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CERN-PH-TH-2017-001 CP3-Origins-2017-001 IFUP-TH/2017 Naturalness of asymptotically safe Higgs Giulio Maria Pelaggia, Francesco Sanninob, Alessandro Strumiaa,c, Elena Vigiania a Dipartimento di Fisica dell’Università di Pisa and INFN, Italy b CP3-Origins and Danish IAS, University of Southern Denmark, Denmark 7 1 0 c CERN, Theory Division, Geneva, Switzerland 2 n a J Abstract 5 We introduce a model that contains a Higgs-like scalar with gauge, ] h p Yukawa and quartic interactions that enter a perturbative asymptot- - p ically safe regime at energies above a scale Λ. The model serves as a e h concrete example to test whether scalars masses unavoidably receive [ quantum correction of order Λ. The answer is that scalars can be 1 v naturally lighter. Although we do not have an answer to whether the 3 5 Standard Model hypercharge coupling growth towards a Landau pole 4 around Λ ∼ 1040GeV can be tamed by non-perturbative asymptotic 1 0 safety, our toy-SM shows that such a possibility is worth exploring. In . 1 0 fact, if successful, it might also offer an explanation for the unbearable 7 1 lightness of the Higgs. : v i X r a 1 Introduction √ The Large Hadron Collider (LHC) data at s = 13TeV confirm the Standard Model (SM) and give strong bounds on supersymmetry, on composite Higgs and on other SM extensions that were put forward to tame the quadratically divergent corrections to the Higgs mass in a natural way. The existence of natural solutions apparently ignored by nature challenges even anthropic approaches. This unsettling situation calls for reconsidering the issue of naturalness. The bulk of the physical corrections to the SM observables are only logarithmically sensi- tive to a potential UV physical scale because they stem from marginal operators. Furthermore, physical corrections to the Higgs mass are naturally small in the SM, and can remain naturally small once it is extended to account for dark matter, neutrino masses [1], gravity, and infla- tion [2]. So, the issue is whether power-divergent corrections imply that the Higgs cannot be naturally light. As well known, the Higgs propagator Π(q2) at zero momentum q = 0 receives a quadratically divergent correction, which is often interpreted as a large correction to the Higgs mass. Writing only the top Yukawa one-loop contribution, one has 1 (cid:90) d4k k2 +m2 Π(0) = −12y2 t +··· (1) t i (2π)4(k2 −m2)2 t The photon too receives at zero momentum a quadratically divergent correction. In QED one has 1 (cid:90) d4k (cid:20) 2k k η (cid:21) Π (0) = −4e2 µ ν − µν . (2) µν i (2π)4 (k2 −m2)2 k2 −m2 e e This is not interpreted as a large photon mass because it is presumed that some unknown physical cut-off regulates divergences while respecting gauge invariance, that forces the photon to be massless. Similarly, the graviton propagator receives a quadratically divergent correction Π (0): in part it can be interpreted as a cosmological constant, in part it breaks µν,ρσ reparametrization invariance. ThefateoftheHiggsmassisnotclear. Someregulators(suchasdimensionalregularization) respect all these symmetries and get rid of all power divergences, including the one that affects the Higgs mass. Other regulators (such as Pauli-Villars and presumably string theory) do not generate a photon mass nor a graviton mass and generate a large Higgs mass, given that it is only protected by scale invariance, which is not a symmetry of the full theory. One possibility is that the SM is (part of) a theory valid up to infinite energy, such that no physical cut-off exists. Then, once that eq. (2) is interpreted to mean zero, the same divergence in eq. (1) must be interpreted in the same way. Furthermore, in a theory with dimension-less (cid:82) parameters only, one can argue that d4k/k2 = 0 by dimensional analysis. Gravity itself could be described by small dimension-less parameters [2,3], such that it makes sense to extrapolate the SM RGE above the Planck scale. In this context, one possibility is devising realistic weak-scale extensions of the SM such that all gauge, Yukawa, and quartic couplings flow to zero at infinite energy [3]. However hypercharge must be embedded into a large non-abelian group, in order to be asymptotically 2 �� λ�� τ �� -�� -� �� |λ| ��-� �τ ��-� �� μ�� Figure 1: Illustration of a possible RGE running in the SM. We assumed central values for all parameters, and solved the 3 loop RGE equations. In order to obtain an asymptotically safe behaviour we artificially removed the bottom and tau Yukawa contributions to the 3-loop term in the RGE. This only affects the running in the non-perturbative region above 1040GeV, where the result cannot of course be trusted. Furthermore we ignored the Yukawa couplings of the lighter generations, and gravity. free: naturalness then demands new vectors at the weak scale, which have not been observed so far. The other possibility is that the SM itself might be asymptotically safe. The hypercharge gauge coupling g becomes non-perturbative at Λ ∼ 1040GeV, hitting a ‘Landau pole’. It is Y not known what it means. It might mean that the SM is not a complete theory and new physics is needed at lower energy. Otherwise g and other couplings might run up to constant non- Y perturbative values as illustrated in fig. 1, such that the SM enters into an asymptotically safe phase. The fate of the SM depends on non-perturbative effects which are presently unknown; see [4–9] for attempts to compute the non-perturbative region and for related ideas. Tavares, Schmaltz and Skiba [10] proposed an alleged no-go argument, according to which Landau makes Higgs obese: i.e. scalars generally receive a mass correction of the order of the would-be-Landau pole scale Λ. In the SM case, this would mean that, whatever happens at 1040GeV, the Higgs mass receives a contribution of order 1040GeV, so that an asymptotically safe Higgs (where asymptotic safety kicks above Λ) cannot be natural. Later, Litim and Sannino (LS) [11] presented the first four-dimensional example of a perturbative quantum field theory where all couplings that are small at low energy flow to a constant value at higher energy persisting up to infinite energy. This model involves a gauge group SU(N ) with large N , a neutral scalar S and vector-like charged fermions, with asymptotically c c safe Yukawa couplings and scalar quartics. The model realises Total Asymptotic Safety (TAS). Another equally relevant property of the model is that without the scalar it cannot be pertur- 3 batively safe [11,12]. Scalars are required to dynamically render the theory fundamental at all scales without invoking supersymmetry. In fact supersymmetry makes it harder to realise an asymptotically safe scenario [13,14] both perturbatively and non-perturbatively. Furthermore the LS model, on the line of physics, connects two fixed points, a non-interacting infrared free one (the theory at low energy is non-abelian QED-like) to an interacting ultraviolet fixed point. Remarkably the model shares the SM backbone since it features gauge, fermion and needed scalar degrees of freedom, albeit it still misses a gauged Higgs-like state. We therefore extend the LS model in section 2 to further feature a Higgs-like charged scalar H. In order to make the model even more similar to the SM we also try to add chiral fermions in section 3. However we could not yet find an asymptotically safe solution in all couplings with the bottle neck being the scalar self-couplings. Having at our disposal a calculable model simil in spirit to the SM, we carefully re-consider the naturalness issue in this class of theories in order to offer an answer to the question: Does the Higgs-like scalar H acquire a mass of the order of the scale Λ? In section 4 we do not find any such contribution, de facto, re-opening the issue. We offer our conclusions in section 5. 2 Asymptotically safe models with an Higgs-like scalar Litim and Sannino considered a model with gauge group SU(N ) and gauge coupling g; N c F vector-likefermionsψ ⊕ψ¯ inthefundamentalplusanti-fundamental,andN2 neutralscalarsS i i F ij ¯ with Yukawa couplings S ψ ψ . The number of flavours N can be fixed to make the one-loop ij i j F gaugebetafunctionβ(1) small. LargeN , N allowstomakeβ(1) arbitrarilysmall, guaranteeing g c F g perturbative control. The new key feature with respect to the analogous construction by Banks and Zaks [15] is that the Yukawa couplings can (non trivially) make the two-loop gauge beta function negative, such that, together with β(1) > 0, g enters into a perturbative fixed point g at large energy. Finally, one verifies that Yukawa couplings and scalar quartics too have a perturbative fixed point. The model satisfies Total Asymptotic Safety (TAS). In general the equations β = β = β = 0 have multiple solutions, that correspond to dif- g y λ ferent global symmetries of the theory. The analysis can be simplified focusing on the maximal global symmetry, U(N ) ⊗ U(N ) , which can be realized with complex scalars S. The field F L F R content is then summarized by the upper box of table 1. The lower box of table 1 shows the fields that we add: one Higgs-like scalar charged under the gauge group. Its introduction does not affect, in the limit of large N , N , the fixed point c F for y and g found in [11]. We also add singlet fermions N , N(cid:48) (see table 1 for the details) in i i order to allow H to have Yukawa couplings, like the SM Higgs. The allowed Yukawa couplings then are L = yS ψ ψ¯ +y(cid:48)S∗N N(cid:48) +y˜Hψ¯N +y˜(cid:48)H∗ψ N(cid:48) +h.c. (3) Y ij i j ij i j i i i i The scalar potential is V = λ (TrS†S)2 +λ Tr(S†SS†S)+λ (H†H)2 +λ (H†H)Tr(SS†), (4) S1 S2 H HS 4 Fields Gauge symmetries Global symmetries Spin SU(N ) U(N ) U(N ) c F L F R ψ 1/2 1 ψ¯ 1/2 1 S 0 1 H 0 1 1 N 1/2 1 1 N(cid:48) 1/2 1 1 Table 1: Field content of the model. The upper box is the original Litim-Sannino model. The lower box are the extra fields that we add in order to get a Higgs-like scalar H. and it is positive if (cid:112) λ +ηλ ≥ 0, λ > 0, λ +2 λ (λ +ηλ ) ≥ 0, (5) S1 S2 H HS H S1 S2 where η = Tr(S†SS†S)/Tr2(S†S) ranges between η = 1 and η = 1/N . The bounds in eq. (5) F need only to be imposed at the extremal values. 2.1 RGE and their fixed points Defining the β-functions coefficients as dX β(1) β(2) = X + X +... , (6) dlnµ (4π)2 (4π)4 the relevant RGE are (cid:18) (cid:19) 11 2N 1 β(1) = g3 − N + F + (7a) g 3 c 3 6 (cid:18)13N N N 34N2 4N 1 (cid:19) (cid:18) y˜2 +y˜(cid:48)2(cid:19) β(2) = g5 c F − F − c + c − −g3 N2y2 +N (7b) g 3 N 3 3 N F F 2 c c (cid:18) 3 (cid:19) y˜2 +y˜(cid:48)2 β(1) = y3(N +N )+g2y −3N +y +yy(cid:48)2 +2y(cid:48)y˜y˜(cid:48) (7c) y c F N c 2 c y˜2 +y˜(cid:48)2 β(1) = N y2y(cid:48) +y(cid:48)3(N +1)+y(cid:48)N +2N yy˜y˜(cid:48) (7d) y(cid:48) c F c 2 c y2 +y(cid:48)2 +2y˜(cid:48)2 (cid:18)N 1(cid:19) 31−N2 β(1) = N y˜ +y˜3 c +N + +g2y˜ c +2N yy(cid:48)y˜(cid:48) (7e) y˜ F 2 2 F 2 2 N F c y2 +y(cid:48)2 +2y˜2 (cid:18)N 1(cid:19) 31−N2 β(1) = N y˜(cid:48) +y˜(cid:48)3 c +N + +g2y˜(cid:48) c +2N yy(cid:48)y˜ (7f) y˜(cid:48) F 2 2 F 2 2 N F c β(1) = 4y2N λ +4y(cid:48)2λ +N λ2 +(cid:0)4N2 +16(cid:1)λ2 +16N λ λ +12λ2 (7g) λS1 c S1 S1 c HS F S1 F S1 S2 S2 β(1) = 4y2N λ −2y4N +8N λ2 +24λ λ −2y(cid:48)4 +4y(cid:48)2λ (7h) λS2 c S2 c F S2 S1 S2 S2 5 (cid:18) (cid:19) (cid:18) (cid:19) 3N 3 3 3 6 β(1) = g4 c − + + +g2 −6N λ +(4N +16)λ2 + λH 4 N 2N2 4 N c H c H c c c +4N λ (y˜2 +y˜(cid:48)2)+N2λ2 −2N (y˜4 +y˜(cid:48)4) (7i) F H F HS F (cid:18) (cid:19) 3 β(1) = 2(y2N +y(cid:48)2)λ +(y˜2 +y˜(cid:48)2)(cid:0)2N λ −4y2 −4y(cid:48)2(cid:1)+g2λ −3N + λHS c HS F HS HS N c c (cid:0)(cid:0) (cid:1) (cid:1) +(4N +4)λ λ +λ 4N2 +4 λ +8N λ +4λ2 −8yy(cid:48)y˜y˜(cid:48) (7j) c H HS HS F S1 F S2 HS Notice that yy(cid:48)y˜∗y˜(cid:48)∗ is left invariant by redefinitions of the phases of all fields, so the model admits one CP-violating phase. Nevertheless CP is conserved at all fixed points, so that the RGE can written in terms of real couplings. For simplicity we therefore assume all couplings to be real. The one-loop gauge beta function can be rewritten as 2N N 11 1 β(1) = g3 c(cid:15), where (cid:15) ≡ F − + (8) g 3 N 2 4N c c can be made arbitrarily small in the limit of large N ,N . In this limit β(1) reduces to c F y (cid:18) (cid:19) 13 β(1) Nc(cid:39)(cid:29)1 N y −3g2 + y2 (9) y c 2 and it vanishes for y2/g2 (cid:39) 6/13, which corresponds to a negative (cid:18) (cid:19) 25 363 β(2)(cid:39) g5N2 1− . (10) g 2 c 325 Thereby the gauge coupling has an IR-attractive fixed point g = 0 and a non-trivial UV- attractive fixed point at 26(4π)2 g2 = g2 (cid:39) (cid:15). (11) ∗ 57N c The scalar quartics λ ,λ admit two fixed points. At leading order in (cid:15): S1 S2 λ 3 (cid:18) √ (cid:113) √ (cid:19) 1 (cid:26) 0.348 − S1 (cid:39) −2 23± 20+6 23 ≈ − (12a) g2 143N N 0.055 + F F λ 3 (cid:16)√ (cid:17) S2 (cid:39) 23−1 ≈ 0.080. (12b) g2 143 The solution with the + (−) sign corresponds to a stable (unstable) potential V(S) as deter- mined in [16]. At the stable solution, the fixed point for both quartics, as well as the fixed point for y, are IR-attractive: this means that their low-energy values are univocally fixed, with respect to g, along the RGE trajectory that reaches infinite energy. So far the new fields that we added just acted as spectators. We must check that they have their own fixed points. By studying the full equations we find that the extra Yukawa couplings y(cid:48),y˜ and y˜(cid:48) have 3 inequivalent fixed points. The fixed points with y(cid:48) = y˜(cid:48) = 0 lead to fixed 6 y/g N λ /g2 λ /g2 y˜/g y(cid:48),y˜(cid:48)/g λ /g2 N λ /g2 V F S1 S2 H F HS 0.138 0 unstable 0 0 UV UV UV UV 1.362 0 unstable −0.348 0.080 IR UV UV IR √ 0.163 −0.076 unstable 1/ 26 0 UV UV (cid:113) IR UV 1.125 −0.301 unstable 6 IR UV 13IR 0.138 0 ≥ 0 0 0 UV IR UV UV 1.362 0 ≥ 0 −0.055 0.080 IR IR IR IR √ 0.163 0.301 ≥ 0 1/ 26 0 UV IR IR UV 1.125 0.076 ≥ 0 IR IR Table 2: Fixed points at leading order in (cid:15). The left panel of the table refers to the Litim- Sannino model; the right panel to the extra couplings. All fixed points have g = g , and the ∗UV extra trivial fixed point with g = 0 is ignored. The pedix denotes an UV-attractive fixed IR UV point; while denotes an IR-attractive fixed point, where the low-energy value of the coupling IR is fixed. The equivalent solutions with y˜, y˜(cid:48) exchanged are not showed. points for the quartics, as listed in table 2. The full potential V(S,H) is stable when V(S) is stable. All these couplings are perturbative for (cid:15) (cid:28) 1, in the sense that higher order corrections are suppressed by powers of (cid:15). An explicit solution to the RGE equations is obtained by assuming that all ratios y/g, λ/g2 run remaining constant up to corrections of relative order (cid:15). Then one obtains an RGE equation for g dg b g3 b g5 2N 19N2 (cid:15)(cid:39)→0 1 − 2 , b = c(cid:15), b = c . (13) dlnµ (4π)2 (4π)4 1 3 2 13 Its solution is µ (4π)2(cid:20) 1 1 (cid:18) (cid:18) 1 1 (cid:19)(cid:19)(cid:21) b ln = − + ln (4π)2b − , g2 = (4π)2 1. (14) Λ 2b g2 g2 1 g2 g2 ∗ b 1 ∗ ∗ 2 which can be used to define in an RGE-invariant way the transmutation scale Λ in terms of µ and of g(µ). In the limit where the second two-loop term is neglected, Λ becomes the Landau pole scale of one-loop RGE. Imposing the boundary condition g(µ ) = g the solution 0 0 becomes [11] g2 g2(µ) = ∗ (15) 1+W[(µ0/µ)2b21/b2(g∗2/g02 −1)eg∗2/g02−1] where W(z) is the Lambert function defined by z = WeW. The fixed point g = g is UV- ∗ attractive: this means that g can become smaller at low energy. Fig. 2 illustrates a typical RGE running. 7 �� -��  � λ� �� � ��λ�� λ�� λ�� -�� -�� (�π)-��μ/Λ Figure 2: Illustration of a possible RGE running with N = 10, (cid:15) = 0.01. c 3 Asymptotically safe models with chiral fermions? In order to build a TAS model that more closely resembles the SM, we try to add one or more families of chiral fermions. The minimal chiral anomaly-free family is made of either one ˜ anti-symmetric ψ and N −4 anti-fundamentals ψ or of one symmetric ψ and N +4 anti- A c S c ˜ fundamentals ψ of SU(N ). The two possibilities coincide in the limit of large N , where no c c other possibility exists: all higher representations of SU(N ) with 3 or more indeces contribute c to β(1) more than vectors at large N , such that β(1) becomes large and positive. Spinors of g c g SO(N ) have the same problem: β(1) cannot be small. c g A look at the relevant SU(N ) group-theorethical factors c representation R dimension d T in Tr(TaTb) = T δab C = d T /d R R R R G R R singlet 1 0 0 fundamental N 1/2 N /2−1/2N c c c (16) adjoint N2 −1 N N c c c anti-symmetric N (N −1)/2 (N −2)/2 N −1−2/N c c c c c symmetric N (N +1)/2 (N +2)/2 N +1−2/N c c c c c shows that adding minimal chiral fermions cannot be a correction which is sub-leading in the limit of large N with respect to the LS model. A non-trivial feature of this model is that the c Yukawa contribution makes β(2) = β(2)| +β(2)| negative. g g gauge g Yukawa In general, it is not easy to satisfy this crucial condition. The generic expressions for the gauge beta functions (cid:20) (cid:21) 11 (cid:88)2 (cid:88) 1 β(1) = g3 − C + T + κ T , (17a) g 3 G 3 F S3 S F S 8 Fields Gauge Global symmetries SU(N ) U(N ) U(N ) U(N ±4) c F L F R c ψ 1 1 ψ¯ 1 1 S 1 1 ψ , 1 1 1 A,S ψ˜ 1 1 S˜ 1 1 H 1 1 ˜ Table 3: A candidate model with anomaly-free chiral fermions ψ⊕ψ and Higgs-like scalars A,S H that might satisfy Total Asymptotic Safety. As in table 1, the upper box represents the original LS model, while in the lower box are listed the extra fields. (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) 34 (cid:88) 10 (cid:88) 2 β(2)| = g5 − C2 + 2C + C T + κ 4C + C T , (17b) g gauge 3 G F 3 G F S S 3 G S F S (where κ = 1, 1/2 for complex or real scalars respectively) show that, choosing a matter S content such that β(1) = 0, β(2)| is positive and minimal if the matter content consists g g gauge only of fermions in the fundamental, as in the LS model, where the total β(2) is negative by a g relatively small amount, see eq. (9). Adding one or more chiral families and a small number ((cid:28) N ∼ N ) of Higgs scalars in the fundamental coupled to the fermions as H∗ψ˜ψ , we c F A,S find that the conditions β(1) = 0 and β(2) < 0 cannot be satisfied together. No perturbative g g UV-interacting fixed point can be found for the gauge and yukawa couplings. We then need to consider more involved models. In general, it is convenient to add as many singlet fermions and/or scalars as possible, as they can allow for extra Yukawa couplings contributing to a negative β(2)| without contributing to β(2)| . Indeed the LS model g Yukawa g gauge introduces many singlet scalars S. Table 3 shows a candidate TAS model, obtained adding one chiral family to the LS model, ˜ some Higgs-like scalars H and some neutral scalars S. A more complex pattern of Yukawa couplings is allowed, making more complicated to compute their possible fixed points. Fixed points correspond to specific values of the couplings such that all their beta-functions vanish. The theories under consideration allow for ∼ N3 Yukawa couplings. In a theory with c many couplings the equations β = 0 can have many different solutions. Each solution seems to correspond to a specific flavour symmetry, because couplings with the same gauge quantum numbers have the same beta functions. Although we don’t know weather this is a generic mathematical result, we proceed by computing the β functions assuming the various possible flavour symmetries, such that the number of independent Yukawa couplings is reduced to a few. For example, the LS model assumed the maximal flavour symmetry allowed by its matter content, see table 1, such that there is only one independent Yukawa coupling y. In table 3 we again assume a quasi-maximal flavour symmetry. The most generic Yukawa interactions 9 allowed by the gauge and global flavour symmetries then are L = y Sψψ¯+y˜ S˜ψψ˜+y˜ H∗ψ ψ˜+h.c. (18) Y 1 1 2 A,S The one-loop RGE for the gauge coupling is (cid:20) (cid:21) 17 2 8 2N N ±4 17 β(1) = g3 − N + N ± = g3 c(cid:15), (cid:15) = F − . (19) g 6 c 3 F 3 3 N 4 c The RGE for the Yukawa couplings in the relevant limit (cid:15) (cid:28) 1 and N (cid:29) 1 are c  β(1) (cid:39) N y [−3g2 + 21y2 + 1y˜2],  y1 c 1 4 1 2 1 β(1) (cid:39) 1N y˜ [−12g2 + 17y2 + 29y˜2 +y˜2], (20)  βy˜(11) (cid:39) 41Ncy˜1[−18g2 + 127y˜12 +52y˜21]. 2 y˜2 4 c 2 2 1 2 They have a few fixed-point solutions. One solution is 684 372 3900 y2 = g2, y˜2 = g2, y˜2 = g2. (21) 1 1259 1 1259 2 1259 This fixed point leads to a negative (cid:18) (cid:19) 53 67443 β(2) Nc(cid:39)(cid:29)1 g5N2 1− (22) g 4 c 66727 such that g has a fixed point, that we compute at leading order in (cid:15) and 1/N : c 2518(4π)2 g2 (cid:39) (cid:15). (23) ∗ 537 N c Again, (cid:15) can be made arbitrarily small, such that g can be perturbative. The scalar potential ∗ that respects the assumed flavour symmetry contains 11 scalar quartics: V = λ Tr[S†S]2 +λ(cid:48) Tr[S†SS†S]+λ Tr[S˜†S˜]2 +λ Tr[S˜†S˜S˜†S˜]+ S S S˜ S˜ +λ (H†H)2 +λ(cid:48) Tr[H†HH†H]+λ Tr[S†S]Tr[S˜†S˜]+λ(cid:48) Tr[SS†S˜S˜†]+ H H SS˜ SS˜ +λ Tr[SS†](H†H)+λ Tr[S˜S˜†](H†H)+λ(cid:48) Tr[S˜HH†S˜†]. (24) HS HS˜ HS˜ We computed their RGE equations, but we don’t find any fixed point for them. There seems to be no particular reason: just a matter of unlucky order one factors in a system of 11 quadratic equations that would fill a page. Adding extra neutral fermions N transforming as anti-fundamentals of SU(N ) ⊗ SU(N ± 4) and N(cid:48) transforming as fundamentals of F R c SU(N ) ⊗ SU(N ± 4) allows for extra Yukawa couplings analogous to the one in eq. (3); F L c however we cannot find other fixed point solutions for the gauge and yukawa couplings apart from the solution of eq. (21). Wetriedtoreducetheglobalsymmetriesinordertosearchformoregenericfixedpoints. For example the symmetric and anti-symmetric components of S transform independently under 10