Thermal gravitino production and collider tests of leptogenesis

Josef Pradler1,2,* and Frank Daniel Steffen2,†

1Institut fur Theoretische Physik, Universitat Wien, Boltzmanngasse 5, A-1090 Vienna, Austria2Max-Planck-Institut fur Physik, Fohringer Ring 6, D-80805 Munich, Germany

(Received 6 September 2006; published 9 January 2007)

Considering gravitino dark matter scenarios, we obtain the full gauge-invariant result for the relicdensity of thermally produced gravitinos to leading order in the standard model gauge couplings. For thetemperatures required by thermal leptogenesis, we find gaugino mass bounds which will be probed atfuture colliders. We show that a conceivable determination of the gravitino mass will allow for a uniquetest of the viability of thermal leptogenesis in the laboratory.

DOI: 10.1103/PhysRevD.75.023509 PACS numbers: 98.80.Cq, 04.65.+e, 12.60.Jv, 95.35.+d

I. INTRODUCTION

The smallness of the neutrino masses can be understoodnaturally in terms of the seesaw mechanism [1] once thestandard model is extended with right-handed neutrinoswhich have heavy Majorana masses and only Yukawacouplings. For a reheating temperature after inflation, TR,which is larger or not much smaller than the masses of theheavy neutrinos, these particles are produced in thermalreactions in the early Universe. The CP-violating out-of-equilibrium decays of the heavy neutrinos generate a lep-ton asymmetry that is converted into a baryon asymmetryby sphaleron processes [2]. This mechanism, known asthermal leptogenesis, can explain the cosmic baryon asym-metry for TR * 3� 109 GeV [3].

One will face severe cosmological constraints on TR ifsupersymmetry (SUSY) is discovered. An unavoidableimplication of SUSY theories including gravity is theexistence of the gravitino ~G which is the gauge field oflocal SUSY transformations. As the spin-3=2 superpartnerof the graviton, the gravitino is an extremely weakly inter-acting particle with couplings suppressed by inversepowers of the (reduced) Planck scale MP � 2:4�1018 GeV [4]. In the course of spontaneous SUSY break-ing, the gravitino acquires a mass m ~G and the couplings ofthe spin-1=2 goldstino which become dominant for smallm ~G. Depending on the SUSY breaking scheme, m ~G canrange from the eV scale up to scales beyond the TeV region[5]. Gravitinos can be produced efficiently in the hotprimordial plasma. Because of their extremely weak inter-actions, unstable gravitinos with m ~G & 5 TeV have longlifetimes, � ~G * 100 sec , and decay after big bang nucleo-synthesis (BBN). The associated decay products affect theabundances of the primordial light elements. Demandingthat the successful BBN predictions are preserved, boundson the abundance of gravitinos before their decay can bederived which imply TR & 108 GeV for m ~G & 5 TeV [6].Thus, the temperatures needed for thermal leptogenesis areexcluded.

We therefore consider SUSY scenarios in which a grav-itino with m ~G * 10 GeV is the lightest supersymmetricparticle (LSP) and stable due to R-parity conservation.These scenarios are particularly attractive for two reasons:(i) the gravitino LSP is a compelling dark matter candidateand (ii) thermal leptogenesis can still be a viable explana-tion of the baryon asymmetry [7].

II. THERMAL GRAVITINO PRODUCTION

Assuming that inflation governed the earliest momentsof the Universe, any initial population of gravitinos mustbe diluted away by the exponential expansion during theslow-roll phase. We consider the thermal production (orregeneration) of gravitinos in the radiation-dominatedepoch that starts after completion of reheating at the tem-perature TR. Gravitinos with m ~G * 10 GeV are not inthermal equilibrium with the primordial plasma after in-flation because of their extremely weak interactions. Athigh temperatures, gravitinos are generated in scatteringprocesses of particles that are in thermal equilibrium withthe hot SUSY plasma. The calculation of the relic densityof these thermally produced gravitinos, �TP

~G, requires a

consistent finite-temperature approach. A result that isindependent of arbitrary cutoffs has been derived forSUSY quantum chromodynamics (QCD) in a gauge-invariant way [8]. Following this approach, we providethe complete SU�3�c � SU�2�L � U�1�Y result to leadingorder in the couplings.

We compute �TP~G

from the Boltzmann equation for thegravitino number density

dn ~G

dt� 3Hn ~G � C ~G: (1)

The term proportional to the Hubble parameter H accountsfor the dilution by the cosmic expansion. The collisionterm C ~G describes the production and disappearance ofgravitinos in thermal reactions in the primordial plasma.Since the phase space density of the gravitino is signifi-cantly below those of the particles in thermal equilibrium,gravitino disappearance processes can be neglected. Thus,

*Electronic address: jpradler@mppmu.mpg.de†Electronic address: steffen@mppmu.mpg.de

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C ~G is given by integrating the thermal gravitino productionrate.

Considering a primordial plasma with the particle con-tent of the minimal SUSY standard model (MSSM) in thehigh-temperature limit, we calculate the thermal produc-tion rate of gravitinos with E * T using the Braaten-Yuanprescription [9] and hard thermal loop (HTL) resummation[10]. With the systematic treatment of screening effects inthe plasma, we obtain a finite result in a gauge-invariantway. Moreover, in contrast to previous estimates [11,12],our result does not depend on arbitrary cutoffs. The explicitform of the thermal gravitino production rate and the de-tailed derivation will be presented in a forthcoming pub-lication [13].

After numerical integration of the thermal gravitinoproduction rate, we obtain the SU�3�c � SU�2�L � U�1�Yresult for the collision term:

C ~G �X3

i�1

3��3�T6

16�3M2P

�1�

M2i

3m2~G

�cig

2i ln

�kigi

�; (2)

where the gaugino mass parameters Mi, the gauge cou-plings gi, and the constants ci and ki are associated with thegauge groups U�1�Y, SU�2�L, and SU�3�c as given inTable I. In expression (2) the temperature T provides thescale for the evaluation of Mi and gi. Note that HTLresummation [9,10] requires weak couplings, gi � 1,and thus high temperatures T � 106 GeV.

Our result k3 � 1:271 for the SU�3�c contribution islarger than 1.163 obtained from [8]. This results from ananalytical disagreement [13]: We find a cancellation of theterm T3�N � nf��Li2�e

E=T� �2=6 given as part ofIBFB in (C.14) of Ref. [8].

Assuming conservation of entropy per comoving vol-ume, the Boltzmann equation (1) can be solved analytically[8,14]. With the collision term (2), we find

�TP~Gh2 �

X3

i�1

!ig2i

�1�

M2i

3m2~G

�ln�kigi

��m ~G

100 GeV

�

�

�TR

1010 GeV

�(3)

with the Hubble constant h in units of 100 km Mpc1 s1

and the constants !i given in Table I. Here Mi and gi areunderstood to be evaluated at TR. With our new k3 value,

we find an enhancement of about 30% of the SU�3�ccontribution to the relic density.

Figure 1 shows �TP~Gh2 as a function of TR for m ~G � 10,

50, and 300 GeV. With m1=2 � 400 GeV, the solid anddashed lines are obtained, respectively, for universal(M1;2;3 � m1=2) and nonuniversal (0:5M1;2 � M3 � m1=2)gaugino masses at the grand unification scale MGUT ’ 2�1016 GeV. The SU�3�c contributions are shown by thedotted lines. The gray band indicates the dark matterdensity [15]

�DMh2 � 0:105�0:007

0:010: (4)

We find that electroweak processes enhance �TP~G

byabout 20% for universal gaugino masses at MGUT. In non-universal cases, M1;2 >M3 at MGUT, the electroweak con-tributions are more important. For 0:5M1;2 � M3 at MGUT,they provide about 40% of �TP

~G.

III. COLLIDER PREDICTIONS OFLEPTOGENESIS

Thermal leptogenesis requires TR * 3� 109 GeV [3].This condition together with the constraint �TP

~G� �DM

leads to upper limits on the gaugino masses. The SU�3�cresult for �TP

~Gimplies limits on the gluino mass [8,16].

TABLE I. Assignments of the index i, the gauge coupling gi,and the gaugino mass parameter Mi to the gauge groups U�1�Y,SU�2�L, and SU�3�c and the values of the associated constants ci,ki, and !i.

Gauge group i gi Mi ci ki !i

U�1�Y 1 g0 M1 11 1.266 0.018SU�2�L 2 g M2 27 1.312 0.044SU�3�c 3 gs M3 72 1.271 0.117

FIG. 1. The relic gravitino density from thermal production,�TP

~Gh2, as a function of TR. The solid and dashed curves show

the SU�3�c � SU�2�L � U�1�Y results for universal (M1;2;3 �

m1=2) and nonuniversal (0:5M1;2 � M3 � m1=2) gaugino massesat MGUT, respectively. The dotted curves show our result of theSU�3�c contribution for M3 � m1=2 at MGUT. The gray bandindicates the dark matter density �DMh

2.

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With our SU�3�c � SU�2�L � U�1�Y result, the limits onthe gluino mass M3 become more stringent because of thenew k3 value and the additional electroweak contributions.Moreover, as a prediction of thermal leptogenesis, weobtain upper limits on the electroweak gaugino mass pa-rameters M1;2. At the Large Hadron Collider (LHC) andthe International Linear Collider (ILC), these limits will beprobed in measurements of the masses of the neutralinosand charginos, which are typically lighter than the gluino.If the superparticle spectrum does not respect thesebounds, one will be able to exclude standard thermalleptogenesis.

Figure 2 shows the gaugino mass bounds for TR � 109,3� 109, and 1010 GeV evolved to MGUT, i.e., in terms oflimits on the gaugino mass parameter m1=2. Here�max

DMh2 � 0:126 is adopted as a nominal 3� upper limit

on �DMh2. With the observed superparticle spectrum, one

will be able to evaluate the gaugino mass parameters M1;2;3

at MGUT using the SUSY renormalization group equations[17–19]. While the determination of M1;2 at low energiesdepends on details of the SUSY model that will be probedat colliders [5], the bounds shown in Fig. 2 depend mainlyon the Mi relation at MGUT. This is illustrated by the solidand dashed curves obtained with M1;2;3 � m1=2 and0:5M1;2 � M3 � m1=2, respectively. The dotted curvesrepresent the SU�3�c limits for M3 � m1=2 at MGUT andemphasize the importance of the electroweakcontributions.

IV. DECAYS OF THE NEXT-TO-LIGHTESTSUPERSYMMETRIC PARTICLE

With a gravitino LSP of m ~G * 10 GeV, the next-to-lightest SUSY particle (NLSP) has a long lifetime of�NLSP * 106 s [20,21]. After decoupling from the primor-dial plasma, each NLSP decays into one gravitino LSP andstandard model particles. The resulting relic density ofthese nonthermally produced gravitinos is

�NTP~Gh2 �

m ~G

mNLSP�NLSPh

2; (5)

where mNLSP is the mass of the NLSP and �NLSPh2 is therelic density that the NLSP would have today, if it had notdecayed. As shown below, more severe limits on m1=2 areobtained with �NTP

~Gh2 taken into account. Moreover, since

the NLSP decays take place after BBN, the emitted stan-dard model particles can affect the abundance of the pri-mordial light elements. Successful BBN predictions thusimply bounds on m ~G and mNLSP [20,21]. From these cos-mological constraints it has been found that thermal lepto-genesis remains viable only in the cases of a chargedslepton NLSP or a sneutrino NLSP [16,22].

Note that the cosmological constraints from BBN canbecome much weaker with entropy production after decou-pling of the NLSP and before BBN. For example, largeparts of the parameter region disfavored by BBN con-straints on charged slepton NLSP scenarios become al-lowed with a moderate amount of entropy production[23]. In addition, such an entropy production dilutes boththe generated baryon asymmetry and the thermally pro-duced abundance of gravitinos. Therefore, upper limits onthe gaugino masses will still allow us to probe the viabilityof thermal leptogenesis. However, the TR labels of thecurves in Fig. 2 will change to higher values.

V. COLLIDER TESTS OF LEPTOGENESIS

Thermal leptogenesis will predict a lower bound on thegravitino mass m ~G once the masses of the standard modelsuperpartners are known. With a charged slepton as thelightest standard model superpartner, it could even bepossible to identify the gravitino as the LSP and to measureits mass m ~G at future colliders [24–26]. Confronting themeasured m ~G with the predicted lower bound will thenallow us to decide about the viability of thermalleptogenesis.

In order to explain our method, we do now consider anexemplary SUSY model. Let us assume that the analysis ofthe observed spectrum [19] will point to the universality ofthe soft SUSY breaking parameters at MGUT and, in par-ticular, to the minimal supergravity (mSUGRA) scenariowith the gaugino mass parameter m1=2 � 400 GeV, thescalar mass parameter m0 � 150 GeV, the trilinear cou-pling A0 � 150 GeV, a positive higgsino mass parame-ter �> 0, and the mixing angle tan� � 30 in the Higgs

FIG. 2. Upper limits on the gaugino mass parameter m1=2 from�TP

~G� �max

DM for the indicated values of TR. The solid anddashed curves show our SU�3�c � SU�2�L � U�1�Y results foruniversal (M1;2;3 � m1=2) and nonuniversal (0:5M1;2 � M3 �

m1=2) gaugino masses at MGUT, respectively. The dotted curvesshow the SU�3�c limits for M3 � m1=2 at MGUT.

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sector. A striking feature of the spectrum will then be theappearance of the lighter stau ~�1 withm~�1

� 143:4 GeV asthe lightest standard model superpartner [27]. In the con-sidered gravitino LSP case, 10 GeV & m ~G < m~�1

, this stauis the NLSP and decays with a lifetime of �~�1

* 106 s intothe gravitino. For the identified mSUGRA scenario and theconsidered reheating temperatures, the cosmological abun-dance of the ~�1 NLSP prior to decay can be computed from�NLSPh

2 � �~�1h2 ’ 3:83� 103, which is provided by

the computer program micrOMEGAs [28]. For given m ~G,this abundance determines �NTP

~Gh2 and the release of

electromagnetic (EM) and hadronic energy in ~�1 NLSPdecays governing the cosmological constraints [20,21].1

Figure 3 allows us to probe the viability of thermalleptogenesis in the considered mSUGRA scenario.2 Fromthe constraint �TP

~G��NTP

~G� �max

DM , we obtain the solidcurves which provide the upper limits on m1=2 for TR �

109, 3� 109, and 1010 GeV. The dashed line indicates them1=2 value of the considered scenario. The vertical solidline is given by m~�1

� 143:4 GeV which limits m ~G fromabove. In the considered scenario, the m1=2 value exceedsthe m1=2 limits for TR * 1010 GeV. Thus, temperaturesabove 1010 GeV can be excluded. Temperatures above 3�109 GeV and 109 GeV remain allowed for m ~G valuesindicated by the dark-shaded and medium-shaded regions,respectively. The m ~G values indicated by the light-shadedregion are excluded by BBN constraints for late ~�1 NLSPdecays.3

Here thermal leptogenesis, TR * 3� 109 GeV, predictsm ~G * 130 GeV and thus a ~�1 lifetime of �~�1

> 1011 s[20,21]. If decays of long-lived ~�1’s can be analyzed atcolliders giving evidence for the gravitino LSP [24–26],there will be the possibility to determine m ~G in the labo-ratory: From a measurement of the lifetime �~�1

governedby the decay ~�1 ! ~G�, m ~G can be extracted using thesupergravity prediction for the associated partial width

�~�1’ �1�~�1 ! ~G�� �

48�m2~GM2

P

m5~�1

�1

m2~G

m2~�1

�4

(6)

as obtained for m� ! 0. Moreover, for m ~G * 0:1m~�1, m ~G

can be inferred kinematically from the energy of the tau E�emitted in the 2-body decay ~�1 ! ~G� [24,26]:

m ~G �������������������������������������������m2

~�1m2

� 2m~�1E�

q: (7)

Whilem ~G within the dark-shaded region will favor thermalleptogenesis, anym ~G outside of the medium-shaded regionwill require either nonstandard mechanisms lowering theTR value needed for thermal leptogenesis or an alternativeexplanation of the cosmic baryon asymmetry.

VI. CONCLUSION

We provide the full SU�3�c � SU�2�L � U�1�Y result forthe relic density of thermally produced gravitino LSPs toleading order in the gauge couplings. Our result is obtainedin a consistent gauge-invariant finite-temperature calcula-tion and thus independent of arbitrary cutoffs. With this

FIG. 3 (color online). Probing the viability of thermal lepto-genesis. The solid curves show the limits on the gaugino massparameter m1=2 from �TP

~G��NTP

~G� �max

DM for TR � 109, 3�

109, and 1010 GeV. The dashed line indicates the m1=2 value ofthe considered scenario. The vertical solid line is given by the ~�1

NLSP mass which limits the gravitino LSP mass from above:m ~G < m~�1

� 143:4 GeV. The m ~G values at which temperaturesabove 3� 109 GeV and 109 GeV remain allowed are indicatedby the dark-shaded and medium-shaded regions, respectively.The m ~G values within the light-shaded region are excluded byBBN constraints.

1Considering bound-state effects of long-lived negativelycharged particles on BBN, it has recently been claimed thatthe stau NLSP abundance for �~�1

* 103–104 s is severely con-strained by the observed 6;7Li abundances [29,30]. If the asso-ciated bounds are confirmed, the considered mSUGRA scenariowill be cosmologically allowed only with late-time entropyproduction [23].

2Thermal leptogenesis requires right-handed neutrinos andthus an extended mSUGRA scenario. This could manifest itselfin the masses of the third generation sleptons [17]. Since theeffects are typically small, we leave a systematic investigation ofextended scenarios for future work.

3We use the conservative BBN bounds considered in [21]. Theaverage EM energy release in one ~�1 NLSP decay is assumed tobe E�=2, where E� is the energy of the tau emitted in thedominant 2-body decay ~�1 ! ~G� (cf. Fig. 16 of Ref. [21]).With an EM energy release below E�=2, the light-shaded bandcan become smaller. For less conservative BBN constraints and/or enhanced EM energy release, the excluded m ~G region be-comes larger.

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result, new gravitino and gaugino mass bounds emerge as aprediction of thermal leptogenesis. If supersymmetry isrealized in nature, these bounds will be accessible at theLHC and the ILC. In particular, with a charged sleptonNLSP, there will be the exciting possibility to identify thegravitino as the LSP and to measure its mass. Confrontingthe measured gravitino mass with the predicted bounds will

then allow for a unique test of the viability of thermalleptogenesis in the laboratory.

ACKNOWLEDGMENTS

We are grateful to T. Plehn, M. Plumacher, G. Raffelt,and Y. Y. Y. Wong for valuable discussions.

[1] P. Minkowski, Phys. Lett. B 67, 421 (1977); T. Yanagida,in Proceedings of the Workshop on Unified Theory andBaryon Number in the Universe, Tsukuba, Japan, 1979,edited by O. Sawada and A. Sugamoto (KEK ReportNo. 79-18, 1979) p. 95; M. Gell-Mann, P. Ramond, andR. Slansky, in Supergravity, edited by P. vanNieuwenhuizen and D. Freedman (North-Holland,Amsterdam, 1979) p. 315; S. L. Glashow, in Proceedingsof the Cargese Summer Institute on Quarks and Leptons(Plenum, New York, 1980) p. 707.

[2] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45(1986).

[3] See W. Buchmuller, P. Di Bari, and M. Plumacher, Ann.Phys. (N.Y.) 315, 305 (2005), and references therein.

[4] J. Wess and J. Bagger, Supersymmetry and Supergravity(Princeton University, Princeton, NJ, 1992).

[5] See S. P. Martin, hep-ph/9709356, and references therein.[6] K. Kohri, T. Moroi, and A. Yotsuyanagi, Phys. Rev. D 73,

123511 (2006).[7] M. Bolz, W. Buchmuller, and M. Plumacher, Phys. Lett. B

443, 209 (1998).[8] M. Bolz, A. Brandenburg, and W. Buchmuller, Nucl. Phys.

B606, 518 (2001).[9] E. Braaten and T. C. Yuan, Phys. Rev. Lett. 66, 2183

(1991).[10] E. Braaten and R. D. Pisarski, Nucl. Phys. B337, 569

(1990).[11] J. R. Ellis, J. E. Kim, and D. V. Nanopoulos, Phys. Lett. B

145, 181 (1984).[12] T. Moroi, H. Murayama, and M. Yamaguchi, Phys. Lett. B

303, 289 (1993).[13] J. Pradler and F. D. Steffen, Report No. MPP-2006-254 (to

be published).[14] A. Brandenburg and F. D. Steffen, J. Cosmol. Astropart.

Phys. 08 (2004) 008; hep-ph/0406021; hep-ph/0407324.[15] W.-M. Yao et al., J. Phys. G 33, 1 (2006).[16] M. Fujii, M. Ibe, and T. Yanagida, Phys. Lett. B 579, 6

(2004).[17] H. Baer, M. A. Diaz, P. Quintana, and X. Tata, J. High

Energy Phys. 04 (2000) 016.[18] G. A. Blair, W. Porod, and P. M. Zerwas, Eur. Phys. J. C

27, 263 (2003).[19] R. Lafaye, T. Plehn, and D. Zerwas, hep-ph/0404282; P.

Bechtle, K. Desch, and P. Wienemann, Comput. Phys.Commun. 174, 47 (2006).

[20] J. L. Feng, S. Su, and F. Takayama, Phys. Rev. D 70,075019 (2004).

[21] F. D. Steffen, J. Cosmol. Astropart. Phys. 09 (2006) 001.[22] D. G. Cerdeno, K. Y. Choi, K. Jedamzik, L. Roszkowski,

and R. Ruiz de Austri, J. Cosmol. Astropart. Phys. 06(2006) 005.

[23] W. Buchmuller, K. Hamaguchi, M. Ibe, and T. T.Yanagida, hep-ph/0605164.

[24] W. Buchmuller, K. Hamaguchi, M. Ratz, and T. Yanagida,Phys. Lett. B 588, 90 (2004).

[25] A. Brandenburg, L. Covi, K. Hamaguchi, L. Roszkowski,and F. D. Steffen, Phys. Lett. B 617, 99 (2005); F. D.Steffen, hep-ph/0507003.

[26] H. U. Martyn, Eur. Phys. J. C 48, 15 (2006).[27] A. Djouadi, J. L. Kneur, and G. Moultaka, hep-ph/

0211331. We use mt � 172:5 GeV for the top quark mass.[28] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov,

Comput. Phys. Commun. 149, 103 (2002); 174, 577(2006).

[29] M. Pospelov, hep-ph/0605215.[30] R. H. Cyburt, J. Ellis, B. D. Fields, K. A. Olive, and V. C.

Spanos, J. Cosmol. Astropart. Phys. 11 (2006) 014.

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