Unparticle & Higgs as Composites
aa r X i v : . [ h e p - ph ] O c t Edinburgh 2008/41IPPP/08/33DCPT/08/66
Unparticle & Higgs as Composites
Francesco Sannino a ∗ & Roman Zwicky b c † a HEP Center, University of Southern Denmark, Campusvej 55, DK-5230 Odense M,Denmark. b School of Physics and Astronomy, University of Edinburgh, Scotland c IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK
Abstract
We propose a generic framework in which the Higgs and the unparticle areboth composite. The underlying theories are four dimensional, asymptotically free,nonsupersymmetric gauge theories with fermionic matter. We sketch a possibleunification of these two sectors at a much higher scale resembling extended tech-nicolor models. By construction our extensions are natural, meaning that thereis no hierarchy problem. The coupling of the unparticle sector to the compositeHiggs emerges as a four-Fermi operator. The bilinear unparticle operator near theelectroweak scale has scaling dimension in the range 1 < d U <
3. We investigate,in various ways, the breaking of scale invariance induced by the electroweak scaleresulting in an unparticle condensate. The latter acts as a natural infrared cut off orhadronic scale. We give the low-energy effective theory valid near the electroweakscale. The unparticle-Higgs mixing is found to be suppressed within our framework. ∗ [email protected] † [email protected] t is an exciting possibility that new strong dynamics could be discovered at the LargeHadron Collider (LHC). The hope is fueled by the fact that some of the best motivatedextensions of the Standard Model (SM) break the electroweak symmetry dynamically[1, 2]. The new models, passing the precision electroweak tests, are summarized in [3]. Itis then interesting to explore the possibility to accommodate the unparticle scenario [4]into a natural setting featuring four dimensional strongly interacting dynamics.Georgi’s original idea is that at high energy there is an ultraviolet (UV) sector coupledto the SM through the exchange of messenger fields with a large mass scale M U . Below thatscale two things happen consecutively. Firstly, the messenger sector decouples, resulting incontact interactions between the SM and the unparticle sector. Secondly, the latter flowsinto a non-perturbative infrared (IR) fixed point at a scale Λ U ≪ M U hence exhibitingscale invariance; L ∼ O UV O SM → O U O SM . (1)The UV unparticle operator is denoted by O UV and it posses integer dimension d UV .When the IR fixed point is reached the operator O IR ≡ O U acquires a non-integer scalingdimension d U through dimensional transmutation |h |O U | P i| ∼ ( √ P ) d U − . (2)This defines the matrix element up to a normalization factor. In the regime of exactscale invariance the spectrum of the operator O U is continuous, does not contain isolatedparticle excitations and might be regarded as one of the reasons for the name “unparti-cle”. The unparticle propagator carries a CP-even phase [6, 7] for space-like momentum.Effects were found to be most unconventional for non-integer scaling dimension d U , e.g.[4, 6] and [8].The coupling of the unparticle sector to the SM (1) breaks the scale invariance of theunparticle sector at a certain energy. Such a possibility was first investigated with naivedimensional analysis (NDA) in reference [9] via the Higgs-unparticle coupling of the form L eff ∼ O U | H | . (3)The dynamical interplay of the unparticle and Higgs sector in connection with the interac-tion (3) has been studied in [10]. It was found, for instance, that the Higgs VEV inducesan unparticle VEV, which turned out to be infrared (IR) divergent for their assumed rangeof scaling dimension and forced the authors to introduce various IR regulators [10, 11].In this work we elevate the unparticle scenario to a natural extension of the SM The resulting CP violation was found to be consistent with the CPT theorem [5].
1y proposing a generic framework in which the Higgs and the unparticle sectors areboth composites of elementary fermions. We use four dimensional, non-supersymmetricasymptotically free gauge theories with fermionic matter. This framework allows us toaddress, in principle, the dynamics beyond the use of scale invariance per se.The Higgs sector is replaced by a walking technicolor model (TC), whereas the un-particle one corresponds to a gauge theory developing a nonperturbative IR fixed point(conformal phase) . By virtue of TC there is no hierarchy problem. We sketch a possibleunification of the two sectors, embedding the two gauge theories in a higher gauge group.The model resembles the ones of extended technicolor and leads to a simple explanationof the interaction between the Higgs and the unparticle sectors.The paper is organized as follows. In section 1.1 we describe the basic scenario.Thereafter we address the formation of the unparticle VEV in section 1.2 and identify theVEV as the natural IR cut off in connection with the dynamical (constituent) fermionmass. The comparison with the IR cut off suggested by NDA is presented in appendixB. In section 2 we give some more details about the unified framework. The low energyeffective Lagrangian, which could also be taken as a starting point, is given in section2.1. The regularized unparticle propagator with IR and UV cut off is discussed in section2.2. The normalization of the unparticle operator to our specific model is discussed inappendix C. In a further section 2.3 we discuss the mixing of the Higgs with the unparticlebased upon the previously given effective Lagrangian. In appendix D we comment on theproposed unparticle limit of the model presented in[28]. The paper ends with an outlookin section 3, where possible future directions of research in collider physics, lattice anddark matter are discussed. For example, we put forward the idea of the Unbaryon as apossible dark matter candidate. We note that the Banks-Zaks [12] type IR points, used to illustrate the unparticle sector in [4], areaccessible in perturbation theory. This yields anomalous dimensions of the gauge singlet operators whichare close to the pertubative ones, resulting in very small unparticle type effects. Strictly speaking conformal invariance is a larger symmetry than scale invariance but we shall usethese terms interchangeably throughout this paper. We refer the reader to reference [13] for an investi-gation of the differences. Only very recently has it been possible to directly investigate, via lattice simulations, the dynamicsof a number of gauge theories[14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] expected to develop or to be veryclose an IR fixed point [25, 26, 27]. The bulk of the lattice results support the theoretical expectations[25, 27]. The Higgs & Unparticle as Composites
Our building block is an extended G T × U ≡ SU ( N T ) × SU ( N U ) technicolor (TC) gaugetheory. The matter content constitutes of techniquarks Q af charged under the representa-tion R T of the TC group SU ( N T ) and Dirac techniunparticle fermions Ψ As charged underthe representation R U of the unparticle group SU ( N U ), where a/A = 1 . . . dim[ R T/U ]and f /s = 1 . . . F/S denote gauge and flavor indices respectively. We will first describethe (walking) TC and (techni)unparticle sectors separately before addressing their com-mon dynamical origin. A graphical illustration of the scenario is depicted in Fig. 1 as aguidance for the reader throughout this section.In the TC sector the number of techniflavors, the matter representation and the num-ber of colors are arranged in such a way that the dynamics is controlled by a near conformal(NC) IR fixed point . In this case the gauge coupling reaches almost a fixed point aroundthe scale Λ T ≫ M W , with M W the mass of the electroweak gauge boson. The TC gaugecoupling, at most, gently rises from this energy scale down to the electroweak one. Thecoupling is said to walk . Around the electroweak scale the TC dynamics triggers thespontaneous breaking of the electroweak symmetry through the formation of the tech-nifermion condensate, which therefore has the quantum numbers of the SM Higgs boson.The associated Goldstone bosons (technipions) then become the longitudinal degrees offreedom of the electroweak bosons in exact formal analogy to the SM. In the simplest TCmodels the technipion decay constant F T is related to the weak scale as 2 M W = gF T ( g is the weak coupling constant) and therefore F T ≃
250 GeV. The TC scale, analogous toΛ
QCD for the strong force, is roughly Λ
T C ∼ πF T .Now we turn our attention to the unparticle sector. Here the total number of masslesstechniunparticle flavors S is balanced against the total number of colors N U in such away that the theory, per se, is asymptotically free and admits a nonperturbative IR fixedpoint. The energy scale around which the IR fixed point starts to set in is indicated withΛ U ≫ M W .It might be regarded as natural to assume that the unparticle and the TC sectors havea common dynamical origin, e.g. are part of a larger gauge group at energies above Λ T There are a number of ways of achieving (near) conformal dynamics as summarized in [29]. Thestate-of-the-art phase diagram [26, 27] and new tools [25] to construct viable NC nonsupersymmetricgauge theory are reported in [3]. Such models are known as walking TC [30]. They are preferred over QCD-like TC models by theelectroweak precision data. In particular, the S-parameter receives a negative contribution for NC models[31]. A large class of phenomenologically viable models have been identified [32, 27, 29, 3] of which MinimalWalking Technicolor (MWT) and Partially Gauged Technicolor (PGT) constitute two relevant examples. (cid:1)(cid:1) (cid:2)(cid:3) (cid:4)(cid:5)(cid:5)(cid:5)(cid:5)(cid:6)(cid:5)(cid:7)(cid:8)(cid:9)(cid:5)(cid:10)(cid:5)(cid:5)(cid:11)(cid:5)(cid:10)(cid:5)(cid:5)(cid:12) (cid:13)(cid:8) (cid:5)(cid:5) (cid:14) (cid:13) (cid:7)(cid:8)(cid:9)(cid:5)(cid:10)(cid:5)(cid:12)(cid:5)(cid:15)(cid:5)(cid:7)(cid:8)(cid:9)(cid:5)(cid:10)(cid:5)(cid:12)(cid:14) (cid:13) (cid:7)(cid:8)(cid:9)(cid:5)(cid:10)(cid:5)(cid:12)(cid:14) (cid:13) (cid:7)(cid:8)(cid:9)(cid:5)(cid:10)(cid:5)(cid:12) (cid:14)(cid:14) (cid:13) (cid:1)(cid:16)(cid:17)(cid:17)(cid:16)(cid:18)(cid:19)(cid:16)(cid:20)(cid:17)(cid:5)(cid:5)(cid:21)(cid:16)(cid:22)(cid:23)(cid:14)(cid:24)(cid:25)(cid:16) (cid:26)(cid:27)(cid:5)(cid:28)(cid:29)(cid:15)(cid:16)(cid:21)(cid:5)(cid:24)(cid:23)(cid:29)(cid:18)(cid:30) (cid:10)(cid:16)(cid:31)(cid:20)(cid:5) (cid:23)(cid:18)(cid:28)(cid:23)(cid:20)!(cid:31)(cid:25) (cid:2)(cid:2) (cid:1)(cid:1) (cid:1)(cid:1) (cid:3) (cid:14) (cid:3) (cid:14) (cid:3) (cid:13) (cid:4) (cid:5) Figure 1:
Schematic scenario. The ordering of the energy scales Λ U and Λ T is not of anyimportance. and Λ U . We would like to point out that the relative ordering between Λ T and Λ U is of noparticular relevance for our scenario. The low energy relics of such a unified-type modelare four-Fermi operators allowing the two sectors to communicate with each other at lowenergy. The unparticle sector will then be driven away from the fixed point due to theappearance of the electroweak scale in the TC sector.The model, of which further details are presented in section 2 resembles models ofextended technicolor (ETC) [3], where the techniunparticles play the role of the SMfermions. We refer to these type of models as Extended Techni-Unparticle (ETU) models .At very high energies E ≫ M U the gauge group G T × U is thought to be embedded ina simple group G TU ⊃ G T × U . At around the scale M U the ETU group is broken to G T U → G T × U and the heavy gauge fields receive masses of the order of M U and playthe role of the messenger sector. Below the scale M U the massive gauge fields decoupleand four-Fermi operators emerge, which corresponds to the first step of the scenario, e.g.Eq. (1) and Fig. 1. Without committing to the specific ETU dynamics the interactionscan be parametrized as: L eff 2. The equation of motion (e.o.m.) for the operator ϕ n is¯ αf n + 2¯ γf n ( X m f m ϕ m ) − M n ϕ n = 0 . (14)A simple recursive relation follows from these relations, ϕ n = n ( d U − / ϕ . (15)Inserting this result into the e.o.m. for ϕ we obtain, h ϕ i = ¯ α b d U ∆ d U − − γ ( b d U Ω ∆ ) , b d U ≡ r B d U π , (16)where Ω ∆ is the sum over the modes,Ω ∆ ≡ ∆ d U − X n (cid:16) f n b d U (cid:17) ϕ n ϕ = ∆ X n ( n ∆ ) d U − . h ϕ n i are then obtained from the recursion relation (15) and the unparticleVEV is the sum of its deconstructed parts (10), hO U i ∆ = X n f n h ϕ n i = ¯ α − γ ( b d U Ω ∆ ) ( b d U Ω ∆ ) . (17)Solving this equation with appropriate UV and IR regularizations is the main goal ofthe rest of this section. The unparticle condensate will be connected with the IR cutoff, which implies that Eq. (17) has to be solved in a self consistent way. Removing thediscrete regularization, the sum Ω ∆ is converted into an integral, which we shall regularizewith an IR and UV regulator for later convenience,Ω(Λ IR , Λ UV ) = lim ∆ → Ω ∆ = Z Λ Λ dss d U − = (cid:16) (Λ ) d U − − (Λ ) d U − d U − (cid:17) . (18)We note that when the quadratic term is removed, i.e. γ → IR , Λ UV ) (18) is sensitive to the UV cut off for d U > d U < 2. The effective theory for the unparticle operator is valid up to the scaleΛ U and is therefore a UV cut off of the theory. Moreover at energies larger than Λ T thereis no ¯ QQ condensate, which implies ¯ α → T do notcontribute to the VEV in Eq. (17)). So effectively the UV cut off is the lower of the twoscales, Λ UV ≃ min(Λ U , Λ T ) . (19)The constituent fermion mass m const provides a natural IR cut off:Λ IR ≃ m const ≃ |hO U i| /d U . (20)For numerical estimates we have chosen the factor two in front of the condensate based onthe crude idea of identifying the IR cut off with a possible lightest meson of mass roughlytwice the constituent mass. However this choice does not affect the qualitative nature of In QCD the condensate induces a dynamical mass, the so-called constituent quark mass. An estimatecan be obtained by extending the definition of the perturbative pole mass to include additional termsfrom the Operator Product Expansion [40]. Adapting the situation to the case of a non-trivial fixedpoint leads to ( m const ) d U ≃ − g U C d U hO U i , where C d U an order one coefficient which is not calculable dueto virulent strong interaction effects. The lowest order QCD result is recovered by setting d U → C d U → { Λ T C , Λ U , Λ T , M U } .and the anomalous dimensions γ U and γ T . The parameters obey the following hierarchies,Λ T C ≪ Λ U , Λ T ≪ M U , γ U < γ T , (21)c.f. Fig. 1 for the scales and section 2 for an explanation concerning the relation of theanomalous dimensions. We investigate Eq. (17) analytically in the follwing three regimesApproximate solution of Eq . (17) for h ˆ O U i valid near sensitive h ˆ O U i ( |h ˆ O U i| γ U − − γ U − C γ ′ ) − C α ′ = 0 (cid:26) d U & γ U . C = b d U γ U − (cid:16) Λ U M U (cid:17)(cid:16) (2Λ T C ) Λ U (cid:17) − γ U C = C (cid:16) Λ T C Λ U (cid:17) γ U w γ T T h ˆ O U i ≃ − α ′ b (cid:16) Λ U Λ T C M U (cid:17) w γ T T log h Λ T C Λ U |h ˆ O U i| i (cid:26) d U = 2 γ U = 1 IR ∼ UV (23) h ˆ O U i ≃ + α ′ b d U − γ U (cid:16) Λ U M U (cid:17)(cid:16) Λ T C Λ U (cid:17) γ U w γ T T (cid:26) d U . γ U & h ˆ O U i ≡ hO U i Λ d U T C , to the chiral symmetry breaking scale of the TC sector. Eqs. (22) and (24) are sensitiveto the IR and UV domain of (18). The solutions are valid in a small neighborhood of d U & d U . d U . d U = 2 strictly for presentationalconvenience only. The γ -term is solely important for (22) or more precisely is of the sameorder as the α -term for typical values of the model parameters. For α ′ , γ ′ ≥ γ U fordifferent γ T up to the bound γ U ≤ γ T (21). The input values, which are thought to be9ypical, are indicated in the caption. L IR (cid:144) L TC Γ U Log @ L IRNDA (cid:144) L IR D - - - - Γ U Figure 2: (left) Λ IR / Λ T C as a function of γ U up to the constraint γ U ≤ γ T . The actual value of γ T can therefore be read-off from the endpoint of the curve. (right) Logarithm of the ratio ofIR cut offs against γ U for γ T = 1. The influence of the γ -term is completely negligible for thechosen input values. The dependence on γ T is very mild and we have chosen somewhat arbitrarily γ T = 3 / 2. Trivial factors, like b d U , are fixed such that equality of Λ IR and Λ NDAIR is reached for γ = 0 in the domain γ U = 2. In both figures we have chosen N U = 4 in the interpolation formula(26). Furthermore the hierarchies of scales (21) are set to Λ T C : (Λ U = Λ T ) : M U = 1 : 10 : 10 and the coefficients α ′ and γ ′ to a value of unity. The breaking of scale invariance, due to the coupling to the Higgs sector, was investi-gated in an earlier reference by the use of na¨ıve dimensional analysis (NDA) [9] . In theappendix B we compare their results with ours. At the parametric level we find,Λ IR & Λ NDAIR , (25)c.f. Fig. 2 (right). The difference being caused by the fact that in the NDA analysisit is implicitly assumed that the unparticle sector scales with the IR cut off whereas inour model the unparticle condensate can also be sensitive to the UV domain. Parametricequality is reached in the region of IR sensitivity, e.g. (22). Needless to say that withNDA factors of 4 π can go unnoticed. In connection with the latter a similar criticismcould apply to our prescription in (20). Nevertheless it appears to us that it is physicallymotivated and to some extent is backed up from our empirical knowledge of QCD.So far we have not specified the normalization factor B d U introduced in Eqs. (10) and(11). In appendix C we motivate the following formula( B d U ) interpol = 2 π ( d U − 1) + (cid:16) N U π − π (cid:17) ( d U − , (26)10s interpolation formula between the value of B , which is determined by the free fermionloop in our model, and the behaviour around B , which is model independent [36]. Aspreviously stated it differs from the normalization factor A d U in reference [4].In the next section we will discuss an ETU model in some more detail. We imagine that at an energy much higher than the electroweak scale the theory isdescribed by a gauge theory L UV = − 12 Tr [ F µν F µν ] + F X F =1 ¯ ξ F ( i / ∂ + g T U / A ) ξ F + .. (27)where A is the gauge field of the SU ( N T + N U ) group and gauge indices are suppressed.( ξ AF ) T = ( Q ...Q N T , Ψ ... Ψ N U ) F is the fermion field unifying the technifermion and TCmatter content. The dots in (27) stand for the SU (3) × SU (2) L × U (1) Y gauge fields andtheir interactions to the SM fermions and technifermions. There is no elementary Higgsfield in this formulation. Unification of the TC and techniunparticle dynamics, as outlinedin section 1, constrains the flavor symmetry of the two sectors to be identical at highenergies. The matter content and the number of technifermions (TC + techniunparticles)is chosen, within the phase diagram in [3], such that the theory is asymptotically free athigh energies. The non-abelian global flavor symmetry is SU L ( F ) × SU R ( F ).At an intermediate scale M U , much higher than the scale where the unparticle andTC subgroup become strongly coupled, the dynamics is such that SU ( N T + N U ) breaksto SU ( N T ) × SU ( N U ). Only two flavors (i.e. one electroweak doublet) are gaugedunder the electroweak group. The global symmetry group breaks explicitly to G F = SU L (2) × SU R (2) × SU L ( F − × SU R ( F − M U there are the Q ic fermions - with i = 1 , . . . , F and c = 1 , . . . , N T - aswell as the Ψ iu ones - with i = 1 , . . . , F and u = 1 , . . . , N U . Assigning the indices i = 1 , . One could for instance unify the flavor symmetry of the unparticles with the technicolor gauge groupinto an ETC group. This would also produce a Lagrangian of the type (4). The TC fermions would be SU ( N T ) and SU ( N U ) have to be arranged such that the formeris NC and the latter is conformal. This enforces the conditions: F ≤ F ∗ N T , F ∗ N U ≤ F − . (28) F ∗ N denotes the critical number of flavors, for a given number of colors N , above whichthe theory develops an IR fixed point. Recall that two unparticle flavors are decoupledand hence F → F − F ∗ N is F ∗ N = 11 Nγ ∗ + 2 , (29)for an SU(N) gauge theory with matter in the fundamental representation. This restricts F ∗ N inasmuch as the critical anomalous dimension has to satisfy the unitarity bound γ ∗ ≤ γ ∗ ≈ N U γ ∗ + 2 + 2 ≤ F ≤ N T γ ∗ + 2 . (30)The anomalous dimension of the mass operator for the unparticle and TC fermions at thefixed point are γ U = 11 N U − F + 4 F − , γ T = 11 N T − FF . (31)They follow from the conjectured all order beta function [25]. For walking TC γ T is, infact, very near γ ∗ and F is very close to the upper bound of equation (30). Conformalityof the unparticle sector requires γ U to be smaller than γ ∗ . Summarizing: γ U < γ ∗ . γ T . (32) Below the scale M U all four-Fermi interactions have to respect the flavor symmetry G F .The most general four-Fermi operators have been classified in [42] and the coefficient of charged under the electroweak group separately. L eff = (cid:16) G L ΣΨ R + h . c . (cid:17) + G ′ M U ( ¯Ψ L Ψ R )( ¯Ψ R Ψ L ) + . . . , (33)the scalar-scalar interactions of Eq. (4). Here Σ is the quark bilinear,Σ ji ∼ ( Q Li ¯ Q jR ) UV , i = 1 , . . . , F . (34)The flavor indices are contracted and the sum starts from the index value 3; the first twoindices correspond to the Ψ’s charged under the electroweak force, which are decoupledat low energy. The fermion bilinear becomes the unparticle operator (7), (cid:0) O U (cid:1) ji = Ψ Li ¯Ψ jR Λ γ U U . (35)The matrix Σ at energies near the electroweak symmetry breaking scale is identified withthe interpolating field for the mesonic composite operators.To investigate the coupling to the composite Higgs we write down the low energyeffective theory using linear realizations. We parameterize the complex F × F matrix Σby Σ = σ + i Θ √ F + √ i Π a + e Π a ) T a , (36)where ( σ , ˜Π) and (Θ,Π) have 0 ++ and 0 − + quantum numbers respectively. The Lagrangianis given by L eff = 12 Tr (cid:2) ( D Σ) † D Σ (cid:3) − k ( c Tr (cid:2) Σ † O U (cid:3) + h . c . ) − k c Tr (cid:2) O U O U † (cid:3) − m ET C F − X a =4 Π a Π a − V (Σ , Σ † ) , (37)where D Σ = ∂ Σ − igW Σ + ig ′ Σ BT R , and W = W a T aL , (38)and Tr[ T aL/R T bL/R ] = δ ab / 2. The coefficients k and k are directly proportional to the α ′ and γ ′ coefficients in (6). The hat on some of the traces indicates that the summation isonly on the flavor indices from 3 to F . Three of the Goldstone bosons play the role13f the longitudinal gauge bosons and the remaining ones receive a mass m ET C froman ETC mechanism. We refer the reader to reference [35] for discussion of differentETC models with mechanisms for sufficiently large mass generation. The first term inthe Lagrangian is responsible for the mass of the weak gauge bosons and the kineticterm for the remaining Goldstone bosons. The VEV’s for the flavor-diagonal part of theunparticle operator, reduces to the computation performed in the previous section. Thepotential term preserves the global flavor symmetry G F . Up to dimension four, includingthe determinant responsible for the η ′ mass in QCD, the terms respecting the globalsymmetries of the TC theory are: V (Σ , Σ † ) = − m (cid:2) Σ † Σ (cid:3) + λ F Tr (cid:2) Σ † Σ (cid:3) + λ Tr[(Σ † Σ) ] − λ ( detΣ + detΣ † ) . (39)The coefficient m is positive to ensure chiral symmetry breaking in the TC sector. TheHiggs VEV enters as follows, σ = v + h , with F T = r F v ≃ 250 GeV . (40) F here is the number of flavors and h the composite field with the same quantum numbersas the SM Higgs. The particles σ, Θ , ˜Π all have masses of the order of v . The Higgs mass,the Higgs VEV and the Θ mass, for instance, are v = m ( λ + λ − λ ) , m h = 2 m , m = 4 v λ , (41)up to corrections of the order of O (Λ T C /M U ) due to contributions from α -terms.The lightest pesudoscalars of the unparticle sector are the pseudo Goldstone bosonsemerging from the explicit breaking of the global flavor symmetry in the unparticle sector.Their mass can be read off from the linear term in O U of the effective Largrangian (37) m U ≃ Λ T C (cid:18) Λ T C M U (cid:19) (cid:18) Λ U Λ T C (cid:19) γ U (cid:18) Λ T Λ T C (cid:19) γ T . (42)14 .2 Regularized unparticle propagator In our model the unparticle propagator to be used for phenomenology, defined from thethe K¨all´en-Lehmann representation (A.11), is∆ U ( q , Λ , Λ ) = − B d U π Z Λ Λ ds s d U − s − q − i s.t. . (43)For d U > s.t. ) are a mnemonic. More precisely, the part which is sensitive to Λ UV is ambiguousdue to the presence of, in principle computable, counterterms, which are expected to beof order one . This will limit, in practice, the predictivity of the theory . Modelingthe UV and IR transition regions by hard cut offs is of course a crude model. Yet thisshould not be relevant as long as q is sufficiently far away from these cut offs. Whereas,for q close to the cut off, the integral has an endpoint singularity which is, to a greatextent, a model artefact. The situation could be ameliorated for instance by smearing themomentum with a smooth probability density. Due to the breaking of the scale invariancethere will be single and multiparticle states appearing in the spectrum, which will affectthe q ∼ Λ behavior. Having made these statements, we now turn to the evaluation ofthe integral in (43). It can be expressed as the difference of an IR and UV part,∆ U ( q , Λ , Λ ) = f d U (Λ , q + i − f d U (Λ , q + i , (44)given by, f d U (Λ , q ) ≡ h B d U π (Λ ) d U − d U − i F (1 , − d U , − d U , q Λ ) | {z } ≡ ¯ f d U ( q / Λ ) . (45)For later convenience we give the behaviour of the function ¯ f d U ( x ) for small and largeargument appropriate for the respective domains:¯ f d U ( x + i 0) = ( a + a x + O ( x ) x ≪ a d U − ( − x − i d U − + a − x + O ( x ) x ≫ , (46) The counterterms are expected to be of order one in a theory which is not fine tuned. This is alsoknown under the term: ’naturalness’. In our model the UV completion is known and the countertermscould in principle be determined, but in practice this is outside the scope of our possibilities. a = 1 , a − = − Γ(1 − d U )Γ(3 − d U )Γ(2 − d U ) a = d U − d U − , a d U − = Γ(3 − d U )Γ( d U − . (47) < d U < < d U < IR → UV → ∞ ∆ U ( q , Λ , Λ ) = B d U d U π ) ( − q − i d U − < d U < , (48)using Γ( z )Γ(1 − z ) sin( πz ) = π . Note that for finite cut offs the UV part of the propagatoris suppressed by (Λ IR / Λ UV ) − d U ) and is therefore of minor importance for d U close to 1. ≤ d U < d U > d U → 3, for example,the UV contribution is formally the same as the fermion loop contribution to the Higgsmass in the SM, e.g. Fig. 3, which is quadratically divergent. The effective theory isvalid for q ≪ Λ and therefore the coefficient a (46) is relevant for the UV part of thepropagator (44). In practice this means that only a single counterterm, the one associatedwith a is relevant. As stated earlier, by naturalness the counterterm are expected to beof order one.Note that the limit d U → d U → ∆ U ( q , Λ , Λ ) = B d U π log (cid:18) − q − i (cid:19) + s.t. , (49)where s.t. stands for subtraction terms (counterterms).16 .3 Unparticle-Higgs mixing We shall now turn to the question of the mixing of the unparticle and the Higgs. Ourfindings resemble results from extra dimensional models. E.g. the model called HEIDI[43], where the continuous spectrum is mimiked by an infinite tower of narrowly spacedKaluza-Klein modes. The difference is that our model is inherently four dimensionaland that the parameters, such as the IR cut off and the strength of the unparticle-Higgscoupling, are related to each other. Our model is also different from the one in reference[10] since, although both are in four dimensions, the Higgs and unparticle coupling emergesdynamically within a UV complete theory.The interaction term from Eq. (37) L mix = − g h O U h O U , g h O U = k ( F − √ F , (50)introduces a mixing between the Higgs and the unparticle. The constant k has massdimension γ U . Its size, on which we will comment below, is crucial for the qualitativenature of the physics. The Higgs propagator is obtained from inverting the combinedHiggs-unparticle system∆ hh ( q ) = 1 q − m h − g h O U ∆ U ( q , Λ , Λ ) . (51)This, of course, results in unparticle corrections controlled by g h O U . The propagator canbe rewritten in terms of a dispersion representation∆ hh ( q ) = − Z ds ρ hh ( s ) s − q − i , (52)where the density, Z dsρ hh ( s ) = 1 , (53)is automatically normalized to unity. The non zero value of the coupling g h O U results solelyin a change of basis (or poles and cuts) of the intermediate particles but does not changethe overall density of states. A direct way to derive (53) is to equate the representations(52) and (51), multiply them by q and take the limit q → ∞ resulting in (53). Pleasenote, this only works in the case where d U < 3, for which the interaction (50) is powercounting renormalizable. If this condition is not fulfilled one could resort to a subtracteddispersion relation. 17he dispersion representation can be split into resonance and continuum contribu-tions, ρ hh ( s ) = X i r i δ ( s − ¯ m i ) + σ ( s ) . (54)The resonance contribution, if present, can then be obtained from the pole equation∆ − hh ( ¯ m i ) = 0 , r i = (cid:12)(cid:12)(cid:12)(cid:12) d ∆ − hh ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) − s = ¯ m i ≤ . (55)The residues r i are smaller (or equal) to one as a consequence of the normalization con-dition Eq. (53). The continuum is simply given by the cut, σ ( s ) = Θ( s − Λ )Im[∆ − hh ( s )] , (56)which corresponds to the imaginary part; most familiar from the optical theorem.To a large extent the spectral function is characterized by the zeros of the pole equationand the onset of the continuum relative to the poles. This will depend on the strength ofthe mixing and the anomalous dimension. Somewhat exotic effects can be obtained whenthe mixing term is made very large [44, 28] . In our model the mixing is determined by k (50). Its parametric value is given by k ∼ α ′ Λ γ U T C (cid:18) Λ T C M U (cid:19) (cid:18) Λ U Λ T C (cid:19) γ U w γ T T , w T = (cid:18) Λ T Λ T C (cid:19) (57)which we have normalized to the TC scaleThe value of k is, of course, suppressed by the large scale M U per se, but receivesenhancements from the powers of the anomalous dimensions. For the maximal allowedanomalous dimensions γ U ≃ γ T ≃ k Λ − γ U T C ≃ α ′ · O (10 − ). We therefore expect the coupling g h O U Λ − γ U T C (50) to be considerably smaller than one.In this case there is generally a unique solution to the pole equation. In the IR region1 < d U < < d U < The pole description is only adequate in the narrow width approimation. The Higgs width is of courserather sizable in a theory of strong interactions. The presentation below is meant to be for illustrativepurposes only. It should also be mentioned that for very large mixing the theory typically becomes unstable. The poleequation has tachyonic solutions and vertices grow in an uncontrolled manner, indicating the appearanceof a new vacuum. It is possible though that interesting effects could arise in a somewhat intermediateregime. a (46).At the qualitative level it is an interesting question of whether the Higgs resonance isbelow or above the threshold [43, 10]. For the values chosen in the caption of Fig. 2 theHiggs resonance is close to the IR cut off. On the other hand it could very well be that thescale M U is closer to the GUT scale which would decrease the IR cut off definitely belowthe composite Higgs mass scale. In appendix D we comment on the unparticle limit ofthe HEIDI models and compare our parameters with the fit of that model to the excessof the Higgs search at LEP. We introduced a framework in which the Higgs and the unparticle are both composite.The underlying theories are four dimensional, asymptotically free, nonsupersymmetricgauge theories with fermionic matter. We sketched a possible unification of these twosectors at a scale much higher than the electroweak scale. The resulting model resemblesextended technicolor models and we termed it extended technicolor unparticle (ETU).The coupling of the unparticle sector to the SM emerges in a simple way and assumes theform of four-Fermi interactions below M U .In our model the unparticle sector is coupled to the composite Higgs. Another possi-bility is to assume that the Higgs sector itself is unparticle-like, with a continuous massdistribution. This UnHiggs [45, 46] could find a natural setting within walking technicolor,which is part of our framework. Of course it is also possible to think of an unparticlescenario that is not coupled to the electroweak sector, where scale invariance is brokenat a (much) lower scale. This could result in interesting effects on low energy physics asextensively studied in the literature.With respect to our model in the future one can: • Study the composite Higgs production in association with a SM gauge boson, bothfor proton-proton (LHC) and proton-antiproton (Tevatron) collisions via the lowenergy effective theory (37). In references [47, 48] it has been demonstrated thatsuch a process is enhanced with respect to the SM, due to the presence of a lightcomposite (techni)axial resonance . The mixing of the light composite Higgs withthe unparticle sector modifies these processes in a way that can be explored at collid-ers. Concretely, the transverse missing energy spectrum can be used to disentanglethe unparticle sector from the TC contribution per se. A similar analysis within an extradimensional set up has been performed in [49]. Use first principle lattice simulations to gain insight on the nonperturbative (near)conformal dynamics. It is clear from our analysis that this knowledge is crucial fordescribing and understanding unparticle dynamics. As a model example we haveconsidered in the main text partially gauge technicolor introduced in [50]. Thesegauge theories are being studied on the lattice [22, 23, 24]. Once the presence ofa fixed point is established, for example via lattice simulations [18, 19, 20, 21], theanomalous dimension of the fermion mass can be determined from the conjecturedall order beta function [3, 25], as done in section 2. Moreover, on the lattice oneshould be able to directly investigate the two-point function, i.e. the unparticlepropagator. • Investigate different models at the ETU level. For example one could adapt somemodels, introduced to generate masses to the SM fermions, in [51, 52, 53, 54, 55,56, 57] to improve on our ETU model. • Study possible cosmological consequences of our framework. The lightest baryon ofthe unparticle gauge theory, the Unbaryon , is naturally stable (due to a protected U (1) unbaryon number) and therefore a possible dark matter candidate. Due to thefact that we expect a closely spaced spectrum of Unbaryons and unparticle vectormesons, it shares properties in common with secluded models of dark matter [58]or previously discussed unparticle dark matter models [59].Within our framework unparticle physics emerges as a natural extension of dynamicalmodels of electroweak symmetry breaking. As seen above the link opens the doors to yetunexplored collider phenomenology and possible new avenues for dark matter, such as theuse of the Unbaryon. Acknowledgments FS thanks Mads T. Frandsen for discussions. RZ is grateful to Johan Bijnens, OliverBrein, Mike Pennington, Tilman Plehn and Thomas Gregoire for discussions. The workof FS is supported by the Marie Curie Excellence Grant under contract MEXT-CT-2004-013510. RZ gratefully acknowledges the support of an advanced STFC fellowship.Further support is offered by the Marie Curie research training networks contract NosMRTN-CT-2006-035482, Flavianet , and MRTN-CT-2006-035505, Heptools .20 γ -induced condensate hO U i In this appendix we intend to sketch how the γ -term, in addition to the α -term in Eq. (1),can induce induce an unparticle VEV. The treatment essentially follows the Nambu − Jona-Lasino model [60]; a simple and concise summary of the latter is given in the appendix ofreference [35]. The γ -term δ L effΛ U = γ ′ Λ γ U U M U O U (A.1)can be rewritten into the following form δ L effΛ U = ( p γ ′ Λ U M U Λ γ U U O U H + h.c ) − Λ U | H | , (A.2)by the purely formal manipulation of introducing an auxiliary field H . The crucial ques-tion is then whether the coupling of the γ -term is large enough to enforce a dynamicalVEV. This will be decided solely by the sign of the | H | -term. One has to integrateout the fermions between the scales Λ U and µ . This is straightforward in the unparticlescenario since the propagator is known, up to UV and IR cut offs. The | H | -term is thensimply given by contracting the unparticle propagator (43) between two O U H interactionpoints; this leads to δ L eff µ = − Λ U | H | (cid:16) γ ′ Λ γ U U M U ∆ U ( − µ , Λ , Λ ) (cid:17) ≡ − m H | H | . (A.3)For m H < | H | field acquires a VEV and induces an unparticle VEV through thegap equation. We remind the reader that the value of γ ′ is expected to be of the orderone. In the range γ U ∼ − m H is negative for γ ′ > M U / Λ U which woulddemand an unnatural enhancement of the γ ′ coefficient. For γ U > γ -term becomesa relevant operator and one could expect a qualitatively change in the picture. Around γ U . γ ′ > (Λ T / Λ U ) / ( M U / Λ U ) / (Λ T C / Λ U ) / and could indeedlead to VEV at a scale comparable to the one from the α VEV (17). To determine thevalue of the VEV we would need to evaluate the coefficient of the | H | -term which is adifficult task per se and beyond the scope of this paper. B Comparison with Na¨ıve Dimensional Analysis In reference [9] it was pointed out that the interaction of a SM operator to the unpar-ticle sector would act as a source of breaking the scale invariance. In the absence of an21nderlying model, the authors resorted to NDA. We will see here that the physics ofthe condensate or the anomalous dimension is, of course, not captured by such a genericapproach as in reference [9].The schematic notation in (1) made more precise [4] reads, L eff = M U (cid:16) Λ U M U (cid:17) d UV − d U (cid:16) O SM M d SM U (cid:17)(cid:16) O U M d U U (cid:17) , (A.4)where d SM / UV are the scaling dimensions of the SM operator and the unparticle operatorin the UV. Assuming that O SM → v d SM acquires a VEV at the electroweak scale v , NDAthen suggests that scale invariance is broken at a scale Λ NDAIR ,(Λ NDAIR ) ≃ L eff ( O U → (Λ NDAIR ) d U , O SM → v d SM ) , (A.5)when the term in Eq. (A.4) is of the same size as a generic four dimensional operator ofthe unparticle sector. This leads toΛ NDAIR ∼ v (cid:16) Λ U M U (cid:17) d UV − d U − d U (cid:16) vM U (cid:17) d SM4 − d U − . (A.6)The above equation reduces to (3.3) in reference [9] for O SM = | H | with d SM = 2.In our work O SM → O TC = ¯ QQ (6) with d SM = 3. The role of the electroweak scale istaken by v → Λ TC . The knowledge of the UV completion settles the question on the UVdimension; d UV = 3. Furthermore the anomalous dimension γ T introduces an additionalmultiplicative factor w γ T T (9) to the Lagrangian density (A.4) as an artefact of walkingtechnicolor. Altogether this yieldsΛ NDAIR → Λ TC ( w T ) γ T γ U (cid:16) Λ U M U (cid:17) γ U γ U (cid:16) Λ TC M U (cid:17) − γ U γ U . (A.7)The crucial question is then how this compares with the IR cut off in (20). We find that,for generic values of the parameters, the condensate IR cut off isΛ IR & Λ NDAIR , (A.8)larger than the IR cut off suggested by NDA. The essential point is that the VEV issensitive to the UV cut off for d U ≥ NDAIR .This suggests that parametric equality (A.8) is reached in the IR sensitive domain22 U & 1. Most reassuringly it is verified that in this domain both IR cut offs scale asΛ TC (Λ U /M U ) / . In the UV domain d U . NDAIR ∼ Λ TC (Λ TC /M U ) and Λ IR ∼ Λ TC (Λ U /M U ) / . In Fig. 2(right) the logarithm of the ratio of the two IRcut offs is plotted against γ U for specified input values and provides an example of thequalitative statement made above. C Normalization factor B d U In this appendix we shall discuss the normalization factor b d U ≡ p B d U / (2 π ) (11) and(16). This is a necessary task in order to extract quantitative results from the unparticleVEV equation (17). Generally we do not know the behaviour of B d U as a function of d U ≡ − γ U , except around d U & d U = 3. Firstly, it is a fact that at d U = 1 theoperator O U is equivalent to a free field [36]. This fixes the normalization factor, B d U = 2 π ( d U − 1) + O (( d U − η ) with η > , for d U & , (A.9)around d U & d U → + ρ d U ( s ) = lim d U → + ( d U − s d U − θ ( s ) = δ ( s ) , (A.10)with unit residue. This is equivalent to Georgi’s [4] requirement that A d U , in the notationof that paper, has to reproduce the 1-particle phase space in that limit. Secondly, in ourmodel at d U = 3 the fermions are free fields and the unparticle propagator, which wewrite in a K¨all´en-Lehmann form,∆ U ( q ) ≡ − i Z d xe i q · x h | T O U ( x ) O †U (0) | i = − Z ds ρ d U ( s ) s − q − i s.t. , (A.11)has to reduce to the free fermion loop depicted in Fig. 3. The letters s.t. denote possiblesubtraction terms which are relevant for d U > B d U (11) at d U = 3 to ρ ( s ) = s N U π ↔ B = N U π . (A.12)This value is different from A = 1 / (256 π ) (A.13)23 ΨΨ¯ΨΨ Figure 3: Fermion bubble with scalar vertices, corresponding to the unparticle propagator inthe limt d U → obtained from the normalization, A d U = 16 π / (2 π ) d U Γ( d U + 1 / d U − d U ) = 12 1(4 π ) d U − d U )Γ( d U − 1) (A.14)in reference [4]. This is not surprising since in this reference it was proposed to adapt A d U as the analytic continuation of the phase space of an integer number of d U masslessparticles. The operator O U = ϕ d U , with ϕ denoting a free massless scalar field, is ofcourse a special realization of the unparticle scenario for integer scaling dimension d U .We would like to emphasize that in reference [4] it was clearly stated that the actualnormalization might be rather different from the one in a concrete model.In the case at hand O U | d U =3 = ¯Ψ Ψ corresponds to two free fermions, instead of threefree boson, which explains the difference. One could in principle generalize this scenarioto higher powers of pairs of free fermion fields and adapt it as the normalization conditionsfor B d U via analytic continuation. Unfortunately it appears that no closed formula canbe written down for this case. In order to obtain some quantitative results we resort tomodel B d U by a quadratic interpolation function,( B d U ) interpol = 2 π ( d U − 1) + (cid:16) N U π − π (cid:17) ( d U − . (A.15)Please not that these interpolation formula is positive as required by a positive spectralfunction (11). We would like to emphasize once more that the only firmly known partsare B (model dependent) and the behaviour around B (model independent).24 Unparticle limit of HEIDI models In the HEIDI model [28] the Higgs-Higgs propagator assumes the∆ hh ( q ) = 1 q − M − c ( m − q − i d − . (A.16)same form as in (51). The letter c denotes a dimensionful constant proportional to themixing parameter, m is the mass of the lowest Kaluza Klein excitations and M is thetree-level Higgs mass. 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