We consider deep inelastic scattering in the 't Hooft model. Being solvable, this model allows us to directly compute the moments associated with the cross section at next-to-leading order in the 1/Q^2 expansion. We perform the same computation using the operator product expansion. We find that all the terms match in both computations except for one in the hadronic side, which is proportional to a non-local operator. The basics of the result suggest that a similar phenomenon may occur in four dimensions in the large N_c limit.
A derivative expansion technique is developed to compute functional determinants of quadratic operators, non diagonal in spacetime indices. This kind of operators arise in general 't Hooft gauge fixed Lagrangians. Elaborate applications of the developed derivative expansion are presented.
We present a number of exact relations for the heavy quark limit and develop an analytical 1/m_Q expansion for heavy mesons in the 't Hooft model. Among the new results are relation 3μ_π^2=\barΛ^2 - m_{sp}^2+β^2, 1/m_Q corrections to the decay constants, to the kinetic expectation values and 1/m_Q^2 nonperturbative corrections to the B->D amplitude at zero recoil. The properties of the IW functions are addressed and the small velocity sum rules are verified.
Pauli Interference in decays of heavy flavor mesons, a genuinely nonleptonic nonperturbative effect, is considered in the 't Hooft model. Analytic summation of the exclusive decay widths yields the same expression as obtained in the OPE-based approach. Threshold-related violations of local duality in PI are found to be suppressed. A novel case is identified where the OPE effect is completely saturated by a single final state.
We address nonperturbative power corrections to inclusive decay widths of heavy flavor hadrons in the context of the 't Hooft model (two-dimensional QCD at N_c->oo), with the emphasis on the spectator-dependent effects sensitive to the flavor of the spectator. The summation of exclusive widths is performed analytically using the `t Hooft equation. We show that the 1/m_Q expansion of both the Weak Annihilation and Pauli Interference widths coincides with the OPE predictions, to the computed orders. Violation of local duality in the inclusive widths is quantified, and the new example is identifi…
We study in detail deep inelastic scattering in the 't Hooft model. We are able to analytically check current conservation and to obtain analytic expressions for the matrix elements with relative precision O(1/Q^2) for 1-x >> β^2/Q^2. This allows us to compute the electron-meson differential cross section and its moments with 1/Q^2 precision. For the former we find maximal violations of quark-hadron duality, as it is expected for a large N_c analysis. For the latter we find violations of the operator product expansion at next-to-leading order in the 1/Q^2 expansion.
YM and QCD are known to be renormalizable, but not ultraviolet finite, order by order in perturbation theory. It is a fundamental question as to whether YM or QCD are ultraviolet finite, or only renormalizable, order by order in the large-N 't Hooft or Veneziano expansions. We demonstrate that Renormalization Group and Asymptotic Freedom imply that in 't Hooft large-N expansion the S-matrix in YM is ultraviolet finite, while in both 't Hooft and Veneziano large-N expansions the S-matrix in confining QCD with massless quarks is renormalizable but not ultraviolet finite. By the same argument it …
A recently proposed variational mass expansion approach to dynamical chiral symmetry breakdown is reviewed. We briefly explain how a specific integral ansatz over the analytically continued, arbitrary Lagrangian mass parameter, resums the usual variational mass expansion (``delta-expansion".) The construction is then generalized to obtain non-perturbative expressions for the order parameters of the $SU(n_f)_L \times SU(n_f)_R$ breakdown ($n_f =$ 2 or 3), in QCD. Emphasis is put on some general aspects as well as possible limitations of this approach.
We derive the quadratic action of the fluctuations around the classical world sheet underlying the jet quenching from AdS/CFT. After obtaining the correspondence partition function, the expansion of the jet quenching parameter of $\mathcal N=4$ super symmetric Yang-Mills theory is carried out to the sub-leading term in the large 't Hooft coupling $λ$ at a nonzero temperature. The strong coupling corresponds to the semi-classical expansion of the string-sigma model, the gravity dual of the Wilson loop operator, with the sub-leading term expressed in terms of functional determinants of fluctuati…
Resurgent structure of the 't Hooft-Polyakov monopole
Charles University in Prague（布拉格查理大学）；Faculty of Mathematics and Physics, Charles University in Prague（布拉格查理大学数学与物理学院）；Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University in Prague（布拉格查理大学数学与物理学院粒子与核物理研究所）
We obtain a formal solution of an integral equation for $q\bar q$ bound states, depending on a parameter ηwhich interpolates between 't Hooft's (η=0) and Wu's (η=1) equations. We also get an explicit approximate expression for its spectrum for a particular value of the ratio of the coupling constant to the quark mass. The spectrum turns out to be in qualitative agreement with 't Hooft's as long as η\neq 1. In the limit η=1 (Wu's case) the entire spectrum collapses to zero, in particular no rising Regge trajectories are found.
The 't Hooft model (two-dimensional QCD in the limit of large number of colors) is used as a laboratory for exploring various aspects of the heavy quark expansions in the nonleptonic and semileptonic decays of heavy flavors. We perform a complete operator analysis and construct the operator product expansion (OPE) up to terms O(1/m_Q^4), inclusively. The OPE-based predictions for the inclusive widths are then confronted with the "phenomenological" results, obtained by summation of all open exclusive decay channels, one by one. The summation is carried out analytically, by virtue of the 't Hoof…
The $η$ production in the $(n,n')$ bottomonium transitions $Υ(n) \to Υ(n') η, $ is studied in the method used before for dipion heavy quarkonia transitions. The widths $Γ_η(n,n')$ are calculated without fitting parameters for $n=2,3,4,5, n'=1$. Resulting $Γ_η(4,1)$ is found to be large in agreement with recent data. Multipole expansion method is shown to be inadequate for large size systems considered.
In the framework of the chiral perturbation theory we computed two- and three-loop corrections to the pion parton distributions which posses $δ$-function singularities at x->0. This calculation explicitly demonstrates that in the region of small x ~ mpi^2/(4 pi F)^2 standard ChPT breaks down and one needs resummation of all orders. We give an example of such resummation.
We address numerical aspects of local quark-hadron duality using the example of the exactly solvable 't Hooft model, two-dimensional QCD with N_c --> infinity. The primary focus of these studies is total semileptonic decay widths relevant for extracting |V_{cb}| and |V_{ub}|. We compare the exact channel-by-channel sum of exclusive modes to the corresponding rates obtained in the standard 1/m_Q expansion arising from the Operator Product Expansion. An impressive agreement sets in unexpectedly early, immediately after the threshold for the first hadronic excitation in the final state. Yet even …
In a recent paper, Hagelstein and Pascalutsa examine the error associated with an expansion of proton structure corrections to the Lamb shift in terms of moments of the charge distribution. They propose a small modification to a conventional parameterization of the proton's charge form factor and show that this can resolve the proton radius puzzle. However, while the size of the "bump" they add to the form factor is small, it is large compared to the total proton structure effects in the initial parameterization, yielding a final form factor that is unphysical. Reducing their modification to t…
Strong background fields require a non-perturbative treatment, which is afforded in QED by the Furry expansion of scattering amplitudes. It has been conjectured that this expansion breaks down for sufficiently strong fields, based on the asymptotic growth of loop corrections with increasing "quantum nonlinearity", essentially the product of field strength and particle energy. However, calculations to date have assumed that the background is constant. We show here, using general plane waves of finite duration, that observables at high quantum nonlinearity scale differently depending on whether …
We study the analytic properties of the 't Hooft coupling expansion of the beta-function at the leading nontrivial large-$N_f$ order for QED, QCD, Super QED and Super QCD. For each theory, the 't Hooft coupling expansion is convergent. We discover that an analysis of the expansion coefficients to roughly 30 orders is required to establish the radius of convergence accurately, and to characterize the (logarithmic) nature of the first singularity. We study summations of the beta-function expansion at order $1/N_f$, and identify the physical origin of the singularities in terms of iterated bubble…
To comply with the critique of the Comment [J. Arrington, arXiv:1602.01461], we consider another modification of the proton electric form factor, which resolves the "proton-radius puzzle". The proposed modification satisfies all the consistency criteria put forward in the Comment, and yet has a similar impact on the puzzle as that of the original paper. Contrary to the concluding statement of the Comment, it is not difficult to find an ad hoc modification of the form factor at low $Q$ that resolves the discrepancy and is consistent with analyticity constraints. We emphasize once again that we …
We quantify a limitation in the usual accounting of the finite-size effects, where the leading $[(Zα)^4]$ and subleading $[(Zα)^5]$ contributions to the Lamb shift are given by the mean-square radius and the third Zemach moment of the charge distribution. In the presence of any non-smooth behaviour of the nuclear form factor at scales comparable to the inverse Bohr radius, the expansion of the Lamb shift in the moments breaks down. This is relevant for some of the explanations of the "proton size puzzle". We find, for instance, that the de Rújula toy model of the proton form factor does not re…
We discuss the bremsstrahlung production of soft real and virtual photons in a quark-gluon plasma at thermal equilibrium beyond the Hard Thermal Loop (HTL) resummation. The physics is controlled by the ratio $Q^2/q_0^2$ of the virtuality to the energy. When $Q^2/q_0^2$ is much smaller than $g^2$, where $g$ is the strong coupling constant, the emission rate is enhanced by a factor $1/g^2$ over the HTL results due to light-cone singularities and the bremsstrahlung is induced by scattering of the quark via both transverse and longitudinal soft gluon exchanges. When $Q^2/q_0^2$ increases, the enha…
A graviton of a nonzero mass and decay width propagates five physical polarizations. The question of interactions of these polarizations is crucial for viability of models of massive/metastable gravity. This question is addressed in the context of the DGP model of a metastable graviton. First, I argue that the well-known breakdown of a naive perturbative expansion at a low scale is an artifact of the weak-field expansion itself. Then, I propose a different expansion -- the constrained perturbation theory -- in which the breakdown does not occur and the theory is perturbatively tractable all th…
In this paper, we investigate operator product expansion for thermal correlation function of the two scalar currents. Due to breakdown of Lorentz invariance at finite temperature, more operators of the same dimension appear in the operator product expansion than at zero temperature. We calculated Wilson coefficients in the short distance expansion and obtain operator product expansion for thermal correlation function in terms of quark condensate, gluon condensate, quark energy density and gluon energy density.
Gauge invariance dictates that a resonance produced from initial state gluons must be produced through a non-renormalizable operator or a loop process. Should such a resonance be discovered, uncovering the dynamics that give rise to its couplings to gluons will be crucial to understanding the nature of the new state. Here we study how the production of this resonance at high transverse momentum in association with one (or more) jets can be used to directly measure the scale of the operator or the mass of the particles in the loop. We use a 750 GeV diphoton resonance as an example application, …
