Single top production via gluon fusion at CERN LHC

CUMQ/HEP 139 Single Top Production via Gluon Fusion at CERN LHC Gad Eilam∗ Technion-Israel Institute of Technology, 32000 Haifa, ISRAEL Mariana Frank† and Ismail Turan‡ arXiv:hep-ph/0601253v3 21 Nov 2006 Department of Physics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, CANADA H4B 1R6 (Dated: November 14, 2018) Abstract We calculate the one-loop flavor violating top quark decay t → cgg in the Minimal Supersymmetric Standard Model. We discuss the branching ratios obtained with minimal flavor violation, as well as with soft-supersymmetry induced general flavor violation. Based on this rate we calculate the cross section for the single top quark production via gluon fusion, gg → tc̄, and evaluate its contribution to the cross section for single top quark production in pp collisions at the Large Hadron Collider. We calculate all contributions coming from the standard model and charged Higgs loops, as well as gluino (and neutralino)-up-type squarks, and chargino-down-type squarks loops. Our numerical results show that the gluino and the chargino contributions are largest over the whole parameter range in the unconstrained Minimal Supersymmetric Standard Model. While in general the gluino contributions dominate the cross section, this result depends on the supersymmetric flavor violating parameters in the up and down squark sector, the relative mass of the gauginos, and whether or not the Grand Unified Theory relationships between gaugino masses are satisfied. In the most promising scenarios, the pp → tc̄ + t̄c + X cross section at the Large Hadron Collider can reach a few hundreds fb. PACS numbers: 12.60.Jv, 11.30.Hv, 14.65.Ha Keywords: Rare Top Decays, Single Top Production, MSSM, Higher-Order Dominance ∗ † ‡ [email protected] [email protected] [email protected] 1 I. INTRODUCTION One of the main goals at the CERN Large Hadron Collider (LHC) is to study the production and decay of top quarks. The importance of studying the physics of the top is obvious. It is the quark which is closest to the scale of electroweak symmetry breaking and is therefore most sensitive to that scale, and thus to New Physics (NP) beyond the Standard Model (SM). One of the important tests of the SM is its predictions for the yield of single tops in hadronic collisions. The measurement of single top production cross sections has turned out to be a challenging task so far [1] and only upper limits are obtained. For instance, the D0 experiment, at Tevatron II with integrated luminosity of 230fb−1 , obtained the following upper limits on the s (t)-channel processes (as defined below): 6.4 (5.0) pb, at 95% C.L. It is expected that increased luminosity and improved methods of analysis will eventually lead to the detection of single top events in Tevatron II and subsequently at the LHC. Single top production in hadronic machines has been thoroughly discussed within the SM where, at lowest order, one has the tree level contributions of s-channel (q q̄ → tb̄ through W exchange), t-channel (ub → td via W exchange) and gb → tW with a top quark exchanged. In [2, 3] one finds the most recent SM results, which include Next to Leading Order (NLO) corrections. These are predicted to be approximately equal to (all the following cross sections are in pb), 6.6 (4.1) for a single t (t̄) production in the s-channel, and 156 (91) for a single t (t̄) production in the t-channel at LHC [2]. The background for single top production in the SM was estimated in [4]. At the same time, there has been an increased interest in studying forbidden or highly suppressed processes as they appear ideal for finding the physics lying beyond the SM. As alluded to before, top quark interactions, in particular, might provide a fertile ground to searches for NP. It is expected that if NP is associated with the mass generation mechanism, it may be more apparent in top quark interactions, rather than in the light fermion sector. Along these lines, there have been suggestions that the Flavor Changing Neutral Currents (FCNC) single top quark production could be rather sensitive to non-SM couplings such as tcV (V = g, γ, Z) and tcH [5]. The advantage in looking for FCNC processes in top physics is that although these exist in the SM, they are minute, leading to tiny, unmeasurable SM effects. In general, any measurable FCNC process involving the top will indicate that one is witnessing the effects of NP. Note that here we are only interested in processes that are 2 driven by FCNC couplings, which are highly suppressed in the SM by the GIM mechanism. Therefore we do not consider NP corrections to SM couplings (like tbW or Zqq) or the contributions of new particles (either external or internal), like Z ′ or W ′ , except those of the supersymmetric partners of SM particles. FCNC effects in top production contribute to the following single top production processes on the partonic level: cg → t, cg → tg, cq(q̄) → tq(q̄), q q̄ → tc̄ and gg → tc̄, as well as all the above with c −→ u. These subprocesses have been investigated in the presence of FCNC effective couplings and in the framework of various NP models [5]. Of all scenarios of physics beyond the SM, supersymmetry is the most popular. A characteristic feature of supersymmetry is that, in addition to the SM FCNC generated by the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix, it can provide large soft supersymmetry-generated FCNC which would enhance rates and cross sections beyond SM values. The proton collider LHC can produce supersymmetric particles, such as squarks and gluinos, with masses up to 3 TeV; as well as potentially lighter ones, such as charginos/neutralinos. Flavor-changing interactions appear in supersymmetry in loops involving these particles, and thus enhancements in FCNC signals are expected at the LHC. Single top quark production generated through FCNC processes has been discussed within the effective Lagrangian formalism in a model independent way [6], as well as in modeldependent scenarios [7]. The purpose of this study is to analyze one such class of rare single quark FCNC production: the gluon fusion gg → tc̄ within the framework of low-energy supersymmetry. This process was analyzed in [8] where QCD-only loops (loops of gluino and squarks), were evaluated in the context of the unconstrained Minimal Supersymmetric Standard Model (MSSM). However it is known from analyses of t → cV that charginos, and sometimes neutralinos, can have a large effect on FCNC. Here we first discuss the rare decay t → cgg and show it to be larger than t → cg over most of the parameter space in certain cases. Then we perform a complete analysis of gg → tc̄ in both the constrained MSSM (where FCNC decays and cross sections are driven by chargino-down-like squark loops) and the unconstrained MSSM (where gluino and neutralino loops contribute as well). We include the SM and charged Higgs contributions, contributions from chargino, neutralinos and gluino loops, as well as interference effects between SM and non-SM effects, in the context of the most general left-left, left-right and right-right intergenerational squark mixings. We also address the observability of these channels at LHC. 3 Our paper is organized as follows: After a description of the FCNC sources in the unconstrained MSSM (Section II), we present our complete analysis of the branching ratio for the top quark t → cgg in MSSM, and compare it to the SM case, where t → cgg was shown to be larger than t → cg [9] (Section III). Section IV is devoted to the calculation of the gluon fusion cross section gg → tc̄, as well as the evaluation of the cross section for pp → tc̄ + X at the LHC through gluon fusion. We include a detailed numerical analysis of the various relative supersymmetric contributions from gluino and chargino loops with or without Grand Unified Theory (GUT) mass relations, in addition to a comparison of the constrained versus the unconstrained MSSM predictions, as well as observability of these channels. Our conclusions and prospects for experimental observations are presented in Section V. II. FCNC IN THE UNCONSTRAINED MSSM In the unconstrained MSSM there are two sources of flavor violation. The first one arises from the different mixing of quarks in the d- and u-sectors in the physical bases, and it is described by the CKM matrix (inherited from the SM). In the minimal version of MSSM (the constrained MSSM) this is the only source of flavor violation. The second source of flavor violation consists of a possible misalignment between the rotations that diagonalize the quark and squark sectors, and it is a characteristic of soft supersymmetry breaking. We work in the most general version of the model and discuss the constrained version as a limit. The superpotential of the MSSM Lagrangian is W = µH 1 H 2 + Ylij H 1 Li ejR + Ydij H 1 Qi djR + Yuij H 2 Qi ujR . (2.1) The part of the soft-SUSY-breaking Lagrangian responsible for the non-minimal squark family mixing is given by Lsquark = −Q̃i† (MQ̃2 )ij Q̃j − ũi† (MŨ2 )ij ũj − d˜i† (MD̃2 )ij d˜j soft 1˜ 2 i ij +Yui Aij u Q̃i H ũj + Yd Ad Q̃i H dj . (2.2) In the above expressions Q is the SU(2) scalar doublet, u, d are the up- and down-quark SU(2) singlets (Q̃, ũ, d˜ represent scalar quarks), respectively, Yu,d are the Yukawa couplings and i, j are generation indices. The flavor-changing effects come from the non-diagonal entries in the bilinear terms MQ̃2 , MŨ2 , and MD̃2 , and from the trilinear terms Au and Ad . 4 Here H 1,2 represent two SU(2) Higgs doublets with vacuum expectation values         v cos β v 1 √ √ 0 0 ≡ , hH 1 i =  2  ≡  2  , hH 2i =  (2.3) v√ sin β v2 √ 0 0 2 2 √ where v = ( 2GF )−1/2 = 246 GeV, and the angle β is defined by tan β ≡ v2 /v1 , the ratio of the vacuum expectation values of the two Higgs doublets and µ is the Higgs mixing parameter. Since we are concerned with top quark physics, we assume that the non-CKM squark mixing is significant only for transitions between the squarks of the second and third generations. These mixings are expected to be the largest in Grand Unified Models and are also experimentally the least constrained. The most stringent bounds on these transitions come from b → sγ. In contrast, there exist strong experimental bounds involving the first squark generation, based on data from K 0 –K̄ 0 and D 0 –D̄ 0 mixing [11]. It is convenient to specify the squark mass matrices in the super-CKM basis, in which the mass matrices of the quark fields are diagonalized by rotating the superfields. Our parameterization of the flavor-non-diagonal squark mass matrices for the up- and downtype squarks, for the MSSM with real parameters, reads as follows,   2 0 0 mu Au 0 0   ML̃u   2 2 2  0 ML̃c (MŨ )LL 0 mc Ac (MŨ )LR      2 2 2  0 (M ) M 0 (M ) m A LL RL t t  Ũ L̃t Ũ 2 , Mũ =    2  mu Au  0 0 MR̃u 0 0     2 2 2  0 mc Ac (MŨ )RL 0 MR̃c (MŨ )RR    2 0 (MŨ2 )LR mt At 0 (MŨ2 )RR MR̃t   2 M 0 0 m A 0 0 d d L̃d     2 2 2  0 ML̃s (MD̃ )LL 0 ms As (MD̃ )LR      2 2 2  0 (MD̃ )LL ML̃b 0 (MD̃ )RL mb Ab  2 , Md˜ =    2   md Ad 0 0 MR̃d 0 0     2 2 2  0 ms As (MD̃ )RL 0 MR̃s (MD̃ )RR    2 0 (MD̃2 )LR mb Ab 0 (MD̃2 )RR MR̃b where 2 2 ML̃q = MQ̃,q + m2q + cos 2β(Tq − Qq s2W )m2Z , 5 (2.4) (2.5) 2 MR̃{u,c,t} = MŨ2 ,{u,c,t} + m2u,c,t + cos 2βQt s2W m2Z , 2 2 MR̃{d,s,b} = MD̃,{d,s,b} + m2d,s,b + cos 2βQb s2W m2Z , (2.6) Au,c,t = Au,c,t − µ cot β , Ad,s,b = Ad,s,b − µ tan β , with mq , Tq , Qq the mass, isospin, and electric charge of the quark q, mZ the Z-boson mass, sW ≡ sin θW and θW the electroweak mixing angle. In the above matrices we assumed that significant mixing occurs between the second and third generations only. We define the dimensionless flavor-changing parameters 23 (δU,D )AB (AB = LL, LR, RL, RR) from the flavor off-diagonal elements of the squark mass matrices Eqs. (2.4) and (2.5) in the following way. To simplify the calculation we assume that all diagonal entries in (MŨ2 )LL , (MŨ2 )LR , (MŨ2 )RL and (MŨ2 )RR and similarly for (MD̃2 )AB , are 2 set equal to the common value MSUSY , and then we normalize the off-diagonal elements to 2 MSUSY [12, 13], (δUij )LL (MŨ2 )ij LL = , 2 MSUSY (δUij )RR (MŨ2 )ij RR , = 2 MSUSY ij (δD )RR (MD̃2 )ij RR = 2 MSUSY (δUij )LR (MŨ2 )ij LR = , 2 MSUSY ij (δD )LR (MD̃2 )ij LR = 2 MSUSY (δUij )RL (MŨ2 )ij RL = , 2 MSUSY ij (δD )RL (MD̃2 )ij RL = 2 MSUSY ij (δD )LL (MD̃2 )ij LL = 2 MSUSY (i 6= j, i, j = 2, 3). (2.7) The matrix M2ũ can further be diagonalized by an additional 6 × 6 unitary matrix ΓU to give the up squark mass eigenvalues M2ũ
diag = Γ†U M2ũ ΓU . (2.8) For the down squark mass matrix, we also can define M2d˜ as the similar form of Eq. (2.8) with the replacement of (MŨ2 )AB (A, B = L, R) by (MD̃2 )AB . Note that while SU(2)L gauge † invariance implies that (MŨ2 )LL = KCKM (MD̃2 )LL KCKM , the matrices (MŨ2 )LL and (MD̃2 )LL are correlated. Since the connecting equations are rather complicated and contain several ij unknown parameters, we proceed by including the flavor changing parameters (δU,D )AB as independent quantities, while restricting them using previously set bounds [11]. 6 Thus, in the super-CKM basis, there are potentially new sources of flavor-changing neutral currents: Chargino-quark-squark couplings, neutralino-quark-squark coupling and gluinoquark-squark coupling, which arise from the off-diagonal elements of (MŨ2 ,D̃ )LL , (MŨ2 ,D̃ )LR and (MŨ2 ,D̃ )RR . Previous considerations of flavor violating decays [14] in the MSSM have shown that both up and down squarks contribute significantly. Our analysis shows that this is the case here too, and which one is dominant depends on the parameters of the model, and in particular on the relative mass hierarchy between the chargino and the gluino. In the super-CKM basis, the quark-up squark-gluino (g̃) interaction is given by Luũg̃ = 3 X √ i=1   2 gs Tstr ūsi (ΓU )ia PL g̃ r ũta − ūsi (ΓU )(i+3)a PR g̃ r ũta + H.c. , (2.9) where T r are the SU(3)c generators, PL,R ≡ (1 ∓ γ5 )/2, i = 1, 2, 3 is the generation index, a = 1, . . . , 6 is the scalar quark index, and s, t are color indices. In the gluino interaction, the flavor changing effects from soft broken terms MQ̃2 , MŨ2 and Au on the observables are introduced through the matrix ΓU . The relevant Lagrangian terms for the quark-down squark-chargino (χ̃± σ ) interaction are given by 2 X 3 n X ∗ ja ˜ + ja ˜ Lud˜χ̃+ = ūi [Vσ2 (Yudiag KCKM )ij ] PL χ̃+ σ (ΓD ) da − ūi [g Uσ1 (KCKM )ij ] PR χ̃σ (ΓD ) da σ=1 i,j=1 o (j+3)a ˜ + H.c. + ūi [Uσ2 (KCKM Yddiag )ij ] PR χ̃+ (Γ ) d D a σ (2.10) diag where the index σ refers to chargino mass eigenstates. Yu,d are the diagonal up- and down- quark Yukawa couplings, and V , U are the usual chargino rotation matrices defined by U ∗ Mχ̃+ V −1 = diag(mχ̃+1 , mχ̃+2 ). The flavor changing effects arise from both the off-diagonal elements in the CKM matrix KCKM and from the soft supersymmetry breaking terms in ΓD . Finally, the relevant Lagrangian terms for the quark-up squark neutralino (χ̃0n ) interaction are Luũχ̃0 3  4 X X ∗ 4 g ∗ √ tan θW PL χ̃0n (ΓU )(i+3)a ũa − ūi Nn4 ūi Nn1 = Yudiag PL χ̃0n (ΓU )ia ũa 3 2 n=1 i=1    1 g 0 ia diag 0 (i+3)a ũa , − ūi √ Nn2 + Nn1 tan θW PR χ̃n (ΓU ) ũa − ūi Nn4 Yu PR χ̃n (ΓU ) 3 2 (2.11) 7 where N is the 4 × 4 rotation matrix which diagonalizes the neutralino mass matrix Mχ̃0 , N ∗ Mχ̃0 N −1 = diag(mχ̃01 , mχ̃02 , mχ̃03 , mχ̃04 ). As in the gluino case, FCNC terms arise only from supersymmetric parameters in ΓU . Most of the previous analyses of FCNC processes in the MSSM concentrated on the mass insertion approximation [15]. In this formalism, the (δ) terms represent mixing between chirality states of different squarks, and it is possible to compute the contributions of the first order flavor changing mass insertions perturbatively, if one assumes smallness of the intergenerational mixing elements (δ’s) when compared with the diagonal elements. However, when the off-diagonal elements in the squark mass matrix become large, the mass insertion approximation is no longer valid [12, 13]. In the general mass eigenstate formalism, the mass matrix in Eq. (2.8) (and the similar one in the down-sector) is diagonalized and the flavor changing parameters enter into our expressions through the matrix ΓU,D . So, in the rare top decays t → cgg, the new flavor changing neutral currents show themselves in both gluino-squark-quark and neutralino-squark-quark couplings in the up-type squark loops and in the chargino-squark-quark coupling in the down-type squark loops. Therefore here, as in our previous work [10], we use the general mass eigenstate formalism as described above. III. t → cgg VERSUS t → cg IN MSSM We present here the comparative analysis of the rare two and three body top quark decays, t → cgg and t → cg, closely following the discussion in our earlier paper [9]. There, we have shown that, within the SM framework, the branching ratio of t → cgg is about two orders of magnitude larger than that of t → cg in SM, a phenomenon which can be dubbed ”higher order dominance”, and which was revealed e.g., in b and c-physics in the past. For the detailed discussion, see [9] and the relevant references therein. Even though the branching ratio for t → cgg dominates the one for the two body decay t → cg, it is of the order of 10−9 and still too small to be detected in collider experiments. Any experimental signal for such decay would indicate physics beyond the SM. So, our aim in this section is to extend the discussion in [9] to a favorable beyond SM framework in which we would expect larger contributions due to extra sources of FCNC – the unconstrained MSSM. Note that we include the SM contributions as well in our calculations. The one-loop Feynman diagrams contributing to t → cgg in the MSSM are given in a 8 set of diagrams Figs. 1, 2, 3, 4, and 5 in the ’t Hooft-Feynman gauge (ξ = 1) representing gluino, chargino, neutralino, Higgs, and ghost contributions, respectively.1 t t u˜ a g g g̃ u˜ a g g u˜ a g̃ g g̃ g g 7 g t g̃ u˜ a u˜ a g̃ c g c 13 g g t g̃ c g̃ 19 c 20 u˜ a c g̃ u˜ a g u˜ a g̃ c 12 t g̃ c g u˜ a u˜ a t g̃ c t c 21 22 u˜ a g t c c u˜ a c g g̃ u˜ a c 18 t g g̃ g c g u˜ a t t g g̃ g u˜ a g 17 c g g c g 16 t g g̃ g c g g t u˜ a c t g̃ t t u˜ a g̃ 11 g t t t g̃ c c 15 g u˜ a c g t u˜ a t u˜ a 14 g 6 u˜ a 10 g̃ g g u˜ a g u˜ a u˜ a g̃ t c u˜ a t u˜ a g g̃ 9 g̃ c u˜ a 5 g̃ t g̃ g g c u˜ a 8 u˜ a g t c 4 c u˜ a t t g̃ g c t t g c u˜ a g̃ g̃ u˜ a u˜ a g g̃ c g̃ t u˜ a 3 c g̃ g̃ c g u˜ a g g̃ 2 u˜ a g g g t t c 1 g g g t u˜ a g c g t g̃ We did not g g̃ c g̃ 23 t 24 t g̃ c g c 25 FIG. 1: The one-loop gluino contributions to gg → tc̄ in the unconstrained MSSM in the ’t Hooft-Feynman gauge. show the SM diagrams here (since they appear in [9]) but we took them into account in the numerical evaluation, for both the decays and the production mode. As in [9], we choose to use the ’t Hooft-Feynman gauge in which the gluon polarization P sum is λ ǫ∗µ (k, λ)ǫν (k, λ) = −gµν . In order to obey unitarity, this simple choice results in the existence of QCD ghost fields whose contributions are shown in Fig. 5. We closely follow the method outlined in [9] and references therein for handling the ghost diagrams. 1 Note that we display the one-loop diagrams for the process gg → tc̄. The diagrams for the decay can be easily obtained by crossing. 9 t d˜ a g χ˜ i d˜ a g g g c g t χ˜ i d˜ a d˜ a g g g c χ˜ i 8 d˜ a t t c t 13 14 χ˜ i χ˜ i g d˜ a t d˜ a g d˜ a t g c c χ˜ i 11 c 12 t χ˜ i c c t g c g χ˜ i c d˜ a χ˜ i t t c g t c 6 t 10 c g c g g g g d˜ a t g t g 9 g c d˜ a c t g χ˜ i c 5 d˜ a c t d˜ a d˜ a g d˜ a g g t g t χ˜ i t g d˜ a c d˜ a c 4 t χ˜ i d˜ a c c d˜ a 3 d˜ a 7 χ˜ i c g c χ˜ i d˜ a χ˜ i d˜ a 2 d˜ a g d˜ a t d˜ a g g t t χ˜ i 1 g g t d˜ a g c 15 16 FIG. 2: The one-loop chargino contributions to gg → tc̄ in the unconstrained MSSM in the ’t Hooft-Feynman gauge. t u˜ a g 0 χ˜ i u˜ a g g g 0 χ˜ i t 0 χ˜ i u˜ a u˜ a g g c g g t g g c u˜ a 13 g t 14 c c u˜ a c g g g g 0 χ˜ i 0 χ˜ i u˜ a t t c g 12 u˜ a c 0 χ˜ i t 0 χ˜ i g 15 t g u˜ a 11 c c c c c g g c t 0 χ˜ i g t 0 χ˜ i u˜ a 6 t ˜0 cχ i t g u˜ a t 10 u˜ a t t c u˜ a c 9 g g 0 χ˜ i u˜ a c t 5 t 0 χ˜ i u˜ a t g g g g t 0 χ˜ i 4 t 8 c c u˜ a 3 u˜ a u˜ a 7 0 χ˜ i g u˜ a c u˜ a c t t c g c 0 χ˜ i u˜ a 0 χ˜ i u˜ a 2 u˜ a g u˜ a u˜ a g 1 g t t c g g t u˜ a c 16 FIG. 3: The one-loop neutralino contributions to gg → tc̄ in the unconstrained MSSM in the ’t Hooft-Feynman gauge. Divergences inherent in the t → cgg calculation are ultraviolet, infrared, and collinear types [9]. In numerical evaluations, we used the softwares FeynArts, FormCalc, and LoopTools [16] to obtain our results. In addition to these, HadCalc [17] is used for deriving the pp process corresponding to the gg fusion discussed in the next section. Using 10 g t di g g g di t H g g t di c g c di 1 t di c 2 di di t H g H c g g c di g t di 7 t c g t H t H c H t c g di 10 c di H g c t c g c 9 H t di di g 8 g c c g t t g di g 6 t g H 5 t H c di c 4 t g H g g g t di c di 3 g t di H g g di c t t di c H di H g 11 12 t g t c H c g g t di c c di 13 di g c 14 15 FIG. 4: The one-loop charged Higgs contributions to gg → tc̄ in the unconstrained MSSM in the ’t Hooft-Feynman gauge. t ug t di g H ug ug u˜ a t g ug 0 χ˜ i 7 g t g ug g̃ u˜ a c ug 8 d˜ a c ug ug t g ug χ˜ i d˜ a ug 4 d˜ a c 9 ug g ug t g ug c t c di u˜ a c ug g H c di 6 t ˜0 cχ i ug 10 ug 5 H c t t χ˜ i c 3 t u˜ a c u˜ a ug t d˜ a g ug g̃ c 2 t u˜ a g ug u˜ a c 1 ug 0 χ˜ i ug c t u˜ a g ug g̃ ug di t g̃ g u˜ a c ug g t g̃ c ug 11 12 t c χ˜ i ug 13 FIG. 5: The one-loop QCD ghost contributions to gg → tc̄ in the unconstrained MSSM in the ’t Hooft-Feynman gauge. utilities offered by FormCalc, we checked ultraviolet finiteness of our results numerically, and introduced phase space cuts to avoid infrared and collinear singularities.2 2 These cuts lead to some uncertainties in our results. A more precise approach requires full consideration of the next-to-leading order corrections to t → cg, similar to the ones in b decays [18]. 11 Having mentioned some qualitative features of the decay t → cgg, we do not present here most of the analytical intermediate results. We do this since the calculations are lengthy and uninspiring. Furthermore, we use well known programs.3 We have also checked our calculations with similar ones, whenever published, as we discuss in the next section. We express the matrix element squared |M|2 as a sum over the various contributions. These include the SM contribution as given in our previous work [9]. From Figs. 1, 2, 3, 4, and 5, we obtain expressions for the following non-SM terms: the gluino contribution, chargino, neutralino, charged Higgs and finally the contribution of the ghosts. The results were expressed in terms of Passarino-Veltman functions [19]. Numerical evaluations of these functions have been carried out with LoopTools, which does not require reduction of Passarino-Veltman functions to the scalars A0 , B0 , C0 and D0 . The analytical expressions are obtained with the use of FeynCalc [20]. The partial width dΓ for the decay t → cgg is given as dΓ(t → cgg) = 1 X |M|2dΦ3 (k1 ; k2 , k3 , k4) 2mt spins d 3 k3 d 3 k4 d 3 k2 (2π)4 δ (4) (k1 − k2 − k3 − k4 ), dΦ3 (k1 ; k2 , k3 , k4 ) = 0 0 0 3 3 3 (2π) 2k2 (2π) 2k3 (2π) 2k4 (3.1) where k1 (k2 ) is the momentum of the top( charm) quark and k3 , k4 the momenta of the gluon pair. The volume element can further be expressed as 1 dΦ3 (k1 ; k2 , k3 , k4) = 32π 3 Z (k30 )max (k30 )min dk30 Z (k20 )max (k20 )min dk20 , (3.2) where the limits are (k20 )min = (k20 )max = (k30 )min = (k30 )max =  σ − |k3 | , Max Cmt , 2 σ + |k3 | (1 − 2C), 2 Cmt , mt (1 − 2C), 2  (3.3) with σ = mt − k30 . In addition, C is the cutoff parameter, chosen nonzero to avoid infrared and collinear singularities [9]. For the numerical calculations in the rest of our study we fix 3 The complete analytical results can be obtained by contacting one of us (I.T.) 12 C = 0.1, which is large enough to be able to reach the jet energy resolution sensitivity of the LHC detector. The results are sensitive to the choice of the C parameter; we find that by decreasing C to 0.01, BR(t → cgg) can increase by a factor of 2-4. The total decay width of the top quark is taken to be Γt = 1.55 GeV. The parameters used in our numerical evaluation are given in Table I. TABLE I: The parameters used in the numerical calculation. αs (mt ) α(mt ) sin θW (mt ) mc (mt ) mb (mt ) mt (mt ) 0.106829 0.007544 0.22 0.63 GeV 2.85 GeV 174.3 GeV The MSSM parameters MSUSY , M2 , mA0 , µ, A, and tan β are chosen as free for the constrained MSSM and the SUSY-GUT mass relations are assumed.4 (This is the first scenario we consider). Inclusion of the flavor violating parameters δ’s among second and third generation squarks (the unconstrained MSSM) adds eight more free parameters. Imposing SUSY-GUT relations favors a heavy gluino, which decreases the gluino contributions for both processes under consideration, t → cg(g) and gg → tc̄, and which enhances chargino contributions, since the lightest chargino becomes much lighter than gluino. As a second scenario we consider the constrained and unconstrained MSSM without imposing SUSY-GUT relations. In this case, we run the U(1) gaugino mass parameter M1 and the gluino mass Mg̃ separately.5 Thus the two scenarios we concentrate on are MSSM with, and MSSM without, SUSY-GUT relations. Given the still large number of parameters in either of these scenarios, the parameter space needs to be reduced by making further assumptions. So, for simplicity, we assume that the soft SUSY-breaking parameters in the squark sector are set to the common value MSUSY . In addition to this, the trilinear linear terms Aui and Adi are chosen to be real and equal to each other and µ is also taken to be real and positive. 4 5 The existence of a GUT theory at Planck scale leads to relations among gaugino mass parameters of the form M1 = (5s2W /3c2W )M2 = (5α/3c2W αs )mg̃ where α and αs are running coupling constants. We still keep the relation between M1 and M2 , rather than fixing them independently, since this does not affect significantly the final results. 13 In the case of flavor violating MSSM, only the mixing between the second and the third generations is turned on, and the dimensionless parameters δ’s run over as much of the interval (0,1) as allowed.6 The allowed upper limits of δ’s are constrained by the requirement that mũi ,d˜i > 0 and consistent with the experimental lower bounds (depending on the chosen values of MSUSY , A, tan β, and µ). We assume a lower bound of 96 GeV for all up squark masses and 90 GeV for the down squark masses [22]. The Higgs masses are calculated with FeynHiggs [21], with the requirement that the lightest neutral Higgs mass is larger than 114 GeV. Other experimental bounds included are [22]: 96 GeV for the lightest chargino, 46 GeV the lightest neutralino, and 195 GeV for the gluino. Throughout the paper, only mA0 and A are fixed globally in the decay and production separately, mA0 = 400 GeV and A = 620 GeV in the decay process t → cgg (and t → cg as well) and mA0 = 500 GeV and A = 400 GeV, respectively, in the single top production process gg → tc̄. 10−5 10−6 t → cg t → cgg 10−6 t → cg t → cgg 10−7 10−7 10−8 BR BR 10−8 10−9 10−9 10−10 10−10 10−11 10−11 10−12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−12 0 0.1 0.2 23 ) (δD LL FIG. 6: Left panel: 0.3 0.4 23 ) (δD LR 0.5 = 0.6 0.7 0.8 0.9 1 23 ) (δD RL Branching ratios of t → cgg and t → cg decays as functions 23 ) of (δD LL with the assumption that GUT relations hold. Right panel: 23 ) 23 ratios as functions of (δD LR = (δD )RL under the same conditions. Branching The parameters are chosen as tan β = 10, MSUSY = 300 GeV, M2 = µ = 200 GeV. The rest of the section is devoted to the presentation of our results for the three body decay t → cgg and the comparison with the two body channel t → cg, both within the MSSM framework. Since the flavor violating parameters δ’s play very important role in 6 Even though δ’s are allowed to be negative, we run them in the positive region. 14 both decays (both are flavor-violating rare top decay channels), we vary them by keeping only a single flavor off-diagonal element non-zero unless otherwise stated. In this section, tan β = 10 is chosen in all figures except for Fig. 8, where the dependence of the BR’s on tan β are shown. Furthermore, the common SUSY scale MSUSY = 300 GeV; M2 = 200 GeV, and µ = 200 GeV are chosen and fixed globally in this section. Since we are only interested in the relative size of the BR(t → cgg) with respect to BR(t → cg), we consider the scenario of MSSM with GUT gaugino mass relations for illustration purposes, and present the case without GUT mass relations in one figure at the end of the section, namely Fig. 9. Fig. 6 shows the branching ratios of the decays t → cgg and t → cg as functions of 23 23 23 (δD )LL on the left panel, and as functions of (δD )LR = (δD )RL on the right panel. Since the flavor off-diagonal δ’s in the up sector are switched off, these figures show chargino-only contributions. As seen from the panels, BR(t → cgg) is almost two orders of magnitude 23 larger than BR(t → cg) in most of the parameter space, and especially for small δD , up to 23 23 δD ∼ 0.4. As δD ’s become larger, BR(t → cg) increases rapidly and becomes larger than 23 23 BR(t → cgg) for (δD )LL ≥ 0.6 for left-panel and for (δD )LR ≥ 0.8 for the right panel. The 23 maximum value reached is around 10−7 for non-zero (δD )LL and 10−8 for the special case 23 23 (δD )LR = (δD )RL . (Note that t → cg can get even larger in this part of the phase space). These two figures demonstrate explicitly that t → cgg is larger than t → cg over most of the parameter space. We have checked the dependence of BR(t → cgg) and BR(t → cg) 23 on (δD )RR and observed that BR(t → cgg) remains two orders of magnitude larger than 23 BR(t → cg) for the most part of the interval, while the sensitivity to (δD )RR variations is not as pronounced as in the (depicted) LL and LR, RL cases. In this case, BR(t → cg) can reach a few times 10−9 . In Fig. 7, the (δU23 )LL and (δU23 )RR dependence of the branching ratios of t → cgg and t → cg decays are shown on the left and right panels, respectively. Since the GUT relations are assumed to hold, the gluino mass is rather heavy, about 600 GeV, when M2 is chosen as 200 GeV. The two orders of magnitude difference between the BR’s for the flavor conserving MSSM disappear once we introduce a small flavor violation (∼ 0.1) between the second and third generations in the up squark sector, which holds for either LL or RR case. The branching ratio of t → cg exceeds that of t → cgg for δU23 ≥ 0.1. The maximum attainable branching ratio for t → cg is around 10−7 , and for t → cgg, 10−8 − 10−7 which represents two orders of magnitude enhancement for t → cgg, and more than 4 orders of magnitude 15 10−6 10−6 t → cg t → cgg 10−7 10−7 10−8 10−8 BR BR t → cg t → cgg 10−9 10−9 10−10 10−10 10−11 10−11 10−12 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 10−12 0 0.05 0.1 0.15 23 ) (δU LL FIG. 7: Left panel: 0.2 0.25 0.3 0.35 0.4 0.45 0.5 23 ) (δU RR Branching ratios of t → cgg and t → cg decays as functions of (δU23 )LL with the assumption that GUT relations hold. Right panel: ratios as functions of (δU23 )RR under the same conditions. Branching The parameters are chosen as tan β = 10, MSUSY = 300 GeV, M2 = µ = 200 GeV. enhancement for t → cg, with respect to the constrained case. The case of (δU23 )LR = (δU23 )RL is very similar to the case with non-zero (δU23 )LL (left panel) or (δU23 )RR (right panel). Fig. 8 shows the tan β dependence of the decays with zero flavor off-diagonal parameters δ = 0 for MSUSY = 300 GeV, M2 = µ = 200 GeV. For the decay t → cgg, the SUSY contribution comes from the chargino sector in the MSSM (there are no gluino or neutralino contributions.) Overall the SM contribution dominates over the MSSM one and the tan β dependence is insignificant, as expected, since the constrained MSSM gives smaller contributions than the SM to FCNC decays at one-loop level. There is a mild dependence on tan β for t → cg decay in the very large tan β region (≥ 25). In addition to that, we an- alyzed the case with non-zero δ’s as well and, for example, for (δU23 )LL = 0.4, we obtain BR(t → cgg) almost two orders of magnitude larger than BR(t → cg) in the entire tan β interval considered. The last figure of the section, Fig. 9, presents the dependence of the branching ratios on the SUSY flavor-violating parameters in the MSSM without SUSY-GUT relations. For illustration, we present the (δU23 )LL dependence of the BR’s for the gluino mass mg̃ = 200 GeV on the left panel, and for mg̃ = 300 GeV on the right panel. The other parameters are chosen the same as before, MSUSY = 300 GeV, M2 = µ = 200. As seen from the figure, 16 10−9 BR 10−10 10−11 t → cg t → cgg 10−12 0 5 10 15 20 25 30 35 40 45 50 tan β FIG. 8: The branching ratios of t → cgg and t → cg decays as functions of tan β with the assumption that GUT relations hold. It is further assumed that all flavor off-diagonal parameters δ’s are zero in both the up and down sectors (constrained MSSM). The other parameters are chosen as MSUSY = 300 GeV, M2 = µ = 200 GeV. the relative difference between the decays not only disappears immediately after switching (δU23 )LL on (more precisely, for (δU23 )LL ≥ 0.01) but also t → cg exceeds t → cgg with a constant factor of 5. This is a gluino dominated case which favors the two-body decay t → cg over the three body decay. The decay t → cg can get as large as 10−5 for mg̃ = 200 GeV and 10−6 for mg̃ = 300 GeV. From the analysis in this section, it is fair to say that the branching ratio for the three body t → cgg decay dominates largely over the one for the two body t → cg mode for the flavor conserving MSSM scenario with SUSY-GUT relations, and remains larger even if non-zero flavor off-diagonal parameters in the down squark sector are turned on. Such dominance is valid only for relatively small flavor violating parameter in the up squark sector ((δU23 )LL < 0.1). Our results here show that the t → cgg channel gives a larger contribution (and may be easier to access) than t → cg channel over most of the parameter space if the flavor violation originated from the down squark sector. The predictions of the constrained MSSM (without intergenerational squark mixings) are similar to the SM ones. Thus the existence of such SUSY FCNC mixings, directly related to the SUSY breaking mechanism, is crucial for the enhancement of the branching ratios. Another motivation for considering t → cgg is the issue of single top quark production, 17 10−4 10−5 10−5 10−6 10−6 10−7 10−7 BR BR 10−8 10−8 10−9 10−9 10−10 10−10 10−11 10−12 10−11 t → cg t → cgg 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 23 ) (δU LL 10−12 t → cg t → cgg 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 23 ) (δU LL FIG. 9: The branching ratios of t → cgg and t → cg decays as functions of (δU23 )LL without GUT relations for mg̃ = 200 GeV, on the left panel, and mg̃ = 300 GeV, on the right panel. The parameters are chosen as tan β = 10, MSUSY = 300 GeV, M2 = µ = 200 GeV. which is one of today’s challenging task at colliders. If t → cgg is a promising channel with respect to t → cg 7 , the next question would be what are the consequences of this for the single top quark searches at colliders. For this purpose, gg → tc̄ + t̄c needs to be considered. Gluons will become very important and abundant at the LHC, which reaches very high energies. Therefore, the rest of the paper is devoted to the consideration of the pp → tc̄ + t̄c + X cross section at LHC, within the flavor-violating MSSM, by assuming only the gluon fusion contribution at partonic level. IV. pp → tc̄ + t̄c + X AT LHC Having discussed the decay mode t → cgg and shown that it is a more promising signal than t → cg in the previous section, we consider here the top-charm associated production via gluon fusion gg → tc̄ + t̄c at the partonic level. Since, at the LHC, TeV or even higherscale energies are going to be probed, gluons inside the proton will become very important. This process, as well as other channels involving light quarks, has been considered by Liu et. al [8] in the unconstrained MSSM driven by SUSY-QCD contributions only. Their results 7 The observability of t → cgg at LHC will be briefly discussed at the end of the Section IV. 18 show clearly that tc̄ production through gluon fusion is the dominant channel over the ones involving light quarks q q̄ ′ , q, q ′ = u, c, d, s. For example, 87% of the total hadronic cross section σ(pp → tc̄+ t̄c+X) comes from the partonic channel gg → tc̄+ t̄c for (δU23 )LL,RR = 0.7 [8]. We agree with their results presented in [8] once we make the required modifications to the input parameters. Here, we present the complete calculation of the hadronic cross section σ(pp → tc̄+ t̄c+X) at LHC by including all one-loop contributions. In addition to the gluino, the chargino, neutralino, and charged Higgs loops as well as the SM part is included. The full set of Feynman diagrams contributing to the process at one-loop level through gluino, chargino, neutralino, and Higgs loops is given respectively in Figs. 1, 2, 3, and 4 in the ’t HooftFeynman gauge. As mentioned in the previous section, we did not display here the SM diagrams available in our previous paper [9] for the t → cgg decay case. Note that, as mentioned before, working in the ’t Hooft-Feynman gauge for this process requires the inclusion of QCD ghost diagrams, represented in Fig. 5. The partonic level differential cross section for gg → tc̄ can be expressed as 1 |p|out |M|2 dΩ3 , 32π 2 ŝ3/2 (ŝ + m2t )2 − m2t (4.1) |p|2out = 4ŝ √ where Ω3 is the angular volume of the third particle and ŝ is the partonic center of mass dσ̂ = energy.8 The matrix squared |M|2 can be calculated by using the expressions, for t → cgg by simply using the crossing symmetry (see for example [23]). Then, the hadronic cross section is obtained by convoluting the partonic cross section with the parton distribution functions (PDF’s), fg/p . So, the total hadronic cross section reads Z 1 dL σ= dξ σ̂(ξs, αs (µR )) dξ ξ0 (4.2) √ √ where σ̂(ξs, αs (µR )) is the total partonic cross section at the center of mass energy ŝ = ξs √ ( s is the hadronic center of mass energy) depending on the renormalization scale µR . Here ξ0 defines the production threshold of the process. The parton luminosity is defined as dL = dξ 8 Z ξ 1 dx fg/p (x, µF )fg/p (ξ/x, µF ) x For simplicity we assumed mc zero in our analytical, but not in numerical, estimates. 19 (4.3) where µF is the factorization scale, which is assumed to be equal to the renormalization scale µR in our numerical analysis. If one needs to sum over all possible partonic subprocesses contributing to the particular final state, there will be a sum over PDF’s in Eq. (4.3). We assume that the top quark in the final state will be reconstructed from events and thus it is a physical observable. Of course, to identify the hadronic final state requires making a series of cuts on the transverse momentum pT of the top and charm quarks, the rapidity η, and the jet separation ∆R34 . The following set is used for the cuts pTc , pTt ≥ 15 GeV ηc , ηt ≤ 2.5 ∆R34 ≥ 0.4. (4.4) Their effect is translated into cuts on the limits of ξ in the calculation of the partonic cross section. For the transformation of the initial partons to initial hadrons, the program HadCalc [17] was used, incorporating the Les Houche Accord Parton Density Function library (LHAPDF) version 4.2 [24] with the recent data set CTEQ6AB [25]. For the numerical calculations, we have chosen as input parameter the values mA0 = 500 √ GeV, A = 400 GeV. In addition the hadronic center of mass energy s = 14 GeV is taken for the LHC. The factorization and renormalization scales are chosen as the production threshold of the process (µF = µR = 174.93 GeV). We discuss the dependence of the total hadronic cross section of gg → tc̄ + t̄c process, σ(pp → tc̄ + t̄c + X), on various MSSM parameters for certain δ values in scenarios with and without GUT relations. Note that for simplicity we assume a common δ parameter in 23 23 23 23 the up and down sector (δU,D )LL = (δU,D )RR = (δU,D )LR = (δU,D )RL and only one (U or D) non-zero at a time. At the end of this section we discuss the relative magnitude of the contributions coming from the gluino, chargino, and the rest. Fig. 10 shows the MSUSY dependence of the total hadronic cross section σ(pp → tc̄+t̄c+X) for tan β = 5, µ = 250 GeV, M2 = 200 GeV, and mg̃ = 300 GeV. On the left panel, there are two curves, for (δU23 )AB = 0, and 0.4. The cross section depends very weakly on MSUSY and, for the constrained MSSM case, the SM contribution is the dominant one. There is an enhancement of more than 6 orders of magnitude in the unconstrained MSSM over the constrained one, and the cross section can be as large as 15 fb for (δU23 )AB = 0.4. In the down sector, shown on the right panel, the sensitivity of σ to MSUSY is quite strong, and there is an enhancement of about two orders of magnitude in the interval 250 − 1000 GeV. There are still 2 − 4 orders of magnitude difference between the constrained MSSM versus 20 102 10−1 23 ) (δD AB = 0 23 ) (δD AB = 0.5 101 σ(pp → tc̄ + t̄c + X)/fb σ(pp → tc̄ + t̄c + X)/fb 10−2 100 10−1 10−2 23 ) (δU AB = 0 23 ) (δU AB = 0.4 10−3 10−3 10−4 10−4 10−5 10−5 10−6 200 300 400 500 600 700 MSUSY /GeV 800 900 10−6 200 1000 300 400 500 600 700 MSUSY /GeV 800 900 1000 FIG. 10: The total hadronic cross section σ(pp → tc̄ + t̄c + X) via gluon fusion as a function of MSUSY for tan β = 5, µ = 250 GeV, M2 = 200 GeV, and mg̃ = 300 GeV. On the left panel, δU23 = 0.4 is chosen and compared with the constrained MSSM case 23 = 0.5. The same is shown on the right panel for δD 102 102 101 101 100 100 σ(pp → tc̄ + t̄c + X)/fb σ(pp → tc̄ + t̄c + X)/fb δU23 = 0. 10−1 10−2 10−3 23 ) (δU AB = 0 23 ) (δU AB = 0.2 23 ) (δU AB = 0.4 10−1 10−3 10−4 10−4 10−5 10−5 10−6 200 250 300 350 400 450 Mg̃ /GeV 500 550 600 23 ) (δU AB = 0, Mg̃ = 200 GeV 23 ) (δU AB = 0.4, Mg̃ = 200 GeV 23 ) (δD AB = 0.5, Mg̃ = 300 GeV 10−2 10−6 0 5 10 15 tan β 20 25 30 FIG. 11: On the left panel, the total hadronic cross section σ(pp → tc̄ + t̄c + X) via gluon fusion as a function of mg̃ for tan β = 5, MSUSY = µ = 250 GeV, and M2 = 200 23 ) GeV at various (δU AB , A, B = L, R. On the right panel, σ(pp → tc̄ + t̄c + X) as a 23 function of tan β at various δU,D and mg̃ values. 23 the unconstrained MSSM scenarios at (δD )AB = 0.5. The maximum cross section is around 0.1 fb at around MSUSY ∼ 250 GeV. In Fig. 11, on the left panel, the total cross section σ(pp → tc̄ + t̄c + X) is shown as a 21 101 101 100 100 10−1 σ(pp → tc̄ + t̄c + X)/fb σ(pp → tc̄ + t̄c + X)/fb 102 23 ) (δU AB = 0 23 ) (δU AB = 0.4 10−2 10−3 10−4 10−2 10−3 10−4 23 ) (δD AB = 0 23 ) (δD AB = 0.5 23 ) (δU AB = 0.4 10−5 10−5 10−6 100 10−1 200 300 400 500 600 µ/GeV 700 800 900 1000 10−6 100 200 300 400 500 600 M2 /GeV 700 800 900 1000 FIG. 12: On the left panel, the total hadronic cross section σ(pp → tc̄ + t̄c + X) via gluon fusion as a function of µ for tan β = 5, MSUSY = 250 GeV, M2 = 200 GeV, and 23 ) mg̃ = 300 GeV at various (δU AB , A, B = L, R. On the right panel, σ(pp → tc̄ + t̄c + X) 23 ) as a function of M2 at various (δU,D AB with GUT mass relations. function of the gluino mass for various (δU23 )AB values. Again the constrained MSSM case is dominated by the SM contribution, while for the unconstrained MSSM, σ ∼ 45 fb for mg̃ = 200 GeV and (δU23 )AB = 0.4, which is more than 7 orders of magnitude larger than for the case with (δU23 )AB = 0. On the right panel, the tan β dependence of the total cross section is shown for µ = 250 GeV, M2 = 200 GeV, and mg̃ = 200, 300, (δU23 )AB = 0, 0.4, 23 and (δD )AB = 0.5. For very large tan β values, the cross section reaches 0.001 fb in the constrained MSSM, while in the unconstrained case, for (δU23 )AB = 0.4 and mg̃ = 200 GeV, a cross section of 60 fb is obtained. For (δU23 )AB = 0.5 and mg̃ = 300 GeV, the cross section under these conditions reaches a few fb. Fig. 12 illustrates the µ (on the left panel) and M2 (on the right panel) dependences of 23 the total hadronic cross section σ for representative values of (δU,D )AB . The parameters are tan β = 5, MSUSY = 250 GeV, M2 = 200 GeV, and mg̃ = 300 GeV for the left panel, and tan β = 5, MSUSY = µ = 250 GeV, and mg̃ = 300 GeV for the right panel. For non-zero (δU23 )AB , the cross-section σ is not sensitive to µ and remains around 15 fb, but it decreases significantly with M2 in the interval M2 ∈ [150 − 1000] GeV, if there is a non-zero δ in either the up or down sector. The cross section ranges between 10 fb to 0.1 fb for M2 = 150 GeV and 1000 GeV, respectively, for (δU23 )AB = 0.4, and between 0.1 fb to 0.001 fb for 23 (δD )AB = 0.5. 22 TABLE II: Relative contributions to the total cross section σ(pp → tc̄ + t̄c + X), in fb, with and without GUT mass relations. is considered. 23 ) (δU23 )AB = (δD AB = 0, 0.2, 0.4, A, B = L, R The rest of the parameters are A = 400 GeV, tan β = 10, MSUSY = µ = 250 GeV and M2 = 200 GeV. For the case without GUT, mg̃ = 300 GeV is used and 23 ) (δU,D AB are given in brackets. 23 ) (δU23 )AB = (δD AB 0 0.2 (No GUT) 0.4 (No GUT) Gluino loop 0 1.09 (4.05) 3.07 (14.13) Chargino loop 1.7710−5 0.0052 0.034 The rest 6.1010−6 0.0025 0.017 Before concluding, we comment on the relative contributions of the gluino, chargino, and the rest (namely, neutralino, charged Higgs, and SM contributions) to the total cross section. In Table II, we show the relative contributions to σ(pp → tc̄ + t̄c + X) from gluino, chargino, and the rest, in the MSSM with GUT mass relations (the case without GUT mass relations is shown for mg̃ = 300 GeV in brackets if different). For simplicity we set 23 (δU23 )AB = (δD )AB , A, B = L, R and the values 0, 0.2, and 0.4 are considered. The other parameters are A = 400 GeV, tan β = 10, MSUSY = µ = 250 GeV and M2 = 200 GeV. The case in which no GUT relations between gaugino masses are imposed corresponds (in our analysis) to the case in which the gluino mass is allowed to be smaller. This is the reason why only the gluino contributions are enhanced in this scenario. The gluino contributions are also dominant for the case in which GUT relations are imposed, and the chargino contributions are two orders of magnitude smaller. However, one could envisage a case in which the SUSY 23 FCNC parameters (δD )AB are allowed to be large, while the ones in the up sector restricted to be small or zero, in which the chargino contribution could be dominant. In the case of the constrained MSSM only chargino loops contribute. So far, we have considered a common flavor violation in the LL, LR, RL, and RR sectors of sfermion matrix. However, one could analyze the relative effects of flavor violation in each sector and determine how large their contribution to the cross section could be. If the flavor violation comes only from the up sector, then (δU23 )LR is the most sensitive parameter, as shown in Fig. 13 where we took the parameter values mA0 = 500 GeV, µ = MSUSY = 400 23 103 σ(pp → tc̄ + t̄c + X)/fb 102 101 gluino chargino 100 neutralino charged Higgs 10−1 SM 10−2 10−3 10−4 10−5 10−6 0 0.2 0.4 0.6 0.8 1 23 ) (δU LR FIG. 13: The total hadronic cross section σ(pp → tc̄ + t̄c + X) via gluon fusion as a function of (δU23 )LR (other flavor violating parameters are assumed zero) for tan β = 30, mA0 = 500 GeV, A = 300 GeV, µ = MSUSY = 400 GeV, M2 = 200 GeV, and mg̃ = 200 GeV. GeV, A = 300 GeV, mg̃ = 200 GeV, and tan β = 30. Of course this is only one of the best possible scenarios. As seen from the figure, the total cross section can be as big as 630 fb and the gluino contribution becomes dominant if there exist a large flavor violation in the up LR sector between the second and third generations. If the flavor violation comes only 23 from the down sector, (δD )LL gives the largest contributions. In this case the cross section is dominated by chargino contribution and can be as large as 0.4 fb with the same parameter values. Finally, we would like to qualitatively comment on the observability of both the decay and the production channels of the top quark considered here. The rare decay t → cgg can in general be treated twofold way: one can either treat it inclusively with t → cg or consider it as a separate channel. The former means that t → cgg is taken as QCD-correction to t → cg by assuring that two of three final state jets are collinear so that only two can be resolved in the detector. The latter can be a competitive possibility if BR(t → cgg) is significantly larger than that of t → cg. In here, and in our previous work [9], we conclude that t → cgg could be potentially more significant than the two-body decay channel in the SM [9] and in some part of the MSSM parameter space. However, this should be taken with a dose of caution. Collinearity should be avoided by applying certain cuts. The unphysical C-parameter introduced in the phase space here plays an essential role to distinguish t → cg 24 from t → cgg. Even though in our explorations we considered several values of C, C must be taken in the range of jet energy resolution of the upcoming LHC detector. For C ≪ 0.1, t → cgg dominates over t → cg for a larger parameter space, thus availability of better jet resolution would give an opportunity to detect t → cgg before t → cg. At LHC, predominantly tt̄ pairs are produced. If we consider one of the top quarks decay mainly as t → bW and the other one exotically as t → cgg, then the signal would be pp → tt̄ → (lν)ggcb̄ (4-jets, a lepton and missing energy), where l = e, µ.9 For the single lepton plus jet topology, it is possible to reconstruct the final state fully, and the b-quark can be tagged to obtain a cleaner signal, which introduces extra selection efficiency. We assume σ(pp → tt̄) = 800 pb at LHC and also that the W boson decays leptonically, not hadronically. Under this conditions, one can calculate roughly the total expected(raw) number of events as N = σ(pp → tt̄) × BR(t̄ → b̄W ) × BR(W → lν) × L × BR(t → cgg) , where L is the integrated luminosity which we take as 100 fb−1 . Therefore we have N = 800 × 103 × 1 × (2/9) × 100 × BR(t → cgg) = (1.77 × 107 ) × BR(t → cgg). So, one expects around (1.8×107 )×BR(t → cgg) lepton+4 jets events for an integrated luminosity 100f b−1 . However, counting a total efficiency including trigger efficiency, selection efficiency, as well as detector geometrical acceptance, one would approximate a total efficiency around 1% [26]. Thus, the number reduces to (1.8 × 105 ) × BR(t → cgg). So, if the flavor violation comes from the down squark sector, then for most part of the parameter space t → cgg would dominate over t → cg by around two orders of magnitude, but both will remain unobserved because lepton+jets events are less than a single event. If the flavor violation comes from up-squark sector, then t → cg dominates and can reach 10−5 level for light gluino scenarios, which might lead an observable event rate around 1.8. Should the integrated luminosity increase at later runs of LHC, one could obtain larger event rates (up to 10 events) if the flavor violation is driven by the up squark sector. If the flavor violation comes from the down sector of the unconstrained MSSM, the event rate remains below the observable level. For the single top production case pp → tc̄ + t̄c + X, we already included cuts for the transverse momentum and rapidity of the charm and top quarks in the final state, as well 9 Considering the W decay hadronically would produce a 6 jet final state, requiring determination of the multi-jet trigger threshold. 25 as a lower cut for jet separation. In this case, if assume that the top quark is going to be reconstructed in the final state one can predict for example 50,000 events for an integrated luminosity 100 fb−1 and σ(pp → tc̄ + X) = 0.5 pb. A similar total efficiency consideration will going to reduce this further but there exist enough events to find a signal under the bestcase scenario. Anything beyond the above qualitative discussion about the observability of the decay and production channels will be considered in more detail in our future paper [27]. V. CONCLUSION In this study we analyzed two related issues in top quark physics. In the first part of the paper, we concentrated on the comparison of two rare top quark decays, t → cgg versus t → cg, within the unconstrained MSSM, driven by mixing between the second and third generations only. To the best of our knowledge, t → cgg decay has been considered only in our recent study [9] within the SM framework, where BR(t → cgg) was found to be two orders of magnitude larger than BR(t → cg). However, in the SM, BR(t → cgg) remains at 10−9 level, and thus too small to be detectable. Any experimental signature of such channel would require the existence of physics beyond the SM which justifies further analyses. Here we studied this decay in the MSSM framework by allowing non-zero flavor off-diagonal parameters. Our conclusion of the dominance of the branching ratio of t → cgg over t → cg in the SM paper remains mostly valid in the MSSM framework, but now the BR’s can become as large as 10−6 − 10−5.10 For the cases in which we impose the GUT relation between gaugino masses bf and assume a flavor violation in down squark sector, the large difference in ratio between the t → cgg and t → cg modes disappears only in the case of very large intergenerational flavor-violating parameters (close to 1, or to their maximally allowed upper values). In that case, t → cg exceeds t → cgg. In the case of non-zero (δU23 )AB , t → cg dominates t → cgg, except in regions of small flavor violation. Once we relax the GUT constraints, there is no longer such a large difference between the two and three body decays, as long as a small flavor violation is turned on. Once the 23 flavor off-diagonal parameters, (δU,D )AB , are introduced, the difference in branching ratios disappears as the parameters are minute and BR(t → cg) becomes around 5 times larger 10 The cutoff parameter C=0.1 is being used. 26 than BR(t → cgg). As expected, if the SUSY-GUT relations hold, both modes cannot exceed 10−7 level (except for flavor violating parameters near their maximum allowed values for t → cg decay), because the gluino mass is large. Once we relax this condition, both t → cgg and t → cg have branching ratios of the order 10−6 − 10−5 and 10−5 , respectively. Having shown that the three body rare decay t → cgg is indeed important (comparable with, or larger than, the two body decay t → cg), we carried out a complete calculation of the single top-charm associated production at LHC via gluon fusion at partonic level within the same scenarios discussed above. This production cross section has been considered before including only the SUSY-QCD contributions [8]. We performed a complete analysis by including all the electro-weak contributions: the chargino-down-type squark, neutralino-uptype squark, charged Higgs, as well as the SM contributions. For simplification, a common 23 23 23 23 SUSY FCNC parameter δ is assumed, (δU,D )LL = (δU,D )RR = (δU,D )LR = (δU,D )RL (in the up and down squark sectors), but most often only one common δ parameter in either sector is allowed to be non-zero each time. We have shown that, in the most promising scenarios (if a common SUSY FCNC parameter δ is assumed), the total hadronic cross section σ(pp → tc̄ + t̄c + X) can become as large as 50 − 60 fb and could reach a few hindered fb, especially if we relax the GUT relations between the gaugino masses and assume a flavor violation from one sector only (LL, LR, RL, or RR). We have shown that the cross section could be as large as 600 − 700 fb if a large flavor violation coming from only (δU23 )LR is allowed. The comparative gluino, chargino and other contributions to the process have been estimated. The gluino contributions dominate over most of the parameter space, when allowing flavor violation in both up and down squark mass matrices to be of the same order of magnitude. However, the chargino contribution is non-negligible and would be dominant in either the constrained MSSM, or if the flavor violation was allowed to be much larger in the down than in the up squark sector. While the chargino loop is two orders of magnitude 23 smaller than the gluino for (δU23 )AB = (δD )AB , the contribution of the “rest” (neutralino, charged Higgs and SM) is around half of the chargino contribution, for all of the SUSY FCNC parameters chosen. Gluon fusion could be more promising than cg → t, q q̄ → tc̄, or cg → gt, which we leave for a further study [27]. The huge differences in prediction between the constrained and the unconstrained MSSM 27 scenarios make LHC a fertile testing ground for the study of SUSY FCNC processes. Any significant rate for the top-charm associated production would be a signal of physics beyond the SM, and in particular, of new flavor physics. VI. ACKNOWLEDGMENTS The work of M.F. was funded by NSERC of Canada (SAP0105354). The work of G.E. was supported in part by the Israel Science Foundation and by the Fund for the Promotion of Research at the Technion. G.E would like to thank J.J. Cao for helpful discussions. I.T. would like to thank Micheal Rauch for his help and suggestions about the use of the HadCalc program. Note added. After submitting the first version of the present paper to the active (hep-ph/0601253.v1) we became aware of an analysis similar to ours [28]. 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