Sum rules (quantum field theory) explained

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,^[1] using the operator product expansion or current algebra.^[2]

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 sum rules

(The list is not exhaustive)

• Adler sum rule.^[3] This sum rule relates the charged current structure function of the proton (here, is the Bjorken scaling variable and is the square of the absolute value of the four-momentum transferred between the scattering neutrino and the proton) to the Cabibbo angle . It states that in the limit, then . The and superscripts indicate that relates to antineutrino-proton or neutrino-proton deep inelastic scattering, respectively.
• Baldin sum rule.^[4] 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. Assuming its validity, the Baldin sum rule provides an important information on our knowledge of electric and magnetic polarizabilities, complementary to their direct calculations or measurements. (See e.g. Fig. 3 in the article^[5] .)
• Bjorken sum rule (polarized).^{[6] [7]} 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. Specifically:, 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. Outside of the Bjorken scaling domain, the Bjorken sum rule acquires QCD scaling corrections that are known up to the 5th order in precision.^[2] The sum rule was experimentally verified within better than a 10% precision.^[2]
• Bjorken sum rule (unpolarized).^[8] The sum rule is, at leading order in perturbative QCD:

\int ^{1} _{0} F ^{p\nu} _{1} (x,Q ^{p\bar{\nu} _{1} }(x,Q^2) dx= \int_0^1 F_1^(x,Q^2)-F_1^(x,Q^2) dx =1-\frac\frac, where F ^{p\nu} _{1} (x,Q ^{p\bar{\nu} _{1} }(x,Q^2) and F ^{n\nu} _{1} (x,Q ^{2)} are the first structure functions for the proton-neutrino, proton-antineutrino and neutron-neutrino deep inelastic scattering reactions, Q ^{2} is the square of the 4-momentum exchanged between the nucleon and the (anti)neutrino in the reaction, and \alpha _{s} is the QCD coupling.

• Burkhardt–Cottingham sum rule.^[9] The sum rule was experimentally verified.^[2] The sum rule is "superconvergent", meaning that its form is independent of

Q ^{2} . The sum rule is: \int ^{1} _{0} g ^{2 )} _{2 (x,Q} dx =0 ,~ \forall~Q ^{2} where g ^{2 )} _{2 (x,Q} is the second spin structure function of the object studied. \delta _{LT} sum rule.^[10]

• Efremov–Teryaev–Leader sum rule.^[11]
• Ellis–Jaffe sum rule.^[12] The sum rule was shown to not hold experimentally,^[2] 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.^[13]
• Gerasimov–Drell–Hearn sum rule (GDH, sometimes DHG sum rule).^{[14] [15] [16]} 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 derivation of the GDH sum rule assumes that the theory that governs the structure of the target particle (e.g. QCD for a nucleon or a nucleus) is causal (that is, one can use dispersion relations or equivalently for GDH, the Kramers–Kronig relations), unitary and Lorentz and gauge invariant. These three assumptions are very basic premises of Quantum Field Theory. Therefore, testing the GDH sum rule tests these fundamental premises. The GDH sum rule was experimentally verified (within a 10% precision).^[2]
• Generalized GDH sum rule. Several generalized versions of the GDH sum rule have been proposed.^[2] 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.^[17]
• Gross–Llewellyn Smith sum rule.^[18] It states that in the Bjorken scaling domain, the integral of the structure function of the nucleon is equal to the number of valence quarks composing the nucleon, i.e., equal to 3. Specifically: . Outside of the Bjorken scaling domain, the Gross–Llewellyn Smith sum rule acquires QCD scaling corrections that are identical to that of the Bjorken sum rule.
• Momentum sum rule:^[19] 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.^[20]
• Proton mass sum rule:^{[21] [22]} It decomposes the proton mass in four terms, quark energy, quark mass, gluon energy and quantum anomalous energy, with each of these terms an integral over 3-dimensional coordinate space.
• Schwinger sum rule.^[23]
• Wandzura–Wilczek sum rule.^[24]

See also

• Quantum chromodynamics
• Proton spin crisis

Notes and References

1. 1805.10482 . 10.1146/annurev-nucl-101917-020843 . Dispersion Theory in Electromagnetic Interactions . 2018 . Pasquini . Barbara . Vanderhaeghen . Marc . Annual Review of Nuclear and Particle Science . 68 . 75–103 . 2018ARNPS..68...75P .
2. 1807.05250 . 10.1088/1361-6633/ab0b8f . The spin structure of the nucleon . 2019 . Deur . Alexandre . Brodsky . Stanley J. . De Téramond . Guy F. . Reports on Progress in Physics . 82 . 7 . 30818290 . 2019RPPh...82g6201D .
3. Adler . Stephen L. . 1966 . Sum Rules Giving Tests of Local Current Commutation Relations in High-Energy Neutrino Reactions . Physical Review . 143 . 4 . 1144–1155 . 10.1103/PhysRev.143.1144. 1966PhRv..143.1144A .
4. Baldin . A. M. . 1960 . Polarizability of nucleons . Nuclear Physics . en . 18 . 310–317 . 10.1016/0029-5582(60)90408-9. 1960NucPh..18..310B .
5. 1401.0140 . 10.1146/annurev-nucl-102313-025555 . Hadron Polarizabilities . 2014 . Holstein . Barry R. . Scherer . Stefan . Annual Review of Nuclear and Particle Science . 64 . 1 . 51–81 . 2014ARNPS..64...51H .
6. Bjorken . J. D. . 1966 . Applications of the Chiral ⁡(6)⊗⁡(6) Algebra of Current Densities . Physical Review . 148 . 4 . 1467–1478 . 10.1103/PhysRev.148.1467.
7. https://inspirehep.net/files/639d14249a23be3f8107142df1e6cd75 J. D. Bjorken (1970) “Inelastic scattering of polarized leptons from polarized nucleons”
8. Broadhurst. D. J. . Kataev. A. L.. 2002. Bjorken unpolarized and polarized sum rules: Comparative analysis of large N(F) expansions. Phys. Lett. B . 544. 1–2 . 154–160. 10.1016/S0370-2693(02)02478-4. hep-ph/0207261. 2002PhLB..544..154B . 17436687 .
9. Burkhardt . Hugh . Cottingham . W.N . 1970 . Sum rules for forward virtual compton scattering . Annals of Physics . en . 56 . 2 . 453–463 . 10.1016/0003-4916(70)90025-4. 1970AnPhy..56..453B .
10. nucl-th/9605031 . 10.1016/0375-9474(95)00217-O . Virtual compton scattering and generalized polarizabilities of the proton . 1995 . Guichon . P.A.M. . Liu . G.Q. . Thomas . A.W. . Nuclear Physics A . 591 . 4 . 606–638 . 1995NuPhA.591..606G .
11. hep-ph/9607217 . 10.1103/PhysRevD.55.4307 . Exact sum rule for transversely polarized DIS . 1997 . Efremov . A. V. . Teryaev . O. V. . Leader . Elliot . Physical Review D . 55 . 7 . 4307–4314 . 1997PhRvD..55.4307E .
12. Sum rule for deep-inelastic electroproduction from polarized protons . 10.1103/PhysRevD.9.1444 . 1974 . Ellis . John . Jaffe . Robert . Physical Review D . 9 . 5 . 1444–1446 . 1974PhRvD...9.1444E .
13. A dispersion theory of symmetry breaking . 10.1007/BF02824674 . 1965 . Fubini . S. . Furlan . G. . Rossetti . C. . Il Nuovo Cimento A . 40 . 4 . 1171–1193 . 1965NCimA..40.1171F .
14. https://inspirehep.net/literature/48796 S. B. Gerasimov (1965) “A sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei”
15. Exact Sum Rule for Nucleon Magnetic Moments . 10.1103/PhysRevLett.16.908 . 1966 . Drell . S. D. . Hearn . A. C. . Physical Review Letters . 16 . 20 . 908–911 . 1966PhRvL..16..908D . 1444298 .
16. Hosoda . Masataka . Yamamoto . Kunio . 1966 . Sum Rule for the Magnetic Moment of the Dirac Particle . Progress of Theoretical Physics . en . 36 . 2 . 425–426 . 10.1143/PTP.36.425 . free. 1966PThPh..36..425H .
17. Sum Rule for High-Energy Electron-Proton Scattering . 10.1103/PhysRevLett.18.1174 . 1967 . Gottfried . Kurt . Physical Review Letters . 18 . 25 . 1174–1177 . 1967PhRvL..18.1174G .
18. Gross . David J. . Smith . C.H.Llewellyn . 1969 . High-energy neutrino-nucleon scattering, current algebra and partons . Nuclear Physics B . en . 14 . 2 . 337–347 . 10.1016/0550-3213(69)90213-2. 1969NuPhB..14..337G .
19. Collins . John C. . Soper . Davison E. . 1982 . Parton distribution and decay functions . Nuclear Physics B . 194 . 3 . 445–492 . 10.1016/0550-3213(82)90021-9. 1982NuPhB.194..445C .
20. Ji. Xiangdong. 1997-01-27. Gauge-Invariant Decomposition of Nucleon Spin. Physical Review Letters. 78. 4. 610–613. 10.1103/PhysRevLett.78.610. hep-ph/9603249. 1997PhRvL..78..610J. 15573151.
21. Ji . X. D. . QCD Analysis of the Mass Structure of the Nucleon . Physical Review Letters . 1995 . 74 . 6 . 1071–1074 . 10.1103/PhysRevLett.74.1071 . 10058927 . hep-ph/9410274 . 1995PhRvL..74.1071J .
22. Ji . X. D. . Breakup of hadron masses and the energy-momentum tensor of QCD . Physical Review D . 1995 . 52 . 1 . 271–281 . 10.1103/PhysRevD.52.271 . 10019040 . hep-ph/9502213 . 1995PhRvD..52..271J .
23. Schwinger . Julian . 1975 . Source Theory Discussion of Deep Inelastic Scattering with Polarized Particles . Proceedings of the National Academy of Sciences of the United States of America . 72 . 4 . 1559–1563. 10.1073/pnas.72.4.1559 . free . 64895 . 16592243 . 432577 . 1975PNAS...72.1559S .
24. Wandzura . S. . Wilczek . F. . 1977 . Sum rules for spin-dependent electroproduction- test of relativistic constituent quarks . Physics Letters B . en . 72 . 2 . 195–198 . 10.1016/0370-2693(77)90700-6. 1977PhLB...72..195W .