Historical research guide · 1004 · 2000

Vacuum structure of a modified MIT bag

Ralf Hofmann · M. Schumann · Thomas Gutsche · Raoul D. Viollier

In one minute

In the MIT bag model a proton is a bubble: quarks move freely inside a small sphere, and the surrounding vacuum presses in on it. That pressure is the bag constant, and for twenty-five years it had been a number fitted to the measured hadron spectrum rather than calculated. Ralf Hofmann, M. Schumann, Thomas Gutsche and Raoul Viollier propose calculating it. Their move is to take asymptotic freedom seriously: fluctuations harder than a chosen separation scale do not feel the low-energy confinement mechanism at all, so they should be allowed to pass straight through the boundary, and only the soft fluctuations left inside carry the bag’s vacuum energy. With that one physical cut replacing the usual renormalisation of fitted parameters, the calculated quark bag constant matches the measured gluon condensate at a bag radius near six tenths of a femtometre, and the vacuum energy reproduces the number hadron phenomenology had been fitting. Adding confined gluons then breaks the agreement, and the authors say so plainly.

Connection

Chapter 2 is the chapter that says the vacuum is a structured medium whose energy depends on the boundaries you put in it, and this is that statement carried inside a proton: the same Casimir accounting that governs two metal plates governs the pressure holding a hadron together. Chapter 13 gets the link the paper draws between that accounting and the gluon condensate of quantum chromodynamics, and this is also the early work of Ralf Hofmann, whose later thermal-ground-state papers are elsewhere in this library.

From the source

The proposal in one sentence: instead of splitting the bag’s vacuum energy into a classical part parametrised by phenomenological quantities and a quantum part whose divergences are then renormalised away, separate the perturbative from the nonperturbative regime physically. Hard fluctuations are treated as noninteracting and unconfined, allowed to traverse the boundary because they are not subject to the low-energy confinement mechanism, and only the soft remainder is counted.Abstract and the paragraph beginning ‘In this paper we propose an alternative’, arXiv hep-ph/0003050
The bag constant is not fitted but defined as the vacuum expectation value of the MIT model’s own quadratic boundary condition, computed from a mode-sum representation of the cavity propagator with a Schwinger parameter and a free-space subtraction. The separation scale enters twice and only twice: it truncates the mode sum and it cuts the lower limit of the Schwinger integration.Equations 1 and 2 and the surrounding text, arXiv hep-ph/0003050
The one-loop trace anomaly of the energy-momentum tensor of quantum chromodynamics ties the calculated quark bag constant directly to the gluon condensate, with a fixed numerical coefficient of 0.302. Taking the renormalisation-scale independent gluon condensate as 0.024 plus or minus 0.012 in units of the fourth power of a giga-electronvolt, agreement is reached for a separation scale of 1.0 giga-electronvolts and a bag radius of 0.6 femtometres — values at which the computed bag constant is stable against changes in the radius.Equations 3 and 4 with Tables I and II, arXiv hep-ph/0003050
At those same values the canonical vacuum energy comes out at 0.597 giga-electronvolts, which corresponds to a value of 1.79 for the dimensionless quantity that hadron spectroscopy fits to about 2 — and the authors reach it with no centre-of-mass contribution included, where earlier work found that contribution to be of order forty per cent. The phenomenological Casimir parameter of the bag model is therefore reproduced rather than fitted.The paragraph following equation 4, with Table II, arXiv hep-ph/0003050
Including confined gluons is where the model runs into trouble, and the authors report it as the paper’s result rather than hiding it. The gluonic bag constant vanishes for radii below 0.7 femtometres and stabilises at a radius of 0.8 femtometres; requiring the total bag constant to reproduce the central value of the gluon condensate then drives the separation scale below 0.8 giga-electronvolts, whereas quantum chromodynamics sum rules put the onset of the perturbative regime at a scale of 1.22 to 1.34 giga-electronvolts. That gap is the open question the paper leaves standing.Equations 5 and 6, Table III, and the closing summary, arXiv hep-ph/0003050
As a consistency check the authors estimate the deconfinement temperature at zero baryon chemical potential from the calculated bag constant and obtain 203.8 mega-electronvolts, against 102.8 mega-electronvolts from the older phenomenological bag constant — a figure they call too low, since a transition there would already have been seen. Lattice simulations of pure SU(3) Yang-Mills theory expect the gluon condensate to fall smoothly near 260 mega-electronvolts.Equation 7 and the paragraph introducing it, arXiv hep-ph/0003050

The way in

https://doi.org/10.1007/s100520000445

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

PUBLICATION. The European Physical Journal C, volume 16, number 4, pages 677 to 681, September 2000. Affiliations as printed: Ralf Hofmann and Thomas Gutsche at the Institute of Theoretical Physics, University of Tuebingen; M. Schumann and Raoul D. Viollier at the Institute of Theoretical Physics and Astrophysics, Department of Physics, University of Cape Town, Rondebosch. The second author is recorded only as M. Schumann in Crossref, OpenAlex and on the preprint, and no fuller form of the given name is registered, so the initial stands. LICENCE, CHECKED 2026-09-08. Crossref registers only Springer’s text-and-data-mining terms, and the arXiv deposit carries no Creative Commons statement, so nothing beyond the work’s own abstract is reproduced here. SOURCE READ. The paper is green open access through the authors’ preprint, arXiv hep-ph/0003050 version 1, dated 6 March 2000; that preprint — the complete Letter with its three tables, two figures and thirty-five references — was retrieved and read in full on 2026-09-08, and every claim below cites a numbered equation or a named passage of it. Equations and table entries are described in words, because the extracted text carries the original typesetting only as characters and the table columns interleave on extraction; the numerical values quoted below are the ones the authors state in their running text. COMPANION PAPER. Reference 21, the authors’ own method paper, is Hofmann, Schumann and Viollier, Calculation of the regularized vacuum energy in cavity field theories, European Physical Journal C 11, 153 (1999), doi 10.1007/s100529900213. RELATED PAGES. Ralf Hofmann’s later work on the thermal ground state is at /library/stm-9a9190b65f, /library/stm-e1f7b72444, /library/stm-dcaf382795 and /library/stm-b06591e288; the Casimir literature this Letter builds on, including Milton and the Bordag-Klimchitskaya-Mohideen-Mostepanenko monograph, is at /library/stm-56c98e4b1d and /library/stm-30ce9fddc2; and hot-plasma Casimir physics at /library/stm-956e5ae2ff.

How to cite it

Ralf Hofmann, M. Schumann, Thomas Gutsche, Raoul D. Viollier (2000) Vacuum structure of a modified MIT bag doi:10.1007/s100520000445

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