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Glueball and meson propagators of any spin in large-N QCD
High Energy Physics - Theory · Submitted 1 May 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that only the TFT satisfies the ultraviolet and infrared constraints.
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We prove an asymptotic structure theorem for glueball and meson propagators of any spin in large-N QCD and in n=1 SUSY QCD with massless quarks, that determines asymptotically the residues of the poles of the propagators in terms of their anomalous dimensions and of the spectral density of the masses. The asymptotic theorem follows by the severe constraints on the propagators in large-N QCD with massless quarks, or in any large-N confining asymptotically-free gauge theory massless in perturbation theory, that arise by perturbation theory in conjunction with the renormalization group and by the OPE on the ultraviolet side. The asymptotic theorem is inspired by a recently proposed Topological Field Theory (TFT) underlying large-N pure YM, that computes sums of the scalar and of the pseudoscalar correlators satisfying the asymptotic theorem and that implies for the large-N joint scalar and pseudoscalar glueball spectrum exact linearity in the masses squared. On the infrared side we test the prediction of the exact linearity in the TFT by Meyer-Teper lattice numerical computation of the masses of the low-lying glueballs in SU(8) YM, finding accurate agreement. Besides, we employ the aforementioned ultraviolet and infrared constraints in order to compare critically the scalar or pseudoscalar glueball propagators computed in the framework of the AdS String/large-N Gauge Theory correspondence with those of the TFT underlying large-N YM. We find that only the TFT satisfies the ultraviolet and infrared constraints.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove an asymptotic structure theorem for glueball and meson propagators of any spin in large-N QCD and in n=1 SUSY QCD with massless quarks, that determines asymptotically the residues of the poles of the propagators in terms of their anomalous dimensions and of the spectral density of the masses. | contribution 0.74 | 0.95 | 0.84 | asserted 1.00 |
| s1 | The asymptotic theorem follows by the severe constraints on the propagators in large-N QCD with massless quarks, or in any large-N confining asymptotically-free gauge theory massless in perturbation theory, that arise by perturbation theory in conjunction with the renormalization group and by the OPE on the ultraviolet side. | method 0.80 | 0.43 | 0.10 | asserted 1.00 |
| s2 | The asymptotic theorem is inspired by a recently proposed Topological Field Theory (TFT) underlying large-N pure YM, that computes sums of the scalar and of the pseudoscalar correlators satisfying the asymptotic theorem and that implies for the large-N joint scalar and pseudoscalar glueball spectrum exact linearity in the masses squared. | background 0.67 | 0.18 | 0.08 | asserted 0.93 |
| s3 | On the infrared side we test the prediction of the exact linearity in the TFT by Meyer-Teper lattice numerical computation of the masses of the low-lying glueballs in SU(8) YM, finding accurate agreement. | result 0.82 | 0.88 | 0.07 | asserted 1.00 |
| s4 | Besides, we employ the aforementioned ultraviolet and infrared constraints in order to compare critically the scalar or pseudoscalar glueball propagators computed in the framework of the AdS String/large-N Gauge Theory correspondence with those of the TFT underlying large-N YM. | contribution 0.62 | 0.92 | 0.07 | asserted 0.96 |
| s5 | We find that only the TFT satisfies the ultraviolet and infrared constraints. | result 0.99 | 0.94 | 0.42 | asserted 1.00 |
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Original abstract
We prove an asymptotic structure theorem for glueball and meson propagators of any spin in large-N QCD and in n=1 SUSY QCD with massless quarks, that determines asymptotically the residues of the poles of the propagators in terms of their anomalous dimensions and of the spectral density of the masses. The asymptotic theorem follows by the severe constraints on the propagators in large-N QCD with massless quarks, or in any large-N confining asymptotically-free gauge theory massless in perturbation theory, that arise by perturbation theory in conjunction with the renormalization group and by the OPE on the ultraviolet side. The asymptotic theorem is inspired by a recently proposed Topological Field Theory (TFT) underlying large-N pure YM, that computes sums of the scalar and of the pseudoscalar correlators satisfying the asymptotic theorem and that implies for the large-N joint scalar and pseudoscalar glueball spectrum exact linearity in the masses squared. On the infrared side we test the prediction of the exact linearity in the TFT by Meyer-Teper lattice numerical computation of the masses of the low-lying glueballs in SU(8) YM, finding accurate agreement. Besides, we employ the aforementioned ultraviolet and infrared constraints in order to compare critically the scalar or pseudoscalar glueball propagators computed in the framework of the AdS String/large-N Gauge Theory correspondence with those of the TFT underlying large-N YM. We find that only the TFT satisfies the ultraviolet and infrared constraints.
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References
How the literature uses it
18 arXiv papers cite this in 90 sentences
Papers citing it · 73 sentences without a confident label
Contests it 1 paper
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We thank Marco Bochicchio for having tried to convince us, unsuccessfully, in the correctness of his proof.] On the other hand, the above considerations based on the tauberian theorems provide an alternative and mathematically precise way to put the results of [ref] on solid grounds.
A tauberian theorem for the conformal bootstrap · Application to large $N$ gauge theories · contest 0.86
Builds on it 1 paper
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The asymptotic solution is written in terms of the unknown spectral density for fixed spin and in terms of the anomalous dimension of the operators, thus extending to three-point scalar correlators the aforementioned asymptotic structure theorem for two-point correlators [ref] .
An asymptotic solution of large-$N$ $QCD$, and of large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$ · Introduction and results · 1.00
Uses its method or tool 3 papers
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In order to do so, we employ an asymptotic structure theorem [ref] for glueball and meson two-point correlators in 't Hooft large-N limit of massless QCD-like theories, and the associated, but much more general, asymptotic estimates [ref] , that hold both in the complete theory and a fortiori in 't Hooft and Veneziano large-N limits.
The large-N Yang-Mills S-matrix is ultraviolet finite, but the large-N QCD S-matrix is only renormalizable · Large-N YM and massless QCD correlators · 0.97
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Alternatively, the general $RG$ estimates in [ref] can be applied.
Yang-Mills mass gap, Floer homology, glueball spectrum, and conformal window in large-N QCD · Constraints on any solution of the problem of Yang-Mills mass gap · 0.81
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Our computation in Sec. [ref] involves the nonperturbative UV asymptotics of the OPE coefficients of $F^2$ in the coordinate representation, which we establish a priori, up to a constant overall normalization, by means of the Callan-Symanzik (CS) equation in App. [ref], following standard methods worked out in [ref] .
OPE and a low-energy theorem in QCD-like theories · Verifying Eq. [ref] by the RG-improved OPE · 0.99
Compares with it 1 paper
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Indeed, Meyer-Teper result [ref] for the ratios $r_s=m_0^++*/m_0^++$, $r_ps=m_0^-+/m_0^++$, $r_s=r_ps=1.42(11)$, agrees sharply [ref] with Eq.([ref]), that implies $r_s=r_ps=√(2)=1.4142 ⋯$ .
Yang-Mills mass gap, Floer homology, glueball spectrum, and conformal window in large-N QCD · Introduction and Conclusions · 0.89
Cites it as support 4 papers
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Hence $RG$-improved perturbation theory does not contain in fact spectral information [ref] , as perhaps expected, and so it does not our asymptotic solution.
An asymptotic solution of large-$N$ $QCD$ · Introduction · 0.96
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Moreover, since it there exists a scheme in which the asymptotic theorem holds exactly [ref] and not only asymptotically, even for the lowest state, i.e. for the mass gap $m_1$, the coupling $g(m^2_1)$ cannot diverge and must occur in a scheme in which it is finite, for the Kallen-Lehmann representation to make sense.
An asymptotic solution of large-$N$ $QCD$, and of large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$ · Conclusions and outlook · 0.97
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In particular, in the $TFT$ a powerful non-trivial check of the correct asymptotic behavior [ref] of the $ASD$ correlators in Eq.([ref]) and Eq.([ref]) occurs as the consequence of the conspiracy [ref] between the fourth power $g^4(k)$ of the running coupling evaluated on shell, that arises by the anomalous dimension of our surface operators, and the linearity of the spectrum of the masses squared (see section [ref]), that arises by the localization on surface operators of $Z_N$ holonomy (see section [ref]).
Yang-Mills mass gap, Floer homology, glueball spectrum, and conformal window in large-N QCD · Introduction and Conclusions · 1.00
Cites it as background 4 papers
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The literature on the subject includes for instance the reviews in [ref] , and the detailed lattice studies in [ref] , besides suggestive ideas on general properties of the glueball spectra (the literature on which is vast, and deserves being reviewed elsewhere, but see for instance [ref] ).
Holographic glueballs from the circle reduction of Romans supergravity · Introduction · 0.97
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Among other frameworks, we mention the topological string model recently proposed in [ref] .
Non-perturbative results for large-N gauge theories · Introduction and motivations · 0.96
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Another quantity of interest may be the two-point correlator of the RGI operator $g^2GG̃$, with $G̃$ the dual of $G$, see [ref] for a thorough analysis of its renormalization properties.
On the gradient flow of the Lagrangian density in non-Abelian gauge theories · Conclusions · 0.93
What it cites · 73 sentences without a confident label
Contests 7 papers
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Indeed, in the framework of the $AdS$ String/Large-$N$ Gauge Theory correspondence all the glueball propagators, for which we could find presently an explicit computation in the literature, behave as $p^4 log^n(p^2/μ^2)$, with $n=1$ for the Hard- and Soft-Wall models [ref] and $n=3$ for Klebanov-Strassler background [ref] , in contradiction with the universal $RG$ estimate [ref] for $C_0$ Eq.([ref]).
Holographic glueball structure · The ultraviolet test · contest 0.77
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Indeed, in the framework of the $AdS$ String/Large-$N$ Gauge Theory correspondence all the glueball propagators, for which we could find presently an explicit computation in the literature, behave as $p^4 log^n(p^2/μ^2)$, with $n=1$ for the Hard- and Soft-Wall models [ref] and $n=3$ for Klebanov-Strassler background [ref] , in contradiction with the universal $RG$ estimate [ref] for $C_0$ Eq.([ref]).
Investigating AdS/QCD duality through scalar glueball correlators · The ultraviolet test · contest 0.77
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Indeed, in the framework of the $AdS$ String/Large-$N$ Gauge Theory correspondence all the glueball propagators, for which we could find presently an explicit computation in the literature, behave as $p^4 log^n(p^2/μ^2)$, with $n=1$ for the Hard- and Soft-Wall models [ref] and $n=3$ for Klebanov-Strassler background [ref] , in contradiction with the universal $RG$ estimate [ref] for $C_0$ Eq.([ref]).
Glueball correlators as holograms · The ultraviolet test · contest 0.77
Builds on 1 paper
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The construction extends to chiral fermion bilinear operators of massless quarks and to an open spin chain [ref] .
New Integrable Structures in Large-N QCD · The large-$N$ integrable sector of Ferretti-Heise-Zarembo · 0.84
Uses the method or tool of 3 papers
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To show this, we proceed writing the ansatz for the propagator in the coordinate representation, expressing the free propagator in terms of the modified Bessel function $K_1$: ⟨𝒪(x) 𝒪(0) ⟩_conn = ∑_n=1^∞ 1/(2 π)^4 ∫ R_n m^2D-4_n ρ^-1(m_n^2) /p^2+m^2_n e^ ip·xd^4p = 1/4 π^2 x^2 ∑_n=1^∞ R_n m^2D-4_n ρ^-1(m_n^2) √(x^2 m^2_n) K_1( √(x^2 m^2_n) ) Approximating the sum by the integral using the Euler-MacLaurin formula [ref] , we get asymptotically: ⟨𝒪(x) 𝒪(0) ⟩_conn ∼1/4 π^2 x^2 ∫_1^∞ R_n m^2D-4_n ρ^-1(m_n^2) √(x^2 m^2_n) K_1( √(x^2 m^2_n) ) dn = 1/4 π^2 x^2 ∫_m^2_1^∞ R(m) m^2D-4 √(x^2 m^2) K_1( √(x^2 m^2) ) dm^2 We introduce now the dimensionless variable $z^2=x^2 m^2$: ⟨𝒪(x) 𝒪(0) ⟩_conn ∼1/4 π^2 x^2 ∫_m^2_1^∞ R(m) m^2D-4 √(x^2 m^2) K_1( √(x^2 m^2) ) dm^2 = 1/4 π^2 x^2 ∫_m^2_1 x^2^∞ R(z/x) (z^2/x^2)^D-2 z K_1(z) dz^2/x^2 = 1/4 π^2 (x^2)^D ∫_m^2_1 x^2^∞ R(z/x) z^2D-3 K_1(z) dz^2 In the coordin
Meromorphization of Large N QFT · The asymptotic structure theorem for glueball and meson propagators of any spin in large-$QCD$ · 0.87
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Following [ref] firstly we change variables in the $LHS$ of Eq.([ref]) from $k$ to $k+ν$: [equation] For the first integral in the last line we get: [equation] From the first integral it follows the leading asymptotic behavior [ref] provided $γ' ≠1$: [equation] For $γ' = 0$ there is nothing to add.
Glueballs propagators in large-N YM · The asymptotic structure theorem for glueball and meson propagators of any spin in large-$QCD$ · 0.82
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Following [ref] firstly we change variables in the $LHS$ of Eq.([ref]) from $k$ to $k+ν$: [equation] For the first integral in the last line we get: [equation] From the first integral it follows the leading asymptotic behavior [ref] provided $γ' ≠1$: [equation] For $γ' = 0$ there is nothing to add.
Ultraviolet asymptotics of glueball propagators · The asymptotic structure theorem for glueball and meson propagators of any spin in large-$QCD$ · 0.82
Uses data or values from 13 papers
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There are presently three lattice computations, in chronological order, by Lucini-Teper-Wenger [ref] , by Meyer-Teper [ref] and by Lucini-Rago-Rinaldi [ref] for the mass ratios, $r_s=m_0^++*/m_0^++$, $r_ps=m_0^-+/m_0^++$ and $r_2= m_2^++/m_0^++$ in $SU(8)$ $YM$.
Glueball masses in the large N limit · The infrared test · 0.89
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There are presently three lattice computations, in chronological order, by Lucini-Teper-Wenger [ref] , by Meyer-Teper [ref] and by Lucini-Rago-Rinaldi [ref] for the mass ratios, $r_s=m_0^++*/m_0^++$, $r_ps=m_0^-+/m_0^++$ and $r_2= m_2^++/m_0^++$ in $SU(8)$ $YM$.
Glueball Regge trajectories and the Pomeron -- a lattice study -- · The infrared test · 0.89
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There are presently three lattice computations, in chronological order, by Lucini-Teper-Wenger [ref] , by Meyer-Teper [ref] and by Lucini-Rago-Rinaldi [ref] for the mass ratios, $r_s=m_0^++*/m_0^++$, $r_ps=m_0^-+/m_0^++$ and $r_2= m_2^++/m_0^++$ in $SU(8)$ $YM$.
Glueball Regge Trajectories · The infrared test · 0.89
Compares with 6 papers
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These operators have mass dimension $D=2L$ and are homogeneous polynomials of degree $L$ in the $ASD$ curvature $F^-$ [ref] (see subsect.(2.3)): [equation] Ferretti-Heise-Zarembo have computed the one-loop anomalous dimension of $𝒪_2L$ for large $L$ [ref] : [equation] The one-loop anomalous dimension computed within the $TFT$ Eqs.([ref]-[ref]-[ref]) agrees with Ferretti-Heise-Zarembo computation asymptotically for large $L$ and exactly for the $L=2$ ground state, that is the $ASD$ operator that occurs in Eq.([ref]), for which $γ_0 (𝒪_4)=2 β_0$ exactly.
New Integrable Structures in Large-N QCD · Anti-selfdual glueball propagators in a Topological Field Theory underlying large-$N$ $YM$ · 0.94
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As we discuss in subsect.(1.5), in the lattice computation that is presently closer to the continuum limit [This means on the presently larger lattice with the smaller value of the $YM$ coupling.] for $SU(8)$ $YM$, Meyer-Teper [ref] found for the mass ratios of the lowest scalar and pseudoscalar states, $r_s=m_0^++*/m_0^++$ and $r_ps=m_0^-+/m_0^++$, $r_s=r_ps=1.42(11)$ in accurate agreement with the $TFT$.
Glueball Regge Trajectories · Anti-selfdual glueball propagators in a Topological Field Theory underlying large-$N$ $YM$ · 0.91
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As we discuss in subsect.(1.5), in the lattice computation that is presently closer to the continuum limit [This means on the presently larger lattice with the smaller value of the $YM$ coupling.] for $SU(8)$ $YM$, Meyer-Teper [ref] found for the mass ratios of the lowest scalar and pseudoscalar states, $r_s=m_0^++*/m_0^++$ and $r_ps=m_0^-+/m_0^++$, $r_s=r_ps=1.42(11)$ in accurate agreement with the $TFT$.
Glueball Regge trajectories and the Pomeron -- a lattice study -- · Anti-selfdual glueball propagators in a Topological Field Theory underlying large-$N$ $YM$ · 0.91
Cites as background 13 papers
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The structure of the two-point correlators of (scalar) local gauge invariant operators in $QCD$ with massless quarks or in any asymptotically free gauge theory with no perturbative mass scale is severely constrained [ref] by perturbation theory in conjunction with the renormalization group [ref] and by the operator product expansion ($OPE$) [ref] : ∫⟨𝒪_D(x)𝒪_D(0) ⟩_conne^-ip·xd^4x= C_0(p^2)+C_1(p^2) <𝒪_D_1(0)>+ ⋯Assuming multiplicative renormalizability of the operator $𝒪_D$, the coefficient functions $C_0, C_1, ⋯$ in the $OPE$ satisfy the Callan-Symanzik equations (see for example [ref] ): [equation] and: [equation] The solution for $C_0$ is [ref] : [equation] and: [equation] with: γ_𝒪_D(g)= - ∂logZ_𝒪_D/∂logμ=-γ_0(𝒪_D) g^2 +⋯and: β(g)= ∂g/∂logμ=-β_0 g^3 - β_1 g^5 +⋯The power of $p$ is implied by dimensional analysis, $𝒢$ is a dimensionless function that depends only on the running coupl
Ultraviolet asymptotics of glueball propagators · Renormalization group and $OPE$ · 0.84
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In the 't Hooft large-$N$ limit of $QCD$ there is a special sector of the theory discovered by Ferretti-Heise-Zarembo [ref] , that is integrable at one-loop for the anomalous dimensions.
New Integrable Structures in Large-N QCD · The large-$N$ integrable sector of Ferretti-Heise-Zarembo · 0.90
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In the framework of the $AdS$/Large-$N$ Gauge Theory correspondence [ref] we examine Witten supergravity background [ref] , that has been proposed to describe large-$N$ $QCD$, and Klebanov-Strassler supergravity background [ref] , that has been proposed to describe large-$N$ cascading $𝒩$ $=1$ $SUSY$ gauge theories.
Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities · The $AdS$/Gauge Theory correspondence versus the Topological Field Theory · 0.97
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