Bjorken scaling - Scholarpedia (2024)

Bjorken Scaling refers to an important simplifyingfeature—scaling—of a large class of dimensionless physicalquantities in elementary particles; it strongly suggests thatexperimentally observed strongly interacting particles (hadrons) behaveas collections of point-like constituents when probed at highenergies. A property of hadrons probed inhigh-energy scattering experiments is said to scalewhen it is determined not by the absolute energy of anexperiment but by dimensionless kinematic quantities,such as a scattering angle or the ratio of the energy to a momentum transfer.Because increasing energy implies potentially improved spatial resolution,scaling implies independence of the absolute resolution scale, andhence effectively point-like substructure.Scaling behavior was first proposed by James Bjorken in 1968for the structure functions of deep inelastic scattering of electrons on nucleons. This idea,along with the contemporaneous concept of partons proposed by Feynman,and the experimental discovery of (approximate) scaling behavior,together inspired the idea of asymptotic freedom, and theformulation of Quantum Chromodynamics (QCD)—the modernfundamental theory of strong interactions. Bjorken scaling is, however,not exact; deviations from strict scaling is required in quantum fieldtheory. The QCD theory can predict the detailed form of violations of the scaling behaviorof the relevant physical quantities through the distinctive quantum effectof dimensional transmutation. These predictions have been fullyconfirmed by modern high energy experiments. This theory provides afirm foundation for the intuitive QCD parton picture of elementaryparticles.

Introduction

In 1968, Bjorken proposed (Bjorken 1969) that the structure functions measured in electron-nucleon deep inelastic scattering (DIS, depicted in Fig.Figure 1), \(W_{i}(Q^{2},\nu )\ ,\) mayexhibit scaling behavior in the asymptotic limit,\[\tag{1}\begin{array}{rcl}\lim_{Q^{2}\rightarrow \infty ,\,\nu /Q^{2}\,\mathrm{fixed}}~\nuW_{2}(Q^{2},\nu ) & = & MF_{2}(x) \\\lim_{Q^{2}\rightarrow \infty ,\,\nu /Q^{2}\,\mathrm{fixed}}~W_{1}(Q^{2},\nu) & = & F_{1}(x)\end{array}\]where \(Q^{2}\) represents the squared 4-momentum-transfer vector \(q\) of theexchanged virtual photon, \(\nu =q\cdot p/M\) the energy loss between thescattering electrons (\(l_{1}\) and \(l_{2}\)), \(M\) the target nucleon (\(p\))mass; and the dimensionless variable \(x=Q^{2}/2M\nu \) is the Bjorken\(x\) scaling variable.

The cross section for inclusive DIS of an electron on a nucleon, depicted inFig.Figure 1, is given in terms of the structure functions as\[\tag{2}\sigma _{\mathrm{DIS}}\sim \sigma _{0}\left[ W_{2}+2W_{1}\tan ^{2}(\frac{\theta }{2})\right] \]

where \(\sigma _{0}\) is the well known Mott cross sections for scattering ofa lepton \(l_{1}\) (say an electron) on a point-like charged particle, and \(\theta\) is the scattering angle of the outgoing lepton \(l_{2}\) inthe laboratory frame. This formula resembles that of elastic scattering ofan electron on a nucleon, with \(W_{1,2}\) taking the place of theelectromagnetic form factors of the nucleon, \(F_{i}(Q^{2}), i=1,2.\)

\(F_{i}(Q^{2})\) had been known to fall rapidly as a function of \(Q^{2}\ ,\)reflecting the finite size of the nucleon charge distribution.Therefore, the general expectation for \(\sigma _{\mathrm{DIS}}\) before itsmeasurement was that it would also be a fast falling function of \(Q^{2}\ .\)Bjorken's scaling proposition, expressed by the \(Q\)-independence of theright-hand side of Eq.(1), would contradict thisexpectation. It would imply that the nucleon target appears as a collectionof point-like constituents when probed at very high energies in DIS(implied by the \(Q^{2}\rightarrow \infty\) limit on the left-hand side ofEq.(1). The possible existence of such point-likeconstituents of hadrons was also proposed by Feynman from a differenttheoretical perspective (and he gave them the name partons).

The well-known SLAC-MIT experiment on DIS, carried out at the StanfordLinear Accelerator Center at about the same time as the theoretical proposalof scaling, discovered that the measured \(\sigma _{\mathrm{DIS}}\) indeedexhibit approximate scaling behavior of Eqs.(1) & (2). Fig.Figure 2 shows some early results of thisexperiment. The DIS data points at three different center-of-mass energies(connected by lines to guide the eye) are plotted against the variable \(Q^{2}\ .\) The approximately \(Q\)-independent behavior is in sharp contrast tothe fast fall-off of the elastic form factor shown in the same plot forcomparison.

Theoretical Origin and Harbinger of Asymptotic Freedom

To describe the theoretical origin of Bjorken scaling, we need to introduce theprecise definition of the structure functions \(W_{i}\ .\) First, the cross section, Eq.(2), is given in terms of the square of the DIS scattering amplitudeof Fig.Figure 1. It has the structure \(\sigma \sim L_{\mu \nu }W^{\mu\nu }\) where \(L_{\mu \nu }\) represents the square of the leptonic vertex(upper part of the diagram) and \(W^{\mu \nu }\) represents the square of thehadronic vertex (the lower part). Because the leptons interact with theelectroweak currents \(J_{\mu }\) (coupled to the vector bosons that arerepresented by the wavy line in Fig.Figure 1) as point-likeparticles, \(L_{\mu \nu }\) can be calculated exactly. The more complex hadronic factor \(W^{\mu \nu }\) is the Fourier transform of the nucleon matrix element of thecommutator of the currents \(J^{\mu }\ ,\)\[\tag{3}W^{\mu \nu }(q,p)=\dfrac{1}{4\pi }\int d^{4}x\,e^{iqx}\langle p|[J^{\mu}(x),J^{\nu }(0)]|p\rangle ~. \]

It embodies the strong interaction dynamics of the target nucleon with thecurrent \(J_{\mu }\ ;\) and, by the optical theorem of scattering theory, it isthe imaginary part of the forward Compton scattering amplitude of the vectorboson (the wavy line in Fig.Figure 1 on the nucleon (\(p\)). Thestructure functions \(W_{i}\) of Eqs.(1) and (2) arerelated to \(W^{\mu \nu }(q,p)\) by:\[\tag{4}W^{\mu \nu }(q,p)=\tilde{g}^{\mu \nu }\,W_{1}(x,Q)\,+\tilde{p}^{\mu }\tilde{p}^{\nu }\,W_{2}(x,Q) \]

where \(\tilde{g}^{\mu \nu }=g^{\mu \nu }-\frac{q^{\mu }q^{\nu }}{q^{2}}\) and\(\tilde{p}^{\mu }=(p^{\mu }-\frac{q\cdot p}{q^{2}}q^{\mu })/M\ .\)

Original derivation of Bjorken scaling

The derivation of Eq. (1) by Bjorken was based on theoretical tools of thatera—current algebra, dispersion relations, and the``infinite-momentum frame" method (pioneered by Gell-Man, Fubini, and manyothers)—applied to amplitudes such as \(W^{\mu \nu }(q,p)\ ,\) usinghis \(q_{0}\rightarrow \infty \) asymptotic limit method (known as the Bjorken Limit). This scaling result was intimately relatedto a number of useful asymptotic sum rules on physically measurablecross sections (bearing names such as Adler, Bjorken, Callan-Gross, etc.) that were derived around the same time period using similartheoretical input (Bjorken 1968). Since the derivation depended on assumptions such as current algebra, and on the existence of the infinitemomentum limits of matrix elements of current commutators at almostequal times, it was regarded as highly suggestive but not necessarilydefinitive.

Breaking of Bjorken scaling

In fact, by explicit calculation invarious perturbation theories (Adler and Tung 1969, Jackiw and Preparata 1969)found that Bjorken limit and Bjorken scaling can not hold exactly inany realistic interacting quantum field theory; scaling-breaking termsappear invariably order-by-order in perturbative calculations. Theimplication of this result was significant: the cumulative effect ofsumming the scale-breaking terms to all orders in perturbation theory (byrenormalization group methods) would lead to gross violation of the proposedscaling behavior in all field theories that were known at that time;however, a mild violation of scaling would be possible in a special class oftheories that are asymptotically free-characterized by effectivecouplings that approach zero as the renormalization scale increasesindefinitely. But, there was no known example of such a theory at that time.

Road to asymptotic freedom and QCD

Thus, the experimentaldiscovery of approximate scaling behavior of the DIS structure functions setoff an urgent search in the theoretical physics community for quantum fieldtheories that are asymptotically free. This effort culminated in thediscovery (Gross and Wilczek 1973, Politzer 1973) ofasymptotic freedom in Quantum Chromodynamics (QCD)-a quantum fieldtheory of quarks and gluons with a fundamental symmetry called color thatwas previously proposed as a possible underlying theory of stronginteractions. The gauge symmetry of this theory, characterized by thenon-abelian group SU(3), is the key to its asymptotic free nature. QCD is,by now, well established as the fundamental theory of strong interactionsfor quarks, gluons and the observed hadron.

Scaling Behavior in QCD-Factorization and Dimensional Transmutation

QCD now provides the theoretical basis for the universally accepted partonpicture language that is used to describe the high energy interactionbetween leptons, vector bosons and hadrons in the physics world, in terms ofthe fundamental interactions between elementary particles-leptons, quarks,gluon, electroweak vector bosons, Higgs particles, and otherbeyond-Standard-Model particles. The connection between the physics world(of hadrons) and the parton world (quarks and gluons) is made possible bythe crucial concept of factorization, which allows the systematicseparation of short-distance interactions (of the partons) fromlong-distance interactions (that are responsible for color confinement and hadron formation).Scale-dependence plays a crucial role in establishing factorization in QCD.This makes it possible for QCD theory to make predictions on the \(Q\)-dependence of physically measurable quantities such as the structurefunctions \(W_{i}(x,Q)\ ,\) and to provide a precise derivation of the Bjorkenproposition (Eq.(1) in its modern form. (We shall refer to \(Q\ ,\) the square root of \(Q^{2}\ ,\) as the scaling variable from this point on.)

In QCD, the limits on the left-hand side of Eq.(1)do exist (i.e. is non-zero), confirming Bjorken's conjecture;however, the resulting functions on the right-hand side are not strictlyscale-independent, hence will be written as \(F_{i}(x,Q)\ ,\) in contrast toEq.(1). The QCD factorization theorem for \(F_{i}(x,Q)\ ,\) which has a clear physical interpretation (to be described below), statesthat:\[\tag{5}\lim_{Q\,\mathrm{large},\,x\,\mathrm{fixed}}F_{i}(x,Q)=\,\,f_{a}\otimes \,\hat{\sigma}_{i}^{a}=\int_{x}^{1}\dfrac{d\xi }{\xi }{\sum_{a}}\,\,f_{a}(\xi ,\mu )\,\hat{\sigma}_{i}^{a}(\dfrac{x}{\xi },\dfrac{Q}{\mu },\alpha _{s}) \]

where the variables (\(x,Q\)) have the same meaning as in the previoussection, \(\alpha _{s}(\mu )\) is the QCD effective (sometimes calledrunning) coupling at the factorization scale \(\mu\ ,\) the index\(a\) is a parton label to be summed over contributingquarks and gluons, \(\otimes\) is called a convolution (defined by theright-hand side of the equation). The two main factors of the equation havethe following simple meaning:

• \(f_{a}(x,\mu )\) is a parton distribution function (sometime also called parton density), which represents the probability of finding a parton \(a\) inside the nucleon target with the momentum fraction \(x\) at the effective scale \(\mu\ ;\)

• \(\hat{\sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) is the hard scattering cross section of the electroweak vector boson \(V\) (wavy line of Fig.Figure 1) on the parton \(a\) (cf.the hadron \(A\)) with the equivalent \(x\) and \(Q\) variables for the partonic process \(V+a\rightarrow X\) (cf.the hadronic \(F_{i}\)). {\(\,\hat{\sigma}_{i}^{a}\)} are sometimes called Wilson coefficients, for historical reasons.

The physical content of this Factorization theorem is: at high energies, thenucleonic structure function \(F_{i}\) becomes a convolution of theprobability of finding a parton \(a\) inside the nucleon \(f_{a}\) with the corresponding partonic structure function \(\hat{\sigma}_{i}^{a}\ ,\) summed over all partons that can participate in the interaction.This structure is made even more explicit in Fig.Figure 3, whichis a graphical representation of Eq.(5). This basic resultcan be generalized to most high energy processes (see Sec.Generalization to Other High Energy Processesbelow); it forms the foundation of the QCD improvedparton picture that is used to describe most modern high energy physicsphenomena nowadays.

The appearance of the factorization scale variable \(\mu\) in Eq.(5) is an essential feature of this formalism. Although thephysical quantity \(F_{i}(x,Q)\) on the left-hand-side of the equation is, inprinciple, independent of \(\mu\ ,\) the two theoretical factors \(f_{a}(x,\mu )\) and \(\hat{\sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) that appear on theright-hand-sidemust depend on the renormalization and factorization scales that are required to give them precise meaning in a quantum field theorysuch as QCD-along with the effective coupling \(\alpha _{s}(\mu )\ .\)(For simplicity, we denote both theoretical scale parametersgenerically by \(\mu\ .\)) The basic factorized structure is independent of\(\mu\ :\) a shift in the scaleparameter merely results in a reshuffling between the theoretical factors \(f_{a}(x,\mu )\) and \(\hat{\sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) inEq.(5) and Fig.Figure 3-the overall convolutionintegral remains invariant.

In DIS, \(\mu\) is commonly chosen to be equal to the physicalvariable \(Q\) for convenience; hence is oftentimes regarded assynonymous with the latter. However, a full understanding of the QCDtheory of scaling behavior in QCD requires making a clear conceptualdistinction between the physical variable \(Q\) and the theoreticalparameter \(\mu\ ,\) as will be made more explicit by the next-level ofdiscussion.

• In Eq.(5), the hard scattering cross section factors \(\hat{\sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) (upper, square blob in Fig.Figure 3) involve only short-distance interactions. They are calculable in an asymptotically free theory, such as QCD, provided the scale \(\mu\) is chosen to be large, so that the effective coupling \(\alpha _{s}(\mu )\) becomes small enough to render the order-by-order perturbative expansion useful. Therefore, in general, one has,

\[\tag{6}\hat{\sigma}_{i}^{a}(x,\frac{Q}{\mu },\alpha _{s})=\hat{\sigma}_{i,0}^{a}+\alpha _{s}\,\hat{\sigma}_{i,1}^{a}+\alpha _{s}^{2}\,\hat{\sigma}_{i,2}^{a}+\dots \ :\]

where the first few coefficient functions \(\hat{\sigma}_{i,n}^{a}\) have been calculated in the existing literature.

• All long-distance interactions of \(F_{i}(x,Q)\) are factored into the parton distribution functions \(f_{a}(x,\mu )\) in Eq.(5), represented by the lower, round blob in Fig.Figure 3. Long distance interactions are not yet calculable in perturbative QCD (the coupling is strong at these scales). However, \(\{f_{a}(x,\mu )\}\) have two very important properties: (i) they are independent of the particular physical process under consideration-they represent the parton structure of the nucleon, hence are universal to all high energy processes; and (ii) although the \(x\)-dependence of \(f_{a}(x,\mu )\) is not calculable in perturbative QCD, its scale (\(\mu\)) dependence can be determined by renormalization theory (Gribov and Lipatov 1972, Altarelli and Parisi 1977, Dokshitzer 1977)

\[\tag{7}\mu \frac{d}{d\mu }f_{a}(x,\mu )=\frac{\alpha _{s}}{2\pi }P_{a}^{b}\otimesf_{b}=\frac{\alpha _{s}}{2\pi }\int_{x}^{1}\dfrac{d\xi }{\xi }\sum_{b}\,P_{a}^{b}(\xi ,\alpha _{s})\,\,f_{b}(\frac{x}{\xi },\mu )\,\ :\]

where the function \(P_{a}^{b}(\xi ,\alpha _{s})\) can be calculated order-by-order in perturbative QCD:\[\tag{8}P_{a}^{b}(x,\alpha _{s})=P_{a,0}^{b}(x)+\alpha _{s}P_{a,1}^{b}(x)+\alpha_{s}^{2}P_{a,2}^{b}(x)+\dots \ :\]

Eq.(7) is called the QCD evolution equation for the parton distribution functions \(\{f_{a}(x,\mu )\}\ ;\) the evolution kernel \(P_{a}^{b}(x,\alpha _{s})\ ,\) which physically represent the probability of finding parton \(b\) in parton \(a\) in the evolution chain, are commonly called the splitting functions. The coefficient functions \(P_{a,i}^{b}(x)\) in the perturbative expansion of \(P_{a}^{b}(x,\alpha _{s})\ ,\) Eq.(8), have been calculated up to \(i=2\ .\)

• Since the physical quantity \(F_{i}(x,Q)\) should be independent of where one chooses to express its factorization structure at high energies, as already mentioned earlier, the \(\mu\)-dependence of \(\hat{\sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) is governed by an equation like Eq.(7), with the opposite sign on the right hand side. The compensation between the two factors when \(\mu\) varies is ensured order by order in perturbation theory. At infinite order, the result will be exactly \(\mu\)-independent. In practice, if the perturbation expansions (such as in Eqs.(6) and (8) are truncated at order \(n\ ,\) then any residual \(\mu\)-dependence of the right-hand side of Eq.(5) will be of one order higher than \(n\ .\) Thus, the predictions for \(F_{i}(x,Q)\) will be insensitive to the choice of \(\mu\ ,\) to the same accuracy of the calculation.

• To produce concrete predictions on physical quantities, such as \(F_{i}(x,Q)\ ,\) from Eq.(8), one needs to make a concrete choice of the theoretical scale variable \(\mu\) on the right-hand side in terms of a physical variable. In order to take advantage of asymptotic freedom, \(\mu\) should be chosen large enough, so that the expansion parameter \(\alpha _{s}(\mu )\) becomes small, and the perturbative expansions convergent. For the DIS process, with \(Q\) being the only large physical scale for fixed \(x\ ,\) one can choose \(\mu=\kappa Q\ ,\) with any constant \(\kappa\) that is neither too small nor too large. The natural choice is \(\kappa =1\ .\) But it is not the unique one, as mentioned in the previous paragraph. A choice of \(\kappa \neq 1\) would result in an answer that differs from the order \(\alpha_{s}^{n} \) calculation by an amount of the order \(\alpha_{s}^{n+1}\,\ln \kappa \ .\) Thus, the relative difference is of order \(\alpha _{s}(Q)\ln \kappa \ ,\) which is small for a reasonable value of \(\kappa\ .\) One obtains

\[\tag{9}F_{i}(x,Q)=\int_{x}^{1}\dfrac{d\xi }{\xi }{\sum_{a}}\,\,f_{a}(\xi,\kappa Q)\,\hat{\sigma}_{i}^{a}(\dfrac{x}{\xi },\kappa ,\alpha _{s})\ :\]

where, we have kept the \(\kappa\) dependence as a reminder that this is an approximate formula, with higher-order ambiguities associated with the choice of scale \(\mu\ .\)

In leading-order approximation (keeping only the leading terms in Eqs.(6) and (8), and adopting the usual scale choice \(\mu =Q\ ,\)Eq.(9) becomes\[\tag{10}F_{i}(x,Q)={\sum_{a}}\,c_{i}^{a\,}\,f_{a}(x,Q) \]

where \(c_{i}^{a}\) is the electroweak coupling of the parton \(a\) to theexchanged vector boson relevant for \(\hat{\sigma}_{i}^{a}\ ;\) it isproportional to the square of the charge of the parton in the case ofelectromagnetic interaction as considered originally by Bjorken. We see thateven in the lowest-order approximation, the QCD parton picture yieldsstructure functions \(F_{i}(x,Q)\) that have non-trivial \(Q\)-dependence, assuggested by pre-QCD field theory calculations. The distinctive feature ofQCD is that this \(Q\)-dependence is computable, since {\(f_{a}(x,Q)\)}satisfy the evolution equation Eq.(7) with known evolutionkernels (splitting functions).

The emergence of computable scale-dependence of dimensionless physicalquantities at high energies in an asymptotically free quantum field theorywith dimensionless coupling, such as QCD, is a remarkable feature ofrelativistic quantum mechanics. The mechanism by which the scale-dependencearises, as sketched in this section, is sometimes referred to as dimensional transmutation.

Scaling Behavior Observed in Modern DIS Experiments

Modern high precision data on deep inelastic scattering andother high energy processes confirm the scaling behavior predicted by QCDtheory over a very wide kinematic range of the variables (\(x,Q\)). This factforms the foundation of the belief that QCD is the fundamental theory ofstrong interactions for elementary particle physics. As an example of thisimpressive agreement between theory and experiment, we show in Fig.<ref>fig:FigD</ref> the combined data of the two experiments, H1 and ZEUS,carried out at the electron-proton collider HERA, along with resultsfrom some fixed-target DIS experiments, spanning five orders ofmagnitude in both variables \((x,Q^{2})\ .\) The lines in this plotrepresent the predicted scale (\(Q\)) dependence from QCD theory. The\(x\)-dependence cannot be predicted in the perturbative QCD theory. In practice, the parton distributionfunctions at certain fixed \(Q=Q_{0}\) \(\{f_{a}(x,Q_{0})\}\) aredetermined phenomenologically from global analyses, in which a widerange of hard scattering measurements, including DIS, are compared toQCD theory according to Eqs. (7) & (9). The fullparton distributions \(\{f_{a}(x,Q)\}\) are then determined for all\((x,Q)\) from Eq.(7).

The predicted scale dependence also allows the accurate determination of thefundamental QCD running coupling \(\alpha _{s}(Q)\) from a wide variety ofphysical processes, including DIS. The universality of \(\alpha _{s}(Q)\ ,\)over the entire measurable energy range of modern particle physics providesan unequivocal confirmation of the QCD theory.

Generalization to Other High Energy Processes

The Bjorken scaling behavior discussed above generalize to all highenergy hard processes (processes that have at least one high energyscale much larger than the typical hadron mass scale, say \(1\) GeV). Inthe modern QCD perspective, this generalization can be exemplified by atypical high energy hadron-hadron scattering process, such as theproduction of a lepton-pairwith high invariant mass \(Q, A+B\rightarrow l_{1}l_{2}X\ .\) See Fig.Figure 5.

The equivalent formula to Eq.(5) is easiest to see in terms ofthe dimensionless ratio of the physical cross section to the correspondingone for point-like scattering particles (analogous to the Mott cross sectionof Eq.(2)\[\tag{11}\sigma (s,\tau ){/}\sigma _{0}(s)=\,\,f_{a}\otimes \,\hat{\sigma}^{ab}\otimes f_{b}=\int \int dx_{1}dx_{2}{\sum_{a,b}}\,\,f_{a}(x_{1},Q)\,\hat{\sigma}^{ab}(x_{1}x_{2}\tau ,\alpha_{s}(Q))\,f_{b}(x_{2},Q) \]

where \(s\) denotes the overall center-of-mass energy squared, \(\tau=\frac{Q^{2}}{s}\) is the scaling variable for this physical process, \(f_{a,b}(x,Q)\) are parton distributions, and \(\hat{\sigma}^{ab}(\tau ,\alpha_{s}(Q))\) is the corresponding dimensionless cross section for scattering ofthe two partons (\(a,b\)) at partonic center-of-mass energy squared equal to \(\hat{s}=x_{1}x_{2}s\ .\) To keep matters as simple as possible, we have alreadyset all the factorization scales (\(\mu\)) equal to the physical large scale \(Q\) in this formula.

If the parton distributions \(f_{a}, f_{b}\) and the QCD coupling \(\alpha_{s}\) were not \(Q\)-dependent, Eq.(11) would imply that \(\sigma/\sigma _{0}\) is a function of the scaling variable \(\tau =Q^{2}/s\)only-i.e.\(\sigma (\frac{Q^{2}}{s},s)/\sigma _{0}\) is independent of \(s\ !\)This would be the equivalent of Bjorken scaling in its original form; and,for this case, it was proposed by Drell and Yan, soon after Bjorken'soriginal paper. In reality, QCD predicts the full \(Q\)-dependence shown onthe right hand side of Eq.(11). These predictions are alsoconfirmed by modern experiments.

Once the parton distributions \(\{f_{a}(x,Q)\}\) are determined from globalQCD analysis, as described in the previous section, one can make predictionsfor all physical cross sections at any energy, even beyond currentlymeasured range, and for new physics processes, even beyond the Standardmodel, according to formulas such as Eq.(11). Thus, the scalingbehavior of the theory of QCD enables the parton picture to form thefoundation of all modern particle phenomenology.

References

• Adler, S. L. and W.-K. Tung (1969). Breakdown of Asymptotic Sum Rules in Perturbation Theory. Phys. Rev. Lett. 22, 978-981
• G.Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298
• Bjorken, J. D. (1968). Current Algebra at Small Distances, in Proceedings of the International School of Physics Enrico Fermi Course XLI, J. Steinberger, ed., Academic Press, New York, pp. 55-81.
• Bjorken, J. D. (1969). Asymptotic Sum Rules at Infinite Momentum. Phys. Rev. 179, 1547- 1553.
• V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438
• D.J. Gross and F. Wilczek (1973) Asymptotically Free Gauge Theories. Phys. Rev. Lett. 30 (1973) 1343; Phys. Rev. D8 3633.
• Jackiw, R. and G. Preparata (1969). Probes for the Constituents of the Electromagnetic Current and Anomalous Commutators. Phys. Rev. Lett. 22, 975-977.
• H.D. Politzer. Reliable Perturbative Results for Strong Interactions? Phys. Rev. Lett. 30 (1973) 1346.
• Yu. L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.

Internal references

• Gerard ′t Hooft (2008) Gauge theories. Scholarpedia, 3(12):7443.

• Guido Altarelli (2009) QCD evolution equations for parton densities. Scholarpedia, 4(1):7124.

External links

• Wu-Ki Tung Home Page

See also

asymptotic freedom, gauge theories, parton, Quantum Chromodynamics, QCD evolution equations for parton densities