Geometry of the Scalar Sector
aa r X i v : . [ h e p - ph ] M a y Prepared for submission to JHEP
CERN-TH-2016-116
Geometry of the Scalar Sector
Rodrigo Alonso, Elizabeth E. Jenkins, , Aneesh V. Manohar ,
1. Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA2. CERN TH Division, CH-1211 Geneva 23, Switzerland
Abstract:
The S -matrix of a quantum field theory is unchanged by field redefinitions, andso only depends on geometric quantities such as the curvature of field space. Whether theHiggs multiplet transforms linearly or non-linearly under electroweak symmetry is a subtlequestion since one can make a coordinate change to convert a field that transforms linearlyinto one that transforms non-linearly. Renormalizability of the Standard Model (SM) doesnot depend on the choice of scalar fields or whether the scalar fields transform linearly ornon-linearly under the gauge group, but only on the geometric requirement that the scalarfield manifold M is flat. We explicitly compute the one-loop correction to scalar scatteringin the SM written in non-linear Callan-Coleman-Wess-Zumino (CCWZ) form, where it hasan infinite series of higher dimensional operators, and show that the S -matrix is finite.Standard Model Effective Field Theory (SMEFT) and Higgs Effective Field Theory(HEFT) have curved M , since they parametrize deviations from the flat SM case. Weshow that the HEFT Lagrangian can be written in SMEFT form if and only if M has a SU (2) L × U (1) Y invariant fixed point. Experimental observables in HEFT depend on localgeometric invariants of M such as sectional curvatures, which are of order 1 / Λ , where Λis the EFT scale. We give explicit expressions for these quantities in terms of the structureconstants for a general G → H symmetry breaking pattern. The one-loop radiative correctionin HEFT is determined using a covariant expansion which preserves manifest invariance of M under coordinate redefinitions. The formula for the radiative correction is simple whenwritten in terms of the curvature of M and the gauge curvature field strengths. We alsoextend the CCWZ formalism to non-compact groups, and generalize the HEFT curvaturecomputation to the case of multiple singlet scalar fields. ontents ⊂ SMEFT ⊂ HEFT 4 O (4) Fixed Point 82.2 SMEFT 102.3 HEFT 11 O ( N ) Model 14 ππ Scattering 18 M M M O ( N ) Model 48B One-Loop Renormalization of HEFT 48C Non-reductive Cosets 50
C.1 Example of a Non-reductive Coset. 53– 1 –
Introduction
Current experimental data is consistent with the predictions of the Standard Model (SM)with a light Higgs boson of mass ∼
125 GeV. The measured properties of the Higgs bosonagree with SM predictions, but the current experimental accuracy of measured single-Higgsboson couplings is only at the level of ∼ v ∼
246 GeV, the EFT has thesame field content as the SM. There are two main EFTs used in the literature, the StandardModel Effective Field Theory (SMEFT) and Higgs Effective Field Theory (HEFT). In thispaper, we make the relationship between these two theories precise.The Higgs boson h of the SM is a neutral 0 + scalar particle. In the SM Lagrangian, itappears in a complex scalar field H , which transforms as / under the SU (2) L × U (1) Y electroweak gauge symmetry. An oft-stated goal of the precision Higgs physics program is totest whether (a) the Higgs boson transforms as part of a complex scalar doublet which mixeslinearly under SU (2) L × U (1) Y with the three “eaten” Goldstone bosons ϕ , or (b) whetherthe Higgs field is a singlet radial direction which does not transform under the electroweaksymmetry. In case (b), the three Goldstone modes ϕ transform non-linearly amongst them-selves under the electroweak symmetry, in direct analogy to pions in QCD chiral perturbationtheory, and do not mix with the singlet Higgs field. In case (a), there are relations betweenHiggs boson and Goldstone boson (i.e. longitudinal gauge boson) interactions, whereas incase (b), no relations are expected in general. An objective of this paper is to explore thedistinction between these two pictures for Higgs boson physics.The properties of the scalar sector of the SM and its EFT generalizations can be clarifiedby studying it from a geometrical point of view [1]. The scalar fields define coordinates ona scalar manifold M . The geometry of M is invariant under coordinate transformations,which are scalar field redefinitions. The quantum field theory S -matrix also is invariantunder scalar field redefinitions, so it depends only on coordinate-independent properties of M .Consequently, experimentally measured quantities depend only on the geometric invariants of M , such as the curvature. Formulating physical observables geometrically avoids argumentsbased on a particular choice of fields. It also allows us to correctly pose and answer thequestion of whether the Higgs boson transforms linearly or non-linearly under the electroweakgauge symmetry. Further, a geometric analysis gives a better understanding of the structureof the theory and its coordinate-invariant properties.– 2 –he UV theory can have additional states, such as massive meson excitations in the caseof theories with strong dynamics. At low energies, the EFT interactions in the electroweaksymmetry breaking sector are described by a Lagrangian with scalar degrees of freedom onsome manifold M , with the Lagrangian expanded in gradients of the scalar fields. Thegeometric description captures the features of the UV dynamics needed to make predictionsfor experiments at energies below the scale of new physics.The geometrical structure of non-linear sigma models has been worked out over manyyears, mainly in the context of supersymmetric sigma models (see e.g. [2–12]). The applica-tions to the SM Higgs sector presented here are new, and they provide a better understandingof the structure of HEFT and the search for signals of new physics through the couplings ofthe Higgs boson.Some of the results in this paper have already been given in Ref. [1]. Here we providemore explanation of the results presented there, as well as details of explicit calculations inthat work. These calculations include the proof of renormalizability of the SM written in non-linear form, and the derivation of the one-loop effective action for a curved scalar manifold M . For most of the paper, we will assume that the scalar sector has an enlarged globalsymmetry, known as custodial symmetry. Also note that we will usually treat the scalarsector in the ungauged case, referring to the scalar fields as Higgs and Goldstone bosons.The gauged version of the theory follows immediately by replacing ordinary derivatives bygauge covariant derivatives. In the gauged case, the Goldstone bosons are eaten via the Higgsmechanism, becoming the longitudinal polarization states of the massive electroweak gaugebosons. Thus, the Higgs-Goldstone boson relations we refer to are in fact relations betweenthe couplings of the Higgs boson and the three longitudinal gauge boson states W ± L and Z L [13–15].The organization of the paper is as follows. The relationship between the SM, SMEFTand HEFT is discussed in Sec. 2 from a geometrical point of view. It is shown that SMEFT isa special case of HEFT when M is expanded about an O (4) invariant fixed point. Further, itis shown that the existence of such an O (4) invariant fixed point is a necessary and sufficientcondition for the existence of a choice of scalar fields such that the Higgs field transformslinearly under the electroweak gauge symmetry. In Sec. 3, a scalar field redefinition is per-formed on the SM Lagrangian to write it in terms of the non-linear exponential scalar fieldparametrization of Callan, Coleman, Wess and Zumino (CCWZ) [16, 17]. In this non-linearparametrization, the SM contains an infinite series of terms with arbitrarily high dimension,but it nonetheless remains renormalizable. We demonstrate renormalizability of the CCWZform of the Lagrangian by an explicit calculation of the one-loop correction to φφ → φφ .The S -matrix is finite, even though Green’s functions are divergent. The one-loop calcula-tions in the linear and non-linear parameterizations only differ by equation-of-motion terms.Both parameterizations have a divergence-free S -matrix at one loop after including the usualcounterterms computed in the unbroken phase. Sec. 4 presents the covariant formalism forcurved scalar field space. We discuss global and gauge symmetries in terms of Killing vectorsof the scalar manifold, and we derive the one-loop correction to the effective action for curved– 3 – . In Sec. 5, the geometric formulation of G / H theories is connected with the standardcoordinates of CCWZ. We give formulæ for the curvature tensor in terms of field strengthsfor a general sigma model. We also discuss the extension of the CCWZ standard coordinatesto non-compact groups. As shown in Ref. [1], the sign of deviations from SM values of Higgsboson-longitudinal gauge boson scattering amplitudes is controlled by sectional curvaturesin HEFT. For G / H theories based on compact groups, these sectional curvatures are typi-cally positive. We compute the sectional curvature, and show that in certain cases, it canbe negative. In Sec. 6, we briefly discuss the SM and custodial symmetry violation, and therelation between the SM scalar manifold and the configuration space of a rigid rotator. Sec. 7generalizes HEFT to the case of multiple singlet Higgs bosons. Finally, Sec. 8 provides ourconclusions. Additional formulae are provided in the appendices, including intermediate stepsin the computation of the one-loop correction to HEFT given in Refs. [1, 18], and discussionof the complications for non-reductive cosets. ⊂ SMEFT ⊂ HEFT
In this section, we discuss the scalar sector of the SM and its EFT generalizations, SMEFTand HEFT, as well as the relationship between these three theories. We begin with a summaryof the scalar sector of the SM.The SM scalar Lagrangian (with the gauge fields turned off) is L = ∂ µ H † ∂ µ H − λ (cid:18) H † H − v (cid:19) . (2.1)This scalar Lagrangian is the most general SU (2) L × U (1) Y invariant Lagrangian with terms ofdimension ≤ H that transforms as / under SU (2) L × U (1) Y .As is well-known, the SM scalar sector has an enhanced global custodial symmetry group O (4) ∼ SU (2) L × SU (2) R . This global symmetry can be made manifest by writing the SMcomplex scalar doublet field H in terms of four real scalar fields, H ≡ √ " φ + iφ φ − iφ . (2.2)Substitution in Eq. (2.1) yields the Lagrangian L = 12 ∂ µ φ · ∂ µ φ − λ (cid:0) φ · φ − v (cid:1) , (2.3)where φ = ( φ , φ , φ , φ ). Lagrangian Eq. (2.3) is invariant under G = O (4) global symmetrytransformations φ → O φ , O T O = . (2.4)– 4 –he scalar field φ transforms linearly as the four-dimensional vector representation of theglobal symmetry group G = O (4). The minimum of the potential is the three-sphere S ofradius v , h φ · φ i = v , (2.5)which is the Goldstone boson vacuum submanifold of the SM. The radius of the sphere, v ∼
246 GeV, is fixed by the gauge boson masses. It is conventional to choose the vacuumexpectation value h φ i = v , (2.6)and expand the Lagrangian about this vacuum state in the shifted fields φ ≡ v + h and φ a ≡ ϕ a , a = 1 , , φ = ϕ ϕ ϕ v + h , H = 1 √ " ϕ + iϕ v + h − iϕ . (2.7)The vacuum expectation value h φ i spontaneously breaks the global symmetry group G = O (4)to the unbroken global symmetry group H = O (3). The Goldstone bosons ϕ a , a = 1 , , h transforms as a singlet.We will refer to both the enlarged global symmetries G = O (4) ∼ SU (2) L × SU (2) R and H = O (3) ∼ SU (2) V as custodial symmetries. The unbroken global symmetry group H leads to the relation M W = M Z cos θ W , which is a successful prediction of the SM. Theexperimental success of this gauge boson mass relation implies that custodial symmetry is agood approximate symmetry of the SM.The Lagrangian Eq. (2.3) in terms of shifted fields Eq. (2.7) becomes L = 12 ∂ µ ϕ · ∂ µ ϕ + 12 ( ∂ µ h ) − λ (cid:0) h + 2 h v + ϕ · ϕ (cid:1) . (2.8)The singlet h is the physical Higgs field with mass m h = 2 λv , (2.9)whereas the Goldstone bosons are strictly massless. In the gauged theory, the three Goldstonebosons ϕ a of the G → H global symmetry breakdown are “eaten” via the Higgs mechanism,becoming the longitudinal polarization states of the massive W ± and Z gauge bosons. Notethat the O (4)-invariant potential V ( h , ϕ ) depends on an O (4)-invariant combination of both h and ϕ . – 5 – ✧❛✈ ❤♥✭✦✮ (cid:0) ❙✸✈ Figure 1 . Two-dimensional depiction of the four-dimensional scalar manifold M = R of the SM. TheSM vacuum is the black dot shown in the figure. The origin (green dot) is an O (4) invariant fixed point.The left and right diagrams show the fields in Cartesian and polar coordinates, respectively. O (4)symmetry acts linearly on the Cartesian coordinates. In polar coordinates, h is O (4)-invariant, andthe angular coordinates n ( π ) transform non-linearly under the O (4) symmetry. The scalar manifold M is flat, so the scale Λ setting the curvature is formally infinite. Equating the scalar kinetic energy term in Eq. (2.8) with L KE = 12 g ij ( φ ) (cid:0) ∂ µ φ i (cid:1) (cid:0) ∂ µ φ j (cid:1) , i, j = 1 , , , , (2.10)defines the scalar metric g SM ij ( φ ) = δ ij for the SM scalar manifold M with coordinates givenby the scalar fields φ i . Distances on M are determined by ds = g ij ( φ ) dφ i dφ j .The four-dimensional SM scalar manifold M = R is shown in Fig. 1. The O (4) symmetryacts by rotations. The minimum of the potential is the solid red curve, and forms the three-dimensional Goldstone boson submanifold S of radius v . The parameterization Eq. (2.7)is a Cartesian coordinate system for M centered on the vacuum (black dot), where h is thehorizontal direction, and ϕ a , a = 1 , ,
3, are the three other directions orthogonal to h . Theangular coordinates of S are ϕ a /v . The O (4) symmetry acts linearly on ( ϕ , ϕ , ϕ , v + h ).In Cartesian coordinates, it seems intuitively clear that ϕ a and h interactions are related,given that the four scalar fields belong to the same Higgs doublet Eq. (2.2). However, theprecise relation is subtle. In order to understand this point better, it is instructive to expressthe SM Lagrangian Eq. (2.3) in polar coordinates as well.In polar coordinates, φ = ( v + h ) n ( π ) , n · n = 1 , (2.11)where ( v + h ) is the magnitude of φ , and n ( π ) ∈ S is a four-dimensional unit vector. Thefour shifted scalar fields consist of the three dimensionless angular coordinates π a = π a /v We use h , ϕ for the fields in Cartesian coordinates, and h, π (or h, n ) in polar coordinates. – 6 –the direction of n ( π ) on S ), and the radial coordinate h . The SM Lagrangian in polarcoordinates is L = 12 ( v + h ) ( ∂ µ n ) + 12 ( ∂ µ h ) − λ (cid:0) h + 2 vh (cid:1) . (2.12)An advantage of expressing the SM Lagrangian in polar coordinates is that the three Gold-stone boson fields of n ( π ) are derivatively coupled. In addition, the scalar potential in polarcoordinates only depends on the radial coordinate h , whereas in Cartesian coordinates itdepends on all four scalar fields.The O (4) symmetry transformations of M in polar coordinates are h → h, n → O n , (2.13)so the Higgs field h is invariant under O (4) transformations, and n transforms linearly byan orthogonal transformation that preserves the constraint n · n = 1. Due to the constraint,however, only three of the four components of n are independent. Without loss of generality,one can take the first three components of n to be the independent components. Then, thefourth component n is a non-linear function of the independent components n , , . Thenon-linear constraint n · n = 1 turns the linear O (4) transformation on n into a non-linear transformation when written in terms of unconstrained fields. Thus, the O (4) transformationon the three independent angular coordinates π a /v is a non-linear transformation.Many different parameterizations of n ( π ) in terms of the independent unconstrainedcoordinates π a /v are possible. Two natural non-linear parameterizations are the square rootparameterization and the exponential parameterization, which are defined by n ( π ) = 1 v π π π √ v − π · π , (2.14)and n ( π ) = exp v π π π − π − π − π , (2.15)respectively. For most of this paper, we use the exponential parameterization for n ( π ) sinceit corresponds to the standard coordinates of CCWZ.Rotations in the 12, 13 and 23 planes act linearly on ( n , n , n ), and leave n invariant.However, rotations in the 14, 24 and 34 planes mix ( n , n , n ) and n . For example, a 14rotation gives δn = δθ n , δn = 0 , δn = 0 , δn = − δθ n . (2.16)– 7 –n terms of the independent unconstrained coordinates π a of the square root parameterization,12, 13 and 23 rotations act linearly, but a 14 rotation gives δπ = δθ p v − π · π , δπ = 0 , δπ = 0 . (2.17)The O (4) transformation Eq. (2.17) is non-linear . Consequently, Eq. (2.13) is called a non-linear transformation, since it is non-linear when written in terms of unconstrained coordi-nates ( π , π , π ).In polar coordinates, n and h are very different objects, and it is not at all obvious that n and h interactions are related. Nevertheless, all we have done is switch from Cartesiancoordinates { ϕ a , h } to polar coordinates { π a , h } while keeping the Lagrangian fixed. Thischange of coordinates does not affect physical observables such as S -matrix elements. Anyrelations that exist amongst physical observables must be present irrespective of the choiceof coordinates.We have summarized the standard analysis of the SM in Cartesian and polar coordi-nates. In Cartesian coordinates, the Higgs field h and the three Goldstone fields ϕ a form afour-dimensional representation which transforms linearly under O (4). In polar coordinates,the Higgs field h is an O (4) singlet or invariant, and the three Goldstone bosons π a param-eterizing the S unit vector n ( π ) transform among themselves under the non-linear O (4)transformation law Eq. (2.13). The Higgs boson field h in polar coordinates is not the samefield as the Higgs boson field h in Cartesian coordinates. The relation between the two Higgsboson fields is ( v + h ) = ( v + h ) + ϕ · ϕ , (2.18)so that h = h + ϕ · ϕ v − h ϕ · ϕ v + . . . (2.19)By the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, h and h give the same S -matrix, and both are perfectly acceptable choices for the Higgs boson field. O (4) Fixed Point
We now return to the question of whether the Higgs field transforms linearly or non-linearlyunder the electroweak gauge symmetry, and whether interactions of the Higgs boson and thethree Goldstone bosons (i.e. longitudinal gauge boson polarizations) are related. As we havejust seen, this question is not well-posed in the SM, since the answer depends on the choice ofcoordinates. However, it is intuitively clear that there is an underlying relationship betweenthe couplings of the Higgs and Goldstone bosons in the SM that does not remain valid inthe general context of HEFT. We need to formulate any coupling relations in a coordinate-invariant way. There are two conditions which make the SM special — (i) there is a point The nomenclature “the Higgs field” is misleading, since there is no unique choice for the Higgs field. – 8 – = (or H = ) of M which is an O (4) invariant fixed point, and (ii) the scalar manifold M is flat, i.e. it has a vanishing Riemann curvature tensor. As we now see, relations inthe SM between the couplings of the Higgs boson and the three Goldstone bosons arise fromthese two conditions which are no longer true in HEFT in general.We first analyze whether the Higgs field is part of a multiplet that transforms linearly under the O (4) symmetry. Even in the SM, the answer to this question depends on the choiceof coordinates. The coordinate-invariant formulation of the question is: Does there exist achoice of coordinates for M such that the Higgs field is part of a multiplet that transforms linearly under the O (4) symmetry? We now show that the answer is yes if and only if M hasan O (4) invariant fixed point. It is clear from the O (4) transformation law Eq. (2.4) for φ that the origin φ = is an O (4) invariant fixed point. Any other theory that can be formulated using fields φ whichtransform linearly under the O (4) symmetry also must have an O (4) invariant fixed point at φ = . Thus, if there exists a choice of coordinates φ which transform linearly under the O (4) symmetry, then the scalar manifold M has an O (4) invariant fixed point.Now, we prove the converse statement. Consider a general scalar manifold M , whichis described by coordinates which transform under O (4) transformations and which containsan O (4) invariant fixed point P . Is there a choice of coordinates such that the scalar fieldstransform linearly under the O (4) symmetry? The key result we need for the proof in thisdirection is the linearization lemma of Coleman, Wess and Zumino [16], which states that if P is an O (4) invariant fixed point, there exists a set of coordinates in a neighborhood of P whichtransform linearly under O (4) transformations in some (possibly reducible) representation of O (4). If this O (4) representation contains the four-dimensional vector representation of O (4),then the four coordinates φ i , i = 1 , , ,
4, which transform as a vector, can be combined intoa Higgs doublet H , as in Eq. (2.2). Thus, the Higgs field is part of a linear representation H if and only if there is an O (4) invariant fixed point whose tangent space transforms under O (4) in a representation that contains the vector representation. In most of our examples,the scalar manifold is four-dimensional, and the tangent space of P automatically transformsas the vector representation, so we will omit the condition that the tangent space transformsas the vector representation.The condition that M contains an O (4) fixed point divides theories into those which canand cannot be written in a form where the Higgs boson is part of a multiplet that transforms linearly under the electroweak gauge symmetry group G gauge = SU (2) L × U (1) Y (or the largerglobal custodial symmetry group G = O (4) = SU (2) L × SU (2) R ). There are theories whichsatisfy the condition that M contains an O (4) invariant fixed point, but which do not haverelations between the couplings of the Higgs boson and the Goldstone bosons. To understandthis point better, we now introduce SMEFT and HEFT. In Cartesian coordinates, g SM ij ( φ ) = δ ij , and it trivially follows that the Riemann curvature tensor vanishes.Since the curvature is coordinate independent, it also vanishes in polar coordinates, even though the metric ismore complicated. In theories without custodial symmetry, the fixed point is SU (2) L × U (1) Y invariant. – 9 – .2 SMEFT SMEFT is an effective theory with the most general Lagrangian written in terms of SMfields, including all independent higher dimension operators with dimension greater than four,suppressed by an EFT power counting scale Λ. The independent operators at dimension six,and their renormalization [19, 20], has been worked out in detail [21–28].In SMEFT, all operators involving scalar fields are written in terms of the Higgs doubletfield H . For simplicity, at present we assume that the custodial symmetry group of SMEFTis G = O (4). The SMEFT scalar kinetic energy term, which consists of all operators builtout of Higgs doublet fields with two derivatives, is L KE = ∂ µ H † ∂ µ H + 1Λ d − X i C i O ( d ) i = ∂ µ H † ∂ µ H + 1Λ C HD (cid:16) H † ∂ µ H (cid:17) ∗ (cid:16) H † ∂ µ H (cid:17) + · · · , (2.20)where the sum in the first line is over all independent mass dimension d operators built outof two derivatives and Higgs doublet fields H † and H , and the second line gives the explicitexpression including the leading d = 6 operator. Using Eq. (2.2) to write the Higgs doublet H in terms of four real scalars φ , yields a scalar kinetic energy term of the form L KE = 12 " A (cid:18) φ · φ Λ (cid:19) ∂ µ φ · ∂ µ φ + B (cid:18) φ · φ Λ (cid:19) ( φ · ∂ µ φ ) Λ , (2.21)where the arbitrary functions A ( z ) and B ( z ) are defined by power series expansions in theirargument z ≡ φ · φ / Λ . In the Λ → ∞ limit, the kinetic energy term of SMEFT reduces tothe SM kinetic energy term, so the functions A ( z ) and B ( z ) satisfy A (0) = 1 and B (0) = 0.Comparison of Eq. (2.21) with Eq. (2.10) yields the SMEFT scalar metric g ij ( φ ) = A (cid:18) φ · φ Λ (cid:19) δ ij + B (cid:18) φ · φ Λ (cid:19) φ i φ j Λ . (2.22)The Riemann curvature tensor R ijkl ( φ ) of the curved scalar manifold M in SMEFT can becalculated from the above metric. The SM is a special case of the SMEFT in which all higherdimension operators with d > → ∞ .From Eq. (2.22), we see that in this limit the SMEFT metric yields the SM scalar metric g SM ij ( φ ) = δ ij in Cartesian coordinates, and M → R becomes flat with vanishing Riemanncurvature tensor.Most composite Higgs models [29, 30] can be written in SMEFT form. A simple exampleis the SO (5) → SO (4) composite Higgs model [31]. The symmetry breaking field lives on asphere of radius f in five dimensions, and can be written as " φ p f − φ · φ . (2.23)– 10 – is the SMEFT field, and the Lagrangian can be written in SMEFT form. In general,composite Higgs theories solve the hierarchy problem by vacuum misalignment. There isa field configuration where the vacuum is “aligned,” so that the electroweak symmetry isunbroken. This is the point φ = of SMEFT, and φ measures deviations from this point,as in Eq. (2.23). In the neighborhood of φ = , φ gives a linear representation of O (4). ForHEFT to reduce to SMEFT form, this representation must transform as the vector of O (4).Composite Higgs models which are consistent with experimental data are of this type [32, 33].The SMEFT is the EFT generalization of the SM where the scalar manifold has an O (4)invariant fixed point, so that the Lagrangian can be written in terms of the Higgs doubletfield H or the four-dimensional vector field φ on which the O (4) symmetry acts linearly.This restriction is not enough to give the same scattering amplitudes of Higgs bosons andGoldstone bosons (longitudinal gauge bosons) as the SM, which can be verified by explicitcomputation using Eq. (2.21). In Refs. [1, 34], it was shown that the high energy behavior ofthe cross sections for W L W L → W L W L and W L W L → hh scattering depend on two sectionalcurvatures which can be obtained from the Riemann curvature tensor R ijkl ( φ ). The one-loop radiative correction in the scalar sector also depends on the Riemann curvature tensor R ijkl ( φ ) [1]. The details of these calculations are presented later in this paper. The importantpoint is that the φφ → φφ scattering cross sections and the one-loop radiative correction inSMEFT are equal to the SM values if and only if M is flat , i.e. the Riemann curvaturetensor of SMEFT vanishes. This statement is a coordinate-independent condition, which istrue in the SM using either Cartesian or polar coordinates. Thus, the intuitive idea that theGoldstone boson and Higgs boson directions in Fig. 1 are related in the SM can be formulatedprecisely as the condition that M in the SM is a four-dimensional flat Euclidean space. HEFT is a generalization of the SM using the polar coordinate form of the SM Lagrangian,Eq. (2.12). The theory is written in terms of three angular coordinates π a /v that parametrizea unit vector n ( π ) ∈ S , and one or more coordinates { h i } . As in the SM, the unit vector n parametrizes the Goldstone bosons directions [35–39]. Here we restrict to one additional h field. The case of multiple { h i } is considered in Sec. 7. The coordinate h is chosen so that h = 0 is the ground state. The HEFT Lagrangian is L = 12 v F ( h ) ( ∂ µ n ) + 12 ( ∂ µ h ) − V ( h ) + . . . (2.24)where F ( h ) is an arbitrary dimensionless function with a power series expansion in h/v [40],normalized so that F (0) = 1 , (2.25)since the radius of S in the vacuum is fixed to be v by the gauge boson masses. TheHEFT manifold is shown schematically in Fig. 2. M has a coordinate h , with an S fiberat each value of h . While h is often called the radial direction by analogy with the polar– 11 – ✦✈❙✸ (cid:0)✵ Figure 2 . The HEFT scalar manifold. There is S for each value of h . An O (4) invariant fixed pointexists if there is a value of h for which the radius of S vanishes. The fixed point φ at h = h ∗ isshown in a dotted region of M since it need not exist. There is no boundary at the transition betweenthe solid and dotted regions, if the dotted region does not exist. Instead, the manifold can extend toinfinity, or is smoothly connected without a point where F ( h ) = 0. SMEFT has a scalar manifoldwhere φ = 0 is an O (4) invariant fixed point that always exists, and are like the HEFT manifoldincluding the dotted section. coordinate form of the SM, in HEFT, h is simply a scalar field, and need not be the radiusof anything. HEFT power counting is discussed in [41], and is a combination of chiral powercounting [42, 43] and naive dimensional analysis [44]. The terms omitted in Eq. (2.24) arethe NLO operators [45–49].The O (4) transformation laws for h and n are given in Eq. (2.13), so h is invariant and n transforms non-linearly. The SM and SMEFT are both special cases of HEFT. In the SM,the radial function is F SM ( h ) = (cid:18) hv (cid:19) . (2.26)The SMEFT kinetic energy term Eq. (2.21) yields the polar coordinate kinetic energy term L = 12 ( v + h ) A ( z ) ( ∂ µ n ) + 12 [ A ( z ) + z B ( z )] ( ∂ µ h ) , z = ( v + h ) Λ . (2.27)This kinetic energy term can be put into the standard form of HEFT by performing a fieldredefinition on h to make the coefficient of the ( ∂ µ h ) term equal to 1 /
2. Thus, the HEFTscalar metric for one singlet Higgs field is g ij ( φ ) = " F ( h ) g ab ( π ) 00 1 , (2.28)where the function F ( h ) is parametrized by coefficients c n , n ≥ F ( h ) = 1 + c (cid:18) hv (cid:19) + 12 c (cid:18) hv (cid:19) + · · · . (2.29)– 12 –he coefficient c is already constrained by experiment to be equal to its SM value c = 1 toa precision of about 10%. The coefficient c is not constrained at present. The HEFT scalarmetric reduces to the SM scalar metric when F ( h ) = F SM ( h ) = 1 + h/v .In the SMEFT, the functions A and B in Eq. (2.22) are expanded out in powers of φ · φ ,whereas in the HEFT literature, they are treated as arbitrary (unexpanded) functions.When is it possible to rewrite HEFT in SMEFT form? We have seen that a necessaryand sufficient condition is that there must exist an O (4) invariant fixed point P on M . Onecan then define φ as coordinates around P and write the Lagrangian in terms of φ . Thegeneral HEFT manifold consists of h and a sequence of spheres of radius vF ( h ) fibered overeach point of h . The HEFT manifold is depicted in Fig. 2. O (4) acts on the point n onthe surface of S by rotation, so that O (4) maps points on the the red curve onto itself. Nopoint of S is invariant under the full O (4) group, so the only way to have an O (4) invariantfixed point is if the sphere has zero radius, i.e. if F ( h ∗ ) = 0 for some h ∗ . Such a pointmay not exist; its existence depends on the structure of the HEFT manifold. For example, if F ( h ) = e h/v cosh(1 + h/f ) the HEFT manifold has no O (4) invariant fixed point. In the SM, F ( h ) is given by Eq. (2.26), and F ( h ∗ ) = 0 at h ∗ = − v . If there is an O (4) fixed point, theHEFT can be written as a SMEFT. Some examples are given in Refs. [49–51].To summarize, HEFT with no O (4) invariant point, i.e. no point where F ( h ) = 0, cannotbe written in SMEFT form, and hence cannot be written using a doublet field H (or equiv-alently, a four-dimensional vector field φ ) which transforms linearly under the electroweakgauge symmetry. This statement answers the question posed in the introduction: when dothe scalar fields of HEFT transform linearly or non-linearly under the gauge symmetry? Theytransform linearly if and only if F ( h ∗ ) = 0 for some h ∗ , so that there is a O (4) fixed point.Thus, we have shown that the relationship of the SM, SMEFT and HEFT is describedby the hierarchy SM ⊆ SMEFT ⊆ HEFT. SMEFT is a special case of HEFT when there isa value of the Higgs field h ∗ where F ( h ∗ ) = 0. The SM is the special case of SMEFT (andHEFT) when there are no higher dimension operators in the theory, and so M is flat.One can convert the SMEFT Lagrangian to HEFT form using Eq. (2.11) to switch fromCartesian and polar coordinates. One can attempt to convert from HEFT to SMEFT formusing φ ( φ · φ ) / = n (2.30)with ( φ · φ ) / some function of h . This substitution gives a Lagrangian L ( φ ) that need notbe analytic in φ . However, if there is an O (4) fixed point, then there is a suitable change ofvariables such that the resulting Lagrangian is analytic in φ .Scattering amplitudes are evaluated in perturbation theory by expanding the action insmall fluctuations about the vacuum (the black dot) in Fig. 2. The curvature of M is a local quantity, given by the metric and its derivatives up to second order, evaluated at thevacuum state. Scattering amplitudes, and hence experimentally measurable cross sections– 13 –epend directly on the curvature [1, 34], so the curvature of the EFT scalar manifold can bedetermined experimentally.Whether there is an O (4) invariant fixed point where F ( h ∗ ) = 0 is a non-perturbativequestion, since F (0) = 1 in the ground state. One has to move a distance of at least h ∼ v away from the ground state to probe the existence of a fixed point where F ( h ) vanishes. O ( N ) Model
One of the main points of Refs. [1, 34] and this paper is that the scalar sector can be studiedin a coordinate-invariant way. Thus, the SM written in the linear Cartesian coordinatesEq. (2.8), and the SM written in non-linear polar coordinates Eq. (2.12), are completelyequivalent formulations of the same theory. In particular, even though Eq. (2.12) is a non-linear formulation of the SM, where the Lagrangian contains operators of arbitrarily highdimension, it is still renormalizable. In this section, we demonstrate this result by explicitcomputation of the one-loop φφ → φφ scattering amplitude. It is instructive to see howthe theory is renormalizable even when written in non-linear form — we find that Green’sfunctions can be divergent but the S -matrix is finite. We compute the scattering amplitudein the O ( N ) theory in the linear and non-linear formulations. The SM is the special case N = 4. Our results are related to the well-known calculations by Longhitano [35, 36] and byAppelquist and Bernard [37, 38] in the non-linear sigma model with no Higgs field, and byGavela et al. [52] in HEFT. The O ( N ) sigma model has an N -component real scalar field φ i = ( φ , . . . , φ N ) with La-grangian L = 12 ∂ µ φ · ∂ µ φ − λ (cid:0) φ · φ − v (cid:1) , (3.1)which is invariant under transformations φ → O φ , O T O = 1 , (3.2)where O is a real N × N orthogonal matrix. The global symmetry group of the theory is G = O ( N ), which has N ( N − / v >
0. The minimum of the potential in Eq. (3.1) is at h φ · φ i = v , so the set of minimaform the surface S N − , the sphere in N -dimensions, with radius v . All points on S N − areequivalent vacua. One can make an O ( N ) transformation so that h φ i ≡ φ = v χ , χ = , (3.3)– 14 –here χ is a unit vector pointing to the North pole of the sphere. The global symmetrygroup G = O ( N ) of the theory is spontaneously broken to the subgroup H = O ( N − φ invariant. The vacuum manifold is G / H = S N − . The number ofbroken generators is N ϕ = ( N − N ϕ Goldstone bosons.The generators of O ( N ) are[ M ab ] i j = − i (cid:0) δ ia δ jb − δ ja δ ib (cid:1) , ≤ a < b ≤ N, (3.4)where the non-zero entries of M ab have − i in row a , column b , and i in row b , column a . It isoften convenient to consider M ab without the restriction a < b , which includes each unbrokengenerator twice, since M ab = − M ba . The matrices have been normalized so thatTr M ab M cd = 2 ( δ ac δ bd − δ ad δ bc ) . (3.5)The broken O ( N ) generators are[ X a ] i j ≡ [ M aN ] i j = − i (cid:0) δ ia δ jN − δ ja δ iN (cid:1) = − i · · · · · · · · · − · · · , a = 1 , . . . , N − , (3.6)The unbroken O ( N ) generators are M ab , 1 ≤ a < b ≤ N −
1, which are the generators of the O ( N −
1) subgroup.The unbroken transformations with the vacuum choice φ are O ( N −
1) rotations thatleave the North pole fixed, i.e. rotations among the first ( N −
1) components of φ . Of course,one could have picked any other vacuum state φ n , a vector of length v pointing in somedirection n , which is invariant under H n , O ( N −
1) transformations that leave n fixed. Since φ can be rotated to φ n by a G = O ( N ) transformation, the two vacua are equivalent and H n is conjugate to H , H n = g H g − , where g ∈ O ( N ) is the transformation that maps φ to φ n , φ n = g φ .In the linear realization, one expands about the classical vacuum φ in Cartesian coor-dinates φ ( x ) = ϕ ( x )... ϕ N ϕ ( x ) v + h ( x ) . (3.7)The Lagrangian Eq. (3.1) with this field parametrization is L = 12 ( ∂ µ h ) ( ∂ µ h ) + 12 ∂ µ ϕ · ∂ µ ϕ − λ (cid:0) h + 2 h ϕ · ϕ + ( ϕ · ϕ ) + 4 v h + 4 v h ϕ · ϕ + 4 h v (cid:1) . (3.8)– 15 –he unbroken global symmetry subgroup H = O ( N −
1) under which ϕ is a vector is manifestin this coordinate system, but the original global symmetry group G = O ( N ) of the underlyingtheory is not obvious. From Eq. (3.8), we see immediately that all ϕ are massless, and h ismassive with m h = 2 λv . (3.9)The masses and couplings in Eq. (3.8) are given in terms of two parameters λ and v , whichis a reflection of the hidden O ( N ) invariance of the theory.We now parameterize the O ( N ) model in a different way, following the non-linear real-ization of CCWZ. Let φ ( x ) = [ v + h ( x )] ξ ( x ) χ , (3.10)where ξ ( x ) ≡ exp ( Π ) = exp 1 v . . . π . . . π ... ... ...0 . . . π N ϕ − π . . . − π N ϕ , Π ≡ iπ a X a v . (3.11)Eq. (3.10) is a polar coordinate system in field space with radial coordinate ( v + h ) and ( N − π /v of the sphere S N − . The field ξ ( x ) is a real orthogonalmatrix, so φ · φ = ( v + h ) . (3.12)The Lagrangian Eq. (3.1) with this field parameterization is L = 12 ( v + h ) χ T ( ∂ µ ξ ) T ( ∂ µ ξ ) χ + 12 ( ∂ µ h ) ( ∂ µ h ) − λ (cid:0) h + 2 hv (cid:1) . (3.13)The potential only depends on the radial coordinate h ; it is independent of the Goldstoneboson fields π , which are massless and derivatively coupled. Expanding the exponential ξ ( x )in a power series gives the leading terms L = 12 (cid:18) hv (cid:19) [ ∂ µ π · ∂ µ π ] + 16 v (cid:18) hv (cid:19) (cid:2) ( π · ∂ µ π ) − ( π · π )( ∂ µ π · ∂ µ π ) (cid:3) + . . . + 12 ( ∂ µ h ) ( ∂ µ h ) − λ (cid:0) h + 2 hv (cid:1) (3.14)The full expression is given in Appendix A. The Lagrangian Eq. (3.14) naively looks like anon-renormalizable theory with an infinite set of higher dimension operators. However, it issimply the renormalizable Lagrangian Eq. (3.1) written using a different parametrization ofthe fields. – 16 –he Lagrangians Eq. (3.1) and Eq. (3.14) correspond to different choices of coordinatesfor the scalar manifold M , and they are related by a field redefinition. Since the S -matrix isinvariant under a field redefinition, the two theories have the same S -matrix. Renormalizabil-ity of Lagrangian Eq. (3.14) is hidden, as is O ( N ) invariance. Treating Eq. (3.14) as an EFTwith the usual power counting rules (for a pedagogical review, see [53]) gives the same S -matrix as Eq. (3.1). In particular, Eq. (3.14) is a renormalizable theory with a finite numberof renormalization counterterms even though it looks superficially non-renormalizable. The linear O ( N ) model including renormalization counterterms is L = 12 Z φ ∂ µ φ · ∂ µ φ − Z λ λµ ǫ (cid:0) Z φ φ · φ − Z v v µ − ǫ (cid:1) . (3.15)In dimensional regularization in 4 − ǫ dimensions, the one-loop counterterms Z φ , Z λ and Z v are given by Z i = 1 + δ i π ǫ , δ φ = 0 , δ λ = λ ( N + 8) , δ v = − λ. (3.16)These renormalization counterterms can be computed using perturbation theory in the un-broken phase, where v <
0. The combinations Z λ and Z λ Z v are the counterterm renormal-izations of the O ( N ) invariant operators ( φ · φ ) and φ · φ , and they are gauge independent.Field theory divergences arise from the short distance structure of the theory. Thusthe renormalization counterterms do not depend on whether the symmetry is unbroken orspontaneously broken; the same counterterms Eq. (3.15) also renormalize the broken theory. In the broken phase, one uses Eq. (3.7) with the replacement v → ( v + ∆ v ) µ − ǫ , φ ( x ) = ϕ ( x )... ϕ N ϕ ( x )( v + ∆ v ) µ − ǫ + h ( x ) , N ϕ = N − . (3.17)The tadpole shift ∆ v has a perturbative expansion in powers of λ , and it is computed orderby order in perturbation theory by cancelling the tadpole graphs to maintain h h i = 0. Attree-level, ∆ v = 0. The Lagrangian Eq. (3.8) including renormalization counterterms is L = 12 Z φ ∂ µ ϕ · ∂ µ ϕ + 12 Z φ ( ∂ µ h ) ( ∂ µ h ) − Z λ λµ ǫ (cid:18) Z φ ϕ · ϕ + Z φ (cid:2) ( v + ∆ v ) µ − ǫ + h (cid:3) − Z v v µ − ǫ (cid:19) (3.18)The Lagrangian Eq. (3.8) gives finite Green’s functions and finite S -matrix elements in thebroken phase. The underlying G -symmetry of the theory ensures that the counterterms in There are subtleties in the gauged case, which are discussed later. – 17 – igure 3 . Some vertices in the Lagrangian Eq. (3.20). Solid lines are h and dashed lines are π . Eq. (3.18) are given in terms of Z φ , Z λ and Z v of the unbroken theory Eq. (3.16), plus a tadpoleshift ∆ v . The ϕ · ϕ term in Eq. (3.18) is a pure counterterm, and keeps the Goldstone bosonsmassless in the presence of radiative corrections. The Higgs mass is m h = 2 λv .In the non-linear realization, one uses φ = (cid:2) ( v + ∆ v ) µ − ǫ + h ( x ) (cid:3) ξ ( x ) χ (3.19)with ξ ( x ) given by Eq. (3.11). Since Eq. (3.19) is simply a different choice of field coordinatesin comparison to Eq. (3.17), the renormalization constants and tadpole shift ∆ v are the same.Note that no Z factor is needed in the exponent of ξ ( x ). The π a /v in Eq. (3.11) are periodicvariables, since a 2 π rotation about some axis is equivalent to the identity transformation,and cannot be multiplicatively renormalized.The renormalized Lagrangian in the non-linear parameterization is L = 12 (cid:2) ( v + ∆ v ) µ − ǫ + h ( x ) (cid:3) Z φ χ T ( ∂ µ ξ ) T ( ∂ µ ξ ) χ + 12 Z φ ( ∂ µ h ) ( ∂ µ h ) − Z λ λ (cid:16) Z φ (cid:2) ( v + ∆ v ) µ − ǫ + h ( x ) (cid:3) − v µ − ǫ (cid:17) , (3.20)with Z φ,λ,v given by Eq. (3.16). This Lagrangian can be expanded in a power series in π andused in perturbation theory. The claim which we wish to prove is that Eq. (3.20) gives finite S -matrix elements (but not necessarily Green’s functions), since it is a field redefinition ofEq. (3.18). ππ Scattering
The finiteness of the S -matrix using Lagrangian Eq. (3.20) seems surprising, and is worthexplaining in some detail. The Lagrangian Eq. (3.20) contains vertices with an arbitrarynumber of fields. For example, it contains the vertices in Fig. 3 which involve five and sixscalar fields. We will use Eq. (3.20) to compute the infinite part of ππ scattering to one loop. In the non-linear case, we will only give the explicit results for the amplitude to O ( p ), but wehave checked that the S -matrix is finite to all orders in p . The skeleton graphs that contributeto the S -matrix for ππ → ππ are shown in Fig. 4. The tree-level amplitude is given by the In our notation, for the linear case, we compute ϕϕ → ϕϕ , and for the non-linear case ππ → ππ . – 18 – a ) ( b ) Figure 4 . Skeleton graphs for the ππ → ππ scattering S -matrix. The shaded blobs are irreduciblevertices and two-point functions. There is also a pion wavefunction correction to the amplitude. ( a ) ( b ) ( c ) Figure 5 . h tadpole graphs. Graph ( c ) includes counterterm and tadpole vertices. skeleton graphs in Fig. 4 with the blobs replaced by tree vertices, and the one-loop correctionto the amplitude is given by using the one-loop irreducible vertex for one blob in each graph,and tree vertices for the rest. We will give the results of the various contributions using thelinear parameterization, Eq. (3.18), and the non-linear one, Eq. (3.20), which will be denotedby subscripts L and N , respectively. In this subsection, we only compute the infinite parts ofthe graphs, and omit an overall factor of i/ (16 π ǫ ).The π tadpole vanishes by O ( N −
1) invariance. The h tadpole graphs are shown in Fig. 5and give the h one-point functionΓ ( h ) L = Γ ( h ) N = 3 λvm h + 0 + (cid:2) − λv ∆ v − λv ( δ φ − δ v ) (cid:3) (3.21)where the three terms are the infinite contributions from the three diagrams. The linear andnon-linear parameterizations give the same result. Using the counterterms from Eq. (3.16), m h = 2 λv , and requiring that Γ ( h ) vanishes gives∆ v = 0 . (3.22)Note that the tadpole shift ∆ v is finite in the non-gauged case, but it develops an infinitepiece when gauge interactions are turned on.The infinite contribution to the h two-point function isΓ ( hh ) L = 3 λm h + 0 + 18 λ v + 2 λ v N ϕ + (cid:2) − λv ∆ v − λv (2 δ λ + 5 δ φ − δ v ) (cid:3) = 0Γ ( hh ) N = 3 λm h + 0 + 18 λ v + p v N ϕ + (cid:2) − λv ∆ v − λv (2 δ λ + 5 δ φ − δ v ) (cid:3) = p v N ϕ − λ v N ϕ = 12 v N ϕ (cid:0) p − m h (cid:1) (cid:0) p + m h (cid:1) (3.23)– 19 – a ) ( b ) ( c )( d ) ( e ) Figure 6 . h propagator graphs. Graph ( e ) includes counterterm and tadpole vertices. ( a ) ( b ) ( c ) ( d ) Figure 7 . π propagator graphs. Graph ( d ) includes counterterm and tadpole vertices. from the individual graphs in Fig. 6. The two forms of the Lagrangian give a different result.In the non-linear parameterization, π is derivatively coupled, so graph (d) is O ( p ); in thelinear parameterization, the graph is O ( p ) since there is a h ϕ · ϕ coupling in the potential.In the non-linear parameterization, the one-loop corrected h propagator in Fig. 4(b) is1 p − m h (cid:20) − π ǫ v N ϕ (cid:0) p + m h (cid:1)(cid:21) (3.24)on expanding out the correction Eq. (3.23), and does not have a double pole in ( p − m h )because Γ ( hh ) N ∝ (cid:0) p − m h (cid:1) . This feature is important for the cancellation of divergences.The π propagator graphs give the infinite contribution to the two-point functionΓ ( ϕϕ ) L = λm h δ ab + 0 + 4 λ v δ ab + (cid:2) − λv ∆ v − λv ( δ φ − δ v ) (cid:3) δ ab = 0 , Γ ( ππ ) N = − v m h p δ ab + 0 + 1 v ( m h + p ) p δ ab + 2∆ vv p δ ab = 1 v p δ ab , (3.25)where a, b are π flavor indices. The loop contribution to the Goldstone boson mass is cancelledby the counterterm and tadpole vertices in the linear parameterization. In the non-linearparameterization, the one-loop correction to the Goldstone boson propagator is O ( p ), by thechiral counting rules. – 20 – a ) ( b ) ( c ) ( d )( e ) ( f ) ( g ) Figure 8 . hππ graphs. Graph ( g ) includes counterterm and tadpole vertices. Graph ( f ) is not presentfor the linear case. The hππ vertex correction from the graphs in Fig. 8 isΓ ( h ϕϕ ) L = 2 λ v ( N ϕ + 2) δ ab + 6 λ vδ ab + 0 + 0 + 8 λ vδ ab + 0 + [2 λ ( − vδ φ − vδ λ − ∆ v )] δ ab = 0 , Γ ( hππ ) N = (cid:26) − v ( N ϕ − p + p ) (cid:2) p + p + 3 p · p (cid:3)(cid:27) δ ab + 6 λv ( p · p ) δ ab − v λ ( p · p ) δ ab + (cid:26) v (cid:0) m h p · p − p p − ( p · p ) − ( p + p ) p · p (cid:1)(cid:27) δ ab + (cid:26) − v (cid:20) m h p · p + 14 p · p ( p + p ) (cid:21)(cid:27) δ ab + 0 − vv ( p · p ) δ ab = 13 v (cid:20) − N ϕ ( p · p ) − (5 N ϕ + 4)( p · p )( p + p ) − ( N ϕ − p ) + ( p ) ) − (2 N ϕ + 10) p p (cid:21) δ ab , (3.26)where the pions have incoming momentum and flavor p , a and p , b , respectively. Graph ( f )does not exist for the linear case.One can already see non-trivial evidence for finiteness of the S -matrix for h → ππ inEq. (3.26). On-shell, only the first term in Γ ( hππ ) N is non-zero, and is precisely cancelled bythe h propagator correction in Eq. (3.24).The π graphs are shown in Fig. 9. The pions have incoming momentum and flavor p i , a i , i = 1 , , ,
4. Let I denote the flavor structure I = δ a a δ a a + δ a a δ a a + δ a a δ a a . (3.27)– 21 – a ) ( b ) ( c ) ( d ) ( e )( f ) ( g ) ( h ) ( i ) Figure 9 . π graphs. Graph ( i ) includes counterterm and tadpole vertices. Graphs ( a, b, c ) do not exist for the linear case. Then, in the linear parameterization, thegraphs giveΓ ( ϕϕϕϕ ) L = 0 + 0 + 0 + 2 λ ( N ϕ + 8) I + 2 λ I + 0 + 0 + 0 + [ − λ (2 δ φ + δ λ )] I = 0 , (3.28)using N ϕ = ( N − O = ( ∂ µ π · ∂ µ π ) ( ∂ ν π · ∂ ν π ) , O = ( ∂ µ π · ∂ ν π ) ( ∂ µ π · ∂ ν π ) ,O = (cid:0) ∂ π · π (cid:1) ( ∂ µ π · ∂ µ π ) , O = (cid:0) ∂ π · ∂ µ π (cid:1) ( π · ∂ µ π ) ,O = (cid:0) ∂ π · π (cid:1) (cid:0) ∂ π · π (cid:1) , O = (cid:0) ∂ π · ∂ π (cid:1) ( π · π ) ,O = − m h (cid:0) ∂ π · π (cid:1) ( π · π ) , O = − m h ( ∂ µ π · ∂ µ π ) ( π · π ) , (3.29)– 22 –he contribution from the loop graphs to Γ ( ππππ ) N is( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) Total O N ϕ − − N ϕ O − O N ϕ − N ϕ + 4 O −
12 4 0 0 0 0 − O N ϕ − − N ϕ O − − O − − O − − / (12 v ). Graph ( i ) is proportional to ∆ v and vanishes by Eq. (3.22).We can now study the finiteness of the S -matrix. In the linear case, all the irreduciblevertices are finite, and so is the S -matrix. The counterterms were chosen to render theirreducible vertices finite in the unbroken sector. Symmetry breaking does not affect theshort distance behavior of the theory, and the same counterterms of the unbroken theory alsomake the broken theory finite.More interesting is the divergence structure of the S -matrix in the non-linear parameter-ization. The infinite part of the total amplitude iA can be written as the matrix element oflocal operators, A = N ϕ v O − v O + 2 N ϕ − v O + 118 v O , (3.31)which vanishes on-shell since ∂ π = 0, so the ππ scattering S -matrix is finite at one-loop,as claimed. Some interesting cancellations are necessary for the on-shell S -matrix to befinite. The operators O , do not vanish on-shell. Eq. (3.30) shows that there is a non-zerocontribution to O in Γ ( ππππ ) N , but not to O . The O contribution is cancelled by the Higgspropagator correction Eq. (3.24) in Fig. 4(b). The Fig. 4(b) amplitude is proportional to thesquare of the tree-level hππ vertex, and only produces the tensor structure O . A bit morealgebra shows that the one-loop S -matrix is finite to all orders in p .The non-covariant terms in the scattering amplitude that vanish on-shell arise from fieldredefinitions in the functional integral. A coordinate transformation is equivalent to a redef-inition of the source, i.e. to a field redefinition in the generators of 1PI graphs Γ(Φ). Startwith an O ( N ) covariant source term L J = J · φ = ( v + h ) ( J · ξ χ ) , (3.32)where ξ χ = " sin | π || π | π a v cos | π | , | π | ≡ π a π a v . (3.33)– 23 – a ) ( b ) Figure 10 . W ( J ) with source term J · φ is covariant, and n -point Green’s functions are given by expanding W ( J ) in powers of J . This expansion is still covariant. On the other hand, the Feynmangraph computation we have done is equivalent to computing with a source J · π and thenLegendre transforming in π . This procedure is not covariant, as explained in detail in thenext section. The covariant version uses the coupling (cid:18) hv (cid:19) (cid:20) J · π − v ( J · π )( π · π ) + . . . (cid:21) . (3.34)The fourth-order contribution to W ( J ) has contributions from the graphs shown in Fig. 10,where graph (b) involves the J π term in Eq. (3.34). Both graphs are the same order in p ,since graph (a) has an extra p from the π interaction, and an extra 1 /p from the extrapropagator relative to graph (b). The two graphs add to give a covariant contribution if oneuses Eq. (3.32) for the source term.Note the following remarks. • Green’s functions are finite in the linear parameterization, but not in the non-linearparameterization. The 4-point function Γ ( ππππ ) N has the divergence Eq. (3.31) which isnot cancelled by any counterterm. • The divergence Eq. (3.31) is not chirally invariant, i.e. it cannot be written in a G = O ( N ) invariant way. Although it explicitly breaks the O ( N ) symmetry, the breakingis unphysical since it does not enter measurable quantities such as on-shell S -matrixelements. • It is not necessary to add a counterterm to cancel Eq. (3.31). One can ignore it, or makea field redefinition (which does not change the S -matrix) to remove it. It is possible toremove it by making a field redefinition because the operator vanishes on-shell. • In the usual computation of hadronic weak decays, one replaces penguin operators (cid:0) ¯ ψT A γ ν P L ψ (cid:1) D µ G Aµν by four-fermion operators using the QCD equations of motion.In this case, penguin graphs are infinite, since there is no counterterm to cancel them,but the weak decay S -matrix is finite. This situation is analogous to the situation we– 24 –re finding in the O ( N ) model. In both cases, the field redefinitions that eliminate theequation of motion terms involve divergent 1 /ǫ terms. • In the O ( N ) non-linear sigma model, i.e. the O ( N ) theory with the h field set to zero,Appelquist and Bernard [37, 38] showed that there were O ( N ) non-invariant countert-erms which vanished on-shell. The reason for these terms is explained in the nextsection. In the Appelquist-Bernard calculation, there were, in addition, O ( p ) diver-gences proportional to O and O at one-loop, as expected in chiral perturbation theory.In our calculation, these higher order in p divergences do not occur because of extragraphs involving h which cancel the divergences. • A similar situation occurs in renormalization of HEFT. Ref. [52] found non-covariantterms 132 π ǫ ((cid:18)
32 + 10 η + 18 η (cid:19) ( ϕ (cid:3) ϕ ) v − c (3 + 10 η ) ϕ (cid:3) ϕ (cid:3) hv ) , which vanish on-shell. Ref. [1] showed that these terms arose from the use of non-covariant perturbation theory due to the non-covariant term in Eq. (4.7).Finally, we return to an important subtlety in the gauged case. As mentioned earlier,the gauged O ( N ) sigma model has different counterterms in the unbroken and broken phases,even though symmetry breaking is an infrared effect and does not change the short distancestructure of the theory. The reason that the counterterms differ is that the O ( N ) theory inthe unbroken phase is quantized using the gauge fixing term L gf = − ξ (cid:0) ∂ µ A aµ (cid:1) (3.35)whereas the broken theory is quantized using the gauge fixing term L gf = − ξ (cid:2) ∂ µ A aµ + iξ ′ gv (cid:0) χ T T a φ − φ T .T a χ (cid:1)(cid:3) . (3.36)Gauge theory counterterms do depend on the gauge fixing term, so the two theories can havedifferent counterterms if they are quantized using different gauge fixing terms. If one usesthe same gauge fixing term for both theories, then the renormalization counterterms are thesame.The renormalizability of the O ( N ) theory in non-linear coordinates will hold to arbitraryloop order, as is obvious from the general arguments given earlier. In this section, we review the well-known geometric formulation of non-linear sigma models [3,4, 11, 43, 54]. The use of functional methods for quantum corrections, combined with a– 25 –ovariant formalism sheds light on a number of technical issues identified in Refs [38, 52].This covariant formalism has wide applicability — the CCWZ phenomenological Lagrangianis a special case of the geometric approach in a particular choice of coordinates, as discussedin Sec. 5. M Consider N real scalar fields φ i which are the coordinates of a curved scalar manifold M .The scalar action for the O ( p ) Lagrangian (with no gauge fields) containing all operatorswith up to two derivatives is S = Z d x L [ φ ( x )] = Z d x (cid:18) g ij ( φ ) ( ∂ µ φ ) i ( ∂ µ φ ) j + I ( φ ) (cid:19) , (4.1)where I ( φ ) is an invariant scalar density on M . The two-derivative terms define the scalarmetric g ij ( φ ) of M . Under a scalar field redefinition or change of scalar coordinates φ ′ ( φ ),the derivative (cid:0) ∂ µ φ i (cid:1) transforms as a contravariant vector ∂ µ φ ′ i = (cid:18) ∂φ ′ i ∂φ j (cid:19) ∂ µ φ j , (4.2)and the metric g ij ( φ ) transforms as a tensor with two lower indices, g ′ ij = (cid:18) ∂φ k ∂φ ′ i (cid:19) (cid:18) ∂φ l ∂φ ′ j (cid:19) g kl . (4.3)Thus, the Lagrangian also is an invariant scalar density. The potential I ( φ ) is non-zero, ingeneral. It is a constant if all the fields φ i are exact Goldstone bosons of an enlarged globalsymmetry.The first variation of the action yields the equation of motion for the field φ . Under aninfinitesimal variation φ → φ + η , the linear in η variation of the action is δS = Z d x (cid:16) − g ij ( D µ ( ∂ µ φ )) i + I , j (cid:17) η j , (4.4)where ( D µ η ) i ≡ ∂ µ η i + Γ ikj ( ∂ µ φ ) k η j (4.5)is the covariant derivative on a vector field η i and Γ ijk ( φ ) is the Christoffel symbol. FromEq. (4.4), one obtains the classical equation of motion E j = g ij ( D µ ( ∂ µ φ )) i − I , j = g ij (cid:16) ∂ φ i + Γ ikj ( ∂ µ φ ) k ( ∂ µ φ ) j (cid:17) − I , j = 0 , (4.6)which is the wave equation for φ on the curved manifold M .The second variation of the action under an infinitesimal variation φ → φ + η is δ S = 12 Z d x (cid:20) g ij ( D µ η ) i ( D µ η ) j − R ijkl η i ( ∂ µ φ ) j η k ( ∂ µ φ ) l − E j Γ jkl η k η l + I ; ij η i η j (cid:21) , (4.7)– 26 –here R ijkl is the Riemann curvature tensor and I ; ij = ∇ i ∇ j I = ∂ I ∂φ i ∂φ j − Γ kij ∂ I ∂φ k . (4.8)Eq. (4.7) is not covariant because of the third term which depends explicitly on the connectionΓ ijk . This term, however, vanishes on shell since it is proportional to the equation of motion E i . The non-covariant term leads to non-covariant divergences in Green’s functions whichvanish in S -matrix elements. Non-covariant terms arose in the explicit calculation in Sec. 3of the O ( N ) model in flat space. Even though they have no physical consequences, theappearance of non-covariant terms is puzzling since the original theory is covariant. Thenon-covariant terms occur because the infinitesimal variation φ → φ + η is not a covariantparameterization of fluctuations to second order in η , as was explained in Ref. [3, 4].The variation of the scalar field η i = δφ i should transform as a vector under a change ofcoordinates. However, under a change of coordinates, φ ′ i ( φ + η ) = φ ′ i ( φ ) + (cid:18) ∂φ ′ i ∂φ j (cid:19) η j + 12 (cid:18) ∂ φ ′ i ∂φ j ∂φ k (cid:19) η j η k + . . . ≡ φ ′ i ( φ ) + η ′ i , (4.9)implies that η ′ i = (cid:18) ∂φ ′ i ∂φ j (cid:19) η j + 12 (cid:18) ∂ φ ′ i ∂φ j ∂φ k (cid:19) η j η k + . . . , (4.10)which is the correct transformation law for a vector at first order in η , but not at secondorder. The solution to this problem is to use geodesic coordinates to parametrize fluctuationsin φ , as shown in Ref. [4]. The equation for a geodesic on M parameterized by λ isd φ i d λ + Γ ijk ( φ ) d φ j d λ d φ k d λ = 0 . (4.11)Solving this equation in perturbation theory, starting at φ i = φ i with tangent vector η i gives φ i = φ i + λη i − λ Γ ijk ( φ ) η j η k + . . . (4.12)Fluctuations in φ are parameterized by picking η i to be tangent vector such that the geodesicreaches φ + δφ at λ = 1, i.e. using the variation φ i → φ i + η i −
12 Γ ijk η j η k + O ( η ) , (4.13)which suffices to restore the correct transformation law for the vector η ′ to second order inthe expansion, η ′ i = (cid:18) ∂φ ′ i ∂φ j (cid:19) η j . (4.14)– 27 –xpanding the action in the geodesic fluctuation η to quadratic order in η yields S [ φ + η ] = S [ φ ] + δSδφ i (cid:18) η i −
12 Γ ijk η j η k (cid:19) + δ Sδφ i δφ j η i η j + O ( η ) , (4.15)which shows that there is a quadratic in η term proportional to the equation of motionoperator E i ≡ (cid:0) δS/δφ i (cid:1) . This contribution exactly cancels the non-covariant term of Eq. (4.7),yielding a second variation of the action which transforms covariantly δ S = 12 Z d x (cid:20) g ij ( D µ η ) i ( D µ η ) j − R ijkl η i ( ∂ µ φ ) j η k ( ∂ µ φ ) l + ( ∇ i ∇ j I ) η i η j (cid:21) . (4.16)An equivalent way to implement the covariant expansion is to promote ordinary functionalderivatives to covariant functional derivatives [11], ∇ i S = δSδφ i , ∇ i ∇ j S = δ Sδφ i δφ j − Γ kij δSδφ k . (4.17)The second variation of the action enters the one-loop correction to the functional integral,Γ one-loop = i (cid:18) − g ik δ Sδη k δη j (cid:19) . (4.18)The one-loop corrections computed using Eq. (4.16) are covariant, since δ S is covariant. Thetwo forms for δ S , Eq. (4.7) and Eq. (4.16), differ in the form for φ ′ , Eq. (4.10) and Eq. (4.13),i.e. by a field redefinition as discussed in Sec. 3.3. Thus the two formulations have the same S -matrix, but different Green’s functions. The covariant form Eq. (4.16) has covariant Green’sfunctions and S -matrix elements, so the non-covariant version Eq. (4.7) has covariant S -matrix elements (since they are not changed by field redefinitions) but non-covariant Green’sfunctions.The one-loop radiative correction can be computed from Eq. (4.18). For renormalizationof the theory at one-loop in dimensional reqularization, we only require the divergent one-loop contribution to the Lagrangian. This contribution can be extracted using the covariantderivative formalism in Refs. [11, 55–58], which gives the same result as an earlier explicitcomputation by ’t Hooft [59]. The results are given in Eq. (4.44), after we have discussedthe gauged version of Eq. (4.18). Since δ S is covariant, the radiative corrections are alsocovariant when computed this way. M We now consider the global symmetries of the ungauged action Eq. (4.1). The global symme-tries of the scalar kinetic energy term are the isometries of M . These isometries are specifiedby a set of vector fields t iα , the Killing vectors of M , where the different isometries are labelledby α . The Killing vectors generate the infinitesimal field transformations δ θ φ i = θ α t iα ( φ ) , (4.19)– 28 –here θ α are infinitesimal parameters. The gradient of φ transforms as δ θ (cid:0) ∂ µ φ i (cid:1) = θ α (cid:18) ∂t iα ∂φ j (cid:19) (cid:0) ∂ µ φ j (cid:1) . (4.20)For the O ( N ) sigma model, the global symmetries of the scalar kinetic energy term are G = O ( N ) transformations on the N -component real scalar field φ . The N ( N − / M are t iab ( φ ) = i [ M ab ] ij φ j = i ( M ab φ ) i , (4.21)where M ab , 1 ≤ a < b ≤ N , are the N × N anti-symmetric Hermitian matrices of Eq. (3.4),and the label α has been replaced by the bi-index ab . The Killing vectors in Eq. (4.21) arelinear in the N Cartesian components of the field φ , but not in the N polar components.The O ( N ) Killing vectors can be divided into the ( N − N − / H = O ( N −
1) and the ( N −
1) Killing vectors which are spontaneouslybroken, t iab ( φ ) = i ( M ab φ ) i = (cid:0) δ ia φ b − δ ib φ a (cid:1) , ≤ a < b < N ,t iaN ( φ ) = i ( M aN φ ) i = (cid:0) δ ia φ N − δ iN φ a (cid:1) , ≤ a < N . (4.22)Restricting to the scalar submanifold S N − such that h φ · φ i = v with h = 0, yields N ϕ =( N −
1) independent real scalar fields ϕ a . The Killing vectors of S N − on the first line ofEq. (4.22) act linearly on the ϕ a in both Cartesian and polar coordinates. Those on thesecond line act non-linearly, since φ N = p v − ϕ · ϕ . Explicitly, t iab ( ϕ ) = i ( M ab ϕ ) i = (cid:0) δ ia ϕ b − δ ib ϕ a (cid:1) , ≤ a < b < N ,t iaN ( ϕ ) = i ( M aN φ ) i = δ ia p v − ϕ · ϕ , ≤ a < N , (4.23)for i = 1 , . . . , N − L t α g = 0 , L t α I = 0 , (4.24)where L t α is the Lie derivative for Killing vector t iα . The first condition in Eq. (4.24) is thedefinition of a Killing vector; it is an isometry of the metric. The second condition is thatthe potential is invariant.The Lie bracket [ t α , t β ] of two isometries is also an isometry since (cid:2) L t α , L t β (cid:3) = L [ t α ,t β ] , (4.25)so the Killing vectors form the symmetry algebra[ t α , t β ] i = f γαβ t iγ . (4.26)– 29 –valuating the Lie bracket gives[ t α , t β ] i = t kα t iβ, k − t kβ t iα, k = f γαβ t iγ . (4.27)Note that the above equation also holds with the ordinary derivative t iα,k replaced by thecovariant derivative t iα ; k = t iα ,j + Γ ikj t jα = ∂t iα ∂φ k + Γ ikj t jα , (4.28)since the Christoffel symbol is symmetric in lower indices, and cancels in the antisymmetricderivative of Eq. (4.27). The Killing vectors in Eq. (4.23) are a non-trivial example of Killingvectors which form a closed set under the Lie bracket.As noted at the beginning of the section, a covariant treatment guarantees that vectors η i transform the same way as ∂ µ φ i under isometries, e.g. δ θ η i = θ α (cid:18) ∂t iα ∂φ j (cid:19) η j , (4.29)which is a linear transformation law. M The global symmetries Eq. (4.19) can be promoted to local symmetries by replacing the globalsymmetry parameters θ α by functions of spacetime θ α ( x ), δ θ φ i ( x ) = θ α ( x ) t iα ( φ ( x )) , (4.30)and introducing gauge fields.The gauge covariant derivative of the scalar field on the curved manifold M is defined by( D µ φ ( x )) i ≡ ∂ µ φ i ( x ) + A βµ ( x ) t iβ ( φ ( x )) , (4.31)where A βµ ( x ) is the gauge field associated with the Killing vector t iβ ( φ ), and the gauge couplingconstant and a factor of i has been absorbed into the gauge field. The gauge covariantderivative of the scalar field should transform the same way as ∂ µ φ i in Eq. (4.20) under the local symmetry, which implies the transformation rule δ θ ( D µ φ ) i = θ α ( x ) (cid:18) ∂t iα ∂φ j (cid:19) ( D µ φ ) j . (4.32)Consequently, the transformation law of A βµ ( x ) under the local symmetry is (cid:16) δ θ A βµ (cid:17) t iβ = − (cid:16) ∂ µ θ β (cid:17) t iβ + θ β A γµ t jγ ∂t iβ ∂φ j − t jβ ∂t iγ ∂φ j ! . (4.33)– 30 –sing the definition of the Lie bracket in Eq. (4.27), this equation yields the usual transfor-mation law for the gauge field δ θ A αµ = − ∂ µ θ α − f αβγ θ β A γµ . (4.34)The gauged version of the Lagrangian Eq. (4.1) is L = 12 g ij ( φ ) ( D µ φ ) i ( D µ φ ) j + I ( φ ) , (4.35)where the partial derivatives of the scalar field have been replaced by gauge covariant deriva-tives Eq. (4.31). The first variation of the Lagrangian gives the gauged generalization of theequation of motion Eq. (4.6), E i = g ij (cid:16) ∂ µ δ jk + A βµ t jβ, k (cid:17) ( D µ φ ) k + g il Γ ljk ( D µ φ ) j ( D µ φ ) k − I , i ≡ g ij ( D µ ( D µ φ )) j − I , i . (4.36)The gauge covariant derivative D µ φ of coordinates φ i is given in Eq. (4.31), and the gaugedcovariant derivative D µ on a vector field η i is( D µ η ) i = (cid:16) ∂ µ η i + Γ ikj ∂ µ φ k η j (cid:17) + A βµ (cid:16) t iβ,j + Γ ijk t kβ (cid:17) η j (4.37)which is the gauged generalization of Eq. (4.5). Eq. (4.37) is the appropriate definition forcovariant derivatives acting on vector fields. It arises in our calculation by a direct calculationto obtain the equations of motion Eq. (4.36) by varying the Lagrangian Eq. (4.35). One canshow that Eq. (4.37) transforms as δ θ ( D µ η ) i = θ α ( x ) (cid:18) ∂t iα ∂φ j (cid:19) ( D µ η ) j . (4.38)The derivation of Eq. (4.38) relies on two useful identities. The first is obtained by differen-tiating Eq. (4.27), f αβγ ∂t iγ ∂φ k ! = " ∂ t iβ ∂φ j ∂φ k ! t jα − (cid:18) ∂ t iα ∂φ j ∂φ k (cid:19) t jβ + " ∂t iβ ∂φ j ! ∂t jα ∂φ k ! − (cid:18) ∂t iα ∂φ j (cid:19) ∂t jβ ∂φ k ! . (4.39)The second relation is that the Lie derivative of the Levi-Civita connection vanishes because t α is a Killing vector. The explicit formula is0 = L t α Γ ikj = t ℓα ∂ Γ ikj ∂φ ℓ + ∂t ℓα ∂φ k Γ iℓj + ∂t ℓα ∂φ j Γ ikℓ − ∂t iα ∂φ ℓ Γ ℓkj + ∂ t iα ∂φ k ∂φ j . (4.40)The first and second variations of the gauged action up to second order give the gaugedversions of Eqs. (4.4) and (4.16), δS = Z d x h − g ij ( D µ ( D µ φ )) i η j + I ,i η i i ,δ S = 12 Z d x h g ij ( D µ η ) i ( D µ η ) j − R ijkl ( D µ φ ) j ( D µ φ ) l η i η k + ( ∇ i ∇ j I ) η i η j i . (4.41)– 31 –he gauge field now appears implicitly in every term except for those involving the potential I . The second variation δ S depends on the curvature R ijkl of M , but it does not have aterm that depends on the gauge curvature (i.e. field-strength) F µν .The divergent one-loop contribution in 4 − ǫ dimensions for quadratic actions such asEq. (4.16) was derived by ’t Hooft in Ref. [59],∆ L − loop = 132 π ǫ (cid:18)
112 Tr [ Y µν Y µν ] + 12 Tr (cid:2) X (cid:3)(cid:19) , (4.42)where [ Y µν ] ij ≡ [ D µ , D ν ] ij , [ X ] ik ≡ − R ijkl ( D µ φ ) j ( D µ φ ) l + g ij I ; jk . (4.43)’t Hooft’s original derivation is valid when the scalar metric is δ ij . Our form Eqs. (4.42) with Y µν and X given by Eq. (4.43) applies for any metric g ij .The matrix X is the mass squared term for the fluctuations η in Eq. (4.41), and Y µν isa field strength tensor constructed from the covariant derivative D . An explicit computationusing the identities (4.39) and (4.40) shows that Y µν is equal to the sum of the curvature of M and the curvature of the gauge field,[ Y µν ] i j = [ D µ , D ν ] i j = R ijkl ( D µ φ ) k ( D ν φ ) l + F αµν t iα ; j . (4.44)For Goldstone bosons, where I is a constant, X and Y µν are both proportional to twoderivatives of φ times the curvature R ijkl , i.e. they are order O ( R p ), where R is a typicalcurvature and p is a typical momentum. Thus, the one-loop correction, which is proportionalto the traces of X and Y µν , is order O ( R p ), and is O ( p ) as one expects in chiral perturba-tion theory. The O ( p ) correction is proportional to the square of the curvature, and vanishesif the manifold is flat, i.e. in a theory such as the SM. Thus, the SM is renormalizable evenin non-linear coordinates; one-loop graphs do not require four-derivative counterterms. The F µν term in Y µν gives the Goldstone boson contribution to the gauge coupling β -function oforder O ( F µν ), and the running of operators involving field strengths of order O ( RF µν p ).The quadratic invariants that enter Eq. (4.42) areTr (cid:2) X (cid:3) = ( ∇ i ∇ j I )( ∇ j ∇ i I ) + R i ( d µ φ ) j ( d µ φ ) R j ( d ν φ ) i ( d ν φ ) − ∇ i ∇ j I ) R i ( d µ φ ) j ( d µ φ ) , (4.45)andTr [ Y µν Y µν ] = R ij ( d µ φ ) ( d ν φ ) R ji ( d µ φ ) ( d ν φ ) + 2 R j i ( d µ φ ) ( d ν φ ) F αµν ( t iα ) ; j + F αµν F βµν ( t iα ) ; j ( t jβ ) ; i . (4.46)Eq. (4.46) is universal and applies to many theories. The one-loop correction in HEFT,which is complicated, and was given previously in Refs. [1, 18], is simply an expansion ofEq. (4.46) into component fields. More details about the expansion are given in Appendix B.As explained in Ref. [1], the same formula Eq. (4.46) applies to HEFT, the SM scalar sector,dilaton theories, and chiral perturbation theory.– 32 –o close this section, we consider spontaneous symmetry breaking in a theory with aninvariant potential L t α I = 0, so that we have exact Goldstone bosons. The fields φ i havevacuum expectation values (cid:10) φ i (cid:11) , and transform as δφ i = θ α t iα ( h φ i ). Thus, broken symmetries t iA satisfy t iA ( h φ i ) = 0, and t iA ( h φ i ) is a vector in the Goldstone boson direction — i.e. motionalong the vector field t iA (for each broken generator) is motion between different vacuum stateswith the same value of the potential I .In the gauged case, the Goldstone bosons are eaten, giving a mass term for the gaugebosons. The Lagrangian of Eq. (4.35) gives the gauge boson mass term L = 12 M BC A Bµ A C µ , M BC ( h φ i ) ≡ g ij ( h φ i ) t iB ( h φ i ) t jC ( h φ i ) . (4.47)The rank of M BC determines the number of massive gauge bosons, which cannot exceed thedimension of the manifold M . If the number of isometries exceeds dim M , then there areunbroken symmetries. This is true in the M = S N theory, where there are N ( N + 1) / G = O ( N + 1), and the unbroken subgroup H = O ( N )has N ( N − / N , which is equal to thedimension of S N . In this section, we connect the geometric formalism with the explicit formulæ of CCWZ [16, 17]for Goldstone boson Lagrangians with symmetry breaking pattern
G → H . We are interestedin applying the formalism to non-compact groups, and to sigma models with non-trivialmetrics on G / H . Our presentation thus parallels the discussion in the original work, whilepointing out differences which arise for the case of non-compact groups.Consider a group G with generators t α , α = 1 , · · · , dim G , satisfying the Lie algebra g [ t α , t β ] = if γαβ t γ , (5.1)and the Jacobi identity f λαβ f σγλ + f λγα f σβλ + f λβγ f σαλ = 0 . (5.2)To allow for negatively curved spaces [34], we do not assume that the group G is compact.Consequently, the Lie algebra Eq. (5.1) implies that the structure constants f γαβ are anti-symmetric in their first two indices, f γαβ = − f γβα , but total antisymmetry of the structureconstants in all three indices, which is true for compact groups, is not assumed.The group G is spontaneously broken to the subgroup H with generators T a , a =1 , · · · , dim H , satisfying the Lie algebra h ,[ T a , T b ] = if cab T c . (5.3)The remaining broken generators of the coset G / H needed to span g are given by X A , A =1 , · · · , dim G / H . The choice of the broken generators X A is not unique. In the familiar– 33 –xample of broken chiral symmetry in QCD, different choices of broken generators lead todifferent parameterizations of the chiral Lagrangian, e.g. by ξ ( x ) which transforms as ξ → Lξh † = hξR † , or by U ( x ) which transforms as LU R † [53].The g commutation relations of the generators t α = { T a , X A } in Eq. (5.1) decomposeinto the following commutation relations for the unbroken and broken generators[ T a , T b ] = if cab T c , (5.4a)[ T a , X B ] = if CaB X C + if caB T c , (5.4b)[ X A , X B ] = if CAB X C + if cAB T c . (5.4c)The first line Eq. (5.4a) is the Lie algebra h of the subgroup H in Eq. (5.3), which is closedunder commutation, so the commutator [ T a , T b ] has no term proportional to the broken gen-erators X C , which implies that the structure constants f Cab = 0.For compact groups, complete antisymmetry of the structure constants then implies that f caB = 0, so Eq. (5.4b) simplifies to[ T a , X B ] = if CaB X C , (5.5)which implies that the broken generators X A form a (possibly reducible) representation R ( π ) ofthe unbroken subgroup H . The generators T a of H in the R ( π ) representation are determinedby the structure constants f CaB , h T R ( π ) a i B C = − if CaB . (5.6)The h commutation relations Eq. (5.4a) in representation R ( π ) , h T R ( π ) a , T R ( π ) b i = if cab T R ( π ) c , (5.7)follow from the Jacobi identity Eq. (5.2).For non-compact groups, Eq. (5.5) need not be satisfied. For now, we restrict our at-tention to symmetry breaking patterns where Eq. (5.5) holds, so f caB = 0. Such cosets arecalled reductive cosets. Non-reductive cosets are discussed in Appendix C. An example ofa reductive coset is the breaking of the Lorentz group down to its rotation subgroup. Forreductive cosets, the broken generators transform as a representation R ( π ) of the unbrokensymmetry group H , just as in the compact case. The coset is reductive if H is compact, evenif G is non-compact.Often, there is a discrete symmetry of the Lie algebra X A → − X A under which thebroken generators change sign. The presence of such a discrete symmetry implies that thestructure constants f caB and f CAB vanish, so the Lie algebra reduces to[ T a , T b ] = if cab T c , [ T a , X B ] = if CaB X C , (5.8)[ X A , X B ] = if cAB T c . – 34 –n example is chiral symmetry breaking in the strong interactions, where the broken genera-tors are odd under parity. Cosets with such a discrete symmetry are referred to as symmetriccosets. Symmetric cosets are automatically reductive.The CCWZ formalism picks elements of G / H cosets using the exponential map of thebroken generators { X A } ξ ( x ) ≡ e iπ · X , π · X ≡ π A ( x ) X A = (cid:18) π A ( x ) F π (cid:19) X A , (5.9)where π A ( x ) are the dimensionless spacetime-dependent parameters describing the Goldstoneboson directions on the vacuum coset G / H . This exponential map gives a unique associationbetween a point in the coset G / H and π A ( x ) in a neighborhood of the identity element e . Anarbitrary group element g ∈ G in the neighborhood of the identity element e can be writtenuniquely as g = e iπ · X e iα · T , (5.10)where α · T ≡ α a ( x ) T a . Left action by an arbitrary group element g ∈ G on G / H is given by T g : ξ ( x ) → g ξ ( x ) , (5.11)which maps a point in coset space to a new point in coset space. The transformation law for ξ ( x ) is g ξ ( x ) = ξ ′ ( x ) h ( ξ ( x ) , g ) , g ∈ G , h ∈ H , (5.12)where ξ ′ ( x ) is a new coset and h ∈ H is an implicit function of g ∈ G and the original coset ξ ( x ). Using the identity g ξ ( x ) = (cid:0) g ξ ( x ) g − (cid:1) g , g ξ ( x ) g − = exp (cid:0) i π ( x ) · (cid:0) g X g − (cid:1)(cid:1) , (5.13)one sees that if g = h ∈ H is an unbroken symmetry transformation, then h ( ξ ( x ) , h ) = h .In addition, ξ ′ = h ξh − , which implies that (since the coset is assumed reductive) π ′ A ( x ) = h D R ( π ) ( h ) i AB π B ( x ) , (5.14)where D R ( π ) ( h ) is the H transformation matrix in the R ( π ) representation. Note that forreductive cosets, if g = h ∈ H , then h ( ξ ( x ) , h ) = h is a constant (i.e. it does not dependon x through ξ ( x )).The CCWZ procedure for building a G -invariant Lagrangian is to map all fields to theorigin of coset space ξ = 1 with π ( x ) = 0 by left-action by g = ξ − , and to define covariantderivatives in terms of this map. Explicitly, one starts with ξ − D µ ξ = ξ − ( ∂ µ + iA αµ t α ) ξ (5.15)– 35 –here the gauge coupling constant has been absorbed into the normalization of the gaugefield A αµ . If only a subgroup G gauge ⊂ G is gauged, then only gauge bosons of G gauge appearin the above equation, or equivalently, the gauge bosons corresponding to global symmetrydirections are set equal to zero. In addition, different factor gauge groups in G gauge can havedistinct gauge coupling constants. Power series expansion of ξ − D µ ξ shows that it can beexpressed in terms of multiple commutators, so it is an element of the Lie algebra g whichcan be decomposed in terms of unbroken and broken generators, ξ − D µ ξ = ξ − D µ ξ (cid:12)(cid:12) T + ξ − D µ ξ (cid:12)(cid:12) X = i V µ + i ( D µ π ) ,ξ − D µ ξ (cid:12)(cid:12) T ≡ i V µ = i V aµ T a ,ξ − D µ ξ (cid:12)(cid:12) X ≡ i ( D µ π ) = i ( D µ π ) A X A . (5.16)The above equations define V µ and ( D µ π ). Usually, one normalizes the generators so thatTr t α t β = δ αβ /
2, and projects out the broken and unbroken pieces of ξ − D µ ξ by taking theappropriate traces. The decomposition of a vector into a linear combination of basis vectorsdoes not require an inner product on the vector space, so Eq. (5.16) is well-defined evenwithout this normalization of generators. An orthogonal normalization of generators is notpossible for non-compact G , but Eq. (5.16) is well-defined. Under an unbroken symmetrytransformation h ∈ H , V µ transforms like a gauge field V µ → h V µ h − − ( ∂ µ h ) h − , (5.17)and ( D µ π ) transforms by adjoint action by H in the representation R ( π ) ,( D µ π ) → h ( D µ π ) h − . (5.18)These last two equations require the reductive coset condition f caB = 0. The generalizationto non-reductive cosets is discussed in Appendix C.The pion covariant derivative can be decomposed into a purely pionic piece and a gaugefield piece, ( D µ π ) A ≡ [ e ( π )] AB (cid:0) ∂ µ π B (cid:1) + F Aα ( π ) A αµ (5.19)where [ e ( π )] AB are vierbeins of the G / H vacuum manifold, and F Aα ( π ) are related to theKilling vectors of G / H .For groups where ( D µ π ) A transforms as a single irreducible representation R ( π ) , as inQCD, the simplest invariant Lagrangian is the O ( p ) term L = 12 F π X A ( D µ π ) A ( D µ π ) A = 12 F π X A [ e ( π )] AB [ e ( π )] AC ( ∂ µ π ) B ( ∂ µ π ) C + · · ·≡ g BC ( π ) ( ∂ µ π ) B ( ∂ µ π ) C + · · · , (5.20)– 36 –here F π is the Goldstone boson decay constant, and the ellipsis denotes terms depending onthe gauge fields. The Lagrangian Eq. (5.20) defines the scalar field metric of the G / H vacuummanifold, g BC ( π ) = F π X A [ e ( π )] AB [ e ( π )] AC . (5.21)If the representation is reducible, the sum in Eq. (5.20) can be divided into sums over the indi-vidual irreducible representations, with arbitrary weights for each irreducible representation.The most general O ( p ) term allowed is L = 12 F π X A,B η AB ( D µ π ) A ( D µ π ) B , (5.22)where η AB is a symmetric tensor invariant under the adjoint action of H , Eq. (5.18). η AB isa positive definite matrix so that the pion kinetic energies have the correct sign. Note that η AB is a constant, i.e. it does not depend on π . One can always define a positive definitekinetic energy if H is a compact subgroup, e.g. by choosing η AB = δ AB . In summary, themost general scalar metric for G / H is g CD ( π ) = F π X A,B η AB [ e ( π )] AC [ e ( π )] BD , (5.23)and the Killing vectors in Sec. 4.2 are given by t Aα ( π ) = (cid:2) e − ( π ) (cid:3) AB F Bα ( π ) , (5.24)where (cid:2) e − ( π ) (cid:3) is the inverse vierbein, which satisfies the identity (cid:2) e − ( π ) (cid:3) A B [ e ( π )] BC = δ AC . (5.25)Eq. (5.24) can easily be derived by looking at the shift π → π + δπ for an infinitesimal G transformation.For the HEFT example, we need to evaluate the curvature tensors at the vacuum fieldconfiguration π A = 0, which requires knowing the metric tensor to quadratic order in π .The curvature at any other point can then be obtained using left-action by G . ExpandingEq. (A.4) and using the most general Lie algebra relations Eqs. (5.4a), (5.4b) and (5.4c), oneobtains ( D µ π ) A = ∂ µ π A + 12 f ACB π C ∂ µ π B + 16 f ADα f αCB π D π C ∂ µ π B + . . . + A Aµ + f ABα π B A αµ + 12 f ACβ f βBα π B π C A αµ + . . . . (5.26)The term A Aµ only involves the broken generators, and it is the square of this term in thekinetic energy which results in the broken gauge bosons acquiring a mass proportional to F π .From Eq. (5.19), the vierbein is[ e ( π )] AB = δ AB + 12 f ACB π C + 16 f ADα f αEB π D π E + . . . (5.27)– 37 –sing Eq. (5.23), the metric g CD ( π ) is1 F π g CD ( π ) = η CD + 12 (cid:0) η AD f AEC + η CB f BED (cid:1) π E + (cid:18) η CB f BEα f αF D + 16 η AD f AEα f αF C + 14 η AB f AEC f BF D (cid:19) π E π F + O ( π ) . (5.28)For a compact group, the structure constants are completely antisymmetric, so the linearterm in π vanishes if η AB ∝ δ AB . However, in some cases, such as the SM with custodialsymmetry violation, the linear term is non-zero.The geometric quantities we need can be computed directly from the metric Eq. (5.28).The Christoffel symbol isΓ ABC = 12 η AG (cid:0) η CE f EBG + η BE f ECG (cid:1) + 14 (cid:0) f EGB η EC + f EGC η EB (cid:1) (cid:0) f ADH η HG + f GDH η AH (cid:1) π D − (cid:0) f GHC f EDB + f GDC f EHB (cid:1) η AH η GE π D + 112 (cid:0) f ACα f αDB + f ABα f αDC (cid:1) π D + 14 (cid:0) f αCG η BE + f αBG η CE (cid:1) η AG f EDα π D + O ( π ) , (5.29)where η AB is the inverse of η AB , and the Jacobi identity has been used to simplify the finalresult. The Riemann curvature tensor is1 F π R ABCD = 14 (cid:0) f αAB f EDα η CE − f αAB f ECα η DE + f αCD f EBα η AE − f αCD f EAα η BE (cid:1) + 14 (cid:0) f GAD f EBC − f GAC f EBD − f GAB f ECD (cid:1) η GE + 14 η GE (cid:2)(cid:0) f HAG η DH + f HDG η AH (cid:1) (cid:0) f IBE η CI + f ICE η BI (cid:1) − (cid:0) f HBG η DH + f HDG η BH (cid:1) (cid:0) f IAE η CI + f ICE η AI (cid:1)(cid:3) , (5.30)where we recall that the sum on α runs over both broken and unbroken generators, whereasthe sums on E , etc. are only over the broken generators. The Ricci curvature is R BD = 14 (cid:0) f αAB f ADα + f αAD f ABα − f αAB f GCα η DG η AC − f αAD f GCα η BG η AC (cid:1) − f GAB f HCD η AC η GH + 14 η GH (cid:0) f RAG η DR + f RDG η AR (cid:1) (cid:0) f RBH η CR + f RCH η BR (cid:1) η AC − (cid:0) f RBG η DR + f RDG η BR (cid:1) f AAH η GH , (5.31)and the scalar curvature is F π R = f αAB f ACα η BC − f CAB f DGH η AG η BH η CD + 12 f BAC f ABD η CD − f AAC f BBD η CD . (5.32)– 38 –he scalar curvature does not have a definite sign unless the group is compact. Eqs. (5.28),(5.29), (5.30), (5.31) and (5.32) are valid even for non-reductive cosets.The results simplify considerably in a number of special cases. For a symmetric coset, f CAB = 0, and the curvatures Eqs. (5.30), (5.31) and (5.32) reduce to1 F π R ABCD = 14 (cid:0) f αAB f GDα η CG − f αAB f GCα η DG + f αCD f GBα η AG − f αCD f GAα η BG (cid:1) ,R BD = 14 (cid:0) f αAB f ADα + f αAD f ABα − f αAB f GCα η DG η AC − f αAD f GCα η BG η AC (cid:1) ,F π R = f αAB f ACα η BC , (5.33)where the sum on α = { a, A } can be restricted to the unbroken generator index a only.Another special case is G compact and η AB = δ AB . For a compact group, the generatorscan be normalized so that Tr t α t β ∝ δ αβ , so the structure constants are completely antisym-metric tensors in their three indices. Writing the structure constants with three lower indicesin the usual notation for compact groups, Eqs. (5.30), (5.31) and (5.32) simplify to1 F π R ABCD = f ABα f CDα − f ABG f CDG = f ABg f CDg + 14 f ABG f CDG ,R BD = f ABg f ADg + 14 f ABG f ADG ,F π R = f ABg f ABg + 14 f ABG f ABG . (5.34)An interesting feature is the relative 1 / G relativeto the sum over unbroken generator index g .If one adds the additional restriction that the coset of the compact group G is symmetric,so f ABC = 0, the formulæ Eqs. (5.34) simplify further to1 F π R ABCD = f ABg f CDg ,R BD = f ABg f ADg = 12 C A ( G ) δ BD ,F π R = 12 C A ( G ) N π , (5.35)where C A ( G ) is the Casimir in the adjoint representation of G , and N π = dim G / H is thenumber of broken generators.Finally, if the gauge group is compact and completely broken, so that G / H = G , and η AB = δ AB , Eqs. (5.34) become 1 F π R ABCD = 14 f ABG f CDG ,R BD = 14 C A ( G ) δ BD ,F π R = 14 C A ( G ) N π . (5.36)– 39 – .1 Matter Fields We refer to all non-Goldstone boson or gauge fields generically as matter fields. The CCWZtransformation for matter fields ψ under the group transformation law Eq. (5.12) is ψ → D ( ψ ) ( h ) ψ , (5.37)where D ( ψ ) ( h ) are the H representation matrices for ψ . Note that D ( ψ ) ( h ) is assumed tobe an irreducible representation, so if it is reducible, one must first decompose it into itsirreducible representations. The different irreducible representation components are thentreated as separate matter fields. One can define a chiral covariant derivative for matter field ψ by D µ ψ → (cid:16) ∂ µ + iT ( ψ ) a V aµ (cid:17) ψ , (5.38)where T ( ψ ) a are the generators of the unbroken subgroup H in the representation D ( ψ ) ( h ) of H . The chiral covariant derivative transforms as( D µ ψ ) → D ( ψ ) ( h ) ( D µ ψ ) . (5.39)The covariant derivative Eq. (5.38) is derived in CCWZ. The argument relies on defining itas the ordinary derivative at ξ = 1, and then using G action to define it for arbitrary ξ . Thekey point (which is not true for non-reductive cosets) is that if g ∈ H , then h in Eq. (5.12) isa constant, so the ordinary derivative transforms the same way as the field, Eq. (5.37). Usingthis result at ξ = 1, the transformation Eq. (5.39) for arbitrary ξ follows.The covariant derivative Eq. (5.38) is based on Eq. (5.16), and hence on the Maurer-Cartan form g − d g . This is the canonical connection on the principal H -bundle G → G / H ,and makes no reference to a metric, i.e. to η AB . One can also define covariant derivativesbased on the metric (Christoffel) connection Eq. (5.29), which does depend on η AB . Thetwo are equivalent if η AB = δ AB , i.e. if the G -invariant metric on G / H is obtained froma G -invariant metric on G . The difference in the connections transforms as a H -invarianttensor [60], so that the change in connection can be compensated by a change in coefficientsof invariant terms in the sigma model Lagrangian. The exponential map ξ ( λ ) = exp( Xλ ) isgeodesic for the Maurer-Cartan connection, but not for a general η AB metric connection. The sectional curvature K ( Y, Z ) in the plane spanned by tangent vectors Y and Z is definedby K ( Y, Z ) = R ABCD Y A Z B Y C Z D h Y, Y i h
Z, Z i − h
Y, Z i (5.40)where the inner product h∗ , ∗i is w.r.t. the metric g AB . The Cauchy-Schwartz inequalityimplies the denominator is positive, so the sign of the sectional curvature depends on the– 40 –ign of the numerator. The sign of the sectional curvature is important, because, as shownin Refs. [1, 34], the sign of deviations in Higgs-gauge boson scattering amplitudes from SMamplitudes is determined by the sign of the sectional curvatures of the HEFT sigma model.From Eq. (5.30),1 F π R ABCD Y A Z B Y C Z D = 12 f αY Z (cid:0) f AZα Y B η AB − f AY α Z B η AB (cid:1) − f AY Z f BY Z η AB + 14 (cid:0) f AY G Z B η AB + f AZG Y B η AB (cid:1) (cid:0) f CY H Z D η CD + f CZH Y D η CD (cid:1) η GH − f AY G f CZH Y B Z D η AB η CD η GH (5.41)and we have used the definition[ Y, Z ] = (cid:2) Y A T A , Z B T B (cid:3) ≡ f αY Z t α (5.42)for f αY Z . The general form Eq. (5.41) does not have a definite sign.For compact groups with η AB ∝ δ AB , antisymmetry of the structure constants implies1 F π R ABCD Y A Z B Y C Z D = f gY Z f gY Z + 14 f GY Z f GY Z ≥ Y, Z . For compact groups with η AB = δ AB , thesectional curvatures need not be positive. A simple example is G = SU (2) completely broken,with η AB = diag( η , η , η ), and Y = (1 , , Z = (0 , , K ( Y, Z ) = 2( η + η ) η + ( η − η ) − η F π η η η (5.44)which is negative for η ≫ η , .In HEFT applications where there is only a single h field, the possible sectional curvaturesare:(a) Both Y and Z are in the Goldstone boson directions. Since the Goldstone boson mani-fold S is a maximally symmetric space, K ( Y π , Z π ) is independent of the choice Y π , Z π ,and is the quantity K ( Y π , Z π ) = R in Ref. [1].(b) Y is in the Goldstone boson direction, and Z is in the h direction. In this case K ( Y π , Z h )is independent of the choice Y π and Z h (since there is only one direction Z h ) and is K ( Y π , Z h ) = R h in Ref. [1].As shown in Ref. [1], deviations in W L W L → W L W L were proportional to r = R ( h = 0),the sectional curvature where Y and Z are in Goldstone boson directions. The longitudinalgauge bosons at high energies are related to the Goldstone bosons, and so probe the Goldstoneboson directions in M . The W L W L → hh scattering amplitudes is proportional to r h = R h ( h = 0), and probes the sectional curvature where Y is in a Goldstone boson direction,and Z in the Higgs direction. If the HEFT is based on a composite Higgs theory [29], where– 41 – is itself a (pseudo) Goldstone boson of some strong dynamics at a scale f > v , then we seefrom Eq. (5.43) that R and R h are both positive if the composite Higgs model is based ona compact group. On the other hand, if the sigma-model group is non-compact, it is possibleto get negative values [34] for these curvatures because Eq. (5.41) has no definite sign.We also consider multi-Higgs theories in Sec. 7. In such theories, the possible sectionalcurvatures are R = K ( Y π , Z π ), R h,I = K ( Y π , Z I ), where Z I runs over the possible Higgsdirections, and K ( Y I , Z J ) over distinct pairs of Higgs directions I = J . The SM sigma model for the custodial symmetric breaking pattern SU (2) L × SU (2) R → SU (2) V can be written in the CCWZ formalism, choosing the broken generators to be T L .Let U ( x ) = e iπ A ( x ) T A (6.1)be a 2 × T A are SU (2) L generators, and π A are dimensionless.The ξ field of the CCWZ formalism given by exponentiating the broken generators is ξ ( x ) = U ( x )0 1 × ! , (6.2)where the first 2 × SU (2) L transformation, and the second is the SU (2) R transformation. From this ξ field, one finds ξ ( x ) − D µ ξ = U ( x ) − × ! " ∂ µ U ( x )0 0 ! + ig W αµ T α U ( x )0 ig B µ T ! = U ( x ) − ∂ µ U ( x ) + U ( x ) − ig W αµ T α U ( x )0 ig B µ T ! = ig B µ T ig B µ T ! + U ( x ) − ∂ µ U ( x ) + U ( x ) − ig W αµ T α U ( x ) − ig B µ T ! , (6.3)where the last line projects onto the unbroken and broken spaces, respectively. Thus, weobtain i ( D µ π ) A T A = U ( x ) − ∂ µ U ( x ) + U ( x ) − ig W αµ T α U ( x ) − ig B µ T , (6.4)and, using the results in Appendix A,( D µ π ) A = (cid:18) sin | π || π | (cid:19) d π A + (cid:18) − cos | π || π | (cid:19) ǫ ABC π B d π C + (cid:18) | π | − sin | π || π | (cid:19) π A ( π · d π )+ g W Aµ cos | π | + g (cid:18) sin | π || π | (cid:19) ǫ ABC π B W Cµ + g (cid:18) − cos | π || π | (cid:19) ( π · W µ ) π A − g B µ δ A (6.5)– 42 –ith | π | = π · π . Decomposing ( D µ π ) A into gauge and non-gauge pieces as in Eq. (5.19)yields ( D µ π ) A = e AB ∂ µ π B + F Aβ W βµ + F AZ Z µ + F Aγ A µ , (6.6)where e AB = (cid:18) sin | π || π | (cid:19) δ AB − (cid:18) − cos | π || π | (cid:19) ǫ ABC π C + (cid:18) | π | − sin | π || π | (cid:19) π A π B ,F Aβ = es W (cid:20) δ Aβ cos | π | + (cid:18) sin | π || π | (cid:19) ǫ ADβ π D + (cid:18) − cos | π || π | (cid:19) π β π A (cid:21) , β = 1 , F AZ = es W c W (cid:20) δ A (cid:0) s W + c W cos | π | (cid:1) + c W (cid:18) sin | π || π | (cid:19) ǫ AB π B + c W (cid:18) − cos | π || π | (cid:19) π π A (cid:21) ,F Aγ = e (cid:20) − δ A (1 − cos | π | ) + (cid:18) sin | π || π | (cid:19) ǫ AB π B + (cid:18) − cos | π || π | (cid:19) π π A (cid:21) , (6.7)with c W = cos θ W and s W = sin θ W . The F Aα can be used to construct the Killing vectorsusing Eq. (5.24). Expanding these equations gives e AB = δ AB − ǫ ABC π C + 16 h π A π B − | π | δ AB i + . . .F Aβ = es W (cid:20) δ Aβ (cid:18) − | π | (cid:19) + ǫ ADβ π D + 12 π β π A (cid:21) + . . . , β = 1 , F AZ = es W c W (cid:20) δ A (cid:18) − c W | π | (cid:19) + c W ǫ AB π B + 12 c W π π A (cid:21) + . . . ,F Aγ = e (cid:20) − | π | δ A + ǫ AB π B + 12 π π A (cid:21) + . . . . (6.8)In unitary gauge, π = 0 and F Aβ = es W δ Aβ , β = 1 , , F AZ = es W c W δ A , F Aγ = 0 , (6.9)so the photon is massless, and W, Z acquire mass.The most general O ( p ) Lagrangian is L = 12 X AB η AB ( D µ π ) A ( D µ π ) B (6.10)where η AB is a H -invariant tensor. For the SM with custodial SU (2) symmetry, the breakingpattern is SU (2) L × SU (2) R → SU (2) V . The tensor η AB must be invariant under the unbroken H = SU (2) V symmetry, so η AB = v δ AB , (6.11)where v ∼
246 GeV is chosen to give the correct gauge boson masses.– 43 –f custodial symmetry is not exact, the breaking pattern is SU (2) L × U (1) Y → U (1) em ,and η AB must be invariant under the unbroken H = U (1) em symmetry. In this case, η AB = v ρ , (6.12)where ρ is the ρ -parameter ρ = M Z c W M W , (6.13)which is no longer equal to one. The experimental constraint on the ρ parameter is anextremely stringent constraint on custodial symmetry violation, since it requires | ρ − | . .
01. A simple example of custodial symmetry violating is the SM with an additional tripletscalar field [61] χ = " √ χ + − χ ++ χ − √ χ + . (6.14)If the doublet and triplet vacuum expectation values are h H i = " v D √ , h χ i = " v T √ , (6.15)then the values of the η AB parameters in Eq. (6.12) are v = v D + 2 v T , ρ = v D + 4 v T v D + 2 v T . (6.16)The geometry of the scalar manifold with metric Eq. (6.10) has been studied in othercontexts [62]. The configuration space of a rigid body with one point fixed is given by therotation matrix R ( θ, φ, ψ ) ∈ SO (3) parameterized by three Euler angles, and, up to Z factors,is the same as the Goldstone boson manifold of the SM. Rotations of the body about space-fixed axes correspond to SO (3) L rotations R → g L R , g L ∈ SO (3), and rotations about thebody-fixed axes correspond to SO (3) R rotations R → Rg − R , g R ∈ SO (3). The body-axisangular momenta are given by ω A T A = R − ˙ R . The kinetic energy for a rigid body is thengiven by the analog of Eq. (6.10), L = 12 X A I A (cid:0) ω A (cid:1) , (6.17)where η AB can be chosen to be diagonal by picking the body axes to coincide with theprincipal axes of the body. The kinetic energy for a spherical top with all three principalmoments of inertia equal, I = I = I is the analog of the SM with custodial symmetry. The– 44 –onfiguration space of the top is the (undeformed) three-sphere S . The custodial symmetryviolating case is analogous to I = I = I , which is the configuration space of a symmetrictop. This space is known as the squashed three-sphere, and also occurs in the metric for theTaub universe [62]. The asymmetric top with all I i different would correspond to the SMwith electromagnetism broken. The HEFT formalism can be extended to the case of multiple singlet (under custodial SU (2))Higgs fields h I , I = 1 , , · · · , which involves adding additional singlet scalars to the SM fieldcontent. The generalization of the HEFT Lagrangian Eq. (2.24) to multiple singlet scalarfields is L = 12 v F ( h ) ( ∂ µ n ) + 12 g IJ ( h ) (cid:0) ∂ µ h I (cid:1) (cid:0) ∂ µ h J (cid:1) − V ( h ) + . . . (7.1)where F ( h ) is an arbitrary function of the dimensionless singlet scalar fields h I /v . Thecoordinates { h I } are chosen so that h = (0 , , . . . ,
0) is the ground state, and the HEFTfunction F ( h ) is normalized so that F (0 , . . . ,
0) = 1 (7.2)since the radius of S in the vacuum is fixed to be v by the gauge boson masses.Consider the O (4) → O (3) symmetry breaking pattern of the SM, with multiple scalarfields h I which are singlets under the unbroken custodial O (3) symmetry. The most generalmetric of the scalar fields Φ i ≡ { π A , h I } has the form g ij (Φ) = " F ( h ) g AB ( π ) 00 g IJ ( h ) , (7.3)where π A /v are coordinates on the coset space G / H = O (4) /O (3) = S , and g AB ( π ) is themetric on the unit 3-sphere. O (4) invariance implies that the off-diagonal metric terms g AI and g IA vanish, and that g IJ ( h ) has no dependence on the π fields. An easy way to provethat the general metric takes the form Eq. (7.3) is to note that a point on S is given by afour-component unit vector n . The entry g IJ ( h ) can depend on n , but not on its derivatives; O (4) invariance then requires it to be function of n · n = 1, and therefore independent of π .Similarly, g IA ∂ µ π A is an O (4) invariant function of n and ∂ µ n with one derivative; the onlyinvariant object is ∂ µ n · n = 0, so the off-diagonal entries vanish. The 11 entry has the form F ( h ) g AB ( π ) because G -invariance requires that h dependence is an overall multiplicativefactor, since there is only one G -invariant metric on S . We will consider the geometry ofthe metric Eq. (7.3), with a general metric g AB ( π ), so the results are valid for a general G / H manifold as long as the off-diagonal terms of g ij (Φ) vanish as in Eq. (7.3).– 45 –sing the metric Eq. (7.3), the Christoffel symbols areΓ ABC = γ ABC , Γ ABK = F ,K F δ AB , Γ AJK = 0 , Γ IBC = − F F ,M g IM g BC , Γ IBK = 0 , Γ IJK = γ IJK , (7.4)where γ ABC and γ IJK are the Christoffel symbols computed from the metrics g AB ( π ) and g IJ ( h ),respectively. Similarly, in the expressions below, r ABCD , r BD and r π are the curvaturescomputed from the metric g AB ( π ), whereas r I JKL , r JL and r h are the curvatures computedfrom the metric g IJ ( h ). The Riemann curvature tensor is R ABCD = r ABCD − g MN F ,M F ,N (cid:0) δ AC g DB − δ AD g BC (cid:1) , R ABCL = 0 ,R ABKL = 0 , R
I JCD = 0 ,R I JKD = 0 , R
I JKL = r I JKL ,R AJCD = 0 , R
AJKL = 0 ,R I BCD = 0 , R
I BKD = − g DB g IM F ; MK ,R I BKL = 0 , R
AJCL = − δ AC F ; JL . (7.5)The covariant derivatives of F are w.r.t. γ IJK . The Ricci tensor is R BD = r BD − g RS F ,R F ,S ( N π − g BD − g BD g RS F F ; RS ,R BL = 0 ,R JL = − N π F ; JL + r JL , (7.6)and the curvature scalar is R = 1 F r π − N π ( N π −
1) 1 F g RS F ,R F ,S − N π F g RS F ; RS + r h . (7.7)If G / H is a maximally symmetric space, r ABCD = 1 F π (cid:0) δ AC g BD − δ AD g BC (cid:1) , r BD = 1 F π ( N π − g BD , r π = 1 F π N π ( N π − . (7.8)The above expressions reduce to the formulæ given in Ref. [1] for one Higgs singlet field h and G / H a symmetric space, which used g IJ ( h ) = 1 , F ( h ) = 1 + c (cid:18) hv (cid:19) + 12 c (cid:18) hv (cid:19) + . . . (7.9)with F π = v .The above expressions can be further simplified if one picks one h field to be the radiusof S , F ( { h } ) = h , in which case F does not depend on h I , I = 1. The radial direction is ingeneral not a mass-eigenstate direction. Letting ρ be the radial direction, with ρ = 1 in the– 46 –acuum, and letting the remaining directions still be called h I (there is one less h now), with I, J, K running over ρ, { h } , one gets a simpler version of the above equations, where F ,K = 1if K = ρ , and zero otherwise. For example, F → ρ, G RS F ,R F ,S → G ρρ , F ; RS → − γ ρRS , (7.10)etc. In this paper, we have discussed the relation between the SM and two of its generalizations,SMEFT and HEFT, and have shown that HEFT can be written in SMEFT form if and onlyif there is an O (4) invariant fixed point of the scalar manifold in a neighborhood of whichthe scalar fields transform as a vector of O (4). We have shown that the SM can be writtenusing scalar fields transforming either linearly or non-linearly under SU (2) L × U (1) Y , and isrenormalizable with either choice. Whether “the Higgs transforms linearly or non-linearly”is not observable; the correct question, which can be resolved experimentally, is whether theSM scalar manifold M is flat or curved.We have discussed the formulation of scalar fields on a curved manifold, including thecase with gauge symmetry, reviewed the computation of one-loop corrections in terms of thecurvature, and applied these known results to the case where the manifold is a coset. Thegeneral expressions were used to obtain the one-loop renormalization of HEFT [1, 18], anddetails of the computation are given here.Deviations of Higgs and longitudinal gauge boson scattering amplitudes from their SMvalues are given by sectional curvatures of the scalar manifold. In simple examples based on G / H symmetry breaking with compact groups, the sectional curvatures are positive, whichfixes the signs of deviations from the SM. We are investigating examples where sectionalcurvatures can be negative, and have given the generalization of the CCWZ formalism tonon-compact groups. AM would like to thank Luis Alvarez-Gaum´e, Luca Merlo, and Vyacheslav Rychkov for forhelpful discussions. This work was partially supported by grants from the Simons Foundation(
Exponential Parametrization of the O ( N ) Model
The real antisymmetric Goldstone boson matrix is given by Π ≡ i ( π · X ) = " π − π T = . . . π . . . π ... ... ...0 . . . π N ϕ − π . . . − π N ϕ , (A.1)where π A ≡ π A /F π . ξ is ξ ≡ e Π = + (cid:18) sin | π || π | (cid:19) Π + (cid:18) − cos | π || π | (cid:19) Π , | π | ≡ π A π A . (A.2)The Mauer-Cartan form is ξ − ∂ µ ξ = (cid:18) sin | π || π | (cid:19) i ( ∂ µ π ) · X + (cid:18) | π | − sin | π || π | (cid:19) (cid:0) π B ∂ µ π B (cid:1) i π · X + (cid:18) − cos | π || π | (cid:19) " − (cid:0) ∂ µ π A (cid:1) π B + π A (cid:0) ∂ µ π B (cid:1)
00 0 , (A.3)where the first two terms are linear combinations of the broken generators, and the last termis a linear combination of the unbroken generators. The indices A, B in the last term are therow and column indices of the N ϕ × N ϕ submatrix in the upper 11 block. Using Eq. (5.16),( D µ π ) A = (cid:18) sin ππ (cid:19) ( ∂ µ π ) A + (cid:18) π − sin ππ (cid:19) (cid:0) π B ∂ µ π B (cid:1) π A , (A.4)and ξ − ∂ µ ξ (cid:12)(cid:12) T = i V µ · T = (cid:18) − cos ππ (cid:19) " − (cid:0) ∂ µ π A (cid:1) π B + π A (cid:0) ∂ µ π B (cid:1)
00 0 . (A.5) B One-Loop Renormalization of HEFT
In this appendix, we provide some intermediate results in the computation of the one-looprenormalization of HEFT [1, 18].The metric for the scalar manifold M in HEFT is g ij ( φ ) = " v F ( h ) g ab ( π ) 00 1 , (B.1)where F ( h ) is a dimensionless function with a power series expansion in h/v , and g ab ( π ) isthe metric on the Goldstone boson manifold G / H = S . The field h has mass dimension one,– 48 – is dimensionless, and i runs over indices a , h . The scalar kinetic term in HEFT is given by L = 12 g ij ( φ ) ∂ µ φ i ∂ µ φ j = 12 v F ( h ) g ab ( π ) ∂ µ π a ∂ µ π b + 12 ∂ µ h ∂ µ h ≡ F ( h ) v ∂ µ n · ∂ µ n + 12 ∂ µ h ∂ µ h, (B.2)where the unit vector n ( π ) is a dimensionless function of the the three independent coordinates π a = π a /v on S . Note that we have chosen to normalize π a to be dimensionless coordinates,which differs from Ref. [1] by a rescaling by v . Eq. (B.2) implies that the S metric g ab ( π ) isgiven in terms of the unit vector n ( π ) by g ab ( π ) ≡ ∂ n ( π ) ∂π a · ∂ n ( π ) ∂π b . (B.3)The Riemann curvature tensor R ijkl ( φ ) obtained from the scalar metric g ij ( φ ) consistsof the non-vanishing components R abcd ( φ ) = (cid:2) − v ( F ′ ( h )) (cid:3) v F ( h ) ( g ac ( π ) g bd ( π ) − g ad ( π ) g bc ( π )) ,R ahbh ( φ ) = − v F ( h ) F ′′ ( h ) g ab ( π ) , (B.4)and components related to these by the permutation symmetry of the Riemann tensor. R abcd ( φ ) is proportional to the tensor ( g ac g bd − g ad g bc ) because S is a maximally symmetricspace.The quantities X and Y µν from Eqs. (4.43) and (4.44) that appear in the one-loop cor-rection Eq. (4.42) contain terms depending on the Riemann curvature tensor. The Riemanncurvature tensor components contributing to [ X ] ik and [ Y µν ] ij , respectively, are R ijkl ( D µ φ ) j ( D µ φ ) l = (cid:2) − v ( F ′ ) (cid:3) (cid:2) ( D µ π ) δ ac − ( D µ π ) a ( D µ π ) c (cid:3) − F ′′ F ( ∂ µ h )( ∂ µ h ) δ ac F ′′ F ( D µ π ) a ( ∂ µ h ) v F F ′′ ( ∂ µ h )( D µ π ) c − v F F ′′ ( D µ π ) ,R ijkl ( D µ φ ) k ( D µ φ ) l = (cid:2) − v ( F ′ ) (cid:3) [( D µ π ) a ( D ν π ) b − ( D ν π ) a ( D ν π ) b ] F ′′ F [( D ν π ) a ( ∂ µ h ) − ( D µ π ) a ( ∂ ν h )] − v F F ′′ [( ∂ µ h )( D ν π ) b − ( ∂ ν h )( D µ π ) b ] 0 . (B.5)The Lagrangian term I ( φ ) containing the potential and Yukawa couplings is I ( φ ) = − V ( h ) + K ( h ) n · W (B.6)where W is a constant, in the notation of Ref. [1]. ∇ i ∇ j I = " g ab (cid:2) v F F ′ ( W · n K ′ − V ′ ) − W · n K (cid:3) F (cid:0) KF (cid:1) ′ W · n ,a F (cid:0) KF (cid:1) ′ W · n ,b − V ′′ + K ′′ n · W (B.7)– 49 –here n ,a = ∂ n /∂π a .The field strength Y µν is[ Y µν ] i j = (cid:2) − v ( F ′ ) (cid:3) [( D µ π ) a ( D ν π ) b − ( D ν π ) a ( D ν π ) b ] F ′′ F [( ∂ µ h )( D ν π ) a − ( ∂ ν h )( D µ π ) a ] − v F F ′′ [( ∂ µ h )( D ν π ) b − ( ∂ ν ) h ( D µ π ) b ] 0 + A βµν ( t iβ ) ; j (B.8)with A βµν ( t iβ ) ; j = − F F ′ ( ∂ b n ) T A µν nv F ′ F g ac ( ∂ c n ) T A µν n g ac ( ∂ c n ) T A µν ( ∂ b n ) (B.9)and A µ = gW µ + g ′ B µ − gW µ gW µ − gW µ − g ′ B µ gW µ gW µ gW µ − gW µ gW µ − g ′ B µ − gW µ − gW µ − gW µ + g ′ B µ (B.10)in terms of the electroweak gauge bosons. The field strength tensor A µν is given by Eq. (B.10)with the replacements W αµ → W αµν , B µ → B µν . The covariant derivative D µ n is given by D µ n = ∂ µ n + A µ n (B.11)treating n as a four-component column vector, and using matrix multiplication. The covariantderivative on π is defined implicitly through D µ n · D µ n = g ab ( π )( D µ π ) a ( D µ π ) b (B.12)Substituting the above equations into Eq. (4.42) gives Eq. (59) in Ref. [1]. C Non-reductive Cosets
In this appendix, we comment briefly on the CCWZ formalism when [ T a , X B ] contains a pieceproportional to the unbroken generators, so that the coset is non-reductive. Such examplesare relevant for constructing G / H theories with negative sectional curvature [1].One can still define the CCWZ ξ field as in Eq. (5.9) which transforms as in Eq. (5.12).The complication for the non-reductive case is in Eq. (5.13). For g ∈ H , g (cid:0) π A ( x ) X A (cid:1) g − (C.1)is no longer a linear combination of the broken generators, but also has a component alongthe unbroken generators, g π A ( x ) X A g − = X A h D R ( π ) ( g ) i A B π B + T a M aB π B , (C.2)– 50 –here D R ( π ) is the R ( π ) transformation matrix constructed out of f CaB , as in Eq. (5.6), and M aB π B is the component in the unbroken direction. The exponential of Eq. (C.2) can beschematically written as e X + T = e X ′ e T ′ (C.3)where X, X ′ are linear combinations of broken generators, and T, T ′ are linear combinationsof unbroken generators, and the primed and unprimed quantities are connected by the Baker-Campbell-Hausdorff formula. Thus one gets Eq. (5.12) with some important changes even inif g ∈ H : (a) The relation between π and π ′ is non-linear. Eq. (5.14) only holds for the linearterm, i.e. for the transformation of the tangent vector to the Goldstone boson manifold atthe origin, and (b) h ′ ( ξ ( x ) , g ) depends on ξ and hence x , even if g ∈ H .The transformation of ( D µ π ) and V µ in Eqs. (5.17) and (5.18) is also changed,( D µ π ) → h ( D µ π ) h − (cid:12)(cid:12) X (C.4) V µ → hV µ h − − ∂ µ h h − + h ( D µ π ) h − (cid:12)(cid:12) T (C.5)( D µ π ) transforms by adjoint action by H in the representation R ( π ) , as before. However, V µ picks up an additional piece and no longer transforms as a gauge field under H . Onecan still define Goldstone boson kinetic terms as before, Eq. (5.20). However, since V µ doesnot transform as a gauge field, it is not possible to define covariant derivatives on matterfields ψ which transform as arbitrary irreducible representations of H , as was done in CCWZ.Nevertheless, some matter fields are allowed in the EFT. For example, if ψ transforms as arepresentation R G of the full group G , ψ → D ( g ) ψ , (C.6)then ( ∂ µ + it α A αµ ) ψ (C.7)is a covariant derivative, where the generators t α are in the R G representation. FollowingCCWZ, we can define new fields χ by χ = D ( ξ † ) ψ (C.8)which transform as χ → D ( h ) χ , (C.9)where h is given by Eq. (5.12). The covariant derivative Eq. (C.7) turns into( ∂ µ + ξ − D µ ξ ) χ = (cid:2) ∂ µ + i ( D µ π ) A X A + iV aµ T a (cid:3) χ (C.10)– 51 –n χ using Eq. (5.17). The sum ( D µ π + V µ ) in the covariant derivative transforms as a gaugefield ( D µ π + V µ ) → h ( D µ π + V µ ) h − − ∂ µ h h − , (C.11)and the covariant derivative Eq. (C.10) is well-defined. For compact groups, where ( D µ π )transforms as ( D µ π ) → h ( D µ π ) h − , (C.12)and does not mix with V µ , one can omit ( D µ π ) in Eq. (C.10) to get the CCWZ covariantderivative. In this case, for the covariant derivative on χ to make sense, it is only necessaryto define the action of the unbroken generators T a on χ , i.e. one can restrict χ to only be inan irreducible representation of H ; it does not have to form a representation of G . Baryons inQCD are an example — they form a representation of the unbroken SU (3) V symmetry, butnot of chiral SU (3) L × SU (3) R . However, for the non-reductive case, it is necessary to retainthe ( D µ π ) term in the covariant derivative, to cancel the extra piece in the transformation of V µ , the last term in Eq. (C.5). In this case, we need to define the action of T a and X A , whichrequires χ to form a representation of the full symmetry G , not just its unbroken subgroup.The main difficulty for sigma models with non-compact H is unitarity. The ψ kineticenergy term for compact groups H is X a ( D µ ψ ) † a ( D µ ψ ) a (C.13)if ψ is a complex scalar. If H is non-compact, then the unitary representations are infinitedimensional. For a finite dimensional non-unitary representation, the kinetic term Eq. (C.13)is not an invariant, since ψ † does not transform as the inverse of ψ . One can construct invariantterms. For example, if H is SO (3 , ψ transforms as the (real) vector representation, X i =1 , , ( D µ ψ i ) ( D µ ψ i ) − ( D µ ψ ) ( D µ ψ ) (C.14)is invariant, as should be familiar from the Lorentz group. Eq. (C.14) has a wrong signkinetic term, and leads to ghosts. We do not know, in general, whether there are finitedimensional representations for a non-compact group H with a positive definite H -invariantkinetic energy term. This is possible for a trivial example: if H is a non-compact U (1), i.e. ofthe form h = exp αT , −∞ ≤ α ≤ ∞ , one can pick the fermion to transform as exp iqα , andthe kinetic energy Eq. (C.13) is H -invariant.One can construct a suitable kinetic energy term if H is compact even if G is non-compact,since ψ transforms under H , not G . An example of this type based on SO (4 , → SO (4) wasstudied in Ref. [34]. In this case, the low energy EFT is unitary. However, implementing a– 52 –nitary UV theory in which G invariance is manifest is problematic, and we do not know ofany examples where this is possible. C.1 Example of a Non-reductive Coset.
A simple example of a non-reductive coset is the 2-parameter group of matrices " x y , y > , (C.15)under multiplication. The generators (absorbing a factor of i ) can be chosen as T = " , X = " − , (C.16)with the commutation relation [ T, X ] =
T . (C.17)If the matrices act on a vector v = " , (C.18)then T v = 0, Xv = 0, so that T is an unbroken generator and X is a broken generator. Thematrices are sufficiently simple that the CCWZ formulæ can be computed explicitly. Theexponential of a Lie algebra element is g = e aT + bX = " ab (1 − e − b ) e − b , (C.19)so that ξ = e πX = " e − π , (C.20)and e uT = " u . (C.21)The CCWZ multiplication rule ge πX = e π ′ X e u ′ T (C.22) A simple argument due to S. Rychkov is to look at G -current correlators (cid:10) J µα J νβ (cid:11) in the UV theory. G invariance requires the correlator to be proportional to the Killing form B αβ , which is not positive definite if G is non-compact, so that unitarity is violated. However, the low-energy EFT correlators are unitary, so itmight be possible to construct theories where the G symmetry of G / H arises only in the low energy limit. – 53 –ith g in Eq. (C.19) gives π ′ = π + b,u ′ = ab (cid:16) e b − (cid:17) e π . (C.23)In the special case where g ∈ H , b = 0 and π ′ ( x ) = π ( x ) ,u ′ ( x ) = ae π ( x ) , (C.24)so that u ′ depends on x through π ( x ). Eq. (C.22) becomes e aT e π X = e πX h, h ( x ) = e ae π ( x ) T , (C.25)and h depends on x even for an unbroken transformation.The Maurer-Cartan form is ξ − d ξ = d π X, ω = ξ − d ξ (cid:12)(cid:12) X , V = ξ − d ξ (cid:12)(cid:12) T , (C.26)so that ω = d π X, V = 0 . (C.27)Under a global unbroken transformation g = exp aT , ξ − d ξ → ξ ′ − d ξ ′ = d π ′ X. (C.28)Using Eq. (C.22), ω ′ = ω, V ′ = 0 . (C.29)The transformation laws are ω ′ = hωh − (cid:12)(cid:12) X V ′ = hωh − (cid:12)(cid:12) T + hV h − − d hh − (C.30)with h in Eq. (C.25). These equations are satisfied because of the extra hωh − term in the V transformation. References [1] R. Alonso, E. E. Jenkins, and A. V. Manohar,
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