Dynamical generation of the weak and Dark Matter scales from strong interactions
DDynamical generation of the weak andDark Matter scales from strong interactions
Oleg Antipin a , Michele Redi a , Alessandro Strumia ba INFN, Sezione di Firenze, Via G. Sansone, 1; I-50019 Sesto Fiorentino, Italy b Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italyand National Institute of Chemical Physics and Biophysics, Tallinn, Estonia
Abstract
Assuming that mass scales arise in nature only via dimensional transmutation, weextend the dimension-less Standard Model by adding vector-like fermions chargedunder a new strong gauge interaction. Their non-perturbative dynamics generatesa mass scale that is transmitted to the elementary Higgs boson by electro-weakgauge interactions. In its minimal version the model has the same number ofparameters as the Standard Model, predicts that the electro-weak symmetry getsbroken, predicts new-physics in the multi-TeV region and is compatible with allexisting bounds, provides two Dark Matter candidates stable thanks to accidentalsymmetries: a composite scalar in the adjoint of
SU(2) L and a composite sin-glet fermion; their thermal relic abundance is predicted to be comparable to themeasured cosmological DM abundance. Some models of this type allow for extraYukawa couplings; DM candidates remain even if explicit masses are added. Contents a r X i v : . [ h e p - ph ] F e b Introduction
The idea that the weak scale could be dynamically generated from strong interactions has along history. Originally, techni-color models were developed as an alternative to the Higgs:the weak interactions of the techni-quarks Q were chosen so that their condensates wouldbreak the SM electro-weak group and the weak scale was the techni-color scale. This scenariowas disfavoured by flavour and precision data even before the first LHC run, where the Higgsand no new physics was observed.Later, strong dynamics was invoked to generate a composite or partially-composite Higgs,although realising complete models is so complicated that model-building is usually substi-tuted by postulating effective Lagrangians with the needed properties.Recently, models where new strong dynamics does not break the electro-weak symmetrynor provide a composite Higgs have been considered in the literature, just because they aresimple, phenomenologically viable and lead to interesting LHC phenomenology [1]. Withabuse of language we use the old name ‘techni-color’. In this paper we show that thesemodels1. provide Dark Matter candidates;2. provide a dynamical origin for the electro-weak scale, if we adopt the scenario of ‘finitenaturalness’ [2, 3, 4].Point 2 amounts to assuming that quadratically divergent corrections to the Higgs masshave no physical meaning and can be ignored, possibly because the fundamental theory doesnot contain any mass term [4]. In this context, dynamical generation of the weak scalevia dimensional transmutation has been realised with weakly-coupled dynamics, in modelswhere an extra scalar S has interactions that drive its quartic λ S | S | negative around or abovethe weak scale: S acquires a vev at this scale, and its interaction λ HS | H | | S | effectivelybecomes a Higgs mass term, m = λ HS (cid:104) S (cid:105) [5]. A related possibility is that the scalar S isinteracting with techni-quarks [6] or charged under a techni-color gauge group [7] and again S acquires a vev or forms a condensate. In all these models (cid:104) S (cid:105) can be pushed arbitrarilyabove the weak scale by making λ HS arbitrarily small, leaving no observable signals.We here consider simple models without any extra scalar S beside the Higgs doublet H .The SM is extended by adding a gauge group G TC (for example SU( N ) ) and techni-quarks Q L charged under the SM, as well as the corresponding Q R in the conjugated representationsof the gauge group G SM ⊗ G TC , so that Q L ⊕ Q R is vectorial. As a consequence the condensate (cid:104)Q L Q R (cid:105) transforms as a singlet of G SM and does not break it.The techni-quarks have no mass terms because of our assumption that only dimension-less couplings exist ; for certain assignments of their gauge quantum numbers, techni-quarks Relaxing this hypothesis allows other interesting possibilities for Dark Matter that will be discussed in aseparate publication [8]. y with the elementary SM Higgs doublet H . The scenario thatwe consider is described by the renormalizable Lagrangian L = L m =0SM − G A µν + ¯ Q iL i /D Q iL + ¯ Q jR i /D Q jR + ( y ij H Q iL Q jR + h.c. ) (1)where L m =0SM describes the SM without the Higgs mass term, and G Aµν is the techni-color fieldstrength. In models where Yukawa couplings y are not allowed (for example techni-quarksin the 3 of SU(2) L ) the number of free parameters is the same as in the Standard Model: allnew physics is univocally predicted. This new physics manifests as: • Strong dynamics generates a dynamical scale Λ TC that can be identified with the mass ofthe lightest vector meson resonance, the techni- ρ , and spontaneously breaks accidentalchiral symmetries conserved by the techni-strong interactions producing light pseudo-Goldstone bosons (GB). Using large N counting m ρ = g ρ f where f is the decay constantof the techni-pions and g ρ ≈ π/ √ N . • In absence of techni-quark masses, the techni-pions π ∼ Q L Q R acquire mass m π ≈ α m ρ / π from the electro-weak gauge interactions that explicitly break the global techni-flavour accidental symmetries. Yukawa couplings also contribute to their masses; inabsence of Yukawa couplings the lightest techni-pions could be a stable SU(2) L tripletproviding a viable DM candidate. • The heaviest new particles are techni-baryons with mass m B ≈ N m ρ . The lightesttechni-baryon is stable and is a natural DM candidate; if it is a thermal relic, the ob-served DM abundance is reproduced for m B ≈
100 TeV [9].The LHC phenomenology of techni-strong dynamics was discussed in [1]. The main newpoint of our work is the possible connection with the weak scale and implications for darkmatter. Assuming that power divergences vanish [2, 4], the techni-strong interactions givea finite negative contribution to the Higgs squared mass term, such that the weak scale isdynamically generated. The Higgs physical mass arises as M h ≈ + α f + y m ρ f m π (2)so that the techni-color scale is predicted to be f ≈ M h /α ≈ few TeV , or smaller in modelswhere y is present and dominant in eq. (2). Unlike ordinary techni-color as a solution to theusual hierarchy problem, where the natural scale for new physics is the weak scale itself, inthis scenario the natural mass scales are m π ∼ , m ρ ∼
20 TeV , m B ∼
50 TeV . (3)New physics effects in accelerator searches and precision experiments are well below thepresent sensitivity. In particular no new effects are generated in flavor physics. Techni-pions [11] and techni-baryons [12], stable due to accidental symmetries of the renormaliz-able Lagrangian, can provide a thermal Dark Matter candidate.3his work is organised as follows. In section 2 we consider the Higgs mass generated bythe SM electro-weak gauge couplings, by the SM strong coupling, and by the Yukawa cou-plings of the Higgs with the techni-quarks, allowed in some models. Dark Matter is discussedin section 3. We conclude in section 4. In the appendix we present the technical details ofthe computation of the potential induced by Yukawa interactions. We write the tree-level potential of the SM Higgs doublet H as V = m | H | + λ | H | . (4)If m ≡ − M h / is negative, the Higgs doublet H develops the vacuum expectation value v = M h / √ λ ≈ . : expanding the potential V around its minimum as H = (0 , ( v + h ) / √ shows that M h ≈
125 GeV is the tree-level mass of the physical Higgs boson h .Under our assumptions, the only mass scale of the theory is set by the dynamical scale ofthe techni-color sector. Through loop corrections it induces other scales and in particular theHiggs mass parameter. Electro-weak interactions of the techni-quarks induce a 2-loop con-tribution, computed in section 2.1, and color charges give a 3 loop contribution to the Higgsmass, computed in section 2.2. If the Higgs couples to the techni-quarks through Yukawainteractions (for example if techni-quarks contain doublets and singlets under the electro-weak interactions) a contribution to the Higgs mass is also generated at 1-loop, computed insection 2.3. Electro-weak gauge interactions give a minimal, quasi-model-independent, contribution tothe Higgs mass, described by the non-perturbative techni-color multi-loop dressing of the two-loop Feynman diagram in fig. 1a (plus the associated seagull diagram): the Higgs interactswith the electro-weak vectors, that interact with the techni-quarks.To leading order in the SM interaction, and to all orders in the techni-strong interactions,the techni-strong dynamics corrects the SM electro-weak gauge bosons propagator as D Y Yµν ( q ) = − i η µν q (1 + g Y Π Y Y ( q )) + iξ Y q µ q ν q (5) D abµν ( q ) = − i η µν q (1 + g Π W W ( q )) δ ab + iξ W q µ q ν q δ ab (6)where ξ V are gauge-fixing parameters. Techni-strong dynamics is encoded in the Π V V ( q ) functions. From the point of view of the techni-strong dynamics, they are the renormalisedtwo-point functions of the currents J aµ = (cid:80) i ¯ Q i γ µ T a Q i Q i (where Q i = ( Q iL , ¯ Q iR ) is a Dirac4 eak coupling H HHW WQQ
Strong coupling at high energy (cid:88) G ΜΝ a (cid:92) H HHW WQ
Strong coupling
H HHW W Ρ Figure 1:
The two loop contribution to the Higgs mass coming from the electro-weak gaugeinteractions of: a) a techni-quark, to be dressed with non-perturbative techni-interactions, ap-proximated as: b) the techni-gluon condensate; c) the techni- ρ . The extra seagull diagram is notexplicitly plotted. spinor and T a are the SM gauge generators) corresponding to the unbroken part of the acci-dental global techni-flavour symmetry, partially gauged by electro-weak interactions: i (cid:90) d x e iq · x (cid:104) | T J Vµ ( x ) J V (cid:48) ν (0) | (cid:105) ≡ δ V V (cid:48) ( q g µν − q µ q ν )Π V V ( q ) . (7)The correction to the Higgs mass is ∆ m = − i (cid:90) d q (2 π ) g Π W W ( q ) + g Y Π Y Y ( q ) q , (8)and, performing the Wick rotation to the Eucliedan Q = − q > , ∆ m = 34(4 π ) (cid:90) dQ (cid:20) g Π W W ( − Q ) + g Y Π Y Y ( − Q ) (cid:21) . (9)In general the integral above is UV-divergent, quadratically and logarithmically. In the caseat hand, the unphysical power divergences are ignored because of our assumption of finitenaturalness, and logarithmic divergences (that describe the RGE running of m ) are absent,because of our assumption that the only mass scale, Λ TC , is generated dynamically. Therebythe generated squared Higgs mass term is finite and scheme independent.We next show that the electro-weak interactions induce a calculable negative Higgs massso that the electro-weak symmetry is spontaneously broken. We proceed in 3 steps: dispersionrelations in section 2.1.1 show in general that ∆ m < , Operator Product Expansion insection 2.1.2 shows that ∆ m is ultra-violet finite, vector meson dominance and/or large N in section 2.1.3 allow to give the estimate ∆ m ≈ − α f .5 .1.1 Dispersion relation Under our assumptions, quadratically divergent terms are zero and we are interested in thedependence on the physical scales of the theory. To extract this we consider the variation ofthe Higgs mass with respect to the dynamical scale of the theory Λ TC , ∂ ∆ m ∂ Λ = 3 g π ) (cid:90) dQ (cid:20) g ∂ Π W W ∂ Λ + g Y ∂ Π Y Y ∂ Λ (cid:21) . (10)The sign of the gauge correction ∆ m can be determined using the dispersion relation [16] ∂ Π V V ( q ) ∂q = 1 π (cid:90) ∞ ds Im Π
V V ( s )( s − q − i(cid:15) ) . (11)where we use the conventions of [17]. The optical theorem relates the cross-sections σ ( s ) to Im Π
V V ( s ) , allowing to show in general that Im Π
V V ≤ . For dimensional reasons, thedimension-less Π V V can only depend on Q / Λ . Thereby ∂ Π V V ∂ Λ = − Q Λ ∂ Π V V ∂Q = Q Λ π (cid:90) ImΠ
V V ( s )( s + Q ) ds < (13)where in the last step we used the dispersion relation. A similar relation holds for the hyper-charge contribution. The integrand in (10) is negative definite corresponding to a negative ∆ m given the boundary condition ∆ m = 0 for Λ TC = 0 . In a theory with a dynamical scale Λ TC , arguments based on Operator Product Expansionallow to show that ∂ ∆ m /∂ Λ is ultra-violet convergent as expected and to compute thehigh-energy tail of Π V V ( q ) . Π V V can be expanded as Π V V ( q ) q (cid:29) Λ (cid:39) c ( q ) + c ( q ) (cid:104) | m Q Q L Q R | (cid:105) + c ( q ) (cid:104) | α TC π G A µν | (cid:105) + · · · . (14)The first term (unity operator) does not contribute to (10). Indeed, at leading order it de-scribes the diagram in fig. 1a with techni-quarks but neglecting their techni-color interactions,such that c = C π ln( − q ) + · · · (15) As a check, replacing techni-color with a perturbative one-loop correction of fermions with explicit mass m Q , one would obtain ∂ Π V V ( − Q ) ∂m Q = − g π Q m Q (cid:90) x (1 − x ) m Q + x (1 − x ) Q . (12)Inserting this into eq. (10) the integrand is negative definite but the integral is logarithmically divergent. Thiscorresponds to a contribution proportional to g m Q in the RG equation for the Higgs mass m . No such UV-divergent RGE effect is present in a techni-color theory that generates dynamically a mass scale Λ TC from adimension-less coupling g TC , given that, in any mass-independent scheme such as Minimal Subtraction, only g TC can appear in the RGE. C > is a model-dependent group theory factor given by C = Tr T a T a in terms of the SU(2) L techni-quark generators (with a similar expression factor for the U(1) Y generators).This high energy tail does not contain any mass scale, so that the associated quadraticallydivergent no-scale integral in eq. (9) vanishes, under our assumptions. The second term alsovanishes, because it is proportional to the techni-quark masses m Q that vanish under ourassumption that the theory does not contain any mass scale.The third term in eq. (14) is represented by the Feynman diagram in fig. 1b, which gives c = − C (cid:48) /q [16], where C (cid:48) > is another order one model-dependent group theory fac-tor. The techni-gluons form a positive condensate (the condensate is positive-defined in theEucliedian path-integral [16], in agreement with QCD lattice computations) (cid:104) | α TC π G A µν | (cid:105) = κ Λ . (16)where κ > is an order-one coefficient. This allows to show that the UV contribution to thesquared Higgs mass term is negative as expected: ∆ m | UV (cid:39) − C (cid:48) g π ) κ Λ (cid:90) ∞ Q dQ Q ≈ − α κ Λ Q . (17)The /Q dependence on the artificial infra-red cut-off Q min ∼ Λ TC shows that the dominanteffects comes from virtual momenta Q around the techni-meson masses. The dominant contribution to the Higgs mass comes from the Q region densely populated bythe techni-meson resonances. A variety of methods have been proposed to approximativelydescribe such region: vector meson dominance, Weinberg sum rules, large N , holographicmodels... As long as the techni-quarks are charged under the electro-weak group, they form,among the various mesons, spin-1 resonances that mix with the SM electro-weak vectors V µ .This is described by the effective Lagrangian L eff = − g V aµν V a µν − g ρ ρ aµν ρ a µν + f V aµ − ρ aµ ) (18)such that the massless eigenstate has gauge coupling /g = 1 /g + 1 /g ρ and the orthogonalheavy state has mass m ρ = f ( g + g ρ ) . Integrating out the ρ at tree-level one finds: Π V V ( q ) = m ρ g ρ ( q − m ρ + i(cid:15) ) . (19)Plugging eq. (19) into eq. (9) we obtain a logarithmically divergent infra-red correction tothe squared Higgs mass term: ∆ m ≈ − g π ) (cid:90) dQ m ρ g ρ ( Q + m ρ ) ∼ − g m ρ (4 π ) g ρ log Λ m ρ ∼ − α f . (20)7 HGG ttQ Q H Htt ttG
Figure 2:
The three loop contribution to the Higgs mass coming from techni-quarks Q that onlyhave color interactions. Similar diagrams can be drawn for graviton contributions. The integrand is negative definite and its size agrees with the naive expectation based onthe Feynman diagram plotted in fig. 1c, including the /g ρ suppression of vector mixing. Thelogarithmic UV divergence here arises because this is only an approximate description, wherean explicit mass term m ρ substitutes the dynamical mechanism of mass generation. An infinitenumber of states would be needed to properly describe the non-perturbative dynamics.In theories with large N this can be made more rigorous: Π V V can be represented exactlyas an infinite sum of poles corresponding to the physical quasi-stable techni-mesons of thetheory: Π V V ( q ) = N π m ρ (cid:88) i c i q − m i + i(cid:15) . (21)where c i are adimensional coefficients. The infinite number of resonances allows to repro-duce the logarithmic divergence, that does not contribute to the Higgs mass zero under ourassumption of finite naturalness.These considerations offer an intuitive argument to understand the sign of ∆ m . Thenet effect of non-perturbative dynamics is creating a mass gap that stops the techni-quarkcontribution to the RGE running of g , g Y below Λ TC , effectively making g , g Y smaller withrespect to the perturbative case. As a consequence the unphysical power divergence presentin the SM, ∆ m ∼ + g ,Y Λ , gets replaced by a finite physical effect ∆ m ∼ − g ,Y Λ . We next consider techni-color models where the techni-quarks have SM color interactions.For example, techni-quarks could be a color octet of
SU(3) c , charged also under the techni-color gauge group. Then techni-quarks cannot have any Yukawa coupling to the SM Higgs:both the Yukawa contribution of section 2.3 and the electro-weak contribution of section 2.18re absent.In these models, the Higgs mass is dominantly generated at three loops: the Higgs inter-acts with the top quark, that interacts with the gluons, that interact with the techni-quarks, asplotted in fig. 2. The computation can be performed along the lines of section 2.1 by defining Π GG ( q ) , the techni-color correction to the gluon propagator. Summing the two diagrams offig. 2, the result is ultraviolet-convergent: ∆ m = − y t g (4 π ) (cid:90) dQ Π GG ( − Q ) ∼ y t g π f . (22)The computation of the sign is analogous to what described in the previous section (with Π W W replaced by Π GG ): in the present case we find a positive ∆ m , such that this contribu-tion does not induce electro-weak symmetry breaking. The sign of the effect also correspondsto the intuitive reasoning presented at the end of the previous section: the sign is opposite tothe known negative sign of the naive quadratic divergence associated with y t , because g andthereby y t are reduced by techni-strong dynamics.We mention a final possibility. The techni-quarks could be completely neutral under thewhole SM gauge group. In this situation only gravity mediates a contribution to the Higgsmass, proportional to the two-point function of the energy momentum tensor. Furthermore, asuper-Planckian techni-color condensate would dynamically generate the Planck mass itself,within a dimensionless extension of Einstein gravity such as agravity [4]. The problem isthat techni-color dynamics, dominated by a single non-perturbative coupling, has no freeparameters and would also generate a large negative cosmological constant, which is at oddwith observations. Finally, we consider the case where the gauge quantum numbers of the techni-quarks allowfor Yukawa couplings to the elementary Higgs. This choice implies the existence of a techni-pion π with the same quantum numbers of the Higgs doublet H , that can then mix with H . The left panel of fig. 3 shows the one-loop corrections to the squared Higgs mass generatedby a weakly coupled techni-quark with Yukawa interactions to the Higgs. At strong couplingthe physical degrees of freedom become bound state techni-hadrons that can be describedusing effective Lagrangian techniques. The techni-quark loop can be matched to an effectivechiral Lagrangian, so that such diagrams collapses to a tree level diagram (right-handed panelof fig. 3) dominated by the lightest techni-mesons, the techni-pions π ≈ Q L Q R . For simplicitywe here consider Yukawa couplings that preserve the Q L ↔ Q R parity of the techni-stronginteractions; a more general discussion can be found in the appendix. Similarly to quarkmasses in QCD, the Yukawa interactions produce the following term in the chiral Lagrangian, y m ρ f Tr[ HU ] + h.c. (23)9 eak coupling H H (cid:42) QU Strong coupling
H H (cid:42) Π Figure 3:
Correction to the Higgs mass coming from the Yukawa coupling with: a) a weaklycoupled massive fermion; b) a massless strongly interacting fermion. where U = exp( iπ ˆ a T ˆ a /f ) is the Goldstone boson matrix. As we discuss in detail in theappendix, upon minimisation of the potential this term induces a mass mixing ≈ ym ρ f Hπ ∗ between the techni-pion and the elementary Higgs. This term also explicitly breaks accidentalsymmetries respected by gauge interactions.What emerges is a two-Higgs doublet system where the extra Higgs doublet π is a heavycomposite doublet with negligible vev. In order to compute the mass eigenstates, we need tocompute the mass matrix. Including effects at tree and one-loop level in the SM couplings g and y , the mass matrix has the structure (cid:32) π ∗ H ∗ π ( O ( g ) ± O ( y )) / (4 π ) O ( y ) √ N / (4 π ) H O ( y ) √ N / (4 π ) −O ( y ) N/ (4 π ) (cid:33) m ρ (24)where we used the fact that the one-loop contribution of weak gauge interactions to m π ≈ g m ρ / (4 π ) is positive (as known from the SM analogous computation of the π + / π massdifference [13]), and added the one-loop Yukawa contribution (absent in the SM ). The HH ∗ entry describes the contribution of composite scalar resonances that can also mix withthe Higgs giving a negative sub-leading contribution to its mass squared, see appendix formore details.We see that the phenomenologically acceptable regime is the one where the Yukawa cou-pling is small, y (cid:28) g , such that: 1) the loop contribution coming from the Yukawa couplingcan be ignored; 2) the heaviest eigenstate is the techni-pion with squared mass m π > ; 3) The literature on composite Higgs models explored linear couplings of SM quarks to composite fermionicstates, finding that they can give a negative contribution to the Higgs mass term. Simple UV completionsrequire extra scalars as in the supersymmetric realisation of [14]. Here instead we compute the techni-pionpotential induced by a bi-linear H Q L Q R Yukawa coupling, involving techni-quarks Q and a scalar H withouttechni-strong interactions. negative squared mass termdominated by the mass mixing term in eq. (24) and given by a see-saw-like formula: ∆ m ∼ − y (4 π ) m ρ Nm π ∼ − y m ρ f m π . (25) The models described in this paper contain two Dark Matter (DM) candidates: techni-baryonsand techni-pions. Their stability is guaranteed by accidental symmetries of the renormalizableLagrangian, techni-baryon number and (possibly) G -parity [11].In fact the presence of stable states is a generic prediction of the framework that impliesrestrictions on the representations of the techni-quarks under the SM gauge group, such thatthe stable states are viable DM candidates. In table 1 we summarise the simplest allowedcharge assignments under the electro-weak group and the resulting DM candidates. Intro-ducing techni-quark masses allows several other possibilities [8].The new matter modifies the running of SM gauge couplings. Adding n weak doubletsand n weak triplets in the N ⊕ ¯ N of SU( N ) TC the beta-function of SU(2) L becomes b = −
196 + 2 N n + 4 n ) (26)such that the SU(2) L gauge coupling does not develop a Landau pole below the Planck scale( b < ∼ ) and possibly remains asymptotically free ( b < ) for small enough n , n , N . Higher SU (2) L lead to Landau poles instead. The trans-Planckian Landau pole for hypercharge canbe naturally avoided in models where hypercharge is embedded in SU(2) R below a fewTeV [10]; a technicolor sector could be used to dynamically break the extended gauge group. If techni-quarks fill N F fundamentals and anti-fundamentals of the SU( N ) TC gauge groupwith N ≥ , the spontaneous symmetry breaking SU( N F ) L ⊗ SU( N F ) R / SU( N F ) of the acci-dental global techni-flavor symmetry produces N F − Goldstone bosons in the adjoint of theunbroken
SU( N F ) . These scalars acquire mass from effects that explicitly break the globalsymmetries. Within finite naturalness the only contribution to their masses is due to SMgauge interactions, and possibly to the techni-quark Yukawa couplings.If Yukawa couplings are forbidden by the fermions quantum numbers, then the model isextremely predictive: it only has one free parameter — the techni-color scale — which isfixed by the Higgs mass under the hypothesis of finite naturalness. All the rest is univocallypredicted: techni-pion masses, Dark Matter and its thermal relic abundance.11umber of N = 3 N = 4 techni-flavors Yukawa TCb TC π TCb TC πN F = 2 2 3 1 3 under TC-flavor SU(2) model 1: Q = 2 Y =0 charged 3 DM, under
SU(2) L N F = 3 8 8 ¯6 8 under TC-flavor SU(3) model 1: Q = 1 Y + 2 Y (cid:48) no no DM, under SU(2) L model 2: Q = 3 Y =0 DM, under
SU(2) L N F = 4 20 15 20 (cid:48) under TC-flavor SU(4) model 1: Q = 4 Y =0 charged DM, under
SU(2) L N F = 5 40 24 50 24 under TC-flavor SU(5) model 1: Q = 2 Y + 3 Y (cid:48) no charged no DM, under SU(2) L model 2: Q = 5 Y =0 DM, under
SU(2) L Table 1:
Dimension-less techni-color models that give viable techni-baryon (TCb) and/or techni-pion (TC π ) Dark Matter candidates with Q = Y = 0 . We consider models with SU( N ) gaugegroup for N = { , } and N F = { , , , } flavours of techni-quarks in its fundamental plusanti-fundamental. The darker rows give the techni-flavour content of the lightest TCb and TC π considering only masses induced by techni-color interactions. The lighter rows consider modelswith viable assignments of electro-weak interactions and show, after including the mass splittingdue to unbroken electro-weak interactions, the SU(2) L content of the lighter DM candidates. The SM gauge interactions give positive squared masses to the gauge-charged techni-pions, while SM singlets remain exact massless Goldstone bosons. If the N F techni-quarkflavors are composed of k irreducible (real or pseudo-real) representations of G SM , then thetechni-pions decompose under G SM as Adj
SU( N F ) = (cid:34) k (cid:88) i =1 r i (cid:35) ⊗ (cid:34) k (cid:88) i =1 ¯ r i (cid:35) (cid:9) (27)so that k − techni-pions are neutral gauge singlets (the extra scalar singlet analog of the η (cid:48) in QCD acquires mass from anomalies with techni-interactions and will not play a role inwhat follows).One combination of singlets corresponds to a global symmetry anomalous under SU(2) L ,so that the corresponding Goldstone boson acquires an axion-like couplings to SM vectors:an almost massless axion with a decay constant f ∼ TeV would be grossly excluded by starcooling and other bounds. In absence of techni-quark Yukawa interactions, these boundssignificantly reduce the space of models favouring the simplest models with k = 1 . Thetechni-quarks should belong to a single irreducible representation j = ( N F − / of SU(2) L and, in order to obtain a neutral lightest techni-baryon, the techni-quark hypercharge shouldvanish. Then the N F − techni-pions lie in the following irreducible representations J of12 U(2) L : Adj
SU( N F ) = N F − (cid:88) J =1 J. (28)Models of this kind were studied in [11], where it was pointed out that a discrete symmetry,“ G -parity” exists in these theories (for zero hypercharge) due to the fact that SU(2) L rep-resentations are real or pseudo-real. G parity acts on techni-quarks as Q → exp( iπT ) Q c ,replacing any SU(2) L representation with its conjugate representation, which is equivalentto the original representation. SM fields are neutral. On techni-pions G parity becomes the ( − J Z symmetry, so that techni-pions with even (odd) isospin (J) are even (odd). Summa-rizing: • Techni-pion singlets under
SU(2) L are G -even, do not acquire masses from SM gaugeinteractions and can have anomalous couplings to SU(2) L vectors: they are excludedin our framework unless Yukawa couplings make them massive. They are absent iftechni-quarks fill a single irreducible representation of SU(2) L . • Techni-pions in the 3 of
SU(2) L are G -odd and could be the lightest stable DM candi-dates. The simplest models are listed in table 1. • Techni-pions in the 5 of
SU(2) L are G -even and are heavier, m π ≈ √ m π : they un-dergo anomalous decays into electro-weak vectors, π → W W . • Techni-pions in higher representations of
SU(2) L , if present, decay into lighter techni-pions respecting G -parity by emitting two SU(2) L vectors, e.g. π → π W W .The situation is different in models where Yukawa couplings y of techni-quarks to theelementary Higgs are present. The Yukawa couplings break explicitly G -parity and accidentalglobal symmetries so that the SM singlet techni-pions η receive non-zero masses given byeq. (56), M η ∼ | y | vm ρ /m π and star cooling bounds are easily avoided. Furthermore, techni-pions can now decay through the Higgs, so that only techni-baryons remain as dark mattercandidates.Models with techni-color gauge group SU(2) ∼ Sp(2) are special: its fundamental repre-sentation is pseudo-real, ∼ ∗ , so that the techni-flavour symmetry is enhanced becoming SU(2 N F ) / Sp(2 N F ) . The extra techni-pions are QQ scalars and there are no stable techni-baryons. Dangerous light techni-pions neutral under SU(2) L are again absent if techni-quarkslie in a single representation of SU(2) L with dimension N F . Within our assumptions how-ever these models do not provide DM candidates because techni-pions are G -even.13 .2 Techni-baryons Techni-baryons are techni-color singlet states constructed with N techni-quarks. The sta-bility of the lightest techni-baryon follows from the accidental techni-baryon number globalsymmetry.Using the non-relativistic quark model, group theory allows to compute the electro-weakquantum numbers of the techni-baryons: their wave-function must be anti-symmetric in thetechni-quarks. The wave function is assumed to be antisymmetric in techni-color, and so mustbe symmetric in spin and flavour for the lightest techni-baryons that have no orbital angularmomentum. Different techni-baryons are split by spin-spin interactions that prefer, as lightesttechni-baryon, the one with smallest spin. As a consequence, the lightest techni-baryons havespin 0 (1/2) for even (odd) N ≥ .In general the SU( N F ) techni-flavour representation of the lightest techni-baryon corre-sponds to a Young diagram with 2 rows having N/ boxes each (for N even) and to a Youngdiagram with 2 rows having ( N + 1) / and ( N − / boxes respectively (for N odd). Inparticular, they are for N = 3 and for N = 4 . (29)This is the end of the story, as long as techni-color interactions are involved.Next, the components of a techni-baryon multiplet are split by SM gauge interactions,and possibly by techni-quark Yukawa interactions. The lightest components are those withsmallest G SM charge.Furthermore, electro-weak symmetry breaking induces extra splitting within the compo-nents of any electro-weak multiplet, with the result that the component with smallest electriccharge is the lightest stable state [18]. Since DM direct detection constraints demand thatDM does not couple at tree level to the Z , the DM hypercharge should be zero, which ispossible for integer isospin. The previous discussion is summarised in table 1, which tells that the simplest TC modelslead to the following viable stable DM candidates: • Techni-baryons, fermions for odd N and scalars for even N . Their annihilation crosssection is estimated to be σv ∼ g / πM , around the unitarity bound [9]. By per-forming a naive rescaling of the QCD non-relativistic p ¯ p cross section, σ p ¯ p v ∼ /m p ,we estimate that the cosmological thermal relic abundance of a techni-baryon equalsthe total DM abundance if its mass is loosely around m B ∼
200 TeV . A cosmologicaltechni-baryon asymmetry can leave a higher abundance, allowing for a lighter m B .14 U X (cid:37) C L b o u n d Ν background (cid:42) techni (cid:45) pion DM (cid:72) MDM triplet (cid:76) t ec hn i (cid:45) b a r yon D M f o r g (cid:61) r g E (cid:61) . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Dark Matter mass in GeV D M c r o sss ec ti on Σ S I i n c m Figure 4:
The signals in Dark Matter direct detection produced by a DM techni-baryon withmagnetic or electric dipole moment (line) or from a Minimal-Dark-Matter-like techni-pion withthermal abundance (star), compared to the present experimental LUX bound [20] and to thebackground due to neutrinos. • Scalar techni-pions, that fill a
SU(2) L triplet with hypercharge Y = 0 . Techni-pionshave small residual techni-color interactions (as well as small quartic couplings) andthereby behave as Minimal Dark Matter [18]. Their cosmological thermal relic abun-dance equals the total DM abundance if their mass is around . [18]. Their spin-independent cross section for direct detection is σ SI ≈ .
12 10 − cm [19, 2], as plottedin fig. 4.As already discussed, both mass scales suggested by the DM cosmological abundance arisenaturally within the context of finite naturalness.Techni-baryons have distinctive features in direct detection experiments: if DM is a neutralcomposite particle made of charged techni-quarks, direct detection can be mediated by thephoton [21]. Any such DM particle can have a non trivial form factor, dominated at lowenergy by the ‘charge radius’ interaction. For a scalar DM S this is the only interaction andcan be written as e Λ ( S ∗ i∂ α S ) ∂ µ F µα . (30)The resulting cross section for direct detection is suppressed by four powers of the TC scale,and is negligible for Λ TC ∼ few TeV. 15he situation is more promising if DM is a fermionic techni-baryon B , which genericallyhas magnetic (and possibly electric) dipole moments, µ and d . They are described by theeffective operator ¯ Bσ µν µ + idγ B F µν . (31)Electro-magnetic dipoles give sizeable direct detection signals with a characteristic testableenhancement at low recoil-energy E R , given that the DM/matter scattering is mediated by themassless photon. Furthermore, in the relevant non-relativistic limit, the cross-section inducedby the magnetic dipole µ is suppressed by two extra power of the relative DM/matter velocity v with respect to the cross section induced by the more speculative electric dipole d [21] dσdE R ≈ e Z πE R (cid:32) µ + d v (cid:33) . (32)For simplicity, we here assumed a nucleus with Z (cid:29) , a recoil energy E R (cid:28) m N v andapproximated the nuclear charge form factor with unity.The magnetic g -factor, defined by µ = ge/ m B , is expected to be of order one for astrongly-coupled particle (while it is loop suppressed for an elementary particle). We alsodefine the electric g -factor as d = g E e/ m B . In terms of such g -factors we find that thepresent direct detection bound is g + 1 . g E < (cid:18) m B . (cid:19) (33)dominated by LUX data [20, 22]. This bound assumes that techni-baryons constitute allgalactic DM, and must be rescaled otherwise. Fig. 4 shows the resulting prediction in theusual plane ( M DM , σ SI ) used to describe spin-independent direct detection of Dark Matter.An electric dipole moment needs CP-violation. In our context, techni-quarks are strictlymassless, such that the CP-violating techni-strong θ term is not physical. A small g E could begenerated if techni-quark masses are included. More quantitative predictions can be given in the QCD-like scenario with N = N F = 3 [12].In this case the spectrum can be obtained by rescaling known QCD results, m B m ρ ≈ . m π m ρ ≈ . (cid:113) J ( J + 1) (34)where m ρ is the mass of the lightest techni-vector resonance and techni-pions π lie in the J representation of SU(2) L . The second estimate is obtained from the electro-magnetic splittingof QCD pions, see the appendix. 16he lightest techni-baryons are an octet of flavour SU(3) and table 1 lists the two possibleviable assignments for techni-quarks under
SU(2) L ⊗ U(1) Y : Q = (cid:40) ∓ / ⊕ ± / . (35)The hypercharges are determined by requiring that the lightest techni-baryon is neutral; inthe first case their overall normalisation is determined by requiring that techni-quarks canhave a Yukawa interaction with the Higgs. For this choice of quantum numbers the lightesttechni-baryon is an electro-weak singlet with Y = Q = 0 , avoiding direct detection con-straints.The lightest technibaryons decompose under SU(2) L as = (cid:40) (p , n) ⊕ (Σ ± , ) ⊕ (Ξ , Ξ − ) ⊕ (Λ ) for Q = 1 ⊕ ⊕ for Q = 3 . (36)In the Q = 1 ⊕ model we used the familiar names of the QCD octet. The lightest techni-baryon is Λ , that is analogous to the QCD state Λ QCD0 ∼ s ( ud − du ) . Its magnetic dipolemoment can be estimated from QCD data: µ Λ QCD0 = 0 . e/ m p [23]. Inserting g = − . ineq. (33) we obtain the bound m Λ > . TeV. The previous QCD-based estimate of the DM an-nihilation cross section becomes exact, such that the cosmological DM density is reproducedfor m Λ ≈
200 TeV . In this model there are no stable techni-pions.In the Q = 3 Y model the lightest technibaryon is a triplet Y of SU(2) L , so that neutralDM is obtained for Y = 0 and Y = ± / : the first possibility is allowed by direct detectionconstraints. Due to the absence of Yukawa and hypercharge interactions, the neutral mem-ber of the techni-pion triplet is the DM candidate, stable thanks to the accidental G-paritydiscussed in section 3.1. Its mass must be smaller than . in order to avoid a thermalrelic density bigger than the observed DM density. This implies that in this model the thermalrelic density of the technibaryon dark matter is subdominant. In conclusion, we presented a new class of models where the Standard Model is madedimension-less by dropping the mass term of the elementary Higgs and extended by addingtechni-quarks with techni-color interactions arranged in such a way that they do not breakthe electro-weak gauge group nor generate a composite Higgs. Within the context of finitenaturalness — the assumption that a QFT with no mass parameters nor power divergencesmight provide a revised concept of weak-scale naturalness and of the origin of mass scales —the simplest models of this type dynamically generate a mass term for the Higgs.The elementary Higgs acquires a squared mass term m entirely determined in terms ofweak interactions of the techni-quarks and of the techni-color scale. Using various approxi-mation techniques that allow to control the techni-color dynamics, in section 2.1 we found17hat the sign of m is negative, such that SU(2) L ⊗ U(1) Y gets broken, and that the ob-served weak scale is obtained for a techni-color scale m ρ ≈ πM h /α ≈ −
20 TeV . This islarge enough that such models do not pose any phenomenological problem. Techni-pions arelighter, as determined by their electro-weak interactions, and could give observable signals atLHC; in particular techni-pions π in the 5 of SU(2) L undergo anomalous decays into pairs ofelectro-weak vectors, π → W W . Such models can have the same number of free parametersas the Standard Model: all new physics is univocally predicted , up to theoretical uncertain-ties in the techni-strong dynamics, that could be reduced with respect to our estimates byperforming dedicated lattice computations. Independently of the assumption of finite naturalness, the models studied in this papercontain two Dark Matter candidates: the lightest techni-baryon B with mass m B ∼
50 TeV (section 3.2) and, in some models, the lightest techni-pion π , a triplet under SU(2) L withmass m π ∼ . m ρ ∼ − (section 3.1). Their thermal relic abundance is also univocallypredicted, with the result that the observed cosmological Dark Matter abundance is naturallyreproduced in the techni-pion case, while the techni-baryon seems more likely to be a sub-dominant Dark Matter component, if a naive rescaling of the QCD p ¯ p cross-section holds,and ignoring possible techni-baryon asymmetries. The direct detection cross section of suchDM candidates is predicted to be − orders of magnitude below present bounds. Magneticmoment interactions of techni-baryons would lead to recoil events with a distinctive energyspectrum (section 3.3).Table 1 offers a panoramic of models that lead to DM candidates. In some models thequantum numbers allow for Yukawa interactions between techni-quarks and the elementaryHiggs. Such Yukawas give extra negative contributions to the squared Higgs mass term (sec-tion 2.3), so that the techni-color scale needed to reproduce the weak scale gets lighter;in such models a singlet techni-pion is especially light. Models where techni-quarks onlyhave QCD interactions or gravitational interactions do not seem to lead to a promising phe-nomenology, as discussed in section 2.2. Acknowledgments
We wish to thank Roberto Franceschini and Giovanni Villadoro for discussions and collaboration at theearly stages of this work. We thank Roberto Contino, Riccardo Rattazzi, Slava Rychkov, Riccardo Torrefor advice about strong dynamics and the authors of [22] (in particular Eugenio del Nobile) for helpabout using their code. The work of OA and MR is supported by the MIUR-FIRB grant RBFR12H1MW. Techni-strong dynamics generates a negative vacuum energy of order − Λ . It can be canceled, compatiblywith the scenario of dynamical mass generation in the SM sector, by adding another sector negligibly coupledto SM particles; this kind of sector is anyhow needed to account for the Planck mass. This cancellation is theusual huge fine-tuning associated with the cosmological constant problem, on which we have nothing to say. Effective potential
The effective potential for the elementary Higgs and the techni-pions receives contributions at treelevel in the Yukawa couplings and at loop level in the gauge and Yukawa couplings. It can be computedusing the techniques reviewed in [15]: the relevant ingredients are the correlation functions of thecomposite operators of the theory. There are three main contributions: from SM gauge interactionsat loop level (section A.1); from the possible Yukawa couplings at tree level (section A.2) and at looplevel (section A.3). Summing these contributions, the full potential is studied in section A.4.
A.1 Gauge contribution at one loop level
At 1-loop the SM gauge interactions induce a techni-pion mass that can be computed in terms ofcorrelators of the vector ( J aµ = (cid:80) Q ¯ Q γ µ T a Q Q ) and axial ( J aµ = (cid:80) Q ¯ Q γ µ T ˆ a Q γ Q ) symmetry currents.On general grounds these have the form, i (cid:90) d x e iq · x (cid:104) | T J Vµ ( x ) J V (cid:48) ν (0) | (cid:105) ≡ δ V V (cid:48) ( q g µν − q µ q ν )Π V V ( q ) ,i (cid:90) d x e iq · x (cid:104) | T J Aµ ( x ) J A (cid:48) ν (0) | (cid:105) ≡ δ AA (cid:48) ( q g µν − q µ q ν )Π AA ( q ) . (37)The one-loop techni-pion potential reads [15]: V g ≈ π ) (cid:88) i g i Tr[
U T i U † T i ] (cid:90) ∞ Q dQ (cid:104) Π AA ( − Q ) − Π V V ( − Q ) (cid:105) (38)where U = e iπ ˆ a T ˆ a /f is the Goldstone boson matrix, g i are the SM couplings and T i their generators.Gauge-charged techni-pions acquire positive squared masses, that, for the SU(2) L interactions, areestimated as m π ≈ g (4 π ) J ( J + 1) m ρ (39)where J is the weak isospin of the techni-pion representation. A.2 Yukawa contribution at tree level
We now consider the potential generated by the Yukawa interactions. For concreteness we here focuson the case where techni-quarks Q = 2 ⊕ fill one doublet and one singlet of SU(2) L with hyperchargesas in section 3.4. The 8 techni-pions decompose under SU(2) L ⊗ U(1) Y as ± / + 3 + 1 . (40)In general there are two Yukawa couplings: yH Q L Q R + y (cid:48) H † Q R Q L + h.c. = H ¯ Q (cid:18) y + y (cid:48)∗ γ − y + y (cid:48)∗ (cid:19) Q + h.c. (41)where on the right hand side we used Dirac spinors Q i = ( Q iL , ¯ Q iR ) . The phases of y and y (cid:48) are notphysical and can be chosen for convenience, for example real and positive. The terms above generatethe tree level effective potential V y = a Tr[
M U ] + h.c. (42) here a ≈ − m ρ f and M = (cid:32) Q R Q R Q L yH Q L y (cid:48) H † (cid:33) . (43)The explicit result for the potential of the doublet ( π ) and singlet ( η ) techni-pions is V y = − √ a Im e − i η √ f sin ∆ /f ∆ [ yH † π + y (cid:48) π † H ] , ∆ = 14 (cid:113) η + 8 π † π . (44) A.3 Yukawa contribution at one loop level
To compute the one-loop Yukawa correction to the effective potential we proceed similarly to thegauge interactions. We formally introduce sources S ¯ ij for the techni-quark bilinears Q iL Q ¯ jR ( x ) (suchthat, in the real theory of interest, it contains yH in some of its components) and write the effectiveLagrangian that describes the Higgs/techni-pion system after having integrated out the heavier techni-strong dynamics. For simplicity we consider vectorial couplings as in these case fewer invariants exist.In a constant techni-pion configuration and to quadratic order in the sources S , the effective actionhas the following structure determined by the symmetries, L QQ eff = a δ ( q )(Tr[ SU ] + h.c. ) + Π QQ ( q )Tr[ SS † ] + Π QQ ( q ) | Tr[ SU ] | . (45)The first term linear in S describes the Q L Q R condensate. The form factors can be obtained integratingover the strong dynamics including techni-pion fluctuations. By construction they encode the twopoint functions of the techni-quark bilinears, (cid:104) | ¯ Q i Q ¯ j ( q ) ¯ Q ¯ k ¯ Q l ( − q ) | (cid:105) = i G QQ Adj ( q ) (cid:18) δ i ¯ k δ l ¯ j − δ i ¯ j δ l ¯ k (cid:19) + i G QQS ( q ) δ i ¯ j δ l ¯ k (46)where G QQS and G QQ Adj correspond to the singlet and adjoint channels (namely, the octet for N F = 3 ).Matching eq.s (45) and (46) (for example choosing U = 1 ) one finds Π QQ = G QQ Adj , Π QQ = G QQS − G QQ Adj . (47)At large N one has G QQ Adj ( q ) = N π m ρ (cid:88) n c n q − m n + i(cid:15) , G QQS ( q ) = N π m ρ (cid:88) n c S n q − m S n + i(cid:15) , (48)where the coefficients c are of order 1 and the sum is over the scalar resonances in the theory. Thesum does not include techni-pions because we only consider vectorial Yukawa couplings that do notgenerate 1 techni-pion states.To obtain the effective action for the scalars we just need to set to zero the non dynamical compo-nents of S and add kinetic terms for the components of S associated to the Higgs. This produces L Heff = a δ ( q )(Tr[ M U ] + h.c. ) + ( q + y Π QQ ( q )) H † H + Π QQ ( q ) | Tr[
M U ] | . (49) he first term describes the tree level contribution discussed above. The second term encodes the treelevel effect of mixing with heavy scalar resonances that gives the HH ∗ entry of the mass matrix ineq. (24), m H = y Π QQ (0) ∼ − y N (4 π ) m ρ . (50)Performing the path integral with respect to H we obtain the one-loop Yukawa contribution to thetechni-pion potential, V y = 12 (cid:90) d Q (2 π ) ln (cid:34) Q − y Π QQ ( − Q ) − y Π QQ ( − Q ) (cid:88) a Tr[ T a U ]Tr[ T a U † ] (cid:35) ≈ v − y π ) (cid:88) a Tr[ T a U ]Tr[ T a U † ] (cid:90) ∞ dQ Π QQ ( − Q ) (51)where v is the contribution to the the vacuum energy and T a are SU(3) matrices derived from (43).One can prove that, similarly to the gauge contribution, the loop integral in (51) is finite: since Π QQ is sensitive to the chiral symmetry breaking, the Operator Product Expansion demands that Π QQ ( q ) q (cid:29) Λ (cid:39) (cid:104) | ( ¯ Q L Γ Q R )( ¯ Q R Γ Q L ) | (cid:105) q (52)where Γ , are appropriate matrices in techni-color and flavour space, see [16].Contrary to the gauge contribution we are not aware of any theorem that guarantees the sign ofthis contribution. As an estimate the contribution above gives δm π ∼ y m ρ (4 π ) . (53)Summing up all the contributions we obtain a mass matrix with the structure of eq. (24). A.4 Minimization of the potential
The vacuum is determined through the minimization of the potential V eff ( π, η, H ) = V g + V y + V y + m | H | + λ | H | (54)where m < is induced by gauge loops (section 2.1). The gauge-charged techni-pions π acquire alarge mass from gauge loops and can be integrated out, leaving an effective potential for the lighterscalars: the elementary Higgs doublet H and the gauge-neutral techni-pion η . In the parameter rangeof interest for us, g (cid:29) y , one has V y ≈ and V g ≈ m π π (1 − η / f ) , where, for simplicity, weexpanded at second order in η/f sufficient to compute the mass of the singlet. We can freely redefinethe phases of the Yukawa couplings y and y (cid:48) so that yy (cid:48) is real and negative. With this choice η = 0 indeed is a local minimum of the effective potential V eff ( η (cid:28) f, H ) ≈ | H | (cid:20) m − m ρ f m π (cid:32) ( | y | + | y (cid:48) | ) − | yy (cid:48) | η f (cid:33) (cid:21) + λ | H | . (55)Around the minimum η acquires a positive squared mass M η ∼ | y | m ρ m π v (56)without mixing with the Higgs, that receives a negative contribution to its m parameter. eferences [1] C. Kilic, T. Okui and R. Sundrum, JHEP 1002, 018 (2010) [arXiv:0906.0577].[2] M. Farina, D. Pappadopulo and A. Strumia, JHEP 1308, 022 (2013) [arXiv:1303.7244]. A. de Gouvea,D. Hernandez and T. M. P. Tait, Phys. Rev. D 89 (2014) 115005 [arXiv:1402.2658].[3] F. Englert, C. Truffin and R. Gastmans, Nucl. Phys. B 117 (1976) 407. W Bardeen, FERMILAB-CONF-95-391-T. C. T. Hill, arXiv:hep-th/0510177.[4] A. Salvio and A. Strumia, JHEP 1406 (2014) 080 [arXiv:1403.4226].[5] R. Hempfling, Phys. Lett. B 379 (1996) 153 [arXiv:hep-ph/9604278]. J. P. Fatelo, J. M. Gerard, T. Ham-bye and J. Weyers, Phys. Rev. Lett. 74 (1995) 492. T. Hambye, Phys. Lett. B 371 (1996) 87 [arXiv:hep-ph/9510266]. W. -F. Chang, J. N. Ng and J. M. S. Wu, Phys. Rev. D 75 (2007) 115016 [arXiv:hep-ph/0701254]. R. Foot, A. Kobakhidze and R. R. Volkas, Phys. Lett. B 655 (2007) 156 [arXiv:0704.1165].R. Foot, A. Kobakhidze, K. L. McDonald, R. R. Volkas, Phys. Rev. D 77 (2008) 035006 [arXiv:0709.2750].S. Iso, N. Okada, Y. Orikasa, Phys. Rev. D 80 (2009) 115007 [arXiv:0909.0128]. S. Iso and Y. Orikasa,PTEP 2013 (2013) 023B08 [arXiv:1210.2848]. C. Englert, J. Jaeckel, V. V. Khoze and M. Spannowsky,arXiv:1301.4224. E.J. Chun, S. Jung, H.M. Lee, arXiv:1304.5815. T. Hambye and A. Strumia, Phys.Rev. D 88 (2013) 055022 [arXiv:1306.2329]. C. D. Carone and R. Ramos, Phys. Rev. D 88 (2013)055020 [arXiv:1307.8428]. R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, Phys. Rev. D 89(2014) 115018 [arXiv:1310.0223]. A. Farzinnia, H. J. He and J. Ren, Phys. Lett. B 727 (2013) 141[arXiv:1308.0295]. C. T. Hill, Phys. Rev. D 89 (2014) 073003 [arXiv:1401.4185. J. Guo and Z. Kang,arXiv:1401.5609. S. Benic and B. Radovcic, Phys. Lett. B 732 (2014) 91 [arXiv:1401.8183]. H. Davoudi-asl, I.M. Lewis, Phys. Rev. D 90, 033003 (2014) [arXiv:1404.6260]. K. Allison, C. T. Hill and G. G. Ross,arXiv:1404.6268. G. M. Pelaggi, arXiv:1406.4104. W. Altmannshofer, W. A. Bardeen, M. Bauer, M. Carenaand J. D. Lykken, arXiv:1408.3429.[6] T. Hur and P. Ko, Phys. Rev. Lett. 106, 141802 (2011) [arXiv:1103.2571]. M. Heikinheimo, A. Racioppi,M. Raidal, C. Spethmann and K. Tuominen, Mod. Phys. Lett. A 29 (2014) 1450077 [arXiv:1304.7006].[7] T. Hambye and M. H. G. Tytgat, Phys. Lett. B 683 (2010) 39 [arXiv:0907.1007]. M. Holthausen, J. Kubo,K. S. Lim and M. Lindner, JHEP 1312 (2013) 076 [arXiv:1310.4423]. See also [24].[8] Work in progress.[9] K. Griest and M. Kamionkowski, Phys. Rev. Lett. 64 (1990) 615. B. von Harling and K. Petraki,arXiv:1407.7874.[10] G. F. Giudice, G. Isidori, A. Salvio and A. Strumia, arXiv:1412.2769.[11] Y. Bai and R. J. Hill, Phys. Rev. D 82, 111701 (2010) [arXiv:1005.0008].[12] R. Pasechnik, V. Beylin, V. Kuksa and G. Vereshkov, Eur. Phys. J. C 74 (2014) 2728 [arXiv:1308.6625].[13] E. Witten, Phys. Rev. Lett. 51, 2351 (1983).[14] D. Marzocca, A. Parolini and M. Serone, JHEP 1403 (2014) 099 [arXiv:1312.5664].[15] R. Contino, arXiv:1005.4269.[16] M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B147 (1979) 385.[17] See e.g. M. Peskin, D.V. Schroeder, “An Introduction to Quantum Field Theory”, pag. 618. Our Π V V isdefined with the same sign as Π in this book.[18] M. Cirelli, N. Fornengo and A. Strumia, Nucl. Phys. B 753 (2006) 178 [arXiv:hep-ph/0512090]. M. Cirelli,A. Strumia and M. Tamburini, Nucl. Phys. B 787 (2007) 152 [arXiv:0706.4071].
19] J. Hisano, K. Ishiwata and N. Nagata, Phys. Rev. D 82 (2010) 115007 [arXiv:1007.2601]. R. J. Hill andM. P. Solon, Phys. Lett. B 707 (2012) 539 [arXiv:1111.0016].[20] LUX Collaboration, arXiv:1405.5906.[21] See e.g. V. Barger, W. Y. Keung and D. Marfatia, Phys. Lett. B 696 (2011) 74 [arXiv:1007.4345].[22] E. Del Nobile, M. Cirelli and P. Panci, arXiv:1307.5955.[23] Particle Data Group Collaboration, Chin. Phys. C 38 (2014) 090001.[24] J. Kubo, K. S. Lim and M. Lindner, arXiv:1403.4262. In the past other authors considered a similar idea, butwithout an elementary Higgs: W. J. Marciano, Phys. Rev. D21 (1980) 2425. D. Lust, E. Papantonopoulos,K. Streng, and G. Zoupanos, Nucl. Phys. B268 (1986) 49.19] J. Hisano, K. Ishiwata and N. Nagata, Phys. Rev. D 82 (2010) 115007 [arXiv:1007.2601]. R. J. Hill andM. P. Solon, Phys. Lett. B 707 (2012) 539 [arXiv:1111.0016].[20] LUX Collaboration, arXiv:1405.5906.[21] See e.g. V. Barger, W. Y. Keung and D. Marfatia, Phys. Lett. B 696 (2011) 74 [arXiv:1007.4345].[22] E. Del Nobile, M. Cirelli and P. Panci, arXiv:1307.5955.[23] Particle Data Group Collaboration, Chin. Phys. C 38 (2014) 090001.[24] J. Kubo, K. S. Lim and M. Lindner, arXiv:1403.4262. In the past other authors considered a similar idea, butwithout an elementary Higgs: W. J. Marciano, Phys. Rev. D21 (1980) 2425. D. Lust, E. Papantonopoulos,K. Streng, and G. Zoupanos, Nucl. Phys. B268 (1986) 49.