EFTA02710832.pdf

Is Higgs Inflation Dead? Jessica L. Cook.' Lawrence M. Krauss,". • Andrew J. Long,'• t and Subir Sabharwal' 'Department of Physics and School of Earth and Space Esplomtion Arizona State University. Tempe. AZ 85827-1404 2Reseatrh School of Astronomy and Astrophysics, Mt. Stromlo Observatory, Australian National University. Canberra, Australia 2611 (Dated: March 19, 2014) We consider the status of Higgs Inflation in light of the recently announced detection of B-modes in the polar- ization of the cosmic microwave background radiation by the BICEP2 collaboration. In order for the primordial B-mode signal to be observable by BICEP2. the energy scale of inflation must be high, Via Az 2 x 101° GeV. Higgs Inflation generally predicts a small amplitude of tensor perturbations. and therefore it is natural to ask if Higgs Inflation might accommodate this new measurement. We find the answer is essentially no. unless one considers either extreme fine tuning. or possibly adding new beyond the standard model fields, which remove some of the more attractive features of the original idea. We also explore the possible importance of a factor that has not previously been explicitly incorporated, namely the gauge dependence of the effective potential used in calculating inflationary otisembles. e.g. ns and r. to sec if this might provide additional wiggle room. Such gauge effects are comparable to effects of lliggs mass uncertainties and other observables already considered in the analysis. and therefore they arc relevant for constraining models. But, they arc therefore too small to remove the apparent incompatibility between the BICEP2 observation and the predictions of Higgs Inflation. I. INTRODUCTION Higgs Inflation (HI) postulates that the Standard Model Higgs field and the inflaton are one in the same PI. (See also The theory of inflation [I-3) successfully addressed the Ref. [8J for a recent review). This powerful assumption allows twentieth century's greatest puzzles of theoretical cosmology. HI to be, in principle much more predictive than many other Over the past 20 years. increasingly precise measurements of models of inflation, as by measuring the masses of the Higgs the temperature fluctuations of the cosmic microwave back- boson and the top quark at the electroweak scale (100 GeV), ground radiation (CMB) also confirmed the nearly scale in- one might predict observables at much larger energy scales ( variant power spectrum of scalar perturbations. a relatively associated with inflation (Viiii 4 < 1016 GeV). generic inflationary prediction. These many successes, how- However, in practice this enhanced predictive power is elu- ever, underscored the inability to probe perhaps the most ro- sive due to a strong sensitivity to quantum effects, unknown bust and unambiguous prediction of inflation, the generation physics, and other technical subtleties in the model. Specifi- of a background of gravity waves associated with what are cally, one connects observables at the electroweak and infla- likely enormous energy densities concomitant with inflation tionary scales using the renormalization group flow (RG) of (e.g.. the SM couplings [9-1,1]. It is reasonable however to expect Recently. the BICEP2 collaboration reported evidence of that there is new physics at intermediate scales, and even if the B-modes in the polarization pattern of the CMB [5]. The B- SM is extended only minimally to include a dark matter can- modes result from primordial gravity wave induced distortions didate [151 or neutrino masses il6-191 this new physics can at the surface of last scattering. If one assumes that these grav- qualitatively affect the connection between electroweak and ity waves are of an inflationary origin, then the BICEP2 mea- inflationary observables. Moreover, penurbative unitarity ar- surement corresponds to an energy scale of inflation: guments require new physics just above the scale of inflation [20, 211, and in addition the unknown coefficients of dimen- Vitilif4 Pt (2 ± 0.2) x 1016 GeV (I) sion six operators can significantly limit the predictive power of HI [22]. The HI calculation also runs into various techni- for a reported tensor-to-scalar ratio of r 0.2-1,:fg (using cal subtleties that arise from the requisite non-minimal grav- also the Planck collaboration's measurement of the amplitude itational coupling (see below) and quantization in a curved of the scalar power spectrum 161). Such a high scale of in- spacetime [23-25]. Finally, it is worth noting that HI is also at flation rules out many compelling models. For the purposes tension with the measured Higgs boson and top quark masses. of this paper, we will assume that the observation r re- 0.2 and an O(2a) heavier Higgs or lighter top is required to evade is valid', and we assess the impact of this measurement on a vacuum stability problems [26]. particular model of inflation, known as Higgs Inflation. Also, as we shall later discuss in detail, there is one addi- tional source of ambiguity in calculations of HI that had not been fully explored. Since the quantum corrections are sig- nificant when connecting the low energy and high energy ob- • Electronic address: [email protected] servables, one should not work with the classical (tree-level) 1FJectronic address: [email protected] I Note that the BICEP2 measurement is in tension with the upper bound. scalar potential, as is done in may models of inflation, but r C 0.11 at 95% C.L.. obtained previously by the Planck collaboration one must calculate the quantum effective potential. It is well- known that in a gauge theory the effective potential explicitly EFTA_R1_02123540 EFTA02710832 depends upon the choice of gauge in which the calculation is non-minimal coupling to be much larger than unity. Specifi- performed [27, 28), and care must be taken to extract gauge- cally one requires (see, e.g., Ref. [81) invariant observables from it [29-32] (see also [33, 34]). This fact can perhaps be understood most directly by recalling 4 st 47000v5 (5) that the effective action is the generating functional for one- particle irreducible Green's functions, which themselves arc which is e 17000 for A .c.,7 0.13. The energy scale of infla- gauge dependent [28]. In practice one often neglects this sub- tion is then predicted to be tlety, fixes the gauge at the start of the calculation, and cal- culates observables with the effective potential as if it were a 14 :v. (0.79 x 1016 GeV)4 (6) classical potential. In the context of finite temperature phase transitions. it is known that when calculated naively in this leading to a tensor-to-scalar ratio, assume scalar density per- way, the predictions for observables depend on the choice of turbations fixed by CMB observations, r •co-- 0.0036. This is gauge used I 34—all. Because of the extreme tension between naively incompatible with the much larger BICEP2 measure- HI models and the data, we assess here the degree to which ment, see Eq. (I). Decrease in HI to attempt to match the this gauge uncertainty might affect the observables in Higgs newly measured value of %1,,r is not workable either, as set- Inflation. We find that the gauge ambiguity introduces uncer- ting e 2000 then produces too little power in scalar density tainties that are comparable to the variation of the physical perturbations. parameters, i.e. the Higgs mass. As a result. this ambiguity Fundamentally then, the problem in obtaining a large value alone cannot resuscitate moribund models. of r in Higgs inflationary models is that the HI potential asymptotes to a constant at large field values where inflation occurs. This flat potential then results in relatively large den- 2, GRAVITY WAVES FROM HIGGS INFLATION sity perturbations, which, in order to then match observations, constrain the magnitude of the potential, resulting in a small The Standard Model Higgs potential, V(h) = Ah4 /4 with tensor contribution. A = O(0.1), is too steeply sloped for successful inflation. The question then becomes whether variations in this The measurement of the Higgs boson mf.. fixes A 0.13, canonical HI, due to quantum effects for example, will allow whereas A a 1 is required to produce the observed ampli- the SM Higgs boson to the be inflaton field while also accom- tude of density perturbations. In the HI model, slow roll is modating the large value of r. achieved by introducing a non-minimal gravitational coupling for the Higgs field. C = ciblibR. where (ff is the Higgs dou- blet and R the Ricci scalar. One can remove the non-minimal 3. SAVING HIGGS INFLATION? coupling term from the Lagrangian by performing a confor- mal transformation, go,„(x) = 51-2(x)li,„(s) where Since it is the non-minimal coupling, e, that flattens out the potential at high scales, one might consider whether there are D2 = 1+ 2e(}141/4/1, (2) other ways to flatten the potential, and so avoid the require- ment for large e values. is the conformal factor and My is the reduced Planck mass. One possibility proposed in this regard [I 3] involves fine By doing so. one passes from the Jordan to the Einstein frame. tuning the Higgs and top masses such that the Higgs self- The scalar potential in the new frame becomes coupling runs very small at the scale of inflation, A 10-4. This allows for relatively small e — 90 and produces r > 0.15 V(h) = 2 (3) that may be compatible with the BICEP measurement. It is 4 (1+ impossible to entirely eliminate the need for the non-minimal coupling. However, as a caveat let us point out that this so- where we have written 4.147, = h2/ 2. At large field values. lution only exists if the theory is first quantized in the Jordan h > ii/p/VZ, the potential asymptotes to a constant frame and then moved to the Einstein frame (so-called "pre- scription I"), and results differ if the operations are reversed Vo s AMP/4e2 (4) ("prescription II"). The apparent disagreement is an artifact of quantizing all the fields except gravity, which results in a This is the appropriate regime for slow roll inflation. different definition of the Ricci scalar in the two frames. A To evince the tension between Higgs Inflation and large ten- full theory of quantum gravity would probably be required to sor perturbations we can first neglect quantum corrections to resolve the problem consistently between frames. Thus, it is V(h), e.g. the running of A, as the energy scale of HI, given not clear if the small £ "prescription I" solution is artificial. by Eq. (4), is insensitive to the quantum corrections, whereas If one goes outside of the Standard Model, then new physics the slope is more sensitive. can affect the running of the Higgs self-coupling or anomalous Since A is fixed by the measured Higgs mass, the scalar dimension, y. For example, one may hope that A or -) acquires potential in Eq. (3) has only one free parameter: 4. It is well- a significant running at high scales so as to give a workable known that to achieve sufficient e-foldings of inflation and the solution consistent with both the measured scalar and tensor correct amplitude for the scalar power spectrum, one needs the power spectra. (See. e.g., [4 1). EFTA_R1_02123541 EFTA02710833  3 As a result, it appears that canonical HI with a non-minimal The Coleman-Weinberg effective action f eff and effective gravitational coupling as the only new physical input appears potential Vof [49) have become standard tools in the study extremely difficult to reconcile with the new observation of a of vacuum structure, phase transitions, and inflation. The large tensor contribution from inflation. If would appear to effective action is the generating functional of one-particle be necessary to add new physics to eliminate the dependence irreducible Green's functions, and therefore it is important on non-minimal coupling entirely and to give the Higgs ef- to recognize that both "off and Kt f are off-shell quantities, fective potential a shape compatible with observations. Such which will carry spurious gauge dependence [28]. When ap- extension ofHI tend to defeat the original purpose of the idea, plying the effective potential to a problem. special care must namely its predictivity, and in any case most such modifica- be taken to extract gauge-invariant information. In particu- tions that have been proposed [42-44) tend to retain the now lar, the Nielsen identities express the gauge invariance of the undesirable feature of small r in any case. effective potential at its stationary points, but derivatives of There are two options that might allow large r consistent the effective potential are not generally gauge invariant [3 I]. with BICEP. One possibility involves tuning the Higgs poten- This suggests that inflationary observables, e.g. ns, r, and tial to form a second local minimum at large scales. i.e., a false dns/dlnk, naively extracted directly from the slow roll pa- vacuum similar to old inflation PSI. To avoid the problems rameters will acquire a spurious gauge dependence. of old inflation, a time dependent tunneling rate is introduced. Ideally one would like to determine the "correct" proce- While most mechanisms to achieve this, however, produce a dure for calculating physical quantities like us from a given small value of r (46), larger can be accommodated by adding model in such a way that the spurious gauge dependence is a new scalar with a non-minimal coupling to gravity, such that canceled. There have been significant efforts made in this di- the Higgs field sees a time dependent Planck mass NM A rection (23. 24]. but a full gauge invariant formalism is yet second possibility uses a non-canonical Galileon type kinetic to be developed. Here we will take a different approach that term for the Higgs field. This model yields an r =-• 0.14 [48]. is more aligned with recent work on the gauge dependence These tuned limits, variants, and extensions of the original of phase transition calculations [34, 38. 39). Specifically, HI model leave the door slightly open for the possibility of we numerically perform the "naive" HI calculation using the connecting the Higgs with the inflation field. However, with- R{ gauge effective potential and RG-improvement to assess out additional scalars or modification of the Higgs potential the sensitivity of the inflationary observables to the spurious via some other mechanism beyond the Standard Model, the gauge dependence. original scenario. i.e. Higgs Inflation with only a non-minimal We begin by reviewing the familiar Higgs Inflation calcu- coupling to gravity, does not appear to be compatible with the lation. After moving from the Jordan to the Einstein frame, BICEP result. as described in Sec. 2. the resulting action contains a non- Before we nail the coffin shut on Higgs Inflation, however, canonical kinetic term for the Higgs field. One cannot, in there is one possible additional source of uncertainty that mer- general, find a field redefinition that makes the kinetic term its further investigation. As we describe below, when one goes canonical globally [21, 50]. At this point, it is customary to beyond the tree level, there are gauge ambiguities involved in move to the unitary 'Inge where the Higgs doublet is written the calculation of effective potentials that need to be consid- as ch(r) = e2"1.1r 1r (0, h(x)/4 2.. Then the kinetic term ered when deriving constraints on parameters. for the radial Higgs excitation can be normalized by the field redefinition x(h) where 4. GAUGE DEPENDENCE AMBIGUITIES 1 3 3/2 0-12/dh.)2 dxI dh - + n2 2 P az (7) When working with a gauge theory, such as the Stan- dard Model electroweak sector, calculations typically involve and now 112 = 1 + h2/3/1,. spurious gauge dependence that cancels when physical ob- Having canonically normalized both the gravity and Higgs servable arc calculated. For example, in a spontaneously kinetic terms, the derivation of the effective potential proceeds broken Yang-Mills theory one may work in the renormaliz- along the standard lines. We calculate the RG improved. one- able class of gauges (RO upon augmenting the Lagrangian loop effective potential as described in the Appendix. After with a gauge fixing term Cgi = —G°Ga/2 where = performing the RG improvement, the parameter A that appears (1/ 1 . ,f)(0,A° - xs) where x, are the would-be in Eq. (3) should be understood at the running coupling eval- Goldstone boson fields and F° = riv, with 711 the sym- uated at the scale of inflation. Generally, A < 0.1 and its metry generators and vj the symmetry-breaking vacuum ex- value depends upon the physical Higgs boson and top quark pectation value. (See. e.g., [33]). A corresponding Fadeev- masses at the input scale. For the best fit observed values, Popov ghost term is also added. Physical or "on-shell" quan- 125 GeV and Mt xr. 173 GeV, the coupling runs neg- tities, such as cross sections and decay rates, may be calcu- ative at h ot 101° —1012 GeV; this is the well-known vacuum lated penurbatively, and any dependence on the gauge fix- stability problem of the Standard Model [26). Successful HI ing parameter, 4f, cancels order-by-order. Unphysical or requires an O(2o) deviation from central values toward either "off-shell" quantities, such as propagators or one-particle irre- larger Higgs boson mass or smaller top quark mass. ducible Green's functions, may harmlessly retain the spurious Gauge dependence enters the calculation at two places: ex- gauge dependence. plicitly in the one-loop correction to the effective potential and EFTA_R1_02123542 EFTA02710834  4 implicitly through the Higgs anomalous dimension upon per- 0.77673 forming the RG improvement. To calculate the slow roll parameters. e.g. I. 0.77672 3— 0.77671 e = Af (W/V)2 1 (8) 2 ki-v .. 0.77670 the derivatives are taken with respect to x, i.e., V'(h(x)) = (OVI0h)(dx1dh)-1. The potential and its derivatives are I 0.77669 0.77668 evaluated at the field value, heath, for which the number of e-foldings, given by — 0.77667 123 124 125 126 127 128 Higgs Mass: µi t GeV = i hna dh ' (9) 111V(h)h (h)fl, FIG. I: The predicted energy scale of inflation. Vm1/11, over a range is Ai = 60. Inflation terminates at h = he„d where of Higgs boson masses (Al,,). for three values of the top quark mass (11/1,/2)(VVV)2 = 1. (Aft). and in the Landau gauge. Co 0. In Fig. I we show the energy scale of inflation. liar= V(hrran) (10) 0.77680 as the the Higgs boson and top quark masses are varied, and 0.77675 the non-minimal coupling, t ar, few x 103, is determined to I 26 match the observed amplitude of scalar perturbations. This 0.77670 demonstrates that the scale of inflation is insensitive to Atif, varying only at the O(10-4) level. It always remains signif- > 0.77665 icantly below 2 x 1016 GeV, which indicates the incompat- 0.77660 124 ( ibility with the BICEP2 measurement. (The corresponding tensor-to-scalar ratio is r r-- 0.003.) .2 To illustrate the gauge dependence, we show in Fig. 2 how a 0.77655 Via varies with $4e. We find that V;,,( also changes at a level 0 77650 comparable to its sensitivity to Ahr or Me as the gauge pa- Gauge Parameter: fq rameter deviates from the Landau gauge (‘ = 0). It is there- fore important to consider this ambiguity for model building FIG. 2: The energy scale of inflation. Via. as the gauge parameter. purposes. Nevertheless, the absolute change in Vim is far too egr. varies. We fix Aft = 170 GeV and show three values of Me. small to reconcile HI with the BICEP2 measurement. Note that at larger vales of Qgt the scale of inflation appears to continue to decrease, but in this limit the perturbative valid- ity of the calculation begins to break down. To resolve this is- ability to explore fundamental physics and the early universe. sue. the unphysical degrees of freedom. the Goldstone bosons If the measurement of r ra 0.2 is confirmed, then it is rea- and ghosts. should be decoupled as the unitary gauge is ap- sonable to expect that, in the not-too-distant future, measure- proached. ments of the spectrum of primordial tensor perturbations will Our numerical results appear consistent with the Nielsen become possible. allowing further tests of inflation. And if identities [31, 32] which capture the gauge dependence of the the measured r can unambiguously be shown to be due to in- effective potential. The relevant identity is flation. then this also substantiates the quantization of gravity [ 0 0 [5 1 ]. ea -4 C(°' — rr(4, =0• (II) Thus, future observations will provide significant con- )601 straints on particle physics and models of inflation. However In the slow roll regime, the gradient of the effective potential is the simple observation of non-zero r already signals the death small, and the gauge dependence is proportionally suppressed. knell for low-scale models of inflation. This includes the class We note that a rigorous gauge invariant calculation could of models captured by the potential in Eq. (3), and among perhaps take Eq. (11) as a starting point. This might be an these apparently Higgs Inflation. We have shown that r re 0.2 interesting avenue for future work, either in the context of HI essentially excludes canonical Higgs Inflation in the absence or other, potentially more viable models of inflation that are of extreme fine tuning. The Higgs field may live on as the embedded in gauge theories. inflaton but only with significant non-minimal variants of HI. In our analysis we have also drawn attention to the issue of gauge dependence in the Higgs Inflation calculation. We find 5. CONCLUSION that the energy scale of inflation acquires an artificial depen- dence on the gauge fixing parameter by virtue of the gauge de- The recent detection of B-modes by the BICEP2 collabo- pendence of the effective potential from it is extracted. How- ration represents a profound and exciting leap forward in our ever, we find this gauge dependence of the scale of inflation EFTA_R1_02123543 EFTA02710835  5 is comparable to the dependence on other physical parameter where 1/2 = 1 + Ch2/1111, was given by Eq. (2). We denote uncertainties. which are themselves small. While this may be the gauge fixing parameter by Car to distinguish it from the important for model building purposes, it does not affect the non-minimal gravitational coupling parameter, robustness of the fact that larger disfavors Higgs Inflation. We implement the RG improvement as per 152-511. (See also the reviews [55, 56]). This consists of (I) solving the RG equations (RGEs) to determine the running parameters as Acknowledgments functions of the RG flow parameter 1, (2) replacing the vari- ous coupling constants in Kg with the corresponding running This work was supported by ANU and by the US DOE un- parameter. and (3) evaluating the RG flow parameter at the dcr Grant No. DE-SC0008016. We would like to thank layden appropriate value t = 1. so as to minimize the would-be large L. Newstead for help with the code. logarithms. For the sake of discussion, let us denote the running parameters collectively as 4(t) Appendix A: Standard Model Effective Potential {43(1), 42(t), oi(t), A(t), fh(t), (1)} where 92 = g and 91 = 9'. Then the RGEs take the form fie, /(1 + -y) = do /dt The Standard Model effective potential is calculated (i) to with the boundary condition M' t = 0) = co. Here 7 is the one-loop order. (ii) working in the MS renormalization the anomalous dimension of the Higgs field. We neglect scheme with renormalization scale p, and (iii) in the tenor- the running of the gauge-fixing parameter. 4r, since it is malizable class of gauges (Re) as follows: self-renormalized. This approximation is reasonable since we focus on 4-1- < 4r; for larger values of est-, perturba- Vergh) = 0 °)(h) + 1,(1)(h) . (Al) tivity becomes an issue. The Higgs field runs according to —7/1 = dh/dt where the anomalous dimension -y(t) is given The tree-level potential is as [571 V(o)(h) = —h4 , (A2) 4 — (4ir)2 1 [ 9 4(1 3 ) 922 4 3 ) 92 + 3Y21 and we can neglect the O(h°) and O(h2) terms for the pur- it271 34 _ a2 poses of studying HI where the field value is large. The one- (4w(4w)°Lk 32 — f 8Csf) 9r2 16" 2 loop correction is [55] (sec also [34] for gauge dependent fac- — 431? _5I 992 + 17_2 + 8_21_2 + 27e ] (AS) tors) 96 2 4 2 I2 YI 12 fit° ( This last equation may be solved immediately along with the I (l)(h) = In —— (A3) 4 1642 p rii22 2/ boundary condition h(t = 0) = Ae to obtain 6 fie rii2 3 theZ~Mrn + 4 167r2 (In w — +— p2 0 4 16;r2 -s\ h(t) = et(t) (A6) + 1 64 on 4 1670 p2 — 3) 2/ 2 114± (in 4± 4 1672 A2 3) 2 where = — f cf, 7(9)/(1 + 7(e)) de and we seek to cal- culate the effective potential as a function of h,.. The beta _ 2 fit°cut ( In 14,,, 3\ I L L er & 3 functions are independent of 4f • but the anomalous dimen- 4 16r2 k p2 2) 416x2 tin p2 5) sion is gauge-variant since the Higgs field is a gauge-variant operator. Finally, the renormalization scale runs according to where we have neglected the light fermions. We also neglect = dµ/dt, which may be solved along with 11(t = 0) = po the contribution from the Higgs mass term. During inflation, to obtain Q(t) = poet. the potential is very flat and this contribution is subdominant. We solve the one-loop beta functions using the Mathe- The remaining SM fields, the massless photon and gluons, do matica code made publicly available by Fedor Bezrukov at not enter the effective potential at the one-loop order. The http: //www. lox .ac ru/ - fedor/SH/ . The code im- effective masses are plements the matching at the electroweak scale to determine raz 44, 21 h2 the couplings, c,,o. at the wale po = Mt in terms of the phys- Top Quark rh2 = art.; h2 ical masses and parameters. The code was extended (I) by W-Bosons generalizing the anomalous dimension to the RE gauge as in IV in , 2 9+ Z-Bosons ntz— 4ni h2 Eq. (A5). and (2) by including the field-dependent factors of 2 I-O 2in J. Higgs Boson mu ?ET 3A/1 ‘ 224.642011,A Neutral Goldstone to A h2 G = riT rI4 1+ 4(0,4102 r p (A7) 1+ (1+64.(0)4(02422 7h2 h2 +11 Charged Goldstones ew Ghosts - 9 nt ?TT = eaffiti2 Ghosts Mew = egantv in the two-loop beta functions, as indicated by 1131. The factor (A4) of s arises because of the non-canonical Higgs kinetic term. EFTA_R1_02123544 EFTA02710836  6 and it appears in the commutator of the Higgs field with its function of the field variable, he. This can be seen by writing conjugate momentum [9]. Finally the RG-improved effective potential is evaluated 1 [Dr(t.)2e2iltah2 ] 1 [figh2 as in Eq. (Al) after making the replacements A —) A(t.). t. = — In — In • (A9) 2 2µ0 2110 g -> §(t.), Mt„), p —> µ(t,), and so on. The RG flow parameter, ts, is chosen to minimize the would-be large logarithm arising from the top quark. This is accomplished by Using Eq. (A8 , the commutator factor in Eq. (A7) is written solving as ut(021402 I (A8) 8 — El + 12tit? Mt? I (A10) 20 + ttlihtio2 let. 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