Single top-quark production by strong and electroweak supersymmetric flavor-changing interactions at the LHC
We report on a complete study of the single top-quark production by direct supersymmetric flavor-changing neutral-current (FCNC) processes at the LHC. The total cross section, , is computed at the -loop order within the unconstrained Minimal Supersymmetric Standard Model (MSSM). The present study extends the results of the supersymmetric strong effects (SUSY-QCD), which were advanced by some of us in a previous work, and includes the computation of the full supersymmetric electroweak corrections (SUSY-EW). Our analysis of in the MSSM has been performed in correspondence with the stringent low-energy constraints from . In the most favorable scenarios, the SUSY-QCD contribution can give rise to production rates of around events per of integrated luminosity. Furthermore, we show that there exist regions of the MSSM parameter space where the SUSY-EW correction becomes sizeable. This could be important, especially if the SUSY-QCD effects would be suppressed. In the SUSY-EW favored regions, one obtains lower, but still appreciable, event production rates that can reach the level for the same range of integrated luminosity. We study also the possible reduction in the maximum event rate obtained from the full MSSM contribution if we additionally include the constraints from . However, we treat these restrictions at a different level from the ones, due to the higher uncertainties inherent in the calculation of the matrix element associated to that mixing. In view of the fact that the FCNC production of heavy quark pairs of different flavors, such as or , is extremely suppressed in the SM, the detection of a significant number of these events could lead to evidence of new physics – of likely supersymmetric origin.
The forthcoming generation of high energy colliders, headed by the Large Hadron Collider (LHC) at CERN, and followed by the future linear collider, depicts an exciting scenario for probing the existence of physics beyond the Standard Model (SM) of strong and electroweak interactions . Among the possible discoveries envisioned for the physics at the LHC (some of them of a rather exotic nature, such as extra dimensions  and black-hole production ), we have the possible confirmation of the fundamental Higgs mechanism of Electroweak Symmetry Breaking. This would be accomplished in practice through the physical production of one or more Higgs boson particles. Undoubtedly, the next-to-most important discovery expected at the LHC is the finding of supersymmetric particles.
Actually, the discovery of Supersymmetry (SUSY) (see  for a comprehensive review) is intimately connected to the structure of the Higgs mechanism. In fact, unearthing supersymmetric particles would be strong evidence that Higgs bosons (in plural) should be around the corner. The opposite, however, is not necessarily true, but if a light Higgs boson of, say, would be found at the LHC, the hopes for SUSY physics would stay high and we would immediately felt encouraged to search for more Higgs bosons and potential supersymmetric particles. It is well-known that a light Higgs boson () is a trademark prediction, if not of SUSY in general, at least of the Minimal Supersymmetric Standard Model (MSSM) in particular, which is after all the canonical scenario for low-energy SUSY phenomenology [5, 6, 7].
If SUSY is realized at the scale (usually taken as the characteristic energy scale to explain the naturalness problem of the SM ), one expects that a few (or even a bunch of) supersymmetric particles of the MSSM spectrum should be well reachable at the LHC. However, the tagging of heavy new particles is not an easy task because of the many decay modes available, most of them carrying invisible neutral species (some of them also of genuine SUSY origin, like sneutrinos and neutralinos) and, therefore, leading to missing energy events – usually hard to interpret. For this reason, one expects to get a complementary clue to the underlying SUSY dynamics from the short-distance quantum corrections on more conventional processes. If these supersymmetric quantum effects can be measured, they can be a solid handle to the properties of the new physics. The idea has been known for a long time and has been applied to the familiar physics of the and gauge bosons, see e.g. [8, 9, 10, 11]. Here we wish to apply this method to the realm of rare processes, namely processes with conventional initial and final states which, although not strictly forbidden, turn out to be highly suppressed within the SM context. Among them, we have the fruitful Flavor-Changing Neutral-Current (FCNC) processes.
The study of the flavor-changing interactions, in particular the FCNC processes, has been a very active field of research for about forty years, namely as of the glorious times when Glashow, Iliopoulos and Maiani (GIM) successfully proposed the existence of a fourth species of quark, the c-quark, to suppress to an acceptable level the strangeness-changing neutral-current effects in rare processes (e.g. ) that otherwise would proceed at the tree-level, and similarly to further suppress the one-loop contributions in e.g. the system. Indeed, it was the experimental evidence that the FCNC processes seemed to be extremely inhibited in nature (actually forbidden at the tree-level and highly suppressed at the one-loop level) the main motivation for the aforementioned GIM mechanism , nowadays embedded in a natural way into the current formulation of the SM – essentially into the unitarity of the CKM matrix. It is remarkable, however, that the degree of suppression at one-loop order can vary from one process to another in a dramatic manner. For instance, in the -quark sector the radiative -meson decay has a branching ratio which, although small, it has been measured experimentally  with quite some accuracy and it is used in practice to constrain models of new physics. In contrast, the FCNC top quark decay becomes radically inhibited in the SM, , namely down to limits far below ever being possibly observed [14, 15, 16]. Amazingly, the top quark decay into the SM Higgs boson is even more unlikely: [15, 17]. In all these cases, it is their highly “expected unobservability” what provides the natural “signature” for potentially unraveling new physics out of their study. In fact, the huge GIM suppression in some rare processes within the SM can be significantly softened if one accounts for possible SUSY virtual contributions. For example, in the case of the extremely rare top quark decay into the SM Higgs boson one can show that if is the lightest CP-even Higgs boson in the MSSM , then can be enhanced times as compared to the SM mode and, thus, bring it to the observable level [19, 20, 21]. Similar results hold for the Higgs boson decay modes into heavy quarks, see e.g. [22, 23, 24, 25, 26]. Actually, not only SUSY can help here; other alternative extensions of the SM, among them the general Two-Higgs-Doublet Model (2HDM) , predict in some cases an enhanced, and often distinctive, FCNC phenomenology [27, 28, 29, 30, 31, 32, 33, 34]. Put another way: by finding experimental evidence of non-standard FCNC processes we can not only enlighten the existence of physics beyond the SM but, in favorable conditions, we can even tell the kind of new physics hiding right there 111For a review, see e.g.  and . See also the interesting flavor mixing studies [36, 37, 38]..
In this paper we wish to further explore the FCNC physics of the top quark, but in this case we focus on the production of single-top quark final states or through gluon fusion () in collisions to take place at the LHC. We denote it by . While it is true that this process is possible within the strict SM, it proceeds through (GIM-supressed) charged-current interactions. We have found the following cross-section for this process at one-loop level (see Section 4):
Obviously, it is so tiny that it amounts to less than one event in the entire lifetime of the LHC! So it is pretty clear that if this kind of FCNC-generated single top quark signatures would ever be detected at the LHC, if only at a level of a few dozen crystal-clear events, then the presence of new physics could perhaps be the only valid explanation for them. We see that the situation with this production process is very similar to the rare top quark decay modes mentioned above; in both cases it is the FCNC physics of the top quark that provides the extreme suppression within the SM. However, it should be clear that the top quark final states in (1) are a particular class of events within the large variety of single top quark processes available in hadron colliders [39, 40, 41, 42].
But this is not the only challenge. The LHC, with all its ability to dig deep beneath the physics of the top quark, could perhaps be sensitive to the class of single top quark final states associated to FCNC processes, , provided of course the underlying mechanism could be sufficiently enhanced by some form of new physics capable to boost its cross-section up to pb level. In this study, we will show that the necessary enhancement (which amounts to a factor of roughly in the total cross-section) could just come from the world of the supersymmetric interactions in the general MSSM.
Interestingly enough, let us remark that for the FCNC process under consideration, , there is no significant competition between the MSSM and the general 2HDM because there is no enhancement to speak of from the latter. This is in contrast to the situation with the rare top quark decays mentioned above, where the 2HDM contributions are non-negligible as compared to the MSSM ones. Moreover, the direct production mechanism is substantially more efficient (typically a factor of ) than the production and subsequent FCNC decay of the heavy Higgs bosons () . In this sense the discovery of a bunch of well-identified and/or events could be strong evidence, not only of new physics, but perhaps of SUSY itself. Some of these features were already emphasized in Ref. , where it was presented a first self-consistent study of this subject (see also ). These references, however, reported on the computation of the SUSY-QCD effects only. Other studies can be found in [45, 46] under different sets of assumptions. In our case we will continue within the general approach initiated in , and continued in . It means that the flavor-mixing coefficients will be allowed only in the purely left-handed part of the sfermion mass matrices in flavor-chirality space, as it is indeed suggested by standard renormalization group (RG) arguments [47, 5]. Within this well motivated setup we provide here a full treatment of the SUSY-EW effects and combine them with the SUSY-QCD ones within the general framework of the MSSM. We wish to remark that, in contradistinction to the aforesaid studies by other authors, we present our MSSM calculation of in combination with the corresponding MSSM effects on the low-energy decay and, therefore, we extract the single top quark FCNC results only in the region of parameter space compatible with the experimental bounds on the radiative B-meson decays. This procedure is, in our opinion, a self-consistent approach to the computation of the FCNC single top quark signal under study.
The paper is organized as follows. Section 2 is devoted to the general formalism for the FCNC processes in the MSSM. In section 3 we summarize the details of our calculation of in this framework. The full numerical analysis is presented in section 4, leaving section 5 to discuss the results and deliver our conclusions.
2 Formalism: FCNC interactions in the MSSM
Apart from the conventional charged-current flavor changing interactions in the SM, the FCNC processes in the MSSM are driven by explicit intergenerational mixing terms arising from the mass sector of the squarks. For a brief review of this topic, our starting point shall be to specify the form of the superpotential, which is the crucial piece of any SUSY theory of particle interactions. In our case we will consider the MSSM with arbitrary soft-SUSY-breaking terms. The most general gauge-invariant form of the superpotential can be cast in terms of chiral superfields (denoted by a hat) as follows [5, 6, 7]:
Indices refer to the components of the doublets, which are combined in a gauge-invariant form through (with ). The set of parameters , and constitute Yukawa coupling matrices in generation space. Although explicit generation labels have been suppressed here, they will be introduced at due time. Let us notice that in more general SUSY theories there are additional pieces of the superpotential inducing violation of baryon or lepton number, but in the MSSM they are set to zero because one assumes that the R-parity symmetry holds.
We also need to settle the piece of the soft SUSY-breaking Lagrangian that takes part in the squark mass matrix:
In this expression, stands for the quark doublets, while denote the corresponding singlets. Let us recall that each of the above mass and trilinear coupling parameters carries a matrix structure in the flavor space, although we shall not keep track of it explicitly.
We can now collect the different pieces contributing to the general form of the squark mass matrix, which come either from the explicit mass terms in the soft-SUSY-breaking Lagrangian (3) or from the couplings triggered by the superpotential (2) after spontaneous symmetry breaking (SSB) of the EW symmetry. If we arrange all such terms in a -dimensional left-right chirality space, we are left with the following mass matrix:
where in the off-diagonal mass terms we have defined and . As usual, , with , defines the ratio of the vacuum expectation values of the two Higgs doublets giving masses to the up and down quarks respectively, while stands for the th component of the weak isospin of the left-handed quark , and denotes its charge. The non-diagonal structure of (4) in the chirality basis requires its diagonalization in order to obtain the physical mass-eigenstates in terms of the electroweak (EW) squark eigenstates with well-defined quantum numbers. If denotes the matrix rotating the th flavor, we can diagonalize the mass matrix as follows: . Notice that each matrix elements in Eq (4) is proportional to the unity matrix in the flavor space. It is worth realizing, however, that such a trivial flavor structure for the mass matrix does not provide the most general realization of the squark mass sector. Indeed, in the MSSM we have two fundamental sources of flavor violation. One of them just mimics the SM one, namely it consists of the flavor mixing among up- and down-like squarks triggered by the charged-current interactions induced by the charged gauge bosons, the charged Higgs bosons and the charginos. The second one is qualitatively new and is caused by the so-called misalignment between the rotation matrices that diagonalize the quark and squark sectors or, in other words, the fact that the squark mass matrices in general need not diagonalize with the same matrices as the quark mass matrices [47, 48, 49, 50]. This is reflected in the existence of the gaugino-fermion-sfermion interactions mediated by gluinos () and neutralinos (). Consider e.g. the gluino-quark-squark interactions
Here are generation indices, are the generators, and and are rotation matrices in generation space which relate the electroweak and the mass-eigenstates; e.g. rotates up-quarks and rotates up-squarks, etc. Notice that in the down sector we only need to rotate squarks through because after SSB of the gauge symmetry the down quark matrix is already diagonal. This follows from the fermion mass matrix structure that emerges from the superpotential (2) in generation space after the Higgs bosons acquire VEV’s and spontaneously break the EW symmetry. Let us consider only the quark sector,
We can rotate and in generation space until the mass matrix for down quarks, , becomes diagonal, but then the mass matrix for up quarks, , will in general be non-diagonal because was already rotated. By inspecting the charged current interaction of quarks, this immediately implies that the ordinary CKM matrix is just . Similarly, from the charged current for squarks we read off the corresponding CKM matrix in the squark sector: . Therefore one finds a relation between the CKM and SCKM matrices:
As a result, in the MSSM we need three unitary matrices to parametrize the flavor changing interactions, one is the ordinary CKM matrix and the other two are associated to the new FCNC gaugino-quark-squark couplings (5). Neglecting this second source of flavor changing interactions (i.e. assuming that the matrices and are unity in flavor space) would be tantamount to assume that quarks and squarks diagonalize simultaneously, i.e. . This is the super-CKM basis approach to the FCNC processes; it assumes that these processes appear at one-loop only through the charged current interactions (from , charged Higgs bosons and charginos ) and with the same mixing matrix elements as in the Standard Model CKM matrix. However, in general we expect that the two sources of FCNC should be active in the MSSM and we will take them both into account in our calculation.
These observations turn out to be crucial for the discussion of the flavor-changing processes in the MSSM because it means that we can extend the simple squark mass matrices in chiral space into mass matrices in (flavor)(chiral) space. We shall comment below on how to parametrize the flavor mixing terms. For the moment we note that, due to the aforementioned flavor mixing, the squark mass matrix diagonalization process must be extended as follows:
where are the square mass matrices for squarks in the EW basis (), the eigenvalues being denoted . Indices run now over -dimensional space vectors with suitable identifications. For example, for up-type squarks , and a similar assignment for down-type squarks. Furthermore, let us notice that the gauge invariance of the MSSM Lagrangian imposes certain restrictions over the up-squark and down-squark soft-SUSY breaking mass matrices, specifically in their LL blocks, as follows:
where stands for the CKM matrix (previously denoted for convenience). It is thus clear, in particular, that both squark matrices cannot be simultaneously diagonal (unless they are proportional to the identity) and, therefore, they cannot be simultaneously diagonal with the up-like and down-like quark mass matrices either. This is again a reflect of the misalignment effect between the mass matrices of quarks and squarks.
Despite what we have just argued above, within the context of Grand Unified Theories (GUT’s), one usually assumes that the parameters should be aligned at the characteristic high energy scale of these theories (that is to say, the quark and squark mass matrices should diagonalize simultaneously at ). But even within such theoretically-motivated scenario, it can be shown that the renormalization group running of these parameters down to the EW scale would again destroy the primeval aligned configuration [47, 48, 5]. It is therefore wiser to take the misalignment into account right from the start in the calculation. The most common way to parametrize it is by defining the following dimensionless quantities, , being the chirality indices and the flavor ones, in such a way that we can set the non-diagonal squark mass matrix elements to be:
where stands for the soft-SUSY breaking parameter of a given chirality and flavor. (No sum over repeated indices here.) It is very common to set all the mass parameters equal to a generic SUSY scale .
As far as we are dealing with FCNC processes involving the top quark, the most relevant mixing parameters are those ones relating the heavy up-like flavors among themselves, thus essentially transitions parametrized by above. In close relation to them we have the transitions controlled by the parameter . Only these mixing parameters are expected to be large in GUT’s and, moreover, they are not significantly constrained by phenomenological considerations. The experimental bounds on the various mixing parameters are derived from the absence of low-energy FCNC processes, which mainly involve the first and second generations. For instance, the measurements of the mass splitting in and phenomena [49, 50].
Regarding heavy flavors, the phenomenological constraints come from the branching ratio of the radiative B-meson decay and also from the mass splitting in mixing effects. Clearly, such two processes can only be sensitive to the down-like heavy-flavor mixing parameter, . However, they can also provide information on the allowed values for since both up and down-like flavor-mixing parameters must necessarily be related through the symmetry (10). As advertised, in our framework we limit ourselves to consider flavor mixings only in the LL-block of the squark mass-matrices, the only ones which are well-motivated by RG arguments. Thus, the relevant piece in our calculation will be the LL sector of the up-type squark mass matrix, which can be rewritten in the following manner:
Similarly for with . Squark mass-eigenstates follow from diagonalization of these matrices through Eq. (2). The mass-eigenstates of (4) are recovered by setting the mixing parameters to zero, as could be expected.
Once we have discussed where the flavor-mixing source is rooted in the MSSM, we must now trace back its role at the Lagrangian level . The misalignment between the diagonalization matrices in the quark and squark mass sectors triggers the presence of couplings of the guise gluino-quark-squark (in the SUSY-QCD part) and neutralino-quark-squark (in the SUSY-EW one) that allow the interaction of quarks having the same charge but belonging to different generations. At the -loop level it is also possible to have this kind of flavor-changing interactions mediated by the ordinary SM charged currents, but in the -odd part of the MSSM we also have the chargino-up-quark-down-squark interactions and the charged Higgs-up-quark-down-squark vertices. In all such cases the SUSY nature of the couplings allow the resulting process to bypass the SM GIM mechanism and provide non-suppressed FCNC events (the charged Higgs piece is an exception, as we shall see). The importance of such effect is correlated with the choice of the MSSM parameters, in particular those specifying the soft-supersymmetry breaking and, of course, the explicit intergenerational mixing , which are the most relevant ones for the flavor-changing dynamics in the MSSM. For the SUSY-QCD coupling one must work out the supersymmetrized gauge interaction piece:
where are the gauge group generators, the indices denote the corresponding gauge quantum numbers (color, weak isospin) of the interacting particles and stands for a generic gaugino field. To extract the FCNC vertices one must include the generation indices in these interactions. For the particular case of the gluino-mediated interactions, this was done in (5). For the practical calculations we will use the extended diagonalization matrices defined in (2). Therefore, by plugging the squark mass-eigenstates in this expression we can rephrase the result in the mass-eigenstate basis and in terms of four-component Dirac spinors (for both the gluino and quarks). In the up quark-squark sector we get
and similarly for the down quark-squark sector. Here we have omitted color indices for gluinos, quarks and squarks. Notice that while the sum over index runs over the whole flavorchirality space, index runs only over generations because we are already using the standard projectors to set the chirality of the quarks.
A similar analysis can be performed to obtain the corresponding Lagrangians describing the flavor-changing interactions in the SUSY-EW sector. The calculations are slightly more involved since such terms arise from the combination of the SUSY-gauge piece (13) together with the higgsino-quark-squark Yukawa couplings dictated by the superpotential (cf. Eq. (2)). Moreover, because of the EW symmetry breaking, the higgsinos and gauginos mix together to give the final physical eigenstates, the neutralinos and charginos . We shall quote here the final result for such interaction Lagrangians (a detailed derivation can be found in  and references therein). For the case of the neutralinos, we get:
where is the weak gauge coupling constant. A few words about notation: index is running over the four neutralino states, being the diagonalization matrix that provides the neutralino mass-eigenstates, , while stands for the corresponding Yukawa coupling and is the weak mixing angle (). Similarly the chargino-up-squark-down-quark interaction Lagrangian can be cast in the following form:
This time the standard CKM matrix also needs to be taken into account because of the charged-current mixing between up-like squarks with down-like quarks. Again, refer to the diagonalization matrices of the chargino mass, such that .
3 Single top-quark production through FCNC processes in the MSSM: computation procedure
In the following we will concentrate on the analysis of the single top-quark production by direct supersymmetric flavor-changing interactions at the LHC, namely the processes leading to or final states. The leading mechanism is the gluon fusion channel: (see Section 4 for a full list of Feynman diagrams). It should be clear that . It was already shown in  that the partonic channel largely dominates over the one at the LHC. Although there are previous studies of this process in the literature within the MSSM and adopting different approximations [43, 46, 45], a closer look is highly desirable from our point of view. This is so because the kind of simplified assumptions made in some of the previous analyses do not shed sufficient light on the possibility that this process could be sufficiently enhanced in the MSSM as to be considered realistically at the LHC. We will comment on the differences among these approaches later on. In the present paper we carry out our calculation within the framework of [43, 44], which was first delineated in .
Throughout the present work we have made use of the standard algebraic and numerical packages Feynarts, FormCalc and LoopTools [52, 53, 54] for the obtention of the Feynman diagrams, the analytical computation and simplification of the scattering amplitudes and the numerical evaluation of the cross section (up to the partonic level). Notice, however, that we also need to address the computation of the total hadronic cross section in order to account for the physical process, a collision, to take place at the LHC. To this aim we have made use of the program HadCalc  222The source code is available on request from the author., while several cross-checks have also been performed with other independent codes implemented by us. Throughout our calculations we have settled both renormalization and factorization scales at a common value, chosen to be half of the production threshold . Concerning the parton distribution functions (PDF’s) involved in the long-distance dynamics of the hadronic process, we have included the recent CTEQ6AB data set  provided by Les Houches Accord Parton Distribution Functions Library (v.5.2) .
The computation of in the MSSM is not straightforward. It involves a number of subtleties that must be carefully handled. To begin with, we must deal with the PDF of a gluon, which exhibits a huge slope in the low-momentum region. To that purpose, we have implemented a logarithmic mesh for the integration over the partonic variables instead of the linear mesh that is provided by default. We must obviously pay the price of adding the corresponding jacobian piece to the original integral. Furthermore, a double call to the integration subroutine (viz. the Vegas routine provided by the Cuba library ) has been implemented. The first call provides an adapted grid for the second one, in such a way that the convergence is much faster. As a result a good and reliable numerical accuracy is achieved (meaning that the values always remain of order ). Last but not least, a second non-trivial subtlety emerges from the fact that one of the final states, the -quark, has a very small mass when compared to the value of the scattering process. We are thus very close to a collinear-divergence regime. Although there is no analytical divergence, the mass of the c-quark is low enough to trigger instabilities in the code when integrating over very small angles. We have carefully studied the problem and have included a tiny angular cut (chosen to be such that ) in order to avoid the aforementioned instability. We have checked the dependence of the final calculation on the choice of the angular cut (the total hadronic cross section can change about a when moving from to ), thus no dramatic changes occur when tuning the cut within reasonable ranges.
In regard to the calculation of the amplitudes contributing to the relevant process under consideration, , let us note that the leading order is the -loop level. This is a common feature when studying FCNC processes in any renormalizable theory (due to the lack of FCNC tree-level interactions). This implies that one need not renormalize the bare parameters nor the Green’s functions as there are no explicit terms in the interaction Lagrangian where to absorb the UV divergences. In other words, the overall amplitude of the process should be already finite as soon as we add up all the diagrams contributing to that process. In order to check the finiteness of the resulting amplitude, we have made use of a standard numerical procedure provided by FormCalc.
In the following section we present our final numerical results. We shall not provide here analytical details of the complicated algebraic structures appearing in the calculation of the many one-loop diagrams involved (see Figures 1-5). We have carried out the computation in a fully automatic fashion by means of the numerical and algebraic tools mentioned above, and of course we have previously submitted our codes to many important tests and non-trivial cross-checks of different nature.
The calculation of has been linked to the one-loop calculation of in the MSSM, so that after enforcing this low-energy observable to stay within the experimental bounds we have obtained the desired cross-section only in this allowed region of the MSSM parameter space. Our computation of contains the complete leading order (one-loop) MSSM computation including the flavor-violating couplings. Specifically, we include the contributions to the high energy operators from the SM ( loops), SUSY-QCD (gluino loops) and SUSY-EW (chargino-neutralinos and Higgs boson loops). The Wilson coefficient expressions have been taken from Ref. 333Reference  contains a computation of in the MSSM including some two-loop parts, but only the one-loop contributions have been used for the present work., and they are evolved using the leading order QCD renormalization group equations down to the bottom mass scale. However, at certain stages of our work we use only a part of these corrections, this will be clearly indicated in the text below.
4 Numerical analysis
To start with, let us present the calculation of the cross-section for the process within the context of the SM. The Feynman diagrams describing the interaction at the partonic level in the t’Hooft-Feynman gauge are shown in Fig. 1. In this gauge, the covariant sum over the polarization states of the gauge bosons yields the relation . Notice that in the present situation it is unnecessary to introduce the Faddeev-Popov ghost-field contributions since the current process involves only external gluon lines and it thus suffices to restrict the above sum to the two physical degrees of freedom carried by the gluons. This is straightforwardly done within the framework of the standard computational tools of Ref. [52, 53, 54] and allows us to get rid of the spurious modes of the quantized gluon field. Also worth emphasizing is the effect of the GIM suppression, which is associated to the SM diagrams of Fig. 1. We take for instance a vertex correction diagram driven by the exchange of a charged boson with a pair of quark lines closing the loop, and then sum over flavors. The result is a form factor of the type
where denote again matrix elements of the standard CKM matrix, and is a flavor index that runs over the down-like quark states . We have also included a standard numerical suppression factor from the one-loop integral. The additional GIM suppression is of dynamical origin within the SM and it stems from the unitarity of the CKM matrix. As a result the overall behavior of the form factor amplitude goes like , where the correspond in this case to down-like quark masses circulating in the loops. The cross-section gets suppressed as . It is thus not surprising that we finally get the value early indicated in Eq. (1), which amounts to about fb. The order of magnitude of the cross-section at the parton level roughly follows from naive power counting and educated guess. Using (17) we get:
where we have included factors from the strong () and weak () interactions. For the estimate we include only the bottom quark contribution (with matrix elements and ) as the other terms are suppressed either by very small CKM matrix elements or very light quark masses 444This is similar to the kind of ansatz made by Gaillard and Lee to predict the charm quark mass , except that here it is the bottom quark that gives the dominant effect because the external quarks are up-like.. At the LHC energies the previous estimate provides the cross-section within the ballpark of the exact result after convoluting with the parton distribution functions. However, in order to understand the dynamical mechanism of enhancement of the SUSY interactions (see below) it will suffice to compare with the partonic contribution (18).
Using the exact (numerically computed) result (1), we find that this cross-section is literally invisible; even assuming a total integrated luminosity of it amounts to one tenth of event during the whole lifetime of the LHC. This result supports quite convincingly the idea that the eventual detection of such kind of FCNC processes could give us important hints of some form of physics beyond the SM.
Regarding the choice of SM parameters, we have taken the heavy-quark masses and coupling constants given by their corresponding renormalization group running values at the renormalization scale of the process, see Section 3. The running masses and coupling constants have been explicitly computed by means of the and functions at the one-loop level (in the case of , we have made use of an specific subroutine from the CERNLIB). The obtained values are displayed in Table 1.
Next we evaluate the SUSY-QCD contribution to . The optimized set of values that we have found for the MSSM parameters is indicated in Table 2. Below we give some details on their determination, which basically follows the method of . However, here we have performed the calculation with a slightly different set of parameters; in particular, the SM parameters have been improved by using their RG running values. The SUSY-QCD corrections are driven by all possible -loop diagrams (vertex corrections, self-energy insertions and box diagrams) involving gluinos and squarks (cf. Fig. 2). In the following we describe the behavior of the SUSY-QCD contribution to the total hadronic cross section as a function of a given parameter at fixed values of the others, taking Table 2 as a reference. The corresponding results are reported in Figs. 6, 7, 8.
To begin with, we consider the curve as a function of . What we find is that the cross section grows steadily until reaching a saturation regime at values of . The shaded region is ruled out by the experimental determination of the branching ratio . The excluded region reflects that the MSSM calculation of in it yields a value of out of the experimental band allowed for this observable in the range at the -level – see  for details. It can be proven that the overall MSSM amplitude for , and the purely-SM one, both must have the same sign . We have included this restriction also in our numerical codes, so that we automatically enforce the different scannings over the MSSM parameter space to be consistent with both the experimental band and the sign criterion.
We should clarify at this point that up to now we are just retaining the SUSY-QCD (gluino-mediated) contributions for the computation of the two observables and . In other words, at the moment we neglect the EW effects both from supersymmetric particles and Higgs bosons in all these processes as if they were exactly decoupled. At due time we will switch them on in combination with the SUSY-QCD effects to evaluate the full MSSM result.
It is worth noting the strong dependence of on the trilinear coupling (cf. Fig. 6b), where we have included the approximate constraint to avoid color-breaking minima. We see that changes around two orders of magnitude along the explored range. The dependence of the cross section as a function of the SUSY-breaking scale () and the gluino mass () is given in Figs. 7a,b. In both cases decreases with the mass scale, as expected from the decoupling theorem, but this feature is more accentuated with the parameter . For instance, becomes times smaller when increasing the gluino mass from to . For the given values of the parameters in Table 2, cannot be smaller that the value indicated there, the reason being that we must respect the lower mass limits on the squark masses. For the latter we just take the LEP limits. This means that we do not exclude from our scanning regions of the parameter space where some physical squark masses can be as light as , hence we assume in our analysis 555For a general overview of the different strategies and up-to-date results concerning the squark mass bounds, see Ref. .
There is also a monotonously decreasing trend when scanning over the higgsino mixing parameter (Fig. 8a), although in this case the variation involves less than one order of magnitude in the allowed range. Concerning phenomenological bounds, values of are excluded by the observable , and also because of the bounds on the lightest squark mass. LEP bounds also exclude since otherwise the chargino mass limit, , would be violated.
The most dramatic dependence of the SUSY-QCD contribution to arises from the explicit flavor-mixing terms. This can be seen at work in Fig. 8b, where we scan over . In this case the cross section grows from (the gluino-quark-squark coupling vanishes in the limit ) to at the maximum allowed value of the flavor-mixing, viz. (see below). Larger values of are excluded by the lower experimental limits on the squark masses (we recall that the flavor-mixing parameters participate in the diagonalization of the squark mass matrix, see Eq. (12)).
It is important to realize that is also constrained by . This was advanced in Section 2. As we shall next argue, the Set I of MSSM parameters (cf. Table 2) does maximize the SUSY-QCD contribution to within the present phenomenological constraints on the MSSM parameter space. We will refrain from writing cumbersome analytical expressions for the exact formulas. However, we can provide the main analytical ingredients of the calculation in a schematic way as they will be useful to understand the physical origin of the SUSY enhancements. Let us illustrate the procedure by calculating the approximate optimal value of . The starting point in this discussion is the general form of the SUSY-QCD contributions to the cross section. It will suffice to consider the partonic cross-section since all the distinctive dynamical features are already contained in it. From the formulae of Section 2, and educated guess, we find