Chiral perturbation theory

Last updated July 26, 2023

Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation.^[1] ChPT is a theory which allows one to study the low-energy dynamics of QCD on the basis of this underlying chiral symmetry.

Goals

In the theory of the strong interaction of the standard model, we describe the interactions between quarks and gluons. Due to the running of the strong coupling constant, we can apply perturbation theory in the coupling constant only at high energies. But in the low-energy regime of QCD, the degrees of freedom are no longer quarks and gluons, but rather hadrons. This is a result of confinement. If one could "solve" the QCD partition function (such that the degrees of freedom in the Lagrangian are replaced by hadrons), then one could extract information about low-energy physics. To date this has not been accomplished. Because QCD becomes non-perturbative at low energy, it is impossible to use perturbative methods to extract information from the partition function of QCD. Lattice QCD is an alternative method that has proved successful in extracting non-perturbative information.

Method

Using different degrees of freedom, we have to assure that observables calculated in the EFT are related to those of the underlying theory. This is achieved by using the most general Lagrangian that is consistent with the symmetries of the underlying theory, as this yields the ‘‘most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry.^[2]^[3] In general there is an infinite number of terms which meet this requirement. Therefore in order to make any physical predictions, one assigns to the theory a power-ordering scheme which organizes terms by some pre-determined degree of importance. The ordering allows one to keep some terms and omit all other, higher-order corrections which can safely be temporarily ignored.

There are several power counting schemes in ChPT. The most widely used one is the $p$ -expansion where $p$ stands for momentum. However, there also exist the $\epsilon$ , $\delta ,$ and $\epsilon ^{\prime }$ expansions. All of these expansions are valid in finite volume, (though the $p$ expansion is the only one valid in infinite volume.) Particular choices of finite volumes require one to use different reorganizations of the chiral theory in order to correctly understand the physics. These different reorganizations correspond to the different power counting schemes.

In addition to the ordering scheme, most terms in the approximate Lagrangian will be multiplied by coupling constants which represent the relative strengths of the force represented by each term. Values of these constants – also called low-energy constants or Ls – are usually not known. The constants can be determined by fitting to experimental data or be derived from underlying theory.

The model Lagrangian

The Lagrangian of the $p$ -expansion is constructed by writing down all interactions which are not excluded by symmetry, and then ordering them based on the number of momentum and mass powers.

The order is chosen so that $(\partial \pi )^{2}+m_{\pi }^{2}\pi ^{2}$ is considered in the first-order approximation, where $\pi$ is the pion field and $m_{\pi }$ the pion mass, which breaks the underlying chiral symmetry explicitly (PCAC).^[4]^[5] Terms like $m_{\pi }^{4}\pi ^{2}+(\partial \pi )^{6}$ are part of other, higher order corrections.

It is also customary to compress the Lagrangian by replacing the single pion fields in each term with an infinite series of all possible combinations of pion fields. One of the most common choices is

U=\exp \left\{{\frac {i}{F}}{\begin{pmatrix}\pi ^{0}&{\sqrt {2}}\pi ^{+}\\{\sqrt {2}}\pi ^{-}&-\pi ^{0}\end{pmatrix}}\right\}

where $F$ is called the pion decay constant which is 93 MeV.

In general, different choices of the normalization for $F$ exist, so that one must choose the value that is consistent with the charged pion decay rate.

Renormalization

The effective theory in general is non-renormalizable, however given a particular power counting scheme in ChPT, the effective theory is renormalizable at a given order in the chiral expansion. For example, if one wishes to compute an observable to ${\mathcal {O}}(p^{4})$ , then one must compute the contact terms that come from the ${\mathcal {O}}(p^{4})$ Lagrangian (this is different for an SU(2) vs. SU(3) theory) at tree-level and the one-loop contributions from the ${\mathcal {O}}(p^{2})$ Lagrangian.)

One can easily see that a one-loop contribution from the ${\mathcal {O}}(p^{2})$ Lagrangian counts as ${\mathcal {O}}(p^{4})$ by noting that the integration measure counts as $p^{4}$ , the propagator counts as $p^{-2}$ , while the derivative contributions count as $p^{2}$ . Therefore, since the calculation is valid to ${\mathcal {O}}(p^{4})$ , one removes the divergences in the calculation with the renormalization of the low-energy constants (LECs) from the ${\mathcal {O}}(p^{4})$ Lagrangian. So if one wishes to remove all the divergences in the computation of a given observable to ${\mathcal {O}}(p^{n})$ , one uses the coupling constants in the expression for the ${\mathcal {O}}(p^{n})$ Lagrangian to remove those divergences.

Successful application

Mesons and nucleons

The theory allows the description of interactions between pions, and between pions and nucleons (or other matter fields). SU(3) ChPT can also describe interactions of kaons and eta mesons, while similar theories can be used to describe the vector mesons. Since chiral perturbation theory assumes chiral symmetry, and therefore massless quarks, it cannot be used to model interactions of the heavier quarks.

For an SU(2) theory the leading order chiral Lagrangian is given by^[1]

{\mathcal {L}}_{2}={\frac {F^{2}}{4}}{\rm {tr}}(\partial _{\mu }U\partial ^{\mu }U^{\dagger })+{\frac {\lambda F^{3}}{4}}{\rm {tr}}(m_{q}U+m_{q}^{\dagger }U^{\dagger })

where $F=93$ MeV and $m_{q}$ is the quark mass matrix. In the $p$ -expansion of ChPT, the small expansion parameters are

{\frac {p}{\Lambda _{\chi }}},{\frac {m_{\pi }}{\Lambda _{\chi }}}.

where $\Lambda _{\chi }$ is the chiral symmetry breaking scale, of order 1 GeV (sometimes estimated as $\Lambda _{\chi }=4\pi F$ ). In this expansion, $m_{q}$ counts as ${\mathcal {O}}(p^{2})$ because $m_{\pi }^{2}=\lambda m_{q}F$ to leading order in the chiral expansion.^[6]

Hadron-hadron interactions

In some cases, chiral perturbation theory has been successful in describing the interactions between hadrons in the non-perturbative regime of the strong interaction. For instance, it can be applied to few-nucleon systems, and at next-to-next-to-leading order in the perturbative expansion, it can account for three-nucleon forces in a natural way.^[7]

Related Research Articles

In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carriers of the theory, just as photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years.

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.

In particle physics, a pion is any of three subatomic particles: π⁰
, π⁺
, and π⁻
. Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more generally, the lightest hadrons. They are unstable, with the charged pions π⁺
and π⁻
decaying after a mean lifetime of 26.033 nanoseconds, and the neutral pion π⁰
decaying after a much shorter lifetime of 85 attoseconds. Charged pions most often decay into muons and muon neutrinos, while neutral pions generally decay into gamma rays.

The Standard Model of particle physics is the theory describing three of the four known fundamental forces in the universe and classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists worldwide, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, proof of the top quark (1995), the tau neutrino (2000), and the Higgs boson (2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.

Technicolor theories are models of physics beyond the Standard Model that address electroweak gauge symmetry breaking, the mechanism through which W and Z bosons acquire masses. Early technicolor theories were modelled on quantum chromodynamics (QCD), the "color" theory of the strong nuclear force, which inspired their name.

In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa.

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, $, between the bodies; thus: in for Newtonian gravity and in for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers.$

In particle physics, the Peccei–Quinn theory is a well-known, long-standing proposal for the resolution of the strong CP problem formulated by Roberto Peccei and Helen Quinn in 1977. The theory introduces a new anomalous symmetry to the Standard Model along with a new scalar field which spontaneously breaks the symmetry at low energies, giving rise to an axion that suppresses the problematic CP violation. This model has long since been ruled out by experiments and has instead been replaced by similar invisible axion models which utilize the same mechanism to solve the strong CP problem.

In nuclear physics, the chiral model, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of mesons in the chiral limit (where the masses of the quarks go to zero), but without necessarily mentioning quarks at all. It is a nonlinear sigma model with the principal homogeneous space of a Lie group $as its target manifold. When the model was originally introduced, this Lie group was the SU(N), where N is the number of quark flavors. The Riemannian metric of the target manifold is given by a positive constant multiplied by the Killing form acting upon the Maurer-Cartan form of SU(N).$

In quantum field theory, a quartic interaction is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field $satisfies the Klein-Gordon equation. If a scalar field is denoted, a quartic interaction is represented by adding a potential energy term to the Lagrangian density. The coupling constant is dimensionless in 4-dimensional spacetime.$

In quantum field theory and statistical mechanics, the 1/N expansion is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It consists in deriving an expansion for the properties of the theory in powers of $, which is treated as a small parameter.$

The QCD vacuum is the quantum vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter.

This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group $SU(3) \times SU(2) \times U(1)$ . The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.

In quantum field theory, the Nambu–Jona-Lasinio model is a complicated effective theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry, paralleling the construction of Cooper pairs from electrons in the BCS theory of superconductivity. The "complicatedness" of the theory has become more natural as it is now seen as a low-energy approximation of the still more basic theory of quantum chromodynamics, which does not work perturbatively at low energies.

In particle physics, chiral symmetry breaking is the spontaneous symmetry breaking of a chiral symmetry – usually by a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction. Yoichiro Nambu was awarded the 2008 Nobel prize in physics for describing this phenomenon.

In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.

In theoretical physics, the BRST formalism, or BRST quantization denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.

In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks. It plays the same role in quantum chromodynamics as the electromagnetic four-potential in quantum electrodynamics – the gluon field constructs the gluon field strength tensor.

The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where $plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and, are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.$

In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics.

References

1 2 Heinrich Leutwyler (2012), Chiral perturbation theory, Scholarpedia, 7(10):8708. doi : 10.4249/scholarpedia.8708
↑ Weinberg, Steven (1979-04-01). "Phenomenological Lagrangians". Physica A: Statistical Mechanics and Its Applications. 96 (1): 327–340. Bibcode:1979PhyA...96..327W. doi:10.1016/0378-4371(79)90223-1. ISSN 0378-4371.
↑ Scherer, Stefan; Schindler, Matthias R. (2012). A Primer for Chiral Perturbation Theory. Lecture Notes in Physics. Berlin Heidelberg: Springer-Verlag. ISBN 978-3-642-19253-1.
↑ Gell-Mann, M., Lévy, M., The axial vector current in beta decay, Nuovo Cim **16**, 705–726 (1960). doi : 10.1007/BF02859738
↑ J Donoghue, E Golowich and B Holstein, Dynamics of the Standard Model, (Cambridge University Press, 1994) ISBN 9780521476522.
↑ Gell-Mann, M.; Oakes, R.; Renner, B. (1968). "Behavior of Current Divergences under SU_{3}×SU_{3}" (PDF). Physical Review. 175 (5): 2195. Bibcode:1968PhRv..175.2195G. doi:10.1103/PhysRev.175.2195.
↑ Machleidt, R.; Entem, D.R. (2011). "Chiral effective field theory and nuclear forces". Physics Reports. 503 (1): 1–75. arXiv: 1105.2919 . Bibcode:2011PhR...503....1M. doi:10.1016/j.physrep.2011.02.001. S2CID 118434586.

External links

Howard Georgi, Weak Interactions and Modern Particle Theory, Benjamin Cummings, 1984; revised version 2008
H Leutwyler, On the foundations of chiral perturbation theory, Annals of Physics, 235, 1994, p 165-203.
Stefan Scherer, Introduction to Chiral Perturbation Theory, Adv. Nucl. Phys. 27 (2003) 277.
Gerhard Ecker, Chiral perturbation theory, Prog. Part. Nucl. Phys. 35 (1995), pp. 1–80.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[:0-1] 1 2 Heinrich Leutwyler (2012), Chiral perturbation theory, Scholarpedia, 7(10):8708. doi : 10.4249/scholarpedia.8708

[2] Weinberg, Steven (1979-04-01). "Phenomenological Lagrangians". Physica A: Statistical Mechanics and Its Applications. 96 (1): 327–340. Bibcode:1979PhyA...96..327W. doi:10.1016/0378-4371(79)90223-1. ISSN 0378-4371.

[3] Scherer, Stefan; Schindler, Matthias R. (2012). A Primer for Chiral Perturbation Theory. Lecture Notes in Physics. Berlin Heidelberg: Springer-Verlag. ISBN 978-3-642-19253-1.

[4] Gell-Mann, M., Lévy, M., The axial vector current in beta decay, Nuovo Cim **16**, 705–726 (1960). doi : 10.1007/BF02859738

[5] J Donoghue, E Golowich and B Holstein, Dynamics of the Standard Model, (Cambridge University Press, 1994) ISBN 9780521476522.

[6] Gell-Mann, M.; Oakes, R.; Renner, B. (1968). "Behavior of Current Divergences under SU_{3}×SU_{3}" (PDF). Physical Review. 175 (5): 2195. Bibcode:1968PhRv..175.2195G. doi:10.1103/PhysRev.175.2195.

[Machleidt-7] Machleidt, R.; Entem, D.R. (2011). "Chiral effective field theory and nuclear forces". Physics Reports. 503 (1): 1–75. arXiv: 1105.2919 . Bibcode:2011PhR...503....1M. doi:10.1016/j.physrep.2011.02.001. S2CID 118434586.

[1]

[2]

[3]

[4]

[5]

[6]

[7]