Scalar color octets and triplets in the SUSY with R-symmetry
SScalar color octets and triplets in the SUSY withR-symmetry
Wojciech Kotlarski
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, PolandInstitute for Nuclear and Particle Physics, TU Dresden, Zellescher Weg 19, 01069 Dresden,GermanyE-mail: [email protected]
Abstract.
In this note we report on the recent progress in the study of the strongly interactingsector of the Minimal R-symmetric Supersymmetric Standard Model (MRSSM). First, wediscuss the limits originating from the search of the sgluon pair production in the same-signlepton final state at the LHC. Next, we present the first available in literature calculation of theNLO SQCD corrections to the same-sign squark pair production in the MRSSM. We discussthe relative size of k -factors compared to the MSSM and some technical details that led to thisresult.
1. Introduction
The Minimal Supersymmetric Standard Model (MSSM), with its simple structure and richphenomenology, has been for many years the primary realization of supersymmetry consideredin the literature. But as the Large Hadron Collider continues its second run with no MSSM insight, the question is being raised whether low-scale supersymmetry exists at all. While it ispossible that this is indeed not the case, it is also possible that supersymmetry is just not realizedin the minimal, MSSM way. This is the reason behind increased attention in the non-minimalSUSY models observed in recent years.One of the models which proved to be very promising in this context is the Minimal R-symmetric Supersymmetric Standard Model (MRSSM) [1]. The model ameliorates the flavorproblem of the MSSM [1, 2] while reducing fine-tuning [3] and exhibiting distinct, interestingphenomenology. Though very promising, it is also much less explored - with the MSSM havinga 30-year head start. Until now, the main effort has been devoted to the study of its electroweaksector, especially in the context of the discovery of the Higgs boson [3–5] and studies of darkmatter [6]. The phenomenology of the strongly interacting sector, which is the most importantfrom the point of view of direct detection at the LHC, was much less investigated. The focusthere has so far been mainly on the sgluon production and LO analyses [7–13]. As the super-QCD (SQCD) k -factors tend to be large, a question arises of how the latter studies change inthe presence of quantum corrections. To fill this gap, in this work, after the recap of the recent Strictly speaking, some of these works do not deal with the MRSSM but with a model in which N = 1 SUSYis extended in the strongly interacting sector by N = 2 gauge companions, which does have a very similar SQCDsector. a r X i v : . [ h e p - ph ] M a r able 1: Strongly interacting superfield content of the MRSSM together with R-charges of thecomponent fields. The superfield in the last line is absent in the MSSM. It comprises of theright-handed component of the Dirac gluino and two real scalar gluons.superfield boson fermionleft-handed (s)quark ˆ Q L ˜ q L q L Q R ˜ q † R q R g g g +1adjoint chiral superfield ˆ O O O −
2. The Model
The guiding principle behind the construction of the MRSSM is the presence of the unbrokenR-symmetry at the low scale. However, with the R-charge assignment mimicking R-parity, theconstruction of phenomenologically viable model does require the extension of the MSSM fieldcontent [1]. In the strongly interacting sector this amounts to adding a color-octet (electroweaksinglet) chiral superfield ˆ O , whose fermionic component allows one to construct a Dirac massterm for the Dirac gluino ˜ g D , ˜ g D ≡ (cid:18) ˜ g ˜ O (cid:19) , (1)where, as shown in table 1, ˜ g and ˜ O are the gluino and the octino Weyl fermions, respectively.The addition of the superfield ˆ O has important phenomenological consequences. For one,its scalar, CP-odd component O A can be arbitrarily light as its tree-level mass is controlledpurely by the soft-breaking parameter. This makes it ideal to search for at the LHC, also in theframework of simplified models. Also, heavier squark pair production is dominated by same-signsquarks, a process mediated by the exchange of a gluino in the t-channel. In the MSSM, dueto gluino’s Majorana nature, it is possible to produce ˜ q L ˜ q L , ˜ q R ˜ q R and ˜ q L ˜ q R final states while inthe MRSSM only the ˜ q L ˜ q R combination is allowed. This lowers the inclusive squark productioncross section, weakening the LHC exclusion limits. This comes at the expense of the gluino masslimits, which are more stringent for Dirac gluinos.To illustrate this discussion, in figure 1 we compare the MRSSM and MSSM cross sections forthe pair production of various strongly interacting SUSY particles present in these models. In theplot we use the MMHT2014lo68cl [14] PDFs interfaced through
LHAPDF6 [15] with factorizationand renormalization scales set equal to arithmetic mean of the masses of final-state particles.As can be seen, assuming the gluino mass of 2 TeV, for masses of squarks of around 1 TeV theigure 1: Comparison of the selected cross sections between the MSSM and the MRSSM infunction of the squark mass. Different quark flavors (excluding stops) and “chiralities” areadded together.MSSM cross section is dominated by the (inclusive) same-sign squark pair production. As statedin the previous paragraph, the analogous cross section in the MRSSM is much smaller while theopposite-sign squark production cross section is quite similar to the MSSM. The sgluon pairproduction cross section, which is an exclusive feature of the MRSSM, is also of the same order.Since we are primarily interested in the differences between the MSSM and the MRSSM, wewill focus our discussion in the next two sections on production of sgluon pairs and same-signsquarks.
3. Sgluon pair production at the 13 TeV LHC
The partonic cross sections for the pair production of scalar or pseudoscalar sgluons are givenby ˆ σ ( q ¯ q → OO ) = 2 πα s s β O , (2)ˆ σ ( gg → OO ) = 3 πα s s (cid:0) β O − β O + 6( − β O + β O ) arctanh β O (cid:1) , (3)where ˆ s is the standard partonic Mandelstam variable and β O ≡ (cid:113) − m O / ˆ s is the sgluon’svelocity in the center-of-mass system of colliding partons. In the MRSSM, the masses ofthe scalar ( S ) and the pseudoscalar ( A ) components of the field O from table 1, where O ≡ ( O S + ıO A ), are related through the gluino mass M DO (in the normalization used inRef. [4]) as m O S = m O A + 4( M DO ) . (4)Since LHC constraints require gluinos to be (cid:38) . m O A (cid:39) .9 1.0 1.1 1.2 1.3 1.4 1.5012345 sgluon mass [ TeV ] nu m be r o f s i gna l e v en t s Figure 2: Predicted number of observed signal events as a function of the sgluon mass (bluepoints). Solid line shows interpolation between these points. Red region is excluded by ATLASfor SR3b of [16] at 95% CL. Interpreted in the context of sgluon production, it corresponds toa lower limit on the sgluon mass m O (cid:38) .
95 TeV.Once produced, pseudoscalar sgluons will decay almost exclusively to top quark pairs (seeAppendix E of [17]). This produces a 4-top quark final state, a signature that is rare in theSM. Using the search of the same-sign lepton production performed by ATLAS [16], we haveconstrained the sgluons’ contribution to this signature [11]. This, as shown in figure 2, directlytranslates to the lower limit on the pseudoscalar sgluon mass of 0.95 TeV.Possibility of the existence of a relatively light color-charged particle that decays directly tostandard model particles is one of the distinguishing features of models with Dirac gluinos. Ifever discovered, it will be a clear falsification of the MSSM.
4. Same-sign squark pair production at the NLO
The second important channel is the same-sign squark pair production. The partonic crosssection for the pair production of left and right squarks of flavor i and equal masses in theMSSM and in the MRSSM is given byˆ σ ( q i q i → ˜ q i,L ˜ q i,R ) = πα s ˆ s (cid:34) − (cid:115) − m q i ˆ s + (cid:32) − − m g − m q i )9ˆ s (cid:33) (5) × ln ˆ s (cid:18) − (cid:113) − m qi ˆ s (cid:19) + 2( m g − m q i )ˆ s (cid:18) (cid:113) − m qi ˆ s (cid:19) + 2( m g − m q i ) , where m ˜ g is (Dirac or Majorana) gluino mass and ˆ s is as in eqs 2 and 3. As noted in section 2, inthe MSSM one also has additional channels for left-left and right-right squark pair production.While these tree-level results are enough for qualitative discussions as done in figure 1, a detailedcomparison of the MSSM and the MRSSM does require going beyond that. We therefore presenthere the calculation of NLO SQCD corrections to the left-right same-sign squark pair productionin the MRSSM.The setup of the calculation is as follows. We created a custom FeynArts [18] MRSSMmodel file containing UV and DREG (dimensional regularization) to DRED (dimensionalreduction) [19, 20] transition countertems. Both the real and virtual amplitudes were generatedsing
FeynArts and evaluated in
FormCalc [21]. The amplitudes were then processed in mathematica and exported to a standalone
C++ code.
The evaluation of the virtual amplitude follows standard procedures, so here we only outlinethe most important details. The amplitude is regularized using the so-called ’t Hooft-Veltmanscheme (HV in the notation of Ref. [22]). In the HV, internal gluons are kept D-dimensional,which breaks supersymmetry due to the mismatch between gluon and gluino number of degreesof freedom. This is corrected by supplementing the result with a set of counterterms thattranslate the UV-poles from DREG to DRED, as DRED was proven to not break SUSY atthe one-loop level (for reviews of checks see e.g. [23, 24]). For the process at hand, only therenormalisation constant of the squark-quark-gluino vertex needs to be changed. Denoting therenormalization constant of this vertex by δ ˆ g s , we find that it has to satisfy δ ˆ g s = δg s + g s π (cid:18) C A − C F (cid:19) , (6)where δg s is the strong coupling constant counterterm in DREG and zero momentum subtractionscheme (more precisely, we use zero momentum subtraction for contributions from particlesheavier than the bottom quark and MS for contributions from bottom and light quarks). Theremaining counterterms are fixed by the on-shell renormalization conditions.After this procedure the amplitude is UV finite and supersymmetric but it still contains theinfrared (IR) singularities. The IR singularities of the virtual amplitude cancel after adding real corrections - soft ones cancelaccording to the Kinoshita-Lee-Nauenberg theorem [25, 26] while collinear ones are removedthrough mass factorization [27, 28]. To extract soft and collinear divergences, we employed thetwo-cut phase space slicing method as documented in [29]. In short, this amounts to splittingthe real emission contribution phase space according to emitted gluon energy into soft regiondefined as E g ≤ δ s √ ˆ s , (7)where δ s (cid:28)
1, and its complement. This region of phase space does contain both soft and/orcollinear divergences. Using the eikonal approximation, the divergences can be extracted inD-dimensions as poles in 4-D.However, the complement of the phase space defined by eq. 7 still contains collineardivergences. These are extracted by slicing the phase space according to collinearity condition.We use the collinear condition from [30] as it allows, contrary to the one used in [29], to decouplethe soft and collinear limits.The IR finiteness of the result was checked by evaluating squared matrix elements for asingle phase space point, showing cancellation of divergences at the level of a double numericalprecision.
The calculation of the real emission corrections poses one more difficulty. Figure 3 shows thediagrams for gu → ˜ u L ˜ u R ¯ u process. This process contains a contribution from the s-channelproduction of a gluino. These kinds of effects, observed already in the SM for single-top � � ˜ �� ˜ ��� � ˜ � � � � ˜ � � ˜ ���� ˜ � �� � ˜ �� ˜ ��� ˜ �� ˜ � �� � ˜ ��� ˜ � ˜ �� ˜ ��� � ˜ �� ˜ � � � ˜ �� ˜ � �� � ˜ � � ˜ ���� ˜ � �� � ˜ � � ˜ ��� ˜ �� ˜ � �� � ˜ �� ˜ � �� ˜ � � ˜ ��� � ˜ � � ˜ ���� ˜ � �� � ˜ �� ˜ � �� ˜ � � ˜ � Figure 3: Feynman diagrams for gu → ˜ u L ˜ u R ¯ u subprocess. For m ˜ g D > m ˜ u L/R , amplitude exhibitan on-shell singularity.production, require a careful definition of the process of interest. The amplitude for diagramsin figure 3 can be split into resonant and non-resonant parts as M tot = M nr + M r . (8)For m ˜ g D > m ˜ u L/R , the M r becomes infinite when ( p ˜ g D + p ˜ u L/R ) = m g D . We therefore employthe diagram removal (DR) procedure of Ref. [31], where resonant diagrams are removed at theamplitude level. As this procedure breaks gauge invariance, it requires a careful choice of thegauge for the external gluon. We choose a light-cone gauge, with the gauge vector η set equalto the momentum of the second incoming parton. For numerical studies, we consider 2 MRSSM scenarios with a different gluino-squark masshierarchy. Both points have common squark masses m ˜ q = 1 . m O A = 5 TeV, m t = 172GeV and differ only in the value of the gluino mass, with 1 TeV mass in the BMP1 and 2TeV one in the BMP2. The scalar sgluon mass m O S is fixed by the eq. 4. We perform all thecalculations for the LHC running at center-of-mass energy of 13 TeV using MMHT2014nlo68cl
PDFs interfaced through
LHAPDF6 .The total, NLO cross section in the BMP1 is σ BMP1MRSSM ( pp → ˜ u L ˜ u R ) = 9 . ± . . (9)This can be compared with the MSSM result obtained using MadGraph5 aMC@NLO [32] and anNLO-capable UFO [33] model
SUSYQCD UFO [34], which is σ BMP1MSSM ( pp → ˜ u L ˜ u R ) = 9 . ± .
003 fb . (10)he difference is of the order of 5%, and results exclusively from the virtual part.Similarly, for the BMP2 and after employing the diagram removal procedure, we get σ BMP2MRSSM ( pp → ˜ u L ˜ u R ) = 3 . ± . σ BMP2MSSM ( pp → ˜ u L ˜ u R ) = 2 . ± . , (12)giving an 8% difference.Since the Born and the real matrix elements are identical for those processes in both models,it is not surprising that differences, though not negligible, do not exceed a few percent.The results for the MRSSM are in full agreement with our second calculation based on MadGraph5 aMC@NLO and
GoSam [35, 36], interfaced using
BLHA2 interface [37]. However, wepostpone the description of this method to further publication.In the appendix, we give the decomposition of numbers in eqs 9 and 11 into tree, virtual, softand/or collinear and hard non-collinear contributions. For phase space slicing parameters δ s and δ c we chose values 10 − and 10 − , respectively. We checked extensively that the final result isindependent of precise numerical values of those (unphysical) parameters. We also verified resultsfor separate partonic channels between our C++ and
MadGraph5 aMC@NLO implementations.The above-mentioned
C++ code, containing pp → ˜ u L ˜ u R and pp → ˜ u L ˜ u ∗ L processes, will becomepublic soon together with a more detailed description of both calculations.
5. Conclusions
The Minimal R-symmetric Supersymmetric Standard Model is a well-motivated and viablealternative to the MSSM. Recent studies have shown that within the MRSSM framework onecan accommodate the Higgs boson discovered at the LHC while staying in agreement withconstraints from electroweak precision observables. Also, the first very promising studies of thedark matter sector were performed.With respect to this progress, the study of the SQCD sector is lagging behind. With thatin mind, we have presented in this note our recent progress in that matter. This included boththe discussion of the recently derived limits on the sgluon mass from the LHC Run 2 data, aswell as the calculation of the NLO SQCD corrections to the same-sign squark pair production.To cross-check the latter results, we did two separate calculations, one based on analytic resultsencapsulated into a standalone
C++ code and the other using a well-established framework of
MadGraph5 aMC@NLO , which are in full agreement.Our calculations showed a moderate difference of 5 - 10% between MSSM and MRSSM for theconsidered benchmark points, which can be traced back to the difference in the virtual matrixelement.The developed
C++ code is already also capable of calculating the pp → ˜ u L ˜ u ∗ L process. Weplan to release the code to the public together with a description of the theoretical setup of thecalculation. This will be followed shorty by a phenomenological analysis. Acknowledgments
I would like to thank Philip Diessner, Jan Kalinowski, Sebastian Liebschner and DominikStoeckinger, with whom the work on which this note is based was done. Work supportedin part by the German DFG grant STO 876/4-1, the Polish National Science CentreHARMONIA project under contract UMO-2015/18/M/ST2/00518 (2016-2019) and the PL-Grid Infrastructure. ppendix A.
Appendix A.1. BMP1
Results for subprocess uu > suLsuR(+X)---------------------------------------------------------------tree: 5.67655e+00 +/- 5.7e-06 fb ( p-value = 0.0e+00 )virtual: 3.53574e+00 +/- 3.5e-05 fb ( p-value = 9.9e-17 )real (soft): -5.71007e+01 +/- 4.4e-05 fb ( p-value = 0.0e+00 )real (hard): 5.78352e+01 +/- 5.8e-04 fb ( p-value = 0.0e+00 )---------------------------------------------------------------sum: 9.94681e+00 +/- 5.8e-04 fbResults for subprocess gu > suLsuR(+X)---------------------------------------------------------------tree: 0.00000e+00 +/- 0.0e+00 fb ( p-value = 0.0e+00 )virtual: 0.00000e+00 +/- 0.0e+00 fb ( p-value = 0.0e+00 )real (soft): -5.03323e-02 +/- 5.0e-09 fb ( p-value = 0.0e+00 )real (hard): 4.66389e-02 +/- 4.7e-07 fb ( p-value = 0.0e+00 )---------------------------------------------------------------sum: -3.69334e-03 +/- 4.7e-07 fbResults for subprocess sum---------------------------------------------------------------tree: 5.67655e+00 +/- 5.7e-06 fb ( p-value = 0.0e+00 )virtual: 3.53574e+00 +/- 3.5e-05 fb ( p-value = 9.9e-17 )real (soft): -5.71511e+01 +/- 4.4e-05 fb ( p-value = 0.0e+00 )real (hard): 5.78819e+01 +/- 5.8e-04 fb ( p-value = 0.0e+00 )---------------------------------------------------------------sum: 9.94312e+00 +/- 5.8e-04 fb
Appendix A.2. BMP2
Results for subprocess uu > suLsuR(+X)---------------------------------------------------------------tree: 2.01180e+00 +/- 2.0e-06 fb ( p-value = 0.0e+00 )virtual: 6.97868e-01 +/- 7.0e-06 fb ( p-value = 0.0e+00 )real (soft): -1.99882e+01 +/- 1.6e-05 fb ( p-value = 0.0e+00 )real (hard): 2.02943e+01 +/- 2.0e-04 fb ( p-value = 0.0e+00 )---------------------------------------------------------------sum: 3.01582e+00 +/- 2.0e-04 fbResults for subprocess gu > suLsuR(+X)---------------------------------------------------------------tree: 0.00000e+00 +/- 0.0e+00 fb ( p-value = 0.0e+00 )virtual: 0.00000e+00 +/- 0.0e+00 fb ( p-value = 0.0e+00 )real (soft): -1.69119e-02 +/- 1.7e-09 fb ( p-value = 0.0e+00 )real (hard): 1.46572e-02 +/- 1.5e-07 fb ( p-value = 0.0e+00 )---------------------------------------------------------------sum: -2.25477e-03 +/- 1.5e-07 fbesults for subprocess sum---------------------------------------------------------------tree: 2.01180e+00 +/- 2.0e-06 fb ( p-value = 0.0e+00 )virtual: 6.97868e-01 +/- 7.0e-06 fb ( p-value = 0.0e+00 )real (soft): -2.00051e+01 +/- 1.6e-05 fb ( p-value = 0.0e+00 )real (hard): 2.03090e+01 +/- 2.0e-04 fb ( p-value = 0.0e+00 )---------------------------------------------------------------sum: 3.01357e+00 +/- 2.0e-04 fb
References [1] Kribs G D, Poppitz E and Weiner N 2008
Phys. Rev.
D78
Preprint )[2] Dudas E, Goodsell M, Heurtier L and Tziveloglou P 2014
Nucl. Phys.
B884
Preprint )[3] Bertuzzo E, Frugiuele C, Gregoire T and Ponton E 2015
JHEP
089 (
Preprint )[4] Dießner P, Kalinowski J, Kotlarski W and Stoeckinger D 2014
JHEP
124 (
Preprint )[5] Diessner P, Kalinowski J, Kotlarski W and Stoeckinger D 2015
Adv. High Energy Phys.
Preprint )[6] Diessner P, Kalinowski J, Kotlarski W and Stoeckinger D 2016
JHEP
007 (
Preprint )[7] Choi S Y, Drees M, Kalinowski J, Kim J M, Popenda E and Zerwas P M 2009
Phys. Lett.
B672
Preprint )[8] Plehn T and Tait T M P 2009
J. Phys.
G36
Preprint )[9] Calvet S, Fuks B, Gris P and Valery L 2013
JHEP
043 (
Preprint )[10] Kotlarski W, Kalinowski A and Kalinowski J 2013
Acta Phys. Polon.
B44
JHEP
027 (
Preprint )[12] Choi S Y, Drees M, Freitas A and Zerwas P M 2008
Phys. Rev.
D78
Preprint )[13] Kribs G D and Martin A 2013 (
Preprint )[14] Harland-Lang L A, Martin A D, Motylinski P and Thorne R S 2015
Eur. Phys. J.
C75
Preprint )[15] Buckley A, Ferrando J, Lloyd S, Nordstr¨om K, Page B, R¨ufenacht M, Sch¨onherr M andWatt G 2015
Eur. Phys. J.
C75
132 (
Preprint )[16] Aad G et al. (ATLAS) 2016
Eur. Phys. J.
C76
259 (
Preprint )[17] Kotlarski W 2016
Analysis of the R-symmetric supersymmetric models including quantumcorrections
Ph.D. thesis U. of Warsaw (
Preprint ) URL http://inspirehep.net/record/1499505/files/arXiv:1611.06622.pdf [18] Hahn T 2001
Comput. Phys. Commun.
Preprint hep-ph/0012260 )[19] Siegel W 1979
Phys.Lett.
B84
Nucl.Phys.
B167
Comput. Phys. Commun.
Preprint hep-ph/9807565 )[22] Signer A and Stoeckinger D 2009
Nucl. Phys.
B808
Preprint )[23] Jack I and Jones D R T 1997 [Adv. Ser. Direct. High Energy Phys.21,494(2010)] (
Preprint hep-ph/9707278 )24] Stoeckinger D 2005
JHEP
076 (
Preprint hep-ph/0503129 )[25] Kinoshita T 1962
J. Math. Phys. Phys. Rev. (6B) B1549–B1562[27] Collins J C, Soper D E and Sterman G F 1985
Nucl. Phys.
B261
Phys. Rev.
D31
Phys. Rev.
D65
Preprint hep-ph/0102128 )[30] Dawson S, Kao C, Wang Y and Williams P 2007
Phys. Rev.
D75
Preprint hep-ph/0610284 )[31] Frixione S, Laenen E, Motylinski P, Webber B R and White C D 2008
JHEP Preprint )[32] Alwall J, Frederix R, Frixione S, Hirschi V, Maltoni F, Mattelaer O, Shao H S, Stelzer T,Torrielli P and Zaro M 2014
JHEP
079 (
Preprint )[33] Degrande C, Duhr C, Fuks B, Grellscheid D, Mattelaer O and Reiter T 2012
Comput. Phys.Commun.
Preprint )[34] Degrande C, Fuks B, Hirschi V, Proudom J and Shao H S 2016
Phys. Lett.
B755
Preprint )[35] Cullen G, Greiner N, Heinrich G, Luisoni G, Mastrolia P, Ossola G, Reiter T andTramontano F 2012
Eur. Phys. J.
C72
Preprint )[36] Cullen G et al.
Eur. Phys. J.
C74
Preprint )[37] Alioli S et al.
Comput. Phys. Commun.
Preprint1308.3462