PRL 110, 241306 (2013)
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PHYSICAL REVIEW LETTERS
Gauge Coupling Unification and Nonequilibrium Thermal Dark Matter Yann Mambrini,1,* Keith A. Olive,2,† Je´re´mie Quevillon,1,‡ and Bryan Zaldı´var3,§ 1
2
Laboratoire de Physique The´orique Universite´ Paris-Sud, F-91405 Orsay, France William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 3 Instituto de Fisica Teorica, IFT-UAM/CSIC, 28049 Madrid, Spain (Received 10 March 2013; published 14 June 2013) We study a new mechanism for the production of dark matter in the Universe which does not rely on thermal equilibrium. Dark matter is populated from the thermal bath subsequent to inflationary reheating via a massive mediator whose mass is above the reheating scale TRH . To this end, we consider models with an extra Uð1Þ gauge symmetry broken at some intermediate scale (Mint ’ 1010 –1012 GeV). We show that not only does the model allow for gauge coupling unification (at a higher scale associated with grand unification) but it can provide a dark matter candidate which is a standard model singlet but charged under the extra Uð1Þ. The intermediate scale gauge boson(s) which are predicted in several E6=SOð10Þ constructions can be a natural mediator between dark matter and the thermal bath. We show that the dark matter abundance, while never having achieved thermal equilibrium, is fixed shortly after 3 4 . As a consequence, we show that the unification of =Mint the reheating epoch by the relation TRH gauge couplings which determines Mint also fixes the reheating temperature, which can be as high as TRH ’ 1011 GeV. DOI: 10.1103/PhysRevLett.110.241306
PACS numbers: 95.35.+d, 12.10.Dm, 98.80.Cq
Introduction.—The standard model (SM) of particle physics is now more than ever motivated by the recent discovery of the Higgs boson at both the ATLAS [1] and CMS [2] detectors. The SM, however, contains many free parameters, and the gauge couplings do not unify. Among the most elegant approaches to understanding some of these parameters is the idea of a grand unified theory (GUT) in which the three gauge couplings 1;2;3 originate from a single gauge coupling associated to a grand unified gauge group [3]. This idea is supported by the fact that quantum numbers of quarks and leptons in the SM nicely fill representations of a GUT symmetry, e.g., the 10 and 5 components of SUð5Þ or 16 component of SOð10Þ. Another issue concerning the SM is the lack of a candidate to account for dark matter (DM) which consists of 22% of the energy density of our Universe. Stable weakly interacting massive particles are among the most popular candidates for DM. In most models, such as popular supersymmetric extensions of the SM [4], the annihilation of weakly interacting massive particles in thermal equilibrium in the early Universe determined the relic abundance of DM. In this Letter, we will show that GUT gauge groups such as E6 or SOð10Þ which contain additional Uð1Þ gauge subgroups and are broken at an intermediate scale, can easily lead to gauge coupling unification [5] and may contain a new dark matter candidate which is charged under the extra Uð1Þ. However, unlike the standard equilibrium annihilation process, or complimentary process of freeze-in [6], we propose an alternative mechanism for producing dark matter through interactions which are 0031-9007=13=110(24)=241306(5)
mediated by the heavy gauge bosons associated with the extra Uð1Þ. While being produced from the thermal bath, these dark matter particles never reach equilibrium. We will refer to dark matter produced with this mechanism as nonequilibrium thermal dark matter or NETDM. The final relic abundance of NETDM is obtained shortly after the inflationary reheating epoch. This mechanism is fundamentally different from other nonthermal DM production mechanisms in the literature (to our knowledge). Assuming that none of the dark matter particles are directly produced by the decays of the inflaton during reheating, we compute the production of dark matter and relate the inflationary reheating temperature to the choice of the gauge group and the intermediate scale needed for gauge coupling unification. As an added benefit, the model naturally possesses the capability of producing a baryon asymmetry through leptogenesis, although that lies beyond of the scope of this work. The Letter is organized as follows: After a summary of the unified models under consideration, we show how the presence of an intermediate scale allows for the possibility of producing a dark matter candidate which respects the WMAP constraint [7] and apply it to an explicit scenario. Unification in SOð10Þ models.—The prototype of grand unification is based on the SUð5Þ gauge group. In an extension of SUð5Þ one can introduce SUð5Þ singlets as potential dark matter candidates. The simplest extension in which singlets are automatically incorporated is that of SOð10Þ. There are, however, many ways to break SOð10Þ down to SUð3Þ SUð2Þ Uð1Þ. This may happen in multiple stages, but here we are mainly concerned with the breaking of an additional Uð1Þ [or SUð2Þ] factor at an
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Ó 2013 American Physical Society
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1/α i
intermediate scale Mint . Here, we will not go into the details of the breaking, but take some specific, well-known examples when needed. Assuming gauge coupling unification, the GUT mass scale MGUT and the intermediate scale Mint can be predicted from the low-energy coupling constants with the use of the renormalization group equation
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PHYSICAL REVIEW LETTERS
di ¼ bi 2i : d
40 1/α 2 20
(1)
The evolution of the three running coupling constants 1 , 2 , and 3 from MZ to the intermediate scale Mint is obtained from Eq. (1) using the functions of the standard model: b1;2;3 ¼ ð41=10; 19=6; 7Þ=2. We note that the gauge coupling gD associated with U0 ð1Þ is related at the pffiffiffiffiffiffiffiffiffiffiffi GUT scale to g1 of Uð1ÞY by gD ¼ ð5=3Þg1 and i ¼ g2i =4. Between Mint and MGUT (both to be determined) the running coupling constants are again obtained from Eq. (1), now using functions associated with the intermediate scale gauge group, which we will label b~i . The matching condition between the two different runnings at Mint can be written ð0i Þ1 þ bi tint ¼ 1 þ b~i ðtint tGUT Þ;
1/α 1
(2)
where tint ¼ lnðMint =MZ Þ, tGUT ¼ lnðMGUT =MZ Þ, 0i ¼ i ðMZ Þ which is measured, and ¼ i ðMGUT Þ is the unified coupling constant at the GUT scale. This gives us a system of three equations, for three unknown parameters: , tint , tGUT . Solving Eq. (2), we obtain 0 1 1 ð3 Þ ð02 Þ1 ð02 Þ1 ð01 Þ1 tint ¼ ; b32 b21 b~2 b~3 b~1 b~2 (3) where bij ðbi bj Þ=ðb~i b~j Þ. To be concrete, we will consider a specific example to derive numerical results for the case of the breaking of SOð10Þ: SOð10Þ!SUð4ÞSUð2ÞL Uð1ÞR !Mint SUð3ÞC SUð2ÞL Uð1ÞY !MEW SUð3ÞC Uð1Þem . When the intermediate symmetry is broken by a 16 component of Higgs bosons, the b~i functions are given by b~1;2;3 ¼ ð5=2;19=6;63=6Þ=2 [5], where the computation was done at 1-loop level. For this case, we obtain Mint ¼ 7:8 1012 GeV and MGUT ¼ 1:3 1015 GeV using ð01;2;3 Þ1 ’ ð59:47; 29:81; 8:45Þ. The evolution of the gauge couplings for this example is shown in Fig. 1. Heavy Z0 and dark matter.—It has been shown in Refs. [8,9] that a stable dark matter candidate may arise in SOð10Þ models from an unbroken Z2B-L symmetry. If the dark matter is a fermion (scalar) it should belong to a 3ðB-LÞ even (odd) representation of SOð10Þ. For example, the 126 or 144 contains a stable component which is neutral under the SM, yet charged under the extra Uð1Þ. As we have seen, to explain the unification of the gauge
1/α 3
10
6
10
10
10
14
µ (GeV)
10
16
FIG. 1 (color online). Example of the running of the SM gauge couplings for SOð10Þ ! SUð4Þ SUð2ÞL Uð1ÞR .
couplings in SOð10Þ one needs an intermediate scale Mint of order 1010 GeV. The dark matter candidate can be 0 produced in the early Universe through s-channel pffiffiffi Z ex0 change: SM SM ! Z ! . Since MZ0 ¼ ð5= 3ÞgD Mint , the exchanged particle is so heavy (above the reheating scale, as we show below) that the DM production rate is very slow, and we can neglect the self-annihilation process in the Boltzmann equation. Thus while the dark matter is produced from the thermal bath, we have a nonequilibrium production mechanism for dark matter, hence NETDM. The evolution of the yield of , Y ¼ n =s follows rffiffiffiffiffiffi dY gs hvi 2 ¼ Y ; (4) pffiffiffiffiffiffi m M 45 g P x2 eq dx where n is the number density of and s is the entropy of the Universe, and g , gs are the effective degrees of freedom for energy density and entropy, respectively, x ¼ m =T, m being the dark matter mass, MP the Planck mass, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 T Z 1 hvin2eq dsd s 4m2 jMj2 K1 ð s=TÞ: 6 2 2048 4m (5) Here neq is the equilibrium number density of the initial state (SM) particles, and K1 is the first order modified Bessel function and the effective degrees of freedom of incoming particles. Since the production of DM occurs mainly at TRH m , we can neglect m in estimating the amplitude for production. In this case, assuming that both and the initial state f are fermions, we obtain jM j2
g4D q2 q2f Ncf ðs MZ2 0 Þ2
½s2 ð1 þ cos2 Þ;
(6)
where is the angle between the two outgoing DM particles, Ncf is number of colors of the particle f, and qi
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TRH (GeV)
is the charge of the particle i under U0 ð1Þ with a gauge coupling gD . Here, q is an effective coupling which will ultimately depend on the specific intermediate gauge pffiffigroup ffi chosen. With the approximations m , mf s and MZ0 TRH , and after integration over and summing over all incoming SM fermions in the thermal bath, we obtain f dY X g4D q2 q2f Nc 45 3=2 1 m3 MP 2f ¼ : pffiffiffiffiffiffi gs g MZ4 0 27 dx x4 f
20
10
10 GeV 15
100 GeV 1 TeV
10
10
10
5
10
(7) 6
10
Solving Eq. (7) between the reheating temperature and a temperature T gives X 45 3=2 MP 32f 3 ½TRH T 3 ; Y ðTÞ ¼ q2 q2f Ncf 4 12507 g M s int f
10
10
14
18
10
10
Mint (GeV)
FIG. 2 (color online). Reheating temperature as function of the SOð10Þ breaking scale for different mass of dark matter: 10, 100, and 1000 GeV.
(8) 0 where pffiffiffi we replaced the mass of the Z by MZ0 ¼ ð5= 3ÞgD Mint and made the approximation g ¼ gs . We note that the effect of Z0 decay on the abundance of is completely negligible due to its Boltzmann suppression in the Universe: the Z0 is largely decoupled from the thermal bath already at the time of reheating. We note several interesting features from Eq. (8). First of all, the number density of the dark matter does not depend at all on the strength of the U0 ð1Þ coupling gD but rather on the intermediate scale (that is determined by requiring gauge coupling unification as we demonstrated in the previous section). Second, the production of dark matter is mainly achieved at reheating. Third, once the relic abundance is obtained, the number density per comoving frame (Y) is fixed, never having reached thermal equilibrium with the bath. And finally, upon applying the WMAP determination for the DM abundance, and assuming instantaneous reheating after inflation, we obtain a constraint on TRH once the pattern of SOð10Þ breaking is known (and thus Mint fixed). In a more complete model of reheating, the resulting relic density may be modified by an order of magnitude (or more) for a given reheating temperature. Thus, given a scheme of SOð10Þ breaking we can determine the reheating temperature very precisely from the relic abundance constraint in the Universe. From 2 crit h 13:5 H02 MP2 0 Y0 ¼ ¼ ; (9) m s0 0:1 163 g0s T03 m
where H is the Hubble parameter and the index ‘‘0’’ corresponds to present-day values, and combining Eqs. (8) and (9) we find 2 56254 h gs 3=2 MP H02 4 3 TRH ¼ M (10) P 45 T03 m g0s int 16q2 f 2f q2f Ncf 0:1 or
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PHYSICAL REVIEW LETTERS
2 1=3 h 100 GeV 1=3 TRH ’ 2 108 GeV m 0:1 4=3 Mint ; 12 10 GeV
(11)
P where we took for illustration q2 f 2f q2f Ncf ¼ 1. We show in Fig. 2 the evolution of TRH as function of Mint for different values of the dark matter mass m . We can thus fix the reheating temperature predicted by different symmetry breaking patterns. (We note that the value obtained for the intermediate scale in different SOð10Þ breaking schemes is not modified by the presence of a dark matter particle which is not charged under the SM gauge group.) We summarize them in Table I, where the values of TRH are given for m ¼ 100 GeV. Note that fixing TRH in this way is reminiscent of fixing particle masses or couplings in other dark matter models necessary to obtain the correct relic density. We note that although we have here and in Fig. 2 shown results for central values of m ¼ 100 GeV, dark matter candidates are possible with mass up to the order of the reheating temperature, which, based on Eq. (11), would allow it to be as large as 106 –107 GeV. Finally, we must specify the identity of the NETDM candidate in the context described above. The DM can be in the 126 or 144 representations of SOð10Þ. There are several mechanisms to render the DM mass light [9], one TABLE I. Possible breaking schemes of SOð10Þ.
A A B B C C
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SOð10Þ ! G [Higgs]
Mint ðGeVÞ
TRH ðGeVÞ
4 2L 1R ½16 4 2L 1R ½126 4 2L 2R ½16 4 2L 2R ½126 3C 2L 2R 1B-L ½16 3C 2L 2R 1BL ½126
1012:9 1011:8 1014:4 1013:8 1010:6 108:6
3 109 1 108 3 1011 5 1010 3 106 6 103
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PHYSICAL REVIEW LETTERS
of which is through a fine tuning of the SOð10Þ couplings contributing with different Clebsch-Gordan coefficients (see for example, Refs. [10,11]) to the masses of the various 126 components. For example, for the group GA 126ðM þ y45 45H þ y210 210H Þ126;
(12)
where M MGUT , and a GA singlet in a linear combination of 210H and 45H has a vev at the GUT scale. m is then given by a linear combination of M and the vev and can be tuned to small values, while all other particles inside the 126 live close to MGUT . This is reminiscent of and no more severe than the doublet-triplet separation problem common to grand unified theories. Discussion.—Unfortunately, the chance of detection (direct or indirect) of NETDM with a massive mediator Z0 is nearly hopeless. Indeed, the diagram for the direct detection process, measuring the elastic scattering off a nucleus, proceeds through the t-channel exchange of the Z0 boson, and is proportional to 1=MZ4 0 yielding a negligible cross section. In addition, due to the present low velocity of dark matter in our Galaxy (’ 200 km=s), the indirect detection prospects from s-channel Z0 annihilation ! Z0 ! ff proportional to s2 =MZ4 0 is also negligible. As we have seen in Eq. (8), the production of dark matter occurs in the very early Universe at the epoch of reheating. A similar mechanism (though fundamentally completely different) where a dark matter candidate is produced close to the reheating time is the case of the gravitino [4,12]. Indeed, in both cases equilibrium is never reached and the relic abundance is produced from the thermal background to attain the decoupling value =H, with H the Hubble constant and ¼ hvinf the production rate. However, in the case of SOð10Þ, the cross section decreases with the temperature like hviZ0 / T 2 =MZ4 0 , whereas in the case of the gravitino the cross section is constant hvi3=2 / 1=MP2 implying YðTÞ / TRH . Gravitinos are produced though the scattering of light particles, whereas in the present context, the dark matter is mediated by a heavy particle (MZ0 > TRH ) which connects the dark and observable sectors. They are physically distinct possibilities. Finally, we note that cases B and C (in Table I) predict reheating temperatures which are larger (case B) or smaller (case C) than the case under consideration. Case A would also be compatible with successful thermal leptogenesis with a zero initial state abundance of right-handed neutrinos [13,14]. However in the cases B and C, the persistence of the SUð2ÞR symmetry would imply that the cancelation in Eq. (12) would leave behind a light SUð2ÞR triplet (for DM inside a 126) or doublet (for DM inside a 144). These would affect somewhat the beta functions for the RGE’s but more importantly leave behind a test of the model. In the triplet (doublet) case, we would expect three (at least two) nearly degenerate states: one with charge 0, being the DM candidate, and also states with electric charge 1 and 2 (or 1 in the doublet case).
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Conclusion.—In this work, we have shown that it is possible to produce dark matter through nonequilibrium thermal processes in the context of SOð10Þ models which respect the WMAP constraints. Insisting on gauge coupling unification, we have demonstrated that there exists a tight link between the reheating temperature and the scheme of the SOð10Þ breaking to the SM gauge group. Interestingly, the numerical values we obtained are quite high and very compatible with inflationary and leptogenesislike models. The authors would like to thank M. Tytgat, T. Hambye, E. Dudas, M. Voloshin, E. Fernandez-Martinez, K. Tarachenko, and J. M. Moreno for very useful discussions. This work was supported by the French ANR TAPDMS ANR-09-JCJC-0146 and the Spanish MICINN’s Consolider-Ingenio 2010 Programme under Grant No. Multi-Dark CSD2009-00064. B. Z. acknowledges Consolider-Ingenio PAU CSD2007-00060, CPAN CSD2007-00042, under Contract No. FPA2010-17747. Y. M. acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). The work of K. A. O. was supported in part by DOE Grant No. DE-FG02-94ER-40823 at the University of Minnesota.
*[email protected] † [email protected] ‡ [email protected] § [email protected] [1] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716, 1 (2012). [2] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 716, 30 (2012). [3] For a review and references see the section on grand unified theory in J. Beringer et al. (Particle Data Group Collaboration), Phys. Rev. D 86, 010001 (2012). [4] H. Goldberg, Phys. Rev. Lett. 50, 1419 (1983); J. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. A. Olive, and M. Srednicki, Nucl. Phys. B238, 453 (1984). [5] M. Fukugita and T. Yanagida, in Physics and Astrophysics of Neutrinos, edited by M. Fukugita and A. Suzuki (Springer-Verlag, Berlin, 1994); Kyoto University, YITPK-1050 (93/12,rec.Feb.94) 248 p. C. [6] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, J. High Energy Phys. 03 (2010) 080; J. McDonald, Phys. Rev. Lett. 88, 091304 (2002); X. Chu, T. Hambye, and M. H. G. Tytgat, J. Cosmol. Astropart. Phys. 05 (2012) 034; C. E. Yaguna, J. Cosmol. Astropart. Phys. 02 (2012) 006. [7] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern et al., arXiv:1212.5226. [8] M. Kadastik, K. Kannike, and M. Raidal, Phys. Rev. D 80, 085020 (2009); 81, 029903(E) (2010). [9] M. Frigerio and T. Hambye, Phys. Rev. D 81, 075002 (2010).
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[10] R. Slansky, Phys. Rep. 79, 1 (1981). [11] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, and N. Okada, J. Math. Phys. (N.Y.) 46, 033505 (2005). [12] M. Bolz, A. Brandenburg, and W. Buchmuller, Nucl. Phys. B606, 518 (2001); B790, 336 (2008); V. S. Rychkov and A. Strumia, Phys. Rev. D 75, 075011 (2007); [13] G. F. Giudice, A. Notari, M. Raidal, A. Riotto, and A. Strumia, Nucl. Phys. B685, 89 (2004); S. Antusch and A. M. Teixeira, J. Cosmol. Astropart. Phys. 02 (2007) 024.
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[14] S. Davidson and A. Ibarra, Phys. Lett. B 535, 25 (2002); E. Nardi, Y. Nir, E. Roulet, and J. Racker, J. High Energy Phys. 01 (2006) 164; A. Abada, S. Davidson, A. Ibarra, F.-X. Josse-Michaux, M. Losada, and A. Riotto, J. High Energy Phys. 09 (2006) 010; R. Barbieri, P. Creminelli, A. Strumia, and N. Tetradis, Nucl. Phys. B575, 61 (2000); M. Raidal, A. Strumia, and K. Turzynski, Phys. Lett. B 609, 351 (2005); 632, 752 (2006); A. Pilaftsis and T. E. J. Underwood, Nucl. Phys. B692, 303 (2004).
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PHYSICAL REVIEW LETTERS
Gauge Coupling Unification and Nonequilibrium Thermal Dark Matter Yann Mambrini,1,* Keith A. Olive,2,† Je´re´mie Quevillon,1,‡ and Bryan Zaldı´var3,§ 1
2
Laboratoire de Physique The´orique Universite´ Paris-Sud, F-91405 Orsay, France William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 3 Instituto de Fisica Teorica, IFT-UAM/CSIC, 28049 Madrid, Spain (Received 10 March 2013; published 14 June 2013) We study a new mechanism for the production of dark matter in the Universe which does not rely on thermal equilibrium. Dark matter is populated from the thermal bath subsequent to inflationary reheating via a massive mediator whose mass is above the reheating scale TRH . To this end, we consider models with an extra Uð1Þ gauge symmetry broken at some intermediate scale (Mint ’ 1010 –1012 GeV). We show that not only does the model allow for gauge coupling unification (at a higher scale associated with grand unification) but it can provide a dark matter candidate which is a standard model singlet but charged under the extra Uð1Þ. The intermediate scale gauge boson(s) which are predicted in several E6=SOð10Þ constructions can be a natural mediator between dark matter and the thermal bath. We show that the dark matter abundance, while never having achieved thermal equilibrium, is fixed shortly after 3 4 . As a consequence, we show that the unification of =Mint the reheating epoch by the relation TRH gauge couplings which determines Mint also fixes the reheating temperature, which can be as high as TRH ’ 1011 GeV. DOI: 10.1103/PhysRevLett.110.241306
PACS numbers: 95.35.+d, 12.10.Dm, 98.80.Cq
Introduction.—The standard model (SM) of particle physics is now more than ever motivated by the recent discovery of the Higgs boson at both the ATLAS [1] and CMS [2] detectors. The SM, however, contains many free parameters, and the gauge couplings do not unify. Among the most elegant approaches to understanding some of these parameters is the idea of a grand unified theory (GUT) in which the three gauge couplings 1;2;3 originate from a single gauge coupling associated to a grand unified gauge group [3]. This idea is supported by the fact that quantum numbers of quarks and leptons in the SM nicely fill representations of a GUT symmetry, e.g., the 10 and 5 components of SUð5Þ or 16 component of SOð10Þ. Another issue concerning the SM is the lack of a candidate to account for dark matter (DM) which consists of 22% of the energy density of our Universe. Stable weakly interacting massive particles are among the most popular candidates for DM. In most models, such as popular supersymmetric extensions of the SM [4], the annihilation of weakly interacting massive particles in thermal equilibrium in the early Universe determined the relic abundance of DM. In this Letter, we will show that GUT gauge groups such as E6 or SOð10Þ which contain additional Uð1Þ gauge subgroups and are broken at an intermediate scale, can easily lead to gauge coupling unification [5] and may contain a new dark matter candidate which is charged under the extra Uð1Þ. However, unlike the standard equilibrium annihilation process, or complimentary process of freeze-in [6], we propose an alternative mechanism for producing dark matter through interactions which are 0031-9007=13=110(24)=241306(5)
mediated by the heavy gauge bosons associated with the extra Uð1Þ. While being produced from the thermal bath, these dark matter particles never reach equilibrium. We will refer to dark matter produced with this mechanism as nonequilibrium thermal dark matter or NETDM. The final relic abundance of NETDM is obtained shortly after the inflationary reheating epoch. This mechanism is fundamentally different from other nonthermal DM production mechanisms in the literature (to our knowledge). Assuming that none of the dark matter particles are directly produced by the decays of the inflaton during reheating, we compute the production of dark matter and relate the inflationary reheating temperature to the choice of the gauge group and the intermediate scale needed for gauge coupling unification. As an added benefit, the model naturally possesses the capability of producing a baryon asymmetry through leptogenesis, although that lies beyond of the scope of this work. The Letter is organized as follows: After a summary of the unified models under consideration, we show how the presence of an intermediate scale allows for the possibility of producing a dark matter candidate which respects the WMAP constraint [7] and apply it to an explicit scenario. Unification in SOð10Þ models.—The prototype of grand unification is based on the SUð5Þ gauge group. In an extension of SUð5Þ one can introduce SUð5Þ singlets as potential dark matter candidates. The simplest extension in which singlets are automatically incorporated is that of SOð10Þ. There are, however, many ways to break SOð10Þ down to SUð3Þ SUð2Þ Uð1Þ. This may happen in multiple stages, but here we are mainly concerned with the breaking of an additional Uð1Þ [or SUð2Þ] factor at an
241306-1
Ó 2013 American Physical Society
PRL 110, 241306 (2013)
1/α i
intermediate scale Mint . Here, we will not go into the details of the breaking, but take some specific, well-known examples when needed. Assuming gauge coupling unification, the GUT mass scale MGUT and the intermediate scale Mint can be predicted from the low-energy coupling constants with the use of the renormalization group equation
week ending 14 JUNE 2013
PHYSICAL REVIEW LETTERS
di ¼ bi 2i : d
40 1/α 2 20
(1)
The evolution of the three running coupling constants 1 , 2 , and 3 from MZ to the intermediate scale Mint is obtained from Eq. (1) using the functions of the standard model: b1;2;3 ¼ ð41=10; 19=6; 7Þ=2. We note that the gauge coupling gD associated with U0 ð1Þ is related at the pffiffiffiffiffiffiffiffiffiffiffi GUT scale to g1 of Uð1ÞY by gD ¼ ð5=3Þg1 and i ¼ g2i =4. Between Mint and MGUT (both to be determined) the running coupling constants are again obtained from Eq. (1), now using functions associated with the intermediate scale gauge group, which we will label b~i . The matching condition between the two different runnings at Mint can be written ð0i Þ1 þ bi tint ¼ 1 þ b~i ðtint tGUT Þ;
1/α 1
(2)
where tint ¼ lnðMint =MZ Þ, tGUT ¼ lnðMGUT =MZ Þ, 0i ¼ i ðMZ Þ which is measured, and ¼ i ðMGUT Þ is the unified coupling constant at the GUT scale. This gives us a system of three equations, for three unknown parameters: , tint , tGUT . Solving Eq. (2), we obtain 0 1 1 ð3 Þ ð02 Þ1 ð02 Þ1 ð01 Þ1 tint ¼ ; b32 b21 b~2 b~3 b~1 b~2 (3) where bij ðbi bj Þ=ðb~i b~j Þ. To be concrete, we will consider a specific example to derive numerical results for the case of the breaking of SOð10Þ: SOð10Þ!SUð4ÞSUð2ÞL Uð1ÞR !Mint SUð3ÞC SUð2ÞL Uð1ÞY !MEW SUð3ÞC Uð1Þem . When the intermediate symmetry is broken by a 16 component of Higgs bosons, the b~i functions are given by b~1;2;3 ¼ ð5=2;19=6;63=6Þ=2 [5], where the computation was done at 1-loop level. For this case, we obtain Mint ¼ 7:8 1012 GeV and MGUT ¼ 1:3 1015 GeV using ð01;2;3 Þ1 ’ ð59:47; 29:81; 8:45Þ. The evolution of the gauge couplings for this example is shown in Fig. 1. Heavy Z0 and dark matter.—It has been shown in Refs. [8,9] that a stable dark matter candidate may arise in SOð10Þ models from an unbroken Z2B-L symmetry. If the dark matter is a fermion (scalar) it should belong to a 3ðB-LÞ even (odd) representation of SOð10Þ. For example, the 126 or 144 contains a stable component which is neutral under the SM, yet charged under the extra Uð1Þ. As we have seen, to explain the unification of the gauge
1/α 3
10
6
10
10
10
14
µ (GeV)
10
16
FIG. 1 (color online). Example of the running of the SM gauge couplings for SOð10Þ ! SUð4Þ SUð2ÞL Uð1ÞR .
couplings in SOð10Þ one needs an intermediate scale Mint of order 1010 GeV. The dark matter candidate can be 0 produced in the early Universe through s-channel pffiffiffi Z ex0 change: SM SM ! Z ! . Since MZ0 ¼ ð5= 3ÞgD Mint , the exchanged particle is so heavy (above the reheating scale, as we show below) that the DM production rate is very slow, and we can neglect the self-annihilation process in the Boltzmann equation. Thus while the dark matter is produced from the thermal bath, we have a nonequilibrium production mechanism for dark matter, hence NETDM. The evolution of the yield of , Y ¼ n =s follows rffiffiffiffiffiffi dY gs hvi 2 ¼ Y ; (4) pffiffiffiffiffiffi m M 45 g P x2 eq dx where n is the number density of and s is the entropy of the Universe, and g , gs are the effective degrees of freedom for energy density and entropy, respectively, x ¼ m =T, m being the dark matter mass, MP the Planck mass, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 T Z 1 hvin2eq dsd s 4m2 jMj2 K1 ð s=TÞ: 6 2 2048 4m (5) Here neq is the equilibrium number density of the initial state (SM) particles, and K1 is the first order modified Bessel function and the effective degrees of freedom of incoming particles. Since the production of DM occurs mainly at TRH m , we can neglect m in estimating the amplitude for production. In this case, assuming that both and the initial state f are fermions, we obtain jM j2
g4D q2 q2f Ncf ðs MZ2 0 Þ2
½s2 ð1 þ cos2 Þ;
(6)
where is the angle between the two outgoing DM particles, Ncf is number of colors of the particle f, and qi
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is the charge of the particle i under U0 ð1Þ with a gauge coupling gD . Here, q is an effective coupling which will ultimately depend on the specific intermediate gauge pffiffigroup ffi chosen. With the approximations m , mf s and MZ0 TRH , and after integration over and summing over all incoming SM fermions in the thermal bath, we obtain f dY X g4D q2 q2f Nc 45 3=2 1 m3 MP 2f ¼ : pffiffiffiffiffiffi gs g MZ4 0 27 dx x4 f
20
10
10 GeV 15
100 GeV 1 TeV
10
10
10
5
10
(7) 6
10
Solving Eq. (7) between the reheating temperature and a temperature T gives X 45 3=2 MP 32f 3 ½TRH T 3 ; Y ðTÞ ¼ q2 q2f Ncf 4 12507 g M s int f
10
10
14
18
10
10
Mint (GeV)
FIG. 2 (color online). Reheating temperature as function of the SOð10Þ breaking scale for different mass of dark matter: 10, 100, and 1000 GeV.
(8) 0 where pffiffiffi we replaced the mass of the Z by MZ0 ¼ ð5= 3ÞgD Mint and made the approximation g ¼ gs . We note that the effect of Z0 decay on the abundance of is completely negligible due to its Boltzmann suppression in the Universe: the Z0 is largely decoupled from the thermal bath already at the time of reheating. We note several interesting features from Eq. (8). First of all, the number density of the dark matter does not depend at all on the strength of the U0 ð1Þ coupling gD but rather on the intermediate scale (that is determined by requiring gauge coupling unification as we demonstrated in the previous section). Second, the production of dark matter is mainly achieved at reheating. Third, once the relic abundance is obtained, the number density per comoving frame (Y) is fixed, never having reached thermal equilibrium with the bath. And finally, upon applying the WMAP determination for the DM abundance, and assuming instantaneous reheating after inflation, we obtain a constraint on TRH once the pattern of SOð10Þ breaking is known (and thus Mint fixed). In a more complete model of reheating, the resulting relic density may be modified by an order of magnitude (or more) for a given reheating temperature. Thus, given a scheme of SOð10Þ breaking we can determine the reheating temperature very precisely from the relic abundance constraint in the Universe. From 2 crit h 13:5 H02 MP2 0 Y0 ¼ ¼ ; (9) m s0 0:1 163 g0s T03 m
where H is the Hubble parameter and the index ‘‘0’’ corresponds to present-day values, and combining Eqs. (8) and (9) we find 2 56254 h gs 3=2 MP H02 4 3 TRH ¼ M (10) P 45 T03 m g0s int 16q2 f 2f q2f Ncf 0:1 or
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2 1=3 h 100 GeV 1=3 TRH ’ 2 108 GeV m 0:1 4=3 Mint ; 12 10 GeV
(11)
P where we took for illustration q2 f 2f q2f Ncf ¼ 1. We show in Fig. 2 the evolution of TRH as function of Mint for different values of the dark matter mass m . We can thus fix the reheating temperature predicted by different symmetry breaking patterns. (We note that the value obtained for the intermediate scale in different SOð10Þ breaking schemes is not modified by the presence of a dark matter particle which is not charged under the SM gauge group.) We summarize them in Table I, where the values of TRH are given for m ¼ 100 GeV. Note that fixing TRH in this way is reminiscent of fixing particle masses or couplings in other dark matter models necessary to obtain the correct relic density. We note that although we have here and in Fig. 2 shown results for central values of m ¼ 100 GeV, dark matter candidates are possible with mass up to the order of the reheating temperature, which, based on Eq. (11), would allow it to be as large as 106 –107 GeV. Finally, we must specify the identity of the NETDM candidate in the context described above. The DM can be in the 126 or 144 representations of SOð10Þ. There are several mechanisms to render the DM mass light [9], one TABLE I. Possible breaking schemes of SOð10Þ.
A A B B C C
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Mint ðGeVÞ
TRH ðGeVÞ
4 2L 1R ½16 4 2L 1R ½126 4 2L 2R ½16 4 2L 2R ½126 3C 2L 2R 1B-L ½16 3C 2L 2R 1BL ½126
1012:9 1011:8 1014:4 1013:8 1010:6 108:6
3 109 1 108 3 1011 5 1010 3 106 6 103
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of which is through a fine tuning of the SOð10Þ couplings contributing with different Clebsch-Gordan coefficients (see for example, Refs. [10,11]) to the masses of the various 126 components. For example, for the group GA 126ðM þ y45 45H þ y210 210H Þ126;
(12)
where M MGUT , and a GA singlet in a linear combination of 210H and 45H has a vev at the GUT scale. m is then given by a linear combination of M and the vev and can be tuned to small values, while all other particles inside the 126 live close to MGUT . This is reminiscent of and no more severe than the doublet-triplet separation problem common to grand unified theories. Discussion.—Unfortunately, the chance of detection (direct or indirect) of NETDM with a massive mediator Z0 is nearly hopeless. Indeed, the diagram for the direct detection process, measuring the elastic scattering off a nucleus, proceeds through the t-channel exchange of the Z0 boson, and is proportional to 1=MZ4 0 yielding a negligible cross section. In addition, due to the present low velocity of dark matter in our Galaxy (’ 200 km=s), the indirect detection prospects from s-channel Z0 annihilation ! Z0 ! ff proportional to s2 =MZ4 0 is also negligible. As we have seen in Eq. (8), the production of dark matter occurs in the very early Universe at the epoch of reheating. A similar mechanism (though fundamentally completely different) where a dark matter candidate is produced close to the reheating time is the case of the gravitino [4,12]. Indeed, in both cases equilibrium is never reached and the relic abundance is produced from the thermal background to attain the decoupling value =H, with H the Hubble constant and ¼ hvinf the production rate. However, in the case of SOð10Þ, the cross section decreases with the temperature like hviZ0 / T 2 =MZ4 0 , whereas in the case of the gravitino the cross section is constant hvi3=2 / 1=MP2 implying YðTÞ / TRH . Gravitinos are produced though the scattering of light particles, whereas in the present context, the dark matter is mediated by a heavy particle (MZ0 > TRH ) which connects the dark and observable sectors. They are physically distinct possibilities. Finally, we note that cases B and C (in Table I) predict reheating temperatures which are larger (case B) or smaller (case C) than the case under consideration. Case A would also be compatible with successful thermal leptogenesis with a zero initial state abundance of right-handed neutrinos [13,14]. However in the cases B and C, the persistence of the SUð2ÞR symmetry would imply that the cancelation in Eq. (12) would leave behind a light SUð2ÞR triplet (for DM inside a 126) or doublet (for DM inside a 144). These would affect somewhat the beta functions for the RGE’s but more importantly leave behind a test of the model. In the triplet (doublet) case, we would expect three (at least two) nearly degenerate states: one with charge 0, being the DM candidate, and also states with electric charge 1 and 2 (or 1 in the doublet case).
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Conclusion.—In this work, we have shown that it is possible to produce dark matter through nonequilibrium thermal processes in the context of SOð10Þ models which respect the WMAP constraints. Insisting on gauge coupling unification, we have demonstrated that there exists a tight link between the reheating temperature and the scheme of the SOð10Þ breaking to the SM gauge group. Interestingly, the numerical values we obtained are quite high and very compatible with inflationary and leptogenesislike models. The authors would like to thank M. Tytgat, T. Hambye, E. Dudas, M. Voloshin, E. Fernandez-Martinez, K. Tarachenko, and J. M. Moreno for very useful discussions. This work was supported by the French ANR TAPDMS ANR-09-JCJC-0146 and the Spanish MICINN’s Consolider-Ingenio 2010 Programme under Grant No. Multi-Dark CSD2009-00064. B. Z. acknowledges Consolider-Ingenio PAU CSD2007-00060, CPAN CSD2007-00042, under Contract No. FPA2010-17747. Y. M. acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). The work of K. A. O. was supported in part by DOE Grant No. DE-FG02-94ER-40823 at the University of Minnesota.
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