Bjorken scaling
| Wu-Ki Tung (2009), Scholarpedia, 4(3):7412. | doi:10.4249/scholarpedia.7412 | revision #137304 [link to/cite this article] |
Bjorken Scaling refers to an important simplifying feature—scaling—of a large class of dimensionless physical quantities in elementary particles; it strongly suggests that experimentally observed strongly interacting particles (hadrons) behave as collections of point-like constituents when probed at high energies. A property of hadrons probed in high-energy scattering experiments is said to scale when it is determined not by the absolute energy of an experiment 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, and hence effectively point-like substructure. Scaling behavior was first proposed by James Bjorken in 1968 for 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 the formulation of Quantum Chromodynamics (QCD)—the modern fundamental theory of strong interactions. Bjorken scaling is, however, not exact; deviations from strict scaling is required in quantum field theory. The QCD theory can predict the detailed form of violations of the scaling behavior of the relevant physical quantities through the distinctive quantum effect of dimensional transmutation. These predictions have been fully confirmed by modern high energy experiments. This theory provides a firm foundation for the intuitive QCD parton picture of elementary particles.
Contents |
Introduction
In 1968, Bjorken proposed (Bjorken 1969) that the structure functions measured in electron-nucleon deep inelastic scattering (DIS, depicted in Fig. Figure 1), \(W_{i}(Q^{2},\nu )\ ,\) may exhibit scaling behavior in the asymptotic limit, \[\tag{1} \begin{array}{rcl} \lim_{Q^{2}\rightarrow \infty ,\,\nu /Q^{2}\,\mathrm{fixed}}~\nu W_{2}(Q^{2},\nu ) & = & MF_{2}(x) \\ \lim_{Q^{2}\rightarrow \infty ,\,\nu /Q^{2}\,\mathrm{fixed}}~W_{1}(Q^{2},\nu ) & = & F_{1}(x) \end{array} \] where \(Q^{2}\) represents the squared 4-momentum-transfer vector \(q\) of the exchanged virtual photon, \(\nu =q\cdot p/M\) the energy loss between the scattering 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 in Fig. 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 of a lepton \(l_{1}\) (say an electron) on a point-like charged particle , and \(\theta\) is the scattering angle of the outgoing lepton \(l_{2}\) in the laboratory frame. This formula resembles that of elastic scattering of an electron on a nucleon, with \(W_{1,2}\) taking the place of the electromagnetic 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 its measurement was that it would also be a fast falling function of \(Q^{2}\ .\) Bjorken's scaling proposition, expressed by the \(Q\)-independence of the right-hand side of Eq. (1), would contradict this expectation. It would imply that the nucleon target appears as a collection of point-like constituents when probed at very high energies in DIS (implied by the \(Q^{2}\rightarrow \infty\) limit on the left-hand side of Eq. (1). The possible existence of such point-like constituents of hadrons was also proposed by Feynman from a different theoretical perspective (and he gave them the name partons).
The well-known SLAC-MIT experiment on DIS, carried out at the Stanford Linear Accelerator Center at about the same time as the theoretical proposal of scaling, discovered that the measured \(\sigma _{\mathrm{DIS}}\) indeed exhibit approximate scaling behavior of Eqs. (1) & (2). Fig. Figure 2 shows some early results of this experiment. 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 to the fast fall-off of the elastic form factor shown in the same plot for comparison.
Theoretical Origin and Harbinger of Asymptotic Freedom
To describe the theoretical origin of Bjorken scaling, we need to introduce the precise definition of the structure functions \(W_{i}\ .\) First, the cross section, Eq. (2), is given in terms of the square of the DIS scattering amplitude of 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 the hadronic vertex (the lower part). Because the leptons interact with the electroweak currents \(J_{\mu }\) (coupled to the vector bosons that are represented by the wavy line in Fig. Figure 1) as point-like particles, \(L_{\mu \nu }\) can be calculated exactly. The more complex hadronic factor \(W^{\mu \nu }\) is the Fourier transform of the nucleon matrix element of the commutator 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 the current \(J_{\mu }\ ;\) and, by the optical theorem of scattering theory, it is the imaginary part of the forward Compton scattering amplitude of the vector boson (the wavy line in Fig. Figure 1 on the nucleon (\(p\)). The structure functions \(W_{i}\) of Eqs. (1) and (2) are related 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 that era—current algebra, dispersion relations, and the ``infinite-momentum frame" method (pioneered by Gell-Man, Fubini, and many others)—applied to amplitudes such as \(W^{\mu \nu }(q,p)\ ,\) using his \(q_{0}\rightarrow \infty \) asymptotic limit method (known as the Bjorken Limit). This scaling result was intimately related to a number of useful asymptotic sum rules on physically measurable cross sections (bearing names such as Adler, Bjorken, Callan-Gross, etc.) that were derived around the same time period using similar theoretical input (Bjorken 1968). Since the derivation depended on assumptions such as current algebra, and on the existence of the infinite momentum limits of matrix elements of current commutators at almost equal times, it was regarded as highly suggestive but not necessarily definitive.
Breaking of Bjorken scaling
In fact, by explicit calculation in various perturbation theories (Adler and Tung 1969, Jackiw and Preparata 1969) found that Bjorken limit and Bjorken scaling can not hold exactly in any realistic interacting quantum field theory; scaling-breaking terms appear invariably order-by-order in perturbative calculations. The implication of this result was significant: the cumulative effect of summing the scale-breaking terms to all orders in perturbation theory (by renormalization group methods) would lead to gross violation of the proposed scaling behavior in all field theories that were known at that time; however, a mild violation of scaling would be possible in a special class of theories that are asymptotically free-characterized by effective couplings that approach zero as the renormalization scale increases indefinitely. But, there was no known example of such a theory at that time.
Road to asymptotic freedom and QCD
Thus, the experimental discovery of approximate scaling behavior of the DIS structure functions set off an urgent search in the theoretical physics community for quantum field theories that are asymptotically free. This effort culminated in the discovery (Gross and Wilczek 1973, Politzer 1973) of asymptotic freedom in Quantum Chromodynamics (QCD)-a quantum field theory of quarks and gluons with a fundamental symmetry called color that was previously proposed as a possible underlying theory of strong interactions. The gauge symmetry of this theory, characterized by the non-abelian group SU(3), is the key to its asymptotic free nature. QCD is, by now, well established as the fundamental theory of strong interactions for quarks, gluons and the observed hadron.
Scaling Behavior in QCD-Factorization and Dimensional Transmutation
QCD now provides the theoretical basis for the universally accepted parton picture language that is used to describe the high energy interaction between leptons, vector bosons and hadrons in the physics world, in terms of the fundamental interactions between elementary particles-leptons, quarks, gluon, electroweak vector bosons, Higgs particles, and other beyond-Standard-Model particles. The connection between the physics world (of hadrons) and the parton world (quarks and gluons) is made possible by the crucial concept of factorization, which allows the systematic separation of short-distance interactions (of the partons) from long-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 structure functions \(W_{i}(x,Q)\ ,\) and to provide a precise derivation of the Bjorken proposition (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 strictly scale-independent, hence will be written as \(F_{i}(x,Q)\ ,\) in contrast to Eq. (1). The QCD factorization theorem for \(F_{i}(x,Q)\ ,\) which has a clear physical interpretation (to be described below), states that: \[\tag{5} \lim_{Q\,\mathrm{large},\,x\,\mathrm{fixed}}F_{i}(x,Q)=\,\,f_{a}\otimes \, \hat{\sigma}_{i}^{a}=\int_{x}^{1}\dfrac{d\xi }{\xi }{\sum_{a}} \,\,f_{a}(\xi ,\mu )\,\hat{\sigma}_{i}^{a}(\dfrac{x}{\xi },\dfrac{Q}{\mu } ,\alpha _{s}) \]
where the variables (\(x,Q\)) have the same meaning as in the previous section, \(\alpha _{s}(\mu )\) is the QCD effective (sometimes called running) coupling at the factorization scale \(\mu\ ,\) the index \(a\) is a parton label to be summed over contributing quarks and gluons, \(\otimes\) is called a convolution (defined by the right-hand side of the equation). The two main factors of the equation have the 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, the nucleonic structure function \(F_{i}\) becomes a convolution of the probability 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, which is a graphical representation of Eq. (5). This basic result can be generalized to most high energy processes (see Sec. Generalization to Other High Energy Processes below); it forms the foundation of the QCD improved parton picture that is used to describe most modern high energy physics phenomena nowadays.
The appearance of the factorization scale variable \(\mu\) in Eq. (5) is an essential feature of this formalism. Although the physical quantity \(F_{i}(x,Q)\) on the left-hand-side of the equation is, in principle, independent of \(\mu\ ,\) the two theoretical factors \(f_{a}(x,\mu )\) and \(\hat{ \sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) that appear on the right-hand-side must depend on the renormalization and factorization scales that are required to give them precise meaning in a quantum field theory such as QCD-along with the effective coupling \(\alpha _{s}(\mu )\ .\) (For simplicity, we denote both theoretical scale parameters generically by \(\mu\ .\)) The basic factorized structure is independent of \(\mu\ :\) a shift in the scale parameter merely results in a reshuffling between the theoretical factors \(f_{a}(x,\mu )\) and \(\hat{\sigma}_{i}^{a}(x,Q/\mu ,\alpha _{s})\) in Eq. (5) and Fig. Figure 3-the overall convolution integral remains invariant.
In DIS, \(\mu\) is commonly chosen to be equal to the physical variable \(Q\) for convenience; hence is oftentimes regarded as synonymous with the latter. However, a full understanding of the QCD theory of scaling behavior in QCD requires making a clear conceptual distinction between the physical variable \(Q\) and the theoretical parameter \(\mu\ ,\) as will be made more explicit by the next-level of discussion.
- 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)
\[\tag{7} \mu \frac{d}{d\mu }f_{a}(x,\mu )=\frac{\alpha _{s}}{2\pi }P_{a}^{b}\otimes f_{b}=\frac{\alpha _{s}}{2\pi }\int_{x}^{1}\dfrac{d\xi }{\xi } \sum_{b}\,P_{a}^{b}(\xi ,\alpha _{s})\,\,f_{b}(\frac{x}{\xi },\mu )\, \ :\]
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 the exchanged vector boson relevant for \(\hat{\sigma}_{i}^{a}\ ;\) it is proportional to the square of the charge of the parton in the case of electromagnetic interaction as considered originally by Bjorken. We see that even in the lowest-order approximation, the QCD parton picture yields structure functions \(F_{i}(x,Q)\) that have non-trivial \(Q\)-dependence, as suggested by pre-QCD field theory calculations. The distinctive feature of QCD is that this \(Q\)-dependence is computable, since {\(f_{a}(x,Q)\)} satisfy the evolution equation Eq. (7) with known evolution kernels (splitting functions).
The emergence of computable scale-dependence of dimensionless physical quantities at high energies in an asymptotically free quantum field theory with dimensionless coupling, such as QCD, is a remarkable feature of relativistic quantum mechanics. The mechanism by which the scale-dependence arises, 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 and other high energy processes confirm the scaling behavior predicted by QCD theory over a very wide kinematic range of the variables (\(x,Q\)). This fact forms the foundation of the belief that QCD is the fundamental theory of strong interactions for elementary particle physics. As an example of this impressive 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 results from some fixed-target DIS experiments, spanning five orders of magnitude in both variables \((x,Q^{2})\ .\) The lines in this plot represent the predicted scale (\(Q\)) dependence from QCD theory. The \(x\)-dependence cannot be predicted in the perturbative QCD theory. In practice, the parton distribution functions at certain fixed \(Q=Q_{0}\) \(\{f_{a}(x,Q_{0})\}\) are determined phenomenologically from global analyses, in which a wide range of hard scattering measurements, including DIS, are compared to QCD theory according to Eqs. (7) & (9). The full parton 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 the fundamental QCD running coupling \(\alpha _{s}(Q)\) from a wide variety of physical processes, including DIS. The universality of \(\alpha _{s}(Q)\ ,\) over the entire measurable energy range of modern particle physics provides an unequivocal confirmation of the QCD theory.
Generalization to Other High Energy Processes
The Bjorken scaling behavior discussed above generalize to all high energy hard processes (processes that have at least one high energy scale much larger than the typical hadron mass scale, say \(1\) GeV). In the modern QCD perspective, this generalization can be exemplified by a typical high energy hadron-hadron scattering process, such as the production of a lepton-pair with 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 of the dimensionless ratio of the physical cross section to the corresponding one for point-like scattering particles (analogous to the Mott cross section of 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 of the 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 already set 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's original paper. In reality, QCD predicts the full \(Q\)-dependence shown on the right hand side of Eq.(11). These predictions are also confirmed by modern experiments.
Once the parton distributions \(\{f_{a}(x,Q)\}\) are determined from global QCD analysis, as described in the previous section, one can make predictions for all physical cross sections at any energy, even beyond currently measured range, and for new physics processes, even beyond the Standard model, according to formulas such as Eq.(11). Thus, the scaling behavior of the theory of QCD enables the parton picture to form the foundation 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
See also
asymptotic freedom, gauge theories, parton, Quantum Chromodynamics, QCD evolution equations for parton densities
