Sum Rules (Quantum Field Theory)

In quantum field theory, a ''sum rule'' is a relation between a static quantity and an integral over a dynamical quantity. Therefore, they have a form such as:

$\int A(x) dx = B$

where $A(x)$ is the dynamical quantity, for example a structure function characterizing a particle, and $B$ is the static quantity, for example the mass or the charge of that particle.

Quantum field theory sum rules should not be confused with sum rules in quantum chromodynamics or quantum mechanics.

Properties

Many sum rules exist. The validity of a particular sum rule can be sound if its derivation is based on solid assumptions, or on the contrary, some sum rules have been shown experimentally to be incorrect, due to unwarranted assumptions made in their derivation. The list of sum rules below illustrate this.

Sum rules are usually obtained by combining a dispersion relation with the optical theorem,B. Pasquini and M. Vanderhaeghen (2018) “Dispersion theory in electromagnetic interactions”
Ann. Rev. Nucl. Part. Sci. 68, 75 using the operator product expansion or current algebra.A. Deur, S. J. Brodsky, G. F. de Teramond (2019) “The Spin Structure of the Nucleon”
Rept. Prog. Phys. 82 076201

Quantum field theory sum rules are useful in a variety of ways. They permit to test the theory used to derive them, e.g. quantum chromodynamics, or an assumption made for the derivation, e.g. Lorentz invariance. They can be used to study a particle, e.g. how does the spins of partons make up the spin of the proton. They can also be used as a measurement method. If the static quantity $B$ is difficult to measure directly, measuring $A(x)$ and integrating it offers a practical way to obtain $B$ (providing that the particular sum rule linking $A(x)$ to $B$ is reliable).

Although in principle, $B$ is a static quantity, the denomination of ''sum rule'' has been extended to the case where $B$ is a probability amplitude, e.g. the probability amplitude of Compton scattering, see the list of sum rules below.

List of QCD sum rules

(The list is not exhaustive)

*Baldin sum rule.A. M. Baldin (1960) “Polarizability of nucleons”
Nucl. Phys. 18, 310 This is the unpolarized equivalent of the GDH sum rule (see below). It relates the probability that a photon absorbed by a particle results in the production of hadrons (this probability is called the photo-production cross-section) to the electric and magnetic polarizabilities of the absorbing particle. The sum rule reads , where is the photon energy, is minimum value of energy necessary to create the lightest hadron (i.e. a pion), is the photo-production cross-section, and and are the particle electric and magnetic polarizabilities, respectively.
* Bjorken sum rule.J. D. Bjorken (1966) “Applications of the chiral U(6)×U(6) algebra of current densities”
Phys. Rev. 148, 1467J. D. Bjorken (1970) “Inelastic scattering of polarized leptons from polarized nucleons”
Phys. Rev. D 1, 1376 This sum rule is the prototypical QCD spin sum rule. It states that in the ''Bjorken scaling'' domain, the integral of the spin structure function of the proton minus that of the neutron is proportional to the axial charge of the nucleon. Specially: , where is the Bjorken scaling variable, is the first spin structure function of the proton (neutron), and is the nucleon axial charge that characterizes the neutron β-decay. The sum rule was experimentally verified within better than a 10% precision.
* Burkhardt–Cottingham sum rule.H. Burkhardt and W. N. Cottingham (1970) “Sum rules for forward virtual Compton scattering”
Annals Phys. 56, 453 The sum rule was experimentally verified.
*$\delta_$ sum rule.P.A.M Guichon, G.Q. Liu and A. W. Thomas (1995) “Virtual Compton scattering and generalized polarizabilities of the proton”
Nucl. Phys. A 591, 606-638
* Efremov–Teryaev–Leader sum rule.A. V. Efremov, O. V. Teryaev and E. Leader (1997) “Exact sum rule for transversely polarized DIS”
Phys. Rev. D 55, 4307
* Ellis–Jaffe sum rule.J. R. Ellis and R. L. Jaffe (1974) “Sum rule for deep-inelastic electroproduction from polarized protons”
Phys. Rev. D 9, 1444 (1974) The sum rule was shown to not hold experimentally, suggesting that the strange quark spin contributes non-negligibly to the proton spin. The Ellis–Jaffe sum rule provides an example of how the violation of a sum rule teaches us about a fundamental property of matter (in this case, the origin of the proton spin).
* Forward spin polarizability sum rule.
* Fubini–Furlan–Rossetti Sum Rule.S. Fubini, G. Furlan, and C. Rossetti (1965) “A dispersion theory of symmetry breaking”
, Nuovo Cim. 40 1171.
* Gerasimov– Drell–Hearn sum rule (GDH, sometimes DHG sum rule).S. B. Gerasimov (1965) “A sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei”
Sov. J. Nucl. Phys. 2, 430 (1966) ad. Fiz. 2, 598 (1965)/ref>S. D. Drell and A. C. Hearn (1966) “Exact sum rule for nucleon magnetic moments”
Phys. Rev. Lett. 16, 908M. Hosoda and K. Yamamoto (1966) “Sum rule for the magnetic moment of the Dirac particle”
Prog. Theor. Phys. 36 (2), 425 This is the polarized equivalent of the Baldin sum rule (see above). The sum rule is: , where is the minimal energy required to produce a pion once the photon is absorbed by the target particle, is the difference between the photon absorption cross-sections when the photons spin are aligned and anti-aligned with the target spin, is the photon energy, is the fine-structure constant, and , and are the anomalous magnetic moment, spin quantum number and mass of the target particle, respectively. The GDH sum rule was experimentally verified (within a 10% precision).
* Generalized GDH sum rule. Several generalized versions of the GDH sum rule have been proposed. The first and most common one is: , where is the first spin structure function of the target particle, is the Bjorken scaling variable, is the virtuality of the photon or equivalently, the square of the absolute value of the four-momentum transferred between the beam particle that produced the virtual photon and the target particle, and is the first forward virtual Compton scattering amplitude. It can be argued that calling this relation ''sum rule'' is improper, since is not a static property of the target particle nor a directly measurable observable. Nonetheless, the denomination sum rule is widely used.
* Gottfried sum rule.K. Gottfried (1967) “Sum rule for high-energy electron-proton scattering”
Phys. Rev. Lett. 18, 1174
* Gross–Llewellyn Smith sum rule.D. J. Gross and C. H. Llewellyn Smith (1969) “High-energy neutrino-nucleon scattering, current algebra and partons”
Nucl. Phys B14 337
* Momentum sum rule:J. C. Collins and D. E. Soper (1982) “Parton distribution and decay functions”
Nucl. Phys. B194 445 It states that the sum of the momentum fraction of all the partons (quarks, antiquarks and gluons inside a hadron is equal to 1.
*Ji Sum rule: Relates the integral of generalized parton distributions to the angular momentum carried by the quarks or by the gluons.
* Schwinger sum rule.J. S. Schwinger (1975) “Source Theory Discussion of Deep Inelastic Scattering with Polarized Particles”
Proc. Natl. Acad. Sci. 72, 1
* Wandzura– Wilczek sum rule.S. Wandzura and F. Wilczek (1977) “Sum rules for spin-dependent electroproduction: Test of relativistic constituent quarks”
Phys. Lett. B 72, 195

See also

*Quantum chromodynamics
*Proton spin crisis

References

Quantum field theory
Quantum chromodynamics

Nuclear physics
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