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 þ: 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 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). (The full, δ 2 ¼ 4 2 δ 2 ¼ 0 from the CS operator is maR mD, maI . In par- diagrammatic calculation can be found in the Appendix ticular, the scalar adjoint becomes a tachyon due to the of Ref. [19].) dominant contribution from LTS. In contrast, the couplings of the fermions are Let us now introduce a novel realization of Dirac unchanged—i.e., there are no relevant higher order terms gauginos, which we will refer to as the “Goldstone in the expansion of the exponential of (6)—and thus the gaugino” (GoGa) mechanism. We begin by promoting A CS operator is still generated, with Ai being replaced with to a nonlinear sigma field, Πi in (1).

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Another perspective on the GoGa mechanism is that the This anomaly is accounted for in the low-energy LTS operator is formally a correction to the holomorphic Lagrangian by the presence of a Wess-Zumino-Witten ð1Þ gauge coupling of U B (even if for us it is not gauged, we (WZW) term [20,21], whose bosonic part, to leading order can in principle gauge it). It is generated only if the adjoints in the πi fields, is given by Π, when treated as background fields, correct the masses of 2 1 1 the messengers at OðΠ Þ [17], which is not the case if Π is a ~ μν ~ μν L ⊃ Tr½πBμνF þ Tr½πFμνF : ð12Þ Goldstone coupling to the messengers as in (6). 16π2f 48π2f A final, simple argument as to why such a scenario is safe from the LTS tachyon is as follows: There is an Interestingly, this supersymmetric chiral anomaly when approximate shift symmetry of Π which is explicitly broken expressed SUSY-covariantly in the superpotential becomes by the gauge coupling g. Therefore, any operators that 1 1 violate the shift symmetry can be generated only if ⊃ α i Πi þ ½ α Π ð Þ W 2 WBWα 2 Tr W Wα ; 13 accompanied by powers of g, such that in the limit g → 16π f 48π f 0 the shift symmetry is restored and such operators vanish. This is the case for CS, which is ∝ g. In contrast, the LTS and we can immediately recognize the first term as the CS operator (1). [In (11)–(13), the gauge coupling can be operator has no connection to the gauge coupling and hence → cannot be generated. recovered by rescaling Aμ gAμ.] Anomalies and classic supersoft.—While we can calcu- The LTS operator, on the other hand, is not related to late the size of the CS operator in a given perturbative gauge anomalies and does not stem from a WZW term. theory, we would also like to understand when the GoGa [This is familiar from ordinary QCD. The neutral pion ð1Þ mechanism occurs more generally. For instance, strongly couples to two U EM currents through a WZW term, coupled models of symmetry breaking are common, but it while a coupling of two neutral pions to these same currents is not immediately obvious whether CS is generated or not. is absent when explicit breaking terms (e.g., quark masses) We can understand the origin of the classic supersoft are taken to vanish.] It is formally related to the holomor- ð1Þ operator in its connection to the UV gauge anomalies of phic gauge coupling of U B, which depends only on ð Þ Π the theory. Setting μ ¼ 0 in the messenger superpotential log det A , and thus it is independent of the Goldstone . Dangerous linear terms involving the singlet S which (3), the approximate global symmetry is enlarged to α SUðFÞ × SUðFÞ × Uð1Þ , with A now promoted to a marries the bino [7,22], namely, SWBWBα, are not for- L R B bidden by a shift symmetry of the pseudoscalar component bifundamental. This global symmetry is only weakly θ ð1Þ ð Þ of S. Such a shift would simply generate a term for U B, broken by the gauging the vector subgroup SU F V, which we identify with (all or part of) the SM gauge group. As A which is a total derivative. Nonetheless, it is not hard to develops a vacuum expectation value hAi¼v × 1 , the make this term absent, for instance, by embedding hyper- F charge into a non-Abelian group, such that S is associated global symmetry is spontaneously broken to SUðFÞ × V with a traceless generator, or by imposing an S → −S Uð1Þ and the messengers T and T¯ can be integrated out. B parity [7]. The remaining low-energy degrees of freedom are given by — Πi A perturbative model. To realize the Goldstone gau- the Goldstone superfields : gino scenario, we must invoke dynamics that can sponta- neously break a large flavor symmetry. This can be ¼ð þ σÞ TiΠi=f Πij ¼ i þ πi ð Þ A v e ; θ;θ¯¼0 s i : 10 achieved by a simple symmetry-breaking sector coupling A to a singlet X: An essential feature of our mechanism is that the axial μi πi current JA associated with the Goldstones is anomalous X ð Þ ð1Þ W ¼ ðdet A − vFÞ: ð14Þ with respect to the gauging of SU F V × U B: ΛF−2 1 1 μi μν ~ i i μν ~ In the vacuum the symmetry-breaking structure is ∂μJ ¼ B Fμν þ Tr½T F Fμν; ð11Þ A 16π2 48π2 ð Þ ð Þ → ð Þ SU F L × SU F R SU F V as the F term for X requires a vacuum expectation value hAi¼v × 1 ; the traceless i μν ð Þ F where Fμν and B are the field strengths of SU F V and components Π therefore become Goldstones of the sponta- ð1Þ U B, respectively. Each term can be understood dia- neously broken symmetry. While this superpotential is grammatically from the following: nonrenormalizable, it captures the dynamics of certain SUSY QCD models which provide a UV completion, under which composite states of the microscopic quarks provide the right-handed gaugino [19]. The Goldstone bosons associated with the broken gen- erators can be parameterized as in Eq. (10). Thus, if A couples to messengers as in Eq. (3) with μ ¼ 0, then this

161801-3 week ending PRL 115, 161801 (2015) PHYSICAL REVIEW LETTERS 16 OCTOBER 2015 realizes the Goldstone gaugino scenario. By inspection one *[email protected] † [email protected] can see that the arguments applied to Eq. (6) apply here as ‡ [email protected] well, and no LTS operator is generated. § Discussion.—It is fair to say at this point that much of the [email protected] MSSM parameter space under consideration a decade ago [1] J. D. Wells, Implications of supersymmetry breaking with a little hierarchy between gauginos and scalars, arXiv:hep-ph/ has been excluded. This has come from a combination of 0306127. direct and indirect searches, as well as from the discovery [2] N. Arkani-Hamed and S. Dimopoulos, Supersymmetric of the Higgs boson and its mass measurement. In light of unification without low energy supersymmetry and signa- this, alternatives that provide qualitatively different expect- tures for fine-tuning at the LHC, J. 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McCullough, We thank Nathaniel Craig, Sergei Dubovsky, Claudia Models of Goldstone gauginos, arXiv:1502.05055. Frugiuele, Brian Henning, Anson Hook, Patrick Fox, Adam [20] J. Wess and B. Zumino, Consequences of anomalous Ward Martin, John Terning, and Jay Wacker for useful discus- identities, Phys. Lett. B 37, 95 (1971). sions. D. S. M. A. is supported by NSF-PHY-0969510 (the [21] E. Witten, Global aspects of current algebra, Nucl. Phys. B223, 422 (1983). LHC theory initiative). N. W. and D. S. M. A. are supported [22] L. M. Carpenter and J. Goodman, New calculations in Dirac by the NSF under Grants No. PHY-0947827 and No. PHY- gaugino models: Operators, expansions, and effects, J. High 1316753. J. G. is supported by the James Arthur Energy Phys. 07 (2015) 107. Postdoctoral Fellowship at NYU.

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