A Short Guide to Some Quantum Field Theoretical Models R. Rosenfelder Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Abstract There is a wide variety of field theoretical models which - although unrealistic - are very useful for investigating characteristic phenomena in particle physics or for testing various approximation schemes. These include, in particular, exactly solvable models mostly in lower space-time dimensions. For convenience the following models are therefore listed and shortly explained (ordered according to complexity and/or physical relevance): I. 0-dimensional models 1. Classical partition function for the anharmonic oscillator II. 0+1-dimensional models 1. Quantum-mechanical anharmonic oscillator III. 3-dimensional non-relativistic models 1. Many-body system with two-body potentials 2. Electron gas (“jellium”) 3. Polaron model IV. 1+1-dimensional relativistic models 1. sine-Gordon model 2. CPN model 3. Gross-Neveu model 4. Thirring model 5. Federbush model 6. Rothe-Stamatescu model 7. Schwinger model (QED1+1 ) 8. ’t Hooft model (QCD1+1 ) V. 2+1-dimensional relativistic models 1. Nonlinear O(3)-model in two space dimensions 2. QED2+1
1
VI. 3+1-dimensional relativistic models 1. Scalar models a) Zachariasen model b) Φ3 -model c) Wick-Cutkosky model d) Φ4 -model 2. Sigma models a) Linear σ-model b) Nonlinear σ-model c) Skyrme model 3. Yukawa model 4. Walecka model 5. Nambu-Jona-Lasinio model 6. Quark models of hadrons a) Bag model b) Lee-Friedberg model c) Dyson-Schwinger approach to QCD 7. Scalar QED 8. Higgs models a) Abelian Higgs model b) Georgi-Glashow model (non-abelian Higgs model) 9. Lee-Wick model Some comments on the importance and applicability of the various models are given and references to the original work as well as to recent reviews or textbooks are supplied.
0. Introduction As far as the models refer to reality, they are not manageable; and as far as they are manageable, they do not refer to reality
Variation of a quote by A. Einstein (1921)
In the shadow of the Standard “Model” (i.e. Theory) an abundance of field theoretical models exists which either are solvable or have certain features which a more realistic theory is also supposed to exhibit. Historically some of these models have been essential to establish or investigate characteristic phenomena in particle physics. Even today they continue to play a role as testing ground for theoretical methods or as phenomenological models. For the beginner as well as for the practitioner of Quantum Field Theory it may be therefore useful to have a simple compendium of the most frequently encountered models – mainly in particle/nuclear physics – together with a summary of their properties and further references 1 . 1
Undoubtedly the experts will find many omissions or other shortcomings in the presentation for which I apologize in advance.
2
This will be provided in the following together with a rating of the importance of each model. The full or half stars ( ? or (?) ) awarded by me should indicate the importance (in particle physics) but are, of course, as subjective and debatable as the ones in the Guide Michelin 2 and don’t take into account the role of the particular model in other fields, e.g. condensed matter physics. The references are neither exhaustive nor supposed to be historically correct but just offer convenient background material. For solvable models they contain details of the solutions which are not presented here. Of course, there are many variants of the presented models and the listing is far from being complete. Not included are the well-known parts of the Standard ”Model” like QED3+1 , QCD3+1 and the electroweak sector (which all should be rated ? ? ? ? ? . . .) and extensions of the Standard Theory, such as Grand Unified “Theories”, supersymmetric models etc. Also omitted are models based on Dirac’s light or front form of relativistic dynamics and discrete/lattice models.
Conventions: Minkowski space-time with diag gµν = (1, −1, −1, . . .) is used throughout as well as units with ~ = c = 1 and other Bjorken-Drell conventions. Scalar fields are denoted by Φ, χ . . . and assumed to be real/uncharged, (Dirac) fermionic fields by Ψ. The extension to charged fields or N -component fields is in most cases straightforward: Φ
2
†
−→ Φ Φ ,
¯ −→ ΨΨ
N X
¯ a Ψa Ψ
etc.
a=1
The summation convention is used for Lorentz indices (µ, ν . . .) but not for internal indices (a, b, . . .).
Acknowledgement: I would like to thank Michael Spira for reading the manuscript and for helpful comments.
2
There is no reason for theoreticians attached to a particular, low-rated model to behave in the same way as some chefs de cuisine whose restaurant has lost a star (i.e. to shoot themselves ...).
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I.
0-dimensional models
I.1
Classical partition function for the anharmonic oscillator
Partition function:
=
+∞
2 dxdp p m exp −β + ω 2 x2 + λx4 2π 2m 2 −∞ Z +∞ 1 2 λ 1 4 √ dΦ exp − Φ − gΦ , g = , ω2 > 0 2 βm2 ω 4 βω 2π −∞
Z Z(β, λ) =
(I.1a) (I.1b)
Properties: 0-dimensional model for the euclidean (vacuum) generating functional of Φ4 -theory. Power series in g or λ diverges but is Borel summable. Exact integral can be expressed in terms of parabolic cylinder or Whittaker functions. Importance: (?) Serves as illustration for large orders of perturbation theory. References: J. W. Negele and H. Orland: Quantum Many-Particle Systems, Addison-Wesley (1988), chapter 2.1, 7.5 [textbook] M. Abramowitz and I. A. Stegun (eds.): Handbook of Mathematical Functions, Dover (1965), ch. 19, eq. 19.5.3 [parabolic cylinder functions]
II.
0+1-dimensional models
II.1
Quantum-mechanical anharmonic oscillator
Lagrangian: 1 L = 2
dΦ dt
2 − V (Φ) ,
V (Φ) =
1 2 2 ω Φ + λΦ4 2
(II.1a)
Properties: Field theory in 0 space- and 1 time-dimension is formally equivalent to one-dimensional stationary quantum mechanics by substituting t → x and Φ → wavefunction. Exactly solvable by (numerical) solution of Schr¨ odinger equation. E(λ) ∼ λ1/3 for large λ. The perturbative series for the ground state energy is divergent for all values of λ but Borel summable. For ω 2 < 0 the potential may be written as 2 |ω 2 | ω4 V (Φ) = λ Φ2 − − (II.1b) 4λ 16λ 4
and instanton solutions (“kinks”) are found. These allow calculation of the energy splitting of the degenerate ground state by tunneling processes ∆E ∼ exp (−constant/λ) for small λ (no spontaneous symmetry breaking is possible in Quantum Mechanics). Importance: ?(?) Serves as testing ground for approximation methods in Φ4 -theory. References: C. M. Bender and T. T. Wu, Phys. Rev. 184 (1969) 1231; Phys. Rev. D 7 (1972) 1620; L. N. Lipatov, J.E.T.P. Lett. 25 (1977) 104 [large orders of perturbation theory] J. W. Negele and H. Orland: Quantum Many-Particle Systems, Addison-Wesley (1988), ch. 7.5 [textbook] E. Gildener and A. Patrascioiu, Phys. Rev. D 16 (1977) 423 effects] Ch.-Sh. Hsue and J. L. Chern, Phys. Rev. D 29 (1984) 643
III.
3-dimensional non-relativistic models
III.1
Many-body system with two-body potentials
[energy splitting by instanton
[numerical values for energies]
Hamiltonian: N N X p2i 1X + V (xi − xj ) 2m 2 i=1 i6=j Z Z ∆ 1 3 † = ˆ d x Φ (x) − Φ(x) + d3 xd3 x0 Φ† (x)Φ† (x0 ) V (x − x0 ) Φ(x0 )Φ(x) . 2m 2
H =
(III.1a) (III.1b)
This ”Standard Model of Many-Body Physics” describes a system of (N in the ”first-quantized” form (III.1a)) non-relativistic, identical particles with mass m interacting via a local two-body potential V (xi − xj ). The particles may be either bosons or fermions (canonical commutation or anticommutation rules for the field operators) and carry additional internal quantum numbers (spin, isospin etc.). Properties: Finite due to nonlocality of interaction term (if potential is sufficiently well-behaved), cf. Φ4 -model. The total number of particles is conserved. Galilei-invariant. Importance: ? ? ? ? ? Applies to nearly all of nuclear, atomic, molecular, solid state physics (with appropriate potential). In some cases additional many-body forces are needed, e.g. 3-body forces for the accurate description of light nuclei. 5
References: A. L. Fetter and J. D. Walecka: Quantum Theory of Many-Particle Systems, McGrawHill (1971) [textbook] J. W. Negele and H. Orland: Quantum Many-Particle Systems, Addison-Wesley (1988) [textbook]
III.2
Electron gas (“jellium”)
Hamiltonian: H =
N N X p2i e2 X exp (−µ|ri − rj |) + 2m 2 |ri − rj | i=1
−e2
i6=j
N V
N Z X i=1
d3 x
exp (−µ|x − ri |) e2 N 2 + |x − ri | 2 V2
Z
d3 x d3 x0
exp (−µ|x − x0 |) . (III.2) |x − x0 |
Describes system of N electrons interacting with each other (second term) and a positively charged uniform background (third term). The last term, the interaction energy of the background, is a c-number. V denotes the quantization volume and µ the parameter for the shielding of the long-range Coulomb interaction (→ 0 at the end). The thermodynamical limit is to be taken: N, V → ∞ with the density N/V = n fixed. Properties: The dimensionless coupling constant rs = me2 (3/(4πn))1/3 is the ratio of interparticle spacing to Bohr radius. The energy per particle is a non-analytic function of this coupling constant: In the high-density limit (rs → 0) the ring diagrams have to be summed to avoid singularities from the Coulomb interaction, in the low-density limit (rs → ∞) the electrons crystallize into a “Wigner solid”. Importance: ?? Describes electrons in metals, storage rings etc. References: A. L. Fetter and J. D. Walecka: Quantum Theory of Many-Particle Systems, McGrawHill (1971), sections 3, 12 [textbook] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45 (1980) 566 [Monte-Carlo solution]
III.3
Polaron model
Hamiltonian: 1 H = p2 + 2
Z
√ 1/2 d3 k † a a + i 2 2πα k (2π)3 k 6
Z
i d3 k 1 h † −ik·x a e − h.c. . k (2π)3 |k|
(III.3)
Describes an electron of unit mass which slowly moves through a polarizable crystal. Lattice distortions produce phonons (of unit frequency and without dispersion, i.e. ωk = 1) which act back on the electron changing its energy and mass. α is the dimensionless coupling constant between the electron and the phonons. Properties: Finite, non-trivial field theory describing a dressed electron = bare electron + cloud of vitual phonons. Not Galilei-invariant since phonon number is not conserved. Electron momentum is limited to |p| < 2 by Cerenkov radiation of real phonons. Weak- and strong-coupling P α→∞ α→0 expansions for the ground state energy E0 −→ −α + k=2 ek αk and E0 −→ −0.108512 α2 + P −2k have been calculated. k=0 fk α Importance: ?? Testing ground for many-body techniques including Feynman’s path-integral approach: integrating out the phonons + variational approximation. Solid state physics: mobility of electrons in ionic crystals, bi-polarons as possible mechanism for high-Tc superconductivity. References: H. Fr¨ ohlich, Philos. Mag. Suppl. 3 (1954) 325 [original] R. P. Feynman, Phys. Rev. 97 (1955) 660 [path integral approach] T. K. Mitra, A. Chatterjee and S. Mukhopadhyay, Phys. Rep. 153 (1987) 91 [review] C. Alexandrou and R. Rosenfelder, Phys. Rept. 215 (1992) 1 [Monte-Carlo solution] R. Rosenfelder, Phys. Rev. E 79 (2009) 016705 (arXiv:0805.4525) [high orders in pert. theory]
IV.
1 + 1-dimensional relativistic models
IV.1
sine-Gordon model
Lagrangian: " 4 1 m L = ∂µ Φ ∂ µ Φ + cos 2 λ
√
λ Φ m
!
# −1
(IV.1a)
Properties: Has soliton solutions. Name comes from a pun referring to the ”Klein-Gordon” equation to which the equation of motion ! √ m3 λ Φ = 0 (IV.1b) Φ + √ sin m λ reduces for small λ . Importance: ?
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References: J. Rubinstein, J. Math. Phys. 11 (1970) 258 [original naming: see footnote 7] R. Rajaraman: Solitons and Instantons, North Holland (1982), ch. 2.5 [textbook]
IV.2
CPN model
Lagrangian: L = (∂µ n)? · (∂ µ n) + (n? · ∂µ n) (n? · ∂ µ n) ,
(IV.2a)
with constraint n(x)? · n(x) =
N +1 X
n?a (x) na (x) = 1
(IV.2b)
a=1
Consists of N + 1 complex scalar fields (forming an N-dimensional Complex Projective space). Properties: Locally gauge invariant, has instanton solutions in euclidean space-time for all N . Importance: ? References: R. Rajaraman: Solitons and Instantons, North Holland (1982), ch. 4.5 [textbook]
IV.3
Gross-Neveu model
Lagrangian: ¯ i∂/ Ψ + 1 g 2 ΨΨ ¯ 2 . L1 = Ψ 2
(IV.3a)
The U (N ) symmetric version is LN
=
N X a=1
1 Ψ i∂/ Ψ + g 2 2 ¯a
a
N X
!2 ¯a
a
Ψ Ψ
.
(IV.3b)
a=1
A version which is invariant under Ψ → exp(iαγ5 )Ψ h i ¯ /Ψ + 1 g 2 ΨΨ ¯ 2 + Ψγ ¯ 5Ψ 2 L01 = Ψi∂ 2 is called the “chiral Gross-Neveu model”.
(IV.3c)
Properties: L1 is invariant under a discrete chiral transformation Ψ → γ5 Ψ. Spontaneous symmetry breaking has been shown to occur in the limit N → ∞. Importance: ?(?) References: D. J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235 [original] M. Thies, J. Phys. A 39 (2006) 12707 (hep-th/0601049) [phase diagram in large-N limit]
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IV.4
Thirring model
Lagrangian: ¯ µΨ ¯ (i∂/ − mF ) Ψ − 1 g Ψγ ¯ µ Ψ Ψγ L = Ψ 2
(IV.4)
Properties: For mF = 0 (massless Thirring model) exactly solvable. The charge zero sector of the massive Thirring model is equivalent to the sine-Gordon model. Importance: ? References: W. Thirring, Ann. Phys. (N.Y.) 3 (1958) 91 [original] S. Coleman, Phys. Rev. D 11 (1975) 2088 [equivalence] R. Rajaraman: Solitons and Instantons, North Holland (1982), ch. 7.3 [textbook] N. Ilieva and W. E. Thirring, hep-th/9808118 [mathematical survey]
IV.5
Federbush model
Lagrangian: ¯ 1 ( i∂/ − m1 ) Ψ1 + Ψ ¯ 2 ( i∂/ − m2 ) Ψ2 − g µν j 1 j 2 L = Ψ µ ν ¯ a (x)γµ Ψa (x) , jµa (x) = Ψ
a = 1, 2
(IV.5a) (IV.5b)
where µν is the total antisymmetric tensor in 2 dimensions. Describes a (vector-)current-current coupling between two fermionic fields Ψ1 , Ψ2 of mass m1 , m2 . Properties: Exactly solvable by regularizing the singular currents with point splitting: 1 h ¯a ¯a i jµa (x) = lim Ψ x+ γµ Ψa x − +Ψ x− γµ Ψa x + →0 2 2 2 2 2
(IV.5c)
Importance: ? References: K. Federbush, Phys. Rev. 121 (1961) 1247 [original] L. Martinoviˇc, Few-Body Syst. 55 (2014) 527 [solution]
IV.6
Rothe-Stamatescu model
Lagrangian: ¯ ( i∂/ − m ) Ψ + 1 ( ∂µ Φ )2 − 1 µ2 Φ2 − g ∂µ Φ Ψγ ¯ µγ5Ψ . L = Ψ (IV.6a) 2 2 Describes fermions (originally massless, in general with mass m) and pseudoscalar ”mesons” (mass µ) interacting via a gradient coupling between the pseudoscalar and the axial-vector of the fermionic field. 9
Properties: Exactly solvable Importance: ? References: K. D. Rothe and I. O. Stamatescu, Ann. Phys. 95 (1975) 202
IV.7
[original]
Schwinger model (QED1+1 )
Lagrangian: 1 1 ¯ (i∂/ − eγµ Aµ − m) Ψ L1 = − Fµν F µν − (∂µ Aµ )2 + Ψ 4 2ξ Fµν = ∂µ Aν − ∂ν Aµ .
(IV.7a) (IV.7b)
Describes electrons and “photons” in 1 + 1 dimensions: Quantum Electro Dynamics. e (with mass dimension 1) is the coupling between electrons and “photons”, ξ the gauge fixing parameter. ¯ µ Aµ (1 + iγ5 )Ψ is called the The massless version where the coupling to the photon field is Ψeγ chiral Schwinger model. Properties: The massless Schwinger model is exactly solvable and an example of dynamical sym√ metry breaking: the “photon” acquires a mass e/ π. For the massive model, weak (e m ) and strong (e m ) expansions are available. Importance: ? ? (?) References: J. Schwinger, Phys. Rev. 128 (1962) 2425 [original] S. Coleman, Ann. Phys. 101 (1976) 239 [massive model] E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B 190 (1981) 734 [Monte Carlo simulation] N. K. Falck and G. Kramer, Ann. Phys. (N. Y.) 176 (1987) 330 [chiral Schwinger model]
IV.8
’t Hooft model (QCD1+1 )
Lagrangian: 1 ¯ (i∂/ + gγµ Aµ ) Ψ L = − tr (Fµν F µν ) + Ψ 4 Fµν = ∂µ Aν − ∂ν Aµ − g [Aµ , Aν ] , Aµ = Aaµ T a
(IV.8a) (IV.8b)
Describes the interaction of Nc coloured, massless quarks and gluons in 1 + 1 dimensions: Quantum Chromo Dynamics. g is the quark-gluon coupling constant, T a denote matrix representations of the generators of the gauge group SU (Nc ). 10
Properties: Non-abelian local gauge theory. Can be solved in the limit Nc → ∞, g 2 Nc = fixed (only planar diagrams contribute). Exhibits confinement and chiral symmetry breaking. The meson spectrum can be determined. Importance: ?? References: G. ’t Hooft, Nucl. Phys. B 72 (1974) 461 [original] G. ’t Hooft, Nucl. Phys. B 75 (1974) 461 [meson spectrum] V. Sch¨ on and M. Thies, Phys. Rev. D 62 (2000) 096002 [hep-th/0003195]
V.
2 + 1-dimensional relativistic models
V.1
Nonlinear O(3)-model in two space dimensions
[finite density]
Lagrangian : L =
3 1 X ∂µ Φa ∂ µ Φa , 2
with constraint
a=1
3 X
Φa Φa = 1 .
(V.1a)
a=1
Importance: ? Describes the statistical mechanics of an isotropic ferromagnet. Properties: Has classical soliton solutions characterized by a topological quantum number for O(3) but not for O(N > 3) (cf. ”CPN -model”). Energy is bounded in each sector by 4π|Q| where Q is the winding number. References: R. Rajaraman: Solitons and Instantons, North Holland (1982), ch. 3.3 [textbook]
V.2
QED2+1
Lagrangian : Nf
X 1 ¯ a ( i∂/ − eγ µ Aµ − ma ) Ψa L = − Fµν F µν + Ψ 4
(V.2a)
a=1
Importance: ?? Effective theory for strongly interacting fermionic systems like graphene and high temperature cuprate superconductors, testing bed for QCD approximations.
11
Properties: Super-renormalizable, confining (plausible since in 2 dimensions the Coulomb potential grows logarithmically with distance). Dynamical chiral symmetry breaking can occur if the number Nf of massless fermions is smaller than a critical value. References: T. Appelquist, D. Nash and L. C. R. Wijewardhana, Phys. Rev. Lett. 60 (1988) 2575 [critical fermion number] J. Braun, H. Gies, L. Janssen and D. Roscher, Phys. Rev. D 90 (2014) 036002 [arXiv:1404.1362] [many flavor phase diagram] O. Vafek, A. Vishwanath, Ann. Rev. Cond. Mat. Phys. vol. 5 (2014)83 [arXiv:1306.2272] [applications in condensed matter physics]
VI.
3 + 1-dimensional relativistic models
VI.1
Scalar models
VI.1.a
Zachariasen model
Lagrangian: 1 1 L = (∂µ Φ)2 − m2 Φ2 + 2 2
Z
∞
ds
s0
Z ∞ 1 1 2 2 ds ρ(s) χs . (∂µ χs ) − s χs − gΦ 2 2 s0
(VI.1a)
Describes a scalar particle Φ decaying into and re-emerging from a (S state) particle-antiparticle √ pair χs with continous mass s . s0 denotes the threshold of pair production and ρ(s) is a (free) spectral function. Properties: Exactly solvable Importance: (?) References: F. Zachariasen, Phys. Rev. 121 (1962) 1851 [original but hard to find there ...] W. Thirring, Phys. Rev. 126 (1962) 1209 [Lagrangian formulation] T. Mannel, T. Ohl and P. Manakos, Z. Phys. A 335 (1990) 341 [bounds on coupling constants] N. V. Krasnikov, Int. J. Mod. Phys. A 22 (2007) 5117 [hep-ph/0707.1419] [connection with ”unparticle” models]
12
VI.1.b
Φ3 -model
Lagrangian: 1 1 g (∂µ Φ)2 − M 2 Φ2 − Φ3 . (VI.1b) 2 2 3! Describes scalar particles with mass M interacting by a 3-particle interaction with coupling constant g. L =
Properties: Renormalizable but unstable since the ”potential” Φ3 is not bounded from below. Asymptotically free in d = 6 dimensions. For g0 → ig0 a PT-symmetric field theory is obtained which seems to be stable and trivial in d = 4 dimensions. Importance: ? Textbook example for renormalization. References J. C. Collins: Renormalization, Cambridge University Press (1984), ch. 3 [textbook] G. Sterman: An Introduction to Quantum Field Theory, Cambridge University Press (1993), ch. 10 [textbook] M. Srednicki: Quantum Field Theory, Cambridge University Press (2007), part 1 [hep-th/0409035] [nice textbook, but ”wrong” metric] C. M. Bender, V. Branchina and E. Messina, Phys. Rev. D 85 (2012) 085001 [arXiv:1201.1244] [PT-symmetric Φ3 quantum field theory]
VI.1.c
Wick-Cutkosky model
Lagrangian: 1 1 1 1 (∂µ Φ)2 − M 2 Φ2 + (∂µ χ)2 − µ2 χ2 + g Φ2 χ . (VI.1c) 2 2 2 2 Describes scalar particles (Φ) with mass M interacting by exchange of scalar particles (χ) with mass µ (very often µ = 0 is taken: the original model). A variant is L1 =
L2 =
1 1 1 1 (∂µ Φ1 )2 − M12 Φ21 + (∂µ Φ2 )2 − M22 Φ22 2 2 2 2 1 1 2 2 2 2 2 + (∂µ χ) − µ χ + g1 Φ1 χ + g2 Φ2 χ 2 2
describing two different species Φ1,2 with different masses and coupling constants. Properties: Super-renormalizable but unstable since the ”potential” Φ2 χ is not bounded from below. The nonrelativistic limit gives an attractive Yukawa potential between the Φ-particles. Importance: ?? Extensively used in relativistic bound state calculations (mostly in the ladder approximation to the Bethe-Salpeter equation). 13
References: G. C. Wick, Phys. Rev. 96 (1954) 1124; R. E. Cutkosky, Phys. Rev. 96 (1954) 1135 [original] G. Baym, Phys. Rev. 117 (1960) 886 [instability] C. Itzykson and J.-B. Zuber: Quantum Field Theory, McGraw-Hill (1980), ch. 10-2 [textbook, relativistic bound states] R. Rosenfelder and A. W. Schreiber, Phys. Rev. D 53 (1996) 3337, 3354 [nucl-th/9504002,954005] [variational approach]
VI.1.d
Φ4 -model
Lagrangian: 1 1 λ (∂µ Φ)2 − M 2 Φ2 − Φ4 . (VI.1d) 2 2 4! Describes scalar particles with mass M interacting by a 4-particle interaction with coupling constant λ. L =
Properties: Renormalizable but not asymptotically free for λ > 0 which is required for stability. Very likely “trivial” when cut-off is sent to infinity. In the non-relativistic limit a repulsive δ 3 potential between the particles emerges which also requires that the physical coupling constant vanishes. Exhibits “spontaneous symmetry breaking” for M 2 < 0. Importance: ? ? ? ? Higgs-physics in the Standard Model, Landau-Ginzburg theory of superconductivity, text-book example for perturbation theory, Feynman diagrams etc. References: M. E. Peskin and D. V. Schroeder: An Introduction to Quantum Field Theory, AddisonWesley (1995), ch. 4.1 [textbook] M.A.B. Beg, R.C. Furlong, Phys. Rev. D 31 (1985) 1370 [non-relativistic limit]
VI.2 VI.2.a
σ-models Linear σ-model
Lagrangian: L =
λ 2 1 1 (∂µ~π · ∂ µ~π + ∂µ σ ∂ µ σ) − µ2 ~π · ~π + σ 2 − ~π · ~π + σ 2 2 2 4
14
(VI.2a)
Describes isotriplets of pions (π a , a = 1, 2, 3) and scalar σ-mesons. The model can be extended to include also the interactions with (initially massless) nucleons (Ψ) ¯ i∂/ Ψ + gπN N Ψ ¯ (σ + iγ5~τ · ~π ) Ψ L0 = L + Ψ where τ a denote the Pauli matrices. An explicit symmetry-breaking term may be added ∆LSB = c σ . Properties: Renormalizable. Invariant under O(4)-rotations (∼ = SU (2) × SU (2) chiral transformations) in the internal ~π , σ space. The model is an example for spontaneous symmetry breaking: For µ2 < 0 the σ-field acquires a vacuum expectation value, the nucleon becomes massive and the pion massless (Goldstone boson). The Goldberger-Treiman relation with gA = 1 follows. Importance: ? ? (?) References: M. Gell-Mann and M. L´evy, Nuov. Cim. 16 (1960) 705 [original] C. Itzykson and J.-B. Zuber: Quantum Field Theory, McGraw-Hill (1980), ch. 11-4
VI.2.b
[textbook]
Nonlinear σ-model
Lagrangian: L =
fπ2 tr ∂µ U † ∂ µ U , 4
h i ~ U (x) := exp i~τ · Φ(x)/f π
(VI.2b)
Properties: Obtained by eliminating the σ-field in the linear σ-model by a chirally invariant constraint σ 2 + ~π 2 = fπ2 (pion decay constant squared). Realized by defining the unitary matrix U ≡ [ σ(x) + i~τ · ~π (x) ] /fπ . Invariant under chiral transformation U → LU R−1 where L, R are ~ SU(2) matrices but non-renormalizable. Expansion for ”small” Φ 2 1 ~ 2 1 ~ ~ + ... L = ∂µ Φ + 2 Φ × ∂µ Φ 2 6fπ shows that it describes massless pions which interact via a derivative coupling so that their interaction vanishes at low momentum. Importance: ? ? ? First term of a systematic expansion in chiral perturbation theory (χPT) for mesons. References: J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142; Nucl. Phys. B 250 (1985) 465 [χPT] J. F. Donoghue, E. Golowich and B. R. Holstein: Dynamics of the Standard Model, Cambridge U. Press (1994) [textbook]
15
VI.2.c
Skyrme model
Lagrangian: L =
h ih i fπ2 1 † † † µ † ν tr U ∂ U, U ∂ U U ∂ U, U ∂ U tr ∂µ U † ∂ µ U + µ ν 4 32e2S
(VI.2c)
where U is an unitary 2 × 2 matrix (see nonlinear σ-model). Properties: Has static solitonic solutions because the fourth-order term stabilizes the energy. Model for (spin 1/2) nucleons made out of (spinless) pions: Topological quantum number = baryon number. Gives reasonable description of nucleonic properties since it may be considered as large Nc -limit of QCD. Importance: ?? References: T. H. R. Skyrme, Proc. R. Soc. (London) A 260 (1961) 127; Nucl. Phys. 31 (1962) 556 [original] E. Witten, Nucl. Phys. B 160 (1979) 57 [baryons for large Nc ] G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552 [phenomenology] T. H. Gisiger and M. B. Paranjape, Phys. Rep. 306 (1998) 109 [review]
VI.3
Yukawa model
Lagrangian: i h ¯ ¯ (i∂/ − mF ) Ψ + 1 (∂µ Φ)2 − m2 Φ2 − g ΨΨΦ L = Ψ S 2
(VI.3)
Properties: Renormalizable. Gives an attractive (Yukawa-) potential between Ψ-particles in the non-relativistic limit. For mF = 0 the Ψ-particles acquire mass if the scalar field has additional self-interaction and develops a vacuum expectation value < Φ > 6= 0 (see Φ4 -model). Importance: ?? References: M. E. Peskin and D. V. Schroeder: An Introduction to Quantum Field Theory, AddisonWesley (1995), ch. 4.7 [textbook]
16
VI.4
Walecka model
Lagrangian: L
Fµν
=
≡
h i ¯ (i∂/ − M ) Ψ + 1 (∂µ Φ)2 − m2S Φ2 − 1 Fµν F µν − 1 m2V Vµ V µ Ψ 2 4 2 ¯ Φ − gV Ψγ ¯ µΨ V µ −gS ΨΨ
(VI.4)
∂µ Vν − ∂ν Vµ .
Properties: Describes nucleons of mass M interacting via the attractive exchange of scalar (σ) mesons Φ and the repulsive exchange of vector (ω) mesons Vµ mimicking the nuclear interaction. Alternative names are σ − ω model or Quantum Hadrodynamics (QHD) (for extended models). Renormalizable despite the massive vector mesons because these are coupled to a conserved current. However, this is mostly irrelevant since the main purpose of the model is an effective description of nuclei and nuclear matter. Importance: ?? By fitting the few parameters of the model in mean-field approximation [violating translational invariance...] a description of closed-shell nuclei is achieved which is as good as with the best nonrelativistic effective interactions. The large spin-orbit interaction and the energy-dependence of the optical potential for nucleon-nucleon scattering arise naturally. References: J. D. Walecka, Ann. Phys. 83 (1974) 491 [original] J. D. Walecka and B. D. Serot, Adv. Nucl. Phys. 16 (1986) 1 [review] B. D. Serot and J. D. Walecka, Int. J. Mod. Phys. E 6 (1997) 515 [progress report]
VI.5
Nambu-Jona-Lasinio model
Lagrangian: ¯ (i∂/ − m) Ψ + g L = Ψ
h
i ¯ 2 + Ψiγ ¯ 5Ψ 2 ΨΨ
(VI.5)
Properties: Non-renormalizable, is only to be considered in one-loop approximation. Spontaneous symmetry breaking for g > gcrit so that a constituent quark mass is obtained from the current quark mass m. However, the quarks are not confined. Extensions with additional six- and eight-quark interactions, more quark families and coupling to the “Polyakov loop” exist as well as nonlocal versions. Importance: ? ? (?) Has been (and still is) used to model mesons and baryons, nuclear and quark matter and to investigate phase diagrams and compositeness.
17
References: Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; V. G. Vaks and A. I. Larin, Sov. Phys. JETP 13 (1961) 192 [original] S. P. Klevansky, Rev. Mod. Phys. 64 (1992) 649 [review] R. Alkofer, H. Reinhardt and H. Weigel, Phys.Rept. 265 (1996) 139 [hep-ph/9501213] [baryons] M. Buballa, Phys. Rept. 407 (2005) 205 [hep-ph/0402234] [quark matter]
VI.6
Quark models of hadrons
VI.6.a
Bag model
Lagrangian: ¯ (i∂/ − mF ) Ψ − B ΘV − 1 ΨΨ∆ ¯ L = Ψ (VI.6a) S . 2 Describes free quarks inside a “bag” of volume V . ΘV is a step function, i.e. takes the value 1 when x ∈ V and 0 otherwise, ∆S a surface delta function. Bag boundary gives quarks an effective (constituent) mass. A version which restores the chiral symmetry violated by the bag boundary is the “chiral bag model” (“cloudy bag”) where the surface term is modified to ~ be the source of the pion field Φ 1¯ 1¯ ~ 5 /fπ Ψ ∆S ∆S −→ − Ψ exp i~τ · Φγ − ΨΨ 2 2 Properties: Equations of motion plus two boundary conditions obtained by varying fields plus bag surface. B is the “bag constant” - the energy density of the vacuum needed to create the bag. Except for 1 + 1 dimensions the bag model has been only considered in the “cavity approximation” assuming a static, spherically symmetric surface. Describes low-lying hadronic spectrum reasonably well. Translational invariance, however, is violated (not to speak of Lorentz invariance ...). Importance: ? Practically dead now, i.e. mostly of historical interest. References: A. Chodos et al., Phys. Rev. D 9 (1974) 3471 [original] T. DeGrand et al., Phys. Rev. D 12 (1975) 2060 [phenomenology] A. Chodos and C. B. Thorn, Phys. Rev. D 12 (1975) 2733 [chiral bag] S. Th´eberge, A. W. Thomas and G. A. Miller, Phys. Rev. D 22 (1980) 2838 [cloudy bag] F. Close: An Introduction to Quarks and Partons, Academic Press (1979), ch. 18 [textbook]
18
VI.6.b
Soliton bag (Friedberg-Lee) model
Lagrangian: L = U (σ) =
1 ¯ ( i∂/ − gσ ) Ψ (∂µ σ)2 − U (σ) + Ψ 2 a 2 b 3 c σ + σ + σ4 + B . 2 6 24
(VI.6b)
Properties: Treated in quasiclassical approximation for the scalar σ-field (not a strict soliton solution). By adjusting the parameters in U (σ) a “bubble” for the quarks is generated representing a hadron. The original bag model is obtained as special limit. Importance: same as bag model References: R. Friedberg and T. D. Lee, Phys. Rev. D 15 (1977) 1694; D 16 (1977) 1096; D 18 (1978) 2623 [original] R. Goldflam and L. Wilets, Phys. Rev. D 25 (1982) 1 [phenomenology]
VI.6.c
Dyson-Schwinger approach to QCD
Lagrangian: LQCD = Lquarks + Lgluons + Lgauge fixing + Lghosts
(VI.6c)
with truncation of infinite hierarchy of Dyson-Schwinger equations (DSEs) for the Green functions of the theory. Properties: Approximative method for solving strong-coupling QCD by using model ans¨ atze for higher-order Green functions, e.g. the ”rainbow approximation”. Truncation scheme introduces model and gauge dependence (usually the Landau gauge is used) which is hard to control. Importance: ? ? ? Closer to QCD than any other hadronic model with good phenomenological results and in fair agreement with lattice data for the gluon and ghost propagators. Still under active research (improved parametrizations, extended truncation schemes) References: F. J. Dyson, Phys. Rev. 75 1736 (1949); J. S. Schwinger, Proc. Nat. Acad. Sci. 37 452, 455 (1951) [origin of DSEs] R. Alkofer and L. v. Smekal, Phys. Rept. 353 (2001) 281 (hep-ph/0007355) [review] C. D. Roberts, arXiv:1203.5341 [lecture notes on QCD + DSEs]
19
VI.7
Scalar QED
Lagrangian: 1 L = − Fµν F µν + (Dµ Φ)† Dµ Φ − V |Φ|2 4 Fµν ≡ ∂µ Aν − ∂ν Aµ , Dµ ≡ ∂µ + ieAµ
(VI.7a) (VI.7b)
Properties: Needs a potential V = λ|Φ|4 /4 for renormalization (see also “Abelian Higgs model”). Importance: ?? Electrodynamics of pointlike pions and kaons, used for description of hadronic atoms References: F. Rohrlich, Phys. Rev. 80 (1950) 666 [renormalization] J. D. Bjorken and S. D. Drell: Relativistic Quantum Mechanics, McGraw-Hill (1964), ch. 9 and app. B [textbook] M. Baig et al., Phys. Rev. D 51 (1995) 5216 [lattice simulation] J. Gasser, V. E. Lyubovitskij and A. Rusetsky, Phys. Rept. 456 (2008) 167 (arXiv:0711.3522) [hadronic atoms]
VI.8 VI.8.a
Higgs models Abelian Higgs model
Lagrangian: 2 1 1 L = − Fµν F µν + (Dµ Φ)† Dµ Φ − λ |Φ|2 − F 2 , 4 4
Dµ ≡ ∂µ + ieAµ
(VI.8a)
Properties: Scalar QED with the particular |Φ4 | interaction which gives mass to the scalar particles in 3 + 1 dimensions (Higgs mechanism); in 1 + 1 dimensions vacuum tunneling through instanton effects takes place. Importance: ? References: R. Rajaraman: Solitons and Instantons, North Holland (1982), ch. 10.2, 10. 3 book]
20
[text-
VI.8.b
Georgi-Glashow (non-abelian Higgs) model
Lagrangian: 3 3 3 X 1 X a a µν X µ a † 1 L = − Gµν G + (D Φ ) Dµ Φa − λ Φa Φa − F 2 4 4 a=1
Gaµν
=
∂µ Aaν
−
∂ν Aaµ
+g
a=1 3 X
!2 (VI.8b)
a=1
abc
Abµ Acν
b,c=1
,
a
a
Dµ Φ = ∂µ Φ + g
3 X
abc Abµ Φc
b,c=1
Properties: Describes a triplet of scalar (”Higgs”) fields Φa (x), a = 1, 2, 3 coupled to a non-abelian gauge field Aaµ . abc is the total antisymmetric tensor in 3 dimension = structure constants of SU (2) . Invariant under local SU (2) gauge transformations. Has static classical soliton solutions: the ’t Hooft-Polyakov monopole. Importance: ?(?) References: M. Georgi and S. L. Glashow, Phys. Rev. Lett. 28 (1972) 1494; G. ’t Hooft, Nucl. Phys. B 79 (1974) 276; A. M. Polyakov, JETP Lett. 20 (1974) 194 [original] R. Rajaraman: Solitons and Instantons, North Holland (1982), ch. 3.4 [textbook] A. Rajantie, JHEP 0601 (2006) 088 (hep-lat/0512006) [lattice calculation of monopole mass]
VI.9
Lee-Wick model
Lagrangian: (for scalar fields) L
= eliminating ΦLW
−→
1 ( ∂µ Φ )2 − 2 1 ( ∂µ Φ )2 − 2
1 2 2 1 m Φ − ΦLW ∂ 2 Φ + M 2 Φ2LW − V (Φ) 2 2 2 1 2 2 1 2 m Φ − ∂ Φ − V (Φ) 2 2M 2
(VI.9a) (VI.9b)
Properties: Attempt to formulate a finite field theory by giving Pauli-Villars regulators physical freedom; equivalent is a higher-derivative theory containing massive ”Lee-Wick particles” ΦLW . The wrong-sign residue of the corresponding propagators causes no problem for unitarity since they decay into ordinary particles but is problematic for causality. This is cured by an ad-hoc modification of the usual contour integration in Feynman diagrams. Importance: (?) resurrected recently References: T. D. Lee and G. C. Wick, Nucl. Phys. B 9 (1969) 209; Phys. Rev. D 2 (1970) 1033; [original for QED] B. Grinstein, D. OConnell and M. B. Wise, Phys. Rev. D 77 (2008) 025012 (arXiv:0704.1845) [Lee-Wick version of Standard Model] 21
Appendix: A Short History of Some Attempts to Put this Paper on the ArXiv
November 27, 2014, 15:46 : submission to arXiv, classification: High Energy Physics – Theory; reply: We have received your submission to arXiv. Your temporary submission identifier is: submit/1124991.
November 28, 2014, 20:02
: arXiv Moderation [
[email protected]]
Your submission has been removed upon a notice from our moderators, who determined it inappropriate for arXiv. Our moderators suggest that you please send your paper to a conventional journal instead. Please do not resubmit this paper without contacting moderation for permission, and obtaining a positive response. Resubmission of removed papers may result in suspension of submission privileges. For more information on our moderation policies see: http://arxiv.org/help/moderation
December 2, 2014, 14:09
: Letter to arXiv moderation
Dear moderator, I was surprised to hear that my submission ”A short guide to some quantum field theoretical models” has been removed from the arXiv since I never had any problems of that kind during nearly twenty years of putting my papers on the arXiv. After checking the information you give on your web page for the arXiv moderation system I only could infer that my - somehow jokingly given - classification ”stars” for the importance of the various models was the main reason to deem my submission as ”inappropriate” for the arXiv. However, it should be clear that my contribution is no ”joke” but a serious guide to the plethora of quantum field theoretical models which are scattered in the literature. I have taken much effort to scan the literature and to give an unbiased - although subjective - overview which certainly will be helpful for the beginner as for the expert. I have removed the controversial rating scheme and deleted certain sentences which might be considered offensive. Therefore I am asking you to give permission to submit the new version (which is attached to this message) to the arXiv. Sincerely yours,
R. Rosenfelder
December 4, 2014, 15:50
: arXiv Moderation [
[email protected]]
Dear R. Rosenfelder, The moderators determined that your submission was in need of significant review and revision before it would be considered publishable by a conventional journal. Please note that arXiv moderators are not referees and provide no feedback nor reasons for removal of submissions. Please submit instead to a conventional journal to receive the requisite feedback. For more information about our moderation policies, please see: http://arxiv.org/help/moderation
22
December 5, 2014, 19:06
: Letter to the arXiv moderators
Dear moderators, I am very disappointed about your reaction: 100 minutes after my personal and detailed letter to you I obtained an (automatic?) terse statement that ”The moderators DETERMINED that your submission was in need of significant review and revision before it would be considered publishable by a conventional journal.” Next you proclaim - somehow contradicting the previous sentence - that ”arXiv moderators are NOT referees and provide no feedback nor reasons for removal of submissions”. This immediately raises the question by which (obviously non-scientific) arguments and judgements do they DETERMINE the need for significant review and REMOVE submissions? I understand that the arXiv needs protection and ”moderation” against misuse and propaganda but this is clearly not the case of a crackpot claiming that Einstein is wrong... I have offered a version of my submission which does not contain any sentences which might be considered offensive (just my guess as I have no hint what makes my submission ”inappropriate”) but I didn’t get any response other than these general and cryptic statements! Has my letter been read at all or should I understand your reaction as actually meaning ”We don’t like your submission but we won’t tell you why”? This would sound like a novel by Kafka where judgments and judges remain hidden and all rational arguments from the defendant are to no avail... I think that by establishing this strict moderation system you do a disservice to the scientific community: it is neither fair nor open and affects adversely the free scientific communication. These were the virtues which I always have associated with the arXiv when I started to submit my papers to it in 1995. Sincerely,
December 5, 2014, 19:59
R. Rosenfelder
: arXiv Moderation [
[email protected]]
Dear R. Rosenfelder, You misunderstand. ”arXiv is an openly accessible, moderated repository for scholarly articles in specific scientific disciplines. Material submitted to arXiv is expected to be of interest, relevance, and value to those disciplines. arXiv reserves the right to reject or reclassify any submission. Submissions are reviewed by expert moderators to verify that they are topical and refereeable scientific contributions that follow accepted standards of scholarly communication (as exemplified by conventional journal articles).” http://arxiv.org/help/primer
23
