Two-measure theory with third-rank antisymmetric tensor for local scale symmetry breaking Eduardo Guendelman * Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel Hitoshi Nishino † and Subhash Rajpoot ‡ Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, California 90840, USA (Received 21 December 2016; published 6 March 2017) We present a new mechanism of local scale symmetry breaking based on the scalar density Φ ≡ ð1=3!Þϵ μνρσ ∂ μ A νρσ ≡ ð1=4!Þϵ μνρσ F ð0Þ μνρσ with an independent third-rank tensor A μνρ , which replaces the scalar density Φ ≡ ϵ μνρσ ϵ abcd ð∂ μ φ a Þð∂ ν φ b Þð∂ ρ φ c Þð∂ σ φ d Þ used in “two-measure theory. ” We apply this function both to globally and locally scale-invariant systems. For local scale invariance, we modify F ð0Þ μνρσ by a certain Chern-Simons term, based on the recently developed tensor-hierarchy formulation. For a locally scale-invariant system with multiple scalars, the minimum value of the potential is realized at exactly zero value, while local scale invariance is broken by some nonzero vacuum expectation values: ∃ hσ i i ≠ 0, ∃ hF mnrs i¼ f 0 ϵ mnrs ≠ 0. For these values, the cosmological constant is maintained to be zero, despite the broken local scale invariance. DOI: 10.1103/PhysRevD.95.065002 I. INTRODUCTION In our paper in 2009 [1], two of the authors (H. N. and S. R) presented the locally scale-invariant generalization of the standard model. In a subsequent paper in 2011 [2], it was further shown that the breaking mechanism of the scale invariance in [1] is nothing but a Proca-Sückelberg-type compensator mechanism. In the model in [1,2], both local scale invariance and SUð2Þ W × Uð1Þ are broken, while the cosmological constant is maintained to be zero. One alternative method to the compensator mechanism [1,2] is the so-called “two-measure theory” (TMT) origi- nally developed by one of the authors (E. G.) and A. Kaganovich [3,4]. The introduction of TMT [3,4] for the breaking of scale invariance was first performed in [5] and subsequently in [6]. TMT was also introduced for the breaking of global scale invariance in our more recent paper [7]. Compared with the compensator method [1,2], TMT [3,4] has a certain advantage, such as the easier control of interactions for the breaking of symmetries. In the present paper, we first consider a globally scale- invariant action, with a peculiar scalar density Φ defined in terms of the third-rank tensor Φ ≡ ð1=3!Þϵ μνρσ ∂ μ A νρσ ≡ ð1=4!Þϵ μνρσ F ð0Þ μνρσ . In other words, Φ is Hodge dual to the field strength F ð0Þ μνρσ ≡ 4∂ ½μ A νρσ . This combination of the closed form within a scalar density is very similar to Φ ≡ ϵ μνρσ ϵ abcd ð∂ μ φ a Þð∂ ν φ b Þð∂ ρ φ c Þð∂ σ φ d Þ used in the TMT [3,4]. In Sec. III, we apply a similar mechanism to local scale invariance, and consider a more general potential with plural scalar fields. For local scale invariance, the original field strength F ð0Þ μνρσ is modified by a Chern- Simons term U∧B with the Weylon field strength U and a second-rank new tensor B, based on the recently developed “tensor-hierarchy” formulation [8,9]. We show that when static solutions are required, local scale invariance is broken, while the cosmological constant is maintained to be zero. In Sec. IV , we consider whether any duality transformation is possible from the third-rank tensor A μνρ . In Sec. V , we consider the generalization of potentials for scalar fields. Section VI is devoted for concluding remarks. II. BREAKING OF GLOBAL SCALE INVARIANCE Consider an action I ≡ R d 4 xL with L ¼ - 1 2 eζσ 2 RðeÞ - eσ 4 F ðe -1 ΦÞþ 1 2 eg μν ð∂ μ σÞð∂ ν σÞ; ð2:1Þ where e ≡ detðe μ m Þ, ζ is a real positive constant, σ is a real scalar field, and F ðe -1 ΦÞ is an arbitrary real scalar function of e -1 Φ, where Φ ≡ 1 3! ϵ μνρσ ∂ μ A νρσ ≡ 1 4! ϵ μνρσ F ð0Þ μνρσ ; F ð0Þ μνρσ ≡ 4∂ ½μ A νρσ ; ð2:2Þ * guendel@bgu.ac.il † h.nishino@csulb.edu ‡ subhash.rajpoot@csulb.edu PHYSICAL REVIEW D 95, 065002 (2017) 2470-0010=2017=95(6)=065002(6) 065002-1 © 2017 American Physical Society