Source-linked AI summary
Automatic evaluation of UV and R2 terms for beyond the Standard Model Lagrangians: a proof-of-principle
Celine Degrande
TL;DR
One-loop computations need UV counterterms and, for OPP methods, R2 rational terms in addition to the tree-level Lagrangian and Feynman rules. The paper automates their determination for renormalizable models with NLOCT, FeynRules, and FeynArts, validates the implementation, and applies it to the 2HDM. The resulting ingredients support automated NLO computations for renormalizable BSM models, although the displayed electroweak UV-counterterm examples are incomplete.
Problem
One-loop calculations require UV counterterms and method-dependent R2 terms that had been added manually, limiting automated NLO treatment of available BSM models.
Method
NLOCT works with FeynRules and FeynArts to derive UV counterterms and R2 terms automatically from renormalizable tree-level Lagrangians.
Results
The code was validated against Standard Model and MSSM results and built-in MadGraph5 versions, while complete generic-2HDM QCD rules were obtained.
Takeaways & Limitations
The generated models provide one-loop ingredients for aMC@NLO, GoSam, and other tools, enabling automated NLO computations for renormalizable BSM models.
Takeaways & Limitations
Only a few electroweak UV-counterterm examples are displayed because the full expressions are too large, although the complete list is publicly available.
Abstract
from arXiv · showhide
The computation of renormalized one-loop amplitudes in quantum field theory requires not only the knowledge of the Lagrangian density and the corresponding Feynman rules, but also that of the ultraviolet counterterms. More in general, and depending also on the methods used in the actual computation of the one-loop amplitudes, additional interactions might be needed. One example is that of the R2 rational terms in the OPP method. In this paper, we argue that the determination of all elements necessary for loop computations in arbitrary models can be automated starting only from information on the Lagrangian at the tree-level. In particular, we show how the R2 rational and ultraviolet counterterms for any renormalizable model can be computed with the help of a new package, which we name NLOCT and builds upon FeynRules and FeynArts. To show the potential of our approach, we calculate all additional rules that are needed to promote a Two Higgs Doublet Model Lagrangian to one-loop computations in QCD and electroweak couplings.
1 Introduction
Accurate loop predictions require ultraviolet counterterms and, for OPP methods, additional rational terms that have traditionally limited automation beyond the Standard Model. The paper proposes automating these ingredients from tree-level Lagrangians using NLOCT with FeynRules and FeynArts, demonstrated for the 2HDM.
- Motivation: Precise predictions test fundamental interactions and constrain both Standard Model measurements and searches for new particles.Accurate Standard Model backgrounds are essential for collider searches, while accurate BSM predictions become important after an excess is found.
- Missing loop ingredients: NLO loop calculations require ultraviolet counterterms and, in OPP-based methods, an additional part of the rational term.Counterterms absorb one-loop ultraviolet divergences, whereas the required rational contribution depends on the tensor-decomposition method.
- Rational terms: Rational terms split into R1 and R2, with R1 obtained from denominator effects and R2 from the d−4 component of the numerator.The passages describe R1 as computable using four-dimensional ingredients with a different scalar-integral set, while R2 is a finite numerator contribution.
- Automated approach: The paper automates UV-counterterm and R2 determination for renormalizable Lagrangians using FeynRules, NLOCT, and FeynArts.The approach requires the model to be written in Feynman gauge and currently applies to operators of dimension four or less.
- Demonstration: The Two Higgs Doublet Model serves as an explicit example for generating the additional rules needed for one-loop QCD and electroweak computations.The 2HDM is presented as a simple extension of the Standard Model for searching for extra scalar particles.
2 Renormalization
The renormalization procedure rewrites bare quantities using renormalized fields and parameters, then extracts UV counterterm vertices from the counterterm Lagrangian. NLOCT automates this process for FeynRules models under specified scheme and model assumptions.
- In dimensional regularization, UV divergences appear as 1/ε poles and are absorbed by redefining free parameters and fields.
- The renormalization constants: External parameters are independent experimentally fixed quantities, whereas internal parameters depend on external ones and inherit renormalization through those dependencies.Internal parameters therefore require no new renormalization constants.
- The renormalization constants: The bare Lagrangian is split into a renormalized Lagrangian and a counterterm Lagrangian linear in the renormalization constants.UV counterterm vertices are extracted from the counterterm Lagrangian.
- The renormalization constants: OnShellRenormalization renormalizes physical fields, masses, and remaining parameters in the on-shell or complex-mass scheme, after expanding the Lagrangian over flavor.Its options can remove unwanted terms and speed computation.
- The renormalization constants: Internal masses can be exchanged for external parameters before renormalization, allowing quantities such as the W-boson mass to be renormalized without changing the initial model.The FR$LoopSwitches variable specifies the parameter exchange.
- The renormalization constants: Yukawa masses are replaced by usual fermion masses before renormalization to ensure a renormalizable Lagrangian and unitary treatment.The two external masses must be equal for these properties to hold.
- Renormalization conditions: The finite parts of counterterms may require renormalization conditions beyond cancellation of ultraviolet poles, chosen to simplify calculations or clarify the physics.
- Renormalization conditions: On-shell conditions fix masses and wave functions through two-point functions, absorb flavor-mixing corrections, and keep renormalized fields as mass eigenstates.For massless fermions, some standard conditions are replaced by zero-momentum conditions.
3 Computation of the R2 and UV counterterms
The paper automates extraction of R2 and UV counterterms from renormalizable Lagrangians using FeynRules, NLOCT, and FeynArts. The algorithm evaluates generic one-loop contributions, resolves renormalization conditions, and produces model-specific counterterm outputs.
- Package workflow: NLOCT uses FeynArts-generated amplitudes and a FeynRules interface to compute R2 terms and UV counterterms from a renormalized Lagrangian.The workflow includes WriteFeynArtsOutput and WriteCT, with outputs written for subsequent UFO use.
- Algorithmic assumptions: The procedure assumes a renormalizable model written in Feynman gauge, which is also used by MadLoop.These assumptions constrain the class of models directly supported by the algorithm.
- Generic amplitudes: R2 and UV parts are computed at generic level, separating model-independent Lorentz structures from model-specific couplings and masses.Actual field insertions later determine masses and couplings for each diagram.
- Integral treatment: The algorithm retains finite one- and two-point contributions for renormalization, while terms for amplitudes with more than two external particles are removed.The MSbar option instead retains only UV divergences.
- Renormalization conditions: UV-divergent counterterms are obtained from one-loop divergences, while UV-finite renormalization constants come from finite renormalized two-point functions.The two pieces are combined to form full UV counterterm vertices; tadpoles are handled directly from the corresponding one-loop amplitudes.
- Validation and performance: Running times for the SM, MSSM, and 2HDM remain within a few hours on a dual-core 2.4 GHz laptop with 4 GB of RAM.The SM QCD expressions agree with published results, and the generated UFO agrees with the built-in aMC@NLO version.
4 2HDM
The 2HDM extends the Standard Model by adding a second scalar doublet, whose potential, vacuum structure, Yukawa couplings, and physical scalar spectrum are parameterized for implementation in FeynRules.
- Model definition: The 2HDM adds one scalar multiplet with the Standard Model Higgs doublet’s gauge transformation properties.Its Lagrangian separates the SM terms without the Higgs doublet, the scalar potential, and fermion-scalar interactions.
- Scalar potential: The most general scalar potential contains complex µ3, λ5, λ6, and λ7 parameters, while the remaining parameters are real.Some parameters are basis-dependent and can change under unitary transformations of the two doublets.
- Vacuum and basis: The vacuum is chosen with aligned vacuum expectation values so that U(1)EM remains unbroken and only one doublet has a nonzero vev in the Higgs basis.A remaining phase freedom can be used to make one selected parameter real or fix the CP transformation.
- Physical parameters: The physical scalar masses and mixing angles are functions of the potential parameters and are used as external parameters in the FeynRules implementation.This choice facilitates on-shell renormalization and makes the potential positive definite; λ2, λ3, and λ7 remain internal exceptions.
- Flavor and CP assumptions: Flavor conservation requires diagonal Yukawa matrices and a diagonal CKM matrix, while CP conservation makes selected potential parameters real and Yukawa matrices hermitian.The implementation treats Yukawa matrices and their real and imaginary parts according to the chosen model restrictions.
5 2HDM R2 counterterms
The paper provides automated 2HDM R2 counterterms for QCD and electroweak interactions, identifying which vertices differ from Standard Model expressions under the stated model assumptions.
- QCD corrections: For QCD, the generic 2HDM is used without imposing flavor or CP conservation, and only quark-scalar and gluon-scalar vertices are modified relative to the SM.These are the only new tree-level interactions involving colored particles.
- QCD R2 vertices: The QCD R2 rules include fermion-fermion-scalar, gluon-gluon-scalar, and gluon-gluon-two-scalar vertices.The scalar index distinguishes the three neutral physical scalars, and CP conservation imposes the expected pseudoscalar γ5 structure and vanishing vertices.
- Electroweak corrections: For electroweak corrections, CP and flavor conservation are assumed, while fermion and vector two-point R2 vertices remain unchanged from the SM.Several vector, Goldstone, and scalar interaction classes are correspondingly SM-like or related by scalar mixing substitutions.
- Electroweak R2 vertices: The electroweak rules cover scalar, fermion, vector, Goldstone, and mixed multi-field R2 interactions, including scalar-scalar-vector, three-scalar, and scalar-scalar-vector-vector vertices.Vertices involving two Goldstone bosons are omitted when identical to their SM values.
- Scope limitation: The four-scalar electroweak R2 vertices were not computed because of their size and low phenomenological relevance.This is the principal stated scope limitation of the displayed electroweak counterterm set.
6 2HDM UV counterterms
The 2HDM electroweak counterterms are derived under CP and flavor symmetries, with mass and scheme assumptions that make the expressions extensive. Only representative electroweak examples are displayed, while the complete results are made available publicly.
- 6.2 EW corrections: Electroweak UV counterterms are computed assuming CP and flavor symmetries, nonzero nondegenerate masses, and the complex mass scheme.Electroweak corrections involve more particles and mass hierarchies than QCD corrections.
- 6.2 EW corrections: The on-shell scheme requires taking only the real part of the l function when masses are real.The other terms are real under this scheme and mass assumption.
- 6.2 EW corrections: Electroweak corrections mix fields, including an extra physical-charged-scalar contribution to photon-Z mixing.
- 6.2 EW corrections: New-field two-point functions receive electroweak corrections; the physical charged scalar provides an example through its wave-function renormalization constant.
- 6.2 EW corrections: Only a few illustrative electroweak UV-counterterm examples are displayed because the expressions are large, while the full list is available on the FeynRules 2HDM webpage.
7 Conclusion
The paper presents an automated workflow for obtaining counterterm vertices in renormalizable models and applies it to NLO computations in BSM theories. The method is validated against established results and supplies comprehensive 2HDM QCD and electroweak ingredients within stated symmetry and renormalizability assumptions.
- 7 Conclusion: Counterterm vertices for any renormalizable Lagrangian with operators of dimension four or less can be obtained automatically using three packages.The packages assume renormalizability rather than checking it.
- 7 Conclusion: The workflow requires a FeynRules model in Feynman gauge, performs renormalization in FeynRules, computes UV and R2 vertices with NLOCT and FeynArts, and exports UFO output.The exported vertices can be used by MadGraph5.
- 7 Conclusion: The implementation enables fully automated NLO computations for renormalizable BSM models and was validated against analytical SM, MSSM, and MadGraph5 comparisons.
- 7 Conclusion: The full QCD R2 and UV counterterms are provided for the generic 2HDM, while electroweak results are supplied under CP and flavor conservation with representative examples displayed.The complete electroweak UV counterterms were obtained automatically and made publicly available.