[Paper Review] Holography and Anomaly Matching for Resonances

D. Son, Naoki Yamamoto|arXiv (Cornell University)|Oct 4, 2010

Nuclear Physics and Applications1 references19 citations

TL;DR

This paper derives a universal holographic relation for the transverse part of triangle anomalies in gauge theories with gravity duals described by Yang-Mills-Chern-Simons theory, linking resonance masses, decay constants, and couplings via sum rules. The key result is a non-perturbative anomaly matching condition for resonances, analogous to the standard anomaly matching for massless modes, which constrains matrix elements of vector and axial currents in soft electromagnetic fields and holds approximately in real QCD.

ABSTRACT

We derive a universal relation for the transverse part of triangle anomalies within a class of theories whose gravity dual is described by the Yang-Mills-Chern-Simons theory. This relation provides a set of sum rules involving the masses, decay constants and couplings between resonances, and leads to the formulas for the matrix elements of the vector and axial currents in the presence of the soft electromagnetic field. We also discuss that this relation is valid in real QCD at least approximately. This may be regarded as the anomaly matching for resonances as an analogue of that for the massless excitations in QCD.

Motivation & Objective

• To establish a universal relation for the transverse part of triangle anomalies in theories with holographic gravity duals, particularly those with chiral symmetry breaking via infrared boundary conditions.
• To derive sum rules connecting resonance parameters—masses, decay constants, and couplings—for vector and axial-vector mesons.
• To extend the concept of anomaly matching beyond massless modes to include massive resonances, providing a holographic analogue of the 't Hooft anomaly matching condition.
• To examine the validity of the derived relation in real QCD, especially at both low and high momentum scales.
• To compute matrix elements of vector and axial currents in the presence of soft electromagnetic fields using the derived sum rules.

Proposed method

• Employing holographic techniques based on the AdS/CFT correspondence, specifically the Yang-Mills-Chern-Simons theory as the gravity dual of a class of strongly coupled gauge theories.
• Deriving the transverse anomaly function $ w_T(Q^2) $ from the holographic action, using the Chern-Simons term to capture triangle anomaly structures.
• Relating $ w_T(Q^2) $ to the difference of vector and axial-vector current correlators $ ilde{ ho}_V(Q^2) - ilde{ ho}_A(Q^2) $, leading to the key equation: $ w_T(Q^2) = rac{N_c}{Q^2} - rac{N_c}{f_ ho^2} ig[ ilde{ ho}_A(Q^2) - ilde{ ho}_V(Q^2)ig] $.
• Applying the relation to compute matrix elements of vector and axial currents in soft electromagnetic fields, yielding expressions proportional to $ g_V/m_V^2 $ and $ g_A/m_A^2 $ for highly excited resonances.
• Using operator product expansion (OPE) and effective field theory techniques to analyze the large-$ Q^2 $ behavior and compare with perturbative and non-perturbative QCD expectations.
• Testing the validity of the relation in the Sakai-Sugimoto model and the bottom-up AdS/QCD model, both of which reproduce low-energy QCD phenomenology.

Experimental results

Research questions

• RQ1 Can a universal relation be derived for the transverse part of triangle anomalies in strongly coupled gauge theories with holographic duals?
• RQ2 How do resonance masses, decay constants, and couplings constrain the transverse anomaly function $ w_T(Q^2) $ in holographic models?
• RQ3 To what extent does the derived anomaly matching relation for resonances hold in real QCD, particularly at low and high momentum scales?
• RQ4 What are the implications of the relation for the matrix elements of vector and axial currents in soft electromagnetic fields?
• RQ5 How does the $ 1/Q^2 $ behavior in the transverse anomaly function affect the structure of sum rules for highly excited resonances?

Key findings

• The paper derives a universal holographic relation: $ w_T(Q^2) = rac{N_c}{Q^2} - rac{N_c}{f_ ho^2} ig[ ilde{ ho}_A(Q^2) - ilde{ ho}_V(Q^2)ig] $, valid for all $ Q^2 $, which generalizes anomaly matching to massive resonances.
• This relation leads to sum rules involving resonance parameters: $ rac{1}{Q^2} ig[ ilde{ ho}_A(Q^2) - ilde{ ho}_V(Q^2) ig] = rac{1}{N_c f_ ho^2} ig( rac{N_c}{Q^2} - w_T(Q^2) ig) $, linking spectral functions to the transverse anomaly.
• For highly excited resonances ($ i,j o ho $), the sum rules take the form $ rac{g_{ ho_i} g_{A_j}^{ ext{eff}}}{m_{A_j}^2 - m_{ ho_i}^2} = 12 rac{g_{ ho_i}}{m_{ ho_i}^2} $ and $ rac{g_{ ho_i}^{ ext{eff}} g_{A_j}}{m_{A_j}^2 - m_{ ho_i}^2} = 12 rac{g_{A_j}}{m_{A_j}^2} $, showing inverse mass-squared dependence.
• Matrix elements of currents in soft electromagnetic fields are derived: $ raket{0|j_ ho^{5a}|V_i^b}^ot = 12 rac{g_{V_i}}{m_{V_i}^2} d^{ab} ilde{F}_{ hoeta} $ for $ i o ho $, with similar form for axial currents.
• The relation is shown to hold approximately in real QCD at both low and high $ Q^2 $, suggesting that the holographic sum rules may describe physical hadron resonances.
• The $ 1/Q^2 $ dependence in the transverse anomaly function leads to distinct sum rule structures compared to perturbative QCD, and the model requires fine-tuning to match the OPE in real QCD at large $ Q^2 $.