One-loop Anomalous Dimension Matrix Research Articles

Revisiting renormalization group equations of the SMEFT dimension-seven operators

In this work, we revisit the renormalization group equations (RGEs) of dimension-seven (dim-7) operators in the Standard Model effective field theory (SMEFT) resulting from mixing among dim-7 operators themselves by means of the background field method. Adopting a recently proposed physical basis for dim-7 operators, we achieve the explicit RGEs of all non-redundant dim-7 operators in the SMEFT for the first time. Together with those originating from the dim-5 and dim-6 operators, these results constitute the complete RGEs of dim-7 operators, and hence can be exploited to study full RG-running effects on some lepton- or baryon-number-violating processes involving dim-7 operators in the SMEFT, such as neutrino masses, neutrinoless double beta decay, meson and nucleon decays. We perform an analysis of the structure and perturbative power counting of the obtained one-loop anomalous dimension matrix, which is consistent with a non-renormalization theorem and the naive dimension analysis. Additionally, a partial check on some results is carried out by means of different tools and quantum field gauges.

Renormalisation group effects on SMEFT interpretations of LHC data

We explore the impact of Renormalisation Group (RG) effects in the Standard Model Effective Field Theory (SMEFT) interpretations of LHC measurements. We implement the RG running and mixing for the Wilson coefficients as obtained from the one-loop anomalous dimension matrix in the SMEFT into a Monte Carlo generator. This allows to consistently predict and combine in global fits collider observables characterised by different scales. As a showcase, we examine the impact of RG running in the strong coupling on the SMEFT predictions for tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} production cross sections and differential distributions as well as on the bounds on the Wilson coefficients that can be obtained from current LHC data.

Anomalous dimensions from Yukawa couplings in SMNEFT: four-fermion operators

The Standard Model Neutrino Effective Field Theory (SMNEFT) is the Standard Model Effective Field Theory (SMEFT) augmented with right-handed neutrinos. Building on our previous work, arXiv:2010.12109, we calculate the Yukawa coupling contributions to the one-loop anomalous dimension matrix for the 11 dimension-six four-fermion SMNEFT operators. We also present the new contributions to the anomalous dimension matrix for the 14 four-fermion SMEFT operators that mix with the SMNEFT operators through the Yukawa couplings of the right-handed neutrinos.

Anomalous dimensions from gauge couplings in SMEFT with right-handed neutrinos

Standard Model Neutrino Effective Field Theory (SMNEFT) is an effective theory with Standard Model (SM) gauge-invariant operators constructed only from SM and right-handed neutrino fields. For the full set of dimension-six SMNEFT operators, we present the gauge coupling terms of the one-loop anomalous dimension matrix for renormalization group evolution (RGE) of the Wilson coefficients between a new physics scale and the electroweak scale. We find that the SMNEFT operators can be divided into five subsets which are closed under RGE. Our results apply for both Dirac and Majorana neutrinos. We also discuss the operator mixing pattern numerically and comment on some interesting phenomenological implications.

Asymptotic behavior of cutoff effects in Yang\u2013Mills theory and in Wilson\u2019s lattice QCD

Discretization effects of lattice QCD are described by Symanzik’s effective theory when the lattice spacing, a, is small. Asymptotic freedom predicts that the leading asymptotic behavior is sim a^{n_{mathrm{min}}}[{bar{g}}^2(a^{-1})]^{hat{gamma }_1} sim a^{n_{mathrm{min}}}left[ frac{1}{-log (aLambda )}right] ^{hat{gamma }_1}. For spectral quantities, {n_{mathrm{min}}}=d is given in terms of the (lowest) canonical dimension, d+4, of the operators in the local effective Lagrangian and hat{gamma }_1 is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix gamma ^{(0)}. We determine gamma ^{(0)} for Yang–Mills theory ({n_{mathrm{min}}}=2) and discuss consequences in general and for perturbatively improved short distance observables. With the help of results from the literature, we also discuss the {n_{mathrm{min}}}=1 case of Wilson fermions with perturbative mathrm{O}(a) improvement and the discretization effects specific to the flavor currents. In all cases known so far, the discretization effects are found to vanish faster than the naive sim a^{n_{mathrm{min}}} behavior with rather small logarithmic corrections – in contrast to the two-dimensional O(3) sigma model.

DsixTools: the standard model effective field theory toolkit

We present DsixTools, a Mathematica package for the handling of the dimension-six standard model effective field theory. Among other features, DsixTools allows the user to perform the full one-loop renormalization group evolution of the Wilson coefficients in the Warsaw basis. This is achieved thanks to the SMEFTrunner module, which implements the full one-loop anomalous dimension matrix previously derived in the literature. In addition, DsixTools also contains modules devoted to the matching to the Delta B = Delta S = 1,2 and Delta B = Delta C = 1 operators of the Weak Effective Theory at the electroweak scale, and their QCD and QED Renormalization group evolution below the electroweak scale.

Renormalization group evolution of dimension-seven baryon- and lepton-number-violating operators

We study dimension-seven operators in standard model effective field theory. These operators are classified into two sets, one violating lepton number but preserving baryon number ($\Delta L=\pm 2$, $\Delta B=0$) and the other violating both but preserving their sum ($-\Delta L=\Delta B=\pm 1$). It has been found in the previous literature that there are respectively 13 and 7 such independent operators. We show that one operator is redundant in each set so that the complete list contains only 12 and 6 operators respectively. We accomplish this by using standard model equations of motion and various Fierz identities. We calculate the one-loop anomalous dimension matrix for the 6 operators in the second set, and illustrate its possible phenomenological implications by working out renormalization group running of the Wilson coefficients that could contribute to the type of proton decays with $-\Delta L=\Delta B=\pm 1$, such as $p\to\nu\pi^+$.

Renormalization group evolution of dimension-six baryon number violating operators

We calculate the one-loop anomalous dimension matrix for the dimension-six baryon number violating operators of the Standard Model effective field theory, including right-handed neutrino fields. We discuss the flavor structure of the renormalization group evolution in the contexts of minimal flavor violation and unification.

Scaling and tuning of EW and Higgs observables

We study deformations of the SM via higher dimensional operators. In particular, we explicitly calculate the one-loop anomalous dimension matrix for 13 bosonic dimension-6 operators relevant for electroweak and Higgs physics. These scaling equations allow us to derive RG-induced bounds, stronger than the direct constraints, on a universal shift of the Higgs couplings and some anomalous triple gauge couplings by assuming no tuning at the scale of new physics, i.e. by requiring that their individual contributions to the running of other severely constrained observables, like the electroweak oblique parameters or $\Gamma(h \rightarrow \gamma\gamma)$, do not exceed their experimental direct bounds. We also study operators involving the Higgs and gluon fields.

Renormalization group evolution of the Standard Model dimension six operators III: gauge coupling dependence and phenomenology

We calculate the gauge terms of the one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory (SM EFT). Combining these results with our previous results for the $\lambda$ and Yukawa coupling terms completes the calculation of the one-loop anomalous dimension matrix for the dimension-six operators. There are 1350 $CP$-even and $1149$ $CP$-odd parameters in the dimension-six Lagrangian for 3 generations, and our results give the entire $2499 \times 2499$ anomalous dimension matrix. We discuss how the renormalization of the dimension-six operators, and the additional renormalization of the dimension $d \le 4$ terms of the SM Lagrangian due to dimension-six operators, lays the groundwork for future precision studies of the SM EFT aimed at constraining the effects of new physics through precision measurements at the electroweak scale. As some sample applications, we discuss some aspects of the full RGE improved result for essential processes such as $gg \to h$, $h \to \gamma \gamma$ and $h \to Z \gamma$, for Higgs couplings to fermions, for the precision electroweak parameters $S$ and $T$, and for the operators that modify important processes in precision electroweak phenomenology, such as the three-body Higgs boson decay $h \rightarrow Z \, \ell^+ \, \ell^-$ and triple gauge boson couplings. We discuss how the renormalization group improved results can be used to study the flavor problem in the SM EFT, and to test the minimal flavor violation (MFV) hypothesis. We briefly discuss the renormalization effects on the dipole coefficient $C_{e\gamma}$ which contributes to $\mu \to e \gamma$ and to the muon and electron magnetic and electric dipole moments.

Renormalization group evolution of the Standard Model dimension six operators II: Yukawa dependence

We calculate the complete order y^2 and y^4 terms of the 59 x 59 one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory, where y is a generic Yukawa coupling. These terms, together with the terms of order lambda, lambda^2 and lambda y^2 depending on the Standard Model Higgs self-coupling lambda which were calculated in a previous work, yield the complete one-loop anomalous dimension matrix in the limit of vanishing gauge couplings. The Yukawa contributions result in non-trivial flavor mixing in the various operator sectors of the Standard Model effective theory.

Renormalization group evolution of the standard model dimension six operators. I: formalism and λ dependence

Abstract We calculate the order λ, λ2 and λy 2 terms of the 59 × 59 one-loop anomalous dimension matrix of dimension-six operators, where λ and y are the Standard Model Higgs self-coupling and a generic Yukawa coupling, respectively. The dimension-six operators modify the running of the Standard Model parameters themselves, and we compute the complete one-loop result for this. We discuss how there is mixing between operators for which no direct one-particle-irreducible diagram exists, due to operator replacements by the equations of motion.

Naive dimensional analysis counting of gauge theory amplitudes and anomalous dimensions

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N=L+Δ, with a shift Δ that depends on the NDA-weight of operator insertions. The NDA weight of an operator is defined in this Letter, and the general NDA formula for perturbative order N is derived. The formula is used to explain why the one-loop anomalous dimension matrix for dimension-six operators in the Standard Model effective field theory has entries with perturbative order ranging from 0 to 4. The results in this Letter are valid for an arbitrary effective field theory, and they constrain the coupling constant dependence of anomalous dimensions and scattering amplitudes in a general effective field theory.

Systematics of quarkonium production at the LHC and double parton fragmentation

In this paper we discuss the systematics of quarkonium production at the LHC. In particular, we focus on the necessity to sum logs of the form log(Q/p_perp) and log(p_perp/m_Q). We show that the former contributions are power suppressed, while the latter, whose contribution in fragmentation is well known, also arise in the short distance (i.e., non-fragmentation) production mechanisms. Though these contributions are suppressed by powers of m_Q/p_perp, they can be enhanced by inverse powers of v, the relative velocity between heavy quarks in the quarkonium. In the limit p_perp >> m_Q short distance production can be thought of as the fragmentation of a pair of partons (i.e., the heavy quark and anti-quark) into the final state quarkonium. We derive an all order factorization theorem for this process in terms of double parton fragmentation functions (DPFF) and calculate the one-loop anomalous dimension matrix for the DPFF.

IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in 𝒩 = 4 SYM theory

We derive a compact all-loop-order expression for the IR-divergent part of the N=4 SYM four-gluon amplitude, which includes both planar and all subleading-color contributions, based on the assumption that the higher-loop soft anomalous dimension matrices are proportional to the one-loop soft anomalous dimension matrix, as has been recently conjectured. We also consider the Regge limit of the four-gluon amplitude, and we present evidence that the leading logarithmic growth of the subleading-color amplitudes is less severe than that of the planar amplitudes. We examine possible 1/N^2 corrections to the gluon Regge trajectory, previously obtained in the planar limit from the BDS ansatz. The double-trace amplitudes have Regge behavior as well, with a nonsense-choosing Regge trajectory and a Regge cut which first emerges at three loops.

Integrable open spin chains and the doubling trick in Script N = 2 SYM with fundamental matter

We demonstrate that the one-loop anomalous dimension matrix in N = 2 SYM with a single chiral hypermultiplet of fundamental matter, which is dual to AdS_5 X S^5 with a D7-brane filling AdS_5 and wrapped around an $^3 in the S^5, is an integrable open spin chain Hamiltonian. We also use the doubling trick to relate these open spin chains to closed spin chains in pure N = 4 SYM. By using the AdS/CFT correspondence, we find a relation between the corresponding open and closed strings that differs from a simple doubling trick by terms that vanish in the semiclassical limit. We also demonstrate that in some cases the closed string is simpler and easier to study than the corresponding open string, and we speculate on the nature of corrections due to the presence of D-branes that this implies.

Integrable spin chain and operator mixing in [formula omitted] supersymmetric theories

We study operator mixing, due to planar one-loop corrections, for composite operators in D=4 supersymmetric theories. We present some N=1,2 Yang–Mills and Wess–Zumino models, in which the planar one-loop anomalous dimension matrix in the sector of holomorphic scalars is identified with the Hamiltonian of an integrable quantum spin chain with SU(3) or SU(2) symmetry, even if the theory is away from the conformal points. This points to a more universal origin of the integrable structure beyond superconformal symmetry. We also emphasize the role of the superpotential in the appearance of the integrable structure. The computations of operator mixing in our examples by solving Bethe ansatz equations show some new features absent in N=4 SYM.

Calculation of the continuum-lattice HQET matching for the complete basis of four-fermion operators: reanalysis of the [formula omitted] mixing

In this work, we find the expressions of continuum HQET four-fermion operators in terms of lattice operators in perturbation theory. To do so, we calculate the one-loop continuum-lattice HQET matching for the complete basis of ΔB = 2 and ΔB = 0 operators (excluding penguin diagrams), extending and completing previous studies. We have also corrected some errors in previous evaluations of the matching for the operator O LL . Our results are relevant to the lattice computation of the values of unknown hadronic matrix elements which enter in many very important theoretical predictions in B-meson phenomenology: B 0− B 0 mixing, Γ B and τ B s lifetimes, SUSY effects in ΔB = 2 transitions and the B s width difference Δγ B s . We have reanlyzed our lattice data for the B B parameter of the B 0− B 0 mixing on 600 lattices of size 24 3 × 40 at β = 6.0 computed with the SW-Clover and HQET lattice actions. We have used the correct lattice-comtinuum matching factors and boosted perturbation theory with tadpole improved heavy-light operators to reduce the systematic error in the evaluation of the renormalization constants. Our best estimate of the renormalization scale independent B-parameter is B ̂ B = 1.29 ± 0.08 ± 0.06 , where the first error is statistical and the second is systematic coming from the uncertainty in the determination of the renormalization constants. Our result is in good agreement with previous results obtained by extrapolating Wilson data. As a byproduct, we also obtain the complete one-loop anomalous dimension matrix for four-fermion operators in the HQET.