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Goldstone Gauginos week ending PRL 115, 161801 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 Goldstone Gauginos † ‡ Daniele S. M. Alves,1,2,* Jamison Galloway,1, Matthew McCullough,3, and Neal Weiner1,§ 1Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 3Theory Division, CERN, 1211 Geneva 23, Switzerland (Received 10 April 2015; published 14 October 2015) Models of supersymmetry with Dirac gauginos provide an attractive scenario for physics beyond the standard model. The “supersoft” radiative corrections and suppressed supersymmetry production at colliders provide for more natural theories and an understanding of why no new states have been seen. Unfortunately, these models are handicapped by a tachyon which is naturally present in existing models of Dirac gauginos. We argue that this tachyon is absent, with the phenomenological successes of the model preserved, if the right-handed gaugino is a (pseudo-)Goldstone field of a spontaneously broken anomalous flavor symmetry. DOI: 10.1103/PhysRevLett.115.161801 PACS numbers: 12.60.Jv The results of LHC7 and LHC8 have directly challenged corrections that are “supersoft,” in that they are cut off the idea that the weak scale is natural. The quadratic without any divergences. The squark mass is then naturally divergences of the Higgs mass associated with the top a factor of ∼4–5 below the gluino mass, where the Yukawa coupling, the Higgs’ gauge interactions, and its contributions to the Higgs soft masses are cut off, and self-interactions all should be cut off not too far above the the logarithmic enhancement is minimal [7]. Higgs mass. The top Yukawa contribution, in particular, If a new standard model (SM) adjoint superfield Ai is should be cut off below the TeV scale for a “natural” theory introduced, a Dirac gaugino state with mass mD can be ð1Þ with at most ∼10% tuning. The possibility that colored generated in the presence of a hidden U B [we label this ð1Þ partners may not be found at the LHC has pushed many to U B as it can be identified as a baryon number in explore alternatives to a natural weak scale [1–6]. supersymmetry (SUSY) QCD models where Goldstone As a general idea, there is nothing wrong with the gauginos can be simply realized] D-term expectation value, B fact that no colored partners have been found. A TeV hWα i¼θαD, by the operator mass top partner need not have been found yet, since a 2 3 2 2 g α correction δm ≈ 2 TeV ∼ ð200 GeVÞ would not sig- ¼ i i ð Þ h 8π WCS WBWαA ; 1 nal an unnatural theory per se. The problem, however, M is that this understates the divergence in many cases. i with Wα the field strength of a SM gauge group with gauge In supersymmetry, in particular, this quadratic divergence 2 coupling g. This operator then marries the SM gaugino to is softened only to a logarithmic divergence, with δm ≈ i Hu the fermionic component of A [7,8]. Such an operator ð3 4π2Þ 2 ðΛ Þ = mstop log =mstop . If this divergence is cut off at preserves an R symmetry which forbids a Majorana 16π2 mstop, the theory is generally tuned at the few per- gaugino mass and generates a mass for the real scalar in cent level. Ai while leaving its imaginary component massless. In typical supersymmetric theories, this tuning is aggra- With purely Dirac gauginos, there are no t-channel vated by the gluinos. Gluinos, while not directly coupling gluino-induced processes such as qq → q~ q~ at the LHC, to the Higgs boson, indirectly contribute to δm2 through making such theories “supersafe” from experimental con- Hu their radiative contributions to the soft masses of squarks. straints [9,10]. Simultaneously, q~ g~ searches are often stronger than q~ q~ There are problems with this setup, however. Unification searches alone, and q~ q~ production is enhanced by a light is constrained to be within SUð3Þ3, as a full adjoint of gluino. SUð5Þ contributes as five flavors [7], inducing a Landau Many of these issues can be ameliorated if gauginos are pole for QCD well below the grand unified theory scale Dirac rather than Majorana. Dirac gauginos yield radiative [11,12]. This is merely aesthetic, however. More pressing is the so-called “lemon-twist” problem. In addition to the “classic” supersoft operator of Eq. (1), hereafter CS, a second supersoft operator is allowed: Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- 1 α i i bution of this work must maintain attribution to the author(s) and W ¼ W W αA A ; ð2Þ the published article’s title, journal citation, and DOI. LTS M2 B B 0031-9007=15=115(16)=161801(5) 161801-1 Published by the American Physical Society week ending PRL 115, 161801 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 “ ” ¼ ΠiTi=f Πij ¼ i þ πi ð Þ which has been termed the lemon-twist supersoft oper- A fe ; θ;θ¯¼0 s i ; 5 ator, hereafter LTS (see, e.g., [13]). This operator generates a Bμ-type term for the scalar adjoints, destabilizing one and setting μ ¼ 0 in (3), so that we have direction of the adjoint and thereby breaking SM sym- 0 ¯ Π metries. If both operators of Eqs. (1) and (2) are generated W ¼ λfTe =fT; ð6Þ at one loop, we encounter a standard μ-Bμ problem where the tachyon is not only present but cannot be removed by where Π ≡ ΠiTi and Ti are SUðNÞ generators. The the contribution to the scalar mass from the classic super- messengers are now massive only as a result of spontaneous soft operator, which is Oð4π=gÞ smaller than the contri- symmetry breaking, whose dynamics will be left unspeci- bution from LTS [7,8,14–17]. One can cancel this operator fied for the moment. with additional soft scalar masses, which may be generated There are a number of ways to see what is achieved with with D-term spurions [18], but typically this comes at the this mechanism. To begin, let us simply study this cost of nullifying the theory’s supersoftness. Alternatively, perturbatively. one can include an explicit Majorana mass term, ΔW ¼ μA2 The model in (6) resembles the one in (3) up to higher order terms in Π, i.e., with μ ∼ ð4π=gÞmD, but this would render the low-energy gluino a Majorana gaugino, with a mass m2 =μ ∼ αμ=4π. λ D W0 ¼ λfTT¯ þ λT¯ ΠT þ T¯ Π2T þÁÁÁ: ð7Þ This reintroduces a large logarithmic contribution to the 2f Higgs soft masses and removes supersafeness (without even beginning to worry about the origin of the scale for μ). The third term becomes important when the messengers are This operator seems impossible to forbid [7], as the integrated out, such that its contribution to the generation of α α 0 presence of W Wα along with W WαA would seem to LTS cancels that of the previous terms. To see this more imply that no symmetry can preclude ðW0αAÞ2. explicitly, consider the (bosonic part of the) potential for In this Letter, we show that if the CS operator originates the pseudoscalar adjoint aI originating from (3): from the spontaneous breaking of an anomalous global scalar ¼ 2λ2ðj j2 þj¯j2Þ 2 ð Þ symmetry, the right-handed gaugino is the fermionic VaI t t aI : 8 component of a Goldstone superfield, and LTS is not ¯ generated. We will argue that CS is connected to a mixed When the scalars t and t are integrated out, a mass is anomaly, while LTS is not, providing a natural framework generated for aI. For the GoGa model described in (7),on within which Dirac gauginos are viable. the other hand, the presence of the additional superpotential ¯ 2 A Goldstone gaugino.—Let us begin by understanding term ∝ TΠ T guarantees that pseudoscalar quartic cou- the μ-Bμ problem for Dirac gauginos in the context of a plings of the form ∝ ðjtj2 þjt¯j2Þπ2 vanish. In fact, there are simple messenger model. We take a superpotential no tree-level scalar potential terms for the pseudoscalar π, scalar ¼ μ ¯ þ λ ¯ ð Þ Vπ ¼ 0; ð9Þ WUV TT TAT 3 and hence no mass is generated for π when t and t¯ are and assume an N-plet of messengers (T;T¯ ), which trans- integrated out. forms nontrivially under the SM, and whose scalars are μ2 Æ We can also see this easily without any expansion: split by a D term such that their masses are D. A simple inspection of the superpotential in (6) reveals Integrating the messengers out, both CS and LTS are that j∂W=∂Tj2; j∂W=∂T¯ j2; j∂W=∂Πj2 are all manifestly generated: independent of π. The potential does depend on the real pffiffiffi scalar s, and so the absence of LTS contributions to m2 is 2λ g λ2 1 s ¼ α i i þ α i i ð Þ less trivial to see. It depends on a cancellation of diagrams WIR 2 WBWαA 2 2 WBWBαA A : 4 16π μ 32π μ generated from sðjtj2 þjt¯j2Þ and s2ðjtj2 þjt¯j2Þ interactions. Nonetheless, as LTS generates masses for both scalar and The LTS operator above contributes to the masses of the δ 2 ¼ −δ 2 2 2 pseudoscalar components satisfying ms mπ and scalar and pseudoscalar adjoints as δm ¼ −ð4π=αÞm ; 2 2 aR D since we can see that δmπ ¼ 0, δm contributions from δ 2 ¼ð4π αÞ 2 s maI = mD, respectively, while the contribution LTS necessarily vanish in the GoGa model of (6).
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