On the MSSM Higgsino mass and fine tuning
DDESY-16-062OUTP-16-06PCERN-TH-2016-072
On the MSSM Higgsino mass and fine tuning
Graham G. Ross ∗ Rudolf Peierls Centre for Theoretical Physics, University of Oxford,1 Keble Road, Oxford OX1 3NP, UK
Kai Schmidt-Hoberg † Deutsches Elektronen-Synchrotron DESY,Notkestraße 85, D-22607 Hamburg, Germany
Florian Staub ‡ Theoretical Physics Department, CERN, Geneva, Switzerland
It is often argued that low fine tuning in the MSSM necessarily requires a rather light Higgsino.In this note we show that this need not be the case when a more complete set of soft SUSYbreaking mass terms are included. In particular an Higgsino mass term, that correlates the µ − termcontribution with the soft SUSY-breaking Higgsino masses, significantly reduces the fine tuning evenfor Higgsinos in the TeV mass range where its relic abundance means it can make up all the darkmatter. I. INTRODUCTION
Our expectation of what to find beyond the StandardModel (SM) of particle physics has largely been shapedby naturalness arguments, and arguably low energy su-persymmetry emerged as the prime candidate for BSMphysics. Fine tuning considerations give us a handle tojudge (i) which classes of models are (more) natural butalso (ii) for a given model which parameter choices arepreferred. It has been realised long ago that the µ termplays a special role in fine tuning considerations. TheRGE evolution is very mild and the fine tuning with re-spect to µ can be estimated as∆ µ ∼ µ M Z (1)which implies that for a natural theory with fine tuning∆ µ <
100 the value of µ should not exceed a few hun-dred GeV. As the Higgsino mass in the usual MSSM isroughly given by µ , this has led to the belief that a nat-ural theory necessarily requires a rather light Higgsino,see e.g. [1–10]. It is of crucial importance to evaluatewhether this conclusion is true, as light Higgsinos startto be established as one of the main tell tale signs fornaturalness within the community and search strategiesare developed accordingly.A light Higgsino typically means that it is the lightestsupersymmetric particle (LSP). As the annihilation crosssection of Higgsinos is sizeable, dark matter is typicallyunderproduced in this case. While this is experimentally ∗ [email protected] † [email protected] ‡ fl[email protected] viable, because dark matter could consist of several com-ponents (there might e.g. be an additional componentof axion dark matter), it would be nice to saturate therelic abundance with Higgsinos only. The Higgsino relicabundance depends on the Higgsino mass m ˜ H and thecorrect relic abundance is achieved for m ˜ H ∼ µ term a SUSY breaking Higgsino mass term can bepresent. As discussed below although such a term canreadily be generated nevertheless it is almost always dis-carded. The reason is that it can be reabsorbed intoother parameters of the model and hence seems superflu-ous. While this is true with respect to the particle spec-trum, the inferred values for the fine tuning can differsignificantly. Accordingly, the conclusions with regardsto the particle spectrum based on fine tuning considera-tions change as well; in particular a TeV scale Higgsinomight well be natural.This letter is organises as follows. In Section II we in-troduce the new ingredients we consider in the context ofthe MSSM and discuss in detail the structure and possi-ble origin of the Higgsino mass term. In Section III wegive an approximate relation between the fine-tuning inthe model and the new soft-terms before we perform, inSection IV, a purely numerical study of the fine-tuning.We conclude in Section V. a r X i v : . [ h e p - ph ] M a r II. THE MSSM WITH THE FULL SET OF SOFTTERMS
Let us consider the MSSM extended by the followingnon-holomorphic soft-terms, “soft” in the sense that theydo not lead to quadratic divergences at radiative order[11], L NH = T (cid:48) u,ij H ∗ d ˜ u ∗ R,i ˜ q j + T (cid:48) d,ij H ∗ u ˜ d ∗ R,i ˜ q j + T (cid:48) e,ij H ∗ u ˜ e ∗ R,i ˜ l j + µ (cid:48) ˜ H d ˜ H u + h.c. (2)The potential origin of these terms can either be spon-taneous SUSY breaking within gravity mediation [12],strongly coupled SUSY gauge theories [13], or they areradiatively-generated in N = 2 and N = 4 SUSY gaugetheories [14–18]. In II A we give an explicit example forthe case of the Higgsino mass term.While µ (cid:48) enters the neutralino mass matrix m ˜ χ = M − g v d g v u M g v d − g v u − g v d g v d − µ (cid:48) − µ g v u − g v u − µ (cid:48) − µ (3)the mass matrices for all scalars as well as the two min-imisation conditions ∂V∂v i = 0 ( i = u, d ) are not changedcompared to the MSSM without the terms given ineq. (2). Thus, the dependence of M Z on the SUSY pa-rameters and tan β = v u v d = t β is as usual M Z µ + m H d + t β (cid:0) − (cid:0) µ + m H u (cid:1)(cid:1) t β − (cid:39) − µ ( Q ) − m H u ( Q ) (4)where in the last step we explicitly show the dependenceon the scale Q at which the parameters are determined.In the following, unless otherwise stated, we take thisto be the SUSY breaking scale. Some phenomenologicalconsequences of the additional soft-terms were analysedin Refs. [19, 20]. A. The soft Higgsino mass
Low fine-tuning requires that there should be no sig-nificant relation between uncorrelated coefficients of thesoft terms. However in specific SUSY breaking schemesthere may be correlations between the coefficients andsuch natural correlations can significantly affect the fine-tuning measure. For example if SUSY breaking leads todegenerate soft scalar masses there is a cancellation be-tween the tree level and radiative contributions to theHiggs mass that leads to a reduction of the sensitivityof the Higgs mass to the initial scalar masses, the so-called “focus point”. As a result the fine-tuning measureis significantly reduced.Thus, when computing fine-tuning, it is important totake care of all possible natural correlations between the coefficients of the soft terms. Here we argue that thesoft Higgsino mass provides one such correlation that hasnot been included in fine tuning estimates and that itcan lead to a significant reduction in fine-tuning. Thereason that it is not included is that it can be elim-inated by a change in the supersymmetric “ µ term”, µH u H d | θθ , together with a change in the Higgs softmasses , m H u | H u | , m H d | H d | : µ (cid:48) ˜ H u ˜ H d ≡ m ˜ H H u H d | θθ − m H ( | H u | | + H d | ) (5)However dropping the Higgsino mass term is inconsis-tent with the determination of the fine-tuning measurebecause, as may be seen from this equation, the Higgsinomass term implies a natural correlation between the co-efficients of the µ term and the Higgs soft masses.Of course it is important to ask whether, in an effec-tive field theory sense, an Higgsino mass can occur with acoefficient uncorrelated with the other soft SUSY break-ing terms of the MSSM. It is straightforward to establishthat this is the case. For example the authors of reference[21] have tabulated all the allowed dimension 5 operatorsin the MSSM that are consistent with R parity. In par-ticular they find the operator O = 1 M (cid:90) d θ [ A ( S, S † ) D α (cid:0) B ( S, S † ) H e − V (cid:1) × D α (cid:0) Γ( S, S † ) e V H (cid:1) + h.c. ] (6)where A, B and Γ are functions of the SUSY breakingspurion S = M s θ where M s is the SUSY breaking scale, V is a combination of the MSSM vector superfields, M isthe mediator mass coming from integrating out massivefields in the underlying theory. For example, this par-ticular operator can be generated by integrating out twomassive SU (2) multiplets that are coupled to the MSSMHiggs supermultiplets and in this case M is the mass ofthese massive doublets. Including the SUSY breaking ef-fects this operator generates a soft Higgsino mass termwith coefficient proportional to the coefficient of the SS † term in A . As this is the only SUSY breaking term pro-portional to this coefficient the soft Higgsino mass is notcorrelated with other SUSY breaking terms, and shouldbe included when calculating fine-tuning in the MSSM. III. THE IMPACT OF THE NEW SOFT-TERMSON THE FINE-TUNING MEASURE
The fine tuning measure which we consider with re-spect to a set of independent parameters, p , is given by[22, 23]∆ ≡ max Abs (cid:2) ∆ p (cid:3) , ∆ p ≡ ∂ ln v ∂ ln p = pv ∂v ∂p . (7) In general also the non-holomorphic trilinear couplings T’ need tobe shifted due to the F -term contribution of the superpotential µ -term. The quantity ∆ − gives a measure of the accuracy towhich independent parameters must be tuned to get thecorrect electroweak breaking scale. In the following wewill concentrate on the contributions of µ and µ (cid:48) on thefine tuning measure.The generic expressions for the Renormalisation GroupEquations (RGEs) in the presence of non-holomorphicsoft-terms are given in Refs. [24, 25]. We have imple-mented them in the Mathematica package
SARAH [26–31]to calculate the β -functions for all relevant terms in theconsidered model. The one-loop results for running ofthe new holomorphic soft-terms as well as the standardsoft-breaking masses are summarised in appendix A. Wecan use these results to find an approximate dependenceof the running m H u as function of all other soft-breakingterms. For this purpose, we assume CMSSM-like bound-ary conditions at the scale M GUT = 2 . × GeV M = M = M ≡ m / , m H d = m H u ≡ m m e = m d = m u = m l = m q ≡ m T i = A Y i , T (cid:48) i = A (cid:48) Y i and expand around m = m / = A = A (cid:48) = µ = µ (cid:48) =1 TeV. For tan β = 50, we find m H u ( Q ) (cid:39) . A b A t − . A b M − . A b M − . A (cid:48) b + 0 . A (cid:48) b A (cid:48) t − . A (cid:48) b µ (cid:48) − . A (cid:48) τ − . A (cid:48) τ µ (cid:48) − . A t + 0 . A t M + 0 . A t M + 0 . A t M + 0 . A (cid:48) t + 0 . A (cid:48) t µ (cid:48) + 0 . M − . M M − . M M + 0 . M − . M M − . M − . m d, − . m d, − . m e, − . m e, + 0 . m H d + 0 . m H u + 0 . m l, + 0 . m l, − . m q, − . m q, + 0 . m u, − . m u, + 0 . µ (cid:48) (8)Here, we neglected first and second generation Yukawacouplings and skipped all terms with coefficients smallerthan 10 − . In addition, we parametrised T ( (cid:48) ) i = A ( (cid:48) ) i Y i for ( i = t, b, τ ). For both µ and µ (cid:48) we obtain the simplerelation µ ( (cid:48) ) (cid:39) . µ ( (cid:48) ) ( M GUT ). Thus, the dependenceof the Z-boson mass at the weak scale on the parametersat the GUT scale is given by M Z (cid:39) − . µ (cid:48) ( M GUT ) − . µ ( M GUT ) + . . . (9)Neglecting mixing effects in the neutralino sector, theHiggsino mass is given by m ˜ H ∼ . µ + µ (cid:48) ). We there-fore expect that the fine tuning can be very mild evenfor rather heavy Higgsinos, if the main contribution toits mass comes from the non-holomorphic soft term. IV. PARAMETER SCAN AND PLOTS
We verified this expectation with an explicit numeri-cal study. As free parameters at the GUT scale we take m , m / , A , tan β, µ, Bµ, µ (cid:48) . The correct electroweakvacuum is ensured via the choice of the Higgs masses m H u , m H d . The overall fine tuning in this setup will typ-ically not be dominated by µ , but this could be changedby considering e.g. non-universal gaugino masses (seee.g. [32, 33]). As these two problems ‘factorise’ we con-centrate on the fine tuning with respect to µ and µ (cid:48) .We performed a scan over the MSSM parameter spaceusing the SARAH generated
SPheno [34, 35] version. InFig. 1 we show the contribution to the fine tuning mea-sure with respect to µ and the non-holomorphic Higgsinomass term µ (cid:48) . We find that the usual approximationfor the fine tuning with respect to µ , ∆ µ ∼ µ M Z is anexcellent approximation. Inspecting the plots we infer
200 400 600 8001000510501005001000 μ [ GeV ] Δ μ
200 400 600 8001000510501005001000 μ ' [ GeV ] Δ μ ' FIG. 1. Top: Contribution of µ to the fine tuning measure vs.the value of µ . The red line corresponds to the rough estimate∆ µ ∼ µ M Z , which we observe to be an excellent approxima-tion. Bottom: Contribution of µ (cid:48) to the fine tuning measurevs. the value of µ (cid:48) . that the lowest fine tuning for a given Higgsino mass willbe achieved for a non-holomorphic contribution to theHiggsino mass which is about 4-5 times larger than thecontribution from the usual µ term. If we aim for a Hig-gsino mass of 1 TeV, particularly interesting from thedark matter perspective as it naturally gives the correctrelic abundance, the ideal combination with respect tofine tuning would therefore be for values µ ∼
200 GeVand µ (cid:48) ∼ µ ∼ ∆ µ (cid:48) ∼
20. Without the soft Higgsino mass termthe fine tuning would be ∆ µ ∼ O (1 TeV)for ∆ (cid:48) µ,µ ≤
20. Depending on the parameter choice thisstate can be the LSP and give the correct relic abundanceto be dark matter. For comparison we also show the finetuning for the case where µ (cid:48) = 0 (red points).
100 200 500 1000 2000151050100 m h ∼ [ GeV ] m a x ( Δ μ , Δ μ ' ) FIG. 2. The maximum ∆ µ,µ (cid:48) contribution to the fine tun-ing measure plotted against the Higgsino mass for varying µ (cid:48) (blue) and µ (cid:48) = 0 (red). V. SUMMARY AND CONCLUSIONS
In this letter we have shown that a heavy Higgsinowith a mass of O (1 TeV) can arise in the MSSM withouthaving a very large contribution to the the fine tuningin the MSSM from the µ term, provided that one al-lows for a non-holomorphic soft SUSY breaking Higgsinomass term. Such a heavy Higgsino has a sufficiently smallannihilation cross section so that it can readily be darkmatter without the need for any additional dark mat-ter component. Although the soft Higgsino mass term isequivalent to a combination of a supersymmetric µ termand soft SUSY breaking Higgs mass terms, it is essentialto keep the Higgsino mass term explicitly when calcu-lating the fine tuning because it naturally correlates themagnitude of the equivalent µ term and soft Higgs massterms in such a way as to largely cancel the fine tuningcontributions of these terms.In order to make the role of the Higgsino mass clear,we have concentrated on the contribution to fine tuningcoming from the µ term and the soft Higgsino mass term.However the significant suppression of these contribu-tions that we find means that other contributions to thefine tuning are likely to be dominant and it will be impor- tant to perform a complete analysis including all fine tun-ing contributions. In this context it will also be impor-tant to include the Higgsino soft mass term in extensionsof the MSSM, such as those with non-universal gauginomasses [32, 33] or the generalised NMSSM [33, 36, 37]that have been shown to reduce fine tuning. We hope toconsider these issues shortly. ACKNOWLEDGEMENT
This work is supported by the German Science Foun-dation (DFG) under the Collaborative Research Center(SFB) 676 Particles, Strings and the Early Universe aswell as the ERC Starting Grant ‘NewAve’ (638528).
Appendix A: Renormalisation group equationsincluding non-holomorphic soft-terms1. RGEs for non-holomorphic soft-terms β (1) T (cid:48) u = +3 T (cid:48) u Y † d Y d + T (cid:48) u Y † u Y u + 2 Y u Y † d T (cid:48) d − µ (cid:48) Y u Y † d Y d + 2 Y u Y † u T (cid:48) u − Y u (cid:16)(cid:16) g + g (cid:17) µ (cid:48) − (cid:16) T (cid:48) u Y † u (cid:17)(cid:17) + T (cid:48) u (cid:16) (cid:16) Y d Y † d (cid:17) − (cid:16) g + g (cid:17) + Tr (cid:16) Y e Y † e (cid:17)(cid:17) (A1) β (1) T (cid:48) d = + T (cid:48) d Y † d Y d + 3 T (cid:48) d Y † u Y u + 2 Y d Y † d T (cid:48) d + 2 Y d Y † u T (cid:48) u − µ (cid:48) Y d Y † u Y u + Y d (cid:16) (cid:16) T (cid:48) e Y † e (cid:17) + 6Tr (cid:16) T (cid:48) d Y † d (cid:17) − (cid:16) g + g (cid:17) µ (cid:48) (cid:17) + 115 T (cid:48) d (cid:16) g + 45Tr (cid:16) Y u Y † u (cid:17) − g (cid:17) (A2) β (1) T (cid:48) e = + T (cid:48) e Y † e Y e + 2 Y e Y † e T (cid:48) e + Y e (cid:16) (cid:16) T (cid:48) e Y † e (cid:17) + 6Tr (cid:16) T (cid:48) d Y † d (cid:17) − (cid:16) g + g (cid:17) µ (cid:48) (cid:17) + T (cid:48) e (cid:16) (cid:16) Y u Y † u (cid:17) − g (cid:17) (A3) β (1) µ (cid:48) = 3 µ (cid:48) Tr (cid:16) Y d Y † d (cid:17) − µ (cid:48) (cid:16) g − (cid:16) Y u Y † u (cid:17) + g (cid:17) + µ (cid:48) Tr (cid:16) Y e Y † e (cid:17) (A4)
2. RGEs for soft-breaking masses β (1) m q = − g | M | − g | M | − g | M | + 2 m H d Y † d Y d + 2 m H u Y † u Y u + 2 T † d T d + 2 T † u T u + 2 T (cid:48) dT T (cid:48) d ∗ + 2 T (cid:48) uT T (cid:48) u ∗ − | µ (cid:48) | Y Td Y ∗ d − | µ (cid:48) | Y Tu Y ∗ u + m q Y † d Y d + m q Y † u Y u + 2 Y † d m d Y d + Y † d Y d m q + 2 Y † u m u Y u + Y † u Y u m q + g σ β (1) m l = − g | M | − g | M | + 2 m H d Y † e Y e + 2 T † e T e + 2 T (cid:48) eT T (cid:48) e ∗ − | µ (cid:48) | Y Te Y ∗ e + m l Y † e Y e + 2 Y † e m e Y e + Y † e Y e m l − g σ (A6) β (1) m Hd = − g µ (cid:48) − g µ (cid:48) − g | M | − g | M | − g σ + 6Tr (cid:16) T (cid:48) u T (cid:48) u † (cid:17) + 6 m H d Tr (cid:16) Y d Y † d (cid:17) + 2 m H d Tr (cid:16) Y e Y † e (cid:17) + 6Tr (cid:16) T ∗ d T Td (cid:17) + 2Tr (cid:16) T ∗ e T Te (cid:17) + 6Tr (cid:16) m d Y d Y † d (cid:17) + 2Tr (cid:16) m e Y e Y † e (cid:17) + 2Tr (cid:16) m l Y † e Y e (cid:17) + 6Tr (cid:16) m q Y † d Y d (cid:17) (A7) β (1) m Hu = − g µ (cid:48) − g µ (cid:48) − g | M | − g | M | + g σ + 6Tr (cid:16) T (cid:48) d T (cid:48) d † (cid:17) + 2Tr (cid:16) T (cid:48) e T (cid:48) e † (cid:17) + 6 m H u Tr (cid:16) Y u Y † u (cid:17) + 6Tr (cid:16) T ∗ u T Tu (cid:17) + 6Tr (cid:16) m q Y † u Y u (cid:17) + 6Tr (cid:16) m u Y u Y † u (cid:17) (A8) β (1) m d = − g | M | − g | M | + 4 m H d Y d Y † d + 4 T (cid:48) d ∗ T (cid:48) dT − | µ (cid:48) | Y ∗ d Y Td + 4 T d T † d + 2 m d Y d Y † d + 4 Y d m q Y † d + 2 Y d Y † d m d + 2 g σ β (1) m u = − g | M | − g | M | + 4 m H u Y u Y † u + 4 T (cid:48) u ∗ T (cid:48) uT − | µ (cid:48) | Y ∗ u Y Tu + 4 T u T † u + 2 m u Y u Y † u + 4 Y u m q Y † u + 2 Y u Y † u m u − g σ β (1) m e = − g | M | + 2 (cid:16) m H d Y e Y † e + 2 T (cid:48) e ∗ T (cid:48) eT − | µ (cid:48) | Y ∗ e Y Te + 2 T e T † e + m e Y e Y † e + 2 Y e m l Y † e + Y e Y † e m e (cid:17) + 2 g σ (A11)with σ = 35 g (cid:16) − (cid:16) m u (cid:17) − Tr (cid:16) m l (cid:17) − m H d + m H u + Tr (cid:16) m d (cid:17) + Tr (cid:16) m e (cid:17) + Tr (cid:16) m q (cid:17)(cid:17) (A12) [1] H. Baer, V. Barger, P. Huang, A. Mustafayev,and X. Tata, Phys. Rev. Lett. (2012), 161802,[1207.3343].[2] H. Baer, V. Barger, P. Huang, and X. Tata, JHEP (2012), 109, [1203.5539].[3] H. Baer, V. Barger, P. Huang, D. Mickelson,A. Mustafayev, and X. Tata, Phys. Rev. D87 (2013),no. 11, 115028, [1212.2655].[4] M. W. Cahill-Rowley, J. L. Hewett, A. Ismail, and T. G.Rizzo, Phys. Rev.
D86 (2012), 075015, [1206.5800].[5] J. L. Feng and D. Sanford, Phys. Rev.
D86 (2012),055015, [1205.2372].[6] Z. Kang, J. Li, and T. Li, JHEP (2012), 024,[1201.5305].[7] H. Baer, V. Barger, and D. Mickelson, Phys. Rev. D88 (2013), no. 9, 095013, [1309.2984].[8] H. Baer, V. Barger, and D. Mickelson, Phys. Lett.
B726 (2013), 330–336, [1303.3816].[9] K. Kowalska and E. M. Sessolo, (2013), 1307.5790.[10] H. Baer, V. Barger, D. Mickelson, and M. Padeffke-Kirkland, Phys. Rev.
D89 (2014), no. 11, 115019,[1404.2277].[11] L. Girardello and M. T. Grisaru, Nucl. Phys.
B194 (1982), 65.[12] S. P. Martin, Phys. Rev.
D61 (2000), 035004, [hep-ph/9907550].[13] N. Arkani-Hamed and R. Rattazzi, Phys. Lett.
B454 (1999), 290–296, [hep-th/9804068].[14] B. de Wit, M. T. Grisaru, and M. Rocek, Phys. Lett.
B374 (1996), 297–303, [hep-th/9601115].[15] D. Bellisai, F. Fucito, M. Matone, and G. Travaglini,Phys. Rev.
D56 (1997), 5218–5232, [hep-th/9706099].[16] M. Dine and N. Seiberg, Phys. Lett.
B409 (1997), 239–244, [hep-th/9705057]. [17] F. Gonzalez-Rey and M. Rocek, Phys. Lett.
B434 (1998), 303–311, [hep-th/9804010].[18] E. I. Buchbinder, I. L. Buchbinder, and S. M. Kuzenko,Phys. Lett.
B446 (1999), 216–223, [hep-th/9810239].[19] D. A. Demir, G. L. Kane, and T. T. Wang, Phys. Rev.
D72 (2005), 015012, [hep-ph/0503290].[20] C. S. ¨Un, S. H. Tanyildizi, S. Kerman, and L. Solmaz,Phys. Rev.
D91 (2015), no. 10, 105033, [1412.1440].[21] I. Antoniadis, E. Dudas, D. M. Ghilencea, and P. Tzivel-oglou, Nucl. Phys.
B808 (2009), 155–184, [0806.3778].[22] J. R. Ellis, K. Enqvist, D. V. Nanopoulos, andF. Zwirner, Mod.Phys.Lett. A1 (1986), 57.[23] R. Barbieri and G. F. Giudice, Nucl. Phys. B306 (1988),63.[24] I. Jack and D. R. T. Jones, Phys. Lett.
B457 (1999),101–108, [hep-ph/9903365].[25] I. Jack and D. R. T. Jones, Phys. Rev.
D61 (2000),095002, [hep-ph/9909570].[26] F. Staub, (2008), 0806.0538.[27] F. Staub, Comput.Phys.Commun. (2010), 1077–1086, [0909.2863].[28] F. Staub, Comput.Phys.Commun. (2011), 808–833,[1002.0840].[29] F. Staub, Computer Physics Communications (2013), pp. 1792–1809, [1207.0906].[30] F. Staub, Comput. Phys. Commun. (2014), 1773–1790, [1309.7223].[31] F. Staub, Adv. High Energy Phys. (2015), 840780,[1503.04200].[32] D. Horton and G. Ross, Nucl.Phys.
B830 (2010), 221–247, [0908.0857].[33] A. Kaminska, G. G. Ross, and K. Schmidt-Hoberg,(2013), 1308.4168. [34] W. Porod and F. Staub, Comput.Phys.Commun. (2012), 2458–2469, [1104.1573].[35] W. Porod, Comput.Phys.Commun. (2003), 275–315,[hep-ph/0301101]. [36] G. G. Ross and K. Schmidt-Hoberg, Nucl.Phys.
B862 (2012), 710–719, [1108.1284].[37] G. G. Ross, K. Schmidt-Hoberg, and F. Staub, JHEP1208