A Higher-Derivative Lee-Wick Standard Model Christopher D. Carone1, and Richard F. Lebed2, ∗ † 1Department of Physics, College of William & Mary, Williamsburg, VA 23187-8795 2Department of Physics, Arizona State University, Tempe, AZ 85287-1504 (Dated: November 2008) Abstract 9 0 0 The Lee-Wick Standard Model assumes a minimal set of higher-derivative quadratic terms that 2 produce a negative-norm partner for each Standard Model particle. Here we introduce additional n a J terms of one higher order in the derivative expansion that give each Standard Model particle two 7 Lee-Wick partners: one with negative and one with positive norm. These states collectively cancel ] h unwanted quadratic divergences and resolve the hierarchy problem as in the minimal theory. We p - show how this next-to-minimal higher-derivative theory may be reformulated via an auxiliary field p e h approach and written as a Lagrangian with interactions of dimension four or less. This mapping [ provides a convenient framework for studies of the formal and phenomenological properties of the 2 v theory. 0 5 1 4 . 1 1 8 0 : v i X r a ∗Electronic address: [email protected] †Electronic address: [email protected] 1 I. INTRODUCTION Extensions of the Standard Model (SM) generally involve mass scales that are much higher thanthescale of electroweak symmetry breaking. If oneviews the SMasa low-energy effective theory, then the Higgs boson squared mass m2 receives radiative corrections that h grow quadratically with the cutoff. This leads to the hierarchy problem: A large separation of scales requires an extremely close cancellation between the bare Higgs boson mass and the cutoff-dependent loop corrections. Within the low-energy effective theory, such a fine tuning has no natural explanation. Solutions to the hierarchy problem can be grouped into three broad categories, distin- guished by their assumptions: (1) models thatassume fine tuning to beextreme and present, but natural from the point of view of the string landscape, as in split-supersymmetric mod- els [1]; (2) models that assume fine tuning is not extreme since no high mass scales are present, as in scenarios with large extra dimensions and a low Planck scale [2]; (3) models that assume fine tuning is not extreme, even when high mass scales are present, because new physics just above the electroweak scale modifies the ultraviolet divergence of m2 from h quadratic to logarithmic. The Minimal Supersymmetric Standard Model (MSSM) is per- haps the most famous example of a model in the last category: Each SM particle has a supersymmetric partner with the same gauge quantum numbers but opposite spin statistics. As fermion and boson loops enter with opposite relative signs, quadratic divergences cancel between Feynman loop diagrams when both particles and their associated superpartners are taken into account. A similar cancellation is achieved in the Lee-Wick Standard Model (LWSM) [3], which has recently been proposed as a theory that solves the hierarchy problem. Each SM particle possesses a Lee-Wick (LW) partner [4] with the same spin statistics, but with opposite- sign quadratic terms. Since the propagators of ordinary and LW particles differ in overall sign, quadratic divergences cancel between pairs of diagrams. A LW partner for a given field arisesviatheinclusionofahigher-derivative(HD)kinetictermwhichgeneratesanadditional pole in the associated two-point function. As reviewed below, the HD Lagrangian can be recast, using auxiliary fields, as a dimension-four Lagrangian that includes partner fields with “wrong-sign” quadratic terms [3]. The cancellation of divergences in this formulation of the theory occurs because HD terms in the original Lagrangian cause propagators to fall 2 off more quickly with momentum, so that loop diagrams become less divergent. While LW particles have wrong-sign kinetic and mass terms (like Pauli-Villars regula- tors) it is nonetheless believed consistent to treat them as physical particles. Neither the LWSM [3], in which all the LW states can decay, nor the O(N) LW model at large N [5] violates causality at a macroscopic level. Moreover, studies of longitudinalgauge-bosonscat- tering in the LWSM indicate that unitarity is not violated provided the HD theory can be mapped to a Lagrangian with interactions of dimension four or less [6]. Taking these obser- vations into account, a number of authors have begun to explore the phenomenology [7, 8] and cosmology [9] of LW extensions of the SM. These studies have assumed the minimal theory, in which the lowest-order HD term for each field is included, and precisely one LW partner accompanies each SM particle. Whiletheminimalscenarioisthesimplesttostudy,onemaywonderwhethertheinclusion of a single HD term, and exactly no others of higher order, represents a natural state of affairs. In this paper we explore a next-to-minimal scenario that includes HD terms of the next order in a derivative expansion, leading to two partners for each SM particle. Our immediate focus is a technical one: What is the generalization of the auxiliary field (AF) formulation introduced in the minimal theory [3], and what form of the HD Lagrangian leads to an auxiliary field theory with interactions of dimension four or less? We address this question in a non-Abelian gauge theory with fermions and complex scalars, so that our results can be immediately applied to the SM. Interestingly, one of the two new LW partners for each SM particle is ordinary (with correct-sign quadratic terms), suggesting that collider signatures and experimental limits on this theory can be qualitatively different from the minimal version. Our results suggest that there is no impediment, in principle, to constructing similar theories with additional LW states via the inclusion of appropriate interactions that are of yet higher order in the number of derivatives. We note that previous work [10, 11] extensively studies a particular O(p6) form for a HD scalar Lagrangian, in which O(p4) terms are absent and gauge couplings are omitted. In particular, this work develops a strongly-interacting Higgs sector that tames ultraviolet corrections and can be studied on the lattice. Reference [10] represents pioneering early work on the consistency of O(p6) scalar theories. By contrast, the thrust here is to study the duality between more general HD theories with O(p6) terms and equivalent theories with operators of dimension four or less, not only in the Higgs sector but including all SM 3 particles, with an eye toward future phenomenological studies. This paper is organized as follows. In the next section we review the LW idea in a simple scalar field theory and show how the AF formulation is applied when HD terms of next-to-lowest order are present. In Section III we extend our approach to non-Abelian gauge theories, focusing on the pure gauge sector; in Section IV we show how fermions are included in the theory. In Section V we discuss the Higgs sector of the theory. In Section VI we discuss the cancellation of one-loop quadratic divergences in an SU(N ) gauge theory c with complex scalars and chiral fermions. In Section VII we summarize our conclusions. II. A SCALAR EXAMPLE Let us begin by reviewing the formulation of a LW theory of a real scalar field. The simplest HD Lagrangian is given by 1 1 1 = φˆ(cid:3)φˆ φˆ(cid:3)2φˆ m2φˆ2 + (φˆ), (2.1) LHD −2 − 2M2 − 2 φ Lint where the last term represents interactions. The HD term leads to an additional pole in the ˆ φ two-point function near the mass M, which corresponds to the LW partner of the usual state with mass eigenvalue near m . The HD term also assures high-momentum falloff of φ the φˆ propagator as 1/p4, improving the convergence of φˆ loop diagrams. Following the approach of Ref. [3], one observes that Eq. (2.1) is equivalent to a Lagrangian including an ˜ auxiliary field, φ and no higher-derivative interactions: 1 1 1 = φˆ(cid:3)φˆ m2φˆ2 φ˜(cid:3)φˆ+ M2φ˜2 + (φˆ). (2.2) LAF −2 − 2 φ − 2 Lint ˜ The φ equation of motion (EOM) is 1 φ˜= (cid:3)φˆ, (2.3) M2 which, upon substitution into Eq. (2.2), reproduces the original Lagrangian of Eq. (2.1). The kinetic terms in Eq. (2.2) can be diagonalized via the substitution ˆ ˜ φ = φ φ, (2.4) − yielding 1 1 1 1 = φ(cid:3)φ+ φ˜(cid:3)φ˜ m2(φ φ˜)2 + M2φ˜2 + (φ φ˜). (2.5) L −2 2 − 2 φ − 2 Lint − 4 The scalar mass matrix can be diagonalized without affecting the form of the kinetic terms via a symplectic transformation: φ coshθ sinhθ φ 0 = , (2.6)  ˜   ˜  φ sinhθ coshθ φ 0      where the subscript 0 indicates a mass eigenstate; one finds 2m2 tanh2θ = − φ . (2.7) M2 2m2 − φ The final Lagrangian takes the form 1 1 1 1 = φ (cid:3)φ + φ˜ (cid:3)φ˜ m2φ2 + M2φ˜2 + [e θ(φ φ˜ )], (2.8) LLW −2 0 0 2 0 0 − 2 0 0 2 0 0 Lint − 0 − 0 where m and M are the mass eigenvalues, and the factor of e θ can be absorbed into 0 0 − ˜ redefinitions of the couplings. The opposite-sign φ and φ propagators following from the 0 0 ˜ quadratic terms in Eq. (2.8), together with the specific relationship between the φ and φ 0 0 couplings in , assures the cancellation of quadratic divergences, as is shown explicitly in int L Ref. [3]. ˆ Indicating by N the number of physical poles in the φ propagator, let us refer to the minimal example just considered as an N = 2 theory. An N=3 model corresponds to a HD Lagrangian of the general form 1 1 1 1 N=3 = φˆ(cid:3)φˆ φˆ(cid:3)2φˆ φˆ(cid:3)3φˆ m2φˆ2 + (φˆ), (2.9) LHD −2 − 2M2 − 2M4 − 2 φ Lint 1 2 where M and M are the LW mass scales, which we assume are comparable. The restriction 1 2 that the φˆ propagator has three physical poles restricts the values of m2, M2 and M2, so φ 1 2 that it is possible to map Eq. (2.9) to a Lagrangian of the form 3 1 N=3 = c φ(i)((cid:3)+m2)φ(i) + ( φ(i) ), (2.10) LLW i −2 i Lint { } i=1 (cid:20) (cid:21) X where the c = 1 or 1, and the m2 are positive. The missing link that connects Eq. (2.9) i − i to (2.10) is an AF Lagrangian, analogous to Eq. (2.2) in the N = 2 theory, and appropriate field redefinitions, analogousto Eq. (2.4). Let us first examine the special case where m = 0 φ [which corresponds to m = 0 in Eq. (2.10)] before stating the general result. The desired 1 AF Lagrangian involves two new scalar fields, χ and ψ: 1 1 1 = φˆ(cid:3)φˆ χ(cid:3)φˆ+m m χψ ψ(cid:3)ψ (m2 +m2)ψ2 + (φˆ). (2.11) LAF −2 − 2 3 − 2 − 2 2 3 Lint 5 ˜ Like the field φ in the N = 2 theory, χ is an auxiliary field; since it occurs linearly in Eq. (2.11), its EOM imposes a constraint that is exact at the quantum level: 1 ψ = (cid:3)φˆ. (2.12) m m 2 3 Substituting Eq. (2.12) into Eq. (2.11), one obtains 1 1 m2 +m2 1 1 = φˆ(cid:3)φˆ 2 3 φˆ(cid:3)2φˆ φˆ(cid:3)3φˆ+ (φˆ), (2.13) LHD −2 − 2 m2m2 − 2 m2m2 Lint (cid:18) 2 3 (cid:19) (cid:18) 2 3(cid:19) which factorizes as 1 = φˆ(cid:3)((cid:3)+m2)((cid:3)+m2)φˆ+ (φˆ), (2.14) LHD −2m2m2 2 3 Lint 2 3 and from which one identifies m = 0, M2 = m2m2/(m2 + m2) and M4 = m2m2 upon φ 1 2 3 2 3 2 2 3 comparison with Eq. (2.9). Showing next that the AF Lagrangian can also be written in the form of Eq. (2.10) is a simple matter of linear algebra. Taking m to be the lighter LW state and substituting the 2 field redefinitions m m φˆ = φ(1) 3 φ(2) + 2 φ(3), (2.15) − (m2 m2)1/2 (m2 m2)1/2 3 − 2 3 − 2 1 χ = m φ(2) m φ(3) , (2.16) (m2 m2)1/2 3 − 2 3 − 2 1 (cid:2) (cid:3) ψ = m φ(2) m φ(3) , (2.17) (m2 m2)1/2 2 − 3 3 − 2 (cid:2) (cid:3) into Eq. (2.11), one obtains 1 1 1 = φ(1)(cid:3)φ(1) + φ(2)((cid:3) + m2)φ(2) φ(3)((cid:3) + m2)φ(3) + (φˆ). (2.18) L −2 2 2 − 2 3 Lint As with Eq. (2.4) in the N =2 theory, Eq. (2.15) leads to a very specific form for the interaction terms in Eq. (2.18). We find that there is no finite field redefinition that takes the AF Lagrangian Eq. (2.11) to the LW form Eq. (2.18) for m = m , so we do not consider 2 3 that possibility further. For completeness, we exhibit the results for m (and m ) non-zero. The AF Lagrangian φ 1 is given by 1 1 = φˆ((cid:3)+m2)φˆ χ((cid:3)+m2)φˆ+(m2 m2)1/2(m2 m2)1/2χψ LAF η −2 1 − 1 3 − 1 2 − 1 1 (cid:20) 1 1 ψ(cid:3)ψ (m2 +m2 m2)ψ2 + (φˆ), (2.19) −2 − 2 2 3 − 1 Lint (cid:21) 6 where η (m2m2 +m2m2 +m2m2)/(m2 m2)(m2 m2). Varying Eq. (2.19) with respect 1≡ 1 2 1 3 2 3 2 − 1 3 − 1 to auxiliary field χ generalizes the EOM Eq. (2.12) to 1 ψ = ((cid:3)+m2)φˆ, (2.20) (m2 m2)1/2(m2 m2)1/2 1 2 − 1 3 − 1 which, when substituted back into Eq. (2.19), yields 1 = φˆ((cid:3)+m2)((cid:3)+m2)((cid:3)+m2)φˆ, (2.21) LHD −2Λ4 1 2 3 where Λ4 m2m2 +m2m2 +m2m2. (2.22) ≡ 1 2 1 3 2 3 Equation (2.21) is equivalent to the HD Lagrangian in Eq. (2.9) with the identifications m2 = (m2m2m2)/Λ4, (2.23) φ 1 2 3 M2 = Λ4/(m2 +m2 +m2), (2.24) 1 1 2 3 M2 = Λ2. (2.25) 2 On the other hand, one can obtain the canonical LW form, Eq. (2.10) with c = c =c =1, 1 2 3 − from Eq. (2.19) by the field redefinitions φˆ = √η φ(1) √ η φ(2) +√η φ(3), (2.26) 1 2 3 − − χ = √ η φ(2) √η φ(3), (2.27) 2 3 − − ψ = √η φ(2) √ η φ(3), (2.28) 3 2 − − where the parameters η are defined by i Λ4 η , (2.29) 1 ≡ (m2 m2)(m2 m2) 2 − 1 3 − 1 Λ4 η , (2.30) 2 ≡ (m2 m2)(m2 m2) 1 − 2 3 − 2 Λ4 η . (2.31) 3 ≡ (m2 m2)(m2 m2) 1 − 3 2 − 3 Noting, for example, that η =1 when m =0, one sees that Eqs. (2.15)–(2.17) immediately 1 1 follow in this case. As before, we assume m > m > m , so that sign(η ) = ( 1)i+1. 3 2 1 i − The remarkable algebraic simplifications that occur in converting the AF Lagrangian are a consequence of simple sum rules that are satisfied by the η : i 3 m2nη = 0 (n = 0,1), (2.32) i i i=1 X 7 3 m2nη = Λ4 (n = 2), (2.33) i i i=1 X m2m2η +m2m2η +m2m2η = Λ4. (2.34) 1 2 3 2 3 1 3 1 2 Our η parameters are equivalent to those introduced by Pais and Uhlenbeck [12] (which we i call ηPU) to describe purely quantum-mechanical theories with HD Lagrangians analogous i to those used here. The mapping m4Λ2N 2 η = i − ηPU (2.35) i Π m2 i j j converts the sum rules of Ref. [12] into Eqs. (2.32) and (2.34) for the case N = 3, while Eq. (2.33) is linearly dependent on the others. ˆ The interaction terms in the general N = 3 theory are functions of φ. Following from Eq. (2.26), (φˆ) √η φ(1) √ η φ(2) +√η φ(3) . (2.36) int int 1 2 3 L ≡ L − − (cid:0) (cid:1) The restriction on the form of the couplings imposed by Eq. (2.36) is necessary for the cancellation of divergences. This fact is illustrated in the following simple example: Let (φˆ) = λφˆ4/4!, or equivalently, int L λ (φˆ) = η η η η φ(i)φ(j)φ(k)φ(l) . (2.37) int i j k l L 4! | | Xijkl q The self-energy for φ(1) (corresponding to the state that is present when the LW particles are decoupled) is given by d4p ( 1)k+1i Π(p2) = λη − η , (2.38) 1 (2π)4 p2 m2 | k| Z k (cid:20) − k (cid:21) X wherethefactor( 1)k+1 yieldstheappropriateoverallsignforeachscalarpropagator. Using − the fact that ( 1)k+1 η = η and formally expanding the integrand, one finds k k − | | d4p η η m2 η m4 Π(p2) = iλη k + k k + k k + . (2.39) 1 (2π)4 p2 p4 p6 ··· Z k (cid:18) (cid:19) X The first two terms vanish as a consequence of the n = 0 and 1 sum rules, Eq. (2.32), respectively; these terms would otherwise be quadratically and logarithmically divergent, respectively. Although the interactions in the LW form of the N = 3 theory are more 8 complicated than in the N = 2 case, the sum rules satisfied by the η always provide the i necessary algebraic miracles that cancel the leading divergences in the theory1. III. PURE YANG-MILLS THEORY We now generalize the approach of the previous section to a pure Yang-Mills theory. The next-to-leading-order HD Lagrangian reads 1 1 1 1 = TrFˆ Fˆµν + TrFˆ DˆµDˆ Fˆαν TrFˆ DˆµDˆ Dˆ[αDˆ Fˆβν], (3.1) LHD −2 µν − m2 m2 µν α −m2m2 µν α β (cid:18) 2 3(cid:19) 2 3 where the superscript brackets indicate antisymmetrization of just the first and last indices: X[α1α2 αN−1αN] Xα1α2 αN−1αN XαNα2 αN−1α1. (3.2) ··· ··· ··· ≡ − Equation (3.1) can be written in the elegant factorized form 1 DˆµDˆ 1 DˆνDˆ = TrFˆ gµ + α gν + β gα (α ν) Fˆβλ. (3.3) LHD µν 2 α m2 2 β m2 λ − ↔ 2 !" 3 ! # The field strength Fˆ, and the covariant derivative Dˆ acting upon a field X transforming in the adjoint representation of the gauge group, are defined in the usual manner: Fˆµν ∂µAˆν ∂νAˆµ ig[Aˆµ,Aˆν], (3.4) ≡ − − DˆµX ∂µX ig[Aˆµ,X]. (3.5) ≡ − This HD Lagrangian may be obtained from the equivalent Lagrangian 1 1 = TrFˆ Fˆµν TrFˆµν(Dˆ χ Dˆ χ ) Tr(Dˆ ω Dˆ ω )2 YM µν µ ν ν µ µ ν ν µ L −2 − − − 2 − 2m m Trχ ων +(m2 +m2)Trω ωµ, (3.6) − 2 3 µ 2 3 µ where the new fields χ and ω transform in the adjoint representation. Integration by parts on the second term leads to a form for in which no derivatives on χ appear, making YM L it an auxiliary field; since χ appears linearly in , it is also a Lagrange multiplier. The YM L constraint imposed by its EOM, Dˆ Fˆνµ m m ωµ = 0, (3.7) ν 2 3 − 1 Despite this example, N >2 LWSMs are not finite theories, but remain logarithmically divergent,as can be shown by a generalization of the power-counting argument given in Ref. [3]. 9 is exact at the quantum level. Using Eq. (3.7) to eliminate ωµ from Eq. (3.6), one finds that the terms proportional to χ cancel, and that the remaining terms reduce to the HD Lagrangian, Eq. (3.1). In order to obtain a Lagrangian in the LW form, we rewrite the three fields Aˆ, χ and ω in terms of three new fields A : 1,2,3 Aµ Aˆµ +χµ, 1 ≡ η η Aµ 2χµ 3ωµ, 2 ≡ −η − η r 1 r 1 η η Aµ 3χµ 2ωµ. (3.8) 3 ≡ η − −η 1 1 r r Under the action of the gauge group, A and A transform as matter fields in the adjoint 2 3 representation, while A transforms as a gauge field, due to the additional shift in Aˆ. The 1 inverse transformations are given by η η Aˆµ = Aµ 2Aµ + 3Aµ, 1 − −η 2 η 3 r 1 r 1 η η χµ = 2Aµ 3Aµ, −η 2 − η 3 1 1 r r η η ωµ = 3Aµ 2Aµ, (3.9) η 2 − −η 3 r 1 r 1 as may be shown by using the sum rule Eq. (2.32). Substituting Eqs. (3.9) into Eq. (3.6) is a laborious but straightforward procedure. Using Eqs. (2.29)–(2.31) to express the parameters η in terms of masses m , and defining the unhatted field strength Fµν and covariant i 2,3 1 derivative Dµ as analogous to Eqs. (3.4)–(3.5) with Aˆµ Aµ, one obtains the Lagrangian → 1 = + + , (3.10) YM,LW 0 1 2 L L L L where the subscript indicates the power of g that appears in the coefficient of each gauge- invariant term. The kinetic and mass terms are contained in 1 1 1 = TrFµνF + Tr(D A D A )2 Tr(D A D A )2 L0 −2 1 1µν 2 µ 2ν − ν 2µ − 2 µ 3ν − ν 3µ m2TrAµA +m2TrAµA , (3.11) − 2 2 2µ 3 3 3µ from which one immediately sees that A is massless (m =0), and only A has wrong-sign 1 1 2 10