(The list is not exhaustive) •
Adler sum rule. This sum rule relates the
charged current structure function of the
proton {{#tag:math|F_2^{\nu p}(Q^2,x)}} (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 {{#tag:math|\int_0^1 F_2^{\bar \nu p}(Q^2,x)-F_2^{ \nu p}(Q^2,x) \frac{dx}{x}= 2(1+\sin^2\theta_c)}}. The and superscripts indicate that relates to antineutrino-proton or neutrino-proton
deep inelastic scattering, respectively. •
Adler-Weisberger sum rule. It relates the
axial charge of the
nucleon to
pion-nucleon scattering quantities: {{#tag:math|1+\frac{F_\pi^2}{2\pi}\int_0^\infty \big[\sigma_{\pi^+p}(E) - \sigma_{\pi^-p}(E) \big]\frac{dE}{E}=g_A^2}}, where MeV is the pion decay constant, and {{#tag:math|\sigma_{\pi^+p} }} and {{#tag:math|\sigma_{\pi^-p} }} are the pion-proton scattering cross-sections for and , respectively. •
Baldin sum rule. 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 {{#tag:math|\int_{\nu_0}^\infty \sigma_{tot}/\nu^2 d\nu = 4\pi^2(\alpha+\beta)}}, where is the photon energy, is minimum value of energy necessary to create the lightest hadron (i.e. a
pion), {{#tag:math|\sigma_{tot} }} 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.) •
Bass-Brodsky-Schmidt sum rule: The sum rule states that the integral of the spin-dependent structure function of a real
photon, {{#tag:math|g_1^{\gamma}(x,Q^2)}} vanishes for any value of the momentum transfer the photon is probed with: {{#tag:math|\int_0^1 g_1^{\gamma}(x,Q^2) dx= 0}}. The relation is non-perturbative and holds for abelian and non-abelian gauge theories. •
Bjorken sum rule (polarized). 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, {{#tag:math|g_1^{p(n)}(x)}} 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. The sum rule is, at
leading order in
perturbative QCD: \int_0^1 F_1^{p\nu}(x,Q^2)-F_1^{p\bar{\nu}}(x,Q^2) dx= \int_0^1 F_1^{p\nu}(x,Q^2)-F_1^{n\nu}(x,Q^2) dx =1-\frac{2}{3}\frac{\alpha_s(Q^2)}{\pi}, where F_1^{p\nu}(x,Q^2),~F_1^{p\bar{\nu}}(x,Q^2) and F_1^{n\nu}(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. The sum rule was experimentally verified. •
Close-Kumano sum rule. states that in the Bjorken limit with finite, the integral over the first tensor structure function of the
deuteron, , vanishes {{#tag:math|\int_0^1 b_1^{\gamma}(x) dx= 0}}. •
Efremov–Teryaev–Leader sum rule. •
Ellis–Jaffe sum rule. The sum rule was shown to not hold experimentally, •
Gerasimov–Drell–Hearn sum rule (GDH, sometimes DHG sum rule). This is the polarized equivalent of the Baldin sum rule (see above). The sum rule is: {{#tag:math|\int_{\nu_0}^\infty (2\sigma_{TT})/(\nu)d\nu = -4\pi^2\alpha\kappa^2S/m_t^2}}, where is the minimal energy required to produce a
pion once the photon is absorbed by the target particle, {{#tag:math|\sigma_{TT} }} 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). The sum rule states that the integral weighted by of the unpolarized
structure function of the proton minus that of the neutron is related to the
flavor asymmetry of the
sea quarks: {{#tag:math|I_{G} = \int_0^1 F_2^{p}(x,Q^2)-F_2^{n}(x,Q^2) \frac{dx}{x}= \frac{1}{3} - \frac{2}{3}\int_0^1 \bar d(x,Q^2) - \bar u(x,Q^2)dx}}. Assuming a flavor symmetric sea yields the Gottfried sum rule proper, {{#tag:math|I_{G} = 1/3 }}, which has been ruled out by measurements, yielding the first clear evidence for flavor asymmetry in the nucleon sea. •
Gross–Llewellyn Smith sum rule. 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. The Gross–Llewellyn Smith integral was measured by the CCFR experiment at
Fermilab. •
Momentum sum rule: 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 (GPD) to the total
angular momentum carried by the
quarks or by the
gluons. It reads {{#tag:math|\frac{1}{2}\int_{-1}^1 x \big[E_{q,g}(x,0,0) + H_{q,g}(x,0,0) \big] dx = J_{q,g} }}, where {{#tag:math|J_{q,g} }} is the quark () or gluon () total angular momentum, {{#tag:math|E_{q,g} (x,\xi,t)}} is the quark or gluon unpolarized helicity-conserving GDP and {{#tag:math|H_{q,g} (x,\xi,t)}} is the quark or gluon unpolarized helicity-flip GDP. The kinematics variables
skewness and
Mandestam's are set to zero. •
Proton mass sum rule: 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. The Schwinger sum rule is a theoretical result involving the scattering of polarized leptons off polarized target particles. It reads: {{#tag:math|\frac{8M^2}{Q^2}\int_0^{x_0} g_1(x,Q^2)+g_2(x,Q^2) dx \to \kappa e \text{ as } Q^2 \to 0}}, where is the
mass of the target particle, the square of the absolute value of the
four-momentum transferred to the target particle during the scattering process, the
Bjorken scaling variable, the -value for the minimal energy required to produce a
pion off the target particle, and and the first and second spin
structure functions of the target particle, respectively. The limit is for , with the
anomalous magnetic moment of the target particle and its
charge. The integrand of the sum rule can also be expressed with the -weighted transverse-longitudinal interference cross-section, {{#tag:math|\sigma_{LT}/Q}}. This makes it similar to the generalized GDH sum rule. and although experimental uncertainties exist, it was found to hold, provided the GDH sum rule also holds. •
Wandzura–Wilczek sum rule. ==See also==