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Mass Gap Without Vacuum Energy Physics Letters B 681 (2009) 287–289 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Mass gap without vacuum energy B. Holdom Department of Physics, University of Toronto, Toronto ON, Canada M5S1A7 article info abstract Article history: We consider soft nonlocal deformations of massless theories that introduce a mass gap. By use of Received 6 July 2009 a renormalization scheme that preserves the ultraviolet softness of the deformation, renormalized Accepted 6 October 2009 quantities of low mass dimension, such as normal mass terms, vanish via finite counterterms. The same Available online 9 October 2009 applies to the renormalized cosmological constant. We connect this discussion to gauge theories, since Editor: T. Yanagida they are also subject to a soft nonlocal deformation due to the effects of Gribov copies. These effects are softer than usually portrayed. © 2009 Elsevier B.V. All rights reserved. We wish to explore the apparently tight relationship between or nonperturbative means. It would usually be expected that the μ 4 the existence of a mass gap and the existence of vacuum energy. mass gap M would imply that Tμ ≈M which in turn implies Consider first the case of a free massless scalar field. By scale in- that at least some combination of dimension four operators de- μ variance Tμ = 0 and Lorentz invariance Tμν ∝ημν we know velop vacuum expectation values. The operator product expansion that the vacuum energy vanishes. There is an apparent contri- translates these vevs into power law corrections to the ultraviolet bution from the zero-point energies that requires regularization, behavior of the theory, with these corrections of the form M4/p4. but a regulator in the form of an ultraviolet cutoff breaks both Thus if we want to contemplate a mass gap that could develop the scale and Lorentz invariance. A regulator that preserves scale without vacuum energy, then the power law corrections that re- invariance, such as dimensional regularization, automatically pro- sult in the ultraviolet will have to be softer than this. duces the expected vanishing of Tμν. This continues to be true in To explore the consistency of this possibility we extend the perturbation theory in an interacting λφ4 theory, as long as there scalar theory through the addition of a few nonlocal terms so that is no dimensionful coupling in the theory. the bare Lagrangian in Euclidean form is The situation changes with the introduction of an explicit mass 1 λ 1 term m2φ2. Now an infinite cosmological constant counterterm L = (∂ φ)2 + φ4 + m2φ2 E μ ! proportional to m4 is required in the renormalization procedure 2 4 2 1 1 1 1 1 (see [1] for a thorough discussion). This occurs even in the free + Λ + Λ φ2 + κ6 φ2. (3) theory where the result for the renormalized cosmological con- 2 2 stant is The nonlocal terms have the effect of damping φ fluctuations in m4 the infrared, while producing small, subleading power law cor- Λr = log(m/μ). (1) rections in the ultraviolet. The Λ notation emphasizes that this 32π 2 parameter has the same dimension as Λ. Λ = 0producesaΛ/p4 The renormalization scale μ also absorbs a renormalization scheme power law correction in the propagator and an infinite contribu- dependence. At any order in perturbation theory in the interacting tion to Λ very similar to the m2 = 0 case. More interesting is the theory there is a similar result in terms of renormalized quantities, softer ultraviolet power law correction κ6/p6 from the last term in 2 = = 4 the case m Λ 0. Then the Euclidean propagator is Λr = m fΛ(λ,m/μ). (2) 1 Vacuum energy and mass appear to be inextricably linked. = (p) 6 , (4) p2 + κ Our focus must therefore remain on interacting theories that p4 have no explicit mass parameters. The scale anomaly in such theo- μ = O which has a massive pole in Minkowski space. We refer to this ries involves only dimension four operators Tμ i βi (λ) i .Sup- pose such a theory develops a mass gap through some dynamical mass gap as ultrasoft, and in this case the resulting contribution to Λ is at most finite. We need a renormalization scheme that ensures that dimen- E-mail address: [email protected]. sionful couplings of mass dimension less than six are not gener- 0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.10.021 288 B. Holdom / Physics Letters B 681 (2009) 287–289 ated. In dimensional regularization, the infinite counterterms that and arise are proportional to integer powers of the couplings in the 1 − theory. If κ6 is the only dimensionful coupling then the countert- J(p) = √ d4xJ(x)e ipx. (14) erms as represented by the bare parameters m2, Λ and Λ remain V V finite. We require that the corresponding renormalized quantities vanish order by order in perturbation theory. At zeroth order, The propagator becomes 1 4 − = ip(x−y) (0) = √π κ + (0) = κ (x y) e κ (p), (15) Λr Λ 0(5)V 2 3 (4π)2 p while at first order, where 2 λ 2 1 d 2 (1) =− √π κ + 2 (1) = (p) = Z (p, j) , (16) mr m 0, (6) κ 2 κ 3 3 (4π)2 Zκ (p, j) dj j=0 2 4 (1) π λκ and Λ = + Λ(1) = 0. (7) r 4 54 (4π) 1 κ 2 1 2 2 The renormalization of Λ can first occur at order λ , Zκ (p, j) = dxexp − p x + jx . (17) 2 (2) = O 2 4 + (2) = − 1 Λr λ κ Λ 0. (8) κ The O(λ2κ4) quantity would be obtained by isolating a κ4/p4 We find 2 ⎧ power law correction to the large p behavior of the 2-point func- ⎨ 1 − 2 p2 +··· 2 2 2 4 , p κ , tion. Clearly this renormalization procedure can be continued to A = 3κ 45κ √ 2 (p) (18) yield m = Λ = Λ = 0atanyorder. κ ⎩ p2 2 r r r 1 (1 − 2 exp(− 1 p ) +···), p2 κ2. This defines an ultrasoft theory. It has vanishing vacuum energy p2 π κ 2 κ2 and a propagator with ultraviolet behavior This propagator (labelled to distinguish it from others below) acts 6 like a massive propagator in the infrared, but it approaches a mass- −1 κ lim p2 p2 = f (λ, p/μ) 1 + O . (9) less propagator in the ultraviolet exponentially quickly. r 6 p2→∞ p Thus restricting the field space to a hypercube has caused an The only infinite renormalization of the 2-point function is the infrared deformation of the massless theory, sufficient to produce standard wave function renormalization as reflected by the μ de- a mass gap. It is an extreme version of the previous ultrasoft ex- pendence of f . Other than the existence of a mass gap our con- ample, where now the approach to massless behavior in the ul- 2 siderations are not constraining the form of the full propagator at traviolet is faster than any negative power of p . For this “infinite small p2, which could differ substantially from the form of the ze- softness” to survive in perturbation theory, finite adjustments of roth order propagator in (4). an infinite number of nonlocal terms quadratic in the fields would We now present another example of a very soft deformation be required. This again defines a renormalization scheme. which may be of more immediate interest for gauge theories. First This seemingly artificial example may be relevant to gauge the- we put the massless interacting scalar field theory in a finite vol- ories. The existence of Gribov copies dictates a similar constraint ume V . The point will be to constrain the discrete set of Fourier on the functional integral of gauge theories, to avoid the mul- amplitudes φ(p), tiple counting of gauge equivalent configurations [2].Thegauge field configuration space must be restricted to the fundamental 1 = √ 4 −ipx modular region (FMR) [3], where this is a bounded convex re- φ(p) d xφ(x)e , (10) μ = V gion within the gauge-fixed hypersurface ∂μ A 0. Each gauge V in-equivalent configuration occurs once and only once in this re- to lie within a hypercube. This corresponds to adding a highly non- gion. The boundary of this region is nontrivial and difficult to work local potential term to the theory. with, but the FMR is known to lie within and share part of its boundary with the first Gribov region defined by positive Fadeev– 0, if (| Re φ(p)| < 1/κ and | Im φ(p)| < 1/κ) ∀p, Popov operator DA 0. This region has an ellipsoidal shape and Vκ (φ) = ∞, otherwise. if translated to our scalar field example would take the form |φ(p)|2/p2 C. A hyperbox that most resembles this region (11) p is given by (| Re φ(p)| < p/κ2 and | Im φ(p)| < p/κ2) for some κ. Alternatively this can be implemented as a constraint on the path Using this hyperbox rather than the hypercube in (11) yields the integral definition of the generating functional following propagator. ⎧ 4 λ δ 4 2 ∗ ∗ p2 p6 − d x ! ( ) − (p φ(p) φ(p)+ J(p) φ(p)) ⎨ − 2 +··· 2 2 Z [ J]=e V 4 δ J(x) Dφ(p) e p , 4 8 , p κ , κ B = 3κ 45κ κ (p) (19) κ ⎩ 1 − 2 p2 − 1 p4 +··· 2 2 2 (1 π 2 exp( 2 4 ) ), p κ . (12) p κ κ When compared to A(p) this propagator is suppressed even more where κ in the infrared, and approaches 1/p2 even more quickly in the ul- 1 1 traviolet.
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