One-loop SQCD corrections to the decay of top squarks to charm and neutralino in the generic MSSM

In this article we calculate the one-loop supersymmetric QCD (SQCD) corrections to the decay ~ u 1 → c ~ χ 01 in the minimal supersymmetric standard model with generic flavor structure. This decay mode is phenomenologically important if the mass difference between the lightest squark ~ u 1 (which is assumed to be mainly stoplike) and the neutralino lightest supersymmetric particle ~ χ 01 is smaller than the top mass. In such a scenario ~ u 1 → t ~ χ 01 is kinematically not allowed and searches for ~ u 1 → Wb ~ χ 01 and ~ u 1 → c ~ χ 01 are performed. A large decay rate for ~ u 1 → c ~ χ 01 can weaken the LHC bounds from ~ u 1 → Wb χ 01 which are usually obtained under the assumption Br ½ ~ u 1 → Wb χ 01 (cid:2) ¼ 100% . We find the SQCD corrections enhance Γ ½ ~ u 1 → c ~ χ 01 (cid:2) by approximately 10% if the flavor violation originates from bilinear terms. If flavor violation originates from trilinear terms, the effect can be (cid:3) 50% or more, depending on the sign of A t . We note that connecting a theory of supersymmetry breaking to LHC observables, the shift from the DR to the on-shell mass is numerically very important for light stop decays.


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is kinematically not allowed and searches for ũ1 → W b χ0 1 and ũ1 → c χ0 1 are performed.A large decay rate for ũ1 → c χ0 1 can weaken the LHC bounds from ũ1 → W bχ 0 1 which are usually obtained under the assumption Br[ũ 1 → W bχ 0  1 ] = 100%.We find the SQCD corrections enhance Γ[ũ 1 → c χ0 1 ] by approximately 10% if the flavour-violation originates from bilinear terms.If flavour-violation originates from trilinear terms, the effect can be ±50% or more, depending on the sign of A t .We note that connecting a theory of SUSY breaking to LHC observables, the shift from the DR to the on-shell mass is numerically very important for light stop decays.

I. INTRODUCTION
Natural Supersymmetry requires light stops in order to cancel the quadratic divergences of the Higgs self-energies involving a top quark while the other supersymmetric partner can be much heavier [1,2].Theoretical motivation for light stops also comes from the fact that when starting at a high scale with universal squark masses, the renormalization group evolution (RGE) (known at the two-loop level [3,4,5]) generically drive the masses of the third generation squarks to lower values as for example in gravity mediated SUSY breaking scenarios (see for example [6]).In addition, light stops are also welcome in order to accommodate for the observed relic density within the MSSM [7,8,9,10,11,12] and to realize baryogensis [13,14,15,16,17,18,19,20,21].
On the experimental side, the bounds on the stop mass are much weaker than the ones on the other strongly interacting SUSY particles, i.e. squarks of the first two generations [22,23] and the gluino (see for example [24] for a recent overview of ATLAS and CMS results).Light stops might even be welcome in the light of recent LHC data for W -pair production where the observed cross section [25,26] is slightly above the SM predictions [27].This can be interpreted as a hint for light sleptons, light chargions and/or light stops [28,29].However, in order to accommodate the measured Higgs mass of around 125 GeV [30,31] rather heavy stops are required.This tension can be solved if the stop mixing angle is large (or even maximal [32]), by promoting the MSSM to the NMSSM/λSUSY [33,34] or by adding Dterm contributions [35].
Concerning the exclusion limits on stop masses from the LHC, there are still regions in parameter space in which light stops are allowed.If the mass splitting between the stop and the neutralino is bigger than the top mass, the main search channel is ũ1 → t χ0 1 and the constraints are stringent [36,37].However, if the mass difference is smaller than m t the limits on the stop mass come from searches for ũ1 → W b χ0 1 and the limits are much weaker [38,39,40,41].If the mass difference between the stop and the neutralino is even smaller than m W + m b the limits are obtained from searches for the flavour-changing decay ũ1 → c χ0 1 [42,43].The decay ũ1 → c χ0 1 has important experimental implications, both for scenarios with minimal and non-minimal flavour-violation: In the case of minimal flavor violation (MFV) [44,45,46,47,48] the decay rate is suppressed leading to a sizable stop decay length, which can be used to determine the flavour structure [49,50] and is in principle measurable at the LHC [51,52,53] 1 .The most plausible scenario with a suppressed stop decay rate ũ1 → c χ0 1 is to assume a flavour-blind SUSY breaking mechanism at some high scale Λ, for example the GUT scale.In this case, flavour off-diagonal elements in the squark mass matrices are induced by the RG for which the decay width has been calculated in Ref. [56] and the finite part of the 1-loop electroweak 1 If the decay rate for ũ1 → c χ0 1 is small, the four body decay ũ1 → b χ0 1 f f ′ [54] (also searched for at the LHC [40]) can have a significant impact on branching ratio for ũ1 → c χ0 1 [55].
In the case of non-minimal flavour violation the decay width for ũ1 → c χ0 1 can be significantly enhanced since the flavour-changing elements in the up-sector are rather poorly constrained from FCNC processes.It has been noticed in Ref. [59] (see also [60,61] for later analysis) that an enhanced branching ratio for ũ1 → c χ0 1 can weaken the bounds from ũ1 → t χ0 1 , for which a branching ration of 100% is commonly assumed in the experimental analysis, allowing for lighter stop masses.We point out that a similar effect occurs concerning the limits extracted from ũ1 → W bχ 0 1 searches.Since ũ1 → W bχ 0 1 is a three body decay, it is kinematically suppressed compared to the two body decay ũ1 → t χ0 1 .Therefore, already a much smaller amount of flavour violation, as the one necessary to affect the limits from ũ1 → t χ0 1 , would be sufficient to significantly weaken the limits extracted from ũ1 → W bχ 0 1 .This observation is especially interesting taking into account that the bounds on the stop mass from ũ1 → W bχ 0 1 are currently anyway the weakest ones.Therefore, very light stop masses for m W < m ũ1 − m χ0 < m t are allowed, especially in the case of non-minimal flavour-violation.
In this article we investigate the 1-loop SQCD corrections to ũ1 → c χ0 1 in the MSSM3 with generic flavour structure.These α s corrections are the leading ones in case of nonminimal flavor violation.Furthermore, assuming a flavour-blind SUSY-breaking mechanism at a high scale Λ the counting of the loop-effects is as follows: The leading order effect is the one-loop electroweak running from Λ to m SUSY .To this leading effect the next-to-leading order (NLO) corrections are the two-loop RGE effects [3,4,5] originating from α s and the one-loop QCD corrections to the decay width at the SUSY scale which we calculate here 4 .
The article is structured as follows: In the next section we establish our conventions and recall the tree-level expression for the decay rate for ũ1 → c χ0 1 .Sec. III describes the calculation as well as the renormalization followed by a numerical analysis IV. Finally we conclude in Sec.V.

II. CONVENTIONS AND TREE-LEVEL DECAY
In this section we define our conventions and discuss the tree-level decay width.First, we denote the term in the Lagrangian for the coupling of an up-quark u i to a up-squark ũs and a neutralino χ0 p as ũ * where P L and P R are chiral projectors.For the coupling of quarks to squarks and gluinos we introduce a similar notation: In the following, we will order the mass eigenstates for the neutralino p = 1 − 4 and of the up-squarks s = 1 − 6 in increasing order and u 3 , u 2 and u 1 correspond to the t, c and u quark, respectively.For the neutralino mass matrix we use the convention The up-squark mass term in the Lagrangian is given by where both ũL and ũR are 3-vectors in flavour space.The squark mass(-squared) matrix is given by with Here A u , m LL2 U and m RR2 U are 3 × 3 matrices in flavour space and we neglected small terms involving electroweak gauge couplings.Here we allowed for complex Yukawa couplings and used LR conventions for them and the A-terms [64].Note that in Eq. ( 5) the Yukawa couplings and not the quark masses enter which is a relevant difference since we are computing 1-loop SQCD corrections in this article5 .
Eq. ( 5) is given in the super-CKM basis which we define to be the basis in which the Yukawa couplings of the MSSM superpotential are diagonal, both for quarks and squarks, so that supersymmertry is manifest: Note that in the literature the super-CKM basis is often defined to be the basis with diagonal quark mass matrices.However, this definition has the disadvantage that the basis changes with every loop order.
We diagonalize the full hermetian 6 × 6 squark mass-squared matrix M 2 ũ and the symmetric 4 × 4 neutralino mass matrix M χ0 as where Z N and W ũ are unitary matrices.With these conventions we get for the squarkquark-neutralino couplings in Eq. ( 1): and for the squark-quark gluino vertex Here e denotes the electric charge and s W ≡ sin θ W , c W ≡ cos θ W , where θ W is the Weinberg angle.The tree-level decay width of the lightest squark into the LSP and a (massless) charm quark is given by: If the LSP is mostly bino like, we can further simplify the expression neglecting very small neutralino mixing and small charm Yukawa couplings: Note that the decay to a right-handed charm quark is enhanced by a factor 16 which can be traced back to hyper-charges.

III. CALCULATION OF THE SQCD CORRECTIONS
In this section we discuss in detail the calculation of the 1-loop SQCD corrections including our renormalization scheme.Our calculation involves the following steps: 1. Renormalization of the quark sector.
2. Renormalization of the squark sector.

Calculation of the gluon contributions to the decay width including real emission
corrections, i.e. the decay ũ1 → c χ0 1 g.

Calculation of the gluino contributions (including the cancellation of ultraviolet (UV) divergences).
We renormalize the fundamental parameters entering the decay width of the stop decay at tree-level, which receive SQCD corrections at the one loop level, in the DR scheme.These quantities are • The Yukawa couplings Y u i of the MSSM superpotential.
• The trilinear A u ij terms.
• The bilinear squark mass terms m LL2 U and m RR2 U .
We write the bare quantities of the Lagrangian (labeled with a superscript (0)) as Since we renormalize all quantities in a minimal renormalization scheme, i.e. the DR scheme, A u , m 2 Q,U and Y u are understood to be the renormalized ones in the DR-scheme.However, in the decay width of the stop, the on-shell squark mass also enters.Therefore, a conversion from the on-shell squark mass to the DR is necessary.In addition, the Yukawa couplings have to be related to the measured quark masses of the SM by running and threshold corrections.
A. Renormalization of the quark sector SQCD corrections to quark masses and Yukawa couplings can be calculated from the quark self-energies (see Fig. 1).The UV renormalization of the Yukawa couplings (in the DR-bar scheme) is given by In our approach we compute only LSZ factors corresponding to flavour-diagonal self-energies, which are also the only UV divergent ones.All other contributions from self-energies can be calculated as one-particle irreducible diagrams [79,80,81].Therefore, the LSZ factor for left and right-handed quarks is Here, the superscript a denotes the flavour-independent gluon piece, while the index b refers to the gluino piece whose finite part is in general flavour dependent: Here ε IR denotes the dimensionally regularized infrared (IR) divergence and ε the UV one while Σ gLL ii and Σ gRR ii are defined in eqs.( 22) and ( 23).

Threshold corrections
In order to determine the actual values of the Yukawa couplings we have to make the connection to the quark masses determined within the SM 6 .The self-energies with heavy virtual particles, in our case the one with squarks and gluinos, lead to threshold corrections modifying the tree-level relation v u Y u i = m u i .In order to write down these corrections we decompose the quark self-energies originating from squark-gluino loops as Since in the decay ũ1 → c χ0 1 we are dealing with external charm-quarks on the mass-shell it is sufficient to evaluate Eq. ( 22) at vanishing external momenta, i.e. neglecting finite terms of the order m 2 c /m 2 SUSY : With these notations the relation between the Yukawa couplings of the MSSM superpotential and the running quark masses of the SM (evaluated at the scale m SUSY ) is given by B. Renormalization of the squark sector As for the quarks, we compute in our approach only LSZ factors corresponding to flavourdiagonal squark self-energies (i.e.ũs → ũs transitions), while all the other contributions from squark-self energies are calculated as one-particle irreducible diagrams.The LSZ factors for the squarks then read δ Zb ũs = ∂Σ g+q ũs ũs (p 2 ) ∂p 2 Like in the quark case a refers to the gluon part and b to the gluino and squark-tadpole part and Σ g+q ũs ũs (p 2 ) denotes the sum of eqs.( 67) and (69).From the eqs.( 67)-( 69) in the appendix we find that the sum of those UV divergent parts of the squark self-energies which are independent of the external momentum (i.e. the mass-like contribution) is given by To Eq. ( 27), the divergent squark mass terms induced by the LSZ factors in eqs.( 25) and ( 26) have to be added, canceling the divergence involving m 2 ũs .In order to see the cancellation of the remaining UV divergences in Eq. ( 27), we consider the bare mass matrix which is given in the super-CKM basis ) Since the squark mixing matrix W ũ diagonalizes the renormalized mass matrix, the bare mass matrix is not diagonal in this basis but rather has the form Comparing Eq. ( 27) and Eq. ( 28) to Eq. ( 30), we observe that the counterterms and cancel the divergences.As required by supersymmetry, Eq. ( 31) equals Eq. ( 16).Therefore, no renormalization of the squark mixing matrices W is necessary in this formalism 7 .
In the numerical analysis, we will use the connection between the on-shell and the DR mass.This relation is given by

C. Gluon contributions
Here we combine the virtual gluon contributions with the real radiation (see singularities are regularized dimensionally; more precisely, we use dimensional reduction and introduce the renormalization scale in the form µ 2 e γ /(4π), where γ = 0.577... is the Euler constant.
For the vertex correction diagram due to gluon exchange (left diagram in Fig. 4) we get ũ1 and L µ = ln(µ/m ũ1 ).ξ denotes the gauge-parameter which is involved in the gluon propagator.As before, poles of the form 1/ε correspond to ultraviolet singularities, while poles of the form 1/ε 2 IR , 1/ε IR are due to infrared and collinear singularites.Finally A 0 is the tree-level amplitude (originating from Eq. ( 1)), reading To get the renormalized result A g for the amplitude, we need to add the contributions induced by the gluon part of the LSZ factors of the (massless) charm quark and the stop squark (see Eq. ( 19) and Eq. ( 25), respectively), as well as the effects induced by the renormalization constants for the coupling constants e and Y u i appearing in the tree-level squarkquark-neutralino vertex.These renormalization constants are written as Z e = 1 + δZ a e + δZ b e for the gauge coupling e and Z Y u i = 1 + δZ a Y u i + δZ b Y u i for the Yukawa coupling Y u i .The parts due gluon corrections, which are relevant in this subsection, read 8 As expected, these expressions are independent of the gauge parameter ξ.Adding up the mentioned contributions, we get the renormalized amplitude This result is, as required by consistency, again independent of the gauge parameter ξ.To get from the renormalized amplitude to the decay width is straightforward.Doing all these manipulations in d = 4 − 2ε dimensions, we get where Γ 0 is the corresponding decay width at order α 0 s given in Eq. ( 13).We now turn to the bremsstrahlung corrections (see Fig 3).Using the information given in section 1 of the appendix on the three-particle phase space and making use of the mathematica package HypExp 2.0 [82], it is straightforward to derive the decay width for ũ1 → c χ0 1 g.We obtain Adding the virtual corrections (39) and the gluon bremsstrahlung corrections (40), we get As expected, the collinear and infrared singularities canceled and the result is finite 9 .
8 The parts δZ b e and δZ b Y u i , which are due to gluino corrections, will be taken into account in the following subsection. 9As the renormalization scheme in Ref. [63] is quite different from ours, a full comparison is difficult.It was, however, possible to compare the gluino vertex correction, the virtual gluon corrections and the

D. Gluino and squark-tadpole contributions
We write the amplitude containing the tree-level and the contribution of loop diagrams involving gluinos and the squark tadpole (right diagram in Fig 2 ) as ũ1 u 2 , given in Eq. ( 57) of the appendix, denotes the genuine vertex correction involving the gluino.Furthermore, X L,R u f u i and Xũsũt originate from quark and squark self-energy diagrams, respectively.The explicit expressions read Let us briefly discuss the ultraviolet singularities in Eq. ( 42) and how they get canceled: All divergences in the off-diagonal elements of Xũsũt are canceled by the counter-terms induced through the renormalization of U ij in the squark mass matrix, while the off-diagonal elements of X L,R u f u i are finite ab initio.Therefore, we are effectively left in Eq. ( 42) with the singularities in the flavour conserving parts of X L,R u f u i and Xũsũt which originate from LSZ factors, and with the singularities present in the vertex correction Λ Using the unitarity of the squark-mixing matrices in Eq. ( 57), the latter singularities read It is straightforward to see that the remaining singularities get cancelled against those which are induced by the gluino parts δZ b e and δZ b Y u i of the renormalization constants of the gluon bremsstrahlung corrections individually.Taking into account that in Ref. [63] the two-particle phase space (and a corresponding part of the three-particle phase space) is in d = 4 dimensions and that the renormalization scale is of the form µ 2ε Γ(1 − ε)/(4π) ε ), we found that the results are in agreement.In our calculation we used a d-dimensional phase space (and introduce the renormalization scale in the form µ 2ε e γε /(4π) ε ).
FIG. 5: Ratio of the DR stop mass over its on-shell mass for m OS ũ1 = 275 GeV as generated by Eq. ( 48) and A t = 1 TeV as a function of the gluino mass for different values of the renormalization scale µ (see text).
gauge coupling e and Y u i present in the tree-level squark-quark-neutralino vertex.These renormalization constants read 10 where Z a e is given in Eq. (37).Therefore, the renormalized version of the amplitude is obtained by just taking the finite part of Eq. (42).The corresponding contribution to the decay width is then obtained by inserting the renormalized amplitude into Eq.( 13) and working out the interference term, i.e. the term proportional to α s . 10δZ b e = −δZ b e verifies that the electric charge is not renormalized by SQCD and the compatibility of δZ a Y u i + δZ b Y u i with Eq. ( 16) shows that SUSY is respected.

IV. NUMERICAL ANALYSIS
In our numerical analysis we investigate the size of the calculated SQCD corrections.For this purpose we consider the following squark mass matrix given in the DR-scheme: Here ∆ ij = δ ij M 2 ũ,ii M 2 ũ,jj parametrizes the flavour change (and is assumed to be small compared to the diagonal elements) and we choose A t = ±1 TeV.In the following, we will consider the case of m RR 33 = m LL 33 (i.e.maximal mixing).For the neutralino, which we assume to be bino like, we choose a mass of 250 GeV and use α s (m SUSY ) = 0.087 as an input.
At tree-level, the scheme for the stop mass is not defined.At the 1-Loop level the quantities of the MSSM superpotential must be renormalized in a process independent way in order to respect supersymmetry, e.g. the Yukawa couplings have to be renormalized in the DR-scheme.For consistency, also all other elements of the squark mass matrix should be renormalized in this scheme as well and should be given at the same renormalization scale.After diagonalization of the squark mass matrix, the eigenvalues correspond to DRmasses which can be translated to on-shell masses if necessary or desired.This is the case for ũ1 → c χ0 1 where the masses entering the decay width in Eq. ( 13) are on-shell masses.The shift between the DR and the on-shell mass (see Eq. 34) turns out to be numerically especially important for our scenario with a light stop because it scales like 11 m 2 g/m 2 q .In Fig. 5 we show the ratio m DR ũ1 /m OS ũ1 as a function of the gluino mass at the 1-loop level for m OS ũ1 = 275 GeV.For this we set all flavour off-diagonal elements ∆ ij in Eq. ( 48) to zero.Note that for large gluino masses the on-shell stop mass is smaller than the DR mass.This has interesting consequences for model building with light stops: Assuming that there is already a splitting between the DR squark masses of the first two generations and the stop squark (for example due to the running from the GUT scale to the SUSY scale) at the SUSY scale, then this splitting is significantly increased for heavy gluinos, making the stop even lighter.Therefore, light stop scenarios, which are interesting for the decay ũ1 → c χ0 1 , can be even generated via finite loop effects.
For the numerical analysis of the SQCD corrections to ũ1 → c χ0 1 , we choose m LL 33 = m RR 33 in Eq. ( 48) in such a way, that a given on-shell mass for ũ1 (275 GeV in our example) 11 Even though the correction is very large, pertubation theory still works, because the parametric enhancement m 2 g /m 2 q can only appear once at any loop-level.Therefore, higher loop-corrections will have the size of ordinary SQCD effects compared to the one-loop result.results after diagonalizing Eq. ( 48) and shifting the so-obtained DR squark masses to the corresponding on-shell masses 12 .This procedure we do for both, the tree-level decay width Γ tree and for the SQCD corrected version Γ 1−loop calculated in this paper.
In Fig. 6 (Fig. 7) we illustrate the effect of the one-loop contributions for positive (negative) A t for the four different sources of flavour-violation: δ RR 23 , δ LL 23 , δ RL 23 and δ LR 23 .Here we defined the ratio R = Γ 1−loop /Γ tree of the partial widths.In each of the four curves in Fig. 6 and Fig. 7 the indicated δ AB ij is put to 0.01, while the other δ's are switched off.Note that the actual numerical values of the mentioned δs drops out in this ratio to a very good approximation.We find that if bilinear terms are the only sources of flavour violation, the SQCD effects are around 10%, while if flavour violations originate from trilinear terms the corrections can reach ±50% or even more.The large corrections in the case of δ RL 23 and δ LR 23 can be traced back to the suppressed decay width for left-handed charm quarks.

V. CONCLUSIONS
In this article we computed the 1-loop SQCD corrections to the decay ũ1 → c χ0 1 in the MSSM with generic sources of flavour violation.This decay is phenomenologically very important if the mass splitting between the neutralino and the lightest stop is smaller than the top mass.In particular, we pointed out that a sizable partial width for ũ1 → c χ0 1 , which is possible in the presence of non-minimal sources of flavour violation, can significantly weaken the LHC exclusion bounds obtained from ũ1 → W bχ 0 1 where usually a branching ratio of 100% is assumed.
Working in the super-CKM basis with diagonal Yukawa couplings and renormalizing all parameters in the DR scheme, we explicitly checked for the cancellation of UV divergences and verified that SUSY relations are satisfied.In particular, in the squark sector all divergences are eliminated by flavour-conserving counter-terms to Yukawa couplings, A-terms and the bilinear terms, meaning that no renormalization of the squark mixing matrices is necessary.Concerning the gluon corrections we regularized all divergences dimensionally and verified their cancellation in a general R ξ gauge.
Numerically, we observe a large shift between the on-shell and the DR mass of the stop.Due to the inherited quadratic divergence, the shift involves a term proportional to m 2 g /m 2 q .Since for large gluino masses the on-shell stop mass is driven to smaller values compared to the DR mass, it is important to take into account this shift for model building.Taking the on-shell stop mass as in input, we find a SQCD enhancement of the decay width compared to the tree-level for ũ1 → c χ0 1 (assuming a bino like LSP) of approximately 10% if the flavour violation is due to bilinear terms and ±50% and more if the single origin of flavour violation are the trilinear terms.
For the future, a NLO SQCD calculation of ũ1 → W bχ 0 1 would be desirable and a phenomenological study of the impact of ũ1 → c χ0 1 on the exclusion bounds from ũ1 → W bχ 0 1 is planed.
The scalar products of the momenta p i , encoded in the quantities s ij = (p i + p j ) 2 /m 2 ũ1 , can be written in terms of the parameters λ 1 and λ 2 as

Loop functions
The one-loop functions which appear at various places in this appendix are defined as  In our approximation where we put m c = 0, the quark self-energy contribution with an internal squark and gluino is only needed at p 2 = 0: Σ gLL For the contribution with an internal quark and gluon we get (for arbitrary p 2 ): Σ gLR,RL f i p 2 = α s 4π C F d m q i B 0 p 2 ; m 2 q i , 0 δ f i .

23 LLFIG. 6 :
FIG.6: Ratio of the decay width including the 1-loop SQCD corrections over the tree-level decay width for different sources of flavour violation as a function of the gluino mass for A t = 1 TeV.The renormalization scale is chosen to be µ = 275 GeV.

23 LLFIG. 7 :
FIG.7: Ratio of the decay width including the 1-loop SQCD corrections over the tree-level decay width for different sources of flavour violation as a function of the gluino mass for A t = −1 TeV.The renormalization scale is chosen to be µ = 275 GeV.