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Standard Model with Spontaneously Broken Quantum Scale Invariance

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Standard Model with Spontaneously Broken Quantum Scale Invariance PHYSICAL REVIEW D 96, 055034 (2017) Standard model with spontaneously broken quantum scale invariance † ‡ D. M. Ghilencea,1,2,* Z. Lalak,3, and P. Olszewski3, 1Theoretical Physics Department, National Institute of Physics and Nuclear Engineering, Bucharest 077125, Romania 2Theory Division, CERN, 1211 Geneva 23, Switzerland 3Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, 5 Pasteura Street, 02-093 Warsaw, Poland (Received 28 July 2017; published 25 September 2017) We explore the possibility that scale symmetry is a quantum symmetry that is broken only spontaneously and apply this idea to the standard model. We compute the quantum corrections to the potential of the Higgs field (ϕ) in the classically scale-invariant version of the standard model (mϕ ¼ 0 at tree level) extended by the dilaton (σ). The tree-level potential of ϕ and σ, dictated by scale invariance, may contain nonpolynomial effective operators, e.g., ϕ6=σ2, ϕ8=σ4, ϕ10=σ6, etc. The one-loop scalar potential is scale invariant, since the loop calculations manifestly preserve the scale symmetry, with the dimensional regularization subtraction scale μ generated spontaneously by the dilaton vacuum expectation value μ ∼ hσi. The Callan-Symanzik equation of the potential is verified in the presence of the gauge, Yukawa, and the nonpolynomial operators. The couplings of the nonpolynomial operators have nonzero beta functions that we can actually compute from the quantum potential. At the quantum level, the Higgs mass is protected by spontaneously broken scale symmetry, even though the theory is nonrenormalizable. We compare the one- loop potential to its counterpart computed in the “traditional” dimensional regularization scheme that breaks scale symmetry explicitly (μ ¼ constant) in the presence at the tree level of the nonpolynomial operators. DOI: 10.1103/PhysRevD.96.055034 I. MOTIVATION that breaks explicitly the scale symmetry that we want to investigate.2 To avoid such breaking, the UV regularization In this paper, we explore the idea that scale symmetry is a must preserve this symmetry. This is done by using a quantum symmetry and study its implications for physics subtraction function μðσÞ, which generates (dynamically) a beyond the standard model (SM). However, this symmetry subtraction scale μðhσiÞ when the field σ acquires a vacuum is broken in the real world. We shall consider here only 1 expectation value (vev) hσi after spontaneous scale sym- spontaneous breaking of this (quantum) symmetry. One metry breaking. For details on this, see Ref. [3] and recent motivation of our study is that scale symmetry plays a role examples at one loop [4–8] and higher loops [9,10]. Here, σ in the UV behavior of the models. In particular, the SM is the Goldstone mode (dilaton) of the spontaneously with a classical Higgs mass parameter mϕ ¼ 0 has an broken scale symmetry3 and is an additional degree of increased symmetry: it is scale invariant at the tree level; freedom of the theory. this was invoked [1] to protect mϕ naturally [2] from The model we consider is a scale-invariant SM, defined large quantum corrections, but a full quantum study is as the SM with classical mϕ ¼ 0 and extended by the needed. dilaton. The goal is to use this scale-invariant regularization Consider a classically scale-invariant theory. One to compute quantum corrections to the scalar potential. The known issue when studying scale symmetry at the quantum quantum result is scale invariant, so it can only have level is that the regularization of the loop corrections spontaneous scale symmetry breaking, with a flat direction introduces a dimensionful parameter (subtraction scale μ) for the dilaton (σ). For clarity, this result is then compared to that in the “usual” dimensional regularization (DR) of *[email protected] † μ ¼ constant scale, which breaks explicitly the scale [email protected] ‡ symmetry at the quantum level. [email protected] 1By quantum scale symmetry, we mean that the full one- particle irreducible (1PI) quantum action is scale invariant. 2One could use a regularization that does not keep manifest Published by the American Physical Society under the terms of scale symmetry and then attempt to restore it “by hand” at the the Creative Commons Attribution 4.0 International license. end, but this misses scale-invariant operators if the theory is Further distribution of this work must maintain attribution to nonrenormalizable (see later, Sec. II). the author(s) and the published article’s title, journal citation, 3To be exact, the mass eigenstates may actually contain a small and DOI. mixing of original ϕ, σ. 2470-0010=2017=96(5)=055034(10) 055034-1 Published by the American Physical Society D. M. GHILENCEA, Z. LALAK, and P. OLSZEWSKI PHYSICAL REVIEW D 96, 055034 (2017) “ ” O λ Let us consider first a simplified scale-invariant tunings bringing acceptable ð jÞ corrections to this (classical) theory (e.g., Refs. [11–27]) of two real scalar relation, relative to the previous perturbation order4; this fields ϕ (Higgs-like) and σ. The potential V is a homo- tuning ensures a vanishing vacuum energy Vðhϕi; hσiÞ ∼ geneous function, having no dimensionful couplings, so Wðρ0Þ¼0 [see conditions (2)]. We stress that the above picture, which builds on – Vðϕ;σÞ¼σ4Wðϕ=σÞ; where Wðϕ=σÞ¼Vðϕ=σ;1Þ: ð1Þ previous studies [3 10], is very different from that obtained in the “traditional” DR scheme (μ ¼ constant scale) that is often used in classically scale-invariant models, e.g., We assume that Vðϕ; σÞ has spontaneous scale symmetry Refs. [18–27]; in such models, scale symmetry is broken breaking i.e., that σ acquires a nonzero vacuum expectation explicitly by the (regularization of) quantum effects, and value hσi ≠ 0. We thus search for such a solution and for then conditions (1) and (2) are not true anymore at the the necessary condition for this spontaneous breaking to quantum level, and the flat direction is lifted by quantum happen. With hσi ≠ 0, it is then easy to see that the corrections (the dilaton is then a pseudo-Goldstone, which minimum conditions Vσ ¼ Vϕ ¼ 0 (Vα ¼ ∂V=∂α) are is light). equivalent to What about the hierarchy problem? In the absence of gravity (not included here), the standard model has no ρ 0 ρ 0 ρ ≡ ϕ σ Wð Þ¼W ð Þ¼ ; = : ð2Þ hierarchy problem. However, this situation is no longer true under the reasonable assumption that there is some “new ρ ≡ ϕ σ These equations can have a common solution 0 h i=h i, physics” beyond SM, e.g. a large vev of a new scalar field if the couplings satisfy a particular condition (constraint); that couples to Higgs, etc. In the model we consider, ϕ σ see below. Then, a flat direction exists in the plane ð ; Þ defined by the scale-invariant version of the SM extended ϕ ρ σ ϕ σ with ¼ 0 . Indeed, if ðh i; h iÞ is a ground state with by the dilaton, we have “new physics” beyond the SM, 0 ϕ σ V ¼ , then so is ðth i;th iÞ, t real. Also, the second represented by the vev σ that spontaneously breaks the α β ϕ σ h i derivatives matrix Vαβ with respect to ; ¼ ; has scale symmetry. hσi can be very large compared to hϕi, 00 02 detðVαβÞ ∝ ð4WW − 3W Þ¼0 on the ground state, so which fixes the electroweak scale. In the Brans-Dicke- a massless state is indeed present corresponding to the flat Jordan theory of gravity, not considered here, one expects5 ρ σ ∼ σ direction. Finally, since 0 is a root of both W and of its h i MPlanck.(h i can then be regarded as a physical cutoff 0 2 derivative W , then Wðϕ=σÞ ∝ ðϕ=σ − ρ0Þ , while if V of the theory.) We simply enforce such hierarchy by depends only on even powers of the scalar fields (our model choosing a very weak coupling of the visible to the hidden below), then the general structure is sector of the dilaton6 [31]. Such hierarchy is, however, stable under quantum corrections, so mϕ ∼ hϕi ≪ hσi 2 2 2 2 Wðϕ=σÞ ∝ ðϕ =σ − ρ0Þ : ð3Þ without tuning at the quantum level [4,8], and we verify this in our model at one loop. This is expected to remain Note that the vanishing vacuum energy Vðhϕi; hσiÞ ¼ 0 true to all orders in perturbation theory since scale follows from the (spontaneously broken) scale symmetry; symmetry is preserved by the regularization and is broken see Eq. (2). A scale-invariant regularization of this theory only spontaneously.7 We thus have an example of a leads to a scale-invariant quantum potential, which thus quantum stable hierarchy, with a vanishing vacuum energy remains of the form shown in Eq. (1). Hence, the above at the loop level, that follows from the demand of sponta- discussion around Eqs. (1), (2), and (3) remains true at the neously broken quantum scale symmetry. quantum level, including the possibility of spontaneous- only breaking of the scale symmetry. 4 λ 1 λ One of the two minimum conditions in (2) fixes the ratio Perturbativity, σ < , is maintained for a weak coupling m between the visible (ϕ) and hidden (σ) sectors; see later, Eq. (9) ρ0 ¼hϕi=hσi in terms of the (dimensionless) couplings of (a) or (11) (a), which fixes λσ < 1 in terms of the other couplings, the theory. Thus, all vevs of such a theory, including hϕi, for a small enough λm. are proportional to hσi ≠ 0, which is a(n) (unknown) 5Such hierarchy can be generated dynamically (Refs. [28,29] parameter of the theory. The second minimum condition, or as in Ref. [30]). 6 λ ≪ λ λ σ after eliminating ρ0 between the two equations in (2),gives One takes j mj ϕ; m: coupling of hidden ( ) to the visible ϕ λ a relation among the couplings of the theory in the order of sector ( ); ϕ: Higgs self-coupling; see later in the paper, Eq.
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