Physics: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: [math]\displaystyle{ \int A(x) dx = B }[/math]
where [math]\displaystyle{ A(x) }[/math] is the dynamical quantity, e.g. a structure function characterizing a particle and [math]\displaystyle{ B }[/math] the static quantity, for example the mass or the charge of that particle. Quantum Field Theory sum rules should not be confused with QCD sum rules, which are different, nor Sum rule in quantum mechanics.
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 [math]\displaystyle{ B }[/math] is difficult to measure directly, measuring [math]\displaystyle{ A(x) }[/math] and integrating it offers a practical way to obtain [math]\displaystyle{ B }[/math] (providing that the particular sum rule linking [math]\displaystyle{ A(x) }[/math] to [math]\displaystyle{ B }[/math] is reliable).
Although in principle, [math]\displaystyle{ B }[/math] is a static quantity, the denomination of sum rule has been extended to the case where [math]\displaystyle{ B }[/math] 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).
- Bjorken sum rule.[4][5] 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: [math]\displaystyle{ \int_0^1 dx g_1^p(x)-g_1^n(x) = g_A/6 }[/math], where [math]\displaystyle{ x }[/math] is the Bjorken scaling variable, [math]\displaystyle{ g_1^{p(n)}(x) }[/math] is the first spin structure function of the proton (neutron), and [math]\displaystyle{ g_A }[/math] is the nucleon axial charge that characterizes the neutron β-decay. The sum rule was experimentally verified within better than a 10% precision.[2]
- [math]\displaystyle{ \delta_{LT} }[/math] sum rule.[7]
- Efremov-Teryaev-Leader sum rule.[8]
- Ellis-Jaffe sum rule.[9] 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.[7]
- Gerasimov-Drell-Hearn sum rule (GDH, sometimes DHG sum rule).[10][11][12] This is the polarized equivalent of the Baldin sum rule (see above). The GDH sum rule was experimentally verified (within a 10% precision).[2] Several generalized versions of the GDH sum rule have been proposed.[2]
- Gottfried sum rule.[13]
- Gross-Llewellyn Smith sum rule.[14]
- Momentum sum rule:[15] It states that the sum of the momentum fraction [math]\displaystyle{ x }[/math] of all the partons (quarks, antiquarks and gluons inside a hadron is equal to 1.
See also
References
- ↑ 1.0 1.1 B. Pasquini and M. Vanderhaeghen (2018) “Dispersion theory in electromagnetic interactions” Ann. Rev. Nucl. Part. Sci. 68, 75
- ↑ 2.0 2.1 2.2 2.3 2.4 2.5 A. Deur, S. J. Brodsky, G. F. de Teramond (2019) “The Spin Structure of the Nucleon” Rept. Prog. Phys. 82 076201
- ↑ A. M. Baldin (1960) “Polarizability of nucleons” Nucl. Phys. 18, 310
- ↑ J. D. Bjorken (1966) “Applications of the chiral U(6)×U(6) algebra of current densities” Phys. Rev. 148, 1467
- ↑ J. D. Bjorken (1970) “Inelastic scattering of polarized leptons from polarized nucleons” Phys. Rev. D 1, 1376
- ↑ H. Burkhardt and W. N. Cottingham (1970) “Sum rules for forward virtual Compton scattering” Annals Phys. 56, 453
- ↑ 7.0 7.1 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
- ↑ A. V. Efremov, O. V. Teryaev and E. Leader (1997) “Exact sum rule for transversely polarized DIS” Phys. Rev. D 55, 4307
- ↑ J. R. Ellis and R. L. Jaffe (1974) “Sum rule for deep-inelastic electroproduction from polarized protons” Phys. Rev. D 9, 1444 (1974)
- ↑ 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)]
- ↑ S. D. Drell and A. C. Hearn (1966) “Exact sum rule for nucleon magnetic moments” Phys. Rev. Lett. 16, 908
- ↑ M. Hosoda and K. Yamamoto (1966) “Sum rule for the magnetic moment of the Dirac particle” Prog. Theor. Phys. 36 (2), 425
- ↑ K. Gottfried (1967) “Sum rule for high-energy electron-proton scattering” Phys. Rev. Lett. 18, 1174
- ↑ D. J. Gross and C. H. Llewellyn Smith (1969) “High-energy neutrino-nucleon scattering, current algebra and partons” Nucl. Phys B14 337
- ↑ J. C. Collins and D. E. Soper (1982) “Parton distribution and decay functions” Nucl. Phys. B194 445
- ↑ J. S. Schwinger (1975) “Source Theory Discussion of Deep Inelastic Scattering with Polarized Particles” Proc. Nat. Acad. Sci. 72, 1
- ↑ 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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