Hierarchy problem and dimension-six effective operators
HHierarchy problem and dimension-6 effective operators
Ambalika Biswas a ∗ , Anirban Kundu b † , and Poulami Mondal b ‡ a Department of Physics, Vivekananda College,269, Diamond Harbour Road, Thakurpukur, Kolkata 700063, India b Department of Physics, University of Calcutta,92 Acharya Prafulla Chandra Road, Kolkata 700009, IndiaJune 25, 2020
Abstract
Without any mechanism to protect its mass, the self-energy of the Higgs boson diverges quadrat-ically, leading to the hierarchy or fine-tuning problem. One bottom-up solution is to postulate someyet-to-be-discovered symmetry which forces the sum of the quadratic divergences to be zero, or al-most negligible; this is known as the Veltman Condition. Even if one assumes the existence of somenew physics at a high scale, the fine-tuning problem is not eradicated, although it is softer thanwhat it would have been with a Planck scale momentum cut-off. We study such divergences in aneffective theory framework, and construct the Veltman Condition with dimension-6 operators. Weshow that there are two classes of diagrams, the one-loop and the two-loop ones, that contribute toquadratic divergences, but the contribution of the latter is suppressed by a loop factor of 1 / π .The Wilson coefficients of these higher-dimensional operators that contribute to the former classplay an important role towards softening the fine-tuning problem. We find the parameter space forthe Wilson coefficients that satisfies the extended Veltman Condition, and also discuss why one neednot bother about the d > The title of this paper may appear to be an oxymoron for two reasons. First, effective theories areknown to be valid up to a certain energy scale, so why should one talk about the hierarchy problem,which essentially is a manifestation of extreme weakness of gravity, or the extremely high value of thePlanck scale ∼ GeV? Second, any calculation of the scalar self-energy involves the evaluation ofloop contributions to the self-energy, and how may one evaluate a loop in an effective theory withhigher-dimensional operators?Both these problems can be easily surmounted (see, e.g. , Ref. [1]). In a sense, it is subjective,depending on what level of fine-tuning one is comfortable with. If the cutoff scale of the theory be Λand the Higgs vacuum expectation value (VEV) be v , the typical fine-tuning is of the order of v / Λ .One can, however, be more quantitative. One may write the physical Higgs mass, m h , in terms of abare mass term m h, and higher-order self-energy corrections: m h = m h, + δm h , (1)where δm h is some function of the masses, v , and Λ. In this case, m h / | δm h | may be taken as anapproximate measure of fine-tuning. For Λ = 2 TeV, just outside the reach of the Large Hadron Collider(LHC), this is about one or a few per cent, not at all uncomfortable, but higher values of Λ definitelybrings back the fine-tuning problem, maybe in a softened way. Also, if one has a renormalisable theory ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] J un elow Λ, loop calculations do not pose any problem, with the understanding of a momentum cut-offat Λ. Cut-off regularisation is not Lorentz invariant, but it is undoubtedly the best way to feel thebadness of a divergence.Very briefly, the hierarchy of fine-tuning problem is why the Higgs mass is at the electroweakscale and not at the Planck scale, when it is not protected by any symmetry. If we use a cut-offregularisation, the Higgs self-energy diverges as Λ while the fermion and gauge boson masses divergeonly logarithmically. Thus, to get a Higgs mass of the order of v from a quantum correction of theorder of Λ, one needs a fine-tuning between the bare mass term and the quantum corrections.In this paper, we will not talk about any possible ultraviolet complete (UVC) theory, like super-symmetry, that may solve the hierarchy problem. We will, rather, demand that perhaps due to someyet-to-be-discovered symmetry, the quadratically divergent contributions to the Higgs mass add up tozero, or a very small value. This is known as the Veltman Condition (VC) [2].Thus, if we confine ourselves to one-loop diagrams only, the renormalised Higgs mass squared isgiven by m h = m h, + δm h = m h, + 116 π f ( g i )Λ + · · · , (2)where the ubiquitous coefficient of 1 / π comes from the evaluation of the loop, and f ( g i ) is a functionof relevant scalar, Yukawa, and gauge couplings. The VC demands that f ( g i ) should be zero, orextremely tiny, so that m h, is not too much away from the electroweak scale. The logarithmicallydivergent as well as the finite terms coming from the loop diagrams have been neglected, and denotedby the trailing ellipses.One may argue that f ( g i ) need not be exactly zero; in fact, f ( g i ) ∼ π m h / Λ should be perfectlyacceptable. However, with all the masses known, the VC fails badly for the SM [3, 4]. There arenumerous attempts in the literature to make f ( g i ) ≈ v to Λ if one considers the renormalisation group (RG) evolution of the couplings. This remains one ofthe major shortcomings of the bottom-up approach.We will take the bottom-up approach to its extreme limit. For us, whatever New Physics (NP)exists there at the high energy scale can be effectively integrated out at the scale Λ to give us the SM,plus some effective operators involving only the SM fields, which is known as the SM Effective FieldTheory (SMEFT). We will not venture to investigate the possible nature of the UVC theory; rather,all the UVC information will be clubbed in the Wilson coefficients (WC) of the effective operators.In SMEFT, the first interesting higher dimensional operators come at d = 6 (the d = 5 Weinbergoperators is not relevant for scalar self-energies). There are many equivalent bases to express thecomplete set of d = 6 operators. We will use the basis given in Ref. [12]. Only a handful among the 59dimension-6 operators contribute to the quadratically divergent part of the scalar self-energy.An n -dimensional operator can at most result in a divergence in Higgs self-energy that goes as Λ n − .As these operators are suppressed by Λ n − , one expects contributions to f ( g i ) from all orders. What,then, is the rationale to consider only d = 4 and d = 6 operators? We have tried to answer this questionin Section 2.Thus, we will focus only on an effective theory with a schematic Lagrangian L = c i O d =4 i + 1Λ c i O d =6 i , (3)2here c i and c i are dimensionless constants. The VC now takes the form F ( c i , c i ) ≈ . (4)Our aim will be to find out the parameter space for the c i coefficients.In Section 2, we discuss why it is enough to take into account only the dimension-6 operators. InSections 3 and 4, we discuss the VC in the SM (with dimension-4 operators) and in SMEFT withdimension-6 operators. In Section 5, we show the allowed parameter space for the c i coefficients anddiscuss our results. Section 6 concludes the paper. d = 8 operators The d = 6 SMEFT has been well-explored, and there are several equivalent bases to express all the d = 6 operators. While the d = 8 operators are not that well-investigated, it is known [13] that thereare 993 such operators with one generation and 44807 operators with three generations. A list of therelevant bosonic operators can be found in Ref. [14].For d = 6 operators, there are two types of diagrams that come with a Λ divergence. Firstare the two-loop diagrams, like the one from (cid:0) Φ † Φ (cid:1) , where Φ is the SM doublet Higgs field. Thesecond class consists of one-loop diagrams but momentum-dependent vertices, like the one coming from( D µ Φ) † ( D µ Φ) Φ † Φ. If the derivatives act on the internal scalar lines, the vertex has a momentum-dependence ∼ k , where k is the loop momentum to be integrated over, and the resulting divergence isagain quartic. However, there is a crucial difference: the first set comes with (16 π ) − , and the secondset only with (16 π ) − , similar to the d = 4 operators. Therefore, it is the one-loop diagrams thatshould be the most relevant in calculating the VC. One can have a similar conclusion with operatorsinvolving the gauge field tensors.Thus, among the d = 8 operators, one should look only for those operators that come with fourderivatives, i.e. , D . There are only three such operators [14], and all of them have a generic structureof ( D Φ) † ( D Φ)( D Φ) † ( D Φ). As two of the derivatives act on the external leg fields and hence givethe square of the external leg momentum, the vertex factor can only have a k dependence, and thedivergence remains only Λ and not Λ . Similarly, operators of the form D (Φ W ), where W is thegeneric gauge tensor, do not generate any Λ divergence. Thus, it is enough, within the limits ofuncertainty, to consider only the d = 6 operators. However, one may note that the argument is notwatertight; the huge number of d = 8 operators may offset the extra suppression coming from 1 / π . We start from the SM Higgs potential with only d ≤ V (Φ) = − µ Φ † Φ + λ (Φ † Φ) (5)where Φ is the SM doublet, with (cid:104) Φ (cid:105) = v/ √
2. The Higgs self-energy receives a quadratically divergentcorrection δm h = Λ π (cid:18) λ + 34 g + 94 g − g t (cid:19) , (6)3here g and g are the U (1) Y and SU (2) L gauge couplings, and g t = √ m t /v is the top quark Yukawacoupling. All other fermions are treated as massless. Dimensional regularisation does not differentiatebetween quadratic and logarithmic divergences, and we get a slightly different correction [4]: δm h ∝ (cid:15) (cid:18) λ + 14 g + 34 g − g t (cid:19) . (7)As our goal is to cancel the strongest divergence, we will use the cut-off regularisation. Two- andhigher-loop diagrams can also contribute to the quadratic divergence, but they are suppressed from theone-loop contributions by a factor of ln(Λ /µ ) / π or more, where µ is the regularisation scale. Wewill, therefore, not consider anything beyond one loop.At this point, let us make some comments on the gauge dependence of the VC. They are gauge-independent, as can be explicitly checked by working out the quadratic divergences in Landau and’t Hooft-Feynman gauge. However, one may ask what happens in the unitary gauge, as the gaugepropagator has a leading momentum dependence of k . While one may question the justification touse the unitary gauge as the VC is relevant only for Λ (cid:29) v where the electroweak symmetry is stillunbroken, all particles are massless, and the condition is formulated in terms of the couplings only, itwould nevertheless be satisfactory to see that nothing catastrophic happens in the unitary gauge. Forexample, one may think of having a quartic divergence, ∼ Λ , coming from the gauge loop, as the gaugepropagator is not momentum suppressed. At the same time, one has to remember that the electroweaksymmetry is broken, and there are generic Higgs-gauge-gauge vertices in the theory. One can have aself-energy contribution with two such vertices, which again is quartically divergent. We have explicitlychecked that these quartic divergences cancel; for the W -loop, the amplitude with the four-point vertexgives g Λ / (128 π m W ), which is exactly cancelled by the amplitude with two three-point vertices. Thelatter also gives a quadratic divergence, which is needed to restore the gauge invariance.One can say that the quadratic divergence is under control if | δm h | ≤ m h , which translates into (cid:12)(cid:12) m h + 2 m W + m Z − m t (cid:12)(cid:12) ≤ π v Λ m h . (8)This inequality is clearly not satisfied in the SM for v / Λ ≤ .
1, or Λ ≥
760 GeV, and onset of NP atsuch a low scale is already ruled out by the LHC. Thus, one needs extra degrees of freedom, like morescalars or fermions. There are a number of such studies in the literature; we refer the reader to, e.g. ,Refs. [5, 8–11].
We will use the SMEFT basis as in Ref. [12]. Keeping in mind that only operators with two or moreHiggs fields are relevant and the divergence should be quartic, the relevant operators are as follows: O W W = Φ † (cid:100) W µν (cid:91) W µν Φ , O BB = Φ † (cid:100) B µν (cid:100) B µν Φ , O GG = Φ † Φ (cid:100) G µν (cid:100) G µν ,O W = ( D µ Φ) † (cid:91) W µν ( D ν Φ) , O B = ( D µ Φ) † (cid:100) B µν ( D ν Φ) , O φ, = ( D µ Φ) † Φ Φ † ( D µ Φ) ,O φ, = 12 ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) , O φ, = 13 (Φ † Φ) , O φ, = ( D µ Φ) † ( D µ Φ) Φ † Φ , (9) The VC can be expressed in terms of the masses only after the electroweak symmetry is broken. (cid:100) B µν = ig (cid:48) B µν , (cid:100) W µν = ig σ a W aµν , (cid:100) G µν = ig s λ a G aµν , (10) g , g (cid:48) being the SU (2) L and U (1) Y gauge couplings respectively, and λ a , σ a are the Gell-Mann andPauli matrices. Note that the mixed gauge operator O BW = Φ † (cid:100) B µν (cid:91) W µν Φ cannot generate a self-energy amplitude, either at one- or at two-loop.With these set of nine operators, we can write the dimension-6 part of Eq. (3) as L = 1Λ (cid:88) i =1 c i O i , (11)and Eq. (6) takes the form δm h = Λ π (cid:18) λ + 34 g + 94 g − g t (cid:19) + Λ π (cid:88) i f i + Λ (16 π ) (cid:88) i g i , (12)where f i and g i terms come respectively from the one-loop and two-loop quartic divergences with theinsertion of the operator O i , any of the nine dimension-6 operators listed above. The contributions aregiven by f φ, = − c φ, , g φ, = − (cid:16) g (cid:48) + 3 g (cid:17) c φ, ,f φ, = − c φ, , g φ, = 0 ,f φ, = 0 , g φ, = 18 c φ, ,f φ, = − c φ, , g φ, = − (cid:16) g (cid:48) + 3 g (cid:17) c φ, ,f W W = − g c W W , g
W W = − g c W W ,f BB = − g (cid:48) c BB , g BB = 0 ,f GG = − g s c GG , g GG = − g s c GG ,f W = − g c W , g W = − g c W ,f B = − g (cid:48) c B , g B = 0 . (13)Eqs. (12) and (13) are the central results of this paper. The extra 1 / π suppression tells us thatwe may neglect the g i terms (and thus will be justified to neglect the dimension-8 and other higher-dimensional operators), unless we deal with pathological cases like (cid:80) f i ≈
0, or all the WCs being zeroexcept c φ, .With only the f i terms, the modified VC reads (cid:18) λ + 34 g + 94 g − g t (cid:19) + (cid:88) i f i ≈ , (14)which immediately tells us that at least one, or perhaps more, WCs should be negative. As the operatorsdo not contain strongly interacting fields (except O GG ), the running between Λ, which is the matchingscale, and the electroweak scale, are controlled by electroweak radiative corrections only (at the leadingorder).Before we go into the next Section, let us enlist once more a couple of important points.5 If we assume the WCs at the matching scale are of order unity (which is expected if the UVC isperturbative in nature), we can safely neglect the two-loop g i terms, as they are suppressed byat least two orders of magnitude coming from 1 / π . This also ensures that we do not have toworry about d > • At what scale should the VC be satisfied? Obviously, it should be at the matching scale Λ. Eq.(8) shows that the cancellation need not be exact, it should be of the order of v / Λ . Thus, itis meaningless to talk about the fine-tuning problem if Λ = 1 TeV, and anyway we already knowthat there is no new physics (at least strongly interacting) at that scale, thanks to LHC. Λ = 100TeV makes the fine-tuning problem come back in a softened avatar, so this should be the correctballpark to study the issue. Even higher values, like Λ = 10 TeV, makes the fine-tuning problemseriously uncomfortable.
Following what has been said just now, we will study the VC for two values of Λ, namely, 100 TeVand 10 TeV. To start with, let us assume that only one of the eight SMEFT operators (neglecting O φ, which does not contribute to f i ) is present at the matching scale. We also need to evolve the SMcouplings to that scale, for which we use the package SARAH v4.14.1 [16], with two-loop RG equations.Taking Λ = 100 TeV, one gets, for exact cancellation of the quadratic divergence,100 TeV | c φ, = c φ, = 2 c φ, = − . , c BB = c B = − . , c W W = c W = − . , c GG = − . , (15)and for Λ = 10 TeV10 TeV | c φ, = c φ, = 2 c φ, = − . , c BB = c B = − . , c W W = c W = − . , c GG = − . . (16)Of course, one may relax these numbers a bit if exact cancellation is not warranted. Note the largevalues for the weak gauge WCs, they stem from the definition of the corresponding f i s in Eq. (13)which contain g or g (cid:48) ; the UVC need not be non-perturbative. On the other hand, if we take Λ = 2TeV only, the corresponding exact-cancellation values are2 TeV | c φ, = c φ, = 2 c φ, = − . , c BB = c B = − . , c W W = c W = − . , c GG = − . . (17)This change is entirely due to the running of the SM couplings.However, there is hardly any UVC theory that generates only one of these eight operators at thematching scale. As the sign of the WCs can be either positive or negative, the eight free parameters donot even give a closed hypersurface in the 8-dimensional plot, and therefore marginalisation is of verylimited use. We thus show in Fig. 1, two distinct cases where only a pair of WCs are nonzero at Λ. Forthe left panel of Fig. 1, we take c φ, , c φ, (cid:54) = 0, while for the right panel, c W W and c BB are taken to benonzero (an identical plot is obtained for c W versus c B ).The narrow lines, as shown in the plot, are obtained with the demand of an exact cancellation.They broaden out to bands if we allow a finite amount of fine-tuning, the bands getting narrower forhigher values of Λ.The SMEFT operators contribute to anomalous trilinear and quartic gauge-gauge and gauge-Higgscouplings, as well as modified wavefunction renormalisation for the bosonic fields. It is indeed hearteningto note that the parameter space that we obtain is consistent with all other theoretical and experimental6igure 1: The parameter space for c φ, and c φ, (left), and c W W and c BB (right) that is needed for anexact cancellation of the quadratic divergence at the scale Λ. The red solid line is for Λ = 100 TeV,while the blue dashed line is for Λ = 10 TeV. For the left plot, the two lines almost coincide.constraints [17]. For other collider signatures of these d = 6 operators, like vector boson scattering andHiggs pair production at the LHC, we refer the reader to, e.g. , Refs. [18] and [19]. In this paper, we have discussed the Veltman condition leading to the cancellation of the quadraticdivergence of the Higgs self-energy in the context of an SMEFT framework. In other words, we assumethe existence of a cut-off scale Λ, below which we have the SM, while the theory above Λ introduceshigher-dimensional operators in the low-energy domain. If Λ is large enough, the low-energy theory isstill plagued by the ∼ Λ divergence, even if it is not as uncomfortable as what one gets with a desertup to the Planck scale.We show that the higher dimensional operators lead to quadratic divergences too, but there aretwo distinct sources of them. For example, with d = 6 operators, such divergences can come from one-loop diagrams with momentum-dependent vertices, or two-loop diagrams with momentum-independentvertices. The latter, however, are suppressed by an extra loop factor of 1 / π and hence can beneglected as a first approximation. The same logic leads to the important point that only d = 6operators are relevant for such one-loop quadratic divergences. (There is a caveat, though: the numberof relevant effective operators increases almost exponentially with d , and the loop suppression may justbe compensated by the large number of such amplitudes.)We find that there are only eight operators that contribute to the Veltman condition. It turns outthat at least one of the WCs has to be negative, but they are all consistent with a high-scale perturba-tive theory. The parameter space that we find is compatible with other theoretical and experimentalconstraints. Thus, this study should set a benchmark for the model builders. 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