Spectral Sum Rules Research Articles

Spectral sum rules and finite volume partition function in gauge theories with real and pseudoreal fermions.

Based on the chiral-symmetry-breaking pattern and the corresponding low-energy effective Lagrangian, we determine the fermion mass dependence of the partition function and derive sum rules for eigenvalues of the QCD Dirac operator in finite Euclidean volume. Results are given for ${\mathit{N}}_{\mathit{c}}$=2 and for the Yang-Mills theory coupled to several light adjoint Majorana fermions. They coincide with those derived earlier in the framework of random matrix theory.

Target independence of the EMC-SMC effect

An approach to deep inelastic scattering is described in which the matrix elements arising from the operator product expansion are factorised into composite operator propagators and proper vertex functions. In the case of polarised μp scattering, the composite operator propagator is identified with the square root of the QCD topological susceptibility √ x′(0) , while the corresponding proper vertex is a renormalisation group invariant. We estimate x′(0) using QCD spectral sum rules and find that it is significantly suppressed relative to the OZI expectation. Assuming OZI is a good approximation for the proper vertex, our predictions, f 0 1 d x g 1 p ( x; Q 2= 10 GeV 2) = 0.143 ± 0.005 and G A 0 = ΔΣ = 0.353 ± 0.052, are in excellent agreement with the new SMC data. This result, together with one confirming the validity of the OZI rule in the η′ radiative decay, supports our earlier conjecture that the suppression in the flavour singlet component of the first moment of gp observed by the EMC-SMC Collaboration is a target-independent feature of QCD related to the U(1) anomaly and is not a property of the proton structure. As a corollary, we extract the magnitude of higher twist effects from the neutron and Bjorken sum rules.

Spectral properties and sum rules of the one-dimensional Hubbard model

We study spectral properties of dynamical structure factors of the one-dimensional Hubbard model at T = 0, using sum rules and the Bethe-ansatz solution. Sum rules help us to obtain information about dynamic quantities from static quantities. We estimate the spectral weight on the lowest-excitation frequency in the dynamical structure factors of spin and charge at small momenta. It is found that there exists a strong or broad continuum spectrum at small momentum above the lowest frequency in the dynamical spin-structure factor of the dilute limit and in the dynamical charge-structure factor near half filling. The spectral weight on the lowest frequency is negligible in these two regions. In other regions, most of the spectral weight concentrates near the lowest frequency.

Slope of the Isgur-Wise function from a QSSR constraint on the [formula omitted] couplings

We derive optimal upper and lower bounds on the slope of the Isgur-Wise function at the zero recoil point in terms of the sum of the YB B couplings, estimated recently from a QCD Spectral Sum Rule (QSSR). The problem is solved by means of a duality theorem in functional optimization. Optimal correlations between the slope and the convexity parameters of the Isgur-Wise function with the same input are also obtained.

SU(3) breaking in [formula omitted] decays and the value of Vcd/Vcs

We determine the SU(3)-breaking effects in the B/D → Klv decays from the ratio of the B/D → Klv over the B/D → πlv semileptonic form factors using a semi-analytic expression based on QCD spectral sum rules. We obtain ƒ+B→K(0)/ƒ+B→π(0) = 1.007 ± 0.020 and ƒ+D→K(0)/ƒ+E→π(0) = 1.102 ±0.007 which combined with the present CLEO data, leads to Vcd/Vcs = 0.323±0.058 and then, to: |Vcs| = 0.72±0.10. The prediction on the ration of mixing angle is (surprisingly) much higher than the value 0.226±0.004 derived from the unitarity constraint on the CKM matrix, while the corresponding value of Vcs is lower. These results can open a window for new physics, though a stringent conclusion needs data with much higher statistics (from, e.g., tau-charm factory) or/and alternative QCD-based estimates of the ratio of the semileptonic form factors.

The spectrum of the Dirac operator near zero virtuality for Nc = 2 and chiral random matrix theory

We study the spectrum of the QCD Dirac operator near zero virtuality for N c = 2. According to a universality argument, it can be described by a random matrix theory with the chiral structure of QCD, but with real matrix elements. Using results derived by Mehta and Mahoux and Nagao and Wadati, we are able to obtain an analytical result for the microscopic spectral density that in turn is the generating function for Leutwyler-Smilga type spectral sum rules.

Spectral sum rules and Selberg's integral formula

Using Selberg's integral formula we derive all Leutwyler-Smilga type sum rules for one and two flavors, and for each of the three chiral random matrix ensembles. In agreement with arguments from effective field theory, all sum rules for N f =1 coincide for the three ensembles. The connection between spectral correlations and the low-energy effective partition function is discussed.

QSSR estimate of the BB parameter at next-to-leading order

We compute the leading α s corrections to the two-point correlator of the O ΔB=2 operator which controls the B 0 B 0 mixing. Using this result within the QCD spectral sum rules approach and some phenomenologically reasonable assumptions in the parametrization of the spectral function, we conclude that the vacuum saturation values B B ⋍ B B ∗ ⋍ 1 are satisfied within 15%.

Formula omitted] couplings, slope of the Isgur-Wise function and improved estimate of Vcb

We estimate the sum of the ϒ B ̄ B couplings using QCD spectral sum rules (QSSR). Our result implies the phenomenological bound ξ′( vv′=1)⩾−1.04 for the slope of the Isgur-Wise function. An analytic estimate of the (physical) slope to two loops within QSSR leads to the accurate value ξ′(vv′=1)⋍−(1.00 ± 0.02) due to the (almost) complete cancellations between the perturbative and non-perturbative corrections at the stability points. Then, we deduce, from the present data, the improved estimate |V cb|⋍( 1.48 ps τ B ) 1 2 (37.3±1.2±1.4)×10 −3 where the first error comes from the data analysis and the second one from the different model parameterizations of the Isgur-Wise function.

Tau decay and spin-one resonances.

The implications of the recent measurement of the decay {tau}{sup {minus}}{r_arrow}{nu}{sub {tau}}{ital {bar K}}{pi}{pi} are analyzed in the resonance dominance picture. With flavor SU(3) symmetry, the observed {ital {bar K}}{sub 1} resonance production in {tau}{sup {minus}}{r_arrow}{nu}{sub {tau}}{ital {bar K}}{pi}{pi} requires {tau}{sup {minus}}{r_arrow}{nu}{sub {tau}}{ital a1}{sup {minus}} dominance in the nonstrange axial-vector channel. Then the spectral function sum rule cannot be saturated with {rho}, {pi}, and {ital a}{sub 1}. The inclusion of the first excited vector meson brings back a consistent picture and improves the {pi}{sup +}-{pi}{sup 0} mass difference calculation.

Hadrons with charm and beauty

By combining potential models and QCD spectral sum rules (QSSR), we discuss the spectroscopy of the $(b\bar c)$ mesons and of the $(bcq)$, $(ccq)$ and $(bbq)$ baryons (${q}\equiv {d}$ or $s$), the decay constant and the (semi)leptonic decay modes of the $B_c$ meson. For the masses, the best predictions come from potential models and read: $M_{B_c} = (6255 \pm 20)$~MeV, $M_{B^*_c} = (6330 \pm 20)$~MeV, $M_{\Lambda(bcu)} = (6.93\pm 0.05)$~GeV, $M_{\Omega(bcs)} = (7.00\pm 0.05)$~GeV, $M_{\Xi^*(ccu)} =(3.63\pm 0.05)$~GeV and $M_{\Xi^*(bbu)} = (10.21\pm 0.05)$~GeV. The decay constant $f_{B_c} = (2.94 \pm 0.21) f_\pi $ is well determined from QSSR and leads to: $\Gamma(B_c \rightarrow \nu_\tau \tau) = (3.0 \pm 0.4)( V_{cb}/0.037 )^2$ $\times 10^{10}$ s$^{-1}$.The uses of the vertex sum rules for the semileptonic decays of the $B_c$ show that the $t$-dependence of the form factors is much stronger than predicted by vector meson dominance. It also predicts the almost equal strength of about 0.30 $\times 10^{10}$ sec$^{-1}$ for the semileptonic rates $B_c$ into $B_s, B^*_s,\eta_c$ and J/$\psi$. Besides these phenomenological results, we also show explicitly how the Wilson coefficients of the $\langle\alpha_s G^2\rangle$ and $\langle G^3\rangle$ gluon condensates already contain the full heavy quark- ($\langle\bar QQ\rangle$) and mixed- ($\langle\bar QGQ\rangle$) condensate contributions in the OPE.}

Spectral sum rules for the Tomonaga-Luttinger model.

In connection with recent publications we discuss spectral sum rules for the Tomonaga-Luttinger model without using the explicit result for the one-electron Green's function. They are usefull in the interpretation of recent high resolution photoemission spectra of quasi-one-dimensional conductors. It is shown that the limit of infinite frequency and band cut\-off do not commute. Our result for arbitrary shape of the interaction potential generalizes an earlier discussion by Suzumura. A general analytical expression for the spectral function for wave vectors far from the Fermi wave vector $k_{F}$ is presented. Numerical spectra are shown to illustrate the sum rules.

Random matrix theory and spectral sum rules for the Dirac operator in QCD

We construct a random matrix model that, in the large- N limit, reduces to the low-energy limit of the QCD partition function put forward by Leutwyler and Smilga. This equivalence holds for an arbitrary number of flavors and any value of the QCD vacuum angle. In this model, moments of the inverse squares of eigenvalues of the Dirac operator obey sum rules, which we conjecture to be universal. In other words, the validity of the sum rules depends only on the symmetries of the theory but not on its details. To illustrate this point we show that the sum rules hold for an interacting liquid of instantons. The physical interpretations is that the way the thermodynamic limit of the spectral density near zero is approached is universal. However, its value, i.e. the chiral condensate, is not.

The value of fB from QSSR

We extract f B from new moments sum rule in the heavy quark effective theory (HQET). We check and improve previous results from Laplace sum rules in HQET by considering all sources of errors including the ones due to the estimated α s 2-term. Averaging these with the ones from the “standard” relativistic sum rules, we deduce the unique QCD spectral sum rules (QSSR) value f B = (1.60 ± 0.32) f π , while in the static limit f B static = (2.07 ± 0.32) f π . The renormalization group invariant coupling in HQET is estimated to be f P ∞ ≅ (0.35 ± 0.07) GeV 3 2 . We find that the linear quark mass dependence of the decay constants is f P √ M P ≅ ( f P √ M P ) ∞ [1 - (0.88 ± 0.18)GeV/ M Q ]. We also show that the moments tend to the Laplace sum rule for M Q →∞.

Baryons with two heavy quarks from QCD spectral sum rules

We study the masses and couplings of baryons made with two heavy quarks using QCD spectral sum rules (QSSR). Coulomb-like 1 ν 3 singularities enhance largely the mixed quark-gluon condensate contributions and modify the usual quark-hadron duality picture in the QSSR analysis.

Determination of the D=2 “operator” from e+t> −data

We present an estimate of the dimension D=2 “operator” induced by large order perturbation theory using two different sum rules for the e + e − into I=1 hadrons total cross-section. We conclude that this “operator” can only affect the existing ϱ-meson spectral sum rules results and the τ-decay semi-inclusive process by an amount respectively less than 12% and 3.6%. This effect is consistent with the estimated perturbative error used previously in the determination of α s from τ-decay.

Tests of the infinite mass effective theory and properties of the charming and beautiful mesons

Using an operator product expansion (OPE) in the inverse of the heavy quark massM Q and a dispersive approach, we rederive theB andB * couplings and the Isgur-Wise function predicted by the infinite mass effective theory (IMET). We find that, at the subtraction pointM Q and using VDM for the spectral function, these observables are governed by the universal $$\left\langle {\bar qq} \right\rangle $$ light quark condensate. We also find that the corrections induced by finite values ofM c andM b are large and are due to the splitting Δb between the ground state and its radial excitation. We review the determinations of the different decay constants, couplings and masses from QCD spectral sum rules (QSSR). Forf B andf D , we conclude the optimum average:f B =(1.59±0.09±0.27)f π andf D =(1.31±0.12)f π, where the main error is due to the quark mass, while the first one inf B is due to the non-perturbative terms and to the choice of the continuum thresholdt c from the onset of sum rule variable stability until thet c -stability regions. We also find that the vector coupling γV satisfies approximately the IMETM Q 3/2 mass dependence scaling law, whilef P obeys in units of GeV:f P/fπ≅(4.6±0.4)M Q −1/2 (1–1.56/M Q +0.88/M Q 2 ), forM Q ≥M c . We also predict the flavour independence:M P -M Q ≅0.6-0.7 GeV andM V -M Q ≅0.7-0.8 GeV of the mass differences between the mesons and the corresponding quark. Finally we find that theB→Dev e form factorf +(0)≅(0.55±0.10) deviates from the naive expectation (f +(0)=1).

Vacuum alignment in composite technicolor standard models

We consider the question of vacuum alignment in the recently proposed “composite technicolor standard models” (CTSMs). In these models, explicit chiral symmetry breaking due to masses of electroweak-singlet “ultrafermions” is communicated to the quarks and leptons by a chiral condensate produced by strong “ultracolor” gauge interactions. In these models the ultracolor vacuum alignment is determined by the competition between ultrafermion masses and “flavor” gauge boson exchange. In order for these models to work, the gauge contribution to the effective potential is assumed to dominate the contribution from ultrafermion masses and stabilize the desired vacuum. Using spectral function sum rules and a vector-meson saturation approximation known to work well in QCD, we determine the gauge contribution to the effective potential for perturbative flavor gauge boson couplings in the limit where the number of ultracolors is large. We find that the flavor gauge interactions do not favor the vacuum required by CTSMs for weak couplings. While realistic models require nonperturbative values of the flavor gauge couplings, we interpret this as evidence against the vacuum required by CTSMs.

Spectra of heavy baryons from QCD spectral sum rules

We study the masses and coupling of the heavy Σ b,c ∗ and Λ b,c baryons within QCD spectral sum rules (QSSR). The Σ b,c ∗ is found to be degenerate with the Σ b,c within the errors of the method, while the Λ b,c is lighter than the Σ b,c.

B semi-leptonic decays from QCD spectral sum rules

We estimate the form factors at zero momentum transfer, the decay rates and the polarization parameters relevant for the B→D, D ∗, π, ϱ semi-leptonic processes, by using two types of relativistic QCD vertex spectral sum rules (QSSR) within stability criteria.