EFTA02710832

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 The theory of inflation [I-3) successfully addressed the twentieth century's greatest puzzles of theoretical cosmology. Over the past 20 years. increasingly precise measurements of the temperature fluctuations of the cosmic microwave back- ground radiation (CMB) also confirmed the nearly scale in- variant power spectrum of scalar perturbations. a relatively generic inflationary prediction. These many successes, how- ever, underscored the inability to probe perhaps the most ro- bust and unambiguous prediction of inflation, the generation of a background of gravity waves associated with what are likely enormous energy densities concomitant with inflation (e.g.. Recently. the BICEP2 collaboration reported evidence of B-modes in the polarization pattern of the CMB [5]. The B- modes result from primordial gravity wave induced distortions at the surface of last scattering. If one assumes that these grav- ity waves are of an inflationary origin, then the BICEP2 mea- surement corresponds to an energy scale of inflation: Vitilif4 Pt (2 ± 0.2) x 1016 GeV (I) for a reported tensor-to-scalar ratio of r 0.2-1,:fg (using also the Planck collaboration's measurement of the amplitude of the scalar power spectrum 161). Such a high scale of in- flation rules out many compelling models. For the purposes of this paper, we will assume that the observation r re- 0.2 is valid', and we assess the impact of this measurement on a particular model of inflation, known as Higgs Inflation. • Electronic address: [email protected] 1FJectronic address: [email protected] I Note that the BICEP2 measurement is in tension with the upper bound. r C 0.11 at 95% C.L.. obtained previously by the Planck collaboration Higgs Inflation (HI) postulates that the Standard Model Higgs field and the inflaton are one in the same PI. (See also Ref. [8J for a recent review). This powerful assumption allows HI to be, in principle much more predictive than many other models of inflation, as by measuring the masses of the Higgs boson and the top quark at the electroweak scale (100 GeV), one might predict observables at much larger energy scales associated with inflation (Viiii( 4 < 1016 GeV). However, in practice this enhanced predictive power is elu- sive due to a strong sensitivity to quantum effects, unknown physics, and other technical subtleties in the model. Specifi- cally, one connects observables at the electroweak and infla- tionary scales using the renormalization group flow (RG) of the SM couplings [9-1,1]. It is reasonable however to expect that there is new physics at intermediate scales, and even if the SM is extended only minimally to include a dark matter can- didate [151 or neutrino masses il6-191 this new physics can qualitatively affect the connection between electroweak and inflationary observables. Moreover, penurbative unitarity ar- guments require new physics just above the scale of inflation [20, 211, and in addition the unknown coefficients of dimen- sion six operators can significantly limit the predictive power of HI [22]. The HI calculation also runs into various techni- cal subtleties that arise from the requisite non-minimal grav- itational coupling (see below) and quantization in a curved spacetime [23-25]. Finally, it is worth noting that HI is also at tension with the measured Higgs boson and top quark masses. and an O(2a) heavier Higgs or lighter top is required to evade vacuum stability problems [26]. 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- servables, one should not work with the classical (tree-level) scalar potential, as is done in may models of inflation, but 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 performed [27, 28), and care must be taken to extract gauge- invariant observables from it [29-32] (see also [33, 34]). This fact can perhaps be understood most directly by recalling that the effective action is the generating functional for one- particle irreducible Green's functions, which themselves arc gauge dependent [28]. In practice one often neglects this sub- tlety, fixes the gauge at the start of the calculation, and cal- culates observables with the effective potential as if it were a classical potential. In the context of finite temperature phase transitions. it is known that when calculated naively in this way, the predictions for observables depend on the choice of gauge used I 34—all. Because of the extreme tension between HI models and the data, we assess here the degree to which this gauge uncertainty might affect the observables in Higgs Inflation. We find that the gauge ambiguity introduces uncer- tainties that are comparable to the variation of the physical parameters, i.e. the Higgs mass. As a result. this ambiguity alone cannot resuscitate moribund models. 2, GRAVITY WAVES FROM HIGGS INFLATION The Standard Model Higgs potential, V(h) = Ah4/4 with A = O(0.1), is too steeply sloped for successful inflation. The measurement of the Higgs boson mf.. fixes A 0.13, whereas A a 1 is required to produce the observed ampli- tude of density perturbations. In the HI model, slow roll is 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 coupling term from the Lagrangian by performing a confor- mal transformation, go,„(x) = 51-2(x)li,„(s) where D2 = 1 + 2e(}141/4/1, (2) is the conformal factor and My is the reduced Planck mass. By doing so. one passes from the Jordan to the Einstein frame. The scalar potential in the new frame becomes V(h) = 2 (3) 4 (1 + where we have written 4.147, = h2/ 2. At large field values. h > ii/p/VZ, the potential asymptotes to a constant Vo s AMP/4e2 (4) This is the appropriate regime for slow roll inflation. To evince the tension between Higgs Inflation and large ten- sor perturbations we can first neglect quantum corrections to V(h), e.g. the running of A, as the energy scale of HI, given by Eq. (4), is insensitive to the quantum corrections, whereas the slope is more sensitive. Since A is fixed by the measured Higgs mass, the scalar potential in Eq. (3) has only one free parameter: 4. It is well- known that to achieve sufficient e-foldings of inflation and the correct amplitude for the scalar power spectrum, one needs the non-minimal coupling to be much larger than unity. Specifi- cally one requires (see, e.g., Ref. [81) 4 st 47000v5 (5) which is e 17000 for A .c.,7 0.13. The energy scale of infla- tion is then predicted to be 14 :v. (0.79 x 1016 GeV)4 (6) leading to a tensor-to-scalar ratio, assume scalar density per- turbations fixed by CMB observations, r •co-- 0.0036. This is naively incompatible with the much larger BICEP2 measure- ment, see Eq. (I). Decrease in HI to attempt to match the newly measured value of %1,,r is not workable either, as set- ting e 2000 then produces too little power in scalar density perturbations. Fundamentally then, the problem in obtaining a large value 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- sity perturbations, which, in order to then match observations, constrain the magnitude of the potential, resulting in a small tensor contribution. The question then becomes whether variations in this canonical HI, due to quantum effects for example, will allow the SM Higgs boson to the be inflaton field while also accom- modating the large value of r. 3. SAVING HIGGS INFLATION? Since it is the non-minimal coupling, e, that flattens out the potential at high scales, one might consider whether there are other ways to flatten the potential, and so avoid the require- ment for large e values. One possibility proposed in this regard [I 3] involves fine tuning the Higgs and top masses such that the Higgs self- coupling runs very small at the scale of inflation, A 10-4. This allows for relatively small e — 90 and produces r > 0.15 that may be compatible with the BICEP measurement. It is impossible to entirely eliminate the need for the non-minimal coupling. However, as a caveat let us point out that this so- lution only exists if the theory is first quantized in the Jordan frame and then moved to the Einstein frame (so-called "pre- scription I"), and results differ if the operations are reversed ("prescription II"). The apparent disagreement is an artifact of quantizing all the fields except gravity, which results in a different definition of the Ricci scalar in the two frames. A full theory of quantum gravity would probably be required to resolve the problem consistently between frames. Thus, it is not clear if the small £ "prescription I" solution is artificial. If one goes outside of the Standard Model, then new physics can affect the running of the Higgs self-coupling or anomalous dimension, y. For example, one may hope that A or -) acquires a significant running at high scales so as to give a workable solution consistent with both the measured scalar and tensor power spectra. (See. e.g., [4 1). EFTA_R1_02123541 EFTA02710833 3 As a result, it appears that canonical HI with a non-minimal gravitational coupling as the only new physical input appears extremely difficult to reconcile with the new observation of a large tensor contribution from inflation. If would appear to be necessary to add new physics to eliminate the dependence on non-minimal coupling entirely and to give the Higgs ef- fective potential a shape compatible with observations. Such extension of HI tend to defeat the original purpose of the idea, namely its predictivity, and in any case most such modifica- tions that have been proposed [42-44) tend to retain the now undesirable feature of small r in any case. There are two options that might allow large r consistent with BICEP. One possibility involves tuning the Higgs poten- tial to form a second local minimum at large scales. i.e., a false vacuum similar to old inflation PSI. To avoid the problems of old inflation, a time dependent tunneling rate is introduced. While most mechanisms to achieve this, however, produce a small value of r (46), larger can be accommodated by adding a new scalar with a non-minimal coupling to gravity, such that the Higgs field sees a time dependent Planck mass NM A second possibility uses a non-canonical Galileon type kinetic term for the Higgs field. This model yields an r =-• 0.14 [48]. These tuned limits, variants, and extensions of the original HI model leave the door slightly open for the possibility of connecting the Higgs with the inflation field. However, with- out additional scalars or modification of the Higgs potential via some other mechanism beyond the Standard Model, the original scenario. i.e. Higgs Inflation with only a non-minimal coupling to gravity, does not appear to be compatible with the BICEP result. Before we nail the coffin shut on Higgs Inflation, however, there is one possible additional source of uncertainty that mer- its further investigation. As we describe below, when one goes beyond the tree level, there are gauge ambiguities involved in the calculation of effective potentials that need to be consid- ered when deriving constraints on parameters. 4. GAUGE DEPENDENCE AMBIGUITIES When working with a gauge theory, such as the Stan- dard Model electroweak sector, calculations typically involve spurious gauge dependence that cancels when physical ob- servable arc calculated. For example, in a spontaneously broken Yang-Mills theory one may work in the renormaliz- able class of gauges (RO upon augmenting the Lagrangian with a gauge fixing term Cgi = —G°Ga/2 where = (1/ 1. ,f)(0,A° - xs) where x, are the would-be Goldstone boson fields and F° = riv, with 711 the sym- metry generators and vj the symmetry-breaking vacuum ex- pectation value. (See. e.g., [33]). A corresponding Fadeev- Popov ghost term is also added. Physical or "on-shell" quan- tities, such as cross sections and decay rates, may be calcu- lated penurbatively, and any dependence on the gauge fix- ing parameter, 4f , cancels order-by-order. Unphysical or "off-shell" quantities, such as propagators or one-particle irre- ducible Green's functions, may harmlessly retain the spurious gauge dependence. The Coleman-Weinberg effective action f eff and effective potential Vof [49) have become standard tools in the study of vacuum structure, phase transitions, and inflation. The effective action is the generating functional of one-particle irreducible Green's functions, and therefore it is important to recognize that both "off and Kt f are off-shell quantities, which will carry spurious gauge dependence [28]. When ap- plying the effective potential to a problem. special care must be taken to extract gauge-invariant information. In particu- lar, the Nielsen identities express the gauge invariance of the effective potential at its stationary points, but derivatives of the effective potential are not generally gauge invariant [3 I]. This suggests that inflationary observables, e.g. ns, r, and dns/dlnk, naively extracted directly from the slow roll pa- rameters will acquire a spurious gauge dependence. Ideally one would like to determine the "correct" proce- dure for calculating physical quantities like us from a given model in such a way that the spurious gauge dependence is canceled. There have been significant efforts made in this di- rection (23. 24]. but a full gauge invariant formalism is yet to be developed. Here we will take a different approach that is more aligned with recent work on the gauge dependence of phase transition calculations [34, 38. 39). Specifically, we numerically perform the "naive" HI calculation using the R{ gauge effective potential and RG-improvement to assess the sensitivity of the inflationary observables to the spurious gauge dependence. We begin by reviewing the familiar Higgs Inflation calcu- lation. After moving from the Jordan to the Einstein frame, as described in Sec. 2. the resulting action contains a non- canonical kinetic term for the Higgs field. One cannot, in general, find a field redefinition that makes the kinetic term canonical globally [21, 50]. At this point, it is customary to move to the unitary 'Inge where the Higgs doublet is written as ch(r) = e2"1.1r 1r (0, h(x)/4 2.. Then the kinetic term for the radial Higgs excitation can be normalized by the field redefinition x(h) where 1 3 3/2 0-12/dh.)2 dx I dh - n2 + P (7) 2 az and now 112 = 1 + h2/3/1,. Having canonically normalized both the gravity and Higgs kinetic terms, the derivation of the effective potential proceeds along the standard lines. We calculate the RG improved. one- loop effective potential as described in the Appendix. After performing the RG improvement, the parameter A that appears in Eq. (3) should be understood at the running coupling eval- uated at the scale of inflation. Generally, A < 0.1 and its value depends upon the physical Higgs boson and top quark masses at the input scale. For the best fit observed values, 125 GeV and Mt xr. 173 GeV, the coupling runs neg- ative at h ot 101° —1012 GeV; this is the well-known vacuum stability problem of the Standard Model [26). Successful HI requires an O(2o) deviation from central values toward either larger Higgs boson mass or smaller top quark mass. Gauge dependence enters the calculation at two places: ex- plicitly in the one-loop correction to the effective potential and EFTA_R1_02123542 EFTA02710834 4 implicitly through the Higgs anomalous dimension upon per- forming the RG improvement. To calculate the slow roll parameters. e.g. e = Af (W/V)21 2 ki-v (8) the derivatives are taken with respect to x, i.e., V'(h(x)) = (OVI0h)(dx1dh)-1. The potential and its derivatives are evaluated at the field value, heath, for which the number of e-foldings, given by = i 111V(h)h hna dh (h)fl, ' (9) is Ai = 60. Inflation terminates at h = he„d where (11/1,/2)(VVV)2 = 1. In Fig. I we show the energy scale of inflation. liar= V(hrran) (10) as the the Higgs boson and top quark masses are varied, and the non-minimal coupling, t ar, few x 103, is determined to match the observed amplitude of scalar perturbations. This demonstrates that the scale of inflation is insensitive to Atif, varying only at the O(10-4) level. It always remains signif- icantly below 2 x 1016 GeV, which indicates the incompat- ibility with the BICEP2 measurement. (The corresponding tensor-to-scalar ratio is r r-- 0.003.) To illustrate the gauge dependence, we show in Fig. 2 how Via varies with $4e. We find that V;,,( also changes at a level comparable to its sensitivity to Ahr or Me as the gauge pa- rameter deviates from the Landau gauge (‘ = 0). It is there- fore important to consider this ambiguity for model building purposes. Nevertheless, the absolute change in Vim is far too 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- sue. the unphysical degrees of freedom. the Goldstone bosons and ghosts. should be decoupled as the unitary gauge is ap- proached. Our numerical results appear consistent with the Nielsen identities [31, 32] which capture the gauge dependence of the effective potential. The relevant identity is [ a- 4 C(°' )60 0 0 e — rr(4, = 0 • 1 (II) In the slow roll regime, the gradient of the effective potential is small, and the gauge dependence is proportionally suppressed. We note that a rigorous gauge invariant calculation could perhaps take Eq. (11) as a starting point. This might be an interesting avenue for future work, either in the context of HI or other, potentially more viable models of inflation that are embedded in gauge theories. 5. CONCLUSION The recent detection of B-modes by the BICEP2 collabo- ration represents a profound and exciting leap forward in our 0.77673 I. 0.77672 3 — 0.77671 .. 0.77670 I 0.77669 0.77668 — 0.77667 123 124 125 126 127 128 Higgs Mass: µi t GeV FIG. I: The predicted energy scale of inflation. Vm1/11, over a range of Higgs boson masses (Al,,). for three values of the top quark mass (Aft). and in the Landau gauge. Co 0. 0.77680 0.77675 0.77670 > 0.77665 I 26 0. 77660 124 ( a .2 0.77655 0 77650 Gauge Parameter: fq FIG. 2: The energy scale of inflation. Via. as the gauge parameter. egr. varies. We fix Aft = 170 GeV and show three values of Me. ability to explore fundamental physics and the early universe. If the measurement of r ra 0.2 is confirmed, then it is rea- sonable to expect that, in the not-too-distant future, measure- ments of the spectrum of primordial tensor perturbations will become possible. allowing further tests of inflation. And if the measured r can unambiguously be shown to be due to in- flation. then this also substantiates the quantization of gravity [5 1 ]. Thus, future observations will provide significant con- straints on particle physics and models of inflation. However the simple observation of non-zero r already signals the death knell for low-scale models of inflation. This includes the class of models captured by the potential in Eq. (3), and among these apparently Higgs Inflation. We have shown that r re 0.2 essentially excludes canonical Higgs Inflation in the absence of extreme fine tuning. The Higgs field may live on as the 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 that the energy scale of inflation acquires an artificial depen- dence on the gauge fixing parameter by virtue of the gauge de- pendence of the effective potential from it is extracted. How- 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 uncertainties. which are themselves small. While this may be important for model building purposes, it does not affect the robustness of the fact that larger disfavors Higgs Inflation. Acknowledgments This work was supported by ANU and by the US DOE un- dcr Grant No. DE-SC0008016. We would like to thank layden L. Newstead for help with the code. Appendix A: Standard Model Effective Potential The Standard Model effective potential is calculated (i) to the one-loop order. (ii) working in the MS renormalization scheme with renormalization scale p, and (iii) in the tenor- malizable class of gauges (Re) as follows: Vergh ) = 0 °)(h) + 1,(1)(h) . (Al) The tree-level potential is V(o)(h) = 4 —h4 , (A2) and we can neglect the O(h°) and O(h2) terms for the pur- poses of studying HI where the field value is large. The one- loop correction is [55] (sec also [34] for gauge dependent fac- tors) I (l)(h) = 4 1642 In — — 12 fit° ( rii22 2/ p 6 fie 2 e + (Inrii w — + 3 — th Z~Mrn -s\ 4 167r2 p2 0 4 16;r2 + 1 64 4 1670 on — 3) 2 2 2 72 16 114± (in 4± 3) p2 2/ 4 A 2 fit° _ cut 4 16r2 (A3) ( In 14,,, 3\ I L er L & 3 k p2 2) 416x2 tin p2 5) where we have neglected the light fermions. We also neglect the contribution from the Higgs mass term. During inflation, the potential is very flat and this contribution is subdominant. The remaining SM fields, the massless photon and gluons, do not enter the effective potential at the one-loop order. The effective masses are Top Quark W-Bosons Z-Bosons Higgs Boson Neutral Goldstone Charged Goldstones Ghosts Ghosts ra z 44,1 2 h2 rh2 = art.; h2 IV in , 2 ntz— 9+ 4ni h2 2 I-O 2in J. u 3A/ m ?ET 1 ‘ 224.642011,A to A h2 rI4 G = riT 7h2 h2 +11 - 9 ?TT nt = eaffiti2 ew Mew = egantv (A4) where 1/2 = 1 + Ch2/1111, was given by Eq. (2). We denote the gauge fixing parameter by Car to distinguish it from the non-minimal gravitational coupling parameter, 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 functions of the RG flow parameter 1, (2) replacing the vari- ous coupling constants in Kg with the corresponding running parameter. and (3) evaluating the RG flow parameter at the appropriate value t = 1. so as to minimize the would-be large logarithms. For the sake of discussion, let us denote the running parameters collectively as 4(t) {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 with the boundary condition M' t = 0) = co. Here 7 is the anomalous dimension of the Higgs field. We neglect the running of the gauge-fixing parameter. 4r, since it is self-renormalized. This approximation is reasonable since we focus on 4-1- < 4r; for larger values of est-, perturba- tivity becomes an issue. The Higgs field runs according to —7/1 = dh/dt where the anomalous dimension -y(t) is given as [571 — 1 [ 9 ( 1 ) 92 ) 92 + 3Y21 (4ir)2 4 3 2 4 3 it271 — 34 _ f 8Csf) a2 (4w(4w)° Lk 32 9r2 16" 2 — 431? _5 I 992 + 17_2 + 8_21_2 + 27e ] (AS) 96 2 4 2 I2 YI This last equation may be solved immediately along with the boundary condition h(t = 0) = Ae to obtain h(t) = et(t) (A6) 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 functions are independent of 4f • but the anomalous dimen- sion is gauge-variant since the Higgs field is a gauge-variant operator. Finally, the renormalization scale runs according to = dµ/dt, which may be solved along with 11(t = 0) = po to obtain Q(t) = poet. We solve the one-loop beta functions using the Mathe- matica code made publicly available by Fedor Bezrukov at http: //www. lox .ac ru/ - fedor/SH/ . The code im- plements the matching at the electroweak scale to determine the couplings, c,,o. at the wale po = Mt in terms of the phys- ical masses and parameters. The code was extended (I) by generalizing the anomalous dimension to the RE gauge as in Eq. (A5). and (2) by including the field-dependent factors of 1+ 4(0,4102 rp 1 + (1 +64.(0)4(02422 (A7) in the two-loop beta functions, as indicated by 1131. The factor 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 conjugate momentum [9]. Finally the RG-improved effective potential is evaluated as in Eq. (Al) after making the replacements A —) A(t.). g -> §(t.), Mt„), p —> µ(t,), and so on. 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