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Composite Higgs Lectures on Composite Higgs Adri´anCarmona Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland E-mail: [email protected] Contents 1 Non-Linear Realizations of a Symmetry1 2 Composite Higgs Models6 2.1 General Picture6 2.2 Minimal Composite Higgs Models7 2.2.1 Minimal Composite Higgs model (the real one)7 2.2.2 Minimal Composite Higgs model (the custodial one) 10 1 Non-Linear Realizations of a Symmetry In this lecture, I will mostly follow Jose Santiago's notes, Pokorski [1], the review by Feruglio [2] as well as the original papers by CWZ [3] and CCWZ [4]. Due to several interesting features that we will review in Section2, we will consider theories where the Higgs boson is identified with the pseudo Nambu-Goldstone bosons (pNGB) associated to the spontaneous breaking of some global symmetry G. One of the key ingredients in order to compute the corresponding low energy effective theory is the use of non-linear σ-models, that you have already encountered throughout this course. In the following we will review and introduce some useful concepts, most of which were first introduced in [3,4]. Let us consider a real analytic manifold M, together with a Lie Group G acting on M ' : G × M −! M (1.1) (g; Φ(x)) 7−! T (g) · Φ(x) which we will assume hereinafter to be compact, connected and semi-simple. We will also assume that ' is analytical on its two arguments. The physical situation that we have in mind is that of a manifold of scalar fields Φ(x), with the origin describing the vacuum configuration Σ0, whereas the Lie group G acting on these fields correspond to the symmetry group of the theory1. Let us call H to the continuous subgroup of G formed by all the elements of G leaving the origin invariant, i.e., −1 H = fh 2 G : T (h) · Σ0 = Σ0g = '( · ; Σ0) (fΣ0g) : (1.2) If we assume that H 6= ?, we will be just dealing with the spontaneous symmetry break- ing G ! H. Our goal is somehow to classify all possible non-linear realizations of this symmetry breaking. This is a very ambitious task that fortunately can be reduced to the study of some particular class of them, where H acts linearly on the manifold M. Those 1We will assume for the moment that this group is global, although we will change this later on. { 1 { for which this happens are said to be in the standard form. The first important step in this direction is provided by the so called Haag's theorem [5], which assures the existence of some particular type of coordinate transformation leading to equivalent physical theories. Theorem 1 (Haag's Theorem) If a theory is defined by a Lagrangian density L = L(φ, @µφ) (1.3) depending on a set of scalar fields φ, and the following local transformation of fields is performed φ = F (φ0); (1.4) then the transformed Lagrangian density: 0 0 0 L (φ ;@µφ ) (1.5) defines a new theory with same S−matrix elements provided the transformation (1.4) has a Jacobian determinant equal to one at the origin. The transformations fulfilling the conditions of the previous theorem will be called allowed ones. It turns out that it is always possible to choose coordinates on M (i.e., scalar field representations) such that the action of H on M is linear. Theorem 2 If H is the subgroup of G leaving the origin invariant, then it is always possible to choose coordinates on M so that T (h) · Φ(x) = D(h)Φ(x); 8h 2 H; (1.6) where D(h) is a linear representation of H. As we already mentioned, this set of coordinates is said to be in standard form. Finally, it can be shown that any non-linear representation of the group G acting on M can be always brought to the standard form by means of an allowed coordinate transformation, so we just need to care about the study of the latter. Theorem 3 Any non-linear realization of G can be put into the standard form by an allowed coordinate transformation. Let be Φ0(x) some field which transforms according to a linear representation of G, i.e., T (g) · Φ0(x) = D(g)Φ0(x); 8g 2 G: (1.7) In some neighborhood of the identity g = e, we can write an element of G as, D(g) = exp −iξa^Xa^ exp −iuiY i (1.8) { 2 { where Xa^ and Y i are the broken and unbroken generators, respectively, which we assume a b to be orthonormal with regard to the Cartan-Killing inner product, i.e., Tr(T T ) = δab with T a;T b 2 fXa^;Y ig. The scalar fields ξa^(x) can be seen as the coordinates of the manifold of left cosets G=H (remember that G=H = f[l]; l 2 Gg, where [l] = flh; h 2 Hg) at each point of space-time. The decomposition (1.8) means that every element of the group g 2 G can be uniquely decomposed as a product g = l(ξ)h where h 2 H and l(ξ) is the representative member of the coset to which g belongs, i.e., g 2 [l] with some l(ξ) 2 G. Let us define a^ a^ Φ0(x) = exp −iξ X Φ(x) ≡ UΦ(x); (1.9) Let us now perform an arbitrary G transformation on the field Φ0 0 a a a^ a^ Φ0(x) = exp (−iαaT )Φ0(x) = exp (−iαaT ) exp −iξ X Φ(x) 0 a 0a^ a^ i i = exp −iαaT Φ(x) = exp −iξ (ξ; α)X exp −iu (ξ; α)Y Φ(x); (1.10) where in the last steps we have just used that gl(ξ) = g0(g; ξ) = l(ξ0)h; (1.11) 0 0 a^ where h = h(g; ξ) and ξ = ξ (g; ξ). We can then parametrize Φ0 with the fields ξ (x) and Φ(x), transforming as follows ξa^(x) ! ξ0a^(x) = ξ0a^(g; ξ(x)); (1.12) Φ(x) ! exp −iui(x)Y i Φ(x) = D(h(g; ξ(x)))Φ(x): (1.13) This means that we can represent Φ0(x) by the couple of fields fξ(x); Φ(x)g, where ξ(x) transforms non-linearly under the global group G and Φ transforms locally under the unbroken subgroup H. What happens if the transformation is under the unbroken group H? In this case 0 i a^ a^ i a^ a^ i Φ0(x) = exp −iαiY exp −iξ X 1 Φ(x) = exp −iαiY exp −iξ X exp iαiY i a^ ^b i · exp −iαiY Φ(x) = exp −iξ Ra^^bX exp −iαiY Φ(x) 0^b ^b i = exp −iξ X exp −iαiY Φ(x); (1.14) where Ra^^b is the matrix representation of the linear transformation of the broken generators under the unbroken group, i a^ i ^b exp −iαiY X exp iαiY = Ra^^bX : (1.15) We can thus see that under global transformations h 2 H, both ξa^(x) and Φ(x) transform linearly and globally ^ ξa^(x) ! RT ξb(x); (1.16) a^^b Φ(x) ! D(h)Φ(x): (1.17) { 3 { We would like to use this non-linear realization of the group to construct G-invariant Lagrangians. However, despite the fact that the symmetry is global, the non-linear realiza- tion involves the Goldstone fields ξa^ which transform locally. Thus derivatives have to be transformed into covariant derivatives. To this end, we will consider the following object2 y iξ · X −iξ · X a^ a^ i i !µ ≡ U @µU = e @µe = idµX + iEµY ≡ idµ + iEµ: (1.18) If we compute h −iξ · X i −iξ · X iξ · X h −iξ · X i @µΦ0 = @µ e Φ = e e @µ e Φ −iξ · X h a^ a^ i i i = e idµX Φ + (@µ + iEµY )Φ ; (1.19) and note that @µΦ0 transforms under a global transformation g in the same was as Φ0 does, one can readily conclude that h a^ a^ i i i h a^ a^ i i i idµX Φ + (@µ + iEµY )Φ ! D(h(g; ξ(x))) idµX Φ + (@µ + iEµY )Φ : (1.20) As different representations of the unbroken group H are not mixed by D(h(g; ξ(x))) it is clear that both terms inside the square brackets transform independently. Therefore, a^ a^ a^ a^ 0a^ a^ 0 dµX Φ ! D(h(g; ξ(x)))dµX Φ = dµ X Φ −1 0a^ a^ = D(h(g; ξ(x)))D(h(g; ξ(x))) dµ X D(h(g; ξ(x)))Φ: (1.21) This means in particular that, 0 −1 ^b −iu(g,ξ(x)) · Y iu(g,ξ(x)) · Y dµ = D(h(g; ξ(x)))dµD(h(g; ξ(x))) = dµe X^be ^b a^ = dµR^ba^(g; ξ(x))X : (1.22) Analogously, i i 0 0 @µ + iEµY Φ ! D(h(g; ξ(x))) (@µ + iEµ) Φ = @µ + iEµ Φ −1 0 = D(h(g; ξ(x))) @µ + iD(h(g; ξ(x))) EµD(h(g; ξ(x))) −1 + D(h(g; ξ(x))) @µD(h(g; ξ(x))) Φ; (1.23) and thus 0 −1 −1 Eµ = D(h(g; ξ(x)))EµD(h(g; ξ(x))) + i [@µD(h(g; ξ(x)))] D(h(g; ξ(x))) −1 −1 = D(h(g; ξ(x)))EµD(h(g; ξ(x))) − iD(h(g; ξ(x))) @µD(h(g; ξ(x))) : (1.24) Summarizing, under a global transformation g 2 G, Φ ! D(h(g; ξ(x)))Φ; (1.25) −1 a^ T ^b dµ ! D(h(g; ξ(x)))dµD(h(g; ξ(x))) ; dµ ! Ra^^b(g; ξ(x)) dµ; (1.26) −1 −1 Eµ ! D(h(g; ξ(x)))EµD(h(g; ξ(x))) − iD(h(g; ξ(x)))@µD(h(g; ξ(x))) : (1.27) 2Technically, this is the so called Maurer-Cartan one form.
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