Neutrino Cosmology after WMAP and LHC7
Abstract
The gaugeextended model has the attractive property of elevating the two major global symmetries of the standard model (baryon number and lepton number ) to local gauge symmetries. The symmetry prevents the generation of Majorana masses, leading to three superweakly interacting righthanded neutrinos. This also renders a symmetry nonanomalous. We show that the superweak interactions of these Dirac states (through their coupling to the TeVscale gauge boson) permit righthanded neutrino decoupling just above the QCD phase transition: . In this transitional region, the residual temperature ratio between and generates extra relativistic degrees of freedom at BBN and at the CMB epochs. Consistency (within ) with both WMAP 7year data and the most recent estimate of the primordial He mass fraction is achieved for . The model is fully predictive, and can be confronted with dijet and dilepton data (or lack thereof) from LHC7 and, eventually, LHC14.
Heavy neutral vector gauge bosons (’s) are ubiquitous in extensions of the standard model (SM) Langacker:2008yv (), often including a gauged symmetry. This symmetry is nonanomalous if the three lefthanded Weyl neutrinos are accompanied by three righthanded neutrinos. The masses are a priori open parameters – not determined by the low energy effective theory – but subject to recent experimental bounds () from searches of dilepton Chatrchyan:2011wq () and dijet Chatrchyan:2011ns () events in the 7 TeV run of the Large Hadron Collider (LHC7). In this Letter we reexamine some critical cosmological issues surrounding the presence of the six additional neutrino degrees of freedom Steigman:1979xp () correlated to the presence of a in our dynamical model which is coupled to . These considerations, when viewed in the context of most recent data collected by the Wilkinson Microwave Anisotropy Probe (WMAP) Komatsu:2010fb (), are found to constrain the mass of the to an interestingly narrow band, which will be directly probed by LHC14.
For a good part of the past two decades, bigbang nucleosynthesis (BBN) provided the best inference of the radiation content of the universe. The timedependent quantity being the neutron abundance at , which regulates the primordial fraction of baryonic mass in He,
(1) 
where is the effective number of extra (nonSM) light neutrino species, is the neutron half life, and is the ratio of the baryon number density to the photon number density Sarkar:1995dd (). The observationallyinferred primordial fractions of baryonic mass in He (, , , and Izotov:2007ed ()) have been constantly favoring Simha:2008zj (). Out of the blue, two recent independent studies determined and Izotov:2010ca (). For and , the updated effective number of light neutrino species is reported as () and (), respectively.
Very recently, in support of these trends, observations of the cosmic microwave background (CMB) anisotropies and the largescale structure distribution have allowed a probe of at the CMB decoupling epoch with unprecedented precision. The relativistic particles that stream freely influence the CMB in two ways: (1) their energy density alters the matterradiation equality epoch, and (2) their anisotropic stress acts as an additional source for the gravitational potential via Einstein’s equations. Hence, the number of light relativistic species becomes a function of the matter density and the redshift of matterradiation equality ,
(2) 
where is the presentday photon energy density (for ) and the scaled Hubble parameter is defined by Kolb:1990vq (). The variation in reads
(3) 
The equality redshift is one of the fundamental observables that one can extract from the CMB power spectrum. More specifically, WMAP data constrain mainly from the height of the third acoustic peak relative to the first peak Komatsu:2010fb (). The fractional error in is determined using external data: the latest distance measurements from the Baryon Acoustic Oscillations (BAO) in the distribution of galaxies Percival:2009xn () and precise measurements of the Hubble constant Riess:2009pu (). The parameter constraints from the combination of WMAP 7year data, BAO, and lead to Komatsu:2010fb ().
In summary, though uncertainties remain large, the most recent cosmological observations show a consistent preference for additional relativistic degrees of freedom (r.d.o.f.) during BBN and the CMB epochs. We take these hints as motivation for our analysis, which consists of the following tasks: (1) to present a model in which the additional r.d.o.f. are three flavors of light righthanded neutrinos which interact with the SM fermions via the exchange of heavy vector fields; (2) to suppress the six additional fermionic r.d.o.f. to levels in compliance with BBN and CMB. This is accomplished by imposing the decoupling of ’s from the plasma early enough so that they undergo incomplete reheating during the QCD phase transition; and late enough so as to leave an excess neutrino density suggested by the data Feng:2011uf (). These requirements strongly constrain the masses of the heavy vector fields. Together with the couplings, which are determined in accord with other considerations, the model is fully predictive, and can be confronted with dijet and dilepton data (or lack thereof) from LHC7 and, eventually, LHC14.
An economic choice of the model to implement the task outlined above is based on the gaugeextended sector Anchordoqui:2011eg (). The resulting content gauges the baryon number [with ], the lepton number , and a third additional abelian charge which acts as the third isospin component of an . The usual electroweak hypercharge is a linear combination of these three charges: . The matter fields consist of six sets (labeled by an index ) of Weyl fermionantifermion pairs: . The field is the righthanded neutrino (and lefthanded antineutrino) accompanying the fields in the set , with mass The gauging of lepton number precludes the presence of a seesaw for generating Majorana neutrino masses. In addition to the SM interactions, these fields experience two gauge interactions mediated by two associated vector bosons ( and ) whose masses lie well above a TeV.
The initially free parameters consist of three couplings These are augmented by three Euler angles to allow for a field rotation to coupling diagonal
in
hypercharge. This diagonalization fixes two of the angles and the orthogonal nature of the rotation introduces one constraint on the couplings . The baryon number coupling is fixed to be of the nonabelian
coupling at the scale of unification, and is therefore determined at all energies
through RG running. This leaves one free angle and two couplings with one constraint.
The two remaining degrees of freedom allow a further rotation leaving to couple to at 90% and to couple to at 99%.
The model as described enjoys distinct advantages: (1) Gauging of the anomalous and its cancellation by generalized GreenSchwarz mechanism (which leaves as a global symmetry) prevents proton decay. (2) The presence of renders nonanomalous. This has been appealing for minimal extension of SM at the TeVscale. For example, the mass growth of can occur via a conventional Higgs mechanism at TeV without relying on possible Planck scale physics. (3) By inspection of Table 1 the charges , , and are mutually orthogonal in the fermion space. This will maintain the othogonality relation to one loop without inducing kinetic mixing Anchordoqui:2011eg ().
We begin by first establishing, in a model independent manner, the range of decoupling temperatures implied by the the BBN and CMB analyses. For this work, the physics of interest will be taking place at energies in the region of the QCD phase transition, so that we will restrict ourselves to the following fermionic fields, and their contribution to r.d.o.f.:
(4) 
This amounts to 22 Weyl fields, translating to 44 fermionic r.d.o.f.
Name  Representation  

Next, in line with our stated plan, we use the data estimate to calculate the range of decoupling temperature. The effective number of neutrino species contributing to r.d.o.f. can be written as
(5) 
therefore, taking into account the isentropic heating of the rest of the plasma between decoupling temperature and the end of the reheating phase,
(6) 
where is the temperature at the end of the reheating phase, and is the effective number of r.d.o.f. at temperature , with for each real vector field and for each spin Weyl field. The coefficient is unity for the lepton and photon contributions, and is the ratio for the quarkgluon plasma. Here is the actual (ideal StefanBolzmann) entropy. Hence We take reflecting . We consistently omit in considering the thermodynamics part of the discussion, but will include it when dealing with expansion. As stated in the introduction
(7) 
so the excess r.d.o.f. will lie within of the central value of each experiment if . From Eqs. (6) and (7), the allowable range for is
(8) 
This is achieved for By comparing to Fig. 8 in Ref. Bazavov:2009zn (), this can be translated into temperature range
(9) 
with the lower temperature coinciding with the region of most rapid rise of the entropy. Thus, the data implies that the decoupling takes place during QCD phase transition.
We now turn to use our model in conjunction with the decoupling condition to constrain its parameters. To this end we calculate the interaction rate for a righthanded neutrino and determine from the plasma via the prescription
(10) 
Let be a single species of Weyl fermion, representing the two r.d.o.f. , where the superscript indicates bins . Similarly for . Notice that the subscripts denote the actual helicities of the massless particles in question, not the chirality of the fields. With this said, we may write the amplitude for scattering
(11) 
The other 3 amplitudes are obtained by the crossing substitutions in the second square bracket; for scattering from
(12)  
The cross sections for the four scattering processes (no average over helicities) are
(13) 
and
(14) 
In addition to these scattering processes, the interacts with the plasma through the annihilation processes: for bins , and , for bins These all yield cross sections due to forward and backward suppression. Assuming all chemical potentials to be zero, the plasma will have an equal number density , for each fermion r.d.o.f. Thus,
(15) 
where is the Moller velocity, is the square of the centerofmass energy, and is the multiplicty of Weyl fields in each bin (e.g., for ). The scattering cross section is given by
(16) 
similarly,
(17) 
Since and , we perform an approximate angular average , followed by a thermal averaging to give
(18) 
(19) 
Each of the is given by the sum of the contributions from and exchange,
(20) 
The Hubble expansion parameter during this time is
(21) 
where is the Planck mass. Since the quarkgluon energy density in the plasma has a similar dependence to that of the entropy (see Fig. 7 in Bazavov:2009zn ()), we take so that . (The first factor provides an average for over the temperature region, and we have now included the six r.d.o.f.) Since and , it is clear that if at some temperature , , the ratio will fall rapidly on further cooling. Thus from (10) and (21) the equation determining depends on: (1) whether we need to preserve the absence of a chemical potential, or (2) whether we need simply to mantain physical equilibrium. The decoupling condition in these two cases is: (1) and (2) ; or numerically: (1)
(22) 
and (2)
(23) 
as determined from these equations must lie in the band (9).
Since all freedom of determining coupling constant and mixing angles has been exercised, there remains only constraints on the possible values of and . Our results are encapsulated in Fig. 1, and along with other aspects of this work are summarized in these concluding remarks:

In this Letter, we develop a dynamic explanation of recent hints that the relativistic component of the energy during the era of last scattering is equivalent to about 1 extra Weyl neutrino.

We work within the context of a specific (string based) model with 3 gauge symmetries, originally coupled to baryon number , lepton number , and a 3rd component of righthanded isospin . We find that rotation of the gauge fields to a basis exactly diagonal in hypercharge , and very nearly diagonal in and fixes all the mixing angles and the gauge couplings. Of course, of most significance for this work, requiring that the current be anomaly free, implies the existence of 3 righthanded Weyl neutrinos.

We then find that for certain ranges of and the decoupling of the ’s occurs during the course of the QCD phase transition, just so that they are only partially reheated compared to the ’s — the desired outcome.

To carry out this program, we needed to make use of some high statistics lattice simulations of a QCD plasma in the hot phase, especially the behavior of the entropy during the changeover.

Since our aim is to match the data, which has lower and upper bounds on the neutrino “excess”, we obtain corresponding upper and lower bounds on the gauge field masses. Roughly speaking, if decoupling requires a freezout of the annihilation channel (loss of chemical equilibrium), then 3 TeV 4 TeV. If thermal equilibrium via scattering is sufficient, then 4.5 TeV 6 TeV. These are ranges to be probed at LHC14.

Finally, a remark about the model: the gauging of allows a global conservation of baryon number. The gauging of brings with it the loss of a heavy Majorana for the seesaw model, as well as for leptogenesis through the decay of this particle. Thus, along with all its companion fields, the neutrino it is a Dirac particle, with the small mass originating through small Yukawa.
L.A.A. is supported by the U.S. National Science Foundation (NSF) under CAREER Award PHY1053663. H.G. is supported by NSF Grant PHY0757959.
Footnotes
 Although not generally appreciated, it is important to note that a 100% coupling of the and to and , respectively, is possible only if the gauge coupling constants are equal.
 Our couplings are consistent with the bounds presented in Williams:2011qb () from a variety of experimental constraints.
References
 P. Langacker, Rev. Mod. Phys. 81, 1199 (2009) [arXiv:0801.1345 [hepph]].
 S. Chatrchyan et al. [CMS Collaboration], JHEP 1105, 093 (2011) [arXiv:1103.0981 [hepex]]; G. Aad et al. [ATLAS Collaboration], arXiv:1108.1582 [hepex].
 S. Chatrchyan et al. [CMS Collaboration], arXiv:1107.4771 [hepex]; G. Aad et al. [ATLAS Collaboration], arXiv:1108.6311 [hepex].
 G. Steigman, K. A. Olive, D. N. Schramm, Phys. Rev. Lett. 43, 239 (1979); K. A. Olive, D. N. Schramm, G. Steigman, Nucl. Phys. B180, 497 (1981); V. Barger, P. Langacker and H. S. Lee, Phys. Rev. D 67, 075009 (2003) [arXiv:hepph/0302066].
 E. Komatsu et al. [ WMAP Collaboration ], Astrophys. J. Suppl. 192, 18 (2011) [arXiv:1001.4538 [astroph.CO]].
 S. Sarkar, Rept. Prog. Phys. 59, 1493 (1996) [arXiv:hepph/9602260].
 Y. I. Izotov, T. X. Thuan and G. Stasinska, Astrophys. J. 662, 15 (2007) [arXiv:astroph/0702072]. M. Peimbert, V. Luridiana and A. Peimbert, Astrophys. J. 666, 636 (2007) [arXiv:astroph/0701580]; G. Steigman, Ann. Rev. Nucl. Part. Sci. 57, 463 (2007) [arXiv:0712.1100 [astroph]].
 V. Simha and G. Steigman, JCAP 0806, 016 (2008) [arXiv:0803.3465 [astroph]].
 Y. I. Izotov and T. X. Thuan, Astrophys. J. 710, L67 (2010) [arXiv:1001.4440 [astroph.CO]]; E. Aver, K. A. Olive and E. D. Skillman, JCAP 1103, 043 (2011) [arXiv:1012.2385 [astroph.CO]]; E. Aver, K. A. Olive and E. D. Skillman, JCAP 1005, 003 (2010) [arXiv:1001.5218 [astroph.CO]].
 E. W. Kolb and M. S. Turner, The Early universe, Front. Phys. 69, 1 (1990).
 W. J. Percival et al. [SDSS Collaboration], Mon. Not. Roy. Astron. Soc. 401, 2148 (2010) [arXiv:0907.1660 [astroph.CO]].
 A. G. Riess et al., Astrophys. J. 699, 539 (2009) [arXiv:0905.0695 [astroph.CO]].
 Other specific models for increasing via freezeout of new dark matter degrees of freedom prior to various reheating stages have very recently been proposed. See J. L. Feng, V. Rentala and Z. Surujon, Phys. Rev. D 84, 095033 (2011) [arXiv:1108.4689 [hepph]]; H. K. Dreiner, M. Hanussek, J. S. Kim and S. Sarkar, arXiv:1111.5715 [hepph]; D. Hooper, F. S. Queiroz and N. Y. Gnedin, arXiv:1111.6599 [astroph.CO].
 L. A. Anchordoqui, I. Antoniadis, H. Goldberg, X. Huang, D. Lüst and T. R. Taylor, arXiv:1107.4309 [hepph].
 M. Williams, C. P. Burgess, A. Maharana and F. Quevedo, arXiv:1103.4556 [hepph].
 A. Bazavov et al., Phys. Rev. D 80, 014504 (2009) [arXiv:0903.4379 [heplat]].