en>Chris Howard: not to be confused with: Born rule

2012-10-31T19:00:31Z

not to be confused with: Born rule

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{{intro too long|date=November 2013}}
High energy photons can transform in [[introduction to quantum mechanics|quantum mechanics]] to lepton and [[quark]] pairs, the latter [[hadronization|fragmented]] subsequently to jets of hadrons, i.e. protons, pions etc. At high energies <big>''E''</big> the lifetime ''<big>t</big>'' of such quantum fluctuations of mass <big>''M''</big> becomes nearly macroscopic: <big>''t'' ≈ ''E/M2''</big>; this amounts to flight lengths as large as 1,000 nanometers for electron pairs in a 100 GeV photon beam, and still 10 fermi, i.e. the tenfold radius of a proton, for light hadrons. High energy photon beams have been generated by
photon radiation off electron beams in <big>''e''-''e''+</big> circular beam facilities such as [[PETRA]] at [[DESY]] in Hamburg and [[Large Electron-Positron Collider|LEP]] at [[CERN]] in Geneva. Exceedingly high photon energies may be generated in the future by shining laser
light on TeV electron beams in a [[international linear collider|linear collider]] facility.

The classical technique for analyzing the [[virtual particle]] content of photons is provided by scattering electrons off the photons. In high-energy, large-angle scattering the experimental facility can be viewed as an
electron microscope of very high resolution <big>''Q''</big>, corresponding to the momentum transfer in the scattering process according to Heisenberg's [[uncertainty principle]]. The intrinsic charge structure of the target photon beam is revealed by observing characteristic patterns of the scattered electrons in the final state.

[[File:Fgamfeyndia.png|thumb|Electron-photon scattering generic [[Feynman diagram]].]] The incoming target photon splits into a nearly collinear quark-antiquark pair. The impinging electron is scattered off the quark to large angles, the scatter pattern revealing the internal quark structure of the photon. Quark and antiquark finally transform to hadrons.]]Most exciting is the theoretical analysis of the quark content of the photon, termed "photon structure function". The analysis can be described quantitatively in quantum chromodynamics ([[Quantum chromodynamics|QCD]]), the theory of quarks as constituents of the strongly interacting elementary particles, which are bound together by gluonic forces. The primary splitting of photons to quark pairs, cf. Fig.1, regulates the essential characteristica of the photon structure function, the number and the energy spectrum of the quark constituents within the photon.<ref>T.F. Walsh and P.M. Zerwas, Two-photon processes in the parton model, Physics Letters B44 (1973) 195</ref> QCD refines the picture <ref>E.Witten, Anomalous cross-section for photon - photon scattering in gauge theories, Nuclear Physics B120 (1977) 189</ref><ref>W.A. Bardeen and A.J. Buras, Higher order asymptotic freedom corrections to photon - photon scattering, Physical Review D20 (1979) 166 [E-ibid. D21 (1980) 2041]</ref> by modifying the shape of the spectrum, to order unity unlike the small modifications naively expected as a result of [[asymptotic freedom]].

Quantum mechanics predicts the number of quark pairs in the photon splitting process to increase logarithmically with the resolution ''<big>Q</big>'', and the momenta (approximately) linearly with ''<big>x</big>'', the fraction of the quark momenta within the photon. The characteristic behavior

:<math> F_{2,B}(x,Q^2) = f_B(x)\log{\frac{ Q^2}{\Lambda^2}} + ... </math>
with
:<math> f_B(x) = \frac{3 \alpha}{2\pi}\sum_{q,\bar{q}}e_q^4 x [x^2+(1-x)^2]</math>

is predicted for the photon structure function in the quark model to leading logarithmic behavior, the Sommerfeld [[fine-structure constant]] denoted by α = 1/137 and the fractional quark charges by ''<big>eq</big>''; the factor 3 counting the quark color degrees. Turning
on the radiation of [[gluon]] quanta off quarks in QCD, the quark momenta are reshuffled partly from large to small ''<big>x</big>'' values with increasing resolution. At the same time the radiation is damped moderately due to asymptotic freedom. The delicate interplay between photon splitting and damped gluon radiation re-normalizes the photon structure function
<math>\textstyle F_{2,B}(x,Q^2)\rightarrow F_2(x,Q^2) = f(x) \log{ Q^2 / \Lambda^2 } </math>
to order unity, leaving the logarithmic behavior in the resolution <math>Q</math> untouched apart from superficially introducing the fundamental QCD scale Λ, but tilting the shape of the structure function <math>\textstyle f_B(x)\rightarrow f(x) </math> by damping the momentum spectrum at large ''<big>x</big>''. These characteristica, dramatically different from the proton [[Parton (particle physics)|parton]] density, are unique features of the photon
structure function within QCD. They are the origin of the excitement associated with the photon structure function.<ref>A.J. Buras, Photon structure functions: 1978 and 2005, Acta physica Polonica B37 (2006) 609, arXiv:hep-ph/0512238v2</ref>

While electron scattering off photons maps out the quark spectra, the electrically neutral [[gluons|gluon]] content of the photons can best be analyzed by jet pair production in photon-proton scattering. Gluons as components of the photon may scatter off gluons residing in the proton, and generate two hadron jets in the final state. The complexity of these scattering processes, due to the superposition of many subprocesses, renders the analysis of the gluon content of the photon quite complicated.

The quantitative representation of the photon structure function introduced above is strictly valid only for asymptotically high resolution <big>''Q''</big>, i.e. the logarithm of <big>''Q''</big> being much larger than the logarithm of the quark masses. However, the asymptotic behavior is approached steadily with increasing <big>''Q''</big> for <big>''x''</big> away from zero as demonstrated next. In this asymptotic regime the photon structure function is predicted uniquely in QCD to logarithmic accuracy.

== Experimental analyses ==
Up to now the photon structure function has only been investigated experimentally by electron scattering off a beam of quasi-real photons. The experiments utilize the so-called [[two-photon physics|two-photon]] reactions at electron-positron colliders <big>''e''-''e''+→ ''e''-''e''++ ''h''</big>, where <big>''h''</big> includes all hadrons of the final state. The kinematics chosen is characterized by the electron scattered at large angles and the positron at very small angles thus providing a calculable flux of quasi-real photons (Weizsäcker-Williams approximation). The cross section for electron-photon scattering is then analyzed in terms of photon structure function quite analogously to studies of the nucleon structure in electron-nucleon scattering.

In order to ensure a small virtual mass of the target photon one uses the so-called anti-tagging. Special forward detectors are arranged down to small angles close to the beam pipe. Events with a positron signal in these detectors are eliminated from the analysis. By contrast, events with the positrons travelling undetected down the beam pipe, are accepted. The energy of the emitted quasi-real target photon is unknown. Whereas the four-momentum transfer squared <big>''Q''2</big> can be determined alone from the energy and angle of the scattered electron, <big>''x''</big> has to be calculated from <big>''Q''2</big> and the invariant mass <big>''W''</big> of the hadronic system using <big>''x=''</big><big>''Q''2</big>/<big>''(Q''2+''W'' 2)</big>.
The experimental situation is thus comparable to neutrino-nucleon scattering where the unknown energy of the incoming neutrino also requires the determination of W for calculating the kinematical parameters of the neutrino quark scattering process.

[[File:F2versusxQ245.png|thumb|Fig.2 Photon structure function versus x for Q2=45 GeV2. See text for more information on data and curve.]]
The hadronic system produced in two-photon reactions has in general a rather high momentum along the beam direction resulting in small hadronic scattering angles. This kinematic feature again requires special forward detectors. A high efficiency in reconstructing hadronic events is now also essential. Nevertheless losses of hadronic energy are practically unavoidable and the real hadronic energy is therefore determined using sophisticated unfolding techniques.<ref>Ch. Berger and W.Wagner, Photon-photon reactions, Physics Reports 146 (1987) 1</ref>

The first measurement of the photon structure function has been performed using the detector PLUTO at the DESY storage ring [[PETRA]] <ref>PLUTO Collaboration Ch. Berger et al., First measurement of the photon structure function, Physics Letters B107 (1981) 168</ref> followed subsequently by many investigations at all large electron-positron colliders. The published data are accessible<ref>http://durpdg.dur.ac.uk/hepdata/online/2gamma/2gammaindex.html</ref> in a web site. It is customary to display the structure function in units of the fine-structure constant <big>α</big>. The basic theoretical features discussed above are impressively verified by the data. A comprehensive discussion<ref>R. Nisius, The photon structure from deep inelastic electron photon scattering, Physics Report 332 (2000) 165</ref> can be found in a recent review. The increase of <math>\textstyle F_2(x,Q^2)</math> with <big>''x''</big>, shown in Fig. 2 at <big>''Q2''</big> = 45 GeV<big>2</big>, is obviously quite different from the behaviour of the proton structure function, which falls with rising <big>''x''</big>, and it demonstrates nicely the influence of the photon splitting to quark pairs. The predicted <big>log</big> <big>''Q²''</big> dependence of <math>\textstyle F_2(x,Q^2)</math> is clearly demonstrated in Fig. 3, here plotted for data with 0.3<<big>''x''</big>< 0.5.

[[File:F2versusQ2atx04GRV.png|thumb|Fig.3 Photon structure function for x=0.4 versus Q2. See text for more information on data and curve]]
In both figures the data are compared to theoretical calculations, the red curves representing the analysis of photon structure function data based on the standard higher-order QCD evolution.<ref>M. Glück, E. Reya and A. Vogt, Parton structure of the photon beyond the leading order, Physical Review D45 (1992) 3986</ref> The numerical values were calculated for Λ = 0.2 GeV and 4 quark flavors using the parton density function library (pdflib) available at the CERN computer centre.<ref>H. Plotow-Besch, pdflib Manual,http://cds.cern.ch/record/239783/</ref> The curves approach values close to asymptotic leading-order QCD <ref>L.E. Gordon and J.K. Storrow, The parton distribution functions of the photon and the structure function ''F2(x,Q2)'', Zeitschrift für Physik C56 (1992) 307</ref> for three light quarks (QCD parameter Λ = 0.2 GeV), supplemented by the quark model charm contribution (mass = 1.5 GeV).

One might be tempted to use the data for a precision measurement of Λ. However, while the asymptotic solution of the evolution equations appears superficially very sensitive to Λ, it is actually not defined properly to leading order. This problem solved at higher orders, spurious singularities at small <big>''x''</big> require either technical ad-hoc regularizations or the switching to the evolution from pre-fixed initial conditions at small <big>''Q2''</big>. Both techniques reduce the sensitivity to Λ. Nevertheless, values of

:<math> \alpha_s(m_Z^2) = 0.1198 \pm 0.0028(ex) \pm 0.0040(th) </math>

in analyses of the [[Quantum chromodynamics|QCD coupling]] along these lines <ref>S.Albino, M.Klasen and S.Söldner-Rembold, Strong coupling constant from the photon structure function, Physical Review Letters 89 (2002) 122004, [hep-ph/0205069].</ref> are in satisfactory agreement with other experimental methods.

Thus, the prediction for the number of quarks and their momentum spectrum in high-energy photons, with characteristics very much different from the proton, are borne out nicely by the experimental analyses—a fascinating success of QCD.

== References ==
{{reflist|33em}}

[[Category:Particle physics]]
[[Category:Quantum chromodynamics]]

Born reciprocity - Revision history

en>Chris Howard: not to be confused with: Born rule