High-Precision Electroweak SUSY Production Cross Sections at e+e- Colliders
IIFT–UAM/CSIC–18-079
High-Precision Electroweak SUSY Production CrossSections at e + e − Colliders
S. Heinemeyer ∗ Instituto de Física Teórica (UAM/CSIC), Universidad Autónoma de Madrid, Cantoblanco,28049, Madrid, SpainCampus of International Excellence UAM+CSIC, Cantoblanco, 28049, Madrid, SpainInstituto de Física de Cantabria (CSIC-UC), 39005, Santander, SpainE-mail:
C. Schappacher
Institut für Theoretische Physik, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany(former address)E-mail: [email protected]
For the search for electroweak (EW) particles in the Minimal Supersymmetric Standard Model(MSSM) as well as for future precision analyses of these particles an accurate knowledge of theirproduction and decay properties is mandatory. We evaluate the cross sections for the chargino,neutralino and slepton production at e + e − colliders in the MSSM with complex parameters(cMSSM). The evaluation is based on a full one-loop calculation of all possible production chan-nels including soft and hard photon radiation. The dependence of the cross sections on the relevantcMSSM parameters is analyzed numerically. We find sizable contributions to many productioncross sections. They amount to roughly 15 % of the tree-level results but can go up to 40 % orhigher in extreme cases. Also the dependence on complex parameters of the one-loop correctionsfor many production channels was found non-negligible. The full one-loop contributions are thuscrucial for physics analyses at a future linear e + e − collider such as the ILC or CLIC. Loops and Legs in Quantum Field Theory (LL2018)29 April 2018 - 04 May 2018St. Goar, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - ph ] J u l W SUSY Production at e + e − Colliders
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1. Introduction
One of the important tasks at the LHC is to search for physics beyond the Standard Model(SM), where the Minimal Supersymmetric Standard Model (MSSM) [1] is one of the leadingcandidates. Supersymmetry (SUSY) predicts two scalar partners for all SM fermions as well asfermionic partners to all SM bosons. Contrary to the case of the SM, in the MSSM two Higgsdoublets are required. This results in five physical Higgs bosons instead of the single Higgs bosonin the SM. These are the light and heavy CP -even Higgs bosons, h and H , the CP -odd Higgsboson, A , and the charged Higgs bosons, H ± . The neutral SUSY partners of the (neutral) Higgsand electroweak gauge bosons are the four neutralinos, ˜ χ , , , . The corresponding charged SUSYpartners are the charginos, ˜ χ ± , . The SUSY partner of the charged leptons are the ˜ e s , ˜ µ s , ˜ τ s ( s = , ν e , ˜ ν µ , ˜ ν τ .If SUSY is realized in nature and the scalar quarks and/or the gluino are in the kinematic reachof the (HL-)LHC, it is expected that these strongly interacting particles are eventually producedand studied. On the other hand, SUSY particles that interact only via the electroweak force, i.e.,the charginos, neutralinos, and scalar leptons, have a much smaller production cross section at theLHC. Correspondingly, the LHC discovery potential as well as the current experimental boundsare substantially weaker [2, 3]. At a (future) e + e − collider charginos, neutralinos and sleptons,depending on their masses and the available center-of-mass energy, could be produced and analyzedin detail [4, 5]. Corresponding studies can be found for the ILC in Refs. [6, 7] and for CLIC inRefs. [7, 8]. (Results on the combination of LHC and ILC results can be found in Ref. [9].) Suchprecision studies will be crucial to determine their nature and the underlying SUSY parameters.In order to yield a sufficient accuracy, one-loop corrections to the various production anddecay modes have to be considered. Full one-loop calculations in the cMSSM of (heavy) scalar taudecays was evaluated in Ref. [10], where the calculation can easily be taken over to other sleptondecays. Similarly, full one-loop calculations for various chargino/neutralino decays in the cMSSMhave been presented in Ref. [11]. Sleptons can also be produced in SUSY cascade decays, wherefull one-loop evaluations in the cMSSM exist for the corresponding decays of Higgs bosons [12].Similarly, the one-loop corrections for chargino/neutralino production from the decay of Higgsbosons (at the LHC or ILC/CLIC) can be found in Ref. [13]. Here we review the predictions forchargino, neutralino and slepton production at e + e − colliders [14, 15] (see also Ref. [16]), i.e. thechannels (with ˜e gs = { ˜ e s , ˜ µ s , ˜ τ s } , ˜ ν g = { ˜ ν e , ˜ ν µ , ˜ ν τ } , generation index g and slepton index s ) σ ( e + e − → ˜ χ ± c ˜ χ ∓ c (cid:48) ) c , c (cid:48) = , , σ ( e + e − → ˜ χ n ˜ χ n (cid:48) ) n , n (cid:48) = , , , , (1.1) σ ( e + e − → ˜e ± gs ˜e ∓ gs (cid:48) ) s , s (cid:48) = , , σ ( e + e − → ˜ ν g ˜ ν ∗ g ) g = , , . (1.2)
2. Calculation of diagrams
In this section we review some details regarding the renormalization procedure and the cal-culation of the tree-level and higher-order corrections to the production of charginos, neutralinosand sleptons in e + e − collisions. The diagrams and corresponding amplitudes have been obtainedwith FeynArts (version 3.9) [17], using our MSSM model file (including the MSSM countert-erms) of Ref. [18]. The further evaluation has been performed with
FormCalc (version 9.5) and
LoopTools (version 2.14) [19]. 1
W SUSY Production at e + e − Colliders
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The cross sections (1.1) - (1.2) are calculated at the one-loop level, including soft, hard andcollinear QED radiation. This requires the simultaneous renormalization of the gauge-boson sec-tor, the fermion/sfermion sector as well as the chargino/neutralino sector of the cMSSM, basedon Refs. [18, 20]. All the relevant details can be found in Refs. [14, 15]. The renormalizationscheme employed is the same one as for the decay of sleptons [10] or charginos/neutralinos [11].Consequently, the predictions for the production and decay can be used together in a consistentmanner. More details and the application to Higgs-boson and SUSY particle decays can be foundin Refs. [10–13, 18, 21–23]. Similarly, the application to Higgs-boson production cross sections at e + e − colliders are given in Refs. [24, 25].Sample diagrams for the process e + e − → ˜e ± gs ˜e ∓ gs (cid:48) and e + e − → ˜ ν g ˜ ν ∗ g are shown in Fig. 1. Di-agrams for chargino/neutralino production can be found in Ref. [14]. Not shown in Fig. 1 are thediagrams for real (hard and soft) photon radiation. We have neglected all electron–Higgs couplingsand terms proportional to the electron mass whenever this is safe, i.e. except when the electron massappears in negative powers or in loop integrals. We have verified numerically that these contribu-tions are indeed totally negligible. Moreover, in general, in Fig. 1 we have omitted diagrams withself-energy type corrections of external (on-shell) particles. While the contributions from the realparts of the loop functions are taken into account via the renormalization constants defined by OSrenormalization conditions, the contributions coming from the imaginary part of the loop functionscan result in an additional (real) correction if multiplied by complex parameters. In the analyt-ical and numerical evaluation, these diagrams have been taken into account via the prescriptiondescribed in Ref. [18].As regularization scheme for the UV divergences we have used constrained differential renor-malization [26], which has been shown to be equivalent to dimensional reduction [27, 28] at theone-loop level [19]. Thus the employed regularization scheme preserves SUSY [29, 30] and guar-antees that the SUSY relations are kept intact. All UV divergences cancel in the final result. For adiscussion on soft photon emission and corresponding problems with the phase space integration,see Refs. [14, 15].
3. Numerical analysis
Here we review two examples for the numerical analysis of chargino/neutralino and sleptonproduction at e + e − colliders in the cMSSM as presented in Refs. [14, 15]. In the figures below weshow the cross sections at the tree level (“tree”) and at the full one-loop level (“full”), which is thecross section including all one-loop corrections. All results shown use the CCN[1] renormaliza-tion scheme [18] (i.e. OS conditions for the two charginos and the lightest neutralino). e + e − → ˜ χ ± c ˜ χ ∓ c (cid:48) and e + e − → ˜ χ n ˜ χ n (cid:48) The SUSY parameters for the evaluation of these production cross sections are chosen accord-ing to the scenario S1, shown in Tab. 1.As an example for chargino/neutralino production the process e + e − → ˜ χ + ˜ χ − is shown inFig. 2. In the analysis of the production cross section as a function of √ s (upper left plot) wefind the expected behavior: a strong rise close to the production threshold, followed by a decreasewith increasing √ s . Away from the production threshold, loop corrections of ∼ − √ s = W SUSY Production at e + e − Colliders
S. Heinemeyer e e ˜ l gs ˜ l gs ′ V ee ˜ l s ˜ l s ′ F e e ˜ l gs ˜ l gs ′ V VS e e ˜ l gs ˜ l gs ′ V VV e e ˜ l gs ˜ l gs ′ V VFF e e ˜ l gs ˜ l gs ′ V VSS e e ˜ l gs ˜ l gs ′ V VUU e e ˜ l gs ˜ l gs ′ V VSVe e ˜ l gs ˜ l gs ′ V VVV ee ˜ l s ˜ l s ′ FFF S ee ˜ l s ˜ l s ′ FFF V e e ˜ l gs ˜ l gs ′ V FFF e e ˜ l gs ˜ l gs ′ V SSS e e ˜ l gs ˜ l gs ′ V SSV e e ˜ l gs ˜ l gs ′ V SVS e e ˜ l gs ˜ l gs ′ V VSS e e ˜ l gs ˜ l gs ′ V SVVee ˜ l gs ˜ l gs ′ VS FF ee ˜ l gs ˜ l gs ′ VF SS ee ˜ l gs ˜ l gs ′ VV FF ee ˜ l gs ˜ l gs ′ VF VV e e ˜ l gs ˜ l gs ′ VS V e e ˜ l gs ˜ l gs ′ V SVe e ˜ l gs ˜ l gs ′ V SS e e ˜ l gs ˜ l gs ′ V VV ee ˜ l s ˜ l s ′ FFS F ee ˜ l s ˜ l s ′ FSF S ee ˜ l s ˜ l s ′ FFV F ee ˜ l s ˜ l s ′ FSF Vee ˜ l s ˜ l s ′ FVF S ee ˜ l s ˜ l s ′ FFS F ee ˜ l s ˜ l s ′ FSF S ee ˜ l s ˜ l s ′ FFV F ee ˜ l s ˜ l s ′ FSF V ee ˜ l s ˜ l s ′ FVF S ee ˜ l gs ˜ l gs ′ F SS ee ˜ l gs ˜ l gs ′ F VV ee ˜ l gs ˜ l gs ′ S FF F ee ˜ l gs ˜ l gs ′ F VV S ee ˜ l gs ˜ l gs ′ S FFF ee ˜ l gs ˜ l gs ′ F VVSee ˜ l s ˜ l s ′ F SS S ee ˜ l s ˜ l s ′ V FF F ee ˜ l s ˜ l s ′ F SS V ee ˜ l s ˜ l s ′ F VF S ee ˜ l s ˜ l s ′ S FV F e e ˜ l gs ˜ l gs ′ V e e ˜ l gs ˜ l gs ′ V e e ˜ l gs ˜ l gs ′ V V ee ˜ l s ˜ l s ′ F ee ˜ l s ˜ l s ′ F ee ˜ l s ˜ l s ′ FF Figure 1:
Generic tree, self-energy, vertex, box, and counterterm diagrams for the process e + e − → ˜ l gs ˜ l gs (cid:48) (˜ l gs = { ˜e gs , ˜ ν g } ; g = , , s , s (cid:48) = , l s . F can be a SM fermion, chargino or neutralino; S canbe a sfermion or a Higgs/Goldstone boson; V can be a γ , Z or W ± . It should be noted that electron–Higgscouplings are neglected. W SUSY Production at e + e − Colliders
S. Heinemeyer
Scen. √ s t β µ M H ± M ˜ Q , ˜ U , ˜ D M ˜ L , ˜ E | A t | A b A τ | M | M M S1 1000 10 450 500 1500 1500 2000 | A t | M ˜ L µ /4 µ /2 2000 Table 1:
MSSM default parameters for the numerical investigation of chargino and neutralino production;all parameters (except of t β ) are in GeV. fulltree e + e − → ˜ χ +1 ˜ χ − σ /fb √ s fulltree e + e − → ˜ χ +1 ˜ χ − σ /fb µ fulltree e + e − → ˜ χ +1 ˜ χ − σ /fb M ˜ L = M ˜ E σ loop /σ tree ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ . . . . . full ϕ A t : treefull ϕ M : tree e + e − → ˜ χ +1 ˜ χ − σ /fb ϕ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Figure 2: σ ( e + e − → ˜ χ + ˜ χ − ) . Tree-level and full one-loop corrected cross sections are shown with param-eters chosen according to S1. The upper plots show the cross sections with √ s (left) and µ (right) varied;the lower plots show M ˜ L = M ˜ E (left) and ϕ M , ϕ A t (right) varied.
500 GeV and ∼ +
14 % at √ s = √ s ≈
575 GeV. The relative size of loop corrections increase with increasing √ s (and decreasing σ ) and reach ∼ +
19 % at √ s = µ in S1 (upper right plot) we find astrong decrease of the production cross section, as can be expected from kinematics. The relativeloop corrections in S1 reach ∼ +
30 % at µ =
240 GeV (at the border of the experimental limit), ∼ +
14 % at µ =
450 GeV (i.e. S1) and ∼ −
30 % at µ = µ = M ˜ L ( = M ˜ E ) is shown in the lower left plot of Fig. 2. This mass parameter controls the t -channelexchange of first generation sleptons at tree-level. First a small decrease down to ∼
90 fb can beobserved for M ˜ L ≈
400 GeV. For larger M ˜ L the cross section rises up to ∼
190 fb for M ˜ L = W SUSY Production at e + e − Colliders
S. Heinemeyer
In scenario S1 we find a substantial increase of the cross sections from the loop corrections. Theyreach the maximum of ∼ +
18 % at M ˜ L ≈
850 GeV with a nearly constant offset of about 20 fb forhigher values of M ˜ L . We find that the phase dependence ϕ M of the cross section in our scenario istiny. The loop corrections are found to be nearly independent of ϕ M at the level below ∼ + . ϕ A t , which enter via final state vertex corrections. While thevariation with ϕ A t is somewhat larger than with ϕ M , it remains tiny and unobservable. However,in Ref. [14] other production channels with an appreciable phase dependence were identified. e + e − → ˜e ± gs ˜e ∓ gs (cid:48) and e + e − → ˜ ν g ˜ ν ∗ g The SUSY parameters for the numerical analysis for slepton production (i.e. in Ref. [15]) arechosen according to the scenario S2, shown in Tab. 2.Scen. √ s t β µ M H ± M ˜ Q , ˜ U , ˜ D M ˜ E A u g A d g | A e g | | M | M M S2 1000 10 350 1200 2000 300 2600 2000 2000 400 600 2000
Table 2:
MSSM default parameters for the numerical investigation; all parameters (except of t β ) are inGeV. Furthermore, M ˜ L = M ˜ E +
50 GeV was chosen for all slepton generations. fulltree e + e − → ˜ τ +1 ˜ τ − σ /fb √ s e + e − → ˜ τ +1 ˜ τ − σ /fb M ˜ E fulltree e + e − → ˜ τ +1 ˜ τ − σ /fb tan β . . . . σ loop /σ tree ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ . . . . . fulltree e + e − → ˜ τ +1 ˜ τ − σ /fb ϕ A e g ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ . . . . . . . . Figure 3: σ ( e + e − → ˜ τ + ˜ τ − ) . Tree-level and full one-loop corrected cross sections are shown with param-eters chosen according to S2. The upper plots show the cross sections with √ s (left) and M ˜ E (right) varied;the lower plots show t β (left) and ϕ A e g (right) varied. All masses and energies are in GeV. W SUSY Production at e + e − Colliders
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As an example of the numerical analysis presented in Ref. [15] we show the process e + e − → ˜ τ + ˜ τ − in Fig. 3. As a function of √ s we find loop corrections of ∼ +
14 % at √ s = √ s ≈
725 GeV (where the one-loop corrections are between ±
10 %for √ s < ∼
900 GeV) and ∼ +
35 % at √ s = M ˜ E (upperright plot) the cross sections are decreasing with increasing M ˜ E as obvious from kinematics andthe full corrections have their maximum of ∼
28 fb at M ˜ E =
100 GeV, more than two times largerthan in S2. The relative corrections are changing from ∼ +
33 % at M ˜ E =
100 GeV to ∼ −
25 %at M ˜ E =
490 GeV with a tree crossing at M ˜ E =
415 GeV. In the lower left row of Fig. 3 we showthe dependence on t β . The relative corrections for the t β dependence vary between ∼ + . t β = ∼ + . t β = ϕ A e g of the cross section in S2 is shown in the lower right plot ofFig. 3. The loop correction increases the tree-level result by ∼ +
14 %. The phase dependence ofthe relative loop correction is very small and found to be below 0 . ϕ M isnegligible and therefore not shown here. Acknowledgements
S.H. thanks the organizers of L&L 2018 for the invitation and the (as always!) inspiring atmo-sphere. The work of S.H. is supported in part by the MEINCOP Spain under contract FPA2016-78022-P, in part by the “Spanish Agencia Estatal de Investigación” (AEI) and the EU “FondoEuropeo de Desarrollo Regional” (FEDER) through the project FPA2016-78022-P, in part bythe “Spanish Red Consolider MultiDark” FPA2017â ˘A ˇR90566â ˘A ˇRREDC, and in part by the AEIthrough the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597.
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