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 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…
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 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.
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.
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 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 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.
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. 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.
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.
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.
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 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 derive exact relations among the polarized gluon and three-gluon distributions in the transversely polarized nucleon which are relevant to single and double spin asymmetries in various hard processes. We also discuss the partonic decomposition of the transverse nucleon spin and point out a potential problem with frame-independence.
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 argue that through the Wilson lines, gauge invariance has as an effect that the hard functions in weighted spin-asymmetries in hadronic scattering processes are given by gluonic pole cross sections, rather than the usual partonic cross sections.
Mechanism for the single transverse spin asymmeties in the pion production, p^\uparrow + p'\to π(\ell_T) + X, and the Λhyperon polarization, p + p' \to Λ^\uparrow (\ell_T)+ X, is investigated in the framework of QCD factorization theorem. We identify all the soft-gluon pole contributions coming from the twist-3 distribution and/or fragmentation functions and show that they give rise to large asymmetries at large x_F.
We discuss the way in which factorisation is partially maintained but nevertheless modified by process-dependent colour factors in hadronic single-spin asymmetries. We also examine QCD evolution of the twist-three gluonic-pole strength defining an effective T-odd Sivers function in the large-x limit, where evolution of the T-even transverse-spin DIS structure function g2 is known to be multiplicative.
Contributions from gluon interactions, which are non-leading in high-energy semi-hard processes, are studied and represented in terms of reggeon exchanges. Unlike the leading gluonic reggeon, related to the BFKL pomeron, the non-leading reggeons are sensitive to the spin and transverse momentum distributions of scattering partons. There are several gluonic reggeons with poles in the vicinity of angular momentum $j=0$ contributing to the perturbative Regge asymptotics of QCD. We extend the high-energy effective action including sub-leading terms which describe these reggeons and their interacti…
We study the question whether the pole-model VMD approach to weak radiative hyperon decays can be made consistent with Hara's theorem and still yield the pattern of asymmetries characteristic of the quark model. It is found that an essential ingredient which governs the pattern of asymmetries is the assumed off-shell behaviour of the parity-conserving 1/2^- 1/2^+ gamma amplitudes. It appears that this behaviour can be chosen in such a way that the pattern characteristic of the quark model is obtained, and yet Hara's theorem satisfied. As a byproduct, however, all parity-violating amplitudes in…
We present the leading unpolarized and single spin asymmetries in inclusive two-hadron production in electron-positron annihilation at the Z pole. The azimuthal dependence in the unpolarized differential cross section of almost back-to-back hadrons is a leading cos(2 phi) asymmetry, which arises solely due to the intrinsic transverse momenta of the quarks. An extensive discussion on how to measure this asymmetry and the accompanying time-reversal odd fragmentation functions is given. A simple estimate indicates that the asymmetry could be of the order of a percent.
We use multi-parton states to examine the leading order collinear factorization of single transverse-spin asymmetries in Drell-Yan processes. Twist-3 operators are involved in the factorization. We find that the so-called soft-gluon-pole contribution in the factorization must exist in order to make the factorization correct. This contribution comes from the corresponding cross-section at one-loop, while the hard-pole contribution in the factorization comes from the cross-section at tree-level. Although the two contributions come from results at different orders, their perturbative coefficient …
A dominant QCD mechanism for the single transverse-spin asymmetry in hard processes is induced by the twist-3 quark-gluon correlations inside nucleon, combined with the soft-gluonic poles to produce the interfering phase for the associated partonic hard scattering. It is shown that the corresponding interfering amplitude can be calculated entirely in terms of the partonic Born cross section which participates in the twist-2 cross section formula for the spin-averaged process.
Recent studies have shown that two widely-used mechanisms for single-transverse-spin asymmetries based on either twist-three contributions or on transverse-momentum-dependent (Sivers) parton distributions become identical in a kinematical regime of overlap. This was demonstrated for the so-called soft-gluon-pole and hard-pole contributions to the asymmetry associated with a particular quark-gluon correlation function in the nucleon. In this paper, using semi-inclusive deep inelastic scattering as an example, we extend the study to the contributions by soft-fermion poles and by another independ…
We study weighted azimuthal single spin asymmetries in hadron-hadron scattering using the diagrammatic approach at leading order and assuming factorization. The effects of the intrinsic transverse momenta of the partons are taken into account. We show that the way in which $T$-odd functions, such as the Sivers function, appear in these processes does not merely involve a sign flip when compared with semi-inclusive deep inelastic scattering, such as in the case of the Drell-Yan process. Expressions for the weighted scattering cross sections in terms of distribution and fragmentation functions f…
We review the theory and present status of the proton spin problem with emphasis on possible gluonic and sea contributions. We discuss the possibility of a J=1 fixed pole correction to the Ellis-Jaffe sum rule for polarized deep inelastic scattering. Fixed poles in the real part of the forward Compton scattering amplitude have the potential to induce subtraction constant corrections to sum rules for photon nucleon scattering.
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…
The most precise measurement of the weak mixing angle sin^2(theta) at LEP is from the forward-backward asymmetry e+e- --> bbbar at the Z-pole. In this note the QED and electroweak radiative corrections to obtain the pole asymmetry from the measured asymmetry for b- and c-quarks have been calculated using ZFITTER, which has been amended to allow a consistent treatment of partial two-loop corrections for the b-quark final asymmetries. A total correction of dAfbb=0.0019+/-0.0002 and dAfbc=0.0064+/-0.0001 has been found, where the remaining theoretical uncertainty is much too small to explain the …
We discuss the hadron tensor of the direct photon production. We study the effects which lead to the soft breaking of factorization by inspection of the corresponding QCD gauge invariance. We emphasize that the special role is played by the contour gauge for gluon fields. We demonstrate that the different prescriptions in the gluonic pole contributions are dictated by the presence of initial or final state interactions in diagrams. Moreover, the different prescriptions that correspond to the initial of final state interactions are needed to ensure the QCD gauge invariance.
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.
We review the determination of the QED coupling at the Z pole, which is a crucial parameter for electroweak theory. We include recent e+e- -> hadron data from Novosibirsk and Beijing to re-evaluate alpha (M_Z^2). We find 1/alpha (M_Z^2) = 128.973 +/- 0.035 or 128.934 +/- 0.040 according, respectively, to whether inclusive or exclusive e+e- -> hadron data are used in the interval 1.4 < \sqrt{s} < 2.1 GeV. The error is mainly due to uncertainties in the data in the low energy region, \sqrt{s} < 2.5 GeV. We find that no advantage is obtained by analytic continuation of the dispersion relation int…
The color gauge-invariant transverse momentum dependent (TMD) quark correlators contain process dependent gauge links in the bilocal matrix elements. In this paper, we split these process dependent correlators into universal TMD correlators, which in turn can be parametrized in universal TMD distribution functions. The process dependence is contained in gluonic pole factors, of which the value is determined by the gauge link. The operator structures of the universal TMD correlators are identified using transverse moments. In this paper, specific results for double transverse weighting of quark…
We examine the effects of the R parity odd renormalizable interactions on flavor changing rates and CP violation asymmetries in the production of fermion-antifermion pairs at $e^-- e^+$ leptonic colliders. The produced fermions may be leptons, down-quarks or up-quarks, and the center of mass energies may range from the Z-boson pole up to $ 1000$ GeV. Off the Z-boson pole, the flavor changing rates are controlled by tree level amplitudes and the CP asymmetries by interference terms between tree and loop level amplitudes. At the Z-boson pole, both observables involve loop amplitudes. The lepton …
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 present a fit to spin asymmetries which gives polarized quark distributions. These functions are closely related to the ones given by the newest Martin, Roberts and Stirling fit for unpolarized structure functions. The integrals of polarized distributions are discussed and compared with the corresponding quantities obtained from neutron and hyperon $β$-decay data. We use the combination of proton and neutron spin asymmetries in order to determine the coefficients of our polarized quark distributions. Our fit shows that phenomenologically there is no need for taking gluonic degrees of freedo…
We investigate the possibility to measure the nonforward gluon distribution function by means of diffractively produced π^+π^- and K^+K^- pairs in polarized lepton nucleon scattering. The resulting cross sections are small and are dominated by the gluonic contribution. We find relatively large spin asymmetries, both for π^+π^- and for K^+K^- pairs.
We compute the transverse single spin asymmetry in light hadron production $p^\uparrow p\to hX$ and $p^\uparrow A\to hX$ including the gluon saturation effect in the unpolarized nucleon/nucleus. In the forward (large-$x_F$) region, the dominant contribution comes from the so-called derivative term associated with the soft gluonic pole. This leads to the cancellation of nuclear effects in $A_N$ which can be tested at RHIC. We also show that the soft fermionic pole disappears in the saturation environment.
Based on the use of contour gauge and collinear factorization, we propose a new set of single spin asymmetry which can be measured in polarized Drell-Yan process by SPD@NICA. We stress that the discussed single transverse spin asymmetries exist owing to the gluon poles manifesting in the twist-3 or twist-2$\otimes$twist-3 parton distributions related to the transverse-polarized Drell-Yan process.
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 consider the pion pole contribution and transversity effects determined by the $H_T$ and $\bar E_T$ Generalized Parton Distributions (GPDs) which are essential in hard pseudoscalar and vector meson leptoproduction. We investigate spin effects in the $ω$ and $ρ^0$ reactions. It is shown that the pion pole contribution is very important in the $ω$ production. Such effects in the $ρ^0$ channel are much smaller. Our results on spin asymmetries and spin density matrix elements in these reactions were found to be in good agreement with HERMES data.
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 prove that twist-3 soft-gluon-pole (SGP) cross section for single spin asymmetries (SSA) is determined by a certain ``primordial'' twist-2 cross section up to kinematic and color factors in the leading order perturbative QCD. In particular, for the processes in which the partonic hard scattering occurs among massless partons, the invariance of the ``primordial'' partonic cross section under scale transformation leads to remarkable simplification of the SGP cross section, reproducing compact form that was recently observed for pion production $p^\uparrow p\to πX$ and direct-photon production…
In the field theoretical description of hadronic scattering processes, single transverse-spin asymmetries arise due to gluon initial and final state interactions. These interactions lead to process dependent Wilson lines in the operator definitions of transverse momentum dependent parton distribution functions. In particular for hadron-hadron scattering processes with hadronic final states this has important ramifications for possible factorization formulas in terms of (non)universal TMD parton distribution functions. In this paper we will systematically separate the universality-breaking part…
We prove that twist-3 soft-gluon-pole (SGP) cross section for single spin asymmetries is determined by a certain ``primordial'' twist-2 cross section up to kinematic and color factors in the leading order perturbative QCD. This unveils universal structure behind the SGP cross sections in a variety of hard processes, and also the special role of the scale invariance in the corresponding primordial cross section, which leads to remarkable simplification of the SGP cross sections for the production of massless particle, such as those for pion production p^\uparrow p \to πX and direct-photon produ…
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 study the single transverse-spin asymmetry for the inclusive direct-photon and single-jet productions in the proton-proton collision based on the twist-3 mechanism in the collinear factorization. Taking into account all the effects from the twist-3 quark-gluon correlation functions inside a transversely polarized proton, we present a prediction for the asymmetries at the typical RHIC kinematics. In both processes we find sizable asymmetries in the forward region of the polarized proton while they are almost zero in the backward region. This implies that if one finds a nonzero asymmetries in…
Unified description of Sivers and Boer-Mulders asymmetries from twist-3 correlations
Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Iowa State University（爱荷华州立大学）
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…
Quark matter at sufficiently high density exhibits color superconductivity, due to attractive gluonic interactions. At lower densities of order $Λ_{QCD}^3$, it has been proposed that instanton generated vertices may play an important role in the Cooper pair formation. We study the renormalization group flow to the Fermi surface of the full set of couplings generated by gluonic and instanton interactions. In earlier work we showed that if the gluonic interactions dominate at the matching scale, their running determines the scale of the Cooper pair formation Delta. Here we consider all possibili…
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.
We study a weighted asymmetry in the azimuthal distribution of photon-jet pairs produced in the process p^\uparrow p --> γjet X with a transversely polarized proton. We focus on the contribution of the Sivers effect only, considering experimental configurations accessible at RHIC-BNL. We show that predictions for the asymmetry, obtained in terms of gluonic-pole cross sections calculable in perturbative QCD, can be tested and clearly discriminated from those based on a generalized parton model, involving standard partonic cross sections. Experimental measurements of the asymmetry will therefore…
We investigate exclusive electroproduction of vector and pseudoscalar mesons at large photon virtuality $Q^2$. These reactions were analyzed within the handbag approach where amplitudes factorize into hard subprocesses and generalized parton distributions (GPDs). The essential role of transversity effects were found in pseudoscalar and light vector meson leptoproduction. These contributions are important at low $Q^2$ and determined by twist-3 effects accompanied by transversity GPDs. The transversity contributions lead to large transverse cross sections for most reactions of pseudoscalar meson…
We study the contribution of the collinear twist-3 quark-gluon correlation to the single transverse-spin asymmetry in direct-photon and single-jet productions in pp collision. At typical RHIC kinematics we find sizable asymmetries in the forward region of the polarized nucleon while they are almost zero in the backward region. We also find the soft-fermion pole contribution is vanishingly small, suggesting the measurement of these asymmetries provides us with an unique opportunity to determine the net soft-gluon pole function.
We analyze the transverse single-spin asymmetry in direct photon production from proton-proton collisions, denoted $A_N^γ$, within collinear twist-3 factorization. We provide a calculation of the contribution due to quark-gluon-quark correlations in the unpolarized proton as well as summarize previous studies on those effects in the polarized proton. Both soft-gluon poles and soft-fermion poles are considered. From this complete result we then estimate $A_N^γ$, including error bands due to uncertainties in the non-perturbative inputs, at kinematics relevant for planned measurements of this obs…
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 constructed two sets of twist-3 correlation functions that are responsible for generating the novel single transverse-spin asymmetry in the QCD collinear factorization approach. We derive evolution equations for these universal three-parton correlation functions. We calculate evolution kernels relevant to the gluonic pole contributions to the asymmetry at the order of $α_s$. We find that all evolution kernels are infrared safe as they should be and have a lot in common to the DGLAP evolution kernels of unpolarized parton distributions. By solving the evolution equations, we explicitly demon…
Transverse momentum dependent (TMD) parton correlators describing the partonic structure of hadrons contain gauge links, required by color gauge invariance. The required gauge links enter in the matrix elements that contain the parton fields and depend on the color flow in the hard process. The correlators are expanded in terms of transverse momentum dependent parton distribution functions, referred to as TMD PDFs, or in short TMDs. In this paper, we introduce gluon TMDs of definite rank, by making an expansion of the TMD gluon correlator with the help of irreducible tensors built from the tra…
Azimuthal asymmetries in high-energy processes, most pronounced showing up in combination with single or double (transverse) spin asymmetries, can be understood with the help of transverse momentum dependent (TMD) parton distribution and fragmentation functions. These appear in correlators containing expectation values of quark and gluon operators. TMDs allow access to new operators as compared to collinear (transverse momentum integrated) correlators. These operators include nontrivial process dependent Wilson lines breaking universality for TMDs. Making an angular decomposition in the azimut…
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…
