construction. Humans choose the P2B + EFT matching framework and specify the physical constraints. The LLM develops neural and analytic projections, contact formulas and their numerical implementation. EFT supplies the Born-dependent singular distribution. The contact is fitted through matching or calculated from finite radiation integrals and the EFT $\delta(\tau_{N})$ coefficient. In the LLM-developed local subtraction, learned control functions reduce the integration variance, with their known integrals added back to preserve the result. The radiation term and Born contact together give the fixed-order prediction.

Ingredient	Input or permitted use	Excluded from the new construction
Projection	Born information, parton kinematics and identities, universal infrared limits and geometric constraints.	Reuse of local counterterms or momentum maps from existing subtraction schemes.
Radiation	The complete calculation with one additional jet at one lower perturbative order, including its internal subtraction.	—
Contact	EFT singular predictions at fixed Born kinematics, including the coefficient of $\delta(\tau_{N})$ , after fixing the projection.	Separate virtual or integrated-counterterm contributions as contact-fitting targets.
Prediction	A common fixed projection and contact; independent reference comparisons.	Observable-dependent changes to the fixed projection or contact.

Table 1: Physical inputs and construction restrictions for AI-developed P2B subtraction. The exclusions concern the new subtraction being constructed. The lower-order radiation input may use its existing subtraction, as indicated by the dash.

Panel of Figure 2 — Figure 2: Common workflow for the two NLO implementations. Humans specify the framework and physical constraints. The LLM develops a projection and checks its kinematic and infrared properties, including training for the neural option. With the projection fixed, the contact is fitted through EFT matching or calculated directly using finite radiation integrals and the EFT $\delta(\tau_{3})$ coefficient. Each implementation uses its own common projection and contact for all final distributions.

Panel of Figure 3 — Figure 3: Modules for projection, radiation integration and contact determination. Panels (a)–(c) describe the neural implementation at NLO. The projection network has two hidden layers with sigmoid linear unit (SiLU) activations. Its 11 outputs comprise three shape parameters and eight coefficients for two orientation vectors. The real-emission integral uses the exact matrix element. Contact fitting uses EFT singular cumulants weighted by functions of the Born variables, in the limit $t\to 0$ . Here $\widetilde{\Phi}_{3}=\widehat{M}_{4}\Phi_{4}$ . Panel (d) shows the additional use of machine learning in local two-jet NNLO contact integration. A control function is trained on separate samples, then held fixed while the exact residual is integrated and the known control integral restored.

Figure 4: Complete three-jet NLO coefficient distributions with strict JADE/E

y_{23}>0.04

and no

y_{34}

cut. The projection and the 127-parameter contact are common to all panels. Blue bands combine P2B Monte Carlo and contact bootstrap errors; black points show EERAD3 with Monte Carlo errors. In all distribution plots,

O

denotes the observable on the horizontal axis. Lower panels show

\Delta/\sigma_{\Delta}

, where

\Delta

is the difference from the reference and

\sigma_{\Delta}

its combined statistical error. The same event sample contributes to different bins, and the fitted contact introduces additional correlations.

Figure 5: Three-jet NLO coefficient distributions from the analytic local subtraction with a directly calculated contact, using the same observables and physical cuts as Fig. 4 . Green, orange and purple curves show the P2B radiation term

F_{\widehat{M}}

, the Born-weighted EFT

\delta(\tau_{3})

coefficient

H_{1}

, and the Born-weighted finite radiation integral

\mathcal{A}_{1}

, respectively. Their sum is the blue result; its band includes the correlation between radiation and contact. Black points reproduce the EERAD3 reference of Fig. 4 . Lower panels compare the total with the reference in combined standard errors, with grey bands at

\pm 2

. The calculation uses the maps of Appendix A and the local contact weights of Appendix D , over the full radiation phase space.

Figure 6: Comparison of three contact determinations for the analytic projection. Each curve shows the Born integral in Eq. ( 8 ), denoted

B_{\mathrm{contact}}

, for the analytic projection and the cuts of Fig. 4 . The black histogram uses the finite radiation integral

\mathcal{A}_{1}

in Eq. ( 52 ) plus the EFT

\delta(\tau_{3})

coefficient

H_{1}

. Blue circles and orange squares use Eq. ( 55 ) for

\mathcal{A}_{1}

, with

\rho=0.1

and

0.01

. Error bars and the black band denote one standard error; markers are displaced within each bin for visibility. Lower panels show the differences from the black result in standard errors, including shared-sample correlations; grey bands indicate

\pm 2

. As defined in Appendix D ,

\rho

is a free rescaling parameter, not a cutoff.

Figure 8: Two-jet NNLO coefficient distributions

\mathcal{B}_{2}

. Red dashed curves use local subtraction with the directly calculated contact. The red bands combine P2B and contact errors, retaining the common contact correlation between Born-supported bins. Green solid curves show the cumulant-matched calculation, and black points the EERAD3 reference. Lower panels compare the red prediction with the reference in combined statistical errors; grey bands indicate

\pm 2

. Shaded upper-panel bins contain Born support. There are no event-selection cuts; EEC includes all ordered parton pairs and self correlations, and JADE/E clustering at

y_{\rm cut}=0.04

includes two-, three-, and four-jet events.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Check	Result
On-shell and momentum conservation	$<4\times 10^{-14}$
Complete four-body round trip	$<5\times 10^{-13}$
Fixed-Born reconstruction	$<3\times 10^{-16}$
Volume / $[Q^{4}/(24576\pi^{5})]$	$1.00006\pm 0.00008$

Appendix

Table 2: Phase-space checks of the analytic

4\to 3

map. Residuals are maximum componentwise errors, with momenta normalized to

Q

and squared masses to

Q^{2}

. The scans use 4096 points per branch and 4096 independent four-body points. The volume test uses

2^{28}

points in 32 independent batches; its uncertainty is one standard error.

Figure 9: Local P2B cancellation at a fixed three-parton Born point. The upper panels show the

q\bar{q}gg

channel with the

qg

pair

(1,3)

generated by Eq. ( 16 ), at

\phi=0.7

. Write

\delta=1-\lambda

and

w=1-\cos\theta

. The collinear, soft and soft-collinear paths use

(\delta,w)=(\epsilon,0.8),(\epsilon,20\epsilon),(\epsilon^{2},2\epsilon)

, respectively. The ordinate

\mathcal{W}_{J}

denotes the logarithmic radiation density for

J=1-T

. Blue and orange show

\omega\mathcal{J}_{h}R_{4}J_{4}/B_{3}

and

\omega\mathcal{J}_{h}R_{4}J_{3}(b)/B_{3}

; green shows their difference. Here

\omega=\delta

in (a) and

\omega=\delta w

in (b,c), corresponding to logarithmic radiation coordinates. The lower panels integrate all six branches and both partonic channels. Their lower strips show

\mathcal{F}_{J}(\eta)/\mathcal{F}_{J}^{\mathrm{ref}}

, with

\mathcal{F}_{J}^{\mathrm{ref}}=\mathcal{F}_{J}(10^{-10})

and correlated statistical errors. Grey bands give the relative one-standard-error uncertainty of the reference integral. The technical cut

\eta

on pair invariants probes the numerical stability of the locally subtracted integral.

Check	Maximum residual
Mass shell	$<7\times 10^{-17}$
Momentum conservation	$<2\times 10^{-16}$
Subtraction inverse, $\widehat{m}_{a}G_{a}^{\mathrm{ant}}=x$	$<3\times 10^{-13}$
Fixed Born point, $\widehat{M}_{5,a}G_{a}^{\mathrm{ant}}G_{4,h}=b$	$<3\times 10^{-16}$

Appendix

Table 3: Kinematic checks of the common fixed-Born phase space at

2E_{i}/Q=(0.8,0.7,0.5)

. Of 600 attempted four-parton configurations, 308 satisfy the selected-chart condition. Each is tested at azimuths

\phi

and

\phi+\pi/2

. Entries are maximum componentwise residuals normalized to

Q

, or

Q^{2}

for the mass shell, using extended-precision geometry.

Figure 10: Local cancellation in the three-jet NNLO radiation contribution at fixed Born kinematics

2E_{i}/Q=(0.8,0.7,0.5)

, for thrust

J=1-T

. The upper and lower rows approach three double-collinear and three double-soft limits, respectively. The curves show the magnitudes of the locally subtracted radiation density (blue), its Born-projected counterpart (orange), and their paired difference (green), including the four-jet NLO counterterms. All densities include the logarithmic phase-space Jacobian and are normalized to the tree-level Born density. The paired difference is evaluated before integration and vanishes along all six paths. No technical cut is applied. The paths, normalization and correlated azimuthal average are specified in the text. These checks concern the radiation contribution and do not include the NNLO contact.

Figure 11: NNLO radiation integrals at the fixed three-parton Born point of Fig. 10 . The ordinate

\mathcal{F}_{J}^{(2)}

is the radiation integral normalized to the tree-level Born density, with the coupling stripped as in Eq. ( 33 ). The upper panels show the paired five-parton integral (blue), the four-parton term containing the finite one-loop and integrated-counterterm weight

K_{4}

(orange), and their sum (green). In each sector

a

, the paired integrand is

R_{a}[J_{5}(p)-J_{3}(b)]-S_{a}[J_{4}(\widehat{m}_{a}p)-J_{3}(b)]

, where

R_{a}

is the real sector weight and

S_{a}

its NLO counterterm, evaluated on the same five-parton state

p

;

b

is the common Born point. The middle panels divide the sum by its mean over

10^{-11}\leq\eta\leq 10^{-7}

, accounting for correlations. The first two rows share aligned axes, with the directly evaluated

\eta=0

point shown separately. The lower panels show the cumulative mean at

\eta=0

over 160 independent batches. Bands and error bars denote one standard error, including the independent four-parton uncertainty in the sum. The parameter

\eta

is a technical cut for numerical-stability checks. These results evaluate the locally subtracted radiation contribution without a small slicing parameter; the NNLO contact is a separate contribution.

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