Light Higgs and Vector-like Quarks without Prejudice
Svjetlana Fajfer, Admir Greljo, Jernej F. Kamenik, Ivana Mustac
LLight Higgs and Vector-like Quarks without Prejudice
Svjetlana Fajfer,
1, 2, ∗ Admir Greljo, † Jernej F. Kamenik,
1, 2, ‡ and Ivana Musta´c § Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: September 27, 2018)Light vector-like quarks with non-renormalizable couplings to the Higgs are a common feature ofmodels trying to address the electroweak (EW) hierarchy problem by treating the Higgs as a pseudo-goldstone boson of a global (approximate) symmetry. We systematically investigate the implicationsof the leading dimension five operators on Higgs phenomenology in presence of dynamical up-and down-type weak singlet as well as weak doublet vector-like quarks. After taking into accountconstraints from precision EW and flavour observables we show that contrary to the renormalizablemodels, significant modifications of Higgs properties are still possible and could shed light on the roleof vector-like quarks in solutions to the EW hierarchy problem. We also briefly discuss implicationsof higher dimensional operators for direct vector-like quark searches at the LHC.
I. INTRODUCTION
The recent LHC discovery of a Higgs boson [1] seems to experimentally complete the standard model (SM) pictureof fundamental interactions. Although the SM works extremely well phenomenologically, the electro-weak (EW)hierarchy problem, as exemplified by the extreme UV sensitivity of the Higgs potential, provides a strong theoreticalmotivation for contemplating beyond SM physics at the TeV scale.On the other hand, direct searches for on-shell production of new degrees of freedom at the LHC together withever more stringent constraints from measurements of flavor, EW and Higgs observables are starting to directlyprobe models addressing the naturalness problem of the SM. At present these experimental null-results are not yetconclusive, but viable new physics (NP) models’ parameter space is becoming significantly reduced. In light of this, ithas become crucial to focus on the minimal light particle content required to fulfil the naturalness conditions (see [2]for examples in supersymmetric theories).Vector-like quarks are expected to be the lightest new degrees of freedom in models addressing the EW hierarchyproblem by treating the light Higgs as a pseudo-goldstone boson of a global symmetry, broken explicitly by the SMgauging and Yukawa couplings [3, 4]. In such models, the dominant quadratic divergences in one-loop corrections tothe Higgs boson mass coming from the top quark loop are canceled by contributions of dimension five operators ofthe form H † H ¯ QQ , where H is the Higgs doublet and Q a vector-like quark weak multiplet. In general, parametrizingsingle- and double-Higgs interactions of an arbitrary number of quark flavors f in the mass eigenbasis L eff h = (cid:18) − y ij h + x ij h v (cid:19) f iL f jR + h . c . , (1)where v = ( √ G F ) − / (cid:39)
246 GeV is the EW condensate, the condition for cancelation of one-loop quadraticdivergences from such interactions can be put into the following simple form (cid:88) i (cid:60) ( x ii ) m i v = (cid:88) i,j | y ij | . (2)The appearance of new operator contributions already at dimension five is a singular feature of effective theories withnew light vector-like fermions and intrinsically connected to the resolution of the SM hierarchy problem within suchscenarios. These dimension five contributions are thus expected to represent the dominant probe of new dynamicsin the UV. At the same time one should keep in mind that unless the vector-like quarks are related to the SMfield content by a symmetry, such cancelation of quadratic divergences is fine tuned in general. Furthermore, if one ∗ Electronic address:[email protected] † Electronic address:[email protected] ‡ Electronic address:[email protected] § Electronic address:[email protected] a r X i v : . [ h e p - ph ] O c t considers the running of the leading vector-like quark operator and the SM couplings, even if tuned to cancel at onescale, the quadratic divergences will not cancel at other scales. Thus, the effective theory discussion concerning EWnaturalness should be understood under the implicit assumption that the relation (2) is enforced by symmetry in theUV complete theory.Furthermore, even though a (symmetry enforced) relation (2) removes the quadratic UV sensitivity of the Higgspotential, logarithmically divergent contributions remain present in the effective theory. The biggest resulting shift tothe bare Higgs mass ( δm h ) is now due to the new heavy quark states with bilinear couplings to the Higgs. Assuminga single such state ( f ) cancelling the one-loop quadratically divergent contribution to δm h of the top quark, thedominant remaining correction is of the form δm h ≈ m t π v m f log Λ m f , (3)where the result was obtained using a hard UV cut-off of the loop momentum integral and equating it with thecut-off scale of the effective theory Λ. We immediately observe that allowing for only moderate fine-tuning (requiringconservatively δm h /m h (cid:46)
10) and the effective theory treatment valid (and thus m f (cid:28) Λ) requires f to be relativelylight ( m f (cid:46) It turns out that they can always beparametrized in a way that preserves the form of gauge interactions of the renormalizable theory. Therefore the onlyway to approach and constrain such terms is by studying their impact on Higgs phenomenology, which is the maintopic of the present work. The paper is structured as follows. In Sec. II we present some general considerations of models with vector-likequarks including generic flavor, electroweak and Higgs constraints on their interactions, stemming from renormalizableas well as leading dimension five non-renormalizable terms in the effective Lagrangian. Then in Secs. III – V we considerthree specific model examples and analyze their viability in light of the derived direct and indirect constraints. Forsimplicity we will consider examples where vector-like quarks appear in existing weak representations of SM chiralquarks, thus allowing for kinetic mixing with chiral quark multiplets, with interesting phenomenological consequences.We summarize our conclusions in Sec. VI. Several supporting derivations and analyses of experimental constraintshave been relegated to the appendices.
II. GENERAL CONSIDERATIONSA. Renormalizable models
Since all vector-like quark models under consideration only contain colored fermions in SM gauge representationsand charges, we can start by considering the mass matrices of the up- and down-type quarks in the weak (chiral)eigenbasis − L mass = ¯ u iL M iju u jR + ¯ d iL M ijd d jR + h . c . , (4)where the indices i, j run over all dynamical quark flavors (including new vector-like generations). The mass matrices M u,d can be diagonalized via bi-unitary rotations as M u,d, diag = U u,dL M u,d U u,d † R . Consequently, the gauge and Higgsinteractions of physical quarks in the mass eigenbasis can be written in the general form (c.f. [8]) L W = − g √ V Lij ¯ u i γ µ P L d j + V Rij ¯ u i γ µ P R d j ) W + µ + h . c . , (5) L Z = − g c W (cid:0) X uij ¯ u i γ µ P L u j − X dij ¯ d i γ µ P L d j + Y uij ¯ u i γ µ P R u j − Y dij ¯ d i γ µ P R d j − s W J µ EM (cid:1) Z µ , (6) L (0) h = − ( X uij − Y uij ) m j v ¯ u i P R u j h − ( X dij − Y dij ) m j v ¯ d i P R d j h + h . c . , (7) For a recent analysis of dimension six operator effects in composite Higgs scenarios without dynamical vector-like fermions see [6]. For recent related studies in the context of explicit composite Higgs model realisations see [7]. where P R,L = (1 ± γ ) / g = 2 m W /v (cid:39) .
65 is the weak coupling, while s W (cid:39) √ .
23 and c W = (cid:112) − s W are thesine and cosine of the weak angle, respectively. J µ EM = (2¯ u i γ µ u i − ¯ d i γ µ d i ) / V L,R , X u,d and Y u,d are all given in terms of U u,dL,R , in particular we can write V Lij ≡ ( U dL ) ∗ jk ( U uL ) ik , where therepeated index runs over all left-handed weak doublets, and V Rij ≡ ( U dR ) ∗ jk ( U uR ) ik , where the repeated index runs overall right-handed weak doublets. Then the (hermitian) flavor matrices entering neutral current and Higgs interactionsare given simply by X u ≡ V L V L † , X d ≡ V L † V L , Y u ≡ V R V R † and X d ≡ V R † V R . Thus, non-standard Higgsinteractions in such renormalizable models with extra quarks are necessarily constrained by charged and neutral weakcurrents among the known three generations of quarks. For example, the departures of V L from 3 × V R are constrained by precisely measured tree level charged current processes.For example, (cid:80) j = d,s,b | V Lij | = 1 − ∆ ui ≤ i = u, c, t ) and (cid:80) j = u,c,t | V Lji | = 1 − ∆ di ≤ i = d, s, b ) areconstrained in absence of V R as ∆ uu < .
001 [9], ∆ uc < .
052 [10], ∆ ut < .
13 (see Appendix A for details) , ∆ dd < . ds < .
08 [10] and ∆ db < − | V Ltb | < .
15 (see Appendix A for details). Note that in models with no extra up-type(down-type) quarks, ∆ di = δX dii (∆ ui = δX uii ), where δX u,dii ≡ − X u,dii . The entries of V R on the other hand,are also constrained at the tree-level by searches for right-handed charged currents (c.f. [11] for a recent analysis).Unfortunately, without information on the matrix elements involving also extra quarks present in the model beyondthe known SM generations, these cannot be directly related to Z and Higgs couplings ( Y u,d ) .In addition, one can obtain tree-level constraints on the off-diagonal entries of X u,d and Y u,d directly from theircontributions to Z -mediated FCNCs of up- or down-type quarks. In all scenarios we consider, either nonstandard X u,dij (cid:54) = δ ij or Y udij (cid:54) = 0 are generated but not both. In this case the bounds on non-diagonal entries of X u,d or Y u,d read | X ucu | , | Y ucu | < . × − [12, 13], | X utu,tc | , | Y utu,tc | < .
14 (see Appendix A for details); Re( X dds ) , Re( Y dds ) < . × − , | X ddb | , | Y ddb | < × − and | X dsb | , | Y dsb | < × − (see Appendix B for details). Finally, electroweak measurementsprovide strong tree-level constraints also on the diagonal entries of X u,d and Y u,d corresponding to the five light quarkflavours (see Appendix C for details).The main consequence of the above discussion is that in renormalizable models with additional vector-like quarks,Higgs couplings to the known three generations of quarks, except possibly the top (i.e. X utt , Y utt ), must remain SM-like,irrespective of the spectrum or interactions of additional heavy quarks. This is because they are rigidly related to thecorresponding Z couplings, and thus subject to severe constraints from charged and neutral weak currents. In orderto possibly obtain more interesting Higgs phenomenology, we are thus led to consider effects of higher-dimensionaloperators. B. Including non-renormalizable Higgs interactions
Treating the SM as an effective field theory with particle content valid below a UV cut-off scale Λ, it is well knownthat the leading higher dimensional operators involving quark fields are of dimension six, a virtue of the chiral natureof weak interactions in the SM. Thus, effects of NP degrees of freedom appearing above Λ in low energy observablesare suppressed by at least two powers of 1 / Λ. On the other hand, in presence of dynamical vector-like quarks, theleading non-renormalizable operators can appear already at dimension five. In general, they are of the form H † H ¯ QQ and H † H ¯ qQ , where q denotes the SM chiral quark multiplets. The main consequences of these new interactions are(i) direct di-Higgs coupling to physical quarks [ x ij in eq. (1)] with possible implications for the SM hierarchy problem;(ii) modifications of single Higgs - quark couplings [ y ij in eq. (1)] not related to weak neutral or charged currents. Inthe quark mass eigenbasis these additional contributions can generally be written as L (1) h = (cid:32) X u (cid:48) ij Λ ¯ u iL u jR + X d (cid:48) ij Λ ¯ d iL d jR (cid:33) (cid:20) vh + h (cid:21) + h . c . , (8)where Λ is the UV cut-off scale of the effective theory encompasing the SM together with a number of additionalquark-like states. First note that the appearance of X u,d (cid:48) ij couplings of the known three generations of quarks tothe Higgs is a manifestation of mixing between chiral and vector-like quarks, which is in general unrelated and thus In the following we do not consider operators of the form ¯ Q ( σ · G ) Q and ¯ q ( σ · G ) Q , where σ µν = i [ γ µ , γ ν ] / G µν ∈{ T a G aµν , τ a W aµν , B µν } stands for the three SM gauge field strengths, since these are not directly related to Higgs phenomenology.If the vector-like quarks mix with the SM generations, they will induce anomalous dipole gauge interactions of SM quarks at order1 /m Q Λ, where m Q is the vector-like quark mass scale, and can be constrained from precision electroweak, flavor and collider observ-ables (c.f. [14]). Modulo fine-tuned cancelations in these constraints, their presence would thus not affect our analysis. unconstrained by charged and neutral weak currents. On the other hand, naturalness of the hierarchical quark massspectrum would require | X q (cid:48) ij X q (cid:48) ji ∗ | v / Λ < m i m j [15]. In order to keep our analysis as general as possible, we shall notimpose such a condition on the parameter space of our models, although one should keep it in mind. The off-diagonalvalues of X u,d (cid:48) are constrained by low energy flavour observables [16]. In the up-sector, | X u (cid:48) uc,cu | v/ Λ < × − and (cid:113) | X u (cid:48) tu,tc | + | X u (cid:48) ut,ct | v/ Λ < .
34 are constrained by D mixing and t → ( c, u ) h decay searches, respectively. Similarly, K , B d and B s mixing measurements require | X d (cid:48) sd,ds | v/ Λ < × − , | X d (cid:48) bd,db | v/ Λ < × − and | X d (cid:48) sb,bs | v/ Λ < × − , respectively. Potentially the most striking tree-level effects on Higgs phenomenology in non-renormalizable vector-like quarkmodels are thus modifications of the flavor diagonal Higgs couplings to lighter quarks. In particular, in the SM thethe total Higgs decay width is dominated by the h → b ¯ b channel, the first hints of which have also been observedat the LHC [17, 18]. The modifications of y bb in eq. (1) can thus have important consequences for all experimen-tally observed Higgs signals. Similarly, while the h → u ¯ u, d ¯ d, s ¯ s or h → c ¯ c decays are very suppressed in the SMand also cannot be reconstructed at the LHC due to the large QCD backgrounds, they can contribute to the to-tal Higgs decay width in case of non-zero X u,d (cid:48) ii . In particular, defining ∆ γ ≡ (cid:80) f = d,u,s,c Γ h → f ¯ f / Γ SM h , we obtain (cid:80) i = d,s (cid:12)(cid:12) X d (cid:48) ii v/ Λ − m i /v (cid:12)(cid:12) + (cid:80) i = u,c | X u (cid:48) ii v/ Λ − m i /v | (cid:39) − ∆ γ showing that sizable enhancement in these decaychannels is possible (although the required values of X u,d (cid:48) ii / Λ would necessarily violate the corresponding quark massnaturalness conditions) and that a non-trivial constraint on X u,d (cid:48) ii can in principle be obtained from the total Higgsdecay width. This rises a question of importance of uu → h (or to a lesser extent ¯ dd → h ) production mechanismcompared to the dominant gg → h mode at the LHC and Tevatron. In the zero-width approximation and at leadingorder in QCD, the ratio of the relevant hadronic cross sections in the two cases can be written as σ ( p p → h ) uu σ ( p p → h ) gg = Γ( h → uu )Γ( h → gg ) L uup p ( τ ) L ggp p ( τ ) , (9)where τ ≡ m h /s with s being the invariant collider energy squared. The relevant luminosity functions at hadronic( p p ) colliders are given by L q q p p ( x ) = 11 + δ q q (cid:90) x dyy [ f p q ( y ) f p q ( x/y ) + f p q ( y ) f p q ( x/y )] , (10)where f p i q j are the corresponding parton distribution functions (pdfs). Using the LO MSTW2008 [19] set, with thefactorization and renormalization scales fixed to m h (cid:39)
125 GeV, the ratio L uupp / L ggpp for √ s = 7 TeV and √ s = 14 TeVis 3 .
9% and 2 .
3% respectively. Thus, even assuming comparable h → u ¯ u and h → gg decay rates, the up-quarkcontribution to Higgs production at the LHC is below the ∼
12% theoretical uncertainties [20] of the dominant gluonfusion production cross section. Conversely, at the Tevatron, we find the relevant ratio L uup ¯ p / L ggp ¯ p for √ s = 1 .
96 TeV tobe sizable 26%. However, given the low statistics in the gluon fusion Higgs production channel at the Tevatron [21],this again gives no relevant constraint on Γ( h → uu ). In the future, enhanced di-Higgs production at the LHC couldpossibly offer a competitive constraint on X u (cid:48) uu / Λ . In particular, the relevant LO hadronic cross section is given by σ ( p p → hh ) X (cid:48) = (cid:90) τ dx ˆ σ X (cid:48) u ¯ u → hh ( xs ) L u ¯ up p ( x ) , (11)where ˆ σ X (cid:48) u ¯ u → hh (ˆ s ) = | X u (cid:48) uu | β h π Λ (cid:18) m h ˆ s − m h (cid:19) , (12)and β h = (cid:112) − m h / ˆ s . Using the same pdf parameters as above we obtain σ ( pp → hh ) X (cid:48) / [( | X u (cid:48) uu | / . / Λ)] (cid:39) σ ( pp → hh ) SM = 4(16) fb,respectively. All bounds from neutral meson mixing apply in absence of large cancellations with the tree-level Z -mediated X u,d , Y u,d contributions. Using instead the NNLO MSTW2008 pdfs with the same scale choices yields ratios of 4 .
0% and 2 .
5% respectively.
C. Impact of existing Higgs data
Most interesting effects involving light vector-like quarks in Higgs phenomenology appear at the one-loop level. Ingeneral, Higgs-fermion interactions of the form (1) will contribute to gluon fusion production and di-photon decay ofthe Higgs at one loop R gg ≡ Γ h → gg Γ SMh → gg (cid:39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:80) i y ii vm i C ( r i ) F / ( τ i ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) F / ( τ t ) + F / ( τ b ) (cid:12)(cid:12) , (13) R γγ ≡ Γ h → γγ Γ SMh → γγ (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) F ( τ W ) + (cid:80) i y ii vm i d ( r i ) Q i F / ( τ i ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) F ( τ W ) + F / ( τ t ) (cid:12)(cid:12) , (14)where d ( r i ) and C ( r i ) are the dimension and index of the color representation of f i , respectively, and Q i is its electriccharge. The relevant loop functions F ( τ ) and F / ( τ ) can be found e.g. in ref. [23], and τ i ≡ m h / (4 m i ). In the limitof large fermion mass, F / ( τ ) → F / (0) = 4 /
3. In the SM, gluon fusion production, gg → h , is dominated by the topquark loop with (1 / F / ( τ t ) = 0 . / F / ( τ b ) = − .
04 + ı . W boson loop yielding F ( τ W ) = − .
34, andinterfering destructively with the top quark contribution of (4 / F / ( τ t ) = 1 .
84. It turns out that lighter quarkcontributions to loop induced Higgs processes are negligible even if their couplings to the Higgs saturate the limitsfrom ∆ γ as discussed below.In order to evaluate the current constraints on modified Higgs interactions, we analyze the latest available Higgsdata, presented in Table I. Our procedure follows closely those of similar previous analyses [24] and we refer the readerto those references for a detailed discussion of the current issues in using the data set.Measurements are given in terms of Higgs signal strengths normalized to SM predictions µ h → BA → h = σ A → h σ SMA → h B h → B B SMh → B , (15)where A → h and h → B stands for different production mode and decay channel, respectively. Experimental best-fitvalues and variances are denoted by ˆ µ i and ˆ σ i , respectively. Experimental collaborations generally provide plots withseparate contribution from VBF plus VH, and from ggF plus ttH production channels for a given decay channel. Inthis case, we take into account their correlation, obtaining a correlation parameter ρ by reproducing the plots. Thecontribution from these decay channels to the total χ function is χ = (cid:88) i (cid:0) µ iGF − ˆ µ iGF , µ iV F − ˆ µ iV F (cid:1) (cid:32) (cid:0) ˆ σ iGF (cid:1) ρ i ˆ σ iGF ˆ σ iV F ρ i ˆ σ iGF ˆ σ iV F (cid:0) ˆ σ iV F (cid:1) (cid:33) − (cid:18) µ iGF − ˆ µ iGF µ iV F − ˆ µ iV F (cid:19) , (16)where GF stands for ggF+ttH, and V F stands for VBF+VH, and index i runs over decay channels. If the separationinto production modes is not provided, we use the data from different search categories for a particular decay channel,which generally target certain production mechanism, but does not imply 100% purity. Inclusive categories aredominated by ggF ( ∼ σ A → h σ SMA → h = ξ ggF σ ggF σ SMggF + ξ V BF σ V BF σ SMV BF + ξ V H σ V H σ SMV H + ξ ttH σ ttH σ SMttH , (17)where ξ i represent contributions of the specified production mechanisms for the given category. We do not assumecorrelations here, and add each search category to χ separately, χ = (cid:88) j (cid:18) µ j − ˆ µ j ˆ σ j (cid:19) , (18) ttH contribution in GF is less then 1%. Decay channel Production mode Signal strength CommentATLAS h → ZZ ∗ Inclusive (87% ggF) 1 . ± . h → bb VH − . ± . h → W W ∗ ggF+ttH 0 . ± .
35 Correlation ρ = − .
3, [17, 26]VBF+VH 1 . ± . h → γγ ggF+ttH 1 . ± .
44 Correlation ρ = − .
4, [17, 27]VBF+VH 1 . ± . h → τ τ ggF+ttH 2 . ± . ρ = − .
5, [17]VBF+VH − . ± . h → bb VH 1 . ± . h → W W ∗ . ± .
21 [28]VBF-tag (20% ggF) 0 . ± . − . ± . h → ZZ ∗ ggF+ttH 0 . ± .
45 Correlation ρ = − .
7, [29]VBF+VH 1 . ± . h → γγ ggF+ttH 0 . ± .
40 Correlation, ρ = − .
5, [30]VBF+VH 1 . ± . h → τ τ . ± .
51 [31]VBF-tag (20% ggF) 1 . ± .
63 [31]VH 0 . ± . and the total χ function is given by χ = χ + χ .We take into account the recent evaluation [20] of ggF production in the SM at approximate N LO in perturbativeexpansion, which exhibits a 17% shift with respect to the values adopted by experimental collaborations, by rescalingcentral values for signal strengths which depend on ggF production by a factor 1 / (1+0 . ξ ggF ). On the other hand, wehave checked that the resulting slightly reduced theory error (from 14% to 12%) has negligible effect on the reportedvariances. Finally, we note that using 7 TeV and 8 TeV data combinations introduces potentially non-negligible effectsdue to the different parton luminosities when constraining NP. We expect however these to be subleading compared tothe overall theoretical uncertainty, especially in light of our ignorance of higher order QCD effects in NP contributions.In the following, we present results of our analysis. The SM is in overall very good agreement with the data. Theassociated χ for 19 observables presented in Table I is χ SM = 16 . .
64. Withinthe vector-like quark scenarios, all the modifications to Higgs signal strengths can be expressed in terms of fourparameters, R gg , R γγ , R bb and ∆ γ , where R bb ≡ Γ h → bb Γ SMh → bb = (cid:18) | y bb | vm b (cid:19) . (19)In particular, one can write µ h → γγGF = R gg ˆΓ R γγ , µ h → ZZ,W W,ττGF = R gg ˆΓ , µ h → γγV F = R γγ ˆΓ µ h → ZZ,W W,ττV F = 1ˆΓ , µ h → bbV H = R bb ˆΓ . (20)The modification of the total Higgs decay width coming from R gg , R bb and ∆ γ is taken into account by writingˆΓ ≡ Γ tot Γ SMtot = 0 . R bb + 0 .
317 + 0 . R gg + ∆ γ , (21)where ∆ γ is constrained to ∆ γ > χ function. Results are presented in the ( R gg , R γγ ) plane, after marginalizing overthe other parameters. We define 68 .
2% (1 σ ) best-fit region to satisfy χ min < χ < χ min + 2 .
3, and 95 .
5% (2 σ ) best-fitregion to satisfy χ min + 2 . < χ < χ min + 6 . (cid:236) (cid:180) (cid:72) h (cid:174) gg (cid:76) R (cid:72) h (cid:174) ΓΓ (cid:76) (cid:236) (cid:180) (cid:72) h (cid:174) gg (cid:76) R (cid:72) h (cid:174) ΓΓ (cid:76) Figure 1:
Left:
Fit of Higgs data taking R gg ≡ R ( h → gg ), R γγ ≡ R ( h → γγ ) and ∆ γ as fitting parameters. The best fitpoint (cross), 1 σ (dark gray) and 2 σ (light gray) regions are shown in the ( R gg , R γγ ) plane after marginalizing over ∆ γ . TheSM reference scenario is marked with a diamond. Results of the fit for ∆ γ fixed to its SM value are given by the darker redcontour (1 σ region) and the lighter orange contour (2 σ region). The resulting prediction from the non-renormalizable modelwith an additional up-like vector-like quark, the model with a down-like vector-like and the model with a doublet vector-likequark in the no-mixing and negligible isospin breaking limit are given by the continuous-blue, dotted-red and dashed-greencurves, respectively. General predictions from the non-renormalizable model with a doublet vector-like quark are given bythe region presented with the dashed-black contour (see text for details). Right:
Fit of Higgs data taking R gg , R γγ , R b and∆ γ as fitting parameters. The best fit point (cross), 1 σ (dark gray) and 2 σ (light gray) regions are shown in the ( R gg , R γγ )plane after marginalizing over R b and ∆ γ . Results of the fit for ∆ γ fixed to its SM value are given by the darker red contour(1 σ region) and the lighter orange contour (2 σ region). The resulting prediction from the non-renormalizable model with adown-like vector-like and the model with a doublet vector-like quark in the no-mixing and negligible isospin breaking limitare given by dotted-red and dashed-green curves, respectively. General predictions from the non-renormalizable model with adoublet vector-like quark are given by the region presented with the dashed-black contour (see text for details). First, we take R gg and R γγ as fitting parameters, while fixing R bb and ∆ γ to their SM values. This applies toscenarios, where the couplings of SM quarks to the Higgs (the Yukawas) are not modified, while R gg and R γγ receivenew loop contributions. In models with vector-like quarks this corresponds to the limit of zero mixing betweenthe chiral and vector-like quarks. The minimum of the χ corresponds to a point ( R gg , R γγ ) = (0 . , .
30) with χ min = 12 . .
76. Second, we take R gg , R γγ and ∆ γ as fitting parameters, while fixing R bb = 1. Thiscorresponds to scenarios where the vector-like quarks do not mix with the b quark, but possibly with the lighterquark generations (see [32] for a recent model example). In this case, the point ( R gg , R γγ , ∆ γ ) = (0 . , . , . χ , with χ min = 12 . .
7. The results for the first two scenarios arepresented on the left plot in Fig. 1. For the first scenario, 1 σ and 2 σ contours are represented by (darker) red and(lighter) orange curves, respectively.In the third case, we take R gg , R γγ and R bb as fitting parameters, while fixing ∆ γ to its SM value. This caseapplies to the most studied scenarios in the literature, where the vector-like quarks only mix with the third generation(c.f. [33] for a recent study). The minimum of the χ corresponds to a point ( R gg , R γγ , R bb ) = (0 . , . , .
97) with χ min = 12 . .
69. In the last case, we take all four parameters to fit the data, corresponding to the mostgeneral case of vector-like quarks mixing with all three SM generations. Here, the minimum of the χ corresponds toa point ( R gg , R γγ , R bb , ∆ γ ) = (0 . , . , . , .
2) with χ min = 12 . .
65. The results for the last twoscenarios are presented on the same plot, Fig. 1 right. For the third scenario, 1 σ and 2 σ contours are represented by(darker) red and (lighter) orange curves, respectively.The main observation at this point is that allowing for a modification of R bb and/or ∆ γ (non-zero mixing of vector-like quarks with some of the SM chiral quarks) significantly increases the allowed range of R gg , while it has muchless of an effect on R γγ . This has important implications for constraining vector-like quark models using Higgs data,since these generically predict much larger effects in R gg . In the following sections we apply these general results toa few simplest SM extensions with a single light vector-like quark state below the effective theory cut-off Λ. III. SINGLET UP-TYPE VECTOR-LIKE QUARKA. Renormalizable model
As a first example, we consider the SM extended by a vector-like quark pair ( U L , U R ) in the / representation ofthe SM electroweak group. In the most general renormalizable model the quark Yukawa interactions and mass termscan be described by the following Lagrangian − L (0) U = y ijd ¯ q iL Hd jR + y iju ¯ q iL ˜ Hu jR + y iU ¯ q iL ˜ HU R + M U ¯ U L U R + h.c. , (22)where ˜ H ≡ iτ H ∗ , H = ( G + , ( v + h + iG ) / √
2) is the SM Higgs doublet, q iL the SM quark doublets and u iR the SMup-type quark singlets. Note that additional kinetic mixing terms of the form U L u iR can always be rotated away andreabsorbed into the definitions of y u,U . Furthermore, one can, without loss of generality, choose a weak interactionbasis where y u is diagonal and real. After EW symmetry breaking (EWSB) the mass matrices for up- and down-typequarks are M u = (cid:32) y u v/ √ y U v/ √ M U (cid:33) , M d = ( y d v/ √ . (23)The weak gauge and Higgs interactions of 4 ( u, c, t, u (cid:48) ) physical up-like and 3 ( d, s, b ) down-like quarks in this (mass)eigenbasis are given by eqs. (5)-(7), where V R = 0, V L is a general 4 × X d = I × . Note that in thismodel X uii = 1 − ∆ ui and that tree level constraints on the entries of X u already severely constrain the admixture of U within the physical u and c quarks. In particular, we find for the 3 × X u describing Z and Higgscouplings to known up-type quark flavours | X u − I | × < .
001 2 . × − . . . . . (24)Loop-level u (cid:48) effects provide better constraints only on the mixing of the vector-like singlet quark with the top quark.Neglecting the small mixing with the first two generations (effectively setting y u,cU = 0) the t − u (cid:48) system can bedescribed by three independent physical parameters: two quark masses ( m t , m u (cid:48) ) and a single (left-handed) mixingangle ( θ tU ), which are defined as [34]tan(2 θ tU ) = √ vy tU M U M U − [( y tu ) + ( y tU ) ] v / , (25) m t m u (cid:48) = M U y tu v √ , m t + m u (cid:48) = M U + v y tu ) + ( y tU ) ] . (26)In terms of these, X utt = c tU , X utu (cid:48) = c tU s tU and X uu (cid:48) u (cid:48) = s tU , where c tU ≡ cos θ tU and s tU ≡ sin θ tU .Presently, the most sensitive observable to nonzero s tU is the ρ parameter, which receives a new contribution of theform [34] ∆ ρ = αN C πs W m t m W s tU (cid:20) − (1 + c tU ) + s tU r + 2 c tU rr − r ) (cid:21) , (27)where r ≡ m u (cid:48) /m t and we have neglected terms of higher order in m Z,b /m t,u (cid:48) . A comparison with the experimentalbound of ∆ ρ exp = 4 +3 − × − [10] yields a constraint on s tU as a function of the u (cid:48) mass as shown in Fig. 2.While the modified top quark coupling to the Higgs boson and the presence of an additional heavy quark can inprinciple impact also loop induced Higgs decays, namely h → gg , h → γγ and h → Zγ , taking into account theabove constraints on X uij these effects turn out severely suppressed making it impossible in practice to distinguish therenormalizable model with a singlet vector-like up type quark from the SM in single Higgs production processes. B. Non-renormalizable models
Extending the above renormalizible model with the leading dimension five operators containing the light SM fieldsand U L,R as the only dynamical degrees of freedom below a UV cut-off scale Λ, Yukawa interactions and mass terms
500 600 700 800 900 10000.00.10.20.30.40.5 m u ' s t U Figure 2: Upper limit at 95% C.L. on t − u (cid:48) (left-handed) mixing angle as a function of the u (cid:48) quark mass in the model withan up-like vector-like quark. The gray region marks the ATLAS experimental search bound on the renormalizable model usingthe u (cid:48) → th decay signature [35]. in eq. (22) receive corrections which result in modified interactions between up-type quarks and the Higgs. Onecan manifestly preserve the exact mass diagonalization procedure of the renormalizable model by parametrizing theleading non-renormalizable contributions in terms of the replacement M U → M U + c v / − | H | Λ , (28)plus an additional Higgs-dependent ‘kinetic mixing’ operator − L (1) U = c i v / − | H | Λ ¯ U L u iR . (29)After EWSB, the flavor structure of gauge interactions (and the associated bounds on X uij ) in the renormalizable modelis preserved and only the Higgs interactions in the mass eigenbasis receive new contributions of the form (8) where X d (cid:48) = 0 (leading to R bb = 1) while X u (cid:48) = U uL . [(0 , , ( c , c )] .U u † R . Interestingly, even though X u (cid:48) has no observableeffects on charged current interactions of quarks, one can derive an indirect bound on the diagonal entries of X u (cid:48) fromCKM unitarity. Following its definition, X u (cid:48) ij = ( U uL ) i (cid:16) c k ( U uR ) ∗ jk + c ( U uR ) ∗ j (cid:17) , we note that | X u (cid:48) ij | is proportionalto | ( U uL ) i | = 1 − X uii , multiplied by at most O (1) coefficients in the effective field theory expansion c i , c and theunitary rotation U uR . Furthermore, the general identity | ( U uR ) i | = ( m i /M U ) | ( U uL ) i | implies that c contributionsto u, c interactions are severely suppressed. Consequently, CKM unitarity constraints on X uuu,cc yield indirect boundson the diagonal elements | X u (cid:48) uu | (cid:46) .
03 max( c i ) and | X u (cid:48) cc | (cid:46) . c i ). Note that sizable contributions to ∆ γ arewell consistent with these indirect CKM unitarity constraints.The relevant modifications to R gg and R γγ in the up-type singlet scenario come from the modified top quarkcoupling and the presence of the additional heavy quark in the loop. In the limit m t,u (cid:48) (cid:29) m h , which is a goodapproximation here, the relevant numerical expressions are given by R gg = | . r y − . | + 0 . . , R γγ = |− . . r y | |− . | , (30)where to order 1 / Λ and including possible small t − u (cid:48) mixing r y ≡ y tt vm t + y u (cid:48) u (cid:48) vm u (cid:48) = 1 + s tU (cid:0) c t c (cid:48) tU − c s (cid:48) tU (cid:1) v Λ m t − c tU (cid:0) c t s (cid:48) tU + c c (cid:48) tU (cid:1) v Λ m u (cid:48) . (31)Above we have used the short-hand notation for c (cid:48) tU = cos θ (cid:48) tU , s (cid:48) tU = sin θ (cid:48) tU , where the (right-handed) mixing angle θ (cid:48) tU is defined via tan θ (cid:48) tU = ( m t /m u (cid:48) ) tan θ tU . Thus, h → γγ and h → gg are highly correlated in this set-up. Note0that at the one loop level and in the large m t,u (cid:48) limit, contributions of renormalizable interactions of t and u (cid:48) cancelexactly , therefore, leading effects appear at order O ( v/ Λ).The resulting predictions in the up-type singlet scenario are presented by the continuous (blue) curve in the( R gg , R γγ ) plain in left plot of Fig. 1. For concreteness we take | ( c t , c ) | / Λ ≤ − and m u (cid:48) ≥
640 GeV assuggested by direct searches [35] (see also the related discussion at the end of this section). Also, we take the t − u (cid:48) mixing angle to be within the 95% C.L. experimental bound discussed above ( s tU (cid:46) . r y at 68% C.L. of r y = 0 . +0 . − . (when marginalizing over ∆ γ ) and r y = 0 . ± .
08 (when fixing ∆ γ = 0). Interestingly, in both cases the fit slightly prefers r y <
1. We note in passingthat after marginalising over r y in this scenario ∆ γ is bounded as ∆ γ < .
75 at 95% C.L. , which in term implies | X u (cid:48) uu + X u (cid:48) cc | v/ Λ < .
022 .Finally, turning to the naturalness condition in (2), for the case of a single vector-like up-type singlet quark mixingwith the top it reads m t c tU + m u (cid:48) s tU v = 1Λ [ m t s tU ( − c t c (cid:48) tU + c s (cid:48) tU ) + m u (cid:48) c tU ( c t s (cid:48) tU + c c (cid:48) tU )] + O (1 / Λ ) . (32)In the zero-mixing limit this leads to the prediction r y = 1 − ( m t /m u (cid:48) ) , independent of the cut-off scale Λ. Interest-ingly, present Higgs data (exhibiting a preference for r y <
1) are perfectly consistent with the naturalness condition.On the other hand, the Higgs fit results can also be interpreted in this context as imposing an indirect bound on the u (cid:48) mass of m u (cid:48) >
360 GeV at 95% C.L. It is instructive to compare the above constraint to results of direct experimental searches for up-type singlet vector-like quarks. Interestingly, the most severe bound on u (cid:48) in the renormalizable model and assuming dominant but small u (cid:48) mixing with the top, m u (cid:48) >
640 GeV [35] is given by the ATLAS experimental search using the u (cid:48) → th decay signature.In the non-renormalizable model the relevant couplings are given by y tu (cid:48) = s tU c tU m u (cid:48) /v + s tU ( s (cid:48) tU c t + c (cid:48) tU c ) v/ Λ and y u (cid:48) t = s tU c tU m t /v − c tU ( c (cid:48) tU c t − s (cid:48) tU c ) v/ Λ. It is then easy to check that compared to u (cid:48) → tZ and u (cid:48) → bW rates, the1 / Λ corrections can in principle enhance the B ( u (cid:48) → th ) in the small s tU limit (in the extreme case B ( u (cid:48) → th ) = 1the present bound is then strengthened to m u (cid:48) (cid:38)
850 GeV [35]) but cannot reduce it significantly below its value inthe renormalizable model. However, if u (cid:48) does not dominantly decay to third generation quarks (but instead to firsttwo quark generations), the current direct search constraints are relaxed dramatically (c.f. [38]) and m u (cid:48) (cid:39)
300 GeVbecomes a possibility. In summary, the Higgs fit already provides an interesting complementary constraint on scenarioswith an up-type singlet vector-like quark cancelling the top-loop quadratic divergence to the Higgs mass. Althoughit is at present only marginally competitive with existing direct search bounds, it is far less sensitive to the hierarchyof mixings with the known three generations of up-type quarks (provided they are small).
IV. SINGLET DOWN-LIKE VECTOR-LIKE QUARKA. Renormalizable model
Next we consider a SM extension with a vector-like quark pair ( D L , D R ) in the − / electroweak representation.The most general renormalizable Lagrangian now contains the Yukawa and mass terms − L (0) D = y ijd ¯ q iL Hd jR + y iju ¯ q iL ˜ Hu jR + y iD ¯ q iL HD R + M D ¯ D L D R + h.c. . (33)The mass matrices of up- and down-type quarks after EWSB have the form (23) with the replacement u ↔ d and U ↔ D . In the mass-eigenbasis of 4 ( d, s, b, d (cid:48) ) physical down-type and 3 ( u, c, t ) up-type quarks the weak gaugeand Higgs interactions are controlled by the general 3 × V Lij (again V R = 0) defined as before leading to X u = I × . On the other hand, now the entries of the hermitian matrix X d are experimentally severily constrainedby their tree-level contributions to CKM non-unitarity and FCNCs in the down-quark sector and already preclude Deviations from the large mass limit, as well as higher order perturbative corrections can upset this cancellation. However a recentstudy of gluon fusion production in the renormalizable model with a singlet vector-like top partner at NNLO in QCD [34] has foundsuch effects to be tiny, only a few percent for maximal mixing. For a comparison with the situation after the first Higgs data see [36]. | X d − I | × < .
004 1 . × − × − .
006 0 . . . (34)We immediately observe that Higgs phenomenology in the renormalizable down-type singlet model is again indistin-guishable from the SM. In particular, considering only the dominant effects due to b − d (cid:48) mixing and thus parametrizing X dbb = c bD , X dbd (cid:48) = c bD s bD and X dd (cid:48) d (cid:48) = s bD , where c bD + s bD = 1, experimental constraints indicate s bD = 0 . h → b ¯ b , h → gg and h → γγ of 0 . .
5% and − . B. Non-renormalizable models
The leading higher dimensional modifications of Higgs interactions can again be most conveniently parametrizedvia the replacement M D → M D + c v / − | H | Λ , (35)plus an additional Higgs-dependent ‘kinetic mixing’ operator − L (1) D = c i v / − | H | Λ ¯ D L d iR , (36)yielding new Higgs interactions in the mass eigenbasis of the form (8), where now X u (cid:48) = 0 and X d (cid:48) = U dL . [(0 , , ( c , c )] .U d † R . Constraints on | ( U dL ) i | = 1 − X dii lead to the following indirect bounds | X d (cid:48) dd | (cid:46) .
06 max( c i ), | X d (cid:48) ss | (cid:46) .
08 max( c i ) (both allowing for sizeable modifications of ∆ γ ) and | X d (cid:48) bb | (cid:46) .
13 max( c i ). In fact, while the Z → b ¯ b anomaly cannot be fully resolved in this model, the data prefers non-zero b − d (cid:48) mixing with s bD = 0 . d (cid:48) mixing with first two generations). This is enough to allow for O (1) modification of y bb andthus R b ¯ b at order 1 / Λ. In particular neglecting also the m b /M D suppressed right-handed b − d (cid:48) mixing one can write y bb (cid:39) m b v + s bD c b v Λ = m b v (cid:18) s bD c b v m b Λ (cid:19) . (37)On the other hand, b − d (cid:48) mixing has negligible effects on the modifications to gluon fusion production and Higgsdecay to two photons R gg = 0 . + | .
65 + 0 . y d (cid:48) d (cid:48) | . , R γγ = |− . . y d (cid:48) d (cid:48) | |− . | , (38)where c contributions to y d (cid:48) d (cid:48) dominate as y d (cid:48) d (cid:48) = − c bD c v / Λ m d (cid:48) . The resulting predictions from the non-renormalizable model with a singlet down-type vector-like quark are presented by the red-dotted curves in the( R gg , R γγ ) plane in Fig. 1. Again we have used | c | / Λ ≤ − and m d (cid:48) >
350 GeV as suggested by directsearches (see the related discussion below). Allowing for R bb (cid:54) = 1 (right plot), the preferred parameter regions for y d (cid:48) d (cid:48) at 68% C.L. are y d (cid:48) d (cid:48) = − . +0 . − . (when marginalizing over ∆ γ ) and y d (cid:48) d (cid:48) = − . +0 . − . (when fixing ∆ γ = 0).Consequently, current Higgs data are not yet very constraining in this context. On the other hand, in absence ofsignificant d (cid:48) mixing with lighter quarks (for R bb = 1 and ∆ γ = 0 in left plot of Fig. 1), the Higgs data already givean interesting constraint (as discussed below) on y d (cid:48) d (cid:48) = − . ± .
08. Marginalizing instead over y d (cid:48) d (cid:48) and R bb in thisscenario, we obtain a bound on ∆ γ < . | X d (cid:48) dd + X d (cid:48) ss | v/ Λ < . y d (cid:48) d (cid:48) and ∆ γ , we get R bb < .
3, implying | . − X d (cid:48) bb v/ Λ | < .
038 .Finally, turning to the naturalness condition in (2), for the case of a single vector-like down-type singlet quark itreads in the small b − d (cid:48) mixing limit m t + m d (cid:48) s bD v = c bD c m d (cid:48) Λ + O (1 / Λ ) , (39)or equivalently y d (cid:48) d (cid:48) = − s bD − ( m t /m d (cid:48) ) again independent of the cut-off scale Λ. Present Higgs data then providean indirect constraint on the d (cid:48) mass, which reads m d (cid:48) >
330 GeV in the zero b − d (cid:48) mixing case and grows stronger2for non-zero s bD . This is to be compared to direct experimental searches [37], which yield m d (cid:48) >
480 GeV for therenormalizable down-type singlet model dominantly mixing with the b . In this case however, the direct constraint isdominated by the d (cid:48) → W t decay signature. Enhancing the d (cid:48) → bh rate in the small b − d (cid:48) mixing limit throughthe coupling y d (cid:48) b (cid:39) − c bD c (cid:48) bD c b v/ Λ can thus naturally relax it to m d (cid:48) (cid:38)
350 GeV; dominant (but small) mixing withthe first two generations possibly even further. In light of this, the Higgs data already provide a complementary andcompetitive handle on such models.
V. DOUBLET VECTOR-LIKE QUARKA. Renormalizable model
As a final example, we consider the SM extended by a vector-like pair ( Q L , Q R ) in the / electroweak represen-tation. The most general renormalizable Lagrangian now contains the Yukawa and mass terms − L (0) Q = y ijd ¯ q iL Hd jR + y iju ¯ q iL ˜ Hu jR + y iD ¯ Q L Hd iR + y iU ¯ Q L ˜ Hu iR + M Q ¯ Q L Q R + h.c. . (40)The mass matrices of both up- and down-like quarks after EWSB now have the form M u = (cid:32) y u v/ √ y U v/ √ M Q (cid:33) , M d = (cid:32) y d v/ √ y D v/ √ M Q (cid:33) . (41)In the quark mass eigenbasis the weak gauge and Higgs interactions of 4 ( u, c, t, u (cid:48) ) physical up-like and 4 ( d, s, b, d (cid:48) )down-like quarks are governed by two 4 × V L and a non-unitary V R . Consequently X u,dij = δ ij ,while Y u,dij are hermitian and constrained as | Y u | × < .
11 2 . × − . .
018 0 . − , | Y d | × < . . × − × − .
21 0 . . . (42)Due to such severe experimental bounds on the mixing of vector-like doublet components with the first two quarkgenerations, and also with the b quark, the dominant effect on Higgs phenomenology could possibly come from themixing in the top sector (via induced Y utt ), which remains unconstrained at the tree-level. However, as shown in [34]the (right-handed) mixing angles in the top ( t − u (cid:48) ) and bottom ( b − d (cid:48) ) quark sectors are related via the mass splittingbetween the two extra quark states u (cid:48) and d (cid:48) as m d (cid:48) [1 − s bD (1 − r bd (cid:48) )] = m u (cid:48) [1 − s tU (1 − r tu (cid:48) )] , (43)where r ij ≡ m i /m j . Left-handed and right-handed mixing angles are now related through tan θ (cid:48) ij = r ij (cid:48) tan θ ij . Atthe one-loop level, the u (cid:48) − d (cid:48) mass splitting (∆ m Q ≡ m u (cid:48) − m d (cid:48) ) is constrained from EW precision measurements.In particular, Z → b ¯ b observables constrain the b − d (cid:48) and t − u (cid:48) mixing angles as shown in Fig. 5 (see appendix Cfor details). Together with a constraint from the ρ parameter, this gives the bound on ∆ m Q as shown in Fig. 3(the narrowest purple bands). Taking all this into account (in particular discarding the fine-tuned solution for largenegative ∆ m Q ), we find (in accordance with [34]) that the vector-like quark doublet with renormalizable couplingshas unobservable effects in single Higgs production and decay processes. B. Non-renormalizable models
At the non-renormalizable level, the doublet vector-like quark model offers a somewhat richer structure thanthe singlet examples. Namely, the introduction of dimension five operators allows to shift the vector-like massindependently for both isospin components of Q via the insertion of an iso-triplet combination of Higgs fields M Q ¯ Q R Q L → M Q ¯ Q R Q L + c +2 Λ ( v / − | H | ) ¯ Q R Q L + c − Λ ¯ Q R ( HH † − ˜ H ˜ H † ) Q L . (44) Note that we do not find a bound as strong as reported in [34]. The resulting implications for Higgs phenomenology remain howeverqualitatively unchanged.
600 700 800 900 1000 1100 (cid:45) (cid:45) m u ' (cid:68) m Q (cid:144) m u ' s tU (cid:174) (cid:76) (cid:174) (cid:165) Figure 3: Allowed region at 95% C.L. for u (cid:48) − d (cid:48) mass splitting as a function of the u (cid:48) quark mass in models with a doubletvector-like quark. The narrowest purple bands apply to the renormalizable model, the middle orange band stands for thenon-renormalizable model in the zero t − u (cid:48) mixing limit, while the broadest green band is for the non-renormalizable modelwith non-zero t − u (cid:48) mixing effects allowed by Z → bb data. The gray area marks the ATLAS experimental search bound onthe renormalizable model using the u (cid:48) → th decay signature [35]. Similarly, one can now introduce two new operators (via iso-singlet and iso-triplet Higgs field insertions) − L (1) Q = ( c +1 ) i Λ ( v / − | H | ) ¯ Q R q iL + ( c − ) i Λ ¯ Q R ( HH † − ˜ H ˜ H † ) q iL . (45)The two isospin breaking corrections (proportional to c − , ) now necessarily induce corrections to quark masses andmixings. In particular, the resulting changes to M u,d can be parametrized as( y u,d ) ij → ( y (cid:48) u,d ) ij ≡ ( y u,d ) ij ¯ M U,D M U,D ∓ v ( y U,D ) i ( c − ) j M U,D , (46)( y U,D ) i → ( y (cid:48) U,D ) i ≡ ( y U,D ) i ¯ M U,D M U,D ± v ( y u,d ) ij ( c − ) j M U,D , (47) M Q → M U,D , (48)where M U,D ≡ (cid:113) ¯ M U,D + v (( c − ) i ( c − ) ∗ i ) / and ¯ M U,D ≡ M Q ± v c − / m Q ≡ m u (cid:48) − m d (cid:48) becomes an independent free parameter, given in the zero-mixing limit (when y (cid:48) U,D = 0)solely by ∆ m Q = v c − / Λ. At the one-loop level it will affect the ρ parameter as∆ ρ (cid:39) − αN C πs W (∆ m Q ) m W , (49)where we have only kept the leading ∆ m Q dependence. The resulting constraint is shown in Fig. 3 (in middle orangeband). However, if we also include non-zero t − u (cid:48) mixing effects, marginalizing over the allowed values of s tU from Z → b ¯ b data we obtain a much weaker bound on ∆ m Q shown in the uppermost (green) band in Fig. 3. In ournumerical evaluation we employ the full one-loop formula for ∆ ρ , which can be found in Appendix D. We thusconclude that after including the contributions of leading higher dimensional operators, the isospin components of aTeV scale vector-like quark doublet can be split by as much as 30%. Although this has no observable consequencesfor Higgs phenomenology, it can have profound implications for direct u (cid:48) searches if the u (cid:48) → d (cid:48) W decay channelbecomes kinematically allowed.The new Higgs interactions in the mass eigenbasis are again of the form (8), where now X u,d (cid:48) = U u,dR . [(0 , , ( c +1 ± c − , c +2 ± c − )] ∗ .U u,d † L or explicitly ( X u,d (cid:48) ) ij = ( U u,dR ) i [( c +1 ± c − ) k ( U u,dL ) ∗ jk + ( c +2 ± c − )( U u,dL ) ∗ j ]. They are thusconstrained indirectly by bounds on Y u,d , since now | ( U u,dR ) i | = Y u,dii . Taking into account also the relation4 | ( U u,dL ) i | = ( m i /M U,D ) | ( U u,dL ) i | , we can safely neglect c ± contributions and obtain X u (cid:48) uu (cid:46) .
35 max[( c +1 + c − ) i ], X u (cid:48) cc (cid:46) .
13 max[( c +1 + c − ) i ], X d (cid:48) dd (cid:46) .
30 max[( c +1 − c − ) i ], X d (cid:48) ss (cid:46) .
40 max[( c +1 − c − ) i ] and X d (cid:48) bb (cid:46) .
11 max[( c +1 − c − ) i ].Significant effects in ∆ γ and R bb are thus possible and can be used to constrain the diagonal entries of X u,d (cid:48) .Additional interesting effects again appear in loop induced Higgs processes. Higgs decays to pairs of gluons orphotons are modified by additional heavy particles in the loop R gg = | . r x + r y ) − . | + 0 . . , R γγ = |− . . r x + 0 . r y | |− . | , (50)where to order 1 / Λ and including t − u (cid:48) mixing r x ≡ y tt vm t + y u (cid:48) u (cid:48) vm u (cid:48) = 1 + s tU (cid:0) ( c +1 + c − ) t c (cid:48) tU − ( c +2 + c − ) s (cid:48) tU (cid:1) v Λ m t − c tU (cid:0) ( c +1 + c − ) t s (cid:48) tU + ( c +2 + c − ) c (cid:48) tU (cid:1) v Λ m u (cid:48) , (51)and r y ≡ y d (cid:48) d (cid:48) vm d (cid:48) = − v Λ m d (cid:48) ( c +2 − c − ) . (52)Taking into account the bound on t − u (cid:48) mixing, a good approximation for r x = 1 + s tU ( c +1 + c − ) t v Λ m t − ( c +2 + c − ) v Λ m u (cid:48) . Thereis no correlation between R gg and R γγ in general, unless one imposes additional constraints on the parameters.Therefore, the resulting predictions from the non-renormalizable model with a doublet vector-like quark are givenby a region (dashed-black contour) in the ( R gg , R γγ ) plane of Fig. 1 left, in the case when modification of ∆ γ isallowed and R bb = 1, and on Fig. 1 right, when sizable modification of R bb is allowed as well. We have assumed (cid:12)(cid:12) c + , − , (cid:12)(cid:12) / Λ ≤ − and m q (cid:48) >
790 GeV as suggested by direct searches [35]. Also, we take the t − u (cid:48) mixing angleto be within the 95% C.L. experimental bound discussed above ( s tU (cid:46) . u (cid:48) and d (cid:48) quarks are degenerate and r y = r x − − v Λ m q (cid:48) c +2 . Allowing for modification of ∆ γ and fixing R bb = 1, the resulting predictions in this scenario are presentedby the green-dashed curve in the ( R gg , R γγ ) plane in Fig. 1 left. Allowing for sizable modification of R bb as well,the predictions are presented by the green-dashed curve in the ( R gg , R γγ ) plane in Fig. 1 right. Again we haveassumed (cid:12)(cid:12) c +2 (cid:12)(cid:12) / Λ ≤ − and m q (cid:48) >
790 GeV. In particular, the preferred parameter regions for r y at 68% C.L.are r y = − . +0 . − . (marginalizing over both R bb and ∆ γ ), r y = − . +0 . − . (marginalising over R bb but fixing ∆ γ ), r y = − . +0 . − . (fixing R bb = 1 and marginalizing over ∆ γ ) and finally r y = − . ± .
04 (fixing both R bb and ∆ γ to their SM values) .Also in the unmixed isospin symmetric doublet scenario we can get ∆ γ < . r y and R bb . This implies | X u (cid:48) uu + X u (cid:48) cc + X d (cid:48) dd + X d (cid:48) ss | v/ Λ < . r y and ∆ γ , we get R bb < . | . − X d (cid:48) bb v/ Λ | < .
036 . We have checked that these results remain stable even in presence ofsmall t − u (cid:48) mixing and isospin breaking.In this scenario, the Higgs mass naturalness condition reads m t v = 2 c +2 m q (cid:48) Λ + O (1 / Λ ) , (53)or equivalently r y = − ( m t / √ m q (cid:48) ) . This condition allows to put an indirect bound on the mass of q (cid:48) to be 390 GeVat 95% C.L. in the case of fixed R bb .Turning to direct searches, the most severe bound on u (cid:48) in the renormalizable model with a doublet vector-likequark assuming dominant but small mixing with the third generation, m u (cid:48) >
790 GeV [35] is given by the ATLASexperimental search using the u (cid:48) → th decay signature. In the non-renormalizable model the relevant couplingsare given by y tu (cid:48) = s tU c tU m u (cid:48) /v + s tU ( s (cid:48) tU ( c +1 + c − ) t + c (cid:48) tU ( c +2 + c − )) v/ Λ and y u (cid:48) t = s tU c tU m t /v − c tU ( c (cid:48) tU ( c +1 + c − ) t − s (cid:48) tU ( c +2 + c − )) v/ Λ. It is then easy to check that again compared to u (cid:48) → tZ and u (cid:48) → bW rates, the 1 / Λcorrections can in principle enhance B ( u (cid:48) → th ) in the small s tU limit but cannot reduce it significantly below its As in the case of the non-renormalizable model with a down-like vector-like quark, y bb can receive O (1) modifications even for small b − d (cid:48) mixing. Coupling Constraint Reference | X ucu | , | Y ucu | < . × − [12, 13] | X utu,tc | , | Y utu,tc | < .
14 Appendix A | X dds | , | Y dds | < . × − Appendix B | X ddb | , | Y ddb | < × − | X dsb | , | Y dsb | < × − δX uuu − . δX ucc − . δX ddd − . δX dss − . δX dbb . δY uuu . δY ucc − . δY ddd . δY dss − . +0 . − . δY dbb − . Z ) couplings to SM quarks in renormalizable vector-likequark models (see Eqs. (6), (7)) from precision flavor and electroweak observables. All upper bounds are given at 95% C.L. Fordiscussion of δX utt , δY utt constraints see Sections III A and V A, respectively. value in the renormalizable model. However, the significant u (cid:48) − d (cid:48) mass splitting allowed by present data whenincluding dimension five contributions, reopens the possibility that the dominant decay channel of u (cid:48) is actually u (cid:48) → d (cid:48) W , in which case the existing direct search constraints are considerably relaxed and dominated by searchesfor the lighter isospin component via d (cid:48) → tW and d (cid:48) → bh decay signatures (in the case of dominant mixing with thethird generation). Consequently, in such scenarios u (cid:48) ( d (cid:48) ) masses as low as m u (cid:48) ( d (cid:48) ) (cid:38)
400 GeV (300 GeV) could still beviable.
VI. CONCLUSIONS
We have systematically investigated the impact of dynamical vector-like quarks, accommodated within SM gaugerepresentations and charges, on Higgs physics. In particular, we have considered the weak singlet up-type, singletdown-type and doublet vector-like quarks, potentially mixing with all three known generations of chiral quarks, andhave updated the most relevant constraints on such scenarios from low energy flavour phenomenology and electro-weak precision measurements. The resulting general constraints on the renormalizable and leading non-renormalizablequark-Higgs interactions are summarised in Tables II and III, respectively.Within the renormalizable SM extended by additional vector-like quarks, we have shown generally that Higgscouplings to the known three generations of quarks need to remain SM-like regardless of the extra quark masses. Thisfeature is a consequence of the fact that precision flavour and electro-weak observables are affected by the mixing ofvector-like and chiral quarks, some of them already at the tree level. Consequently, Higgs decay widths to light quarkpairs, gg and γγ cannot deviate significantly from their SM predictions.A singular feature of models with vector-like fermions is that non-renormalizable contributions sensitive to physicsat the cut-off scale of the effective low energy theory appear already at dimension five. The inclusion of such higher-dimensional operators is essential in models that aim to cancel dominant quadratic divergences to the Higgs bosonmass coming from top quark loops with new fermionic contributions. Contrary to the renormalizable models (seehowever [39]), they also predict interesting effects in Higgs phenomenology. We have investigated such contributionsfor all three types of additional vector-like quarks mixing with SM generations (see Fig. 1). The most importantconsequences for Higgs physics are possible significant enhancements in Higgs decay rates to pairs of light ( u ¯ u, d ¯ d, s ¯ s, c ¯ c )quarks, which may still account for a significant fraction of the total Higgs decay width. This is possible due to asimultaneous strong modification of the Higgs production by gluon fusion gg → h (up to 50% deviation from SMpredictions, while the h → γγ decay width can receive modifications up to 10%). Such a possibility could thus betested by future more accurate determinations of vector boson fusion and ( W , Z or t ¯ t ) associated Higgs production.Inaddition, a possible modification of the Higgs coupling to b quarks in models containing down-type vector-like quarks6 Coupling Constraint Reference | X u (cid:48) uc,cu | v/ Λ < × − [16] (cid:113) | X u (cid:48) tu,tc | + | X u (cid:48) ut,ct | v/ Λ < . | X d (cid:48) sd,ds | v/ Λ < × − | X d (cid:48) bd,db | v/ Λ < × − | X d (cid:48) sb,bs | v/ Λ < × − (cid:112) | X u (cid:48) uu | + | X u (cid:48) cc | v/ Λ < .
022 Section III B (cid:113) | X d (cid:48) dd | + | X d (cid:48) ss | v/ Λ < .
027 Section IV B (cid:113) | X u (cid:48) uu | + | X u (cid:48) cc | + | X d (cid:48) dd | + | X d (cid:48) ss | v/ Λ < .
025 Section V B | . − X d (cid:48) bb v/ Λ | < .
038 Section IV B < .
036 Section V BTable III: Compilation of constraints on additional Higgs couplings to SM quarks due to dimension five operators in modelswith vector-like quarks (see Eq. (8)). The off-diagonal couplings are bounded by low energy flavor observables, whereas upperlimits on the diagonal ones were estimated from the fit to current Higgs data. All bounds are given at 95% C.L. For discussionof δX u (cid:48) tt constraints within up-type singlet and doublet vector-like quark scenarios see Sections III B and V B, respectively. will be probed by future precise measurements of the h → b ¯ b decay. Interestingly, current Higgs data are perfectly consistent with (and even exhibit a slight preference for) the possi-bility that vector-like quarks contribute to the cancellation of the top loop quadratic divergence to the Higgs mass.Conversely in some cases, a fit to existing Higgs measurements under such an assumption already offers competitiveand robust constraints on vector-like quark masses in comparison with results of direct experimental searches whichneed to rely on particular decay signatures. In the example of a weak doublet of vector-like quarks, we have shownthat dimension five contributions allow to relax the stringent bounds on the mass splitting between the two isospinstates. Consequently the decay of the heavier doublet component to the lighter one with the emission of a W bosonmay become kinematically allowed, affecting the relevant experimental signatures. More generally in presence ofdimension five contributions, vector-like quark decay widths are naturally dominated by decay channels involving theHiggs. Future dedicated experimental searches for such particular signatures (as exemplified in [35], see also [40])could thus shed light on the relevance of vector-like quarks in the solution to the SM hierarchy problem. Acknowledgments
This work was supported in part by the Slovenian research agency and the Ad futura Programme of the SlovenianHuman Resources and Scholarship Fund.
Appendix A: CKM non-unitarity and Z mediated FCNCs in top quark production and decays In absence of right-handed charged currents, experimental constraints on (cid:80) i = d,s,b | V Lti | = 1 − ∆ ut ≤ | V Ltb | can be obtained from the measurements of the ( t -channel) single top production cross-section ( σ t ) at the LHCand the fraction of top decays to W b pairs in t ¯ t production ( R ). Assuming t → W q channels dominate the topdecay width, to a very good approximation R is given by R (cid:39) | V Ltb | / ( (cid:80) i = d,s,b | V Lti | ) and we can use the recent CMSresult R exp = 0 . ± .
04 [41]. Note that this measurement alone requires that | V Ltb | (cid:29) | V Lts | , | V Ltd | . The relevant t -channel single top production cross-section can then be written as σ t (cid:39) σ SM t | V Ltb | R , where the SM prediction of σ SM t = 64 . +2 . − . pb [42] is obtained with | V Ltb | = 1 and R = 1. We compare this to the weighted average of the recentATLAS [43], and CMS [44] results ( σ exp t = 68 . ± . χ fit of the two experimental quantities interms of | V Ltb | and (cid:80) i = d,s,b | V Lti | (and taking into account theoretical constraints (cid:80) i = d,s,b | V Lti | ≤ | V Lti | > A direct determination of Higgs decays to charmed and light jets would also be enlightening in this respect. | V Ltb | > . , (cid:88) i = d,s,b | V Lti | > . . (A1)In models with up-type weak singlet vector-like quarks, | X ut | and | X ct | will contribute to FCNC top decays.Combined with σ t and R , the experimental bounds on B ( t → Zq ) can then be used to constrain (cid:112) | X ut | + | X ct | ≡| X tu,tc | . The presence of these new decay channels in principle also needs to be accommodated in σ t and R by writing σ t = σ SM | V Ltb | (cid:18) R + ρ W Z | X tu,tc | | V Ltb | (cid:19) − , (A2)where ρ W Z = 12 (2 M Z + m t ) (cid:16) − M Z m t (cid:17) (2 M W + m t ) (cid:16) − M W m t (cid:17) , (A3)takes into account the dominant phase-space difference in t → W q and t → Zq decays. In addition, searches for t → Zq typically assume B ( t → W b ) + B ( t → Zq ) = 1 and | V Ltb | = 1. In presence of | V Ltb | <
1, the experimental resultsshould instead be compared to B ( t → Zq ) = (cid:18) | V Ltb | ρ | X tu,tc | (cid:19) − . (A4)Including the recent ATLAS result B ( t → Zq ) exp < .
73% [45] in our fit, we first observe that the presence of FCNCshas no observable effect on the results in eq. (A2). On the other hand, we can obtain an upper bound on | X tu,tc | < . | Y tu,tc | and also | V Rtb | can nonetheless be obtained. To a good approximation namely in this case R = | V Ltb | + | V Rtb | (cid:80) i = d,s,b ( | V Lti | + | V Rti | ) , (A5)and thus experimentally | V Ltb | + | V Rtb | (cid:29) | V Lts | , | V Rts | , | V Ltd | , | V Rtd | . In addition, the presence of V Rtb will affect singletop production as σ t = σ SM ( | V Ltb | + κ R | V Rtb | ) (cid:18) R + ρ W Z | Y tu,tc | | V Ltb | + | V Rtb | (cid:19) − , (A6)where κ R (cid:39) .
92 [46] . Finally, V Rtb contributes to the positive W helicity fraction ( F + ) in top decays as F + = | V Ltb | | V Ltb | + | V Rtb | F SM+ + η R + | V Rtb | | V Ltb | + | V Rtb | x (1 + 2 x ) , (A7)where F SM+ = 0 . x = ( m W /m t ) and η R + = 0 .
93 parametrizes NLO QCD corrections [48]. Using therecent determination of F exp+ = 0 . | V Rtb | < . | V Ltb | . Thus we may conservatively usethe results of the previous paragraphs also to constrain | V Ltb | > .
85 and | Y tu,tc | < .
14 .
Appendix B: Bounds on down-type quark mixing with vector-like weak singlets and doublets from rare K and B q processes Model independent bounds on the off-diagonal entries of X d , Y d in models with additional vector-like down-typeweak singlet and weak doublet quarks respectively can be obtained from their tree-level ( Z -mediated) contributionsto FCNCs involving down-type quarks. For example, a bound on X dds , Y dds can be extracted from the K L → µ + µ − decay. We use a conservative estimate for the pure short distance branching fraction B ( K L → µ + µ − ) exp SD < . × − ,8obtained using dispersive techniques [50], as a 1 σ upper bound. Neglecting the much smaller SM contributions, the X dds contribution can be written as B ( K L → µ + µ − ) X = G F π f K m K τ K L m µ (cid:115) − m µ m K Re( X sd ) . (B1)Using the inputs for G F , masses and lifetimes from [10] and also f K = 155 . X sd ) < × − (the same result applies also to Y sd ) . Since much stricter bounds are expected on Im( X sd ) (Im( Y sd )) from the preciseknowledge of (cid:15) K , we interpret the above values as conservative constraints also on the moduli of X sd and Y sd .In the B d sector, X bd contributes at the tree-level both to B d → µ + µ − , as well as in B − ¯ B mixing. Neglecting X bd loop level corrections due to CKM non-unitarity and dynamical vector-like quarks running in the loop, the B d → µ + µ − branching fraction can be written as B ( B d → µ + µ − ) = G F π f B d m B d τ B d m µ (cid:115) − m µ m B d (cid:12)(cid:12)(cid:12)(cid:12) λ tbd C SM dB =1 + X bd √ (cid:12)(cid:12)(cid:12)(cid:12) , (B2)with λ tbd = V Ltd V L ∗ tb the relevant CKM combination and the SM Wilson coefficient given by C SM dB =1 = α √ πs W η Y Y (cid:18) m t m W (cid:19) . (B3)Here Y is the relevant Inami-Lim loop function [51] Y ( x ) = x (cid:20) − x − x + 3 x (1 − x ) ln x (cid:21) , (B4)while η Y = 1 .
01 [52] parametrizes higher order QCD corrections. For the values of SM input parameters (in particular α em , s W and m t ) we follow the prescription of [52] , and we use f B = 190 . ± . B − ¯ B system, the twomost relevant observables are the mass-difference between the two B d mass eigenstates (∆ m d ) and the CP violatingphase in the mixing ( β d ). Since the corresponding width difference (∆Γ d ) is small | ∆Γ d | (cid:28) | ∆ m d | , one can write∆ m d (cid:39) | M d | , sin 2 β d = Im( M d ) | M d | , where M d = G F m B d f B d B B d (cid:18) λ tbd C SM dB =2 + X bd √ (cid:19) . (B5)with the SM Wilson coefficient given by C SM dB =2 = α √ πs W η B S (cid:18) m t m W (cid:19) . (B6)Here again S is the relevant Inami-Lim loop function [51] S ( x ) = x (cid:20)
12 + 32 1 − x (1 − x ) − x (1 − x ) ln x (cid:21) , (B7)while η B = 0 .
939 [55] parametrizes higher order QCD corrections. In addition we use f B d B B d = 0 . ± . X bd but also the CKM combination λ tbd . For thispurpose we use the radiative rate B → X d γ , which is unaffected by X bd at the tree-level. Although it receivesnon-standard contributions proportional to X bd due to CKM non-unitarity and extra down-type quarks in the loops,these are parametrically (loop) suppressed compared to tree-level effects in B d → µ + µ − and B − ¯ B mixing. Usingthe SM recent evaluation [56] B ( B → X d γ ) SM E γ > . = (cid:12)(cid:12)(cid:12)(cid:12) λ tbd . (cid:12)(cid:12)(cid:12)(cid:12) . × − , (B8)and comparing it to the experimental result B ( B → X d γ ) exp E γ > . = 1 . × − [57] we extract | λ tbd | =8 . +1 . − . × − . Plugging this into eq. (B2) we observe that compared to the experimental 95% C.L. upper limitof B ( B d → µ + µ − ) exp < . × − [58] the λ tbd contribution can be safely neglected and we obtain a bound on | X bd | < × − . Note that the measurements of ∆ m exp d = 0 . − and sin 2 β exp = 0 . λ tbd and X bd can always be arranged such that cancellationsbetween these contributions weaken the prospective bounds above the one by B ( B d → µ + µ − ) exp .Finally in the B s sector we can proceed similarly, with obvious replacements λ tbd → λ tbs and B d → B s . Weagain employ B → X s γ to extract | λ tbs | (modulo loop-suppressed X bs effects) as | λ tbs | = 0 . f B s = 227 . ± . f B s B B s = 0 . m exp s = 17 . − , 2 β exp s = − . . ◦ [13] and B ( B s → µ + µ − ) exp = (3 . +1 . − . ) × − [58] we thus obtain abound on | X bs | < × − , which is again dominated by the muonic B s decay rate.The presence of vector-like weak doublet quarks induces right handed charged and neutral currents among SMquarks. The resulting flavor phenomenology is very rich and deserves a dedicated study. For our purpose however, itsuffices to show that to a first approximation, one can actually neglect all terms coming from right handed chargedcurrent operators as well as the ones containing extra u (cid:48) , d (cid:48) quarks in the loops contributing to quark FCNCs. Thisrequires some knowledge of the mixing matrices, which can be expressed through the parameters of the Yukawa sectorin terms of an expansion in ratios of mass parameters, which enables us to connect the left and right handed mixingmatrices. However, the approximation only holds if such an expansion is justified, as we will check a posteriori.First recall that the up- and down-like quark mass matrices in presence of a single vector-like quark doublet can bewritten in the form (41). Diagonalizing the products MM † and M † M one obtains the left and right mixing matrices,respectively, which we write in the form (c.f. [59]) U u,dL,R = (cid:32) K u,dL,R R u,dL,R S u,dL,R T u,dL,R (cid:33) . (B9)The mixing matrices between the vector-like and chiral quarks can most easily be obtained by starting from a basis ofright handed chiral quarks, where y u,d y † u,d v / y U,D v/ √ v , whereas the Dirac mass m Q is experimentallyrequired to be larger. Taking thus v/M Q as the expansion parameter, it can be shown that, to first order in thisexpansion, both 4 × U u,dR = × y † U,D v √ M Q − y U,Dv √ M Q + O (cid:32) v M Q (cid:33) , (B10)so the right handed charged currents are suppressed with V Rll ∼ O ( v /M Q ) and V Rlh ∼ O ( v/M Q ), where l, h stand forlight three generations and the extra heavy quarks, respectively.Turning to the left handed sector, we can choose e.g. a basis of left handed quarks where y u is diagonal and realand use a unitary transformation to diagonalize y d . Then we proceed in a similar way as before, and we obtainthat, to first nonvanishing order in v/M Q , K u,dL from (B9) equal those matrices (no corrections to unity or unitarity).Combining the up and down rotations, the 3 × V L takes the form V Lij = ( K uL ) ik ( K dL ) ∗ jk + ( R uL ) i ( R dL ) ∗ j , i, j, k = 1 , , . (B11)In order to obtain the matrix elements, one has to solve the equations M u,d M u,d † U u,dL = U u,dL D u,d , where D q =diag( m q , m q (cid:48) ) are the diagonal mass matrices. In the process we require that the entries in U u,dL mixing chiral andvector-like components ( R u,dL and S u,dL ) are smaller than the remaining ones ( K u,dL and T u,dL ) in terms of v/M Q scaling(similar to the case of U u,dR ). Also, we assume that the corrections to the masses are small enough so that m q , m q (cid:48) are of order v and M Q , respectively. Consequently the equations for R u,dL read v ( y u,d y † u,d R u,dL + y u,d y † U,D T u,dL ) = 2 m u (cid:48) ,d (cid:48) R u,dL . (B12)With the mentioned assumptions, one can neglect the first term in the above equation, obtaining R u,dL = v y u,d y † U,D / m u (cid:48) ,d (cid:48) . Thus, R u,dL are one order higher than R u,dR (see B10), the off-diagonal elements in the fourthrow and column are of higher order as well, which makes K dL a unitary 3 × v/M Q .Consequently all right handed contributions, as well as corrections in the left handed sector are doubly suppressed inthe loops of B s → µ + µ − and we can deduce a conservative bound on the FCNC’s by neglecting the afore mentionedcorrections, as we have done it in the vector-like singlet quark case. In fact, with the approximations made, the upperbound on | Y bs | is the same as the one on | X bs | in the down-type singlet case.0 Observable Measured value SM prediction Reference R b . . A b . . A bFB . . σ had [nb] 41 . . Z [MeV] 2495 . .
3) 2495 . .
0) [10, 63] R c . . A c . . A cFB . . Appendix C: Constraining Z → q ¯ q The appearance of Z-mediated FCNC’s is generically connected to modifications of diagonal Zf ¯ f couplings. In thequark sector, such effects can be probed by direct measurements of the hadronic Z decay width, its heavy flavor decays,such as Z → b ¯ b , but also at low energies by e.g. atomic parity violation (APV) measurements. In general we canparametrize the chiral Zq ¯ q couplings in terms of G qM ( M = L, R ) as the coefficients multiplying − g cos θ W ( q M γ µ q M ) Z µ and δG qM = G qM − ( G qM ) SM where ( G qM ) SM are given in ref. [60]. In our scenarios δG qL = δX uqq / δG qL = − δX dqq / δG qR = Y uqq / δG qR = − Y dqq / X uuu and X ddd can be derived from APV measurements in Cs. The tree level modificationof the Z boson couplings to first generation quarks will modify the nuclear weak charge as [61] δQ W ( Z, N ) = 2(2 Z + N )( δG uL + δG uR ) + 2( Z + 2 N )( δG dL + δG dR ) , (C1)where the measured value deviates from the SM expectation by 1 . σ [62] Q W − Q SMW ≡ δQ W = 0 . . (C2)For the singlet up-like vector-like quark, this constrains δX uuu = 0 . δX ddd = − . δG uR and δG dR at the tree level,and only a certain linear combination can be constrained, namely Y uuu − . Y ddd = 0 . Z → q ¯ q measurements presented in Table IV. In particular, thetotal Z decay width is given by Γ Z = Γ inv + Γ lep + Γ had . (C3)It incorporates the decays to leptons, where Γ lep = 251 . inv = 501 . had = (cid:80) q Γ q . The partial Z -decay width tolight quarks is given byΓ q ≡ Γ( Z → q ¯ q ) = N C G F M Z √ π (cid:16) R qV | G qL + G qR | + R qA | G qL − G qR | (cid:17) + ∆ qEW,QCD , with radiator factors R qV and R qA and non-factorizable radiative correction parameters ∆ qEW,QCD given in [60]. Thefractions of hadronic Z decays involving b quark pairs and c quark pairs are defined as R b = Γ b Γ had , R c = Γ c Γ had , (C4)respectively. The associated bottom and charm quark left-right asymmetries ( A b and A c ), and forward-backwardasymmetries ( A bF B and A cF B ) can be written as A f = 2 (cid:112) − z f G fL + G fR G fL − G fR − z f + (1 + 2 z f ) (cid:16) G fL + G fR G fL − G fR (cid:17) , A fF B = 34 A e A f , (C5)1 (cid:180) (cid:45) (cid:45) (cid:45) ∆ G Rb ∆ G L b (cid:180) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ∆ X bb d ∆ X ss d Figure 4:
Left:
Fit of Z-pole data taking δG qL,R for q = u, d, c, s, b as fitting parameters. Best fit point (cross), 1 σ (dark gray)and 2 σ (light gray) regions are shown in the ( δG bR , δG bL ) plane after marginalizing over the other parameters. The results ofthe fit for fixed δG u,d,c,sL,R = 0 are given by the red contour (1 σ region) and the orange contour (2 σ region). Right:
The fit ofZ-pole data in the model with a singlet down-like vector-like quark. Best fit point (cross), 1 σ (dark gray) and 2 σ (light gray)regions are shown in ( δX dbb , δX dss ) plane. where f = b, c and z f = m f ( m Z ) /m Z . The electron asymmetry parameter is fixed to its SM value, ( A e ) SM = 0 . e + e − cross section at the Z pole ( σ had ). It can be written as σ had = 12 πM Z Γ e Γ had Γ Z , (C6)where Γ e = 84 .
005 MeV and M z = 91 . χ fit of the data presented in Table IV in termsof δG qM as fitting parameters. The results are presented in the ( δG bR , δG bL ) plane, after marginalizing over all othervariables, in Fig. 4 left. At the best fit point χ min = 3 .
7, and the most important observables in the fit are R b , A bF B and σ had , which contribute to χ SM − χ min by 5 .
6, 4 . .
3, respectively. In the SM reference scenario χ SM = 14 . .
06. The largest contributions to χ SM come from R b , A bF B and σ had and are 5 .
6, 5 . . χ fit only with δG bR and δG bL while putting other parameters to zero. In thiscase χ min = 7 . .
25. The corresponding 1 σ and 2 σ regions are presented in the left Fig. 4 by red andorange curves, respectively. As expected, the data is mainly sensitive to bottom quark couplings leading to modelindependent bounds of δG bL = 0 . δG bR = 0 . δG uL and δG cL is possible. Taking into account the severe constraint on δX uuu = − . δG uL has nosignificant influence in the fit. Therefore, we use the data to extract the best current constraint on δX ucc = − . R b , σ had and Γ z , which are more constraining than direct Z → c ¯ c measurements R c and A cF B . We calculate the individual contribution to ∆ χ from each observable in thepoints which are ± σ away from the best fit point. Contributions to ∆ χ − σ from R b , σ had , Γ z , R c and A cF B are − . − .
7, 2 .
5, 0 . − .
1, respectively, while contributions to ∆ χ σ from R b , σ had , Γ z , R c and A cF B are 1 .
4, 1 . − . − . .
1, respectively.In the singlet down-type vector-like quark model, tree level modifications of δG dL , δG sL and δG bL are possible.APV puts a strong constraint on δX ddd , so that the Z data can be fitted with only two parameters, δX dss and δX dbb .Results are presented in Fig. 4 right. The best fit point now corresponds to ( δX dbb , δX dss ) = (0 . , − . χ min = 8 .
7. The fit is statistically better than in the SM with a p-value of 0 .
2. The most relevant observable in the fitis R b and its contribution to χ SM − χ min is 5 .
6, while the contribution of σ had is 1 .
0. Analyzing one-dimensional χ functions for each observable, we find 1 σ preferred regions to be δX dss = − . δX dbb = 0 . δX ddd in the fit and taking into account the APV constraint, the preferred region for δX dss is reduced to δX dss = − . δX dbb is unaffected.In the doublet vector-like quark model, tree level modification of all right handed couplings of light quarks withZ boson is possible. Therefore, we fit Z-pole observables together with the APV constraint on Y uuu − . Y ddd withfive parameters Y uuu , Y ucc , Y ddd , Y dss and Y dbb . Marginalizing over four parameters, we get the preferred range for theremaining one. Modification of charm and bottom quark couplings can be constrained to a percent level, Y dbb = − . Y ucc = − . Y dbb is mainly driven by R b and A bF B , which contribute to χ SM − χ min by 5 . .
5, respectively. In the case of the light quark couplings, observables given in the table can2 (cid:45) (cid:45) (cid:45) (cid:45) s bD s t U Figure 5: Fit of Z-pole data in the model with a doublet vector-like quark mixing predominantly with the third generation.Best fit point, 1 σ (dark gray) and 2 σ (light gray) regions are shown in the ( s bD , s tU ) plane. not distinguish between different light flavors, giving very poor constraints on one coupling after marginalizing overothers. Therefore, we include new observables into the fit which are poorly measured but can distinguish between lightquark flavours, namely asymmetries associated with the strange quark, with experimental values ( A s ) exp = 0 . A sF B ) exp = 0 . A s and A sF B have large experimental uncertainties. We obtain the following bounds on Y uuu = 0 . Y ddd = 0 . Y dss = − . +0 . − . .Finally, in the model with one vector-like quark doublet mixing predominantly with the third generation, modifi-cation of δG bR is induced at tree level δG bR = − (1 / s bD , while modification of δG bL is induced at one-loop level δG bL = g π s tU (cid:18) f ( x, x (cid:48) ) r + f ( x, x (cid:48) ) (cid:19) , (C7)where x ≡ m t /m W , x (cid:48) ≡ m u (cid:48) /m W and r ≡ m u (cid:48) /m t , and f and f are loop functions given in refs. [34, 65]. Theabove expression is given in the x, x (cid:48) (cid:29) Z -pole data to fit s bD and s tU parameters, we present the results in the ( s bD , s tU )plane in Fig. 5 for fixed m u (cid:48) = 800 GeV. We have checked, however, that the results are mostly insensitive to theprecise value of the u (cid:48) mass in the interesting region (500 GeV < m u (cid:48) < s bD , we obtain | s tU | < .
35 at 95% C.L. .
Appendix D: Doublet vector-like quark contributions to ρ parameter in presence of / Λ corrections In the renormalizable doublet vector-like quark model, the divergences in the loop calculation of the ρ parametervanish only after imposing the connection between masses and mixing angles in the up- and down-quark sectorsnamely M D = M U ≡ M Q , (D1)with M D = c bD m d (cid:48) + s bD m b ,M U = c tU m u (cid:48) + s tU m t . (D2)Relation D1 no longer holds after the inclusion of non-renormalizable operators. However, one can still relate theparameters in the up and down sector through the identities (see Sec. V) M U,D ≡ (cid:113) ¯ M U,D + v (( c − ) i ( c − ) ∗ i ) / , ¯ M U,D ≡ M Q ± v c − / . (D3)Expanding to O (1 / Λ), the divergences in ρ again cancel. Building upon known oblique corrections from vector-likequarks in the renormalizable model [66], the new physics contribution to the ρ parameter including leading dimension35 non-renormalizable operators can then be written as∆ ρ = αN C πs W (cid:40) (cid:88) i = t,u (cid:48) ,j = b,d (cid:48) (cid:104)(cid:16) | ˜ V Lij | + | ˜ V Rij | (cid:17) g ( x i , x j ) + 2Re (cid:16) ˜ V Lij ˜ V R ∗ ij (cid:17) g ( x i , x j ) (cid:105) − s bD c bD g ( x d (cid:48) , x b ) − s tU c tU g ( x u (cid:48) , x t ) − g ( x t , x b ) + g nr (cid:41) , (D4)where x i ≡ m i /m W . The corresponding mixing matrices are defined as ( ˜ V L ) ij ≡ ( ˜ U Lu ) ik ( ˜ U Ld ) ∗ jk and ( ˜ V R ) ij ≡ ( ˜ U Ru ) i ( ˜ U Rd ) ∗ j , where ˜ U Ru,d = (cid:32) c tU,bD s tU,bD − s tU,bD c tU,bD (cid:33) , ˜ U Lu,d = (cid:32) c (cid:48) tU,bD s (cid:48) tU,bD − s (cid:48) tU,bD c (cid:48) tU,bD (cid:33) . 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