Mass Gap Without Vacuum Energy

Physics Letters B 681 (2009) 287–289

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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. κ κ Since the hyperbox in field space is suppressing the field fluc- Dφ(p) = d Re φ(p) d Im φ(p) , (13) B 2 tuations we have the relation κ (p)<1/p .Wemayusethesere- p κ − 1 − 1 sults to constrain the behavior of the propagator in the case when κ κ B. Holdom / Physics Letters B 681 (2009) 287–289 289 the constraint in configuration space is not completely known or away from the boundaries located at ∼ p2/κ4.Sincethesefluc- too difficult to specify, as is the case for the FMR of a gauge theory. tuations have close to a Gaussian distribution the effects of the In particular there is a smallest possible value of κ that would still boundaries on these fluctuations are expected to be exponentially have the hyperbox completely contained within the FMR. Upon suppressed, in agreement with what we have seen and in contrast moving the boundary outward from the hyperbox to the convex to the GZ prescription. FMR boundary, the field constraint is being relaxed and the fluc- The softness of the effects of Gribov copies raises a question. tuations are less suppressed. Thus the corresponding propagator Could a pure gauge theory, confining and asymptotically free, be an FMR(p) (the scalar factor defining the gauge field propagator) can example of an infinitely soft theory with a mass gap? Apparently only be closer to 1/p2 and we have not, since vacuum energy receives a nonperturbative contribution

B 2 in a pure gauge theory, from instanton effects, and we have argued κ (p)< FMR(p)<1/p . (20) that vacuum energy is not consistent with an ultrasoft theory. But

This is a very tight constraint on FMR(p) in the ultraviolet. it is also known that the addition of massless fermions can have There is a complementary approach to the problem of Gribov the effect of removing this instanton contribution to vacuum en- copies. Rather than attempting to constrain the gauge field config- ergy [6]. uration space to the FMR, one can consider an equivalent definition Thus consider a confining QCD-like theory with massless of the generating functional of a gauge theory, quarks, which also experiences chiral symmetry breaking. The mass gap is again not infinitely soft due to the appearance of the iS[A, J] μ 1 quark condensate, which introduces power law corrections in var- Z[ J]= [dA]e δ ∂μ A det(DA ) . (21) 1 + N(A) ious operator product expansions. But since the quark condensate As first introduced by Gribov, no restriction to the FMR is needed is chiral, it must appear squared in operator products that do not here since the counting factor N(A) accounts for the number of carry chirality. In particular for the gluon propagator gauge equivalent copies of a configuration. By explicitly counting 2 −1 qq Gribov copies [4] for a class of configurations it is found that the lim p2G p2 = f (p/μ) 1 + c +··· . (23) G 6 last factor strongly suppresses field configurations outside a certain p2→∞ p region in configuration space. Within this region, which in turn This has the same behavior as the ultrasoft scalar field theory lies within the FMR, N(A) identically vanishes. The corresponding discussed above. That discussion then has a bearing on the self- bound on the pth Fourier amplitude is found to be proportional consistency of this form for the gluon propagator. The suggestion to p, which agrees with the shape of the ellipsoid or hyperbox is that the nonperturbative physics is producing a mass gap, and described above. Outside this region N(A) is roughly proportional that it is doing so as softly as consistently possible. to the Fourier amplitude divided by p all raised to some power A by-product of this picture is a vanishing vacuum energy and a > 3. Thus even though the 1/(1 + N(A)) factor does not produce a vanishing gluon condensate. It leads to the question of whether a sharp cutoff in field space, it does have a similar effect on the the gluon condensate that is thought to exist in QCD would survive propagator. The result from [4] is ⎧ the limit of vanishing current quark masses. The experimental and − 2 ⎨ 1 a 1 p +···, p2 2, lattice-based evidence for a gluon condensate in this limit was re- 3 a−3 κ4 κ C (p) = (22) viewed in [7] and the evidence was found to be lacking. It appears κ ⎩ 2 4 1 (1 − a 2 κ exp(− 1 p ) +···), p2 κ2. to be a question that deserves more attention, given the relation it p2 π p2 2 κ4 has to our understanding of vacuum energy in theories of funda- These considerations are showing that the nonlocal effects of mental interest. Gribov copies in gauge theories are severe in the infrared while being infinitely soft in the ultraviolet. But this ultraviolet soft- Acknowledgements ness is not apparent in the well-known Gribov–Zwanziger (GZ) approach [1,5]. In that semi-perturbative approach the constraint I thank E. Poppitz for his comments. This work was supported on the gauge fields is implemented through a modification of the in part by the Natural Sciences and Engineering Research Council action and the result is the appearance of a new nonlocal term, of Canada. which happens to be of the form of our Λ term. The resulting propagator is similar to B (p) or C (p) in the infrared, but it κ κ References differs in the ultraviolet where it instead has a power law correction. We note that the modified action in the GZ approach is [1] L.S. Brown, Quantum Field Theory, Cambridge University Press, Cambridge, 1994. arrived at by focusing on the first Gribov region rather than the [2] V.N. Gribov, Nucl. Phys. B 139 (1978) 1; FMR, and then by requiring that the functional integral is self- R.F. Sobreiro, S.P. Sorella, arXiv:hep-th/0504095. consistently dominated by field configurations at the boundary of [3] M.A. Semenov-Tyan-Shanskii, V.A. Franke, Pubs. LOMI Seminar 120 (1982) 159, this region. But some correlation functions are not dominated by J. Sov. Math. 34 (1986) 1999. 2 [4] B. Holdom, Phys. Rev. D 79 (2009) 085013, arXiv:0901.0497. such configurations, most notably those evaluated at high p .The [5] D. Zwanziger, Nucl. Phys. B 323 (1989) 513. 2 relevant fluctuations are perturbative and their variance is ∼ 1/p , [6] G. ’t Hooft, Phys. Rev. D 14 (1976) 3432. and when p2 κ2 these fluctuations are much smaller and far [7] B. Holdom, New J. Phys. 10 (5) (2008) 053040, arXiv:0708.1057.