Ladder Schwinger-Dyson Equation Research Articles

On the Anomalous Dimension in QCD

The anomalous dimension γm=1 in the infrared region near the conformal edge in the broken phase of the large Nf QCD has been shown by the ladder Schwinger–Dyson equation and also by the lattice simulation for Nf=8 and for Nc=3. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, γm=1, by comparing the pion matrix element of the trace of the energy-momentum tensor π(p2)|(1+γm)·∑i=1Nfmfψ¯iψi|π(p1)=π(p2)|θμμ|π(p1)=2Mπ2 (up to trace anomaly) with the estimate of π(p2)|2·∑i=1Nfmfψ¯iψi|π(p1)=2Mπ2 through the Feynman–Hellmann theorem combined with an assumption Mπ2∼mf characteristic of the broken phase. We show that this is not justified by the explicit evaluation of each matrix element based on the dilaton chiral perturbation theory (dChPT): π(p2)|2·∑i=1Nfmfψ¯iψi|π(p1)=2Mπ2+[(1−γm)Mπ2·2/(1+γm)]=2Mπ2·2/(1+γm)≠2Mπ2 in contradiction with his estimate, which is compared with π(p2)|(1+γm)·∑i=1Nfmfψ¯iψi|π(p1)=(1+γm)Mπ2+[(1−γm)Mπ2]=2Mπ2 (both up to trace anomaly), where the terms in [] are from the σ (pseudo-dilaton) pole contribution. Thus, there is no constraint on γm when the σ pole contribution is treated consistently for both. We further show that the Feynman–Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the σ pole contribution is absent, and with Mπ2∼mf2/(1+γm) instead of Mπ2∼mf, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of f0(500) to ππ for Nf=2. It is shown to be consistent with the reality, when f0(500) is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance.

Walking on the ladder: 125 GeV technidilaton, or Conformal Higgs

The walking technicolor based on the ladder Schwinger-Dyson gap equation is studied, with the scale-invariant coupling being an idealization of the Caswell-Banks-Zaks infrared fixed point in the "anti-Veneziano limit", such that $N_C \rightarrow \infty$ with $N_C \cdot \alpha(\mu^2)=$ fixed and $N_F/N_C=$ fixed ($\gg 1$), of the $SU(N_C)$ gauge theory with massless $N_F$ flavors near criticality. We show that the 125 GeV Higgs can be naturally identified with the technidilaton (TD) predicted in the walking technicolor, a pseudo Nambu-Goldstone (NG) boson of the spontaneous symmetry breaking of the approximate scale symmetry. Ladder calculations yield the TD mass $M_\phi$ from the trace anomaly as $M_\phi^2 F_\phi^2= -4 \langle \theta_\mu^\mu \rangle = - \frac{\beta(\alpha (\mu^2))}{\alpha(\mu^2)}\, \langle G_{\lambda \nu}^2(\mu^2)\rangle \simeq N_C N_F\frac{16}{\pi^4} m_F^4$, independently of the renormalization point $\mu$, where $m_F$ is the dynamical mass of the technifermion, and $F_\phi={\cal O} (\sqrt{N_F N_C}\, m_F)$ the TD decay constant. It reads $M_\phi^2\simeq (\frac{v_{\rm EW}}{2} \cdot \frac{5 v_{\rm EW}}{F_\phi})^2 \cdot [\frac{8}{N_F}\frac{4}{N_C}]$, ($v_{\rm EW}=246$ GeV), which implies $F_\phi\simeq 5 \,v_{\rm EW} $ for $M_\phi \simeq 125\, {\rm GeV}\simeq \frac{1}{2} v_{\rm EW}$ in the one-family model ($N_C=4, N_F=8$), in good agreement with the current LHC Higgs data. The result reflects a generic scaling $ M_\phi^2/v_{\rm EW}^2\sim M_\phi^2/F_\phi^2 \sim m_F^2 /F_\phi^2 \sim 1/(N_F N_C) \rightarrow 0 $ as a vanishing trace anomaly, namely the TD has a mass vanishing in the anti-Veneziano limit, similarly to $\eta^\prime$ meson as a pseudo-NG boson of the ordinary QCD with vanishing $U(1)_A$ anomaly in the Veneziano limit ($N_F/N_C \ll 1$).

Study of the conformal hyperscaling relation through the Schwinger-Dyson equation

We study corrections to the conformal hyperscaling relation in the conformal window of the large ${N}_{f}$ QCD by using the ladder Schwinger-Dyson (SD) equation as a concrete dynamical model. From the analytical expression of the solution of the ladder SD equation, we identify the form of the leading mass correction to the hyperscaling relation. We find that the anomalous dimension, when identified through the hyperscaling relation neglecting these corrections, yields a value substantially lower than the one at the fixed point ${\ensuremath{\gamma}}_{m}^{*}$ for large mass region. We further study finite-volume effects on the hyperscaling relation, based on the ladder SD equation in a finite space-time with the periodic boundary condition. We find that the finite-volume corrections on the hyperscaling relation are negligible compared with the mass correction. The anomalous dimension, when identified through the finite-size hyperscaling relation neglecting the mass corrections as is often done in the lattice analyses, yields almost the same value as that in the case of the infinite space-time neglecting the mass correction, i.e., a substantially lower value than ${\ensuremath{\gamma}}_{m}^{*}$ for large mass. We also apply the finite-volume SD equation to the chiral-symmetry-breaking phase and find that when the theory is close to the critical point such that the dynamically generated mass is much smaller than the explicit breaking mass, the finite-size hyperscaling relation is still operative. We also suggest a concrete form of the modification of the finite-size hyperscaling relation by including the mass correction, which may be useful to analyze the lattice data.

Scalar decay constant and Yukawa coupling in walking gauge theories

We propose an approach for the calculation of the Yukawa coupling through the scalar decay constant and the chiral condensate in the context of the extended technicolor . We perform the nonperturbative computation of the Yukawa coupling based on the improved ladder Schwinger-Dyson equation. It turns out that the Yukawa coupling can be larger or smaller than the standard model value, depending on the number ${N}_{D}$ of the weak doublets for each technicolor (TC) index. It is thus nontrivial whether or not the huge enhancement of the production of the scalar via the gluon fusion takes place even for a walking TC model with a colored techni-fermion. For the typical one-family TC model near conformality, it is found that the Yukawa coupling is slightly larger than the standard model one, where the expected mass of the scalar bound state is around 500 GeV. In this case, the production cross section via the gluon fusion is considerably enhanced, as naively expected, and hence such a scalar can be discovered/excluded at the early stage of the LHC.

Technidilaton at the conformal edge

Techni-dilaton (TD) was proposed long ago in the technicolor (TC) near criticality/conformality. To reveal the critical behavior of TD, we explicitly compute the nonperturbative contributions to the scale anomaly $<\theta^\mu_\mu>$ and to the techni-gluon condensate $<G_{\mu\nu}^2>$, which are generated by the dynamical mass m of the techni-fermions. Our computation is based on the (improved) ladder Schwinger-Dyson equation, with the gauge coupling $\alpha$ replaced by the two-loop running one $\alpha(\mu)$ having the Caswell-Banks-Zaks IR fixed point $\alpha_*$: $\alpha(\mu) \simeq \alpha = \alpha_*$ for the IR region $m < \mu < \Lambda_{TC}$, where $\Lambda_{TC}$ is the intrinsic scale (analogue of $\Lambda_{QCD}$ of QCD) relevant to the perturbative scale anomaly. We find that $-<\theta^\mu_\mu>/m^4\to const \ne 0$ and $<G_{\mu\nu}^2>/m^4\to (\alpha/\alpha_{cr}-1)^{-3/2}\to\infty$ in the criticality limit $m/\Lambda_{TC}\sim\exp(-\pi/(\alpha/\alpha_{cr}-1)^{1/2})\to 0$ ($\alpha=\alpha_* \to \alpha_{cr}$) ("conformal edge"). Our result precisely reproduces the formal identity $<\theta^\mu_\mu>=(\beta(\alpha)/4 \alpha) <G_{\mu\nu}^2>$, where $\beta(\alpha)=-(2\alpha_{cr}/\pi) (\alpha/\alpha_{cr}-1)^{3/2}$ is the nonperturbative beta function corresponding to the above essential singularity scaling of $m/\Lambda_{TC}$. Accordingly, the PCDC implies $(M_{TD}/m)^2 (F_{TD}/m)^2=-4<\theta_\mu^\mu>/m^4 \to const \ne 0$ at criticality limit, where $M_{TD}$ is the mass of TD and $F_{TD}$ the decay constant of TD. We thus conclude that at criticality limit the TD could become a "true (massless) Nambu-Goldstone boson" $M_{TD}/m\to 0$, only when $m/F_{TD}\to 0$, namely getting decoupled, as was the case of "holographic TD" of Haba-Matsuzaki-Yamawaki. The decoupled TD can be a candidate of dark matter.

CONFORMAL HIGGS, OR TECHNI-DILATON — COMPOSITE HIGGS NEAR CONFORMALITY

In contrast to the folklore that Technicolor (TC) is a "Higgsless theory", we shall discuss existence of a composite Higgs boson, Techni-Dilaton (TD), a pseudo-Nambu-Goldstone boson of the scale invariance in the Scale-invariant/Walking/Conformal TC (SWC TC) which generates a large anomalous dimension γm ≃ 1 in a wide region from the dynamical mass [Formula: see text] of the techni-fermion all the way up to the intrinsic scale ΛTC of the SWC TC (analogue of ΛQCD), where ΛTC is taken typically as the scale of the Extended TC scale Λ ETC : Λ TC ≃ Λ ETC ~ 103 TeV (≫ m). All the techni-hadrons have mass on the same order [Formula: see text], which in SWC TC is extremely smaller than the intrinsic scale Λ TC ≃ Λ ETC , in sharp contrast to QCD where both are of the same order. The mass of TD arises from the non-perturbative scale anomaly associated with the techni-fermion mass generation and is typically 500-600 GeV, even smaller than other techni-hadrons of the same order of [Formula: see text], in another contrast to QCD which is believed to have no scalar [Formula: see text] bound state lighter than other hadrons. We discuss the TD mass in various methods, Gauged NJL model via ladder Schwinger-Dyson (SD) equation, straightforward calculations in the ladder SD/ Bethe-Salpeter equation, and the holographic approach including techni-gluon condensate. The TD may be discovered in LHC.

Holographic technidilaton

Technidilaton, a pseudo--Nambu-Goldstone boson of scale symmetry, was predicted long ago in the scale-invariant/walking/conformal technicolor (SWC-TC) as a remnant of the (approximate) scale symmetry associated with the conformal fixed point, based on the conformal gauge dynamics of ladder Schwinger-Dyson (SD) equation with nonrunning coupling. We study the technidilaton as a flavor-singlet bound state of technifermions by including the technigluon condensate (tGC) effect into the previous (bottom-up) holographic approach to the SWC-TC, a deformation of the holographic QCD with ${\ensuremath{\gamma}}_{m}\ensuremath{\simeq}0$ by large anomalous dimension ${\ensuremath{\gamma}}_{m}\ensuremath{\simeq}1$. With including a bulk scalar field corresponding to the gluon condensate, we first improve the operator product expansion of the current correlators so as to reproduce gluonic $1/{Q}^{4}$ term both in QCD and SWC-TC. We find in QCD about 10% (negative) contribution of gluon condensate to the $\ensuremath{\rho}$ meson mass. We also calculate the oblique electroweak $S$-parameter in the presence of the effect of the tGC and find that for the fixed value of $S$ the tGC effects dramatically reduce the flavor-singlet scalar (technidilaton) mass ${M}_{\mathrm{TD}}$ (in the unit of ${F}_{\ensuremath{\pi}}$), while the vector and axial-vector masses ${M}_{\ensuremath{\rho}}$ and ${M}_{{a}_{1}}$ are rather insensitive to the tGC, where ${F}_{\ensuremath{\pi}}$ is the decay constant of the technipion. If we use the range of values of tGC implied by the ladder SD analysis of the nonperturbative scale anomaly in the large ${N}_{f}$ QCD near the conformal window, the phenomenological constraint $S\ensuremath{\simeq}0.1$ predicts the technidilaton mass ${M}_{\mathrm{TD}}\ensuremath{\sim}600\text{ }\text{ }\mathrm{GeV}$ which is within reach of LHC discovery.

S Parameter in the Holographic Walking/Conformal Technicolor

We explicitly calculate the S parameter in entire parameter space of the holographic walking/conformal technicolor (W/C TC), based on the deformation of the holographic QCD by varying the anomalous dimension from γm ≃ 0 through γm ≃ 1 continuously. The S parameter is given as a positive monotonic function of ξ which is fairly insensitive to γm and continuously vanishes as S ∼ ξ2 → 0 when ξ → 0, where ξ is the vacuum expectation value of the bulk scalar field at the infrared boundary of the 5th dimension z = zm and is related to the mass of (techni-) ρ meson (Mρ) and the decay constant (ƒπ) as ξ ∼ ƒπ zm ∼ ƒπ / Mρ for ξ ≪ 1. However, although ξ is related to the techni-fermion condensate 〈TT〉, we find no particular suppression of ξ and hence of S due to large γm, based on the correct identification of the renormalization-point dependence of 〈TT〉 in contrast to the literature. Then we argue possible behaviors of ƒπ/Mρ as 〈TT〉 → 0 near the conformal window characterized by the Banks-Zaks infrared fixed point in more explicit dynamics with γm ≃ 1. It is a curious coincidence that the result from ladder Schwinger-Dyson and Bethe-Salpeter equations well fits in the parameter space obtained in this paper. When ƒπ/Mρ → 0 is realized, the holography suggests a novel possibility that ƒπ vanishes much faster than the dynamical mass m does.

Topcolor breaking through boundary conditions

The nontrivial boundary conditions (BC's) for the topcolor breaking are investigated in the context of the TeV-scale extra dimension scenario. In the gauge symmetry breaking mechanism via the BC's we do not need to incorporate a dynamical mechanism for the topcolor breaking into the model. Moreover, the topcolor breaking can be realized without introducing explicitly a (composite) scalar field. We present a six dimensional model where the top and bottom quarks in the bulk have the topcolor charge while the other quarks in the bulk do not. We also put the electroweak gauge interaction in the six dimensional bulk. The bottom quark condensation is naturally suppressed owing to the powerlike running of the bulk $\mathrm{U}(1{)}_{Y}$ interaction, so that only the top condensation is expected to take place. We explore such a possibility based on the ladder Schwinger-Dyson equation and show the cutoff to make the model viable.

Gauged Nambu-Jona-Lasinio model with extra dimensions: Phase structure and renormalizability

We investigate phase structure of the D (> 4)-dimensional gauged Nambu-Jona-Lasinio (NJL) model with $\delta(=D-4)$ extra dimensions compactified on TeV scale, based on the improved ladder Schwinger-Dyson (SD) equation in the bulk. We assume that the bulk running gauge coupling in the SD equation for the SU(N_c) gauge theory with N_f massless flavors is given by the truncated Kaluza-Klein effective theory and hence has a nontrivial ultraviolet fixed point (UVFP). We find the critical line in the parameter space of two couplings, the gauge coupling and the four-fermion coupling, which is similar to that of the gauged NJL model with fixed (walking) gauge coupling in four dimensions. It is shown that in the presence of such walking gauge interactions the four-fermion interactions become ``nontrivial'' even in higher dimensions, similarly to the four-dimensional gauged NJL model. Such a nontriviality holds only in the restricted region of the critical line (``nontrivial window'') with the gauge coupling larger than a non-vanishing value (``marginal triviality (MT)'' point), in contrast to the four-dimensional case where such a nontriviality holds for all regions of the critical line except for the pure NJL point. In the nontrivial window the renormalized effective potential yields a nontrivial interaction which is conformal invariant. The exisitence of the nontrivial window implies ``cutoff insensitivity'' of the physics prediction in spite of the ultraviolet dominance of the dynamics. In the formal limit D -> 4, the nontrivial window coincides with the known condition of the nontriviality of the four-dimensional gauged NJL model, $9/(2N_c) < N_f - N_c < 9/2 N_c$.

Chiral Phase Transition of Bulk Abelian Gauge Theories in the Randall-Sundrum Background

The chiral phase transition of strong-coupling Abelian gauge theories is investigated in the brane world. It is assumed that gauge boson propagates in an extra dimension, i.e. bulk gauge theories. The phase structure is analytically evaluated by using the low-energy effective theories. We also numerically solve the ladder Schwinger-Dyson equation for the full fermion propagator including Kaluza-Klein (KK) excitation modes of the gauge field. It is found that the chiral phase transition is of the second order, and the critical value of the coupling constant is obtained. The extra dimension has a large influence on the chiral phase transition for the Randall-Sundrum (RS) brane world. It is studied how the number of KK modes affect the chiral phase transition.

Top mode standard model with extra dimensions

We critically examine a version of the top mode standard model recently cast in extra dimensions by Arkani-Hamed, Cheng, Dobrescu, and Hall, based on the (improved) ladder Schwinger-Dyson (SD) equation for the D-(=6,8-)-dimensional gauge theories. We find that the bulk QCD cannot have larger coupling beyond the non-trivial ultraviolet (UV) fixed point, the existence of which is supported by a recent lattice analysis. The coupling strength at the fixed point is evaluated by using the one-loop renormalization group equation. It is then found that, in a version with only the third family (as well as the gauge bosons) living in the D-dimensional bulk, the critical (dimensionless) coupling for dynamical chiral symmetry breaking to occur is larger than the UV fixed point of the bulk QCD coupling for D=6, while smaller for D=8. We further find that the improved ladder SD equation in D dimensions has an approximate scale-invariance due to the running of the coupling and hence has an essential-singularity scaling of the ``conformal phase transition,'' similar to Miransky scaling in the four-dimensional ladder SD equation with a nonrunning coupling. This essential-singularity scaling can resolve the fine-tuning even when the cutoff (``string scale'') is large. Such a theory has a large anomalous dimension \gamma_m=D/2 - 1 and is expected to be free from the flavor-changing-neutral-current problem as in walking technicolor for D=4. Furthermore, the induced bulk Yukawa coupling becomes finite even at infinite cutoff limit (in the formal sense), similarly to the renormalizability of the gauged Nambu-Jona-Lasinio model. Comments are made on the use of the ``effective'' coupling, which includes finite renormalization effects, instead of the \bar{MS} running coupling in the improved ladder SD equation.

INTRODUCTION TO THE NON-PERTURBATIVE RENORMALIZATION GROUP AND ITS RECENT APPLICATIONS

We introduce the basic ideas and the framework of the non-perturbative renormalization group particularly for pedestrians using elementary examples. First we briefly review the history of the renormalization theory and the renormalization group. We will make it clear that the modern renormalization theory is constructed on the idea of the renormalization group and it is quite a new type of theory in physics. Then we derive the exact non-perturbative renormalization group equation and set up its systematic approximation method. The lowest order approximation called the local potential approximation is applied to scalar theories with the ferromagnetic transition and quantum mechanics with tunneling. We compare our results with other methods, and will show that the non-perturbative renormalization group method is promising since it gives fairly good results already in the lowest order approximation and it does not suffer any divergent series expansion. As a typical application in high energy physics, we analyze the dynamical chiral symmetry breaking in gauge theories and investigate the chiral phase structures. Our new method improves results by the ladder Schwinger–Dyson equation so that the physical results might be less gauge dependent.

Non-Ladder Extended Renormalization Group Analysis of the Dynamical Chiral Symmetry Breaking

The order parameters of dynamical chiral symmetry breaking in QCD, the dynamical mass of quarks and the chiral condensates, are evaluated by numerically solving the Non-Perturbative Renormalization Group (NPRG) equations. We employ an approximation scheme beyond ``the ladder'', that is, beyond the (improved) ladder Schwinger-Dyson equations. The chiral condensates are enhanced compared with the ladder ones, which is phenomenologically favorable. The gauge dependence of the order parameters is fairly reduced in this scheme.

Wilson Renormalization Group Equations for the Critical Dynamics of Chiral Symmetry

The critical dynamics of the chiral symmetry breaking induced by gauge interaction is examined in the Wilson renormalization group framework in comparison with the Schwinger-Dyson approach. We derive the beta functions for the four-fermi couplings in the sharp cutoff renormalzation group scheme, from which the critical couplings and the anomalous dimensions of the fermion composite operators near criticality are immediately obtained. It is also shown that the beta functions lead to the same critical behavior found by solving the so-called ladder Schwinger-Dyson equation, if we restrict the radiative corrections to a certain limited type.

Finite quark mass effects in the improved ladder Bethe-Salpeter amplitudes

We study the finite quark mass effects of the low-energy QCD using the improved ladder Schwinger-Dyson and Bethe-Salpeter equations which are derived in the manner consistent with the vector and axial-vector Ward-Takahashi identities. The non-perturbative mass-independent renormalization allows us to calculate the quark condensate for a non-zero quark mass. We explicitly show that the PCAC relation holds. The key ingredients are the Cornwall-Jackiw-Tomboulis effective action, the generalized Noether current and the introduction of the regularization function to the Lagrangian. The reasonable values of the pion mass, the pion decay constant and the quark condensate are obtained with a rather large $\Lambda_{QCD}$. The pion mass square and the pion decay constant are almost proportional to the current quark mass up to the strange quark mass region. It suggests that the chiral perturbation is applicable up to the strange quark mass region. We study the validity of the approximation often used in solving the Bethe-Salpeter equations too.

Finite quark mass effect in the improved ladder approximation of QCD

We study the finite quark mass effects of the low-energy QCD using the improved ladder Schwinger-Dyson and Bethe-Salpeter equations. We find that the first order chiral perturbation is good in rather wide range of the current quark mass for the low-lying pseudoscalar meson mass.

Solving the Dynamical Chiral Symmetry Breaking by Non-Perturbative Renormalization Group

We introduce the basic notions and the framework of the non-perturbative renormaliza­ tion group (NPRG) particularly for pedestrians using elementary examples. We derive an exact NPRG equation and set up its approximation method. The lowest order approxima­ tion called the local potential approximation (LPA) is applied to scalar theories with the ferromagnetic transition and quantum mechanics with tunneling. We compare our results with other methods, and will show that NPRG is very promising since it does not suffer any divergent series expansion. Then we analyze the dynamical chiral symmetry breaking in gauge theories and get the chiral phase structures. The LPA NPRG improves results by the ladder Schwinger-Dyson equation in a gauge independent way.

Chiral symmetry breaking in the gauged NJL model in curved spacetime.

Using the renormalization group (RG) approach and the equivalency between the class of gauge-Higgs-Yukawa models and the gauged Nambu--Jona-Lasinio (NJL) model, we study the gauged NJL model in curved space-time. The behavior of the scalar-gravitational coupling constant \ensuremath{\xi}(t) in both theories is discussed. The RG-improved effective potential of the gauged NJL model in curved spacetime is found. The curvature at which chiral symmetry in the gauged NJL model is broken is obtained explicitly in a remarkably simple form. The powerful RG-improved effective potential formalism leads to the same results as ladder Schwinger-Dyson equations which have not been formulated yet in curved spacetime. \textcopyright{} 1996 The American Physical Society.

Gauged NJL-model at strong curvature

We investigate the gauged NJL-model in curved space-time using the RG formulation and the equivalency with the gauge Higgs-Yukawa model in a modified 1 N c expansion. The strong curvature induced chiral symmetry breaking is found in the non-perturbative RG approach (presumably equivalent to the ladder Schwinger-Dyson equations). The dynamically generated fermion mass is explicitly calculated and inducing of Einstein gravity is briefly discussed. This approach shows the way to the non-perturbative study of the dynamical symmetry breaking at external fields.