An Exactly Solvable Model for Dimension Six Higgs Operators and h→γγ
aa r X i v : . [ h e p - ph ] M a y An Exactly Solvable Model for Dimension Six Higgs Operators and h → γγ Aneesh V. Manohar
Department of Physics, University of California at San Diego, La Jolla, CA 92093
An exactly solvable large N model is constructed which reduces at low energies to the Stan-dard Model plus the dimension six Higgs-gauge operators g H † HB µν B µν , g H † HW aµν W aµν , g g H † τ a HW aµν B µν , and ǫ abc W aµ ν W bν ρ W cρ µ . All other dimension six operators are suppressed bypowers of 1 /N . The Higgs-gauge operators lead to deviations from the Standard Model h → γγ and h → γZ rates. A simple variant of the model can be used to also generate the Higgs-gluon operator g H † HG Aµν G Aµν which contributes to the Higgs production rate via gluon fusion.
A scalar boson with a mass of ∼
125 GeV has recentlybeen discovered at the LHC, and it is important to studyits properties in a model-independent way. The StandardModel provides a good description of the LHC data sofar, with no evidence for any new particles with massesbelow ∼ H (6) = −L (6) = c G O G + c B O B + c W O W + c WB O WB (1)generated by new physics at some scale Λ, where O G = g H † HG Aµν G Aµν , O B = g H † HB µν B µν , O W = g H † HW aµν W aµν , O WB = g g H † τ a HW aµν B µν . (2)using the notation of Refs. [2, 3]. These operators giveamplitudes which can interfere constructively or destruc-tively with the Standard Model amplitudes for gg → h , h → γγ , h → Zγ , etc.. The phenomenology of the oper-ators in Eq. (2), including constraints from recent LHCmeasurements of the Higgs decay rates, and from preci-sion electroweak constraints, was studied in Ref. [1].This paper constructs an exactly soluble model whichgenerates the dimension six Higgs operators in Eq. (2),with arbitrary coefficients consistent with the effectivetheory power counting. It also provides an explicit re-alization of the Lagrangian given in the appendix ofRef. [4].The set of all dimension-six operators in the StandardModel was classified in Ref. [5]. There are 59 indepen-dent ones after redundant operators are eliminated bythe equations of motion. The operators not involvingfermions are the ones listed in Eq. (2), their CP -oddpartners e O G , e O B , e O W , e O WB , four pure gauge operators of which two are CP even and two are CP odd, O G = f ABC G Aµ ν G Bν ρ G Cρ µ , e O G = f ABC e G Aµ ν G Bν ρ G Cρ µ ,O W = ǫ abc W aµ ν W bν ρ W cρ µ , e O W = ǫ abc f W aµ ν W bν ρ W cρ µ , (3)and three more operators involving the Higgs field, O H = (cid:0) H † H (cid:1) ,O H (cid:3) = (cid:0) H † H (cid:1) ∂ (cid:0) H † H (cid:1) ,O HD = (cid:0) H † D µ H (cid:1) † (cid:0) H † D µ H (cid:1) . (4)The exactly soluble model given here is a large- N ver-sion of that constructed in Ref. [3]. It produces the CP conserving operators in Eq. (2) with arbitrary order onecoefficients, and the operator O W , and does not gener-ate any other dimension six operators at leading order in1 /N .Consider the Standard Model plus an additional scalarfield S α which is a weak SU (2) doublet with hypercharge Y S , and transforms as the N dimensional representationof an internal SU ( N ) global symmetry. S α is a two com-ponent column vector, and α = 1 , . . . , N . The theory isgiven by L = L SM + D µ S † α D µ S α − V, (5)the usual Standard Model Lagrangian L SM , the S α ki-netic energy term, and the potential V = m S S † α S α + λ N H † H S † α S α + λ N H † τ a H S † α τ a S α + λ N S † α S α S † β S β + λ N S † α τ a S α S † β τ a S β (6)where the Standard Model Higgs potential λ (cid:0) H † H − v / (cid:1) is part of L SM . The Lagrangianis the most general renormalizable one consistent withthe symmetries. Yukawa couplings of S to the StandardModel fermions are forbidden by SU ( N ) invariance.We assume that m S >
0, so that SU ( N ) is not spon-taneously broken and the scalar mass m S is largerthan the electroweak scale v ∼
246 GeV, so that thenew interactions can be treated as higher dimensionoperators at the electroweak scale. The large- N limit of the theory is taken in the standardway [6]. One treats the theory in a perturbative expan-sion in the electroweak couplings g ∼ g , , i.e. one firstexpands in powers of g and then takes the N → ∞ limit.This is the usual method of computing weak decays inQCD using the 1 /N expansion [7]. The method of Refs. [6, 8, 9] is used to solve the theory.Add to V the dimension two auxiliary fields Φ and Ψ a which are real SU (2) singlet and triplet fields with Y = 0, V → V − λ N (cid:18) N S † α S α + m S λ + λ λ N H † H − Φ λ (cid:19) − λ N (cid:18) N S † α τ a S α + λ λ N H † τ a H − Ψ a λ (cid:19) (7)The auxiliarly field equations of motion areΦ = m S + λ N H † H + + 2 λ N S † α S α Ψ a = λ N H † τ a H + 2 λ N S † α τ a S α (8)which can be used to eliminate them and give back theoriginal Lagrangian Eq. (6).In weak coupling, the scalar mass m S is h Φ i . Wewill therefore use h Φ i as the scale of new physics Λ inEq. (1) and to normalize the operators in Eq. (2). Using( H † τ a H ) = ( H † H ) , the new potential Eq. (7) is V = N λ m S Φ − N λ Φ − N λ Ψ a Ψ a + λ λ H † H Φ+ λ λ H † τ a H Ψ a + Φ S † α S α + Ψ a S † α τ a S α − N m S λ − λ λ m S H † H − (cid:0) H † H (cid:1) (cid:18) λ λ N + λ λ N (cid:19) (9)The last term, which is subleading in 1 /N , as well as thecosmological constant term, can be dropped.The field S α is now integrated out. This can be doneexactly in the large- N limit [6] to give an effective actionwhich is an expansion in powers of H , Φ and Ψ a . TheHiggs field H does not couple directly to S in Eq. (9), sothe S functional integral generates terms which only de-pend on Φ and Ψ a . The effective action has a derivativeexpansion in inverse powers of m S , which will turn intoa derivative expansion in inverse powers of h Φ i . The in-frared divergences are controlled by h Φ i , since the theoryis in the phase where the SU ( N ) symmetry is unbrokenand the S -sector is massive.At zero derivatives, one gets the Coleman-Weinbergeffective potential [10] in the MS scheme V CW = 164 π Tr (cid:0) M (cid:1) (cid:20) −
32 + log M µ (cid:21) (10) The expansion has terms of order ( g N ) r , so we take the limit g → N → ∞ . Equivalently, N ≫
1, and g N ≪ where (cid:2) M (cid:3) ab = ∂ V∂φ a ∂φ b (11)and φ a = Re S α,i , Im S α,i are the scalar fields. For theinteraction in Eq. (9), M has eigenvalues Φ ± Ψ , Ψ =Ψ a Ψ a , each twice, so that V = N λ (cid:18) m S + λ N H † H (cid:19) Φ − N λ Φ − N λ Ψ a Ψ a + λ λ H † τ a H Ψ a + X ± N π (Φ ± Ψ ) (cid:20) −
32 + log Φ ± Ψ µ (cid:21) (12)on integrating out the S α field. This potential is exactin the large N limit. V is quadratic in Ψ . From therenormalization group (RG) equation for V , one findsthatΦ , Ψ a , λ (cid:18) m S + λ N H † H (cid:19) , λ λ H † τ a H (13)are µ independent and the exact β -functions are µ d λ d µ = λ π , µ d λ d µ = λ π . (14)Introduce the parameters Λ , in place of λ , ( µ ),1 λ ( µ ) = 14 π log Λ µ , λ ( µ ) = 14 π log Λ µ , (15)which are RG invariant. They are the scales at which λ , have a Landau pole, and at which the scalar the-ory breaks down. For consistency, we need Λ , > h Φ i .New physics has to enter below Λ , for the theory to bevalid to arbitarily high energies. For example S α couldbe scalar fields generated by strong dynamics, or new in-teractions could enter which make the scalar couplingsasymptotically free. We also define [9] m S = m S λ (16)which is RG invariant, from Eq. (13).The effective action can be computed exactly in thelarge N limit [11–15] in a derivative expansion. Theterms which generate operators with dimension ≤ N → ∞ limit are L S = N π (Φ) [ ∂ µ Φ ∂ µ Φ + D µ Ψ a D µ Ψ a ]+ N π (cid:18) log Φ µ (cid:19) (cid:20) W aµν W aµν + 4 Y S B µν B µν (cid:21) + N π (Φ) Y S Ψ a W aµν B µν + O ( g ) (17) RG invariance refers to the dynamics of S α . The Standard Modelfields are treated as external background fields. where the gauge fields have been normalized so that thecovariant derivative is D µ = ∂ µ + iW aµ T a + iB µ Y . Φ isa gauge singlet, and has an ordinary derivative. Ψ a is inthe I = 1 representation of SU (2) W with Y = 0, and hasa covariant derivative D µ Ψ a = ∂ µ Ψ a + i ( T c ) ab W cµ Ψ b , ( T c ) ab = − iǫ cab (18)It is instructive to analyze Eq. (17) at weak coupling.Let Φ = m S + 4 √ πm S √ N σ, Ψ a = 4 √ πm S √ N Σ a (19)The σ and Σ a are dimension one fields with a canonicallynormalized kinetic energy term, and L S = 12 ∂ µ σ∂ µ σ + 12 D µ Σ a D µ Σ a + N π (cid:18) log m S µ (cid:19) (cid:20) W aµν W aµν + 4 Y S B µν B µν (cid:21) + √ N √ πm S σ (cid:20) W aµν W aµν + 4 Y S B µν B µν (cid:21) + √ NY S √ πm S Σ a W aµν B µν + O ( g ) . (20)The effective action can be computed at weak couplingfrom the graphs in Fig. 1, which add up to give the gaugeinvariant structure in Eq. (20).The one-loop effective action generates kinetic energyterms for σ, Σ a , given in the first line. The second linegives the threshold correction between the gauge cou-plings g h in the theory above m S and g l in the theorybelow m S , − g l, ( µ ) = − g h, ( µ ) + N π log m S µ , − g l, ( µ ) = − g h, ( µ ) + N π Y S log m S µ , (21)for the SU (2) and U (1) coupling constants. The discon-tinuity in coupling matches the S α contribution to the β -functions, which exists above m S , but not below. The re-maining terms are σW aµν W aµν , σB µν B µν and Σ a W aµν B µν interactions.The scalar potential Eq. (12) becomes V = − π m S (cid:18) σ λ + Σ a Σ a λ (cid:19) + 2 √ π √ N (cid:18) λ λ σH † H + λ λ Σ a H † τ a H (cid:19) (22)which has mass terms for σ and Σ a (with the wrong sign,but it does not matter, they are auxiliary fields), andcouplings of σ and Σ a to the Higgs doublet. Integratingout the auxilary fields generates the operators Eq. (2) viathe graph in Fig. 2 We can now integrate out Φ , Ψ a exactly, by doingthe functional integral using the method of steepest de-scent [11]. The minimum h Φ i is atΦ log Φ µ − Φ + 4 π λ (cid:18) m S + λ N H † H (cid:19) − π λ Φ = 0 . (23)Dropping the 1 /N term, and using Eq. (15,16), gives h Φ i e Λ log h Φ i e Λ + 4 π m S e Λ = 0 (24)in terms of RG invariant parameters. Instead of choosing m S as the Lagrangian parameter, and solving Eq. (24)for h Φ i , we can use h Φ i as the free parameter and thendetermine m S from Eq. (24). m S → h Φ i →
0. As h Φ i increases, so does m S , and 2 π m S → Λ as h Φ i → Λ .Eq. (24) implies that m S decreases again for h Φ i > Λ [8,9], but this is above the Landau pole, and the theory isnot valid in this regime.Evaluating the functional integral around h Φ i gives L S = N π (cid:20)(cid:18) log h Φ i µ (cid:19) (cid:0) W aµν W aµν + 4 Y S B µν B µν (cid:1)(cid:21) + λ λ (cid:0) λ m S − h Φ i (cid:1) H † H + λ h Φ i λ log Λ h Φ i H † H (cid:0) W aµν W aµν + 4 Y S B µν B µν (cid:1) + λ Y S h Φ i λ log Λ h Φ i H † τ a HW aµν B µν + N g π h Φ i ǫ abc W aµ ν W bν ρ W cρ µ . (25)There are also terms at higher order in the derivativeexpansion which have not been computed here. The firstterm in Eq. (25) gives the strong-coupling version of thethreshold correction Eq. (21). The second term is a shiftin the Higgs mass proportional to the S α mass, and canbe absorbed into the v term in the Higgs potential in L SM .Using Λ = h Φ i ∼ m S > v as the scale in Eq. (2), we seethat we have generated the Standard Model Lagrangianplus the three CP -even dimension six operators in Eq. (1)with coefficients c W = ( λ /λ )48 log Λ h Φ i ,c B = ( λ /λ ) Y S
12 log Λ h Φ i ,c WB = ( λ /λ ) Y S
24 log Λ h Φ i , (26) e = 2 . . . . . FIG. 1. Graphs contributing to the dimension six effective action. The internal lines are S α scalar fields. The external linesare Φ, Ψ a and gauge fields. HH σ , Σ FIG. 2. Graph generating the h → γγ amplitude. and the O W operator with coefficient c W = N g π h Φ i . (27)All other dimension six operators are subleading in 1 /N .The ratios ( λ /λ ) and ( λ /λ ) are RG invariant under S α dynamics with the Standard Model fields treated asbackground fields, from Eq. (13).The linear combinations of coefficients relevant for h → γγ and h → γZ decays are c γγ = c W + c B − c WB ,c γZ = c W cot θ W − c B tan θ W − c WB cot 2 θ W . (28)The operator c WB is constrained by the S -parameter [16–19], S = − π v c WB Λ . (29)From Eq. (26), we see that we can get order unity valuesof c W , c B and c WB . The phenomenology of the Higgs-gauge operators was discussed in detail in Refs. [1–3]. There is one relation that follows from Eq. (26), c B = 4 Y S c W , (30)if we restrict to the model considered here with a singlescalar multiplet with hypercharge Y S . One can constructtrivial generalizations of the large N model with multi-ple heavy scalar fields, which can have different hyper-charges, and can also be colored. In this case, one canalso generate the gluon term c G , as in the octet scalarmodel of Ref. [3], and the c B − c W relation no longerholds.The large N calculation drops terms of order 1 /N ,as well as higher order radiative corrections of order g N/ (16 π ). For finite N , the neglected terms are smallif 1 ≪ N ≪ m S is below the Landau pole,to determine the allowed region for c W , c B and c WB .The S α interactions break custodial SU (2) symmetry,since S α is in a complex representation of SU ( N ), andthe real and imaginary parts of S α cannot be combinedto form an O (4) vector, as is possible for the Higgs field.This does not affect the standard relations such as M W = M Z cos θ W that follow from custodial SU (2) symmetry inthe Higgs sector. Custodial SU (2) symmetry violationdue to S α interactions only arise from higher dimensionoperators. One can also study variants of the theorywith SO ( N ) symmetry, or double the S α fields to havea O (4) × SU ( N ) symmetry. In these variants, custodial SU (2) can be incorporated in the S potential.The model has threshold corrections to the gauge cou-plings, Eq. (21), which affects gauge unification. This wasstudied in Ref. [2]. In general, all theories that introducenew dynamics will modify the standard unification sce-nario, which has perturbative unification with a desertup to the GUT scale.Finally, the model needs a fine tuning of order 1% tokeep m H small compared to the scale Λ ∼ m S , sincethere is contribution to m H ∝ m S in Eq. (25). While notdesirable, this is not worse than fine-tunings required inmany models proposed to solve the hierarchy problem.The H mass term and dimension six operators have dif-ferent dependence on the RG invariant parameters, so theories with additional S multiplets can cancel the m H contribution without cancelling the Higgs-gauge opera-tors, if the parameters satisfy X i λ ,i m S ,i − λ ,i λ ,i h Φ i i = 0 . (31)This cancellation condition is not adjusted order-by-order in perturbation theory, since Eqs. (25,31) are exactat leading order in 1 /N . The Higgs mass is then lightbecause it is 1 /N suppressed.I would like to thank E. Jenkins and M. Trott for help-ful discussions. [1] C. Grojean, E. E. Jenkins, A. V. Manohar, and M. Trott,(2013), arXiv:1301.2588 [hep-ph].[2] A. V. Manohar and M. B.Wise, Phys.Lett. B636 , 107 (2006),arXiv:hep-ph/0601212 [hep-ph].[3] A. V. Manohar and M. B.Wise, Phys.Rev.
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