Attracting the Electroweak Scale to a Tachyonic Trap Sokratis Trinopoulos1and Miguel Vanvlasselaer2 3 4y 1Center for Theoretical Physics Massachusetts Institute of Technology Cambridge MA 02139 USA

2025-05-02 0 0 551.59KB 7 页 10玖币

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Attracting the Electroweak Scale to a Tachyonic Trap

Sokratis Triﬁnopoulos1, ∗and Miguel Vanvlasselaer2, 3, 4, †

1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2Theoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel,

& The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium

3SISSA International School for Advanced Studies, Via Bonomea 265, 34136, Trieste, Italy

4INFN, Sezione di Trieste, SISSA, Via Bonomea 265, 34136, Trieste, Italy

We propose a new mechanism to dynamically select the electroweak scale during inﬂation. An

axion-like ﬁeld φthat couples quadratically to the Higgs with a large initial velocity towards a

critical point φcwhere the Higgs becomes massless. When φcrosses this point, it enters a region

where the Higgs mass is tachyonic and this results into an explosive production of Higgs particles.

Consequently, a back-reaction potential is generated and the ﬁeld φis attracted back to φc. After

a series of oscillations around this point it is eventually trapped in its vicinity due to the periodic

term of the potential. The model avoids transplanckian ﬁeld excursions, requires very few e-folds of

inﬂation and it is compatible with inﬂation scales up to 105GeV. The mass of φlies in the range

of hundreds of GeV to a few TeV and it can be potentially probed in future colliders.

I. INTRODUCTION

In recent years, the idea that the electroweak scale

could be dynamically determined by the cosmological

evolution of a (pseudo-)scalar ﬁeld sparked a paradigm

shift in theories of naturalness. The ﬁrst model of this

kind [5] features an axion-like ﬁeld φ, called the relax-

ion, which couples to the Higgs Hvia a term of the type

gΛφH2with tiny g. The relaxion slow-rolls during in-

ﬂation and scans the Higgs mass m2

H(φ) = −Λ2+gΛφ,

where Λ is the scale that New Physics (NP) is expected

to appear. Electroweak symmetry breaking occurs after

the ﬁeld crosses the critical point φc= Λ/g and a peri-

odic back-reaction potential for φis generated via non-

perturbative eﬀects of a conﬁning sector at scale M. The

height of the potential barriers grows with the increasing

Higgs vacuum expectation value (VEV), eventually stop-

ping the relaxion and trapping it into a local minimum

at the electroweak scale vEW. No new degrees of free-

dom at the TeV scale charged under the Standard Model

(SM) are required and as a result experimental strate-

gies motivated by naturalness are radically diﬀerent in

this framework.

The original proposal was not without some theoret-

ical shortcomings such as the requirement M.vEW,

which implies that the conﬁning sector is hidden (i.e. not

charged under the SM symmetries) and its scale coincides

with the electroweak scale without any a priori reason.

Moreover, transplanckian ﬁeld excursions of the relaxion

∆φ∼Λ/g are necessary as well as an enormous number

of e-folds that have to be produced by low-scale inﬂation,

which raises concerns of cosmological ﬁne-tuning [6,7].

Various model-building attempts to address these issues

have appeared in the literature [6,8–25], albeit at the

price of introducing non-minimal setups. Beyond the

∗triﬁnos@mit.edu

†miguel.vanvlasselaer@vub.be

relaxion framework, recent works [26–28] have consid-

ered scenarios in which the electroweak scale is also de-

termined due to the interplay between a scalar and the

Higgs, but instead of a dynamical relaxation there is en-

vironmental and anthropical selection related to the vac-

uum energy in diﬀerent patches of the inﬂationary uni-

verse.

In this letter we present a model of cosmological relax-

ation of the electroweak scale which is free of the above-

mentioned pathologies while at the same time remains

economical introducing only one new ﬁeld at the eﬀective

theory level. In particular, it utilizes a stopping mecha-

nism that relies on the extremely rapid production of ex-

citations of a scalar ﬁeld, in our case the Higgs ﬁeld, that

couples quadratically to another (pseudo-)scalar ﬁeld φ.

The particle production takes place when the Higgs be-

comes massless at a critical point of the classical tra-

jectory of φ, i.e. the symmetry breaking point (SBP)

φc. The produced particles generate an eﬀective back-

reaction potential that attracts the ﬁeld φ, which we

will call attraxion, back to the SBP. If the production

is strong enough, the global minimum of the potential

is now φ=φcand the ﬁeld starts to oscillate around

it. Hubble expansion causes a decrease of the oscillation

amplitude and eventually the ﬁeld is trapped in the vicin-

ity of the SBP. A similar trapping mechanism was ﬁrst

envisioned as a possible solution to the cosmological mod-

uli problem [29] and then exploited in models of trapped

inﬂation [30,31] as a method to obtain slow-rolling con-

ditions for the inﬂaton even in a non-ﬂat potential. More

recently it has been used in the context of quintessential

inﬂation in order to freeze the inﬂaton dynamics until

later times [32,33].

In contrast to the slow-rolling relaxion, the mechanism

is eﬀective in the high initial velocity regime of the pa-

rameter space, which additionally enables a fast scanning

of the Higgs mass requiring only very few e-folds of in-

ﬂation. The attraxion potential also has a periodic term

which is initially not interfering with the fast rolling, but

after the kinetic energy is depleted, the ﬁeld is eventually

arXiv:2210.13484v2 [hep-ph] 16 May 2023

trapped in one of its valleys. The process occurs before

the Higgs number density is diluted due to inﬂation or

the Higgs bosons decay removing the back-reaction term.

It is worth noticing that the periodic potential does not

depend on the Higgs VEV disentangling in principle the

scale of the conﬁning sector from the electroweak scale.

Furthermore, the size of the coupling grequired by the

mechanism is much larger than the one in relaxion mod-

els which implies that ﬁeld excursions are always smaller

than the Planck scale.

II. THE ATTRAXION MODEL

Eﬀective potential - The eﬀective potential at tree-

level reads

Vtree(H, φ) = g2φ2−φ2

2|H|2+λ

4|H|4+Vφ(φ),(1)

where φc≡Λ/g. The potential has two SBPs at φ=

±φc. In this letter, we study the case of a quadratic

attraxion-Higgs coupling (e.g. see Ref. [9]).

We assume that φdoes not couple at tree level to the

NP at scale Λ. Despite that, closing the Higgs loop pro-

vides the leading loop-level correction

Vloop(φ)∼g4φ2

16π2φ2=g2Λ2

16π2φ2.(2)

Finally, we assume that φobeys a shift symmetry bro-

ken at scale fand couples to a hidden conﬁning sector

at scale M, which yields the periodic potential

Vφ(φ) = M4cos φ

f.(3)

Unlike in the traditional relaxion model, this term does

not depend on the Higgs VEV and is present even before

the stopping mechanism is triggered. This term allows

for the existence of local minima close to the SBPs when

M4&gΛ3f

8π2.(4)

A concrete ultraviolet (UV) completion is beyond the

scope of this letter, but we mention that variations of the

constructions laid out in Refs. [5,9,34] and in particular

the clockwork framework of Ref. [10,11] could match to

our model in the low-energy limit.

The electroweak symmetry is broken in the region φ <

φc, where the minimum of the potential in the Higgs

direction is situated at

H(φ) = g2

λ(φ2

c−φ2).(5)

The minimum of the potential in the attraxion direction

is at φ= 0.

Trapping mechanism - The rolling of the attraxion

starts during the inﬂation era at large negative ﬁeld val-

ues φi −φc(the choice of the sign is free) and with a

large initial velocity towards the origin. As the attrax-

ion comes close to the ﬁrst SBP φ=−φcwith veloc-

ity ˙

φc, the Higgs becomes massless. The Higgs modes

with momentum kand frequency ωk=pk2+m2

H(φ)

for which the non-adiabatic parameter ˙ωk/ω2

kbecomes

large, are excited and resonant particle production takes

place [35]. After it crosses the SBP, the mass parame-

ter becomes negative and the modes with k2<|m2

H(φ)|

will be exponentially ampliﬁed via a process called tachy-

onic resonance [33,36–39]. The particle production oc-

curs throughout the non-adiabatic region between the

two SBPs |φ|< φcand it peaks at φ= 0, where the

maximal number of modes become tachyonic.

The Higgs quartic self-interaction λh4reintroduces an

eﬀective mass term m2

H+3λhH2ieﬀ , which suppresses the

particle production. Taking this eﬀect into consideration,

in Ref. [33] the authors derive an analytic approximation

for the total particle number density after the exit from

the non-adiabatic region at the second SBP φ=φc,

nH≈

qg˙

φc

2π



eπΛ2

g˙

φc×e−3πλ(Λ)hH2i(0)

eff

g˙

φc,(6)

where

hH2i(0)

eﬀ =g˙

φc

2π3sπ/2

1−Q/2

eQ/2−1, Q ≡πΛ2

g˙

φc

.(7)

The production is favored for smaller values of the quar-

tic. Notice that λis evaluated at scale Λ, because this

is the relevant energy scale of the Higgs potential at the

point of maximum production. In the following and un-

less explicitly mentioned otherwise, we will abbreviate

λ=λ(Λ) and consider it as a free parameter.

The corresponding energy density stored in Higgs ex-

citations is [29]

ρH≈nH|mH(φ)| ≈ (√2gnH|∆φ|,|∆φ|  φc

gnHp2φc|∆φ|,|∆φ|  φc

(8)

where ∆φ=φ−φc.

For a wide range of model parameters we have

V(0) > V (φc)⇒nH|mH(0)| − Λ4

4λ>Λ4

16π2,(9)

which implies that ρHacts as a back-reaction potential

and the SBP φ=φcbecomes the new global minimum

attracting φback to it. In fact, as the attraxion moves

away from the SBP its kinetic energy is transferred to the

Higgs energy density and when ρH∼˙

φ2

c/2 at ∆φ=A

with

A≡





√2˙

φ2

4gnH,|∆φ|  φc

φ4

8gn2

HΛ,|∆φ|  φc

,(10)

it stops and returns back to the SBP. As it crosses this

point again (with practically the same velocity) it trig-

gers a second burst of particle production and the newly

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AttractingtheElectroweakScaletoaTachyonicTrapSokratisTrinopoulos1,andMiguelVanvlasselaer2,3,4,y1CenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA2TheoretischeNatuurkundeandIIHE/ELEM,VrijeUniversiteitBrussel,&TheInternationalSolvayInstitutes,Pleinlaan2,B-1050Bruss...

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Attracting the Electroweak Scale to a Tachyonic Trap Sokratis Trinopoulos1and Miguel Vanvlasselaer2 3 4y 1Center for Theoretical Physics Massachusetts Institute of Technology Cambridge MA 02139 USA

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