We use the on-shell S-matrix and form factors to compute anomalous dimensions of higher dimension operators in the Standard Model Effective Field Theory. We find that in many instances, these computations are made simple by using the on-shell method. We first compute contributions to anomalous dimensions of operators at dimension-six that arise at one-loop. Then we calculate two-loop anomalous dimensions for which the corresponding one-loop contribution is absent, using this powerful method.
We study axion effective field theories (EFTs), with a focus on axion couplings to massive chiral gauge fields. We investigate the EFT interactions that participate in processes with an axion and two gauge bosons, and we show that, when massive chiral gauge fields are present, such interactions do not entirely originate from the usual anomalous EFT terms. We illustrate this both at the EFT level and by matching to UV-complete theories. In order to assess the consistency of the Peccei--Quinn (PQ) anomaly matching, it is useful to introduce an auxiliary, non-dynamical gauge field associated to t…
We consider two-loop renormalization of high-dimensional Lorentz scalar operators in the gluonic sector of QCD. These operators appear also in the Higgs effective theory obtained by integrating out the top quark loop in the gluon fusion process. We first discuss the classification of operators and how to construct a good set of basis using both off-shell field theory method and on-shell form factor formalism. To study loop corrections, we apply efficient unitarity-IBP strategy and compute the two-loop minimal form factors of length-3 operators up to dimension sixteen. From the UV divergences o…
In Effective Field Theories (EFTs) with higher-dimensional operators many anomalous dimensions vanish at the one-loop level for no apparent reason. With the use of supersymmetry, and a classification of the operators according to their embedding in super-operators, we are able to show why many of these anomalous dimensions are zero. The key observation is that one-loop contributions from superpartners trivially vanish in many cases under consideration, making supersymmetry a powerful tool even for non-supersymmetric models. We show this in detail in a simple U(1) model with a scalar and fermio…
We give a short overview of the theory of the muon anomalous magnetic moment with emphasis on the hadronic light-by-light and the pion loop contribution. We explain the difference between the hidden local symmetry and full VMD pion loop and discuss leading logarithms in the anomalous sector of 2-flavour chiral perturbation theory.
The pion contributions to the coupling $C_0$ of pionless EFT are studied via both non-relativistic and relativistic forms of chiral effective field theory for nuclear forces. A definite item in the $2N$-reducible component of the box diagram is shown to be dominant over the $2N$-irreducible (potential) ones due to the pinching of low-lying nucleon poles, and this anomalous mapping between pionless and pionfull EFT's occurs right within the non-relativistic regime. A natural strategy for renormalization of the pionfull theory emerges as a byproduct through the interactive analysis of the box di…
The Legendre flow equation, a version of exact Wilsonian renormalization group (WRG) equation, is employed to consider the power counting issues in Nuclear Effective Field Theory. A WRG approach is an ideal framework because it is nonperturbative and does not require any prescribed power counting rule. The power counting is determined systematically from the scaling dimensions of the operators at the nontrivial fixed point. The phase structure is emphasized and the inverse of the scattering length, which is identified as a relevant coupling, is shown to play a role of the order parameter. The …
We study the impact of anomalous gauge boson and fermion couplings on the production of $W^+W^-$ pairs at potential future LHC upgrades and estimate the sensitivity at $\sqrt{S}=14$ TeV with $3~ab^{-1}$ and $\sqrt{S}=27$ TeV with $15~ab^{-1}$. A general technique for including NLO QCD effects in effective field theory (EFT) fits to kinematic distributions is presented, and numerical results are given for $\sqrt{S}=13$ TeV $W^+W^-$ production. Our method allows fits to anomalous couplings at NLO accuracy in any EFT basis and has been implemented in a publicly available version of the POWHEG-BOX…
We present a combined analysis of LHC Higgs data (signal strengths) together with LEP-2 WW production measurements. To characterize possible deviations from the Standard Model (SM) predictions, we employ the framework of an Effective Field Theory (EFT) where the SM is extended by higher-dimensional operators suppressed by the mass scale of new physics $Λ$. The analysis is performed consistently at the order $Λ^{-2}$ in the EFT expansion keeping all the relevant operators. While the two data sets suffer from flat directions, together they impose stringent model-independent constraints on the an…
Anomalous Dimension of a General Effective Gauge Theory II: Fermionic Sector
PSI Center for Neutron and Muon Sciences（瑞士保罗谢勒研究所中子与缪子科学中心）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学伽利略物理与天文学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
The closed-form Brükner $G$ matrix for nuclear matter is computed in the $^1S_0$ channel of EFT($\not\!\!π$) and renormalized in nonperturbative context. The nuclear medium environment yields additional constraints that are consistent with off-shell $T$ matrix renormalization, keeping the power counting intact and simplifying the running behaviors of the EFT couplings. With the $G$ obtained we computed the energy per particle for neutron and symmetric nuclear matter to demonstrate the physical relevance of certain 'physical' parameters that arise from nonperturbative renormalization. We also e…
The renormalization of EFT for nucleon-nucleon scattering in nonperturbative regimes is investigated in a compact parametrization of the $T$-matrix. The key difference between perturbative and nonperturbative renormalization is clarified. The underlying theory perspective and the 'fixing' of the prescriptions for the $T$-matrix from physical boundary conditions are stressed.
We propose a parametrization of anomalous Higgs-boson couplings that is both systematic and practical. It is based on the electroweak chiral Lagrangian, including a light Higgs boson, as the effective field theory (EFT) at the electroweak scale $v$. This is the appropriate framework for the case of sizeable deviations in the Higgs couplings of order $10\%$ from the Standard Model, considered to be parametrically larger than new-physics effects in the sector of electroweak gauge interactions. The role of power counting in identifying the relevant parameters is emphasized. The three relevant sca…
We calculate the process $pp\to W^+W^-\to e^+ ν_e μ^-\barν_μ$ at NLO QCD, including also effective field theory (EFT) operators mediating the $ggW^+W^-$ interaction, which first occur at dimension eight. We further combine the NLO and EFT matrix elements produced by GoSam with the HERWIG7/MATCHBOX framework, which offers the possibility to study the impact of a parton shower. We assess the effects of the anomalous couplings by comparing them to top-mass effects as well as uncertainties related to variations of the renormalisation, factorisation and hard shower scales.
Finding better ways to prove the Standard Model Effective Field Theory is a very important direction of research. This paper focuses on measurements of Electroweak triple gauge couplings, paying special attention on the regime of validity of the Effective Field Theory (EFT). In this regard, one of our goals is to find measurements leading to a large increase of the interference between the SM amplitude and the contribution of irrelevant operators in the EFT. We propose two such distributions that will lead to a better accuracy. Improvements compared to the traditional methods as well as LHC hi…
In this document we study the effect of anomalous Higgs boson couplings on non-resonant pair production of Higgs bosons ($HH$) at the LHC. We explore the space of the five parameters $κ_λ$, $κ_{t}$, $c_2$, $c_g$, and $c_{2g}$ in terms of the corresponding kinematics of the final state, and describe a partition of the space into a limited number of regions featuring similar phenomenology in the kinematics of $HH$ final state. We call clusters the sets of points belonging to the same region; to each cluster corresponds a representative point which we call a benchmark. We discuss a possible techn…
The study of the anomalous non-resonant $HH$ production is presented from the point of view of effective field theory (EFT) parameterization. The searches for $pp \rightarrow HH \rightarrow γγWW$, $pp \rightarrow HH \rightarrow γγbb$ and $pp \rightarrow HH \rightarrow bbbb$ processes by ATLAS experiment at the Large Hadron Collider (LHC) are reproduced. The selection efficiencies and exclusion limits on the $HH$ production cross sections for different Beyond Standard Model (BSM) EFT benchmark models are obtained using the fast Monte-Carlo simulation of the ATLAS detector.
The low energy $S$-matrix which describes non-relativistic scattering arising from finite-range forces has UV/IR symmetries that are hidden in the corresponding effective field theory (EFT) action. It is shown that the $S$-matrix symmetries are manifest as geometric symmetries of the RG flow of coupling constants in the EFT action in both three and two spatial dimensions. An example is given demonstrating that UV/IR symmetry breaking in the $S$-matrix implies strong constraints on the RG flow of the coefficients of the corresponding symmetry-breaking operators in the EFT.
On-shell amplitude methods have proven to be extremely efficient for calculating anomalous dimensions. We further elaborate on these methods to show that, by the use of an angular momentum decomposition, the one-loop anomalous dimensions can be reduced to essentially a sum of products of partial waves. We apply this to the SM EFT, and show how certain classes of anomalous dimensions have their origin in the same partial-wave coefficients. We also use our result to obtain a generic formula for the one-loop anomalous dimensions of nonlinear sigma models at any order in the energy expansion, and …
We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix. To do so we define an EFT classification based on four parameters characterizing 1) the number of derivatives per interaction, 2) the soft properties of amplitudes, 3) the leading valency of the interactions, and 4) the spacetime dimension. Carving out the allowed space of EFTs, we prove that exceptional EFTs like the non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie precisely on the boundary of allowed theory space. Using on-shell m…
Lepton Flavor Violating (LFV) observables such as $μ\to eγ$, $μ\to 3e$ and $μN \to eN$ are among the best probes for new physics at the TeV scale. In the near future the bounds on these observables will improve by many orders of magnitude. In this work we use the SM EFT to understand the impact of these measurements. The precision reach is such that the interpretation of the bounds requires an analysis of the dimension-six operator mixing up to the two-loop level. Using on-shell amplitude techniques, which make transparent many selection rules, we classify and calculate the different operator …
We discuss the implications of dimension-six operators of the Effective Field Theory (EFT) framework in the study of Vector Boson Scattering (VBS) in the $pp \to Z Z j j $ channel. We show that operators of dimension-six should not be neglected in favour of those of dimension-eight. We observe that this process is very sensitive to some of the operators commonly fit using LEP and Higgs data, and that it can be used to improve the bounds on the former. Further we show that other operators than the ones generating anomalous triple and quartic gauge couplings (aTGCs/aQCGs) can have a non-negligib…
We discuss an alternative scheme for including effective range corrections in pionless effective field theory. The standard approach treats range terms as perturbative insertions in the T -matrix. In a finite volume this scheme can lead to singular behavior close to the unperturbed energies. We consider an alternative scheme that resums the effective range but expands the spurious pole of the T -matrix created by this resummation. We test this alternative expansion for several model potentials and observe good convergence.
We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it to Hilbert series -- partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated pheno…
The ACME collaboration has recently reported a new bound on the electric dipole moment (EDM) of the electron, $|d_e|< 1.1 \times 10^{-29}\, {\rm e\cdot cm}$ at 90$\%$ confidence level, reaching an unprecedented accuracy level. This can translate into new relevant constraints on theories beyond the SM laying at the TeV scale, even when they contribute to the electron EDM at the two-loop level. We use the EFT approach to classify these corrections, presenting the contributions to the anomalous dimension of the CP-violating dipole operators of the electron up to the two-loop level. Selection rule…
An Effective Field Theory (EFT) analysis of the magnetic moment of the muon is discussed. The expression for the dipole moment is given in terms of operator coefficients of the low-energy effective field theory (LEFT) and the Standard Model effective field theory (SMEFT), where one-loop renormalization group improved perturbation theory, the one-loop matching from SMEFT onto LEFT as well as one-loop lepton matrix elements of the effective-theory operators were taken into account. Interestingly only a very limited set of the SMEFT operators is able to explain the current deviation of the magnet…
The s-wave nucleon-nucleon (NN) scattering matrix ($S$-matrix) exhibits UV/IR symmetries which are hidden in the effective field theory (EFT) action and scattering amplitudes, and which explain some generic features of the phase shifts. These symmetries offer clarifying interpretations of existing pionless EFT expansions, and suggest starting points for novel expansions. The leading-order (LO) $S$-matrix obtained in the pionless EFT with scattering lengths treated exactly is shown to have a UV/IR symmetry which leaves the sum of s-wave phase shifts invariant. A new scheme, which treats effecti…
We present new results for the light-quark connected part of the leading order hadronic-vacuum-polarization (HVP) contribution to the muon anomalous magnetic moment, using $2+1+1$ staggered fermions. We have collected more statistics on previous ensembles, and we added two new ensembles. This allows us to reduce statistical errors on the HVP contribution and related window quantities significantly. We also calculated the current-current correlator to next-to-next-to-leading order (NNLO) in staggered chiralperturbation theory, so that we can correct to NNLO for finite-volume, pion-mass mistunin…
The resummation of soft gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions, which has been computed through two loops. The two-loop matrix is proportional to the one-loop matrix. Recently there have been proposals that this proportionality extends to higher loops. One can test such proposals by computing the dependence of this matrix on the matter content in a generic gauge theory. It is shown that for the matter-dependent part the proportionality extends to three loops for arbitrary massless processes.
We propose a new approach to the LHC dark matter search analysis within the effective field theory (EFT) framework by utilising the K-matrix unitarisation formalism. This approach provides a reasonable estimate of the dark matter production cross section at high energies, and hence allows reliable bounds to be placed on the cut-off scale of relevant operators without running into the problem of perturbative unitarity violation. We exemplify this procedure for the effective operator D5 in monojet dark matter searches in the collinear approximation. We compare our bounds to those obtained using …
In this paper, we present the computation of the Higgs boson pair production cross section, both inclusive as well as differential on the invariant mass distribution, at next-to-next-to-leading order (NNLO) in QCD including effects of new physics beyond the standard model. We parametrize the effects of new physics with the relevant dimension 6 operators in a standard model effective field theory (EFT) approach and examine their phenomenology. The dependence of the NNLO $K$-factor on the EFT couplings is analysed, finding that, while rather flat for a number of EFT coefficients, it can consider…
We describe a Python-based computer program, DEFT, for manipulating operators in effective field theories (EFTs). In its current incarnation, DEFT can be applied to 4-dimensional, Poincaré invariant theories with gauge group $SU(3)\times SU(2) \times U(1)$, such as the Standard Model (SM), but a variety of extensions (e.g. to lower dimensions or to an arbitrary product of unitary gauge groups) are conceptually straightforward. Amongst other features, the program is able to: (i) check whether an input list of Lagrangian operators (of a given dimension in the EFT expansion) is a basis for the sp…
We calculate all contributions $\propto T_F$ to the polarized three-loop anomalous dimensions in the M-scheme using massive operator matrix elements and compare to results in the literature. This includes the complete anomalous dimensions $γ_{qq}^{(2),\rm PS}$ and $γ_{qg}^{(2)}$. We also obtain the complete two-loop polarized anomalous dimensions in an independent calculation. While for most of the anomalous dimensions the usual direct computation methods in Mellin $N$-space can be applied since all recurrences factorize at first order, this is not the case for $γ_{qg}^{(2)}$. Due to the neces…
Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where $S$-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the $S$-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies assoc…
We present the results of two-loop calculations of the anomalous dimension matrix for the Wilson twist-2 operators in the N=4 Supersymmetric Yang-Mills theory for polarized and unpolarized cases. This matrix can be transformed to a triangle form by the same similarity transformation as in the leading order. The eigenvalues of the anomalous dimension matrix are expressed in terms of an universal function with its argument shifted by integer numbers. In the conclusion we discuss relations between the weak and strong coupling regimes in the framework of the AdS/CFT correspondence.
We report on the calculation of the three-loop polarized and unpolarized flavor non-singlet and the polarized singlet anomalous dimensions using massless off-shell operator matrix elements in a gauge-variant framework. We also reconsider the unpolarized two-loop singlet anomalous dimensions and correct errors in the foregoing literature.
We develop tools for performing effective field theory (EFT) calculations in a manifestly gauge-covariant fashion. We clarify how functional methods account for one-loop diagrams resulting from the exchange of both heavy and light fields, as some confusion has recently arisen in the literature. To efficiently evaluate functional traces containing these "mixed" one-loop terms, we develop a new covariant derivative expansion (CDE) technique that is capable of evaluating a much wider class of traces than previous methods. The technique is detailed in an appendix, so that it can be read independen…
Debate topic for Effective Field Theory (EFT) is the choice of a "basis" for $\mrdim = 6$ operators Clearly all bases are equivalent as long as they are a "basis", containing a minimal set of operators after the use of equations of motion and respecting gauge invariance. From a more formal point of view a basis is characterized by its closure with respect to renormalization. Equivalence of bases should always be understood as a statement for the S-matrix and not for the Lagrangian, as dictated by the equivalence theorem. Any phenomenological approach that misses one of these ingredients is sti…
We calculate the unpolarized and polarized three--loop anomalous dimensions and splitting functions $P_{\rm NS}^+, P_{\rm NS}^-$ and $P_{\rm NS}^{\rm s}$ in QCD in the $\overline{\sf MS}$ scheme by using the traditional method of space--like off shell massless operator matrix elements. This is a gauge--dependent framework. For the first time we also calculate the three--loop anomalous dimensions $P_{\rm NS}^{\rm \pm tr}$ for transversity directly. We compare our results to the literature.
We show that chiral effective field theory (EFT) two-body currents provide important contributions to the quenching of low-momentum-transfer Gamow-Teller transitions, and use chiral EFT to predict the momentum-transfer dependence that is probed in neutrinoless double-beta decay. We then calculate for the first time the neutrinoless double-beta decay operator based on chiral EFT currents and study the nuclear matrix elements at successive orders. The contributions from chiral two-body currents are significant and should be included in all calculations.
We identify the effective theory describing inflationary super-Hubble scales and show it to be a special case of effective field theories appropriate to open systems. Open systems allow information to be exchanged between the degrees of freedom of interest and those that are integrated out, such as for particles moving through a fluid. Strictly speaking they cannot in general be described by an effective lagrangian; rather the appropriate `low-energy' limit is instead a Lindblad equation describing the evolution of the density matrix of the slow degrees of freedom. We derive the equation relev…
Anomalous self-interactions of non-abelian gauge fields can be described by higher dimensional operators featuring gauge-invariant combinations of the field strengths. In the case of QCD, the gluon self-interactions start to be modified at dimension six by operators of the type $GGG$, with $G$ the gluon field strength tensor, possibly leading to deviations in all observables and measurements that probe strong interactions at very small distances. In this work we consider the sensitivity to the triple gluon operator of a series of observables at the LHC in di-jet, three- and multi-jet final sta…
We compare two existing approaches to calculating the decay of molecular quarkonium states to conventional quarkonia in effective field theory, using $X(3872) \to χ_{cJ} π^0$ as an example. In one approach the decay of the molecular quarkonium proceeds through a triangle diagram with charmed mesons in the loop. We argue this approach predicts excessively large rates for $Γ[X(3872) \to χ_{cJ}π^0]$ unless both charged and neutral mesons are included and a cancellation between these contributions is arranged to suppress the decay rates. This cancellation occurs naturally if the $X(3872)$ is prima…
We complete the coordinate space calculation of the three-parton correlation in the two-loop massive soft anomalous dimension matrix. The full answer agrees with the result found previously by a different approach. The coordinate space treatment of renormalized two-loop gluon exchange diagrams exhibits their color symmetries in a transparent fashion. We compare coordinate space calculations of the soft anomalous dimension matrix with massive and massless eikonal lines and examine its nonuniform limit at absolute threshold.
Transforming a $μ$ to a $τ$, then the $τ$ to to an $e$, results in $μ\to e$. In an EFT framework, we explore the sensitivity of $μ\to e$ observables to products of $(μ\to τ)\times (τ\to e)$ interactions, and show that the exceptional sensitivity of upcoming $μ\to e$ experiments could allow to probe parameter space beyond the reach of upcoming $τ\to l$ searches in Higgs, $τ$ and $B$ decays. We describe the $τ\to l$ interactions as dimension six operators in the SM EFT, identify pairs of them giving interesting contributions to $μ\to e$ processes, and obtain the anomalous dimensions mixing those…
We calculate the nuclear matrix element for the two-neutrino $ββ$ decay of $^{136}$Xe into the first excited $0^+$ state of $^{136}$Ba. We use different many-body methods: the quasiparticle random-phase approximation (QRPA) framework, the nuclear shell model, the interacting boson model (IBM-2), and an effective field theory (EFT) for $β$ and $ββ$ decays. While the QRPA suggests a decay rate at the edge of current experimental limits, the shell model points to a half-life about two orders of magnitude longer. The predictions of the IBM-2 and the EFT lie in between, and the latter provides syst…
