arXiv subjects

Maximilian Stahlhofen

Publications and source records attributed to Maximilian Stahlhofen.

At least 19 recordsLinked to original sources

arXiv2026

Integrating out a heavy Higgs singlet: on the edge between SMEFT and HEFT

We use a functional approach based on the background-field formalism and the expansion by regions to integrate out the heavy Higgs field (associated with the mass eigenstate H) at the one-loop level in a singlet extension of the SM, which features an additional real scalar singlet and a spontaneously broken $Z_2$ symmetry. We obtain an effective Lagrangian to $O(1/M_H^2)$ in the limit of large Higgs mass ($M_H\gg M_h=125$GeV) providing a consistent treatment of effects from Higgs mixing and the renormalization of the underlying model. The scaling behaviour of the model parameters in the large-$M_H$ limit determines whether the effective Lagrangian can be accommodated in the SM Effective Field Theory (SMEFT) or involves non-SMEFT operators within the more general Higgs Effective Field Theory (HEFT) framework. We choose a limit that ensures decoupling of beyond-SM (BSM) effects at $O(M_H^0)$ by demanding that the Higgs mixing angle $α$ is of $O(M_h/M_H)$ and putting minimal constraints on the other input parameters. The considered model is restricted to massless fermions. Bottom-up (diagrammatic) matching with only bosonic SMEFT operators at $O(1/M_H^2)$ necessarily fails, although the BSM sector does not directly couple to massless fermions. However, it is possible to match SMEFT with bosonic and fermionic operators to the SESM in the considered large-$M_H$ limit. For the emerging Effective Field Theory (EFT) we give two alternative Lagrangians: one that includes only bosonic BSM EFT operators, but necessarily involves operators of non-SMEFT type, and another one that is of SMEFT form, but involves also fermionic EFT operators. We validate our results for the effective Lagrangians at NLO in the coupling expansion by verifying that the difference between EFT and full-theory predictions vanishes faster than $1/M_H^2$ for several electroweak observables in the large-$M_H$ limit.

hep-ph↗

arXiv2026

Effective Lagrangians from functional matching

We briefly review a variant of functional matching to derive an Effective Field Theory (EFT) for heavy particles at the one-loop level in the top-down approach. The method integrates out heavy fields that correspond to mass eigenstates, i.e. after removing mixing effects by diagonalizing mass matrices. Tree- and loop-level effects are separated by employing the background-field method, hard and soft modes are separated with the use of the expansion by regions. The method is exemplified for the Higgs Singlet Extension of the Standard Model where the mass $M_\mathrm{H}$ of the additional Higgs boson is considered large, and the Higgs mixing angle $α$ is assumed to scale like $1/M_\mathrm{H}$, in order to guarantee decoupling in the large-$M_\mathrm{H}$ limit. Our calculation is agnostic w.r.t. the type (SMEFT vs. HEFT) of the emerging EFT. Eventually the emerging EFT Lagrangian can be transformed into SMEFT form, but only at the cost of introducing fermionic EFT operators, although no such operators are directly generated upon solving the functional integral over the heavy Higgs field.

hep-ph↗

arXiv2026

Three-loop jet function for boosted top quarks

We present the calculation of the inclusive jet function for highly energetic heavy quarks at order $\mathcal{O}(α_s^3)$ using boosted Heavy-Quark Effective Theory (bHQET). This jet function describes the effect of collinear radiation emitted by energetic heavy quarks on observables dependent on the jet invariant mass $M$. In particular, we focus on the regime $M^2 - m^2 \ll m^2$, which is relevant for boosted top quark production at high-energy colliders in the resonance region. Our results are consistent with non-Abelian exponentiation and reproduce the known cusp and non-cusp anomalous dimensions up to three loops. We also verify that the $n_\ell^2 α_s^3$ contribution, with $n_\ell$ denoting the number of light quark flavors, agrees with predictions from renormalon calculus. This calculation completes the list of ingredients required for the N$^3$LL$^\prime$ resummed (self-normalized) thrust distribution, an essential component for calibrating the top quark mass parameter in parton-shower Monte Carlo generators. It likewise contributes to the invariant-mass distribution of reconstructed top quarks, enabling precise mass determinations at future lepton colliders. Finally, we determine the relation between the pole and two short-distance jet-mass schemes at $\mathcal{O}(α_s^3)$ and provide an estimate of the non-logarithmic part of the four-loop jet function based on renormalon dominance.

hep-ph↗

arXiv2025

NNLO electron structure functions (PDFs) from SCET

We calculate the electron structure functions, aka parton distribution functions (PDFs), to NNLO in QED. The calculation is based on the definition of the PDFs in terms of operator matrix elements in soft collinear effective theory (SCET) and directly performed in momentum space. The electron PDFs describe the universal effects of collinear initial state radiation (ISR) off the electron (or positron) in DIS or $e^+ e^-$ collision events. The parton collinear to the electron that enters the hard scattering process governed by the energy scale $Q$ can be a photon or a (anti)fermion with mass $m_f \sim m_e \ll Q$. Our SCET momentum-space calculation confirms earlier results obtained from a QED computation in Mellin space and extends them by taking more than one massive fermion flavor into account. The well-known factorization of sufficiently inclusive cross sections into PDFs with massive fermions and hard partonic cross sections with massless fermions at leading order in $m_e/Q$ is discussed from the SCET perspective. Analogies to the nonperturbative PDFs and collinear factorization in QCD are pointed out. Based on this factorization DGLAP-type resummation of large logarithms of the ratio $m_e/Q$ is straightforward.

hep-ph↗

arXiv2025

Three-loop jet function for boosted heavy quarks

We compute the inclusive jet function for boosted heavy quarks to $\mathcal{O}(α_s^3)$. The jet function is defined and calculated in the framework of boosted Heavy-Quark Effective Theory (bHQET). It describes the effect of radiation collimated in narrow jets arising from energetic heavy quarks on observables probing the jet invariant mass $M$ in the region where $M^2 - m^2 \ll m^2$, with $m$ the heavy quark mass. This kinematic situation is relevant e.g. in boosted top (pair) production at high-energy colliders. We have verified that our result satisfies non-Abelian exponentiation and checked that our calculation reproduces the known cusp and non-cusp anomalous dimensions of the jet function to $\mathcal{O}(α_s^3)$. We also confirmed that the $n_\ell^2α_s^3$ contribution, where $n_\ell$ is the number of massless quark flavors, agrees with the prediction from renormalon calculus. Our computation provides the last missing piece to obtain the N$^3$LL$^\prime$ resummed (self-normalized) thrust distribution used for the calibration of the top quark mass parameter in parton-shower Monte Carlo generators. Our result also contributes to the invariant mass distribution of reconstructed top quarks at N$^3$LL$^\prime$, which can be employed for a precise top mass determination at future lepton colliders. As a by-product, we obtain the relation between the pole and short-distance jet-mass schemes at $\mathcal{O}(α_s^3)$. Finally, we estimate the non-logarithmic contribution to the four-loop jet function based on renormalon dominance.

hep-ph↗

arXiv2024

Three-Loop OPE Wilson Coefficients of Dimension-Four Operators for (Axial-)Vector and (Pseudo-)Scalar Current Correlators

We calculate the three-loop Wilson coefficients of all physically relevant dimension-four operators, i.e. $G_{μν}^a G^{a,μν}$, $m_i\bar q_j q_j$ and $m_i m_j m_k^2$, in the short-distance expansion of the time-ordered product of a pair of gauge-singlet vector, axial-vector, scalar and pseudo-scalar currents. The results are given for a general non-Abelian gauge theory with arbitrary (compact semi-simple) gauge group and $n_f$ light fermion flavors (quarks) in a common arbitrary representation of the gauge group, which includes QCD as a special case. In particular, we allow for arbitrary flavor contents of each of the currents. For the axial-vector current the included contributions from so-called singlet diagrams are consistent with the one-loop axial anomaly.

hep-ph↗

arXiv2023

Two-loop bottom mass effects on the Higgs transverse momentum spectrum in top-induced gluon fusion

We compute bottom mass ($m_b$) corrections to the transverse momentum ($q_T$) spectrum of Higgs bosons produced by gluon fusion in the regime $q_T \sim m_b \ll m_H$ at leading power in $m_b/m_H$ and $q_T/m_H$, where the gluons couple to the Higgs via a top loop. To this end we calculate the quark mass dependence of the transverse momentum dependent gluon beam functions (aka gluon TMDPDFs) at two loops in the framework of SCET. These functions represent the collinear matrix elements in the factorized gluon-fusion cross section for small $q_T$. We discuss in detail technical subtleties regarding rapidity regulators and zero-bin subtractions in the calculation of the virtual corrections present for massive quarks. Combined with the known soft function for $m_b \neq 0$ our results allow to determine the resummed Higgs $q_T$ distribution in the top-induced gluon fusion channel at NNLL$^\prime$ (and eventually N$^3$LL) with full dependence on $m_b/q_T$. We perform a first phenomenological analysis at fixed order, where the new corrections to the massless approximation lead to percent-level effects in the peak region of the Higgs $q_T$ spectrum. Upon resummation they may thus be relevant for state-of-the-art precision predictions for the LHC.

hep-ph↗

arXiv2023

Two-Loop Massive Quark Jet Functions in SCET

We calculate the $\mathcal O(α_s^2)$ corrections to the primary massive quark jet functions in Soft-Collinear Effective Theory (SCET). They are an important ingredient in factorized predictions for inclusive jet mass cross sections initiated by massive quarks emerging from a hard interaction with smooth quark mass dependence. Due to the effects coming from the secondary production of massive quark-antiquark pairs there are two options to define the SCET jet function, which we call universal and mass mode jet functions. They are related to whether or not a soft mass mode (zero) bin subtraction is applied for the secondary massive quark contributions and differ in particular concerning the infrared behavior for vanishing quark mass. We advocate that a useful alternative to the common zero-bin subtraction concept is to define the SCET jet functions through subtractions related to collinear-soft matrix elements. This avoids the need to impose additional power counting arguments as required for zero-bin subtractions. We demonstrate how the two SCET jet function definitions may be used in the context of two recently developed factorization approaches to treat secondary massive quark effects. We clarify the relation between these approaches and in which way they are equivalent. Our two-loop calculation involves interesting technical subtleties related to spurious rapidity divergences and infrared regularization in the presence of massive quarks.

hep-ph↗

arXiv2021

Integrating out heavy fields in the path integral using the background-field method: general formalism

Building on an older method used to derive non-decoupling effects of a heavy Higgs boson in the Standard Model, we describe a general procedure to integrate out heavy fields in the path integral. The derivation of the corresponding effective Lagrangian including the one-loop contributions of the heavy particle(s) is particularly transparent, flexible, and algorithmic. The background-field formalism allows for a clear separation of tree-level and one-loop effects involving the heavy fields. Using expansion by regions the one-loop effects are further split into contributions from large and small momentum modes. The former are contained in Wilson coefficients of effective operators, the latter are reproduced by one-loop diagrams involving effective tree-level couplings. The method is illustrated by calculating potential non-decoupling effects of a heavy Higgs boson in a singlet Higgs extension of the Standard Model. In particular, we work in a field basis corresponding to mass eigenstates and properly take into account non-vanishing mixing between the two Higgs fields of the model. We also show that a proper choice of renormalization scheme for the non-standard sector of the underlying full theory is crucial for the construction of a consistent effective field theory.

hep-ph↗

arXiv2021

Three-loop soft function for energetic electroweak boson production at hadron colliders

We calculate the three-loop soft function for the production of an electroweak boson (Higgs, $γ$, $W^\pm$, $Z$) with large transverse momentum at a hadron collider. It is the first time a soft function for a three-parton process is computed at next-to-next-to-next-to-leading order (N$^3$LO). As a technical novelty, we perform the calculation in terms of forward-scattering-type loop diagrams rather than evaluating phase space integrals. Our three-loop result contains color-tripole contributions and explicitly confirms predictions on the universal infrared structure of QCD scattering amplitudes with three massless parton legs. The soft function is a central ingredient in the factorized cross section for electroweak boson production near the kinematic endpoint (threshold), where the invariant mass of the recoiling hadronic radiation is small compared to its transverse momentum. Our result is required for predictions of the near-threshold cross sections at N$^3$LO and for the resummation of threshold logarithms at primed next-to-next-to-next-to-leading logarithmic (N$^3$LL$^\prime$) accuracy.

hep-ph↗

arXiv2020

The Fully-Differential Gluon Beam Function at NNLO

The fully-differential beam function (dBF) is a universal ingredient in resummed predictions of hadron collider observables that probe the full kinematics of the incoming parton from each colliding proton -- the virtuality and transverse momentum as well as the light-cone momentum fraction $x$. In this paper we compute the matching coefficients between the unpolarized gluon dBF and the usual parton distribution functions (PDFs) at the two-loop order. For observables probing both the virtuality and transverse momentum of incoming gluons, our results provide the part of the NNLO singular cross section related to collinear initial-state radiation, and are required for the resummation of large logarithms through N$^3$LL. Further to this, the dBF is closely linked to the beam function appearing in a generalized version of threshold factorization, via a simple integration. By performing this integration for the two-loop gluon matching coefficients, we also obtain the corresponding quantities for the generalized threshold beam function.

hep-ph↗

arXiv2020

Two-Loop Beam and Soft Functions for Rapidity-Dependent Jet Vetoes

Jet vetoes play an important role in many analyses at the LHC. Traditionally, jet vetoes have been imposed using a restriction on the transverse momentum $p_{Tj}$ of jets. Alternatively, one can also consider jet observables for which $p_{Tj}$ is weighted by a smooth function of the jet rapidity $y_j$ that vanishes as $|y_j| \to \infty$. Such observables are useful as they provide a natural way to impose a tight veto on central jets but a looser one at forward rapidities. We consider two such rapidity-dependent jet veto observables, $\mathcal{T}_{Bj}$ and $\mathcal{T}_{Cj}$, and compute the required beam and dijet soft functions for the jet-vetoed color-singlet production cross section at two loops. At this order, clustering effects from the jet algorithm become important. The dominant contributions are computed fully analytically while corrections that are subleading in the limit of small jet radii are expressed in terms of finite numerical integrals. Our results enable the full NNLL' resummation and are an important step towards N3LL resummation for cross sections with a $\mathcal{T}_{Bj}$ or $\mathcal{T}_{Cj}$ jet veto.

hep-ph↗

arXiv2020

Three-loop soft function for heavy-to-light quark decays

We compute the 1-jettiness soft function for the decay of a heavy quark into a light quark jet plus colorless particles at three-loop order in soft-collinear effective theory. The 1-jettiness measurement fixes the total small light-cone momentum component of the soft radiation with respect to the jet direction. This soft function is a universal ingredient to the factorization of heavy-to-light quark decays in the limit of small 1-jettiness. Our three-loop result is required for resummation at the N$^3$LL$^\prime$ level, e.g. near the endpoint in the photon energy spectrum of the $B \to X_s γ$ decay. It is also a necessary ingredient for future calculations of fully-differential heavy-to-light quark decay rates at N$^3$LO using the $N$-jettiness subtraction method, e.g. for semileptonic top decays. Using our result for the soft anomalous dimension we confirm predictions on the universal infrared structure of QCD scattering amplitudes with a massive external quark at three loops.

hep-ph↗

arXiv2019

Matter dependence of the four-loop cusp anomalous dimension

We compute analytically the matter-dependent contributions to the quartic Casimir term of the four-loop light-like cusp anomalous dimension in QCD, with $n_f$ fermion and $n_s$ scalar flavours. The result is extracted from the double pole of a scalar form factor. We adopt a new strategy for the choice of master integrals with simple analytic and infrared properties, which significantly simplifies our calculation. To this end we first identify a set of integrals whose integrands have a dlog form, and are hence expected to have uniform transcendental weight. We then perform a systematic analysis of the soft and collinear regions of loop integration and build linear combinations of integrals with a simpler infrared pole structure. In this way, only integrals with ten or fewer propagators are needed for obtaining the cusp anomalous dimension. These integrals are then computed via the method of differential equations through the addition of an auxiliary scale. Combining our result with that of a parallel paper, we obtain the complete $n_{f}$ dependence of the four-loop cusp anomalous dimension in QCD. Finally, using known numerical results for the gluonic contributions, we obtain an improved numerical prediction for the cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills theory.

hep-ph↗

arXiv2019

Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles

We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the corresponding conjectured expressions with full angle dependence. Taking the limit of large Minkowskian angle, we extract novel analytic results for certain terms of the light-like cusp anomalous dimension. They agree with the known numerical results. Finally, we study the anti-parallel lines limit of the cusp anomalous dimension. In a conformal theory, the latter is proportional to the static quark-antiquark potential. We use the new four-loop results to determine parts of the conformal anomaly term.

hep-ph↗

arXiv2018

Exclusive Top Threshold Matching at Lepton Colliders

The threshold scan at future lepton colliders is the most precise known method to determine the top quark mass (well below 100 MeV), a fundamental parameter of the Standard Model that co-determines the stability properties of the electroweak vacuum. We present a new method to match the continuum next-to-leading order QCD corrections with the next-to-leading logarithmic resummation of the Coulomb singularities of the quasi-toponium bound state at threshold where fixed-order perturbation theory is invalid. This matching is performed at the level of the fully exclusive $W^+bW^-\bar{b}$ final state. It allows to study all kinds of differential distributions at or close to threshold. The top mass dependence of these distributions opens up new possibilities for the top mass determination that might be competitive with the inclusive threshold scan.

hep-ph↗

arXiv2018

Potential NRQCD for unequal masses and the $B_c$ spectrum at NNNLO

We determine the $1/m$ and $1/m^2$ spin-independent heavy quarkonium potentials in the unequal mass case with $\mathcal O(α^3)$ and $\mathcal O(α^2)$ accuracy, respectively. We discuss in detail different methods to calculate the potentials, and show the equivalence among them. In particular we obtain, for the first time, the manifestly gauge invariant $1/m$ and $1/m^2$ potentials in terms of Wilson loops with next-to-leading order (NLO) precision. As an application of our results we derive the theoretical expression for the $B_c$ spectrum in the weak-coupling limit through next-to-next-to-next-to-leading order (N$^3$LO).

hep-ph↗

arXiv2018

The Three-loop Quark Jet Function

We calculate the massless quark jet function to three-loop order. The quark jet function is a universal ingredient in SCET factorization for many collider and decay processes with quark initiated final state jets. Our three-loop result contributes to the resummation for observables probing the invariant mass of final state quark jets at N$^3$LL$^\prime$. It represents the first complete three-loop result for a factorization ingredient describing collinear radiation. Furthermore it constitutes a major component of the $N$-jettiness subtraction/slicing method at N$^3$LO, which eventually may enable the calculation of fully-differential cross sections with a colorful final state at this order.

hep-ph↗