HHiggs Potential from Derivative Interactions
A. Quadri ∗ Dip. di Fisica, Universit`a degli Studi di Milano,via Celoria 16, I-20133 Milano, Italyand INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy (Dated: May 8, 2017)
Abstract
A formulation of the linear σ model with derivative interactions is studied. The classical theoryis on-shell equivalent to the σ model with the standard quartic Higgs potential. The mass ofthe scalar mode only appears in the quadratic part and not in the interaction vertices, unlikein the ordinary formulation of the theory. Renormalization of the model is discussed. A nonpower-counting renormalizable extension, obeying the defining functional identities of the theory,is presented. This extension is physically equivalent to the tree-level inclusion of a dimension sixeffective operator ∂ µ (Φ † Φ) ∂ µ (Φ † Φ). The resulting UV divergences are arranged in a perturbationseries around the power-counting renormalizable theory. The application of the formalism to theStandard Model in the presence of the dimension-six operator ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) is discussed.
PACS numbers: 11.10.Gh,11.15.Ex,11.30.Rd ∗ Electronic address: [email protected] a r X i v : . [ h e p - t h ] M a y . INTRODUCTION The discovery of the Higgs boson at the LHC [1, 2] has by now firmly established theexistence of a scalar particle as a fundamental ingredient of the spontaneous symmetrybreaking (SSB) mechanism for electroweak theory [3–6].On the other end, further investigation is required in order to understand the propertiesof the SSB potential. In addition to the Standard Model (SM) quartic Higgs potential, manyother possibilities can be considered. The model-independent approach based on the effectivefield theory (EFT) technique allows to disentangle the phenomenological consequences ofhigher dimensional operators [7, 8]. The one-loop anomalous dimensions of dimension-sixoperators have been studied in [9–13]. Prospects of measuring anomalous Higgs couplingsat the LHC and at future colliders have been considered e.g. in [14–16].In this paper we discuss the field-theoretical properties of a formulation of the SSBpotential based on higher derivatives interactions [17] that, at the classical level, is physicallyequivalent to the quartic Higgs potential.The main advantage of this formulation is that the mass of the physical Higgs excitationonly enters in the mass term of the physical field and not in the coupling constant (unlikein the ordinary quartic potential).This has a number of consequences. The functional equations governing the theoryinclude the equation of motion for the physical massive mode. Let us denote it by X . Inaddition to the quadratic mass term, one can include in the action a kinetic term for X (cid:90) d x z ∂ µ X ∂ µ X . Once such a term is introduced into the classical action, the equation of motion for X is modified by a contribution linear in X and still survives quantization (due to the factthat the breaking is linear in the quantized fields). On the other hand, power-countingrenormalizability is lost, since new divergences arise, vanishing at z = 0.This remark leads to a perturbative definition of the non power-counting renormalizabletheory as a series expansion around z = 0, i.e. the coefficients of the expansion in z of theamplitudes are fixed in terms of amplitudes of the renormalizable theory at z = 0.By going on-shell and eliminating X via the relevant equation of motion we obtain the lin-ear σ model with a quartic Higgs potential plus the dimension six operator ∼ z∂ µ Φ † Φ ∂ µ Φ † Φ.2n this model the mass of the physical scalar particle is given by M phys = M z . The limitwhere the mass scale of the theory M goes to infinity while keeping M phys fixed coincideswith the strongly interacting regime z → ∞ .The X -equation allows one to control the UV divergences of the one-particle-irreducible(1-PI) Green’s functions with X -legs in terms of amplitudes with insertions of externalsources with a better UV behaviour.This allows one to disentangle some new relations between 1-PI amplitudes involving X -external lines that are not apparent on the basis of the power-counting of the underlyingeffective field theory.One might then conjecture that such relations will translate into consistency conditionsbetween the Green’s functions in the ordinary Higgs EFT once one eliminates X throughthe equations of motion of the auxiliary fields. This is currently under investigation [18].The paper is organized as follows. In Sect. II we introduce our notation and discuss thetheory at z = 0. The BRST symmetry is given and the tree-level on-shell equivalence withthe linear σ model in the presence of a quartic Higgs potential is discussed. In Sect. IIIthe mechanism guaranteeing the on-shell equivalence of the derivative interactions with theusual quartic Higgs potential is elucidated on some sample tree-level computations. InSect. IV the functional identities of the theory are presented and the renormalization of themodel at z = 0 is carried out. The one-loop divergences are computed and the on-shellnormalization conditions are presented. Sect. V describes the Standard Model (SM) actionin the derivative representation of the Higgs potential. The BRST symmetry of the SM isprovided. In Sect. VI the non power-counting renormalizable theory at z (cid:54) = 0 is consideredand the differential equation defining the Green’s functions of the theory as an expansionaround z = 0 is introduced. In Sect. VII the UV subtraction of the model at z (cid:54) = 0 is studied.The ambiguities in the choice of the finite parts of the counterterms at z (cid:54) = 0 are relatedto the insertion at zero momentum of the quadratic operator X (cid:3) X in the amplitudes ofthe power-counting renormalizable theory at z = 0. In Sect. VIII we analyze how non-localsymmetric deformations of the X -propagator can induce a UV completion of the theory,restoring power-counting renormalizability also at z (cid:54) = 0, as well as the UV completionrealized through the addition of further physical scalars, while preserving locality of theclassical action. Conclusions are presented in Sect. IX.3 I. FORMULATION OF THE THEORY
We start from the following action written in components S = (cid:90) d x (cid:104) ∂ µ σ∂ µ σ + 12 ∂ µ φ a ∂ µ φ a − M X + 1 v ( X + X ) (cid:3) (cid:16) σ + vσ + 12 φ a − vX (cid:17)(cid:105) . (2.1)The fields ( σ, φ a ) belong to a SU(2) doubletΦ = 1 √ (cid:16) iφ + φ σ + v − iφ (cid:17) , (2.2) X is a SU(2) singlet. v is the scale of the spontaneous symmetry breaking.The equation of motion of X is δS δX = 1 v (cid:3) (cid:16) σ + vσ + 12 φ a − vX (cid:17) . (2.3)Going on-shell with X this yields (neglecting zero modes of the Laplacian ) X = 12 v σ + σ + 12 v φ a , (2.4)which, substituted into Eq.(2.1), gives S | on − shell = (cid:90) d x (cid:104) ∂ µ σ∂ µ σ + 12 ∂ µ φ a ∂ µ φ a − M v (cid:16) σ + vσ + 12 φ a (cid:17) (cid:105) , (2.5)i.e. at tree level one finds the ordinary SU(2) linear σ model with a quartic Higgs potentialof coupling constant λ = M v .It should be remarked that the right sign of the potential, triggering the spontaneoussymmetry breaking in Eq.(2.5), is dictated by the sign of the mass term of the field X ,which in turn is fixed by the requirement of the absence of tachyons in the theory.Notice that by going on-shell with X , asymptotically σ coincides with X , as can be seenby taking the linearization of the r.h.s. of Eq.(2.4). In particular, σ acquires a mass M , asin Eq.(2.5). A rigorous argument of why this is legitimate will be given in Subsection II A after Eq.(2.16). . Off-shell Formalism The off-shell implementation of the constraint in Eq.(2.4) can be realized `a la
BRST [19–21]. The construction works as follows [17]. One introduces a pair of ghost c and antighost¯ c such that the BRST variation of the antighost is the constraint: s ¯ c = Φ † Φ − vX − v σ + vσ + 12 φ a − vX . (2.6)The Lagrange multiplier field X , enforcing the constraint in the action S , pairs with theghost c into a BRST doublet [22–24] sX = vc , sc = 0 . (2.7)A set of variables u, v such that su = v, sv = 0 is known as a BRST doublet or a trivialpair [25] since it does not affect the cohomology H ( s ) of the BRST differential s . We recallthat H ( s ) is the set of local polynomials in the fields and their derivatives such that twopolynomials I and I are equivalent if and only if they differ by a s -exact term: I = I + s K for some K . H ( s ) identifies the local physical observables of the theory [22, 24]. If oneconsiders local functionals (e.g. the action) and allows for integration by parts, one definesin a similar way the cohomology H ( s | d ) of s modulo the exterior differential d [22, 24]. H ( s | d ) controls the local deformations of the classical action (including counterterms) aswell as (in the sector with ghost number one) the potential anomalies of the theory.By Eq.(2.7) X and c are not physical fields of the theory (as expected, since X is aLagrange multiplier and c is required in the algebraic BRST implementation of the off-shellconstraint but should not affect the physics itself). All other fields are BRST invariant: sσ = sφ a = sX = 0 . (2.8) s is nilpotent.One recovers BRST invariance of the action by adding to S the ghost-dependent term S ghost = − (cid:90) d x ¯ c (cid:3) c (2.9)so that the full action of the theory is S = S + S ghost . (2.10)5otice that the ghost is a free field. It should be stressed that the BRST symmetry s isnot associated with a local gauge symmetry of the theory. It implements algebraically the(SU(2)-invariant) constraint in Eq.(2.4).We remark that also the pair ¯ c, F = σ + vσ + φ a − vX forms a BRST doubletaccording to Eq.(2.6) and therefore drops out of the cohomology H ( s ). The cohomology H ( s ), respecting all the relevant symmetries of the theory, is thus given by Lorentz-invariant,global SU(2)-invariant polynomials constructed out of the doublet Φ and derivatives thereof, X being cohomologically equivalent to Φ † Φ according to Eq.(2.6). This is the cohomologyof the linear σ model, as expected, since the introduction of the fields X , X , ¯ c, c in orderto enforce the constraint in Eq.(2.4) should not alter the physics of the theory.Since the BRST transformation of the antighost ¯ c is non-linear in the quantized fields, oneneeds one external source ¯ c ∗ in order to control the renormalization of the BRST variation s ¯ c . The latter need to be coupled to the source ¯ c ∗ , known as an antifield [22, 25].Thus the tree-level vertex functional of the theory is finally given byΓ (0) = (cid:90) d x (cid:104) ∂ µ σ∂ µ σ + 12 ∂ µ φ a ∂ µ φ a − M X − ¯ c (cid:3) c + 1 v ( X + X ) (cid:3) (cid:16) σ + vσ + 12 φ a − vX (cid:17) + ¯ c ∗ (cid:16) σ + vσ + 12 φ a − vX (cid:17)(cid:105) . (2.11)Notice that the second line can be rewritten as a s -exact term as follows: S constr = (cid:90) d x (cid:104) − ¯ c (cid:3) c + 1 v ( X + X ) (cid:3) (cid:16) σ + vσ + 12 φ a − vX (cid:17)(cid:105) = (cid:90) d x s ( 1 v ¯ c (cid:3) ( X + X )) . (2.12)We see that the first line of Eq.(2.11) describes the (SU(2)-invariant) action of the linear σ -model (in the derivative representation of the potential), the second (BRST-exact) linethe off-shell implementation of the constraint in Eq.(2.4), while the third line contains theantifield-dependent sector.BRST invariance can be translated into the following Slavnov-Taylor (ST) identity (cid:90) d x (cid:16) vc δ Γ δX + δ Γ δ ¯ c ∗ δ Γ δ ¯ c (cid:17) = 0 (2.13)which holds for the full vertex functional Γ (the generator of the 1-PI amplitudes, whoseleading order in the loop expansion coincides with Γ (0) ).6he ghost field c has ghost number +1, the antighost field ¯ c has ghost number −
1. Allother fields and the external source ¯ c ∗ have ghost number zero. Γ has ghost number zero.Some comments are in order. According to Eq.(2.6) we haveΦ † Φ − v vX + s ¯ c (2.14)and thus when X = 0 the operator Φ † Φ − v is physically equivalent to the null operator. Inthis case the theory reduces to the non-linear σ model [17], enforcing off-shell the non-linearconstraint Φ † Φ − v = 0.When X is different than zero, the theory has the same degrees of freedom as the linear σ model. Notice in particular that the X -equation of motion δS δX = 1 v (cid:3) (cid:16) σ + vσ + 12 φ a − vX (cid:17) = 0 (2.15)yields the most general solution X = 12 v σ + σ + 12 v φ a + χ (2.16)with χ a massless degree of freedom satisfying the free Klein-Gordon equation (cid:3) χ = 0.Compatibility between Eqs.(2.14) and (2.16) entails that χ must be cohomologically equiv-alent to the null operator and thus one can safely set χ = 0 when going on-shell with theauxiliary field X .Since X is a scalar singlet, one can add any polynomial in X and ordinary derivativesthereof to Γ (0) without breaking BRST symmetry. With the conventions adopted, σ and X have zero vacuum expectation value. This prevents to add the X -tadpole contribution tothe action. The simplest term is then a mass term for X as in Eq.(2.1). This turns out tobe compatible with power-counting renormalizability [17]. B. Propagators
Diagonalization of the quadratic part of Γ (0) is achieved by setting σ = σ (cid:48) + X + X .Then the propagators are∆ σ (cid:48) σ (cid:48) = ip , ∆ φ a φ b = iδ ab p , ∆ ¯ cc = ip ∆ X X = − ip , ∆ X X = ip − M . (2.17)7otice the minus sign in the propagator of X . This entails that the combination X = X + X has a propagator which falls off as p − for large momentum:∆ XX = iM p ( p − M ) . (2.18)The physical states of the theory are identified by standard cohomological methods [25,26]. The asymptotic BRST charge Q acts on the fields as the linearization of the BRSTdifferential s . X is not invariant under Q and thus it does not belong to the physical space H = Ker Q/ Im Q , as well as σ (cid:48) (since [ Q, σ (cid:48) ] = [
Q, σ − X − X ] = − vc ). The combination σ (cid:48) + X is BRST invariant, however it is BRST exact, since it is generated by the variationof the antighost field ¯ c : [ Q, ¯ c ] + = σ (cid:48) + X (2.19)and thus it does not belong to H . The only physical modes are the scalar X and the fields φ a , namely the degrees of freedom of the linear σ model. III. TREE-LEVELA. On-shell amplitudes
We check in this Section the on-shell equivalence between the theory and the linear σ model on the tree-level 3- and 4-point amplitudes.
1. 3-point amplitude
The off-shell 3-point amplitude is A X X X = − iv (cid:88) i =1 p i (3.1)where the sum is over the momenta of the particles. By going on shell p i = M we get A X X X | on − shell = − iv M , (3.2)which coincides with the amplitude of the SU(2) linear sigma model for the coupling constant λ = M v . 8 . 4-point amplitude The situation is more involved here. Diagrams contributing with an exchange of a σ (cid:48) and X of momentum q sum up to cancel out the unphysical pole at q = 0, yielding a finitecontribution A (1) X X X X (cid:12)(cid:12)(cid:12) on − shell = i M v . (3.3)Diagrams where a X particle is exchanged give in turn A (2) X X X X (cid:12)(cid:12)(cid:12) on − shell = − i M v − i M v (cid:16) s − M + 1 t − M + 1 u − M (cid:17) , (3.4)as a function of the usual Mandelstam variables.The sum is A X X X X | on − shell = − i M v − i M v (cid:16) s − M + 1 t − M + 1 u − M (cid:17) . (3.5)The first term is the one arising in the linear sigma model from the contact four-point vertex,the last three are those generated by diagrams with the exchange of a propagator and twotrilinear couplings. Notice that the contact interaction contribution is controlled by the sumof two terms, originating both from A (1) and A (2) . IV. RENORMALIZATIONA. Functional identities
In addition to the ST identity in Eq.(2.13), the theory obeys a set of functional identitiesconstraining the 1-PI Green’s functions and their UV divergences: • the ghost and the antighost equations δ Γ δ ¯ c = − (cid:3) c , δ Γ δc = (cid:3) ¯ c . (4.1)These equations imply that the ghost and the antighost fields are free to all order inthe loop expansion. • since the ghost is free, one can take a derivative w.r.t. c of the ST identity and usethe first of Eqs.(4.1) to get the X equation for the full vertex functional δ Γ δX = 1 v (cid:3) δ Γ δ ¯ c ∗ . (4.2)9 the shift symmetryThe shift symmetry δX ( x ) = α ( x ) , δX ( x ) = − α ( x ) , (4.3)gives δ Γ δX − δ Γ δX = − (cid:3) ( X + X ) + M X + v ¯ c ∗ . (4.4)The r.h.s. is linear in the quantized fields and therefore the classical symmetry can beextended at the full quantum level. By using the X equation (4.2) into Eq.(4.4) weobtain the X -equation δ Γ δX = 1 v (cid:3) δ Γ δ ¯ c ∗ + (cid:3) ( X + X ) − M X − v ¯ c ∗ . (4.5) • global SU(2) invariance (cid:90) d x (cid:104) − α a φ a δ Γ δσ + (cid:16)
12 ( σ + v ) α a + 12 (cid:15) abc φ b α c (cid:17) δ Γ δφ a (cid:105) = 0 . (4.6)In the above equation α a are constant parameters and Γ denotes the full 1-PI vertexfunctional (the generator of 1-PI amplitudes). B. Power-counting
The potentially dangerous interaction terms are the ones involving two derivatives arisingfrom the fluctuation around the SU(2) constraint, namely (cid:90) d x v ( X + X ) (cid:3) (cid:16) σ + 12 φ a (cid:17) . (4.7)1-PI Green’s functions involving external X and X legs are not independent, since they canbe obtained through the functional identities Eqs. (4.2) and (4.5) in terms of amplitudes onlyinvolving insertions of σ (cid:48) , φ a and ¯ c ∗ . For these amplitudes the dangerous interaction verticesin Eq.(4.7) are always connected inside loops to the combination X . Since the propagator∆ XX falls off as 1 /p for large momenta, it turns out that the theory is still renormalizableby power counting.The UV indices of the fields and the external source ¯ c ∗ are as follows: σ (cid:48) and φ a have UVdimension 1, ¯ c ∗ has UV dimension 2. 10 . Structure of the counterterms We consider the action-like sector independent of X and X , since amplitudes involvingthese latter fields are controlled by Eqs.(4.2) and (4.5).Eq.(4.6) entails that the dependence on σ and φ a can only happen through action-likefunctionals invariant under global SU(2) symmetry, namely L ct, = − Z (cid:16) ∂ µ σ∂ µ σ + 12 ∂ µ φ a ∂ µ φ a (cid:17) − M (cid:16) σ + vσ + 12 φ a (cid:17) − G (cid:16) σ + vσ + 12 φ a (cid:17) − R ¯ c ∗ (cid:16) σ + vσ + 12 φ a (cid:17) . (4.8)There also two invariants depending only on ¯ c ∗ , i.e. L ct, = −R ¯ c ∗ − R (¯ c ∗ ) . (4.9)The most general counterterm Lagrangian at X = X = 0 is thus given by L ct = L ct, + L ct, . (4.10)Notice the appearance of a quartic potential term absent in the classical action (2.1). Itcan be introduced from the beginning into the classical action without violating power-counting renormalizability, as was done in [17]. Notice that if one adds the invariant (cid:82) d x G (0) (cid:16) σ + 2 vσ + φ a (cid:17) at tree level, the physical content of the theory does notchange ( G (0) is not an additional physical parameter). Indeed this term can be rewritten as (cid:90) d x G (0) (cid:16) σ + 2 vσ + 12 φ a (cid:17) = (cid:90) d x s (cid:104) G (0) ¯ c (cid:16) σ + 2 vσ + 12 φ a + vX (cid:17)(cid:105) + G (0) v X (4.11)and this amounts to a redefinition of the mass parameter M → M − G (0) v plus a s -exactterm that does not affect the physics. The relevant deformations of the functional identitiescontrolling the theory when G (0) is non-zero has been given in [17]. D. One-loop divergences
The one-loop divergences are controlled by the six coefficients Z (1) , M (1) , G (1) , R ( j ) . Am-plitudes are dimensionally regularized in D dimensions.11he amplitude Γ (1)¯ c ∗ fixes R (2) = − π M − D . Γ (1)¯ c ∗ ¯ c ∗ fixes R (3) = π − D . Γ (1)¯ c ∗ σ (cid:48) fixes R (1) = − π M v − D .Moreover Γ (1) φ a φ b (cid:12)(cid:12)(cid:12) UV div = 18 π M v δ ab − D . (4.12)The divergent part has no momentum dependence. This implies that Z (1) = 0 and M (1) = π M v − D . Finally from the amplitude Γ (1) σ (cid:48) σ (cid:48) one obtains the coefficient G (1) = π M v − D .Let us now compute the divergences of the one- and two-point functions of X and X .This requires to use Eqs.(4.2) and (4.5). One finds (we denote the external momenta asarguments of the fields) Γ ( n ) X (0) = Γ ( n ) X (0) = 0 , n ≥ . (4.13)Notice that Eqs.(4.2) and (4.5) holds in the σ − X − X (canonical) basis. If one performsexplicit computations in the most convenient diagonal σ (cid:48) − X − X basis one needs to takeinto account the contributions arising from the field redefinition from σ (cid:48) to σ , namely on theexample of the one-point functions (we denote by an underline the Green’s functions in thediagonal basis whenever they differ from those in the canonical basis) (cid:90) d x (cid:16) Γ (1) σ (cid:48) σ (cid:48) + Γ (1) X X + Γ (1) X X (cid:17) = (cid:90) d x (cid:104) Γ (1) σ (cid:48) σ + (Γ (1) X − Γ (1) σ (cid:48) ) X + (Γ (1) X − Γ (1) σ (cid:48) ) X (cid:105) (4.14)so that one finds by explicit computationsΓ (1) σ (cid:48) (0) = Γ (1) X (0) = Γ (1) X (0) = 116 π M v A ( M ) (4.15)in terms of the standard Passarino-Veltman scalar function A ( M ) (we use the conventionsof [27, 28]). Hence from Eq.(4.14)Γ (1) X (0) = Γ (1) X (0) − Γ (1) σ (cid:48) (0) = 0 , Γ (1) X (0) = Γ (1) X (0) − Γ (1) σ (cid:48) (0) = 0 , (4.16)consistent with Eq.(4.13). We denote by a subscript the fields and external sources w.r.t. which one differentiates, e.g. Γ ¯ c ∗ = δ Γ δ ¯ c ∗ .It is understood that we set all fields and external sources to zero after differentiation. ( n ) X ( − p ) X ( p ) = Γ ( n ) X ( − p ) X ( p ) = Γ ( n ) X ( − p ) X ( p ) = 1 v p Γ ( n )¯ c ∗ ( − p )¯ c ∗ ( p ) , n ≥ , (4.17)so that the common UV divergent part is1 v p Γ ( n )¯ c ∗ ( − p )¯ c ∗ ( p ) (cid:12)(cid:12)(cid:12)(cid:12) UV div = 14 π p v − D .
E. Normalization conditions
The two-point functions of X and X in the diagonal basis readΓ ( n ) X ( − p ) X ( p ) = Γ ( n ) X ( − p ) X ( p ) = Γ ( n ) X ( − p ) X ( p ) = p v Γ ( n )¯ c ∗ ( − p )¯ c ∗ ( p ) − Γ ( n ) σ (cid:48) ( − p ) σ (cid:48) ( p ) . (4.18)The finite part of the coefficient R ( n )3 is chosen order by order in the loop expansion so thatΓ ( n ) X ( − p ) X ( p ) (cid:12)(cid:12)(cid:12) p = M = Γ ( n ) X ( − p ) X ( p ) (cid:12)(cid:12)(cid:12) p = M = 0 . (4.19)This ensures that there is no mixing between X and X at the pole p = M . By Eq.(4.18)this also implies that Γ ( n ) X ( − p ) X ( p ) (cid:12)(cid:12)(cid:12) p = M = 0 , (4.20)i.e. one is performing an on-shell renormalization, fixing the position of the physical pole ofthe X mode at its tree level value M .The finite part of the coefficient R ( n )2 is chosen order by order in the loop expansion insuch a way to ensure the absence of a tadpole contribution to X :Γ ( n ) X (0) (cid:12)(cid:12)(cid:12) p = M = − p v Γ ( n )¯ c ∗ (0) (cid:12)(cid:12)(cid:12) p = M + Γ ( n ) σ (cid:48) (0) (cid:12)(cid:12)(cid:12) p = M = 0 . (4.21)The finite part of the coefficient R ( n )1 is adjusted so that the mixing between σ (cid:48) and X vanishes on the pole p = M :Γ ( n ) X ( − p ) σ (cid:48) ( p ) (cid:12)(cid:12)(cid:12) p = M = − p v Γ ( n ) σ (cid:48) ( − p )¯ c ∗ ( p ) (cid:12)(cid:12)(cid:12) p = M + Γ ( n ) σ (cid:48) ( − p ) σ (cid:48) ( p ) (cid:12)(cid:12)(cid:12) p = M = 0 . (4.22)Eqs.(4.19), (4.21) and (4.22) ensure that the massive physical scalar mode is described bythe field X to all orders in the loop expansion.The finite parts of the remaining coefficients M ( n ) , G ( n ) and Z ( n ) are chosen in orderto ensure the absence of a tadpole for σ (cid:48) and the on-shell normalization conditions for σ (cid:48) ,namely Γ ( n ) σ (cid:48) (0) (cid:12)(cid:12)(cid:12) p =0 = 0 , Γ ( n ) σ (cid:48) ( − p ) σ (cid:48) ( p ) (cid:12)(cid:12)(cid:12) p =0 = 0 , ddp Γ ( n ) σ (cid:48) ( − p ) σ (cid:48) ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 = 1 . (4.23)13 . INCLUSION IN THE STANDARD MODEL We now discuss how the Higgs potential in the derivative representation is included intothe Standard Model (SM). X , X and the ghosts c, ¯ c are invariant under the electroweak gauge group SU L (2) × U Y (1)of weak isospin and hypercharge. The SU L (2) doublet Φ transforms as usual under aninfinitesimal gauge transformation δ Φ = (cid:16) − i α Y + i σ a α a (cid:17) Φ (5.1)where α a and α Y are the gauge parameters of SU L (2) and U Y (1) respectively and σ a are thePauli matrices. This implies that the couplings with the gauge fields and the fermions arethe same as in the SM.The SM action can be written as the sum of five terms: S SM = S Y M + S H + S F + S g.f. + S ghost . (5.2) S Y M , S H , S F , S g.f. , S ghost are respectively the Yang-Mills, the Higgs, the fermion, the gauge-fixing and the ghost parts. S Y M and S F are the same as in the ordinary formulation of thetheory and are given for the sake of completeness in Appendix A, where we also collect ournotations.The Higgs part S H is obtained from Eq.(2.1) upon replacement of ordinary derivativeswith the covariant ones and by adding the Yukawa sector as in the ordinary formulation ofthe SM: S H = (cid:90) d x (cid:104) ( D µ Φ) † D µ Φ − M X − (cid:88) i,j (cid:16) g ij ¯Ψ Li Ψ Rj, − Φ + ˜ g ij ¯Ψ Li Ψ Rj, + Φ C + h.c. (cid:17) + 1 v ( X + X ) (cid:3) (cid:16) Φ † Φ − v − vX (cid:17)(cid:105) . (5.3)Φ C is the charge conjugated field Φ C = iσ Φ ∗ .Spontaneous symmetry breaking induces a mixing between φ a and ∂A a . With the choiceof the doublet as in Eq.(2.2) the mixed bilinear terms read (cid:90) d x (cid:16) g v φ a ∂A a + δ a g v ∂Bφ (cid:17) . (5.4)14 is coupled to the divergence of the Z field, obtained from the Weinberg rotation as ( A µ is the photon) A µ = c W B µ − s W A µ , Z µ = s W B µ + c W A µ . (5.5)The sine and cosine of the Weinberg angle are given by c W = g (cid:112) g + g , s W = g (cid:112) g + g . (5.6)It is also convenient to introduce the charged combinations W ± = 1 √ A µ ∓ iA µ ) , φ ± = 1 √ φ ∓ iφ ) (5.7)The masses of the gauge bosons W ± , Z are M W = g v , M Z = v (cid:112) g + g . The mixings inEq.(5.4) are cancelled in a renormalized ξ -gauge by choosing S g.f. = (cid:90) d x (cid:16) b + F − + b − F + + b Z F Z + b A F A + ξ W b + b − + ξ Z b Z ) + ξ A b A ) (cid:17) (5.8)with the gauge-fixing functions F ± = ∂W ± + ξ W M W φ ± , F Z = ∂Z + M Z ξ Z φ , F A = ∂A . (5.9)Finally one constructs the ghost dependent part by summing the SM ghost sector andEq.(2.9) S ghost = (cid:90) d x (cid:16) − ¯ c + s F − − ¯ c − s F + − ¯ c Z s F Z − ¯ c A s F A − ¯ c (cid:3) c (cid:17) . (5.10)In the above equation s is the BRST differential associated with the SU L (2) × U Y (1) elec-troweak gauge group presented in Appendix A 1.The full BRST symmetry of the theory is given by ˜ s = s + s . ˜ s = 0 since both s and s are nilpotent and s and s anticommute, as a consequence of the fact that the constraint inEq.(2.6) is invariant under s .The physical states of the theory can then be identified as those belonging to the space15 phys = Ker ˜ Q / Im ˜ Q , where ˜ Q is the asymptotic charge associated with ˜ s :[ ˜ Q , A µ ] = ∂ µ c A , [ ˜ Q , Z µ ] = ∂ µ c Z , [ ˜ Q , W ± ] µ = ∂ µ c ± , [ ˜ Q , σ ] = 0 , [ ˜ Q , φ ± ] = M W c ± , [ ˜ Q , φ ] = M Z c Z , [ ˜ Q , X ] = 0 , [ ˜ Q , X ] = vc , [ ˜ Q , Ψ Li ] + = 0 , [ ˜ Q , Ψ Ri,σ ] + = 0 , [ ˜ Q , ¯ c ] + = v ( σ − X ) , [ ˜ Q , ¯ c A ] + = b A , [ ˜ Q , ¯ c Z ] + = b Z , [ ˜ Q , ¯ c ± ] + = b ± , [ ˜ Q , b A ] = [ ˜ Q , b Z ] = [ ˜ Q , b ± ] = 0 , [ ˜ Q , c ] + = [ ˜ Q , c A ] + = [ ˜ Q , c Z ] + = [ ˜ Q , c ± ] + = 0 . (5.11)One sees that the physical states are the physical polarizations of the gauge fields W ± µ , A µ , Z µ ,the fermion fields and the scalar X . Notice that σ is ˜ Q -invariant, however it belongs tothe same cohomology class of X , since X = σ − v [ ˜ Q , ¯ c ] + . (5.12)The antighosts and the Nakanishi-Lautrup fields drop out of H phys being BRST doublets,the ghosts and the pseudo-Goldstone bosons do not belong to H phys , since they also formBRST doublets. The standard quartet mechanism [20, 29–32] is at work.The equations (4.1), (4.2) and (4.5) controlling the dependence of the vertex functional Γon X , X , ¯ c and c do not change.We remark that the sector spanned by X , X , ¯ c and c respects custodial symmetry pro-vided that X , X , ¯ c and c do not transform under the global SU L (2) × SU R (2) group. Thiscan be seen by introducing the matrix Ω = (Φ C , Φ) . (5.13)Since Φ † Φ = Tr(Ω † Ω), we see that the last line of Eq.(5.3) is invariant under the custodialtransformation Ω (cid:48) = V L Ω V † R , V L ∈ SU L (2), V R ∈ SU R (2). VI. THE z -MODEL There is a unique term that can be added to the classical action in order to preserveEq.(4.5) at the quantum level by deforming its r.h.s. by a linear term in the quantized16elds, namely a kinetic term for X (cid:90) d x z ∂ µ X ∂ µ X . (6.1)By the same arguments leading to Eq.(2.5), upon eliminating X by imposing the X -equation of motion one obtains at tree-level the dimension-six operator (cid:90) d x zv ∂ µ Φ † Φ ∂ µ Φ † Φ . (6.2)The functional identities controlling the theory are unchanged with the exception ofEq.(4.5), which becomes δ Γ δX = 1 v (cid:3) δ Γ δ ¯ c ∗ + (cid:3) ( X + (1 − z ) X ) − M X − v ¯ c ∗ . (6.3)Notice that this equation is still valid for the SM with the inclusion of the kinetic term forthe X -field in Eq.(6.1). The propagator of X is modified as follows∆ X X = i (1 + z ) p − M . (6.4)Power-counting renormalizability is lost since the cancellation mechanism between the prop-agator of X and X is no more at work at z (cid:54) = 0. Indeed the propagator for the combination X = X + X is now ∆ XX = i ( − zp + M ) p [(1 + z ) p − M ] , (6.5)so that at z (cid:54) = 0 the propagator falls off as p − for large momentum p and therefore cannotcompensate the contributions from the derivative interaction vertices.The dependence on z can be controlled by the following differential equation ∂ Γ ∂z = (cid:90) d x δ Γ δR ( x ) (6.6)where R ( x ) is the source coupled in the classical action to the composite operator O ( x ) = − X (cid:3) X . Notice that the insertion of the operator O ( x ) happens at zero momentum (dueto the integration over d x ).This in turn entails that the Green’s functions at z (cid:54) = 0 can be defined as a perturbativeexpansion around the power-counting renormalizable model at z = 0.A comment is in order here. Eq.(6.3) relates the Green’s functions with the insertion of X -lines with those with the insertion of the source ¯ c ∗ , which has a better UV behaviour.17he simplest example is the two-point function for X , which is obtained by differentiatingEq.(6.3) w.r.t. X and then by replacing Γ X ¯ c ∗ by using once more Eq.(6.3), this time afterdifferentiation w.r.t. ¯ c ∗ . In momentum space we findΓ ( n ) X ( − p ) X ( p ) = 1 v p Γ ( n )¯ c ∗ ( − p )¯ c ∗ ( p ) , n ≥ . (6.7)The above equation states that the divergence of the two-point function Γ ( n ) X ( − p ) X ( p ) in thecanonical basis does not have a constant or a p -term. Therefore there cannot be any mixingbetween the operator X (cid:3) X and the operator X or X (cid:3) X (off-shell and by forbiddingfield redefinitions). This goes along the same bulk of patterns as those observed in the one-loop anomalous dimensions studied in [9–13], although one cannot establish an immediateand straightforward correspondence, due to the fact that field redefinitions are used in [9–13]and moreover those results are valid for one-loop on-shell matrix elements (while Eq.(6.7)holds off-shell and to all orders in the loop expansion). We stress the fact that Eq.(6.7) isvalid at any value fo z .A systematic study of the mixing constraints arising from Eq.(6.3) in the full SM withthe dimension-six operator induced by the kinetic term in Eq.(6.1) is beyond the scope ofthe present paper and is currently under investigation [18]. VII. EXPANSION AROUND z = 0 Eq.(6.6) states that the contribution of the order z n of a 1-PI Green’s function γ isobtained by a repeated insertion of the integrated operator (cid:82) d x O ( x ) in the X -lines ofeach diagram contributing to γ in the theory at z = 0.This can be easily understood since the propagator ∆ X X can be Taylor-expanded ac-cording to∆ X X = i (1 + z ) p − M = i ( p − M ) (cid:104) zp p − M (cid:105) = ip − M ∞ (cid:88) n =0 ( − n (cid:16) zp p − M (cid:17) n . (7.1)The term of order n in the r.h.s. of the above equation is generated by the insertion of n vertices izp on the X -line, namely n insertions of the operators (cid:82) d x O ( x ) at zero externalmomenta (see Fig. 1). 18 IG. 1: Mass insertions (the dots) on a X propagator (drawn as a solid line)FIG. 2: Diagrams contributing to Γ (1) σ (cid:48) σ (cid:48) (thin solid lines denote the X propagator, thick solid linesthe σ (cid:48) propagator. A dashed line denote a X propagator) We remark that the decomposition of the propagator ∆ X X at z (cid:54) = 0 can be expressedas the result of repeated mass insertions according to the following identity∆ X X = ∞ (cid:88) n =0 ( − n z n (cid:104) − δ n, ) n (cid:88) k =1 (cid:18) nk (cid:19) M k k ∂ k ∂ ( M ) k (cid:105) ip − M . (7.2) A. Two-point function
The z -dependence of the diagrams can be derived according to a simple prescription,namely one multiplies each diagram in the power-counting renormalizable theory at z = 0,with N internal X -lines and contributing to γ , by a prefactor (1 + z ) − N and replaces M → M / (1 + z ). Then the UV divergences are related directly to those of the amplitudesevaluated at z = 0.We consider here as an example the two-point amplitude Γ (1) σ (cid:48) σ (cid:48) in the linear σ model.There are five diagrams contributing to this amplitude, depicted in Fig. 2.The first two diagrams are UV convergent, the third and the fourth contain one X -line,the fifth two internal X -lines. This dictates the behaviour of the prefactors 1 / (1 + z ) infront of each amplitude. 19he physical mass of the canonically normalized X field is M phys = M z . (7.3)If one fixes M phys (which, when the model is embedded as the scalar sector of the electroweaktheory, corresponds to the measured Higgs mass), then for amplitudes expressed in terms of M phys there are no further sources of z -dependence other than the prefactors 1 / (1 + z ).Considering M as the mass of new physics, the limit M → ∞ is equivalent to z → ∞ . Inthis limit the diagrams involving the exchange of internal X -lines go to zero and one getsback the non-linear σ model described in [17].In terms of M phys the one-loop divergence of Γ (1) σ (cid:48) σ (cid:48) at finite z isΓ (1) UVσ (cid:48) σ (cid:48) = M phys π v (4 − D ) (cid:104) M phys − z (1 + z ) − p z (1 + z ) (cid:105) . (7.4)For large z it is of order 1 /z , as expected (since the diagrams 1 and 2 are not UV divergentand the remaining three contain at least one X -line). At z = 0 one recovers the divergenceof the power-counting renormalizable theory. B. Finite renormalizations
Consider now in the theory at z = 0 an amplitude γ with superficial degree of divergence δ ( γ ). Let us denote by γ R ( q ) ...R ( q n ) (cid:12)(cid:12) q k =0 the diagram with n insertions of the operator O atzero momentum. The superficial degree of divergence δ ( γ R ( q ) ...R ( q n ) (cid:12)(cid:12) q k =0 ) = δ ( γ ), since eachof the terms in the series of the r.h.s. of Eq.(7.1) tends to 1 for large p and thus does notalter the overall behaviour as 1 /p of the propagator ∆ X X .This implies that in the power-counting renormalizable theory at z = 0 the externalsource R has UV dimension zero.Consequently when one allows in the theory at z = 0 for the insertion of the operator (cid:82) d x O ( x ), the most general form of the counterterms is no more given by Eqs.(4.8) and(4.9).The problem of identifying the counterterms in the power-counting renormalizable theoryat z = 0 in the presence of the source R ( x ) amounts to find all possible Lorentz-covariant,global SU(2)-invariant local monomials in the fields, the external sources and their deriva-tives of dimension ≤
4. 20e can safely disregard all monomials involving derivatives of R ( x ), since we are onlyinterested in zero momentum insertions. Then Eq.(4.10) is modified as follows: L ct,R = A − Z R (cid:16) ∂ µ σ∂ µ σ + 12 ∂ µ φ a ∂ µ φ a (cid:17) − M R (cid:16) σ + vσ + 12 φ a (cid:17) − G R (cid:16) σ + vσ + 12 φ a (cid:17) − R ,R ¯ c ∗ (cid:16) σ + vσ + 12 φ a (cid:17) − R ,R ¯ c ∗ − R ,R (¯ c ∗ ) , (7.5)where Z R , M R , G R and R j,R , j = 1 , , K l,R , with K l standing for the corresponding quantities in Eqs.(4.8) and (4.9)) become analytic functionalsof the integrated source R ( x ): K l,R = K l + ∞ (cid:88) k =1 k ! (cid:90) k (cid:89) i =1 d x i K kl R ( x ) . . . R ( x k ) (7.6)while A = ∞ (cid:88) k =1 k ! (cid:90) k (cid:89) i =1 d x i A k R ( x ) . . . R ( x k )controls the renormalization of the O -insertions into vacuum amplitudes.The finite parts of the coefficients A k , K kl are unconstrained by the symmetries of thetheory and have to be chosen by an (infinite) set of normalization conditions. This is thecounterpart at z = 0 of the ambiguities induced by the loss of power-counting renormaliz-ability at z (cid:54) = 0. VIII. UV COMPLETION
Under the assumption that all scalar fields should obey at the classical level asymptoti-cally Klein-Gordon equations of motion, the most general X -equation is given by Eq.(6.3)and includes the kinetic term controlling the violation of power-counting renormalizability.The fact that X is a SU(2) singlet entails however on symmetry grounds that more generalquadratic terms in the tree-level action can be considered without violating the definingfunctional identities of the theory.These terms might capture some features of the UV completion of the model from a morefundamental theory, valid at a much higher scale Λ. This possibility is a peculiar feature ofthe higher derivative formulation where the physical parameters of the theory are embodiedin the two-point sector. 21s an example, the bad UV behaviour of the X propagator at z (cid:54) = 0 can be regularizedby the following choice of the X -propagator∆ X X = i (1 + z ) p − M − i Λ (cid:104) exp (cid:16) − zz + 1 Λ p (cid:17) − (cid:105) . (8.1)For large momenta ∆ X X goes as∆ X X ∼ p →∞ ip + i M − Λ z (1 + z ) p (8.2)so that ∆ XX goes as p − , making the derivative interaction terms harmless. Of coursesuch a deformation is not unique (for instance the replacement exp ( − zz +1 Λ p ) → exp ( − z Λ (1+ z ) p − M ) into Eq.(8.1) would also work).This implies that one can regularize the propagator of the X field in the UV in such away to preserve power-counting renormalizability without destroying the symmetries of themodel and without introducing new physical poles in the spectrum.Power-counting renormalizability at z (cid:54) = 0 can also be restored by adding new physicalSU(2)-invariant modes. For that purpose consider the following propagator for the field X (now a field with a non-trivial K¨allen-Lehmann spectral density):∆ X X = i (1 + z ) p − M + N (cid:88) j =1 iz j p − M j , (8.3)where z j >
0. The propagator in Eq.(8.3) describes a set of physical resonances at mass M / (1 + z ) and M j /z j . A suitable choice of the z j allows to cancel the UV behaviour in1 /p of ∆ XX , therefore re-establishing power-counting renormalizability. For simplicity wechoose all the z j to have the same common value z c . Then the leading order in 1 /p of thepropagator ∆ XX is cancelled provided that one chooses z c = N zz = N M M − M phys . (8.4)The positivity condition on z c yields M > M phys .Both with a UV-regularized propagator and with the addition of further resonances athigher masses, the X -equation now takes the form (written for convenience in the Fourierspace) δ Γ δX ( p ) = − v p δ Γ δ ¯ c ∗ ( p ) − p X + K ( p ) X ( p ) − v ¯ c ∗ ( p ) , (8.5)22here K − ( p ) = − i ∆ X X ( p ) . (8.6)The r.h.s. of Eq.(8.5) is still linear in the quantized fields.It is suggestive that such pieces of information about the UV completion can be embodiedinto the two-point function of the X scalar, without the need to modify the interaction sectorof the theory.Linearity of Eq.(8.5) imposes strong constraints on the allowed potential. As an example,let us consider the theory with two physical massive scalars, i.e. X = χ + χ where χ has mass M/ √ z and χ has mass M / √ z . Since both χ and χ are BRST-invariant,bilinear mixing terms involving these fields can be removed and the quadratic part of theaction reads S = (cid:90) d x (cid:16) − σ (cid:48) (cid:3) σ (cid:48) + 12 X (cid:3) X − z χ (cid:3) χ − M χ − z χ (cid:3) χ − M χ (cid:17) . (8.7)The propagator of X is then given by Eq.(8.3). The interaction terms are obtained byreplacing X = χ + χ . The inclusion of the field χ allows to reproduce the Higgs singletmodel (HSM), where in addition to the Higgs doublet the real scalar singlet χ is introduced[33–40]. The most general tree-level potential compatible with SU(2) symmetry and ofdimension ≤ V = − vµ X + M X + vµ Φ χ X χ + v λ Φ χ X χ + µ χ χ + M χ + µ (cid:48) χ χ + λ χ χ (8.8)Going on-shell with X one again finds X = v Φ † Φ and upon substitution in Eq.(8.8) weobtain the standard formulation of the HSM potential. The coefficient of the quartic term(Φ † Φ) is λ Φ = M v .Imposing linearity of the X -equation yields the constraint λ Φ χ = 0 (8.9)Provided that λ χ >
0, positivity of M and Eq.(8.9) ensures the fulfillment of the vacuumstability condition, namely λ Φ > , λ χ > , λ Φ λ χ > λ χ [40]. IX. CONCLUSIONS
The linear σ model can be formulated in such a way that the mass of the physical scalarparticle only appears in the quadratic part of the action (and not also in the potential23oupling) by introducing suitable derivative interactions and additional unphysical fieldspairing into BRST doublets.In the present paper we have given all technical tools required to study such a model andits extension to the SM.Power-counting renormalizability has been established and the BRST symmetry guaran-teeing the cancellation of the unphysical degrees of freedom has been given.The 1-PI amplitudes involving the physical massive scalar field X are determined interms of external sources with a better UV behaviour by a functional identity implementingat the quantum level the equation of motion for X .Remarkably, such an identity admits a unique deformation, given by a kinetic term for X whose coefficient we have denoted by z .Once such a kinetic term is introduced into the classical action, power-counting renormal-izability is lost. Violation of power-counting renormalizability is controlled by the parameter z . At z = 0 we recover the original power-counting renormalizable model.The amplitudes of the full theory can be expanded as a power series in z with coefficientsgiven by diagrams of the original theory at z = 0. Each order in z is associated to anadditional zero-momentum insertion of the kinetic operator into X -lines.This property can be used to express the structure of the counterterms of the non power-counting renormalizable theory at z (cid:54) = 0 in terms of analytic functionals of the integratedsource R ( x ) coupled to the kinetic term of the X field.The theory studied in this paper provides a novel example of a non power-counting renor-malizable model, defined as a series expansion around the power-counting renormalizabletheory at z = 0.The proposed representation of the Higgs potential does not spoil any of the SM sym-metries as well as the custodial symmetry, in the limit where the U Y (1) coupling constantvanishes.The X -functional equation at z (cid:54) = 0 holds in the SM deformed by the kinetic term for X . This paves the way for a rigorous all-orders algebraic study of the renormalizationproperties of the effective operator ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) in the Higgs EFT.24 cknowledgments
The Author wishes to thank the Service de Physique Th´eorique et Math´ematique at theULB Brussels, where part of this work was carried out, for the warm hospitality. Usefulcomments by D. Binosi are gratefully acknowledged.
Appendix A: Electroweak SM Lagrangian
The generators of the weak isospin SU L (2) group are I a = σ a where σ a , a = 1 , , S Y M of the SM Lagrangian is S Y M = (cid:90) d x (cid:16) − G aµν G µνa − F µν F µν (cid:17) (A1)where the field strength G aµν is given in terms of the non-Abelian gauge fields A aµ by ( f abc are the SU L (2) structure constants) G aµν = ∂ µ A aν − ∂ ν A aµ + g f abc A bµ A cν (A2)and the Abelian U Y (1) field strength F µν is ( B µ is the hypercharge U Y (1) vector field) F µν = ∂ µ B ν − ∂ ν B µ . (A3) g , g are the U Y (1) and SU L (2) coupling constants respectively.The fermionic part S F is S F = (cid:90) d x (cid:16) i (cid:88) i ¯Ψ Li /D Ψ Li + i (cid:88) i,σ ¯Ψ Ri,σ /D Ψ Ri,σ (cid:17) (A4)where the sum is over all left fermionic doublets and the right singlets.The index σ runs over the right fermion fields corresponding to the two components ofthe associated left doublet, namely if Ψ L = (cid:16) ν L e L (cid:17) , then Ψ R, + = ν R , Ψ R, − = e R .The covariant derivative is defined by D µ = ∂ µ − ig I a A aµ + ig Y B µ (A5)where Y is the hypercharge. The electric charge is related to Y by the Gell-Mann-Nishijimaformula Q = I + Y . 25 . BRST Symmetry We collect here the BRST symmetry of the SM. c a are the SU L (2) ghosts, c is the U Y (1)ghost: s A aµ = ∂ µ c a + g f abc A bµ c c , s B µ = ∂ µ c , s Φ = (cid:16) − ig c + ig τ a c a (cid:17) Φ , s Ψ Li = (cid:16) − ig Y Li c + ig τ a c a (cid:17) Ψ Li , s Ψ Ri,σ = − ig Y Ri,σ c Ψ Ri,σ , s X = s X = s ¯ c = s c = 0 , s c a = − g f abc c b c c , s c = 0 . (A6)The ghosts in the physical basis of the fields W ± µ , A µ , Z µ are obtained by c ± = 1 √ c ∓ ic ) , c A = c W c − s W c , c Z = s W c + c W c , (A7)where c W , s W are the cosine and sine of the Weinberg angle (see Eq.(5.6)). The Nakaniski-Lautrup fields in Eq.(5.8) form BRST doublets with the antighosts: s ¯ c ± = b ± , s b ± = 0 , s ¯ c A = b A , s b A = 0 , s ¯ c Z = b Z , s b Z = 0 . (A8) [1] S. Chatrchyan et al. (CMS), Phys. Lett. B716 , 30 (2012), 1207.7235.[2] G. Aad et al. (ATLAS), Phys. Lett.
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