A Phase of Confined Electroweak Force in the Early Universe

FERMILAB-PUB-19-269-T PITT-PACC-1903 A phase of conﬁned electroweak force in the early Universe

Joshua Berger,1 Andrew J. Long,2, 3 and Jessica Turner4 1Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 2Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 3Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA 4Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510, USA (Dated: September 6, 2019) We consider a modiﬁed cosmological history in which the presence of beyond-the-Standard-Model physics causes the weak gauge sector, SU(2)L, to conﬁne before it is Higgsed. Under the assumption of chiral symmetry breaking, quark and lepton weak doublets form condensates that break the global symmetries of the Standard Model, including baryon and lepton number, down to a U(1) subgroup under which only the weak singlet fermions and Higgs transform. The weakly-coupled gauge group SU(3)c×U(1)Y is also broken to an SU(2)c× U(1)Q gauge group. The light states include (pseudo-)Goldstone bosons of the global symmetry breaking, mostly-elementary fermions primarily composed of the weak singlet quarks and leptons, and the gauge bosons of the weakly-coupled gauge group. We discuss possible signatures from early Universe cosmology including gravitational wave radiation, topological defects, and baryogenesis.

I. INTRODUCTION and the baryon asymmetry of the Universe. Our work shares some common elements with several pre- vious studies of SM exotic phases, their associated phase tran- Quarks and leptons can interact via the weak force, medi- sitions, and their imprints on cosmology. As we have al- ated by the W - and Z-bosons. The Standard Model of ele- ready mentioned above, SU(2) confinement was studied in mentary particles (SM) describes the electroweak force by a L Refs. [4–8], and we present a detailed comparison of our work quantum field theory that is constructed from the gauge group with theirs in Sec.VI. The authors of Refs. [12, 13] studied a SU(2) × U(1) and is in the Higgs phase [1–3]. Higgs- L Y variation of the SM in which the Higgs field is absent, and the ing screens weak isospin at the weak scale and leaves only electroweak force is Higgsed by the quark chiral condensates, weakly-coupled electromagnetic long-range interactions. The and Ref. [14] points out a dark matter candidate in a related theory is weakly coupled in the infrared, which agrees ex- model. Delaying the electroweak phase transition (EWPT) to tremely well with measured quark and lepton interactions. By coincide with the chiral phase transition of QCD was stud- contrast the strong force, which mediates interactions among ied as a means of generating the baryon asymmetry of the the quarks, is strongly coupled in the infrared, leading to Universe [15–17] (see also Ref. [18]) and a stochastic back- the the confinement of the constituent quarks into compos- ground of gravitational waves [19]. Likewise, an early pe- ite states, namely mesons and baryons. In this article, we riod of QCD confinement, which triggers a first order EWPT, study the weak-confined Standard Model (WCSM) in which was proposed as a possible mechanism of baryogenesis [20]. the SU(2) component of the electroweak force is strongly L Moreover, early QCD phase transitions have been studied in coupled and weak isospin is confined in composite particles. order to open the axion parameter space [21, 22]. Confinement of SU(2) was studied in the pioneering work L We will see that SU(2)L confinement in the WCSM causes of Abbott, Farhi, and others in the 1980’s [4–7] (also [8–10]) baryon number and lepton number to be spontaneously bro- as an alternative to the SM. Assuming that confinement oc- ken, which is signaled by the formation of lepton-quark and curs without chiral symmetry breaking, they argued that the quark-quark condensates, hlqi and hqqi. The violation of known “elementary particles” are actually composite particles baryon number is a necessary condition to explain the cos- and that the predicted low-energy fermion spectrum agrees mological excess of matter over antimatter, and therefore arXiv:1906.05157v2 [hep-ph] 5 Sep 2019 well with the measured quark and lepton masses. However, one may expect that the WCSM can naturally accommodate this scenario is now ruled out by precision measurements of baryogenesis. However, when we discuss the cosmological the electroweak sector [11], and it would appear that the weak implications of weak confinement in Sec.V, we will see that force is not confining in our present day universe. the viability of baryogenesis remains unclear. Nevertheless, when we study particle physics in a cosmo- This article is structured as follows. In Sec.II we pro- logical context, the system can exist in different phases at vide an example of the new physics that could lead to weak different times, corresponding to different temperatures of confinement in the early universe. In order to develop intu- the primordial plasma. We explore a scenario in which the ition for confinement in an SU(2) gauge theory, we next study SU(2)L weak force was strongly coupled in the early Uni- three simplified models in Sec. III. Turning to the WCSM in verse, when the plasma temperature was T 100 GeV, but Sec.IV, we investigate the symmetry breaking pattern, cal- became weakly coupled and Higgsed well before the epoch culate the spectrum of composite particles, and discuss their of nucleosynthesis, T ∼ MeV. Although we are unable to interactions. We discuss the possible cosmological implica- access the weak-confined phase in the laboratory today, e.g. tions in SectionV and highlight directions for future work. at collider experiments, the Universe may contain relics from We contrast the WCSM with Abbot and Farhi’s earlier work this period in its cosmic history, such as gravitational waves on SU(2)L confinement in Sec.VI and conclude in Sec. VII. 2

II. CONFINEMENT OF THE WEAK FORCE SM running WCSM What new physics might allow the SU(2)L weak force to confine in the early Universe? In the context of QCD, early 30 M color confinement has been studied previously in Refs. [21, ∼ ϕˆ = 2 h i g 22] (see also Ref. [20]), and we can straightforwardly adapt 1 − that mechanism for our purposes. The essential idea is to link W α the strength of the weak force with the expectation value of a scalar field that experiences a phase transition in the early deconfinement ϕˆ 0 Universe. h i → Let ϕˆ(x) be a real scalar field, which we call the modulus

field, and suppose that it interacts with the weak force through weak coupling: a dimension-five operator. The relevant Lagrangian is 1 1 ϕˆ µν L = − 2 − Tr Wµν W − V (ϕ ˆ) , (2.1) 2 g M SU(2)L is confined 1/4π where g is the SU(2)L gauge coupling, M is an energy scale parameter, W is the SU(2)L field strength tensor, and V is the v ΛW scalar potential of ϕˆ. In the limit M → ∞ the modulus field cosmological plasma temperature: T decouples from the Standard Model. We will consider M >

TeV in order to avoid disrupting the remarkable agreement Figure 1. The strength of the SU(2)L weak force is parametrized 2 between electroweak-precision theory and measurement. by the fine structure constant αW = geff /4π, which varies with the In general the scalar modulus may have a nonzero expec- temperature of the cosmological plasma, T , according to the renor- tation value, hϕî, and in this background, the strength of the malization group flow. The solid-green line illustrates the Standard −1 Model prediction: the weak force grows stronger (smaller αW ) as SU(2)L interaction is controlled by the effective coupling −1 the plasma cools (smaller T ), but it remains weak (αW 1/4π) 1 1 hϕî when the theory is Higgsed at T ∼ v ∼ 100 GeV. Instead, we are interested in the weak-confined Standard Model, corresponding 2 = 2 − . (2.2) geff g M to the dashed-green line: the nonzero modulus field makes the weak −1 force stronger (smaller αW ) at early times (large T ) such that αW 2 2 If hϕî M/g then geff ≈ g, but if hϕî ∼ M/g then the reaches ∼ 4π at T ∼ ΛW , and the theory enters the SU(2)L-confined weak force is stronger, geff & g. The value of hϕî need not re- phase. main fixed throughout the cosmic history, and we demonstrate this point in the paragraphs below. Thus we will assume that hϕî was large in the early Universe, corresponding to g > g, eff large value, g2 /4π = O(4π), while the Higgs field’s expec- but that it evolved to a small value, where g ≈ g, at some eff eff tation value remains zero. In that case the Standard Model point before the epoch of big bang nucleosynthesis. In this enters the confined-SU(2) phase, which we call the WCSM, way, laboratory probes of the weak force only access g ≈ g, L eff and we study the properties of this phase in the remainder whereas cosmological probes of the pre-nucleosynthesis era of this article. The proposed modification to the gauge cou- may uncover evidence for g > g in the early Universe. eff pling’s running is schematically illustrated in Fig.1. As discussed, the effective strength of the weak interaction, One can imagine several different explanations for the non- g , varies with time through the scalar field modulus, hϕî, eff trivial dynamics of ϕˆ. On the one hand, the potential V (ϕ ˆ) but it also varies through the cosmological plasma tempera- may have a local minimum at hϕî 6= 0, which traps the field ture, T . At temperatures T > 100 GeV, the hot Standard until it is either released, for instance by the changing tem- Model plasma contains SU(2) -charged quarks and leptons, L perature of the cosmological plasma, or it escapes through a which interact via the weak force. As the Universe expands first order phase transition. Alternatively, the potential V (ϕ ˆ) and the cosmological plasma cools, these scatterers carry less may simply be very shallow such that the field ϕˆ is “frozen kinetic energy, and they probe the weak force at larger-and- by Hubble drag” at its nonzero value until a time such that larger length scales. This scale dependence of the weak force the Hubble parameter, H, decreases below its effective mass, is described by the equations of renormalization group flow, i.e., H < m ∼ |V 00|1/2.1 If the Universe is radiation- and its effects are captured by treating g as a running coupling ϕˆ dominated at this time, then T ∼ pm M where M is that is tiny at small length scales in the high-energy ultravio- ϕˆ pl pl let (UV) regime and grows bigger at large length scales in the low-energy infrared (IR) regime. In the Standard Model the weak force was Higgsed at the electroweak phase transition 1 It is worth remarking that the modulus field’s effective potential is a sum of 4 when the plasma temperature was T ∼ v ∼ 100 GeV, corre- V (ϕ ˆ) and a correction induced by confinement, which is ∆V ∼ ΛW . The 2 2 2 2 4 4 −8π /geff sponding to g /4π 1. However, if hϕî ∼ M/g at early confinement scale can be written as ΛW ∼ ΛUV e where ΛUV is times, then the weak coupling may run to a non-perturbatively a high-energy input scale and geff depends on ϕˆ through Eq. (2.2). The 3

−5 the reduced Planck mass. For instance if mϕˆ ∼ 10 eV then and three lepton doublets, but here we keep the analysis more T ∼ 100 GeV. Once ϕˆ is released, it can roll toward the min- general. imum of its potential, located at hϕî = 0, where it oscillates The theory of Eq. (3.3) has a U(2Nf )ψ flavor symmetry and decays. under which the ψi transform in the fundamental representa- In the remainder of this article, we do not assume any par- tion and Wµ transforms as a singlet. Using the isomorphism ticular model for inducing confinement of the weak force. In- U(2Nf )ψ = SU(2Nf )ψ × U(1)ψ, we can assign ψ a charge stead we assume that the SU(2)L weak interaction becomes qψ under the U(1)ψ group whose conserved charge is axial- non-perturbatively large at a scale ΛW & TeV in the early ψ-number. The SU(2)L gauge interactions induce an anomaly Universe at a time before the electroweak interaction is Hig- in U(1)ψ [25], breaking it to the Z2Nf subgroup. The trans- gsed and before color is confined. formation properties of the various fields are summarized in TableI. It is illustrative to draw a contrast with the flavor symme- III. BUILDING UP TO THE WCSM try of QCD [26]. In the limit that the up, down, and strange quarks and antiquarks are massless, we would have Nf = 3 Our goal is to study the weak-confined Standard Model, flavors of left-chiral Weyl fermions transforming in the fun- which has the same particle content as the SM, but in which damental of SU(3)c color, 3, and Nf = 3 additional fla- the SU(2)L weak force is confined. As weak confinement vors (of left-chiral Weyl fermions) transforming in the anti- leads to a dramatic departure from the usual SM dynamics, fundamental, 3¯. The flavor symmetry is U(Nf ) × U(Nf ) = we use this section to put together the pieces of the SU(2)L- SU(Nf )V ×U(1)V ×SU(Nf )A ×U(1)A where the conserved confined SM bit-by-bit. First, we consider an SU(2) gauge charge associated to U(1)V is baryon number and the anoma- theory with doublet fermions followed by the addition of a lous charge associated with U(1)A is axial-baryon number. doublet scalar, and we finally consider singlet fermions. The situation in the theory with gauged SU(2)L is different because the fundamental representation is pseudo-real, i.e. 2 = 2¯, which allows the 2Nf fields to be collected into a sin- A. Model 1: Include only SU(2)L-doublet fermions gle flavor multiplet with the larger U(2Nf ) flavor symmetry instead. We consider a gauge theory based on the SU(2) symme- The SU(2)L force in Eq. (3.3) is not Higgsed, as in the SM, but rather may become strongly coupled in the IR. To an- try group, which we denote by SU(2)L, anticipating the connection with SM weak isospin. The force carriers are rep- alytically understand the evolution towards strong coupling, one can calculate the gauge coupling’s beta function β , and a resented by the vector field Wµ(x), which transforms in the g value β < 0 implies that g grows in the IR. Calculating β at adjoint of SU(2)L, and the matter is represented by 2Nf fla- g g 2 3 one-loop order in perturbation theory gives β = (b/16π2)g3 vors of left-chiral Weyl fermion fields, denoted by ψi(x) for g with b = −22/3 + 2N /3 [27]. A theory with N < 11 has i = 1, 2, ··· , 2Nf , which separately transform in the funda- f f β < 0, and this perturbative analysis suggests that such a mental representation of SU(2)L. The fermions are assumed g 2 to have no mass terms, and the Lagrangian is simply theory is strongly coupled in the IR; that is, g /4π grows as large as 4π as the energy scale decreases toward ΛW. This per- 2Nf turbative analysis is confirmed by numerical lattice studies for X † µ 1 µν L1 = ψ iσ¯ Dµψi − Tr Wµν W , (3.3) small Nf while Nf = 6 is marginal and near the conformal UV i 2g2 i=1 window [28–32]. The strong coupling is assumed to imply that SU(2)L where Dµψi = ∂µψi − iWµψi is the covariant derivative of charge is confined, in analogy with QCD’s color confine- ψi, g is the running SU(2)L gauge coupling at the UV scale, ment.5 Consequently the degrees of freedom in the low- 4 and Wµν is the SU(2)L field strength tensor. In SectionIV energy confined phase are composite states, analogs of the we will take 2Nf = 12 to count the SM’s nine quark doublets mesons and baryons of QCD. Specifically, the spectrum consists of several massless Goldstone bosons Πa, associated with the assumed SU(2Nf )ψ global flavor symmetry, a single massive pseudo-Goldstone boson η0, associated with the 2 2 corresponding mass correction, ∆mϕˆ ∼ 4π ΛW /M, may be larger than 2 anomalous U(1)ψ global flavor symmetry, and a tower of H ∼ T /Mpl at the time of confinement when T ∼ ΛW , which would cause ϕˆ to be released from its Hubble drag before the system spends much time in the confined phase. This outcome can be avoided by taking M > Mpl or possibly by tuning V (ϕ ˆ) against ∆V . 2 An even number of fermions (2Nf ) is required to avoid an anomaly in to perform a field redefinition and remove the topological term from the the SU(2)L gauge group. We assume Nf > 1, but see Ref. [23] for an Lagrangian, implying that θ cannot affect any observable. Analogously analysis of Nf = 1. the weak vacuum angle can be removed from the SM Lagrangian through 3 Here and throughout the text we suppress the spinor and SU(2)L gauge the baryon- or lepton-number anomalies [24]. Therefore, without loss of indices when appropriate, showing instead only the flavor indices. generality, we will work in the basis where the topological term is absent. 4 5 In general the Lagrangian will also contain a topological term, Lθ = If the theory instead has a strongly-coupled, IR-stable fixed point, as sug- 2 µν −(θ/16π ) Tr[Wµν W˜ ] where θ is the vacuum angle. However, as we gested by Ref. [31], then there is no charge confinement, and a different will discuss momentarily, this theory has an anomaly in axial ψ-number analysis is required to determine the properties and interactions of the low- and no other explicit breaking of this symmetry, which makes it possible energy degrees of freedom. 4 heavy resonances with masses around the confinement scale, we have discussed above, the antisymmetric complex scalar mres ∼ ΛW. These heavy resonances include meson-like field Σij(x) parametrizes the massless Goldstone boson fields a 0 scalars, vectors, and glueball-like bound states of the Wµ bo- Π (x) and the pseudo-Goldstone boson field η (x). We make son. As these resonances are expected to be heavy and un- this connection explicit by writing stable, we can assume that they decay shortly after SU(2)L is 0 p X a a confined, and we focus on the properties of the light states. Σ = exp iη / Nf f exp 2iX Π /f Σ0 , (3.7) Collectively the lowest-lying, meson-like states Πa and η0 a are encoded in the scalar fields Σij(x), which have the same a 2 symmetry properties as the fermion bilinears: where Xij are the 2Nf − Nf − 1 broken generators of SU(2Nf ), and the vacuum expectation value Σ0 satisfies p Σij ∼ ψiεψj . (3.4) Eq. (3.5). The factors of 2 and Nf are included to ensure canonical normalization of kinetic terms and the generators Here the totally-antisymmetric tensor, ε, with ε = 1, con- 12 are normalized such that Tr[XaXb] = δab/2. tracts the (suppressed) SU(2) indices such that Σ is an L ij The last term of Eq. (3.6) arises from instanton effects SU(2) singlet. Under the flavor symmetries, Σ transforms L via the chiral anomaly of U(1) . The anomaly explicitly as an antisymmetric rank-2 tensor of SU(2N ) with charge ψ f ψ breaks the U(1) symmetry to its subgroup and thereby 2q under U(1) . We assume that Σ acquires a nonzero vac- ψ Z2Nf ψ ψ lifts the mass of the η0 through instanton effects, in analogy uum expectation value, and the global SU(2N ) × U(1) f ψ ψ with the U(1) problem of QCD [37]. One calculates the η0 symmetry is spontaneously broken. A mass by substituting Eq. (3.7) into Eq. (3.6) and expanding to The symmetry breaking pattern cannot be determined 2 2 quadratic order; doing so gives m 0 = 4κN Λ . through a perturbative calculation, since after all, symmetry η f W breaking is a consequence of non-perturbative charge confinement. However, this simple model has been studied us- B. Model 2: Add an SU(2) -doublet scalar ing non-perturbative, numerical lattice techniques [33, 34] for L Nf = 2 (see also Refs. [28–32]). Those studies conclude that SU(2Nf )ψ is broken to Sp(2Nf )ψ, which is the naive Let us next consider an extension of the first model that is a expectation from chiral symmetry breaking in analogy with closer approximation to the Standard Model particle content, QCD [35, 36]. Therefore, we will adopt the results of the by introducing an analog of the Higgs doublet field. Consider lattice studies, and assume that the unbroken global flavor the complex scalar field φ(x) that transforms under SU(2)L symmetry is the Sp(2Nf )ψ group. This pattern of symmetry in the fundamental representation. The Lagrangian of the UV breaking is obtained if the antisymmetric field Σij(x) acquires degrees of freedom can be written as (see also Ref. [38]) a vacuum expectation value hΣ i = (Σ ) that satisfies ij 0 ij 2 ∗ L2 = L1 + Dµφ − U(φ φ) , (3.8) † † UV UV Σ0Σ0 = Σ0Σ0 = 1 ⇔ SU(2Nf )ψ / Sp(2Nf )ψ . (3.5) where Dµφ = ∂µφ − iWµφ is the covariant derivative, and U is a potential for the scalar φ. We require the Lagrangian Without loss of generality, we work in the basis where Σ0 is a to be invariant under SU(2)L, which forces U to be a func- ∗ block diagonal matrix with Nf blocks of ε along the diagonal. tion of φ φ only, since φεφ is identically zero, and forbids Due to the spontaneous symmetry breaking in Eq. (3.5), renormalizable interactions between φ and the ψi. Goldstone’s theorem implies that there will be massless Gold- The complex SU(2)L-doublet field φ consists of 4 real de- stone bosons in the spectrum of low-lying, meson-like states. grees of freedom, and in fact Eq. (3.8) respects an SO(4) 2 Note that SU(2Nf )ψ has 4Nf − 1 generators, which we de- symmetry. It is useful to apply the isomorphism SO(4) = a a a 2 SU(2) × SU(2) note by Tij and Xij such that Tij for a = 1, 2, ··· , 2Nf +Nf L φ where the first factor is identified as the are the generators of the unbroken subgroup Sp(2Nf )ψ and gauged subgroup, and the second factor is a new global fla- a 2 vor symmetry which is the analog of custodial SU(2) in the Xij for a = 1, 2, ··· , 2Nf − Nf − 1 are the generators Standard Model. Thus the global flavor symmetry is enlarged of the broken symmetries. The U(1)ψ factor is also broken by Eq. (3.5). Thus we expect the spectrum to contain to U(2Nf )ψ × SU(2)φ. It is useful to recall that an SU(2)L- 2 a doublet field φ(x) can also be represented as ϕa(x) with [39] 2Nf − Nf − 1 massless Goldstone bosons, denoted by Π , 0 and one massive pseudo-Goldstone boson, denoted by η ; the ∗ ϕ1 = εφ and ϕ2 = −φ , (3.9) notation is chosen to draw an analogy with the pions and η0 0 meson of QCD. Since U(1)ψ is anomalous, the η mass is where ϕ transforms as 2 under each factor of SU(2)L × lifted by SU(2)L-instanton effects (see below). SU(2)φ. Using this equivalent representation of the SU(2)L- In the confined phase, the low-energy physics can be de- doublet scalar, we can write the Lagrangian as scribed with a non-linear sigma model, 1 X 2 2 2 L2 = L1 + Dµϕa − U(|ϕ| /2) , (3.10) f † µ 2 2 UV UV 2 L1 = Tr ∂µΣ ∂ Σ + κΛ f Re[det Σ] , (3.6) a=1,2 IR 4 W where f is the decay constant, κ is an O(1) dimension- where Dµϕa = ∂µϕa + iWµϕa, and we have used the short- 2 † † less number, and ΛW ∼ 4πf is the confinement scale. As hand notation |ϕ| ≡ ϕ1ϕ1 + ϕ2ϕ2. 5

From the low energy perspective, we have extended the SO(4) U(2Nf )ψ U(Nξ)ξ spectrum by additional low-lying states, corresponding to a SU(2)L SU(2)φ SU(2Nf )ψ U(1)ψ SU(Nξ)ξ U(1)ξ new real scalar ﬁeld Φ(x) and a set of new Weyl fermions ψ 2 1 2Nf qψ 1 0 Ψia(x) for i = 1, 2, ··· , 2Nf and a = 1, 2, which have the W 3 1 1 0 1 0 same symmetry properties as the following bilinears µ φ/ϕ 2 2 1 0 1 0 ∗ Φ ∼ φ φ (3.11a) ξ 1 1 1 0 Nξ qξ 2 Ψia ∼ ψiϕa , (3.11b) Σ 1 1 2Nf − Nf 2qψ 1 0 Φ 1 1 1 0 1 0 which are all singlets under SU(2)L. The composite scalar, Ψ 1 2 2Nf qψ 1 0 Φ, is a total singlet, while the composite fermions, Ψia, trans- ξ 1 1 1 0 Nξ qξ form as fundamentals under the SU(2Nf )ψ ﬂavor symmetry, carry charge q under the U(1) symmetry, and transform as ψ ψ Table I. This table summarizes the charge assignments for the se- a 2 under the SU(2) custodial symmetry as summarized in 2 φ quence of models in Section III. Note that the N- and (N − N)- TableI. Note that Φ is a singlet under the custodial SU(2), dimensional representations of SU(N) are the fundamental and anti- and there does not exist an analogous triplet state; only the symmetric representations, respectively. When applied to the WCSM custodial anti-symmetric combination is allowed. in SectionIV we take 2Nf = 12 and Nξ = 21. The low energy physics is described by

2Nf X X † µ transform in the fundamental representation of the U(Nξ)ξ L2 = L1 + Ψiaiσ¯ ∂µΨia (3.12) IR IR and the ψi, Wµ, and φ are singlets; see TableI. This sym- a=1,2 i=1 metry forbids masses such as µ ξ ξ and interactions such as 1 ij i j µ 2 2 λijφψiξj. Therefore the Lagrangian for the high-energy de- + ∂ Φ∂µΦ − ΛWf V (Φ/f) 2 grees of freedom is simply 2Nf X X ab † N − y0ΛWε ΨiaΣijΨjb + h.c. ξ X † µ i,j=1 a,b=1,2 = + ξ iσ¯ ∂ ξ . (3.13) L3 UV L2 UV i µ i 2Nf i=1 X X 0 ab † − (y0ΛW/f)ε ΨiaΣijΨjbΦ + h.c. , Moreover, these SU(2)L-singlet fermions do not feel the i,j=1 a,b=1,2 strong SU(2)L force and therefore do not participate in con- where f is the decay constant, ΛW ∼ 4πf is the confinement finement. Thus the Lagrangian for the low-energy degrees of scale, V is a potential for the composite scalar Φ, and y0 and freedom is also simply 0 y0 are Yukawa couplings. The 4π counting is performed us- Nξ ing naïve dimensional analysis [40–42], which implies that y0, X † µ 0 L3 = L2 + ξ iσ¯ ∂µξi . (3.14) y0, and the dimensionless coefficients in V are O(1) numbers. IR IR i We assume that the potential U in Eq. (3.8) is a small pertur- i=1 bation on V , such that the scalar’s mass is of order mΦ ∼ ΛW. These singlet fermions are entirely inert and decoupled. In the The potential V may also induce a nonzero vacuum expecta- next section we find that breaking the large global flavor sym- tion value for Φ, which is expected to be hΦi ∼ f, but since metry allows for Yukawa interactions, such as ∆L ∼ Ψξ and Φ is a singlet, this vacuum expectation value does not signal ΨξΦ. These interactions will have interesting consequences any spontaneous symmetry breaking. The y0 term, which is for the spectrum in the IR. a Yukawa interaction, respects the U(2Nf )ψ × SU(2)φ flavor symmetry, and when Σij acquires a nonzero vacuum expectation value (3.5), it induces a mass term for the Ψia. Together IV. THE WEAK-CONFINED STANDARD MODEL Ψia combine in pairs to form 2Nf Dirac fermions, which are all degenerate and have a mass m = y Λ . Ψ 0 W Let us now investigate weak isospin confinement in the WCSM. The UV particle content and symmetries of the WCSM are identical to the SM, shown in TableII. In par- C. Model 3: Add SU(2)L-singlet fermions ticular the SM symmetry group is

As a final iteration, we introduce a set of SU(2)L-singlet, SU(3)c × SU(2)L × U(1)Y gauge (4.15) left-chiral Weyl fermion fields, denoted as ξi(x) for i = × U(1) × U(1)3 , 1, 2, ··· ,Nξ. Note that Nξ may be either even or odd, since B+L anomaly B/3−Li global these fields do not contribute to the SU(2)L anomaly. As we build up to the SM, these fields will correspond to the where the U(1)B+L factor is anomalous under the SU(2)L × Nξ = 21 SU(2)L-singlet quark and lepton fields. U(1)Y gauge interactions. The key difference between the We require the theory to respect an extended global sym- WCSM and the SM has to do with the strength of the SU(2)L metry group, U(2Nf )ψ × SU(2)φ × U(Nξ)ξ where the ξi weak force, which is assumed to be larger in the WCSM, 6

gauge global 12 and Nξ = 21; speciﬁcally, the SU(2)L-doublet fermions transform as a ψi ∼ 12 of U(12)ψ, the SU(2)L-doublet Higgs SU(3)c SU(2)L U(1)Y U(1)B+L U(1)B/3−Lj transforms as a ϕa ∼ 2 of the “custodial” SU(2)φ, and the qi 3 2 1/6 1/3 1/9 SU(2) -singlet fermions transform as a ξ ∼ 21 of U(21) . U 3¯ 1 -2/3 -1/3 -1/9 L i ξ i Using the multiplets in Eq. (4.17), the WCSM Lagrangian Di 3¯ 1 1/3 -1/3 -1/9 0 (for gs = g = λYuk = 0) is written compactly as li 1 2 -1/2 1 -δij 12 21 Ei 1 1 1 -1 δij X X = ψ†iσ¯µD ψ + ξ†iσ¯µ∂ ξ (4.18) H 1 2 1/2 0 0 LWCSM UV i µ i i µ i i=1 i=1 gµ 8 1 0 0 0 1 µν X 1 µ † Wµ 1 3 0 0 0 − Tr W W + D ϕ D ϕ 2g2 µν 2 a µ a Bµ 1 1 0 0 0 eff a=1,2

1 2 2 1 4 Table II. This table shows the WCSM particle content and charge − µ |ϕ| − λ|ϕ| . 2 4 assignments. The index i ∈ {1, 2, 3} labels the three generations of fermions. There are four global charges, corresponding to U(1) B+L Here we have written the SU(2)L gauge coupling as geff > g, and U(1) for j = 1, 2, 3. All fermions are represented by B/3−Lj which distinguishes the WCSM from the SM. Additionally, left-chiral Weyl spinor fields in the two-component notation [43]. Dµψi = ∂µψi − iWµψi is the covariant derivative of the doublet fermions, Dµϕa = ∂µϕa +iWµϕa is the Higgs field’s covariant derivative, W is the SU(2) field strength tensor, µ2 g → g g, such that weak isospin is confined in the µν L eff is the Higgs mass parameter, λ is the Higgs self-coupling, and IR. Therefore, whereas the SM predicts that SU(2)L is Hig- 2 † † gsed by the vacuum expectation value of H, the WCSM pre- |ϕ| ≡ ϕ1ϕ1 +ϕ2ϕ2. We note that the gluon and hypercharge bosons are present but decoupled due to the assumption that dicts that the SU(2)L weak force is confined in the IR and the their gauge couplings are zero valued. SU(2)L-doublet fields (qi, li, and H) form composite states. This section contains the main results of our paper. To sim- By neglecting the strong, hypercharge, and Yukawa inter- plify the analysis, we will begin in Sec.IVA by setting to zero actions, we can write the Lagrangian as in Eq. (4.18), which 0 now bears a marked similarity to the models that we have the strong coupling gs = 0, the hypercharge coupling g = 0, already discussed in Sec. III for Nf = 6 and Nξ = 21. and the Yukawa couplings λYuk = 0. We reintroduce these couplings in Sec.IVB where they are treated as perturbations As before, we assume that the weak force confines, and the around an enhanced symmetry point. chiral symmetry is spontaneously broken. By applying what we learned in Sec. III, we can immediately infer the effects of SU(2)L-charge confinement in the WCSM. The SU(2)L- A. Turn off strong, hypercharge, and Yukawa couplings doublet fermions condense in pairs to form composite scalars, Σij ∼ ψiεψj; the doublet Higgs condenses with the doublet fermions to form composite fermions, Ψ ∼ ψ ϕ ; the Upon taking g = g0 = λ = 0, the symmetry group of ia i a s Yuk Higgs condenses in pairs to form another composite scalar, the SM Lagrangian is enlarged from Eq. (4.15) to P † † Φ ∼ a ϕaϕa ∼ H H; and the SU(2)L-singlet fermions, ξi, naturally do not confine but (most of them) acquire masses in- SU(2)L × U(1)ψ (4.16) gauge anomaly directly through weak-isospin confinement as outlined in Sec- × SU(12)ψ × SU(2)φ × U(21)ξ global , tionIVB. The symmetry transformation properties of these low-energy degrees of freedom are also shown in TableI. where the U(1)ψ factor is anomalous under the SU(2)L gauge By carrying over the results from Sec. III, the Lagrangian interaction. To make these symmetries manifest in the La- describing the low-energy degrees of freedom of the WCSM grangian, it is useful to collect the fields together into the fol- can be written as lowing multiplets 12 2 21 X X X R G B = Ψ† iσ¯µ∂ Ψ + ξ†iσ¯µ∂ ξ (4.19) ψ = {l1 , q1 , q1 , q1 , ··· , ···}, (4.17) LWCSM IR ia µ ia i µ i ϕ = {εH , −H∗}, i=1 a=1 i=1 12 2 R¯ G¯ B¯ R¯ G¯ B¯ X X ab † ξ = {E1,U1 ,U1 ,U1 ,D1 ,D1 ,D1 , ··· , ···} , − y0ΛWε ΨiaΣijΨjb + h.c. i,j=1 a,b=1 where ψi for i = 1, 2, ··· , 12 contains the SU(2)L-doublet 12 2 fermions, ϕa for a = 1, 2 contains the SU(2)L-doublet X X 0 ab † − (y0ΛW/f)ε ΨiaΣijΨjbΦ + h.c. Higgs, and ξi for i = 1, 2, ··· , 21 contains the SU(2)L- i,j=1 a,b=1 singlet fermions. The subscript 1 indicates first-generation 2 fields, the dots correspond to the second and third genera- f † µ 2 2 + Tr ∂µΣ ∂ Σ + κΛWf Re[det Σ] tions, the SU(2)L index is suppressed, and the SU(3)c indices 4 are {R, G, B} and {R,¯ G,¯ B¯}. The symmetry transformation 1 µ 2 2 + ∂ Φ∂µΦ − ΛWf V Φ/f . properties of these fields are shown in TableI where 2Nf = 2 7

0 Recall that y0, y0, and κ are O(1) dimensionless coefficients metric scalar condensate’s vacuum expectation value satisfies † and ΛW ∼ 4πf is the scale at which SU(2)L weak isospin Σ0Σ0 = 1. To be more explicit, we work in a basis where confines. This Lagrangian describes an effective field theory   of the low-energy degrees of freedom where we have inte-   0 1 0 0 A grated out a tower of resonances with masses mres ∼ ΛW, and −1 0 0 0 we do not show all the non-renormalizable operators that they Σ0 =  A  ,A =   , (4.22)    0 0 0 1 generate. The anomalous U(1)ψ part of Eq. (4.16) is broken A   by the determinant term, for κ 6= 0, which arises from non- 0 0 −1 0 perturbative instanton effects in the confined phase. As we have discussed already in Sec. III, confinement is where the three blocks correspond to the three generations of fermions. Recall the mapping from the scalar condensate to accompanied by the spontaneous breaking of the U(12)ψ = the SM quark and lepton doublets, Σij ∼ ψiεψj with ψi given SU(12)ψ × U(1)ψ global symmetry from Eq. (4.16). The scalar condensate field Σ(x) acquires a nonzero vacuum ex- by Eq. (4.17). Therefore the symmetry breaking pattern implies pectation value, Σ0 = hΣi, and in order to achieve the an- ticipated symmetry breaking pattern, SU(12)ψ/Sp(12)ψ, we R G B † hliεqi i= 6 0 and hqi εqi i= 6 0 , (4.23) should have Σ0Σ0 = 1 [33, 34, 44]. As SU(12)ψ has 143 generators and Sp(12)ψ has 78 generators, then the number for each of the three generations i ∈ {1, 2, 3}. Thus we see of broken generators is 65, which equals the number of mass- that the vacuum state carries a nonzero baryon number and a less Goldstone boson fields, Π (x). The vacuum expectation lepton number, and consequently the particles in Eq. (4.19) value Σ0 6= 0 also breaks the U(1)ψ ⊂ U(12)ψ, but since this will experience B- and L-violating interactions that allow symmetry is anomalous, the would-be Goldstone boson has its mass lifted by electroweak instantons, in analogy with the 3∆B = ∆L = ±1 and ∆B = ±2/3 . (4.24) η0 meson of QCD. Thus after confinement and spontaneous symmetry breaking, the symmetry group from Eq. (4.16) is The violation of baryon number is a requirement for any broken to model of baryogenesis, and its natural emergence here is a tantalizing hint for a connection between the WCSM and the Sp(12) × SU(2) × U(21) , (4.20) ψ φ ξ global cosmological excess of matter over antimatter; we discuss this a possibility further in Sec.V. where Sp(12)ψ acts (nonlinearly) on the 65 “pions,” Π . Due to spontaneous symmetry breaking, Σ0 6= 0, the Yukawa interaction, ∼ ΨΣ†Ψ in Eq. (4.19), becomes a Dirac B. Turn on strong, hypercharge, and Yukawa couplings mass for the composite fermions. In particular, the 24 composite fermions, Ψia, form 12 degenerate Dirac fermions with Let us finally consider SU(2)L confinement with the Stan- mass mΨ ∼ y0ΛW, on the order of the confinement scale. dard Model particle content at the measured values for the Meanwhile the 21 singlet fermions, ξi, remain massless and noninteracting to all orders, since the relevant operators are strong, hypercharge, and Yukawa couplings. Since these forbidden by the residual global symmetry (4.20). nonzero couplings explicitly break the large global flavor The composite scalar Φ has a potential V , which induces symmetry, we will find that the spectrum of pseudo-Goldstone its mass and self-interaction. An analytical calculation of V pions is lifted. Moreover, because the condensates (4.23) is hindered by the strongly-coupled nature of the theory, but spontaneously break color and hypercharge, we will see that dimensional analysis arguments suggest that Φ will acquire a some of the gluons and hypercharge boson acquire mass from nonzero vacuum expectation value, hΦi ∼ f. Since Φ is a the Higgs mechanism by “eating” the would-be Goldstone singlet under the residual symmetry (4.20) its vacuum expec- bosons. The reader may be familiar the style of analysis that tation value does not signal any symmetry breaking. we employ in this section from studies of Little Higgs theo- 0 ries [45]. In summary, the spectrum of the WCSM (with gs = g = Comparison with QCD. Before beginning the determina- λYuk = 0) in the confined phase is tion of the effect of these explicit chiral symmetry breaking mΠ = 0 65 Goldstone bosons (4.21a) terms in the WCSM, we contrast the symmetry breaking to that in confining SM QCD. The quark masses and electric cou- mη0 ∼ ΛW 1 pseudo-Goldstone (4.21b) plings are both sources of explicit chiral symmetry breaking in m ∼ Λ 12 composite Dirac fermions (4.21c) Ψ W the SM. Since the quark masses only involve confining states, mΦ ∼ ΛW 1 composite scalar (4.21d) they contribute to chiral symmetry breaking terms at tree-level mξ = 0 21 elementary Weyl fermions (4.21e) in the Lagrangian. On the other hand, the electric coupling, which involves the non-confining photon, only contributes a mres ∼ ΛW heavier resonances . (4.21f) small correction at loop level. Furthermore, the same com- The heavy resonances are not included in Eq. (4.19). bination of the non-confining SU(2)L × U(1)Y gauge group Finally, we investigate the vacuum structure of this the- that would get broken by confinement of QCD is already bro- ory more carefully. We have seen that the symmetry break- ken at a much higher scale by the Higgs mechanism, so all of ing pattern SU(12)ψ/Sp(12)ψ is obtained when the antisym- the pions remain in the spectrum rather than being eaten. On 8 the other hand, there are no chiral symmetry breaking opera- As such, the SU(3)c and U(1)Y generators are written as tors involving only the weak-charged fermions in the WCSM, so all chiral symmetry breaking for the pions occurs at loop La = diag(0, λa, 0, λa, 0, λa) for a = 1, ··· , 8 (4.27) level stemming from the SU(3)c × U(1)Y gauge and Yukawa 1 1 1 1 1 1 1 1 1 1 1 1 Y = diag − , , , , − , , , , − , , , , couplings. There is no higher scale breaking of the weakly- 2 6 6 6 2 6 6 6 2 6 6 6 coupled SU(3)c × U(1)Y gauge group, so it is broken only by the confinement of SU(2)L, leading to both some eaten respectively. The generators are orthonormal such that Goldstone modes and non-negligible corrections due to the Tr[LaLb] = 3δab/2, Tr[LaY ] = 0, and Tr[YY ] = 1. 0 massive gauge bosons that are not completely decoupled at Explicit global symmetry breaking. By setting gs = g = the confinement scale. λYuk = 0 previously, we saw that the SM global symme- Treat small couplings perturbatively. A key assumption try of Eq. (4.15) was enhanced to Eq. (4.16). Since the in the following analysis is that the nonzero strong, hyper- nonzero Yukawa couplings are typically much smaller than charge, and Yukawa couplings can be treated as perturbations the SU(3)c × U(1)Y gauge couplings, it is also useful to con- 0 on the preceding analysis of Sec.IVA. This is a reasonable sider the intermediate case in which gs, g 6= 0 and λYuk = 0. expectation as most of these couplings are O(1), with the For vanishing Yukawa couplings, the global symmetry group largest being the top Yukawa coupling and the strong cou- of the theory is pling. Specifically, we assume that the scalar condensate, Σij, still acquires a vacuum expectation value Σ0 that leads to the SU(3)q × SU(3)U × SU(3)D × SU(3)l (4.28) SU(12)ψ/Sp(12)ψ symmetry breaking pattern as in Eq. (3.5). × SU(3)E × U(1)U × U(1)D × U(1)E × U(1)H , Introduce gauge interactions. We implement the SU(3)c × U(1)Y gauge interactions by promoting partial where each factor acts nontrivially on only one field, indi- derivatives in the WCSM’s IR Lagrangian (4.19) to covariant cated by the subscript. In other words the nonzero gauge cou- derivatives: DµΨia, Dµξi, DµΣij, and DµΦ. Since the plings explicitly break the global symmetry from Eq. (4.16) to scalar Φ is a singlet under SU(3)c × U(1)Y , we have simply Eq. (4.28). This explicit breaking lifts several of the would- DµΦ = ∂µΦ. Let us look more closely at DµΣij, as it be Goldstone bosons, as we demonstrate below. The choice reveals how the gauge boson masses arise through the Higgs of the definition of the U(1) factors here is not unique; any mechanism. The covariant derivative of the scalar field linear combinations of these with U(1)Y will also be a sym- Σij(x), is written as metry. We choose these combinations for later convenience. Two combinations of the residual U(1) global symmetry factors are anomalous under SU(3)c×U(1)Y , but this anomalous 8 X explicit breaking is extremely weak and we neglect it in this D Σ = ∂ Σ − ig Ga LaΣ + ΣLaT (4.25) µ µ s µ work. a=1 0 Spontaneous symmetry breaking. The nonzero vacuum − ig Bµ Y Σ + ΣY , expectation value, hΣi = Σ0 in Eq. (4.22), indicates that the gauge symmetry is broken as follows: a where gs is the SU(3)c coupling, Gµ(x) for a = 1, ··· , 8 are a 0 the gluon fields, L are the generators of SU(3)c, g is the { SU(3)c × U(1)Y }gauge → {SU(2)c × U(1)Q}gauge . U(1)Y coupling, Bµ(x) is the hypercharge boson field, and (4.29) Y is the generator of U(1)Y . It is convenient to define the Specifically, SU(3)c is broken to an SU(2)c subgroup, and a SU(3)c generators in a slightly unconventional way using the linear combination of SU(3)c and U(1)Y is broken to a U(1)Q following basis subgroup. The generators of the SU(2)c are simply the Pauli matrices, τ a = σa/2 for a = 1, 2, 3, and the generator of     U(1)Q is 0 0 0 0 0 0 1 1 λ1 = 0 0 1 , λ2 = 0 0 −i , 1 2   2   Q = √ L8 − Y (4.30) 0 1 0 0 i 0 3     1 1 1 1 1 1 0 0 0 0 1 0 = diag , − , 0, 0, , − , 0, 0, , − , 0, 0 . 1 1 2 2 2 2 2 2 λ3 = 0 1 0  , λ4 = 1 0 0 2   2   0 0 −1 0 0 0 Note that Q is not SM electromagnetism, but we use this nota-     (4.26) tion to draw a parallel between Eq. (4.29) and SM electroweak 0 −i 0 0 0 1 1 1 symmetry breaking. By counting the broken symmetry gener- λ5 = i 0 0 , λ6 = 0 0 0 , 2   2   ators, we identify the gauge bosons that acquire mass through 0 0 0 1 0 0 the Higgs mechanism. This will be discussed in detail shortly,     but the result is that five gauge bosons acquire a mass and 0 0 −i −2 0 0 1 1 “eat” five of the would-be pions. λ7 = 0 0 0  , λ8 = √  0 1 0 . 2   2 3   Nonzero Σ0 also signals spontaneous breaking of the global i 0 0 0 0 1 symmetry acting on the SU(2)L-charged fermions. In the 9

regime where the Yukawa couplings are neglected, we observe States SU(2)c U(1)Q U(1)r that the global symmetry of Eq. (4.28) is broken to li 6 1 1/2 0

qSi 6 1 -1/2 0 SO(3)g × SU(3)U × SU(3)D × SU(3)E (4.31) qDi 12 2 0 0 × U(1)U × U(1)D × U(1)E × U(1)H . Ei 3 1 -1 -1 The SO(3)g factor corresponds to a real rotation among the USi 3 1 1 1 three generations. There are thus 13 spontaneously broken UDi 6 2 1/2 1 global symmetry generators, and correspondingly we find DSi 3 1 0 -1 13 massless Goldstone bosons remain after inclusion of the DDi 6 2 -1/2 -1 SU(3) × U(1) gauge interactions, while the remaining 47 c Y ϕ± 4 1 ±1/2 ±1 pions are lifted at this level. 1,2,3 Charges under SU(2) × U(1) . G 6 3 0 0 c Q Let us take a brief aside 0 to study how the residual gauge symmetry (4.29) acts on the A 2 1 0 0 W ±0 2 ±1/2 various fields. For the SU(2)L-doublet fermions, we find that 12 0 0 li ∼ 1 for i = 1, 2, 3 transforms as a singlet under SU(2)c, Z 3 1 0 0 while qR ∼ 1 also transforms as a singlet and (qG, qB) ∼ 2 0 0c i i i Ψ1, Ψ1 6 1 0 ±1 transforms as a doublet. To emphasize this distinction be- ± Ψ1 6 1 ±1 ±1 tween the SU(2)c-singlet and doublet quark fields, we intro- ± Ψ2 12 2 ±1/2 ±1 duce the following notation: 0 Ξ1 3 1 0 -1 R G B ± qSi = qi and qDi = (qi , qi ), (4.32) Ξ1 6 1 ±1 1 ± Ξ2 12 2 ±1/2 1 where i = 1, 2, 3 labels the generation. Additionally, ± Π1 6 1 ±1 0 Eq. (4.30) reveals that li and qSi carry charges +1/2 and 0 Π1 13 1 0 0 −1/2, respectively, under U(1)Q, while qDi is not charged Π± 32 2 ±1/2 0 under U(1)Q. A similar decomposition is also possible for the 2 0 SU(2)L-singlet quarks, Ui and Di, while the SU(2)L-singlet Π3 9 3 0 0 0 leptons, Ei, are singlets under the SU(2)c and carry charge η 1 1 0 0 −1 under the U(1)Q. Finally, the Higgs doublet, as repre- Φ 1 1 0 0 + sented by ϕa for a = 1, 2, can also be denoted as ϕ = ϕ2 − and ϕ = ϕ1, where the superscript indicates the sign of the Table III. This table summarizes how the WCSM particle content U(1)Q charge. These charge assignments are summarized in transforms under the SU(2)c ×U(1)Q subgroup of SU(3)c ×U(1)Y Table III. and the global U(1)r. Particles above the double bar correspond with Composite scalars under SU(2) × U(1) . We have seen states in TableII, and those below the bar correspond to composite c Q 1 that Σ (x) represents 66 pseudoscalar fields: η0(x) and particles. The “States” column counts for each Weyl spinor degree ij of freedom and 1 for each real boson. Πa(x) for a = 1, ··· , 65. To clarify how these scalar fields transform under SU(2)c × U(1)Q, it useful to map them to fermion bilinears as follows: Composite fermions under SU(2)c × U(1)Q. The composite fermion fields, Ψ (x), can also be decomposed into ir- Π+ ∼ lεl − q† εq† (4.33a) ia 1 S S reducible representations of the residual gauge symmetry. Us- − † † Π1 ∼ qSεqS − l εl (4.33b) ing the same notation as above, the composite fermions (3.11) 0 † † † † correspond to Π1 ∼ lεqS − qSεl , qDεεcqD − qDεεcqD (4.33c) + † † Ψ0 ∼ l ϕ− (4.34a) Π2 ∼ lεqD − qSεqD (4.33d) 1 − † † + + Ψ1 ∼ l ϕ (4.34b) Π2 ∼ qSεqD − l εqD (4.33e) − − 0 a † a † Ψ1 ∼ qS ϕ (4.34c) Π3 ∼ qDεσ qD − qDεσ qD, (4.33f) 0c + Ψ ∼ qS ϕ (4.34d) 2 a 1 where εc = iσ and σ are the Pauli matrices with SU(2)c − − Ψ ∼ qD ϕ (4.34e) indices, and there is an implicit contraction over SU(2)L in- 2 + + dices such that each Π is an SU(2)L singlet. Note that these Ψ2 ∼ qD ϕ , (4.34f) Π’s correspond to a total of 66 states, which includes the η0, the 5 Goldstone bosons eaten by the broken gauge fields and the 60 would-be Goldstone bosons.6 ± anti-symmetric for each of ± which results in six Π1 states (equivalently Ng(Ng − 1) where Ng is the number of generations). Likewise, for the 0 Π3 pion there are three combinations of generational indices each with 6 We provide a few examples of counting these degrees of freedom. For three possible SU(2)c states which result in nine (3Ng(Ng − 1)/2) pion ± 0 the Π1 there are three possible combinations of i and j as they must be states of Π3. 10 and their charge assignments are shown in Table III. Gauge boson spectrum. The scalar field Σij(x) acquires Massless gauge boson interactions. We write the interac- a nonzero vacuum expectation value, given by Eq. (4.22), and tions of the four massless force carriers, corresponding to the the Higgs mechanism splits the spectrum of gauge bosons. unbroken symmetry group SU(2)c × U(1)Q, as follows. The The mass terms can be derived from Eqs. (4.35) and (4.39) by covariant derivative from Eq. (4.25) contains using Σ = Σ0 from Eq. (4.22). The gauge boson spectrum is 3 summarized as follows X a a aT DµΣ ⊃ −igs Gµ L Σ + ΣL (4.35) mG1,2,3 = 0 (4.43a) a=1 m 0 = 0 − ie A0 QΣ + ΣQ . A (4.43b) Q µ p mW 0± = 3/2 gsf (4.43c) For the SU(2)c interactions, gs is the SU(3)c gauge coupling 1,2,3 p p 2 02 that also determines the SU(2)c coupling strength, Gµ are mZ0 = 2/3 3gs + g f . (4.43d) 1,2,3 the force carriers of SU(2)c, and L from Eq. (4.27) are 1,2,3 0 the generators of SU(2)c. For the U(1)Q interaction The four massless vector fields, Gµ and Aµ, are the force SU(2) × U(1) 0 0 carriers of c Q, while the two complex vectors eQ = g cos θQ ≈ g , (4.36) 0± 0 Wµ and the single vector Zµ acquire mass through the Higgs is the gauge coupling and the mixing angle is given by mechanism. Symmetry breaking including Yukawa interactions. Fi- 0 g nally we discuss the ultimate spectrum of pions. Recall that sin θQ = (4.37) p 2 02 3gs + g for vanishing strong, hypercharge, and Yukawa couplings, the a √ scalar Σij(x) encodes 65 massless Goldstone bosons Π (x), 0 which is θQ ≈ g / 3gs 1 in the regime of interest where which we called “pions,” and one massive pseudoscalar η0(x), 0 g gs. The massless force carrier is which is associated with an anomalous U(1)ψ and acquires A0 = cos θ G8 + sin θ B , (4.38) mass from SU(2)L instantons. Taking the strong, hyper- µ Q µ Q µ charge, and Yukawa couplings to be nonzero breaks the large 3 and Q from Eq. (4.30) is the generator of U(1)Q. global symmetry of Eq. (4.16) down to the U(1) of B/3−Li Massive gauge boson interactions. The five remaining Eq. (4.15). gauge bosons correspond to the broken symmetry generators Thus we expect that all but at most three of the pions will 0 of Eq. (4.29). These states can be arranged into a pair of have their masses lifted by the effect of nonzero gs, g , and 0± complex vector fields Wµ (x), and a single real vector field λYuk. Since the gauge groups SU(3)c and U(1)Y are sub- 0 0± 0 Zµ(x). We use the notation W and Z to highlight the anal- groups of SU(12)ψ × SU(21)ξ, the gauge generators are spu- ogy with the Standard Model weak gauge bosons, convention- rions of SU(12)ψ transforming as adjoints. The Yukawas are ally denoted by W and Z. Their interactions are written as spurions transforming as anti-fundamentals of SU(12)ψ, as well as fundamentals under SU(12) as summarized in Ta- gs X X 0i± i± i∓ ξ DµΣ ⊃ −i√ Wµ (L Σ + ΣL ) (4.39) bleIV. 2 ± i=1,2 The global symmetry is also spontaneously broken, but e Q 0 there is an unbroken U(1)r symmetry acting only on the weak − i Zµ JΣ + ΣJ , sQcQ singlet fermions and Higgs in the UV or, correspondingly, where we define the new notation in the remainder of this only on the fermion states in the IR. The charge can be written as the following combination of U(1) charges paragraph. The SU(3)c factor has five broken genera- 4,5,6,7 tors, which are L , and the corresponding gauge fields, 4,5,6,7 X B Gµ (x), pair up to form two complex, massive vector r = −2Y + − Li = −2Y + B − L . (4.44) 0± 3 fields, Wµ (x) corresponding to i L± = (L4 ± iL5,L6 ± iL7) . (4.40) This symmetry is an exact global symmetry of the Lagrangian 0± and is not anomalous under the surviving gauge symmetries The interaction strength of the W vectors is inherited from in the IR. It has important consequences for the fermion spec- the SU(3)c gauge coupling, gs. The one final broken genera- trum and baryogenesis. The charges of the spectrum under it tor is are shown in Table III. There are thus two broken global sym- 1 metry generators and, correspondingly, two massless pions in J = √ L8 − s2 Q, (4.41) 3 Q the WCSM. The 11 other pions that were massless after inclusion of the SU(3) × U(1) gauge interactions (see below s = sin θ c = cos θ c Y where Q Q and Q Q, and the corresponding Eq. (4.31)) get lifted, such that a total of 58 of the 60 physical Z0 (x) massive vector field, µ , can be written as pions are massive. Z0 = − sin θ G8 + cos θ B . (4.42) Gauge corrections to pion masses. The strong and hyper- µ Q µ Q µ 0 charge gauge interactions (for gs 6= 0 and g 6= 0) induce 0± The W transform as a charged doublet under SU(2)c × radiative corrections that split the pion mass spectrum. Work- 0 U(1)Q while Z is a singlet; see Table III. ing in the mass basis of the vector bosons, where the leading 11

SO(4) U(12) U(21) 2 2 ψ ξ Pion # Squared Mass ×16π /ΛW SU(2)L SU(2)φ SU(12)ψ U(1)ψ SU(21)ξ U(1)ξ 3 −4C g2 Π0 G s T a,Y 1 1 143 1 0 3 2 4 4 0 6 −4CGgs + 2CYuk|Vtb| yt λu, λd, λe 1 2 12 -qψ 21 -qξ − 3 C g2 − 1 C e2 + 1 C g2 + C |V |4y4 8 2 G s 2 A Q 2 Z s Yuk tb t −pC2 g4 + C2 |V |8y8 Table IV. In the spurion analysis of SectionIVB, this table shows the W s Yuk tb t ± 3 2 1 2 1 2 4 4 a Π − CGg − CAe + CZ g + CYuk|Vtb| y charge assignments of the SU(3)c generators T , the U(1)Y gener- 2 8 2 s 2 Q 2 s t p 2 4 2 8 8 ator Y , and the Yukawa couplings λu, λd, and λe. + CW gs + CYuk|Vtb| yt 3 2 1 2 1 2 2 12 − 2 CGgs − 2 CAeQ + 2 CZ gs + CW gs 3 2 1 2 1 2 2 4 − 2 CGgs − 2 CAeQ + 2 CZ gs − CW gs contribution for each gauge boson eigenstate sums up with- ± 2 2 2 Π1 6 −2CAeQ − 3 CZ gs out mixing, the relevant mass-correction operators are written 2 4 2 2CYuk|Vtb| yt as [46] 2 2 2 4 2CYuk|Vtb| yt yτ 2 4 4 2 2 gs X aT † a 1 2CYuk|Vcs| yc 0 ∆L = CGΛWf 2 Tr[L Σ L Σ] (4.45) 1 2 2 2 16π Π1 2 − CYukRe[Vts] yt yτ a=1,2,3 2 1 −2C Re[V ]2y2y2 2 Yuk ts t τ 2 2 eQ † Re[Vcs]Re[Vtd]−Re[Vcd]Re[Vts] 2 2 1 6CYukRe[Vcs]Re[Vtd] Re[V ]2 yc yµ + CAΛWf 2 Tr[QΣ QΣ] ts 16π 2 0 g2/2 X X + C Λ2 f 2 s Tr[Li±Σ†Li±Σ] W W 16π2 ± i=1,2 Table V. Approximate mass spectrum for the pions. To simplify the expressions we assume that all the C’s are O(1) and show only the e2 /s2 c2 + C Λ2 f 2 Q Q Q Tr[JΣ†JΣ] , leading contributions. The second column indicates the multiplicity Z W 16π2 of each state, although some degeneracies are split by higher order contributions. Five would-be pions, which are “eaten” by the massive 0 a 0± 0 where eQ ≈ g was given by Eq. (4.36), L was given by W and Z bosons, are not shown in the table. Eq. (4.27), Q was given by Eq. (4.30), L± was given by

Eq. (4.40), J was given by Eq. (4.41), and ΛW ≈ 4πf. Each term corresponds to a one-loop correction to the pion’s self- states, but choosing CG < 0, CA < 0, 1 . CZ < 0, 1,2,3 energy; the first term arises from a G loop, the second |CW | . 1, and CYuk > 0 leads to only four slightly tachy- 0 0± 0 from an A loop, the third from a W loop, and the fourth onic Π1 with masses suppressed by the small Yukawas yτ and 0 term arises from a Z loop. The dimensionless operator co- yc. This indicates that the assumed symmetry breaking pat- efficients – CG, CA, CW , and CZ – encode the details of tern, SU(2Nf )/Sp(2Nf ), is radiatively stable modulo small the loop calculation, and they are expected to be O(1) num- corrections. bers [47, 48] with CA = CG since these two operators both Fermion mass spectrum As we have discussed in Sec- correspond to massless vectors in the loop. tionIVA, the composite fermions Ψia get a mass of order Yukawa corrections to pion masses. Yukawa interactions ΛW from the confinement process. However, once we include between the pions and fermions (for λYuk 6= 0) also con- the Yukawa interactions, the composite fermions acquire a tribute to the pion’s self-energy at one-loop order. The leading small mixing with the SU(2)L-singlet fermions, ξj, and the non-trivial contributions due to the Yukawas come in at fourth spectrum is split. To see this explicitly, we construct the La- power, taking the form grangian for the fermions. In the UV the Yukawa interactions take the form 2 2 2 ΛWf X X † † ∗ T 12 21 2 ∆L = CYuk Tr[(λkaλ )Σ (λk0bλk0b)Σ] , X X X 16π2 ka ∆ = (λ + λ + λ ) ϕ ψ ξ . k,k0 a,b=1 LYuk UV u d e ija a i j (4.46) i=1 j=1 a=1 (4.47) where λu is the up-type quark Yukawa matrix, λd is the down- type quark Yukawa matrix, λe is the charged lepton Yukawa matrix and the indices k, k0 run over u, d, e. These are 12×12 In the IR the SU(2)L-doublet fermions are confined with the complex matrices in the basis of Eq. (4.17). Higgs, and this Lagrangian is mapped to Pion mass spectrum. To calculate the pion mass spec- 12 21 2 a X X X trum, we parametrize the pions Π in terms of Σij using ∆ = C f (λ + λ + λ ) Ψ ξ , LYuk IR λ u d e ija ia j Eq. (3.7), evaluate Eqs. (4.45) and (4.46), and read off the i=1 j=1 a=1 2 a b quadratic terms, ∆L = −(1/2)(MΠ)abΠ Π . The leading (4.48) masses of the pions are shown in TableV. Including all of the mass-correction operators, the pion spectrum contains only where Cλ is an O(1) constant. This operator allows the com- two massless states, corresponding to the Goldstone bosons posite Ψia to mix with the elementary ξj, and the resultant associated with U(1) × U(1)3 /U(1) × U(1) . If Y B/3−Li Q r mass eigenstates are mostly-composite fermions with masses all C’s are nonzero then one cannot avoid some tachyonic ∼ ΛW and mostly-elementary fermions, which we denote 12

by Ξ, with masses ΛW. The mostly-elementary fermion SU(2)c ×U(1)Q gauge interactions. The gauge interac- mass eigenstates correspond to the following elementary con- tions explicitly break the global symmetry SU(12)ψ × 5 4 stituents: SU(2)φ × U(21)ξ/SU(3) × U(1) , which gets spontaneously broken in the conﬁned phase to a global sym- 0 3 Ξ1 ∼ DS (4.49a) metry SO(3)g × U(3) × U(1), leaving 13 massless + Goldstone bosons after their inclusion; Ξ1 ∼ US (4.49b) − • 11 pseudo-Goldstone bosons of the spontaneous break- Ξ1 ∼ E (4.49c) + ing of SU(12)ψ/Sp(12)ψ get lifted by the Yukawa in- Ξ2 ∼ UD (4.49d) teractions only. The Yukawas explicitly break SU(3)5× − U(1)4/U(1)3 , which gets spontaneously broken Ξ2 ∼ DD . (4.49e) B/3−Li to U(1)r. There are only 2 massless Goldstones bosons The masses for these fermions arise only in the presence of after their inclusion; non-zero Yukawa couplings. The charged fermions get Dirac • 9 mostly-elementary Dirac fermions that get lifted by masses via a seesaw mechanism as follows the Yukawa couplings. They correspond mostly to the 2 2 f remaining weak-singlet fermions; m ± ≈ C λeλu (4.50a) Ξ1 λ y0ΛW • 5 massive gauge bosons of the broken SU(2)c × U(1)Q 2 gauge group. These eat 5 of the would-be pions; 2 f m ± ≈ C λuλd . (4.50b) 0 Ξ2 λ y0ΛW • 1 η pion corresponding to the anomalous U(1)B+L global symmetry and acquires a mass of order ΛW; Note that the CKM matrix should appear in this mass matrix, • 1 Higgs-Higgs composite scalar that is not protected by but can be rotated to generate an off-diagonal W 0 coupling, any symmetry and gets a mass of order Λ ; as in the SM. The mass of each of generation of composite W fermions is then determined by the product diagonal Yukawa • 12 Dirac fermions corresponding mostly to fermion- matrices of that generation as in (4.50). The mostly composite Higgs composite states and they receive Dirac masses heavy fermions also get some very small correction due to the of order ΛW; 0 Yukawa couplings. The Ξ1 fermions are protected from get- • Additional scalar, radial, and spin excitations that are ting a mass by the U(1)r symmetry and remain exactly mass- not included in the non-linear sigma model. less Weyl fermions. The light, mostly-elementary fermions mix with the heavy, mostly-composite fermions with a mixing angle given by V. POTENTIAL COSMOLOGICAL IMPLICATIONS

Cλλu,d,ef θLH ∼ , (4.51) How would we know if the Universe passed through a phase y0ΛW of confined SU(2)L? In this section we briefly discuss the and the mass eigenstates are partially composite [49]. Since possible implications for various cosmological relics. the Yukawas are typically λ 1, this is a small mixing. Cosmological phase transition. After cosmological infla- tion evacuates the Universe of matter, we suppose that reheat- ing produces a plasma of Standard Model particles at temper- C. Summary of the WCSM atures above the scale of weak confinement. As the Universe grows older and expands, the cosmological plasma cools to We have studied the symmetries and the lightest states of the confinement scale, and the confinement of the weak force the WCSM. After all interactions and spontaneous symmme- is accomplished through a cosmological phase transition. This try breaking, the symmetry group of the WCSM is transition is expected to be first order, meaning that bubbles of the confined phase nucleate in a background of the uncon- fined phase, and the transition completes as the bubbles grow SU(2)c × U(1)Q × U(1)r . (4.52) gauge global and merge, filling all of space. The first order nature of the The lightest states in the spectrum include: weak-confining phase transition follows from a more general argument by Pisarski and Wilczek (1983) [50], who consid- • 2 massless Goldstone bosons corresponding to 2 com- ered a general Yang-Mills gauge theory with Nf flavors of binations of spontaneous B/3 − Li breaking; massless vector-like fermions, and showed that it will con- • 3 mostly-elementary massless Weyl fermions corre- fine through a first order phase transition if Nf ≥ 3. As we have discussed already in Sec.IV, the Standard Model cor- sponding to the SU(2)c × U(1)Q singlets charged un- responds to 2Nf = 12 massless flavors of SU(2)L doublets, der U(1)r. They are mostly SU(2)L-singlet down-type fermions; and therefore we expect the weak-confining phase transition to be first order. A first order cosmological phase transition • 4 massless gauge bosons of SU(2)c × U(1)Q. typically provides the right environment for the creation of • 47 pseudo-Goldstone bosons of the spontaneous break- cosmological relics, such as topological defects, gravitational ing of SU(12)ψ/Sp(12)ψ are dominantly lifted by the wave radiation, and the matter-antimatter asymmetry. 13

Topological defects. When spontaneous symmetry break- asymmetry in r cannot be generated by the weak sphaleron. ing is accomplished through a cosmological phase transition, All other global charges are violated by interactions in the La- causality arguments require the symmetry-breaking scalar grangian or by the quark-lepton condensates. Since an asym- field to be inhomogeneous on the Hubble scale, and this in- metry in a non-conserved global charge can be washed out homogeneity may allow the field to form topological defects, in the confined phase, this observation suggests that WCSM such as cosmic strings and domain walls [51]. In the context does not have all the necessary ingredients for baryogenesis of QCD’s chiral phase transition, where the SU(3)A × U(1)A during the confining phase transition. symmetry is spontaneously broken by the chiral quark con- Nevertheless, one can easily imagine extending the WCSM densate, these defects have been called pion and eta-prime by heavy Majorana neutrinos so as to accommodate the ob- strings [52]. As the nonzero quark masses break the SU(3)A× served neutrino masses and mixings. The associated lepton- U(1)A symmetry explicitly, this may prevent the QCD defects number violation could provide a means of generating r- from forming at all, but if they do form, then they are expected charge during the confining phase transition, which is later to be unstable and decay soon after the QCD phase transi- distributed into baryon number when the system de-confines tion [53]. Similarly we expect that spontaneous symmetry and enters the standard low-temperature Higgs phase. breaking in the WCSM will generate a string-wall defect net- Alternatively, it may be possible to accomplish baryogene- work with a rich structure. For instance, spontaneously break- sis during the deconfining phase transition. For this scenario ing the global symmetry U(1)3 in Eq. (4.15) is expected B/3−Li to work, the system must pass directly from the confined phase to generate multiple networks of stable cosmic strings, which into the Higgs phase where the sphalerons are inactive; if the could survive until SU(2)L is de-confined. While the defect system passes instead through the Coulomb phase then the network is present, it may play a role in primordial magneto- sphalerons will come back into equilibrium and wash out any genesis [54] or baryogenesis [55]. baryon asymmetry. This constraint has the interesting impli- Gravitational wave radiation. A first order cosmological cation that the system must remain in the confined phase even phase transition produces a stochastic background of gravita- at a time where the cosmological plasma temperature is quite tional waves [56] due to the collisions of bubbles and their low, . 100 GeV. If the weak-confined phase results from the interactions with the plasma. For a weakly-coupled theory, nontrivial dynamics of a modulus field, such as we discussed it is straightforward to calculate the spectrum of this gravita- in the model of Eq. (2.1), then this constraint implies a low tional wave radiation [57], and gravitational waves have been mass for the modulus field. studied in strongly-coupled theories as well [58]. Since grav- ity is extremely weakly interacting, the gravitational wave radiation is expected to free stream to us today where it VI. COMPARISON WITH THE ABBOTT-FARHI MODEL might be detected. For instance, gravitational waves pro- duced at an electroweak-scale phase transition could be de- We explored the idea that the SU(2)L weak force may tected by space-based gravitational wave interferometer ex- have been confined in the early Universe, only to become periments like LISA [59]. Measuring the appropriate spec- weakly coupled and Higgsed at later times. Confinement of trum of gravitational wave radiation would provide one of the the weak force was first studied in the pioneering work by most direct probes of weak confinement in the early Universe. Abbot and Farhi [4,5], which developed into a theory [6–8] The matter-antimatter asymmetry. In general a first-order that we will call the Abbott-Farhi Model (AFM). The AFM cosmological phase transition also provides the right environ- describes the SM particle spectrum as composite states that ment to explain the cosmological excess of matter over anti- arise from the√ confinement of the SU(2)L weak force at a matter through the physics of baryogenesis [60]. At the weak- scale ΛW = GF ∼ 300 GeV. These ideas were motivated confining phase transition in particular, weak sphalerons are in part by the complementarity between a Yang-Mills gauge active in front of the bubble wall where they mediate baryon- theory in the Higgs phase and the confined phase [62–65] (see number-violating reactions, and they become suppressed be- also Refs. [66, 67]). While the AFM was able to reproduce the hind the bubble wall where SU(2)L is confined and the observed mass spectrum, interactions, and even weak boson weak-magnetic fields are screened. This observation sug- phenomenology, it was eventually found to be in tension with gests that it may be possible to generate a baryon asymmetry electroweak precision measurements at the Z-pole [11, 68], at the confined-phase bubble walls through dynamics similar leaving the (Higgs-phase) SM as the preferred description of to electroweak baryogenesis [61] or spontaneous baryogene- weak-scale phenomenology. sis [15, 16, 20]. A key assumption of the AFM, and the essential distinc- However, although sphalerons may be suppressed in the tion with our work here, has to do with the assumed pat- confined phase, this does not imply that B + L is effectively tern of symmetry breaking. The AFM assumes that the chiral conserved, as we find in the SM. As discussed in Sec.IV, symmetry remains unbroken after confinement (in the regime baryon number is not a good quantum number in the con- where the hypercharge, QCD, and Yukawa interactions can be fined phase where the quarks and leptons condense and ac- neglected). In our notation, this assumption corresponds to quire a vacuum expectation value, spontaneously breaking hΣi = 0. If the chiral symmetry is unbroken, then the AFM’s U(1) × U(1)3 . Instead there is only one conserved B+L B/3−Li composite fermions, corresponding to the familiar SM parti- global charge in the confined phase, r = −2Y + (B − L) cle content, must be massless. In this way, the AFM’s fermion from Eq. (4.44), which is not anomalous, and consequently an masses are induced by the nonzero hypercharge, QCD, and 14

Yukawa interactions, which explains why these particles are Model and new physics may allow the WCSM phase to be much lighter than the confinement scale ΛW ∼ 300 GeV. By realized in the early Universe at a time when the cosmolog- contrast, we are motivated by recent numerical lattice studies ical plasma temperature was sufficiently high, T ∼ ΛW. By to assume an SU(2Nf )/Sp(2Nf ) pattern of chiral symmetry assuming a pattern of chiral symmetry breaking that it moti- breaking (3.5), which leads to a massive spectrum of compos- vated by numerical lattice studies, we calculate the spectrum ite fermions with mΨ ∼ ΛW. of low-lying composite particles, discuss their interactions, Which of these two symmetry-breaking patterns is the “cor- and compare this exotic phase with strong color confinement. rect” one? Since the physics controlling symmetry breaking Our analysis of the WCSM differs notably from earlier work is inherently non-perturbative, it is prohibitive to determine on weak confinement, where it was assumed that confinement the symmetry breaking pattern through analytical techniques. did not lead to chiral symmetry breaking. Although direct Arguably, the assumed absence of chiral symmetry breaking tests of the weak-confined phase will be challenging, given that underlies the AFM did not follow from any fundamen- the richness of the WCSM spectrum there are several possi- tal physical principle, but rather it was taken as a desirable ble cosmological implications of this exotic phase in the early assumption in order to reproduced the known particle mass Universe. These cosmological observables, including topo- spectrum. However, recent numerical lattice studies [33, 34] logical defects, gravitational wave radiation, and baryogene- have provided evidence against the AFM’s assumed absence sis, may provide fruitful avenues for further study. of symmetry breaking and in favor of an SU(2N)/Sp(2N) symmetry-breaking pattern instead. Motivated by these lattice results, we adopt the SU(2N)/Sp(2N) breaking in our work ACKNOWLEDGMENTS (3.5), and all of our results for the spectrum and interactions of composite particles in the early-universe weak-confined phase We are grateful to Seyda Ipek whose thought-provoking are predicated on this assumption. colloquium at the Aspen Center for Physics stimulated this work. We would also like to thank Bogdan Dobrescu, Nima Arkani-Hamed, Anson Hook, William Jay, Aaron Pierce, Chris Quigg, Tim Tait, and Jesse Thaler for valuable discus- VII. SUMMARY sions and encouragement. J.B. is supported by PITT PACC. A.J.L. is supported at the University of Michigan by the US We have studied a scenario in which the weak force is Department of Energy under grant DE-SC0007859. This strong and the strong force is weak. In other words, we as- manuscript has been authored by Fermi Research Alliance, sume that new physics allows the SU(2)L weak force to be- LLC under Contract No. DE-AC02-07CH11359 with the U.S. come strongly coupled and confine at a scale ΛW > TeV Department of Energy, Office of Science, Office of High En- where the QCD strong force is weakly coupled, and we re- ergy Physics. This work was initiated and completed at the fer to this scenario as the weak-confined Standard Model Aspen Center for Physics, which is supported by National Sci- (WCSM). We describe how interactions between the Standard ence Foundation grant PHY-1607611.

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