Effective squark/chargino/neutralino couplings: MadGraph implementation

Abstract

We have included the effective description of squark interactions with charginos/neutralinos in the MadGraph MSSM model. This effective description includes the effective Yukawa couplings, and another logarithmic term which encodes the supersymmetry-breaking. We have performed an extensive test of our implementation analyzing the results of the partial decay widths of squarks into charginos and neutralinos obtained by using FeynArts/FormCalc programs and the new model file in MadGraph. We present results for the cross-section of top-squark production decaying into charginos and neutralinos.

Eur. Phys. J. C (2013) 73:2368DOI 10.1140/epjc/s10052-013-2368-z

Regular Article - Theoretical Physics

Effective squark/chargino/neutralino couplings: MadGraph implementation

Arian Abrahantes1,a, Jaume Guasch2,b, Siannah Pearanda1,2,c, Ral Snchez-Florit3,d

1Departamento de Fsica Terica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

2Departament de Fsica Fonamental, Institut de Cincies del Cosmos, Universitat de Barcelona, Mart i Franqus 1, 08028 Barcelona, Catalonia, Spain

3Departament dEstructura i Constituents de la Matria, Institut de Cincies del Cosmos, Universitat de Barcelona, Mart i Franqus 1, 08028 Barcelona, Catalonia, Spain

Received: 3 October 2012 / Revised: 20 February 2013 / Published online: 4 April 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 2013

Abstract We have included the effective description of squark interactions with charginos/neutralinos in the Mad-Graph MSSM model. This effective description includes the effective Yukawa couplings, and another logarithmic term which encodes the supersymmetry-breaking. We have performed an extensive test of our implementation analyzing the results of the partial decay widths of squarks into charginos and neutralinos obtained by using FeynArts/FormCalc programs and the new model le in Mad-Graph. We present results for the cross-section of top-squark production decaying into charginos and neutralinos.

1 Introduction

The Standard Model (SM) of the strong and electroweak interactions is the present paradigm of particle physics. Its validity has been tested to a level better than one per mile at particle accelerators [1]. Nevertheless, there are arguments against the SM being the fundamental model of particle interactions [2], giving rise to the investigation of competing alternative or extended models, which can be tested at high-energy colliders, such as the Large Hadron Collider (LHC) [3, 4], or the 5001000 GeV e+e International

Linear Collider (ILC) [5, 6]. One of the most promising possibilities for physics beyond the Standard Model is Supersymmetry (SUSY) [710], which leads to a renormalizable eld theory with precisely calculable predictions to be tested in present and future experiments. The simplest

a e-mail: [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

c e-mail: [email protected]

d e-mail: [email protected]

model of this kind is the Minimal Supersymmetric Standard Model (MSSM). Among the most important phenomenological consequences of SUSY models, is the prediction of new particles, the SUSY partners of SM particles, sometimes called sparticles. There is much excitement for the possibility of discovering these new particles at LHC [4, 11]. The LHC collaborations are already performing searches on these particles, and excluding portions of the SUSY parameter space, see e.g. [1227]. Recently, the CMS and ATLAS collaborations have reported a 5 standard deviations signal on a new boson particle, at a mass m 125 GeV, which is

compatible with the interpretation of a SM Higgs boson[28 31]. The CDF and D0 collaborations at the Tevatron also found (less signicant) signals compatible with this interpretation [32, 33]. This signal is also compatible with the lightest neutral Higgs boson of the MSSM (see e.g. [34]). Precision measurements and precision computations are both mandatory nowadays. Under the huge avalanche of new data at present, an accurate prediction of sparticles couplings to other particles and their production cross-section is needed. In this work we focus on the properties of the squarks the SUSY partners of SM quarks. In particular, we concentrate on the squark decay channels involving charginos and neutralinosthe fermionic SUSY partners of the electroweak gauge and Higgs bosons.

Once produced, squarks will decay in a way dependent on the model parameters (see e.g. [35]). If gluinos (the fermionic SUSY partners of the SM gluons) are light enough, squarks will mainly decay into gluinos and quarks ( q q g) [36, 37], which proceeds through a coupling con

stant of strong strength. If the mass difference among different squarks is large enough, some squarks can decay via

Page 2 of 21 Eur. Phys. J. C (2013) 73:2368

a bosonic channel into an electroweak gauge boson and another squark ( qa q b(Z, W)), and if Higgs bosons are

light enough, also the scalar decay channels are available ( qa q b(h0, H0, A0, H)) [3841], which can be dom

inant for third generation squarks due to the large Yukawa couplings. Otherwise, the main decay channels of squarks are the fermionic ones: chargino/neutralino and a quark ( q q ) [4244]. Some of those channels are expected

to be always open, given the large mass difference between quarks and squarks, and that the charginos/neutralinos are expected to be lighter than most of squarks in the majority of SUSY-breaking models. In the few cases in which these channels are closed, the squarks will decay through avour changing neutral channels [4548], or through three- or four-body decay channels involving a non-resonant SUSY particle [4954].

Here we will concentrate on the squark decay channels involving charginos and neutralinos. Their partial decay widths were computed some time ago, including the radiative corrections due to the strong (QCD) [5557], and the electroweak (EW) [42, 43, 58, 59] sectors of the theory. These radiative corrections are large in certain regions of the parameter space [42], and their complicated expressions are not suitable for their introduction in the Monte-Carlo programs used in experimental analysis. Recently Ref. [44] presented an improved description of squark/chargino/neutralino couplings, more simple to write and to introduce in computer codes. This computation combines the effective description (which includes higher order terms) with the complete one-loop description (which includes all kinetic and mass-effects factors) and denes a new effective coupling. It includes a non-decoupling logarithmic gluino mass term, which implies a deviation of the higgsino/gaugino and Higgs/gauge couplings equality predicted by exact SUSY. This deviation is important and has to be taken into account in the experimental measurement of SUSY relations. Reference [44] showed that the effective description approximates the improved description within a 10 % precision, except in special uninteresting corners of the parameter space, where the corresponding branching ratios are practically zero. Reference [44] applies the description only to squark decays. The present work expands the results of Ref. [44] by applying those results to the production cross-section of squarks at the LHC. To that end we have implemented this effective description in Mad-

Graphs [6062] MSSM framework [63], we have applied it to the partial decay widths of squarks into charginos and neutralinos and we have computed the corresponding cross-section.

In Sect. 2 we present the theoretical framework, by introducing the notation for particles and couplings. Section 3 summarizes the results for the one-loop QCD-corrections to the squark partial decay widths into charginos and neutralinos. A few details of the effective description approach described in [44] are given in this section. This will allow readers to know what we have coded into our effective MSSM model for squarks within MadGraph. In Sect. 4 we describe our Monte-Carlo tool implementation of the MSSM effective couplings. Section 5 presents the numerical analysis: numerical setup and parameter choices (Sect. 5.1), partial decay widths analysis (Sect. 5.2), and cross-section results (Sect. 5.3). Finally Sect. 6 shows our conclusions.

2 Tree-level relations and parameter denitions

Here we introduce our notation for SUSY particles and couplings. Throughout this work we will use a third-generation notation to describe quarks and squarks, but the analytic results and conclusions are completely general, and can be used for quarkssquarks of any generation. We will show numerical results only for top-squarks (t), since their signals

are the most phenomenologically interesting.

To describe the computation of the partial decay widths, we will follow the conventions of Ref. [64]. We will study the partial decay widths of sfermions into fermions and charginos/neutralinos,

f f . (1)

We denote the two sfermion-mass eigenvalues by m

fa

(a = 1, 2), with m f1 < m f2 . The sfermion-mixing angle

f is dened by the transformation relating the weak-interaction ( f a = fL, fR) and the mass eigenstate ( fa =

f1, f2) sfermion bases:

fa = R(f)ab f b; R(f) =

. (2)

By this basis transformation, the sfermion mass matrix,

cos f sin f

sin f cos f

M

fL

M2f =

2 +m2f + c2(T3 Q s2W ) M2Z mf MLRf

mf MLRf M fR

, (3)

2 + m2f + Q c2 s2W M2Z

Eur. Phys. J. C (2013) 73:2368 Page 3 of 21

becomes diagonal: R(f)M2f R(f) = diag{m2f1 , m2f2 }. M fL

is the soft-SUSY-breaking mass parameter of the SU(2)L doublet,1 whereas M

fR

2 is the soft-SUSY-breaking mass parameter of the singlet. T3 and Q are the usual third component of the isospin and the electric charge, respectively, mf is the corresponding fermion mass, MZ is the electroweak Z boson mass, and sW is the sinus of the weak mixing angle.2 The mixing parameters in the non-diagonal entries read

MLRb = Ab tan , MLRt = At / tan .

Ab,t are the trilinear soft-SUSY-breaking couplings, is the higgsino mass parameter, and tan is the ratio between the vacuum expectation values of the two Higgs doublets, tan = v2/v1. The input parameters in the sfermion sector

are then

(M

fL , MbR , MtR, Ab, At, , tan ), (4) for each sfermion doublet. From them, we can derive the masses and mixing angles:

(m

b1, mb2, b), (mt1, mt2, t). (5) For the trilinear couplings, we require the approximate (necessary) condition

A2q < 3 m2 t

+m2b + M2H + 2 , (6)

to avoid colour-breaking minima in the MSSM scalar potential [6568]. Here m

q is of the order of the average squark masses for q = t, b, and MH is the Soft-SUSY-breaking

mass parameter of the Higgs elds, see [6568] for details.

Although the tree-level chargino (+)-neutralino (0) sector is well known, we give here a short description, in order to set our conventions. We start by constructing the following set of Weyl spinors:

+ i W+, H+2 ,

i W, H1 ,

0 i B0, i W03, H01, H02 .

(7)

2 where we have dened

M =

M 2MW s2MW c

,

M0 =

M 0 MZcsW MZssW

0 M MZccW MZscW

MZcsW MZccW 0

MZssW MZscW 0

,

(9)

with M and M the SU(2)L and U(1)Y soft-SUSY-breaking gaugino masses. The four-component mass-eigenstate elds are related to the ones in (7) by

+i =

Vij +j

Uij

, i = C

i+T =

Uij j

V ij

,

j

+j

= C

0 =

N 0

N

0T ,

0

where U, V and N are in general complex matrices that diagonalize the mass-matrices (9):

UMV = MD = diag(M1, M2) (0 < M1 < M2), NM0N = M0D = diag M01, M02, M03, M04

0 < M01 < M02 < M03 < M04 .

(10)

Using this notation, the tree-level interaction Lagrangian between fermionsfermion(chargino or neutralino) reads [42]

L f f =

a=1,2

r

Lr faf + h.c.,

Lr faf = g fa

(11)

Here we have adopted a compact notation, where f is either f or its SU(2)L partner for r being a neutralino or a chargino, respectively. Roman characters a, b . . . are reserved for sfermion indices and i, j, . . . for chargino indices, Greek indices , , . . . denote neutralinos, Roman indices r, s . . . indicate either a chargino or a neutralino. For example, the top-squark interactions with charginos are obtained by replacing f t, f b, r r, r = 1, 2. The cou

pling matrices that encode the dynamics are given by

A(t)+ai = R(t)a1V i1 tR(t)a2V i2,

A(t)ai = bR(t)a1Ui2,

A(t)+a =

r A(f

)

+ar PL + A(f)arPR f .

The mass Lagrangian in this basis reads

LM

=

1

2 +,

0 MT M 0

+

1

2 1, 2, 3, 4 M0

1 2 3 4

+

h.c., (8)

12 R(t)a1 N2 + YLtW N1 +

2t R(t)a2N4 ,

1With MtL = MbL due to SU(2)L gauge invariance.

2We abbreviate trigonometric functions by their initials, like sW

sin W , c2 cos(2), tW sW /cW , etc.

A(t)a =

1 2

2t R(t)a1N4 Y tRtW R(t)a2N1 ,

Page 4 of 21 Eur. Phys. J. C (2013) 73:2368

A(b)+ai = R(b)a1Ui1 bR(b)a2Ui2, (12)

A(b)ai = tR(b)a1Vi2,

A(b)+a =

12 R(b)a1 N2 YLtW N1

2bR(b)a2N3 ,

A(b)a =

1 2

2bR(b)a1N3 + Y bRtW R(b)a2N1 ,

with YL and Y t,bR the weak hypercharges of the left-handed SU(2)L doublet and right-handed singlet fermion, and t =

mt/(2MW sin ) and b = mb/(2MW cos ) are the

Yukawa couplings normalized to the SU(2)L gauge coupling constant g. Note that each coupling is formed by two parts: the gaugino parts (G), formed exclusively by gauge couplings, and the higgsino part (H), which contains fac

tors of the quark masses, and each of these parts will receive different kinds of correction. The rst line from Eq. (12) can be rewritten as

A(t)+ai = H+ + G. (13)

The tree-level partial decay width reads

Treear = Tree fa f r

=

meffbv1

mb(Q) v1(1 + mb)

,

(15)

where mq(Q) is the running quark mass and mq is the nite threshold correction. The SUSY-QCD contributions to mq are

mSQCDb =

efft

mefftv2

mt(Q) v2(1 + mt)

2s

3 mg tan I (mb1, mb2, mg),

mSQCDt =

2s

3 mg

tan I (mt1, mt2, mg)

g2 16m3fa

fa , M2r, m2f

m2fa M2r m2f A(f

m2

(16)

where I (a, b, c) is the scalar three-point function at zero momentum transfer:

I (a, b, c)

=

a2b2 ln (a2/b2) + b2c2 ln (b2/c2) + a2c2 ln (c2/a2)

(a2 b2)(b2 c2)(a2 c2)

) 2 + A(f

)

ar

2

+ar A(f)ar (14)

with (x2, y2, z2) = [x2 (y z)2][x2 (y + z)2].

3 QCD corrections

It is known that QCD corrections to the squark partial decay widths into charginos and neutralinos can be numerically large, specially in certain regions of the parameter space [42]. An effective description of squark/chargino/neutralino couplings, simple to write and to introduce in computer codes, was given in [44]. The complete one-loop corrections to squark partial decay widths are already available [42, 69],3 but their complicated expressions are not suitable for the introduction in Monte-Carlo programs used for experimental analysis. In this article we present the results

3Reference [70] provides a computer program to compute the sfermion two-body partial decay widths at the one-loop level in the DR renormalization scheme. The comparison with Ref. [69] is not straightforward due to the use of different renormalization schemes.

of the implementation of the effective description in Mad-Graphs MSSM framework. While the EW corrections can be important in some regions of the parameter space, they do not admit a simple effective description as the one described here, and therefore one cannot compute an approximation to the cross-section as the one performed in the present work.

In the following we present a few details of the effective description approach as given in [44]. In this approach, following hints from Higgs-boson physics, an effective Yukawa coupling is dened as

effb

.

(17)

The effective description of the squark interaction consist of replacing the tree-level quark masses in the couplings (12) by the effective Yukawa couplings of Eq. (15), and use this Lagrangian to compute the partial decay width, schematically Yuk-eff = Tree(meffq) (see [44] for details). This ex

pression contains the large one-loop corrections from the nite threshold corrections (15), but it also contains higher order corrections. At this point one can make a computation that combines the higher order effects (which ignore the effects of external momenta) and the xed one-loop (which ignore the higher order effects). At the same time, this will allow us to quantify the degree of accuracy obtained by the effective description. A Yukawa-improved decay width computation have been dened in [44] and they showed that the effective description using just the Yukawa threshold corrections (15) is not enough for the squark decay widths description. The one-loop corrections develop a term which grows as the gluino mass m

g [55], which is absent in the effective Yukawa couplings (15). Therefore, the QCD corrections to squark decay widths produce explicit non-decoupling terms

4mf Mr A(f

)

Eur. Phys. J. C (2013) 73:2368 Page 5 of 21

of the sort log m

g. To understand those terms a renormalization group analysis is in order [44]. They constructed an effective theory below the gluino mass scale, which contains only squarks, quarks, charginos, neutralinos and gluons in the light sector of the theory, and integrate out the gluino contributions. Then, they found the renormalization group equations (RGE) of the gaugino and higgsino couplings, and performed the matching with the full MSSM couplings at the gluino mass scale m

g. Only the logarithmic RGE effects have been considered, neglecting the possible threshold effects at the gluino mass scale. Since the effective theory does not contain gluinos, only the contributions from the gluon has to be taken into account. We do not present here details of the renormalization group analysis and we restrict ourselves to the results that are relevant for our purpose.

The squarkchargino running coupling constant as a function of the gauge and Higgs boson couplings at the renormalization scale Q, up to O(s), is given by [44]

A(t)+ai(Q) H+ m

q (Q) 1 +

s(Q)

log

Q m

g

+G 1

s(Q)

. (18)

log

Q m

g

g term arises as a factor to each higgsino (H) or

gaugino (G) term, Eq. (13). Note that the expressions for the higgsino and gaugino couplings are different due to the different running of the gauge and Higgs-boson couplings between the scales m

g and Q. The same can be safely extended to the rest of the couplings in (12).

Then, the renormalization group running of the coupling constant can be summarized as follows: we can use effective gaugino and higgsino couplings given by [44]

geff(Q) = g

The log m

s(Q) s(mg)

2 0

1

s(Q)

,

g

log

Q m

g

s(Q) s(mg)

2 0

(19)

effb,t(Q) = effb,t(Q)

s(Q)

effb,t(Q)

1 +

,

log

Q m

g

where eff(Q) are the effective Yukawa couplings dened in (15), and 0 is the QCD -function. At this point, a simple expression for the effective description of squark/chargino/ neutralino couplings is given by Eqs. (15), (16), (19). Reference [44] showed that the effective description (19) approximates the improved computation to within 25 % for large enough gluino masses (m

g [greaterorsimilar] 1 TeV). The effects of the log terms are better visible in the gaugino-like channels, where the Yukawa couplings play no role, and the bulk of the corrections corresponds to the log terms. In the higgsino-like

channels their importance is less apparent. After introducing these expressions in computer codes a reasonable description for squark decays into charginos and neutralinos is accomplished. They can be used in Monte-Carlo generators and other computer programs that provide predictions for the LHC and the ILC to improve their accuracy, requiring little computational costs.

4 Monte-Carlo tool

The effective couplings of Sect. 3 have been implemented in a standard Monte-Carlo tool to allow their use in multiple processes. We have chosen to include them in the Mad-Graph/MadEvent framework [6062, 71].

The program MadGraph [60] automatically generates Fortran code to calculate arbitrary tree-level helicity amplitudes. Its later implementation, the MadGraph/MadEvent package [61, 62], allows the computation of physical processes at the partonic or hadronic level, and includes interfaces to hadronization routines. Recent additions allow also to compute the partial decay widths of unstable particles [71]. A number of physical models are supplied with the standard version of MadGraph/MadEvent, aside from the SM, we will be using the MSSM implementation [63]. For our computation, we use MadGraph/MadEvent 5.1.4.8 [62], and we make heavy use of its possibility to extend/modify internal physical models. MadGraph accepts inputs in the form of SUSY Les Houches Accord (SLHA) [72] le format, which allows easy interfacing with programs computing SUSY spectra and SUSY particles decay widths.

We have modied MadGraphs standard MSSM le, changing the squarkquarkneutralino/chargino vertices by the ones of expression (19). Using our model le, we are able to compute any physical processes involving these vertices, including the leading radiative corrections.

In Sect. 5.2 we compute the partial decay widths of squarks, using the recent addition to MadGraph implemented in [71]. These results are checked against an independent implementation of the partial decay widths in programs based in the FeynArts/FormCalc/LoopTools [64, 73, 74] (FAFCLT) chain, which were used in Ref. [44]. Our set of FAFCLT-based programs prepares a standard SLHA [72] input le, which contains the SUSY inputs, as well as the SUSY spectrum and mixing matrices. This SLHA le is then used as input for both, the MadGraph-based and the FAFCLT-based computation. In this way we can make meaningful comparisons of the output of both computations. We nd that both computations agree among them (see Sect. 5.2) and with Ref. [44].

In Sect. 5.3 we use our MadGraph-based programs to compute the production cross-section of squarks decaying into chargino/neutralino. We generate an SLHA input le

Page 6 of 21 Eur. Phys. J. C (2013) 73:2368

with our set of FAFCLT-based programs, which we use as input for MadGraph. MadGraph needs the full decay width of on-shell particles for the treatment of resonant propagators, while our decay programs (FAFCLT- or MadGraph-based) can only compute the fermionic decay channels. We use SDECAY 1.1a [75] for the computation of the bosonic decay channels partial decay widths. All along the computation a single SLHA le is used for all steps, ensuring consistency among the different inputs/outputs. The computing ow is the following:

1. we use the FAFCLT-based programs to write a standard SLHA input le, which includes the SUSY inputs as well as the SUSY particle masses and mixing matrices;

2. we use this le as input for SDECAY to obtain the squark decay tables at the tree-level;

3. then, we use our set of FAFCLT-based programs to replace, in the above le, the squark partial decay widths into charginos/neutralinos, and the full decay width, by using the effective coupling approach;4

4. this SLHA le is used as an input for the MadGraph/

MadEvent cross-section computation, which uses the input section for its internal parameter setup, and the full decay width in the resonant particles propagators.

In this way a cross-section at the parton level in the nal state is obtained. MadGraph/MadEvent interfaces may process further this output through parton showers, hadronization and detector simulation packages, however, this analysis is beyond the scope of the present work.

5 Numerical analysis

5.1 Numerical set up

Our programs are able to perform computations for any MSSM parameter space point. They admit SLHA [72] input for easy interaction with other programs/routines. As an example of the effects of the new included terms, we will show numerical analysis for xed values of the SUSY parameters, and make plots by changing one parameter at a time. On one side, we choose a set of SUSY parameters as described below (see Eqs. (20) and (22)) and, on the other side, SUSY parameters are chosen from the Snowmass Points and Slopes (SPS) [76] set. It should be noted that some of the SPS scenarios are already in conict with LHC data [77, 78], but we have chosen them as reference points to perform our computation since they have been studied at length in the literature.

4The result would be the same if we used our MadGraph-based programs, since we have checked that we obtain the same results. It is a matter of practical convenience.

The results for the decay widths serve as a test of correct implementation, they are checked against previous computer programs [44].

For the SM parameters we use mt = 172 GeV, mb =

4.7 GeV, s(MZ) = 0.1172, s2W = 0.221, MZ =

91.1875 GeV, 1/ = 137.035989. The renormalization

scale Q is taken to be the physical mass of the decaying squark.

First, we take for the central values of the parameters:

tan = 5, = 300 GeV, M = 200 GeV, M

fL = 800 GeV, mg = 3000 GeV, MSUSY M fR = 1000 GeV,

At = Ab = 2M fL + / tan = 1660 GeV,

(20)

where we have introduced a parameter MSUSY as a shortcut for all the SUSY mass parameters which are not explicitly given. We use the grand unication relation M =

5/3 tW 2 M for the bino mass parameter. The value of the trilinear couplings Ab,t is given by the algebraic expression, the given numerical value corresponds to the default values of the other parameters, this numerical value will change in the plots, the chosen expression allows to show plots with a signicant parameter variation avoiding colour-breaking-vacuum conditions (6). The gluino mass is chosen to be large to enhance the effects of the logarithmic terms. Note also that if the gluino decay channel is open, it will be the dominant decay channel for squarks, rendering the chargino/neutralino channels phenomenologically irrelevant. Therefore our region of interest is

m

g

+mq > mq . (21) With these input parameters, the central values for the physical SUSY particle masses are

M+ = (170.40, 337.50) GeV,M0 = (89.52, 172.28, 305.46, 338.58) GeV, m

b

= (802.05, 1000.30) GeV, mt = (720.00, 1084.87) GeV.

(22)

Of course, the parameters in Eq. (20) are just an example for illustrative purposes. We denote this parameter choice Def in the numerical analysis below. In addition we will be analyzing the following parameter space points from the SPS [76] collection: 1b, 3, 7 (see below). The corresponding values of the parameters are reproduced in Table 1. We have checked that our conclusions hold for a wide range of the parameter space.

At present the ATLAS collaboration has already put some limits on the top-squark mass from pair production [1921,

Eur. Phys. J. C (2013) 73:2368 Page 7 of 21

Table 1 Snowmass Points and Slopes parameters from Ref. [76]. Mass parameters in GeV

SPS mg MA tan M1 M2 At Ab M fL MbR MtR MqL M dR MuR

1b 916.1 495.6 525.5 30 162.8 310.9 729.3 987.4 762.5 780.3 670.7 836.2 803.9 807.5

3 914.3 508.6 572.4 10 162.8 311.4 733.5 1042.2 760.7 785.6 661.2 818.3 788.9 792.6

7 926.0 300.0 377.9 15 168.6 326.8 319.4 350.5 836.3 826.9 780.1 861.3 828.6 831.3

24, 25], the largest excluded top-squark mass at 95 % C.L. is 465 GeV [19] assuming BR(t1 t01) = 1. Our top

squark mass parameter choices are larger than the excluded ones, moreover the exclusion limits would be loosened by allowing the existence of several decay channels for the topsquark.

5.2 Partial decay widths

Our aim is the computation of the production cross-section, however, some of the numerical features of the production cross-section are traceable to the numerical behaviour of the partial and total decay widths. For this reason we show in the present section a short numerical analysis of the partial decay widths of top-squarks into charginos and neutralinos, following the computation and setup of the previous sections. We show detailed results only for the SPS scenarios, since the Def (20) scenario has been largely explored in [44].

First of all, we have made a complete check of our FAFCLT-based and MadGraph-based computations, both at the tree-level and effective approximations, and we have found perfect agreement between both methods. We have checked explicitly that for all possible squarks decay channels into charginos and neutralinos the relative deviations between both methods of calculations is within the precision of MadGraphs Monte-Carlo numerical integrations (smaller than 0.08 %, and typically of the order of 0.02 %).

The analysis of the accuracy of the effective approximation was performed in Ref. [44], using the parameter set Def (20) as an example. We have additionally checked that the same conclusions hold for the SPS scenarios analyzed in the present work, namely that the effective approximation provides a good description of the radiative-corrected partial decay widths, if the gluino mass is heavier than the top-squark mass (m

g [greaterorsimilar] 800 GeV).

For future reference, we comment briey on the partial decay widths in the Def (20) scenario without showing any plots, details are available in Ref. [44]. In this scenario, in the explored range of gluino masses, the corrections shrink in more than a 10 % the full decay width of squarks compared to the tree-level result, and the effect is highly noticed for tan > 35 growing up to a 40 % in some regions of the parameter space. As expected, the largest difference with

respect to tree-level calculation is achieved at the largest gluino mass evaluated.

We have made a complete numerical analysis for the Snowmass Points and Slopes parameters: SPS1b, SPS3, SPS5 and SPS7. SPS5 provides a very narrow range of useful SUSY parameters for our investigation due to a relatively light scalar top quark, therefore we are not showing these results in the present paper. The other SPS parameters are away of the range of SUSY parameters interesting for our analysis. Notice that the gluino mass is around 1 TeV for the three points mentioned above, and therefore the effects of the logarithmic terms is not as enhanced as in the Def (20) scenario, where m

g

3 TeV. We explore the SUSY param

eter space for SPS1b, 3 and 7 rst by scanning m

g in the

interval [400, 5000] GeV while keeping tan xed, then by

scanning tan between 3 and 50 with m

g xed. We remind the reader that the best accuracy of the effective description of squarks interactions for these points is obtained when m

g [greaterorsimilar] 800 GeV [44]. Figures 1 and 2 show a summary of the analysis. We analyzed all possible decay channels, however, in these gures we present results for the most interesting channels in the analysis of the cross-section in the next section, i.e. the decays of the squarks into the lightest chargino and the two lightest neutralinos. Figures 1, 2 show the partial decay widths in the effective approximation (labelled Eff ), results for the tree-level prediction (Tree) and xed-order one-loop prediction (1-loop) are not shown in the gures, but will be commented in the text if necessary. The effective description follows the logarithmic behaviour of the full one-loop corrections. In addition to the partial decay widths ( ) we also show the relative corrections dened as

correc =

correc Tree(mq)

Tree(mq) (23)

where the label correc is, in general, Eff for the effective description computation and 1-loop for the full one-loop corrections. Results for the tree-level computation, Tree, and the relative corrections, Eff and 1-loop, for the scenarios analyzed in this section are shown in Table 2.

Figures 1 and 2 show the effective approximation prediction for the partial decay widths, , and relative corrections, (23), of the top-squarks t1 and t2 decaying into the

Page 8 of 21 Eur. Phys. J. C (2013) 73:2368

Fig. 1 Effective approximation prediction (Eff ) for top-squarks, t1

and t2, (a) partial decay widths into 01 (solid lines), 02 (dashed lines)

and +1 (dotted lines); and (b) relative corrections, Eq. (23), in %, as

a function of mg, for SPS1b (rst row), SPS 3 (second row) and SPS 7

(third row) scenarios

lightest chargino (t1,2 b +1) and the two lightest neu

tralinos (t1,2 t 01,2), as a function of mg and tan , re

spectively. The results are presented for the SPS1b, SPS3 and SPS7 scenarios. In general, one expects the effective approximation to be valid when the gluino mass is much larger than the squark mass scale. We found that the shape

of the effective and one-loop approximation can deviate signicantly for light gluino masses 400 < m

g < 700 GeV, and in this case the effective approximation is not appropriate.

Since the study presented in this work is only relevant when the gluino is heavy, we can use the effective approximation since it is valid for m

g [greaterorsimilar] 900 GeV.

Eur. Phys. J. C (2013) 73:2368 Page 9 of 21

Fig. 2 Same as Fig. 1, but as a function of tan

Figure 1 shows that, for the SPS1b and SPS3 scenarios, (t1 t01) has positive corrections for mg [greaterorsimilar] 850 GeV.

The largest difference with respect to tree-level calculation is obtained at the largest value of the gluino mass m

g= 5 TeV (10 %). For t1 t02 and t1 b+1 decay

channels the corrections are negative and slightly decreasing (in absolute value) with m

g. Opposite to the SPS1b

and SPS3 situation, in the SPS7 scenario, 01,2 and +1 are mostly of higgsino-type and 03,4 and +2 are mostly of gaugino-type. Then, the SPS7 scenario has larger partial decay width values into the lightest chargino and neutralinos, and the radiative corrections show a different behaviour. In this scenario all decay channels receive negative corrections in the whole analyzed range of gluino mass.

Page 10 of 21 Eur. Phys. J. C (2013) 73:2368

While the (absolute value) corrections in the t1 t01 chan

nel decrease with m

g, the t1 t02 and t1 b+1 channels

have an opposite behaviour, becoming more negative as m

g

increases. For all the SUSY parameter choices, the largest partial decay width values correspond to the decay channel

t1 b+1.

Figure 2 shows the same partial decay widths and relative corrections as before, for SPS1b, SPS3 and for SPS7, as a function of tan . In SPS1b, SPS3 t1 t01 receives posi

tive corrections in almost the whole analyzed range of tan , with the exception of the region of small tan ([lessorsimilar] 6, 13 for

SPS1b, SPS3, respectively). For t1 t02, b+1 decay chan

nels, the radiative corrections are always negative. We found that, for all SPS at tan 10, Tree(t1 b+1) has a mini

mum and a steep increase for larger tan . The radiative corrections are large and negative, compensating this increase.The one-loop corrections are proportional to tan , and grow faster than the tree-level value, the effective computation, on the other hand, resums the large corrections obtained at large tan , providing a smoother behaviour. For example, in the SPS1b scenario, while Tree(t1 b+1) increases a

25 % in the range tan = 2050, the effective prediction

Eff(t1 b+1) increases only a 0.4 %.

For a gluino mass around 900 GeV, corresponding to

the nominal value of the chosen SPS, the corrections to (t1 t01) in SPS1b and SPS3 scenarios are small, be

low 1 % in absolute value in the whole tan range (see Fig. 2b). For SPS7 is larger (in absolute value), evolving from 9 % close to 5 %. For the 02 decay chan

nel, the deviations are negative: 10 % < < 7 % for

SPS1b and SPS3 and 22 % for SPS7. For 1 the situ

ation changes, Eff reaches 36 % for SPS7 and 21 % for

SPS1b and 3, in absolute values they always increase at a faster rate with tan if compared to the neutralino channels.We want to stress that neutralino channels show a nearly at behaviour for because the couplings only involve terms of t strength (1/ tan ), a suppression which wipes out

new dependence with tan introduced by mSQCDt and log terms beyond certain tan ( 10) for the effective ap

proximation. Meanwhile chargino channels contain terms of b strength ( tan ) which enhance higher order cor

rections. As pointed out above, the SPS7 scenario has the largest radiative corrections. In this scenario, 01, 02 and +1 are of higgsino-type, enhancing higgsino terms (H+)

in the generic couplings of expression (18). It is clear that large part of the dependence in tan is encoded in H+, and

those are also the terms most affected by radiative corrections. SPS1b and SPS3 scenarios have the opposite situation, 01, 02 and +1 are of gaugino-type, therefore gaugino terms (G)depending much less on tan in the effective approximationare amplied over H+.

For the heavy top-squark (t2) decaying into the light

est neutralinos and chargino (01,2, +1) as a function of the

gluino mass (m

g) and tan , Fig. 1 shows that the largest partial decay widths correspond to the chargino channel (+1)

for SPS1b/SPS3 and the second neutralino channel (02) for SPS7. The corrections show the characteristic logarithmic behaviour. For SPS1b/SPS3 the corrections are moderate to large (up to 20 %) and they are positive, except for the chargino channel in SPS1b. For SPS7 the corrections are negative, they are around 10 % for 01, 30 % for 02 and 60 % for +1. Figure 2 shows that tan has its largest impact in the chargino +1 channel, both in the partial decay width and in the radiative corrections. We found that the effective approximation moderates the impact of the radiative corrections (see Table 2). While the one-loop corrections may surpass the 100 % value in some cases (e.g. the +1

channel in SPS7 for tan > 41), meaning that the one-loop correction is not reliable, the effective computation predicts always values that, although quite large, are compatible with perturbation theory (e.g. 75 % in that case), and we can

rely on this computation.

Finally, Table 2 summarizes the results for the tree-level computation, Tree [GeV], and the relative radiative corrections, Eff [%] and 1-loop (23), of all surveyed partial de

cay widths. Numbers not shown correspond to kinematically closed channels. The results are shown for the three SPS scenarios analyzed in this paper SPS1b, SPS3, SPS7, and Def, Eq. (20). We include two sets of Def parameters: one with m

g

= 3 TeV as in Eq. (20), and one with mg = 1 TeV.

The reason is a fair comparison among the various SUSY scenarios: since the radiative corrections grow as the logarithm of m

g, the corrections in the Def parameter set are enhanced over those of the SPS, which have a m

g

1 TeV. A

quick look at Table 2 shows that the radiative corrections are quite different in each channel, which means that they will have an impact on the decay branching ratios (and production cross-sections). As an example, in SPS1b the tree-level prediction is that the largest branching ratio of the heavy top-squark corresponds to the second chargino (t2 b+2),

followed by 04 and +1, however, after computing the radiative corrections one nds that the largest branching ratio corresponds to (t2 t04) 2.8 GeV, followed by

(t2 b+1) 2.6 GeV, and (t2 b+2) 2.4 GeV

in third place.

5.3 Cross-section computation

After successfully implementing and checking our effective MSSM couplings in MadGraph,5 we have computed the cross-section of top-squark pair production6 for the Def,

5Files are available on request.

6Gauge invariance requires one to consider pair production together with single production and non-resonant diagrams, see below.

Eur. Phys. J. C (2013) 73:2368 Page 11 of 21

Table 2 Tree-level decay width, Tree [GeV], and relative radiative corrections, Eff and

1-loop [%] (23), for the various

SPS scenarios and Def (with two values of mg = 1, 3 TeV)

Decay SPS1b SPS3

Tree Eff 1-loop Tree Eff 1-loop

t1 t01 0.7927 0.8 0.2 0.7769 0.8 0.2 t1 t02 0.6577 7.7 7.3 0.6193 7.7 7.2 t1 t03 t1 t04 t1 b+1 2.099 13.7 18.0 1.903 9.4 11.5 t1 b+2 0.9378 39.9 47.8 0.4941 30.9 31.9 t2 t01 0.3262 4.4 3.7 0.3075 4.9 4.2 t2 t02 1.026 5.1 4.4 1.043 4.9 4.3 t2 t03 1.489 29.0 28.7 1.362 29.0 28.5 t2 t04 3.764 25.6 27.6 3.509 25.7 27.6 t2 b+1 2.946 11.0 21.5 2.343 6.5 3.6 t2 b+2 4.900 51.7 73.1 2.472 28.8 33.8

SPS7 Def mg = 1 TeV Def mg = 3 TeV

Tree Eff 1-loop Tree Eff 1-loop Eff 1-loop

t1 t01 1.481 5.9 6.5 0.1264 11.5 7.8 22.9 15.2 t1 t02 3.323 22.0 22.1 1.174 2.1 3.2 3.6 1.9 t1 t03 5.776 27.5 28.4 2.963 29.1 24.8 33.9 27.5 t1 t04 0.5845 40.2 40.2 5.298 23.7 20.9 26.7 22.4 t1 b+1 8.754 23.3 25.4 1.642 16.1 10.1 29.4 20.2 t1 b+2 1.493 43.4 47.3 3.737 20.5 19.3 22.0 19.7 t2 t01 0.1435 8.2 13.1 2.103 5.5 8.7 1.2 1.4 t2 t02 2.730 25.9 27.6 3.299 16.9 17.6 16.7 14.1 t2 t03 4.723 29.4 30.2 9.944 29.2 28.2 33.1 29.2 t2 t04 5.624 19.0 20.5 4.515 34.7 31.9 40.4 35.2 t2 b+1 1.371 54.9 69.6 7.531 16.9 18.6 16.8 15.0 t2 b+2 6.262 16.7 19.6 8.662 34.5 33.4 40.0 36.5

SPS1b, SPS3 and SPS7 scenarios. We have focussed on reactions where top-squarks decay into the two lightest neutralinos (01,2) and the lightest chargino (1), since these are the search channels used at the LHC [3, 4, 11]. The ATLAS collaboration has already used them to analyze LHC data and set limits on top-squark pair-production [1921, 24, 25].

We have simulated protonproton (pp) collisions at14 TeV using the CTEQ6L set of parton distribution functions. Our aim is to use the standard MadGraph input les as far as possible. By default, MadGraph sets renormalization and factorization scales dynamically. For the production of a pair of heavy particles, as is our case, these scales are set equal to the geometric mean of M2 + p2T of both particles,

where M is the mass of the particle and pT is its transverse momentum. The scale Q of the effective couplings (19) is xed at the mass of the internal top-squark for each process.

In this section we present the results for the top-squark pair production cross-section in pp collisions, followed by the decay into a quark and charginos and neutralinos, in-

cluding the effect of the radiative corrections in the effective coupling approximation.

5.3.1 (pp (q r)( q s))

The cross-section we consider in this work is7

pp qa, qa qa q r q s ,

qa = t1,2, q , q = t, b, r,s = 1, 01,2,

(24)

that is, chargino or neutralino production, associated with top and bottom quarks, which proceeds through a single or double top-squark in the intermediate states. Figure 3 shows the partonic Feynman diagrams. Figures 3ad show the double-resonant diagrams contributing to the process

7Our programs are prepared to compute any nal state r,s =

1,2, 01,2,3,4.

Page 12 of 21 Eur. Phys. J. C (2013) 73:2368

Fig. 3 Generic partonic Feynman diagrams contributing to (pp (q)( q)), (a)(d) double resonant diagrams ((pp qa qa (q)( q))),

(e)(h) single resonant diagrams, (i) non-resonant diagram

we are interested in: (pp qa qa (q)( q)), how

ever, this set of diagrams is not gauge invariant. The complete set of gauge-invariant diagrams includes also single-resonant diagrams (Figs. 3eh) and non-resonant diagrams (Fig. 3i). We have checked that quarkanti-quark channels (q q, Figs. 3d, h) are three times smaller than the gluon

gluon channel (gg), i.e. they represent less than a 30 % of the total cross-section. Both channels are included in our computation. Since all top-squarks contribute to the same nal state, one should add up all the amplitudes in Fig. 3 for t1

and t2, however, in squark search studies it is customary to

separate the t1 and t2 channels, and look for each signal sep

arately. For this reason we will show results for the cross-section with only one of the top-squarks in the intermediate states of Fig. 3. This separation is possible because we have checked that the interference effects between t1 and t2 chan

nels is less than 0.8 %. We will denote this process as

pp [ qa] X

where qa is the squark appearing in the intermediate state,

and X is the nal state, under the understanding that all diagrams (double-resonant, single-resonant, and non-resonant) contribute.

Before performing the full simulation it is useful to analyze approximations to the quantity under study. Under the narrow width approximation the total cross-section (24) can be computed as the squark pair production cross-section followed by the decay branching ratios,

pp [ qa] q r q s

pp qa qa BR qa q r

BR qa q s . (25)

In this approximation only the double-pole diagrams (Figs. 3a-d) contribute. In the present process the rst part (pp qi qi) proceeds purely through standard QCD cou

plings, the only SUSY parameters being the squark masses. All the information on SUSY couplings is in the second part of the expression, that is, in the decay branching ratios or partial decay widths, which have been analyzed in Sect. 5.2. The top-squark production cross-section has been computed to NLO-SUSY-QCD [79] and NLL-SUSY-QCD [80], the radiative corrections increase the production cross-section at the 14 TeV LHC by 2550 % depending on the topsquark mass, with a mild dependence on other SUSY parameters [79, 80].

Eur. Phys. J. C (2013) 73:2368 Page 13 of 21

Fig. 4 (a) Cross-section of pp [t1] t01 t01 and (b) KSUSY, as a function of tan and mg for the input parameters as dened in SPS1b, 3, 7

and Def (20)

In order to asses the effects of the radiative corrections, it is useful to dene a factor KSUSY as

KSUSY =

Effective

Tree , (26)

which under the narrow-width approximation of Eq. (25) can be computed as a ratio of partial decay widths:

KSUSY =

Kpartial

Kfull

Kpartial =

Eff( q q r) Tree( q q r)

Eff( q q s) Tree( q q s)

(27)

2

Kfull =

Efffull

Treefull

where full is the full decay width and Eff and Tree stand for the effective approximation and the tree-level computation. For the parameter space points of the present study, all bosonic t1 decay channels are kinematically closed,

and therefore the full decay width consists only of the

chargino/neutralino (inos) channels. Bosonic channels have some impact on the t2 full decay width. KSUSY is composed

of two factors: Kpartial, regarding the partial decay widths of the produced squarks decaying into a quark and the selected nal particles (01,2 or 1 analyzed in Sect. 5.2) and

Kfull the radiative correction to the total decay width. For the parameter space explored in the present work we have found that the radiative corrections to the total decay width are always negative (Kfull < 1). Of course, using the narrow-width approximation (25) one can not compute angular distributions or correlations, something that it is possible to perform with the MadGraph implementation.

The results for the cross-sections are summarized in Figs. 49, where we show the tree-level and effective approximation cross-section computations, as well as the KSUSY (26) factor, as a function of tan and mg.

Figures 46 present the cross-section corresponding to the lightest top-squark t1. Figure 4a shows the results for

(pp [t1] (t01)(t01)) as a function of mg and tan

for the different scenarios of the SUSY input parameters analyzed in this work. The ratio KSUSY is shown in Fig. 4b.

Page 14 of 21 Eur. Phys. J. C (2013) 73:2368

Fig. 5 (a) Cross-section of pp [t1] t02 t01 and (b) KSUSY, as a function of tan and mg for the input parameters as dened in SPS1b, 3, 7

and Def (20)

The radiative corrections are positive in all scenarios, this is due to the combination of two factors: rst, the radiative corrections to the partial decay width (t1 b01) are positive

in the scenarios SPS1b, SPS3 and Def (Figs. 1, 2 and Table 2), providing a Kpartial > 1, but, more importantly, the radiative corrections to the total decay width are negative, providing a Kfull < 1, and therefore a KSUSY > 1 (see Eq. (27)), as can be seen in Table 2. For example, in SPS1b the leading decay channels (at the tree-level) are b+1 and b+2, but they have negative radiative corrections of 13.7 %

and 39.9 %, respectively, in the end, when taking into ac

count all channels one obtains Treefull = 4.49 GeV, Efffull =

3.78 GeV, providing Kfull = (0.84)2 0.71 1/1.41, so a

Efffull = 26 % radiative correction to the total decay width

translates to a 41 % enhancement of the production cross-section. The increase of the radiative corrections KSUSY with m

g is given by the log(mg) terms (19) of the effec

tive computation. This gure shows the importance of the radiative corrections, which can enhance the cross-section by a large factor (between 1.2 and 5 in Fig. 4), and also of the newly included log(m

g) terms, the log terms pro-

duce an increase of the KSUSY-factor (and hence the cross-section) of 1.6/1.4 = 1.14, 1.4/1.2 = 1.17, 2/1.5 = 1.33,

2.4/1.7 = 1.41 in the range mg = 10005000 GeV for

the scenarios SPS1b, SPS3, SPS7, Def, respectively. The largest cross-sections are obtained in the SPS1b and SPS3 scenarios, which have a similar behaviour of the corrections. At tan = 30, the cross-section values in the ef

fective approximation are 4.08 103 pb (SPS1b) and

5.38 103 pb (SPS3), a factor 1.43 and 1.37 larger than

the tree-level prediction respectively. The SPS7 scenario provides an intermediate value of the cross-section in our analysis. For tan = 15 and mg = 926 GeV (as xed in

this scenario) the effective prediction for the cross-section is 1.82 104 pb, a factor 1.56 larger than the tree-level

prediction (see Fig. 4b). The tree-level cross-sections decrease as a function of tan in the region tan > 10, this behaviour is softened by the positive radiative corrections. We recall that the squark masses (and hence phase-space factors) also change with tan , i.e. mt1 changes from 772 GeV

to 780 GeV. The smaller values of the cross-section are obtained in the Def scenario (20), e.g. Eff = 6.54 106 pb

Eur. Phys. J. C (2013) 73:2368 Page 15 of 21

Fig. 6 (a) Cross-section (pp [t1] b+1 t01) + (pp [t1] b1t01) and (b) KSUSY, as a function of tan and mg for the input

parameters as dened in SPS1b, 3, 7 and Def (20)

for the effective computation prediction at tan = 5, a factor

2.25 larger than the tree-level prediction. The largest radiative corrections are obtained at large tan , with KSUSY =

4.81. We recall that in this scenario m

g= 3 TeV, enhancing

the radiative corrections.The results for (pp [t1] t02t01) are presented in

Fig. 5. For SPS1b/SPS3 the cross-sections are slightly larger than the (t01)(t01) channel (Fig. 4), whereas they are much larger for SPS7/Def. We can trace back this behaviour to the relative value of the partial decay widths (Figs. 1, 2, Table 2) and combinatorial factors. First of all, there is a factor 2 combinatorial enhancement factor because now the particles in the nal state are different. Then in SPS1b/SPS3 there is a slight suppression because BR(t1 t02) < BR(t1

t01), whereas in SPS7/Def there is a large enhancement because BR(t1 t02) BR(t1 t01). The radiative cor

rections (Fig. 5b) are smaller than in the (t01)(t01) channel (Fig. 4b), this is due to the fact that the radiative corrections are smaller (more negative) for the partial decay width (t1 t02) than for (t1 t01) (Table 2). This channel

in the Def scenario has a cross-section one order of magnitude larger than the (t01)(t01) channel.

To nish with the t1 production analysis, Fig. 6 shows

the cross-sections and radiative corrections for the lightest neutralino (01) and chargino (1) production mediated by t1. Since the experimental analysis does not per

form charge identication, both charge-conjugate channels (pp [t1] t01 b1) and (pp [t1] t01b+1)

contribute to the same experimental signal, therefore Fig. 6a shows the sum of both cross-sections.8 This channel provides the largest cross-section for t1-pair production in all

studied scenarios. For example, at tan = 30, the cross-

section values in the effective approximation are 1.89

102 pb (SPS1b), 2.29 102 pb (SPS3), 1.85 103 pb

(SPS7) and 3.08 104 pb (Def ). The radiative corrections

(Fig. 6b) are slightly smaller in this channel for the SPS sce-

8Bottom-squark mediated channels also contribute to this nal state, we do not take into account their contribution because they have different kinematical resonances, and the interferences with t channels

are small.

Page 16 of 21 Eur. Phys. J. C (2013) 73:2368

Fig. 7 (a) Cross-section of pp [t2] t01 t01 and (b) KSUSY, as a function of tan and mg for the input parameters as dened in SPS1b, 3, 7

and Def (20)

narios, providing KSUSY factors in the range KSUSY 1.1

1.4 (1040 % increase of the cross-section). The Def scenario has the largest radiative corrections for the reference value tan = 5 (20), but they have a different behaviour

from the previous channels (Figs. 4, 5) at large tan : they decrease instead of growing at large tan . In this scenario the leading t1 decay channels are 03,4 and +2, all of them

having negative corrections (Table 2) which grow with tan .

This provides a growing contribution to KSUSY through 1/Kfull (27). The other contribution, Kpartial, has two distinct factors: the radiative corrections to (t1 t01) have

a nearly at behaviour with tan at a value around 1822 %,

but the radiative corrections to (t1 b+1) have a strong

decreasing behaviour from 29 % (tan = 5) to 52 %

(tan = 50). This decrease in Kpartial partially compensates

for the increase due to 1/Kfull resulting in a more at behaviour with tan as compared with the t0t0 channels.

The results for the t2 mediated cross-sections are shown

in Figs. 7, 8 and 9. The values of these cross-sections are smaller than for the t1 mediated cross-sections. Fig

ure 7 presents the cross-section (pp [t2] t01t01)

and the corresponding radiative corrections. Radiative corrections are positive (except for a small corner at low tan for SPS7), enhancing the cross-section, which stays in the 107105 pb range. The largest results for the cross-section are obtained in the SPS1b and SPS3 scenarios, which have similar values, and they overlap in the plots as function of tan . At tan = 30, the corresponding val

ues in the effective approximation are 1.094 105 pb

(SPS1b) and 1.098 105 pb (SPS3). The different re

sults between these points (when plotted as a function of m

g) arise from the different nominal values of tan (Table 1). The same similarities occur in the other channels (Figs. 8, 9), but not as pronounced as the present one. In scenario Def we obtain an intermediate value of the cross-section (3.37 106 pb), and the lowest value is obtained in

the SPS7 scenario (7.29 107 pb). Note the quite distinct

behaviour of the radiative corrections (Fig. 7b) as compared with the t1 channels (Fig. 4b): in the present channel the

Def scenario has smaller radiative corrections, and a atter evolution with tan . Quite opposite, the SPS scenarios have larger radiative corrections and a steeper slope as a function

Eur. Phys. J. C (2013) 73:2368 Page 17 of 21

Fig. 8 (a) Cross-section of pp [t2] t02 t01 and (b) KSUSY, as a function of tan and mg for the input parameters as dened in SPS1b, 3, 7

and Def (20)

of tan . The radiative corrections tend to soften the slopes of the cross-sections as a function of tan . We note also a small region of negative radiative corrections for SPS7 at low tan < 7.

Figure 8 shows the results for (pp [t2] t02t01).

The cross-sections are larger than in the 0101 channel in all scenarios, being in the range of 106105 pb. The largest values are again obtained for the SPS1b and SPS3 scenarios.

In these two scenarios, the partial decay widths of t2 decay

ing into 01,2 have always positive radiative corrections in the whole explored region of gluino masses and tan (Figs. 1b, 2b), providing an enhancement to KSUSY, additional to the negative correction to the total decay width (Kfull < 1). The cross-sections for SPS7 and Def have a mild evolution with tan , and the radiative corrections are smaller than in the 0101 channel.

Finally, we present the results for the chargino-neutralino channel in Fig. 9. Again, the plots in Fig. 9 contain the sum of the two charge-conjugate modes: (pp

[t2] b+1t01) + (pp [t2] b1t01). This channel

provides the largest cross-section for t2 production in the

SPS1b, SPS3 and Def scenarios. In these scenarios the radiative corrections are smaller than previous channels, and they have a quite at evolution with tan and m

g. In con

trast, in the SPS7 scenario we nd a steep evolution with tan of both: the cross-section and the radiative corrections. This evolution is inherited from the partial decay width (t2 b+1) (Fig. 2). The radiative corrections are nega

tive in a wide region of parameters, producing a decrease of the cross-section.

Let us note that the main t2 decay channels correspond to

03,4 in the neutralino sector and +2 in the chargino sector (Table 2), together these channels are responsible for more than 60 % of the t2 decay branching ratio in all studied sce

narios. At the same time, these leading channels have large negative radiative corrections (|Eff| > 20 %), providing a

large 1/Kfull (27). For the SPS1b, SPS3, Def scenarios the radiative corrections to (t2 b+1) are quite moderate

and negative (Figs. 1, 2, Table 2) producing a small reduction of KSUSY through Kpartial. But for SPS7 the radiative corrections are larger, and have a very strong tan dependence Eff (80 %, 40 %, 75 %) for tan = (2, 5, 50)

Page 18 of 21 Eur. Phys. J. C (2013) 73:2368

Fig. 9 (a) Cross-section (pp [t2] b+1 t01) + (pp [t2] b1t01) and (b) KSUSY, as a function of tan and mg for the input

parameters as dened in SPS1b, 3, 7 and Def (20)

(Fig. 2), for tan [greaterorsimilar] 10 the negative corrections to (t2

b+1) (negative contribution to KSUSY) overcompensate the negative corrections to the total decay width full (positive contribution to KSUSY), providing an overall negative contribution to the cross-section radiative corrections.

To complete the discussion in this section, Table 3 shows the radiative correction factor KSUSY (26) for all the production channels cross-sections and all the scenarios presented in this work. Results are shown for three different values of tan , to make the comparison more meaningful. For the same reason the Def scenario is shown also for a light gluino mass m

g= 1 TeV. In general, the radiative corrections are

positive, increasing the production cross-section. As analyzed above, this is mainly (but not only) due to the fact that the radiative corrections decrease the top-squark total decay width. For m

g= 1 TeV the positive corrections pro

vide a KSUSY factor between 1.05 and 3.91, depending on the channel and scenario. Negative radiative corrections are only obtained for (pp [t2] (b+1)(t01)) in the SPS7

scenario and the Def scenario at m

g= 1 TeV, due to large

negative corrections to (t2 b+1) (Fig. 2 and Table 2),

they provide a KSUSY between 0.62 and 0.99 depending on the scenario. The largest radiative corrections are obtained at large tan . We recall that the corrections grow with the gluino mass (e.g. KSUSY = 4.81 for tan = 50, mg = 3 TeV

in the (0101) channel and Def scenario).

6 Conclusions

We have implemented and tested an effective description of squark interactions with charginos and neutralinos in the MSSM [44] into MadGraph [6062]. A careful check of our implementation has been done by comparing the computation of the partial decay width of squarks into charginos and neutralinos with FeynArts/FormCalc/LoopTools-based programs [64, 73, 74]. We nd perfect agreement. We have reproduced previous results in the literature [44]. This implementation allows to perform any Monte-Carlo computation taking into account the leading SUSY radiative corrections to squarkchargino/neutralino couplings.

Eur. Phys. J. C (2013) 73:2368 Page 19 of 21

Table 3 Effects of the radiative corrections to production cross-sections in the effective approximation, KSUSY Eq. (26), for SPS1b, SPS3, SPS7 and Def

SUSY parameters choice at different tan

KSUSY SPS7 Def

mg = 926.0 GeV mg = 1 TeV mg = 3 TeV

tan = 10 30 50 10 30 50 10 30 50

pp [t1] (t01)(t01) 1.53 1.69 1.95 1.83 2.46 3.91 2.20 2.98 4.81

pp [t1] (t02)(t01) 1.27 1.38 1.60 1.66 2.32 3.71 1.94 2.70 4.40

pp [t1] (b+1)(t01) 1.24 1.25 1.27 1.77 1.62 1.62 2.15 1.93 2.00

pp [t2] (t01)(t01) 1.28 2.00 3.31 1.20 1.28 1.40 1.37 1.47 1.62

pp [t2] (t02)(t01) 1.06 1.57 2.59 1.05 1.11 1.22 1.15 1.22 1.34

pp [t2] (b+1)(t01) 0.87 0.62 0.78 0.99 0.98 0.93 1.08 1.06 1.01

SPS1b SPS3

mg = 916.1 GeV mg = 914.3 GeV

tan = 10 30 50 10 30 50

pp [t1] (t01)(t01) 1.27 1.42 1.78 1.24 1.37 1.66

pp [t1] (t02)(t01) 1.16 1.30 1.63 1.13 1.25 1.52

pp [t1] (b+1)(t01) 1.14 1.22 1.37 1.11 1.18 1.29

pp [t2] (t01)(t01) 1.46 1.86 2.66 1.42 1.81 2.58

pp [t2] (t02)(t01) 1.45 1.88 2.69 1.42 1.84 2.62

pp [t2] (b+1)(t01) 1.48 1.56 1.66 1.44 1.55 1.69

Using this implementation, we have computed the partial decay widths of top-squarks into charginos and neutralinos in the effective coupling approximation, for sets of SUSY input parameters as dened in the Snowmass Points and Slopes SPS1b, SPS3 and SPS7, which correspond to scenarios in which the effective approximation can be applied, Eq. (21). We have also analyzed a particular scenario which we denote as Def, Eq. (20). We have checked that the effective approximation provides a good description of the radiative corrections if the gluino mass is larger than the squark mass (m

g [greaterorsimilar] 800 GeV for the chosen scenarios).

For the rst time we have computed SUSY particle production cross-sections at the 14 TeV LHC using the effective description of squark interactions. We have analyzed the cross-sections performing a comprehensive scan of the SUSY parameter space around the scenarios SPS1b, SPS3, SPS7 and Def. We have focussed on reactions involving topsquarks, giving rise to a nal state involving the lightest neutralinos and chargino (01,2, +1). These are the channels used by the CMS and ATLAS collaborations to perform squark searches. The radiative corrections are positive in most of the explored parameter space, producing an increase of the SUSY production cross-section. For a gluino mass of m

g 1 TeV they provide up to a factor 4 enhance

ment of the cross-section (Table 3). This enhancement factors are mostly (but not only) driven by the negative radiative corrections to the top-squark total decay width Kfull (27).

The corrections grow with the gluino mass (m

g). This leads to the lucky situation that, if the gluino is heavy (and hence, has a small production rate at the LHC) the radiative cor-

rections to the squarkchargino/neutralino couplings will be large, and easier to study at the LHC. On the other hand, if the gluino is light the radiative corrections to the squark chargino/neutralino couplings will be small, but then the gluino production rate at the LHC will be quite large, and will be easy to study! all in all, it is a win-win situation (provided SUSY exists at all). We leave the proof or rejection of SUSY to our experimental colleagues, and give them a new tool to explore with better precision the SUSY parameter space at the LHC or the future ILC.

Acknowledgements J.G. and R.S.F. have been supported by MICINN (Spain) (FPA2010-20807-C02-02); J.G. also by DURSI (2009-SGR-168) and by DGIID-DGA (FMI45/10); S.P. and A.A. by grant (FPA2009-09638); S.P. also by a Ramn y Cajal contract from MICINN (PDRYC-2006-000930), DGIID-DGA (2011-E24/2) and DURSI (2009-SGR-502); A.A. by an nimo-Chvere project from Erasmus Mundus Program of the European Commission and by a SANTANDER Scholarship Program for Latinoamerican students. The Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042) has supported this work. J.G. and R.S.F. wish to thank the hospitality of the Universidad de Zaragoza. A.A. wishes to thank the hospitality of the Universitat de Barcelona.

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