In this paper we study the renormalization of the product of two operators $O_1=-\frac{1}{4} G^{μν}G_{μν}$ in QCD. An insertion of two such operators $O_1(x)O_1(0)$ into a Greens function produces divergent contact terms for $x\rightarrow 0$.
In the course of the computation of the operator product expansion (OPE) of the correlator of two such operators $i\int\!\mathrm{d}^4x\,e^{iqx} T\{\,O_1(x)O_1(0)\}$ to three-loop order we discovered that divergent contact terms remain not only in the leading Wilson coefficient $C_0$, which is just the VEV of the correlator, but also in the Wilson coefficient $C_1$ in front of $O_1$. As this correlator plays an important role for example in QCD sum rules a full understanding of its renormalization is desireable.
This work explains how the divergences encountered in higher orders of an OPE of this correlator should be absorbed in counterterms and derives an additive renormalization constant for $C_1$ from first principles and to all orders in perturnbation theory. The method to derive the renormalization of this operator product is an extension of the ideas of a paper by Spiridonov and can be generalized to other cases.
Instanton contributions to the Laplace sum-rules for correlation functions of scalar gluonic currents are calculated. The role of the constant low-energy theorem term, whose substantial contribution is unique to the leading Laplace sum-rule $L_{-1}$, is shown to be diminished by instanton contributions, significantly increasing the resulting mass bounds for the ground state of scalar gluonium and improving compatibility with results from higher-weight sum-rules.
Evanescent operators are a special class of operators that vanish classically in four-dimensional spacetime, while in general dimensions they are non-zero and are expected to have non-trivial physical effects at the quantum loop level in dimensional regularization. In this paper we initiate the study of evanescent operators in pure Yang-Mills theory. We develop a systematic method for classifying and constructing the $d$-dimensional Lorentz invariant evanescent operators, which start to appear at mass dimension ten. We also compute one-loop form factors for the dimension-ten operators via the …
We provide the all-loop structure of gauge-variant operators required for the renormalisation of Green's functions with insertions of twist-two operators in Yang-Mills theory. Using this structure we work out an explicit basis valid up to 4-loop order for an arbitrary compact simple gauge group. To achieve this we employ a generalised gauge symmetry, originally proposed by Dixon and Taylor, which arises after adding to the Yang-Mills Lagrangian also operators proportional to its equation of motion. Promoting this symmetry to a generalised BRST symmetry allows to generate the ghost operator fro…
We derive the analogue of the QCD low energy theorems for the scalar and pseudoscalar gluonic correlators in nuclear matter. We find that the scalar correlations are depleted while the pseudoscalar correlations are enhanced to leading order in the nuclear matter density. We briefly discuss the consequences of these findings on the QCD spectrum.
As a topic of "quantum color dynamics", we study various mass generation of colored particles and gluonic dressing effect in a non-perturbative manner, using the Schwinger-Dyson (SD) formalism in (scalar) QCD. First, we review dynamical quark-mass generation in QCD in the SD approach as a typical fermion-mass generation via spontaneous chiral-symmetry breaking. Second, using the SD formalism for scalar QCD, we investigate the scalar diquark, a bound-state-like object of two quarks, and its mass generation, which is clearly non-chiral-origin. Here, the scalar diquark is treated as an extended c…
We calculate the $O(α_s^2)$ gluonic operator matrix elements for the twist--2 operators, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region $Q^2 \gg m^2$, up to the linear terms in the dimensional parameter $\varepsilon$, ($D= 4 + \varepsilon$). These quantities are required for the description of parton distribution functions in the variable flavor number scheme (VFNS). The $O(α_s^2 \varepsilon)$ terms contribute at the level of the $O(α_s^3)$ corrections through renormalization. We also comment on additional terms, which have to …
We discuss the recent observation of f_0(980) in charmless B-decays and in gluon jets which hints toward a gluonic coupling of this meson similar to eta'. Further predictions on B-decays into scalar particles are presented. Charmless B decays also show a broad K Kbar (and possibly pi pi) S-wave enhancement which we relate to the 0^{++} glueball. These gluonic mesons represent a sizable fraction of the theoretically derived decay rate for b -> sg.
Extending previous QCD Hamiltonian studies, we present a new renormalization procedure which generates an effective Hamiltonian for the gluon sector. The formulation is in the Coulomb gauge where the QCD Hamiltonian is renormalizable and the Gribov problem can be resolved. We utilize elements of the Glazek and Wilson regularization method but now introduce a continuous cut-off procedure which eliminates non-local counterterms. The effective Hamiltonian is then derived to second order in the strong coupling constant. The resulting renormalized Hamiltonian provides a realistic starting point for…
Gaussian QCD sum-rules are used to analyze all possible two-point correlation functions of scalar gluonic and quark currents. The independent predictions of the masses and relative coupling strengths from the different correlators are remarkably consistent with a scenario of two scalar states that couple to nearly-maximal mixtures of quark and gluonic currents.
Gaussian QCD sum-rules are ideally suited to the study of mixed states of gluonium (glueballs) and quark ($q\bar q$) mesons because of their capability to resolve widely-separated states of comparable strength. The analysis of the Gaussian QCD sum-rules (GSRs) for all possible two-point correlation functions of gluonic and non-strange ($I=0$) quark scalar ($J^{PC}=0^{++}$) currents is discussed. For the non-diagonal sum-rule of gluonic and $q\bar q$ currents we show that perturbative and gluon condensate contributions are chirally suppressed compared to non-perturbative effects of the quark co…
Analytic structure in the strong coupling constant that emerges for some observables in QCD after duality averaging of renormalization group improved amplitudes is discussed. It is shown that perturbation theory calculations are justified for the proper observables related to the two-point correlators of hadronic currents the analytic properties of which are well-established. A particular case of gluonic current correlators is discussed in detail.
The existence of glueballs is predicted in QCD, the lightest one with quantum numbers J^{PC}=0^{++}, but different calculations do not well agree on its mass in the range below 1800 MeV. Several theoretical schemes have been proposed to cope with the experimental data which often have considerable uncertainties. Further experimental studies of the scalar meson sector are therefore important and we discuss recent proposals to study leading clusters in gluon jets and charmless B-decays to serve this purpose.
Evanescent operators are a special class of operators that vanish in four-dimensional spacetime but are non-zero in $d=4-2ε$ dimensions. In this paper, we continue our systematic study of the evanescent operators in the pure Yang-Mills theory and focus on their two-loop renormalization. We develop an efficient strategy to compute the two-loop divergences of form factors of high-dimensional and high-length operators by combining the $d$-dimensional unitarity method and the improved tensor reduction techniques. Two-loop anomalous dimensions are obtained for the mass-dimension-10 basis in the pla…
We explore connections between high energy QCD spin physics and $CP$-odd scalar gluonic operators $\tilde{F}^{μν}F_{μν}$ and $\tilde{F}_{μν}F^{μα}F^ν_α$, the latter being called the Weinberg operator in the context of the nucleons' electric dipole moment. We first introduce the twist-four generalized parton distribution (GPD) associated with the topological operator $F_{μν}\tilde{F}^{μν}$. This has interesting applications in spin physics which go beyond the standard framework in terms of twist-two and twist-three distributions. In the second part, we show that the off-forward matrix element o…
The mixing of the Standard Model Higgs boson with a ttbar scalar bound state (arising from a new stronger-than-QCD interaction) can enhance the Higgs boson's coupling to two gluons. It is possible that the two-gluon decay mode of a light Higgs boson may even dominate. This alters the Higgs boson search strategy at Tevatron Run 2 in a significant way. We present a simple model in which such a scenario may be realized. The modifications to the Higgs search signal, as well as further experimental tests of this scenario are also discussed.
We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and $\bar R$ of SU($N_c$). The cases in which $R$ is the fundamental representation and the symmetric and antisymmetric rank-2 tensor representation are considered. We calculate Padé approximants to the beta functions for these theories in the $\overline{\rm DR}$ scheme up to four-loop order for the gluonic theory and up to three-loop order for the theories with ma…
The Dirac structure of confinement is shown to be of timelike-vector nature in the heavy quark limit of QCD. This stands in contradiction with the phenomenological success of the Dirac scalar confining potential. A resolution is achieved through the demonstration that an effective scalar interaction is dynamically generated by nonperturbative mixing between ordinary and hybrid $Q \bar Q$ states. The resolution depends crucially on the collective nature of the gluonic degrees of freedom. This implies that dynamical gluonic effects are vital when attempting to incorporate fine structure in model…
The indirect evidence for gluonic objects in gluon jets is recalled, more accurate tests at the LHC are possible. Estimates of gluonic and quarkonic components of scalar mesons are presented, which could be improved by further studies, in particular of B and B_s decays. The results are consistent with the scalar glueball mixing into f_0(500) and f_0(1500). The study of symmetry relations for 2-body decay rates of charmonium χ_c is suggested as crucial test in the identification of q\bar q multiplets and glueballs. Further related results are discussed in the recent review \cite{Ochs:2013gi}.
We study glueball G production in gluonic penguin decay B -> X_s G, using next-to-leading order b -> s g^* gluonic penguin interaction and an effective coupling of a glueball to two gluons. The effective coupling allows us to study the decay rate of a glueball to two pseudoscalars in the framework of chiral perturbation theory. Identifying the f_0(1710) to be a scalar glueball, we then determine the effective coupling strength with the branching ratio of f_0(1710) -> K KBar. We find that the charm penguin to be important and obtain a sizable branching ratio for Br(B -> X_s G) in the range of (…
We study glueball $G$ production in gluonic penguin decay $B\to G + X_s$, using the next-to-leading order $b\to s g^*$ gluonic penguin interaction and effective couplings of a glueball to two perturbative gluons. Subsequent decays of a scalar glueball are described by using techniques of effective chiral Lagrangian to incorporate the interaction between a glueball and pseudoscalar mesons. Mixing effects between the pure glueball with other mesons are considered. Identifying the $f_0(1710)$ to be a scalar glueball, we find that both the top and charm penguin to be important and obtain a sizable…
We study the influence of strong external magnetic fields on gluonic and fermionic observables in the QCD vacuum at zero and nonzero temperatures, via lattice simulations with N_f=1+1+1 staggered quarks of physical masses. The gluonic action density is found to undergo magnetic catalysis at low temperatures and inverse magnetic catalysis near and above the transition temperature, similar to the quark condensate. Moreover, the gluonic action develops an anisotropy: the chromo-magnetic field parallel to the external field is enhanced, while the chromo-electric field in this direction is suppress…
We review standard holographic derivations of the scalar glueball spectra and emphasize computational assumptions which rely strongly on the original AdS/CFT correspondence, and those which are more arbitrary. In the soft-wall model, we show explicitly that the dual of the spectral mass constraint in the Kallen-Lehmann representation of the 2-point scalar gluonic correlator is an eigenvalue constraint on the motion equation of the bulk field corresponding this correlator.
The vNRQCD Lagrangian for colored heavy scalar fields in the fundamental representation of QCD and the renormalization group analysis of the corresponding operators are presented. The results are an important ingredient for renormalization group improved computations of scalar-antiscalar bound state energies and production rates at next-to-next-to-leading-logarithmic (NNLL) order.
The framework of the electroweak chiral Lagrangian with a light Higgs is extended by an additional scalar and then generalized to N scalars in the Higgs sector. Divergences from scalar fluctuations are renormalized up to one loop using the background field method. The results are crosschecked against the case of one scalar. A subset of the divergences is demonstrated and crosschecked diagrammatically. Together with the complete one-loop renormalization of the electroweak chiral theory with one light Higgs conducted previously, this constitutes a renormalization framework of any pure scalar ext…
We consider the renormalization of theories with many scalar fields. We discuss at the one-loop level some simple, non-gauge models with an arbitrary number of scalars and fermions both in mass-shell and MS schemes. In MS scheme we give a detailed qualitative analysis of the RG flow of dimensionless couplings in flavor space.
We study $\bar qq$-hybrid mixing for the light vector mesons and $\bar qq$-glueball mixing for the light scalar mesons in Monte-Carlo based QCD Laplace sum rules. By calculating the two-point correlation function of a vector $\bar qγ_μq$ (scalar $\bar q q$) current and a hybrid (glueball) current we are able to estimate the mass and the decay constants of the corresponding mixed "physical state" that couples to both currents. Our results do not support strong quark/gluonic mixing for either the $1^{--}$ or the $0^{++}$ states.
The complete QCD corrections to the scalar top quark pair production via $γγ$ fusion in parent $e^{+}e^{-}$ collider are calculated in the Supersymmetric Model. The various origins of corrections are discussed. We find that the correction due to gluino exchange is smaller than that due to conventional gluonic contribution, and the contribution due to squark quartic interactions is much smaller. Moreover, the dependences of the corrections on the masses and the mixing angle of the supersymmetric particles are also investigated.
We have computed the two-loop, electroweak corrections to the production of a light and a heavy neutral, scalar Higgs-boson through the important gluon fusion process in the Two-Higgs-Doublet Model. We provide our results in various renormalization schemes for different scenarios and benchmark points, which will be valuable for experimental studies at the LHC. We describe the technicalities of our two-loop calculation and augment it by a phenomenological discussion. Our results are also applicable to the gluonic neutral, scalar Higgs-boson decays.
We match the electroweak chiral Lagrangian with two singlet scalars to the next-to-minimal composite Higgs model with $ SO(6)/SO(5) $ coset structure and extract the scalar divergences to one loop. Assuming the additional scalar to be heavy, we integrate it out and perform a matching to the well-established electroweak chiral Lagrangian with one light Higgs.
The renormalization of the scalar diquark operator and its anomalous dimension is calculated at two-loop order in QCD, enabling higher-order QCD studies of diquarks. As an application of our result, the two-loop diquark anomalous dimension in the $\bar{\rm MS}$ scheme is used to study the QCD renormalization scale dependence of diquark matrix elements of the $ΔS=1$ effective weak Hamiltonian.
Quantum field theories containing scalar fields with equal quantum numbers allow for a mixed kinetic term in the Lagrangian. It has been argued that this mixing must be taken into consideration when performing renormalization group (RG) analyses of such a theory. However, from the fact that scalar kinetic mixing does not correspond to a physical observable, we show that no extra parameters need to be introduced. Using a toy model, we explicitly derive the 1-loop RG equations (RGEs) in three different renormalization schemes to demonstrate how this issue can be dealt with. In schemes without ki…
Self-consistent new renormalization group flow equations for an O(N)-symmetric scalar theory are approximated in next-to-leading order of the derivative expansion. The Wilson-Fisher fixed point in three dimensions is analyzed in detail and various critical exponents are calculated.
The five-loop effective potential and the associated summation of subleading logarithms for O(4) globally-symmetric massless $λϕ^4$ field theory in the Coleman-Weinberg renormalization scheme $\frac{d^4V}{dϕ^4}|_{ϕ= μ} = λ$ (where $μ$ is the renormalization scale) is calculated via renormalization-group methods. An important aspect of this analysis is conversion of the known five-loop renormalization-group functions in the minimal-subtraction (MS) scheme to the Coleman-Weinberg scheme.
We consider the Resonance Chiral Theory with one multiplet of scalar and pseudoscalar resonances, up to bilinear couplings in the resonance fields, and evaluate its beta-function at one-loop with the use of the background field method. Thus we also provide the full set of operators that renormalize the theory at one loop and render it finite.
We consider an Abelian model with a CP-conserving Higgs potential spanned by two complex Higgs fields. The CP invariance of the Higgs potential is then broken explicitly beyond the Born approximation by introducing soft-CP-violating Yukawa interactions. Based on the non-renormalization theorem, we derive the consistency conditions under which a CP-odd counterterm exists and, at the same time, renders the one-loop-induced mixing of a CP-even Higgs boson with a CP-odd Higgs scalar ultra-violet finite. The novel CP-odd tadpole renormalization may then be determined from the minimization constrain…
The divergent part of the generating functional of the Resonance Chiral Theory is evaluated up to one loop when one multiplet of scalar an pseudoscalar resonances are included and interaction terms which couple up to two resonances are considered. Hence we obtain the renormalization of the couplings of the initial Lagrangian and, moreover, the complete list of operators that make this theory finite, at this order.
We employ renormalization group (RG) summation techniques to obtain portions of Laplace QCD sum rules for scalar gluon currents beyond the order to which they have been explicitly calculated. The first two of these sum rules are considered in some detail, and it is shown that they have significantly less dependence on the renormalization scale parameter $μ^2$ once the RG summation is used to extend the perturbative results. Using the sum rules, we then compute the bound on the scalar glueball mass and demonstrate that the 3 and 4-Loop perturbative results form lower and upper bounds to their R…
We discuss renormalization in a toy model with one fermion field and one real scalar field phi, featuring a spontaneously broken discrete symmetry which forbids a fermion mass term and a phi^3 term in the Lagrangian. We employ a renormalization scheme which uses the MSbar scheme for the Yukawa and quartic scalar couplings and renormalizes the vacuum expectation value of phi by requiring that the one-point function of the shifted field is zero. In this scheme, the tadpole contributions to the fermion and scalar selfenergies are canceled by choice of the renormalization parameter delta_v of the …
We provide a renormalization procedure for Phi-derivable approximations in theories coupling different types of fields. We illustrate our approach on a scalar phi^4 theory coupled to fermions via a Yukawa-like interaction. The non-perturbative renormalization amounts to fixing the scalar coupling via a set of nested Bethe-Salpeter equations coupling fermions to scalars.
We discuss the renormalization of Φ-derivable approximations for scalar field theories. In such approximations, the self-energy is obtained as the solution of a self-consistent equation which effectively resums infinite subsets of diagrams of perturbation theory. We show that a consistent renormalization can be carried out, and we provide an explicit construction of the counterterms needed to eliminate the subdivergences. The counterterms are calculated from the solution of an auxiliary gap equation which determines the leading asymptotic part of the self-energy. This auxiliary gap equation ma…
Multi-scale renormalization group (RG) methods are reviewed and applied to the analysis of the effective potential for radiative symmetry breaking with multiple scalar fields, allowing an extension of the Gildener & Weinberg (GW) method beyond the weak coupling limit. A model containing two interacting real scalar fields is used to illustrate multi-scale RG methods and the multi-scale RG functions of this model are calculated to one-loop order for the $β$ function and two-loop order for the anomalous mass dimension. The introduction of an extra renormalization scale allows the mapping of the e…
The correlator of the square of the Yang-Mills field-strength tensor corresponds to a scalar glueball, i.e., to a bound-state formed by gluonic ingredients only. It has quantum numbers 0++ and its mass, as predicted by different theoretical approaches, is expected to lie between 1 and 2 GeV. Here we restrict our considerations to the Born level, that is, we consider the correlator to zeroth order in the coupling. Gluonic self-interaction is taken into account indirectly by using non-perturbative gluon propagators. The employed closed expressions are motivated by lattice and Dyson-Schwinger stu…
We present a renormalized computational framework for the evolution of a self-interacting scalar field (inflaton) and its quantum fluctuations in an FRW background geometry. We include a coupling of the field to the Ricci scalar with a general coupling parameter $ξ$. We take into account the classical and quantum back reactions, i.e., we consider the the dynamical evolution of the cosmic scale factor. We perform, in the one-loop and in the large-N approximation, the renormalization of the equation of motion for the inflaton field, and of its energy momentum tensor. Our formalism is based on a …
An Aoki-Denner form of the renormalization scheme is suggested for the physical amplitudes in MSSM. The Higgs sector is reparameterized with the mass of the CP odd scalar, and the mass of the heavy CP even one instead of $tanβ$ in our scheme. The counterterms of $tanβ$ is fixed perturbatively on mass shell just within Higgs sector. The counterterms of gauge-scalar mixings are defined with Ward-Takahashi identit ies from scalar-scalar mixings. The effect of the reparameterization is also probed on the radiative correction of the mass of the lightest Higgs.
We show how to renormalize Phi-derivable approximations in a theory with a fermionic field coupled to a self-interacting scalar field through a Yukawa interaction. The nonperturbative renormalization concerns the self-interaction coupling of the scalar field which is renormalized via a set of nested Bethe-Salpeter equations for the scalar and fermionic four-point functions. We use this information to construct explicit finite equations of motion in the symmetric phase. We work in the context of equilibrium quantum field theory and show that the renormalization can be carried out without introd…
We study the dynamics in phases with vector condensates of gluons (gluonic phases) in dense two-flavor quark matter. These phases yield an example of dynamics in which the Higgs mechanism is provided by condensates of gauge (or gauge plus scalar) fields. Because vacuum expectation values of spatial components of vector fields break the rotational symmetry, it is naturally to have a spontaneous breakdown both of external and internal symmetries in this case. In particular, by using the Ginzburg-Landau approach, we establish the existence of a gluonic phase with both the rotational symmetry and …
We address the problem of solution of the QCD three-particle evolution equations which govern the Q-dependence of the chiral-even quark-gluon-quark and three-gluon correlators contributing to a number of asymmetries at leading order and the transversely polarized structure function g_2(x). The quark-gluon-quark case is completely integrable in multicolour limit and corresponds to a spin chain with non-periodic boundary conditions, while the pure gluonic sector contains, apart from a piece in the Hamiltonian equivalent to XXX Heisenberg magnet of spin s = - 3/2, a non-integrable addendum which …
The QCD energy-momentum tensor on the lattice: non-perturbative renormalization with $N_f=3$
School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin（三一学院都柏林数学与汉密尔顿数学研究所）；Department of Physics “Giuseppe Occhialini”, University of Milano-Bicocca（米兰比可卡大学“朱塞佩·奥恰利尼”物理系）；INFN Milano-Bicocca（米兰比可卡意大利国家核物理研究所）；Theoretical Physics Department, CERN（欧洲核子研究中心理论物理部）
The quadratically divergent scalar mass is subtractively renormalized unlike other divergences which are multiplicatively renormalized. We re-examine some technical aspects of the subtractive renormalization, in particular, the mass independent renormalization of massive $λϕ^{4}$ theory with higher derivative regularization. We then discuss an unconventional scheme to introduce the notion of renormalization point $μ$ to the subtractive renormalization in a theory defined by a large fixed cut-off $M$. The resulting renormalization group equation generally becomes inhomogeneous but it is transfo…
