PHYSICAL REVIEW D 88, 057703 (2013) Low-scale - unification

Pavel Fileviez Pe´rez1 and Mark B. Wise2 1Particle and Astro-Particle Division, Max Planck Institute for Nuclear Physics (MPIK), Saupfercheckweg 1, 69117 Heidelberg, Germany 2California Institute of Technology, Pasadena, California 91125, USA (Received 6 August 2013; published 30 September 2013) We investigate the possibility that and are unified at a low energy scale much smaller than the grand unified scale. A simple theory for quark-lepton unification based on the gauge group SUð4ÞC SUð2ÞL Uð1ÞR is proposed. This theory predicts the existence of scalar leptoquarks which could be produced at the Large Hadron Collider. In order to have light without fine-tuning, their masses are generated through the inverse seesaw mechanism.

DOI: 10.1103/PhysRevD.88.057703 PACS numbers: 12.10.Dm, 14.80.Sv

I. INTRODUCTION our model and outline its main phenomenological consequences. In the (SM) there are two types of In Sec. II we present the simplest model with quark- matter fields: the lepton and quark fields. The SM with lepton unification at a low scale that is consistent with the right-handed neutrinos describes all the measured properties experimental properties of quarks and leptons. In Sec. III of quarks and leptons. Grand unified theories based on we discuss the properties of the vector and scalar lepto- ð10Þ gauge groups, such as SU(5) or SO , provide one avenue quarks. Finally, we briefly summarize our main results for unifying the properties of quarks and leptons since in Sec. IV. quarks and leptons are part of the same representation of the gauge group. However, in this case the scale of 15 16 II. QUARK-LEPTON UNIFICATION unification is very high, MGUT 10 – GeV. An appealing approach to quark and lepton unification In models with QL unification based on the idea that was proposed by Pati and Salam in Ref. [1]. They used an leptons have the fourth color [1] the SM quarks and leptons ð Þ ð Þ SUð4ÞC gauge symmetry and the quarks and leptons are can be unified in the same multiplets: QL;‘L , uR;R , together in the fundamental representation of the gauge and ðdR;eRÞ. Therefore, naively one finds the following group. In this framework the leptons are the fermions with relations between quark and lepton masses: the fourth color. This idea also played a major role in grand M ¼ MD and M ¼ M ; (1) unification since the Pati-Salam gauge group is the maxi- u e d ð10Þ D mal subgroup of SO . This theory also predicts the where Mu, M , Me and Md are the up-quark, Dirac , existence of right-handed neutrinos needed for the seesaw charged lepton and down-quark masses, respectively. As is mechanism [2–6] of neutrino masses. well known, these relations are not consistent with In this paper we revisit the idea of quark-lepton (QL) experiment. unification based on the Pati and Salam paper where the We now construct the simplest model of quark-lepton leptons have the fourth color. We find a very simple ex- unification based on the gauge group tension of the SM based on the gauge group SUð4ÞC GQL ¼ SUð4Þ SUð2Þ Uð1Þ ; (2) SUð2ÞL Uð1ÞR, where quarks and leptons are unified in C L R the same representation. In addition to the SM fermions the which is consistent with experimental results on the prop- model contains three right-handed singlet fermions needed erties of quarks and leptons evading the unacceptable mass to generate Majorana neutrino masses through the inverse relations discussed above, and which allows the scale for seesaw mechanism. the symmetry breaking GQL ! GSM ¼ SUð3ÞSUð2ÞL Assuming near alignment of the quark and lepton Uð1Þ to be much smaller than the grand unification scale. generations the experimental limit on the branching ratio Y 0 ! SUð4Þ The fermion matter fields are in the representations for KL e implies that the scale of C breaking ! must be greater than 1000 TeV. See for example the studies u FQL ¼ ð4; 2; 0Þ; (3) in Refs. [7,8] for the constraints coming from meson de decays. This theory predicts the existence of vector and scalar leptoquarks. While the vector leptoquarks must be F ¼ uc c ð4; 1; 1=2Þ; (4) heavy, the scalar leptoquarks could be at the TeV scale u and give rise to exotic signatures at the Large Hadron ¼ c c ð4 1 1 2Þ Collider. In this article we discuss the spectrum of Fd d e ; ; = : (5)

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Here we have chosen to work only with left-handed v1 1 v2 M ¼ Y3 pffiffiffi þ pffiffiffi Y4 pffiffiffi ; (14) fermion fields, as is common in discussions of grand d 2 2 6 2 unified theories. The gauge group GQL is spontaneously broken to GSM v1 3 v2 M ¼ Y3 pffiffiffi pffiffiffi Y4 pffiffiffi : (15) by the of the scalar field, e 2 2 6 2 ¼ 0 ð4 1 1 2Þ u R ; ; = : (6) Here the vacuum expectation values that break GSM are v1 v2 Without loss of generality the vacuum expectation value hH0i¼pffiffiffi and hH0i¼pffiffiffi : (16) h 0 i¼ 2 2 2 canp beffiffiffi taken to be only in the fourth component, R 2 v= . The SM hypercharge Y is given by Since there are four independent Yukawa coupling pffiffiffi 6 matrices in the above equations we can generate acceptable ¼ þ masses for all the quarks and leptons. However, in order to Y R 3 T4; (7) achieve light Dirac neutrino masses one needs a severe D where T4 is the properly normalized SUð4ÞC generator, that fine-tuning between the two terms contributing to M when acting on the fundamental 4 representation is the in Eq. (13). See Ref. [11] for an alternative model using diagonal the Pati-Salam symmetry. 0 1 It is useful to note that the renormalizable couplings of 100 0 B C the model contain an automatic global U(1) fermion matter 1 B 010 0C symmetry Uð1Þ where the matter charges of the fermion ¼ pffiffiffi B C F T4 B C: (8) fields F ¼ 1, F ¼ F ¼1. The scalar fields do not 2 6 @ 001 0A FQL Fu Fd 0003 transform under this symmetry. Although this model can be consistent with experiment

To break the gauge group down to the low-energy SUð3ÞC the fine-tuning needed to get very light Dirac neutrinos is Uð1ÞY gauge group in a way that can give acceptable not attractive. A modest extension of the fermion content fermion masses we add two more scalar representations, a of the model allows us to avoid this fine-tuning. Higgs doublet We can generate small Majorana masses for the light neutrinos if we add three new singlet left-handed fermionic T þ 0 H ¼ H H ð1; 2; 1=2Þ; (9) fields N and use the following interaction terms: ð15 2 1 2Þ 1 and the scalar ; ; = , L ¼ þ þ H c ! QL Y5FuN NN : : (17) 2 ¼ 8 3 þ T4H2; (10) For simplicity, we now discuss the neutrino sector in the 4 0 one-generation case. The discussion generalizes easily to which contains a second Higgs doublet H2. The new the three-generation case. Then, there are three left-handed scalars in are the second Higgs doublet, the color octet neutrino fields (one each of , c and N) and the neutrino with the same weak quantum numbers as the Higgs dou- mass matrix reads as ð8 2 1 2Þ 0 10 1 blet, 8 ; ; = SM, studied by Manohar and Wise in 0 D 0 M Ref. [9], and the scalar leptoquarks 3 ð3; 2; 1=6ÞSM B CB C c B ð DÞT 0 D CB c C and 4 ð3; 2; 7=6ÞSM. These scalar leptoquarks do not N @ M M A@ A: (18) give rise to proton decay [10] at the renormalizable level D T 0 ðM Þ N since they do not couple to a quark pair. Proton decay D occurs at the dimension-six level. Here M is given by Eq. (13) and The Yukawa interactions in this theory are given by v D pffiffiffi M ¼ Y5 : (19) Y y 2 LQL ¼ Y1FQLFuH þ Y2FQLFu þ Y3H FQLFd y By assigning the N fermion charge F ¼ 1 only the term þ Y4 FQLFd þ H:c:; (11) N proportional to breaks the matter symmetry. Hence it is which after symmetry breaking give rise to the following natural to take this parameter to be much smaller than the mass matrices for the SM fermions: other entries in the mass matrix. D D 1 In the limit when Mqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, M , the model has a heavy pv1ffiffiffi pffiffiffi pv2ffiffiffi Mu ¼ Y1 þ Y2 ; (12) D 2 D 2 c 2 2 6 2 Dirac neutrino with mass ðM Þ þðM Þ , where is paired with the linear combination v1 3 v2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ pffiffiffi pffiffiffi pffiffiffi M Y1 Y2 ; (13) ¼ð D þ D Þ ð DÞ2 þð DÞ2 2 2 6 2 h M N M = M M : (20)

057703-2 BRIEF REPORTS PHYSICAL REVIEW D 88, 057703 (2013) ¼ 3 þ The orthogonal linear combination is the light Majorana leave color and the electromagnetic charge Q TL Y neutrino unbroken, i.e., only the neutral components of , H and H2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi get expectation values. D D D 2 D 2 l ¼ðM M NÞ= ðM Þ þðM Þ : (21) III. VECTOR AND SCALAR LEPTOQUARKS It would be massless in the limit ! 0 since then Majorana masses are forbidden by the fermion matter symmetry. This theory predicts the existence of vector and scalar So for this neutrino to have the same properties as in the leptoquarks. The vector leptoquarks, X ð3; 1; 2=3ÞSM, D D SM we need M M , which is reasonable when the have the following interactions: scale of SM symmetry breaking is much smaller than g4 SUð4Þ Uð1Þ L pffiffiffi X ðQ ‘ þ u þ d e þ u Þ the scale of C R symmetry breaking. When 2 L L R R R R R R the parameters in the neutrino mass matrix follow the D þ H:c: relation M M, the Majorana light neutrino (25) masses are given by The gauge coupling g4 is equal to the strong coupling 2 2 SUð4Þ m ¼ ðMDÞ =ðMDÞ ; (22) constant evaluated at the C scale. The vector lepto- 0 ! quarks contribute to the rare meson decays, KL e , which is the usual relation in the inverse seesaw mecha- which give a lower bound on the SUð4ÞC scale. This issue nism [12,13]. Therefore, we can have light neutrinos with- has been studied by several groups (see for example D 102 GeV D 106 GeV 3 out fine-tuning. If M and M , Refs. [7,8]), and the bound is MX 10 TeV.Noticethat 8 the neutrino mass satisfies m 10 . Thus, has there is freedom in the unknown mixings between quarks to be very small—smaller than 0.1 GeV—but it is protected and leptons and one can have a lower scale. For simplicity by the matter symmetry. we assume that there is no suppression mechanism. If we use the usual seesaw mechanism the scale for QL The scalar leptoquarks 3 ð3; 2; 1=6ÞSM and 4 14 unification has to be close to the seesaw scale 10 GeV. ð3; 2; 7=6ÞSM present in the field ð15; 2; 1=2Þ have the See for example Ref. [14]. However, the inverse seesaw following interactions: mechanism allows us to have a low scale for QL unification L c þ c þ y c as we have discussed above. Y2QL 3 Y2‘L 4u Y4QL 4 e The SUð4Þ gauge boson, A ð15; 1; 0Þ, can be y c C þ Y4‘L3 d þ H:c: (26) written as 0 pffiffiffi 1 It is important to mention that the only coupling con- 2 0 ! @ G X= A 0 strained by KL e is Y4, but this coupling is A ¼ pffiffiffi þ T4B : (23) 2 small—below 10 —to be in agreement with fermion X= 2 0 masses. The coupling Y2 is less constrained and can be small as well. Therefore, the scalar leptoquarks 3 and 4 Here G ð8; 1; 0ÞSM are the gluons and X ð3;1;2=3ÞSM are the new massive vector leptoquarks. The different trans- can be at the TeV scale and can be produced at the Large Hadron Collider. As is well known, one can have the QCD formation properties under color of X and 3 arise from our conventions for how the two 15’s A and transform under pair production of leptoquarks and the decays into a jet and lepton can give a unique signal. For a recent discussion SUð4ÞC gauge transformations U. Neglecting the space-time y of the leptoquark signatures at the LHC see Refs. [15–18]. dependence of the transformations, A ! UA U , while A detailed analysis of the constraints coming from meson ! UUT. decays, the lepton-number-violating decays, and the col- We have mentioned before that the Higgs sector is com- lider signatures is beyond the scope of this paper. posed of three Higgses: H ð1; 2; 1=2Þ, ð4; 1; 1=2Þ and ð15; 2; 1=2Þ. Therefore, the scalar potential can be IV. CONCLUDING REMARKS written as 2 y 2 y 2 y y y In this article we have proposed a simple theory where V ¼ m H H þ m þ m Trð Þþ1H H H the Standard Model quarks and leptons are unified using þ y TrðyÞþ y TrðyÞ 2H H 3 the gauge symmetry SUð4ÞC SUð2ÞL Uð1ÞR. The neu- y y y y trinos are Majorana fermions and their masses are gener- þð4H þ H:c:Þþ5H Trð ÞH ated through the inverse seesaw mechanism. The quark and y y y 2 y 2 3 þ 6 þ 7ðH HÞ þ 8ð Þ lepton unification scale can be as low as 10 TeV. The y 2 y 2 þ 9 Trð Þ þ 10ðTr Þ : (24) main constraints on the QL breaking scale are coming from the rare meson decays mediated by the vector leptoquark. Here the trace is only in the SUð4Þ space. We assume that This theory predicts the existence of scalar leptoquarks, C the parameters of the potential can be chosen so that these 3 ð3; 2; 1=6ÞSM and 4 ð3; 2; 7=6ÞSM, which could additional scalars get vacuum expectation values that be at the TeV scale and give rise to exotic signatures at the

057703-3 BRIEF REPORTS PHYSICAL REVIEW D 88, 057703 (2013) Large Hadron Collider. Subjects for further work include ACKNOWLEDGMENTS studying the correlation between the collider signals and P. F. P. thanks the theory group at Caltech for their great the different constraints coming from flavor violation, and hospitality. This paper is funded by the Gordon and Betty constructing the supersymmetric version of the theory to Moore Foundation through Grant No. #776 to the Caltech solve the hierarchy problem. We would like to mention that Moore Center for Theoretical Cosmology and Physics. The this theory could have a UV completion based on the work of M. B. W. was supported also in part by the U.S. Pati-Salam gauge group or it could arise from a grand Department of Energy under Contract No. DE-FG02- unified theory based on SU(6). 92ER40701.

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