Abstract
Effective field theory provides a powerful framework to exploit a separation of scales in physical systems. In these lectures, we discuss some general aspects of effective field theories and their application to few-body physics. In particular, we consider an effective field theory for non-relativistic particles with resonant short-range interactions where certain parts of the interaction need to be treated nonperturbatively. As an application, we discuss the so-called pionless effective field theory for low-energy nuclear physics. The extension to include long-range interactions mediated by photon and pion-exchange is also addressed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note there are often more than two scales, which complicates the power counting. Here we focus on the simplest case to introduce the general principle.
- 2.
- 3.
- 4.
We stress, however, that this really is a model and not a proper EFT describing QCD.
- 5.
Note that Eq. (4.45) is very simple because we have not included a coupling of ψ to the electromagnetic field. If we had done that, the \(\boldsymbol{\nabla }\) would be a covariant derivative, \(\mathbf{D} =\boldsymbol{ \nabla } +\mathrm{ i}e\mathbf{A}\), and Eq. (4.45) would generate, among other terms, the magnetic spin coupling \(\vec{\sigma }\cdot \mathbf{B}\).
- 6.
This is assuming m π < m σ .
- 7.
Note that in the nonrelativistic case there is no “Feynman propagator.” Particles and particles are decoupled, and the denominator in Eq. (4.64) only has a single pole at p 0 = p 2∕(2m) − iɛ. Flipping the sign of the iɛ term gives the advanced Green’s function.
- 8.
A static (time-independent) potential, as it is more common in quantum mechanics, would be a function only of x and y, and all fields in the interaction term would be evaluated at the same time t.
- 9.
See for example [11, Sect. 2.1.1] for a rigorous discussion of the required transformation properties.
- 10.
We alert the reader that in the literature this is sometimes referred to as the “cutoff of the EFT.” We do not use that language to avoid confusion with an (arbitrary) momentum cutoff introduced to regularize divergent loop integrals (discussed).
- 11.
This separation would be quite clear if we had not set c = 1, which would in fact be more appropriate for a nonrelativistic system. The reason we still do it that it allows us to still energies and momenta in the same units, e.g., in MeV, following the standard convenient in nuclear physics.
- 12.
See [15] for a similar discussion with a focus on applications in ultracold atoms.
- 13.
Note that coupling constants scale with the particle mass as 1∕m in nonrelativistic theories. This can be seen by rescaling all energies as \(q_{0} \rightarrow \tilde{ q_{0}}/m\) and all time coordinates as \(t \rightarrow \tilde{ t}m\), so that dimensionful quantities are measured in units of momentum. Demanding that the action is independent of m, it follows that the coupling constants must scale as 1∕m.
- 14.
If the calculation was carried out in a frame in which the total momentum of the two scattering particles was nonzero, the simple cutoff \(\vert \boldsymbol{q}\vert <\varLambda\) would give a result that does not respect Galilean invariance. To obtain a Galilean-invariant result requires either using a more sophisticated cutoff or else imposing the cutoff \(\vert \boldsymbol{q}\vert <\varLambda\) only after an appropriate shift in the integration variable \(\boldsymbol{q}\).
- 15.
Note that this amplitude is well defined even if a < 0 and there is no two-body bound state. In this case particle lines must be attached to the external dimer propagators to obtain the 3-particle scattering amplitude.
- 16.
For a detailed discussion of the Efimov effect for finite scattering length and applications to ultracold atoms, see [15].
- 17.
As discussed in Sect. 4.5.3 this actually has to be the exchange of two or more pions, as one-pion exchange does not contribute to S-wave scattering at zero energy.
- 18.
The shallow deuteron bound-state pole is within the radius of convergence of the effective range expansion. The deuteron binding momentum is γ t = 1a t + ⋯ , where the ellipses include corrections from the effective range (and higher-order shape parameters).
- 19.
The minus sign in Eq. (4.134) is a consequence of the convention we use here for the T-matrix.
- 20.
We work in the isospin-symmetric theory here, but in the absence of electromagnetic interactions (discussed in Sect. 4.5.1), the nucleon here should be thought of as a neutron.
- 21.
Unlike what is done in Sect. 4.3, we also read diagrams from left (incoming particles) to right (outgoing particles). Both conventions can be found in the literature.
- 22.
Note that this vector is one part of the more general full off-shell amplitude, which is a 2 × 2 block matrix including the two combinations of dibaryon legs that do not appear in Fig. 4.12.
References
K.G. Wilson, Rev. Mod. Phys. 55, 583 (1983)
G.P. Lepage, What is renormalization? (1989). Hep-ph/0506330
S. Weinberg, Physica A 96, 327 (1979)
D.B. Kaplan, Effective field theories (1995). Nucl-th/9506035
D.B. Kaplan, Five lectures on effective field theory (2005). Nucl-th/0510023
J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984)
J. Gasser, H. Leutwyler, Nucl. Phys. B 250, 465 (1985)
J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model (Cambridge University Press, New York, 1992)
E. Jenkins, A.V. Manohar, Phys. Lett. B 255, 558 (1991). doi: 10.1016/0370-2693(91)90266-S
A. Gardestig, K. Kubodera, F. Myhrer, Phys. Rev. C 76, 014005 (2007). doi: 10.1103/PhysRevC.76.014005
P.R. Hagen, Effective field theory for halo nuclei. Ph.D. thesis, Bonn University (2014)
S.R. Beane, P.F. Bedaque, W.C. Haxton, D.R. Phillips, M.J. Savage, From hadrons to nuclei: crossing the border, in At The Frontier of Particle Physics. Handbook of QCD, vol. 1 (World Scientific, River Edge, 2001), p. 133
P.F. Bedaque, H.W. Hammer, U. van Kolck, Phys. Lett. B 569, 159 (2003)
E. Braaten, H.W. Hammer, Phys. Rep. 428, 259 (2006)
U. van Kolck, Nucl. Phys. A 645, 273 (1999)
D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 534, 329 (1998)
D.B. Kaplan, Nucl. Phys. B 494, 471 (1997)
G.V. Skorniakov, K.A. Ter-Martirosian, Sov. Phys. JETP 4, 648 (1957)
G. Danilov, Sov. Phys. JETP 13, 349 (1961)
V.F. Kharchenko, Sov. J. Nucl. Phys. 16, 173 (1973)
P.F. Bedaque, H.W. Hammer, U. van Kolck, Phys. Rev. Lett. 82, 463 (1999)
V. Efimov, Phys. Lett. B 33 (8), 563 (1970)
E. Braaten, D. Kang, L. Platter, Phys. Rev. Lett. 106, 153005 (2011)
F. Ferlaino, R. Grimm, Physics 3, 9 (2010)
J. Ulmanis, S. Häfner, R. Pires, F. Werner, D.S. Petrov, E.D. Kuhnle, M. Weidemüller, Phys. Rev. A 93, 022707 (2016)
P.F. Bedaque, H.W. Hammer, U. van Kolck, Nucl. Phys. A 646, 444 (1999)
S. König, H.W. Grießhammer, H.W. Hammer, J. Phys. G 42, 045101 (2015)
P.F. Bedaque, H.W. Hammer, U. van Kolck, Nucl. Phys. A 676, 357 (2000)
M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Boulder, 1995)
S. Weinberg, Phys. Lett. B 251, 288 (1990)
S. Weinberg, Nucl. Phys. B 363, 3 (1991)
D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 478, 629 (1996)
D.B. Kaplan, M.J. Savage, M.B. Wise, Phys. Lett. B 424, 390 (1998)
S. Fleming, T. Mehen, I.W. Stewart, Nucl. Phys. A 677, 313 (2000)
P.F. Bedaque, U. van Kolck, Annu. Rev. Nucl. Part. Sci. 52, 339 (2002)
B. Long, Y. Mei, Phys. Rev. C 93, 044003 (2016)
S. Scherer, M.R. Schindler, Lect. Notes Phys. 830, 1 (2012)
Acknowledgements
We thank Dick Furnstahl and Bira van Kolck and for various stimulating discussions. Moreover, Sebastian König is grateful to Martin Hoferichter for the insights into Coulomb-gauge quantization presented in Sect. 4.5.1.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Hammer, HW., König, S. (2017). General Aspects of Effective Field Theories and Few-Body Applications. In: Hjorth-Jensen, M., Lombardo, M., van Kolck, U. (eds) An Advanced Course in Computational Nuclear Physics. Lecture Notes in Physics, vol 936. Springer, Cham. https://doi.org/10.1007/978-3-319-53336-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-53336-0_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53335-3
Online ISBN: 978-3-319-53336-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)