Anomaly driven signatures of new invisible physics at the Large Hadron Collider
Ignatios Antoniadis, Alexey Boyarsky, Sam Espahbodi, Oleg Ruchayskiy, James D. Wells
aa r X i v : . [ h e p - ph ] J a n CERN-PH-TH/2009-006
Anomaly driven signatures of new invisible physicsat the Large Hadron Collider
Ignatios Antoniadis a,b , Alexey Boyarsky c,d , Sam Espahbodi a,e ,Oleg Ruchayskiy f , James D. Wells a,e ( a ) CERN PH-TH, CH-1211 Geneva 23, Switzerland ( b ) CPHT, UMR du CNRS 7644, ´Ecole Polytechnique, 91128 Palaiseau Cedex, France ( c ) ETHZ, Z¨urich, CH-8093, Switzerland ( d ) Bogolyubov Institute for Theoretical Physics, Kiev 03680, Ukraine ( e ) MCTP, University of Michigan, Ann Arbor, MI 48109, USA ( f ) ´Ecole Polytechnique F´ed´erale de Lausanne, FSB/ITP/LPPC, BSP, CH-1015,Lausanne, Switzerland Abstract
Many extensions of the Standard Model (SM) predict new neutral vectorbosons at energies accessible by the Large Hadron Collider (LHC). We studyan extension of the SM with new chiral fermions subject to non-trivial anomalycancellations. If the new fermions have SM charges, but are too heavy to becreated at LHC, and the SM fermions are not charged under the extra gaugefield, one would expect that this new sector remains completely invisible atLHC. We show, however, that a non-trivial anomaly cancellation between thenew heavy fermions may give rise to observable effects in the gauge bosonsector that can be seen at the LHC and distinguished from backgrounds. ontents SU (2) × U Y (1) × U X (1) Model 115 Phenomenology 12 X boson . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Decays of X boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Collider Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 It is well known that theories in which fermions have chiral couplings with gaugefields suffer from anomalies – a phenomenon of breaking of gauge symmetries ofthe classical theory at one-loop level. Anomalies make a theory inconsistent (inparticular, its unitarity is lost). The only way to restore consistency of such a theoryis to arrange the exact cancellation of anomalies between various chiral sectors ofthe theory. This happens, for example, in the Standard Model (SM), where thecancellation occurs between quarks and leptons within each generation [1, 2, 3].Another well-studied example is the Green-Schwarz anomaly cancellation mecha-nism [4] in string theory. In this case the cancellation happens between the anomalouscontribution of chiral matter of the closed string sector with that of the open string. Particles involved in anomaly cancellation may have very different masses – forexample, the mass of the top quark in the SM is much higher than the massesof all other fermions. On the other hand, gauge invariance should pertain in thetheory at all energies, including those which are smaller than the mass of one orseveral particles involved in anomaly cancellation. The usual logic of renormalizabletheories tells us that the interactions, mediated by heavy fermions running in loops,are generally suppressed by the masses of these fermions [6]. The case of anomalycancellation presents a notable counterexample to this famous “decoupling theorem”– the contribution of a priori arbitrary heavy particles should remain unsuppressedat arbitrarily low energies. As was pointed out by D’Hoker and Farhi [7, 8], this is Formally, the Green-Schwarz anomaly cancellation occurs due to the anomalous Bianchi identityfor the field strength of a 2-form closed string. However, this modification of Bianchi identity arisesfrom the 1-loop contribution of chiral fermions in the open string sector. A toy model, describingmicroscopically Green-Schwarz mechanism was studied e.g. in [5]. U (1) subgroup is not anomalousby definition. However, the mixed triangular hypercharge U Y (1) × SU (2) anomaliesand gravitational anomalies are non-zero for a generic choice of hypercharges. If onetakes the most general choice of hypercharges, consistent with the structure of theYukawa terms, one sees that it is parametrized by two independent quantum numbers Q e (shift of hypercharge of left-handed lepton doublet from its SM value) and Q q (corresponding shift of quark doublet hypercharge). All the anomalies are then pro-portional to one particular linear combination: ǫ = Q e + 3 Q q . Interestingly enough, ǫ is equal to the sum of electric charges of the electron and proton. The experimentalupper bound on the parameter ǫ , coming from checks of electro-neutrality of matteris rather small: ǫ < − e [38, 39]. If it is non-zero, the anomaly of the SM has tobe cancelled by additional anomalous contributions from some physics beyond theSM, possibly giving rise to some non-trivial effects in the low energy effective theory.In the scenario described above the anomaly-induced effects are proportionalto a very small parameter, which makes experimental detection very difficult. Inthis paper we consider another situation, where anomalous charges and therefore,anomaly-induced effects, are of order one. To reconcile this with existing experi-mental bounds, such an anomaly cancellation should take place between the SM and“hidden” sector, with the corresponding new particles appearing at relatively highenergies. Namely, many extensions of the SM add extra gauge fields to the SM gaugegroup (see e.g. [42] and refs. therein). For example, additional U (1)s naturally appearin models in which SU (2) and SU (3) gauge factors of the SM arise as parts of uni-tary U (2) and U (3) groups (as e.g. in D-brane constructions of the SM [43, 44, 45]).In this paper, we consider extensions of the SM with an additional U X (1) factor,so that the gauge group becomes SU (3) c × SU (2) W × U Y (1) × U X (1). As the SMfermions are chiral with respect to the EW group SU (2) W × U Y (1), even choosing thecharges for the U X (1) group so that the triangular U X (1) anomaly vanishes, mixedanomalies may still arise: U X (1) U Y (1) , U X (1) U Y (1), U X (1) SU (2) . In this workwe are interested in the situation when only (some of these) mixed anomalies withthe electroweak group SU (2) × U Y (1) are non-zero. A number of works have alreadydiscussed such theories and their signatures (see e.g. [11, 12, 43, 46, 47, 48, 49]).The question of experimental signatures of such theories at the LHC should beaddressed differently, depending on whether or not the SM fermions are charged withrespect to the U X (1) group: • If SM fermions are charged with respect to the U X (1) group, and the mass ofthe new X boson is around the TeV scale, we should be able to see the corresponding3esonance in the forthcoming runs of LHC in e.g., q ¯ q → X → f ¯ f . The analysis of thisis rather standard Z ′ phenomenology, although in this case an important questionis to distinguish between theories with non-trivial cancellation of mixed anomalies,and those that are anomaly free. • On the other hand, one is presented with a completely different challenge if theSM fermions are not charged with respect to the U X (1) group. This makes impossiblethe usual direct production of the X boson via coupling to fermions. Therefore, thequestion of whether an anomalous gauge boson with mass M X ∼ U X (1) SU (2) anomaly occurs be-tween some heavy fermions and Green-Schwarz (i.e. tree-level gauge-variant ) termswas considered in [49]. The leading non-gauge invariant contributions from the tri-angular diagrams of heavy fermions, unsuppressed by the fermion masses, cancelsthe Green-Schwarz terms. The triangular diagrams also produce subleading (gauge-invariant) terms, suppressed by the mass of the fermions running in the loop. Thisleads to an appearance of dimension-6 operators in the effective action, having thegeneral form F µν / Λ X , where F µν is the field strength of X , Z or W ± bosons. Suchterms contribute to the XZZ and
XW W vertices. As the fermions in the loopsare heavy, such vertices are in general strongly suppressed by their mass. However,motivated by various string constructions, [49] assumed two things: (a) these ad-ditional massive fermions are above the LHC reach but not too heavy (e.g., havemasses in tens of TeV); (b) there are many such fermions (for instance Hagedorntower of states) and therefore the mass suppression can be compensated by the largemultiplicity of these fermions.In this paper we consider another possible setup, in which the anomaly can-cellation occurs only within a high-energy sector (at scales not accessible by currentexperiments), but at low energies there remain contributions unsuppressed by massesof heavy particles. A similar setup, with completely different phenomenology, hasbeen previously considered in [11, 12].The paper is organized as follows. We first consider in section 2 the general theoryof D’Hoker-Farhi terms arising from the existence of heavy states that contribute toanomalies. We illustrate the theory issues in the following section 3 with a toy model.In section 4 we give a complete set of charges for a realistic SU (2) × U (1) Y × U (1) X theory. In section 5 we bring all these elements together to demonstrate expectedLHC phenomenology of this theory. In this Section we consider an extension of the SM with an additional U X (1) field.The SM fields are neutral with respect to the U X (1) group, however, the heavy fieldsare charged with respect to the electroweak (EW) U Y (1) × SU (2) group. This leads4o a non-trivial mixed anomaly cancellation in the heavy sector and in this respectour setup is similar to the work [49]. However, unlike the work [49], we show thatthere exists a setup in which non-trivial anomaly cancellation induces a dimension4 operator at low energies. The theories of this type were previously consideredin [11, 12].At energies accessible at LHC and below the masses of the new heavy fermions,the theory in question is simply the SM plus a massive vector boson X : L = L SM − g X | F X | + M X | Dθ X | + L int (1)where θ X a pseudo-scalar field, charged under U X (1) so that Dθ X = dθ X + X remainsgauge invariant (St¨uckelberg field). One can think about θ X as being a phase of aheavy Higgs field, which gets “eaten” by the longitudinal component of the X boson.Alternatively, θ X can be a component of an antisymmetric n -form, living in the bulkand wrapped around an n -cycle. The interaction term L int contains the verticesbetween the X boson and the Z, γ, W ± : ǫ µνλρ Z µ X ν ∂ λ Z ρ , ǫ µνλρ Z µ X ν ∂ λ γ ρ , ǫ µνλρ W + µ X ν ∂ λ W − ρ , (2)We wish to generalize these terms into an SU (2) × U Y (1) covariant form. One possibleway would be to have them arise from ǫ µνλρ X µ Y ν ∂ λ Y ρ and ǫ µνλρ X µ ω νλρ ( A a ) (3)where ω νλρ ( A a ) is the Chern-Simons term, built of the SU (2) fields A aµ : ω νλρ ( A a ) = A aν ∂ λ A aρ + 23 ǫ abc A aν A bλ A cρ (4)However, apart from the desired terms of eq. (2) they contain also terms like ǫ µνλρ X µ γ ν ∂ λ γ ρ which is not gauge invariant with respect to the electro-magnetic U (1) group, andthus unacceptable.To write the expressions of (2) in a gauge-covariant form, we should recall that itis the SM Higgs field H which selects massive directions through its covariant deriva-tive D µ H . Therefore, we can write the interaction term in the following, explicitly SU (2) × U Y (1) × U X (1) invariant, form: L int = c H † DH | H | Dθ X F Y + c HF W DH † | H | Dθ X (5)The coefficients c , c are dimensionless and can have arbitrary values, determinedentirely by the properties of the high-energy theory. In eq. (5) we use the differentialform notation (and further we omit the wedge product symbol ∧ ) to keep expressions5 ψ χ χ ψ L ψ R ψ L ψ R χ L χ R χ L χ R U (1) A e e e e e e e e U (1) B q − q − q q q q − q − q Table 1: A simple choice of charges for all fermions, leading to the low-energy ef-fective action (8). The charges are chosen in such a way that all gauge anomaliescancel. The cancellation of U (1) A and U (1) B anomalies happens for any value of e i , q i . Cancellation of mixed anomalies requires q = q ( e − e )2( e − e ) .more compact. We will often call the terms in eq. (5) as the D’Hoker-Farhi terms [7,8]. What can be the origin of the interaction terms (5)? The simplest possibilitywould be to add to the SM several heavy fermions, charged with respect to SU (2) × U Y (1) × U X (1). Then, at energies below their masses the terms (5) will be generated.Below, we illustrate this idea in a toy-model setup.Consider a theory with a set of chiral fermions ψ , and χ , , charged with respectto the gauge groups U (1) A × U (1) B . As the fermions are chiral, they can obtainmasses only through Yukawa interactions with both Φ and Φ scalar fields. Φ ischarged with respect to U (1) B , and Φ is charged with respect to U (1) A : L Y ukawa = i X i =1 , ¯ ψ i D / ψ i + ( f v ) ¯ ψ e iγ θ B ψ + ( f v ) ¯ ψ e − iγ θ B ψ + i X i =1 , i ¯ χ i D / χ i + ( λ v ) ¯ χ e iγ θ A χ + ( λ v ) ¯ χ e − iγ θ A χ + h . c . (6)Here we have taken Φ in the form Φ = v e iθ B , where v is its vacuum expectationvalue (VEV) and θ B is charged with respect to the U (1) B group with charge 2 q , andΦ = v e iθ A , where θ A is charged with respect to U (1) A group with charge e − e .The structure of the Yukawa terms restricts the possible charge assignments, sothat the fermions ψ , should be vector-like with respect to the group U (1) A andchiral with respect to the U (1) B (and vice versa for the fermions χ , ). The choiceof the charges in Table 1 is such that triangular anomalies [ U (1) A ] and [ U (1) B ] cancel separately for the ψ and χ sector for any choice of e i , q i . The cancellation ofmixed anomalies occurs only between ψ and χ sectors. It is instructive to analyze itat energies below the masses of all fermions. The terms in the low-energy effectiveaction, not suppressed by the scale of fermion masses are given by S cs = Z ( e − e ) q π θ B F A ∧ F A + ( e − e )(2 q )16 π θ A F A ∧ F B + αA ∧ B ∧ F A (7)The diagrammatic expressions for the first two terms are shown in Fig. 1, while theChern-Simons (CS) term is produced by the diagrams of the type presented in Fig. 2.6he contribution to the CS term A ∧ B ∧ F A comes from both sets of fermions. Onlyfermions ψ contribute to the θ B terms and only fermions χ couple to θ A and thuscontribute to θ A F A ∧ F B . Notice that while coefficients in front of the θ A and θ B terms are uniquely determined by charges, the coefficient α in front of the CS termis regularization dependent . As the theory is anomaly free, there exists a choice of α such that the expression (7) becomes gauge-invariant with respect to both gaugegroups. Notice, however, that in the present case α cannot be zero, as θ A F A ∧ F B and θ B F A ∧ F A have gauge variations with respect to different groups. For the choiceof charges presented in Table 1, the choice of α is restricted such that expression (7)can be written in an explicitly gauge-invariant form: S cs = Z κDθ A ∧ Dθ B ∧ F A (8)where the relation between the coefficient κ in front of the CS term and the fermioncharges is given by α ≡ κ = q ( e − e )16 π (9)For the anomaly cancellation, it is also necessary to impose the condition q = q ( e − e )2( e − e ) (10)as indicated in table 1.The term (8) was obtained by integrating out heavy fermions (Table 1). Theresulting expression is not suppressed by their mass and contains only a dimensionlesscoupling κ . Unlike the case of [7, 8], the anomaly was cancelled entirely amongthe fermions which we had integrated out. The expression (8) represents thereforean apparent counterexample of the “decoupling theorem” [6]. Note that the CSterm (8) contains only massive vector fields. This effective action can only be validat energies above the masses of all vector fields and below the masses of all heavyfermions, contributing to it. However, masses of both types arise from the same Higgsfields. Therefore a hierarchy of mass scales can only be achieved by making gaugecouplings smaller than Yukawa couplings. On the other hand, the CS coefficient κ isproportional to the (cube of the) gauge couplings. Therefore we can schematicallywrite a dimensionless coefficient κ ∼ ( M V /M f ) , where M V is the mass of the vectorfields and M f is the mass of the fermions (with their Yukawa couplings ∼ M f is sent to infinity, while keeping M V finite, the decoupling theoremholds, as the CS terms get suppressed by the small gauge coupling constant. However,a window of energies M V . E . M f , at which the term (8) is applicable, alwaysremains and this opens interesting phenomenological possibilities, which are absentin the situation when the corresponding terms in the effective action are suppressedas E/M f (as in [6]) and not as M V /M f . 7 l ll A λ ( k ) A µ ( k ) θ h ¯ ψγ ψ i γ µ γ λ (cid:2) l l A µ ( k ) l A λ ( k ) θ h ¯ ψγ ψ i γ λ γ µ Figure 1: Anomalous contributions to the correlator h ¯ ψγ ψ i . (cid:1) X λ ( k ) p pp Y ν ( k ) Y µ ( k ) γ λ γ µ P L γ ν P L (cid:2) X λ ( k ) p p Y µ ( k ) p Y ν ( k ) γ λ γ ν P L γ µ P L Figure 2: Two graphs, contributing to the Chern-Simons termsFinally, it is also possible that the fermion masses are not generated via the Higgsmechanism, (e.g. coming from extra dimensions) and are not directly related to themasses of the gauge fields. In this case, the decoupling theorem may not hold andnew terms can appear in a wide range of energies (see e.g. [9, 10] for discussion).
Let us now generalize this construction to the case of interest, when one of the scalarfields generates mass for the chiral fermions and is the SM Higgs field, while at thesame time the masses of all new fermions are higher than about 10 TeV.Note that previously, in the theory described by (6) the mass terms for fermionswere diagonal in the basis ψ, χ and schematically had the form m ¯ ψψ + m ¯ χχ . Tomake both masses for ψ and χ heavy (i.e., determined by the non-SM scalar field),while still preserving a coupling of the fermions with the SM Higgs, we consider anon-diagonal mass term which (schematically) has the following form: L mass = m ¯ ψψ + M ( ¯ ψχ + ¯ χψ ) (11)Computing the eigenvalues of the mass matrix, we find that the two mass eigenstateshave masses M ± m (in the limit m ≪ M ).Now, we consider the case when the mass terms, similar to those of Eq. (11) aregenerated through the Higgs mechanism. We introduce two complex scalar Higgs8 ψ χ χ ψ L ψ R ψ L ψ R χ L χ R χ L χ R Q X x x x x x − x + 1 x + 1 x − Q Y y y + 1 y y y + 1 y y y Table 2: Charge assignment for the U Y (1) × U X (1) with 4 Dirac fermions. Chargesof the scalar fields H and Φ are equal to (1,0) and (0,1), respectively.fields: H = H + iH and Φ = Φ + i Φ . H is charged with respect to the U Y (1)only (with charge 1), while Φ is U Y (1) neutral, but has charge 1 with respect to the U X (1). We further assume that both Higgs fields develop non-trivial VEVs: h H i = v ; h Φ i = V ; v ≪ V (12)Then, we may write H = ve iθ H ; Φ = V e iθ X (13)neglecting physical Higgs field excitations ( H ( x ) = ( v + h ( x )) e iθ H , etc.).Let us suppose that the full gauge group of our theory is just U Y (1) × U X (1).Consider 4 Dirac fermions ( ψ , ψ , χ , χ ) with the following Yukawa terms, leadingto the Lagrangian in the form, similar to (11): L Yukawa = m ¯ ψ e iγ θ H ψ + M ( ¯ ψ e iγ θ X χ + c.c. ) + M ( ¯ ψ e − iγ θ X χ + c.c. ) (14)Here we introduced masses m = f v and M , = F , V , with f and F , the corre-sponding Yukawa couplings.The choice of fermion charges is dictated by the Yukawa terms (14). The ψ fermions are vector-like with respect to U X (1) group, although chiral with respectto the U Y (1). The fermions ψ , χ (and ψ , χ ) have charges with respect to U Y (1)group, such that Q Y ( ψ L ) = Q Y ( χ R ) and Q Y ( ψ R ) = Q Y ( χ L ) (15)and similarly for the pair ψ , χ . Unlike ψ , the fermions ψ do not have Yukawaterm m ¯ ψ e iǫγ θ H ψ , as this would make the choice of charges too restrictive anddoes not allow us to generate terms similar to (8). The resulting charge assignmentis shown in Table 2.It is clear that the triangular anomalies XXX and
Y Y Y cancel as there is equalnumber of left and right moving fermions with the same charges. Let us consider themixed anomaly
XY Y . The condition for anomaly cancellation is given by A XY Y = X Q LX ( Q LY ) − Q RX ( Q RY ) = y + y − − y − y = 0 (16)The other mixed anomaly XXY is proportional to A XXY = 1 − y + y + 2 x ( − y + y + y −
1) = 0 (17)9 ψ χ χ ψ L ψ R ψ L ψ R χ L χ R χ L χ R Q X Q Y − − U Y (1) × U X (1) of the 4 Diracfermions. The anomaly coefficient κ (Eq. (21)) is nonzero and equal to 6.and should also cancel.In analogy with the toy-model, described above, Table 3 presents an anomaly freeassignment for which the mixed anomalies cancel only between the ψ and χ sectorsand lead to the following term in the effective action (similar to (8)): L A = κDθ H ∧ Dθ X ∧ F Y (18)Here the parameter κ is defined by the XY Y anomaly in the ψ or χ sector, in analogywith Eq. (9): κ = − x ( − y + y + 2 y + 1)32 π (19)To have κ = 0 we had to make two mass eigenstates in the sector ψ , χ degenerateand equal to M . The charges x, y become then arbitrary, while y , should satisfythe constraints (16) and (17). It is easy to see that indeed this can be done togetherwith the inequality κ = 0. The solution gives: y = 4 yx − yx − x − y x + 1 ; y = 4 yx + 4 x + 4 yx − y − x + 1 (20)The choice (20) leads to the following value of κ : κ = − x (4 x −
1) ((8 y + 4) x + 8 y ( y + 1) x − y − x + 1) (21)One can easily see that κ is non-zero for generic choices of x and y . One such achoice is shown in Table 3 (recall that all U X (1) charges are normalized so that θ X has Q X ( θ X ) = 1 and all U Y (1) charges are normalized so that Q Y ( θ H ) = 1).To make the anomalous structure of the Lagrangian (14) more transparent, wecan perform a chiral change of variables, that makes the fermions vector-like. Namely,let us start with the term m ¯ ψ e iθ H γ ψ . We want to perform a change of variablesto a new field ˜ ψ , which will turn this term into m ¯˜ ψ ˜ ψ . This is given by (cid:18) ψ L ψ R (cid:19) → e − i θ H ˜ ψ L e i θ H ˜ ψ R ! or ψ → e − i γ θ H ˜ ψ (22)10 ψ χ χ ψ a L ψ R ψ L ψ a R χ L χ a R χ a L χ R Q X x x x x x − x + 1 x + 1 x − Q Y y y + 1 y y y + 1 y y y Table 4: Charge assignment for the SU (2) × U Y (1) × U X (1) gauge group. Fermions,which are doublets with respect to the SU (2) are marked with the superscript a .Charges of the SM Higgs field H and of the heavy Higgs Φ are equal to (1,0) and(0,1) with respect to U Y (1) × U X (1).so that the Yukawa term becomes m ¯˜ ψ ˜ ψ . The field ˜ ψ has vector-like charge x withrespect to U X (1) and vector-like charge y + with respect to U Y (1). As the change ofvariables is chiral, it introduces a Jacobian J ψ [50]. The transformation (22) turnsthe term M ( ¯ ψ e iγ θ X χ + c.c. ) into M ( ¯˜ ψ e iγ ( θ X − θH ) χ + c.c. ). By performing achange of variables from χ to ˜ χ , χ → e − i γ ( θ X − θH ) ˜ χ , (23)we make the sector ˜ ψ , ˜ χ fully vector-like, and generate two anomalous Jacobians J ψ and J χ . Similarly, for the last term in eq. (14), we perform the change of variables χ → e iθ X / χ and ψ → e iθ X / ψ , generating two more Jacobians. By computingthe Jacobians, one can easily show that performing the above change of variables forall 4 fermions, we arrive to a vector-like Lagrangian with the additional term (18). SU (2) × U Y (1) × U X (1) Model
The above example shows us how to construct a realistic model of high-energy theory,whose low-energy effective action produces the terms (5). We consider the followingfermionic content (iso-index a = 1 , SU (2) doublets): two left SU (2) doublets ψ a L and χ a L , two right SU (2) doublets ψ a R and χ a R , as well as two left SU (2) singlets ψ L and χ L , and two right SU (2) singlets ψ R and χ R . The corresponding chargeassignments are shown in Table 4.The Yukawa interaction terms have the form: L Yukawa = f ( ¯ ψ a L H a ) ψ R + F (cid:16) ¯ ψ a L (Φ − iγ Φ ) χ a R + c . c . (cid:17) + F (cid:16) ¯ ψ a R (Φ + iγ Φ ) χ a L + c . c . (cid:17) + ˜ F (cid:16) ¯ ψ R (Φ − iγ Φ ) χ L + c . c . (cid:17) + ˜ F (cid:16) ¯ ψ L (Φ + iγ Φ ) χ R + c . c . (cid:17) (24)where H is the SM Higgs boson and Φ , are SU (2) × U (1) Y singlets. Here again h H i = v ≪ h Φ i , and all states have heavy masses ∼ F h Φ i (plus possible correctionsof order O ( f v )). 11 ψ χ χ ψ a L ψ R ψ L ψ a R χ L χ a R χ a L χ R Q X − − − − −
76 56 56 − Q Y
12 32 − −
76 32 12 − − Table 5: Explicit charge assignment for the SU (2) × U Y (1) × U X (1) gauge group. (cid:1) X µ ( k ) Z ν ( k ) Z ρ ( k ) (cid:2) X µ ( k ) Z ν ( k ) γ ρ ( k ) Figure 3: Γ
XZZ and Γ
XZγ interaction vertices, generated by (25)The anomaly analysis is similar to the one performed in the previous section. Theonly difference being of course two isospin degrees of freedom in the SU (2) doublets.The resulting choice of charges is shown in Table 5 (we do not write the generalexpression as it is too cumbersome and provides only an example when x = − Q H / y = Q Φ / c , in the interaction terms (5) are non-zero, which leads to interesting phenomenologyto be discussed in the next section. The analysis of the previous sections puts us in position to now discuss the phe-nomenology of the X boson. To do this, we first detail the relevant interactions ithas with the SM gauge bosons.The first term in (5) generates two interaction vertices: XZZ and
XZγ (Fig.3).In the EW broken phase one can think of the first term in expression (5) as beingsimply L XZY = c ( dθ Z + Z ) F Y Dθ X + O (cid:18) ∂hv (cid:19) (25)where we parametrized the Higgs doublet as H = e i ( τ + θ + ( x )+ τ − θ − ( x )+( + τ ) θ Z ) (cid:18) v + h ( x ) (cid:19) (26)12ere the phases θ ± , θ Z will be “eaten” by W ± and Z bosons correspondingly, v isthe Higgs VEV and the real scalar field h is the physical Higgs field.The vertices Γ XZZ and Γ
XZγ are given correspondingly byΓ µνρXZZ ( k , k | k ) = 12 c sin θ w ǫ µνλρ ( k λ − k λ )Γ µνρXZγ ( k , k | k ) = c cos θ w ǫ µνλρ k ρ (27)Similarly to above one can analyze the second term in (5). It leads to the inter-action XW + W − : Γ µνρXW + W − ( k , k | k ) = c ǫ µνλρ ( k λ − k λ ) (28)The most important relevant fact to phenomenology is that the X boson is pro-duced by and decays into SM gauge bosons. We shall discuss in turn the productionmechanisms and the decay final states of the X boson and then estimate the discoverycapability at colliders. X boson Producing the X boson must proceed via its coupling to pairs of SM gauge bosons.One such mechanism is through vector-boson fusion , where two SM gauge bosons areradiated off initial state quark lines and fused into an X boson: pp → qq ′ V V ′ → qq ′ X or V V ′ → X for short , (29)where V V ′ can be W + W − , ZZ or Zγ . This production mechanism was studied inref. [49]. One of the advantages is that if the decays of X are not much different thanthe SM, the high-rapidity quarks that accompany the event can be used as “taggingjets” to help separate signal from the background. This production mechanism isvery similar to what has been exploited in the Higgs boson literature.A second class of production channels is through associated production : pp → qq ′ → V ∗ → XV ′ (30)where an off-shell vector boson V ∗ and the final state V ′ can be any of the SMelectroweak gauge bosons: XZ , XW ± or Xγ . It turns out that this production classhas a larger cross-section than the vector boson fusion class. This is opposite to whatone finds in SM Higgs phenomenology, where V V ′ → H cross-section is by O (10 )greater than HV ′ associated production. The reason for this is that both vectorbosons can be longitudinal when scattering into H , thereby increasing the V V ′ → H cross-section over HV ′ . This is not the case for the X boson production, in whichonly one longitudinal boson can be present at the vertex. This leads to a suppressionby ∼ ( √ s/M V ) of the process (29) as opposed to the similar process for the Higgsboson. For LHC energies ( √ s ∼
10 TeV) this suppression is of the order 10 − .13
00 120 140 160 180 200M X (GeV)10 -2 -1 σ ( pb ) ZX γ X 0 200 400 600 800 1000M X (GeV)10 -4 -3 -2 -1 σ ( pb ) W + XW - XZX γ X (a) (b)Figure 4: Production cross-section for XZ and Xγ at LEP (left) and for XZ , Xγ , XW ± at Tevatron (right panel) vs. the X boson mass. For LEP √ s = 200 GeV,and for Tevatron √ s = 2 TeV. In both cases c = c = 1.Without special longitudinal enhancements, the two body final state XV ′ dominatesover the three-body final state qq ′ X , which makes the associated production (30)about 2 orders of magnitude stronger than the corresponding vector-boson fusion.As we shall see below, the decays of the X boson are sufficiently exotic in naturethat background issues do not change the ordering of the importance of these twoclasses of diagrams. Thus, we focus our attention on the associated production XV ′ to estimate collider sensitivities.In figs. 4 and 5 we plot the production cross-sections of XV for various V = W ± , Z, γ at √ s = 14 TeV pp LHC, √ s = 2 TeV p ¯ p Tevatron and √ s = 200 GeV e + e − LEP. X boson The X boson decays primarily via its couplings to SM gauge boson pairs. The im-portant decay channels are computed from the interaction vertices computed above.14 X (GeV)10 -6 -5 -4 -3 -2 -1 σ ( pb ) W + XW - XZX γ X Figure 5: Production cross-section at √ s = 14 TeV LHC of XV ′ for various V ′ = W ± , Z, γ vs. the X boson mass with c = c = 0 . X → ZZ = c sin θ w M X πM Z (cid:18) − M Z M X (cid:19) / ≈ c (45 GeV) (cid:18) M X TeV (cid:19) + . . . , Γ X → W + W − = c M X πM W (cid:18) − M W M X (cid:19) / ≈ c (1 .
03 TeV) (cid:18) M X TeV (cid:19) + . . . (31)Γ X → Zγ = c cos θ w M X πM Z (cid:18) − M Z M X (cid:19) (cid:18) M Z M X (cid:19) ≈ c (307 GeV) (cid:18) M X TeV (cid:19) + . . . , , where . . . denote corrections of the order ( M V /M X ) . The interaction term of eq. (25)also allows interaction of the X boson with γH and ZH , which are generically small.At leading order in M Z /M X the decay width into Zγ exceeds that of ZZ byΓ X → Zγ Γ X → ZZ = 2 cos θ w sin θ w ≈ . W W is the largest over much of parameter space where c ∼ > c , and exceeds that of ZZ byΓ X → W + W − Γ X → ZZ = 4sin θ w c c ≈ . c c . (33)15 c /c B ( X ) Z γ ZZW + W - Figure 6: Branching fractions of X boson decays into W + W − (blue), ZZ (yellow-green) and Zγ (purple) as a function of c /c assuming M X ≫ M Z .This ratio depends on the a priori unknown ratio of couplings c /c . In Fig. 6 we plotthe branching fractions of X into the W W (blue), Zγ (purple) and ZZ (yellow-green)as a function of c /c .Let us compare decay widths (31) with analogous expressions from [49]. Schemat-ically, decay widths can be obtained in our case asΓ X → V V ∼ c , M X M V (34)where we denote by V both Z and W ± vector bosons and M V = { M Z , M W } . Incase of setup of Ref. [49] the interaction is the dimension 6 operators, suppressed bythe cutoff scale Λ X . Therefore, the decay width is suppressed by Λ X and the wholeexpression is given by Γ X → V V ∼ M X Λ X M X M V M V M X = M X M V Λ (35)The presence of the factor M V M X , appearing in the first equation of (35), can be ex-plained as follows. The vector boson current is conserved in the interaction, generatedby the higher-dimensional operators of Ref. [49]. Therefore the corresponding prob-ability for emitting on-shell Z or W boson is suppressed by the ( M V E ) where theenergy E ∼ M X . In case of the interaction (5) the vector current is not conserved inthe vertex and therefore such a suppression does not appear.16 .3 Collider Searches Combining the various production modes and branching fractions yields many per-mutations of final states to consider at high energy colliders. All permutations, aftertaking into account X decays, give rise to three vector boson final states such as ZZZ , W + W − γ , etc. The collider phenomenology associated with these kinds offinal states is interesting, and we focus on a few aspects of it below.Our primary interest will be to study how sensitive the LHC is to finding thiskind of X boson. The limits that one can obtain from LEP 2 and Tevatron are wellbelow the sensitivity of the LHC, and so we forego a more thorough analysis of theirconstraining power. Briefly, in the limit of no background, the Tevatron cannot dobetter than the mass scale at which at least a few events are produced. This impliesfrom fig. 4b that M X ∼ >
750 GeV (for c i = 1) is inaccessible territory to the Fermilabwith up to 10 fb − of integrated luminosity. The LHC can do significantly betterthan this, as we shall see below.Moving to the LHC, the energy is of course an important increase as is the plannedluminosity. After discovery is made a comprehensive study programme to measureall the final states, and determine production cross-sections and branching ratioswould be a major endeavor by the experimental community. However, the first stepis discovery. In this section we demonstrate one of the cleanest and most uniquediscovery modes to this theory. As has been emphasized earlier and in ref. [43], the X → γZ decay mode is especially important for this kind of theory. Thus, we studythat decay mode. Consulting the production cross-sections results for LHC, we findthat producing the X in association with W ± gives the highest rate. Thus, we focusour attentions on discovering the X boson through XW ± production followed by X → γZ decay.The γZW ± signature is an interesting one since it involves all three electroweakgauge bosons. If the Z decays into leptons, it is especially easy to find the X bosonmass through the invariant mass reconstruction of γl + l − . The additional W is alsohelpful as it can be used to further cut out background by requiring an additionallepton if the W decays leptonically, or by requiring that two jets reconstruct a W mass.In our analysis, we are very conservative and only consider the leptonic decays ofthe Z and the W . Thus, after assuming X → γZ decay, 1 . γZW ± turninto γl + l − l ′± plus missing E T events. These events have very little background whencut around their kinematic expectations. For example, if we assume M X = 1 TeVwe find negligible background while retaining 0 .
82 fraction of all signal events whenwe making kinematic cuts η ( γ, l ) < . m l + l − = m Z ± p T ( γ ) >
50 GeV, p T ( l + , l − , l ′ ) >
10 GeV, missing E T greater than 10 GeV and m γl + l − >
500 GeV.Thus, for 10 fb − of integrated luminosity at the LHC, when c i = 1 ( c i = 0 .
1) we get atleast five events of this type, γl + l − l ′ plus missing E T , if M X > M X > ∆ R ) ∆ d ( σ d σ Figure 7: Distribution of ∆ R of e + e − in the Z decays of W X production followedby X → γZ followed by Z → e + e − . The distributions are for M X = 500 GeV (red), M X = 1 TeV (blue) and M X = 2 TeV (green).resonance, the X boson.One subtlety for this signal is the required separation of the leptons from the Z decay in order to distinguish two leptons and be able to reconstruct the invariantmass well. The challenge arises because the Z is highly boosted if its parent particlehas mass much greater than m Z , and thus the subsequent leptons from Z decaysare highly boosted and collimated in the detector. One does not expect this to be aproblem for Z → µ + µ − decays, as muon separation is efficient. Separation of electronand positron in the electromagnetic calorimeter in highly boosted Z → e + e − finalstates is expected to be more challenging. We do not attempt to give precise numbersof separability for e + e − . Instead, we only make two relevant comments. First, oneis safe restricting to muons. Second, once separability of e + e − is better understood,it can be compared with the kinematic distributions of this example to estimate thenumber of events that are cut out due to the inability to resolve e + e − . In Fig. 7 weshow the ∆ R separation of e + e − for a parent M X = 500 GeV, 1 TeV and 2 TeV. Forexample, if it turns out that ∆ R > . .
1) is required, then one can expect about2 / /
4) of the e + e − events are cut out by this separation criterion.After discovery, in addition to doing a comprehensive search over all possiblefinal states, each individual final state will be studied carefully to see what evidenceexists for the spin of the X boson. The topology of γZW ± exists within the SMfor HW ± production followed by H → γZ decays. However, the rate at whichthis happens is very suppressed even for the most optimal mass range of the Higgsboson [51]. A heavy resonance that decays into γZ would certainly not be a SM Higgsboson, but nevertheless a scalar origin would be considered if a signal were found.Careful studying of angular correlations among the final state particles can helpdetermine this question directly. For example, distinguishing between the scalar andvector spin possibilities of the X boson is possible by carefully analyzing the photon’s18os θ γ distribution with respect to the X boost direction in X → γZ decays in therest frame of the X . If X is a scalar its distribution is flat in cos θ , whereas if itis a vector it has a non-trivial dependence on cos θ . With enough events (severalhundred) this distribution can be filled in, and the spin of the X resonance can bediscerned among the possibilities. Acknowledgments
We thank J. Kumar, J. Lykken, F. Maltoni, A. Rajaraman, A. De Roeck for helpfuldiscussions. I.A. was supported in part by the European Commission under the ERCAdvanced Grant 226371. O.R. was supported in part by the Swiss National ScienceFoundation.
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