JHEP06(2019)110 , pos- L Springer June 3, 2019 June 10, 2019 June 20, 2019 : : : a,b February 22, 2019 : Revised Accepted Published , Received and Daniele Teresi a Published for SISSA by https://doi.org/10.1007/JHEP06(2019)110 [email protected] , [email protected] Alessandro Strumia , c SO(4) composite Higgs with fermions in the fundamental / . 3 Michele Redi, 1902.05933 The Authors. a,b Cosmology of Theories beyond the SM, Global Symmetries, Spontaneous c

We show that the potential of Nambu-Goldstone bosons can have two or more , [email protected] [email protected] INFN, Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa,INFN Italy Sezione di Firenze, Via G. Sansone 1, I-50019E-mail: Sesto Fiorentino, Italy Dipartimento di Fisica, Universit`adi Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy b c a Open Access Article funded by SCOAP Keywords: Symmetry Breaking, Technicolor and Composite Models ArXiv ePrint: els, such as therepresentation, minimal the SO(5) two minima areOtherwise, degenerate an giving unusual cosmological domain-wall cosmologyholes; problems. arises, to that vacuum or can thermalsibly lead interesting decays; to for to supermassive baryogenesis. a primordial high-temperature black phase of broken SU(2) Abstract: local minima e.g. atmodels antipodal of positions composite in Higgsto the and unusual vacuum of features, manifold. composite such as Dark This symmetry Matter. happens non-restoration at in Trigonometric high potentials many temperature. lead In some mod- Luca Di Luzio, Coset cosmology JHEP06(2019)110 – G 9 2 is the pseudo- √ / ) , h = (0 H 8 22 6 12 4 16 2 18 8 23 – 1 – by some strong dynamics, analogously to what 23 14 f ]) and composite models of Dark Matter (see e.g. [ angle 24 6 8 θ – 5 1 at a scale 10 H 1 20 13 2.1.1 Gauge2.1.2 contribution Yukawa contribution B.1 Multiple minimaB.2 from the Examples of coset potentials 4.1 Compositness4.2 phase transition (Quasi)degenerate minima4.3 and domain walls Bubbles filling the observable Universe 3.1 Composite3.2 Higgs with SO(6)/SO(5) QCD-like theories 3.3 Theories with the Higgs and composite scalars 2.1 SM loop contributions to the Higgs potential 2.2 General parametrization of the Higgs potential ]) received recent attention. In the introduction, for concreteness, we focus on composite broken to a sub-group happens with pions invalley QCD. of the Thenon-trivial field energy topology space potential e.g. of of a compositeinto the sphere. scalars account full As some describes a theory, effects the result, beyond well bottom the those approximated low-energy of by effective low-energy field a renormalizable theory coset theories: takes with the a Higgs Composite Higgs models (see e.g.17 [ Higgs models but ourglobal results symmetries. apply In inComposite order general Higgs to to models partially theoriesNambu-Goldstone assume with justify boson that spontaneous the of the breaking smallness some Higgs of of approximate doublet the (possibly electro-weak accidental) scale, global symmetry 1 Introduction A On the (in)equivalence of HiggsB configurations Potential in QCD-like examples 5 Conclusions 4 Cosmology 3 Potential of multiple pseudo-Goldstone bosons Contents 1 Introduction 2 Higgs potential in composite Higgs models JHEP06(2019)110 . 5 (1.1) comes v 2 h dominates,  1 f V , and an anti-SM f where quantum and  πf v 4 must be somehow smaller needed to get the desired = ∼ < v h T H comes at a price of a tuning of / G v we focus on the Higgs potential f 2 nh dominates, giving rise to two nearly- . Conclusions are given in section as low as possible, as a 4 2 f cos V n V : both minima remain present in the thermal cannot dominate and higher order terms can =0 ∞ – 2 – can be written as a Fourier series in X n fπ 1 V ∼ πf ) = h h ( . Given the bounds from LHC, we will not limit our we extend the discussion studying the potential in the V 2 ) 3 . Then f larger than the Higgs vev f/v f with period 2 ], focusing on the electro-weak phase transition for applications 2 , as a much larger scale could arise for anthropic reasons. / 27 1 f ) – . H 18 † fπ H ≈ h . Therefore, there is a range of temperatures = (2 2 h /v = 0 ‘burning’ the vacuum at 2 f h The paper is structured as follows. In section An unusual feature specific of composite models is that thermal corrections do not A compositeness scale theorist can assumephenomenological the outcome. pattern of Often,other global symmetries complicated symmetries are constructions proposed withwith in a extra order fine-tuning custodial to ofstudy and keep to order a ( TeV-scale 2 Higgs potential in compositeComposite Higgs Higgs models models areas often studied present from experiments a only low-energy effective offer theory bounds perspective, from this limited point of view. An effective Goldstone bosons. (including thermal corrections)includes in extra composite scalars. Higgs Infull models. section coset, considering In bothCosmological most composite implications models, Higgs are models discussed the in and coset section composite Dark Matter models. consequences. Various groups studiedGoldstone cosmological implications boson of [ theto Higgs as baryogenesis. a Our pseudo- workweak differs phase transition from around these theexisting references tuned as at vacuum we and temperatures we do below study not the the modify implications confinement the of phase electro- minima transitions that gives rise to the allow us to compute thethe new Higgs interesting potential. cosmological effects related to the twoselect minima in potential. As we will see in the following, this leads to a number of interesting cosmological antipodal local minima inminimum at the coset: the SM minimum at order thermal corrections are computable in terms of low-energy degrees of freedom. This will with the higher-order terms being sub-leading.the potential If the has lowest-periodicity a term models, single minimum: instead, the this happens Higgsthan for vacuum the pions expectation compositeness in value scale generate QCD. (vev) extra In local composite minima. Higgs In many models trigonometric functions; the potential,direction restricted for simplicity along the physical Higgs gauge and Yukawa interactions present in the Standard Model (SM) are corrected by JHEP06(2019)110 ⊗ 2 L (2.1) ) (2.2a) F SO(5), ]. This ) gauge ) where ). For / c N 8 F /f N and, as we ] proposed N Π 4 i W , ··· Sp( 3 / M 3 + ) = SU(5) F  f h N 2 ) for SO( W H µ θ F = exp (2 / 2 sin Z N G µ f U 2 Z ), SU( g F 2 cos SO( N / = ]: no known confining gauge ) + couplings, refs. [ 4 F µ W , SO( − L 2 N / or to make it non-degenerate. b M ) ] based on F 1 W + N to generate the spontaneous break- µ fπ SO(4) structure, when described by an W ≈  / DC h ) G h ), SU( ( F This restricts the possible accidental global 2 W SO(4) coset [ N 1 / ‘flavours’ of techni-fermions give SU( M – 3 – · · · ⇒ SU( + F / . + R 2 N ) ) gauge groups; SU( ) c       F h 3.1 ) gauge groups. Other composite particles do not lie in . 2 c µ . N N . ··· ··· ··· ∂ we show that these are distinct points in field space, like N ( . . . SU( A . . . 0 0 0 0 h h ] = ⊗ = SU( . . . 0 h h ], as they have double periodicity. As such models are subject L U 2 periodic in the coset but different periodicities for )       µ DC For symmetric cosets the Nambu-Goldstone bosons (that include ) for Sp( F 2 D F G N πf 1 † √ ]: for example N 2 8 2, as e.g. in section ) can be parametrised with the unitary matrix U [ is 2 / µ ) for H Sp( . In appendix h Φ) F D / µ Π = ) H N fπ 1) the pseudo-Goldstones can be parametrised by a vector Φ with fixed length and kinetic D / F Tr[ ( · G − = N 2 4 SU( f are the broken generators. Their gauge-covariant kinetic term is h N Φ) / µ ˆ a R π D ) SO( ( ˆ a / 2 F ) T f While models are sometimes complicated, their final results needed for our study can A possible UV realisation of composite Higgs models has been proposed in [ The minimal model assumes an SO(5) N The weak scale is as unnatural as in the SM,Such within terms the might assumption vanish that ifThis quadratic one divergences demands indicate applies that all to mass scales are SU( dynamically generated. N 1 2 3 = 0 and contributions of order of the Planck scale. SO( term one finds This Higgs boson will see, for the potential are possible. In the model of [ be understood in a simple way, asGauge we interactions. now discuss. the Higgs doublet Π = SU( groups; SU( arbitrary representations. Furthermore (as indark-quark QCD) the masses global giving symmetry can a bepotential specific broken that by UV-dominated allows contribution to to remove the the minimum pseudo-Goldstone at construction employs a newing gauge of interaction globalture symmetries, through partially and compositesymmetries elementary fermions. scalars to obtain the needed flavour struc- the North and South polespinorial of representations the [ Earth. Theto two strong vacua constraints are from non-degenerate electro-weakmodels in precision models based tests on of with a 5possible representation: domain-wall in problems these in modelslow-energy cosmology. minima global are symmetries Lifting degenerate, are the giving gauge rise degeneracy symmetries to in is the difficult, warped because extra dimensions. theory in 3 + 1energy dimensions models provides such might symmetry lie structure.dimensions in reduce One a to can 4-dimensional effective wonder swampland. theories ifobserver with Constructions living the on SO(5) with low- a warped 4-dimensional brane. extra h As we will see, these models generate two vacua at JHEP06(2019)110 are and (2.4) (2.3) πf/N (2.2b) U gauge . ). The above is diagonal with 2.1 ··· is given by power- + U  V ). In the next sections 2 top h f in eq. ( 1.1 as well as, crucially, extra )Λ h sin πf ( ) with the coset-generalized 2 f t . By using formulas such as h 2 2 2 W,Z,t ( ] g M 0 and have periodicities 2 ] 6 M ] 8 , ] = 2 4 28 → 8 − ln 2.1.2 W,Z,t , are sizeable and explicitly break the h W 4 M t , in [ in [ in [ y M 2 ). 4 W,Z,t 2 gauge h h f f 2 4 M 2.4 h f ))Λ h sin sin ( sin = 0 and, in the second (third) possibility, also 2 2 2 Z f f 2 f h – 4 – t t √ √ t y y M √ 4 y 2 . At the latter point the matrix              ) + } · · · ⇒ h ( ) = + 2 , fπ W generating at quantum level the SM Higgs potential. The h 0 (    1, giving rise to the periodicity { t M . G . , and the top Yukawa − = 0. In other models [ (2 M . = ··· ··· ,Y 2 3 h . . . h 2 0 h  The top Yukawa interaction depends on extra group theory and g . . . 2 0 h )    1 π 2 the potential is brought to the form of eq. ( 1 2 ) reduce to the SM expression for (4 / h ) ( ≈ x ], replacing the SM expressions for ) given in the previous section, and introducing two cut-offs Λ W ) . Other periodicities might arise in other models. Π = 2), with implications for quantum and thermal potentials. h vanishes at = 0 only at } h 29 ]. The top mass vanishes at ( M ( 2 8 cos 2 , W W V fπ/ 1 − ( { M M W,Z,t generalizes to denote smaller low-energy terms of order = M fπ ¯ t = (1 t N ) of the order of the compositeness scale: = ··· x h ( 2 h t top breaking effects unrelated to theto low-energy fermion SM kinetic couplings terms. such These asdivergent give higher-order quantum significant corrections contributions corrections, because assin discussed in section we discuss the approximations that lead to eq. ( Λ The The SM gauge couplings, approximate global symmetry (often) dominant part of thenew Higgs computation, potential from the can quadratically be divergentSM roughly part potential of estimated, the [ without one-loop doing Coleman-Weinberg masses any ties might be possible in fundamentalgauge composite group Higgs [ theories, depending onat the confining 2.1 SM loop contributions to the Higgs potential where, in each given model and coset, only one term is usually present. Different periodici- model details, such asnecessary the to obtain embedding the of topsimply top Yukawa interaction. correspond quarks, At and to the end, howM the the many lowest various insertions possibilities coefficients of again in a Fourier series. The SM expression of elements equal to either 1two functions or with Yukawa interactions. such that such that JHEP06(2019)110 , ) h 19 ( ) is such (2.7) (2.5) (2.6a) (2.8a) (2.8b) (2.6b) 2.7 h limit as W,Z,t T M ). E
 . − ) h = 0. (  (Λ h 2 θ Z 2 t 2  , ∝  T M M 2 ) ν 2 12 π  − 24 π E A 2 F +  − p J 4 ν ··· 4 2 p 45 π 12 π p µ + 360 7 p − together with ' ' − − + 3 ln  2
2 gauge Z 2 ) µν dx dx fπ T h ) ) g M )Λ   ( 2  =  ) + + 2 ) can be obtained at one-loop from its SM W 2 2 µ h h B h x x ( ( A M J √ √ Z  T − − ) − V e e M h + 3 functions can be expanded in the high- 2 – 5 – ( p ). − + 2 J  A ) originate from the gauge kinetic Lagrangian and 2 ) 2 2 M W 2.5 h 2.7 T 4 ln(1 + ln(1 6 ln ( M +  2 2 2 W  x x 4 ) and, as such, are sub-leading with respect to the tree- p ) p M B h 4 ∞ ∞ π − ( J 0 0  d (2 A 6 (2 Z Z 2 1 2  ). The Coleman-Weinberg potential obtained from eq. ( ) M 2 π Z 4 ≈ 3 π ) = ) = 2.1 i 2 T   2 ( ( eff 2 (4 ]) with the same trick of promoting the SM expressions for F B L ), in a well-defined and calculable way, since the momentum integrals J J Λ, obtaining eq. ( ≈ ≈ − ) = 30 ). The finite-temperature part of the potential can be obtained from the h (  2.8a T 2.4 f V gauge V . ). Higher-order corrections, including Higgs-dependent wave-functions, can be = 0. In many models this includes and reabsorbed Higgs-independent terms into the renormalization of the gauge ]. T 2.1 5 32 , W,Z,t 31 M , The composite-Higgs thermal potential For related studies of finite-temperature effects in the presence ofMore pseudo precisely, Goldstone we bosons approximate see the also form [ factors with a step function Π( 27 4 5 – analogous of eq. ( are cut by 25 which gives eq. ( fields. The leading contributionsfrom to eq. ( ( absorbed into the function level contribution in eq. ( where we neglected the momentumdynamics dependence of form factors originating from the strong 2.1.1 Gauge contribution We now discuss moregauge, only precisely the the transverse part of gauge theLagrangian contribution effective in gauge to Lagrangian momentum contributes. space the The (keeping quadratic potential. only the In transverse part) the can Landau be written as reducing to the usual thermalsee mass. that This the is thermal athat good corrections approximation to around the the potential minima: give we a minimum at all values of The usual bosonic and fermionic thermal expression (see e.g. [ to their coset-generalized extensions: JHEP06(2019)110 4 top (2.9) (2.13) (2.12) . Cor- can be (2.10a) (2.10b) h Q,U . . Z i ) in this limit ] . ··· 8 ) (2.11) n 2 t + 1.1 V ) 4 + h.c. M U n ( ). We thus have: Z F are required to obtain 2 =1 J ∞ QU ) are model-dependent X n f ) 4 h 2 2  t 4 2 h ( Q,U 2 t T ( T )(1+ , with respect to the term 1 f π U t M ( Z 6 6 Q f M O M Z ,Z − h ) = ···  ) h ' − ( − U + T (1+  Q Z ] small 4 ) 2 Z . Expanding eq. ( h ) 4 / ). In general, the term of order Λ U 2 top , DU f π 4 λ . . In some models in [ 2 , λ Z 2 n )(1+ f (4 6Λ 2.4 Ui + Λ  V Q )] 2 2 / − 2 Z  t 2 h h i h n ( ) )(1 + f 2 h v M U U Q 4 =1 ∞ (1+ M X n Z Z Z Λ, ). 2 – 6 – / 2 p h 2 − (  F f t m (1 + = M (0)  + [1 + 2 h V )+ln F )+ln(1+ Q J ' . In the models of [ Q / Z DQ 4 ) Z 2 this reduces to eq. ( 2 top T h π Qi ( Λ 6 ) is ,M Λ we can neglect the momentum-dependence of the wavefunc- / )] V − n 2 Q,U ln(1+ h V f ( 2.9   Z = Q 2ln(1+ 4 ∼ < h =0 ) E ∞ p Z X n 4 π = 1I)the potential reduces to the SM form 2 to top quark kinetic terms can also be relevant (in the fundamental d ) 4 top Q,U and can be generically neglected for (2 π U ). Therefore, for the thermal correction we only need to consider the U Z [1 + Yukawa 4 , that contains thermal-mass contributions | Z (4 Z 2 t 12Λ T (0) = T 2.5 i ≈ V 6 M V eff and ) in a well-defined way. The terms depending on only the wavefunctions are ≈ − ≈ ] this happens when their dark-Yukawa couplings are large enough), such that L 8 Q Z 2.10a Yukawa Again, the finite-temperature part of the potential can be obtained from the analogous V with 2.2 General parametrization ofWe assume the Higgs that potential the(corresponding SM-like to minimum lies at that gives eq. ( different possibilities for the function neglected being further suppressed by of eq. ( proportional to depending on In the limit of negligible dominates, unless the phenomenologically interesting situation sense if they arepotential sufficiently obtained smaller from than ( 1. At zero temperature, the Coleman-Weinberg In the limit of interest tions, as above.renormalization In this of approximation the we fermiontrigonometric absorbed fields. functions, the multiplied Higgs-independent The by effects functions possibly in the small coefficients. The expansion makes A more precise estimatesidering of the top-Yukawa power-divergent general corrections form can ofrections be the obtained effective Lagrangian con- fortheory the of top-quark [ sector and 2.1.2 Yukawa contribution JHEP06(2019)110 . . 4 is = n 4 V V 1 MS . n V ∗ H (2.15) (2.14) = 8 g 1) / by SM γ − 2 G ( ) ), is ignored) π 4 V → λ in the Higgs (4 / . The general ) and typically + 4 } n 2 ∗ n 4 g 2 V , V 2 2 V 2.3 2 g . The top Yukawa , 2 1 ∼ ], e.g. a 4 if = 2( ) { 4 π β 34 , = ] it arises proportionally (4 1 /f 8 / ), SM gauge interactions 1 V n 4 2 V g = )). In other models it can be α 2.2b ∼ 2.4 4 /f , with ) is fully characterised by its values 4 h V . ( 4 h/f . = 2, while for Yukawa couplings it is 4 V 4 ]. In general the functional form of the parametrizes the breaking of the global i ) and eq. ( V λf  , sin 8 33 i / γ ]), the potential including the lowest Fourier modes   1 5 2.2a and . In models with partial compositeness 2 4 + V 4 086 when renormalized at 2 TeV in the 2 π . f is given by eq. ( V ∗ 2. In order to reproduce the Higgs mass either − gives the same potential with 4 – 7 – ∗ 0 /f g 16 h/f /g 2 ∼ ), so that = W Λ 2 ≈ h ∗ πf + 16 Q M SM g X sin − g 2 λ ( , a generic kinetic term can be made canonical through M = β V ] (see eq. ( h V = 1 − and higher-order terms in the potential. h ∼  4 + 4 4 V 1 ) = h/f and subleading terms ] where V /f is generated with large coefficient 2 ∗ 1 πf 8 cos 1 g V , 2 V α ) 4 + 2 , π 2 h ) = Focusing on (4 ( h / ( 6 V 2 2 V ). g domain. is the number of insertions required to generate the contribution to the ]. This puts some pressure on composite Higgs models that often favour 1.1 ) = ∼ ]. i fπ 2 h i.e. the tuning 35 4 ( 2 ≤ V /f f h 2 is the relevant strong sector coupling, V is small or some deviation from naive scaling must be assumed.  ≤ ∗ SO(4) [ 2 ∗ g / the SM Higgsscheme quartic [ equals larger values. Already thewhile gauge contribution the gives top a Yukawa indicates toog large quartic unless v The form of the tuned potential (up to an overall rescaling, if the value of Phenomenological considerations impose that: In the models of [ The lowest frequency Different models generate different combinations of the coefficients Matching to more standard notations (as e.g. in [ • • 6 is determined by the free parameter can be also written as coupling contributes in similar ways,dominates depending numerically. on its periodicity in eq. ( to dark-fermion masses as different from zero only inSO(5) the presence of specific representations [ generate potential of eq. ( a field redefinition that affects gauge interactions in the model ofgenerated [ with small coefficients: in the fundamental theories of [ symmetry and potential. For gaugemodel couplings dependent. Inmore particular, couplings for than modelscontributions in with to the partially the composite SM potentialour fermions cannot that purposes be there break related it are the to is global SM sufficient Yukawa symmetry to couplings include in and general. the consequently Fourier For the terms with associated to the Higgs boson.the natural If size generated of to each leading contribution order to in the the potential strong is sector coupling, where Notice that in the 0 structure of the potentialpotential can is be fully found determined e.g. by in the [ couplings that explicitly break the global symmetry Expanding around the antipode JHEP06(2019)110 . 7 η 4 V and ∼ and f 2 h V . 1 V , and the coset is h -���� �� ���� ��� ���� �� ���� � can be described by a 2. ) (3.1) that depend on η ϕ we consider the next-to- ψ fπ/ cos and = 3.1 and reduces to h ψ, H ϕ H ��� sin ], however we focus on particles belonging = 0 it is degenerate with the extra ϕ . The electro-weak symmetry group 27 , f ��� X sin 26 we consider fundamental models based on ψ, ��� 3.2 cos 0. For – 8 – ϕ such extra minimum shifts towards smaller ��� sin ). X < ������/� X , 0 2.15 , ��� 0 ]. The 5 Goldstone bosons , ����� ���� ����� �� ���� (0 37 , 1. Minima remain degenerate when the Higgs mass is not tuned ��� f 36 Φ = X > . For growing ��� defined in eq. ( = 0 the potential is symmetric around � � fπ �� �� ��

-� X

X

SO(5) [

� � � ��������� / = / h . Possible composite Higgs potentials with lowest-frequency terms for different values of shows the various possibilities. The SM minimum is the global minimum for = SO(6) ) strong gauge interactions. 1 0 and is a local minimum for c In conclusion, an interesting structure of non-degenerate minima arises for N A light dilaton could play a similar role as discussed in [ H 7 / conveniently parametrised in terms of two spherical angles to the coset. 3.1 Composite Higgs withThe SO(6)/SO(5) next-to-minimal composite HiggsG model is basedreal on the vector Φ symmetry breaking withacts pattern 6 on components its and first fixed four length components. In the unitary gauge etc. Singlets neutral under theTheir SM presence gauge and interactions can potential beminimal is especially composite model light Higgs and dependent. model. relevant. InSU( In section section 3 Potential of multiple pseudo-GoldstoneThe bosons previous discussionpseudo-Goldstone considered bosons the set potential toclusions: along zero. the connecting If Higgs the present, Higgs direction, they minima with can through extra qualitatively different change trajectories; the give con- new minima, minimum at finally disappears for to zero, since for Figure 1 the free parameter Figure X > JHEP06(2019)110 2. the = 1 π/ i (3.2) (3.3) (3.4) c ψ by the = 3 ψ c ) for 3.4  ϕ + 2
. Increasing . ψ ) in eq. ( cos  2 , π ϕ α h, η 2 sin ( . = 0 α cos V 2 2 ϕ α + sin cos ψ 2 by top right couplings; − the angle one finds + sin 2 α α ψ c sin 2 ψ . ϕ sin 2 sin cos ϕ sin ϕ ϕ 2 α sin 2 sin α sin f directions are indicated on the North pole, which 2 – 9 – cos 2 2 c cos g  η i + ψ  = 2 ψ ψ and the potential is identical to the potential of the and 4 the barrier disappears along the direction 2) in the left (right) panel. The potential increases going 2 W cos h ). Contour lines of the potential cos π/ π/ M ϕ cos can couple to two different singlets corresponding to the ϕ h/f , f 2 = = ϕ 0 R , t 2 = α α ). 0 sin sin , 3 ϕ 2 0 f sin 4 ( c t , 1 √ 0 y c π/ − , 2.10b . For = α ≈ = scalars. The t ) α η is anomalous under QCD behaving as an electro-weak axion (unless M exists while η ϕ, ψ ( and L t V = 0 we have h ψ couplings is generated by gauge and top left couplings; . The coset of the next-to-minimal composite Higgs model forms a sphere parameterized R 1 t ). Therefore a breaking of its shift symmetry is phenomenologically necessary; a c v The singlet Along  dependent parameter In this limit the singlet isthrough an a exact valley Goldstone of boson minima, and as the antipodal shown points in are the connected f left panel of figure where top Yukawa. Thus, theand Yukawa contributions third correspond terms respectively in to eq. the ( first, second minimal composite Higgs, withpotential its barrier two gets anti-podal parametrically minima smaller, at by an amount that depends on the model- The potential generated by SM gauge interactions has the form Fermion masses are model dependent.embedding of For composite fermionsfifth in and the sixth 6 components of of SO(6) a a vector. unique Denoting with and for from blue to brown to white. so that Figure 2 by the Higgs corresponds to Φ = diag(0 JHEP06(2019)110 b a g ]). by 8 ; Q , the . We Q (3.5c) ‘dark- M G Q F ). Up to Yukawa F N V with N Q . M F ]: we here assume ), and (3.5a) 8 [ ) F forming a vector-like + cte N iθ/N S Q e 2 V SM M = G Yukawa ]. Furthermore, theories of V Tr Q 2 11 2 ˜ + ,...,M ) M f 1 2 ∗ ) gauge group with π ), collectively denoted as Q c g c can be parametrised by a unitary (4 3 angle. Its effects can be included 4, as shown in the right panel of N N M gauge V V θ ) π/ (3.5b) F ] = + b and the new strong dynamics does not 6= N T Q † α is the effective strong coupling. = diag( mass M SU( + h.c.] U V is proportional to the squared mass matrix c b ( / † Q N T R background. The generators of the SM gauge − ) M U √ – 10 – ] U F Q † gauge π/ U N V M Tr[ 4 U 2 F b µ g ∼ SU( D ∗ b iθ/N g ⊗ determined by the SM gauge quantum numbers of e X U L b µ 2 can dominantly tunnel along the singlet direction rather ) 4 ) Tr[ T D f F 3 π η 2 ∗ N f g ∗ Tr[ 3 2(4 g 2 4 f − = = ≈ − = SU( matrices typically acquire potential barriers due to gauge loops, so they are not F eff (as needed to get SM fermion masses in theories of composite Higgs [ H SM N mass / L gauge S V G G V × F ) gauge theory can have a non-vanishing to the dark quark mass matrix, that becomes c with unit determinant and thereby has the same topology as SU( . These theories have been studied to construct models where dark matter is an N the low energy effective Lagrangian is N θ SM 2 . The two minima are degenerate: a small splitting can be obtained breaking the U ) is generated by constituent masses G 2 of gauge bosons generated by the The coset This example shows a general phenomenon: in the presence of extra pseudo-Goldstone symmetry, for example coupling the SM fermions to a 4 of SO(6). E/f 2 V ( 2 mass rotating diagonal matrix with positiveM entries. group are are the SM gauge couplings, and V Yukawa interactions in thecolored fundamental scalars theory, either withThe the SU( SM Higgs or with dark- composite Higgs are obtained inthat the these presence do of not extra lead ‘dark to scalars’ extra Goldstone bosons. matrix O flavours’ of dark-quarks in the (anti)fundamentalassume of that SU( dark-quarks are chargedrepresentation under such the that SM they gauge group canbreak have masses accidentally stable bound state of the new strong dynamics [ multi-field treatment. 3.2 QCD-like theories In order to extend thewe discussion focus to fundamental on theories with those multiple based Goldstone bosons, on a ‘dark-color’ strong SU( expected to change thenew qualitative features local of minima. thecouplings Higgs barriers to Singlets but SM possibly on fermionslocal introducing are the minima model other connected dependent: handthan by if can through they the have (approximately) Higgs a preserve direction barrier. is small large The potential enough. opposite since is The their obtained intermediate situation if with the comparable barrier barriers along requires the a singlet figure Z bosons the topology of the vacuum cancharged change, under connecting minima along new paths. Bosons barrier between the two minima is present for JHEP06(2019)110 ) ), 0 f πf η 3 (and (3.7) (3.8) (3.6) √ = / . The Q 8 η η , and fo- M centers of . Starting Q iηλ 3 . F M } 2 1 2 the minima N ) since → − π − π π/ . Q 3. Since a shift 3 − = exp( F M θ π/  ( . The 4 U θ 3 , Q Q 3 − M < θ < 3 ,...,N π/ η 2 0 f f 2 { ∗ . g π/  = 2 is shown in figure with period 2 = π ∼ -th singlet being the heavy θ ), whereas there is a maximum θ n reducible representations under r +2 V θ r for a derivation and more details. η/f 3.8 . For + cos = 0, a local minimum at  we explicitly compute models with η (normalised as ) has two degenerate minima. These f  η 8 B (blue and yellow lines) gets closer to π B.1 θ 3 3 ’s (the λ 3.7 2 representation in η η + − η f SM η consists of  G  Q – 11 – mass . 2 cos V  3 because of eq. ( associated to Q 1) for integer = the potential ( mass is minimal for configurations that do not break the corresponds to a shift in the compact field π/ η V , with the two points remaining distinct. On the other θ M π 4 ) 3 π , π η f = ( includes multiple copies of the same representation. 1 singlets, named ,..., ∗ = θ g gauge ). Keeping all other dark pions at the origin, its potential is = 0 − θ + 2 Q 2 V r mass θ πf − . In addition if V the two minima get split by ∆ diag(1 η/f = ), for → π F θ gauge ) θ 2, which is a saddle point along ≈ V η = 0 has a global minimum at mass ]. This implies that the minima of the gauge potential correspond to ( block diagonal over each θ V π/ θ πin/N 38 [ 2 U e = mass V → − θ SM = angle in the dark sector, see appendix ) for 3. For G n θ corresponds to flipping the sign of the constituent masses U mass 3.7 π/ π V are true minima of the full = 0 the energy of the two minima along = 2 + 3 η . Extra singlets exist if θ θ , the dark pions include The potential of the stationary points as a function of For what concerns the minima generated by mass terms the discussion depends cru- Mass terms and Yukawa couplings can lift the degeneracy of singlets leading to local To study the vacuum of the theory we consider first the gauge contribution, as it = 2 and 3 flavours, finding a variety of behaviours: there is only one minimum in some η/f → F SM gauge from each other until they crosshand, at the local minimumsingle (yellow point line) at andalong one of the maxima (green line) merge into a and two degenerate maximaθ at the othertherefore two centers are shifted to the centers at Therefore a rotation potential ( such that its periodicity is 2 Although not manifest at first sight, physics is periodic in potential barriers. cially on the Let us consider, for simplicity, acus theory on with the 3 pseudo-Goldstone flavours boson and degenerate masses minima separated by potential barriers.N In appendix models for some values ofare their valleys parameters of (this minima, is in the some case other of cases QCD); in there some are cases multiple there local minima separated by are always minima of G whose generators are block diagonalV traceless matrices, and that do not receive mass from satisfies general properties: gauge group unitary matrices the flavour group JHEP06(2019)110 ]. SM . they 42 G (3.9) , the , θ (3.10) (3.11) π H 2 . As a 39 2 [ / 2 | π 3 f QQ H N ∗ | g 2 ··· ) |  N | ]. If the mixing (blue and yellow < θ < 2 φ 2 π η 2 41 has the same – M has two degenerate π/ cos N 1 the light Higgs is ··· mixes with 39 − + h.c. + [ N Q 2 c L H = mass  ⊕Q π Q V M L 6  L V + Q Q and L L = . In the presence of gauge or + h.c.) + φ Q H π 5 Q H M † 2 6= cos π have Yukawa couplings + L ( θ 2 Q 4 c N Q f ∗ M Q g ) N ∗ and of a right-handed neutrino Q = 3 where ˜ angle y 3 + 2( N θ- L F 2 − with the same quantum numbers as the Higgs. Q f N y 2 2 – 12 – M 2 ∗ π ) 2 2( g π 2 2 + √ g i L 9 4(4 Q − 1 2 ≈ c N | 2 2 Q 2 π † π | 2 0 2 π yH 0 2 4 6 6 4 2

M

- - -

+ ˜

Q * mass Potential f M g V /( ) − 3 c L ,M so that the ground state has a cusp singularity. For 2 2 = Q π 2 π f N ∗ eff = M g Q , which become non-degenerate for ) θ L together with pseudo-Goldstone bosons. We will see that the Higgs can ∗ π ˜ y yH H ), in a theory with 3 degenerate flavours. The two minima along = − = θ y 3.7 L 2 ( . Energy of the stationary points of the pseudo-Golstone potential generate by dark quark √ i For example, we consider a model with In models where the new constituent fermions Summarising, for negligible gauge and Yukawa contributions ≡  controls the degreemostly of elementary, compositeness and of the the mixing light contributes Higgs. to its For mass, ∆ Expanding the low-energy effective Lagrangian around the origin, The mixing parameter is large, it has a phenomenology similar to aquantum composite numbers Higgs. of the SMThe lepton fundamental doublet Lagrangian contains participate in theirdifferently possible the multiple electro-weak group. minima, giving rise topseudo-Goldstone multiple bosons include vacua a scalar thatThe break light Higgs doublet is in general a linear combination of Yukawa interactions, multiple minima (degenerate or not) can3.3 exist for any value Theories of withFinally, the it Higgs is and interesting to compositeHiggs study scalars doublet what happens in theories that feature the SM elementary lines) level-cross at are minima of the full potential. Green and red lines are always maxima. minima for Figure 3 masses, eq. ( JHEP06(2019)110 , , 1 X 18 πf  4  ∼ Λ ∼ cr T in the whole coset, and at , this is indeed what happens in f f ]. ∼ > particles that interact with the . This is sufficient to study the fate of T f 27 , ∼ < 26 T ) can have the same multiple minima, W, Z, t 2.5 and/or the top quark become massless. In . Assuming that inflation starts from a cooling . = 0 as the only minimum, as this is the vac- – 13 – T infl W, Z h V Thermal corrections to the potential due to a particle 8 . ), which has to compete with the zero-temperature potential πf that selects h 4 ( 2 2 X ∼ h 2 M ) Λ 2 can only be tuned around a single minimum of the strong sector; at T ); therefore, they are significant at T ∼ < ( h f ( O T . 2 X  T M v V 2 ) driven by a vacuum energy π ]. Due to the light dilaton, in some models there is no sharp distinction between 4 locally around the SM minimum where the curvature is tuned to be small. / infl 43 h , H (Λ Different cosmologies are possible, mainly depending on whether the compositeness The case of a pseudo-Goldstone boson is special: in the symmetric limit all points in Usually, thermal corrections to the SM Higgs potential are roughly approximated by We focus on lower temperatures, where the possible presence of two or more inequiv- A phase transition is expected to happen at a critical temperature M It is possible that the critical temperature is numerically around 23 ∼ 8 , ∼ > the minima of the potential at finite temperature. phase transition happened before,stant after or during cosmological inflation with Hubble con- QCD. In such a case, the thermal potential is calculable for part of the potential induced bycorrections. interactions with light SM Focusing, particles for receivesaround special simplicity, thermal those on field the valuessuch Higgs at a direction, which case multiple the theseparated minima thermal by can potential increasing arise given barriers in at large eq. ( Higgs boson. the coset are equivalent, anda interactions more that complex break structure thegenerated accidental that by global interactions allows symmetry with for heavy have extra particles receives local negligible minima. thermal corrections. The The part of the potential V T a thermal mass uum expectation value that makes massless the the electro-weak and confinement phase transitions [ alent local minima inpotential the at Higgs finite potential temperatureto can is temperatures leave reliably cosmological computableare signatures. in of the order The effective Higgs theory roughly up below which the Higgs existsdynamics, as which is a model composite dependent.weakly particle. first In This QCD-like order physics models or depends the cross-over.be on transition A controlled the is special through strong expected class holography to21 of that be models indicates features a a strong light first dilaton order and phase can transition [ In the previous sections wetrivial found topology, the that, potential in can theoriesnot; have where separated multiple the by local potential scalar minima barriers field which or space can not. has be a We degenerate here non- or explore the consequent cosmology. the other minima the weakthe gauge light symmetry Higgs can is bematrix mostly preserved has composite or a and badly negative the broken.minimum, eigenvalue. electro-weak while For symmetry the Also is other in broken induces this if a case the second the local mass tuning4 minimum can of the be composite enforced Cosmology Higgs. around a consequence JHEP06(2019)110 0 ). infl R and T 4.2 (4.2) (4.3) (4.4) (4.1) cr close to T and the τ ξ − . This is not cr t σ . ; ; smaller than infl /M infl given by 2 1 . T . / 1 < f µ ) ≤ infl ∼ /H − T p at the phase transition, 0 infl infl   RH T R < f < T µ H cr f /H H T ≤ ν cr RH  infl T − 1 + T f < T RH 1 Γ f T , T 2 = Pl =  M p 1 f π 8  2 infl min(1 . The three main cases correspond to ∼ H . This gives the first term in eq. ( dt , p ) dT 3 infl t infl  ( T H τ = 1 HT H f T ≈ , − − infl p  V = RH – 14 –  T 1 ν f = f H − ∼   that describe how the correlation length 4 infl ) 1 f cr dT/dt T τ µ T 2  must be below the Hubble scale 1 30 cr ]) to a generic Hubble constant , π − 0 is smaller than T − ∗ ν g 46 cr R min T t ). Regions have characteristic size ( infl  ξ ∼ T can be described by the effective field theory. The Universe is massless at the critical temperature (when global symmetry ( 1 0 f ≈ σ σ f R 0 ∼ M and thereby R degrees of freedom, it begins at temperature ∼ ) t ∗ ( ] (see also [ /a g T ξ formally diverge close to the phase transition at temperature 1 is an important quantity which depends on details of the strong phase 45 τ , 0 ∝ 44 R T : field with mass cr t inflation after the compositeness phase transition if inflation before the compositeness phase transition if inflation during the compositeness phase transition if σ The size 1. 2. 3. having used Due to lack of causal contact, Sufficiently far from thesystem phase evolves transition by the aHubble relaxation expansion sequence time the of is phase transition equilibrium microscopic,the is phase states. so crossed transition that at correlations a Because the freeze finite of and rate. determine the At the a cooling correlation time length due to the depends on criticalrelaxation exponents time time becomes broken) if thelimited phase by the transition time is variationcomputation of of [ the second cosmological order. temperature.we Adapting The find the that size Kibble-Zurek the of size domains of is bubbles then separated by domain walls. transition that leads to the appearancelike of composite scalars. Itnecessarily can be microscopic: described by a QCD- A case-by-case discussion wouldarise be at lengthy. the compositeness We phase prefer transition, to and that highlight can the be4.1 new specialised to features the that various Compositness cases. phaseWhat transition happens at randomly splits into domains of the various vacua forming domains with typical size and ends giving a reheatingif temperature the inflaton decay width Γ thermal bath with JHEP06(2019)110 : 2 / N 1 , in ) /H (4.6) (4.5) f 1 H 2, such Pl after  /  0. For a 0 M ( N H R ]) = 1 − → ∼ e = 0 gives the µ 48 . In conclusion p , ˙ infl S infl = M T 47 T R ) a bubble of false ν ) this only happens ≈ . . ] whenever S 2 4.2 4.2 , the time evolution of T 3 /R f /H 1 πGσ 1 ∼ 6 ∼ We next study how domains A more interesting situation σ −  T is obtained for 0 0 out R ), as it means that the SM vacuum ρ /f R , assuming that the vacuum decay 1 1 − . /f  in Pl ρ becomes smaller than [ /f : according to eq. ( 3 /M 3 is the anomalous dimension of the squared R 2 / 3 2 γ πR /H T ) 4 1 needs either a low-scale inflation model (e.g. 1, such that ∼ /f – 15 – ∼ + f Pl 2 H 0 ˙ is the Newton constant. Imposing M R p < R ∼ ( ) where 2 Pl γ ∼ > remains constant, becoming kinetic energy of walls. In is positive (cf. figure H 1 + 0 − which soon becomes relativistic, and thereby on a time- 4 /M R p V M ˙ 8 (2 σ R 2 / / : microscopic black holes. = 1 1 : macroscopic black holes. V . πR G = 1 − /H H /H 1 ν = . Ignoring accretion, such black hole quickly evaporate in a time = 4 1 and proceeds with an exponential cooling, /f  evolve. We here consider the case of microscopic domains, X ∼ < M f 2 0 0 Pl 0 R R M R ∼ > ] for a similar mechanism in the case of domain walls. is larger than the fundamental scale 1 emitting Hawking radiation with temperature is dictated by the conservation of its mass/energy (see e.g. [ 3. In reasonable models 49 M / 3 assuming that it starts from a thermal phase with temperature R . Such a possibility is realised if the compositeness phase transition occurs during 9 M f GM 2 = 1 2 ∼ G p In conclusion, small (sub-horizon) bubbles of false vacuum just disappear, and the The bubble can either disappear or form a black hole, if its Schwarzschild radius A first-order phase transition and a microscopic ≡ See also [ H ∼ 9 S -folds of inflation. Notice that ev for inflation, much larger than e Universe gets filled bypotential the parameter true vacuum.is An the acceptable deepest cosmology vacuum. is obtained whenBubbles the with happens if domains have horizon size with mass t R rate is negligible, andsuch that that walls shrink all loosing thesuch negligible initial a energy energy case to a matter blackblack in hole holes the forms are plasma, when formed its for radius classical equation ofvacuum motion: energies the have negative deeperspecial pressure. vacuum case of expands Unless quasi-degenerate the intovacuum vacua energy shrinks the will difference with false be velocity is considered vacuascale very in because much small smaller section than (the The first term combines the surfacecosmological and kinetic scales); energy the (we second neglect term anenergy is extra term the of relevant mass on the excess, wall, the latter where term is the gravitational with typical size their evolution can bebubble studied neglecting of cosmology. false vacuum Assuming,its with for radius thin-wall simplicity, a with spherical surface tension second-order transition mass of the orderthat parameter. In the ‘classical’particular Ginzburg-Landau during a limit thermal phase with Bubbles with JHEP06(2019)110 , 4 / 1 1 GeV (4.8) (4.9) (4.7) V ] SO(4) of the 13 / 52 ∼ ℘ 10 s T 2 ∼ ∼ / as the inflaton 9 However, rare H H  01 , too small to make 10 . . This is also the 2 Pl 0 H/f π 2  /M : as discussed in the 2 2 / H/ 5 7 eV is the temperature . /H  HM 0 1 ∼ ∼ < ∼ ∼ 0 f δh M R just because nearby regions can -folds before the compositeness eq ) and depends exponentially on -folds. At the end of inflation, e 10 TeV ], due to the change in T e ξ  keeps inflating down to 4.7 51 2 ,

0 ) p 1 − 50 after R /ξ M 1 0 N R ! 50) ( 4 ) where + α − much above the weak scale, if / , the bubble inflates following the de Sitter 1 − f N 1 f e RH V 3( /H e ∼ /T – 16 –

1 before the number of and have mass: ) ∼ 0 N ∼ > . s R is given in eq. ( /f 4 3 /H ( f infl = R H f ℘ T ℘ infl (10 TeV 3 T after N 13 N initially does not shrink because of thermal friction. The  . If ln e 4 − 0 / f H 1 1 ∼ 10 ∼ R  R , whereas the pressure to the energy difference between the min- ∼ N , such that the dynamics of composite scalars is dominated by ). Ignoring accretion, black holes make a fraction f/V some composite Higgs models (such as the minimal SO(5) 4 3 0 e f 3 before ˙ R 4.1 RH larger than the correlation length RT ∼ N 0 3 ∼ /T ∼ and R R s eq 2 4 with radius T R f TeV) or a compositeness scale ∼ < cr ∼ t × ℘ . Therefore, a bubble with initial size 1 M few V 0. Let us estimate the mass and density of the population of such primordial black 1. ∼ ∼ ∼ → f Another effect helps some bubbles to reach Hubble size: a bubble formed during infla- In such a case most bubbles have sub-horizon size α Black holes formed during inflation expands only mildly [ p ∼ 10 degenerate minima, finding that thecussed deeper in minimum sections expands into the false vacua. As dis- rolls down itsthem potential. macroscopic. We estimate their mass increase to be ∆ density for at matter/radiation equality, and model parameters. 4.2 (Quasi)degenerate minimaIn and the domain previous walls discussion we assumed that composite scalars have a potential with non- having assumed eq.inflationary ( energy density, such that their density is below the present dark matter but not much smaller thanfor this, with again theholes. first-order phase Denoting transition as casephase recovered transition, inflationbubbles lasts inflated to radius geometry. At the end oftrue inflation, vacuum it expands, can reach and a thewith large non-negligible bubble cosmological (but shrinks size. suppressed) to After probability a inflation if macroscopic the black hole. This happens tion at time friction pressure is ima is growing to size previous section they shrinkbubbles have forming size small blackaccidentally holes fluctuate in that the evaporate. same way with exponentially suppressed probability [ for as in simplest inflationaryis models. mildly We smaller actually assume than classical that motion, the rather inflationarysituation than Hubble that rate by leads to inflationary black-hole fluctuations signals compatible with existing data, as we now discuss. H JHEP06(2019)110 ] , ]. 3 ) 55 58 . /f (4.10) (4.11) 2 SO(5) . The /  Pl -walls dis- 2 /M between the 2 f Z / (10 TeV . 1 1 ) 10 TeV × V  tσ ( f the temperature of , which normalized 30 2 ∼ − T 100 s / ) = 2 10 TeV . The contribution of 3 ] based on the coset 2 10 Pl cr ∼ 29 πf /t 4 − > /ρ Pl ] with fermions in the fun- /M = /σ 2 10 3 M 2 36 Pl h  wall 2 > ( σt / ρ M 1 Pl V , which must overcome the wall f σ ∼ Pl 1 = ∼ M . Assuming radiation domination 2 − V 2 1 MeV): BBN ∼ )  F t T fM ) p ( ∼ wall wall t is the radius of curvature of the wall > t ( R >

∼ = 0) R

4 1 T 4 1 BBN V f h p ( σ/R

V f T

is the wall velocity and V SO(5) model of [ ). The region over which the wall is smoothed ∼ yields Ω with the surrounding medium. The former is v / 2 Pl 2 ⇒ F – 17 – due to wall tension (which tends to minimise the p wall /t , with ρ T 2 RM Pl ⇒ p 2 ( / 2 M BBN  2 , where 4 Pl /T ∼ σt f . We here explore what happens in this situation. Pl M vT ] composite Higgs models non-degeneracy can be achieved cr ∼ 1  ρ M ∼ is enough to remove the domain wall issue. In both the min- 36 Pl 2 v BBN 4 | T F ∼ f 4 M 3 p is the wall tension and f H ∼ /f 3 1 R f ∼ 2 V Pl 2 | σ 11 ∼ H M ] σ and hence it grows with time as R σ 57 ) this yields > ]. Since the system is above the percolation threshold there is a single – 2 > vt σ R /t 55 1 54 is 2 V , Pl ∼ ] with fermions in the 6, are excluded by their unacceptable cosmology, unless t ] and the next-to-minimal SO(6) 53 M T , where 36 4 ∼ SO(4) with fermions in the 5 of SO(5), as well as the next-to-minimal SO(6) > p ] and next-to-minimal [ ρ / σ/R 4 1 A possible way out is a small difference These considerations imply that the minimal model of [ The compositeness phase transition leads to domain walls similar to the ∼ V This simple estimate is supported by numerical simulations of the domain wall network evolution [ ∼ = 0 in the notation of figure 4 11 T T In conclusion, a small imal [ by introducing almost-decoupledSO(6), fermions respectively. in the spinorial representation of SO(5) and before nucleosynthesis ( the condition [ A parametrically different bound is obtained requiring that the domain wall disappears inflation occurs at very low scale, after the compositeenergy phase densities transition. of thedisappears. two The vacua, corresponding pressure sotension on that before the the the wall domain is deepest wall 2 vacuum starts dominates to dominate and the the energy wall density. Hence, one obtains domain wall start dominating thein energy contradiction density with at observations. SO(5) model of [ is determined by the( balance of the twoout forces at time the walls to theto energy the density critical is then energy density while the number density ofof finite the wall closed is walls governed iswall by exponentially the area) pressure suppressed. and by The thep evolution frictional pressure structure, and the latter is the surrounding medium. At very early times the walls are over-damped and their velocity damental representation) predict degenerateX vacua, which corresponds to the parameter cussed in [ domain wall per horizon much larger than the correlation length of the phase transition [ model of [ JHEP06(2019)110 , so ) to (4.12) , such λf V √ nh/f / 400. µ x ∼ < (˜ 4 ˜ h , as defined in S 4 f V 8 / ) = 1 µ V ). The O(4)-invariant ) x ) obtaining the bounce − ( X λ ( h 1.0 2.12 = F ) depends on cos 4.12 X ≡ and 1.1  µ ˜ h x 0.9 n cos 4 ]. This extends straightforwardly to a 0.05 n 0.8

= V λf λ 59 1, vacuum decay is cosmologically fast for . 0 =0 ∞ X n RatioX only depends on ∼

0.1 is the Euclidean action of the bounce, a classical + = 0.7 f – 18 – 4 λ λ 2 S ]. Another possibility is then that our observable )  ˜ h 2 47 µ v 0) can also be long-lived enough to be cosmologically ˜ ∂ SM minimum tuned to zero ( 0.6 T=0 ). Cosmologically fast rates are obtained for  . The Euclidean action becomes , where ˜ x µ 4 X < 4 x The space-time density of vacuum decay probability is ap- d 2.15 /f 4 . Given that Z 0.5 S 5 4 3 2 and ˜ 4 1 −

λ

10 10 10 10

e 4 ˜ oneaction Bounce h S equal to the Higgs quartic coupling of eq. ( = ∼ 4 λ x S 4 defined in eq. ( dp/d X 1, which corresponds to a small enough potential barrier. In this restricted we can live in a large bubble of false vacuum that decayed fast enough to the plotted in figure ≤ is a free parameter that can be used to get rid of the overall scale in X . Action of the bounce for quantum vacuum decay at zero temperature as function of the ). We choose 4 S λ X ∼ < 2.15 A metastable SM vacuum ( Let us consider the composite Higgs. We can conveniently work in the basis where has a canonical kinetic term: the Higgs potential of eq. ( 98 . acceptable. action 0 range of SM vacuum. where that the Higgs potentialeq. with ( tuned bounce solution is computed numerically and plugged into eq. ( h that it is convenient todimension-less rescale variables the space-time coordinates Quantum vacuum decay. proximated by solution that interpolates betweenGoldstone the boson minima as long [ as we are careful to canonically normalize the kinetic term. inflation) true vacuum expandsAccording in to them: general relativity the anvacuum, interior observer is and outside not forms affected sees a by aUniverse classical baby expansion black Universe was of hole [ inside the remnant. true athermal tunnelling false (ignored vacuum so bubble. far)model-dependent. towards the This true is vacuum is possible fast provided enough. that The answer quantum is or free parameter 4.3 Bubbles fillingAs the discussed observable previously, Universe false vacuum bubbles can become exponentially large when (after Figure 4 JHEP06(2019)110 0 ]. fπ fπ 60 ' , so = f = h (4.15) (4.14) (4.13) h h ∼ = 0 false ]): T 30 ) in the Euclidean T ; 2 shows an example / 1 X 5 /T ( T T λ (see e.g. [ h 6= 0 at the h f f f F 2 2 2 /T ≡ W 3 sin S  sin ] M 2 4 4 2 T f 2 f V 2 8 t λf 2 t y 95 the second minimum disappears y . this becomes the deepest minimum, + 0 = = ˜ h , the case of a pseudo-Goldstone boson f 2 & t 2 n t = 0 but 4 ) can have multiple minima, separated by ∼ X t cos T 2.5 M becomes the global minimum and populates 4 ), such that at finite temperature the n f V ,M λf ] where ,M – 19 – ]) without requiring a large number of fields. h f h f 8  , 2.10b =0 2 19 ∞ 2 2 in composite Higgs) the dominant configuration is X n v sin + f sin = 2 1 2 2 . f ) 0 4 f h 2 4 2 ˜ h 2 2 2 g µ g the electro-weak gauge bosons have large masses & ˜ ∂ . Details are model-dependent and figure ( = = ] (see also [ T  T ( πf 2 ˜ W x 2 63 W 3 – The rate for vacuum decay at finite temperature is computed from f M d ' M 61 Z h  . Moreover, in this case for 2 . If at high temperature / T where our computation is trustable. This results from the competition 6 1 f πf f T λ = 0. Another possibility is an unsuppressed wave-function contribution ∼ ) in the first term of eq. ( 15 . h = 0 h 3 is the thermal contribution to the potential. Depending on the shape of the po- h/f ∼ T S T , while at T V cos( f Consider, for example, a model where The false vacuum can even become the true vacuum at finite temperature. This hap- Usually a faster thermal decay arises at temperatures above the mass scale in the po- ∼  U the compositeness phase transition willshrinking populate fast). it (with the At that microscopic bubbles the of sphalerons arecomposite suppressed phase and transition electro-weaktemperatures is baryogenesis the first-order can SM and take minimum sufficient place, CP if violation the is present. At low but not at Z minimum is favoured. Thisabove the can weak in scale principle [ realise the idea ofvacuum electro-weak at baryogenesis as shown in figure in a small temperaturethan interval, tunnelling. and the field reaches the truepens, vacuum by for rolling, instance, rather in the presence of new particles that become massless e.g. at creases, however in our casetunnelling the rate thermal is barriers obtained, slow for the example, tunnelling in growth a too. model A with faster [ we see that thermal tunnelingatures can become cosmologically fast inof a two narrow factors: range of as temper- usual, thermal tunnelling tends to become faster as the temperature in- tential. Furthermore, thermal corrections tovacuum. the As potential discussed usually at tend theis to beginning special: remove the of the false section thermalincreasing potential barriers given at in large eq. ( in this direction. Considering models [ where tential, more complicated configurations can be relevant at intermediate temperatures [ the thermal potential,time finding direction. a Thereby bounce theT O(4)-invariant solution bounce with remains the periodicity dominantO(3)-invariant 1 configuration and at constant in Euclidean time, with action Thermal tunneling. JHEP06(2019)110 0.9 0.96 0.9 0.96 X X 3 TeV. . Below f 130 GeV & 0.4 0.4 .  f action of the T T /T 3 S ). Exploring this 0.3 0.3 6 : – ). 4 4.15 Right 0.2 0.2 minimum until temperatures Quarticλ=0.1 Quarticλ=0.1 TemperatureT/f TemperatureT/f πf 0.1 0.1 = h f=10 TeV f=10 TeV 0.0 0.0 3 2 3 2 10 10 10 10

: thermal potential.

3 3 / action bounce ) ( O / action Bounce T S 3 T S , for the model of eq. ( 5 Left – 20 – ). 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 0 T/f T/f 4.14 6 6 04 for the phenomenologically acceptable values . 5 5 . Same as in figure 0 1 or does not occur altogether (see figures . 0 . 4 4 & T/f 3 3 Figure 6 Higgsh/f T/f Higgsh/f ], the advantage of having electro-weak baryogenesis well above the weak 2 63 2 – RatioX=0.8,λ=0.1 quartic RatioX=0.8, quarticλ=0.1 61 1 1 . We consider the model of eq. ( 0 130 GeV, i.e. 0 However, we find it difficult to realize this scenario quantitatively in the models con-

2.0 1.5 1.0 0.5 0.0 0 4 3 2 1

. -0.5 -1

/ Potential / Potential f V f V

4 4 5 Conclusions Pseudo-Goldstone bosons, such as thecan composite scalars be arising described from by newnon-trivial low-energy strong dynamics, effective topology. theories We where found theSelecting that scalar one their fields scalar, form scalar its a potential coset field often with space admits is multiple minima. a circle along which the kinetic term can be made either occurs for possibility in more complicated models,the scope possibly of involving more this scalar work fields, and goes might be beyond done elsewhere. scale is that thechallenged required models CP at the violation weak is scale. much less constrainedsidered than in in this the work:T increasingly the Higgs field mustInstead, be for at the the models considered here we find that, according to parameters, tunnelling the Universe, either by rolling orelectro-weak tunnelling. sphalerons If remain this frozen happensargued at and a in the temperature [ baryon asymmetry is not washed out. As Figure 5 O(3) bounce (full curves).the The dashed dot line shows tunnelling the is action cosmologically of fast the during O(4) radiation bounce domination. relevant at JHEP06(2019)110 F N = 2 F N angle of θ for 3 S vanish is the rea- t bosons and with the M SO(4): the potential has / W, Z and/or ) topology, equal to W,Z ) new strong gauge interaction and F c M ) cannot be dominated by the term N N , finding a variety of behaviours which 1.1 ) by their model-dependent periodicities. B 2.3 – 21 – ) and eq. ( 2.2 0 is the global minimum of the potential, thus restricting ' h = 3. When dark quarks are charged under the SM gauge group, F N ) and along singlet directions. Dark quark masses and/or the we explored the cosmological consequences. An interesting feature of F for we considered the potential of a composite Higgs boson: for phenomeno- we considered the potential along the full coset in different models, relevant N 5 4 2 3 S × . We also considered models where an SU( 3 2 S bosons can be massive inside the false vacuum: this new kind of minima could have In section In section In section flavours of ‘dark quarks’ leads to a coset with SU( F into the false vacuumA only related after different inflation, possibility leavingfast is supermassive enough that macroscopic through we black thermal liveW, holes. Z or inside quantum a tunnelling false toimplications vacuum the for bubble, electro-weak true which baryogenesis. vacuum. decays In general, the transition occurs strictlyonly before if or the after minimumparameters inflation, at of an the acceptable modelThe potentially cosmology shrinking of coming is false-vacuum bubbles from can obtained UV leaveand black physics evaporate hole remnants, quickly and which unless inaccessible are the microscopic leading otherwise. compositeness to phase transition Hubble-sized happens domains during that inflation, inflate. In such a case the true vacuum expands finite temperature and withare generated higher by interactions barrier, with particles because (theare SM thermal light vector at bosons, corrections multiple the to top point quark,domain the in etc.) the wall potential that coset issues, field unless space.degenerate: the Degenerate the minima degeneracy lead deeper can to minimum be problematic expands lifted. into In the general false the vacua. minima Therefore, are if not the phase the strong gauge groupcases and/or have Yukawa been interactions explicitly can computedincludes break multiple in their local appendix degeneracy. minima. Various pseudo-Goldstone bosons potentials is that multiple minima tend to remain present at in figure N and to the gauge contribution tocenters the of pseudo-Goldstone SU( bosons potential has minima at the are introduced. for composite Higgs and/or compositethe Dark composite Matter. Higgs We model foundget based a connected on by variety SO(6)/SO(5) another of scalar: behaviours. the its two In potential minima can of have the or not Higgs have potential a barrier, as exemplified potential generated by low-energytop Higgs quark, interactions parameterized with in eq. the The ( presence of multipleson points for in the field presencepresent space of in where multiple the minimal minima, composite even Higgsone at model based SM finite on minimum temperature. the degenerate SO(5) No with extra an scalars anti-SM are minimum, unless spinorial representations metric’ terms that go beyond the polynomial terms oflogical low-energy reasons renormalizable the theories. trigonometricwith potential largest of period, eq. ( minima. such that We provided terms simple with expressions smaller for periodicities the quantum can and give thermal rise corrections to to the multiple canonical: in this basis its interactions with SM particles and its potential contain ‘trigono- JHEP06(2019)110 ~α ). 12 . . has h  !
A.2 (A.3) (A.4) (A.1) h f 1 0 h/f  in ( 1 0 0 cos SO(2), closer ∓ πf /
+ h f 1 0 0 0 0 0 1 00 00 0 0 0 0 0 0 0 0 0 h − that connects the

→ sin i 2 = SO(3) h ) H − / πα ) (A.2) H G = α / sin( G 3 h/f L,R α 3 = 1. To summarize, − cos = 0 considering the composite α ,
SO(4) ,T  h 0 h f 5 i / ! δ a j appearing in the double-covering relation δ 1 0 h/f, 2  ) sin 1). The Higgs boson is given by the − Z 1 0 0 , sin 5 j 0 − πα , δ , 0 a 1 0 0 0 i 0 , δ . This is not the case. The SO(3) manifold is the ∓ ,  SO(4), which is a higher-dimensional sphere. 0 0 0 0 01 00 0 0 0 0 0 0 0 0 0 cos ( fπ 2 / , i

√ = (0 – 22 – = (0 h f i 2 0 − h /f − a = is not equivalent to h sin = a = Σ ij T ) ) 2 fπ 0 a πα L,R √ 2 ~ Σ T i = α = 0 with ( − sin( e h h 1 0 ,T is a gauge transformation with α ! h
h f 1 0 gauge transformations of the Higgs field with gauge parameters  Σ = Σ → − generators, embedded in SO(4) are L sin h ; therefore, their generators are not in the coset 1 0 0 0 ) − could identify L,R πα H 1 0 0 0 0 00 00 0 0 10 0 0 0 0 0 0 2 α ∓ Z sin( /

L , so that the 5th component is left unchanged. This also shows that, instead, 2 | ~ α T i 2 · ~α , with the gauge symmetry imposing that the potential is an even function of − − π SU(2) iπ~α  ≡ | 2 = ∼ − = α e The two points would be the same point if a gauge transformation existed, that con- This conclusion is also reached considering the simpler analogous case L,R 1 ~ Σ. The SU(2) Σ 12 T the transformation period 2 to our geometricalSO(3) intuition. One might worry that the give with Since they vanish on the 5thFor component, instance, they cannot perform SU(2) the shift having exploited part of theof gauge redundancy to align the doublet in the 3rd component with the alignmentexcitations of along the the vev coset, i.e. unbroken group two minima. As atransformation, consequence, so that the they two arelet minima two us are distinct consider points. connected the by To broken see a generators this global in argument but the more coset not explicitly, local SO(5) We here explicitly show that Higgs model with the minimal coset, SO(5) nected them. However, this is not the case. Gauge transformations are embedded in the This work was supported byDelle the Rose, ERC Ramona Maxim grant Gr¨ober, Khlopov, NEO-NAT. Geraldine Servant, WeTetradis Andrea thank for Tesi Roberto and discussions. Nikolaos Contino, Luigi A On the (in)equivalence of Higgs configurations Acknowledgments JHEP06(2019)110 . ) 3.3 being c/N (B.3) (B.4) (B.6) (B.1) π singlet  0  η 2 1 µ 2 ) is obtained 0 = 0 corresponds η ≤ , in section h 2 2 N µ # 2 ) θ ⊕Q , L − 2 i Q  . Dashen equations take ) (B.5) φ ) 1 1 † are distinct: = , reducing to what written φ φ i θ U X ) (B.2) Q − fπ ( = F 2 , with the two rotations of c . = N , i N π φ ) ) φ h iφ 2 i 2 ≤ i + φ ln(det φ i − θ ) supplemented by the anomaly term 2 P φ i θ − ≤ ( X . In view of the determinant, we can ) − , . . . , e c π c and a phase 3.5a 1 cos N θ − = 0 and U − N i 2 2 iφ θ h e = µ ( φ sin( c 1 c 2 1 φ N X µ – 23 – ln(det 8 ]. The low energy Lagrangian is described by a diag ( = sin =  angle F − i c 66 2 1 angle it is convenient to include the heavy 2 " φ – µ c θ φ N θ 2 iθ/N 4 corresponds to the two identified points on the boundary of the 64 2 f sin − 16 f sin = 4 e 2 i 2 2 πf ≈ 2 − µ µ = φ 4 0 implies = = h where the energy is degenerate breaking CP spontaneously as ≈ U sin 1 π 2 2 φ anomaly µ becomes light at large = 4 V − c anomaly θ 2 + identification. The two points ) matrix with the action eq. ( V φ . The extrema of the potential correspond to the solutions of Dashen F 2 i 2 c/N in 3 dimensions, with antipodal points identified. This is because any 3-dimensional Z restricting to the diagonal ansatz 3 Q N mass − π V θ ≈ fM . These equations admit multiple solutions, for certain range of masses. 0 ∗ 2 η g anomaly m 3.2 = V 2 i µ solid ball of radius rotation is uniquely determinedthe by same: an axis this and isto an the the angle centre ofball the along ball, a whereas given direction. These equations can be solved numericallyThe and solutions lead cross to at multiple vacuawe for now show. An analyticby approximation noting (equivalent to that integrating out the Assuming, for example, two degeneratewe flavors and look a for singlet ( athe solution form with two equal phases equations In the limit ofin small section masses these equations impose obtaining where such that compute In general to discuss thein effect of the the effective lowunitary energy matrix U( theory [ In this appendix wetheories. discuss in detail potential of pseudo-GoldstoneB.1 bosons in QCD-like Multiple minima from the B Potential in QCD-like examples JHEP06(2019)110 , . } = = π 11 2 λ , 2 ) = ) = 1 have (B.7) (B.8) (B.9) equals U a = , /f θ 0 a π 6 a ij ranging ( { λ π λ σ a , 3 a = π 22 P iπ λ n = and the three = 2 5 , a + 1) λ σ of SU(2): 1Ifor , 2 1 n n 2 2 ), with 3 i = exp( µ 12 2 1 µ π σ 2 2 3 (2 2 λ 2 /f µ µ π 3 U 2 − πiN/ = σ , one is redundant, merging + 2 − 4 3 3 e = = λ θ 3 θ , 2 2 iπ = f . φ φ 33 1Icorrespond to Π = 0 and ∗ = = f n g λ n the solutions are ∗ 2 2 ) U g 1 , a sphere in 4 dimensions. For φ φ = 2 1) ) 3 φ 3 µ 1 − = exp( S λ φ 2 Pauli matrices , = cos × 23 U = ( 1 1 cos ) gauge group and lowest number of λ µ c n Q 1 = N Q = 1I. Furthermore, extra points such as U M 2 Its centers . The coset matrix λ M ,  U and . For f : + 2 13 + 2 π . ∗ 13 2 5 φ = 0 while the second is the minimum at g λ 2 , π φ 2 quarks charged under electro-magnetism (gauge reduce to the Pauli matrices S φ θ – 24 – = φ direction as a × 1 3) on = 0 in terms of the 2 cos 3 λ λ 3 cos u, d √ θ 2 cos π S 2 ij / 2 = 0. We adopt the standard parametrisation Q such that the ‘dark-pion’ Goldstone boson λ 8 Q Q so that the higher minimum becomes a saddle point of θ = 0= 0 or or cos cos M M ab ( M 2 2 δ π 4 3 φ φ πinλ 0. can be computed analytically, in terms of Π + 1) and and a charged 6= = 2( = = 4 3)). The = 2 The case of two dark-quarks charged under an U(1) gauge → 0 n θ / 2 3 2 η /f 1 2 π π n θ b 2 K -th position. Among the 9 generators of SU(2) (2 m j − λ 3. m π π 3 m , a 1. 4 √ 3 : sin λ + / / ) − ⊕ π θ 31 Π) sin Π θ 3 − λ 1) can be special for specific gauge and Yukawa interactions. = 0 : sin = / = 2 and 3 and . , a + θ θ 1 = = π F = 1 ( πf 1 1 . The two elements of the center 32 −          a , λ N φ φ − = diag(2 1 the solution is simple for . The usual Gell-Mann basis is ) with Tr = 2 the coset group is SU(2) with topology π Q iσ 5    = ( − 2 + S T F /f + 8 µ π a λ N λ , 6= /f a in a + 2 21 1 = 2, 2 with extra zeroes in the λ iπ For 2 µ = 3 the SU(3) coset has topology and they are connected along the This can be seen defining 9 generators i ) = diag( = σ 0 F 13 F 7 π πf between 0 and to SU(2) λ interaction is realised in QCD withgenerator the dark-pions form a neutral 1Icos Π ( N can be reached actingU as exp(2 N We now turn todark some quarks, explicit theoriesexp( with SU( canonical normalization at the origin. the first solution is the globalThe minimum two at vacua are splitthe for potential approaching B.2 Examples of coset potentials the Goldstone bosons where each solution corresponds to two physical points. Considering the mass matrix of For which leads to an algebraic equation for sin JHEP06(2019)110 , 5 0 8 = ~π π π has mass with ⊕ V 3 Q (B.14) (B.12) (B.13) (B.11) L mass (B.10a) ~π (B.10b) V 5 = nor the gauge ⊕ possibly charged U . masses, = 0 it equals to scalars is model- f Q Π containing a stable 5 2 in correspondence of Q Q ~π } ; the two minima are 7 } leads to a dark-matter sin 1  πf , π , − L 5 in models where its shift f 0 8 . Neither 2 3 2 π , 3 } π 0. ~π 0 1 7 , π f Π + 2 √ 2 π π p > the potential does not depend , λ π 8 2 {− 5 { Q 5 f π 2 f 3 cos ~π = , λ M e f = Π 2 2 + cos 2 √ n component. The potential along g λ 2 3 = , f 4 { 2 8 8 ~π 3 ) 2 f f γ Π can form a doublet under SU(2) sin π cos π 2 = ∗ π ). triplet with zero hypercharge. Thereby √ 2 2 2 2 g M b 3 g Q g 2 (4 6 L and T 4 4 2 ) setting degenerate 3 with 2 ) cos ) − f f ) } 2 2 2 √ ∗ π 2 cos ∗ 8 π Q g g  is a triplet of SU(2) (4 3 – 25 – (4 , π M Q 12 B.10a 6 πn/ with Q = M + − = 0) = , π 3 direction, and are smoothly connected along the 1 4 = 2 f 5 = Q , such that the only minimum is at Π = 0. We consider ∗ vanishes at Π = 0 and Π =  , π 8 n g gauge , ~π 2 γ M π 3 2 π 3 V = 0. Turning on only ( generators are 3 ~π have zero hypercharge and decompose as 3 M − gauge , π gauge 3 ( f 2 1 2 ), separated by potential barriers. A numerical study shows L V V a ~π gauge ∗ ) = f π π g V 2 ∗ π { /f 2 g 8 gauge generated by possible Yukawa couplings of . (4 3 − λ V = mass Y 8 n Assuming that V = = 5 iπ ) can be written in an useful closed form. For . ~π 5 Alternatively, the fermions Yukawa . It splits the two minima, possibly removing one of them if , ~π mass V gauge 3 V gauge /f V ~π V = 3. ( = exp( . The dark pions form a SU(2) , with dominates over n = 2. L Y Q U cos Π gauge , and has different periodicity along its Q mass ]. The dark-pions V 2 Q 5 V ~π 11 M 3 = 3, f = 2, The potential The resulting potential is well known ∗ g F F 4 makes the origin deeper than the other two centers for has three degenerate minimathe at centers that these are the only local minima. The potential due to constituent masses only on with all other components vanishing zero hypercharge). The SU(2) potential and is minimal at the origin dominates over N model [ under SU(2) dark-matter candidate (dark-baryons provide an extra dark-matter candidate, if which contains two inequivalent degeneratemass minima potential separated by is− potential barriers. obtained The from eq. ( symmetry corresponds to an U(1) accidental symmetryN of the Yukawa interactions. hypercharge hypercharge does not contribute to the gauge potential a more general range ofunder parameters, hypercharge realised U(1) as darkseparated color by with a singlet barrierdirection along (analogously the to the left panel of figure dependent. It can generate barriers, and it is flat along In QCD JHEP06(2019)110 , 1) B ]. , 1 . . The − (B.15) 1 913 3. The L , JHEP Phys. Q 1 SPIRE , , − b ¯ M IN [ = 0 and at πn/ Nucl. Phys. , ), given that Zb η Nucl. Phys. 1) is obtained iα = 2 , , 2 1 − − = diag( , e η/f , (2010) 111701 symmetry between 1 iα 3 U 1 under SU(2) − Z , e Physics of the large and Lect. Notes Phys. η ⊕ f iα , D 82 2 ¯ 2 e , in ⊕ ]. arXiv:1005.4269 1) and cos 2 , , = diag( 1 67 3 (non-canonically normalized ]. 1 ]. ⊕ Q has a minimum at , √ U = diag( / M 8 3 Phys. 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