Virtual Photon Structure Function Research Articles

Photon structure function in supersymmetric QCD revisited

We investigate the virtual photon structure function in the supersymmetric QCD (SQCD), where we have squarks and gluinos in addition to the quarks and gluons. Taking into account the heavy particle mass effects to the leading order in QCD and SQCD we evaluate the photon structure function and numerically study its behavior for the QCD and SQCD cases.

Squark contributions to photon structure functions and positivity constraints

Photon structure functions in supersymmetric QCD are investigated in terms of the parton model where squark contributions are evaluated. We calculate the eight virtual photon structure functions by taking the discontinuity of the squark massive one-loop diagrams of the photon-photon forward amplitude. The model-independent positivity constraints derived from the Cauchy-Schwarz inequalities are satisfied by the squark parton model calculation and actually the two equality relations hold for the squark contribution. We also show that our polarized photon structure function ${g}_{1}^{\ensuremath{\gamma}}$ for the real photon leads to the vanishing first moment sum rule, and the constraint $|{g}_{1}^{\ensuremath{\gamma}}|\ensuremath{\le}{F}_{1}^{\ensuremath{\gamma}}$ is satisfied by the real photon. We also discuss a squark signature in the structure function ${W}_{TT}^{\ensuremath{\tau}}$.

Top quark effects on the virtual photon structure function at ILC

We investigated top quark effects on virtual photon structure functions by pQCD. We include the top quark mass effects on the virtual photon structure function with the quark parton model and with the operator product expansion up to the next-to-leading order in QCD. We also consider the threshold effect on the running coupling constant in the calculation to the effective photon structure function with a matching condition. The numerical calculations are investigated in the kinematical region expected at the future international linear collider.

Application of FIRE for the calculation of photon matrix elements

The next-to-next-to-leading order (the order αα s 2 ) corrections to the first moment of the polarized virtual photon structure function g 1 γ (x,Q 2, P 2) are studied in perturbative QCD for the kinematical region Λ2 ≪ P 2 ≪ Q 2, where −Q 2 (−P 2) is the mass square of the probe (target) photon and Λ is the QCD scale parameter. In order to evaluate the two-loop Feynman diagrams for the photon matrix element of the gluon operator, I apply the recently developed algorithm FIRE which reduces a complicated sum of scalar Feynman integrals to a linear combination of a few master integrals. The details of the calculation are presented.

Virtual Photon Structure Functions to NNLO in QCD

The unpolarized virtual photon structure functions $F_2^\gamma(x,Q^2,P^2)$ and $F_L^\gamma(x,Q^2,P^2)$ are investigated in perturbative QCD for the kinematical region $\Lambda^2 \ll P^2 \ll Q^2$, where $-Q^2(-P^2)$ is the mass squared of the probe (target) photon and $\Lambda$ is the QCD scale parameter. In the framework of operator product expansion supplemented by the renormalization group method, the definite predictions are derived for the moments of $F_2^\gamma(x,Q^2,P^2)$ up to the next-to-next-to-leading order (the order $\alpha\alpha_s$) and for the moments of $F_L^\gamma(x,Q^2,P^2)$ up to the next-to-leading order (the order $\alpha\alpha_s$)

NNLO corrections to the first moment sum rule of the polarized virtual photon structure function g^γ_1

NNLO corrections to the first moment sum rule of the polarized virtual photon structure function g^γ_1

ααs2corrections to the first moment of the polarized virtual photon structure functiong1γ(x,Q2,P2)

We present the next-to-next-to-leading order ({alpha}{alpha}{sub s}{sup 2}) corrections to the first moment of the polarized virtual photon structure function g{sub 1}{sup {gamma}}(x,Q{sup 2},P{sup 2}) in the kinematical region {lambda}{sup 2}<<P{sup 2}<<Q{sup 2}, where -Q{sup 2}(-P{sup 2}) is the mass squared of the probe (target) photon and {lambda} is the QCD scale parameter. In order to evaluate the three-loop-level photon matrix element of the flavor singlet axial current, we resort to the Adler-Bardeen theorem for the axial anomaly and we calculate in effect the two-loop diagrams for the photon matrix element of the gluon operator. The {alpha}{alpha}{sub s}{sup 2} corrections are found to be about 3% of the sum of the leading order ({alpha}) and the next-to-leading order ({alpha}{alpha}{sub s}) contributions, when Q{sup 2}=30{approx}100 GeV{sup 2} and P{sup 2}=3 GeV{sup 2}, and the number of active quark flavors n{sub f} is three to five.

Sum rules of polarized photon structure functions revisited

We present the next-to-next-to-leading order ( α α s 2 ) corrections to the first moment of the polarized virtual photon structure function gγ1 ( x , Q 2 , P 2 ) in the kinematical region Q 2 ≫ P 2 ≫ Λ 2 in QCD. We find that the α α s 2 corrections are about 3% of the sum of the leading (α) and the next-to-leading ( α α s ) contributions, when Q 2 = 30 ∼ 100 GeV 2 and P 2 = 3 GeV 2 .

The polarised photon [formula omitted] sum rule at the linear collider and high luminosity B factories

The sum rule for the first moment of the polarised (virtual) photon structure function g 1 γ ( x , Q 2 ; K 2 ) is revisited in the light of proposals for future e + e − colliders. The sum rule exhibits an array of phenomena characteristic of QCD: for real photons ( K 2 = 0 ) electromagnetic gauge invariance constrains the first moment to vanish; the limit for asymptotic photon virtuality ( m ρ 2 ≪ K 2 ≪ Q 2 ) is governed by the electromagnetic U A ( 1 ) axial anomaly and the approach to asymptopia by the gluonic anomaly; for intermediate values of K 2 , it reflects the realisation of chiral symmetry and is determined by the off-shell radiative couplings of the pseudoscalar mesons; finally, like many polarisation phenomena in QCD, the first moment of g 1 γ involves the gluon topological susceptibility. In this paper, we review the original sum rule proposed by Narison, Shore and Veneziano and extend the relation with pseudoscalar mesons. The possibility of measuring the sum rule in future polarised e + e − colliders is then considered in detail, focusing on the International Linear Collider (ILC) and high luminosity B factories. We conclude that all the above features of the sum rule should be accessible at a polarised collider with the characteristics of SuperKEKB.

Target mass effects in polarized virtual photon structure functions

We study target mass effects in the polarized virtual photon structure functions $g_1^\gamma (x,Q^2,P^2)$, $g_2^\gamma (x,Q^2,P^2)$ in the kinematic region $\Lambda^2\ll P^2 \ll Q^2$, where $-Q^2 (-P^2)$ is the mass squared of the probe (target) photon. We obtain the expressions for $g_1^\gamma (x,Q^2,P^2)$ and $g_2^\gamma (x,Q^2,P^2)$ in closed form by inverting the Nachtmann moments for the twist-2 and twist-3 operators. Numerical analysis shows that target mass effects appear at large $x$ and become sizable near $x_{\rm max}(=1/(1+\frac{P^2}{Q^2}))$, the maximal value of $x$, as the ratio $P^2/Q^2$ increases. Target mass effects for the sum rules of $g_1^\gamma$ and $g_2^\gamma$ are also discussed.

Virtual photon structure functions

We discuss the perturbatively calculable virtual photon structure functions. First we present the framework for analyzing the structure functions of the virtual photon and derive a first moment of g 1 γ of the virtual photon. We then investigate the three positivity constraints satisfied by the eight structure functions of the virtual photon.

Virtual photon structure functions and positivity constraints

We study the three positivity constraints on the eight virtual photon structure functions, derived from the Cauchy-Schwarz inequality and which are, hence, model independent. The photon structure functions obtained from the simple parton model show quite different behaviors in a massive or a massless quark case, but they satisfy, in both cases, the three positivity constraints. We then discuss an inequality which holds among the unpolarized and polarized photon structure functions ${F}_{1}^{\ensuremath{\gamma}},$ ${g}_{1}^{\ensuremath{\gamma}},$ and ${W}_{\mathrm{TT}}^{\ensuremath{\tau}},$ in the kinematic region ${\ensuremath{\Lambda}}^{2}\ensuremath{\ll}{P}^{2}\ensuremath{\ll}{Q}^{2},$ where $\ensuremath{-}{Q}^{2}(\ensuremath{-}{P}^{2})$ is the mass squared of the probe (target) photon, and we examine whether this inequality is satisfied by the perturbative QCD results.

Polarized virtual photon structure functiong2γand twist-3 effects in QCD

We investigate the twist-3 effects in the polarized virtual photon structure. The structure functions $g_1^\gamma$ and $g_2^\gamma$ of polarized photon could be experimentally studied in the future polarized $ep$ or $e^+e^-$ colliders. The leading contributions to $g_1^\gamma$ are the twist-2 effects, while another structure function $g_2^\gamma$, which only exists for the virtual photon target, receives not only the twist-2 but also twist-3 contributions. We first show that the twist-3 effects actually exist in the box-diagram contributions and we extract the twist-3 part, which can also be reproduced by the pure QED operator product expansion. We then calculate the non-trivial lowest moment ($n=3$) of the twist-3 contribution to $g_2^\gamma$ in QCD. For large $N_c$ (the number of colors), the QCD analysis of the twist-3 effects in the flavor nonsinglet part of $g_2^\gamma$ becomes tractable and we can obtain its moments in a compact form for all $n$.

Jet cross sections in [formula omitted] at e+e− colliders in NLO QCD

Recent results from NLO QCD calculations for inclusive jet cross sections in γ ∗γ- scattering at e + e − colliders, especially for LEP, are reported. The virtuality Q 2 of the virtual photon is non-zero and can be unlimited large. The virtuality of the second photon is zero and the spectrum is calculated with the Weizsäcker-Williams approximation. Four components of the cross sections have to be distinguished, involving direct and resolved real and virtual photon contributions. Since Q 2 is non-zero, the virtual photon structure function is needed to calculate the contributions involving a resolved virtual photon.

Inclusive jet production with virtual photons in next-to-leading order QCD

We present a next-to-leading order calculation for the virtual photoproduction of one and two jets inep collisions. Soft and collinear singularities are extracted using the phase space slicing method. The collinear photon initial state singularity depends logarithmically on the mass of the virtual photon and is absorbed into the virtual photon structure function. An % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaie% aacaWFnbGaa83uaaaaaaa!37A9! $$\overline {MS} $$ factorization scheme is defined similarly to the real photon case. Numerical results are presented for HERA conditions using the Snowmass jet definition for inclusive single jet and dijet cross sections. We study the dependence of these cross sections on the transverse energies and rapidities of the jets. Finally, we compare the ratio of the experimentally defined resolved and direct cross sections with recent ZEUS data as a function of the photon virtualityP 2.

Overview of the discussions of the photoproduction group

In this résumé we wish to provide an overview of the discussions of the photoproduction working group at the workshop. The first point we wish to make about the five articles which follow and summarize the discussions of the group is that they should be read in conjunction with this note and with the accounts of the three keynote talks, by Maxfield, Kramer and Sjöstrand, which also appear in these proceedings. These three talks, on `Photoproduction events' (Sjöstrand), `Photoproduction of jets' (Kramer) and `Experimental highlights' (Maxfield) provided the initial focus for the discussions of the group. The written versions serve the same purpose for the summaries and also incorporate, where appropriate, some of the results of our discussions. They also include some results from further work which arose out of these discussions. Much of our time was devoted to discussions of the parton distribution functions (PDFs) of the photon, both real and virtual, and how they can be obtained from HERA data. This had many ramifications. The report by Sjöstrand, Storrow and Vogt reviews the current state of knowledge of the photon PDFs and discusses the prospects of improving this knowledge by experiments at HERA. One conclusion that emerges is that a comparison of HERA jet data with NLO QCD calculations could provide tests of the PDFs of the real photon competitive with structure function measurements at LEP2, certainly in the large x region. Examples of the first signs of this are given in Kramer's article. Another conclusion is that the forward electron tagging capabilities at the HERA experiments provide a unique opportunity to study the structure of virtual photons. There have been some theoretical efforts here, which are discussed, but until now, no experimental information. The data that will be available from HERA (some were presented at the workshop) will be of much higher statistical accuracy than LEP2 data from double tag experiments on the virtual photon structure function. Also, as regards the gluon content of the photon, either real or virtual, structure function measurements provide essentially no information, whereas jet production does. One theoretical problem that arises in fixing the photon PDFs is the lack of a momentum sum rule comparable to the hadronic case, where it is invaluable in constraining the gluon density. There is a recent proposal by Frankfurt and Gurvich that there is a momentum sum rule, albeit of a different type, and that, even more interestingly, existing photon PDFs do not satisfy it. We had discussions on this, summarized in the article by Frankfurt and Gurvich. Another problem that arises when trying to obtain photon PDFs from data on the photoproduction of jets is that of multiple interactions, which are discussed in the article by Butterworth et al. As energies are increased, regions of low x and high parton density are explored and at some energy multiple parton interactions must occur in order that unitarity is not violated by the jet cross sections. This article discusses the underlying physics of the models used to incorporate multiple interactions and also discusses how these models are interfaced with Monte Carlo programs of hadronization. The effects on both the total cross sections and jet cross sections are discussed and it is suggested that both models and data are hinting that the effects of multiple interactions might be present at HERA energies. Turning to the physics of the soft pomeron, the article by Donnachie reviews the diffractive production of vector mesons and concludes that we have a consistent picture for the photoproduction of ρ,ω and ϕ, and also for electroproduction of these mesons at fixed target energies. However, cross section data on electroproduction of ρ0 and ϕ at HERA show a steeper energy dependence than that expected from a soft diffractive process, but consistent with QCD based calculations. We now turn to the photoproduction of heavy flavour vector mesons, and in particular the J/ψ. The steeper energy dependence than expected from the soft pomeron of the electroproduction cross section of ρ0 and ϕ is seen also in the photoproduction of J/ψ. While the hard scale in the case of the electroproduction is provided by the virtuality of the photon, Q2, it is the mass of the J/ψ that serves as a hard scale here, large enough to be considered as a hard interaction calculable in QCD. This is discussed in the article of Frankfurt and Levy. We would like to end this introduction by noting that photoproduction at HERA has turned out to be a source to study both soft and hard hadronic interactions and the interplay between them.

Moments of the virtual photon structure function

The photon structure function is a useful testing ground for QCD. It is perturbatively computable apart from a contribution from what is usually called the hadronic component of the photon. There have been many proposals for this nonperturbative part of the real photon structure function. By studying moments of the virtual photon structure function, we explore the extent to which these proposed nonperturbative contributions can be identified experimentally.

QCD calculation of the hadronic part of the photon structure function

The structure function of virtual photon is calculated in QCD by using the operator expansion in the photon virtualityp2. A model is proposed which has correct analytical properties inp2 and represent the results of QCD calculation in terms of contributions of physical states. Using this model to extrapolate to the pointp2=0, the photon structure function as well as its hadronic part is found in the region of intermediatex.