Renormalization Group Evolution Research Articles

A collinear perspective on the Regge limit

The high energy (Regge) limit provides a playground for understanding all loop structures of scattering amplitudes, and plays an important role in the description of many phenomenologically relevant cross-sections. While well understood in the planar limit, the structure of non-planar corrections introduces many fascinating complexities, for which a general organizing principle is still lacking. We study the structure of multi-reggeon exchanges in the context of the effective field theory for forward scattering, and derive their factorization into collinear operators (impact factors) and soft operators. We derive the structure of the renormalization group consistency equations in the effective theory, showing how the anomalous dimensions of the soft operators are related to those of the collinear operators, allowing us to derive renormalization group equations in the Regge limit purely from a collinear perspective. The rigidity of the consistency equations provides considerable insight into the all orders organization of Regge amplitudes in the effective theory, as well as its relation to other approaches. Along the way we derive a number of technical results that improve the understanding of the effective theory. We illustrate this collinear perspective by re-deriving all the standard BFKL equations for two-Glauber exchange from purely collinear calculations, and we show that this perspective provides a number of conceptual and computational advantages as compared to the standard view from soft or Glauber physics. We anticipate that this formulation in terms of collinear operators will enable a better understanding of the relation between BFKL and DGLAP in gauge theories, and facilitate the analysis of renormalization group evolution equations describing Reggeization beyond next-to-leading order.

Understanding parton evolution in matter from renormalization group analysis

We perform a renormalization group (RG) analysis of collinear hadron production in deep inelastic scattering on nuclei. We consider the limit where the parent parton energy E is large, while the medium opacity L/λg remains small. We identify the fixed order and leading ln⁡(E/ξ2L) enhanced medium contributions to the semi-inclusive cross sections and derive RG equations that resum multiple emissions near the endpoints of the splitting functions at first order in opacity. These evolution equations treat the same type of radiation enhancement in matter as the modified Dokshitzer-Gribov-Lipatov-Altarelli-Parisi approach, but differ in the way one regulates the collinear divergences. They provide a unique analytic insight into the problem of resummation and a faster and more efficient path to phenomenology. The new RG evolution framework is applied to study fragmentation in eA reactions.

Duality between the quantum inverted harmonic oscillator and inverse square potentials

In this paper we show how the quantum mechanics of the inverted harmonic oscillator (IHO) can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential (ISP). We demonstrate this by relating both of these systems to the Berry–Keating system with Hamiltonian H=(xp+px)/2 . It has long been appreciated that the quantum mechanics of the ISP has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the IHO system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalisation group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the ISP is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the IHO. However, unlike the ISP where the continuous scaling symmetry is explicit, in the case of the IHO it is hidden and occurs because the Hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.

New physics in the third generation. A comprehensive SMEFT analysis and future prospects

We present a comprehensive analysis of electroweak, flavor, and collider bounds on the complete set of dimension-six SMEFT operators in the U(2)5-symmetric limit. This operator basis provides a consistent framework to describe a wide class of new physics models and, in particular, the motivated class of models where the new degrees of freedom couple mostly to the third generation. By analyzing observables from all three sectors, and consistently including renormalization group evolution, we provide bounds on the effective scale of all 124 U(2)5-invariant operators. The relation between flavor-conserving and flavor-violating observables is analyzed taking into account the leading U(2)5 breaking in the Yukawa sector, which is responsible for heavy-light quark mixing. We show that under simple, motivated, and non-tuned hypotheses for the parametric size of the Wilson coefficients at the high scale, all present bounds are consistent with an effective scale as low as 1.5 TeV. We also show that a future circular e+e− collider program such as FCC-ee would push most of these bounds by an order of magnitude. This would rule out or provide clear evidence for a wide class of compelling new physics models that are fully compatible with present data.

Back-to-Back Inclusive Dijets in Deep Inelastic Scattering at Small x: Complete NLO Results and Predictions.

We compute the back-to-back dijet cross section in deep inelastic scattering at small x to next-to-leading order (NLO) in the color glass condensate effective field theory. Our result can be factorized into a convolution of the Weizsäcker-Williams gluon transverse-momentum-dependent distribution function (WW gluon TMD) with a universal soft factor and an NLO coefficient function. The soft factor includes both double and single logarithms in the ratio of the relative transverse momentum P_{⊥} of the dijet pair to the dijet momentum imbalance q_{⊥}; its renormalization group (RG) evolution is resummed into the Sudakov factor. Likewise, the WW TMD obeys a nonlinear RG equation in x that is kinematically constrained to satisfy both the lifetime and rapidity ordering of the projectile. Exact analytical expressions are obtained for the NLO coefficient function of transversely and longitudinally polarized photons. Our results allow for the first quantitative separation of the dynamics of Sudakov suppression from that of gluon saturation. They can be extended to other final states and provide a framework for precision tests of novel QCD many-body dynamics at the Electron-Ion Collider.

Exploring dark matter-gauge boson effective interactions at current and future colliders* *RD is Supported in part by the National Key R&D Programme of China (2021YFC2203100). YZ is Supported in part by the National Natural Science Foundation of China (11805001) and the Fundamental Research Funds for the Central Universities (JZ2023HGTB0222)

In this study, we systematically investigate collider constraints on effective interactions between Dark Matter (DM) particles and electroweak gauge bosons. We consider the simplified models in which scalar or Dirac fermion DM candidates couple only to electroweak gauge bosons through high dimensional effective operators. Considering the induced DM-quarks and DM-gluons operators from the Renormalization Group Evolution (RGE) running effect, we present comprehensive constraints on the effective energy scale Λ and Wilson coefficients from direct detection, indirect detection, and collider searches. In particular, we present the corresponding sensitivity from the Large Hadron Electron Collider (LHeC) and Future Circular Collider in the electron-proton mode (FCC-ep) for the first time, update the mono-j and mono-γ search limits at the Large Hadron Collider (LHC), and derive the new limits at the Circular Electron Positron Collider (CEPC).

Indirect constraints on top quark operators from a global SMEFT analysis

We perform a model-independent analysis of top-philic New Physics scenarios, under the assumption that only effective operators involving top quarks are generated at tree level. Within the SMEFT framework, we derive indirect constraints on Wilson Coefficients by combining a large set of low-energy observables: B-meson and kaon decays, meson mixing observables, precision electroweak and Higgs measurements, anomalous magnetic moments, lepton flavour violating processes, lepton flavour universality tests, and measurements of the Cabibbo angle. We consider the renormalization group evolution of the operators and use the one-loop matching of the SMEFT onto the LEFT. The global analysis is then used to perform one-parameter, two-parameter, and global fits, as well as applications to explicit ultraviolet models. We find that the inclusion of measurements from different physics sectors reveals a strong interplay and complementarity among the observables. The resulting constraints are also compared to direct bounds provided by top quark productions at the LHC.

Renormalization group evolution with scalar leptoquarks

Leptoquarks are theoretically well-motivated and have received increasing attention in recent years as they can explain several hints for physics beyond the Standard Model. In this article, we calculate the renormalisation group evolution of models with scalar leptoquarks. We compute the anomalous dimensions for all couplings (gauge, Yukawa, Higgs and leptoquarks interactions) of the most general Lagrangian at the two-loop level and the corresponding threshold corrections at one-loop. The most relevant analytic results are presented in the appendix, while the notebook containing the full expressions can be downloaded at https://github.com/SumitBanikGit/SLQ-RG. In our phenomenological analysis, we consider some exemplary cases with focus on gauge and Yukawa coupling unification.

ALP-LEFT Interference and the Muon (g − 2)

The low-energy effective field theory (LEFT) provides the appropriate framework to describe particle interactions below the scale of electroweak symmetry breaking, μw ~ v. By matching the Standard Model onto the LEFT, non-zero Wilson coefficients of higher-dimensional operators are generated, suppressed by the corresponding power of 1/v. An axion or axion-like particle (ALP) with mass ma ≪ μw that interacts with the Standard Model via classically shift-invariant dimension-five operators would also contribute to the LEFT Wilson coefficients, since it can appear as a virtual particle in divergent Green’s functions and thus has an impact on the renormalization of the LEFT operators. We present the full set of one-loop ALP-induced source terms modifying the renormalization-group evolution equations of the LEFT Wilson coefficients up to dimension-six order. Our framework allows for model-independent ALP searches at low energies from current bounds on LEFT Wilson coefficients. As a concrete application, we present an improved prediction for ALP effects on the anomalous magnetic moment of the muon.

New Physics in CP violating and flavour changing quark dipole transitions

We explore CP-violating (CPV) effects of heavy New Physics in flavour-changing quark dipole transitions, within the framework of Standard Model Effective Field Theory (SMEFT). First, we establish the relevant dimension six operators and consider the Renormalisation Group (RG) evolution of the appropriate Wilson coefficients. We investigate RG-induced correlations between different flavour-violating processes and electric dipole moments (EDMs) within the Minimal Flavour Violating and U(2)3 quark flavour models. At low energies, we set bounds on the Wilson coefficients of the dipole operators using CPV induced contributions to observables in non-leptonic and radiative B, D and K decays as well as the neutron and electron EDMs. This enables us to connect observable CPV effects at low energies and general NP appearing at high scales. We present bounds on the Wilson coefficients of the relevant SMEFT operators at the high scale Λ = 5 TeV, and discuss most sensitive CPV observables for future experimental searches.

A global analysis of axion-like particle interactions using SMEFT fits

In the presence of axions or axion-like particles (ALPs) that couple to the Standard Model via dimension-five interactions, dimension-six SMEFT interactions are generated via renormalization-group evolution. As many of these SMEFT contributions are experimentally tightly constrained, this “ALP-SMEFT interference” can be used to derive indirect bounds on the ALP couplings to the Standard Model particles. We present a global analysis of the Wilson coefficients of the ALP effective Lagrangian based on Higgs, top, and low-energy data. The obtained bounds are model independent and are competitive or even stronger than direct bounds in the GeV to TeV ALP-mass range.

Back-to-back inclusive dijets in DIS at small x: gluon Weizsäcker-Williams distribution at NLO

In [1], we performed the first complete computation of the back-to-back inclusive dijet cross-section in Deeply Inelastic Scattering (DIS) at small xBj to next-to-leading order (NLO) in the Color Glass Condensate effective field theory (CGC EFT). We demonstrate here that for dijets with relative transverse momentum P⊥ and transverse momentum imbalance q⊥, to leading power in q⊥/P⊥, the cross-section for longitudinally polarized photons can be fully factorized into the product of a perturbative impact factor and the non-perturbative Weizsäcker-Williams (WW) transverse momentum dependent (TMD) gluon distribution to NLO accuracy. The impact factor can further be expressed as the product of a universal soft factor which resums Sudakov double and single logs in P⊥/q⊥ and a coefficient function given by a remarkably compact analytic expression. We show that in the CGC EFT the WW TMD satisfies a kinematically constrained JIMWLK renormalization group evolution in rapidity. This factorization formula is valid to all orders in Qs/q⊥ for q⊥, Qs ≪ P⊥, where Qs is the semi-hard saturation scale that grows with rapidity.

Multi-collinear splitting kernels for track function evolution

Jets and their substructure play a central role in many analyses at the Large Hadron Collider (LHC). To improve the precision of measurements, as well as to enable measurement of jet substructure at increasingly small angular scales, tracking information is often used due to its superior angular resolution and robustness to pile-up. Calculations of track-based observables involve non-perturbative track functions, that absorb infrared divergences in perturbative calculations and describe the transition to charged hadrons. The infrared divergences are directly related to the renormalization group evolution (RGE), and can be systematically computed in perturbation theory. Unlike the standard DGLAP evolution, the RGE of the track functions is non-linear, encoding correlations in the fragmentation process. We compute the next-to-leading order (NLO) evolution of the track functions, which involves in its kernel the full 1 → 3 splitting function. We discuss in detail how we implement the evolution equation numerically, and illustrate the size of the NLO corrections. We also show that our equation can be viewed as a master equation for collinear evolution at NLO, by illustrating that by integrating out specific terms, one can derive the evolution for any N -hadron fragmentation function. Our results provide a crucial ingredient for obtaining track-based predictions for generic measurements at the LHC, and for improving the description of the collinear dynamics of jets.

Can leptonic mixing matrix have a Wolfenstein form?

We analyze the possibility of the leptonic mixing matrix having a Wolfenstein form at the Grand Unified Theory scale. The renormalization group evolution of masses and mixing angles from the high scale to electroweak scale, in certain new physics scenarios, can significantly alter the form of the leptonic mixing matrix. In the past it was shown that such significant enhancement implies that the leptonic mixing matrix at high scale can be the same or similar in structure to the quark one. We thoroughly analyze this hypothesis in the light of the latest neutrino oscillation data as well as other constraints such as those coming from neutrinoless double beta decay. We show that such an ansatz, at least within the context of minimal supersymmetric models, is no longer compatible with the latest experimental data.

Shift-Invariant Orders of an Axionlike Particle.

It is generally believed that global symmetries, in particular, axion shift symmetries, can only be approximate. This motivates one to quantify the breaking of the shift invariance that characterizes the flavorful effective couplings of an axionlike particle to standard model fermions. We identify a minimal set of Jarlskog-like flavor invariants that vanish if and only if the axion is shift symmetric. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate properties of the invariants by matching to a UV model, studying the CP transformation of the invariants, calculating their renormalization group evolution, and investigating similar conditions in the low-energy effective field theory below the electroweak scale. Finally, we discuss the order parameter associated to the nonperturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.

RGESolver: a C++ library to perform renormalization group evolution in the Standard Model Effective Theory

Renormalization group evolution above the electroweak scale is a crucial ingredient in the phenomenology of the Standard Model Effective Theory. The RGESolver open-source C++ library performs the evolution at leading order for dimension-six operators in the most general flavour scenario (assuming lepton and baryon number conservation). Given its efficiency, RGESolver can be used to include the effects of renormalization group evolution in extensive phenomenological analyses in the framework of the Standard Model Effective Theory.

Gauge Couplings Evolution from the Standard Model, through Pati–Salam Theory, into E8 Unification of Families and Forces

We explore the potential of ultimate unification of the Standard Model matter and gauge sectors into a single E8 superfield in ten dimensions via an intermediate Pati–Salam gauge theory. Through a consistent realisation of a T6/(Z6×Z2×Z2) orbifolding procedure and renormalisation group evolution of gauge couplings, we establish several novel benchmark scenarios for New Physics that are worth further phenomenological exploration.

Stronger C -odd color charge correlations in the proton at higher energy

The non-forward eikonal scattering matrix for dipole-proton scattering at high energy obtains an imaginary part due to a $C$-odd three gluon exchange. We present numerical estimates for the perturbative Odderon amplitude as a function of dipole size, impact parameter, their relative azimuthal angle, and light-cone momentum cutoff $x$. The proton is approximated as $\psi_\mathrm{qqq}|qqq\rangle + \psi_\mathrm{qqqg}|qqqg\rangle$, where $\psi_\mathrm{qqq}$ is a non-perturbative three quark model wave function while the gluon emission is computed in light-cone perturbation theory. We find that the Odderon amplitude increases as $x$ decreases from 0.1 to 0.01. At yet lower $x$, the reversal of this energy dependence would reflect the onset of universal small-$x$ renormalization group evolution.

Comprehensive analysis of charged lepton flavour violation in the symmetry protected type-I seesaw

The type-I seesaw model is probably the most straightforward and best studied extension of the Standard Model that can account for the tiny active neutrino masses determined from neutrino oscillation data. In this article, we calculate the complete set of one-loop corrections to charged lepton flavour violating processes within this model. We give the results both using exact diagonalisation of the neutrino mass matrix, and at leading order in the seesaw expansion (i.e. mathcal{O} (v2/ {M}_R^2 )). Furthermore, we perform the matching onto the SU(2)L invariant Standard Model Effective Field Theory at the dimension 6 level. These results can be used as initial conditions for the renormalisation group evolution from the right-handed neutrino scale down to the scale of the physical processes, which resums large logarithms. In our numerical analysis, we study the inverse seesaw limit, i.e. the symmetry protected type-I seesaw, where the Wilson coefficient of the Weinberg operator is zero such that sizeable neutrino Yukawas are permissible and relevant effects in charged lepton flavour violating observables are possible. We correlate the different charged lepton flavour violating processes, e.g. ℓ → ℓ′γ, ℓ → 3ℓ′, μ → e conversion and Z → ℓℓ′, taking into account the constraints from electroweak precision observables and tests of lepton flavour universality.

The stability analysis of the extended singlet scalar model with two high scale minima

We study the simplest viable dark matter (DM) model with a real singlet scalar, vector-like singlet, and doublet fermions. The Yukawa couplings associated with the fermion sector are important in obtaining the current DM relic density through Freeze-out and Freeze-in mechanisms. In addition to the standard model-like high scale minima along the Higgs field, we find other high scale minima along the singlet scalar field direction. The other minima appears due to the renormalization group evolutions of the couplings involving the gauge sector, Higgs portal, and the new Yukawa sector. A detailed study of the parameter space identifying the region of electroweak vacuum stability and metastability along different directions of the scalar field is carried out using different phase diagrams.