Bound on Z' Mass from CDMS II in the Dark Left-Right Gauge Model II
aa r X i v : . [ h e p - ph ] F e b UCRHEP-T486February 2010
Bound on Z ′ Mass from CDMS IIin the Dark Left-Right Gauge Model II
Shaaban Khalil a,b , Hye-Sung Lee c , and Ernest Ma da Centre for Theoretical Physics, The British University in Egypt,El Sherouk City, Postal No. 11837, P.O. Box 43, Egypt b Department of Mathematics, Ain Shams University,Faculty of Science, Cairo 11566, Egypt c Department of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA d Department of Physics and Astronomy, University of California,Riverside, California 92521, USA
Abstract
With the recent possible signal of dark matter from the CDMS II experiment, the Z ′ mass of a new version of the dark left-right gauge model (DLRM II) is predicted tobe at around a TeV. As such, it has an excellent discovery prognosis at the operatingLarge Hadron Collider. ntroduction : One year ago, we proposed [1] that dark-matter fermions (scotinos) are nat-urally present in an unconventional left-right gauge extension of the standard model (SM)of particle interactions, which we call the dark left-right model (DLRM). It is a nonsuper-symmeric variation of the alternative left-right model (ALRM) discussed already 23 yearsago [2, 3]. One important difference of both the DLRM and the ALRM with the conventionalleft-right model (LRM) [4] is the fact that tree-level flavor-changing neutral currents [5] arenaturally absent so that the SU (2) R breaking scale may easily be at around a TeV, allow-ing both the charged W ± R and the extra neutral Z ′ gauge bosons to be observable at thelarge hadron collider (LHC). Interesting phenomenology of Z ′ decay into scalar bosons inthe DLRM has just recently been discussed [6].In this paper, we propose a new variant of this extension which we call DLRM II. [Othermore exotic variants are also possible [7].] Instead of having Majorana scotinos as darkmatter, we now have Dirac scotinos. Their interactions with nuclei through the Z ′ are thusrelevant for understanding the recent result of the dark-matter direct-search experimentCDMS II [8]. It will be shown that the Z ′ mass may indeed be around a TeV, and itsdiscovery prognosis at the LHC is excellent. Model : Consider the gauge group SU (3) C × SU (2) L × SU (2) R × U (1). The conventionalleptonic assignments are ψ L = ( ν, e ) L ∼ (1 , , , − /
2) and ψ R = ( ν, e ) R ∼ (1 , , , − / ν and e obtain Dirac masses through the Yukawa terms ψ L Φ ψ R and ψ L ˜Φ ψ R , whereΦ = ( φ , φ − ; φ +2 , φ ) ∼ (1 , , ,
0) is a Higgs bidoublet and ˜Φ = σ Φ ∗ σ = ( φ , − φ − ; − φ +1 , φ )transforms in the same way. Both h φ i and h φ i contribute to m ν and m e , and similarly m u and m d in the quark sector, resulting thus in the appearance of tree-level flavor-changingneutral currents.Suppose the term ψ L ˜Φ ψ R is forbidden by a symmetry, then the same symmetry may beused to maintain h φ i = 0 and only e gets a mass through h φ i 6 = 0. At the same time, ν L ν R are not Dirac mass partners, so they could in fact be completely different particleswith independent masses of their own. Whereas ν L is clearly the neutrino we observe in theusual weak interactions, ν R can now be something else entirely. Here we rename ν R as n R and show that it may in fact be a scotino, i.e. a fermionic dark-matter candidate.In our previous proposal [1], we imposed a new global U (1) symmetry S in such a way thatthe spontaneous breaking of SU (2) R × S will leave the combination L = S − T R unbroken.We then showed that L is a generalized lepton number, with L = 1 for the known leptons,and L = 0 for all known particles which are not leptons. Here we consider instead the case L = S + T R . Our model is nonsupersymmetric, but it may be rendered supersymmetric bythe usual procedure which takes the SM to the MSSM (minimal supersymmetric standardmodel). Under SU (3) C × SU (2) L × SU (2) R × U (1) × S , the fermions transform as shown inTable 1. Note the necessary appearance of the exotic quark h , which will turn out to carrylepton number as well. Table 1: Fermion content of proposed model.Fermion SU (3) C × SU (2) L × SU (2) R × U (1) Sψ L = ( ν, e ) L (1 , , , − /
2) 1 ψ R = ( n, e ) R (1 , , , − /
2) 3 / ν R (1 , , ,
0) 1 n L (1 , , ,
0) 2 Q L = ( u, d ) L (3 , , , /
6) 0 Q R = ( u, h ) R (3 , , , / − / d R (3 , , , − /
3) 0 h L (3 , , , − / − φ φ +2 φ − φ ! , Φ L = φ + L φ L ! , Φ R = φ + R φ R ! . (1)3able 2: Scalar content of proposed model.Scalar SU (3) C × SU (2) L × SU (2) R × U (1) S Φ (1 , , , − / σ Φ ∗ σ (1 , , ,
0) 1 / L (1 , , , /
2) 0Φ R (1 , , , /
2) 1 / S are listed in Table 2.The Yukawa terms allowed by S are then ψ L Φ ψ R , ψ L ˜Φ L ν R , ψ R ˜Φ R n L , Q L ˜Φ Q R , Q L Φ L d R ,and Q R Φ R h L , whereas ψ L ˜Φ ψ R , n L ν R , Q L Φ Q R , and h L d R are forbidden. Hence m e , m u comefrom v = h φ i ; m ν , m d come from v = h φ L i ; and m n , m h come from v = h φ R i . Thisstructure shows clearly that flavor-changing neutral currents are guaranteed to be absent attree level.As it stands, both the neutrino ν and the scotino n are Dirac fermions, and lepton number L is conserved. If we now introduce a soft term ν R ν R which breaks L by two units, then ν L gets a Majorana mass through the canonical seesaw mechanism, as is usually assumed.As for n , it remains a Dirac fermion, being protected by a residual global U (1) symmetry,under which n , W + R transform as 1, and h , φ , − transform as − Gauge sector : Since e has L = 1 and n has L = 2, the W + R of this model must have L = S + T R = 0 + 1 = 1. This also means that unlike the conventional LRM, W ± R does notmix with the W ± L of the SM at all. This important property allows the SU (2) R breakingscale to be much lower than it would be otherwise, as explained already 23 years ago [2, 3].Let e/g L = s L = sin θ W and s R = e/g R , with c L,R = q − s L,R , then g B = e/ q c L − s R and4he neutral gauge bosons of the DLRM (as well as the ALRM) are given by AZZ ′ = s L s R q c L − s R c L − s L s R /c L − s L q c L − s R /c L q c L − s R /c L − s R /c L W L W R B . (2)Whereas Z couples to the current J L − s L J em with coupling e/s L c L as in the SM, Z ′ couplesto the current J Z ′ = s R J L + c L J R − s R J em (3)with coupling g Z ′ = e/s R c L q c L − s R .The masses of the gauge bosons are given by M W L = e s L ( v + v ) , M Z = M W L c L , M W R = e s R ( v + v ) , (4) M Z ′ = e c L s R ( c L − s R ) ( v + v ) − s L s R M W L c L ( c L − s R ) , (5)where zero Z − Z ′ mixing has been assumed, using the condition [3] v / ( v + v ) = s R /c L . Direct search constraint from CDMS II : The Z ′ couplings to u , d , n (in units of g Z ′ ) aregiven by u L = − s R , u R = 12 c L − s R , u V = 14 c L − s R , (6) d L = − s R , d R = 13 s R , d V = 112 s R , (7) n L = 0 , n R = 12 c L , n V = 14 c L . (8)The effective Lagrangian for elastic scattering of the scotino n off quarks is then given by L = g Z ′ n V M Z ′ (¯ nγ µ n )( u V ¯ uγ µ u + d V ¯ dγ µ d ) . (9)In the original DLRM [1], n is a Majorana scotino, so it does not contribute to the s-waveelastic spin-independent scattering cross section in the nonrelativistic limit. Here n is a Diracscotino, so it will contribute. Let f P = g Z ′ n V (2 u V + d V ) /M Z ′ , f N = g Z ′ n V ( u V + 2 d V ) /M Z ′ , (10)5hen its elastic cross section per nucleon is given by [9] σ = 4 m r π [ Zf P + ( A − Z ) f N ] A , (11)where Z and A are the atomic and mass numbers of the target nucleus, and m r = m n m P / ( m n + m P ) ≃ m P . The CDMS II collaboration [8] observed two possible signal events with an ex-pected background of 0 . ± .
1. Using Ge, i.e. Z = 32 and A − Z = 41, as a representativeestimate of σ , this result could also be considered as an upper bound, i.e. σ = πα m P (105 c L − s R ) (146) s R ( c L − s R ) M Z ′ < . × − pb , (12)which occurs at m n = 70 GeV. Phenomenological analysis : We consider the range e < s R < c L − e , where the lower boundcorresponds to g R = 1 and the upper bound to g B = 1. The values of g Z ′ and Γ Z ′ /M Z ′ areplotted in Fig. 1(a) and (b), where Z ′ is assumed to decay only into SM fermions.(a) (b)Figure 1: (a) g Z ′ vs s R . (b) Γ Z ′ /M Z ′ vs s R for SM fermions decay products only in the cases M Z ′ = 500 GeV (blue solid) and M Z ′ → ∞ (red dashed).We compute the production and decay of Z ′ to e + e − at the Tevatron as a function of M Z ′ for various values of s R and compare it to data [10] at E cm = 1 .
96 TeV and an integrated6uminosity of 2.5 fb − in Fig. 2(a). We then plot the exclusion limits on M Z ′ from boththe new CDMS II data and the Tevatron as a function of s R in Fig. 2(b). Note that theCDMS II bound is stronger than the Tevatron bound for s R < .
5. Note also that due tothe accidental cancellation in the numerator of σ in Eq. (12), the observed events at CDMSII cannot be interpreted as signals of dark matter in this model if s R > .
5, because theywould be excluded by the Tevatron data.(a) (b)Figure 2: (a) Lower bound on the Z ′ mass in this model from Tevatron dielectron search.(b) M Z ′ vs s R from the CDMS II (blue dashed) and Tevatron (red solid) bounds. The dottedsegments assume a simple extrapolation of the Tevatron data.Given that M Z ′ is allowed to be in the TeV range, its discovery prognosis is excellent atthe LHC. We show in Fig. 3 its discovery reach (assuming E cm = 14 TeV) by 10 dileptonevents (either dielectron or dimuon) which satisfy the following basic cuts on their transversemomenta, rapidities, and invariant mass: p T >
20 GeV (each lepton), | η | < . | M ℓ ¯ ℓ − M Z ′ | < Z ′ .Using these cuts, the dominant SM background from γ/Z (Drell-Yan) is negligible. Withan integrated luminosity of 1 fb − , the Z ′ of DLRM II with M Z ′ ∼ Z ′ discovery by 10 dielectron events at LHC. Small circles areTevatron limits. Dark-matter relic abundance : In this model, the dark-matter relic abundance is presumablydetermined by the annihilation n ¯ n → Z ′ → SM fermions. The thermally averaged crosssection multiplied by relative velocity is approximately given by h σv rel i Z ′ = g Z ′ c L m n P f ( f L + f R )32 π (4 m n − M Z ′ ) , (13)where the sum over fermions should include a factor of 3 for quarks and an overall factorof 3 for families. Fixing the above at 1 pb as a typical value to satisfy the requirement ofdark-matter relic abundance, it can easily be shown that for m n = 70 GeV, the required M Z ′ is very much below the CDMS II bound. [For example, for s R = 0 . M Z ′ = 267 GeVwould be required.] In other words, the n ¯ n → Z ′ annihilation cross section would be toosmall to account for the observed dark-matter relic abundance. To remedy this situation,the mechanism proposed in the original DLRM may be invoked, i.e. n ¯ n → l − l + through ∆ + R exchange. However, this requires adding the SU (2) R scalar triplet (∆ ++ R , ∆ + R , ∆ R ), which isnot necessary in our present version and thus not very much motivated. The alternative isto consider a larger value of m n . 8he CDMS II bound on σ is very well approximated in the range 0 . < m n < . σ < . × − pb ( m n / . . (14)Using this on the right-hand side of Eq. (12), we plot in Fig. 4(a) and (b) the M Z ′ boundsfor m n = 400 and 600 GeV, as well as the solutions of M Z ′ (with M Z ′ > m n ) to Eq. (13)for 1 pb. We see that there are indeed consistent solutions (where the solid curve is higherthan the dash curve) for a range of s R in each case. If m n falls below 300 GeV, then thereis no solution because M Z ′ would then be excluded by the Tevatron bound. We note alsothat only a modest resonance enhancement is needed from the denominator of Eq. (13).The n ¯ n annihilation to l + l − through W R exchange also contributes to the dark-matter relicabundance, but its value is an order of magnitude less, i.e. h σv rel i W R = 3 g R m n π ( m n + M W R ) . (15) m n =400 GeV M Z ’ ( G e V ) s R2 m n =600 GeV M Z ’ ( G e V ) s R2 (a) (b)Figure 4: (a) For m n = 400 GeV, the CDMS II bound on M Z ′ (blue dashed) and the valueof M Z ′ (red) from h σv rel i Z ′ = 1 pb vs s R ; (b) same as in (a) for m n = 600 GeV.9 epton flavor violation : Unlike the original DLRM, where a scalar triplet (∆ ++ R , ∆ + R , ∆ R )may mediate lepton flavor violating processes such as µ → eee at tree level, and must beforbidden by hand, the DLRM II is safe because it has no such interactions. Nevertheless,lepton (as well quark) flavor violation occurs in one loop in the SU (2) R sector, in completeanalogy to that of the SM in the SU (2) L sector. The branching fraction of µ → eγ is then B ( µ → eγ ) = 3 α | δ R | π s L M W L s R M W R ! < . × − , (16)where the experimental upper bound has also been displayed, and δ R is the analog of the well-known suppression factor δ L = P i U ∗ ei U µi ( m ν i /M W L ) in the SM. For s R = s L , we have M W R =1 . | δ R | < . µ → eγ may be imminent. The same holds for other lepton flavor violating processes suchas µ − e conversion in nuclei. Note that the contribution to the muon anomalous magneticmoment here is about 10 − , well below the experimental sensitivity. A more comprehensivestudy, including contributions to D − ¯ D mixing [11], will be given elsewhere. Connecting the Z ′ and dark-matter searches : As the LHC begins its operation, one of itsfirst possible discoveries could be a Z ′ through the process q ¯ q → Z ′ → l + l − . There are many Z ′ models, and some of them could also be invoked [12] to explain the CDMS II results.However, the coupling of the dark matter to the Z ′ in these models is in general not relatedto the Z ′ leptonic couplings. Here they are intimately connected and predicted as a functionof only s R . In fact, if we assume s R = s L (i.e. left-right symmetry), then there is no freeparameter. Our numerical analysis in this paper is only a rough estimate for illustration,but it points to the important assertion that the Z ′ interactions in this model are fixedwith respect to direct dark-matter search and the detection of Z ′ itself at an accelerator. Inthese exciting times of having both the functioning LHC and ongoing dark-matter searchexperiments, the dark-matter mystery in astroparticle physics may be near a solution.10 cknowledgements : The work of S.K. was partially supported by the Science and TechnologyDevelopment Fund (STDF) Project ID 437 and the ICTP Project ID 30. This work wasalso supported by the U. S. Department of Energy Grants No. DE-AC02-98CH10886 (HL)and No. DE-FG03-94ER40837 (EM). References [1] S. Khalil, H. S. Lee, and E. Ma, Phys. Rev.
D79 , 041701(R) (2009).[2] E. Ma, Phys. Rev.
D36 , 274 (1987).[3] K. S. Babu, X.-G. He, and E. Ma, Phys. Rev.
D36 , 878 (1987).[4] For a comprehensive treatment, see for example N. G. Deshpande et al. , Phys. Rev.
D44 , 837 (1991).[5] S. L. Glashow and S. Weinberg, Phys. Rev.
D15 , 1958 (1977).[6] A. Aranda, J. L. Diaz-Cruz, J. Hernandez-Sanchez, and E. Ma, arXiv:1001.4057 [hep-ph].[7] E. Ma, Phys. Rev.
D79 , 117701 (2009).[8] Z. Ahmed et al. [CDMS Collaboration], arXiv:0912.3592 [astro-ph].[9] See, for example, Q.-H. Cao, I. Low, and G. Shaunghnessy, arXiv:0912.4510 [hep-ph].[10] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. , 031801 (2009).[11] E. Ma, Mod. Phys. Lett. A3 , 319 (1988).[12] See, for example, H. S. Lee, AIP Conf. Proc.1078