CComposite electroweak sectors on the lattice
Vincent Drach ∗ School of Engineering, Computing and Mathematics, University of Plymouth2-5 Kirkby Place, Drake Circus, PL4 8AA Plymouth, United KingdomE-mail: [email protected]
In the post-Higgs discovery era, the primary goal of the Large Hadron collider is to discover newphysics Beyond the Standard Model. One fundamental question is does new beyond the StandardModel composite dynamics provides the origin of the Higgs field and potential. After reviewingthe main motivations to consider composite models based on a new strongly interacting sector, wesummarise the efforts of the lattice community to investigate the viability of models featuring acomposite Higgs sector. We argue that first principle calculations are necessary in view of the fastimprovements in accuracy of experimental measurements in the Higgs sector. We stress the im-portance for lattice calculations to provide a testing benchmark for non perturbative mechanisms.It is highlighted that the rich phenomenology of non-abelian gauge theories raises a number ofquestions that can be explored using lattice calculations. First principle results therefore providecrucial insights in the theory landscape that could guide the next generation of Composite Higgsmodels. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] M a y omposite electroweak sectors on the lattice Vincent Drach
1. Introduction
The success of the Standard Model (SM) of particle physics to describe a huge amount ofexperimental results is undeniable. The discovery of the Higgs boson in 2012 by the ATLAS andCMS experiments[1, 2], after an endeavour that started half a century ago, marks the beginningof a new era where finding deviations from the Standard Model is crucial to answer fundamen-tal questions about our Universe. A profusion of approaches to the physics Beyond the StandardModel (BSM) must be explored and scrutinised in view of the latest experimental results. Bearingthis in mind the Higgs discovery is undoubtedly a major step in our understanding of the interac-tions at the fundamental level, and confirms our effective understanding of the origin of mass atthe electroweak scale. Since 2012, the combined measurements of the Higgs’ mass have been im-proved to the subpercent accuracy and read m H = . ± . ( Total ± . ( Stat . only ) GeV[3].The on-going efforts to determine quantum numbers of the Higgs greatly favoured the ones of theStandard Model, i.e a CP-even scalar particle[4, 5]. A lot of attention has also been given to thetests investigating the coupling of the Higgs boson to the fermions and gauge bosons. Parametris-ing the strength of the coupling between the Higgs and vector bosons (V) or fermions (F), using theratio κ i = V , F = g Hi g SM Hi , the Higgs coupling to gauge bosons is measured with a 10 −
15% error whilethe Higgs coupling to the third generation of fermions is currently measured with a 20 to 30%error[6] (68% CL). These results are summarised in Fig. 1. Investigations of the Higgs potentialremain extremely challenging at the LHC, but a first bound on the Higgs cubic self-coupling hasbeen derived by considering the coupling contribution at EW contributions at NLO[7]. So far, allthe current experimental results suggest a very Standard Model like Higgs boson.The experimental results put severe constraints on new physics scenarios and in particular, inthe context of composite Higgs models reviewed here, the precision of the Higgs mass determina-tion is such that it should not be ignored as a probe to rule out scenario involving a non-standardHiggs. Until now, the coupling measurements κ V , F provide less stringent constraints. While theexperimental results are constantly pushing the limit of the predictive power of the Standard Model,numerous experimental evidence and theoretical puzzles that calls for BSM physics are still lackingan explanation.It is well-known that a number of theoretical and phenomenological facts question our under-standing of the interactions at the fundamental level and calls for New Physics beyond the StandardModel. From the theoretical point of view the naturalness problem, the hierarchy problem or thestrong CP problem have attracted a lot of attention. From the phenomenological point of view, theDark Matter density, the neutrino masses, or the origin of the Matter-Antimatter asymmetry, areexamples of limitations of the Standard Model. Over the years a large number of mechanisms andtheories have been proposed to explain some or several of these issues and are under investigations.Composite Models based on gauge theories with a number of fermions in various irreduciblerepresentations of the gauge group, are particularly interesting because they are known to generatescales dynamically and evade the naturalness problem. The fact that they exhibit a non trivial nonperturbative dynamics at low energy is also source of a rich phenomenology that can be exploitedto build extensions of the Standard Model. Composite Models face the challenge that robust quan-titative predictions require to resort to expensive lattice simulations to ultimately compare withexperimental data. Lattice calculations can furthermore provide inputs to guide model builders, to1 omposite electroweak sectors on the lattice Vincent Drach - - -
68% CL: 95% CL: = 0
BSM B = 88% SM p < 1 V k = 96% SM p off k = on k = 95% SM p ATLAS -1 = 13 TeV, 24.5 - 79.8 fb s | < 2.5 H y = 125.09 GeV, | H m Parameter value - - - Z k W k t k b k t k g k g k inv B undet B BSM B Figure 1:
Best-fit values and uncertainties for Higgs boson coupling modifiers per particle type with effectivephoton and gluon couplings assuming no BSM contributions (black). More details can be found in [6]. test mechanisms, and to suggest new experimental signatures that are sensitive to the underlyingdynamics. Over the years many scenarios relying on a new strong dynamics BSM composite mod-els have been proposed and received considerable attention. From technicolor models, to walkingtechnicolor and to Pseudo-Nambu Goldstone Composite Higgs, the idea of confining new funda-mental degrees of freedom into the known particles is among the most intriguing possibilities toaddress the flaws of the Standard Model.Although compositeness can be used in other context, for instance in the context of Dark Mat-ter, we limit ourselves to models related to the dynamical breaking of the electroweak symmetry.2 omposite electroweak sectors on the lattice
Vincent Drach
2. Model building: Pseudo-Nambu Goldstone Composite Higgs and Pseudo-DilatonHiggs
Broadly speaking, the underlying theories that are used to design models featuring a compositeHiggs are non abelian gauge theories parametrised by a a gauge group G , and a number N f of Diracmassless fermions in a representation R . Depending on the number of fermions and on the gaugegroup, the large distance behaviour of the theory is expected to change drastically. While for N f large enough, it is known that asymptotic freedom is lost [8], for small enough N c and N f , forinstance in QCD, asymptotic freedom, confinement and spontaneous chiral symmetry breaking areexpected to occur. In between this two regimes, it is expected that a conformal window exists,where the β -function exhibit an infrared fixed point (IRFP) for the gauge coupling g . In thatregime, the theory is conformal at long distances, and all the states of the theory are thereforemassless. We refer the reader to Fig. 2 for a summary of the perturbative expectations[9]. Thedetermination of the boundaries of the conformal window is however a non-perturbative question,crucial for model building, that can be studied using lattice field theory.. A brief update on thelatest lattice calculations that address that issue will be discussed at a later stage. Just below,the critical number of flavour for which the theory becomes conformal, the system might developinteresting features for model building, among which the possibility that the lightest scalar boundstate becomes a pseudo-nambu Goldstone boson associated to the spontaneous breaking of dilationsymmetry, which mass is controlled by the distance to the conformal window. Lattice studies areunderway to determine if this is occurring or not, but a number of challenges render the conclusionsunclear at this stage.In this section, we focus on the model building point of view. We will assume that we have atour disposal a theory featuring the following properties. First, the theory features asymptotic free-dom, confinement and spontaneuous chiral symmetry breaking. Second, the mass of the scalar stateis close to the spontaneous symmetry breaking scale F PS in the chiral limit. This last assumptionis related to to the possibility that the scalar state is a pseudo-nambu Goldstone boson associatedwith restoration of the conformal symmetry when N f is increased.In the following, pseudo Nambu Goldstone Bosons (pNGB) will refer to the degrees of free-dom associated with the spontaneous symmetry breaking of chiral symmetry, while pseudo-Dilatonwill refer to the light scalar state associated to the closeness of the lower bound of the conformalwindow. The pNGB Composite Higgs and pseudo-Dilaton Composite Higgs scenarios share com-mon features that we start by discussing. The scale of New Physics (NP) will be referred as Λ NP .The Electroweak (EW) scale will be denoted Λ EW . Models of composite Higgs can be summarisedas follow. At the New Physics scale, assume that the physics is described by the effective La-grangian of the Standard Model without a Higgs sector and with massless fermions augmented bythe Lagrangian of a carefully chosen Strongly Coupled Theory L SCT which confinement scale isgiven by Λ NP . At the energy Λ EW (cid:28) Λ NP , the new strongly interacting sector is replaced by itslow energy effective field theory (EFT), exactly like QCD is described by chiral perturbative the-ory at low energy, so that the effective Lagrangian is the one of the Standard Model with masslessfermions plus corrections suppressed by powers of 1 / Λ NP . The origin of the mass of the weakbosons and of the Higgs mass will differ in the two scenarios and will be discussed below. Fermionmasses pose a problem common to all these scenario that we will discuss later on. Note that the3 omposite electroweak sectors on the lattice Vincent Drach SU ( N ) N n f Figure 2:
Pertubative (4-loops ¯ MS) calculation of the conformal window for SU ( N ) groups in the fun-damental representation (upper light-blue), two-index antisymmetric (next to the highest light-green), two-index symmetric (third window from the top light-brown) and finally the adjoint representation (bottomlight-pink)[9]. Lagrangian at the scale Λ EW is dictated by the chiral symmetry breaking pattern and the quantumnumbers of the bound states of the theory. This explains why many effective models can be studiedwithout even specifying an underlying strongly interacting sector at Λ NP . When a gauge theory L SCT is known in terms of its underlying fermions content, the model is said to have a UV com-pletion. The choice of the quantum number of the underlying fermions is crucial to engineer arealisation such a scenario.We now turn our attention to the differences between the two scenarios namely PNGB Com-posite Higgs and the pseudo-dilaton Composite Higgs scenarios.In the PNGB Composite Higgs case, the underlying gauge theory L SCT is assumed to have aglobal symmetry group G F spontaneously broken down to a subgroup H F . To preserve the custodialsymmetry G cust = SU ( ) L × SU ( ) R of the Standard Model, the quotient group G F / H F must besuch that H F ⊇ G cust and the quantum numbers of the underlying fermions should be chosen so thatat least one of the Goldstone bosons have the quantum numbers of the Higgs particle. The Higgsparticle being identified with a Nambu-Goldstone boson is naturally light [10, 11, 12, 13, 14, 15].The effective theory at the scale Λ EW describes massless SM fermions. The Higgs mass isgenerated by the EW interaction exactly like the pions would acquire a mass if the electromag-netic interaction is switched on in massless QCD. A vacuum expectation for the Higgs field is notgenerated, and electroweak symmetry remains unbroken. To trigger EW symmetry breaking theinteractions with the SM fermions must be taken into account at the scale Λ EW . Depending on thefull Higgs potential, the Higgs potential might trigger electroweak symmetry breaking (EWSB).We assume the potential to be such that EWSB occurs, and we will denote V the minimum of thepotential. Then, setting the scale so that F PS sin V / F PS = v EW = F PS is the pseu-4 omposite electroweak sectors on the lattice Vincent Drach doscalar decay constant of the strongly interacting sector, the mass of the vector bosons is by con-struction the Standard Model one at tree-level. Furthermore it can be shown that κ V , F = + O ( ξ ) where ξ = ( v EW / F PS ) which therefore guarantees small deviations from the SM couplings if theso-called vacuum misalignement ξ is small.In the near-conformal framework, the scale is set by F PS = v EW = m σ = m H and so that Goldstone bosons become the longitudinaldegrees of freedom of the vector bosons (like in a technicolor scenario). While in the case ofthe pNGB scenario, the low energy effective theory can be systematically written, the question ofthe existence of an effective theory describing a light scalar is very challenging and has stemmedconsiderable efforts by the community.The question of the fermion mass generation, and in a first step, of the heaviest fermion,the top quark, is a long-standing issue. Several mechanisms have been proposed. Typically, theyrequire a new sector at an even higher energy Λ UV , this new sector typically generates new effectiveoperators at the scale Λ NP . Two classes of models have been proposed: • models that effectively generate operator of the form Λ q ¯ qO SCT , where q stands for thea Standard Model quark field and O SCT a bilinear operator of the new fermions fields at Λ NP . Typically, nothing prevent effective operators that generate Flavour Changing NeutralCurrent which are tightly constrained by the experiments. Introducing fermion masses intothese models is therefore a challenge. • models that effectively generate operator of the form Λ dim ( O ) − ¯ q a O a SCT , where a stands fora SU ( ) c colour indices. This scenario is referred to as Partial compositeness mechanism,introduced in [16], and requires that the strongly interacting sector at Λ NP to have QCDcharged bound states. Candidate UV completions for partial compositeness scenarios havebeen extensively studied, first by [17, 18]. Other studies have followed [19, 20]Fermion mass generation in composite scenario is a challenge for theorists and lattice simu-lations start be used to test fermion mass generation mechanisms and to guide the model buildingcommunity. ⇤ NP : L (SM Higgs)massless + L SCT + . . . ????y ⇤ EW : L (SM)massless + O ✓ NP ◆ Figure 3:
A sketch of composite Higgs scenarios with massless fermions (at LO). omposite electroweak sectors on the lattice Vincent Drach
3. Lattice investigations of near-conformal scenarios
Many lattice groups are accumulating evidence that near-conformal dynamics gives rise to alight scalar state. The focus has been on models based on SU ( ) and SU ( ) gauge groups. Thelattice calculations of the lightest scalar singlet are demanding because of the costly disconnectedcontributions associated with the scalar bilinear interpolating field.The interpretation of lattice calculation is difficult because the prediction of the spectrum inthe chiral limit depends strongly of the choice of the effective theory describing the low-lyingstates. For instance, chiral perturbation theory based on the chiral lagrangian is expected to receivelarge contributions from the scalar state. Models and effective theories have been designed andconsistency checks are now being performed to conclude about their ability to describe the latticedata.At finite fermion mass, several groups observe signs of chiral symmetry breaking and of ascalar states that have a mass very similar or even below the mass of the Goldstone bosons. Thisjustifies, a posteriori, the use of two-point functions to obtain the mass of the state without havingto consider a full-fledged finite volume calculation of the two Goldstone bosons scattering process.In order to improve our understanding of near-conformal gauge theories, further investigations bothat the numerical level by reducing the systematics and at the theoretical level by progressing in ourunderstanding of the effective description are required.To date, the theories that observe candidates for light scalar states are: SU ( ) with N f = SU ( ) with N f = SU ( ) with N f = A number of models and effective theories have been proposed to describe theories close tothe lower bound of the conformal window which include the scalar field as a degree of freedom[22,23, 24, 25, 26, 27, 28, 29, 30, 31]. The goal of this section is not to review them but to provide anup-to-date list of the low energy descriptions, and to highlight their main features. The discussionof the latest tests of these low energy description using lattice data is postponed to the next section.
Bound state model: H oldom and Koniuk consider a Hamlitonian-based bound state model wherethe pseudo-scalar, scalar, vector and axial-vectors are included[22]. They study the relation be-tween the spectrum and the relevant form factors. Chiral perturbation theory with a flavor-singlet scalar: the central idea is to augment thechiral Lagrangian by an iso-singlet scalar in the most general way[32, 33]. The corresponding lowenergy description does not rely on the closeness from the lower bound of the conformal window,but it thought to be able to capture a variety of underlying dynamics where the iso-singlet scalarplays a role.
Generalised Linear sigma model based on an approximate infrared conformal invariance, thescalar potential breaks chiral symmetry spontaneously[27].6 omposite electroweak sectors on the lattice
Vincent Drach M X / F π N f = 4 N f = 8 πa ρNσ m f /F π Figure 4:
Comparison of the spectroscopy of SU ( ) gauge theory for N f = (left) and N f = (right)fundamental fermions with. Hadron masses (vertical axis) and the fundamental fermion mass (horizontalaxis) are both shown in units of the pion decay constant F π ; the chiral limit m f = is at the center of theplot for both theories. The major qualitative difference between the two values of N f is the degeneracy ofthe light scalar σ with the pions at N f = . [21] Effective theory of a pseudo-Nambu-Goldstone boson associated to the spontaneous breakingof the conformal invariance: several authors build on the idea that the scalar state is associatedto the spontaneous breaking of the dilation symmetry. There is a long history of effective descrip-tion of the dilaton Lagrangian depending on the focus of the authors [23, 24, 34, 25].
Appelquist,Ingoldby and Piai add Nambu Goldstone bosons associated to the spontaneous breaking of chiralsymmetry to build their effective lagrangian[26].
Golterman and Shamir derive an effective the-ory by introducing a systematic expansion in terms of a parameter controlling the distance to thelower bound of the conformal window in their Dilation-pion low-energy effective theory[25]. Theauthors also derived a systematic expansion in which the fermion mass is not small relative to theconfinement scale. In the large-mass regime of the Dilaton low-energy theory, they have shownthat at leading order hyper-scaling relations are expected [28].
Complex conformal field theory: inspired by the work of
Gorbenko et al. [35, 36], and inparticular of their analysis of the two-dimensional Q − state Potts model with Q > K uti arguesthat the slow running of the coupling constant in near conformal theory might be related to thepresence of two fixed points at complex coupling, referred as complex CFTs[31]. He suggests another avenue to explore other types of low energy descriptions.7 omposite electroweak sectors on the lattice Vincent Drach
Several theories are being extrapolated using the low energy descriptions discussed in theprevious section. The latest findings will be reviewed here.The Lattice Strong Dynamics Collaboration presented new preliminary results including pseudo-scalar decay constant and mass, the mass of the scalar state and scattering length of the Goldstonebosons for the SU ( ) gauge theory with N f = ε -regime to test further the consistency of their dilatonic Lagrangian.In Fig. 5, the chiral condensate obtained from the GMOR relation and from Random Matrix Theoryin the ε -regime are compared and show good agreement. -0.5 0 0.5 1 1.5 2 2.5 3 fermion mass m -3 c h i r a l c onden s a t e ( m ) -3 linear fit connecting (m) in the p-regime and -regime(0) fitted = 0.00062(7)(0) GMOR = 0.00072(5)
GMOR fit /dof = 1.4 = 3.20 -0.5 0 0.5 1 1.5 2 2.5 3 fermion mass m -3 c h i r a l c onden s a t e ( m ) -3 linear fit connecting (m) in the p-regime and -regime(0) fitted = 0.00035(4)(0) GMOR = 0.00030(3)
GMOR fit /dof = 1.1 = 3.25 Figure 5:
Comparison of the chiral condensate Σ ( m ) obtained from simulations in the ε -regime and by theGMOR relation in the p-regime for two sextet (Dirac) fermions of SU ( ) gauge theory. More details can befound in [31]. omposite electroweak sectors on the lattice Vincent Drach preliminary ˜m ℓ / ˜m h M p s / F ℓℓ p s M ℓℓ ps / F ℓℓ ps M hhps / F ℓℓ ps M h ℓ ps / F ℓℓ ps am h = . , β = . h = . , β = . h = . , β = . preliminary ˜m ℓ / ˜m h M v t / F ℓℓ p s M ℓℓ vt / F ℓℓ ps M hhvt / F ℓℓ ps M h ℓ vt / F ℓℓ ps am h = . , β = . h = . , β = . h = . , β = . preliminary ˜m ℓ / ˜m h M s c / F ℓℓ p s M ℓℓ sc / F ℓℓ ps M hhsc / F ℓℓ ps M h ℓ sc / F ℓℓ ps am h = . , β = . h = . , β = . h = . , β = . preliminary ˜m ℓ / ˜m h M a x / F ℓℓ p s M ℓℓ ax / F ℓℓ ps M hhax / F ℓℓ ps M h ℓ ax / F ℓℓ ps am h = . , β = . h = . , β = . h = . , β = . Figure 6:
Low-lying connected meson spectrum obtained from light-light, heavy-light, or heavy-heavytwo- point correlator functions as a function of the ratio of light over heavy flavour mass to highlighthyperscaling[45]. SU ( ) with light and heavy fundamental flavours Motivated by the experimental fact that the Higgs boson is light and that no other heavierresonances have been observed, describing a composite Higgs boson requires a system with alarge separation of scales. An alternative approach to near-conformal theory is based on mass-splitmodels which feature a number of light and heavy flavours. Those theories are chirally broken inthe IR but conformal in the ultraviolet (UV). Such a class of model features interesting propertiesfrom the model building point of view but fails to explain the origin of the two well separated massscales in a first place.Mass-split systems can be used to design models involving a light dilaton or Composite Higgsmodels based solely on the spontaneous breaking of chiral symmetry scenarios [39, 40, 41, 17]. Inthe UV, the number of flavours is chosen to lie in the Conformal Window and the theory is thereforedriven by the conformal fixed point. In the IR, the heavy flavours decouple and the system thereforeexhibits spontaneous chiral symmetry breaking.It has been shown that in such mass-split system, dimensionless ratios exhibit two importantfeatures[42, 43, 44]. First, dimensionless ratios of physical observables exhibit hyper-scaling :they are function of the ratio m ; / m h . Such ratios are therefore independent of the mass of theheavy flavours. Second, the authors argue that the range of slowly evolving (“walking”) couplingincreases when the mass of the heavy flavours is reduced. The walking region can therefore betuned at the price of introducing explicitly two scales.Mass-split systems have been investigated previously with 4 light and 8 heavy fundamentalflavours of SU ( ) and it seems that the lightest scalar is the lightest massive particle in the chirallimit[43].Results for 4 light and 6 heavy flavours of SU ( ) have been presented[45]. They show that theconnected meson spectrum exhibits hyperscaling. The calculations of the scalar meson have notbeen performed yet. 9 omposite electroweak sectors on the lattice Vincent Drach
4. Lattice investigations of Pseudo-Nambu Goldstone Boson Higgs scenarios
We review here the questions addressed by lattice collaborations to test PNGBs scenarios inview of the latest experimental results and to provide first principle predictions to guide phenome-nologists.The first information that lattice collaborations can provide regarding a UV completion is itsspectrum in isolation of the Standard Model. A related question is to determine if the spectrum issimilar to the one observed in QCD or on the contrary rather different. Identifying common featuresof different UV completion allow to understand better what could be the experimental signaturesof such models.Other low energy properties of UV completion in isolation can also provide insights: formfactors, properties of the candidate Higgs boson, the Higgs potential itself, and scattering propertiesof the PNGBs are as many possible interesting observables that can bring essential information inthe search for experimental signatures of a composite electroweak sector at the LHC.A third class of problems that is addressed is to use lattice calculations as a laboratory to testmechanisms for the generation of the top quark mass or to pave the way for a calculation of theHiggs potential. SU ( ) = Sp ( ) gauge theory with N f = fundamental fermions The most minimalistic known UV completion of PNGBs Higgs boson scenario is based on SU ( ) gauge theory with two fermions in the fundamental representation. The theory is far from theconformal window and is therefore QCD-like. Because the fundamental representation of SU ( ) is pseudo-real, the flavour symmetry of the classical massless theory is upgraded to SU ( ) . Clas-sically, the mass term breaks the flavour symmetry down to Sp ( ) . A number of publications haveconfirmed non-perturbatively that the chiral breaking pattern is SU ( ) −→ Sp ( ) leading to fiveGoldstone Bosons. The EW embedding has been proposed in [46], and the model is phenomeno-logically viable [47]. The fact that this theory is used as UV completion by model builders makesit ideal to explore the underlying dynamics in details. For the first time, a full non-perturbativecalculation of the vector meson width has been presented[48]. The phase-shift is calculated usingthe energies of two-pions in finite volume and are related to infinite-volume scattering amplitudes S ( E ) = e i δ ( E ) via rigorous Lüscher’s formalism [49, 50]. The resonance parameters ( g ρππ , M ρ )are obtained from the phase shift δ ( p ) using the parametrisation: p ∗ cot δ E CM = π g ρππ (cid:16) M ρ − E CM (cid:17) (4.1)as illustrated in Fig. 7.The calculation shows that the preliminary value of the coupling constant that control thevector meson width is g ρππ ∼ ( ) , a value somewhat larger than in QCD where g ρππ ∼
6. Thisresult does not include chiral or continuum extrapolations needed to extract the physical valueof g ρππ . The number can be compared to the phenomenological relation obtained using vectormeson dominance by Kawarabayashi & Suzuki [51] and by Riazuddin & Fayyazuddin[52] (KSRF)stating that g KSRFVPP = m V / √ F PS . Using numberes from[53], the KSRF estimate of the couplingread g KSRFVPP = ( ) . The preliminary result, obtained at finite quark mass and finite lattice spacing,10 omposite electroweak sectors on the lattice Vincent Drach suggests that the KSRF turns out to be satisfied for SU ( ) gauge theory with N f = Figure 7:
Plot of a p ∗ cot δ / E CM as a function of the squared centre-of-mass energy. Points with error barscorrespond to energy levels from different ensembles and/or total momenta P. A linear fit allows to determinethe resonance parameters. Sp ( ) gauge theory with N f = fundamental fermions The latest results on the on-going research programme started in [54] focusing on Sp ( ) with N f = SU ( ) → Sp ( ) as SU ( ) = Sp ( ) with two fundamental flavours andtherefore shed light on the dependence of the physical observables on the gauge dynamics. Onevery interesting aspects of the Sp ( ) gauge dynamics is that when the two fundamental flavours aresupplemented by three antisymmetric fermions, the theory features top-partner candidate and canbe shown to be a UV completion of partial composite models. More details concerning this lineof research can be found in a recent paper[55]. Finally, the model is also relevant in the context ofstrongly interacting massive particles (SIMP) as dark matter candidates[56].Performing unquenched simulations with Wilson fermions, extensive studies of the spectrum(chiral behaviour, discretisation error) have been presented. The low-energy description that ex-plicitly includes the vector and axial-vector states proposed in[54], and based on the idea of hiddenlocal symmetry[57], is used to determine 10 low energy constants using the continuum-extrapolatedresults of the decay constants and masses in the pseudo-scalar, vector and axial-vector channels.The effective model describes well the continuum extrapolated lattice data of the masses and decayconstant squared as a function of the pseudo-scalar mass squared as shown by Fig. 8. The coupling g VPP appears in the effective Lagrangian, and the value turns out to be compatible with the KSRF11 omposite electroweak sectors on the lattice
Vincent Drach □□□ □□□ □□□□ □□□□ □□□ □□□□ □ ��� ��� ��� ��� ��������������������������� □□□ □□□ □□□□ □□□□ □□□ □□□□ □□□□ □□□ □□□□ □ ��� ��� ��� ��� ���������������������������
Figure 8:
Continuum-extrapolated meson masses and decay constants squared as a function of the pseudo-scalar mass squared. Black, blue and red colours are for PS, V and AV mesons. The global fit results aredenoted by shaded bands with their widths representing the statistical errors. estimate and close to the value obtained in QCD. The author argue that it provide empirical supportfor the KSRF relation.
As mentioned earlier, an important issue with models of composite Higgs is to provide mass tothe SM fermions and in particular to the top quark. A mechanism used by phenomenologists in thecontext of Composite Higgs models is referred as partial compositeness. The mechanism providesmass to the top quark by introducing a coupling between the top quark and a top-quark partner atthe scale L SCT . This effective operator at the composite scale is generated by an unspecified newsector. A practical requirement for the PNGB UV completion is therefore to have a spin 1 / SU ( ) c . UV completion that features such top partners have beenextensively studied[17, 18]. The authors consider theories involving fermions in mixed represen-tations of the gauge group, and one of the minimal theory that exhibits top partner candidates isbased on an SU ( ) gauge theory with five Weyl fermions in the antisymmetric representation andthree Dirac fermions in the fundamental representation.One collaboration undertook instead the extensive numerical investigation of SU ( ) gaugetheory with two Dirac fermions in the antisymmetric representation (denoted Q ) and two Dirac inthe fundamental (denoted q ). Such a gauge theory is not a UV completion of PNGBs CompositeHiggs model, but features spin 1 / Qqq ), called chimera baryons, that would correspond tothe top partner in the original theory proposed by
Feretti et al . In terms of number of Weyl fermions,the simulated theory is close to Ferreti’s UV completion and might serve as a laboratory to test themechanism. Any phenomenological conclusion obtained must therefore acknowledge the strongassumption that the predictions do not change by adding one Weyl fermion in the antisymmetricrepresentation, and one Dirac in the fundamental. Several results have been obtained: the massesand decay constants for the pseudo-scalar and vector mesons[58], the baryons masses constitutedof four q operators and six Q operators, and of the more interesting Qqq “chimera” baryons[59].The necessary chiral extrapolations have been performed using the chiral perturbation theory ofa two-representation theory derived in [60]. More recently, another collaboration has started toinvestigate the same gauge theory. While they perform an extensive investigation of the spectrum,12 omposite electroweak sectors on the lattice
Vincent Drach the focus is more on the algorithmic side of the simulations of gauge theories involving mattercontent in multiple representations[61].In summary, while the phenomenological lessons should be taken with caution, the followingresults are certainly a first interesting step forward in our understanding of simulations with mixedrepresentations and of partial compositeness.The study of the matrix element relevant in determining the strong sector contribution to themass of the top quark has been presented in[62] and summarises the recent results obtained in [63].At the scale L SCT , the linear mixing between the chimera baryon and the massless top quark iscontrolled by two coupling constant G L , R generated by the extended sector that can be thought as“Fermi” constants. At at Λ EW scale, the resulting top Yukawa coupling can be written : y t ≈ G L G R Z L Z R M B F P (4.2)where M B F P is a combination of the mass of a chimera baryon and of pseudo-scalar decay constantof the Q fields, already computed in previous studies, and Z L , R are matrix elements of a top partneroperator between the vacuum and the top partner state. The results of the renormalised matrixelement Z L , R is shown in Fig. 9. The renormalisation is performed perturbatively. The authorsargue that imposing y t ∼
1, implies that Λ UV cannot be much larger than F as the “model” wouldrequire it. We refer to [62] and to [63] for more details. Figure 9:
Chimera baryon matrix elements as a function of of the quartet mass.
The Higgs potential is not guaranteed to trigger EWSB: the potential is induced by loopsof gauge bosons and top quarks. The top contribution requires to estimate a 4-point function,as shown in [64] and in [65]. Vector boson contribution is controlled by a low energy constant,similar to the one that controls the pion mass spitting in QCD denoted C : LR . The low energyconstant has been computed for the first time in [66] by using a current-current correlator. Theauthors take the continuum and chiral limit and find that the value of the low energy constant is13 omposite electroweak sectors on the lattice Vincent Drach similar to its QCD counterpart in unit of the pseudo-scalar decay constant. The results do not allowto draw any phenomenological conclusion because the theory is not an UV completion of a PNGBsmodel but shows that such calculation is possible and pave the way to calculations in more realisticmodels. Taking the continuum and chiral limits, the authors of [66] find that the C : LR in units ofthe pseudo-scalar decay constant is roughly of the same size as its QCD counterpart. The resultsare summarised as a function of the mass of the fermions in the antisymmetric representation inFig. 10. m C L R = C L R t Figure 10:
Fit of the C LR data, the continuum prediction is shown by a green band
5. The theory landscape
We summarise in this section the recent developpement that are less directly related to thephenomenology but which are crucial to clarify our understanding of non perturbative gauge dy-namics. β functions A long standing task for the lattice community is to show the existence of a conformal window,and in particular the critical number of flavour marking the onset of the conformal window. Thetask turns out to be very challenging.Concerning the gauge group SU ( ) with fundamental fermions, extensive investigations sug-gest no IRFP for N f = , ,
4, while a IRFP is developed for N f = , , N F = , / , SU ( ) gauge group, results are controversial. The latest results for 12 fundamentalfermions, using domain-wall fermions, support the existence of an IRFP[76]. while staggeredcalculations obtained for 12 flavours find no such evidence [77]. More details on the latest resultscan be found in these proceedings. For the antisymmetric representation, a similar contradictionbetween lattice results obtained using the staggered discretisation[77] confirms previous published14 omposite electroweak sectors on the lattice Vincent Drach results[78] and results obtained using Wilson fermions[79] which find an IRFP. Note that a anindependent spectrum study performed using Wilson fermions is compatible with the conformalscenario[80]. m V / F PS If a new strongly interacting sector gives rise to a composite Higgs, the most obvious ex-perimental signature would be the observation of new composite particles associated to quantumnumbers allowed by the underlying dynamics. It is expected that, for a large class of stronglyinteracting theories, the lightest of such particles would be the vector meson resonance. It is there-fore relevant to determine the mass of such a state in as many theories as possible. An interestingwork tackled the issue by studying flavour dependence of the ratio m ρ / f π below the onset of theconformal windows, i.e in QCD-like theories. The calculation is performed for the SU ( ) gaugegroup with N f = , , , , ρ is stable. Continuum extrapolations are performedfor each number of flavours. A careful analysis of the finite volume effects have been performedand is crucial to obtain the results shown in Fig. 11. The result reads m V / F PS = . ( ) with nosignificant N f -dependence. The result relies on extrapolations below the two pion threshold andon the assumption that ignoring the resonant nature of the particle does not affect significantly theprediction. The authors also provide a compilation of that ratio for a number of theories usingvarious gauge groups and fermions in different representations. Once the leading gauge-group de-pendence is factored out, only a mild gauge group dependence is observed, except for SU ( ) gaugetheory. Assuming that the KSRF relations holds, such ratio can be used to estimate the value ofthe coupling constant g ρππ . The value obtained suggests that g ρππ is constant when the numberof flavour is changed below the conformal window. Under the assumptions discussed, the studysuggests that the vector resonance coupling is not a quantity sensitive to the underlying dynamics.The interested reader is referred to[81] and to a more recent paper [82]. One collaboration is investigating a novel alternative non-perturbative mechanism for elemen-tary particle mass generation [83, 84]. The framework is based on a conjectured non-perturnativeobstruction to the recovery of broken fermionic chiral symmetries which give rise to a dynam-ically generated fermion mass term. For the first time, the authors provide numerical evidenceof the conjectured phenomenon by using lattice simulations[85]. The authors simulate within thesimplest 4-dimensional model in which the phenomona is supposed to occur: an SU ( ) gauge the-ory with two fundamental fermions augmented by a colourless scalar doublet, Yukawa terms anda Wilson-like term. Tuning the bare parameters to restore the fermionic chiral symmetry in theeffective Lagrangian and performing a continuum extrapolation, they show that the pseudoscalarmeson mass is significantly different from zero in the phase where the the exact symmetry actingon fermions and scalars is spontaneously broken. From a phenomenological standpoint, the authorsargue the EW interactions can be included without introducing tree-level flavour changing neutralcurrents. The non-pertubative mechanism would then generate weak bosons mass terms and pro-vide an alternative to the Higgs mechanism. In such an approach, the observed Higgs boson wouldbe a bound state of the new interaction appearing in the vector bosons scattering channel.15 omposite electroweak sectors on the lattice Vincent Drach m ρ / f π N f Chiral - continuum limitmethod 1method 2
Figure 11:
The ratio m ρ / f π in the chiral-continuum limit for each N f [82].
6. Conclusions
The fascinating possibility that composite solutions to the puzzles of Beyond the StandardModel physics could benefit from first principle calculations is driving the lattice community tostudy non pertubative phenomena in various gauge theories.In the case of near-conformal dynamics, the presence of light scalars is a challenge both fromthe numerical and theoretical points of view. The studies of PNGBs models is now going beyondspectroscopy and new calculations provide insight in quantities relevant for the experiments.By exploring the dynamics of such theories, the lattice simulations contribute to deepen ourunderstanding of non-perturbative effects in Quantum Field Theory. There is an impressive amountof results on the behavior of quantum field theory, when the gauge group or the matter content isvaried. The results provide quantitative insights which guide the model building community todesign more realistic models. The lattice simulations can also test mechanisms relying on nonperturbative phenomena in quantum field theory.Interestingly, the various aspects of the lattice simulations as a tool to address Physics Beyondthe Standard Model raise a number of challenges. Often, the investigations question the algorithmsdevelopped in the context of Quantum Chromodynamics and their applicability to other theories. Inother cases, the problems require new methods to be developped. The interpretation of the latticeresults often require a thorough and critical examination in theories that cannot be compared toactual experimental measures. 16 omposite electroweak sectors on the lattice
Vincent Drach
Acknowledgments
I would like to thank the organizers for inviting me and for their kind hospitality. I wouldlike to thank V. Afferrante, G. Fleming, R. Frezzotti, A. Hasenfratz, K. Holland, T. Janowski, W.Jay, J. Kuti, M. Golterman, J.-W. Lee, D. Lin, D. Nogradi, B. Svetitsky, O. Witzel, C. H. Wong foruseful discussions and for providing material prior to the conference. VD received support from theScience and Technology Faculty Council (STFC) grant and from the DiRAC data intensive systemat the University of Cambridge and Leicester, operated on behalf of the U.K. STFC DiRAC HPCFacility, funded by the Department of Business, Innovation and Skills national e-infrastructure andSTFC capital grants and STFC Dirac operations grants.
References [1] ATLAS, G. Aad et al. , Phys. Lett.
B716 , 1 (2012), 1207.7214.[2] CMS, S. Chatrchyan et al. , Phys. Lett.
B716 , 30 (2012), 1207.7235.[3] ATLAS, M. Aaboud et al. , Phys. Lett.
B784 , 345 (2018), 1806.00242.[4] CMS, V. Khachatryan et al. , Phys. Rev.
D92 , 012004 (2015), 1411.3441.[5] ATLAS, G. Aad et al. , Eur. Phys. J.
C75 , 476 (2015), 1506.05669, [Erratum: Eur. Phys.J.C76,no.3,152(2016)].[6] ATLAS, G. Aad et al. , Phys. Rev.
D101 , 012002 (2020), 1909.02845.[7] ATLAS, T. A. collaboration, (2019).[8] T. Banks and A. Zaks, Nucl. Phys.
B196 , 189 (1982).[9] C. Pica and F. Sannino, Phys. Rev.
D83 , 035013 (2011), 1011.5917.[10] D. B. Kaplan and H. Georgi, Phys. Lett.
B136 , 183 (1984).[11] D. B. Kaplan, H. Georgi, and S. Dimopoulos, Phys. Lett.
B136 , 187 (1984).[12] T. Banks, Nucl. Phys.
B243 , 125 (1984).[13] H. Georgi, D. B. Kaplan, and P. Galison, Phys. Lett. , 152 (1984).[14] H. Georgi and D. B. Kaplan, Phys. Lett. , 216 (1984).[15] M. J. Dugan, H. Georgi, and D. B. Kaplan, Nucl. Phys.
B254 , 299 (1985).[16] D. B. Kaplan, Nucl. Phys.
B365 , 259 (1991).[17] G. Ferretti and D. Karateev, JHEP , 077 (2014), 1312.5330.[18] G. Ferretti, JHEP , 142 (2014), 1404.7137.[19] D. Buarque Franzosi and G. Ferretti, SciPost Phys. , 027 (2019), 1905.08273.[20] H. Gertov, A. E. Nelson, A. Perko, and D. G. E. Walker, JHEP , 181 (2019), 1901.10456.[21] Lattice Strong Dynamics, T. Appelquist et al. , Phys. Rev. D99 , 014509 (2019), 1807.08411.[22] B. Holdom and R. Koniuk, JHEP , 102 (2017), 1704.05893.[23] W. D. Goldberger, B. Grinstein, and W. Skiba, Phys. Rev. Lett. , 111802 (2008), 0708.1463. omposite electroweak sectors on the lattice Vincent Drach[24] S. Matsuzaki and K. Yamawaki, Phys. Rev. Lett. , 082002 (2014), 1311.3784.[25] M. Golterman and Y. Shamir, Phys. Rev.
D94 , 054502 (2016), 1603.04575.[26] T. Appelquist, J. Ingoldby, and M. Piai, JHEP , 035 (2017), 1702.04410.[27] LSD, T. Appelquist et al. , Phys. Rev. D98 , 114510 (2018), 1809.02624.[28] M. Golterman and Y. Shamir, Phys. Rev.
D98 , 056025 (2018), 1805.00198.[29] M. Golterman and Y. Shamir, PoS
LATTICE2018 , 202 (2018), 1810.05353.[30] D. Floor, E. Gustafson, and Y. Meurice, Phys. Rev.
D98 , 094509 (2018), 1807.05047.[31] Z. Fodor, K. Holland, J. Kuti, and C. H. Wong, Dilaton EFT from p-regime to RMT in the ε -regime,in , 2020, 2002.05163.[32] J. Soto, P. Talavera, and J. Tarrus, Nucl. Phys. B866 , 270 (2013), 1110.6156.[33] M. Hansen, K. LangÃ˛eble, and F. Sannino, PoS
Confinement2018 , 222 (2019), 1810.11993.[34] M. Golterman and Y. Shamir, Phys. Rev.
D95 , 016003 (2017), 1611.04275.[35] V. Gorbenko, S. Rychkov, and B. Zan, JHEP , 108 (2018), 1807.11512.[36] V. Gorbenko, S. Rychkov, and B. Zan, SciPost Phys. , 050 (2018), 1808.04380.[37] LSD collaboration, G. Fleming, Constraining EFTâ ˘A ´Zs in a Theory with a Light Scalar, in , 2019.[38] M. Golterman and Y. Shamir, Fits of SU ( ) N f = , 2019, 1910.10331.[39] M. A. Luty and T. Okui, JHEP , 070 (2006), hep-ph/0409274.[40] D. D. Dietrich and F. Sannino, Phys. Rev. D75 , 085018 (2007), hep-ph/0611341.[41] L. Vecchi, JHEP , 094 (2017), 1506.00623.[42] R. Brower, A. Hasenfratz, C. Rebbi, E. Weinberg, and O. Witzel, J. Exp. Theor. Phys. , 423(2015), 1410.4091.[43] R. C. Brower, A. Hasenfratz, C. Rebbi, E. Weinberg, and O. Witzel, Phys. Rev. D93 , 075028 (2016),1512.02576.[44] A. Hasenfratz, C. Rebbi, and O. Witzel, Phys. Lett.
B773 , 86 (2017), 1609.01401.[45] O. Witzel and A. Hasenfratz, Constructing a composite Higgs model with built-in large separation ofscales, in , 2019, 1912.12255.[46] G. Cacciapaglia and F. Sannino, JHEP , 111 (2014), 1402.0233.[47] A. Arbey et al. , Phys. Rev.
D95 , 015028 (2017), 1502.04718.[48] T. Janowski, V. Drach, and S. Prelovsek, Resonance Study of SU(2) Model with 2 FundamentalFlavours of Fermions, in , 2019, 1910.13847. omposite electroweak sectors on the lattice Vincent Drach[49] M. Luscher, Nucl. Phys.
B354 , 531 (1991).[50] K. Rummukainen and S. A. Gottlieb, Nucl. Phys.
B450 , 397 (1995), hep-lat/9503028.[51] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. , 255 (1966).[52] Riazuddin and Fayyazuddin, Phys. Rev. , 1071 (1966).[53] R. Arthur et al. , Phys. Rev. D , 094507 (2016), 1602.06559.[54] E. Bennett et al. , JHEP , 185 (2018), 1712.04220.[55] E. Bennett et al. , (2019), 1912.06505.[56] Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky, and J. G. Wacker, Phys. Rev. Lett. , 021301(2015), 1411.3727.[57] M. Bando, T. Kugo, S. Uehara, K. Yamawaki, and T. Yanagida, Phys. Rev. Lett. , 1215 (1985).[58] V. Ayyar et al. , Phys. Rev. D97 , 074505 (2018), 1710.00806.[59] V. Ayyar et al. , Phys. Rev.
D97 , 114505 (2018), 1801.05809.[60] T. DeGrand, M. Golterman, E. T. Neil, and Y. Shamir, Phys. Rev.
D94 , 025020 (2016), 1605.07738.[61] G. Cossu, L. Del Debbio, M. Panero, and D. Preti, Eur. Phys. J.
C79 , 638 (2019), 1904.08885.[62] B. Svetitsky et al. , Towards a Composite Higgs and a Partially Composite Top Quark, in , 2019, 1911.10867.[63] V. Ayyar et al. , Phys. Rev.
D99 , 094502 (2019), 1812.02727.[64] M. Golterman and Y. Shamir, Phys. Rev.
D91 , 094506 (2015), 1502.00390.[65] L. Del Debbio, C. Englert, and R. Zwicky, JHEP , 142 (2017), 1703.06064.[66] V. Ayyar et al. , Phys. Rev. D99 , 094504 (2019), 1903.02535.[67] A. Amato, V. Leino, K. Rummukainen, K. Tuominen, and S. TÃd’htinen, (2018), 1806.07154.[68] V. Leino, K. Rummukainen, J. M. Suorsa, K. Tuominen, and S. TÃd’htinen, PoS
Confinement2018 ,225 (2019), 1811.12438.[69] A. Athenodorou, E. Bennett, G. Bergner, and B. Lucini, Int. J. Mod. Phys.
A32 , 1747006 (2017),1507.08892.[70] J. Rantaharju, Phys. Rev.
D93 , 094516 (2016), 1512.02793.[71] J. Rantaharju, T. Rantalaiho, K. Rummukainen, and K. Tuominen, Phys. Rev.
D93 , 094509 (2016),1510.03335.[72] L. Del Debbio, B. Lucini, A. Patella, C. Pica, and A. Rago, Phys. Rev.
D93 , 054505 (2016),1512.08242.[73] G. Bergner et al. , JHEP , 119 (2018), 1712.04692.[74] G. Bergner, C. Løspez, and S. Piemonte, Study of thermal SU(3) supersymmetric Yang-Mills theoryand near-conformal theories from the gradient flow, in , 2019, 1911.11575.[75] Z. Bi et al. , Lattice Analysis of SU ( ) with 1 Adjoint Dirac Flavor, in , 2019, 1912.11723. omposite electroweak sectors on the lattice Vincent Drach[76] A. Hasenfratz and O. Witzel, Continuous β function for the SU(3) gauge systems with two andtwelve fundamental flavors, in , 2019, 1911.11531.[77] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. Wong, Case studies of near-conformal β -functions, in , 2019, 1912.07653.[78] Z. Fodor et al. , JHEP , 039 (2015), 1506.06599.[79] A. Hasenfratz, Y. Liu, and C. Y.-H. Huang, (2015), 1507.08260.[80] M. Hansen, V. Drach, and C. Pica, Phys. Rev. D96 , 034518 (2017), 1705.11010.[81] D. Nogradi and L. Szikszai, The model dependence of m ρ / f π , in , 2019, 1912.04114.[82] D. Nogradi and L. Szikszai, JHEP , 197 (2019), 1905.01909.[83] R. Frezzotti and G. C. Rossi, Phys. Rev. D92 , 054505 (2015), 1402.0389.[84] R. Frezzotti and G. Rossi, PoS
LATTICE2018 , 190 (2018), 1811.10326.[85] S. Capitani et al. , Phys. Rev. Lett. , 061802 (2019), 1901.09872., 061802 (2019), 1901.09872.