Regge theory

The Regge theory , named after the Italian physicist and mathematician Tullio Regge , is a theoretical attempt to research aspects of elementary particle physics. In this experiment the essential new aspect is the continuation of the angular momentum into the complex. As a function of the energy, the complex angular momentum then describes a path called the regge trajectory . If one looks at the scattering of one particle on another and calculates the scattering amplitude, the mass and lifetime of a resonance state result from the real part and the imaginary part of a certain point on a regge trajectory, which is a pole of the scattering amplitude, called the regge pole .

example

The simplest example provides the quantum mechanical treatment of the binding or scattering of an electron with mass and charge to a proton with mass and charge . In contrast to elastic scattering, the binding energy of the electron to the proton is negative. The corresponding formula is: ${\ displaystyle m}$ ${\ displaystyle -e}$ ${\ displaystyle M}$ ${\ displaystyle + e}$ ${\ displaystyle E}$

{\ displaystyle E \ to E_ {N} = - {\ frac {2m '\ pi ^ {2} e ^ {4}} {h ^ {2} N ^ {2} (4 \ pi \ varepsilon _ {0 }) ^ {2}}} = - {\ frac {13 {,} 6 \, \ mathrm {eV}} {N ^ {2}}} = - {\ frac {R_ {y}} {N ^ { 2}}} {\ frac {m '} {m}}, \ qquad m' = {\ frac {mM} {M + m}},}

where the Rydberg energy , and the Planck's constant is. This formula was first set up in a more general form by Niels Bohr and about 10 years later it was justified quantum mechanically correctly in wave mechanics or by solving the Schrödinger equation for the Coulomb potential with the potential energy . In this calculation the principal quantum number is given by where is the radial quantum number and the secondary quantum number of the orbital angular momentum, and is the permittivity of the vacuum. Dissolve according to, one obtains the equation ${\ displaystyle R_ {y}}$ ${\ displaystyle N = 1,2,3, \ dots}$ ${\ displaystyle h}$ ${\ displaystyle V (r) = - e ^ {2} / (4 \ pi \ varepsilon _ {0} r)}$ ${\ displaystyle N}$ ${\ displaystyle N = n_ {r} + l + 1,}$ ${\ displaystyle n_ {r} = 0,1,2,3, \ dots}$ ${\ displaystyle l = 0,1,2,3, \ dots}$ ${\ displaystyle \ varepsilon _ {0}}$ ${\ displaystyle E_ {N}}$ ${\ displaystyle l}$

{\ displaystyle l \ mapsto l (E) = - n + g (E), \ qquad g (E) = - 1+ \ mathrm {i} {\ frac {\ pi e ^ {2}} {(4 \ pi \ varepsilon _ {0}) h}} \ left ({\ frac {2m '} {E}} \ right) ^ {1/2}.}

This expression - viewed as a complex function of - describes a path in the complex plane, which is called the Regge trajectory. The orbital angular momentum can thus assume any complex values on the Regge trajectory. Regge trajectories can be calculated for many other potentials, especially for the Yukawa potential , which describes the exchange of a meson in nucleon-nucleon scattering . Regge poles for Yukawa potentials, or the corresponding eigenvalues of the radial Schrödinger equation, were derived from perturbation theory by Harald JW Müller-Kirsten . ${\ displaystyle E}$

In quantum mechanics, scattering, like that of the electron on the proton, is described by the S matrix . In the case of the above example, this S matrix is given by the following expression:

{\ displaystyle S = {\ frac {\ Gamma (lg (E))} {\ Gamma (l + g (E))}} e ^ {\ mathrm {-} \ mathrm {i} \ pi l},}

where the gamma function is, i.e. the generalization of the faculty . This gamma function is a non-vanishing function with simple poles at The expression has poles (here the gamma function in the numerator), called regge poles, which are precisely given by the expression for the regge trajectories. ${\ displaystyle \ Gamma (x)}$ ${\ displaystyle n!}$ ${\ displaystyle x = 0, -1, -2, -3, \ dots.}$ ${\ displaystyle S}$

It was Regge who had the idea to consider the scattering amplitude or S-matrix of a particle reaction as an analytical function of the (above orbital) angular momentum.

Regge poles played an important role in the development of theoretical elementary particle physics. An important role of Regge trajectories found himself in the nucleon-nucleon scattering . If the antiparticle belonging to the nucleon designates , the high-energy behavior of this process, also called the -channel, is determined by the dominant regge trajectory of the so-called crossed process, also called the -channel . ${\ displaystyle N_ {1} + N_ {2} \ mapsto N_ {1} + N_ {2}}$ ${\ displaystyle N '}$ ${\ displaystyle N}$ ${\ displaystyle s}$ ${\ displaystyle t}$ ${\ displaystyle N_ {1} + N_ {2} '\ mapsto N_ {1} + N_ {2}'}$

Developing the idea of the Regge trajectories, GF Chew and SC Frautschi discovered around 1960 that the known mesons can be divided into families in which the spin (intrinsic angular momentum) is proportional to the square of the mass. Chew thus developed his “bootstrap theory” of elementary particles, in which there are no elementary particles according to a nuclear democracy. Although unsuccessful, this idea led to the Veneziano formula of an S matrix, named after Gabriele Veneziano (analogous to the above S matrix expressed in Euler's beta function), and this formula was fundamental for the further path to today's string theory . One can express this more concretely. One defines and analog and for the two possible crossed channels. Then the Veneziano formula has the form ${\ displaystyle s = (p_ {1} + p_ {2}) ^ {2}}$ ${\ displaystyle t}$ ${\ displaystyle u}$

{\ displaystyle S = {\ frac {\ Gamma (a-bs) \ Gamma (c-dt)} {\ Gamma (a + c-bs-dt)}},}

where are constants. By measuring in units of , and in units of , and renaming the constants, the formula is obtained in its standard form ${\ displaystyle a, b, c, d}$ ${\ displaystyle s}$ ${\ displaystyle 1 / b}$ ${\ displaystyle t}$ ${\ displaystyle 1 / d}$

{\ displaystyle S = {\ frac {\ Gamma (as) \ Gamma (bt)} {\ Gamma (a + bst)}}.}

This function has simple poles at the points

{\ displaystyle s = a, a + 1, a + 2, ..., \; \; \; t = b, b + 1, b + 2, ...}

.

The residual at the pole is and corresponds to a pole with zero spin. The residual at the pole is a linear function of ${\ displaystyle s = a}$ ${\ displaystyle 1}$ ${\ displaystyle s = a + 1}$ ${\ displaystyle t,}$

{\ displaystyle {\ frac {\ Gamma (bt)} {\ Gamma (bt-1)}} = bt-1,}

which corresponds to poles with spin and zero. In general, the residual of the pole corresponds to poles with spin . ${\ displaystyle 1}$ ${\ displaystyle s = n + a}$ ${\ displaystyle J = 0,1,2, ...}$

literature

Euan J. Squires: Complex Angular Momenta and Particle Physics . WA Benjamin, 1963.

Steven Frautschi : Regge-Poles and S-Matrix Theory . WA Benjamin, 1963.

Geoffrey F. Chew: S-Matrix Theory of Strong Interactions . WA Benjamin, 1962.

Tullio Regge : Introduction to Complex Orbital Momenta . In: Nuovo Cimento . tape 14 , 1959, pp. 951 ( physics.princeton.edu [PDF]).

Virendra Singh: Analyticity in the Complex Angular Momentum Plane of the Coulomb Scattering Amplitude . In: Phys. Rev. Band 127 , 1962, pp. 632 , doi : 10.1103 / PhysRev.127.632 .

Gabriele Veneziano: Elementary particles . In: Physics Today . tape September 22 , 1969, p. 31 , doi : 10.1063 / 1.3035780 .

Harald JW Müller-Kirsten: Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral . World Scientific, 2012, p. 395-414 .

Individual evidence

↑ HJW Müller: Regge poles in nonrelativistic potential scattering. In: Ann. d. Phys. (Leipz.) 15, 1965, pp. 395-411; HJW Müller, K. Schilcher: High-energy scattering for Yukawa potentials. In: J. Math. Phys. 9, 1968, pp. 255-259.

[1] HJW Müller: Regge poles in nonrelativistic potential scattering. In: Ann. d. Phys. (Leipz.) 15, 1965, pp. 395-411; HJW Müller, K. Schilcher: High-energy scattering for Yukawa potentials. In: J. Math. Phys. 9, 1968, pp. 255-259.