Heavy Quark Sum Rules Research Articles

[formula omitted], QCD power corrections and αs from e+e− → Hadrons

In this talk, I review the results obtained recently in Ref. [1]. First, we estimate the LO hadronic vacuum polarization contribution to the muon and τ anomalous magnetic moments to be: aμ|l.ohvp=(7036.5±38.9)×10−11, aτ|l.ohvp=(3494.8±24.7)×10−9 (see Table 1) leading to: Δaμ≡aμexp−aμSM=(143±42th±22exp)×10−11 which is about 3σ discrepancy between the SM predictions and experiment. One also finds: α(5)(MZ)|had=(2766.3±4.5)×10−5. Second, we estimate the QCD power corrections up to dimension 20 from the ratio of Laplace sum rule and from τ-like decay high moments (see Table 3). We do not observe any exponential growth of their size which may not favour a duality violation of the spectral function. We obtain 〈αsG2〉=(7.8±3.5)×10−2GeV4 in agreement with the more precise one from heavy quark sum rules, while ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4GeV6 confirms a violation of the four-quark condensate factorization by a factor ρ≃6. Third, using the previous values of the condensates, we re-extract αs from the lowest τ-decay Braaten-SN-Pich (BNP) moment and find to order αs4:αs(Mτ)=0.3081(86)[resp.0.3260(79)]⟶∘αs(MZ)=0.1170(7)[resp.0.1192(7)] for Fixed Order (FO) [resp. Contour Improved (CI)] PT series. We also show that the contributions beyond the Shifman-Vainshtein-Zakharov (SVZ)-expansion are negligible.

QCD parameters and SM-high precisions from e+e−→ Hadrons: Summary

In this talk, I summarize the results obtained recently in Ref. [1] using the PDG 22 compilation of the e+e−→ Hadrons ⊕ the recent CMD3 data for the pion form factor. Using the gluon condensate 〈αsG2〉=(6.49±0.35)×10−2 GeV4 from heavy quark sum rules, the extracted QCD four-quark and dimension eight condensate condensates values are: ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4 GeV6 and d8=(4.3±3.0)×10−2 GeV8 from the ratio R10 of Laplace sum rules to order αs4. Inversely using these estimated values of the condensates, we obtain from R10: 〈αsG2〉=(6.12±0.61)×10−2 GeV4 which leads to the average (6.40±0.30)×10−2 GeV4. Using the lowest τ-like decay moment, the mean result of Fixed Order (FO) and Contour Improved (CI) PT series within the standard OPE is: αs(Mτ)=0.3385(50)(136)syst [resp. 0.3262(37)(78)syst] to order αs4 [resp. αs5] leading to αs(MZ)=0.1207(17)(3) [resp. 0.1193(11)(3)], while the sum of the non-perturbative contribution at Mτ is: δNPV(Mτ)=(2.3±0.2)×10−2. Using the same data, one also obtains the LO hadronic vacuum polarization to the muon and τ anomalous magnetic moments: aμ|l.ohvp=(7036.5±38.9)×10−11,aτ|l.ohvp=(3494.8±24.7)×10−9 which leads to: Δaμ≡aμexp−aμth=(143±42th±22exp)×10−11 indicating about 3σ discrepancy between the SM predictions and experiment. One also finds: α(5)(MZ)|had=(2766.3±4.5)×10−5.

Zc,b-like states from QCD Laplace sum rules at NLO

We review our results on Zc-like states [1] which we complete with the ones on Zb-like states by using relativistic QCD Laplace Sum Rules (LSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections and Lowest Order (LO) QCD condensates up to 〈G3〉. We emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass in the analysis. Our estimates are compiled in Tables 3 and 4. From our results, the observed Zcs(3983) state are good candidate for being Tcs tetramole (superposition of nearly degenerated molecules and tetraquark states having the same quatum numbers JPC and with almost the same couplings to the currents). The Zcs bump around 4100 MeV can be interpreted as a combination of D0⁎Ds1 and Ds0⁎D1 molecules. The physical states Zcs(4000) and Zcs(4220) found by LHCb are too low to be considered as the first radial excitations of Zcs(3983). For the future Zb, Zbs and Zbss, we suggest to scan the region around (10.3 ∼ 10.9) GeV while the 1st radial excitations are about 2.4 GeV above the ground states.

Modern status of heavy quark sum rules in QCD

We briefly report the modern status of heavy quark sum rules (HQSR) based on stability criteria by emphasizing the recent progresses for determining the QCD parameters (αs, mc,b and gluon condensates) where their correlations have been taken into account. The results: αs(MZ) = 0.1181(16)(3), m‾c(m‾c)=1286(16)MeV, m‾b(m‾b)=4202(7)MeV, 〈αsG2〉 = (6.49±0.35)×10−2 GeV4, 〈g3G3〉 = (8.2±1.0)×GeV2〈αsG2〉 and the ones from recent light quark sum rules are summarized in Table 2. One can notice that the SVZ value of 〈αsG2〉 has been underestimated by a factor 1.6, 〈g3G3〉 which is much bigger than the instanton liquid model estimate, while the four-quark condensate which mixes under renormalization is incompatible with the vacuum saturation by a factor (2 ∼ 4). The uses of HQSR for molecules and tetraquarks states are commented.

Doubly hidden 0++ molecules and tetraquarks states from QCD at NLO

Motivated by the LHCb-group discovery of exotic hadrons in the range (6.2 ∼ 6.9) GeV, we present new results for the masses and couplings of 0++ fully heavy (Q‾Q)(QQ‾) molecules and (QQ)(QQ‾) tetraquaks states from relativistic QCD Laplace Sum Rule (LSR) within stability criteria where Next-to-Leading Order (NLO) Factorized (F) Perturbative (PT) corrections is included. As the Operator Product Expansion (OPE) usually converges for d ⩽ 6–8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to 〈G3〉. We also emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass value in the analysis. We compare our predictions in Table 3 with the ones from ratio of Moments (MOM). The broad structure arround (6.2 ∼ 6.9) GeV can be described by the η‾cηc, J/ψ‾J/ψ and χc1‾χc1 molecules or/and Sc‾Sc, Ac‾Ac and Vc‾Vc tetraquarks lowest mass ground states. The narrow structure at (6.8 ∼ 6.9) GeV if it is a 0++ state can be a χc0‾χc0 molecules or/and its analogue Pc‾Pc tetraquark. The χc1‾χc1 predicted mass is found to be below the χc1χc1 threshold while for the beauty states, all of the estimated masses are above the ηbηb and ϒ(1S)ϒ(1S) threshold.

Zc -like spectra from QCD Laplace sum rules at NLO

We present a global analysis of the observed Z_c, Z_cs and future Z_css-like spectra using the inverse Laplace transform (LSR) version of QCD spectral sum rules (QSSR) within stability criteria. Integrated compact QCD expressions of the LO spectral functions up to dimension-six condensates are given. Next-to-Leading Order (NLO) factorized perturbative contributions are included. We re-emphasize the importance to include PT radiative corrections (though numerically small) for heavy quark sum rules in order to justify the (ad hoc) definition and value of the heavy quark mass used frequently at LO in the literature. We also demonstrate that, contrary to a na\"ive qualitative 1/N_c counting, the two-meson scattering contributions to the four-quark spectral functions are numerically negligible confirming the reliability of the LSR predictions. Our results are summarized in Tables III to VI. The Z_c(3900) and Z_cs(3983) spectra are well reproduced by the T_c(3900) and T_cs(3973) tetramoles (superposition of quasi-degenerated molecules and tetraquark states having the same quantum numbers and with almost equal couplings to the currents). The Z_c(4025) or Z_c(4040) state can be fitted with the D*_0D_1 molecule having a mass 4023(130) MeV while the Z_cs bump around 4.1 GeV can be likely due to the (D^*_s0D_1+ D^*_0D_s1) molecules. The Z_c(4430) can be a radial excitation of the Z_c(3900) weakly coupled to the current, while all strongly coupled ones are in the region (5634-6527) MeV. The double strange tetramole state T_css which one may identify with the future Z_css is predicted to be at 4064(46) MeV. It is remarkable to notice the regular mass-spliitings of the tetramoles due to SU(3) breakings M_{T_cs}-M_{T_c}= M_{T_css}-M_{T_cs= (73- 91) MeV.

Doubly-hidden scalar heavy molecules and tetraquarks states from QCD at NLO

Alerted by the recent LHCb discovery of exotic hadrons in the range (6.2 -- 6.9) GeV, we present new results for the doubly-hidden scalar heavy $(\bar QQ) (Q\bar Q)$ charm and beauty molecules using the inverse Laplace transform sum rule (LSR) within stability criteria and including the Next-to-Leading Order (NLO) factorized perturbative and $\langle G^3\rangle$ gluon condensate corrections. We also critically revisit and improve existing Lowest Order (LO) QCD spectral sum rules (QSSR) estimates of the $({ \bar Q \bar Q})(QQ)$ tetraquarks analogous states. In the example of the anti-scalar-scalar molecule, we separate explicitly the contributions of the factorized and non-factorized contributions to LO of perturbative QCD and to the $\langle\alpha_sG^2\rangle$ gluon condensate contributions in order to disprove some criticisms on the (mis)uses of the sum rules for four-quark currents. We also re-emphasize the importance to include PT radiative corrections for heavy quark sum rules in order to justify the (ad hoc) definition and value of the heavy quark mass used frequently at LO in the literature. Our LSR results for tetraquark masses summarized in Table II are compared with the ones from ratio of moments (MOM) at NLO and results from LSR and ratios of MOM at LO (Table IV). The LHCb broad structure around (6.2 --6.7) GeV can be described by the $\overline{\eta}_{c}{\eta}_{c}$, $\overline{J/\psi}{J/\psi}$ and $\overline{\chi}_{c1}{\chi}_{c1}$ molecules or/and their analogue tetraquark scalar-scalar, axial-axial and vector-vector lowest mass ground states. The peak at (6.8--6.9) GeV can be likely due to a $\overline{\chi}_{c0}{\chi}_{c0}$ molecule or/and a pseudoscalar-pseudoscalar tetraquark state. Similar analysis is done for the scalar beauty states whose masses are found to be above the $\overline\eta_b\eta_b$ and $\overline\Upsilon(1S)\Upsilon(1S)$ thresholds.

B→D*at zero recoil revisited

We examine the B -> D* form factor at zero recoil using a continuum QCD approach rooted in the heavy quark sum rules framework. A refined evaluation of the radiative corrections as well as the most recent estimates of higher order power terms together with more careful continuum calculation are included. An upper bound on the form factor of F(1)< 0.93 is derived, based on just the positivity of inelastic contributions. A model-independent estimate of the inelastic contributions shows they are quite significant, lowering the form factor by about 6% or more. This results in an unbiased estimate F(1) \approx 0.86 with about three percent uncertainty in the central value.

New contributions to heavy quark sum rules

We analyse new contributions to the theoretical input in heavy quark sum rules and we show that the general theory of singularities of perturbation theory amplitudes yields the method to handle these specific features. In particular we study the inclusion of heavy quark radiation by light quarks at O(alpha_s^2) and non-symmetric correlators at O(alpha_s^3). Closely related, we also propose a solution to the construction of moments of the spectral densities at O(alpha_s^3) where the presence of massless contributions invalidates the standard approach. We circumvent this problem through a new definition of the moments, providing an infrared safe and consistent procedure.

Low-energy gluon to the vacuum polarization of heavy quarks

We calculate a correction to the electromagnetic current induced by a heavy quark loop. The contribution of this correction to the vacuum polarization function appears at the O(alpha_s^3) order of perturbation theory and has a qualitatively new feature -- its absorptive part starts at zero energy in contrast to other contributions where the absorptive parts start at the two-particle threshold. Our result imposes a constraint on the order n of the moments used in the heavy-quark sum rules, n<4.

New exact heavy quark sum rules

Considering nonforward scattering amplitude off the heavy quark in the Small Velocity limit two exact superconvergent sum rules are derived. The first is the sum rule for spin of the light cloud and leads to the lower bound ϱ 2>3/4 for the slope of the Isgur–Wise function. It also provides the rationale for the fact that the vector heavy flavor mesons are heavier than the pseudoscalar ones. A spin-nonsinglet analogue Σ ̄ of Λ ̄ =M B−m b is introduced.

ASPECTS OF HEAVY-QUARK THEORY

▪ Abstract Recent achievements in heavy-quark theory are critically reviewed. The emphasis is put on those aspects that either did not attract enough attention or cause heated debates in the current literature. Among other topics we discuss (a) basic parameters of the heavy-quark theory; (b) a class of exact QCD inequalities; (c) new heavy quark sum rules; (d) the virial theorem; (e) applications (|Vcb| from the total semileptonic width and from the B → D* transition at zero recoil). In some instances new derivations of previously known results are given, or new aspects addressed. In particular, we dwell on the exact QCD inequalities. Furthermore, a toy model is considered that may shed light on the controversy regarding the value of the kinetic energy of heavy quarks obtained by different methods.