Sum rules (quantum field theory)

Short description: Relation between static and dynamic quantities

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:

∫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.

## Contents

* 1 Properties
* 2 List of sum rules
* 3 See also
* 4 References

## 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,[1] see the list of sum rules below.

## List of sum rules

(The list is not exhaustive)

* Baldin sum rule.[3] 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 ∫ν0∞σtot/ν2dν=4π2(α+β), where ν is the photon energy, ν0 is minimum value of energy necessary to create the lightest hadron (i.e. a pion), σtot is the photo-production cross-section, and α and β 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[4].)
* Bjorken sum rule (polarized).[5][6] 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: ∫01g1p(x)−g1n(x)dx=gA/6, where x is the Bjorken scaling variable, g1p(n)(x) is the first spin structure function of the proton (neutron), and gA 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).[7] The sum rule is, at leading order in perturbative QCD: ∫01F1pν(x,Q2)−F1pν¯(x,Q2)dx=∫01F1pν(x,Q2)−F1nν(x,Q2)dx=1−23αs(Q2)π, where F1pν(x,Q2),F1pν¯(x,Q2) and F1nν(x,Q2) are the first structure functions for the proton-neutrino, proton-antineutrino and neutron-neutrino deep inelastic scattering reactions, Q2 is the square of the 4-momentum exchanged between the nucleon and the (anti)neutrino in the reaction, and αs is the QCD coupling.
* Burkhardt–Cottingham sum rule.[8] The sum rule was experimentally verified.[2] The sum rule is "superconvergent", meaning that its form is independent of Q2. The sum rule is: ∫01g2(x,Q2)dx=0,∀Q2 where g2(x,Q2) is the second spin structure function of the object studied.
* δLT sum rule.[9]
* Efremov–Teryaev–Leader sum rule.[10]
* Ellis–Jaffe sum rule.[11] 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.[9]
* Fubini–Furlan–Rossetti Sum Rule.[12]
* Gerasimov–Drell–Hearn sum rule (GDH, sometimes DHG sum rule).[13][14][15] This is the polarized equivalent of the Baldin sum rule (see above). The sum rule is: ∫ν0∞(2σTT)/(ν)dν=−4π2ακ2S/mt2, where ν0 is the minimal energy required to produce a pion once the photon is absorbed by the target particle, σTT is the difference between the photon absorption cross-sections when the photons spin are aligned and anti-aligned with the target spin, ν is the photon energy, α is the fine-structure constant, and κ, S and mt 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: ∫01g1(x,Q2)dx=Q2S1(0,Q2)/8, where g1 is the first spin structure function of the target particle, x is the Bjorken scaling variable, Q2 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 S1(ν,Q2) is the first forward virtual Compton scattering amplitude. It can be argued that calling this relation sum rule is improper, since S1(ν,Q2) 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.[16]
* Gross–Llewellyn Smith sum rule.[17] It states that in the Bjorken scaling domain, the integral of the F3(x) structure function of the nucleon is equal to the number of valence quarks composing the nucleon, i.e., equal to 3. Specifically: ∫01F3(x)dx=3. 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:[18] It states that the sum of the momentum fraction x 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.[19]
* Proton mass sum rule:[20][21] 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.[22]
* Wandzura–Wilczek sum rule.[23]

## See also

* Quantum chromodynamics
* Proton spin crisis

## References

1. ↑ 1.0 1.1 B. Pasquini and M. Vanderhaeghen (2018) “Dispersion theory in electromagnetic interactions” Ann. Rev. Nucl. Part. Sci. 68, 75
2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 A. Deur, S. J. Brodsky, G. F. de Teramond (2019) “The Spin Structure of the Nucleon” Rept. Prog. Phys. 82 076201
3. ↑ A. M. Baldin (1960) “Polarizability of nucleons” Nucl. Phys. 18, 310
4. ↑ “Hadron Polarizabilities” Ann.Rev.Nucl.Part.Sci. 64 (2014) 51-81
5. ↑ J. D. Bjorken (1966) “Applications of the chiral U(6)×U(6) algebra of current densities” Phys. Rev. 148, 1467
6. ↑ J. D. Bjorken (1970) “Inelastic scattering of polarized leptons from polarized nucleons” Phys. Rev. D 1, 1376
7. ↑ 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. doi:10.1016/S0370-2693(02)02478-4. Bibcode: 2002PhLB..544..154B.
8. ↑ H. Burkhardt and W. N. Cottingham (1970) “Sum rules for forward virtual Compton scattering” Annals Phys. 56, 453
9. ↑ 9.0 9.1 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
10. ↑ A. V. Efremov, O. V. Teryaev and E. Leader (1997) “Exact sum rule for transversely polarized DIS” Phys. Rev. D 55, 4307
11. ↑ J. R. Ellis and R. L. Jaffe (1974) “Sum rule for deep-inelastic electroproduction from polarized protons” Phys. Rev. D 9, 1444 (1974)
12. ↑ S. Fubini, G. Furlan, and C. Rossetti (1965) “A dispersion theory of symmetry breaking” , Nuovo Cim. 40 1171.
13. ↑ 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) [Yad. Fiz. 2, 598 (1965)]
14. ↑ S. D. Drell and A. C. Hearn (1966) “Exact sum rule for nucleon magnetic moments” Phys. Rev. Lett. 16, 908
15. ↑ M. Hosoda and K. Yamamoto (1966) “Sum rule for the magnetic moment of the Dirac particle” Prog. Theor. Phys. 36 (2), 425
16. ↑ K. Gottfried (1967) “Sum rule for high-energy electron-proton scattering” Phys. Rev. Lett. 18, 1174
17. ↑ D. J. Gross and C. H. Llewellyn Smith (1969) “High-energy neutrino-nucleon scattering, current algebra and partons” Nucl. Phys B14 337
18. ↑ J. C. Collins and D. E. Soper (1982) “Parton distribution and decay functions” Nucl. Phys. B194 445
19. ↑ Ji, Xiangdong (1997-01-27). "Gauge-Invariant Decomposition of Nucleon Spin". Physical Review Letters 78 (4): 610–613. doi:10.1103/PhysRevLett.78.610. Bibcode: 1997PhRvL..78..610J.
20. ↑ Ji, X. D. (1995). "QCD Analysis of the Mass Structure of the Nucleon". Physical Review Letters 74 (6): 1071-1074. doi:10.1103/PhysRevLett.74.1071.
21. ↑ Ji, X. D. (1995). "Breakup of hadron masses and the energy-momentum tensor of QCD". Physical Review D 52: 271-281. doi:10.1103/PhysRevD.52.271.
22. ↑ J. S. Schwinger (1975) “Source Theory Discussion of Deep Inelastic Scattering with Polarized Particles” Proc. Natl. Acad. Sci. 72, 1
23. ↑ S. Wandzura and F. Wilczek (1977) “Sum rules for spin-dependent electroproduction: Test of relativistic constituent quarks” Phys. Lett. B 72, 195

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