aa r X i v : . [ h e p - ph ] N ov The Physics of Heavy Z ′ Gauge Bosons
Paul Langacker
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540email: [email protected]
The U (1) ′ symmetry associated with a possible heavy Z ′ would have profound implications forparticle physics and cosmology. The motivations for such particles in various extensions of thestandard model, possible ranges for their masses and couplings, and classes of anomaly-free modelsare discussed. Present limits from electroweak and collider experiments are briefly surveyed, as areprospects for discovery and diagnostic study at future colliders. Implications of a Z ′ are discussed,including an extended Higgs sector, extended neutralino sector, and solution to the µ problem insupersymmetry; exotic fermions needed for anomaly cancellation; possible flavor changing neutralcurrent effects; neutrino mass; possible Z ′ mediation of supersymmetry breaking; and cosmologicalimplications for cold dark matter and electroweak baryogenesis. PACS numbers: 12.60.Cn, 12.60.Fr, 14.70.Pw
Contents
I. Introduction II. Basic Issues Z ′ Couplings 3B. Masses and Mass Mixings 4C. Anomalies and Exotics 4D. Kinetic Mixing 5E. One and Two Higgs Doublets, Supersymmetry, and the µ Problem 61. Higgs Doublets 62. Non-Holomorphic Terms 73. The µ Problem 7
III. Models T R and B − L
73. The E models 9B. Anomaly-Free Sets 10C. TeV Scale Physics Models 111. Little Higgs, Twin Higgs, and Un-Unified Models 122. Extra Dimensions 123. Strong Dynamics 13D. Non-Standard Couplings 131. Decoupled Models 132. St¨uckelberg Models 143. Family Nonuniversal Models 14E. U (1) ′ Breaking Scales 141. A Massless Z ′ Z ′ U (1) ′ U (1) ′ IV. Experimental Issues e + e − Colliders 19C. Diagnostics of Z ′ Couplings 19
V. Implications µ Problem and Extended Higgs/NeutralinoSectors 221. The µ Problem 222. Extended Higgs Sector 223. Extended Neutralino Sector 22B. Exotics 23C. The Z ′ as a Factory 24D. Flavor Changing Neutral Currents 24E. Supersymmetry Breaking, Z ′ Mediation, and theHidden Sector 25F. Neutrino Mass 25G. Cosmology 261. Cold Dark Matter 262. Electroweak Baryogenesis 263. Cosmic Strings 27
VI. Conclusions and Outlook Acknowledgments References I. INTRODUCTION
Additional U (1) ′ gauge symmetries and associated Z ′ gauge bosons are one of the best motivated extensionsof the standard model (SM). It is not so much that theysolve any problems as the fact that it is more difficult toreduce the rank of an extended gauge group containingthe standard model than it is to break the non-abelianfactors. As a toy example, consider the gauge group G = SU ( N ), with N − G can be brokenby the vacuum expectation value (VEV) of a real adjointHiggs representation Φ, which can be represented by aHermitian traceless N × N matrixΦ = N − X i =1 ϕ i L i , (1)where the ϕ i are the real components of Φ and the L i arethe fundamental ( N × N ) representation matrices. WhenΦ acquires a VEV h Φ i , SU ( N ) is broken to a subgroupassociated with those generators which commute with h Φ i . Without loss of generality, h Φ i can be diagonalizedby an SU ( N ) transformation, so that the N − SU ( K ) subgroups(when K diagonal elements are equal), but the unbrokensubgroup always contains at least U (1) N − .Soon after the proposal of the electroweak SU (2) × U (1) Y model there were many suggestions for extendedor alternative electroweak gauge theories, some of whichinvolved additional U (1) ′ factors. An especially compelling motivation came from the de-velopment of grand unified theories larger than the orig-inal SU (5) model (Georgi and Glashow, 1974), such asthose based on SO (10) or E (See, e.g., (Langacker et al. ,1984; Robinett and Rosner, 1982a,b). For reviews, see(Hewett and Rizzo, 1989; Langacker, 1981).). These hadrank larger than 4 and could break to G SM × U (1) ′ n , n ≥
1, where G SM = SU (3) × SU (2) × U (1) Y is the stan-dard model gauge group. However, in the original (non-supersymmetric) versions there was no particular reasonfor the additional Z ′ masses to be at the electroweak orTeV scale where they could be directly observed. Simi-larly, superstring constructions often involve large gaugesymmetries which break to G SM × U (1) ′ n in the effectivefour-dimensional theory (Cvetic and Langacker, 1996a),where some of the U (1) ′ are non-anomalous. In bothstring theories and in supersymmetric versions of grandunification with extra U (1) ′ s below the string or GUTscale, both the U (1) ′ and the SU (2) × U (1) Y breakingscales are generally tied to the soft supersymmetry break-ing scale (Cvetic and Langacker, 1996a,b, 1997). There-fore, if supersymmetry is observed at the LHC there is astrong motivation that a string or GUT induced Z ′ wouldalso have a mass at an observable scale. (An exceptionto this is when the U (1) ′ breaking occurs along a flatdirection.)In recent years many TeV scale extensions to theSM have been proposed in addition to supersymme-try, often with the motivation of resolving the finetuning associated with the quadratic divergence inthe Higgs mass. These include various forms of dy-namical symmetry breaking (Chivukula et al. , 2004; Some examples include (del Aguila and Mendez,1981; Barger et al. , 1980, 1982a,b; Barger and Phillips,1978; Barr, 1983; Barr and Zee, 1980; Davidson, 1979;Deshpande and Iskandar, 1980; Fayet, 1977, 1980;Georgi and Weinberg, 1978; de Groot et al. , 1980;Kim and Zee, 1980; Li and Marshak, 1982; Masiero, 1980;Mohapatra and Sidhu, 1978; Rizzo, 1980; Rizzo and Senjanovic,1981). More complete lists of early references canbe found in (Hewett and Rizzo, 1989; Langacker et al. ,1984; Robinett and Rosner, 1982a,b). Previous reviewsinclude (del Aguila, 1994; Cvetic and Godfrey, 1995;Cvetic and Langacker, 1997; Hewett and Rizzo, 1989; Leike,1999; Rizzo, 2006).
Chivukula and Simmons, 2002; Hill and Simmons, 2003)and little Higgs models (Arkani-Hamed et al. , 2001a;Han et al. , 2003, 2006; Perelstein, 2007), which typicallyinvolve extended gauge structures, often including new Z ′ gauge bosons at the TeV scale. Some versions of theo-ries with large extra dimensions allow the standard modelgauge bosons to propagate freely in the extra dimensions,implying Kaluza-Klein excitations (see, e.g., (Antoniadis,1990; Appelquist et al. , 2001; Appelquist and Yee,2003; Barbieri et al. , 2004; Casalbuoni et al. , 1999;Cheng et al. , 2002; Cheung and Landsberg, 2002;Delgado et al. , 2000; Gogoladze and Macesanu, 2006;Masip and Pomarol, 1999)) of the Z and other standardmodel gauge bosons, with effective masses of order R − ∼ × (10 − cm /R ), where R is the scale ofthe extra dimension. Such excitations can also occur inRandall-Sundrum models (Randall and Sundrum, 1999)(see, e.g., (Agashe et al. , 2003, 2007; Carena et al. ,2003a; Hewett et al. , 2002)).Other motivations for new Z ′ bosons, e.g., associatedwith (approximately) hidden sectors of nature, are de-tailed in Sections III and V. Extensions of the SMmay also involve new TeV scale charged W bosons (see,e.g., (Rizzo, 2007)), which could couple either to left orright handed currents, but the focus of this article willbe on Z ′ s.The experimental discovery of a new Z ′ would be ex-citing, but the implications would be much greater thanjust the existence of a new vector boson. Breaking the U (1) ′ symmetry would require an extended Higgs (andneutralino) sector, with significant consequences for col-lider physics and cosmology (direct searches, the µ prob-lem, dark matter, electroweak baryogenesis). Anomalycancellation usually requires the existence of new exoticparticles that are vectorlike with respect to the stan-dard model but chiral under U (1) ′ , with several possibil-ities for their decay characteristics. The expanded Higgsand exotic sectors can modify or maintain the approx-imate gauge coupling unification of the minimal super-symmetric standard model (MSSM). In some construc-tions (especially string derived) the U (1) ′ charges arefamily nonuniversal, which can lead to flavor changingneutral current (FCNC) effects, e.g., in rare B decays.Finally, the decays of a heavy Z ′ may be a useful pro-duction mechanism for exotics and superpartners. Theconstraints from the U (1) ′ symmetry can significantly al-ter the theoretical possibilities for neutrino mass. Finally, U (1) ′ interactions can couple to a hidden sector, possiblyplaying a role in supersymmetry breaking or mediation.Section II of this review discusses basic issues, suchas the Z ′ interactions and properties, U (1) ′ breaking,anomalies, and ordinary and kinetic mixing between Z and Z ′ . Section III surveys the large range of modelsthat have been proposed, including the U (1) ′ -breakingscale; GUT-inspired models; sets of exotics and chargesconstructed to avoid anomalies; and more exotic pos-sibilities such as ultra-weak coupling, low mass, hid-den sector, leptophobic, intermediate scale, sequential,family nonuniversal, and anomalous U (1) ′ models. Sec-tion IV briefly outlines the existing constraints from pre-cision electroweak and direct collider searches, as well asprospects for detection and diagnostics of couplings atfuture colliders. Finally, Section V is a survey of thetheoretical, collider, and cosmological implications of apossible Z ′ . II. BASIC ISSUESA. Z ′ Couplings
In the standard model the neutral current interactionsof the fermions are described by the Lagrangian − L SMNC = gJ µ W µ + g ′ J µY B µ = eJ µem A µ + g J µ Z µ , (2)where g and g ′ are the SU (2) and U (1) Y gauge couplings, W µ is the (weak eigenstate) gauge boson associated withthe third (diagonal) component of SU (2), and B µ is the U (1) Y boson. The currents in the first form are J µ = X i ¯ f i γ µ [ t i L P L + t i R P R ] f i J µY = X i ¯ f i γ µ [ y i L P L + y i R P R ] f i , (3)where f i is the field of the i th fermion and P L,R ≡ (1 ∓ γ ) / t i L ( t i R )is the third component of weak isospin for the left (right)chiral component of f i . For the known fermions, t u L = t ν L = + , t d L = t e − L = − , and t i R = 0. The weakhypercharges y i L,R are chosen to yield the correct electriccharges, t i L + y i L = t i R + y i R = q i , (4)where q i is the electric charge of f i in units of the positroncharge e > SU (2) × U (1) Y to the electromagnetic subgroup U (1) em (Sec-tion II.B), the mass eigenstate neutral gauge bosons inEq. 2 are the (massless) photon field A µ and the (mas-sive) Z µ ≡ Z µ , where A µ = sin θ W W µ + cos θ W B µ Z µ = cos θ W W µ − sin θ W B µ , (5)and the weak angle is θ W ≡ tan − ( g ′ /g ). The new gaugecouplings are e = g sin θ W and g ≡ g + g ′ = g / cos θ W . (6) We largely follow the formalism and conventions in(Durkin and Langacker, 1986; Langacker and Luo, 1992).
The currents in the new basis are J µem = X i q i ¯ f i γ µ f i J µ = X i ¯ f i γ µ [ ǫ L ( i ) P L + ǫ R ( i ) P R ] f i , (7)with the chiral couplings ǫ L ( i ) = t i L − sin θ W q i , ǫ R ( i ) = t i R − sin θ W q i . (8)In the extension to SU (2) × U (1) Y × U (1) ′ n , n ≥ L NC becomes − L NC = eJ µem A µ + n +1 X α =1 g α J µα Z αµ , (9)where g , Z µ , and J µ are respectively the gauge cou-pling, boson, and current of the standard model. Simi-larly, g α and Z αµ , α = 2 · · · n + 1, are the gauge couplingsand bosons for the additional U (1) ′ s. The currents inEq. 9 are J µα = X i ¯ f i γ µ [ ǫ αL ( i ) P L + ǫ αR ( i ) P R ] f i = 12 X i ¯ f i γ µ [ g αV ( i ) − g αA ( i ) γ ] f i . (10)The chiral couplings ǫ αL,R ( i ), which may be unequal fora chiral gauge symmetry, are respectively the U (1) α charges of the left and right handed components offermion f i , and g αV,A ( i ) = ǫ αL ( i ) ± ǫ αR ( i ) are the corre-sponding vector and axial couplings.Frequently, it is more convenient to instead specify the U (1) α charges of the left chiral components of both thefermion f and the antifermion (conjugate) f c , denoted Q αf and Q αf c , respectively. The two sets of charges aresimply related, ǫ αL ( f ) = Q αf , ǫ αR ( f ) = − Q αf c . (11)For example, in the SM one has Q u = − sin θ W and Q u c = + sin θ W .The additional gauge couplings and charges, as wellas the gauge boson masses and mixings, are extremelymodel dependent. The gauge couplings and charges arenot independent, i.e., one can always replace g α by λ α g α provided the charges Q α are all simultaneously scaledby 1 /λ α . Usually, the charges are normalized by someconvenient convention.The three and four point gauge interactions of a com-plex SU (2) scalar multiplet φ can be read off from thekinetic term L kinφ = ( D µ φ ) † D µ φ . The diagonal (neutralcurrent) part of the gauge covariant derivative of an in-dividual field φ i is D µ φ i = ∂ µ + ieq i A µ + i n +1 X α =1 g α Q αi Z αµ ! φ i , (12)where q i and Q αi are respectively the electric and U (1) α charges of φ i . For the SM part, t i = 0 , , , · · · labels the SU (2) representation, t i is the third component of weakisospin, the weak hypercharge is y i = q i − t i , and Q i = t i − sin θ W q i . Thus, for the neutral component φ ofthe Higgs doublet φ = (cid:18) φ + φ (cid:19) one has t φ = − t φ = y φ = . B. Masses and Mass Mixings
We assume that electrically neutral scalar fields φ i ac-quire VEVs, so A µ remains massless, while the Z αµ fieldsdevelop a mass term L massZ = M αβ Z αµ Z µβ , where M αβ = 2 g α g β X i Q αi Q βi |h φ i i| . (13) M ≡ M Z would be the (tree-level) Z mass in the SMlimit in which the other Z ’s and their mixing can beignored. If the only Higgs fields are SU (2) doublets (orsinglets), as in the SM or the MSSM, then M Z = 12 g X i |h φ i i| = 14 g ν = M W cos θ W , (14)where ν = 2 P i |h φ i i| ∼ ( √ G F ) − ∼ (246 GeV) is the square of the weak scale and G F is the Fermiconstant. The observed Z mass strongly constrains ei-ther higher-dimensional Higgs VEVs or Z − Z ′ mix-ing (Yao et al. , 2006), but in principle they could com-pensate and should both be considered. Allowing a gen-eral Higgs structure, one has M Z = g √ G F ρ = M W ρ cos θ W , (15)where ρ ≡ P i ( t i − t i + t i ) |h φ i i| P i t i |h φ i i| −−−−−−−−−−−→ doublets , singlets . (16)Diagonalizing the mass matrix Eq. 13 one obtains n +1(usually) massive eigenstates Z αµ with mass M α , Z αµ = n +1 X β =1 U αβ Z βµ , (17)where U is an orthogonal mixing matrix. It is straight-forward to show that the mass-squared eigenvalues arealways nonnegative. From Eq. 9 and Eq. 17 Z αµ couplesto P β g β U αβ J µβ .The most studied case is n = 1. Writing Q i ≡ Q i , themass matrix is M Z − Z ′ = (cid:18) g P i t i |h φ i i| g g P i t i Q i |h φ i i| g g P i t i Q i |h φ i i| g P i Q i |h φ i i| (cid:19) ≡ (cid:18) M Z ∆ ∆ M Z ′ (cid:19) . (18) As an example, many U (1) ′ models involve an SU (2)singlet, S , and two Higgs doublets, φ u = (cid:18) φ u φ − u (cid:19) , φ d = (cid:18) φ + d φ d (cid:19) , (19)with U (1) ′ charges Q S,u,d . Then M Z = 14 g ( | ν u | + | ν d | )∆ = 12 g g ( Q u | ν u | − Q d | ν d | ) M Z ′ = g ( Q u | ν u | + Q d | ν d | + Q S | s | ) , (20)where ν u,d ≡ √ h φ u,d i , s = √ h S i , and ν = ( | ν u | + | ν d | ) ∼ (246 GeV) .The eigenvalues of a general M Z − Z ′ are M , = 12 (cid:20) M Z + M Z ′ ∓ q ( M Z − M Z ′ ) + 4∆ (cid:21) , (21)and U is the rotation U = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) , (22)with θ = 12 arctan (cid:18) M Z − M Z ′ (cid:19) . (23) θ is related to the masses bytan θ = M Z − M M − M Z . (24)An important limit is M Z ′ ≫ ( M Z , | ∆ | ), which typi-cally occurs because an SU (2) singlet field (such as S inthe example) has a large VEV and contributes only to M Z ′ . One then has M ∼ M Z − ∆ M Z ′ ≪ M , M ∼ M Z ′ (25)and θ ∼ − ∆ M Z ′ ∼ C g g M M with C = − P i t i Q i |h φ i i| P i t i |h φ i i| . (26) C is model dependent, but typically | C | . O (1). FromEq. 24-26 one sees that both | θ | and the downward shift( M Z − M ) /M Z are of order M /M . Generaliza-tions of these results for n > U (1)’s are givenin (Langacker, 1984). C. Anomalies and Exotics
A symmetry is chiral if it acts differently on the leftand right handed fermions, and non-chiral (or vector)otherwise. Thus, a chiral U (1) ′ has Q αf = − Q αf c for atleast one f , which is also referred to as chiral. Even for achiral symmetry, some of the fermions may be non-chiral.If a given fermion pair is non-chiral with respect to all ofthe symmetries then an elementary mass term − L m = m f ¯ f L f R + h.c. is allowed, where m f could be arbitrarilylarge. Such a term is forbidden for a chiral fermion, whosemass is only generated when the symmetry is broken. Forexample, if the symmetries allow the Yukawa coupling − L Y uk = λ f ϕ ¯ f L f R + h.c., (27)where ϕ is charged under the symmetry, then an effec-tive mass λ f h ϕ i is generated when ϕ acquires a VEV.Assuming λ f . m f cannot be larger than the symme-try breaking scale h ϕ i . In the SM, the ordinary quarksand leptons are chiral under both SU (2) and U (1) Y , and ϕ is the Higgs doublet. Similar constraints apply to newfields occurring in U (1) ′ models, which are frequently quasi-chiral , i.e., non-chiral under the SM but chiral un-der U (1) ′ .Consistency of a low-energy gauge theory requires theabsence of triangle anomalies, including mixed gauge-gravitational ones . For the SM, the non-trivial condi-tions are X f Y f = 0 , X f Y f = 0 , X f ∈ , ∗ Y f = 0 , X f ∈ Y f = 0 , (28)where the sum extends over all left-handed fermions( Y f = y f L ) and antifermions ( Y f c = − y f R ). The firstcondition is the mixed anomaly; the sum is over colortriplets and antitriplets in the third [ SU (3) Y ] condition;and the sum is over SU (2) doublets in the last [ SU (2) Y ]condition. The sum includes counting factors of 3 forfamilies, 3 for color triplets, and 2 for SU (2) doublets,since SU (3) and SU (2) commute with hypercharge andadditional U (1) ′ s. For example, the second [ Y ] condi-tion is 3[6 Y Q + 2 Y L + 3 Y u c + 3 Y d c + Y e c ] = 0 , (29)where Q and L refer respectively to quark and leptondoublets. This is satisfied by a cancellation betweenquark and lepton terms. One also requires the absenceof an [ SU (3) ] anomaly. In the SM this is achieved au-tomatically because there equal numbers of quarks andantiquarks. With an additional U (1) ′ with charge Q ,there are additional conditions obtained from Eq. 28 byreplacing Y by Q . There are also mixed [ Y U (1) ′ ] con-ditions P f Y Q = P f Y Q = 0. For n > U (1) ′ there are similar conditions for every Q β , β ≥ P f Y Q α Q β = P f Q α Q β Q γ = 0. All of these See Section III for the role of anomalous U (1) ′ s. sums include any extra chiral fermions in the theory, suchas the superpartners of Higgs scalars in supersymmetry.Non-chiral fermion pairs cancel.Even for a single U (1) ′ it is easy to show thatthe anomaly conditions cannot be satisfied by the SMfermions alone if the U (1) ′ charges are the same for allthree families, except for the trivial case Q = 0. ( Q = cY is also possible, but this is equivalent to Q = 0after performing a rotation on B µ and Z µ .) Thus, al-most all U (1) ′ constructions involve additional fermions,known as exotics. These may be singlets under the SMgauge group, such as a singlet right-handed neutrino, orthey may carry nontrivial SM quantum numbers. Preci-sion electroweak constraints strongly restrict, but don’tentirely exclude, the possibilities of new fermions chiralunder SU (2) × U (1) Y (Yao et al. , 2006), so such exoticsare usually assumed to be quasi-chiral, e.g., both leftand right-handed components might be SU (2) doublets,or both might be singlets. A typical example is a new SU (2)-singlet heavy down-type quark D with q D = − / D c .One can introduce vector pairs that are charged butnon-chiral under both the SM and U (1) ′ . These do notcontribute to the anomaly conditions, but contribute tothe renormalization group equations (RGE) for the gaugecouplings, and may also be relevant to the decays of ex-otics.If two U (1) ′ charges Q α,β (one of these can be Y ) areboth generators of a simple underlying group, then oneexpects them to be orthogonal, i.e., P f Q αf Q βf = 0for α = β , with a corresponding condition for the scalarcharges. However, this condition need not hold withoutsuch an embedding or for a more complicated one, or itcould be violated due to kinetic mixing (Section II.D).Furthermore, all fermions, including the non-chiral ones,contribute to the orthogonality condition. In particular,an apparent violation of orthogonality could be due to thefact that the contributions of a very heavy vector pair (orof heavy scalars) have not been taken into account. Thereis always some freedom to perform rotations on the gaugefields Z αµ , e.g., to make the U (1) ′ charges orthogonal (atleast with respect to the fermions, in a nonsupersym-metric theory). However, such a rotated basis may notcoincide with either the mass or kinetic eigenstates. D. Kinetic Mixing
The most general kinetic energy term for the two gaugebosons Z αµ and Z βµ in U (1) α × U (1) β is L kin → − c α F µνα F αµν − c β F µνβ F βµν − c αβ F µνα F βµν , (30)where F αµν = ∂ µ Z αν − ∂ ν Z αµ . One can put the firsttwo terms into canonical form c α = c β = 1 by rescal-ing the fields, and take c αβ = sin χ . Since U (1) fieldstrengths are invariant, the cross (kinetic mixing) termdoes not spoil the gauge invariance (Holdom, 1986). Evenif χ = 0 at tree level, it can be generated by loop ef-fects if there are particles in the theory that are simul-taneously charged under both U (1)’s (del Aguila et al. ,1988, 1995b; Babu et al. , 1996, 1998; Foot and He, 1991;Holdom, 1986; Matsuoka and Suematsu, 1986). Themixing term can be cast as a cross term in the renormal-ization group equations (RGE) for the gauge couplings,with a coefficient proportional to P m f <µ Q αf Q βf , where µ is the RGE scale, with corresponding contributionsfrom scalars. Even for orthogonal charges the sum atlower mass scales may be nonzero due to the decouplingof heavy particles. Such RGE effects are usually of ordera few % in χ , but could be larger if there are many de-coupled states (Babu et al. , 1996). A non-zero χ can alsobe generated by string loop effects in superstring theory.These contributions are small in the heterotic construc-tions considered in (Dienes et al. , 1997). However, if oneof the U (1) factors is broken in a hidden sector at a largescale, the associated D terms could propagate this scaleto the ordinary sector by kinetic mixing, destabilizingthe supersymmetry breaking scale and leading to nega-tive mass-square scalars (Dienes et al. , 1997).Now, consider the consequences of kinetic mixing for asingle extra U (1) ′ , i.e., α = 1 , β = 2. L kin can be put incanonical form (for c , = 1 , c = sin χ ) by defining (cid:18) Z µ Z µ (cid:19) = (cid:18) − tan χ / cos χ (cid:19) (cid:18) ˆ Z µ ˆ Z µ (cid:19) ≡ V (cid:18) ˆ Z µ ˆ Z µ (cid:19) , (31)where V is non-unitary. In the new ˆ Z basis, the massmatrix in Eq. 18 becomes V T M Z − Z ′ V , which can bediagonalized by an orthogonal matrix U T . Similarly, theinteraction term in Eq. 9 becomes( g J µ g J µ ) (cid:18) Z µ Z µ (cid:19) ≡ J T (cid:18) Z µ Z µ (cid:19) → J T V (cid:18) ˆ Z µ ˆ Z µ (cid:19) = J T V U T (cid:18) Z µ Z µ (cid:19) , (32)where Z , are the mass eigenstates. These transforma-tions are analyzed in detail, in, e.g., (Babu et al. , 1998).The essential feature can be seen for ∆ = 0 in Eq. 18,for which V T M Z − Z ′ V = (cid:18) M Z − M Z tan χ − M Z tan χ M Z tan χ + M Z ′ / cos χ (cid:19) . (33)One sees immediately that for M Z = 0 there is a zeroeigenvalue, even for large χ , i.e., any shift in the lightermass induced by kinetic mixing is proportional to thelight mass and therefore small. In fact, for | M Z | ≪| M Z ′ | , ∆ = 0, and | χ | ≪ M ∼ M Z − M Z χ /M Z ′ , a negligible shift. The only significant effectin this limit is that the couplings become g J µ Z µ + ( g J µ − g χJ µ ) Z µ , (34) i.e., the coupling of the heavy boson is shifted to includea small component proportional to J . The light bosoncouplings are not affected to this order. One must stillinclude the further effects of mass mixing (∆ = 0). For | ∆ | . | M Z | ≪ | M Z ′ | and | χ | ≪ M ∼ M Z − (∆ − M Z χ ) M Z ′ ∼ M Z − ˆ θ M Z ′ , (35)where ˆ θ ≡ ( − ∆ + M Z χ ) M Z ′ . (36)This is of the same form as Eq. 25 except that the ef-fective mixing angle is shifted from Eq. 26 by the kineticmixing. The interactions are just the rotation by ˆ θ ofthose in Eq. 34.In a supersymmetric theory the charges in the U (1) D terms are also shifted, g Q → g Q − g χQ . Therecan also be kinetic mixing between the U (1) ′ gaugi-nos (Suematsu, 1999), with consequences analogous tothose for the gauge bosons. E. One and Two Higgs Doublets, Supersymmetry, and the µ Problem
1. Higgs Doublets
The standard model involves a single Higgs doublet φ = (cid:18) φ + φ (cid:19) , which has Yukawa couplings (ignoring fam-ily indices) − L Y uk = h d ¯ Q L φd R + h u ¯ Q L e φu R + h e ¯ L L φe − R + h ν ¯ L L e φν R + h.c., (37)where Q L ≡ (cid:18) u L d L (cid:19) , L L ≡ (cid:18) ν L e − L (cid:19) , and ν R is the right-handed (SM-singlet) neutrino. The tilde field is definedby e φ ≡ iσ φ ∗ = (cid:18) φ ∗ − φ − (cid:19) , (38)where σ is the second Pauli matrix. It is essentially theHermitian conjugate of φ , but transforms as a ratherthan a ∗ under SU (2), and has y = − /
2. A singledoublet suffices for the SM, but in many extensions, in-cluding supersymmetry and many U (1) ′ models, the e φ couplings are not allowed. One must introduce a second(independent) doublet φ u , as in Eq. 19, which plays therole of e φ , while φ d plays that of φ .In supersymmetric models it is convenient to workentirely in terms of (left) chiral superfields, such as Q, L, u c , d c , e + , and the SM singlet ν c which is conju-gate to ν R (we do not distinguish between chiral super-fields and their components in our notation – the contextshould always make the meaning clear). Furthermore,supersymmetry (and anomaly constraints) require twoHiggs doublets H u = (cid:18) H + u H u (cid:19) and H d = (cid:18) H d H − d (cid:19) with Y H u,d = ± /
2, defined so that the MSSM superpotentialis W = µH u H d − h d QH d d c + h u QH u u c − h e LH d e + + h ν LH u ν c . (39)Doublets are contracted according to H u H d ≡ ǫ ab H ua H db , etc., where ǫ = − ǫ = 1. The two sets ofHiggs doublets are related by H u,d = ∓ e φ u,d . The super-potential in Eq. 39 assumes that R -parity is conserved.Some U (1) ′ models enforce this automatically, as will bementioned in Section III.B.
2. Non-Holomorphic Terms
In some U (1) ′ extensions of the MSSM, some ofthe Yukawa couplings in Eq. 39 may be forbidden bythe U (1) ′ gauge symmetry. In some cases, however,the operators involving the wrong Higgs field, such as Q e H u d c or L e H u e + , may be U (1) ′ invariant. Such non-holomorphic operators are not allowed in W by super-symmetry, but could be present in the K¨ahler potential,where they would lead to corresponding non-holomorphicsoft terms (Borzumati et al. , 1999) for the scalar squarksand sleptons. These then lead to fermion masses at oneloop by gluino or neutralino exchange. However, in mostsupersymmetry breaking schemes it is difficult to gener-ate a large enough effective Yukawa (Martin, 1997), be-cause the non-holomorphic soft terms have an additionalsuppression (compared to the usual soft SUSY break-ing scale of M SUSY ∼ M SUSY /M med , where M med ≫ M SUSY is the SUSY mediation scale (such asthe Planck scale for supergravity mediation).
3. The µ Problem
One difficulty with the MSSM is the µ prob-lem (Kim and Nilles, 1984), i.e., the supersymmetricHiggs mass µ in Eq. 39 could be arbitrarily large, butphenomenologically needs to be of the same order as thesoft supersymmetry breaking terms. In many supersym-metric U (1) ′ models this problem is solved because anelementary µ term is forbidden by the U (1) ′ , but a tri-linear W µ = λ S SH u H d is allowed, where S is a singletunder the SM but charged under the U (1) ′ . Then, adynamical effective µ eff = λ S h S i is generated that is re-lated to the scale of U (1) ′ breaking (Cvetic et al. , 1997;Cvetic and Langacker, 1996a; Suematsu and Yamagishi,1995), as will be further discussed in Section V.A.1.This mechanism can also be associated with discrete orother symmetries (Accomando et al. , 2006). An alterna-tive solution, the Giudice-Masiero mechanism, generates µ through a nonrenormalizable operator in the K¨ahlerpotential (Giudice and Masiero, 1988). It is especiallyuseful when an elementary µ term is allowed by the lowenergy symmetries of the theory, but is forbidden by theunderlying string construction. This mechanism can alsobe used to generate mass for vector pairs in U (1) ′ theo-ries. III. MODELS
There are enormous numbers of U (1) ′ models, and itis only possible to touch on the major classes and is-sues here. The models are distinguished by: (a) the cou-pling constants g α , which are often assumed to be of elec-troweak strength, but could be larger or smaller. (b) The U (1) ′ breaking scale. In some scenarios this is arbitrary,with no good reason to expect it to be around the TeVscale. However, in supersymmetric models it is usually atthe TeV scale, unless the breaking is associated with an F and D flat direction, when it could be much larger. TheTeV scale is also expected when the U (1) ′ is associatedwith alternative models of electroweak breaking. Stringconstructions usually imply some Z ′ s close to the stringscale, and often involve lighter ones as well. Finally, a Z ′ could actually be lighter than the electroweak scaleif its couplings to the SM fields are small. (c) Othercritical issues are the charges of the SM fermions andHiggs doublet, and whether the fermion charges are fam-ily universal; the type of scalar responsible for the U (1) ′ breaking; whether additional exotic fields are needed tocancel anomalies; whether the theory is supersymmetric(so that the Higgs superpartners must be included in theanomaly considerations); whether the Yukawa couplingsof the ordinary fermions are allowed by the U (1) ′ sym-metry; and whether other couplings, such as those asso-ciated with the supersymmetric µ parameter, R -parityviolation, and Majorana neutrino masses are allowed. A. Canonical Examples
1. The sequential model
The sequential Z SM boson is defined to have the samecouplings to fermions as the SM Z boson. The Z SM isnot expected in the context of gauge theories unless it hasdifferent couplings to exotic fermions, or if it occurs asan excited state of the ordinary Z in models with extradimensions at the weak scale. However, it serves as auseful reference case when comparing constraints fromvarious sources.
2. Models based on T R and B − L One of the simplest and most common classes ofmodels involves SU (2) × U (1) R × U (1) BL , where the U (1) R generator T R is for ( u R , ν R ), − for ( d R , e − R ),and 0 for f L ; and the U (1) BL generator is T BL ≡ ( B − L ), where B ( L ) is baryon (lepton) number; and ν R are right-handed neutrinos. (See Table I.) T R and T BL are related to weak hypercharge by Y = T R + T BL . T R occurs in left-right symmetric modelsbased on the group G LR ≡ SU (2) L × SU (2) R × U (1) BL (for a review, see (Mohapatra, 2003)) and in SO (10)models (which contain G LR ) (Hewett and Rizzo, 1989;Langacker, 1981). The Higgs doublet φ can be assigned T R = and T BL = 0. However, in the G LR or SO (10)embeddings (or in supersymmetric versions), there aretwo Higgs doublets, φ u,d , as defined in Eq. 19, with T R = ∓ . All of these versions are anomaly-free afterincluding the three ν R .For these models, the fermion neutral current couplingsare − L NC = gJ L W L + g R J R W R + g BL J BL W BL , (40)where J L ≡ J , J R and J BL are the currents corre-sponding to T R and T BL , the g ’s and W ’s are the cou-pling constants and gauge bosons, and the Lorentz in-dices have been suppressed.We anticipate that U (1) R × U (1) BL will be brokento U (1) Y at a scale M Z ′ ≫ M Z , so it is convenient tofirst transform the gauge bosons W R and W BL to a newbasis B and Z , where B is identified with the SM U (1) Y boson, − L NC = gJ L W L + g ′ J Y B + g J Z = eJ em A + g J Z + g J Z , (41)as in Eq. 9. Let us first assume that the gauge kineticterms are canonical, i.e., with unit strength and no ki-netic mixing, so that orthogonal transformations on thethree gauge bosons will leave the kinetic terms invariant.Taking B ≡ cos γ W R + sin γ W BL and choosing γ sothat B couples to g ′ Y , one finds 1 /g ′ = 1 /g R + 1 /g BL ,and that the orthogonal gauge boson Z = sin γ W R − cos γ W BL couples to the current J associated with thecharge Q LR = r (cid:20) αT R − α T BL (cid:21) , (42)where α = g R g BL = p κ cot θ W − , (43)with κ ≡ g R /g . The coupling has been normalized to g = r g tan θ W ∼ .
46 (44)for later convenience.One interesting case is when G LR survives down tothe TeV scale. This is usually studied assuming a left-right symmetry under the interchange of the two SU (2)factors (Mohapatra, 2003), in which case g R = g and α ∼ .
53 for sin θ W ∼ .
23. Two forms of the modelare often considered. In both cases, the Higgs doublets φ u,d responsible for fermion mass transform as (2 , under SU (2) L × SU (2) R , where the subscript is the T BL charge. In one class, an additional doublet pair δ R,L transforming as (1 , / + (2 , / is introduced,with the VEV of δ R breaking G LR to the SM. In theother, one instead introduces a triplet pair ∆ R,L trans-forming as (1 , + (3 , . The ∆ R VEV not onlybreaks G LR but also leads to a large Majorana massfor the ν R and therefore a small ν L mass by the see-saw mechanism (Mohapatra and Senjanovic, 1980). Thelow-scale left-right model also implies a new W ± R whichcouples to right-handed currents and can mix with theSM W ± . Strategies for determining the symmetry break-ing pattern were described in (Cvetic et al. , 1992), andlimits on the charged sector masses and mixings forgeneral models without left-right symmetry are givenin (Langacker and Uma Sankar, 1989; Yao et al. , 2006).The simple forms of the (supersymmetric) left-rightmodel are not consistent with gauge unification unlessthe SU (2) R breaking occurs at a much higher scale(e.g., 10 GeV). Such a large scale is also requiredby current allowed ranges for the neutrino masses inthe triplet versions. In some cases, the initial break-ing can leave U (1) R × U (1) BL unbroken. Realistic SO (10) breaking patterns suggest α in the range 0 . − . χ model, which occurs when SO (10)breaks directly to SU (5) × U (1) χ . This corresponds toEq. 42 for κ = 1 and sin θ W = 3 / SU (5) at the unification scale ), leading to α = p / ∼ . SU (2) × U (1) Y × U (1) , where Q is a linear combination Q Y BL = aY + bT BL ≡ b ( zY + T BL ) , (45)where b = 0. It is convenient to normalize b so thatthe coupling g is given by Eq. 44, or alternatively onecan choose b = 1 and take g to be arbitrary. The U (1) R × U (1) BL limit in Eq. 42 corresponds to choos-ing b z (1 + z ) = − / α ≡ p / bz . Q Y BL isanomaly free for the standard model fermions (including ν R ) (Weinberg, 1995). Y and Q Y BL are non-orthogonal(i.e., P f Y f Q Y BLf = 0 when summed over a family of theknown left-handed fermions and antifermions), except forthe special case of U (1) R × U (1) BL , but it could comeabout by a more general embedding of the generators orby (possibly large) kinetic mixing, as discussed in Sec-tion II.C. The pure B − L model ( z = 0) is often stud-ied phenomenologically, and has the property that theordinary Higgs doublets do not induce Z − Z ′ mixing.The models in this class have been systematically dis-cussed in (Appelquist et al. , 2003), including generaliza-tions with an arbitrary number of ν R with nonuniversalcharges.This entire class of models based on T R and T BL (or Y and T BL ) is perhaps less interesting in a supersymmetriccontext, because the two supersymmetric Higgs doublets H u,d form a vector pair with T R = ± and T BL = 0.Therefore, an elementary µ term in Eq. 39 is not for-bidden by the extra U (1) ′ . Similar difficulties apply tothe SM singlet supermultiplets that are needed to breakthe U (1) ′ , since they would most likely be introduced asnon-chiral vector pairs to avoid anomalies. (One couldinstead give large VEVs to the scalar partners of the ν c ,but this would break R -parity and would be challengingfor neutrino phenomenology.) TABLE I Charges of the left-chiral components of thefermions in the models based on T R and T BL = ( B − L ) / g = q g tan θ W . Q LR is defined in Eq. 42, and Q Y BL ≡ b ( zY + T BL ). Q LR is aspecial case of Q Y BL for b z (1 + z ) = − / α and ( b, z ) arefree parameters, with α = 1 .
53 for left-right symmetry and α ∼ . − . SO (10) models. T R T BL Y q Q LR b Q Y BL Q
16 16 − α ( z + 1) u cL − − − − α + α − z − d cL −
16 13 α + α z − L L − −
12 12 α − ( z + 1) e + L
12 12 α − α z + ν cL −
12 12 − α − α
3. The E models Many Z ′ studies focus on the two extra U (1) ′ swhich occur in the decomposition of the E GUT (Hewett and Rizzo, 1989; Langacker et al. , 1984;Robinett and Rosner, 1982a), i.e., E → SO (10) × U (1) ψ and SO (10) → SU (5) × U (1) χ . We consider them onlyas simple examples of anomaly-free U (1) ′ charges andexotic fields, and do not assume a full underlyinggrand unified theory. In E , each family of left-handedfermions is promoted to a fundamental -plet, whichdecomposes under E → SO (10) → SU (5) as → + + → ( + ∗ + ) + ( + ∗ ) + , (46)as shown in Table II. In addition to the standard modelfermions, each -plet contains two standard model sin-glets, ν c and S (which may be charged under the U (1) ′ ).The ν c may be interpreted as the conjugate of theright-handed neutrino. There is also an exotic color-triplet quark D with charge − / D c , both of which are SU (2) singlets, and a pair ofcolor-singlet SU (2)-doublet exotics, H u = H + u H u ! and H d = H d H − d ! with y H u,d = ± / H d transforms the same way as H cu ≡ e H u , the (tilde) conjugate of H u un-der the SM. The exotic fields are all therefore singlets ornon-chiral under the standard model, but may be chiralunder the U (1) ′ .The E models can be considered in both non-supersymmetric and supersymmetric versions. In the su-persymmetric case, the scalar partners of the S and ν c can develop VEVs to break the U (1) ′ symmetry, thoughthe latter (as well as a VEV for the scalar partner ofthe ν ) would break R -parity and may be problematicfor neutrino phenomenology. Similarly, the scalar part-ners of one H u,d pair can be interpreted as the two Higgsdoublets of the MSSM. The two additional H u,d familiesmay be interpreted either as additional Higgs pairs or asexotic-leptons ( H d has the same SM quantum numbers asan ordinary lepton doublet, while H u would be conjugateto a right-handed exotic doublet).Table II also lists the U (1) χ and U (1) ψ charges of the -plet. By construction, the fields in an irreducible rep-resentation of SO (10) ( SU (5)) all carry the same ψ ( χ )charges. Most studies assume that only one Z ′ , couplingto the linear combination Q ( θ E ) = cos θ E Q χ + sin θ E Q ψ , (47)where 0 ≤ θ E < π is a mixing angle, is relevant atlow energies. (One can also include a kinetic mixingcorrection − ǫY to the effective charge, as in Eq. 34).As discussed in Section III.A.2, the χ model ( θ E = 0)is a special case of the T R and B − L models, sup-plemented with additional exotic fields in the + of SO (10). Since the latter are non-chiral in this casethey may be omitted, or one or more ’s may be in-troduced as Higgs fields. The ψ model ( θ E = π/ -plets. Using Eq. 11 one sees that the currentsof the fields in the and have purely axial cou-plings to the Z ψ (this only holds for the ν if it pairs withthe ν c to form a Dirac fermion). Another commonlystudied case is the η model, Q η = q Q χ − q Q ψ = − Q ( θ E = π − arctan p / ∼ . π ), which occurs inCalabi-Yau compactifications of the heterotic string if E breaks directly to a rank 5 group (Witten, 1985) viathe Wilson line (Hosotani) mechanism. The inert model, Q I = − Q ( θ E = arctan p / ∼ . π ), has a charge or-thogonal to Q η and follows from an alternative E break-ing pattern (Robinett and Rosner, 1982a). In the neu-tral N model ( θ E = arctan √ ∼ . π ) (Barger et al. ,2003; Kang et al. , 2005b; King et al. , 2006; Ma, 1996),the ν c has zero charge, allowing a large Majorana massor avoiding big bang nucleosynthesis constraints for aDirac ν , as discussed in Section V.F. It essentially inter-changes the assignments of the S and ν c and of the two ∗ representations (which have the same standard modelquantum numbers) with respect to the χ model, andis basically the same as the alternative left-right modelin (Babu et al. , 1987; Ma, 1987). The secluded sec-tor model ( θ E = arctan( √ / ∼ . π ) (Erler et al. ,02002) will be discussed in Section III.E.3.The E models allow the Yukawa couplings neededto generate masses for the standard model and exoticfermions. In particular, in the supersymmetric case thesuperpotential terms W = − h d QH d d c + h u QH u u c − h e LH d e + + h ν LH u ν c + λ S SH u H d + λ D SDD c (48)are all allowed, where family indices have been neglected.(In the non-SUSY case, two Higgs doublets, analogous to H u and H d , are required.) From Eq. 48 we see that the E models all allow a dynamical µ eff , while an elemen-tary µ is forbidden in all but the χ model.The supersymmetric E model with three -plets canincorporate one or more pairs of Higgs doublets H u,d inthe + ∗ pairs. However, that version of the model isnot consistent with the simple form of gauge unificationobserved in the MSSM for the SM subgroup. That is be-cause the complete extra + ∗ multiplets give equal con-tributions to the SU (3), SU (2), and U (1) Y β functions atone loop, so the unification conditions are similar to theMSSM with 3 families but no Higgs pair. Unification canbe restored by introducing an H u and H cu pair from an in-complete + ∗ representation (Langacker and Wang,1998). (The physical H u could either be this one or fromthe complete -plets.) This pair is completely non-chiral, so it does not introduce any anomalies, but atthe cost of introducing a rather arbitrary aspect to themodel. Also, there is no obvious reason (except perhapsthe mechanism in (Giudice and Masiero, 1988)) why thisextra pair should be at the electroweak or TeV scale,reintroducing a form of the µ problem. Nevertheless, theunification of the SM gauge couplings and the unifica-tion scale M X are then the same as in the MSSM at oneloop, though the value of the gauge coupling at M X isincreased because of the extra exotics.If the U (1) ′ really derives from an E -type GUTwhich breaks directly to SU (3) × SU (2) × U (1) Y × U (1) ′ ,one expects that g = q g ′ at the unification scale,where q g ′ is the GUT-normalized hypercharge cou-pling. Running down to the TeV scale, this implies g = r g tan θ W λ / g , (49)where λ / g ∼ θ E -dependent) correction of afew % due to the U (1) ′ charge of the incomplete + ∗ .Eq. 49 can be taken as a definition of λ g for an arbi-trary model. It is typically of order unity even for morecomplicated E breaking patterns (Robinett and Rosner,1982a), and was taken to be unity by construction for the G LR model.In a full E grand unified theory the exotic D, D c part-ners of the Higgs doublets would have diquark Yukawacouplings such as W DQ ∼ DQQ or D c u c d c , as well as leptoquark couplings W LQ ∼ Du c e c or D c QL , whichare related by E to the ordinary Higgs Yukawa cou-plings. These would lead to rapid proton decay medi-ated by the D and D c unless their masses (and thereforethe U (1) ′ breaking scale) is comparable to the unifica-tion scale. A TeV-scale Z ′ therefore requires that theGUT Yukawa relations are not respected, so that eitherthe leptoquark or the diquark couplings (or both) areabsent. This could come about in a string constructionif the fields in the multiplet are not directly related toeach other in the underlying theory (see, e.g., (Witten,2001)). See (Howl and King, 2008; King et al. , 2006) fora detailed study of complete E models with a low energy U (1) ′ . Alternatively, one can simply view the charges andexotics as an example of an anomaly-free construction. B. Anomaly-Free Sets
Many authors have described classes of U (1) ′ mod-els by requiring the cancellation of anomalies andother criteria (Appelquist et al. , 2003; Barr et al. , 1986;Batra et al. , 2006; Carena et al. , 2004; Cheng et al. ,1998, 1999; Cvetic et al. , 1997; Demir et al. , 2005;Erler, 2000; Joshipura et al. , 2000; Kang et al. , 2008;Langacker et al. , 2008; Lee et al. , 2008b; Ma, 2002;Morrissey and Wells, 2006). Usually some conditions areapplied on the types of exotics. It is usually assumedthat any exotic fermions are non-chiral under the stan-dard model, i.e., that they occur in vector pairs ψ + ψ c .This avoids the introduction of any SM anomalies andalso reduces the sensitivity to precision electroweak con-straints (Yao et al. , 2006). One can then constrain theexotic representations with respect to the SM and their U (1) ′ charges from the mixed SM- U (1) ′ conditions X f ∈ , ∗ Q f = X f ∈ Q f = X f Y Q f = X f Y Q f = 0 . (50)The pure U (1) ′ conditions P f Q f = P f Q f = 0 fur-ther restrict the charges. Alternatively, some authorsignore the latter because they can be satisfied by addingSM singlets to the model. This can always be donewith rational charges if the mixed anomaly solutions arerational (Batra et al. , 2006; Morrissey and Wells, 2006).However, because of its cubic nature the singlet structureis sometimes complicated.In non-supersymmetric models it is often assumed thatthe only chiral fermions are the three ordinary familiesand 3 corresponding families of exotics. This assumptionis often not valid in supersymmetric models, where onemust also take into account the fermionic partners of theHiggs doublets and of the SM singlets which break the One expects the charges to be rational if the U (1) ′ is embed-ded in a simple group, but this need not be the case for morecomplicated embeddings, such as the SM couplings in Eq. 8. TABLE II Decomposition of the E fundamental representation of left-handed fermions under SO (10) and SU (5), andtheir U (1) χ , U (1) ψ , U (1) η , inert U (1) I , neutral- N U (1) N , and secluded sector U (1) S charges. A general model in this class hascharge Q = cos θ E Q χ + sin θ E Q ψ − ǫY , where ǫ can result from kinetic mixing, and coupling g = q g tan θ W λ / g , where λ g is usually of O (1). SO (10) SU (5) 2 √ Q χ √ Q ψ √ Q η Q I √ Q N √ Q S
16 10 ( u, d, u c , e + ) − − − / ∗ ( d c , ν, e − ) 3 1 1 − ν c − − −
510 5 (
D, H u ) 2 − − ∗ ( D c , H d ) − − − − /
21 1 S − − / U (1) ′ . These, as well as other exotics, often do not occurin three families (exceptions are the E models, wherethey do occur in three families, and the T R , B − L mod-els, where the Higgs doublets and singlets are usuallynon-chiral).Another issue in the supersymmetric models is whetherthe MSSM unification of the SM gauge couplings is pre-served. The simplest way for this to occur is for thethree SM families, which transform as + ∗ under SU (5), and the two Higgs doublets H u,d , are supple-mented by exotics which transform as + ∗ and/or + ∗ . It is not necessary for the fields in a SU (5)multiplet to have the same U (1) ′ charges (e.g., they mayhave different origins in an underlying string theory), andin fact under minimal assumptions they must be differ-ent (Morrissey and Wells, 2006). An alternative is to al-low non-chiral exotics, as in the E models, reintroducinga form of the µ problem.Other conditions are often employed along with theanomaly and unification constraints. These may in-volve the existence of quark and lepton Yukawa cou-plings for one or two Higgs doublets, constraints on neu-trino mass, Yukawas that can lead to masses for the ex-otics, operators that can allow exotic decays, whether thecharges are family universal, whether the U (1) ′ solvesthe supersymmetric µ problem, whether it forbids R -parity violating operators (Erler, 2000; Joshipura et al. ,2000; Ma, 2002) or other operators relevant to protondecay (Chamseddine and Dreiner, 1995; Coriano et al. ,2008; Lee et al. , 2008a,b), whether it plays the role ofa family symmetry relevant to the fermion masses andmixings (Joshipura et al. , 2000; Kaplan et al. , 1999), andmany other possible conditions.The Q Y BL models of Section III.A.2, which donot require any exotics other than ν R , are discussedin (Appelquist et al. , 2003). Four one-parameter familiesof models with three families of exotics were constructedin (Carena et al. , 2004). Two of these, referred to as q + xu and + x ∗ are equivalent to the Q Y BL and the E model Q ( θ E ), respectively, while the others ( B − xL and d − xu ), have not emerged from other considerationsfor general x . The + x ∗ and d − xu would require two Higgs doublets to have normal quark and charged leptonYukawas.The most systematic classification of the supersym-metric models is given in (Erler, 2000), which requiresanomaly cancellation, minimal gauge unification with nonon-chiral states, exotic masses, and the absence of rapidproton decay or fractional electric charges. Classes of so-lutions were found, which required that more than oneSM singlet participates in the U (1) ′ breaking. A par-ticularly simple one is the Q ˜ ψ model. It involves two + ∗ pairs ( D i + L i ) and ( D ci + L ci ) , i = 1 ,
2, whichare analogous to the (
D, H u ) and ( D c , H d ) of the E model, along with H u,d and the three SM families. The U (1) ′ symmetry is broken by the VEVs of two singlets, S and S D , which also generate masses for the H u,d and L i , L ci ( h S i ) and for D i , D ci ( h S D i ). Additional singletsare needed for the U (1) ′ anomalies. The Q ˜ ψ charges arelisted in Table III. The fermion currents are purely axial.It is straightforward to generalize the Q ˜ ψ model to Q ∗ ,which allows non-axial charges and n ∗ pairs of + ∗ .Three distinct chiral singlets must acquire VEVs to gen-erate all of the exotic masses, except for n ∗ = 2 or 3.Additional SM singlets are needed for the U (1) ′ anoma-lies and to generate singlet masses. The gauge coupling g is arbitrary.It was shown in (Demir et al. , 2005) that anomaly-free supersymmetric models can be constructed withoutany exotics (not even ν c ) and only one singlet S (whichgenerates a dynamical µ eff ) provided one allows familynonuniversal charges (an early example was also givenin (Cvetic et al. , 1997)). It is possible to choose thecharges to avoid flavor changing neutral current (FCNC)effects (see Section V.D). However, the U (1) ′ forbidssome of the quark and lepton Yukawa interactions in thesuperpotential. These could possibly be generated bynon-holomorphic soft terms, as described in Section II.E. C. TeV Scale Physics Models
In this section we briefly consider various models in-volving new TeV scale physics, especially those motivated2
TABLE III Examples of supersymmetric models consistentwith minimal SM gauge unification. n ∗ is the number ofpairs of + ∗ . Q S is taken to be 1. The free parameters are Q H u ≡ x, Q Q ≡ y, Q D ≡ z (which only affects the exotics),and the gauge coupling g . Kinetic mixing can be added. The Q ˜ ψ model is a special case with axial charges and n ∗ =2. Additional SM singlets are not displayed. The ν c chargeallows a Dirac ν mass term. Q ∗ Q ˜ ψ Q ∗ Q ˜ ψ Q y / H u x − / u c − x − y / H d − − x − / d c x − y / S D /n ∗ / L − y / D i z − / e + x + 3 y / D ci − /n ∗ − z − / ν c − − x + 3 y / S L /n ∗ S L i − n ∗ n ∗ + x + 3 y + 3 z/ − / L ci − /n ∗ − Q L i − / as alternatives to the elementary Higgs for electroweaksymmetry breaking.As a preliminary, consider a direct product of two iden-tical gauge group G ≡ G × G , with generators ~T , andassociated currents ~J , . Then − L = g ~J · ~W + g ~J · ~W . (51) G can be spontaneously broken to the diagonal subgroup G D with generators ~T D = ~T + ~T if there is a Higgs fieldwhich transforms equivalently under both groups. An ex-ample is SU ( N ) × SU ( N ), with a Higgs ϕ αa transformingas N ∗ × N , with h ϕ αa i = cδ αa . It is then straightforwardto show that − L = g L ( ~J + ~J ) · ~W L + g L (cot δ ~J − tan δ ~J ) · ~W H , (52)where ~W L = sin δ ~W +cos δ ~W is the massless boson, W H is the massive orthogonal combination, tan δ = g /g ,and g L = g sin δ . W L can acquire mass and W L,H canmix due to additional Higgs fields. A simple illustrationis the SM breaking of U (1) T × U (1) Y to U (1) em by theordinary Higgs doublet.
1. Little Higgs, Twin Higgs, and Un-Unified Models In Little Higgs models (Arkani-Hamed et al. , 2001a)the Higgs is a pseudo-Goldstone boson of an approx-imate global symmetry. (For reviews, see (Han et al. ,2003, 2006; Perelstein, 2007).) The one-loop (and some-times two-loop) quadratic divergences in the Higgs mass-square are cancelled by new TeV gauge bosons, fermions,and scalar particles related to those of the SM. Thereare a wide class of models, all of which involve heavyneutral and charged gauge bosons. For example, in the Littlest Higgs models (Arkani-Hamed et al. , 2002) theelectroweak gauge group is [ SU (2) × U (1)] , which is asubgroup of a larger global group. The SM left-handedfermions are charged under only the first SU (2). The SU (2) symmetry is broken by a condensate charged un-der both factors to an unbroken diagonal subgroup, andthe U (1) charges are chosen to yield U (1) Y × U (1) H ,where Y is the normal hypercharge. Thus, the residualgauge group is SU (2) L × U (1) Y × SU (2) H × U (1) H . FromEq. 52, the heavy charged W ± H and neutral W H couple tothe left-handed SM quarks and leptons with the SU (2) L generators ~τ / g cot δ . The neutral U (1) H boson is lighter, with model dependent couplings.Precision electroweak constraints are rather severe, un-less one pushes the Little Higgs scale to be uncomfortablylarge compared to the original motivation. However, thedifficulties can be reduced if the U (1) H is not gauged.The precision electroweak constraints are greatly weak-ened (they are only generated at loop level) if one in-troduces a discrete symmetry, T -parity (Cheng and Low,2003; Hubisz et al. , 2006). This is analogous to R -parityin supersymmetry, and requires that the heavy states,such as the new gauge bosons, only couple in pairs to theordinary particles. This also means that they must bepair produced at colliders. The lightest could be stableand possibly be a dark matter candidate. However, ithas recently been argued that the T -parity may be bro-ken by anomalies (Hill and Hill, 2007), leading to decays,e.g., into ZZ .In the Twin Higgs model (Chacko et al. , 2006) theHiggs quadratic divergences are canceled by particlesfrom a hidden sector that is a mirror of the SM and whichmainly communicates by an extended Higgs sector. Thegauge bosons in the hidden sector may essentially decou-ple from the SM particles and could even be massless,while in other versions there may be kinetic mixing withthe photon.In the
Un-unified model (Georgi et al. , 1990), the left-chiral SM quarks and leptons transform under distinct SU (2) groups SU (2) q and SU (2) l with gauge couplings g q,l , i.e., they are not unified. There is a single con-ventional U (1) Y . After diagonal breaking, one recov-ers the SM along with heavy W ± , H which couple to g (cot δ ~J q − tan δ ~J l ) using Eq. 52. For small tan δ theheavy bosons couple mainly to quarks.
2. Extra Dimensions
The existence of extra dimensions is suggested bystring models (Antoniadis, 1990). There are a wide vari-ety of models, depending on their number, size, whetherthey are flat or warped, whether the SM fields are al-lowed to propagate in the extra dimensions (i.e., in the bulk ), etc. For a review, see (Yao et al. , 2006).The simplest case involves a single extra dimensionof radius R , implying the existence of Kaluza-Klein ex-citations of the states that can propagate in the bulk,3with mass ∼ n/R, n = 1 , · · · . If only gravitons prop-agate, then R can be large enough to probe in labo-ratory gravity experiments. However, if the SM gaugebosons are also allowed to propagate, then R − must belarger than O (TeV) ( R . − cm). If the SM fermionsand Higgs are not allowed to propagate (i.e., confined tothe brane ), then the excitations of the SM gauge bosons( W ± , Z, A, gluon) couple to the same currents as theirSM counterparts, but with a coupling constant largerby √ et al. , 1999; Masip and Pomarol,1999). Current experimental limits require R − & et al. , 2004; Cheung and Landsberg,2002). The limits are much weaker ( O (300 GeV)) in universal extra dimension models, in which all of the SMfields propagate uniformly in the bulk (Appelquist et al. ,2001; Appelquist and Yee, 2003; Cheng et al. , 2002;Gogoladze and Macesanu, 2006). Similar to R or T -parity, there is a KK -parity so that the n = 1 states canonly be pair produced and only contribute to electroweakobservables in loops. The lightest is stable. In variants inwhich the various quarks and leptons are localized in dif-ferent parts of the extra-dimensional space (with implica-tions for the flavor problem) the couplings of the Kaluza-Klein excitations are family nonuniversal (since the over-lap of the wave functions depends on location). Thisleads to the possibility of FCNC effects (Delgado et al. ,2000), as discussed in Section V.D.Models involving warped extra dimen-sions (Randall and Sundrum, 1999) may have all ofthe SM fields confined to the infrared brane. However,much attention has been devoted to the possibility thatthe SM fields other than the Higgs can also propagatein the bulk (Agashe et al. , 2003, 2007; Carena et al. ,2003a; Hewett et al. , 2002), e.g., because in that casethe theory is related to technicolor models by theAdS/CFT correspondence (Maldacena, 1998). It isthen useful to enhance the electroweak gauge sym-metry to SU (2) L × SU (2) R × U (1) B − L to providea custodial symmetry to protect the electroweak ρ parameter (Agashe et al. , 2003). The Kaluza-Kleinexcitations of the gauge bosons couple mainly to the t and b due to wave function overlaps, and decays to W W and Z + Higgs are also possible due to mixings (see,e.g., (Agashe et al. , 2007)).
3. Strong Dynamics
There have been many models in which strong dynam-ics is involved in electroweak symmetry breaking, whichoften involve additional elementary gauge bosons or com-posite spin-1 states, which may be strongly coupled.Dynamical symmetry breaking (DSB) modelsin which the Higgs is replaced by a fermion con-densate are reviewed in (Chivukula et al. , 2004;Chivukula and Simmons, 2002; Hill and Simmons,2003). For example, topcolor models (Hill, 1995)typically involve new gluons and a new Z ′ that cou- ple preferentially and with enhanced strength to thethird generation and which assist in forming a topcondensate. Nonuniversal extended technicolor mod-els (Chivukula and Simmons, 2002) also feature newgauge interactions preferentially coupled to the thirdfamily.The BESS (Breaking Electroweak Symmetry Strongly)models (Casalbuoni et al. , 1985, 1987) are effective La-grangian descriptions of models with a strongly interact-ing longitudinal gauge boson sector, such as one expectsin the large M H limit of the SM or in some forms ofDSB. There are vector and axial bound states which canmix with the W ± , Z , and A . They interact with the SMparticles directly and by mixing. The possibility thatthe electroweak bosons could be composite has also beenconsidered (Baur et al. , 1987).Another interesting model with no elementary or com-posite Higgs (Csaki et al. , 2004) is a variant on thewarped extra dimension scenario. However, instead of in-cluding a Higgs field the electroweak symmetry is brokenby boundary conditions. The Kaluza-Klein excitations ofthe gauge bosons unitarize the high energy scattering oflongitudinal gauge bosons. More general classes of Hig-gless models may involve fermiophobic Z ′ which may beproduced and detected by their couplings to the W and Z (He et al. , 2008). D. Non-Standard Couplings
Most of the canonical Z ′ models assume electroweakscale couplings, and that the Z ′ couplings to most or allof the SM fermions are of comparable strength and familyuniversal, in which case existing experimental constraintsrequire masses not too much below 1 TeV (Section IV).However, there are many models with different assump-tions concerning the gauge couplings, charges, and scales.
1. Decoupled Models
Leptophobic Z ′ s (del Aguila et al. , 1987a) do not cou-ple to ordinary neutrinos or charged leptons, and there-fore most direct electroweak and collider searches areinsensitive to them. They could emerge, e.g., in the E η model in Table II, combined with a (large) ki-netic mixing (Babu et al. , 1996) ǫ ∼ − / √
15; in aflipped SU (5) model (Lopez and Nanopoulos, 1997); orin models in which the Z ′ couples to baryon num-ber (Carone and Murayama, 1995). Approximatelyleptophobic models were once suggested by apparentanomalies in Z → b ¯ b decays (see (Rosner, 1996b;Umeda et al. , 1998) for references), but are still an in-teresting possibility for allowing Z ′ masses much smallerthan a TeV. A purely leptophobic Z ′ is still constrainedby Z − Z ′ mixing effects (Umeda et al. , 1998), andcould be inferred by collider signals such as the pro-duction of t ¯ t pairs, exotics (Rosner, 1996b), or the4same-sign dilepton decays of a pair of heavy Majo-rana neutrinos (del Aguila and Aguilar-Saavedra, 2007;Duncan and Langacker, 1986). They could even be lightenough to be produced in Υ decays (Aranda and Carone,1998).Limits are also weak, e.g., if the Z ′ couples only to thesecond and third family leptons (Foot et al. , 1994) or if itcouples only to third family fermions (Andrianov et al. ,1998). In fermiophobic models (Barger et al. , 1980;Donini et al. , 1997) there are no direct couplings of the Z ′ to the SM fermions, although they may be induced byordinary or kinetic mixing. An interesting possibility isthat such fermiophobic Z ′ may couple to a hidden sec-tor (Chang et al. , 2006; Kumar and Wells, 2006), such asmay be associated with supersymmetry breaking. Mix-ing effects could therefore possibly be a means of probingsuch a sector (direct Z ′ couplings to a hidden sector areconsidered in Section V.E). Finally, a Z ′ with canonicalcharges could still be much lighter than a TeV if its gaugecoupling is sufficiently small (Fayet, 1980, 2007; Freitas,2004; Nelson and Walsh, 2008).
2. St¨uckelberg Models
It is possible to write a U (1) gauge invariant theorywith a massive gauge boson C µ by the St¨uckelberg mech-anism (Stueckelberg, 1938). The Lagrangian is L = − C µν C µν −
12 ( mC µ + ∂ µ σ )( mC µ + ∂ µ σ ) , (53)where C µν is the field strength tensor. Under a gaugetransformation, ∆ C µ = ∂ µ β , while the field σ is shifted,∆ σ = − mβ , analogous to the shift in an axion field.A gauge-fixing term can be added to Eq. 53 which can-cels the cross term between C and σ , leaving a mas-sive C field and a decoupled σ . This is analogous tothe Higgs mechanism, but there is no field with a VEVand no physical Higgs boson. This mechanism has re-cently been applied to a U (1) ′ extension of the SM orthe MSSM (Feldman et al. , 2007; Kors and Nath, 2004).For example, if one replaces the second term in Eq. 53by − ( M C µ + M B µ + ∂ µ σ ) / M /M ≡ ǫ ≪ C will mix with the A and Z , but there willremain a massless photon. The new Z can be relativelylight (e.g., several hundred GeV), so ǫ must be small. Ifthe C has no direct couplings to matter, the new Z willdecay only to SM particles via the mixing and will bevery narrow. If the C does couple to exotic matter, thenthe mixing with the photon will induce tiny (generally)irrational electric charges of O ( ǫ ) for the exotic particles.Such mixing with the photon is never induced by ordi-nary Higgs-type mixing if U (1) em is unbroken, but canalso be induced by kinetic mixing with another masslessboson (Section III.E.1). Other applications, such as todark matter, are reviewed in (Feldman et al. , 2007).
3. Family Nonuniversal Models
Another variant is the possibility of family nonuniver-sal charges (e.g., (Demir et al. , 2005)). A number of ex-amples of Z ′ coupling preferentially to the third family orto the t quark were mentioned in Sections III.C.2, III.C.3,and III.D.1. These could have enhanced gauge couplings,and could be observed as a resonance in t ¯ t production.String-derived Z ′ s often have nonuniversal couplings aswell (Section III.F), as do the Kaluza-Klein excitations inextra-dimensional theories in which the fermion familiesare spatially separated (Section III.C.2). Possible FCNCeffects are considered in Section V.D. E. U (1) ′ Breaking Scales
Most attention is given to possible electroweak or TeVscale Z ′ s, but there are other possibilities. Here we de-scribe massless, TeV scale, and intermediate scale mod-els. Models involving the GUT or string scales are de-scribed in Section III.F.
1. A Massless Z ′ A Z ′ would be massless if the U (1) ′ symmetry is un-broken. This would imply an unacceptable long rangeforce if it coupled to ordinary matter unless the couplingwere incredibly small (Dobrescu and Mocioiu, 2006). Itwould be allowed if the primary coupling were to a hid-den sector and communicated only by higher-dimensionaloperators (Dobrescu, 2005) or by kinetic mixing with thephoton (Holdom, 1986). The latter scenario would in-duce a small fractional electric charge for hidden sectorparticles.
2. Electroweak/TeV Scale Z ′ Models in which the U (1) ′ is involved in electroweaksymmetry breaking, such as in Section III.C, typicallyinvolve U (1) ′ breaking at the electroweak or TeV scale.In the U (1) ′ extension of the MSSM with a single S field (Cvetic et al. , 1997; Cvetic and Langacker, 1996a,1997; Keith and Ma, 1997; Langacker and Wang, 1998),the part of the superpotential involving S and H u,d is W = λ S SH u H d , where we have assumed Q S = 0 and Q S + Q u + Q d = 0. Like the MSSM, the minimum ofthe tree-level potential always occurs along the charge-conserving direction with only h H u,d i 6 = 0 (this assumesthat the squark and slepton VEVs vanish). The potentialis then V = V F + V D + V soft , (54)5where V F = λ S (cid:0) | H u | | H d | + | S | | H u | + | S | | H d | (cid:1) V D = g (cid:0) | H u | − | H d | (cid:1) + g (cid:0) Q u | H u | + Q d | H d | + Q S | S | (cid:1) V soft = m u | H u | + m d | H d | + m S | S | − (cid:0) λ S A S SH u H d + h . c . (cid:1) . (55)If S acquires a VEV, then the effective µ parameteris µ eff = λ S h S i , the corresponding effective Bµ is( Bµ ) eff = λ S A S h S i , and the Z − Z ′ mass matrix isgiven by Eq. 18 and Eq. 20. One can define the fieldsso that λ S A S and therefore the VEVs ν u,d and s definedafter Eq. 20 are real and positive. There is no analog ofthe first (second) term in V F ( V D ) in the MSSM.For generic parameters one expects ν u,d and s tobe comparable. For example, for λ S A S large com-pared to the soft masses and Q u = Q d = − Q S / et al. , 1997) ν u ∼ ν d ∼ s , with negligible Z − Z ′ mixing and M Z ′ /M Z ∼ g Q u /g , which is typ-ically of order 1. This case is excluded unless the modelis leptophobic or something similar.A more likely scenario is that the soft parameters( | m u,d,S | , | A S | ) are of O (1 TeV), with m S <
0. Then s ∼ − m S /g Q S and M Z ′ ∼ − m S . One can havea smaller EW scale ν u,d ≪ s by accidental cancella-tions, which are not excessive provided M Z ′ is not toomuch larger than a TeV. In most supersymmetry medi-ation schemes m S is positive at a large scale such as thePlanck scale. The running m S can be driven negativeat low scales radiatively provided it has sufficiently largeYukawa couplings, such as λ S and/or couplings to ex-otics such as in Eq. 48. This is analogous to the MSSMin which m u can be driven negative by its large Yukawacoupling to the top.
3. Secluded Sector and Intermediate Scales
In the single S model in Eq. 54 and (55) there is sometension between the electroweak scale and developing alarge enough M Z ′ . These can be decoupled without tun-ing when there are several S fields. For example, in the secluded sector model (Erler et al. , 2002) there are fourstandard model singlets S, S , , that are charged undera U (1) ′ , with W = λ S SH u H d + λS S S . (56)(Structures similar to this are often encountered in het-erotic string constructions.) µ eff is given by λ S h S i , butall four VEVs contribute to M Z ′ . The only couplingsbetween the ordinary ( S, H u,d ) and secluded ( S , , ) sec-tors are from the U (1) ′ D term and the soft masses (spe-cial values of the U (1) ′ charges, which allow soft mixingterms, are required to avoid unwanted additional global symmetries). It is straightforward to choose the soft pa-rameters so that there is a runaway direction in the limit λ →
0, for which the ordinary sector VEVs remain finitewhile the S i VEVs become large. For λ finite but small,the h S i i and M Z ′ scale as 1 /λ . For example, one can find M Z ′ in the TeV range for λ ∼ . − .
1. The secludedmodel can be embedded in the E context (Table II). Intermediate Scale models (Cleaver et al. , 1998;Morrissey and Wells, 2007) are those in which the U (1) ′ breaking is associated with a F and D flat direction, suchas the secluded model in Eq. 56 with λ = 0. However,let us consider a simpler toy model with two fields S , with Q S Q S <
0. If there are no terms in W like S i S j or S i S j S k , then the potential for S , is V ( S , S ) = m | S | + m | S | + g Q S | S | + Q S | S | ) . (57)The quartic term vanishes for | S | / | S | = − Q S /Q S .For simplicity, take Q S = − Q S , and assume that atlow energies m S < m S >
0, as would typicallyoccur by the radiative mechanism if W contains a term h D S DD c . If m ≡ m S + m S > h S i 6 = 0 , h S i = 0. If there is also a λ S S H u H d term in W then h S i and M Z ′ will be at the EW scale( . S . On the otherhand, for m <
0, the potential along the F and D flatdirection S = S ≡ S is V ( S ) = m S , (58)which appears to be unbounded from below. In fact, V ( S ) is typically stabilized by one or both of two mech-anisms: (a) The leading loop corrections to the effec-tive (RGE-improved) potential result in m → m ( S ),leading to a minimum slightly below the scale at which m ( S ) goes through zero, which can be anywhere in therange 10 − GeV. (b) Another possibility is thatthe F -flatness is lifted by higher-dimensional operators(HDO) in W , such as W = ( S S ) k /M k − , where M is the Planck or some other large scale. This would leadto h S i ∼ √ mM ∼ GeV for k = 2, m ∼ M the Planck scale. In such models, HDO suchas LH d e + ( S/M ) p or SH u H d ( S/M ) q could also be im-portant for generating small effective Yukawa couplings(and therefore fermion mass hierarchies) or µ eff ≪ h S i terms (Cleaver et al. , 1998). Implications for neutrinomass are considered in Section V.F. F. Grand Unification, Strings, and Anomalous U (1) ′
1. Grand Unification
In some full grand unified theories (Hewett and Rizzo,1989; Langacker, 1981), such as E , the extra U (1) ′ smust be broken at or near the GUT unification scale toavoid rapid proton decay. This typically occurs if the6Higgs doublets (and their Yukawa couplings to ordinaryfermions) are related by the GUT symmetry to chiralexotics, which cannot be much heavier than the U (1) ′ breaking scale. However, as mentioned in Section III.A.3this can be evaded in models which respect the GUTquantum numbers but not the Yukawa relations, or inmodels, such as the E χ model, in which the Higgs dou-blets are non-chiral.
2. String Theories
Most semi-realistic superstring constructions yieldeffective four-dimensional field theories that includethe SM gauge group ( not a full four-dimensionalGUT), as well as additional gauge group factorsthat often involve additional U (1) ′ s. (Examples in-clude (Anastasopoulos et al. , 2006b; Braun et al. , 2005;Cleaver et al. , 1999; Coriano et al. , 2008; Cvetic et al. ,2001; Faraggi, 1993; Faraggi and Nanopoulos,1991; Giedt, 2001; Lebedev et al. , 2008). Forreviews, see (Blumenhagen et al. , 2005, 2007;Cvetic and Langacker, 1997).) Heterotic construc-tions often descend through an underlying SO (10) or E in the higher-dimensional space, and may therefore leadto the T R and B − L (i.e., Q LR ) or the E -type charges.Additional or alternative U (1) ′ structures may emergethat do not have any GUT-type interpretation andtherefore have very model dependent charges. Similarly,intersecting brane constructions often descend throughPati-Salam type models (Mohapatra, 2003), yielding Q LR . Other branes can lead to other types of U (1) ′ charges. For example, the construction in (Cvetic et al. ,2001) involves two extra U (1) ′ s, one coupling to Q LR and the other only to the Higgs and the right-handedfermions.Constructions often have one or multiple SM singletswhich can acquire VEVs to break the extra U (1) ′ . How-ever, that is not always the case. For example, insome of the Q LR models (see, e.g., (Braun et al. , 2005;Cvetic et al. , 2001)) the only fields available to break theenhanced gauge symmetry are the scalar partners ˜ ν R ofthe right-handed neutrinos (Cvetic et al. , 2002). Theseact like the δ R defined in Section III.A.2, but it is difficultto reconcile the Z ′ constraints with neutrino phenomenol-ogy. This also occurs in the simpler supersymmetric ver-sions of the χ model.The U (1) ′ in string constructions may couple tohidden sector particles, and in some cases they cancommunicate between the ordinary and hidden sec-tors (Langacker et al. , 2008; Verlinde et al. , 2008). Thenon-standard string U (1) ′ often have family nonuniver-sal charges. This can occur if the fermion families havedifferent embeddings in the underlying theory. A simplefield-theoretic example is a variant on the E model inTable II. One could assign, e.g., the first two families( d ci , L i ) , i = 1 ,
2, to the ∗ from the of SO (10), andthe third to the ∗ from the .
3. Anomalous U (1) ′ The effective four-dimensional field theories arisingfrom the compactification of a string theory usually con-tain anomalous U (1) ′ factors (see (Kiritsis, 2004) for areview). There is typically one anomalous combinationin heterotic constructions. In intersecting brane mod-els (Blumenhagen et al. , 2005) there are stacks of branesyielding U ( N ) ∼ SU ( N ) × U (1), in which the U (1) isusually anomalous. Since the underlying string theoryis anomaly free, these anomalies must be cancelled bya generalized Green-Schwarz mechanism. In particular,the Z ′ associated with the U (1) ′ acquires a string-scalemass by what is essentially the St¨uckelberg mechanism inEq. 53, with the axion field σ associated with an antisym-metric field in the internal space (this sometimes appliesto nonanomalous U (1) ′ as well). The U (1) ′ still acts asa global symmetry on the low energy theory, restrictingthe possible couplings and having possible implications,e.g., for baryon or lepton number. In addition, effec-tive trilinear vertices may be generated between the Z ′ and the SM gauge bosons (Anastasopoulos et al. , 2006a;Coriano et al. , 2006). It is possible that the string scaleis actually very low (e.g., TeV scale) if there is a largetotal volume of the extra-dimensional space (a realiza-tion of the large extra dimension scenario). This wouldallow TeV scale Z ′ s associated with anomalous (or some-times nonanomalous) U (1) ′ s, without any associatedHiggs scalar and with anomalous decays into ZZ , W W ,and Zγ (Armillis et al. , 2008; Berenstein and Pinansky,2007; Ghilencea et al. , 2002; Kumar et al. , 2008).Anomalous U (1) ′ s in heterotic constructions lead toFayet-Iliopoulos (FI) terms, which are effectively con-stant contributions to the U (1) ′ D terms that are closeto the string scale. Smaller FI terms may also appearin intersecting brane constructions which break super-symmetry. In many cases, FI terms trigger scalar fieldsin the low energy theory to acquire VEVs to cancelthem. These VEVs in turn may lead to the break-ing of gauge symmetries (such as other nonanomalous U (1) ′ s) and the generation of masses for some of theparticles at the FI scale, a process known as vacuumrestabilization (see (Cleaver et al. , 1999) for an exam-ple). Family nonuniversal U (1) ′ s may be used to gen-erate fermion textures using the Froggatt-Nielsen mech-anism (Binetruy and Ramond, 1995; Chankowski et al. ,2005; Ibanez and Ross, 1994; Jain and Shrock, 1995).The elements are associated with higher-dimensional op-erators allowed by the symmetry, and involve powers ofthe ratio of the FI and Planck scales. IV. EXPERIMENTAL ISSUES
There are limits on Z ′ masses and Z − Z ′ mixingfrom precision electroweak data, from direct and indirectsearches at the Tevatron, and from interference effectsat LEP 2. In this section we briefly review the existing7limits and future prospects for discovery and diagnos-tics. FCNC effects for family nonuniversal couplings andastrophysical/cosmological constraints are touched on inSection V. A. Constraints from Precision Electroweak
1. Parametrization
Precision electroweak data include purely weak νe and ν -hadron weak neutral current (WNC) scattering;weak-electromagnetic interferences in heavy atoms andin e ± e − , l ± -hadron, and ¯ pp scattering; precision Z polephysics; and associated measurements of the W and topmass. They have verified the SM at the level of radiativecorrections and strongly constrained the possibilitiesfor new physics below the TeV scale (Yao et al. , 2006).There have been a number of global analyses of theconstraints from precision electroweak on a possible Z ′ (del Aguila et al. , 1987a, 1992; Amaldi et al. , 1987;Cacciapaglia et al. , 2006; Chivukula and Simmons,2002; Cho et al. , 1998; Costa et al. , 1988;Durkin and Langacker, 1986; Erler and Langacker,1999, 2000; Gonzalez-Garcia and Valle, 1991;Langacker and Luo, 1992; Langacker et al. , 1992;London and Rosner, 1986). Because of the number ofdifferent chiral fermions involved, it is difficult to dothis in a model independent way, so most studies havefocussed on specific classes of models, such as describedin Section III.A, and have emphasized electroweak scalecouplings and family universal charges.Low energy WNC experiments are affected by Z ′ ex-change, which is mainly sensitive to its mass, and by Z − Z ′ mixing. Prior to the Tevatron and LEP 2 theyyielded the best limits on the Z ′ mass. The Z -pole exper-iments at LEP and SLC, on the other hand, are mainlysensitive to Z − Z ′ mixing, which lowers the mass of the Z relative to the SM prediction, and also modifies the Z ¯ f f vertices.The effective four-Fermi Lagrangian for the WNC ob-tained from Eq. 9 is − L eff = 4 G F √ n +1 X α =1 ρ α n +1 X β =1 g β g U αβ J µβ , (59)where ρ α ≡ M W / ( M α cos θ W ), M α are the mass eigen-values, U is the orthogonal transformation defined inEq. 17, and the currents are given in Eq. 10 (kineticmixing can be added). Specializing to the n = 1 case,this is − L eff = 4 G F √ ρ eff J + 2 wJ J + yJ ) , (60)in the notation of (Durkin and Langacker, 1986; Langacker and Luo, 1992), where ρ eff = ρ cos θ + ρ sin θw = g g cos θ sin θ ( ρ − ρ ) y = (cid:18) g g (cid:19) ( ρ sin θ + ρ cos θ ) , (61)with the mixing angle θ defined in Eq. 22. For small ρ and θ , these are approximated by ρ eff ∼ ρ , w ∼ b θ, y ∼ b ρ , (62)where b θ ≡ g g θ = C b ρ , b ρ ≡ (cid:18) g g (cid:19) ρ . (63) C is the Higgs-dependent mixing parameter of O (1) de-fined in Eq. 26. In the same limit, from Eq. 24, ρ ∼ ρ (1 + ρ o θ /ρ ) −−−→ ρ =1 θ /ρ = 1 + C b ρ , (64)where ρ , defined in Eq. 16, is 1 if there are only Higgssinglets and doublets.At the Z pole, in addition to the shift in M be-low the SM value, any mixing will affect the current P β g β U β J µβ /g that couples to the Z . For n = 1, thevector and axial couplings V i and A i of the Z to fermion f i , which determine the various Z pole asymmetries andpartial widths (Yao et al. , 2006), become V i = cos θg V ( i ) + g g sin θg V ( i ) ∼ g V ( i ) + b θg V ( i ) A i = cos θg A ( i ) + g g sin θg A ( i ) ∼ g A ( i ) + b θg A ( i ) , (65)where g αV,A ( i ) are defined in Eq. 10. It should be notedthat the S , T , U formalism (Peskin and Takeuchi, 1990)only describes propagator corrections and is not appro-priate for most Z ′ s.
2. Radiative Corrections
The expressions for the electroweak couplings inEq. 10, 61, and 65 and for M Z in Eq. 15 are valid at treelevel only. One must also apply full radiative corrections.In practice, since one is searching for very small tree-leveleffects from the Z ′ it is a reasonable approximation touse the SM radiative corrections (Yao et al. , 2006) andneglect the effects of the Z ′ in loops . However, some The largest effects are from Z loops in µ decay, which modifiyslightly the relation between the extracted Fermi constant andthe W and Z masses (Degrassi and Sirlin, 1989). Z loops canalso modify the relation between µ and β decay and thereforeaffect the CKM universality tests (Marciano and Sirlin, 1987).However, these effects are small for the currently allowed masses. θ W , to ensure that they are not sig-nificantly affected by Z ′ effects (Chankowski et al. , 2006;Degrassi and Sirlin, 1989).
3. Results
The results from precision electroweak and other dataare shown for some specific models in Table IV and Fig-ure 1. One sees that the precision data strongly constrainthe Z − Z ′ mixing angle θ . They also give lower limits on M , but these are weaker than the Tevatron and LEP 2limits. The precision limit on the Z ψ mass is low due toits weak coupling to the neutrino and its purely axial cou-pling to the e − . There is no significant indication for a Z ′ in the data (although the NuTeV anomaly could possiblybe explained by a Z ′ coupling to B − L (Davidson et al. ,2002)). The precision results are presented for two cases: ρ free is for an arbitrary Higgs structure, while ρ = 1 isfor Higgs doublets and singlets with unrestricted charges(i.e., C is left free). There is little difference between thelimits obtained. The precision electroweak constraintsare for the g value in Eq. 44 (except for the sequentialmodel, which uses g = g ∼ . θ and M scale as g − and g , respectively.The stringent mixing limits from (mainly) the Z poledata lead to strong indirect limits on the Z ′ massfor specific theoretical values of C , as can be seenfrom the theoretical curves labeled 0 , , , ∞ in Figure1 (Langacker and Luo, 1992). For the χ and LR modelsthe label refers to the value of | x | / ( | ν u | + | ν d | ), where x/ √ L doublet for χ or like the δ L defined in Section III.A.2 for the LR). Themost commonly studied cases are for x = 0, which yield M Z χ > , M Z LR > ψ and η models, the label represents tan β ≡ | ν u | / | ν d | ,with x = 0 assumed. B. Constraints from Colliders
1. Hadron Colliders
The primary discovery mode for a Z ′ at a hadron col-lider is the Drell-Yan production of a dilepton resonance pp (¯ pp ) → Z ′ → ℓ + ℓ − , where ℓ = e or µ (Aaltonen et al. ,2007; del Aguila et al. , 1989; Barger et al. , 1987;Carena et al. , 2004; Dittmar, 1997; Dittmar et al. ,2004; Godfrey, 2002; Kang and Langacker, 2005;Langacker et al. , 1984; Leike, 1999; Weiglein et al. ,2006; Yao et al. , 2006). Other channels, such as Z ′ → jj where j = jet (Weiglein et al. , 2006), ¯ tt (Han et al. ,2004b), eµ (Abulencia et al. , 2006), or τ + τ − , arealso possible. The forward-backward asymmetry for pp (¯ pp ) → ℓ + ℓ − (as a function of rapidity, y , for pp ) due to γ, Z, Z ′ interference below the Z ′ peak isalso important (Aaltonen et al. , 2007; Dittmar, 1997;Langacker et al. , 1984; Rosner, 1996a).The cross section for hadrons A and B at CM energy √ s to produce a Z α of mass M α at rapidity y is, in thenarrow width approximation (Langacker et al. , 1984), dσdy = 4 π x x M α × X i ( f Aq i ( x ) f B ¯ q i ( x ) + f A ¯ q i ( x ) f Bq i ( x ))Γ( Z α → q i ¯ q i ) , (66)where f A,Bq i , ¯ q i are the structure functions of quark (or an-tiquark) q i (¯ q i ) in hadrons A or B , and the momentumfractions are x , = ( M α / √ s ) e ± y . (67)Neglecting mixing effects the decay width into fermion f i is Γ αf i ≡ Γ( Z α → f i ¯ f i ) = g α C f i M α π (cid:0) ǫ αL ( i ) + ǫ αR ( i ) (cid:1) , (68)where the fermion mass has been neglected. C f i is thecolor factor (1 for color singlets, 3 for triplets). Formulasincluding fermion mass effects, decays into bosons, Ma-jorana fermions, etc., are given in (Kang and Langacker,2005).To a good first approximation, Eq. 66 leads to the Z ′ total production cross section (Leike, 1999) σ Z ′ = 1 s c Z ′ CK exp( − A M Z ′ √ s ) , (69)where C =600 (300) and A =32 (20) for pp ( p ¯ p ) collisions,and K ∼ . M Z ′ . The details of the Z ′ model are collectedin c Z ′ , which depends on M Z ′ , the Z ′ couplings, and themasses of the decay products, c Z ′ ≡ π Z ′ M Z ′ (cid:18) B u + 1 C ud B d (cid:19) , (70)where C ud = 2 (25), Γ Z ′ is the total Z ′ width, and B f =Γ f / Γ Z ′ is the branching ratio into f ¯ f . It is also usefulto define σ fZ ′ ≡ σ Z ′ B f = N f / L , (71)where N f is the number of produced f ¯ f pairs for inte-grated luminosity L . More detailed estimates for theTevatron and LHC are given in (Carena et al. , 2004;Dittmar et al. , 2004; Fuks et al. , 2008; Godfrey, 2002;Leike, 1999), including discussions of parton distributionfunctions, higher order effects, width effects, resolutions,and backgrounds.9 TABLE IV 95% cl lower limits on various extra Z ′ gauge boson masses (GeV) and 90% cl ranges for the mixing sin θ from precision electroweak data (columns 2-4), Tevatron searches (assuming decays into SM particles only), and LEP 2.From (Aaltonen et al. , 2007; Alcaraz et al. , 2006; Erler and Langacker, 1999; Yao et al. , 2006). The Tevatron numbers inparentheses are preliminary CDF results from March, 2008 based on 2.5 fb − (CDF note CDF/PUB/EXOTIC/PUBLIC/9160). ρ free ρ = 1 sin θ ( ρ = 1) Tevatron LEP 2 χ
551 545 ( − . − (+0 . ψ
151 146 ( − . − (+0 . η
379 365 ( − . − (+0 . LR
570 564 ( − . − (+0 . − . − (+0 . The production cross sections, widths, and branchingratios are considered in detail in (Barger et al. , 1987;Gherghetta et al. , 1998; Kang and Langacker, 2005;Langacker et al. , 1984). For the E models, the to-tal width is close to 0.01 M Z ′ assuming decays intoSM fermions only and g ∼ p / g tan θ W . However,Γ Z ′ would be larger if superpartners and/or exotics arelight enough to be produced in the Z ′ decays, andcould therefore be as large as 0.05 M Z ′ in the E mod-els (Kang and Langacker, 2005). The rates for a givenchannel, such as σ eZ ′ , decrease as Γ − Z ′ in that case. Onthe other hand, for smaller Γ Z ′ but fixed branching ra-tios (e.g., from some of the decoupled models described inSection III.D.1) the leptonic rate would decrease and thepeak could be smeared out by detector resolution effects.The Tevatron limits from the CDF and D0 collabo-rations (Aaltonen et al. , 2007; Yao et al. , 2006) (domi-nated, at the time of this writing, by the CDF e + e − search using 1.3 fb − of data) are given in Table IV.Figure 2 shows the sensitivity of the Tevatron and LHCto the E bosons as a function of θ E for L = 1 or 3fb − (Tevatron), and 100 or 300 fb − (LHC), requiring10 events in the combined e + e − and µ + µ − channels. TheTevatron sensitivity is in the 600-900 GeV range for de-cays into standard model fermions only, but lower by asmuch as 200 GeV in the (extreme) case of unsuppresseddecays into sparticles and exotics. The LHC sensitivityis around 4-5 TeV, but can be lower by ∼ e + e − Colliders Z ′ s much heavier than the CM energy in e + e − colli-sions above the Z pole would manifest themselves as newfour-fermion interactions analogous to Eq. 59, but withthe α sum starting at 2. These would interfere with thevirtual γ and Z contributions for leptonic and hadronicfinal states (see, e.g., (Cheung, 2001)).The ALEPH, DELPHI, L3, and OPAL collabora-tions at LEP2 have measured production cross sectionsand angular distributions or asymmetries for e + e − → e + e − , µ + µ − , τ + τ − , ¯ cc , and ¯ bb , as well as hadronic cross sections, at CM energies up to ∼
209 GeV (Alcaraz et al. ,2006). They saw no indication of new four-fermi interac-tions, and the combined lower limits for typical modelsare given in Table IV and Figure 1.Similarly, a future linear collider would have sensitivityto M Z ′ well above the CM energy by interference withthe γ and Z (del Aguila and Cvetic, 1994; Babich et al. ,1999; Godfrey et al. , 2005; Leike and Riemann, 1997;Richard, 2003; Weiglein et al. , 2006). Observablescould include production cross sections, forward-backward (FB) asymmetries, polarization (LR) asym-metries, and mixed FB-LR asymmetries for e + e − → e + e − , µ + µ − , τ + τ − , ¯ cc, ¯ bb , and ¯ tt ; τ polarization; andcross sections and polarization asymmetries for ¯ qq . Highluminosity, e − polarization, and efficient tagging ofheavy flavors are important. For example, the Inter-national Linear Collider (ILC) with √ s = 500 GeV, L = 1000 fb − , and P e − = 80% would have 5 σ sensi-tivity to the E and LR bosons in the range 2 − ∼ √ s = 1 TeV (Weiglein et al. ,2006). There is some chance that a Z ′ could be observedfirst at the ILC, e.g., if its mass were beyond the LHCrange or its couplings weak, in which case only M /g could be determined for large M . More likely, the Z ′ would be discovered first and M Z ′ determined indepen-dently at the LHC or Tevatron. A GigaZ ( Z -pole) optionfor the ILC would be extremely sensitive to Z − Z ′ mix-ing (Weiglein et al. , 2006). C. Diagnostics of Z ′ Couplings
Following the discovery of a resonance in the ℓ + ℓ − channels, the next step would be to establish its spin-1 nature (as opposed, e.g., to a spin-0 Higgs resonanceor a spin-2 Kaluza-Klein graviton excitation). This canbe done by the angular distribution in the resonance restframe, which for spin-1 is dσ fZ ′ d cos θ ∗ ∝
38 (1 + cos θ ∗ ) + A fF B cos θ ∗ , (72)where θ ∗ is the angle between the incident quark orlepton and fermion f . Of course, for a hadron col-0 -0.01 -0.005 0 0.005 0.01sin θ θ [GeV] M [GeV]
00 00 Z χ Z χ Z χ Z ψ -0.01 -0.005 0 0.005 0.01sin θ θ [GeV] M [GeV]
00 00 Z η Z LR FIG. 1 Limits on the Z ′ mass M and the Z − Z ′ mixing angle θ for the χ , ψ , η , and LR ( α = 1 .
53) models. The solid(dashed) contours are 90% cl exclusions from precision electroweak data for ρ = 1 ( ρ = free). A cross, x , is the best fit. Thehorizontal solid line is the 95% cl Tevatron lower limit, assuming decays into SM particles only. The horizontal dotted line isthe 95% cl lower limit from LEP 2. The contours marked 0 , , , ∞ are for various theoretical relations between the mass andmixing and are defined in the text. Updated from (Erler and Langacker, 1999). lider one does not know which hadron is the sourceof the q and which the ¯ q on an event by event ba-sis, but the ambiguity washes out in the determina-tion of the 1 + cos θ ∗ distribution characteristic of spin1 (Dittmar, 1997; Langacker et al. , 1984). The spincan also be probed in e + e − by polarization asymme- tries (Weiglein et al. , 2006).One would next want to determine the chiral couplingsto the quarks, leptons, and other particles in order to dis-criminate between models. (The gauge coupling g canbe fixed to the value in Eq. 44, or alternatively can betaken as a free parameter if the charges are normalized1 Z ’ d i sc o v e r y li m i t a t t he T e v a t r on ( G e V ) SM (3 fb -1 ) SM (1 fb -1 ) ALL (3 fb -1 ) ALL (1 fb -1 )Secluded N 0.0 0.5 1.0 1.5 2.0 2.5 3.03000320034003600380040004200440046004800 SM (300 fb -1 ) SM (100 fb -1 ) ALL (300 fb -1 ) ALL (100 fb -1 ) Z ’ d i sc o v e r y li m i t a t t he L HC ( G e V ) Secluded N
FIG. 2 Discovery limits for an E Z ′ as a function of θ ≡ θ E corresponding to a total of 10 e + e − or µ + µ − events using σ Z ′ from Eq. 69. In each panel the top two curves assume decays into SM fermions only, while the bottom two assume that decaysinto exotics and sparticles are unsuppressed. The different shapes of the Tevatron and LHC curves is because the u quarkdominates at the Tevatron, while the u and d are more comparable at the LHC. From (Kang and Langacker, 2005). by some convention.) This should be possible for massesup to ∼ − . pp → Z ′ → ℓ + ℓ − ( ℓ = e, µ ), one would be able to measure themass M Z ′ , the width Γ Z ′ and the leptonic cross section σ ℓZ ′ = σ Z ′ B ℓ . By itself, σ ℓZ ′ is not a useful diagnos-tic for the Z ′ couplings to quarks and leptons: while σ Z ′ can be calculated to within a few percent for given Z ′ couplings, the branching ratio into leptons, B ℓ , de-pends strongly on the contribution of exotics and spar-ticles to Γ Z ′ (Kang and Langacker, 2005). However, σ ℓZ ′ would be a useful indirect probe for the existence ofthe exotics or superpartners. Furthermore, the product σ ℓZ ′ Γ Z ′ = σ Z ′ Γ ℓ does probe the absolute magnitude ofthe quark and lepton couplings.The most useful diagnostics involve the relativestrengths of Z ′ couplings to ordinary quarks and leptons.The forward-backward asymmetry as a function of the Z ′ rapidity, A fF B ( y ) (Langacker et al. , 1984), avoids the ¯ qq ambiguity in Eq. 72. For AB → Z ′ → ¯ f f , define θ CM as the angle of fermion f with respect to the directionof hadron A in the Z ′ rest frame, and let F ( B ) be thecross section for fixed rapidity y with cos θ CM > < A fF B ( y ) ≡ ( F − B ) / ( F + B ), with F ± B ∼ " / i (cid:0) f Aq i ( x ) f B ¯ q i ( x ) ± f A ¯ q i ( x ) f Bq i ( x ) (cid:1) × (cid:0) ǫ L ( q i ) ± ǫ R ( q i ) (cid:1) (cid:0) ǫ L ( f ) ± ǫ R ( f ) (cid:1) . (73) Clearly, A fF B ( y ) vanishes for pp at y = 0, but can benonzero at large y where there is more likely a valence q from the first proton and sea ¯ q from the other. Theleptonic forward-backward asymmetry is sensitive to acombination of quark and lepton chiral couplings and is apowerful discriminant between models (Langacker et al. ,1984).There are a number of additional probes. The ratio ofcross sections in different rapidity bins (del Aguila et al. ,1993) gives information on the relative u and d cou-plings. Possible observables in other two-fermion fi-nal state channels include the polarization of produced τ ’s (Anderson et al. , 1992) and the pp → Z ′ → jj cross section (Rizzo, 1993a; Weiglein et al. , 2006). Thereare no current plans for polarization at the LHC,but polarization asymmetries at a future or upgradedhadron collider would provide another useful diagnos-tic (Fiandrino and Taxil, 1992).In four-fermion final state channels the rare decays Z ′ → V f ¯ f , where V = W or Z is radiated from the Z ′ decay products, have a double logarithmic enhance-ment. In particular, Z ′ → W ℓν ℓ (with W → hadronsand an ℓν ℓ transverse mass >
90 GeV to separate fromSM background) may be observable and projects outthe left-chiral lepton couplings (Cvetic and Langacker,1992a; Hewett and Rizzo, 1993; Rizzo, 1987). Sim-ilarly, the associated productions pp → Z ′ V with V = ( Z, W ) (Cvetic and Langacker, 1992b) and V = γ (Rizzo, 1993b) could yield information on the quarkchiral couplings.Finally, decays into two bosons, such as Z ′ → W + W − , Zh, or W ± H ∓ , can occur only by Z − Z ′ mixing or with amplitudes related to the mix-ing. However, this suppression may be compen-sated for the longitudinal modes of the W or Z by the large polarization vectors, with compo-nents scaling as M Z ′ /M W (del Aguila et al. , 1987b;Barger and Whisnant, 1987; Deshpande and Trampetic,1988). For example, Γ( Z ′ → W + W − ) ∼ θ , which ap-pears to be hopelessly small to observe. However, the en-hancement factor is ∼ ( M Z ′ /M W ) . Thus, from Eq. 26,these factors compensate, leaving a possibly observablerate that in principle could give information on the Higgscharges. In the limit of M Z ′ ≫ M Z one hasΓ( Z ′ → W + W − ) = g θ M Z ′ π (cid:18) M Z ′ M Z (cid:19) = g C M Z ′ π . (74)Global studies of the possible LHC diagnostic pos-sibilities for determining ratios of chiral charges in amodel independent way and discriminating models aregiven in (del Aguila et al. , 1993; Cvetic and Godfrey,1995). The complementarity of LHC and ILC observa-tions is especially emphasized in (del Aguila and Cvetic,1994; del Aguila et al. , 1995a; Cvetic and Godfrey, 1995;Weiglein et al. , 2006). V. IMPLICATIONSA. The µ Problem and Extended Higgs/Neutralino Sectors
1. The µ Problem
As described in Section II.E.3, the µ problem of theMSSM can be solved in singlet extended models in whicha symmetry forbids an elementary µ term, but allowsa dynamical µ eff = λ S h S i . There are a number ofrealizations of this mechanism (see (Accomando et al. ,2006; Barger et al. , 2007c) for reviews). The best knownis the next to minimal model (NMSSM), in which adiscrete Z symmetry forbids µ but allows the cu-bic terms λ S SH u H d and κS / et al. , 1989). The original form of the NMSSMsuffers from cosmological domain wall problems be-cause of the discrete symmetry. This can be reme-died in more sophisticated forms involving an R sym-metry (Accomando et al. , 2006). A variation on thatapproach yields the new minimal model (nMSSM), inwhich the cubic term and its soft analog are replacedby tadpole terms linear in S with sufficiently small co-efficients (Panagiotakopoulos and Tamvakis, 1999). A U (1) ′ symmetry, which is perhaps more likely toemerge from a string construction, is another possi-bility (Cvetic et al. , 1997; Cvetic and Langacker, 1996a;Suematsu and Yamagishi, 1995). This avoids the domainwall problem by embedding the discrete symmetry of theNMSSM into a continuous one.
2. Extended Higgs Sector
Conventional U (1) ′ models necessarily involve ex-tended Higgs sectors associated with the SM singlet fieldswhose VEVs break the U (1) ′ symmetry. Especially in-teresting in this respect are those supersymmetric mod-els involving a dynamical µ eff = λ S h S i . If one ignoresHiggs sector CP violation , then there will be an ad-ditional Higgs scalar associated with S , that can mixwith the two MSSM scalars from H u,d . (There is alsoan additional pseudoscalar in the models involving a dis-crete symmetry.) Since the S does not couple directlyto the SM fermions or gauge bosons, the LEP lower lim-its on the Higgs mass ( m H > . et al. , 2006). Conversely, the theoretical upper limit on the lightest Higgs is also relaxed, from ∼
130 GeV in the MSSM to around 170 GeV in the sim-plest U (1) ′ model, due to the new F and D term contri-butions to the potential in Eq. 55. (One must include theloop corrections to these estimates (Barger et al. , 2006).)These relaxed limits allow lower values for tan β ≡ ν u /ν d in the U (1) ′ models than are favored for the MSSM.The UMSSM is the U (1) ′ with a single S , with the po-tential in Eq. 55. In the decoupling limit, h S i → ∞ with µ eff fixed, the UMSSM reduces to the MSSM. Existingconstraints favor this limit (unless the Z ′ is leptophobic,with small Z − Z ′ mixing due to a cancellation of the twoterms in ∆ in Eq. 20). For large h S i the extra Higgsis heavy and mainly singlet (Barger et al. , 2006), so theHiggs sector is MSSM-like. However, more general U (1) ′ models such as the secluded model in Section III.E.3, aswell as other models such as the nMSSM, can yield sig-nificant doublet-singlet mixing, light singlet-dominatedstates, etc (Erler et al. , 2002; Han et al. , 2004a). (In fact,the secluded model reduces to the nMSSM in an appro-priate limit (Barger et al. , 2006).) This may yield suchnonstandard collider signatures as light weakly coupledHiggs, multiple Higgs with reduced couplings, and invis-ible decays into light neutralinos (Barger et al. , 2007a).
3. Extended Neutralino Sector
The neutralino sector of the MSSM (the bino, ˜ B , andthe wino, ˜ W , with soft masses M ˜ B, ˜ W ; and two neutralHiggsinos ˜ H u,d ) is extended in U (1) ′ models by one ormore singlinos, ˜ S , and by the Z ′ -gaugino, ˜ Z ′ , with softmass M ˜ Z ′ (Barger et al. , 2005, 2007b; Choi et al. , 2007;Hesselbach et al. , 2002; Suematsu, 1998). (There couldalso be soft mass or kinetic ˜ B − ˜ Z ′ mixing terms.) Loop effects may generate significant CP effects, especially forthe heavier Higgs states (Demir and Everett, 2004). B, ˜ W , ˜ H d , ˜ H u , ˜ S, ˜ Z ′ ) basis, the mass matrix for the six neutralinos in the UMSSM is M χ = M ˜ B − g ′ ν d / g ′ ν u / M ˜ W gν d / − gν u / − g ′ ν d / gν d / − µ eff − µ eff ν u /s g Q d ν d g ′ ν u / − gν u / − µ eff − µ eff ν d /s g Q u ν u − µ eff ν u /s − µ eff ν d /s g Q S s g Q d ν d g Q u ν u g Q S s M ˜ Z ′ . (75)In the decoupling limit with g Q S s ≫ M ˜ Z ′ thesinglino and the Z ′ -gaugino will combine to forman approximately Dirac fermion with mass g Q S s ∼ M Z ′ and little mixing with the four MSSM neu-tralinos. For large M ˜ Z ′ ≫ g Q S s , on the otherhand, there will be a heavy Majorana ˜ Z ′ , and amuch lighter singlino with a seesaw type mass ∼− M Z ′ /M ˜ Z ′ . For smaller s there can be significant mix-ing with the MSSM neutralinos. One can easily ex-tend to secluded models (Erler et al. , 2002; Han et al. ,2004a), models with multiple U (1) ′ s (Hesselbach et al. ,2002), or singlet extended models with discrete sym-metries (Accomando et al. , 2006; Barger et al. , 2005,2007c). In many of these cases there are light singlino-dominated states (which can be the lightest supersym-metric particle (LSP)) and/or significant mixing effects.These can lead to a variety of collider signatures verydifferent from the MSSM (Barger et al. , 2007b). For ex-ample, in some cases there are four MSSM-like neutrali-nos with production and cascades similar to the MSSM.However, the lightest of these may then undergo an ad-ditional decay to a singlino LSP, accompanied e.g., byan on-shell Z or Higgs. Enhanced rates for the decayof chargino-neutralino pairs to three or more leptons arepossible. It is also possible for the lightest Higgs to de-cay invisibly to two light singlinos. Cold dark matterimplications are described in Section V.G. B. Exotics
Almost all U (1) ′ models require the addition of newchiral exotic states to cancel anomalies (Section II.C).Precision electroweak constraints favor that these are quasi-chiral , i.e., vector pairs under the SM but chiral un-der U (1) ′ . Examples are the SU (2)-singlet D, D c quarkswith charge − / E model (Table II); the SU (2)doublet pairs in E which may be interpreted either asadditional Higgs pairs H u.d or as exotic lepton doublets;or SM singlets. Realistic models must provide means ofgenerating masses for such exotics, e.g., by coupling tochiral (or nonchiral) singlets which acquire VEVs, suchas SDD c , and also for their decays. Consider the example of the exotic D quarks, whichcan be pair-produced by QCD processes at a hadroncollider, and their scalar supersymmetric partners e D ,produced with an order of magnitude smaller crosssection. (The rates are smaller for exotic leptons.)Once produced, there are three major decay possibili-ties (Kang et al. , 2008): • The decay may be D → u i W − , D → d i Z ,or D → d i H , if driven by mixing with alight charge − / et al. , 1986). The current limit is m D &
200 GeV (Andre and Rosner, 2004), which shouldbe improved to ∼ E model if R -parity is conserved. • One may have e D → jj if there is a small di-quark operator such as u c d c D c , or e D → jℓ fora leptoquark operator like LQD c . (They cannotboth be present because of proton decay.) Suchoperators do not by themselves violate R parity( R = +1 for the scalar), and therefore allow astable lightest supersymmetric particle. They arestrongly constrained by the K L − K S mass differ-ence and by µ − e conversion, but may still be sig-nificant (Kang et al. , 2008). If the scalar e D is heav-ier than the fermion, then it may decay resonantlyinto the fermion pair, or into a D and neutralino(or gluino). The lighter fermion D can decay intoa neutralino and nonresonant fermion pair via avirtual e D or via a real or virtual squark or slep-ton. A heavier fermion will usually decay into anon-shell e D and a neutralino (or gluino), with the e D decaying to fermions. The signals from these de-cays, especially for a heavier scalar, may be difficultto extract from normal SUSY cascades, especiallyfor diquarks. However, there are some possibilitiesbased on missing transverse energy, lepton multi-plicities and p T , etc (Kang et al. , 2008). • They may be stable at the renormalizable level dueto the U (1) ′ , or to an accidental or other sym-metry, so that they hadronize and escape from or4stop in the detector (Kang et al. , 2008), with signa-tures (Kraan et al. , 2007) somewhat similar to thequasi-stable gluino expected in split supersymme-try (Arkani-Hamed and Dimopoulos, 2005). Theycould then decay by higher-dimensional operatorson a time scale of . − −
100 s, short enoughto avoid cosmological problems (Kawasaki et al. ,2005). These operators could allow direct decaysto SM particles, or they could involve SM singletswith VEVs which could induce tiny mixings withordinary quarks.Exotics carrying SM charges significantly modify therunning of the SM couplings, and therefore can affectgauge unification unless, e.g., they occur in SU (5)-typemultiplets. Examples of U (1) ′ constructions which pre-serve the MSSM running at tree level are described inSections III.A.3 and III.B. C. The Z ′ as a Factory The decays of a Z ′ could serve as an efficient source ofother particles if it is sufficiently massive. This has beenexplored in detail for slepton production, pp → Z ′ → e ℓ e ℓ ∗ ,with e ℓ → ℓ + LSP, assuming that M Z ′ is already knownfrom the conventional ℓ + ℓ − channel (Baumgart et al. ,2007). This can greatly extend the discovery reach of the e ℓ and may give information on the identity of the LSP.Decays of the Z ′ could also be a useful production mecha-nism for pairs of exotics (Rosner, 1996b) or heavy Majo-rana neutrinos (del Aguila and Aguilar-Saavedra, 2007;Duncan and Langacker, 1986). The latter could lead tothe interesting signature of like sign leptons + jets. Thetotal width Γ Z ′ , in combination with other constraints onthe quark and lepton charges, would also give some infor-mation on the exotic/sparticle decays (Gherghetta et al. ,1998; Kang and Langacker, 2005). D. Flavor Changing Neutral Currents
In Section II it was implicitly assumed that the U (1) ′ charges were family universal. That implies that the Z ′ couplings are unaffected by fermion mixings and re-main diagonal (the GIM mechanism). However, manymodels involve nonuniversal charges, as described in Sec-tion III.D.3. Let us rewrite the U (1) ′ current in Eq. 10as J µα = ¯ f L γ µ ǫ αfL f L + ¯ f R γ µ ǫ αfR f R , (76)where f L is a column vector of weak-eigenstate left chi-ral fermions of a given type (i.e., u L , d L , e L , or ν L ), andsimilarly for f R . The ǫ αf are diagonal matrices of U (1) ′ charges. The f L,R are related to the mass eigenstates f L,R by f L = V f † L f L , f R = V f † R f R , (77) where V fL,R are unitary. In particular, the CKM andPMNS matrices are given by V uL V d † L and V νL V e † L , respec-tively. In the mass basis, J µα = ¯ f L γ µ B αfL f L + ¯ f R γ µ B αfR f R , (78)where B αfL ≡ V fL ǫ αfL V f † L , B αfR ≡ V fR ǫ αfR V f † R . (79)For family universal charges, ǫ fL,R are proportional tothe identity, and B fL,R = ǫ fL,R . However, for thenonuniversal case, B fL,R will in general be nondiagonal.As a simple two family example, if ǫ = ! and V isa rotation of the same form as Eq. 22 then J µ = sin θ ¯ f γ µ f + cos θ ¯ f γ µ f + sin θ cos θ ( ¯ f γ µ f + ¯ f γ µ f ) . (80)The formalism for FCNC mediated by Z ′ , andalso by off-diagonal Z couplings induced by Z − Z ′ mixing, was developed in (Langacker and Plumacher,2000), and limits were obtained for a number of treeand loop level mixings and decays. The limits from K − ¯ K mixing (including CP violating effects)and from µ − e conversion in muonic atoms are suf-ficiently strong to exclude significant nonuniversaleffects for the first two families for a TeV-scale Z ′ with electroweak couplings. However, nonuniversalcouplings for the third family are still possible andcould contribute (Baek et al. , 2006; Barger et al. ,2004a,b,c; Chen and Hatanaka, 2006; Cheung et al. ,2007; Chiang et al. , 2006; He and Valencia, 2006;Langacker and Plumacher, 2000; Leroux and London,2002) to processes such as B ¯ B and D ¯ D mixing, B → µ + µ − , or b → ss ¯ s (such as in B → φK ). Since the Z ′ effects are at tree level, they may be important evenfor small couplings since they are competing with SMor MSSM loop effects. The possible anomaly observedin the Z → ¯ bb forward-backward asymmetry (Yao et al. ,2006) could possibly be a (flavor diagonal) result of anonuniversal Z ′ coupling (Erler and Langacker, 2000).Collider processes such as single top production couldpossibly be observable as well (Arhrib et al. , 2006).The nonuniversal couplings could also be relevant toloop effects, such as b → sγ or µ → eγ , or intrinsic mag-netic or electric dipole moments. One interpretation ofthe possible anomaly (Yao et al. , 2006) suggested by theBNL experiment for the anomalous magnetic moment ofthe µ involves the vertex diagram with a Z ′ exchange(see, e.g., (Cheung et al. , 2007)). The flavor-diagonal di-agram with an internal µ is too small to be relevant (un-less M Z ′ ∼
100 GeV or the couplings are large). How-ever, an internal τ enhances the effect by m τ /m µ , andthe anomaly could be accounted for by a TeV scale Z ′ with large µ − τ mixing.Mixing between the ordinary and exotic fermions canalso lead to FCNC effects (Langacker and London, 1988).5For example, a small d c − D c mixing in the E modelof Table II would induce off diagonal couplings of the Z to the light and heavy mass eigenstate, while a d − D mixing (i.e., between an SU (2) doublet and singlet)would generate similar effects for the ordinary Z . Off-diagonal vertices between the light mass eigenstates, suchas Z α ¯ bs , would be induced as second order effects. E. Supersymmetry Breaking, Z ′ Mediation, and theHidden Sector U (1) ′ s have many possible implications for supersym-metry breaking and mediation, and for communicationwith a hidden sector. For example, one limit of the sin-gle S scenario of Section III.E.2 requires large (TeV scale)soft masses in the Higgs sector, suggesting the possibilityof heavy sparticles as well (Everett et al. , 2000).Another implication is the U (1) ′ D term contributionto the scalar potential (Kolda and Martin, 1996), V D = 12 D ≡ − g X i Q i | φ i | ! . (81) V D of course contributes to the minimization conditionsand Higgs sector masses. Assuming a value D min = 0 for D at the minimum, it gives a contribution to the masses m i of the squarks, sleptons, and exotic scalars∆ m i = ( − D min )( g Q i ) . (82)For a single S field, one has − D min = g ( Q u | ν u | + Q d | ν d | + Q S | s | ) / m i can be of either sign, and must be added toother supersymmetric and soft contributions. Whenthe U (1) ′ scale is large, there is a danger of over-all negative mass-squares which de-stabilize the vac-uum. However, in that case there is the possibilityof breaking along a D flat direction in which V minD issmall, as in the secluded models (Erler et al. , 2002).Positive D term contributions to the slepton masseshave been suggested as a means of compensating thenegative ones from anomaly mediated supersymme-try breaking (Anoka et al. , 2004; Murakami and Wells,2003). The D term quartic interactions also con-tribute to the RGE equations for the soft masses (see,e.g., (Cvetic et al. , 1997; Langacker and Wang, 1998)). U (1) ′ s have been invoked in many models of supersym-metry breaking or mediation. For example, many mod-els of gauge mediation involve a U (1) ′ which may helptransmit the breaking by loop effects and/or D terms inthe hidden or ordinary sectors (Cheng et al. , 1998, 1999;Dobrescu, 1997; Kaplan et al. , 1999; Langacker et al. ,1999). The Fayet-Iliopoulos terms (Section III.F.3) as-sociated with anomalous U (1) ′ s in string constructionsmay also help trigger and transmit supersymmetry break-ing (Dvali and Pomarol, 1998; Mohapatra and Riotto,1997). In many string constructions particles in both the or-dinary and hidden sector may carry U (1) ′ charges, allow-ing for the possibility of Z ′ mediation (Langacker et al. ,2008). The simplest case is that the U (1) ′ gauge symme-try is not broken in the hidden sector, but the Z ′ gauginoacquires a mass from the SUSY breaking. The Z ′ − ˜ Z ′ mass difference induces ordinary sector scalar masses atone loop, and SM gaugino masses at two loops. Requir-ing the latter to be in the range 10 − GeV implies M ˜ Z ′ & TeV (for electroweak couplings), with thesparticles, exotics, and Z ′ around 10 −
100 TeV and theelectroweak scale obtained by a fine-tuning, i.e., a versionof split supersymmetry (Arkani-Hamed and Dimopoulos,2005). String embeddings of this scenario are addressedin (Verlinde et al. , 2008). It can also be combinedwith other mediation scenarios, allowing a lower Z ′ scale (Nakayama, 2008). A Z ′ communicating with ahidden sector could also allow the production and decaysinto SM particles of relatively light hidden valley parti-cles (Han et al. , 2008). F. Neutrino Mass
The seesaw model (see, e.g., (Mohapatra et al. , 2007))leads to a small Majorana mass m ν ∼ − m D /M ν c forthe ordinary doublet neutrinos ν , where m D is a Diracmass (generated by the VEV of a Higgs doublet), and M ν c ≫ m D is the Majorana mass of the heavy singlet ν c , − L ν = m D ¯ ν L ν R + 12 M ν c ¯ ν cL ν R + h.c., (83)where ν R is the conjugate of ν cL . For m D ∼
100 GeVand M ν c ∼ GeV one obtains | m ν | in the observed0.1 eV range. However, if the ν c is charged under a U (1) ′ then M ν c cannot be much larger than the U (1) ′ scale. One possibility for a TeV scale Z ′ is that the ν c is neutral, as in the N model (Barger et al. , 2003;Kang et al. , 2005b; King et al. , 2006; Ma, 1996). Then, aconventional seesaw (Kang et al. , 2005b; Keith and Ma,1996; King et al. , 2006; Ma, 1996) and leptogene-sis (Hambye et al. , 2001) scenario can be possible if alarge M ν c can be generated. For other models withTeV scale M Z ′ one must invoke an alternative to theseesaw. For example, small Majorana masses can begenerated using the double seesaw mechanism (involv-ing an additional power of M − ν c ), or by invoking a Higgstriplet (Kang et al. , 2005b; Mohapatra et al. , 2007).Another possibility, which can lead to eithersmall Dirac or Majorana masses, involves higher-dimensional operators (HDO) (Arkani-Hamed et al. ,2001b; Borzumati and Nomura, 2001; Chen et al. ,2007; Cleaver et al. , 1998; Demir et al. , 2008;Gogoladze and Perez-Lorenzana, 2002; Kang et al. ,2005b; Langacker, 1998). For example, a superpo-tential operator W = SLH u ν c /M could generate asmall Dirac mass in the correct range for h S i ∼ M ∼ GeV. Such a VEV can easilyoccur in intermediate scale models (Cleaver et al. ,1998; Langacker, 1998) or in the Z ′ mediation sce-nario (Langacker et al. , 2008). Higher powers couldoccur for a larger h S i or smaller M , associated, e.g., withan anomalous U (1) ′ (Gogoladze and Perez-Lorenzana,2002). Non-holomorphic (wrong Higgs) terms (seeSection II.E.2) can also lead to naturally small Diracmasses, suppressed by the ratio of the SUSY breakingand mediation scales (Demir et al. , 2008). In somecases, a Z ′ -gaugino is needed to generate a fermionmass at loop level from a non-holomorphic soft term.In all of these mechanisms, some low energy symmetrysuch as a U (1) ′ must forbid a renormalizable level Diracmass term W = LH u ν c , while allowing the HDO. (Therenormalizable level term is allowed in the E models.)Discrete gauge symmetries (i.e., remnants of a gaugesymmetry broken at a high scale), may also help restrictthe allowed operators (Luhn and Thormeier, 2008).Some mechanisms (Langacker, 1998; Ma, 1996) alsoallow the generation of light sterile neutrino masses andordinary-sterile mixing, as suggested by the LSND ex-periment.Recently, it was shown (Nelson and Walsh, 2008) thatthe LSND and MiniBooNE results could be reconciled ina model involving ordinary and sterile neutrinos if thereis a very light ( ∼
10 keV) Z ′ coupled to B − L witha very weak coupling ( . − ). In analogy with theMSW effect (Mohapatra et al. , 2007) the Z ′ generates apotential in matter that is different for the ordinary andsterile neutrinos and strongly energy dependent.The right handed components of light Dirac neutri-nos could upset the successful predictions of big bangnucleosynthesis if they were present in sufficient num-bers. Mass and Yukawa coupling effects are too smallto be dangerous. However, couplings of the ν c to aTeV-scale Z ′ could have kept them in equilibrium un-til relatively late (Olive et al. , 1981). A detailed es-timate (Barger et al. , 2003) found that too much He would have been produced for light Dirac neutrinos formost of the E models unless M Z ′ & − N model. This is especially relevant for aparameter range of the generalized E model (with two U (1) ′ s) in which the Z N is much lighter than the or-thogonal boson, but nevertheless no Majorana massesare allowed (Kang et al. , 2005b). G. Cosmology
1. Cold Dark Matter U (1) ′ models (Barger et al. , 2004d, 2007d;Belanger et al. , 2008; de Carlos and Espinosa,1997; Hur et al. , 2008; Lee et al. , 2007;Nakamura and Suematsu, 2007; Pospelov et al. , 2008), as well as other singlet extended models with a dy-namical µ term (Accomando et al. , 2006; Barger et al. ,2007c,d; Menon et al. , 2004), have many implicationsfor cold dark matter (CDM). For example, the ex-tended neutralino sector in Eq. 75 allows the possibilityof a light singlino as the LSP (Barger et al. , 2004d,2007d; de Carlos and Espinosa, 1997; Menon et al. ,2004; Nakamura and Suematsu, 2007), with efficientannihilation into a light Z ′ or into the Z (via smalladmixtures with the Higgsinos). More generally, theLSP may contain admixtures of ˜ S or ˜ Z ′ with the MSSMneutralinos, or allow a modified MSSM composition forthe LSP. The models also have enlarged Higgs sectorsand different allowed ranges, extending the possiblemechanisms for Higgs-mediated LSP annihilation. Mostof the interesting cases should be observable in directdetection experiments (Barger et al. , 2007d).There are other LSP candidates in U (1) ′ models. Forexample, the scalar partners ˜ ν c of the singlet neutrinosbecome viable thermal CDM candidates due to the pos-sibility of annihilation through the Z ′ (Lee et al. , 2007).Other possibilities include a neutral exotic particle ormultiple stable particles (Hur et al. , 2008), a heavy Diracneutrino (Belanger et al. , 2008), or a semi-secluded weaksector (Pospelov et al. , 2008) coupled via a Z ′ .
2. Electroweak Baryogenesis
The seesaw model of neutrino mass allows the possibil-ity of explaining the observed baryon asymmetry by lep-togenesis, i.e., the decays of the heavy Majorana neutrinogenerate a small lepton asymmetry, which is partiallyconverted to a baryon asymmetry by the electroweaksphaleron process (Mohapatra et al. , 2007). As discussedin Section V.F, however, an additional U (1) ′ symmetryoften forbids the seesaw model. Some of the alternativesdiscussed there allow other forms of leptogenesis (Chun,2005).However, the U (1) ′ (Ham and OH, 2007; Ham et al. ,2007; Kang et al. , 2005a) and other singlet ex-tended models (Barger et al. , 2007c; Menon et al. , 2004;Profumo et al. , 2007) open up the possibility of a com-pletely different mechanism, electroweak baryogenesis . Inthis scenario, the interactions of particles with the ex-panding bubble wall from a strongly first order elec-troweak phase transition lead to a CP asymmetry, whichis then converted to a baryon asymmetry by sphaleronprocesses. However, the SM does not have a strong firstorder transition or sufficient CP violation; the MSSMhas only a small parameter range involving a light stopfor the transition, and there is tension between the CPviolation needed and electric dipole moment (EDM) con-straints (Carena et al. , 2003b). In the extended models,however, there is a tree-level cubic scalar interaction (the λ S A S SH u H d term in Eq. 55) which can easily lead to theneeded strong first order transition. There are also pos-sible new sources of tree-level CP violation in the Higgs7sector (Ham and OH, 2007; Kang et al. , 2005a), whichcan contribute to the baryon asymmetry but have negli-gible effect on EDMs.
3. Cosmic Strings
A broken global or gauge U (1) ′ can lead to cosmicstrings , which are allowed cosmologically for a wide rangeof parameters and which could have interesting implica-tions for gravitational waves, dark matter, particle emis-sion, and gravitational lensing. For a recent discussion,with emphasis on breaking a supersymmetric U (1) ′ alongan almost flat direction, see (Cui et al. , 2008). VI. CONCLUSIONS AND OUTLOOK
A new U (1) ′ gauge symmetry is one of the best mo-tivated extensions of the standard model. For example, U (1) ′ s occur frequently in superstring constructions. Ifthere is supersymmetry at the TeV scale, then both theelectroweak and Z ′ scales are usually set by the scaleof soft supersymmetry, so it is natural to expect M Z ′ in the TeV range. (One exception is when the U (1) ′ breaking occurs along an approximately flat direction,in which case a large breaking scale could be associ-ated with fermion mass hierarchies generated by higher-dimensional operators.) Similarly, TeV-scale U (1) ′ s (orKaluza-Klein excitations of the photon and Z ) frequentlyoccur in models of dynamical symmetry breaking, LittleHiggs models, and models with TeV − -scale extra dimen-sions. Other constructions, such as non-supersymmetricgrand unified theories larger than SU (5), also lead to ex-tra U (1) ′ s, but in these cases there is no particular reasonto expect breaking at the TeV scale (and breaking belowthe GUT may lead to rapid proton decay).The observation of a Z ′ would have consequences farbeyond just the existence of a new gauge boson. Anomalycancellation would imply the existence of new fermions.These could just be right-handed neutrinos, but usu-ally there are additional particles with exotic electroweakquantum numbers. There must also be at least one newSM singlet scalar whose VEV breaks the U (1) ′ symme-try. This scalar could mix with the Higgs doublet(s)and significantly alter the collider phenomenology. The Z ′ couplings could be family nonuniversal, allowing newtree-level contributions, e.g., to t , b , and τ decays.In the supersymmetric case the U (1) ′ could solve the µ problem by replacing µ by a dynamical variable linkedto the U (1) ′ breaking, and the allowed MSSM param-eter range would be extended. The singlets and exoticswould be parts of chiral supermultiplets, and there wouldbe extended neutralino sectors associated with the newsinglino and gaugino, modifying the collider physics andcold dark matter possibilities. Gauge unification couldbe maintained if the exotics fell into SU (5)-type mul-tiplets. The U (1) ′ symmetry would also constrain the possibilities for neutrino mass and might be related toproton stability and R -parity conservation. A Z ′ mightalso couple to a hidden sector and could play a role insupersymmetry breaking or mediation. Finally, a dynam-ical µ would allow a strong first order electroweak phasetransition and new sources of CP violation in the Higgssector, making electroweak baryogenesis more likely thanin the SM or the MSSM, with the ingredients observablein the laboratory.There are large classes of Z ′ models, distinguished bythe chiral charges of the quarks, leptons, and Higgs fields,as well as the Higgs and exotic spectrum, gauge coupling, Z ′ mass, and possible mass and kinetic mixing. In stringconstructions, for example, U (1) ′ s that do not descendthrough SO (10) or left-right symmetry can have seem-ingly random charges. There is no simple classificationor parametrization that takes into account all of the pos-sibilities. One (model independent) approach, valid forfamily universality, is to take a conventional value forthe new gauge coupling, and regard the charges of theleft-handed quarks ( Q L ), leptons ( L L ), and antiparticles u cL , d cL , and e + L , as well as M Z ′ , Γ Z ′ and the mixing angle θ as free parameters relevant to experimental searches.However, 8 parameters are too many for most purposes,so one must resort to specific models or lower-dimensionalparametrizations to illustrate the possibilities. A recom-mended set are those summarized in Tables I, II, andIII.Table I lists the U (1) R × U (1) BL model, which is aone-parameter (not counting the Z ′ mass and mixing)set of models based on various forms of SO (10) and left-right symmetry, and a two parameter generalization mo-tivated by more general embeddings or by kinetic mixing.It requires no exotics other than ν cL . However, the super-symmetric version requires non-chiral Higgs doublets and(probably) vector pairs of SM singlets, and does not solvethe µ problem.Table II lists popular E -motivated models. A wholeclass of interesting models involves one free parameter, θ E , or a two parameter generalization with kinetic mix-ing (or a third parameter if the gauge coupling is varied).These models illustrate typical exotics, and (with the ex-ception of the χ model) the supersymmetric version in-volves a dynamical µ term. However, supersymmetricgauge unification requires an additional vector pair ofHiggs-like doublets.The models in Table III are examples of supersymmet-ric models with a dynamical µ that are consistent withgauge unification without additional vector pairs. Threeparameters, including the gauge coupling, are relevant tothe non-exotic sector.If there is a Z ′ with typical electroweak scale couplingsto the ordinary fermions, it should be readily observableat the LHC for masses up to ∼ − ∼ −
900 GeV. Significantdiagnostic probes of the Z ′ couplings would be possibleup to ∼ − . Z ′ could completely alter the paradigm ofhaving just the MSSM at the TeV scale, with a desert upto a scale of grand unification or heavy Majorana neu-trino masses, and would suggest a whole range of newlaboratory and cosmological consequences. In the non-supersymmetric case, a Z ′ might be one of the first ex-perimental manifestations of a new TeV scale sector ofphysics. Acknowledgments
I am extremely grateful to Vernon Barger, MirjamCvetiˇc, Jens Erler, and all of my other collaboratorson work related to this article, and Hye-Sung Lee fora careful reading of the manuscript. This work was sup-ported by the Friends of the IAS and by NSF grant PHT-0503584.
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