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In physics, a coupling constant or gauge coupling parameter , is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two static bodies to the "charges" of the bodies divided by the distance squared, r 2 {\displaystyle r^{2}} , between the bodies; thus: G {\displaystyle G} in F = G M m / r 2 {\displaystyle F=GMm/r^{2}} for Newton's gravity and k e {\displaystyle k_{\text{e}}} in F = k e q 1 q 2 / r 2 {\displaystyle F=k_{\text{e}}q_{1}q_{2}/r^{2}} for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers.
A modern and more general definition uses the Lagrangian L {\displaystyle {\mathcal {L}}} of a system. Usually, L {\displaystyle {\mathcal {L}}} of a system describing an interaction can be separated into a kinetic part T {\displaystyle T} and an interaction part V {\displaystyle V} : L = T − V {\displaystyle {\mathcal {L}}=T-V} .In field theory, V {\displaystyle V} always contains 3 fields terms or more, expressing for example that an initial electron interacted with a photon producing the final state of the electron. In contrast, the kinetic part T {\displaystyle T} always contains only two fields, expressing the free propagation of an initial particle into a later state.The coupling constant determines the magnitude of the T {\displaystyle T} part with respect to the V {\displaystyle V} part. For example, the electric charge of a particle is a coupling constant that characterizes an interaction with two charge-carrying fields and one photon field. Since photons mediate the electromagnetic force, this coupling determines how strongly electrons feel such a force, and has its value fixed by experiment. By looking at the QED Lagrangian, one sees that indeed, the coupling sets the proportionality between the kinetic term T = ψ ¯ ψ − 1 4 μ 0 F μ ν F μ ν {\displaystyle T={\bar {\psi }}\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }} and the interaction term V = − e ψ ¯ ψ {\displaystyle V=-e{\bar {\psi }}\psi }.