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PHYSICAL REVIEW D 100, 055005 (2019)

Phase of confined electroweak in the early

Joshua Berger,1 Andrew J. Long,2,3 and Jessica Turner4 1Department of and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 2Leinweber Center for , 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

(Received 18 June 2019; published 6 September 2019)

We consider a modified cosmological history in which the presence of beyond-the-Standard-Model 2 physics causes the weak gauge sector, SUð ÞL, to confine before it is Higgsed. Under the assumption of chiral breaking, quark and weak doublets form condensates that break the global symmetries of the , including and lepton number, down to a U(1) subgroup under which only the weak singlet and Higgs transform. The weakly coupled gauge group 3 1 2 1 SUð Þc ×Uð ÞY is also broken to an SUð Þc ×Uð ÞQ gauge group. The states include (pseudo-) Goldstone of the global symmetry breaking, mostly elementary fermions primarily composed of the weak singlet quarks and , and the gauge bosons of the weakly coupled gauge group. We discuss possible signatures from early Universe cosmology including radiation, topological defects, and .

DOI: 10.1103/PhysRevD.100.055005

I. INTRODUCTION composite particles and that the predicted low-energy spectrum agrees well with the measured quark Quarks and leptons can interact via the weak force, and lepton masses. However, this scenario is now ruled out mediated by the W- and Z-bosons. The Standard Model of by precision measurements of the electroweak sector [11], elementary particles (SM) describes the electroweak force and it would appear that the weak force is not confining in by a quantum theory that is constructed from the our present day Universe. gauge group SUð2Þ ×Uð1Þ and is in the Higgs phase L Y Nevertheless, when we study in a [1–3]. Higgsing screens at the weak scale and cosmological context, the system can exist in different leaves only weakly coupled electromagnetic long-range phases at different times, corresponding to different temper- . The theory is weakly coupled in the infrared, atures of the primordial plasma. We explore a scenario in which agrees extremely well with measured quark and which the SUð2Þ weak force was strongly coupled in lepton interactions. By contrast the strong force, which L the early Universe, when the plasma temperature was mediates interactions among the quarks, is strongly coupled T ≫ 100 GeV, but became weakly coupled and Higgsed in the infrared, leading to the confinement of the constitu- well before the epoch of nucleosynthesis, T ∼ MeV. ent quarks into composite states, namely mesons and Although we are unable to access the weak-confined phase . In this article, we study the weak-confined in the laboratory today, e.g., at collider experiments, the Standard Model (WCSM) in which the SUð2Þ component L Universe may contain relics from this period in its cosmic of the electroweak force is strongly coupled and the weak history, such as gravitational waves and the baryon asym- isospin is confined in composite particles. metry of the Universe. Confinement of SUð2Þ was studied in the pioneering L Our work shares some common elements with several work of Abbott, Farhi, and others in the 1980’s [4–7] previous studies of SM exotic phases, their associated (also [8–10]) as an alternative to the SM. Assuming that phase transitions, and their imprints on cosmology. As we confinement occurs without chiral symmetry breaking, they have already mentioned above, SUð2Þ confinement was argued that the known “elementary particles” are actually L studied in Refs. [4–8], and we present a detailed compari- son of our work with theirs in Sec. VI. The authors of Refs. [12,13] studied a variation of the SM in which the Published by the American Physical Society under the terms of Higgs field is absent, and the electroweak force is Higgsed the Creative Commons Attribution 4.0 International license. by the quark chiral condensates, and Ref. [14] points out a Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, candidate in a related model. Delaying the and DOI. Funded by SCOAP3. electroweak (EWPT) to coincide with the

2470-0010=2019=100(5)=055005(17) 055005-1 Published by the American Physical Society BERGER, LONG, and TURNER PHYS. REV. D 100, 055005 (2019) chiral phase transition of QCD was studied as a means of remarkable agreement between electroweak-precision generating the baryon asymmetry of the Universe [15–17] theory and measurement. (see also Ref. [18]) and a stochastic background of In general the scalar modulus may have a nonzero gravitational waves [19]. Likewise, an early period of expectation value, hφˆ i, and in this background, the strength 2 QCD confinement, which triggers a first order EWPT, of the SUð ÞL is controlled by the effective was proposed as a possible mechanism of baryogenesis coupling, [20]. Moreover, early QCD phase transitions have been 1 1 φˆ studied in order to open the axion parameter space [21,22]. − h i 2 2 ¼ 2 : ð2:2Þ We will see that SUð ÞL confinement in the WCSM geff g M causes and lepton number to be sponta- φˆ ≪ 2 ≈ φˆ ∼ 2 neously broken, which is signaled by the formation of If h i M=g then geff g, but if h i M=g then the ≳ φˆ lepton-quark and quark-quark condensates, hlqi and hqqi. weak force is stronger, geff g. The value of h i need not The violation of baryon number is a necessary condition to remain fixed throughout the cosmic history, and we explain the cosmological excess of matter over , demonstrate this point in the paragraphs below. Thus we and therefore one may expect that the WCSM can naturally will assume that hφˆ i was large in the early Universe, accommodate baryogenesis. However, when we discuss the corresponding to geff >g, but that it evolved to a small ≈ cosmological implications of weak confinement in Sec. V, value, where geff g, at some point before the epoch of big we will see that the viability of baryogenesis remains bang nucleosynthesis. In this way, laboratory probes of the ≈ unclear. weak force only access geff g, whereas cosmological This article is structured as follows. In Sec. II we provide probes of the prenucleosynthesis era may uncover evidence an example of the new physics that could lead to weak for geff >gin the early Universe. confinement in the early Universe. In order to develop As discussed, the effective strength of the weak inter- intuition for confinement in an SU(2) , we next action, geff, varies with time through the scalar field study three simplified models in Sec. III. Turning to the modulus, hφˆ i, but it also varies through the cosmological WCSM in Sec. IV, we investigate the symmetry breaking plasma temperature, T. At temperatures T>100 GeV, the 2 pattern, calculate the spectrum of composite particles, and hot Standard Model plasma contains SUð ÞL-charged discuss their interactions. We discuss the possible cosmo- quarks and leptons, which interact via the weak force. logical implications in Sec. V and highlight directions for As the Universe expands and the cosmological plasma future work. We contrast the WCSM with Abbot and cools, these scatterers carry less kinetic energy, and they ’ 2 Farhi s earlier work on SUð ÞL confinement in Sec. VI and probe the weak force at larger-and-larger length scales. This conclude in Sec. VII. scale dependence of the weak force is described by the equations of group flow, and its effects are captured by treating g as a running coupling that is tiny at II. CONFINEMENT OF THE WEAK FORCE small length scales in the high-energy ultraviolet (UV) 2 What new physics might allow the SUð ÞL weak force to regime and grows bigger at large length scales in the low- confine in the early Universe? In the context of QCD, early energy infrared (IR) regime. In the Standard Model the color confinement has been studied previously in weak force was Higgsed at the electroweak phase transition Refs. [21,22] (see also Ref. [20]), and we can straightfor- when the plasma temperature was T ∼ v ∼ 100 GeV, cor- wardly adapt that mechanism for our purposes. The responding to g2=4π ≪ 1. However, if hφˆ i ∼ M=g2 at early essential idea is to link the strength of the weak force times, then the weak coupling may run to a nonperturba- with the expectation value of a scalar field that experiences 2 4π O 4π ’ tively large value, geff= ¼ ð Þ, while the Higgs field s a phase transition in the early Universe. expectation value remains zero. In that case the Standard Let φˆ ðxÞ be a real scalar field, which we call the 2 Model enters the confined-SUð ÞL phase, which we call modulus field, and suppose that it interacts with the weak the WCSM, and we study the properties of this phase in the force through a dimension-five operator. The relevant remainder of this article. The proposed modification to the Lagrangian is gauge coupling’s running is schematically illustrated in Fig. 1. 1 1 φˆ μν One can imagine several different explanations for the L ¼ − − Tr½WμνW − Vðφˆ Þ; ð2:1Þ 2 g2 M nontrivial dynamics of φˆ . On the one hand, the potential Vðφˆ Þ may have a local minimum at hφˆ i ≠ 0, which traps 2 where g is the SUð ÞL gauge coupling, M is an energy the field until it is either released, for instance by the 2 scale parameter, W is the SUð ÞL field strength tensor, and changing temperature of the cosmological plasma, or it V is the scalar potential of φˆ . In the limit M → ∞ the escapes through a first order phase transition. Alternatively, modulus field decouples from the Standard Model. We the potential Vðφˆ Þ may simply be very shallow such that will consider M>TeV in order to avoid disrupting the the field φˆ is “frozen by Hubble drag” at its nonzero value

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early Universe at a time before the electroweak interaction is Higgsed and before color is confined.

III. BUILDING UP TO THE WCSM Our goal is to study the weak-confined Standard Model, which has the same particle content as the SM, but in which 2 the SUð ÞL weak force is confined. As weak confinement leads to a dramatic departure from the usual SM dynamics, 2 we use this section to put together the pieces of the SUð ÞL- confined SM bit-by-bit. First, we consider an SU(2) gauge theory with doublet fermions followed by the addition of a doublet scalar, and we finally consider singlet fermions.

SU A. Model 1: Include only ð2ÞL-doublet fermions We consider a gauge theory based on the SU(2) 2 symmetry group, which we denote by SUð ÞL, anticipating the connection with SM weak isospin. The force carriers are represented by the vector field WμðxÞ, which transforms 2 in the adjoint of SUð ÞL, and the matter is represented by 2 2 3 FIG. 1. The strength of the SUð ÞL weak force is parametrized 2N flavors of left-chiral Weyl fermion fields, denoted by α 2 4π f by the fine structure constant W ¼ geff = , which varies with ψ iðxÞ for i ¼ 1; 2; …; 2Nf, which separately transform in the temperature of the cosmological plasma, T, according to the 2 flow. The solid-green line illustrates the fundamental representation of SUð ÞL. The fermions the Standard Model prediction: the weak force grows stronger are assumed to have no mass terms, and the Lagrangian is α−1 simply (smaller W ) as the plasma cools (smaller T), but it remains weak (α−1 ≫ 1=4π) when the theory is Higgsed at T ∼ v ∼ 100 GeV. W 2 Instead, we are interested in the weak-confined Standard Model, XNf † μ 1 μν L1 ψ iσ¯ Dμψ − WμνW ; : corresponding to the dashed-green line: the nonzero modulus jUV ¼ i i 2 2 Tr½ ð3 1Þ α−1 i¼1 g field makes the weak force stronger (smaller W ) at early times (large T) such that α reaches ∼4π at T ∼ Λ , and the theory W W ψ ∂ ψ − ψ enters the SUð2Þ -confined phase. where Dμ i ¼ μ i iWμ i is the covariant derivative of L ψ 2 i, g is the running SUð ÞL gauge coupling at the UV scale, 2 4 and Wμν is the SUð ÞL field strength tensor. In Sec. IV we until a time such that the Hubble parameter, H, decreases 2 12 ’ 00 1=2 1 will take Nf ¼ to count the SM s nine quark doublets below its effective mass, i.e., H

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2 TABLE I. This table summarizes the assignments for the The strong coupling is assumed to imply that SUð ÞL sequence of models in Sec. III. Note that the N- and ðN2 − NÞ- charge is confined, in analogy with QCD’s color confine- dimensional representations of SUðNÞ are the fundamental and ment.5 Consequently the d.o.f. in the low-energy confined antisymmetric representations, respectively. When applied to the phase are composite states, analogs of the mesons and 2 12 21 WCSM in Sec. IV we take Nf ¼ and Nξ ¼ . baryons of QCD. Specifically, the spectrum consists of several massless Goldstone bosons Πa, associated with the SO(4) Uð2NfÞψ UðNξÞξ assumed SUð2NfÞψ global flavor symmetry, a single 2 2 2 1 1 SUð ÞL SUð Þϕ SUð NfÞψ Uð Þψ SUðNξÞξ Uð Þξ massive pseudo-Goldstone boson η0, associated with the 1 ψ 212Nf qψ 1 0 anomalous Uð Þψ global flavor symmetry, and a tower of Wμ 31 1 0 1 0 heavy resonances with masses around the confinement ϕ=φ 22 1 0 1 0 scale, m ∼ Λ . These heavy resonances include meson- ξ 11 1 res W 0 Nξ qξ like scalars, vectors, and -like bound states of the Σ 112 2 − 2 1 Wμ boson. As these resonances are expected to be heavy Nf Nf qψ 0 Φ 11 1 0 1 0 and unstable, we can assume that they decay shortly after Ψ 122 1 2 Nf qψ 0 SUð ÞL is confined, and we focus on the properties of the ξ 11 1 0 Nξ qξ light states. Collectively the lowest-lying, mesonlike states Πa 0 and η are encoded in the scalar fields ΣijðxÞ, which have the same symmetry properties as the fermion isomorphism Uð2NfÞψ ¼ SUð2NfÞψ ×Uð1Þψ , we can bilinears, assign ψ a charge qψ under the Uð1Þψ group whose conserved charge is axial-ψ-number. The SUð2Þ gauge L Σ ∼ ψ εψ interactions induce an anomaly in Uð1Þψ [25], breaking it to ij i j: ð3:2Þ Z the 2Nf subgroup. The transformation properties of the various fields are summarized in Table I. Here the totally antisymmetric tensor, ε, with ε12 ¼ 1, 2 It is illustrative to draw a contrast with the flavor contracts the (suppressed) SUð ÞL indices such that Σ 2 Σ symmetry of QCD [26]. In the limit that the up, down, ij is an SUð ÞL singlet. Under the flavor symmetries, and strange quarks and antiquarks are massless, we would transforms as an antisymmetric rank-2 tensor of 2 2 1 have Nf ¼ 3 flavors of left-chiral Weyl fermions trans- SUð NfÞψ with charge qψ under Uð Þψ. We assume that 3 3 3 forming in the fundamental of SUð Þc color, , and Nf ¼ Σ acquires a nonzero , and additional flavors (of left-chiral Weyl fermions) transform- the global SUð2NfÞψ ×Uð1Þψ symmetry is spontaneously ing in the antifundamental, 3¯. The flavor symmetry is broken. 1 1 UðNfÞ ×UðNfÞ¼SUðNfÞV ×Uð ÞV ×SUðNfÞA ×Uð ÞA The symmetry breaking pattern cannot be deter- 1 where the conserved charge associated to Uð ÞV is baryon mined through a perturbative calculation, since after all, 1 number and the anomalous charge associated with Uð ÞA is symmetry breaking is a consequence of nonperturbative axial-baryon number. The situation in the theory with charge confinement. However, this simple model has 2 gauged SUð ÞL is different because the fundamental been studied using nonperturbative, numerical lattice tech- ¯ 2 – representation is pseudoreal, i.e., 2 ¼ 2, which allows niques [33,34] for Nf ¼ (see also Refs. [28 32]). Those 2 2 the 2Nf fields to be collected into a single flavor multiplet studies conclude that SUð NfÞψ is broken to Spð NfÞψ , with the larger Uð2NfÞ flavor symmetry instead. which is the naive expectation from chiral symmetry 2 breaking in analogy with QCD [35,36]. Therefore, we will The SUð ÞL force in Eq. (3.1) is not Higgsed, as in the SM, but rather may become strongly coupled in the IR. To adopt the results of the lattice studies and assume that the 2 analytically understand the evolution towards strong cou- unbroken global flavor symmetry is the Spð NfÞψ group. pling, one can calculate the gauge coupling’s This pattern of symmetry breaking is obtained if the Σ βg, and a value βg < 0 implies that g grows in the IR. antisymmetric field ijðxÞ acquires a vacuum expectation Σ Σ Calculating βg at one-loop order in perturbation theory value h iji¼ð 0Þij that satisfies 2 3 gives βg ¼ðb=16π Þg with b ¼ −22=3 þ 2Nf=3 [27]. A theory with N < 11 has β < 0, and this perturbative † † f g Σ0Σ0 ¼ Σ0Σ0 ¼ 1 ⇔ SUð2NfÞψ =Spð2NfÞψ : ð3:3Þ analysis suggests that such a theory is strongly coupled in the IR; that is, g2=4π grows as large as 4π as the Λ 5If the theory instead has a strongly coupled, IR-stable fixed energy scale decreases toward W. This perturbative analysis is confirmed by numerical lattice studies for point, as suggested by Ref. [31], then there is no charge 6 confinement, and a different analysis is required to determine small Nf while Nf ¼ is marginal and near the conformal the properties and interactions of the low-energy degrees of window [28–32]. freedom (d.o.f.).

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Σ 2 Without loss of generality, we work in the basis where 0 is transforms under SUð ÞL in the fundamental representa- a block diagonal matrix with Nf blocks of ε along the tion. The Lagrangian of the UV d.o.f. can be written as (see diagonal. also Ref. [38]) Due to the spontaneous symmetry breaking in Eq. (3.3), L L ϕ 2 − ϕϕ Goldstone’s theorem implies that there will be massless 2jUV ¼ 1jUV þjDμ j Uð Þ; ð3:6Þ Goldstone bosons in the spectrum of low-lying, mesonlike 2 4 2 − 1 ϕ ∂ ϕ − ϕ states. Note that SUð NfÞψ has Nf generators, where Dμ ¼ μ iWμ is the covariant derivative, a a a ϕ which we denote by Tij and Xij such that Tij for a ¼ and U is a potential for the scalar . We require the 2 2 1; 2; …; 2N þ N are the generators of the unbroken Lagrangian to be invariant under SUð ÞL, which U f f 2 to be a function of ϕ ϕ only, since ϕεϕ is identically subgroup Spð2N Þ and Xa for a¼1;2;…;2N −N −1 f ψ ij f f zero, and forbids renormalizable interactions between ϕ are the generators of the broken symmetries. The Uð1Þψ and the ψ i. factor is also broken by Eq. (3.3). Thus we expect the The complex SUð2Þ -doublet field ϕ consists of 2 2 − − 1 L spectrum to contain Nf Nf massless Goldstone 4 real d.o.f., and in fact Eq. (3.6) respects an SO(4) bosons, denoted by Πa, and one massive pseudo- symmetry. It is useful to apply the isomorphism SOð4Þ¼ η0 Goldstone boson, denoted by ; the notation is chosen SUð2Þ ×SUð2Þϕ where the first factor is identified as the η0 L to draw an analogy with the pions and meson of gauged subgroup, and the second factor is a new global 1 η0 QCD. Since Uð Þψ is anomalous, the mass is lifted flavor symmetry which is the analog of custodial SU(2) in 2 by SUð ÞL- effects (see below). the Standard Model. Thus the global flavor symmetry is In the confined phase, the low-energy physics can be enlarged to Uð2NfÞψ ×SUð2Þϕ. It is useful to recall that an described with a nonlinear sigma model, 2 ϕ φ SUð ÞL-doublet field ðxÞ can also be represented as aðxÞ 2 with [39] L f ∂ Σ†∂μΣ κΛ2 2 Σ 1jIR ¼ Tr½ μ þ Wf Re½det ; ð3:4Þ 4 φ1 ¼ εϕ and φ2 ¼ −ϕ ; ð3:7Þ κ O 1 where f is the decay constant, is an ð Þ dimensionless φ 2 2 Λ ∼ 4π where transforms as under each factor of SUð ÞL× number, and W f is the confinement scale. As we SUð2Þϕ. Using this equivalent representation of the have discussed above, the antisymmetric complex scalar 2 SUð ÞL-doublet scalar, we can write the Lagrangian as field ΣijðxÞ parametrizes the massless Goldstone boson a fields Π ðxÞ and the pseudo-Goldstone boson field η0ðxÞ. 1 X L L φ 2 − φ 2 2 We make this connection explicit by writing 2jUV ¼ 1jUV þ 2 jDμ aj Uðj j = Þ; ð3:8Þ   a¼1;2 pffiffiffiffiffiffi X 0 a a Σ ¼ exp½iη = N f exp 2iX Π =f Σ0; ð3:5Þ f where Dμφa ¼∂μφa þiWμφa, and we have used the short- a 2 † † hand notation jφj ≡ φ1φ1 þ φ2φ2. a 2 2 − − 1 From the low energy perspective, we have extended where Xij are the Nf Nf broken generators of the spectrum by additional low-lying states, correspond- SUð2NfÞ, and the vacuum expectation value Σ0 satisfies pffiffiffiffiffiffi ing to a new real scalar field ΦðxÞ and a set of new Weyl Eq. (3.3). The factors of 2 and N are included to f fermions Ψ ðxÞ for i ¼ 1; 2; …; 2N and a ¼ 1, 2, which ensure canonical normalization of kinetic terms and the ia f have the same symmetry properties as the following a b δab 2 generators are normalized such that Tr½X X ¼ = . bilinears: The last term of Eq. (3.4) arises from instanton effects 1 via the of Uð Þψ . The anomaly explicitly Φ ∼ ϕϕ 1 Z ð3:9aÞ breaks the Uð Þψ symmetry to its 2Nf subgroup and η0 thereby lifts the mass of the through instanton effects, Ψ ∼ ψ φ ; ð3:9bÞ 1 ia i a in analogy with the Uð ÞA problem of QCD [37].One calculates the η0 mass by substituting Eq. (3.5) into 2 which are all singlets under SUð ÞL. The composite Eq. (3.4) and expanding to quadratic order; doing so Φ 2 2 scalar, , is a total singlet, while the composite fermions, gives m 0 4κN Λ . η ¼ f W Ψia, transform as fundamentals under the SUð2NfÞψ flavor symmetry, carry charge qψ under the Uð1Þψ sym- SU B. Model 2: Add an ð2ÞL-doublet scalar metry, and transform as a 2 under the SUð2Þϕ custodial Let us next consider an extension of the first model symmetry as summarized in Table I. Note that Φ is a that is a closer approximation to the Standard Model singlet under the custodial SU(2), and there does not particle content, by introducing an analog of the Higgs exist an analogous triplet state; only the custodial anti- doublet field. Consider the complex scalar field ϕðxÞ that symmetric combination is allowed.

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The low energy physics is described by confinement. Thus the Lagrangian for the low-energy d.o.f. is also simply 2 XNf X L L Ψ† σ¯ μ∂ Ψ 2jIR ¼ 1jIR þ iai μ ia XNξ † i¼1 a¼1;2 L L ξ σ¯ μ∂ ξ 3jIR ¼ 2jIR þ i i μ i: ð3:12Þ 1 i¼1 ∂μΦ∂ Φ − Λ2 f2V Φ=f þ 2 μ W ð Þ 2 These singlet fermions are entirely inert and decoupled. XNf X ab † In the next section we find that breaking the large global − ½y0Λ ε Ψ Σ Ψ þ H:c: W ia ij jb flavor symmetry allows for Yukawa interactions, such as i;j¼1 a;b 1;2 ¼ ΔL ∼ Ψξ and ΨξΦ. These interactions will have interest- 2 XNf X ing consequences for the spectrum in the IR. − 0 Λ εabΨ Σ† Ψ Φ ½ðy0 W=fÞ ia ij jb þ H:c:; i;j¼1 a;b¼1;2 IV. THE WEAK-CONFINED STANDARD MODEL ð3:10Þ Let us now investigate weak isospin confinement in Λ ∼ 4π the WCSM. The UV particle content and symmetries of where f is the decay constant, W f is the confine- ment scale, V is a potential for the composite scalar Φ, the WCSM are identical to the SM, shown in Table II. 0 In particular the SM symmetry group is and y0 and y0 are Yukawa couplings. The 4π counting is performed using naïve dimensional analysis [40–42], 0 fSUð3Þ ×SUð2Þ ×Uð1Þ g which implies that y0, y0, and the dimensionless coef- c L Y gauge ficients in V are Oð1Þ numbers. We assume that the 1 1 3 × fUð ÞBþLganomaly × fUð ÞB=3−L gglobal; ð4:1Þ potential U in Eq. (3.6) is a small perturbation on V, i such that the scalar’s mass is of order mΦ ∼ Λ . The 1 W where the Uð ÞBþL factor is anomalous under the potential V may also induce a nonzero vacuum expect- 2 1 SUð ÞL ×Uð ÞY gauge interactions. The key difference ation value for Φ, which is expected to be hΦi ∼ f,but between the WCSM and the SM has to do with the strength since Φ is a singlet, this vacuum expectation value does 2 of the SUð ÞL weak force, which is assumed to be larger in not signal any spontaneous symmetry breaking. The y0 → ≫ the WCSM, g geff g, such that weak isospin is term, which is a , respects the confined in the IR. Therefore, whereas the SM predicts Uð2N Þψ ×SUð2Þϕ flavor symmetry, and when Σ 2 f ij that SUð ÞL is Higgsed by the vacuum expectation value of 2 acquires a nonzero vacuum expectation value (3.3),it H, the WCSM predicts that the SUð ÞL weak force is Ψ Ψ 2 induces a mass term for the ia. Together ia combine in confined in the IR and the SUð ÞL-doublet fields (qi, li, and pairs to form 2Nf Dirac fermions, which are all degen- H) form composite states. Λ erate and have a mass mΨ ¼ y0 W. This section contains the main results of our paper. To simplify the analysis, we will begin in Sec. IVA by setting SU 0 C. Model 3: Add ð2ÞL-singlet fermions to zero the strong coupling gs ¼ , the 2 As a final iteration, we introduce a set of SUð ÞL-singlet, left-chiral Weyl fermion fields, denoted as ξiðxÞ for TABLE II. This table shows the WCSM particle content and i ¼ 1; 2; …;Nξ. Note that Nξ may be either even or odd, charge assignments. The index i ∈ f1; 2; 3g labels the three 2 generations of fermions. There are four global charges, corre- since these fields do not contribute to the SUð ÞL anomaly. sponding to Uð1ÞB L and Uð1ÞB 3−L for j ¼ 1, 2, 3. All fermions As we build up to the SM, these fields will correspond to þ = j 21 2 the Nξ ¼ SUð ÞL-singlet quark and lepton fields. are represented by left-chiral Weyl spinor fields in the two- We require the theory to respect an extended global component notation [43]. 2 2 symmetry group, Uð NfÞψ ×SUð Þϕ ×UðNξÞξ where the Gauge Global ξi transform in the fundamental representation of the 3 2 1 1 Uð1ÞB 3−L UðNξÞξ and the ψ i, Wμ, and ϕ are singlets; see Table I. SUð Þc SUð ÞL Uð ÞY Uð ÞBþL = j μ ξ ξ 321 61319 This symmetry forbids masses such as ij i j and inter- qi = = = ¯ 1 −2 3 −1 3 −1 9 actions such as λijϕψiξj. Therefore the Lagrangian for the Ui 3 = = = ¯ 1 1 3 −1 3 −1 9 high-energy d.o.f. is simply Di 3 = = = li 12−1=2 1 −δij XNξ E 111 −1 δ L L ξ† σ¯ μ∂ ξ i ij 3jUV ¼ 2jUV þ i i μ i: ð3:11Þ H 121=2 00 i¼1 gμ 8100 0 13 2 Wμ 00 0 Moreover, these SUð ÞL-singlet fermions do not feel the 11 2 Bμ 00 0 strong SUð ÞL force and therefore do not participate in

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0 0 λ 0 2 † † coupling g ¼ , and the Yukawa couplings Yuk ¼ . Higgs self-coupling, and jφj ≡ φ1φ1 þ φ2φ2. We note that We reintroduce these couplings in Sec. IV B where the and hypercharge bosons are present but they are treated as perturbations around an enhanced decoupled due to the assumption that their gauge couplings symmetry point. are zero valued. By neglecting the strong, hypercharge, and Yukawa A. Turn off strong, hypercharge, interactions, we can write the Lagrangian as in Eq. (4.4), and Yukawa couplings which now bears a marked similarity to the models that we have already discussed in Sec. III for N ¼ 6 and Nξ ¼ 21. Upon taking g ¼ g0 ¼ λ ¼ 0, the symmetry group of f s Yuk As before, we assume that the weak force confines, and the the SM Lagrangian is enlarged from Eq. (4.1) to chiral symmetry is spontaneously broken. By applying 2 1 what we learned in Sec. III, we can immediately infer fSUð ÞLggauge × fUð Þψ ganomaly 2 the effects of SUð ÞL-charge confinement in the WCSM. 12 2 21 2 × fSUð Þψ ×SUð Þϕ ×Uð Þξgglobal; ð4:2Þ The SUð ÞL-doublet fermions condense in pairs to form composite scalars, Σij ∼ ψ iεψ j; the doublet 1 2 where the Uð Þψ factor is anomalous under the SUð ÞL condenses with the doublet fermions to form composite Ψ ∼ ψ φ gauge interaction. To make these symmetries manifest in fermions, ia i a; the Higgs boson condensesP in pairs Φ ∼ φ†φ ∼ † the Lagrangian, it is useful to collect the fields together into to form another composite scalar, a a a H H; 2 ξ the following multiplets: and the SUð ÞL-singlet fermions, i, naturally do not con- fine but (most of them) acquire masses indirectly through R G B ψ ¼fl1;q1 ;q1 ;q1 ; …; …g; weak-isospin confinement as outlined in Sec. IV B. The symmetry transformation properties of these low-energy φ ¼fεH; −Hg; d.o.f. are also shown in Table I. R¯ G¯ B¯ R¯ G¯ B¯ ξ ¼fE1;U1 ;U1 ;U1 ;D1 ;D1 ;D1 ; …; …g; ð4:3Þ By carrying over the results from Sec. III, the Lagrangian describing the low-energy d.o.f. of the WCSM can be ψ 1 2 … 12 2 where i for i ¼ ; ; ; contains the SUð ÞL-doublet written as φ 1 2 fermions, a for a ¼ , 2 contains the SUð ÞL-doublet ξ 1 2 … 21 X12 X2 X21 Higgs boson, and i for i ¼ ; ; ; contains the † μ † μ 2 L j ¼ Ψ iσ¯ ∂μΨ þ ξ iσ¯ ∂μξ SUð ÞL-singlet fermions. The subscript 1 indicates first- WCSM IR ia ia i i generation fields, the dots correspond to the second and i¼1 a¼1 i¼1 third generations, the SUð2Þ index is suppressed, and the X12 X2 L ab † ¯ ¯ ¯ − y0Λ ε Ψ Σ Ψ : : SUð3Þ indices are fR; G; Bg and fR; G; Bg. The symmetry ½ W ia ij jb þ H c c 1 1 transformation properties of these fields are shown in i;j¼ a;b¼ X12 X2 Table I where 2N ¼ 12 and Nξ ¼ 21; specifically, the f − 0 Λ εabΨ Σ† Ψ Φ 2 ψ ∼ 12 ½ðy0 W=fÞ ia ij jb þ H:c: SUð ÞL-doublet fermions transform as a i of i;j¼1 a;b¼1 Uð12Þψ , the SUð2Þ -doublet Higgs boson transforms as L 2 a φ ∼ 2 of the “custodial” SUð2Þϕ, and the SUð2Þ -singlet f † μ 2 2 a L Tr ∂μΣ ∂ Σ κΛ f Re det Σ þ 4 ½ þ W ½ fermions transform as a ξ ∼ 21 of Uð21Þξ. i 1 Using the multiplets in Eq. (4.3), the WCSM Lagrangian μ 2 2 þ ∂ Φ∂μΦ − Λ f VðΦ=fÞ: ð4:5Þ 0 λ 0 2 W (for gs ¼ g ¼ Yuk ¼ ) is written compactly as 0 X12 X21 Recall that y0, y0, and κ are Oð1Þ dimensionless coef- L ψ † σ¯ μ ψ ξ† σ¯ μ∂ ξ ficients and Λ ∼ 4πf is the scale at which SUð2Þ WCSMjUV ¼ i i Dμ i þ i i μ i W L i¼1 i¼1 weak isospin confines. This Lagrangian describes an X of the low-energy d.o.f. where we 1 μν 1 μ † − Tr½WμνW þ D φ Dμφ 2g2 2 a a have integrated out a tower of resonances with masses eff a¼1;2 ∼ Λ mres W, and we do not show all the nonrenormalizable 1 2 2 1 4 operators that they generate. The anomalous Uð1Þψ part of − μ jφj − λjφj : ð4:4Þ 2 4 Eq. (4.2) is broken by the determinant term, for κ ≠ 0, which arises from nonperturbative instanton effects in the 2 Here we have written the SUð ÞL gauge coupling as confined phase. geff >g, which distinguishes the WCSM from the SM. As we have discussed already in Sec. III, the confine- ψ ∂ ψ − ψ Additionally, Dμ i ¼ μ i iWμ i is the covariant deriva- ment is accompanied by the spontaneous breaking of φ ∂ φ φ tive of the doublet fermions, Dμ a ¼ μ a þ iWμ a is the the Uð12Þψ ¼ SUð12Þψ ×Uð1Þψ global symmetry from ’ 2 Higgs field s covariant derivative, Wμν is the SUð ÞL field Eq. (4.2). The scalar condensate field ΣðxÞ acquires a 2 strength tensor, μ is the Higgs mass parameter, λ is the nonzero vacuum expectation value, Σ0 ¼hΣi, and in order

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† to achieve the anticipated symmetry breaking pattern, value satisfies Σ0Σ0 ¼ 1. To be more explicit, we work † SUð12Þψ =Spð12Þψ , we should have Σ0Σ0 ¼1 [33,34,44]. in a basis where As SUð12Þψ has 143 generators and Spð12Þψ has 78 0 1 0 1 0100 generators, then the number of broken generators is 65, A B C which equals the number of massless Goldstone boson B C B −10 0 0C a Σ0 ¼ @ A A;A¼ B C; ð4:8Þ fields, Π ðxÞ. The vacuum expectation value Σ0 ≠ 0 also @ 0001A breaks the Uð1Þψ ⊂ Uð12Þψ , but since this symmetry is A 00−10 anomalous, the would-be Goldstone boson has its mass lifted by electroweak , in analogy with the η0 meson of QCD. Thus after confinement and spontaneous where the three blocks correspond to the three generations symmetry breaking, the symmetry group from Eq. (4.2) is of fermions. Recall the mapping from the scalar condensate Σ ∼ ψ εψ ψ broken to to the SM quark and lepton doublets, ij i j with i given by Eq. (4.3). Therefore the symmetry breaking 12 2 21 pattern implies fSpð Þψ ×SUð Þϕ ×Uð Þξgglobal; ð4:6Þ l εqR ≠ 0 qGεqB ≠ 0; : a h i i i and h i i i ð4 9Þ where Spð12Þψ acts (nonlinearly) on the 65 “pions,” Π . Σ ≠ 0 Due to spontaneous symmetry breaking, 0 , the for each of the three generations i ∈ f1; 2; 3g. Thus we see ∼ΨΣ†Ψ Yukawa interaction, in Eq. (4.5), becomes a Dirac that the vacuum state carries a nonzero baryon number mass for the composite fermions. In particular, the 24 and lepton number, and consequently the particles in Ψ composite fermions, ia, form 12 degenerate Dirac fer- Eq. (4.5) will experience B- and L-violating interactions ∼ Λ mions with mass mΨ y0 W, on the order of the confine- that allow ment scale. Meanwhile the 21 singlet fermions, ξi, remain massless and noninteracting to all orders, since 3ΔB ¼ ΔL ¼1 and ΔB ¼2=3: ð4:10Þ the relevant operators are forbidden by the residual global symmetry (4.6). The composite scalar Φ has a potential V, which induces The violation of baryon number is a requirement for any its mass and self-interaction. An analytical calculation of V model of baryogenesis, and its natural emergence here is a is hindered by the strongly coupled nature of the theory, tantalizing hint for a connection between the WCSM and but dimensional analysis arguments suggest that Φ will the cosmological excess of matter over antimatter; we acquire a nonzero vacuum expectation value, hΦi ∼ f. discuss this possibility further in Sec. V. Since Φ is a singlet under the residual symmetry (4.6) its vacuum expectation value does not signal any symmetry B. Turn on strong, hypercharge, breaking. and Yukawa couplings In summary, the spectrum of the WCSM (with Let us finally consider SUð2Þ confinement with the 0 λ 0 L gs ¼ g ¼ Yuk ¼ ) in the confined phase is Standard Model particle content at the measured values for the strong, hypercharge, and Yukawa couplings. Since 065 mΠ ¼ Goldstone bosons ð4:7aÞ these nonzero couplings explicitly break the large global flavor symmetry, we will find that the spectrum of pseudo- ∼ Λ 1 mη0 W pseudo-Goldstone ð4:7bÞ Goldstone pions is lifted. Moreover, because the conden- sates (4.9) spontaneously break color and hypercharge, we ∼ Λ 12 mΨ W composite Dirac fermions ð4:7cÞ will see that some of the and hypercharge boson acquire mass from the by “eating” the ∼ Λ 1 would-be Goldstone bosons. The reader may be familiar mΦ W composite scalar ð4:7dÞ the style of analysis that we employ in this section from studies of little Higgs theories [45]. mξ ¼ 021elementary Weyl fermions ð4:7eÞ

∼ Λ 1. Comparison with QCD mres W heavier resonances: ð4:7fÞ Before beginning the determination of the effect of these The heavy resonances are not included in Eq. (4.5). explicit chiral symmetry breaking terms in the WCSM, we Finally, we investigate the vacuum structure of this contrast the symmetry breaking to that in confining SM theory more carefully. We have seen that the symmetry QCD. The quark masses and electric couplings are both breaking pattern SUð12Þψ =Spð12Þψ is obtained when the sources of explicit chiral symmetry breaking in the SM. antisymmetric scalar condensate’s vacuum expectation Since the quark masses only involve confining states, they

055005-8 PHASE OF CONFINED ELECTROWEAK FORCE IN THE EARLY … PHYS. REV. D 100, 055005 (2019) 0 1 0 1 contribute to chiral symmetry breaking terms at tree-level 000 00 0 1B C 1B C in the Lagrangian. On the other hand, the electric coupling, λ1 ¼ @001A; λ2 ¼ @00−iA; which involves the nonconfining , only contributes a 2 2 010 0 i 0 small correction at loop level. Furthermore, the same 0 1 0 1 combination of the nonconfining SUð2Þ ×Uð1Þ gauge 00 0 010 L Y 1B C 1B C group that would get broken by confinement of QCD is λ3 ¼ @01 0A; λ4 ¼ @100A already broken at a much higher scale by the Higgs 2 2 00−1 000 mechanism, so all of the pions remain in the spectrum 0 1 0 1 rather than being eaten. On the other hand, there are no 0 −i 0 001 1B C 1B C chiral symmetry breaking operators involving only the λ5 ¼ @ i 00A; λ6 ¼ @000A; weak-charged fermions in the WCSM, so all chiral sym- 2 2 000 100 metry breaking for the pions occurs at loop level stemming 0 1 0 1 from the SUð3Þ ×Uð1Þ gauge and Yukawa couplings. 00−i −200 c Y 1B C 1 B C There is no higher scale breaking of the weakly coupled λ7 ¼ @00 0A; λ8 ¼ pffiffiffi@ 010A: ð4:12Þ 3 1 2 2 3 SUð Þc ×Uð ÞY gauge group, so it is broken only by the 2 i 00 001 confinement of SUð ÞL, leading to both some eaten Goldstone modes and non-negligible corrections due to As such, the SUð3Þ and Uð1Þ generators are written as the massive gauge bosons that are not completely c Y decoupled at the confinement scale. La ¼ diagð0;λa;0;λa;0;λaÞ for a ¼ 1;…;8 1 1 1 1 1 1 1 1 1 1 1 1 − − − 2. Treat small couplings perturbatively Y ¼ diag 2;6;6;6; 2;6;6;6; 2;6;6;6 ; ð4:13Þ A key assumption in the following analysis is that the nonzero strong, hypercharge, and Yukawa couplings respectively. The generators are orthonormal such that can be treated as perturbations on the preceding analysis Tr½LaLb¼3δab=2,Tr½LaY¼0, and Tr½YY¼1. of Sec. IVA. This is a reasonable expectation as most of these couplings are ≪ Oð1Þ, with the largest being 4. Explicit global symmetry breaking the top Yukawa coupling and the strong coupling. By setting g ¼ g0 ¼ λ ¼ 0 previously, we saw that Σ s Yuk Specifically, we assume that the scalar condensate, ij, the SM global symmetry of Eq. (4.1) was enhanced to still acquires a vacuum expectation value Σ0 that leads Eq. (4.2). Since the nonzero Yukawa couplings are typi- 12 12 3 1 to the SUð Þψ =Spð Þψ symmetry breaking pattern as cally much smaller than the SUð Þc ×Uð ÞY gauge cou- in Eq. (3.3). plings, it is also useful to consider the intermediate case in 0 ≠ 0 λ 0 which gs, g and Yuk ¼ . For vanishing Yukawa couplings, the global symmetry group of the theory is 3. Introduce gauge interactions 3 1 3 3 3 3 We implement the SUð Þc ×Uð ÞY gauge interactions SUð Þq ×SUð ÞU ×SUð ÞD ×SUð Þl ’ by promoting partial derivatives in the WCSM sIR 3 1 1 1 1 ×SUð ÞE ×Uð ÞU ×Uð ÞD ×Uð ÞE ×Uð ÞH; ð4:14Þ Lagrangian (4.5) to covariant derivatives: DμΨia, Dμξi, Σ Φ Φ Dμ ij, and Dμ . Since the scalar is a singlet under where each factor acts nontrivially on only one field, 3 1 Φ ∂ Φ SUð Þc ×Uð ÞY, we have simply Dμ ¼ μ . Let us look indicated by the subscript. In other words the nonzero more closely at DμΣij, as it reveals how the gauge couplings explicitly break the global symmetry from masses arise through the Higgs mechanism. The covariant Eq. (4.2) to Eq. (4.14). This explicit breaking lifts several of derivative of the scalar field ΣijðxÞ, is written as the would-be Goldstone bosons, as we demonstrate below. The choice of the definition of the U(1) factors here is not 8 1 X unique; any linear combinations of these with Uð ÞY will a a aT 0 DμΣ ¼ ∂μΣ − igs GμðL Σ þ ΣL Þ − ig BμðYΣ þ ΣYÞ; also be a symmetry. We choose these combinations for later a¼1 convenience. Two combinations of the residual U(1) global 3 1 ð4:11Þ symmetry factors are anomalous under SUð Þc ×Uð ÞY, but this anomalous explicit breaking is extremely weak, 3 a 1 … 8 where gs is the SUð Þc coupling, GμðxÞ for a ¼ ; ; are and we neglect it in this work. a 3 0 the gluon fields, L are the generators of SUð Þc, g is the 1 5. Spontaneous symmetry breaking Uð ÞY coupling, BμðxÞ is the hypercharge boson field, and 1 Σ Σ Y is the generator of Uð ÞY. It is convenient to define the The nonzero vacuum expectation value, h i¼ 0 in 3 SUð Þc generators in a slightly unconventional way using Eq. (4.8), indicates that the gauge symmetry is broken as the following basis: follows:

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3 1 → 2 1 2 fSUð Þc ×Uð ÞYggauge fSUð Þc ×Uð ÞQggauge: ð4:15Þ the SUð ÞL-singlet leptons, Ei, are singlets under the 2 −1 1 SUð Þc and carry charge under the Uð ÞQ. Finally, 3 2 φ 1 Specifically, SUð Þc is broken to an SUð Þc subgroup, and the Higgs doublet, as represented by a for a ¼ , 2, can 3 1 φþ φ φ− φ a linear combination of SUð Þc and Uð ÞY is broken to a also be denoted as ¼ 2 and ¼ 1, where the 1 2 1 Uð ÞQ subgroup. The generators of the SUð Þc are simply superscript indicates the sign of the Uð ÞQ charge. These the Pauli matrices, τa ¼ σa=2 for a ¼ 1, 2, 3, and the charge assignments are summarized in Table III. 1 generator of Uð ÞQ is SU U 7. Composite scalars under ð2Þc × ð1ÞQ 1 8 Q ¼ pffiffiffi L − Y Σ x 3 We have seen that ijð Þ represents 66 pseudoscalar fields: η0ðxÞ and ΠaðxÞ for a ¼ 1; …; 65. To clarify how 1 1 1 1 1 1 2 1 ¼ diag ; − ; 0; 0; ; − ; 0; 0; ; − ; 0; 0 : ð4:16Þ these scalar fields transform under SUð Þc ×Uð ÞQ,it 2 2 2 2 2 2 useful to map them to fermion bilinears as follows:

Note that Q is not SM , but we use this þ † † Π1 ∼ lεl − q εq ð4:19aÞ notation to draw a parallel between Eq. (4.15) and SM S S electroweak symmetry breaking. By counting the broken − † † Π1 ∼ qSεqS − l εl ð4:19bÞ symmetry generators, we identify the gauge bosons that acquire mass through the Higgs mechanism. This will be 0 † † † † Π ∼ lεq − q εl ;q εε q − q εε q ð4:19cÞ discussed in detail shortly, but the result is that five gauge 1 S S D c D D c D “ ” bosons acquire a mass and eat five of the would-be pions. þ † † Π2 ∼ lεqD − q εq ð4:19dÞ Nonzero Σ0 also signals spontaneous breaking of the S D 2 global symmetry acting on the SUð ÞL-charged fermions. In the regime where the Yukawa couplings are neglected, TABLE III. This table summarizes how the WCSM particle we observe that the global symmetry of Eq. (4.14) is 2 1 content transforms under the SUð Þc ×Uð ÞQ subgroup of 3 1 1 broken to SUð Þc ×Uð ÞY and the global Uð Þr. Particles above the double bar correspond with states in Table II, and those below the bar 3 3 3 3 “ ” SOð Þg ×SUð ÞU ×SUð ÞD ×SUð ÞE correspond to composite particles. The States column counts 1 for each Weyl spinor d.o.f. and 1 for each real boson. 1 1 1 1 ×Uð ÞU ×Uð ÞD ×Uð ÞE ×Uð ÞH: ð4:17Þ 2 1 1 States SUð Þc Uð ÞQ Uð Þr 3 The SOð Þg factor corresponds to a real rotation among li 6 1 1=2 0 the three generations. There are thus 13 spontaneously 1 −1 2 qSi 6 = 0 2 broken global symmetry generators, and correspondingly qDi 12 00 we find 13 massless Goldstone bosons remain after Ei 3 1 −1 −1 3 1 1 inclusion of the SUð Þc ×Uð ÞY gauge interactions, while USi 3 11 2 1 2 the remaining 47 pions are lifted at this level. UDi 6 = 1 DSi 3 1 0 −1 D 6 2 −1=2 −1 SU U Di 6. Charges under ð2Þc × ð1ÞQ φ 4 1 1=2 1 Let us take a brief aside to study how the residual G1;2;3 6 3 00 gauge symmetry (4.15) acts on the various fields. For the A0 2 1 00 0 2 1 2 SUð2Þ -doublet fermions, we find that l ∼ 1 for i ¼ 1,2,3 W 12 = 0 L i 0 1 2 R ∼ 1 Z 3 00 transforms as a singlet under SUð Þc, while qi also G B ∼ 2 Ψ0; Ψ0c 6 1 0 1 transforms as a singlet and ðqi ;qi Þ transforms as 1 1 Ψ 1 1 1 a doublet. To emphasize this distinction between the 1 6 Ψ 2 1 2 1 SUð2Þ -singlet and doublet quark fields, we introduce 2 12 = c Ξ0 1 −1 the following notation: 1 3 0 1 1 Ξ1 6 1 2 1 2 R G B Ξ2 12 = 1 qSi ¼ qi and qDi ¼ðqi ;qi Þ; ð4:18Þ 1 1 Π1 6 0 Π0 1 where i ¼ 1, 2, 3 labels the generation. Additionally, 1 13 00 1 2 Π 32 2 1=2 0 Eq. (4.16) reveals that li and qSi carry charges þ = 2 −1 2 1 Π0 9 3 00 and = , respectively, under Uð ÞQ, while qDi is not 3 1 η0 1 1 00 charged under Uð ÞQ. A similar decomposition is also Φ 1 2 1 00 possible for the SUð ÞL-singlet quarks, Ui and Di, while

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2 1 − † † Eq. (4.13) are the generators of SUð Þc. For the Uð ÞQ Π2 ∼ q εq − l εq ð4:19eÞ S D D interaction, Π0 ∼ εσa − † εσa † 3 qD qD qD qD; ð4:19fÞ 0 0 eQ ¼ g cos θQ ≈ g ; ð4:22Þ ε σ2 σa 2 where c ¼ i and are the Pauli matrices with SUð Þc 2 is the gauge coupling and the mixing angle is given by indices, and there is an implicit contraction over SUð ÞL Π 2 indices such that each is an SUð ÞL singlet. Note that 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig these Π’s correspond to a total of 66 states, which includes sin θQ ¼ ; ð4:23Þ 3 2 02 the η0, the five Goldstone bosons eaten by the broken gauge gs þ g 6 fields and the 60 would-be Goldstone bosons. pffiffiffi 0 which is θQ ≈ g = 3gs ≪ 1 in the regime of interest where 0 ≪ SU U g gs. The massless is 8. Composite fermions under ð2Þc × ð1ÞQ The composite fermion fields, Ψ ðxÞ, can also be 0 8 ia Aμ ¼ cos θ Gμ þ sin θ Bμ; ð4:24Þ decomposed into irreducible representations of the residual Q Q gauge symmetry. Using the same notation as above, the and Q from Eq. (4.16) is the generator of Uð1Þ . composite fermions (3.9) correspond to Q

0 − Ψ1 ∼ lφ ð4:20aÞ 10. Massive gauge boson interactions The five remaining gauge bosons correspond to the þ þ Ψ1 ∼ lφ ð4:20bÞ broken symmetry generators of Eq. (4.15). These states can 0 be arranged into a pair of complex vector fields Wμ ðxÞ, Ψ− ∼ φ− 0 1 qS ð4:20cÞ and a single real vector field ZμðxÞ. We use the notation W0 and Z0 to highlight the analogy with the Standard 0c þ Ψ1 ∼ qSφ ð4:20dÞ Model weak gauge bosons, conventionally denoted by W and Z. Their interactions are written as Ψ− ∼ φ− 2 qD ð4:20eÞ X X gs 0i i i∓ DμΣ ⊃−i pffiffiffi Wμ ðL Σ þ ΣL Þ 2 þ þ i¼1;2 Ψ2 ∼ qDφ ; ð4:20fÞ eQ 0 − i ZμðJΣ þ ΣJÞ; ð4:25Þ and their charge assignments are shown in Table III. sQcQ

9. Massless gauge boson interactions where we define the new notation in the remainder of this 3 paragraph. The SUð Þc factor has five broken generators, We write the interactions of the four massless force 4 5 6 7 which are L ; ; ; , and the corresponding gauge fields, carriers, corresponding to the unbroken symmetry group 4;5;6;7 SU 2 ×U 1 , as follows. The covariant derivative from Gμ ðxÞ, pair up to form two complex, massive vector ð Þc ð ÞQ 0 fields, Wμ x corresponding to Eq. (4.11) contains ð Þ 4 5 6 7 X3 L ¼ðL iL ;L iL Þ: ð4:26Þ a a aT DμΣ ⊃−igs GμðL Σ þ ΣL Þ 0 a¼1 The interaction strength of the W vectors is inherited 0 from the SUð3Þ gauge coupling, g . The one final broken − ieQAμðQΣ þ ΣQÞ: ð4:21Þ c s generator is For the SU 2 interactions, g is the SU 3 gauge ð Þc s ð Þc 1 coupling that also determines the SUð2Þ coupling strength, ffiffiffi 8 − 2 c J ¼ p L sQQ; ð4:27Þ 1;2;3 2 1;2;3 3 Gμ are the force carriers of SUð Þc, and L from where s ¼ sin θ and c ¼ cos θ , and the corresponding 6 Q Q Q Q We provide a few examples of counting these d.o.f. For the 0 massive vector field, ZμðxÞ, can be written as Π1 there are three possible combinations of i and j as they must Π be antisymmetric for each of which results in six 1 states 0 8 Zμ ¼ − sin θ Gμ þ cos θ Bμ: ð4:28Þ [equivalently NgðNg − 1Þ where Ng is the number of genera- Q Q 0 tions]. Likewise, for the Π3 pion there are three combinations of 2 0 generational indices each with three possible SUð Þc states which The W transform as a charged doublet under 3 − 1 2 Π0 2 1 0 result in nine [ NgðNg Þ= ] pion states of 3. SUð Þc ×Uð ÞQ while Z is a singlet; see Table III.

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11. Gauge boson spectrum correspondingly, only on the fermion states in the IR. The charge can be written as the following combination of The scalar field ΣijðxÞ acquires a nonzero vacuum expectation value, given by Eq. (4.8), and the Higgs U(1) charges: mechanism splits the spectrum of gauge bosons. The mass X B terms can be derived from Eqs. (4.21) and (4.25) by −2 − L −2 B − L r ¼ Y þ 3 i ¼ Y þ : ð4:30Þ using Σ ¼ Σ0 from Eq. (4.8). The gauge boson spectrum i is summarized as follows: This symmetry is an exact global symmetry of the

1 2 3 0 mG ; ; ¼ ð4:29aÞ Lagrangian and is not anomalous under the surviving gauge symmetries in the IR. It has important consequences for the fermion spectrum and baryogenesis. The charges of mA0 ¼ 0 ð4:29bÞ pffiffiffiffiffiffiffiffi the spectrum under it are shown in Table III. There are thus two broken global symmetry generators and, correspond- m 0 ¼ 3=2g f ð4:29cÞ W s ingly, two massless pions in the WCSM. The 11 other pions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 pffiffiffiffiffiffiffiffi that were massless after inclusion of the SUð Þc ×Uð ÞY 2 02 mZ0 ¼ 2=3 3gs þ g f: ð4:29dÞ gauge interactions (see below Eq. (4.17)) get lifted, such that a total of 58 of the 60 physical pions are massive. 1;2;3 0 The four massless vector fields, Gμ and Aμ, are the force 2 1 carriers of SUð Þc ×Uð ÞQ, while the two complex vectors 13. Gauge corrections to pion masses 0 0 Wμ and the single vector Zμ acquire mass through the The strong and hypercharge gauge interactions (for 0 Higgs mechanism. gs ≠ 0 and g ≠ 0) induce radiative corrections that split the pion mass spectrum. Working in the mass basis of the 12. Symmetry breaking including Yukawa interactions vector bosons, where the leading contribution for each gauge boson eigenstate sums up without mixing, the Finally we discuss the ultimate spectrum of pions. Recall relevant mass-correction operators are written as [46] that for vanishing strong, hypercharge, and Yukawa cou- Σ 2 X plings, the scalar ijðxÞ encodes 65 massless Goldstone g Πa “ ” ΔL C Λ2 f2 s Tr LaTΣ†LaΣ bosons ðxÞ, which we called pions, and one massive ¼ G W 16π2 ½ pseudoscalar η0ðxÞ, which is associated with an anomalous a¼1;2;3 1 2 2 Uð Þψ and acquires mass from SUð ÞL instantons. Taking e C Λ2 f2 Q Tr QΣ†QΣ the strong, hypercharge, and Yukawa couplings to be þ A W 16π2 ½ nonzero breaks the large global symmetry of Eq. (4.2) g2=2 X X 3 Λ2 2 s iΣ† iΣ down to the Uð1ÞB 3−L of Eq. (4.1). þ CW Wf 2 Tr½L L = i 16π Thus we expect that all but at most three of the pions will i¼1;2 have their masses lifted by the effect of nonzero g , g0, and e2 =s2 c2 s Λ2 2 Q Q Q Σ† Σ λ 3 1 þ CZ Wf 2 Tr½J J ; ð4:31Þ Yuk. Since the gauge groups SUð Þc and Uð ÞY are 16π subgroups of SUð12Þψ ×SUð21Þξ, the gauge generators ≈ 0 a are spurions of SUð12Þψ transforming as adjoints. The where eQ g was given by Eq. (4.22), L was given by Yukawas are spurions transforming as antifundamentals of Eq. (4.13), Q was given by Eq. (4.16), L was given by Λ ≈ 4π SUð12Þψ , as well as fundamentals under SUð12Þξ as Eq. (4.26), J was given by Eq. (4.27), and W f. Each ’ summarized in Table IV. term corresponds to a one-loop correction to the pion s self- 1;2;3 The global symmetry is also spontaneously broken, energy; the first term arises from a G loop, the second 1 from an A0 loop, the third from a W0 loop, and the fourth but there is an unbroken Uð Þr symmetry acting only on the weak singlet fermions and Higgs in the UV or, term arises from a Z0 loop. The dimensionless operator coefficients—CG, CA, CW, and CZ—encode the details of the loop calculation, and they are expected to be Oð1Þ TABLE IV. In the spurion analysis of Sec. IV B, this table numbers [47,48] with C ¼ C since these two operators 3 a A G shows the charge assignments of the SUð Þc generators T , the 1 λ λ λ both correspond to massless vectors in the loop. Uð ÞY generator Y, and the Yukawa couplings u, d, and e.

SO(4) Uð12Þψ Uð21Þξ 14. Yukawa corrections to pion masses 2 2 12 1 21 1 Yukawa interactions between the pions and fermions (for SUð ÞL SUð Þϕ SUð Þψ Uð Þψ SUð Þξ Uð Þξ λ ≠ 0) also contribute to the pion’s self-energy at one- Ta, Y 1 1 143 0 1 0 Yuk loop order. The leading nontrivial contributions due to the λ , λ , λ 12 12 −qψ 21 −qξ u d e Yukawas come in at fourth power, taking the form,

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2 2 2 Λ X X symmetry breaking pattern, SUð2NfÞ=Spð2NfÞ, is radia- Wf † † T ΔL C Tr λ λ Σ λ 0 λ 0 Σ ; ¼ Yuk 16π2 ½ð ka kaÞ ð k b k bÞ tively stable modulo small corrections. k;k0 a;b¼1 ð4:32Þ 16. Fermion mass spectrum As we have discussed in Sec. IVA, the composite λ λ where u is the up-type quark Yukawa matrix, d is the fermions Ψ get a mass of order Λ from the confinement λ ia W down-type quark Yukawa matrix, e is the charged lepton process. However, once we include the Yukawa inter- 0 Yukawa matrix and the indices k, k run over u, d, e. These actions, the composite fermions acquire a small mixing are 12 × 12 complex matrices in the basis of Eq. (4.3). 2 ξ with the SUð ÞL-singlet fermions, j, and the spectrum is split. To see this explicitly, we construct the Lagrangian for 15. Pion mass spectrum the fermions. In the UV the Yukawa interactions take the To calculate the pion mass spectrum, we parametrize form, Πa Σ the pions in terms of ij using Eq. (3.5), evaluate X12 X21 X2 Eqs. (4.31) and (4.32), and read off the quadratic terms, ΔL j ¼ ðλ þ λ þ λ Þ φ ψ ξ : ð4:33Þ ΔL − 1 2 2 ΠaΠb Yuk UV u d e ija a i j ¼ ð = ÞðMΠÞab . The leading masses of the i¼1 j¼1 a¼1 pions are shown in Table V. Including all of the mass- 2 correction operators, the pion spectrum contains only two In the IR the SUð ÞL-doublet fermions are confined with massless states, corresponding to the Goldstone bosons the Higgs, and this Lagrangian is mapped to 1 1 3 1 1 associated with Uð ÞY ×Uð ÞB 3−L =Uð ÞQ ×Uð Þr. If all = i X12 X21 X2 C’s are nonzero then one cannot avoid some tachyonic ΔL j ¼ Cλf ðλ þ λ þ λ Þ Ψ ξ ; 0 0 1 ≲ 0 Yuk IR u d e ija ia j states, but choosing CG < , CA < , CZ < , i¼1 j¼1 a¼1 ≲ 1 0 jCWj , and CYuk > leads to only four slightly 0 ð4:34Þ tachyonic Π1 with masses suppressed by the small Yukawas yτ and yc. This indicates that the assumed where Cλ is an Oð1Þ constant. This operator allows the composite Ψia to mix with the elementary ξj, and the TABLE V. Approximate mass spectrum for the pions. To resultant mass eigenstates are mostly composite fermions ∼Λ simplify the expressions we assume that all the C’s are Oð1Þ with masses W and mostly elementary fermions, which Ξ ≪ Λ and show only the leading contributions. The second column we denote by , with masses W. The mostly elementary indicates the multiplicity of each state, although some degener- fermion mass eigenstates correspond to the following acies are split by higher order contributions. Five would-be pions, elementary constituents: which are “eaten” by the massive W0 and Z0 bosons, are not shown in the table. 0 Ξ1 ∼ DS ð4:35aÞ Pion # Squared mass ×16π2=Λ2 W Ξþ ∼ 0 2 1 US ð4:35bÞ Π3 3 −4CGgs 6 −4C g2 2C V 4y4 − G s þ Yukj tbj t Ξ1 ∼ E ð4:35cÞ 3 2 1 2 1 2 4 4 Π2 8 − C g − C e þ C g þ C jV j y 2 G s p2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA Q 2 Z s Yuk tb t Ξþ ∼ − 2 4 2 8 8 2 UD ð4:35dÞ CW gs þ CYukjVtbj yt − 3 2 − 1 2 1 2 4 4 − 8 2 CGgs 2 CAe þ 2 CZgs þ CYukjVtbj yt Ξ ∼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ 2 DD: ð4:35eÞ 2 4 2 8 8 þ CWgs þ CYukjVtbj yt − 3 2 − 1 2 1 2 2 The masses for these fermions arise only in the presence of 12 2 CGgs 2 CAeQ þ 2 CZgs þ CWgs non-zero Yukawa couplings. The charged fermions get 4 − 3 C g2 − 1 C e2 þ 1 C g2 − C g2 2 G s 2 A Q 2 Z s W s Dirac masses via a as follows: Π −2 2 − 2 2 1 6 CAeQ 3 CZgs f2 0 2 4 2 Π 2 mΞ ≈ Cλλ λ ð4:36aÞ 1 2 CYukjVtbj yt 1 e u Λ 2 2 2 2 y0 W 4 CYukjVtbj yt yτ 2 4 4 2 1 CYukjVcsj yc f 1 2 2 2 ≈ 2λ λ 2 − C Re½V y yτ mΞ Cλ u d : ð4:36bÞ 2 Yuk ts t 2 y0Λ −2 2 2 2 W 1 CYukRe½Vts yt yτ − 1 6 Re½VcsRe½Vtd Re½VcdRe½Vts 2 2 Note that the CKM matrix should appear in this mass CYukRe½VcsRe½Vtd 2 ycyμ Re½Vts matrix, but can be rotated to generate an off diagonal W0 20 coupling, as in the SM. The mass of each of generation of

055005-13 BERGER, LONG, and TURNER PHYS. REV. D 100, 055005 (2019) composite fermions is then determined by the product (ix) one Higgs-Higgs composite scalar that is not pro- Λ diagonal Yukawa matrices of that generation as in (4.36). tected by any symmetry and gets a mass of order W; The mostly composite heavy fermions also get some very (x) 12 Dirac fermions corresponding mostly to fermion- 0 small correction due to the Yukawa couplings. The Ξ1 Higgs composite states and they receive Dirac 1 Λ fermions are protected from getting a mass by the Uð Þr masses of order W. symmetry and remain exactly massless Weyl fermions. The (xi) Additional scalar, radial, and spin excitations that are light, mostly elementary fermions mix with the heavy, not included in the nonlinear sigma model. mostly composite fermions with a mixing angle given by V. POTENTIAL COSMOLOGICAL λ IMPLICATIONS θ ∼ Cλ u;d;ef LH Λ ; ð4:37Þ y0 W How would we know if the Universe passed through 2 a phase of confined SUð ÞL? In this section we briefly and the mass eigenstates are partially composite [49]. Since discuss the possible implications for various cosmological the Yukawas are typically λ ≪ 1, this is a small mixing. relics.

C. Summary of the WCSM A. Cosmological phase transition We have studied the symmetries and the lightest states of After cosmological evacuates the Universe of the WCSM. After all interactions and spontaneous symm- matter, we suppose that reheating produces a plasma of metry breaking, the symmetry group of the WCSM is Standard Model particles at temperatures above the scale of weak confinement. As the Universe grows older and expands, the cosmological plasma cools to the confinement SU 2 ×U 1 × U 1 : 4:38 f ð Þc ð ÞQggauge f ð Þrgglobal ð Þ scale, and the confinement of the weak force is accom- plished through a cosmological phase transition. This The lightest states in the spectrum include: transition is expected to be first order, meaning that bubbles (i) two massless Goldstone bosons corresponding to two of the confined phase nucleate in a background of the B 3 − L combinations of spontaneous = i breaking; unconfined phase, and the transition completes as the (ii) three mostly elementary massless Weyl fermions bubbles grow and merge, filling all of space. The first 2 1 corresponding to the SUð Þc ×Uð ÞQ singlets order nature of the weak-confining phase transition follows 1 2 charged under Uð Þr. They are mostly SUð ÞL- from a more general argument by Pisarski and Wilczek singlet down-type fermions; (1983) [50], who considered a general Yang-Mills gauge (iii) four massless gauge bosons of SUð2Þ ×Uð1Þ . c Q theory with Nf flavors of massless vectorlike fermions and (iv) 47 pseudo-Goldstone bosons of the spontaneous showed that it will confine through a first order phase breaking of SUð12Þψ =Spð12Þψ are dominantly lifted transition if Nf ≥ 3. As we have discussed already in by the SUð2Þ ×Uð1Þ gauge interactions. The gauge c Q Sec. IV, the Standard Model corresponds to 2Nf ¼ 12 interactions explicitly break the global symmetry massless flavors of SUð2Þ doublets, and therefore we 12 2 21 3 5 1 4 L SUð Þψ ×SUð Þϕ ×Uð Þξ=SUð Þ ×Uð Þ , which expect the weak-confining phase transition to be first order. gets spontaneously broken in the confined phase to a A first order cosmological phase transition typically pro- 3 3 3 1 global symmetry SOð Þg ×Uð Þ ×Uð Þ, leaving 13 vides the right environment for the creation of cosmologi- massless Goldstone bosons after their inclusion; cal relics, such as topological defects, gravitational wave (v) 11 pseudo-Goldstone bosons of the spontaneous radiation, and the matter-antimatter asymmetry. breaking of SUð12Þψ =Spð12Þψ get lifted by the Yukawa interactions only. The Yukawas explicitly B. Topological defects break SUð3Þ5 ×Uð1Þ4=Uð1Þ3 , which gets spon- B=3−Li When spontaneous symmetry breaking is accomplished 1 taneously broken to Uð Þr. There are only two through a cosmological phase transition, causality argu- massless Goldstones bosons after their inclusion; ments require the symmetry-breaking scalar field to be (vi) nine mostly elementary Dirac fermions that get lifted inhomogeneous on the Hubble scale, and this inhomoge- by the Yukawa couplings. They correspond mostly neity may allow the field to form topological defects, such to the remaining weak-singlet fermions; as cosmic strings and domain walls [51]. In the context of 2 (vii) five massive gauge bosons of the broken SUð Þc × QCD’s chiral phase transition, where the SUð3Þ ×Uð1Þ 1 A A Uð ÞQ gauge group. These eat five of the would- symmetry is spontaneously broken by the chiral quark be pions; condensate, these defects have been called pion and eta- (viii) one η0 pion corresponding to the anomalous prime strings [52]. As the nonzero quark masses break the 1 3 1 Uð ÞBþL global symmetry and acquires a mass of SUð ÞA ×Uð ÞA symmetry explicitly, this may prevent the Λ order W; QCD defects from forming at all, but if they do form, then

055005-14 PHASE OF CONFINED ELECTROWEAK FORCE IN THE EARLY … PHYS. REV. D 100, 055005 (2019) they are expected to be unstable and decay soon after the condensates. Since an asymmetry in a nonconserved global QCD phase transition [53]. Similarly we expect that charge can be washed out in the confined phase, this spontaneous symmetry breaking in the WCSM will gen- observation suggests that WCSM does not have all the erate a string-wall defect network with a rich structure. necessary ingredients for baryogenesis during the confining For instance, spontaneously breaking the global symmetry phase transition. Uð1Þ3 in Eq. (4.1) is expected to generate multiple Nevertheless, one can easily imagine extending the B=3−Li networks of stable cosmic strings, which could survive WCSM by heavy Majorana neutrinos so as to accommo- 2 date the observed neutrino masses and mixings. The until SUð ÞL is deconfined. While the defect network is present, it may play a role in primordial magnetogenesis associated lepton-number violation could provide a means [54] or baryogenesis [55]. of generating r-charge during the confining phase transition, which is later distributed into baryon number C. Gravitational wave radiation when the system deconfines and enters the standard low- temperature Higgs phase. A first order cosmological phase transition produces a Alternatively, it may be possible to accomplish baryo- stochastic background of gravitational waves [56] due to genesis during the deconfining phase transition. For this the collisions of bubbles and their interactions with the scenario to work, the system must pass directly from the plasma. For a weakly coupled theory, it is straightforward confined phase into the Higgs phase where the to calculate the spectrum of this gravitational wave radi- are inactive; if the system passes instead through the ation [57], and gravitational waves have been studied in Coulomb phase then the sphalerons will come back into strongly coupled theories as well [58]. Since is equilibrium and wash out any baryon asymmetry. This extremely weakly interacting, the gravitational wave radi- constraint has the interesting implication that the system ation is expected to free stream to us today where it might must remain in the confined phase even at a time where the be detected. For instance, gravitational waves produced at cosmological plasma temperature is quite low, ≲100 GeV. an electroweak-scale phase transition could be detected by If the weak-confined phase results from the nontrivial space-based gravitational wave interferometer experiments dynamics of a modulus field, such as we discussed in like LISA [59]. Measuring the appropriate spectrum of the model of Eq. (2.1), then this constraint implies a low gravitational wave radiation would provide one of the most mass for the modulus field. direct probes of weak confinement in the early Universe. VI. COMPARISON WITH THE D. The matter-antimatter asymmetry ABBOTT-FARHI MODEL In general a first-order cosmological phase transition 2 We explored the idea that the SUð ÞL weak force may also provides the right environment to explain the cosmo- have been confined in the early Universe, only to become logical excess of matter over antimatter through the physics weakly coupled and Higgsed at later times. Confinement of of baryogenesis [60]. At the weak-confining phase tran- the weak force was first studied in the pioneering work by sition in particular, weak sphalerons are active in front Abbot and Farhi [4,5], which developed into a theory [6–8] of the bubble wall where they mediate baryon-number- that we will call the Abbott-Farhi Model (AFM). The AFM violating reactions, and they become suppressed behind the describes the SM particle spectrum as composite states that bubble wall where SUð2Þ is confined and the weak- 2 L arise from theffiffiffiffiffiffiffi confinement of the SUð ÞL weak force at a magnetic fields are screened. This observation suggests Λ p ∼ 300 scale W ¼ GF GeV. These ideas were motivated that it may be possible to generate a baryon asymmetry in part by the complementarity between a Yang-Mills at the confined-phase bubble walls through dynamics gauge theory in the Higgs phase and the confined phase similar to electroweak baryogenesis [61] or spontaneous [62–65] (see also Refs. [66,67]). While the AFM was able baryogenesis [15,16,20]. to reproduce the observed mass spectrum, interactions, and However, although sphalerons may be suppressed in the even weak boson phenomenology, it was eventually found B L confined phase, this does not imply that þ is effectively to be in tension with electroweak precision measurements conserved, as we find in the SM. As discussed in Sec. IV, at the Z-pole [11,68], leaving the (Higgs-phase) SM as the baryon number is not a good quantum number in the preferred description of weak-scale phenomenology. confined phase where the quarks and leptons condense A key assumption of the AFM, and the essential dis- and acquire a vacuum expectation value, spontaneously tinction with our work here, has to do with the assumed 3 breaking Uð1ÞB ×Uð1Þ . Instead there is only one þL B=3−Li pattern of symmetry breaking. The AFM assumes that the conserved global charge in the confined phase, r ¼ chiral symmetry remains unbroken after confinement (in the −2Y þðB − LÞ from Eq. (4.30), which is not anomalous, regime where the hypercharge, QCD, and Yukawa inter- and consequently an asymmetry in r cannot be generated actions can be neglected). In our notation, this assumption by the weak . All other global charges are violated corresponds to hΣi¼0. If the chiral symmetry is unbroken, by interactions in the Lagrangian or by the quark-lepton then the AFM’s composite fermions, corresponding to the

055005-15 BERGER, LONG, and TURNER PHYS. REV. D 100, 055005 (2019) familiar SM particle content, must be massless. In this way, the Standard Model and new physics may allow the the AFM’s fermion masses are induced by the nonzero WCSM phase to be realized in the early Universe at a hypercharge, QCD, and Yukawa interactions, which time when the cosmological plasma temperature was ∼ Λ explains why these particles are much lighter than the sufficiently high, T W. By assuming a pattern of chiral Λ ∼ 300 confinement scale W GeV. By contrast, we are symmetry breaking that it motivated by numerical lattice motivated by recent numerical lattice studies to assume an studies, we calculate the spectrum of low-lying composite SUð2NfÞ=Spð2NfÞ pattern of chiral symmetry breaking particles, discuss their interactions, and compare this (3.3), which leads to a massive spectrum of composite exotic phase with strong color confinement. Our analysis ∼ Λ fermions with mΨ W. of the WCSM differs notably from earlier work on weak Which of these two symmetry-breaking patterns is the confinement, where it was assumed that confinement did “correct” one? Since the physics controlling symmetry not lead to chiral symmetry breaking. Although direct breaking is inherently nonperturbative, it is prohibitive to tests of the weak-confined phase will be challenging, determine the symmetry breaking pattern through analyti- given the richness of the WCSM spectrum there are cal techniques. Arguably, the assumed absence of chiral several possible cosmological implications of this exotic symmetry breaking that underlies the AFM did not follow phase in the early Universe. These cosmological observ- from any fundamental physical principle, but rather it was ables, including topological defects, gravitational wave taken as a desirable assumption in order to reproduced the radiation, and baryogenesis, may provide fruitful avenues known particle mass spectrum. However, recent numerical for further study. lattice studies [33,34] have provided evidence against the AFM’s assumed absence of symmetry breaking and in ACKNOWLEDGMENTS favor of an SUð2NÞ=Spð2NÞ symmetry-breaking pattern We are grateful to Seyda Ipek whose thought-provoking instead. Motivated by these lattice results, we adopt the colloquium at the Aspen Center for Physics stimulated this SUð2NÞ=Spð2NÞ breaking in our work (3.3), and all of our work. We would also like to thank Bogdan Dobrescu, Nima results for the spectrum and interactions of composite Arkani-Hamed, Anson Hook, William Jay, Aaron Pierce, particles in the early-Universe weak-confined phase are Chris Quigg, Tim Tait, and Jesse Thaler for valuable predicated on this assumption. discussions and encouragement. J. B. is supported by PITT PACC. A. J. L. is supported at the University of VII. SUMMARY Michigan by the US Department of Energy under Grant We have studied a scenario in which the weak force is No. DE-SC0007859. This manuscript has been authored by strong and the strong force is weak. In other words, we Fermi Research Alliance, LLC under Contract No. DE- 2 assume that new physics allows the SUð ÞL weak force to AC02-07CH11359 with the U.S. Department of Energy, Λ become strongly coupled and confine at a scale W > TeV Office of Science, Office of High Energy Physics. This where the QCD strong force is weakly coupled, and work was initiated and completed at the Aspen Center we refer to this scenario as the weak-confined Standard for Physics, which is supported by National Science Model (WCSM). We describe how interactions between Foundation Grant No. PHY-1607611.

[1] S. Glashow, Nucl. Phys. 22, 579 (1961). [12] S. Samuel, Nucl. Phys. B597, 70 (2001). [2] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). [13] C. Quigg and R. Shrock, Phys. Rev. D 79, 096002 [3] A. Salam, Conf. Proc. C680519, 367 (1968). (2009). [4] L. F. Abbott and E. Farhi, Phys. Lett. B 101B, 69 (1981). [14] Y. Bai and A. J. Long, J. High Energy Phys. 06 (2018) 072. [5] L. F. Abbott and E. Farhi, Nucl. Phys. B189, 547 (1981). [15] V. Kuzmin, M. Shaposhnikov, and I. Tkachev, Phys. Rev. D [6] M. Claudson, E. Farhi, and R. L. Jaffe, Phys. Rev. D 34, 873 45, 466 (1992). (1986). [16] G. Servant, Phys. Rev. Lett. 113, 171803 (2014). [7] G. ’t Hooft, in From the Planck length to the Hubble radius, [17] B. von Harling and G. Servant, J. High Energy Phys. 01 Proceedings of the, International School of Subnuclear (2018) 159. Physics, Erice, Italy, 1998 (World Scientific, Geneva, [18] S. A. R. Ellis, S. Ipek, and G. White, J. High Energy Phys. Switzerland, 1998), pp. 216–236. 08 (2019) 002. [8] X. Calmet and H. Fritzsch, Phys. Lett. B 496, 161 (2000). [19] S. Iso, P. D. Serpico, and K. Shimada, Phys. Rev. Lett. 119, [9] X. Calmet and H. Fritzsch, Phys. Lett. B 525, 297 (2002). 141301 (2017). [10] X. Calmet and H. Fritzsch, Phys. Lett. B 526, 90 (2002). [20] S. Ipek and T. Tait, Phys. Rev. Lett. 122, 112001 (2019). [11] F. D’Eramo, Phys. Rev. D 80, 015021 (2009). [21] G. R. Dvali, arXiv:hep-ph/9505253.

055005-16 PHASE OF CONFINED ELECTROWEAK FORCE IN THE EARLY … PHYS. REV. D 100, 055005 (2019)

[22] K. Choi, H. B. Kim, and J. E. Kim, Nucl. Phys. B490, 349 [45] M. Perelstein, Prog. Part. Nucl. Phys. 58, 247 (2007). (1997). [46] S. R. Coleman and E. J. Weinberg, Phys. Rev. D 7, 1888 [23] A. Francis, R. J. Hudspith, R. Lewis, and S. Tulin, J. High (1973). Energy Phys. 12 (2018) 118. [47] T. Das, G. S. Guralnik, V. S. Mathur, F. E. Low, and J. E. [24] P. F. Perez and H. H. Patel, Phys. Lett. B 732, 241 (2014). Young, Phys. Rev. Lett. 18, 759 (1967). [25] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976). [48] V. Ayyar, M. F. Golterman, D. C. Hackett, W. Jay, E. T. Neil, [26] S. Scherer, Adv. Nucl. Phys. 27, 277 (2003). Y. Shamir, and B. Svetitsky, Phys. Rev. D 99, 094504 [27] D. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). (2019). [28] T. Karavirta, J. Rantaharju, K. Rummukainen, and K. [49] D. B. Kaplan, Nucl. Phys. B365, 259 (1991). Tuominen, J. High Energy Phys. 05 (2012) 003. [50] R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984). [29] M. Hayakawa, K. I. Ishikawa, S. Takeda, M. Tomii, and N. [51] T. Kibble, J. Phys. A 9, 1387 (1976). Yamada, Phys. Rev. D 88, 094506 (2013). [52] X. Zhang, T. Huang, and R. H. Brandenberger, Phys. Rev. D [30] A. Amato, T. Rantalaiho, K. Rummukainen, K. Tuominen, 58, 027702 (1998). and S. Thtinen, Proc. Sci., LATTICE2015 (2016) 225. [53] M. Eto, Y. Hirono, and M. Nitta, Prog. Theor. Exp. Phys. [31] V. Leino, K. Rummukainen, J. M. Suorsa, K. Tuominen, 2014, 033B01 (2014). and S. Thtinen, Phys. Rev. D 97, 114501 (2018). [54] R. H. Brandenberger and X.-m. Zhang, Phys. Rev. D 59, [32] V. Leino, K. Rummukainen, and K. Tuominen, Phys. Rev. D 081301 (1999). 98, 054503 (2018). [55] R. H. Brandenberger and A.-C. Davis, Phys. Lett. B 308,79 [33] R. Lewis, C. Pica, and F. Sannino, Phys. Rev. D 85, 014504 (1993). (2012). [56] C. J. Hogan, Mon. Not. R. Astron. Soc. 218, 629 (1986). [34] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica, and F. [57] M. Kamionkowski, A. Kosowsky, and M. S. Turner, Phys. Sannino, Phys. Rev. D 94, 094507 (2016). Rev. D 49, 2837 (1994). [35] J. Preskill, Nucl. Phys. B177, 21 (1981). [58] A. J. Helmboldt, J. Kubo, and S. van der Woude, arXiv: [36] D. A. Kosower, Phys. Lett. 144B, 215 (1984). 1904.07891. [37] G. ’t Hooft, Phys. Rep. 142, 357 (1986). [59] C. Caprini et al., J. Cosmol. Astropart. Phys. 04 (2016) 001. [38] T. Hambye and M. H. G. Tytgat, Phys. Lett. B 683,39 [60] E. W. Kolb and M. S. Turner, The Early Universe (CRC (2010). Press, Baco Raton, Florida, USA, 1990). [39] P. Sikivie, L. Susskind, M. B. Voloshin, and V. I. Zakharov, [61] A. G. Cohen, D. Kaplan, and A. Nelson, Annu. Rev. Nucl. Nucl. Phys. B173, 189 (1980). Part. Sci. 43, 27 (1993). [40] S. Weinberg, Physica (Amsterdam) 96A, 327 (1979). [62] K. Osterwalder and E. Seiler, Ann. Phys. (N.Y.) 110, 440 [41] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1978). (1984). [63] E. H. Fradkin and S. H. Shenker, Phys. Rev. D 19, 3682 [42] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Phys. Lett. B (1979). 412, 301 (1997). [64] T. Banks and E. Rabinovici, Nucl. Phys. B160, 349 (1979). [43] H. K. Dreiner, H. E. Haber, and S. P. Martin, Phys. Rep. 494, [65] S. Dimopoulos, S. Raby, and L. Susskind, Nucl. Phys. 1 (2010). B173, 208 (1980). [44] H. E. Haber, Notes on the spontaneous breakng of suðnÞ and [66] M. Grady, arXiv:1502.04362. soðnÞ via a second-rank tensor multiplet, http://scipp.ucsc [67] E. Seiler, arXiv:1506.00862. .edu/haber/webpage/sun_son.ps. [68] E. Sather and W. Skiba, Phys. Rev. D 53, 527 (1996).

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