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The variances of the quantum-mechanical noise in a two-input-port Michelson interferometer within the framework of the Loudon-Ni model were solved exactly in two general cases: (1) one coherent state input and one squeezed state input, and (2) two photon number states inputs. Low intensity limit, exponential decaying signal and the noise due to mixing were discussed briefly.
Here we discuss our present knowledge of $\alpha_s$, the fundamental running coupling or effective charge of Quantum Chromodynamics (QCD). A precise understanding of the running of $\alpha_s(Q^2) $ at high momentum transfer, $Q$, is necessary for any perturbative QCD calculation. Equally important, the behavior of $\alpha_s$} at low $Q^2$ in the nonperturbative QCD domain is critical for understanding strong interaction phenomena, including the emergence of mass and quark confinement. The behavior of $\alpha_s(Q^2)$ at all momentum transfers also provides a connection between perturbative and nonperturbative QCD phenomena, such as hadron spectroscopy and dynamics. We first sketch the origin of the QCD coupling, the reason why its magnitude depends on the scale at which hadronic phenomena are probed, and the resulting consequences for QCD phenomenology. We then summarize latest measurements in both the perturbative and nonperturbative domains. New theory developments include the derivation of the universal nonperturbative behavior of $\alpha_s(Q^2)$ from both the Dyson-Schwinger equations and light-front holography. We also describe theory advances for the calculation of gluon and quark Schwinger functions in the nonperturbative domain and the relation of these quantities to $\alpha_s$. We conclude by highlighting how the nonperturbative knowledge of $\alpha_s$ is now providing a parameter-free determination of hadron spectroscopy and structure, a central and long-sought goal of QCD studies.
We present a first practical implementation of a recently proposed hadron structure oriented (HSO) approach to transverse momentum dependent (TMD) phenomenology applied to Drell-Yan-like processes, including lepton pair production at moderate 𝑄 2 and 𝑍 0 boson production. We compare and contrast general features of our methodology with other common practices and emphasize aspects that we view as improvements. Separately, we discuss the importance of treatments that preserve various theoretical properties of the nonperturbative parts of TMD parametrizations for future applications that have the extraction of hadron structure as a primary goal. These include the HSO’s preservation of a basic TMD parton-model-like framework even while accounting for full TMD factorization and evolution, explicit preservation of the integral relationship between TMD and collinear parton density functions, and the ability to meaningfully compare different theoretical models of nonperturbative TMD parton distributions. In our examples, we show that there is significant sensitivity at moderate 𝑄 2 to both the form of the nonperturbative transverse momentum dependence and the parametrization of collinear parton densities. However, we also find that evolving to 𝑄 2 =𝑀$^2_𝑍$, without fitting, results in a satisfactory postdiction of existing data for 𝑍0 production, nearly independently of the modeling of nonperturbative transverse momentum behavior. We argue that this demonstrates that moderate 𝑄 measurements should be given greater weight than high 𝑄 measurements in extractions of nonperturbative transverse momentum dependence. We also obtain new extractions of the nonperturbative Collins-Soper kernel within the HSO approach. We discuss its features and compare with some earlier extractions.
The jet angularities are a class of jet substructure observables which characterize the angular and momentum distribution of particles within jets. These observables are sensitive to momentum scales ranging from perturbative hard scatterings to nonperturbative fragmentation into final-state hadrons. We report measurements of several groomed and ungroomed jet angularities in pp collisions at $\sqrt{s}$ = 5.02 TeV with the ALICE detector. Jets are reconstructed using charged particle tracks at midrapidity (|η| < 0.9). The anti-$k_T$ algorithm is used with jet resolution parameters R = 0.2 and R = 0.4 for several transverse momentum $p^{ch jet}_{T}$ intervals in the 20–100 GeV/c range. Using the jet grooming algorithm Soft Drop, the sensitivity to softer, wide-angle processes, as well as the underlying event, can be reduced in a way which is well-controlled in theoretical calculations. We report the ungroomed jet angularities, λ α , and groomed jet angularities, λ α,g , to investigate the interplay between perturbative and nonperturbative effects at low jet momenta. Various angular exponent parameters α = 1, 1.5, 2, and 3 are used to systematically vary the sensitivity of the observable to collinear and soft radiation. Results are compared to analytical predictions at next-to-leading-logarithmic accuracy, which provide a generally good description of the data in the perturbative regime but exhibit discrepancies in the nonperturbative regime. Moreover, these measurements serve as a baseline for future ones in heavy-ion collisions by providing new insight into the interplay between perturbative and nonperturbative effects in the angular and momentum substructure of jets. They supply crucial guidance on the selection of jet resolution parameter, jet transverse momentum, and angular scaling variable for jet quenching studies.
Important aspects of quantum chromodynamics (QCD) factorization theorems are the properties of the objects involved that can be identified as universal. One example is that the definitions of parton densities and fragmentation functions for different types of hadrons differ only in the identity of the nonperturbative states that form the matrix elements, but are otherwise the same. This leads to independence of perturbative calculations on nonperturbative details of external states. It also lends support to interpretations of correlation functions as encapsulations of intrinsic nonperturbative properties. These characteristics have usually been presumed to still hold true in fragmentation functions even when the observed nonperturbative state is a small-mass cluster of n hadrons rather than simply a single isolated hadron. However, the multidifferential aspect of cross sections that rely on these latter types of fragmentation functions complicates the treatment of kinematical approximations in factorization derivations. That has led to recent claims that the operator definitions for fragmentation functions need to be modified from the single hadron case with nonuniversal prefactors. With such concerns as our motivation, we retrace the steps for factorizing the unpolarized semi-inclusive e + e − annihilation cross section and confirm that they do apply without modification to the case of a small-mass multihadron observed in the final state. In particular, we verify that the standard operator definition from single hadron fragmentation, with its usual prefactor, remains equally valid for the small-mass n -hadron case with the same hard parts and evolution kernels, whereas the more recently proposed definitions with nonuniversal prefactors do not. Our results reaffirm the reliability of most past phenomenological applications of dihadron fragmentation functions. Published by the American Physical Society 2025
We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact “ghost instantons”, which are extrema of the one-eigenvalue effective potential on the “unphysical sheet” of the spectral curve and have been discussed in detail recently by Mariño, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the ’t Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.
We investigate Wilsonian effective field theory as a model for the construction of the tachyon potential and nonperturbative vacua in closed string field theory. In a number of cases we are able to find the effective potential exactly, and observe what appear to be universal features. We find that the effective field theory contains the same nonperturbative vacuum structure as the bare Lagrangian, though this information is encoded less efficiently as the distance scale of the effective field theory is increased. The implication is that closed string field theory plausibly contains information about the nonperturbative vacuum structure of string theory, in spite of its similarities to effective field theory. We also truncate the effective potential at a fixed power of the field and investigate how the global structure of the effective potential may be approximated via Padé resummation. Qualitative comparisons suggest that computation of the eighth to sixteenth order closed string vertex should be enough to obtain reliable results for the closed string field theory action evaluated on the tachyon field.
We present predictions for the DIS 1-jettiness event shape $τ^b_1$, or DIS thrust, using the framework of Soft Collinear Effective Theory (SCET) for factorization, resummation of large logarithms, and rigorous treatment of nonperturbative power corrections, matched to fixed-order QCD away from the resummation region. Our predictions reach next-to-next-to-next-to-leading-logarithmic (N 3 LL) accuracy in resummed perturbation theory, matched to $\mathcal{O}(α^2_s)$ fixed-order QCD calculations obtained using the program NLOJet++. We include a rigorous treatment of hadronization corrections, which are universal across different event shapes and kinematic variables x and Q at leading power, and supplement them with a systematic scheme to remove $\mathcal{O}$(Λ QCD ) renormalon ambiguities in their definition. The framework of SCET allows us to connect smoothly the nonperturbative, resummation, and fixed-order regions, whose relative importance varies with x and Q, and to rigorously estimate theoretical uncertainties, across a broad range of x and Q covering existing experimental results from HERA as well as expected new measurements from the upcoming Electron- Ion-Collider (EIC). Our predictions will serve as an important benchmark for the EIC program, enabling the precise determination of the QCD strong coupling α s and the universal nonperturbative first moment parameter Ω 1 .
We present nonperturbative results for beyond the standard model kaon mixing matrix elements in the isospin symmetric limit ( m u = m d ) of QCD, including a complete estimate of all dominant sources of systematic error. Our results are obtained from numerical simulations of lattice QCD with N f = 2 + 1 flavors of dynamical domain wall fermions. For the first time, these quantities are simulated directly at the physical pion mass m π ∼ 139 MeV for two different lattice spacings. We include data at three lattice spacings in the range a = 0.11 – 0.07 fm and with pion masses ranging from the physical value up to 450 MeV. Compared to our earlier work, we have added both direct calculations at physical quark masses and a third lattice spacing making the removal of discretization effects significantly more precise and eliminating the need for any significant mass extrapolation beyond the range of simulated data. We renormalize the lattice operators nonperturbatively using RI-SMOM off-shell schemes. These schemes eliminate the need to model and subtract nonperturbative pion poles that arises in the RI-MOM scheme and, since the calculations are performed with domain wall fermions, the unphysical mixing between chirality sectors is suppressed. Our results for the bag parameters in the MS ¯ scheme at 3 GeV are B K ≡ B 1 = 0.5240 ( 17 ) ( 54 ) , B 2 = 0.4794 ( 25 ) ( 35 ) , B 3 = 0.746 ( 13 ) ( 17 ) , B 4 = 0.897 ( 02 ) ( 10 ) and B 5 = 0.6882 ( 78 ) ( 94 ) , where the first error is from lattice uncertainties and the second is the uncertainty due to the perturbative matching to MS ¯ . Published by the American Physical Society 2024
The study of quantum chromodynamics (QCD) at ultrarelativistic energies can be performed in a controlled environment through lepton-hadron deep inelastic scatterings. In such collisions, the high-energy partonic emissions that follow from the ejected hard partons are accurately described by perturbative QCD. However, the lower energy scales at which quarks and gluons experience color confinement, i.e., hadronization mechanism, fall outside the validity regions for perturbative calculations, requiring phenomenological models tuned to data to describe it. As such, hadronization physics cannot be currently derived from first principles alone. Monte Carlo event generators are useful tools to describe these processes as they simulate both the perturbative and the nonperturbative interactions, with model-dependent energy scales that control parton dynamics. This work employs jets—experimental reconstructions of final-state particles likely to have a common partonic origin—to inspect this transition further. Although originally proposed to circumvent hadronization effects, we show that jets can be utilized as probes of nonperturbative phenomena via their substructure. The charge correlation ratio was recently shown to be sensitive to hadronization effects. Our work further improves this sensitivity to nonperturbative scales by introducing a new selection based on the relative placement of the within the clustering tree, defined as the unclustering that resolves the jet’s leading charged particles. Published by the American Physical Society 2025
We investigate the energy-energy correlator (EEC) of hadrons produced on the same side in e + e − annihilation or in leading jets in p p collisions. We observe a remarkable universality of the correlator. Using a nonperturbative transverse momentum dependent (TMD) fragmentation function to model the transition from the “free-hadron” region to the perturbative collinear region, we are able to describe the near-side shapes and peaks over a wide range of energy for both the e + e − annihilation and the p p jet substructure measurements in terms of just two parameters. We present further predictions for the ratio of the projected three-point energy correlator to the EEC. The excellent agreement between our calculations and the experimental data may provide new insights into the role of nonperturbative physics for EECs, and suggests the possibility of exploring nonperturbative TMDs using theoretical tools developed for the energy correlators. Published by the American Physical Society 2025
We study the radiative properties—the Lamb shift, Purcell decay rate, and spontaneous emission dynamics—of an artificial atom coupled to a long, multimode cavity formed by an array of Josephson junctions. Introducing a tunable coupling element between the atom and the array, we demonstrate that such a system can exhibit a crossover from a perturbative to a nonperturbative regime of light-matter interaction as one strengthens the coupling between the atom and the Josephson-junction array. As a consequence, the concept of spontaneous emission as the occupation of the local atomic site being governed by a single complex-valued exponent breaks down. This breakdown, we show, can be interpreted in terms of formation of hybrid atom-resonator modes with radiative losses that are nontrivially related to the effective coupling between individual modes. We develop a singular function expansion approach for the description of the open quantum system dynamics in such a multimode nonperturbative regime. This modal framework generalizes the normal-mode description of quantum fields in a finite volume, incorporating exact radiative losses and incident quantum noise at the delimiting surface. Our results are pertinent to recent experiments with Josephson atoms coupled to high-impedance Josephson-junction arrays.
We derive a nonperturbative, convergent operator product expansion (OPE) for null-integrated operators on the same null plane in a CFT. The objects appearing in the expansion are light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. For example, a product of average null energy (ANEC) operators has an expansion in the light-ray operators that appear in the stress-tensor OPE. An important application is to collider event shapes. The light-ray OPE gives a nonperturbative expansion for event shapes in special functions that we call celestial blocks. As an example, we apply the celestial block expansion to energy-energy correlators in N = 4 Super Yang-Mills theory. Using known OPE data, we find perfect agreement with previous results both at weak and strong coupling, and make new predictions at weak coupling through 4 loops (NNNLO).
We propose to utilize angularity distributions in Higgs boson decay to probe light quark Yukawa couplings at e + e - colliders. Angularities $τ$ a are a class of 2-jet event shapes with variable and tunable sensitivity to the distribution of radiation in hadronic jets in the final state. Using soft-collinear effective theory (SCET), we present a prediction of angularity distributions from Higgs decaying to quark and gluon states at e + e - colliders to NNLL + $\mathcal{O}$(α s ) accuracy. Due to the different color structures in quark and gluon jets, the angularity distributions from H → $q\overline{q}$ and H → gg show different behaviors and can be used to constrain the light quark Yukawa couplings. We show that the upper limit of light quark Yukawa couplings could be probed to the level of ~ 15% of the bottom quark Yukawa coupling in the Standard Model in a conservative analysis window far away from nonperturbative effects and other uncertainties; the limit can be pushed to ≲ 7 – 9% with better control of the nonperturbative effects especially on gluon angularity distributions and/or with multiple angularities.
We study the angle between i) the standard jet axis, ii) the axis of a jet which has been groomed using soft drop, with reduced sensitivity to soft radiation, iii) the jet axis obtained with the winner-take-all recombination scheme, which is insensitive to soft radiation at leading power. We calculate the distributions for these angles at next- to-leading logarithmic accuracy, including non-global logarithms. The angle between the standard and groomed jet axis directly probes soft wide-angle radiation, leading to a novel factorization formula. This angle is also very sensitive to nonperturbative physics, which is directly connected to nonperturbative contribution to the rapidity anomalous dimension for transverse momentum distributions. Comparing our predictions to Pythia we find good agreement, and we foresee applications to jet substructure in proton-proton and heavy ion collisions.
We compute the soft-drop jet-mass distribution from pp collisions to NNLL accuracy while including nonperturbative corrections through a field-theory based formalism. Using these calculations, we assess the theoretical uncertainties on an αs precision measurement due to higher order perturbative effects, nonperturbative corrections, and PDF uncertainty. We identify which soft-drop parameters are well-suited for measuring αs, and find that higher-logarithmic resummation has a qualitatively important effect on the shape of the jet-mass distribution. We find that quark jets and gluon jets have similar sensitivity to αs, and emphasize that experimentally distinguishing quark and gluon jets is not required for an αs measurement. We conclude that measuring αs to the 10% level is feasible now, and with improvements in theory a 5% level measurement is possible. Getting down to the 1% level to be competitive with other state-of-the-art measurements will be challenging.
We develop a novel nonperturbative approach to a class of three-point functions in planar $ \mathcal{N} $ N = 4 SYM based on Thermodynamic Bethe Ansatz (TBA). More specifically, we study three-point functions of a non-BPS single-trace operator and two determinant operators dual to maximal Giant Graviton D-branes in AdS 5 × S 5 . They correspond to disk one-point functions on the worldsheet and admit a simpler and more powerful integrability description than the standard single-trace three-point functions. We first introduce two new methods to efficiently compute such correlators at weak coupling; one based on large N collective fields, which provides an example of open-closed-open duality discussed by Gopakumar, and the other based on combinatorics. The results so obtained exhibit a simple determinant structure and indicate that the correlator can be interpreted as a generalization of g -functions in 2d QFT; an overlap between an integrable boundary state and a state corresponding to the single-trace operator. We then determine the boundary state at finite coupling using the symmetry, the crossing equation and the boundary Yang-Baxter equation. With the resulting boundary state, we derive the ground-state g -function based on TBA and conjecture its generalization to other states. This is the first fully nonperturbative proposal for the structure constants of operators of finite length. The results are tested extensively at weak and strong couplings. Finally, we point out that determinant operators can provide better probes of sub-AdS locality than single-trace operators and discuss possible applications.
The extraction of nonperturbative TMD physics is made challenging by prescriptions that shield the Landau pole, which entangle long- and short-distance contributions in momentum space. The use of different prescriptions then makes the comparison of fit results for underlying nonperturbative contributions not meaningful on their own. We propose a model-independent method to restrict momentum-space observables to the perturbative domain. This method is based on a set of integral functionals that act linearly on terms in the conventional position-space operator product expansion (OPE). Artifacts from the truncation of the integral can be systematically pushed to higher powers in Λ QCD /k T . We demonstrate that this method can be used to compute the cumulative integral of TMD PDFs over k T ≤ k$^{cut}_{T}$ in terms of collinear PDFs, accounting for both radiative corrections and evolution effects. This yields a systematic way of correcting the naive picture where the TMD PDF integrates to a collinear PDF, and for unpolarized quark distributions we find that when renormalization scales are chosen near k$^{cut}_{T}$, such corrections are a percent-level effect. We also show that, when supplemented with experimental data and improved perturbative inputs, our integral functionals will enable model-independent limits to be put on the non-perturbative OPE contributions to the Collins-Soper kernel and intrinsic TMD distributions.