Symmetry breaking solutions are investigated in the $N\to \infty$ limit for
the ground state of a system consisting of a Lorentz-scalar, N component
``phantom'' field and an O(N) singlet. The most general form of O(N) x Z_2
invariant quartic interaction is considered. The non-perturbatively
renormalised solution demonstrates the possibility for Z_2 symmetry breaking
induced by phantom fluctuations. It becomes also evident that the strength of
the ``internal'' dynamics of the N-component field tunes away the ratio of the
Higgs condensate and the Higgs mass from its perturbative (nearly tree-level)
expression.Comment: 9 pages, uses elsart.cls, version to appear in Phys. Lett. 

A. Patkós

Barbieri

Bardeen

Blaizot

Calmet

Cerdeno

Chivukula

Coleman

Davoudiasl

Hill

Ignatiev

Patt

Rivers

Shaposhnikov

van der Bij

Zs. Szép

Physics Letters B

English

arXiv

Crossref

Symmetry breaking solutions are investigated in the N->infinity limit for the ground state of a system consisting of a Lorentz-scalar, N component "phantom" field and an O (N) singlet. The most general form of O(N) x Z(2) invariant quartic interaction is considered. The non-perturbatively renormalised solution demonstrates the possibility for Z(2) symmetry breaking induced by phantom fluctuations. It becomes also evident that the strength of the "internal" dynamics of the N-component field tunes away the ratio of the Higgs condensate and the Higgs mass from its perturbative (nearly tree-level) expression

Patkós, A

Szép, ZS

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-th/0607143v2  21 Sep 2006Phantom field fluctuation induced HiggseffectA. Patkós a,baDepartment of Atomic Physics, Eötvös University, H-1117 Budapest, HungarybResearch Group for Statistical Physics of the Hungarian Academy of Sciences,H-1117 Budapest, HungaryZs. SzépResearch Institute for Solid State Physics and Optics of the Hungarian Academyof Sciences, H-1525 Budapest, HungaryAbstractSymmetry breaking solutions are investigated in the N → ∞ limit for the groundstate of a system consisting of a Lorentz-scalar, N component “phantom” field andan O(N) singlet. The most general form of O(N)×Z2 invariant quartic interactionis considered. The non-perturbatively renormalised solution demonstrates the pos-sibility for Z2 symmetry breaking induced by phantom fluctuations. It becomes alsoevident that the strength of the “internal” dynamics of the N-component field tunesaway the ratio of the Higgs condensate and the Higgs mass from its perturbative(nearly tree-level) expression.Key words: Higgs mass, phantom field, Dyson–Schwinger equations, large NapproximationPACS: 11.10.Wx, 11.10.Gh, 12.38.Cy1 IntroductionScalar fields are proposed at present with renewed intensity as minimal, phenomeno-logically motivated complements to the Standard Model. The simplest mechanism forelectroweak symmetry breaking is realised by introducing a renormalisable biquadraticEmail addresses: patkos@ludens.elte.hu (A. Patkós), szepzs@achilles.elte.hu (Zs.Szép).Preprint submitted to Elsevier 26 October 2018coupling of the Higgs field to a “hidden scalar” (the phantom field) which acquires nonzerovacuum expectation value [1,2]. Postulation of two extra scalars with appropriate dynam-ics might provide the minimal framework for the particle physics interpretation of thecosmologically well-established facts of missing luminous mass and of inflationary densityfluctuations [3,4]. A scalar inflaton coupled to sterile right-handed neutrinos offers anothercosmologically motivated minimal extension of the standard model, which accounts alsofor neutrino oscillation phenomena and the fermion-asymmetry of the universe [5].A general background to this evolution is provided by low energy field theories constructedon string vacua showing maximal compatibility with the particle content of the StandardModel. Appearance of extra scalars is characteristic for all these constructions. A con-sequence of possible physical significance is that the natural cut-off scale to be appliedfor the top quark contribution to the Higgs mass would be raised considerably [6]. Con-sequences of the existence of a phantom sector on the precision electroweak tests of theStandard Model are being systematically explored [7]. The history of such investigationsgoes back to the 80’s [8].The goal of this letter is to illustrate the substantial influence of an extended scalarsector on the symmetry breaking vacuum features through the results of an exact, non-perturbatively renormalised large N analysis. We propose the following embedding ofthe Higgs sector of the Standard Model into a wider set of elementary scalar fields. Thegauged self-interacting scalar SU(2) doublet is coupled also to a set of scalar fields whichform an N -component vector under a hidden internal O(N) symmetry. The couplingpreserves the SU(2) symmetry of the standard gauge model. After fixing 3 components ofthe complex Higgs doublet to zero a Z2 invariance of the Higgs field is left. The phantomO(N) vector field has some extra O(N) invariant non-linear interaction on its own. Withappropriately chosen N -scaling of the quartic couplings the renormalised N = ∞ solutionswill be presented. We shall demonstrate that the non-perturbative renormalisation of thesolution leads to a non-trivial Z(2) symmetry breaking Higgs condensate in a large rangeof the renormalised couplings of the model even if no negative renormalised squared massis introduced. It will be shown that also essential deviations might appear in the relationof the Higgs mass to the strength of the Higgs condensate, relative to the leading orderlarge N estimate derived in the O(N) symmetric linear sigma model.The present model is a generalisation of the O(N + M) symmetric model studied byChivukula et al. in 1991-1993 [9]. Another extension of the Higgs sector was investigatedin [10], where consequences of the mirror world assumption were studied. Here the generalmodel will be solved by taking the large N limit of the renormalised Dyson-Schwingerequations for the exact n-point functions. The renormalisation process will follow anapproach advocated recently in the context of 2PI approximate solution of scalar fieldtheories [11], but we emphasise that it represents a rather non-trivial implementation ofthe strategy of nonperturbative renormalisation.22 Leading order formal solutionThe Lagrangian to be solved at N = ∞ describes the most general renormalisable SU(2)×O(N) symmetric interaction of an N -component (and SU(2) singlet) field ψi, i = 1, .., Nwith a gauged SU(2) doublet (and O(N) singlet) Φ:L[ψi, φ] =12(DµΦ)†DµΦ +12(∂µψi)2 −12m22ψ2i −12m23Φ†Φ−λ124N(ψ2i )2 −λ224N(Φ†Φ)2 −λ312Nψ2iΦ†Φ. (1)The present parametrisation is a generalisation of that used in [9], where only a uniquequartic coupling ∼ (ψ2i + Φ†Φ)2 was considered.We shall investigate the usual symmetry breaking pattern, where the lower real partof the complex SU(2) doublet gets a nonzero classical value√Nv, and the other threecomponents are gauge transformed away (unitary gauge). The shift in the Higgs field thenbreaks the Z2 symmetry related to the remaining reflection symmetry of the Higgs field.When gauge fields are set to zero the resulting shifted Lagrangian density is the following:Lfree[ψi, σ] =12(∂µσ)2 +12(∂µψi)2 −12m22ψ2i −12m23(σ2 + 2σ√Nv +Nv2),Lint[ψi, σ] =−λ124N(ψ2i )2 −λ224N(σ2 + 2σ√Nv +Nv2)2−λ312Nψ2i (σ2 + 2σ√Nv +Nv2). (2)The large N solutions will be constructed with help of the Dyson-Schwinger (DS) equa-tions which are generated by the master equationδΓδϕU=δSclδϕU[ϕA +GABδδϕB], (3)and its functional derivatives [12]. The right hand side is evaluated by replacing in the ex-pression of the functional derivative of the classical action the field ϕU by ϕU+GUV δ/δϕVand the whole expression is applied to the unity in the field space. Γ is the effective action,ϕU is a generic field variable, GAB is the two-point function of the generic fields ϕA andϕB (deWitt’s convention is used for the index A,B, ...). Note, that one should includeinto the intermediate steps of the derivation all possible (A,B) pairs of indices for GAB,the corresponding external currents are set to zero only at the very end.Below we give the unrenormalised equations which determine the vacuum condensatev, the propagators Gσσ(x, y) =: Gσ(x, y), Gψmψn(x, y) =: δnmGψ(x, y) and the relevant33-point function Γψnψmσ(x, y, z) = δnmΓψψσ(x, y, z). The LO (N = ∞) equation for thevacuum expectation value v looks rather simple:−δΓδσ |ϕU=0=√Nv[m23 +λ26v2 +λ36Tψ]= 0, Tψ =∫kGψ(k). (4)The LO ψ propagator is obtained as δ2Γ/δψδψ =: iG−1ψ (p) in the following formiG−1ψ (p) = p2 −m22 −λ36v2 −λ16Tψ. (5)Since the tadpole is momentum independent the propagator can be parametrised asGψ(p) = i/(p2 −m2ψ) where m2ψ is determined by the gap-equationm2ψ = m22 +λ36v2 +λ16Tψ. (6)Finally, the closed system determining the propagator Gσ and the 3-point function Γψψσis derived by taking the corresponding second and third functional derivatives of theeffective action:iG−1σ (p) = p2 −m23 −λ22v2 −λ36Tψ−i√Nvλ36∫kGψ(p− k)Gψ(k)Γψψσ(p− k, k,−p),Γψψσ(p, q,−p− q) =−λ3v3√N−iλ16∫kGψ(p+ q − k)Gψ(k)Γψψσ(p+ q − k, k,−p− q). (7)We note that Γψψσ was calculated from the master equation of δΓ/δψ. In the last equationthe expression of the right hand side shows that Γψψσ(p, q,−p − q) on the left dependsonly on the momentum of σ. From this it promptly follows for the 3-point function thatΓψψσ(p, q,−p− q) = −λ3v3√N11− λ16Iψ(p+ q), Iψ(p) = −i∫kGψ(p− k)Gψ(k), (8)which leads (by exploiting Eq.(4)) toiG−1σ (p) = p2 −λ23v2 −λ2318v2Iψ(p)1− λ16Iψ(p). (9)The renormalisation algorithm of the equations (4), (5) and (7) is rather non-trivial. Theresult can be stated, however, very simply: i) one should replace in these equations the barecouplings m2i , λi by the corresponding renormalised ones; ii) the quadratically divergenttadpole integral Tψ and the logarithmically divergent Iψ should be replaced by the finite4parts Tψ,F , Iψ,F arising after subtracting the divergent pieces. It is worth noting, that bythe renormalisation of λi the 3-point function Γψψσ is actually a finite asymptoticallydecreasing function of the σ-momentum. As a consequence no divergence proportional top2 appears in G−1σ , therefore there is no need for infinite field renormalisation.The derivation of the non-trivial relation of the renormalised couplings to the unrenor-malised ones will be outlined in the next section. We apply cut-off regularisation to thedivergent integrals Tψ(M) and Iψ(p,M), which are evaluated after Euclidean continuationof the variables:Tψ(M)≡∫kik2 −M2=Λ216π2−M216π2lneΛ2M20+ Tψ,F (M),Iψ(p,M)≡∫ki(k2 −M2)((p− k)2 −M2)= −116π2lneΛ2M20+ Iψ,F (p,M), (10)where the scale M0 is introduced to eliminate the M-dependence of the cut-off dependentterms. The renormalisation will be achieved by compensating the cut-off dependenceexplicitly written out above by appropriately chosen coupling counterterms.3 RenormalisationThe ultraviolet renormalisation of the equations (4), (5) and (7) requires a fresh analysisof the counter-term resummation, since the unrenormalised equations do not suggest anytransparent formula similar to the simpler case of the O(N) model (see e.g. [13]).Below we shall apply the iterative renormalisation of [11] to our case. The renormali-sation of the equation of state and pion propagator (gap-equation) is the usual super-daisy renormalisation described in details in [11]. The renormalisation of the system (7)is a little bit more involved. First one rewrites the previous equations by making ex-plicit the counterterms. Since the derivation of the DS equations starts from the classi-cal action, there are explicit Lagrangian parameters present which we have to split asm2i = m2i,R + δm2i , λi = λi,R + δλi. But the effect of the counterterm does not reducemerely to this. Their consistent application implies that the 3-point function entering theDS equation of the sigma propagator is the renormalised one. This statement can be easilychecked when using in the equation of the sigma propagator the iterative solution of theequation of the renormalised 3-point function:Γψψσ(p, q,−p− q) =−λ3,R + δλ33√Nv (11)−iλ1,R + δλ16∫kGψ(p+ q − k)Gψ(k)Γψψσ(p+ q − k, k,−p− q).5At a given order of the iteration we recover the chain of bubble diagrams including thecounterterm diagrams of the usual perturbation theory at the corresponding order of theloop expansion.Next we subtract Eq. (4) from the first equation of (7) to obtain for the renormalisediG−1σ the expressioniG−1σ (p) = p2−λ2,R + δλ23v2−i√Nvλ3,R + δλ36∫kGψ(p−k)Gψ(k)Γψψσ(p−k, k,−p). (12)The iterative renormalisation consists of assuming infinite series expansion for the coun-terterms and determining them by requiring the cancellation of the divergencies generatedat each iteration step of the solution of the coupled equations of Gσ and Γψψσ.In the first iteration of the vertex function one substitutes into the propagator equationthe tree-level value of the 3-point vertex which yields the requirement:δλ(1)23v2 +λ23,R18v2(−i∫kGψ(p− k)Gψ(k))div= 0. (13)The same sort of first iteration is done also in the integral equation of Γψψσ and the twotogether lead toδλ(1)2 = −λ23,R6Iψ,div, δλ(1)3 = −λ1,Rλ3,R6Iψ,div. (14)With this step also the finite parts are determined, which are substituted into the seconditeration. This step produces on the right hand side of the expression iG−1ψ the followingdivergent piece in the full expression, which has to vanish:δλ(2)23+ 2λ3,Rδλ(1)318Iψ,div +λ3,R9Iψ,F (p)(δλ(1)3 +λ1,Rλ3,R6Iψ,div)+λ1,Rλ23,R108I2ψ,div v2 = 0.(15)It is important that the subdivergence proportional to the finite (possibly temperatureor density dependent) Iψ,F automatically vanishes in view of the previous iteration step.This feature of subdivergence cancellation is generally valid for each iteration step. Theremaining divergent terms determine the second term in the infinite series of δλ2:δλ(2)2 =λ23,Rλ1,R36I2ψ,div. (16)The second iteration of the equation of Γψψσ determines δλ(2)3 and δλ(1)1 = −λ21,RIψ,div/6.With simple induction the following general recursions are found for the subsequent iter-ative terms of the counterterm series:δλ(n)1 = −δλ(n−1)1λ1,R6Iψ,div, δλ(n)2 = −δλ(n−1)3λ3,R6Iψ,div, δλ(n)3 = −δλ(n−1)1λ3,R6Iψ,div,(17)6where n ≥ 1 and δλ(0)1 := λ1,R, δλ(0)3 := λ3,R.These equations imply the following non-perturbative renormalisation formulae:1λ1=1λ1,R+16Iψ,div,λ3λ1=λ3,Rλ1,R, λ2 −λ23λ1= λ2,R −λ23,Rλ1,R. (18)In addition to these relations between bare and renormalised couplings we also obtain therenormalised sigma self-energy which is nothing but the expansion of the last term in (9)in powers of λ1 up to the given order of the iteration (of course, the couplings and thebubble integral have to be replaced by their renormalised expressions).It is easy to check that provided we would have known a priori these relations, they indeedensure the simple form of the renormalised expression of (9) announced at the end of theprevious section. The mass of the Higgs particle is determined by the pole of equation (9)(iG−1(p2 =M2σ)=0)M2σ =v23[(λ2,R −λ23,Rλ1,R)+λ23,Rλ21,R1λ−11,R − Iψ,F (p2 =M2σ)/6]. (19)The renormalised quartic couplings are used also in the iterative renormalisation of theequations for the order parameter v and of Gψ in which the tadpole is calculated with aself-consistent propagator. The mass counterterms δm22 and δm23 are determined as:δm2(n)2 = −δλ(n−1)16Λ216π2+m22,Rδλ(n)3λ3,R= −δλ(n−1)16Λ216π2+m22,Rδλ(n)1λ1,R,δm2(n)3 = −δλ(n−1)36Λ216π2+m22,Rδλ(n)2λ3,R= −δλ(n−1)36Λ216π2+m22,Rδλ(n)1 λ3,Rλ21,R(20)implyingm22λ1+Λ296π2=m22,Rλ1,R, m23 −m22λ3λ1= m23,R −m22,Rλ3,Rλ1,R. (21)The second equalities in both equations of (20) follow when one exploits (17). Makinguse of these renormalised relations in (4) and (5) one easily derives for the renormalisedexpression of the vacuum expectation value the following formula:16v2 =[−m23,R +(m22,R −m2ψ) λ3,Rλ1,R](λ2,R −λ23,Rλ1,R)−1. (22)4 DiscussionThe expressions for the Higgs mass (19) and the Higgs condensate (22) show that theinfluence of the phantom sector might be quite strong on both. In order to make the7discussion more transparent below we shall chooseλ2,R = λ3,R ≡ λ, λ1,R ≡ λ+ λ′. (23)In this case the gap equation (6) which determines mψ reads after the elimination of v2asm2ψ = m22,R −m23,R +λ′6Tψ,F , Tψ,F =m2ψ16π2ln(em2ψM20). (24)which has the approximate solution (for small λ′)m2ψ ≈m22,R −m23,R1− λ′96π2lne(m22,R−m23,R)M20. (25)For sufficiently large renormalisation scale M0 one has 0 < m2ψ < m22,R−m23,R. The Higgscondensate has the simple equationλ6v2 = −m23,R +λλ′(m22,R −m23,R −m2ψ). (26)Depending on the ratio of the two quartic couplings and the differences in the renor-malised mass parameters we see that the phantom fluctuations might produce a stable Z2symmetry breaking condensate. This mechanism is similar to the dynamical symmetrybreaking of Coleman and Weinberg [14]. The role of the electromagnetic field in theirexample is played here by ψi, λ′ and the dynamically determined m2ψ. Also the HiggsmassM2σ =v23λλ+ λ′[λ′ +λ1− λ+λ′6Iψ,F (p2 =M2σ)](27)depends in a substantial way on both couplings which would give more flexibility in Higgsphenomenology.In conclusion, we have presented an O(N) symmetric model of the phantom world in-teracting with the SM Higgs field, which was solved in the N = ∞ limit. The solutionobtained with iterative renormalisation represents a non-trivial generalisation of the well–known example of the pure O(N) model. The method can be applied to a large varietyof models containing several multiplets with large number of components.The existence of symmetry breaking solutions stabilised by the phantom sector fluctu-ations was pointed out. In this respect we emphasise that Eqs. (24), (26) and (27) arevalid also at finite temperature, if Iψ,F and Tψ,F are evaluated accordingly. They can servefor investigating the nature of finite temperature symmetry restoration in the generalisedHiggs sector.Non-perturbative (lattice) studies at finite N could clarify if our findings are genericallyvalid also for finite number of phantom field components.8AcknowledgementsWe thank the participants of a Croatian-Hungarian Bilateral Workshop in TheoreticalPhysics for valuable discussions on the first version of this material. Work supported byHungarian Scientific Research Fund (OTKA) under contract number T046129. Zs. Sz. issupported by OTKA Postdoctoral Grant no. PD 050015. Support by the Hungarian Re-search and Technological Innovation Fund, and the Croatian Ministry of Science, Educa-tion and Sports is gratefully acknowledged. The authors acknowledge valuable commentsof the Referee on the paper.References[1] B. Patt and F. Wilczek, hep-ph/0605188[2] X. Calmet and J. F. Oliver, hep-ph/0606209[3] H. Davoudiasl, R. Kitano, T. Li and H. Murayama, Phys. Lett. B609, 117 (2005)[4] J. J. van der Bij, Phys. Lett. B636, 56 (2006)[5] M. Shaposhnikov and I. Tkachev, Phys. Lett. B639, 414 (2006)[6] R. Barbieri and L. J. Hall, hep-ph/0510243[7] D. G. Cerdeno, A. Dedes, T. E. J. Underwood, hep-ph/0607157[8] A. Hill and J. J. van der Bij, Phys. Rev. D 36, 3463 (1987)[9] R. S. Chivukula, M. Golden, Phys. Lett. 267, 233 (1991)[10] A. Yu. Ignatiev and R. R. Volkas, Phys. Lett. B487, 294 (2000)[11] J.-P. Blaizot, E. Iancu and U. Reinosa, Nucl. Phys. A736, 149 (2004)[12] R. J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge University Press1987, Cambridge[13] W. A. Bardeen and M. Moshe, Phys. Rev. D 28, 1372 (1983)[14] S. R. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973)9

Phantom field fluctuation induced Higgs effect

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Phantom field fluctuation induced Higgs effect

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