Witten-Veneziano Formula Research Articles

CP -odd sector and θ dynamics in holographic QCD

The holographic model of V-QCD is used to analyze the physics of QCD in the Veneziano large-N limit. An unprecedented analysis of the CP-odd physics is performed going beyond the level of effective field theories. The structure of holographic saddle-points at finite $\theta$ is determined, as well as its interplay with chiral symmetry breaking. Many observables (vacuum energy and higher-order susceptibilities, singlet and non-singlet masses and mixings) are computed as functions of $\theta$ and the quark mass $m$. Wherever applicable the results are compared to those of chiral Lagrangians, finding agreement. In particular, we recover the Witten-Veneziano formula in the small $x\to 0$ limit, we compute the $\theta$-dependence of the pion mass and we derive the hyperscaling relation for the topological susceptibility in the conformal window in terms of the quark mass.

Non-perturbative test of the Witten-Veneziano formula from lattice QCD

We compute both sides of the Witten-Veneziano formula using lattice techniques. For the one side we perform dedicated quenched simulations and use the spectral projector method to determine the topological susceptibility in the pure Yang-Mills theory. The other side we determine in lattice QCD with $N_f=2+1+1$ dynamical Wilson twisted mass fermions including for the first time also the flavour singlet decay constant. The Witten-Veneziano formula represents a leading order expression in the framework of chiral perturbation theory and we also employ leading order chiral perturbation theory to relate the flavor singlet decay constant to the relevant decay constant parameters in the quark flavor basis and flavor non-singlet decay constants. After taking the continuum and the SU$(2)$ chiral limits we compare both sides and find good agreement within uncertainties.

Renewed look atη′in medium

We revisit the question of whether the ${\mathrm{U}}_{A}(1)$ symmetry is effectively restored in hot and dense medium. In particular, by generalizing the Witten-Veneziano formula to finite temperature, we investigate whether the mass of ${\ensuremath{\eta}}^{\ensuremath{'}}$-meson will change in medium due to the restoration of chiral symmetry.

In-Medium η′ Mass Reduction

In this talk, we present the reduction of the mass of η′-meson at finite temperature which leads to the enhancement of η′ contributions to the dilepton spectra from relativistic heavy ion collisions. QCD low energy theorem, together with the Witten–Veneziano formula, provides the relation of the η′ mass with gluon condensates.

Pseudoscalar meson decay constants and couplings, the Witten–Veneziano formula beyond large [formula omitted], and the topological susceptibility

The QCD formulae for the radiative decays η , η ′ → γ γ , and the corresponding Dashen–Gell-Mann–Oakes–Renner relations, differ from conventional PCAC results due to the gluonic U ( 1 ) A axial anomaly. This introduces a critical dependence on the gluon topological susceptibility. In this paper, we revisit our earlier theoretical analysis of radiative pseudoscalar decays and the DGMOR relations and extract explicit experimental values for the decay constants. This is our main result. The flavour singlet DGMOR relation is the generalisation of the Witten–Veneziano formula beyond large N c , so we are able to give a quantitative assessment of the realisation of the 1 / N c expansion in the U ( 1 ) A sector of QCD. Applications to other aspects of η ′ physics, including the relation with the first moment sum rule for the polarised photon structure function g 1 γ , are highlighted. The U ( 1 ) A Goldberger–Treiman relation is extended to accommodate SU ( 3 ) flavour breaking and the implications of a more precise measurement of the η and η ′ -nucleon couplings are discussed. A comparison with the existing literature on pseudoscalar meson decay constants using large- N c chiral Lagrangians is also made.

Low Energy Hadron Physics in Holographic QCD

We present a holographic dual of four-dimensional, large N_c QCD with massless flavors. This model is constructed by placing N_f probe D8-branes into a D4 background, where supersymmetry is completely broken. The chiral symmetry breaking in QCD is manifested as a smooth interpolation of D8 - anti-D8 pairs in the supergravity background. The meson spectrum is examined by analyzing a five-dimensional Yang-Mills theory that originates from the non-Abelian DBI action of the probe D8-brane. It is found that our model yields massless pions, which are identified with Nambu-Goldstone bosons associated with the chiral symmetry breaking. We obtain the low-energy effective action of the pion field and show that it contains the usual kinetic term of the chiral Lagrangian and the Skyrme term. A brane configuration that defines a dynamical baryon is identified with the Skyrmion. We also derive the effective action including the lightest vector meson. Our model is closely related to that in the hidden local symmetry approach, and we obtain a Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin-type relation among the couplings. Furthermore, we investigate the Chern-Simons term on the probe brane and show that it leads to the Wess-Zumino-Witten term. The mass of the \eta' meson is also considered, and we formulate a simple derivation of the \eta' mass term satisfying the Witten-Veneziano formula from supergravity.

The holographic life of the η′

In the string holographic dual of large-Nc QCD with Nf ∞avours of (1), the · 0 meson is massless at inflnite Nc and dual to a collective ∞uctuation of Nf D6-brane probes in a supergravity background. Here we identify the string diagrams responsible for the gen- eration of a mass of order Nf=Nc, consistent with the Witten-Veneziano formula, and show that the supergravity limit of these diagrams corresponds to mixings with pseudoscalar glueballs. We argue that the dependence on the µ-angle in the supergravity description occurs only through the combination µ + 2 p Nf· 0 =f…, as dictated by the U(1)A anomaly. We provide a quantitative test by computing the linear term in the · 0 potential in two independent ways, with perfect agreement.

Witten-Veneziano from Green-Schwarz

We consider the U(1) problem within the AdS/CFT framework. We explain how the Witten-Veneziano formula for the eta' mass is related to a generalized Green-Schwarz mechanism. The closed string mode, that cancels the anomaly of the gauged U(1) axial symmetry, is identified with the eta' meson. In a particular set-up of D3-branes on a C3/(Z3xZ3) orbifold singularity, the eta' meson is a twisted-sector R-R field.

Free energy and θ dependence of SU( N) gauge theories

We study the dependence of the free energy on the CP violating angle θ, in four-dimensional SU( N) gauge theories with N ≥ 3, and in the large- N limit. Using the Wilson lattice formulation for numerical simulations, we compute the first few terms of the expansion of the ground-state energy F(θ) around θ = 0, F(θ) − F(θ) = A 2θ 2(1 + b 2θ 2 + ...). Our results support Witten's conjecture: F(θ) − F(0) = Aθ 2 + O(1/ N) for θ < π. We verify that the topological susceptibility has a nonzero large- N limit χ ∞ = 2A with corrections of O(1/N 2) , in substantial agreement with the Witten-Veneziano formula which relates χ ∞ to the η′ mass. Furthermore, higher order terms in θ are suppressed; in particular, the O(θ 4) term b 2 (related to the η′ − η′ elastic scattering amplitude) turns out to be quite small: b 2 = −0.023(7) for N = 3, and its absolute value decreases with increasing N, consistently with the expectation b 2 = O(1/ N 2).

θ dependence of SU(N) gauge theories

We study the $\theta$ dependence of four-dimensional SU($N$) gauge theories, for $N\geq 3$ and in the large-N limit. We use numerical simulations of the Wilson lattice formulation of gauge theories to compute the first few terms of the expansion of the ground-state energy $F(\theta)$ around $\theta=0$, $F(\theta)-F(0) = A_2 \theta^2 (1 + b_2 \theta^2 + ...)$. Our results support Witten's conjecture: $F(\theta)-F(0) = {\cal A} \theta^2 + O(1/N)$ for sufficiently small values of $\theta$, $\theta < \pi$. Indeed we verify that the topological susceptibility has a nonzero large-N limit $\chi_\infty=2 {\cal A}$ with corrections of $O(1/N^2)$, in substantial agreement with the Witten-Veneziano formula which relates $\chi_\infty$ to the $\eta^\prime$ mass. Furthermore, higher order terms in $\theta$ are suppressed; in particular, the $O(\theta^4)$ term $b_2$ (related to the $\eta^\prime - \eta^\prime$ elastic scattering amplitude) turns out to be quite small: $b_2=-0.023(7)$ for N=3, and its absolute value decreases with increasing $N$, consistently with the expectation $b_2=O(1/N^2)$.

The UA(1) problem on the lattice with Ginsparg–Wilson fermions

We show how it is possible to give a precise and unambiguous implementation of the Witten–Veneziano formula for the η′ mass on the lattice, which looks like the formal continuum one, if the expression of the topological charge density operator, suggested by fermions obeying the Ginsparg–Wilson relation, is employed. By using recent numerical results from simulations with overlap fermions in 2 (abelian Schwinger model) and 4 (QCD) dimensions, one obtains values for the mass of the lightest pseudo-scalar flavour-singlet state that agree within errors with theoretical expectations and experimental data, respectively.

The U A(1) problem on the lattice

If the expression of the topological charge density operator, suggested by fermions obeying the Ginsparg-Wilson relation, is employed, it is possible to prove on the lattice the validity of the Witten—Veneziano formula for the η′ mass. Recent numerical results from simulations with overlap fermions in 2 (abelian Schwinger model) and 4 (QCD) dimensions give values for the mass of the lightest pseudo-scalar flavour-singlet state that agree with theoretical expectations and/or experimental data.

Some more remarks on the Witten–Veneziano formula for the η′ mass

We discuss some subtleties in connection with the new attempts to provide a firm basis for ths Witten-Veneziano formula.

The mass of the η′ meson in the chiral limit due to instantons

Using the trace anomaly and low energy relations, as well as the Witten–Veneziano formula for the mass of the η′ meson, the chiral topology of the QCD nonperturbative instanton vacuum has been numerically evaluated. Our formalism makes it possible to express the topological susceptibility and the mass of the η′ meson as a functions of the instanton number density in the chiral limit. We have explicitly shown that the topological susceptibility in this case is approximately one fours and one half of its phenomenological value at instanton number densities in the chiral limit, 0.5 fm −4 and 1.0 fm −4 , respectively. Thus the instanton contributions substantially underestimate the mass of the η′ meson which still remains large in the chiral limit. With the help of the above-mentioned Witten–Veneziano formula, we derived an absolute lower bounds for the pion decay constant and the mass of the η′ meson in the chiral limit.

Erratum: Topological structure of a chiral QCD vacuum [Phys. Rev. D61, 036006 (2000)

Using the trace anomaly relation, low-energy theorem and Witten-Veneziano formula, we have developed an analytical formalism which allows one to calculate the gluon condensate, the topological susceptibility and the mass of the $\eta'$ meson in the chiral limit as functions of the non-perturbative vacuum energy density. It is used for numerical evaluation of the chiral QCD topology within the QCD vacuum model consisting mainly of the quantum component given by the recently proposed zero modes enhancement (ZME) model and the classical component given by the the random instanton liquid model (RILM). We sum up both contributions into the total, nonperturbative vacuum energy density. A very good agreement with the phenomenological values of the topological susceptibility, the mass of the $\eta'$ meson in the chiral limit and the gluon condensate has been obtained. This puts the above mentioned QCD vacuum model on a firm phenomenological ground.

η′ → γγ and the topological susceptibility

The radiative decays η′(η) → γγ are discussed. The modifications of the conventional PCAC formulae due to the gluonic contribution to the flavour singlet axial anomaly are given. The decay constants satisfy a modified Dashen formula which generalises the Witten-Veneziano formula for the mass of the η′. It is shown how the topological susceptibility in QCD with massive, dynamical quarks may be extracted from measurements of η′(η) → γγ.

Chiral effects of quenched η′ loops

Preliminary results of a study of quenched chiral logarithms at β = 5.7 are presented. Four independent determinations of the quenched chiral log parameter δ are obtained. Two of these are from estimates of the η′ mass, one from the residue of the hairpin diagram and the other from the topological susceptibility combined with the Witten-Veneziano formula. The other two determinations of δ are from measurement of virtual η′ loop effects in m π 2 vs. quark mass and in the chiral behavior of the pseudoscalar decay constant. All of our results are consistent with δ = .080(15). The expected absence of quenched chiral logs in the axial-vector decay constant is also observed.

Radiative η′ decays, the topological susceptibility and the Witten–Veneziano mass formula

The formulae describing the radiative decays η ′( η)→ γγ in QCD beyond the chiral limit are derived. The modifications of the conventional PCAC formulae due to the gluonic contribution to the axial anomaly in the flavour singlet channel are precisely described. The decay constants are found to satisfy a modified Dashen formula which generalises the Witten–Veneziano formula for the mass of the η′. Combining these results, it is shown how the topological susceptibility in QCD with massive, dynamical quarks may be extracted from measurements of η′( η)→ γγ.

Topological structure of a chiral QCD vacuum

Using the trace anomaly relation, low-energy theorem, and Witten-Veneziano formula, we have developed an analytical formalism which allows one to calculate the gluon condensate, the topological susceptibility, and the mass of the ${\ensuremath{\eta}}^{\ensuremath{'}}$ meson in the chiral limit as functions of the nonperturbative vacuum energy density. It is used for numerical evaluation of the chiral QCD topology within the QCD vacuum model consisting mainly of the quantum component given by the recently proposed zero modes enhancement model and the classical component given by the random instanton liquid model. We sum up both contributions into the total, nonperturbative vacuum energy density. A very good agreement with the phenomenological values of the topological susceptibility, the mass of the ${\ensuremath{\eta}}^{\ensuremath{'}}$ meson in the chiral limit, and the gluon condensate has been obtained. This puts the above-mentioned QCD vacuum model on a firm phenomenological ground.

Topology of the chiral QCD vacuum

Using the trace anomaly relation, low-energy theorem and Witten-Veneziano formula, we have developed an analytical formalism which allows one to calculate the gluon condensate, the topological susceptibility and the mass of the η′ meson in the chiral limit as functions of the non-perturbative vacuum energy density. It is used for numerical evaluation of the chiral QCD topology within the QCD vacuum model consisting mainly of the quantum component given by the recently proposed zero modes enhancement (ZME) model and the classical component given by the random instanton liquid model (RILM). We sum up both contributions into the total, non-perturbative vacuum energy density. A very good agreement with the phenomenological values of the topological susceptibility, the mass of the η′ meson in the chiral limit and the gluon condensate has been obtained. This puts the above mentioned QCD vacuum model on a firm phenomenological ground.