The Standard Model Higgs Boson As the Inflaton

arXiv:0710.3755v2 [hep-th] 9 Jan 2008 osdrtospee ev clrfil ihamass a with known field well scalar ∼ the heavy fixed, a not prefer are considerations fields in- the matter of with interaction the flaton the in and mass distribution the matter Though universe. the Microwave and Cosmic (CMB) the Background of fluctuations the of by observations constrained di- are extra properties The- Inflaton theory, different etc. Unified mensions, string in Grand supersymmetry, way involving (GUTs), natural SM, ories so the not of or extensions natural may a in particle mod- additional hypothetical appear an present This of of “inflaton”. introduction majority an scalar—the The require 6]. inflation in- 5, of 4, by els 3, solved 2, elegantly [1, most flation (almost) are of formation, for necessary structure generation perturbations, of of puzzles, spectrum these invariant problem scale Indeed, the elemen- complete. strong with of not together a (SM) is as Model particles Standard considered tary the often that is indication isotropic and neous Introduction 1. words: Key requir is theory electroweak non-min the the in present to is lead already requirement can besides essential Model Standard An the observations. of boson with Higgs the that argue We Abstract Shaposhnikov). Bezrukov), PACS: rpitsbitdt Elsevier to submitted Preprint mi addresses: Email h atta u nvrei lotflt homoge- flat, almost is universe our that fact The 10 13 a 88.q 14.80.Bn 98.80.Cq, b ntttd heredsPhénomènes Physiques, des Théorie de Institut GeV nttt o ula eerho usa cdm fScienc of Academy Russian of Research Nuclear for Institute [email protected] nain ig ed tnadMdl aibePac mass Planck Variable Model, Standard field, Higgs Inflation, n xrml ml efitrcigquar- self-interacting small extremely and [email protected] h tnadMdlHgsbsna h inflaton the as boson Higgs Model Standard The eo Bezrukov Fedor (Fedor (Mikhail cl oyehiu ´deaed asne H11 Lausa CH-1015 Lausanne, Fédérale Polytechnique de Ecole ´ a , b ed. ihi Shaposhnikov Mikhail , mlculn fteHgssaa edt rvt;n e part new no gravity; to field scalar Higgs the of coupling imal s rset6-eiaOtara7,Mso 132 Russia 117312, Moscow 7a, Oktyabrya 60-letiya Prospect es, nainadpouecsooia etrain naccord in perturbations cosmological produce and inflation o-iia coupling Non-minimal , lnkscale Planck ftesaa edi uvdsaetm background space-time curved a in If [9]. field properties scalar renormalization the the by of required fact in is later. (1) fixed be to constant unknown ssi ob mnml.Then “minimal”. be to said is R 1 where L gravity, to coupled non-minimally SM [8]. observations SM the the for within parameters fall these inflation; for spectral from models possibility be other this The distinguish can to inflation. perturbations used be tensor to and of predicted rise amplitude the give and can index itself SM the new con- scales. of Planck often existence and of is electroweak favour the which in between scale, physics argument GUT an the as to sidered close is mass constant coupling tic tot nornttostecnomlculn is coupling conformal the notations our In stesaa curvature, scalar the is oepanormi da osdrLgaga fthe of Lagrangian consider idea, main our explain To that demonstrate to is Letter present the of aim The = ξ L L SM 0 = SM steS part, SM the is h opigo h ig edt gravity to field Higgs the of coupling the , − M 1 M σ P 2 ofiec otuso h WMAP-3 the of contours confidence 2 eae oteNwo’ osatas constant Newton’s the to related R λ a − ∼ H ξH 10 steHgsfil,and field, Higgs the is M † − R, HR M 13 ssm asparameter, mass some is a eietfidwith identified be can 7.Ti au fthe of value This [7]. n,Switzerland nne, 1 ξ h hr emin term third The = eebr2007 December 4 − 1 / 6 . ξ san is ance icle (1) −1/2 18 MP = (8πGN ) = 2.4 10 GeV. This model cussed later in Section 3. Then the Lagrangian has the has “good” particle physics phenomenology× but gives form:

“bad” inflation since the self-coupling of the Higgs field 2 2 4 M + ξh is too large and matter fluctuations are many orders of SJ = d x√ g R magnitude larger than those observed. Another extreme − − 2 Z ( (2) µ is to put M to zero and consider the “induced” gravity ∂µh∂ h λ 2 2 2 [10, 11, 12, 13, 14], in which the electroweak symmetry + h v . 2 − 4 − ) breaking generates the Planck mass [15, 16, 17]. This 17 happens if √ξ 1/(√GN MW ) 10 , where MW This Lagrangian has been studied in detail in many pa- 100 GeV is the∼ electroweak scale.∼ This model may give∼ pers on inflation [14, 19, 20, 24], we will reproduce here “good” inflation [12, 13, 14, 18, 19, 20] even if the the main results of [14, 19]. To simplify the formulae, scalar self-coupling is of the order of one, but most we will consider only ξ in the region 1 √ξ ≪ 1017, ≪ probably fails to describe particle physics experiments. in which M MP with very good accuracy. Indeed, the Higgs field in this case almost completely It is possible≃ to get rid of the non-minimal coupling decouples from other fields of the SM 2 [15, 16, 17], to gravity by making the conformal transformation from which corresponds formally to the infinite Higgs mass the Jordan frame to the Einstein frame m . This is in conflict with the precision tests of the 2 H ξh electroweak theory which tell that m must be below gˆ =Ω2g , Ω2 =1+ . (3) H µν µν M 2 285 GeV [21] or even 200GeV [22] if less conservative P point of view is taken. This transformation leads to a non-minimal kinetic term These arguments indicate that there may exist some for the Higgs field. So, it is convenient to make the intermediate choice of M and ξ which is “good” for change to the new scalar field χ with particle physics and for inflation at the same time. In- 2 2 2 2 deed, if the parameter ξ is sufficiently small, √ξ ≪ dχ Ω +6ξ h /MP 17 = 4 . (4) 10 , we are very far from the regime of induced grav- dh r Ω ity and the low energy limit of the theory (1) is just Finally, the action in the Einstein frame is the SM with the usual Higgs boson. At the same time, if ξ is sufficiently large, ξ 1, the scalar field be- 2 µ 4 MP ˆ ∂µχ∂ χ haviour, relevant for chaotic inflation≫ scenario [7], dras- SE = d x gˆ R + U(χ) , (5) − ( − 2 2 − ) tically changes, and successful inflation becomes pos- Z p sible. We should note, that models of chaotic inflation where Rˆ is calculated using the metric gˆµν and the with both nonzero M and ξ were considered in litera- potential is ture [12, 14, 19, 20, 23, 24, 25], but in the context of 1 λ 2 either GUT or with an additional inflaton having noth- U(χ)= h(χ)2 v2 . (6) ing to do with the Higgs field of the Standard Model. Ω(χ)4 4 − The Letter is organised as follows. We start from dis- For small field values h χ and Ω2 1, so the poten- cussion of inflation in the model, and use the slow-roll tial for the field χ is the same≃ as that for≃ the initial Higgs approximation to find the perturbation spectra parame- field. However, for large values of h M /√ξ (or ters. Then we will argue in Section 3 that quantum cor- P χ √6M ) the situation changes a lot.≫ In this limit rections are unlikely to spoil the classical analysis we ≫ P used in Section 2. We conclude in Section 4. M χ h P exp . (7) ≃ √ξ √6M P 2. Inflation and CMB fluctuations This means that the potential for the Higgs field is ex- ponentially flat and has the form Let us consider the scalar sector of the Standard 2 λM 4 2χ − Model, coupled to gravity in a non-minimal way. We U(χ)= P 1+exp . (8) 4ξ2 −√6M will use the unitary gauge H = h/√2 and neglect all P gauge interactions for the time being, they will be dis- The full effective potential in the Einstein frame is pre- sented in Fig. 1. It is the flatness of the potential at 2 χ M which makes the successful (chaotic) infla- This can be seen most easily by rewriting the Lagrangian (1), ≫ P given in the Jordan frame, to the Einstein frame, see also below. tion possible.

2 U(χ) 1.0 WMAP 50 60 λM4/ξ2/4 0.8 λφ4 mφ2 2 0.6 SM+ξh R

0.002 HZ r 0.4

λ v4/4 4 2 λM /ξ /16 0.2 0 0 v 0 0.0 χ χ χ 0 end COBE 0.90 0.95 1.00 1.05 ns

Fig. 1. Effective potential in the Einstein frame. Fig. 2. The allowed WMAP region for inflationary parameters (r, n). The green boxes are our predictions supposing 50 and 60 e– 3 Analysis of the inflation in the Einstein frame can foldings of inflation. Black and white dots are predictions of usual be performed in standard way using the slow-roll ap- chaotic inflation with λφ4 and m2φ2 potentials, HZ is the Har- proximation. The slow roll parameters (in notations of rison-Zeldovich spectrum. [28]) can be expressed analytically as functions of the field h(χ) using (4) and (6) (in the limit of h2 λ NCOBE √ mH 2 2 ≫ ξ 2 49000 λ = 49000 . (13) MP /ξ v ), 3 0.027 √ ≫ ≃ r ≃ 2v 2 M 2 dU/dχ 4M 4 Note, that if one could deduce ξ from some fundamen- ǫ = P P , (9) 2 U ≃ 3ξ2h4 tal theory this relation would provide a connection be- 2 2 2 tween the Higgs mass and the amplitude of primordial 2 d U/dχ 4MP η = MP , (10) perturbations. The spectral index n =1 6ǫ +2η cal- U ≃− 3ξh2 − 3 3 4 culated for N = 60 (corresponding to the scale k = 2 4 (d U/dχ )dU/dχ 16MP 0.002/Mpc) is n 1 8(4N + 9)/(4N + 3)2 0.97. ζ = MP 2 2 4 . (11) U ≃ 9ξ h The tensor to scalar≃ perturbation− ratio [8] is r =≃ 16ǫ 192/(4N +3)2 0.0033. The predicted values are well≃ Slow roll ends when ǫ 1, so the field value at ≃ ≃ 1/4 within one sigma of the current WMAP measurements the end of inflation is hend (4/3) MP /√ξ ≃ ≃ [8], see Fig. 2. 1.07MP /√ξ. The number of e-foldings for the change of the field h from h0 to hend is given by

h0 2 2 2 3. Radiative corrections 1 U dχ 6 h0 hend N = 2 dh −2 .(12) MP dU/dh dh ≃ 8 MP /ξ hendZ An essential point for inflation is the flatness of the scalar potential in the region of the field values We see that for all values of √ξ ≪ 1017 the scale of h 10M /√ξ, what corresponds to the Einstein ∼ P the Standard Model v does not enter in the formulae, frame field χ 6MP . It is important that radiative so the inflationary physics is independent on it. Since corrections do∼ not spoil this property. Of course, any interactions of the Higgs boson with the particles of discussion of quantum corrections is flawed by the non- the SM after the end of inflation are strong, the re- renormalizable character of gravity, so the arguments heating happens right after the slow-roll, and Treh we present below are not rigorous. 2λ 1/4 15 ∗ ≃ ( 2 ∗ ) M /√ξ 2 10 GeV, where g = 106.75 There are two qualitatively different type of correc- π g P ≃ × is the number of degrees of freedom of the SM. So, tions one can think about. The first one is related to the the number of e-foldings for the the COBE scale enter- quantum gravity contribution. It is conceivable to think ing the horizon NCOBE 62 (see [28]) and hCOBE [29] that these terms are proportional to the energy den- ≃ ≃ 9.4MP /√ξ. Inserting (12) into the COBE normaliza- sity of the field χ rather than its value and are of the 4 4 2 tion U/ǫ = (0.027MP ) we find the required value for order of magnitude U(χ)/MP λ/ξ . They are small ξ at large ξ required by observations.∼ Moreover, adding 4+2n 2n non-renormalizable operators h /MP to the La- 3 The same results can be obtained in the Jordan frame [26, 27]. grangian (2) also does not change the flatness of the

3 potential in the inflationary region. 4 4. Conclusions Other type of corrections is induced by the fields of the Standard Model coupled to the Higgs field. In In this Letter we argued that inflation can be a nat- one loop approximation these contributions have the ural consequence of the Standard Model, rather than structure an indication of its weakness. The price to pay is very m4(χ) m2(χ) modest—a non-minimal coupling of the Higgs field to ∆U log , (14) gravity. An interesting consequence of this hypothesis is ∼ 64π2 µ2 that the amplitude of scalar perturbations is proportional where m(χ) is the mass of the particle (vector bo- to the square of the Higgs mass (at fixed ξ), revealing son, fermion, or the Higgs field itself) in the back- a non-trivial connection between electroweak symme- ground of field χ, and µ is the normalization point. try breaking and the structure of the universe. The spe- 2 2 cific prediction of the inflationary parameters (spectral Note that the terms of the type m (χ)MP (related to quadratic divergences) do not appear in scale-invariant index and tensor-to-scalar ratio) can distinguish it from subtraction schemes that are based, for example, on di- other models (based, e.g. on inflaton with quadratic po- mensional regularisation (see a relevant discussion in tential), provided these parameters are determined with [30, 31, 32, 33]). The masses of the SM fields can be better accuracy. readily computed [14] and have the form The inflation mechanism we discussed has in fact a general character and can be used in many exten- 2 m(v) h(χ) 2 d U sions of the SM. Thus, the νMSM of [36, 37] (SM plus m (χ)= , m (χ)= (15) ψ,A v Ω(χ) H dχ2 three light fermionic singlets) can explain simultane- ously neutrino masses, dark matter, baryon asymmetry for fermions, vector bosons and the Higgs (inflaton) of the universe and inflation without introducing any field. It is crucial that for large χ these masses approach additional particles (the νMSM with the inflaton was different constants (i.e. the one-loop contribution is as considered in [30]). This provides an extra argument in flat as the tree potential) and that (14) is suppressed by favour of absence of a new energy scale between the the gauge or Yukawa couplings in comparison with the electroweak and Planck scales, advocated in [32]. tree term. In other words, one-loop radiative corrections do not spoil the flatness of the potential as well. This argument is identical to the one given in [14]. Another important correction is connected with running of the non-minimal coupling ξ to gravity. The cor- Acknowledgements responding renormalization group equation is [34, 35] The authors thank S. Sibiryakov, V. Rubakov, I. Tkachev, O. Ruchayskiy, H.D. Kim, P. Tinyakov, and 2 9 2 3 ′2 A. Boyarsky for valuable discussions. This work was dξ 1 12λ + 12yt 2 g 2 g µ = ξ + − − , (16) supported by the Swiss National Science Foundation. dµ 6 16π2

where yt = mt/v is the top Yukawa coupling, g and g′ are SU(2) and U(1) couplings of the Standard Model and µ is the characteristic scale. The renormalization of References ξ from µ M to the Planck scale is considerable, [1] A.A. Starobinsky, JETP Lett. 30 (1979) 682. ∼ W ξ(MP ) 2ξ(MW ). At the same time, the change of ξ [2] A.A. Starobinsky, Phys. Lett. B91 (1980) 99. in the inflationary≈ region is small, δξ/ξ 0.2. Thus, the [3] V.F. Mukhanov and G.V. Chibisov, JETP Lett. 33 logarithmic running of ξ does not change≈ the behaviour (1981) 532. of the potential required for inflation. [4] A.H. Guth, Phys. Rev. D23 (1981) 347. There is also the induced one-loop pure gravitational [5] A.D. Linde, Phys. Lett. B108 (1982) 389. term of the form ξ2R2/64π2. During the inflationary [6] A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 2 epoch it is smaller than the tree term MP R by the Higgs 48 (1982) 1220. self-coupling λ/64π2 and does not change the conclu- [7] A.D. Linde, Phys. Lett. B129 (1983) 177. sion. [8] D.N. Spergel et al., ApJS, 170, (2007) 377. [9] N.D.Birrell andP.C.W. Davies, QuantumFields in 4 Actually, in the Jordan frame, we expect that higher-dimensional Curved Space (Cambridge, UK: Univ. Pr., 1982). 2 2 operators are suppressed by the effective Planck scale MP + ξh . [10] A. Zee, Phys. Rev. Lett. 42 (1979) 417.

4 [11] L. Smolin, Nucl. Phys. B160 (1979) 253. [12] B.L. Spokoiny, Phys. Lett. B147 (1984) 39. [13] R. Fakir and W.G. Unruh, Phys. Rev. D41 (1990) 1792. [14] D.S. Salopek, J.R. Bond and J.M. Bardeen, Phys. Rev. D40 (1989) 1753. [15] J.J. van derBij, Acta Phys.Polon.B25 (1994)827. [16] J.L. Cervantes-Cota and H. Dehnen, Nucl. Phys. B442 (1995) 391. [17] J.J. van der Bij, Int.J.Phys. 1 (1995) 63. [18] D.I. Kaiser, Phys. Lett. B340 (1994) 23. [19] D.I. Kaiser, Phys. Rev. D52 (1995) 4295. [20] E. Komatsu and T. Futamase, Phys. Rev. D59 (1999) 064029. [21] ALEPH, Phys. Rept. 427 (2006) 257. [22] Particle Data Group,W.M. Yao et al., J. Phys. G33 (2006) 1, and 2007 partial update for the 2008 edition. [23] T. Futamase and K. Maeda, Phys. Rev. D39 (1989) 399. [24] R. Fakir and W.G. Unruh, Phys. Rev. D41 (1990) 1783. [25] M.V. Libanov, V.A. Rubakov and P.G. Tinyakov, Phys.Lett. B 442 (1998) 63. [26] N. Makino and M. Sasaki, Prog. Theor. Phys. 86 (1991) 103. [27] R. Fakir, S. Habib and W. Unruh, Astrophys. J. 394 (1992) 396. [28] D.H. Lyth and A. Riotto, Phys. Rept. 314 (1999) 1. [29] A.D. Linde, Phys. Lett. B202 (1988) 194. [30] M. Shaposhnikovand I. Tkachev, Phys. Lett. B639 (2006) 414. [31] K.A. Meissner and H. Nicolai, Phys. Lett. B648 (2007) 312. [32] M. Shaposhnikov, (2007), arXiv:0708.3550 [hep- th]. [33] K.A. Meissner and H. Nicolai, (2007), arXiv:0710.2840 [hep-th]. [34] I.L. Buchbinder, D.D. Odintsov and I.L. Shapiro, Effective Action in Quantum Gravity (Bristol, UK: IOP, 1992). [35] Y.Yoon and Y.Yoon, Int.J.Mod.Phys.A 12 (1997) 2903. [36] T. Asaka, S. Blanchet and M. Shaposhnikov, Phys. Lett. B631 (2005) 151. [37] T. Asaka and M. Shaposhnikov, Phys. Lett. B620 (2005) 17.