4 pages, 2 figures: To appear in the proceedings of 16th International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2008), London, U.K., 7-11 Apr 2008
We use a spectator framework to investigate the spectral properties of quark-quark-gluon correlators and use this to study gluonic pole matrix elements. Such matrix elements appear in principle both for distribution functions such as the Sivers function and fragmentation functions such as the Collins function. We find that the contribution of the gluonic pole matrix element in fragmentation functions vanishes. This outcome is important in the study of universality for fragmentation functions.
We investigate the gluonic pole matrix element contributing to the first $p_T$ moment of the distribution and fragmentation functions in a spectator model. By performing a spectral analysis, we find that for a large class of spectator models, the contribution of gluonic pole matrix elements is non-zero for the distribution correlators, whereas in fragmentation correlators they vanish. This outcome is important in the study of universality for fragmentation functions.
The non-vanishing of gluonic pole matrix elements can explain the appearance of single spin asymmetries in high-energy scattering processes. We use a spectator framework approach to investigate the spectral properties of quark-quark-gluon correlators and use this to study gluonic pole matrix elements. Such matrix elements appear in principle both for distribution functions such as the Sivers function and fragmentation functions such as the Collins function. We find that for a large class of spectator models, the contribution of the gluonic pole matrix element in fragmentation functions vanishe…
We investigate the spectral properties of quark-quark-gluon correlators and use this to study gluonic pole matrix elements. Such matrix elements appear in principle both for distribution functions such as the Sivers function and fragmentation functions such as the Collins function. We find that for a large class of spectator models, the contribution of the gluonic pole matrix element for fragmentation functions vanishes. This result is important in the study of universality for fragmentation functions.
The gauge-links connecting the parton field operators in the hadronic matrix elements appearing in the transverse momentum dependent distribution functions give rise to T-odd effects. Due to the process-dependence of the gauge-links the T-odd distribution functions appear with different pre-factors. A consequence is that in the description of single spin asymmetries the parton distribution and fragmentation functions are convoluted with gluonic pole cross sections rather than the basic partonic cross sections. In this paper we calculate the gluonic pole cross sections encountered in single spi…
We discuss the effects of so-called gluonic poles in twist-three hadronic matrix elements, as first considered by Qiu and Sterman, in the Drell-Yan process. These effects cannot be distinguished from those of time-reversal odd distribution functions, although time-reversal invariance is not broken by the presence of gluonic poles. Both gluonic poles and time-reversal odd distribution functions can lead to the same single spin asymmetries. We explicitly show the connection between gluonic poles and large distance gluon fields, identify the possible single spin asymmetries in the Drell-Yan proce…
We explore the electromagnetic gauge invariance of the hadron tensor of the Drell-Yan process with one transversely polarized hadron. The special role is played by the contour gauge for gluon fields. The prescription for the gluonic pole in the twist 3 correlator is related to causality property and compared with the prescriptions for exclusive hard processes. As a result of we get the extra contributions, which naively do not have an imaginary phase. The single spin asymmetry for the Drell-Yan process is enhanced by the factor of two.
We explore the electromagnetic gauge invariance of the hadron tensor of the Drell-Yan process with one transversely polarized hadron. Due to the special role of the contour gauge for gluon fields, the prescription for the gluonic pole in the twist 3 correlator can be related to the causality prescriptions for exclusive hard processes. Because of this, we find the extra contributions, which naively do not have an imaginary phase. The single spin asymmetry for the Drell-Yan process is accordingly enhanced by the factor of two.
We investigate the emergence of single spin asymmetries (SSA) in hard processes using transverse momentum dependent (TMD) distribution and fragmentation functions. Specifically, the description of SSA involves time reversal-odd functions. Process-dependence (non-universality) in measurements of SSA can be attributed to the non-trivial gauge link structure in the TMD correlator. Finding the appropriate gauge links, however, also enables us to characterize the non-universality
We explore the electromagnetic gauge invariance of the hadron tensor of the Drell-Yan process with one transversely polarized hadron. The special role is played by the contour gauge for gluon fields. The prescription for the gluonic pole in the twist 3 correlator is related to causality property and compared with the prescriptions for exclusive hard processes. As a result we get the extra contributions, which naively do not have an imaginary phase. The single spin asymmetry for the Drell-Yan process is accordingly enhanced by the factor of two.
Non-perturbative QCD mechanisms are of fundamental importance in strong interaction physics. In particular, the flavor singlet axial anomaly leads to a gluonic pole mechanism which has been shown to explain the $η^{\prime}$ mass, violations of the OZI rule and more recently the proton spin. We show here that the interaction derived from the gluonic pole exchange explains the high momentum transfer behavior of the photoproduction cross sections of vector mesons at JLab energies.
Gluonic pole matrix elements explain the appearance of single spin asymmetries (SSA) in high-energy scattering processes. They involve a combination of operators which are odd under time reversal (T-odd). Such matrix elements appear in principle both for parton distribution functions and parton fragmentation functions. We show that for parton fragmentation functions these gluonic pole matrix elements vanish as a consequence of the analytic structure of scattering amplitudes in Quantum Chromodynamics. This result is important in the study of the universality of transverse momentum dependent (TM…
We discuss the use of time reversal symmetry in the classification of parton correlators. Specifically, we consider the role of (small) intrinsic transverse momenta in these correlators and the determination of the proper color gauge link. The transverse momentum weighted correlators in hard processes can be expressed as a product of universal gluonic pole matrix elements and gluonic pole cross sections.
We discuss single transverse spin asymmetries in the Drell-Yan process originating from so-called gluonic poles in twist-three hadronic matrix elements, as first considered by Qiu and Sterman. Even though time-reversal invariance is not broken, the effects of such poles cannot be distinguished from those of time-reversal odd distribution functions. We show the connection between gluonic poles and large distance gluon fields, in particular we focus on boundary conditions. We identify the possible single spin asymmetries in the Drell-Yan process.
We argue that weighted azimuthal single spin asymmetries in back-to-back jet or pion production in polarized proton-proton scattering can be written as convolutions of universal distribution and fragmentation functions with gluonic pole cross sections as hard functions. Gluonic pole cross sections are gauge-invariant weighted sums of Feynman diagrams. The weight factors are a direct consequence of the (diagram-dependence of) gauge links. The best known consequence of the gauge links is the generation of the Sivers effect that is a source for single-spin asymmetries. Moreover, due to the depend…
We calculate the $O(α_s^2)$ gluonic operator matrix elements for the twist--2 operators, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region $Q^2 \gg m^2$, up to the linear terms in the dimensional parameter $\varepsilon$, ($D= 4 + \varepsilon$). These quantities are required for the description of parton distribution functions in the variable flavor number scheme (VFNS). The $O(α_s^2 \varepsilon)$ terms contribute at the level of the $O(α_s^3)$ corrections through renormalization. We also comment on additional terms, which have to …
We reexamine the scale dependence of the gluonic pole tri-gluon correlations relevant for the transverse single spin asymmetries. Our results differ from those of all previous similar calculations. The corresponding asymptotic behavior of tri-gluon correlations at small x is also briefly discussed.
The $O(α_s^3 n_f T_F^2 C_{A,F})$ terms to the massive gluonic operator matrix elements are calculated for general values of the Mellin variable $N$. These twist-2 matrix elements occur as transition functions in the variable flavor number scheme at NNLO. The calculation uses sum-representations in generalized hypergeometric series turning into harmonic sums. The analytic continuation to complex values of $N$ is provided.
The $O(α_s^3 T_F^2 C_F (C_A))$ contributions to the transition matrix element $A_{gg,Q}$ relevant for the variable flavor number scheme at 3--loop order are calculated. The corresponding graphs contain two massive fermion lines of equal mass leading to terms given by inverse binomially weighted sums beyond the usual harmonic sums. In $x$-space two root-valued letters contribute in the iterated integrals in addition to those forming the harmonic polylogarithms. We outline technical details needed in the calculation of graphs of this type, which are as well of importance in the case of two diffe…
Recent results on the calculation of 3-loop massive operator matrix elements in case of one and two heavy quark masses are reported. They concern the $O(n_f T_F^2 C_{F,A})$ and $O(T_F^2 C_{F,A})$ gluonic corrections, two-mass quarkonic moments, and ladder- and Benz-topologies. We also discuss technical aspects of the calculations.
Applying perturbative QCD methods recently seen to give a good description of the two body hadronic decays of the B meson, we address the question of bound-state effects on the decay B into K* + gamma. Consistent with most analyses, we demonstrate that gluonic penguins, with photonic bremsstrahlung off a quark, change the decay rate by only a few percent. However, explicit off-shell b-quark effects normally discarded are found to be large in amplitude, although in the standard model accidents of phase minimize the effect on the rate. Using an asymptotic distribution amplitude for the K* and ju…
Perturbative QCD relates the single transverse-spin asymmetries (SSAs) for hard processes at large transverse-momentum of produced particle to partonic matrix elements that describe interference between scattering from a coherent quark-gluon pair and from a single quark, generated through twist-3 quark-gluon correlations inside a hadron. When the coherent gluon is soft at the gluonic poles, its coupling to partonic subprocess can be systematically disentangled, so that the relevant interfering amplitude can be derived entirely from the Born diagrams for the scattering from a single quark. We e…
We examine the large-x QCD evolution of the twist-three gluonic-pole strength defining an effective T-odd Sivers function, where evolution of the T-even transverse-spin DIS structure function g2 is multiplicative. The result corresponds to a colour-factor modified spin-averaged twist-two evolution.
We calculate the two-mass QCD contributions to the massive operator matrix element $A_{gg,Q}$ at $\mathcal{O} (α_s^3)$ in analytic form in Mellin $N$- and $z$-space, maintaining the complete dependence on the heavy quark mass ratio. These terms are important ingredients for the matching relations of the variable flavor number scheme in the presence of two heavy quark flavors, such as charm and bottom. In Mellin $N$-space the result is given in the form of nested harmonic, generalized harmonic, cyclotomic and binomial sums, with arguments depending on the mass ratio. The Mellin inversion of the…
It is shown that conventional mass renormalization, when applied to photonic or gluonic corrections to unstable particle propagators, leads to non-convergent series in the resonance region. A solution of this problem, based on the concepts of pole mass and width, is presented. In contrast with the Z case, the conventional on-shell definition of mass for W bosons and unstable quarks contains an unbounded gauge dependence in next-to-leading order. The on-shell and pole definitions of width are shown to coincide if terms of O(Γ^2) and higher are neglected, but not otherwise.
We study meson-meson interactions using an extended $q^2\bar{q}^2(g)$ basis that allows calculating coupling of an ordinary meson-meson system to a hybrid-hybrid one. We use a potential model matrix in this extended basis which at quark level is known to provide a good fit to numerical simulations of a $q^2\bar{q}^2$ system in pure gluonic theory for static quarks in a selection of geometries. We use a combination of resonating group method formalism and Born approximation to include the quark motion using wave functions of a $q\bar{q}$ potential within a cluster. This potential is taken to be…
We study the properties of time evolution of the $K^{0}-\bar{K}^{0} $ system in spectral formulation. Within the one--pole model we find the exact form of the diagonal matrix elements of the effective Hamiltonian for this system. It appears that, contrary to the Lee--Oehme--Yang (LOY) result, these exact diagonal matrix elements are different if the total system is CPT--invariant but CP--noninvariant.
We consider diagrams with up to four t-channel gluons in order to specify gluonic twist-four contributions to deep inelastic structure functions. This enables us to extend the method developed by R.K.Ellis, W.Furmanski, and R.Petronzio (EFP) to the gluonic case. The method is based on low-order Feynman diagrams in combination with a dimensional analysis. It results in explicitly gauge invariant expressions for the factorization of twist-four gluon-operator matrix elements and the corresponding coefficient functions.
We investigate the relative size of various twist-3 quark-gluon correlation functions relevant to single transverse spin asymmetries (SSAs) in a quark-diquark model of the nucleon. We calculate the quark-gluon correlation function $T_{q,F}(x, x)$ that is responsible for the gluonic pole contribution to the SSAs, as well as $T_{q,F}(0, x)$ and $T_{Δ{q}, F}(0, x)$ responsible for the fermionic pole contributions. We find in both cases of a scalar diquark and an axial-vector diquark that at the first nontrivial order only the $\tq(x, x)$ is finite while all other quark-gluon correlation functions…
High energy hadronic scattering processes can be described in terms of partonic scattering processes and parton distribution and fragmentation functions. These are bilocal matrix elements of quark operators. Colour gauge-invariance requires inclusion of a gauge link, which is characterized by its path. For transverse momentum dependent distribution and fragmentation functions, appearing in single spin asymmetries, different paths appear depending on the process. The difference is related to a time-reversal odd gluonic pole matrix element. We discuss briefly the path structure of the gauge link…
We present a consistent analysis of $Υ$ sum rules and $B$-meson semileptonic width in the next-to-next-to-leading order in the strong coupling constant. The analysis is based on the analytical result for the heavy quark vector current correlator near threshold in the second order in perturbative and relativistic expansion around the nonrelativistic Coulomb approximation.
Distribution functions in hard processes can be described by quark-quark correlators, nonlocal matrix elements of quark fields. Color gauge invariance requires inclusion of appropriate gauge links in these correlators. For transverse momentum dependent distribution functions, in particular important for describing T-odd effects in hard processes, we find that new link structures containing loops can appear in abelian and non-abelian theories. In transverse moments, e.g. measured in azimuthal asymmetries, these loops may enhance the contribution of gluonic poles. Some explicit results for the l…
The glueball-to-vacuum matrix elements of local gluonic operators in scalar, tensor, and pseudoscalar channels are investigated numerically on several anisotropic lattices with the spatial lattice spacing ranging from 0.1fm - 0.2fm. These matrix elements are needed to predict the glueball branching ratios in $J/ψ$ radiative decays which will help identify the glueball states in experiments. Two types of improved local gluonic operators are constructed for a self-consistent check and the finite volume effects are studied. We find that lattice spacing dependence of our results is very weak and t…
K-matrix analysis of the $00^{++}$ wave is performed in the channels $ππ,~K\bar K,~ηη$ and $4π$ in the mass region up to 1550 MeV. The fit is based on the following data: $p\bar p ~(at~rest)\to π^0π^0π^0$, $π^0π^0η,~π^0ηη$ [1,2], $πN\to ππN$ [3,4], $πN\to K\bar K N$ [5] and the inelastic cross section of the $ππ$ interaction [6]. Simultaneous analysis of these data confirms the existence of the scalar resonances: $f_0(980),~f_0(1300)$ and $f_0(1500)$, the poles of the amplitude being at the following complex masses (in MeV): $(1008\pm 10)- i(43\pm 5)$, $(1290\pm 25)-i(120\pm 15)$, and $(1497\p…
We study the transition widths of $Υ(10753)$ and $Υ(11020)$ into standard bottomonium under the hypothesis that they correspond to the two lowest laying $1^{--}$ hybrid bottomonium states. We employ weakly coupled potential NRQCD an effective filed theory incorporating the heavy quark and multipole expansions. We consider the transitions generated by the leading order and next-to-leading order singlet-octet operators. In the multipole expansion the heavy quark matrix elements factorize from the production of light-quark mesons by gluonic operators. For the leading order operator we compute the…
We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and $\bar R$ of SU($N_c$). The cases in which $R$ is the fundamental representation and the symmetric and antisymmetric rank-2 tensor representation are considered. We calculate Padé approximants to the beta functions for these theories in the $\overline{\rm DR}$ scheme up to four-loop order for the gluonic theory and up to three-loop order for the theories with ma…
A model-independent analysis of collinear three-parton correlation functions for fragmentation is performed. By investigating their support properties it is shown, in particular, that the so-called partonic pole matrix elements vanish. This sheds new light on the understanding of transverse single spin asymmetries in various hard semi-inclusive reactions. Moreover, it gives additional strong evidence for the universality of transverse-momentum-dependent fragmentation functions.
In covariant gauges, the gluonic mass gap in Yang-Mills theory manifests itself in the basic observation that the massless pole in the perturbative gluon propagator disappears in nonperturbative calculations, but the origin of this behavior is not yet fully understood. We summarize a recent study of the respective dynamics with Dyson-Schwinger equations in Landau-gauge Yang-Mills theory. We identify the parameter that distinguishes the massive Yang-Mills regime from the massless decoupling solutions, whose endpoint is the scaling solution. Similar to the PT-BFM scheme, we find evidence that ma…
We determine the gluonic spectral function $SU(3)$ Yang-Mills theory as well as we found fermionic spectral functions in the strong quenched QED where we found new solutions. Our novel technique provides solutions with the usual branch cut for propagators while not showing any pole structure at the first Riemann sheet (identical with entire complex plane) of complex square of momentum. Implications and further utilizations are briefly adressed for QCD and Standard model calculations.
We develop a formalism for computing inclusive production cross sections of heavy quarkonia based on the nonrelativistic QCD and the potential nonrelativistic QCD effective field theories. Our formalism applies to strongly coupled quarkonia, which include excited charmonium and bottomonium states. Analogously to heavy quarkonium decay processes, we express nonrelativistic QCD long-distance matrix elements in terms of quarkonium wavefunctions at the origin and universal gluonic correlators. Our expressions for the long-distance matrix elements are valid up to corrections of order $1/N_c^2$. The…
Isospin-breaking effects, in particular those associated with electroweak-penguin contributions and pi0-eta,eta' mixing, have long been known to affect the Standard Model prediction of Re(epsilon'/epsilon) in a significant manner. We have found an heretofore unconsidered isospin-violating effect of importance; namely, the u-d quark mass difference can spawn |Delta I|=3/2 components in the matrix elements of the gluonic penguin operators. Using chiral perturbation theory and the factorization approximation for the hadronic matrix elements, we find within a specific model for the low-energy cons…
We derive the scale dependence of twist-3 quark-gluon operators, or ETQS matrix elements, at one-loop. These operators are used to factorize transverse single spin asymmetries, which are studied intensively both in experiment and theory. The scale dependence of two special cases are particularly interesting. One is of soft-gluon-pole matrix elements, another is of soft-quark-pole matrix elements. From our results the evolutions in the two cases can be obtained. A comparison with existing results of soft-gluon-pole matrix elements is made.
We compute the relativistic corrections to the color-singlet contribution to gluon fragmentation into a J/psi at relative order v^4, making use of the nonrelativistic QCD (NRQCD) factorization approach. The corresponding full-QCD process exhibits infrared divergences that manifest themselves as single and double poles in epsilon in 4-2epsilon dimensions. We isolate the infrared-divergent contributions and treat them analytically. In the matching of full QCD to NRQCD, the pole contributions are absorbed into long-distance NRQCD matrix elements. The renormalizations of the ultraviolet divergence…
The use of Pade approximants for the description of QCD matrix elements is discussed in this talk. We will see how they prove to be an extremely useful tool, specially in the case of resonant amplitudes. It will allow the inclusion of high-energy Euclidian data to improve the determination of low-energy properties, such as the quadratic vector radius. This does not mean that the rational approximations can be arbitrarily employed for the extraction of any desired hadronic parameter. A discussion about the validity, limitations and possible issues of the Pade analysis is carried on along the pa…
We consider the exclusive $P_{\ell 4}$ decays, $P\to (P_1P_2)_V\ell ν_{\ell}$, where the subindex $V$ means that the invariant mass of the pseudoscalar pair is taken within a small window around the mass of the vector meson $V$. Pole contributions beyond the dominant $P\to V(\to P_1P_2)\ellν_{\ell}$ amplitude of $P_{\ell 4}$ decays are identified, which, in turn, affects the determination of the CKM matrix elements $|V_{qq'}|$. We evaluate the effects of those contributions in the extraction of bottom and charm quark mixings. An application to $B\to (ππ)_ρ\ell ν_{\ell}$ data from Belle collabo…
The key quantity of the heavy quark theory is the quark mass $m_Q$. Since quarks are unobservable one can suggest different definitions of $m_Q$. One of the most popular choices is the pole quark mass routinely used in perturbative calculations and in some analyses based on heavy quark expansions. We show that no precise definition of the pole mass can be given in the full theory once non-perturbative effects are included. Any definition of this quantity suffers from an intrinsic uncertainty of order $\Lam /m_Q$. This fact is succinctly described by the existence of an infrared renormalon gene…
We present a calculation of the next-to-leading-order QCD and electroweak corrections to WWW production with leptonically decaying $\mathrm{W}$ bosons at the LHC, fully taking into account off-shell contributions, intermediary resonances, and spin correlations. The contributions of the quark-antiquark-induced electroweak correction to typical fiducial cross sections at the LHC are of the order of 5-8% and grow to tens of percent in the high-energy tails of distributions. We observe strong cancellations among the positive quark-photon and negative quark-antiquark-induced electroweak corrections…
We calculate the matrix elements of the gluonic contributions to the energy-momentum tensor for a pion of mass 600 < Mpi < 1100 MeV in quenched lattice QCD. We find that gluons contribute (37 +/- 8 +/- 12)% of the pion's light cone momentum. The bare matrix elements corresponding to the trace anomaly contribution to the pion mass are also obtained. The discretizations of the energy-momentum tensor we use have other promising applications, ranging from calculating the origin of hadron spin to QCD thermodynamics.
We report on our latest results in the calculation of the two--mass contributions to 3--loop operator matrix elements (OMEs). These OMEs are needed to compute the corresponding contributions to the deep-inealstic scattering structure functions and to generalize the variable flavor number scheme by including both charm and bottom quarks. We present the results for the non-singlet and $A_{gq,Q}$ OMEs, and compare the size of their contribution relative to the single mass case. Results for the gluonic OME $A_{gg,Q}$ are given in the physical case, going beyond those presented in a previous public…
We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,μ^2)$ and $ΔA_{gg,Q}(x,μ^2)$, at three-loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$-space and iterated integrals over square root-valued alphabets in momentum fraction $x$-space. We derive the necessary analytic relations for the…
We formulate the basic points of the pseudo-PDF approach to the lattice calculation of polarized gluon PDFs. We present the results of our calculations of the one-loop corrections for the bilocal $G_{μα}(z) \widetilde G_{λβ}(0)$ correlator of gluonic fields. Expressions are given for a general situation when all four indices are arbitrary, and also for specific combinations of indices corresponding to three matrix elements that contain the twist-2 invariant amplitude related to the polarized PDF. We study the evolution properties of these matrix elements, and derive matching relations between …
We calculate parton and generalized parton distributions in Minkowski space using a scalar propagator with a pair of complex conjugate poles. Correct spectral and support properties are obtained only after careful analytic continuation from Euclidean space. Alternately the quark distribution function can be calculated from modified cutting rules, which put the intermediate state on its complex mass shells. Distribution functions agree with those resulting from the model's Euclidean space double distribution which we calculate via non-diagonal matrix elements of twist-two operators. Thus one ca…
We explore the role of soft gluon exchange in heavy quarkonium production at large transverse momentum. We find uncanceled infrared poles at NNLO that are not matched by conventional NRQCD matrix elements. We show, however, that gauge invariance and factorization require that conventional NRQCD production operators be modified to include nonabelian phases or Wilson lines. With appropriately modified operators, factorization is restored at NNLO. We also argue that, in the absence of special cancellations, infrared poles at yet higher orders may require the inclusion of additional nonlocal opera…
We study the transition of a heavy quark pair from octet to singlet color configurations at next-to-next-to-leading order (NNLO) in heavy quarkonium production. We show that the infrared singularities in this process are consistent with NRQCD factorization to all orders in the heavy quark relative velocity v. This factorization requires the gauge-completed matrix elements that we introduced previously to prove NNLO factorization to order v ^2.
We calculate results for K to pi and K to 0 matrix elements to next-to-leading order in 2+1 flavor partially quenched chiral perturbation theory. Results are presented for both the Delta I=1/2 and 3/2 channels, for chiral operators corresponding to current-current, gluonic penguin, and electroweak penguin 4-quark operators. These formulas are useful for studying the chiral behavior of currently available 2+1 flavor lattice QCD results, from which the low energy constants of the chiral effective theory can be determined. The low energy constants of these matrix elements are necessary for an und…
