Three-body gravitino decays in the MSSM

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Published for SISSA by Springer

Received: June 7, 2013

Accepted: July 11, 2013

Published: August 12, 2013

Helmut Eberla and Vassilis C. Spanosb

aInstitut fr Hochenergiephysik dersterreichischen Akademie der Wissenschaften, A1050 Vienna, Austria

bInstitute of Nuclear and Particle Physics, NCSR Demokritos, GR-15310 Athens, Greece

E-mail: [email protected], mailto:[email protected]

Web End [email protected]

Abstract: We present results from a new calculation of two- and three-body decays of the gravitino including all possible Feynman graphs. The work is done in the R-parity conserving Minimal Supersymmetric Standard Model, assuming that the gravitino is unstable. We calculate the full width for the three-body decay process

eG ~01 X Y , where ~01 is the lightest neutralino that plays the role of the lightest supersymmetric particle, and X, Y are standard model particles. For calculating the squared amplitudes we use the packages FeynArts and FormCalc, with appropriate extensions to accommodate the gravitino with spin 3/2. Our code treats automatically the intermediate exchanged particles that are on their mass shell, using the narrow width approximation. This method considerably simplies the complexity of the evaluation of the total gravitino width and stabilizes it numerically. At various points of the supersymmetric parameter space, we compare the full two-body and the three-body results with approximations used in the past. In addition, we discuss the relative sizes of the full two-body and the three-body gravitino decay widths. Our results are obtained using the package GravitinoPack, that will be publicly available soon, including various handy features.

Keywords: Cosmology of Theories beyond the SM, Supersymmetric Standard Model, Supergravity Models

ArXiv ePrint: 1305.6934

Three-body gravitino decays in the MSSM

JHEP08(2013)055

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP08(2013)055

Web End =10.1007/JHEP08(2013)055

Contents

1 Introduction 1

2 The calculation 32.1 Gravitino interactions within the MSSM 32.2 Two-body decays 62.3 Three-body decays 7

3 Numerical results 10

4 Conclusions 19

1 Introduction

The gravitino, the supersymmetric partner of the graviton, is part of the particle spectrum of extensions of the Standard Model (SM) that incorporate the local version of supersymmetry, the supergravity. Depending on the mass hierarchy of these models, in the case of R-parity conservation, the gravitino can be either the stable lightest supersymmetric particle (LSP) that plays the role of the dark matter particle, or it is heavier than the LSP and thus unstable. In the latter case, it is important to calculate precisely the width of the gravitino decays to the lightest neutralino (~

01), that naturally is the LSP in this case, and other SM particles. The precise value of the gravitino lifetime is probably the most important parameter that a ects the primordial Big-Bang Nucleosynthesis (BBN) constraints, assuming that it decays during or after the BBN time. Indeed, the abundances of the light nuclei, like D, 4He, 3He and 7Li produced during the BBN epoch, provide very stringent constraints on the decays of unstable massive particles during the early Universe [120].

The astrophysical measurements of the abundances of various light elements agree well with those predicted by the homogeneous BBN calculations. Thereby, it is assumed that the baryon-to-photon ratio required for these calculations [21] agrees very well with that inferred [22] from observations of the power spectrum of uctuations in the cosmic microwave background. On the other hand, the value of that they provide is now quite precise, reducing the main uncertainty in the BBN calculation. The decays of massive unstable particles, like the gravitino, could have destroyed this concordance because the electromagnetic and/or hadronic showers they produced might have either destroyed or created the light nuclear species. In this calculation, probably the most important parameter is the lifetime of the unstable gravitino, because it dictates the cosmic time when these injections will occur [18, 19].

In the previous studies the dominant two-body decays of the gravitino have been considered, that is the processes

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eX Y , where

eX is a sparticle, and Y a SM particle.

These two-body decays dominate the total gravitino width, and in particular the channel

eG ~01 , which is kinematically open in the whole region of the neutralino LSP model, m[tildewide]

G > m~01 . This channel is also important because it provides the bulk of the electromagnetic energy produced by the gravitino decays. On the other hand, the decay

eG ~01 Z is the dominant channel that produces hadronic products. Since, these decays are of gravitational nature, the gravitino width is proportional to M2P, where MP = 1/8GN is the reduced Planck mass, and its lifetime is of the order of few seconds or more. Afterwards, the produced sparticle

eX decays almost instantaneously in comparison to the gravitational

decay, like

eX Y EW ~01 X Y , (1.1)

where, as it is indicated, the rst decay is of gravitational and the second one of electroweak nature. If the primary two-body gravitino decay channel is not open, m [tildewide]

G < m [tildewide]

X + mY , the

three-body decay

is still possible, if m [tildewide]

G > m~01 + mX + mY , through o -shell sparticles exchange. Therefore, it is important to include the three-body channels, like

eG ~01 q q, which inject hadrons in the cosmic plasma even below the threshold for the on-shell Z-boson production. This channel, together with the

eG ~01 W + W , usually have been taken into account in the previous studies. In particular, the channel ~01 W + W and its high energy behavior has been already studied in detail [18, 19, 23], due to the tricky cancellation of the unitarity violating terms that are associated with the longitudinal components of the W -bosons.

The purpose of this paper is to present the complete calculation of the gravitino two-body decays, as well as all the possible three-body decay channels

eG ~01 X Y . This calculation provides us with a comprehensive picture of the gravitino decays above and below the kinematical thresholds of the two-body decays. For the computation of the gravitino decay amplitudes we use the FeynArts (FA) and FormCalc (FC) packages [24 26]. In order to be able to generate the amplitudes with FA we have extended its generic structures to interactions with spin-3/2 particles and built a model le with all possible gravitino interactions with the particles of the Minimal Supersymmetric Standard Model (MSSM). FC has been extended so that it automatically generates a Fortran code for the numerical calculation of the squared amplitudes. As there are many gamma matrices involved we have done this within FC by using the Weyl-van-der-Waerden formalism [27, 28], as implemented into FC from [29].

Moreover, the code treats automatically the intermediate exchanged particles that are on their mass shell as unstable, using the narrow width approximation. By doing so, one avoids the double counting among the two- and three-body channels, but also the cpu-time consuming phase space integrations over the unstable exchanged particles in the propagators. It also leads to a considerable simplication of the numerical evaluation of the total three-body gravitino width.

eX ~01 X, where X is a SM particle. So, assuming that the dominant two-body channel is kinematically possible, m [tildewide]

G > m [tildewide]

X + mY , the gravitino decays to the LSP ~01 as

grav

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grav

01 X Y (1.2)

Our computer code will become publicly available soon as part of GravitinoPack [30]. This package supports the SUSY Les Houches Accord 2 format [31, 32] and works within the general avour conserving MSSM with possibly complex input parameters. It incorporates handy Fortran and Mathematica routines that evaluate two- and three-body gravitino decay widths. The code returns all partial decay widths (or branching ratios) for the individual decay channels.

As basis for our numerical study we use various benchmark points from supersym-metric models with di erent characteristics. In particular, we choose points based on the phenomenological MSSM (pMSSM), the Constrained MSSM (CMSSM) and the Non-Universal Higgs Model (NUHM). These points satisfy all the recent high energy constraints from the LHC experiments on the superpartner lower mass bounds, the Higgs boson mass

126 GeV, as well as the LHCb measurement on the rare decay Bs + that constrains supersymmetric models at the large tan regime. In addition, these points conform with the cosmological constraints on the amount of the neutralino relic density, measured by the Planck and the WMAP experiments.

Numerically we have found that the calculation of the full three-body decay amplitudes is important in the computation of the gravitino width for any gravitino mass, above and below the mass thresholds for the dominant two-body decay channels. Furthermore, we have compared the total gravitino decay width, up to the three-body level, with a previously used approximation. We have found that the full result can di er from the approximation by more than a factor of 2, especially for m [tildewide]

G [greaterorsimilar] 2 TeV. This a ects considerably the gravitino lifetime, which is an important parameter in constraining models with unstable gravitinos, using the BBN predictions. Interestingly enough, we have found that both goldstino (12) and the pure gravitino (32) spin states contribute comparably, even for

large gravitino masses, up to 6 TeV or larger.

The paper is organized as: in section 2 the details of our calculation are explained. In particular, we analyse the gravitino interactions in the MSSM and we demonstrate, using specic examples, our implementation in the FA/FC code. Moreover, the calculation of the gravitino two- and three-body decay width is discussed. In section 3 we present the numerical results, using as basis various benchmark points from di erent supersymmetric models. Finally in section 4, we summarize our ndings and give a perspective of our future work in this direction.

2 The calculation

2.1 Gravitino interactions within the MSSM

Our starting point is the Lagrangian that describes the gravitino interactions in the MSSM [33],

L()[tildewide]G,int =

i 8MP

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i2MP hD()

eGiL D()iiL

eG[, ]()aF ()a , (2.1)

with the covariant derivative given by

D()i = i + igA()aT ()a,ij j , (2.2) and the eld strength tensor F ()a reads

F ()a = A()a A()a gf()abcA()bA()c . (2.3)

The index corresponds to the three groups U(1)Y , SU(2)L, and SU(3)c with a up to 1, 3, 8, respectively, and i = 1, 2, 3. i, j, Aa, and a are the scalar, spin 1/2, gauge and gaugino elds of the MSSM, respectively, as given in the tables 1 and 2 in [33].

We write the four polarisation states of the gravitino eld in the momentum space in terms of spin-1 and spin-1/2 components as

k, 3 2

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= u

k, 12 (k, 1) ,

k, 12 =

r13 u

k, 1 2

(k, 1) +

r23 u

k, 12 (k, 0)

k, 1 2

r13 u

k, 12 (k, 1) +

r23 u

k, 12 (k, 0)

k, 32 = u

k, 12 (k, 1) , (2.4)

with k being the spatial part of the four momentum k = (k0, |k|e), and e the unit vector in the ight direction of the gravitino.

From the eld equation for the spin-3/2 particle, the so called Rarita-Schwinger equation [34], we get three equations in momentum space, which we have to fulll by the right choice of the spin-1 and spin-1/2 eld components [35],

(k, ) = 0 , (2.5)

k(k, ) = 0 , (2.6)

k m

[tildewide]

G)(k, ) = 0 , (2.7)

where m [tildewide]

G is the mass of the gravitino. We have checked that our implementation of into FC indeed fullls these three equations.

All possible structures for the gravitino interactions with the MSSM particles in eq. (2.1), are depicted in gure 1. We have extended the FA generic Lorentz le with these structures. The 78 couplings given by the coe cient vectors C[. . .], have been appended to the FA MSSM model le. Further technical details will be provided in [30].

As an illustrative example, we will exhibit how we obtain the ~

eG W + W interaction Lagrangian, and the corresponding structure struc1 in gure 1 used in our numerical calculation. For this, we will need the term in the second line of eq. (2.1) and from eq. (2.3) the part that is proportional to the SU(2)L gauge coupling constant g. In particular, we get (2)a = 3, f(2)abc = abc with 123 = 1, and A(2)1,2 = W 1,2. The relevant Lagrangian

can be written as

L =

ig 8MP

eG[, ] 3 W 1W 2 W 2W 1

. (2.8)

struc1 :

[, ] PL

[, ] PR

[, ] PL

[, ] PR

.C[F |

eG, V 1, V 2]

V 1

V 2

[, /

p ] PL [, /

p ] PR PL

[, /

p ] PL

[, /

p ] PR

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struc2 :

eG, V ]

.C[F |

struc3 :

p PL /

p PR /

p PL

p PR

eG, S]

.C[F |

struc4 :

.C[F |

eG, V , S]

Figure 1. Possible structures of gravitino interactions with MSSM particles, detailed explanation is given in the text. As all momenta are dened incoming in FA, ip.

We write the eld 3 in terms of the physical neutralino eld as, 3 = Zk,2PL ~

0k +

Zk,2PR ~

0k and the W 1,2 components in terms of the physical W -boson elds as, W 1 = (W + + W )/2, W 2 = i (W + W )/2. Thus, we get

g4MP W +W

eG[, ](Zi,2PL + Zi,2PR)~0i . (2.9)

i g

4MP [, ](Zi,2PL + Zi,2PR)

Figure 2. Feynman rule for

eG ~0i W + W four-particle interaction that corresponds to the

Lagrangian in eq. (2.10).

Making use of the reality of the Lagrangian, from the hermitian conjugate part one gets

g4MP W +W ~

0i[, ](Zi,2PL + Zi,2PR)

eG . (2.10)

From this formula we calculate the Feynman rule for the vertex in gure 2. Finally, to obtain the FA structure struc1, given in gure 1, we write the vertices for an incoming and outgoing gravitino by adding the terms from eqs. (2.9) and (2.10) as

C[~

0i,

eG, W +, W ](1) [, ]PL + C[~ 0i, eG, W +, W ](2) [, ]PR +

C[~ 0i,

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eG, W +, W ](3) [, ]PL + C[~ 0i,

eG, W +, W ](4) [, ]PR , (2.11)

with the coupling vector

C[~

0i,

eG, W +, W ](1) = i g4MP Zi,2 ,

C[~ 0i,

eG, W +, W ](2) = i g4MP Zi,2 ,

C[~ 0i,

eG, W +, W ](3) = i g4MP Zi,2

C[~ 0i,

eG, W +, W ](4) = i g4MP Zi,2 . (2.12)

Note, that C[ , , V1, V2](i) = C[ , , V2, V1](i).

Similarly, we have calculated all the other coe cient vectors C[. . .] for the struc1,2,3,4 shown in gure 1, in order to get the full list of all possible gravitino interactions in the MSSM.

2.2 Two-body decays

The gravitino decays into a fermion F , and a scalar S or a vector particle V are

eG F S f~fi, ~0j (h0, H0, A0), ~+k H ,

eG F V g, ~0j (Z0, ), ~+k W , (2.13)

where f denotes a SM fermion, quarks (u, d, c, s, t, b) and leptons (e, , , e, , ).

Its corresponding superpartners are the squarks ~fi, i = 1, 2. The four neutralino states are ~

0j, j = 1, . . . , 4 and the two charginos ~

k, k = 1, 2. With g we denote the gluon and with its superpartner, the gluino. Furthermore, in the MSSM we have three neutral Higgs bosons (two CP -even: h0 and H0, and one CP -odd: A0) and two charged Higgs bosons: H. The vector bosons are the photon , the Z-boson Z0 and the W -bosons W .

The two-body matrix element M2 is auto-generated with the help of the FA/FC pack

ages [24, 25]. The decay width for the processes in eq. (2.13) is given by

14 |M2|2 , (2.14)

where X = S, V and Nc is the color factor (3 for q ~q, 8 for the g channel, and 1 otherwise). Please note the factor 14 being the spin average of the incoming massive gravitino with spin 3

2 . As usual, the kinematic function is dened as 2(x, y, z) = (x y z)2 4yz.

In parallel, we also calculated these two-body decay processes analytically and included them into GravitinoPack. This enables us to cross-check the auto-generated FA/FC code. Details will be published in [30].

2.3 Three-body decays

The gravitino decays into a neutralino and a pair of SM particles are

eG ~0i ff ,

eG ~0i V V , V V = Z0Z0 , Z0 , W +W

eG ~0i V S , V S = (Z0, )(h0, H0, A0), W +H, W H+

eG ~0i SS , SS = (h0, H0, A0)(h0, H0, A0), H+H . (2.15) These are 19 three-body decay channels. All possible three-body decay channels for the process

eG ~0i X Y are given in table 1. It is worth to note that

and its charge conjugated process

(

m2[tildewide]G, m2F , m2X

m3[tildewide]G

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G F X

= Nc

eG ~0i W H+

eG ~0i W + H count as individual contributions, but

eG ~0i W H+) = ( eG ~0i W + H).

As before, the three-body matrix element M3 is auto-generated with the help of the

FA/FC packages. The corresponding four-momenta for the decay

eG ~0i X Y are xed by the relation k1 = k2 + k3 + k4, where X and Y are given in eq. (2.15). In our code, the calculation can be performed in three di erent center of mass systems. This option was used as an additional check for our numerical results. For illustration, we show here only the particular system xed by the kinematical relation: ~k1 ~k2 = ~q = ~k3 + ~k4 = 0. The

Mandelstam variables M21 and T are dened as

(k1 k2)2 = M21 = (k3 + k4)2 ,(k1 k3)2 = T = (k2 + k4)2 . (2.16)

The decay width for the processes in eq. (2.15) is given in di erential form by

d2 (

eG ~0i X Y )dM21 dT= Nc256 3 m3[tildewide]G14 |M3|2 . (2.17)

process number rst decay possible

eG ~0iXY of graphs eG ~XY resonances

~0if f 7 ~0i(h0, H0, A0, , Z0), f ~fl, ~fl f h0, H0, A0, Z0, ~fl

0iZ0Z0 4 ~

0i(h0, H0), Z0 ~

0k, ~

0kZ0 H0, ~

0iZ0 1 ~

0k ~

0iW +W 6 + 4pt ~

0i(h0, H0, , Z0), W + ~

j, ~

+jW H0, ~

0iZ0h0 4 + 4pt ~

0i(A0, Z0), Z0 ~

0k, ~

0kh0 A0, ~

0iZ0H0 4 + 4pt ~

0i(A0, Z0), Z0 ~

0k, ~

0kH0 A0, ~

0iZ0A0 4 + 4pt ~

0i(h0, H0), ~

0kZ0, A0 ~

0k H0, ~

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0ih0 1 ~

0k ~

0iH0 1 ~

0k ~

0iA0 1 ~

0k ~

0iW +H 5 + 4pt ~

0i(h0, H0, A0), W + ~

j, ~

+jH H0, A0, ~

0iW H+ 5 + 4pt ~

0i(h0, H0, A0), W ~

+j, ~

jH+ H0, A0, ~

0ih0h0 4 ~

0i(h0, H0), h0 ~

0k, ~

0kh0 H0, ~

0iH0H0 4 ~

0i(h0, H0), H0 ~

0k, ~

0kH0 ~

0ih0H0 4 ~

0i(h0, H0), h0 ~

0k, ~

0kH0 ~

0iA0A0 4 ~

0i(h0, H0), A0 ~

0k, ~

0kA0 H0, ~

0ih0A0 3 ~

0i(A0, Z0), h0 ~

0k ~

0iH0A0 3 ~

0i(A0, Z0), H0 ~

0k ~

0iH+H 6 ~

0i(h0, H0, , Z0), H+ ~

j, ~

+jH H0, ~

Table 1. All possible three-body decays channels of the gravitino into a neutralino and a pair of SM particles; 4pt denotes a Feynman graph with four-point interaction, like graph a) in gure 3. The indices are i = 1, . . . , 4; k = 2, 3, 4; j, l = 1, 2.

With p = (M21, m2[tildewide]G, m2~0

i )/(2M1) and k = (M21, m2X, m2Y )/(2M1), E21 = m2[tildewide]G + p2 and E23 = m2X + k2 we can write the lower and upper bound of T as

T min,max = m2[tildewide]G + m2X 2E1E3 2pk . (2.18)

We get

G ~

= Nc

256 3 m3[tildewide]G

1 4

(m [tildewide]

Gm~~0

0iXY

i )2

(mX+mY )2 dM21 Z

T max

T min dT |M3|2 . (2.19)

In general, the matrix element M3 is a sum of di erent amplitudes. As in the subsec

tion 2.1, we are using the process

eG ~0i W W + as an example to demonstrate how one calculates the resonant, the non-resonant and the interference part of eq. (2.19). For this particular process there exist 7 Feynman graphs, plotted in gure 3, which are 9 individual amplitudes. In this case only the H0 and ~

1,2 propagators can become resonant. We

~0i

W +

~0i

W +

~0i

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W +

Figure 3. Feynman graphs for the decay

eG ~0iW W +.

assume that we have H0 resonant in Mc)3: m

[tildewide]

G > m~0i + mH

0 and mH0 > 2mW . We write

M3 as

M3 =

Xi6=cMi)3+Mc)3 = Mrest3+M2,1(M21)

iM21m2H0 +imH0 H0 M2,2(M21, T ) (2.20)

with M2,1 = M(

eG ~0i +H0) and M2,2 = M(H0 W W +) and H0is the total decay

width of H0. We get

|M3|2 = |Mrest3|2 + |M2,1|2 | M2,2|2

M21 m2H0

2 + m2H0 2H0

+ 2Re Mrest3

M2,1

M2,2 i M21 m2H0

+ mH0 H0

M21 m2H0

. (2.21)

2 + m2H0 2H0

We use

limm 0

m (s m2) (2.22) and apply the narrow width approximation (NWA) for H0, H0 mH

1 (s m2)2 + m2 2

0 . We have three

terms in eq. (2.21). The rst one, nonreso, leads to the remaining non-resonant three-

body decay width. To get this, we just have to substitute M3 in eq. (2.19) by Mrest3. The

second one corresponds to the resonant part. Using dT = 2pkd cos and M21 m2H0

we get

reso(

eG ~0i W W +) = 1 4

1128 3 m3[tildewide]G

mH0 H0 pk|M2,1|2 Z

1 d cos |M2,2|2 . (2.23)

M2,2 must be independent on cos because H0 is at rest. Using eq. (2.14) we have

(

eG ~ 0iH0)= 1 4

mH0 8

pm3[tildewide]G |M2,1|2

and (H0 W W +)=

1 8

km2H0 |M2,2|2

. (2.24)

Note that (mH0 /m [tildewide]

G) p = (m2[tildewide]G, m2H0, m20

i )/(2m [tildewide]

G) is the momentum of the outgoing particles in the restframe of the gravitino. We can write eq. (2.23) as

reso(

eG ~0iW W +) = ( eG ~0iH0) (H0 W W +) H0 . (2.25)

The third term in eq. (2.21) denotes the interference. It can be written in the NWA as

if (

eG ~0i W W +) = 1 4

1 642m3[tildewide]G

1 d cos Re (Mrest3)M2,1M2,2

. (2.26)

We can generalize our formulas to include n resonances, for example, see the last column in table 1. The total matrix element is a sum of non-resonant and a sum of resonant matrix elements. The sum of the non-resonant matrix elements is denoted as before as Mrest3. Moreover, we split the sum of the resonant matrix elements, Mn3, into the

matrix element with the particular resonance j, Mj3, and the other ones, M!j3 =

Pk6=j Mk.The result for the non-resonant part is the same like in the case with only one resonance.

The resonant part is

reso(

eG ~0iXY ) = 1512 2 m3[tildewide]G

Xjpkmj j |Mj2,1|2 Z11 d cos |Mj2,2|2 , (2.27)

and the interference term reads

if (

eG ~0iXY ) = 1256 2 m3[tildewide]G

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Xjpk

1 d cos Re h Mrest3

+ 12M!j3 Mj2,1Mj2,2i

(2.28)

The factor 1/2 in front of M!j3 is necessary in order to avoid double counting of the in

terference terms within |Mn3|2. These NWA formulas, eqs. (2.27) and (2.28), are only

applicable when there is no overlap of resonances, j + k |mj mk| for all possible couples of j and k.

3 Numerical results

In our numerical analysis we choose representative points from various supersymmetric models, assuming di erent mechanism for the supersymmetry breaking. In particular, we study points based on the pMSSM [36], the CMSSM [3748] model and the NUHM [4952]. These points satisfy the sparticle mass bounds from LHC [5355] and the cosmological constraints on the amount of the dark matter as measured by the WMAP [56] and Planck [57] experiments, assuming that the dark matter is made of neutralinos. In addition, we use the branching ratio for the process Bs + as it is measured from LHCb and other

experiments [5865], Higgs mass at 126 GeV [66, 67], and the recent XENON100 [68]

direct detection bound.

As discussed in the previous section, the total three-body decay result consists of three parts: the resonant part calculated, the non-resonant part, and the interference term. The resonant part, naturally, vanishes below the threshold of the on-shell production of the

Parameters Coannihilation point A0-funnel point tan = hH02i/hH01i 30 20, Higgsino mixing parameter 2200 GeV 700 GeV

MA, A0 Higgs boson mass 1100 GeV 770 GeV(M1, M2, M3), Gauginos masses (399,1000,3000) GeV (400,800,2400) GeV At, top trilinear coupling 2300 GeV 2050 GeV

Ab, bottom trilinear coupling 2300 GeV 2050 GeV

A, tau trilinear coupling 2300 GeV 1000 GeV

m[tildewide]

qL, 1st/2nd family QL squark mass 1500 GeV 1500 GeV m[tildewide]

uR, 1st/2nd family UR squark 1500 GeV 1500 GeV m[tildewide]

dR, 1st/2nd family DR squark 1500 GeV 1500 GeV m[tildewide]

L, 1st/2nd family LL slepton 800 GeV 600 GeV m[tildewide]

eR, 1st/2nd family ER slepton 2500 GeV 2500 GeV m[tildewide]

Q3L, 3rd family QL squark 1100 GeV 800 GeV m[tildewide]

tR, 3rd family UR squark 1100 GeV 1000 GeV m[tildewide]

bR, 3rd family DR squark 1100 GeV 1500 GeV m[tildewide]

L3L, 3rd family LL slepton 400 GeV 800 GeV m[tildewide]

R, 3rd family ER slepton 2000 GeV 2000 GeV

Table 2. The pMSSM parameters used as input for the two points studied in our analysis.

intermediate particle. The interference term is numerically always very small in the studied scenarios and negligible compared to the non-resonant and resonant parts. Therefore, we will not show explicitly this term in the gures with three-body gravitino decays.

First we study two pMSSM points, one in the stau-coannihilation region (m~1 m~

01 ), with the input parameters given in table 2. At the former point, we get a neutralino dark matter relic density compatible with WMAP data by increasing the neutralino pair annihilation cross-section by adding ~

01-~1 coannihilation processes, while at the latter one by using the rapid pair annihilation of the ~

01s through the A0 resonant exchange. For the coannihilation point we get the masses m~01,2,3,4 = (399, 998, 2201, 2203) GeV, m~

+1,2 = (998, 2203) GeV, and mH

+1,2 = (674, 830) GeV, and mH+ = 774 GeV. These values will help us understand the various features appearing in the gures be discussed below.

For the funnel point we get m~01,2,3,4 = (397, 675, 703, 830) GeV, m~

In gure 4 left, the decay width for the processes

eG ~01 W + W is shown. The solid blue line is the total width (resonant and non-resonant part), while the dashed green line is the non-resonant part only. As can be seen in table 1, for this three-body decay channel, we can have a resonant behavior through the chargino (~

+1,2) or H0 exchange.

G = m~01 + mH0 , one expects to see a structure, being proportional to the resonant contribution as given in eq. (2.27), following the NWA. The size of the NWA term, for example in the case of the ~

+1 resonance, is proportional to the partial width ~

+1 ~

01 W +, and inversely proportional

JHEP08(2013)055

01 ),

and the other one in the A0-funnel region (mA0 2m~

+ = 1103 GeV.

This means, that for a gravitino mass m [tildewide]

G = m~+

j + MW and m [tildewide]

[GeV]

x [GeV]

10[Minus]30

total non-reso

10[Minus]32

total non-reso

10[Minus]34

10[Minus]36

10[Minus]38

[Minus]40

10 500 1000 1500 2000 2500 3000

10[Minus]40

600 800 1000 1200 1400

m [GeV]

JHEP08(2013)055

Figure 4. The decay width for the process

eG ~01 W + W left (right), for the stau-coannihilation (A0-funnel) pMSSM point as discussed in the text. The solid blue line is the total width (resonant and non-resonant part) of this decay, while the dashed green line is the non-resonant part only.

to the total width of ~

+1. In gure 4 left, there are very small structures that correspond to chargino thresholds m [tildewide]

G = m~+

1,2 + MW . For this case the total chargino widths are

+1,2 = (8, 144) GeV. The partial chargino width to ~

01 W + is of the order 104 GeV because 01 is almost a pure bino state, and for a chargino-neutralino-W interaction one needs a wino-wino or a higgsino-higgsino combination of chargino-neutralino. Thus the e ects are suppressed in this scenario. In gure 4 right, we show the corresponding widths for the A0-funnel point. For this point we have the total widths ~+

1,2 = (0.7, 5) GeV, and

the partial width ~

01 W + is almost three orders of magnitude bigger (O(0.1) GeV)

because the charginos are mixed states of gauginos and higgsinos and 01 is again a bino state but with non-negligible admixtures of higgsinos and winos. This makes the partial gravitino width quite large after the ~

+1W threshold.

Before we discuss other gravitino decay channels some comments are in order. First, it is important to note that below the rst threshold (the ~

+1 for this case) the knowledge of the full three-body amplitude is important in the computation of the gravitino decay width.

Looking at gure 4 one notices that the size of the width below and above the threshold is comparable, especially for the coannihilation point. Thus, using only the two body decays (or the NWA terms) one gets, in general, inadequate results. A second comment is related to the relative size of the resonant and the non-resonant contributions. If the partial width of the intermediate unstable particle, like the ~

+1 ~

01 W + at the coannihilation point, is small these two contributions are comparable. In this case the non-resonant contribution dominates and therefore in gure 4 left the blue and the green curves are very close. For example, in gure 4 left at m [tildewide]

G = 2 TeV we get reso = 1.8 1033 GeV and nonreso = 3.8 1033 GeV. This underlines the relevance of the full three-body calculation.

We now turn to gure 5, where we display the width for the channel

eG ~01 Z0 Z0.In this case, the possible resonant particles, as can be seen in table 1, are the heavier neutralinos ~

02,3,4 and the heavy CP -even Higgs boson H0. The rst kink in gure 5 left (coannihilation point) corresponds to the threshold of the two body process

+1 ~

~02 Z0, where m~02 = 998 GeV. The corresponding increase in gure 5 right, for m~02 =

[GeV]

x [GeV]

10[Minus]30

10[Minus]32

total

10[Minus]34

non-reso

total non-reso

10[Minus]36

10[Minus]38

10 500 1000 1500 2000 2500 3000

[Minus]40

10[Minus]40

600 800 1000 1200 1400

m [GeV]

JHEP08(2013)055

Figure 5. The decay width for the process

eG ~01 Z0 Z0 left (right), for the stau-coannihilation (A0-funnel) pMSSM point as discussed in the text. The lines are as in gure 4.

10 500 1000 1500 2000 2500 3000

[GeV]

x [GeV]

10[Minus]30

10[Minus]32

total

10[Minus]34

total

10[Minus]36

non-reso

10[Minus]38

non-reso

10[Minus]38

[Minus]40

10[Minus]40

600 800 1000 1200 1400

m [GeV]

Figure 6. The decay width for the process

eG ~01 h0 h0 left (right), for the stau-coannihilation (A0-funnel) pMSSM point as discussed in the text. The lines are as in gure 4.

675 GeV, is much stronger because the partial width for the channel ~

02 ~

01 Z0 is three

order of magnitudes bigger, while the total width ~02 is almost ten times smaller. For a neutralino-neutralino-Z interaction both neutralino states must have a higgsino component.

As already stated, in the coannihilation scenario the lightest neutralino is almost a pure bino state which suppresses the ~

02 ~

01 Z0 coupling. In addition, the e ects of the

thresholds of the neutralinos ~

03,4 with masses 702 and 829 GeV (see gure 5 right), can be seen also in the dashed line denoting the non-resonant contribution.

In gure 6, we study the channel

eG ~01 h0 h0. The possible resonant exchanged

particles are the same as in

eG ~01 Z0 Z0, namely the heavier neutralinos ~02,3,4 and the heavy CP -even Higgs boson H0. In the coannihilation point (left) the e ect of the enhancement after the threshold of the ~

02 resonance at m [tildewide]

1100 GeV is quite suppressed,

because the ~

02 ~

01 h0 partial width is small as the ~

02 ~

01 Z0 before. On the other

hand, the ~

03,4 ~

01 h0 widths are much bigger, producing this large structure in the gravitino decay width across the almost degenerate ~

03,4 thresholds at m [tildewide]

2300 GeV. At

the A0-funnel point the situation is di erent. The threshold e ect across the ~

02 resonance

at m [tildewide]

G = 801 GeV is much bigger due to the larger ~

02 ~

01 h0 decay width. Moreover, the thresholds of the heavier neutralinos are also visible.

[GeV]

x [GeV]

10[Minus]30

10[Minus]32

total

total non-reso

10[Minus]34

non-reso

10[Minus]36

10[Minus]38

[Minus]40

10 500 1000 1500 2000 2500 3000

10[Minus]40

600 800 1000 1200 1400

m [GeV]

JHEP08(2013)055

Figure 7. The decay width for the process

eG ~01 Z0 h0 left (right), for the stau-coannihilation (A0-funnel) pMSSM point as discussed in the text. The lines are as in gure 4.

10 400 600 800 1000 1200 1400

10[Minus]28 x

1 x

total

0.5

~01

Gwidth[GeV]

~0 1XY)/

10[Minus]30

total

W +W

0.2

Z0Z0

0.1

10[Minus]32

0.05

10[Minus]34

~01

0.02

W +W

Z0Z0

0.01

[Minus]36

0.005 400 600 800 1000 1200 1400

m [GeV]

Figure 8. The three-body decay widths of the gravitino decaying into ~

01 and qq ,ll, W -pairs, and Z-pairs at the A0-funnel point; total denotes total which is the full two-body width plus the sum of the non-resonant part of three-body decay widths with ~

01; qq stands for the sum over all six quark avors and ll for the sum over the three charged lepton and three neutrino avors. The red dotted lines denote the two-body decay

eG ~01. In the right gure we display the corresponding branching ratios for the decay channels plotted in the left gure.

Figure 7 shows the width for the decay

eG ~01 Z0 h0 as a function of m [tildewide]

G. As in gure 4, the e ect of the thresholds is not important for the coannihilation point (left).

In contrast, especially the ~

02 threshold can be seen at the A0-funnel point (right). Note again, that at the coannihilation point (left) ~

01 is a pure bino state. Therefore, the decays

01 Z0 are suppressed and the non-resonant contribution is dominant.

Closing the discussion on the gravitino widths at these two representative pMSSM points, we show in gure 8 the three-body decay widths of the gravitino decaying into ~

01 together with qq, ll, and W, Z-boson pairs at the A0-funnel point. We also show the two-body decay channel

eG ~01 as a reference, because it is dominant for small m [tildewide]

G. In the left gure we also show total which is the full two-body width plus the sum of the non-

resonant part of three-body decay widths with ~

01, denoted by total. In the right gure we show the relative quantities in terms of total; qq stands for the sum over all six quark

avors,

Pi=1,6 reso + nonreso(

eG ~01qiqi) and ll, qi = u, d, c, s, t, b, and ll the sum of the

02,3,4 ~

1 x

total

~01

0.5

total

0.5

~0 1XY)/

~01

0.1

0.05

W +W

Z0Z0

0.05

W +W

Z0Z0

0.01

0.005

0.001 500 1000 1500 2000 2500 3000

m [GeV]

JHEP08(2013)055

Figure 9. The branching ratios for the CMSSM (left) and NUHM point (right). The curves are as in gure 8 right.

three charged lepton and three neutrino avors,

Pj=1,3

nonreso + reso(

eG ~ 01l+jlj) +

nonreso + reso(

eG ~01lj lj)

, lj = e, , . The decay width summing up the decays into all fermion pair can reach 38%, into the W -boson pair 5.6% and into the Z-boson pair1.5% in the shown range. The analogous plots for the coannhilation point look similar but the decays into W - and Z-boson pairs are suppressed by about two orders of magnitude due to the pure bino state of the LSP as already discussed before.

In addition to these two pMSSM points, we study two other representative points: one based on the CMSSM and another based on the NUHM. The CMSSM is characterized by ve parameters: the universal soft mass for scalar particles m0, the universal soft gaugino mass m1/2, the universal trilinear coupling A0, the ratio of the two vacuum expectation values of the Higgs doublets tan = v2v1 and the sign of the Higgsino mixing mass . In the so-called NUHM1 model [69] there is one extra free parameter, the soft mass of the Higgs doublets. In our analysis, this mass is xed by the absolute value of the Higgsino mixing mass at the electro-weak symmetry breaking scale, that we use as input parameter. The values of the parameters m0, m1/2 and A0 for both models are dened at the GUT scale.

The CMSSM point we are using is dened as: m0 = 1150 GeV, m1/2 = 1115 GeV, tan = 40, A0 = 2.5 m0 = 2750 GeV and sign()> 0. Using the package SPheno [70, 71]

this point yields mh 126 GeV. Quite similar results are obtained by using the FeynHiggs

package [72, 73]. Furthermore, this point is compatible with the WMAP bound on the neutralino relic density and it delivers acceptable values for the decay Bs + and the dark matter direct detection cross-section. This point belongs to the stau-coannihilation region of the CMSSM and has been discussed in [74]. On the other hand, the NUHM1 point is dened as m0 = 1000 GeV, m1/2 = 1200 GeV, tan = 10, A0 = 2.5 m0 = 2500 GeV

and = 500 GeV. At this point the light Higgs mass is mh = 126.5 GeV and it is also compatible with other cosmological and accelerator constraints [69].

In gure 9 we present the various branching ratios for the important gravitino decay channels, as functions of the gravitino mass, as we have done in gure 8 right. At the CMSSM point the four neutralino masses are m~01,2,3,4 = (485, 919, 1824, 1826) GeV. The ~

01 is almost a pure bino with its higgsino components less than 0.1%. We notice that the

0.1

0.08

0.06

all ~

~0i

all ~

~0i

0.04

~01

0.02

~01

0. 500 1000 1500 2000 2500 3000

m [GeV]

Figure 10. The ratio nonreso(

eG ~01 X Y )/ 2B (green curve) and

Pi=1,...,4 nonreso(

~0i X Y )/ 2B (red curve), for the CMSSM (left) and NUHM (right) point. 2B is the total gravitino two-body decay width.

branching ratios for the channels ~

01 ll and ~

01 qq are approximately of the order of 10% after the mass threshold of the 500 GeV. The two-body channel ~

01 clearly dominates in the range of small gravitino masses, but after 2000 GeV it becomes comparable to the ~

01 ll/qq channels. On the other hand, the branching ratio for the ~

01W +W channel is almost an order of magnitude smaller, and for the channel ~

01Z0Z0 a further order of

magnitude smaller.

Figure 9 right displays the branching ratios for the NUHM point. At this point m~01,2,3,4 = (470, 511, 539, 1025) GeV and there ~

JHEP08(2013)055

01 has a signicant higgsino component.

The two higgsino components (01,2) of the LSP amount to approximately 25% each. In comparison to the CMSSM point, the bino component decreased from almost 100% to about 50%. The wino component of ~

01 is very small. Therefore, the channels which couple to the bino and wino components, like ~

01 ll/qq except that with stops, are suppressed by a factor of 2 or more, in comparison to the CMSSM case. The channels that depend on the higgsino content of ~

01, like ~

01W +W /Z0Z0 are enhanced by approximately a factor of 2. For both the CMSSM and NUHM cases displayed in gure 9, the not shown two-body channels, as ~

+jW , ~

0iZ0 andg, provide the bulk of the remaining contribution to the total gravitino decay width. We notice that also in this case for m [tildewide]

G [greaterorsimilar] 2000 GeV the

channels ~

01 ll/qq are comparable to the two-body channel ~

01.

In gure 10 we present the ratio nonreso(

eG ~01 X Y )/ 2B (green solid curve) and the

Pi=1,...,4 nonreso(

eG ~0i X Y )/ 2B (red solid curve) as function of the gravitino mass for the CMSSM (left) and NUHM (right) point, we discussed before. 2B denotes the total two-body width of the gravitino. These gures give us an idea of the contribution of the heavier neutralino states in comparison to ~

01, especially for the non-resonant contribution of the three-body decay channels ~

0i X Y . The spikes in both cases, just before m [tildewide]

G = 600 GeV

are due to the threshold of the two-body decay

eG ~01 Z0. Especially for the NUHM case at right, the second smaller spike corresponds to the almost degenerate threshold for the processes

eG ~02,3 Z0. The structures in the region m [tildewide]

G 1000 GeV correspond to

threshold

eG ~02 Z0, for the CMSSM case and to

eG ~04 Z0 for NUHM.

10[Minus]28

full approx.

3-body non-reso

approx.

full approx.

3-body non-reso

approx.

[GeV]

10[Minus]30

[GeV]

10[Minus]32

10 500 1000 1500 2000 2500 3000

[Minus]34

m [GeV]

Figure 11. Comparison between the full gravitino width, that is the contribution from the two-body decays and the three-body non-resonant part (black curve) and the approximation (red dashed curve). The green curve is the three-body non-resonant part only and the dark green dashed curve the corresponding approximation. The left (right) gure correspond to the CMSSM (NUHM) point as described in the text.

We see that for m at 3 TeV the ratio

Pi=1,...,4 nonreso(

eG ~01 X Y )/ 2B, both for the CMSSM and the NUHM point. On the other hand, for the CMSSM point for m [tildewide]

G up to 1 TeV the ratio

nonreso(

eG ~01 X Y )/ 2B dominates over the other neutralino contributions. For the NUHM point (right gure) the dominance of the ~01 ends just before m [tildewide]

G = 600 GeV.

Furthermore, from gure 10 one can see the size of the total non-resonant contribution of the three-body decay channel

eG ~0i X Y relative to the dominant two-body decay width. For the CMSSM (NUHM) point, in the region up to m [tildewide]

G [lessorsimilar] 1 TeV (650 GeV) the

non-resonant part of the three-body decays into ~

JHEP08(2013)055

eG ~0i X Y )/ 2B is four

times larger than the ratio nonreso(

01 is about 6% and decreases to 2% (1%)

for m [tildewide]

G > 2 TeV.

An important comment, however, is in order. The numerical smallness of the contribution of the non-resonant three-body decay width can be misinterpreted to consider the three-body decays to be as unimportant in comparison to the two-body decays. Discussing the details of various individual three-body channels in the gures 47 we have seen that below the two-body mass thresholds, the exact knowledge of the gravitino three-body decay widths is important in the calculation of the corresponding branching ratios in the whole range of m [tildewide]

G. Thus, it is essential to calculate all possible three-body decay channels for the process

eG ~0i X Y , in order to have the complete information both for the total gravitino width and the individual branching ratios, for any gravitino mass.

In gure 11 we compare the full result for the gravitino decay width (black curve), that corresponds to the sum of the two-body decays and the non-resonant part of the three-body decays

eG ~0i X Y , with a commonly used approximation (red dashed curve). In this approximation one calculates the gravitino width using the two-body channels ~

0i(, Z0, h0, H0, A0) and the three-body channels ~

01 f f, where f can be either lepton or quark. The approximation is relatively good up to m [tildewide]

G = 1000 GeV, di ering from the full result by 20%, for the CMSSM point. For the NUHM point this di erence is 25%.

But for larger gravitino masses (m [tildewide]

3000 GeV), where almost all two-body channels

1010 x

108

106

104

total[s]

full approx.

500 1000 1500 2000 2500 3000

Figure 12. The gravitino lifetime as a function of the gravitino mass for the CMSSM point. The blue curve is the lifetime as calculated including all the two-body and three-body channels in our study and the red dashed curve corresponds to the approximation described in the text to gure 11.

contribute the di erence is quite important. The approximation underestimates the full result by a factor of 2.7 for the CMSSM point and by 2.9 for the NUHM. Comparing the non-resonant contribution of the three-body decays we get a di erent picture. For the CMSSM point, the approximation (dark green dashed curve) describe quite well the full non-resonant part (light green solid curve). The reason for this is that the dominant bino content of ~

01 leads to the dominance of the three-body decay channel ~

01 f f, as we have seen also in gure 9. On the other hand, at the NUHM point the approximation are equally important, the approximation is almost a factor of 2 smaller than the complete result for m[tildewide]

G = 3 TeV. This is due to the fact that the three-body channels ~

01 W + W /Z0 Z0 etc. are not included in the approximation but play an important role in this scenario.

The relevance of the calculation of the full three-body gravitino decay width is also underlined in gure 12. For the CMSSM point, we display the gravitino lifetime in seconds using the full result (blue curve) and the approximation described above (red dashed curve). The corresponding gure for the NUHM point is quite similar. Reecting the behavior of gure 11 the approximation up to m [tildewide]

G = 1 TeV is relatively good, but for larger gravitino masses overestimates the gravitino lifetime by a factor of 2 and more. As already mentioned, the precise calculation of the gravitino lifetime especially in the range m [tildewide]

2 4 TeV,

is important because one thinks that the models with unstable gravitinos can explain the so-called Lithium cosmological puzzle [18, 19]. This is due to the fact that the lifetime of the gravitino for these gravitino masses is 104 s and shorter, time scales that can be quite

relevant to BBN predictions for the primordial light elements abundances, and especially for the 6Li and 7Li isotopes. Thus, our calculation is of importance in the phenomenology of supersymmetric models with unstable gravitinos.

The use of the Weyl-van-der-Waerden formalism for the calculation of the gravitino width allows one to explicitly compute the individual contributions of the four gravitino spin states to the total width. It is known that the gravitino in broken superysymmetric models as studied here, acquires its mass by absorbing the goldstino 12 modes.

JHEP08(2013)055

m [GeV]

1. x

0.8

spin 32

spin 12

Figure 13. The relative contribution of the 12 gravitino states (green region up to the black curve)

to the total gravitino width, as a function of the gravitino mass. The remaining part (salmon color) corresponds to the relative contribution of the 32 spin states to the total gravitino decay width.

For this gure the CMSSM point parameters are used.

In gure 13, we display the sum of the contributions of the 12 gravitino spin states

(green region), relatively to the sum of the 32 contributions (salmon region) for the

CMSSM point. Again the NUHM case is very similar and thus it is not shown. We see that for m [tildewide]

G up to about 3 TeV the 12 states contribute up to 40% to the total decay

width. The remaining 60% is due to the 32 spin states. Eventually, for larger gravitino

masses the 12 spin states contribute about 30% of total width, this contribution remaining

constant for much larger values of the gravitino mass. The reason for this is that in the gravitino decay amplitudes the center of mass energy s of the process is actually the gravitino mass m [tildewide]

G. Hence, we are far from the high-energy limit (s m

[tildewide]

0.6

0.4

0.2

0. 1000 2000 3000 4000 5000 6000

JHEP08(2013)055

m [GeV]

G), in which

one expects the goldstino (12) components to be dominant.

4 Conclusions

We have presented results from a new calculation of the gravitino decay width in the context of supersymmetric models, where R-parity is conserved. This calculation includes all two-body decay channels

eX Y where

eX is a sparticle decaying as

eX ~01 X. X and Y are

SM particles, and ~01 is the lightest neutralino that plays the role of the LSP. In addition, we have also calculated the contributions of the three-body decay channels

eG ~0i X Y .Especially the computation of the three-body decays is quite a complicated task. For this purpose we have used the packages FeynArts and FormCalc that perform symbolic analytic calculations. In those packages we have added appropriate modules in order to accommodate them for particles with spin 3/2 like the gravitino. Furthermore, we have used the Weyl-van-der-Waerden formalism which considerably simplies the treatment of the complex spin structure of the three-body gravitino decay amplitudes and enables us to calculate the individual contributions of each gravitino spin state to the total decay width.

An important advantage of our calculation of the gravitino three-body decay amplitudes, is that it treats automatically the intermediate exchanged particles that are on their

mass shell by using the narrow width approximation. In such a way, we not only avoid the double counting among the two- and three-body channels, but we also improve the performance of the numerical phase space integration over the unstable exchanged particles in the propagators.

In our numerical analysis, we have chosen four representative points: two points based on the pMSSM model and two other based on the CMSSM and NUHM1 models. These points satisfy all the recent high energy experimental constraints from LHC and LHCb, but also the cosmological constraints from WMAP, Planck, and XENON100.

We have found that the knowledge of the complete three-body decay amplitude with a neutralino is important in order to compute precisely the gravitino width for any gravitino mass, not only below the mass thresholds of the dominant two-body decay channels, but also above them. Moreover, comparing the full gravitino decay width, up to the three-body level, with a previously used approximation we have demonstrated that the full result can be quite di erent, especially for m [tildewide]

G in the range 24 TeV. This di erence is reected also in the gravitino lifetime that we know is an important parameter to constraint models with unstable gravitinos adopting the BBN predictions. In addition, we have found that all the gravitino spin states (12 and 32) contribute at the same order of magnitude, even for

quite large gravitino masses.

Our results have been calculated using the code GravitinoPack that will be publicly released soon. This package incorporates various Fortran and Mathematica routines that compute all the discussed two- and three-body gravitino branching ratios and the gravitino decay width. In addition, GravitinoPack will calculate analogous decay widths for the complementary case where the gravitino is stable and the LSP.

Acknowledgments

This work is supported by the Fonds zur Frderung der wissenschaftlichen Forschung (FWF) of Austria, project No. I 297-N16. The authors thank Thomas Hahn for helpful discussions regarding the implementation of particles with spin 3/2 into FeynArts and they are grateful to Walter Majerotto for the careful reading of the manuscript. V.C.S. was supported by Marie Curie International Reintegration grant SUSYDM-PHEN, MIRGCT-2007-203189.

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We present results from a new calculation of two- and three-body decays of the gravitino including all possible Feynman graphs. The work is done in the R-parity conserving Minimal Supersymmetric Standard Model, assuming that the gravitino is unstable. We calculate the full width for the three-body decay process ... X Y , where ... is the lightest neutralino that plays the role of the lightest supersymmetric particle, and X, Y are standard model particles. For calculating the squared amplitudes we use the packages FeynArts and FormCalc, with appropriate extensions to accommodate the gravitino with spin 3/2. Our code treats automatically the intermediate exchanged particles that are on their mass shell, using the narrow width approximation. This method considerably simplifies the complexity of the evaluation of the total gravitino width and stabilizes it numerically. At various points of the supersymmetric parameter space, we compare the full two-body and the three-body results with approximations used in the past. In addition, we discuss the relative sizes of the full two-body and the three-body gravitino decay widths. Our results are obtained using the package GravitinoPack, that will be publicly available soon, including various handy features.

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Title

Three-body gravitino decays in the MSSM

Author

Eberl, Helmut; Spanos, Vassilis C

Pages

1-25

Publication year

2013

Publication date

Aug 2013

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP08(2013)055

ProQuest document ID

1652927667

SISSA, Trieste, Italy 2013

Three-body gravitino decays in the MSSM

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