Mass gap

Mass gap

In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.

Since exact energy eigenstates are infinitely spread out and are therefore usually excluded from a formal mathematical description, a stronger definition is that the mass gap is the greatest lower bound of the energy of any state which is orthogonal to the vacuum.

Contents

Mathematical definitions

For a given real field ϕ(x), we can say that the theory has a mass gap if the two-point function has the property

\langle\phi(0,t)\phi(0,0)\rangle\sim \sum_nA_n\exp\left(-\Delta_nt\right)

with Δ0 > 0 being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. It was proved in this way that Yang-Mills theory develops a mass gap.[1][2] The corresponding time-ordered value, the propagator, will have the property

\lim_{p\rightarrow 0}\Delta(p)=\mathrm{constant}

with the constant being finite. A typical example is offered by a free massive particle and, in this case, the constant has the value 1/m2. In the same limit, the propagator for a massless particle is singular.

Examples from classical theories

An example of mass gap arising for massless theories, already at the classical level, can be seen in spontaneous breaking of symmetry or Higgs mechanism. In the former case, one has to cope with the appearance of massless excitations, Goldstone bosons, that are removed in the latter case due to gauge freedom. Quantization preserves this property.

A quartic massless scalar field theory develops a mass gap already at classical level. Let us consider the equation

\Box\phi+\lambda\phi^3=0.

This equation has the exact solution

\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}\left(p\cdot x+\theta,-1\right)

-- where μ and θ are integration constants, and sn is a Jacobi elliptic function -- provided

p^2=\mu^2\sqrt{\frac{\lambda}{2}}.

At the classical level, a mass gap appears while, at quantum level, one has a tower of excitations and this property of the theory is preserved after quantization in the limit of momenta going to zero.[3]

While lattice computations have suggested that Yang-Mills theory indeed has a mass gap and a tower of excitations, a theoretical proof is still missing. This is one of the Clay Institute Millennium problems and it remains an open problem. Such states for Yang-Mills theory should be physical states, named glueballs, and should be observable in the laboratory.

Källén-Lehmann representation

If Källén-Lehmann spectral representation holds, at this stage we exclude gauge theories, the spectral density function can take a very simple form with a discrete spectrum starting with a mass gap

\rho(\mu^2)=\sum_{n=1}^NZ_n\delta(\mu^2-m_n^2)+\rho_c(\mu^2)

being ρc2) the contribution from multi-particle part of the spectrum. In this case, the propagator will take the simple form

\Delta(p)=\sum_{n=1}^N\frac{Z_n}{p^2-m^2_n+i\epsilon}+\int_{4m_N^2}^\infty d\mu^2\rho_c(\mu^2)\frac{1}{p^2-\mu^2+i\epsilon}

being 4m_N^2 approximatively the starting point of the multi-particle sector. Now, using the fact that

\int_0^\infty d\mu^2\rho(\mu^2)=1

we arrive at the following conclusion for the constants in the spectral density

1=\sum_{n=1}^NZ_n+\int_0^\infty d\mu^2\rho_c(\mu^2).

This could not be true in a gauge theory. Rather it must be proved that a Källén-Lehmann representation for the propagator holds also for this case. Absence of multi-particle contributions implies that the theory is trivial, as no bound states appear in the theory and so there is no interaction, even if the theory has a mass gap. In this case we have immediately the propagator just setting ρc2) = 0 in the formulas above.

References

  1. ^ Lucini, Biagio; Teper, Michael; Wenger, Urs (2004). "Glueballs and k-strings in SU(N) gauge theories : calculations with improved operators". Journal of High-Energy Physics 0406: 012. arXiv:hep-lat/0404008. Bibcode 2004JHEP...06..012L. doi:10.1088/1126-6708/2004/06/012. .
  2. ^ Chen, Y.; Alexandru, A.; Dong, S. J.; Draper, T.; Horvath, I.; Lee, F. X.; Liu, K. F.; Mathur, N. et al. (2006). "Glueball Spectrum and Matrix Elements on Anisotropic Lattices". Physical Review D 73: 014516. arXiv:hep-lat/0510074. Bibcode 2006PhRvD..73a4516C. doi:10.1103/PhysRevD.73.014516. .
  3. ^ Frasca, Marco (2006). "Strongly coupled quantum field theory". Physical Review D 73: 027701. arXiv:hep-th/0511068. Bibcode 2006PhRvD..73b7701F. doi:10.1103/PhysRevD.73.027701. 

External links

See also


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Yang–Mills existence and mass gap — The Clay Mathematics Institute has offered the prize of USD 1,000,000 for each of 7 great problems in mathematics. One of them is a proof that Yang Mills theory exists according to the rigorous standards of mathematical physics (i.e. constructive …   Wikipedia

  • Mass — This article is about the scientific concept. For the substance of which all physical objects consist, see Matter. For other uses, see Mass (disambiguation). Classical mechanics …   Wikipedia

  • mass media, sociology of — A medium is a means of communication such as print, radio, or television. The mass media are defined as large scale organizations which use one or more of these technologies to communicate with large numbers of people (‘mass communications’).… …   Dictionary of sociology

  • Mass index — The mass index is an indicator, developed by Donald Dorsey, used in technical analysis to predict trend reversals. It is based on the notion that there is a tendency for reversal when the price range widens, and therefore compares previous… …   Wikipedia

  • Mass Market Retailer — A company that sells affordably priced products that appeal to a wide variety of consumers. Mass market retailers are not necessarily known for selling durable, high quality merchandise or for having exceptional customer service, but they do meet …   Investment dictionary

  • Mountain-gap wind — A mountain gap wind, gap wind or gap flow is a local wind blowing through a gap between mountains. Gap winds are low level winds and can be associated with strong winds of 20 40 knots and on occasion exceeding 50 knots. Gap winds are generally… …   Wikipedia

  • Knowledge gap — Die Hypothese der wachsenden Wissenskluft wurde erstmals 1970 von Phillip J. Tichenor, George A. Donohue und Clarice N. Olien in dem Artikel Mass Media Flow and Differential Growth in Knowledge in der Zeitschrift Public Opinion Quarterly als… …   Deutsch Wikipedia

  • Myelin sheath gap — Drawing of a peripheral nerve axon (labeled axis cylinder ), showing a node of Ranvier along with other features …   Wikipedia

  • Canada and weapons of mass destruction — Weapons of mass destruction …   Wikipedia

  • History of the Mass Rapid Transit — The history of the Mass Rapid Transit system of Singapore commences with its planning in the 1960s, leading to its opening in 1987 with the launch of the 6 km section of the North South Line from Yio Chu Kang Station to Toa Payoh Station. It now… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”