Naturalness in the dark at the LHC

Journal of High Energy Physics, Jul 2015

We revisit the Twin Higgs scenario as a “dark” solution to the little hierarchy problem, identify the structure of a minimal model and its viable parameter space, and analyze its collider implications. In this model, dark naturalness generally leads to Hidden Valley phenomenology. The twin particles, including the top partner, are all Standard-Model-neutral, but naturalness favors the existence of twin strong interactions — an asymptotically-free force that confines not far above the Standard Model QCD scale — and a Higgs portal interaction. We show that, taken together, these typically give rise to exotic decays of the Higgs to twin hadrons. Across a substantial portion of the parameter space, certain twin hadrons have visible and often displaced decays, providing a potentially striking LHC signature. We briefly discuss appropriate experimental search strategies.

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Naturalness in the dark at the LHC

Received: March Naturalness in the dark at the LHC Nathaniel Craig 0 1 3 7 8 9 Andrey Katz 0 1 2 3 5 6 8 9 Matt Strassler 0 1 2 3 8 9 Raman Sundrum 0 1 3 4 8 9 0 Universit ́e de Gen`eve, 24 quai E. Ansermet , CH-1211 Geneva 4 , Switzerland 1 CH-1211 Geneva 23 , Switzerland 2 Department of Physics, Harvard University 3 Santa Barbara , CA 93106 , U.S.A 4 Department of Physics, University of Maryland 5 Theory Division , CERN 6 Department of Theoretical Physics and Center for Astroparticle Physics , CAP 7 Department of Physics, University of California 8 College Park, MD 20742 , U.S.A 9 Cambridge , MA 02138 , U.S.A We revisit the Twin Higgs scenario as a “dark” solution to the little hierarchy problem, identify the structure of a minimal model and its viable parameter space, and analyze its collider implications. In this model, dark naturalness generally leads to Hidden Valley phenomenology. The twin particles, including the top partner, are all Standard-Model-neutral, but naturalness favors the existence of twin strong interactions - an asymptotically-free force that confines not far above the Standard Model QCD scale - and a Higgs portal interaction. We show that, taken together, these typically give rise to exotic decays of the Higgs to twin hadrons. Across a substantial portion of the parameter space, certain twin hadrons have visible and often displaced decays, providing a potentially striking LHC signature. We briefly discuss appropriate experimental search strategies. Nonperturbative Effects; Global Symmetries; Effective field theories 4 Fraternal color 5 Twin hadron phenomenology Perturbative considerations Fraternal confinement Kinematic regions Couplings to the visible sector 5.5 Twin hadron production 6 LHC phenomenology New Higgs decays with displaced vertices New Higgs decays without displaced vertices Precision Higgs measurements 7 Conclusions A Quarkonium mass spectrum and decays A.1 Spectrum of the quarkonium states B Twin hadron production in more detail B.1 Nonperturbative effects in twin gluon production B.2 Nonperturbative effects in twin bottom production 1 Introduction 2 The minimal or “Fraternal” Twin Higgs The central mechanism Minimal particle content 2.3 Summary of Fraternal Twin Higgs model 3 Electroweak breaking and tuning Matching to SM effective field theory Estimating electroweak tuning 3.4 Twin Higgs effective potential approximation 1 5 5 C Search strategies C.1 Comments on triggering C.2 Benchmark models C.2.1 Single displaced vertex search Exclusive double displaced vertex search C.2.3 Inclusive double displaced vertex search D Precision electroweak The effect of twin hypercharge The principle of Naturalness, the notion that the weak scale should be insensitive to quantum effects from physics at much higher mass scales, necessitates new TeV-scale physics beyond the Standard Model (SM). It has motivated a broad program of searches at the LHC, as well as lower-energy precision/flavor/CP experiments. The absence thus far of any signals in these experiments has disfavored the most popular scenarios, including supersymmetry (SUSY), Composite Higgs and Extra Dimensions, unless their mass scales are raised above natural expectations, leading to sub-percent level electroweak fine-tuning in complete models. The puzzle over why Nature should be tuned at this level has been dubbed the “Little Hierarchy Problem”, and it has led to a major rethinking of naturalness and its implications for experiments. In a bottom-up approach to this problem, one may take a relatively agnostic view of very high energy physics, and focus instead on naturalness of just the “little hierarchy”, from an experimental cutoff of about 5–10 TeV or so down to the weak scale. Unlike naturalness considerations involving extremely high scales, such as the Planck scale, which are tied to multi-loop virtual effects on the Higgs sector of all SM particles, the bottom-up little hierarchy problem is simpler, relating predominantly to one-loop effects of just the heaviest SM particles, i.e. those coupling most strongly to the Higgs. In Little Higgs, (Natural) SUSY, and extra-dimensional models of gauge-Higgs unification (including warped models that are dual to Higgs Compositeness via the AdS/CFT correspondence), large one-loop radiative corrections to the Higgs from the heaviest SM particles cancel algebraically against those of new symmetry “partners” of these heavy particles, thereby ensuring stability of The most significant of these corrections is associated to the top quark. Naturalness requires that this must be substantially canceled by a corresponding correction from the top’s partner(s), to which it is related by an ordinary global symmetry or supersymmetry. 500 GeV, easily within the kinematic reach of the LHC. Such particles also have significant LHC production cross-sections, since a top partner generally carries the same color charge as the top quark. Through this logic, the search for top partners in the above incarnations, under a variety of assumptions about possible decay modes, has become a central pursuit But must the top partner be colored? The answer is obviously critical to experimental exploration. Naively the answer is yes, because the algebraic cancellations depend at oneloop on the top Yukawa coupling to the Higgs, and this coupling is itself corrected at one higher loop by QCD. In a rare exception to the “one-loop rule”, such two-loop radiative corrections to the Higgs are still quantitatively important for the little hierarchy problem because of the strength of QCD. It would seem then that the top partner should also be colored so as to parametrically “know” about this QCD effect in symmetry-enforced Yet, remarkably, there do exist solutions to the little hierarchy problem, “Twin Higgs” being the first and prime example, in which the top partners are uncolored [1–3].1 Here the cancellation among radiative corrections is enforced by a discrete Z2 symmetry that exchanges SM particles with new “twin” states. One way to assure naturalness cancellation of QCD two-loop effects is to have the twin symmetry exchange SM color and its gluons with a distinct twin color gauge group and its twin gluons, which couple to and correct the twin top partner just as QCD does the top quark. We will focus on theories of this type, specifically ones in which all twin particles are “dark”, with no SM quantum numbers. Colorless twin tops are vastly more difficult to produce at the LHC than top partners of more popular theories; indeed this is true for all twin sector particles. Twin particle production can only proceed through a “Higgs portal”, a modest mixing between the SM and twin Higgs sectors that is a necessary consequence of the twin Higgs mechanism for addressing the little hierarchy. Not only is the production rate small, the hidden particles barely interact with ordinary matter, and (at least naively) one would expect they escape the detectors unobserved. The resulting missing energy signature with a very low crosssection would pose great difficulties at the LHC. How else can the twin sector be detected? Higgs mixing and virtual twin top loops can also subtly affect the SM-like Higgs, making precision tests of its properties extremely important. But with expected LHC precision, the visibility of the twin sector in this manner at the LHC is limited [5]. The Higgs of the twin sector can also potentially be produced, but it may be too wide to observe as a resonance. At best, it is heavy and has a low cross-section, and is far from being excluded or discovered. For these reasons, Twin Higgs remains a viable resolution of the little hierarchy problem, a well-hidden outpost of In this paper, we re-examine the Twin Higgs scenario as an important and distinctive case study in “dark” naturalness, and we identify exciting new experimental opportunities We develop minimal Twin Higgs models addressing the little hierarchy problem, roughly paralleling the way in which “Natural SUSY” [6–11] has emerged as a minimal phenomenological approach to the little hierarchy problem in the SUSY 1Another known solution to the little hierarchy problem which involves uncolored top partners is “Folded the twin tops into a pair of twin gluons, which subsequently hadronize to produce various twin While some glueballs are stable at the collider scale, G0+ decay to Standard Model particles is sufficiently fast to give LHC-observable effects, including possible displaced vertices. The hgˆgˆ coupling, indicated by a black dot, is generated by small mixing of the Higgs and the twin paradigm. In both cases, minimalism can be viewed as a tentative organizing principle for doing phenomenology, starting with searches for the minimal natural spectrum and then “radiating outwards” to include searches complicated by non-minimal states. Also in parallel to SUSY, we find that the two-loop relevance of QCD interactions to the little hierarchy problem leads to some of the most promising experimental signals. In SUSY, the symmetry cancellation at two loops requires the presence of a gluonpartner, the gluino. With large color charge and spin, the gluino is phenomenologically striking over much of motivated parameter space, almost independent of its decay modes [12–14]. In Twin Higgs models, the analogous two-loop role is played by twin gluons, which can again give rise to striking signatures over a large part of parameter space, not because of large cross-sections but because they, along with any light twin matter, are confined into bound states: twin hadrons. Together with the Higgs portal connecting the SM and twin sectors, the presence of metastable hadrons sets up classic “confining Hidden Valley” phenomenology [15–21], now in a plot directly linked to naturalness. A prototypical new physics event is illustrated in figure 1. The scalar line represents the recently discovered 125 GeV Higgs scalar. This particle is primarily the SM Higgs with a small admixture of twin Higgs; it is readily produced by gluon fusion. But because of its twin Higgs content, it has at least one exotic decay mode into twin gluons, induced by twin top loops, with a branching fraction of order 0.1%. The twin gluons ultimately hadronize While most twin glueballs have very long lifetimes and escape the detector as missing energy, the lightest 0++ twin glueball has the right quantum numbers to mix with the SM Higgs, allowing it to decay back to the SM on detector timescales. The first excited 0++ state also may have this property. This type of effect was first studied, in the context of a Hidden Valley/quirk model [15, 17, 21, 22] by Juknevich [23]. If the lightest 0++ glueball, which we call G0+, has a high mass, then its decay is prompt, and its production rate in Higgs decays may be too rare for it to be observed machines, providing strong motivation for ILC, FCC-ee, or CEPC. of the G0+ may be macroscopically displaced from the interaction point. Such displaced decays are a striking signature, spectacular enough to compensate for the relatively low production rate, and represent an excellent opportunity for the LHC. Moreover, the signal may be enhanced and/or enriched if there is a sufficiently light twin bottom quark. Depending on this quark’s mass, twin bottom production may lead to a larger overall twin hadron production rate; the resulting final states may include twin glueballs, twin bottomonium, or both. In some portions of parameter space, all twin the Higgs just like the G0+, though possibly with a small branching fraction. There is also the potential for displaced vertices from this state, though our calculations of lifetimes and branching fractions suffer from large uncertainties. With a branching fraction of the Higgs to twin hadrons of order 0.1% or greater, our minimal Twin Higgs model should motivate further experimental searches for this signal ATLAS and at CMS during Run I, and we can expect many more at Run II. But triggering inefficiencies threaten this signal. Despite the low backgrounds for highly displaced vertices, triggering on such events can be a significant challenge [16, 24–31]. There is urgency as we approach Run II, since it is not trivial to design triggers that will efficiently capture all variants of the displaced vertex signature arising within the parameter space of the model. In particular, since twin hadron lifetimes are a strong function of their mass, and since displaced decays in different subdetectors require quite different trigger and analysis methods, a considerable variety of approaches will be required for efficient coverage of the parameter space of the model. Further complicating the matter is that precise theoretical predictions of twin glueball and especially twin quarkonium production in Higgs decays is extremely difficult. Hadronization in the twin sector is a complex and poorly understood process, and considerably more investigation will be needed before predictions of glueball and quarkonium multiplicity and kinematic distributions could be possible. Thus this solution to the naturalness problem requires further work on both experimental and theoretical fronts. We should note that every element that goes into this story has appeared previously in the literature. Hidden glueballs appear in quirk models and other Hidden Valley models [15, 21, 22]; they specifically arise in Folded Supersymmetry [4] and the Quirky Little Higgs [32], which attempt to address the hierarchy problem; the mixing of the G0+ with the Higgs to generate a lifetime that is short enough to be observed at the LHC but long enough to often be displaced has appeared in a study [23] of a Hidden Valley model with quirks [21]. However we believe that this is the first time these elements have all been assembled together, giving the striking observable signal of exotic Higgs decays, possibly to long-lived particles, as a sign of hidden naturalness. Returning from the phenomenology to broader considerations, we note that the Twin Higgs shares with Little Higgs and Composite Higgs models the realization of the Higgs as a pseudo-Goldstone boson, while representing a significant break with earlier thinking and phenomenology in having a colorless top partner. In the far UV, it is possible that the Twin Higgs structure might match on to more conventional SUSY [33–35] or Composite dynamics [36, 37], or perhaps to something quite novel [38, 39]. We hope that studying the UV matching is facilitated by our bottom-up exploration of minimal Twin Higgs structure and phenomenology. We also hope that this work broadens our perspective on naturalness, motivates more careful investigations of other “darkly” natural mechanisms, such as “Folded Supersymmetry” [4], and perhaps inspires entirely new mechanisms. Ultimately, we hope to broaden the scope of experimental strategies motivated by naturalness, perhaps even leading to a discovery unanticipated by theory. The paper is organized as follows. Section 2 reviews the basic Twin Higgs structure and develops the minimal model. We show in section 3 that the electroweak tuning in the minimal Twin Higgs model is very mild. Section 4 shows quantitatively that the twin sector likely contains twin QCD in order to maintain naturalness, and its confinement and hadrons are discussed. Section 5 derives some properties of the resulting twin hadrons, and discusses their production and decays. Some subtle points on hadron decays and hadron production are left for appendices A and B. Section 6 synthesizes the earlier considerations and discusses LHC phenomenological implications. Possible experimental strategies for long-lived particle searches are briefly considered in section 6.1 and in more detail in appendix C. Precision Higgs measurements are considered in section 6.4, and precision electroweak constraints are discussed in appendix D. Our conclusions appear in section 7. In appendix E, the phenomenological effect of gauging twin hypercharge, as a non-minimal extension of the model, is considered. The minimal or “Fraternal” Twin Higgs In this section we construct the minimal Twin Higgs model, starting by reviewing the basic symmetries and Higgs structure and then justifying each addition to the twin sector based on the need to maintain naturalness and internal consistency. Because the minimal model does not duplicate all SM states in the twin sector (in contrast to the original mirror Twin Higgs model and its descendants [1–3], in which the twin sector and its couplings are an exact copy of the SM), we will refer to this construction as the “Fraternal Twin Higgs”. The model is summarized in subsection 2.3. The central mechanism At its heart, the Twin Higgs mechanism involves realizing the SM-like Higgs as a pseudoGoldstone boson of an approximate global symmetry, namely SU(4). An SU(4)-fundamental complex scalar H with potential, to seven Goldstone bosons, of which one is ultimately identified as the SM-like Higgs scalar. The SU(4) is explicitly broken by the gauge and Yukawa couplings of the Standard Model. Without additional recourse, this explicit breaking would lead to quadratic sensitivity to higher scales at one loop. In the context of conventional global symmetry protection, this explicit breaking and UV sensitivity can be ameliorated by extending Standard Model gauge bosons and fermions into representations of the full global symmetry, at the price of introducing additional partner states charged under the SM. In the Twin Higgs, the key insight is that SM states need not be extended into full representations of the global symmetry, and instead are merely related to partner states by a Z2 exchange symmetry. This Z2 is then promoted, at the level of quadratically divergent radiative corrections, to an accidental SU(4) symmetry. The partner particles are no longer related to SM states by a continuous symmetry, and so need not carry SM gauge quantum numbers. As in any global symmetry mechanism that stabilizes the weak scale, the Twin Higgs does not address the big hierarchy problem all the way to the Planck scale, but merely a the “big” hierarchy problem, such as supersymmetry or compositeness, kicks in to provide protection against yet higher scales. A and B. We identify A with the SM Higgs doublet of gauged electroweak SU(2)L (and charged under U(1)Y of course), while B is a doublet of a different SU(2) symmetry, which we will call twin SU(2). At this stage, the twin SU(2) group can be either global or gauged. The Z2 symmetry acts by exchanging A ↔ B. By far the largest source of explicit breaking from the Standard Model will be the top Yukawa, so to see the magic of the twin mechanism let us also introduce twin top multiplets: three species of the twin fermion Qˆa transforming Z2 symmetry implies a twin top Yukawa coupling L ⊃ yˆtBQˆauˆa, implies at least a global SU(3) symmetry acting on fermions Qˆ and uˆ. The one-loop radiative potential for the A and B multiplets coming from loops of top and twin top quarks takes the form 16π2V (1-loop) = −6yt2Λ2|A|2 − 6yˆt2Λ2|B|2 + 3yt4|A|4 log Λ2/yt2|A|2 + 3yˆt4|B|4 log Λ2/yˆt2|B|2 break the SU(4) but preserve the Z2. This is the magic of the twin mechanism: if the SMlike Higgs can be identified with a pseudo-Goldstone of the spontaneously broken SU(4), its mass will be insensitive to the SU(4)-symmetric quadratic divergences at one loop (or more) provided the Z2 relates yt and yˆt. will be replaced by Z2-symmetric physical thresholds. A second central issue is vacuum alignment. The vacuum expectation value (vev) of f 2 = vA2 + vB2 . need perturbations stabilizing v Minimal particle content In general this vev breaks both SU(2)L ×U(1)Y and twin SU(2), such that most components of H are eaten if twin SU(2) is gauged. The remaining physical scalar states consist of linear combinations of the radial mode and a single uneaten Goldstone boson. We will identify the observed SM-like Higgs with the uneaten Goldstone boson of SU(4)/SU(3). The mass of the radial mode (corresponding to |H|2 fluctuations) is √2λf . (Alternately, we could work purely in terms of the non-linear sigma model of SU(4)/SU(3), in which case there is theory. We do not follow this approach here.) To obtain a realistic vacuum, we must pseudo-Goldstone Higgs is an equal mixture of A and B, where recall only A carries SM Higgs quantum numbers. Since we have observed a SM-like Higgs experimentally, we will vB2, so that the pseudo-Goldstone Higgs is primarily Thus far we have seen how the essential structure of the twin mechanism operates at the level of the Higgs and the largest source of explicit SU(4) breaking, the top Yukawa coupling, and understood the new twin states thereby required. We continue this process of deducing the minimal ingredients for a realistic and natural Twin Higgs model. We begin with the top yukawa itself. We have seen schematically that the Higgs is But how much can the top Yukawa Z2 be relaxed while preserving the naturalness of the weak scale? When the coupling (2.2) is introduced, by eq. (2.3) the physical mass of the pseudo-Goldstone boson Higgs gets a quadratically divergent radiative correction at the This precisely cancels out when the Z2 symmetry is exact. We can picture this in figure 2, where the pseudo-Goldstone Higgs acquires an effective coupling to the twin top upon integrating out the heavy radial Higgs mode. The cancellation in figure 2 is very similar to that in Little Higgs theories [40–42],3 with the difference that the top partner is uncolored. But without exact Z2 symmetry, the naturalness demand that these corrections are not much larger than the observed SM-like Higgs mass-squared of (125 GeV)2, translates into 3See also [43] for review and references therein. −yˆft second diagram arises upon integrating out the heavy radial mode. Ordered by the size of tree-level couplings to the Higgs doublet, the next ingredients to consider are the potential twin sector equivalents of SU(2)L × U(1)Y gauge bosons. The subdominant to top loops. This suggests gauging the twin SU(2) global symmetry acting on B and Qˆ. Introducing twin weak gauge bosons with coupling gˆ2 translates to the quadratic cutoff sensitivity in the pseudo-Goldstone Higgs, in analogy with eq. (2.5). Demanding this not significantly exceed the observed Higgs massbosons of SU(4) breaking, except for the SM-like Higgs itself, are now longitudinal weak bosons of the visible and twin sectors. In contrast, the contribution to m2h from U(1)Y loops in the SM is comparable to m2h does not require twin hypercharge, although it was included in the original Twin Higgs [1]. This is analogous to the statement that in natural supersymmetry there is no need for the Bino to be light; its presence in the low-energy spectrum is non-minimal from the bottomup point of view. Given that our principle in this paper is to seek the most economical version of the twin Higgs that is consistent with the naturalness of the little hierarchy, we do not include twin hypercharge in the minimal twin Higgs model, assuming instead that of completeness, we will briefly discuss the significant phenomenological consequences of a light twin hypercharge boson in appendix E. Next we turn to the twin analogue of QCD. Of course the Higgs does not couple to SU(3) at tree level, but rather at one loop via its coupling to the top quark. This nonetheless leads to sizable two-loop corrections to the Higgs mass from physics around the cutoff. As we will discuss in detail in section 4.1, the contribution to the Higgs mass-squared from QCD on similar footing as the weak gauge group. Gauging the twin SU(3) global symmetry with coupling gˆ3 gives quadratic cutoff sensitivity in the pseudo-Goldstone Higgs it suffices that yˆb This is a key observation that will drive the phenomenology of a viable Twin Higgs model: naturalness and minimality favor a confining gauge symmetry in the hidden sector, “twin glue”. This twin glue has a coupling close to the QCD coupling — we will see how close in SM QCD confinement scale. Once the twin SU(2) and SU(3) are gauged, we should include a variety of twin fermions in addition to the twin top quark to cancel anomalies. To render the twin SU(3) anomalyfree we should include a twin RH bottom quark ˆb; symmetries then admit the hidden sector bottom Yukawa coupling L = yˆbBQˆˆb . bottom Yukawa has a much weaker effect on the SM Higgs mass at one loop. At this stage yt in order to avoid creating a hierarchy problem from the bottom Similarly, canceling the twin SU(2) anomaly requires an additional doublet neutral under the twin SU(3): Lˆ, left-handed twin tau. Although not required for anomaly cancelyˆt. The twin neutrino may be rendered massive in the same way as the SM neutrinos; its mass plays no role in naturalness and is essentially a free parameter of the model. Finally, twin light-flavor (first and second generation) fermions are totally unnecessary for naturalness, as their Yukawa couplings too small to meaningfully disturb the Higgs potential, and are therefore absent in our minimal “Fraternal” model. Summary of Fraternal Twin Higgs model Thus we arrive at the ingredients of our minimal Twin Higgs model: 1. An additional twin Higgs doublet and an approximately SU(4)-symmetric potential. 2. Twin tops and a twin top Yukawa that is numerically very close to the SM top Yukawa. 3. Twin weak bosons from the gauged SU(2) with gˆ2(Λ) ≈ g2(Λ). 4. Twin glue, a gauged SU(3) symmetry with gˆ3(Λ) ≈ g3(Λ). This gauge group is 5. Twin bottoms and twin taus, whose masses are essentially free parameters so long as they remain much lighter than the twin top. 6. Twin neutrino from the twin tau doublet, which may have a Majorana mass, again a free parameter as long as it is sufficiently light. As an aside, we note that in contrast to a model with a perfect twin of the Standard Model, this model is cosmologically safe; with at worst one massless particle (the twin tau neutrino), and fewer degrees of freedom than the visible sector, the effective number of degrees of freedom during nucleosynthesis and recombination is very small. Not accidentally, the most crucial ingredients for the naturalness of the theory — a twin Higgs, twin tops, and twin glue — strongly resemble the ingredients of natural supersymmetry (Higgsinos, stops, and the gluino). But the key difference here is the likely existence of a new, confining gauge group in the minimal twin sector. Although twin glue does not impact the Higgs directly at one loop, its contributions at two loops make it a key component of a viable twin Higgs model. Electroweak breaking and tuning In this section, we study the effective potential of the Fraternal Twin Higgs model outlined above. We show how realistic electroweak symmetry breaking can be achieved, accompanied by a 125 GeV Higgs scalar, and estimate the tuning of couplings needed. In subsection 3.1, we write down the effective potential for the full Higgs sector of the model at one-loop order. In subsection 3.2, we integrate out the heavier Higgs to get an effective potential for just the SM-like Higgs. In subsection 3.3, we determine that the degree of fineFinally, in subsection 3.4 we more fully justify the form of our starting effective potential in subsection 3.1, by showing that it is free from other types of fine-tuning. The Twin Higgs effective potential is given, to good approximation, by where for concrete estimates we will take the UV cutoff of the Twin Higgs theory to be consistent with the SM and twin gauge symmetries. Line (3.1) is just our starting point, eq. (2.1), the subset of terms respecting the global SU(4) symmetry, under which (A, B) transform in the fundamental representation. Line (3.2) consists of the extra terms allowed by breaking SU(4) but preserving the discrete Z2 global subgroup, A ↔ B. (The other Z2 invariant, |A|2|B|2, is equivalent to this, modulo SU(4)-invariant terms.) Line (3.3) consists of the remaining extra terms which respect only the gauge symmetries. Line (3.4) is the dominant one-loop radiative correction that cannot be absorbed into a redefinition of the in section 4, we will justify this as a good approximation because of the gauging of twin color. While the logarithmic cutoff dependence above can be removed by renormalization Matching to SM effective field theory Before fully justifying the above approximate structure for the effective potential, we will first work out its consequences, in particular matching it to a SM effective field theory at lower energies. This will allow us to choose the rough sizes of the different couplings needed for realism, and to then self-consistently check our approximation. Line (3.1) contains the dominant mass scale, set by f . We take the SU(4)-symmetric f = few × vA , vA = v ≡ 246 GeV, so that the pseudo-Goldstone Higgs is primarily SM-like. We will also use this small hierarchy to work perturbatively in powers of v/f . rise to a spontaneous symmetry breaking SU(4) → SU(3), where the breaking VEV is in hB0i = f / 2 . mhˆ = √ |B|2 = 2 − |A|2. This breaks twin SU(2) gauge symmetry, with three of the seven Nambu-Goldstone bosons being eaten in the process. The remaining four real Nambu-Goldstone bosons form the complex A doublet, namely the SM Higgs doublet of electroweak gauge symmetry. The fourth uneaten B scalar is the radial mode of the potential, a physical heavy exotic Higgs Later, O(v2/f 2) perturbations will mix these boson identifications to a small extent. in the theory that break Z2 are the small SM hypercharge coupling and the very small b, also taken to be ∼ g14/(16π2), where g1 is the SM hypercharge coupling. This is so small that we neglect it in what follows.4 scales as assumed above. At energies well below the heavy Higgs mass and heavy twin gauge boson masses, set by f , |A|2 + |B|2 is rigidly fixed at the bottom of the potential in line (3.1), an effective potential for just the lighter A degrees of freedom, y2 | | t where this equation is expressed in terms of the mass of the twin top, or “top partner”, mtˆ = yt √ , in analogy with the top quark mass. Eq. (3.9) has the form of a SM effective potential, with the tree-like |A|2, |A|4 terms, the tuning involved, we can neglect the modest ln |A| modulation of |A|4, and just set the logarithm to its expectation value, Veff ≈ y2 | | t ln(mtˆ/mt) − 4 analogous corrections in supersymmetry, is expected to reduce this radiative correction, the central feature of which can be captured by using a top-Yukawa coupling renormalized 4 at several TeV, where yt ≈ 1/2. In any case, the rough sizes of the radiative corrections comparable magnitude we are able to successfully and naturally fit the observed physical As an aside, it is instructive to compare the form of the logarithmically divergent (twin) top-loop contributions in eq. (3.9) with the analogous contributions from top/stop symmetry, the quadratic divergence in the top-loop contribution to the Higgs potential has been canceled by a top partner mechanism. However, there remains a logarithmically divergent contribution to the Higgs mass-squared. This has precisely the same form as in divergent contributions to the Higgs quartic self-coupling from the (twin) top loop. Again, there are analogous contributions in supersymmetry of the same magnitude, but the stop contribution has the opposite sign due to its Bose statistics. Thus in supersymmetry the logarithmic divergence cancels out here, but there is still a finite logarithmic enhancement factor of ln(ms2top/mt2) to the Higgs quartic correction. Unlike the MSSM where the treelevel Higgs quartic is set by electroweak gauge couplings, here we have an unconstrained will affect electroweak tuning, as discussed below.) Estimating electroweak tuning Electroweak tuning involves the quadratic terms of the potential. Fortunately, the same combination of parameters that dominates the quartic self-coupling, in the second line of eq. (3.11), also appears in the quadratic terms of the potential, in the first line of eq. (3.11). Then, matching the quadratic terms to the SM form on the last line, we have 2mm2t2 = Twin Higgs effective potential approximation Finally, we justify our starting approximation for the Twin Higgs effective potential upon which our analysis of electroweak breaking and tuning is based. To do this, we examine each term of eqs. (3.1)–(3.4) and ask whether its coefficient is radiatively stable, and also whether this coupling itself significantly radiatively corrects other couplings. diative corrections to the quadratic terms in line (3.1). We have taken the SU(4)-symmetric self-interaction to be weakly coupled enough that the leading quadratic radiative corrections come from the (twin) top loop and the large Yukawa coupling, The other couplings that can similarly contribute at one loop are the electroweak gauge ensuing discussion), and loops using the SM Higgs quartic coupling are also considerably tuning to keep f If we take for example f ∼ 3v ∼ 750 GeV, we see that there is essentially no tuning. loop (eq. (3.12) and ensuing discussion), so this Z2-preserving but SU(4)-violating coupling is radiatively stable and natural. As explained earlier, we can naturally take the hard of soft breaking. To tune to realistic electroweak symmetry breaking we saw that we needed σ ∼ λSM/2 ∼ 1/16, therefore we can easily have σf 2 dominates the mass scales of the Twin Higgs potential as our analysis presumes. With all couplings in our theory being perturbative, our one-loop analysis suffices for demonstrating the successful matching to a realistic SM effective field theory and for estimating the tuning required. We now discuss the dynamics of twin color in more detail. We first calculate how close the twin and visible sector color couplings must be to preserve naturalness. Then we estimate the confinement scale of twin color and discuss the associated twin hadrons, including glueballs and quarkonia. Perturbative considerations As we have seen above, the twin Higgs mechanism for naturalness, at one-loop order, requires the top and twin-top Yukawa couplings to be very nearly identical close to the cutoff. Here we show that when QCD effects are taken into account at two-loop order, naturalness favors having a twin QCD, with a gauge coupling similar to that of QCD near To see this, consider the one-loop RG analysis for the dimensionless Wilsonian running = −2x + where g3 is the SM QCD coupling and gˆ3 is the twin QCD coupling. Note that g3 and gˆ3 run differently because the SM has six QCD quark flavors while the minimal model has just two twin QCD flavors. If we neglect the running of yt, yˆt, g3, gˆ3, then the solution to the first of these equations is simply given by 3(yt2 − yˆ2) matched onto the one-loop result of eq. (2.5), which stresses the importance of having yt and yˆt be very nearly the same. But taking account of the running of yt, yˆt, g3, gˆ3 by solving all of eqs. (4.1) then gives an RG-improved result, which allows us to explore the role of twin color in maintaining yt ∼ yˆt as the couplings run. The simplest calculation arises by seeing what happens when we have no twin color, deviation will feed into m2h. To focus on just this effect we will drop the g3-independent (Keeping these effects would be subleading in the running of m2h compared to those of the working to first order in yˆt − yt, the solution to eqs. (4.1) is given by Two loop fine tuning ∼ 3yt2(Λ)g32(Λ) Λ2 ≈ 0.25 . Of course, it is not reasonable for two couplings, yt, yˆt, that run differently to be exactly they differ by at least the running of yt due to QCD over an e-folding of running, In other words, we expect a comparable quadratic divergence in m2h,IR from the splitting in Yukawa couplings and from the explicit O(g32yt2) divergence. Thus a better estimate of fine-tuning in the absence of twin QCD is . 10 percent. The tuning due to QCD at two-loop order will clearly become very mild if we do be determined by the UV couplings. Roughly, eq. (4.4) is then replaced by combines the O(g32yt2) contribution from running and the threshold correction in eq. (4.6) in quadrature to reflect the unknown relative sign of the threshold correction. We will take this to be the case in what follows and study the effects of twin confinement from this twin QCD sector on the twin spectrum and phenomenology. Finally, of course even g3 and gˆ3 run differently because the particle content of the minimal Twin sector differs from that of the SM, so one may wonder how close they can which is easily consistent with the requirement of naturalness discussed above. governs the infrared phenomenology of the twin sector. If all fermions carrying twin SU(3) quantum numbers are much heavier than the confinement scale, the infrared physics is that of pure SU(3) gauge fields, and the lightest states in the confined twin sector will be glueballs, whose rich spectrum includes states with different angular momentum J , charge conjugation C and parity P . As shown in lattice studies of SU(3) pure glue [44, 45], at least a dozen glueballs are stable against decay data provides the ratios of these glueballs’ masses to each other and to the confinement Once the twin b becomes sufficiently light that twin glueballs and twin bottomonium states (which we will refer to generically as “[ˆbˆb]”) have comparable masses, the situation becomes more complex. We will not explore this regime carefully in this paper, leaving its still applies approximately. fewer quark flavors, faster running (i.e. a more negative beta function), and therefore a modestly higher confinement scale. This is illustrated in figure 3, which shows the strong ∆ g-0.05 L3 @GeVD, Minimal Twin Higgs on yˆb through its impact on the twin QCD beta function.5 We may now estimate the mass scale of twin glueballs. Using lattice estimates of the glueball mass spectrum in units of the inverse force radius [44, 45] and the zero-flavor SU(3) MS confinement scale in units of the inverse force radius [46], we find the mass m0 of the m0 determined by running couplings down from the cutoff carries a combined uncertainty of O(10%) from the lattice estimates, primarily due to uncertainty in the inverse force radius [46]. Given the mass of the G0+, the masses of higher glueball excitations in terms of m0 are known to good precision. The next highest states in the glueball spectrum are well separated from the G0+, with the closest states being the G2+, a 2++ glueball with spectrum is illustrated on the left-hand side of figure 4. Meanwhile, the twin bottomonium states form a rich spectrum, whose lowest lying states are narrow if rapid decays via twin glueballs are inaccessible. As is familiar from SM quarkonium data, and as sketched on the right-hand side of figure 4, the spectrum to be the MS coupling. Figure 4. Sketch of the twin hadron spectrum in the regime where m0 < 2mˆb < 2m0. In addition to the G0+, of mass m0, about a dozen other glueballs, with mass splittings of order m0, are stable against twin strong decays. Numerous twin bottomonium states, including a tower of 0++ states quarkonia, can dominantly decay via annihilation through an s-channel off-shell Higgs to the SM. are no “open twin bottom” mesons analogous to the SM’s Bu, Bd mesons. Thus the towers of narrow quarkonium states extend much further up than in the SM, potentially up to a scale of order 2m0, as sketched in figure 4, or m0 + 2mˆb. The reader should note that figure 4 is only illustrative, and must be interpreted with caution. Its details change dramatically as one raises or lowers the quarkonium masses relative to the glueball masses, and it omits the many narrow higher-spin quarkonium states, along with various other phenomenological details. Twin hadron phenomenology Thus far we have seen that viable Fraternal Twin Higgs models include twin glue, with couplings that favor confinement roughly an order of magnitude or so larger than the SM QCD scale, producing relatively light twin glueballs and/or twin quarkonia. Both twin gluons and twin quarks are connected to the Standard Model via low-dimensional portals, and this can lead to observable and even spectacular twin hadron phenomenology. As in Folded Supersymmetry [4], where twin glueballs also arise, we thus find a connection between dark naturalness and twin hadrons. In our case, this connection manifests itself as new and exciting opportunities for discovery at the LHC. The model’s phenomenology changes significantly as we move around in the parameter space, and in most regions it is rather complicated. But the most promising and dramatic LHC signals arise even in the conceptually simplest region, namely where mˆb > 12 mh (i.e., parameter space; see figure 5. Details are explained in subsequent sections. Solid lines indicate kinematic boundaries. Common final states are indicated in italics. At low glueball mass, decays of the G0+ are displaced; see figure 6. Here it is assumed that there are light twin leptons, so one regions C and D, and is displaced at low mass. LHC phenomenology By far the most spectacular signal that can arise from our minimal Twin Higgs model is the displaced decays of twin glueballs and quarkonia. We describe the phenomenology of this signal, as it arises from h decays, in section 6.1; further details on search methods are given in appendix C. If no displaced decays are observable, h decay signals may be brief discussion of how a twin hypercharge U(1) would affect the phenomenology is given in appendix E. Section 6.3 covers signals from a heavier Higgs hˆ. Finally, section 6.4, explores the order-(v/f )2 effects on SM Higgs production rates. New Higgs decays with displaced vertices cally forbidden. Because the number of Higgs bosons produced at LHC in Run II will be of order 107, and because displaced vertices are spectacular signals when identified, these numbers represent a very promising opportunity. Already hundreds or thousands of events with displaced vertices may have been produced, though in many parts of parameter space they would clearly have evaded existing LHC Run I searches [27–29, 50, 51]. As figure 7 suggests, the model exhibits a great diversity of displaced vertex phenomenology. Rather than address the full story here, we mainly discuss the regions with the simplest phenomenology. These regions, it turns out, produce most of the possible Higgs decay signatures, and are therefore sufficient to motivate the most important searches, which are sensitive to effects in more complicated regions that we will not discuss in detail. The simplest region is the portion of region A with mˆb > mh/2, where the irreducible rate applies and only glueballs can be produced. As we move across this region from large to small m0, taking f ∼ 3v, we find the following twin hadron phenomena: • For m0 & 40 GeV, h → G0+G0+ dominates and G0+ decays are prompt. • For 10 GeV . m0 . 40 GeV, the G0+ decays are displaced; decays to other glueballs, and to higher multiplicities of glueballs, become more common for smaller m0. The decay h → G0+G00+ is of particular note, since the G00+ decays visibly. • Below about 10 GeV the G0+ lifetime is so large that decays in the detector are rare. This is partly compensated by higher glueball multiplicity per event. Consider next small mˆb . 15 GeV. In region B, where m0 is small, the irreducible process produces the same phenomena as in region A, but h → ˆbˆb enhances the rate for per event. In the low-mˆb portion of region C, the glueballs instead decay to bottomonium, leading to states that may all be invisible; alternatively (see section 5.4) final states may With all of these different subregions with different phenomenological details, many of which cannot be calculated, one may rightly worry that experimental coverage of this model, and others like it, will be extremely difficult. However, it is possible to bring the challenges under some control by focusing on simple search strategies that cover multiple regions. For the displaced decays, just a few strategies are potentially sufficient. 1. Search(es) for single vertex production, h → G0+ + . . . , perhaps separated into (a) h → G0+ + E/ T where the E/ T is due to twin hadrons that decay invisibly and/or outside the detector. (b) h → G0+ + jet(s) where a promptly decaying twin hadron produces the jet(s). This type of search may only be feasible when requiring the presence of associated objects that may accompany the h, such as a lepton or a pair of vector boson fusion 2. Search for exclusive di-vertex production production: as in h → G0+G0+. 3. Search for inclusive di-vertex production: h →≥ 3 twin hadrons, of which at least two decay visibly and displaced. In contrast to exclusive di-vertex production, here the pair of observed twin hadrons generally have invariant mass below mh, and need not be back-to-back in the h rest To a limited degree, each of the three search strategies has been explored by ATLAS [27, 28, 50, 52–55], CMS [29, 56–58] and/or LHCb [51]. However, due often to trigger limitations or analysis gaps, even the most sensitive of these searches do not yet put significant bounds on this model, and a broader and deeper program of searches is needed in Run II. We will discuss these issues further in appendix C. We note also that although our minimal model has specific relationships between masses, lifetimes and production mechanisms, these relationships will not necessarily hold in other Twin Higgs and Twin Higgs-like models. It is therefore preferable that the above searches be carried out with the masses and lifetimes of the long-lived particles, and characteristics of the E/ T (if any), treated as free parameters. In appendix C we suggest benchmark models for these searches and consider some important triggering and analysis issues. New Higgs decays without displaced vertices When the G0+ is heavy, its decays are prompt. Prompt non-SM decays of the Higgs such as h → G0+G0+ → (b¯b)(b¯b), (b¯b)(μ+μ−), or h → G0+ + E/ T → (b¯b) + E/ T , (τ +τ −) + E/ T , Not all cases have yet been investigated for a 13–14 TeV machine, but it appears that a or non-perturbative effects, as discussed in section 5.5, could bring these processes within reach of the LHC. At the other extreme, one can discuss completely invisible decays. These can dominate Again, an e+e− collider would do much better. in region C, and can become important in regions A and B if the G0+ has an extremely twists on the model (such as the presence of a massless twin hypercharge boson with small kinetic mixing, see appendix E) can cause all twin hadrons to decay to invisible hidden particles. Detecting an invisible decay rate much smaller than 10% is very difficult at the LHC, and thus can only be done if there is significant enhancement by the h → ˆbˆb process. Heavy Higgs decays Not only the h but also the heavy twin Higgs hˆ may serve as a portal, if the twin SU(2) is linearly realized. The existence of a second perturbative Higgs, while not guaranteed, is favored by precision electroweak data [60]. If present it provides additional opportunities to uncover the twin mechanism. The mass of this second Higgs is ∼ lies around the TeV scale. It possesses a v/f -suppressed coupling to top quarks and thus to an SM Higgs of equivalent mass — falling at √ 300 GeV to 10 fb for a mass of 1 TeV. is produced through gluon fusion, albeit with a (v/f )2-suppressed cross-section compared s = 14 TeV from 1000 fb at a mass of On the lower end of this mass range, decays into W W , ZZ, and hh dominate, with branching ratios roughly proportional to 2 : 1 : 1. Once hˆ → Wˆ Wˆ , ZˆZˆ is kinematically allowed, these processes become at most comparable to W W , ZZ, and hh; although couplings to the SM bosons are suppressed by v/f due to mixing, the longitudinal coupling scales as m3hˆ/v2 and entirely compensates. This is inevitable, since in the Twin Higgs mechanism the h and longitudinal W , Z, Wˆ , Zˆ are all Goldstone modes under the same The decays hˆ → hh, ZZ are common in many BSM models, and searches for these promising signals are already underway [61–64]. The hˆ will appear as a resonance with width suppressed by (v/f )2 compared to a SM Higgs of the same mass, times 43 73 to account for the channel hˆ → hh (and channels hˆ → Wˆ Wˆ , ZˆZˆ, if kinematically allowed). Observation of this resonance at equal rates in hˆ → ZZ and hˆ → hh, and measurement of its width, could therefore allow for a test of the model. In Run II, CMS expects to 3000 fb−1 at √ This corresponds to an exclusion reach of 900 GeV or a discovery reach of 750 GeV for a hˆ s = 14 TeV, or to discover σ × Br(pp → hˆ → ZZ) & 30 (10) fb at 5σ [65]. with f = 3v. ¯ The hˆ may also give rise to twin hadrons, either via hˆ → tˆtˆ (followed by tˆ → ˆbτ ν) or hˆ → ZˆZˆ (with Zˆ → ˆbˆb). Although the rate is less than the perturbative irreducible rate for h → twin hadrons, it can easily happen that the trigger efficiency for h decays is low while that for hˆ decays is high, so that hˆ decays may provide an easier signal. Moreover the rate to produce twin hadrons via Zˆ → ˆbˆb is larger by a factor of 10 or more than h → ZZ → `+`−`+`− and hˆ → hh → b¯bγγ, the cleanest hˆ → SM processes. Decays of ˆ hˆ may often produce displaced glueballs, including ones too heavy to appear in h decays and decaying via an off-shell h [21, 23]. Even if all twin hadron decays are prompt, the events might be observable at LHC if the rate, multiplicity, and total (or missing) energy are sufficient [20, 66]. On the other hand, if the twin SU(2) is non-linearly realized and the hˆ is as indistinct enhancements of gg → ZˆZˆ and VBF production of Zˆ pairs. Precision Higgs measurements Here we consider the role of the canonical signature of the Twin Higgs [1], namely O(v2/f 2) changes in Higgs couplings due to the misalignment between the electroweak vacuum expectation value and the pseudo-Goldstone Higgs. This leads to a suppression of all Higgs couplings by an amount 1 − 2vf22 relative to Standard Model predictions. There may also be a shift in branching ratios due to the additional partial width of decays into the twin sector, but as we have seen this can be much less than 10%. Assuming that Br(h → twin sector) 10%, then the sole effect of the twin sector on SM Higgs measurements is a reduction in all production rates, relative to SM predictions, by a factor of ` yb‘ ` yb‘ combined fit to Higgs coupling measurements. Solid, dashed, and dotted black lines denote the 1-, measurements. The grey lines correspond to the perturbative calculation of the Higgs branching branching ratio may differ significantly from the perturbative result for a given value of yˆb. performed a combined fit of the most recent ATLAS and CMS Higgs measurements [67– 75] using the profile likelihood method [76]. The resulting bounds on v/f are shown in figure 8 as a function of v/f and the Higgs branching ratio into the twin sector.9 also show contours corresponding to the perturbative calculation of Br(h → twin sector) of bottomonium production suggest that the actual branching ratio is potentially much smaller than the perturbative value for sufficiently large yˆb/yb, while the irreducible rate For f ∼ 3v the shift may be detected definitively before the end of the LHC, but not soon — certainly not within Run II. Future projections are a somewhat delicate matter, as measurements in certain channels will become systematics-limited and naive combinations neglecting correlated systematics are no longer appropriate. However, the collaborations quote appropriate coupling projections taking these effects into account. For example, 9This fit does not include implicit precision electroweak bounds from infrared contributions to S and T . However, as we will discuss more in appendix D, in contrast to composite Higgs models where the infrared here the infrared contribution is cut off by the mass of the heavy Higgs. For mhˆ . TeV these corrections to S and T are comfortably compatible with current precision electroweak bounds and do not strongly influence the coupling fit. 0.35 (0.31) assuming the invisible branching ratio gives a sub-leading correction to the observed Higgs couplings. The remaining high-luminosity run should improve sensitivity bounds of v/f . 0.31 (0.25). It is therefore unlikely that Higgs coupling measurements at the LHC could be used to substantially constrain the parameter space of Twin Higgs of the high-luminosity run. (Direct limits on invisible decays are likely to fare even worse, well short of the typical rate for decays into the hidden sector.) In any case, an overall coupling reduction is purely a sign of mixing and is nondiagnostic; while observation of this reduction would be revolutionary, no unique interpretation could be assigned to it. Only discovery of the twin hadrons and/or discovery and study of the mostly-twin heavy Higgs would allow for a clear interpretation of such a The “Twin Higgs” mechanism provides an existence proof for the unsettling possibility that the solution to the hierarchy problem involves a sector of particles that carry no Standard Model quantum numbers, and are therefore difficult to produce at the LHC. The existence of dark matter already motivates us to consider hidden sectors, and it is important that the possibility of hidden naturalness be thoroughly considered. Fortunately, hidden sectors are often not as hidden as they first appear. As we have seen, a Fraternal Twin Higgs model, one whose hidden sector is not a precise twin of the Standard Model, but which contains the minimal ingredients for the theory to address the hierarchy, naturally leads to Hidden Valley phenomenology. Specifically, the Higgs sector requires a twin top, which in turn favors twin color; and the lack of light twin quarks in this minimal model then leads to confinement and twin hadrons that include twin glueballs, the lightest of which necessarily mixes with the Standard Model-like Higgs, and twin quarkonium, whose tower of 0++ states has analogous mixing with the Higgs. We have shown that the resulting mixing can often cause these glueballs, and in some cases the lightest quarkonium states, to decay on an observable timescale, leading to a new source of visible non-Standard-Model Higgs decays. The branching fraction for these measurements. For heavier twin hadrons with prompt decays, the visible final states — four heavy-flavor fermions, or two heavy-flavor fermions plus missing energy — are among those summarized in a recent overview of non-SM Higgs decays [59]. Searches for these final states and for invisible decays at LHC are possible if the branching fractions approach 10%; if much smaller, a lepton collider may be needed. But for moderately light glueballs (and perhaps quarkonia), these decays can produce a potentially spectacular signal of one or more highly displaced vertices, often accompanied by moderate amounts of missing energy from other twin hadrons that escape the detector. While several experimental searches for similar vertices have already taken place, a wider array of more powerful search strategies is required if the parameter space for this model is to be fully covered. The minimal Twin Higgs model we have presented is only mildly tuned, and no more unnatural than bottom-up modeling of Higgs compositeness or (natural) supersymmetry. This model also has no obvious cosmological problems, flavor problems, or other glaring issues. Thus we see no reason from the bottom up, given present knowledge, to treat Twin Higgs as less motivated than, say, composite Higgs models. Meanwhile, in addition to its cousin Folded Supersymmetry, the Twin Higgs model has recently been generalized, by recognizing it as an orbifold model [38, 39]. Variants of these generalized models, by their very construction, share features with our minimal model, though they are different in details. This shows that the model space of sibling Higgses has not yet been fully explored, and should provide additional motivation for considering seriously and more generally the possibility of a hidden-sector solution to naturalness. On general grounds, we expect anything vaguely resembling a Twin Higgs model with a hidden sector to require, as part of its solution to the naturalness problem, Higgs mixing via a “portal”-type interaction. This feature easily leads to additional Higgs-like resonances, new sources of missing energy, and exotic phenomenology of hidden-valley type, including non-SM Higgs decays to multi-body final states and/or displaced vertices. The challenges are that no individual model is required to produce any or all of these signals, and that production rates for these phenomena are not determined by known interactions (in contrast to gluino or stop production) and can be small. Among the most motivated places to search for new signals are in decays of known particles, whose production rates are known and large. New decays of the Higgs may not even be rare. In non-minimal models there may also be opportunities in rare decays of the Z. Searches for new phenomena generated by rarely produced heavy particles (such as the heavy Higgs in our model) must Our field has tended to assume that the solution to the hierarchy problem lies in particles that resemble the ones that we know. While hidden sectors are often found in string theory vacua and required in models of supersymmetry breaking, their role has been limited to higher energy or purely gravitational interactions, leaving them, as far as the LHC is concerned, out of sight and out of mind. But with the possible scale of supersymmetry receding upwards, and with no sign of Higgs compositeness or of the colored top partners that were widely expected, the possibility of something more radical, such as a hidden sector around the weak scale that communicates with our sector through a portal, cannot be ignored. Our searches must move beyond the easier and more obvious lampposts, for the secrets of nature may lie hidden in the dark. We thank Christopher Brust, Zackaria Chacko, Patrick Draper, Roni Harnik, Simon Knapen, Pietro Longhi, Colin Morningstar, and Agostino Patella for useful conversations. The work of M.J.S. was also supported by NSF grant PHY-1358729 and DOE grant DESC003916. R.S. was supported by NSF grant PHY-1315155 and by the Maryland Center for Fundamental Physics. Quarkonium mass spectrum and decays Spectrum of the quarkonium states As explained in section 4.2, the most phenomenologically interesting quarkonium states decay back to the SM. We can make a crude but useful analysis of these states assuming they are governed by non-relativistic quantum mechanics; this approximation holds for be modeled as a combination of a (logarithmically corrected) Coulomb potential and a long-distance linear potential. G0+G0+ is kinematically forbidden, since otherwise the twin hadronic decay will dominate. This requires mχˆ < 2m0 ≈ 13.6Λˆ 3. One may then check that (remembering a p-wave state is larger than an s-wave state) the Coulomb potential would imply an the Coulomb approximation is very poor. This motivates using a purely linear potential, an approximation which improves for heavier states. Taking the potential as linear in r with a string tension σ ≈ 4Λˆ 23, the Schr¨odinger equation for the p-wave radial function is: − r2 dr with reduced mass μ = m2ˆb . No exact solutions of this equation are known, but an excellent repulsive potential at large r, and then matching the two asymptotic oscillating solutions. Making this approximation we find the energy states En ≈ 2 and associated wave functions. This approximate solution is quite close to the exact numerical solution and may be used as a basis for more accurate estimates, though we have not done so here. In any case we believe that this formula correctly captures the parametric behavior of the energy levels and masses. In the Standard Model, highly excited bottomonium states above 10.56 GeV promptly decay to a pair of B mesons; a similar story applies for charmonium. But in this model there are no light twin quarks, and correspondingly no twin B mesons, so the twin bottomonium states remain narrow until their masses are above 2m0 or 2mˆb + m0. In most parts of parameter space these are very heavy scales, so that the towers of narrow states extend to much higher n than in the SM. For the physically interesting values of m0 and mˆb, our approximation does not survive to high n, because relativistic effects become large at rather small n. Still, at small n and some portion of regions D and C. Full study of the spectrum with Coulombic and relativistic corrections is beyond our scope, but we expect that many of the results that we describe here will be parametrically valid (with order-one corrections) in the relativistic state will dominantly decay to the SM, with a lifetime obtained from eq. (5.8): displaced decays occur in low-mˆb portions of both regions C and D. as (mχˆ − mΥˆ )7. While we could attempt to compute this splitting in the non-relativistic approximation, which would not be unreasonable for the higher n states, we know from SM However, we may use a trick to get a good estimate in a phenomenologically important part of parameter space, namely the part of region D (where both bottomonium and glueballs may be produced, often in the same event) that lies close to region B. This “BDboundary” lies at mG0+ ≈ mˆb ≈ 21 mχˆ, a mass ratio very similar to the SM charmonium system, for which lattice studies give a SM glueball mass of 1.5–1.7 GeV [45, 78, 79], and Using this scaling argument and the formulas of section 5.4, we see from figure 9 that remind the reader that in this part of region D one expects the Higgs branching rate to twin bottomonium, from the h → ˆbˆb process, to be well enhanced above the irreducible rate. Consequently even a rather small branching fraction can be interesting. However, we must emphasize that our estimates are highly uncertain due to our crude methods, which include the fact that we have not computed the numerical coefficient in (5.9). One should therefore only conclude that there is good reason to believe that the As we increase m0, moving away from the BD-boundary across region D and into C, we cannot reliably compute the m0/mˆb dependence of the χˆ − Υˆ splitting and corresponding twin weak decay rates. However, both our approximations and reasoning from QCD data suggest the mass splitting rapidly increases with m0, at least as fast as Λˆ 3 ∝ m0 itself. We near unity, across the right side of region D and all of region C. Finally let us note that we are not aware of any other bottomonium states that have Twin hadron production in more detail Twin hadrons are produced when h → gˆgˆ or ˆbˆb, but twin hadronization is complicated and poorly understood. Despite many years of experimental study of QCD hadron production, theoretical understanding of hadronization is still limited. Moreover a twin sector with no light twin quarks has a very different spectrum and dynamics from SM QCD. Perturbation theory applies for inclusive calculations, such as Br(h → twin hadrons), small. (A similar example is the hadronic branching fraction of the Z.) However, for large just above the QCD resonance region, where non-perturbative effects can be important. Decays of the h can be enhanced relative to perturbative estimates if mh lies close to a narrow excited 0++ glueball or quarkonium resonance, and suppressed if it lies between Nonperturbative effects in twin gluon production First consider h → gˆgˆ for mˆb > mh/2. For small m0, the glueball resonances near 125 GeV are likely to be both numerous and wide, blending into a continuum. In this case the irreducible rate is given by the perturbative result (5.10) for h → gˆgˆ. For larger m0, resonance effects are potentially important. Based on large-Nc counting excited 0++ glueball resonances have Γ/m ∼ 1/Nc2 = 1/9, perhaps with an additional minor suppression factor. There is also likely a phase-space effect increasing the width for higher excitations, but we ignore this for our conservative estimate. Meanwhile the spacing among others) but it is surely no larger; lattice evidence for the second 0++ glueball [47] expect the worst possible non-perturbative suppression factor is about 0.1, bringing the worst-case rate for glueball production down no further than 10−4. Nonperturbative effects in twin bottom production If the Higgs can decay to ˆb quarks and thus to [ˆbˆb] states, the rate for twin hadron production is often enhanced. As before, the [ˆbˆb] production rate is given by the perturbative rate for h → ˆbˆb for sufficiently small mˆb and m0. As discussed in section 5.5, the perturbative branching fraction for h → [ˆbˆ¯b] + X grows as yˆb2, contradicting existing Higgs measurements for yˆb & 1.25yb. But at high yˆb, the perturbative rate often gives the wrong answer. The widths of the m0. If mh lies between two resonances, then there can be a strong non-perturbative suppression compared to the perturbative rate. We may make an estimate of the maximal suppression factor as we did for glueballs for somewhat below mh/2, specifically those where mh lies between two excited bottomonium states, are probably not excluded by data. ulated by excited glueballs for the same reason as h → gˆgˆ is modulated by excited glueball resonances. We therefore expect that the non-perturbatively allowed portion of region A extends to lower mˆb than that of region B. However, we have no reliable computational methods in this regime. Here we briefly discuss some triggering and analysis issues with regard to the search for the displaced decays of long-lived twin hadrons. For brevity, we refer to the visible decay products of a long-lived particle as a “displaced vertex” (DV) even when it occurs in regions where tracks are not actually reconstructed. We also refer collectively to the decay products of a W or Z in W h and Zh production, and to the vector boson fusion (VBF) jets in VBF Higgs production, as “associated objects” (AO). Comments on triggering Triggering on Higgs decays to long-lived neutral particles requires three classes of trigger strategies, which respectively focus on • the presence of one or two DVs; this method is largely independent of how h is produced and is sensitive to non-Higgs production of the DVs. Displaced decay objects can include jets with displaced tracks, trackless jets (possibly including a muon), narrow trackless jets with little electromagnetic calorimeter (ECAL) deposition, and unusual clusters of hits or tracks in the muon system. • the presence of AOs that accompany the Higgs, including VBF jets or daughters of a W or Z (leptons, neutrinos, jets); this method is relatively independent of the details of the h decay and can therefore be used for any exotic h decay mode. • the presence of both; for instance, in an event with VBF jets along with a trackless jet. Requiring both may be used to lower pT thresholds on the trigger objects, or to access DVs that would be unusable on their own. However, this powerful method is specifically optimized for Higgs decays to long-lived particles. Triggers of the first type were used at ATLAS [26–28, 50, 53–55, 81] and CMS [29, 58, 82] in Run I; lepton and E/ T triggers are standard, while a VBF trigger was used in 2012 parked data at CMS [83]; and no triggers of the third type were used in Run I, to our knowledge. Depending on the lifetimes of the long-lived particles and on their masses and other kinematics, any one of the three approaches to triggering can work for any of the three search strategies outlined in section 6.1. We leave the appropriate studies to our experiNext we point out possible benchmarks that may be used as straw-man targets. Although the parameters of the benchmarks are correlated in the Fraternal Twin Higgs — for instance the mass and lifetime of the G0+ are highly correlated, eq. (5.5) — it is important to account for the many uncertainties in twin hadronization, and even more important, to retain model-independence. Therefore, notwithstanding the particular features of the Fraternal Twin Higgs, it seems best to ignore these correlations and study the benchmark models across their entire parameter space. In section 6.1 we suggested three possible search strategies, and we now discuss suitable benchmarks appropriate for each of them. To keep things simple, we discuss them in the context of phenomena that occur in region A. However, with a little thought the reader may verify that the same benchmarks would be useful for these and other processes that occur in other regions of parameter space. Single displaced vertex search To look for a single hadronic DV from h → G0+ + . . . (and similar decays) is challenging, because of large, difficult-to-measure backgrounds. A CMS search [29] that required only one DV was not able to set limits on decays of a 125 GeV particle. Relevant ATLAS searches used single DV events to obtain background estimates on double DV events. 10 m (as for m0 . 10 GeV in region A), or if hadronization assures that particles making DVs are rare among twin hadrons, the number of DVs rarely exceeds one per event. To date no search for a DV has exploited the AOs, the VBF jets and/or lepton(s), that sometimes accompany the Higgs. To obtain background estimates, it may be enough to measure the rates for events with neither an AO nor a DV, and with either an AO or a DV; then if the AO and DV are uncorrelated one may predict the rate to have both an Also, searches for a single DV can demand a second object from the twin hadrons that do not produce a DV. This object could be E/ T (relevant for very long-lived G0+) or prompt jets, possibly b-tagged (relevant for h → G0+G0+0, where G0+0 → b¯b promptly). In this context we suggest the following benchmark models (in which all Xi of mass mi decay to SM final states with the same branching fractions as a Higgs of mass mi, unless otherwise noted). These benchmarks do not cover all the kinematic possibilities but will serve as a useful initial target. It is important to consider a range of masses for m1 and m2. For the first benchmark, m1 6= m2 helps account for the possibility of h → many glueballs, of which only one large invariant mass. The signal of the second benchmark may arise from decays such as undertaken at ATLAS [27, 28, 50] and at LHCb [51]. Also interesting is h → G0+G0+0; here one has particles of two different lifetimes and masses. In this context we suggest benchmark models of the form In our minimal model it is sufficient to consider m1 < m2 < 2m1 since otherwise X2 → X1X1 decays occur. This will not be true in all models however. C.2.3 Inclusive double displaced vertex search > 2 glueballs per event is larger, making it more common to have two G0+ DVs along with one or more invisible glueballs. One key difference from the exclusive double DV search is that the two DVs may not be well-separated in the lab frame, and may even tend to be found in the same angular region of the detector. Also, the invariant mass of the two (or more) DVs may be well below mh. A suitable benchmark highlighting this difference would be distribution in angular separation and in the momentum that they carry in the h rest The one serious caveat is the possibility (perhaps remote in the our minimal model, but not necessarily in other models) that the DVs are often clustered [15, 16, 20, 52, 84]. A special benchmark model may be needed if this clustering is sufficiently common that isolation requirements on DV candidates typically fail. We suggest • h → X0X2, followed promptly by X0 → X1X1, X1 decays as usual with lifetime relativistic and the two X1 decays are correlated in angle. mh, one assures that X0 is Here we briefly consider precision electroweak constraints on the Twin Higgs mechanism. In any theory where coupling deviations of the SM-like Higgs are due to mixing with heavier states there are infrared contributions to the S and T parameters whose coefficients depend on the reduced coupling of the SM-like Higgs to electroweak gauge bosons. The logarithmic contributions to S and T scale like log(mh/mZ ) + (1 − a2) log(Λ/mh), where from both the SM-like Higgs and the heavy twin Higgs hˆ. The small (large) blue “×” shows the mass mh = 125 GeV. Higgs models the coupling deviations come from mixing with heavy resonances, which must lie in the multi-TeV range due to tree-level contributions to S and T . The cutoff on infrared contributions is then of order Λ = 4πv/√1 − a2 ∼ 4πf , leading to extremely strong constraints on a [60]. In the Twin Higgs mechanism, the O(v/f ) deviations in Higgs couplings come from Higgs models, but in general the heavy twin Higgs can be much lighter thanks to a custodial symmetry; there are no tree-level contributions to S and T associated with hˆ. The complete contributions to S and T from the SM-like Higgs and heavy Higgs take the form relative to the precision electroweak contributions of a theory with a SM Higgs of mass mh. (5 TeV). In contrast to composite Higgs models, precision electroweak constraints do not substantially constrain Higgs coupling deviations in the Twin Higgs as long as the heavy Higgs is in the TeV range. In our minimal model, for reasons described in section 2, we have assumed twin hypercharge is absent. What if it is present? Then the most natural values of the mass for the twin ultraviolet that we do not specify. It is thus a free parameter of the model. the visible sector, and the τˆ and ˆb acquire electric charges ε ≡ κeˆ/e and ε/3, respectively, in units of e. The twin photon is stable and invisible. All twin hadrons may decay via radiation or annihilation to on-shell or off-shell twin photons — twin bottomonium through tree-level graphs and twin glueballs through a loop of twin bottom quarks. The rates for these decays far exceed rates for decays through the off-shell Higgs, and so most decays to the hidden sector are invisible. seen at LEP. The detailed limits are far too complex for us to work out here. − 10−1 range, h → γ + E/ T and rare h → γγ + E/ T decays may occur, especially if the twin hadron production rate is large (regions B, C, and D with mˆb not too small). Moreover, annihilation of spin-one states via an off-shell SM photon may produce a visible final state, e.g. Υˆ → γ∗ → `+`−. This prompt resonant decay will be subleading by ∼ ε2 compared to the invisible decay Υˆ → γˆ∗ → τˆτˆ as long and may even dominate. This type of decay is probably common in regions C and D. In region B, where bottomonium decays to glueballs, this decay can perhaps still occur for the best bet but are very rare. In region B, the larger production rate offers the possibility invisible and partly visible decays of the Higgs are highly motivated; see section 6.2 above. lem if Higgsed, since one more Higgs field S is required and its expectation value and mass In either case, we view this possibility as lying beyond the minimal Twin Higgs story. contrast to the massless case, here we define the massless SM photon as decoupled from twin particles; the twin photon is identified as the mass eigenstate, which couples to SM can decay to it, twin hadron production generically (and not just for 0++ hadrons) leads photon, the twin photons then decay to lepton pairs and jet pairs (or simply hadron pairs, such as K+K− and π+π−, if mγˆ ∼ few GeV or below). The decays of the twin photon constraints; for a recent summary, see [88]. If prompt, however, limits on multilepton final 1 GeV) [89–92] and on prompt “lepton-jets” (for mγˆ . 1 GeV) [93–98] are strong constraints. 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Nathaniel Craig, Andrey Katz, Matt Strassler. Naturalness in the dark at the LHC, Journal of High Energy Physics, 2015, 105, DOI: 10.1007/JHEP07(2015)105