LSP Squark Decays at the LHC and the Neutrino Mass Hierarchy

LSP Squark Decays at the LHC and the Neutrino Mass Hierarchy

Zachary Marshall, Burt A. Ovrut, Austin Purves and Sogee Spinner
Physics Division, Lawrence Berkeley National Laboratory
Berkeley, CA 94704
Department of Physics, University of Pennsylvania
July 12, 2019
Abstract

The existence of -parity in supersymmetric models can be naturally explained as being a discrete subgroup of gauged baryon minus lepton number (). The most minimal supersymmetric model triggers spontaneous -parity violation, while remaining consistent with proton stability. This model is well-motivated by string theory and makes several interesting, testable predictions. Furthermore, -parity violation contributes to neutrino masses, thereby connecting the neutrino sector to the decay of the lightest supersymmetric particle (LSP). This paper analyzes the decays of third generation squark LSPs into a quark and a lepton. In certain cases, the branching ratios into charged leptons reveal information about the neutrino mass hierarchy, a current goal of experimental neutrino physics, as well as the neutrino mixing angle. Furthermore, optimization of leptoquark searches for this scenario is discussed. Using currently available data, the lower bounds on the third generation squarks are computed.

\@footnotetext

zlmarshall@lbl.gov,  ovrut@elcapitan.hep.upenn.edu,  apurves@sas.upenn.edu,  sogee@sas.upenn.edu

I Introduction

The upgrade to the Large Hadron Collider (LHC) will soon be completed, providing us with an exciting opportunity to probe the next energy frontier. Among the many candidates for new physics in that frontier, supersymmetry (SUSY) stands out as a rich and compelling framework. SUSY not only addresses the gauge hierarchy problem, a puzzle that has driven many model building efforts over several decades, but can also speak to other outstanding issues in the standard model (SM). This includes dark matter and a mechanism for radiative electroweak symmetry breaking. As we wait for the next LHC run to begin, the interim is a good period to reconsider the phenomenology of low energy supersymmetric models. Among other things, it is of interest to investigate if they can yield any signals that have not yet been seriously considered, especially in well-motivated alternatives to the -parity conserving minimal supersymmetric standard model (MSSM).

Despite their theoretically pleasing aspects, generic SUSY particle physics models potentially have a serious problem regarding proton decay. This follows from the fact that the most general MSSM superpotential allows for baryon and lepton number violating terms at tree level and, therefore, rapid proton decay. The typical, yet ad hoc, solution is to impose -parity, where is the spin of the particle. This discrete symmetry forbids violation of baryon number () minus lepton number () by one unit. Accepting -parity conservation, however, severely narrows one’s view of the SUSY phenomenological landscape. This is because the lightest supersymmetric particle (LSP) in -parity conserving theories is stable and, therefore, must be neutral due to cosmological considerations.

Perhaps the most appealing candidates for a deeper origin for -parity, models with gauged , are based on the observation that -parity is a discrete subgroup of . In such models, -parity is a good symmetry as long as is. However, once is broken, the number of the field that breaks determines the fate of -parity: an even field leads to automatic -parity conservation (RPC) Mohapatra:1986su (); Krauss:1988zc (); Font:1989ai (); Martin:1992mq () (for more recent studies see  Aulakh:1999cd (); Aulakh:2000sn (); Babu:2008ep (); Feldman:2011ms (); FileviezPerez:2011dg ()) , while an odd field triggers spontaneous -parity violation (RPV) Aulakh:1982yn (); Hayashi:1984rd (); Mohapatra:1986aw (); Masiero:1990uj ()111 See also recent studies of explicit -parity violation assuming minimal flavor violation oai:arXiv.org:0710.3129 (); oai:arXiv.org:1111.1239 (). . Typically, spontaneous -parity violation is safe in the sense that only lepton number violation is generated at tree level, leaving the proton as stable as it would be with RPC.

As one might expect, the approach in these early studies was to introduce a new “Higgs” sector (that is, superfields with a charge) with which to spontaneously break the symmetry. However, the anomaly cancellation conditions provide a subtle, and more minimal, alternative to this approach. Note that the three generations of right-handed neutrino superfields required to cancel these anomalies contain right-handed sneutrinos. Remarkably, the right-handed sneutrinos have the correct quantum numbers to spontaneously break in a phenomenologically acceptable way. Specifically, they are neutral under the SM, carry no baryon number and, of course, have a charge of one. Therefore, anomaly cancellation defines the most minimal extension of the MSSM. This model has exactly the MSSM particle content plus three generations of right-handed neutrino supermultiplets, and it does not require a new Higgs sector. This minimal theory was proposed in FileviezPerez:2008sx (); Barger:2008wn (); FP:2009gr (); Everett:2009vy (), arguing for it’s appeal from a “bottom up” point of view.222 Such a minimal model was outlined as a possible low energy manifestation of grand unified theory (GUT) models in Mohapatra:1986aw (). The same theory was found from a “top down” approach within the context of a class of vacua of heterotic -theory Evans:1986ada (); Lukas:1998yy (); Braun:2005ux (); Braun:2005nv (); Braun:2006ae (); Ambroso:2009jd (). Due to the odd charge of the sneutrino, the minimal model must always spontaneously break -parity. However, because the right-handed sneutrino has no baryon number, it’s vacuum expectation value (VEV) does not introduce proton decay at tree level. In addition, this model has several potentially testable and interesting predictions:

• -parity violation is manifest though lepton number violating operators, which could lead to lepton number violating signatures at the LHC, e.g. FileviezPerez:2012mj (); Perez:2013kla ().

• The existence of two neutral light fermions (sterile neutrinos), in addition to the usual three neutrinos Mohapatra:1986aw (); Ghosh:2010hy (); Barger:2010iv (). These may play a role in cosmology Ghosh:2010hy (); Ade:2013zuv (); Perez:2013kla ().

• A neutral gauge boson, , whose mass is proportional to the soft mass of the right-handed sneutrino. This gauge boson must be at the TeV scale and, therefore, detectable at the LHC.

• The right-handed sneutrino VEV directly links the neutrino sector to lepton number violation by one unit. This generates tree-level Majorana contributions to the neutrino masses.

This last statement is significant, since it specifies the size of the RPV. It follows from the upper bound placed on this contribution by the neutrino masses that the RPV is only relevant for the decay of the LSP, which would otherwise be stable under RPC. All other SUSY processes will effectively be -parity conserving. The last bullet point is also crucial because it relates neutrino masses to collider physics through -parity violation, an exciting synergy. It suggests that one may be able to infer information about the neutrino sector from LSP decays. Finally, it is worthwhile to note that despite RPV, a gravitino LSP, while unstable, may live long enough to be the dark matter of the universe  Borgani:1996ag (); Takayama:2000uz (); Buchmuller:2007ui ().

This model of spontaneous RPV is, therefore, a well-motivated alternative to RPC. As with all SUSY models, its phenomenology will be highly depended on the choice of the LSP333 While the complete model would include a gravitino LSP as the dark matter of the universe, throughout this paper we shall use LSP to refer to the lightest supersymmetric particle relevant for collider physics. . -parity violation plays an important role from this perspective because it allows the LSP to decay. This liberates the LSP to be any superpartner, including those that have color and charge. One example, of this type, is a charged slepton LSP. However, this will decay like a charged Higgs, an element that already exists in the MSSM. Squark LSPs, on the other hand, offer an opportunity for a whole new set of signals since they act as leptoquarks; that is, scalar particles that are pair produced and decay into a quark and a lepton. Among the squarks, the third generation is perhaps the most interesting LSP candidate since these are generally expected to have the lowest masses due to renormalization group effects, e.g. Martin:1997ns (). Furthermore, since the lower generations must be fairly degenerate due to the SUSY flavor problem, they would be produced more readily and, therefore, have stronger bounds. Finally, stops are the most engaging of all the squarks because of their substantial radiative contribution to the Higgs mass and the role they play as a measure of fine-tuning in SUSY; that is, the little hierarchy problem.

Motivated by this discussion, this paper extends the study of our earlier paper Marshall:2014kea (), by analyzing the prompt decays of third generation squark LSPs within the context of a minimal extension of the MSSM. One of the aims of this paper is to highlight the relationship between stop and sbottom LSP decays and the neutrino sector. Especially striking is the fact that one may infer information about the neutrino mass hierarchy from the -parity violating LSP decays. Just as important are the leptoquark signals, which are typically not associated with SUSY. Experimentally, they have not yet been analyzed with data from the latest LHC run. As we will show in this paper, the leptoquark searches that have previously been conducted allow stop LSP masses as low as 420 GeV and sbottom LSP masses as low as 500 GeV.

The rest of this paper is organized as follows. Section II introduces the details of the model as well as specifying the -parity violating sector. The consequences in terms of -parity violation are discussed in Section III and their influence on neutrino masses are illustrated in Section IV. Section V contains the results for both the stops and sbottoms, including lower bounds and the connection between squark decays and the neutrino sector. This connection is explored through a numerical scan, but the results can be understood analytically, an is done in Section VI. Section VI also attempts to frame the results in terms of a bigger picture, investigating how this scenario can be distinguished from scenarios with similar signatures. Finally, Section VII summarizes our results. Throughout this work, many references will be made to technical calculations discussed in Appendix A, making this a potentially important section for the reader. The remaining three Appendices, B, C and D, briefly discuss the chargino sector, the third generation squark sector and the Feynman rules used in the calculations of the squark decays.

Ii The Minimal SUSY B-L and Spontaneous R-parity Violation

There are several possible minimal extensions of the MSSM of the form , characterized by different choices of the two factors. If these are remnants of a GUT theory, such as SO(10), then these possibilities are all physically equivalent, but will be characterized by different kinetic mixing between the two factors. Among these possibilities, as shown in Ovrut:2012wg (), there is a unique choice that will have vanishing kinetic mixing–not only at the GUT scale, but at any lower scale. This choice of factors is , where is the third component of right-handed isospin. The fact that this basis has no kinetic mixing greatly simplifies the present analysis. Therefore, in this paper, we proceed using the specific minimal extension gauge group

 SU(3)C⊗SU(2)L⊗U(1)3R⊗U(1)B−L . (1)

We will comment later in the paper on how our results apply to the other similar extensions. The gauge structure in this case is such that the hypercharge, , is related to the and third component of right-handed isospin charges by

 Y=I3R+B−L2, (2)

analogous to the relationship between the electric charge, hypercharge and third component of left-handed isospin in the SM.

The matter content and its charges is given by three copies of

 Q∼(3,2,0,1/3), uc∼(¯3,1,−1/2,−1/3), dc∼(¯3,1,1/2,−1/3), (3)
 L∼(1,2,0,−1), ec∼(1,1,1/2,1), νc∼(1,1,−1/2,1), (4)

while the MSSM Higgs sector is

 Hu∼(1,2,1/2,0),  Hd∼(1,2,−1/2,0).

The superpotential is similar to that of the MSSM but contains an additional Yukawa coupling to the right-handed neutrino superfield

 W=YuQHuuc−YdQHddc−YeLHdec+YνLHuνc+μHuHd, (5)

where the Yukawa couplings are three-by-three matrices in family space and are in general complex. The soft SUSY breaking Lagrangian is

 −Lsoft=m2~νc|~νc|2+m2~L|~L|2+m2Hu|Hu|2+m2Hd|Hd|2+(MR~W2R+M2~W2+MBL~B′2+M3~g2+aν~LHu~νc+bHuHd+h.c.)+⋯, (6)

where the ellipses refer to terms which also exist in the MSSM and are not crucial here. The fields and are the fermion superpartners of the third component of right-handed isospin, left-handed isospin, and color gauge bosons respectively. The is the soft trilinear analogue of and is, therefore, also a three-by-three matrix in family space. The superpotential and Lagrangian are valid in the energy regime between the GUT scale and the TeV scale. Here we continue by analyzing physics at the TeV scale.

The notation for the VEVs of the fields phenomenologically allowed to acquire sizable VEVs is

 ⟨~νc3⟩≡1√2vR,  ⟨~νi⟩≡1√2vLi,  ⟨H0u⟩≡1√2vu,  ⟨H0d⟩≡1√2vd, (7)

where is the generational index and . The generation of a right-handed sneutrino superfield is not identifiable through its interactions, unlike a left-handed electron neutrino which couples to the electron through the gauge interactions. As a result, there is freedom to rotate the right-handed neutrino fields into any basis and specifically to a basis in which only one generation of right-handed sneutrino acquires a VEV. Here this will be chosen, without loss of generality, to be the third generation. Electroweak symmetry breaking will induce VEVs in the remaining two right-handed sneutrino generations. However, these will be on the order of the neutrino masses and, therefore, are neglibible. Note that is in general complex.

Substituting the VEVs from Eq. (7) into the -term, -term and soft potentials yields

 ⟨VF⟩=12|μ|2v2+14|Yνi3vLi|2v2R+14|Yνi3|2v2uv2R+14|YνijvLi|2v2u−12√2(μYνi3vdvLivR+h.c.), (8) ⟨VD⟩ =g2232(v2u−v2d−|vLi|2)2+g2BL32(v2R−|vLi|2)2+g2R32(v2u−v2d−v2R)2, (9) ⟨Vsoft⟩ =12m2~Li|vLi|2+12m2~νc3v2R+12m2Huv2u+12m2Hdv2d+bvdvu+12√2(aνi3vuvLivR+h.c.), (10)

where repeated generational indices are summed and and are the third component of right-handed isospin, left-handed isospin and gauge couplings respectively.

Equations (8)-(10) can be simplified by considering some general phenomenological features of this model. For example, neutrino masses are roughly proportional to the and parameters and, hence, and . With this in mind, the complete potential energy has the following minimization conditions:

 v2R= −8m2~νc3+g2R(v2u−v2d)g2R+g2BL (11) 18(g22+g2R)v2= −|μ|2+M2Hutan2β−M2Hd1−tan2β (12) 2bsin2β= 2|μ|2+m2Hu+m2Hd (13)
 vLi= vR√2(Y∗νi3μvd−a∗νi3vu)m2~Li−g228(v2u−v2d)−g2BL8v2R (14)

where and

 M2Hu ≡ m2Hu−18g2Rv2R (15) M2Hd ≡ m2Hd+18g2Rv2R . (16)

These conditions necessarily mean that the soft mass of the sneutrino that acquires a VEV, the third generation here, must have a tachyonic soft mass. Radiative mechanisms for achieving such a mass have been discussed in references Ambroso:2009jd (); Ambroso:2009sc (); Ambroso:2010pe ().

Prior to electroweak symmetry breaking, breaking leaves one linear combination of the third component of right-handed isospin and gauge bosons massless– the hypercharge gauge boson. The other linear combination, , becomes massive. Including electroweak symmetry breaking effects, the mass of is

 M2ZR≃14(g2R+g2BL)v2R⎛⎝1+g4R(g2R+g2BL)2v2v2R⎞⎠. (17)

See reference Everett:2009vy () for more details. Current bounds on are at around 2.5 TeV ATLAS:2013jma (); CMS:2013qca ().

Iii R-parity Violation

-parity violation in this model is best parameterized by the two flavorful parameters– and

 ϵi≡1√2Yνi3vR . (18)

The superpotential expanded around the vacuum now contains the -parity violating terms

 W⊃ϵiLiHu−1√2YeivLiH−deci , (19)

which is similar to the so-called bilinear RPV scenario Hirsch:2000ef (). In addition, the Lagrangian contains various other bilinear terms, generated by and , from the super-covariant derivative:

 L⊃−12vL∗i[g2(√2ei~W++νi~W0)−gBLνi~B′]−12vR[−gRνc3~WR+gBLνc3~B′]+h.c. (20)

The results and analysis in the paper will be carried out using the Lagrangian based on Eqs. (19) and (20). However, it is worthwhile to note that it is sometimes useful to rotate away the term in favor of the so-called trilinear -parity violating terms. This is true when comparing to given bounds on various low-energy constraints on RPV, such as lepton number violating processes, and it makes approximating decays widths more straightforward. An example of each of these will be given in this section. Rotating away generates the following terms in the superpotential:

 WTRPV=λijkLiLjeck+λ′ijkQiLjdck, (21)

where is antisymmetric under the interchange of and .444Note that each is an doublet. Hence, is antisymmetric in . This is accomplished by considering as a fourth generation lepton. In this case, the - and -terms can be combined to read , where , , , and . The term can be perturbatively rotated so that only is nonzero. This requires the rotation with

 Rμ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1−ϵ1μ−ϵ2μ−ϵ3μϵ1μ100ϵ2μ010ϵ3μ001⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (22)

Implicit in this is that , which follows from the fact that contributes to neutrino masses, as we shall see later. The rotation leaves only one bilinear between and a linear combination of , which is, of course, mostly composed of . This rotation must also be applied to in the down-type quark Yukawa term, , and the charged lepton Yukawa coupling term, , see Eq. (5). The parameterization of and can be read off from this rotation:

 λijk =12Yeikϵjμ−12Yejkϵiμ (23) λ′ijk =Ydikϵjμ. (24)

Because the charged lepton and down quark Yukawa matrices are dominated by the three-three component which gives mass to the tau lepton and bottom quark respectively, those matrices can be calculated to be and . This means that the largest elements in the trilinear RPV Yukawas are and .

As an application of this rotation, consider the lepton number violating decay . This places the following approximate bound on the trilinear -parity violating couplings Barbier:2004ez ():

 |λ23kλ13k|≲2×10−4(m~ν3100 GeV)−2 . (25)

Using Eq. (23) yields

 ∣∣∣ϵ1ϵ2μ2∣∣∣≲2.5×10−3(m~ν3100 GeV)−2 (26)

as the most stringent constraint. This corresponds to , approximately the upper bound on that keeps perturbative up to the GUT scale. The dependence on is due to the fact that the SUSY Yukawa coupling , where is the tau mass. This is negligible due to the suppression of the lepton Yukawa coupling and the term. One would expect values much lower than this bound due to constraints from neutrino masses, as we shall see later. It is worth noting that contributions to also arise from the term in Eq. (20). However, this is further suppressed due to the -charged lepton mixing, which is proportional to lepton masses. See the approximate value in Eq. (119).

Using Eq. (24), the decay width of the stop LSP into a bottom quark and a charged lepton (henceforth, referred to as a bottom–charged lepton) is given by

 Γ~t1→bℓ+i ∼116πY2b∣∣∣ϵiμ∣∣∣2m~t1, (27)

where indicates the lightest of the two physical stop states (SUSY mass eigenstates are typically numbered from lightest to heaviest). While this neglects order one factors and the contributions from , it is useful for getting an impression of how the stop lifetime depends on the strength of -parity violation. At any rate, it will be shown later that is typically larger than . An order of magnitude approximation for the lifetime can be simply attained from the largest value, denoted , by

 τ~t1∼1×10−14(ϵmax/μ10−5)−2(1001+tan2β)(500 GeVm~t1)seconds. (28)

Taking representative values of GeV and , the lifetimes can be divided up into the following interesting regimes:

• Cosmologically significant ( GeV): The decays of squarks with lifetimes greater than about 100 seconds would disrupt the predictions of big bang nucleosynthesis, see reference Kusakabe:2009jt () for example, and would therefore be ruled out.

• Collider stability (): In this regime, the decay length of the squark is longer than the radius of the LHC detectors, about ten meters in size. Such squarks would hadronize and are referred to as -hadrons. These states would be detectable through their activity in the hadronic calorimeter of the detectors and have been studied in references Raby:1997pb (); Berger:2003kc (); Buckley:2010fj (); Aad:2011yf (); Aad:2012zn (); Aad:2012pra (), for example.

• Displaced vertices (): Squark decays inside an LHC detector with a decay length greater than a millimeter have a large enough displaced vertex from the squark origin to be measured. Such vertices, in a phenomenologically similar scenario, were discussed in Graham:2012th (). Experimentally, some searches for displaced vertices have been performed in references Aad:2011zb (); Aad:2012zx (); Chatrchyan:2012jwg ().

• Prompt decays (): Decays in this case occur at an indistinguishable distance from the collision point at an LHC detector.

The physics associated with non-prompt decays is mostly dependent on the mass of the squark (through its production) and its decay length (displaced vertices or collider stable squarks). Such probes would not be the ideal way of studying the specific branching ratios of the squarks predicted in the model under consideration. In addition such signals have already been analyzed in the references above. We therefore continue this paper considering prompt squark LSP decays only. As we shall see, this will intimately relate the neutrino sector to -parity violation.

The existence of this relationship is already suggested by Eqs. (19) and (20). These RPV bilinear terms mix fields with different -parity number but the same spin and SM quantum numbers. Specifically, the neutrinos now mix with the neutralinos, Eq. (64), the charged leptons mix with the charginos, Eq. (106) and the Higgs fields mix with the sleptons. The neutrino/neutralino mixings are crucial because they generate tree-level Majorana neutrino masses through a seesaw mechanism. As a result of this, the bilinear -parity violating terms cannot be too large. All -parity violating effects will therefore be negligible compared to the -parity conserving effects, except for the LSP, which now decays via RPV.

Since -parity violation simultaneously determines both the neutrino sector and the decays of the LSP, it is possible that some of the information from the neutrino sector will be revealed in the LSP decay. This is an exciting and rare opportunity to relate these two fields.

Iv Neutrino Masses and R-parity Violation

Any model with right-handed neutrinos allows for Dirac neutrino masses through the Yukawa coupling between left- and right-handed neutrinos. In this model, Majorana masses are also possible due to the VEV of the right-handed sneutrino. As mentioned above, only one generation of right-handed sneutrino can attain a significant VEV Mohapatra:1986aw (); Ghosh:2010hy (); Barger:2010iv (). This means that lepton number is only significantly violated (TeV-scale violation) in one generation of the right-handed neutrinos. It is only that generation of right-handed neutrinos that will attain a TeV-scale mass. This gives rise to a system of neutrinos with three layers: a TeV scale right-handed Majorana neutrino, the three active neutrinos and two light sterile neutrinos555 Sterile neutrinos are typically sub-MeV fermions without SM quantum numbers. In this model, their masses must be at or below those of the left-handed, or active, neutrinos since their masses arise from Dirac Yukawa couplings to the left-handed neutrinos. Models with two sterile neutrinos are sometimes called 3+2 models in the literature, where the three represents the active neutrinos. .

Majorana masses for the active neutrinos are generated through an effective type I seesaw mechanism Minkowski:1977sc (); Yanagida (); GellMann:1980vs (); Mohapatra:1979ia () where the seesaw fields include the one heavy right-handed neutrino and the neutralinos. Once the heavy seesaw fields are integrated out, the Majorana contribution to the neutrino mass matrix is

 mνij=AvL∗ivL∗j+B(vL∗iϵj+ϵivL∗j)+Cϵiϵj . (29)

The non-flavored parameters, , and , are the results of integrating out the heavy fields. They, and more details, are given in Appendix A. The Dirac neutrino mass contributions are simply given by the product of the up-type Higgs VEV and the neutrino Yukawa couplings that do not couple to the third generation right-handed neutrino: .

One of the main tools at our disposal for probing the neutrino sector is the observation of neutrino oscillations. Such oscillations between two neutrinos are determined by the amount of mixing between the two neutrinos and their mass difference. In a purely Dirac neutrino case, the active-sterile mixing is maximal but the mass difference is zero and, therefore, no active-sterile oscillations result. Here, in the pure Majorana case, the mass difference is significant but the mixing is negligible. A situation in which both Dirac and Majorana mass contributions are comparable would lead to large active-sterile oscillations which have not been observed and are therefore ruled out, e.g. deGouvea:2009fp (); Antonello:2012pq ().

The question then remains, should this analysis assume that neutrinos receive their masses dominantly from Dirac or Majorana mass terms? Here, already, the connection to -parity becomes important. Prompt LSP decays, which were argued to be of interest in the last section, will allow significant Majorana masses. Since these cannot coexist with significant Dirac masses, neutrinos must receive their masses dominantly from Majorana mass terms. This makes further study of the Majorana mass matrix, Eq. (29), fruitful.

As a first step, it is important to notice that the determinant of the neutrino mass matrix in Eq. (29) is zero. This is a consequence of the flavor structure and is independent of the and parameters. Closer observation reveals that only one eigenstate is massless. This constrains the neutrino masses to be either in the normal hierarchy (NH):

 m1=0

or in the inverted hierarchy (IH):

 m1∼m2∼50 meV>m3=0 (31)

where only the squared mass differences are measured in neutrino oscillation experiments.

The relevant seesaw contributions from and are also informative. For example, the term proportional to in Eq. (29) is a contribution associated with the VEVs of the left-handed sneutrinos. It arises from neutrino-gaugino mixing such as in Eq. (20). The gauginos are naturally Majorana due to their soft masses and, therefore, integrating them out directly leads to Majorana mass terms for the neutrinos. One can therefore conclude that

 A∼1msoft, (32)

where is some combination of gaugino and Higgsino masses. This conclusion can be verified with the full analytic expression for in Appendix A. The parameter , on the other hand, arises through neutrino-Higgsino mixing because of the term. Higgsinos are not Majorana particles before electroweak symmetry breaking and only their electroweak mixings with the gauginos gives them a Majorana nature. Therefore, must include at least two factors of Higgsino-gaugino mixing terms, each of which is proportional to the ratio of an electroweak VEV to :

 C∼v2m3soft. (33)

A similar argument yields that at lowest order. All of these conclusions can be verified with the full expressions in Appendix A.

The neutrino mass matrix is diagonalized by the so-called PMNS matrix:

 VPMNS=⎛⎜⎝c12c13s12c13s13e−iδ−s12c23−c12s23s13eiδc12c23−s12s23s13eiδc13s23s12s23−c12c23s13eiδ−c12s23−s12c23s13eiδc13c23⎞⎟⎠×diag(1,eiα/2,1), (34)

where . There are Majorana phases associated with N Majorana neutrinos. This translates into only one Majorana phase, , in this case because one of the neutrinos is massless and, therefore, does not have a Majorana mass. The CP phase corresponds to the freedom in the three-by-three matrix. In models that predict a massless neutrino, such as the one discussed here, the neutrino masses in terms of the mass squared differences in the normal hierarchy are

 m1=0,m2=√Δm221,m3=√Δm231, (35)

while in the inverted hierarchy one has

 m1=√Δm231,m2=√Δm231+Δm221,m3=0. (36)

The current values for the parameters in (34) and (35), (36) are given in Tortola:2012te (); GonzalezGarcia:2012sz (); Fogli:2012ua (). We use the most recent values nufit () from the collaboration of reference GonzalezGarcia:2012sz (), which at one sigma are given by

 sin2θ12=0.306+0.012−0.012,sin2θ23=0.446+0.007−0.007 \ or% \ 0.587+0.032−0.037,sin2θ13=0.0229+0.0020−0.0019,Δm221(10−5 eV2)=7.45+0.19−0.16,Δm231(10−3 eV2)=2.417+0.013−0.013,δ(∘)=265+56−61. (37)

Note that at three sigma, spans its full range of and that has not been measured. The two values of represent a degeneracy in the best fit to the data.

One can solve for the flavorful parameters and by requiring that the diagonalization of the neutrino mass matrix, Eq. (29), yields the correct neutrino data specified in Eq. (37). A procedure for this is outlined in Appendix A in terms of a new set of variables and , where

 vLi =VPMNSilV∗l, (38) ϵi =V∗PMNSilEl. (39)

These imply that and should be on the order of magnitude of and respectively–where and are the largest of and –since the elements of are mostly of order one. In the normal hierarchy and Eqs. (96), (97), and (98) are used to calculate and in terms of . Together, they imply that , where the coefficients are of order one as long as there are not finely tuned numerical cancellations between terms. The same conclusion holds in the inverted hierarchy. This in turn means that . Based on the approximations made above for , and in Eqs. (32) and (33), it follows that

 |ϵi|∼msoftv|vLi| . (40)

Quantitatively is verified through the scan specified in Table 1, which is used to generate the numerical results in the next section. Indeed, we find that for of the points for all and that the largest value is larger than the largest value () in of the points. Points that do not satisfy these conditions correspond to finely tuned cancellations between terms which, although unlikely, nevertheless arise randomly in the scan. This indicates that typically approximates the amount of -parity violation and that is a good approximation. This will be useful to obtain an analytic understanding of the numerical results.

V Third Generation Squark LSP’s

The previous two sections have reviewed various aspects of the minimal SUSY model, RPV and the neutrino sector. It was shown that there is an interesting region of parameter space where the 1) strength of RPV corresponds to prompt LSP decays and 2) where the LSP decays might reveal information about the neutrino sector. This paper plans to study these properties under the assumption that the LSP is a third generation squark; that is, for both a stop and sbottom LSP. In addition, we will place lower bounds on the masses of these sparticles using current publicly available LHC results.

Squark LSPs are interesting in RPV for various reasons. First, they are not possible in RPC, so this provides an opportunity to look beyond the typical SUSY LSP candidates and beyond the typical SUSY signatures. Specifically, squark LSPs behave like leptoquarks, meaning they are scalar particles that are pair produced and decay into a quark and a lepton. The stops and sbottoms have the following possible decays:

 ~t1→t νi,  or  ~t1→b ℓ+i , (41) ~b1→b νi,  or  ~b1→t ℓ−i , (42)

where and are the lightest physical stop and sbottom respectively.

Colored particles are, furthermore, more abundantly produced at the LHC, so more aggressive bounds can be placed on them. Generally, one expects a third generation squark to be lighter than the first two generations on the basis of the renormalization group equations. However, this only holds true if one starts with fairly degenerate squarks in all three generations at some high scale associated with soft SUSY breaking. From a phenomenological point of view, the first two generation of squarks should be relatively degenerate to avoid large disallowed contributions to flavor physics processes. This is known as the SUSY flavor problem. Light degenerate first and second generation squarks effectively double the expected number of events for a given process and will consequently have stronger bounds. Furthermore, the first two generations have additional contributions to their production cross section due to the presence of light quarks in the proton. This can, once again, increase the number of events. For these reasons, we continue our analysis focusing on the third generation squarks. Some general comments about the branching ratios of the first two generations will be made in the discussion.

Stop LSPs are especially compelling because of the central role they play in SUSY. Before discussing this further, we briefly review some basic stop phenomenology. More details can be found in Appendix C. In the gauge eigenstate basis, the stop sector contains the field, which is the superpartner of the left-handed top and part of the squark doublet . Since it is a scalar, the stop has no actual chiral properties. The stop sector also contains the superpartner of the right-handed top, , which is an singlet. Both have unrelated soft squared masses and are mixed through mass mixing terms. Diagonalization yields the physical stops and , which are traditionally labeled so that . The mass mixing term leads to what is usually referred to as the left-right mixing angle in the stop sector, , with the convention used here that () corresponds to a purely left-handed (right-handed) lightest stop, . A purely left-handed cannot be the LSP because its partner, the left-handed sbottom, will always be lighter. This is because they share the same SUSY-breaking soft mass squared term and both get -term contributions from their SM partner mass squared. That is, the sbottom mass gets a bottom mass squared contribution and the stop gets a top mass squared contribution. Since the top is much heavier than the bottom, the left-handed stop will always be heavier than the left-handed sbottom.

The stops in SUSY are important because they couple most strongly to the Higgs. This means they contribute most to the little hierarchy problem and provide a measure of the fine-tuning required in SUSY models. In RPC, stop decays can involve complicated decay chains with multi-particle final states making determination of the stop mass from the observation of such a decay difficult. As an LSP with -parity violation, stop decays are very clean in the sense that each stop decays to only two particles. Therefore, such decays can be used to deduce the stop mass in a relatively straightforward way. This is especially true for the bottom–charged lepton channel, whose final states are both detectable. Neutrinos, on the other hand, escape the detector as missing energy. As we shall see, typically the bottom–charged lepton channel dominates the stop decays.

The issue of the little hierarchy problem is also strongly linked to the Higgs mass. In SUSY, the Higgs tree-level mass must be less than the mass. This can be increased at the loop level by radiative corrections to the Higgs mass which grow as the logarithms of the stop masses and also increase with stop mixing angle. This leads to a conflict between the heavy stops masses needed to make SUSY compatible with the recent Higgs discovery and the desire to keep the stops light so as to minimize fine-tuning in SUSY. The former seems to be an argument against a stop LSP. However, it is possible that only one stop is quite heavy while the second remains light–which will indeed be the case when the stop mixing angle is relatively large. This translates into an LSP stop that is composed of significant left- and right-handed components. Since the Higgs mass is not altered in this model, one can consult the MSSM literature to explore the possibilities, e.g. Carena:2011aa ().

The stop partial widths into top neutrino and bottom–charged lepton are

 Γ(~t1→tνi) = 116π(|GL~t1tχ06+i|2+|GR~t1tχ06+i|2)m~t1(1−m2tm2~t1) ⎷1−2m2tm2~t1+m4tm4~t1 (43) Γ(~t1→bℓ+i) = 116π(|GL~t1bχ±2+i|2+|GR~t1bχ±2+i|2)m~t1, (44)

where the parameters are the coefficients of the relevant vertices, and . They, as well as more details, can be found in Appendix D. Parametrically, the parameters contain the elements of the matrix that diagonalize the neutrino-neutralino sector and the parameters contain the elements of the matrix that diagonalize the lepton-chargino sector and are, therefore, proportional to some combination of and . Also encoded in the parameters is information about the stop left-right mixing angle, .

Before tackling a numerical study of stop LSP phenomenology, it is instructive to approximate the relative sizes of the different branching ratios. This can be done by perturbatively diagonalizing the neutrino-neutralino and charged lepton-chargino mass matrices, as is done in Appendices A and B and applied in Appenedix D. For ease of comparison, the leading squared amplitudes for the different final states are given in the approximation that . This is a phenomenologically relevant approximation because bounds on are much higher than electroweak gaugino and Higgsino bounds and both are above the electroweak scale itself. We also employ the results of the last section, . The leading contributions to the square of the vertex amplitude, , are then

 |A(~t1→bℓ+i)|2 ∼c2tY2b∣∣∣ϵiμ∣∣∣2 (45) |A(~t1→tνi)|2 ∼⎡⎣18c2t(g22M2−g2BLg2R3M~Y)2+118s2tg4BLg4RM2~Y⎤⎦∣∣∣VPMSNij(vdϵjμ+vL∗j)∣∣∣2, (46)

where () is (), and there is an implicit sum over . The top–neutrino channel is suppressed compared to the bottom–charged lepton channel both by helicity suppression to the term proportional to and suppression by when the lightest stop is not purely right-handed. When the lightest stop is purely right-handed, the leading order bottom–charged lepton amplitude vanishes and the next order term becomes important:

 |A(~t1→bℓ+i)|2∣∣∣θt∼90∘ ∼Y2t∣∣∣mℓivLiμvd∣∣∣2. (47)

This term is suppressed by both and the mass of the charged lepton in the final state, , indicating that, for the mostly right-handed stop, only the top–neutrino and bottom-tau channels are significant. The stop branching ratios, where branching ratio is defined as the partial width normalized to the total width, falls into two regimes of interest depending on the composition of the stop:

• Admixture stop LSP: Stop decays into into bottom–charged leptons dominate, . We therefore approximated the total width as coming completely from the charged leptons, and the decays of the stop can be described by three branching ratios, which must satisfy

 Br(~t1→be+)+Br(~t1→bμ+)+Br(~t1→bτ+)=1. (48)
• Right-handed stop LSP: Only the top–neutrino and bottom-tau channel are significant. We therefore approximate the width as coming completely from these two channels and the decays can be described by two branching ratios, which must satisfy:

 Br(~t1→bτ+)+Br(~t1→tν)=1. (49)

Let us qualitatively understand these results, which may be a bit counterintuitive. Since mixes with , one would expect the leading contributions to be proportional to the , since it couples the stops to and through it to the parameter. However, such decays are helicity suppressed by a factor of (in Eq. (46)) and are, therefore, subdominant. The dominant channel to RPV then usually goes through and, therefore, includes a factor of . This explains Eq. (45). The top–neutrino channel cannot, however, be accessed through and must, therefore, suffer the helicity suppression or be suppressed by , as are the two terms in Eq. (46). The right-handed stop also cannot access . Its decay into bottom–charged lepton must go through mixing and finally through , which is the reason that Eq. (47) depends on the lepton mass.

With these guidelines in mind, we proceed to our numerical study.

v.1 Stop LSP Decays and the Neutrino Spectrum

The numerical procedure starts with the process in Appendix A, which takes as input the unmeasured CP violating phases of the neutrino sector, the neutralino spectrum, the parameters, any one of the parameters, and two signs. It yields values for and the other two that are consistent with neutrino physics. These values are then used to numerically diagonalize the neutrino/neutralino and charged lepton/chargino mass matrices. These rotation matrices are then inputted into the Feynman rules in Appendix D, which can be used in Eqs. (43) and (44) to calculate the partial widths. Because of the dependence on a variety of parameters, full analytic relationships between the input parameters and the stop decay branching ratios are complicated and not very illuminating. However, random scans in the space of the input parameters yield fairly simple behavior.

The parameters of our scan and their ranges are specified in Table 1. As mentioned above, the neutrino sector specifies all but one -parity violating parameter, which we choose to be and we randomly choose the generation, , of to avoid any bias in the scan. The sign factors, and are further discussed in Appendix A. While only the gluino mass range is shown, we use the GUT inspired gaugino mass relation for the gaugino masses Ovrut:2012wg (). This is based on the ratio of the gauge couplings at the TeV scale. The lower ranges on and roughly correspond to the lower bounds on those particles, while roughly corresponds to the mass of one of the physical chargino states. The lower and upper bounds on are based on keeping all Yukawa couplings perturbative to the GUT scale. Meanwhile, the bounds on follow from requiring no fine-tuning in the neutrino sector, the conditions for which are described in Appendix A. This fine-tuning depends on the actual parameter point and we find that non fine-tuned points lie in the range , which is used in the scan.

In addition, the uncertainties on the neutrino parameters themselves can quantitatively alter the results. We, therefore, also scan over the three sigma range of the neutrino parameters based on their values and uncertainties given in Eq. (37). To do this, we need a probability distribution to describe the uncertainty in these parameters. A simple Gaussian will not do, because the uncertainties in some of the neutrino parameters are asymmetric. Instead we randomly select, with probability one half, which side of the central value a parameter will be on. Then a value for that parameter is randomly generated based on a Gaussian distribution whose standard deviation is equal to the 1 uncertainty on the chosen side of that parameter’s central value. The Gaussian distribution is curtailed a distance of three standard deviations away from the central value. No correlations between neutrino parameter ranges are taken into account here. Furthermore, the CP-violating phases, and , are scanned over their full range and the central value of used is randomly chosen between the two ambiguous experimental values.

Since we are studying a stop LSP, points in the scan at which one of the neutralinos or charginos end up being lighter than the stop are rejected. It is also possible that some points in the scan may have a nearly purely left-handed lightest stop, which may be unable to be the LSP (see Appendix C). A criterion for excluding such points from the scan would depend on parameters that do not effect the physics of this paper, so we do not impose it here. Such a criterion would have no impact on the overall trends displayed by our scan, so it would not effect the conclusions of this paper.

We note that due to the extra suppression in the decays of the right-handed stop, Eq. (47), the LSP stop lifetime increases by a significant amount when it approaches a purely right-handed stop composition. Using the scan from Table 1, we plot the decay length of the stop LSP versus stop mixing angle in Fig. 1. The figure shows that for a pure right-handed stop LSP, a significant number of points in the scan yield lifetimes long enough for displaced vertices (decay length greater than a millimeter). We continue our analysis focusing on prompt decays.

Figure 2 shows how , where , depends on the stop mixing angle. This verifies the relationship between the stop mixing angle and branching ratios into bottom–charged lepton and top–neutrino derived from Eqs. (45) - (47). Figures 1 and 2 both show that the right-handed stop-like behavior, significant top–neutrino channel and longer lifetimes, turns on around .

Perhaps the most striking result from this scan is the connection between the stop decays and the neutrino hierarchy. This connection is evident in Fig. 3 where the possible branching ratios are displayed in the - plane and where, for simplici