Light Higgs Boson from Multi-Phase Criticality in Dynamical Symmetry

Physics Letters B 816 (2021) 136241

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Physics Letters B

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Light Higgs boson from multi-phase criticality in dynamical symmetry breaking ∗ Kristjan Kannike a, , Luca Marzola a, Martti Raidal a, Alessandro Strumia b a NICPB, Rävala 10, 10143 Tallinn, Estonia b Università di Pisa, Dipartimento di Fisica, Italy a r t i c l e i n f o a b s t r a c t

Article history: The Coleman-Weinberg mechanism can realise different phases of dynamical symmetry breaking. In each Received 9 February 2021 phase a combination of scalars, corresponding to the pseudo-Goldstone boson of scale invariance, has Received in revised form 7 March 2021 a loop-suppressed mass. We show that additional scalars, beyond the pseudo-Goldstone bosons, can Accepted 19 March 2021 become light at critical points in the parameter space where two different phases co-exist. We present a Available online 23 March 2021 minimal implementation of the mechanism in multi-scalar models, detailing how loop-suppressed masses Editor: J. Hisano and mixings can be computed. We discuss realisations of the resulting multi-phase criticality principle Keywords: and its relevance to the case of the Higgs boson. Multi-phase criticality © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license Coleman-Weinberg (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Effective potential Higgs boson Pseudo-Goldstone boson

1. Introduction than the Higgs boson or mixes negligibly with it. At the same time, alternative approaches aimed at identifying the Higgs boson with Experimental results from the Large Hadron Collider (LHC) indi- the dilaton have failed to single out the underlying mechanism. cate that the Higgs boson [1,2]is not accompanied by new physics In this work we consider dynamical symmetry breaking in a responsible for stabilising the weak scale against effects arising regime where additional scalars become as light as the pseudo- from higher scales present in Nature, such as the Planck scale. Goldstone bosons. This happens for special values of the parame- This renews the interest in alternative ideas of naturalness and ters such that two different phases of dynamical symmetry break- approaches aimed at understanding the origin of the electroweak ing classically co-exist, and quantum effects smoothly connect scale. them selecting the true vacuum. In the vicinity of this multi-phase One of the ideas is combining classical scale invariance, i.e., the critical point, the Higgs boson can be much lighter than the typical absence of explicit mass terms in the scalar potential, with dy- scale of new physics involved. This finding highlights the impor- namical symmetry breaking. Gildener and Weinberg [3], following tance of critical phenomena in physics, in this particular case for the work by Coleman and Weinberg [4], showed how to approxi- explaining the smallness of given mass scales when compared to mate the dynamical symmetry breaking. Their method identifies a the typical scale of symmetry breaking. flat direction that arises in the scalar potential after a given com- To investigate this framework, we go beyond the Gildener- bination of quartic couplings has crossed a critical condition. The Weinberg approximation [3] and compute the loop-contributions quantum correction, corresponding to the renormalisation group that usually bear a negligible impact in dynamical symmetry (RG) running, then dominate the tree-level contribution along such breaking. To the contrary, when the multi-phase criticality con- direction and effectively shape the potential. As a result, the scalar dition is realised, these corrections are crucial to capture the field combination aligned with the flat direction in field space – physical consequences of the framework. Our computations show the dilaton pseudo-Goldstone boson of the broken classical scale that extra scalar fields, not corresponding to pseudo-Goldstone invariance – acquires a loop suppressed mass. However, no dila- bosons, acquire loop-suppressed masses while maintaining a loop- ton has been observed in data so far and the current experimental suppressed mixing with the dilaton. The effect can be understood bounds are satisfied only in scenarios where the dilaton is heavier as a consequence of a small misalignment that quantum correction induces between the particular tree-level flat direction indicated * Corresponding author. by the Gildener-Weinberg method (along which only the dilaton E-mail address: kannike@cern.ch (K. Kannike). develops a vacuum expectation value) and the actual direction https://doi.org/10.1016/j.physletb.2021.136241 0370-2693/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. K. Kannike, L. Marzola, M. Raidal et al. Physics Letters B 816 (2021) 136241

where the minimum is generated by radiative corrections (result- the multi-critical point is trivially given by λ = λ = 0. Thereby, all ing in non-vanishing vacuum expectation value for all the fields). components of S become light with the exception of those eaten Technically, near the multi-phase critical point, the scalar fields by possible SU(N) gauge bosons. that develop non-vanishing vacuum expectation values, acquire Less trivial situations arise in more complicated models. For ex- masses that are independently suppressed by different β-functions, ample, one scalar field in an n-index representation of G has n allowing for the natural emergence of a mass hierarchy. When ap- independent quartics that allow for n inequivalent breaking pat- plied to the Standard Model, the framework predicts the existence terns; when two of them merge, one quartic can be non-vanishing, of at least one new scalar field in addition to the Higgs boson, but possibly leaving massless some components of S involved in the the mechanism allows for different mass hierarchies that can ac- transition. Alternatively, one can also consider models with more commodate states lighter or heavier than the Higgs boson without than one scalar representation, as we shall show next. particular tuning of the involved parameters. Because of the suppressed mixing supported by the multi-phase criticality scenario, 3. Implementation of the multi-phase criticality principle no other signatures are expected to appear at the electroweak scale, in agreement with the LHC results. 3.1. Phases in two-scalar set-up The paper is organised as follows. In Section 2 we present the concept of multi-phase criticality principle (MPCP) in models of A light Higgs boson arises at the intersection of two phases dynamical symmetry breaking. The implementation of the multi- with broken and unbroken SU(2)L . To exemplify this scenario, con- phase criticality in minimal scenario with two scalars is presented sider a minimal√ model with two scalar fields: the Higgs doublet in Section 3, together with the computation of scalar masses and H = (0, h/ 2) and a neutral singlet scalar s, with biquadratic po- mixing. In Section 4 we focus on the Higgs boson and present tential simple models that implement the proposed multi-phase criticality principle. Finally, we conclude in Section 5. Technical de- s2 s4 V = λ |H|4 + λ |H|2 + λ tails related to our work are collected in appendices. Appendix A H HS S 2 4 (2) presents a matrix formalism useful to study more involved scenar- 1 1 1 = 4 + 2 2 + 4 ios, whereas Appendix B details the computation of masses and λH h λHSh s λS s . 4 4 4 mixing angle for the minimal model considered in the main. The couplings λH , λHS, λS depend on the RG scale μ¯ according to = = ¯ 2 ¯ 2 2 2. Multi-phase criticality in dynamical symmetry breaking the β-functions βX dX/dt with t ln(μ /μ0)/(4π) . We leave the β-functions generic, in order to consider more generic models Coleman-Weinberg symmetry breaking in theories with generic with extra couplings. scalar quartics occurs when the RG running crosses the critical The possible phases of dynamical symmetry breaking are: boundary such that the tree-level scalar potential satisfies V ≥ 0 = = for all field values. Positivity implies a restriction on the quartic s) s 0 and h 0arises when the critical boundary couplings [5], that generally is an involved intersection of multi- = ple conditions imposed on these quantities. Each condition corre- λS 0(3) sponds to a pattern of symmetry breaking, which when realised is crossed, while λHS > 0gives a tree-level positive squared brings the theory into some Higgsed phase. In each phase, a dila- mass to the Higgs boson. In this phase the two scalars are ton (the combination of scalars proportional to the vacuum expec- not mixed. Dynamical symmetry breaking happens if βλS > 0 tation values) has a loop-suppressed mass. If RG running crosses along the critical boundary. a point which is simultaneously critical for two different phases, h) h = 0 and s = 0arises when λH = 0 and λHS > 0. In this phase extra light scalars arise if the two phases are smoothly connected, the two scalars are not mixed. This is the case originally con- rather than through a first order phase transition. sidered by Coleman and Weinberg, now excluded by the Higgs This phenomenon occurs, more in general, when different mass measurement that implies λH ≈ 0.12. phases co-exist. Dynamical symmetry breaking restricts which sh) s, h = 0arises when the critical boundary phases can be realised and how they are connected. It is thereby interesting to study if light particles can arise at multi-phase points 2 λH λS + λHS = 0(4) in this specific context. The following concrete example illustrates that the answer is is crossed, while λHS < 0, λH,S ≥ 0. The flat direction is given 1/4 positive. Let us consider a theory with symmetry group G (for by s/h = (λH /λS ) . Dynamical symmetry breaking happens concreteness, a SU(N) group) and one scalar S in a two-index sym- if metric representation. The dimensionless scalar potential, = + − βcrit λS βλH λH βλS λHSβλHS/2 > 0(5) V = λ(Tr SS†)2 + λ TrSS† SS†, (1) along the critical boundary. In this phase the two mass eigen- contains two quartic couplings. The two different critical condi- states are admixtures of the original scalar fields. tions that correspond to two distinct patterns of dynamical symmetry breaking are [6,7]: Fig. 1 shows the three critical boundaries geometrically. The phases h) and s) are not smoothly connected. They corre- • SU(N) → SU(N − 1). This breaking takes place if the RG flow spond to two different flat directions separated by the tree-level = 2 2 ≥ crosses the critical condition λ + λ = 0. potential V λHSh s /4 0. At their intersection (represented by • SU(N) → SO(N). This breaking takes place if the RG flow the red line in Fig. 1) the potential has two disjoint local minima crosses the critical condition λ + λ /N = 0. with h = 0 and with s = 0, corresponding to a first-order phase transition with no extra light scalars. The two phases identified above result in different dilatons s and On the other hand, Fig. 1 shows that the phases s) and sh) are s , originally contained in S, which are light compared to the smoothly connected along the dashed green line. Indeed, they cor- scale at which the breaking occurs. In particular, in this model respond to a common flat direction, the field s. The squared Higgs

2 K. Kannike, L. Marzola, M. Raidal et al. Physics Letters B 816 (2021) 136241

2 2 2 2s0 βcrit 2 s0λHS m = , m =− , (8) s 2 1/2 h 1/2 (4π) e λH + λS − λHS e where s is the dilaton, and h is the combination orthogonal to s . Within the Coleman-Weinberg approximation, h receives a dominant tree-level mass contribution, and the approximation does not allow to include loop-suppressed corrections. However, at the multi-critical point under examination, the tree-level contribution 2 to mh is also vanishing or small enough that the one-loop corrections must necessarily be retained. In regard of this, the general form of the one-loop potential is

V = V (0) + V (1), (9)

with the tree-level part V (0) given in Eq. (2), having omitted terms involving other possible scalars. The one-loop term, V (1), is given by 1 M2 3 V (1)| = Tr M4 ln S − + (10) MS 4(4π)2 S μ¯ 2 2 Fig. 1. Phase structure of the model of Eq. (2). No symmetry breaking arises in the un-shaded region, where V ≥ 0for all field values. Dynamical symmetry breaking 2 2 4 M F 3 4 MV 5 arises when RG flow of the couplings (λH , λS , λHS) crosses its boundary. The two − 2M ln − + 3M ln − . F 2 V 2 phases s) and sh) are smoothly connected along the green dashed line. The phases μ¯ 2 μ¯ 6 s) and h) intersect along the red line. Here, μ¯ indicates the RG scale introduced by the regularisation. mass changes sign between the phases s) and sh). Therefore, the We used the MS dimensional regularisation scheme. The parameters in the tree-level potential also depend on μ¯ as dictated by Higgs boson is light around their intersection. Furthermore, as the their RGEs (this is sometimes called RG improvement). This can- scalar mixing vanishes in the phase s), it must be small near the cels the dependence of the potential on the arbitrary parameter multi-phase point. The two conditions in Eqs. (3) and (4)intersect μ¯ , up to higher-loop orders and up to wave-function renormalisa- at tion. The symbols M S,F ,V denote the usual field-dependent masses ¯ = ¯ = of generic scalars, fermions and vectors, respectively. For example, λS (μ) λHS(μ) 0, (6) 2 = 2 2 + 2 2 MV ghh gs s is the mass of the U(1) gauge boson in a model that trivially implies a massless Higgs boson. where h and s have corresponding gauge charges gh and gs. In 2 The running of the Higgs quartic coupling in the standard more general models M S,F ,V are mass matrices and their eigenval- | ≈− ues (needed to compute Eq. (10)) often do not have a useful closed model (SM) alone, where βλH SM 2around the weak scale, does not cross the boundary condition for sh) in the direction analytic form. For example, complicated expressions for M F easily that gives dynamical symmetry breaking. Consequently, in mod- arise in models where h and s have Yukawa couplings to fermions. els where the SM running is dominant, the Higgs can acquire a Rather than resorting to model-dependent numerical methods, we next derive simple analytic expressions. vacuum expectation value only for small values of λS close to the multi-phase critical point. Eq. (10)can be simplified taking into account that h s at the multi-phase critical point, so that all field-dependent masses M 3.2. Computing scalar masses and mixing acquire a common form, + 2 +··· Having made clear the gist of the scenario, we now compute Mi ci s cih /s . (11) the vacuum expectation values, masses and mixing angles of the The above expansion fails in regions of the parameter space where scalars involved in the two-field model of Eq. (2). A more general couplings have values that undo the h s hierarchy. In such a and abstract computation is presented in Appendix A. case the Higgs mass is still loop-suppressed but it is given by a We start by summarising the usual Coleman-Weinberg com- more complicated and model-dependent expression, therefore we putation for the phase sh), as it provides an example about how will not consider this possibility. Expanding V (1) up to quartic or- dynamical symmetry breaking can be approximated using the RG- der in h shows that the full potential (not restricted to the flat improved tree-level potential alone, omitting the more complicated direction) is well approximated by the tree-level potential with the and model-dependent full one-loop contribution. Along the tree- λ and λ couplings replaced by level flat direction the potential can be approximated by replacing S HS 2 2 the tree-level potential with field-dependent effective couplings βλ s βλ s λeff(s) = S ln ,λeff (s) = HS ln , (12) expanded at the first order in the beta functions: S 4 2 2 HS 4 2 2 ( π) sS ( π) sHS = + 2 2 λeff(s ) λ βλ ln(s /s0), (7) where βλS and βλHS are the β-functions. The contributions to the β-functions coming from wave-function renormalisations of s and 2 = 2 + 2 where s0 is the typical scale of the problem and s s h is h, not included in the one-loop potential, vanish as we are expand- the distance in field space. We stress that this approximation is ing around λS = λHS = 0. The sS and sHS parameters are computed appropriate along the flat direction, rather than in all field space. by expanding the one-loop potential, obtaining the usual logarith- The minimum approximately lies on the flat direction, at field mic running plus non-logarithmic terms, that can be traded for the 2 = 2 1/2 dependent scale s s0/e determined by the flat direction scale precise scales sS and sHS (as opposed to a generic typical scale s0) 2 s0. The eigenvalues of the field-dependent mass matrix evaluated in the logarithmic terms only. They effectively encode how the full at the minimum of the potential are one-loop result deviates from the multi-phase critical point hit by

3 K. Kannike, L. Marzola, M. Raidal et al. Physics Letters B 816 (2021) 136241 the RG running. Given that the one-loop potential in the approximation of Eq. (11) becomes similar to the running potential, the two parameters sS and sHS loosely correspond to the RGE scales at which λS (μ¯ ) and λHS(μ¯ ) vanish, respectively. The precise order- = −1/2 2 2 one value of R e sS /sHS encodes how much the full result deviates from a naïve running potential. Assuming that the β-functions of λS and λHS are comparable and much smaller than λH , the potential has a minimum at non- vanishing s and h −1/4 − −1/4 e sS βλHS ln R s ≈ e sS , h ≈ , (13) 4π 2λH − provided that βλHS ln R > 0(otherwise only s acquires a vacuum expectation value). The mass eigenvalues are both loop- suppressed: 2 − 2 2 2s βλS 2 s βλHS ln R 2 m ≈ , m ≈ = 2λH h . (14) s (4π)2 h (4π)2 Their mixing angle is also loop-suppressed, Fig. 2. In the minimal model, the RG flow towards low energy that starts from λ β3 ln R + S λHS 1 ln R λ 1is attracted towards the multi-phase point (dot) where λ and λ are θ ≈ − , (15) HS S HS + small. 2λH 4π(2βλS βλHS ln R) unless the two scalars are nearly degenerate. No combinations of starting from λS |λHS| at high energy, which can be justified as Mh, Ms, θ are univocally predicted. All these results get multiplica- follows. tively corrected by the wave-function renormalisation effects that Assuming that s arises as a massless composite scalar corre- can be neglected. sponding to an operator O in some fundamental theory, one can Here we have outlined the computation in a specific model introduce a separate field s0 for the composite state by adding with the minimal field content. Similar computations along these to the fundamental Lagrangian renormalised at the compositeness lines can be performed in more general models. In Appendix A we energy scale M the term show how the results get extended. In Appendix B we show how UV the general formalism can be applied to recover the results for the 2 2 Z0(∂μs0) /2 + (s0 − O) (19) specific model considered in this section. with a vanishing kinetic term Z0 = 0such that s0 is a Lagrange 4. Simple models multiplier [8]. Interactions√ generate a small Z, such that the canon- ical field s = s0 Z obtained from the bare field s0 has non- In the minimal model with just the singlet scalar s in addition perturbatively large to the Higgs boson H and the rest of the SM, the RG equations are ∝ 2 ∝ given by λS (MUV) 1/Z and λHS(MUV) 1/Z (20) while the quartic λ of the Higgs is unconstrained, as no com- β = 9λ2 + λ2 , (16a) H λS S HS posite particles are involved in |H|4. Among the many composite ˜ models that one can consider, we mention the possibility that s βλ = λHS Zh + 3λS + 2λHS , (16b) HS could be the “conformal mode of the graviton” obtained reduc- 1 ing f (R) theories to the Einstein basis. In theories where f (R) = β = βSM + λ2 , (16c) λH λH HS − 1 ¯ 2 + 2 2 = 4 2 MPl R R /3 f0 the scalar s has renormalisable quartics, λS 2 = 2 + where the β-function f0 , λHS f0 (ξH 1/6), where ξH is the non-minimal coupling of the Higgs boson to gravity, and f0 becomes non-perturbative at 4 4 2 2 large energy [9]. SM ˜ 4 9g2 27g1 9g2 g1 β = 2ZhλH − 3y + + + (17) λH t 16 400 40 However, reducing λHS down to a loop-suppressed value of order 1/(4π)2 needs a long running of about (4π)2 orders of mag- ˜ = + is the SM contribution to the running of λH . Here Zh Zh 6λH , nitude (thereby, much above the Planck scale and even above the and scale where hypercharge gets strongly coupled), near to a fixed flow of the RGE system [10] which is not infrared attractive. 2 9 2 3 2 Zh = 3y − g − g (18) t 4 2 4 Y In the minimal model RG evolution does not change the sign of λHS. Crossing λHS = 0 becomes possible adding extra vectors un- is the one-loop wave-function renormalisation for the Higgs boson. der which H and s are charged (but introducing extra Z vectors As expected, it cancels out in Eq. (5). risks forbidding the SM Higgs Yukawa couplings), or extra fermions Of course, one possible reason for being near the multi-phase (but this tends to contribute as βλHS < 0) or extra scalars. Follow- critical point of Eq. (6)is just a fine-tuning of the parameters. Here ing the latter option, we introduce a second scalar s . The most we discuss one possible dynamical motivation for such tuning, general quartic potential symmetric under Z2 ⊗ Z (that respec- namely that the RG flow converges towards λS λHS 0. Tak- 2 tively act as s →−s and as s →−s ) is ing into account the λS and λHS couplings only, Fig. 2 shows that their RGE flow towards low energy can bring them nearer to the 4 λS 4 λS 4 multi-phase critical point λ = λ = 0. A near-approach can arise V = λH |H| + s + s + (21) S HS 4 4

4 K. Kannike, L. Marzola, M. Raidal et al. Physics Letters B 816 (2021) 136241

2 2 2 s 2 s λSS 2 2 The new physics associated with the dynamical symmetry break- +λHS|H| + λHS |H| + s s . 2 2 4 ing can be around the weak scale and weakly coupled to the Higgs, in agreement with collider bounds. The additional new scalar, the In this model the β-functions are dilaton, must be discovered to test our framework.

SM 1 2 2 βλ = β + (λ + λ ), (22a) H λH 4 HS HS Declaration of competing interest

2 1 2 2 βλ = 9λ + λ + λ , (22b) S S 4 SS HS The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to 2 1 2 2 βλ = 9λ + λ + λ , (22c) influence the work reported in this paper. S S 4 SS HS ˜ 1 βλ = λHS(Zh + 3λS + 2λHS) + λSS λHS , (22d) Acknowledgements HS 2 ˜ 1 = Z + 3 + 2 + (22e) This work was supported by the ERC grant 669668 NEO-NAT, by βλHS λHS ( h λS λHS ) λSS λHS, 2 European Regional Development Fund through the CoE program β = λ (2λ + 3λ + 3λ ) + 2λ λ . (22f) λSS SS SS S S HS HS grant TK133, the Mobilitas Pluss grants MOBTT5, MOBTT86, and the Estonian Research Council grants PRG434, PRG803 and PRG356. For simplicity we only consider the phases where s = 0, for which it is sufficient to have λS , λHS , λSS > 0. We then have a positive Appendix A. General matrix formalism contribution to βλHS , so that λHS can run negative at low energy. As a final aside comment, we mention another possible ratio- nale for being near to the multi-phase critical point λH = λHS = 0: In this appendix we extend the multi-phase point computation an RG flow with λHS that runs much faster than λS , down to neg- presented in section 3.2 in a specific simple model to a generic ative values at low energy. Then the symmetry breaking critical scalar field content φi , adapting the matrix formalism of [12,13]. = = (0) condition in Eq. (4) could be crossed only near to λH λHS 0. The effective potential is again given by Eq. (9), where V is However we do not know how to realise this flow without gener- the tree-level and V (1) is the one-loop contribution. We define ating other problems.1 the scalar field vector ={φ1, ..., φn}. We consider a tree-level quartic potential V (0) biquadratic in the fields, such that quartic 5. Conclusions couplings form a symmetric matrix as

Dynamical symmetry breaking is a specific form of symmetry ◦ ◦ V (0) = ( 2)T 2. (A.1) breaking in 3 + 1 space-time dimensions that: leads to a pseudo- Goldstone boson (the dilaton) with loop-suppressed mass; restricts ◦2 ={ 2 2} ◦ Here φ1 , ..., φn is the Hadamard square of the the parameters of possible broken phases; predicts how they are vector.2 This simplification allows to write a necessary and suf- connected. We studied how the dynamical symmetry breaking be- ficient condition for the stability of the tree-level scalar poten- haves near the multi-phase critical boundaries, where different tial, V (0) ≥ 0for all . The condition is that the matrix be patterns of symmetry breaking may co-exist. We found that some copositive [5]. According to the Cottle-Habetler-Lemke (CHL) theo- phases are smoothly connected, so that extra scalars become light rem [14], is copositive if when such phases merge. We explored the possibility that the lightness of the Higgs boson follows from the multi-phase criti- 1) the principal submatrices of order n − 1of are copositive. cality in dynamical symmetry breaking. We found that it can be These are the matrices obtained from deleting one row and realised in the example models considered in this work, around a the corresponding column, specific multi-phase critical point where specific scalar couplings are vanishingly small, λ = λ = 0. As a result, not only the Higgs S HS and boson acquires a loop-suppressed mass similarly to the dilaton, but also its mixing with the dilaton is loop-suppressed, allowing to sat- 2) either isfy collider bounds. 2a) det() ≥ 0; We have shown how the simple Gildener-Weinberg approxima- or tion can be extended to obtain simple expressions for the loop- 2b) at least one element of adj() is negative. The adjugate is suppressed masses and mixing. The original Gildener-Weinberg flat − related to the matrix inverse as 1 = adj()/ det(), but direction in the multi-field space is misaligned from the axis by exists even when det() = 0. a small tilt — an effect that is usually negligible. In this case, however, it contributes to the mass of the Higgs boson and to its This iterative procedure implies that all diagonal elements of mixing with extra particles. must be ≥ 0, and for n ≥ 2 adds a complicated set of extra condi- In this framework the Higgs boson and the dilaton masses are tions, corresponding to the various phases of the theory. both loop-suppressed, being proportional to square roots of differ- The one-loop potential can be written as ent combinations of β-functions. As couplings beyond the SM are unknown, this scenario gives no univocal prediction, allowing for M2 different mass orderings between the Higgs boson and the dilaton. V (1) = A + B ln , (A.2) μ¯ 2

1 where For example, one can reduce the symmetry of the model to a single Z2 (under 2 which both s and s flip sign), such that the extra coupling λHSS |H| ss is allowed. If larger than other couplings, λHSS provides the desired flow. The problem is that a 2 dominant quartic λHSS modifies the dynamical symmetry breaking conditions [11]: In general, the Hadamard product is the component-wise product of two matri- by itself it implies vacuum expectation values ss < 0. ces: (A ◦ B)ij = AijBij.

5 K. Kannike, L. Marzola, M. Raidal et al. Physics Letters B 816 (2021) 136241 1 M2 0, to ensure that the potential is bounded from below at higher A = Str M4 ln − C , (A.3) 64π 2 M2 scales. Then Eq. (A.11), expanded to linear order in t, takes the form 1 B = Str M4. (A.4) 64π 2 dV (0) 0 ≈ 4 V (t0) + t + 2B, (A.12) The matrix M 2 comprises the field-dependent matrices of scalars, dt vectors and fermions that appear in the theory, while C (in the MS M2 = scheme) is a constant diagonal matrix with entries 3/2for scalars which recovers the familiar Gildener-Weinberg relation −1/2M2 and fermions and 5/6for vector bosons. In order to split the e 0 between the pivot scale at a stationary point and at the 2 one-loop contributions, we introduced an arbitrary field-dependent flat direction, i.e. t = t0 − 1/[2(4π) ]. “pivot scale” M with the dimension of mass [15]. We choose Using Eq. (A.5), we can write Eq. (A.9)as 2 T ◦2 M = eM , where eM is a constant vector, to be conveniently ◦2 eM chosen. 0 = ◦ 4(t) + 2B , (A.13) M2 The RG scale μ¯ multiplies the B term only, and it cancels with ∇ the running of the couplings in the tree-level potential. Indeed, the where we have momentarily ignored the A term for simplicity. Callan-Szymanzik equation tells that This correction, in fact, does not alter the analytical form of the solution and its importance depends on the choice made for the (0) dV 2 ◦2 T ◦2 T (0) pivot scale which enters in A. We will return to this point below = (4π) B = ( ) β − γ ∇ V (A.5) dt and provide an explicit example of the treatment of these corrections for the two-fields model discussed in the main text. where We search the stationary point at = 0. Using the radial 2 2 d ln(μ¯ /μ¯ ) Eq. (A.11), the factor in parentheses in Eq. (A.13)gives β = , t = 0 (A.6) dt (4π)2 ◦2 V (t) = eM, (A.14) is the matrix of β-functions. The anomalous dimension matrix M2 γ , which is diagonal in the biquadratic case, accounts for wave- solved by function renormalisation. By setting ¯ = M, so that M = ¯ , the effective potential μ 0 μ0 ◦2 1 V = adj((t))eM. (A.15) becomes det((t)) M2 (0) V (t) = V (t) + A, (A.7) Approximating V by V (0) this becomes where V (0)(t) is the tree-level potential with field-dependent ef- M2 ◦2 fective couplings (t), but we neglect anomalous dimensions.3 The = adj((t))eM. (A.16) eT adj((t))eM running parameter is now also M-dependent: M The mass matrix around the minimum is then given by 1 M2 t = ln . (A.8) (4 )2 M2 2 =∇ ∇ T = 2 + 2∇ ∇ T π 0 mS V M S (4π) B t + 2∇ ∇ T + 2 ∇ ∇ T The minimisation procedure can be simplified by considering (4π) t B (4π) B t only the submatrix in selected by the subspace of fields which T +∇∇ A, (A.17) acquire non-vanishing vacuum expectation values, ignoring the others. The stationary point equation then is 2 where M S is the tree-level scalar mass matrix. Notice that the dV term proportional to B is cancelled by a similar term resulting =∇ = ◦ ◦2 +∇ + ∇ ∇ ∇ T 0 V 4 (t) A t. (A.9) from A. dt With the above solution at hand, it is possible to incorporate 2 We have ∇M = 2eM ◦ and consequently the corrections previously neglected by expanding A around the stationary point. This results in finite corrections that shift the val- 2 1 ∇ t = eM ◦ . (A.10) ues of quartic couplings near the minimum: (t) → (t) + . It 2 M2 (4π) is then possible to use Eq. (A.16)to compute the corrected mini- The radial minimisation equation is obtained projecting Eq. (A.9) mum solution which, generally, is not exactly aligned with the flat along the field vector: direction indicated by the Gildener-Weinberg approximation.

T (0) 0 = ∇ V = 4V (t) + 4A + 2B = 4V + 2B, (A.11) Appendix B. Detailed calculation of the potential minimum T ∇ = where we used A 4A, which holds because A is a homo- We now show how the general formalism can be applied, re- geneous function of order four. computing the two-field model used in the main text, where Let us now require that V (t ) = 0at a scale t (where the tree- 0 0 = h s and the scalar quartic coupling matrix is (0) ( , ) level flat direction along which V = 0is recovered) and that B > 1 λ (t) 1 λ (t) = H 2 HS (t) 1 , (B.1) 4 λHS(t)λS (t) 3 In general, fields in Eq. (A.7)must be scaled with anomalous dimensions as 2 t (t) = exp((t))(t0), where (t) =− γ (s)ds. Then, to avoid ambiguity, one t0 Positivity conditions can be recovered from det() ≥ 0 and from can denote ≡ (t0) and take the anomalous dimensions into account by scal- ing → exp(2(t)) exp(2(t)) and β → exp(2(t))β exp(2(t)). However, multi- the adjugate of the scalar quartic matrix phase conditions usually demand the vanishing of the relevant couplings multiplica- 1 λ − 1 λ tively corrected by wave function renormalisation. In addition, the second term in adj() = S 2 HS . (B.2) Eq. (A.5)that depends on γ is proportional to h2 and therefore small. − 1 4 2 λHS λH

6 K. Kannike, L. Marzola, M. Raidal et al. Physics Letters B 816 (2021) 136241

We choose M = s along the flat direction i.e. contribution with running couplings (amended by corrections from A) does not vanish: 0 eM = es = . (B.3) 1 2 2 1 2 m = 3λH (t)h + λHS(t)s h 2 = 2 2 2 Then t ln(s /s0)/(4π) . We identify s0 with the field value at 2 (B.10) −s βλ ln R which the potential crosses the critical boundary V = 0of the full ≈ HS = 2λ h2. 2 H potential, i.e., the scale corresponding to the tree-level flat direc- (4π) tion. Around the multi-phase point we can neglect wave-function T 2 +∇ ∇ T = Since (M S A) 12V , the scalon mass is renormalisations, i.e. the ∇A in Eq. (A.9). The minimum scale is 2 − 1 2 given by the usual Gildener-Weinberg relation, s = e 2 s . 1 βλS 0 m2 ≈ T m2 ≈ 2s2. (B.11) In order to account for the one-loop corrections, we can expand s s2 (4π)2 A in the parameter h2/s2, small in a region of the field space close We have thereby recovered the expressions obtained in the main = = = to the identified flat direction. By defining tS t(s sS ) and tHS text from direct computations. t(s = sHS), the result of the expansion can be incorporated in the definition of the field-dependent couplings, yielding References

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