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No Strong C P Violation up to the One-Loop Level in a Two-Higgs

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No Strong C P Violation up to the One-Loop Level in a Two-Higgs Eur. Phys. J. C (2019) 79:552 https://doi.org/10.1140/epjc/s10052-019-7053-4 Regular Article - Theoretical Physics No strong CP violation up to the one-loop level in a two-Higgs-doublet model P. M. Ferreira1,2,a, L. Lavoura3,b 1 Instituto Superior de Engenharia de Lisboa-ISEL, 1959-007 Lisboa, Portugal 2 Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 3 Universidade de Lisboa, Instituto Superior Técnico, CFTP, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Received: 29 April 2019 / Accepted: 17 June 2019 / Published online: 1 July 2019 © The Author(s) 2019 Abstract We put forward a two-Higgs-doublet model, fur- Lmass =−¯pL Mp pR −¯nL Mnn R + H.c., (1) nished with a Z3 symmetry, wherein CP is conserved in the dimension-four terms of the Lagrangian and is softly bro- where Mp and Mn are 3 × 3 matrices in flavour space. Then, ken in the scalar potential. The new particles of our model θQFD = arg det (Mp Mn). are one neutral scalar H, one neutral pseudoscalar A, and ± There are two general approaches to solving the strong CP two charged scalars H . In our model the only locus of problem. In the first approach it is claimed that θ has no sig- CP violation is the CKM matrix. Strong CP violation is nificance or physical consequences; theories with different absent both at the tree and one-loop levels. We work out the values of θ are equivalent and one may set θ to zero without phenomenological constraints on our model, which features loss of generality. This may happen either because one of the flavour-changing neutral Yukawa interactions, showing that quarks is massless2 or because of the presence in the theory the new scalar particles may in some cases be lighter than of a Peccei–Quinn symmetry [3]; there are also claims that 500 GeV. QCD dynamics itself cures the strong CP problem.3 The second approach, which we shall follow, acknowledges the strong CP problem and tries to find some symmetry that 1 Introduction naturally leads to the smallness of θ. One firstly assumes the dimension-four part of the Lagrangian to be either CP- Non-perturbative effects in Quantum Chromodynamics symmetric or P-symmetric; this assumption sets θQCD to (QCD) may lead to P and CP violation, characterized by a zero. The CP or P symmetry must be either softly or spon- parameter θ, in hadronic processes. The experimental upper taneously broken; one performs this breaking in such a way bound on the electric dipole moment of the neutron neces- that θQFD turns out to be zero at the tree level, because of some −9 1 sitates θ 10 . The presence in the Lagrangian of this peculiar form of Mp and Mn. Still, it is difficult to avoid loop unnaturally small parameter is known as the ‘strong CP contributions to θQFD arising from the quark self-energies ; problem’. they add to the tree-level mass matrices M and then The angle θ is the sum of two terms, θQCD and θQFD. θ − Here, QCD is the value of a P- and CP-violating angle in arg det (M + ) ≈ Im[tr(M 1)] (2) the QCD vacuum, and θQFD originates in the chiral rotation of the quark fields needed to render the quark masses real and is in general nonzero. Artful models are able to obtain positive. Let p and n denote the three up-type quarks and the Im[tr(M−1 )]+Im[tr(M−1 )] equal to zero at the one- three down-type quarks, respectively, in a weak basis. Let the p p n n loop level and sometimes even at the two-loop level. mass terms of those quarks be given by There are various ways to achieve quark mass matrices displaying arg det (Mp Mn) = 0. Most of those ways, col- 1 For a recente estimate of the maximum possible value of θ,seeRef. [1]. 2 For a recent speculation that some quarks may be massless, see Ref. a e-mail: [email protected] [2]. b e-mail: [email protected] 3 For a recent instance of such a claim, see Ref. [4]. 123 552 Page 2 of 17 Eur. Phys. J. C (2019) 79 :552 4 lectively known as Barr–Nelson-type models ,employextra there are softly-broken CP and Z3 symmetries such that quarks. There are also many models for solving the strong h and H do not mix with A. The interaction Lagrangian CPproblem that use extra gauge symmetries, especially the between a scalar and a pair of gauge bosons is ( ) × ( ) × ( ) − left–right symmetry SU 2 L SU 2 R U 1 B L .Inthis g − ξ+ m Z ξ LSVV = mW Wξ W + Zξ Z (v1ρ1 + v2ρ2) paper we propose a simple extension of the Standard Model v 2cθw (SM), with gauge group SU(2)L × U(1) and without any (9a) extra fermions, that partially solves the strong CP problem. =− − ξ+ + m Z ξ + , Our model is a two-Higgs-doublet model (2HDM) [6]. g mW Wξ W Zξ Z hsβ−α Hcβ−α 2cθw In a 2HDM the quark Yukawa Lagrangian is (9b) 3 2 ¯ where θw is Weinberg’s angle. Because of the LHC data we LYukawa =− QLj now know that sβ−α ≈ 1. j,k=1 a=1 The quark mass matrices are × ( ) + ˜ ( ) + . a a jk n Rk a a jk pRk H c (3) iℵ iℵ v1e 1 1 + v2e 2 2 Mn = √ , (10a) + 0 T ˜ 0∗ − T where a = φ ,φ and a = φ , −φ for a = 2 a a a a − ℵ − ℵ ¯ v i 1 + v i 2 1, 2 are scalar doublets of SU(2)L. Furthermore, QLj = 1e 1 2e 2 Mp = √ . (10b) p¯Lj, n¯ Lj , and the a and a are four 3 × 3 matrices in 2 flavour space containing the Yukawa coupling constants. We n,p Let the unitary matrices U bi-diagonalize M and M as expand the scalar doublets as L,R n p n† n = ≡ ( , , ) , + UL MnUR Md diag md ms mb (11a) ℵ φ = i a a √ , p† p a e (4) U MpU = Mu ≡ diag (mu, mc, mt ) . (11b) (va + ρa + iηa) 2 L R √ The CKM matrix is v ( ℵ ) = φ0 v where a exp i a 2 0 a 0 and the a are non- p† v = v2 + v2 = V = U U n. (12) negative real by definition. We define 1 2 L L 2mW /g = 246 GeV and tan β ≡ v2/v1; then, We define v v 1 2 iℵ iℵ = cos β, = sin β, (5) v2e 1 1 − v1e 2 2 v v Nn = √ , (13a) 2 − ℵ − ℵ where the angle β is in the first quadrant. There is one phys- v i 1 − v i 2 = 2e 1 1e 2 , ical pseudoscalar A and one unphysical (Goldstone boson) N p √ (13b) 2 pseudoscalar G0: and 0 η G cβ sβ 1 = n† n , = . (6) Nd UL NnUR (14a) A sβ −cβ η2 = p† p. Nu UL N pUR (14b) (From now on, sξ ≡ sin ξ and cξ ≡ cos ξ for any needed Then, the Yukawa interactions in the physical basis are given angle ξ). There is a pair of physical charged scalars H ± and by a pair of unphysical (Goldstone bosons) charged scalars G±: iA L = u¯(N P − N † P )u (15a) physical v u R u L ± φ± G cβ sβ 1 iA † ± = ± . (7) + ¯( − ) − φ d Nd PL Nd PR d (15b) H sβ cβ 2 v h † + u¯[(sβ−α M − cβ−α N )P There are two physical neutral scalars h and H: v u u L +(sβ−α Mu − cβ−α Nu)PR]u (15c) h sα −cα ρ1 = . (8) h ¯ † H −cα −sα ρ + d[(sβ−α M − cβ−α N )P 2 v d d L +(sβ−α M − cβ−α N )P ]d (15d) The neutral scalar h is chosen to coincide with the LHC- d d R observed particle with mass 125GeV. In our specific 2HDM H † + u¯[(cβ−α M + sβ−α N )P v u u L 4 For a recent example of one such model, see Ref. [6] +(cβ−α Mu + sβ−α Nu)PR]u (15e) 123 Eur. Phys. J. C (2019) 79 :552 Page 3 of 17 552 H ¯ † be justified through an ultraviolet completion, viz. amore + d[(cβ−α Md + sβ−α N )PL v d complete model, with extra fields active at higher energies, + (cβ−α M + sβ−α N )P ]d (15f) √ d d R which effectively mimics at low energy scales the model with + the softly-broken symmetry. Unfortunately, such a ultravio- + 2H ¯( † − ) u Nu VPL VNd PR d (15g) √ v let completion is often quite difficult to construct explicitly 2H − – we attempted such a construction by adding singlet scalar + d¯(V † N P − N †V † P )u, v u R d L (15h) fields to the theory, who would develop vevs at some high scale and then be “integrated out”, leaving the desired low = ( − γ )/ = ( + γ )/ where PL 1 5 2 and PR 1 5 2 are the projec- energy potential with only two doublets. However, we were tors of chirality. Also, u and d are column vectors subsum- unable to build such extensions that left both the CPand Z3 ing the fields of the physical up-type and down-type quarks, symmetries intact at low energies – which of course does not respectively.
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